foundation may provide compared to a conventional pile foundation, ...... Simplified representation of piled raft unit as defined by Randolph..............44. 2.24.
Numerical Modelling of Vertically Loaded Piled Raft Foundation
by Amr Ahmed Bakry Hemaida, B.Sc., M.Sc.
A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In Civil Engineering (Structural)
FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2007
Numerical Modelling of Vertically Loaded Piled Raft Foundation
by Amr Ahmed Bakry Hemaida, B.Sc., M.Sc.
A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In Civil Engineering (Structural)
Under the supervision of Prof. Dr. Adel Yehia Akl Professor of analysis and mechanics of structures Structural Engineering Departement Faculty of Engineering, Cairo University
FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2007
ii
Numerical Modelling of Vertically Loaded Piled Raft Foundation by Amr Ahmed Bakry Hemaida, B.Sc., M.Sc.
A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In Civil Engineering (Structural)
Approved by the Examining Committee: Prof. Dr. Adel Yahia Akl
Thesis Main Supervisor
Prof. Dr. Abdel-Rahman Sadek Bazaraa
Member
Prof. Dr. Nadia Shenouda Guirguis
Member
FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2007
iii
DEDICATION
TO
MY PARENTS, MY WIFE & SONS
iv
ACKNOLEDGMENTS
The author wishes to express his sincere appreciation and gratitude to Prof. Dr. Adel Yehia Akl, who advised and guided him during the course of the study.
Deep thanks are due to Dr. Tarek Thabet Abdel-Fattah who provided valuable suggestions and useful advices which helped make this work possible.
Appreciation is also extended to Prof. Dr. Amira Abdel-Rahman, Prof. Dr. Rawia El-Sakhawy, Prof. Dr. Khalid El-Zahaby, Prof. Dr. Khadiga Abdelghani, Prof. Dr. Suzan Sad, and Dr. Hisham Amin of the Housing and Building National Research Centre, for their continuous encouragement.
Words are not enough to express the role played by the author's family. The author proudly dedicates this work to his family members to express his appreciation for their role.
Last but not least, the author wishes to express his deep appreciation to his friend Mr. Mohamed Farid for his support and willingness to help at any time.
v
ABSTRACT In the last few decades the concept of piled raft foundation became popular in practical use, as it provides economical solution for the circumstances where the raft alone can not satisfy design requirements. In spite of the economy that a piled raft foundation may provide compared to a conventional pile foundation, it represents a complex foundation system that needs qualified understanding of the different soil structure interactions involved in such type of foundation. The nature of the piled raft foundation system dictates that a suitable method of analysis should be a three dimensional method (Poulos et al., 1997, El-Mossallamy, 2000, Reul and Randolph, 2003 and others). The present work aims primarily at the detailed investigation of the three dimensional finite element modelling aspects of the elements of piled raft foundation system which are; raft, piles and supporting soil. Various alternatives of the material constitutive modelling, geometrical modelling as well as soil structure interface modelling, are studied through out a series of two and three dimensional finite element test problems. All the finite element analyses through out the current research are carried out using the general purpose program DIANA® 9.1 (TNO DIANA BV. 2005). Illustrative comparisons are presented for the results of analyses. A proper modelling procedure for the problem in hand was presented and verified. Some practical techniques for reducing the computational demands associated with large three dimensional piled raft problems were suggested, so that large problems can be handled using personal computers in reasonable CPU time and without violating the solution accuracy. These techniques which include employment of three dimensional interface elements for modeling of far field bottom soil layers as well as a simplified method for handling secondary lateral loads, are presented and verified. For calculation of pile loads from the internal stresses output from the general purpose program DIANA® , a special purpose finite element program PILELO has been adopted by modifying the available finite element code DIATUN developed by Abdel-Fattah (2004). The use of this program is verified through solving a three dimensional test problem of a uniformly loaded single pile. Extensive research program has been conducted to study the behavior and performance of piled raft foundation. The finite element elasto-plastic analyses were
vi
performed on a hypothetical square piled raft foundation in different structural and geotechnical conditions. In this study, the soil stratification, number of piles, pile position and pile length were varied. The effect of the above parameters on loadaverage settlement behavior, load-differential settlement behavior, raft contact pressure (part of the applied load taken by the raft), raft bending moments and behavior of individual piles beneath the raft as well as the distribution of load among piles within the group, is presented. The results of analyses were compared to that of corresponding single pile, conventional pile group and unpiled raft. Finally, two well documented case histories of piled raft foundation in stiff over consolidated clay were simulated using the proposed finite element modelling. The obtained results are verified against the field measurements recorded during and after the construction of the original buildings. Also these results are compared to available results obtained numerically by other researchers for those problems. The results obtained compare well with their observed counterparts, indicating the suitability of the present finite element analysis for handling such problems.
vii
TABLE OF CONTENTS Page DEDICATION.......................................................................................... iv ACKNOWLEDGMENT ..................................................................................... v ABSTRACT ..............................................................................................................vi TALE OF CONTENTS...................................................................................... viii LIST OF TABLES ...............................................................................................xiv LIST OF FIGURES ..............................................................................................xv CHAPTER 1
INTRODUCTION
1.1 General......................................................................................................................1 1.2 Background ...............................................................................................................2 1.3 Objectives of the research.........................................................................................3 1.4 Outlines of the research ............................................................................................3
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction...............................................................................................................5 2.2 Load response of axially loaded single pile and pile group......................................5 2.2.1 General............................................................................................................5 2.2.2 Methods based on theory of elasticity ............................................................6 2.2.2.1 Load response of single pile ...............................................................6 2.2.2.2 Load response of pile groups............................................................10 2.2.3 Original load transfer method .......................................................................12 2.2.4 Theoretical load transfer curves....................................................................14 2.2.5 Numerical methods .......................................................................................18 viii
2.2.5.1 Plane strain finite element analysis ..................................................18 2.2.5.2 Axisymmetric finite element analysis ..............................................19 2.2.5.3 Three dimensional finite element analysis .......................................22 2.3 Alternative design philosophies of piled raft..........................................................23 2.3.1 General..........................................................................................................23 2.3.2 The Conventional approach philosophy .......................................................25 2.3.3 The Creep piling philosophy.........................................................................27 2.3.4 Differential settlement control philosophy (some times called optimizing pile-raft analysis) ...................................................................................................29 2.4 Some aspects of piled raft behaviour .....................................................................33 2.4.1 Different interactions affecting behaviour of piled rafts ..............................33 2.4.2 Factors affecting behaviour of piled rafts .....................................................35 2.4.3 Coefficients quantifying performance of piled rafts.....................................39 2.5 Methods of analysis of piled raft foundation ..........................................................40 2.5.1 Simplified calculation methods ....................................................................40 2.5.1.1 Poulos and Davis, (1980)'s method ..................................................41 2.5.1.2 Randolph, (1994)'s method (flexibility matrix method)...................41 2.5.1.3 Burland's approach ...........................................................................45 2.5.1.4 Equivalent raft and equivalent pier methods ....................................45 2.5.2 Approximate computer-based methods ........................................................48 2.5.2.1 Methods employing a strip on springs approach..............................48 2.5.2.2 Methods employing a plate on springs approach .............................49 2.5.2.3 Method developed by Clancy and Randolph, (1993) .......................49 2.5.3 More rigorous computer based methods.......................................................50 2.5.3.1 Full boundary element method proposed by Butterfield and Banerjee, (1971) .................................................................................... 51 2.5.3.2 Mixed boundary and finite element methods proposed by ElMossallamy and Franke, (1997)...................................................................51 2.5.3.3 Finite element method ......................................................................52
ix
2.5.4 Comparison of some methods of analysis of piled rafts. ..............................56
CHAPTER 3 MAIN
ASPECTS
OF
FINITE
ELEMENT
MODELLING OF VERTICALLY LOADED PILED RAFTS 3.1 Introduction........................................................................................................... 58 3.2 Fundamentals of finite element modelling of piled raft foundation system ...... 58 3.2.1 General..........................................................................................................58 3.2.2 Modelling of foundation elements (raft, piles) ............................................ 59 3.2.3 Modelling of supporting soil ........................................................................60 3.2.3.1 General..............................................................................................60 3.2.3.2 Theoretical background of selected soil constitutive models...........60 3.2.3.2.1 Mohr-Coulomb model .......................................................60 3.2.3.2.2 Modified Mohr-Coulomb model .......................................62 3.2.3.2.3 Enhanced Delft Clay model...............................................64 3.2.3.3 Evaluation of the selected soil constitutive models..........................66 3.2.3.3.1 Test problem 1: Back analysis of monitored pile load test in stiff clay ............................................................................................66 3.2.3.3.2 Test problem 2: Back analysis of monitored pile load test in non cohesive soil. ............................................................................68 3.2.3.3.3 Test problem 3: Analysis of piled raft foundation in stiff clay....................................................................................................71 3.2.4 Modelling of soil structure interface.............................................................73 3.2.4.1 General..............................................................................................73 3.2.4.2 The importance of using interface elements in the analysis of piled rafts...............................................................................................................74 3.2.4.3 Characteristics of the Coulomb friction model incorporated in DIANA program...........................................................................................77 3.2.4.4 Material parameters for interface modeling .....................................78 3.2.4.5 Geometric definition of interface elements ......................................79
x
3.2.4.6 Using soil-soil interface elements ....................................................80 3.2.5 Calculation of pile load from the out put stresses of the finite element analyses .................................................................................... 81 3.3 Suggested practical techniques for handling of large three dimensional finite element problems of piled raft foundation system........................................................81 3.3.1 General..........................................................................................................81 3.3.2 Employment of three dimensional interface elements for modelling of far field bottom soil layer - A proposed technique ....................................................82 3.3.2.1 Description of the proposed technique .............................................82 3.3.2.2 Verification of the proposed method ................................................83 3.3.3 Modelling of raft using shell elements .........................................................86 3.4 Simplified method for solving piled raft foundations subjected to lateral loading ..............................................................................................................88 3.4.1 Description of the proposed simplified method............................................88 3.4.2 Verification of the proposed method ............................................................89
CHAPTER 4 NUMERICAL INVESTIGATION OF PILED RAFT FOUNDATION BEHAVIOUR 4.1 Introduction.............................................................................................................93 4.2 Subsoil conditions...................................................................................................93 4.3 Geometric and material properties of the foundation elements (raft and piles) .....95 4.4 Soil structure interface modeling ............................................................................98 4.5 Methodology ...........................................................................................................99 4.6 Behaviour of un piled rafts, piles and conventional pile groups ..........................100 4.6.1 Unpiled raft .................................................................................................100 4.6.2 Single piles..................................................................................................104 4.6.3 Conventional pile group..............................................................................106 4.7 Behaviour of piled raft foundation........................................................................106 4.8 Results of analyses................................................................................................110 4.8.1 Load settlement behaviour of piled raft......................................................110
xi
4.8.1.1 Load - average settlement behaviour of piled raft..........................110 4.8.1.2 Load – differential settlement behaviour of piled raft....................111 4.8.2 Behaviour of piled raft compared to other alternative foundation types ....121 4.8.3 Load settlement behaviour of average individual pile beneath piled raft...134 4.8.4 Load distribution between raft and piles ....................................................148 4.8.5 Raft bending moment..................................................................................158 4.9 Effect of pile shaft length on piled raft foundation behaviour..............................167 4.9.1 Effect on load – average settlement behaviour...........................................167 4.9.2 Effect on load – differential settlement behaviour......................................171 4.9.3 Effect on load distribution between raft and piles. .....................................174
CHAPTER
5
THREE
DIMENSIONAL
FINITE
ELEMENT
SIMULATION FOR VERTICALLY LOADED PILED RAFT FOUNDATION, (CASE STUDY) 5.1 Introduction...........................................................................................................178 5.2 Methodology .........................................................................................................178 5.3 Soil structure interface modelling.........................................................................179 5.4 The Stonebridge Park building .............................................................................179 5.4.1 Geometric and material properties..............................................................179 5.4.2 Subsoil conditions.......................................................................................181 5.4.3 Detailed three dimensional nonlinear finite element model .......................182 5.4.4 Results of analysis ......................................................................................186 5.4.4.1 Vertical displacements....................................................................186 5.4.4.2 Raft contact pressure distribution...................................................190 5.5 The Torhaus building............................................................................................193 5.5.1 Geometric and material properties..............................................................193 5.5.2 Subsoil conditions.......................................................................................193 5.5.3 Detailed three dimensional nonlinear finite element model .......................195 5.5.4 Results of analysis ......................................................................................199 xii
5.5.4.1 Vertical displacements....................................................................199 5.5.4.2 Raft contact pressure distribution ..................................................203 5.5.4.3 Load distribution between piles......................................................206
CHAPTER 6 CONCLUSIONS .......................................................................209 REFERENCES...................................................................................... 212
xiii
LIST OF TABLES Table 2.1
Page Maximum Differential and Average Settlement of The Raft (Kim et al. 2001) ........................................................................................................31
2.2
Material parameters used in the finite element analyses of the Westend 1 building. (Reul and Randolph, 2003) .............................................................55
2.3
Construction sequence followed in the finite element analyses of the Westend 1 building. (Reul and Randolph, 2003).................... 55
3.1
Soil Material parameters used in the finite element analyses of test problem 1 .......................................................................................................67
3.2
Soil Material parameters used in the finite element analyses of test problem 2 ........................................................................................................69
3.3
Soil Material parameters used in the finite element analyses of test problem 3 .......................................................................................................71
3.4
Constitutive law for frictional behaviour of interface elements .....................77
4.1
The program of the parametric study..............................................................97
4.2
Soil Material parameters used in the finite element analyses for the parametric study...............................................................................................98
4.3
Allowable working load for un piled raft .....................................................103
4.4
Ultimate load for single piles........................................................................105
4.5
Total number of finite elements and nodes in the finite element mesh for the different pile configurations ..........................................................................106
5.1
Soil Material parameters used in the finite element analyses - Stonebridge park building..................................................................................................181
5.2
Soil Material parameters used in the finite element analyses - Tourhaus building ..........................................................................................................194
xiv
LIST OF FIGURES Figure
Page
2.1
Axially loaded pile............................................................................................6
2.2
Assumed variation of soil shear modulus.........................................................9
2.3
Charts for calculation of exponent e for efficiency of pile groups.................11
2.4
Load transfer method......................................................................................12
2.5
Non-linear load transfer curves based on hyperbolic stress-strain response of soil ..................................................................................................................17
2.6
Basic ideas of the MC (top), the SS (middle) and the HS (bottom) models; p'=1/3(σ'1+σ'2+σ'3)m.......................................................................................20
2.7
Mesh dependency with and without interface ................................................21
2.8
Results of pile load test, the MC and the HS models for the base and the shaft resistance ........................................................................................................22
2.9
Concept of piled raft system in the control of settlement...............................24
2.10
Details of foundations for Stonebridge park building ....................................26
2.11
Effect of number of piles on settlement and load sharing for Stonebridge park building ...........................................................................................................27
2.12
Settlement performance of two buildings.......................................................28
2.13
Central piles to reduce settlement...................................................................30
2.14
Schematic design approach for settlement .....................................................30
2.15
Optimal pile arrangement ...............................................................................32
2.16
Soil-structure interaction of piled rafts...........................................................34
2.17
Hypothetical example used to compare results of various methods of piled raft analysis.....................................................................................................35
2.18
Effect of number of piles on piled raft behaviour...........................................36
2.19
Effect of raft thickness on piled raft behaviour ..............................................37
2.20
Distribution of pile load and the skin friction along the pile shaft - raft with 13 piles............................................................................................................38
xv
2.21
Load-settlement
relationship
of
a
single
pile
compared
with
thecorresponding average behaviour of a pile group and the piles of a piled raft foundation ................................................................................................38 2.22
Simplified approach for calculation of undrained load-settlement curves .....41
2.23
Simplified representation of piled raft unit as defined by Randolph..............44
2.24
Simplified load settlement curve ....................................................................44
2.25
Burland’s simplified design concept ..............................................................45
2.26
Equivalent raft approach for pile groups ........................................................46
2.27
Numerical representation of piled raft after Clancy and Randolph................50
2.28
The Numerical representation developed by El-Mossallamy and Franke......52
2.29
Westend 1 ......................................................................................................54
2.30
Comparison of methods of analysis................................................................57
3.1
Topology and local axes of the used finite elements......................................60
3.2
Mohr-Coulomb yield condition (in π-plane and rendulic plane) ...................61
3.3
Modified Mohr-Coulomb model ....................................................................63
3.4
Cam Clay models............................................................................................65
3.5
Layout of the pile load test and the measured points .....................................67
3.6
Predicted and observed load-settlement relation-ship ....................................68
3.7
General test conditions of the pile load test....................................................69
3.8
Predicted and observed load-settlement relation-ship ....................................70
3.9
Hypothetical example .....................................................................................72
3.10
Load-settlement relation-ship at the center of the raft....................................73
3.11
Coulomb friction criterion used for interface elements..................................74
3.12
Finite element mesh configurations................................................................75
3.13
Effect of pile shaft-soil interface on the load-settlement relation-ship ..........76
3.14
Effect of tip interface on the load-settlement relation-ship ............................76
3.15
3-D interface element CQ48I..........................................................................79
3.16
Methodology used for construction of pile soil interface elements................80
3.17
Using soil-soil interface elements under pile tip level ...................................80
xvi
3.18
Test problem for verification of PILELO program ........................................81
3.19
Finite element mesh........................................................................................84
3.20
Load-settlement(m) relationship for the center of the raft .............................85
3.21
Contours of raft bending moment My .............................................................85
3.22
Distribution of contact pressure (kPa) between raft and soil along a line joining the raft center and the raft mid edge...................................................86
3.23
Finite element mesh of raft and piles..............................................................87
3.24
Load-settlement(m) relationship for the center of the raft .............................87
3.25
Finite element mesh of the exact model .........................................................90
3.26
Finite element mesh of the simplified model .................................................91
3.27
Load distribution within the pile group (kN)..................................................91
3.28
Bending moment in the raft Mx (kN.m/m) .....................................................92
4.1
General Soil stratification and cross section of foundation............................94
4.2
Structural system of building’s typical floor ..................................................95
4.3
Piled raft configurations for the parametric study ..........................................96
4.4
Column loads per floor including own weight ...............................................99
4.5
Finite element mesh for unpiled raft.............................................................101
4.6
Load settlement relationship of un piled raft for different soil profiles .......102
4.7
Log/log plot for estimation of ultimate load for un piled rafts.....................102
4.8
Axi-symmetric finite element mesh for single piles.....................................104
4.9
Load settlement relationship of single piles for different soil profiles – Lp=10 m ...................................................................................................................104
4.10
The finite element mesh for different pile configurations ............................107
4.11
Raft and piles finite element mesh for different pile configurations ............108
4.12
Pile shaft – soil interface elements for different pile configurations............109
4.13
Load settlement relationship of the piled raft for soil profile ‘MC-MC’ .....113
4.14
Load settlement relationship of the piled raft for soil profile ‘SC-SC’ ........114
4.15
Load settlement relationship of the piled raft for soil profile ‘MS-MS’ ......114
4.16
Load settlement relationship of the piled raft for soil profile ‘DS-DS’........115
xvii
4.17
Load settlement relationship of the piled raft for soil profile ‘MC-MS’......115
4.18
Load settlement relationship of the piled raft for soil profile ‘MC-DS’ ......116
4.19
Load settlement relationship of the piled raft for soil profile ‘SM-MC’......116
4.20
Effect of pile configuration on load carried by the piled raft corresponding to allowable settlement (S/Sall=1)....................................................................117
4.21
Effect of pile configuration on load carried by the piled raft corresponding to allowable settlement (S/Sall=1.5).................................................................117
4.22
Load - differential settlement relationship of the piled raft for soil profile ‘MC-MC’......................................................................................................118
4.23
Load - differential settlement relationship of the piled raft for soil profile ‘SC-SC’.........................................................................................................118
4.24
Load - differential settlement relationship of the piled raft for soil profile ‘MS-MS’.......................................................................................................119
4.25
Load - differential settlement relationship of the piled raft for soil profile ‘DS-DS’ ........................................................................................................119
4.26
Load - differential settlement relationship of the piled raft for soil profile ‘MC-MS’ ......................................................................................................120
4.27
Load differential - settlement relationship of the piled raft for soil profile ‘MC-DS’ .......................................................................................................120
4.28
Load - differential settlement relationship of the piled raft for soil profile ‘SM-MC’ ......................................................................................................121
4.29
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config A – 36 piles’ .............................................................122
4.30
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config B – 25 piles’..............................................................123
xviii
4.31
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config C – 16 piles’..............................................................123
4.32
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config D – 16 piles’ .............................................................124
4.33
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’ .............................................................124
4.34
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config B– 25 piles’...............................................................125
4.35
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’..............................................................125
4.36
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config D – 16 piles’ .............................................................126
4.37
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config A – 36 piles’ .............................................................126
4.38
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config B – 25 piles ’.............................................................127
4.39
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config C - 16 piles’ ..............................................................127
xix
4.40
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config D - 16 piles’ ..............................................................128
4.41
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config A – 36 piles’ .............................................................128
4.42
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config B – 25 piles’..............................................................129
4.43
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config C – 16 piles’..............................................................129
4.44
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config D – 16 piles’ .............................................................130
4.45
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’ .............................................................130
4.46
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config B – 25 piles’..............................................................131
4.47
Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’..............................................................131
4.48
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config D – 16 piles’ .............................................................132
xx
4.49
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config A - 36 piles ’ .............................................................132
4.50
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config B - 25 piles ’ .............................................................133
4.51
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config C - 16 piles’ ..............................................................133
4.52
Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config D - 16 piles’ ..............................................................134
4.53
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’ – Pile configuration ‘Config A – 36 piles’ ...............136
4.54
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -MC’ – Pile configuration ‘Config B – 25 piles’...............136
4.55
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -MC’ – Pile configuration ‘Config C – 16 piles’...............137
4.56
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -MC’ – Pile configuration ‘Config D – 16 piles’ ..............137
4.57
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config A – 36 piles’...................138
4.58
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config B – 25 piles’ ...................138
4.59
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config C – 16 piles’ ...................139
4.60
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config D – 16 piles’ ..................139
xxi
4.61
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ – Pile configuration ‘Config A – 36 piles’..................140
4.62
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS- DS’ – Pile configuration ‘Config B – 25 piles’.................140
4.63
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS- DS’ – Pile configuration ‘Config C – 16 piles’.................141
4.64
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS- DS’ – Pile configuration ‘Config D – 16 piles’.................141
4.65
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config A – 36 piles’................142
4.66
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config B – 25 piles’ ................142
4.67
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -MS’ – Pile configuration ‘Config C – 16 piles’ ...............143
4.68
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -MS’ – Pile configuration ‘Config D – 16 piles’...............143
4.69
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -DS’ – Pile configuration ‘Config A – 36 piles’ ...............144
4.70
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -DS’ – Pile configuration ‘Config B – 25 piles’................144
4.71
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC -DS’ – Pile configuration ‘Config C – 16 piles’................145
4.72
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config D – 16 piles’ ................145
4.73
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC - MC’ .................................................................................146
4.74
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ ......................................................................................146
xxii
4.75
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’......................................................................................147
4.76
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’....................................................................................147
4.77
Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ ....................................................................................148
4.78
Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config A1 – 64 piles’ (Units in kPa – m) .....150
4.79
Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config A – 36 piles’ (Units in kPa – m) .......151
4.80
Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config B – 25 piles’ (Units in kPa – m)........152
4.81
Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config C – 16 piles’ (Units in kPa – m)........153
4.82
Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config D – 16 piles’ (Units in kPa – m) .......154
4.83
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-MC’.............................155
4.84
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘SC-SC’ ...............................155
4.85
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MS-MS’..............................156
4.86
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘DS-DS’ ...............................156
4.87
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-MS’ .............................157
4.88
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-DS’ ..............................157
xxiii
4.89
Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘SM-MC...............................158
4.90
Variation of normalized maximum raft bending moment for soil profile ‘MCMC’...............................................................................................................159
4.91
Variation of normalized maximum raft bending moment for soil profile ‘SC-SC’......................................................................................... 159
4.92
Variation of normalized maximum raft bending moment for soil profile ‘MS-MS’ ....................................................................................... 160
4.93
Variation of normalized maximum raft bending moment for soil profile ‘DS-DS’ ........................................................................................ 160
4.94
Variation of normalized maximum raft bending moment for soil profile ‘MC-Ms’ ....................................................................................... 161
4.95
Variation of normalized maximum raft bending moment for soil profile ‘MC-DS’ ....................................................................................... 161
4.96
Variation of normalized maximum raft bending moment for soil profile ‘SM-MC’....................................................................................... 162
4.97
Contours of bending moment for ‘MC-MC’ – ‘Config C’ at P/Pwr=1........162
4.98
Contours of bending moment for ‘SC-SC’ – ‘Config C’ at P/Pwr=1...........163
4.99
Contours of bending moment for ‘MS-MS’ – ‘Config C’ at P/Pwr=1.........163
4.100
Contours of bending moment for ‘DS-DS’ – ‘Config C’ at P/Pwr=1 ..........163
4.101
Contours of bending moment for ‘MC-MS’ – ‘Config C’ at P/Pwr=1 ........164
4.102
Contours of bending moment for ‘MC-DS’ – ‘Config C’ at P/Pwr=1 .........164
4.103
Contours of bending moment for ‘SM-MC’ – ‘Config C’ at P/Pwr=1 ........164
4.104
Contours of bending moment for ‘MC-MC’ – ‘Config D’ at P/Pwr=1........165
4.105
Contours of bending moment for ‘SC-SC’ – ‘Config D’ at P/Pwr=1 ..........165
4.106
Contours of bending moment for ‘MS-MS’ – ‘Config D’ at P/Pwr=1 ........165
4.107
Contours of bending moment for ‘DS-DS’ – ‘Config D’ at P/Pwr=1..........166
4.108
Contours of bending moment for ‘MC-MS’ – ‘Config D’ at P/Pwr=1 ........166
4.109
Contours of bending moment for ‘MC-DS’ – ‘Config D’ at P/Pwr=1.........166
xxiv
4.110
Contours of bending moment for ‘SM-MC’ – ‘Config D’ at P/Pwr=1 ........167
4.111
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’.........................168
4.112
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ .........................168
4.113
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’ ...........................169
4.114
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’ .......................169
4.115
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’ .......................170
4.116
Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config E – 9 piles’..........................170
4.117
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’......171
4.118
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ ......172
4.119
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’ ........172
4.120
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’ ....173
4.121
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’ ....173
4.122
Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config E – 9 piles’.......174
4.123
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’.........................175
4.124
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ .........................175
xxv
4.125
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’ ...........................176
4.126
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’.........................176
4.127
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ .........................177
4.128
Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’ ...........................177
5.1
Stonebridge park building: Foundation plan of the quarter showing positions of instruments and cable runs (Cooke et al. 1981)..............................................180
5.2
Stonebridge park building: Soil stratification and position of foundation .....182
5.3
Finite element mesh - Stonebridge park building ...........................................183
5.4
Piled-raft system mesh - Stonebridge park building.......................................184
5.5
Pile-soil interface elements - Stonebridge park building................................184
5.6
soil-soil interface elements below soil-pile interface elements - Stonebridge park building ...................................................................................................185
5.7
Raft layout showing key-node numbers - Stonebridge park building ............185
5.8
Contours of vertical settlement (in meters) of the raft – Undrained condition Stonebridge park building...............................................................................187
5.9
Contours of vertical settlement (in meters) of the raft – Drained condition Stonebridge park building...............................................................................187
5.10
Load-vertical settlement relation ship for raft corner, mid edges and center points - Stonebridge park building.................................................................188
5.11
Distribution of vertical settlement (in meters) in longitudinal (X) direction through the center of the raft (through nodes 99656 – 139) – Stonebridge park building ...........................................................................................................188
5.12
Distribution of vertical settlement (in meters) in (Y) direction through the center of the raft (through nodes 99656 – 12) - Stonebridge park building ...188
xxvi
5.13
Predicted and observed load settlement relationship for the center of the raft Stonebridge park building...............................................................................189
5.14
Distribution of vertical stresses (in kPa) in (X) direction through the center of the raft at a distance of 1m below the bottom of raft (through nodes 99656 – 139) - Stonebridge park building ....................................................................191
5.15
Distribution of vertical stresses (in kPa) in (Y) direction through the center of the raft at a distance of 1m below the bottom of raft (through nodes 99656 – 12) - Stonebridge park building ......................................................................192
5.16
Torhaus building: (a) Profile view of the building; (b) foundation plan showing positions of instruments (Reul and Randolph 2003) ......................................193
5.17
Torhaus building: Soil stratification and position of foundation....................195
5.18
Finite element mesh - Torhaus building .........................................................196
5.19
Piled-raft system mesh - Torhaus building .....................................................197
5.20
Pile-soil interface elements - Torhaus building ..............................................197
5.21
soil-soil interface elements below soil-pile interface elements - Torhaus building ........................................................................................... 198
5.22
Raft layout showing key-node numbers - Torhaus building...........................198
5.23
Contours of vertical settlement (in meters) of the raft – Undrained condition Torhaus building .............................................................................................200
5.24
Contours of vertical settlement (in meters) of the raft – Drained condition Torhaus building .............................................................................................200
5.25
Load-vertical settlement relation ship for raft mid edges and corners - Torhaus building ...........................................................................................................201
5.26
Distribution of vertical settlement (in meters) in (X) direction through the raft inner mid-edge and corner, (through nodes 1130 – 365) – Torhaus building 201
5.27
Distribution of vertical settlement (in meters) in (Y) direction through the center of the raft (through nodes 99656 – 275) - Torhaus building................202
5.28
Predicted and observed load settlement relationship for the center of the raft Torhaus building .............................................................................................202
xxvii
5.29
Distribution of vertical stresses (in kPa) in (X) direction through the raft midedges at a distance of 0.5m below the bottom of raft (through nodes 1130 – 275) - Torhaus building ..................................................................................204
5.30
Distribution of vertical stresses (in kPa) in (Y) direction through the raft inner mid-edge and corner, at a distance of 0.5m below the bottom of raft (through nodes 1130 – 365) - Torhaus building ............................................................205
5.31
Load distribution among piles at 75% of total load (kN) - Torhaus building 207
5.32
Load distribution among piles at full load (kN) - Torhaus building...............207
5.33
Predicted and observed pile loads at 75% of total load - Torhaus building ...208
5.34
Predicted and observed pile loads at full load - Torhaus building..................208
xxviii
CHAPTER ONE INTRODUCTION
1.1 General In big crowded cities with limited areas available for horizontal extension, the need for vertical extension (i.e. high rise buildings) becomes essential. Usually piled foundations are used to support such heavy high rise buildings, mainly to control settlements and or differential settlements. According to existing building codes, the contribution of pile cap in direct contact with the ground is neglected and piles are designed to carry the total structural load with adequate factor of safety against bearing failure. In the last few decades, the concept of piled raft foundation system was developed and proved to provide economic foundations for high rise buildings, in cases at which geotechnical conditions allow raft to develop reasonable bearing capacity. Field observations from well instrumented piled rafts available in the literature contributed to better understanding of the mechanics of behaviour of piled rafts. Also now-a-days, there exist a number of approximate design methods and tools for analysis and design of piled rafts. However, both the above mentioned records and approximate design methods are not enough to establish standards and regulations that can be included in building codes for this type of foundations. Regarding the high cost of instrumentations and the time it requires as well as the simplifications inherited in the approximate solutions, the advanced three dimensional finite element method may be considered as the most suitable tool for researchers to model such complex problem and investigate various geotechnical and structural conditions. The limited capabilities of computers in the past restrained the use of finite element method to either non realistic two dimensional models or linear three dimensional models, thus neglecting some issues that may significantly influence the behaviour. With the rapid advances in the computer technology over the past few years, the use of three dimensional finite element method to model the different aspects of the problem has become affordable.
1
1.2 Background Poulos et al., (1997) stated that the following capabilities are desirable to have in a method of analysis to fulfil all design requirements for a piled raft: 1. Consideration of pile-raft-soil interaction in a logical manner, including the interactions between raft-raft, pile-pile, raft-pile and pile-raft. 2. Variation of the number, location and characteristics of the piles. 3. Consideration of realistic soil profiles. 4. Calculation of load-sharing between piles and raft. 5. Ability to allow for the development of ultimate loads in the piles, in both compression and tension, and non-linear load-deflection behaviour of the foundation. 6. Calculation of settlement and differential settlement over the entire foundation. 7. Calculation of raft bending moments and shears for the structural design of the raft. 8. The ability to be used by practising engineers without excessive training and effort. It is clear that the above requirements can only be satisfied by using a three dimensional model. Randolph (1994) has defined three different design philosophies for piled raft foundation; these are: 1. The conventional approach philosophy is defined as one where the foundation is designed essentially as a pile group, with regular spacing of piles over the complete foundation area but where allowance is made for the load transmitted directly from the pile cap to the ground. 2. The creep piling philosophy, Randolph (1994), stated two principles behind the original approach of creep piling which are: Each pile is designed to operate at a working load at which significant creep starts to occur, typically at about 70-80% of its ultimate bearing capacity. Sufficient piles are included to reduce the net contact pressure between raft and soil to below the pre-consolidation pressure of the clay.
2
3. The Differential settlement control philosophy, Randolph (1994), suggested that placing piles in the central area of a raft could effectively reduce differential settlements. Assuming that the structural load is relatively uniformly distributed over the plan area of the building, then there will be a tendency for an un piled raft to dish in the centre. A few piles added over the central region of the foundation, probably loaded to close to their ultimate capacity, will reduce that tendency, and thus minimise differential settlements.
It can be seen that the above three approaches lead to economic design of piled foundation, with raft in direct contact with ground surface.
1.3 Objectives of the research The prime objective of this research is to contribute to three dimensional finite element non linear modelling of vertically loaded piled rafts and to investigate its behaviour and design concepts in different geotechnical and structural conditions, to enhance such approaches to a logical procedure that may be implemented in building codes. This will be done as follows: • Studying the aspects of finite element modelling of piled raft foundation. • A wide numerical investigation performed on a hypothetical square piled raft covering many parameters affecting the stability and performance of piled raft foundations. • A three dimensional finite element model for the simulation of two well instrumented case histories of buildings founded on piled raft, to verify the accuracy of the model against published field measurements
1.4 Outlines of the research Chapter two presents a brief overview of the previous studies concerning Load response of axially loaded single piles and pile groups, alternative design philosophies of piled raft foundation, some aspects of piled raft behaviour from including factors quantifying performance of piled raft foundation, methods of analysis of piled raft foundation and a comparison of some of these methods.
3
Chapter three includes investigations of some aspects of piled raft foundation modelling. The first part deals with the fundamentals of finite element modelling of piled raft foundation system then some suggested practical techniques for handling large three dimensional finite element problems of piled raft foundations are proposed and verified. Chapter four presents a wide numerical investigation performed using finite element code DIANA® 9.1. A complete three dimensional finite element analysis of various piled raft configurations in different soil conditions were carried out. Mohr-coulomb elastoplastic material model with non associative flow rule was used. A summary of output results and necessary comparisons are presented. Chapter five presents three dimensional finite element analyses for the back analysis of two well documented case histories of piled raft foundation. Chapter six provides the conclusions drawn from the present investigation.
4
CHAPTER TWO LITERATURE REVIEW 2.1 Introduction Piled raft foundation can be defined as a composite construction consisting of piles, raft and soil. In the past few years, there has been an increasing recognition that the use of piles to reduce raft settlements and differential settlements can lead to considerable economy without compromising the safety and performance of the foundation (Poulos, 2001). This chapter presents a brief overview of the previous studies concerning Load response of axially loaded single piles and pile groups under vertical loading, alternative design philosophies of piled raft foundation, some aspects of piled raft behaviour including factors quantifying performance of piled raft foundation and methods of analysis of piled raft foundation; a comparison of some of these methods is held and discussed. 2.2 Load response of axially loaded single piles and pile groups 2.2.1 General The vertical load carried by a pile may be regarded as the sum of two load components. The first component is the resultant of shear stresses developed along the pile shaft where as the second component is the resultant of the normal stresses generated at the base of the pile as shown by figure (2-1). Kezdi, (1975) showed that the ratio of the above two components depends on the stratification and physical properties of the soil, the dimensions of the pile, the method of installation, and the magnitude of the total load. The difference between the load deformation characteristics of the two components of pile resistance is important in determining the settlement of a pile. In practice most of piled foundations consist of a group of piles not of a single pile. This group of piles act in the dual role of reinforcing the soil, as well as carrying the applied load down to deeper, stronger soil strata (Fleming et al., 1992). There are simplified methods to predict the settlement of a single pile and pile groups. These methods are either based on the use of 5
conventional one dimensional theory of Terzaghi, or on empirical correlations based on observations as those proposed by Meyerhof, (1959) and Focht, (1967). Generally simplified methods which are capable of predicting pile response under working load are important, as it gives first estimate concerning the required number of piles, pile length and diameter. With development of computers, more advanced analytical and numerical methods of analysis have become available. These methods include, methods based on theory of elasticity, load transfer methods and numerical methods (e.g. finite element and boundary element methods). Q
2ro=d τS
l
Q S = π d l τS
2
Q b = π d qb 4
qb
Figure (2-1) Axially loaded pile 2.2.2 Methods based on the theory of elasticity 2.2.2.1 Load response of single pile Many researchers proposed approximate methods based on the theory of elasticity for the analysis of axially loaded pile response. The most commonly used methods are those due to Poulos and Davis (1980), and Randolph and Wroth (1978). Herein the approximate method developed by Fleming et al. (1992), based on the work of Randolph and Wroth (1978), is briefly presented. In this solution, the manners in which load is transferred to
6
the soil through the pile shaft and the pile base, were first examined separately, and then combined to predict the pile response. The soil is treated as an elastic material, characterized by an appropriatevalue of secant elastic modulus, which may vary along the pile length. The pile is considered as a cylinder of diameter d (or radius ro). Considering the case of a rigid pile, the deformation of the soil around the pile shaft may be considered as shearing of concentric cylinders, (Cooke, 1974). The settlement of the pile shaft ws can be calculated as:
ws = ζ
τoro
,
(2.1)
⎛ rm ⎞ ⎟ ⎝ ro ⎠
(2.2)
G
where
ζ = ln⎜
In which,
τo : Shear stress on the pile shaft G: Soil shear modulus ro : Pile radius rm : Influence radius at which shear stress becomes negligible. This radius has been
found to be of the order of the pile length. Randolph (1994) reported that, for typical pile slenderness ratios the value of ζ will lie in the range of 3.5 to 4.5, and it is often sufficient to adopt a value of 4. Also he recommended using the initial (low strain) shear modulus. The overall load taken by the pile shaft is Ps = 2πlroτ o where τ o is the average shear stress mobilized at the pile shaft. Thus the load settlement ratio (or stiffness of the pilesoil system) may be expressed as: Ps 2πlG , = ws ζ
(2.3)
where G is the average shear modulus of the soil over the pile penetration depth.
7
The pile base may be treated as a rigid punch acting at the surface of a soil medium. The deflection wb is obtained from the standard solution of Timoshenko and Goodier, 1970 as:
wb =
Pb (1 − ν ) , rbGb 4
(2.4)
where the subscript b denots to the pile base. Combining the shaft and base response for a rigid pile, the base settlement and shaft settlement will be similar to the settlement of pile head, wt. The total load, Pt, may thus be written as: ⎛ Pb Ps ⎞ Pt = Pb + Ps = wt ⎜ + ⎟ , ⎝ wb ws ⎠
(2.5)
It is convenient to introduce a load settlement relationship for the pile. Thus, equation (2.5) may be rewritten, making use of equations (2-3) and (2-4), as: 4rbGb 2πG l Pt , = + wtroGl (1 − ν )roGl Glζ ro
(2.6)
Considering pile compressibility and variation of shear modulus with depth as shown in figure (2-2), closed form expressions for the pile stiffness and the load at pile base may be written as follows: 4η 2πρ tanh( µl ) l + Pt (1 − ν )ξ ζ µl ro = , 4η tanh( µl ) l wtroGl 1+ πλ (1 − ν )ξ µl ro
(2.7)
4η 1 + Pb (1 − ν )ξ cosh(µl ) , = 4η 2πρ tanh( µl ) l Pt + (1 − ν )ξ ξ µl ro
(2.8)
8
where,
η=
rb ro
ratio of underream for underreamed piles,
ξ=
Gl Gb
ratio of end-bearing for end-bearing piles,
ρ=
G Gl
variation of soil modulus with depth,
λ=
Ep Gl
pile-soil stiffness ratio,
⎛ rm ⎞ ⎟ ⎝ ro ⎠
ζ = ln⎜
measure of radius of influence of pile,
where, rm = (0.25 + ξ [2.5 ρ (1 − ν ) − 0.25])l , ⎛ 2 ⎞⎛ l ⎞ ⎟⎟ ⎜ ⎟ ⎝ ζλ ⎠ ⎝ ro ⎠
µl = ⎜⎜
measure of pile compressibility.
Gl/2
Gl
shear modulus
Gl/2
Gl
Gb
shear modulus
Pile l/2
l/2
l depth
ρ=
Gl/2 Gl
depth
(a) Floating pile
Gl/2 Gl G ξ= l Gb ρ=
(b) End bearing pile
Figure (2-2) Assumed variation of soil shear modulus (Fleming et al. 1992)
Fleming et al. (1992), compared the pile stiffness values obtained by solution outlined above with those obtained by the more rigorous alternative elastic solution described by
9
Poulos and Davis (1980). The two sets of results were found in good agreement with each other. 2.2.2.2 Load response of pile groups
The response of vertically loaded pile groups is normally predicted using either the equivalent raft or equivalent pier approaches; (these two approaches are described later in this chapter). The response of a pile group may be estimated based on the response of a single pile, using the principal of elastic interaction between piles. The settlement of a pile in a group may be estimated using either the group settlement ratio proposed by Poulos and Davis, (1980) or the settlement efficiency factor proposed by Fleming et al. (1992), or alternatively by extending the solution for single piles described in 2.2.2.1 to deal with pile groups. However this approach has some limitations when the soil behaviour is non linear, since the portion of the non linear settlement of a single pile is localised around the pile, and does not contribute to interaction with the neighbouring piles (Randolph, 1994). A brief description of the efficiency factor method proposed by Fleming et al., (1992) is presented as follows: The group stiffness can be written as:
K = ηwnk ,
(2.9)
ηw = n − e ,
(2.10)
where, K : stiffness of pile group,
ηw : efficiency factor n : number of piles, k : stiffness of individual pile, e : ranges from 0.4 to 0.6 for most pile groups. The actual value of e depends on: Pile slenderness ratio, l/d, Pile stiffness ratio, λ=Ep/Gl, Pile spacing ratio, s/d,
10
Homogeneity of soil, characterised by ρ, and Poisson's ratio, ν. For a given combination of the above factors, the value of e may be estimated using the charts presented in figure (2-3). The upper part of the chart allows a base value of e to be chosen whereas the bottom part plots four correction factors that should be multiplied by the base value of e. An example illustrating the use of these curves can be found in Fleming et al., (1992)
Figure (2-3) Charts for calculation of exponent e for efficiency of pile groups (Fleming et al. 1992)
11
2.2.3 Original load transfer method
Original load transfer method, was first proposed by Coyle and Reese (1966) and is based on data compiled from pile load tests on instrumented piles. The required data is presented in curves relating load transfer ratio (ratio of mobilized skin friction) to the pile shaft settlement. These curves are sometimes called t-z curves which present the non linear behaviour of the pile shaft resistance. An example of these curves and the method of analysis is given by figure (2-4). It should be noticed that the single curve shown in figure (2-4) may be inadequate for predicting actually the behaviour of an axially loaded pile and a family of load transfer curves at different depths is required to model the pile soil response reasonably. Q1
Q1
L1
ρ0 τ1
ρ1
L2
Pile
l
Q2
τ2
ρ2
L3
Q3
τ3
ρ3
Pt ( a ) M e th o d o f a n a ly s is
Load transfer/soil shear strength
1 .0
0 .5
0 0
0 .4
0 .8
1 .2
1 .6
2 .0
P ile m o v e m e n t ( in ) ( b ) T y p ic a l lo a d tr a n s fe r c u r v e
Figure (2-4) Load transfer method (Coyle and Reese, 1966)
12
As illustrated by figure (2-4), the pile is divided into segments (3 segments are chosen for simplicity) guided by the load transfer curves. A small tip displacement ρt is assumed and pile point resistance Pt caused by this displacement is calculated using theory of elasticity. Using the suitable load transfer curve, the value of the adhesion factor corresponding to the assumed displacement can be evaluated. From the segment equilibrium the load Q3 can be calculated as:
Q 3 = Pt + τaL3πd ,
(2.9)
where,
τa : adhesion factor * soil shear strength The elastic deformation at mid point of the pile segment (assuming linear variation of load within the segment) is calculated as:
ρ 3 = ρt +
(Q3 + Pt )L3 ,
(2.10)
A3 Ep
Where A3 : area of segment 3 Ep : pile modulus The calculated displacement is used to obtain a new adhesion factor. If the calculated adhesion is not in tolerable range with the assumed value, more cycles are required until satisfactory convergence is obtained between the assumed and calculated displacements. The above procedure is repeated for all segments and the sum of the resulting forces and displacements for all segments will give the pile head load and corresponding settlement respectively. Repeating the above procedure to increase the pile head load and corresponding settlement, a complete load response for the pile may be computed. More details of the above procedure may be found in Coyle and Reese (1966), Poulos and Davis (1980) and Bowles (1996). The following notes should be considered: 1. Better results may be obtained using a larger number of segments if there are sufficient load transfer curves and the data are of good quality. (Bowles 1996)
13
2. In using load transfer curves, it is inherently assumed that the displacement of the pile at any point is related only to the shear stress at that point and is independent of the stresses else where of the pile, thus no proper account is made for the continuity of the soil mass. (Poulos and Davis, 1980) 3. For the above reason the method can not be used directly to study the behaviour of pile groups. 4. Load transfer curves must be used with caution and for cases where the conditions are similar to the tests on which these relations are based. (ElMosallamy 1997) 2.2.4 Theoretical load transfer curves
Due to the high cost of pile load tests on instrumented piles, necessary for construction of load transfer curves, theoretical load transfer curves were proposed. In general the original stress strain response of soil may be fitted using hyperbolic approximation, that leads to the following relation:
τ G = 1 − Rf , Go τs
(2.11)
where,
τ : shear stress τs : failure shear stress G : secant shear modulus (corresponding to shear stress τ) Go : initial shear modulus Rf : a parameter that dictates the curvature of the stress strain curve.(ranges from 0.9-1) Randolph (1994), reported that in practice, real soils often show a more rapid decrease of secant modulus with shear stress level, which may be described better by a relation of the form (Fahey and Carter, 1993):
14
G = 1− Go
⎛τ ⎞ f⎜ ⎟ , ⎝ τs ⎠ g
(2.12) where, f : a parameter that dictates the curvature of the stress strain curve.(ranges from 0.9-1) g : a parameter that may be taken 0.25 for natural soils and 0.70 to 1 for remoulded soil samples or for samples that suffered minor disturbance according to (Fahey and Carter, 1993). The load response of a single pile can be approximated by a hyperbolic function. It is common to use the hyperbolic model proposed by Chin (1970; 1972) to extrapolate the results of pile load tests to estimate the ultimate load capacity of the pile. Chin expressed the load settlement relationship of the tested pile in the form: w = mw + C , P
(2.13)
where, w : pile vertical head settlement P : applied load C : constant, equals the intercept on the w/P axis m : slope of the relationship of w versus w/P Fleming (1992), modelled the shaft and base response by separate hyperbolic functions and then the two components are combined with consideration of elastic shortening of the pile. Randolph (1994), commented that modelling pile base response by a hyperbolic function appears an acceptable approach, at least for cast in-situ piles, since the soil displacement field is relatively localised beneath the pile tip, and large displacements are required to develop the ultimate load. However, some caution is necessary before applying a similar technique to the shaft response, where a significant proportion of the shaft displacement occurs far from the pile, in soil at relatively low shear stress levels, and the ultimate condition is reached rather abruptly, at displacements of the order of 1%
15
of the pile diameter. For a hyperbolic stress strain response, the load transfer parameter of equation (2.2) becomes: ⎛ rm ⎞ ⎜ −Ψ⎟ ⎟, ζ = ln⎜ ro ⎜ 1− Ψ ⎟ ⎜ ⎟ ⎝ ⎠
(2.14)
where, Ψ = Rf
τo , τs
(2.15)
where, τs is the limiting shaft friction, and Rf is a parameter that dictates the curvature of the stress strain curve. A load transfer curve obtained from a parabolic stress strain response can be written as follows: 1/ m ⎡ ⎤ τ ⎛ ⎞ τs ⎢ w = mζ 1 −⎜1− ⎟ ⎥⎥ , ⎢ ro Go ⎢⎣ ⎝ τs ⎠ ⎥⎦
(2.16)
where, m is a constant that ranges from 2 to 3. Figure (2-5) shows theoretical non linear load transfer curves obtained from a hyperbolic stress strain response (according to equation 2.11), for Rf values of 0.95 and 1. For comparison a parabolic curve (equation 2.16) is also shown. Another load transfer curve based on the modified form of equation 2.12 is presented. It can be seen that the parabolic curve compares well with those curves derived from the hyperbolic stress strain response. The curve based on the modified form of equation 2.12 is more non linear than other curves, and it is usually more close to the actual soil stress strain response. However, the actual load transfer curves are more linear due to the variation of shear stress around the pile shaft, corresponding to hyperbolic shapes with values of Rf of 0.5 or lower (Poulos 1989). Randolph (1994), reported based on experimental work available in literature, that the parabolic representation of equation 2.16, with m in the range 2 to 3, gives closer fit to the experimental load transfer curves than does the use of hyperbolic curve with low values of Rf.
16
Figure (2-5) Non-linear load transfer curves based on hyperbolic stress-strain response of soil (Randolph, 1994)
Castelli and Maugeri, (2002) used hyperbolic load transfer functions to predict the non linear single pile response (shaft and base responses) and they modified the single pile stiffness to fit that of a pile group idealized as an equivalent pier interacting with surrounding soil. Reasonable agreement between this simple approach and field measurements reported in literature was observed. Guo (2000), proposed a visco-elastic load transfer model for axially loaded piles operating at high load levels. In this case creep can lead to significant pile head movement at constant load, and even gradual reduction in shaft capacity. He modified the formula suggested by Randolph and Wroth (1978), which was previously discussed, and obtained closed form solution for the pile load response considering creep effect. The shaft displacement can be expressed as:
ws = ζ 1ζc
τoro G1
,
(2.17)
where,
ζ 1 : non linear instantaneous (before creep) measure of the influence of load transfer. ζc : non dimensional creep modification factor or function (can be considered constant up to a working load of about 70% of the ultimate pile capacity)
17
G1 : instantaneous (before creep) initial elastic shear modulus (before creep)
τo : Shear stress on the pile shaft ro : Pile radius
Guo (2000), also obtained Similar expressions for the base response and for the part of load transferred to pile base considering creep effect. More details can be found in Guo (2000). 2.2.5 Numerical methods
The numerical analyses are generally accepted techniques for solving geotechnical problems. Numerical analysis methods include, the finite element method, the finite difference method and the boundary element method. In the present thesis, only the finite element method is discussed and used for analysis of problems in the parametric study. Detailed description of the finite element method is available in many references (e.g. Zienkiewiez and Taylor, 1987, Yang, 1986 and many others). The finite element method is very useful in considering more complex constitutive laws, dimensions and boundary conditions. When dealing with non linear behaviour, it may be considered to be the more accurate method in comparison with other methods. (ElMossallamy and Franke, 1997) The finite element method is considered as one of the most powerful approaches for analyses of the load response of single piles and pile groups. Three types of finite element analysis will be discussed in the following subsections, the plane strain finite element analysis, the axisymmetric finite element analysis and three dimensional finite element analysis. 2.2.5.1 Plane strain finite element analysis
Plane strain finite element analysis was applied by some researchers (e.g. Prakoso and Kulhawy, 2001) to the approximate analysis of vertically loaded pile groups. The group in this approximation is condensed into a strip of piles, such that the axial stiffness of an
18
in-plane row of piles has to be simplified to an equivalent plane strain pile. This method has the advantage of analysis time saving and relatively low required computer memory. There are several shortcomings for using this method for the analysis of pile groups among which: •
The problem of vertically loaded pile group doesn't satisfy the plane strain condition.
•
The stress path followed by a soil element in plane strain and constrained between two equivalent piles (actually between two embedded walls) is different from that followed by a soil element within a real pile group. This can affect to some unknown extent the results obtained.
•
This type of analysis is restricted to uniform type of loading.
•
Solving two perpendicular strips may lead to incompatible displacement at the same point.
2.2.5.2 Axisymmetric finite element analysis
Axisymmetric finite element analysis is a special case of the general two dimensional plane strain finite element analysis. This type of analysis may be applied to perform stress analysis of
symmetrically loaded cylindrical and annular type of structures.
Axisymmetric finite element analysis was successfully used by many researchers to obtain the load response of single piles (e.g. Ottaviani, 1975, El-Mossallamy and Franke, 1997, Wehnert and Vermeer, 2004 and many others). A good example that shows the procedure and capabilities of this type of finite element analysis, is the analysis performed by
Wehnert and Vermeer, (2004). They applied an axisymmetric finite
element analysis to perform a back analysis of a load test of a bored pile. The tested pile has a diameter of 1.3 m and a length of 9.5 m, the subsoil is stiff over consolidated clay. In the analysis the pile is assumed to be linear elastic and for the subsoil three different constitutive models are used, i.e. the elastic plastic Mohr-Coulomb model (MC), the soft soil model (SS), which is based on the modified Cam-clay model, and the advanced hardening soil model (HS). The models result in different stress strain curves. The MC model is an elastic perfectly-plastic model, while in the SS model a logarithmic relation between the volumetric strain εv and the mean effective stress p' is introduced and the HS 19
model shows a hyperbolic stress-strain relationship between the axial strain and the deviatoric stress in primary triaxial loading. The basic ideas of these models are shown in figure (2-6). As a first step the influence of the mesh and of interface elements is examined and the results are presented in figure (2-7). They concluded that the interface elements are important. Especially for the shaft resistance the results of a calculation without interface elements were heavily mesh dependent. When using interface elements the mesh dependency is negligible for the base resistance. One needs at least two or three elements at the pile tip to get rid of the mesh dependency. The results of the three models are compared with those obtained from pile load test. Figure (2-8) presents a comparison between the results of pile load test and the used constitutive models. The authors concluded the following: •
For base resistance the differences between computational results appeared to be remarkably small. For all three models, the shape of computed load-displacement curves is more or less the same. The choice of the constitutive model is thus not that important for the base resistance.
•
For the shaft resistance, it appears to depend significantly on the choice of the constitutive model. For small settlements the MC and the SS models lead to the same curve and behave too stiff. The peak value for the SS model is a bit bigger than for MC and a bit smaller than for HS. σ1-σ3
σ1-σ3
σ1-σ3
σ1-σ3
σ1-σ3
Figure (2-6) Basic ideas of the MC (top), the SS (middle) and the HS (bottom) models; p'=1/3(σ'1+ σ'2+ σ'3) (Wehnert and Vermeer, 2004)
20
(a) Very fine and very coarse mesh
(b) Results of Mohr-Coulomb model
(c) Results of HS model Figure (2-7) Mesh dependency with and without interface (Wehnert and Vermeer, 2004)
21
Figure (2-8) Results of pile load test, the MC and the HS models for the base and the shaft resistance (Wehnert and Vermeer, 2004)
Some attempts were made to model pile groups using axisymmetric finite element analysis (e.g. Hooper and Wood, 1977). This means that the piles are represented in the analysis as concentric annuli with an equivalent stiffness. The accuracy of such approach depends very much on the pile configuration in the group, another shortcoming is the introduction of un realistic radial and tangential stiffness (El-Mossallamy and Franke, 1997). 2.2.5.3 Three dimensional finite element analysis
In general three dimensional finite element analyses are considered to be the most powerful and accurate method for the analysis of most of engineering problems. The problem of analysis of pile groups subjected to vertical loading, is essentially a three dimensional problem. Three dimensional finite element analysis can model non linear soil behaviour with a variety of constitutive models, also the complete history of pile construction procedure can be simulated, time effects, and it can account for all types of interaction between all elements of the problem. There are two main obstacles limiting the wide use of this type of analysis for practical purposes:
22
• It needs high computational requirements (i.e. a lot of time is required to prepare,
run and overcome the difficulties that may be encountered during analysis of the problem, also capabilities of personal computers may be not enough for the computational demands associated with some problems). • As all types of analysis of geotechnical engineering problems, the evaluation of
the properties of the sub soil and assigning appropriate parameters still remains the most uncertain step in the analysis. Ottaniani, (1975) seems to be the first to apply such an analysis to piled foundations. He used three dimensional finite element analysis to study the behaviour of vertically loaded pile groups in a homogenous linearly elastic medium and both, the piles and the soil are assumed to be weightless to reduce computer cost. 3*3 and 5*3 pile groups were analysed with and without pile cap being in contact with the ground and the stress distribution in piles and in the soil mass was obtained. This analysis confirmed that the cap in contact with the soil has a considerable effect on the pile-soil load transfer mechanism. Many others used this method for research purposes to validate simplified approaches (e.g. Randolph, 1977) or to achieve better understanding of the behaviour of single piles, pile groups and piled rafts (e.g. Reul and Randolph, 2003). It is used for analysis and back analysis of important structures founded on piles (e.g. analysis of the foundations of the 300 m high Commmerzbank tower in Frankfurt am Main by Katzenbach et. al., 1997). Three dimensional finite element analysis of piled raft foundations available in literature will be discussed in more detail later in this chapter. 2.3 Alternative design philosophies for piled rafts 2.3.1 General
When a foundation engineer is designing foundation to support a building or a structure where competent soil exists close to the ground surface, a shallow raft will provide adequate bearing capacity. If estimated settlements and/or differential settlements are not
23
tolerable, it is common to use piles to reduce settlements. However, once the decision has been made to use piles, the piles are designed to carry the total structural load with an adequate factor of safety against bearing failure. The nature of load transfer between piles and soil, particularly where shaft friction provides a sufficient component of total pile capacity, will then automatically lead to small (and may be negligible settlements), Randolph, (1994). From the economic point of view the use of a limited number of piles strategically located under the raft may improve both the ultimate load capacity, the settlements and differential settlement performance of the raft, which is known as piled raft foundation, such a foundation makes use of both the raft and the piles leading to considerable economy without compromising the safety and performance of the foundation, Poulos (2001). The existence of the raft directly casted on the ground enforces a block type failure even at large pile spacing compared with free standing pile groups, however as the pile soil block behaviour is approached the contribution of each additional pile to settlement reduction becomes smaller and once block behaviour can be presumed, there is no advantage in installing more piles at closer spacing, Cooke (1986). Figure (2-9) illustrates the concept of piled raft system in the control of settlement to acceptable values. Randolph (1994) has defined three different design philosophies for piled raft foundation, these philosophies will be discussed in the following sections.
Figure (2-9) Concept of piled raft system in the control of settlement (Poulos, 2001)
24
2.3.2 The conventional approach philosophy
Randolph (1994) defined the conventional approach as one where the foundation is designed essentially as a pile group, with regular spacing of piles over the complete foundation area, but where allowance is made for the load transmitted directly from the pile cap to the ground. The principal benefit is a reduction in the total number of piles, due to perhaps (60-75)% of the total structural load being carried by the piles. One of the best documented case histories of the performance of a piled raft of this type has been described by Cooke et al. (1981), the foundations for Stonebridge park building, which is a sixteen storey block of flats built on London clay consisting of 351 bored piles, each 0.45m in diameter and 13.0m long capped by a raft of 0.9m in thickness in direct contact with ground, see figure (2-10). The raft and piles were extensively instrumented to measure contact stresses between the raft and the ground, and the loads going into typical piles in the group, the measurements extended to six years period covering the erection and early life of the building. The building overall load on the foundation was reported by Cooke et al. (1981) as 15560 kN. The foundation was planned essentially as a standard pile group (with a factor of safety for piles greater than 3) carrying the overall load of the building and neglecting the raft contribution, as this was the design approach widely adopted when the building was planned in 1973. Results of the field study show that the raft share of the total structural load reached a long term value of about 23% which corresponds to total long term central settlement of 25 mm. Randolph (1983), comments on the above results that the piles in this foundation design were only contributing about 5% of their possible stiffness as individual piles and halving the number of piles under this foundation would have reduced the pile group stiffness by only 13% giving an estimated average settlement of 29 mm with some 31% of the total structural load being taken by the raft. Other researchers (Padfield and Sharrock, 1983; Randolph and Clancy, 1993) were interested in the above case study due to the existence of detailed
25
measurements of performance. They discussed alternative design of foundation with number of piles reduced to less than 100, with only marginal increase in average settlement. Viggiani (1998), carried out an interesting theoretical exercise of reducing the number of piles progressively, and observing the effects on the settlement and load sharing between the raft and piles. The results of this exercise are shown on figure (2-11). For the original foundation, virtually all the load was carried by the piles. Reducing the number of piles had very little effect on either the settlement or amount of load carried by the raft until the number of piles become less than 200. Even with 117 piles (one-third of the original number), the settlement increase was only about 50%, while the factor of safety was reduced by about 58%. ( Poulos, 2001)
Figure (2-10) Details of foundations for Stonebridge park building (cooke et al., 1981)
26
Figure (2-11) Effect of number of piles on settlement and load sharing for Stonebridge park building (Viggiani, 1998)
It is clear that improved understanding of the behaviour and performance of single piles, pile groups and piled rafts developed in the last decades will lead to a great reduction in the cost of foundations. 2.3.3 The creep piling philosophy
Randolph (1994), stated two principles behind the original approach of creep piling which are: •
Each pile is designed to operate at a working load at which significant creep starts to occur, typically at about 70-80% of its ultimate bearing capacity.
•
Sufficient piles are included to reduce the net contact pressure between raft and soil to below the pre-consolidation pressure of the clay.
The foundation is treated as raft foundation in which a limited number of piles are added to reduce settlements. This concept was suggested by Burland et al. (1977), the piles are distributed uniformly beneath the raft. Fleming et al. (1992) stated that maintained load tests on piles generally show that the creep or consolidation settlement of the pile starts to increase significantly once the load reaches about 70% of the ultimate capacity. At creep load level, piles are allowed to move plastically relative to the surrounding soil. The choice of the creep load as a working load of each pile prevents high loads developing in
27
piles at edges of the foundation as the stiffness of such piles will start to decrease at higher load levels. In this philosophy piles are usually not required to ensure over all stability of the foundation. Hansbo (1993), has presented a case history involving two similar residential buildings supported by piles in Sweden. The first was designed using a traditional approach, with a factor of safety of 3 for the piles. A total of 211 piles, 28 m long were used. The second was designed using the creep pile concept, in which the piles are designed as settlement reducers with a factor of safety for the piles of 1.25. This building was supported on only 104 piles, 26 m long. Figure (2-12) shows the measured settlement contours for each building, and the measured relationship between average settlement and time. Despite the fact that the second building was supported on less than half the number of piles, it settled no more (in fact slightly less) than the first building. This case clearly demonstrates the potential economy that may be achieved by the use of the piled raft concept, without significant sacrifice of foundation performance. (Poulos 2001)
Figure (2-12) Settlement performance of two buildings (Hansbo, 1993)
28
It should be known that this approach may be more suitable for relatively soft cohesive soils in relatively uniform ground conditions, rather than for non cohesive soils. For non cohesive soils Hansbo (1993), claimed that essentially the piles will operate below their creep load, and the performance of the foundation can be handled conventionally. 2.3.4 Differential settlement control philosophy (some times called optimizing pile-raft analysis)
Differential settlements may have negative effects on a super structure as well as on a raft foundation, thus it should be controlled to allowable limits. Randolph (1994), suggested that placing piles in the central area of a raft could effectively reduce differential settlements. Figure (2-13) and (2-14) show schematically the principles behind the design of piles to reduce differential settlements. Assuming that the structural load is relatively uniformly distributed over the plan area of the building, then there will be a tendency for an un piled raft to dish in the centre. A few piles added over the central region of the foundation, probably loaded to close to their ultimate capacity, will reduce that tendency, and thus minimise differential settlements. Two key questions must be answered, the first is how many piles are required to reduce the differential settlements to an acceptable level, the second question over which region of the raft should the piles be installed to minimize the differential settlement. Horikoshi and Randolph (1997) conducted extensive parametric study to answer the above questions. They concluded the following guide lines: 1. Piles should be distributed over the central 16 to 25% of the raft area. 2. The pile group stiffness should be approximately equal to the stiffness of the raft alone. 3. Total pile capacity should be designed for between 40 and 70% of the total design load depending on the pile group area ratio and Poisson's ratio for the soil. 4. The degree of mobilization of pile capacity, should be less than or equal to 0.8 to avoid significant increase in differential settlements. Padfield and Sharrock (1983), discussed the use of central piles to reduce differential settlement. They suggested an alternative design for the Stonebridge building (see figure (2-10)) using 40 piles situated near the centre of the raft, at a spacing of 3.2 m (7.1 pile 29
diameters). Such arrangement of piles would result in elimination of differential settlements, even for the case of flexible raft.
Figure (2-13) Central piles to reduce settlement (Randolph, 1994)
Figure (2-14) Schematic design approach for settlement (Randolph, 1994)
30
Reul et al. (2004), performed a parametric study that included 256 different piled raft configuration. They performed the analysis using 3-D elasto-plastic finite element analysis. In the study, the pile positions, the pile number, the pile length and raft soil stiffness ratio as well as load distribution on the raft have been varied. They concluded that differential settlements are much more sensitive to the raft soil stiffness ratio and the load configuration than the average settlements. For a raft under uniform loading or core-edge loading, the differential settlements can be most efficiently reduced by installation of piles only under the central area of the raft. Kim et al. (2001), proposed an optimization scheme to minimize differential settlements of a piled raft foundation. They demonstrated their proposed scheme for three different load configurations as shown by figure (2-15). Although they used an approximate analysis model for the piled raft foundation which has limitations in the consideration of the interaction between elements of the piled raft system, their method presents a rigorous engineering tool to determine optimum pile arrangement. It can be seen from figure (2-15) that, the arrangement of piles required to reduce differential settlements depends on the load distribution. Table (2-1) presents the reduction in differential settlements due to using the optimal pile arrangement for the three different load configurations. Table 2-1:Maximum Differential and Average Settlement of The Raft (Kim et al. 2001)
Example 1 (uniformly distributed load) Example 2 (line loads) Example 3 (4 concentrated loads)
Initial arrangement
Optimal arrangement
Reduction
(mm)
(mm)
ratio in differential
Differential
average
Differential
average
3.2
16.2
0.20
17.0
94
13.2
17.0
2.2
20.6
83
14.9
16.7
3.3
18.9
78
31
settlement(%)
(a) Initial Pile Arrangement And Loading
(b) Case of Uniform Distributed Load
Conditions
(c) Case of Line Load
(d) Case of Concentrated Load
Figure (2-15) Optimal pile arrangement (Kim et al., 2001)
32
2.4 Some aspects of piled raft behaviour 2.4.1 Different interactions affecting behaviour of piled rafts
The piled raft foundation represents a complex foundation system, which requires a qualified understanding of the soil structure interactions. The super structure load is transferred to the soil by skin friction, end bearing and contact pressure between the raft and the soil. Many researchers discussed the different interactions affecting behaviour of piled raft foundations (e.g. Hain and Lee, 1978, El-Mossallamy and Franke, 1997 and Katzenbach et al. 1998 and 1999). Figure (2-16) summarizes the different interactions in the piled raft foundation system, which are:
1. The pile-soil interaction, which consists of two components, base resistance-soil and shaft resistance-soil interactions. 2. Pile-pile interaction, which depends on pile spacing and the shaft load level. 3. Raft-soil interaction, which depends on the raft settlement level and causes the increase in both, vertical and horizontal effective stress states in the soil, thus increasing the ultimate shear strength of the soil. 4. Pile-raft interaction, as the shear strength of the soil increases (interaction 3), the ultimate shaft and base capacity increase consequently. To insure sufficient accuracy of the analysis, the analysis method should take the above interactions into account. The requirements of these interactions may only be satisfied by using a three dimensional model.
33
Figure (2-16) Soil-structure interaction of piled rafts (Katzenbach et al., 1999)
34
2.4.2 Factors affecting behaviour of piled rafts
Characteristics of piled raft behaviour had been examined through many extensive parametric studies available in literature (e.g. Hain and Lee, 1978, Poulos, 2001 and ElMossallamy and Franke, 1997 and many others). Poulos, (2001) studied a hypothetical case shown in figure (2-17) using the computer program GARP.
Figure (2-17) Hypothetical example used to compare results of various methods of piled raft analysis (Poulos, 2001)
The studied parameters are, the number of piles, the nature of the loading (concentrated versus uniform), raft thickness and the applied load level. Poulos observed the following: 1. The effect of increasing the number of piles, while generally of benefit, does not always produce the best foundation performance, and there is an upper limit to the
35
number of piles beyond which very little additional benefit is obtained as shown by figure (2-18). 2. The raft thickness affects differential settlement and bending moments, but has little effect on load sharing between raft and piles or maximum settlement as shown by figure (2-19). 3. The nature of the applied loading (concentrated or uniform) is important for differential settlement and bending moment, but is generally not very important for maximum settlement or load sharing between the raft and the piles, although it influences the distribution of load among the piles. 4. Increasing the load level increases the settlements and beneficial effects of adding piles as the design load level increases are obvious.
Figure (2-18) Effect of number of piles on piled raft behaviour (Poulos, 2001).
36
Figure(2-19) Effect of raft thickness on piled raft behaviour (Poulos, 2001).
Katzenbach et al. (1998), carried out three dimensional elasto-plastic finite element analysis of various piled raft configurations. They analyzed a square raft containing from 0 to 49 piles. They examined the effect of number of piles and the relative length of piles (length/diameter) on the relative settlement (settlement of the piled raft divided by the settlement of the raft alone). They developed interaction diagrams as shown in figure (220). The diagram shows that for a given number of piles, the relative settlement is reduced as the relative length increases. It also supports the results of Poulos, (2001) concerning that there is an upper limit for the number of piles beyond which, increasing number of piles has negligible benefits. Another important aspect of piled raft behaviour obtained by Katzenbach et al. (1998), is that the ultimate shaft friction developed by piles
37
within a piled raft can be significantly greater than that for a single pile or a pile in a conventional group. This is due to interactions 3 and 4 described in section 2.4.1. Figure (2-20) represents an example of the results obtained by Katzenbach et al. (1998). These results was also obtained by El-Mossallamy and Franke, 1997 as shown in figure (2-21).
Figure(2-20) Distribution of pile load and the skin friction along the pile shaft - raft with 13 piles (Katzenbacch et al., 1998)
Figure(2-21) Load-settlement relationship of a single pile compared with the corresponding average behaviour of a pile group and the piles of a piled raft foundation (El-Mossallamy and Franke, 1997).
38
2.4.3 Coefficients quantifying performance of piled rafts
To quantify the performance of piled rafts, normalized coefficients should be introduced. These coefficients must answer key questions that arise in the design of piled rafts concerning the relative proportion of the load carried by the raft and piles, and the effect of additional pile support on absolute and differential settlements (Randolph, 1994). As an example of these coefficients are those three coefficients to quantify the performance of piled rafts proposed by Reul and Randolph, (2003): 1. The piled raft coefficient, αpr, describes the ratio of the sum of all the pile loads,
ΣPpile, to the total load on the foundation Ptot: ΣPpile , Ptot
αpr =
(2.17)
a piled raft coefficient of unity indicates a free standing pile group, whereas a piled raft coefficient of zero describes an unpiled raft. 2. The coefficient of maximum settlement, ξs, is defined as the ratio of the maximum settlement of the piled raft, spr, to the maximum settlement of the corresponding un piled raft, sr :
ξs =
spr , sr
(2.18)
3. The coefficient of differential settlement, ξ∆s, is defined correspondingly. Unless other wise mentioned, this is the differential settlement between the centre and the middle of the shorter side of the raft.
ξ∆s =
s∆pr , s∆r
(2.19)
These coefficients which reflect the performance of piled rafts, should be controlled by local standards and practical experience, that give the limiting criteria to judge the performance of piled rafts.
39
2.5 Methods of analysis of piled raft foundations
The main methods of analysis of piled raft foundations are reviewed by Poulos et al., (1997) and El-Mossallamy and Franke, (1997). To fulfil all design requirements of piled raft foundations, Poulos et al., (1997) stated the following capabilities that are desirable to have in a method of analysis: 1. Consideration of pile-raft-soil interaction in a logical manner, including the interactions between raft-raft, pile-pile, raft-pile and pile-raft. 2. Variation of the number, location and characteristics of the piles. 3. Consideration of realistic soil profiles. 4. Calculation of load-sharing between piles and raft. 5. Ability to allow for the development of ultimate loads in the piles, in both compression and tension, and non-linear load-deflection behaviour of the foundation. 6. Calculation of settlement and differential settlement over the entire foundation. 7. Calculation of raft bending moments and shears for the structural design of the raft. 8. The ability to be used by practising engineers without excessive training and effort.
The methods of analysis of piled raft foundations can be classified into three categories as proposed by Poulos et al., (1997): 1. Simplified calculation methods. 2. Approximate computer-based methods. 3. More rigorous computer-based methods.
2.5.1 Simplified calculation methods
These methods, which can be performed using hand calculations, are useful to predict approximate load response of piled rafts in the first design stage to give quick estimation of the foundation elements. Some examples of methods falling in this category will be briefly described in the following sub sections.
40
2.5.1.1 Poulos and Davis, (1980)'s method
In this method Poulos and Davis, (1980) proposed a simplified method of obtaining loadsettlement curve to failure for a piled raft. Figure (2-22), illustrates the un-drained loadsettlement curve of the piled raft system which, consists of two linear sections. The method assumes that for un-drained conditions, purely elastic conditions prevail up to the load at which the piles would fail (PA) if no raft were present (line OA). Thereafter it is assumed that any additional load is taken entirely by the raft and that the additional settlement of the system is then given by the settlement of the raft only (line AB). This method neglects the raft flexibility, and is only valid for either perfectly rigid or perfectly flexible rafts. More details of this method and for accounting for consolidation settlement may be found in Poulos and Davis, (1980).
Load
PB
B
Pw PA
o
C A
SA
Sw Undrained or immediate settlement
Figure(2-22) Simplified approach for calculation of undrained load-settlement curves (Poulos and Davis, 1980) 2.5.1.2 Randolph, (1994)'s method (flexibility matrix method).
In this method Randolph, (1994) has developed very convenient equations for the stiffness of a piled raft system and the load sharing between the piles and the raft. The
41
definition of the pile problem considered by Randolph is shown in figure (2-23). The stiffness of the piled raft foundation can be estimated as follows:
Kpr =
Kp + Kr (1 − αcp ) , 1 − αcp 2 KrKp
(2.18)
where, Kpr: Stiffness of piled raft Kp : Stiffness of the pile group which can be estimated using elastic theory (e.g. using the solutions of Poulos and Davis, 1980 or Fleming et al. 1992). Kr : Stiffness of raft alone which, can be estimated using elastic theory (e.g. using the solutions of Mayne and Poulos, 1999). The proportion of total applied load carried by the raft : Kr (1 − αcp ) Pr = , Pt Kp + Kr (1 − αcp )
(2.19)
where, Pr : Load carried by the raft Pt : Total applied load. The raft-pile interaction factor, αcp
αcp = 1 −
ln (rC /rO )
ζ
,
(2.20)
where, rc : average radius of pile cap (corresponding to an area equal to the raft area divided by number of piles). ro : radius of pile ⎛ rm ⎞ ⎟ ⎝ ro ⎠
ζ = ln⎜
(measure of radius of influence of pile)
42
where, rm = (0.25 + ξ [2.5 ρ (1 − ν ) − 0.25])l
ξ=
Esl Esb
ρ=
Esav Esl
ν
: Poisson's ratio of soil.
l
: Pile length.
Esl : Soil Young's modulus at level of pile tip. Esb : Soil Young's modulus of bearing stratum below pile tip. Esav : Average soil Young's modulus along pile shaft. Although, Randolph's method is restricted to linear behaviour of piled raft system, Poulos, (2001) used Randolph's approximate equations to develop a tri-linear load settlement curve for the piled raft up to failure as shown by figure (2-24). First the stiffness of the piled raft is computed from equation (2-18). This stiffness will remain operative until the pile capacity is fully mobilised. Making the simplifying assumption that the pile load mobilisation occurs simultaneously, the total applied load, P1, at which the pile capacity is reached is given by:
P1 =
Pup , Pr 1− Pt
(2.21)
where, Pup : Ultimate load capacity of the piles in the group. Pr : Proportion of the load carried by the raft (equation 2-19)) Pt
43
Figure(2-23) Simplified representation of piled raft unit as defined by Randolph, 1994 - (Poulos, 2001)
Figure(2-24) Simplified load settlement curve (Poulos, 2001)
44
2.5.1.3 Burland’s approach.
The Burland’s approach developed by Burland (1995), is an answer to the key question, how much piles are required to reduce the settlements to an acceptable level. This means that piles are designed to act as settlement reducers. The basic concept of this method can be summarized as follows: 1. estimate the long term load settlement relationship for the raft without piles as shown in figure (2-25), such that the design load P0 gives a total settlement S0. 2. define an allowable settlement Sa, which should include a margin of safety, such that P1 is the load carried by the raft corresponding to Sa. 3. the difference (P0- P1) is proposed to be carried by the settlement-reducing piles. The piles are supposed to develop its full geotechnical capacity. However, Burland (1995), suggests a mobilization factor of about 0.90 to be applied to the ultimate capacity.
Figure(2-25) Burland’s simplified design concept (Poulos, 2001) 2.5.1.4 Equivalent raft and equivalent pier methods.
These methods are based on the idea of transforming the pile group or piled raft foundation to an analogous form. This approach treats the foundation as a whole unit, thus, eliminating the difficulties of the detailed treatment of the interaction between each foundation unit. 45
Randolph, (1994), described these approaches. Traditionally, the settlement of a pile group has been estimated by considering an equivalent raft situated two-thirds of the way down the part of the piles that penetrate the main founding stratum, or at the level of the pile bases for end-bearing piles as shown in figure (2-26). The average settlement at the ground level is then calculated as: wavg = wraft + ∆w ,
(2.22)
where, ∆w : the elastic compression of the piles above the level of the equivalent raft, treated as
free standing columns.
Figure(2-26) Equivalent raft approach for pile groups ( Randolph, 1994)
46
The raft settlement, wraft may be calculated using elastic theory. The area of the equivalent raft may be estimated using a load spread of 1 in 4 as indicated by figure (2-26). The main advantage of the equivalent raft approach is that it enables due account to be taken of variations in soil stiffness below the level of the raft. This is of particular relevance in situations where a layer of softer soil exists at some level below the base of the piles. An alternative to the equivalent raft approach, is to consider the region of soil in which the piles are embedded as an equivalent continuum, effectively replacing the pile group by an equivalent pier (Poulos and Davis, 1980). For a pile group of a plan area Ag, the diameter of the equivalent pier may be taken as:
de = 1.13 Ag ,
(2.23)
and the young's modulus of the pier as: ⎛ Ap ⎞ Ee = Es + (Ep − Es )⎜ ⎟ , ⎝ As ⎠
(2.24)
where, Ep : Young's modulus of the piles. Es : Average Young's modulus of the soil penetrated by the piles. Ap : Total cross sectional area of the piles in the group. The load settlement response of the equivalent pier may be calculated using solutions for the response of single pile such as that presented by Randolph and Wroth, (1978). As for the case of the equivalent raft approach, the equivalent pier will furnish an estimate of only the average settlement of the pile group, not the differential settlement. Randolph and Clancy, (1993) proposed an appropriate parameter (R) to categorise pile groups:
R=
ns , l
(2.25)
47
where, n : Number of piles in the group. s : Spacing between piles. l : Pile embedded length.
For values of R which are greater than 4, the pattern of differential settlement (assuming flexible raft) is very similar to that for raft foundation. An equivalent raft would therefore be a logical analogue for analysis. For smaller values of R, and certainly for values less than 2, it would seem at the outset that an equivalent pier approach is more logical, at least for estimating average settlements, Randolph, (1994).
Horikoshi and Randolph, (1999) proposed a method for estimating the overall (average) settlement of a piled raft by combining the equivalent pier method (Poulos and Davis, 1980) and the flexibility matrix method (Randolph, 1994). The piled raft system is simplified to an equivalent capped pier system. The approach is simple and gives satisfactorily accurate stiffness of piled rafts without any long computations.
2.5.2 Approximate computer-based methods.
These methods can approximately account for some of the interactions between foundation elements. These approaches require the use of computer and have the advantage that distribution of settlement, pile load, and raft straining actions can be obtained. Some examples of methods falling in this category will be briefly described in the following sub sections.
2.5.2.1 Methods employing a strip on springs approach.
A section of the raft is presented by a strip, and the supporting piles by springs. Approximate allowance for the interaction between foundation parts is taken into consideration. Poulos et al. (1997) described a method of this category in which the effects of parts of the raft outside the considered strip, are taken into account by computing the free-field soil movements due to these parts and interacting these with the strip section. The method has shown to give reasonable agreement with more complete
48
analyses. Torsional moments within the raft can't be considered, also consistency of results at a point, if strips in two directions through that point are analysed, is not guaranteed.
2.5.2.2 Methods employing a plate on springs approach.
The raft is presented by an elastic plate and the piles are modelled using spring elements that support the plate. Poulos, (1994) described a method of this category using finite difference method for the plate and allowed for various interactions via approximate elastic solutions. Allowance was also made for piles reaching their ultimate capacity, the development of bearing capacity failure below the raft, and the presence of free-field vertical soil movements acting on the foundation system.
2.5.2.3 Method developed by Clancy and Randolph, (1993).
Clancy and Randolph, (1993), have described an approximate numerical procedure that takes full account of interaction between the various foundation elements. They developed a computer code called HyPR for the solution of this problem. Figure (2-27) represents the used numerical procedure. The subsoil is considered as a homogeneous elastic layer of finite thickness. A load transfer approach (Randolph and Wroth, 1978) is used to express the relation between local tractions and displacement of each pile, together with an elastic continuum analysis of the additional displacements due to the tractions acting on other elements. HyPR takes into account to some extent the non linear behaviour of the piles with a load cut-off for the elements where pile-soil slip is computed to occur, using an incremental elastic analysis.
Russo and Viggiani, (1997), discussed the main limits of HyPR, which are the assumption of elastic homogeneous subsoil, and the possibility of tensile forces developing at the raft-soil interface.
49
Figure(2-27) Numerical representation of piled raft after Clancy and Randolph, (1993)
2.5.3 More rigorous computer-based methods.
More rigorous computer-based methods are soil continuum based solutions that simulate the original problem with the least amount of simplifications. These solutions provide an efficient means of retaining the essential aspects of pile interaction through soil continuum and hence a more realistic representation of the problem. These methods include the full boundary element method, the approach of mixed boundary element and finite element method and the finite element method. Some examples of methods falling in this category will be briefly described in the following sub sections.
50
2.5.3.1 Full boundary element method proposed by Butterfield and Banerjee, (1971).
In this approach, the raft in contact with the soil surface is discretised and each pile within the foundation system is also discretised. The soil is modelled as an elastic material. The analysis is based on the Mindlin's solution for a point load in the interior of a semi-infinite ideal elastic homogeneous half space. The main advantages of this approach as summarized by El-Mossallamy and Franke, (1997): •
The disturbance of the continuity of the elastic half space due to the presence of the piles is simulated more accurately.
•
Piles of differing length and diameter within the group can be simulated.
•
Slip at the pile-soil interface as well as nonlinear soil response can be directly considered in the analysis.
The main shortcomings of this approach are: •
Limited to the case of rigid raft and incompressible piles.
•
The solution needs considerable computer resources.
2.5.3.2 Mixed boundary and finite element methods proposed by El-Mossallamy and Franke, (1997).
El-Mossallamy and Franke, (1997) proposed a mixed technique using the boundary and the finite element methods to represent the three dimensional nature of the problem as shown by figure (2-28). The raft is modelled by the finite element method as plate in bending and the pile group embedded in a finite half space is simulated by a complete boundary element structure, taking into account the raft contact pressure. Non linear pile responses are modelled at the pile-soil interface and at the pile base. An incremental type of nonlinear calculation with iterative adaptation of finite element method applied to the raft and boundary element method applied to the pile group, is applied. This mixed technique is used to reduce the computer memory required to simulate this three dimensional problem taking into account the non linear behaviour. The difference in stress strain behaviour between primary loading and reloading caused by stress relief due to pit excavation as well as locked pile stresses caused by the
51
installation sequence, can be considered. Good agreement with monitored case histories and with other rigorous analysis methods was observed.
Figure(2-28) The Numerical representation developed by El-Mossallamy and Franke, (1994, 1997) 2.5.3.3 Finite element method.
The three dimensional finite element method is the most rigorous approach for the analysis of piled raft foundations. All elements of interaction can be handled, also construction sequence, non linear behaviour of the soil and actual soil stratification can be considered. Few complete three dimensional finite element analyses of real piled rafts are available in the literature. As an example, the analysis performed by Reul and Randolph, (2003), is briefly described. They applied three dimensional elasto-plastic 52
finite element analyses to perform detailed back analyses of three case histories of piled rafts founded on Frankfurt overconsolidated clay. The soil and the piles are represented by first order solid finite elements of hexahedron (brick) and triangular prism (wedge shape). The raft is modelled using first order shell elements of square and triangular shape with reduced integration. The circular piles are replaced by square piles with the same shaft circumference. The description of the problem is shown in figure (2-29). The soil non linear behaviour was modelled using elasto-plastic cap model. The piles and the raft are considered to behave linear-elastically. The contact between structure and soil was described as perfectly rough. This means that no relative motion between the nodes of the finite element mesh that represents the structure and those representing the soil (i.e. no special interface elements were used to model the contact zone between raft and soil and between soil and piles). The material parameters used in the finite element analyses are summarized in table (2-2). The distribution of the Young's modulus of the Frankfurt clay with depth is described by the following empirical formulation proposed by Reul: ⎡ ⎛ z − 30 ⎞ ⎤ E = 45 + ⎢ tanh⎜ ⎟ + 1⎥ × 0.7 z , ⎝ 15 ⎠ ⎦ ⎣
(2.26)
where, E : Young's modulus (MPa). z : the depth below the surface of the tertiary layers (m). Table (2-3) outlines the construction sequence followed in the finite element analyses. The analyses were carried out using commercially available ABAQUS® program. Reasonable agreement between the settlements obtained from the finite element analysis and the field measurements, was observed. The calculated loads carried by piles were overestimated compared with the measured values. The above analyses by Reul and Randolph, (2003) can be considered sufficient for design purposes and back analyses to control the performance of piled rafts, but, for research purpose, more investigations have to be done to adjust the finite element analysis model. The following items need to be considered:
53
•
The contact behaviour between pile shaft and soil have to be investigated and considered in the analyses of piled rafts.
•
The actual circular cross section of piles should be considered in the analysis instead of the equivalent square cross section, which can be handled using second order finite elements to model the problem instead of the first order elements used in there analyses.
Figure(2-29) Westend 1: (a) cross section; (b) foundation plan; (c) finite element mesh of the system; (d) finite element mesh of the raft (Reul and Randolph , 2003)
54
Table 2-2 Material parameters used in the finite element analyses of the Westend 1 building. (Reul and Randolph, 2003)
Frankfurt clay
Frankfurt limestone
Raft
Piles
Equation (2-26)
2000
34000
22000
0.15
0.25
0.2
0.2
Total unit weight of moist soil, γ: kN/m3
19
22
25
25
Buoyant unit weight of moist soil, γ': kN/m3
9
12
15
15
0.5
-
-
15
-
-
37.67
29.53
-
-
20
1000
-
-
42.42
2114
-
-
0
0.001
-
-
0.795
0.841
-
-
0.1
0.01
-
-
Parameter Young's modulus, E: MPa Poisson's ratio, ν
0.7, (0 ≤ z p 25) 0.57, ( z ≥ 25) 20
Coefficient of earth pressure at rest, Ko Angle of internal friction, φ' : degrees Slope of the conical yield surface in the p-t plane, β: degrees Cohesion, c': kPa Intersection of the conical yield surface with the t-axis, d: kPa Shape parameter of the transition surface between cone and cap, α Shape parameter of the cone, K Shape parameter of the cap, R
Table 2-3 Construction sequence followed in the finite element analyses of the Westend 1 building. (Reul and Randolph, 2003)
Applied load, Peff: MN
Mean vertical effective stress at foundation level, σ'v: kPa
1. In situ stress state
-
192
2. Excavation to a depth of 7 m below ground level
-
66
3. Installation of piles
-
66
-
0
61.9
21.9
6. Installation of raft
61.9
21.9
7. Loading of raft
956.9
338
Step
4. Excavation to a depth of 14.5 m below ground level 5. Application of weight of raft minus uplift due to pore pressures as uniform load on subsoil (zero stiffness of raft)
55
2.5.4 Comparison of some methods of analysis of piled rafts
Poulos et al., (1997) have compared some of the methods of the three categories described before for the analysis of piled rafts. They applied these methods to solve two problems, the hypothetical example previously shown in figure (2-17) and a case history of the well instrumented foundations of the Westend 1 tower, which is a 208 m high building in Frankfurt described by Franke et al., (1994). The methods used for comparison were: 1. Poulos and Davis, (1980) 2. Randolph (1983) 3. Strip on springs approach, using the program GASP (Poulos, 1991) 4. Plate on springs approach, using the program GARP (Poulos, 1994) 5. Finite element method of Ta and Small, (1996) 6. Finite element and boundary element of Sinha (1997) 7. Finite element and boundary element of Franke et al. (1994) for the case history only 8. Measurements reported by Franke et al. (1994) for the case history only
The results of the comparison are shown in figure (2-30). From the comparison they concluded that while the piles are behaving elastically, i.e. their axial load capacity is not fully utilized, the average settlements predicted by the various methods are similar. However when the capacity of a significant number of piles is fully utilized, the differences between the methods tend to become more pronounced. Simple non linear hand calculation methods, such that of Poulos and Davis (1980), tend to over predict the settlement under these circumstances, compared to the more complete numerical analyses. There is considerable variability in the computed maximum bending moments in the raft between the methods that can estimate straining actions in the raft. Generally the available methods appear to provide reasonable basis for the overall behaviour of piled raft foundations, but more researches are required to increase reliability of prediction of the detailed behaviour of piled rafts, especially distribution of bending moments.
56
(a) Case A: raft with 15 piles (P=12MN)
(b) Case B: raft with 15 piles (P=15MN)
(c) Case C: raft with 9 piles (P=12MN)
(d) Westend building
Figure(2-30) Comparison of methods of analysis (Poulos et al., 1997)
57
CHAPTER THREE MAIN ASPECTS OF THREE DIMENSIONAL FINITE ELEMENT MODELLING OF PILED RAFT FOUNDATION 3.1 Introduction The problem of complete analysis of piled raft foundation system is essentially a three dimensional problem as concluded by many researchers e.g. Poulos et al. (1997), ElMossallamy (2000), Reul and Randolph (2003) and many others. This chapter is divided into three main parts, which are; fundamentals of finite element modelling of piled raft foundation system, suggested practical techniques for handling of large three dimensional finite element problems of piled raft foundation system, and simplified method for solving piled raft foundations subjected to lateral loading. The first part deals with the requirements for the finite element modelling of the different parts of piled raft foundation system which are, the raft, piles and supporting soil as well as the modelling of pile soil interface. A brief theoretical back ground and detailed investigation of different modelling alternatives with illustrative comparisons and small test problems are presented for verification of the chosen procedures. The second part presents some techniques for reducing the computational demands associated with the three dimensional piled raft problems, so that large problems can be handled with personal computers in reasonable CPU time and without violating the solution accuracy. The third part is a proposed simplified approach to handle lateral loads, to fulfil design requirements and code provisions for that type of foundation which, usually supports symmetric high rise buildings. To verify the use of the proposed method, a three dimensional test problem is solved and the results are compared to those resulted from the detailed three dimensional model. 3.2 Fundamentals of finite element modelling of piled raft foundation system 3.2.1 General In this part, a detailed investigation was carried out to study the fundamentals of three dimensional finite element modelling of the different parts of piled raft foundation system. The behaviour of real piled raft foundation system is complex and involves many
58
types of interactions that affect its behaviour, such as pile-soil interaction, pile-pile interaction, raft-soil interaction and pile-raft interaction. Proper material model and type of finite elements for modelling of each part of the system, to satisfy the requirements of simulation of the complex behaviour of real piled raft foundation system are investigated, in order to propose a methodology for three dimensional finite element modelling of this problem.
3.2.2 Modelling of foundation elements (raft and piles) Raft and piles are often made of reinforced concrete sections with relatively high strength compared with supporting soil. Stresses in pile sections are not expected to exceed allowable working stresses of its material. Thus, in the present research as well as in previous ones (e.g Katzenbach et al. 1997, Reul and Randolph 2003 and 2004, Shabana et al. 2003 and 2004 and others), the material behaviour of foundation elements have been simulated as isotropic linear elastic in the finite element analyses. Both raft and piles are modelled with three dimensional iso-parametric solid elements of the second order to allow modelling the cylindrical shape of piles. It should be clear that first order elements can not model the cylindrical shape of piles, thus in previous work researchers who used first order elements (e.g Reul and Randolph 2003 and 2004, Shabana et al. 2003 and 2004 and others) replaced the circular piles with square piles having the same shaft circumference. This approximation results in changing the area of pile base and consequently the toe resistance. In addition the use of such approximation may cause problems such as stress singularity, to arise at the corners of the pile shaft-soil interface, if used. The CHX60, brick, 20 nodes and/or the CTP45, pyramid, 3 sides, 16 nodes, three dimensional solid elements available in the DIANA® element library are used for modelling of both raft and piles. The topology and local axes of the above elements are shown in figure (3-1). The above solid elements may be loaded with different types of loading such as concentrated load, distributed load over one or more of its faces, load acting on the entire element volume and change of temperature as well as initial stress situation may be assigned to element nodes.
59
(a) CHX60
(b) CTP45
Figure (3-1) Topology and local axes of the used finite elements (DIANA® 9.1 users manual, 2005) 3.2.3 Modelling of supporting soil 3.2.3.1 General The behaviour of real soil is essentially non linear and the importance of considering soil non linearity has been satisfactorily addressed through the literature. A brief theoretical overview is presented for three commonly used elasto-plastic constitutive models (incorporated into DIANA® program), namely the Mohr-Coulomb model, Modified Mohr-Coulomb model (cap model) and the Enhanced Delft-clay model. It is evident that the advanced material models like the modified Mohr-coulomb for modelling non cohesive soils and the Enhanced Delft-Clay model for modelling of clay like materials, have more capabilities to capture the behaviour of real soils compared to simple models. On the other hand the problem of assigning appropriate material parameters from routine soil investigation data, still remains the main obstacle that limits the use of such models and push geotechnical engineers to use simple models such as Mohr-Coulomb model. The above constitutive models are compared through obtaining finite element solutions for test problems using these models, in order to evaluate the use of simple MohrCoulomb model for soil modelling in piled raft foundation problems. 3.2.3.2 Theoretical background of selected soil constitutive models 3.2.3.2.1 Mohr-Coulomb model The well known Mohr-Coulomb criterion proposed by Coulomb in 1773, is the oldest failure criterion, that was originally presented for soils. Although it was not intended to 60
represent a yield surface, it has been used frequently in engineering practice in finite element modelling of soils and it is included in most of general purpose finite element programs. The parameters required for Mohr-Coulomb model are the internal friction angle φ’ , the cohesion c’ and the diltancy angle ψ. The above parameters have the advantage of being simple and quite well known by most of geotechnical engineers. Figure (3-2) presents the Mohr-Coulomb yield condition in both π-plane and Rendulic plane. The DIANA program gives two alternatives for Mohr-Coulomb plasticity, the first is ideal plasticity (i.e. no hardening) and the second is strain hardening plasticity. Both associative plasticity (ψ=φ) and non associative plasticity (ψ≠φ) can be modelled. The formulation of the yield function f(σ,κ) and the plastic potential function g(σ,κ) expressed in the principal stress space (σ1≥σ2≥σ3) as reported in the DIANA users manual are as follows: f(σ,κ)=½ (σ1−σ3) + ½( σ1+σ3)sinφ’(κ) –c’(κ)cosφ0
(3-1)
g(σ,κ)=½ (σ1−σ3) + ½( σ1+σ3)sinψ(κ)
(3-2)
where: κ : the internal state variable defining the hardening behaviour. φ’(κ) : angle of internal friction which is a function of the internal state variable c’(κ) : cohesion which is a function of the internal state variable φ0 : initial angle of internal friction
Figure (3-2) Mohr-Coulomb yield condition (in π-plane and Rendulic plane) (DIANA® 9.1 users manual, 2005) 61
3.2.3.2.2 Modified Mohr-Coulomb model DIANA offers the Modified Mohr–Coulomb capped plasticity model which is particularly useful to simulate the behaviour of frictional (e.g. non cohesive soils) materials. This plasticity model has been developed at Delft University of Technology. Unlike the regular linear elastic perfect plastic Mohr-Coulomb model, the following features are included in this model: - Nonlinear elasticity - A smooth shear yield surface capable of modelling shear hardening or softening. - An elliptical compression yield surface (cap) capable of modelling isotropic hardening. - A dilatancy angle which is optionally related to the friction angle via Rowe’s dilatancy rule, which reads: sinψ =
sin φ − sin φcv 1 − sin φ sin φcv
(3-3)
in which sin φcv a constant value which can be conceived as the friction angle at constant volume. In DIANA, the non linear elastic behaviour is presented by the tangent bulk modulus denoted by Kt, which is pressure dependent and its pressure dependency can be defined by two alternative formulations namely: •
The Power Law nonlinear elasticity for which: 1− m
⎛ p′ ⎞ ⎟⎟ Kt = Kref ⎜⎜ ⎝ p ′ref ⎠
(3.4)
in which Kref and p’ref respectively are the reference compression modulus and the reference pressure; m is a floating point value which is in the order of 0.5 for sand. •
Kt =
The Exponential nonlinear elasticity for which:
1+ e
κ
p′
(3.5)
In which, e is the void ratio, κ is a material parameter presenting the unloading reloading and p’ is the current hydrostatic pressure.
62
Figure (3-3) presents the Modified Mohr-Coulomb yield condition in both π-plane and pq space.
Figure (3-3) Modified Mohr-Coulomb model (DIANA® 9.1 users manual, 2005)
The failure surface of the Modified Mohr–Coulomb model is a so-called double hardening model in which the shear failure and the compressive failure are uncoupled. The combined failure surface is given by the formulation in the p–q space as follows: f1=
q 6 sin φ − ( p + ∆p) = 0 R1(θ ) 3 − sin φ (3-6) 2
2 2 ⎛ q ⎞ ⎟⎟ − pc = 0 f 2 = ( p + ∆p) + α ⎜⎜ ⎝ R 2(θ ) ⎠
in which φ is the friction angle in triaxial compression, ∆p a constant which models cohesive material behaviour, and pc the preconsolidation pressure and α is the cap shape parameter. The functions R1(θ) and R2(θ) model the differences in strength in triaxial compression and in triaxial extension and are functions of Lode’s angle θ. The direction of the inelastic strain rate is determined by the plastic potential surfaces, where in case of the Modified Mohr–Coulomb model the following two surfaces are applied:
63
g1 = q −
6 sin φ ( p + ∆p) 3 − sin φ (3-7) 2
g 2 = ( p + ∆p) + α q − pc 2
2
which implies an associative behaviour in the p–q space and a nonassociated flow in the deviatoric space. The dilatancy angle ψ is related to the friction angle φ by the assumption of Rowe’s stress dilatancy theory. For the hardening behaviour of the double hardening, Modified Mohr-Coulomb model, DIANA requires two evolution of functions: the evolution of sinφ, the sine of the friction angle, and the evolution of the preconsolidation pressure pc. The evolution of the dilatancy angle sinψ is implicitly given by the assumption of Rowe’s stress dilatancy model (equation 3-3).
3.2.3.2.3 Enhanced Delft Clay model
The Enhanced Delft clay model is member of the Cam clay models family to simulate the behaviour of clay and clay-like materials. It is an enhancement for the Egg Delft clay model (Van Eeklen and Van den Berg 1994), by incorporating the effect of soil tensile strength into both the tangential bulk modulus Kt and to the expression for the yield surface f to be as follows:
Kt =
1+ e
κ
( p′ + pt ),
(3-8)
in which pt = tensile pressure; a numerical artifice introduced to incorporate the effect of the tensile stresses when the initial pressure is equal to zero.
f = q2 +
M2
β2
(( p′ + ∆p )( p′ + ∆p − 2a ) + a (1 − β )) = 0, 2
2
Where,
64
(3-9)
M=
6 sin φ , 3 − sin φ
(3-10)
′ ∆p = reference pressure to model cohesion, and can be obtained from: ∆p = c . tan φ
β = parameter calculated as: β = γ , for p′ + ∆p ≤ a , and β=
1
α
, for p′ + ∆p > a ,
where
γ = an optional shape factor for the dry side of the yield surface, and a = yield function parameter obtained from a = Pc′
α 1+α
.
It is assumed that the Egg Cam-clay model is an associated plasticity model. Thus the above yield surface also defines the plastic potential g. Figure (3-4) shows the yield condition in p-q space for Cam clay models. For more details about models, interested reader may refer to DIANA® users manual and Groen, (1995).
Figure (3-4) Cam Clay models (DIANA® 9.1 users manual, 2005)
65
3.2.3.3 Evaluation of the selected soil constitutive models
Finite element two dimensional and three dimensional test problems are solved using the selected advanced models in comparison with the simple Mohr-Coulomb model to validate the use of latter for modelling of piles and piled raft problems in cohesive and non cohesive soils.
3.2.3.3.1 Test problem 1: Back analysis of monitored pile load test in stiff clay.
This test problem is a back analysis of a pile load test in Frankfurt, reported by Sommer and Hambach (1974). The tested pile has a diameter of 1.30 m and a length of 9.5 m and is located completely in the over consolidated Frankfurt clay. In this test the loads were applied to failure in increments and maintained constant till the settlement rate was negligible (i.e drained condition). Figure (3-5) gives the lay out of the pile load test arrangement. This test problem aims at studying the efficiency of Mohr-Coulomb model in modelling of clayey soils compared to the Enhanced Delft Clay model. Two dimensional axi-symmetric finite element analysis of the problem is performed twice, once with Mohr-Coulomb model and the other with the Enhanced Delft Clay model. Table (3-1) summarizes, the estimated soil parameters for both the Mohr-Coulomb model and Enhanced Delft Clay model. Results obtained from both models are compared to field measurements as shown in figure (3-6). It can be seen that the results obtained using the Mohr-Coulomb model compares well with those obtained using the Enhanced Delft-clay model. Also the good agreement between field measurements and the results obtained from drained analyses for both models can be recognized.
66
Figure (3-5) Layout of the pile load test and the measured points, Sommer and Hambach (1974) Table (3-1) Soil Material parameters used in the finite element analyses of test problem 1:
Mohr Coulomb model
Enhanced Delft-Clay model
60
-
0.30/-
-/0.20
Total unit weight of moist soil, γ: kN/m3
20
20
Buoyant unit weight of moist soil, γ': kN/m3
10
10
Coefficient of earth pressure at rest, Ko
0.80
0.80
Angle of internal friction, φ' : degrees
20
20
Cohesion, c': kPa
20
20
Initial void ratio, e0
-
0.585
Slope of isotropic unloading line, κ
-
0.00158
Slope of isotropic compression line, λ
-
0.00475
Over consolidation ratio, OCR
1
1
Effective Drained/Undrained
Effective Drained/Undrained
Parameter
Young's modulus, E: MPa Poisson's ratio, ν'/ν'ur
Analysis Type
67
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
0 5 10
Settlement (mm)
15 20 25 30 35 40 45 50
Observed, Sommer & Hambach (1974) Mohr-Coulomb-Drained Mohr-Coulomb-Udrained Enhanced Clay ModelDrained Enhanced Clay ModelUdrained
55
Figure (3-6) Predicted and observed load-settlement relation-ship
3.2.3.3.2 Test problem 2: Back analysis of monitored pile load test in non cohesive soil.
This test problem is a back analysis of a pile load test in San Francisco, reported by Briaud et. al. (1989). The tested pile is a close ended steel pipe pile: 27.3 cm outer diameter, 0.93 cm wall thickness and embedded length of 9.15 m and is located in medium dense sand. Figure (3-7) gives the general test conditions of the pile load test. This test problem aims at studying the efficiency of Mohr-Coulomb model in modelling of frictional soils (sand) compared to the Modified Mohr-Coulomb model. Two dimensional axi-symmetric finite element analysis of the problem is performed twice, once with
Mohr-Coulomb model and the other with the Modified Mohr-Coulomb
model. Table (3-2) summarizes, the estimated soil parameters for both the MohrCoulomb model and Modified Mohr-Coulomb model. Results obtained from both 68
models are compared to field measurements as shown in figure (3-8). It can be seen that the results obtained using the Mohr-Coulomb model compares well with their observed counterparts as well with those obtained using the Modified Mohr-Coulomb model.
5.0 ft
Q
Sandy Gravel (Fill) 5.0 ft 8.0 ft
GWL PIPE PILE
Medium Sand
Figure (3-7) General test conditions of the pile load test, Briaud et. al. (1989)
Table (3-2) Soil Material parameters used in the finite element analyses of test problem 2: Parameter
Young's modulus, E: MPa Poisson's ratio, ν'/ν'ur Total unit weight of moist soil, γ: kN/m3 Buoyant unit weight of moist soil, γ': kN/m3 Coefficient of earth pressure at rest, Ko Angle of internal friction, φ' : degrees Cohesion, c': kPa
Mohr Coulomb model
Modified Mohr Coulomb model
51
-
0.30/-
-/0.20
19.25
19.25
9.25
9.25
0.82
0.82
35.5
35.5
2
2
69
Parameter of elasticity, m
power
low
-
0.50
Over consolidation ratio, OCR
-
Equation (3-11)*
Slope of isotropic unloading line, κ
-
0.0033
Tensile pressure, Pt (kPa)
-
2
Effective-Drained
Effective-Drained
Analysis Type *
K o = OCR. K nc − (OCR − 1).
ν ur′ ′ 1 − ν ur
(3-11)
Where, Knc = the K-ratio for normally consolidated soil.
Load (kN) 0
100
200
300
400
500
600
0.0
Settlement (mm)
5.0
10.0
15.0
20.0
Observed, Briaud et al. 1989
Mohr-Coulomb w ithout tip interface
25.0 Modified MohrCoulomb
30.0
Figure (3-8) Predicted and observed load-settlement relation-ship 3.2.3.3.3 Test problem 3: Analysis of piled raft foundation in stiff clay.
70
In order to examine the validity of Mohr-Coulomb model for modelling of cohesive soils, in three dimensional finite element analysis of piled raft foundation, a hypothetical example of piled raft foundation shown in figure (3-9), that will be dented by reference through out this chapter, is considered. It is also evident that using Mohr-Coulomb non associative plasticity model (ψ≠φ) in finite element analysis results in non-symmetric stiffness matrices. In large three dimensional finite element analysis of piled raft foundation system, non associative plasticity may lead to dramatic increase in computational demands and CPU time. The reference problem was solved three times, the first using Mohr-Coulomb model (ψ=φ), the second using Mohr-Coulomb model (ψ=0) and the third using the Enhanced Delft-clay model. The finite element mesh only presents one eighth of the foundation due to its three fold symmetry. Table (3-3) summarizes, the estimated soil parameters for both the Mohr-Coulomb model and Enhanced Delft-clay model. The results of the load settlement at the centre of the raft were plotted for the two models in figure (3-10). It can be seen that both models almost resulted in the same response until a certain point, which may be defined as the system yield point, after which the Mohr-Coulomb model starts to violate the actual behaviour, which is expected from such open (uncapped) models as it is not able to describe plastic deformations inside the yield surface. In spite of this, the Mohr-Coulomb model (ψ=φ) in this type of problems, proved to reproduce accurately enough the soil behaviour up to the above defined yield point.
Table (3-3) Soil Material parameters used in the finite element analyses of test problem 3: Parameter
Mohr Coulomb
Enhanced Delftclay model
Young's modulus, E: MPa
Equation (3-2)**
-
0.30/-
-/0.20
20
20
10
10
Poisson's ratio, ν'/ν'ur Total unit weight of moist soil, γ: kN/m3 Buoyant unit weight of moist soil, γ': kN/m3
71
Coefficient of earth pressure at rest, Ko Angle of internal friction, φ' : degrees
1
1
25
25
Cohesion, c': kPa
5
5
Initial void ratio, e0
-
0.82
-
0.02
-
0.15
-
Equation (3-12)
Effective-Drained
Effective-Drained
Slope of isotropic unloading line, κ Slope of isotropic compression line, λ Over consolidation ratio, OCR Analysis Type **
E = 48 + 3.456 Z ,
(3-12)
GWT
1.50 1.50 1.50
NGL
0.50
5.50
10.00
0.50 1.50 1.50 1.50
0.50
1.00
where, E : Young's modulus (MPa). z : the depth below the ground surface (m).
0.50
Dia.=0.50 m
5.50
Figure (3-9) Hypothetical example, (a) Foundation plan, (b) Soil stratification and position of foundation
72
Load (kN) 0
4000
8000
12000
16000
0.0
5.0
Settlement (mm)
10.0
15.0
20.0
25.0
30.0
Mohr-Coulomb model ψ=∅
Enhanced Clay model
Mohr-Coulomb model ψ=0
35.0
Figure (3-10) Load-settlement relation-ship at the center of the raft
3.2.4 Modelling of soil structure interface 3.2.4.1 General
An interface may be defined as the contact plane between two bodies of different material properties. Modelling of the contact behaviour is a problem of great importance in soilstructure interaction problems. In the present research, friction type interface elements are used to model the contact behaviour between pile shafts and surrounding soil. This frictional behaviour can be modelled with the Coulomb friction model shown in figure (3-11), as a constitutive relation for the interface elements, which has close resemblance with Mohr-Coulomb plasticity model for continuum elements. In general interface elements should accurately predict stresses along and across the interface. However, care must be exercised, when using interface elements in the finite element model, because there are many difficulties that may occur such as, ill-conditioning of the stiffness matrix, 73
poor convergence of the nonlinear solution algorithm, steep stress oscillation and possible pre-processing problems. Abdel-Fattah (2004), discussed these difficulties and suggested some useful techniques for overcoming them. For more details, interested reader may refer to Abdel-Fattah (2004).
Figure (3-11) Coulomb friction criterion used for interface elements (DIANA® 9.1 users manual, 2005)
3.2.4.2 The importance of using interface elements in the analysis of piled rafts
In any soil–structure interaction situation, relative movement of the structure with respect to the soil can occur. The use of continuum elements prohibits relative movement at the soil–structure interface as the nodal compatibility of the finite element method constraints the adjacent structural and soil elements to move together. Many methods have been proposed to model discontinuous behaviour at the soil–structure interface. Of these methods, the use of zero-thickness interface elements has become the most popular (DIANA® 9.1 Geotechnical manual, 2005). Particularly in piled raft foundation problems, interface elements are generally required in three positions, pile shaft-soil, raft-soil and pile tip-soil. To discuss the importance of the interface elements between pile shaft and surrounding soil, the problem of the pile load test described in 3.2.3.3.2 is resolved without interface elements between pile shaft and surrounding soil. Two configurations for the finite element mesh were studied, which are a fine mesh and a coarse mesh as shown in figure (3-12), and the results are plotted on figure (3-13). It can be seen that for fine mesh the benefit of using interface element doesn’t worth the difficulties associated with interface elements while for coarse mesh, it is clear that using interface elements
74
enhanced the solution such that it is in good agreement with the correct solution. This means that interface elements reduce mesh dependency in such problems. The above findings agree well with those obtained by Wehnert and Vermeer (2004). In case of three dimensional piled raft foundation problems, since fine meshing is not affordable, thus using interface elements between pile shaft and surrounding soil becomes necessary. The contact between raft and soil beneath it as well as the contact between piles’ tips and the soil beneath it, can be modelled as perfectly rough, i.e. no relative movement between the nodes of the finite elements representing the raft or piles’ tips and those of the finite elements representing soil beneath it. The above assumption is valid for mainly vertically loaded piled rafts with small horizontal forces, typically wind or earthquake loads but for piled rafts that is mainly subjected to horizontal forces (out of scope of the present research), interface elements should be added between raft and soil. To show the validity of the above assumption, the problem of the pile load test described in 3.2.3.3.2 is resolved using interface between the pile tip and soil beneath it. The results are compared to the case without tip interface as shown in figure (3-14). It is clear that the tip interface almost has no effect on the pile response.
(a) Fine mesh
(b) Coarse mesh
Figure (3-12) Finite element mesh configurations
75
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
0.0
5.0
Settlement (mm)
10.0
15.0 Mohr-Coulomb w ith interface - fine mesh
20.0
25.0
Mohr-Coulomb w ithout interface fine mesh Mohr-Coulomb w ith interface Coarse mesh Mohr-Coulomb w ithout interface Coarse mesh
30.0
Figure (3-13) Effect of pile shaft-soil interface on the load-settlement relation-ship Load (kN) 0
100
200
300
400
500
600
0.0
Settlement (mm)
5.0
10.0
15.0 Observed, Briaud et al. 1989
20.0 Mohr-Coulomb w ithout tip interface
25.0 Mohr-Coulomb w ith tip interface
30.0
Figure (3-14) Effect of tip interface on the load-settlement relation-ship
76
3.2.4.3 Characteristics of the Coulomb friction model incorporated in DIANA® program
The friction model incorporated in DIANA is characterized by the following features: 1. Both associated and non associated plasticities can be modelled. 2. Both cohesion hardening and friction hardening can be modelled. 3. The gap criterion can be modelled, this means that a gap arises if the tensile traction normal to the interface exceeds a specified tensile strength value. Either brittle shear behaviour or constant shear stiffness reduction can be modelled after gap opening and if gap closure follows, the interface behaviour is governed by the Coulomb friction criterion again. Table (3-4) summarizes the Coulomb constitutive law of the frictional contact between structure and soil. Table (3-4): Constitutive law for frictional behaviour of interface elements (DIANA® 9.1 users manual, 2005)
77
3.2.4.4 Material parameters for interface modelling
The parameters are divided into: 1.
Elastic parameters which are, the shear stiffness component denoted by Kt and the
normal stiffness component denoted by Kn, where, the initial elastic shear stiffness may be calculated according to DIANA user manual as follows: Es A2 . Kt = t int 2(1 + ν s ) ,
(3-13)
Where, A= a reduction factor to make the pile-soil interface more flexible than the surrounding soil = 0.67 tint = virtual thickness of the interface = 0.01 m
Es = Young’s modulus for the soil,
ν s = Poisson’s ratio for the soil. The initial elastic normal stiffness may be calculated according to the following equation: K n ≈ 100 E s ,
(3-14)
The normal behaviour is set as zero stiffness in tension. The value of the initial elastic shear stiffness of the interface (Kt) can be practically taken as 0.10 Kn. The magnitude of the above elastic stiffness components should be large enough to model the initial continuous geometry (i.e. as if interface elements don’t exist) and the influence of interface is limited to the case of true plastic slip, where it enhances the flexibility of the model as reported by Van Langen and Vermeer (1991). 2.
Strength parameters which are, the parameters representing the roughness of the
interaction and can be estimated by selecting suitable reduced values for the soil strength parameters (c’ and φ’). The reduction factors (Rc and Rφ) should be evaluated experimentally, but in the absence of experimental data, the values of the reduction factors are usually dependent on engineering judgement. It is common to use an average value of 0.67 for Rc and 0.67 and 0.50 for Rφ in case of steel-sand and steel clay interfaces respectively, whereas slightly higher values are used for interaction with rough concrete. Moorman (2002) performed experiments with stiff over consolidated clay and
78
rough concrete surfaces typical for bored piles. He found that there is no considerable reduction for both friction and cohesion. 3.2.4.5 Geometric definition of interface elements
The interface elements suitable to model the contact plane between pile shafts and surrounding soil in three dimensional configuration should be surface to surface interface elements. The CQ48I, plane quadrilateral, 16 nodes three dimensional interface elements available in the DIANA element library are used in the present research. The formulation of these elements is fully isoparametric, allowing modelling of both straight and curved contact surfaces. Figure (3-15) shows the topology, local axes and displacements for these elements. One of the great problems that limited the use of interface elements in three dimensional finite element problems is that the real thickness of the interface is zero and thus the elements in normal direction can be seen as single lines which is rather inconvenient when checking connections, assigned properties, etc., thus the user can’t determine the source of error in the interface elements. To overcome this draw back, in the present research, the piles are initially given a smaller diameter than it is, allowing for a virtual gap between the pile and the surrounding soil. This gap is filled with interface elements with virtual thickness (equal to gap thickness). Once the model has been fully constructed and all the checks have been done, the thickness of the interface elements is reduced to zero by moving pile shafts against the surrounding soil to retrieve the original pile diameter again as illustrated in figure (3-16). Care should be given when modelling of pile soil interface in layered soil at the boundary between two soil layers where, sudden change in soil stiffness and consequently in the interface parameters take place. This may result in either solution divergence or unreliable results, therefore the interface mesh should be refined in the vertical direction around the boundary of the two layers.
Figure (3-15) 3-D interface element CQ48I (DIANA® 9.1 users manual, 2005)
79
Pile element
Interface element 2' Virtual Thickness of Interface Element
2 3'
3
1 4
1' Point 1 is moved to coincide with 1'
4'
Reduced Pile Diameter Original Pile Diameter (a)
(b)
Figure (3-16) Methodology used for construction of pile soil interface elements.
3.2.4.6 Using soil-soil interface elements
Soil-soil interface elements, are interface elements used in the three dimensional finite element analyses of the present research for the purpose of mesh compatibility. These elements overcome the problem of mesh discontinuity of the soil mass below pile tips, which resulted from pile soil interface elements as illustrated by figure (3-17). This type of elements was successfully used by other researchers such as Abdel-Fattah, (2004), who emphasised that the planes modelled using soil-soil interface elements are not zones of weakness since they have the same strength properties as the rest of the material. Care should be exercised in case of ending the interface elements just by the pile toe, this may result in distorted soil elements just below pile tip, thus un reliable results are obtained for toe resistance. Q
P ile
S o il p ile
Figure (3-17) Using soil-soil interface elements under pile tip level
80
3.2.5 Calculation of pile load from the out put stresses of the finite element analyses
Presentation of some out put results requires the nodal element forces rather than the nodal element stresses outputted by the original general purpose program DIANA. For the calculation of vertical pile load from internal stresses outputted from the DIANA results, a special purpose finite element program PILELO has been adopted by modifying the available finite element code DIATUN developed by Abdel-Fattah (2004). To verify the use of the program PILELO for calculation of the pile load, a three dimensional test problem is solved using the DIANA program, under the effect of a uniform pressure of 10 kN/m2 applied at the top surface of the pile as shown in figure (3-18). The output nodal vertical stresses are then inputted to the PILELO program, in order to evaluate the equivalent nodal vertical forces. at the top surface of the pile. The sum of the nodal forces at the top surface of the pile, was found equal to the applied load.
(a) Finite element mesh (b) Vertical stresses Figure (3-18) Test problem for verification of PILELO program 3.3 Suggested practical techniques for handling of large three dimensional finite element problems of piled raft foundation system 3.3.1 General
In most cases, the problem of detailed three dimensional finite element modelling of piled raft foundation system is a large one, that may be either beyond the capabilities of 81
advanced personal computers or it may require a large un acceptable CPU time from the practical view point, thus limiting the use of this method to research purposes only. To overcome the above drawbacks, it is necessary to propose some techniques to reduce the required computational demands associated with this type of problems, without violating the solution accuracy. The following sections present the proposed techniques with the necessary test problems to verify the validity of using such techniques. 3.3.2 Employment of three dimensional interface elements for modelling of far field bottom soil layer - A proposed technique 3.3.2.1 Description of the proposed technique
In large three dimensional finite element models of piled raft foundations, the depth of the soil mesh under pile tips is carefully selected, using either engineering judgement of expert geotechnical engineer, or using trials to determine this depth, such that no part of the soil contributing to foundation settlements is neglected. The mesh borders are best modelled using infinite elements to reduce the size of the mesh effectively without violating the accuracy of the results (Abdel-Fattah, et. al, 1998). Unfortunately, the majority of commercially available general purpose finite element programs (including DIANA program used to perform the finite element analyses of the current research), neither contain infinite elements, nor allow user defined elements. On the other hand, using finite elements for parts at mesh boundaries may in some problems, either increase the size of the mesh such that it may exceed the computer capabilities, or it may dictate using a coarse mesh and consequently lead to un reliable results. It is evident from previous studies, (e.g. Randolph, 1994, El-Mossallamy, 2000 and others) as well as from the results of the analyses performed in the current research, that far field zones of the finite element mesh mainly behave linear elastically and don’t suffer any plastic deformations. In the current research it is proposed to use three dimensional interface elements for modelling of far field bottom soil layers instead of the unavailable infinite elements. The normal stiffness of the interface elements equivalent to the replaced part of the finite element mesh of the soil layers (Kn), may be calculated according to the following:
82
•
For the case of simple elastic-perfect plastic models (e.g. Mohr-Coulomb model):
Kn =
E 50 Hr
(3.15)
Where, E50 = primary loading stiffness Hr = thickness of the soil layer modelled using interface elements •
For the case of advanced hardening plasticity models (e.g. Modified MohrCoulomb model and Enhanced Delft Clay model):
Eur Hr Where, Kn =
(3.16)
Eur = unloading/reloading stiffness For clayey soils, (Kn), may be estimated based on values of the Cam Clay elastic parameter (κ) of the bottom soil layer as follows: Kn =
3(1 + eo)( p ′o + pt )(1 − 2vur ) κ .Hr
(3.17)
In which, eo = initial voids ratio, p0′ = insitu mean stress, Pt = tensile pressure,
ν ur = unloading/reloading Poisson’s ratio = 0.25,
κ = rebound modulus =
Cr ln 10
where, Cr = elastic reloading modulus resulted from the Oedometer test.
3.3.2.2 Verification of the proposed technique
In order to verify the use of the proposed technique for piled raft foundation, a test problem of the reference piled raft, previously presented in 3.2.3.3.3 (The total number of nodes in this model is 10543 and the total number of elements is 2428), is resolved using the proposed technique by replacing the far field bottom layer (Hr=10m) by interface
83
element as shown by figure (3-19) (The total number of nodes in this model is 8582 and the total number of elements is 2004). The normal stiffness of the interface elements is calculated as 11712 kN/m3 according to equation (3-15), with Eur of 117 MPa. The out put central load settlement relationship, bending moments and distribution of contact pressure between raft and soil beneath it, resulting from the finite element analyses for both the proposed technique and the reference problem are plotted in figures (3-20) through (3-22) respectively. From these figures, it can be seen that the two sets of results are almost the same. This proves the validity of the proposed technique.
(a) reference
(b) proposed method
Figure (3-19) Finite element mesh
84
(a) reference
(b) proposed method
Figure (3-20) Load-settlement(m) relationship for the center of the raft
(a) reference
(b) proposed method
Figure (3-21) Contours of raft bending moment My (kN.m/m)
85
(a) reference (b) proposed method Figure (3-22) Distribution of contact pressure (kPa) between raft and soil along a line joining the raft center and the raft mid edge
3.3.3 Modelling of raft using shell elements
The raft may be modelled using shell elements instead of the solid brick or wedge elements. This has two advantages, the first advantage is to reduce the size of the finite element mesh and the second advantage is the ability to directly output the straining actions of the raft (bending moments, shearing forces and axial forces if any) without the need to integrate the nodal element forces as in the case of solid elements. To test the effect of replacement of solid elements by shell elements in modelling of raft a test problem in which the raft is modelled using solid elements as shown in figure (3-23) (The total number of nodes in this model is 7531 and the total number of elements is 1614) is solved and compared to the results of the reference piled raft, previously presented in 3.2.3.3.3 (The total number of nodes in this model is 7027 and the total number of elements is 1590).
86
(a)
(b)
Figure (3-23) Finite element mesh of raft and piles, (a) raft modeled using shell elements (b) raft modeled using solid elements
(a)
(b)
Figure (3-24) Load-settlement(m) relationship for the center of the raft, (a) raft modeled using shell elements (b) raft modeled using solid elements
87
3.4 Simplified method for analysis of piled raft foundations subjected to lateral loading 3.4.1 Description of the proposed simplified method
In most cases the structural geometry of high rise buildings is almost symmetric in order to provide adequate resistance to lateral loading. As a result of symmetry, under vertical loading, only a part of the building foundation usually, a quarter or one eighth is modelled via detailed three dimensional finite element analysis, (this part will be denoted here as the reduced model) to reduce computational demands. On the other hand, complete analysis of such foundations dictates considering the problem under the potential of lateral loading due to wind or earthquakes in addition to vertical one. This requires considering the full model or half of the model in case of three fold symmetry, which is usually beyond the capabilities of advanced personal computers. Herein a simplified method for the finite element analysis is proposed and verified. The proposed method requires two stages of analysis. The first stage is a detailed three dimensional non linear finite element analysis of the reduced model under vertical loads as outlined in section 3.2 in order to evaluate the following: •
The portion of the total vertical working load transferred by the raft directly to soil beneath it (Pr).
•
The portion of the total vertical working load transferred to soil through piles (Pp).
•
The average settlement of the foundation under total vertical working load (Savg).
The second stage is a two dimensional finite element analysis of the raft, with both soil and piles modelled using interface elements with normal stiffness estimated from the first stage as follows:
Ks =
Pr ( Ar − Ap ) * Savg
(3.18)
Kp =
Pp Ap * Savg
(3.19)
Where,
88
Ks = normal stiffness of interface elements representing soil Kp = normal stiffness of interface elements representing piles Ar = area of the raft Ap = area of the pile group Care should be taken that, the above proposed method has the following assumptions and limitations: 1. The method is applicable for lateral loads that are not expected to have considerable potential on the stiffness matrix of the model (e.g. the magnitude of the lateral loads due to wind and earthquakes on piled raft foundations of high rise buildings, which are usually in the order of about 2% of the vertical loads). 2. Both stiffness of soil and piles obtained from the results of first stage of analysis, are secant values at the level of vertical working load. 3. The foundation elements in the second stage behave linearly elastic. 4. The effect of direct horizontal loading (base shear) on piles can be neglected.
3.4.2 Verification of the proposed method
In order to verify the use of the proposed method for piled raft foundation, a test problem of the reference piled raft, previously presented in 3,2.3.3.3 is solved considering lateral loads, once using detailed three dimensional non linear finite element analysis (exact model) on half the full model due to three fold symmetry and then resolved using the proposed two stage simplified method using one eighth of the full model for stage one as the reduced model. Figure (3-25) presents the finite element mesh of the exact model (the total number of nodes in this model is 20221 and the total number of elements is 5068), used for verifying the simplified method. The finite element mesh of the reduced model was previously shown in figure (3-19a) (The total number of nodes in this model is 7027 and the total number of elements is 1590), while the finite element mesh of the simplified model of the second stage is shown in figure (3-26) (The total number of nodes in this model is 1310 and the total number of elements is 364). From the out put results obtained from the detailed three dimensional finite element analysis of the reduced model, the distribution of load within the pile group was
89
calculated using the PILELO program and plotted on figure (3-27) and the total load on piles was assembled. Pp = 15112 kN Ptotal = PSuper structure + Wraft = 16000 + 741.92 = 16741.92
kN
Pr = 1125 kN Savg = 0.0178 m The normal stiffness of soil and piles are estimated according to equations (3-18) and (319) respectively as follows: Ks = 3389 kN/m3 Kp = 270241 kN/m3 The bending moments of both the exact and simplified methods are shown in figure (328), which reflects good agreement of the results of both methods.
Figure (3-25) Finite element mesh of the exact model
90
Interface elements are given thickness for clarity Figure (3-26) Finite element mesh of the simplified model
1.50 1.50 1.50
0.50
5.50
0.50 1.50 1.50 1.50
0.50
0.50
5.50
Figure (3-27) Load distribution within the pile group (kN)
91
(a) Detailed FE analysis
(b) Simplified method
Figure (3-28) Bending moment in the raft Mx (kN.m/m)
92
CHAPTER FOUR NUMERICAL INVESTIGATION OF PILED RAFT FOUNDATION BEHAVIOUR
4.1 Introduction In order to study the behaviour and performance of piled raft foundation in different structural and geotechnical conditions, a detailed numerical investigation was carried out on a hypothetical square building, founded on piled raft foundation. The structural supporting system of the building consists of a central core and perimeter columns, which is a common system for high rise buildings. An elasto-plastic three dimensional finite element model was used for the analyses of the piled raft foundation system. Various configurations of piles under the raft were studied with number of piles ranging from 9 to 64 piles. The corresponding single pile, conventional pile group and un piled raft were also investigated for the purpose of comparison. Different soil profiles were investigated and the main construction stages were modelled. The effect of relative pile length was also studied. Table (4-1) presents the cases studied in the parametric study. Results of the parametric study are presented for the case after installation of the raft. All the finite element analyses are carried out using the general purpose finite element program DIANA® 9.1 (TNO DIANA BV. 2005). Also program PILELO is used for calculation of equivalent nodal forces from the DIANA out put stresses, in order to evaluate pile loads as described in 3.2.5.
4.2 Subsoil conditions The parametric study included five soil types which are dense sand (DS), medium sand (MS), stiff over consolidated clay (SC), medium clay (MC) and soft to medium clay (SM). The non linear soil behaviour was simulated using the elasto-plastic MohrCoulomb model with non associative flow rule. The soil parameters used in the finite element analyses had been estimated according to the guide soil parameters presented by the Egyptian code of practice for soil mechanics, design and construction of foundations (202/3) for these soil types as well as from available data in literature (e.g. Wehnert and 93
Vermeer, 2004 and El-Kadi and Abdel-Fattah, 1998). Table (4-2) summarizes, the estimated soil parameters for the Mohr-Coulomb model. Combinations of the above soils are arranged in seven idealized soil profiles, each of a two layer soil system as shown by figure (4-1). Abbreviations used to identify different two layer soil profiles are written as ‘top soil layer – bottom soil layer’ (e.g. MC-DS means that the top soil layer is medium clay and the bottom soil layer is dense sand). Both of the foundation level of the raft and the ground water table are located at the same level of two meters below original ground surface. The piles penetrate the bearing layer (bottom soil layer) with a length of three times the pile diameter (1.5 m), which is practically common.
13.00m 1.50 m
2.00 m
GWT
(Lp-1.5) m
F.L.
NGL
1.50 m
Top soil layer
15.00 m
Dia.=0.50 m
Bottom soil layer
Bottom of the FE mesh
Figure (4-1) General Soil stratification and cross section of foundation
94
4.3 Geometric and material properties of the foundation elements (raft and piles) The structural system of the building on which the parametric study will be performed is shown in figure (4-2). The foundations of the building consists of reinforced concrete piles ranging from 9 to 64 piles arranged in five different configurations as shown in figure (4-3), each pile of diameter 0.50 meter which is held constant through out the present study and three pile lengths (Lp) of 10, 15 and 20 meters were investigated. The piles are capped by a square reinforced concrete raft 13 * 13 meter in plan, 1.5 meter in thickness and in direct contact with the ground (except for the case of conventional pile group, the raft rests on piles only) as shown in figure (4-1). Both the reinforced concrete piles and raft are considered to behave linearly elastic with Young’s modulus of 21000 MPa and Poisson’s ratio of 0.20. As a result of three fold symmetry of the square piled raft, the model only represents one eighth of the analysed piled raft.
B
A
C
D
12.50 0.25
4.00
0.25
4.00
4.00 1
2
2 12.50
1
STAIRS
3
3
4
4
A
B
C
D
Figure (4-2) Structural system of building’s typical floor
95
C
D
B
A
2.40
3
4
4
2.40 2.40
D
1
3
3
4
4
4.00
Y axis
C
D 0.50
4.00
4.00
6.00
1
6.00 0.50 6.00
4.00 2
2
3
4
4
X axis 3
4.00
3
2
6.00
X axis
4.00
X axis
2.40
D
(e) Config D - 16 piles
B
C
(f) Config E - 9 piles
Figure (4-3) Piled raft configurations for the parametric study 96
4
0.50 A
Y axis
C
4.00
0.50
2.90
Y axis
B
D
0.50
1
2.90
4
A
C
0.50
0.50
2.40
13.00
3
4
13.00
13.00
2.90
2.40
2
X axis
B
A
4.00
2.40
2.40
B
0.50
1 2.40
X axis
0.50
0.50
A
D
1 2.90
0.50
(d) Config C - 16 piles
C
4.00
1
3
0.50
4.00
4.00
2
D
13.00 0.50
4.00
4.00
3.00
Y axis
B
4.00
2
X axis
(c) Config B - 25 piles
A
4.00
4.00
3.00
2
3.00 C
4.00
4.00
0.50
0.50 B
Y axis
A
0.50
4.00
4.00
0.50
13.00
4
D
0.50
1
X axis
3
C
0.50
0.50
3.00
2
X axis
D
13.00
4.00
3.00
C
4.00
4.00
3.00
B
A 0.50
3.00
3.00
B
Y axis
D
1 0.50
3
Y axis
C
0.50
4.00
X axis
(b) Config A - 36 piles
13.00 0.50
2
A
13.00
Y axis
B
0.50
Y axis
C
(a) Config A1 - 64 piles A
2.40
0.50
Y axis
B
2.40
4
12.00 A
2.40
2.40
4.00 0.50
0.50
4
2.40
2.40
3 6.00
3
X axis
4.00
13.00
X axis
13.00
2
2
1
0.50 6.00
2
0.50
0.50
1
1
6.00 0.50
X axis
D
2.40
6.00
4.00
1 0.50
C
13.00 0.50
13.00
Y axis
Y axis
B
A
D
Table(4-1) : The program of the parametric study Config (A1)1 Config (A)1 Pile Soil4 64 piles 36 piles Single Stratific Length pile (m) ation PR2 PG3 PR2 PG3 Lp=10 MC-DS
MC-MC
SC-SC
x
x
x
Config (B)1 25 piles
Config (C)1 16 piles
Config (D)1 16 piles
Config (E)1 9 piles
PR2
PG3
PR2
PG3
PR2
PG3
PR2
x
x
x
x
x
x
x
x
Lp=15
x
x
x
Lp=20
x
x
x
Lp=10
x
x
x
x
x
x
x
x
x
x
x
Lp=10
x
x
x
x
x
x
x
x
x
x
x
Lp=15
x
x
x
Lp=20
x
x
x
MS-MS
Lp=10
x
x
x
DS-DS
Lp=10
x
x
x
x
x
x
x
x
x
MC-MS
Lp=10
x
x
x
x
x
x
x
x
Lp=10
x
x
x
x
x
x
x
x
SM-MC
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Lp=20
x
x
x
97
x
x
Lp=15
1 pile configurations illustrated in figure (4-3) 2 PR = piled raft 3 PG = conventional pile group 4 top soil layer – bottom soil layer (for abbreviations refer to table 4-2)
PG3
Un piled Raft
x
Table (4-2) Soil Material parameters used in the finite element analyses for the parametric study Parameter
DS1
MS2
SC3
MC4
SM5
Young's modulus, E: MPa
100
60
45
10
2
Poisson's ratio, ν'
0.3
0.3
0.3
0.3
0.3
Total unit weight of moist soil, γ: kN/m3
18.5
17.5
19.5
18.5
17
Buoyant unit weight of moist soil, γ': kN/m3
8.5
7.5
9.5
8.5
7
Coefficient of earth pressure at rest, Ko
0.38
0.44
0.8
0.58
0.50
Angle of internal friction, φ' : degrees
38
34
20
25
30
Cohesion, c': kPa
1
1
20
1
1
Material model
Elasto-Plastic Mohr-Coulomb
Analysis Type
Effective drained
1. 2. 3. 4. 5.
Dense sand Medium sand Stiff over-consolidated clay Medium clay Soft to medium clay
4.4 Soil structure interface modelling To model the non linear soil-pile interface behaviour, Mohr-Coulomb friction type interface elements are used for modelling the interface between pile shafts and surrounding soil. The initial elastic stiffness of the interface is calculated according to equations (3-13) and (3-14). The reduction factors of the interface parameters (Rc and Rφ) between rough concrete piles and soil are taken as one according to the experimental work performed by Moorman (2002). The contact between raft and soil beneath it was modelled as perfectly rough, i.e. no relative movement between the nodes of the finite elements representing the raft and those of the finite elements representing soil beneath it. The contact between piles’ tips and the soil beneath it was also modelled as perfectly rough. For conventional pile groups, the nodes of the finite elements representing the raft
98
and those of the finite elements representing soil beneath it are fully separated i.e. the raft only rests on piles.
4.5 Methodology Through out the current parametric study, only drained analyses are carried out as the purpose of the finite element modelling is to predict the maximum values of settlement and the straining actions. In order to simulate the main stages of construction of piled raft foundation system, The analyses are performed in three phases using mesh adaptation technique as follows: Phase 1: presents the soil before execution i.e. only finite elements representing soil layers are activated to calculate the in-situ stresses. Phase 2: this phase represents the excavation to foundation level by deactivating the finite elements representing soil layer above foundation level. Phase 3: this phase represents the case after construction of the piles and/or the raft, i.e. the finite elements representing both piles and/or the raft are activated and those representing soil replaced by piles (excavated soil) are deactivated. The loads due to own weight of the piles and/or the raft are applied incrementally, then the super structure loads shown in figure (4-4) are applied at column positions and are increased in steps. In case of single piles, loads are applied at the pile top and increased incrementally up to failure. A
B
C
D 0.50
4.00
4.00
4.00
0.50
0.50
1
4.00
1
2 4.00
2
3
4
0.50
4.00
3
4
A
B
C
D
Figure (4-4) Column loads per floor including own weight
99
4.6 Behaviour of un piled rafts, piles and conventional pile groups The piled raft foundation system is a composite which consists of piles, raft and supporting soil. For the purpose of comparison, in the following subsections, the behaviour of each of single piles, conventional pile groups and un piled rafts, is simulated separately with the sub soil conditions outlined in 4.2.
4.6.1 Unpiled raft The raft which is in direct contact with the ground is a main element of piled raft foundation system. In the present research, behaviour of un piled raft is firstly investigated for the subsoil conditions outlined in 4.2. Figure (4-5) presents the three dimensional finite element mesh of the problem. The load settlement relationships of the un piled rafts for the different soil profiles analysed in the present parametric study are plotted in figure (4-6). The ultimate load of the foundation may be estimated from the load settlement relationship according to the ultimate load criterion suggested by De Beer (1967), which defines the ultimate load at the point of break of the load settlement curve in a log/log plot (figure 4-7). According to the Egyptian code of practice for soil mechanics, design and construction of foundations (202/3), the allowable working load (Pwr) for un piled raft should satisfy the following two conditions: 1. The maximum allowable total settlement for raft foundation ranges between 100-150 mm for clayey soils and ranges between 70-100 mm for sandy soils. 2. A minimum factor of safety against bearing capacity failure of supporting soil, not less than 2.5. In the present research, the allowable total settlement (Sall) is defined as the settlement value corresponding to the previously defined allowable working load. Table (4-3) presents the allowable working load for the un piled raft (Pwr) and the corresponding settlement (Sall), for the different soil profiles investigated in the present study. From figure (4-6) and table (4-3), there are some points to note about the range of allowable total settlement for un piled raft foundation provided by the Egyptian code of practice for soil mechanics, design and construction of foundations (202/3):
100
•
In the case of sandy soils and medium clay soils, this range of settlement is appropriate and ensures adequate factor of safety against bearing capacity failure.
•
In case of stiff over consolidated clayey soils, allowable total settlements should be reduced to ensure adequate factor of safety against bearing capacity failure.
•
In case of soft to medium clayey soils, it can be seen that even at total settlement level of more than twice the allowable total settlement, there still an adequate factor of safety against bearing capacity failure. From economic point of view, the allowable total settlement may be increased provided that, this increase has no considerable effect on the building safety and function in addition to the water supply and sanitary drainage connections.
(a) Complete mesh
(b) Raft mesh
Figure (4-5) Finite element mesh for unpiled raft
101
0 100 200 300
Settlement (mm)
400
SM -M C M C-M C M C-DS M C-M S SC-SC M S-M S DS-DS
500 600 700 800 900
1000 1100 1200 0
50
100
150
200
250
300
350
Load (MN)
Figure (4-6) Load settlement relationship of un piled raft for different soil profiles
10
S ettlem en t (m m )
100
1000
SM-MC
MC-MC
MC-DS
10000
MC-MS
SC-SC
MS-MS
DS-DS 100000 10
100
1000
Load (MN)
Figure (4-7) Log/log plot for estimation of ultimate load for un piled rafts
102
Table (4-3) Allowable working load for un piled raft Soil profile
Ultimate load (Pu) MN
Allowable load (Pwr) MN
Allowable settlement (Sall) mm
MC-MC
›58*
23.3
150
MC-MS
≈80
33.3
150
MC-DS
≈80
34.2
150
SC-SC
120
50
70
MS-MS
›192*
77
100
DS-DS
›350*
140
100
SM-MC
››15.6*
6.25
150
* 2.5 times the allowable load (Pwr) which corresponds to the maximum allowable total settlement defined by the Egyptian code of practice for soil mechanics, design and construction of foundations (202/3) 4.6.2 Single piles To simulate the behaviour of single piles, two dimensional axi-symmetric elasto-plastic finite element analyses are carried out for the subsoil conditions outlined in 4.2. Both piles and soil are modelled using eight node isoparametric axi-symmetric solid elements. The interaction between pile and surrounding soil is modelled using six node friction type interface elements as outlined in 4.4. Figure (4-8) presents the finite element mesh of the problem. The load settlement relationships of the single piles for the different soil profiles analysed in the present parametric study are plotted in figure (4-9). The ultimate load for the single piles are estimated according to the Egyptian code of practice for soil mechanics, design and construction of foundations (202/4), such that the ultimate pile load corresponds to a settlement equal 10% of the pile diameter. Table (4-4), summarizes the estimated ultimate load for single piles.
103
(a) Complete mesh
(b) Pile mesh
Figure (4-8) Axi-symmetric finite element mesh for single piles
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.00
M C-DS M C-M C M C-M S SC-SC DS-DS M S-M S SM -M C
Settlement (m)
0.05
0.10
0.15
0.20
0.25
0.30
Figure (4-9) Load settlement relationship of single piles for different soil profiles – Lp=10 m.
104
Table (4-4) Ultimate load for single piles Soil profile
Ultimate pile load (KN)
MC-MC
345
MC-MS
730
MC-DS
1150
SC-SC
880
MS-MS
645
DS-DS
1650
SM-MC
260
4.6.3 Conventional pile group In this section, selected cases listed in table (4-1) of conventional pile groups (raft is not in direct contact with soil beneath it), are studied. The finite element mesh for those problems are the same as for the corresponding piled raft foundations except that the nodes of the finite elements representing the raft and those of the finite elements representing soil beneath it are disconnected. Later in this chapter, the results of analyses of conventional pile groups, is plotted on graphs against the results of analyses of both, the corresponding un piled raft and piled raft to detect the differences in behaviour between these foundation systems.
4.7 Behaviour of piled raft foundation The three dimensional finite-element mesh for the piled-raft model for different pile configurations are shown in Figure (4-10). The raft is modelled using 8-noded and 6noded curved shell elements. The piles are modelled using 20-noded brick and 15-noded wedge elements. Figure (4-11) shows the meshing for the raft and piles for the different pile configurations. The soil layers are modelled using 20-noded brick and 15-noded
105
wedge elements. The interaction between piles’ shafts and soil is presented using 16noded interface elements. The interface elements between the pile shaft and soil are shown in Figure (4-12). Table (4-5) presents the total number of finite elements and nodes in the finite element mesh for the different pile configurations.
Table (4-5) Total number of finite elements and nodes in the finite element mesh for the different pile configurations. Pile configuration
Number of finite elements
Number of nodes
Config A1
5218
20749
Config A
5426
22559
Config B
4506
19653
Config C
4095
17819
Config D
4274
18569
Config E
4317
19407
106
(a) Config A1
(b) Config A
(c) Config B
(d) Config C
(e) Config D
(f) Config E
Figure (4-10) The finite element mesh for different pile configurations
107
(a) Config A1
(b) Config A
(c) Config B
(d) Config C
(e) Config D
(f) Config E
Figure (4-11) Raft and piles finite element mesh for different pile configurations
108
(a) Config A1
(b) Config A
(c) Config B
(d) Config C
(e) Config D
(f) Config E
Figure (4-12) Pile shaft – soil interface elements for different pile configurations
109
4.8 Results of analyses
4.8.1 Load settlement behaviour of piled raft In general the settlement consideration plays a main role in foundation design and the choice of appropriate factor of safety for the foundation depends to large extent on how much settlement, the supporting soil can tolerate. In this section, the effect of soil stratification and number of piles on the average and differential load settlement behaviour of the piled raft is presented.
4.8.1.1 Load - average settlement behaviour of piled raft Figures (4-13) through (4-21) present the effect of number of piles on average load settlement behaviour of the piled raft for different soil profiles outlined in table (4-1). The reference behaviour of unpiled raft is also presented. The average settlement of raft (Sav) and the applied load (P) are normalized by the allowable settlement for unpiled raft (Sall) and the allowable working load for unpiled raft (Pwr) respectively, (refer to 4.6.1). Generally, the average settlement decreases with the increase of number of piles and the amount of reduction in average settlement depends on soil stratification. It can be seen from these figures that the load settlement behaviour of piled raft foundation may be classified according to soil stratification into two main cases: 1. The case of uniform soil stratification, (floating pile group). In the current study this type is thoroughly investigated for various soil profiles namely MC-MC, SC-SC, MS-MS and DS-DS. In this case a relatively low rate of improvement in average settlement may be observed with increasing number of piles. The column chart in figure (4-20) shows that for Sav/Sall=1, the differences in the percent of increase in the load carried by the piled raft with increasing number of piles for all the studied soil profiles in this case, is within a narrow range, (percent of improvement ranged from 6% to 15% for a group of 9 piles up to 51% to 58% for a group of 64 piles). 2. The case of two layered soil stratification, in which the strength of soil layer beneath pile tip is greater than the strength of the top layer containing the pile shafts and supporting the raft, (point bearing pile group). In the current study this type is thoroughly investigated for various soil profiles namely SM-MC, MC-MS and MC-
110
DS. In this case a relatively higher rate of improvement in average settlement is observed compared to the previous case. The column chart in figure (4-20) shows that for Sav/Sall=1, the percent of increase in the load carried by the piled raft with increasing number of piles depends on the strength of the point bearing layer. The percent of improvement in case of a group of 9 piles were 49%, 32% and 63% for soil profiles SM-MC, MC-MS and MC-DS respectively. For a group of 64 piles the corresponding percent of improvement were about 242%, 200% and 375% for soil profiles SM-MC, MC-MS and MC-DS respectively. In addition to the above classification, another important issue can be read from the above mentioned figures. Allowing for 50% increase in the allowable settlement of the piled raft (Sav/Sall=1.5) compared to the allowable settlement of unpiled raft previously discussed in 4.6.1, still ensures adequate factor of safety against bearing capacity failure for all soil stratifications except for cases of uniform medium clay (MC-MC) and uniform stiff clay (SC-SC) and pile spacing is greater than five times the pile diameter. For the case of MCMC and SC-SC, pile spacing in excess of five times the pile diameter, no significant improvement of the load settlement behaviour of piled raft, can be recognized. Figure (421) shows significant increase in the load carried by the piled raft in case of (Sav/Sall=1.5) compared to the case of (Sav/Sall=1) shown in figure (4-20). The above issue may be useful for cases where, piles are required to carry the major part of the applied load rather than reducing settlements. The above mentioned figures also show that for both of pile configuration ‘Config C’ with uniformly distributed 16 piles and the pile configuration ‘Config D’ with central 16 piles almost yield the same average settlement. This means that for the same number of piles, the variation of the pile arrangement has almost no effect on the average settlement of the piled raft. 4.8.1.2 Load – differential settlement behaviour of piled raft The differential settlement (∆S) considered in the current research is the difference between the total settlement at the centre and the corner of the raft, which is believed to be the maximum difference. The results of analyses, show that the deformed shape of an 111
unpiled raft from the beginning of loading and up to loads in excess of the allowable working load (Pwr), is dish-shaped but at higher load levels, (depending on soil stratification) negative values of differential settlement (the settlement at raft corner exceeds that at raft centre) can be observed. Although, addition of a limited number of piles to unpiled raft significantly reduces the average settlement, it doesn’t necessarily reduce differential settlement. As can be seen from Figures (4-22) through (4-28), uniformly distributed piles (pile configurations A1, A, B, C and E) the differential settlement of piled raft is generally greater than that of the corresponding unpiled raft. In addition, it can be observed that increasing number of piles reduces the differential settlement and no negative values are observed. The efficiency of central piles was investigated by many researchers (e.g. Horikoshi and Randolph, 1998 and Poulos, 2001 and others). They reported that for control of differential settlement, a relatively small number of piles located at the centre of the raft is more effective than using a large number of piles uniformly distributed over the raft area. In the current research, the central pile configuration (pile configuration D), generally reduced the differential settlement. Negative values of differential settlement can be observed, at high load levels for the case of uniform soil profiles and from the beginning of loading for the case of two layered soil profile, in which the strength of the soil layer beneath pile tip is greater than that of the top layer containing pile shafts. Therefore the value of differential settlement depends on the soil stratification and the following conclusions can be read from results: •
The differential settlement is inversely proportional with soil strength or in other words it is directly proportional to raft soil stiffness ratio. This means that softer soil stratifications, (e.g. SM-MC and MC-MC) undergo smaller differential settlements. This conclusion supports increasing allowable total settlements of piled raft foundation in case of soft soils because the corresponding differential settlements are relatively small and are not expected to cause damage to the super structure.
•
Good results were obtained in the case of uniform soil profiles (floating pile group) rather than in the case of two layered soil profile, in which the strength of the soil layer beneath pile tip is greater than that of the top layer containing pile shafts (end bearing pile group). For point bearing piles, at high load levels the absolute value of
112
differential settlement may exceed that of corresponding uniformly distributed pile group. In this case increasing number of uniformly distributed piles tends to be more reliable for control of differential settlement. •
Care should be taken when using central piles to control differential settlement because the deformed shape of the raft in this case is totally different compared to the corresponding uniformly distributed piles. As a result, the magnitude and distribution of bending moments in the raft are expected to change as will be discussed later in this chapter.
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0
1
Sav/Sall
2
3
4
5
6
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
7
Figure (4-13) Load settlement relationship of the piled raft for soil profile ‘MC-MC’
113
0
0.5
1
1.5
P/Pwr 2
2.5
3
3.5
4
0
1
Sav/Sall
2
3
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
4
5
6
7
Figure (4-14) Load settlement relationship of the piled raft for soil profile ‘SC-SC’
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1
Sav/Sall
1.5 2 2.5 3 3.5 4
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
Figure (4-15) Load settlement relationship of the piled raft for soil profile ‘MS-MS’
114
P/Pwr 0
0.5
1
1.5
2
2 .5
0
0.5
Sav/Sall
1
1.5
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
2
2.5
Figure (4-16) Load settlement relationship of the piled raft for soil profile ‘DS-DS’
0
0.5
1
1.5
2
2.5
3
P/Pwr 3.5
4
4.5
5
5.5
6
6.5
7
0
1
Sav/Sall
2
3
4
5
6
7
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
Figure (4-17) Load settlement relationship of the piled raft for soil profile ‘MC-MS’
115
0
1
2
3
P/Pwr 4
5
6
7
8
0
1
Sav/Sall
2
3
4 Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
5
6
7
Figure (4-18) Load settlement relationship of the piled raft for soil profile ‘MC-DS’
0
1
2
3
P/Pwr 4
5
6
7
8
0 1
Sav/Sall
2 3 4 5 6
Config A1 (64 piles) Config A (25 piles) Config B (16 piles) Config C (36 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
7
Figure (4-19) Load settlement relationship of the piled raft for soil profile ‘SM-MC’
116
5.0
4.75
Config A1 (64 piles) 4.5
Config A (36 piles) 4.0
3.5
Config B (25 piles)
Config C (16 piles)
3.42
Config D (16 piles) 3.00
P/P wr
3.0
Config E (9 piles)
2.5
2.0 1.63 1.49 1.5
1.58
1.56
1.51
1.56
1.32 1.09
1.08
1.15
1.06
1.0
0.5
0.0 SM-MC
MC-MS
MC-DS
SC-SC
MS-MS
DS-DS
MC-MC
Soil Profile
Figure (4-20) Effect of pile configuration on load carried by the piled raft corresponding to allowable settlement (S/Sall=1) 8.0
Config A1 (64 piles) 7.0
6.75
Config A (36 piles) Config B (25 piles)
6.0
Config C (16 piles) Config D (16 piles)
5.0
Config E (9 piles)
4.50
P/P wr
4.18 4.0
3.0
2.0
2.15
1.95
2.26
2.15
1.98
1.82
1.70
1.32
1.46
1.40
1.48
1.0
0.0 SM-MC
MC-MS
MC-DS
SC-SC
MS-MS
DS-DS
MC-MC
Soil Profile
Figure (4-21) Effect of pile configuration on load carried by the piled raft corresponding to allowable settlement (S/Sall=1.5)
117
-0.03
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
-0.02
∆ S/Sall
-0.01
0
0.01
0.02
0.03 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-22) Load - differential settlement relationship of the piled raft for soil profile ‘MC-MC’
-0.05
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
-0.04 -0.03 -0.02
∆ S/Sall
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-23) Load - differential settlement relationship of the piled raft for soil profile ‘SC-SC’
118
-0.04 -0.03 -0.02 -0.01
∆ S/Sall
0 0.01 0.02 0.03
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
0.04 0.05 0.06 0.07 0.08 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-24) Load - differential settlement relationship of the piled raft for soil profile ‘MS-MS’
-0.08 -0.06 -0.04 -0.02
∆ S/Sall
0 0.02 0.04 0.06
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
0.08 0.1 0.12 0.14 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-25) Load - differential settlement relationship of the piled raft for soil profile ‘DS-DS’
119
-0.03
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
-0.02
∆ S/Sall
-0.01
0
0.01
0.02
0.03 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-26) Load - differential settlement relationship of the piled raft for soil profile ‘MC-MS’
-0.05
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled Raft
-0.04 -0.03 -0.02
∆ S/Sall
-0.01 0
0.01 0.02 0.03 0.04 0.05 0
0.25
0.5
0.75
1
P/Pwr
1.25
1.5
1.75
2
Figure (4-27) Load differential - settlement relationship of the piled raft for soil profile ‘MC-DS’
120
-0.02
Config A1 (64 piles) Config A (25 piles) Config B (16 piles) Config C (36 piles)
-0.01
∆ S/Sall
Config D (16 piles) Config E (9 piles) Unpiled Raft 0
0.01
0.02 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-28) Load - differential settlement relationship of the piled raft for soil profile ‘SM-MC’
4.8.2 Behaviour of piled raft compared to other alternative foundation types In this section, the load settlement behaviour of the piled raft is compared to both corresponding, unpiled raft and conventional pile group. Figures (4-29) through (4-52) present this comparison for different soil profiles as well as for various pile configurations (Lp=10 m) as outlined in table (4-1). In general these figures clarify that the load carrying capacity of a piled raft exceeds that of the corresponding pile group which proves the contribution of the raft. In addition, the ratio of load carried by piled raft and that carried by corresponding pile group is directly proportional to settlement value. The behaviour of piled raft foundation compared to other alternative foundation types may be described in the light of the classification outlined in 4.8.1.1 as follows: 1. The case of uniform soil stratification, (floating pile group). In this case the raft should provide adequate load capacity and piles are mainly required to control settlement and/or differential settlement. In the current study this type is thoroughly investigated for various soil profiles namely MC-MC, SC-SC, MSMS and DS-DS. It can be easily concluded that in such cases, neglecting the
121
contribution of raft in the analysis (using conventional pile group), as required by existing building codes may lead to a dramatic increase in the number of piles and consequently the cost of foundations. 2. The case of two layered soil stratification, in which the strength of soil layer beneath pile tip is greater than the strength of the top layer containing the pile shafts and supporting the raft, (end bearing pile group). In this case the unpiled raft may not be able to provide significant load capacity or stiffness and piles are required to carry the major part of the load. In the current study this type is thoroughly investigated for various soil profiles namely SM-MC, MC-MS and MC-DS. It can be read from the results of this case that, piled raft still have the advantage of providing higher load carrying capacity and/or reducing settlement compared to conventional pile group. The above conclusion becomes more clear with reducing the number of piles, (e.g. for soil profile MC-DS, the percent of increase in the load carrying capacity of piled raft compared to the corresponding conventional pile group for Sav/Sall=1 was 35%, 48% and 90% for number of piles of 36, 25 and 16 respectively). P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2
3
4
5
6
7
Figure (4-29) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config A – 36 piles’ 122
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-30) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config B – 25 piles’ P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-31) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config C – 16 piles’ 123
0
0.5
1
P/Pwr
1.5
2
2.5
3
3.5
4
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-32) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MC’ – Pile configuration ‘Config D – 16 piles’ P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-33) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’ 124
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
0
Piled raft 1
Pile group Unpiled Raft
2
Sav/Sall
3 4 5 6 7
Figure (4-34) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config B– 25 piles’ P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3
4 5
6 7
Figure (4-35) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ 125
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
0
Piled raft 1
Pile group Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-36) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SC-SC’ – Pile configuration ‘Config D– 16 piles’
P/Pwr 0
0.5
1
1.5
2
0
Sav/Sall
0.5
1
1.5
Piled raft 2
Pile group Unpiled Raft
2.5
Figure (4-37) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config A – 36 piles’ 126
0
0.5
P/Pwr 1
1.5
2
0
Sav/Sall
0.5
1
1.5
Piled raft 2
Pile group Unpiled Raft
2.5
Figure (4-38) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config B – 25 piles’
P/Pwr 0
0.5
1
1.5
2
0
Piled raft Pile group
0.5
Sav/Sall
Unpiled Raft 1
1.5
2
2.5
Figure (4-39) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config C – 16 piles’ 127
P/Pwr 0
0.5
1
1.5
2
0
Piled raft Pile group
0.5
Sav/Sall
Unpiled Raft 1
1.5
2
2.5
Figure (4-40) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘DS-DS’ – Pile configuration ‘Config D – 16 piles’
P/Pwr 0
1
2
3
4
5
6
7
0
Piled raft
Pile group
1
Unpiled Raft
S av/S all
2 3 4 5 6 7
Figure (4-41) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config A – 36 piles’ 128
P/Pwr 0
1
2
3
4
5
6
7
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-42) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config B – 25 piles’
P/Pwr 0
1
2
3
4
5
6
7
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-43) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config C – 16 piles’
129
0
1
2
P/Pwr
3
4
5
6
7
0
Piled raft 1
Pile group Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-44) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-MS’ – Pile configuration ‘Config D – 16 piles’ P/Pwr 0
1
2
3
4
5
6
7
8
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-45) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’ 130
0
1
2
3
P/Pwr 4
5
6
7
8
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-46) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config B – 25 piles’
0
1
2
3
P/Pwr 4
5
6
7
8
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-47) Load settlement relationship of the piled raft compared with those of corresponding unpiled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’ 131
P/Pwr 0
1
2
3
4
5
6
7
8
0
Piled raft Pile group
1
Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-48) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘MC-DS’ – Pile configuration ‘Config D – 16 piles’
0
1
2
3
P/Pwr
4
5
6
7
0 Piled raft 1
Pile group Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-49) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config A – 36 piles’ 132
P/Pwr 0
1
2
3
4
5
6
7
0
Piled raft Pile group Unpiled Raft
1
Sav/Sall
2 3 4 5 6 7
Figure (4-50) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config B – 25 piles’
0
1
2
3
P/Pwr
4
5
6
7
0
Piled raft 1
Pile group Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-51) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config C – 16 piles’ 133
0
1
2
3
P/Pwr
4
5
6
7
0
Piled raft 1
Pile group Unpiled Raft
Sav/Sall
2 3 4 5 6 7
Figure (4-52) Load settlement relationship of the piled raft compared with those of corresponding un piled raft and pile group, for soil profile ‘SM-MC’ – Pile configuration ‘Config D – 16 piles’
4.8.3 Load settlement behaviour of average individual pile beneath piled raft In this section, the load settlement behaviour of the average individual pile beneath piled raft is compared to that of both corresponding, single pile and average individual pile of conventional pile group. Figures (4-53) through (4-72) present these comparisons for different soil profiles as well as for various pile configurations (Lp=10m). It can be seen from these figures that for all studied cases, the initial elastic stiffness of single pile is greater than the average individual pile of both corresponding pile group and piled raft. After this stage in the majority of the studied cases, it may be seen that the load carried by average individual pile of a piled raft is greater than that of the corresponding average individual pile of conventional pile group, which is greater than that of single pile. This may be attributed to the raft-pile interaction in case of pile group and to both pile-raft and raft-soil interactions in case of piled raft. The ratio of load carried by average individual pile of a piled raft and that carried by corresponding average individual pile of a conventional pile group is directly proportional to settlement level and this ratio increases with the reduction of number of piles due to the decrease of negative group action. It may 134
be concluded that the working pile load obtained from traditional pile load test may lead to conservative design of piled raft. However, from the economic point of view, the results of pile load test may be used to obtain soil parameters (e.g. interface stiffness) through back analysis of the pile load test using finite element method and then using the calibrated parameters resulting from this analysis for the detailed analysis of piled raft foundation. The behaviour of average individual pile of a piled raft foundation compared to corresponding average individual pile of a conventional pile group may be described in the light of the classification outlined in 4.8.1.1 as follows: 1. The case of uniform soil stratification, (floating pile group). In this case the average individual pile load capacity of a piled raft exceeds that of the corresponding average individual pile of a pile group especially in case of clayey formations. 2. the case of two layered soil stratification, in which the strength of soil layer beneath pile tip is greater than the strength of the top layer containing the pile shafts and supporting the raft, (end bearing pile group). In this case, improvement in the average individual pile load capacity of a piled raft compared to that of the corresponding average individual pile of a pile group can be recognized. It can seen also that the increase in the average individual pile load capacity of a piled raft is relatively smaller than in the case of floating pile group. Figures (4-73) through (4-77) present the effect of number of piles on the load settlement behaviour of the average individual pile of the piled raft. The increase in load carrying capacity with the reduction of number of piles, which was observed in the previous paragraph became more clear except for the case of ‘Config C – 16 piles’. This may be attributed to the geometry of ‘Config C ’ with a pile directly located below each column as shown in figure (4-3), which initially causes most of the column load to be transformed to the pile below the column and consequently increasing the settlement of these piles and consequently, reducing their stiffness. Therefore a redistribution of load sharing between piles and raft may occur (depending on the rigidity of the raft), resulting in reducing the load transferred through piles.
135
0
200
400
Load (KN)
600
800
1000
0
0.05
Settlement (m)
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-53) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’ – Pile configuration ‘Config A – 36 piles’
Load (KN) 0
200
400
600
800
1000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-54) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’ – Pile configuration ‘Config B – 25 piles’
136
Load (KN) 0
200
400
600
800
1000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-55) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’ – Pile configuration ‘Config C – 16 piles’
Load (KN) 0
200
400
600
800
1000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-56) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’ – Pile configuration ‘Config D – 16 piles’
137
Load (kN) 0
500
1000
1500
2000
2500
0.00
Settlement (m)
0.05
0.10
0.15
0.20
Av. Pile of Piled Raft Av. Pile of Pile Group
0.25
Single pile 0.30
Figure (4-57) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config A – 36 piles’
0
500
1000
Load (kN)
1500
2000
2500
0.00
Settlement (m)
0.05
0.10
0.15
0.20
Av. Pile of Piled Raft 0.25
Av. Pile of Pile Group Single pile
0.30
Figure (4-58) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config B – 25 piles’
138
Load (kN) 0
500
1000
1500
2000
2500
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-59) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘config C – 16 piles’
Load (kN) 0
500
1000
1500
2000
2500
3000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-60) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config D – 16 piles’
139
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft Av. Pile of Piled Group
0.25
Single pile 0.3
Figure (4-61) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ – Pile configuration ‘Config A – 36 piles’
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-62) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ – Pile configuration ‘Config B – 25 piles’
140
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft Av. Pile of Piled Group
0.25
Single pile 0.3
Figure (4-63) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ – Pile configuration ‘Config C – 16 piles’
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-64) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ – Pile configuration ‘Config D – 16 piles’ 141
Load (KN) 0
500
1000
1500
2000
2500
0
0.05
Settlement (m)
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-65) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config A – 36 piles’
Load (KN) 0
500
1000
1500
2000
2500
0
0.05
Settlement (m)
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-66) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config B – 25 piles’
142
Load (KN) 0
500
1000
1500
2000
2500
0
0.05
Settlement (m)
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-67) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config C – 16 piles’
Load (KN) 0
500
1000
1500
2000
2500
0
Settlement (m)
0.05
0.1
0.15
0.2
Av. Pile of Piled Raft 0.25
Av. Pile of Piled Group Single pile
0.3
Figure (4-68) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’ – Pile configuration ‘Config D – 16 piles’
143
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.00
Settlement (m)
0.05
0.10
0.15
0.20
Av. Pile of Piled Raft 0.25
Av. Pile of Pile Group Single pile
0.30
Figure (4-69) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.00
Settlement (m)
0.05
0.10
0.15
0.20
"Av. Pile of Piled Raft" 0.25
Av. Pile of Pile Group Single pile
0.30
Figure (4-70) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config B – 25 piles’
144
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
0.00
Settlement (m)
0.05
0.10
0.15
0.20
Av. Pile of Piled Raft Av. Pile of Pile Group
0.25
Single pile 0.30
Figure (4-71) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.00
Settlement (m)
0.05
0.10
0.15
0.20
Av. Pile of Piled Raft 0.25
Av. Pile of Pile Group Single pile
0.30
Figure (4-72) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config D – 16 piles’
145
Load (KN) 0
200
400
600
800
1000
1200
0
Settlement (m)
0.05
0.1
0.15
Config A - 36 piles Config B - 25 piles
0.2
Config C - 16 piles Config D - 16 piles
0.25
Config E - 9 piles Single pile
0.3
Figure (4-73) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MC’
Load (kN) 0
500
1000
1500
2000
2500
3000
0.00
Settlement (m)
0.05
0.10
0.15
Config A - 36 piles 0.20
Config B - 25 piles Config C - 16 piles
0.25
Config D - 16 piles Config E - 9 piles Single pile
0.30
Figure (4-74) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘SC-SC’
146
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
Settlement (m)
0.05
0.1
0.15
Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Single pile
0.2
0.25
0.3
Figure (4-75) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘DS-DS’ Load (KN) 0
500
1000
1500
2000
2500
3000
0
0.05
Settlement (m)
0.1
0.15
Config A - 36 piles Config B - 25 piles
0.2
Config C - 16 piles Config D - 16 piles
0.25
Config E - 9 piles Single pile
0.3
Figure (4-76) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-MS’
147
Load (kN) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.00
Settlement (m)
0.05
0.10
0.15
Config A - 36 piles Config B - 25 piles
0.20
Config C - 16 piles Config D - 16 piles
0.25
Config E - 9 piles Single pile
0.30
Figure (4-77) Load settlement relationship of average individual pile beneath piled raft for soil profile ‘MC-DS’
4.8.4 Load distribution between raft and piles In general a vertically loaded piled raft transmits load to supporting soil partly by contact pressure between raft and soil and partly through the piles. The contact pressure distributions along a section joining centre and mid-edge of the raft (section X-X shown in figure 4-3) at different applied load levels for selected soil stratifications, are presented in figures (4-78) through (4-82). It can be seen that for relatively larger number of piles, (e.g. config A1) the pressure distribution for all soil stratification is almost uniform except for a narrow strip near the perimeter of the raft, where pressure increase may be observed. The width of this narrow strip increases with the increase of applied load level. Also higher contact pressures are observed in cases where raft is in contact with competent soil layers, (e.g. MS-MS and SC-SC) compared to the cases of relatively soft soil layers (MC-DS and SM-MC). Significant increase in contact pressure can be recognized with the reduction in number of piles, which supports the concept of piles
148
reducing settlement (Burland, 1977 and 1995), in which piles capacity becomes fully utilized at first then further increase in the applied load is equilibrated by the raft contact pressures. The portion of the load taken by the raft contact pressure can be represented by the coefficient, αrp which is the ratio between the part of the load carried directly by the raft to the total applied load. The relationship between this coefficient and the applied load level for different soil profiles as well as for various pile configurations as outlined in table (4-1), are presented in figures (4-83) through (4-89). The same remarks outlined in the previous paragraph concerning the increase in contact pressure with the reduction in number of piles and consequently supporting the concept of piles reducing settlement, can be also observed from this group of figures. In addition it can be seen from these figures that, in the case of the central pile arrangement ‘Config D – 16 piles’, the portion of the load taken by the raft contact pressure is considerably less than the corresponding uniformly distributed pile arrangement ‘Config C – 16 piles’. This agrees with the discussions previously presented in section 4.5.3. However, this conclusion was not valid for the case of soil profile ‘SMMC’ at the early stages of loading which may be due to the nature of the uniformly distributed own weight of the raft, representing most of the applied load in these stages.
149
(a) MC-DS
(b) SC-SC
(c) MS-MS
(d) SM-MC
Figure (4-78) Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config A1 – 64 piles’ (Units in kPa – m)
150
(a) MC-DS
(b) SC-SC
(c) MS-MS
(d) SM-MC
Figure (4-79) Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config A – 36 piles’ (Units in kPa – m)
151
(a) MC-DS
(b) SC-SC
(c) MS-MS
(d) SM-MC
Figure (4-80) Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config B – 25 piles’ (Units in kPa – m)
152
(a) MC-DS
(b) SC-SC
(c) MS-MS
(d) SM-MC
Figure (4-81) Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config C – 16 piles’ (Units in kPa – m)
153
(a) MC-DS
(b) SC-SC
(c) MS-MS
(d) SM-MC
Figure (4-82) Distribution of vertical stresses along section X-X, 0.375m below the bottom of raft for pile configuration ‘Config D – 16 piles’ (Units in kPa – m)
154
0
0.5
P/Pwr
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
0.6 0.7 0.8 0.9 1
Figure (4-83) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-MC’
P/Pwr 0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9 1
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1 S/Sall=1.5
Figure (4-84) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘SC-SC’
155
0
0.5
P/Pwr
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9 1
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
Figure (4-85) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MS-MS’
0
0.5
P/Pwr
1
1.5
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9 1
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
Figure (4-86) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘DS-DS’
156
2
P/Pwr 0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
0.7 0.8 0.9 1
Figure (4-87) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-MS’
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9 1
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
Figure (4-88) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘MC-DS’
157
0
0.5
1
1.5
P/Pwr
2
2.5
3
3 .5
0
0.1
α rp
0.2
0.3
0.4
0.5
0.6
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) S/Sall=1
Figure (4-89) Relationship between applied load and the ratio of the load transferred by the raft directly to supporting soil for soil profile ‘SM-MC’
4.8.5 Raft bending moment Figures (4-90) through (4-96) present the variation of maximum raft bending moment with the applied load level. In general, for uniformly distributed pile groups, the maximum raft bending moment of the piled raft is slightly greater than that of the corresponding unpiled raft and the peak values of bending moment are located at column positions. For the case of the central pile arrangement ‘Config D – 16 piles’, considerable reduction in maximum raft bending moment can be observed. For the purpose of comparison, the contours of bending moment of the raft in both directions (shown on figure 4-3) at P/Pwr=1 for the case of central pile arrangement ‘Config D – 16 piles’, as well as for the corresponding uniformly distributed pile arrangement ‘Config C – 16 piles’ are presented for different soil profiles, in figures (4-97) through (4-110). The distribution bending moment in the raft in the case of central pile arrangement differs from the case of uniformly distributed piles and the peak values of bending moment are located away from column positions. This may be attributed to the difference in deformed shape between the two cases as shown before in figures (4-22) through (4-28). Therefore 158
care should be exercised in the design of the raft reinforcement, to the position of peak bending moment. 0
0.5
P/Pwr
1
1.5
2
0
Maximum bending moment (kN.m)
200
400
600
800
1000
1200
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
1400
Figure (4-90) Variation of maximum raft bending moment for soil profile ‘MC-MC’ P/Pwr 0
0.5
1
1.5
2
Maximum bending moment (kN.m)
0 250 500 750 1000 1250 1500 1750 2000 2250 2500
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
2750
Figure (4-91) Variation of maximum raft bending moment for soil profile ‘SC-SC’
159
0
0.5
P/Pwr
1
1.5
2
Maximum bending moment (kN.m)
0 500 1000 1500 2000 2500 3000 3500
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
4000
Figure (4-92) Variation of maximum raft bending moment for soil profile ‘MS-MS’
0
0.5
P/Pwr 1
1.5
2
Maximum bending moment (kN.m)
0 1000 2000 3000 4000 5000 6000 7000
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
8000
Figure (4-93) Variation of maximum raft bending moment for soil profile ‘DS-DS’
160
P/Pwr 0
0.5
1
1.5
2
0
Maximum bending moment (kN.m)
200 400 600 800 1000 1200
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
1400 1600 1800 2000
Figure (4-94) Variation of maximum raft bending moment for soil profile ‘MC-MS’
P/Pwr 0
0.5
1
1.5
2
0
Maximum bending moment (kN.m)
250 500 750 1000 1250 1500 1750 2000 2250
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
2500
Figure (4-95) Variation of maximum raft bending moment for soil profile ‘MC-DS’
161
0
0.5
P/Pwr 1
1.5
2
Maximum bending moment (kN.m)
0
100
200
300
400
500
600
Config A1 (64 piles) Config A (36 piles) Config B (25 piles) Config C (16 piles) Config D (16 piles) Config E (9 piles) Unpiled raft
Figure (4-96) Variation of maximum raft bending moment for soil profile ‘SM-MC’
My
Mx
Figure (4-97) Contours of bending moment for ‘MC-MC’ – ‘Config C’ at P/Pwr=1
162
Mx
My
Figure (4-98) Contours of bending moment for ‘SC-SC’ – ‘Config C’ at P/Pwr=1
Mx
My
Figure (4-99) Contours of bending moment for ‘MS-MS’ – ‘Config C’ at P/Pwr=1
My
Mx
Figure (4-100) Contours of bending moment for ‘DS-DS’ – ‘Config C’ at P/Pwr=1
163
Mx
My
Figure (4-101) Contours of bending moment for ‘MC-MS’ – ‘Config C’ at P/Pwr=1
My
Mx
Figure (4-102) Contours of bending moment for ‘MC-DS’ – ‘Config C’ at P/Pwr=1
Mx
My
Figure (4-103) Contours of bending moment for ‘SM-MC’ – ‘Config C’ at P/Pwr=1 164
Mx
My
Figure (4-104) Contours of bending moment for ‘MC-MC’ – ‘Config D’ at P/Pwr=1
Mx
My
Figure (4-105) Contours of bending moment for ‘SC-SC’ – ‘Config D’ at P/Pwr=1
Mx
My
Figure (4-106) Contours of bending moment for ‘MS-MS’ – ‘Config D’ at P/Pwr=1
165
Mx
My
Figure (4-107) Contours of bending moment for ‘DS-DS’ – ‘Config D’ at P/Pwr=1
Mx
My
Figure (4-108) Contours of bending moment for ‘MC-MS’ – ‘Config D’ at P/Pwr=1
Mx
My
Figure (4-109) Contours of bending moment for ‘MC-DS’ – ‘Config D’ at P/Pwr=1 166
Mx
My
Figure (4-110) Contours of bending moment for ‘SM-MC’ – ‘Config D’ at P/Pwr=1 4.9 Effect of pile shaft length on piled raft foundation behaviour To investigate the effect of pile shaft length (Lp), three different lengths of 10m, 15m, and 20m were studied for the cases outlined in table (4-1). The relative pile length (Lr) can be defined according to Reul and Randolph (2004) as follows:
Lr =
Lp B.L
(4-1)
π Where: Lp : pile length B : raft width L : raft length 4.9.1 Effect on load - average settlement behaviour
The load – average settlement relationship for the selected cases outlined in table (4-1), is presented in figures (4-111) through (4-116). This figures clearly show that, for case of uniform soil stratification (floating pile group) which is presented here by soil profile ‘SC-SC’, the average settlement considerably decrease with the increase of pile shaft length and the rate of decrease is directly proportional to the number of piles in the group. On the other hand, for the case of end bearing pile group which is presented here by soil
167
profile ‘MC-DS’, the pile shaft length almost has no effect on the load – average settlement behaviour of piled raft foundation. 0
0.5
1
P/Pwr 1.5
2
2.5
3
0
1
Sav/Sall
2
3
Lr=1.363 4
Lr=2.045 Lr=2.727 Unpiled Raft
5
Figure (4-111) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’ P/Pwr 0
0.5
1
1.5
2
2.5
3
0
Sav/Sall
1
2
3
Lr=1.363 4
Lr=2.045 Lr=2.727 Unpiled Raft
5
Figure (4-112) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’
168
P/Pwr 0
0.5
1
1.5
2
2.5
3
0
Sav/Sall
1
2
3
Lr=1.363 Lr=2.045
4
Lr=2.727 Unpiled Raft 5
Figure (4-113) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
1
Sav/Sall
2
3 4
5 6
7
Lr=1.363 Lr=2.045 Lr=2.727 Unpiled Raft
Figure (4-114) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’
169
P/Pwr 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
1
Sav/Sall
2
3
4
5
Lr=1.363 Lr=2.045 Lr=2.727 Unpiled Raft
6
7
Figure (4-115) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’
0
0.5
1
1.5
2
P/Pwr 2.5
3
3.5
4
4.5
5
0
1
Sav/Sall
2
3
4
5
6
Lr=1.363 Lr=2.045 Lr=2.727 Unpiled Raft
7
Figure (4-116) Effect of pile length on load settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config E – 9 piles’
170
4.9.2 Effect on load differential - settlement behaviour
The load – differential settlement relationship for the selected cases, is presented in figures (4-117) through (4-122). The picture of differential settlement is more complicated than the above case of average settlement. However, it can be seen from these figures that, for case of uniform soil stratification (floating pile group) which is presented here by soil profile ‘SC-SC’, the differential settlement increase with the increase of pile shaft length. On the other hand, for the case of end bearing pile group which is presented here by soil profile ‘MC-DS’, the pile shaft length almost has no effect on the load – differential settlement behaviour of piled raft foundation except for the case of ‘Config E – 9 piles’, a relatively considerable increase in differential settlement with increase of pile shaft length can be observed.
-0.04
Lr=1.363 Lr=2.045 Lr=2.727 Unpiled Raft
-0.03 -0.02 -0.01 0
∆ S/Sall
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-117) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’
171
-0.04
Lr=1.363 Lr=2.045
-0.03 -0.02
Lr=2.727 Unpiled Raft
-0.01 0 0.01
∆ S/Sall
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-118) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’ -0.06
Lr=1.363 Lr=2.045 Lr=2.727 Unpiled Raft
-0.04 -0.02 0
∆ S/Sall
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
(4-119) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’
172
-0.05 -0.04
Lr=1.363 Lr=2.045
-0.03
Lr=2.727 Unpiled Raft
-0.02
∆ S/Sall
-0.01 0
0.01 0.02 0.03 0.04 0.05 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-120) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’
-0.05 -0.04
Lr=1.363 Lr=2.045
-0.03
Lr=2.727 Unpiled Raft
-0.02
∆ S/Sall
-0.01 0
0.01 0.02 0.03 0.04 0.05 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-121) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’
173
-0.04
Lr=1.363 Lr=2.045
-0.03 -0.02
Lr=2.727 Unpiled Raft
-0.01 0
∆ S/Sall
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
P/Pwr
Figure (4-122) Effect of pile length on load differential settlement relationship of the piled raft for soil profile ‘MC-DS’ – Pile configuration ‘Config E – 9 piles’
4.9.3 Effect on load distribution between raft and piles
The portion of the load transferred directly to supporting soil which is presented by the coefficient, αrp which was previously defined in 4.8.4 is plotted against the applied load level for the selected cases outlined in table (4-1), are presented in figures (4-123) through (4-128). This figures clearly show that, increasing the relative pile shaft length, reduces the part of the load transferred directly by the raft for the soil profiles studied in this section which are soil profile ‘SC-SC’ and soil profile ‘MC-DS’.
174
P/Pwr 0.5
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9
Lr=1.363 Lr=2.045 Lr=2.727
1
Figure (4-123) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config A – 36 piles’
0.5
1
P/Pwr
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9
Lr=1.363 Lr=2.045 Lr=2.727
1
Figure (4-124) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config C – 16 piles’
175
0.5
1
P/Pwr
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8 0.9
Lr=1.363 Lr=2.045 Lr=2.727
1
Figure (4-125) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘SC-SC’ – Pile configuration ‘Config E – 9 piles’ P/Pwr 0.5
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7
Lr=1.363
0.8
Lr=2.045
0.9
Lr=2.727
1
Figure (4-126) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘MC-DS’ – Pile configuration ‘Config A – 36 piles’
176
P/Pwr 0.5
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7
Lr=1.363
0.8
Lr=2.045 0.9
Lr=2.727
1
Figure (4-127) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘MC-DS’ – Pile configuration ‘Config C – 16 piles’ P/Pwr 0.5
1
1.5
2
0 0.1 0.2 0.3
α rp
0.4 0.5 0.6 0.7 0.8
Lr=1.363 Lr=2.045
0.9
Lr=2.727
1
Figure (4-128) Effect of pile length on load transferred directly by the raft to supporting soil profile ‘MC-DS’ – Pile configuration ‘Config E – 9 piles’
177
CHAPTER FIVE THREE DIMENSIONAL FINITE ELEMENT SIMULATION FOR VERTICALLY LOADED PILED RAFT FOUNDATION, (CASE STUDY)
5.1 Introduction In this chapter, two well documented case histories of piled raft foundation have been studied. The first is the Stonebridge Park building, which is a sixteen storey block of flats built on the well known London clay, and was originally designed as a pile group. The second is the 130m high Torhaus building constructed on Frankfurt clay and was the first building in Germany with foundation designed originally as a piled raft. Finite element method was used for the three dimensional simulation of piled raft foundations of the above mentioned case histories. An elasto-plastic three dimensional finite element model, capable of representing different types of interactions affecting behaviour of piled raft foundation system, such as pile-soil interaction, pile-pile interaction, raft-soil interaction and pile-raft interaction, is proposed and used to predict foundation settlements, load sharing between raft and piles as well as the distribution of load between piles. The obtained results are verified against the field measurements recorded during and after the construction of the original buildings. Also these results are compared to available results obtained numerically by other researchers for those problems. The work with these case studies was extended to illustrate some proposed techniques used to handle large three dimensional piled raft foundations.
5.2 Methodology Both undrained and drained analyses are performed in order to predict the short term and long term settlements, the contact stress between the raft and the soil and the load sharing between raft and piles. It is evident that the real foundation behaviour should lie between the above two extreme conditions. The analyses are performed in two phases as follows:
178
Phase 1: presents the soil before construction of foundation i.e. only soil layers are modelled to calculate the in-situ stresses. The soil above foundation level is not modelled using finite elements to reduce the number of equations and it is only considered through its own weight. Phase 2: this phase represents the case after construction of the piles and the raft. Then super structural dead and live loads (136.8 MN in case of the Stonbridge building and 173.2 MN/raft in case of the Torhaus building) are uniformly applied to the raft and are increased in steps. Mesh adaptation technique is used to present the finite element mesh in the two stages mentioned above, in which the mesh configuration is altered to suit the geometry in each construction stage. This is done by activating/deactivating parts of the mesh as required.
5.3 Soil structure interface modelling To model the non linear soil/pile interface behaviour, Mohr-Coulomb friction type interface elements are used for modelling the interface between pile shafts and surrounding soil. The use of the above interface elements permits modelling of an interface condition ranging between the smooth (no friction) condition and the perfect rough condition. The initial elastic stiffness of the interface is calculated according to equation (3-13). The reduction factors of the interface parameters (Rc and Rφ) between rough concrete piles and soil are taken as unity according to the experimental work performed by Moorman (2002). The contact between raft and soil beneath it is modelled as perfectly rough, i.e. no relative movement between the nodes of the finite elements representing the raft and those of the finite elements representing soil beneath it. The contact between piles’ tips and the soil beneath it was also modelled as perfectly rough.
5.4 The Stonebridge Park building
5.4.1 Geometric and material properties The piled raft case study chosen for performing the finite element analyses is that known as the Stonebridge Park building, constructed in north London, UK, reported by Cooke et al. (1981). The construction of the building started in 1973 and ended in 1975. The
179
sixteen storey building is 43.3 m long by 19.2 m wide. The foundation consists of a heavily reinforced concrete raft 0.90 m thick linking the heads of 351 bored, cast in situ concrete piles 0.45 m in diameter and 13 m long, formed on a sensibly square grid at a pitch of 1.6 m. Details of foundation system were previously shown in figure (2-10). The layout of the measurement devices, which consists of eight instrumented piles, eleven raft contact pressure cells and one multi point magnet extensometer, is shown in figure (5-1). The monitoring continued for six years’ period covering the erection and early life of the building. Both the reinforced concrete piles and raft are considered to behave linearly elastic with total unit weight of 25 KN/m3, and buoyant unit weight of 15 KN/m3, Young’s modulus of 21000 MPa and Poisson’s ratio of 0.20.
Figure (5-1) Stonebridge park building: Foundation plan of the quarter showing positions of instruments and cable runs (Cooke et al. 1981)
180
5.4.2 Subsoil conditions The soil profile at the site of the Stonebridge Park building consists of highly over consolidated London stiff clay extending up to ground surface. Both of the foundation level of the raft and the ground water table are located at the same level of two and half meters below original ground surface. The soil profile and position of foundation are shown in figure (5-2). The non linear soil behaviour was simulated using the elatsoplastic Mohr-Coulomb model with associative flow rule (φ=ψ). The soil parameters for the well known London clay used in the finite element analyses had been taken according to Cooke et al. (1981) and Adenbrooke et al. (1997). Table (5-1) summarizes, the estimated soil parameters for the Mohr-Coulomb model with associative flow rule. The distribution of the Young's modulus of the London clay with depth is described by the following empirical equation deduced from Cooke (1981) and Fleming et al. (1992): E = 48 + 3.456 z ,
(5.1)
where, E : Young's modulus (MPa). z : the depth below the ground surface (m). Table (5-1) Soil Material parameters used in the finite element analyses – Stonebridge park building: Parameter
London clay
Young's modulus, E: MPa
Equation (5.1)
Poisson's ratio, ν'
0.20
Total unit weight of moist soil, γ: kN/m3
20
Buoyant unit weight of moist soil, γ': kN/m3
10
Coefficient of earth pressure at rest, Ko
1
Angle of internal friction, φ' : degrees
25
Cohesion, c': kPa
5 Elasto-Plastic Mohr Coloumb Effective Drained/Undrained
Material model Analysis Type
181
(-2.50)
GWT
F.L.
13.00 m
0.90 m 1.60 m
(0.00)
NGL
(-15.50)
Dia.=0.45 m
20.00 m
STIFF OVER CONSOLIDATED LONDON CLAY EXTENDING UP TO GROUND SURFACE
(-35.50)
Figure (5-2) Stonebridge park building: Soil stratification and position of foundation
5.4.3 Detailed three dimensional nonlinear finite element model A three dimensional nonlinear finite-element analysis is used to model the behaviour of the piled raft system under vertical loading. The analysis is carried out as a two phased analysis to simulate the stages of the construction. As a result of symmetry of the foundation, the model represents only one quarter of the analysed piled raft. The purpose of this model is to predict the settlement, the contact stress between the raft and the soil and the load sharing between raft and piles.
182
The finite element analyses are carried out using the general purpose finite element program DIANA® version 9.1 (TNO DIANA BV. 2005). The three dimensional finite-element mesh for the proposed model is shown in Figure (53). The raft is modelled using 8-noded and 6-noded shell elements. The piles are modelled using 15-noded wedge elements. Figure (5-4) shows the meshing for the raft and piles. The soil layers are modelled using 20-noded brick and 15-noded wedge elements. The interaction between piles’ shafts and soil is presented using 16-noded interface elements. Figure (5-5) presents the interface elements modelling soil/pile interface. For the purpose of mesh compatibility soil/soil interface elements are used below pile tips as shown by figure (5-6). The total number of nodes in this model is 114882 and the total number of elements is 35512. Figure (5-7) presents the layout of the raft showing the key-node numbers.
Figure (5-3) Finite element mesh - Stonebridge park building 183
Figure (5-4) Piled-raft system mesh - Stonebridge park building
Figure (5-5) Pile-soil interface elements - Stonebridge park building
184
Figure (5-6) soil-soil interface elements below soil-pile interface elements Stonebridge park building
Figure (5-7) Raft layout showing key-node numbers - Stonebridge park building
185
5.4.4 Results of analysis The results of the analyses obtained here for both undrained and drained conditions compares well with their counterpart observed ones as will be shown in the next subsections.
5.4.4.1 Vertical displacements The deformed shape of the raft is the dish shape as can be seen from the contours of vertical displacement in figure (5-8) for the undrained condition and figure (5-9) for drained condition, which is expected for this uniformly distributed pile configuration. Figure (5-10) presents the load settlements relationship for central, mid-edges and corner of the raft, which is almost linear relationship. This linear response means that the foundations of the Stonebridge building is conservatively designed. The deformed shape of the raft with increasing the applied load is shown in figure (5-11) and figure (5-12). It can be seen from these figures that the dishing of the raft increases with the increase of loading with maximum settlement at full load of 10.2 mm in undrained condition which increased to 19.3 mm in drained condition (including the part of settlement resulted from the own weight of the raft). Figure (5-13) plots the predicted (drained and undrained conditions) and observed load-vertical settlement relationships. Also shown in this figure the results of undrained analysis obtained by Viggiani, 1998 using NAPARA program. It can be seen that, at the beginning of loading the predicted response in undraind condition agrees well with observations until a load of about 45 MN. The load was maintained for about five months allowing for drainage to occur, and the predicted drained settlement matched the observed settlement at the end of this constant load interval. With the steady increase of load, the predicted response in undraind condition tends to be closer to the observed settlements. At the full load the predicted response in undraind condition was about 9 mm compared to an observed value of about 10 mm . Four years after the end of construction, the predicted drained response (about 18 mm) is in very good agreement with the observed settlement (about 17 mm). The above results proves the efficiency and reliability of elasto-plastic three dimensional finite element method in the analysis of piled raft foundation.
186
Figure (5-8) Contours of vertical settlement (in meters) of the raft – Undrained condition - Stonebridge park building
Figure (5-9) Contours of vertical settlement (in meters) of the raft – Drained condition - Stonebridge park building 187
(a) Undrained condition
(b) Drained condition
Figure (5-10) Load-vertical settlement relation ship for raft corner, mid edges and center points - Stonebridge park building
(a) Undrained condition
(b) Drained condition
Figure (5-11) Distribution of vertical settlement (in meters) in longitudinal (X) direction through the center of the raft (through nodes 99656 – 139) – Stonebridge park building 188
(a) Undrained condition
(b) Drained condition
Figure (5-12) Distribution of vertical settlement (in meters) in (Y) direction through the center of the raft (through nodes 99656 – 12) - Stonebridge park building Load (MN) 0
20
40
60
80
100
120
140
160
0.0 2.0
Settlement (mm)
4.0 6.0 8.0 10.0 12.0
Observed Cooke et. al, 1981
14.0
Mohr-Coulomb model - drained
16.0
Mohr-Coulomb model - Undrained
18.0
NAPARA - Undrained Viggiani, 1998
20.0
Figure (5-13) Predicted and observed load settlement relationship for the center of the raft - Stonebridge park building 189
5.4.4.2 Raft contact pressure distribution The raft contact pressure as obtained from the results of finite element analyses is almost uniform for un drained condition. For drained condition the same uniform distribution was obtained, except for a narrow strip at the perimeter of the raft where contact pressure slightly increases. as can be seen from figure (5-14) and figure (5-15). The output results predicted using finite element analysis-drained condition, for the contact stress between the raft and the soil beneath it, show that at full load, the raft transferred about 30% of the total structural load directly to soil, compared to an observed value of 25%. At 50% of the applied load, the raft transferred about 48% of the applied load, compared to an observed value of about 45%. Reasonable agreement between predicted and observed load share of the raft can be seen.
190
(a) Undrained condition
(b) Drained condition Figure (5-14) Distribution of vertical stresses (in kPa) in (X) direction through the center of the raft at a distance of 1m below the bottom of raft (through nodes 99656 – 139) - Stonebridge park building
191
(a) Undrained condition
(b) Drained condition Figure (5-15) Distribution of vertical stresses (in kPa) in (Y) direction through the center of the raft at a distance of 1m below the bottom of raft (through nodes 99656 – 12) - Stonebridge park building
192
5.5 The Torhaus building 5.5.1 Geometric and material properties The Torhaus building was constructed on a narrow site bounded by a triangular intersection of railway bridges within the Frankfurt fair area, in Germany as reported by Sommer et al. (1985). The construction of the building started in 1983 and ended in 1986. The thirty storey building is founded on two separate piled rafts 10 m apart, each of 24.5 m long by 17.5 m wide and 2.50 m thick with 42 bored, concrete piles, 0.90 m in diameter and 20 m long, with pile spacing varying from 3 to 3.5 times the pile diameter. A general view of the building and details of foundation system including the layout of the measurement devices, which consists of six instrumented piles, eleven raft contact pressure cells and three multi point borehole extensometers, are shown in figure (5-16). Both the reinforced concrete piles and raft are considered to behave linearly elastic with total unit weight of 25 KN/m3, and buoyant unit weight of 15 KN/m3, Young’s modulus of 21000 MPa and Poisson’s ratio of 0.20.
Figure (5-16) Torhaus building: (a) Profile view of the building; (b) foundation plan showing positions of instruments (Reul and Randolph 2003)
193
5.5.2 Subsoil conditions The soil profile at the site of the Torhaus building consists of quarternary terrace gravel starting from ground surface, down to 2.5 m below the raft bottom, underlain by layers of Frankfurt clay extending to great depth. Within the Frankfurt clay, thin calcareous sand and silt inclusions, as well as isolated floating limestone layers, are embedded. The foundation level of the raft is located three meters below original ground surface. The ground water table is located at the top of the Frankfurt clay layer. The soil profile and position of foundation are shown in figure (5-17). The non linear soil behaviour was simulated using the elasto-plastic Mohr-Coulomb model with associative flow rule. The soil parameters used in the finite element analyses had been taken according to Reul and Randolph (2003). Table (5-2) summarizes, the estimated soil parameters for the MohrCoulomb model with associative flow rule. The distribution of the Young's modulus of the Frankfurt clay with depth is described by the following empirical equation proposed by Reul (2000): ⎡ ⎛ z − 30 ⎞ ⎤ E = 45 + ⎢ tanh⎜ ⎟ + 1⎥ × 0.7 z , ⎝ 15 ⎠ ⎦ ⎣
(5-2)
where, E : Young's modulus (MPa). z : the depth below the surface of the clay layers (m). Table (5-2) Soil Material parameters used in the finite element analyses – Torhaus building: Parameter
Frankfurt clay
Quarternary layers
Young's modulus, E: MPa
Equation (5.2)
75
0.15
0.25
Total unit weight of moist soil, γ: kN/m3
22
18
Buoyant unit weight of moist soil, γ': kN/m3
12
-
Coefficient of earth pressure at rest, Ko
0.72, (0 ≤ z < 25) 0.57, ( z ≥ 25)
0.46
Angle of internal friction, φ' : degrees
20
32
Cohesion, c': kPa
20
0
Poisson's ratio, ν'
194
Elasto-Plastic Mohr Coloumb Effective Drained/Undrained
Material model Analysis Type
Elasto-Plastic Mohr Coloumb Effective Drained/Undrained
0.50
(0.00)
2.50
NGL
(-3.00)
F.L.
QUARTERNARY TERRACE GRAVEL
(-5.50)
20.00
GWT
(-23.00)
Dia.=0.90 m
STIFF OVER CONSOLIDATED FRANKFURT CLAY
Figure (5-17) Torhaus building: Soil stratification and position of foundation
5.5.3 Detailed three dimensional nonlinear finite element model
A three dimensional nonlinear finite-element analysis is used to model the behaviour of the piled raft system under vertical loading. The analysis is carried out as a two phased analysis to simulate the stages of the construction. As a result of symmetry of the foundation, the model represents only one quarter of the analysed foundation.
195
The purpose of this model is to predict the settlement, the contact stress between the raft and the soil and the load sharing between raft and piles as well as the distribution of load between piles. The finite element analyses are carried out using the general purpose finite element program DIANA® version 9.1 (TNO DIANA BV. 2005). The three dimensional finite-element mesh for the proposed model is shown in Figure (518). The raft is modelled using 8-noded shell elements. The piles are modelled using 20noded brick elements. Figure (5-19) shows the meshing for the raft and piles. The soil layers are modelled using 20-noded brick. The interactions between piles’ shafts and soil are presented using 16-noded interface elements. Figure (5-20) presents the interface elements modelling soil/pile interface. For the purpose of mesh compatibility soil/soil interface elements are used below pile tips as shown by figure (5-21). The total number of nodes in this model is 76359 and the total number of elements is 19836. Figure (5-22) presents the layout of the raft showing the key-node numbers.
Figure (5-18) Finite element mesh - Torhaus building
196
Figure (5-19) Piled-raft system mesh - Torhaus building
Figure (5-20) Pile-soil interface elements - Torhaus building
197
Figure (5-21) soil-soil interface elements below soil-pile interface elements - Torhaus building
Figure (5-22) Raft layout showing key-node numbers - Torhaus building
198
5.5.4 Results of analysis
The results of the analyses obtained here for both drained and un drained conditions compares well with their counterpart observed ones as will be shown in the next subsections. 5.5.4.1 Vertical displacements
The deformed shape of the raft is the dish shape as can be seen from the contours of vertical displacement in figure (5-23) for the undrained condition and figure (5-24) for drained condition, (dishing includes both rafts) which is expected for this uniformly distributed pile configuration. Figure (5-25) presents the load settlements relationship for the raft mid-edges and corners. It can be seen from this figure that the behaviour of the piled raft is non linear at working load level and the non linearity is more clear in the drained condition. This may be attributed to the full mobilization of pile capacity at working load level (creep piling philosophy). The design of the Torhaus building resulted in a major saving in the cost of foundation, with little effect (tolerable) on the performance of the structure. The deformed shape of the raft with increasing the applied load is shown in figure (5-26) and figure (5-27). It can be seen from these figures that the dishing of the raft increases with the increase of loading with maximum settlement at full load of about 58 mm in undrained condition which increased to about 150 mm in drained condition. Figure (5-27) shows that high settlement levels resulted in the region between the two rafts. Although structures like the Torhaus building are rigid enough to tolerate such settlement levels, care should be given to the effect of these settlements on neighbouring buildings. Figure (5-28) plots the predicted (drained and undrained conditions) and observed load-vertical settlement relationship. It can be seen that the available observed settlements, which started after the construction of the raft in 1983 and extended two years after the end of the building construction in 1988, lies between the results of the drained and undrained conditions. The above finding proves the accuracy of the present analyses. In addition, for the purpose of verification, this problem was resolved using the proposed technique outlined in 3.3.2 of employing three dimensional interface elements for modelling of far field bottom soil layer. The results are shown in figure (5-28) in comparison with the predicted and the results of full model. The good
199
agreement between the simplified method and both the predicted and full model response can be seen.
Figure (5-23) Contours of vertical settlement (in meters) of the raft – Undrained condition - Torhaus building
Figure (5-24) Contours of vertical settlement (in meters) of the raft – Drained condition - Torhaus building
200
(a) Drained condition
(b) Undrained condition
Figure (5-25) Load-vertical settlement relation ship for raft mid edges and corners Torhaus building
(a) Undrained condition
(b) Drained condition
Figure (5-26) Distribution of vertical settlement (in meters) in (X) direction through the raft inner mid-edge and corner, (through nodes 1130 – 365) – Torhaus building
201
(a) Undrained condition
(b) Drained condition
Figure (5-27) Distribution of vertical settlement (in meters) in (Y) direction through the center of the raft (through nodes 99656* – 275) - Torhaus building * Node on the intersection of the two axes of symmetry in the plane of the raft Load (MN) 0
25
50
75
100
125
150
175
200
0
20
Settlement (mm)
40
60
80
100 Observed Sommer et. al, 1985 120
Full model - drained Full model - Undrained
140
Simplified model - Undrained Simplified model - Drained
160
Figure (5-28) Predicted and observed load settlement relationship for the center of the raft - Torhaus building
202
5.5.4.2 Raft contact pressure distribution
In the X-direction (section through nodes 1130 - 365) almost a linear increase of the raft contact pressure from the raft centre towards raft edges as can be observed from figure(529) with little change at the edges. For the Y-direction (section through nodes 1130 275), the raft contact pressure as obtained from the results of finite element analyses is almost uniform except for a narrow strip at the perimeter of the raft where pressure increases as can be seen from figure (5-30).
The output results predicted using finite element analyses, for the contact stress between the raft and the soil beneath it show that: •
At 75% of full load the raft transferred about 15% of the full load (20% of applied load) directly to soil in undrained condition and 15.4% of the full load (20.53% of applied load) in drained condition, compared to an observed value of 15% of full load in July 1984 (Sommer et al., 1985).
•
At full load, the raft transferred about 26% of the full load directly to soil in undrained condition and 30% in drained condition, compared to an observed value of 33% at the end of construction in February 1986 (Sommer, 1991).
The above results shows that the load carried by the raft is directly proportional to the applied load level as concluded from the numerical investigation presented in 4.8.4.
203
(a) Undrained condition
(b) Drained condition Figure (5-29) Distribution of vertical stresses (in kPa) in (X) direction through the raft inner mid-edge and corner, at a distance of 0.5m below the bottom of raft (through nodes 1130 – 365) - Torhaus building
204
(a) Undrained condition
(b) Drained condition Figure (5-30) Distribution of vertical stresses (in kPa) in (Y) direction through the raft mid-edges at a distance of 0.5m below the bottom of raft (through nodes 1130 – 275) - Torhaus building
205
5.5.4.3 Load distribution between piles
To investigate the load distribution between piles, the program Pilelo previously described in 3.2.5, was used to calculate the vertical pile loads from the internal vertical stresses obtained from the output results of the finite element analyses. Figures (5-31) and (5-32) demonstrates the distribution of load among piles at 75% and 100% of total applied load respectively. It can be seen from these figures that corner piles carried the highest loads in the pile group, also the edge piles carried higher loads compared to inner piles. The ratio of maximum to minimum pile loads was about 3.4 in undrained condition and about 2.6 in drained condition and these ratios are slightly affected by the applied load level. The pile loads predicted using the present finite element analyses for both drained and undrained conditions are compared to the corresponding observed loads as well as available results of finite element analyses obtained by Reul and Randolph (2003). Figures (5-33) and (5-34) show the above mentioned comparison at 75% and 100% of total applied load respectively. It can be seen from these figures that there is some differences between observed and predicted pile loads which may be attributed to that finite element analyses used loads uniformly distributed over the whole raft area rather than using the actual discrete column loads as it is not available in the published data. However, the predicted pile loads compares reasonably with there corresponding observed ones.
206
Instrumented pile
3600
4860
2 1896
1940
2160
3610
1492
1580
1930
3470
1488
1580
1960
3530
1 1488
3220
2250
3820
3300
3560
5030
6
3590
1908
1950
2260
3620
1556
1610
1930
3390
1572
1640
1970
2
4 1960
3360
3320
3
3
1 1572
3240
5
4580
3440
17.50
3420
17.50
3400
Instrumented pile
4 2020
2330
3720
3310
3620
4690
6
5
Southern raft
Southern raft
12.25
12.25
(a) Undrained condition
(b) Drained condition
Figure (5-31) Load distribution among piles at 75% of total load (kN) - Torhaus building
Instrumented pile
4280
4300
4460
2
6100
4000
3
2
3960
3900
5080 3
2350
2650
4580
2640
2560
2350
4120
1912
2000
2280
4460
2220
2230
2380
4080
1940
1980
2370
2060
2160
2500
1 1940
4420
4500
17.50
2400
4 2480
2710
4260
3210
4710
6540
6
1 2060
3480
5
4260
17.50
Instrumented pile
4 2360
2610
3720
4130
6
4200
5330 5
Southern raft
Southern raft
12.25
12.25
(a) Undrained condition
(b) Drained condition
Figure (5-32) Load distribution among piles at full load (kN) - Torhaus building
207
8000
Observed, Sommer et al. 1985
7000
Present analyses Undrained 6000
Present analyses Drained
Pile load (kN)
5000
4000
3000
2000
1000
0 Pile 1
Pile 2
Pile 3
Pile 4
Pile 5
Pile 6
Figure (5-33) Predicted and observed pile loads at 75% of total load - Torhaus building
8000
Observed, Sommer, 1991 Present analyses Undrained
7000
Present analyses Drained 6000
Pile load (kN)
Analyses by Reul and Randolph, 2003
5000
4000
3000
2000
1000
0 Pile 1
Pile 2
Pile 3
Pile 4
Pile 5
Pile 6
Figure (5-34) Predicted and observed pile loads at full load - Torhaus building
208
CHAPTER SIX CONCLUSIONS
6.1 Conclusions Based on the results of the present detailed numerical investigation carried out in this study, a number of conclusions may be summarized as follows: 1.
The finite-element method is an efficient and reliable tool for the three dimensional analyses of piled raft foundations, provided that the following conditions are satisfied: • The non-linear behaviour of soil is modelled using representative material model. • Proper finite elements are used for modelling the different elements of the problem (second order elements in case of circular piles). • Pile shaft-soil interface is modelled using friction type interface elements allowing for slip.
2.
The proposed technique of using three dimensional interface elements to model the far field soil layer in large three dimensional finite element problems, has proved its accuracy and efficiency in reducing the size of such problems and consequently in reducing the required computational demands.
3.
To fulfil the design requirements for vertically loaded symmetrical piled raft foundation where analyses are carried out on part of the model to reduce computational demands, the potential of lateral load due to wind or earthquake which violates the structural symmetry should be considered. A simplified method for analysis of lateral loads, that doesn’t need to model the full structure was proposed and verified.
4.
In contrast to common belief that piled raft foundation system is only suitable for the situations where, competent soil layers exist near the foundation level, the present research has shown that piled foundation with raft in contact with any soil type beneath it, behaves as a piled raft.
5.
In general, piled raft foundation system has two main advantages over corresponding conventional pile group; the first is the contribution of raft in carrying part of the applied load directly to supporting soil and the second is increasing the load capacity of the piles. The percentage of improvement depends on the supporting soil stratification.
209
6.
The behaviour of piled raft foundation system greatly depends on the stratification of the supporting soil, which may be classified into two broad categories as follows: • The case of uniform soil stratification, (floating pile group), which is the most favourable condition for piled raft foundation system. In this case the raft can support a considerable part of the applied load and greatly improves the behaviour of the piles, such that relatively small number of piles can control both average and differential settlements as well as supporting a significant part of the applied load. • The case of two layered soil stratification, (point bearing pile group). In this case although the piles are required to support a greater portion of the applied load, a great saving in foundation cost may be achieved by considering the improvement of the load carrying capacity of piles.
7.
A pile spacing in the order of 3.5-5 times pile diameter is recommended for case of uniform soil stratification, as for pile spacing greater than 5 times pile diameter, no considerable improvement in behaviour of piled raft can be recognized. On the other hand, for the case of end bearing piles, no limitations on maximum pile spacing is required.
8.
The working pile load obtained from traditional pile load test may lead to conservative design of piled raft. However, from the economic point of view, the results of pile load test may be used to obtain soil parameters (e.g. interface stiffness) through back analysis of the pile load test using finite element method and then using the calibrated parameters resulting from this analysis for the detailed analysis of piled raft foundation.
9.
The piled raft foundation system can tolerate higher settlements compared to unpiled raft foundation without violating the overall factor of safety of the foundation against bearing capacity failure. Therefore, it is recommended from the economic point of view, to increase allowable settlements for piled raft foundation provided that the settlement has no destructive effect on super structure and/or building utilities.
10.
The portion of the load directly transferred by the raft contact pressure is directly proportional to the following: • The stiffness of the soil layer beneath the raft (i.e. higher contact pressures may be observed in cases where raft is in contact with competent soils such as stiff clays and medium to dense sands).
210
• The applied load level (settlement level) • Pile spacing (i.e. higher contact pressures may be observed in case of fewer piles, supporting the concept of piles reducing settlement, Burland, 1977 and 1995). 11.
To control differential settlement, few piles strategically located at the central area of the raft (region of maximum settlement) are more efficient in reducing differential settlement rather than a larger number of uniformly distributed piles. However, it should be noted that central piles result in a different raft deformed shape compared to uniformly distributed pile group, and may cause negative differential settlements (settlement at raft corner greater than that at raft center).
12.
The central pile arrangement effectively reduces raft bending moments but it changes the distribution of the bending moment within the raft such that the position of maximum bending moment may occur away from column positions.
13.
Increasing number of piles is less effective in enhancing the behaviour of piled raft foundation rather than using piles with longer pile shaft length provided that the sum of pile shaft lengths is the same in both cases.
211
REFERENCES 1.
Abdel-Fattah, T.T. (1998). “Development of novel infinite elements for structural and geotechnical problems”, M.Sc. thesis, Cairo University, Egypt.
2.
Abdel-Fattah, T.T. (2004). “Material modelling and 3-D finite-element simulation for shield tunnelling in soft clay”, Ph.D. thesis, Cairo University, Egypt.
3.
Amar Bouzid, DJ., Tiliouine, B., and Vermeer, P.A., (2004). "Exact formulation of stiffness matrix for axisymmetric bodies under nonaxisymmetric loading". Computers and Geotechnics, vol. 31, pp 75-87.
4.
Basile, F. (2003). "Analysis and design of pile groups ". Extract from "Numerical analysis and modeling in geotechnics", spon press (eds, J.W bull), London, 2003, chapter 10, pp 278-315.
5.
Bergdahl, U. and Hult, G. (1981). "Load tests on friction piles in clay". Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, vol. 2, pp 625-630.
6.
Bowles, J. E. (1996). "Foundation analysis and design". McGraw-Hill, New York.
7.
Briaud, J.L., Tucker, L.M., NG, E. (1989). "Axially loaded 5 pile group and single pile in sand ". Proceedings of 12th International Conference on Soil Mechanics and Foundation Engineering, Rio de janeiro, vol. 2, pp 1121-1124.
8.
Burland, J.B. (1995). "Piles as Settlement Reducers". Keynote Address, 18th Italian Congress on Soil Mechanics, Pavia, Italy.
9.
Burland, J.B., Broms, B.B. and de Mello, V.F.B. (1977). "Behaviour of Foundations and Structures". Proc. 9 ICSMFE, Tokyo, 2, 495-546.
10.
Butterfield, R. and Banerjee, P.K. (1971). "A Rigid Disc Embedded In An Elastic Half Space". Journal of Geotechnical Engineering, ASCE, Vol. 2, November, pp 35-49
11.
Butterfield, R. and Banerjee, P.K. (1971). "The Elastic Analysis of Compressible Piles and Pile Groups". Geotechnique, vol. 21, No. 1, pp 43-60.
12.
Butterfield, R. and Banerjee, P.K. (1971). "The Problem of Pile Group-Pile cap Interaction". Geotechnique, vol. 21, No. 1, pp 135-142.
٢١٢
13.
Butterfield, R. and Banerjee, P.K. (2000). "Prediction of the horizontal loaddisplacement curves of pile groups based on the results of single pile tests". Can. Geotech, vol. 37, pp 951-962.
14.
Castelli, F. and Maugeri, M. (2002). "Simplfied Nonlinear Analysis for settlement prediction of Pile groups". Journal of Geotechnical and Geoenvironmental Engineering, ASCE, January, pp 76-82.
15.
Chan, S.I. and Tan, Y.K. (2003). "Improving pile raft performance". Proceedings of 12th Asian regional Conference on Soil Mechanics and Geotechnical Engineering, Leung et al. (eds) World Scientific publishing , vol. 1, pp 577-580.
16.
Chin, F. K. (1970). "Estimtion of the ultimate load of piles from tests not carried to failure". Proc. 2nd . SE Asian Conf. Soil Eng., Singapore, pp 81-92.
17.
Chin, F. K. (1972). "The inverse slope as a prediction of ultimate bearing capacity of piles". Proc. 3nd . SE Asian Conf. Soil Eng., Hong Kong, pp 83-91.
18.
Clancy, P. and Randolph, M.F. (1993). Analysis and Design of Piled raft Foundations. Int. J. NAM Geomechs.
19.
Cooke, R.W. (1974). "The settlement of friction pile foundation". Proc. Conf., Tall Buildings, Kuala Lumpur, vol. 3, pp. 1-16.
20.
Cooke, R.W. (1986). "Piled raft foundations on stiff clays- a contribution to design philosophy". Geotechnique, vol. 36, No. 2, pp 169-203.
21.
Cooke, R.W., Bryden-Smith, D.W., Gooch, M.N. and Sillett, D.F. (1981). "Some observations of the foundation loading and settlement of a multistory building on a piled raft foundation in London clay." Proc. Inst. Civ. Engn., London, Part. 1, No.107, pp. 433- 460.
22.
Coyle, H.M. and Reese, L.C. (1966). "Load transfer for axially loaded piles in clay ". Journal of the Soil Mechanics And Foundations Division, ASCE, Vol. 92, March, No. SM2, pp 1-25.
23.
Cunha, R.P., Poulos, H.G. and Small, J.C. (2001). "Investigation of design Alternatives for A piled raft case history". Journal of Geotechnical and Geoenvironmental Engineering, ASCE, August, pp 635-641.
24.
Dalerci, G. and DeL Grosso, A. (1981). "Non linear finite element analysis of piles in cohesionless soils". Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, vol. 2, pp 681-702.
٢١٣
25.
De Beer, E. E. (1967). "Proefondervindelijke bijdrage tot de studie van het gransdraagvermogen van zand onder funderingen opstaal; Bepaling von der vormfactor sb". Annales des Travaux Publics de Belgique, 68, No. 6, pp 481506.
26.
Desai, C.S., Kuppusamy, T. and Alameddine, A.R. (1981). "Pile cap-Pile Group-Soil Interaction". Journal of Structural Division, ASCE, Vol. 107, May, No. ST5, pp 817-835.
27.
DIANA®, (2005). A finite element analysis program, Users manual release 9.1, TNO DIANA BV., Delft, The Netherlands.
28.
El-Kadi, F.A. and Abdel-Fattah, T.T. (1998). “Structural and geotechnical studies for retrofitting the inclination of a structure built on non homogeneous soil”, Proc. of Arab Conf. for repair and retrofitting of structures, Cairo, Egypt, Vol. 1, pp. 365-382.
29.
EL-Mossallamy, Y. and Franke, E. (1997), "Numerical modeling to simulate the behaviour of piled raft foundations". Published by the authors, Darmstadt, Germany.
30.
EL-Mossallamy, Y., (2000), "Behaviour of large diameter single pile and pile groups
in
overconsolidated
clay".
Proceedings
of
4th
International
Geotechnical Engineering Conference, Egypt, pp 451-464. 31.
EL-Mossallamy, Y., (2003). "Economic application of piled raft foundations; Case histories". Proceedings of 10th International Colloquium on structural and Geotechnical Engineering, Egypt, pp 1-14.
32.
EL-Nahas, F., EL-Kadi, F., Khafagy, A. and Mazen, H. (1992). "Settlement of piled-raft foundation of a tall building ". Proceedings of International Colloquium on structural Engineering, Egypt, pp 181-193.
33.
Fahey, M. and Carter, J. P. (1993). "A finite element study of the pressuremeter test in sand using a non-linear elastic plastic model". Can. Geot. J., April 1993.
34.
Fjellerup, F.E. (1981) "Load tests on large bored piles in sand ". Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, vol. 2, pp713-716.
35.
Fleming, W.G. K., Weltman, A. J., Randolph, M. F. and Elson, W. K. (1992), "Piling Engineering", Blackie and sons LTD., London.
٢١٤
36.
Focht, J. A., (1967). "Discussion to paper by Coyle and Reese". J.S.M.F.D., ASCE, Vol. 93: SM1, pp 133-138.
37.
Franke, E., Lutz, B. and El-Mossallamy, Y. (1994). Measurements and Numerical Modelling of High-Rise Building Foundations on Frankfurt Clay’. Geot. Spec. Pub. 40, ASCE, 2: 1325-1336.
38.
Georgiadis, Y., Pitilakis, K., Tsotsos, S., and Valalas, D. (1989). "Settlement of a liquid storage tank founded on piles". Proceedings of 12th International Conference on Soil Mechanics and Foundation Engineering, Rio de janeiro, pp 1056-1060
39.
Groen, A. E., (1995), "Two elastoplastic models for the behaviour of soils", Technical report 03.21.0.31.12, Delft university of technology.
40.
Guo, W.D., (2000). "Visco-elastic load transfer models for axially loaded piles". Int. J. Numer. Anal. Meth. Geomech., 24, pp 135-163.
41.
Hain, S.J. and Lee, I.K. (1978). "The analysis of flexible raft-pile systems". Geotechnique, vol. 28, No. 1, pp 65-83.
42.
Hain, S.J. and Lee, L.K. (1974). "Rational Analysis Of Raft Foundation". Journal of Geotechnical Engineering Division, ASCE, Vol. 100, July, No. G17, pp 845-860.
43.
Hansbo, S. (1993). "Interaction problems related to the installation of pile groups". Proc. 2nd Seminar, Deep foundation, Ghent, pp 59-66.
44.
Hooper, J.A. and Wood, L.A., (1977). "Comparative Behaviour of Raft and Piled Foundations". Proceedings of 9th International Conference on Soil Mechanics and Foundation Engineering, Tokyo, vol. 1, pp 545-550.
45.
Hooper, J.A. and Wood, L.A., (1981). "Behaviour of flexible piled Foundations". Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, vol. 2, pp 735-740.
46.
Horikoshi, K. And Randolph, M. F. (1997). "On The Definition of Raft-Soil Stiffness Ratio for Rectangular Raft". Geotechnique, Vol. 47, No. 5, Pp 10551061.
47.
Horikoshi, K. and Randolph, M. F. (1997). "Optimum Design of Piled Raft Foundations". Proceedings of 14th International Conference of Soil Mechanic and Foundation Engineering, Hamburg, vol. 1, pp 1073-1076.
48.
Horikoshi, K. and Randolph, M. F. (1998). "A Contribution to Optimum Design of Piled Raft". Geotechnique, vol. 1, No. 1, pp 301-317. ٢١٥
49.
Horikoshi, K. And Randolph, M. F. (1999). "Estimation Of Overall Settlement Of Piled Rafts". Soils and Foundations, Vol. 39, No. 2, Apr. 1999, Pp 59-68.
50.
Katzenbach, R., Arslan, U., and Holzhauser, J. (1997). "Soil-StructureInteraction of the 300m high Commerzbank tower in Frankfurt am Main Measurements and numerical studies". Proceedings of 14th International Conference of Soil Mechanic. and Foundation Engineering, Hamburg, vol. 1, pp 1081-1084.
51.
Katzenbach, R., Arslan, U., Moorman, C. and Reul, O. (1998). "Piled raft foundation: interaction between piles and raft". Darmstadt geotechnics. (Darmstadt university of technology), No. 4, pp 279-296.
52.
Katzenbach, R., Moorman, C. and Reul, O. (1999). "Ein beitrag zur klarung des tragverhaltens von kombinierten pfahl-plattengrundungen". In Rodatz, W. (ed.), Pfshl-symposium, Mitteilung des institutes fur grundbau und Bodenmechanik, TU Braunschweig, 60, pp 263-299.
53.
Kezdi, A., "Pile foundation", Foundation Engineering Handbook, Winterkorn, H. F. and Fang, H. (eds.), Van Nostrand Reinhold publisher, N.Y., pp. 556600.
54.
Kim, K.N., Lee, S.H., Chung, C.K., Kim, M.M. and Lee, H.S., (2001). "Optimal pile arrangement for minimizing differential settlement in piled raft foundations". Computers and Geotechnique, vol. 28, pp 235-253.
55.
Konder, R.L. (1963). "Hyperbolic stress-strain response:cohesive soils". Journal of the Soil Mechanics And Foundations Division , ASCE, Vol. 89, February, No. SM1, pp 115-143.
56.
Kraft, L.M. (1982). "Effective Stress Capacity Model For Piles in Clay ". Journal of Geotechnical Engineering, ASCE, Vol. 108, November, No. GT11, pp 984-997.
57.
Lee, C.Y. (1993). "Pile Group Settlement Analysis By Hybrid Layer Approach". Journal of Geotechnical Engineering, ASCE, Vol. 119, June, No. 6, pp 984-997.
58.
Lee, K.M. and Xiao, Z.R. (2001). "A simplified nonlinear approach for pile group settlement analysis in multilayered soils" Can. Geotech, vol. 38, pp 1063-108
59.
Mandolini, A. and Viggani, C., (1997). "Settlement of piled foundations". Geotechnique, vol. 47, No. 4, pp 791-816. ٢١٦
60.
Maybaum, G., Vittinghoff, T. and Rodatz, W. (1999). "Proof of bearing capacity and the serviceability of piled rafts". Institute for foundation Engineering and soil mechanics, Technical university of Braunschweig, Germany.
61.
Mayne, P.W. and Poulos, H.G. (1999). "Approximate Displacement Influence Factors for Elastic Shallow Foundations". Journal of Geotechnical and Geoenvironmental Engineering, ASCE, June, Pp 453-460.
62.
Meyerhof, G., G., (1959). "Compaction of sands and bearing capacity of piles". J.S.M.F.D., ASCE, Vol. 85: SM6, pp 1-29.
63.
Moorman, C. (2002). "Trag-und verformungsverhalten tiefer baugruben in bindigen boden unter besonderer berucksichtigang der baugrund-tragwerk-und der baugrund-grundwasser-interaktion". Mitleilungen des institutes und der versuchsantalt fur geotechnik der technischen uni-versitat, Darmstadt, Heft, 59.
64.
O’Neill, P.W., Hawkins, R.A and Mahar, L.J. (1982). "Load transfer mechanisms in piles and pile groups". Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 108, December, No.GT12, Pp 1605-1621.
65.
O’Neill, P.W., Hawkins, R.A. and Audibert, J.M.E. (1982). "Installation of Pile Group In Overconsolidated Clay". Journal of Geotechnical Engineering Division, ASCE, Vol. 108, November, No. GT11, pp 1369-1385.
66.
Ottaviani, M., (1975). "Three-dimensional finite element analysis of vertically loaded pile groups". Journal of Geotechnique, Vol. 25, No.2, pp 159-174.
67.
Padfield, C.J. and Sharrock, M.J. (1983). "Settlement of structures on clay soils". CIRIA special publication, No. 27.
68.
Plazek, D., and Jentzsch, E. (1997). "Pile-raft-foundation under exceptional vertical loads-Bearing behaviour and settlements". Proceedings of 14th International Conference of Soil Mechanic. and Foundation Engineering, Hamburg, vol. 1, pp 1115-1118.
69.
Poulos, H. G. (1989). "Twenty-ninth Rankine lecture: Pile behaviour-theory and application". Geotechnique, vol. 39, No. 3, pp 363-415.
70.
Poulos, H. G. (1994). "An Approximate Numerical Analysis of Pile-Raft Interaction". Int. J. NAM Geomechs., 18: 73-92.
٢١٧
71.
Poulos, H. G. (2001). "Piled Raft Foundations: Design and Application". Geotechnique, vol. 51, No. 2, pp 95-113.
72.
Poulos, H.G., and Davis, E.H. (1980). "Pile foundation analysis and design". John Wiley and sons, Inc. New York.
73.
Poulos, H.G., Small, J.C., Ta, L.D. Sinha, J. and Chen, L. (1997). "Comparison of Some Methods for Analysis of Piled Rafts". Proceedings of 14th ICSMFE, Hamburg, vol. 1, pp 69-72.
74.
Poulos, H.G. (1991). "Analysis of Piled Strip Foundations". Comp. Methods & Advances in Geomechs., ed. Beer et al, Balkema, Rotterdam, 1: 183-191.
75.
Poulos, H.G. (2001). "Methods of Analysis of Piled Raft Foundations". A Report Prepared on Behalf of Technical Committee TC18 on Piled Foundations
76.
Prakoso, W.A. and Kulhawy, F.H. (2001). "Contribution to piled foundation design". Journal of Geotechnical and Geoenvironmental Engineering, January, pp 17-24.
77.
Randolph, M.F. (1977). "A theoretical study of the performance of piles". PhD thesis, Cambridge University, England.
78.
Randolph, M.F. (1983). "Design of Piled Raft foundations". Proc. Int. Symp. on recent developments in laboratory and field tests and analysis of geotechnical problems, Bangkok, pp 525-537.
79.
Randolph, M.F. (1994). "Design Methods for Pile Groups and Piled Raft". XIII ICSMFE, 1994, New Delhi, India.
80.
Randolph, M.F. and Clancy, P. (1993). "Efficient design of piled rafts". Proc. 2nd Int. Seminar, Deep foundation, Ghent, pp 119-130.
81.
Randolph, M.F. and Wroth, C.P. (1978). "Analysis of Deformation of Vertically Loaded Piles". Journal of Geotechnical Engineering Division, ASCE, December, 1978, pp 1465-1488.
82.
Reul, O. and Randolph M.F. (2004), "Design Strategies for Piled Raft Subjected to Nonuniform Vertical Loading". Journal of Geotechnical and Geoenvironmental Engineering, ASCE, January, 2004, pp 1-13.
83.
Reul, O. and Randolph M.F. (2003), "Piled Raft in Overconsolidated Clay: Comparison
of
In
Situ
Measurements
Geotechnique, vol. 53, No. 3, pp 301-315.
٢١٨
and
Numerical
Analyses".
84.
Russo, G., and Viggiani, C. (1997). "Some aspects of numerical analysis of piled rafts ". Proceedings of 14th International Conference of Soil Mechanic. and Foundation Engineering, Hamburg, vol. 1, pp 1125-1128.
85.
Shabana, W. A., El-Meligy, M.M, and Bayoumy, A.H. (2003). "Nonlinear analysis of raft foundation partially supported by piles". Proceedings of 10th International Colloquium on Structural and Geotechnical Foundation Engineering, Egypt, pp. E03GE18-1- E03GE18-21.
86.
Shabana, W. A., El-Meligy, M.M, and Bayoumy, A.H. (2004). "Interactive nonlinear behavior of piled-raft foundation ". Proceedings of International Conference on Structural and Geotechnical Engineering and Construction Technology, Mansura, Egypt, pp 995-1014.
87.
Shen, W.Y., Chow, Y.K. and Yong, K.Y., (2000). "Practical method for settlement
analysis
of
pile
groups".
Journal
of
Geotechnical
and
Geoenvironmental Engineering, ASCE, Vol. 126, No. 10, October, pp 890897. 88.
Sinha, J. (1997). Piled Raft Foundations Subjected to Swelling and Shrinking Soils. PhD Thesis, Univ. of Sydney, Australia.
89.
Sommer, H., and Hammbach, P. (1974). "Groβpfahlversuche im Ton fur die Grundung der Talbrucke Alzey". Der Bauingenieur, Vol. 49, pp 310-317.
90.
Sommer, H., Wittmann, P. and Ripper, P. (1985). "Piled raft foundation of a tall building in Frankfurt clay". Proceedings of 11th ICSMFE, Sanfransisco, Vol. 4, pp 2253-2257.
91.
Ta, L.D. and Small, J.C. (1996). Analysis of Piled Raft Systems in Layered Soils. Int. J. NAM Geomechs., 2: 57-72.
92.
Timoshinko, S.P. and Goodier, J. N. (1983). "Theory of elasticity". McGrawHill, New York.
93.
Van Eekelen, S. J. M. and Van den Berg, P., (1994), "The Delft egg model, a constitutive model for clay", G. M. A. Kuslers and M. A. N. Hendriks (eds.), DIANA computational mechanics’94, Kluwer academic publisher , pp. 103116.
94.
Van Langen, H. and Vermeer, P.A., (1991), "Interface elements for singular plasyicity points", International journal for numerical and analytical methods in geomechanics, vol. 15, pp. 301-315.
٢١٩
95.
Viggiani, C. (1998). "Pile groups and piled rafts behaviour". In deep foundations on bored and auger piles (eds Van Impe and Haegman), Rotterdam: Balkema, pp 77-90.
96.
Wehnert, M. and Vermeer, P.A. (2004). "Numerical analysis of load tests on bored piles ". Institute for Geotechnical Engineering, University of Stuttgart, Germany, August, pp 1-6.
97.
Wong, S. H. and Poulos, H. G. (2006). "Approximate Pile-to-Pile Interaction Factors between Two Dissimilar Piles". Computers and Geotechnics.
98.
Yamashita, K., Kakurai, M. and Matsuyama, M. (1989). "Settlement analysis of large-diameter bored pile groups". Proceedings of 12th ICSMFE, Rio de Janeiro, pp 1079-1082.
99.
Yamashita, K., Kakurai, M. and Yamada, T. (1994). "Investigation of a piled raft foundation on stiff clay " XIII ICSMFE, New Delhi, pp 543-546.
100.
Yang, T.Y. (1986). "Finite element structural analysis". Printice-Hall, Inc. New Jersy.
101.
Zienkiewicz, O.C. and Taylor, R. L. (1987). "The finite element method". 4th edition, McGraw-Hill.
٢٢٠
ﺍﻟﺘﻤﺜﻴل ﺍﻟﻌﺩﺩﻱ ﻷﺴﺎﺱ ﺍﻟﻠﺒﺸﺔ ﺍﻟﺨﺎﺯﻭﻗﻴﺔ ﺘﺤﺕ ﺘﺄﺜﻴﺭ ﺍﻷﺤﻤﺎل ﺍﻟﺭﺃﺴﻴﺔ ﺇﻋﺩﺍﺩ
ﻋﻤﺭﻭ ﺃﺤﻤﺩ ﺒﻜﺭﻱ ﺤﻤﻴﺩﻩ ﺒﻜﺎﻟﻭﺭﻴﻭﺱ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ -ﻤﺎﺠﺴﺘﻴﺭ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻹﻨﺸﺎﺌﻴﺔ
ﺭﺴﺎﻟﺔ ﻤﻘﺩﻤﻪ ﺇﻟﻰ ﻜﻠﻴﻪ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﻪ ﺍﻟﻘﺎﻫﺭﺓ ﻜﺠﺯﺀ ﻤﻥ ﻤﺘﻁﻠﺒﺎﺕ ﺍﻟﺤﺼﻭل ﻋﻠﻲ ﺩﺭﺠﺔ ﺍﻟﺩﻜﺘﻭﺭﺍﻩ ﻓﻲ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ )ﺍﻨﺸﺎﺀﺍﺕ(
ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﺔ ﺍﻟﻘﺎﻫﺭﺓ ﺍﻟﺠﻴﺯﺓ ،ﺠﻤﻬﻭﺭﻴﺔ ﻤﺼﺭ ﺍﻟﻌﺭﺒﻴﺔ ٢٠٠٧
ﺍﻟﺘﻤﺜﻴل ﺍﻟﻌﺩﺩﻱ ﻷﺴﺎﺱ ﺍﻟﻠﺒﺸﺔ ﺍﻟﺨﺎﺯﻭﻗﻴﺔ ﺘﺤﺕ ﺘﺄﺜﻴﺭ ﺍﻷﺤﻤﺎل ﺍﻟﺭﺃﺴﻴﺔ ﺇﻋﺩﺍﺩ
ﻋﻤﺭﻭ ﺃﺤﻤﺩ ﺒﻜﺭﻱ ﺤﻤﻴﺩﻩ ﺒﻜﺎﻟﻭﺭﻴﻭﺱ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ -ﻤﺎﺠﺴﺘﻴﺭ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻹﻨﺸﺎﺌﻴﺔ
ﺭﺴﺎﻟﺔ ﻤﻘﺩﻤﻪ ﺇﻟﻰ ﻜﻠﻴﻪ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﻪ ﺍﻟﻘﺎﻫﺭﺓ ﻜﺠﺯﺀ ﻤﻥ ﻤﺘﻁﻠﺒﺎﺕ ﺍﻟﺤﺼﻭل ﻋﻠﻲ ﺩﺭﺠﺔ ﺍﻟﺩﻜﺘﻭﺭﺍﻩ ﻓﻲ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ )ﺍﻨﺸﺎﺀﺍﺕ(
ﺘﺤﺕ ﺇﺸﺭﺍﻑ
ﺃ.ﺩ .ﻋﺎﺩل ﻴﺤﻴﻰ ﻋﻘل
ﺃﺴﺘﺎﺫ ﺘﺤﻠﻴل ﻭﻤﻴﻜﺎﻨﻴﻜﺎ ﺍﻻﻨﺸﺎﺀﺍﺕ ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ – ﺠﺎﻤﻌﺔ ﺍﻟﻘﺎﻫﺭﺓ
ﻜﻠﻴﻪ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﺔ ﺍﻟﻘﺎﻫﺭﺓ ﺍﻟﺠﻴﺯﺓ ،ﺠﻤﻬﻭﺭﻴﺔ ﻤﺼﺭ ﺍﻟﻌﺭﺒﻴﺔ ٢٠٠٧
ﺍﻟﺘﻤﺜﻴل ﺍﻟﻌﺩﺩﻱ ﻷﺴﺎﺱ ﺍﻟﻠﺒﺸﺔ ﺍﻟﺨﺎﺯﻭﻗﻴﺔ ﺘﺤﺕ ﺘﺄﺜﻴﺭ ﺍﻷﺤﻤﺎل ﺍﻟﺭﺃﺴﻴﺔ ﺇﻋﺩﺍﺩ
ﻋﻤﺭﻭ ﺃﺤﻤﺩ ﺒﻜﺭﻱ ﺤﻤﻴﺩﻩ ﺒﻜﺎﻟﻭﺭﻴﻭﺱ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ -ﻤﺎﺠﺴﺘﻴﺭ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻹﻨﺸﺎﺌﻴﺔ
ﺭﺴﺎﻟﺔ ﻤﻘﺩﻤﻪ ﺇﻟﻰ ﻜﻠﻴﻪ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﻪ ﺍﻟﻘﺎﻫﺭﺓ ﻜﺠﺯﺀ ﻤﻥ ﻤﺘﻁﻠﺒﺎﺕ ﺍﻟﺤﺼﻭل ﻋﻠﻲ ﺩﺭﺠﺔ ﺍﻟﺩﻜﺘﻭﺭﺍﻩ ﻓﻲ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ )ﺍﻨﺸﺎﺀﺍﺕ(
ﻴﻌﺘﻤﺩ ﻤﻥ ﻟﺠﻨﺔ ﺍﻟﻤﻤﺘﺤﻨﻴﻥ: ﺍﻷﺴﺘﺎﺫ ﺍﻟﺩﻜﺘﻭﺭ /ﻋﺎﺩل ﻴﺤﻴﻰ ﻋﻘل
ﺍﻟﻤﺸﺭﻑ ﺍﻟﺭﺌﻴﺴﻰ
ﺍﻷﺴﺘﺎﺫ ﺍﻟﺩﻜﺘﻭﺭ /ﻋﺒﺩ ﺍﻟﺭﺤﻤﻥ ﺼﺎﺩﻕ ﺒﺎﺯﺭﻋﻪ
ﻋﻀﻭﹰﺍ
ﺍﻷﺴﺘﺎﺫ ﺍﻟﺩﻜﺘﻭﺭ /ﻨﺎﺩﻴﺔ ﺸﻨﻭﺩﻩ ﺠﺭﺠﺱ
ﻋﻀﻭﹰﺍ
ﻜﻠﻴﻪ ﺍﻟﻬﻨﺩﺴﺔ ،ﺠﺎﻤﻌﺔ ﺍﻟﻘﺎﻫﺭﺓ ﺍﻟﺠﻴﺯﺓ ،ﺠﻤﻬﻭﺭﻴﺔ ﻤﺼﺭ ﺍﻟﻌﺭﺒﻴﺔ ٢٠٠٧
ﻣﻠﺨﺺ اﻟﺮﺳﺎﻟﺔ ﻟﻘﺪ أﺻﺒﺢ ﻣﻔﻬﻮم اﻷﺳﺎﺳﺎت اﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﻣﻌﺮوﻓﺎ ﻣﻨﺬ ﺳﻨﻮات ﻣﻦ اﻟﻨﺎﺣﻴﺔ اﻟﺘﻄﺒﻴﻘﻴﺔ ،ﺣﻴﺚ أﻧﻪ ﻼ اﻗﺘﺼﺎدﻳ ًﺎ ﻓﻲ اﻟﺤﺎﻻت اﻟﺘﻲ ﻻ ﺗﻔﻲ ﻓﻴﻬﺎ اﻟﻠﺒﺸﺔ وﺣﺪهﺎ ﺑﻤﺘﻄﻠﺒﺎت اﻟﺘﺼﻤﻴﻢ .وﻟﻜﻦ ﻣﻘﺎرﻧﺔ ﻳﻤﺜﻞ ﺣ ً ﺑﺎﻷﺳﺎﺳﺎت اﻟﺨﺎزوﻗﻴﺔ اﻟﺘﻘﻠﻴﺪﻳﺔ ﻓﺈن اﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﺗﻌﺘﺒﺮ ﻧﻈﺎم ﺗﺄﺳﻴﺲ ﻣﻌﻘﺪ ﻳﺤﺘﺎج إﻟﻲ ﻓﻬﻢ دﻗﻴﻖ ﻟﺠﻤﻴﻊ اﻟﺘﻔﺎﻋﻼت اﻟﻤﺘﺒﺎدﻟﺔ ﺑﻴﻦ اﻟﺘﺮﺑﺔ واﻷﺳﺎس واﻟﺘﻲ ﺗﺘﺤﻜﻢ ﻓﻲ ﺳﻠﻮك ﻣﺜﻞ هﺬا اﻟﻨﻮع ﻣﻦ اﻷﺳﺎﺳﺎت .آﻤﺎ ﺗﻘﺘﻀﻲ ﻃﺒﻴﻌﺔ هﺬﻩ اﻟﺘﻔﺎﻋﻼت أن ﻃﺮق اﻟﺘﺤﻠﻴﻞ اﻹﻧﺸﺎﺋﻲ اﻟﻤﻨﺎﺳﺒﺔ ﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﻳﺠﺐ أن ﺗﻜﻮن ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد )ﺑﻮﻟﺲ ،١٩٩٧اﻟﻤﺴﻠﻤﻲ ،٢٠٠١راؤول وراﻧﺪوﻟﻒ ٢٠٠٣ وﺁﺧﺮون(. ﻳﻬﺪف اﻟﺒﺤﺚ إﺑﺘﺪا ًء إﻟﻲ ﻋﻤﻞ دراﺳﺔ ﺗﻔﺼﻴﻠﻴﺔ ﻟﻠﻤﺘﻄﻠﺒﺎت اﻷﺳﺎﺳﻴﺔ ﻟﻠﺘﻤﺜﻴﻞ اﻟﻌﺪدي ﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ واﻟﺘﻲ ﺗﺘﻜﻮن ﻣﻦ اﻟﻠﺒﺸﺔ ،اﻟﺨﻮازﻳﻖ واﻟﺘﺮﺑﺔ اﻟﺤﺎﻣﻠﺔ وذﻟﻚ ﺑﻄﺮﻳﻘﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد. ﺗﻤﺖ دراﺳﺔ واﺧﺘﺒﺎر اﻟﺒﺪاﺋﻞ اﻟﻤﺨﺘﻠﻔﺔ ﻟﻜﻞ ﻣﻦ ﻧﻤﺎذج ﺗﻤﺜﻴﻞ اﻟﻤﺎدة و اﻟﺘﻤﺜﻴﻞ اﻟﻬﻨﺪﺳﻲ ﻟﻤﻜﻮﻧﺎت اﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﺑﺎﻹﺿﺎﻓﺔ إﻟﻲ أﺳﻠﻮب ﺗﻤﺜﻴﻞ اﻟﻌﻼﻗﺔ ﺑﻴﻦ اﻟﺘﺮﺑﺔ واﻷﺳﺎس وذﻟﻚ ﻣﻦ ﺧﻼل ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﻤﺴﺎﺋﻞ ﺛﻨﺎﺋﻴﺔ وﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد واﻟﺘﻲ ﺗﻢ ﺣﻠﻬﺎ ﺑﻄﺮﻳﻘﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد. ﺗﻢ ﻋﻤﻞ اﻟﻤﻘﺎرﻧﺎت اﻟﺘﻮﺿﻴﺤﻴﺔ ﻟﻨﺘﺎﺋﺞ هﺬﻩ اﻟﺪراﺳﺔ ﺣﻴﺚ ﺗﻢ اﺳﺘﺨﻼص اﻷﺳﻠﻮب اﻟﻤﻨﺎﺳﺐ ﻟﻠﺘﻤﺜﻴﻞ اﻟﻌﺪدي ﻟﻤﺜﻞ هﺬا اﻟﻨﻮع ﻣﻦ اﻷﺳﺎﺳﺎت ﺑﻄﺮﻳﻘﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد .ﺗﻢ ﺑﻌﺪ ذﻟﻚ اﻗﺘﺮاح ﺑﻌﺾ اﻟﻮﺳﺎﺋﻞ اﻟﻌﻤﻠﻴﺔ ﻟﺘﻘﻠﻴﻞ ﺣﺠﻢ اﻟﺬاآﺮة وﺳﻌﺔ اﻟﺘﺨﺰﻳﻦ وزﻣﻦ اﻟﺤﻞ ﻓﻲ ﻣﺜﻞ هﺬﻩ اﻟﻨﻮﻋﻴﺔ ﻣﻦ اﻟﻤﺴﺎﺋﻞ اﻟﻜﺒﻴﺮة ﺑﺤﻴﺚ ﻳﻤﻜﻦ ﺣﻠﻬﺎ ﺑﺎﺳﺘﺨﺪام اﻟﺤﺎﺳﺒﺎت اﻟﺸﺨﺼﻴﺔ اﻟﻤﺘﻄﻮرة وﻣﻦ هﺬﻩ اﻟﻮﺳﺎﺋﻞ، ﺗﻤﺜﻴﻞ اﻟﻄﺒﻘﺔ اﻟﺴﻔﻠﻴﺔ ﻣﻦ اﻟﺘﺮﺑﺔ ﻓﻲ ﺷﺒﻜﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺑﺎﺳﺘﺨﺪام ﻋﻨﺎﺻﺮ ﺗﻤﺎس ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد، آﻤﺎ ﺗﻢ اﻗﺘﺮاح ﻃﺮﻳﻘﺔ ﻣﺒﺴﻄﺔ ﻟﺘﺤﻠﻴﻞ اﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﺗﺤﺖ ﺗﺄﺛﻴﺮ أﺣﻤﺎل ﺟﺎﻧﺒﻴﺔ ﺛﺎﻧﻮﻳﺔ .وﺗﻢ اﻟﺘﺤﻘﻖ ﻣﻦ ﻓﺎﻋﻠﻴﺔ ودﻗﺔ اﻷﺳﺎﻟﻴﺐ اﻟﻤﻘﺘﺮﺣﺔ ﻋﻦ ﻃﺮﻳﻖ اﺳﺘﺨﺪاﻣﻬﺎ ﻓﻲ ﺣﻞ ﻣﺴﺎﺋﻞ ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد وﻣﻘﺎرﻧﺔ اﻟﻨﺘﺎﺋﺞ ﺑﻨﺘﺎﺋﺞ ﻃﺮق اﻟﺘﻤﺜﻴﻞ اﻟﺪﻗﻴﻘﺔ. ﻟﺤﺴﺎب أﺣﻤﺎل اﻟﺨﻮازﻳﻖ ﻣﻦ اﻹﺟﻬﺎدات اﻟﺪاﺧﻠﻴﺔ اﻟﻨﺎﺗﺠﺔ ﻋﻦ اﻟﺘﺤﻠﻴﻞ اﻟﻌﺪدي ﺑﻄﺮﻳﻘﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد ،ﺗﻢ ﺗﻄﻮﻳﺮ ﺑﺮﻧﺎﻣﺞ ﺧﺎص ﻟﻬﺬا اﻟﻐﺮض وذﻟﻚ ﺑﺘﻌﺪﻳﻞ ﺑﺮﻧﺎﻣﺞ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة اﻟﻤﺘﺎح "دﻳﺎﺗﻦ" واﻟﻤﻌﺪ ﺑﻮاﺳﻄﺔ ﻋﺒﺪاﻟﻔﺘﺎح ) .(٢٠٠٤ﺗﻢ اﺧﺘﺒﺎر اﻟﺒﺮﻧﺎﻣﺞ ﺑﺤﻞ ﻣﺴﺄﻟﺔ ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد ﻟﺨﺎزوق ﻣﻔﺮد ﺗﺤﺖ ﺗﺄﺛﻴﺮ ﺣﻤﻞ ﻣﻮزع ﺑﺎﻧﺘﻈﺎم. ﺗﻢ ﺑﻌﺪ ذﻟﻚ ﺗﻨﻔﻴﺬ ﺑﺮﻧﺎﻣﺞ ﺑﺤﺜﻲ ﺗﻔﺼﻴﻠﻲ ﻟﺪراﺳﺔ ﺳﻠﻮك وﺗﻘﻴﻴﻢ أداء اﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ وذﻟﻚ ﺑﺘﻄﺒﻴﻖ أﺳﻠﻮب اﻟﺘﻤﺜﻴﻞ اﻟﻌﺪدي ﺛﻼﺛﻲ اﻷﺑﻌﺎد ﺑﺎﺳﺘﺨﺪام ﻃﺮﻳﻘﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة اﻟﺬي ﺗﻢ دراﺳﺘﻪ ﺳﺎﺑﻘ ًﺎ ﺣﻴﺚ ﺗﻤﺖ اﻟﺪراﺳﺔ ﻋﻠﻲ ﻧﻤﻮذج اﻓﺘﺮاﺿﻲ ﻟﻠﺒﺸﺔ ﺧﺎزوﻗﻴﺔ ﻣﺮﺑﻌﺔ .وﺷﻤﻞ اﻟﺒﺤﺚ دراﺳﺔ اﻟﺘﺮآﻴﺐ
اﻟﻄﺒﻘﻲ ﻟﻠﺘﺮﺑﺔ ،ﻋﺪد اﻟﺨﻮازﻳﻖ ،ﺗﻮزﻳﻊ اﻟﺨﻮازﻳﻖ وﺗﺄﺛﻴﺮ زﻳﺎدة ﻃﻮل اﻟﺨﺎزوق وﺗﺄﺛﻴﺮ هﺬﻩ اﻟﻌﻮاﻣﻞ ﻋﻠﻲ اﻟﻌﻼﻗﺔ ﺑﻴﻦ اﻟﺤﻤﻞ وآﻞ ﻣﻦ اﻟﻬﺒﻮط وﻓﺮق اﻟﻬﺒﻮط وآﺬﻟﻚ ﺗﻮزﻳﻊ ﺿﻐﻂ اﻟﺘﻼﻣﺲ أﺳﻔﻞ اﻟﻠﺒﺸﺔ )اﻟﺠﺰء ﻣﻦ اﻟﺤﻤﻞ اﻟﺬي ﺗﻨﻘﻠﻪ اﻟﻠﺒﺸﺔ ﻣﺒﺎﺷﺮة ﻟﻠﺘﺮﺑﺔ( ﺑﺎﻹﺿﺎﻓﺔ إﻟﻲ ﻋﺰوم اﻹﻧﺤﻨﺎء ﺑﺎﻟﻠﺒﺸﺔ وﺳﻠﻮك اﻟﺨﻮازﻳﻖ اﻟﻤﻨﻔﺮدة أﺳﻔﻞ اﻟﻠﺒﺸﺔ وﺗﻮزﻳﻊ اﻷﺣﻤﺎل ﺑﻴﻦ اﻟﺨﻮازﻳﻖ .آﻤﺎ ﺗﻤﺖ دراﺳﺔ اﻟﺤﺎﻻت اﻟﻤﻨﺎﻇﺮة ﻟﻠﺒﺸﺔ ﻏﻴﺮ ﺧﺎزوﻗﻴﺔ ،ﻣﺠﻤﻮﻋﺔ ﺧﻮازﻳﻖ ﺗﻘﻠﻴﺪﻳﺔ وﺣﺎﻟﺔ اﻟﺨﺎزوق اﻟﻤﻔﺮد وﺗﻤﺖ ﻣﻘﺎرﻧﺔ اﻟﻨﺘﺎﺋﺞ .أﺛﺒﺘﺖ هﺬﻩ اﻟﺪراﺳﺔ ﺑﺼﻔﺔ ﻋﺎﻣﺔ اﻟﻔﺎﻋﻠﻴﺔ اﻻﻗﺘﺼﺎدﻳﺔ ﻟﻠﺒﺸﺔ اﻟﺨﺎزوﻗﻴﺔ ﻣﻘﺎرﻧﺔ ﺑﺄﺳﺎﻟﻴﺐ اﻟﺘﺄﺳﻴﺲ اﻟﻤﻨﺎﻇﺮة ﻓﻲ ﺟﻤﻴﻊ أﻧﻮاع اﻟﺘﺮﺑﺔ اﻟﺘﻲ ﺗﻤﺖ دراﺳﺘﻬﺎ. وأﺧﻴﺮًا ﺗﻢ ﻋﻤﻞ دراﺳﺔ ﻟﺤﺎﻟﺘﻴﻦ ﻷﺳﺎﺳﺎت ﻟﺒﺸﺔ ﺧﺎزوﻗﻴﺔ ﺣﻘﻴﻘﻴﺔ ﻟﻬﻤﺎ ﻗﻴﺎﺳﺎت ﺣﻘﻠﻴﺔ ﻣﺘﺎﺣﺔ ،ﺣﻴﺚ ﺗﻢ ﻋﻤﻞ ﻣﺤﺎآﺎﻩ ﻟﻬﺎﺗﻴﻦ اﻟﺤﺎﻟﺘﻴﻦ ﺑﺎﺳﺘﺨﺪام أﺳﻠﻮب اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد اﻟﻤﻘﺘﺮح .ﺗﻤﺖ ﻣﻘﺎرﻧﺔ ﻧﺘﺎﺋﺞ اﻟﺘﺤﻠﻴﻞ اﻟﻌﺪدي ﺑﺎﻟﻘﻴﺎﺳﺎت اﻟﺤﻘﻠﻴﺔ اﻟﻤﺘﺎﺣﺔ أﺛﻨﺎء وﺑﻌﺪ ﺗﻨﻔﻴﺬ اﻟﻤﺒﺎﻧﻲ اﻷﺻﻠﻴﺔ ،آﻤﺎ ﺗﻤﺖ ﻣﻘﺎرﻧﺔ اﻟﻨﺘﺎﺋﺞ آﺬﻟﻚ ﺑﻨﺘﺎﺋﺞ ﻣﺘﺎﺣﺔ ﻟﺘﺤﻠﻴﻞ ﻋﺪدي ﻟﺒﺎﺣﺜﻴﻦ أﺧﺮﻳﻦ ﻟﻨﻔﺲ اﻟﺤﺎﻟﺘﻴﻦ .وﻗﺪ ﻇﻬﺮ أن ﻧﺘﺎﺋﺞ اﻟﺘﺤﻠﻴﻞ اﻟﻌﺪدي اﻟﺤﺎﻟﻴﺔ ﺗﺘﻔﻖ إﻟﻲ ﺣﺪ آﺒﻴﺮ ﻣﻊ اﻟﻘﻴﺎﺳﺎت اﻟﺤﻘﻠﻴﺔ اﻟﻤﻨﺎﻇﺮة ﻣﻤﺎ ﻳﺆآﺪ دﻗﺔ وآﻔﺎءة أﺳﻠﻮب اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺪدة ﺛﻼﺛﻴﺔ اﻷﺑﻌﺎد اﻟﻤﺴﺘﺨﺪم ﻓﻲ هﺬﻩ اﻟﺪراﺳﺔ.
