january 1986 prof.dr.ir. a. verruijt

x2-ct2. I

January 1986

T H Delft Delft University of Technology

Offshore soil mechanics

Prof.dr.ir. A . Verruijt

z:

Afdeling der Civiele Techniek Vakgroep Waterbouwkunde Sectie Geotechniek

:::

TECHNISCHE HOGESCHOOL DELFT

Laboratori"LllII voor Geotechniek

OFFSHORE SOIL MECHANICS

A. Verru.ijt

Delft, 1985

Acknowledaements :

uitgave april 'as

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Prlntstar, copyrlaht by Mlcrostar Inc., Emeryville, California

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1

86 xZ -ctz.I 156000 f 9,50

Offshore Soil Mechanics

CONTENTS

1. 2. 3.

4.

5.

6.

7.

8.

9.

10. ll.

Introduction .............. . Subgrade modulus of soils Linear flexibility of piles 3.1 Axially loaded pile 3.2 Laterally loaded pile Dynamic response of piles 4.1 Axially loaded pile 4.2 Laterally loaded pile Elasto-plastic axial response 5.1 Description of the model Elasto-plastic lateral response 6.1 Active and passive soil pressure 6,2 API 6.3 Generalized elasto-plastic model A numerical model for axially loaded piles 7.1 Basic equations .... 7.2 Computer program 7.3 Example A numerical model for laterally loaded piles 8.1 Basic equations 8.2 Computer program 8, 3 Example .....•••.•..•. 8.4 Approximate solution Damping ............. , ... . 9.1 Viscoelastic damping 9.2 Numerical model 9.3 Damping in lateral loading Pile driveability Gravity foundations

11.1 Bearing capacity 11.2 Cyclic loading

References .............. . Appendices

API Reconunended practice, Foundation design DNV Rules, Appendix F, Foundations J. Brinch Hansen, Bearing capacity F.P. Smits, Geotechnical design of gravity structures

1 2 4 4 6 8 8 9

10 10 13 13 14 16 19 19 23 26 28 28 32 36 38 41 41 45 47 52 53 53 54 58


1, INTRODUCTION

The recent development of offshore engineering, in particular the construction of platforms for drilling and production of oil and gas, leads to a growing demand for realistic predictions of the behaviour of the foundation, and of the sea bottom. Even though the soils usually encountered on the sea bottom are of the same nature as the soils on land (mainly sand and clay) there are various factors that lead to a difference in the approach and in the type of problems to be considered. The main factors are: - Investigation of the soil in situ is much more difficult offshore than it

is on land, - One of the main loading conditions to be considered is of a cyclic nature,

due to wave action. - The bearing capacity of piles consists mainly of friction. In this report these factors will be discussed, for three types of structures: pile foundations, gravity foundations, and pipelines. Special attention will be paid to the determination of parameters describing the soil behaviour suitable for the analysis of the structures as a whole.

1


2. SUBGRADE MODULUS OF SOILS

Many problems of foundation engineering can be formulated in a rather simple form when using a so-called subgrade modulus to characterize the soil response. In this section this subgrade modulus is correlated with other elastic properties of the soil.

Consider a square footing, with dimensions b*b. The pressure at the base of the footing is denoted by p. The stresses in the soil under the footing will decrease with depth, because of the spatial spreading. It can be expected that at a depth z the stresses will be distributed over an area b+az, where the dimensionless constant a is of the order of magnitude of 1. Thus the stresses at depth z are

(J' = p (l+az/b) 2

Assuming a linear relationship between stresses and strains, with a modulus of elasticity E, the strains are

Because t=dw/dz, where w is the vertical displacement, one now obtains, after integration from z=O to z=oo, the following expression for the displacement of the footing,

w = ~ aE

If this is compared with the usual expression in terms of a subgrade modulus c,

w = p/c

it follows that the parameters are related by

c = aE/b

Because the value of a is about 1 this means that the subgrade modulus can be considered to be approximately equal to the modulus of elasticity, divided by the dimension b. In the case of a bemn on elastic foundation the subgrade constant k is usually defined as

k = cb

This means that k and E are related as follows.

k.= aE

It should of course be noted that this correlation is only justified as a first approximation. In engineering practice it is advisable to determine

2


the actual value·on the basis of a site investigation. The formulas given above can serve as an estimate, however, if no further information is available.

The modulus of elasticity of a soil is sometimes correlated with the compression constant C in Terzaghi's logarithmic formula

t = _cl log( L ) <To

where <ro is the original stress and <r the actual stress. small stress increment one may write <r=<ro+dcr,, and then

In the case of a

t = _cl log(l + d<r) ~ d<r <To C<ro

If this is compared with the elastic relation t=d<r/E it follows that

E = C<ro ..

Thus the modulus of elasticity of a soil can be constant C and the stress level <ro are known. C=200 for sand, and C=lO to C=50 for clay,

3

estimated if the compression Common values are C=50 to


;L___iINEAR FLEXIBILITY OF PILES

Although the response of a foundation pile to a load applied at its top is in general strongly non-linear, it is of some interest to consider the linear case first. The purpose of these considerations is to serve as a reference for later non-linear calculations, and to provide some .simple approximate formulas.

3.1 Axially loaded pile

Consider a pile of constant cross section, consisting of a homogeneous linear elastic material, with modulus of elasticity Ep, The cross-sectional area is denoted by A, and the circumference by o. The normal force Nin the pile can be related to the friction .r by the equation of equilibrium

dN + n> = O dz

where. is the shear stress acting on the pile. Assuming a linear relationship between this shear stress and the displacement w of the pile one may write

T = -cw

where the constant c has the character (and dimension) of a subgrade modulus. The minus sign has been introduced to express that the shear stress acting upon the pile is opposed to the direction of the displacement. The normal force Nin the pile can be expressed into the displacement w by Hooke's law,

dw N = EA --dz

where EA is the extension stiffness of the pile, defined as

Combination of the three equations given above leads to the following differential equation

EA dZw - cOw = 0 dz2

The solution of the differential equation, for the case of a pile of infinite length, loaded at its top by a force P is

4


Ph w = EA exp(-z/h)

where his a parameter defined by

h = v(EA/cO)

"The normal force in the pile is

N = -P exp(-z/h)

and the shear stress distribution is

p r = Oh exp(-z/h)

The maximum shear stress occurs at the top of the pile, and its magnitude is P/Oh.

The value of the parameter h determines whether a pile can be considered as infinitely long. For a tubular pile the cross sectional area A is about Od, where d is the wall thickness of the pile, If furthermore the subgrade modulus c is expressed as c=Es/D, where Es is the modulus of elasticity of the soil, and Dis the pile diameter, one obtains

h = v(DdEp/Es)

In general the pile will be much stiffer than the soil, say by a factor 2000. For a pile having a diameter of 1 meter and a wall thickness of 50 nun the value of his about 10 meter. Foundation pile of offshore structures are usually much longer, so that they can indeed be considered, as far as their axial flexibility is concerned, as infinitely long.

The flexibility of the pile as a whole can be expressed by a single spring constant K relating the applied force P and the displacement w,

w = P/K

where

K = EA/h = v(EAcO)

For a tubular pile this can also be written as

K = nv(EpEsDd)

where again the subgrade modulus c has been replaced by Es/D, circumference O of the pile has been expressed as ~D, and nDd,

5

and where the the area as


3,2 Laterally loaded pile

For a laterally loaded pile the basic differential equation is the well known equation for a beam on elastic foundation,

EI d4

!! + ku = f dz4

where EI is the bending stiffness, k the subgrade constant (k=cD), and f the lateral load. The solution for a beam of infinite length, loaded by a lateral force Q at its top is

where}. is a characteristic length, defined by

>.4 = 4EI/k

For a tubular pile one may write I=¼nD3 d, and therefore, if k is replaced by Es (see chapter 1),

If Ep/Es=2000 and d/D=0.05 this gives >./D=3.5. From the theory of beams on elastic foundation· it is known that the "wave length" of the deflection curve is 2n>.. In this case this is about 22D. Thus a pile can be considered as infinitely long, as far as lateral flexibility is concerned, if it is longer than about 20 pile diameters. This will often be the case.

The response of the pile top can be expressed by a spring constant K such that

u = Q/K

where

For the more general case of loading by a lateral load Q and a moment Tone may write

where now

u = Q/K-T/R

du/dz = -Q/R+T/L

6


It is interesting to note that the lateral stiffness factor K is mainly determined by the subgrade modulus, whereas the rotational stiffness factor L is mainly determined by the bending stiffness EI. This can be seen by noting that K=kA/2, where A is defined by (4EI/k) 1 1 4 , which means that K is proportional to k3 1 4 , and to (EI) 1 1 4 , This means that the dependence on k is 3 times stronger than that on EI. On the other hand the value of L is proportional to (EI) 3 1 4 and k1 1 4 , which means that the rotational flexibility depends mainly on the bending stiffness EI of the beam. This difference· in behaviour can be explained on an intuitive basis by notating that for rotation the main deformation characteristic is the curvature of the beam, whereas for the lateral displacement of the pile top the main phenomenon is the generation 9f soil resistance. It will be seen later, see paragraph 8, that these conclusions are not valid if the soil is considered as an elasto-plastic material. Then the soil strength will be found to be the principal factor,

7


4. DYNAMIC RESPONSE OF PILES

The external loads on an offshore structure consist for a large part of dynamic forces, due to wave action. On the North Sea, and in many other locations in the world, these dynamic loads have a typical period of the order of magnitude of 10 seconds. In this chapter the importance of certain inertia effects for a pile foundation are investigated.

4.1 Axially loaded pile

In the case of an axially loaded pile, supported by friction of the soil, the basic differential equation for the axial displacement w of the pile is, when ,inertia of the pile is taken into account

EA::: - cOw = pA :::

where p is the density of the pile material. It is asswned that the load is periodic, with angular frequency w, so that the response of the pile is also periodic, with the same frequency, Then one may write

w = W(z)exp(iwt)

The differential equation for W(z), the complex amplitude of the displacement is

EA d2_W - cO[l- pAcOw2 )W = 0 dz2

The relative importance of inertia of the pile can be investigated by estimating the value of the dimensionless factor pAw2 /cO. If this is a small quantity inertia effects can be disregarded. If it is greater than 1 the basic behaviour changes from exponentially damped to sinusoidal fluctuations. If this factor is equal to 1 resonance can be considered to occur, As mentioned before the usual type of an offshore pile is a steel tubular pile. For such a pile the area divided by the circumference is about equal to the wall thickness (A/O=d). Furthermore the subgrade modulus c can be estimated as c=Es/D, where Es is the modulus of elasticity of the soil, and D the pile diameter, The modulus of.elasticity of the soil can be expressed, approximately, in terms of Terzaghi's compression factor C by the correlation

Es = C<T

where <T is the stress level in the soil. This can be estimated to be

<T = ps gz

where ps is the (effective) soil density, g is the gravity constant, and z is the local depth, One now obtains

8


pAw2 = Q!. dDw2

cO p Cgz

Thus the resonance frequency wo is

wo = v((ps/pp)Cgz/(dD)]

This seems to be a large quantity, In order to obtain a low estimate one may assume that ps/pp=0.5, that C=50, z=D, anq that d=O,l m. Then one obtains wo=50/s. The major frequency to be considered is the one corresponding to a wave having a period of about 10 seconds, The angular frequency w then is about 0.6/s, This is much smaller than the critical frequency fJo,

The conclusion can be that loading conditions with a period greater than about 0.5 seconds can be considered as quasi-static, i.e. inertia effects of the pile can be disregarded.

4,2 Laterally loaded pile

A similar consideration for a laterally loaded pile leads to a formula for the critical (resonance) frequency fJo of the following form

wo = v[k/(ppA)

Using the same estimations as used in the previous paragraph this can also be written as

wo = v[(ps/pp)Cgz/(2ndD)J

With C=50, z=D and d=O.l m the critical frequency is now found to be about 28/s, which means that for lateral loading inertia effects of the pile can also be disregarded.

The general conclusion from this paragraph can be that for the analysis of the response of a pile foundation of an offshore structure to wave action, inertia effects of the pile can usually be disregarded. The analysis can be static or quasi-static.

9


5. ELASTO-PLASTIC AXIAL RESPONSE

For a realistic analysis of the load transfer from the top of a pile to the soil it is necessary to include the main non-linear effect, which is the possibility of local slip of the pile relative to the soil. The skin friction between pile and soil in general is limited in both directions, and these limiting values are usually denoted as positive and negative skin friction. The positive friction can be determined in a relatively easy way by the cone penetrometer test (CPT). In this chapter a simple model incorporating elasto-plastic deformations for an axially loaded pile is presented.

!h_L_pescription of th~ model

The simplest elasto-plastic model assumes a linear branch between two limiting values, as shown in figure 5.1.

tp ........... ,,----~--

DisplaceMent

-------· · · · · · · · · · · · ta

Figure 5,1. Elasto-plastic model.

The response is characterized by three parameters the maximum positive friction (tp), the maximum negative friction (ta), and a spring constant. This spring constant can conveniently be described by the length of the elastic branch. This represents the displacement necessary to generate the maximum (positive or negative) friction. This value, often denoted as the quake, can often be estimated more easily from a soil investigation than the spring constant itself, The use of this quantity has the additional advantage that the rigidity of the soil will automatically increase with depth if the quake is constant, and the maximum friction is increasing with depth, which is often observed in practice. It may be expected that the length of the linear branch varies within narrower bounds than the spring constant, and therefore spring constants will be defined through parameters such as

10


the quake in this report,

The values of the maximum positive and negative friction are usually almost equal. It is sometimes observed, however, that the negative friction is somewhat smaller than the positive friction, and this may be attributed to the fact that negative friction is usually associated with tension in the pile, during which the pile contracts, whereas positive friction in the pile occurs ·during compression, with the pile dilating laterally. In order to take this possibility into account the soil will be characterized by independent parameters for positive and negative friction, see figure 5.2.

Shear stress

tp ........... ·------

wa DisplaceMent

WP

ta

Figure 5.2. Asymmetric elasto-plastic model.

In figure 5.2 the positive and negative quake (wp and wa) are considered to be completely independent. A mathematical formulation of this elastoplastic behaviour is as follows (in PASCAL).

if (w>O) and (w~wp) then t:=-tp*w/wp; if (w>=wp) then t:=-tp; if (w<O) and (w>-wa) then t:=ta*w/wa; if (w<=-wa) then t:=ta;

The sign convention used here is that the displacement w is positive in downward direction (the positive z-direction), and that the shear stress t acting on the pile is also positive when acting in downward direction,

A complete formulation of the model should also be applicable during unloading and subsequent reloading, and therefore this behaviour should also be specified, The formulation given above, when incorporated in a computer program, merely describes the behaviour in monotonic loading, and does not correctly describe the possibility of permanent plastic displacements. The general notion of plastic deformations is that the response in unloading (and in subsequent reloading up to the previous stress level) follows a branch parallel to the elastic branch. This can most conveniently be formulated by introducing a possible shift of the origin, which occurs during any plastic deformation. In order to take into account the different behaviour

~

11


in the positive and negative branches one may consider the a summation of two independent elasto-plastic springs, one tp and an initial value wo, and the other described by wa, complete fonnulation now is as follows.

if (w-wo>O) and (w-wo<wp) then t:=-tp*(w-wo)/wp; if (w-wo>=wp) then

begin t:=-tp; wo:=w-wp

end; if (w-wo<O) and (w-wo>-wa) then t:=ta*(w-wo)/wa; if (w-wo<=-wa) then

begin t: =ta; wo:=w+wa

end;

soil response as described by wp,

ta and wo. A

As can be seen from this algorithm the value of wo (which is initially zero) is changed whenever plastic slip occurs, so that the length of the elastic branch remains constant during the deformation history,

12


6 ELASTO-PLASTIC LATERAL RESPONSE

In this paragraph the response of the soil to a laterally loaded pile is considered. Two possibilities of describing this response will be presented, and these will be correlated, The mathematical description of the models will be given in PASCAL.

6.1 Activ~ and passive soil pressures

An elementary way to describe the soil response is by using quantities such as active, passive and neutral soil pressure, which are widely used in classical soil mechanics, In this model it is postulated that a displacement of a pile element, say towards the right, will generate a local response on both sides of the pile. On the right side a force will act towards the left, and this force cannot be smaller than the active pressure (fa), and not greater than the passive pressure (fp). Similarly a force acting towards the right will be acting on the left side of the pile, and this force cannot be smaller than fa, and not greater than fp, Furthermore it is assumed that the soil pressures in the case of zero displacement are fn (neutral soil pressure) on both sides. For intermediate values of the displacement the relationship between force and displacement is assumed to be linear, The force acting on the right side (fr) can now be described as follows.

if (u>-ua) and (u<up) then fr:=-fn-(fp-fa)*u/du; if (u>=up) then fr:=-fp; if (u<=-ua) then fr:=-fa;

in which O<fa<fn<fp, du>O, and

ua:=du*(fn-fa)/(fp-fa); up:=du*(fp-fn)/(fp-fa);

The fundamental parameters are fa,fn,fp and du, where du is the total length of the linear branch, to be denoted as the stroke, This is the displacement necessary to pass from active to passive pressure. The minus sign in the expressions for fr indicate that this force is acting in negative x-direction.

In a similar way the force on the left side is described by the following algorithm.

if (u<ua) and (u>-up) then fl:=fn-(fp-fa)*u/du; if (u<=-up) then fl:=fp; if (u>ua) then fl:=fa;

This force acts in positive x-direction.

In order to complete the formulation it is again necessary to also specify the behaviour during unloading and reloading, Again it is assumed that

13


during unloading the non-linear behaviour is such that the soil reacts elastic (stiff), This is described by introducing two additional parameters ur and ul, which express the pennanent displacements of the right and left side springs, respectively. The complete formulation is now as follows.

if (u-ur>-ua) and (u-ur<up) then fr:=-fn-·(fp-fa)*(u-ur)/du; if (u-ur>=up) then

begin fr: =-fp; ur:=u-up

end; if (u-ur<=-ua) then

begin fl: =-fa; ur:=u+ua

end; if (u-ul<ua) and (u-ul>-up) then fl:=fn-(fp-fa)*(u-ul)/du; if (u-ul<=-up) then

begin fl: =fp; ul:=u+up

end; if (u-ul>=ua) then

begin fl: =fa; ul:=u-ua

end; f: =fr+fl;

The physical meaning of the parameters ul and ur is the value of the displacement for which the soil pressure acting on the left or right side of the pile is equal to the neutral value. These values constantly change during plastic deformation. In a complete numerical model, in which the response of a pile to a varying lateral load is calculated, the values of ul and ur must be updated after each loading step.

§..2 API

A description which more closely follows the Recommendations of the American Petroleum Institute (API) is to assume that the total response of the soil can be described by ·a force-displacement relation consisting of two linear branches and a constant maximum value. Such a relation can be characterized by the location of two points in the force displacement diagram, see figure 6. l. The bi-linear elasto-plastic relation ·is characterized by the coordinates of the points 1 and 2: ul and fl, respectively u2 and f2. It should be noted that the directions of displacement and force in general are opposed; hence in a formal mathematical formulation such as used in the previous paragraph, a minus sign will appear. If it is now assumed that the total response is generated by a physical model as described in the previous paragraph, it follows that

14


ul:=ua; u2:=up; fl:=-fa+fn+(fp-fa)*ua/du; f2:=fp-fa;

These relations can be established by noting that the discontinuity in point 1, assuming a displacement to the right, must be due to the fact that on the

NorMal stHss

Displacei.ent

Figure 6.1. API-model.

left side the minimum (active) soil pressure is reached. The second discontinuity, in point 2, corresponds to reaching the maximum (passive) soil pressure on the right side. Thus the API model seems to be a straightforward representation of the physical model used in developing the algorithms in the previous paragraph.

Relations between parameters

The two models seem to be fully equivalent, both characterized by four parameters: fa, fn, fp and du for the first model, and fl, f2, ul and u2 for the second model. This is not the case, however, as can be seen from a more detailed inspection of the properties of the two modeis.

It follows from the description of the first model that

(fn-fa)=(fp-fa)*ua/du,

from which it can be derived that

fl=2*(fp-fa)*ul/du,

f2=fp-fa.

It now follows that

15


2*(f2-fl)/(u2-ul)=fl/ul.

This means that the slope of the first branch of the force-displacement relation is twice the slope of the second branch. Actually this is n9t so surprising, taking into account the physical model underlying the basic formulas, which consists of two elasto-plastic springs. In the first branch both springs are elastic, and in the second branch only one elastic spring remains. It is now assumed that the basic (physical) model is valid, so that the relationship obtained above must indeed be satisfied. This implies a restriction in the API model such that the first slope is exactly twice the second slope, which entails that only three parameters remain, say ul, fl and f2. The value of the displacement u2 now follows from the condition

u2:=ul+2*ul*(f2-fl)/fl;

It remains to explain the apparent occurrence of four parameters in the original first model. This can be done by first noting that it follows from the description of the two models that

ua=ul,

up=u2,

du=up-ua.

Furthermore it can be seen that

fl/ul=2*(fp-fa)/du,

which means that the value of (fp-fa) can be determined from fl and ul,

(fp-fa)=0.5*fl*du/ul.

The value of (fn-fa) can also be determined, because (fn-fa)=(fp-fa)*ua/du.

One of the three values fa, fn or fp can be chosen arbitrarily (e.g. fa=O) without affecting the total force on the pile. Actually this is physically fully acceptable: the first model indeed contains only three parameters, because the total soil response is composed of a superposition of the forces left and right, which means that a constant value may be added to the force on both sides without affecting the resultant force. It can be concluded that only the differences (fn-fa) and (fp-fa) are physically meaningful.

6.3 Generalized elasto-plastic model

The elasto-plastic model introduced in section 6,1 has the advantage that its parameters have a physical meaning which closely follows classical soil mechanics, using parameters such as active and passive soil pressure. This model can be generalized so that it covers all the possibilities of the API model in the following way.

16


A model with an elastic branch which is limited on both sides is in general a schematization of a curved relation between displacement and soil pressure. A possible refinement in the schematization is to assume that the slope of the linear branch is different in the active and passive sections, see figure 6.2. The model illustrated in this figure contains four basic parameters, namely the values of ua, up, fp-fn, and fn-fa. As stated before the actual value of the parameter fn can be considered to be irrelevant, because in the neutral state this stress acts on both sides of the pile.

NofMal stress

'tp·''' ''' '''

ta DisplaceMent

M ~ Figure 6.2, Generalized elasto-plastic model.

The model illustrated in figure 6.2 can be described mathematically as follows.

if (u-ur>=O) and (u-ur<up) then fr:=-fn-(fp-fn)*(u-ur)/up; if (u-ur>=up) then

begin fr:=-fp; ur:=u-up

end; if (u-ur<=O) and (u-ur>-ua) then fr:=-fn-(fn-fa)*(u-ur)/ua; if (u-ur<=-ua) then

begin fr:=-fa; ur:=u+ua

end; if (u-ul<=O) and (u-ul>-up) then fl:=fn-(fp-fn)*(u-ul)/up; if (u-ul)<=-up) then

begin fl:=fp; ul:=u+up

end; if (u-ul>=O) and (u-ul<ua) then fl:=fn-(fn-fa)*(u-ul)/ua; if (u-ul>=ua) then

begin

17

fl:=fa; ul:=u-ua

end; f:=fl+fr;


The relationship with the API model can be established by noting (see also the previous section) that

ul=ua,

u2=up,

fl=(fn-fa)+(fp-fn)*ua/up,

f2=fp-fa.

The value given for fl can be derived from the description given above, with u-ur=ua and u-ul=ua. In that case fl=fa fn)*ua/up. By adding these values (and changing the sign) for fl is obtained.

The inverse relations are

ua=ul,

up=u2,

(fp-fn)=(f2-fl)*up/(up-ua),

(fn-fa)=(up*fl-ua*f2)/(up-ua).

of the model and fr=-fn-(fpthe expression

All these quantities can be shown to be positive (as required), provided that the general shape of the API curve is as shown in paragraph 5.2, with the slope of the seconq branch being smaller than the slope of the first branch, which is almost self-evident.

The generalized model presented in this section has been found to be fully equivalent to the API model. Because its parameters have been defined in terms of familiar soil properties such as active, passive and n~utral soil properties, it has been found to be possible to formulate the behaviour of the model for a general type of deformation, including the possibilities of repeated loading and unloading.

18


7. A NUMERICAL MODEL FOR AXIALLY LOADED PILES

In this section complete numerical model for the transfer of axial loads to the soil is developed. Load transfer takes place through friction along the shaft of the pile, and at the point by point resistance. The characteristics of both forms of stress transfer are in general non-linear. An elementary computer program will be presented.

7.1 Basic equations

Consider a pile of length L, loaded at its top by a force P, see figure.7.1. The pile is subdivided into N small elements, each having a length D(I), and axial stiffness EA(I). The elements are numbered from 1 to N, and the nodal points are numbered from Oto N, so that element I is located between points I-1 and I. The basic equations can be developed in the following way.

1-1

f(l-1) ... (--

( R(I)

--) f(I)

Figure 7.1. Pile with axial load.

The forces acting on element I are the normal forces F(I-1) and F(I) at the two.ends, and a friction force R(I), acting along the shaft. Equilibrium requires that

F(I)-F(l-1)-R(I)=O, (I=l. .. N).

The strain in this element is determined by the average normal force N.( I),

N(I)=(F(I-l)+F(I))/2.

With Hooke's law the increment of the length of this element is

DW(I)=N(I)*D(I)/EA(I),

19


where D(I) is the length of the element, and EA(!) its axial stiffness. Because DW(I)=W(I)-W(I-1), where W(I) is the displacement of point I, one may write

(EA(I)/D(I))*(W(I)-W(I-l))=N(I)=(F(I)+F(I-1))/2,

(EA(I+l)/D(I+l))*(W(I+l)-W(I))=N(I+l)=(F(I+l)+F(I))/2,

Subtraction of these two equations gives, after multiplication by -2,

where

A(I+l)*W(I+l)+A(I)*W(I)+A(I-l)*W(I-l)=-F(I+l)+F(I-1),

A(I+l)=-2*EA(I+l)/D(I+l),

A(I)=2*EA(I+l)/D(I+l)+2*EA(I)/D(I),

A(I-1)=-2*EA(I)/D(I).

The forces F(I+l) and F(I~l) can be related to the friction forces R(l+l) and R(I) by the equations of equilibrium

F(I+l)-F(I)=R(I+l),

F(I)-F(I-l)=R(I),

It follows from these two equations that

F(I+l)-F(I-l)=R(I)+R(I+l),

The basic equation can now be written as follows.

A(I+l)*W(I+l)+A(I)*W(I)+A(I-l)*W(I-1)=-R(I)-R(I+l).

This will lead to a linear system of equations, if the forces R(I) are given. In general these friction forces depend upon the local displacement, as described in section 4. It is assumed that the friction force is determined by the average displacement V(I), defined as

V(I)=(W(I-l)+W(I))/2.

Furthermore it is assumed that in the linear branch the friction force is proportional to the displacement difference V(I)-WM(I), where WM(I) is a memory function, initially zero, which represents the permanent displacements that have occurred during plastic deformations, The formulation of the model is as follows.

if (V(I)-WM(I)>-B(I)) and (V(I)-WM(I)<B(I)) then begin

S(l):=C(l)/B(I); T(I):=O

end;.

20


if (V(I)-WM(I)<=-B(I)) then begin

S(I):=O; T(I):=-C(I)

end; if (V(I)-WM(I)>=B(I)) then

begin S(I):=O; T(I):=C(I)

end; R(I):=S(I)*(V(I)-WM(I))+T(I);

Here C(I) is the maximum value of the friction (here assumed to be independent of the direction of the displacement), B(I) is the quake, i.e. the displacement necessary to generate the maximum friction force. For reasons of simplicity this parameter has also been assumed to be independent of the direction of the displacement. The parameter S(I) is a spring constant, and T(I) is a possible given force.

After completion of each lpading step the memory function WM(!) should be updated as follows.

if (V(I)-WM(I)<=-B(I)) then WM(I):=V(I)+B(I); if (V(I)-WM(I)>=B(I)) then WM(I):=V(I)-B(I);

In this way it is ensured that during unloading after plastic deformation the response is again elastic, and that in further reloading the response is elasto-plastic.

The general expression for the shear force R(I) is

R(I)=S(I)*(V(I)-WM(I))+T(I),

or

where

R(I)=S(I)*V(I)+TT(I),

V(I)=(W(I-l)+W(I))/2,

TT(I)=T(I)-S(I)*WM(I).

The system of equation now becomes

where

AA(I+l)*W(I+l)+AA(I)*W(I)+AA(I-l)*W(I-1)=-TT(I)-TT(I+l), (I=l •.• N-1),

AA(I+l)=A(I+l)+S(I+l)/2,

AA(I)=A(I)+S(I)/2+S(I+l)/2,

AA(I-l)=A(I-l)+S(I)/2.

21

. I


The system of equations is now completely defined for all interior points. The boundary conditions will next be specified,

It is assumed that at the pile top the external force is given. With Hooke's law for the first element one may write

(EA(l)/D(l))*(W(l)-W(O))=(F(O)+F(l))/2.

The normal force F(l) can be eliminated from this equation by using the equilibrium pondition for element 1,

F(l)-F(O) =R(l).

One now obtains, with F(O)=-P, and with R(l)=S(l)*V(l)+TT(l),

(2*EA(l)/D(l)+S(l)/2)*W(0)+(-2*EA(l)/D(l)+S(l)/2)*W(l)= =2*P-TT(l).

This equation can be added to the system of equations. It can be considered as equation number 0,

Pile point

At the point of the pile a resistance is assumed to be generated by any downward displacement. This resisting force is written as

F(N)=-CP*W(N)-FP,

where CP=FM/WM is a spring constant, and where FP is either zero (in the elastic branch, for small displacements) or FM (a given constant) if the displacement is greater than \'JM, the quake. The mathematical formulation of this elasto-plastic response is as follows.

if(W(N)-WP>O) and (W(N)-WP<\'JM) then begin

CP:=FM/\'JM; FP:=O

end; if (W(N)-WP<=O) then

begin CP:=O; FP:=O

end; if (W(N)-WP)=\'JM) then

begin CP:=O; FP: =FM

end;

Here it has been assumed that no tension can be transmitted at the pile point.

22


After each loading step the memory function WP should be updated, to account for the permanent plastic deformations,

if (W(N)-WP<=O) then WP:=W(N); if (W(N)-WP)=\'tM) then WP:=W(N)-\'tM;

The boundary condition can be introduced by starting from a consideration of the deformation of the last element,

(EA(N)/D(N))*(W(N)-W(N-l))=(F(N-l)+F(N))/2.

The normal force F(N-1) can be eliminated by using the equation of equilibrium of the last element,

F(N)-F(N-l)=R(N).

This gives

(2*EA(N)/D(N))*(W(N)-W(N-1))=2*F(N)-R(N).

Now by using the relation F(N)=-CP*W(N)-FP and expressing R(N) into W(N-1) and W(N) the following result is obtained

(2*EA(N)/D(N)+2*CP+S(N)/2)*W(N)+ +(-2*EA(N)/D(N)+S(N)/2)*W(N-l)=-2*FP-TT(N).

This is the final formulation of the boundary condition at the pile point. Again an equation in terms of the basic variables W(I) has been obtained, with given forces on the right hand side. This equation can be considered to be equation number N.

The system of equations, for I=O ... N, can be solved numerically if all the coefficients are known. Because of the non-linear spring behaviour this must be done iteratively, by first assuming elastic or plastic behaviour in each element, and subsequent correction, if necessary.

7.2 Computer program

An elementary computer program (ALP-1,0) that performs the calculations described above, is given in,this section, The program has been kept as simple as possible, by a restriction to a homogeneous pile in a homogeneous soil. The handling of input and output is also very simple, without any possibilities of correction of typing errors, or possible output on a printer. In the program all input is performed interactively, with the user responding to question prompts from the program. As these questions are self-explanatory no further manual is needed. Output is given on the screen of the computer, in the form of a list of values for each node, After presentation of output data the program will ask for a new value of the axial load. The program can only be stopped by an external interrupt, for instance by BREAK or AC. The program has been written in Microsoft BASIC, and has been tested ·and used successfully on various computers. No responsibility for any errors can be accepted, however.

23


Program ALP-1.0

1000 REM===== ALP-1.0 -----1010 GOSUB 2000:DEFINT I-N:B$=" Impossible":Z=4 1020 DIM X(lOO),W(lOO),F(lOO),D(lOO),S(lOO),T(lOO),WM(lOO) 1030 DIM EA(lOO),SS(lOO),TA(lOO),TB(lOO),K%(100,4),P(l00,4) 1040 DIM DA(lOO') ,DB(lOO) ,NP(lOO) 1050 INPUT"Length of the pile (m) . . . . • "; WL 1060 IF WL<=O THEN PRINT B$:GOT0 1050 1070 INPUT"Axial stiffness EA (kN) . , .• ";EA 1080 IF EA<=O THEN PRINT B$:GOT0 1070 1090 INPUT"Circumference (m) •••••••••• ";00 1100 IF OO<=O THEN PRINT B$:GOT0 1090 1110 INPUT"Maximum friction (kN/m"2) , . "; FR 1120 IF FR<O THEN PRINT B$:GOT0 1110 1130 INPUT" Quake (m) .. .. • .. .. .. .. .. "; QR 1140 IF QR<=O THEN PRINT B$:GOT0 1130 1150 INPUT"Point resistance (kN) ,, .... ";FM 1160 IF FM<O THEN PRINT B$:.GOT0 1150 1170 INPUT" Quake (m) .............. , "; UM 1180 IF UM<=O THEN PRINT B$:GOT0 1170 1190 INPUT"Nwnber of elements ...•..... ";N 1200 IF N<lO THEN N=lO ELSE IF N>lOO THEN N=lOO 1210 DA=WL/N:TA=DA*OO*FR:SA=TA/QR:CP=FM/UM:X(O)=O 1220 FOR I=l TO N:D(I)=DA:EA(I)=EA:SS(l)=SA:T(I)=TA 1230 TA(I)=TA:TB(I)=TA:DA(I)=QR:DB(I)=QR:X(I)=X(I-l)+DA:NEXT I 1240 GOSUB 2000:INPUT"Force (kN) •••.••••.•... ";FA 1250 KP=O:NQ=O:GOSUB 2000 1260 IF IQ()O AND ABS(FA)>=FB THEN 1290 1270 MP=O:CP=FM/UM:FP=O 1280 FOR I=l TO N:S(I)=SS(I):NP(I)=O:T(I)=-SS(I)*WM(I):NEXT I 1290 PRH~T"Force (kN) •...... , ..... : "; FA: FB=ABS(FA) 1300 FOR I=O TO N:FOR J=l TO Z:K%(I,J)=O:P(I,J)=O:NEXT J,I 1310 IQ=IQ+l:FOR I=l TO N-1 1320 K%(I,l)=I:K%(I,2)=I-l:K%(I,3)=I+l:K%(I,Z)=3:NEXT I 1330 K%(0,l)=O:K%(0,2)=1:K%(0,Z)=2:K%(N,l)=N:K%(N,2)=N-l:K%(N,Z)=2 1340 PRINT:PRINT"Generation of matrix ••• : +"; 1350 FOR I=l TO N-l:Al=2*EA(I)/D(I):A2=2*EA(I+l)/D(I+l) 1360 Bl=S(I)/2:B2=S(I+l)/2:PRINT 11+11

;

1370 P(I,l)=Al+A2+Bl+B2:P(I,2)=-Al+Bl:P(I,3)=-A2+B2 1380 P(I,Z)=-T(I)-T(I+l) :NEXT !:PRINT "+" 1390 Al=2*EA(1)/D(l):B1=S(l)/2 1400 P(O,l)=Al+B1:P(0,2)=-Al+Bl:P(O,Z)=2*FA-T(l) 1410 Al=2*EA(N)/D(N):B1=S(N)/2:Cl=2*CP 1420 P(N,l)=Al+Bl+Cl:P(N,2)=-Al+B1:P(N,Z)=-2*FP-T(N) 1430 PRINT: PRINT"Elimination •. , •..••. , .. : "; 1440 FOR I=O TO N:PRINT "+";:KC=K%(I,Z):IF KC=l .THEN 1570 1450 FOR J=2 TO KC:C=P(I,J)/P(I,l):JJ=K%(I,J) 1460 P(JJ,Z)=P(JJ,Z)-C*P(I,Z):L=K%(JJ,Z) 1470 FOR JK=2 TO L:IF K%(JJ,JK)=I THEN 1490 1480 NEXT JK 1490 K%(JJ,JK)=K%(JJ,L):K%(JJ,L)=O 1500 P(JJ,JK)=P(JJ,L):P(JJ,L)=O:L=L-l:K%(JJ,Z)=L 1510 FOR 11=2 TO KC:FOR IJ=l TO L:IF K%(JJ,IJ)=K%(I,I1) THEN 1540

24


1520 NEXT IJ:L=L+l:IJ=L 1530 K%(JJ,Z)=L:K%(JJ,IJ)=K%(I 1 II) 1540 P(JJ,IJ)=P(JJ,IJ)-C*P(I,II) 1550 NEXT II 1560 NEXT J 1570 C=l/P(I,l):FOR J=l TO KC:P(I,J)=C*P(I,J):NEXT J 1580 P(I,Z)=C*P(I,Z):NEXT I 1590 PRINT:PRINT:PRINT"Back substitution ....•. : "; 1600 FOR I=O TO N:J=N-I:PRINT "+";:L=K%(J,Z):IF L=l THEN 1620 1610 FOR K=2 TO L:JJ=K%(J,K):P(J,Z)=P(J,Z)-P(J,K)*P(JJ,Z):NEXT K 1620 NEXT I:PRINT:F(O)=-FA:FOR I=O TO N:W(I)=P(I,Z):NEXT I 1630 FOR I=l TO N:F(I)=2*EA(I)*(W(I)-W(I-l))/D(I)-F(I-l):NEXT I· 1640 PRINT:PRINT"Check springs";:LL=O 1650 FOR I=l TO N:A=(W(I)+W(I-1))/2:AA=A-WM(I):IF NP(I)=O THEN 1690 1660 IF NP(I)=l THEN 1720 1670 IF AA<=-DB(I) THEN 1740 1680 S(I)=SS(I):T(I)=-SS(I)*WM(I):NP(I)=O:LL=LL+l:GOTO 1740 1690 IF AA>=-DB(I) AND AA<=DA(I) THEN 1740 1700 LL=LL+l:IF AA>=DA(I) THEN NP(I)=l:S(I)=O:T(I)=TA(I):GOTO 1740 1710 NP(I)=-1:S(I)=O:T(I)=-TB(I):GOTO 1740 1720 IF AA>=DA(I) THEN 1740 1730 S(I)=SS(I):T(I)=-SS(I)*WM(I):NP(I)=O:LL=LL+l 1740 NEXT I:AA=W(N)-WP:IF MP=O THEN 1790 1750 IF MP=l THEN 1780 1760 IF W(N)<=WP THEN 1820 1770 MP=O:CP=FM/UM:FP=-CP*WP:LL=LL+l:GOTO 1820 1780 IF AA>=UM THEN GOTO 1820 ELSE GOTO 1770 1790 IF AA<=UM AND W(N)>=WP THEN 1820 1800 LL=LL+l:IF AA>=UM THEN MP=l:CP=O:FP=FM:GOTO 1820 1810 MP=-1:CP=O:FP=O 1820 I=O:IF LL=O THEN GOTO 1870 ELSR IF NQ=lOO THEN GOTO 1860 1830 NQ=NQ+l:LP=O:FOR I=O TO N:IF NP(I)<>O THEN LP=LP+l 1840 NEXT I:GOSUB 2000:PRINT"Plastic springs •.••..•• :";LP 1850 PRINT:GOTO 1290 1860 GOSUB 2000:PRINT"No convergence":GOSUB 1980:CLS:END 1870 GOSUB 2000: PRINT" I Z W N" 1880 A$="### ####,### ####,#### ######,##":PRINT 1890 FOR I=O TO N:PRINT USING A$;I,X(I),W(I),F(I):NEXT !:PRINT 1900 A$= 11####,####11 :PRINT"Top: F = ";:PRINT USING A$;FA 1910 PRINT" W = 11

; : PRINT USING A$; W(O): PRINT 1920 GOSUB 1980:FOR I=l TO N:A=(W(I)+W(I-1))/2:AA=A-WM(I) 1930 IF AA>=DA(I) THEN WM(I)=A-DA(I) 1940 IF AA<=-DB(I) THEN WM(I)=A+DB(I) 1950 NEXT I:IF W(N)>=WP+UM THEN WP=W(N)-UM 1960 IF W(N)<=WP THEN WP=W(N) 1970 GOTO 1240 1980 LOCATE 25,25,0:COLOR 0,7:PRINT" Touch any key to continue"; 1990 COLOR 7,0:A$=INPUT$(1):RETURN 2000 CLS:LOCATE 1,31,0:COLOR 0,7 2010 PRINT" ALP-1.0 11 :COLOR 7,0:PRINT:PRINT:RETURN

25


7.3 Example

As an example the following case will be considered. A pile of 50 m length has been installed in a homogeneous soil. The soil properties are a maximum friction of 50 kN/m2 , with a corresponding quake of 0.01 m. Point resistance is neglected. The pile diameter is 1,00 m, and its wall thickness is 0.05 m. This means that the orea of the pile section is 0.1492 m2 , and that the circumference of the pile is 3.1416 m. The load-displacement curve for this pile, as calculated by a computer program of the type presented above, is shown in figure 7.2.

I

·FMax Forice I

DisplaceMent , ·

Figure 7.2. Example,

It is interesting to note that plastic deformations start to occur only for large forces, close to failure. Actually this could have been expected by considering the load transfer just before failure. The maximum force is 50 kN/m2*50 m*3.1416 m = 7854 kN. Failure will occur if in the last spring, at the bottom of the pile, the displacement reaches the quake, i.e. 0.01 m. Then the average force in the pile is 3927 kN, and the elastic deformation of the pile corresponding to this force is 0.00627 m. This means that the displacement just before failure is 0,01627 m, whereas the displacement necessary to cause the first spring to fail is 0.01 m, which is reached only for a force of 6000 kN. Apparently this pile is relatively stiff. In engineering practice it is important to compare the.deformation of the pile itself (at failure) with the soil deformations at failure. If the pile deformations are relatively small the pile behaves almost as an infinitely stiff body, and in a homogeneous soil failure may occur rather suddenly. In cases like this, with a relatively stiff pile, the analysis of the stress distribution can directly be done, without using a computer program. The displacement of the pile will be such that the total soil reaction, generated all along the pile, equals the total load. Another interesting observation from the example is that the behaviour in unloading and subsequent re-loading is linear. In the terminology of plasticity theory this can be called shakedown. The permanent deformations are such that after initial plastic deformations no further plastic deformations

26 ·


occur, provided that the maximum load does not exceed the previous maximum value. It should be realized that this is a feature of the present model, in nature this phenomenon is usually not observed, and therefore it is a shortcoming of the model,

It should be noted that the program presented above is restricted to a pile in a homogeneous soil. In a more advanced version of the program (ALP-1.3) the soil may consist of a number of layers with different properties. Input is stored in a datafile on diskette, which enables to inspect the data, and if necessary, to correct or modify them. Input data can be created and modified interactively. Output can be given either on the screen of the terminal, or on a line printer. This program is written, for the IBM Personal computer or a similar computer using MS-DOS, in compiled Basic, using Microsoft's BASCOM compiler. The program is distributed by the Geotechnical Laboratory of the Delft University, Stevinweg 1, Delft.

27


8. A NUMERICAL MODEL FOR 1ATERALLY L9ADED PILES

In this section a complete numerical model for the transfer of lateral loads to the soil is developed. Load transfer takes place along the shaft of the pile to the soil on either side. The characteristics of this form of stress transfer will be considered non-linear, as described in paragraph 5.3. An elementary computer program will also be presented,

8.1 Basic equations

Just as for the case of axial loading the pile 'is subdivided into N small elements, see figure 8.1.

Pm l ~<l+ll Q( I) ... , ... · ,1:, :·

I\ ~·.,\':/ ,:,i., ,,.._~,~·:· ~lo: ----------+-i..~~;,++---------.lio~ z '•i· .. .: .. ,,. ,· Q( I 1) . ,::,,:.,,<H'f I +

:c.~ ... ·.i,·?.'

t(l+l)

Figure 8.1. Element of beam.

The element located between points Z(I) and Z(I+l) is loaded by a distributed load, the resulting force of which is denoted by F(I+l). It is assumed that the elements are small enough that this force can be considered to be acting in the center of the element. Another possible load is a concentrated force P(I), acting immediately to the right of point I. The soil reaction is a distributed load, which is assumed to be acting in the center of the element, and the magnitude of which depends upon the average displacement of the element. Equilibrium of forces in the direction perpendicular to the axis of the pile now requires, for element I+l,

Q(I+l)-Q(I)+F(I+l)+P(I)=R(I+l),

where R(I+l) represents the soil reaction. equilibrium is

Q(I)-Q(I-l)+F(l)+P(I-l)=R(I).

28

For element I the equation of


Equilibrium of moments for element I+l requires that

M(I+l)-M(I)=(Q(I+l)+Q(I)-P(I))*D(I+l)/2.

Similarly for element I

M(I)-M(I-l)=(Q(I)+Q(I-1)-P(I-l))*D(I)/2.

It follows from these last two equations that

Q(I+l)-Q(I-l)=P(I)-P(I-1)+2*(M(I+l)-M(I))/D(I+l)-2*(M(I)-M(I-1))/D(I).

Similarly from the first two equations one obtains, by addition of the two equations

Q(I+l)-Q(I-l)=R(I+l)+R(I)-F(I+l)-F(I)-P(I)-P(I-1).

The shear forces can now be eliminated from the equations, which leads to the following equation in terms of bending moments only

A(I+l)*M(I+l)+A(I)*M(I)+A(I~l)*M(I-l)+R(I+l)+R(I)=F(I+l)+F(I)+2*P(I), (8,1)

where A(I+l)=-2/D(I+l),

A(I)=2/D(I+l)+2/(D(I),

A(I-1)=-2/D(I).

Equation (8.1) is the numerical equivalent of the equation of equilibrium for the moments in the analysis of bending of a beam,

d2 M/dZ2 =-f,

If all intervals are of the same length the standard molecule finite differences (-1,2,-1) is obtained, because then A(I)=4/D, and A(I+l)=-2/D,

for central A(I-1)=-2/D,

In eq. (8.1) the soil resistance R will be dependent upon the lateral displacement U. In general this will be a non-linear relation, which will be specified later. At this point it suffices to note that in each node there are two basic variables, U(I) and M(I), and therefore a second equation is needed to complete the model. This second equation can be obtained from a consideration of the deformation of the beam itself. If the beam material is linear elastic this leads to the classical bending equation

d2 U/dZ2 =-M/EI, (8.2)

where EI is the bending stiffness of the beam: This equation will next be approximated, taking special care to include the possibility of irregular intervals, because these intervals may be determined by the natural variations in the soil properties.

In order to derive a numerical expression for the bending equation a part of the beam between nodes I-1 and I+l is considered, see figure 8.2.

29


u

2 1-1 1+1

Figure 8.2. Bending of beam.

For simplicity it is assumed that the origin Z=O is located in node I. The deflection of the beam is now assumed to be as follows.

Z(I-l)<Z<Z(I) U=U(I)+A*Z+B*Z*Z,

Z(I)<Z<Z(I+l) U=U(I)+A*Z+c*Z*Z,

where A, Band Care constants. It can be seen that this approximation ensures that the displacement U and its first derivative are continuous in Z=O. For Z=Z(I-1) and Z=Z(I+l) one now obtains

U(I-l)=U(I)-A*D(I)+B*D(I)*D(I),

U(I+l)=U(I)+A*D(I+l)+C*D(I+l)*D(I+l),

and hence, after elimination of A,

(U(I+l)-U(I))/D(I+l)-(U(I)-U(I-1))/D(I)=B*D(I)+c*D(I+l),

On the other hand the second derivative in the interval between I-1 and I is 2*8, and in the interval between I and I+l is 2*C, and these can be related to the average bending moment in these intervals as follows,

Z(I-l)<Z<Z(I) d2 U/dZ2 =2*B=-(M(I-l)+M(I))/(2*EI(I))

Z(I)<Z<Z(I+l) d2 U/dZ2 =2*C=-(M(I)+M(I+l))/(2*EI(I+l))

Using these expressions for Band C the final equation is

where

B(I-l)*U(I-l)+B(I)*U(I)+B(I+l)*U(I+l)= =-C(I-l)*M(I-1)-C(I)*M(I)-C(I+l)*M(I+l),

30

(8.3)


B(I-1)=-l/D(I),

B(I)=l/D(I)+l/D(I+l),

B(I+l)=-1/D(I+l),

C(I-l)=D(I)/(4*El(l)),

C(I)=D(I)/(4*EI(I))+D(I+l)/(4*EI(I+l)),

C(I+l)=D(I+l)/(4*EI(I+l)).

Equation (8.3) is the numerical equivalent of the bending equation (8,2). A simplified version (used in earlier editions of this report) is to use only the value of M(I) in the right hand side, with a coefficient 2*C(I). This seems to be somewhat less accurate, especially for non-homogeneous piles.

Boundary conditions

The boundary conditions deserve some special attention, especially because it is preferable that in each point an equation can be formulated for each of the two variables: the lateral displacement U and the bending moment M. It is then avoided that in the matrix describing the system of equations a zero coefficient appears on the main diagonal, which would be numerically inconvenient. In the numerical model it is assumed that the upper end of the beam (the top of the pile) is the nodal point I=O, and that the lower end of the beam (the point of the pile) is the nodal point I=N. The simplest boundary condition would be a hinged support, because then both the bending moment and the lateral displacement are directly given, which can inunediately be incorporated in the numerical model, Unfortunately this is not the most realistic boundary condition. The most realistic boundary condition is a free end, with a given shear force and a given moment. The given moment (zero or a prescribed value) can inunediately be incorporated, but the given shear fore~ should be transformed into a condition for the lateral displacement. This can be done, for example for the point I=O, by starting from the equation of equilibrium of moments in the first element,

M(l)-M(O)=(Q(l)+Q(O)-P(O))*D(I)/2,

The shear force Q(O) in this equation can be assumed to be zero, with the external load being introduced as P(O). The shear force Q(l) can be eliminated by using the equation of horizontal equilibrium for element 1,

Q(l)-Q(O)+F(l)+P(O)=R(l),

With Q(O)=O it now follows that

R(l)+(2/D(l))*(M(O)-M(l))=F(l)+2*P(O),

Because the soil reaction R(l) will depend upon the lateral displacements W(O) and W(l) this can be considered as an equation formulated in terms of W(O).

31


Soil response

The response of the soil has been introduced above as a force R(I). In accordance with the considerations of chapter 5 this should be related to the local lateral displacement, taking into account that the soil pressure cannot be greater than in the passive state, and not smaller than in the active state. The precise formulation will not be repeated here. The reader may also inspect the computer program presented on the next pages.

An elementary computer program (LLP-1.0) is reproduced below. Again the program has been kept as simple as possible, with interactive input, and output as a list on the screen. For simplicity the pile has been assumed to be homogeneous (EI constant), and the soil properties have also been assumed as constant. This does not mean that the elastic and plastic soil reactions are constant, however, because the parameters describing the soil reaction are the vertical stress (which increases with depth), active and passive coefficients, and a value for the total stroke (i.e. the displacement difference between the generation of active and passive soil pressures). In this program no difference is made between the stiffnesses in the active and passive branches, as suggested in paragraph 6,3.

32


Program LLP-1.0

1000 REM===== LLP-1.0 -----1010 GOSUB 2210:DEFINT I-N:B$= 11 Impossible":Z=4 1020 DIM X(lOO),R(lOO),D(lOO),F(lOO),U(lOO),Q(lOO) 1030 DIM NP(lOO),NQ(lOO),BM(lOO),FF(lOO) 1040 DIM K%(100,3),A(l00,3,l,l),UM(l00),UN(l00),G(l,1) 1050 INPUT"Length of the pile (m) •.•...•.. ";UL 1060 IF UL<=O THEN PRINT 8$:GOTO 1050 1070. INPUT"Width of the pile (m) , •. , • , , .• , "; BB 1080 IF BB<=O THEN PRINT 8$:GOTO 1070 1090 INPUT"Unit weight of soil (kN/m"3) ,,, ";DG 1100 IF DG<=O THEN PRINT 8$:GOTO 1090 1110 INPUT"Active soil pressure,,,,,,,,,,, ";CA 1120 IF CA<=O OR CA>l THEN PRINT B$:GOT0 1110 1130 INPUT"Passive soil pressure ...•. , .. ,, ";CP 1140 IF CP<l THEN PRINT B$:GOT0 1130 1150 INPUT"Neutral soil pressure , . , , ..•.• , "; CK 1160 IF CK<=CA OR CK)=CP THEN PRINT B$:GOT0 1150 1170 INPUT" Total stroke (m) • , , , .. , . , .• , "; UU 1180 IF UU<=O THEN PRINT B$:GOT0 1170 1190 INPUT"Bending stiffness EI (kN*m"2) .. ";EI 1200 IF EI<=O THEN PRINT B$:GOT0 1190 1210 INPUT"Number of elements (10,,.100) ,. ";N 1220 IF N<lO THEN N=lO ELSE IF N>lOO THEN N=lOO 1230 IF CA=l AND CP=l THEN UC=O:UP=O:GOTO 1250 1240 UC=(CK-CA)*UU/(CP-CA):UP=(CP-CK)*UU/(CP-CA) 1250 DG=DG*BB:ZZ=(CP-CA)/UU:Z=ZZ*DG*UL/2:FT=O 1260 A=UL/N:FOR K=l TO N:X(K)=X(K-l)+A:NEXT K 1270 GOSUB 2210:INPUT"Lateral force (kN) ,,,,, ";FL 1280 IF ABS(FL)(FT THEN FOR K=l TO N:NP(K)=O:NQ(K)=O:NEXT K 1290 FOR K=l TO N:GOSUB 1880:NEXT K 1300 NP(O)=O:NQ(O)=O:LI=O:GOSUB 2210:FT=ABS(FL) 1310 FOR I=O TO N:FOR J=O TO 3:K%(I,J)=O 1320 FOR K=O TO l:FOR L=O TO l:A(I,J,K,L)=O:NEXT L,K,J,I 1330 PRINT"Force (kN) •.•. , . , .. , , , . : "; FL: PRINT 1340 PRINT"Generet ion of matrix • ·• , : "; : FOR I= 1 TO N-1 1350 K%(I,O)=I:K%(I,l)=I-l:K%(I,2)=I+l:K%(I,3)=2:NEXT I 1360 K%(0,1)=1:K%(0,3)=1:K%(N,O)=N:K%(N,l)=N-l:K%(N,3)=1 1370 I=O:PRINT "+";:FOR I=l TO N-1:PRINT "+"; 1380 AA=l/(D(I)*D(I+l)):Al=AA*D(I):A2=AA*D(I+l) 1390 A(I,O,O,O)=-Al-A2:A(I,l,O,O)=A2:A(I,2,0,0)=Al 1400 A(I,O,O,l)=-(R(I)+R(I+l))/4:A(I,l,O,l)=-R(I)/4 1410 A(I,2,0,l)=-R(I+l)/4 1420 A(I,3,0,0)=-(F(I)+F(I+l))/2:A(I,0,1,l)=-Al-A2:A(I,l,1,l)=A2 1430 A(I,2,l,l)=Al:A(I,0,1,0)=(D(I)+D(I+l))/(4*EI) 1440 A(I,l,l,O)=D(I)/(4*EI):A(I,2,1,0)=D(I+l)/(4*EI) 1450 NEXT I: PRINT "+" 1460 A(O,O,O,O)=l:A(0,3,0,0)=0:A(O,O,l,l)=R(l)/4 1470 A(O,l,l,l)=R(l)/4:A(0,0,1,0)=l/D(l):A(O,l,1,0)=-l/D(l) 1480 A(0,3,l,l)=F(l)/2+FL:A(N,O,O,O)=l 1490 A(N,3,0,0)=0:A(N,O,l,l)=R(N)/4:PRINT 1500 A(N,l,1,l)=R(N)/4:A(N,O,l,O)=l/D(N):A(N,l,l,O)=-l/D(N) 1510 A(N,3,l,l)=F(N)/2:PRINT"Elimination . .. .. .. .. .. . ";

33

..


1520 H=3:FOR I=N TOO STEP -1 1530 KC=K%(I.,H):PRINT "+";:FOR KV=O TO l:CC=l/A(I,O,KV,KV) 1540 FOR II=O TO KC:FOR LV=O TO l:A(I,II,KV,LV)=CC*A(I,II,KV,LV) 1550 NEXT LV,II 1560 A(I,H,KV,KV)=CC*A(I,H,KV,KV):FOR LV=O TO l:IF LV=KV THEN 1600 1570 CC=A(I,O,LV,KV):FOR II=O TO KC:FOR IJ=O TO 1 1580 A(I,II,LV,IJ)=A(I,II,LV,IJ)-CC*A(I,II,KV,IJ):NEXT IJ,II 1590 A(I,H,LV,LV)=A(I,H,LV,LV)-CC*A(I,H,KV,KV) 1600 NEXT LV,KV:IF KC=O THEN 1750 1610 FOR J=l TO KC:JJ=K%(I,J):L=K%(JJ,H) 1620 FOR JK=l TO L:IF K%(JJ,JK)=I THEN 1640 1630 NEXT JK 1640 FOR KV=O TO l:FOR LV=O TO l:G(KV,LV)=A(JJ,JK,KV,LV) 1650 NEXT LV,KV:K%(JJ,JK)=K%(JJ,L):K%(JJ,L)=O 1660 FOR KV=O TO 1:FOR LV=O TO l:A(JJ,JK,KV,LV)=A(JJ,L,KV,LV) 1670 A(JJ,L,KV,LV)=O 1680 A(JJ,H,LV,LV)=A(JJ,H,LV,LV)-G(LV,KV)*A(I,H,KV,KV) 1690 NEXT LV,KV:L=L-l:K%(JJ,H)=L:FOR II=l TO KC 1700 FOR IJ=O TO L:IF K%(JJ,IJ)=K%(I,1I) THEN 1720 1710 NEXT IJ:L=L+l:IJ=L:K%(JJ,H)=L:K%(JJ,IJ)=K%(I,I1) 1720 FOR KV=O TO l:FOR LV=O TO l:FOR JV=O TO 1 1730 A(JJ,IJ,KV,LV)=A(JJ,IJ,KV,LV)-G(KV,JV)*A(I,II,JV,LV) 1740 NEXT JV,LV,KV,II,J 1750 NEXT I 1760 PRINT:PRINT:PRINT"Back substitution ...... : "; 1770 FOR J=O TO N:PRINT "+";:L=K%(J,H):IF L=O THEN 1810 1780 FOR K=l TO L:JJ=K%(J,K):FOH KV=O TO l:FOR LV=O TO 1 1790 A(J,H,KV,KV)=A(J,H,KV,KV)-A(J,K,KV,LV)*A(JJ,H,LV,LV) 1800 NEXT LV,KV,K 1810 NEXT J:FOR I=O TO N:U(I):::A(I,H,1,1):BM(I)=A(l,H,O,O):NEXT I 1820 Q(O)=-FL:FF(O)=O:FOR I=l TO N:AA=(BM(I)-BM(I-1))/D(I) 1830 Q(I)=-Q(I-1)+2*AA:FF(I)=(Q(I-l)-Q(I))/D(I):NEXT I 1840 LI=LI+l:GOSUB 2010:IF LI<lOO THEN 1860 1850 CLS:PRINT"After 100 iterations no convergence":PRINT:END 1860 IF LL>O THEN 1310 1870 GOSUB 2060:GOSUB 2190:GOTO 1270 1880 REM Calculation of stresses and springs 1890 A=X(K)-X(K-l):X=X(K)-A/2:D(K)=A:U=(U(K)+U(K-1))/2 1900 VM=UM(K):VN=UN(K) 1910 SZ=DG*X:IF U-VM>UC THEN SX=CA*SZ:R(K)=O:NR=l:LP=LP+l:GOTO 1940 1920 IF U-VM<-UP THEN SX=CP*SZ:R(K)=O:NR=-1:LP=LP+l:GOTO 1940 1930 R(K)=ZZ*SZ:SX=CK*SZ+R(K)*VM:NR=O 1940 F(K)=SX*A:R(K)=R(K)*A:IF NP(K)<>NR THEN NP(K)=NR:LL=LL+l 1950 IF U-VN<-UC THEN SX=CA*SZ:RR=O:NR=l:LP=LP+l:GOTO 1980 1960 IF U-VN>UP THEN SX=CP*SZ:RR=O:NR=-1:LP=LP+l:GOTO 1980 1970 RR=ZZ*SZ:SX=CK*SZ-RR*VN:NR=O 1980 F(K)=F(K)-SX*A:R(K)=R(K)+RR*A 1990 IF NQ(K)<>NR THEN NQ(K)=NR:LL=LL+l 2000 RETURN 2010 REM Ver~fication of springs 2020 PRINT: PRINT: PRINT"Check springs" 2030 LL=O:LP=O:FOR K=l TO N:GOSUB 1880:NEXT K:PRINT:GOSUB 2210 2040 PRINT"Plastic springs ........ :";LP:PRINT 2050 RETURN 2060 REM Output

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2070 PRINT C$:FOR J=l TO N:U=(U(J)+U(J-1))/2 2080 IF U-UM(J)>=UC THEN UM(J)=U-UC 2090 IF U-UM(J)<=-UP THEN UM(J)=U+UP 2100 IF U-UN(J)<=-UC THEN UN(J)=U+UC 2110 IF U-UN(J)>=UP THEN UN(J)=U-UP 2120 NEXT J 2130 PRINT" Z (m) U (m) M (kNm) Q (kN) F(kN/m)" 2140 A$="###,### ###,### #####,### #####,### #####,###":PRINT 2150 FOR J=O TO N:PRINT USING A$;X(J);U(J);BM(J);Q(J);FF(J):NEXT J 2160 A$="####.######":PRINT:PRINT"Top: F = "; 2170 PRINT USING A$;FL:PRINT" U = ";:PRINT USING A$;U(O):PRINT 2180 RETURN 2190 LOCATE 24,25,0:COLOR 0,7:PRINT" Touch any key to ~ontinue "; 2200 COLOR 7,0:A$=INPUT$(1):RETURN 22]0 CLS:LOCATE 1,31,0:COLOR 0,7:PRINT" LLP-1.0 11

2220 COLOR 7,0:PRINT:PRINT:RETURN

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In the computer program presented above the elasticity of the soil has been characterized by a value for the displacement difference between the generation of active and passive lateral stress, denoted as the stroke. By defining the elastic properties of the soil in this way the spring constant is automatically increasing with depth, which is not unrealistic. As mentioned in paragraph 2 the spring constant in a soil is often correlated to the modulus of elasticity (Vesic [1961]) by writing

c = aE/D,

where a is a coefficient which describes the spreading of stresses, of the order of magnitude of 1, Eis the modulus of elasticity of the soil, and D is the pile diameter. The modulus of elasticity of a soil can be expressed in terms of the compressibility C in Terzaghi's [1941) logarithmic formula, by assuming a small stress increment. This gives

E = Ccr,

where cr is the stress level. Thus one obtains

c = aCcr/D

In the present analysis the spring constant in the elastic branch is defined as

c = Kcr/Au

where K is a pressure coefficient, for example K=3. By equating these two expressions one obtains

Au/D = 1/aKC

The possibility to correlate the stroke Au to such a familiar soil parameter as the compressibility may serve to further illustrate the relevance of the description of the elastic branch in this way. Conunon values for the compressibility of sandy soils are 50-200. This means that the stroke Du is of the order of 1 % of the pile diameter.

8,3 Example

As an example a steel pile of 50 meter length, and l meter diameter will be considered. The wall thickness is supposed to be 0,05 m, so that the bending stiffness is EI=4.12 GNm2 • The soil is homogeneous, with an effective volumetric weight of 10 kN/1113, and with a response characterized by a minimum stress equal to 0.3333 times the vertical (effective) stress, a maximum stress equal to 3 times that vertical stress, and a displacement difference between these two extremes of 0.01 m. The displacement caused by a variable lateral load, up to 2000 kN, is shown in figure 8.3. It should be noted that, in contrast with the case of an axial load (see the previous paragraph), plastic deformations start to occur for small loads. This must be due to the fact that near the soil surface the effective stresses, and therefore the soil strength, are very small. Another important factor may be

36


that the pile, although of considerable dimensions, is relatively flexible in bending, Another interesting feature of the behaviour shown in figure 8.3

r---===::::=----------------:.--Q

11 Figure 8.3. Example.

is that plastic deformations continue to occur during unloading and subsequent reloading, thus leading to a continuous loss of energy. The deformation corresponding to the maximum force also tends to increase in each loading cycle, All these seemingly realistic characteristics of the loaddisplacement diagram have been obtained using the simple local elastoplastic stress-strain relation of paragraph 6, and have not been introduced as such.

By varying the elastic range of the soil deformations (the stroke 6u) it is found that this parameter has only limited influence, especially for large lateral forces, see figure 8.4. The three curves shown were calculated using

11

37


Figure 8.4. Influence of soil elasticity. ~u=0.001, 0.01, and 0.1. It appears that a ten-fold increase or decrease in the soil elasticity hardly influences the response of the pile as a whole, This must be due to the fact that, whatever the value of the soil stiffness, the soil near the upper part of the pile is always in the plastic zone. The main resistance now is from the bending stiffness of the pile, and thus the actual displacements are determined by the stiffness of 'the pile, and not by the stiffness of the soil. This conclusion is of practical interest, because it means that no great effort needs to be put into the determination of the soil stiffness.

8.4. Approximate solution

The behaviour observed in the numerical calculations suggests that an approximative analysis may be developed in the following way. It is assumed that the lateral load is relatively large, so that plastic deformations are generated along a considerable part at the top of the pile. In that case the soil reaction in the plastic region, say from z=O to z=h, is fully known, and the pile is a beam with a given distributed load of triangular shape, with its maximum value at z=h, and a concentrated force at the top z=O. It is now assumed, to keep the problem as simple as possible, that at the depth z=h the displacement u as well as its first two derivatives are zero. This is suggested by the circumstance that the pile is well clamped, provided that the length of the pile is sufficiently large, of course, compared to the distance h. Another motivation for the assumption that the second derivative is zero for z=h is the notion that a beam on elastic foundation (i.e. the lower part of the pile) is less stiff against bending than it is against a lateral force, Thus the bending moment at the lower end will be small. This statement is also supported by the results of numerical calculations. It may be added that the schematization used here resembles a schematization often used for the analysis of sheet pile walls, a problem of classical soil mechanics. There it is often assumed that at the bottom of the wall a concentrated force may be acting (due to soil resistance), with the bending moment vanishing.

The problem now is a standard problem of applied mechanics: a beam with given lo'ads and boundary conditions. The solution can be summarized by the following expression for the lateral displacement at the pile top

_ (Kp-Ka))1Dh5

u - 45EI

where Kp and Ka are the coefficients of passive and active soil pressure, y is the (effective) volumetric weight of the soil, Dis the diameter of the pile, and where the value of h should be determined from the condition that

For given soil parameters and a given force the value of h can be determined from the second equation. This value (which should of course be considerably smaller than the total length of the pile for the approximation to be applicable) then can be used to determine the displacement from the first equation.

38


The approximate relation between the force F and the lateral displacement at the top u is shown in figure 8.5, which was drawn on the same scale as figure 8.4. The agreement appears to be rather good.

u Figure 8,5. Approximate solution.

In contrast with the elasticity of the soil, which was found to have little influence on the lateral response, the strength of the soil is of great influence on the behaviour of the pile, as is illustrated in figure 8.6, which

Figure 8.6. Influence of soil strength.

shows the relation between force and displacement for three values of the coefficients of lateral soil pressure: Kp=l.5, 3 and 6, and the corresponding values for Ka=l/Kp, The data used for figure 8.6 were calculated by a computer program similar to program LLP-1.0. The soil strength is seen to have a considerable influence, which is of course understandable. If the

39


soil is much softer the soil reaction is much smaller, and the pile deformations will be larger. The lower curve in figure 8,6, which represents a large flexibility, of course corresponds to the weakest soil.

It is interesting to note that the approximate solution given above also predicts a great influence of the soil strength. The results calculated from

Figure 8.7. Approximate solution.

this approximation, using the same data, are shown in figure 8.7. Again the correspondence with the results of the numerical calculations is striking.

The approximate solution indicates that the flexibility of the system is directly proportional to the bending stiffness EI of This has also been verified by numerical calculations.

pile-soil the pile.

It should be noted that all results in this section apply only to the particular situation considered here, of a long pile in a homogeneous sandy soil. The approximate solution derived above applies only if the lateral load is rather large, so that plastic deformations are dominant, and provided that the pile is sufficiently long. For other cases, such as layered soils, or cohesive materials, the behaviour of a single pile may be analyzed by an appropriate computer program. Input data in such a program usually include simple coefficients for the maximum and minimum lateral effective stress, expressed into the vertical stress. The behaviour of the soil is greatly simplified in this way. Stress transfer in the horizontal plane is neglected, for instance. This is sometimes taken into account by increasing the passive pressure coefficient. A more refined approach would be to model the soil mass by finite elements.

40


9. DAMPING

In this section the description of damping in a structure due to losses of energy in the pile foundation is considered. In order to conform to the usual concepts in the design of offshore structures, it is attempted to use similar variables, or to translate the behaviour of the pile foundation into parameters used in dynamic analysis. A brief survey of viscoelastic properties is included for reference purposes.

In the dynamic analysis of offshore structures the usual approach is to consider the structure as being composed of three types of elements masses, springs and viscoelastic dampers (dashpots). The simplest of these is the system of a single mass-spring-dashpot, shown in figure 9.1.

F

Figure 9.1. Mass-spring-dashpot system.

The mass may be the foundation of the structure, the spring and dashpot represent the soil response. The basic equation for this system is

d2 u du m. dt2 + c dt + ku = F(t)

where m is the mass, k the spring constant, c the viscosity, u the displacement of the mass, and F(t) the external force, a function of time t. This equation can also be written as

d2 u/dt2 + 2(adu/dt + a2 u = F(t)/m

where a is the circular frequency of the undamped system, defined by

41


a=v(k/m), and t is the damping factor, t=c/(ma). If t=l the system is said to be critically damped.

The response of the system under cyclic loading can be investigated in a simple form by assuming that the load function F(t) is sinusoidal,

F(t) = Psin(<Jt)

The solution for large values of time now is

where the amplitude A is defined by

and where the phase angle IP is defined by

tan(!;') = A~L!L 1-..,2;a2

Of particular interest is the amount of energy dissipated during a full cycle. This can be determined from the amount of work done by the force F during a full cycle, i.e. integration of Fdu=Fvdt over one full cycle. This is found to be

W = nPAsin(lf)

Because the duration of a cycle is 2n/<J it follows that the average dissipation rate (the dissipation per second) is

Wr = -½PA<Jsin('I')

This formula expresses that the dissipation rate is proportional to the amplitudes of force and displacement, to the frequency (a higher frequency means more cycles per second, and therefore more dissipation per second) and to the sine of the phase angle. If 1;'=0 there is no dissipation at all. It can be seen from the formula for 1P that this is the case if the viscosity of the damper is zero, i.e. if there is no damping.

The response of the system can be illustrated in a phase diagram, see figure 9.2, in which the force and the displacement are represented, with time t as a running parameter. In this figure the phase angle has been assumed to be 30 degrees. If the phase angle is zero, the force and displacement are directly proportional, and the ellipse in the figure reduces to a straight line. Maximum dissipation occurs if l;'=t7ri then the ellipse becomes a circle, which also explains the factor n in the formula for the dissipation of energy per cycle. It is interesting to compare the shape of the response curve in figure 9.2 with the curve in figure 8.4 for a pile supported by elasto-plastic springs. The present curve is smoother, but the general shape is similar.

.. 42


Q

u

Figure 9.2. Phase diagram.

It may be illustrative to consider some special cases in some more detail.

If the dashpot is absent t=O, and then one obtains

A= P/k l-w2/,x2

1/1 = 0

W = 0

Thus the system is indeed not damped. If w<a (i.e. for slow fluctuations), force and displacement have the same direction. If w>a (i.e. for very rapid fluctuations), the displacement is opposed to the force. If w=a the solution degenerates to very large displacements, this is called resonance.

The limiting behaviour for zero spring constant can best be investigated by rewriting the formulas in terms of the original parameters k, c and m, because both a and t have kin the denominator. One may even return to the original differential equation, and solve it for k=O. The resulting values for the response parameters are

A'_ P/cw - v' [ 1+m2 w2 / c2 ]

tan(!;')= -c/mw

W = 1rPAsi·n(I/I)

The phase angle 1/1 is now in the range f1r<l/l<1r.

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No mass

In the absence of mass the parameters reduce to

_ P/k A - v [ l+c2 w2 /k2 ]

tan(l/f) = cw/k

W = nPAsin(l/1)

This simplification is applicable if w/a is small compared to 1. For offshore structures the significan~ frequency usually corresponds to a wave period of about 10 seconds, hence w is about 0.6/s. On the other hand the value of a can be estimated by considering an equivalent elastic problem of a wave propagating in a beam of length L. Then one obtains a=v/L, where vis the velocity of propagation of an elastic wave. In soils this velocity is about 100 m/s, Thus if L=l m, then a is about 100/s, which means that w/a is of the order of magnitude of 0.01. Hence it seems justified to disregard the soil mass compared to the elastic deformations, at least for waves with periods of about 10 seconds. ·or course this applies only to the mass of the soil, not to the mass of the foundation and the superstructure.

A case of considerable practical interest is that of a system with a relatively small damping, say less than 10 % of the critical damping, i,e. (<O.l. It can be expected that under those circumstances the phase angle 1/f will be small, so that

tan(l/f) = sin(l/f) = 1/1

The dissipation per cycle now becomes

W = 6PA

where

or, in terms of the original parameters,

6 = 1(~~

l-w2 /a.2

The above expression for the dissipation per cycle enables to simulate other forms of damping by an apparent viscosity. If the response of a system under cyclic loading, assuming a certain characteristic frequency w, has been determined, this cyclic response may be of the form as shown in figure 8.3, in case of an elasto-plastic system. From this response both the apparent spring constant k and a dissipation per cycle can be determined. This can then be translated into an apparent viscosity by using the expression just derived. This viscosity can then be used in a further dynamic analysis of

44


the structure, It should be noted that the plastic dissipation is independent of the frequency of loading, whereas the viscous dissipation is dependent upon this frequency. This means that the approach can be applied only if the frequency is given beforehand, As dynamic analysis is often done in the frequency domain it follows that the approach can be applied, provided that a new apparent viscosity is determined for each frequency. The apparent viscosity will be approximately inversely proportional to the frequency w.

9.2 Numerical model

In order to conclude this chapter a simple numerical model will be constructed, Therefore the basic differential equation is written as

du/dt=v

dv/dt=[F(t)-cv-ku]/m

where u is the displacement and v the velocity. The simplest way to solve this system of equations numerically is by explicit finite differences, or, in other words, by the so-called Euler method, in which a forward finite difference approximation is used to determine values of u and vat the end of a time step from the values at the beginning of this step. A better approximation can be obtained by using a so-called Runge-Kutta method, see Abramowitz & Stegun (1964, p. 896]. A program using such a method is listed below. This is a program in BASIC, in which the amplitude and phase angle are calculated in the tenth cycle, with the system initially being at rest, This assumes that a steady state has been reached after ten cycles, which need not be the case, especially near resonance, The values calculated numerically are compared with the analytic results presented above. Agreement is usually rather good, except for systems with a very small spring constant. In the program the amplitude of the force P and the frequency of the load ware assumed to be 1.

45


100 CLS:PRINT"------- Mass-Spring-Dashpot -------":PRINT 110 DEF FNF(T)=P*SIN(W*T):DEF FNA(T,X,Y)=Y:P=l:W=l:PI=4*ATN(l) 120 DEF FNB(T,X,Y)=FNF(T)/M-2*ZZ*A*Y-A*A*X 130 INPUT"Spring constant ";K:INPUT"Viscosity ";C 140 INPUT"Mass ";M:A=SQR(K/M):ZZ=C/(2*M*A):WA=W/A 150 UM=(P/K)/SQR((l-WA*WA)*(l-WA*WA)+(2*ZZ*WA)*(2*ZZ*WA)) 160 IF WA=l THEN PHI=PI/2:GOTO 180 170 TP=(2*ZZ*WA)/(l-WA*WA):PHI=ATN(TP):IF PHI<O THEN PHI=PHI+PI 180 PHI=l80*PHI/PI:TT=2*PI/W:NN=l00:H=TT/NN:DD=l80*H/Pl:PRINT 190 FOR J=l TO 9:PRINT"Period :";J:GOSUB 290:NEXT J 200 T=O:PRINT"Period :";J 210 FOR I=l TO NN:F=FNF(T):GOSUB 310:T=T+H:IF U>AM THEN AM=U:TM=T 220 IF T>'IM AND U*UU<O THEN PSI=T-PI-H*U/(U-UU) 230 UU=U:NEXT I:PSI=l80*PSI/PI:PRINT:A$="####.####" 240 PRINT "Amplitude : "; :PRINT USING A$;AM; 250 PRINT" - Exact: ";:PRINT USING A$;UM 260 PRINT "Phase angle : "; :PRINT USING A$;PSI; 270 PRINT" - Exact: ";:PRINT USING A$;PHI 280 END 290 REM------- One Period---------300 FOR I=l TO NN:F=FNF(T):GOSUB 310:T=T+H:NEXT I:RETURN 310 REM------- Runge-Kutta --------320 Kl=H*FNA(T,U,V):Ll=H*FNB(T,U,V) 330 K2=H*FNA(T+H/2,U+Kl/2,V+Ll/2):L2=H*FNB(T+H/2,U+Kl/2,V+Ll/2) 340 K3=H*FNA(T+H/2,U+K2/2,V+L2/2):L3=H*FNB(T+H/2,U+K2/2,V+L2/2) 350 K4=H*FNA(T+H,U+K3,V+L3):L4=H*FNB(T+H,U+K3,V+L3) 360 DU=(Kl+2*K2+2*K3+K4)/6:DV=(Ll+2*L2+2*L3+L4)/6 370 U=U+DU:V=V+DV:RETURN

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9,3 Damping in lateral loading

One of the mnjor characteristics of the response to lateral loading presented in the previous section is the curved shape of the relationship between force and displacements, which indicates that a considerable amount of energy is dissipated. In order to further investigate this phenomenon of damping one may consider a pile loaded by a sinusoidal load, i.e. a load that varies between equal maximum values in both directions. A typical response is shown in figure 9.3, which was obtained using the same data for

u

Figure 9.3. Response to cyclic load.

the pile and soil properties as used in the previous section, i.e. a pile of 50 m length, 1 m diameter, with a flexural rigidity EI of 4.12 GNm2, in a cohesionless soil, characterized by a friction angle~ of 30 degrees, and an effective volumetric weight of 10 kN/m3 • The amplitude of the cyclic load has been assumed to be 1000 kN. The figure actually shows the response in 5 consecutive cycles. It appears that there is a strong non-linear behaviour, but after the first cycle has been completed, a constant response is generated, which remains the same in each cycle. In each cycle a large a.mount of energy seems to be dissipated, as is indicated by the area within the loop. This form of damping is usually called hysteretic damping, or plastic damping.

The elasticity of the soil again has little influence on the response function, as is illustrated in figure 9.4, which shows the response in a full cycle for three values of the stroke, 6u=0.001, 0.01 and 0.1, that is a variation of the soil stiffness by factors in the ratio 1:10:100, Only for the largest value (which is probably unrealistic) the damping is reduced, because then the displacements are not large enough to generate plasticity. For the smaller values of the stroke the response is practically independent of this parameter, It may be mentioned here that this insensitivity for the soil elasticity is a consequence of the non-linear (plastic) local stress-strain relationship. For a beam on a completely elastic foundation it can be shown, on the basis

47


Figure 9.4. Influence of soil elasticity.

of the standard solutions for these problems (Hetenyi [1946]), that the lateral flexibility depends mostly on the subgrade modulus, even more than on the bending stiffness of the beam, see also paragraph 3. Here it is found that, when the possibility of plastic soil deformations is also taken into account, the influence of soil elasticity appears to be small,

The soil strength has already been found to have a considerable influence on the displacements, see figure 8.6. This does not necessarily mean that the damping is influenced, as is illustrated in figure 9.5, which shows the

u

Figure 9.5. Influence of soil strength.

response for Kp = 6, 3 and 1.5, on different scales, such that the maximum displacements coincide in the graph, Although there is a definite influence

48


of the soil strength, with the largest dissipation for the weakest soil, which seems natural, the general shape of the curves is very similar. This suggests to compare these numerical results with the damping if the simple approximate formula of the previous paragraph is used as a standard curve. The cyclic response resulting from a double ·application of this function, in positive and negative sense, each time starting from the previous extreme, is shown.in figure 9.6. As the approximation is based upon a schematization in which the elastic deformations of the soil are completely

u

Figure 9.6. Damping in approximate solution.

disregarded, it seems understandable that the initial slopes of the two branches of the curve are practically horizontal, The shape of the curves in figures 9.5 and 9.6 seems to be sufficiently similar to conclude that the simplified analytical analysis can be used as a first estimation.

The availability of an approximate formula for the relation between force and displacement also enables to determine the dissipation of energy in a full cycle. By evaluating the integral

f Fdu = J F(du/dh)dh

over a full cycle, one obtains, using the expressions given in the previous section for F and u as functions of h,

where w· is the dissipation of energy, Fm is the amplitude of the force, and Um is the amplitude of the cyclic displacements. It should be noted that the maximum dissipation would give a factor 4 instead of 12/7, for rigid plastic behaviour. Apparently the dissipation is about 40 % of the maximum, It is interesting to compare the behaviour in cyclic loading, as determined above, with the behaviour of a simple mechanical element, namely a system of spring and a dashpot, placed parallel (a Kelvin body), such as considered in section 9.1 above. The response of such a system is shown in figure 9,7.

49


u

Figure 9.7. Viscoelastic schematization.

Although the general shape of the curve is quite different from that shown in the previous figures one might use equivalent coefficients such that the amplitude of the displacements as well as· the total damping are the same, This can be done as follows, The spring constant k determines the relation between the two amplitudes,

k = Fm/Um

Thus this equivalent spring constant can be determined if the displacement for a given force is known, even if the path is non-linear. The viscosity c of the dashpot can be determined by noting that the dissipation in a full cycle is

W = 1£FmUmsin(1p)

where 1/1 is the phase angle, defined by

tan(111) = cw/k

where w is the frequency of the applied force, which is given. Comparison of this result with the previous one, approximation, gives

nsin(t;1) = 12/7

supposed to be obtained from the

It now follows that sin(I/I) = 0.54567, from which one obtains 1/1 = 33°, and thus tan(I/I) = 0.65, This means that the behaviour of the pile can be simulated by a Kelvin body having n viscosity c determined by

C = 0,65w/k

Here w is the frequency of the load, and k is the spring constant. Actually figure 9.7 has been drawn for the value 1/1 = 33°, in order to facilitate corn-

50


parison with figure 9.6. It is interesting to note that the equivalent phase angle is independent of all soil properties, provided of course that the approximate solution is applicable.

The approximation presented in this section may be useful for the analysis of the structure, beacuse the dynamic analysis of offshore structures is usually done by using a mechanical model of the structure involving discrete elements such as springs, dashpots and masses, It has been shown here that a foundation pile can be incorporated in such a model by a representation in terms of a· spring and a dashpot. First order approximations for the equivalent spring constant and the viscosity have been indicated above, for a homogeneous sandy deposit. These parameters depend upon the load itself, because the relation between force and displacement is non-linear, and also upon the frequency, This is due to the fact that the damping in a viscous element depends upon the frequency, whereas in a plastic element the damping per cycle is independent of the frequency. If the dominant frequency is known, as is usually the case, this is only a minor complication. It should be noted that the method presented here can also be used in more complicated cases than the simplified example of a homogeneous sand deposit. In such a case the response of the pile can be determined by a numerical method, and the resulting flexibility and damping can be represented by an equivalent spring constant and a viscosity. ·

51


10. PILE DRIVEABILITY

The driving of piles in the seabottom requires special equipment, and is in general very expensive. It is therefore imperative to be able to predict the behaviour of the pile during installation, so that the appropriate equipment can be selected, and so that the pile will indeed reach its design depth without too many difficulties. For the prediction of pile driveability computer programs have been developed on the basis of an analysis of the propagation of an elastic wave in the pile, taking into account the friction along the pile shaft, and the point resistance near the tip of the pile. This is usually called a wave equation analysis (Smith [1962], Bowles [1974], Smith [1982)), The computer programs use a finite difference or a finite element discretization, or may use an integration procedure along characteristics. The production of such a computer program is not as simple as it may seem. The main part of the model is a one-dimensional elastic element: the pile, but the soil-pile interaction is an essential part of the model, and this should be introduced very carefully, because of its non-linear character. Another difficulty, which arises in the spatial discretization, is the numerical accuracy, or numerical dispersion. If this aspect is not treated well in the numerical algorithms it is possible that waves propagate faster in the numerical model than they do in the physical model, which makes the numerical models unreliable (Verruijt (1984]).

The most important part in the numerical analysis is the description of the soil behaviour, As in the static case the soil itself is usually not modelled at all, but only the soil-pile interaction at the interface is taken into account. In general the friction resistance along the pile shaft is modelled by an elasto-plastic model, usually also including a linear or perhaps a non-linear viscous element. This enables to relate the dynamic response, as observed during pile driving, to the static response. One of the main differences between the static and dynamic response is of course the vanishing of the influence of the viscous velocity-dependent term in the static case. Other differences are also observed, however, in engineering practice, such as the gradual degradation of the shearing resistance during pile driving, and its possible later recovery. The quantitative correlation of static and dynamic properties is still subject to much controversy, but some general agreement seems to exist that there is a reasonable agreement between the (undisturbed) properties of the non-linear friction spring, which is the most important factor contributing to the bearing capacity and the flexibility of the pile.

The rational approach to pile driving and its analysis is especially important because it is based upon an interpretation of pile driving in terms·of parameters of a physical nature, such as soil strength. This enables to use basically the same soil parameters in all problems pile driving, pile flexibility, and bearing capacity. One of the practical advantages of this fundamentally sound procedure is that the initial predictions can be updated after pile driving, using the actual field observations and measurements. This can best be done of course if the pile has been provided with recording gauges, such as strain gauges or velocity transducers. This will enable to compare the actual behaviour of the pile in the field with the predictions, and so to adjust the soil parameters, and possibly even to reconsider the previous predictions of bearing capacity and pile flexibility.

52


11. GRAVITY FOUNDATIONS

Because of the very stiff nature of most of the soil deposits in the North Sea an alternative to pile foundations has been developed in the form of gravity foundations (Bjerrum [1973)), These consist of heavy concrete footings, upon which the actual platform is constructed. The capacity of such a structure to carry a dead load, and to withstand storm loading conditions, is due to the great strength of the soil and the relatively high compressive stresses generated in the soil by the weight of the foundation. The major design conditions again refer to the bearing capacity of the platform, and its behaviour during cyclic storm wave loading.

It should be mentioned that the installation of a gravity foundation involves many problems besides the main problems of bearing capacity and cyclic behaviour. These include the penetration of the skirts and the stresses in the neighbourhood of these skirts, local stress concentrations in an uneven seabottom, the injection of grout in the space between the bottom of the structure and the soil surface, and the stability of the structure during transport and installation (Bjerrum [1973], Smits [1980), Vos [1980]).

).1.1 Bearin_g___~apaci ty

The static bearing capacity of a gravity foundation is usually determined by using the same method as used for strip footings in classical soil mechanics. This method, developed by Prandtl, Terzaghi, Buisman, Caquot and Brinch Hansen (to mention only the main contributions), is based upon the theory of limit analysis for elasto-plastic materials. The theory has gradually been developed to include the influence of the weight of the soil itself, surcharge loads, inclined and eccentric loading, and the shape of the footing. Part of this development has been theoretical, another part is empirical, for instance on the basis of model.tests or field observations. In recent years the results have been verified by numerical calculations using the finite element method for elasto-plastic materials. Furthermore the method has been used extensively to design the foundations of actual structures, and thus the geoteclmical community has obtained much experience with this method.

The application of the classical methods of bearing capacity (Brinch Hansen (1970]) to the large gravity foundations of offshore structures extends the scope of common engineering practice by at least one order of magnitude, and this is justified only if it is accompanied by a careful consideration of eventual risks due to the larger scale. One of the difficulties is that for a large structure possible inhomogeneities in the soil are of relatively great importance. This applies not only to the variations in the horizontal direction, but also to the variations in the vertical direction, because the region in which the soil strength is mobilized is about as deep as it is wide. Unfortunately the classical bearing capacity formulas are applicable only in homogeneous materials. Therefore natural inhomogeneities can be taken into account only in the more sophisticated numerical methods, if sufficiently reliable soil data are available, of course. A simpler approach

"

53


might be to just reduce the soil strength parameters, but this may lead to an uneconomical design, if bearing capacity is the critical design criterion.

The classical notion of bearing capacity is that the loads are applied so slowly that no excess pore pressures are being developed. This is called a drained analysis in soil mechanics. For an offshore structure this may be the case for the dead weight of the platform and its installations, but not for the large horizontal force due to for instance the 100-year wave, which forms part of the design load. Because loading time of this component of the force on the platform is very short (about 5 seconds) drainage can only tal<e place if the permeability is sufficiently high, which is usually not the case, not. even for sands. The consolidation time can be estimated, as a first approximation, by (Verruijt [1984])

tc = 7L2y/kE

where )' is the volumetric weight of water, k is the permeability of the soil, E its elasticity, and L the drainage length, For a sandy deposit the modulus of elasticity can be estimated to be E=Ccr=CyL, where cr is the stress level and C is Terzaghi's compressibility coefficient. Thus one obtains

tc = 71/kC

If C=lOOO, L=lO m and k=l0- 4 m/s this gives a value of about 2 hours, which indicates that during the period of 5 seconds the soil must be considered as undrained, lt'or cohesive materials the concept of an undrained strength is well known, but for sands this is an unusual quantity, Therefore the undrained bearing capacity analysis of sandy soils must be done by introducing the reduction of shear strength by excess pore pressures into the analysis, or by performing a finite element analysis. Preferably both, because the introduction of pore pressures into a bearing capacity analysis requires some additional assumptions, and experience with the finite element method for this type of limit analysis problem is still not very large.

_11. 2 __ Cytlic _loadi_ng

The soils in the North Sea usually have a high relative density, This means that they are rather stiff and strong (which is the main reason for the possib:ility of gravity foundations), and that they do not exhibit the undesirable property of a large contraction during shearing, such as loose soils often do, The cyclic nature of storm loading may give rise to problems associated with the development of p_ore pressures, however, and this aspect requires careful investigation. As is well known one of the main characteristics of many soils is that a shear deformation may be accompanied by a volume change, even if the isotropic stress remains constant. This behaviour can easily be demonstrated by shearing a sample of dense sand, which will tend to expand (dilatancy), or a sample of loose sand, which will tend to contract. Rapid shear deformations of satut·ated loose deposits may lead to large excess pore pressures, possibly leading to a complete loss of effective stress, resulting in a total failure (liquefaction). Because the seabed in the North Sea usually consists

54


of densely packed soils it is not to be expected that a loss of stability will occur if the structure is suddenly loaded by a large shearing force. This is the reason that in a bearing capacity analysis volume changes due to shear can usually be disregarded. The loading conditions of offshore structures due to wave action are of a cyclic nature, however, and experimental evidence shows that practically all soils, even dense sands, tend to compact in a full cycle of loading and unloading (see e.g. Pantle and Zienkiewicz [1982]), This phenomenon has been observed in many laboratory tests, and data collected in situ seem to support these observations, This means that the analysis of the behaviour of a gravity foundation under the continuous application of wave loads should include the tendency for volume decrease, which may lead to a generation of excess pore water pressures, As the time span of a great number of waves is relatively long it can be expected that at least part of these pore pressures will be dissipated as a result of consolidation, at least in sandy soils, so that both the generation and dissipation of pore pressures should be considered. The classical analysis of pore pressure generation and dissipation is by Biot's theory of consolidation (Biot [1941]). Many analytical and numerical methods of solution for consolidation problems have been developed (Schiffman [1984]), In this theory, in which the soil is represented by an elastic material, pore pressures are generated only by changes of the isotropic stress. Volume changes at constant isotropic effective stress do not occur in an isotropic elastic medium, The theory of consolidation does constitute a consistent basic theory, however, with proofs of existence and uniqueness of solutions, and with numerical models being available, usually in the form of finite element programs. Thus the theory of consolidation seems to give the.basic tools, and the only problem remaining is to introduce the possibility of volume changes due to cyclic loading. This can be done by considering one of the basic equations of consolidation, the storage equation (see e.g. Verruijt [1984)),

Here e is the volume strain (positive for extension), p is the excess pore water pressure, n is the porosity of the soil, bis the compressibility of the pore fluid and q1,1 is the divergence of the flow·rate of the pore water with respect to the grain skeleton (qi is usually called the specific discharge vector). This equation expresses that a volume change of the soil skeleton must be accompanied by either a compression of the pore fluid, or a net outflow of pore water. The theory of consolidation is completed by equations describing equilibrium of the soil mass, compatibility of the deformations, and a constitutive relation. These equations will not be reproduced here. The storage equation retains its validity also for complicated stress-strain relationships, for instance including dilatancy of the soil. This means that an additional volume change can easily be incorporated into the consolidation process by an addition to the volume strain e in this equation. Of course the equations of equilibrium and compatibility are not affected by a modification of the stress-strain relationship, so that most of the consolidation equations remain unchanged, and the starting point thus can be an existing (linear) consolidation program. Cyclic laboratory tests on sands are most often performed on saturated samples, usually undrained. In such a test any change of volume must be

55


accompanied by a change of the pore water pressure. If there is no drainage the last term in the storage equation vanishes, and this can then be integrated to give

6p = - 6e/nb

The major contribution to the volume strain in a full cycle is the contraction from the cyclic shear deformation. On the basis of the results of undrained cyclic tests Bjerrum [1973] proposed to express the pore pressure generated in such a test as follows.

6p/cr = PN

where cr is the (effective) stress level, N is the number of cycles, and P is a (small) parameter, which in its turn is strongly dependent upon the ratio of the maximum shear stress to the isotropic stress (Smits [1980)), The formula expresses that, as a first approximation, each cycle produces about the same increase in pore water pressure. For soils of low permeability there will be no drainage during the storm considered, and in that case the formula given above is sufficient to determine the pore pressures in situ, if the value of p is known for various shear stress levels, if the wave spectrum is known, and if the shear stresses corresponding to a certain wave have been calculated, for instance by using the theory of elasticity. This then results in a prediction of the pore pressures generated in situ, and the corresponding reduction of shear strength and bearing capacity. For a soil of relatively high permeability (sand) it may be necessary to include the dissipation of pore water pressures by consolidation. This can be done as follows. If the value of p is measured in a laboratory test, and the compressibility of the flu~d is known, it is now possible to interprete the experimental data in terms of a tendency to contract, using the equations given above. The additional volume change can be introduced in the consolidation process, by adding the volume change produced by cyclic shear in the storage equation. This is not a simple matter, however. Experience with the numerical solution of the consolidation equations shows that this is a sensitive process, with a tendency to exhibit local instabilities, even in the linear case, in time and in space (Vermeer and Verruijt [1981)). Thus a careful analysis of the discretization scheme is needed. On the other hand, it is impractical to follow the process in hundreds of time steps, so that some smoothing of the loads is necessary, with the volume changes due to cyclic shear superimposed onto n constant or slowly varying average load. As the volume change is, as a first approximation, proportional to the number of cycles, the additional volume changes can be considered to give an additional rate of volume change, 3e/3t. As this is precisely the form in which volume changes appear in the storage equation, this additional volume change can be introduced without much difficulty, Another complication is that the value of the generation parameter P, which is extremely important in the process, is strongly dependent on the effective stress level, as already mentioned above, Because this stress level changes during consolidation it· may be necessary to adjust the value of p during the numerical calculations. Thus the computational process contains several non-linearities: volume changes due to cyclic shear, and increased shear deformations for large shear stresses to simulate the behaviour near failure. It seems logical that this requires a careful consideration of all the possible errors involved, and an investigation of the sensitivity for

56


the various parameters, The strong dependence of p on the levels of shear stress and isotropic stress has a positive effect as well. It can be expected that before the design storm will act upon the structure smaller storms will have occurred, and during these storms pore pressures have been developed, with the corresponding consolidation, This will lead to a reduction of the value of p, or in other words: preshearing reduces the tendency for volume changes, This is the same phenomenon that is usually considered to be responsible for the high densities observed in the sands on the North Sea bottom.

57


REFERENCES

American Petroleum Institute, API Recommended practice for plf111ning, de. signing and constructing fixed offshore platforms, API, Dallas, 1981.

Biot, M.A., General theory of three-dimensional consolidation, .~ Appl. Phys., 12, p, 155-164, 1941.

Bjerrum, L., Geotechnical problems involved in foundations of structures in the North Sea, Geotechnique, 23, p. 319-358, 1973,

Bowles, J.E., Analytical and computer methods in foundation engineering, McGraw-Hill, New York, 1974.

Brinch Hansen, J., A revised and extended formula for bearing capacity, Bulletin of the Danish Geotechnical Institute, 28, p. 5-11, 1970.

De Ruiter, J. and Beringen, F.L., Pile foundation for large North Sea structures, Marine Geotechnology, 1978.

Det Norske Veritas, Rules for the design, construction and inspection of offshore structures, Det Norske Veritas, Oslo, 1977.

Heerema, E.P. and De Jong, A., An advanced wave equation computer program, Conf. Num. methods in Offshore Piling, London, 1979.

Hetenyi, M., Beams on elastic foundation, Univ. of Michigan Press, Ann Arbor, 1946,

Pande, G.N. and Zienkiewicz, O.C., Soil mechanics - transient and cyclic loads, Wiley, Chichester, 1982.

Schiffman, R.L., A bibliography of consolidation, in: Fundamentals of transport phenomena in porous media, edited by J. Bear and M.Y. Corapcioglu, p, 617-669, Martinus Nijhoff, Dordrecht, 1984.

Selvadurai, A. P. S., Elastic analysis of soil-foundation interaction, Elsevier, Amsterdam, 1979.

Smith, E. A. L., Pile driving analysis by the wave equation, Trans. ASCE, vol. 127, pt. 1, p. 1145-1193, 1962.

Smith, I.M. Programming the fi11.ite element method, Wiley, Chichester, 1982. Smith, I.M. and Molenkamp, F., Dynamic displacements of offshore structures

due to low frequency sinusoidal loadin~, Geotechnique, 30, p. 179-205, 1980. Smits, F.P., Geotechnical design of gravity structures, LGM-Mededelingen,

vol. 21, no. 4, p. 283-318, 1980. Terzaghi, K., Theoretical soil mechanics, Wiley, New York, 1943. Vermeer, P.A. and Verruijt, A., An accuracy condition for consolidation by

finite elements, Int. J, Numer. and Anal. Methods in Geomechanics, 5, p. 1-14, 1981.

Verrui,jt, A., Numerical verification of dynamic pile testing analysis, Proc. 2nd Int. Conf. Appl. Stress-wave theor.v on piles, Stockholm, 1984.

Verruijt, A., The theory of consolidation, in : Fundamentals of transport phenomena in porous media, edited by J. Bear and M,Y. Corapcioglu, p. 349-368, Martinus Nijhoff, Dordrecht, 1984.

Vesic, A.B., Beams on elastic subgrade and Winkler's hypothesis, Proc. Sth Int. Conf. Soil Mech, and Found. Eng., 1, p. 845-850, 1961.

Vos, Ch. J., Gravity structures, practical design and construction aspects regarding the foundation, L<lM-Mededelingen, vol. 21, no. 4, p.267-282, 1980.

58

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API

RECOMMENDED PRACTICE

FOR

...... , ..... , ..... -.... , ..... API RP2A

Twelfth Edition January 1981

PLANNING, DESIGNING,

and CONSTRUCTING

FIXED OFFSHORE PLATFORl\tIS

Of"FICIAL. PUBLICATION

ll&G. U.S. PATENT Ol"P'ICIC

AMERICAN PETROLEUl'vl INSTITL'TE

Washington, D.C.

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Issued by

AMERICAN PETROLEUM INSTITUTE Production Department

211 N. Ervay, Suite 1700

Dallas TX 75201

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Cop):right (, 1981 American Petroleum Institute Users or this publication should become completely familiar with its ~ope and cont.enL This document is intended lo supplement rather than replace individual engineering judgment.

2 American Petroleum Institute

TABLE OF CONTENTS

Page FORE'\VORD .....................................................•.. 3 DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 SECT. 1: Planning.................................................. 4

General ......................................•.•...... 4 Operational Data ............................... : . . . . . . 4 Environmental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Foundations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . 7 Selecting the Design Environmental Conditions . . . . . . . 8 Platform Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Safety Considerations. . . . . .. . . . .. . . . . .. . . . . . . . . . . . . . . . . 9 Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

SECT. 2: Design Criteria Procedures ................................ 10 G-eneral ............................................... 10 Design Loading Conditions .................•.......... 10 Design Loads .......................................... 11 Installation Forces .................................... 22 Structural Steel Design ............................... ·. 24 Foundation Design .................................... 38 Allowable Cement Grout Bond Stress ................... 48 11:aterial .............................................. 48 C-0rrosion Protection ................................... 51 Drawings and Speoifications ........................... 51

SECT. 3: Welding ................................................... 54 G-eneral ........................................ , ...... 54 Qualification .......................................... 54 \\'elding ..................•........................•.. 54

SECT. 4: Fabric;ation .....................................•......... 55 • .\.ssem bly .............................................. 55 Coatings .............................................. 58

SECT. 5: Installation ................................................ 59 General ..........................................•.... 59 Transportation ........................................ 59 Removal of Jacket from Transport Barge ............... 59 Erection .............................................. 60 Pile Installation ....................................... 60 Superstructure Installation ...... , ..................... 63 Grounding of Installation Welding Equipment .......... 63

SECT. 6: Inspection ..... · ............................... , ............ 64 G-eneral ................................. , . , ........... 64 Personnel ............................................. 64 Method and Extent of Inspection ................ , ...... 64 Quality of \Velds ........................... , ........... 64 Magnetic Particle Inspection ......... , ................. 64 Ultrasonic Inspection .................................. 64

SECT. 7: Surveys ............................ , ...................... 65 General .............. · .......................•......... 65 Yearly Surveys ........................................ 65 Additional Surveys .... .' ............................... 65 Records ................................................ 65

Commentary: Earthquake Criteria ...................................... 66 Allowable Stresses for Structural Steel ..................... 75 F01Jndations ..... : ............... : ......................... 89

r

/'"-· I \ \ i ·--~,/

38 American Pttroleum lnstitut.e

2.6 FOL'~DATIO~ DESIGN

Tht rerommended mteria of Par. :!.6.J through Par. :!.ti.JO art dt1-otcd tc, pile fo1111dations. and mort SJ>t"rifica/Jy to ilttl rylindrical (pipe) pile .foundations .. Tht n-rommtnded mteria of Par. :!.6.11 thro1i9h Par. :Z.6.J 6 arc dn-oted to Mallow fo11ndati011,S.

2.6.1 General. The foundation ~hould be designed to carry static. cyclic and transient loads without excessive deformations or ,·ibra:ions in the platform. Special attention should be ~fren to the effects of cyclic and transient loading or. the strength of the supporting soils as weil a.~ on the structural response of piles. The po~ibility of mo,·ement of the ~floor against .the foundation memb,;~ should be im·1!$tigated and the forc-es cau~d br !f.Uch movements. if ar,ticipated. should be considered in the design.

2.6.2 Pile Foundations. Types of pile founaations used to support offshore structures are as follows:

2..6.2a. Drh·en Piles. Open ended piles are common· )y used in foundations for offshore platforms. These piles are usually driven into the sea-floor with impact hammers which use steam, diesel fuel, or hydraulic power as the ~urce of energy. The pipe

· wall thickness should be adequ:i.te to resist axial and lateral loads as well as the stresses during . pile drh·ing. It is possible to predict approximately the i:tresses during pile drh'ing using the principles of one-dimensional ela.c;tic stres~ wave transmission by carefull~· selecting the parameters that :,ro,·ern the behavior of !',Oil. pile. cu!-hion~ capblock and ham·. mer. For a more df'tailed study of these principles. refer to E. A. L Smith's paper Pile-Driting Analym by tht Wart Ec;vation, Transactions ASCE. Vol. 127, 1962. Part 1. Paper Ne,. 3306. pp. 1145-1193. The above approach may a)M> be ui;ed to optimize the pile-hammer-t'ushion and czpblock with the aid of computer analy;es (commonly known as the Wave Equation· Ana!rsesl. The de~ign penetration of driven piles s:iould be determined in accordance with the principles outlined in Par. 2.6.3 through 2.6.6 and 2.6.~ rather than upon any correlation of pile capacity -..:-i~h the number of blows required to drive the pile a certain distance into the seafloor.

When hard driving is encountered before the pile reaches design penetration, one of the following procedures can be used to aid pile driving.

1. Plug Remo\'al. The soil plug inside the pi.le is removed by jetting and air lifting or by drilling to reduce pile drh-ing resistance. If plug remo,·al re~ults in inadequate pile capacities, the removed soil plug should be replaced b~· a grout or concrete plug having sufficient load-carrying capacity to replace that of the removed soil plug.

2. Soil Removal Below Pile Tip. Soil below the pile tip ii: remcwed either by drilling an undersized hole or by jetting and possibly air lifting. The drilling or jetting equipment is lowered through the pile which acts as the casing pipe for the operation. The effect on pile capacity of drilling an undersized hole is unpredictable unless there has been previous experience under similar conditions. Jetting below the pile tip should in general be a,·oided because of the unpredictability of the results.

3. T_wo-St.age Driven Piles. A first stage or outer pile is driven to a predetermim,d depth. the soil plug is remo,•ed, and a second stage or inner pile 1s driven inside the first st.age pile. The annulus between the two piles is grouted to permit load transfer and develop composit.e action.

2.6.2b. Drilled and Grouted Piles. Drilled and grouted piles can be used in soils which will hold an c,pen hoie with or without drilling mud. Load transfer between grout and pile should be designed in accordance with Par. 2.i.2, 2.i.3 and 2.i.4. There are two types of drilled and grouted piles. as follows:

1. Single-Stage. For the single-staged, drilled and j?routed pile. an oversized hole is drilled to the required penetration. a pile is lowered into the hole and the annulus between the pile and the $Oil is grouted. This type pile can be installed only in soils which will hold an open hole to the surface. As an alt.ernati\'e method. the pile with expendable cutting tools attached to the tip can be used as part of the drill stem to a\'oid the time required to remo"e the drill bit and insert a pile.

2. Two-Stage. The two-staged, drilled and grout• ed pile consists of two concentrically placed piles grouted to become a composite section. A pile is dri,·en to a penetration which has been determined to be achic,·able with the availzble equipment and below which an ooen hole can be maintained. This outer pile becomes the casing for the r,ext operation which is to drill throuJ?h it to the required penetration for the

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RP 2A: Planning, Designing, and C-Onstructing Fixed Offshore Platforms

inner or winsert" pile. The insert pile is then lowered into the drilled hole and the annuli between the insert pile and the soil and between the two piles are grouted. Under certain soil conditions, the drilled hole is stopped above required penetration, and the insert pile is driven' to required penetration. The diameter of the drilled hole should be at least 6 inches (150mm) larger than the pile diameter.

2.6.2c. Belled Piles. Bells may be constructed at the tip of piles to give increased bearing and uplift capacity through direct bearing on the soil. Drilling of the bell is carried out through the pile by underreaming with an expander tool. A pilot hole may be drilled below the bell to act as a sump for unrecoverable cuttings. The bell and pile are filled with concrete ,to a height sufficient to develop necessary load transfer between the bell and the pile. Bells are connected to the pile to transfer full uplift and bearing loads using steel reinforcing such as structural members with adequate shear lugs, deformed rt!inforcement bars or pre-stres.~ ten· dons. Load transfer into the concrete should be designed in accordance with ACI 318. The steel reinforcing should be enc)o$ed for their full length below the pile with spiral reinforcement meeting the requirements of ACI 318. Load transfer between the concrete and the pile i;hould be designed in accordance with Par. 2.7.2. 2.i.3 and 2.i.4.

2.6.3 Pile Design

2.6.3a. Foundation Size. When sizing a pile foundation. the following items should be considered: diameter, penetration. wall thickness. t)'pe of tip. spacing, number of piles. geometry, location, mudline restraint, material strength, installation method. and other parameters as may be considered appropriate.

2.6.3b. Foundation Re~ponse. A number of different analysis procedures may be utilized to determine the requirements of a foundation. At a minimum. the procedure used should properly simulate the nonlinear response behavior of the soil and assure loaddeflection compatibility between the structure and the pile-soil system.

2.6.3c. Deflections and Rotations. Deflections and rotations of individual piles and the total foundation system should be checked at all critkal locations which may include pile tops. points of contrafiecture, mudline. etc. Denections and rotations $hould not exceed serviceabilitr limits which would render the structure inadequate for its intended function.

2.6.3d. Pile Penetration. The desiirn pile penetration should be sufficient to develop adequate capacity to resist the maximum computed axial bearing and ·

pullout loads with an appropriate factor of safety .. The ultimate pile capacities can be computed in accordance with Par. 2.6.4 and 2.6.5 or by other methods which are supported by reliable comprehensive data. The allowable pile capacities are determined by dividing the ultimate pile capacities by appropriate factors of safety which should not be less than the following values:

Factors Load Condition of Safety

1. Design environmental conditions with appropriate drilling loads 1.5

2. Operating environmental conditions during drilling operations 2.0

3. Design environmental conditions with appropriate producing loads Ur

4. Operating environmental conditions during producing operations 2.0

5. Design environmental conditions with minimum loads (for pullout) 1.5

2.6.3e. Alternative Design :',lethods. The provisions of this recommended practice for sizing the foundation pile are based on an allowable stress (working stress) method except for pile penetration per Par. 2.6.3d. In this method. the foundation piles should conform to the requirt-ments of Par. 2.5.2 and 2.6.9 in addition to the provisions of Par. 2.6.3. Any alternative method supported by sound engineering methods and empirical e\·idence may also be utilized. Such alternative methods include the limit state design approach or ultimate strength design of the total foundation system.

2.6.4 Pile Capacity for Axial Bearing Loads

2.6.4a. Ultimate Bearing Capacity. The uldmattbearing capacity of piles. including belled piles, Qd shoµld be determined by the equation:

~d = Qr+ Qp = fA5 + qAp •••.• (2.6.4-li

where:

Qr "' skin friction resistance. lb (kN) QP = total end bearing. lb (kNI f = unit skin friction capacity. lbtit2

(kPa) A, = side surface area of pile. ftt (m2)

q = unit end bearing capacitr, lb'ft2

(kPa) Ap = gross end area of pile, ftz (m=)

Total end bearing, Qp· should not exceed the capacity of the internal plug. In computing pile loadin11: and capacity the weight of the pile-~il plug system and hydrostatic uplift should be considered.

In determining the load capacity or a pile, con$idera· ·tion should be given to the relative deformations

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40 American Petroleum Institute

between the soil and the pile as well as the compressibility of the soil pile system. The ultimate skin· friction increments along the pile are not necessarily directly additive, nor is the ultimate end bearing additive to the ultimate skin friction. For additional discussion of these effects refer to Par. 2.6.6 and ASCE Journal of the Soil Mechanics and Foundations Division for LoaiJ Tran,sfer for Aria.Uy Loadtd Pi/e.s in Clay. by H. M. Coyle and L. C. Reese, Vol. 92, No. 1052, March 1966.

The foundation configurations should be based on those that experienet" has shown can ·be installed consistently,· practical!)' and economically under similar conditions whh the pile size and installation equipment being u.<.ed. Alternatives for possible remf'<lia! action in the event design objectives cannot be obtained during in!'tallation should also be investipted and del'1~ prior to const!'Uction.

For the pile-bell system.' the factors of safety should be those given in Par. 2.6.3d. The allowable skin friction values on the pile section should be those given in this section a:id in Par. 2.6.5. Skin friction on the upper bell surface and possibly above the bell on the pile should be discounted in computing skin friction resistance, Q!, The end bearing area of a pilot hole. if drillc'l'.i. should be discounted· in computing total C>P.aring are:. of the bell.

2.6.4b. Skin Friction and End Bearing in Clay. For piles dri"en through ~lay. f may be equal to or less than. but should not exceed the undrained shear strength of the clay, c. as determined in accordance with AST.\[ ,Uethods of Testfl for Unconfined Com· pres..~ion Strrngth of C'.lhufre Soil, ASTM Designator D-2166-63T. or as determined by miniature vane shear tests.

Unless test data indicate otherwise, f should not exceed c or the follc~ng limits:

1. For highly p)a.::""tic clays such as found in the Gulf o! Mexico f may be equal to c for underconsolidated and norr.ially consoiidated clays. For O\'er-consoliciated clays f should not exceed 11 ton per square foot (4S kPa) for shallow penetrations or c equivalent to a normally conrolidated ciay for deeper penetrations, whichever is gn,ater.

2. For other tn,es of clay, f should be taken equal to c for c less than or equal to ~~ ton per

- square foot (24 kPal. For c in excess of !/4 ton per square foot r24 kPa) but less than or equal to ~I ton per square foot (i2 kPa) the ratio f to c should decrea..<e linearly from unity at c equal to ¼ ton per square foot (24 kPa) to ½ at c equal to ~; ton per square foot (72 kPa), For c in excess of ~. ton per square foot (72 kPal, f should be taken as ½ of c.

For piles drh·en in undersized drilled or jetted holes or drilled and grouted piles in normally or under· consolidated clay, f should be determined by some reliable method ba..<.ed on the amount of soil disturbance resulting from installation, but f should not exceed nlues given for dri\'en piles. For drilled and grouted piles in O\'er-consolidated clay, the value o( f may exceed \'alues given for driven piles. In determining ! for drilled and grouted piles, th, strength of the soil-grout interfa...--e should be considered. The soil-grout interface strength may be reduced if excess drilling mud is presenL The limiting \'alue for this type pile may be the allowable bond stress between the pile steel and the grout as recommended in Par. 2. 7.3.

For piles end bearing in clay, q in lb/ft2 (kPa) should be equal to ~. If the strength profile below the pile tip is not uniform, then the c utilized should reflec\ appropriate adjustment.

2.6.4c. Skin Friction and End Bearing in Sand and Silt. Fer piles dri\'en through sand or silt. f in lb.'ft2 (kPal should be computed by the equation:

f =Kp, tan 6 ........................ (2.6.4-2) where: K = ('O(,fficient cf lateral earth pressure p~ = effecti\'e ove!burden pressure, lb/ft'

(kPa) 6 = angle of soil friction on pile wall, deg.

For piies driven in undersized drilled or jetted holes in sand or silt. f should be determined by some reiiable method based on the amount of soil disturbance from installation. but f should not exceed the \'alues gi\'en for driven piles. The same \'alues of r ~hould apply to drilled and grouted piles. with the exception that (1 l ,·alues of f for drilled and grouted pil<?S in calcareous·t)-pe sand or silt may exceed th~ for drh·en piles. and (2) the strength of ;,(,;l-grout interface should be considered in establishing \'alues of f. Tne value of r for piles driven into calcareous i:ands and silts will usuall~· be substantial· ly less than that indicated by Eq. 2.6.4·2 and should be determined for the local conditions.

For piles end bearing in sand or silt. q should be computed by the equation:

q = p0

Nq •••.•••••••••••••••.••..• (2.6.4-3j where: ?-;q = bearing capacity factor

The following \'alues are considered applicable for medium-denS<! to dense granular formations:

K = 0.5 to 1.0 for axial compressive loads Soil Type tp' ! N• Clean sand 35° 300 40 Silty sand 30° 25° 20 Sandy silt 25° 20° 12 Silt 20° 15° 8 where: q,' = angle of internal friction of soil, deg.

RP 2A: Planning, Designing, and Conrtructing Fixed Offshore Platforms

For deep foundations, limiting values of f and q may f. be less than indicated by Eq. 2.6.4-2 and 2.6.4-3. For

layered systems, the sand bearing capacity factor, Nq, may be less than in the table above if adequate penetration into the sand layer is not ob:ained. These limiting · values should be determined for local conditions.

2.6.4d. Skin Friction and End Bearing of Grouted Piles in Rock. The unit skin friction of grouted piles in jetted or drilled holes in rock should not exceed the triaxial shear strength of the rock or grout. but in general should be much less than this value based on the amount of reduced shear strength from installation. For example the st~ngth of dry compacted shale may be greatly reduced when exposed to water from jetting or drilling. The sidewall of the hole may develop a layer of slaked mud or clay which will never regain the strength of the rock. The limiting value for this type pile may be the allowable bond stress between the pile steel and the grout as rec?mmended in Par. 2.i.3.

The end bearing capacity of the rock should be determined from the triaxial shear strength o( the rock and an appropriate bearing capacity factor based on sound engineering practice for the rock materials but should not exceed 100 ton.s per square foot (9.58 MPa).

2.6.5 Pile Capacity for Axial Pullout I..cads

The ultimate pile pullout capacity may be equal to or less than but should not excl!t!d Qr. the tot.al skin friction resistance. The effective weight of the pile including hydrostatic uplift and the soil plug shall be considered in the analysis to determine the ultimate pullout capacity. For clay, f should be the same as stated in Par. 2.6.4b. For sand and silt. f should be computed according to Par. 2.6.4c. All "l"alues shown there are applicable. except K = 0.5 should be used. For rock, f should be the same as stated in Par. 2.6.4d.

The allowable pullout capacity should be determined by applying the factors of safety in Par. 2.6.3d to the ultimate pullout capacity.

2.6.6 Axial Pile Performance

Piling axial deflections should be 1,i;ithin accept.able · serviceability limits and these deflections should be compatible with the structural fore~ and mo1.·ements. An analytical method for detenninm~ axial pile performance is provided in Com1>11ter PredictiOM of Axially Loaded Pile., with .Von-linear Supports, by P. T. Meyer. et al .. OTC 218'3 .. May 1975. Pile response is affected by load directions. load types, load rates. time of loading-. inst.allation technique, and other parameters.

Some of these effects for cohesive soils have been observed in both laboratory and field tests.

Detailed information on load deflection relationships for sand are provided in Predicted Belunwr of A:r:ia.lly Loo.d.ed Piles in Sand by I. H. Sulaiman and H. M. Coyle, OTC 1482, April. 1971. Load deflection relationships for grouted piles are discussed in Criteria far Design of A.rially Loaded Drilled Shafts, by L. C. Reese and 11. O'Neill, Center for Highway Research Report. Unh·ersity of Texas, August 1971. Other information may be used, provided such information can be shown to result in adequate safeguards against excessive deflection and rotation.

2.6.7 Soil Reaction for Laterally-Loaded Piles

2.6.7a. General. The pile foundation should be designed to sustain lateral loads, whether static or cyclic. Additionally, the designer should consider overload cases in which the desig-n lateral loads on the platform foundation are increased by an appropriate safety factor. The designer should satisfy himself that the overall structural foundation system will not fail under these overloads. The lateral resistance of the soil near the surface is significant to pile design, and the effects on this resistance of scour and soil .disturbance during piie installation should be considered. Generally, under lateral loading, clay soils behave as a plastic material which makes it necessarv to relate ·oile-soil deformation to soil resistan;e. To facilitat~ this pr<X!edure. lateral soil resistance deflection lr-Y) curves should be con· structed using stre~·st:-ain data from laboratory soil samples. The ordinate for these curves is soil resistance. p, and the absciss:. is soil deflection. y. By iterative procedures. a compatible set of load· deflection values for the pile-soil system can be developed.

For a more detailed study of the construction of p-y curves refer to the following Offshore Technology Conference Papers.

Soft Clay: OTC 1204, Corrflations /or Dtsign of Laterally Loaded Piles in Soft Clay, by H. ~latlock. April 1970.

Stirr Clay: OTC 2312, Field Testing 11nd Analysi., of Laterally Loaded Piles in Stiff Clay, by L. C. Reese and W. R. Cox. April 1975.

Sand: OTC 2080, Aria!y.~ of Laterally Lorided Piles i11 Sand. by L. C. Rees.?. W. R Cox. and F. D. Koop, May 197-1.

In the absence of more definitive criteria, p~edures recommended in Par. 2.6.7b and 2.6.7c may be used for constructing ultimate lateral bearing capacity curves and p-y curves.

2.6. ib. Lateral Bearing Capacity for Soft Clay. For static lateral ioads the ultimate unit lr.teral bearing capacity of soft clay Pu has ~n f:iunci to vary between 8<! and 12c except at shallow depths ..

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42 American Petroleum I nstitut.e

where failure occurs in a different mode due to minimum overburden pressure. Cyclic loads cause deterioration of .lateral bearing capacity below that for static loads.' In the absence of more definitive criteria.. the following is recommended:

Pu increases from 3c to 9c as X increases from O to X R according to:

Pu= . 3c + 'YX • J Xc ................ (2.6.7·1) D .

and Pu = 9c for X ;;;i, :XR ..................... (2.6.7-2) where: Pu = ultimate resistance. psi (kPa) c = undrainecl sheu strength f undisturbed

clay soil samples, psi (kPa) D = pile diame:er, in. tmm) 'Y = eff€'Ctive unit weight of soil. lb/in.~

IM!'..m3J J = dimensionless empirical constant with

\'alues ranging from .25 to .5 having been det.erminc>d by field testing. A \·alue of .5 is appropriatt- for Gulf of Mexico clays.

X = depth belr,\,· soil surface, in. (mm) . XR = depth below soil i:urface to bottom of

reduced resistance zone in in. (mm). For a condition of constant strength with depth, equations 2.6.'i-l and 2.6.i-2 are solved simultaneously to give:

XR= ...:G=D __ _ 'YD , I -+J / C / Where the strength \'aries with depth. equations 2.6.i-l and 2.6.7-2 may be 5".>l\'ed by plotting the two equations, i.e., p u vs. depth. The point of first intersection of the two equations is taken to be XR. These empirical relationships may not apply where strength \'liriations are erratic. In general. minimum \'alues of XR should be about 2.5 pile diameters.

2.6.ic. Load-Deflection (p·y) Curves for Soft Clay. Lateral wil resistanct--deflection relationships for piles ir. wft clay arc generally non-linear. The p-y curve, for lhe short-term static load case may be ireneral.tc-d from the following table:

PiP.J r-0.5 0.72 1.00 1.00

where: p = y = Ye= •.=

y/y~ 0 1.0 3.0 8.0 00

actual lateral resistance. psi (kPai actual lateral deflection. in. (mm) 2.5 £ < D. in. (mm) strain which occurs at one-half the · maximum stress on laboratory undrained compression tests of _undisturbed soil samples

For the case where equilibrium has been reached under cyclic loading, the p-y curves may be generated from the following table:

.X>Xll X<X• p/pa yty. p/pa y/y. -0- -0- 0 0 0.5 1.0 0.5 1.0 0.72 3.0 0.72 3.0 0.12 c:o 0.72X/Xll 15.0

0.72X/Xll co

2.6.id. Lateral Bearing Capacity for Stiff Clay. For static lateral loads th.e ultimate bearing capacity Pu of stiff clay 1c > 1 Tsf or 96 kPa) as for soft clay would vary between & and 12c. Due to rapid deterioration under cyclic loadings the ultimate resistance will be reduced to something considerably less and should be so con~idered in cyclic design.

2.6.7e. Load-Deflection (p-y) Curves for Stiff Clay, While stiff clays also ha,·e non-linear stress-strain relationships, they are generally more brittle than soft clays. In developing stress-strain curves and subsequent p-y curves for cyclic loads, good judgment should refie-ct the r:.pid deterioration of load capacity at large deflections for stiff clays.

2.6.if. Lateral Bearing Capacity for Sand. The ultimate lateral bearing cap11city for sand has been found to \'ary from a value at shallow depths determined by Eq. 2.6.i-3 to a \'alue at deep depths determine<i by Eq·. 2.6.7-4. A numerical solution of the two equations for se\·eral depths for specific sand strata properties will result in a depth of transition, X1, which s.eparates shallow depths from deep depths.

_ A j YH [ K.H t.an "'' 1in p + Pus - I D tan (P-?') coa cz

tan P ' tan (/3 _ ~·) (D + H tan P tan cz) +

K.,H tan /3 (t.an IS"sin P-ta.n cz) - K..D J { .. (U.7-3)

Pud= A [ K. y'H (tan~tl -1) + K. '(H tan t/>'tan'/J J · · · · · • • · · · • ·, · •, ·,,,,,.,, .•... , ..•.....•. (2.6.7-4)

where: Pu= ultimate resistance (force/ unit area), psi

(kPa) (s = shallow, d = deep) A = empiric.al adjustment factor y' = effective soil weight, lb/in.3 (MN /m3)

H = depth, in. (mm) Ku= earth pressure at rest coefficient (0.4) "''= angle of internal friction o! sand, deg,

/3 = 45• + ~/2 cz= f/2 D = pile diameter, in. (mm)

Ka= Rankine zrµnimum active earth pressure coefficient Ltanz (45• - ¥/2)]

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RP 2A: 1-'iannin::-. DE,,.ie:ninir. and Constructing F'ixed Offshore Plat:=s

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The coefficient A is an empirical adjustment factor which accounts for differences in static and cyclic behavior. Fig. 2.6.7-1 is a recommended variation of the factor, A. with non-dimensional depth, H/D.

0

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ULTIMATE SOIL RESISTA:'.'\CE VERSl."S DEPTH

2.6.7g. Load-Deflection (p-y) Curves for Sand. The lateral soil resistance-deflection (p-y) relationships for sand are also non-linear and in the absence of more definitive information may be approximated at any specific depth by the four se~ent curve of Fig. 2.6.7-2. The values Cot points u, m. and k may be computed as follows:

Point u: p _ j Eq. 2.6.7-3at depths < X,

u - 1 Eq. 2.6. 7-4at depths > X, 3

Tu= 80D

Where: p = lateral bearing resistance, psi (kPa) y = lateral pile deflection, in. (mm)

Point m: B

Pm=7Pu 1

7m=60D

Where: B = non-dimensional empirical adjustment factor to account for difference in static and cyclic behavior from Fig. 2.6. 7-3.

Point le: H

P• =D. Jc, • .,,

( Dp. ) •

7•= k.Hy.v. .::::r

Where: p. (y.-y.)

n = y,. (p ... p..)

k. = initial soil modulus. For submerired sand subjected to Static or cyclic loading, the following is reeommended:

'1

FIG. 2.6.7-:? RESISTA:'-i'C'E - DEFLECTiO:'.'\

RELATIO:-.;SHIP FOR SA:-.;D

Relative k. Jc. Deruiity lb'' • ,m kPa/mm

Loose 21) 5.43 Medium 60 16.28. Dense 12;; 3.'3.93·

The (p-y) eurve between point.S k and m is a parabola with intermediate point.$ calcuiable from:

p = (.,~-:,.) T"'

For some combinations Qf sand parameter at depth (approximately 100 feet or :{II meter.;) the k: ,·alue selected may result in a denection. Yk pcater than Ym· in which ca.~e the parab<ilic portion of the curve should be omitted,

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1.0

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,., t >so, a,•o ss ,. 0.5

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FOR SOIL RESJSTA~CE \'ERSUS DEPTH

2.6.8 Pile Group Action

2.6.8a. General. C,msideration should be given to the ef!~ts of closely spaced adjace:,t piles on the load a!ld deflection characteristics of pile groups. Generallr. for pile spacing less than eight (8) diameters, group eife-=ts may ha\'e to be evaluated. For more detailed di!oC'JSSions refer to the following four papers: &hati(lr of Pile Groups S11bjer:t t.o Latrro! Load, a. thesis b)' Prakish. S., presented to llniversitv of f:linois at Urbana, Illinois. 1962: The A11a/µ1<i.s ·of F1aihl€ Ra.ft·Pile System by Han, S. J. and Lee. I. K .. Geo~hnique 28. No. 1. 1978: Soil .lfrr:hanies, F.;,.:ndatfons a11d Earth Strurture, Naval Facilities En~neering Command Publication NA v. F AC D~i-7, 1971: and Offshore Technology Confer· ence paper number OTC 2838. Analysi.s of Three· Dimr11sional Pile Gro11ps with No11-Li11ear Soil Re· itpon.'lt and Pilt·Soil Interaction by M. W. O'Neill, et al .. 19ii.

2.6.Sb. Axial Beha,ior. For piles embedded in clays. the group capliCity may be less than a singlc.iFoOlat.ed pile capacity multiplied by the number of piles in the group: con\'ersely. for piles embedded in sands the group capacit.,• may be higher than the sum of the capacitie!' in the isolated piles. The group

· settlement in either clay or sand would normally be larger than that of a single pile subjected to the avera11"e pile load of the pile group. See Load Tran.•f<r. Lalrrol [.,<,Gd.• and Gro11p Arlion of Deep Fovndations by Vesic. A. S., Performance of Deep Foundation. ASTM 444. 1969.

Jn general, group effects depend considerably on pile group geometry and penetrations, and thickness of any bearing strata underneath the pile tips. Refer to Analysis of the SeN/rmenl of Pi'le Group., by Poulos, H. G .. Geote<:hnique 18, 1968. ·

2.6.Sc. Lateral Behavior. For piles embedded in either cohesh·e or cohesionless soils. the pile group · would normally experience greater lateral deflection than that of a single pile under the a\'erage pile load of the corresponding group. The major factors influencin(Z the group deflections and load distribu· tion among the piles are the pile spacing, the ratio of pile penetration to the diameter. the pile flexibility relative to the soil. and the dimensions of the group.

For more discussion, refer to the papers entitled Behaiior of Lotrrally Loadrd Pi/('.s II-Pile Group~ by Poulos, H. G .. Journal of ASCE. \'ol. 97, No. SM5, May 19il: and Ratio>1ol Analy.sis of the Laterlil Performan('t? of 0/jshore Pile Group.5 by Focht and Koch, OTC 1896. May 1973 .

2.6.9 Pile Wall Thickness

2.6.9a. General. The wall thickness of the pile may vary aiong its length and may be controlled at a particular point by any one of several loading conditions or requir<?ments which are discussed in the paragraphs below.

2.6.9b. Ailowable Pile Stre.sses. The allowable pile stresses should be the same as those ~rmitted by the. AISC speciiicatior. for a compact hot rolled section, giving due consideration to Par. 2.5.l and 2.5.3. A rational analysis considering the restraints placed upon the pile by the i:tructure and the soil should be used to determine the allowable stresses for the portion of the pile which is not laterally restrained by the soil. General column buckling of the portion of the pile below the mudline need not be considered unless the pile is belie,·ed to be laterally unsupported because of extremely low soil shear strenirths, large computed lateral deflections. or for some other rea.""On.

2.6.9c. Dei;ign Pile Strei:ses. The pile wall thickness in the \'icinity of the mudline. arid possibl~· at other points. is normally controlled by the combined axial load and bending moment which results from the design loading conditions for the platform. The moment cur\'e for the pile may be computed with· soil reactions determined in accordance with Par. 2.6.7, g-h'ing due consideration to possible soil remo,·al by scour. It may be assumed that the axial load is removed from the pile by the soil at a rate equal to the ultimate soil·pile adhesion divided by the appropriate pile safety factor from Par. 2.6.3d. When lateral deflections associated with cyclic loads at or near the mudline are relatively large (e.g .. exceeding Yt as defined in Par. 2.6.7c for soft clay),

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RP.2A: Planning, Designing, and Constructing Fixed Offshore Platforms

consideration should be given to reducing or neglecting the soil-pile adhesion through this zone.

2.6.9d Stresses Due to Weight of Hammer During Hammer Placement. Each pile or conductor section on which a pile hammer (pile top drilling rig, etc.) will be placed should be checked for streSS'.!s due to both placing and operating the equipment. Often, these loads are the limiting factors in establishing maximum length ot add-on sections. This is particularly true in cases where piling will be driven or drilled on a batter. The most frequent effects include: static bending, axial loads, and arresting lateral loads generated during initial hammer placement.

Experience indicates that reasonable protection from failure of the pile wall due to the abon• loads is provided if the static stresses are calculated as follows:

1. The pile projecting section should be considered as a freestanding column with a minimum effective length factor K of 2.

2. Bending moments and axial loads should be calculated using the full weight of the pile hammer, cap, and leads acting through the cent.er o! gravity of their combined masses,· and the weight of the pile add-on section with due consideration to pile batter eccentricities. The bending moment so determined should not be less than that corresponding to a load equal to 10 percent oi the combined weight of the hammer, cap, and leads applied at the pile head and perpendicular to its centerline.

3. No increase in A.I.S.C. allowable stresses should be permitted.

2.6.9e Stresses During Dri\'ing. Consideration should also be gh'en to the stresses that occur in the free standing pile section during driving. A dynamic analysis may be used to determine the maximum dynamically induced stress level that occurs. In general, it may be assumed that column buckling will not occur as a result of the dynamic portion of the drivin~ stresses. , In the absence of reliable data regarding the maximum dynamically induced stresses, the static portion of the· stress should be· limited to one-half the yield strength of the pile material. The static stress during drh·ing may be taken to be the stress resulting from the weight of the pile above the point of evaluation plus the pile hammer components actually supported by the pile during the hammer blows. including any bending stresses resulting therefrom. The pile hammers evaluated for use during driving should be noted by the designer on the installation drawings or specifications.

2.6.9f. Minimum Wall Thickness. The Dit ratio of the entire length of a pile should be small -enough to

preclude local buckling at stresses up to the yield strength of the pile material. Consideration should be given to the different loading situations occurring during the installation and the service life of a piling. For inservice conditions, and for those installation situations where normal pile-driving is anticipated or where piling installation will be by means other than drh·ing, the limitations of Par. 2.5.2 should be considered to be the minimum requirements. For piles that are to be installed by driving where sustained hard driving (250 blows per foot (820 blows per meter] with the largest size hammer to be used) is anticipated, the minimum piling wall thickness used should not be less than

t = 0.25 + l~O ) Metric Form~la ........................ (2.6.9)

t = 6.35 + 100 where:

t = wall thickness, in. (mm) D = diameter, in. (mm)

Minim.um wall thickness for normally used pile sizes should be as listed in the following table:

MINIMUM PILE WALL THICKXESS

Pile Diameter Nominal Wall Thickness. t

in. mm in. mm

24 610 ½ 13 30 i62 !. 14 36 914 ¾ 16 42 1067 H li 48 1219 ¾ 19 60 1524 ¼ 22 i2 1829 1 25 84 213.t l¼ 28 96 2438 H~ 31

108 2743 1¾ 34 120 30.t8 1½ 37

The preceding requirement for a lesser D't ratio when hard driving is expected may be relaxed when it can be shown by past experience or by detailed analrsis that the pile will not be damaged during its in!:tallation.

2.6.9g. Allowance fol" Underdrive and Onrdrh:e. With piles having thickened sections at the mudline, consideration should be given to pro\'iding an extra length of hea\j' wall material in the vicinity of the mudline so the pile will not be overstressed at this point if the design penetration is not reached. The amounl of underdrive allowance provided in the design will depend on the degree of uncertainty regarding the penetration that can be obtained. In some instances an overdrive allowance should be provided in a similar• manner in the event an

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expected bearing str:i.tum is not encountered at the anticipated depth.

2.6.9h. Driving Shoe. Pro,;sion of a drh·ing shoe at the pile tip at least one diameter in length with a minimum· wall thickness of 1.5 times the value established by Par. 2.6.9f should be considered. A dri\'ing hea.d should be considered where hard driving is expected.

2.G.10 Length (If , Pile Sections. In selecting pile section lengths consideration should be given to: 1) the capability of the lift equipment to raise, lower and stab the sections: 2) the capability of the lift equipment to place the pile drh·ini hammer on the sections to be dri\'en: 3l the possibility of a large amount of downward pile mo,·ement immediately following the penetration of a jacket leg closure: -I I stresses developed in the pile section while lifting; 51 the wall thickness and material properties at field weids: 6) a\'oiding interference with. the planned concurrent dri\'ing of neighborini piles; and 7) the type of soil in which the pile tip is positioned durinl! driving interruptions for field welding to attach additional sections. In addition, static and dynamic stresses due to the hammer weight and operation should be considered as discussed in Par. 2.6.9d and Par. 2.6.9e.

Each pile ~tion on which dri\'ing is required should contain a cut.off allowance to permit the removal of material damaged by the impact of the pile driving hammer. The normal allowance is 2 to 5 ft. (.5 to 1.5 meters) per section. Where ;>ossible the cut for the remo\'al of the cut.off allowance should be made at a conveniently accessible elevation;

2.6.11 Shallow Foundations. Shallow foundations are those foundations for which the depth of embcdment is less than the minimum lateral dimension of the foundation l.'!ement. The desi~ of i:hallow foundationi,. sho11ld include, where appropriate to the intended application. consideration of the following:

1. Stability, including failure due to overturning, bearing, sliding or combinations thereof.

2. Static foundation deformations, including possible damage to components of the structure and its foundation or attached facilities;

3. Dynamic foundation characteristics, including the inOuence of the foundation on structural response and the performance of the foundation itself under dynamic loading.

4. Hydraulic instability such as scour or piping due to wave pressures, including the potential for damaie to the structure and for foundation

. instability.

5. Installation and removai. including penetration and pull out of shear skirtS or the foundation base

itself and the effects of pressure build up or draw down of trapped water underneath the base.

Recommendations pertaining to these aspects of shallow foundation design are gi,·cn in Par. 2.6.12 thru 2.6.16.

2.6.12 Stability of Shallow Foundations. The equations of this paragraph should be considered in evaluating the stability of shallow foundations. These equations are applicable to idealized conditions, and a discussion of the lir.iit.ations and of alternate approaches is gi\·en in the Commentary. Where use of these equations is not justified, a more refined analysis or specia! considerations should be considered.

2.6.12a. l"ndrained Bearing Capacity (o = 0). The maximum gros.;. \'ertical load which a footing can support under undrained conditions is

Q = {c~c ~ + 'Y D)A' .......... (2.6.12-1) -----·-· .. - ·- ... _ _,.

where: Q C

= =

maximum verticai ·1oad at failure undraine<! shear strength of soil

Ne = a dimensioniess constant. 5.14 for~= O o · = undrained friction angle :0

~ 'Y = total unit weight of soil. D = depth of embedment of foundation A' = eifecti\'e area of the foundation de

pending on the load eccentricity Kc = correction factor which accounts for

load inclination, footing shape, depth of embedment, inclination of base, and inclination of the ground surface.

A method for determining the correction facwr and the effecth·e area is gi\'en in the Commentary, Two special cases of Eq. 2.6.12-1 are frequently encoun· tered. For a ,·ertical centric load applied to a foundation at ground )e\'e} where both the foundation base and ground are horizontal, Eq. 2.6.12-1 is reduced beiow for two foundation shapes. ·

1. Infinite!~· Long Strip Footing. ~ = 5.14c.~ .................. (2.6.12-2) where: Q~ = maximum \'ertical (load per unit

length of footing .A0 = actual foundation area per unit l~ngth

2. Circular or Square Footing, Q = 6.17cA .................... (2.6.12-3) where: A = actual foundation area

2.6.12b. Drained Bearing Capacity. The maximum net \'ertical load which a footing can support under drained conditions is

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i C.- . ,.:-..• , ·,._,,

-· ·-·-------··-· ·-···-------------

RP 2A: Planning, Designing, and Constructing Fixed Offshore Platforms

Q' = (c'NcKc + qNqKq + ½ -Y'BN-yK-y)A ......................... (2.6.12-4)

where: Q'" = <I =

I =

'Y' = q :::

B =

A' =

Kc,.Kq.K-y =

maximum net vertical load at failure effective cohesion intercept of Mohr ·Envelope (Exp [ ·rr tand1) (tan1(,15°~'/2)), a dimensionless function of e' (Nq-1) co~'. a dimensionless function oft an empirical dimensionless function of rf that can be approximated by 2(Nq + 1) tanf effective friction angle of :\!ohr Envelope effective unit weight 'Y 'D. where D = depth of embedment of foundation minimum lateral foundation dimension effective area of the foundation de-- . pending on the load eccentricity

correction factors which account for load inclination, footing shape. depth of embe<lment. inclination of base. and inclination of the ground surface, respectively. The subscript. c: q. and · -y:-efer to the particular term in the equation.

A complete description of the K factors. as well as curves showing the numerical \'alues of };q, Xc. and N,y as a function of~· are given in the Commentary.

Two special cases of Eq. 2.6.12-4 for c' = 0 (u,;ually sand) are frequently encountered. For a vertical. centric )ol\d applied to a foundation at ground level where both the foundation ba..<:e and ground are horizontal, Eq. 2.6.12-4 is reduced below for two foundation shapes.

1. Infinitely Long Strip Footing. Qo = 0.5 "Y' BN-y~ ........•..• (2.6.12·5)

2. Circular or Square Footing. Q = 0.3 ">'' BN-y A ............... (2.6.12-6)

2.6.12c, Sliding Stability. The limiting conditions of the bearing capacity equations· in Par. 2.6.12a and b. with respect to inclined loading. repr~nt sliding failure and result in the following equations:

1. Undrained Analysis: H = cA ...................... : (2.6.12-7)

where: H = horizontal load ~t failure

2. Drained Analysis: H = c'A + Q tan f,' ........... (2.6.12-8)

2.6. l 2d. Safety Facton;. Foundations should have an adequate margin of safety against failure 1Jnder the design loading conditions. The following factors

of safety should be used !or the specific failure modes indicated:

Failure Mode

Bearing Failure Sliding Failure

Safety Factor 2.0 1..5

These values should be u..~ after cyclic loading effects have been taken into accounL Where geot.ech· nical data are sparse or site condition~ are particu· larly uncertain, increases i:i these values mar be warranted. See the Comme::tary for further discussion of safety factors.

2.6.13 Static Deformation of Shallow Foundations. The maximum foundation deformation under static or equivalent static loading 2..ffe<:ts the structural integrity of the platform. its !'en;ceahi!:~·. ar,d its components. Equations for evaluating the s~tic deformation of shallow foundations are gi"en in Par. a and b belov.·. These equations are applic::;.ble to idealized conditions. A discussion of the limitarions and o! alternate approaches is given in the Co!r.menta.ry.

2,6.13a. Short Term Deformation. For foundation materials which· can be a.ss~:ne<i to be isotropic and homogenous and for the eo:-.dition where the struc· ture ba.c:e is circular. rigid. and rests on the soil surface, the deformations o! the base under ,·arious loads are as follows:

Vertical: Ur "' ( .!:.!) Q ............ (2.6.13-1) 4GR

Horizontal: uh"' ( i-Sr )' H .•.••• (2.6.13-2) 32(1-~·IGR

Rocking: · er =. (_ 3<l-~·1 _\ M •••••••. (2.6.13·3l ~ B9R3

/

Torsion: &1 = /_!_) T ......... (2.6.134) \16GR3

.

whe!'e: Uv,Un=

Q,H = 8r,8t = M.T= G = JI "'

R =

vertical and horiz.onul displaeemtinU vertical and hnrizont.al loads overturning 2:,d torsional rotation~ overturning a..~d torsional moments elastic: 1-hear modulus of the soil pois..<on's r-cttio of the soil · radius of the base

These solutions can also- be used for approximating the response of a square b ... ~ of equal are:i..

2.6.13b. Long Tenn Deformation. An ~:im:ite of the vertical settlement oi a roil layer under an imposed vertical load can be dE:termined by the following equation:•

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u,.. = h" =. et, = C =

Qo = .. ~q =

American Petroleum lnstitu\.e

hC J .... ·tlo+bq ........... (2.6.13-5) l ·•10-~. c.,

,·ertia.! senlement la)•er '!,!1.ickness initial TOid ratio or the soil comp~ion index of the soil over the load n.."l:-e considered i~itia.l effective vertical stress added crecth·e \"ertical stress

Where the ,·ertical i:=-ess varies within· a thin layer, a.s in the a..~ of a di:::inishinr stress, estima.tes may

· be det.erminl'd by us::.g the stress at the midpoint of the laj·er. Thick h~,:nogeneous layers should be subdhidtd for anal),:.s. \\nere more than one layer is in,·olved. the esti:::.te is simply the sum of the settlement or the laye:-s. Ccmpression characteristics

; . or the soil are dt-~~ined from one--dimensional

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:?.6.14 D,-namic Beharior of Shallow Foundations. Dynamic· loads a.re im~ or. a structure-foundation S)'SlA!rr, br current. 'lt'a,·E.. ice. wind. and earthquakes.·.

· Both the influen~ or ti..t foundation on the structural r!!ponf.l! :uid the in~ty of the foundation itself sho~Jd ~ COn!idered.

2.6.15 H,-drliulic Inst.a.bility of Shallow Foundations

2.6.15a.. Scour. Posit.~e m!a;.ures should be taken to pre'\.·en! trosion and t:::ciercutting of the roil beneath or near tht st.ructt:rt' ~"" cue lo ~ur. Examples of such measura art' 11 l ~-:our skirts penetrating throu~h l?rodible laye!"S into s.cour resist.ant materials or to such depths ~ ;., elimina.t.e t~e scour hazard, or (2) riprap emplac~ a."'Dund the edres of the founda· _tion. 2.6.15b. Piping. Tr.ee foundation E:hould be so desil:Md to p""·en: the creation of exc~sive ·hydraulic graditnts 1:;:,ipini; conditions) in the soi! due to en,·ironment.i} loadings or operations carried out during or su~'jtnt t.o nructur! in!'\.allation.

2.6. J 6 I n.,;u,llation and Remonl of Shallow Founda• tions. lnsullation !'ho-.::.: be planned to ensure the foundation an be properly seated at the intendP.d site

·without excessh•e ciistur.:=ince to the supporting soil . . \\'here J'emo\"a.l is ::nti::pat.ed a.n anaiysis should be . made of the forces g-enented during remo\·al to ensure that remo'\.'al can be .._--:x,mplished with. the means 1\·ailable.

c·;

l

RULES FOR THE DESIGN CONSTRUCTION AND INSPECTION OF OFFSHORE STRUCTURES

1977

APPENDIX F

FOUNDATIONS REPRINT WITH CORRECTIONS (1980)

DET NORSKE VERITAS HEAD OFFICE: VERITASVEIEN 1, 1322 H0VIK P.0.BOX: 300, 1322 H0VIK, NORWAY TELEGRAMS: VERITAS, OSLO TELEX: 16192 TELEPHONE (International Nos.): + 47 2 12 99 00 TELEPHONE (National Nos.): (02) 12 99 00

FOREWORD

The purpose of the appendices to the DnV Rules for offshore structures is to provide recommended practice, methods and procedures for design, construction and inspection of offshore structures.

The guidance, methods and procedures given in the appendices are non-mandatory, thus the engineer is free to use other methods and procedures than those recommended provided an equivalent standard of quality and safety is obtained.

Each appendix is selfcontained and the procedures and methods given may be used independent of the Rules although the content of the appendices is directly related to the Rules.

In the appendix text reference to specific paragraphs in the Rules is made by giving the paragraph number marked with the letter (R), e.g. see 5.4.2 (R).

REPRINT WITH CORRECTIONS (1980)

A few corrections have been made in this reprint, mainly of linguistic or typographic nature.

TABLE OF CONTENT

APPENDIX F FOUNDATIONS

Fl BEARING CAPACITY OF FOUNDATlONS .... F 7 Fl.l General ............................. F 7 F l.2 Effective foundation area ................. F 7 F 1.3 Bearing capacity factors .................. F 8 F 1 .4 Load inclination factors .................. F 9 F 1.5 Shape factors ......................... F 9 F 1.6 Depth factors ......................... F 9 F l. 7 Simplified bearing capacity fonnula

for end resistance of piles ................. F 9 References ........................... F 9

F2 ANALYSIS OF AXIAL PILE RESISTANCE .... F l 0 F2.1 General ............................. FlO F2.2 Resistance of piles in compression ........... Fl 0 F2.3 Use of dynamic fonnulae ................. Fl l F2.4 Resistance of piles in tension ............... F 12 F2.5 Group effects ......................... F 12

References ........................... F12

F3 ANALYSIS OF LATERALLY LOADED PILES . ,Fl3 F3.l General ............................. FI 3 F3.2 Construction of p-y curves ................ FI 4 F3.3 Lateral resistance in sand ................. Fl4 F3.4 Load-<leflection characteristics for sand ........ Fl 5 F3 .5 Lateral resistance in clay .................. F 15 F3 .6 Load-deflection characteristics for clay . . . . . . . F 16 F3.7 Modification due to scour ................. Fl6 F3.8 Modification due to group effects ........... Fl6 F3.9 Modification due to reloading ............. FI6 F3. l O Modification due to long tenn loading . . . . . . . . F 17 F3 .11 Plastic analysis of piles . . . . . . . . . . . . . . . . . . F l 7

References . . . . . . . . . . . . . . . . . . . . . . . . . . F 17

F4 ANALYSIS OF PENETRATION RESISTANCE .. FI8 F4. l General ............................. Fl 8 F4.2 Method of calculation .................. .1FI8 F4.3 Penetration resistance of steel skirts .......... FI 8 F4.4 Penetration resistance of dowels ............. F 19 F4.5 Penetration resistance of concrete skirts ........ F 19 F4.6 Penetration resistance of ribs ............... Fl 9

References ........................... F 19

F5 LOCAL SOIL REACTION STRESSES ........ F20 FS.1 General ............................. F20 F5.2 Elasto-plastic analysis .................... F20 F5.3 Elasticity factors ....................... F22 F5.4 Empirical expressions for Young's Modulus E .... F22

References ........................... F23

F7

APPENDIX F FOUNDATIONS

Fl BEARING CAPACITY OF FOUNDATIONS

F 1.1 General

F 1. 1.1 Under certain conditions as explained in Section 9 (R) the stability of foundations and the point bearing capacity of piles may be analysed by means of bearing capacity fonnulae. In such cases, the procedure outlined below, in principle based on /1/, may be used.

F 1.1.2 In the special case of a footing with horizontal base founded on a horizontal ground surface the design bearing capacity with respect to vertical load may be determined from the following fonnula.

qd = ~y'b'N,,s,,d,yi,y + p0 'Nqsqdqiq + CtlNcscdcic (Fl-1) 2

where

N')',Nq,Nc

S,,,Sq ,Sc d1 ,dq ,de i')')q Jc

design bearing capacity effective (submerged) unit weight of soil effective foundation width (see F 1.2) effective overburden pressure at base level design cohesion (c'hmc) or design undrained shear strength (cuhmc) assessed on the basis of the actual shear strength profile, load configuration, and estimated depth of potential failure surface. bearing capacity factors, see F 1.3 shape factors, see F 1.5 depth factors, see Fl.6 load inclination factors, see F 1.4

In the general case with inclined base and ground surface each term in Eq. F 1-1 are multiplied with base and ground inclination factors, according to /1/.

The calculation is to be based on design shear strength parameters cd and tan<fJd defined as follows

c' cd=-

1'mc

tan</)' tan<fJd = __

1'mf

where

Cu ) (orcd = -1'cm

c' characteristic cohesion, see 9.6.4.2 (R).

(F 1-2)

(F 1-3)

Cu characteristic undrained shear strength, see 9 .6.4.3 (R).

</>' characteristic angle of shearing resistance at the appropriate stress level, see 9.6.4.2 (R).

1'mc,1'mf material coefficients associated with tire actual type of analysis, see 9.5.5 (R) and 9.6.4 (R).

F 1.1.3 Since the q- and c-factors in Eq. F 1- 1 are interrelated, see Eq. Fl-8, the formula can be written in a somewhat simpler form. When the design angle of shearing resistance <Pd * 0, the design bearing capacity is then:

(Fl-4)

For the special case that <f>d = 0 (undrained failure in clay), additive constants are used instead of factors and we get:

(F 1-5)

F 1.1.4 Mathematical expressions and numerical values for the various factors are given in F 1.3 through F 1.6. These are based on the assumption that an ideal plastic failure with full mobilization of the shear strength in the entire plastic zone governs the bearing capacity (general shear failure). For loose soils or at high stress levels, the shear strength may not be fully mobilized (local or punching shear failure). The bearing capacity may then be considerably smaller than that calculated for ideal plastic conditions. In calculations of bearing capacity, the effect of soil compressibility should be considered when appropriate.

Fl.2 Effective foundation area I .

Fl.2.1 In the calculation of bearing capacity according to /I/- the effective foundation area is used. This area is defined as follows;

The resultant of all horizontal and vertical forces acting from above upon the base of the foundation are combined into a resultant force. Each force is multiplied by a loacf coefficient, ')'f, according to 4.4.4.3 (R). The point where the resultant intersects the base, is called the load centre.

A rectangular "effective foundation area" is now determined. The geometrical centre of this area coincides with the load centre, and it follows as closely as possible the nearest contour of the actual base area. Two examples are shown in Figure F 1.1. The width of the effective foundation area is b' and the length .f.

~ Fv l'

-b' I

-1

Figure Fl.l. Effective foundation area A'= b'l '.

F 1.3 Bearing capacity factors

Fl.3.1 The expression for Nc(<f>d=.o) from /2/ is

Ne= rr + 2== 5.14

b

(Fl-6)

F 1.3 .2 The bearing capacity factor, Nq, from /3/ is

Nq = errtan<f>dtan2 (45 + <Pd ) 2

(Fl-7)

Fl.3.3 The interrelationship between q- and c-factors is given by

(Fl-8)

For an infinitely long surface footing subjected to a vertical, centric load this expression simplifies to

Ne = (Nq - 1 )cot<f>d (Fl-9)

Fl.3.4 For N,,, the following expression is recommended /1/

N"Y = 1.5 (Nq - 1 )tan<f>d

while according to /4/

(Fl-10)

The latter relationship for N"Y is proposed for the calculation of local soil reaction stresses on the foundation structure as described in F5.

Fl.3.5 Numerical values of Ne, Nq and N"Y are tabulated in

Table Fl.l.

F8

Table FI.I Bearing capacity factors Ne, Nq and N"Y

<l>d N-y

Caquot Degrees Ne Nq and Brinch-

Kerisel Hansen /4/ /1/

0 5.14 1.00 0.00 0.00 1 5.38 1.09 0.07 0.00 2 5.63 1.20 0.15 0.01 3 5.90 1.31 0.24 0.02 4 6.19 1.43 0.34 0.05 5 6.49 1.57 0.45 0.07 6 6.81 1.72 0.57 0.11 7 7.16 1.88 0.71 0.16 8 7.53 2.06 0.86 0.22 9 7.92 2.25 1.03 0.30

10 8.35 2.47 1.22 0.39 11 8.80 2.71 1.44 0.50 12 9.28 2.97 1.69 0.63 13 9.81 3.26 1.97 0.78 14 1037 3.59 2.29 0.97 15 10.98 3.94 2.65 1.18 16 11.63 4.34 3.06 1.43 17 12.34 ·4.77 3.53 1.73 18 13.10 5.26 4.07 2.08 19 13.93 5.80 4.68 2.48 20 14.83 6.40 5.39 2.95 21 15.82 7.07 6.20 3.50 22 16.88 7.82 7.13 4.13 23 18.05 8.66 8.20 4.88 24 19.32 9.60 9.44 5.75 25 20.72 10.66 10.88 6.76 26 22.25 11.85 12.54 7.94 27 23.94 13.20 14.47 9.32 28 25.80 14.72 16.72 10.94 29 27.86 16.44 19.34 12.84 30 30.14 18.40 22.40 15.07 31 32.67 20.63 25.99 17.96 32 35.49 23.18 30.22 20.79 33 38.64 26.09 36.19 24.44 34 42.16 29.44 41.06 28.77 35 46.12 · 33.30 48.03 33.92 36 50.59 37.75 56.31 40.05 37 55.63 42.92 66.19 4:'7.38 38 6135 48.93 78.03 56.17 39 67.87 55.96 92.25 66.75 40 75.31 64.20 l 09.41 79.54 41 83.86 73.90 130.22 95.05 42 93.71 85.37 155.54 113.95 43 105.11 99.01 186.54 137.10 44 118.3 7 115.31 224.64 165.58 45 133.88 134.87 271.76 200.81

F 1 .4 Load inclination factors

F 1.4. l The expression for iq and i-y are:

ir =~ -

where

design horizontal load= rrFH design vertical load = rrFv

(Fl-12)

(Fl-13)

characteristic horizontal and vertical load, respectively, compatible with the loading condi· tion under consideration

'Yf appropriate load coefficients according to 4.4.4 (R).

A' effective foundation area

Fl.4.2 The expression for ica in Eq. Fl-5 is

ica = 0.5 - 0.5~ 1 - l:Hdl A'cd

(Fl-14)

F 1.4.3 The results of the calculations should be used with care when the ratio F11 ctlfv d approaches or becomes greater than 0.4.

F 1.5 Shape factors

F 1.5. l The expression for sq and 5r are:

. b' s.. = 1 + ~ sin<l>d 'I /'

i b' lv., = 1-0.4 ~ - , I'

F 1.5 .2 For the case of <Pd = 0, Sea is:

. b' Sea = 0.2(1 - 2 lea) -

I'

Fl.6 Depth factors

(Fl-15)

(Fl-16)

(Fl-17)

F 1.6.1 For shallow foundations, especially those of offshore gravity structures, the depth factor has almost negligible effect on the calculated bearing capacity. In this context we therefore use

dq =de= 1.0 (Fl-18)

which implies that dca = 0 in Eq. Fl-5.

The depth factor dr is per definition equal to unity, thus

d-y = 1.0 (Fl-19)

F9

Fl.6.2 In special cases values dq > 1.0 and dca > 0 may still be used, provided that the foundation installation procedure and other critical aspects allows for the mobiliza. tion of resisting shear stresses in the soil above the foundation level. In such cases the following expression for dq, valid for d < b', defines an upper limit for this contribution

d 2 dq = 1 + 1.2 b' tan<l>d ( 1 - sin<l>d ) (Fl-20)

The corresponding expression for dca is

, d dca = 0.3 arc tan (b') (Fl-21)

which approaches a limit value dca = 0.47 for large depths.

Fl.7 Simplified bearing capacity formulae for end resistance of piles

F 1. 7 .1 The expressions given in F 1. 7 .2 and F 1. 7 .3 are valid for circular or square footings founded at depths d > 4b '. The load is assumed to be centric and vertical.

F 1.7 .2 For piles in mainly cohesionless soils, the following expression for the design unit end resistance qdp may be used:

(Fl-22)

F 1.7 .3 For piles in mainly cohesive soils, the design unit end resistance may be expressed Js:

(Fl-23)

F 1.7 .4 For limitations in the use of the expressions given in Fl.7.2 and Fl.7.3 reference is made to F2.2.S, F2.2.7 and F2.2.9.

References

/1/ Brinch-Hansen, J.: "A Revised and Extended Formula for Bearing Capacity". The Danish Geotechnical Institute, Bulletin No. 28. Copenhagen, 1970.

/2/ Prandtl, L.: "Uber die harte plastischer Korper". Nachrichten der Gesellschaft der Wissenschaften. Gottingen, 1920.

/3/ Reissner, H., "Zurn Erddruckproblem". Proc. 1st Intern, Congr. Appl. Mech. Delft, 1924.

/4/ Caquot, A. and Kerisel, J.: "Sur la terme de surface dans le calcul des fondations en milieu pulverult". Proc. 3rd Intern. Conf. Soil Mech. and Found. Engng., Vol. 1, Ziirich, 195 3.

F2 ANALYSIS OF AXIAL PILE RESISTANCE

F2. l General

F2. l. l Axial pile resistance is composed of two parts, one part is the accumulated skin resistance and the other part is the end resistance.

F2 .1.2 Piles carrying their loads mainly through mobilized end bearing resistance are called end bearing piles while the term friction piles is used for piles carrying their loads mainly through mobilized shaft friction.

F2.1.3 Structural behaviour of piles is greatly affected by the method of installation, see 10.4 (R). Various pile installation procedures such as preboring and jetting tend to decrease the shaft friction while hard driving can leave large residual stresses in both the pile and the soil.

F2. l.4 Open-ended steel cylindrical (pipe) piles are the most common piles for support of offshore structures.

F2.2 Resistance of piles in compression

F2.2.l For piles in mainly cohesive soils, the average unit skin friction f5 may be calculated according to one of the following methods;

the a-method (total stress method) the /3-method (effective stress method) the A-method (combined total/effective stress method)

Other methods may be considered provided that the results are shown to be on the conservative side.

Irrespectively of the method applied for calculation of the skin resistance the influences of factors such as procedure of pile installation (driven or drilled piles), type of drilling mud and grout, length and geometry of pile (cylindrical or with increased base diameter), etc. have to be considered.

F2.2.2 In the a-method the average unit skin friction in layer i is given by

(F2-l)

where a is a multiplier which is correlated with the undrained shear strength Cu and may vary between 0.2 and 1.2, decreasing with increasing Cu. Considerable judgement and experience is required to make a realistic choice of a-values to be applied to the different soil layers along the pile shaft. For guidance in this choice reference is made to rele','.ant literature, e.g. /1/.

FlO

F2.2.3 In the /3-method, see e.q. /2/, the unit skin friction fsi is related to the effective stress parameters K and o, see Eq. 9-2 (R), through

fsi = Ktano p~ = /31'~ (F2-2)

where

K

tano

Po'

average coefficient of earth pressure on pile shaft average coefficient of friction between soil and pile shaft effective overburden pressure

For piles in normally consolidated clays inducing no appreciable change in lateral ground stress conditions it may be assumed that

K = 1 - sinq>' (F2-3)

If it is further assumed that failure takes place in the remoulded soil close to the shaft surface the remoulded, drained angle of shearing resistance may be used for rp' along with o = ,p'. Under these assumptions Eq. F2-2 gives /3-values between 0.2 and 0.3 for a reasonable range of 4>'. For pile lengths exceeding about 15 m results from fullscale tests suggest /3-values in the range 0.1 to 0.25.

In the case of piles driven into overconsolidated clays the scatter in /3 is considerable since it reflects the uncertainty in the lateral earth pressure coefficient K which usually only can be estimated within wide limits. The decrease in /3 with increasing pile length is even more pronounced for overconsolidatec,l clays than for normally consolidated clays.

F2.2.4 In the A-method /3/ the total shaft resistance R5 is calculated from the expression

or

where

(F2-4)

(F2-5)

average unit skin friction along pile shaft mean effective overburden pressure between the mudline and the pile tip mean undrained shear strength along the pile shaft pile shaft area dimensiQn!ess coefficient

Analysis of load tests on pipe piles has given A-values in the range 0.4 to 0.2 for piles penetrating less than 15 m decreasing to about 0.1 for piles embedded about 60 m. It should, however, be noted that A-values exceeding 0.18 refer to pile load tests only in overconsolidated clays.

This method is applied to piles driven at least 15 m into normally consolidated or lightly overconsolidated clays but should be applied with caution when the depth of penetration in these clays is less than 15 m.

F2 .2 .5 The unit end resistance, qp, of piles in mainly cohesive soils may as an average be taken equal to 9 times the undrained shear strength of the soil at the level of the pile tip, provided that the installation process has not reduced the strength. The end resistance may, however, be limited by the capacity of an internal plug in the pile. Recent experience, e.g. /4/, also indicates that size effects may be of importance, i.e. that large diameter piles develop a smaller unit end resistance than small diameter piles in the same soil.

F2.2.6 For piles in mainly cohesionless soils the skin friction is dependent on

- pile-soil adhesion, which may be neglected for design purpose,

- effective overburden pressure, p0 '

- coefficient of lateral earth pressure, K, the value of which depends significantly on factors such as those mentioned in F2.2.l, the coefficient of friction, tano, between the soil and the pile shaft, which may be taken smaller or equal to tarn/>', i.e. the characteristic coefficient of friction of the soil in terms of effective stresses.

F2.2.7 The unit end resistance of piles in mainly cohesionless soils may be computed by means of bearing capacity formulae (see Appendix Fl). The end resistance may be limited by the capacity of an internal plug in the pile.

F2.2.8 For piles under compression in mainly cohesive soils the design resistance R<J of a single pile is calculated from the expression

1 Rd=-

'Ym (F2-6)

where for the shaft resistance a material coefficient 'Ym = 'Ymc = 1.3 is applied in connection with the O:· or the A-method whereas rm = 'Ymf = 1.2 is applied when the µ.method is used, see 9.5.5.2 (R).

Fl 1

F2.2.9 For piles under compression in mainly cohesionless soils neglecting the pile-soil adhesion, the design resistance of a single pile is calculated from the expression

I R<J =-'Ymf

where

'Ymr= 1.2 ,Pd design angle of shearing resistance at the appropriate

stress level related to the characteristic angle of shearing resistance ,P' through tan,Pd = (tan,P')hmr where 'Ymf = 1.1, see 9.5.5.2 (R).

Nqd bearing capacity factor Nq in Table F 1.1 detern1ined for ,Pd,

I

The application of Eq. F2-7 has to consider the existence of limiting values for both unit end resistance and unit skin resistance, see 9.5.2.5 (R) and 9.5.2.6 (R). The material coefficients related to these limiting values are given in 9.5.5.2 (R), i.e. 'Ymf = 1.3 on the limiting unit end resistance and 'Ymf = 1.2 on the limiting unit skin resistance.

F2.2.10 In the calculation of resistance of compression piles with enlarged base area it is good practice to neglect the skin resistance over a length of three pile diameters above the enlarged section.

F2.2.l l The reliability of the above methods of calculation is to a great extent dependent on the quality of the geotechnical data available at the time of the analyses. Current experience clearly points out the advantage of supporting the laboratory determinations of shear strength parameters by the results of in situ measurements e.g. by means of tests such as the cone penetration test.

F2.2.12 In order to evaluate the reasonableness of the calculated pile resistance a pile drivability study is often required. The inost critical input parameters in this analysis are those defining the pile-soil interaction behaviour. Here, again, in situ testing may provide valuable information for the analysis, see also 9.2 (R) and F2.3.

F2.3 Use of dynamic fonnulae

F2.3.l Dynamic pile formulae, preferably those based on the wave progagation theory, may be used to relate the penetration resistance of the pile during driving to the static pile capacity and to check on stresses in the pile and soil elements during and after driving.

F2.3.2 The application of the wave equation fonnulae requires as input significant parameters such as

- type and size of pile-driving hammer - energy output of hammer (continuous recording during

driving desired) driving assemblies (cushion blocks, anvil, etc.) type and size of pile soil conditions pile-soil interaction parameters.

F2.4 Resistance of piles in tension

F2.4.l For piles under tensile loads the load transfer mechanism is different from that in compression. In general this is reflected in the choice of a lower value on the unit skin friction in comparison with that used for piles in compression, see also 9.5.3 (R).

F2.S Group effects

F2 .5. I The group resistance of piles depends on factor such as pile spacing, type and strength of soils, sequence of soil layers, pile installation method etc. The knowledge of the behaviour of full-scale pile groups relative to the behaviour of individual piles in the same group is limited and conservative assumptions are therefore recommended for the calculation of pile group resistance.

F2.5.2 For homogeneous soil conditions the end resistance of a pile group can be taken equal to the sum of the end resistances of individual piles.

F2.5.3 The skin resistance of pile groups in cohesionless soils may at least be taken equal to the sum of the skin resistance of individual piles times the ratio of the outer perimeter of the group to the sum of perimeters of individual piles. The possible increase of this minimum resistance depends on the initial density of the soil, e.g. is considerable larger for piles in loose sand than for piles in dense sand.

Fl2

F2.5 .4 The skin resistance of pile groups in mainly cohesive soils should not be taken larger than the sum of the skin resistance of individual piles. Reductions may be required when the pile spacing s is small or when the overlapping zones of shearing defonnation influence the skin resistance of individual piles over a significant length of the embedded part of the piles. The reduction factor may be expressed as ·the ratio of the outer perimeter of the group to the sum of the perimeters of individual piles. According to this approach no reduction is required if the relative spacing of the piles s/d, d being the pile diameter, is greater than vn + I for square piles and 0.785 (yn + I) for circular piles, where n is the number of piles in the group.

F2.5.5 In addition to the above mentioned general app· roach to the problem of estimating pile group resistance from a calculated single pile resistance special considerations are required in each case in order to account for

- method of pile installation - weak deposit underlying a bearing layer of limited

thickness negative skin friction along pile shaft etc.

Some of these considerations may lead to group effects which will change the general view given in F2.5.2 through F2.5.4. For guidelines reference is made to /2/.

References

/1/ "API RP 2A, Recommended Practice for Panning, Designing and Constructing Fixed Offshore Platforms". American Petroleum Institute, Eighth Edition. Dallas, Texas, 1977.

/2/ Meyerhof, G.G.: "Bearing Capacity and Settlement of Pile Foundations". Proc. Amer. Soc. of Civil Engineers, Vol. 102, No. GT3, March, 1976.

/3/ Vijayvergiya, V.N. and Focht, J.A. Jr.: "A New Way to Predict Capacity of Piles in Clay", Proc. 4th Annual Offshore Techn. Conf., Vol. 2, Paper No. 1718. Houston, Texas, 1972.

/4/ De Beer, E.E.: "Scale Effects In Results of Penetration Tests Performed in Stiff Clays". Proc. European Symp. on Penetration Testing, Vol. 2 : 2. Stockholm, 1974.

F3 ANALYSIS OF LATERALLY LOADED PILES

F3.l General

F3 .1.1 This Appendix deals with problems concerning the design of laterally loaded piles for offshore platforms. For such piles having diameters typically between 1.0 and l .5 m the most severe loading conditions arise from cyclic wave loads. Special problems related to other types of piled structures subjected to lateral loads will be mentioned only briefly.

F3. l.2 The general method for analysis of laterally loaded piles proposed here is based upon a modified Winkler hypothesis. It is postulated that at each depth there is a unique relationship between the deflection of the pile and the soil pressure reacting against the face of the pile.

F3. l.3 The pile is divided in elements and each element will be characterized by a load-deflection curve, the so-called p-y curve, see Figure F3. l.

6

la}LOAOING lb}OISPLACEMENT ltl SOIL PRESSURE DISPLACEMENT CONDITIONS CHARACTERISTICS

Figure F3.1 P·Y curves for a pile.

F3. l.4 The total force F on the element due to the soil pressure p reacting against a certain pile element is obtained by multiplying the average value of that pressure by the exposed area. Thus

F = pb .11 (F3-l)

where

F force on element p average soil pressure b width of pile .11 length of pile element

F3.l.5 Taking into account the pile properties and the boundary conditions of the pile, a numerical solution of the load-deflection and the sectional forces and moments in the pile can be found.

FI3

F3. l.6 .In the following it will be assumed that a satisfactory numerical analysis procedure is available. This Appendix will thus concentrate on the selection of parameters required for the construction of the p-y curves. Special effects such as scour and group interaction will also be mentioned.

F3.1.7 The design with respect to lateral loads is based upon semi-empirical methods supported by the results of only a few well documented full-scale tests. Due to this limited knowledge about the behaviour of laterally loaded piles the method for construction of p-y curves should be as simple as possible considering also the uncertainty related to the soil data, etc.

F3. 1.8 In the following the initial portion of the p-y curve (between the origin and point "a" in Figure 3.2) will be approximated by a hyperbola. This shape is fully described by three parameters. The parameters are functions of factors determined by the pile-soil system and are defined in more detail in F3.2.

INITIAi. SLOPE

Figure F3.2 Characteristic shape of the p-y curve.

F3.1.9 The deterioration of the lateral resistance for deflections beyond point "a", as indicated in Figure F3.2, may be severe for clay while the resistance of sand is much less affected. The scarcity of full-scale test results makes, however, the prediction of p-y curves uncertain, especially the shape of the curve for defleptions beyond point "a" in Figure F3.2. This necessitates conservative assumptions · until our knowledge has been substantiated on this point based on future experiences.

F3.1.10 The development of p-y curves have to consider the effects of dynamic loading, reloading after extreme loading, scour, as well as group effects. Details about these effects on the p-y curves are given in F3.3 through F3.1 l.

F3.1.1 l Dynamic loading may have the nature of impact loading (e.g. ship impact) or repeated loading in the form of earthquake loading or storm loading. Earthquakes are of short durations (a few seconds) while storms have durations of several hours.

F3.l.12 Special piles, e.g. for moorings, may be subjected ·to impact forces, which may often be considered as static in the design.

F3.1.13 Two types of analyses are considered herein:

- Analysis for design extreme lateral load using appropriate load and material coefficients to find maximum stresses and deflections (ultimate limit state), see F3.2 through F3.8.

- Analysis of reloading to find stresses and deflections for lateral loads succeeding the design extreme load (fatigue limit state and serviceability limit state), see F3.9.

Both these analyses are carried out with material coefficients 'Ymc = 'Ymf = 1.0 for the soil.

F3.1.14 Under certain conditions as stated in 6.2.4 (R) and 9.5.4.2 (R) the lateral resistance of a pile or a pile group may in the ultimate limit state and the progressive collapse limit state be calculated based on the theory of plasticity. In this case material coefficients for the soil are taken in accordance with 9.5.5.3 (R). Plastic analysis of piles is treated in F3 .11.

F3.l.15 It should be stressed that the scope of the site investigations and laboratory testing should be sufficiently extensive to reveal possible variations of the soil properties in both lateral and vertical directions down to a depth of at least 10 pile diameters. A thorough knowledge of the properties of these upper soil layers is of the greatest importance for the design of laterally loaded piles.

F3. l.16 An evaluation of the depth of scour to be considered in the analysis requires special knowledge about the properties of the upper soil layer. Where scour is expected, p-y curves should be modified so as to take the scour into account, see also F3.7.

F3.2 Construction of p-y curves

F3.2.1 The p-y curve representative for the load-deflection characteristics of a given pile element is shown in Fig. F3.2. The curve consists of three parts.

For p ~ Pd and y ~ 13b the curve is described by the hyperbola

t = _!_ + L (F3-2) p k1 Cl'.Pd

where

y p

Pd Cl'.

Plim k1 {J

pile deflection lateral soil resistance design lateral resistance

Plimf Pd = failure ratio, always greater than 1.0, related to coefficient {J according to Eq. F3-3 assymptotic value of the hyperbola for y --+co

initial slope of the curve coefficient related to type of soil and loading condition under consideration, see Table F3.1 for sand and Table F3.2 for clay.

Fl4

F3.2.2 The location of the point of intersection between the hyperbola and the design resistance, Pd (point "a" in Figure F3.2), is dependent on the value given to the coefficient a, which for y = {Jb and p = Pd in Eq. F3-2 has the following form (valid for {Jb > Pdfk 1 ):

a=---

1-..RL k1{Jb

(F3-3)

The y-value at point "a" has then been related to the pile width by the coefficient {J which is a parameter of major importance for the shape of the p-y curve.

For {Jb ~ Pdfk1 the hyperbola may be replaced by a straight line. Numerical values for {J are given in Table F3.1 for sand and in Table F3.2 for clay.

F3.2.3 Special effects may lead to deterioration of the lateral resistance for deflections beyond point "a". A residual lateral resistance pa' may then be defined based on the results from representative laboratory tests and a certain amount of engineering judgement. The approach of this residual resistance for large deflections is evident in clay and the deflection required to mobilize this resistance is termed y = {J'b. Numerical tentative values for coefficient {J' are given in Table F3.2 for clay.

For sand there is no evidence that the soil resistance deteriorates beyond point "a". Thus in sand for y > /3b a constant resistance p = Pd = Pd' is assumed.

F3.2.4 For deflections y > /3'b no further reduction is assumed for lateral resistance in clay, i.e. p = Pd' for y > {J'b.

F3.2.5 Principles for the calculation of design lateral resistance are given for sand in F3.3 and for clay in F3.5 while the load-deflection characteristics for piles in sand and clay are given in F3.4 and F3.6, respectively.

F3.3 Lateral resistance in sand

F3.3.l The design lateral resistance, Pd, of a single pile in sand is dependent on the type of loading and the depth of scouring.

F3.3.2 For static loading the design lateral resistance, termed Pds, can be taken as

Pds = 4KpPo'

where

(F3-4)

1 + simPd

1 - simPd

p~ effective overburden pressure (after scouring)

4>d design angle of shearing resistance (tan</>d = 1

-- tan</>) 'Ymf

</> characteristic angle of shearing resistance

'Ym r material coefficient according to 9 .5 .5 .3 (R) for plastic analysis, otherwise 'Ymf = 1.0, see F3.l.13 and F3.1.14.

F3.3.3 For cyclic loading the design lateral resistance, tenned Pde, can be taken as

Pde= 3Kpp~ (F3-5)

for depths x > 2b.

For depths x ~ 2b Eq. F3-5 is replaced by

Pde = 3~ KpPo' 2b

(F3-6)

F3.3.4 At large depths Eqs. F3-4 and F3-5 probably underestimate the pile resistance. Being on the conservative side and considering the small shear stress mobilization at these depths this expression may still be used.

F3.3.5 In static loading the lateral resistance tends to be higher than in cyclic loading, in some cases twice as high resistance has been recorded. Few test results are, however, available and conservative assumptions are therefore recommended, see Eq. F3-4.

F3.3.6 The assessment of characteristic shear strength parameters for the soil should be based on the results of representative laboratory tests. Of almost equal importance is a correct detennination of the unit weight of the foundation soil.

F3.4 Load-deflection characteristics for sand

F3.4.1 The characteristic p-y curve in Figure F3.2 can be applied to sand by introducing Pd = Pct and

(F3-7)

where

flh coefficient of subgrade reaction, see Table F3 .1 x depth below mudline (after scouring).

F15

Table F3. l Recommended values for nh and ~ for static and cyclic loading in sand.

Parameter Relative density for sand

Loose Medium Dense

nh (MPa/m) 5.0 12.0 18.0

~ 0.04 0.04 0.04

F3.5 Lateral resistance in clay

F3 .5 .1 The design lateral resistance of a single pile in clay can be taken as

N Cu Pd= p-

'Yme (F3-8)

where

Cu

'Ymc

Np

characteristic undrained shear strength representative for the loading condition under consideration material coefficient according to 9.5.5.3 (R) for plastic analysis, otherwise 'Ymc = 1.0, see F3.l.13 and F3.1.14. semi-empirical constant linearly increasing from 1.0 (static loading) or O (cyclic loading) at mudline (after scouring) to 8 at a depth x = Nrb below mudline

Nr = 10, for nonnally consolidated clays = 5 for overconsolidated clays.

F3.5.2 For offshore piles which shall withstand cyclic waveloads the characteristic undrained shear strength should be found from specially designed laboratory tests considering the actual static and dynamic stress conditions in situ. It is recommended to perform undrained cyclic triaxial or simple shear tests on soil samples consolidated to in situ stress conditions and to run a static undrained test on the same sample after completion of the cyclic load test. The static shear strength thus obtained is tenned Cuc which substituted for Cu in Eq. F3-8 will give the design lateral resistance, Pde, for cyclic loading.

F3.5.3 For piles subjected to pure static loading the cyclic · part of the laboratory tests described in F3.5 .2 is omitted. The undrained shear strength thus oba tined is termed Cu which introduced in Eq. F3-8 gives the design lateral resistance, Pds for static loading.

F3.S.4 It is important that the static laboratory test is run to strains large enough to define a residual value of the undrained shear strength, Cur, which substituted for Cu in Eq. F3-8,gives the residual (static) lateral resistance Pcts or the residual (cyclic) lateral resistance, Pctc·

F3.5.5 In lack of residual undrained shear strength from laboratory tests a value of 0.5 at depth x = Nrb may be assumed for the ratios Cur/ Cu (static loading) and Cur/Cuc (cyclic loading) decreasing linearly to 0.25 at depth x = 0. This may, however, be very conservative for some clays and on the unsafe side for other clays which indicates the need for laboratory detennination of the residual value when appropriate.

F3.6 Load-deflection characteristics for clay

F3 .6.1 The characteristic p-y curve in Figure F3 .2 can be applied to clay by introducing for the initial slope.

where

Pd Pd ~

Pds for static loading Pd c for cyclic loading empirical coefficient

(F3-9)

vertical strain at one-half the maximum principal stress difference in a static undrained triaxial compression test on undisturbed soil sample, see F3.S .2 and F3.S.3.

F3.6.2 Numerical values of the parameters {3 and {3', defined in F3.2 and Figure F3.2, and ~ are given in Table F3.2. It has been made a differentiation between static and cyclic loading for normally consolidated and overconsolidated clays, respectively.

Table F3.2 Recommended values for ~, {3 and P' for clay

Type of load Type of clay

Parameter Normally Over-

consolidated consolidated

~ Static 10 30

{3 Static 20Ec Sec

w Static 80ec 8Ec

~ Cyclic 10 30 {3 Cyclic 7.Sec 2.Sec {3' Cyclic 20Ec Sec

F3.6.3 For slightly overconsolidated clays the set of parameters in Table F3.2 giving the most conservative results should be chosen.

F3.7 Modification due to scour

F3.7. l Scour will lead to complete loss of lateral resistance down to the depth of scour and should be considered so in the construction of the p-y curves for the soil layer susceptible to scour, see Figure F3.3.

F3.7.2 Scour will also reduce the effective stress, p0 ',

further down which should be considered by using the scoured base as mudline in the construction of p-y curves. This has been demonstrated in Figure F3.3. In sand this will reduce the value both of the k1 -parameter and the design l~teral resistance, Pd, defining the p-y curve for a certain pile element. .,

F16

ORIGINAL MUDllNE

SCOURED \ 'l· = 0 AFTER SCOUR DEPTH,••

Po NEW MUDLINE AFTER SCOUR

'----EFFECTIVE STRESSES WITHOUT SCOUR

Figure F3.3 Modification of effective overburden pressure, p0 ' due to scour.

F3.8 Modification due to group effects

F.3.8.l The influence of one pile on the behaviour of another in a group of piles should be considered when the centre to centre distance between the piles (pile spacing) is 8 pile diameters or less.

F3.8.2 This analysis may be run as a single pile analysis as outlined herein provided that the p-y curves are corrected for "shadow" effects on the p-values and displacement effects on the y-values as a result of the group action.

Modification of the y-values to account for group effect, may be done by superimposing the interaction effects calculated according to the theory of elasticity.

F3.8.3 For further details on pile group analysis we refer to relevant literature, e.g. /1/, /2/.

F3.9 Modification due to reloading

F3 .9 .I This modification is for analysis in the fatigue limit state and the serviceability limit state and is applied to the p-y cutves developed for the ultimate limit state analysis described in F3.2 through F3.8.

F3.9.2 The modification due to reloading is based on the assumption that the design extreme lateral load generates a space between the pile and the surrounding soil. For subsequent loading the effect on the pile response of this space should be considered by introducing an initial deflection Yv for subsequent loads, see Figure F3.4. Conservatively the slope of the unloading branch in the extreme load cycle is taken equal to the initial slope, k1 , of the loading branch. This gives for reloading the load-deformation curve shown in Figure F3.4b.

Figure F3.4 p-y curves for extreme load (a) and subsequent loads (b).

F3.10 Modification due to long term loading

F3.l0.1 Structures as piled anchors can be subjected to long term static loads. Tests in both clay and sand has shown that long term loads can give deformation 2-3 times greater than for short term static conditions. The increase will be greater with higher stress level.

F3.,11 Plastic analysis of piles

F3.11.1 In a plastic analysis of piles plastic hinges are assumed to develop in the pile (Figure F3.5) along with a fully mobilized earth pressure between the two hinges. This analysis may be used for the ultimate limit state and the progressive collapse limit state under certain conditions, see 6.2.4 (R) and 9.5.4.2 (R).

The analysis should be supplemented with a load-deflection analysis according to the method described in F3.9 for analysis of the serviceability limi.t state.

tN

LATERAL

0

FORCE, F

MUDLINE

p

s ...

i N,

My 1

n

)(

t N z

N X

F

Figure F3.5 Plastic hinges in pile and fully mobilized earth pressure

F17

F3.1 l.2 The ultimate situation is illustrated in Figure F3.5. Plastic hinges are assumed to develop in the upper fixed end and at some depth below mudline. It is assumed that the pile is sufficiently long that the lower end is prevented from rotating. The fully mobilized earth pressure may be calculated according to F3.3 and F3.5 for the sand and clay, respectively.

F3.11.3 The equilibrium of the pile can be descnbed by

F=P+S

My1 = Px1 + Sx2 + My2

where (see also Figure F3.5)

F p

s My1 My2 Xl x2

design load resultant of soil resistance shear force in plastic hinge no. 2 yield moment in plastic hinge no. 1 yield moment in plastic hinge no. 2 distance from hinge no. 1 to resultant P distance between the two hinges

(F3-10)

(F3-11)

(F3-12)

(F3-13)

F3.11.4 When evaluating the yield moments, Myt and My 2, the axial force N (Figure F3.5) must be taken into account. A reduction in the axial force due to skin friction (N1-N2 in Figure F3.5) can be found according to F2.

F3.1 l.5 When the yield moments My 1 and My2 and the magnitude and distribution of the soil reaction are known, Eqs. F3-10 through F3-13 may be used to solve the unknowns (P, S, x1 and x2). F must be given the lowest value for which the equations have real solutions.

In clay the second hinge will be situated at a depth shallower than Nrb.

References

/1/ Focht, J.A. and Koch, J,J.: "Rational Analysis of the Lateral Performance of OffsJlore Pile Groups". Proc. 5th Annual Offshore Techn. Conf., Vol. 2, Paper No. 1896. Houston, Texas, 1973.

/2/ Poulos, H.G.: "Lateral Load-Deflection Prediction for Pile Groups". Proc. Amer. Soc. of Civil Engineers, Vol. 101, No. GTl, January 1975.

F4 ANALYSIS OF PENETRATION RESISTANCE

F4.I General

F4.1.1 With the aim to improve foundation stability and to serve as a means for scour protection, skirts are often required around the perimeter of the foundation in addition to skirts under the central part of the platform.

The skirts may be of steel and/or concrete. In this context it is assumed that steel skirts are thin, 20 to 30 mm in thickness, while concrete skirts (when used together with steel skirts) may have widths ranging from 0.3 to 1.2 m.

F4.1.2 As an aid during "touch-down", especially for positioning and orientational operations, dowels projecting a few metres below the skirts are used. These dowels are often hollow pipes of large diameter, 1 to 3 m.

F4.1.3 Furthermore, so-called ribs being shallower than the skirts but of about the same thickness, may be used.

F4. l .4 The penetration resistance of the members mentioned in F4.l.l through F4.l.3 is the sum of skin resistance and end resistance. The principles for the calculation of these two contributions to penetration resistance are outlined below. Normally it is necessary to make two calculations of penetration resistance, a "most probable" and a "highest expected" resistance. The latter will govern the requirements to penetration force, while the first should be combined with the analysis of local soil reaction stresses against the foundation structure, see FS. All calculations are based on material coe£ficients 'Ymc = 'Ymf = 1.0. See also 9.6.4.6 (R) and 9.6.4.7 (R).

F4.1.5 Due to inhomogeneities in the foundation soil the penetration resistance may vary across the foundation area. By combining the "most probable" and the "highest expected" penetration resistance reasonable criteria can be developed for the design of the ballasting system.

F4.1.6 The calculation of the penetration resistance should be based on the results of in situ testing supported by the results of relevant laboratory tests. The best field test presently available for this purpose is the cone penetration test, since it gives a continuous record of the cone penetration resistance with depth. Consequently, the most fruitful approaches to calculate the penetration resistance of steel skirts or dowels have been those based on the cone penetration resistance, Still there remain several uncertainties regarding the conversion from one type of penetration resistance to the other, e.g. the effects of different rate of penetration, excess pore pressures during cone penetration testing etc. Therefore, a consistent set of correlation factors for various penetration condition can never be developed. This emphasizes even more the need for sufficiently extensive site investigations for each platform site.

F18

F4.2 Method of calculation

F4.2.l The approach described below is applicable primarily to steel skirts, dowels and ribs whereas the penetration resistance of concrete skirts should be c~culated according to the principles laid down in F4.5.

F4.2.2 The method is based on the results of a number of cone penetration tests (cpt's) distributed over the target area for the platform. These test results are interpreted as follows:

1. Identify soil strata from soil borings and cpt's. 2. Determine for each cpt an average cone penetration

resistance, qc,av, at even interval, for example 0.2 m. 3. Determine for each depth an average cone penetration

resistance, termed ifc, of a selected number of individual qc,av representing certain identified strata.

F4.2.3 The penetration resistance is calculated from the following basic expression

d R = kp(d)Apqc(d) + As J kr(z)qc(z)dz

0

where

d depth of tip of penetrating member, m

(F4-I)

kp(z) empirical coefficient relating qc to end resistance kr(z) empirical coefficient relating qc to skin friction qc(z) cone resistance, MPa Ap tip area of penetrating member, m2

As side area of penetrating member, per unit penetration depth, m2 /m.

F4.3 Penetration resistance of steel skirts

F4.3.l Based on Eq. F4-l a "most probable" and a "highest expected" penetration resistance, Rprob and Rmax, respectively, can be calculated. For North Sea conditions the co~fficients kp and kr for dense sand and stiff clay can be introduced with the tentative values given in Table F4.1.

Table F4.I Numerical values of coefficients kp and kr for sand and clay

Type of soil Most probable (Rprob) Highest expected ( Rm ax)

kp kr kp kr

CLAY 0.4 0.03 0.6 0.05

SAND 0.3 0.001 0.6 0.003

Note: Experience has shown that values of kp and kr for the upper 1-1.5 m should be 25 to 50 percent lower than those given in the table due to local "piping" or lateral movement of the platform. Use of skirts with increased tip area or with stiffeners will reduce the kr-values.

Highly stratified soils or sand/clay mixtures will receive kp and kr values in between those given aboveo

F4.3.2 The cone resistance qc = cfc to be introduced in Eq. F4-l is derived as outlined in F4.2.2. Combined with the coefficients given in Table F4.1 for the most probable penetration resistance, Rprob, and the highest expected penetration resistance, Rmax, a range of penetration resistance may be defined. The range obtained for the most unfavourable combination of penetration resistances across the foundation area indicates the capacity of the ballasting system required during installation.

4.4 Penetration resistance of dowels

F4.4. l The approach described in F4.3 is· in principle applicable to dowels, as well. However, when dowels with friction reducers both inside and outside are used the skin resistance coefficient kr for sand given in Table F4.1 should be divided by a factor of 2. The calculations are made with the additional assumption that no plugging occurs.

F4.4.2 The assumption of no plugging behaviour has to be verified by separate calculations. If plugging occurs the approach in F4.3 is applicable with respect to the skin resistance contribution while the end resistance should be calculated as for a large diameter closed-end pile accounting for the scale effects when converting from cone penetration resistance to dowel end resistance.

F4.4.3 The dowel end resistance which limits the plug bearing resistance throughout the depth of penetration may as an alternative be evaluated from the simplified bearing capacity formulae given in F 1.7. Due to the non-linear stress-strain behaviour of soil and the effect of soil compressibility the effective angle of shearing resistance around the dowel tip will decrease with increasing depth of penetration into a homogeneous sand deposit. This effect will have to be considered when the dowel end resistance is calculated based on bearing capacity formulae.

F4.5 Penetration resistance of concrete skirts

F4.5.I The end resistance qp of concrete skirts should be evaluated from bearing capacity formulae as presented in Fl. For clay the end resistance is then

where the value of Ne is given in Fl.

For sand th~ end resistance is

q =_! ...,'b'N + p' N p 2' 'Y O q

(F4-2)

(F4-3)

where Nrvalues according to /1/ should be used for the calculation of the "most probable" end resistance and those according to /2/ for obtaining the "highest expected" end resistance, see Table Fl.1.

Fl9

F4.5.2 As an aid in the assessment of the bearing capacity of wide concrete skirts the results of plate loading tests carried out within the platform. foundation area may be useful.

F4.5 .3 The skin resistance of wide concrete skirts designed for small depths of penetration will ii:i most cases contribute little to the total penetration resistance and can therefore be neglected. However, if wedge-shaped concrete skirts or skirts designed for penetration depths exceeding 0.5 mare used this contribution should be considered.

F4.5.4 For wedge-shaped concrete skirts a "most probable" skin resistance may be calculated from the second term in Eq. F4-l with higher kr values than those given in Table F4.l. The maximum kr value for this type of calculation and wedge angles exceeding 5 degrees should be for sand kr = 0.006 and for clay kr = 0.08. For wedge angles between O and 5 degrees linear interpolation between these values and those given in Table F4. I is recommended.

F4.5.5 The "highest expected" unit skin friction, f5 ,

against wedge-shaped skirts penetration in sand may be obtained by assuming that passive earth pressure is mobilized against the skirt, thus

f5 = Kp Po 'tanli (F4-4)

The passive earth pressure coefficient Kp in Eq. F4-4 is a function of the characteristic angle of shearing resistance rp' and the wall friction angle Ii. The values for Kp given in Table F4.2 are for curved surfaces of failure and a coefficient of wall friction tanli = (2/3)tantf>' which may be considered high but still reasonable. Due to scale effects the values given for Kp may be somewhat large for concrete skirts penetrating more than one metre in dense sand.

Table F4.2 Passive earth pressure coefficient Kp

tan</>' 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

</)', degrees 26.6 28.8 31.0 33,0 35.0 36.9 38.7 40.4 42.0 43.5 45.0

Kp 4.1 4.7 5.5 6.3 7.3 8.6 9.9 10.7 13.2 15.3 17.8

F4.6 Penetration resistance of ribs

F4.6.1 The penetration resistance of ribs may be calculated according to either of the procedures described in F4.3 and F4.5 for steel and concrete skirts, respectively, whichever is the most representative for the ac.tual rib geometry. References

/1/ Brinch-Hansen, J.: "A Revised and Extended Formula for Bearing Capacity". The Danish Geotechnical Institute, Bulle· iin No. 28. Copenhagen. 1970.

/2/ Caquot, A. and Kerisel, J.: "Sur la terme de surface dans le calcul des fondations en milieu pulverult." Proc. 3rd Intern. Conf. Soil Mech. and Found. Engng., Vol. l. Ziirich, 1953.

FS LOCAL SOIL REACTION STRESSES

FS.l General

FS. I. I The unevenness of the sea bed and the shape of the foundation structure may lead to the development of high local contact stresses on the base of a gravity structure. The magnitude and the distribution of these stresses are highly dependent on the stress - strain - strength properties of the soil and their local variations, the degree of unevenness of the sea bed and the geometry of the foundation structure. The influence of these conditions on the contact stresses can be analysed by means of the procedure outlined in FS.2.

FS.1.2 The unevenness assigned to the sea bed in the design should be assessed on the basis of results from a sea bottom survey, see 9 .5 .5 .4 (R), taking account of the accuracy of the survey methods and the density of the topography measurements over the foundation site.

A prediction of the distribution and number of contact areas for different depths of platform penetration may be established based on a statistical evaluation of the bottom

. topography measurements.

Due consideration should be given to the influence of significant sea bed formations and obstructions of different kinds, see 9.2.2.5 (R).

FS.1.3 Soil properties of primary importance for the calculation of local contact stresses are:

ip' angle of shearing resistance } shear strength c cohesion/undrained shear strength properties E Young's modulus } stress-strain v Poisson's ratio properties

The characteristic shear strength properties should be chosen with due consideration to the uncertainties involved in the determination of these parameters. The material coefficients are taken equal to unity, 'Ymc = 'Ymf = 1.0, see 9.6.4.7 (R).

Young's modulus, E, should be derived from relevant field and laboratory tests or be based on empirical formulae taking into account the possible lateral and vertical variations in the soil properties, the depth of influence and the stress history of the soil.

FS.1.4 The non-linear stress-strain behaviour of soils indicates that a non-linear elasto0plastic.analysis will serve as a good approximation for the calculation of contact stresses. A pure plastic analysis will in most cases be too conservative as the increase in contact area due to elastic deformations is not taken into account. It will, however, indicate the upper limits of contact stresses calculated by a pure elastic method.

F20

FS.2 Elasto-plastic analysis

FS .2.1 The non-linear load-displacement behaviour of soils can often be approximated by a hyperbola as shown in Figure FS.l.

F

ELASTIC SOLUTION

..,,,,,,,,,,..-----PLASTIC SOLUTION

ELASTO·PLASTIC SOLUTION

6 Figure FS.1 Hyperbolic load-displacement curv_e.

The hyperbolic load-displacement curve is defined by the following expression (valid for F,;;; Fd):

0 F=----

!_ +-oki Fum

where

0 =----

!_ +-oki o:Fd

(FS-1)

initial slope of the load-displacement curve, defined in FS.2.2.

Fum asymptotic value of hyperbola for o -+-' 00

design resistance

a Flim failure ratio always greater than I, Fd , '

assume a = 1.25.

FS.2.2 For small loads the elastic solution will be valid and thus define the value of ki

F = Ei A o = k10 K(l-v2 ) b

(FS-2)

where

o vertical displacement (penetration) at the centre b width of contact area A contact area K load shape factor v Poisson 's ratio Ei Young's modulus for initial loading

This formula is valid only for constant b and homogeneous soil conditions.

FS.2.3 Inserting Eq. FS-2 into Eq. FS-1 gives the following expression for the hyperbolic load - displacement curve for F,;;; Fd

«'l A F=

K(l - 112

) b + ...!_ A (FS-3)

Ei aFd

In general b, Ei, 11 and Fd will vary with depth and the above expression will thus be valid only for infinitely small increments in load, dF, and penetration, d«'l.

FS.2.4 The design resistance, Fd, defining the upper limit of the load, may be calculated from the following simplified bearing capacity formulae

Fd = 1/21' b N1 s,., A (cohesionless soils)

Fd = c Ne (1 + Sca)A (cohesive soils)

(FS-4)

(FS-5

'Y' b A C

N1,Nc S,.,, Sea

effective (submerged) unit weight of soil width of contact area (function of penetration) contact area cohesion or undrained shear strength,

. bearing capacity factors, see F 1 shape factors, see F l

FS.2.5 From the assumption of a horizontal, spherical, conical or another shape of the sea bed under the contact points combined with the information about the geometry of the base (horizontal, spherical, etc.) it is possible to express the width b and the contact area A as functions of the penetration «'l.

b = b(«'l)

A= A(«'l)

{FS-6)

(FS-7)

FS.2.6 The initial value of Young's modulus Ei will vary with depth and stress conditions. The average value over a depth of influence equal to b may then be used as an approximation, thus

(FS-8)

or, since b = b(«'l)

(FS-9)

FS.2.7 An incremental penetration . .M of the base structure into the soil (see Figures FS.2 and FS.3) will lead to an increase in total load, contact stress and contact area. For a rigid structure the penetration will be the same over the whole contact area. Referring to the theory of elasticity and assuming a circular contact area the value of K in Eq. FS-3 will then be

TT K =-

4 (FS-10)

The factor r, in Figure FS.2 is defined in FS.2.10 and FS.3.

F21

J ; ) ) ~~~ r-~½ 7/ 7) ......... ...._ ______ .,.,,,

1. b • I

.11 I muoo ::::"'" . , ., .. Figure FS.2 Increase in contact stress due to increase in penetration All.

t.61 -~-J_~--

1b1,avj

+ c.:.==---::; +

t.q 2

t.F 2

INCREMENTS

=~ F

TOTAL

Figure FS.3 Incremental changes in contact stress at different depths of penetration and the resulting stress distribution after summation

FS.2.8 An incremental penetration Ao will cause an increase in load AF which can be expressed as

(FS-11)

where

kt tangential slope of the load-displacement curve, see Figure FS.1.

For a hyperbolic load-displacement curve as defined in Eq. FS-1 the expression for kt will be

I

= Ei A (1 - _!::_ / (FS-12) K(l-v2 )b Flim

Inserting Eq. FS-12 into Eq. FS-11 gives

(FS-13)

FS.2.9 Based on the above expression an incremental procedure can be used for calculation of the total load and the contact stress as a function of the penetration. In this procedure the j'th increment of penetration, A«'lj, will cause an increase in load, AFj, which can be expressed as

(FS-14)

The value of AFj, which is contained implicitely in Eq. FS-14, may be obtained by solving a quadratic equation or by iteration, and the total load after j'th increment is then obtained by adding AFj to the previous load Fj-1

(FS-15)

F 5 .2 .10 The increase in load will result in an increase in contact stress as illustrated in Figure FS.3. The distribution of the contact stress over the contact area will in general be non-uniform. In cases where the area increases with increasing penetration, the maximum value will develop at the centre of the contact area.

(FS-16)

According to the theory of elasticity the increase in contact stress under the centre due to an incremental penetration Ao j will be

AF· Aqcentrej = T/ ~

Aj (FS-17)

Where T/ is a factor depending on the rigidity of the base structure, see Figure FS.2. After the j'th increment the contact stress under the centre will be

qcentrej = qcentrej-1 + ~qcentrej (FS-18)

The average contact stress, qav, can be expressed as

F· qavj =t

J The factors K and T/ are discussed in FS.3.

FS.3 Elasticity factors

(FS-19)

FS.3.1 According to the theory of elasticity infinite stresses will develop at tbe edge of the contact area of a rigid base construction. The non-linear stress-strain behaviour of soils will, however, limit the stresses at the edge and lead to higher stresses under the centre of the contact area as a result of redistribution of the stresses when yielding takes place under the edge, This is illustrated in Figure FS.2.

FS.3.2 The load shape factor K will thus increase, but it is proposed to use the conservative value according to Eq. FS-10.

FS.3.3 For the interaction between a linear elastic soil and a rigid base structure the rigidity factor T/ will be 1/ = 0.5 (FS-20)

FS .3 .4 Due to the conditions mentioned in FS .3 .1 the value of T/. will increase with increasing ratios F /F d and the following value of T/ is proposed for general use

1/ = 2/3 (FS-21)

F22

FS .4 Empirical expressions for Young's modulus E

FS.4.1 For the choice of a characteristic value of Young's modulus for sand the results of cone penetration tests (cpt's) will be of great value.

For moderate contact stresses (say about 1/3 of the ultimate pressure) against footings founded on normally consolidated sand the constrained modulus M is commonly related to the unit cone penetration resistance qc by

(FS-22)

The coefficient 1/J increases from about 2 in loose sand to about 4 in dense sand /1/, /2/. The initial constrained modulus Mi compatible with the concept in Figure F5.l will then be about SO% larger, thus allowing an estimate of Young's modulus Ei which is related to Mi through Poisson 's ratio v. For drained conditions and v = I /4 to l /3 an average value of Ei will then be

(FS-23)

Thus, for loose sand

Ei = 2.25 qc (FS-24)

and for dense sand

(FS-25)

where qc should be taken as the average value ifc (derived as outlined in F4.2.4) within the depth of influence (here assumed equal to b ).

FS.4.2 On the basis of a set of cpt profiles certain lateral variations in penetration resistance can be detected. By identifying certain strata and their lateral extension and inclination it is possible to define a reasonable "lower bound" and "upper bound" cpt profile which used in the proposed calculation method will define a range of contact stresses to be considered in the design.

In layered soil due consideration must be given to the deformation and failure mechanism relevant for the actual contact area when choosing soil parameters for a penetration increment.

FS.4.3 For stresses within the preconsolidation range substantially larger values are to be assumed, see e.g. /1/. Tentatively the following values are proposed.

For loose sand

(FS-26)

and for dense sand

(FS-27)

FS.4.4 Another commonly used expression for normally consolidated sands which taJ<es the stress conditions into account is /3/

(FS-28)

a~ average effective . vertical stress within depth of influence

Pa atmospheric pressure ~ 100 kPa m factor depending on the density of the soil

FS.4.S For saturated normally consolidated sands m may be in the range of 100 to SOO, increasing with increasing density.

Thus, for loose sand

Ei = 110 Pa ~ {aJ' "Pa

and for dense sand

(FS-29)

F23

(FS-30)

FS.4.6 For clays the following expression is proposed for both normally consolidated and overconsolidated clays

Ei = 400 Cu (FS-31)

where Cu = characteristic undrained shear strength based on in situ and laboratory tests.

References

/1/ Dahlberg, R.: "Settlement Characteristics of Preconsolidated Natural Sands". Thesis, Royal Institute of Technology (KTH), Swedish Council for Building Research, Document Dl: 1975. Stockholm, 1975.

/2/ Vesic, A.S.: "Tests on Instrumented Piles, Ogeechee River Site", Proc. Amer. Soc. of Civil Engineers, Vol. 96, No. SM2, March, 1970.

/3/ Janbu, N.: "Soil Compressibility as Determined by Oedometer and Triaxial Tests". European Conf. on Soil Mech. and Found. Engineering, Vol. I, Wiesbaden, 1963.

A Revised and Extended Formula for Bearing Capacity

by J. Brinch Hansen

(Reprint of Lecture in Japan, October 1968)

1. Original Formula. A simple formula for the bearing capacity of a

shallow foundation was developed around 1943 by Buisman, Caquot and Terzaghi. With the latter's

notations it reads:

Q/B = 1

'f'BN, + qNq + cN" 2

(1)

This formula is developed for an infinitely long foundation of width B, placed upon the horizontal surface of soil with an effective unit weight y, a friction angle r; and a cohesion c. q is a unit surcharge acting upon the soil surface outside the foundation, and Q is the ultimate bearing capacity per unit length of this foundation, provided that it is loaded centrally and vertically.

2. Bearing Capacity Factors. The exact formulas for N q and Ne were indicated

already by Prandtl:

Nq = e;i tnn rr tan2 (45° + <p/2)

Ne = (Nq - I) cot If

(2)

(3)

Fig. I Lundgren-Mortensen rupture figure for calculation of N1. Vertical load on heavy earth (no surface load).

The best available calculations of N, were made, first by Lundgren-Mortensen, and later by Odgaard and N. H. Christensen, using the rupture-figure shown in fig. 1. The results correspond closely to the empir

ical formula:

N, = I.5 (Nq-1) tan <f

Curves for all 3 factors are shown in fig. 2.

(4)

0 ~

lOO

1IO

110 100

IO

0

0

)0

20

I

1

IO

• I

4

J

2

• ;,;

!

t

~

~

h

1,

Ii

~

~ ~

~

~

~ j

N, ' 1 ( ~ ~ ~ ~ ~ ~ ~ ~ ~

Fig. 2 Bearing capacity factors Nq• Ne• and N, as functions of rp.

Since N q and N c are calculated for one rupturefigure, and N, for another, the simple superposition implied by equation (1) must actually be an approximation. However, it is always on the safe side, and the error is usually less than 20 °/o (Lundgren-Mor

tensen).

3, Practical Cases, Actual foundations deviate in several respects from

the simple case considered above. Thus, the load may be eccentric or inclined or

both. The base of the foundation is usually placed at a depth D below the soil surface. The foundation has always a limited length L, and its shape may not even be rectangular. Finally, both the foundation base and the ground surface may be inclined.

Apart from the eccentricity, which is best taken

5

into account by considering the so-called effective foundation area, all the other influences can be expressed by means of suitable factors to the 3 terms in the original formula.

The different factors can be found by considering rather simple cases, in which only one complication occurs at a time. When these factors are then used together for more complicated cases this will, of course, be an approximation.

For the different new factors we shall use the following symbols:

s shape factors. d depth factors. i inclination factors. b base inclination factors. g ground inclination factors.

4. Effective Foundation Area. All loads acting from above upon the base of the

foundation are combined into one resultant. It has a component V normal to the base, a component H in the base, and intersects the base in a point called the load centre.

Now, a so-called effective foundation area of rectangular shape is determined in such a way, that its geometric centre coincides with the load centre, and that it follows as closely as possible the, nearest contour of the actual base area. A few examples are shown in fig. 3.

~v L

-~

Fig. 3 Equivalent and effective foundation areas.

The short side of this equivalent rectangle is called B and the long one L. The effective foundation area is A = BL. For a strip foundation the effective width B will simply be twice the distance from the load centre to the nearest edge of the base.

Meyerhof, the writer and others have shown that the actual bearing capacity of an eccentrically loaded foundation will be very nearly equal to the bearing capacity of the centrally loaded effective foundation area. Consequently, in the following we shall only consider central loading, and the symbols B, L and A will always refer to the effective rectangle.

6

5, Extended Formulas, Applying the 5 new kinds of factors to the original

equation (1), we get the following formula (Brinch Hansen):

Q/A = -~yBN;,s;,dyiyb;,gy + qNir5q<lqi,/Jqgq - 2

+ CNcScdcicbcgc (5)

q is now to be understood as the effective overburden pressure at base level.

In the special case of a horizontal ground surface. the ground inclination factors g disappear, of course, and the equation can then be written somewhat simpler:

Q/A = J-yBNysydyi;,by + (q + ccot rp) Nir5q<lqiqbq 2 ~)

-ccot <p

In the other speical case of <p = 0 ( - undrained failure in clay), it will theoretically be more correct to introduce additive constants instead of factors. Since the c-term is usually dominant, we may write:

6. Load Inclination Factors. An inclination of the load will always mean a

reduced bearing capacity, and the reduction is often very considerable.

Exact formulas for iq and ic have been derived by sfveral authors, using the rupture-figure shown in fig. 4. The rather complicated results can be approximated by means of simple empirical formulas, however (Brinch Hansen).

Fig. 4 Rupture figure for calculation of iq and i c·

Inclined load on weightless earth (with vertical surface load).

.. [\Iii

u .\ I

[\\_ ' ~ t\. •

f\_"' .. "r,._"" • "'-~

' " ~ I .~ N ~ ' ""' -~ .. u .. tJ u ,, u u u .. .. ..

Fig. 5 Inclination factor i q for q-term.

In the case of ,1, = 0 we get:

(8)

For 1p = 30° and 45° the results are shown in fig. 5. The dotted line corresponds to the formula:

iq = (1-0.5 H: (V + Ac cot g:,)]5 (9)

Calculations of i r have been made by Odgaard and N. H. Christensen, using the rupture-figure shown in fig. 6. Their results for 'I' = 30° and 45° arc shown in fig. 7. The dotted line corresponds to the formula:

i;, = (1 - 0. 7 H: (V + Ac cot p))5 (10)

Fig. 6 Rupture figure for calculation of i1. Inclined load on heavy earth (no surface load).

,~ i . \1 . \~ '

•• ..

\\ \\ .~~ . ~

l

' I ,, ., ~ --_ ...

-1:1._ v,~=• ..... II U U IU I.I U 11 U H U

Fig. 7 Inclination factor i1 for y-term.

As we shall see later, iy must be modified if the foundation base is inclined.

To avoid misunderstandings it should be mentioned that (9) and (10) must not be used, if the quantity inside the bracket becomes negative. The bearing capacity will in such cases be negligible.

The above inclination factors are valid for a horizontal force H = H

8, acting parallel with the short

sides B of the equivalent effective rectangle. If we substitute H by H

8 in the equations (8-10), the cor

responding factors may be termed i~8 , iqB and i18 respectively.

In the more general case, where there is also a horizontal force component H L' acting parallel with the long sides L, we can find another set of factors i~u iqL and ii'L by substituting H by H L in (8-10).

The first set of factors (with second subscript B)

should be used for investigating the usual failure along the long sides L, occuring when H

8 is dominant. The

second set (with second subscript L) are used for investigating a possible failure along the short sides B, which may occur when H L is dominant.

7. Base and Ground Inclination. The general case is shown in fig. 8. The slope angle

is called p, and since the ground will usually be sloping away from the foundation, /J shall be defined as positive in this case .. The foundation depth D is measured vertically.

Fig. 8 General case of base and front of retaining wall with inclination of base and ground .

The effective width of the inclined base is called B, whereas V indicates the foundation load normal to the base and H the load in the base. v is the angle between base and horizon .

Fi1;. 9 Rupture figure for calculation of be, bq, Kc and Kq· Frictionless earth or weightless earth with vertical surface load.

For the theoretical case of frictionless or weightless earth we get the rupture-figure shown in fig. 9. In the case of cp = 0 it gives the exact formulas:

ba_ ..l!__~ C - ll + 2 - 147° ( 11)

(12)

For other friction angles we can find the following exact formula for bq:

bq = e-2 I' tan tp (13)

7

For b .. we can develop an empirical formula by I

utilizing the fact that, for a vertical wall, we must have Kl' = K P according to Coulomb. This gives:

b., = e-2.1 I' tan If' I

(14)

With regard to gq, exact formulas can be developed. A closer inspection reveals that g is exactly the same

q

function of tan fi, as i q is of H: (V +Ac cot q). Hence we can write approximately:

g = [I -0.5 tan p]5 = g., (15) q , I

We have here assumed g., equal tog in accordance I q

with the fact that, for a vertical wall and any ground inclination, we must have K., = K according to Cou-

' p lamb.

The above formulas should only be used for positive values of v and fi, the latter being smaller than (f!,

Also, ,, + fi must not exceed 90°. In the case that the foundation base becomes a

vertical wall (1• = 90°), we shall, according to Coulomb's earth pressure theory, have K., = K , inde-

' p pendent of the roughness of the wall. Therefore it will, in the general case, be necessary to let iy depend upon v in such a way, that for ,, = 90° we get i., = i . This is easily achieved by writing:

I q

two sets of inclination factors. The following formulas arc proposed (Brinch Hansen):

lo = 0.2 i~8 B/L (20)

lL = 0.2 i~L L/B (21)

SqB = I + sin er. Biqo/L (22)

SqL = I+ sin cp • LiqL/B (23)

S;•n = I - 0.4 (Biyo): (Li;·L) (24)

s;'L = I - 0.4 (Li;•L) : (Biy8 ) (25)

For the last two factors the special rule mu,st be followed, that the value exceeding 0.6 should always be used.

9, Depth Effects. Actual foundations are always placed at a certain

depth D below the surface. One consequence of this is, that we must take into account the effective weight of the soil above base level. This is done by putting:

(26)

ii,= [I - (0. 7 - 110 /450°) H: (V + Ac cot gi)]5 ( 16) where y m is the average effective weight of the soil above base level.

8, Shape Factors, Theoretical values of the shape factors can hardly

be indicated at present, since their calculation would require a 3-dimensional theory of plasticity. Thus, experimental evidence must be used.

Extensive and careful plate loading tests on sand have led de Beer to propose the empirical formulas:

Sy= l -0.4B/L

Sq= 1 + sin IJ) , B/L

(17)

(18)

As a result of loading tests on undrained clay Skempton found the value:

s~ = 0.2B/L (19)

which can be shown to be a limiting case of ( 18). The shape factors indicated above are actually only

valid for vertical loading. ln° the other limiting case of the foundation sliding on its base, the shape can have little influence. For inclined loads we must, therefore, modify the formulas by introducing the inclination factors. And since failure can take place either along the long sides, or along the short sides, we shall need

8

Another effect is due to the fact that the soil above base level has a certain shearing strength. In the following this soil is assumed to be identical with the soil below base level. When it is inferior (or possesses no strength at all) the indicated depth effect will have to be reduced (or neglected completely).

At least one rupture-line will always pass through the soil layer above base level, and the effect will be an increase of the bearing capacity. This we express by means of a depth factor, which is used for foundations with mainly vertical loads.

If an upper Rankine zone is continued through the soil above the base without changing the rupturefigure below base level, the result will be an extra. horizontal force. Therefore, on foundations subjected to considerable horizontal loads (e.g. retaining walls), the simplest way to take the depth effect into consideration is to assume a passive earth pressure acting upon the side of the foundation.

10, Depth Factors, The depth factor di' presents no problem, because

we have always, according to definitions:

di'= I (27)

For small values of D/B it is easy to calculate dq or d0 because we can just extend the known rupturefigure for D = 0 up to the actual surface. In this way the following approximate formulas have been found (Brinch Hansen):

d~=0.4D/B (28)

dq = I + 2 tan <p(I - sin <p)2 D/B (29)

These formulas may be used for D < B. For greater depth it is difficult to calculate the depth factors, but we know that they must ultimately approach an asymptotic value. I have, therefore, tentatively proposed the following formulas:

d~ = 0.4 arc tan D/B (30)

dq = I+ 2 tan <f(I - sin rp)2 arc tan D/B (31)

We can actually 'test these formulas by applying them, together with the formulas for the shape factors, to a square pile base at great depth (D -+ oo). This gives for '/1 = 0:

QI A = (:r + 2}cu(I + 0.2 + 0.4 a/2) = 9.4 Cu (32)

which is a well-known result for pile point resistance in clay.

For other friction angles we get:

Q/A·= qNq(l + sin <p) (I+ n tan rp(l - sin rp)2) (33)

For values of <p between 30° and 40° this formula gives the result 2.2 q Nq in very good agreement with Danish experience for pile point resistances in sand, provided that rp is taken as the friction angle in plane strain.

The above formulas are valid for the usual case of failure along the long sides L of the base, and formulas (30)-(31) give the corresponding depth factors

d~8 and dqB' For the investigation of a possible failure along the

short sides B we must use another set of depth factors:

d~L = 0.4 arc tan D/L (34)

dqL!~ I+ 2 tan <p(l - sin gi)2 arc tan D/L (35)

11, General Formulas, In the general case, where the horizontal force has

both a component H 9

parallel with the short sides B, and a component H L parallel with the long sides L of the equivalent effective rectangle, we must use the following formulas:

-ccotcp (36)

This formula should be used in the following way. Of the two possibilities for the y-term, the upper one should be used when Bi"/8 :c:;'. lii'L' whereas the lower one should be used when Bi,.

8 > Li

1L. A check on

the right choice is that s ,' > 0.6. Of the two possibilities for the q-term we must always choose the one giving the smallest numerical value.

In the special case of rp = 0 we must choose the smallest of the following two values:

12, Passive Earth Pressure, For foundations subjected to considerable horizon

tal forces it has always been a question, whether it is permissible to assume a passive earth pressure acting on one vertical side of the foundation. The fear has been expressed that this would require too great horizontal movements of the foundation.

0

®

I~ t, .

R .

©

Fig. 10 Statically admissible rupture figures. Frictionless or weightless earth. Inclined load with fully developed passive eiuth pressure.

9

Fig. 10 shows first, for the case of weightless earth, the rupture-figure for a surface foundation (a). For foundations below the ground surface two statically possible rupture-figures are shown, one for a smooth side (b) and another for a rough side (c).

In the considered case of weightless (or frictionless) earth the stresses in the outermost rupture-line at base level are exactly the same as for a surface foundation. The reaction R on the base is, consequently, also the same, but in addition we have now an earth pressure P acting on the vertical side.

Therefore, due to the depth D, the foundation can actually take up a total load L, determined so as to be in equilibrium with the two forces R and P. The simplest way to take this into account is to add the earth pressure P (vectorially) to the main foundation load L. The resultant R (components V and H) can then be treated in the usual way, Q being calculated by means of the appropriate bearing capacity formula. It should be noted that depth factors must not be used, when the passive earth pressure is taken into account as proposed.

For load inclinations up to a certain value the rupture-figure shown in fig. 10c will occur. It indicates that in this case the passive earth pressure can be calculated as for a perfectly rough wall. For greater load inclinations the rupture-figure shown in fig. 1 Od will occur, and to this will correspond a slightly lower earth pressure. A limiting case occurs, when the base starts to slide horizontally. To this corresponds the passive earth pressure on a rough wall, which is translated horizontally.

As will appear from fig. 10, it is one and the same rupture-figure, which is responsible for both the bearing capacity and the passive earth presure. Therefore, the passive earth pressure does not require other or greater movements of the foundation than does the bearing capacity.

On the other hand, we must limit the movements to allowable values. As regards the bearing capacity, this is usually ensured by dividing the ultimate bearing capacity with a safety factor of e.g. 2.0. For exactly the same reason we must also divide the ultimate passive earth pressure with a safety factor, e.g. 1.4.

I have already mentioned that for mainly vertical loads we use depth factors but assume no passive pressure on the side of the foundation, whereas for great horizontal loads we do the opposite.

In intermediate cases we may, as proposed by Bent Hansen, do the following. If the horizontal force H is smaller than the horizontal component of passive pressure on the whole height D, we calculate the height DO necessary to develop a passive pressure just balancing the horizontal force. We can then calculate the foundation for vertical load only, and with depth factors corresponding to the remaining height D - D 0 •

10

13, Soil Parameters, In case of cpu = 0 (saturated clay) we must, of

course, for c u use the relevant undrained shear strength. For long term calculations we must use the effective parameters 'ip and c as found in drained triaxial tests.

For sand we can usually, on the safe side, assume c = 0. As regards cp, it has been common practice to use the friction angle rp

1, found in ordinary triaxial

tests. However, plate loading tests in several laboratories have shown that this leads to a severe underestimation of the bearing capacities. (Fig. 11).

Since the theoretical expressions for N and N., arc q I

developed for the case of plane strain, we must actually use the friction angle cp pi found in a plane

200 160 N

120 100

80

60 50

40

30 I

-20

16

12

I/

10 ~ 3v·

411

.,.,,.,.,,.

•

/

w

/ ill N,. / .,.

I/ - ,v //

~ ., , .,.., /

cp triax.

35°

Fig. 11 Comparison of plate loading tests with calculated bearing capacities using friction angles from triaxial tests.

2QO 160

120 100

80

60 50

40

30

20 16

12 10

N

~

~ / • I/ ....

"

30•

/

• I/ .,. "

V

. " • /"'

, , •

fptane

40°

Fig. 12 Comparison of plate loading tests with calculated bearing capacities using friction angles corrected for effect of plane strain.

strain test. Such tests have been made in several laboratories with somewhat different results. In Denmark we use always in our bearing capacity calculations the following value:

</'p1= 1.1 CJ>,, (38)

which should be on the safe side, since w,e have found factors up to 1.15.

This procedure has been found to give reasonable agreement between plate loading tests and the theoretical formulas. (Fig. 12).

14. Safety Factors, The safety in bearing capacity problems is usually

introduced as a total or overall safety factor F. This means that the bearing capacity Q should be calculated for the actual loads and actual soil parameters, and then the actual vertical load V on the foundation must not exceed QI F.

In Denmark we prefer the system of partial safety factors. This means that we calculate a nominal bearing capacity Qn for nominal loads and for nominal soil parameters defined by:

Pn=P0 ·fp

tan <pn = tan cp/ff/J (39)

The foundation should then be designed so that Vn :S; Qn. The following values are used in Denmark:

! = 1.5 p !,,= 1.75 (40)

15. Example, As a simple example we shall consider a couple of

full-scale tests with foundation blocks, made by H. Muhs in Berlin.

The blocks had the dimensions L = 2 m, B = 0.5 m and D = 0.5 m. Base area A = 1 m2 • The soil was dense sand with ji = 0.95 t/m3, and ground water level coincided with the ground surface. We can assume c = 0. ·

The first block was loaded centrally and vertically, and failed for Q = 190 t. If we estimate <p pi = 47° we find:

Ny=300 Nq=190 dqn = 1 + 2, l.07-(1-0.732)2·arc tan (0.5/0.5) = 1.12 Syn = 1-0.4 · 0.5/2.0 = 0.90 SqB = 1+0.732,0,5/2.0= 1.18

We can now calculate the ultimate bearing capacity:

Q = !yBNi'dy8 syn + yDNqdq8 sq8 = 2

1 · 0.95. 0.5. 300, 1, 0.90 + 0.95, 0.5 · 190, 1.12 · 1.18 =

2 64+ 120= 184t(N 190)

If our formulas are correct, this shows that the plane friction angle was about 47°. To this corresponds a triaxial friction angle of 40°-42°, which is quite realistic for dense sand. In fact, Muhs measured <f!,, = 40°.

A second block was also loaded centrally, but the load was inclined in the direction of the long sides L. At failure the forces Q = V = 108 t and H L = 39 t. Passive pressure on the vertical side of the foundation was so small, that it could be neglected. Instead we reckon with depth factors.

Since H8 = 0 we find:

dqn=l.12 Sqn=l.18 ii'8 =iqn=1 Biyn=0.5 dqnSqniqn= 1.12,1.18,1 = 1.32.

For the other direction we find:

dqL = 1 + 2, 1.07(1-0, 732)2 • arc tan (0,5/2.0) = J.04 iyL = (l -0.7, 39/108)5 = 0.235 LiyL = 0.47 iqL =(1-0.5-39/108)5 =0.370 LiqL=0.74

Since Biy8 > LiyL, we must use (25) and the lower y-term in (36):

SyL = 1 -0.4, 0.47 /0,5 = 0.625 SqL = 1 + 0.732,0.74/0,5 = 2.09 dqLSqdqL = 1.04 · 2.09 • 0.370 = 0.805

Since dqLsqdqL < dq8 sq8 iq8 , we must use the lower q-term in (36). Using the same values of Ny and Nq as above, we get the ultimate, vertical bearing capacity:

Q = -~yLN;,dyLsyLi,'L + yDNqdqLSqLiqL = 2

_!_, 0.95, 2.0, 300, 1, 0.625 ,0.235 + 2 0.95,0.5, 190, 1.04,2.09,0.370 =

42+72= 114t(Nl08)

It will be seen that the new formulas explain the results quite well in this case.

11

Geotechnical design of gravity structures

by ir. F. P. Smits Delft Soil Mechanics Laboratory

SUMMARY

This paper presents a review of the foundation design problems and the methods of analysis of offshore gravity structures. The first part of the paper deals with the installation phase, the second part with the operation phase. The main geotechnlcal problems covered are: foundation stability, settlements and cyclic displacements, effects of repeated loading, penetration of skirts, reaction forces on structural elements and instrumentation for performance monitoring. Particular attention is paid to those methods of analysis which are unconventional for land based structures and associated with the.development of gravity platforms. Design data of the Dunlin platform are discussed to elucidate the calculation methods.

1. INTRODUCTION

In 1973 the first concrete gravity structure was installed at the Ekofisk field in the central part of the North Sea at a water depth of 70 m. From then until 1978 twelve more structures have been built and Installed at water depths increasing from approximately 100 m in the Frigg area to 150 m in the Northern North Sea at 61 • latitude. The Delft Soil Mechanics Laboratory has been involved in the geotechnical design of the majority of these structures and some thirty foundation studies have been carried out for different platforms and locations.

With the application of gravity structures for offshore production and storage of oil and gas special foundation design problems have been encountered, which· were unconventional for structures onshore and which required the development of adapted and new design methods. Among them the following are most characteristic:

The unprecedented size of a continuous footing, subjected to highly inclined and eccentric loads. The experience with the bearing capacity of such foundations is very limited. A critical examination of all possible failure modes is necessary, taking into account the nonhomogeneity of the soil over the extensive foundation zone.

The dynamic nature of the wave loading. This unfits especially the classical methods of stability analysis for cohesionless soils and requires the calculation of bearing capacity under undrained or partially drained loading conditions. Also it necessitates the analysis of soil-structure interaction.

The repeated wave loading during storms. The effects of repeated loading of soils are a reduction of the stiffness and strength parameters. With the exception of the liquefaction potential of loose sands, the behaviour of soils under cyclic loading was largely unknown at the early stage of development of sea structures. Since then the understanding has been improved by basic research and methods of analysis have been developed to account for the effect of storm loading on stability and displacements.

The limited information on the foundation soil properties. Although new techniques are being developed rapidly, the available methods for soil investigation, offshore today are by far not equivalent to those used onshore. Besides geophysical investigations, such as topographic survey and seismic profiling, the in-situ

283

Table 1. Concrete platforms in the North Sea

North Sea Structure Site data Base Grouting Installations Experience

through the Type Name Operator Year Water Foundalion Slab Skirts Dowels base

installed depth soil m

Doris Ekolisk tank Phillps 1973 70 Dense line sand Flat 0.4 m None No Pore pressure Skidding during A:7400m' concrete ribs probes touch do~n

Condeep Beryl A Mobil 1975 120 Dense find sand Conical domes 3.0 m steel 3 85% lully ballasted Pore pressure High dilferential over clay A=6200 m• 0.5 m concrete belore grouting probes. skirt penetration

Settlemenl resistance casing Deep drainage Flat seafloor wells

Condeep Brent B Shell 1975 140 Stiff clay Conical domes 4.0 m steel 3 Fully ballasted belore Pore pressure High dillerentiai , interbedded sand A=6200 m• 0.5 m concrete grouting probes skirt penetration

Settlement resistance casing Flat seafloor Deep drainage wells

Doris Frigg CDPI Elf 1976 98 Dense fine sand Flat. ringshaped None None No Pore pressure -A=~600 m' probes.

Settlement casing

Sea Tank Frigg TPI Elf 1976 104 Dense find sand Flat 2.0 m concrete None 50% fully ballasted Fore pressure High dilferential skirt over clay A:5600 m• before grouting probe penetration

resistance ~ Sloping seafloor 1/100

Doris Frlgg- Total 1976 94 Dense find sand Flat, rlngshaped None None No Not known -Scotland A:5600 m' Manifold

Condeep Brent D Shell 1976 140 Still clay Conical domes 4.5 m steel 3 75% fully ballasted Pore pressure Significant dillerential interbedded sand A:6300 m• 0.5 m concrete before grouting probes skirt penetration

Settlement resistance. casing. Deep Flat seafloor drainage wells

Condeep Statfjord A Mobil 1977 145 Still clay Conical domes 3.0 m steel 3 75% fully ballasted Pore pressure Low contact stresses A:7800 m• 0.5 m concrete befole grouting probes on the domes

Settlement casing

Andoc Dunlin A Shell 1977 153 Still clay Flat 4.0 m steel 4 35% fully ballasted Pore pressure Low differential skirt interbedded sand A:10600m' before grouting probes penetration resistance

Condeep Frigg TCP2 Elf 1977 102 Oense fine sand Conical domes 1.2 m steel 3 55% fully ballasted Pore pressure High differential skirt over clay A:9300 m' 0.5 m concrete before grouting probes penetration resistance

Sloping seafloor 1/100

Doris Ninian Chevron 1973 136 Still clay Flat 3.75 m steel None - Not k.nown -interbedded sand A=15400m'

Sea Tank Brent C Shell 1978 140 Still clay Flat 3.0 m concrete None - Pore pressure -interbedded sand A:10300m' probes

Sea Tank Comorant A Shell 1978 150 Still clay Flat 3.0 m concrete None 50% fully ballasted Pore pressure -interbedded sand A:9700 m' before grouting probes

Settlement casing

investigation is mainly limited to cone penetration testing and percussion or push sampling from bore holes. Cone penetration testing is considered to provide the most reliable data at present. The strength and stiffness of clay samples is usually reduced by the hammering and by the sudden pressure relief after sampling, whereas sampling in sand disturbes the in-situ stress condition and density. The limited quality and amount of data has created the need for selection of characteristic values of the soil parameters as mean values on the safe side, i.e. low estimates for displacement and stability calculation and high estimates for determination of reaction forces.

The installation of structures on scarcely prepared seabeds. This has contributed to the common application of skirts combined with sub-base grouting. This design principle makes the structure adaptable to various locations and the safety of the foundation quite Independent of the actual seabed topography and the near-surface soil condition. It also provides the necessary flexibility for control of the installation process.

A summary of concrete platforms Installed until 1978 with some additional data regarding structural and geotechnical design is given in Table 1 (Ref. 1, 1979).

2. INSTALLATION DESIGN

The installation of a gravity platform on the seabed requires extensive prior analysis and careful monitoring, to ensure that neither the structure nor the foundation soil will be damaged. The installation phase reaches from the moment of touch-down until the installation of conductors and performance monitoring instruments. The geotechnical analysis includes the following major subjects:

skirt penetration

platform levelling

base contact stresses

stability during installation

reaction forces on dowels and skirts

The analysis deals in particular with the load carrying capability of the soil and the structural elements. Usually it reveals that some limiting conditions and contingencies must be carefully observed during installation. To this end a platform needs to be instrumented temporarily beyond the extent required to monitor the operational performance. The installation must be achieved in the shortest possible time, as the stability at this stage is insufficient to resist severe storm loading. Accurate positioning is required because the characteristic soil condition may change rapidly over a short distance.

2.1 Dowels and skirts

In general platforms with steel skirts need to be equipped with dowels, extending below the skirt tips. Steel skirts might be damaged easily due tp a residual horizontal movement of the platform during touch-down. Therefore, dowels serve for final positioning and also prevent possible damage of the seabed by skidding of the skirts, as has been experienced with some structures without dowels.

Most of the platforms installed up to now have been equipped with steel or concrete skirts. There are many reasons for this:

Often the top layer of the seabed is soft. Skirts, if sufficiently strong and adequately spaced, transfer the loads to deeper layers.

In the case of a top layer of sand skirts provide an efficient protection against erosion of soil underneath the structure and allow some depth of scour before the stability of the structure is endangered.

285

During installation skirts may carry a substantial part of the platform weight. Consequently they contribute to prevent high local contact stresses on the base slab due to seabed uneveness.

If a platform must be installed on a sloping seabed skirts allow the structure to be levelled.

Skirts provide separate compartments for sub-base grouting after installation.

2.2 Grouting

In conjunction with the use of skirts underbase grouting has become more or less common practice now. When the seabed is uneven, especially with a top layer of sand, it may be uneconomic to design the base structure and the bottom slab for the most unfavourable load distribution and for maxim.um local contact stresses. By grouting a more uniform distribution of the soil reaction on the base will be achieved.

If a platform must be installed on a sloping seabed and if the skirts do not penetrate fully, grouting is necessary to prevent tilt of the structure by uneven penetration, due to eccentric loading or due to variable skirt penetration resistance.

With an uneven seabed voids with entrapped water will remain after complete penetration. Fluctuating loads due to the rocking motions of the platform will then be balanced mainly by water pressure variations in the skirt compartments. Grouting of the voids reduces the differential water pressures across the skirts and thus the danger of piping failure by excessive hydraulic gradients at the base periphery.

If a platform cannot be ballasted fully before grouting, as occurred with the Ounlin platform, its stability after final penetration Is still poor and, therefore, grouting must be completed rapidly. However, the grouting pressures must be kept sufficiently low to avoid piping and to prevent tilting of the structure. Serious piping failure may be detected by monitoring the pressure in the skirt compartments.

2.3 Skirt penetration

If the platform loads must be transferred to deeper layers by the skirts, the required penetration depth depends on the strength profile of the soil and the magnitude of the loads. If skirts are desired to avoid erosion by expelling of entrapped water from underneath the structure, the required penetration depth depends mainly on the local seabed uneveness and the seabed slope. In both cases the design crlterium is that the expected maximum penetration resistance does not exceed the ballasting capacity.

Usually at the early design stage when skirts have to be selected, there is no detailed information on seabed topography. This has induced severe skirt design requirements, such as to account for a 1 % seabed slope and an uneveness of 1 m double amplitude. However, in many cases now topographic survey has shown surprisingly flat surfaces.

If large local contact stresses on the base have to be prevented, the skirt penetration resistance must be sufficiently large to use up the major part of the installed platform weight. On the other hand, again a minimum weight is required to ensure stability during installation.

All these design criteria must be matched within the limits of adjustment of the installation weight by water ballast in the oil storage tanks. Favourable in this respect are skirts with a slender steel tip section, passing into a blunt concrete wedge near the platform base, as they provide a penetration resistance increasing exponentially with depth and a large lateral load capability.

When skirt penetration is terminated the foundation soil surrounding the skirts is in a critical equilibrium state. An increment of lateral load due to wave action tends to decrease the vertical load carrying capability and to increase the penetration depth. Of course, this cannot be considered as a reliable contribution to reach the necessary penetration depth, however, it

286

increases the danger of overstresslng the base slab. Partial debaliasting after final penetration may improve the vertical stability.

It has been experienced that the discussed skirt design aspects are related to some extent to building site conditions. A platform built in shallow water, having the same storage capacity as one built at a deep water site, has a smaller height to base width ratio, smaller maximum horizontal loads and overturning moments, and usually smaller weight. Where the same skirt penetration depth is required, a shallow water platform needs smaller total skirt length or the skirt strength requirements are less severe.

In general a shallow water building site leads to a smaller weight to skirt periphery ratio and a higher variable to dead load ratio. Therefore, with such a platform it usually takes more care to reach the necessary penetration depth, but there is a larger margin for control of the penetration process and to keep the contact stresses low.

The skirt penetration resistance is calculated as the sum of the tip resistance and the accumulated wall friction. With steel skirts and open-ended pipes the calculation is preferably based on the results of cone penetration tests. The unit tip resistance q and the unit wall friction f may be related to the characteristic cone resistance qc and the characteristic sleeve friction fc from a representative number of tests:

in sand:

q qc

= 1 + sin qi (1)

= f C or }

0.004 qc ,., (2)

in clay:

q = qc (3)

= f C or }

0.04 qc "" (4)

With concrete skirts and closed-end or plugged pipes the calculation of penetration resistance is preferably based on bearing capacity and earth pressure theory. It is recommended that undrained shear strength and friction angle values are obtained from cone penetration tests, supplemented by laboratory investigations. The penetration resistance per unit length of skirt Fp may by expressed:

Fig. 1 Geometry of concrete skirts.

287

in clay:

in sand:

where:

Ne = n+2

Ncp = 1 + (1 + .l1!_ - 213) cot 13 2

cot2µ exp ( n tan q>)

1.5 (Nq-1) tan q>

cosl3-sinq>cos(2e+l3) sln(l3+e-µ)•

sin2 l3 cos 2 q>

COS q>0 · {---sin(µ+e-q>0 ) + cosµexp(3etanq>) cos (j)

COS (j)0 + --- sin (q>0 - µ) exp (3 & tan q>)) cos (j)

µ = n/4 - q>/2

& = it-13-µ

(5)

(6)

q>0 = arctan (3 tan q>) --- penetration resistance ( MN)

Fig. 2 Skirt penetration resistance at Dunlln.

288

O O . 0 5 I '.

e w

.c -a. <I,

'O

C 0

g <1> 2.0 C <I, a.

I '-. ' "·, ' '·

\ \

3.0 -

' ·, I

\ ::--

''-i.

\

"

~ \

'"

\

·,

\ \

\ \

\ \

10

measured , \ most \\probable

4.0 ~-~--i-~-~--- . _ _J __

maximum expected

\

15

-· ~,. - --~

In view of the previously discussed skirt design criteria and the uncertainty involved in selecting a representative soil strength profile from a few cone penetration tests, it has become a recommended practice to calculate a 'maximum expected' and a 'most probable' penetration resistance. The maximum expected resistance, based on a high characteristic value of the soil strength, serves to verify that the ballasting capacity is sufficient to reach the required penetration depth. The most probable resistance, based on the best estimate of the average soil strength, may be input for the calculation of base contact stresses.

Figure 2 shows the calculated and measured skirt penetration resistance for the Dunlin platform. The site investigation at Dunlin indicated the presence of boulders in a thin top layer of sand, which resulted in a few exceptionally high cone resistances. This has caused the notable difference between the 'maximum expected' and 'most probable' penetration resistance.

2.4 Base contact stresses

The platform base may be designed to resist the maximum possible contact stresses, to resist only a predetermined limited stress level, or not to contact the seabed at all. In the last case ballasting must be stopped well before full penetration of skirts. In the case of a stress limit it will be required to monitor the loads on the base during the final penetration stage. For the actual loading of the base slab the contact stresses always add to the differential water pressure across the base.

The maximum contact stress to be expected depends on the seabed topography, the soil stiffness and strength, the geometry of the base, and with a top layer of sand also on the installed platform weight. Large boulders on the seabed may cause significant contact stresses too.

Unless the topographic survey of the seabed indicates regular patterns, for instance ridges with a prevailing direction, a design uneveness may be selected which consists of local high spots of spherical shape, such that its radius of curvature R fits as close as possible the actual geometry of the uneveness (Figure 3a)

R:..!!.. + 02 2 Bh

Fig. 3a Design geometry of seabed uneveness.

Contact stresses may be elastic or plastic, the plastic stresses providing an upper limit. The average elastic stress q may be calculated .from Hertz's theory of contact as:

q = 2d E 3nR 1-u2

(7)

in which d is contact area diameter.

289

The plastic stresses are expressed by the bearing capacity formulae:

in sand

in clay

with bearing capacity factors for vertical loading on a circular contact area:

NY= 0.9!exp(ntan<P)tan2 (45+(1)/2)-1 Jtan(j)

Ne= 6.3

(8)

(9)

From the above expressions it becomes evident that plastic stresses will not occur in sand with a slightly undulating seabed if:

(10)

and in clay for sufficiently small contact area if:

3 1-u2

d < -nRc N--=d 2 u C E 0 (11)

If a clay deposit is covered by a top layer of sand punching through the sand layer may occur. The ultimate stress will be somewhat higher than according to equation (9) due to stress dispersion.

The contact area diameter of the stress q may be expressed as a function of the base contact force 6F:

for elastic stresses:

d = 16 R 6F 1 - u2

n E

for plastic stresses in sand:

for plastic stresses in clay:

d =

with 6F is the total ballast weight minus skirt penetration resistance and n is the number of simultaneous contacts.

(12a)

(12b)

(12c)

To obtain an impression of the response to Increasing ballast weight during the final stage of penetration, the vertical displacement after initial base contact may be estimated:

d2 6Z = -- .

4R (13)

It may be difficult to establish the number of simultaneous contacts or the residual penetration force 6F. A conservative value of the contact stresses is then obtained by assuming full impression of the high spots. The maximum contact area diameter is according to Figure 3a:

d(max) = D == v'8Rh. (14)

290

q

'

l sand layer on

top of clay " " ' / ' / ' / ........ ....... / -- //

Cu Ne - - - - - - - - -

-----1 .... d

Fig. 3b Base contact stresses on clay.

q

Fig. 3c Base contact stresses on sand.

d

Therefore, the maximum expected average contact stresses are:

q(max) = _J_fil!._ _E_ 3nD 1-u2

(15)

for an elastic distribution, and according to equations (8) and (9) with d = D for a plastic stress distribution. The maximum stresses are elastic If:

D > 16h E on clay (16a) 3n Cu Ne 1 -u2

DZ> 3.8 h E on sand (16b) ----y Nr 1-u2

291

It is recommended that high characteristic values of the stiffness and strength parameters of the soil are used for calculation of the base contact stresses. Figure 4 Shows the allowable and the maximum expected contact stresses for the Dunlin platform. It may be observed that, for a maximum uneveness h = 0.2 m, high spots with a base diameter between 3 and 6 m, if fully impressed, might cause overstressing of the base slab.

2.5 Platform levelling

Non-uniform soil conditions, such as varying thickness of a top sand layer, may cause uneven skirt penetration resistance and consequently tilt of the platform during installation. Levelling capability must be provided to correct for this tilt, but also to ensure level penetration on a sloping seabed. A platform equipped with skirts can be levelled by adjusting the water ballast In the oil storage tanks and to some extent by regulating the water pressure within the skirt compartments.

Whereas an average soil strength profile is used to calculate the total skirt penetration resistance, for determining the required levelling moment the cone penetration tests have to be interpreted based Ofl their actual position with respect to the target location and orientation of the platform. The calculation of the required levelling moment must always be directed towards a prediction on the safe side, which rather overestimates the uneven skirt penetration resistance. However, the unit resistance distribution, used for the moment calculation, when integrated over the total skirt length should aim at the most probable penetration resistance.

maximum average contact stress q (MN/ m2) ---0oi<:""' ____ ...,.1 _____ --=2;:--------'i-3 _____ .,;;,4

0 11ow?-~~ ------· 2 i-------......... ~-----,:.._· --------+------!

E 0

II 4 ,:, .. c,, -\\! E 0

'O

~ 6 ... 0

-V 0 -C: 0 V

l 8

0 II

.i:

Fig. 4 Base contact stresses calculated for the Dunlln platform.

292

I

J

t

I

t

I

At the start of skirt penetration no differential water pressures across the skirts can be allowed to avoid piping. Levelling must be achieved by eccentric ballasting only. The available levelling moment by eccentric ballasting increases with added ballast w~ight and, therefore, with depth of penetration. Also the required levelling moment increases with depth of penetration. So at this stage it is important to ensure that the rate of increase of the levelling moment by eccentric ballasting is sufficient. If towards the end of penetration the capacity of the ballasting system becomes too small to keep the platform level, an additional levelling moment can be provided by an excess pressure or suction in the outer skirt compartments.

A safe criterlum for the allowable differential pressures to prevent piping is:

~P < 2yz (17)

where z is the local minimum penetration depth.

The minimum penetration depth may be considerable less than the average penetration depth, due to the seabed uneveness and the seabed slope and due to erosion occurring in the touchdown phase.

2.6 Stability during installation

After final penetration it usually takes about two weeks before grouting is completed. Although platforms are Installed in calm weather periods, It must be faced that a moderate storm might occur during the grouting period.

The design horizontal load and overturning moment are calculated from the probable maximum sea state. The horizontal load must be carried primarily by the skirts. Only if considerable base contact stresses are allowed, a contribution by base friction can be taken into account. The overturning moment can be balanced usually by water pressure differences in the skirt compartments.

The horizontal load capacity is obtained by frictional resistance of longitudinal skirts and passive earth pressure resistance of transverse skirts, however, only as far as these resistances do not exceed the structural yield limit of the skirt. The passive earth pressure resistance has an uplift component, which must be subtracted from the weight of the platform to calculate the potential contribution by base friction.

Mobilization of base friction produces a state of failure under inclined loading in the co·ntact zone and, therefore, tends to increase the contact area and to reduce the contact stresses. As in general there is only local base contact, the contact load must not be interpreted as a surcharge for calculation of the passive resistance of the soil.

The differential water pressures in the skirt compartments, caused by the overturning moment, need not be considered for determination of the lateral load capacity, except when the passive earth pressure resistance at failure exceeds the yield limit of the skirt, which frequently occurs with steel skirts. Due to the differential water pressure variations, which are approximately in phase with the horizontal load, the inner skirts usually will be loaded heavier than the outer skirts. Consequently t.he inner skirts may reach the structural yield limit, whilst the outer skirts still resist a smaller passive earth pressure.

For the analysis it. is recommended to consider the excess water pressure due to cyclic overturning moments as an overburden pressure, because of the short rise time of the pressure wave. The allowable pressure differences across the skirts to avoid piping under such undrained loading condition is expressed:

~p < yz(Nq-1)

~P < 5c

with Nq as in equation (6).

"

In sand

in clay

(18)

(19)

293

This criterium may be somewhat unsafe in case of a top layer of coarse sand. A conservative approach would then be to assume full drainage, in which case equation (17) applies.

If the allowable differential water pressures are insufficient to balance the design overturning moment, it is recommended to deballast partly after final penetration. Deballasting turns the state of critical equilibrium of the soil near the skirts into an elastic state. The overturning moment must then produce at least a reloading of the skirts to the previous level, before it will cause any further penetration and tilt of the platform.

Usually after final penetration the exhaust valves of the skirt compartments are closed to prevent further penetration. A continuous cyclic horizontal loading tends to decrease the penetration resistance and the platform weight may become balanced partly by an average excess water pressure in the skirt compartments. To avoid piping this average excess pressure must certainly not exceed the level indicated by the criterium for drained loading in equation (17).

2.7 Reaction forces on dowels and skirts

The dowels must resist the lateral force encountered during touch-down and positioning of the platform and the axial force during penetration afterwards.

The lateral force is the passive earth pressure resistance at failure. Its actual magnitude depends on the strength of the soil, the direction and magnitude of the speed at touch-down and the penetration depth when the platform comes to rest. The touch-down speed is particular important to determine the passive earth pressure resistance in sand under partially drained loading conditions.

her i zon ta 1 force ( k N )

0 2 3 4 5 6 0

E 0.5

.J:.

a. Cl> 1.0 'U

C .2 -C ... ,. s ai C Cl> a.

t 2.0

2.5

Fig. 5 Lateral resistance of dowels on sand under partially drained loading conditions.

Figure 5 shows plots of dowel resistance versus depth of penetration at various touch-down speeds for the Dunlin platform. Such plots provide the necessary information for the analysis of the deceleration of the platform. The final penetration depth and corresponding lateral force may be determined as a function of the touch-down speed and the load contributions from wind, current and tugs .. The axial force during further penetration of the dowels may be determined as described in section 2.3 for open-ended or plugged pipes.

294

The skirts must be designed to resist at least the bending moments and shear forces caused by the environmental loading during the installation period. The bending moments and shear forces reach a maximum at the fixed end. It appears that usually the loading condition during installation tends to be more unfavourable for the skirts than the extreme loading during lifetime operation, because after grouting the loads are mainly transferred by the base.

The horizontal wave forces and the cyclic differential water pressures caused by the overturning moment are the major environmental loads. In addition, lateral loads arise from inclined penetration of skirt walls. Average excess water pressures may occur due to the rate of ballasting, the levelling procedure and the reduction of the penetration resistance by cyclic horizontal loading. Average excess pressures tend to become most unfavourable for the outer skirts, whereas cyclic differential pressures cause heavier loading of the inner skirts. The latter is due to the phase relation between horizontal forces and overturning moments.

In section 2.6 it has been pointed out that the lateral load capacity may be governed by the yield limit of the skirt before the ultimate strength of the soil is mobilized. However, if the design horizontal load is considerably smaller than the soil or skirt capacity, the actual reaction forces will depend on the stiffness of the soil and the skirts. The stiffness of the soil may be assumed proportional to the ultimate passive earth pressure. From the stiffness distribution the heaviest loaded skirt element is identified. Then the fixed end bending moment and shear force of this element are determined at unit average lateral resistance and, subsequently, multiplied by the design horizontal load. The maximum bending moment and shear force, thus calculated, must not exceed the structural yield limit of the skirt.

2.8 Instrumentation for installation monitoring

To control the installation process the following observations have to be made:

bottom clearance from touch-down till final penetration;

cyclic and average excess water pressures In all skirt compartments;

buoyancy and added ballast status;

platform inclination and direction of tilt;

ballasting moment and direction of moment;

axial force and bending moment in dowels;

base contact forces if full penetration is anticipated.

Echo sounders with high resolution or closed hydraulic systems with pressure transducers on the sea floor may take over the draught measurements. shortly before touch-down for observation of the depth of dowel and skirt penetration.

Excess water pressures in the skirt compartments are measured by differential pressure transducers or by level indicators In piezometer tubes. Monitoring of the excess water pressures, which cause an uplift force on the platform, is required to keep track of the net ballast weight and levelling moment. Control of the excess water pressures by adjusting the rate of ballasting or by relieving the water exhaust valves may be necessary to avoid piping during penetration or during a summer storm.

Electronic inclinometers or U-tube systems are commonly used to monitor the platform tilt.

Ballast level transmitters are installed to provide a continuous record of total and eccentric ballast weight during skirt penetration.

Base contact forces and touch-down forces on the dowels are measured by strain gauge sensors. If dowels are overloaded during touch-down, the skirts may be damaged and submarine inspection may be necessary before penetration is continued.

295

3. OPERATION DESIGN

The main geotechnical problems related to the operational performance of the structure are:

stability under extreme loads;

settlements and cyclic displacements.

In both the stability and the displacement analysis due consideration must be given to the effect of repeated loading on the stiffness and the strength of the soil and to the dynamic nature of the wave loading.

To control the foundation performance the platform needs to be instrumented for long term monitoring of at least settlement, pore pressures and caisson motions.

3.1 Stability

The objective of stability analysis is to assure the capability of the foundation soil to carry the maximum load transmitted by the structure. A well established design procedure is then to prove that the soil in a limit state of failure is capable to resist the maximum load with an adequate safety margin. This is quantified by safety factors on the major sources of uncertainty: a load factor on the characteristic environmental load and a material factor on the characteristic soil strength.

The design loading criterium is the extreme storm with a recurrence frequency of once in 100 years. The characteristic load is the probable largest wave force in that storm, superimposed on the wind and current forces and on the minimum or maximum permanent vertical load. Although the wave forces are dynamic in nature, they may be treated as quasi-static forces if multiplied by appropriate dynamic amplification factors. For the stability analysis this may seem a conservative approach in view of the short duration of peak cycle loading. However, the design loading condition must also account. for the repeated approach of the ultimate resistance by successive wave cycles, which may result in large cumulative displacements.

The design soil strength profile is obtained from in situ and laboratory tests. The characteristic value of the strength parameter may be selected in principle as the mean value. However, a conservative interpretation of test data is required, as the amount and quality of data is usually low due to the high costs and faulty techniques of offshore soil Investigation. In situ cone penetration tests may be considered more reliable for determination of the soil strength than laboratory tests on partly disturbed or reconstituted samples. But supplementary laboratory tests are required to investigate the effect of variable stress conditions imposed by the structure. Also the effect of cyclic loading is studied in the laboratory. Repeated shear stress reversals may reduce the undrained strength of the soil. Such a reduction, caused by the design storm loading, must be expressed by the characteristic value of the strength.

In view of the average wave period and the base dimensions of platforms the loading within a wave cycle may be assumed to be undrained for both cohesive and cohesionless soils. Therefore, an undrained loading analysis is appropriate to determine the ultimate bearing capacity. However, partial drainage should be allowed for to evaluate the effect of a great number of cycles during the build-up of a major storm.

An undrained analysis may be based on total or effective stresses and strength parameters.

After installation of a platform on a cohesive soil the weight has caused an excess pore pressure in the foundation zone, whereas the mean effective stresses, which were not known very well before, have not really changed. The initial undrained shear strength, as obtained from cone penetration tests, is probably the most reliable strength parameter. Therefore, a total stress analysis is appropriate to calculate the stability In the early platform life: the short term stability.

Once sufficient time has passed for the foundation soil to be adjusted to the platform weight, i.e. for the initial excess pore pressures and those arising from short term storms to be

296

dissipated, the effective stress distribution is reasonably well defined. The stability calculation is more reliably based on effective strength parameters in undrained loading: the long term stability.

In sand only the long term stability is relevant, as the initial excess pore pressures and those generated during a storm will dissipate in a few hours. However, in clay complete dissipation of the initial pore pressures may last several years. As especially overconsolidated clays may lose a considerable part of their initial strength by cyclic deterioration, both short and long term stability need to be considered in clay.

3.2 Short term stability

It has been experienced that in general the loading conditions of gravity structures cause the critical mode of failure to be sliding or a slight subsidence. Therefore, the strength of the superficial soil layers is most important. A graphical method has been developed whi,ch allows to establish the safety level under the most critical failure mode.

Consider a caisson footing placed on the horizontal surface of a uniform cohesive soil and subjected to an inclined eccentric load. Then the bearing capacity may be expressed from plasticity theory:

Oy = Cu(1 +.E..+ 2P + sin2P + Sc) 2

(20)

in which:

= arccos ~ , Cu

Sc "" 0.4 ~ (2P + sin 2P) ,

Cu is the undrained shear strength,

(21)

ov, tH are the ultimate average normal and shear stresses over the effective foundation area,

p is the angle between the surface and the direction of subsidence at failure,

Sc is a shape factor, with B and L the effective width and length of the footing.

Curve a in figure 6 shows the bearing capacity and demarcates the domain of safe toads. If the horizontal axis is assumed to coincide with the surface, a vector from a point on the curve to the origin of coordinates, such as OS represents the direction and magnitude of the unit inclined load that Just causes failure.

Curve b In figure 6 represents the design loading condition, i.e. the variable horizontal and uplift forces per unit area over the period of the maximum wave (~FH/ A and ~Fv/ A), superimposed on the average vertical stress due to the platform weight (OQ = Fv/ A). Usually there is a phase-lag of approximately 90 degrees between the horizontal and uplift forces, resulting in an ellipticalshaped curve whose axes roughly coincide with the tH and ov directions.

Safety factors and corresponding critical failure modes for the short term state may now be obtained from figure 6. Suppose that the bearing capacity and loading curves refer to the characteristic strength and the characteristic load. An actual undrained shear strength smaller than the characteristic value would then result in a proportional decrease of the distances to the origin of all points on the bearing capacity curve. A state of failure would be reached If the reduced bearing capacity curve becomes tangent to the loading curve at point X. A material safety factor 'Ym may be expressed as the ratio between the characteristic shear

297

@-I I

I

I

: r I ... : I I : . I i : . I . . I .

I ~ I

I \ \ \ \

' I \ \ .J \ . ' ' I ', \ \

OS ym=OX

s

:: QT QY

--- \/ \

I

I

I

X .-· T ....... -·. ,Y , L ' /

I I

I \ I

/ I I

/ :

0\ C

"O

-Ill

__ ..1..-...1,._, ____ ..i,_ ___ ...1,._,_-i.... __ 1H 0

Fig. 6 Short term stability behaviour.

298

strength (cu) and that reduced or mobilized value (cm), which causes the design load to become just critical:

Cu OS Ym = -- =-=-

cm ox (22)

An actual environmental load exceeding the characteristic value would cause an expansion of the loading curve. A state of failure would be reached if the loading curve becomes tangent to the design bearing capacity curve at point T. A load factor y1 may be expessed as the ratio between the ultimate and the characteristic .load, if that ultimate load would cause the structure to become unstable at the design soil strength:

QT Yr=-=-.

QY (23)

The bearing capacity formula shows a relation between stress ratio av/,H and mode of failure:

failure by sliding if:

(P= O) , (24a)

failure by subsidence If:

(P> O) . (24b)

From figure 6 it IT'ay be observed that the critical mode of failure not only depends on the weight of the platform, i.e. the position of the loading curve on the av-axis, but also on the definition of safety. For instance, if in the case of figure 6 the safety is measured by an overall material factor, then the critical failure mode is subsidence. If the safety is measured by an overall load factor, sliding failure is most critical. If partial safety factors are used, the critical mode will correspond to a point on the bearing capacity curve somewhere between T and S.

If a subsidence failure is most critical, the safety factor decreases with increasing platform weight. Therefore, if the platform weight is variable, for instance due to oil storage, the short term stability must be determined for maximum weight.

The horizontal wave force always acts at some distance above the caisson base, thus causing an overturning moment ~M which is about in phase with the horizontal force. Moment equilibrium is satisfied by assuming the design load to act at the base level with an eccentricity e = ~MI Fv and on a reduced effective foundation area A= ( B0 - 2e) L0 , with L0 and B0 the base length and width for centric loading. If the bearing capacity and loading curves in figure 6 refer to the effective foundation area in eccentric loading, the load factor must be corrected for eccentricity:

<X (25) Y1 =

e 1+2-(a-1) Bo

in which

<X = QT

QY

A stability analysis for eccentric loading, based on a reduced effective area as indicated above, underestimates the safety under undrained loading conditions. The effect of load eccentricity on the bearing capacity reduces as the ratio of horizontal to vertical load increases. Therefore, a revised formulation of safety factors rm and r 1 for undrained eccentric loading has been proposed (Smits, 1Q76):

299

fm= Ym•--------6 + tl__av + ( 6._av )2

Oy Ov

r, = Y 6.av c N y 6.av

6 + _1 __ (1 __ u _c) + (-1 __ )2

av 2 av av

in which:

(26)

(r, s Y1) , (27)

Ym, y1 are the safety factors for equivalent centric loading as described previously,

W is the moment of resistance of the base ,

-av = Fv/A ,

6.av = 6.M /W .

Usually the soil strength is non-uniform. In that case the bearing capacity may be determined approximately by selecting an average shear strength over the actual depth of the plastic zone of deformation. The depth of the plastic zone z0 depends on the ratio of horizontal to vertical stress at failure:

Zo = B sin IL (28)

Often the shear strength increases with depth z, and its variation may be approximated by a linear equation, at least for the most relevant failure modes from sliding to shallow subsidence:

For such strength profiles the bearing capacity may be expressed approximately by:

av = c0 (1 + .I!.. + 213 + sin 213 + Sc) + m B ( sin p + . ~ )2

2 '\/2

with

TH 213 = arccos c

0

as shown by the dotted curve c in figure 6.

The effects of repeated wave loading will be discussed in more detail in a later section.

(29)

(30)

At this stage it is mentioned that the shear strength reduction in short term loading is mainly a function of the cyclic stress ratio tcfcu and the number of cycles. The storm loading will primarily affect the strength of superficial layers. An initially uniform strength profile tends to become a profile of which the shear strength increases gradually with depth to the original strength. The initial bearing capacity will reduce to a value as indicated by curve d in figure 6.

3.3 Long term stability

The long term state, as defined in section 3.1, is considered to start by the end of primary consolidation. The foundation soil may or may not have experienced previous storms, but now it has a stationary effective stress distribution for some time past when a new storm comes on. Again the intention is to investigate the stability under the maximum wave force, including the effect of the storm loading.

300

As in undrained loading to failure the effective stress level is not affected by a change of the octahedral total stress, it is assumed that to each soil element an undrained shear strength cu may be assigned, which only depends on the initial effective mean normal stress a' and the effective friction angle at failure (jl1

Cu = a' sin (jlf. (31)

Essentially such an assumption is only true if 6.a2 = (6.01 + 6.03)/2 and if there is no plastic volumetric strain at failure. Dense sands and overconsolidated clays tend to dilate at failure and to develop negative pore pressures, thus mobilizing a much higher strength. Normally, however, this involves large shear strains. In a real storm these strains become even larger due to partial drainage between successive peak cycle loadings. Therefore, it is recommended not to rely on this additional strength.

The mean effective stresses are those resulting from the static loading by the weight of the soil and the structure and by wind and current forces. In sand and normally consolidated clays they are most adequately calculated by a drained bearing capacity analysis, assuming partial but constant mobilization of strength. The average effective mean normal stress a' at the base level is determined by the equation;

a' =

in which

MI is the average excess pore pressure,

(jlm is the mobilized angle of internal friction,

ifv = Fv/A,

sin 6 2T] = 6 + arcsin --- with sin <Pm

FH 6 = arctan F

. v - 6.u A

(32)

The mobilized angle of internal friction <Pm may be obtained from the bearing capacity equation:

- - 1 B N -N 3 Ov - D.U = 2 Y y - 6.U q ( 3)

in which N1 and Nq are functions of (jlm and TI only (Smits, 1978).

Usually the wind and current forces are small compared to the weight of the platform, in which case T'l "'0 and the bearing capacity factors in equation (33) become:

(34)

As the effective mean normal stress increases with depth due to the weight of the soil, the undrained shear strength for the long term state becomes approximately:

_ _ sin <Pt • Cu = (av + y z - 6.u) .

· 1 + Sin <Pm (35)

Therefore, the long term bearing capacity is as for a cohesive soil whose shear strength increases linearly with depth, as expressed previously by equation (30).

301

..

Safety factors are determined by the same procedure as for short term stability. As the weight of the platform affects the long term strength (av in equation (35)), it is obvious that in the case of a variable weight the long term stability must be determined at minimum platform weight. The requirement for long term stability also reduces the domain of safe loads in the short term state. This is illustrated by figure 7.

Fig. 7 Approximate domain of safe loads for short and long term stability on clay, with Cu the undrained shear strength in short term loading, q,1 = 35' the effective friction angle at failure and no long term effect of cyclic loading (Liu = 0) .

302

For stiff clays, which are overconsolidated with respect to the present overburden pressure av + y z - Liu, this method of stability analysis may be too conservative. If the initial undrained shear strength is not reduced by short term storm loading, long term stability analysis is not required. This will be the case if the load intensity, expressed by the inverse of the safety factor, is below a critical level. This critical level decreases with increasing overconsolidation ratio and, hence, with decreasing overburden pressure. For storm load intensities above the critical level, the undrained shear strength according to equation (35)defines a lower limit to the reduction of the initial strength. The relevancy of long term safety factors must be judged from the degree of strength reduction by short term storm loading.

The effect of repeated wave loading in the long term state is a pore pressure generation as a function of the cyclic stress ratio i:0 / a~ and the number of cycles. In sand simultaneously a partial dissipation occurs during a storm. The average net excess pore pressure Liu at the end of the design storm is inserted into equations (33) and (35) to calculate the mobilized friction angle <i>m and the redl!ced undrained shear strength Cu.

As the stress ratio i:0 / a~ decreases with depth a less conservative approach will be to determine the average excess pore pressure Mi at base level and an approximate value of the gradient autaz. The undrained shear strength is then calculated by:

au sin <p1 Cu = (ay + yz + -az Z - Liu)----+ sin<pm

with <pm obtained from the equation

_ _ 1 ( arr l 8 N Oy - £iU : - y + -- y •

2 az

(36)

(37)

The methods of short and long term stability analysis, described in this and the previous section, which assume a continuous plastic stress field, are well suited and preferred above other methods for strength profiles which are fairly uniform to a depth of about 20% of the base width of the platform. However, if the shallow foundation zone is interrupted by one or more distinct soft layers, the stability must be checked also by a limiting equilibrium method which allows for an arbitrary choice of the shape and location of the failure surface, as for instance the Bishop method of slices.

The top layer of the seabed may be soft or the maximum shear stress by base friction may be smaller than the shear strength of the top layer. Therefore, it must be checked also if the horizontal resistance by sliding along the base, including the contribution of the skirts, is sufficient to Justify the safety factors determined for short and long term stability. The contribution of transverse skirts is the force which causes failure of the skirt or failure of the soil, whichever is the smallest. Due to the constrained horizontal movement of the base the ultimate lateral resistance of the soil Is that according to a failure mode as indicated by figure Sa.

A lower limit of the lateral pressure aH on the skirt is calculated by assuming undrained loading and constant volume deformation:

(38)

with Cu the undrained shear strength of a· constant strength profile or the strength c0 at skirt tip level if the strength increases linearly with depth (Fig. Sb). For other frequently occurring strength profiles, such as indicated by figure Sc, the stress distribution may be calculated by:

(39)

If the contribution of the skirts is required to provide adequate safety against sliding at the base slab, it must be verified also that the yield limit of the skirts is not exceeded by the design environmental load. The base friction tends to produce a simple shear type deformation of the soil enclosed by the

303

Cu = Co

base slab I" .,I 0 ------.-------11-c u

h

h = s k i r t he i g h t z Cu = c O • m ( z - h )

®

0 0 ,-----.---Cu

h

© Co

z z

{m = m1

Cu = Co+ m ( z - h) m = m2 if z ~ h

if z .~ h

Fig. 8 Failure mode along skirts with sliding at the base. Superficial strength profiles. ·

skirts. In general it may be assumed that the deflection of steel skirts will follow the simple shear pattern, as long as the stresses required to bend the skirt are small compared to the ultimate stresses to cause rotational failure. The transfer ratio of the horizontal load between the base and the skirts is then obtained from the relative stiffness of the soil and skirts. Concrete skirts must preferably be designed to resist the ultimate soil resistance according to equations (38) or (39).

3.4 Strength reduction by cyclic loading

If a soil element is subjected to cyclic shear stresses, each cycle produces a small plastic volume decrease. If the loading is undrained this volume change is recovered by elastic expansion of the soil skeleton, causing a pore pressure increase to the extent of the elastic unloading. By a great number of cycles the effective stress may be reduced so far that the soil element fails in cyclic loading. Such an event is shown by the stress path 012 in figure 9, for an

.. 304

element which is first consolidated to the mean stress level a~ and then cyclically loaded with a ,shear stress amplitude 'c.

II ,... 1/1 1/1 CIJ ... -Ill ... 0 Cl)

.c 1/1

Fig. 9. Undrained cyclic loading behaviour.

(;~

effective mean stress I I

= G"J + G"3 2

Normally, with sufficiently high safety factors, this state of failure in cyclic loading will be reached only by few elements of the foundation soil. Let the effective mean normal stress a' at an intermediate point 3 represent the state of an element at the end of a design storm, then subsequent undrained static loading to failure occurs along the stress path 34. If the stress path 015 refers to direct static loading, L1, is the reduction of the undrained shear strength due to cyclic loading.

This interpretation of cyclic loading behaviour in terms of effective stresses applies to sand and clay in long term loading states. The pore pressure build-up by a design storm must be determined.

The pore pressure generation per cycle is a function of the cyclic stress ratio tc/cr~, which can be expressed:

13= t1U •c , L1N = exp (a0 + a1 -,-).

Oo Oo (40)

Figure 10 shows 13-values obtained from cyclic triaxial tests on Dunlin clay. Such tests are preferably carried out at consolidation stresses equal to the overburden pressure caused by the weight of the soil and the structure a~ = y z + av, Al this consolidation stress level the estimated overconsolidatlon ratio of the Dunlin clay was about 1.5. The test results also indicate that preshearing at a~ = y z level may cause a slight damage, whereas preshearing at a~ = y z + av level tends to improve the resistance against cyclic loading. Such information is important to evaluate the fatigue behaviour of the foundation soil.

To determine the actual pore pressure generation by the design storm the cyclic stress ratios , 0 /cr~, and the corresponding number of cycles must be defined. · For North Sea conditions the design storm for cyclic loading is usually selected as the 8-hour peak period of the extreme.storm, which includes approximately 2000 cycles with an average frequency f = 0.07 Hz. The number of cycles at subsequent stress ratios , 0 /a~ may be obtained from a Rayleigh probability distribution of wave heights:

P(t:! >H) = exp(-8H 2 /H~ax) (41)

305

,o-3 I

0 virgin loading at a'o=YZ•vv

(I) preshear i ng at rJ~ = yz

~ preshear i ng at c;'~ : y Z+ a'v

10- 5 +----+---"--t----+----t---+-------+-0 0.1 0.2 0.3 0.4

Fig. 10 Generation of pore water pressure of Dunlln clay.

0.5 0.6

'tc/ er' s i n \Pf

0.7

--

and by assuming wave loads proportional to wave heights. If the storm loading is divided into 10 equidistant cyclic stress levels, the following numbers of cycles at each level are calculated:

a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

N 325 450 475 350 225 110 45 15 5 1

Table 2: Number of cycles N at level 'c = a. 'c (max) .

The maximum cyclic stress level, which occurs only once in the design storm, corresponds to the probable maximum wave force.

306

For an approximate analysis the actual distribution of cyclic stress ratios in the foundation soil may be replaced by a representative average value fJo~, which is based on the stress variations in the central plastic zone below the caisson. A general procedure for selection of cyclic stress ratios is described by Smits (1978). If the wind and current forces are of minor significance, the ratio is formulated:

't'H ----(1 + sin<Pm) ov + y z

't'c =

£\F with 'fH is the average shear stress amplitude at base level due to wave forces AH

and <Pm calculated by equations (33) and (34) with £\u = 0.

(42)

By selecting the number of cycles at subsequent cyclic stress ratios from table 2 and corresponding p-values from equation (40), the accumulated average excess pore pressure at the end of the design storm is calculated:

10

£\IT = 00 • !: P; N1 • (43) 1=1

This is the excess pore pressure which Is inserted into equations (35) or (36) to calculate the undrained shear strength for a cohesive soil In the long term state.

With foundations on sand a partial dissipation during the design storm must be taken into account. When conservatively assuming radial drainage only and a storm duration suffiently long to reach steady state pore pressures, the average excess pore pressure at the end of the storm becomes:

Yw .!. R2

= --u--kD 8 '

for a circular base with radius R, and

Yw .!. 92 £\IT = u

RD 3Q·'

for a square base of width B ,

where . TI =

and in which

is the duration of the storm,

IT is the average rate of pore press.ure generation,

k is the permeability,

D is the constrained modulus of recompression.

(44)

(45)

(46)

If an adequate skirt drainage system is provided, the base dimensions In equations (44) or (45) may be replaced by those of the skirt compartments. It may be necessary to Install deep relief wells, which can be operated during storms and in conjunction with deep pore pressure measurements. The expected average excess pore pressure is then calculated:

1w • R2 R £\IT = -IT-(41n--3) - AU

kD 8 a P (47)

307

in which

a is the radius of the wells,

A is the equivalent radius of the drainage zone,

~up is the reduction of the water pressure in the wells with respect to the hydrostatic level.

The initial undrained shear strength of clays in the short term state has no relation to the shear strength according to point 5 in figure 9. If the initial strength is plotted against the in-situ overburden pressure, the stress points generally lie above the failure line in this diagram.

The reduction of the initial strength by repeated loading is investigated by cyclic laboratory tests on samples, which are consolidated vertically to the in-situ overburden pressure yz, with due precaution to avoid swelling. The effect of repeated loading in terms of total stresses may be interpreted by the development of cyclic shear strains with number of cycles as a function of the cyclic stress ratio tc/ Cu, This is shown by the strain contour diagram for Dun I in clay in figure 11 (solid curves).

0.6

0.5

l 0.4

::,

u 0.3

" u p

0.2

0.1

0

y = 0.75 1.5 3.0

0.4 .... ·~--::.~~~~~

·- ·- · - · - · -·-· -· - ·-·-· -+- · - · -~-=-~·:-~· -=·-~. ,..,,,_~----------·-·-· 01_ __ , __________ ·-·-·-·.-·-·-·-·.-·-·~---- - - - - - - -

10 100 number of cycles N

1000

Fig. 11 Strain contour diagram of Dunlin clay.

Total stress Interpretation.

The accumulated cyclic shear strain of a soil element due to storm loading with distinct numbers of cycles at subsequent cyclic stress levels may be plotted In the contour diagram too (dotted curve). When assuming average cyclic stress ratios "fclcu for an approximate analysis, point E in figure 11 represents an average cumulative cyclic strain fc at the end of the design storm.

It appears from cyclic tests that the strain amplitude, once having reached a critical value (Ye == 3 - 5% ), increases rapidly with the number of cycles. Also it has been experienced, that the undrained shear strength in static loading to failure decreases with increasing strain amplitude reached by prior cyclic loading. The strength reduction may be about 25% at cyclic shear strains of 3%.

308

The actual reduction of the initial strength is preferably obtained from static tests after cyclic loading. If insufficient data are available the strength reduction may be estimated:

in which Ye is the average cumulative cyclic shear strain.

3.5 Settlements and cyclic displacements

Settlements can be distinguished In: - initial settlement due to installation of the structure, - consolidation settlement, - cyclic loading settlement.

(48)

Consolidation and cyclic loading settlements may cause stresses in conductors due to negative friction, which add to the stresses by temperature and pressure changes. Differential settlements may occur due to non-uniform soil conditions.

Cyclic displacements concern vertical, horizontal and rotational motions of the structure due to the maximum environmental load, i.e. initial displacements and displacements after short and long term storm loading. The analysis must account for the dynamic soil-structure interaction.

To obtain a first impression of the cyclic displacements they may be calculated by elastic halfspace theory:

6.Fv 6vc = (1 - u)

4GR ,

7-8u

32(1 - u)

= 3(1-u)~. 8GR3

(49)

(50)

(51)

E For undrained loading Poisson's ratio may be assumed u = 0.5. The shear modulus G = ---

2(1 + u) where

E = 400 Cu for clay , (52)

E = 500 y a'v Pa for dense sand

in which Pa is the atmospheric pressure ,

a'y is the effective overburden pressure .

When selecting an average shear modulus for the above equations, due consideration must be given to the extent of the elastic stress field and to the contribution of subsequent layers in the total displacement.

The natural frequencies for the 3 modes of vibration may be estimated from the mass of the structure and the spring constants obtained from equations (49), (50) and (51). The natural frequencies are usually several times larger than the average wave frequency, so that the magnification factors for the undamped vibrations in general are less than 10%. In such cases It is not important to consider the damping.

The initial settlement is determined by equation (49) with 6.Fv replaced by the platform weight Fv and with the same values of u and G. The consolidation settlement only concerns clays and as far as the effective vertical stress

309

a(,, = yz + Fv/ A exceeds the preconsolidatlon stress. Calculation may occur by the same methods as used for land based structures, which requires determination of the compressibility and the coefficient of consolidation. A simpler method may be by using the equation for the initial settlement calculation with moduli for drained condition: u = 0.2 - 0.3 and E = 250 cu.

Cyclic loading settlement and cyclic displacements due to storm loading need separate consideration. Repeated loading not only reduces the undralned shear strength but also the stiffness, as may be observed directly from figure 11. Figure 12 shows the reduction of the shear modulus Gas a function of the cyclic stress ratio , 0 /cu, for equal numbers of cycles in cyclic tests and also for an undrained static test. The approximation E = 400 Cu, used for the calculation of initial settlement and cyclic displacements, corresponds in this plot with a stress level t=•0.25 Cu in the static test.

200

100

! 50 \ .'5',..

::, 'f,:,. . (' u y('j " C) .'5',,..

20 \

\

10 ~ N~ ~ r-. ~ CD a,

I I I I

LO lO LO lO

CD IO IO CD

I I I 5

0 0.1 0.2 0.3 0.4 0.5 0.6

t c/Cu

Fig. 12 Normalized shear modulus of Dunlin clay.

Total stress interpretation.

In figure 13 the shear modulus, obtained from cyclic triaxial tests on Dunlin clay at initial consolidation stress a~ = yz + av, is plotted as a function of the cyclic stress ratio ,

0/ a'.

It may be observed that the shear moduli, normalized with respect to the effective mean normal stress a', are approximately the same in cyclic and static tests. This, of course, points towards a unique stress-strain relation in terms of effective stresses for cyclic and static loading, as is illustrated by figure 14. The data of cyclic tests in this plot refer to peak cycle conditions.

Now to determine the cyclic loading settlements and the cyclic displacements two different procedures may be followed: i) an elastic analysis based on reduction of the shear modulus by storm loading,

ii) a plastic analysis based on increasing strength mobilization and cumulative cyclic strains.

310

100

75

! 50

... b

.......... (!)

25

0 0 0.2

Iii

G) <;]$

$1. QI·

'C' I f.'

$t

0.4

tc/v' sin 'Pt

6.

6. V g

Fig. 13 Normalized shear modulus of Dunlln clay. Effective stress interpretation.

• B 3 - 9/ 7 o B3-6/4 0 63 - 5/4 V B4 b-¥4/6 A BS-10/4/6 0 BS- 7/4/6

'vl 6.

0.6 0.8 1.0

The elastic analysis may be an approximate calculation by defining average shear moduli and inserting them into equations (49), (50) and (51), or a finite element analysis with shear moduli based on local stress variations and soil conditions. The reduction of the shear modulus In a short term storm can be obtained by tracing in figure 12 the path of accumulated reductions due to series of cycles of increasing stress level, similar as the strain path in figure 11. The reduction of the shear modulus in a long term storm is obtained by first calculating the net excess pore pressure .6.u, as described in section 3.4, and the effective mean normal stress a', as described in section 3.3. The shear modulus is then selected from figure 13, by taking 'c equal to the maximum shear stress or mobilized strength cm due to the characteristic load. (Note that here 'c ,;. tH. The abscissa In figure 13 denotes the degree of strength mobilization, whereas 'e = tH was adopted previously as the representative cyclic shear stress level that governs the pore pressure generation). The mobilized strength cm is calculated approximately by the equation:

4 -2 Ov Cm = Ne (4. Ov - tH Ne) ' (53)

TH 2 else if-- :s: --

' av Ne

Cm = 'H (54)

with Ne = n+2

311

1.0

0.8

0.6

b~ '-c ~ 'iii 0.4

0.2

tpf = 32°

• BJ -9/7 •BJ-6/4 oB3-6/7 + B3 -5/4 o B3 -5/7 x B4b- 3/4/6 t,.B4-3/7 , B5 -10/4/6 V 95 -10/7 + B5 - 7/4/5 OBS-7/7

0-------------------------1 2 6 8 10

y ( .,. )

Fig. 14 Stress-strain behaviour of Ounlln clay at o~ = y z + civ .

In the plastic analysis a partial but uniform strength mobilization by peak cycle loading is assumed. The shear strain associated with increasing strength mobilization during storm loading is then obtained from the stress-strain relation. To this shear strain is added the accumulated strain by repeated loading at subsequent cyclic stress levels. The direction of incremental base displacements due to increasing strength mobilization Is defined by the flow rule. The magnitude of incremental displacements is then calculated approximately by assuming average shear strain increments, based on a simple shear mode of deformation over the depth of the plastic stress zone. This method of analysis is outlined in more detail by Smits (1976, 1978).

3.6 Pore pressure observations on the Dunlin platform

Deep pore pressure sensors have been installed at Dunlln for long term monitoring of the foundation performance. Two steel bars, each containing 5 piezometers spaced at distances of 4 m, were pushed into the soil below two opposite towers, so that the deepest piezometer is at 20 m below the seabed. The installation was carried out from the 75 m level In each tower. The first string of piezometers was installed after final penetration of the skirts and before grouting, the second string about 3 weeks later after full ballasting of the platform. All sensors installed are still performing well to date, three years after installation of the platform.

Figure 15 shows the pore pressures measured during installation of the second string and figure 16 the cumulative penetration resistance. The pore pressure increases between 1.5 and 8 m depth are approximately equal for all gauges, indicating a uniform clay layer. At greater depths the first gauge passing by generally shows the highest pressure, the rate of increase with depth is less than in the upper clay layer and there are large fluctuations. This corresponds to the occurrence of sand and silty clay layers below 8 m depth as observed in the majority of the borings. The decreases at 8.5-9 m and 10.5-12 m may be due to a local sand layer, whereas the increases at 10 and 13 m indicate the presence of clay. The pore pressures tend to converge to a value of 155 tf / m2 at seabed level, which is approximately the hydrostatic water pressure. A small decrease of the pressures between O and

312

Fig. 15 Pore water pressures measured during Installation

E

.,, .,

.a Cl ., II)

~ 0 ,; .a .t: a. ., .,,

----pore water pressure ( tt/m2)

012ro;..__-,-_..,,.~r?"-~--2,o~o __ ,--__ 2,4_0 __ ,--__ 2,so

4

8

I I ,__ __ ....._ _ ___.1--

gauge number

(. )

( 0 )

3 (.)

4 ( 0).

5 ( . ) .

final depth

4 m

8 m

12 m 16 m

20m

·- t ---------1

I

I " 16 ._ ___ _,__ __ ,___ ·--+---

I I I I

of plezometer string 2. 2 O L __ 1__ __ 1__ _ __L1__ __ 1___-'--1-~===~..1.---

Fig. 16 Penetration resistance

---- penet_ration resistance ( t f )

0 On~~1§r-'1ro __ l ___ 2ro __ l ___ 3rO __

E

.,, .,

.a 0 ., II)

~ 8 0 ., .a .t:

c. ., .,, ---i--------i..--+----+--

1

f---1 I

of plezometer string 2. 20 L __ __J ___ ___1_ ___ --1.._ ___ .L. __ __JL __ ...::r==...J

313

Fig. 17 Pore water pressures during calm sea.

314

Ul I...

:, 0 ..c

4

195 190 185 180 175 170 pore waterpressure (tf/m2)----

Fig. 18 Pore water pressures during storm period.

.. 315

1.5 m depth corresponds to a top layer of sand with boulders. This is also confirmed by the sudden variations of the penetration resistance in figure 16, whereas the dip in the curve between 13 and 14 m depth is probably due to a softer clay layer as also indicated by the pore pressures.

Figures 17 and 18 show typical pore pressure records during calm and rough sea states. During calm sea periods the pressure variations due to the tide, about 1.5 tf / m2 in figure 17, are clearly observed and are approximately equal for all piezometers. During a heavy storm the tidal pressure variations cannot be distinguised anymore. Pressure variations due to wave forces of about 5 tf/m2 have been recorded. Short storms of 1 or 2 days generally did not affect the average pore pressures. However, heavy storms of longer duration and moderate storms of a few weeks have caused at least a stagnation of the pore pressure dissipation and sometimes an Increase of the average pore pressure of 1 to 2 tf/m2•

Such trends may be observed in figure 19, which shows the pore pressure changes during the frrst 11 months after installation. Th~ pore pressures plotted are daily averages corrected for short and long term tidal variations. (Pore pressures during the first month are not shown in this diagram, as they could not be averaged for the two strings due to the different installation· dates).

The effect of storms is observed most clearly on the records of the piezometers at 4 and 8 m depth. Moderate and heavy storms occurred in mid-September 1977, between 8 and 26 November 1977, between 8 and 16 December 1977, from end December 1977 until early February 1978 and from mid-March until mid-April 1978.

---------19'1'1---------1,.,-----1978 -------l Sept Oct Nov Dec Jan Febr March A ril

depth below seabed 1 = 4 m 2 = 8 m

5 3 = 12 m 4 = 16 m

N

~ 185 - 4

5 = 20m

CII .... :, 1/) 1/)

CII ... a. 180

3 ... CII

0 2 ::t CII .... 0 a. I 175

170

165~-~---'----'---__,_ __ __. ___ _,__ __ _,__ __ -'-----'--~.d

Fig. 19 Pore water:pressures during the first year.

316

Figure 20 shows the excess pore pressures on 1 August 1977 and on 1 May 1978. It is difficult to distinguish the initial excess pore pressures due to installation of the structure from those due to penetration of the piezometers. In view of the average vertical pressure after ballasting of about 14.5 tf/m2, the excess pressures on 1 August 1977 may be reasonably representative for the initial condition. From the initial and final excess pressure distributions in figure 20 it may be concluded, that generally downward and radial drainage occurred for the 10 to 20 m layer, whereas vertical drainage prevailed for the top clay layer. From consolidation calculations and the experienced dissipation in the first year the time required for 90% consolidation is expected to be about 3.5 years at the 4 m level and about 20 years at the 10 m level.

E

"O a, .0 0 tl Ill

~ 0

a, .0

.c -Q. a,

"O

---excess pore pressure ( tt/m2)

0 5 10 15 0.--------,----------,---------,

78-05-01

4

8

77-08-01

I I

I I

\

I

I \ I I I.

I I

l " I I

' I I

16

I I

I

I I

I

I I I

20 ~------~--------Hf---------~

Fig. 20 Excess pore pressures Initially and eleven months after Installation.

Figure 21 shows the relative displacement between the two piezometer strings and the platform. It represents a partial settlement, in particular the contribution to the settlement by a top layer that causes negative friction on the piezometer rod. The height of this layer is estimated from the measured penetration resistance and is approximately 15 m. The plot shows the settlement of the second string starting at the point of initial settlement reached by the first string when the second string was installed (33 mm). The skirt drainage system has been operated twice, which caused an increase of settlement. This settlement appears to have occurred almost directly after opening of the drainage system, which might indicate that some free water was expelled. There has been no noticeable response of the piezometers to this drainage operation.

317

- 1977

0 June July Aug Sept QC t Nov Dec Jan ·Fe br

E E 20

C 111 E 111 .... .... C!I C/1

60

80

I

I

\ ~ ~ string 2

I

~ r---- r---i---

----

Fig. 21 Settlement of the base relative to plezometer strings.

References

Federation Internet/one/ de le Precontrslnte.

drai l"]~e ~r1

+r--r--

r---

---

State of Art Report. Foundations of concrete gravity structures In the North Sea. FIP, Slough, August 1979. Smlts,F.P. Stability and deformation analysis for gravity platforms.

r---

1978

Mor

r--

BOSS '76. Proceedings of the First International Conference on Behaviour of Offshore Structures, Trondheim, 1976, Vol. 2, pp. 85·112. Smits,F.P.

Apr May

drainage period ioi 14

IL\ i---

----

Excess pore pressures and displacements due to wave Induced loading of a caisson foundation as predicted by plasticity analysis. Symposium on Foundation Aspects of Coastal Structures, Delft, 1978, Vol. 1, 1114.

318

january 1986 prof.dr.ir. a. verruijt

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