Investigations of nonlinear p-y piles and pile groups in soft clay subjected to static loading-distributed parameter sensitivity analysis.

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University of Windsor University of Windsor

Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

1-1-2006

Investigations of nonlinear p-y piles and pile groups in soft clay

subjected to static loading-distributed parameter sensitivity

analysis.

Marcia Regina Mora

University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation

Mora, Marcia Regina, "Investigations of nonlinear p-y piles and pile groups in soft clay subjected to static loading-distributed parameter sensitivity analysis." (2006). Electronic Theses and Dissertations. 7139.

https://scholar.uwindsor.ca/etd/7139

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INVESTIGATIONS OF NONLINEAR

p-y

PILES AND

PILE GROUPS IN SOFT CLAY SUBJECTED TO

STATIC LOADING-DISTRIBUTED PARAMETER

SENSITIVITY ANALYSIS

BY

MARCIA REGINA MORA

A Thesis

Submitted to Faculty of Graduate Studies and Research

through Civil and Environmental Engineering in Partial

Fulfillment of the Requirements for the Degree of Master of

Applied Science at the University of Windsor

W indsor, O ntario, C anada

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MARCIA REGINA MORA

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ABSTRACT

For the design o f a pile foundation, the engineer must take into account the axial loads

and overturning moments. Although some structures deal with pile under axial load it is

very common to see the utilization o f piles subjected to lateral loads, especially in areas

suitable for earthquakes or offshore platforms, bridges, high-rise structures, piers and so

on.

Laterally loaded piles embedded in soft clay bellow the water table under static load is

presented in this study under the theoretical formulation and the numerical investigation

o f sensitivity analysis. In this study both single piles and pile groups are analyzed under

the sensitivity approach.

A wide range o f lengths and a large group o f different boundary conditions were applied

on single piles under static loadings, addressed to the pile head. In terms o f structure, here

the single piles are considered as one dimensional beam and the soil supporting selected

is a soft clay and is defined by p -y relationship once under lateral load, deflection o f the pile is directly dependent on the soil response and in this case the soil response is a

nonlinear function o f pile deflection and the depth o f the soil below the surface. The p-y

model mentioned above for soft clay was developed by Cox, Reese and Grubbs and has

been used widely all over the world and is applied in the neighbourhood o f the pile.

Under the broad variation o f the boundary conditions the group o f piles are also analyzed

through the sensitivity approach and in this case the cap is considered as a plate and the

pile members are recognized as one dimensional beam. The p -y relationship once more is the more appropriate model to understand and represent the behaviour o f the pile-soil

system. For the case o f a group o f piles, the utilization o f a specific m o d i f i e r f a c t o r had

to be applied once the group o f piles change with respect to spacing o f the piles and also

the location o f the piles inside o f the group.

The physical parameters o f the soft clay and the stiffness o f the pile used for the

description o f the p -y relationships are taken as the design variables o f the continuous type. They are space dependent.

To be able to analyze the sensitivity o f the single pile and the group o f pile subjected to

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functional o f maximum deformations as a result o f the changes o f the design variables is

formulated based on the virtual work principle.

The variations o f maximum generalized deflection located at the pile head as a result o f

the changes o f the design variables are determined by sensitivity integrands and the

design variables related. The sensitivity integrands are integrated and the numerical

assessment o f the outcomes are presented and discussed in details.

Although some specific design variable appeared in the sensitivity analysis o f the

deformation o f the pile soil system each one as expected had demonstrated particular

differences and significance through the analysis, and theses will be also part o f the final

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Dedicated to

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ACKNOWLEDGMENTS

I would like to address my sincere gratitude to my advisor Dr. B. B. Budkowska for her

relentless support, guidance and important theoretical and technical insights through out

the process to accomplish this thesis.

I would like to express also my profound gratitude to my Committee Members Dr. S.

Cheng and Dr. M. Ahmadi for their constructive and valuable suggestions to improve this

thesis.

I would like to give a special and huge thank you to my wonderful and dear friends at

University o f Windsor: Dahlia Hafez, Sharefah Al-Shammari and Sunghan Lee for

always being there to help me out with everything. I am forever grateful! Many thanks!

There are also a great number o f friends that helped me through this work with their time,

their technical support, and most o f all, with their encouragement. Thanks to all; Lebing

Liu, Zain-Al-Abedin, Andre Bom, Vrushali Trickle, Cindy Kumalas, T.I.M. Nazmur

Rahman, Li Li, Wafa Polies, Family Radjul.

In particular, I would like also to express enormous love, appreciation and gratitude to my

family members for their loving support, endless encouragement and countless hours that

I was away from them; Rubens Wachockier, Kim Wachockier, Alba Mora and Francisco

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TABLE OF CONTENTS

ABSTRACT iv

DEDICATION vi

ACKNOWLEDGEMENTS vii

LIST OF TABLES xii

LIST OF FIGURES xiii

CHAPTER 1 INTRODUCTION 1

1.1 Problem Statement 1

1.2 Objectives 2

1.3 Procedures 3

1.4 Methodology and significance o f sensitivity analysis 4

1.5 Study organization 4

CHAPTER 2 LITERATURE REVIEW 6

2.1 Methods o f analysis o f laterally loaded single piles 6

2.1.1 General 6

2.1.2 Models for use in analysis o f a single piles 7

2.1.2.1 Sub grade-reaction model 7

2.1.2.2 Elastic pile and elastic soil 8

2.1.2.3 Elastic pile and finite element for soil 8

2.1.2.4 Rigid pile and plastic soil 9

2.1.2.5 Characteristic load method 10

2.1.2.6 Nonlinear approach for pile and p-y model for soil 11

2.1.3 Field testing performed over piles under lateral loading in soft 14

clay

2.2 Methods o f analysis o f laterally loaded piles groups 15

2.2.1 Overview 15

2.2.2 Simple static analysis method 16

2.2.3 Equivalent beam method 17

2.2.4 Elastic continuum analysis o f pile behaviour 18

2.2.5 Group reduction factor method 20

2.2.6 Group amplification method 20

2.2.7 The p-multipliers method 22

2.2.8 Field testing performed over a group o f piles under lateral 26

loading in soft clay

2.3 Literature review on a sensitivity analysis o f pile foundation 29

2.3.1 Classification o f sensitivity methods 31

2.3.2 Sensitivity analysis applied previously 33

CHAPTER 3 THEORETICAL FORMULATION 46

3.1 Single Piles 46

3.1.1 General 46

3.1.1.1 Pile and soft clay 47

3 .1 .2 p -y curve for soft clay 48

3.1.2.1 p -y curve theory for soft clay 48

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3.1.3 Theoretical formulation o f sensitivity analysis o f laterally loaded piles embedded in soft clay below the water table

54

3.1.3.1 Introduction 54

3.1.3.2 Primary and adjoint structure 55

3.1.3.3 Sensitivity operators and factors 58

CHAPTER 4 THEORETICAL FORMULATION OF PILE GROUP 71

4.1 Overview 71

4.2 Laterally loaded pile group 72

CHAPTER 5 SINGLE PILE - NUMERICAL INVESTIGATION 76

5.1 General 76

5.2 Load and constraints type 76

5.3 Determination o f the design parameters 77

5.3.1 Soil properties 77

5.3.2 Pile’s physical properties 78

5.4 Determination o f the piles length 79

5.4.1 Determination o f relative stiffness factor T 80

5.5 Long and short piles - Considerations 90

5.6 Load-deflection relationship 90

5.7 The adjoint structure concept - stresses and deformations 93

5.8 COM624P - Computer program 95

5.9 Results o f sensitivity analysis o f laterally loaded pile 96

5.10 Results o f single piles - Method o f verification 96

CHAPTER 6 PILE GROUP - NUMERICAL INVESTIGATION 99

6 .1 Introduction 99

6 .2 Loads and constrains 99

6.3 Determination o f lateral forces Pg and Pgi - Lateral force used in study lease and 2

101

6.3.1 Primary structure 10 1

6.3.2 Adjoint structure 103

6.4 The determination o f bending moments Mg and Mgi bending

moments used in study 3 case

109

6.4.1 Primary structure 109

6.4.2 Adjoint structure 1 1 2

6.5 Results o f the sensitivity analysis o f the laterally loaded piles groups

115

CHAPTER 7 PROGRAMMING OF SENSITIVITY ANALYSIS FOR THE

PILE GROUPS

116

7.1 Overview 116

7.2 Perform sensitivity analysis on single piles 119

7.3 Perform sensitivity analysis on pile groups 124

CHAPTER 8 DISCUSSION 128

8 .1 Introduction 128

8 .2 General discussion on the sensitivity analysis results 128

8.2 .1 Discussion on the lateral deflections and bending moments o f the primary structure

129

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p

8.2.3 Discussion o f the sensitivity operators C^.y. ^ 131

8.2.4 Discussion on the lateral deflections and bending moments o f 135

the adjoint structure subjected to bending moment M = 1

p

8.2.5 Discussion o f the sensitivity operators C^e# ^ 136

8.2.6 Discussion on the quantitative assessment o f the sensitivity 138

factors

8.2.7 Discussion on the relative sensitivity factors F 139

8.3 Error analysis 141

8.4 Quantitative estimation o f the sensitivity factors A 145

8.5 Assessment o f error o f lateral deflection based on comparative 149

analysis o f exact solution and sensitivity analysis solution

CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS FOR

FUTURE WORK

9.1 Conclusions 152

9.2 The application o f this study 160

9.3 Recommendations for future work 160

REFERENCES 162

APPENDIX A 169

Derivations o f formulas o f sensitivity operators

APPENDIX B 179

Sensitivity analysis o f top lateral displacement 6yt for single free head piles with length L=4T subjected to lateral concentrated forces

APPENDIX C 190

APPENDIX D 201

APPENDIX E 221

Sensitivity analysis o f top lateral displacement 8ye for single fixed head piles with length L=4T subjected to lateral concentrated forces

APPENDIX F 232

Sensitivity analysis o f top lateral displacement 8ye for single fixed head piles with length L=5T subjected to lateral concentrated forces

APPENDIX G 243

Sensitivity analysis o f top lateral displacement Sye for single fixed head piles with length L=10T subjected to lateral concentrated forces

APPENDIX H 254

Sensitivity analysis o f top lateral displacement 8y0 for single free head piles with length L=4T under bending moment

APPENDIX I 265

Sensitivity analysis o f top lateral displacement Sye for single free head piles with length L=5T under bending moment

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APPENDIX J

Sensitivity analysis o f top lateral displacement 5ye for single free head piles with length L=10T under bending moment

APPENDIX K 296

Sensitivity analysis for Pile B (1st tailing row) in a pile group embedded in the soft clay below the ground water table subjected to a lateral concentrated force Pg applied to the cap o f the pile group, with pile members pinned to the pile cap, and with pile spacing between piles equal 2D. Length o f the pile equal to 10T

APPENDIX L

Input and output data for laterally loaded pile analysis 307

L .l Typical example o f input data and output data for single free 307 head single pile subjected to a lateral concentrated load embedded in a soft clay below the ground water table

L. 1.2 Output file from COM624P - version 2.0 307

L.2 Typical example o f input data and output data for pile group 309 with the piles pinned to the cap subjected to a lateral concentrated load

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LIST OF TABLES

Table 2.1 Comparison o f p-multiplier values from various experimental tests 25 and field tests for pile groups with pile spacing equal 3 piles width, from Ilyas et al (2004), in Liu (2004)

Table 5.1 The typical value o f £50 (after Wang and Reese 1993) 78

Table 5.2 Values o f exponents m and n (Evans and Duncan, 1982) 81

Table 5.3 The lengths o f piles used in the sensitivity analysis 89

Table 5.4 The relative stiffness factor T for different boundary conditions 90

Table 8.1 Relative error o f the sensitivity factors A for the sensitivity analysis 144 o f a free head single pile subjected to lateral concentrated load embedded in a soft clay below the water table, and length = 10 T (24.50m)

Table 9.1 The average value o f the relative sensitivity factors F for the free 155 head piles subjected to the lateral concentrated load embedded in soft clay below the ground water table

Table 9.2 The average value o f the relative sensitivity factors F for the fixed 155 head pile subjected to the lateral concentrated load embedded in soft clay below the ground water table

Table 9.3 The average values o f the relative sensitivity factors F for the free 155 head pile subjected to the bending moment embedded in soft clay below the ground water table

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LIST OF FIGURES Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5

Figure 2 . 6

Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7

The model for a pile under lateral load with p-y curves (Reese and 13 Van Impe (2001))

Principle o f equivalent-bent approach (Liu (2004)) 1 7

The concept o f p-multiplier (fm) (Brown e tal. 1988) 23

The p-y used by Budkowska and Suwamo (2002) 3 9

The p-y relationship used by Priyanto (2002)-in investigation o f 4 0 piles embedded in soft clay below the ground water table subjected to lateral cyclic loadings

The p-y curves for soft clay below the ground water table 4 1 (Matlock, (1970) model) o f laterally pile group subjected to a cyclic load used by Priyanto (2002)

The p-y curves for stiff clay below the ground water table (Reese et 4 2 al. (1975) model) employed in Suwamo (2003) studies

The p-y curve used by Liu (2004) in “Sensitivity analysis o f 4 3 laterally loaded piles embedded in stiff clay above the water table”

The p-y relationship investigated by Abedin (2004) 4 4

The p-y curves for sand below the ground water table subjected to 4 5 cyclic lateral loadings used by (Rahman 2004)

The p -y model 4 9

Characteristic shape o f p -y curve for soft clay below water table 5 0 subjected a static loading

Variability o f ultimate soil resistance pu along the pile axis 5\

A pile element modeled by a beam supported by nonlinear p -y 5 4 springs

A primary structure subjected to a lateral force Pt and the 5 5 corresponding adjoint structure subjected to a lateral unit force

P = 1 applied at the pile top

The primary subjected to a lateral force Pt and the corresponding 5 5 adjoint structure subjected to a unit bending moment M = 1 applied at the pile top

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Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 4.1 Figure 4.2 Figure 4.3 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

A primary structure subjected to a bending moment Mt and the 5 7 corresponding adjoint structure subjected to a unit bending moment

M = 1 applied to the pile top

Physical interpretation o f the normalized sensitivity 5 3

integrands/operators C(„.) and sensitivity factors A(...),after Liu (2004)

Convention for the notation o f sensitivity operators/integrands and 5 7 sensitivity factors, after Liu (2004)

Numerical integration using Simpson’s rule 5 9

A typical view o f the pile group system under a lateral load Pg 7 1

Nomenclature used to describe pile group arrangements 7 3

The p-multiplier design curves proposed by Mokwa and Duncan 7 4 (2 0 0 1b)

Pile’s properties o f the pile used in the sensitivity analysis 7 9

Load-deformation for curves for Free Head Pile in clay - static 83 loading (after Evans and Duncan 1982)

Load-deformation curves for Fixed Head Pile in clay - static g4 loading (after Evans and Duncan 1982)

Moment-deformation curves for Free Head Pile in clay - static g5 loading (after Evans and Duncan 1982)

Load-moment relationships curves for Free Head Pile in clay - g6 static loading (after Evans and Duncan 1982)

Load-moment curves for Fixed Head Pile in clay - static loading g7 (after Evans and Duncan 1982)

Pile head deflection yt versus lateral force Pt applied to the top o f 91 the pile head o f free head pile embedded in a soft clay below water table. Pile length L = varies from 3T to 10T

Pile head deflection yt versus lateral force Pt applied to the top o f 9 2 the pile head for a fixed head pile embedded in a soft clay below water table. Pile length L = varies from 3T to 10T

Pile head deflection yt versus bending moment Mt applied to the 9 2 top o f the pile head for a free head pile embedded in a soft clay below water table. Pile length L = varies from 3T to 10T

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Figure 6 .1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6 .6

Figure 6.7

Figure 6 .8

Figure 6.9

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Pile group geometry used in the pile group analysis jqq

Determination o f the force Pg applied to the cap o f the piles pinned to the cap (with variable spacing) subjected to lateral concentrated force. Pile group spacing “s” is equal to 2D, 3D, 4D and 5D. The ^ pile length L is equal to 10T (24.5 m)

Force Pgi o f Pile A (2nd trailing row), Pile B (1st trailing row), Pile jq5 C (Leading row) o f group o f 3x3 piles with the spacing 2D and the length L=10T versus the applied lateral concentrated force Pg when the piles are pinned to the cap.

Force Pgi o f Pile A (2nd trailing row), Pile B (1st trailing row), Pile jo6 C (Leading row) o f group o f 3x3 piles with the spacing 3D and the length L=10T versus the applied lateral concentrated force Pg when the piles are pinned to the cap.

Force Pgi o f Pile A (2nd trailing row), Pile B (1st trailing row), Pile iq7

C (Leading row) o f group o f 3x3 piles with the spacing 4D and the length L=10T versus the applied lateral concentrated force Pg when the piles are pinned to the cap.

Force Pgi o f Pile A (2nd trailing row), Pile B (1st trailing row), Pile j 08 C (Leading row) o f group o f 3x3 piles with the spacing 5D and the length L=10T versus the applied lateral concentrated force Pg when the piles are pinned to the cap.

Method used to determine the force Pgi applied on the adjoint \Qg

structure and the temporary structures for the pile groups under lateral load Pg

Determination o f the bending moment Mg applied to the cap o f the \ \ \

piles pinned to the cap subjected to bending moment at the pile head. Pile group spacing “s” is equal to 2D, 3D, 4D and 5D. The pile length L is equal to 10T (23.5 m)

Method used to determine the force Pgi applied to the adjoint pile \ \ 4

group when the primary pile groups is loaded by bending moment Mg applied at the pile head o f members is a pile group

Flow chart o f sensitivity analysis performed for the laterally loaded m

piles (Part 1)

Flow chart o f sensitivity analysis performed for the laterally loaded j j g piles (Part 2)

MATLAB file “gendirectory.m” \ \ g

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Figure 7.5 Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 7.10 Figure 7.11 Figure 7.12 Figure 7.13 Figure 7.14 Figure 8.1 Figure 8.2 Figure 8.3

Figure 8.4

Figure B .l

Figure B.2

MATLAB sub function called “readdatal.m ” \2Q

Part o f the MATLAB file called “plotfig4pppl.m ” where a \2\

function is created to plot the data from COM624P analysis performed for primary and adjoint structure

Part o f the MATLAB file called “sananalysisl.m ” that performs \22

the actual sensitivity analysis, and utilizes two sub functions, simpsonquadl.m and plotbarl.m

MATLAB sub function called “plotbar.m” \23

MATLAB sub function called “simpsonquad.m” \23

MATLAB file “groupgendirectory.m” 1 2 5

Part o f the MATLAB file called MMgroupgeninputprocpdeltas.m” 1 2 5

that generates the input files producing Pg, and Mg versus lateral deflection

Part o f the MATLAB file called “MMgroupcalculate.m” that 1 2 6

calculates the input files produced by

“MMgroupgeninputprocpdeltas.m”

Part o f the MATLAB file called “M Mgroupcalculate.m” that 1 2 7 calculates the Pg and Mg and presentes them in a graphic format

Part o f the MATLAB file called “groureaddata.m” that is a sub 1 2 2

function that reads the data o f “MMgroupplotdeltas.m”

Primary structure - Pile head deflection yt versus lateral static load \29

Pt applied to the pile head - Free head - Length o f the pile L=10T

Adjoint Structure - Pile head deflection yt versus lateral static load j3 0 Pt applied to the pile head - Free head - Length o f the pile L=10T

The exact sensitivity o f lateral deflection expressed in (m) 147

caused by changes o f the design variables b when applied force Pt have values Pi

The pile head lateral deflection ytop versus the ration (c/co) o f 1 5 0 design variable c with respect to the initial value o f the design variable Co, for the case study with free head pile subjected to a concentrated lateral force Pt = 270 kN, where the pile length considered is equal L = 10T

Lateral deflection o f primary structure for free head pile under 1 7 9 variable lateral force - Pile length L = 4T

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Figure B.3

Figure B.4

Figure B.5

Figure B.6

Figure B.7

Figure B.8

Figure B.9

Figure B.10

Figure B . l l

Distribution o f lateral deflection y a (P) o f the adjoint structure 180

subjected to P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

Distribution o f bending moments M (P ) o f adjoint structure 180

subject to P = 1 when the primary structure is subject to variable lateral force Pt - Free head pile - Pile length L = 4T

Distribution o f soil reaction pa o f primary structure subjected to ^ 8 1 P = 1 when the primary structure is subjected to variable lateral

force Pt - Free head pile - Pile length L = 4T

Distribution o f soil reaction pa o f adjoint structure subject to jg i

P = 1 when the primary structure is subject to variable lateral force Pt - Free head pile - Pile length L = 4T

Distribution o f sensitivity operators affecting the changes o f * ^2

the pile head lateral deflection 8y^ due to the changes o f bending

stiffness El when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 4T

Distribution o f sensitivity operators C gy affecting the changes o f

the pile head lateral deflection 8y^ due to the changes o f cohesion

c when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 4T

Distribution o f sensitivity operators affecting the changes o f ^

Y

the pile head lateral deflection 8^ due to the changes o f the

submerged soil unit weight y' when the pile structure is subjected

to variable concentrated lateral force Pt - Free head pile - Pile length = 4T

Distribution o f sensitivity operators C^y affecting the changes o f ^ 3

the pile head lateral deflection 8yt due to the changes o f the pile’s

width b when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 4T

P 10 / 1

Distribution o f sensitivity operators C gJo affecting the changes o f

the pile head lateral deflection 8y^ due to the changes o f 8 50 when

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Figure B.12

Figure B.13

Figure B.14

Figure B.15

Figure B.16

Figure B.17

Figure B.18

Figure B.19

Figure B.20

The quantitative assessment o f sensitivity factor affecting the 184

top lateral deflection yt due to the changes o f bending stiffness El o f the pile when the pile is subject to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor A^y affecting the 185

top lateral deflection yt due to the changes o f cohesion c when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T

top lateral deflection yt due to the changes o f width b o f the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

top lateral deflection yt due to the changes o f the submerged soil unit weight y ' when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor A ^y affecting the 186

top lateral deflection yt due to the changes o f e50 when the pile is

subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity ]g7

factor F^ y affecting the top lateral deflection yt due to the changes

o f width b o f the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity j 3 7

factor F^y affecting the top lateral deflection yt due to the changes

o f cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor jgg

Fg^ affecting the top lateral deflection yt due to the changes of

bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor jgg F^y affecting the top lateral deflection yt due to the changes o f the

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Figure B.21

Figure C .l

Figure C.2

Figure C.3

Figure C.4

Figure C.5

Figure C.6

Figure C.7

Figure C.8

Figure C.9

The quantitative assessment (in %) o f relative sensitivity factor

affecting the top lateral deflection yt due to the changes of

£ 50 when the pile is subjected to variable concentrated lateral

force Pt - Free head pile - Pile length L = 4T

Lateral deflection o f primary structure for free head pile under 1 9 0 variable lateral force Pile length L = 5T

Distribution o f bending moments o f primary structure for free head 1 9 0 pile under variable lateral force - Pile length L = 5T

Distribution o f lateral deflections y a (P) o f the adjoint structure 191

subject to P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T

Distribution o f soil reaction pa o f primary structure subject to 1 9 2 P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T

Distribution o f soil reaction pa o f adjoint structure subject to P = 1 192 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T

p

Distribution o f sensitivity operators affecting the changes o f 193

the pile head lateral deflection 8 due to the changes o f bending

stiffness E l when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T

p

Distribution o f sensitivity operators C cy affecting the changes o f 193

the pile head lateral deflection 8 due to the changes o f cohesion c

when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T

p

Distribution o f sensitivity operators C y,y affecting the changes o f 194

the pile head lateral deflection 8 due to the changes o f the

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Figure C.10

Figure C . l l

Figure C.12 Figure C.13 Figure C.14 Figure C.15 Figure C.16 Figure C.17 Figure C.18 p

Distribution o f sensitivity operators C by affecting the changes o f 194

width b when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T

p

Distribution o f sensitivity operators C E^ affecting the changes o f 195

the pile head lateral deflection 8yt due to the changes o f e50 when

the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T

The quantitative assessment o f sensitivity factor A affecting the 195

top lateral deflection yt due to the changes o f bending stiffness El o f the pile when the pile is subject to variable concentrated lateral force Pt - Free head pile - Pile length L = 5T

The quantitative assessment o f sensitivity factor A Py affecting the 196

top lateral deflection yt due to the changes o f cohesion c when the pile structure are subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 5T

top lateral deflection yt due to the changes o f the submerged soil unit weight y ' when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T

The quantitative assessment (in %) o f relative sensitivity J9g factor Fby affecting the top lateral deflection yt due to the changes

o f width b o f the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T

The quantitative assessment ( in %) o f relative sensitivity J9g factor FcPy affecting the top lateral deflection yt due to the changes

(22)

Figure C.19

Figure C.20

Figure C.21

Figure D .l

Figure D.2

Figure D.3

Figure D.4

Figure D.5

Figure D.6

Figure D.7

The quantitative assessment (in %) o f relative sensitivity factor 1 9 9 Fgjy affecting the top lateral deflection yt due to the changes o f

The quantitative assessment (in %) o f relative sensitivity factor 1 9 9 Fyy affecting the top lateral deflection yt due to the changes o f the

submerged unit weight o f the soil y ' when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 5T

The quantitative assessment (in %) o f relative sensitivity factor 200 FePy affecting the top lateral deflection yt due to the changes o f S50

when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L =5T

Lateral deflection o f primary structure for free head pile under 201 variable lateral force Pile length L = 10T

Distribution o f bending moments o f primary structure for free head 201 pile under variable lateral force - Pile length L = 10T

Distribution o f lateral deflections y a (P ) o f the adjoint structure 2 0 2

Distribution o f bending moments M ( P ) o f adjoint structure 202

Distribution o f soil reaction pa o f primary structure subject to 203 P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f soil reaction pa o f adjoint structure subject to P = 1 203 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f sensitivity operators affecting the changes o f ^04

the pile head lateral deflection 8 due to the changes o f bending

(23)

Figure D.8

Figure D.9

Figure D.10

Figure D . l l

Figure D.12

Figure D.13

Figure D.14

Figure D.15

Distribution o f sensitivity operators C^y affecting the changes o f ^04

the pile head lateral deflection 8y( due to the changes o f cohesion c

when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T

the pile head lateral deflection 5 yt due to the changes o f the

to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

the pile head lateral deflection 8y( due to the changes o f the pile’s

width b when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

top lateral deflection yt due to the changes o f bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

top lateral deflection yt due to the changes o f cohesion c when the pile structure are subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment o f sensitivity factor A{jy affecting the 207

The quantitative assessment o f sensitivity factor A y,y affecting the 208

(24)

Figure D.16

Figure D.17

Figure D.18

Figure D.19

Figure D.20

Figure D.21

Figure D.22

Figure D.23

Figure D.24

top lateral deflection yt due to the changes o f 8 50 when the pile is

The quantitative assessment (in %) o f relative sensitivity 209

factor F Py affecting the top lateral deflection yt due to the changes

o f width b when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment ( in %) o f relative sensitivity 209 factor FcPy affecting the top lateral deflection yt due to the changes

o f cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor 210 FPjy affecting the top lateral deflection yt due to the changes o f

The quantitative assessment (in %) o f relative sensitivity factor 210 FPy affecting the top lateral deflection yt due to the changes o f the

submerged unit weight o f the soil y1 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor 211 FePy affecting the top lateral deflection yt due to the changes o f e50

when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f lateral deflections y a (M) o f the adjoint structure 211

subjected to M = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f bending moments M (M) o f the adjoint structure 212

subjected to M = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f the soil resistance p a o f the adjoint structure 212

(25)

Figure D.25

Figure D.26

Figure D.27

Figure D.28

Figure D.29

Figure D.30

Figure D.31

Figure D.32

the pile top angle o f flexural rotation 80t due to the changes o f the

bending stiffness El when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f sensitivity operators C^e affecting the changes o f ^13

the pile top angle o f flexural rotation 8 0 t due to the changes o f

cohesion c when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f sensitivity operators Cy° affecting the changes o f ^14

the pile top angle o f flexural rotation 80^. due to the changes o f the

to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f sensitivity operators C£9 affecting the changes o f ^14

the pile top angle o f flexural rotation 80t due to the changes o f the

pile’s width b when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Distribution o f sensitivity operators Cg® affecting the changes o f ^15

the pile top angle o f flexural rotation S0t due to the changes o f

e 50when the pile structure is subjected to variable concentrated

lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment o f sensitivity factor A™ affecting the 215

top angle o f flexural rotation 0t due to the changes o f e50 when

the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

top angle o f flexural rotation 0t due to the changes o f cohesion c

when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment o f sensitivity factor A96 affecting the 216

top angle o f flexural rotation 0t due to the changes o f the

(26)

Figure D.33

Figure D.34

Figure D.35

Figure D.36

Figure D.37

Figure D.38

Figure D.39

Figure E .l

Figure E.2

top angle o f flexural rotation 0t due to the changes o f width b o f

the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment o f sensitivity factor A^ 0 affecting the 217

top angle o f flexural rotation 0t due to the changes o f bending

stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor 218

D A

F8 ” affecting the top angle o f flexural rotation 0t due to the

changes o f 8 50 when the pile is subjected to variable concentrated

lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment ( in %) o f relative sensitivity 218 PA

factor Fg affecting the top angle o f flexural rotation 0t due to the

changes o f cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

F? 0 affecting the top angle o f flexural rotation 0t due to the

7 ^

changes o f the submerged unit weight o f the soil y1 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor F0 0 219

affecting the top angle o f flexural rotation 0t due to the changes o f

width b o f the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T

F0 0 affecting the top angle o f flexural rotation 0t due to the

changes o f bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T

Lateral deflection o f primary structure for fixed head pile under 221 variable lateral force Pile length L = 4T

(27)

Figure E.3

Figure E.4

Figure E.5

Figure E.6

Figure E.7

Figure E.8

Figure E.9

Figure E.10

Figure E . l l

subject to P = 1 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 4T

Distribution o f bending moments M ( P ) o f adjoint structure 222

Distribution o f soil reaction pa o f primary structure subject to 223 P = 1 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 4T

Distribution o f soil reaction pa o f adjoint structure subject to P = 1 223 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 4T

p

the pile head lateral deflection 8y( due to the changes o f bending

stiffness El when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

p

Distribution o f sensitivity operators

Ccy

affecting the changes o f 224 the pile head lateral deflection 8yj due to the changes o f cohesion c

when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

p

C y,y

affecting the changes o f

225

the pile head lateral deflection 8yt due to the changes o f the

to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

p

C by

225

the pile head lateral deflection 8 due to the changes o f the pile’s

width b when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

p

Cey

affecting the changes o f 226 the pile head lateral deflection 8 due to the changes o f e50 when

(28)

Figure E.12

Figure E.13

Figure E.14

Figure E.15

Figure E.16

Figure E.17

Figure E.18

Figure E.19

Figure E.20

top lateral deflection yt due to the changes o f bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

top lateral deflection yt due to the changes o f cohesion c when the pile structure are subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor A{jy affecting the 227

top lateral deflection yt due to the changes o f width b o f the pile when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor Ayy affecting the 228

top lateral deflection yt due to the changes o f the submerged soil unit weight y1 when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor Aj^y affecting the 228

subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T

factor F^y affecting the top lateral deflection yt due to the changes

o f width b o f the pile when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T

o f cohesion c when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor 230 Fgjy affecting the top lateral deflection yt due to the changes o f

bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor 230 F^y affecting the top lateral deflection y, due to the changes o f the

(29)

Figure E.21

Figure F .l

Figure F.2

Figure F.3

Figure F.4

Figure F.5

Figure F.6

Figure F.7

Figure F.8

Figure F.9

The quantitative assessment (in %) o f relative sensitivity factor 231 affecting the top lateral deflection yt due to the changes o f

th e e50 when the pile is subjected to variable concentrated lateral

force Pt - Fixed head pile - Pile length L = 4T

Distribution o f bending moments o f primary structure for fixed 232 head pile under variable lateral force - Pile length L = 5T

Distribution o f soil reaction pa o f primary structure subject to 234 P = 1 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 5T

Distribution o f soil reaction pa o f adjoint structure subject to P = 1 234

when the primary structure is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 5T

p

Distribution o f sensitivity operators C E^ affecting the changes o f 235

the pile head lateral deflection Syt due to the changes o f bending

stiffness E l when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

p

C cy

235

the pile head lateral deflection 8yt due to the changes o f cohesion c

p

Distribution o f sensitivity operators C Y.y affecting the changes o f 236

(30)

Figure F.10

Figure F . l l

Figure F.12 Figure F.13 Figure F.14 Figure F.15 Figure F.16 Figure F.17 Figure F.18 p

width b when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

p

the pile head lateral deflection Sy{ due to the changes o f e50 when

the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

The quantitative assessment o f sensitivity factor A by affecting the 238

top lateral deflection yt due to the changes o f the submerged soil unit weight y1 when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 5T

The quantitative assessment o f sensitivity factor A ^y affecting the 239

factor Fb y affecting the top lateral deflection yt due to the changes

(31)

Figure F.19

Figure F.20

Figure F.21

Figure G .l

Figure G.2

Figure G.3

Figure G.4

Figure G.5

Figure G.6

Figure G.7

The quantitative assessment (in %) o f relative sensitivity factor 241 FEIy affecting the top lateral deflection yt due to the changes o f

bending stiffness El o f the pile when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

The quantitative assessment (in %) o f relative sensitivity factor 241 FPy affecting the top lateral deflection yt due to the changes o f the

submerged unit weight o f the soil y1 when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

The quantitative assessment (in %) o f relative sensitivity factor 242 F Py affecting the top lateral deflection yt due to the changes o f S50

when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T

Distribution o f bending moments o f primary structure for fixed 243 head pile under variable lateral force - Pile length L = 10T

subject to P = 1 when the primary structure is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 10T

Distribution o f soil reaction pa o f primary structure subjected to 245 P = 1 when the primary structure is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 10T

Distribution o f soil reaction pa o f adjoint structure subject to P = 1 245 when the primary structure is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 10T

p

Distribution o f sensitivity operators C Ey affecting the changes o f 246

the pile head lateral deflection 8yt due to the changes o f bending

stiffness E l when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L =

(32)

Figure G.8

Figure G.9

Figure G.10

Figure G . l l

Figure G.12

Figure G.13

Figure G.14

Figure G.15

p

C by

246

the pile head lateral deflection Sy( due to the changes o f cohesion c

p

C y.y

247

to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 10T

p

C by

247

the pile head lateral deflection Syt due to the changes o f the pile’s

width b when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L =

10T

p

Distribution o f sensitivity operators C 6yQ affecting the changes o f 248

the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 10T

The quantitative assessment o f sensitivity factor A b| affecting the 248

The quantitative assessment o f sensitivity factor A y,y affecting the 250

(33)

Figure G.16

Figure G.17

Figure G.18

Figure G.19

Figure G.20

Figure G.21

Figure H .l

Figure H.2

Figure H.3

Figure H.4

factor FbPy affecting the top lateral deflection yt due to the changes

factor FcPy affecting the top lateral deflection yt due to the changes

o f cohesion c when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 10T

Fg^ affecting the top lateral deflection yt due to the changes o f

bending stiffness E l o f the pile when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor 252 F Py affecting the top lateral deflection yt due to the changes o f the

submerged unit weight o f the soil y ' when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 10T

The quantitative assessment (in %) o f relative sensitivity factor 253 F Py affecting the top lateral deflection yt due to the changes o f S50

when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 10T

Distribution o f lateral deflection o f primary structure for Free head 254 pile under variable bending moment Mt Pile length L = 4T

Distribution o f bending moments o f primary structure for Free head 254 pile under variable bending moment Mt - Pile length L = 4T

Distribution o f lateral deflections y a (M) o f the adjoint structure 255

subject to M = 1 when the primary structure is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

Distribution o f bending moments M (M ) o f adjoint structure 255

(34)

Figure H.5

Figure H.6

Figure H.7

Figure H.8

Figure H.9

Figure H.10

Figure H . l l

Figure H.12

Distribution o f soil reaction pa o f primary structure subject to 256 M = 1 when the primary structure is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

Distribution o f soil reaction pa o f adjoint structure subject to 256 M = 1 when the primary structure is subjected to variable bending moment M t - Free head pile - Pile length L = 4T

Distribution o f sensitivity operators C ^ e affecting the changes o f 257

the pile head lateral deflection 8y( due to the changes o f bending

stiffness El when the pile structure is subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

Distribution o f sensitivity operators C^ 10 affecting the changes o f 257

the pile head top flexural angle o f rotation 80t due to the changes o f cohesion c when the pile structure is subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

Distribution o f sensitivity operators C^f6 affecting the changes o f 258

the pile head top flexural angle o f rotation S0t due to the changes o f the submerged soil unit weight y' when the pile structure is

subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

Distribution o f sensitivity operators C ^ ° affecting the changes o f 258

the pile head top flexural angle o f rotation 80t due to the changes o f the pile’s width b when the pile structure is subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

the pile head top flexural angle o f rotation 80t due to the changes o f

850when the pile structure is subjected to variable concentrated

bending moment Mt - Free head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor A™8 affecting the 259

(35)

Figure H.13

Figure H.14

Figure H.15

Figure H.16

Figure H.17

Figure H.18

Figure H.19

Figure H.20

The quantitative assessment o f sensitivity factor A f9 affecting the 259

top flexural angle o f rotation 0t due to the changes o f cohesion c when the pile structure are subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

The quantitative assessment o f sensitivity factor A^10 affecting the 260

top flexural angle o f rotation 0t due to the changes o f width b o f the pile when the pile is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

top flexural angle o f rotation 0t due to the changes o f the submerged soil unit weight y ' when the pile is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

top flexural angle o f rotation 0t due to the changes o f e50 when the

pile is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor FbM9 261

affecting the top flexural angle o f rotation 0t due to the changes o f width b o f the pile when the pile is subjected to variable bending moment M t - Free head pile - Pile length L = 4T

factor Fc 9 affecting the top flexural angle o f rotation 0t due to the

changes o f cohesion c when the pile is subjected to variable bending moment Mt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor 262 F^° affecting the top flexural angle o f rotation 0t due to the

changes o f bending stiffness El o f the pile when the pile is subjected to variable concentrated bending moment Mt - Free head pile - Pile length L = 4T

The quantitative assessment (in %) o f relative sensitivity factor 263 F^ 9 affecting the top flexural angle o f rotation 0t due to the

Investigations of nonlinear p-y piles and pile groups in soft clay subjected to static loading-distributed parameter sensitivity analysis.

Scholarship at UWindsor

Scholarship at UWindsor

Investigations of nonlinear p-y piles and pile groups in soft clay

Investigations of nonlinear p-y piles and pile groups in soft clay

subjected to static loading-distributed parameter sensitivity

subjected to static loading-distributed parameter sensitivity

analysis.

analysis.

INVESTIGATIONS OF NONLINEAR

p-y

PILES AND

PILE GROUPS IN SOFT CLAY SUBJECTED TO

STATIC LOADING-DISTRIBUTED PARAMETER

SENSITIVITY ANALYSIS

BY

MARCIA REGINA MORA

A Thesis

Submitted to Faculty of Graduate Studies and Research

through Civil and Environmental Engineering in Partial

Fulfillment of the Requirements for the Degree of Master of

Applied Science at the University of Windsor

W indsor, O ntario, C anada

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