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

Transcription

1 University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations 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: 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 This online database contains the full-text of PhD dissertations and Masters theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via or by telephone at ext

2 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, Ontario, Canada 2006

3 1*1 Library and Archives Canada Published Heritage Branch 395 Wellington Street Ottawa ON K1A0N4 Canada Bibliotheque et Archives Canada Direction du Patrimoine de I'edition 395, rue Wellington Ottawa ON K1A0N4 Canada Your file Votre reference ISBN: Our file Notre reference ISBN: NOTICE: The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. AVIS: L'auteur a accorde une licence non exclusive permettant a la Bibliotheque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par Nntemet, preter, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. L'auteur conserve la propriete du droit d'auteur et des droits moraux qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette these. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. i * i Canada

4 MARCIA REGINA MORA 2006 All Rights Reserved

5 ABSTRACT For the design of 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 of 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 of sensitivity analysis. In this study both single piles and pile groups are analyzed under the sensitivity approach. A wide range of lengths and a large group of different boundary conditions were applied on single piles under static loadings, addressed to the pile head. In terms of 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 of the pile is directly dependent on the soil response and in this case the soil response is a nonlinear function of pile deflection and the depth of 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 of the pile. Under the broad variation of the boundary conditions the group of 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 of the pile-soil system. For the case of a group of piles, the utilization of a specific modifierfactor had to be applied once the group of piles change with respect to spacing of the piles and also the location of the piles inside of the group. The physical parameters of the soft clay and the stiffness of the pile used for the description of the p-y relationships are taken as the design variables of the continuous type. They are space dependent. To be able to analyze the sensitivity of the single pile and the group of pile subjected to lateral loads the adjoint method for nonlinear system is used. The first variation of the IV

6 functional of maximum deformations as a result of the changes of the design variables is formulated based on the virtual work principle. The variations of maximum generalized deflection located at the pile head as a result of the changes of 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 of the deformation of the pile soil system each one as expected had demonstrated particular differences and significance through the analysis, and theses will be also part of the final discussion. v

7 Dedicated to my parents Alba and Francisco Mora vi

8 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 of 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 of friends that helped me through this work with their time, their technical support, and most of 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 Mora. Many thanks!!!! vii

9 TABLE OF CONTENTS ABSTRACT DEDICATION ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES iv vi vii xii xiii CHAPTER 1 INTRODUCTION Problem Statement Objectives Procedures Methodology and significance o f sensitivity analysis Study organization 4 CHAPTER 2 LITERATURE REVIEW Methods o f analysis o f laterally loaded single piles General Models for use in analysis o f a single piles Sub grade-reaction model Elastic pile and elastic soil Elastic pile and finite element for soil Rigid pile and plastic soil Characteristic load method Nonlinear approach for pile and p-y model for soil Field testing performed over piles under lateral loading in soft 14 clay 2.2 Methods of analysis o f laterally loaded piles groups Overview Simple static analysis method Equivalent beam method Elastic continuum analysis o f pile behaviour Group reduction factor method Group amplification method The p-multipliers method 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 Classification o f sensitivity methods Sensitivity analysis applied previously 33 CHAPTER 3 THEORETICAL FORMULATION Single Piles General Pile and soft clay p-y curve for soft clay p-y curve theory for soft clay p-y curve theory for soft clay below the water table 50

10 3.1.3 Theoretical formulation of sensitivity analysis of laterally 54 loaded piles embedded in soft clay below the water table Introduction Primary and adjoint structure Sensitivity operators and factors 58 CHAPTER 4 THEORETICAL FORMULATION OF PILE GROUP Overview Laterally loaded pile group 72 CHAPTER 5 SINGLE PILE - NUMERICAL INVESTIGATION General Load and constraints type Determination o f the design parameters Soil properties Pile s physical properties Determination o f the piles length Determination o f relative stiffness factor T Long and short piles - Considerations Load-deflection relationship The adjoint structure concept - stresses and deformations COM624P - Computer program Results o f sensitivity analysis o f laterally loaded pile Results o f single piles - Method o f verification 96 CHAPTER 6 PILE GROUP - NUMERICAL INVESTIGATION Introduction Loads and constrains Determination of lateral forces Pg and Pgi - Lateral force used 101 in study lease and Primary structure Adjoint structure The determination of bending moments Mg and Mgi bending 109 moments used in study 3 case Primary structure Adjoint structure Results of the sensitivity analysis of the laterally loaded piles 115 groups CHAPTER 7 PROGRAMMING OF SENSITIVITY ANALYSIS FOR THE 116 PILE GROUPS 7.1 Overview Perform sensitivity analysis on single piles Perform sensitivity analysis on pile groups 124 CHAPTER 8 DISCUSSION Introduction General discussion on the sensitivity analysis results Discussion on the lateral deflections and bending moments of 129 the primary structure Discussion on the lateral deflections and bending moments of the adjoint structure 129

11 CHAPTER 9 p Discussion o f the sensitivity operators C^.y. ^ Discussion on the lateral deflections and bending moments o f 135 the adjoint structure subjected to bending moment M = 1 p Discussion o f the sensitivity operators C^e# ^ Discussion on the quantitative assessment o f the sensitivity 138 factors Discussion on the relative sensitivity factors F Error analysis Quantitative estimation o f the sensitivity factors A Assessment o f error o f lateral deflection based on comparative 149 analysis o f exact solution and sensitivity analysis solution CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 9.1 Conclusions The application o f this study Recommendations for future work 160 REFERENCES 162 APPENDIX A 169 Derivations of formulas of sensitivity operators APPENDIX B 179 Sensitivity analysis of top lateral displacement 6yt for single free head piles with length L=4T subjected to lateral concentrated forces APPENDIX C 190 Sensitivity analysis of top lateral displacement 8yt for single free head piles with length L=5T subjected to lateral concentrated forces APPENDIX D 201 Sensitivity analysis of top lateral displacement 8yt for single free head piles with length L=10T subjected to lateral concentrated forces APPENDIX E 221 Sensitivity analysis of top lateral displacement 8ye for single fixed head piles with length L=4T subjected to lateral concentrated forces APPENDIX F 232 Sensitivity analysis of top lateral displacement 8ye for single fixed head piles with length L=5T subjected to lateral concentrated forces APPENDIX G 243 Sensitivity analysis of top lateral displacement Sye for single fixed head piles with length L=10T subjected to lateral concentrated forces APPENDIX H 254 Sensitivity analysis of top lateral displacement 8y0 for single free head piles with length L=4T under bending moment APPENDIX I 265 Sensitivity analysis of top lateral displacement Sye for single free head piles with length L=5T under bending moment 276 x

12 APPENDIX J Sensitivity analysis of 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 of the pile group, with pile members pinned to the pile cap, and with pile spacing between piles equal 2D. Length of 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 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 L.2.2 Output file from FB-Pier - version 3 (just part) 311 xi

13 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 of 50 (after Wang and Reese 1993) 78 Table 5.2 Values of 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 of 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 Table 9.4 The average values o f the relative sensitivity factors F for all the 156 piles subjected to both the lateral concentrated load and bending moment embedded in soft clay below the ground water table xii

14 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)) 17 The concept of p-multiplier (fm) (Brown etal. 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) of 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 of 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 55 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 A primary structure subjected to a bending moment Mt and the 5 7 corresponding adjoint structure subjected to a lateral unit force P = 1 applied at the pile top xiii

15 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 of the normalized sensitivity 53 integrands/operators C(.) and sensitivity factors A(...),after Liu (2004) Convention for the notation of sensitivity operators/integrands and 5 7 sensitivity factors, after Liu (2004) Numerical integration using Simpson s rule 5 9 A typical view of the pile group system under a lateral load Pg 71 Nomenclature used to describe pile group arrangements 7 3 The p-multiplier design curves proposed by Mokwa and Duncan 7 4 ( b) 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 loading (after Evans and Duncan 1982) Moment-deformation curves for Free Head Pile in clay - static loading (after Evans and Duncan 1982) Load-moment relationships curves for Free Head Pile in clay - static loading (after Evans and Duncan 1982) Load-moment curves for Fixed Head Pile in clay - static loading (after Evans and Duncan 1982) Pile head deflection yt versus lateral force Pt applied to the top of 91 the pile head of 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 of 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 of the pile head for a free head pile embedded in a soft clay below water table. Pile length L = varies from 3T to 10T The method used to calculate various components o f internal forces 9 4 of the adjoint structure when subjected to unit generalized load, according to Liu (2004) g4 g5 g6 g7 xiv

16 Figure 6.1 Pile group geometry used in the pile group analysis jqq 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 Determination of the force Pg applied to the cap of 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 of Pile A (2nd trailing row), Pile B (1st trailing row), Pile C (Leading row) of group of 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 of Pile A (2nd trailing row), Pile B (1st trailing row), Pile jo 6 C (Leading row) of group of 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 of Pile A (2nd trailing row), Pile B (1st trailing row), Pile C (Leading row) of group of 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 of Pile A (2nd trailing row), Pile B (1st trailing row), Pile j 08 C (Leading row) of group of 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 structure and the temporary structures for the pile groups under lateral load Pg Determination of the bending moment Mg applied to the cap of 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 of members is a pile group Flow chart of sensitivity analysis performed for the laterally loaded piles (Part 1) Flow chart of sensitivity analysis performed for the laterally loaded piles (Part 2) MATLAB file gendirectory.m Part of the MATLAB file called calculate l.m where a function is 120 created to read the data from COM624P analysis performed on primary and adjoint structure ^ jq5 iq7 \Qg m j j g \\g xv

17 Figure 7.5 MATLAB sub function called readdatal.m \ 2Q 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 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 125 Part of the MATLAB file called MMgroupgeninputprocpdeltas.m 125 that generates the input files producing Pg, and Mg versus lateral deflection Part o f the MATLAB file called MMgroupcalculate.m that calculates the input files produced by MMgroupgeninputprocpdeltas.m Part o f the MATLAB file called MMgroupcalculate.m that calculates the Pg and Mg and presentes them in a graphic format Part of the MATLAB file called groureaddata.m that is a sub 122 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 of the pile L=10T Adjoint Structure - Pile head deflection yt versus lateral static load j 3 0 Pt applied to the pile head - Free head - Length of the pile L=10T The exact sensitivity o f lateral deflection expressed in (m) 147 caused by changes of the design variables b when applied force Pt have values Pi The pile head lateral deflection ytop versus the ration (c/co) of design variable c with respect to the initial value of 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 variable lateral force - Pile length L = 4T Distribution of bending moments of primary structure for free head pile under variable lateral force - Pile length L = 4T xvi

18 Figure B.3 Figure B.4 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 Figure B.5 Figure B.6 Distribution of soil reaction pa of primary structure subjected to P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T Distribution of soil reaction pa of adjoint structure subject to P = 1 when the primary structure is subject to variable lateral force Pt - Free head pile - Pile length L = 4T ^81 jg i Figure B.7 Distribution o f sensitivity operators affecting the changes o f * ^2 the pile head lateral deflection 8 y^ due to the changes of bending stiffness El when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 4T Figure B.8 Distribution of sensitivity operators C gy affecting the changes of the pile head lateral deflection 8 y^ 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 Figure B.9 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 Figure B.10 Distribution of sensitivity operators C^y affecting the changes of ^ 3 the pile head lateral deflection 8 yt 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 Figure B.ll P 10/1 Distribution o f sensitivity operators C gjo affecting the changes o f the pile head lateral deflection 8 y^ due to the changes of 8 50 when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 4T xvii

19 Figure B.12 Figure B.13 Figure B.14 Figure B.15 The quantitative assessment of sensitivity factor affecting the 184 top lateral deflection yt due to the changes of bending stiffness El of 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 of cohesion c when the pile is subjected 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 of width b of the pile 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 of the submerged soil unit weight y ' when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T Figure B.16 The quantitative assessment o f sensitivity factor A ^y affecting the 186 top lateral deflection yt due to the changes of e50 when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T Figure B.17 Figure B.18 The quantitative assessment (in %) o f relative sensitivity ]g7 factor F^ y affecting the top lateral deflection yt due to the changes of width b of 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 of cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 4T Figure B.19 Figure B.20 The quantitative assessment (in %) of relative sensitivity factor Fg^ affecting the top lateral deflection yt due to the changes of bending stiffness El of the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T The quantitative assessment (in %) of relative sensitivity factor F^y affecting the top lateral deflection yt due to the changes of the submerged unit weight of the soil y 1 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T jgg jgg xviii

20 Figure B.21 The quantitative assessment (in %) of 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 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 Lateral deflection of primary structure for free head pile under variable lateral force Pile length L = 5T Distribution of bending moments of primary structure for free head pile under variable lateral force - Pile length L = 5T Distribution of lateral deflections ya (P) of 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 bending moments M (P) o f 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 P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T Distribution of soil reaction pa of adjoint structure subject to P = when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T p Distribution of sensitivity operators affecting the changes of 193 the pile head lateral deflection 8 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 = 5T p Distribution of sensitivity operators C cy affecting the changes of 193 the pile head lateral deflection 8 due to the changes of cohesion c when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T p Distribution of sensitivity operators C y,y affecting the changes of 194 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 = 5T

21 Figure C.10 p Distribution of sensitivity operators C by affecting the changes of 194 the pile head lateral deflection 8 yt 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 = 5T Figure C.ll p Distribution of sensitivity operators C E^ affecting the changes of 195 the pile head lateral deflection 8 yt due to the changes o fe 50 when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length = 5T Figure C.12 Figure C.13 Figure C.14 The quantitative assessment o f sensitivity factor A affecting the 195 top lateral deflection yt due to the changes of bending stiffness El of 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 of cohesion c when the pile structure are subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 5T The quantitative assessment o f sensitivity factor A^y affecting the 196 top lateral deflection yt due to the changes of width b of the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T Figure C.15 The quantitative assessment o f sensitivity factor A Py affecting the 197 top lateral deflection yt due to the changes of the submerged soil unit weight y ' when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T Figure C.16 The quantitative assessment o f sensitivity factor A Py affecting the 197 top lateral deflection yt due to the changes of e50 when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T Figure C.17 Figure C.18 The quantitative assessment (in %) o f relative sensitivity J9 g factor Fby affecting the top lateral deflection yt due to the changes of width b of 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 J9 g factor FcPy affecting the top lateral deflection yt due to the changes of cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 5T x x

22 Figure C.19 Figure C.20 The quantitative assessment (in %) of relative sensitivity factor Fgjy affecting the top lateral deflection yt due to the changes of bending stiffness El of the pile 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 Fyy affecting the top lateral deflection yt due to the changes o f the submerged unit weight of the soil y ' when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 5T Figure C.21 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 Figure D.l Figure D.2 Figure D.3 Lateral deflection of 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 of lateral deflections y a (P) o f the adjoint structure subject to P = 1 when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T Figure D.4 Figure D.5 Figure D.6 Figure D.7 Distribution of bending moments M(P) of adjoint structure 202 subject to 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 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 = when the primary structure is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T Distribution of sensitivity operators affecting the changes of ^04 the pile head lateral deflection 8 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 L = 10T

23 Figure D.8 Figure D.9 Distribution of sensitivity operators C^y affecting the changes of the pile head lateral deflection 8 y( due to the changes of cohesion c when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 4T Distribution of sensitivity operators C^y affecting the changes of the pile head lateral deflection 5 yt due to the changes o f the ^04 ^05 Figure D.10 Figure D.ll submerged soil unit weight y' when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Distribution of sensitivity operators C^y affecting the changes of the pile head lateral deflection 8 y( 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 affecting the changes o f ^06 ^05 the pile head lateral deflection 8 yt due to the changes of e50 when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Figure D.12 Figure D.13 Figure D.14 Figure D.15 The quantitative assessment o f sensitivity factor affecting the 206 top lateral deflection yt due to the changes of bending stiffness El of the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T The quantitative assessment o f sensitivity factor A^y affecting the 207 top lateral deflection yt due to the changes of 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 top lateral deflection yt due to the changes of width b of the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T The quantitative assessment of sensitivity factor A y,y affecting the 208 top lateral deflection yt due to the changes of the submerged soil unit weight y 1 when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T xxii

24 Figure D.16 The quantitative assessment of sensitivity factor affecting the 208 top lateral deflection yt due to the changes of 8 50 when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T Figure D.17 Figure D.18 Figure D.19 Figure D.20 The quantitative assessment (in %) o f relative sensitivity 209 factor FPy affecting the top lateral deflection yt due to the changes of 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 of cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T The quantitative assessment (in %) of relative sensitivity factor 210 FPjy affecting the top lateral deflection yt due to the changes of bending stiffness El of 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 210 FPy affecting the top lateral deflection yt due to the changes of the submerged unit weight of the soil y 1 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Figure D.21 The quantitative assessment (in %) o f relative sensitivity factor 211 FePy affecting the top lateral deflection yt due to the changes of e 50 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Figure D.22 Distribution of lateral deflections ya (M) of 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 Figure D.23 Figure D.24 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 pa 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 xxiii

25 Figure D.25 Distribution o f sensitivity operators affecting the changes o f ^13 the pile top angle of flexural rotation 8 0 t due to the changes of the Figure D.26 Figure D.27 Figure D.28 Figure D.29 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 the pile top angle o f flexural rotation 8 0 t due to the changes of cohesion c when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Distribution of sensitivity operators Cy affecting the changes of the pile top angle of flexural rotation 8 0 ^. due to the changes of the submerged soil unit weight y' 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 9 affecting the changes o f the pile top angle of flexural rotation 8 0 t due to the changes of 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 the pile top angle of flexural rotation S 0 t due to the changes of e50when the pile structure is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T ^13 ^14 ^14 ^15 Figure D.30 The quantitative assessment o f sensitivity factor A affecting the 215 top angle of flexural rotation 0 t due to the changes of e 50 when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T Figure D.31 The quantitative assessment o f sensitivity factor A affecting the 216 top angle of flexural rotation 0 t due to the changes of cohesion c when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T Figure D.32 The quantitative assessment o f sensitivity factor A 96 affecting the 216 top angle o f flexural rotation 0 t 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 = 10T xxiv

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 The quantitative assessment o f sensitivity factor A affecting the 217 top angle of flexural rotation 0 t due to the changes of width b of 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 0 t due to the changes o f bending stiffness El of 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 0 t due to the changes of 8 50 when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T The quantitative assessment ( in %) of relative sensitivity 218 PA factor Fg affecting the top angle o f flexural rotation 0 t due to the changes of 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 219 F? 0 affecting the top angle of flexural rotation 0 t due to the 7 ^ changes of the submerged unit weight of the soil y 1 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 F affecting the top angle of flexural rotation 0 t due to the changes of width b of the pile when the pile is subjected to variable lateral force Pt - Free head pile - Pile length L = 10T The quantitative assessment (in %) of relative sensitivity factor 220 F0 0 affecting the top angle o f flexural rotation 0t due to the changes of bending stiffness El of the pile when the pile is subjected to variable concentrated lateral force Pt - Free head pile - Pile length L = 10T Lateral deflection of primary structure for fixed head pile under 221 variable lateral force Pile length L = 4T Distribution of bending moments of primary structure for fixed 221 head pile under variable lateral force - Pile length L = 4T xxv

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.ll Distribution of lateral deflections ya (P) of the adjoint structure 222 subject to P = 1 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 4T Distribution of bending moments M(P) of adjoint structure 222 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 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 = when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 4T p Distribution of sensitivity operators affecting the changes of 224 the pile head lateral deflection 8 y( due to the changes of bending stiffness El when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T p Distribution of sensitivity operators Ccy affecting the changes of 224 the pile head lateral deflection 8 yj due to the changes of cohesion c when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T p Distribution of sensitivity operators Cy,y affecting the changes of 225 the pile head lateral deflection 8 yt due to the changes o f the submerged soil unit weight y' when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T p Distribution of sensitivity operators Cby affecting the changes of 225 the pile head lateral deflection 8 due to the changes of 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 Distribution o f sensitivity operators Cey affecting the changes o f 226 the pile head lateral deflection 8 due to the changes of e 50 when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T xxvi

28 Figure E.12 Figure E.13 Figure E.14 Figure E.15 The quantitative assessment of sensitivity factor affecting the 226 top lateral deflection yt due to the changes of bending stiffness El of the pile when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T The quantitative assessment of sensitivity factor A^y affecting the 227 top lateral deflection yt due to the changes of 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 of width b of 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 of the submerged soil unit weight y 1 when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T Figure E.16 The quantitative assessment o f sensitivity factor Aj^y affecting the 228 top lateral deflection yt due to the changes of 8 50 when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T Figure E.17 Figure E.18 Figure E.19 Figure E.20 The quantitative assessment (in %) o f relative sensitivity 229 factor F^y affecting the top lateral deflection yt due to the changes of width b of the pile 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 229 factor FcPy affecting the top lateral deflection yt due to the changes of cohesion c when the pile is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 4T The quantitative assessment (in %) of 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 of the submerged unit weight of the soil y 1 when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T xxvii

29 Figure E.21 The quantitative assessment (in %) o f relative sensitivity factor 231 affecting the top lateral deflection yt due to the changes o f thee50 when the pile is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 4T Figure F.l Figure F.2 Figure F.3 Lateral deflection of primary structure for fixed head pile under 232 variable lateral force Pile length L = 5T 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 lateral deflections y a (P) o f the adjoint structure 233 subject to P = 1 when the primary structure is subject to variable lateral force Pt - Fixed head pile - Pile length L = 5T Figure F.4 Figure F.5 Figure F.6 Figure F.7 Figure F.8 Figure F.9 Distribution o f bending moments M (P) o f adjoint structure 233 subject to 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 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 of soil reaction pa o f adjoint structure subject to P = when the primary structure is subjected to variable lateral force Pt - Fixed head pile - Pile length L = 5T p Distribution of sensitivity operators C E^ affecting the changes of 235 the pile head lateral deflection Syt 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 = 5T p Distribution of sensitivity operators Ccy affecting the changes of 235 the pile head lateral deflection 8 yt due to the changes of cohesion c 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 C Y.y affecting the changes o f 236 the pile head lateral deflection 8 yt due to the changes o f the submerged soil unit weight y' when the pile structure is subjected to variable concentrated lateral force Pt - Fixed head pile - Pile length L = 5T xxviii