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Single Piles and Pile G ro u p sUnder Latera l Loading

2 n d Edi t ion

Ly m o n C. ReeseAcademic Chai r Emer i tusDepartment o f Civi l Engineer ingThe Universi ty o f Texas t Aus t in

W il l i a m Van Im p eFull Professor o f Civi l Engineer ingDirector Laboratory f o r Soi l MechanicsGhent Univers i ty Belg iumProfessor Cathol ic Universi ty Leuven Belgium

tg CRC PressTaylor Francis Gro up

Boca Raton Londo n New York Leiden

CRC Press is an imp rin t of theTaylor Francis Group, a n i n f o r m a business

B L K E M B O O K

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CO PYR IGH T N OT ICE ; nothing in the text may be photocop ied or reproduced with ou t the writt en permission of

the authors.

CRC Press/Balkema is an imprint of the Taylor Francis Groupan informa business

© 201 I Tay l o r F ranc i s Gr ou p , Lon don , UK

Typ ese t by MPS Lim i ted , a Macm i l lan Co mp any , Ch enn ai , IndiaP r i n t ed and bound i n Grea t B r i t a i n by An t on y R owe (a C P I Gr ou p C om pany ) ,C h i p p e n h a m , Wi l t s h i r e

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British Library Cataloguing in Publication DataA cata logue rec ord fo r th i s boo k is avai lab le f ro m the Bri t i sh Libra ry

Library of Congress Cataloging-in-Publication DataR eese , Ly m on C , 19 17 -

Single pi les and pi le groups under lateral loading / Lymon C. Reese,W i l l i a m Van Im pe .

p. cm.Includes b ib l iographical references and index .IS BN 9 7 8 -0 -415 -46 988 -3 (ha rdback )I . P i l ing (Civ i l engine er ing) 2 . Latera l loads . I. Van lm pe ,W . F.

II . Ti t l e .TA780 .R 44 20106 2 4 . l / 5 4 — d c 2 2

2 1 364 1

Published by: CR C Press/BalkemaP.O. Box 447,23 00 A K L eiden,The Netherla ndse-mail: Pub.NL@taylorandfrancis.comwww.crcpress.com - www.taylorandfrancis.co.uk - www.balkema.nl

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r e f a c e

The information presented here is in the emerging field of soil structure interaction.Not only must the engineer compute the loading at which a foundation will collapse,but the deformation at the soil-structure interaction boundary must be found for theexpected loading. Further, the interaction of the foundation with the superstructuremust be consistent with the details of the design and construction. The methods presented here would not have been possible except for the performance of full-scale testson single piles and pile group s where rem ote-reading instrumen ts allowed the responseof the piles to be measured. Further, the computer is necessary to solve the complexmodels that were developed for predicting the response of the piles.

W hen offshore p latform s w ere being installed in significant nu mb ers in the 1 95 0 s,engineers quickly realized that correct solutions required that ways be found to linkthe soil response to the lateral deflection of a pile. A number of experiments wereperformed with full-sized piles, instrumented for the measurement of bending momentalong their length. Static and cyclic loading was employed. Those experiments, andothers performed in later years, allowed experim ental p y curves to be produced. Soilmechanics and structural mechanics were used to develop methods of predicting p ycurves for various soils that yielded excellent agreement with the response of the piles.

Solution s of the relevant differential equ ation s in finite difference form became po ssible with the emergence of the main-frame digital compu ter for routine com puta tionsin the 19 50 s. Solutions on the personal com puter can be obtained tod ay with relativeease, allowing the sensitivity analysis of many significant parameters. Improvements of

computer codes are occurring at a rapid pace, and experimental data on the responseof single piles and pile groups to lateral loading is appearing in the technical literaturewith regularity.

The p y method can be used to attack a wide variety of problems encountered inpractice, as demonstrated by the examples given in Chapter 6. While the computercodes can be used to attack such problems and vast quantities of tabulated results andgraphical results can be prod uced with ease, the impo rtance of comp etent engineeringjudgement cannot be over emphasized. For example, the computer codes will lead tothe required penetration of the pile supporting an overhead sign subjected to windloading; however, the knowledgeable engineer will have to note that in many cases

too few data are available to validate such a result and will take appropriate steps toensure a safe design. One such step could be the performance of a controlled set ofexperiments.

Besides of the requirement of good quality and relevant soil data, im por tant know ledge can be gained from field experiments, particularly with the use of instrumented

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xvi Preface

piles, as presented in Chapter 8 or from recording the performance of completed structures. Modern methods of monitoring make entirely feasible the acquisition, underservice c ond itions, of such information as pile-head deflection and other such data.

The writers wish to acknowledge many who contributed to various aspects of the

book. Of particular im portance are those authors who have made contributions of thetechnical literature that is referenced. The writers express thanks to Professor HeinzBrandi, Dr. William Cox and Professor Hudson Matlock for reading the text andmaking important comments. Professor Michael W. O Neill, University of Houston,used a draft of the book in teaching a course to graduate students. Thanks a re extendedto him and his students for helpful suggestions.

Appreciation is extended to Dr. Robert Gilbert and Dr. Alexander Avram for usefulsuggestions on Chapter 9. In Austin, Dr. Shin-Tower Wang, Dr. William Isenhower,and Mr. Jose Arrellaga were very helpful in reviewing parts of the text, in makingnumerous computer runs, and in preparing graphical output. Appreciation is extendedto Nanc y Reese, Cheryl Waw rzynowicz and Suzanne B urns for dedication and diligencein formatting an d editing the text.

In Ghent, the writers appreciate the assistance of Mr. Etienne Bracke for some ofthe graphical output support and the help of Mrs. Linda Van Cauwenberge for theformatting and editing of the text version prepared in Ghent. Moreover the authorsacknowledge Prof. J. De Rouck, Prof. R. Van Impe and Mr. Ch. Bauduin for puttingavailable some of the information on issues related to some European developmentsfor structural design.

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C o n t e n t s

V reface xv List of Symbols xvii

Techniques for design1.1 Introduction 1 1.2 Occu rrence of laterally loaded piles 2 1.3 N atu re of the soil respon se 3 1.4 Respo nse of a pile to kinds of loadin g 7

1.4.1 Introduction 7 1.4.2 Static loadin g 7 1.4.3 Cyclic loadin g 8

1.4.4 Sustained loadin g 10 1.4.5 Dyn amic loading 10

1.5 M od els for use in analy ses of a single pile 11 1.5.1 Elastic pile an d elastic soil 11 1.5.2 Elastic pile an d finite elements for soil 13 1.5.3 Rigid pile and plas tic soil 13 1.5.4 Characteristic load meth od 14 1.5.5 No nlin ear pile and p-y model for soil 15

1.6 M ode ls for gro ups of piles und er lateral loadin g 17 1.7 Status of curr ent state-of-the-art 20

Homework problems for chapter 1 20

2 De rivation of equations and methods of solution 23 2.1 Introduction 23 2.2 Deriv ation of the differential equ ation 23

2.2 .1 Solution of reduced form of differential equat ion 26 2.2.2 Solution of the differential equ ation by difference equ ation s 32

2.3 Solution for Epy = kPyx 38 2.3.1 Dim ensional analysis 39 2.3.2 Equ ations for Epy = kPyx 44 2.3.3 Exam ple solution 46 2.3.4 Discussion 50

2.4 Validity of the mech anics 51 Homework problems for chapter 2 52

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viii Contents

Models for response of soil and weak rock 53 3.1 Introduction 53 3.2 M echan ics conce rning respon se of soil to lateral loadin g 54

3.2.1 Stress-deform ation of soil 54 3.2.2 Prop osed mo del for decay of £s 55 3.2.3 Va riatio n of stiffness of soil (£s and Gs) with depth 56 3.2.4 Initial stiffness and ult im ate resistan ce of

p-y curves from soil properties 57 3.2.5 Subgrade mo dulu s related to piles under lateral loading 64 3.2.6 Theoretical solution by Skem pton for subgrade

modulus and for p-y curves for saturated clays 66 3.2.7 Practical use of Ske mp ton's equ ation s and values of sub grad e

modulus in analyzing a pile under lateral loading 67 3.2.8 Ap plication of the Finite Element M eth od (FEM)

to obtaining p-y curves for static loading 68 3.3 Influence of diam eter on p-y curves 69

3.3.1 Clay 69 3.3.2 Sand 69

3.4 Influence of cyclic loadin g 69 3.4.1 Clay 69 3.4.2 Sand 71

3.5 Experim ental meth ods of obtainin g p-y curves 72 3.5.1 Soil respon se from direct me asure me nt 72 3.5.2 Soil respon se from exp erim ental m om ent curves 72 3.5.3 No ndim ension al meth ods for obtainin g soil response 73

3.6 Early recom men dations for com puting p-y curves 73 3.6.1 Terzaghi 74 3.6.2 M cCle lland Focht for clay (1958) 75

3.7 p-y curves for clays 75 3.7 .1 Selection of stiffness of clay 76 3.7.2 Response of soft clay in the presen ce of free w ate r 78 3.7.3 Response of stiff clay in the presence of free w ate r 81 3.7.4 Response of stiff clay w ith no free wa ter 88

3.8 p-y curves for sands above and below the water table 91 3.8.1 Detailed proced ure 91 3.8.2 Rec om me nded soil tests 94 3.8.3 Examp le curves 94

3.9 p-y curves for layered soils 94 3.9.1 M etho d of Georgiadis 95 3.9.2 Example p-y curves 96

3.10 p-y curves for soil with both cohesion and a friction angle 99 3.10.1 Background 99 3.10.2 Recomm endations for computing p-y curves 99

3.10.3 Discussion 103 3.11 Other recommen dations for computing p-y curves 104

3.11.1 Clay 104 3.11.2 Sand 105

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Contents ix

3.12 Mo difications to p-y curves for sloping ground 105 3.12.1 Introduction 105 3.12.2 Eq uatio ns for ultim ate resistance in clay 106 3.12.3 Eq uatio ns for ultim ate resistance in sand 107

3.13 Effect of ba tter 108 3.14 Shearing force at bo tto m of pile 108 3.15 p-y curves for weak rock 109

3.15.1 Introduction 109 3.15 .2 Field tests 110 3.15.3 Interim recom men dation s for com puting

p-y curves for weak rock 110 3.15.4 Com men ts on equation s for predicting

p-y curves for rock 114 3.16 Selection of p-y curves 114

3.16.1 Introduction 114 3.16.2 Factors to be considered 114 3.1 6.3 Specific sugg estions 115 Homework problems for chapter 3 117

Structural characteristics of piles 9 4.1 Introduction 119 4.2 Co mp utatio n of an equivalent diameter of a pile

with a noncircular cross section 120 4.3 Mech anics for com putat ion of M

u\t and

pI

p as

a function of bending moment and axial load 121 4.4 Stress-strain curves for norm al-w eigh t concre te and stru ctura l steel 125 4.5 Imp lem entatio n of the m eth od for a steel h-section 126 4.6 Imp lem entatio n of the m eth od for a steel pipe 129 4.7 Imp lem entatio n of the m eth od for a reinforced-con crete section 130

4.7.1 Exam ple com puta tions for a square shape 130 4.7.2 Exam ple com puta tions for a circular shape 132

4.8 Ap prox ima tion of mo men t of inertia for a reinforced-concrete section 133 Homework problems for chapter 4 134

Analysis of groups of piles subjected to inclinedand eccentric loading 35 5.1 Introduction 135 5.2 Ap pro ach to analysis of gro up s of piles 136 5.3 Review of theories for the respon se of grou ps of piles

to inclined and eccentric loads 137 5.4 Ra tion al equ ation s for the respon se of a gro up of piles

under generalized loading 139 5.4.1 Introduction 139

5.4.2 Equa tions for a two-dim ensional group of piles 143 5.5 Laterally loade d piles 147

5.5.1 M ovem ent of pile head due to applied loading 147 5.5.2 Effect of ba tter 147

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x Contents

5.6 Axially loade d piles 148 5.6.1 Introduction 148 5.6.2 Relevant param eters concerning deformation of soil 148 5.6.3 Influence of m etho d of installatio n on soil characteristics 150 5.6.4 M eth od s of form ulating axial-stiffness curves 150 5.6.5 Calculation meth ods for load-settlement behaviour on the

basis of in-situ soil tests 153 5.6.6 Differential equa tion for solu tion of finite-difference

equation for axially loaded piles 157 5.6.7 Finite difference equ ation 159 5.6.8 Load -transfer curves 160

5.7 Closely-spaced piles und er lateral loadin g 165 5.7.1 M odifica tion of load-trans fer curves for closely spaced piles 165 5.7.2 Con cept of interaction under lateral loading 166 5.7.3 Pro posals for solving for influence coefficients for

closely-spaced piles under lateral loading 167 5.7.4 Desc ription and analysis of expe rimen ts with closely-spaced

piles installed in-line and side-by-side 170 5.7.5 Prediction equ atio ns for closely-spaced piles installed

in-line and side-by-side 174 5.7.6 Use of modified pred iction equ ation s in developing

p-y curves for analyzing results of experiments withfull-scale groups of piles 175

5.7.7 Discussion of the me tho d of predictin g the interac tionof closely-spaced piles under lateral loading 190

5.8 Pro posals for solving for influence coefficients forclosely-spaced piles under axial loading 190 5.8.1 M odifica tion of load-trans fer curves for closely spaced piles 190 5.8.2 Con cept of interaction under axial loading 190 5.8.3 Review of relevant literatu re 191 5.8.4 Interim reco mm end ation s for com put ing the efficiency

of groups of piles under axial load 194 5.9 Analysis of an exp erim ent wi th batte r piles 195

5.9.1 Description of the testing arrangem ent 195 5.9.2 Prop erties of the sand 196 5.9.3 Prop erties of the pipe piles 198 5.9.4 Pile gro up 199 5.9.5 Exp erim ental curve of axial load versus settlement for

single pile 200 5.9.6 Results from expe rimen t and from analysis 200 5.9.7 Com ments on analytical method 202 Homework problems for chapter 5 202

Analysis of single piles and groups of pilessubjected to active and passive loading 205 6.1 N atu re of lateral loadin g 205 6.2 Active load ing 205

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Contents xi

6.3

6.4

6.5

7 Case7.17.2

7.3

7.4

7.5

6.2.1 W ind loading6.2.2 Wave loading6.2.3 Cu rrent loading6.2.4 Scour

6.2.5 Ice loadin g6.2.6 Ship imp act6.2.7 Load s from miscellaneo us sourcesSingle piles or groups of piles subjected to active loading6.3.1 Ov erh ead sign6.3.2 Breasting dolp hin6.3.3 Pile for an chor ing a ship in soft soil6.3.4 Offshore platfo rmPassive loading6.4.1 Ear th pressures6.4.2 M ovi ng soil6.4.3 Thru sts from dead loading of structuresSingle piles or groups of piles subjected to passive loading6.5.1 Pile-supported retaining wall6.5.2 An chored bulkh ead6.5.3 Pile-supported ma t at the Pyramid Building6.5.4 Piles for stabilizing a slope6.5.5 Piles in a settling fill in a slop ing valleyHomework problems for chapter 6

: studiesIntroductionPiles installed into cohesive soils with no free water7.2.1 Bagnolet7.2.2 Houston7.2.3 Brent Cros s7.2.4 JapanPiles installed into cohesive soils with free water above groundsurface

7.3.1 Lake Austin7.3.2 Sabine7.3.3 ManorPiles installed in cohesionless soils7.4.1 M usta ng Island7.4.2 Garston7.4.3 Ark ansas river7.4.4 Roosev elt bridgePiles installed into layered soils7.5.1 Talisheek

7.5.2 Alcâcer do Sol7.5.3 Florida7.5.4 Apapa7.5.5 Salt Lake Internation al Airp ort

205207213214

215216218218218222226232243243244246246246251257266272279

281281282282285288290

291

291294296298298301301306307307

311312315315

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xii Contents

7.6 Piles installed in c φ soil 319 7.6.1 Kuwait 319 7.6.2 Los Angeles 320

7.7 Piles installed in we ak rock 322 7.7.1 Islamorada 322 7.7.2 San Francisco 324

7.8 Analy sis of resu lts of case studie s 327 7.9 Com men ts on case studies 328

Homework problems for chapter 7 331

8 Testing of full sized piles 333 8.1 Introduction 333

8.1.1 Scope of pres entat ion 333 8.1.2 M eth od of analysis 333 8.1.3 Classification of tests 334 8.1.4 Features uniq ue to testing of piles und er lateral loadin g 334

8.2 Designing the test pro gra m 335 8.2.1 Plann ing for the testing 335 8.2.2 Selection of test pile an d test site 335

8.3 Subsurface investigation 336 8.4 Ins tallat ion of test pile 339 8.5 Testing techniq ues 340 8.6 Loading arrangem ents and instrum entation at the pile head 341

8.6.1 Loading arrangem ents 341 8.6.2 Instrumentation 344

8.7 Testing for design of pro du ctio n piles 348 8.7.1 Introduction 348 8.7.2 Inter pre tatio n of dat a 348 8.7.3 Example Com putation 348

8.8 Exam ple of testing a research pile for p y curves 350 8.8.1 Introduction 350 8.8.2 Prep aratio n of test piles 350 8.8.3 Test setup and loadin g equ ipm ent 352

8.8.4 Instrumentation 353 8.8.5 Ca libra tion of test piles 357 8.8.6 Soil borin gs and lab ora tor y tests 360 8.8.7 Installatio n of test piles 363 8.8.8 Test proc edu res and details of loadin g 366 8.8.9 Pene trom eter tests 368 8.8.10 G rou nd settlement due to pile driving 371 8.8.11 Gr oun d settlement due to lateral loading 371 8.8.12 Rec alibratio n of test piles 373 8.8.13 Graph ical presentation of curves showing bending mo men t 373

8.8.14 Interpretation of bending mo men t curves to obtainp y curves 375

8.9 Summary 379 Homework problems for chapter 8 379

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Contents xiii

9 Implem entation of factors of safety 381 9.1 Introduction 381 9.2 Limit states 381 9.3 Con sequence s of a failure 382 9.4 Philoso phy conce rning safety coefficient 384 9.5 Influence of na tur e of struc ture 385 9.6 Special pro blem s in characte rizing soil 385

9.6.1 Introduction 385 9.6.2 Ch aracteristic values of soil par am eters 387

9.7 Level of quality con trol 387 9.8 Two general app roa che s to selection of factor of safety 388 9.9 Glob al ap pro ach to selection of a factor of safety 388

9.9.1 Introductory comm ents 388 9.9.2 Reco mm endation s of the Am erican Petroleum Institute

(API) 389 9.10 M eth od of parti al factors (psf) 390

9.10.1 Preliminary Con siderations 390 9.10.2 Suggested values for part ial factors for

design of laterally loaded piles 390 9.10.3 Example comp utations 391

9.11 M eth od of load and resistance factors (LRFD) 393 9.11.1 Introduction 393 9.11.2 Loads addressed by the LRFD specifications 393 9.11 .3 Resistances addressed by the LRFD specifications 394 9.11.4 Design of piles by use of LR FD specifications 394

9.12 Concluding comm ent 395 Homework problems for chapter 9 395

10 Suggestions for design 397 10.1 Introduction 397 10.2 Ran ge of factors to be consid ered in design 397 10.3 Validatio n of results from com pu tatio ns for single pile 398

10.3.1 Introduction 398

10.3.2 Solution of example problem s 398 10 .3.3 Check of echo prin t of inp ut dat a 399 10.3 .4 Investigation of length of w or d emp loyed in intern al

computations 399 10 .3.5 Selection of toleran ce and length of increm ent 399 10.3 .6 Check of soil resistance 400 10.3 .7 Check of mechanics 400 10.3.8 Use of nond imensio nal curves 400

10.4 Validatio n of results from com pu tatio ns for pile gro up 401 10.5 Ad dition al steps in design 401

10.5.1 Risk man agem ent 401 10.5 .2 Peer review 401 10.5.3 Technical contrib ution s 402 10.5 .4 The design team 402

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xiv Contents

APPENDICES

A Brom s me tho d for analysis of single piles und er lateral load ing 403 B No nd im ens ion al coefficients for piles with finite length, no axial load,

constant Ep/Ip, and constant Epy 419 C Difference equ ation s for step-tap ered beam s on fou nda tion s havingvariable stiffness 429

D Computer Program CO M 622 441 E No n-d im ensi ona l curves for piles un der lateral loadin g for case wh ere

Py — f^py^ HO 1

F Tables of values of efficiency measu red in tests of gro up s of piles un de rlateral loading 461

G Ho rizo nta l stresses in soil near shaft during installation of a pile 465 H Use of data from testing uninstru men ted piles under lateral loading to

obtain soil response 471 I Eur oco de principles related to geotechn ical design 477 J Discussion of factor of safety related to piles und er axial load 481

REFERENCES 485

AUTHOR INDEX 501

SUBJECT INDEX 505

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Chapter I

Te c h n i q u e s f o r d e s ig n

I . I I N T R O D U C T I O N

After selecting ma terials for the pile fou nda tion to ma ke sure of durability, the d esignerbegins with the components of loading on the single pile or the group. With the axialload, lateral load, and overturning moment, the engineer must ensure that the singlepile, or the critical pile in the group, is safe against collapse and does not exceedmovements set by serviceability. If the loading is purely axial, the design of a pile canfrequently be accomplished by solving the equations of static equilibrium. The designof a single pile or a group of piles under lateral loading, on the other hand, requiresthe solution of a nonlinear differential equation.

Linear solutions of the differential equation for single piles are available and implemented in some codes of practice, but are of limited value. Another simplification isto assume that raked piles in a group do not deflect laterally; the equations that resultcan be solved readily, but such solutions are usually seriously in error. The followingsections show that treating the soil, and sometimes the material in the pile, as nonlinear complicates the mathematics for the single pile and pile group, but solutionscan be made by numerical methods that are both rational and in close to reasonableagreement with results from full-scale experiments.

The traditional technique of limit analysis, so useful in finding the ultimate capacityof many foundations, has only a marginal application to assessing the behavior of alaterally loaded pile. As will be demonstrated, acceptable solutions are only possible if

explicit, nonlin ear relatio nship s are emplo yed th at give soil stiffness and resistance as afunction of pile deflection, poi nt by poi nt, along the length of a pile. The solu tions of theresulting equations can then be made to satisfy the required conditions of equilibriumand compatibility.

The problem involves the interaction of the soil and the pile, is one of the class ofsoil-structure-interaction problems, and is classified as Geotechnical Category 3 by theEurocode 7 (Reference). The resistance of the soil, in force per unit length at pointsalong the pile, depends on the deflection of the pile, and the soil resistance must be athand in order to solve the relevant equations. Therefore, iteration is necessary to finda solution. In this model, the pile is taken as a free body and the soil is simulated by

a series of Winkler-type mechanisms, which we discuss further in a later chapter. Theequations to be solved for the pile response come directly from ordinary mechanics.

A presentation of the methods of design is made initially for isolated or single piles.A later chapter will deal with the design of pile groups. Two classes of pile-group

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2 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur I.I Installation of piles for n offshore platform.

problems can be identified: (1) the distribution of loading to the heads of the variouspiles in the group; and (2) the efficiency of each of the piles in the group, a problemin pile-soil-pile interaction. Both problems are addressed herein; the first is solvedsatisfactorily by numerical procedures; and the second is discussed fully with respectto available, empirical information.

1.2 O C C U R R E N C E O F L A T E R A L LY L O A D E D PIL E S

With regard to their use in practice, horizontally loaded piles may be termed activeor passive. An active pile has loading applied principally at its top in supporting asuperstructure, such as a bridge. A passive pile has loading applied principally along

its length due to earth pressure, such as for piles in a moving slope or for a secant-pilewall. Chapter 6 will present examples of the design of both active and passive piles.

Figure 1.1 shows the installation of piles for an offshore platform. These activepiles must sustain lateral loading from storm-driven w aves and wind . With the ad vent

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Te c h n i q u e s f o r d e s i g n 3

Figure 1.2 Sketch of a pile - sup ported bridge abutmen t.

of offshore structures, the design of such piles was a primary concern and prompteda number of full-scale field tests. The results of some of these tests are studied inChapter 7.

The design of the piles for an offshore platform presents interesting problems insoil-structure interaction. An example in Chapter 6 shows the elements in the designprocess.

Other examples of active piles are found in foundations for bridges, high-rise structures, overhead signs, and piers for ships. Active piles must be designed for mooringdolph ins, breasting dolp hins, and pile group s that protect the bridge foundations fromship impact. This last application's importance is emphasized by the unfortunate lossof life in the United States during failure of a portion of the Sunshine Skyway Bridgeat Tampa Bay because of destruction of a bridge pier by an out-of-control ship.

The sketch in Fig. 1.2 shows an example of a passive pile. While the pile will besubjected to lo adings a t its top , the prim ary co ncern is the influence of the sliding of thesoil in the slope. Ch apte r 6 presents an app roa ch for the design of such piles. In add itionto being used to stabilize a slope, passive piles are also used for the construction of

tangent- or secant-pile walls, for soldier beams, and for supporting the base of retainingwalls.

1.3 N A T U R E O F T H E S O I L R E S P O N S E

The main parameter from the soil in a pile under lateral loading is a reaction modulus,defined as the resistance from the soil at a point along the pile (F/L) divided by thedeflection of the pile at that po int (L). The reactio n m od ulu s is a function bo th of d epthbelow the ground surface z and the deflection of the pile y. The reaction modulus can

be defined in various ways; concepts that lead to a convenient solution of the relevantequations are presented in Fig. 1.3. The sketch in Fig. 1.3a shows a cylindrical pileunder lateral load with a thin slice of soil shown at the depth below the ground lineof z\. (Note: the symbol z is used to show depth below the ground surface, and the

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Figure 1.3 D istr ibu tio n o f unit stresses against a pile before and after lateral defle ction .

symbol x is used to show distance measured from the top of the pile.) The uniformdistribution of unit stresses normal to the wall of the pile in Fig. 1.3b is correct for thecase of a pile that has been installed without bending. If the pile is caused to deflecta distance y\ (exaggerated in the sketch for clarity), the distribution of unit stresses

would be similar to that shown in Fig. 1.3c. The stresses will have decreased on theback side of the pile and increased on the front side. Some of the stresses will haveboth a normal and a shearing component.

Integration of the unit stresses will result in the quantity /?i, which acts in oppositedirection to y\. Th e dimension s of/? are load per unit length alon g the pile. These un itsare identical to those to those found in the solution of ordinary equations for a beamon an elastic soil bed. The reaction modulus can be defined as the slope of the secantto a p-y curve.

A typical p-y curve is sho wn in Fig. 1.4a, dra wn in the first qu ad ran t for conven ience.The curve is one member of a family of curves that show the soil resistance p) as a

function of depth z) . The curve in Fig. 1.4b depicts the value Epy tha t is con stan t forsmall deflections for a particular depth, but decreases with increased deflection. Epy isproperly termed reaction mod ulus for a pile under lateral loading. While Epy will varywith the properties of the particular soil, the term does not uniquely represent a soil

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Te c h n i q u e s f o r d e s i g n 5

property. Rather Epy is simply a parameter for convenient use in computations. For aparticular practical solution, the term is modified point-b y-poin t along the length of thepile as iteratio n occ urs. The iteration leads to compa tibility b etween pile deflection andsoil resistance, according to the nonlinear p-y curves that have been selected, takingsoil properties and pile dimensions into account.

A numb er of authors w ho first wrote ab ou t piles under lateral loading used the term£ s for Epy, but the term £s is used herein as describing a characteristic of the soil.

Chapter 3 presents recommendations for formulating equations for p-y curves.Experiments are cited where lateral-load tests were performed on piles that wereinstrumented for the measurement of bending moment in the pile as a functionof depth. Differentiation and integration of those curves yielded experimental p-ycurves. Correlations were then developed between these experimental curves and thecharacteristics of the soil, taking pile diameter into account.

Examples of p-y curves obtained from a full-scale experiment involving piles with adiam eter of 641 mm an d a pen etrati on of 15.2 m are show n in Figs. 1.5 and 1.6 (Reese,et al., 1975 ). The piles were instrum ented at close spacings to measure bending mo me ntand were tested in overconsolidated clay.

The portion of the curve in Fig. 1.4a from points a to b shows that the value of pis increasing at a decreasing rate with increasing deflection y. This behavior undoubtedly reflects the nonlinear portion of the in situ stress-strain curve. Many suggestionshave been made for predicting the a-b portion of a p-y curve, but there is no widelyaccepted analytical procedure. Rather, that part of the curve is empirical and based onresults of full-scale tests of piles in a variety of soils with both monotonie and cyclicloading.

The straight-line, horizontal portion of the p-y curve in Fig. 1.4a implies that the insitu soil is behaving plastically w ith n o loss of soil resistance with increasing deflection.W ith that assu mp tion, analytical models can be used to com pute the ultimate resistance

pult as a function of pile dimension s an d soil pro pert ies. These mo dels are discussed inChapter 3.

A mo re direct approac h to formulating p-y curves wo uld be to consider the responseof the soil, rather than the pile. Fig. 1.3d shows an element to suggest that a solution

Figure 1.4 Typical p y curve and resulting soil mod ulus.

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may be obtained by the finite-element method (FEM). An appropriate solution withthe FEM requires a three-dimensional model at and near the ground surface becausethe responses of the upper soils have a dominant effect on pile behavior. The nonlinearstress-strain characteristics of the soil must be modeled, taking into account largestrains. Properties must be selected for the various layers of soil around the pile. In

additio n, non linear geometry must be considered, particularly near the grou nd surface,where gaps in cohesive soils will occur behind a pile while upward bulging will occurin front. For cohesionless soils, there will be settlement of the ground surface due todensification, especially under repeated loading.

A solution with the FEM must start with the constitutive modeling of the in situ soil,then the influence of the installation of the pile or piles must be modeled. Finally, themodeling must address the influence of the various kinds of loading (discussed in thenext section). Some solutions have been proposed with a three-dimensional FEM, forexample by Shie & Brown (1991). Homogeneous soils were studied, and p-y curveswere developed for single piles and piles in a group. The solutions were made with a

super computer, where computer time was considerable, and some analytical difficulties were encountered. While the results were instructive, particularly with respect topile groups, the FEM can make only a limited contrib ution to obtainin g p-y curves forthe generalized problem described above.

Figure 1.5 p y curves developed from static load test of 641 mm -diame ter pile from Reese, et al.,1975).

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Te c h n i q u e s f o r d e s i g n 7

Figure 1.6 p y curves developed from cyclic load test of 641 mm -diame ter pile from Reese, et al.,1975).

1.4 R E S P O N S E O F A P I L E T O K I N D S O F L O A D I N G

1 . 4 . 1 I n t r o d u c t i o n

The na tur e of the loadin g, plus the kind of soil aro un d the pile, are of major im po rtan cein predicting the response of a single pile or a group of piles. With respect to active

loadings at the pile head, four types can be identified: short term or static, cyclic,sustained, and dynamic. In addition, passive loadings can occur along the length of apile from moving soil, such as when a pile is used as an anchor. Another problem to beaddressed is when existing piles are in the vicinity of pile driving or earth work. Briefgeneral discussions are presented below of the pile response to the various loadings.Analyses are presented in later chapters to illustrate the influence of some of the kindsof loading.

1.4.2 S t a t i c l o a d i n g

The curve in Fig. 1.4a represents the case for a particular value of z where a short-term monotonie loading was applied to a pile. This case, called static loading forconven ience, will seldom, if ever, be encou ntered in practice. How ever, static curves areuseful because (1) analytical pro cedu res can be used to develop expressions to correla te

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Figure 1.7 Effect of num ber o f cycles on the p y behavior at very low cyclic strain loading.

with some portions of the curves, (2) the curves serve as a baseline for demonstratingthe effects of other types of loading , and (3) the curves can be used for sustained loadin gfor some clays and sands.

The curves in Fig. 1.5 resulted from static loading of the pile. Several items are ofinterest: (1) the initial stiffness of the curves increases with depth; (2) the ultimate resistance increases with depth; and (3) the scatter in the curves illustrates errors inherentin the process of analyzing numerical results from measurements of bending momentwith depth. Points 1 and 2 demonstrate that analyses employing soil properties canbe correlated with the experimental results, emphasizing the need to do static loadingwhen performing tests of piles.

1.4.3 C y c l i c l o a d i n g

Figure 1.7a shows a typical p-y curve a particular depth. Point b represents the valueof puit for static loadin g an d pu\t is assumed to remain constant for deflections largerthan that for Point b. The shaded portion of Fig. 1.7a indicates the loss of resistancedue to cyclic loading. For the case shown, the static and cyclic curves are identicalthrough the initial straight-line portion to Point a and to a small distance into the

nonlinear portion at Point c. With deflections larger than those for Point c, the valuesof p decrease sharply due to cyclic loading to a value at Point d. In some experiments,the value of p remained constant beyond Point d. The loss of resistance shown by theshade d are a is, for a given soil, plainly a function of the num ber of cycles of load ing. Asma y be seen, for a cons tant value of deflection, the value ofEpy is low ered significantlyeven at relatively low strain levels, due to cyclic loading.

A comparison of the curves in Figs. 1.5 and 1.6 demonstrates dramatically theinfluence of cyclic loading, at least at a site where there is stiff clay with a given setof characteristics. As might be expected, at low magnitudes of deflection, the initialstiffnesses are only moderately affected. However, at large magnitudes of deflection,

the p-values show considerable decreases. The values of pu\t are also decreased. Whilethe results of static loading of a pile ma y be correlated with soil pro pert ies, it is clear theresults of cyclic loading will not easily yield to analysis. Discussions in the followingparagraphs will indicate the direction of some research. Of most importance are the

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Te c h n i q u e s f o r d e s i g n 9

Figure 1.8 Simplified response of piles in clay due to cyclic loading from Long, 1984).

results from carefully performed tests of full-sized piles under lateral loading in a varietyof soils. Formulations for taking cyclic loading into account will be presented in a laterchap ter w here the m eth ods are based on th e available results of testing full-scale, fullyinstrumented piles.

The cyclic loading of laterally loaded piles occurs with offshore structures, bridges,overhead signs, breasting and mooring dolphins, and other structures. For stiff claysabove the water table and for sands, the effect of cyclic loading is important, butfor saturated clays below water, which includes soft clays, the loss of resistance in

comparison to that from static loading can be considerable. Experiments have shownthat stiff clay remains pushed away near the ground surface when a pile deflects, suchas shown in Fig. 1.8, where two-way cyclic loading is assumed. The re-application of aload causes wat er to be forced from the ope ning a t a velocity related to the frequency ofload ing. Typically, as a result, scour of the clay occurs with an ad ditio nal loss of lateralresistance. In the full-scale experiments with stiff clay that have been performed, thescour of the soil during cyclic loading is readily observed by clouds of suspension nearthe front and back faces of the pile (Reese, et al., 1975). The gapping around a pile isnot as prominent in soft clay, probably because the clay is so weak to collapse whenthe loading is cycled. The clouds of suspension were not observed while testing piles

in soft to medium clays, but the cycling caused a substantial loss in lateral resistance(Matlock, 1970).

As seen in Fig. 1.8, the soil resistance near the mudline would be zero, up to a givendeflection. No failure of the soil has occurred because the resistance is transferred to

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the lower portion of the soil profile. There will be an increase in the bending momentin the pile, of course, for a given value of lateral loading.

1.4.4 S u s t a i n e d l o a d i n g

Figure 1.7b shows an increasing deflection with sustained loading. The decreasingvalue of p implies the shifting of resistance to lower elements of soil. The effect of sustained loadin g is likely to be negligible for o verco nsolid ated clays and for granu lar soils.

Sustained loadin g of a pile in soft clay wo uld likely result in a significant am ou nt oftime-related deflection. Analytical solutions could be made using the three-dimensionaltheory of consolidation, b ut the formu lation of the equations depen ds on a large number of param eter s not clearly defined phy sically. The generalization of such a pro cedu reis not yet available in the literature.

The influence of sustained loading, in some cases, can be solved with reasonable

accuracy by experiment. At the site of the Pyramid Building in Memphis, Tennessee, alateral-load was applied to a CFA pile with a diameter of 430 mm in a silty clay withan average value of undrained shear strength over the top several diameters of the pileof 35 kPa (Reuss, et al., 1992). A load of 22 kN , correspon ding appro xim ately to theworking load, was held for a period of 10 days, and deflection was measured. Someerrors in the data occurred because the load was maintained by ma nua l adjustment ofthe hydraulic pressure rather than by a servo-mechanism. However, it was possible toanalyze the data and show that soil-response curves could be stretched by increasingthe deflection 20% over that expected for static loading to predict the behavior ofthe pile under sustained loading. At the Pyramid Building site, some thin strata of silt

in the near-surface soils are believed to have promoted the dissipation of excess porewater pressure.

1.4.5 D y n a m i c l o a d i n g

With respect to dynamic loading, the greatest concern is that some event will causeliquefaction to occur in the soil at the pile-supported structure. It is important to knowthat liquefaction can occur in loose granular soil below the water table, although thepresentation of liquefaction in this text will remain brief.

Pile-supported structures can be subjected to dynamic loads from machines, traffic,

ocean waves, and earthquakes (Hadjian, et al., 1992). The frequency of loading fromtraffic and waves is usually low enough so that p-y curves for static or cyclic loadingcan be used. Brief discussions are presented below about loadings from machineryand from earthq uake s. In additio n, some discussion is given to vibrations an d perha pspermanent soil movement, as a result of the vibrations, due to installing piles in thevicinity of an existing pile-supported structure.

As noted earlier, soil resistance for static loadings can be related to the stress-straincharacteristics of the soil; however, if the loadin g is dyn am ic, an inertia effect m ust beconsidered. Not only are the stress-strain characteristics necessary for formulating p-ycurves for dynamic loading, but the mass of the soil must be taken into account. Use of

the finite element meth od appears pro mising , but if the FEM h as not proven completelysuccessful for static loading, the application to the dynamic problem appears to bedoubly complex. Thus, unproven assumptions must be made if the p-y method isapplied directly to solving dynamic problems.

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If the loading is due to ro tatin g m achinery, the deflection is usually small, and a valu eof soil modulus may be used for analysis. Experimental techniques (Woods & Stokoe,1985; Woods, 1978) have been developed for obtaining the soil parameters that areneeded. Analytical techniques for solving for the response of a pile-supported structure

have been presented by a number of writers. Roesset (1988) and Kaynia & Kausel(1982) have developed techniques that are quite effective in dealing with machine-induced vibrations.

If the loadin g is a result of a seismic event, the analysis of a pile-su ppo rted struc turewill be com plex (G azetas & M ylo nak is 1 99 8). The free-field m otio n of the near surfacesoils at the site must be computed, or selected, taking micro zonation into account.A standard earthquake may be used with an unkno wn degree of appro xima tion. Theresponse of the piles, neglecting the superstructure, must be considered. If the soilmovement is constant with depth, the piles will move with the soil without bending.Such an assumption, if valid, simplified the computations. The distributed massesof the superstructure must be employed in solving for the motion of the piles and themotion of the superstructure. Of course, p -y curves must be available with app rop riatemodification of the inertia effects. Not much experimental data is available on whichto base a method of computation.

In the absence of comprehensive information on the response with depth of pile-supported structures that either have failed or have withstood an earthquake andtaking into account the enormous amount of computations that are needed, fullyrational analyses are currently unavailable.

Various simplifying assumptions are being used: (1) pseudo horizontal load andavailable p-y curves are sometimes employed as a means of simulating the effectsof an earthquake. If the assumption is made that the lateral soil movement duringan earthquake is constant with depth, and if existing p-y curves or curves perhapsmodified empirically for inertia effects are used, the movements of elements of thesuperstructure can be computed by equations of mechanics.

Engineers are aware that the installation of pil s near a pile-supported structure cou ldlead to movements of the existing structure. If the site of construction is near, the lossof ground from installing bored piles or the heave from installing driven piles can bedetrimental. If the pile driving is some distance away, information from the technicalliterature can be helpful (Ram shaw et al., 1998 ; Dra bkin & Lacy 199 8). Prudent engi

neers can establish measurement points on existing structures and have observationsmade as pile installation proceeds. In cases of sensitive machinery near new construction, the installation of transducers for measurement of time-related movements canbe helpful.

1.5 M O D E L S F O R U S E I N A N A LY S E S O F A S I N G L E P I L E

A number of models have been used for the design of piles under lateral loading, andsome of them can be used as supplements to the principal method proposed herein.

1 5 1 E la s t i c p i l e an d e l a s t i c s o i l

The m odel show n in Fig. 1.9a depicts a pile in an elastic soil. A similar model has beenwidely used. Terzaghi (1955) suggested values of the so-called subgrade modulus that

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Figure 1.9 Mode ls fo r a pile und er lateral loading.

can be used to solve for deflection and bending moment, but he went on to qualifyhis recommendations. The standard beam equation was employed in a manner thathad been suggested earlier by such writers as Hetenyi (1946). Terzaghi stated that thetabulated values of subgrade modulus could not be used for lateral loads larger thanthe one that resulted in a soil resistance of one-half of the bearing capacity of thesoil. However, no recommendation was included in regard to the computation of

the bearing capacity under lateral load, nor were any comparisons given between theresults of computations and experiments.Poulos and his colleagues have contributed extensively to developments for piles

under lateral loading using the elastic model and several variations of the model(Poulos & Davis, 1980, Poulos & Hull 1989). Solutions have been presented for avariety of cases of loading of single piles and for the interaction of piles with closespacings. The solutions have gained considerable attentio n but can not readily be usedto compute the larger deformation or collapse of the pile in nonlinear soil.

The differential equation presented by Terzaghi required the use of values of moduliwith a different format than used herein, but conversion is easily made. Values in

terms of the format used in this text are presented in Chapter 3 for the benefit of thereader. The recommendations of Terzaghi have proved useful and provide evidence thatTerzaghi had excellent insight into the problem. However, in a private conversationwith the senior writer, Terzaghi said that he had not been enthusiastic about writing

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Te c h n i q u e s f o r d e s i g n 13

the paper and only did so in response to numerous requests. The method illustratedby Fig. 1.9a serves well in obtaining the response of a pile under small loads, in illustrating the various interrelationships in the response of piles, and in giving an overallinsight into the nature of the problem. The method cannot be employed without some

modification to solve for the loading at which yielding will develop in a pile.

1.5.2 E l a s t i c p i l e a n d f i n i t e e l e m e n t s f o r s o i l

The case shown in Fig. 1.9b is the same as the case in Fig. 1.9a except that the soilhas been modeled by finite elements. No attempt is made in the sketch to indicate anapp ropr iate size of the ma p, bo und ary constrain ts, special interface elements, favoredshape an d size of elements, or other details. The finite elemen ts may be axially symm etric with non-symmetric loading, or fully three dimensional. Additionally, the elements

may be selected as linear or nonlinear.In view of computational power that is available, the model shown in Fig. 1.9b

appears to represent an ideal way to solve the pile problem. The elements can be fullythree-dimensional and nonlinear, and nonlinear geometry can be employed. However,in addition to the problem of selecting the basic nonlinear element for the soils, someother challenges are coding to disregarding tensile stresses, modeling layered soils,accounting for the separation between pile and soil during repeated loading, codingfor the collapse of sand against the back of a pile, and accounting for the changes insoil characteristics associated with the various types of loading. All of these problemscurrently have no satisfactory solution.

Yegian & Wright (1973) and Thompson (1977) did interesting studies using two-dimensional finite elements. Thompson used a plane-stress model and obtained soil-response curves that agreed well with results from full scale experiments near theground surface. Portugal & Sêco e Pinto (1993) used the finite element method basedon p-y curves to obtain a good prediction of the observed lateral behavior of thefoundation piles of a Portuguese bridge. Kooijman (1989) and Brown, et al., (1989)used three-dimensional finite elements to develop p-y curves. Research is continuingwith three-dimensional, nonlinear finite elements; for example, Brown & Shie (1991)but no proposals have been made for a practical method of design. However, a finite-element model likely will be developed that will lead to results that can be used inpractice.

1.5.3 R i g i d p i l e a n d p l a s t i c s o i l

Broms (1964a, 1964b, 1965) employed the model shown in Fig. 1.9c, or a similarone, to derive equations for predicting the loading that develops the ultimate bendingmoment. The pile is assumed to be rigid, and a solution that puts the pile in equilibriumis found by using the equations of statics for the distribution of ultimate resistance ofthe soil tha t put s the pile in equ ilibrium . The soil resistance show n h atche d in the figure

is for cohesive soil, and a solution was developed for cohesionless soil as well. Afterthe ultimate loading is computed for a pile of particular dimensions, Broms suggeststhat the deflection for the working load may be computed by using the model shownin Fig. 1.9a.

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The Broms method obviously makes use of several simplifying assumptions but canbe useful for the initial selection of a pile. A summary of the Broms equations withexamples is presented in Appendix A for the convenience of the user.

The engineer may wish to implement the Broms equations at the start of a designif the pile has constant dimensions and if uniform characteristics can reasonably beselected for the soil. Solution of the equations will yield the size and length of the pilefor the expected loading. The pile can then be employed at the starting point for the p-ymethod of analysis. Further benefits from the Broms method are: (1) the mechanics ofthe problem of lateral loading is clarified, and (2) the method may be used as a checkfor some of the results from the p-y method of analysis.

It is of interest to note that the computer code for the p-y method of analysis,implemented in Appendix D, is so efficient that many trial solutions can be made in ashort period of time. An experienced engineer can use the computer m odel to hon ein rapidly on a correct solution for a particular application w itho ut the limitationsimposed by the Broms equations.

1 5 4 C h a r a c t e r i s t i c l o a d m e t h o d

Duncan et al. (1994) presented the characteristic-load method (CLM), following theearlier wo rk of Evans & D unca n (1982). A series of solutions were made with nonlinearp-y curves for a range of soils and for a range of pile-head co ndi tion s. The results wereanalyzed with the view of obtaining simple equations that could be used for rapid

prediction of the response of piles under lateral loading. Dimensionless variables wereemployed in the prediction equations. The authors state that the method can be usedto solve for: (1) ground-line deflections due to lateral load for free-head conditions,fixed-head con ditio ns, and the flagpo le con ditio n; (2) grou nd-lin e deflections due tomoments applied at the ground line; (3) maximum moments for the three conditionsin (1); and (4) the location of the point of maximum moment along the pile. The soilmay be either a clay or a sand, both limited to uniform strength with depth.

The prediction equations take on the general form of that used for clay, shown inEq. 1.1.

(1.1)

where P c = characteristic load, b = diameter of pile, E p = modulus of elasticity of material of pile, Ri = ratio of moment of inertia of the pile to that of a solid pile of the samediameter, cu = undrained shear strength of clay.

For a given problem of applied lateral load Pt , for a pile in clay with a constantshear strength, a value of P c is computed by the equation above. The ratio P t/P c isfound and becomes the argument for entering a nonlinear curve for the value of yt/b

for free-head or fixed head cases for clay.An equation similar to Eq. 1.1 was developed for piles in sand. Also, equations and

nonlinear curves were developed for computing the value of the maximum bendingmoment and where it occurs along the pile.

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Te c h n i q u e s f o r d e s i g n 15

Figure 1.10 Mod el fo r a pile under lateral loading w ith p y curves.

Duncan and his co-workers were ingenious in developing equations and curves thatgive useful solutions to a number of problems where piles must sustain lateral loads.The limitations in the method with respect to applications were noted by the authors.

Endley et al. (1997) began with recommendations for the formulating p-y curvesand developed equations similar to those of Duncan, et al. for the prediction of pilesin various soils. The Endley equations were designed to deal with piles that penetratedonly a short distance into the ground surface as well as with long piles.

1.5.5 N o n l i n e a r p i l e a n d p y m o d e l f o r s o i l

Interest in the model shown in Fig. 1.10 developed in the late 1940's and 1950's whenenergy companies built offshore structures that were designed to sustain relativelylarge horizontal loads from waves. About the same time offshore structures were built

in the United States for military defense. R utledge (195 6). The relevant differentialequations were stated by Timoshenko (1941) and by other writers. Hetenyi (1946)presented solutions for beams on a foundation with linear response. In 1948, Palmerand Thompson presented a numerical solution to the nonlinear differential equation.

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16 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

In 1953, the American Society for Testing and Materials sponsored a conference on thelateral loading of piles, and pape rs by Gleser, and M cC am mo n a nd Asherm an notablyemphasized full-scale testing.

The offshore industry embarked on a program of full-scale testing of fully instrumented piles in the 1950's. Fortuitously, the digital computer became widely availableab ou t the same time, and as a result, full-scale testing and th e digital com put er allowe dthe development of the method emphasized in this document; contributions continuefrom engineers in many countries.

As a m atter of historical interest, Terzaghi (1955) w rot e If the hor izon tal load ingtests are made on flexible tubes or piles - values of soil resistance - can be estimated forany d epth , if the tube or pile is equip ped w ith fairly closely spaced strain gauges and if,in addition, provisions are made for measuring the deflections by means of an accuratedeflectometer. The strain-gauge readings determine the intensity and distribution ofthe bending moments over the deflected portion of the tube or the pile, and on thebasis of the mo me nt d iagram the intensity and distribution of the horizontal loads canbe ascertained by an analytical or grap hic pro ced ure . ... If the test is repeate d fordifferent horizontal loads acting on the upper end of the pile, a curve can be plottedfor different depths showing the relationship between p and y. Terzaghi goes on towrite , How ever, errors in the com puta tion of the deflections are so imp ortan t that theproced ure cann ot be recom men ded. (Meaning unch anged , but some terms changedto agree with terminology used herein. Authors).

Matlock and his associates devised an extremely accurate method of measuringthe bending moments and formal procedures for interpreting the data. (Matlock &Ripperger 1956; Matlock & Ripperger 1958). Two integrations of the bending-m om ent d ata yielded accurate values of deflection, b ut special techniq ues were requ iredfor the two differentiations to yield adequate values of soil resistance. The result wasthe first set of comprehensive recommendations for predicting the response of a pileto lateral loading. Terzaghi visited the test site while participating in the Eighth TexasConference on Soil Mechanics and Foundation Engineering in 1956 and, in commentsto Matlock and the senior author, appeared to be interested in the direction of theresearch.

As shown in Fig. 1.10, loading on the pile is general for the two-dimensional case;no torsion or out-of-plane bending is assumed. The horizontal lines across the pile are

meant to show that it is made up of different sections; for example, steel pipe could beused with th e wall thickness varied along the length. The difference-equation m etho dpresented in detail in Chapter 2, as employed for the solution of the beam-columnequation, allows the different values of bending stiffness {EpIp) to be considered.Furthermore, the method of solution allows p I p to be nonlinear and a function of thecom puted values of bending mo me nt. For m any solutions it is unnecessary to vary thebending stiffness, even though the loading is carried to a point where a plastic hingeis expected to develop.

An axial load is indicated and is considered in the solution with respect to its effecton bending a nd no t in regard to comp uting the required length to sup port a given axial

load. As shown later, the computational procedure allows to determine the rare caseof the axial load at which the pile will buckle.

The soil around the pile is replaced by a set of mechanisms that merely indicate thatthe soil resistance p is a no nlinea r functio n of pile deflection y. The mechanism s, and the

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Te c h n i q u e s f o r d e s i g n 17

corresponding curves that represent their behavior, are widely spaced in the sketch butare considered to be varying continuously with depth. As may be seen, the p-y curvesare fully variable with respect to distance x along the pile and pile deflection y. Thecurve for x = x\ is dra wn to indicate th at the pile may deflect a finite distance with no

soil resistance. T he curve at x = X is dra wn to sho w th at the soil is deflection-softening.There is no reasonable limit to the variations that can be employed to represent thesoil response to the lateral deflection of a pile.

The p-y method is versatile and provides a practical means for design. The methodwas suggested over thirty years ago (McClelland & Focht 1958; Reese & Matlock1956). Two developments during the 1950's made the method possible: the digitalcomputer for solving the problem of the nonlinear, fourth-order differential equationfor the beam-column; and the remote-reading strain gauge for use in obtaining soil-response {p-y) curves from experiment.

The p-y meth od evolved principally from research sponso red by the petroleum industry when faced with the design of pile-supported platforms subjected to exceptionallylarge horizontal forces from waves and wind. Rules and recommendations for usingthe p-y method for the design of such piles are presented by the American PetroleumInstitute (1987) and Det Norske Veritas (1977).

The use of the method has been extended to the design of onshore foundations,as exemplified by publications of the Federal Highway Administration (USA) (Reese1984). T he procedure is being cited broadly, for exam ple by Jam iolko wsk i, 19 77,Baguelin et al., 1978, George & Wood, 1977, and Poulos & Davis, 1980. The methodhas been used with success for the design of piles; however, research is continuingand improvements, particularly in the characterization of a variety of special soils,are expected. At the Fou ndatio n Engineering Congress, ASCE, Evanston, Illinois, July2 5 - 2 9 , 1989, one of the keynote papers and 14% of the 125 papers dealt with someaspect of piles subjected to lateral loading.

1.6 M O D E L S F O R G R O U P S O F P IL E S U N D E R L A T E R A LL O A D I N G

Piles are most often used in groups as illustrated in Fig. 1.11, and a practical example,

an offshore platform, is shown in Fig. 1.1. The models that are used for the group ofpiles must address two problems: the efficiency of closely-spaced piles under lateralloading (and axial loading); and the distribution of the loading to each of the piles inthe group, a problem in mechanics.

The efficiency of a particular pile is defined as the ratio of the load that it cansustain in close spacing to the load that could have been sustained if the pile hadbeen isolated. Because of the variability of soil and the complex nature of constitutivemodels, theoretical solutions are currently unavailable for computing the efficiencyof a particular pile. Methods for finding the efficiency, both under lateral and axialloading, are based on the results of experiments, most of which are from the laboratory.

In contrast, if one can assume that the procedures are accurate for analyzing a singlepile under lateral loading (and under axial loading), the problem of the distributionof the loading for each of the piles in a group can be solved exactly. A model forthe solution of the problem in mechanics is shown in Fig. 1.12. As may be seen in

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18 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

\

Figure 1.12 Simplified structu re show ing coord inate systems and sign conventions: a) w ith piles show n;b) w ith piles represe nted as springs after Reese and Matloc k 1966).

Fig. 1.12a a global coordinate system is established for the loadings on the structureand for identifying the positions of each of the pile heads and their angle of rake.Then, a local coordinate system is utilized for each of the piles with axial and lateralcoordinates.

Figure l.l I Structure su ppo rted by a group of piles.

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Te c h n i q u e s f o r d e s i g n 19

Figure 1.13 Mod el of a pile under axial load.

Figure 1.12b shows that each of the piles is replaced by nonlinear mechanisms thatgive the resistance to axial movement, lateral movement, and rotation as a function ofpile-head m ove me nt. Also show n in Fig. 1.12b is a set of mo vem ents of the origin or theglobal coordinate system. With these movements, the lateral and vertical movementsand the rotation can be found at each pile head. The forces generated at the pile headsserve to put the structure into equilibrium. Because of nonlinearity, iteration is requiredto find the unique movements of the global coordinate system.

The model for the pile under lateral loading, already described, is used for findingthe pile forces as a function of a lateral deflection and a pile-head rotation. Fig. 1.13shows the model that can be employed to find the axial force on the pile as a functionof settlement. As may be seen, nonlinear mechanisms are used to represent the soilresistance in skin friction and in end bearin g as a function of axial mo vem ent. A lso, thespring repre senting the stiffness of the pile can be non linear if necessary and desirable.

A more detailed description of the method of solving for the distribution of loading

to piles in a group is presented in Chapter 5.As noted in the description presented above, all of the loading on the superstructureis assumed to be taken by the piles and none by the cap or raft supported by thepiles. The sketch in Fig. 1.11 shows that the cap is resting on the ground surface, andsettlement would cause some of the loading to be taken by the cap. The problem offinding the distribution of axial load to the cap (or raft) and to the piles has beenaddressed by a number of authors. While the proposed solutions are limited to thefound ation system response to axial loading, a brief introdu ction to the technology iswarranted because extensions will likely occur to the general problem, where lateralloading is an important parameter.

Van Impe & De Clercq (1994) reviewed the work of a number of authors andelected to extend the solution proposed by Randolph & Wroth (1978). Those authorsemp loyed a two-layer system for the soil. Each of the layers wa s characterized by a shearmodulus G and a Poisson's ratio v. Using these param eters, equ ations w ere developed

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20 Single Piles and Pile Gr oup s Un der Late ral L oading

for the settlement of a rigid pile due to load transfer along the pile in skin friction, dueto load in end bearing and to load on an element of the raft. The equations were solvedto find the identical settlements for the elements of the system in order to obtain thedistrib utio n of load to the piles and to the cap. An im po rtan t analytical difficulty w as to

assume a fixed rad ius for influence of distribu tion of stresses (or distrib utio n influenceof stresses) in the continuum to limit the magnitude of the computed settlements.

The contributions of Van Impe & De Clercq (1994) included using a decay curveto decrease the shear modulus with strain, developing a multi layered soil model,and introducing an improved method of. Results from the extended method werecompared with experimental results from tests of an instrumented, full-sized, pile-supp orted bridge pier. Excellent agreement was found between com puted and observedsettlement, and good agreement was found between computed and observed loads tothe elements of the system after an assumption was made about the distribution of theload due to the placement of concrete.

The results from the case studies suggest that benefits can be derived in extendingthe general method to the case of both axial and lateral loading.

1.7 S TA T U S O F C U R R E N T S TA T E O F T H E A RT

As presented in Appendix D, computer programs are readily available for solvingthe different equations, describing the behavior of a single pile and a group of piles,efficiently and in a user-friendly fashion. The use of computer codes will be demonstrated in Chapters 6 and 7; however, it is useful here to write briefly about the currentstate-of-the-art.

The com puter codes allow the engineer to make solutions rapidly in order to investigate the influence of a variety of param eters. Upp er-bound and lo wer-bou nd solutionscan be done with relative ease. Guidance can be obtained in most cases with respectto the desirability of performing additional tests of the soil or performing a full-scale,lateral-load test at the site.

The principal advances in com puta tional procedures in the future relate to p-y curves.Better information is needed for piles in rock of all kinds, in soils with both cohesionand a friction angle, and in silts. For piles in closely-spaced groups, relevant information is needed on pile-soil-pile interaction. In spite of these limitations, the technologypresented herein is believed to represent a signal advance in engineering practice withrespect to previously available methods.

H O M E W O R K P R O B L E M S F O R C H A P T E R I

1.1 (a) W ith regard to the analysis of a single pile und er lateral loading , w ha t are thetwo principal weaknesses of the models described in Sections 1.51, 1.53, and1.54?

(b) W ha t prob lem s are enco unte red in the use of the model sho wn in Figure 1.5.2?

1.2 Co m par ing the curves in Figures 1.5 a nd 1.6 show s that the loss of lateral resistance in overconsolidated clay with water above the ground surface due to the

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Te c h n i q u e s f o r d e s i g n 21

cyclic case is great, particu larly after a deflection of the pile of 5 m m . Th e tw otests represented by the figures were performed in identical soils.

(a) Make a sketch showing the pile and the overconslidated clay at and near the

gro un d surface durin g cyclic loadin g with a gro und line deflection of 5 mm ormore and after the load had been released.(b) Considering the flow of water during cyclic loading, explain whether or not

you think a theory can be developed to predict the loss of resistance. (Hint:You can assume a theory has been developed to predict the amount of scourof an overconsolidated clay as a function of time and the velocity of waterflow.)

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Chapter 2

D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d so f s o l u t i o n

2 .1 INTRODUCTIONThe equation for the beam-column must be solved for implementation of the p-ymethod, and a derivation is shown in this chapter. An abbreviated version of theequation is shown and can be solved by a closed-form method for some purposes, buta general solution can only be made by a numerical procedure. Both of these kinds ofsolution are presented.

Also presented is the use of dimensional analysis to develop nondimensional expressions for the case where Epy = kPyx, a solutio n for linearly increasing reaction th at ismo st useful for clays. An exam ple pro blem is wo rke d to sh ow th e relevance of this case

to practical applications. The solution with the linearly increasing reaction is also helpful in demonstrating the nature of the nonlinear method of analysis, and the methodcan be used in checking computer solutions that are presented later.

2.2 D E R I VA T I O N O F T H E D I F F E R E N T I A L E Q U A T I O N

In most instances, the axial load on a laterally loaded pile has relatively small or littleinfluence on bending moment. However, there are occasions when it is desirable tofind the buckling load for a pile; thus, the axial load is needed in the derivation. The

derivation for the differential equation for the beam-column on a foundation was givenbyH e teny i (1 9 4 6 ) .

The assum ption is mad e that a bar on an elastic foundatio n is subjected to horizon talloading and a pair of compressive forces P x acting in the center of gravity of the endcross-sections of the bar.

If an infinitely small unloaded element, bounded by two horizontals a distance dxap art , is cut out of this bar (see Fig. 2.1) , the equilib rium of mo me nts (ignoring secon d-order terms) leads to the equation

(2.1)

2.2)

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igure 2 1 Element fro m beam-column after Hetenyi 1946).

Differentiating Eq. 2.2 with respect to x, the following equation is obtained

2.3)

The following identities are noted:

And making the indicated substitutions, Eq. 2.3 becomes

2.4)

The direction of the shearing force Vv is sho wn in Fig. 2 .1 . The shea ring force in theplane normal to the deflection line can be obtained as

(2.5)

Because S is usually sm all, cos S = l and sin S = tan S = dy/dx. Th us, Eq. 2.6 is obtain ed.

(2.6)

24 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 25

Vn will mostly be used in com puta tion s, but Vv can be com puted from Eq. 2.6 wh eredy/dx is equal to the rotation S.

The ability to allow a distributed force W per unit of length along the upper portio nof a pile is conve nient in solving a num ber of practical p rob lem s. Th e differential

equation then is given by Eq. 2.7.

Ass um ption 8 can be addressed by including m ore terms in the differential equ atio n,but errors associated with the omission of these terms are usually small. The numerical

2.7)

where P x = axial load on the pile; y = lateral deflection of the pile at a point x alongthe length of the pile; p = soil reaction per unit length; E pI p = bending stiffness; andW = distributed load along the length of the pile.

Other beam formulas that are needed in analyzing piles under lateral loads are:

(2.8)

(2.9)

(2.10)

where V = shear in the pile; M = bending moment of the pile; and S = the slope of the

elastic curve defined by the axis of the pile.Except for the axial load Px , the sign conventions are the same as those usuallyemployed in the mechanics for beams, with the axis for the pile rotated 90° clockwisefrom the beam axis. The axial load P x does not normally appear in the equations forbeams. The sign conventions are presented graphically in Fig. 2.2. A solution of thedifferential equ ation yields a set of curves such as sho wn in Fig. 2. 3. The ma them atica lrelationships for the various curves that give the response of the pile are shown in thefigure for the case where no axial load is applied.

The assumptions that are made when deriving the differential equation are asfollows.

1. The pile is straigh t and has a uniform cross section,2. The pile has a long itudin al plane of symm etry; loads and reactions lie in tha t

plane,3. The pile material is hom ogeneo us and isotropic,4. The pr op or tio na l limit of the pile ma terial is no t exceeded,5. The mo du lus of elasticity of the pile ma terial is the same in tension and

compression,6. Tran sverse deflections of the pile are small,7. The pile is not subjected to dyn am ic load ing, and

8. Deflections due to shearing stresses are small.

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26 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

method presented later can deal with the behavior of a pile made of materials withnonlinear stress-strain properties.

2.2.1 S o l u t i o n o f r e d u c e d f o r m o f d i f f e r e n t i a l e q u a t i o n

A simpler form of the differential equation results from Eq. 2.4 if the assumptionsare made that no axial load is applied, that the bending stiffness E pI p is constantwith depth, and that the soil reaction Epy is a constant and equal to a. The first two

assumptions can be satisfied in many practical cases; however, the last of the threeassumptions is seldom, if ever, satisfied in practice.

The solution shown in this section is presented for two important reasons: (1) theresulting equations dem onstrate several factors tha t are com mo n to any solution; thus ,

Note: All of the responses of the pile and soil are shown in the positivesense: F = force; L = length

igure 2 2 Sign conv ention s.

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 27

igur 2 3 Form of the results obtained from a com plete solution.

the nature of the problem is revealed; and (2) the closed-form solution allows for acheck for accuracy of the numerical solutions that are given later in this chapter.

If the assumptions shown above and the identity shown in Eq. 2.11 are employed,a reduced form of the differential equation is shown as Eq. 2.12.

The solution to Eq. 2.12 may be directly written as:

(2.11)

(2.12)

(2.13)

The coefficients Xi, X ? X3? and X4 mu st be evaluated for the various bou nda ry conditions that are desired. If one considers a long pile, a simple set of equations can bederived. An exam ination of Eq. 2.13 sho ws that χι and X2 mu st appro ach ze ro for along pile because the term eßx will become large with large values of x.

The boundary conditions for the top of the pile that are employed for the reducedform solution of the differential equation are shown by the simple sketches in Fig. 2.4.A more complete discussion of boundary conditions is presented in the next section.The boundary conditions at the top of the long pile that are selected for the first case

are illustrated in Fig. 2.4a and in equation form are:

(2.14)

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28 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

(2.15)

(2.16)

(2.17)

Eqs. 2.16 and 2.17 are used, and expressions for deflection y, slope 5, bendingmoment M, shear V, and soil resistance p for the long pile can be written in Eqs. 2.18through 2.22.

(2.18)

(2.19)

igure 2 4 Boundary conditions at to p of pile.

From Eq. 2.13 and substituting of Eq. 2.14 one obtains for a long pile:

The substitutions indicated by Eq. 2.15 yield the following:

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 29

p— sin ßx + Mf (sin ßx + cos ßx)ß

(2.20)

(2.21)

(2.22)

It is convenient to define some functions for simplifying the written form of the aboveequations:

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

Values for Ai ,£> i,C i, and D i , are shown in Table 2.1 as a function of thenondimensional distance /3x along the long pile.

For a long pile whose head is fixed against rotation, as shown in Fig. 2.4b, thesolution may be obtained by employing the boundary conditions as given in Eqs. 2.32and 2.33.

2.32)

2.33)

Using these functions, Eqs. 2.18 through 2.22 become:

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30 Sing le P i les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 2 1 Non dimen sional coefficients fo r elastic piles wi th infinite le ngth, no axial load, constant E p /Ip constant E s .

ßx

0 00 20 40 812162 0304 06080

10. 015. 020. 0

A

1. 0000000. 9650670. 8784410. 6353790. 3898650. 1959150. 066741

0. 0422630. 0258330. 0016870. 000283

0. 0000630. 0000000. 000000

ßi

1. 0000000. 6397540. 356371

0. 0092780. 1715850. 2077060. 1793790. 0563150. 0018890. 003073

0. 000381

0. 0000130. 0000000. 000000

c

1. 0000000. 8024110. 6174060. 3130510. 109140

0. 0058950. 0563190. 0482890. 0119720. 002380

0. 000049

0. 0000380. 0000000. 000000

D

1. 0000000. 1626570. 2610350. 3223290. 2807250. 2018100. 1230600. 007026

0. 0138610. 0006930. 000382

0. 0000250. 0000000. 000000

Using the procedures as for the first set of boundary conditions, the results are asfollows:

2.34)

2.35)

2.36)

2.37)

2.38)

2.39)

It is sometimes convenient to have a solution for a third set of boundary conditions,as shown in Fig. 2.4c. These boundary conditions are given in Eqs. 2.40 and 2.41.

2.40)

The solution for long piles is finally given in Eqs. 2.35 through 2.39.

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 31

(2.41)

2.42)

2.43)

(2.44)

(2.45)

Employing these boundary conditions for the long pile, the coefficients X and X4 wereevaluated, and the results are shown in Eqs. 2.42 and 2 .43 . For convenience in writing,the rotational restraint M t/S t is given the symb ol kg.

These expressions can be substituted into Eq. 2.13, with differentiation performed asappropriate, and substitution of Eqs. 2.23 through 2.26 will yield a set of expressionsfor the long pile similar to those in Eqs. 2.27 th roug h 2 .31 and Eqs. 2.35 th roug h 2.3 9.

Timoshenko (1941) stated that the solution for the long pile is satisfactory whereßL > 4; how ever, there are occasions w hen the solutio n of the reduced differentialequ ation is desired for piles th at have a non dim ensio nal length less tha n 4. The s olutionfor any length of pile L can be obtained by using the following boundary conditionsat the tip of the pile:

(M is zero at pile tip)

0, (V is zero at pile tip)

at x

at x

When the above boundary conditions are fulfilled, along with a set for the top ofthe pile, the four coefficients χι,Χ 2,Χ 3, and X4 can be evaluated. T he solutions arenot sh ow n h ere, but Ap pend ix B includes a set of tables that w ere derived for the caseshow n in Fig. 2.4a. N ew values of the para me ters Ai,£>i, Q , and D\ were computedas a function of ßL.

In order to d em on strate the effect of length on the respo nse of a pile to lateral l oadin g,Table 2.1 may be employed. Referring to Eq. 2.27, a pile with a lateral load appliedat its top is strongly dependent on the parameter C\. Only selected values in the tablewere printed to conserve space, and an expan ded version of the table shows that pointsof zero deflection occur at nondimensional lengths close to 1.5+, 4.7+, 7.8+, 10.9+,and 17.2+. The deflection of the piles below those lengths would oscillate betweenpositive and negative values, with the deflections being extremely small. By comparingthe values in App end ix B with those in Table 2. 1 , the influence of the length of thepile can be readily seen. If the loading at the groundline is only a lateral load Pi 5 thedeflection along the length of the pile is given by a constant times the parameter C\.

Table 2.1 and the table for ßL = 10 in Appen dix B show that there will be points ofzero deflection at /3x = 1.5+ , 5. 2+ , and 8. 4+ , and th at the deflection at the tip ofthe pile will be very small. The tables in App end ix B for /3L = 4.0 , 3.5 , 3.0, 2 .8 , 2.6 ,2.4, 2.2, and 2.0 show that the number of points of zero deflection is reduced to one,

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32 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 2 5 Solving fo r the critica l length of a pile.

and that the deflection of the bottom of the pile represents a significant portion of thedeflection of the top.

The influence of the length of a pile on the groundline deflection is illustrated inFig. 2.5. Fig. 2.5a shows a pile with loads applied; an axial load is shown, but theassu mp tion is ma de th at the loa d is small, so the length of the pile will be con trolled bythe lateral load P t and the moment Mt. Com putations are made with constant loadingand con stan t pile cross section, as well as with an initial length th at will be in the long-pile range. The computations proceed with the length being reduced in increments;the groundline deflection is plotted as a function of the selected length, as shown inFig. 2.5 b. Th e figure show s that the gro und line deflection is unaffected until the criticallength is approached. At this length, only one point of zero deflection will occur in thecomputations. There will be a significant increase in the groundline deflection as thelength in the solution is made less than the critical. The engineer can select a lengththat will give an appropriate factor of safety against excessive groundline deflection.The accuracy of the solution will depend, of course, on how well the soil-responsecurves reflect the actual situation in the field.

The reduced form of the differential equation will not normally be used for thesolution of problems encountered in design; however, the influence of pile length, pilestiffness, and other parameters is illustrated with clarity.

2.2.2 S o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n by d i f f e r e n c ee q u a t i o n s

The solution of Eq. 2.7 is desirable and necessary for analyses that are encountered inpractice. The formulation of the differential equation in numerical terms and a solution

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 33

igure 2 6 Representation of deflected pile.

by iteration allows improvements in the solutions shown in the previous section. Theresulting equations form the basis for a computer program that is essential in practice.

• The effect of the axial load on deflection and bending m om ent will be consid ered,and problems of pile buckling can be solved.

• Th e ben ding stiffness E pI p of the pile can be varied along th e length of the pile.• And perhap s of mo re imp ortanc e, the soil reaction Epy can vary with pile deflection

and with distance along the pile. The concept of the soil reaction will be discussedfully in a later section, as the introduction here is presented in a generic sense.

If the pile is subdivided in increments of length h, as shown in Fig. 2.6, Eq. 2.7 indifference form is as follows:

(2.46)

where Rm = (EpIp)m ben din g stiffness of pile at po int m.The assumption is implicit in Eq. 2.46 that the magnitude of P x is constant with

depth. Of course, that assumption is seldom true. However, experience has shownthat the maximum bending moment usually occurs a relatively short distance belowthe groundline at a point where the value of P x is virtually undiminished. The valueof Px , except in cases of buckling, has little influence on the magnitudes of deflection

and bending moment, and leads to the conclusion that the assumption of a constantP x is generally valid.

If the pile is divided in to n incre me nts, n + 1 equ ation s of the sort as Eq. 2.4 6 canbe written. If two equations giving boundary conditions are written at the bottom,

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34 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

and tw o e quations are written at the top , there will be n + 5 equation s to solvesimu ltaneou sly for the n + 5 un kn ow ns . The set of algebraic equatio ns can be solvedby matrix methods in any convenient way.

The two boundary conditions that are employed at the bottom of the pile are basedon the moment and the shear. If the existence of an eccentric axial load that causes amoment at the bottom of the pile is discounted, the moment at the bottom of the pileis zero. The assumption of a zero moment is believed to produce no error in all casesexcept for short rigid piles that carry their loads in end bearing. The case where thereis a moment at the pile tip is unusual and is not treated by the procedure presentedherein. Thus, one of the boundary equations at the pile tip is

with Ro = flexural rigidity at the pile tip.Eq. 2.48 expresses the condition EpIp(d3y/dx3)-\- Px(dy/dx) = Vo at x = L. The value

of Vo should be set equal to zero for long piles with two or more points of zerodeflection.

As presented earlier, two boundary equations are needed at the top of the pile.Equations have been derived for four sets of boundary conditions, each with twoequations. The engineer can select the set that best fits the physical problem.

2.2.2.1 Shear and mom ent at pile head

Case 1 of the boundary conditions at the top of the pile is illustrated graphically inFig. 2.7. The axial load P x is not shown in the sketches, but P x is assumed to beacting for each of the four cases of boundary conditions at the top of the pile. For thecondition where the shear at the top of the pile is equal to Pi 5 and Rt is the ben dingstiffness at the top of the pile, the following difference equation is employed.

2.47)

Eq. 2.47 expresses the condition that EpIp(d2

y/dx2

) = 0 at x = L.The second boundary condition at the bottom of the pile involves the shear. Theassumption is made that soil resistance due to shearing stress can develop at the bottomof a short pile as deflection occurs. It is further assumed that information can bedeveloped that will allow Vo, the shear at the bottom of the pile, to be known as afunction of yo. Thus, the second equation for the boundary conditions at the bottomof the pile is

2.48)

(2.49)

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 35

Note: P tar\6 M tare know n; they are shown in the positive sensein the sketches

igure 2 7 Case I of bound ary conditions at to p of pile.

Note: P tar\6 S tare know n; they are shown in the positive sense

igure 2 8 Case 2 of boundary con ditions at to p of pile.

For the condition where the m om ent at the top of the pile is equal to Mi5 the followingdifference equation is employed.

(2.50)

2 2 2 2 Shear and rotation at pile head

Case 2 of the boundary conditions at the top of the pile is illustrated graphically inFig. 2.8. The pile is assumed to be embedded in a concrete foundation for which therotation is known. In many cases, the rotation can be assumed to be zero, at least forthe initial solutions. Eq. 2.49 is the first of the two equations that are needed. The

second of the two equations reflects the condition that the slope St at the top of thepile is known.

(2.51)

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36 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Note: P tar\6 M t/S t are shown; they are shown in the positive sensein the sketches

igure 2 9 Case 3 of boundary con ditions at to p of pile.

2.2.23 Shear and rotational restraint at pile head

Case 3 of the boundary conditions at the top of the pile is illustrated in Fig. 2.9.The pile is assumed to continue into the superstructure and become a member of aframe. The solution for the problem can proceed by cutting a free body at the bottomjoint of the frame. A moment is applied to the frame at that joint, and the rotationof the frame is computed, or estimated, for the initial solution. The moment dividedby the rotation, M t/S u is the rot atio na l stiffness provid ed by the sup erstru cture a ndbecomes one of the boundary conditions. The boundary condition has proved to bevery useful in some designs. An initial solution may be necessary in order to obtainan estimate of the moment at the bottom joint of the superstructure, then to analyzethe superstructure, and then to re-analyze the pile. One or two iterations should besufficient in most instances.

Eq. 2.49 is the first of the two equations that are needed for Case 3. Eq. 2.52expresses the condition that the rotational stiffness M t/S t is kno wn .

(2.52)

2.2.2.4 Mom ent and deflection at pile head

Case 4 of the boundary conditions at the top of the pile is illustrated in Fig. 2.10.The pile is assumed to be embedded in a bridge abutment that moves laterally for agiven amount; thus, the deflection yt at the top of the pile is known. If the embedmentis small, the bending moment is frequently assumed to be zero. The two equationsneeded at the pile head for Case 4 are Eqs. 2.50 and 2.53

(2.53)

The four sets of boundary conditions at the top of a pile should be adequate forvirtually any situation, but other cases can arise. However, the boundary conditions

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 37

Note: P tar\6 y tare show n; they are shown in the positive sensein the sketches

igure 2 10 Case 4 of boundary con ditions at to p of pile.

igure 2 11 Solving fo r the axial load tha t causes a pile to buckle .

that are available, with a small amount of effort, can produce the required solutions.For example, it can be assumed that P t and yt are known at the top of a pile andconstitute the required boundary conditions (not one of the four cases). The Case 4equations can be employed with a few values of Mt being selected along wit h the givenvalue of yt. The computer output will yield values of P t. A simple plot will yield therequired value of Mt that will produce the given boundary condition P t.

The applic ation of the finite-difference-equation techn ique to the solution of the axialload at which a pile will buckle is illustrated in Fig. 2.11. The pile, with a projectionabove the groundline (as shown in Fig. 2.11a), has been designed for the working or

service loads that are shown. The factor of safety against buckling is found by holdingP t constant and incrementing P x. As shown in Fig 2.11b, the increase of P x will causevirtually no increase in yt until the buckling load is approached. This analysis cannotbe treated as an eigenvalue problem, so the investigator must approach the buckling

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38 Single Piles and Pile Gr oup s Un der Late ral L oading

load with small changes in P x. The equations become unstable at axial loads beyondthe critical, with nonsensical results, and the engineer must be careful not to select anaxial load that is excessive.

Th e com pu ter pr og ram t o solve the finite-difference equ ation s for the respon se ofa pile to lateral loading will be demonstrated in a subsequent chapter. The solutionsof a number of example problems will be presented. Also, case studies will be shownin which the results from computer solutions are compared with experimental results.Because of the obv ious app rox im atio ns th at are inher ent in the difference-equationmethod, a discussion will be given of techniques for the verification of the accuracyof a solution, essential to the proper use of the numerical method. The discussion willdeal with the number of significant figures to be used in the internal computationsand with the selection of increment length h. Another approximation is related to thevariation in the bending stiffness.

The bending stiffness Epip, changed to R in the difference equations, is correctlyrepresented as a constant in the second-order differential equation, Eq. 2.9.

(2.9)

In finite-difference form, Eq. 2.9 becomes

(2.54)

and, in building up the higher ordered terms by differentiation, the value of R is madeto correspond to the central term for y in the second-order expression. The errorsthat are involved in using the approximation where there is a change in the bendingstiffness along the length of a pile are thought to be small, but must be investigated asnecessary.

A derivation has been made for the case where there is an abrupt change in flexuralstiffness and is show n in Append ix C. The formulation show n in App endix C has notbeen incorporated into a computer program for distribution, but the method may bereadily implemented if desirable.

The coding for the equations show n above is implemented in the com puter pro gram

presented in Appendix D that solves the different equations. Iteration is required toachieve a solution to reflect the nonlinear response of the soil with pile deflection.

2.3 S O L U T I O N F O R Epy = kp yx l

The previous section presents a brief exposition of the application of numerical methods to the solution of a nonlinear, fourth-order differential equation. The applicationof the solution will be presented in later chapters. The remainder of this chapter will

1 The reaction modulus for the soil is referenced to the ground surface by the symbol z; however,the symbol x may be used here to reflect the ground surface and for distance along the pile. Theorigin for the case discussed here must be the same for the top of the pile and the ground surface.

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 39

show the use of the finite-difference-equation techn ique to pro du ce n ond ime nsio naltables or curves that facilitate hand computations.

A solution of the reduced form of the differential equation was presented earlier forthe case wh ere E py is constant (piles in heavily overconsolidated soils). Nondimensionaltables were presented that sh ow w ith relevance the nature of the pile problem and thatallow for checking finite-difference-equation solu tion s. How ever, the tables are ofalmost no use in solving practical pro blem s.

Dim ension al analysis, along with so lution s by finite-difference techn ique s, can leadto nondimensional curves to accommodate a wide variety of variations of soil moduliwith depth, such as Epy = k\ + k^x or Epy = kpyx. (Matlock & Reese 1962 ). Solutions that assume that Epy = kPyx do have some practical utility. Cohesionless soil andnormally consolidated clay are two cases where the stiffness is zero at the ground-line and increases rather linearly with depth . Furth erm ore, experience has show n thatuseful solutions can be obtained for some cases of overconsolidated clays. The following paragraphs show the development and implementation of the method where

y — fzpyx»

2 3 1 D im en sion al analysis

Considering the nonlinearity of p-y relations at various depths, the reaction modulusfor the soil Epy is a function of both x and y . T herefore , the form of the Epy-versus-depthrelationship will change if the loading is changed. However, it may be assumed temporarily (subject to adjustment of Epy values by successive trial) that the soil reactionis some function of x only, or that

(2.55)

For solution of the problem, the curve y(x) of the pile must be determined, togetherwith various derivatives that are of interest. The derivatives yield values of slope,moment, shear, and soil reaction as functions of depth. The principles of dimensionalanalysis may be used to establish the form of nondimensional relations for the laterallyloaded pile. With the use of model theory, the necessary relations will be determinedbetween a pro tot yp e having any given set of dimensions and a similar mo del forwhich solutions may be available.

For very long piles, the length L loses significance because the deflection may benearly zero for much of the length of the pile. It is convenient to introduce somecharacteristic length as a substitute. A linear dimension T is therefore included in thequantities to be considered. The specific definition of T will vary with the form of thefunction for soil reaction versus depth. However, for each definition used, T expressesa relation between the stiffness of the soil and the flexural stiffness of the pile and iscalled the relative stiffness fac tor.

For the case of a shear P t and a moment Mt at the top of the pile, the solution fordeflections of the elastic curve will include the relative stiffness factor and other terms(see Fig. 2.12).

(2.56)

Other boundary values can be substituted for P t and Mt.

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4 0 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 2 12 Arran gem ent for dimensional analysis.

If the assumption of linear behavior is introduced for the pile, and if deflectionsremain small relative to the pile dimensions, the principle of superposition may beemployed. Thus, the effects of an imposed lateral load P t and imposed moment Mt

may be considered separately. If y A represents the deflection caused by the lateral loadP t and if ys is the deflection caused by the moment Mi 5 the total deflection is

(2.57)

The ratios of y A to P t and of ys to Mt are sought in reaching generalized solutions forthe linearly-behaving pile. The solutions may be expressed for Case A as

(2.58)

(2.59)

The values /A and /g represent two different functions of the same terms. In each casethere are six terms and two dimensions (force and length). There are therefore fourindependent nondimensional groups which can be formed. The arrangements chosenare, for Case A,

(2.60)

and for C ase B as

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 41

(2.61)

To satisfy conditions of similarity, each of these groups must be equal for both modeland prototype, as shown below.

(2.62)

(2.63)

(2.64)

(2.65)

(2.66)

A group of nondimensional parameters may be defined which will have the samenumerical value for any model and its prototype. These are shown below.

Depth coefficient, Z = —

Maximum depth coefficient, ZnL

T

Ep T4

Soil reaction function,/ (Z) = ζγ

Eplp

Case A deflection coefficient, y =

Case B deflection coefficient, Bv

(2.67)

(2.68)

(2.69)

(2.70)

(2.71)

Th us, from definitions 2. 67 thr ou gh 2 .7 1 , for (1) similar soil-pile stiffness, (2) similarpositions along the piles, and (3) similar pile lengths (unless lengths are very great andneed not be considered), the solution of the problem can be expressed from Eq. 2.57and from Eqs. 2.70 and 2.71, as

2.72)

and for Case B,

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42 S i n g l e P i le s a n d P i l e G r o u p s U n d e r L a t e r a l L o a d i n g

By the same type of reason ing, o ther form s of the solution can be expressed as sho wnbelow.

(2.73)

(2.74)

(2.75)

(2.76)

A particu lar set of A and B coefficients mu st be obtain ed as functions of the dep thparameter, Z , by a solution of a particular mo del. However, the above expressions areindependent of the characteristics of the model except that linear behavior and smalldeflections are assumed. The parameter T is still an undefined, characteristic lengthand the variation of Epy with depth, or the corresponding form of / (Z), has not beenspecified.

From beam theory, as presented earlier, the basic equation for a pile is:

2.77)

W here an applied lateral load P t and an applied mom ent M t are considere d separatelyaccording to the principle of superposition, the equation becomes, for Case A,

(2.78)

(2.79)

Substituting the definitions of nondimensional parameters contained in Eqs. 2.67throu gh 2 .7 1 , a nondim ensional differential equatio n can be written for Case A as

2.80)

(2.81)

To produ ce a part icula r set of no ndi me nsio nal A and B coefficients, (1) f(Z) mustbe specified, including a convenient definition of the relative stiffness factor T, and

Slope, S = SA + SB

Mo ment, M = MA + MB

Shear, V = VA + VB = [P^A ,

Soil reaction, p = PA+PB

and for Case B,

and for C ase B as

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 43

(2) the differential equations (2.80 and 2.81) must be solved. The resulting A and Bcoefficients may then be used, with Eqs. 2.72 through 2.76, to compute deflection,slope, moment, shear, and soil reaction for any pile problem which is similar to thecase for which nondimensional solutions have been obtained.

To obtain the A and B coefficients tha t are needed to mak e solutions with the n ond imensional method, Eqs. 2.80 and 2.81 can be solved by use of difference equations,presented earlier in this chapter.

In solving problems of laterally loaded piles by using nondimensional methods, theconstants in the expressions describing the variation of soil reaction Epy with depth xare adjusted by trial until reasonable compatibility is obtained. The selected form ofthe soil reaction with depth should be kept as simple as possible so that a minimumnumber of constants needs to be adjusted.

A general form of Epy with depth is a power form,

(2.82)

The form Epy = kpyx is seen to be a special case of the powe r form . Th e relativ e stiffnessfactor T can be defined for any particular form of the soil reaction-depth relationship. Itis convenient to select a definition that will simplify the corresponding nondimensionalfunctions.

From the theory given above, the equa tion tha t defines the nondim ensiona l functionfor soil reaction is

(2.69)

If the form Epy = kpyxn is sub stituted in Eq. 2.6 9, the result is

(2.83)

It is convenient to define the relative stiffness factor T by the following expression.

(2.84)

(2.85)

(2.86)

Because γ Z, the general nondimensional function for soil reaction is

Substituting this definition into Eq. 2.83 gives

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4 4 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

The above expression contains only one arbitrary constant, the power n. Therefore,for each value of n which may be selected, one complete set of independent, nondi-mensional solutions may be obtained from the solution of Eqs. 2.80 and 2.81. For thecase wh ere Epy = kPyx, n is equal to 1 and

2.3.2 E q u a t i o n s f o r Ep y kp yx

While the form of the equations has already been shown, a tabulation here isconvenient. The common equations are:

(2.87)

(2.88)

(2.89)

W here only a shear P t is applied at the mudlin e, the following equ ation s are for deflection y, rotation (slope) S, moment M, and shear V, respectively. (The soil resistance pis equal to Epy times y. ) The nondimensional coefficients, obtained by employing thedifference-equation methods, shown in the equations, can be found in Appendix E.The coefficients are shown as a function of the nondimensional depth Z and thenondimensional length of the pile Z max .

(2.90)

(2.91)

(2.92)

(2.93)

Similarly, the respective equations where only a moment Mt is applied at the top ofthe pile are:

(2.94)

The nond imensio nal techn ique, along with a solution of the difference eq uation s, allowcoefficients to be computed for a variety of variations of the soil reaction with depth.However, only the case of Epy = kPyx will be shown here.

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D e r i v a t i o n o f e q u a t i o n s and m e t h o d s o f s o l u t i o n 45

(2.95)

(2.96)

(2.97)

The derivation of the fixed-head case is not shown here; however, the aboveprinciples are followed. The equation for deflection is:

The moment may be found from the following equatio n:

(2.98)

(2.99)

where F mt = 1.06 for Z max -for all higher values of Z max .

For the case where there is a rotational restraint kg at the top of the pile, the precisevalue of the parameter is a matter of some importance. Article 5.4.1.1 presents adiscussion of the manner in which the restraint against rotation of the pile head canvary. If the pile head is fixed against rotation in one case and free to rotate in anothercase, the bending mom ent will be maximum a t the head of the pile in the first case andzero in the second case. If the engineer cou ld achieve a particular pile-head restraint, thevalue of negative bending moment at the pile head wo uld be equal to the value of thepositive bending moment at some distance along the length of the pile, an interestingpossibility for achieving maximum economy.

The following equation shows the rotatio nal restraint tha t is used in the examplethat follows.

(2.100)

The sign for approximately equal may be used more pro perly in m any cases becauseof the num ber of factors tha t affect th e paramete r. In the example solution, the rotatio nof the entire structure will affect the slope at the pile head . The slope (rotation) at thetop of the pile can be found from the following equatio n.

(2.101)

where A st and B st are the nondimensional coefficients at the top of the pile and can be

found from the appropriate curves in Appendix E as a function of Z max . With thesevalues of the nondimensional coefficients for slope, the selection of a trial value of Tallows M t to be found; then the solution can proceed w ith the boundary values of P t

and M t.

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4 6 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

2.3.3 E x a m p l e s o l u t i o n

The example selected for analysis is for a marine structure. A jacket or template,consisting of welded, tubular-steel members, is constructed onshore, transported bybarg e, and set in place at an offshore locati on. On e of the legs of the jacket is show n inFig. 2.13a with the bracing for the lowest panel point at the mudline. A pile is drivenafter spacers are welded inside the jacket leg to ensure co ntacts between the pile and thejacket. Loading may be considered to arise from wave action during a storm. At thisstage of the analysis, the jacket is assumed to move laterally but not rotate, which is asatisfactory assumption if the piles are relatively rigid under axial loading.

The p-y curves to be used are shown in Fig. 2.14. The curves are typical of thosefor sand or normally-consolidated clay. The distance below the mudline to each of thecurves is shown, and three points may be noted:

1. the p-y curve for zero depth shows zero soil resistance for all deflections;2. the initial slopes to the p-y curves are linear and increase with depth; and3. the ultim ate resistance for each curve app roac hes a limiting value tha t increases

with depth.

While the curves are for no particular soil, the character of the curves is such that

the nature of the following solution is clearly indicative of a more exact solution bycomputer.

The portion of the pile above the mudline and within the jacket is shown in theupper sketch in Fig. 2.13b. As may be seen, the pile will behave as a continuous beam

igure 2 13 Pile at a leg of an offshore pla tform .

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 47

igure 2 14 Soil-resistant curves for example solution.

with loading at its lower end. W ith the pile passing beyond the upper p anel poin t in thesketch, its condition at that point is between fixed against rotation and free to rotate.Therefore, as a reasonable approximation, the relationship between Mt and St is givenby Eq. 2.102.

where ^ = the distance between panel po ints (6. 1m in this case); and E pI p = thebending stiffness of the pile within the jacket leg.

The pile is a steel pipe wit h an outsid e diam eter of 7 62 m m, a wall thickness of25.4 mm, and a length below the panel point of 50 m. The steel has a yield strength of414,000 kN/m2 . The E pI p is 800,000 m2-kN, and the ultimate bending moment wascomputed to be 5,690 m- kN , assuming no axial load. The assum ption is mad e th atno restriction exists on the deflection of the pile, and the desired solution is to findthe load that will cause a plastic hinge (the ultimate bending moment). The ultimateload can then be factored to achieve a safe load. (A more comprehensive approach to

achieving a given factor of safety will be discussed in Chapter 9).Examination of the relevant equations show that there are two unknowns that must

be solved by iteration. Firstly, a value of P t must be selected and the appropriatedeflected shape found by varying the value of T. After selecting the value of T, trial

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48 Sing le P i les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 2 2

x m )

0.000.761.522.293.816.10

Example

Z

0.000.1900.3800.5730.9531.525

computations for nondimensional m ethod .

A y

2.402.071.771.500.990.40

YA m)

0.07680.06620.05660.04800.03170.0128

B y

1.601.301.020.780.390.06

rß m)

-0 .0381- 0 .0309-0 .0243- 0 .0186-0 .0093- 0 .0014

y m)

0.03870.03530.03230.02940.02240.0114

p kN/m)

0.0021.340.063.069.562.0

E py kN/m2 )

0.006031238214331035439

solutions must be made by use of the p-y curves, the data for the pile, the respectiveequations, and the nondimensional curves in Appendix E. In using the curves in the

appendix, a value of Z max of 10 is selected with the view tha t the length of the pile isestablished from axial loading and is relatively long.The value of P t selected for the initial computations is 400 kN. The value of T is

related to the loading, w ith T being small when the loading is relatively light, and noguidelines are possible for selecting an initial value. However, convergence is rapid. Avalue of T equal to about 5 pile diam eters (4 m) is selected for the first trial.

A value of M t of —1,190 m-kN is found by solving Eqs. 2.10 1 and 2.10 2. Equations 2 .90 and 2.94 can then be used to solve for values of y. T he value of Zma x(L/T) isneeded t o get values of the non dim ensio nal coefficients from the curves in Appendix E.That value is 50/4 or 12.5 , so the curves for Z max of 10 or more are used.

The referenced equations, after substitutions, yield the following expression. Thecomputations with this expression are shown in the Table 2.2 that follows.

The selection of the starting value of T of 4.0 turned out to be fortuitous, but it isnecessary to converge to a closer result.

The second trial could have been done with a T of 3.92 m, but a better plan is to

adopt a smaller value of T to obtain a point that would plot on the opposite sideof the equality line see Fig. 2.15b). Thus, the second trial wa s made with a T tr ied °f3.5 m, which yielded a Z max of 14 .3 . The compu tat ions for this case, n ot shown here,proceeded as before, with a computed value of M t of —1,012 m- kN . Using that value,

The values of E py are plotted in Fig. 2 .15 and the best straight line, passing throughthe origin, is fitted thro ugh the points by eye. As show n for Trial 1, the solid line passesthrough 6,000 and 6.95 and the following value of k py was obtained.

(2.103)

The value of the relative stiffness factor T 0t, t can no w be computed.

(2.104)

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 49

igure 2 15 Convergence plotting for P t = 400 kN assuming £ s = kx

the relevant equation s, and the p-y curves, the results are sho wn as Trial 2 in Fig. 2.15 a.The dashed line in the figure passes through 6,000 and 5.7. The values of kpy and T0 ^may be computed.

(2.105)

(2.106)

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50 Sing le P i les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 2.16 Plots of deflection and bending m ome nt fo r example problem fo r £ s = kx.

The point for the second trial plots across the equality line, as shown in Fig. 2.15 b.A straight line can be used to connect the plotted p oints, yielding a final value of T of3.9 meters. It is unlikely th at the line connec ting the two plotted points is straight butthe assumption is satisfactory for this method.

With the value of T of 3.9 m, E qs. 2.101 and 2.102 are solved and the value of M t

was found to be —1,154 m-k N. With values of P i 5 M i5 and T, values of deflection andbending moment may be computed along the pile. Using the nond imen sional curvesfor Z max of 10, Eqs. 2.9 0 and 2.94 are used to comp ute deflection, and Eqs. 2.92 and2.96 are used to compute bending moment. The plots are shown in Fig. 2.16.

In continuing with the assigned problem, other trials were made to find the P t that

would cause a plastic hinge to develop; that is, to result in a computed value of M maxequal to M u \ t. These trials, following the procedures already dem onstra ted, yielded aP t of 1185 kN and a value of k Py of 1 0 3 k N / m3 . The relative stiffness factor T wascomputed to be 6.0 m eters. The max imum bending moment occurred at the pile he adand was — 5690 m- kN . The Z max for this value of T is 8.33 , so the curves in Appendix Eof Z max of 10 are approp riate. The com puted values of deflection and bending m omen tcan now be found by following the proced ure indicated for the P t of 400 kN computedfor the relevant equatio ns. The plots a re shown in Fig. 2.16, w ith the curves previouslyplotted for the lateral load of 400 kN.

2 . 3 . 4 D i s c u s s i o n

A number of interesting features of the solutions can be seen. The no nline ar respo nse ofthe pile to lateral load can be surmised by examining the p-y curves, but a comparison

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D e r i v a t i o n o f e q u a t i o n s a n d m e t h o d s o f s o l u t i o n 51

of the computation results reinforce this point. An increase in the lateral load from40 0 to 1 18 5 kN , a factor of less tha n 3 , caused an increase of the grou ndlin e deflectionfrom 0.038 to 0.348 m, a factor of over 9, and an increase in the maximum bendingmo men t from 1170 to 5 690 m-k N, a factor of almost five.

The point is made, then, that a proper design requires the computation of the loadthat will cause a failure, either in excessive deflection or bending moment. In thisparticular case, applying a load factor of 3.0 yields a safe load on the pile of 395 kN,which is close to the first trial that was made.

A soil failure is not possible except for a short pile under lateral loading where thevalue of T would be 2 or less. For the long pile shown in the example computations,if the soil resistance near the groundline is reduced due to cyclic loading, the soilresistance merely increases at a greater depth, which then increases the deflection andbending moment.

An examination of the curves in Fig. 2.16 for both of the loads shows that all valuesbecome close to zero at 25 to 30 m along the pile; therefore, the b otto m 20 to 2 5 mof the 50-m long pile offers little or no resistance to lateral loading. Thus, if a pilecarries only lateral loading, the designer may wish to give attention to the necessarypene tration. However, the required penetratio n is found from the factored load and notthe unfactored load. The nondimensional length of the pile in the example decreasedfrom 12.8 to 8.33 as the load was increased from 4 00 to 1,185 kN . A conv enientrule-of- thum b is tha t a lo ng pile is one wh ere there are at least tw o points of zerodeflection along its length.

Another interesting point is that the plastic hinge will develop at the top of the pile,the point where it joins the superstructure. The computed negative moment at thatpoint is mo re than twice the maxim um positive mom ent th at occurs at depths of abo ut8 m for the P t of 400 kN an d ab out 12 m for the P t of 11 85 kN . Thus a larger lateralload could have been sustained had the distance between the spacers in the jacket beenincreased above 6.1 m, as shown in Fig. 2.1 3. However, this adjustment may n ot havebeen possible in the structure.

The limitations in the solutions shown above are: (1) the pile has a constant wallthickness over its entire length; (2) no axial loading is applied; and (3) the hand computa tions are time-consum ing. These restrictions are removed w ith the com puter code,demonstrated in Chapter 6, and the designer can have much more freedom in finding

an optimum solution. For one thing, the wall thickness of the pile can be increased inthe zone of maximum bending moment. For another, the jacket leg can be extendedsome distance below the mudline in weak soil so that both the wall thicknesses of thepile and the jacket are effective in resisting bending moment.

2.4 VA L I D I T Y OF T H E M E C H A N I C S

The most serious criticism directed against the p-y method is that the soil is nottreated as a continuum but as a series of discrete, uncoupled resistances (the Winkler

approach). Several comments can be given in response to the valid criticism of themethod.

The recommendations for the prediction of p-y curves for use in the analysis of piles,given in a subsequent chapter, are based for the most part on the results of full-scale

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52 Single Piles and Pile Gr oup s Un der Late ral L oading

experiments in which the continuum effect was explicitly satisfied. Further, Matlock(1970) performed some tests of a pile in soft clay where the pattern of pile deflectionwas varied along the length of the pile by restraining the pile head in one test andallowing it to rotate in another test. The p-y curves that were derived from each of

the loading conditions were essentially the same. Thus, the experimental p-y curvesthat were obtained from experiments with fully instrumented piles will predict withinreasonable limits the response of a pile whose head is free to rotate or fixed againstrotation.

The methods of predicting p-y curves have been used in a number of case studies,shown in Chapter 7, and the agreement between results from experiment and fromcomputations ranges generally from good to excellent.

Finally, technology may advance so that the soil resistance for a given deflection ata particular point along a pile can be modified quantitatively to reflect the influence ofthe pile deflection above and below the point in question. In such a case, multi-valuedp-y curves can be developed at every point along the pile. The analytical solution thatis presented herein can be readily modified to deal with the multi-valued p-y curves.

H O M E W O R K P R O B L E M S F OR C H A P T E R 2

2.1 List and discuss briefly the adv antag es and disadva ntages of including a term ofaxial load in Equation 2.4.

2.2 (a) W rite the 2n d order differential for a beam and show the equ ation indifference form.

(b) Divide a cantilever beam five increments (not counting the two imaginarypoints) and show the solution for the beam with a concentrated load at itsend. Show the error in the solution you obtain.

2.3 Explain why a numerical solution to the prob lem of pile buckling, showing inFigure 2.11, must be solved numerically, and why an explicit solution is notpossible.

2.4 H ow w ou ld the solutio n to the pro blem of the analysis of a pile as one leg ofan offshore platform be changed if the space between the pile had been groutedinstead of the use of spacers at the panel points?

2.5 (a) For the exam ple solutio n sho wn in Section 2 .3 .3 , make an additional trialusing a value of T of 3.9 m.(b) Recognizing that the computer code can make an iterative solution of

the nonlinear problem show in the example in seconds, which allows theengineer to learn the effect of modifying a parameter. Explain when youthink the engineer might wa nt to use the han d-so lutio n presented in theexample.

2.6 Study the example p -y curves show n in Figure 2.14 for the example problem andsuggest ways that soil mechanics may be used to obtain such curves for a givensoil instead of employing the expensive field tests described later.

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Chapter 3

M o d e l s f o r r e s p o n s e o f s o i la n d w e a k r o c k

3 .1 INTRODUCTION

The presentation herein deals principally with the formulation of expressions forp-y curves for soils and rock under both static and cyclic loading. A number offundamental concepts are presented that are relevant to any method of analyzingpiles. Chapter 1 demonstrated the concept of the p-y method, and this chapter willpresent details to allow the computation of the behavior of a pile under a variety ofconditions.

More than in any other deep foundation problem, the solution for a pile underlateral loading is arduous because a successful analysis is sensitive to the stress-straincharacteristics of the soil around the pile shaft. Among the concepts presented inChapter 1 was the principle that the soil-reaction modulus is not a soil parameter,but that it depends on soil resistance and pile deflection. For a given soil profile, thesoil-reaction modulus is influenced intrinsically by the following variables:

- pile type and flexural stiffness,- sho rt term , long term , or cyclic load ing,- pile geometry,- pile tip and pile cap con ditio ns,- slope of the gro un d at the pile,- pile installation proce dure, and

- pile batter.

In spite of the complexities noted above, the soil-reaction m odu lus has the advan tageof analytical simplicity and has been validated worldwide through well-documentedcase records. Furthermore, and perhaps of most importance, the method has advancedto become an accepted procedure for describing the nonlinear behavior of theinteraction between the piles and the supporting soil.

This chapter will provide recommendations for selecting a family of p-y curves forvarious cases of soils and loadings. The resulting curves are intended to reflect aswell as possible the deflection and the bending moment as a function of pile depth

under lateral loading. Case studies presented in Chapter 7 compare experimental andcomputational results for a range of soils and landing.

The results of the comparisons in Chapter 7 show that bending moment with lengthalong a pile can generally be computed more accurately than deflection. Thus, if the

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54 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

engineer must design a pile-supported structure that is sensitive to deflection, a fieldload test may be indicated.

The next section of the chapter presents a detailed discussion of the relevance of soilparameters and shows that any solution of a problem requires a thorough discussionof the soil profile. The correspondence of, and differences between, £s for the soil andEpy for the pile are discussed in detail.

3.2 M E C H A N I C S C O N C E R N I N G R E S P O N SE O F S O I LT O L AT E R A L L O A D I N G

3.2.1 S t r e s s - d e f o r m a t i o n o f s o i l

Some discussion was presented in Chapter 1 concerning models for the soil responseto the lateral deflection of a pile. A more detailed discussion is presented here. Plainly,any solution to a problem of lateral loading requires the determination of a range ofsoil properties. In the general discussion that follows and in specific recommendationsat the end of this chapter, the measuring of the relevant properties of soil is an integralstep in the process.

Finding the applicable properties of soil in the laboratory to direct the solutionof a particular problem requires attention to numerous details: characterization of thesite; selection of representative samples; employing techniques in sampling to minimizedisturbanc e; protecting and transp orting sam ples to avoid loss of m oisture; p repa rationof specimens; selection of appropriate testing procedures; and performance of testswith precise controls.

An appropriate cylindrical specimen is presumed to have been obtained and testedto represent the properties of the soil at a particular point in the continuum wherean analysis is to be performed. Further, the assumption is made that any influenceof the proposed construction is negligible. The specimen is initially subjected to anall-aroun d stress as where as is selected to reflect properties at a particular point in thecon tinuu m . The load is applied to the top to cause a stress increase equal to Δ σ , withthe principal stress a\ on a horizontal plane becoming as + Δ σ . The strain ε is defined

as the shortening of the specimen Ah divided by the original height of the specimenh. Loading is assumed to continue in increments until the resulting curve of principalstress versus strain becomes asymptotic to a line parallel to the strain axis, as shownin Fig. 3.1a.

A series of such tests for each of the strata that are encountered will provide data onthe strength of soil for use in design. Laboratory tests are usually complemented within situ tests in the field.

Returning to a discussion of Fig. 3.1a, two lines are drawn from the origin to a pointon the stress-strain curve. The slope of the lines, termed £s , is called the soil-reactionmodulus and represents the stiffness of the soil. The magnitude of £s is plainly related

to the value of strain to which th e line is dra w n, with th e largest value determ ined froma line that is tangent to the initial portion of the curve. In making some computation,investigators have used the largest value of £s , £ s m a x, or an average value, dependingon individual preference or related to some particular usage.

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 55

figure 3. / Results fro m testing soil specimen in the labo rato ry.

Careful measurements of lateral strain must be made, sometimes with special instrumentation, to obtain values of Poisson's ratio v. Of course, the value of v, similar tothat of £s , will vary with the loading and vertical strain.

3.2.2 P r o p o s e d m o d e l f o r d e c a y o f E s

The relationship between the stress-strain curves for a particular soil and p-y curves

is undoubtedly close; therefore, some discussion of the change in the stiffness of aparticular specimen with strain is useful. The values of £s decrease with increasingstrain, as shown in Fig. 3.1b, similarly as do the values of EPy , shown conceptually inFig. 1.4b. Van Impe (1991) proposed a model for the decay of £s , as shown in Fig 3.2.The figure shows G s /G smax where Gs£ s /( 2( 1 + v)) = and also indicates app rox ima tevalues of Poisson's ratio for sand and clay.

While the values of the properties as shown in Fig. 3.2 are generalized and do notprecisely represent the values at a specific site, the values do allow for computationsthat reflect the deformations of the soil in a continuum. Furthermore, the decay ofEpy is shown to reflect the same phenomena, but with different units, as in Fig. 1.4b.

Therefore, in general, the decay of £s with increasing strain is similar to the decayof Epy due to pile deflection. Additionally, the decay in both instances is certainlydue to the same phenomenon: a decrease in stiffness of a soil element with increasedstrain.

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56 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

3.2.3 V a r i a t i o n o f s t i f f n e s s o f s o i l E s a n d G s ) w i t h d e p t h

The previous parag raph s have show n close conceptual relationships between values of£ s and Epy; therefore, a discussion of the variation of £s with depth is desirable. The

values of Es m a x of sand and norm ally consolidated clay are zero at the ground line andincrease in some fashion with depth. The values of £Smax of some overconsolidatedclays and some rocks are approximately constant with depth.

The variation of the maximum tangent shear modulus with depth z and related tothe zero deflection of the pile has been proposed as shown below.

(3.1)

where Gsmax0 = the ma xim um soil-reaction modu lus at ground level (zero for normallyconsolidated soils); and a z = gradient of the maximum tangent soil-reaction modulus

with depth z.As will be dem on strat ed by examples an d case studies in later cha pte rs, in mo st cases,

the properties of the soil between the ground surface and a depth of 6 to 10 diameterswill govern the behavior of a pile subjected to lateral loading. The parameters that

igure 3.2 Mo del for Poisson ratio change and decay of £ s (Van Impe 1991).

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 5 7

govern the stiffness of near-surface soils have been investigated extensively during thepast two decades. Of particular significance are the works of C. C. Ladd, et al. (1977)and C. P. W rot h, et al. (197 9), that were pub lished in the early eighties, and th e w ork sof K. H. Stokoe, et al. (1985, 1989, 1994) and M. Jamiolkowski, et al. (1985, 1991,

1993), that were published from the eighties until the present.For cohesive soils, progress has been made in describing the ratio of ESmax/£> where

c is equal to the undrained shear strength. In these investigations, the results are dependent on the type of test, the over-consolidation ratio, and the index properties of thesoil. For cohesionless soils, studies have been aimed at finding the ratio of Esm2LX/p

f,where p is the mean effective stress.

Some data have been reported in the literature for cohesive soils for values ofEsmax u where researchers have observed considerable scatter in the results. Valuesof Esmâx/cu in the range of 40 to 200 were reporte d by M atloc k, et al., 195 6; andReese, et al., 1968. Observed values probably would have been reported as muchhigher had very careful attention been given to the early part of the laboratory curves.Stokoe (1989) reported that values of 2,000 were routinely found for very small strains.Johnson (1982) performed tests with the self-boring pressuremeter and found valuesof E smâx/cu th at rang ed from 1,440 to 2,8 40 . These high values are for extremely smallstrains.

3.2.4 I n i t i a l s t i f f n e s s a n d u l t i m a t e r e s i s t a n c eo f p y c u r v es f r o m s o i l p r o p e r t i e s

The typical p-y curve for some depth z\ below the ground surface, shown in Fig. 1.4a,is characterized by a straight line from the origin to point a and a value pu\t for theultimate resistance beyond point b. Elementary solutions, based on soil properties,are presented in this section for these two important aspects of p-y curves. As shownlater, recommendations for the formulation of p-y curves are strongly based on resultsfrom full-scale testing of piles, but analytical ex pressions are helpful in interp reting theexperiments.

3.2.4.1 Initial stiffness of p y curves

Relevance. As may be seen in Fig. 1.4b, the po rtio n of the tota l p-y curves occupiedby this initial portion of the p -y curve is small and may have little consequence in mostanalyses. Employing the concepts emphasized herein, the design of a pile under lateralloading is based primarily on limit-states, with loads limited by bending or combinedstress or by deflection. The deflection of the pile where the initial portion of the p-ycurves would be effective would occur at a considerable distance below the groundline, and the resulting horizontal forces in the soil would have only a minor effect onthe response. In most designs, stress will control.

The computation of the deflection under the working load would, of course, makemore important the initial portion of the p-y curves. However, even in such cases it is

unlikely that the initial portion of the p-y curves would play an important role.There are some cases, on the other ha nd, w here the early part of the p -y curves needs

careful consideration. Two cases can be identified: the prediction of behavior undervibratory loading, and the design of piles in brittle soils. Rock is frequently a brittle

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m aterial, and a sudden loss of resistance can be postulated wh en the deflection reachesPoint a in Fig. 1.4b. Thus, the p-y curve would reflect a much different response thanthat shown in the figure.

Theoretical considerations. As a mea ns of establishing the param eters tha t mu st beevaluated in employing linear-elastic concepts in finding equations for the slope valuesof the initial portions of p-y curves, some elementary concepts of mechanics can beused. The following equation is from the theory of elasticity (Skempton, 1951), and itprovides a basis for deriving a simple expression giving the approximate slope of theinitial portion of the p-y curve.

(3.2)

whe re p = mea n settlemen t of a fou ndation width b; q = founda tion pressure;Ip = influence coefficient; v = Poisson's ratio of the solid; and £s = Young's modulus ofthe solid.

The equation pertains to vertical loading, but if the soil resistance p against the piledue to lateral deflection is assumed to be equal to qb and the deflection of the pile isequal to p, the equation can be rewritten as follows, with Ip and v taken as constants,as suggested by Skempton.

where ξ is a constant. While the above equation suggests that Epy has a defined relationship with £s , (discussed below with a fuller discussion of Skempton's suggestions),the relationship would appear to relate more accurately to the initial portion of thevalues of moduli.

where Epymax and £s m a x are the initial slopes of the p-y curves and the stress-straincurves, respectively, and ξ ι is given a subscript to indicate the initial value.

The values of Epymâx from the p-y curves in Figs. 1.5 and 1.6 are plotted in Fig 3.3,and the influence of the ground surface is obvious. These data suggest that the valueof z must reflect the location of the ground surface. The differences in the values of£pymax for static and cyclic loading are striking.

3.3)

3.2.4.2 Com putation of values of p u\t

The prediction of the ultimate values of p as a function of the kind of soil and thedepth below ground surface is of obvious importance. Analytical methods, as appliedhere, are important for allowing comparisons with pu\t values that are determined byexperiment, such as shown in Fig. 1.5.

Some serious mistakes were made during early years in the analysis of piles underlateral loading when engineers failed to heed Terzaghi's warning that the ultimateresistance against a pile could not exceed one-half the bearing capacity of soil. Inthe absence of more rational methods for computing a limiting value of the ultimate

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 59

igure 3 3 Values of Ε Ργ Ύ]ΛΧ from experiment (see Figs. 1.5 and 1.6).

resistance /?uit, which may be considered as the bearing capacity, two simple modelshave been employed for solving the problem by limit equilibrium.

The first of the models is shown in Fig. 3.4. The force Fp may be computed by

integrating the horizo ntal co mp one nts of the resistances on the sliding surfaces, takin ginto account the weight of the wedge. Integration of Fp with respect to the depth zbelow the ground surface will yield an expression for the ultimate resistance alongthe pile, pu\t. The simplicity of the model is obvious; however, the resulting solutionswill illustrate the equation form because the diameter of the pile, the depth below theground surface, and the properties of the soil enter into the solution.

If three-dimensional constitutive relationships become available for all soils androcks, data such as that shown in Fig. 3.5 will be helpful in developing a more realisticmod el than th at show n in Fig. 3.4. The contou rs show the heave of the groun d surfacein front of a steel-pipe pile wit h a diam eter of 64 1 m m in over-con solidated clay (Reese,

et al., 1968). With a lateral load of 596 kN , ground-surface mo vem ent occ urred at adistance from the pile axis of abo ut 4 m (Fig. 3.5a). W hen the load w as rem oved, theground surface subsided somewhat, as shown in Fig. 3.5b. The p-y curves that werederived from the loading test are shown in Fig. 1.5.

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60 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 3.4 Mode l of soil at groun d surface fo r comp uting puit.

igure 3.5 (a) G rou nd heave due to static loading of a pile; (b) Residual heave.

One can reason that, at some depth, the resistance against a wedge will increase to

the point th at ho rizontal m ovem ent of the soil will occur. The second mo del, shown inFig. 3.6a, depicts a cylindrical pile and five blocks of soil. The as sum ption is m ade tha tthe movement of the pile will cause a failure of Block 5 by shearing. Soil movementwill cause Block 4 to fail by shearing, Block 3 to slide, and Blocks 2 and 1 to fail

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 61

igure 3 6 Assumed mode of failure of soil by lateral flow around a pile (a) section through pile;(b) Mohr-Coulomb diagram for pile in sand.

by shearing. The ultimate resistance pu\t can be found by observing the difference inthe stresses σ and σ \. The model is simplistic but again should indicate the form ofthe equation for pu\t. The Mohr-Coulomb diagrams shown in Fig. 3.6 will be used todevelop equations used with flow-around failure for the soil for piles in cohesive soiland in cohesionless soil.

Cohesive soil. Two assump tions are ma de: (1) the soil is assumed to be saturated ,and (2) the undrained-strength approach will yield useful answers. Partially saturated

clays can change in water co ntent w ith time, so satura tion appe ars to be justified. Theintroduction of drainage of clays into the analysis, as noted in Chapter 1, introducesthe necessity of formulating a three-dimensional model for consolidation and will notbe addressed.

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62 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Figure 3 7 Ultimate lateral resistance for cohesive soils.

Equations for forces on the sliding surfaces in Fig. 3.6, assuming the angle a to bezero, are written and solved for F p, and F p is differentiated with respect to z to solvefor the soil resistance pc\ per unit length of the pile.

3.4)

where KC = a reduc tion factor for the shearin g resistance alo ng th e face of the pile;z = depth below the gro und surface, a nd ca = the average und rained shear strengthover the depth of the wedge.

The value of KC can be set to zero w ith some logic for th e case of cyclic load ingbecause one can reason that the relative movement between pile and soil would besmall under repeated loads. The value of ß can be taken as 45° if the soil is assumed

to behave in an undrained mode. With these assumptions, Eq. 3.4 becomes

(3.5)

Thompson (1977) differentiated Eq. 3.4 with respect to z and evaluated the integralsnumerically. His results are shown in Fig. 3.7 with the assumption that the value ofthe term y/ca is negligible. Plots are sho wn for the case wh ere KC is assumed equal tozero or equal to 1.0. Fig. 3.7 also shows a plot of Eq. 3.5 under the same assumptionwith respect to y/ca. As may be seen, the differences in the plots are no t great abov e anondimensional depth of about 3.2.

The second of the two models for computing the ultimate resistance pu\t is investigated , as sho wn below. The ultimate soil resistance pel can be found from the differencebetween σ^ and σ \ in Fig. 3.6b. Four of the blocks are assumed to fail by shear andthat resistance due to sliding is assumed to occur on both sides of Block 3. The value

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of (σ< — σ \) is found to be 10c. Other work, not shown here, shows that there isjustification of using the following equation for pc2.

The value from Eq. 3.6 is also shown plotted in Fig. 3.7 below an intersection withEq. 3.5. Solving for the intersection between Eqs. 3.5 and 3.6, ignoring the influence ofthe term y/ca and assuming that the undrained shear strength is constant with depth,Eq. 3.5 will control to where z/b is equal to abo ut 3 .2. In practice, the imp ortanc e ofthe reduced ultimate resistance at the groundline is significant.

Tho mp son (1977) noted that Hansen (1961a, 1961b) formulated equ ations for computing the ultimate resistance against a pile at ground surface, at moderate depth, andat great depth . Ha nsen considered the roughn ess of the pile wall, the friction angle, andunit weight of the soil. He suggested that the influence of the unit weight be neglected,

and he proposed the following equation for the 0 = 0 case for all depths.

(3.7)

(3.6)

Equ ation 3.7 is also show n plotted in Fig. 3.7. The agreement with the bloc k solutions is satisfactory near the ground surface, but the difference becomes significantwith depth.

Equations 3.5 and 3.6 were used in analyzing the results from full-scale experimentsand the form of the expressions appears to be valid. The recommended methods ofcomputing the p-y curves for clays are presented later in this chapter.

Cohesionless soil. Full drain age is assum ed in the analyses tha t follow. For mo stgranu lar soils, this assum ption is valid. The tw o mod els presented earlier are employedby following a similar procedure to that used for clay, except that no reduction factorwas considered for shearing resistance along the face of the pile. The ultimate soilresistance near the ground surface per unit length of the pile is obtained by finding thetotal force against an upper portion of the pile and by differentiating the results with

respect to z.

(3.8)

where Ko = coefficient of earth pressure at rest; Ka = m inim um coefficient of activeearth pressure; and as = value of a for sand.

Bowman (1958) performed some laboratory experiments with careful measurementsand suggested values of as from 0/2 to 0/3 for loose sand and up to φ for dense sand.The value of β is approximated by the following equation.

(3.9)

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64 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

(3.10)

The model for computing the ultimate soil resistance at some distance below theground surface was implemented. The stress σ \ at the back of the pile must be equal orlarger than the minimum active earth pressure; if this is not the case, the soil could failby slump ing. The assum ption is based on two -dime nsional behavior; thu s, it is subjectto some uncertainty. If the states of stress shown in Fig. 3.6c are assumed, the ultimatesoil resistance for horizontal movement of the soil is

The equations for (pu\t)Sa a n d (Puk)sb a r e admittedly approximate because of theelementary nature of the models. However, the equations serve a useful purpose inindicating the form, if not the magnitude, of the ultimate soil resistance.

3.2.5 S u b g r a d e m o d u l u s r e l a t e d t o p i l e s u n d e r l a t e r a ll o a d i n g

The concept of the subgrade modulus has been presented in technical literature fromearly days, and values have been tabulated in textbooks and other documents. Engineers performing analyses of piles unde r lateral loading, prior to developments repo rtedherein, sometimes relied on ta bulate d values of the subgrad e m odu lus in getting the soilresistance. Numerical values of the subgrade modulus are certainly related to valueso f £ s; and to Epy in some ways; therefore, a brief elementary explanation of the termsubgrade modulus by wa y of a simple experim ent is desirab le.

At the outset in the discussion of subgrade modulus, the influence of the groundsurface, where compressive stresses are zero, on the response to lateral loading may bediscerned by referring to Figs. 1.5 and 1.6. The effect of the ground surface is revealedfor all of the curves that were shown, to a depth of almost 5 diameters. Soil-resistancecurves for the depths shown in those figures have a dominant effect on the responseof a pile to lateral loading. Therefore, the following discussion, while of academicinterest, has relevance to the behavior of a pile only at some distance below the groundsurface.

Figure 3.8a shows a plan view of the plate with m and n indicating the lengths ofthe sides. If a concentrated vertical load is applied to the plate at the central point, the

resulting settlement is shown by Section A-A in Fig. 3.8b, along with an assumed uniform distributed load. If increasingly larger loads are applied, a unit load-settlementcurve is subsequently developed, as shown by the typical curve in Fig. 3.8c. The figure indicates that the magnitude of the unit load reached a point where settlementcontinued without any increase in load.

If a plate with dimensions larger or smaller than given by m and n is employed in thesame soil, one could expect a different result. Further, the stiffness of the plate itselfcan affect the results, because the plate wo uld deform in a horizon tal plan e, d ependingon the method of loading. Also, soils with a friction angle will exhibit an increasedstiffness with depth. As can be understood, with exception to some special cases,

values of subgrade moduli have limited value in solutions for soil structure interactionpro ble m s. Instead , they are m ost useful in merely differentiating the stiffness of variou soils and rocks such as soft clay, stiff clay, loose sand, dense sand, sound limestone, orweathered limestone.

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figure 3.8 Descrip tion o f experim ents leading t o defintion o f subgrade modulus.

A line is drawn in Fig. 3.8c from the origin of the curve to a point corresponding tothe ultimate load. The slope of the line with units of F/L2 is defined as the sub grad emodulus, and is a measure of the stiffness of the soil under the particular loading. Thema xim um value of the subgrade modulu s wo uld be obtain ed from a line draw n th rou ghthe initial portion of the curve, with other lines, such as the one drawn, yielding lowervalues. The tabulated values of subgrade modulus shown in some publications mustrefer to the maximum values or an average value.

More recent research on in situ testing has revealed the possibility of obtaining thesubgrade mo dulu s from M ena rd pressuremeter tests (Y. Ikeda, et al., 19 98 , T. Imai,1970) and from Marchetti dilatometer tests.

From the work of Baldi, et al. (1986) and Robertson, et al. (1989), data from theflat dilatometer tests (DMT) can be used to estimate a value of Epy at a given depthfor displacement piles by using the following equation EDMT = 34.7 (pi —po) wherepi and po are readings from the dilatometer (Fig. 3.8d). A simple equation can bedeveloped to obtain a value of the modulus for the analaysis of displacement piles.

(3.11)

with F = 2 for N .C . sands; F = 5 for O .C. dense sands ; and F = 10 for N.C. clays.While the reasoning in the development of Eq. 3.11 is valid, the implementation of

the equation in design of displacement piles must await further study.

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3.2.6 T h e o r e t i c a l s o l u t i o n b y S k e m p t o n f o r s u b g r a d em o d u l u s a n d f o r p y c u r v e s f o r s a t u r a t e d c l a y s

Skem pton (1951) wr ote tha t simple theoretical con sidera tions were employed to

develop a prediction for load-settlement curves. Even a limited solution, such as forsaturated clays, is useful to reflect the practical application of theory. The theory hassome relevance to p-y curves because the resistance to the deflection of a loaded areais common to both a horizontal plate and a pile under lateral loading.

As noted in Section 3.2.4.1, the mean settlement of a foundation, p, of width b onthe surface of a semi-infinite solid, based on the th eor y of elasticity is given by Eq. 3 .2.

where (σ — oi)f = failure stress.Equations 3.12 and 3.13b show that, for the same ratio of applied stress to ultimate

stress, the strain in the footing test (or pile under lateral loading) is related to the strainin the laboratory compression test by the following equation:

(3.2)

where q = found ation pressure; Ip = influence coefficient; v = Poisson's rati o of thesolid; and £s = Young's modulus of the solid.In Eq. 3.2, Poisson's ratio can assumed to be 1/2 for saturated clays if there is no

change in water content. For a rigid circular footing on the ground surface Ip canbe taken as π /4 and the failure stress qf may be taken as equal 6.8 c, where c is theundrained shear strength. Making the substitutions indicated and setting p = p\ forthe particular case

(3.12)

Skempton noted that the influence value Ir decreases with depth below the surface butthe bearing capacity factor increases; therefore, as a first approximation Eq. 3.12 isvalid for any depth.

In an undrain ed com pression test, the axial strain is given by the following equ ation .

(3.13a)

where £s = Young's modulus at the stress (σ^ — σ ^).For saturated clays with no change in water content, Equation 3.13a may be

rewritten as follows, considering that (σ^ — o^)f = 2cu.

(3.13b)

(3.14)

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 6 7

Skempton indicated that the value of Ip for a footing reaches a maximum value of9cu. A curve of resistance as a function of deflection could be obtained for a long stripfooting, then , by taking po ints from a laborato ry stress-strain curve and using Eq. 3.15to obtain deflection an d 4.5 Δ σ to obta in soil resistance.

3.2 .7 P r a c t i c a l u s e o f S k e m p t o n s e q u a t i o n s a n d v a l u e so f s u b g r a d e m o d u l u s in a n a l y z i n g a p i l e u n d e r

l a t e r a l l o a d i n gSkem pton's equa tions ap pear attractive with respect to solving the problem of the laterally loaded pile in saturate d clays. However, two factors need addressing, particularlyin respect to soils in general: (1) the applicability of the equations, and tabulated values of the subgrade modulus, to portions along a real pile; and (2) accounting for thedifference in units between q and p. These two factors are discussed below, and whilesaturated clay is used principally in the discussion, the concepts presented are general.

The first of the above problems can be dealt with by referring to Fig. 3.9. The sketchin Fig. 3.9 dep icts a vertical cut that h as been ma de int o the clay with sufficient streng th

so that the clay will stand without support. A side view of two loaded areas is shownin Fig. 3.9a and a plan view in Fig. 3.9b. The lower area is a considerable distancebelow the ground surface. If the areas are loaded until failure occurs, sliding surfaceswill develop in the clay similar to that shown by the dotted lines in Fig. 3.9a. Soilmechanics, and simple logic, will show that the load to cause failure for the upperarea is much lower than for the lower area. The simple presentation, along with theearlier discussion referring to Figures 1.5 and 1.6, show that p-y curves are affected bythe distance from the curve to the ground surface. A more formal implementation ofthe concept is presented in later sections where equations are presented for the criticaldistance where the ground surface no longer has an influence on the magnitude of the

quantity p.The logic presented by Professor Skempton, while unrelated to the response of the

soil at and near the ground surface, had a significant influence on the formulation ofthe recomm ended equation s for the design of piles unde r lateral loading. For exam ple,

Skem pton's argum ents based on the theory of elasticity and also on the actual behaviorof full-scale foundations led to the following conclusion:

Thus, to a degree of approximation (20 percent) comparable with the accuracy

of the assumptions, it may be taken that Eq. 3.14 applies to a circular or squarefooting.

As may be seen in the analyses shown a bove , Skemp ton allowed the Young's m odu lusof the soil, £s , to be nonlinear and to assume values from £smax to much lower valueswhen the soil was at failure. The assumption of a nonlinear value of Ep is remarkablebecause of varying stress states of elements below the footing.

Skem pton poin ted ou t that the value of Ip for a footing w ith a length to wid th rat io of10 was reported by Terzaghi (1943) and Timoshenko (1934) to be 1.26. If the bearingcapacity factor is taken as 5.3 cm Eq. 3.14 can be written as follows.

(3.15)

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igure 3.9 Co ncep t showing importan ce of ground surface to response of soil to lateral loading.

Equation 3.15 is reproduced in Equation 3.23 for the response of piles installed in clay.Equation 3.15 is general while Equation 3.23 refers to a specific deflection.

Experimental results, and elementary theory, show that p-y curves at and near theground surface have a dominant affect on the computation of pile deflection andbending moment. The question becomes, then, what is the effect on pile responseof the soil where elements move only horizontally due to pile deflection? The questioncannot be answered in general terms and requires the solution of specific problems.Therefore, the subgrade modulus and similar theories should not be implemented orotherw ise used w ith care in solving for pile behavior, even in the regions unaffected bythe presence of the ground surface.

3.2.8 A p p l i c a t i o n o f t h e F i n i t e E l e m e n t M e t h o d F E M )t o o b t a i n i n g p y c u r v e s f o r s t a t i c l o a d i n g

The above discussion, illustrating the importance of the soils at and near the groundsurface, leads to the suggestion of applying the finite element method to developingp-y curves for static loading. The topic was discussed in Chapter 1 and re-visitedhere because of the relevance of the FEM. The problems of characterizing the soil ata test site; selecting the appropriate nonlinear, three-dimentional constitutive model;performing the three-dimensional analyses for the full range of pile deflection for anelastic pile; and taking the nonlinear soil and nonlinear geom etry into account is beyond

current capabilities.However, the predicting of p-y curves by the FEM is not beyond the capabilities

of a comprehensive research effort. Computing power is growing rapidly and toolsare available for performing the physical research. Of particular importance is that

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several sites are available w here p-y curves have been determined experimentally, suchas shown in Fig. 1.5, giving the researchers the data needed to confirm the analyticalsolutions. Improved and more advanced methods of soil investigation will likely be athand and additional soil testing at sites of previous tests of piles under lateral loading

may be required.

3.3 I N F L U E N C E O F D I A M E T E R O N p y C U RV E S

3 3 1 C l a y

Analytical expressions for p -y curves indicate tha t the term for the pile diam eter app earsto the first power. Reese, et al. (1975) describe tests of piles with diameters of 152 mmand 641 mm at the Manor site. The p-y formulations developed from the results of the

larger piles were used to analyze the behavior of the smaller piles. The computationof bending moment led to good agreement between analysis and experiment, but thecomputation of groundline deflection showed considerable disagreement, with thecomputed deflections being smaller than the measured ones. No explanation could bemade to explain the disagreement.

O'Neill & Dunnavant (1984) and Dunnavant & O'Neill (1985) report on testsperformed at a site where the clay was overconsolidated and where lateral-loadingtests were performed on piles with diam eters of 27 3 m m, 1220 mm , and 1830 m m.They found that the site-specific response of the soil could best be characterized by anonlinear function of the diameter. These studies and subsequent studies can perhaps

provide a basis for specific recommendations.There is good reason to believe that the diameter of the pile should not appear as a

linear function in p-y curves for cyclic loading of piles in clays below the water table.The influence of cyclic loading on p-y curves is discussed in the next section.

3.3 .2 Sand

No special studies have been reported on investigating the influence of diameter theinfluence of diameter on p-y curves. Case studies of piles, some of which are of largediameter, do not reveal any particular influence of the diameter. However, virtually allof the tests that have been performed in sand used only static loading.

3.4 I N F L U E N C E O F C Y C L I C L O A D I N G

3 4 1 C l a y

Cyclic loading is used in the design of piles for some of the structures mentioned inChapter 1; a notable example is an offshore platform. Therefore, a number of thefield tests employing fully instrumented piles have employed cyclic loading. The first

such tests were preformed by Matlock (1970). Cyclic loading has invariably resultedin increased deflection and bending moment above the respective values obtained inshort-term loading. A dramatic example of the loss of soil resistance due to cyclicloading may be seen by comparing the two sets of p-y curves in Figs. 1.5 and 1.6.

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70 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

The following paragraphs deal with the phenomena of gapping and scour of claybelow the water surface. As noted in Chapter 1, clouds of suspension were observedin testing of piles in stiff clay under cyclic loading, but scour was not observed byMatlock (1970) in cyclic tests in soft to medium clays. However, at the conclusion

of one set of cyclic-loading tests, Matlock placed pea-sized gravel beneath the waterand around the pile. Cycling was continued and a considerable quanity of the gravelworked down around the pile, indicating that the clay near the wall of the pile, withno pronounced gap, was weakened to allow the gravel to penetrate by gravity.

W ang (19 82) an d Lon g (198 4) did extensive studies of the influence of cyclic loadin gon p-y curves for clays. Some of the results of those studies were reported by Reese,et al. (1989). Two reasons can be suggested for the reduction in soil resistance fromcyclic loading: the subjection of the clay to repeated strains of large magnitude, andscour from the enforced flow of wate r in the vicinity of the pile. Long (1984) studied thefirst of these factors by perform ing t riaxia l tests with r epeated load ing, using specimen sfrom sites where piles had been tested. The second of the effects is present when wateris above the ground surface, and its influence can be severe.

Welch & Reese (1972) report some experiments with a bored pile under repeatedlateral loading in an overco nsolid ated clay with no free water. Durin g the cyclic loadin g,the deflection of the pile at the grou ndlin e wa s in the orde r of 25 m m . After a loadwas released, a gap was revealed at the face of the pile where the soil had been pushedback. Also, cracks a few millimeters in width radiated away from the front of the pile.Had water covered the ground surface, it is evident that water would have penetratedthe gap and the cracks. With the application of the loads, the gap would have closed,and the water carrying soil particles would have been forced to the ground surface.This process was dramatically revealed during the soil testing in overconsolidated clayat Manor (Reese, et al., 1975) and at Houston (O'Neill & Dunnavant, 1984).

The sketch in Fig. 3.10 illustrates the phenomenon of scour. A space has openedin the overconsolidated clay in front of the pile and has filled with water as load isreleased. With the next excursion of the pile, the water is forced upward from thespace at a velocity that is a function of the rate of load application. The water exitswith turbulence, and particles of clay are scoured away.

Wang (1982) constructed a laboratory device to investigate the scouring phenomenon. A specimen of undisturbed soil from the site of a pile test was brought

to the laboratory , placed in a mold , and a vertical hole abou t 25 mm in diameter w ascut in the specimen. A rod, of the same size as the hole, was placed and attached to ahinge at the base to the specim en. Water, a few m illimeters deep , wa s kept over the surface of the specimen, and the rod was pushed and pulled by a machine at a given periodand at a given deflection for a measured period of time. The soil that was scoured tothe surface of the specimen was carefully collected, dried, and w eighed . The deflectionwas increased, and the process was repeated. A curve was plotted showing the weightof soil that was removed as a function of the imposed deflection. The characteristicsof the curve were used to define the scour potential of that particular clay.

The Wang device was found to be far more discriminating about scour potential of

a clay than was the pinhole test (Sherard, et al., 1976), but the results of the Wang testcould not fully explain the differences in the loss of resistance experienced at differentsites where lateral-load tests were performed in clay with water above the groundsurface. At one site where the loss of resistance due to cyclic loading was relatively

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 71

igure 3.10 Illustra tion o f scour aroun d a pile in clay du ring cyclic loading.

small, it was observed that the clay included some seams of sand. The sand would nothave been scoured readily and particles of sand could have partially filled the space

that was developed around the pile. As noted earlier, a field study by Matlock (1970),showed that pea gravel placed around a pile during cyclic loading was effective inrestoring most of the loss of resistance; however, O'Neill & Dunnavant (1984) reporttha t placin g concrete sand in the pile-soil gap formed du ring previous cyclic loadin gdid no t pro du ce a significant regain in lateral pile-head stiffness (p. 28 2) .

While the work of Long (1984) and Wang (1982) developed considerable information about the factors that influence the loss of resistance in clays under free waterdue to cyclic loading, their work did not produce a definitive method for predictingthis loss of resistance. The analyst, thus, should use the numerical results for cyclicloading presented herein with caution. Full-scale experiments with instrumented piles

at a particular site are indicated for those cases where behavior under cyclic loading isa critical feature of the design.

3.4 .2 Sand

Very few tests of piles unde r cyclic lateral loadin g have been rep orte d. The re is evidencethat the repeated loading on a pile in predominantly one direction will result in apermanent deflection in that direction. When a relatively large load is applied, the topof the pile will deflect a significant am ou nt and allow the particles of cohesionless soilto fall into the back of the pile, preven ting the pile from r etur nin g to its initial posi tion.

Observations of the behavior of the mass of sand near the ground surface duringcyclic loadin g sup po rt the idea th at the void ratio of this sand is app roa chi ng th e criticalvalue. That is, dense sand a pparen tly loosens during cycling and loose sand apparen tlydensifies.

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72 Single Piles and Pile Gr oup s Un der Late ral L oading

3.5 E X P E R I M E N TA L M E T H O D S O F O B TA I N I N Gp y C U RV E S

Methods of getting p-y curves from field experiments with full-sized piles will be presented prior to discussing the use of analysis in obtaining soil response. The strategythat has been employed for acquiring design criteria is to make use of theoreticalmetho ds, to obtain p-y curves from full-scale field experiments, to derive such empirical factors as necessary so th at ther e is close agreem ent b etween results from adjustedtheoretical solutions and those from experiments, and finally to test the proposeddesign criteria against results of tests of full-sized piles in similar soil, as shown inChapter 7. Thus, an important procedure is obtaining experimental p-y curves.

3.5.1 S o i l r e s p o n s e f r o m d i r e c t m e a s u r e m e n t

A number of attempts have been made to make direct measurement of p and y in thefield. Measuring the deflection involves the conceptually simple process of sightingdown a hollow pile from a fixed position at scales that have been placed at intervalsalong the length of the pile. The method is cumbersome in practice and has not beenvery successful.

The measurement of soil resistance directly involves the design of an instrument thatwill integrate the soil stress at a point along the pile. The design of such an instrumenthas been proposed but none has yet been built. Some attempts have been made tomeasure the soil pressure at a few points around the exterior of a pile with the view

that the soil pressures at other points can be estimated. This method has met with littlesuccess.

3.5.2 S o i l r e s p o n s e f r o m e x p e r i m e n t a l m o m e n t c u r v e s

Alm ost all of the successful ex perim ents th at yielded p-y curves have involved the measurem ent of bending m om ent by the use of strain gaug es. The deflection can be ob tain edwith co nsiderable accuracy by two integrations of the mo men t curves. The deflectionand the slope at the groundline have to be measured accurately and it is helpful if

the pile is long enough so that there are at least two points of zero deflection alongthe pile.The computation of soil resistance involves two differentiations of a bending

moment curve. Matlock (1970) made extremely accurate measurements of bendingmoment and was able to do the differentiations numerically (Matlock & Ripperger,1958). However, most other investigators have fitted analytical curves through thepoints of experimental bending moment and have performed the differentiationsmathematically.

With families of curves showing the distribution of deflection and soil resistance,p-y curves can be plo tted . A check may be mad e of the accuracy of the analyses by

using the experimental p-y curves to compute bending-moment curves. The computedbending mo men ts should agree closely with those from experimen t.

Examples of p-y curves that were obtained from a full-scale experiment with pipepiles with a diameter of 641 mm and a penetratio n of 15 .2 m are shown in Figs. 1.5

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 73

The selection of one of the two equations for the analysis of the results of a particulartest depends on the soil at the site of the test. If the soil is a sand or a soft clay wherethe resistance of the soil at the ground surface is expected ot be zero, Equation 3.17 isselected. If the soil is an overconsolidated clay, for example, Equation 3.16 is selected.(Note: depth below the groun d surface is denoted by the symbol £, but nond imensio nalcurves have the origin at the top of the pile; hence, the symb ol x may be used for depth,as noted in Chapter 2.)

In addition to knowledge of dimensions of the test pile and properties of the soil at

the test site, four parameters for a particular loading must be measured (or determined)at the ground surface: shear, moment, deflection, and rotation. With an expression for the modulus of soil reaction (Equations 3.16 or 3.17), the two unknownparameters in the equations may be computed employing equations from the non-dimensional method (Matlock & Reese, 1962). Thus, for a particular load, the soilresistance and deflection may be computed along the length of the pile. The procedure is repeated for each of the loadings and p-y curves may be plotted at selecteddepths.

Reese & Cox (1968) applied the method to two cases involving the testing of un-instrume nted piles with useful results. Wh ile the meth od is app rox im ate, the p-y curves

com pu ted in this fashion d o reflect the measu red b ehavio r of the pile head . Soil responsederived from a sizable nu m ber of such experim ents will add significantly to the existinginformation.

As previously ind icated, the major field experim ents tha t have led to the dev elopm entof the current criteria for p-y curves have involved the acquisition of experimentalmoment curves. However, nondimensional methods of analyses have assisted in thedevelopment of p-y curves in some instances.

3.6 EA RLY R E C O M M E N D A T I O N S F OR C O M P U T I N G

p y C U RV E S

The early methods were based on intuition and insight into the problem of the pileunder lateral loading. Also, some experimental results may have been available to

(3.16)

(3.17)

and 1.6 (Reese, et al., 1975). The piles were instrumented for measurement of bendingmoment at close spacing along the length and were tested in overconsolidated clay.

3.5.3 N o n d i m e n s i o n a l m e t h o d s f o r o b t a i n i n g s o i lr e s p o n s e

Reese & Cox (1968) described a method for obtaining p-y curves for those instanceswhere only pile-head measuremen ts were made during lateral loading. They noted thatnondimensional curves can be obtained for many variations of soil-reaction moduluswith depth. Equations for the soil-reaction modulus involving two parameters wereemployed, such as shown in Eqs. 3.16 and 3.17.

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74 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Terzaghi (1955). McClelland & Focht (1958) had results from a test, but the resultswere less than complete.

3.6 .1 Terzaghi

In a notable paper, and one that is still being used, Terzaghi (1955) discussed a numberof important aspects of subgrade reaction, including the resistance of the soil to lateralloading of a pile. Unfortunately, wh ile his numerical recom men dation s reveal that hisknowledge of the problem of the pile was extensive, he failed to give any experimentaldata or analytical procedure to validate his recommendations.

The units of the quantity in the basic differential equation (Eq. 2.7) are force perunit of length (F/L) and the definition of/? is presented graphically in Fig. 1.3. Terzaghi(1955) addressed the problem of units by introducin g the quan tity \/b on the left-handside of his differential equation. However, the fundamental nature of the problem ofthe pile under lateral loading was not changed; that is, the solution of the differentialequation required that the quantity p have the units of force per unit length, whereforce is defined as shown by Fig. 1.3c; therefore, Terzaghi's formulation implicitlyassumes that p = qb.

3.6.1.1 Stiff clay

Terzaghi's recommendations for the coefficient of subgrade reaction for piles in stiffclay were based on his notion that the deformational characteristics of stiff clay are

mo re or less independ ent of depth. Th us, he prop osed, in effect, that the p-y curves

should be constant with depth and that the ratio between p and y should be defined bythe constant aj . Therefore, his family of p-y curves (though not defined in so specificterms) consists of a series of straigh t lines, all with the same slop e, and passing thr ou ghthe origin of the coordinate system.

Terzaghi recognized, of course, that the pile could not be deflected to an unlimitedextent with a linear increase in soil resistance. He stated that the linear relationshipbetween p and y was valid for values of p that were smaller than about one-half of theultimate bearing stress.

Table 3.1 presents Terzaghi's recommendations for stiff clay. The units have beenchanged to reflect current practice. The values of aj are independent of pile diameter,which is consistent with theory for small deflections.

3.6.1.2 Sand

Terzaghi's recommendations for sand are similar to those for clay in that his coefficients probably are meant to reflect the slope of secants to p-y curves rather than theinitial soil-reaction modulus. As noted above, Terzaghi recommended the use of his

Table 3.1 Terzaghi s recom men dations fo r soil modulus a T for laterally loaded piles in stiff clay.

Con sistency of Clay Stiff Very Stiff Ha rd

q y,kPa 100-200 200-400 >4 00ofT,MPa 3.2-6.4 6.4-1 2.8 12.8 up

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 75

coefficients up to the point where the computed soil resistance was equal to aboutone-half of the ultimate bearing stress.

In terms of p-y curves, Terzaghi recommends a series of straight lines with slopesthat increase linearly with depth, as indicated in Eq. 3.18.

EPy = kTz (3.18)

where kj = con stant giving variation of soil-reaction m odu lus with depth ; andz = depth below ground surface.

Terzaghi's recommended values in terms of the appropriate units are given inTable 3.2.

3.6.2 M c C l e l l a n d F o c h t f o r c l a y ( 1 9 5 8 )

One of the first papers, giving the concept of p-y curves, was presented and curveswere included from the analysis of the results of a full-scale, instrumented, lateral-loadtest. The paper showed conclusively that Epy is not just a soil property but is also afunction of pile diameter, deflection, and soil properties.

The paper recommended the performance of the consolidated-undrained triaxialtest, with confining pressure equal to the overburde n p ressure, at various depths belowthe groundline. The soil curves could be converted to p-y curves, point by point, byuse of the following equations.

p = 5.5bAa (3.19)

y = 0 5bs (3.20)

where b = diameter of pile; Δ σ = (σ \ — σ ^ ) or de viator stress from the stress-straincurve; and ε = strain from the stress-strain curve. The above equations are similar inform but different in magnitude from those that can be derived from the work ofSkempton.

In a discussion of the paper, Reese (1958) pointed out that the ultimate value of pis similar to that for bearing capacity well below the ground surface. Limit-analysiswas used to derive an equation for clays for the ultimate resistance of p at and nearthe ground surface. The influence of the ground surface will be implemented in the

recommendations for p-y curves given later.

3.7 p y C U RV E S F O R C L AY S

Three sets of recommendations are presented for obtaining p-y curves for clay. All arebased on the analysis of the results of full-scale experiments with instrumented piles. A

able 3 2 Terzaghi s re comm endations for values of k T

Relative Density of Sand

Dry or moist , k T , MN /m3

Submerged sand,/c T, MN/m3

Loose

0.95-2.80.57-1.7

for laterally loaded piles in

Medium

3.5-10.92.2-7.3

sand.

Dense

13.8-27.78.7-17.9

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76 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

comprehensive soil investigation was performed at each site, and the best estimate ofthe undrained shear strength of the clay was found. The dimensions and stiffness ofthe piles were determined accurately. Experimental p-y curves were obtained by oneor more of the techniques given earlier. Theory was used to the fullest extent andanalytical expressions were developed for p-y curves which, when used in a computersolution, yielded curves of deflection and bending moment versus depth that agreedwell with the experimental values.

Loading in all three cases was both short-term (static) and cyclic. The p-y curvesthat result from the two tests performed with water above the ground surface havebeen used extensively in the design of offshore platforms.

3.7.1 S e l e c t i o n o f s t i f f n e s s o f c l ay

A review of the recommendations for p-y curves for clay soils reveals that the stiffnessof the curves is dependent on the value of £50, the strain at one-half the maximumdifference in princ ipal stresses of the specim en of clay. The sketch in Fig. 3.11a depictsan und raine d com pressive test of a specimen of saturated clay where strain ε is definedas Ahlh. Typical stress-strain curves are plot ted in Fig. 3.11b for both ove rcon solida ted(O.C.) clay and normally consolidated (N.C.) clay.

The u ndra ined strength of the clay, cu, is indica ted for bo th of the tests as one-half ofthe maximum difference in principal stresses (σ \ — σ ). The cha racteris tic s train , £50, isthe strain corresponding to cu and is indicated in Fig. 3.11b. Also shown in Fig. 3.11bare secants drawn to the stress-strain curves, with the slopes of the secants defined as£

s. Values of strain £, in percent, are plotted in Fig. 3.11c as a function of £

s divided

by cu. The value of cu for a particular test is a constant; therefore, when £s is largeat the beginning of a stress-strain curve, the value of ε is small. The sharp increase instrain ε as £s becomes smaller is evident in the figure. The decrease in the parameterEs/cu with increasing values of strain ε is shown in Fig. 3.11c, where experimentalplots are given for both normally consolidated and over-consolidated clay. Because cu

is a constant in a particular case, the curves reflect the decay in £s .Analytical studies presented earlier revealed the relevance of the stress-strain curve

to p-y curves and £50 was selected as the single parameter to characterize the stiffnessof the stress-strain curves. The value appears in the following recommendations for

formulation of p-y curves, and should be found from laboratory data when possible.As shown in the recommendations given below, and as would be expected, the p-ycurves are closer to the /?-axis for smaller values of £50. Th us, com puted values of piledeflection will be smaller for smaller values of £50, especially at relatively lighter load s.For the relatively heavier loads, the values of the ultimate soil resistance, based onvalues of cu, are controlling the results so tha t the comp uted value of ultimate bendingm om ent is affected no t at all or only slightly by the selection of £50. Th erefore , theselection of the value of £50 has not been considered a matter of much importance ifthe engineer is mainly interested in the com puta tion of the max im um bending m om ent,as is the case for the design of an offshore platform. However, the reverse is true if

the engineer is mainly interested in the computation of the pile deflection, especiallyat relatively small loads.

If triaxia l tests are unav ailable on the clay for a pa rticu lar site, there is some guid ancein the literature on the selection of values of £50. Skempton (1951) presented a studyof the settlement of footings on clay and noted that Es/cu, correspo nding to the results

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igure 3.11 Examples of und rained stiffness to undra ined shear streng th for clays w ith low plasticity.

from testing a number of footings ranged from 50 to 200. Because £s = cu/sso asused by Skempton, his values of £5 range from 0.02 to 0.005. Skempton's plot of the

settlement of footings from various experiments showed, of course, that the settlementwas less for Es/cu in the range of 20 0 (£50 of 0.005) tha n wh en Es/cu wa s in the rang eof 50 (£50 of 0.02). These values of £50 are consistent with the experimental resultsshown in Fig. 3.11c.

Matlock (1970) performed experiments in soft clays and found from laboratory teststha t values of £5 may be assumed between 0.005 and 0.20, with the smaller value beingapplicable to brittle or sensitive clays and the larger value applicable to disturbed orremolded soil or unconsolidated sediments. An intermediate value of 0.01 is probab lysatisfactory for most purposes. Reese, et al. (1975) performed experiments on piles instiff clays and recommended somewhat smaller values of £5 than the values suggested

by Matlock. Based on experimental results and the work of Skempton (1951), valuesof £50 are recommended for normally consolidated clays, as shown in Table 3.3, andfor overconsolidated clays, as shown in Table 3.5.

In sum mary , the stiffness of clay is an im po rta nt pa ram eter and sh ould be d eterm inedspecifically by laboratory tests or by appropriate in situ tests for each site. In the

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78 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Table 3.3 Repre sentative values of £ 50 for normally consolidated clays.

Co nsistency of Clay Average value of kPa* £ 50

Soft < 48 0.020Medium 48 -96 0.010Stiff 96 -19 2 0.005

* Peck, R. B.,W. E. Hanson,T. H. Th orn bu rn, oundation Engineering 2nd Ed., 1974, pg. 20.

absence of specific data, values of £5 may be obtained from Tables 3.3 and 3.5 fromvalues of undrained shear strength. Proper selection of values of £5 is important whencomputing deflection under lateral loading, especially for relatively small values ofload. Errors in £50 are less impo rtant in computin g the value of the maxim um bendingmoment.

3.7.2 R e s p o n s e o f s o f t c l a y i n t h e p r e s e n c eo f f r e e w a t e r

3.7.2.1 Detailed procedure

The procedure described below is for short-term loading, for cyclic loading, and forafter-cyclic loading as illustrated by Figs. 3.12a, 3.12b, and 3.12c (Matlock, 1970).

The procedure for after-cyclic loading is shown in Fig. 3.12c. The value of/? is zerofrom the origin to the point of the previous maximum deflection and then the second

branch of the curve rises to intersect with the curve for cyclic loading. As shown in thesketch, the slope of the second branch is parallel to a secant through the early part ofthe static curve. The ability to formulate p-y cu rves for after-cyclic load ing is imp ort an tfor a structure, such as an offshore platform, that has undergone a storm. The stiffnessof the foundation wo uld be reduced and th e structure wou ld be more likely to v ibrateunder loads from wind or from some machines.

Th e tests from wh ich the criteria were developed w ere don e at sites wh ere the clay ha dbeen submerged for some time and where the clay was only slightly overconsolidated.The clay extended to several diameters below the ground surface. The data shownwith the case studies for Lake Austin and Sabine in Chapter 7 provided the basis for

the formulation of the p-y curves.A later section will present a method of formulating p-y curves for layered soils,

taking into account the interaction between layers with varying values of pu\t. H owever, special attention must be given to thin layers at the ground surface with waterand cyclic loading. For example, if the profile consists of a layer of sand over clay, theengineer would need to use discretion in formulating the p-y curves for cyclic loadingfor the soft clay or for stiff clay. Undoubtedly, p-y curves for cyclic loading of claysbelow the water surface are affected considerably by the kind of soil at the groundsurface and above the clay. Future experimental data undoubtedly will provide valuable guidance in formulating p-y curves for cyclic loading for clays below water. Thefollowing procedure is for static loading and is illustrated in Figure 3.12a.

1. Ob tain the best possible estimate of the varia tion of un dra ined shear streng th cu

and submerged unit weight with depth. Also obtain the value of £50, the strain

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igure 3.12 Ch arac teristic shapes of p y curves for soft clay in the presence of free water, (a) staticloading; (b) cyclic loading; (c) after cyclic loading (a fter Ma tlock 1970).

corresponding to one-half the maximum principal stress difference. If no stress-strain curves are available, typical values of £5 are given in the Table 3.3.

2. Co mp ute the ultim ate soil resistance per unit length of pile, using the smaller ofthe values given by the equations below.

(3.21)

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80 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Pult = 9cub (3.22)

where γ ' = average effective unit w eight from gro un d surface to p-y curve;z = depth from the grou nd surface to p-y curve; cu = shear strength at depth z;

and b = width of pile.M atloc k (1970) stated tha t the value of/ wa s determined exp erimentally to be0.5 for a soft clay and about 0.25 for a medium clay. A value of 0.5 is frequentlyused fo r/ . The value of pu\t is com puted at each depth where a p-y curve is desired,based on shear strength at that depth.

3. Co m pu te the deflection, 3/50, at one-half the ultim ate soil resistance from th efollowing equation:

y50 = 2.5s50b (3.23)

4. Points describing the p-y curve are now co mp uted from the following relation ship.

(3.24)

The value of p remains constant beyond y = 83/50. Th e follow ing proc ed ure is for cyclicloading and is illustrated in Fig. 3.12b.

1. Con struct the p-y curve in the same manner as for short-term static loading forvalues of p less than 0.72pu.

2. Solve Eqs. 3.21 and 3.22 sim ultane ously to find the dep th, zr, where the transitionoccurs. If the unit weight and shear strength are constant in the upper zone, then

(3.25)

If the unit weigh t and shear strength vary w ith d epth, the value of zr should becomputed with the soil properties at the depth where the p-y curve is desired.

3. If the dep th to the p-y curve is greater than or equal to zn then p is equal to0.72/?uit for all values of y greater than 3yso.

4. If the dep th of the p-y curve is less than zr, then the value of p decreases from the0.72 pu\t at y = 3yso to the value given by the following expression at y = 15yso-

(3.26)

The value of p remains constant beyond y = 15y5o.

3.7.2.2 Recommended soil tests

For determining the various shear strengths of the soil required in the p-y construction,Matlock (1970) recommended the following tests in order of preference.

1. In-situ vane -shear tests with parallel sam pling for soil identification,2. Un conso lidated-u ndrain ed triaxial com pression tests having a confining stress

equal to the overburden pressure with cu being defined as one-half the totalmaximum principal-stress difference,

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 81

igure 3.13 Shear strength profile fo r example p y curves for soft clay.

3. M inia tur e vane tests of samples in tub es,4. Unconfined compression tests, and5. Unit weight determ ination.

3.7.2.3 Example curves

An example set of p-y curves was computed for soft clay for a pile with a diameterof 610 m m . The soil profile tha t was used is show n in Fig. 3.1 3. The subm erged unitweight was assumed to be 6.3kN/m3 . In the absence of a stress-strain curve for thesoil, £50 was taken as 0.020 for the full depth of the soil profile. The loading wasassumed to be static. The p-y curves were computed for the following depths belowthe mu dline : 0, 1.5, 3 , 6, and 12 m. The plo tted curves are sho wn in Fig. 3.14.

3.7.3 Response o f s t i ff c l ay i n t h e p res enc e o f f r ee w a t e r

3.7.3.1 Detailed procedure

The following procedure is for short-term static loading and is illustrated by Fig. 3.15(Reese, et al., 1975). As may be seen from a study of the p-y curves that are recomme nded for cyclic loading , the results for the M an or site showed a very large loss of soilresistance. The data from the tests (see case study for Manor tests in Chapter 7) have

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82 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 3.14 Example p y curves for soft clay with presence of free water, static loading.

igure 3.15 Cha racteristic shape of p y curves for static loading in stiff clay in the presence of freewater.

been studied carefully, and the recommended p-y curves for cyclic loading accurately

reflect the behavior of the soil at the site. Nevertheless, the loss of resistance due tocyclic loading at Manor is much more than has been observed elsewhere, probablybecause the Manor soil was expansive and continued to imbibe water as cycling progressed. Therefore, the use of the recommendations in this section for cyclic loading

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 83

will yield conservative results for many clays. The work of Long (1984) was unable toshow precisely why the loss of resistance during cyclic loading occurred. One clue wasthat the clay from Manor was found to lose volume by slaking when a specimen wasplaced in fresh water; thus, the clay was quite susceptible to erosion from the hydraulicaction of the free water as the pile was pushed back and forth.

1. Ob tain values of und rained shear strength cm soil submerged unit weight / , pilediam eter &, and select depth z at which p-y curve is desired.

2 . Co mp ute the average und rained shear strength ca over the depth z and value ofcu at depth z.

3. Co mp ute the ultim ate soil resistance per unit length of pile using the smaller ofthe values given by the equations below:

igure 3.16 Values of constants As nd Ac.

(3.27)

(3.28)

4 . Choo se the app rop riate value of As (static case) from Fig. 3.16 for the particularnon-dimensional depth.

5. Establish the initial straight-line po rtio n of the p-y curve,

(3.29)

Use the appropriate value of ks from Table 3.4.

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84 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 3 4 Representative

kPys (static) MN/m3

/ ^ ( c yc l i c ) MN /m3

values of/c py for overconsolidated clay

Average undrained shear <

kPa

50-10013555

s.

strength

100-200270110

300-400540540

*T he average shear strength should be computed from the shear strength of the soil to a depth of pile diameters.It should be defined as half the total maximum principal stress difference in an unconsolidated undrained triaxialtest .

Table 3 5 Rep resen ta t ive va lues o f ε5ο f o r o v e r c o n s o l i d a t e d c l a y s .

Average undrained shear strengthkPa

^50

50-100 100-2000.007 0.005

300-4000.004

6. Co m pute the following:

(3.30)

Use an app rop riate value of £50 from results of labo rat ory tests or, in the abse nceof laboratory tests, from Table 3.5.

7. Establish the first pa rab olic po rtio n of the p-y curve, using the following equa tionand obtaining pc from Eqs. 3.27 or 3.28.

(3.31)

Equation 3.31 should define the portion of the p-y curve from the point ofthe intersection with Eq. 3.29 to a point where y is equal to Asyso (see note inStep 10).

8. Establish the second parab olic portio n of the p-y curve,

(3.32)

Equation 3.32 should define the portion of the p-y curve from the point wherey is equal to Asyso to a point where y is equal to 6Asyso (see note in Step 10).

9. Establish the next straight-line po rtio n of the p-y curve,

(3.33)

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 85

igure 3.1 7 Cha racteristic shape of p y curves for cyclic loading in stiff clay in the presence of freewater.

Equation 3.33 should define the portion of the p-y curve from the pointwhere y is equal to 6Asyso to a point where y is equal to 18Asyso (see note inStep 10).

10. Establish the final straight-line po rtio n of the p-y curve,

(3.34)

or

(3.35)

Equation 3.34 should define the portion of the p-y curve from the point wherey is equal to 18 Asyso and for all larger values of y (see following note). Note:

The step-by-step procedure is outlined, and Fig. 3.15 is drawn, as if there is anintersection between Eq. 3.29 and 3.31. However, there may be no intersectionof Eq. 3.29 with any of the other equations defining the p-y curve. Eq. 3.29defines the p-y curve until it intersects with one of the other equations or, if nointersection occurs, Eq. 3.29 defines the complete p-y curve.

The following procedure is for cyclic loading and is illustrated in Fig. 3.17.

1. Steps 1, 2, 3, 5 and 6 are the same as for the static case.4. Choo se the app rop riate value of Ac from Fig. 3.16 for the particular non-

dimensional depth. Compute the following:

(3.36)

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8 6 S i n g l e P i le s a n d P i l e G r o u p s U n d e r L a t e r a l L o a d i n g

7. Establish the parabo lic portio n of the p-y curve,

( 3 . 3 7 )

Equation 3.39 should define the portion of the p-y curve from the point wherey is equal to 1 8;yp and for all larger values of y (see following note).

Note: The step-by-step procedure is outlined, and Fig. 3.16 is drawn, as if there is anintersection between Eq. 3.29 and Eq. 3.37. There may be no intersection of Eq. 3.29with any of the other equations defining the p-y curve. If there is no intersection, theequation that give the smallest value of p for any value of y should be employed.

3.7 .3 .2 Recom mend ed soi l tes ts

Triaxial compression tests of the unconsolidated-undrained type with confining pressures conforming to in situ pressures are recommended for determining the shearstrength of the soil. The value of £5 should be taken as the strain during the test corresponding to the stress equal to one-half the maximum total principal stress difference.The shear strength, cm should be interpreted as one-half of the maximum total-stressdifference. Values obtained from triaxial tests might be somewhat conservative butwould represent more realistic strength values than other tests. The unit weight of thesoil must be determined.

3.7.3.3 Exam ple cu rves

An example set of p-y curves was computed for stiff clay for a pile with a diameterof 610 mm . The soil profile tha t was used is sho wn in Fig. 3.18. The sub merg ed unitweight of the soil was assumed to be 7.9 kN/m3 for the entire depth.

In the absence of a stress-strain curve, £50 was taken as 0.005 for the full depth of

the soil profile. The slope of the initial portion of the p-y was established by assuminga value of k o f 135MN /m3 . The loading was assumed to be cyclic. The p-y curveswere comp uted for the following depths below the mud line 0.6, 1.5, 3, and 12 m. Th eplotted curves are shown in Fig. 3.19.

Equation 3.38 should define the portion of the p-y curve from the point of theintersection with Eq. 3.29 to where y is equal to 0.6yp (see note in step 9).

8. Establish the next straight-line po rtio n of the p-y curve,

Equation 3.38 should define the portion of the p-y curve from the point wherey is equal to 0.6yp to the point where y is equal to 1 8;yp (see note on Step 9).

9. Establish the final straight-line po rtio n of the p-y curve,

(3.39)

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M o d e l s f o r r e s p o n s e o f s o i l a n d w e a k r o c k 8 7

igure 3.19 Example p y curves for cyclic loading in stiff clay in the presence of free water.

igure 3.18 Soil pro file used fo r example p y curves for stiff clay.

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C h a p t e r

S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l e s

4 .1 INTRODUCTION

The engineer must solve the beam-column equation, presented in Chapter 2 by nondi-mensional methods or by a computer program . The response of the soil is characterizedby equations shown in Chapter 3 or by similar methods. The diameter of a cylindricalpile as a function of depth, or the equivalent diameter of a pile with a non-circular crosssection, must be known to solve the soil-response equations. An approach for computing the equivalent diameter of a pile with a non-circular cross section is presentedin this chapter.

The solution of the beam-column equation requires values of the bending stiffnessEpip. If the engineer is interested in small deflections, a constant value for E pI p maybe employed along the pile. However, in many instances, the engineer is required tofind a loading that produces failure, which may be defined as excessive deflection orthe formation of a plastic hinge. In the latter case, the value of the ultimate bendingmoment Muit must be found, and in both instances, the value of E pI p at each crosssection must be found as a function of the applied loading.

As noted earlier, and as emphasized in this chapter, the nonlinear behavior of soilrequires that the load that causes a pile to fail must be found in order to find the safeload that can be applied. In nearly all cases of practical design, the value of Muit is

required. The value of E p Ip in the nonlinear range of the materials of which a pile isconstructed is a function of the axial load and the bending moment.Methods for the computat ion of nonlinear values of E pI p are presented in this chap

ter. A linear value of E p Ip may be used for piles made of structural steel. Iteration onpile stiffness is usually not required; rather, failure by formulation of a plastic hinge isassumed to occur when increasing the loading causes the maximum stress to reach avalue equal to the proportional l imit of the steel. A failure of the steel pile in deflection,if not achieved at a lower level of loading, is assumed to occur at the loading requiredto cause the plastic hinge.

For a pile of reinforced concrete, the value of E pI p is nonlinear beginning with a

low level of stress. Therefore, as the loading on the pile is increased, the diagram forthe nonlinear E pI p must be implemented, requiring a second level of iteration in thesolution of the differential equ ation for response of the pile. The use of nonlinear valuesof Epip is discussed in some detail in Chapters 5 and 6.

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120 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

4.2 C O M P U TA T I O N O F A N E Q U I VA L E N T D I A M E T E R O F AP IL E W I T H A N O N C I R C U L A R C R O SS S E C T I O N

The primary experiments, and most of the case studies, have been performed with pileswith a circular cross section. However, piles with other shapes are often employed, andan equivalent circular diameter for the various shapes is needed in order to employ therecommendations for p-y curves given in Chapter 3.

The sketch in Fig. 4.1a shows conceptually the stresses from the soil that would actagainst a pile with a circular cross section when the pile is deflected from left to right.If the ultim ate resistance pu is assumed to have been developed, the arrows on the rightside of the section indicate that the soil is in a failure condition. The arrows are drawnto indicate normal stresses and shearing stresses on the front half of the section. Thestresses on the back half of the section are reversed in direction and are indicated tobe small, showing that the earth pressures are reduced with deflection.

As a first approximation, one could assume that a pile with a rectangular crosssection with the same width as a circular section, one diameter, and with half thedepth of a circle, one-half diameter, would behave the same as a pile with a circularcross section.

Then, if the section in Fig. 4.1b is considered, which has a width and a depth of onepile diameter, the resistance to deflection would be greater because shearing stressescould act along the back half of the sides of the section. With the concepts presentedabove, the following equation can be written to solve for the equivalent diameter beq

of a rectangular section.

(4.1)

igur 4 1 Sketches to indicate influence of shape of cross section of pile on p u

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l es 121

where Kz = lateral earth pressure coefficient; γ ζ = effective vertical soil stress at depthz; and φζ = shear angle (between the soil and the wall of the pile) at the relevant levelof shear strain.

The value of Kz will be related to the manner in which the pile is installed, and thevalue of φζ will likely be somewhat lower than the shear angle.

The use of the above eq uation s in com puting the equivalent diame ters for the shapes

shown in Fig. 4.1 will be considered. If the dimensions d and w in Fig. 4.1b are thesame as the diameter &, the equa tions ab ove will show a some wh at larger equivalentdiameter beq. The flat shape in Fig. 4.1c will yield a smaller equivalent diameter. Thestructural shape in Fig. 4. Id , w ith the direction of loading as show n, can be treated bythe equations with the value of a taken as unity, and the value of φζ taken as equal tothe shear angle.

The equivalent diameter for a rectangular section, as computed by the above equations, will vary with the shear strength at the site and with the depth being selected.The engineer may wish to select a few depths and compute values of beq and thenaverage these values when making a solution.

As an example of how to use the equations, a square section is selected, as shownin Fig. 4.1b , with a width and depth of 300 m m. T he soil is assumed to be a sa turatedclay with an undrained shear strength of 50 kPa. The value of a is taken as 0.5; usingEqs. 4.1 and 4.2, the value of beq was co m puted to be 31 7 mm or an increase of lessthan 6% over the use of only 300 millimeters.

4.3 M E C H A N I C S F O R C O M P U T A T I O N OF Mu , t A N D p lp A S AF U N C T I O N OF B E N D I N G M O M E N T A N D A X I A L L O A D

Some manuals include values of the section modulus I for steel members at which aplastic hinge will develop; however, the influence of the axial load is not included. Anumerical procedure must be used in the computations.

where w = width of section; d = depth of section; p uc = ultimate resistance of a circularsection with a diameter b equal to w and fz = shearing resistance along the sides ofthe rectangular shape at the depth z below the ground surface.

For the undrained strength approach for cohesive soils, the shearing resistance maybe computed with the following equation.

4.2)

where a = shear strength reduction factor; and cu = undrained shear strength.The value of cu will depend on the depth z and on the best estimate of a. In obtaining

a value of of, the engineer may wish to use a value derived from equations used todescribe the behavior of piles under axial loading.

For cohesionless soils, the shearing resistance m ay be com puted with the followingequation.

4.3)

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122 Sin gle Pi les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Equations for the behavior of a slice from a beam or from a beam-column underbending or axial load are formulated. A reinforced-concrete section is assumed in thepresentation, but the concepts can be applied to a structural-steel shape. The E pI p

of the concrete member will experience a significant change when cracking occurs.

In the procedure described herein, the assumption is made tha t the tensile strength ofconcrete is minimal and that crack s will be closely spaced when they appears. Actually,such cracks will initially be spaced at some distance apart, and the change in the E pI p

will not be so drastic. Therefore, in respect to the cracking of concrete, the E pI p for abeam will change more gradually than is given by the method presented.

Because the nonlinear stress-strain curves for steel and concrete do not indicate acondition for fracture, values of the ultimate strain of these materials are selectedto reflect their failure. For concrete, the ultimate value of strain is 0 .003 ; for steel,the ultimate value of strain is 0.015. These values appear to be consistent with thosefrequently used in practice.

The method shown here can be applied to any member with a symmetrical crosssection composed of a combination of concrete and steel. However, for the purposesof this texbook, only standard shapes of sections of reinforced concrete, steel pipe,and structural shapes are con sidered.

The following derivation adopts the concept that plane sections in a beam or beam-column remain plane after loading. Thus, an axial load and a moment can be appliedto a section with the result that the neutral axis will be displaced from the center ofgravity of a symmetrical section.

The equations to be solved are as follows:

(4.4)

(4.5)

A convenient procedure of computation, then, is to: select the angle of rotation

for a section; estimate the position of the neutral axis; compute the strain acrossthe section; use numerical method s to solve for the distribution of stresses across thecross section; compute the m agnitude of the axial load by summing the forces acrossthe section as indicated in Eq. 4.4; modify the position of the neutral axis if the computed value of axial load does no t agree with the applied load ; repeat the computationsuntil convergence is achieved; solve for the bending moment by numerical method s inimplementing Eq. 4.5 ; and obtain the bending stiffness.

The equations for implementing the procedure are shown below; the equations forthe mechanics of a beam under pure bending are presented first.

The derivation shown is elementary but is included here for clarity and for a defini

tion of terms. An element from a beam with an unloaded shape of abed is shown bythe dashed lines in Fig. 4.2. The beam is subjected to pure bending and the elementchanges in shape as shown by the solid lines. The relative rotatio n of the sides of theelement is given by the small angle dû, and the radius of curvature of the elastic element

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l e s 123

igur 4 2 Po rtion of a beam subjected to bend ing

is signified by the length p. The unit strain εχ along the length of the beam is given byEq. 4.6.

4.6)

whe re Δ = deforma tion at any distance from the neutral axis; and dx = length of theelement.

From similar triangles

where η = distance from neutral axis.Equation 4.8 is obtained from Eqs. 4.6 and 4.7, as follows:

4.7)

(4.8)

From Hooke's law

(4.9)

where σχ = unit stress along the length of the beam; and £ = Young's modulus.

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124 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

Therefore,

From beam theory

where M = applied moment; and I = moment of inertia of the section.From Eqs. 4.10 and 4.11

Rewriting Eq. 4.12

Continuing with the derivation, it can be seen that dx = rdq and

For convenience, the symbol φ is substituted for d /dx; therefore, from this substitution and Eqs. 4.13 and 4.14, the following equation is found, using the subscriptsto indicate application to a pile.

Also, because Δ = h ά θ and εχ = A/dx then,

The co m puta tion for a reinforced-concrete section, or a section consisting partly orentirely of a pile, proceeds by selecting a value of φ β and estimating the position ofthe neutral axis. The strain at points along the depth of the beam can be computed byuse of Eq. 4.16, which in turn will lead to the forces in the concrete and steel. In thisstep, the assumption is made that the stress-strain curves for concrete and steel are asshown in the following section.

With the magnitude of the forces, both tension and compression, the equilibrium ofthe section can be checked, taking into account the external compressive loading. Ifthe section is not in equilibrium, a revised position of the neutral axis is selected, and

iterations proceed until the neutral axis is found.The bending moment is found from the forces in the concrete and steel by taking

moments about the centroidal axis of the section. Thus, the externally-applied axialload does not enter the equ ations . Then, the value of p I p is found from Eq. 4.15. The

(4.15)

(4.16)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l e s 125

maximum strain is tabulated, and the solution proceeds by incrementing the value ofφ β . The computations continue until the maximum strain selected for failure, in theconcrete or in a steel pipe, is reached or exceeded. Th us, the ultimate m om ent t hat canbe sustained by the section can be found.

4.4 S T R E S S - S T R A I N C U RV E S F O R N O R M A L - W E I G H TC O N C R E T E A N D S T R U C T U R A L S T EE L

Any number of models can be used for the stress-strain curves for normal-weightconcrete and structural steel. For the purposes of the computations presented herein,some relatively simple curves are used.

Fig. 4.3 shows the nonlinear stress-strain curve for concrete employed herein(Hognestad, 1951). Other formulations for the stress-strain curves for concretehave been presented by Eurocode 2, (1992); Comité Euro-International du Béton,(1978); and Todeschini, (1964). The following equations apply to the branches of theHo gnesta d curve. The value of f c is the characte ristic value of the comp ressive s trengthof the concrete, measured on concrete cylinders 150 mm in diameter and 800 mm inheight, and is specified by the engineer. The other symbols are defined below or shownin the figure.

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

igur 4 3 Stress-strain curve for con crete .

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126 Single Piles and Pile G rou ps U nd er La tera l Loa ding

igur 4 4 Stress-strain curve for steel.

where Ec = initial modulus or tangent modulus presumably of the concrete and theunits of £c,/ r, and f c a re kPa.Fig. 4.4 shows the idealized elastic-plastic stress-strain curve for steel, and, as may

be seen, there is no limit to the amount of plastic deformation. The curves for tensionand compression are identical. The yield strength of the steel fy is selected accordingto the material being used, and E is the initial modulus of the steel; the followingequations apply.

4.22)

4.23)

The models and the equations shown here are employed in the derivations that areshown subsequently.

4.5 I M P L E M E N TA T I O N O F T H E M E T H O D F O R A S T EE LH - S E C T I O N

The assumption can be made without significant error that a constant value of E pIpcan be used in the computation of the bending stiffness for all ranges of loading. Thereduced values of the steel modulus after some of the fibers have reached yield willaffect th e com pu ted v alue of the mo me nt o nly slightly. If deflection con trols th e design,the engineer may wish to modify the equation after the yield stress is reached at theextreme fibers of the structural shape to reflect the nonlinear behavior.

With regard to the ultimate bending moment, handbooks include tabulated valuesof the plastic modulus. The values are based on the distribution of stresses as shownin Fig. 4. 5. Therefore, the comp utatio n of Muit is done by a simple equation.

4.24)

where Zx = tabu lated value of plastic mo dulus (bending is assumed to be abo ut thex-axis).

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l e s 127

igur 4 5 Sketch for co mpu ting ultimate m ome nt of a structural shape

The relationship in Eq. 4.24 holds whenever effects of local buckling do not preventthe development of plastic stress across the entire section, and hence prevent the development of the plastic bending moment (classes 1 and 2 cross-sections in Eurocode 3,part 1.1).

As an example for the computation of the plastic moment, a section is selectedwith a depth of 351 m m , a width of 373 mm , a flange thickness of 15.6 m m, and aweb thickness of 15.6 m m. T he yield strength of the steel is taken as 23 5 N /m m2. Thecross-sectional area is 16,900 mm2 and the plastic mo dulus Ζ χ ab out the major axisis 2.39 x 106 m m3. The ultimate plastic moment, assuming bending about the major

axis, is:

Muit = (235)(2.39 x 106) = 5.61 x 108 N m m

Assuming that plastic behavior is developed over the full depth of the section, thevalue of Muit may be computed approximately as:

Mu l t = (2)(235)[(373)(15.6)(167.7) + (159.9)(15.6)(79.95)]

= 5.52 x 108

N m m .

The cross-sectional area used in the above computations is slightly less than the totalarea which accounts for the slightly smaller value by the approximate method.

Home (1971) presents an equation for solving for the effect of axial loading wherethe neutral axis is in the web, as follows:

(4.25)

(4.26)

where tw = thickness of web of beam.

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128 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Home presented a figure for I-sections that shows the reduction of the ultimatemoment due to axial loading (see Fig. 4.6). The abscissa is the applied axial load as afunction of the squash load PM, where P u is found by Eq. 4.27.

4.27)

where A = cross-sectional area of pile.No column action is assumed in Eq. 4.27; that feature in the design of a pile is

discussed in Chapter 6.As may be seen in Fig. 4. 6, the redu ction in the plastic mom ent is quite small when the

axial load is in the range of 5 to 10% of the squash load. This range is encountered inmo st designs. Whe n the axial load is relatively large, Fig. 4.6 m ay be used in p relim inarydesign. Alternately, the designer may work out a curve for the particular section byusing the equations of mechanics.

For the section selected above, the squash load is

P u = (235 ,000)(0 .01690) = 3 ,9 70 kN .

In most cases, a pile that is designed principally to sustain lateral loading would besubjected to an axial load of not more than 5 to 10% of the squash load as computedabove; therefore, the reduction of the allowable plastic moment would be small.

igur 4 6 Effect of axial loading on ultimate bending mo me nt in l-Sections after Hör ne I9 7 I) .

u — fyA

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S t r u c t u ra l c h a r a c t e r i s t i c s o f p i l e s 129

4.6 I M P L E M E N TA T I O N O F T H E M E T H O D F O R A S T E EL P I PE

As for the structural shape, the E pI p for elastic behavior may be used without mucherror in computing the bending moment. The moment of inertia is computed by the

familiar equation.

(4.28)

The ultimate bending moment may be computed with simple expressions if thedistribution of stresses in the pipe is as shown in Fig. 4.5.

Mult = fyZp (4.29)

where

4.30)

A numerical procedure can be used to investigate the influence of axial loading withresults that are similar to those obtained by Ho m e (1971). The cases that w ere studiedwere for a value of fy of 235,000 kPa, and the ratios of diameter b to wall thicknesst ranged from 12 to 48; the results are shown in Fig. 4.7 in a plot similar to that inFig. 4.6.

Figure 4.7 Effect of axial loading on ultimate bending mo me nt in pipes.

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130 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

A pipe wa s selected for study with an o utside d iameter of 838 m m an d an insidediameter of 78 2 mm £ = 28 mm ). The yield strength of the steel is assum ed to be235 .000 kPa. The following com putations w ere made.

A = J ( 0 . 8 3 82 - 0.7822) = 7.125 x 10~2 m 2

I = -^-(0 .8384 - 0.7824) = 5.85 x 10~3 m 4

64

Eplp = (5.85 x 10-3)(2 x 108) = 1.17 x 106 kN m2

Zp = ^ ( 0 . 8 3 83 - 0 . 7 8 23 ) = 1.838 x 10-2 m 3

P u = (235,000)(7.125 x 10~2) = 1 6 , 7 4 0 kN

Muit = (235,000)(1.838 x 10~2) = 4,320 kNm (with no axial load)

If it is assum ed t ha t an axial load of 250 k N is applied w hile the pile is being subjectedto lateral load ing, the value of P x/P u is 0.0 15 . Fig. 4.7 show s tha t a negligible correctionis needed for Muit to account for the presence of the axial loading.

4.7 I M P L E M E N T A T I O N O F T H E M E T H O D F O R AR E I N F O R C E D - C O N C R E T E S E C T I O N

The computation of the necessary parameters for the analysis under lateral load ofstruc tura l shapes or steel pipes is facilitated by the availability of simple equa tion s. O nthe other hand, the analysis of a reinforced-concrete section must depend principallyon numerical analysis for developing the needed parameters. The variables that mustbe addresse d include the geom etry of the section, the strength s of the concrete an d steel,and the percentage of steel. The number of parameters argue against the presentationof charts or tables that can be entered; rather, the designer should have on hand oneof the available computation procedures for the needed values.

The paragraphs that follow present the basics of the computations, and furtherinformation is given in Chapter 6.

The value of E pI p can be taken as that of the gross section. How ever, the cracking ofthe concrete will occur early in the loadin g with a significant red uctio n in Eplp. Furtherreductions occur as the bending moment is increased; therefore, a modification in Eplpmay be needed for accurate computations, especially if deflection will control theloading.

4.7.1 E x a m p l e c o m p u t a t i o n s f o r a s q u a r e s h a p e

Fig. 4.8 sh ow s the cross section of a reinforced-con crete pile tha t is 40 0 mm squ are.The strengths of the concrete and steel, the number and size of bars, and the distance

from the center of the bars to the outside of the section are indicated.A computer code was employed, and Fig. 4.9 was prepared, showing the ultimate

bending moment as a function of the magnitude of the axial load. The squash load,collapse with only axial loading, was computed to be 5 374 kN. Curves such as those

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i le s 131

shown in the figure may be used to obtain the moment at which a plastic hinge willdevelop. As shown in the figure, Muit increases with axial loading up to a value of P x

about 2,000 kN. If there is some question about the magnitude of the axial loading,the lower value should be selected in order to be conservative.

Fig. 4.10 shows computed values of the bending stiffness E pI p for a pile section as afunction of the applied moment. Three values of axial load are assumed. Also plotted

in the figure is the value of E pI p , called gross E pI p , for the con crete only. Becausethe computed bending moment is affected only slightly by variations in the value ofEpJp, the gross E pIp may be used without much error for the relatively small valuesof bending m om ent . Ex cept for the axial load of 2,000 kN , the bending stiffness is

igur 4 9 Interaction diagram fo r square reinforced-con crete pile.

igur 4 8 Dimensions of square reinforce d-con crete pile.

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132 Sin gle Pi les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 4 11 Cylindrical reinforce d-con crete pile.

larger than that of the cracked section for the smaller values of bending momen t. Thisincrease is due of course, to the influence of the steel in the section.

The curves include a section where the E pI p changes dramatically for a given valueof bending moment. To some extent, the sudden drop in E p Ip is an artifact of theparticular code that is used. The code is written to indicate a sudden loss of bendingstiffness with the cracking of the concrete. In practice, the loss of Eplp is likely to bemuch more gradual than indicated in the sketch.

4.7.2 E x a m p l e c o m p u t a t i o n s f o r c i r c u l a r s h a p e

Similar computations related to those for the square shape are given for a reinforced-concrete circular shape, as shown in Figs. 4. 11 , 4.12 and 4 .13 . The curves are similarin shape to those shown earlier, and a similar discussion applies.

igur 4 10 E p lp as a function of M for pile with rectangular cross-section.

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S t r u c t u r a l c h a r a c t e r i s t i c s o f p i l e s 133

igur 4 12 Interaction diagram for cylindrical reinforced -conc rete pile.

igur 4 13 E p lp as a function of M for pile w ith cylindrical cross-section.

4.8 A P P R O X I M A T I O N O F M O M E N T O F I N E RTI A F O R AR E I N F O R C E D - C O N C R E T E S E C T I O N

As noted abo ve, the procedures given in the above parag raph s result in a sharp decreasein the value of E pI p because the mechanics predict continuous cracking at a giventensile strain of the concrete. Observing the behavior of reinforced-concrete sections

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134 Single Piles and Pile G rou ps U nd er La tera l Loa ding

has yielded an empirical equation (Equation 4.31) that gives, as a function of theapplied bending moment, values of bending stiffness that reduce more gradually thanthe values from mechanics (American Concrete Institute, 1989).

(4.31)

4.32)

4.33)r = 7.5 Jfç (for no rm al weig ht concrete)

where

Ie = effective moment of inertia for computation of deflection, Ig = moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement,yc = distance from the centroidal axis of the gross section, neglecting reinforcement,to the extreme fibers in tension, Icr = m om ent of inertia of cracked sec tion, andMa= maximum moment in pile. (Note: Eq. 4.33 is units-dependent; therefore, theuser should enter the value of f z in lb s/in2, compute the value of/ r in lbs/in2, and thenconvert the value of fr into appropriate units of kPa.)

The value of Icr may be computed by the analytical method, using standard mechanics, presented earlier. In computing bending stiffness, the value of Ep is assumed to

remain constant.The absence of a term for axial load in Eq. 4.31 means that the method is limitedin scope. However, plotting of the results for no axial load, along with results fromthe analytical method for no axial load, will reveal a trend that should prove useful insolving a practical problem.

H O M E W O R K P R O B L E M S F O R C H A P T E R 4

4.1 (a) D iscuss the logic in the developm ent of Eq uatio n 4.1 and suggest an alter nate

solution.(b) Apply E qua tion 4.1 to the solutio n for a plate with a wi dth of 0.4 m a nda thickness of 0.2 m, being push ed against the larger side in a clay with acompressive strength of 75 kPa.

(c) Repeat the solution for if the pile is in sand with a friction angle of 35 degrees.

4.2 Use technical literature and find another recommendation for the shape of thestress-strain curve for concrete to compare with the curve in Figure 4.1. Whichcurve do you prefer? Why?

4.3 You are requir ed to perfor m a pu sh-o ver analysis of a steel pile und er lateral

loading. Show a sketch of the more detailed stress-strain curve for steel. On animportant job, would you use the more complex curve or the simple one shownin Figure 4.4?

4.4 Verify the com pu tatio ns for the pro pertie s of the steel pipe show n in Section 4.6 .

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Chapter 5

A n a l y s i s o f g r o u p s o f p i l es s u b j e c t e dt o i n c l i n e d a n d e c c e n t r i c l o a d i n g

5.1 I N T R O D U C T I O N

The objective of this chapter is to develop a procedure for computing the movement ofa cap for a group of piles when subjected to axial load, lateral load, and overturningmoment. Several elements of the procedure must be addressed. The mechanics mustbe addressed, taking into account the nonlinear deflection of each pile head, for bothvertical and battered (raked) piles, due to the imposition of an axial load, a lateralload, and a moment.

The mechanics of the problem are discussed first. A brief review is given of someof the relevant literature, and a set of equilibrium equations is presented that can besolved by iteration. A framework is established for the input of the nonlinear responseof individual piles.

A brief reference is made to the response of individual piles under lateral loading.The early chapters of the text address lateral loading in some detail. A computercode (subroutine) for individual piles under lateral loading can readily be attachedto a global program on pile groups if the influence of close spacing can be explicitlydefined.

A review is made of technical literature concerning the deformation of individualpiles under axial load. While the objective of the book relates principally to lateralloading, the load versus deflection for axially loaded piles must be addressed. If

the cap for a pile group is subjected to an inclined and eccentric load, some of thepiles in the group certainly must sustain an axial force. Thus, nonlinear relationshipsmust be selected to define axial load versus deflection for a variety of kinds and sizesof piles.

If piles under lateral loading are spaced close to each other, the piles will influenceeach other due to pile-soil-pile interac tion . Th e interactio n of closely spaced piles underlateral loading is discussed in detail and recommendations are given for modifying thep-y curves to account for close spacing.

Close spacing of piles under axial loading must also be addressed as is piles underlateral loading. The problem of pile-soil-pile interaction for axial loading is discussed

and recommendations are made for formulating influence coefficients to account forclose spacing.

Finally, a comprehensive study is analyzed where a pile group, including batterpiles, was subjected to inclined and eccentric loading. The computed values of pile-cap

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136 Single Piles and Pile G rou ps U nd er L ate ral Loa ding

movements were compared with values obtained from experiment. A computer code,based on the compu ted response of individual piles and o n the mechanics noted above,yielded results that agreed well with the experimental values.

5.2 A P P R O A C H T O A N A LY S I S O F G R O U P SOF P IL E S

The first four chapters are directed primarily at single or isolated piles under lateralloading; however, most piles are installed in groups and the response of the group toloading is addressed herein. Further, as noted in the title of this chapter, most groupsmust support loadings that are both lateral and axial. Therefore, the mechanics of apile under axial loading must be presented, but the design of single piles under purelyaxial loading is discussed only briefly.

The behavior of a group of piles may be influenced by two forms of interaction:(1) interaction between piles in close proximity, termed efficiency'; and (2) interactionby distribution of loading to individual piles from the pile cap. In the first instance therelevant forces are transmitted through the soil, while in the second instance, the forcesare transmitted by the superstructure (assumed to be the pile cap in this presentatio n).If the piles are widely spaced, th e pile-soil-pile intera ction is insignificant and a so lutionis made in order to reveal lateral load, axial load, and bending moment to each of thepiles in the group.

Equations for the efficiency of closely-spaced piles, under lateral loading as wellas under axial loading, are based largely on experimental data. Methods that treatthe soil as an elastic, isotropic, and homogeneous material are useful in giving insightinto the mechanics of pile-soil-pile interaction but soils behave far differently than theidealized material. Experimental results are not definitive because the various sets ofexperiments do not isolate each of the many parameters and reveal the influence ofeach particular parameter. However, a review of a number of relevant studies leads torecommendations for analytical procedures.

The steps in the analysis of a group of piles under generalized loading, axial, lateral,and overturning, are discussed herein and are: (1) employ a rational method for com

puting the movements of the pile cap and the loads to each of the piles in the group,reflecting properly the difference in response of piles that are vertical and those thatare battered; (2) account the reduced efficiency of each pile in the group due to closespacing; and (3) use equations for the stiffness of each pile under axial and lateralloading.

A literature survey is presented, dealing principally with the mechanics of theresponse of a group of piles. Then, a rational formulation of the mechanics ofresponse is presented which is the basis of the analytical method presented for usein practice. A comprehensive study of the problem of closely-spaced piles is givento provide the basis for degrading the response of some of the piles in the group.

The stiffness of each pile under axial and lateral loading is needed in the analyticalmethod and discussed briefly. Finally in the chapter, an example is solved for a casewhere a group of piles are closely spaced and where the loading comes through thepile cap.

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Analysis of groups of pi les sub jected to incl in ed and ecc entric loading 137

5.3 R E V I E W O F T H E O R I E S F O R T H E R E S P O N S E O F G R O U P SO F P IL E S T O I N C L I N E D A N D E C C E N T R I C L O A D S

The developm ent of comp utatio nal m ethod s has been limited because of lack of know ledge about single-pile behavior. In order to meet the practical needs of designingstructures with grouped piles, various computational methods were developed bymaking assumptions that would permit analysis of the problem.

The simplest way to treat a grouped-pile found ation is to assume that both the structure and the piles are rigid and that only the axial resistance of the piles is considered.Under these assumptions, Culmann (Terzaghi 1956) presented a graphical solution in1866. The equilibrium state of the resultant external load and the axial reaction ofeach group of similar piles was obtained by drawing a force polygon. The application of Culmann's method is limited to the case of a foundation with three groups ofsimilar piles. A supplemental m ethod to this graphical solution w as propose d in 1930by Brennecke & Lohmeyer (Terzaghi 1956). The vertical component of the resultantload is distributed in a trapezoidal shape in such a way that the total area equals themagnitude of the vertical component, and its center of gravity lies on the line of thevertical component of the resultant load. The vertical load is distributed to each pile,assuming that the trapezoidal load is separated into independent blocks at the top ofthe piles, except at the end piles. Unlike Culm ann's m etho d, the latter meth od can h andle more than three groups of similar piles. But the method of Brennecke & Lohmeyeris restricted to the case where all of the pile tops are on the same level.

The elastic displacement of pile tops was first taken into consideration byWestergaard in 1917 (Karol 1960). Westergaard assumed linearly elastic displacement of pile tops under a compressive load, but the lateral resistance of the piles wasnot considered. He developed a method to find a center of rotation of a pile cap.With the center of rotation known, the displacements and forces in each pile could becomputed.

Nokkentved (Hansen 1959) presented in 1924 a method similar to that of Westergaard. He defined a point that was dependent only on the geometry of the pilearrangement, so that forces which pass through this point produce only unit verticaland horizontal translations of the pile cap. The method was also pursued by Vetter(Terzaghi 1956) in 193 9. Vetter introduced the dum my pile technique to simulate

the effect of the lateral restraint and the rotational fixity of pile tops. Dummy piles areproperly assumed to be imaginary elastic columns.Later, in 1953, Vamdepitte (Hansen, 1959) applied the concept of the elastic center

in developing the ultimate-design method, which was further formulated by Hansen(1959). The transitional stage in which some of the piles reach the ultimate bearingcapacity, while the remain der of the piles in a foun datio n are in an elastic rang e, can becomputed by a purely elastic method if the reactions of the piles in the ultimate stagesare regarded as con stan t forces on th e cap. The failure of the cap is reached after successive failures of all bu t the last two piles. Th en the cap can rota te aro un d the intersectionof the axis of the two elastic piles. Vamdepitte resorted to a graphical solution to com

pute directly the ultimate load on a two-dim ensional ca p. Hansen extended the m ethodto the three-dimensional case. Althou gh the plastic-design m ethod is unique a nd ra tional, the assumptions to simplify the real soil-structure system may need examination.It was assumed that a pile had no lateral resistance, and no rotational restraint of

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138 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

the pile tops on the cap was considered. The axial load versus displacement of eachindividual pile was represented by a bilinear relationship.

A comprehen sive, mo dern structu ral treatment was presented by Hrennikoff (1950)for the two-dimensional case. He considered the axial, transverse, and rotational

resistance of piles on the cap. The load-displacement relationship of the pile top wasassumed to be linearly elastic. One restrictive assumption was that all piles must havethe same load-displacement relationship. Hrennikoff substituted a free-standing elasticcolumn for an axially loaded pile. A laterally loaded pile was regarded as an elasticbeam on an elastic foundation with uniform stiffness. Even with these crude approximations of pile behavior, the method is significant in the sense that it presents thepotential for the analytical treatment of the soil-pile-interaction system. Hrennikoff'smethod consisted of obtaining influence coefficients for cap displacements by summingthe influence coefficients of individual piles in terms of the spring co nsta nts w hich re present the pile-head reaction s ont o the pile cap . Alm ost all the subseq uent w ork followsthe approach taken by Hrennikoff.

Radosavljevic (1957) also regarded a laterally loaded pile as an elastic beam in anelastic medium with a uniform stiffness. He advocated the use of the results of tests ofsingle piles under axial loading. In this way a designer can choose the most practicalspring constant for the axially-loaded-pile head, and nonlinear behavior can also beconsidered. Radosavljevic showed a slightly different formulation than Hrennikoff inderiving the coefficients of the equations of the equilibrium of forces. Instead of usingunit disp lacemen t of a cap , he used an arbi trary set of displacemen ts. Still, his struc tura lapproach is essentially analogous to Hrennikoff's method. Radosavljevic's method isrestricted to the case of identical piles in identical soil conditions.

Turzynski (1960) presented a formulation by the matrix method for the two-dimensional case. Neglecting the lateral resistance of pile and soil, he considered onlythe axial resistance of piles. Further, he assumed piles as elastic columns pinned at thetop and at the tip. He derived a stiffness matrix and inverted it to obtain the flexibilitymatrix. Except for the matrix method, Turzynski's method does not serve a practicaluse because of its oversimplification of the soil-pile-interaction system.

Asplund (1956) formulated the matrix method for both two-dimensional and three-dimensional cases. His method also starts out from calculations of a stiffness matrixto obtain a flexibility matrix by inversion. In an attempt to simplify the final flexi

bility matrix, Asplund defined a pile-group center by which the flexibility matrix isdiagonalized. He stressed the importance of the pile arrangement for an economicalgrouped-pile foundation, and he contended that the pile-group-center method helpedto visualize better the effect of the geometrical factors. He employed the elastic-centermethod for the treatment of laterally loaded piles. Any transverse load through theelastic center causes only transverse displacement of the pile head, and rotational loadaround the elastic center gives only the rotation of the pile head. In spite of the elaborate structural formulation, there is no particular correlation with the soil-pile system.Laterally loaded piles are merely regarded as elastic beams on an elastic bed with auniform spring constant.

Francis (1964) com pu ted the two -dim ensio nal case using the influence-coefficientmethod. The lateral resistance of soil was considered either uniform throughout orincreasing in proportion to depth. Assuming a fictitious point of fixity at a certaindepth, elastic columns fixed at both ends are substituted for laterally loaded piles.

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A n a l y s i s o f g r o u p s o f p i l e s s u b j e c t e d t o i n c l i n e d a n d e c c e n t r i c l o a d i n g 139

The axial loads on individual piles are assumed to have an effect only on the elasticstability without causing any settlement or uplift at the pile tips.

Aschenbrenner (1967) presented a three-dimensional analysis based on the influence-coefficient method. This analysis is an extension of Hrennikoff's method to the three-

dimensional case. Aschenbrenn er's m ethod is restricted to pin-con nected piles.Saul (1968) gave the most general formulation of the matrix method for a three-

dimensional foundation with rigidly connected piles. He employed the cantilevermethod to describe the behavior of laterally loaded piles. He left it to the designer toset the soil criteria for determining the settlement of axially loaded piles and the resistance of laterally loaded piles. Saul indicated the possible application of his method todynamically loaded foundations.

Reese & Matlock (1960, 1966) and Reese, et al (1970) presented a method forcoupling the analysis of the grouped-pile foundation with the analysis of laterallyloade d piles by the finite-difference m eth od . Their m eth od p resum es the use of a digitalcomputer. The formulation of equations giving the movement of the pile cap is doneby the influence-coefficient method, similar to Hrennikoff's method. Reese & Matlockdevised a convenient way to represent the pile-head moment and lateral reaction byspring forces only in terms of the lateral displacement of a pile top. The effect of pile-head ro tati on on the pile-head reactio ns is included implicitly in the force-displacementrelationship.

Using Reese & Matlock's method, example problems were worked out by Robertson(1961) and by Parker & Cox (1969). Robertson compared the method with Vetter'smethod and Hrennikoff's method. Parker and Cox integrated into the method typicalsoil criteria for laterally loaded piles.

Reese & O'Neill (1967) developed the theory for the general analysis of a three-dimensional group of piles using matrix formulations. Their theory is an extensionof the theory of Hrenn ikoff (195 0), in wh ich springs are used to represen t the piles.Representation of piles by springs imposes the superposition of two independ ent mod esof deflection of a laterally loaded pile. The spring constants for the lateral reactionand the moment at the pile top must be obtained for a mode of deflection, wherea pile head is given only transitional displacement without rotation and also for amode of deflection where a pile head is given only rotation without translation. Whilethe soil-pile-interaction system has highly nonlinear relationships, the pile material

also exhibits nonlinear characteristics when it is loaded near its ultimate strength.The method of analysis used herein is based on the concept presented by Reese &Matlock (1966), but the solution of the relevant equations is done more convenientlyby special techniques (Awoshika & Reese 1971).

5.4 R A T I O N A L E Q U A T I O N S F O R T H E R E S P O N S E O FA G R O U P O F P IL E S U N D E R G E N E R A L I Z E D L O A D I N G

5 . 4 . 1 I n t r o d u c t i o n

A structural theory, principally following the work of Awo shika (1971) an d Awo shikaand Reese (1971), is formulated herein for computing the behavior of a two-dimensional pile foundation with arbitrarily arranged piles that possess nonlinear

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140 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

igure 5.1 Basic structur al system A fo r pile groups.

force-displacement characteristics. Coupled with the structural theory of a pile capare the theories of a laterally loaded pile and an axially loaded pile. In this chaptereach theory is developed separately. Solutions for all of the theories depend on the use

of digital computers for the actual computations.

5.4.1.1 Basic structural systems

Fig. 5.1a illustrates the general system of a two-dimensional pile foundation. Threepiles, with arbitrary spacing and arbitrary inclination, are connected to an arbitrarilyshaped pile cap. Such sectional properties of a pile as the diameter, the cross-sectionalarea, and the moment of inertia can vary, not only from pile to pile, but also alongthe axis of a pile. The pile materia l ma y be different from pile to pile but it is assum edthat the same material is used within a pile.

The structural system at the pile cap is illustrated in Fig. 5.1b. The axial load, lateralload, and bending moment at each pile head must put the pile cap into equilibrium.Also, the individual pile-head loads must be consistent with the movements of each ofthe pile heads.

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igure 5.3 Typical pile head in a pile cap (af terAw os hik a 1971)

There are three conceivable cases of pile connection to the pile cap. Pile 1 inFigs. 5.1a & 5.1c illustrates a pinn ed con necti on. Pile 2 shows a fixed-head pile with itshead clam ped by the pile cap . And Pile 3 represents an elastically restrained pile, whichis the typical case of an offshore structure, shown in Fig. 5.2. In Fig. 5.2a, the pilesare extended and form a part of the superstructure. In Fig. 5.2b, the piles rest againstknife-edge supports and can deflect freely between these supports. Elastic restraint is

provided by the flexural rigidity of the pile itself The treatment of a laterally loadedpile with an elastically-restrained top, discussed in Chapter 2, gives a useful tool forhandling the real foundation.

Piles are frequently embedded into a monolithic, reinforced-concrete pile cap withthe assumption that complete fixity of the pile to the pile cap is obtained (Fig. 5.3).However, the elasticity of the reinforced concrete and local failure due to stress concentrations allows the rotation of a pile head within the pile cap. The magnitude ofthe restraint on the pile from the pile cap is indeterminate, but a range of values maybe computed by estimating the p-y curves for concrete and solving for the response ofthe portion of the pile within the cap by use of the equations for a pile under lateral

loading.The pile cap is subjected to two-dimensional external loads. The line of action of

the resultant external load may be inclined and may assume any arbitrary positionwith respect to the structure (Fig. 5.1a). The external loads cause displacement of the

igure 5.2 Typical offshore stru ctu re (afterAw osh ika 1971).

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pile cap which results in axial, lateral, and rotational displacements of each individualpile. The displacements of individual piles in turn results in loads on the pile cap(Fig. 5.1b). These pile reactions are highly nonlinear in nature. They are functions ofpile properties, soil properties, and the boundary conditions at the pile top.

The structural theory for the pile grou p uses a numerical meth od to seek com patibledisplacement of the pile cap, which satisfies the equilibrium of the applied externalloads and the nonlinear pile reactions.

5.4.1.2 Assumptions

Some of the basic assumption employed for the treatment of the pile group arepresented below.

Two-dimensionality. The first assum ption is the two-d imension al arrangem ent of

the bent cap and the piles. The usual design practice is to arrange piles symmetricallywith a plane or planes with loads acting in this plane of symmetry. The assumption ofa two-dimensional case reduces considerably the number of variables to be handled.However, there is no essential difference in theory between the two-dimensional caseand th e three-dim ensio nal case. If the validity of the theory for the former is established,the theory can be extended to the latter by adding more components of forces anddisplacements with regard to the new dimension (Reese & O'Neill 1967).

Nondeformab ility of pile cap. The second major assumption is the non de-formability of the pile cap. A pile head encased in a monolithic pile cap (Pile 2, Fig.5.1), or su ppo rted by a pair of knife edges (Pile 3, Fig. 5.1) can ro tate or deflect w ithinthe pile cap. But the shape of the pile cap itself is assum ed t o be always the same for theequation s show n h erein. Th at m eans that the relative positions of the pile tops rem ainthe same for any pile-cap displacement. If the pile cap is deformable, the structuraltheory of the pile group must include the compatibility condition of the pile cap itselfW hile no treatm ent of a foun datio n with a deform able pile cap is included in this study,the theo ry can be extend ed to such a case if the pile cap consists of a structura l me mb ersuch that the analytical computation of the deformation of the pile cap is possible.

Wide spacing of piles. The equ ation s developed here are for the case wh ere theindividual piles are so widely spaced that there is no influence of one pile on another.However, there are many pile designs where the piles are close enough so that pile-soil-pile interaction does occur, and such interaction is discussed in detail later in thischapter. The effects of pile-soil-interaction can be intro duc ed in to the analytical m etho dwithout difficulty.

Behavior under lateral load and under axial load are independent. Theassumption is made that there is no interaction between the axial-pile behavior andthe lateral-pile behavior. That is, the relationship between axial load and displacement

is not affected by the presence of lateral deflection and vice versa. The validity ofthis assumption is discussed by Parker & Reese (1971). The argument is made, andgenerally accepted, that the soil near the ground surface principally determines lateralresponse and the soil at depth principally determine axial response. If overconsolidated

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clay exists at the ground surface and pile deflection is sizable, the recommendation ismade that the soil above the first point where lateral deflection is zero be discountedin computing axial capacity.

5.4.2 E q u a t i o n s f o r a t w o d i m e n s i o n a l g r o u p o f p i l es

To deal with the nonlinearity in the system, the equilibrium of the applied loads andthe pile reactions on a pile cap are sought by the successive correction of pile-capdisplacements. After each correction of the displacement, the difference between theload and the pile reaction is calculated. The next correction is obtained through thecalculation of a new stiffness matrix at the previous pile-cap position. The elements ofa stiffness matrix are obtained by giving a small virtual increment to each componentof displacement, one at a time. The proper magnitude of the virtual increment may beset at 1 x 10- 5 times a unit displacement to attain acceptable accuracy.

5.4.2.1 Coordinate systems and sign conventions

Fig. 5.4a shows the coordinate systems and sign conventions. The superstructure andthe pile cap are referred to the global structural coordinate system (X, Y) where the Xand Y axes are vertical and horizontal, respectively. The resultant external forces areacting at the origin 0 of this global stru ctura l coo rdin ate system. The positive direction s

*Note: There can be one or more piles in and individual group. Each pile in the

group must have identical load-displacement characteristics; or anothergroup must be identified

igure 5.4a Co ord inat e systems fo r analysis of pile groups showing positive direction s of displacements(after A w os hi ka Reese 1971).

(X, Y) Structural coordinate system(x j, yl) Local structural coordinate systemxh yl) Member coordinate system

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of the components of the resultant load Po Qo> and Mo are show n by the arro ws . Thepositive curl of the mom ent w as determined by the usual right-hand rule. The pile headof each individual pile in a group is referred to the local structural coordinate system(x·,)/), whose origin is the pile head and with axes running parallel to those of theglobal structural coordinate system. The member coordinate system (x^ji) is furthe rassigned to each pile. The origin of the member coordinate system is the pile head. ItsXi axis coincides with the pile axis and the yi axis is perpendicular to the Xj axis. TheXi axis makes an angle λ; with the vertical. The angle λ; is positive when it is measuredcounterclockwise.

Fig. 5.4b shows the positive directions of the forces, Pj, Qj, and Mj exerted from thepile cap onto the top of an individual pile in the ith individual pile group. The forcesPi and Qi are acting along the { and yi axes, respectively, of the member coordinatesystem.

5 4 Transformation of coordinates

Displacement. Fig. 5.5a illustrates the pile-head displace me nt in the stru ctu ral, thelocal structu ral, and the mem ber coo rdina te systems. Due to the pile-cap displacem entfrom poi nt 0 to poin t 0' with a rota tio n of, the i-th pile moves from the original positionP to the new po sition Pr. The rotation of the pile head depends on the way it is fastenedto the ca p. The co mp one nts of pile-cap displacement are expressed by (17, V, a) withregard to the structural coordinate system. The pile-head displacement is denoted by(u b,v b, a) in the local coordinate system and by {u^v^ a) in the member coordinate

system.The coordinate transformation between the structural and the local structuralcoordinate system is derived from the simple geometrical consideration:

igur 5 4b Positive dire ctio n of forces on single pile in pile gro up analysis (after Aw osh ika Reese1971).

(5.1)

(5.2)

where (X*·, Yi) = location of /-th pile head in the structural coordinate system.

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The transformation of pile-head displacement from the local structural coordinatesystem to the memb er coo rdinate system is obtained from the geometrical relationship(Fig. 5.5b).

Ui = vl i cos Xj + v\ sin Xj

vi = v\ cos Xj — vl i sin Xj

(5.3)

(5.4)

Substitution of Eqs. 5.1 and 5.2 into Eqs. 5.3 and 5.4 yields the transformationrelationship between the pile-cap displacement in the structural coordinate systemand the corresponding pile-top displacement of the i-th individual pile in the membercoordinate system.

(5.5)

(5.6)

Eq. 5.7 presents in matrix notation the case where the pile head is fixed to the capso that the cap and the pile head rotate the same amount.

(5.7)

The expressions for Uj and V* w ill rema in th e same for the cases wh ere the pile headis free to rotate or is partially restrained, but the expression for otj must be modified.The matrix expression above is written concisely

«i = T Dji U (5.

where Uj = displacem ent vector a t head of pile; T? j = displacement transform ationmatrix of the pile; and U = displacement vector of the pile cap.

Force. Fig. 5.4 illustrates the action of the load on the pile cap and the pile reactio ns.The load is expressed in three components (?o5 Qo5 Mo) with regard to the structuralcoordinate system. The reactions in the ith individual pile are expressed in terms ofthe member coordinate system (Pi5 Qj,Mj). Decom position of the reactions of the i-th.pile with respect to the structural coordinate system gives the transformation of thepile reaction from the member coordinate system to the structural coordinate system.

(5.9)

(5.10)

(5.11)

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Ma t r i x n o t a t i o n e x p re s s e s t h e e q u a t i o n s a b o v e ,

where P· = react ion vector of the p i le of i-th. ind iv idua l p i l e in the s t ruc tu ra l coo rd ina tes y s t e m; Tp j = fo rce t ran s fo rm at io n m at r ix of the p i l e ; an d Pj = react ion ve ctor of th ep i le in the member coord ina te sys tem.

I t i s obser ved tha t the fo rce t ran s fo rm at io n m at r ix Tpj i s o b t a i n e d b y t r a n s p o s i n gt h e d i s p l a c e me n t t r a n s fo rma t i o n ma t r i x Tpj. T h u s ,

TF,i = T TD ? , (5.14)

5.4.2.3 Solution of equilibrium equa tions

With the fo rce and d i sp lacemen t charac te r i s t i cs o f each p i l e a t hand , the equ i l ib r iumequa t ions fo r the g loba l s t ruc tu re can now be so lved . These a re :

(5 .15)

(5 .16)

(5 .17)

w her e / = va lues f rom any i-th ind iv idua l p i l e ; Ji = the n um be r of p i les in th at p i leg r o u p .

These equ i l ib r ium equa t ions a re so lved by any conven ien t manner. The s t i ffnesste rms ( fo rce versus d i sp lacemen t ) fo r the p i l es a re non l inear ; the re fo re , the equa t ionsmus t be so lved by i t e ra t ion .

Aw o s h i k a (1 9 7 1 ) p e r fo rm e d a c o m p re h e n s i v e s et o f e x p e r i m e n t s o n a x i al l y l o a d e dp i les , l a te ra l ly loaded p i l es , and p i l e g roups wi th ba t t e r p i l es . The exper imen ts a l lowedthe equa t ions fo r the d i s t r ibu t ion o f loads to p i l es in a g roup to be va l ida ted . The

exper imen ts wi l l be descr ibed and ana lyzed a t the end o f th i s chap te r. The in te rven ingmater ia l on l a te ra l load , ax ia l load , and the in te rac t ion o f c lose ly spaced p i l es wi l l bei mp l e me n t e d . C o mp a r i s o n s o f v a l u e s f ro m e x p e r i me n t a n d f ro m a n a l y s i s a l l o w t h eu t i l i ty o f the method shown above to be eva lua ted .

(5.12)

(5.13)

or more concisely,

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5.5 L A T E R A L L Y L O A D E D P I L E S

5.5.1 M o v e m e n t o f p i l e h e a d d u e t o a p p l i e d l o a d i n g

The several earlier chapte rs add ress the analysis of single piles to ob tain pile-head m ovements for various boundary conditions at the pile head. The p-y method is consideredto be the best method currently available for making the required computations. Thelateral load and moment at the pile head can be computed readily for the particularlateral deflection that is desired. Thus, the lateral stiffnesses for load and moment canbe found by solution of the nonlinear equations.

5 .5 .2 Effe c t o f b a t t e r

The effect of batter on the behavior of laterally loaded piles was investigated in a test

tank (Awoshika & Reese, 1971). The lateral, soil-resistance curves of a vertical pilewere modified by a constant to express the effect of the pile inclination. The valuesof the modifying constant as a function of the batter angle were deduced from modeltests in sands and also from full-scale, pile-loading tests that are reported in technicalliterature (Kubo, 1965). The criterion is expressed by a solid line in Fig. 3.34.

Plotted po ints in Fig. 3.34 sh ow the mod ification factors for the batter piles tested inthe experiments by Awo shika & Reese (1971). The modification factors were ob tainedfor two series of tests independently.

(b) Transformation of displacement between local structrualcoordinate system and member coordinate system

igure 5.5 Trans forma tion of displacements (after Aw oshika Reese 1971).

(a) Pile head displacement due to pile cap displacement

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Fig. 3.34 indicates that for the out-batter piles, the agreement between the empirical curve and the experiments is good, while the in-batter piles, for the batter anglesthat were investigated experimentally, did not show any effect of the batter. However, Kubo's experiments show greater batter angles and are used in establishing the

recommended curve for use.The values from Fig. 3.34 can be used to modify values of pu\t which in turn will

cause a modification of all of the values of p in the p-y curves. An analytical methodmay be used to compute the value of pu\t by computing the forces on a wedge of soilwhose shape is modified to reflect the batter, in or out.

5.6 A X I A L L Y L O A D E D P IL E S

5 . 6 . 1 I n t r o d u c t i o n

The stiffness of individual piles under axial loading is needed in order to solve theequ ation s presente d in Section 5.4. Th e topic was discussed in detail by Van Impe in hisGeneral Report to the Tenth European Conference on Soil Mechanics and FoundationEngineering (1991). Much of the following material is derived from that paper. Whilethe specific reference is axial loa ding of the individual pile, the topics presented here willserve to elucidate the discussion presented early on lateral loadin g and w ill serve furtheras pile-soil-pile interaction is dealt with later.

5.6.2 R e l e v a n t p a r a m e t e r s c o n c e r n i n g d e f o r m a t i o no f s o i l

A pile under axial load will impose a complex system of stresses in the supportingsoil and a corresponding system of deformation will occur in the elements that areaffected. Assuming no slip at the interface of the soil and the pile, integration ofthe deformation of the affected elements in the vertical direction will yield the movement at the pile head. Assuming axisymmetric behavior, mapping of the distributionof stresses in the soil considering the dimensions of the pile, the method of installation, the magnitude of the axial load, and the kinds of soil remains elusive. While

such a theoretical method of computing the axial movement of a pile head is currently unfeasible, specific consideration of the stress-strain characteristics of a soilis imperative. Van Impe (1991) refers to Wroth (1972) and writes, Poor or evendangerous geotechnical design may quite often be blamed to ignoring of the strictinterrelationship between soil param eters, their method of determina tion, the modelof the soil strain behavior, the method of analyzing the foundation engineering problem and the corresponding choice of safety factors. Fig. 5.7 illustrates the detailedinformation tha t is im por tant in und erstandin g the deformational characteristicsof soil.

The importance of in situ methods of determining the deformational properties

of soils cannot be overemphasized. The importance is shown graphically in Fig. 5.8(Ward, 1959). Van Impe (1991) presented a detailed discussion of the applicability ofvarious in situ methods concerning deformability characteristics of soils. The followingbrief summ ary gives information presented by Jam iolko wsk i, et al. (1985).

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igure 5.7 Young's modulus versus q c from the cone test (Van Impe 1988).

figure 5.8 Effect of meth od o f sampling on behavior of lab orat ory specimens (W ard et al., 1959)(fromVan Impe 1991).

Plate loading test. Application is limited to shallow depths but results show that theaverage drained Young stiffness can be measured within the depth of influence of the

plate. Test is limited in several respects.Self-boring pressuremeter test. The test has great potential for the measurement of

the shear modulus in the horizontal direction and small unloading-reloading cyclesmay be useful.

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Dilatometer test. Empirical correlations will yield values of the tangent constrainedmodulus of sands and clays.

Cone penetration test. Empirical correlations of deformational characteristics of soilare not generally valid except for normally consolidated sand. Fully drained conditions

must be assured.Shear-wave-velocity measurement. The method can yield deformational values of

soil under small strain but assumptions must be made concerning the constitutivemodel, the stress-strain path, and the soil homogeneity.

5.6.3 I n f l u e n c e o f m e t h o d o f i n s t a l l a t i o n o n s o i lc h a r a c t e r i s t i c s

Driven piles must displace a volume of soil equivalent to the volume of the pile thatpenetrates. A heave of the ground around the pile normally results for piles in clay. Adepression around the pile is sometimes noted when piles are driven into sand becauseof the densification of the sand due to the vibrations of driving. The excavation forbored piles allows the soil to deform laterally toward the borehole. Continuous-flight-auger piles also cause changes in the characteristics of the soil near the pile.

Excess porewater pressures develop around piles driven into saturated clays. If thepiles are driven near each other, the zones of pressure will overlap in a com plex man ner.The pressures dissipate with time with a consequent decrease in water content andincrease in soil strength at the wall of the pile and outward.

No comprehensive attack has been mounted by geotechnical engineers to allow theprediction of the effects of pile installation on soil properties, including deformationalcharac teristics, b ut som e studies have been made th at give insight into these effects. V anWeele (1979) obtained experimental data on the effects of driving a displacement pile(Fig. 5.9). Van Impe (1988) studied the effects of installing a continuous-flight-augerpile (Fig. 5.10). De Beer (1988) presented data on the effects of installing a bored pile(Fig. 5.11). Robinsky & Morrison (1964) present a detailed picture of the effects on therelative density of driving a pile into sand (Fig. 5.12). The data are principally derivedfrom the results of cone penetration tests performed before and after pile driving. Thegathering of additional such data, compilation of the data, and detailed analyses willserve to develop me tho ds of predicting th e effects of pile installatio n o n soil prope rties.

5.6.4 M e t h o d s o f f o r m u l a t i n g a x i a l s t i f f n e s s c u r v e s

The stiffness is entered as a constant in the equilibrium equations, solutions of theequations are found, the axial movement of each pile is noted, curves giving axial loadas a function of movement are entered, and a new stiffness for each pile is computed.Similar procedures are used for the lateral stiffness.

There are basically two analytical methods to compute the load-versus-settlementcurve of an axially loaded pile. One method makes use of the theory-of-elasticity. Themethods suggested by D'Appolonia & Romualdi (1963), Thurman & D'Appolonia

(1965), Poulos & Davis (1968), Poulos & Matte s (1969 ), M attes & Po ulos (1969), andPoulos & Davis (1980) belong to the theory-of-elasticity method. All of the theoriesresort to the Mindlin equation, which can be used to find the deformation as a functionof a force at any point in the interior of semi-infinite, elastic, and isotropic solid.

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igure 5.9 Effects of installation o f a displacem ent pile (fro m Van Impe 99 ).

The displacement of the pile is computed by superimposing the influences of theload transfer (skin friction) along th e pile and the pile-tip resistance at the point in thesolid. The compatibility of those forces and the displacement of a pile are obtained by

solving a set of simultaneou s eq uation s. This meth od takes the stress distribution withinthe soil into consideration; therefore, the elasticity method presents the possibility ofsolving for the beha vior of a gro up of closely-spaced piles und er axial load ings (Poulos1968; Poulos & Davis 1980).

The drawback to the elasticity method lies in the basic assumptions which must bemade. The actual ground condition rarely if ever satisfies the assumption of uniformand isotropic material. In spite of the highly nonlinear stress-strain characteristicsof soils, the only soil properties considered in the elasticity method are the Young'smodulus £ and the Poisson's ratio v. The use of only two constants, £ and v, torepresent soil characteristics rarely agrees with field conditions.

The other method to compute the load-versus-settlement curve for an axiallyloade d pile may be called the finite-difference m eth od . Finite-difference equ ation s areemployed to achieve compatibility between pile displacement and load transfer alonga pile and between displacement and resistance at the tip of the pile. This method was

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152 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

igure 5.10 Effects of installation of continuous flight auger pile (from Van Impe 99 )

first used by Seed & Reese (1957 ); othe r studies are repo rted by Coyle & Reese (1966 ),

Coyle & Sulaiman (1967), and Kraft et al. (1981).Th e finite-difference m etho d assum es that the Winkler co ncept is valid, which isto say that the load transfer at a certain pile section is independent of the pile displacement elsewhere. Because the curves employed for load transfer as a function ofpile movement have been developed principally by experiment, where interaction isexplicitly satisfied, the Winkler concept can be used with some confidence.

Close agreement between results from analysis and from experiment for piles inclays has been found (Coyle & Reese, 1966), but the results for piles in sandsshow considerable scatter (Coyle & Sulaiman, 1967). The effects on the soil ofthe driving of piles may be more severe in sands than in clays in terms of load-

transfer characteristics. In spite of limitations, the finite-difference method can dealwith any complex composition of soil layers with any nonlinear relationship ofdisplacement versus shear force and can accommodate improvements in soil criteria with no mod ifications of the basic theory. Im prov eme nts in the finite-difference

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igure 5.11 Effects of installation of a bore d pile (from Van Impe 99 ).

method can be expected as results from additional high quality experiments becomeavailable.

The axial-lo ad-settlem ent curve for a pile is com pu ted by the finite-differencemethod, described below, by employing curves of load-transfer versus pile movementfor poin ts alon g the sides of a pile and for the end of the pile. The u ltim ate capacity of a

pile is found for some specific m ove men t of the top of the pile. The technical literatu reis replete with proposals for computing axial capacity and is more voluminous thanthat for lateral loading. However, the following sections present only specific methodsfor obtaining the axial stiffness of a driven pile and a bored pile. While the methods have been found to yield results that agree reasonably well with experiment, acomprehensive treatment of axial capacity is not presented.

5.6.5 C a l c u l a t i o n m e t h o d s f o r l o a d s e t t l e m e n t b e h a v i o u ro n t h e b a s i s o f i n s i t u s o i l t e s t s

The Dutch piling code N E N 67 43 (1993) provides a me thod to define the design valueof the pile head displacement as a function of the mobilised base resistance and shaftresistance, as calculated on the basis of CPT The method is semi-graphical and basedon 2 charts, one for base resistance and one for shaft resistance. Each charts contains

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igure 5.12 Dis tribu tion of areas w ith equal relative density after pile driving (Robinsky M orris on1964) (from Van Impe 1991).

3 normalised load-displacement curves for displacement piles (without making anydistinction between e.g. driven piles or screwed piles), for CFA piles and for boredpiles respectively.

The Germ an bored piling code DIN 401 4 com prises 4 tables, giving values of experience of mobilisation curves for bored piles in non-cohesive soils (based on CPT cone

resistances) and in cohesive soils (based on cM-values).The French code Fascicule 62-V contains technical rules for the design of foundations of civil engineering structures. It also describes a method to determine theload-displacement curve of a single pile under axial loading, based on bilinear elasto-plastic mobilis ation curves, whereb y the stiffness factors k and ks for respectively baseresistance and shaft resistance result from the work of Frank and Zhao (1982) andare function of the PMT pressure meter modulus EM and the diameter D of the pile.The functions are different for non-cohesive and cohesive soils, but the pile type doesnot interfere.

In particular with regard to the load-settlement prediction of displacement auger

piles, one also refers to the former work of Van Impe (1988) published in the first BAPIll-seminar.

W hen it comes to a single pile axial load settlem ent cu rve, the use of the single hyperbolic function for back-analysis of the pile load-settlement curve is quite convenient.

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The shaft resistance flexibility factor Ks (M/kN) also correspond to the tangent slopeat the origin of a qs - ss diagram. Fleming states that Ks is proportional to the

The graphical inverse slope method, as suggested by Chin (1970), allows in manycases a satisfying curve fitting. The main purpose of this curve fitting is to extrapolatethe measured load-settlement curves and to allow for a mathematical estimate of theultimate or asymptotic pile resistance. The method is semi-graphical and based on

the conversion of the mea sured Q — s values into a s/Q versus s diagram. In fact,the basis equation

(5.18)

(5.19)

can be transformed into:

which corresponds to a straight line in the s/Q versus s plane. Further refinementsconsist of the search of 2 separate hyperbolic equations (Chin & Vail, 1973) to becombined for the curve fitting and backanalysis of pile loading tests. The work ofFleming (1972) in the 90's contributed to a better understanding of the hyperbolictransfer functions and the soil parameters defining these functions, cfr. also Caputo(2003) in the 4th BAP-conference.

The transfer functions for respectively base resistance R and shaft resistance Rs as afunction of base displacement Sb and the shaft displacement ss are expressed as follows:

The base flexibility factor Kb (m/kN) corresponds to the tangent slope at the origin ofthe hyperbolic curve. It also gives, multiplied with 1M, the base displacement at 50%mobilisation of Rbu- On the basis of the settlement formula for circular footings, Kbmay be related to the secant modulu s Eu (considered at 25% of the ultimate stress) by:

(5.20)

(5.21)

With v = 0A andf = 0 .85One consequently can write as the relation between the base displacement at 50%

of Rbu and Kb or Eu:

(5.22)

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It appears from the above mentioned equation s that only a few parame ters are requiredto define the various hyperbolic functions:

- R}yU or q\j U \ ultimate pile base resistance, total or unit value- Eu: secant modulus (at 25% of ultimate stress) of the soil beneath the pile base- Rsu or q su j: ultim ate pile shaft resistance, total or un it value in the different layers

around the pile shaft- Ms: shaft flexibility factor- Ec or £ s : pile material modulus (concrete, steel, grout, ...)

Some commen ts with regard t o the param eter choice (F. De Co ck, 200 9):

1 The ultim ate pile base resistances and pile shaft resistances are in mo st casesobtained by calculation. An overview of the wide panoply of methods used inEurope resulted from the ERTC3 work in the period 1994-2004. (De Cock 1998,De Cock, et al., 2003). It should be mentioned that the required value shouldbe the asymptotic ultimate value at large displacements. However, one can alsouse the hyperbolic law on the basis of another ultimate value, e.g. at 10% of thepile base, and by deducing R u from the next correlation at s = 1 0 % of the pilebase diameter D^:

pile's shaft diameter Ds and inversely proportional to the ultimate shaft friction Rsu,it means:

(5.23)

(5.24)

with Ms a dimensionless flexibility factor in the nature of an angular rotation. Fromthis one deduces that 50% of the ultimate shaft friction is mobilised at a pile shaftdisplacement of:

(5.25)

2 For the secant mo dulus in non-cohesive soils, Ca pu to (2003) found a correlationfactor of 10 between Eu and the average CPT-cone resistance qc in the proximityof the pile base. According to the author, the correlation should also depend on thesoil stress history (e.g. geological overc onso lidation ) and sho uld also be pile typ erelated. There is in fact enough evidence that the soil stiffness may be influenced bythe executio n m eth od of the pile: e.g. a bored p ile may resu lt in some soil relectanceat the pile base, while a driven pile leads to a densification and prestressing of th epile base layer; the latter results in a much higher deformation modulus. Further

literature survey and backanalysis are needed to define suitable correlations, butthe following correlations appear to be quite promising in non-cohesive soils:Eu = 4 to 6 x qc for bored piles in NC-sandsEu = 6 to 8 x qc for bored piles in OC-sands

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Eu = 8 to 12 x qc for screw pilesEu = 15 to 20 x qc for driven pilesFor cohesive soils (stiff OC-clays) we would propose:40 qc in case of steel piles

orEu = 80 qc in case of concrete displacement pilesEu ~ 500 cu in case of steel pilesEu ~ 1000 cM in case of concrete displacement pilesFor the shaft flexibility factor, Caputo's statistical analysis (Caputo, 2003) confirmed the findings from Fleming that this factor generally is in the order of0.001-0.002 for soil displacement pilesThe material moduli of elasticity for concrete and steel are supposed to be wellknown. For concrete several empirical formula relate the elasticity modulus to thecompressive strength, as for example:

On the other hand, the non-linearity of the material modulus at high concreteor steel stresses should be considered. In particular in the case of tension piles,the question rises whether and when the fissuring of the concrete under tensiondegrades during the tension test.

5.6.6 D i f f e r e n t i a l e q u a t i o n f o r s o l u t i o n o f f i n i t e - d i f f e r e n c ee q u a t i o n f o r a x i a l l y l o a d e d p i l e s

A graphical model for a pile under axial loading w as show n in Fig. 5.1 3. The m odel canbe generalized so that nonlinear load-transfer functions can be input, point by point,along the length of the pile. Also, the stiffness AE of the pile can be nonlinear withlength along the pile. Figure 5.13 shows the mechanical system for an axially loadedpile. The pile head is subjected to an axial force Pio p, and the pile head undergoes adisplacement zt. The pile-tip displacement is Ztip and the pile displacement at the depthx is z. Displacement z is positive downward and the compressive force P is positive.

Considering an element dx (Fig. 5.13a) the strain in the element due to the axialforce P is computed by neglecting the second order term dP.

(5.26)

or

where P = axial force in the pile (do wn w ard positive); Ep = Young's modulus of pilematerial; and Ap = cross-sectional area of the pile.

The total load transfer through an element dx is expressed by using the modulusm in the load transfer curve (Fig. 5.14a). The maximum load transfer is indicated the

3

4

5.27)

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158 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

igure 5.13 Illustration of mechanics fo r an axially loaded pile.

(a) Load transfer curve for side resistance

igure 5.14 Load tran sfe r in side resistance and in tip resistance fo r a pile under axial loading.

(a) (b) (c) (d)Mechanical Discretized Displacement Force

system system

(b) Load transfer curve for tip resistance

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Eq. 5.30 constitutes the basic differential equation which must be solved. Boundaryconditions at the tip and at the top of the pile must be established. The boundary condition at the tip of the pile is given by Eq. 5.31. At the top of the pile theboundary condition may be either a force or a displacement.

symbol fmax a value which must be determined point by point along the length of apile. The conceptual shape of a curve for load transfer if the pile is subjected to upliftmay or may not differ from the curve for compressive loading.

(5.28)

(5.29)

wh ere C = circumference of a cylindrical pile or the perimeter for a pile wit h a p rism aticcross section.

Eq. 5.27 is differentiated with respect to x and equated with Eq. 5.29 to obtain

Eq. 5.30.

(5.30)

(5.31)

The pile-tip resistance is given by the product of a secant modulus v and the pile-tipmovement zup (Fig. 5.14b). The maximum load transfer in end bearing is given as q^a value that also must be determined at the tip of a pile.

5.6.7 F i n i t e d i f f e r e n c e e q u a t i o n

Eq. 5.32 gives in difference-equation form the differential equation (Eq. 5.30) forsolving the axial pile displacement at discrete stations.

(5.32)

(5.33)

(5.34)

(5.35)

and where b — increment length or dx (Fig. 5.7a and 5.8a).

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Equations of the form of 5.32 can be written for every element along the pile, appropriate boundary conditions can be used, and the equations can be solved in any suitablemanner. A recursive solution is convenient.

5.6.8 L o a d t r a n s f e r c u r v e s

The acquisition of load-transfer curves from a load test requires that the pile beinstrumental internally for the measurement of axial load with depth. The numberof such experiments is relatively small and in some cases the data are barely adequate;therefore, the amount of information of use in developing analytical expressions islimited.

There will undoubtedly be additional studies reported in technical literature fromtime to time. Any improvements that are made in load-transfer curves can be readilyincorporated into the analyses.

5.6.8.1 Side resistance in cohesive soil

The curves for fma% may be found by a num ber of me tho ds fou nd in technical literature. The American Petroleum Institute (API) (1993) has proposed a method forcohesive soil using Eq. 5.36 through Eq. 5.38.

where cuz = the undrained shear strength of the clay at depth z, and p = the effectiveoverburden pressure. The constraint in Eq. 5.37 is that a < 1.0. W ith values of fmax

that vary from point to point along the length of the pile, the load transfer curves forside resistance can be computed.

Coyle & Reese (1966) examined the results from three instrumented field tests

and rod tests in the laboratory and developed a recommendation for a load-transfercurve. The curve was tested by using results of full-scale experiments with uninstru-mented piles. The comparisons of computed load-settlement curves with those fromexperiments showed agreements that were excellent to fair. Table 5.1 presents thefundamental curve developed by Coyle & Reese.

An examination of the Table 5.1 shows that the movement to develop full loadtransfer is quite small. Furthermore, the curve is independent of soil properties andpile diameter.

Reese & O'Neill (1987) made a study of the results of several field-load tests ofinstrumented bored piles and developed the curves shown in Fig. 5.15. An examina

tion of Fig. 5.15 shows that the maximum load transfer occurred at approximately0.6% of the diameter of a bored pile. Because the piles tested had diameters of 0.8to 0.9 m, the mov eme nt at full load transfer wo uld be in the ord er of 5 m m , which islarger than the 2 mm show n in Table 5. 1.

(5.36)

(5.37)

(5.38)

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able 5.1 Load transfe r vs pile movem ent for cohesive soil

Ratio of load transferto maximum load transfer

Pile Movementin.

Pile movementmm

0.00.180.380.790.971.000.970.930.930.93

00.010.020.040.060.080.120.160.20

>0.20

00.250.511.021.522.033.054.065.08

>5.08

igure 5.15 No rm alized curves showing load transfe r in side resistance versus settlem ent for bore dpiles in clay (after Reese O 'N ei ll 1987).

Kraft et al. (1981) studied the theory related to the transfer of load in side resistanceand noted that pile diameter, axial pile stiffness, pile length, and distribution of soilstrength and stiffness along the pile are all factors that influence load-transfer curves.Equations for computing the curves were presented. Vijayvergiya (1977) also presenteda method for obtaining load-transfer curves.

5.6.8.2 End bearing in cohesive soil

The work of Skempton (1951) (cited also during discussion of lateral loading) wasemployed and a method was developed for predicting the load in end bearing of a pile

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in clay as a function of the movement of the tip of the pile. The laboratory stress-strain curve for the clay at the base of the pile must be obtained by testing, or may beestimated from values given by Skempton for strain, £50, at one-half of the ultimatecompressive strength of the clay. Skempton reported that £50 ranged from 0.005 to0.02, and further used the theory of elasticity to develop approximate equations forthe settlement of a footing (base of a pile). His equations are as follows.

(5.39)

(5.40)

where qb = failure stress in bearing at base of footing; Of = failure c om pressive stressin the laborato ry unconfined-com pression or quick triaxial test; Nc = bearing capacity

factor (Skempton recommended 9.0); b = diam eter of footing o r equivale nt lengthof a side for a square or rectangular shape; ε = strain mea sures from unconfined-compression or quick-triaxial test; and w\, = settlement of footing or base of pile.

The value of £50 can be selected in consideration of whether the clay is brittle orplastic. In the absence of a laboratory stress-strain curve, the shape of the curve canbe selected from experience and made to pass through £50. Equations 5.39 and 5.40may be used and the load-settlement curve for the tip of the pile can be obtained. Theassumption is made that the load will not drop as the tip of the pile penetrates the clay.

Reese O'N eill (1987 ) studied the results of a nu m ber of tests of bore d piles in claywhere measurements yielded load in end bearing versus settlement. Fig. 5.16 resulted

igur 5 1 6 No rma lized curves showing load transfer in end bearing versus settlem ent for b ored pilesin clay (after Reese O 'N eil l 1987).

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from the studies. Examination of the mean curve shows that a settlement of about30 mm will result at the ultimate bearing stress for a pile with a diameter of 0 5 m

The movement of a pile to cause the full load transfer in end bearing is several timesthat which is necessary to develop full load transfer in skin friction. The largest strain

in skin friction occurs within several millimeters away from the wall of a pile whenthe pile is loaded to failure. End bearing mobilizes the strain on many elements of soilin the zone beneath the tip of a pile. Hen ce, the movement of a pile to develop the fullload transfer in end bearing is a function of the diameter of the pile and can be manytimes the movement to develop full load transfer in skin friction.

where K = the lateral earth -press ure coefficient; p = the effective overburden pressure;and <5 = the friction angle between the soil and the pile wall. A value of K of 0 8is recommended for open-ended pipe piles and a value of 1 0 is recommended forfull-displacement piles.

The procedure presented in Eq 5 41 is relatively simplistic and can yield good correlations with experim ental results if the values of K are selected correctly. Two valuesare given, indicating that K is a constant with depth, but the value of K is expectedto vary with depth and many other factors, including the details of the methods ofinstallation.

Another limitation of Eq. 5 41 concerns calcareous or carb ona te soils, consisting ofsoft grains from remains of sea life. Such soils are frequently cemented and open-endedpiles have been known to penetrate to many meters in such soils und er self weight. Thesoft grains are crushed by the steel shell and cementation prevents the developmentof lateral pressure with the result that side resistance will be extremely small. TheAmerican Petroleum Institute suggests the use of the term siliceous in describing soils

to which Eq 5 41 is app licable.Equation 5 41 indicates that f max will increase without limit, but the AmericanPetroleum Institute suggests the limiting values shown in Table 5 2

able 5.2 Guideline f or side frictio n in siliceous soil (from American Petroleum Institute, 1993).

Limiting f m x

Soil 8, degrees kPa

Very loose t o medium sand t o silt 15 47 8

Loose t o dense sand t o silt 20 67 0Medium t o dense sand t o sand-silt 25 83 1

Dense t o very dense sand t o sand-silt 30 95 5

Dense t o ve ry dense gravel t o sand 35 I 14 8

(5.41)

5.6.8.3 Side resistance in cohesionless soil

An equation for computing the max imu m value side resistance fmax in cohesionless soil

was suggested by the American Petroleum Institute (1993), and presented in Eq 5 41

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igur 5 17 No rma lized curves showing load transfer in side resistance versus settlem ent for bore dpiles in cohesionless soil (after Reese O 'N ei ll 1987).

Experiments have been reported in literature where tests have been run to solve forthe value of δ . A suggestion has been made that the value of δ should be taken as 5degrees less than the value of φ , the friction angle for the granular soil.

Coyle Sulaim an (1967) studied the load transfer in skin friction of steel pilesdriven into sand and obtained curves for piles with diameters ranging from 330 mmto 400 mm and w ith a penetratio n of abo ut 15 m. The wa ter table was near theground surface and the sand had a friction angle of 32 degrees. An examination of the shape of the curves shows that they can be fitted with the following

equation.

(5.42)

Reese O'N eill (1987) exam ined the results of load tests on a num ber of full-sizedbored piles that were instrumented for the measurement of axial load with respectto depth. The results of this study showed that the curves for cohesionless soils weresimilar to those for cohesive soils and that Fig. 5.17 can be used for cohesionlesssoils.

Mosher (1984) studied the problem of the transfer of load in skin friction (sideresistance) of axially loaded piles in sand. He recommended the use of an equationthat includes a term for the soil reaction modulus for the sand, a value that will varywith confining pressure and thus is a complex term to evaluate.

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able 5.3 Guideline for tip resistance fo r siliceous soil (from Am erican Petroleum Institute , 1993).

Limiting q b

Soil Nq MPa

Very loose to medium sand-silt 8 1.9Loose to dense sand to silt 12 2.9Medium to dense sand to sand-silt 20 4.8Dense to very dense sand to sand-silt 40 9.6Dense to ve ry dense gravel t o sand 50 12.0

where w = settlement, m; q = applied lo ad, kPa; Dr = relative density; B = diameter oftip, m; q\j = ultim ate base resistance, kPa; and Cw = settlement coefficient (the authorfound values as follows: 0.00372 for driven piles; 0.00465 for jacked piles; and 0.0167for buried piles).

(Note: Eq. 5.44 is not dimensionally homogeneous so values are dependent on thesystem of units being used. Values of Cw were recomputed for the SI system.)

Reese & O'Neill (1987) studied the results of experiments with bored piles anddeveloped Fig. 5.18. The information in Fig. 5.18 was developed from a relativelysmall amount of data and, as with other methods presented in this chapter, should beused with appropriate discretion.

5.7 C L O S E L Y S PA C E D P IL E S U N D E R L AT E R A L L O A D I N G

5.7.1 M o d i f i c a t i o n o f l o a d t r a n s f e r c u r v e s f o r c l o s e l ys p a c e d p i l e s

The method of analysis employed for the behavior of single piles is to employ load-

transfer curves, p-y curves for lateral loading. The method is extended to the analysisof piles in a gro up . If the piles are spaced widely ap art , the p-y curves presented earlierfor single piles may be used without modification. As the piles are installed close toeach other, their efficiency will decrease and the lateral resistance from the soil will

6 SA nd bearing curves in cohesion less soi l

A procedure for obtaining the value of q^ th e u n u~ e n d bearing in cohesionless soil, ispresented by the American Petroleum Institute.

(5.43)

(5.44)

where pt = the effective overburden pressure at the tip of the pile, and Nq = a bearingcapacity factor. Table 5.3 presents values of Nq and limiting values of q^.

Vesic (1970) studied the available literature and perform ed some careful experim entsand proposed an equation for computing the load versus tip settlement for piles in sand.

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166 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

igure 5.18 No rma lized curves showing load transfer in end bearing versus settleme nt for bore d pilesin cohesionless soil (after Reese O 'N ei ll 1987).

decrease. The decision was made that the most effective way to reflect the loss ofefficiency for such piles is to develop procedures for reducing the value of pu\t to reflectthe close spacing, which in turn will reduce all p-values in the p-y curves.

The procedure has the distinct advantage on allowing the solution of the nonlineardifferential equation for the individual piles in a group when the group is subjected toan inclined and eccentric load. The following sections is this chapter will demonstratea rational approach, strongly dependent on experimental data, for reducing the valuesof p to reflect close spacing.

5.7.2 C o n c e p t o f i n t e r a c t i o n u n d e r l a t e r a l l o a d i n g

The influence of the spacing between piles can be illustrated by referring to Fig. 5.19.The assumption is made that all of the piles are fastened to a cap or to a supersturctureand th at th e lateral deflection of all of the piles will be the same or nearly so. Fig. 5.19 ashows three closely spaced piles that are in line. It is evident, without resorting toanalysis, that the resistance of the soil against Pile 2 is less than that for an isolatedpile because of the presence of Piles 1 and 3. Pile 2 may be considered to be in the

sh ad ow of Pile 3; the shado w-effect on soil resistance is obviou sly related to pile

spacing.Similarly, the soil resistance against Pile 2 in Fig. 5.19b is influenced by the pres

ence of Piles 1 and 3. The edge-effect on soil resistan ce is again influenced by pilespacing.

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168 S in g le P i l e s an d P i l e G ro u p s U n d e r L a te ra l L o ad in g

where R = the rati o of the gro und line deflection of a single pile com pu ted by thep-y curve method to the deflection pi computed by the Poulos method for elastic soil,Hj = lateral load on pile j aPFkj tne coefficient t o get the influence of pile j on pile k incomputing the deflection p (curves from the work of Poulos for obtaining the values

of the af-coefficients are not shown here), H = lateral load on pile k; and m = numberof piles in group (the subscript F pertains to the fixed-head case and is used here forconvenience; Poulos also presented curves for influence coefficients where shear isapplied, apHkji a n d where moment is applied, a pMkj-

The above equation can be used to solve for group deflection, Yg5 and loads onindividual piles. With the known group deflection, Yg, the p-y curves at each depthfor a single pile can be mu ltiplied by a factor, termed the Y factor, to m atch th epile-head deflection of a single pile with the group deflection, Yg, by repeated trials.Th e Y factor is a con stan t mu ltiplier emp loyed to increase the deflection values ofeach point on each p-y curves; thus, generating a new set of p-y curves that include thegroup effects. The modification of p -y curves, as described ab ove for piles in the g ro up ,allows the computation of deflection and bending moment as a function of depth.

From a theoretical viewpoint, group effects for the initial part of p-y curves can beobt aine d from elastic theory. The u ltima te resistance of soil on a pile is also affected bythe adjacent piles due to the interference of the shear-failure planes, called shadowingeffects. Focht & Koch suggested a p-factor may need to be applied to the p-y curvesin cases where shadowing effects occur. The p-factor should be less than one and themagnitude depends on the configuration of piles in a group.

Other approaches regarding the modification of the coefficient of subgrade reactionwere also used for pile-group analyses. The Can adian F oun dation Engineering M anu al(1978) recommends that the coefficient of subgrade reaction for pile groups be equalto that of a single pile if the spacing of the piles in the group is eight diameters. Forspacings smaller than eight diameters, the following ratios of the single-pile subgradereaction were recommended: six diameters, 0.70; four diameters, 0.40; and threediameters, 0.25.

The Japan Road Association (1976) is less conservative. A slight reduction in thecoefficient of horizontal subgrade reaction is considered to have no serious effect withregard to bending stress and the use of a factor of safety should be sufficient in designexcept in the case where th e piles get quite close together. W hen piles are closer tog ether

than 2.5 diameters, the following equation is suggested for computing a factor m tomultiply the coefficient of subgrade reaction for the single pile.

(5.46)

where s = center-to-center distance betw een piles; and b = pile diameter.Bogard & Matlock (1983) present a method in which the p-y curve for a single pile

is modified to take into account the group effect. Excellent agreement was obtainedbetween their computed results and results from field experiments (Matlock et al.

1980).As a part of a study where a 3-by-3 group of full-sized piles was loaded laterally,

Brown and Reese (1985) reviewed the common approaches for the analysis of pilegroups, and concluded that none of the methods was effective in predicting the results

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A n a l y s i s o f g r o u p s o f p i l e s s u b j e c t e d t o i n c l i n e d a n d e c c e n t r i c l o a d i n g 169

igure 5.20 Mo dification of soil resistance fo r a p-y curve for a single pile to for interaction of piles ina group.

that were obtained. The most logical approach appeared to be one that would use theinteraction factors for the modification of the p-y curves, as was proposed by Fochtand Koch (1973). However, the use of interaction factors from the theory of elasticitywas unproductive, even for small deflections. The marked difference in the behaviorof soils under tension and compression severely limits the application of the theory ofelasticity in obtaining the interaction of closely spaced piles under lateral loading.

Modification of p-y curves is attractive if some general rules can be found to allowthe adjustment of recommendations for p-y curves for single piles for the various soilsand nature of loading, as presented in Chapter 3. Such modifications would allowloading to each pile in the group to be computed, the effects of cyclic loading tobe accommodated, and deflection and bending moment with depth to be computedfor each pile in the group. Modifications can be done as shown in Fig. 5.20 withp-values multiplied by (a\) and y-values multiplied by (ai). As shown in the materialthat follows, interaction can be accomplished conveniently by the use of only one ofthese curve stretching parameters.

Scott (1995) performed a comprehensive study of the results of experiments withclosely-spaced piles. Experiments were reviewed that were performed both in the field

and in the laboratory, and only those tests were analyzed where a single pile wasloade d in addit ion to the pile gr ou p. The efficiency is defined as the load o n individ ualpiles in the group divided by the load on the single pile at the same deflection. Scottnoted that the efficiency varied throughout the range of loading; therefore, a referencedeflection had to be selected so the results from the various tests could be compared.The reference deflection was selected as l/50th of the pile diameter.

The loads on the piles, corr espo nd ing to the reference deflection, corre spo ndsapproximately to the working load, or the ultimate load divided by a factor of safetyranging from about two to three. A large number of studies were evaluated and theones which were judged to have the most complete information are described below.

The stipulation was made, of course, that the single pile and the piles in the group hadto have the same diameter, had to have been installed in soils with the same characteristics, had to have been installed with the same techniques and had to have beenloaded with the same head conditions, whether fixed-head or free-head. Batter piles

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Chapter 6

A n a l y s i s o f s i n g l e p i l es a n d g r o u p s o fp i l e s s u b j e c t e d t o a c t i v e a n d p a s s i v e

l o a d i n g

6.1 N A T U R E O F L A T E R A L L O A D I N G

Lateral loads on piles can be derived from many sources; a convenient characterization of the sources is to term them active or passive, as suggested by De Beer(1977). As employed herein, active loading is considered to be time-dependent or liveloading. Passive loading, on the other hand, is principally time independent or deadloading.

Active loading may come from wind, waves, current, ice, traffic, ship impact, andmooring forces. Passive loading is derived principally from earth pressures or potentially moving soil, but may also come from dead loading as from an arch bridge. Inthe following sections, some details are presented on the nature of the various kindsof active and passive loading.

The solution of a number of examples is presented in this chapter where the piles aresubjected either to active loading or to passive loading. The solutions are presented insufficient detail to provide guidance in addressing similar problems.

6 . 2 A C T I V E L O A D I N G

6.2.1 W i n d l o a d i n g

6.2.1.1 Introduction

The first step in the usual practice of designing of a pile-supported structure is tocollect data on wind velocities from appropriate sources. The second step is to translatethe wind velocities into forces on a structure employing shape factors for the particular geometrical element. The forces may be time-dependent; therefore, a dynamicanalysis of the structure may be indicated. The problem is simplified greatly bysome agencies. The American National Standards Institute (ANSI, 1982) suggeststhat a minimum static pressure of 0.72 kPa may be used in designing the bracing fora masonry wall.

The velocity may be classified by gusts which a re averaged over less tha n o ne m inut eor sustained velocities which are averaged over one minute or longer. The data may beadjusted to a specified distance above the ground or water and may be averaged overa given time.

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 07

Invariably, the wind will gust and equations are available for finding the velocityof a gust in terms of the average velocity of the wind over a period of one hour atsome distance above the grou nd surface. A formulation is available for com puting theturbulence intensity, the standard deviation of the velocity of the wind normalized by

the velocity of the win d over on e hour. The fluctu ations of the velocity of the win d canbe described by a spectrum, and equations are available for computing the spectraldensity as a function of the standard deviation of the wind velocity.

Wind gusts have spatial scales related to their duration. Three-second gusts arecoherent over shorter distances than 15-second gusts. While the shorter-term gustsare appropriate for investigating the forces on individual members, 5-second gusts areappropriate for obtaining the maximum total loads on structures whose horizontaldimen sion is less tha t 50 m. Fifteen-second gusts are ap pro pri ate for the total staticwind load on larger structures. If a dynamic analysis is unnecessary, the one-hour sustained wind is usually appropriate for the total static wind forces on the superstructureof an offshore structure associated with the maximum wave forces.

Force of Wind as a Function of Wind Velocity. The wind force on an object may becomputed by use of the following equation (API, 1993).

6.2.2 W a v e l o a d i n g

6 2 2 Introduction

The following factors must be considered for each offshore structure: wave heightand period, marine growth, and hydrodynamic coefficients for computation of waveforces. In addition, the forces from the wind, discussed above, and the current,discussed below, must be taken into account. Fu, et al. (1992) presented the flowchart, Fig. 6.1, to illustrate the required procedure. For the specific site, the compu

tation procedure is initiated with the current forces and the wave description, waveheight and period, usually for the 100-year storm. As shown in the figure, the current forces are modified, according to the structural axis selected for analysis, andconsidering the factors affecting the blockage of forces against structural elements

(6.1)

where F = w ind for ce, N , w = weight density of air, N/m3 , g = gravitational acceleration, m/s2, V = wind speed, m/s, Cg = shape coefficient, and A = area of object, m2 .

In the absence of specific data on shape coefficients, the following values may beused for Cg.

BeamSides of buildingsCylindrical sectionsProjected areas of miscellaneous shapes

1.51.50.51.0

Adjustments must be made in the equations for the computation of wind force onsurfaces that are not perpendicular to the direction of the wind.

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2 08 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 6.1 Diagram showing steps in the computation of forces from waves and current (afterFu,etal., 1992).

behind other elements. W ith wave height and period, a wave-height-reduction factoris selected, depending on the relative direction of the wave. Then, a wave theory is

selected, along with the wave kinematics and the wave-kinematics factor, and waveforce computations are made. The wave forces take into account the character of themember, the amount of marine growth, and the shielding factor for the particularmember.

In the design of pile-supported structures, the loads originating from the wavemotion are to be taken into account both in respect of the loading of the individualpiles, as well as of the superstru cture. The superstructure should be located above thecrest of the design wave if possible. O ther wis e, large hor izon tal an d vertical loads fromthe direct wave action can affect the structure, whose determination is not addressedherein. The elevation of the crest of the design wave is to be determined in consideration of the simultaneously occurring highest still-water level, as the case may be,taking into account also the wind-raised water level, the influence of the tides, and theraising and steepening of the waves in shallow water.

6 2 22 Exam ple o f wave height and period

Oceanographers have collected data of storm-induced waves in the oceans around theworld and have developed predictions for wave heights and periods for given locations.An example of the presentation of such a prediction is presented in Table 6.1. Thesite is off the coast of Austra lia, for a water depth of appro xim ately 100 m, an d for

the 100-year storm. The period of the waves Tz rang es from 5.8 sec and 10.6 sec,the characteristic wave height Hs ranges from 3.6 m to 12.0 m, and the total num berof waves is slightly above 18,000. Table 6.1 shows the 35.5 hours of the storm beingdivided into seven periods as the wave heights build up and decay.

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 09

able 6.1 Parameters fo r <

Period(hours)

—21.5 to — 1.0- 1 1.0 t o - 5 . 5- 5 .5 t o - 2 . 0- 2 0 to 1.51.5 to 4.04.0 to 8.08.0 to 14.0

Totals

i hydrograph

Durat ion(hours)

10.55.53.53.52.54.06.0

35.5

of storm waves.

H s

(m)

3.66.0

11.1612.011.166.03.6

Tz

(sec)

5.807.49

10.2210.6010.227.495.80

No . of WavesN

6517264412331189881

19233724

I 8 I I I

For each of the seven periods of the storm, the distribution of wave height H isassumed to be controlled by a spectrum given by the Rayleigh equation:

(6.2)

Employing Eq. 6.2 and the number of waves for each of the seven periods, the values inTable 6.2 were computed. The relatively small number of waves for the greatest waveheights is of interest, even though the predictions are hypothetical. The equations for

predictin g the response of the soil to cyclic load ing, presen ted in Ch apte r 3 , are strong lybased on experimental results where the piles were cycled under a specific load untilstability was achieved. The number of cycles of load necessary to achieve stability wasin the order of 50 to 100. Designs would certainly be conservative if only a few cyclesof load are app lied becau se the difference in the wave forces will be consid erable for awave height of 18 m and com pared to a height of 23 m.

6 2 2 3 Kinematics for two-dimensional waves

A conve nient ap pro ach t o the selection of a part icula r theor y for obta inin g the velocitiesand accelerations of water particles as a function of time and position in the wave isgiven in Fig. 6.2 (Barltrop, et al., 1990; API, 1993). The symbol T in Fig. 6.2 is theapparent wave period which is the period seen by an observer who moves with thevelocity of the current. The actual period and the apparent period are the same for azero current; for the usual values of the current, the apparent period is within about5 to 10% or the actual wave period. The other symbols are as follow: H is waveheight, Hb is height of breaking wave, d is mean water depth, and g is the accelerationof gravity. Entering the curve with data for the maximum wave from Tables 6.1 and6.2, the particle motions may be computed from the Stokes 5 theory or from StreamFunction 3. Atkins (1990) presents procedures for computing the particle motions.

6 2 2 4 Forces from waves on pile-supported structures

General. In respect to the me tho ds of co m pu tati on of the forces from waves, twoappro aches are noted: the meth od of superposition according to Mo rison , et al. (1950)

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2 10 S i n g l e P i l es a n d P i l e G r o u p s U n d e r L a t e r a l L o a d i n g

bl e 6.2

Individual

Num ber of waves of specified height H with

Wave Height

H m )

0-11-22-33-44-55-66-77-88-9

9-1010-111-12

12-1313-1414-1515-1616-1717-1818-1919-2020-2121-2222-23Total

Period hours)

-21 .5 t o - 1 1.0 - 1 1.0 t o -5 .5 -5 .5 t o -

968 145 202132 389 571917 519 901063 521 114398 429 129104 301 134

19 183 1313 97 120

45 105

19 88 907 71 752 54 611 40 47

29 3620 26 1413 18 98 12 65 8 43 5 22 3 11 2 11 1

16604 2658 1235

lin each time

-2.0 -2.0 to

16487698

112119119113103

6351392920

1189

period du

1.5 1.5 to

144164819296938675

1352

-

881

iring the sto rm.

4.0 4.0 to

1052833773793122191337033

1931

8.0 8.0 to 14.0

55312181095607227

59

1-

3771

for slender structural members, and a method based on diffraction theory accordingto MacCamy & Fuchs (1954) for wider structures.

Some detail is presented here on the meth od of super position (CER C, 1984) whichis applicable for non-breaking waves (Fig. 6.2). An approximate method for breakingwaves is proposed in a later section.

The method according to Morison gives useful values if the following conditionis met for the individual pile or member (most pile-supported structures meet thecondition):

where D = effective diameter for circular cylindrical member, or width of non-circularmember, m; L = length of the design wave, m; and L = C-T; C = wave celerity, m/sec;

and T = wave period , sec.

Method of computation according to Morison 1950) for non-breaking waves.The computation of the force exerted by waves on a cylindrical object (or another

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 21 I

shape) may be computed as the sum of a drag force and an inertia force asfollows:

(6.3)

For a pile with circular cross-section, the equation becomes:

(6.4)

where p = hydrodynamic force vector per unit length acting normal to the axis of themember, kN/m; pr = drag force vector per unit length acting normal to the axis ofthe member in the plane of the member axis, kN/m; pM = inert ia force vector per

unit length acting normal to the axis of the member in the plane of the memberaxis, kN/m; CD = drag coefficient yw = unit weight of water, kN/m3; g = accelerationof gravity, m/sec2; D = effective diam eter of circular cylindrical mem ber, includingmarine growth, or width of noncircular member, m; u = comp onen t of the velocity

igure 6.2 Regions of applicability of equations fo r kinem atics of waves (from API 1993, Ba rltrop , et al.,1990).

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2 12 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 6 3 Ac tion of a wave on a vertical pile

vector normal to the axis of the member, m/sec; \u\= absolute value of u m/sec;CM = inertia coefficient; A = projected area norm al to the axis of the member, m2; anddu/dt ~ du/dt = horizo ntal com pon ent of the local acceleration vector of the w aternormal to the axis of the member, m/sec2.

A sketch of a pile with forces from drag and acceleration shown at an element dzat a point along the pile is shown in Fig. 6.3. Most of the terms in the sketch aredefined above ; the terms x, z and η appe ar in the com puta tion of particle velocity a ndacceleration with equations noted in Section 6.2.2.3.

Forces from breaking w aves. At present, there is no accepted m ethod for com puting the forces on a pile from breaking waves; therefore, the Morison formula isemployed as an expedient in making computations. The assumption is that the waveacts as a water mass with high velocity but without acceleration. Thus, the inertiacoefficient is set to CM = 0, w here as the drag coefficient is increased to CD = 1.75CERC, 1984).

Safety Requirements. Th e design of pile structu res again st wav e action is stronglydependent on the selection of the design wave, the wave theory in computations andthe selection of the coefficients CD and CM. The selection of the design wave is amatter to be considered by the owner of the structure.

Factors to be considered are the anticipated life of the structure and the chance thata storm of a given magnitude will occur. Statistics are available that give the frequencythat a storm of a given magnitude will occur in a specific body of water, such as the

Gulf of Mexico. However, data may be quite sparse when a specific site in the Gulfis selected. Therefore, for a particular structure at a particular location in a particularbody of water, the selection of the design wave may require a decision from the highestlevel of management.

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An alys is of single piles and gro up s of piles 213

6.2.3 C u r r e n t l o a d i n g

6 2 3 Introduction

While the effects of current must be considered in the design of offshore structures,the current plays a major role in the design of foundations for bridges. A guidingconcept in the design of offshore platforms is that the platform is relatively austere inthe region of the maximum height of the wave, with the deck designed to be abovethe maximum height of wave. Furthermore, scour is not as severe a problem withoffshore structures as for bridges. Therefore, current loading will be treated separatelyfor offshore structures and for bridges.

6 2 3 2 Current loading for offshore structures

Oceanographers have developed data for various geographical areas with respect to

currents generated by hurricanes. For example, in most of the Gulf of Mexico, API(199 3), the directio n of the curren t in shallow -wat er (45 m and below) is specifiedgeographically w ith the ma xim um current given as 2.1 kno ts. For deeper water (90 mand above) the maximum current is also specified as 2.1 knots with the direction of thecurrent the same as the direction of the waves. The direction of the maximum waveis specified, and coefficients are presented (0.70 to 1.00) for modifying the wave andthe magnitu de of the current as well, depending on direction. For intermediate depthsof water, interpolation is used to find the maximum current. The maximum current isspecified to occur at the water surface, to remain constant for a considerable depth,and to reduce to 20% of the maximum at the mudline (API, 1993).

The st orm tide is also specified by API as a function of the depth of wa ter for certaingeographic areas of the Gulf of Mexico. Site-specific studies are required for otherareas.

In the absence of recommendations by other agencies for areas of the world's oceansother than the Gulf of Mexico, oceanographers must perform site-specific studies.

6 2 3 3 Current loading for bridges

Predicting the maximum flow of a stream at a bridge during the period of recurrenceselected for design depends on the availability of statistics for storms in the watershed

area. With such statistics, a hydrologie study can be made of the factors that affectthe concentration of flow at the bridge, and a prediction can be made. Such predictions usually cover a limited period of time because of construction in the watershed.With the height of the stream, along with a prediction of scour (discussed below),computations can be made of the forces on bridge bents and consequently on the pilessupporting the bents.

The Standard Specifications of the American Association of State Highway andTransportation Officials (AASHTO, 1992) presents a simple, nonhomogeneousequation for the force of stream current on piers.

6.5)

where P = unit force, kPA, V = velocity of stream flow, m/s, K = a constant, dependingon the shape of the structural element.

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2 14 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

The values of the constant K was converted for English units to SI units with thefollowing results: K is equal to 0.71 for members with a square end, is equal to 0.26for angle ends where the angle is 30 degrees or less, and 0.34 for circular piers.

6 .2 .4 S co u rScour is not a condition of loading; however, the scour and erosion of soils at pile-supported foundations, if unanticipated, can create serious instabilities and need tobe addressed along with the various kinds of active loading. The lack of supportingsoil along a portion of a pile, perhaps even a small portion, will lead to increaseddeflections and increased bending moments. Therefore, the amount of scour must bepredicted, or measures must be taken to prevent the scour.

The theory of sediment transport can be used to predict the velocity of flow thatwill move a single-grain particle of soil (silt, sand, gravel, cobble, boulder). Boulders

are moved downstream by swift-flowing water in the mountains; silt and fine sand arecarried by slow-moving river water and deposited in deltas. The theory applies lesswell to the erosion of clays, where cohesion and complex structures exist (Moore &Masch, 1962; Gularte, et al., 1979). Complexities arise when the water must flowaround an obstruction, such as a bridge pier, and velocities increase.

Erosion is prevented by creating a scour-resistant surface. Special blankets, extending well into a river, can be effective to prevent loss of soil at river banks.Reverse filters can be placed, based on the grain-size distribution of the materialto be protected (de Sousa Pinto et al., 19 59 ). T he layers of the filter, startingfrom the soil to be protected, become successively larger in size until the size at

the mudline is judged to be large enough to resist movement. Each layer that isplaced, the filter, should have the following relationships with the layer below, thebase. Disßlter/Dssbase > 5; 4 < D15filter/D15base < 20 ; andD50filter/Dsobase < 25 .The subscripts for the D terms in the equations refer to the particular percentageby weight as determined from a grain-size-distribution curve. The specifications areknown in the United States as TV grading, because of work by Terzaghi at theWaterways Experiment Station in Vicksburg, Mississippi (Posey, 1963; Posey, 1971).

With respect to the suggestions noted above, the engineer should be aware the uniformity coefficient of the filter m aterial sho uld be consid ered, alon g with th e m ax im umsize. A uniform filter m aterial sh ould n ot be allowed , and each layer of the filter shou ldhave self-stability.

6 2 4 Scour at offshore structures

The informal approach of a number of designers for offshore structures is to assumea minim um a mo un t of erosion, perhaps 1.5 to 2 m, and to institute an observation alprogram with the view that anti-scour measures would be instituted if necessary. Suchan approach may be risky if the structure is founded in cohesionless soil, where bottomcurrents can occur.

In 1960, a diver-survey was made of scour around an offshore platform in the Gulf

of Mexico near Padre Island. The structure was installed in water with a depth ofappro xim ately 12 m, and some lateral instability was repo rted even in min or w aves.The soils near the mudline consisted of fine sand with a few shells and some stringersof clay. A bowl-shape depression was found to have formed around the structure;

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 15

the depth of the depression was 2.5 m. The structure was stabilized, using the reversefilters, described above. Three layers of slag from a steel mill were used; the top layerconsisted of particles ranging in size from 50 to 20 0 m m . A survey performed after ahurricane has passed through the area showed that little of the filter material had been

lost (Sybert, 1963).Einstein and Wiegel (1970) performed a comprehensive review of technical literature

on erosion and deposition of sediment near structures in the ocean. Many aspectsof the overall problem were addressed, including sediment, flow condition, and thestructure. A total of 415 references were cited but, as could be expected, a generalapproach was forced on the investigators because of the large number of parametersinvolved. Furthermore, the absence of cases where all details of observed scour wereknown prevented the application of specific equations for design.

The American Petroleum Institute (API, 1993, p.71) includes the following statement under the topic of Scour in the section on the Hydraulic Instability of ShallowFoundations: Positive measures should be taken to prevent erosion and undercuttingof the soil beneath or near the structure base due to scour. Exam ples of such measuresare 1) scour skirts penetrating through erodible layers into scour resistant materialsor to such depths as to eliminate the scour hazard, or 2) riprap emplaced around theedges of the foundation. Sediment transport studies m ay be of value in planning anddesign.

6 2 4 2 S cour at bridges

Scour has been judged to be the cause of the failure of numerous bridges. Laursen(1970) presents photographs of 12 failed bridges and noted that many other examplescould be added. While special mats may be used to prevent erosion at the banks ofa river, the erosion of the soil at the stream bed is usually unavoidable. The use ofriprap or other techniques for a relatively short stretch of a river, in the vicinity of abridge, is usually ineffective because the scour will start at the upstream edge of theprotection and proceed to erode the entire zone of protection . The num bers of factorsthat influence the depth of scour are so numerous, and frequently time-dependent,that predictions of the depth may vary widely. At a proposed bridge for a major cityin the United States, officials predicted that a flood would erode all of the alluvial soildown to the bedrock. A portion, if not all, of the stream bed is re-deposited with theabeyance of the flood so that measurements of the depth of scour must be made duringthe flood, which could be difficult.

The Standard Specifications for Highway Bridges, American Associations of StateHighway and Transportation Officials (AASHTO, 1992, p. 92) requires that the probable depth of scour be determined by subsurface exploration and hydraulic studies.The American Association of State Highway and Transportation Officials has issuedguidelines for the performance of the hydraulic studies (AASHTO, 1992).

6.2 .5 I c e l o a d i n g

The lateral loads from sheets or masses of ice may be a critical factor in the design ofsome structures. Locks and dams along the northern sections Mississippi River in theUnited States, as well as along other rivers, must be designed to withstand forces from

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2 16 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

ice that can occur in various forms and with various characteristics. The ice may passthro ugh the locks or lodge against the dams and apply loads as a function of nu mero usfactors, including the current if the river is in the flood stage.

With the discovery of deposits of petroleum in Cook Inlet on the southern coast of

Alas ka, em phasis was given to the geom etry of the super struc ture and the design of pilesto withstan d the loading from sheets of ice that com mon ly flowed dow n the w aterw ayin winter tim e. Experience had show n the vulnerability of waterfront structures to theforces from the ice. Designs were developed that presented only a vertical column atthe depth of the ice. Some dramatic movies that film the performance of the structuresshowed the fracturing of the sheets of ice in a ratcheting manner. As the ice movedagainst the structure, the lateral force would increase until the ice fractured, and theforce would then decrease to build up again.

N um ero us factors are associated with th e lateral force of ice on a stru ctur e, in cludingthe strength of the ice, the geometry of the moving mass, the velocity of approach, andthe nonlinear force-deflection characteristics. Such factors are investigated on a site-specific basis. How ever, AA SH TO (19 92) suggest a simple equ ation for the ho rizo nta lforces resulting from the pressure of mo ving ice (pg. 26 ).

Inclination of nose to vertical Cn

0° to 15° 1.0015° to 30° 0.7530° to 45° 0.50

AASHTO recommends a careful evaluation of the local conditions governing theparameters in Eq. 6.6 before making using the equation for design. The strength of theice, for example, is expected to range from about 700 kPa to 3000 kPa, depending on

whether the ice is breaking up or is at a temperature significantly below the meltingpoint.

6.2.6 S h i p i m p a c t

An important feature in the design of a bridge or other structures along navigablewaterways is to protect the structure against severe damage if a ship loses control. Inmid-December 1996, the Bright Field, a 234-m-long freighter, loaded with 51,0 00 tonsof grain, lost steerage, crashed into the Riverwalk, N ew Orlean s, Louisiana, destroyedcommercial facilities, and injured 116 people Austin Am erican-Statesman, 1996).

Th e use of rock-filled barr iers, as is done for som e bridge piers, is no t possible at N ewOrleans because of the many docks for river boats and tourist ships. Piles under lateralloading, however, have found a useful role as breasting dolphins in the protection ofmarine facilities when controlled docking is possible.

6.6)

where F = h oriz ont al ice force o n pier, kN , p = effective ice strength, kPa, t = thicknessof ice in contact with pier, m, w = width of pier or diameter of circular-shaft pier atthe level of the ice action, m, Cn = coefficient from the following table.

F = C nptw

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 17

A single pile, driven a sufficient distance belo w the m udl ine, an d w ith several m etersof unbraced length above the mudline, can deflect a considerable distance withoutdamage. The relative flexibility of such a pile is an advantage when the pile is used asa breasting dolphin.

Some breasting dolphins have been constructed of timber piles, which are driven in acircular pa ttern wit h a slight batter. The pile heads are lashed tog ether w ith steel cables.Such a design may resist a relatively large lateral force, but deflection is relatively smallbefore damage occurs. A series of such timber-pile dolphins were in place at a dock inthe Gulf of Mexico when a large tanker was docked for the first time. Even though thevelocity of the tanker had been reduced by tugs to a fraction of a meter per second,many of the timber dolphins broke with loud popping sounds as the ship came againstthe dolphins.

A ship being berthed approaches the dock with a given velocity and attitude, suchthat one or more dolphins are contacted. Harbor and other authorities may establishrules that must be followed by the operators of the ship. Factors that are considered arethe nature of the harbor, the weather at docking, the displacement of the vessel, andperhaps the nature of the soils near the mudline. Table 6.3 shows stipulated berthingvelocities for the design of facilities for various ports.

A perusal of Table 6.3 is of interest. The average of the stipu lated berth ing velocitiesis about 0.20 m/sec; the lowest value is 0.08 m/sec and the highest value is 0.30 m/sec.

able 6.3 Approach velocities assumed for design of Dock-and-Harbor facilities.

Harbor

Kitimat

Rotterdam

Thames Haven

Finnart OilTerminal

Amsterdam

Point Jetty,Davenport

Singapore

Sumatra

ParkingTerminal

Isle ofGrain, Kent

Displacementof designvessel(tons)

24,000

45,000

60,000

100,00065,000

60,000

40,000

20,000

50,000

32,000

Berthing velocitynormal(meters

0.15

0.25

0.30

0.180.22

0.15

0.15

0.30

0.08

0.24

to dockper second) Remarks

Bernups I960

Committee Report Dec. 1958

Fendering system designed for 25% ofmaximum kinetic energy of vessel.Committee Report Aug. 1959

Committee Report Sept. 1959

Risselada 1954

One-half the weight of vessel is used incomputing energy to be absorbed.Little 1955

Velocities considered upper limits.Approach velocity of 0.12 m per secondis usual. Ridehalgh 1955

Silverton I960

Committee ReportJuly 1954

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2 18 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Th e data suggest tha t a value is to be set for each dock, b ut the reason able ran ge s how nin the table indicates berthing practice over a wide geographical area.

A single dolphin may be designed by techniques presented herein to sustain anamount of energy. In addition to breasting dolphins, there are many auxiliary devices

for absorbing the energy of a docking vessel. The example computation, presented inthis chapter, is not intended to suggest a comprehensive approach to the design of adocking system.

6.2.7 L o a d s f r o m m i s c e l l a n e o u s s o u r c e s

The sections above present the principal sources of active loads. There can be a numberof other so urces, and the designer can use creativity to ensure th at all of the active lo adsare taken into account. Temperature effects can cause members to shrink and expand

with resulting lateral load s on piles. Traffic on bridges can cause lateral loads on cu rvedroad way s and m ay cause lateral loads by sudden brak ing. Of course, for piles that areinstalled on a batter, a component of the vertical loads will cause lateral deflection ofa pile. Therefore, a careful evaluation of the vertical loading is usually necessary todesign properly piles under lateral loading.

6.3 S I N G L E P IL E S O R G R O U P S O F P I L E S S U B J E C T E D T OA C T I V E L O A D I N G

6.3.1 O v e r h e a d s i g n

6 3 Introduction

Large numbers of overhead signs are constructed for advertising, where allowed, toproviding information to highway drivers. The larger ones have two columns thatsupport a structure for holding the sign. A smaller sign can be supported by a singlecolumn. The foundations for the columns can be supported by a single pile or bymultiple piles. In the latter case, the piles can be analyzed as a group.

6 3 2 Exam ple for solution

Th e examp le is a sign wi th a single column and w ith a foun datio n consisting of a singlesteel-pipe pile. The problems to be solved are: the diameter and bending stiffness ofthe pile; the required penetration of the pile; and the expected deflection of the signduring the storm. The sign is patterned after advertising signs along some highways:dimen sions of sign, 4 m wid e by 3 m hig h; height of m idp oin t of sign, 8.5 m; velocityof wi nd at sign, 32 m/sec; soil at site, overc onso lidated clay; un dra ined shear streng thof clay, 70 kPa; water tab le, 10 m b elow gro und surface; total unit weigh t, 19 kN /m3 ;and number of cycles of loading, 1000. The value of £50 was selected as 0.007 fromTable 3.5.

The forces from the wind against the sign are computed from Eq. 6.1, with Cs

selected as 1.5 and w as 11.88 N/m3 . The force is computed to be 11.15 kN.

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 19

6 3 3 Step-by-step solution

The first step is to arrive at an approximate diameter and stiffness of the pile near theground surface. The assumption is made that the building authority for the region has

specified a factor of safety of 2.2 for the design. Further, the factor of safety is used toupgrade the load because of the nonlinear nature of the response of the soil. Therefore,the ultimate bending moment can now be estimated. Experience has shown that themaximum moment will occur near the ground surface when the loading is dominatedby an applied moment. Thus, the moment arm is selected as 8.5 m plus 1.0 additionalm, leading to an approximate value of MM/i5 of (9.5)(11.15)(2.2) or 233 .0 m-k N. Forthe purposes of these computations, the axial load is assumed to be negligible until afinal check of stresses is made.

The value of the stren gth of the steel in the pipe is selected as 25 0 M N /m2 andEq. 4.29, Muit =/Zp, is used to solve for the value of Zp as of 9.32 x 10- 4 m 3 . Equa

tion 4.30, Zp = 1/6 CIQ — df), can no w be used to find a pipe section th at will satisfythe requirement of the maximum moment. However, the designer will normally querythe local suppliers of steel pipe to learn the sizes that are readily available. Proceedingwith Eq. 4.30, the selection of a steel pipe with an outside diameter of 300 mm leads toa wall thickness of 11.17 mm. A wall thickness of 12 mm is selected without regard tothe availability of such a size. Using the equations noted above, the ultimate bendingmoment for the pipe section is 249.0 m-kN, somewhat higher than the value required.

With the selection of a trial pile, the next step is to make solutions with the professional version of Computer Program LPILE (see Appendix D). The stiffness of theselected steel pipe is 22.55 MN-m2 . Assuming the properties of the soil as given above

and tha t the loadin g will be cyclic, the pile wa s mod eled as a single mem ber. The extension above the gro un d line is 8.5 m, and the pene tratio n is assum ed to be 20 m to ensu rethat long-pile behavior will hold.

The lateral load is applied in increments at the midheight of the sign to a magnitudegreater tha n the factored load of (11.15)(2 .2) or 24. 53 kN . Figure 6.4 presents a plotof the maximum bending moment, deflection at the ground line, and deflection at themidheight of the sign, all as a function of the applied lateral load. Using the valueof Muit of 249.0 m- kN , the value of P t that would cause a plastic hinge was foundto be 29.1 kN , as sho wn in the figure. The factor of safety was then c om put ed t o be29.1/11.15 or 2 .61 .

Referring to Fig. 6.4 and assuming linear behavior between load increments, thelateral deflection at the midheight of the sign at the ultimate load was computed to be45 3 mm , and the deflection at the ground line was 16 mm . At the compu ted w ind loadof 11.15 kN, the sign was computed to deflect 149 mm, the ground-line deflection was3 mm , and the ultimate bending mo ment w as 95.3 m-kN .

The next step in the solution is to compute the necessary penetration of the steelpile. The required procedure is to gradually reduce the penetration and to computethe deflection. When the penetration becomes inadequate, the deflection will increase,which indicates that the tip of the pile is deflecting. If there are two or more pointsof zero deflection along the pile, the tip of the pile will not deflect, and the deflectionwill be almost u nchang ed. The load used in the comp utatio n w as 29.1 kN , whichincluded the factor of safety of 2 .61 . The results of the computations with the reducedpenetration of the pile are shown in Table 6.4 and Fig. 6.5.

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2 20 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 6.4 Co mp uted response of an overhead sign to w ind loading.

able 6.4 Influence of Pene tration o n the D eflection at Sign.

Penetrationm

15108654.543.532.8

Deflectionm

0.47960.47960.47960.47980.48050.48030.49200.57730.98021.429

at sign Number of points ofzero deflection

421772221111

Figure 6.5 show s tha t the rule -of- thu mb of tw o po ints of zero deflection as necessary to establish the critica l pen etrat ion is app aren tly valid, because the deflection atthe sign remains co nsta nt b eyon d a pen etrati on of 4.5 m. How ever, the value selectedor computed for £50 strongly influences the computed values of deflection. For example, computations with the use of a value of £50 of 0.02, not shown here, led to a

suggested penetration of 6 m. Therefore, a prud ent designer will perform param etricstudies using a range of values of £50 and make a selection for the penetration on thebasis of the results. For th is partic ular case, a penetr atio n of 6 m is suggested, eventhough factored loads were used in the computations.

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 21

igure 6.5 Influence of pen etration of pile on de flection of overhead sign at a constant load.

6 3 4 Discussion of results

The results of the computations showed that the maximum bending moment did, infact, occur very close to the ground line, and the approximate method of finding theinitial dimensions of the cross section of the pile was valid. However, as the momentarm for a particular design becomes relatively small, initial computations with thecomputer program may be necessary to obtain a trial size. Because of the nonlinearnature of the response of the soil, no alternative method can be suggested to obtainthe necessary penetration of the pile other that shown in Fig. 6.5, except that a rangeof values of £5 must be investigated, depending on the conditions at the site.

The engineer must ascertain that the design shown above will meet the standardsand specifications of the governmental body that has cognizance over the area wherethe construction is to be done. Additional computations could reduce the cost of thefoundation by either reducing the size of the steel-pipe pile or by finding a structural

member that would satisfy the loadings. However, the savings to be gained may be lessthan the cost of the computations.The criteria for cyclic loading were used in characterizing the response of the soil.

However, the engineer may wish to examine the ground line deflection with respect tothe soil response to gain information on whether or not continued repeated loadingcould cau se a significant re duc tion of soil resistance. At the expected lateral lo ad at th esign of 11 .15 kN , the deflection at the grou nd line is ab ou t 3 m m . A small gap cou ldopen in the stiff clay and , if there is prec ipitat ion , there co uld be some loss of resistancebecause of soil erosion. Therefore, some improvement of the ground surface would beadvisable. For example, the response of the soil can be improved by casting a concrete

slab at the ground line; however, analyses would still be necessary with a proper set ofp-y curves assigned to the concrete.

Data on the gusting of the wind fails to show that the gusts will come at regularintervals; therefore, excitatio n to develop reson ance in the system is no t likely. Further,

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 22 3

would be in order. Such computations will follow the procedures set down herein andare not shown in the interest of brevity.

The tide at the location of the pier on the Gulf of Mexico is almost negligible,and th e assum ption is ma de tha t the dockin g vessel will strike the dolphin at 14 .6 m

abov e the mu dline (3.3 m abov e the wa ter surface). The design of a breasting dolp hinbecomes complicated if the depth of water varies significantly with time. For example,a design was made for the docking of vessels along the Mississippi River, where thewater surface can vary several meters during the year.

The energy that can be developed by the breasting dolphin is related directly to theload-deflection curve, but the deflection is a function of the bending stiffness of the dolphin (pile). Because the ultimate bending stress and maximum allowable deflection arefunctions of the strength of the steel, a high-strength steel is nearly always preferable.For the example shown here, the strength of the steel was selected as 345,000 kPa. Anim por tant further consideration is that th e pile should be tapered by reducing the wallthickness in zones of lower bending stress. The deflection will be increased to increasethe developed energy without causing excessive stress in the steel.

Rough computations are difficult because finding the lateral load Pi 5 factoredupward for safety, involves too many parameters. Therefore, a pile with an outsidediam eter of 1.4 m an d wit h a wall thickness of 60 m m in the region of m ax im umstress was selected with only some computations on an earlier design for guidance.The adequacy of the selection, considering the tapering to yield the max im um am ou ntof deflection at the point of application of the load, is to be determined by repeatedtrials with Computer Program LPILE (See Appendix D).

6 3 2 3 Step-by-step solution

The first step is to make a trial solution with the following pile: length = 50 m; outsidediam eter = 1.4 m; and wall thickness = 60 m m . The p-y curves for the soil specifiedabove were modeled by the soft-clay (plastic) criteria assuming cyclic loading.

The curve of load versus deflection at the pile head (14.6 m abo ve the m udline) isshown in Table 6.5. A preliminary integration of the area under the load-deflectioncurve , no t show n here, was necessary to ensure tha t sufficient load wa s applied to yieldan am ou nt of energy to balance that of the ship at touching (231 kN -m) . N ume rical

integration of the data in Table 6.5 for an energy of 231 kN -m yielded a lateral lo adof 10 97 kN and a deflection at the point of applicatio n of the load of 373 m m.The bending moment for the load of 1097 kN was computed with the computer

program and the maximu m m omen t was found to be 0.1733 kN-m . Using the Ip ofthe 60-mm section, 0.0568062 m4 , the maximum bending stress was computed to be213,600 kPa. If the strength of the steel is 345,000, the factor of safety at first yieldof the steel is 1.62. However, the factor of safety will increase if the pile is tapered,that is, if the wall thickness is reduced in regions where the bending moment is lessthan the maximum. The reduced bending stiffness along the dolphin will result in anincrease in the deflection at the head of the dolphin and, hence, an increase in the

energy that can be offset. The ben ding -mo me nt curve for a lateral load of 109 7 kNwas examined, and wall thicknesses were selected so that in no point along the pilewould the bending moment be less than 1.62. The results of the analysis are shown inTable 6.6.

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2 24 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

Table 6.5 Trial com putations for energy developed by dolphinassuming constant wall thickness o f 60 mm .

Deflection at point of

Lateral load application of loadkN m

0 0100 0.0195200 0.0441300 0.0721400 0.1027500 0.1359600 0.171 I700 0.2083800 0.2475

900 0.28831000 0.3307MOO 0.37491200 0.4203

able 6.6 Schedule of wall thickness and mo me nt o f inertia along the length of thedolphin t o yield an increased value of co mpu ted energy.

De pth W all Thickness Mo me nt of Inertiam mm m 4

0 to 7 25 0.02553007 to I 1.5 42 0.0413451

1.5 to 22 60 0.056806222 to 25 42 0.041345125 to 50 25 0.0255300

The values shown in Table 6.6 and then used to obtain data for finding the lateralload and corresponding deflection that will yield an energy of 231 kN-m.

With a revised schedule of pile stiffness, a new curve of load versus deflection at thepile head can be computed. The results are shown in Table 6.7. Numerical integrationof the data in Table 6.7 for an energy of 23 1 kN -m yielded a lateral load of 102 9 kNand a deflection at the point of application of the load of 391 mm.

The next step is to solve for a bending-moment curve for a lateral load of 1029 kNand to check the computed bending stresses against the allowable bending stressesto determine the factor of safety along the dolphin. The results of the computationsare shown in Fig. 6.6 and Table 6.8. In the table the values of the computed bendingmoment, using Computer Program LPILE, at the changes in wall thickness are tabulated along with the computed values of allowable bending moment at first yield and

allowable bending moment assuming a plastic hinge. The computed factors of safetyare also shown in Table 6.8.

The final step in the analysis, assuming the factors of safety are satisfactory, is touse the computer program and gradually reduce the penetration of the dolphin to find

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 25

Table 6.7 Computations for energy developed by dolphin usingstep-tapered w all thickness.

Lateral loadkN

Deflection at point of

application of loadm

0

100

200

300

400

500

600

700

800

9001000

MOO

0

0 0219

0 0490

0 0799

0 1140

0 1511

0 1911

0 2337

0 2787

0 32640 3764

0 4285

igure 6.6 Plots of bending-m ome nt curve fo r an applied load of l.0 29 kN and allowable bendingmom ent for breasting dolphin.

the critical penetration. Excessive deflection was found to occur at a length of the pileof between 33 and 34 m. T he deflection for the large-diam eter pile wa s found to besensitive to small changes in penetration; therefore, a pile with a total length of 40 mis warranted, giving a penetration below the mudline of 25.4 meters.

6 3 2 4 Discussion of results

Whether or not the dolphin with a diameter of 1.4 m is satisfactory with the factors ofsafety show n in Table 6.8 is a m atter for discussion by the engineer in con sulta tion with

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2 26 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 6.8 Co mp uted bending mom ents and allowable bending momen ts for first yield of steel and fora plastic hinge, along wi th the respective facto rs of safety.

Depthm

7.011.517.022.025.0

* Maximum

Computed

bendingmomentkN-m

7,20011,80016,20010,6005,310

Allowable

bending momentat first yieldkN-m

12,58020,38028,00020,38012,580

Factor ofsafety

1.751.731.731.922.37

Allowable

bending momentor plastic hingekN-m

16,31026,73037,19026,73016,310

Factor ofsafety

2.272.272.30*2.523.07

the ow ner of the project. An oth er steel-pipe section could be selected, the strength of thesteel could be varied according to locally-available material, and consideration givento the use of auxiliary material for absorbing energy, and the computations repeatedto provide more information on which to make a decision. A number of materials ordevices are available for the use as auxiliary energy absorbers. As an example, about80 kN -m of energy can be absorb ed by using five cylinder bu m per s, 0.5 m by 0.25 m,so that they are loaded radially (Q uinn 19 61 , p. 29 0). The amo un t of com puta tiona ltime could be considerable to work out a variety of solutions, considering the n um berof important parameters that are involved.

As noted earlier, the dolphin will behave with more stiffness under the initial lateralloads than indicated by the computations because of the use of the recommendations for the response of the soil under cyclic loading. Therefore, some computationsassuming the response of the soil for static loading might be useful.

A breasting dolphin can be readily tested experimentally by applying increments oflateral load at the head of the dolphin and measuring the deflection of the dolphin foreach of the individual loads. If there is concern about scour or about deposition of soilnear the dolphin, the influence of these factors can be studied by modifying the soil atthe mudline.

While the computations presented herein are critical to the proper design of a breast

ing dolph in, a numb er of other factors mu st be considered. For examp le, con siderationmu st be given to the use of rub bin g block s to distribute the loadin g against the sideof docking vessel to prevent damage of the vessel itself

The designer must consider the response of the vessel and the breasting dolphinsafter the do lphin s have been deflected by the moving vessel. The energy in the deflecteddolphins would be expended by pushing the vessel away from the dock. However, linesare tied to mooring dolphins, and the damping of the energy in the dolphins wouldoccur rapidly.

6.3.3 P i l e f o r a n c h o r i n g a s h i p i n s o f t s o i l

6 3 3 Introduction

One or more ship's anchors are commonly used to allow the ship to remain in relativethe same position in the ocean. Flukes will cause the anchor to sink below the mudline,

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 27

igure 6.7 An ch or pile at an offshore location.

depending on the strength of the soil, and resistance (usually termed holding power) is

derived from the soil as the anchor acts as a kind of a plow. If the soil at the mudlineis weak, the resistance is low, and a number of anchors may be necessary depending,of course, on the lateral forces that will be applied to the ship.

A drilling ship is sometimes used in offshore operations, and the ship must remainin virtually the same horizo ntal po sition for a considerable period to tim e. A substituteanchor, consisting of a driven pile, can find a useful role if the near surface soil at themudline is quite soft. As shown in Fig. 6.7, the pile can be driven with a follower sothat its top is a desirable distance below the mudline. The anchor chain is attachedto a bracket at some point along the pile, and a tensile force will cause the chain todepart at some angle from the pile and assume a curved shape to the mudline. The

components of the tensile force in the chain at the pile will cause a lateral force, anup wa rd force, and possibly a mo men t, depend ing on the details of the bracket. The pileshould have an appropriate factor of safety against being pulled upward and againstthe development of a plastic hinge.

Before the design of the pile, equations must be developed to predict the configuration of the chain. The following section presents the relevant equations and thesolution of an example problem.

6 3 3 2 Configuration of anchor chain

The concepts presented for computing the position of an anchor chain as its lowerend moves from the vertical, along the driven pile, to a position of equilibrium canbe applied to any soil. However, the equations presented here are applicable onlyto cohesive soil (Reese, 1973). The principal assumptions in the development of the

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2 28 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

equations are: (1) after being subjected to a tensile load, the chain lies in a verticalplane; (2) the position of the chain is a curve formed by a succession of arcs of a circle;(3) the soil surrounding the chain reaches a limiting state of stress where the ultimateresistance presents further movement of the chain; (4) the chain become horizontal at

the mudline; (5) the undrained strength of the clay is constant along each of the arcsof a circle; and (6) the tensile force in the chain remains constant.

W ith respect to (6) above, the assum ption is mad e that the horizon tal loading on thedrilling ship is repeated, and increasing to the final value of tension. With the alternateincrease in load and then the relaxation of load, the axial movement of the chain isexpected to be small at the final load. Thus, the tension is not expected to be reducedwith distance below the mudline. This assumption is undoubtedly conservative. Theanalysis can be extended to the case of a decrease in tension with distance from theanchor (Bang and Taylor, 1994). Because the tension will depend on the relative axialmovement of the chain with respect to the soil, the change in the tension is a matterof some question.

An element from the chain is shown in Fig. 6.8, and the equilibrium of the elementis satisfied by solution of the following equations. An angle is chosen to define an arcof a circle, and an expression is derived to compute the value of R, the radius of thecircle.

igure 6.8 Segment of an anchor chain showing geom etry and applied forces.

(6.5)

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2 30 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 6.9 Pro pertie s of soft clay at site of anchor p ile.

Depthm

09.810.220.1

30.550

Undrainedshear strengthkN/m2

2.9012.4817.2317.2322.122.1

Total unitkN/m3

10.9I 1.812.612.6I4.II4.I

weigh t Strain at 50% ofultimate strength

0.010.010.010.01

0.010.01

ultimate capacity of the chain. The computed position of the chain for a tensile loadof 3580 kN is shown in Fig. 6.10.

W ith a value of tension of 358 0 kN and a value of theta of 55 degrees, the com pon ent

of axial load was 205 3 kN , and the com pon ent of lateral load was 29 33 kN . Using thevalues of shear strength in Table 6.9 and th e dimension of the anch or pile, the resistanceto uplift w as com pu ted t o be 26 12 kN , yielding a factor of safety against pu llou t of1.27. Computer Program LPILE (see Appendix D) was used, and the resulting curves

igure 6.9 Sketch show ing layout of problem of design of an anc hor pile.

able 6.10 Computedin the anchor chain.

Tension,TkN

5001500

20002220267031203580

values of theta as a func tion of tension

Thetadegrees

1335

4346505455

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 31

igure 6.10 Co mp uted position of anchor chain for a tensile load of 3.580 k N .

of deflection a nd b endin g mo me nt for a lateral load of 29 33 kN at the mid heigh t ofthe anchor pile are shown in Fig. 6.11. The maximum deflection was computed tobe about 40 mm , which should be acceptable. The maxim um bending mom ent w ascomputed to be 5695 m-kN. Using the dimensions of the cross section of the pile andassuming a strength of steel of 250,000 kN/m2 , the bending moment at first yield ofthe steel was com puted to be 10 ,800 , and the bending m om ent to develop a full plastichinge was computed to be 14,000 m-kN. Thus, the factor of safety in bending at firstyield of the steel was 1.90.

6 3 3 5 Discussion of results

The so lution of the pro blem of a pile used as an an cho r for a floating vessel is p erform edin a straightforward manner. The technology for the analysis of a pile under lateralloading was applied to find the deflection and bending moment in the anchor pile withno particular difficulty.

As noted earlier, the principal uncertainty in the solution resides with the magnitudeof the tensile load at the wall of the pile. The assumption that no reduction in tensionis observed along the anchor line may be excessively conservative. If design of suchanchor piles is a frequent occurrence, a useful procedure would be to install remote-reading load cells in the anchor line at appropriate intervals. Attention should be given

not only to the amount or reduction in the tensile load from point to point along theline but also to the load-deflection characteristics of the unit load transfer.

While the results show the pile that was selected to be satisfactory, the expenditureof additional time by the engineer would be useful. The parameters that should be

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2 32 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 6.11 Curves showing bending mo me nt and deflection of anchor pile.

investigated are: the position of the top of the anchor pile, the diameter and wall

thickness as a function of depth, and the optimum position for attaching the anchorline to the pile. Attention should also be given to appropriate techniques for recoveringthe anchor piles after they are no longer needed.

6.3.4 O f f s h o r e p l a t f o r m

6 3 4 Geom etry of platform and method of construction

The upper deck of the platform is square, with a dimension of 19.81 m. The platform isalso square at the level of the bo ttom panel poi nt, with a dimension of 24.4 9 m. There

are four main piles, one at each corner of the deck, that are driven on an outwardbatter. Twelve conductor pipes are driven to provide the initial casings for drillingthe wells. The conductor piles are driven through close-fitting openings (slots) so asto move laterally with the deflection of the platform. Therefore, the conductors canprovide lateral resistance but do not take any of the axial load from the platformproper. However, each conductor pipe is designed to sustain an axial load of 1,112 kNthat could arise from drilling operations.

The marine contractor endeavors to position an offshore platform so that its principal axes are parallel and perpendicular to the expected direction of the maximumstorm . The analysis that follows is based on the assumption that the ma xim um forces

from the storm hits the platform on one of these axes. Plainly, the maximum waves andwinds could produce the maximum force from some oblique direction, in which casethe platform could twist about its vertical axis. Thus, the two-dimensional equationsfor analyzing a grou p of piles wo uld be inap pro priate . The two-dimen sional equation s

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 23 3

igure 6.12 Elevation view o f one ben t in an offshore p latform , showing loadings and dimensions.

have been extended to the three-dimensional case, but the assumption is made for theexample shown here that three-dimensional analysis is unnecessary.

An elevation view of one of the two bents in the platform is shown in Fig. 6.12. Thesketch defines one of two bents that are assumed to behave in an identical manner.Thus, six of the conductor pipes can resist lateral loading. The loadings on the bent are

show n by horizontal or vertical arrows and are applied at panel points. The h orizonta lor lateral loads are derived from waves, currents, and wind as discussed in a previoussection. In normal p ractice, the horizontal loads derive from the storm th at is assumed,frequently the so-called 100-year storm. The vertical forces are due to loads fromequipment and supplies on the deck of the platform. The resulting loads and momentare shown by the heavy arrows and are used to analyze the foundation.

Construction is accomplished by placing a jacket, or template, on the ocean floorwith a height of 17.06 m, a few m eters greater tha n the wa ter d epth of 14 .33 m. Aderrick barge is used to drive the piles and conductor pipe. The tops of the piles arewelded to the top of the jacket. A deck section with a height of 10.98 m, fabricated to

close tolerances is lifted, and its legs are stabbed into the tops of the piles. The legs ofthe deck section are welded into place and construction proceeds by placing the drillingequipment and supplies on the deck. The sketch in Fig. 6.12 shows that the geometryof the platform is relatively austere above the jacket with the view that the minimal

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2 34 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 6.11 Values of facto rs of safety fo r pile pen etra tion (API, 1993).

Load co nd ition Factors of safety

1. Design environm ental cond itions w ith app ropriate drilling loads 1.52. Op erating environm ental conditions during drilling opera tions 2.03. Design environm ental cond itions w ith app ropriate produc ing loads 1.54. Op erating environm ental conditions during produ cing operations 2.05. Design environm ental cond itions w ith minimum loads (for pullout) 1.5

area of the structure will limit the lateral forces from waves. The maximum unit forcesfrom waves occur at the wave crest. Oceanographers make careful studies of all ofthe factors affecting the maximum wave height to be expected to ensure that the deck

itself will be above the height of the maximum wave.

6 3 4 2 Factors of safety

The three agencies that have been most active in establishing criteria for the designof offshore platforms are: the American Petroleum Institute, Det Norske Veritas, andLloyd's Register. Inform ation from th e API (1993) will be presen ted here as an e xam pleof the guidelines that have been established. Table 6.11 gives the factors of safety thatare recommended for the penetration of piles.

The factors of safety with respect to the response of the piles under lateral loading

are not presented in such a formal manner but the above factors provide guidelines forjudging the adequacy of a design.

6 3 4 3 Interaction of piles with superstructure

The designer of the platform has several options with respect to the manner in whichthe piles will behave under lateral loading. The first option is to decide if the jacketlegs are to be extended and, if so, the amount of the extension. If soft clay exists atthe mudline, the weight of the platform is usually sufficient to cause the jacket legsto penetrate a few meters. Mud mats are frequently placed under the bottom level

of braces to ensure that the entire jacket does not penetrate into the soft clay. In thecurr ent case, as sho wn in Fig. 6.12 , the jacket legs were designed to pe netr ate a d istanceof 1.52 m, the same depth at which scour is expected to occur after some time.

Two options are available with the extended jacket leg: (1) a gasket may be placedat the bottom of the jacket leg and the entire annular space between the outside ofthe pile and the inside of the jacket leg can be filled with grout, as is planned for thecurrent design; or (2) a permanent shim or spacer can be fabricated into the bottomof the jacket leg to ensure close contact between the pile and the jacket; shims are alsoplaced at each panel point in the jacket. With either of the two options noted above,some of the bending moment near at the top of the pile is transferred into the jacket.

Some designers, however, prefer not to have a jacket-leg extension or to grout the pileto the jacket in order to minimize the bending stress at the joint at the lower panelpoint. That joint is vulnerable to cracking due to repeated loading. A design with nogrouting and with shims or spacers at each of the panel points will cause the pile itself

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 35

igure 6.13 Sketches of portion s of supe rstructure of offshore platform to allow comp utation o frotational restraint at pile heads.

to sustain all of the bending stresses. Such a procedure, not employed for the presentdesign, is more costly with respect to the amount of steel, but the structure is lessvulnerable to fatigue cracking.

6 3 4 4 Pile-head conditions

Elementary mechanics can be used to obtain pile-head conditions (boundary conditions) tha t are expected to be close to the values one wo uld obt ain by mo re soph isticatedanalysis. Figure 6.13 shows models of the main piles and the conductor pipe with theassumption that the braces at the panel points provide only support and do no affectthe bending moments. The relationships between the pile-head moment Mt) and theslope at the pile head St) may be found as shown in the figure by assuming that thepile, above the mudline, will behave as a continuou s beam . The point should be made,however, that the slope of the piles at the bottom panel point (mudline usually) cannotbe found directly from the equations in Fig. 6.13 but must be modified to reflect the

rotation of the superstructure under the combined loadings.Because of the nonlinearity of the soils, and sometimes of the material in the piles,

a more accurate solution for the pile-head conditions requires iteration between thefoundations and the superstructure for each set of loadings. For the present problem,

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the equations shown in Fig. 6.13, modified to reflect the rotation of the structure, canbe used for the initial analyses with the further assumption that the pile heads remainin the same plane after loading as before. The resulting loads and moments at each pilehead can be applied to the superstructure to allow an analysis of the superstructure.

The n the pos itions and ro tatio ns at each pile head can be com pare d to the ones from thefoundation analysis. If necessary, modifications are made in the pile-head-boundaryconditions, and the cycle can be repeated. Only a few cycles are usually necessary toconverge to a correct solution for a given set of loadings. Such a procedure may or notbe necessary, depending on the particular case.

The equations in Fig. 6.13 are used as initial boundary conditions for a computercode, and the program automatically modifies the boundary conditions for each pileby employing the com puted slope of the superstructure. T hat proce dure was employedto obtain the results shown below.

6 3 4 5 Soil conditions at the site

A comprehensive investigation of the soils at the site was u nde rtaken . The explo rationequipm ent was mo unted on a barge or a boa t, and the sampling employed the wire-linetech niqu e. An oil-field-supply bo at of the 40 to 50 m class is frequently used at th edrilling platfo rm (Emrich, 1 97 1). The bo at can include all of the facilities th at allow anefficient ma rine o per atio n. A center well is installed of less tha n o ne meter in diam eter,and th e drilling derrick is set abov e the well. A hole with a diameter of abou t 1 80 mm isadvanced with rotary-drilling tools, employing drilling mud as necessary. The interiorof the drill pile, with a diameter of 89 m m , is flush-jointed, allowin g the dep loym ent of

a 55-m m O.D . sampling tube. When the hole has been drilled to the desired depth, thedrill pipe is lifted a few meters and held at the rotary table with slips. The derrick anda thin wire line are then used to lower the sampling tube to the bottom of the drilledhole, with depth being monitored with a wire-line revolution counter. The wire lineis used to lift a sliding, center-hole weight about 1.5 m.; then the weight is dropped asufficient number of times to achieve a penetration of 0.6 m. The number of blows iscounted to drive the sampler, and the sampler with soil is retrieved for some testingat the site. Portions of the samples are protected against loss of moisture and takento the laboratory for further testing. The wire-line procedure allows the completionof the sampling at a particular borehole without retrieving the drill pile, leading to a

practical procedure.The results of the tests at the site of the offshore platform are shown in four tables.

Table 6.12 shows descriptions of the soils encountered and the results of AtterbergLimit tests; Table 6.13 shows the results of unconfined compression tests of bothundistu rbed and rem olded soil; Table 6.14 show s the results of consolid ated-un drainedtriaxial tests of both undisturbed and remolded soil; and Table 6.15 shows the resultsfrom vane tests performed to a limited depth at the site.

Emrich (1971) reported on studies aimed at arriving at the degree of disturbancecaused by driving the sampling tube as opposed to the recommended procedure ofseating the sampler with a steady push (Hvorslev, 1949)*. Three borings were drilled

*The comprehensive report by the late Dr. Hvorslev, while published some time ago, containsmuch information of lasting value.

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A n a l y s i s o f s i n g l e p i l e s a n d g r o u p s o f p i l es 2 37

bl e 6.12

Depth, m

Summary

Soil

of test results,

description

soil descriptionand Atterberg limits (water depth 14.33 m ).

At terberg

PL

im its

LL

14.94 Soft gray sandy clay w ith some decaying wo od fragments 17 3817.37 Soft clay, silty w ith silt layers, trac e of tan clay 23 6820.42 Firm gray clay w ith silt layers 30 8823.47 Firm gray clay w ith silt lenses and silt pock ets 23 7126.37 Firm to stiff gray clay w ith some seams of silt and fine sand 27 7829.57 Gray clay wi th occasional silt lenses below 29 m 26 8138.71 27 8943.59 Stiff gray clay47.24 Stiff gray clay w ith sandy silt layers49.68 Stiff gray clay w ith sandy silt layers

57.30 26 7961.87 Red clay cuttings no ted62.79 Layered stiff clay and sand64.62

bl e 6.13 Summary of test results, unconfined com pression tests (w ater d epth 14.33 m)

Depthm

14.9317.3720.4223.4726.3729.5738.7148.1657.3071.32

Undisti

w

27.342.455.867.547.453.947.728.142.821.1

irbed

YdkN/m3

15.0512.2110.3411.5511.6711.0111.06

11.69

<JukPa

38.136.665.976.494.492.975.8

42.6

ef

0 092

0.1410 075

0 079

0.0710 045

0 063

0.129

Remolded

w

27.440.051.043.744.342.543.8

39.6

1

YdkN/m3

15.1912.39.10.8911.8811.7812.1111.83

12.52

<JukPa

34.830.431.930.338.638.748.7

61.8

ef

0.020.1320.1070.1270.1520.200.149

0.150

bl e 6.14 Summary of test results, cons olidated-und rained triaxial tests, offshore site w ater depth14.33 m .

Depth

14.7817.2220.2723.32

26.2129.4138.56

w,

31.846.554.944.6

39.946.549.2

Undisturbed

Wc

30.546.054.243.5

38.545.048.2

YdkN/m3

14.0611.4010.2311.55

12.0811.3110.92

<JckPa

80.473.1

112.3117.7

118.5169.4152.1

s f

0 080

0 093

0 047

0 060

0 0400 047

0 040

A V

4.01.42.63.5

4.92.62.7

Confining

PressurekPa

3.821.141.262.2

82.3103.4123.5

w,

31.146.355.442.0

40.347.147.1

Wc

28.545.152.138.5

33.041.843.9

Remolded

YdkN/m3

14.2211.6610.3412.14

12.5011.3911.31

ΦkPa

57.444.047.362.3

67.382.394.1

s f

0 200

0.1500.1190 086

0 0920.1510 080

A V

3.83.04.64.5

8.69.05.3

57.45 40.2 35.1 11.78 163.7 0.150 6.1 289 .2 40.7 34.8 12.35 161.5 0.120 5.3

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2 38 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

able 6.15 Summary of test results, strength for vane tests wa ter depth 14.33 m.

Depthm

14.917.420.423.5

Maximum strengthkPa

42.654.965.5

109.6

Minimum strengthkPa

15.222.824.333.5

to a depth of 91 m at an onshore site near the Mississippi River. The soils were typicalof those that occur over a considerable area of the Gulf of Mexico. Samples weretaken with the wire-line sampler described above, with a wire-line sampler with adiameter of 76 mm , with an open push -sampler w ith a diameter of 76 mm , and withfixed-piston sampler with a diameter of 76 mm . Unconfined compression tests wereperformed on specimens from each of the methods of sampling. In addition, field-vanetests were performed and miniature-vane tests were performed at the ends of the tubesamples. Emrich reported th at the strengths from the various unconfined-comp ressiontests and the vane tests were plotted on the same graph with the results from the fixed-piston sampler assumed to yield correct results. Compared to results from the fixed-piston sampler, the following results were obtained: 57-mm wire-line sampler, 64%:76-m m wire-line sampler, 7 1 % ; and the open-push sampler, 9 5 % . Scattered resultswere obtained from the vane tests, but the results were generally higher that the resultsfrom the fixed-piston tests.

An exhaustive study of the data that are presented on the strength of the clay atthe sites is unwarranted for several reasons. With respect to Table 6.14, the investigators presumably expected that subjecting specimens to a confining pressure equal tothe overburden pressure would reflect a gain in strength that offsets the strength lossin sampling. However, using data from Leonards (1971), the value of {c/p)n for theun dra ined shear strength of norm ally con solida ted clay c a s a function of the effectiveoverburden pressure p is 0.299. Thus, the clay at the site is substantially under consolidated, and consolidated-undrained tests, shown in Table 6.14, would be expected

to reflect shear strength considerably larger than the actual strength. Such an interpreta tion, however, is not confirmed by the mod erate loss of water con tent d uring theconsolidation phase of the tests, as shown in Table 6.14. Nevertheless, the logic ofusing the results from the consolidated-undrained tests is unsubstantiated.

Strong dependence could be placed on the results from the vane tests, as shown inTable 6.15, which show strengths that are substantially higher than those from theunconfined-compression tests. However, nearly all of the experimental data on theresponse of piles to axial and lateral loading have depended strongly on results fromunconfined-compression tests rather than results from vane tests.

Even though the work of Emrich, reviewed above, suggests that the strength of

the unconfined-compression tests of wire-line samples from a sampling tube with adiameter of 57mm (presumable the size used for the results shown in Table 6.13)can be multiplied by a factor of 1.5 to achieve a strength that would be obtainedfrom a fixed-piston sampler, two points argue against such a procedure. Firstly, unless

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 39

able 6.16 Values of shear strength accepted for analysis (depth measured from mudline).

De pth Und rained shear strength Angle of internal frictio nm kPa deg

0 01.52 01.52 22.113.10 48.322.86 40.050.29 40.050.29 3662.00 36

much m ore data can be obtain ed on the comparativ e results of unconfined compressivestrength from various sampling techniques, the application of Emrich's results to allsites is unwise. Secondly, the driving of a pile into soft, sat urat ed clay causes rem oldin gand excess pore-water pressures at the wall of the pile. Some attempts have beenmade to quantify the initial loss of strength of the clay and the subsequent regain ofstrength (Seed & Reese 1957; Reese, 1990), but no method has been accepted by thegeotechnical engineering community. Therefore, to reduce the possibility that a failurecould occur if a pile is loaded axially too soon after installation, a lower value ratherthan a substantially higher value of undrained shear strength is selected.

The result from the above discussion is that the undrained shear strength fromthe unconfined compression tests, shown in Table 6.13, is accepted. A review of theresults in the table indicates two anomalies: (1) the remolded strength at a depth of57.3 m is greater than the undisturb ed strength; and (2) the data on com pression testsis very limited from a depth of 38 .7 m to 64.6 m where the sand w as encoun tered.No explanation was given for the scarcity of results in the lower part of the boring.Table 6.16 shows the values of undrained shear strength that are employed in theanalyses that follow. The clay will probab ly be considerably stronger tha n th at show nin the table after the piles have been in place for several weeks.

In view of the factors note d ab ove a bo ut the testing of the soil, the value of £50* used

in the analyses was selected as 0.02 by referring to Table 3.3. The argument couldbe made that a somewhat higher value could be used for the deeper soils; however,(1) the properties of the soils near the mudline will dominate the results, and (2) thehigher value of £50* will be conservative with respect to deflection. The computation ofthe maximum bending moment, of principal interest in the present analyses, is hardlyaffected by the value employed for £50.

63.4.6 Preliminary dimensions of piles and axial capacities

The lengths of the various segments of the main piles (Piles 1 and 2) and the conductor

pipe (Pile 3) were selected on the basis of some trial computations and previous experience (see Fig. 6.14). Also considered was the necessary mass of the pile for driving.The conductor piles, with lengths of 30.48 m, were in the relatively soft clay for theirentire length and offered no particular problem in installation. However, the main

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2 40 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

piles, with lengths of 54.86 m were designed to penetrate 4.57 m into the sand. If thepiles could not be driven to the designated depth in the sand, the thick-walled sections necessary for sustaining bending moment would be out of place. Therefore, the

contractor made some preliminary analyses to obtain reasonable assurance the mainpiles could be driven to the penetration of 54.86 m. Do wn w ard capacity can usually beobtained by considering end bearing, but the resistance to the expected uplift can be aproblem. If a pile fails to penetrate the proper distance into the sand, the use of jettingto loosen the soil is unacceptable. Many offshore designs have resorted to drilling andgrouting in order to place the piles to the required depth.

Using the data presented ab ove, load-settlement curves were com puted for the m ainpiles with the following results: dow nw ard capacity, 15,970 kN , settlement, 51 m m ;upw ard capacity, 6,430 kN , uplift, 20 mm . In actuality, the do wn wa rd capacity isexpected to increase with increasing depth, but the settlement of 51 mm was judged to

be a limiting value for the present com puta tion s. The cond uctor pipe were comp uted tohave a capacity of 2,360 kN, fully adequate to sustain the load from drilling operationsof 1,112 k N , a load that w ould be on only one condu ctor at a time. Settlement was n ota problem because the conductor pipe would not be fastened rigidly into the jacket.

igure 6.14 Trial dimensions and bending stiffness of main piles and conductor pipe for offshoreplatform.

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A n a ly s i s o f s i n g l e p i l e s a n d g r ou ps o f p i l e s 2 41

able 6.17

Pile 1Pile 2Pile 3

Computed

mm

6.224.3

movements

Vtmm

45.843.345.1

and loads at each pile

arad

0.255 x I0 2

0.246 x I0 2

0.140 x I0 2

head (first loadi

k N

564524322

ng)·

m-kN

- 2 , 5 3 0- 2 , 4 2 0- 1 , 2 1 0

k N

3,2409,370

6 3 4 7 Results of analyses

The loads and other data outlined above were entered into a computer code, using thetechnology described in Chapter 5 . The assump tion w as mad e that the cond uctor pipeswere far enough apart that no reduction was necessary for pile-soil-pile interaction.The following results, from the 100-year storm without factoring, for the movements of the origin on the coordinate system (point where loads were applied) were:dow nw ard movement, 1 5.3 1m m ; horizontal movement, 45.12 mm ; and rotation,4 .377 x 10- 4 radia ns. The loads and m ovemen ts at each of the pile heads are shown inTable 6.17.

A brief examination of the results in Table 6.17 reveals some interesting facts. Thesix conductor pipes with a smaller diameter than the main piles sustain the majorportion of the lateral load, illustrating the importance of construction details. Whilethe diameter of the conductor piles is smaller, the rotational restraint is significantbecause of the short distance between ships above the top of the pile. Further, if thelateral loads are summed and compared with the applied lateral load, the influence ofthe batter of the main piles can be seen. That is, the lateral component of the axialload for Pile 2 is important in resisting the lateral load on the platform.

Whether Pile 1 or Pile 2 has the greatest stress is not immediately apparent from thedata in Table 6.17. The axial load on Pile 2 of 9,370 kN, almost three times that ofPile 1, but still substantially less than the ultimate load of 15,970 kN. The results forPile 2 will be analyzed to compare computed stresses with allowable values.

Figure 6.15 shows a plot of computed values of bending moment and combinedstress, /r, as a function of length of Pile 2, along with the stress that can be sustained

at first yield of the steel. The axial load is assumed to remain unchanged in computingthe combined stress. The influence on the results of the change in wall thickness of thepile is evident. The m ost critical stress is below 2 5.5 m w here the stress is altog etherfrom the axial load. The assumption that the axial load is unchanged with depth isobviously incorrect; therefore, axial stress will not control. Also, the same argumenthold s for the large stress at a po int just below 15 m. A further exa mi natio n of thevalues of combined stress in Fig. 6.15 suggests that the wall thickness on the pile issized rather well over its length, and that the most critical stress will occur at or nearthe top of the pile.

Additional runs were made with a computer code by increasing the loads at the

origin of the global coordinate system by an equal percentage. The results for the casewhere the global loads were factored upward by 1.50 are presented. If one can saythat the factor of safety should reside in the factoring of the loadings, the use of 1.50is consistent with the API factors for pile penetration. The movements of the origin on

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2 42 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 6.15 Plots of comp uted values of bending mom ent and combined stress along the length ofPile 2 of offshore platfor m.

able 6.18 Co mp ute d m ovem ents and loads at each pile head (global loads facto red by 1.5).

Pile 1Pile 2Pile 3

xtmm

9.2104.0-

Ytmm

105.095.6

104.0

arad

0.643 x0 . 6 l 7 x0.473 x

io -2

io -2

IO 2

Ptk N

835718496

Mt

m-kN

- 3 , 8 8 0- 3 , 5 7 0- 1 , 9 5 0

PxkN

8,35014,100-

the coordinate system were: downw ard movement, 57.0 mm ; horizontal movem ent,104.0 mm ; and rotation, 0.3194 x 1 0- 2 radians. The loads and movements at each ofthe pile heads are shown in Table 6.18.

With the factoring of the global loads by 1.50, the nonlinear response of the system,due to the nonlinearity of the soil, is apparent. The origin of the coordinates movedvertically by a factor of 3.72 and moved horizontally by a factor of 2.3. Referringto Table 6.18, a principal factor in the nonlinear movement is that Pile 2 is in thenonlinear rang e of axial response. The pile moved do wn wa rd 104 mm w ith an axialload of 14 ,100 kN . The axial load is less tha n the ultim ate axial load of 15 ,970 kN ;

however, because of the lack of stiffness in the load-settlement curve, little additionalload can be applied to the group before collapse is indicated. The ma xim um comb inedstress that w as comp uted for the factored loads was 219 ,00 0 m-kN , less that the valueof 250,000 m-kN that can be sustained at first yield.

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Chapter 7

C a s e s t u d i e s

7 .1 INTRODUCTION

The procedures for the analysis of single piles under lateral loading, presented in thepreceding chapters, are employed herein to enable comparisons to be made betweenresults from experiments and from computations. Not only will the comparisons provide information on the accuracy of the analytical methods, but the techniques ofanalysis will also be demonstrated.

M an y tests have been repo rted in literature on th e results of field-testing of full-scalepiles; however, in some instances critical infor matio n is missing. The following data areeither necessary or desirable. Where some data are missing, estimates can sometimes

be made to achieve a comparison. Examples of such estimates are given in the casesthat follow.

Pile- Length and penetratio n into the soil- Detailed descriptio n of each cross section as a function of pen etrati on

Occurrence and location of steel, concrete, and any other materialStrength f'c and modulus of elasticity of the concreteYield strength fy and modulus of elasticity of the steelSimilar values for any other materials in the cross sections

Soil- Classification of soils with Atterber g Limits and with any othe r necessary soil tests- Ident ification of rock and classification by RQD and other data- Position of the wa ter table- Un drain ed shear strength of clays and stiffness from sso^somzy be estim ated if

necessary)- Friction angle for cohesionless soils (or data from pen etra tion tests th at can be

correlated with the value of 0), unit weight, and information on the structure ofthe grains (e.g., resistance of grains to crushing)

- Com pressive strength of rock and data on secondary structure

Loading and Pile Head Restraint- Ar rang em ent for applyin g load and po int of app lication of lateral load wit h respect

to the ground surface- N atu re of load ing, wh ethe r static, cyclic, or sustained

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2 82 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

- Sufficient ma gni tud e of loadin g to achieve a non linear response- Free-head are partially restrain edInstrumentation- M etho ds and details of measuring loading, pile-head deflection, and rotation- Na tur e and details of internal instrum entation in pileResults- Presented in tabula r form or such tha t data can be tabula ted- Any special obs ervatio ns

Th e above inform ation will allow the ben ding stiffness {EpIp) of the pile to be com put edas a function of applied moment and will also allow the computation of the bendingmoment at which the pile will develop a plastic hinge Mu\t) or will just reach plasticbehavior at the extreme fibers My) in the case of metal piles. The p-y curves can bedeveloped according to the methods of prediction that were presented earlier.

The cases of more importance are those where the piles were instrumented so thatthe bending moment can be found along the length of the piles and where both staticand cyclic loading were employed. Cyclic loading has proven to be of considerableimportance when the soil is cohesive and water is above the ground surface. A limitednumber of such cases are available. Some of these provided the data on which therecommendations for p-y curves are based, but it is important to determine how wellthe results from the use of the p-y criteria agree with the results from the experiments.

The cases are separated into categories: clays, sands, layered soils, c-φ soils, andweak rock. Unfortunately, only a few cases are available from the last two categories.

Where data are available on bending moments along the pile, curves are presentedcomparing the values of the maximum bending moments from experiment and fromcomputat ion.

The computations were done with the computer program described in inAppendix D. The student version of Computer Program LPILE can be used to makeco m pu tatio ns for several of the cases; the professional version of the pro gra m is neededfor the more complex cases.

7.2 P IL E S I N S TA L L E D I N T O C O H E S I V E S O I L S W I T HN O F R EE W A T E R

7 . 2 . 1 B a g n o l e t

Kerisel (1965) reported the results of three, short-term, static lateral-load tests of aclosed-ended bu lkh ead cai sson . The cross section of the pile is sho wn in Fig. 7 .1 .Two sheet-pile sections were welded together to form the pile. The three tests wereperformed on the same pile which was recovered and reinstalled for all tests followingthe first one. The bending stiffness E p I p was given as 25 ,500 kN -m2 an d, if £ is selectedas 200,000 MPa, the value of I is 0.0001275 m4 .

Insufficient information is available from which to compute the ultimate bendingmoment. However, if the assumption is made that the steel has a yield strength of248 MPa, the bending moment at which the extreme fibers of the pile will just reachyielding My) is at 204 kN-m.

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2 84 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.2 Comparison of experimental and computed values of maximum bending moment anddeflection, Case I, Bagnolet.

igure 7.3 Comparison of experimental and computed values of maximum bending moment andde flection , Case 2, Bagnolet.

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Case s tu d ie s 285

igure 7.4 Comparison of experimental and computed values of maximum bending moment anddeflection, Case 3, Bagnolet.

gro un d surface will result in larger values of pile-head deflection for the same value ofmaximum bending moment.

A review of the figures, from the results at Bagnolet, gives some insights into th e comparative behavior of piles under lateral loading. Even thou gh the groun d-line mo me ntis smaller for Case 1, the ground-line deflection is larger at the same final value ofshear than for the other two cases. The deflected shape of the pile (not shown here)reveals that the small penetration allows the bottom of the pile to deflect and is accompanied by only one point of zero deflection. However, as may be seen, there was not

a corresponding increase in the maximum moment for the pile in Case 1, as might beexpected.In comparing the results for the three cases, as the ground-line moment increases,

due to the increased mom ent arm , the ma xim um bending mo men t goes up for the samefinal value of ground-line shear. These results suggest that, if a lateral load is appliedto a pile at a great distance above the ground-line, the behavior of the pile will dependto a lesser degree on the soil characteristics.

7 . 2 . 2 H o u s t o n

Reese & Welch (1975) reported the results from a test of a bored pile with a diameterof 0.762 m and a penetration of 12.8 m. An instrumented steel pipe, with a diameter of0.26 0 m an d a wall thickness of 6.35 m m , formed the core of the pile. A rebar cage, witha diam eter of 0.610 m, consisted of 20 bars with diam eters of 44 .5 m m wa s placed in

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2 86 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

able 12

Depthm

00.41.046.112.8

Reported properties

Water

1818222015

content

of soil at Houston.

Undrained shearkPa

76.076.0105.0105.0163

strength£ 5 0 *

0.0050.0050.0050.0050.005

Total unit weightkN/m3

19.419.418.819.119.9

* Values from review of laboratory stress-strain curves

the bored pile. The yield strength of the steel wa s 276 M Pa a nd the comp ressive stren gthof the concrete was 24.8 MPa. The value of the bending stiffness E pI p was measuredduring the testing by reading the output from strain gauges on opposite sides on theinstrumented pipe and was computed to be 4.0 x 105 k N - m2 . The bending momentat which a plastic hinge would occur was computed to be 2,030 kN-m. The test wasperformed in Houston, Texas, under the sponsorship of the Texas Department ofTransportation and the Federal Highway Administration.

The soil was overconsolidated clay, called Beaumont clay locally, and had a well-developed sec ond ary stru ctur e. The wa ter table wa s at a depth of 5.5 m at the timeof the field tests. Tube sam ples with a diam eter of 10 0 m m we re taken , observingthe necessary precautions to reduce sampling disturbance. The properties of the clayare show n in Table 7.2. The und rained shear strength was measured by uncon solidated-undrained triaxial compression tests with confining pressure equal to the overburdenpressure. The values of £50 shown in the table were obtained from a review of thelaboratory tests and agree with values shown in Table 3.5 except that the top 0.4 m isshow n to be slightly stiffer tha t presented in Chap ter 3 . Some samples were subjected torepeated loading and the effect on the stress-deformation relationships was observed.

The lateral loads were applied at 0.076 m above the gro und surface, and loads w ereboth static and cyclic. The same pile was used without redriving to obtain results forboth types of loading. The successive loads were widely separated in magnitude so thatthe cycling at the previous load was assumed to have no effect on the first cycle at the

next lo ad. At each increm ent of lateral loa d, reading s were take n at on e cycle, 5 cycles,10 cycles, and 20 cycles for the larger lo ads. Th e results from the cycling were analyzed ,and a method of predicting the effect of cyclic loading was developed, as shown inChapter 3, based on the stress level and the number of cycles. For computation of theresponse of the pile to static lateral loading the p-y curves were developed based onthe criteria for stiff clay with no free water.

Comparisons of the pile-head deflection and maximum bending moment for staticloading are shown in Fig. 7.5. The comparison for deflection is excellent and theanalysis is conservative in the computation of bending moment. The conservatismin the bending-moment computation is also reflected in Fig. 7.6, which shows the

bend ing mo me nts as a function of depth for the lateral load of 44 5 kN for staticloading. The depth to the point of maximum bending moment agrees well betweenexperiment and computation; however, the depth to the point of zero bending momentis underpredicted by one or two meters.

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Case s tu d ie s 287

igure 7.5 Com parison of experime ntal and com puted values of maximum bending mo me nt and pile-head deflection, static loading, Ho usto n.

igure 7.6 Comparison of curves of bending moment versus depth for P t of 445 kN, static loading,Houston.

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2 88 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

7.2.3 B r e n t C r o s s

Price & Wardle (1981) reported the results of a test of a steel pipe in London Clay.The diam eter of the pile was 0 .406 m, and its penetration was 16.5 m. In add itionto the analysis of the test by the original authors, Gabr et al. (1994) did a furtherstudy. The moment of inertia, Jp, of the pile was reported as 2.448 x 10- 4 m 4 ; thebending stiffness, E pIp used in the analyses that follow was 5.14 x 104 kN -m2 . Thebending moment at which the extreme fibers would reach yield was computed to be301 kN-m , and the ultimate bending m om ent, at which a plastic hinge wo uld develop,was computed to be 392 kN-m.

The data on the properties of the London Clay at the site was obtained from the

testing of specimens taken w ith thin-walled tubes w ith a diameter of 98 mm . The w atertable was presumably some distance below the ground surface. The values, shown inTable 7.3, of undrain ed shear strength were scaled from a plot presented by the aut ho rs.The strength of the clay near the ground surface seems low for over-consolidated clay.

igure 7.7 Com parison of experime ntal and com puted values of maximum bending mo me nt and pile-

head deflection, cyclic loading, Hous ton.

Comparisons of the pile-head deflection and maximum bending moment for cyclicloading are show n in Fig. 7.7. The results are for 20 cycles. The com paris on for deflection is excellent, except for the larger loads, and the analytical method is conservativein the computation of bending moment.

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Case s tu d ie s 289

able 7.3 Reported properties of soil at Brent Cross.

Depthm £ 5 0

Undrained shear strengthkPa

04 66 219

0 0070 0070 0070 005

44.185.280.6

133.3

* From Table 3.5

igure 7.8 Com parison of experimen tal and com puted values of pile-head deflection, Brent Cross.

Data on the stiffness of the soil were not reported so computations were done withvalues of sso that were obtained from Table 3.5. The suggested values of sso are in theranges of values obtained experimentally by Jardine, et al. (1986) for low plasticity

clays. The p-y curves for the analyses were obtained by using the criteria for stiff claywith no free water.The lateral load was applied at 1.0 m above the grou ndline, and both static and

cyclic loads were applied. The static loads were of a larger magnitude than thecyclic loads and were applied in a re-loading state after cycling had been done. Theassumption is made that the cycling with the smaller loads did not affect the subsequent static results. Only the results from the static loading are reported below.The results from the experiment and from computations, with the methods presented herein, are shown in Fig. 7.8. The agreement between the experiment andanalysis is reasonable with the analytical method yielding results that are somewhat

conservative.The value of max imu m bending mo men t for the largest lateral load of 100 kN was

com puted to be 198 kN- m, w hich is significantly below the comp uted value of bendingmoment at first yield.

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able 7.4 Rep orted prop erties o f soil at Japan.

De pth Undraine d shear strength Submerged unit weigh tm kPa ε 50 kN /m3

0 27.3 0.02 4.95.18 43.1 0.02 4.9

* Obtained from able 3.3

7.2.4 Japan

The Committee on Piles Subjected to Earthquake (1965) reported the results fromtesting a steel-pipe pile with a closed end that was jacked into the soil. The pile was305 m m in outside diameter with a wall thickness of 3.18 mm , and its penetratio n was5.18 meters. The mom ent of inertia, Jp, was 3.43 x 1 0- 5 m 4 , and the ben ding stiffness,Eplp, was 6,868 kN -m2 . The bending moment My at which yielding of the extremefibers would occur was computed to be 55.9 kN -m, and the ultimate bending mom entM uit was computed to be 71.8 kN-m.

The soil at the site was a soft, medium to highly plastic, silty clay with a highsensitivity. The undrained shear strength and stiffness of the soil was obtained fromun dra ined triaxi al shear tests. The strains at failure were generally less tha t 5 % , andfailure was by brittle fracture. The properties of the soil are shown in Table 7.4. Thevalues of the undrained shear strength are typical of those for normally consolidatedclay. Therefore, in the absence of stress-strain curves from the laboratory, values of £5

show n in the table were taken from Table 5.3. The p-y curves for static loading for theanalyses were obtained by using the criteria for soft clay with free water. However, thep-y curves for static loading using the criteria for stiff clay with no free water yieldedalmost identical results.

The loading was applied at 0.201 m above the groun dline; the maxim um lateral loadwas a moderate value of 14.24 kN which produced a pile-head deflection of 4.83 mmand a maxim um bending mo me nt of 17.34 m- kN . Thu s, with respect to a failure dueto yielding of the extreme fibers, the factor of safety was 3.2 and the factor of safety

against a failure in plastic yielding was 4.1.The loading was static. A plot of the comparison between experimental and computed deflections is shown in Fig. 7.9. The computations were carried to a maximumlateral load of 45 kN, a load that caused a maximum bending moment approximatelyequal to the first yield of the extreme fibers of the steel. Good agreement was foundbetween experimental and computed values of pile-head deflection for the range ofloads that were applied.

Bending moments were measured at the site but information is unavailable on thetechniques that were used. A plot of the comparison between experimental and computed maximum bending moment is also shown in Fig. 7.9. Again, agreement was

good for the range of loads that were applied.Even though the length to diameter ratio was relatively small at 17, examination

of the results for deflection and bending moment along the pile showed that the pilewould have failed in bending rather than by excessive deflection.

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Case s tu d ie s 291

igure 7.9 Com parison o f experim ental and com puted values of maximum bending mo me nt and pile-

head de flection, Japan.

7.3 P I L E S I N S T A L L E D I N T O C O H E S I V E S O I L S W I T H F R E EW A T E R A B O V E G R O U N D S U R F A C E

7.3.1 L a k e A u s t i n

M atlo ck (1970) presente d results from latera l-load tests employing a steel-pipe pile tha twa s 31 9 m m in diameter, wi th a wall thickness of 12.7 m m , and a length of 12.8 m.

The bending stiffness was 28730 kN-m2

. The bending moment at which the extremefibers wo uld first yield was compu ted to be 23 1 kN -m, and the bending m om ent forthe formation of a fully plastic hinge was computed to be 304 kN-m.

The pile was driven into clays near Lake Austin, Texas, that were slightly overcon-solidated by desiccation, slightly fissured, and classified as CH accord ing to the UnifiedSystem. The undrained shear strength was measured with a field vane; was found tobe almost constant with depth, and (cu)vane averaged 38.3 kPa.

A comprehensive investigation of the soil was undertaken and the computationsshown herein are based on tests with the field vane. The vane strengths were modifiedto obtain the undrained shear strength of the clay. The values of the soil properties

employed in the following computations are shown in Table 7.5. The value of ss wasfound from triaxial tests and averaged 0.012. In view of the almost constant valueof cu with depth, a constant value of £50 appears reasonable. The submerged unitweight was d etermined at several points below the mudline an d the average value was

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2 94 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.12 Comparison of experimental and computed values of maximum bending moment andpile-head d eflec tion, cyclic loading, Lake Aus tin.

In a historical com me nt of interest, Terzaghi comm ented in 195 5 that the use of straingauges to get the response of the soil to the lateral loading of the pile was not possible;therefore, emphasis was placed on finding an alternate method, but the investigatorsdecided to the strain gauges. However, Terzaghi visited the test site during a trip tothe University of Texas in 1956 and appeared to be impressed with the progress of theresearch.

7 .3 .2 S ab in e

The pile tested at Lake Austin was removed and installed at Sabine where the soil was

a soft clay. As before, the pile was tested both under static and cyclic loading. Also,testing was conducted with the pile head free to rotate and restrained against r otat ion .Meyer (1979) analyzed the results of testing the soil at Sabine and reported that the

clay was a slightly overconsolidated marine deposit, had an undrained shear strengthof 14.4 kN/m2 , and a submerged unit weight of 5.5 kN/m3 . Computations were madewith values of £50 of 0.02, as suggested in Table 3.3. The p-y curves for the analyseswere obtained by using the criteria for soft clay with free water.

The lateral loads were applied at 0.3 05 m above the gro und line. Com parison s ofthe pile-head deflection and maximum bending moment for static loading are shownin Fig. 7.13. The results from the analytical method for deflection are conservative,

and the agreement is satisfactory for maximum bending moment.Comparisons of the pile-head deflection and maximum bending moment for cyclic

loading are shown in Fig. 7.14. The comparisons show excellent agreement for bothdeflection and maximum bending moment.

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Case s tu d ie s 295

igure 7.13 Comparison of experimental and computed values of maximum bending moment andpile-head d eflec tion, static loading, Sabine.

igure 7.14 Comparison of experimental and computed values of maximum bending moment andpile-head d eflec tion, cyclic loading, Sabine.

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2 96 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

able 7.6

Pile

1

2

able 7.7

Depthm

00.91.524.116.559.1420.00

Mechanical properties

Sectionm

Top 7.01Bottom 8.23Top 7.01Bottom 8.23

of piles at Manor.

/m4

0.002335—0.002335—

Reported properties of soil at Manor.

Water content

_3727222219-

Undrained shearkPa

2570

163333333

MOOMOO

El

kN-m2

493,700168,400480,400174,600

strength£ 5 0 *

0.0070.0070.0050.0040.0040.0040.004

MykN-m

1,757—1,757—

Mu

kN-m

2,322—2,322—

Total unit weightkN/m3

_18.119.420.320.320.8-

* From Table 3.5 and in general agreement w ith experiment

7.3.3 M a n o rReese et al. (1975) describe lateral-load tests employing two steel-pipe piles that were15.2 m long, w ith a diameter of the upper section of 0.6 41 m and of the lower of0.610 m. The piles were driven into stiff clay at a site near Manor, Texas. The pileswere calibrated prior to installation and the mechanical properties of each of thepiles are shown in Table 7.6. The bending moment, My 5 when yield stress develops at the extreme fibers and the ultimate bending moment, Mui t, are shown onlyfor the top sections, where the ultimate bending moment occurs during loading. Theexperimental p-y curves in Figs. 1.5 and 1.6 were derived from data from the tests

at Manor.The clay at the site was strongly overconsolidated, and there was a well developed secondary structure. The undrained shear strength of the clay was measured byunconsolidated-undrained triaxial tests with confining pressure equal to the overburden pressure. The properties of the clay are shown in Table 7.7. The site was excavatedto a depth of about 1 m, and water was kept above the surface of the site for severalweeks prior to obtaining data on soil properties.

The values of £5 were found from experiment, but scatter was great, likely becauseof the secondary structure of the clay. Values of £50 were found from Table 3.5; theresults in Table 7.7 are generally in agreement with experimental values.

Both of the piles were instrumented with electrical-resistance strain gauges for measurement of bending moment. The gauge-readings were taken with an electronicdata-acquisition system, and a full set of readings could be taken in about one minute.The point of application of the load for both piles was 0.305 m above the groundline.

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2 98 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.16 Comparison of experimental and computed values of maximum bending moment andde flection , cyclic loading, Manor.

the analytical method being slightly conservative. The maximum bending moment of1,271 kN-m that was measured was substantially less than the yield moment and theultimate moment.

The comparisons of groundline deflection and maximum bending moment for Pile2 , for cyclic loading, are shown in Fig. 7.16. Excellent agreement was found betweenresults from experiment and from analysis, with computations slightly conservative forboth ground-line deflection and maximum bending moment. The maximum bendingmoment of 1,385 kN-m that was measured was less than the yield moment, and theultimate moment.

In total, the experimental and computed values for the series of tests at Manor agreewell. In some other cases, the analytical method appears to yield a conservative resultfor piles in overconsolidated clay that is under water. The erosion due to cyclic loadingis a critical matter and other results are needed because the soil at Manor may havehad some characteristic that made it more erodible than other overconsolidated clays.However, a series of laboratory studies, not reported here, failed to reveal the natureof any such characteristic.

7.4 P IL E S I N S TA L L E D I N C O H E S I O N L E S S S O I L S

7.4.1 M u s t a n g I s l a n d

Cox et al. (1974) describe lateral-load tests employing two steel-pipe piles that were21 m lon g. The piles were driven into s and a t a site on an island near Co rpu s

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Case s tu d ie s 299

Ch risti, Te xas. Th e piles were identical in design and had diam eters of 0.610 m.They were calibrated prior to installation and had the following mechanical properties: Ip = 8.0845 x l 0 - 4 m 4 ; E pI p = 163,000 kN-m2; M3, = 6 4 0 k N - m ; a n d Mu l t =828 kN -m . Both piles were instru me nted internally with electrical-resistance straingauges for the measurement of bending moment. The piles were loaded separately;Pile 1 was subjected to static loading, and Pile 2 to cyclic loading. The load on bothpiles was applied at 0.305 m above the mudline.

The soil at the site was a uniformly graded, fine sand with a friction angle of 39degrees. The submerged unit weight was 10.4 kN /m3, and the relative density averaged

abo ut 0 .9. The water surface was m aintained at 150 mm or so above the mu dlinethroughout the test program. The piles were driven open-ended and the modificationof the sand was perhaps less than what would have occurred if full-displacement pileshad been installed.

The data were analyzed by use of the criteria for cohesionless soils (Reese et al.1974). The comparisons of groundline deflection and maximum bending moment forPile 1, for static loading, are shown in Fig. 7.17. The agreement in both instances isexcellent.

The engineer is not only interested in the ground-line deflection and maximum bending moment but also on the accuracy of the distribution of the computed bending

moment with depth. Such information will allow a possible reduction, below a particular depth, in the wall thickness of a driven pile or in the number of rebars in thereinforced-concrete pile. For the Mustang Island experiment, the comparison is presented in Fig. 7.18 for a static, lateral load of 210 kN, which would reflect a factor

igure 7.1 7 Com parison o f experim ental and com puted values of maximum bending mo me nt anddeflection, static loading, Mustang Island.

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300 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.18 Com parison o f curves of bending mo me nt versus depth for Pt of 2 l 0 k N , static loading,Mustang Island.

igure 7.19 Comparison of experimental and computed values of maximum bending moment andde flection , cyclic loading, Mustang Island.

of safety of about 1.5 with respect to first yield. The curves agree well and show that

the special requirement for bending strength no longer exists after a depth of about 5meters.

The comp arisons of ground line deflection an d ma xim um bending mo me nt for Pile 2,for cyclic loading , are show n in Fig. 7.1 9. The ag reeme nt in bo th instan ces is excellent.

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Case s tu d ie s 301

able 7.8

Depthm

0-0.360.36-3.53.5-6.56.5-9.59.5-12.5

Reported properties of soil at Garston.

Description

FillDense sandy gravelCoarse sand and gravelWeakly cemented sandstoneHighly weathered sandstone

NSPT

18^ 6 5

30

« 1 4 0

Unit weightkN/m3

21.59.7

11.7

Friction angledegrees

433743

In total, the experimental and computed values for the series of tests at MustangIsland agree extremely well. It is of interest to note, however, the characteristics ofthe sand and the method of installation of the pile. Other sands and other methods ofinstallation could well produce a different set of results.

7 . 4 . 2 G a r s t o n

Price & W ardle (1987 ) repo rted th e results of lateral-loa d tests of a bored p ile, identifiedas TP 15 , with a length of 12.5 m an d a diam eter of 1.5 m. T he locatio n of the testswas not given and is listed as the location of the Building Research Establishmentfor convenience. The reinforcemen t consisted of 36 rou nd bar s, 50 m m in diameter,on a 1.3-m-diameter circle. The yield strength of the steel was 425N/mm2 . The cubestrength of the concrete was 49 .75 N /m m2 . The bending moment (Mu\t) at which aplastic hinge wo uld occur was compu ted to be 15,900 k N-m at concrete strain or 0.00 3,the value of strain explicitly defined as corresponding to the failure of the concrete.

The authors installed highly precise instruments along the length of the pile. Thereadings allowed the determination of bending moment with considerable accuracy.

The properties of soil reported by the authors, and the interpretations used for thefollowing analyses, are shown in Table 7.8. The fact that granular soil has been shownto increase in stiffness with an increase in strain could well influence the values thatare shown

The lateral load w as applied at 0.9 m abov e the grou nd line. Each load wa s helduntil the rate of mov ement w as less than 0.05 mm in 30 minutes. The load was reducedto zero in stages and held at zero for one hour.

The p-y curves for static loading for the analyses were obtained by using the criteriafor sand. The comparisons of pile-head deflection and maximum bending moment areshow n in Fig. 7.20. The curves for deflection sh ow tha t the com put ation is abo ut 2 0 %unconservative for the larger loads and in good agreement for the smaller loads. Thereduction of the shear strength with the increase in strain was not implemented in theanalyses and could well account for the slight unconservatism. T he max imu m bendingmo men t from the experiment is abo ut 1 2 % higher than the com puted value at the samelateral load. The computer yielded a lateral load of 4,520 kN to cause a plastic hinge.

7.4.3 A r k a n s a s r i v e r

Mansur & Hunter (1970), and Alizadeh & Davisson (1970) reported the results oflateral-load tests for a number of piles in connection with a navigation project. Pile 2

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30 2 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.20 Com parison of experimen tal and com puted values of maximum bending mo me nt andpile-head deflection, static loading, Ga rston.

igure 7.21 Cross se ction of pile at Arkansas River.

with a penet ratio n of 15 m wa s selected for analysis. The pile wa s formed of a steel pipewith a diameter of 0.406 m an d a wall thickness of 8.153 mm . As show n in Fig. 7. 21 ,four steel angles were added to the pile at equal spacings to carry instruments, givingthe pile an effective diameter of 0.48 m, a moment of inertia of 3.494 x 10- 4 m 4 , anda bending stiffness of 69,900 kN-m2 . Estimating the value of yield strength of the steelat 248,000 kPa, the value of the moment at first yield of the steel (My) was computedto be 361 kN-m.

Several borings were made at the site, and there was a considerable variation in thepro pertie s across the site. The soil in the top 5.5 m was a poorly g raded sand wi thsome gravel and with little or no fines. The underlying soils were fine sands with someorgan ic silt. The wa ter tab le was at a dep th of 0.3 m. T he total un it weight abov e

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C a s e s t u d i e s 303

Figure 7 22 Ratio o f q c N S PT as a funct ion o f D5 0 af ter Rober tson et al. , 1983 .

the water table was 20.0 kN/m3 , and below the water table was 10.2 kN /m3 . A studyof the soil borings indicated that the water table was at a depth of 1.5 m. Data fromthe site showed that the site had been preconsolidated due to the presence of 6 m ofoverburden that was removed prior to testing.

Data and analysis of the sand at the site are given in Table 7.9. The first threecolumns in the table show the depth below ground surface, the computed value of

vertical overburden pressure, and the blow counts from the Stand ard Pen etration Test.The contributions of Robertson, shown in Fig. 7.22, were consulted, and the fourthcolumn was obtained, showing values of qc in MPa from the static cone test based oncorrelations with the values of Ν ρ τ> Values of qJcr'vO were then computed and used

able 7.9

Depthm

00.612.44.04.65.57.08.510.011.620.0

^(estimated)

Penetration

< oMPa

00.0120.0390.0560.0620.0710.0860.1020.1170.1330.219

resistance and

NSPT

1212142017252818272929*

analysis of soil

<JcMPa

5.05.05.5

10.08.0

13.014.012.015.015.015.0

at test site,

4c/<o

_41718317912918316311812811368

Arkansas River.

Φdeg

_45424241424240414036

£s

MPa

15151522.519.52728.519.5303030

Π £

444333332.52.52.5

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304 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 7.23 Proposed values of friction angle as a function of results from cone tests con sideringoverb urden pressure and coefficient of lateral earth pressure (after Durgu noglu Mitchell1975).

to obtain values of φ from a chart (Fig. 7.23) proposed by Durgunoglu and Mitchell(1975). The con tributio ns of Van Impe (1986) (see Fig. 7.24) and Jamio lkow ski (1993)were used to obtain values shown in the last two colu mn s, estimates of the m agnitud eof the soil modulus £s in MPa and a value of «£, a multiplier of £s , based on thedegree of overconsolidation. These last two columns served to give an insight intothe selection of the initial slopes of the p-y curves. The p-y curves for static loading forthe analyses were obtained by using the criteria for sand.

The loading was applied at the groundline, and the loading was static. The comparison of the results from the experiment and from the computations are shown in

Fig. 7.25. Both cases of shear angle gave results that are somewhat conservative inthe higher ranges of loading, but the computations with the higher value of φ gave,in general, very good agreement. The lateral load to cause the first yield of the steel atthe extreme fiber was computed to be 324 kN.

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Case s tu d ie s 305

igure 7.24 Proposed values of modulus of deform ation fro m e xperime ntal results (from Van Impe1985).

igure 7.25 Com parison o f experim ental and com puted values of groun d-line deflection, static loading,Arkansas River.

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306 S i n g l e P i le s a n d P i l e G r o u p s U n d e r L a t e r a l L o a d i n g

bl e 7.10

Depth*m

0.501.001,502.002.503.003.504.004.50

5.005.506.007.008.008.75

Results of soil tests performed at Roosevelt

Con e Penetro meter Test

QcM n m 2

1.891.674.795.543.265.374.90

20.5433.79

25.5818.4910.61

Fsk N m 2

2.504.94

17.3616.7310.9424.6515.2061.28

117.53

80.4355.3452.35

F r

0.130.290.360.300.330.460.310.300.35

0.310.300.49

Bridge.

DilatometerTest

£dbar

6387

10996

108133206536578

605——

Kd

10.15.45.24.04.44.65.4

19.8II.1

12.8——

Pressurem eter Test

E Mbar

616584

466595

—3,330—

900—1,370

409241841

PLbar

1.51.61.62.12.8

—23.0—21.0—25.0

7.05.5

19.0

Std. Pen. TestN

7—

8—

3—

2—14—1335I I16

*From mudline

7.4.4 R o o s e v e l t b r i d g e

Ruesta and Townsend (1997) present the results of a study at the Roosevelt Bridgereplacement in Florida. A single pile and a group of piles were tested under lateralloading under sponsorship to the Florida Department of Transportation. The pileswere 16.5 m in length and were pre-stressed. The process of pile installation consistedof jetting to 7.6 m, placing the pile, and subsequently driving the pile with an impacthammer. Static loading was used.

Th e test pile had a square s hap e, with a n area of 0.76 m2 . A 3 5 cm d iamete r steelpipe with a 9.5 mm wall thickness, for the placement of strain gauges, was groutedinto a 45 cm diam eter void after the pile wa s driven. T he au tho rs stated th at th e

cracking moment was about 850 kN-m and the ultimate mom ent was about 1,400 kN-m. Appreciation is extended to Mr. Frank Tow nsend for providing special in formation .Several testing methods were used to obtain the properties of the soil at the site.

The results of the testing are shown in Table 7.10. The depth of water at the site was2.08 m. The authors studied the information in Table 7.10 and selected the followingcharacteristics for two layers of soil.

2.08 m to 6.08 m. Sand, friction angle 32 deg, submerged unit weight y' 8.9 kN/m3 ,and lateral earth pressure coefficient K 16 .3 MN/m3 .

6.08 m to 16.08 m. Cemented sand, friction angle 42 deg, submerged unit weight y'11.1 kN/m3 , and lateral earth pressure coefficient K 3 4 M N / m3 .

The load was applied at 0.45 m below the top of the pile and m easuremen ts ofdeflection were me asured at 2.1 m below the po int of applicatio n of lateral load ing.Analy tical com pu tatio ns of the respon se of the pile wa s made by Dr. W illiam Isenho wer,employing Computer Program LPILE.

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C ase s tu d ies 30 7

A comparison of the experimental and analytical results are shown in Fig. 7.26.The agreement is good to excellent with the analytical method yielding results that areslightly conservative.

7.5 P IL E S I N S TA L L E D I N T O L A Y E R ED S O I L S

7 5 1 T a l i sh ee k

Gooding et al. (1984) describe experiments for the Louisiana Power and Light Company where a steel-pipe pile was tested at Bogalusa, Louisiana, under conditions tosimulate the found ation for a transmission tower. A sketch of the loading arrangem entis shown in Fig. 7.27. As may be seen, the pile at the groundline was subjected tolateral load, bending moment, and axial load.

The outside diameter of the pipe was 0 .9144 m, its wall thickness was 0 .009 525 m,and its penetratio n w as 4.27 m. The mo me nt of inertia, I, was com puted as0.002772 m4 , its bending stiffness, E p I p was 55 4,400 kN-m2 , its bending momentat first yield of the steel, My, was 1,516 kN-m, and its ultimate bending moment, Mu\u

was 1,950 kN-m.The upper layer of soil was classified as a stiff sandy clay and consisted principally of

clay-sized particles b ut includ ed som e gran ular particles. It wa s classified as a CL withthe Unified System (see Table 7.11). The shear streng th was found from the uncon finedcom pression tests of specimens th at were either 76 m m or 12 7 m m in diameter. In theabsence of data on £50 a value of 0.007 was selected by reference to Table 3.5.

The second layer was a dense fine sand where the uncorrected values of N from theStand ard P enetration Test averaged 7 1 . The shear angle was based on values from theStandard Penetration Test and the overburden pressure, leading to a value of φ = 50° .Employing the procedure illustrated with the tests at Arkansas River to confirm thevalue of 0, the following analysis can be made. Starting with NSPT = 71 , the followingvalue can be computed for the penetration resistance at 60% of the driving energy:NSPT 60) = 71/0.6 = 120. Using the correlations for dense sand presented earlier, thevalue of 120 corresponds to a cone value (CPT) of qc ~ 48 MPa (for normally consolidated, clean, very dense sand). Then, the value of q c /VvQ = 48,0 00 /6 0 = 800 leads toa value of 0 ~ 50° from Mitchell's correlations.

The p-y curves for static loading for the upper layer of clay were obtained by usingthe criteria for stiff clay with no free water, and for the layer of sand by using thecriteria for sand. The procedure for layered soil was implemented. The penetration ofthe pile was relatively small, and the anticipation was correctly made th at the bo ttomof the pile would undergo a sensible deflection. Therefore, the decision was made tointroduce a set of data at the base of the pile, giving resistance in force as a function ofthe deflection of the tip. The ultimate force was estimated by multiplying the verticalstress at the tip by the tangent of the friction angle and by the area of the base ofthe pile. Th at force was com puted to be 218 kN . The force-displacement relationshipwas estimated by use of the data from load-transfer curves for skin friction for axially

loaded piles. The relationships were computed and are shown in Table 7.12.As noted in Fig. 7.27, the loading arrangement could apply simultaneously lateral

load, bending moment, and axial load, all in the positive direction. Table 7.13 showsthe set of loadings and the observed lateral deflections at the groundline.

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308 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.26 Comparison of experimental and computed values of ground-line deflection (a), andmaximum bending mo me nt (b), static loading, Roosevelt Bridge.

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Case s tu d ie s 309

igure 7.27 Test arrangement atTalisheek.

able 7.11 Rep orted prop erties of soil at Talisheek.

Depthm

01.831.836.0

Watercontent

17.317.32I.62I.6

Totalunit weightkN/m3

18.718.720.120.1

UndrainedshearkN/m2

59.259.2——

strength£ 5 0 *

0.0070.007——

Frictionangledegrees

—5050

From Table 3.5

The results of the test at Talisheek are interesting for a number of reasons: a combinati on of loads were em ployed , the pile wa s sho rt, the pile failed un der lo ad, a resistingforce was assumed at the base of the pile in the computations, and the computationsindicate excellent agreement with the experiment. A comparison of the results for pile-head deflection from experiment and from computation is shown in Fig. 7.28. Thedeflection is shown as ordinate and the abscissa shows the number of the load. A

significant difference in the curves is show n at L oad 6, proba bly because in the experiment, all loads were reduced to zero after Load 5. The computations for deflection areconservative at the larger loads. The selection of the friction angle from data from theStand ard P enetration Test leads to significant app rox ima tion s. H ad a smaller value of

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310 S i n g l e P i le s a n d P i l e G r o u p s U n d e r L a t e r a l L o a d i n g

bl e 7.12 Computed force-displacement relationship for lateraldeflection of the tip of the pile,Talisheek.

Force

k N

0.095133157173184194203215217

218218

Displac

m

0.000.0010 002

0 003

0 004

0 005

0 006

0 007

0 009

0.0100 0127

1.000

bl e 7.13 Set of loads applied at tests atTalisheek.

Load number

1

2345678910I I12131415161718

Lateral loadPtk N

44.1

66.179.488.297.0

000

13.226.439.652.866.179.388.192.596.9

103.5

Bending momentMt

kN-m

456.8

685.0822.3913.6

1004.9328.1641.3939.5

1076.41213.31350.21487.11624.11761.01852.21897.91943.52012.2

Axial loadPxk N

4.5

8.98.9

13.313.3

107.2211.9309.8314.3314.3318.7318.7323.1323.1323.1327.6327.6327.6

DeflectionYtmm

9.1

16.822.927.430.515.219.825.925.930.535.142.748.856.4---73.2*

* ailure by plastic buckling

friction angle been estimated, the agreement at the larger loads would have been notas good as shown in Fig. 7.28.

A further point of interest is that the computations showed that significant values ofshear developed at the tip of the pile because of the com pu ted values of deflection. Theshear was equal or close to the max im um value of 218 kN for L oads 10 throu gh 16.Th e shear at the tip of the pile influenced behav ior for even smaller load s. The ob viou s

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31 2 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

able 7.14 Rep orted p rope rties of soil at Alcâcer do Sol.

Wa t e r+ Total UndrainedDep th con tent unit weight shear strength Friction angle

m % kN/m3

kN/m2

ε 50 deg

03. 50

3. 50

8. 50

8. 50

23 0

23 0

40 0

62 5

62 5

28 6

28 6

62 5

62 5

28 6

28 6

16 0

16 0

19 0

19 0

16 0

16 0

19 0

19 0

20 0

20 0

——32 0

32 0

——

0. 020

0. 020

——0. 020

0. 020

——

3030

3535

+ Computed *FromTab le 3 .3

The pile was 40 m in length and had a diameter of 1.2 m. It was reinforced with 35bars with a diam eter of 25 mm . The strength s of the concrete and steel were repo rted tobe 33.5 M Pa and 40 0 M Pa, respectively. The cover of the rebars was taken as 50 mm.The bending stiffness was computed, and a value of 3.29 x 106 kN-m2 was selected foruse in the analyses. The ultimate bending moment was computed to be 3370 kN-m.The pile was instrumented for the measurement of bending moment along itslength.

From the gro und surface do wn wa rd, the soil is described as silty mu d, sand , m uddycomplex, and sandy complex. The properties of the soil were found from SPT, CPT,and vane tests, and the values that were selected for use in the analyses are shownin Table 7.14. The p-y curves for the upper layer of clay were obtained by using thecriteria for stiff clay with no free water. The subsequent layers used the criteria forsand, for stiff clay with no free water, and for sand. The criteria for layered soils wereimplemented.

The lateral load was applied at 0.2 m above the ground line. Bending moment wasmeasured along the length of the pile but information is unavailable on the techniquesthat were used. Ground-line deflection and maximum bending moment were reportedfor three values of lateral load: 100, 200, and 300 kN.

The position of the water table was not reported, but it is assumed that the watertable was close to the ground surface. The data were analyzed by use of the criteriafor clay with no free water and the criteria for sand. The comparisons of ground-line deflection and maximum bending moment for Pile 2 are shown in Fig. 7.29. Theanalytical metho d over-predicts deflection, bu t the ma xim um bending mo me nt is computed with appropriate accuracy. The maximum bending moment of 1,007 kN-m thatwas measured was much less than the ultimate moment.

7 . 5 . 3 F l o r i d a

Davis (1977) described the testing of a steel-pipe pile that had a diameter of 1.42 mand a penetration of 7.92 m. The tube was filled with concrete to a depth of 1.22 m,and a utility pole wa s embe dded so th at the lateral loads were app lied at 1 5.54 m

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Case s tu d ie s 313

igure 7.29 Comparison of experimental and computed values of maximum bending moment andpile-head deflection, static loading, Alcâcer do Sal.

above the ground line. Meyer (1979) analyzed the results of the test and reportedthe bending stiffness to be 5,079,000 kN-m2 in the top 1.22 m, and 2,525,000 kN-m2

below. The ultimate bending m om ent was reported to be 6,280 kN -m in the top 1.22 m,and 4,410 kN-m in the lower portion.

The soil profile consisted of 3.96 m of sand ab ove sa tura ted clay. The sand h ad atotal un it weight of 19.2 kN /m3 , and a friction angle of 38 degrees. The water tablewas at a depth of 0.61 m. The undrained shear strength of the clay was 120 kPa, andits submerged un it weight was 9.4 kN /m3 . A value of ss of 0.005 was selected for theanalyses, following values shown in Table 3.5.

The p-y curves for static loading for the upper layer were obtained by using thecriteria for sand. Some discussion is desirable about selecting the criteria for the layerof clay. Because the clay is below sand, no loss of resistance would occur becauseof gapping. Then, for static loading, the options are stiff clay with no free water(no erosion will occur) or stiff clay with free water. The latter criteria were selected.However, in similar designs, the engineer might try both sets of recommendationsto gain some insight into possible differences in response. The computed curves fordeflection with depth are not presented, but the pile deflections in the zone of the claywould undoubtedly be quite small. The criteria for layered soil were implemented.

A comparison of the experimental and computed values of pile-head deflection

is shown in Fig. 7.30. The curves agree well for the early loads but start to deviate strongly abo ve a lateral load of 16 0 k N (and a mo me nt at the gro un d line of2,48 6 kN -m , considering the point of the application of the load). The ou tpu t fromthe computer code was examined, and it was noted that the bottom of the relatively

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Case s tu d ie s 315

able 7.16 Mechanical prop erties of piles atAp apa .

Section Diameter, E l

m m kN-m2

0-2.44 0.442 22 4002.44-6.10 0 417 20 1006.10-15.3 0 391 18 700

7.5.4 A p a p a

Coleman (1968), and Coleman & Hancock (1972) describe the testing of Raymondstep-tapered piles near Apapa, Nigeria. The results were analyzed by Meyer (1979).

Two piles, identical in geometry, were driven and capped with concrete blocks. Ahydraulic ram was placed between the caps and the piles were loaded by being pu shedapart. The reported properties of the piles are presented in Table 7.16.

The soil at the site consisted of 1.52 m of dense sand underlain by a thick stratumof soft organic clay. The friction angle of the sand was obtained in the laboratory bytriaxial tests of reconstituted specimens and was found to be 41 degrees. The strengthof the soft organic clay was obtained from in situ-vane tests and was found to be23.9 kPa; the value of £5 for the clay was assumed to be 0.02. The water table was ata depth of 0.91 m. The unit weight of the sand above the water table was 18.9 kN/m3

and the submerged unit weight of the clay was 4.7kN/m3 .

The p-y curves for static loading for the upper layer were obtained from the criteriafor sand and for the lower layer from the criteria for soft clay. The procedure forlayered soil was implemented.

The lateral load s were applied at 0.61 m abov e the gro un d line, and th e deflectionwas measured at that point. A comparison of the experimental and computed valuesof pile-head deflection is shown in Fig. 7.31. The computed curve agrees well withthe experimental results. The difference between the experimental results for the twopiles, which presumably were identical, probably reflects differences in the structuralcharacteristics of the piles, and possibly differences in the effective properties of thesoil due to effects of installation.

7.5.5 S a l t L a k e I n t e r n a t i o n a l A i r p o r t

Rollins et al. (1998) describe the testing under static loading of a 3 x 3 group and ofa single pile. The single pile was tested at a distance away from the group to not beinfluenced by the loading of the group; only the results of the single-pile test will beanalyzed here.

The pile was 0.305 m I.D. and with a wall thickness of 9.5 mm . The pile was drivenwith a closed end to a penetration of approximately 11m. Prior to conductin the

lateral loading, an inclinometer casing and strain guages were placed inside the pile.The pile was then filled w ith a six-b ag, pea-gravel co ncrete . Tests of concrete cylindersshowed the strength of the concrete at the time of testing to be 20.7 MPa and the elasticmodulus to be 17.5 GPa.

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316 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

igure 7.31 Com parison of experim ental and comp uted values of pile-head deflection , static loading,Apapa.

able 7.17 Soil de scrip tions and classifications at Salt Lake.

Depth interval, m

0-1.61.6 2.29

2.29-3.763.76-4.634.63-6.366.36-6.656.65-6.946.94-7.537.53-8.388.38-9.839.83-11.28

Soil description and classification

Compacter sandy gravel fill (excavated)Silt w ith sand, MLLean gray clay, CLLight gray sandy silt, MLPoorly graded light brown sand, SPFat gray clay, CHLean gray clay, CLLight gray silt, MLLight brown sandy silt, MLLight gray silty sand, SMGray silt, ML

The authors characterized the soil as shown in the following table (Table 7.17).As shown in Table 7.18, a variety of methods were used to obtain values for the

strength of soil. The Standard Penetration Test was used to obtain properties (N)^oof the strate of sand and a variety of tests (unconfined compression, vane shear, andpressuremeter) were used to obtain p roperties of the undrain ed strength (su) of th e clay.

The authors made an interpretation of the data shown in Table 7.18 and preparedTable 7.19 showing a detailed soil profile. A conservative soil profile was also givenbut was not employed in the following analyses. The detailed soil profile was used in

the analyses that follow with the view that the detailed soil profile is a more realisticrepresentation of the soil profile as it exists.

The gravel fill was removed prior to testing the pile. The top of the pile, whereload was applied, was scaled as 1.23 m below the original ground surface and 0.4 m

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Ca se s t u d i e s 317

able 7.19 Values of soil prop erties f or detailed soil pro file by interp reta tion at Salt Lake.

De pth interval, m Kind of soil s u o r φ effective unit of weight

0.4 to 0.870.87 t o 1.121.12 t o 1.501.50 to 2.272.27 to 3.273.27 to 5.075.07 to 5.475.47 to 6.076.07 to 6.776.77 to 9.77

clayclayclayclayclaysandclayclayclaysand

46 kN/m 2

4 l . 6 k N / m2

95 kN/m 2

50 kN/m 2

41 kN/m 2

38 kN/m 2

52.3 kN /m 2

25 kN/m 2

51 kN/m 2

36 degrees

8.186 kN/m 3

8.828 kN/m 3

8.828 kN/m 3

8.828 kN/m 3

9.671 kN/m 3

10.055 kN /m 3

6.302 kN /m 3

10.055 kN /m 3

10.055 kN /m 3

9.315 kN/m 3

above soil surface as excavated. Deflection and load were measured at the point whereload was applied. The water table was at the point of load application. As noted

earlier, the 1.6 m of gravel was remo ved , and the load w as applied at 0.4 abov ethe new ground surface. The deflection was measured at the point of application ofthe load.

In the w ork tha t follows, the soil starts at a distance of 0.4 m b elow the po int ofload a ppl icatio n. T hu s, in the analyses that follow, 0.4 m of the pile exists abov e theground surface. Dr. William Isenhower employed a computer code and the proceduresdescribed in Chapter 4 and computed the analytical curves of bending curvature as afuntion of the bending moment. The non-linearity of the bending stiffness as a functionof bending moment is evident as shown in Fig. 7.32 and was taken into account in theanalyses.

The results presented by the authors on the testing and analyses for computation oflateral load versus deflection at the point of application of load are shown in Fig. 7.33.Computer program LPILE was used in making the computations. As may be seen,good to excellent agreement was obtained for the pile, when the detailed soil profile

able 7.18

(N)60

Depth,

0.170.621.241.534.615.356.308.409.30

m

Results from field and laboratory investigations at

Value

353742452645173230

Depth, m

1.82.22.22.73.13.63.74.66.36.77.3

10.5

s u kPaU-U

104

46

58

76

VST

50

108

58

24

Salt Lake.

PMT

29-38

38-49

30-38

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Ca se s t u d i e s 319

was employed in the analyses. When the conservative soil profile was employed, thecomputed deflection was about 60% greater than the experimental values.

7.6 P I L E S I N S TA L L E D I N c - 0 S O I L

7.6.1 K u w a i t

A field test for beh avior of laterally-lo aded, bored piles in cemen ted sand s c-φ soil) w ascon duc ted in Kuw ait (Ismael, 199 0). Twelve bored piles tha t were 0.3 m in diame terwere tested . Piles 1 to 4 we re 3 m lo ng, while piles 5 to 12 w ere 5 m lo ng. Th e stu dywas directed at the behavior of both single piles and piles in a group. Curves showingmeasured load-versus-deflection at the pile head for 3-m-long single piles and 5-m-long single piles are presented in the paper. Only results from loadin g the 5-m piles are

studied by using the soil criteria for c-φ soil.The 5-m piles were reinforced with a 0.25 m-diameter cage made of six 22 mm bars,

and a 36mm-diameter reinforcing bar was positioned at the center of each pile. Thepiles were instrumented with electrical-resistance strain gauges. After lateral-load testswere completed, the soil was excavated to a depth of 2 m to expose strain gauges. Thepile was reloaded and the curvature was found from readings of strain. The flexuralrigidity was calculated from the initial slope of the moment-curvature curves as 20.2M N - m2 . The experimental value of E p I p is significantly larger than values computedfrom mechanics for reasons that cannot be identified. Because of the inability to applymechanics to the analysis of the cross section of the pile, the bending moment to causea plastic hinge to develop could not be computed. The experimental value was judgedto be superior and is used in the analyses shown below.

The subsurface consisted of two layers as shown in Table 7.20. The first layerdescribed as me diu m dense cemented silty sand , was ab ou t 3.5 m in thickn ess. Th evalues of c and φ for this layer were found by drained triaxial compression tests andwere 20kPa and 35°, respectively. The unit weight averaged 17.9kN/m3 . Using Eq.3.67 and Figs. 3.31 and 3.34, kc was found to be 90,000 kN /m3 (at the beginning ofthe curve) and ^ w a s found to be 80,000 kN /m3 (at the end of the curve) yielding avalue of kpy of 170,000 kN/m3.

able 7.20 Properties of soil at Kuwait

Average properties of soil at site

Depth w U n it w t. S.L Sand Siltm Desc. N SPT kN /m3 L L R l. Clay

0 -3 Med . dense 21 3.0 17.9 20.4 3.1 14.6 80.0 12.6 7.4silty sand

3-5.5 Med dense to 75 3.7 19.1 No ne N.P - 82.9 17.1 0

very densesilty sandwith cementedlumps

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320 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 7.34 Co mpa rison o f experimen tal and compu ted values of pile-head deflection , static loading,5-m-long pile, Kuw ait.

Th e first layer wa s under lain by me dium dense to very dense silty sand with cemen tedlumps. The values of c and φ were zero and 4 3° , respectively. The un it weight averaged19.1 kN/m3 . The value of kPy for the soil was found from Fig. 3.30 and Eq. 3.67 as80,000 kN/m3 . The p-y curves for static loading for the upper layer were obtainedfrom the criteria for c — φ soils and for the lower layer from the criteria for sand. Theprocedure for layered soil was implemented.

A computer code, employing c — φ criteria, was used to predict curves of load versus deflection at the pile head for the 5-m pile. Good agreement was found betweenmeasured and predicted behavior, as shown in Fig. 7.34. Because of the inability toapply mechanics to the analysis of the cross section of the pile, the bending momentto cause a plastic hinge to develop could not be computed.

7.6.2 L o s A n g e l e s

A field test for behavior of laterally loaded, bored piles in mostly c — φ soil was conducted in Los Angeles in 1986 by Caltrans (California Dep artm ent of Tr ansp ortatio n).The pile was 1.22 m in diameter and h ad a penetration of 15.85 m. T he load w asapplied at 0.61 m above the g round line. T he pile was instrum ented with electrical-resistance strain gauges and with Carlson cells for the measurement of bendingmoment; however, information on the calibration of the instruments was not given.

The compressive strength of the concrete was 24 ,80 0 kPa and the tensile strengthof the reinforcing steel wa s 413 ,70 0 kPa. The area of the steel reinforcem ent wa s 2% ofthe area of the pile. A total of 24 rebars were used, each with an area of 0.001065 m2 .

Th e distance between the outside of the rebar cage and the wall of the pile wa s 76.2 mm .The kinds of soil and the properties are given in Table 7.21. The water table was

well below the ground surface. The p-y curves for static loading for the first layerwere obtained by using the criteria for stiff clay with no free water; for the second,

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Ca se s t u d i e s 321

able 7.21 Kind of soil and prop ertie s o f soil at Los Angeles.

Depthm Soil

CohesionkPa

Friction angledeg

Unit weightkN/m3

0-1.51.5-7.4

7.4-10.510-5-13.413.4-20

Plastic claySand and some claySandy claySandy siltPlastic clay

1794.8

19.219.2

110

303521

19.219.819.818.918.4

igure 7.35 Com parison of experimen tal and com puted values of maximum bending mo me nt andpile-head de flectio n, static loading, Los Ange les.

third, and fourth layers by using the criteria for c — φ soil; and for the fifth layer byusing the criteria for stiff clay with no free water. The procedure for layered soil wasimplemented.

The lateral loads were applied at 0.61 m above the groundline, and the loads wereapplied in increments. The analysis was done by modifying E p I p according to themagnitude of the bending moment as presented in Chapter 4. Computations withthe analytical technique shown there, as noted earlier, reveal a sudden and dramaticdecrease in E p I p when the first tension crack appears in the concrete. For the presentanalyses, the precipitous decrease occurs in the range of the applied loads, leadingto the computations of very large deflections. The computations for comparison with

experimental results (Fig. 7.35) were made with values of E p I p that were believed to beappropriate for progressive cracking of the concrete in tension. The ultimate bendingmom ent was computed to be 4,400 m-kN and the maximu m applied mom ent w ascomputed to be close to the ultimate.

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Case s tu d ie s 323

and Eplp

the rock in the zone near the rock surface where the deflection of the pile was mostsignificant.

The b ored pile was 1.22 m in diameter and penetrated 13.3 m into the limestone.A layer of sand over the rock was retained by a steel casing, and the lateral load was

applied at 3.51 m above the surface of the rock. A maximum lateral load of 667 kN wasapplied. The curve of load versus deflection was nonlinear, but there was no indicationof failure of the rock.

In the absence of details on the strengths of the concrete and steel and on theamount and placement of the rebars, the bending stiffness of the gross section wasused. The following values were used in the equations for p-y curves presented inChapter 3: ^ r = 3 .45mpa; E ir = 7,240 M Pa; krm = 0 .0005; b = 1.22 m; L = 15.2 m;

3 . 7 3 x l 06 k N - m 2 .The comparison of pile-head deflection for results from experiment and from analy

sis (Reese, 1997) is shown in Fig. 7.36. The figure shows excellent agreement betweenvalues from experiment and from analysis for lateral loads up to about 350 kN, usingunm odified values of the bending stiffness. A sharp ch ange in the load-deflection curveoccurs at 350 kN . A possible reason is the decrease in bend ing stiffness E pI p at the largerloads.

The use of values of E pI p from mechanics would be desirable; however, analysisshows a sudden decrease in Eplp when strain in the concrete reaches the point wherethe concrete crack s in tensio n. H owev er, if a pile (or beam) is consid ered, cra cking doesnot occur at every point along the member, but instead initially at wide spacing. Thus,the net effect is that Eplp reduces gradually for the section, as a function of bendingmoment, and not suddenly as from analysis.

As shown in Fig. 7.36, values of Eplp were reduced gradually to find deflections thatwould agree fairly well with values from experiment. The following combination of

igure 7.36 Com parison of experim ental and compu ted values of pile-head de flection, static loading,Islamorada (after Reese 1997).

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324 Sing le Pi les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 7 37 Initial modulus of rock from pressuremeter, San Francisco (after Reese 1997).

values of load and bending stiffness were used in the analyses; in units of kN andkN -m2 , respectively: 400, 1.24 x 10 6 ; 467, 9.33 x 10 5 ; 534, 7.46 x 10 5 ; 601,6.23 x 10 5 ; and 667 5.36 x 10 5 . The assumption that the decrease in slope of thecurve of y t versus P t at Islamorada can be explained by reduction in values of E pI p

is reasonable. Also, a gradual reduction in values of E p Ip as shown, yields valuesof deflection by computation that agree closely with measured values. However, theIslamorada example gives little guidance to the designer of piles in rock except for theearly loads . The example from San Francisco that follows is more instructive.

7.7.2 S an F r a n c i s c o

The California Department of Tran sporta tion p erformed lateral-load tests of two boredpiles near San Francisco and the results of the tests, while unpublished, have beenprovided through the courtesy of Caltrans (Speer, 1992).

As is often typical in the investigation of the engineering properties of rock, thesecondary structure led to difficulty in sampling. The sandstone was found to bemedium to fine grained 0.10 to 0.5 mm ), well sorted and thinly b edded 25 to 75 mmthick). In most of the corings, the sandstone was described as very intensely to moderately fractured with bedding joints, joints, and fracture zones. Cores of insufficientlength were available for compression tests. Pressuremeter tests were performed atthe site and the results, as might be expected, were scattered. The plotted results ofthe values obtained for the moduli of the rock are shown in Fig. 7.37. The averagesthat were used for analysis are shown as a function of depth by the dashed lines. Thefollowing values were estimated for the compressive strength of the rock: 0 to 3.9 m,

372 kPa; 3.9 to 8.8 m, 1290 kPa; and below 8.8 m, 3,210 kPa.Two piles, 2.25 m in diameter, were tested simultaneously, and the results for Pile

B will be analyzed. Pile B exhibited a large increase in deflection for the last load,probably signaling a failure of the pile due to a plastic hinge. High-strength steel bars

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326 Sing le Pi les and P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igure 7 39 Comparison of experimental and computed values of pile-head deflection fo r differentvalues of be nding stiffness, static loading, San Francisco (after Reese 1997).

begin from 35.15 x lO6 kN-m2 (value from the analytical metho d was som ewh at largerbecause of the presence of the steel). The values of Ifrom the ACI method w ere multiplied by a constant value of E of 28.05 x 106 kPa to get values of E p Ip . The analyticalmethod is based on the assumption that all concrete is cracked when the first crackappears; thus, it is surprising that values from the ACI method and the experimentalmethod fall below values from analysis for over half the range of loading. The logicalexplanation for such a result is not imm ediately available.

The curves of deflection as a function of lateral load, using the values of E p Ip fromFig. 7 .38, are shown in Fig. 7.39. The experimental values yielded precise agreement,as was ensured by the fitting technique. The values from computations using the E pI p

from the ACI procedure fits the experimental values better than do the values from

the analytical meth od (Reese, 1997 ). How ever, as will be demonstrated later, if a loadfactor of 2.0 is selected, app lied to the load that causes a plastic hinge, the deflectionsfrom experiment and from analysis would range from about 2 mm to 4 mm . Suchdifferences are thought to be unimportant in regard to the deflection at service loads.

Also plotted in Fig. 7.39 is a curve showing the deflections th at were com puted withno reduction in the value of EpIp. Wh ile deflections at service loads may be computedwith little significant error, the use of the unmo dified EpIp is unacceptable because thecomputat ion of the load to cause a plastic hinge wou ld be grossly in error.

The values of EpIp from the various procedures were used to compute the maximum bending moment as a function of applied lateral load. The results are shown in

Fig. 7.40. As may be seen, the experimental values of max imum bending mom ent werepredicted quite well with any of the method s. Assuming the computed value of M u \ t iscorrect, as computed from the properties of the cross section, the analytical methodspredict the ultimate bending moment with good accuracy.

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Case s tu d ie s 327

igure 7.40 Comparison of experimental and computed values of maximum bending moment fordiffere nt values o f be nding stiffness, static loading, San Francisco (after Reese 1997).

7.8 A N A L Y S I S O F R E S U L T S O F C A S E S T U D I E S

The results of case studies are analyzed here in a manner to provide guidance to theengineer who wishes to design a pile to sustain lateral loading. The steps the engineertakes in making a design are presented to guide an evaluation of the results of thestudies.

(1) A pile is selected and its cross section is analyzed to obt ain bend ing stiffness asa function of bending moment and axial load.

(2) The ultimate bending mo men t Muit is found during the computations in (1),along with the bending moment My that will cause the extreme fibers of a steel

shape to just reach yielding.(3) The prop erties of the soil are evalu ated, p-y curves are computed for the kind

of loading to be applied, the point of application of the loads is specified, andother boundary conditions are selected for the service loads.

(4) The loads are increm ented in steps and com put er solutions are ma de to find theloads that cause failure in bending {Mu\t or My), or in rare cases the loads arefound that cause excessive deflection.

(5) The service loads are checked , emp loying partial-safety factors or a global factor of safety, and a revised section is selected, if necessary, for a new set ofcomputations.

The above steps were implemented in the analysis of the results of the case studieswith the assumption that only the ultimate bending moment will control; that is,deflection is assumed not to control the design. A review of the combined results

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Case s tu d ie s 329

able 7.23 Cases of piles under lateral loading tha t were analyzed.

Case

BagnoletCase 1Case 2Case 3

Houston

Japan

Lake Austin

Sabine

Manor

Mustang Island

GarstonLos AngelesSalt LakeSan FranciscoRoosevelt Bridge

Diameter, m

0.43

0.762

0.305

0.319

0.319

0.6410.610.61

1.51.22

0.324

2.250.76

Kind of Soil

Soft clay

Stiff clay

Soft clay

Soft clay

Soft clay

Stiff clay h2o

Sand

LayersLayersLayersWeak rockSand

Pile

Material

Steel

Rein. Cone.

Steel

Steel

Steel

Steel

Steel

Rein. Cone.Rein. Cone.CompositeRein. Cone.Composite

Nature of

Loading

Static

StaticCyclicStatic

StaticCyclic

StaticCyclicStaticCyclicStaticCyclicStaticStaticStaticStaticStatic

UltimateMoment,

KN-m

204

20302030

55.9

231231

231231

17571757640640

159004400

335177401,400

Remarks

Defl. failure

Dataextrapolated

the results show larger computed values (on the left of the straight line) as smallercomputed values.

With regard to deflection, the comparative values are all so small to be negligible,or are relatively close, except for the test at Lake Austin under cyclic loading. Therethe experimental value was more than twice as large as the computed value and largeenough to be of some concern from a practical point of view. A similar result wasobtained for the test under cyclic loading at Sabine, where the experimental valuewas 1.52 times as large as the computed value. On the basis of the results from Lake

Austin and Sabine for cyclic loading, the engineer might wish to specify field tests insoft clay under cyclic loading if deflection is a critical parameter. A review of all ofthe curves showing computed and experimental deflection shows that, in general, thecomputation yields acceptable results.

Figure 7.42 present a comparison of the values of experimental and computed pile-head deflection at service load. The agreement is fair with about as many showinglarger computed values as smaller computed values. The test for cyclic loading at LakeAustin shows the poorest agreement. Except possibly for that test, the differencesprobably would not lead to experimental difficulties.

The test at Talisheek is the only one where an axial load was applied along with the

lateral load. While the data do not yield a computed value of the factor of safety, areview of the results in Fig. 7.28 shows that the analytical method was able to predict the results from the experiment with reasonable accuracy. The analytical methodappears to account appropriately for axial loading. Because most axial loads in practice

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330 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

able 7.24 Results of analysis of data from tes ts of piles u

Case

BagnoletCase 2staticCase 3static

Houstonstaticcyclic

Japan

staticLake Au st n

staticcyclic

Sabinestaticcyclic

Manorstaticcyclic

Mustang Is.staticcyclic

Roosevelt Br.static

Garstonstatic

Los Angelesstatic

Salt Lakestatic

San Francis.static

M u it

kN-m

204

204

20302030

55.9

231231

231231

17571757

640640

1450

15900

4400

335

17740

P t failkN

138

130

950900

50

145113

9972

693543

324295

189

4520

1779

174

8670

P t serv.kN

76.7

72.2

432409

28

8163

5540

385302

180164

105

2055

809

97

3940

nder lateral loading.

Values at service load

yt compmm

9.6

9.4

20.226

22

3522

4927

I I13.1

1615

5.6

33

21

23

2

Yt expmm

9.6

9.5

2634

28**

3546

3641

9.710.2

1615

6.2

40

22

22

3

M comkN-m

104

105

702742

19.6

11079

10368

760710

305320

480

6600

1640

65

7030

M expkN-m

95

112

600642

2 | 9**

106110

9682

715610

305320

3.85

7500

1890

75

6640

Fac. saf.

2.34

1.82

3.383.16

2.55

2.182.18

2.42.82

2.462.88

2.12

2.21

2.12

2.33

3.5

2.67

* By extrapolation

are compressive, the beh avior of bor ed piles will be improv ed by the axial loading w hilesteel piles will be affected adversely.

Tw o tests, the ones at Talisheek and at Florid a, were of sh or t piles wh ere the tipof the piles deflected in an opposite direction to the pile head. While short piles areto be avoided in practice, if possible, the facility to use a curve at the pile tip showinglateral load as a function of pile deflection is useful.

A review of all of the tests that were analyzed shows that several parameters are

critical with respect to the response of a pile to lateral loading. Such a review suggeststhe desirability of having a larger data set of experiments, particularly where bendingm om ent is me asured and w here the loadin g is cyclic such as is enco unte red frequentlyin practice.

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Case s tu d ie s 331

igure 7.41 Comparison of experimental and computed values of maximum bending moment atservice load for various tests.

igure 7.42 Com parison of experimen tal and com puted values of pile-head deflection at service loadfor various tests.

H O M E W O R K P R O B L E M S FO R C H A P T E R 7

7.1 You are an employee at a large com pan y engaged in variou s projects involving thepractice of Civil Engineering. Explain the reasons you would research technical

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33 2 S in g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d in g

literature presenting new cases where a pile has been subjected to lateral loadingsand the results are reported in detail.

7.2 M ak e a list of the factors that m ust be presented for you to use the Student V ersionof LPILE to perform computations to compare with the experimental results that

are presented.7.3 (a) Use the Student Version of LPILE and ma ke com pu tatio ns to check the plot s

of pile-head deflection an d ma xim um bending m om ent, sho wn in the text, forthe test in Japan, using static loading.

(b) M ak e additio nal com pu tati on s as necessary with the Student Version of LPILEand find the penetration of the pile that gives two points of zero deflection,assum ing the soil pro pertie s at a depth of 10 m are the same as the depth of5 .18m.

(c) For a pile subjected only to lateral loading, under what conditions would youas the responsible engineer insist that the pile has sufficient penetration toyield two points of zero deflection, recognizing that a considerable expensemay be involved in achieving the required penetration?

7.4 (a) Use the Student Version of LPILE and ma ke com pu tatio ns to check the plot sof pile-head deflection an d ma xim um bending m om ent, show n in the text forstatic loading, for the test at Sabine.

(b) Make additional computations with the Student Version of LPILE, assuming cyclic loading, with the shear strength of the clay reduced to 75 ofexisting value, the same as sho wn , and at 1 2 5 % of exciting value, all oth er values remaining unchanged. Make plots of pile-head deflection and maximum

bending moment as a function of shear strength of the clay.(c) What do the plots in (b) above say to you about the importance, or lack ofit, in performing the appropriate tests to determine the shear strength of claysoil that support piles under lateral loading.

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Chapter 8

Te s t i n g o f f u l l s i z ed p i l e s

8 .1 INTRODUCTION

8.1.1 Scop e o f p r es en t a t i o n

The pr esen tation here is limited to the testing of single, full-sized piles but ackn ow ledgment is made that numerous investigators have obtained valuable results from testingmodel piles in the laboratory. For examp le, Chap ter 5 details the contribu tions of several workers who provided data that allowed numerical values to be assigned to theeffects of pile-soil-pile interaction. In regard to pile groups under lateral loading, thework in the laboratory of Franke (1988), Prakash (1962), and Shibata, et al. (1989)is worthy of note.

The centrifuge has become a popular tool for the investigation of problems in soil-structure interaction and is used by many agencies in many countries. The centrifugehas been used to investigate the pro blem of single piles and grou ps of piles under lateralloading. Among the investigators who have used the centrifuge to study the responseof piles to lateral loading are Kotthaus & Jessberger (1994), Terashi, et al. (1989).Bouafia & Gamier (1991), and McVay, et al. (1998). The results of centrifuge testshave provided useful information, particularly with respect to groups of piles, and thetechnique is expected to continue to contribute to the understanding of piles underlateral loading.

8 1 2 M e t h o d o f a n a l y s i s

The model employed presented herein for design of piles under lateral loading requiresan equation solver, a computer code, experimental data on response of soil, and proposals for the response of various kinds of soil to lateral loading. The method hasbeen presented in detail in the preceding chapters. Because of the complexity of theinteraction between a pile and the supporting soil, the experimental data must comefrom load testing of full-sized piles in the field. Valid predictions could not have been

made for the soil resistance as a function of the lateral deflection of a pile without well documented data from the testing of instrumented piles in the field, calledresearch piles. The testing oi research piles and proof piles is discussed in the follow ingparagraphs.

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334 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

8 1 3 Class i f i ca t ion o f t es t s

Research piles are those with comprehensive instrumentation to improve currentlyavailable p y curves and to obtain data for p y curves for soil where recommendations

are absent or for p y curves to improve existing predictions.Proof piles are those whe re lateral-load tests are emp loyed to obt ain d ata for site-specific designs. The assumption is made that the piles to be used to support theproposed structure have been selected, and may be termed production piles. The testing of proof piles will normally require only minor instrumentation. The pile shouldbe installed at a representative soil profile in the same manner as is planned for theproduction piles.

8.1.4 F e a t u r e s u n i q u e t o t e s t i n g o f p i l e s u n d e r

l a t e r a l l o a d i n gA feature special to lateral loading is that the replication of conditions at the head of aproduction pile may not be possible in the field. The production pile may be subjectedto a range of lateral loads, axial loads, and bending moments, some of which may benegative in the usual sense. Further, while deflections may be measured with accuracy,failure is more often dependent on the value of the maximum bending moment.

If the production piles are to be subjected to purely axial load, a test pile of thesame dimensions and penetration can be installed in a representative soil profile andsubjected to some mu ltiple of the design loa d. If the axial settlem ent is less tha t allowe din specifications, the pile is judged to be adequate for the proposed structure, assuming that pile-soil-pile interaction is taken properly into account. Thus, a proof testof an axially loaded pile may provide sufficient information for design and the specific evaluation of the load transferred in side resistance and end bearing need notbe made.

Because a test pile subjected to lateral loading cannot practically be loaded withan axial load and a moment (usually negative if the pile head is restrained againstro tat ion ), the perform ance of a proo f test as for axial load ing is no t feasible. The refore,the lateral-load test must be aimed at gathering data on the response of the soil.

The test pile is not required to be exactly like a production pile. A desirable plan

is to use data on soil and on the test pile and to make a prediction of the lateralload versus deflection, using a computer program. The prediction would reveal thelateral load at which a plastic hinge would develop and allow the engineer to selectappropriate increments of load. The proof test for lateral loading consists of comparingthe experimental values of deflection with predicted values. Differences in the valuesare assumed to be due to properties of the soil, because the other parameters in theprediction method can be controlled. The properties of the soil can be modified tobring agreement in the measured and computed deflections and those properties canbe used to design the production piles. An example of the testing of a proof pile isgiven later. A detailed example is also presented of a test of a research pile to obtain

p y curves.It is of interest to note that the so-called continuum effect, where the resp onse of

the soil at an element is influenced by the response at all other elements, is explicitlysatisfied during field testing. That is, even though the response of the soil is presented

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Te s t i n g o f f u l l s i z e d p i l e s 335

at discrete locations by p y curves, the experimental p y curves correctly reflect theresponse of the soil as a continuum or a continuous body.

8.2 D E S I G N I N G T H E T E S T P R O G R A M

8.2.1 P l a n n i n g f o r t h e t e s t i n g

The planning for the test of a proof pile may be minimal. For example, the senior writerand his colleagues tested two bored piles that were installed to be production piles butconstruction was delayed. The test consisted of placing a strut between the two piles,including a load cell and hydraulic ram in series with the strut, and applying the loadin increments. Deflection and rotation were measured at each pile head. Predictionsof pile-head deflections could be made because soil properties were determined at thelocation of each of the piles. The Owner required that the piles not be damaged soloading was discontinued when prediction showed the bending moment to be wellbelow the computed value of ultimate moment. The piles had been installed in softrock and the results reveal two important facts: (1) the difference that can be expectedexperim entally w hen piles are installed nea rby in the same soil; and (2) the ability of thecurrent recommendations to predict the early portion of the p y curves for soft rock.

The performance of a test of a research pile, and the testing of some proof piles,requires a major effort and involves the participation of a number of specialists. Thefollowing pages present a number of steps that must be completed successfully. Theplan ning is critical because a misfo rtune at the site can render useless the whole effort.

For example, the senior writer was told of a case where the inadvertent application ofa very large load by an automatic loading system spoiled an entire test.The recognition of the factors involved in a test of large bored piles and suggestions

for testing were presented by Franke (1973). In many test programs, an importantconsideration is the satisfaction of the requirements of a standards association. Somesuch standards are referenced later but certain provisions may not be applicable whenthe response of the pile is to be analyzed by the p y method, and especially when aresearch-oriented program in undertaken.

8 2 2 S e l e c t i o n o f t e s t p i l e a n d t e s t s i t e

The site selection is simplified if a test is to be performed in connection with the designof a particular structure. However, even in such a case, care should be taken in theselection of the precise location of a test pile. In general, the test location should bewhere the soil profile reveals the weakest condition. In evaluating a soil profile, thesoils from the ground surface to a depth equal to five to ten pile diameters are ofprincipal importance.

The selection of the site where a fully instrumented pile is to be tested for researchpurposes is usually difficult. The principal aim of such a test is to obtain experimentalp y curves that can be employed in developing predictions of response of soil in a

well defined subsurface. Thus, the soil at the site must be relatively homogeneous andrepresentative of a soil type for which predictive equations are needed.

After a site has been selected, attention must be given to the moisture content of thenear-surface soils. If cohesive soils exist at the site and are partially saturated, steps

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Te s t i n g o f f u l l - s i z e d p il e s 3 37

The subsurface investigation for axially loaded piles usually extends to a depth of1.2 times the length of the pile. The sam e rule of thu m b can be app lied for the testing oflaterally loaded piles; however, the soils near the ground surface dominate the responseof the pile. Other principles that apply to the investigation of the soil when piles are

tested under lateral load are: (1) for long and short piles in soil, the properties ofthe soil of the zone five to ten diameters below the ground surface is of predominantimportance; (2) for short piles in soil, characteristics of the soil at the tip of the pilemust be determined to allow the formulation of a curve showing lateral load versusdeflection at the tip of the pile; (3) for both long and short piles in rock, whatever thedepth of the rock, the stiffness of the rock is im po rtan t; an d (4) the correlatio ns th at a reproposed for obtaining p y curves from soil properties must be based on proceduresthat are available to practicing engineers.

The first set of procedures noted above is necessary in allowing a definition of thevario us strata in the found ing zo ne and in allow ing the acquisition of data for classification . Dat a can also be obt ained on the undr ain ed stren gth an d stiffness of cohesive soils.However, the difficulties in obtaining undisturbed samples are substantial. Hvorslev(1949) did a classic study on procedures for exploration and sampling and presenteddetailed information on disturbance due to sampling of cohesive soils. Photographsof specimens of cohesive soil showed distortion due to resistance against the walls ofthe sampler. Such resistance was reduced or eliminated by using a sampler with metalfoils (Kjellman, et al., 1950). However, even with minimum disturbance due to thefriction of sampling, disturbance due to the changes in the state of stress due to sampling and to the removal of the specimen cannot normally be eliminated. With respectto accounting for the effects of sampling disturbance, Ladd & Foott (1974) proposeda laboratory testing procedure for obtaining the undrained strength of soft, saturatedclays.

Even though advances have been made in sampling and laboratory testing of cohesive soils, ma ny investigation s are carried ou t with sta nd ar d techn ique s. Inevitably,some subsurface investigations of cohesive soils used the standard, readily availabletechniques where p y curves were correlated with soil properties, and future experiments may be performed in a like manner. Therefore, the engineer must be carefulin studying the soil test that were performed to obtain p y curves with respect tothe methods used to obtain the undrained strength of clays. Techniques employed at

sites where designs are to be made must take into account methods used at the testsites.Subsurface investigation by the use of in situ tests has received intensive stud y an d is

growing in popularity. A detailed discussion of the various tools and their application isgiven in papers and repo rts by numero us au thor s in technical journals and proceedingsof conferences and seminars (Stokoe, et al., 1988 & 1994). A summary of the variousmethods and their application for Belgium in the design of axially loaded piles is givenby Holeyman, et al. (1997).

Some of the methods are mentioned here and discussed briefly. The cone penetrom-eter test (CPT) was developed in Europe and is used worldwide to gain information on

foundation soils, in particular. Because of the extensive use of cone testing in Europeand its increasing use elsewhere in the world, some details are presented here onthat testing technique. The CPT is performed with a cylindrical penetrometer witha conical tip, or cone (see Fig. 8.1), and is pushed into the ground at the end of a

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338 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 8.1 Piezocone-CPTU probe.

series of push ro ds at a con stan t rate of 20 mm /sec. A load of up to 2 50 kN can beapplied at the top of the push rods with a hydraulic ram reacting against a specialtruck or against an anchor. Forces on the cone and the friction sleeve are measuredduring penetration by internal load sensors. Not shown in Fig. 8.1 is that an inclinometer can be placed inside the device to measure any tilt at the time of performinga test.

The penetrometer for performing the cone test may be fitted with devices at oneor several positions along its length for measurement of pore water pressure and istermed the piezocone. Cone tests with pore pressure measurements are designatedCPTU. Guidelines for the cone and other penetrometers were given by the ISSMGETechnical Committee on Penetration Testing (Information 7, Swedish GeotechnicalInstitute, 1989).

The results from cone testing can be interpreted, in principle, to obtain the followinginformation:

stratification,soil type,

soil density,mechanical soil properties,

shear-strength parameters,deformation and consolidation characteristics.

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Te s t i n g o f f u l l s i z e d p i l e s 339

Later in this chapter, whe re the testing of a research pile is described, details on conetesting are presented where the cone was employed in obtainin g changes in prop ertiesof sand as the result of pile installation and pile testing.

The pressuremeter test (PMT) is performed by measuring the pressure and deformation as a membrane is expanded against the walls of a borehole. Various methods offoundation design have been based on the results of such tests. The self-boring pressuremeter was designed to eliminate the disturbance due to the relaxation of the soilin the borehole and is used in special circumstances.

The dilatometer test (DMT) is performed by inserting a metal wedge with a membrane built into its side. The membrane is caused to move laterally against the soilwhile force and deflection are measured. The in situ lateral stress in the soil can bemeasured and the stiffness of the soil at low strains can be measured.

The Standard Penetration Test (SPT) was developed in the United States and is usedthere and elsewhere to obtain information on the strength of cohesionless soils. Athick-walled sampler is driven in the soil at the base of a borehole, seated by blows toa depth of 150 mm , and the blows are counted to drive the tool 300 mm , and designatedthe N-value. Correlations have been made between the N-value and characteristics ofgranular soil. The disturbed soil in the thick-walled sampling tube can be used to getmoisture contents and classify the soil. The results from the SPT can vary widely withthe techniques employed in the testing; thus, the metho d is used with some cautio n bymany practicing engineers.

A combination of two procedures, boring and laboratory testing and in situ testing,is required for the testing of research piles. The particular tests to be performed willdepend on site-specific conditions and in consideration of methods generally availableto correlate with p y curves that are developed. Shear-strength characteristics for cohesive soils can be determined from high quality sampling and laboratory testing. Fornon-cohesive soils, however, and even for cohesive materials, the relevant small-strainparameters for the soil are found with difficulty from laboratory testing and, thus,modern in situ testing finds an important role. The methods of investigation for thetesting of proof piles should be correlated with methods proposed for the design of theproduction piles.

The installation of sensors to measure the changes in stress in the soil due to piledriving and to lateral loading is highly desirable when research piles are to be tested.

Such sensors are readily available and can be installed in the walls of the pile as well asin the surrounding soil. The increase and decay of pore-water pressure in fine-grainedsoils due to pile driving and due to the subsequent lateral loading provides valuableinformation related to the response of the soil.

8.4 I N S T A L L A T I O N O F T E S T P IL E

For cases wh ere info rma tion is requ ired o n pile respon se at a part icula r site, the installation of the research pile should agree as closely as possible to the procedure proposed

for the production piles. It is well known that the response of a pile to a load isaffected considerably by the installation procedure; thus, the detailed procedure usedfor pile installation, including gathering of relevant data on pile driving or on relatedprocedures, is of utmost importance.

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340 Single Piles and Pile G rou ps Un de r Late ral L oad ing

With regard to the effects of installation, soil stresses from the installation of pilescan be studies by use of the dilatometer. Appendix G presents data from Van Impe(1991) on horizontal soil stresses near piles during installation.

For the case of a test pile in cohesive soil, the placing of the pile can cause excessporewater pressures to occur. As a rule, these porewater pressures should be dissipatedbefore testing begins; therefore , the use of piezometers at the test site may be im po rta nt.The use of the cone penetrometer with pore-pressure measurement can be considered.The cone can be left in place during some phases of the installation and the loading ofthe test pile.

The installation of a pile that has been fully instrumented for the measurement ofbending moment along the length of the pile must consider the possible damage of theinstrumentation due to pile driving or other installation effects. Pre-boring or somesimilar procedure may be useful. However, the installation must be such that it isconsistent with methods used in practice. In no case would water jetting be allowed.

It would be desirable to know how the installation procedures had influenced thesoil properties at the test site. The use of almost any sampling technique would causesoil disturbance and would be undesirable. The use of non-intrusive methods may behelpful. Van Impe & Peiffer (1997) described the use of the dilatometer in obtainingthe effects of pile installation. Testing of the near-surface soils close to the pile wall atthe com pletion of the load test is essential and can be don e with ou t un desirab le effects.

8.5 T E S T I N G T E C H N I Q U E S

Excellent guidance for the procedures for testing a pile under a lateral loading is givenby the Stand ard D 39 66 , Stan dard M etho d of Testing Piles Under Lateral Loa ds, ofthe American Society of Testing and M aterials (ASTM). Eurocode 7, Pile Foun datio ns,also presents information on testing. In respect to standards, Bergfelder and Schmidt(1989) discuss testing to comply with the requirements of the German Standard DIN4014 (1989). Recent recommendations (Holeyman, et al., 1997; and Van Impe &Peiffer, 1997) about procedures and principles for horizontal loading show agreementbetween Eurocode 7 and DIN 4020, 4054, and 4014.

For the standard test as well as for the instrumented test, two principles should guide

the testing procedure: (1) the loading (static, repeated with or without load reversal,sustained, or dynamic) shou ld be consistent with tha t expected for the prod uctio n pilesand (2) the testing arrangement should allow deflection, rotation, bending moment,and shear at the groundline (or at the point of load application) to be measured or ableto be computed.

With regard to loading, even though static (short-term) loading is seldom encountered in practice , the respon se from th at load ing is usually desirable so tha t c orrela tion scan be made with soil properties. As noted later, it is frequently desirable to combinestatic and repeated loading. A load can be applied, readings taken, and the same loadcan be re-applied a number of times with readings taken after specific numbers of

cycles. Then, a larger load is applied and the procedure repeated. The assumption ismade that the readings for the first application at a larger load are unaffected by therepetitions of a smaller load. W hile that im por tant assum ption may no t be strictly true ,errors are on the conservative side.

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Te s t i n g o f f u l l s i z e d p i l e s 341

As noted earlier, sustained loads will probably have little influence on the behaviorof granular materials or on over-consolidated clays if the factor of safety is two orlarger (soil stresses are well below ultimate). If a pile is installed in soft, inorganic clayor other compressible soil, sustained loading will obviously influence the soil response.

However, loads would probably have to be maintained a long period of time and aspecial testing program would have to be designed.

The application of a dynamic load to a single pile is feasible and desirable if theproduction piles sustain such loads. The loading equipment and instrumentation forsuch a testing prog ram w ou ld have to be designed to yield results tha t wou ld be relevantto a particular application and a special study would be required. The design of pilesto withstand the effects of an earthquake involves several levels of computation. Soil-respon se curves mu st include an inertia effect an d the free-field m otio n of the earth mu stbe estimated. Therefore, p y curves that are determined from the tests described hereinhave only a limited application to the earthquake problem. No method is currentlyavailable for performing field tests of piles to gain information on soil response thatcan be used directly in design of piles to sustain seismic loadings.

The testing of battered piles is mentioned in ASTM D 3966. The analysis of a pilegroup, some of which are batter piles, is discussed in Chapter 5. As shown in thatchapter, information is required on the behavior of battered piles under a load that isnormal to the axis of the pile. An approximate solution for the difference in responseof battered piles and vertical piles is presented in Chapter 3. Figure 3.30 may be usedto modify p y curves as a function of the direction and amount of the batter.

The testing of pile groups, also mentioned in ASTM D 3966, is desirable but isexpensive in time, material, and instrumentation. If a large-scale test of a group ofpiles is proposed, detailed analysis should precede the design of the test in order thatmeasurements can be made that will provide critical information. Such analysis mayreveal the desirability of internal instrum entation to measure bending m om ent in eachof the piles.

It is noted in ASTM D 3966 that the analysis of test results is not covered. The argume nt can be mad e, as presented earlier in this chapter, tha t test results can fail to revealcritical information unless combined with analytical me thod s. A subsequent section ofthis chapter suggests procedures that dem onstrate the close connection between testingand analysis. A testing progra m should not be initiated unless preceded an d followed

by analytical studies.

8.6 L O A D I N G A R R A N G E M E N T S A N D I N S T R U M E N TA T I O NA T T H E P IL E H E A D

8.6.1 L o a d i n g a r r a n g e m e n t s

A wide variety of arrangements for the test pile and the reaction system are possible.The arrangement to be selected is the one that has the greatest advantage for the partic

ular design. There are some advantages, however, in testing two piles simultaneouslyas shown in Fig. 2 of ASTM D 3966. A reaction system must be supplied and a second pile can supply that need. Furthermore, and more importantly, a comparison ofthe results of the two tests performed simultaneously will give the designer some idea

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34 2 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 8.2 Arran gem ent for testing tw o piles simultaneously under tw o-d ire ctio n lateral loading.

of the natural variations that can be expected in pile performance. It is important tonote, however, that spacing between the two piles should be such that the pile-soil-pileinteraction is minimized.

Drawings of the two-pile arrangements are shown in Figs. 8.2 and 8.3. In bothinstances the head is free to rotat e and th e loads are applied as near the grou nd surfaceas convenient. In both instances, free water should be maintained above the groundsurface if that situation can exist during the life of the structure.

The details of a system where the piles can be shoved apart or pulled together areshown in Fig. 8.2. This two-way loading is important if the production piles can beloaded in this manner. The lateral loading on a pile will be predominantly in onedirection, termed the forward direction here. If the loading is repeated or cyclic, asmaller load in the reverse direction could conceivably cause the soil response to bedifferent than if the load is applied only in the forward direction. As noted earlier, itis important that the shear and moment be known at the groundline; therefore, the

loading arrangement should be designed as shown so that shear only is applied at thepoint of load application.Figure 8.3 shows the details of a second arrangement for testing two piles simul

taneously. In this case, however, the load can be applied only in one direction. Asingle bar of high-strength steel that passes along the diameter of each of the piles isemployed in the arrangement shown in Fig. 8.3a. Two high-strength bars are utilizedin the arrangement shown in Fig. 8.3b. Not shown in the sketches are the means tosupport the ram and load cell that extend horizontally from the pile. Care must betaken in employing the arrangement shown in either Figs. 8.2 or 8.3 to ensure that theloading and measuring systems will be stable under the applied loads.

The photograph in Fig. 8.4 is of a test of two large-diameter bored piles that weretested by the arrangement shown in Fig. 8.3a (Long & Reese 1984).

The m ost convenient w ay to apply the lateral load is to employ a hydraulic pressuredeveloped by an air-operated or electricity-operated hydraulic pump. The capacity of

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Te s t i n g o f f u l l - s i z e d p i le s 343

igur 8.3 Arrangement for testing two piles simultaneously under one-direction lateral loadinga) elevation view, b) plan view.

a ram is computed by multiplying the piston area by the maximum pressure. Somerams, of course, are double acting and can apply a forward or reverse load on the testpile or piles. The preliminary computations should ensure that the ram capacity and

the piston travel are ample.If the rate of loading is important (and it may be if the test is in clay soils beneathwater and erosion at the pile face is im por tant) , the ma xim um rate of flow of the pu m pis important along with the volume required per inch of stroke of the ram. The sealson the pump and on the ram, along with hydraulic lines and connections, must bechecked ahead of time and spare parts should be available.

High pressures in the operating system constitute a safety problem and can causeoperating difficulties. On some projects, the use of an automatic controller for thehydraulic system is justified. A backup control must be available to allow the overrideof the autom atic system in case of malfunction. There was one imp ortan t project w here

the malfunction of the hydraulic system caused a large monetary loss.The system shown in Fig. 8.2 will require that the load cell and the ram be attached

together rigidly and that bearings be placed at the face of each of the piles so that noeccentric loading is applied to the ram or to the load cell. The arrangement shown in

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344 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 8.4 Photograph of testing tw o bore d piles using one -direction lateral loading in California.

Fig. 8.2 may require that the points of application of load be adjustable to prevent theapplic ation of torsio n to one or bot h of the piles. The loadin g system shown in Fig. 8.3awill ensure that no eccentricity will be applied to the load cell and the hydraulic ram. Ifthe two-bar system shown in Fig 8.3b is employed, care will be necessary to maintainstability in the system.

8 . 6 . 2 I n s t r u m e n t a t i o n

The investigator has available a wide variety of instrumentation to be used for mea

surements along the pile and at the pile head. Seco e Pinto & De Sousa Coutinho(1991) describe the use of electrical-resistance strain gauges and a slope indicator formeasurements along the pile. An inclinometer, with a sensitivity of two sexagesimalseconds, was used to measure the rotation of the pile head. While strain gauges on thewall of the pile is the normal instrumentation for measurement of bending moment(Long, et al., 1993), innovative techniques have been developed. Matlock, et al. (1980)devised strain-measuring devices that could be lowered into a pipe pile and locked intoplace by the penetration of sharp-pointed bolts.

The sketches in Figs. 8.2 and 8.3 show how a load cell may be used for the measurement of applied load (shear at the pile head). Also, a knowledge of the point of

application of the load with respect to the groundline will yield the moment at the pilehead. The arrangement shown in Fig. 8.5 provides a concept for obtaining deflectionand rotation of the portion of the pile above the groundline. The device shown inFig. 8.6 has been used successfully for obtaining rotation at the point of application

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igur 8.5 Schematic draw ing of deflection -me asurin g system fo r lateral-load testing of piles.

igur 8.6 Schematic draw ing of device fo r mea suring pile-head rota tio n for latera l load testin g of piles.

of load. Because of the difficulty of applying load exactly at the groundline, analysescan be done more directly by using the data at the point of application of the load andby taking the distance from the load to the groundline into account.

Electronic load cells are available for routine purchase. These cells can be used witha minimum of difficulty and can be tied into a high-speed data-acquisition system ifdesired.

The motion of the pile head can be measured with dial gauges but a more convenientway is to employ electronic gauges. In either case, gauges with sufficient travel shouldbe obtained or difficulty will be encountered during the test program. Two types ofelectronic motion transducers are in common use: linear potentiometers, and LVDT's

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346 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur 8.7 Plan and elevation views of the to p o f a pile showing placement of instrume ntation forlateral-load test.

(linear variable differential transformers); in either case the motion transducer shouldbe attached so that there is no binding as the motion rod moves in and out.

An other possible arrangeme nt for the measu rement of pile-head deflection a nd rotation, as discussed in the European work-group recommendations of September, 1997(Chr. Baudouin, 199 7), is show n in Fig. 8.7. The recom men dations of the wo rk gro up ,with language modified slightly to agree with terminology herein, are given in thefollowing paragraphs.

The supports of the girder, where instruments for the measurement of deflection

and rot ation of the pile head are attached, mu st be separated a minim um distanceof 1.5 times the pile diameter, and the piles must be a clear distance apart of4.0 times the pile diameter. In case the pile-head movements are registered by anelectronic dial gauge, it is mandatory to control the measurements by mechanical

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Te s t i n g o f f u l l - s i z e d p il e s 3 47

or optical measuring devices. The precision of measurement should be 0.01 mmfor electronic dial gauges and 0.10 mm for optical measuring devices. In principle,these measurem ents should be mad e by two totally indepen dent m etho ds. At least,the reference points for the measurement of the pile-head movements must be

controlled by an independen t meth od before beginning and after com pletion of thetest and while the ma xim um load is acting. For pile-head rota tion s, a m easuremen trange of 1.0 degree and an accuracy of 0.05% is adequate.

The deflection curve can be measured by an inclinometer. However, these testsrequire additional time and extend the test period.

The bending moments can be determined by measuring the strains of the pileshaft both in the compressive zone and in the tensile zone.

For measurement of strains at specific points, electric strain gauges are suitable which can be directly attached to the steel pile or the pile reinforcement, orin the case of cast-in-place concrete piles, they are attached to special measurement casings which are located inside the reinforcement cage. Through pairwisearrangement and proper staggering of the measurement sections, pile deformations and their distribution can be determined sufficiently accurately in pile shaftswith constant cross-section and constant modulus of elasticity.

In case of cast-in-place con crete piles, the cross-section as well as the mod ulu s ofelasticity vary frequently along the d epth w ithin a definite b and w idt h. In this case,representative results can be obtained from integral measurement elements, whichmeasure compressive strains for pile sections at about 1.0 m length. For example,strain gauges are arranged inside measurement casings which are attached only atthe ends of the measurement sections, and otherwise are not connected to the pile.Longer measurement sections are not advisable considering the variation of thebending moment with depth.

Measurements with micrometers are relatively expensive because of the necessary preparations and the procedure of measurements which also requires accessabove the pile head during the test loading. Measurements after completion ofconstruction are not possible for foundation piles with this method.

For the evaluatio n of effective stresses and bend ing m om ents from the pile deformations, it is important that pile cross-section and deformation characteristics ofthe pile material are known accurately. The uniformity of the concrete quality can

be controlled by ultrasonic testing.In the case of cast-in-place concrete piles, the analysis will be complicated asa result of deviations from the theoretical cross-section due to bulb formation,variability of concrete quality or from cracking of concrete in the tension zone.

Decisions have to be made in each case on the method for which realistic valuesof flexural stiffness can be obtained. The uncertainties of these analyses requirelimit considerations, which primarily will affect the results more than the otherfactors.

Two final comments about the instrumentation are important. The verification ofthe output of each instrument should be an important step in the testing program.Also, the instruments should be checked for temperature sensitivity. In some cases itmay be necessary to perform tests at night or to protect the various instruments fromall but minor changes in temperature.

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8.7 T E S T I N G F O R D E S I G N O F P R O D U C T I O N P IL E S

8 . 7 . 1 I n t r o d u c t i o n

With regard to the design of production piles, three courses of action are dictated,depen ding on th e num ber of piles to be installed. (1) If the num ber of piles is small an dthe soil profile is similar to one of those for which criteria are available, the designermay proceed with no testing and with the selection of an appropriate factor of safety.(2) If a large number of piles are to be installed and particularly if the soil profile isunusual, the designer may wish to test one or more instrumented piles, as describedin the a section that follows. (3) In many cases, the designer may elect to test a pilethat is essentially uninstrumented. The piles may or may not become a part of thefoundation system after testing. The magnitude of the lateral load is relatively modest,compared to the axial load; therefore, the reaction may be accomplished by a simple

arrangement. A convenient solution is to install two identical piles and to test themsimultaneously.Special problem s with ins trum entation are encountered w hen the piles in an offshore

structure are to be instrumen ted (Kenley & S harp, 1 993 ). Vibrating-wire strain gaugesand electrical-resistance strain gauges were installed at a few points near the tops ofthe piles. The bending moments that were measured served to confirm the methodsemployed to predict the response of the piles based on published p y curves.

8 7 2 In ter pr e ta t io n of data

The interpretation of data from a test of an uninstrumented pile is a straightforwardprocess. The pile of piles will be tested in the free-head condition with the loadingapplied close to the groundline. No attempt is required to produce a loading to agreewith that to be encountered by the production piles. Plots are made for the point ofload application of deflection versus applied load and pile-head rotation.

A compu ter code is then used and com puta tions are mad e of deflection and rotatio nfor the same loads that were applied experimentally. A convenient procedure is tohave a computer in the field and to make the computations simultaneously with theapplication of the experimental load. If the two sets of data do not agree, the propertiesof the soil are changed in the computations to reach agreement. Usually, only the shearstrength needs to be changed; undrained shear strength for clay and the friction anglefor granular soil. The procedure is repeated for each load and average values of shearstrength can be selected. The modified values of shear strength can then be used todesign the produ ction piles.

8.7.3 E x a m p l e C o m p u t a t i o n

The test selected for study was performed by Capozzoli (1968) near St. Gabriel,Louisiana. The pile and soil properties are shown in Fig. 8.8. The loading was short

term. The soil at the site was a soft to medium, intact, silty clay. The natural moisturecon tent of the soil varied from 35 to 46 percent in the uppe r 3 m of soil. The un dra ine dshear strength, shown in Fig. 8.8, was obtained from triaxial tests. The unit weight ofthe soil was 17.3 kN /m3 above the water table and 7.5 kN/m3 below the water table.

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igur 8.8 Infor ma tion fo r analysis of results at St. Ga briel.

igur 8.9 Com parison of experimen tal and com puted values of pile-head deflection fo r St. Gab rieltest.

The results from the field experiment and computed results are shown in Fig. 8.9.The experimental results are shown by the open circles; the results from a computercode with the reported shear strength of 28.7kPa and with an £5 of 0.01 are shown bythe solid line. The soil properties w ere varied by trial and th e best fit to the expe rime ntal

results wa s found for an und rain ed shear stren gth of 42.5 kPa and an sso of 0.009.These values of the modified soil properties can be very useful in design computationsfor the pro du ctio n piles if the prod uc tion piles are to be identical with the one em ployedin the load test.

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A computer code was employed and an ultimate bending moment for the sectionthat is shown was computed to be 157kN-m. In making the design computationswith the modified soil properties, the computed maximum bending moment should beno greater than the ultimate moment (157kN-m) divided by an appropriate factor of

safety. In com puting the max imu m bending mo men t, the rotatio nal restraint at the pilehead must be estimated as accurately as possible. If it is assumed that the pile will beunrestrained against rotation and that the load is applied one ft above the groundline,a load of 93.4 kN will cause the ultimate bending moment to develop. The deflectionof the pile must be considered because deflection can control some designs rather thanthe design being controlled by the bending resistance of the section.

Two other factors must also be considered in design. These are: the nature of theloading and spacing of the piles. The experiment employed short-term loading; if theloading on the production piles is to be different, an appropriate adjustment must bemad e in the p y curves. Also, if the prod uc tion piles are to be in a closely spaced g ro up ,consideration must be given to pile-soil-pile interaction.

8.8 E X A M P L E O F T E S T I N G A R E S E A R C H P I L EF O R p y C U RV E S

8 . 8 . 1 I n t r o d u c t i o n

The performance of tests to obtain experimental p y curves requires planning andthe selection of numerous details. Each such test will be unique, of course, and theestablishment of general procedures is not possible. However, an example is presentedto illustrate the methods employed in a particular case. Tests were performed, usingpipe piles with diam eter of 610 mm , to o btain p y curves in sand at Mustang Island(Cox, et al., 1974). The tests were successful and the details presented in the followingparagraphs should be instructive in the planning and performance of similar tests.

The tests were comm issioned by the petroleum industry w ith particular reference topile supporting offshore structures. An obvious requirement was that the water tablewas above the ground surface.

Two piles were tested, Pile 1 under static loads and Pile 2 under cyclic loads. Theresults from the loading of Pile 1 could be correlated with the properties of the soilwithout any reduction in soil resistance due to the effects of cycling. The loading ofPile 2 simulated that to be expected from wave loadings on an offshore platform. Themajor load was applied by compression in the system for loading and measurement.Then, for each cycle a minor load of about 25% of the major load was applied bytension in the system for loading and measurement. Loading in the opposite directionto the major load simulated the back pressures from a wave as the crest passed throughthe platform.

8.8.2 P r e p a r a t i o n o f t e s t p i l e s

Each test pile consisted of a 11.58-m unin strum ented section, a 9.75 m instrum entedsection, and a 3.05 m uninstrumented section as shown in Fig. 8.10. All pile sectionswere 61 0 mm in diameter with a 9.53 mm wall thickness. The wall thickness and the

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T e s t i n g o f f u l l s i z e d p i l e s 351

Figu re 8.10 Sketch of ar ran gem ent for t es t ing p iles un der l a tera l loading a t Mustang Is land .

lengths of the three different sections were studied by (1) estimating the p y curves

that were expected, and (2) the performance of num erous com puter ru ns to obtain theexpected deflections and bending moments. Such preliminary steps are critical to thesuccess of similar experiments in other soils.

Conn ecting flanges, 914 x 508 x 38 mm , were welded to the instrum ented sectionand to the 3.05-m section. During driving and testing, the 3.05-m section was boltedto the 9.75-m section at the flange by seven 25-mm diameter bolts.

Twenty-five-mm long pieces of 38 x 38 x 3.2-mm angles were welded on 305-mmcenters along the inside of the 9.75-m instrumented section. These angles supportedsteel straps to which the strain-gauge cables were clamped. Electrical resistance straingauges were selected for use in determining bending moment along the piles.

Just below the flange on the 9.75-m section an annular ring was welded inside thepile to which a pressure plate was bolted. A rubber gasket of 3.2-mm thickness wasplaced between the ring and pressure plate. Strain-gauge cables were broug ht throu gh0-ring packing-nuts screwed into the pressure plate.

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35 2 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

To install the strain gauges, technicians could slide into the horizontal pipesection while lying face down on a specially made crawler supported by rollers.After installation of strain gauges, a 38-mm thick diaph ragm was welded 152 mmabove the bottom of the 9.75-m section. The bottom diaphragm and top pressure-

plate seal prevented moisture from entering the 9.75-m instrumented section. Excessmoisture in the piles could cause damage to the strain gauges.

A 76-m m d iameter relief pipe was installed in each of the 9.75-m sections after installation of strain gauges and cables. This relief pipe extended through the diaphragm inthe bottom, through the top pressure plate and out the side of the 3.05-m section, asshown in Fig. 8.10. The purpose of this pipe was to relieve water pressure created atthe bottom diaphragm of the 9.75-m closed section during pile driving. Plans calledfor removal of the relief pipe after pile driving.

Also shown in Fig. 8.10 are parts of the loading and in strum entation systems, whichwill be discussed later.

Th e decision was ma de to lo ad the pile to as high a stress as possible; therefore , tensiletests were perfo rmed on specimen s cut from the test piles. Analysis of the results of thetesting led to the selection of 186 MPa as the highest stress that should be allowed.

8.8.3 Te s t s e t u p a n d l o a d i n g e q u i p m e n t

8 8 3 Description of test setup

A drawing of the two test piles and related equipment is shown in Fig. 8.10. The load

cell and hydraulic ram were placed in series between the reaction frame and the pilebeing tested. Loads to the free-head piles were applied at the connecting flange betweenthe 3.05 - and 9 .75-m sections. The connecting flange w as located 0.30 m above themudline. LVDT's were used to measure the pile deflection at two points along theunstrained 3.05-m section. Micro-switches, which would shut off the hydraulic flowto the ram when activated, were placed on either side of the piles to prevent accidentaloverloading.

8 8 3 2 Hydraulic equipment

The hydraulic equipment was manually operated by controls mounted in a consoleinside a porta ble bu ilding. Loads were applied to the pile by a 305-m m s troke, d ouble-acting ram th at h ad a capacity of 334 kN in compression an d 2 78 kN in tension.Hyd raulic fluid w as supplied to the controls by two 5.1-horsepow er pum ps each havinga maximum flow rate of 0.7m3 per hour at a maximum pressure of 186 MPa.

The tw o electrically-driven h ydraulic pum ps were connected together so that one orboth pumps could be used. A cooling system in the pump reservoirs kept the oil fromoverheating.

The drawing in Fig. 8.11 shows the details of the hydraulic system and its controls.A solenoid-operated four-way valve was used to change the direction of loading of the

double-acting ram. Pressure to the ram, and therefore applied load, was regulated byrelief valves m ou nte d in the con trol co nsole. Th e relief valves had very fine adjustm entswhich resulted in a stable load application for either static or cyclic tests. In addition,there were relief valves on the pump reservoirs. These pressure-relief valves not only

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354 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

13 mm . The nom inal gauge resistance was 120 ohms ± 0 .5 percent and the gage factorwas 2.02.

The gauges were bonded to the pile with a two-part epoxy that maintains bondunder dynamic strains. Immediately after laying a gauge with the epoxy, a weight was

placed on the gauge which resulted in a norm al pressu re of 34 kPa . After on e ho urof curing at roo m te mp erature , heat lamps w ere used to heat the gauge area to 400 F.The epoxy was cured for two hours at this temperature and the weight was removed.To prevent absorptio n of moisture, the gauges were then coated with a synthetic-resinwaterproofing compound.

After so ldering sho rt pigtails to the terminal str ip, the gauge installation w as checkedby measuring its resistance to ground and by testing the bond between the gauge andthe pile. A volt-ohmmeter was used to check resistance to ground and continuity. Allgages checked 1000 megohms or higher with 500 megohms considered a minimumacceptable value. By wiring the gauge as one arm of a Wheatstone bridge, the gaugebond was checked by rolling with light pressure a rubber eraser across the gauge gridand noting the resulting strain and zero stability on a strain indicator. Gauges thatshowed high strains or instability due to these pressure tests were not securely bondedand were removed and replaced.

After check-out of the gauges, a second coat of waterproofing compound, a two-part epoxy, was applied. Final waterproofing and mechanical protection was ensuredby covering the entire gauge installation with a 3-mm thick neoprene pad bonded tothe pile surface with rubber to metal cement.

8 8 4 3 Installation of strain gaug e cables

Cables were installed inside the piles by clam ping th em to brack ets placed o n 0.3 mintervals, as shown in Figs 8.12a and 8.12b. The cable used was an 8-conductor wire,covered by shielding and by a tough neoprene-rubber exterior. Each cable carried output from four strain gauges, resulting in a total of five cables along opposite diametersof the piles. A total of 33 0 m of wire was installed in both piles. In order t o reach fromthe pile to the instrument van, each cable extended 12 m beyo nd the to p pressure plate.

With long lead wires, problems are frequently encountered with temperature effectsand instability of the strain-gauge signal. To reduce the effects of temperature changes,

cables of equal length were installed to diametrically opposed gages. Theoretically,with cables of equal length which are wired into adjacent arms of a Wheatstone bridge,changes in cable resistance due to temperature effects are canceled.

Long lead-wire effects which could cause attenuation of the strain-gauge signalswere also considered. Diametrically opposed gauges were wired as adjacent arms of aW heatsto ne bridge in a half-bridge arran gem ent as wo uld be done during field testing.Each gauge was then sh unted separately first with a 15 0,00 0 ohm and then w ith a60,4 00 ohm resistor. The appa rent strains due to these shunts were measured w ith astrain indicator. After the long lead wires were attached, the same process of shuntingthe gages was repeated. The difference in apparent strains for the two conditions was

the amount of attenuation caused by the long lead wires. The average decrease inapparent strain after adding the cables was about 1 percent of the strain measuredwithout the cables and, therefore, was not considered a problem in the recording ofoutput from the gauges.

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Te s t i n g o f f u l l - s i z e d p i le s 355

igur 8.12 View of inside of pile installation of strain-gauge cables for tests of piles at MustangIsland.

8 8 4 4 Recording equipm ent for strain g auges

The selection of a data-acquisition system was given careful consideration. Effects ofchanges in pile strains due to possible creep of the soil and recording of data duringcyclic loading led to the consideration of an instantaneo us reco rding instrum ent. Suchan instrument was rejected, however, because of its low precision and the involved

process for data reduction. The decision was made to use a digital system that featuredhigh recording speed, high precision, and convenient record of data.A 20-channel digital-data-acquisition system was used for recording the output of

the strain gages, deflection gages, and the load cell. The equipment was mounted invertical racks inside the instrument van. The equipment scanned all 20 channels ofinformation and printed the strain data on paper tape in units of micro-m per m. Thetime for the system to balance and print was from 0.4 to 1.5 seconds per channel,depending upon the range change or variation between readings. During lateral-loadtests, the system was clocked at approxim ately 17 seconds for scanning and recordingall 20 channels of data.

Resolution of the system was to the nearest microstrain and the quoted accuracywas 0.1 percent. However, when the range of the system had to be changed, i.e., forstrains greater than 999 micro-m per m, a multiplier had to be used which decreasedthe resolution.

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8 8 4 5 Circuits

Strain gauges in the pile were wired into conventional Wheatstone-bridge circuits.Diametrically opposed strain gauges in the piles were wired as adjacent arms of a

bridge that was completed with two strain-gauge dummies inside the digital-strainindicator. This bridge arrangement has the advantage that measured strains are twicethe actual pile strains and temperature effects are compensating. It should be notedthat each gauge was wired separately so that if one gauge became inoperative, oneof the six unstrained dummies could be used as a substitute and data could still beobtained from that particular location.

The LVDT's for measuring deflection were also used in a bridge arrangement sothat data on voltage ou tpu t could be obtained w ith a recording system. The deflectiongauges were wired into one arm of an external half-bridge arrange men t.

8 8 4 6 Reference bridge

Due to the expected long duration of tests, an unstrained reference bridge was usedto check possible zero-drift of the balancing and recording equipment. Two straingauges out of the same lot used in the piles were mounted on a piece of steel whichwas positioned inside a protective box . A precision resistor and ten-tu rn poten tiom eterwas wired in to the half-bridge circuit with the unstraine d g auges. Two separate bridgeswere made up in case one became defective.

The reference bridge was used in the following manner to check instrument drift.After all data channels had been null-balanced before a test and the balance controls

locked in place, the input channels were disconnected. Then the reference bridge wasplugged into the switch-and-balance unit and the ten-turn potentiometer was turneduntil the circuit was null-balanced and the number on the dial of the potentiometerwa s recorde d. By repea ting this for each channel the drift in the instru me nt cou ld bechecked by disconnecting any input channel at any time and substituting the referencebridge with the proper potentiometer setting for that channel. If the same setting onthe potentiometer did not produce balance of the bridge, then the instrument haddrifted or the balance-control knob for the channel had inadvertently been turned.The balance-control knob then could be used to return to the original datum.

Stability and repeatability of the reference bridge were verified by checks made in

the laboratory and therefore a drift in the reference bridge during the above procedureswas not considered probable.

8 8 4 7 Measurement of load and deflection

Load s were measure d in the calibration and field tests by a universal load cell of 44 5 kNcapacity. Accuracy of the load cell was quoted by the manufacturer at 0.25 percentof the full-scale range of 4000 micro-m per m. The accuracy and the manufacturer'scalibration constant were checked on a 534-kN testing machine. The load cell had afull four-arm bridge made up of 120-ohm strain gauges.

Deflections during the field tests were measured at two points above the connectingflange on the unstrained 3.05 m section. The gauges used were LVDT's with 150-mmstrokes capable of measuring displacements to 2.5 x 1 0- 5 m m . For these tests, however,the resolution was reduced to 0.025 mm. Because the transducers could not be placed

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Chapter 9

I m p l e m e n t a t i o n o f f a c t o r s o f s af et y

9 . 1 I N T R O D U C T I O N

Paramount in the mind of the engineer is to make a safe design; secondly, and withnearly as much importance, is to have the design perform well without excessive cost.The following sections will direct attention to the many of factors that enter into theselection of pile dimensions. The engineer may be guided by codes and standards of abuilding authority; even so, the engineer is left with considerable latitude. For example,in evaluating the data from a subsurface investigation, a very conservative or a lessconservative set of values could be selected.

The global approach, where an overall factor is selected, is the usual method for theselection of a safety factor. A more recent method, the component app roac h, is where

separate factors are given to various properties required for the design. Both methodswill be discussed in some detail. However, a number of topics will first be discussedthat are common to both methods.

M od ern appro aches to the design of pile found ations emphasize deformation as wellas ultim ate capacity. Deform ation of a pile or a gro up of piles is an im po rtan t aspect ofsome of the methods presented, which parallels to some extent the procedures in thisbook. Examples are presented where successive solutions are made by incrementing theloading, nonlinear load transfer mechanisms are employed, and sometimes nonlinearpile m ateria l, and the load is found tha t causes collapse or excessive deform ation . Sucha procedure is valid, regardless of the method employed to select safety coefficients.

9.2 L IM IT STA TES

The design concept using limit states introduced in the early 1960's, is aimed at implementing mo re rationality regard ing safety and is characterized principally by emp hasison allowable deformation. Thus, the concept of limit states broadened the view ofthe appropriate response of a structure to loading and to the environment. Table 9.1is presented in consideration of the ways a pile under loading may fail to perform.The limit states in the table are related to piles under both lateral and axial load

ing, because in several respects there is a close relationship between the two types ofbehavior. Chapter 5 demonstrates, for example, that a group of piles under inclinedand eccentric loading cannot be designed properly without considering the responseunder both lateral and axial loading.

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able 9.1 Lim it states fo r a pile subjected to lateral and axial loading*.

Ultimate limit states Most probable conditions

Sudden punching failure un der axial loadingof individual piles

Progressive failure under axial loadingof individual piles

Failure under lateral loading ofindividual piles

Structural failure of individual piles

Sudden failure of foundation of structure

Pile bearing on thin stratum of hard material

Overloading of soil in side resistance and bearing

Development of a plastic hinge in pile

Overstressing due to a combination of loads;Failure in buckling due principally to axial load

Extreme loading due to earthquake causing liquefactionor othe r large deformations; loading on a marine fr ommajor st orm , or fro m an underw ater slide

Serviceability lim it states Mo st probable cond itions

Excessive axial defo rm ation

Excessive lateral deformation

Excessive rotation of foundation

Excessive vibra tion

Heave of foundation

Deterioration of piles in foundation

Loss of esthetic characteristics

Design of large diameter pile in end bearingFoundation on compressible soils

Design with incorrect p y curves; incorrec t assumptionabout pile-head restraint

Failure to account for effect of inclined and eccentricloading

Foundation to flexible for vibratory loads

Installation in expansive soils

Failure to account for aggressive water; poorconstruction

Failure to perform maintenance

*Peck, 1975; Feld, l968;Szechy, 1961; Wrigh t, 1977.

Considering the variety of loads to which a pile is subjected and the combinationsth at dictate design, there are a num ber of reason s tha t a single pile of a pile f oun datio n

will fail to perform properly. Table 9.1 gives some limit states under two categories;ultimate limit states and serviceability limit states. The table is not intended to becomprehensive but meant to indicate categories of catastrophic failures in the firstinstance and failures of adequate performance in the second instance. In a particulardesign, the engineer either sets do wn a formal of limit states that w ill con trol th e designor will implicitly have the condition in mind as the design proceeds.

9.3 C O N S E Q U E N C E S O F A F A I L U R E

Th e previous section presen ts some of the way s a fou nda tion can fail. It is of interest toconsider the consequences of a failure should one occur. A question usually addressed:will a failure cause a m inor m on etary loss with no possible loss of life, or will the failurebe catastrophic with a large monetary loss and loss of life? On a particular design,

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 383

igure 9. / Histo rical relationship of risks and consequences fo r engineered structu res after, W hi tm an ,1984).

the answer selected by the engineer will clearly guide the steps in the planning andcom puta tions . Sometimes, but n ot alway s, guidance for a design is given by codes andstandards; even so, the engineer has a considerable measure of personal responsibility.

Some quantitative data, based on an estimation of historical events, is given inFig. 9.1 (slightly modified from Whitman, 1984, and based on a private communication to him by G. B. Baecher). An interesting point shown in the figure is that theconseq uence of failure of a dam and a mobile drill rig is roug hly th e same, bu t a greaterprobability of failure is acceptable for the drill rig. Many more people will be affectedby the failure of the dam than by the failure of the drill rig; thus, the reaction of thepublic to the failure of a structure is important.

A comparison of the factors related to the design of a dam and a mobile drill rigshows the following: subsurface investigation far more effort goes into the dam thanthe drill rig; loading the loads on the dam are more predictable; design methods those

for the dam are more straightforward; length of service far longer for the dam; anddesign team the number of engineers on each team might be roughly the same. Thelist of factors above is not comprehensive but does serve to illustrate the complexityof selecting a numerical value of the probability of failure for a given installation.

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9.4 P H I L O S O P H Y C O N C E R N I N G S A F E T Y C O E F F I C I E N T

With respect to loads characterized by the curve on the left in Fig. 9.2, the actualloadings and frequencies for each structure are unique and time-dependent. Thus,if one considers an offshore platform for the production of oil, different curves forloadings could be conceived for the construction period when resistances could below, for the production period, and for the time after de-commissioning. With respectto the platform, the dead loads presum ably could be com puted w ith accuracy; however,the workers operating the platform have sometimes added steel members to facilitatesome activity not originally anticipated.

With respect to live loads, operations could result in a wide range of loads onthe deck with uncertain frequency. Live loads from a storm are predictable assumingthe storm occurs with an assumed frequency of once in 100 years. In the Gulf ofMexico, for example, a considerable amount of data is available on the magnitudesof hurricanes that have occurred. However, the data become sparse in relation toa particular structure at a specific location. The characterization of the loading is

complex when considering sea-floor slides, ship impact, marine growth, scour, andunanticipated events.Characterizing the resistance to the loading on an offshore structure should present

less difficulty tha n o utlined abov e becaus e, in the contex t of this book , the resistance isprovided by deep foundations. However, the example computations presented earlierreveal that many factors enter in to the computation of the response of a pile, eitherto axial or lateral loading. In general, the factors are the properties of the pile, theproperties of the soil, and the theories for behavior of the pile. Some of the weaknessesof the theories for computing pile behavior were discussed, and improvements are tobe expected. The properties of a pile can be predicted in many cases with accuracy,

but piles of cast-in-place reinforced concrete can be expected to vary considerably incharacter from point to point along the length.

W ith respect to the pro perties of the soil, a discussion related to the offshore struct ureis pertinent. Many sites are investigated with wire-line techniques, where the sampling

igure 9.2 Prob ability frequen ces of loads and resistance.

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 385

tool is driven with a dr op ham mer. Samples of clay are kno wn to exhibit an unk no wnamount of loss of strength by such sampling. Sands may not be sampled at all, exceptin a very disturbed state, with the strength of the sand determined from the number ofblows required to drive the sampling tool a given distance. Thus, at best, the properties

determined from a wire-line investigation lie within a wide range of values. Then theproperties of the soil are modified by the installation of a pile, and, with clay, suchchanges are strongly time-dependent. The time-dependent response of the soil is furtheraffected by the nature of the loading, whether short-term, repeated, or dynamic, andby such things as scour and soil deposition. Therefore, a family of curves, such asthe one on the right in Fig. 9.2 will be required with a multitude of factors takeninto account.

With respect to selecting a coefficient of safety, the nonlinear response of the pilefoundation to loading is of primary importance. In some approaches in structuralengineering, an appropriate method is to select a value of strength of steel that includesan appropriate amount of safety. In foundation engineering, the nonlinear nature ofthe soil, and frequently the pile, requires that the safety must reside in the magnitudeof the load because of the nonlinearity of response.

With respect to philosophy, the formalization of the process for computing the safetyof a particular structure with mathematical equations is possible but a preferableapproach is to rely on competent engineering. A competent engineer does not onlybuild a foundation that will not suffer damage in the normal course of events but willalso not fail by requiring far more expensive construction than necessary.

9.5 I N F L U E N C E O F N A T U R E O F S T R U C T U R E

The selection of a factor of safety to be implemented will depend on the type andpurpose of the foundation (Wright, 1977). When failure would possibly result in theloss of life, Meyerhof (1970) has suggested that a probability of failure of less than1 0 - 2 percent will be acceptable. The problem, not addressed herein, is to translate aprobability of failure, certainly a useful concept, into the selection of a global factorof safety or partial safety factors.

In terms of the nature of structure with respect to the length of time of satisfactory

performance, Pugsley (1966) made the proposal that follows. (1) monumental: life20 0- 50 0 y ears, e.g., churches and large bridges, (2) perm anen t: life 7 5- 10 0 years, e.g.large buildings, ordinary rail and road bridges, and (3) temporary: life 25-50 years,e.g. industrial buildings. The list is useful in that some guidance is given regarding thetype of structure being designed. The list is not comprehensive, and the engineer willneed to expand the list to include a wider variety of structures.

9.6 S P E C I A L P R O B L E M S I N C H A R A C T E R I Z I N G S O I L

9 . 6 . 1 I n t r o d u c t i o n

The shear stren gth an d the stiffness-strain of soil are the princip al characteristics neededin the design for lateral loading. If a single method of subsurface investigation is

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386 S in g le P i l e s an d P i l e G ro u p s U n d er L a t e ra l L o ad in g

employed, some considerable scatter may occur in the estimation of values of significant parameters. With intelligent and careful evaluation, the scatter will be lesspronounced from a wealth of data, from a combination of common and special tests,such as triaxial in the laboratory, and such field tests as cone penetrometer, cone pen-

etrometer with pore-pressure measurement, seismic cone, field-vane, spectral analysisof surface waves, in situ vane, dilatometer, and pressuremeter. The engineer shouldcarefully consider the details of the tests performed before developing recommendations for p-y curves. Later on, in addition to assigning a factor for the quality of thesoil-testing methods, another factor should account for the technique for evaluatingthe soil data itself. Safety coefficients, indeed, are not only related to the soil-testingquality and quantity, but are also linked to the method of design and the specific wayof implementing the required, specific soil parameters.

The derivation of a relevant stress-strain level in selecting soil parameters for designfrom results of in-situ testing and laboratory testing has been the topic of manykeynote lectures, reports from technical committees, and remarkable contributionsfrom geotechnical-testing specialists all over the world, (Jamiolkowski et al., 1985,1991 , 1994; Robertson, 1983, 1990, 1993; Stokoe et al., 1985 and 1989; and Baldiet al., 1989). Further, with respect to geotechnical parameters, blending (statistically)results of tests from several locations, even on the same site, at an early stage in thedesign, may mask the crucial and essential variability of a geotechnical parameter and,thus, no t allow detection of the most imp ortan t phen om ena of soil-structure interaction(Bauduin, 1997).

An additional and even more relevant problem with respect to the selection of soildata is linked to the method of taking into account the influence of pile installation,because installation can affect significantly soil properties close to the pile and pilegroup (see Chapter 8 and Appendix G). Excess porewater pressures are caused bydriving piles into fine-grained and cohesive soils and will dissipate with time, cohe-sionless soils can be densified due to the driving of piles, non-silica sands can becrushed during a displacement-pile installation. Thus, for driven piles, the initial-stress conditions and stress history change continuously during installation and perhap sfor a significant tim e after driving. For bo red piles or continuo us-flight-au ger (CFA)piles, stresses change due to the excavation and placement of concrete. The effects ofinstallation are more pronounced for piles that are spaced closely together. Many

comments are presented in technical literature about the effects of the pile installation on soil parameters, and suggestions are given for design (Van Impe, 1991,1993 , 1994, 1997 et al., and 1998). Consequently, the computation of the capacity of piles from soil-test data should include an installation coefficient a? ? for th e ti pcapacity an d a coefficient ^ for the shaft capacity . Both coefficients can be eitherlower than 1 in case of bored or CFA-type of piles, and either higher or lower than1 in case of soil-displacement type of piles. A thorough evaluation of those installation parameters for the given pile type at the test site can also be the outcomeof dilatometer testing during installation, as described in Chapter 8 (Van Impe andPeiffer, 1997).

Th e geotechnical engineer, from a review of the technical literatu re, from experienc e,and from engineered observations at construction sites, should evaluate the expectedeffects of each type of installation of piles in assigning soil data relevant to a single pileor a pile group.

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 387

As geotechnical parameters are always positive in value, they are in fact as suchnot normal-distributed values, very seldom are enough tests available to make a final,reliable choice of the relevant d istrib utio n. Large values of the varia tion coefficienttherefore require the choice of a log-normal distribution.

For man y geotechnical d esigns, one usually has no m ore th an a few field investigationtests (cone penetrometer, or Standard Penetration, for example) or a boring log withsome classification tests. Characteristic values can be obtained using tables (regionalexperience) in which the field measurement (e.g. cone resistance) or the classification-test results are used as an input to obtain the value of the soil property required. The

numbers in such tables are, of course, conservative estimates. Further steps are tocompare the value of the soil parameter with existing geotechnical map and/or to useexperience with similar soils in the area.

The choice of characteristic values becomes much more complicated for complexgeotechnical problems in which sophisticated codes and models are used. Standardcharts in such case provide only a first estimate, and usually leads to a conservativedesign.

Assessing a characteristic value from the test-derived data for soil parameters shouldcover the uncertainties related to stochastic variations, taking into account the soilvolume that was tested, the nature of the building or structure, the soil-structure-

interaction stiffness, the type of sampling, and the overall engineering practice.

9.7 L E V E L O F Q U A L I T Y C O N T R O L

The degree of responsibility accepted by the engineer weighs heavily in view of alitigious society. The assurance of quality of the work of each of the contractors anddesigners plays an important role in accomplishing a successful project. Related to asuccessful project are the amount and quality of the external inspection of the work.

The first aim mu st always be to avoid om issions. Liability can never remed y the da m

age caused during design and implementation by failure to address all of the necessaryand relevant elements of the project. Strong professional management by a responsible party, and the participation of experts from all required disciplines (forminginterdisciplinary design teams) are seen as two very important factors for success.

9.6.2 C ha ra cte r i s t ic valu es of so i l p ar am et er s

The commonly applied student-t distributions or standard-deviation estimates areestablished by assuming a normal distribution of the relevant geotechnical parameter.

How ever, in most cases the log-norm al distribu tion (i.e. logarithm of the p aram eterin a norm al distribution) is a more reliable p rocedu re:

9.1)

9.2)

9.3)

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388 S in g le P i l e s an d P i l e G ro u p s U n d er L a t e ra l L o ad in g

On e app roa ch frequently u sed is to tender on th e basis of specifications an d a description of functions instead of on a bill of quantities. The so-called systems approachencourages the contractor to bid alternative technical and economic solutions.

9.8 T W O G E N E R A L A P P R O A C H E S T O S E L E C T I O NO F FA C T O R O F S A F E T Y

The first of the two general methods is termed the global approach. The engineer willconsider all of the factors at ha nd , including such thing s as the quality of the subsurfaceinvestigation, the statistical nature of the loading, and the expected competence ofthe contractor, and an overall factor of safety is selected for individual piles and forthe group of piles.

The second of the tw o m ethod s is termed the component

app roac h. For a particulardesign, the components of loads and resistances are identified, and an independentfactor of safety is selected for each. The independent factors can be combined to yieldan overall factor. Two examples are presented for the comp onen t app roach : the m ethodof partial safety factors and the method of load and resistance factors. The first of thesetwo methods has been used informally in Europe for many years, and discussions arecurrently underway relative to formal acceptance. The second of the two methodswas accepted formally in 1994 by the American Association of State Highway andTransportation Officials (AASHTO) as a standard.

The global approach to selection of the factor of safety, and the two component

approaches will be discussed in the following sections.

9.9 G L O B A L A P P R O A C H T O S E L E C T I O N O F A FA C T O RO F S A F E T Y

9.9.1 I n t r o d u c t o r y c o m m e n t s

Engineers have trad ition ally used a global factor of safety for the design of piles, givingcareful consideration to all pertinent parameters influencing behavior. The value in the

use of such an overall factor is that the engineer may use judgment to select relevantparameters. For example, the shear strength of the soil may be chosen more liberally ormore conservatively, depending on the entire character of the design. Examples of theuse of global factors of safety for various geotechnical structures have been discussedby many geotechnical engineers (Feld, 1968; Meyerhof 1970; Peck, 1965; Pugsley,1966; Szechy, 19 61 ; Terzaghi, 1962 ; De Beer, 19 61 , 19 65 , 1976 , 19 81 ; Frank e, 1 96 4,1990; and Costanzo-Lancellotta, 1997).

Plainly, the engineer aims to prevent a failure of the structure. However, the precisedefinition of failure may be difficult, leading to possible misunderstandings in communicating with the owner and others. Therefore, the need for the structure to perform

as expected by the owner over its life needs to be understood by all relevant parties.W ith the par ticip ation of the own er, risk of failure shou ld be reduce d to an

acceptable level. The model in Fig. 9.2 shows, in elementary form, the distribution ofloads and resistances. The probab ility of failure is governed by the overlapping zone.

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 389

One has to select the global factor of safety F c in order to keep the probability of afailure to an acceptable level.

(9.4)

where ra# = mean value of resistance R, and ms = the mean value of load S.Eurocode presents a number of principles related to the design of geotechnical facil

ities, as shown in Appendix I. Even though the material in Appendix I is marginallyrelated to the design of single piles and pile groups under lateral loading, the ideasthere are useful in considering the selection of a factor of safety for a particular design.

Examples can be given of many agencies that use a global factor of safety. The APIwas selected in the presentation shown below because the use of piles is so importantin their offshore operations. Furthermore, petroleum companies sponsored research

in the United States that led to some of the methods used in design.

9.9.2 R e c o m m e n d a t i o n s o f t h e A m e r i c a n P e t r o l e u mI n s t i t u t e A P I )

9 9 2 Design considerations

The kinds of loading to be employed in design were mentioned in Chapter 6, and abrief discussion was given of means of computing the magnitude of the various kinds.The API (1993) suggests that estim ated interv al for the recurren ce of the stor m selected

for computation of loadings should be several times the expected life of a structure.Such experience as available in the Gulf of Mexico, for example, suggests that thestorm expected to occur once in 100 years (the 100-year storm) be used for design.Thus, after a particular location for a structure is selected, the engineer is faced withestimating the maximum wave height for the 100-year storm, the likely directions ofthe storm, an d other factors to allow the com puta tion of the magnitud e of the vertical,horizontal, and overturning forces, as a function of time.

The storm loadings allow the engineer to formulate a collection of loadings forthe life of the structure: fabrication, transportation, installation, normal operation,special operation, and removal. Considering the suite of loadings, various piles in the

foundation may have critical loads at one time or another. In some instances, a riskanalysis for the structure is recommended.

The API recommends an appropriate soil investigation, including the necessaryborings and laboratory testing. For both axial and lateral loading on a pile, thedevelopment of data showing load versus deflection is required.

9 9 2 2 Design of piles under axial loading

The design of piles under axial loading follows the usual procedures of computingload transfer in skin friction and end bearing from pile dimensions and soil properties.

Limiting values of the load-transfer coefficients are required, probably because ofmaximum values developed in full-scale experiments of axially loaded piles. Withrespect to the penetration of piles to develop the required capacity in compression andtension, the factors of safety in the following table are given.

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390 S in g le P i l e s an d P i l e G ro u p s U n d er L a t e ra l L o ad in g

Factor ofLoad co nd it io n safety

Design env iron me ntal con dit ion s w it h ap pro pria te dri l l ing loads 1.5Op erat in g env ironm ental cond i t ions dur ing d r i l l ing opera t ions 2 .0Design env iron me ntal cond i t ions w i th appro pr ia te p rodu cing loads 1.5Op erat in g env ironm ental cond i t ions dur ing p rodu cing operat ion s 2 .0Design env iron me ntal con dit ion s w it h m inim um loads for pullo ut) 1.5

The above presentation of the design of axially loading piles according to the recommended practice of API is by no means comprehensive but does serve as an exampleof the use of global factors of safety. Presumably, the factors are applied to augment

the axial loads.

9 9 2 3 Design of piles under lateral loading

The use of p-y curves is recommended in solving for the capacity of piles under lateralloading and details on several sets of curves are presented in the API manual. Withregard to stresses in the piles, API recommends the use of the equations from theAmerican Institute of Steel Construction. The equations include a factor of safety,based on the yield strength of the particul ar steel being used and based on th e pa rticu larcombination of stresses due to axial and lateral loading.

The writers have called attention to nonlinear nature of the response of soil; thus,relying on a safe level of stress in some cases can lead to a quite low factor of safety. Inmany of the examples presented herein, loads are factored to find the failure condition,usually a plastic hinge, which leads to a better idea of the global factor of safety forthe particular design. Experience shows that many designers are adopting this latterapproach, in addition to achieving compliance with allowable stresses.

9.10 M E T H O D O F PA RT I A L FA C T O R S P SF )

9.10.1 P r e l i m i n a r y C o n s i d e r a t i o n s

A discussion of safety factors im plem ents, as noted earlier, implies a prelimina ry agreement on the definition of failure. The suggestion of Franke (1991) can be introducedby considering, for example, the design of axially loaded single piles. A comprehensivediscussion is presented in Appendix J, in which failure is related to a collapse loadand to excessive settlement. While partial safety factors are not used by Franke inAppendix J, the emphasis of pile settlement is consistent with the general aim whenpartial factors are used.

9.10.2 S u g g e s t e d v a l u e s f o r p a r t i a l f a c t o r s f o r d e s i g no f l a t e r a l l y l o a d e d p i l es

Table 9.2 presents a list of suggested coefficients for partial safety factors for the designof piles under lateral load and under axial load. Earthquake loadings are omitted

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 391

because of their special natur e. For design of founda tion s in seismic zones, the engineerwill follow th e codes and specifications regard ing designs tha t are resistant to th e effectsof earthquakes.

In terms of the kind of structure that is being designed, the assumption is made thatthe list applies to permanent or temporary structures only. If a monumental structureis being designed, the presumption is made that the engineer will undertake specialstudies regarding loading and the selection of materials, and that construction will bedone under extra precautions. The columns of poor control, normal control, and goodcontrol are assumed to reflect the quality of the soil investigation and the quality ofconstruction.

With regard to partial safety factors, the resistance R*) for design is givenby Eq. 9.2.

9.5)

where rm = the mean resistance or strength; ym = partial safety factor to reduce thematerial to a safe value; yf = partial safety factor to account for fabrication and construction; and γρ = partial safety factor to account for inadequacy in theory or modelfor design.

With regard to partial load factors, the design load (S*) is given by Eq. 26.

9.6)

where sm = mean value of load; γ \ = p artial lo ad factors to estim ate the safe level of the

loading; γ ι = implementing modifications during construction that cause an increasein the loads; effects due to temperature and creep; and other similar reasons.

Employing the partial safety factors shown in Eqs. 9.5 and 9.6, a global factor ofsafety may be computed as:

9.10.3 E x a m p l e c o m p u t a t i o n s

To illustrate the use of Eq. 9.7, the design of a pile to support an overhead sign may beconsid ered, w here the lateral load is due to wind forces. The axial load is assum ed to benegligible. The assumption is made that the factor γ \ has been studied and assigned avalue of 1.5 to account for the increase in wind loading above the values recommendedin codes and that the factor γ has been selected as 1.0. The assumptions are furthermade that a steel pile will be erected in stiff clay above the water table, that the soilinvestigation was meager, and that construction will be inspected by a technician;thus, poor control is assumed (relating to the selection of Yf). Because deflection ofthe sign would not control, only the collapse of the structure is to be considered.With regard to collapse, the engineer made certain that the pile would penetrate a

sufficient distance tha t long-pile behavior w ou ld occur. In additio n to a value for γ \, thefollowing values have been selected from Table 9.2: γ \ = 1.5 ; m p = 1.7. Therefore,the value of F c = that w ou ld be selected for the design of the overhe ad sign wo uld be(1.5)(1.0)(1.5)(1.7) = 3.8 3.

(9.7)

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39 2 S in g le P i l e s an d P i l e G ro u p s U n d er L a t e ra l L o ad in g

Table 9.2 Suggested partial coefficien ts fo r analysis of piles

Partial safety factors fo r

Dead weight, wa ter pressure,water loadsBulk goods in silos, fluctuatingwater pressureBraking or equipment forcesWind loads, wave loadsAnalysis of specific loads

Designation

Y\

Y\

Y\Y\

POORC O N T R O L

1.0

1.3

1.31.5

N O R M A LC O N T R O L

1.0

1.3

1.31.5

G O O DC O N T R O L

1.0

1.15

1.151.25

72 value based on uncertainty o f occurrenc e and fo runforeseen change in desi£ n assumptions)

artial safety factors fo r design for lateral loading

p y curves for soft clay, stiffclay above water, collapsep y curves for soft clay, stiffclay above water, d eflectionp y curves for stiff clay, subjectto erosion, collapsep y curves for stiff clay, subjectto erosion, deflectionp y curves for sand, collapsep y curves for sand, deflection

Mu /t for steel piles

Mu /t for reinforced concrete pilesEl for steel pilesEl fo r reinforced concre te piles

7m 7p

7m 7p

7m 7p

7m 7p

7m 7p

7m 7p

7m

7m7m7m

1.7

1.85

2.2

1.8

1.71.85

1.5

1.91.52.1

1.65

1.8

2.0

1.7

1.651.85

1.4

1.851.42.0

1.6

1.7

1.8

1.6

1.61.7

1.3

1.81.31.9

artial safety factors fo r design for axial loading

Unit side resistance from load testUnit side resistance, no load testEnd bearing

AE for steel pilesAE fo r co ncre te piles

Collapse load for steel pilesCollapse load for concrete piles

7m 7p

7m 7p

7m 7p

7m 7p

7m 7p

7m 7p7m 7p

1.71.851.77

1.52.0

1.52.0

1.61.71.62

1.41.9

1.41.9

1.51.631.54

1.31.8

1.31.8

The prob lem of the overhead sign is now presented with an other set of assu mp tions.Wind loading has been studied in the area, and the value of γ \ can be selected as1.25; the value of 72 will remain at 1.0. An excellent soil investigation is assumed, aconcrete slab will be placed at the ground surface (not given a p-y curve) to protectagainst environmental changes, and the construction will be closely inspected by anengineer. T hu s, >y can be selected as 1.0 an d th e value of m p can be reasonably

selected as 1.5. Therefore, the value of the overall safety factor can be computed as(1.25)(1.0)(1.0)(1.5) = 1.88.

The two computations presented above are not intended to reflect the solution toany particular problem but to indicate briefly the kinds of analyses necessary to select

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I m p l e m e n t a t i o n o f f a c t o r s o f s a f e t y 393

the size of pile to perform a certain function. The PSF method is a useful tool but, atpresent, not a sufficient one.

9.11 M E T H O D O F L O A D A N D R E S I S TA N C E FA C T O R S L R F D )

9.1 I.I I n t r o d u c t i o n

As with the method of partial safety factors, the LRFD specifications of AASHTOpresent methods of modifying the component loads and the component resistances.The basic equation is shown below.

(9.8)

where r\i = factors to accoun t for ductility, redun dancy and opera tional importance; i = load factor; Qi = force effect, stress or r esu ltan t; φ = resistance factor;Rn = nominal (ultimate) resistance; and Rr = factored resistance.

As may be seen, several features of the method are similar to the method of partialsafety factors. The engineer, in obtaining a solution to Eq. 9.8, must estimate the loadsand load com bination s that may be imposed on the structure, and estimate the ultimateresistance available to resist the loading.

9.1 1.2 L o a d s a d d r e s s e d b y t h e L R F D s p e c i f i c a t i o n s

A large number of types of loads are considered in the LRFD specifications, including, dead load of structural components and nonstructural attachments; dead load ofwearing surface and utilities; horizontal load from earth pressure; load from earth surcharge; vertical load from earth fill; load from collision of a floating vessel; load fromcollision of vehicles; load from an earthquake; ice load; vertical load from dynamicsof vehicles; load from the centrifugal force of vehicle traveling on a curve; load fromthe braking of vehicles; live load from vehicles; live load from surcharge; live loadfrom pedestrians; load from water pressure in fill; load from currents in stream; loadsdue to changes in temperature of structure; wind load on structure; and wind loadon vehicles. Each of the types of loads is discussed (NHI, 1998) and some guidance is

given in making the computation of magnitude of the load.A number of basic load combinations (called limit states) are identified for use indesign. The combinations are shown below.

Strength I, the basic load combination related to the normal vehicular use of a bridgewithout wind.

Strength II, the load co mb ina tion relating to the use of the bridge by owner-specifiedspecial design vehicles and/or evaluation-permit vehicles, without wind.

Strength III, the load combination relating to the bridge exposed to wind velocityexceeding 9 0 km /hr w itho ut live loads.

Strength IV, the load combination relating to very high dead load to live load force-

effect rat ios , exceeding ab ou t 7.0 (e.g., for spans greater th an 75 m).Strength V, the load co mb ination relating to norm al vehicular use of the bridge w ith

wind velocity of 90 km/hr.Extreme Event I, the load combination including earthquake.

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394 S in g le P i l e s an d P i l e G ro u p s U n d er L a t e ra l L o ad in g

Extreme Event II, the load combination relating to ice load or collision by vesselsand vehicles.

Service I, the load combination relating to the normal operational use of the bridgewith a wind velocity of 90 km/hr.

Service II, the load co m bin atio n intend ed to co ntro l the yielding of steel in structu resand the slip of slip-critical connections due to vehicular live load.

Service III, the load combination relating only to tension in prestressed-concretestructures with the objective of control of cracking.

Fatigue, the combination of loads relating to fatigue and fracture from repetitivegravitational live load from vehicles and the dynamic responses under a single designtruck.

Construction, the combination of loads relating to the live load from constructionequipment during the installation/erection of structures.

While all of the load combinations noted above are considered in the design offoundations, the ones that usually control are the Strength I and the Service I. The twocondition s are related to the com puta tion of (1) ultimate capacity and (2) deformationas would be done if the engineer used specifications based on allowable stress design.

9.1 1.3 R e s i s t a n c e s a d d r e s s e d b y t h e L R F D s p e c i f i c a t i o n s

The principal emphasis of the LRFD specifications in regard to resistance resides inthe determ ination of values of geotechnical param eters. The process for plannin g andexecuting a program of surface investigation is described (NHI, 1998); the sources ofvariability in estimating the properties of soil and rock are described; and the statisticalpar am eters are identified tha t can lead to the selection of a resistance factor. The variou sitems are discussed that pertain to the selection of the magnitude of the parameter φ .

An example is presented below for the design of a bored pile under axial load thatillustrates the procedure in selecting the parameters used in design.

9.1 1.4 D e s i g n o f p i l e s b y u s e o f L R F D s p e c i f i c a t i o n s

With regard to lateral load, the specifications note that the piles must be designed toavoid structural failure and to be without excessive deflection. The method used in

allowable stress design can also be used in design using load-and-resistance factors.The usual proc edures, principally those used in the previous chapters of this boo k, arenoted.

With regard to axial loading, an example that follows will demonstrate the application of the method presented in the LRFD specifications (NHI, 1998). The casewas a bridge pier supported by steel piles. The dead load of structural componentsand non-structural attachments (DC) was 4,600 kN; the dead load of wearing surfaceand utilities (DW) was 3,900 kN, and the live load from vehicles (LL) was 3,450 kN.Referring to the specifications, the value of γ was selected as 1.0 and the various loadfactors were selected as follows: y^c = 1.25; y^w = 1.50; and : YLL = 1.75. Therefore,

the factored load is 1.00[(1.25)(4,600) + (1.50)(3,900) + (1.75)(3,450)] = 17,638 kN.The axial capacity of a single pile was estimated from results from the Standard

Penetration Test. The length of the pile was 1 1 m . The com puted value of load inend bearing was 1.460 kN and in side resistance was 445 kN. The tabulated value

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Imp lem entat ion of facto rs of safety 395

of the factor for end bearing φ wa s 0.45 an d the factor for side resistance j)qs

wa s also 0.4 5. Th us , the factored axial resistance of the single pile Q wa s 0.45(1,460 + 445) = 85 7 kN . Disc oun ting the loss of resistance due t o close spacing, th enum ber of piles required for the foundation of the bridge pier was 17,6 38/8 57 = 20 .6;

use 21. It is of interest to note that the global factor of safety for the example was(21)(1,460 + 445)/(4,600 + 3,900 + 3,450) = 4.95. If the capacity of a pile is based onthe results of load tests, the global factor of safety would have been much smaller.

Only a portion of the procedure for design of the piles for supporting a bridge pieris presented; for example, the check of the settlement of the piles under service load isomitted. However, the example does show the factoring of both load and resistance.

9.12 C O N C L U D I N G C O M M E N T

An important point must be made about the design of pile foundations, where theresponse of the soil and the pile material are both nonlinear with load. Fundamentally,the best estimate of the nonlinear response of the soil and of the pile material mustbe employed in analyses and the factor of safety must reside only in the loading.That is, the best estimate of load-distribution curves must be used to find the loadon a foundation, perhaps inclined and eccentric, that will cause collapse or will causedeflection that is intolerable. If combined loading is to be sustained by a single pile orby a group of piles, the engineer must exercise judgment in using a factor to increaseeach of the independent loads.

W hile a con sider ation of the concep ts involved in the selection of factors from th e PSFm eth od o r from th e LRFD me tho d is useful and d esirable, the best estimates of responseof single piles and groups of piles can be made by use of p-y, t-z, and q-w curves andfrom factors giving the interaction between closely spaced piles. Therefore, emphasisin research is recommended in finding the nonlinear curves along full-scale piles in thefield in a variety of soils and rocks that account for the influence of installation andthe influence of the nature of loading.

H O M E W O R K P R O B L E M S F OR C H A P T E R 9

9.1 Team Project. Assign qualifications, on ly a few lines will be requ ired, to stud entsin the team students for the design of the foundations for a major wind-turbineproject (20 turbines). For example: Edouard, expert on performing tasks andacquiring data related to the subsurface investigation of soils. Identify all of theteam, show necessary qualifications, and prepare short statements on each of thetasks, show necessary interactions with other specialists of the project. List otherinformation you wo uld need in order to estimate the cost of the foundation design.

9.2 You are in negotiation w ith an owner regarding the performance of a major jobinvolving complex engineering. List the reasons you have showing the cost of a

peer review when the project is almost complete.9.3 You are having a meeting with prospective custom er, assume that you have some

experience in performing the expected work, and list the reasons your firm shouldbe selected for the work.

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Chapter 1

S u g g e s t i o n s f o r d e s i g n

10 1 I N T R O D U C T I O N

As presented in earlier chapters, a single pile or a group of piles under lateralload responds nonlinearly to applied load, and previous chapters have demonstratedthe imp orta nce of incremen ting load ing to find the load tha t will cause collapse or causeexcessive defection. Further, a number of case studies were presented to demonstratethat the models employed for pile and soil give answers that agree with experimentclosely or within reasonable limits.

Two areas of design need further discussion: (1) the bro ad ran ge of factors tha t m ustbe considered; and (2) ensuring the validity of results from the computer code. Thesetwo topics will be addressed in the following paragraphs.

10 2 R A N G E O F FA C T O R S T O B E C O N S I D E R E D I N D E S I G N

The brief discussion in Chapter 9 presented a number of limit states that must beconsidered. In connection with a design approach that is beyond being merely computational, Professor Ralph B. Peck, former President of the International Society of SoilMechanics and Foundation Engineering, made an important contribution in the opening lecture at a meeting aimed principally at com puta tiona l m ethod s (1967 ). ProfessorPeck made four points that will be presented with a brief discussion.

1 The assum ed loadin g may be erro neo us; 2. The soil con ditio ns may differ fromthose on which the design is based; 3. The theory upon which the calculations arebased may be inaccurate or inadequate; and 4. Construction defects may invalidatethe design.

Professor Peck gave examples of each of the points that will be reviewed here. Awarehouse was being designed for holding petroleum products. When the cans werestacked with a mechanical loader, the floor load was 2.22 kPa which was selected fordesign. However, when the necessary aisles were taken into account, the load wasreduced to 1.78 kPa, a significant reduction.

Professor Peck noted that by and large thro ug ho ut the wo rld, soil conditions are

erratic rather than homogeneous. Yet, the implications of the heterogeneity of soildeposits are still not properly appreciated. Few soil deposits are uniform enough tow arra nt an elaborate investigation of their prop erties. Further, the mere process ofsampling causes changes in properties. At best, the engineer is likely to be confronted

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398 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

with a considerable range of scattering from the average values of strength and compressibility. The response of the engineer to the erratic value of soil properties mustreflect the nature of the design.

Professor Peck noted that in many cases the quality of the theory is better than

our ability to predict the properties of some soils. However, he mentioned someaspects of theory (1967) that may not have been addressed fully or to some degree: (1)behavior of soils under cyclic loading of rather high frequency, (2) behavior of soilsunder random loads as from earthquakes, (3) response of soils from extremely large,rapidly applied loads as from blasts; and (4) prediction of extremely small motions offoundations under cyclic loads.

The problem of the mistakes in the construction of foundations, Professor Pecknot ed, can void the best efforts of the most able designers, even if their k now ledgeof load s, soil con ditio ns, and theor y is virtually perfect. In regard to the emphasisProfessor Peck placed on the quality of construction, the senior author has had a closeconnection with the bored-pile industry in the United States. Several failures haveoccurred in recent years. With one possible exception, the several failures that haveoccurred have been due to construction deficiencies. The possible exception is that theexcavation for the bored pile allowed water to penetrate to a stratum of expansive clayat the base of the pile. High quality construction probably could have prevented thatfailure.

10 3 VA L I D A T I O N O F R E SU LT S F R O M C O M P U TA T I O N S

F O R S I N G L E P I L E

10 3 1 I n t r o d u c t i o n

At the outset, the engineer must accept that the solution for problems of single pilesund er lateral load ing as well as pile gro ups of piles under inclined an d eccentric load ingare complex. Even though com puter codes are available that yield outp ut w ith relativeease, mos t design will requir e man y trials. Sufficient time mu st be made available in thedesign office, in spite of the speed of the computer, to (1) refer to case studies (Chapter7) for solutions of problems similar to the one at han d; (2) mak e addition al com putersolutions with varied parameters, for example, with upper-bound and lower-boundvalues of shear strength; (3) check computer output to see that boundary conditionsare satisfied; (4) run eno ugh solu tions that a feel is developed for the validity ofoutput for a particular problem; (5) make a hand solution for the problem usingnondimensional curves (Chapter 2); (6) verify the accuracy of the computer output byuse of mechanics and other means; and (7) establish a program of peer review. Thelast two points are discussed in the following paragraphs.

10 3 2 S o l u t i o n o f e x a m p l e p r o b l e m s

Most computer codes, as a matter of standard practice, include example problemswith input and output. The examples should be coded, and solutions obtained shouldbe compared with the output. Then, the engineer will have some valid output forstudy. Some input parameters can be changed. For example, the influence of varying

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Su gg e s t i o ns f o r de s ig n 399

the bending stiffness on bending moment can easily be investigated. Thus, informationwill be gained on the importance of determining some parameters with precision.

10 3 3 C h e c k o f e c h o p r i n t o f i n p u t d a t aMost computer codes will include echo print of the input. A good idea is to examinethe listing of the input on the computer screen or to print the input for careful study.Experience has shown that entering incorrect data is a frequent error; the coding ofthe program to allow for echo printing should prevent such errors.

10 3 4 I n v e s t i g a t i o n o f l e n g t h o f w o r d e m p l o y e d i n i n t e r n a lc o m p u t a t i o n s

The assumption is made that the computer being used is capable of double-precisionarithmetic, yielding about 10 or 12 significant figures or more. The user will wish toestablish that the machine being used has a sufficient length of word before makingcomputations because the difference-equation method requires that a relatively largenumber of significant figures be employed in order to avoid serious errors.

10 3 5 S e l e c t i o n o f t o l e r a n c e an d l e n g t h o f i n c r e m e n t

The tolerance is a number that is usually part of the input to be used in making aparticular run. For example, values of deflection for successive iterations are retainedin memory, and the differences at corresponding depths are computed. All of the if-

ferences must be less than the tolerance to conclude a particular run with the computer.Most codes will include a default value.

The user has con trol over the tolerance an d the default value needs to be investigated.A large value of tolerance will lead to inaccurate computations; a very small value willcause a significant increase in the number of iterations and could prevent convergence.The engineer can easily investigate the influence of the magnitude of the tolerance.For most problems, accuracy appears adequate with a value of tolerance that leads to12 to 15 iterations.

The user m ust select the length of the subdivisions into w hich th e pile is divided. Th etota l length of the pile is the emb edde d length plus the por tion of the pile above gro un d,if any. The first step in the selection of the length of increm ent is to examin e prelim inaryoutput and shorten the length of the pile to limit the points of zero deflection to two orthree. The behavior of the upper portion of the pile is usually unaffected as the lengthis reduced as indicated.

A possible exception to shortening the pile to facilitate the computations may occurif the lower portion of the pile is embedded in rock or very strong soil. In such acase, small deflection could generate large values of soil resistance which in turn couldinfluence the behavior of the upper portion of the pile.

With the length of the pile adjusted so that there is no exceptionally long portion atthe bottom where the pile is oscillating about the axis with extremely small deflectionsand soil resistances, the engineer may wish to make a few runs with the pile subdivided into various numbers of increments and examine the output, say the pile-head

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4 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

deflection. Every solution is unique, so rules for the number of sub-divisions are complex, but the value of yt becomes virtually constant for many problems with the pilesubdivided into 50 increments or more. Errors may be introduced if the number ofincrements is 40 or less.

10 3 6 C h e c k o f s o i l r e s i s t a n c e

With the computer output for a particular problem at hand, a check of the correct soilresistance p for the computed deflection is suggested and can be readily done. Valuesof deflection y and soil resistance p are usually tabulated on the computer output. Avalue of deflection should be selected where a p y curve has been input (or printed).

A useful exercise is to make a hand computation for the ultimate resistance pu whichis tabulated as a part of the output for a p y curve. The engineer merely needs to refer

to the procedures given in Chapter 3.

10 3 7 C h e c k o f m e c h a n i c s

The values of soil resistance that are listed in a computer output may be plotted onengineering paper as a function of depth along the pile. The squares under the curvesfor negative values and positive values can be counted and multiplied by the value in(kN/m)m for each square. A check regarding the forces in the horizontal direction canbe made by employing the value of pile-head shear.

The next step can be to make a check of the position of the point of the maximummoment. The point of zero shear can be found by finding the area under the shearcurve that equates to the applied horizontal load at the top of the pile. The depthfound can be compared with the comp uter o utp ut as a further check of the mechanics.

A rough check of the maximum bending moment can be found by estimating thecentroid of the area of the p y curve above the point to zero shear. Then moments canbe com puted . Th us, a rough co mp utatio n should reveal the correctness of the value ofmaximum bending moment in the computer output.

The next step in verifying the mechanics is to make an approximate solution forthe deflection. The assumption can be made that the slope is zero midway betweenthe first two points of zero deflection below the top of the pile. The deflection at thetop of the pile can be computed by taking moments of the M/EI diagram aboutthe top and down to the point of zero slope. The moment diagram can be basedon the concentrated loads and points of load application found in the plotting of thecurve of soil resistance versus depth.

10 3 8 U s e o f n o n d i m e n s i o n a l c u r v e s

Another type of verification can be made by using the p y curves as tabulated by thecomputer or as found by hand computations. Then nondimensional curves can be

employed to solve the problem. The use of the curves is illustrated in Chapter 2 andis limited in several respects. First, a straight line passing through the origin must passthrou gh the points for Epy versus depth . Then, no axial load is allowed and the bendingstiffness El must be constant. Even with the limitations noted, the nondimensional

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S u g g es t io n s fo r d es ig n 401

method will yield, with careful work, a result that is surprisingly close to that from acomputer code.

10 4 VA L I D A T I O N O F RE SU LTS F R O M C O M P U T A T I O N SF O R P I L E G R O U P

The results from the analysis of a group of piles by computer can readily be checkedby tabulating the axial load, shear, and bending moment on each pile. The directionsof the loads can then be reversed and placed on the pile cap. The equilibrium of the capcan be checked with the three equations of statics using the magnitudes and positionsof the applied loads.

If desirable, the response of a single pile can be checked by making use of theprocedures outlined in the previous section.

10 5 A D D I T I O N A L S T E PS I N D E S I G N

10 5 1 R i s k m a n a g e m e n t

The design and construction of a significant structure involves many important steps,and the total project may be complex. The engineer, the owner, and the contractordo not wish to encounter unforeseen difficulties, especially those that lead to legalconflicts. Thus, all of the parties in a project have an interest in managing the risks

that are involved, with the engineer taking a leading role.The engineer should confer with the owner to eliminate unnecessary constraints orunusual conditions that would add undesirable risks to the job. For example, withmost contractual arrangements, the owner will provide the subsurface investigation.In some instance, a complex investigation is required, which must be done with theconcurrence of the owner.

The engineer is required to provide the contractor with a set of plans and a volumeof specifications that are unambiguous and without error. A series of pre-constructionmeetings with the contractor are useful in eliminating uncertainties.

Many engineering firms establish risk-management teams to ensure, as far as possi

ble, that the design of a project is wi tho ut error and th at the construction can proceedwithout questions or delays.

10 5 2 P e e r r e v i e w

The analysis of a single pile or a group of piles under lateral load can be done readilywith computer codes that are available. However, the problems are complex, andthe engineer has to make numerous decisions about loading, soil properties, pilecharacteristics, and analytical procedures. Furthermore, a significant number ofcomputer runs are necessary as loading is augmented to find collapse or some

limiting deflection. A number of trials may be necessary in selecting the best kindand size of pile for the application.

The responsible engineer could very well develop a peer review process so that eachcritical step in the analysis is checked by another knowledgeable engineer. Cost, of

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4 2 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

course, is added to the design but owners need to be aware that the design of pilesunder lateral load cannot be treated casually. The senior author recently was asked tocomment on a legal case where, for an extremely small sum, an engineer undertookthe responsibility to provide information on the response of an existing pile group

under lateral load. The job was treated lightly, incomplete information was provided,with the result that the engineer's firm was required to make a large effort to defendthemselves.

10 5 3 T e c h n i c a l c o n t r i b u t i o n s

Technical articles are being published regularly on piles under lateral loading. If anengineering office is regularly designing piles under lateral load, such articles must bereviewed. Of particular value are articles that include the testing of piles. The resultsof the tests may be compared with results from in-house computations as a means ofvalidating the models proposed herein.

In the course of designs of piles under lateral load for a large project, the opportunity may arise to recommend and participate in the performance of a field test. Such atest may well be economically feasible. The careful planning, ensuring good construction, acquiring data of high quality, and performance of detailed analyses can be ofbenefit to the owner. Such information, with the permission of the owner, can be asignificant contribution to the technical literature.

10 5 4 T h e d e s i g n t e a m

The design of piles under lateral loading will usually involve the contributions of anumber of specialists. While the design under lateral loading may be a small part ofthe overall design, free and willing cooperation of those involved can lead to an optimum solution. In some instances, geotechnical engineers have been asked to providep y curves for a particular design and are eliminated from any other activity. The complexity of most deposits of soil argues for the inclusion of the geotechnical engineerthroughout the design process. A similar argument can be made for the inclusion ofthe structural engineer, representatives of the owner, representatives of the contractor,and for the participation of a number of others. Such cooperation will usually lead to

the best decisions in the design of piles or pile groups under lateral loading.

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n s w e r s t o H o m e w o r k P ro b le m s

Note: The plan is to bind the solutions of the hom ewo rk problem s in a suitable m annerand to make the solutions available to any faculty member who elects to use the book

in a class in the engineering school.)

1.1 a) W ith regard to the analysis of a single pile und er lateral load ing, wh at arethe two principal weaknesses of the models described in Sections 1.51, 1.53,and 1.54?

The models can only be used to compute the lateral load at failure; and the modelsare only useful for static loading.

b) W hat p roblem s are encountered in the use of the model show n in Figure 1 5 2

Even for static loading, using nonlinear soil properties, the making of a mapis time-consuming and tedious and the computations will be very lengthy,particularly if nonlinear geometry is taken into accou nt as the loading prog resses.

9.1 Answer:Edouard. 1. Determine the precise sites for each of the turbines. 2. Study theengineering geology of the area and acq uire such report s as are available. 3. O bta inany results of soils investigations p erform ed in the area tha t are available. 4. Visitthe site. 5. Get preliminary information from a fellow team member on requireddepth of boring for subsurface characteristics of the soil.

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i s t o f Sym bo l s

A = cross sectional area of the socketAc = em pirical coefficient used in eq uati on for p-y curves for stiff clays below

water surface, cyclic loadingAs = em pirical coefficient used in equ atio ns for p-y curves for stiff clays below

water surface, static loadingAc = em pirical coefficient used in equ atio ns for p-y curves for sand, cyclic loadingAs = em pirical coefficient used in equ atio ns for p-y curves for sand, static loadingai = horizo ntal coord inates of global axis system in pile grou p analysisB = wid th of footing, pile diam eterBe = em pirical coefficient used in equ atio ns for p-y curves for sand, cyclic loadingBs = em pirical coefficients used in equ atio n for p-y curves for sand, static loadingb = pile diam eterC = coefficient rela ted to stress level used in p-y curves for stiff clay above water

surfaceCD = dra g coefficientCM = ine rtia coefficientQ = shape coefficientC = coefficient used in equ atio ns for p-y curves for sandc = cohe sion, und raine d shear strength of soilca = average und raine d shear strengthcu = average und raine d shear strength of clay

cx = und raine d shear strength at depth xD = pile diam eterDr = relative densityE = m od ulu s of elasticity; the secant Young 's m odu lus at the stress {σ \ — σ

Ec = Young mo dulu s of concreteEj = Young's mo dulu s of the recovered, intact core ma terialEm = ma ss m od ulu s of elasticity, m od ulu s of the in situ rockEp = an arbitra ry mo dulu s of deform ation as related to the pressuremeterEplp = the bend ing stiffness of the pileEpy = modulus of p-y curve or a parameter that relates p and y

Es = soil m odu lus (secant to p-y curve)El = flexura l rigidity of pilee = grou p reduction factorF = shearin g forceF p = force aga inst a pile in clay from wedg e of soil

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xviii List of Symbols

F pt = force against pile in sand from wedge of soilF s = glob al factor of safetyf = unit load transfer in skin-frictionf c = comp ressive strength of concrete

fr = fracture streng th of concrete/max = peak soil friction (taken as the me an un dra ine d shearing strength)fs = ultim ate unit side resistancefsz = ultim ate unit side resistance in sand at dep th zfy = yield poi nt of steel reba rfz = shear resistance in clay at depth z, measured from ground surfaceG s = specific gravity of the min eral grainsGs,max = m axim um tangent shear mo dulusH = tota l thicknessh = pile increm ent lengthAh = horizon tal translation in global coord inateI = mo men t of inertiaIp = influence coefficient/ = factor used in equ atio n for ultim ate soil resistance near gro un d surface

for soft clayJ m = M t/yt mod ulus for compu ting Mt from yt

J x = P x/xt mo dulus for com puting P x from xt

J y = Pt/yt modulus for computing P t from yt

K = Mt/St, rotatio nal restraint of pile topKo = coefficient of eart h pres sure at restKir = dimension less con stan t for wea k rock criteriaKrm = dimension less con stan t for wea k rock criteriak = consta nt giving variation of soil mo dulu s with depthkc = coefficient used in equa tio n for p-y curves forstiff clays below water

surface, cyclic loadingkß = Mt/Su sprin g stiffness of rest rain ed pile hea dL = Length of pile; pen etrati on of pile below gro un d surfaceM = bending mom entM m a x = max imum positive bending mom ent

Mt = applied mo me nt at the pile headMt/St = rotatio nal restraint constan t at pile topm = slope used in defining po rtio n of p-y curve for sandniR = me an value of resistance Rms = me an value of loads SN = num ber of cycles of load app lication used in p-y curves for stiff clay

above water surfaceNc = bearing capacity factorNq = bearing capacity factorNSPT = corrected blow coun t from Stand ard Penetration Test

N-value = the sum of the nu mb er of blow s to drive the samp ler thr ou gh thesecond and third intervals

n = num ber of segments; expo nent used in equation s for p-y curves for sandP = force appliedPI = plasticity index

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i s t o f Sy m bo ls x ix

P c = cha ract erist ic load; ulti ma te soil resistan ce for pile in stiff clay belo wwater surface

P cd = ult im ate soil resistan ce at dep th for pile in stiff clay belo w w ate r surfaceP ct = ul tim ate soil resistan ce nea r grou nd surface for pile in stiff clay

without waterP s = ultim ate soil resistance for pile in sandP sd = ultim ate soil resistance at depth for pile in sandP st = ultim ate soil resistance near gro un d surface for pile in sandP t = lateral load (shear) at pile headP x = axia l forceP u = ultim ate pile capacityP uh = the failure load(P u)ca = ultim ate soil resistance near gro un d surface for pile in clay(P u)cb = ult im ate soil resistan ce at dep th for pile in clay(P u)sa = ultim ate soil resistance near gro un d surface for pile in sand(P u)sb = ultim ate soil resistance at depth for pile in sandPI = plasticity inde x for clayp = th e reactio n from the soil due to the deflection of the pilepo = overbu rden pressure(Quit)G = ultim ate axial capacity of the gro up(Quh)p = ultim ate axial capacity of an individ ual pile^max = unit end- bearin g capacityqu = uncon fined comp ressive streng th, uniax ial comp ressive strength of the

rock or concreteRt = E tI t flexural rigidity at pile topRQD = rock quality designationR* = resistancerm = the me an resistance or streng thSPT = Stand ard Penetration TestS = slopeS* = loadingsm = me an value of loadT = relative stiffness factor

TB or Tp = transformation matrixV = shearVo = v olum e of the me asurin g po rtio n of the pro be at zero readin g of

the pressureVm = corrected volum e readin g at the center of the straight line po rtio n

of the pressuremeter curvev = shearv = Poisson s ratioW = distrib uted load along the length of the pilew = pile mo vem ent; settlemen t of the base of the drilled shaft

x = coordinate along the pile measured from the pile measured from the topxr = transition depth at intersection of equation s for com puting ultimate soil

resistance against a pile in clayxt = pile head; transition depth at intersection of equation s for com puting

ultimate soil resistance against a pile in sand

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xx ist of Sym bol s

Yg = deflection of pile groupy = deflectionyc = deflection coo rdin ate for p-y curves for stiff clay ab ove w at er surface ,

cyclic loading

yp = pile deflection with pile hea d fixed agai nst r ot at io nyk = a specific deflection on p-y curves for sandym = pile deflec tion at node m; a specific deflection on p-y curves for sandyp = a specific deflection on p-y curves for stiff clay below w at er surface ,

cyclic loadingys = deflection coo rdin ate for p-y curves for stiff clay above water table,

static loadingyt = deflection at pile headyu = a specific deflection on p-y curves for sandy = a specific deflection on p-y curves for clayZ = plastic mo dulu s; X / T depth coefficient in elastic-pile theoryz = dep th , or p ile movem ent in t-z curvesa = dimensionless factor dep end ent on the depth-w idth re la t ionship of the

pile; cons tan t of proport ional i tyotp = pile hea d rotat ionas = rotation of structurea t = soil modulus for laterally loaded piles by Terzaghia ppkj = the coefficient to get the influence of pile / on pile k

o

β = inclination of the g ro u n d ; i — rela tive stiffness fac tor

y ElAP/AV = slope of the straight-line portion of the pressur e meter curveΦ€ = friction angle at interface of concrete and soilah = no rma l componen t of stress at pile-soil interfaceΔ σ ι = change in majo r pr incipal tota l stressΔ σ3 = change in mi nor p rincipal to tal stressε = axial strain of soilεζ = strain at any dep th z benea th a loade d a rea£5 0 = axial strain of soi l correspondin g to one-half the ma xi mu m principal

stress differenceΘ = angle of rota t ion6j = the inclined angle be tw een ve rtical line and pile axi s of the

i-th batte r pileγ = unit weig ht of soilγ = effective unit we igh t of soilγ \ = partial load factor to assure a safe level of loading72 = partial load factor to account for any modif icat ions d uring construct ion ,

to account for effects of t empe ra tu re , and to account for effects of creepYc = unit weig ht of the concreteYf = partial safety factor to account for deficiencies in fabricat ion or

construct ionYm = partial safety factor to reduce the s trength of the mater ia l to a safe valueYp = partial safety factor to account for inadequacies in the theory or model

for designYw = unit weight of water

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i s t o f Sy m bo ls xx i

λ = coefficient th at is a function of pile pe net ratio nμ = coefficient of frictionφ = friction ang le from effective stress ana lysisφ = friction anglep = rad ius of deform ed sectionσ = norm al stressσ κ or σ ρ = deflection in Po ulo s' equ atio nσ = effective stressσν = stress in the vertical directionr = shea r stress

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R e f e r e n c e s 4 9 9

Van Impe, W.R 1986. Evaluation of deformation and bearing capacity parameters of foundations, from static CPT-results. Proceedings of the IVNTI International Geotechnical Seminar,Singapore, Asia: 51 -70 .

Van Impe, W.R 19 91 . Analysis of CFA pile behavior with D M T results at Geel test site. 4th DFI

International Conference on Piling and Deep Founda tions, Stresa, Italy 101-105 .Van Impe, W.R 1988. Considerations on the auger pile design. Proceedings of the I International Geotechnical Seminar on Deep Foundations on Bored and Auger Piles, Ghent, Belgium7-10 June 1988: 193-218 .

Van Impe, W.R 1991a. Deformations of deep foundations. Proceedings of the Session 3b-X European Conference on Soil Mech anics and Founda tion Engineering, Florence, Italy,26-30 May 1991: 1031-1062 .

Van Impe, W.R 1991b. Developments in pile design. General Report, Proceedings of the session 4-IV Deep Foundations Institute (DFI) International Conference on Piling and DeepFounda tions, Stresa, Italy, 7-12 April 1991.

Van Impe, W.R 1994. Influence of screw pile installation parameters on the overall pile

behaviour. Wo rkshop: Piled Founda tions: Full Scale Investigations, Analysis and Design,Naplesjtaly, 12-15 Decem ber 1994.

Van Impe, W.R, 2004. Two decades of full scale research on screw piles, an overview.Van Impe, W.R & Y. De Clerq 199 4. A piled raft interactio n m odel. Proceedings of the DFI 94,

V International Conference and Exhibition on Piling and Deep Foundations, Bruge, Belgium.Van Impe, W.R & H. Peiffer 1997. Influence of screw pile installation on the stress state in the

soil. International Seminar ERTC3, 17-18 April 1997, Brüssel, Belgium: 3 -19 .Van Weele, A.R 1 979 . Some consideration s with rega rd to bearing capacity of foundation piles.

Geologie en Mijnbouw, 58 : 405 -416 .Vesic, A.S. 1970. Tests on instrumented piles, Ogeechee River site. Journal of Soil Mech anics

and Foundations Division, ASCE 96(SM2): 561 -58 4.Vesic, A.S. 1961a. Bending of beams resting on isotropic elastic solids. Journal of Engineering

Mechanics Division, ASCE 87(EN2): 35-53 .Vesic, A.S. 1961b. Beams on elastic subgrade and the Winkler's hypothesis. Proceedings of the

V International Conference on Soil Mech anics and Foundation Engineering, Paris, France,1:545-550.

Vijayvergiya, V.N. 1977. Load-movement characteristics of piles. Proceedings of the Ports 77Conference, Long Beach, California, ASCE.

Wang, S.T. 1982. Development of a laboratory test to identify the scour potential of soils atpiles supporting offshore structures. Thesis, University of Texas, Austin.

Wang, S.T. 1986. Analysis of drilled shafts employed in earth-retaining structures. Dissertation,

University of Texas, Austin.Wang, S.T. & L.C. Reese 1986. Study of design method for vertical drilled shaft retainingwalls. Research Report 415-2R Center for Transportation Research, Bureau of EngineeringResearch. University of Texas, Austin.

Wa rd W.H., S.G. Samuels & M .E. Butler 1 959 . Further studies of the properties of Londo n clay.Geotechnique, 14(2): 33 -38 .

Welch, R.C. & L.C. Reese 1972. Laterally loaded behavior of drilled shafts. Research Report3-5-65-89. Center for Highway Research. University of Texas, Austin.

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Casagrande Volume, In R.C. Hirschfeld & S.J. Poulos (eds). New York: Wiley.Woods, R.D. 1978. Measurement of dynamic soil properties. Proceedings of the Geotechnical

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5 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

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Wright, S.J. 1977. Limit-state design of drilled shafts. Thesis, University of Texas, Austin.

Wroth, C.R et al. 1979. Stress changes around a pile driven into cohesive soil. Proceedings of theConference o n Resent Developments in the Design and Construction of Piles, ICE, London ,U.K. : 255-264 .

W roth , C.R 197 2. Some aspects of the elastic behavior of overconsolidated clay, Proceedings ofthe Roscoe Memorial Symposium, Foulis: 3 4 7 - 3 6 1 .

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Append ix A

B r o m s m e t h o d f o r an a l y s i s o f s i n g l ep i l es u n d e r l a t e r a l l o a d i n g

The method was presented in three papers published in 1964 and 1965 (Broms 1964a,19 64b , 196 5). As show n in the following para grap hs, a pile can be designed to sustaina lateral load by solving some simple equations or by referring to charts and graphs.

A l P IL E S I N C O H E S I V E S O I L

A I I U l t i m a t e l a t e r a l l o a d f o r p i l e s i n c o h e s i v e s o i l

Broms adopted a distribution of soil resistance, as shown in Fig. A.l, that allows theultimate lateral load to be com puted by equation s of static equilibrium. The eliminationof soil resistance for the top 1.5 diameters of the pile is a result of lower resistance

in that zone because a wedge of soil can move up and out when the pile is deflected.The selection of nine times the undrained shear strength times the pile diameter as theultimate soil resistance, regardless of depth, is based on calculations with movementof soil from the front toward the back of the pile.

A / . / . / Short free-head piles in cohesive soil

For short piles that are unrestrained against rotation, the patterns that were selectedfor behavior are shown in Fig. A.2. The following equation results from the integration

igur A. I Assumed distrib ution of soil resistance fo r cohesive soil.

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 4 5

A.I.1.2 Lo ng free-head piles n cohes ive so i l

As the pile in cohesive soil with the un restrai ned head becomes longer, failure willoccur with the formation of a plastic hinge at a depth of 1.50b-\-f. Eq. A.3 can thenbe used directly to solve for the ultimate lateral load that can be applied. The shapeof the pile under load will be different th an th at sho wn in Fig. A.2 but the equationsof mechanics for the upper p ortion of the pile remain unchan ged.

A plastic hinge will develop when the yield stress of the steel is attained over theentire cross-section. For the pile that is used in the example, the yield moment is430 m-kN if the yield strength of the steel is selected as 276 M Pa.

Substituting into Eq. A.3

As an example of the use of the equations, assume the following:

b = 305 mm assume 305-m m O .D. steel pipe by 19 mm wall),I p = 1.75 x 1 0 -4 m 4 , = 0.61 m, L = 2.44 m, and

cM =

47 .9kPa .Eqs. A.2 through A.5 are solved simultaneously and the following qua dratic

equation is obtained.

? 2t 2083P* 67,900 = 0

F t = 9 A k N

Substituting into Eq. A.3 yields the maximum moment

= 77 kN-m.

Assuming no axial load, the max imum stress is

The computed maximum stress is tolerable for a steel pipe , especially wh en a factorof safety is applied to P i ui t. The com putations, then, show that the short pile wouldfail due to a soil failure.

Broms presented a convenient set of curves for solving the problem of the short-pilesee Fig. A.3). E ntering the curves with L/b of 8 and e/b of 2, one obtains a value of

Puit of 60 kN, which agrees with the results comp uted above.

P i u l t = 2 2 4 k N

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4 06 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur A3 Cu rves for design of sh or t piles und er lateral load in cohesive soil.

Broms pre sented a set of curves for solving the p rob lem of the long pile (see Fig. A.4 ).Entering the curves with a value of My/cub3 of 316A one obtains a value of Piu it ofabout 220 kN.

A.I.1.3 Influence of pile length free-head piles in cohesive soil

Consideration may need to be given to the pile length at which the pile ceases to bea short pile. The value of the yield moment may be computed from the pile geometryand material properties and used with Eqs. A.2 through A.5 to solve for a criticallength. Longer piles will fail by yielding. Or a particular solution may start with useof the short-pile equations; if the resulting moment is larger than the yield moment,the long-pile equations may be used.

For the example problem, the length at which the short-pile equations cease to bevalid may be found by substituting a value of Piu i t of 224 kN into Eq. A.2 and solvingfor f and substituting a value of Mm a x of 43 0 m -kN into Eq. A.4 and solving for g.

Eq. A.5 can then be solved for L. The value of L wa s found to be 5.8 m. T hu s, forthe example problem the value of Piui t increases from zero to 224 kN as the length ofthe pile increases from 0.46 m to 5.8 m, and above a length of 5.8 m the value of Piui t

remains constant at 224 kN.

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 4 07

igur AA Curv es for design of long piles under lateral load in cohesive soil.

A.LI A Short fixed-head piles in cohesive soil

For a pile that is fixed against rotation at its top, the mode of failure depends on thelength of the pile. For a shor t pile, failure con sists of a horizo ntal mo vem ent of the pilethr ou gh the soil with the full soil resistance develop ing over the length of the pile exceptfor the top one and one-half pile diameters, where it is expressly eliminated. A simpleequation can be written for this mode of failure, based on force equilibrium.

(A.7)

(A.6)

A / . / . 5 Intermediate length fixed-hea d piles in cohesive soilAs the pile becomes longer, an intermediate length is reached such that a plastic hingedevelops at the top of the pile. Rotation at the top of the pile will occur and a point ofzero deflection will exist somewhere along the length of the pile. Fig. A.5 presents thediagrams of mechanics for the case of the restrained pile of intermediate length.

The equation for moment equilibrium for the point where the shear is zero (wherethe positive moment is maximum) is:

Substituting a value of / ,

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4 08 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur A.5 Diagrams of deflection soil resistance shear and moment for intermediate-length pilein cohesive soil fixed against rotatio n.

Employing the shear diagram for the lower portion of the pile,

(A.8)

(A.9)

(A.10)

Eqs. A .7 thro ug h A . 10 can be solved for the beh avior of the restrained pile ofintermediate length.

A.l.1.6 Long fixed-head piles in cohesive soil

The mechanics for a long pile that is restrained at its top is similar to that shown inFig. A.5 except that a plastic hinge develops at the point of the maximum positivemoment. Thus, the M^x in Eq. A.7 becomes My and the following equation results

(A.11)

A.I.1.7 Influence of pile length fixed-head piles in cohesive soil

The example problem will be solved for the pile lengths where the pile goes from onemode of behavior to another. Starting with the short pile, an equation can be written

The other equations that are needed to solve for Piui t are:

and

Eqs. A. 10 an d A . 11 can be solved to o btai n Piui t for the long pile.

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 4 09

for moment equilibrium for the case where the yield moment has developed at the topof the pile and where the moment at its bottom is zero. Referring to Fig. A.5, but withthe soil resistance only on the right-hand side of the pile, taking moments about thebottom of the pile yields the following equation.

In sum mary , for the exam ple prob lem th e value of Piu i t increases from zero to 281 kNas the length of the pile increases from 0.46 m to 2 .6 m, increases from 28 1 kN to41 9 kN as the length increases from 2.6 m to 7.3 m, a nd abov e a length of 7.3 m th evalue of Piu i t remains constant at 419 kN.

In his presentation, Broms showed a curve in Fig. A.3 for the short pile that was

restrained against rotati on at its top . That curve is omitted here because the com putation can be made so readily with Eq. A.6. Broms' curve for the long pile that is fixedagainst rotation at its top is retained in Fig. A.4 but a note is added to ensure properuse of the curve. For the example prob lem, a value of 415 kN was o btained for Piui t,

Summing forces in the horizontal direction yield the next equation.

The simultaneous solution of the two equations yields the desired expression.

(A.12)

Eqs. A.6 and A.12 can be solved simultaneously for Piui t and for L, as follows

from Eq. A.6, P tu k = (9)(47.9)(0.305)(L - 0.45 75),from Eq. A.12, P tuk = 430/(0.5L + 0.229),then L = 2.6 m and P tuk = 281 kN.

For the determination of the length where the behavior changes from that of the pileof interme diate length to th at of a long pile, Eqs. A.7 th rou gh A. 10 can be used withM m a x set equal to My as follows:

from Eq. A.7, P i u l t

from Eq. A.8, g

from Eq. A.9,1

from Eq. A. 10, /

then L = 7.27 m and P i u l t = 419 kN.

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410 Sing le P i les n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

where a soil reaction modulus and; E p I p = pile stiffness.

Broms presented equations and curves for computing the deflection at the ground-line. His presentation follows the procedures presented elsewhere in this text.With regard to values of the reaction modulus, Broms used work of himself and

Vesic (1961a, 1961b) for selection of values, depending on the uncon fined comp ressivestrength of the soil. The work of Terzaghi (195 5) and other with respect to the reactionmodulus have been discussed fully in the text.

Broms suggested that the use of a constant for the reaction modu lus is valid o nly fora load of one-half to one-third of the ultimate lateral capacity of a pile.

A . I . 3 E ff ec t s o n a t u r e o l o a d i n g on p i l e s in c o h e s i v e s o il

The values of reaction modulus presented by Terzaghi are apparently for short-termloading. Terzaghi did not discuss dynamic loading or the effects of repeated loading.Also, because Terzaghi's coefficients were for overconsolidated clays only, the effectsof sustained loading would probably be minim al. B ecause the nature of the loading isso important in regard to pile response, some of Broms' remarks are presented here.

Broms suggested th at the increase in the deflection of a pile under lateral loading dueto consolidation can be assumed to be the same as would take place with time for spreadfootings and rafts found ed on the ground surface or at some distance below the groundsurface. Broms suggested that test data for footings on stiff clay indicate that the coef-

ficient of subgrade reaction to be used for long-time lateral deflections should be takenas 1/2 to 1/4 of the initial reaction modulus. The value of the coefficient of subgradereaction for normally consolidated clay should be 1/4 to 1/6 of the initial value.

Broms suggested that repetitive loads cause a gradual decrease in the shear strengthof the soil located in the immediate vicinity of a pile. He stated that unpublished dataindicate that repetitive loading can decrease the ultimate lateral resistance of the soilto about one-half its initial v alue.

A 2 P IL E S N C O H E S I O N L E S S S O IL S

A 2 1 U l t i m a t e l a t e r a l lo a d f o r p i l e s i n c o h e s i o n l e s s s o i l

As for the case of cohesive soil, tw o failure modes were considered; a soil failure anda failure of the pile by the formation of a plastic hinge. With regard to a soil failure in

(A.13)

which agrees well with the computed value. No curves are presented for the pile ofintermediate length.

A . I . 2 D e f l e c t i o n o p i l e s in c o h e s i v e s o il

Broms suggested that for cohesive soils the assumption of a coefficient of subgradereaction that is constant with depth can be used with good results for predicting thelateral deflection at the groundline. He further suggests tha t the coefficient of subgradereaction a should be taken as the average over a depth of 0.8/3L, where

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 4 13

The maximum positive bending moment can then be computed by referring toFig. A.7.

Or, by sub stitutin g expression for Eq. A. 19 into the above equ atio n, the followingexpression is obtained for the maximum moment.

(A.21)

As an exam ple of the use of the equa tion s, the pile used previously is consid ered. Th efriction angle of the sand is assumed to be 34 degrees and the unit weight is assumedto be 8.64 kN/m3 (the water table is assumed to be above the ground surface). AssumeMt is equal to ze ro. Eqs. A. 13 an d A . 15 yield the following:

Kp = tan2

Punt

The distance f can be computed by solving Eq. A.20.

f = 0.816

The max im um positive bending mom ent can be found using Eq. A .2 1.

M m a x = (22.2)(0.61 + 1.259

Assuming no axial load, the maximum bending stress is

The computed maximum stress is undoubtedly tolerable, especially when a factorof safety is used to reduce Piui t. Broms presented curves for the solution of the casewh ere a sho rt, un restra ined pile undergo es a soil failure; however, Eqs. A. 16 and A . 19are so elementary that such curves are unnecessary.

A.2.1.2 Lon g free-head pi les in coh esionless soi l

As the pile in cohesionless soil with the unrestrained head becomes longer, failure

will occur with the formation of a plastic hinge in the pile at the depth f below theground surface. It is assumed that the ultimate soil resistance develops from the groundsurface to the point of the plastic hinge. Also, the shear is zero at the point of m ax im ummoment. The value of f can be obtained from Eq. A.20 shown above. The maximum

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4 14 S i n g l e P i le s a n d P i le G r o u p s U n d e r L a t e r a l L o a d i n g

igur A.8 Curv es for design of long piles und er lateral load in cohesionless soil.

positive moment can then be computed and Eq. A.21 is obtained as before. Assumingthat Mt is equal to zero, an expression can be developed for Piui t as follows:

(A.22)

For the example problem, Eq. A.22 can be solved, as follows:

Broms presented a set of curves for solving the problem of the long pile in cohesion-less soils (see Fig. A.8). Entering the curves with a value of M y/b 4yK p of 192 6, oneobtains a value of Piui t of about 160 kN. The logarithmic scales are somewhat difficultto read and it may be desirable to make a solution using Eq. A.22. Eqs. A.21 an d A.22must be used in any case if a moment is applied at the top of the pile.

A.2.13 Influence of pile length free-head piles in cohesionless soil

There may be a need to solve for the pile length where there is a change in behavior

from the sh ort-p ile case to the long-p ile case. As for the case of the pile in cohesive so ils,the yield mom ent may be used with Eqs. A. 16 thr ou gh A. 18 to solve for th e criticallength of the pile. Alternatively, the short-pile equations would then be compared withthe yield moment. If the yield moment is less, the long-pile equations must be used.

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 41 5

Eqs. A.25 and A.26 can be solved to obtain Piu it for the long pile.

(A.26)

Taking moments at point f leads to the following equation for the ultimate lateralload on a long pile that is fixed against rotation at its top.

For the example problem, the value of Piui t of 15 3 kN is subs tituted into Eq. A. 18and a value of L of 6.0 m is computed. Thus, for the pile that is unrestrained againstrotatio n th e value of Piui t increases from z ero when L is zero to a value of 15 3 kN wh enL is 6.0 m. For larger values of L, the value of Piu it remains constant at 153 kN.

A.2.1A Short f ixed-head pi les in coh esionless soi l

For a pile that is fixed against rotation at its top, as for cohesive soils, the mode offailure for a pile in cohesionless soil depend s on th e length of the pile. For a short p ile,the mode of failure will be a horizontal movement of the pile through the soil, withthe ultim ate soil resistance developing ov er the full length of the pile. The eq uatio n forstatic equilibrium in the horizontal direction leads to a simple expression.

(A.23)

A.2.1.5 Intermediate length f ixed -head pi les in coh esionless soi l

As the pile becomes longer, an intermediate length is reached such that a plastic hingedevelops at the top of the pile. Rotation at the top of the pile will occur, and a pointof zero deflection will exist somewhere along the length of the pile. The assumed soilresistance will be the same as shown in Fig. A.7. Taking moments about the toe of thepile leads to the following equation for the ultimate load.

(A.24)

Eq. A.24 can be solved to obtain Piui t for the pile of intermediate length.

A.2.1.6 Lon g f ixed-head pi les in coh esionless soi l

As the length of the pile increases more, the mode of behavior will be that of a longpile. A plastic hinge will form at the top of the pile where there is a negative bendingmoment and at some depth f where there is a positive bending moment. The shear atdepth f is zero and the ultimate soil resistance is as shown in Fig. A.7. The value of

f may be determined from Eq. A.20 but that equation is re-numbered and presentedhere for convenience.

(A.25)

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416 Single Piles and Pile Grou ps U nde r La teral Load ing

then L = 6 .25 m, Piuit

In sum mary, for the examp le prob lem th e value of Piu it increases from zero to 180 kNas the length of the pile increases from zero to 3.59 m, 180 kN to 251 kN as the lengthincreased from 3.59 m to 6.25 m, and above 6.25 m the value of Piu i t remains constantat 251 kN.

In his presentation, Broms showed curves for short piles that were restrained againstrotation at their top. Those curves are omitted because the equations for those casesare so easy to solve. Broms' curve for the long pile that is fixed against rotation at itstop is retained in Fig. A.8 but a note is added to ensure proper use of the curve. For theexample problem, a value of 300 kN was obtained for Pi ui t, which agrees poorly withthe computed value. The difficulty probably lies in the inability to read the logarithmicscales accurately. N o curves are presented for the pile of interm ediate length wit h fixedhead.

A .2 .2 D e f l e c t i o n o f p i l e s in c o h e s i o n l e s s s o i l

Broms noted that Terzaghi (1955) has shown that the reaction modulus for a cohesionless soil can be assumed to increase approximately linearly with depth. As noted

A.2.1.7 Influence of pi le length f ixed-head pi les in coh esionless soi l

The example problem will be solved for the pile lengths where the pile goes from onemode of behavior to another. An equation can be written for the case where the yield

moment has developed at the top of the short pile. The equation is:

(A.27)

Eqs. A.24 and A.27 are, of course, identical but the repetition is for clarity. Equations A.23 and A.27 can be solved for Piuit and for L, as follows:

from Eq. A.23, P tuk = 14 .0L2 ,430

from Eq. A.27, P tuk = —— + 15.3L2

,then L = 3.59 m and P tuk = 180 kN .

For the determination of the length where the behavior changes from that of a pileof intermediate length to that of a long pile, the value of Piu it from Eq. A.24 may beset equal to that in Eq. A.26. It is assumed that the pile has the same yield momentover its entire length in this example.

from Eq. A.24, Piui t

from Eq. A.26, Piui t

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B r o m s m e t h o d f o r a n a l y s i s o f s i n g l e p i l e s u n d e r l a t e r a l l o a d i n g 4 17

earlier, and using the formulations of this work, Terzaghi recommends the followingequation for the soil modulus.

A.28)

Broms suggested that Terzaghi's values can be used only for computing deflectionsup to the working load and that the measured deflections are usually larger than thecomputed ones except for piles that are placed with the aid of jetting.

Broms presented equations and curves for use in computing the lateral deflection ofa pile; however, the methods presented herein are considered to be appropriate.

A .2 .3 E f f e c t s o f n a t u r e o f l o a d i n g o n p i l e s in c o h e s i o n l e s sso i l

Broms noted that piles installed in cohesionless soil will experience the majority of thelateral deflection under the initial application of the load. There will be only a smallamount of creep under sustained loads.

Repetitive loading and v ibration, on the other h and , can cause significant addition aldeflection, especially if the relative density of the cohesionless soil is low. Broms notedthat work of Prakash (1962) shows that the lateral deflection of a pile group in sandincreased to twice the initial deflection after 40 cycles of load. The increase in deflectioncorresponds to a decrease in the soil modulus to one-third its initial value.

For piles subjected to repeated loading, Broms recommended for cohesionless soilsof low relative density that the reaction modulus be decreased to 1/4 its initial valueand that the value of the reaction modulus be decreased to 1/2 its initial value for soilsof high relative density. He suggested that these recommendations be used with cautionbecause of the scarcity of experimental data.

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AUTHOR INDEX

Index Terms Links

A

Alizadeh, M. 301

Allen, J. 95

American Association of State Highway

and Transportation Officials 206 213 215 216 388

393

American Concrete Institute 134 325 326

American National Standards Institute 205

American Petroleum Institute 17 105 160 163 165

206 207 209 213 215

234 241 389 390

American Society of Civil Engineers 206

American Society for Testing

and Materials 16 261 340 341

Appel, G.C. 244

Aschenbrenner, R. 139

Asherman, J.C. 16

Asplund, S.O. 138

Atkins Engineering Services 209

Audibert, J.M.E. 105

Austin American-Statesman 216

Awoshika, K. 108 139 141 143 144

146 147 195 196 197

198 199 200 201 202

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B

Baecher, G.B. 383Baguelin, F. 17

Baldi, G. 65 386

Bang, S. 228

Banerjee, P.K. 167

Barltrop, N.D.P. 209 211

Bauduin, C. 386

Bergfelder, J. 340

Bieniawski, Z.T. 111 112

Bhushan, K. 105

Bogard, D. 168 176 245

Bolton, M.D. 193

Bouafia, A.A. 333

Boughton, N.O. 273

Bowles, J.E. 191

Bowman, E.R. 63

Bransby, M.F. 245

Briaud, J.-L. 105

Broms, B.B. 13 14 245 403 405

406 407 409 410 411

413 414 415 416 417 Brown, D.A. 6 13 168 178 181

182 183 245

C

Canadian Geotechnical Society 109 168

Capozolli, L. 348

Caputo 156 157

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Carter, J.P. 109 193

CERC 210 212

Chameau, J.L. 267Charles, J.A. 273

Chen, L.T. 244

Chow, Y.K. 193

Coleman, R.B. 315

Comité Euro-International du Béton 125

Committee on Piles Subjected

to Earthquake 290 472 Costanzo, D. 388

Cox, W.R. 73 139 170 298 350

377 461 462

Coyle, H.M. 152 160 164

D

D’Appolonia, E. 150

Davies, T.G. 167

Davis, E.H. 12 17 150 151 167

192

Davis, L.H. 312

Davisson, M.T. 301De Beer, E.E. 150 153 205 305 388

De Clerq, Y. 19 20

De Cock 156

Deere, D.V. 110 111 112

Desai, C.S. 244

De Sousa Coutinho, A.G.F. 344

de Sousa Pinto, N.L. 214Det Norske Veritas 17 234

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Dixon, D.A. 170

Drabkin, S. 11

Drnevich, P. 193Duncan, J.M. 14 15 100 101

Dunnavant, T.W. 69 70 71 105

E

Einstein, H.A. 215

Emrich, W.J. 236 237 238

Endley, S.N. 15

Eurocode 2 125

Eurocode 3 127

Eurocode 7 1 340 477 483

Evans, Jr., L.T. 14 100 101

F

Fahey, M. 193

Feld, J. 382 388

Fleming, W.G.K. 155 157 192 193

Focht, Jr., J.A. 17 74 75 167 168

169 245 Foott, R. 337

Francis, A.J. 138

Franke, E. 170 171 333 335 388

390 462

Fu, S.L. 207 208

Fuchs, R.A. 210

Fukouka, M. 267

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G

Gabr, M.A. 288Garnier, J. 333

Gazetas, G. 11

Gazioglu, S.M. 105

George, P. 17

Georgiadis, M. 95 96

Gleser, S.M. 16

Gooding, T.J. 307

Griffis, L. 258

Gularte, R.C. 214

H

Ha, H.S. 245Hadjian, A.H. 10

Haliburton, T.A. 252

Hancock, T.G. 315

Hansen, B. 137

Hansen, J.B. 62 63

Hanson, W.E. 78

Hardin, B.O. 193Hassiotis, S. 267

Hetenyi, M. 12 15 23 24

Hill, R. 245

Hognestad, E. 125

Holeyman, A.C. 337 340

Horne, M.R. 127 128 129

Horvath, R.G. 110 111

Hrennikoff, A. 138 139

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Hull, T.S. 12

Hunter, A.H. 301

Hvorslev, M.J. 236 337

I

Ingram, W.B. 176

Ismael, N. 100 319

J

Jamiolkowski, M. 17 57 148 304 386

Japan Road Association 168

Jessberger, H.L. 333

Johnson, G.W. 57

K

Karol, R.H. 137

Kausel, E. 11

Kaynia, A.M. 11

Kelley, A.E. 176

Kenley, R.M. 348

Kenny, T.C. 110

Kerisel, J.L. 282

Kimura, M. 173

Koch, K.J. 167 168 169 245

Kooijman, A.P. 13

Kotthaus, M. 333

Kraft, Jr., L.M. 152 161 193

Kubo, K. 108 147 148

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Meyer, B.J. 294 313

Meyerhof, G.G. 385 388

Mitchell, G.M. 304 307Moore, W.L. 214

Morison, J.R. 209 210 212

Morrison, C.E. 150 154

Morrison, C.S. 183 184 185 186

Mosher, R.L. 164

Murchison, J.M. 105

Murphy, B.S. 170Mylonakis, G. 11

N

National Highway Institute 393 394

Nyman, K.J. 110 322

O

Oakland, M.W. 267

O’Neill, M.W. 69 70 71 105 116

139 142 160 161 162

164 165 166 167 178 193 194 195 245

P

Palmer, L.A. 15

Parker, Jr., F. 139 142

Peck, R.B. 78 110 111 194 245

252 382 388 397 398

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Stevens, J.B. 105

Stewart, D.P. 244

Stokoe II, K.H. 11 57 337 386Sulaiman, I.H. 152 164

Sullivan, W.R. 105

Sybert, J.H. 215

Szechy, C. 382 388

T

Taylor, R.J. 228

Terashi, M. 333

Terzaghi, K. 11 12 16 58 67

74 75 137 214 245

246 249 294 388 410

416 417

Thompson, G.R. 13 62 63

Thompson, J.B. 15

Thornburn, T.H. 78

Thurman, A.G. 150

Timoshenko, S.P. 15 31 67

Todeschini, C.E. 125

Tomlinson, M.J. 191Townsend, F.C. 188 189 306

Turzynski, L.D. 138

V

Van Impe, W.F. 19 20 55 56 148

149 150 151 152 153

154 175 191 193 304

305 340 386

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Vanneste, G. 116

Van Weele, A.F. 150 151

Vesic, A.S. 165 410Viggiani, C. 193

Vijayvergiya, V.N. 161

W

Wang, S.T. 9 70 71 115 173

174 244 245 254 463

Ward, W.H. 111 148 149

Wardle, I.F. 288 301

Welch, R.C. 70 88 89 115 285

376 377

Whitman, R.V. 383

Wiegel, R.L. 215

Wilson, S.D. 274

Wood, D. 17

Woods, R.D. 11

Wright, S.G. 13

Wright, S.J. 382 385

Wroth, C.P. 19 57 148 193

Y

Yamashita, K. 193

Yashimi, A. 173

Yegian, M. 13

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SUBJECT INDEX

Index Terms Links

A

Active pile 2

Anchored bulkhead 251

Anchoring pile for a ship 226

Axially loaded single piles 148

Analytical model 158

Differential equation for analysis 157

End bearing in cohesionless soils 165

End bearing in cohesive soil 161

Side resistance in cohesionless soil 163

Side resistance on cohesive soil 160

Stiffness curves for soil 152

Load-settlement behavior 153

Shaft resistance flexibility factor 155

Base flexibility factor 155

B

Bending-moment curves 4

Differentiation and integration 4 374

Examples from experiment 372

Bending moment, ultimate M ult 121

Reinforced-concrete section 130

Steel H-section 126

Steel pipe 129

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Bending stiffness, E p I p 121 360

Reinforced-concrete section 130

Reinforced-concretesection, approximation 132

Boundary conditions at pile head

Shear and moment 34

Shear and rotation 35

Shear and rotational restraint 36

Moment and deflection 57

Breasting dolphin 3 222Bridge foundations 3

Broms method 13

C

Calibration of test piles 357 373

Case studies

Cohesive soils with no free water 282

Bagnolet 282

Brent Cross 288

Houston 285

Japan 290

Cohesive soil with free water 291Lake Austin 291

Manor 296

Sabine 294

Cohesionless soils 298

Arkansas River 301

Garston 301

Mustang Island 298

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Case studies ( Cont. )

Layered soils 307

Alcácer do Sol 311 Apapa 315

Florida 312

Talisheek 307

Soil with cohesion and friction 319

Kuwait 319

Los Angeles 320

Weak Rock 322 Islamorada 322

San Francisco 324

Centrifuge 333

Cone penetrometer 337

Consequences of the failure of a foundation 382

Constitutive modeling of in situ soil 5

Current Loading 213

Cyclic loading influence

Clay 69

Sand 71

D

Decay of modulus of soil, E s 54

Design factors 397

Diameter effect 69

Difference-equation solution 32 33

Differential equation for beam-column 23

Dilatometer 338

Dimensional analysis 39

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E

Earth Pressures 243Equivalent diameter for non-circular

cross section 119

F

Factors of safety 388

Global approach 388

Load and resistance factors 393

Partial safety factors 390

Failure of a foundation 269

Finite-element method 5 13 68

Forces from moving soil 244

G

Global approach to safety 388

Global loading 1

Ground settlement 371

Lateral loading 371

Pile driving 371

Groups of piles under axial load 190

Classical form of interaction factors 192

Equivalent pier method 192

Equivalent raft method 191

Influence coefficients 190

Interim recommendations 194

Modified interaction factors 193

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Groups of piles under axial load ( Cont. )

Review by O’Neill 193

Review by Van Impe 193Groups of piles under lateral load, distribution

of load to individual piles 135

Method of prediction 139

Review of theories

Aschenbrenner 139

Asplund 138

Culmann 137 Francis 138

Hrennikoff 138

Nokkentved 137

Radosovljavic 138

Reese & Matlock 139

Reese & O’Neill 139

Saul 139

Turzynski 138

Vamdepitte 137

Wintergaard 137

Groups of piles under lateral load, efficiency

of closely spaced piles 2 165

Comparisons of experiments

with theory 175

Experiments 173

Method of prediction 166 173

Groups of piles under lateral load, experiment

with batter piles 193

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H

High-rise structures 3

I

Ice Loading 215

Initial stiffness of p- y curves 57

Installation of piles 339 363

Influence on soil properties 150

Instrumentation for piles 282 353

Instrumentation for testing 344

Interaction with superstructure 45

L

Limit analysis 1

Limit-state conditions 1 381

Length of pile, influence 32

Loading from waves 2

Loading from wind 2

M

Mat foundation supported by piles 256

Models for single piles under lateral load

Elastic pile and elastic soil 11

Rigid pile and plastic soil 13

Characteristic load method 14 Nonlinear pile and p- y model for soil 15

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Mooring dolphin 3

N

Nondimensional coefficients 44

Nondimensional solution 46

O

Offshore platform 2 232

Overhead signs 3 218

P

Partial safety factors 390

Passive pile 2

p- y Curves

Effect of installation on a batter 108 147

Examples from field experiment 4

Experimental methods for acquisition 72

Layered soils 94

McClelland & Focht for clay 75

Nondimensional methods for acquisition 73

Sand above and below the water table 91

Sloping ground 105

Soft clay in presence of free water 78

Soil with both cohesion and friction angle 99

Stiff clay in presence of free water 81

Stiff clay with no free water 88

Stiffness of clay, ε 50 76

Terzaghi recommendations 74

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p- y Curves ( Cont. )

Typical curve 4

Typical set 47 Weak rock 109

Peer review 387 401

Penetrometer 334 368

Piles in a settling fill 272

Pressuremeter 338

Program for testing under lateral load 334

Proof piles 334Production piles 334

Q

Quality control 387

R

Raked (batter) piles 1

Reaction modulus 3

Relative stiffness factor 43

Retaining wall supported by piles 246

Risk management 387

S

Safety be method of load and resistance factors 393

Safety coefficient 384

Scour of clay due to cyclic loading 8

Scour of soil (erosion) 214

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Thi h b f tt d b K l t id i i ti

Secant-pile wall 3

Serviceability load 1

Shearing force at bottom of pile 108Ship impact 3 216

Slope stabilization with piles 3 266

Sign conventions 25 26

Soil characterization 385

Soil resistance (reaction) 1 4

Soil stiffness 1

Soil-structure interaction 1Standard penetration test 338

Stress-deformation of soil 54

Stress-strain curve for concrete 125

Stress-strain curve for structural steel 125

Structural collapse 1

Subgrade modulus 64 68

Theoretical solution 66

Subsurface investigation 336 359

T

Tangent-pile wall 3

Techniques for testing under lateral load 340 366Tests of piles under lateral loading 3 340 350

Types of lateral loading of piles

Cyclic 8

Dynamic 10

Seismic 11

Static 7

Sustained 10