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NUMERICAL SOLUTION OF CYLINDRICAL CAVITY EXPANSION IN SANDS:

EFFECTS OF FAILURE CRITERIA AND FLOW RULES

By

BRENTON RYAN COOK

A thesis submitted in partial fulfillment of

the requirements for the degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING

WASHINGTON STATE UNIVERSITY

Department of Civil and Environmental Engineering

December 2010

ii

To the Faculty of Washington State University:

The members of the Committee appointed to examine the thesis of BRENTON

RYAN COOK find it satisfactory and recommend that it be accepted.

___________________________________

Balasingam Muhunthan, Ph.D., Chair

___________________________________

William Cofer, Ph.D.

___________________________________

Haifang Wen, Ph.D.

iii

ACKNOWLEDGEMENTS

I would first like to thank Dr. Balasingam Muhunthan for his support, guidance

and enthusiasm concerning this study and my future in the field of geotechnical

engineering. To my family, thank you for keeping me going over the past six years;

financially and motivationally. Thank you to my fellow graduate students for their

support, specifically Gonzalo Montalva for his frequent and much appreciated technical

guidance. Last, but not least, I want to thank the Department of Civil Engineering at

Washington State University for the financial support provided during the course of this

study while also providing me with a strong academic foundation for future endeavors.

iv

NUMERICAL SOLUTION OF CYLINDRICAL CAVITY EXPANSION IN SANDS: EFFECTS OF

FAILURE CRITERIA AND FLOW RULES

Abstract

by Brenton Ryan Cook, M.S.

Washington State University

December 2010

Chair: Balasingam Muhunthan

Cavity expansion in soil is an important area of interest within the field of

geotechnical engineering. Detailed understanding of this boundary value problem would

allow engineers to better interpret in-situ test data from devices such as the pressuremeter

and the cone penetrometer. Cavity expansion theory can also be extended to important

and complex design problems commonly encountered in practice such as driven/cast in

place lateral pile capacities, in addition to applications for tunneling.

Early research into cavity expansion in soils involved developing closed form

solutions for clays assuming linear elastic, perfectly plastic behavior. Later, rigorous

analytical solutions for the cavity expansion problem were introduced which made a

concerted effort to capture the effects of confinement pressure and density to characterize

the behavior of sands. More recently, finite difference models have been presented that

are more versatile for conducting studies on the influence of various parameters on cavity

expansion.

A recent finite difference model proposed by researchers from Purdue University

discretized the plastic region around an expanding cavity into thin shell elements.

Iterations across each shell were carried out using an empirical friction angle relation in

conjunction with a flow rule and failure criterion. Recognizing that there exist numerous

friction angle models, flow rules and failure criteria, it is of interest to investigate how

their combinations influence results when implemented into a cavity expansion analysis.

v

This study modified the finite difference model with the objective of investigating

the influence of different friction angle models, flow rules and failure criteria within a

cylindrical cavity expansion algorithm for sand assuming drained conditions. Results

from the proposed algorithm were evaluated including limit pressures for different

combinations of depth and relative density. It was found that the choice of flow rule had

little impact on predicted results, whereas the choice of failure criterion did have

influence. Recognition of such modeling nuances allows for algorithms, such as the one

presented in this study, to predict soil conditions in the field with increased confidence.

vi

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTSiii

ABSTRACT.. iv

LIST OF TABLES...... viii

LIST OF FIGURES. ix

CHAPTER 1: INTRODUCTION 1

1.1 Introduction... 1

1.2 Organization of Thesis.. 4

CHAPTER 2: SHEAR BEHAVIOR OF SANDS... 5

2.1 Critical State Soil Mechanics 5

2.2 Measures of Density.. 10

2.3 Correlations between State Parameters and the Internal

Friction Angle... 13

2.4 Failure Criteria.. 14

2.5 Energy Dissipation in Soils and Flow Rules. 17

CHAPTER 3: CAVITY EXPANSION IN SANDS 25

3.1 Cavity Expansion Problem 25

3.2 Outline of Numerical Analysis.. 28

3.3 The Limit Pressure 28

vii

3.4 Stress and Strain Relations for Discrete Shell Analysis in

the Plastic Zone. 30

3.5 Boundary Conditions at Elastic-Plastic Interface.. 35

3.6 Flow Chart for Cavity Expansion Algorithm 37

3.7 Step-by-Step Progress through the Cavity Expansion

Algorithm.. 42

3.8 Soil Properties... 46

CHAPTER 4: RESULTS AND INTERPRETATION.. 47

4.1 Comparisons between Predicted and Measured Limit

Pressures... 47

4.2 Comparison between Models.... 48

4.3 Parametric Study... 51

4.4 Results from Analysis Using Parameter... 51

4.5 Flow Rule and Failure Criterion Combinations 52

CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS.... 56

5.1 Conclusions... 56

5.2 Recommendations for Future Research 58

APPENDIX .. 59

REFERENCES... 72

viii

LIST OF TABLES

Page

Table 2-1: Flow rules chosen for analysis. 23

Table 4-1: Properties assumed for Ticino sand. 48

ix

LIST OF FIGURES

Page

Figure 2-1: Critical state line in v-lnp and q-p space 6

Figure 2-2: Normal, critical and elastic compression lines

in v-lnp space. 9

Figure 2-3: -lines in v-lnp space.. 9

Figure 2-4: -lines in stress space. 10

Figure 2-5: State parameter in v-lnp space 12

Figure 2-6: Relationship between , and in v-lnp space 12

Figure 2-7: versus for Ticino sand (after Lancellotta 1995).. 14

Figure 2-8: Mohr-Coulomb and SMP failure envelopes

(modified after Griffiths & Huang 2009).. 15

Figure 2-9: Flow number versus angle of internal friction 16

Figure 2-10: Dilation under direct shear box conditions

(modified after Lancellotta 1995).. 17

Figure 2-11: Associated flow rule. 19

Figure 2-12: Theoretical dilation using Muhunthan & Sasiharan (2007)

with varying ... 23

Figure 2-13: Variation in theoretical dilation 24

x

Figure 3-1: Cavity expansion diagram.. 25

Figure 3-2: Zones in cavity expansion process. 27

Figure 3-3: Displacement pressuremeter plot

(modified after Houlsby & Carter 1993)... 29

Figure 3-4: Schematic of plastic zone illustrating the termination

Condition... 30

Figure 3-5: Inner and outer radii and displacements for a single

incremental shell 33

Figure 3-6: Variations in shear modulus .. 35

Figure 3-7: Parameter changes across associated zones

(modified after Salgado & Prezzi 2007)... 37

Figure 3-8: Flow chart showing iteration segment to obtain ,

and 39

Figure 3-9: Flow chart showing iteration segment to obtain G,

and . 40

Figure 3-10: Flow chart showing iteration segment across plastic zone... 41

Figure 4-1: Linear regression analysis showing measured versus

predicted values of .... 47

Figure 4-2: Shear modulus at elastic-plastic interface

xi

(modified after Salgado & Prezzi 2007)... 49

Figure 4-3: Initial lateral stress versus limit pressure

(modified after Salgado & Prezzi 2007)... 50

Figure 4-4: Influence of friction angle models on the limit pressure

and R/a for = 29.. 52

Figure 4-5: Influence of flow rules on the limit pressure for = 29. 53

Figure 4-6: Influence of failure criteria on the limit pressure for

= 29. 54

Figure 4-7: Minimum and maximum output of limit pressure for

= 29. 54

Figure A-1: Calibration chamber plots of cavity pressure versus

cavity strain (after Schnaid 1990).. 60

Figure A-2: Pressuremeter variations

(modified after Clough et al. 1990)... 62

Figure A-3: Pressuremeter and laterally loaded pile analogy

(modified after Briaud et al. 1984) 63

Figure A-4: Piezo-cone with pressuremeter (after Withers et al. 1989) 64

Figure B-1: Triaxial loading.. 66

Figure B-2: Bi-axial loading.. 67

1

CHAPTER 1

INTRODUCTION

1.1 Introduction

The expansion of cylindrical or spherical cavities in an infinite medium is one of

the basic boundary value problems in solid mechanics (Carter et al. 1986, Palmer 1972).

The solution to such problems has been of interest in the area of geomechanics where

they have been used to develop approximate analyses of the s

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