Behaviour of Single Piles under Axial Loading Analysis of Settlement and Load Distribution
Joana Gonçalves Sumares Betencourt Ribeiro
Thesis to obtain the Master of Science Degree in Civil Engineering
Examination Committee Chairperson:
Prof. José Manuel Matos Noronha da Câmara
Supervisor:
Prof. Jaime Alberto dos Santos
Member of the Committee:
Prof. Peter John Bourne-Webb
May 2013
To my parents Aos meus pais
May the God of hope fill you with all joy and peace in believing, so that by the power of the Holy Spirit you may abound in hope. Romans 15:13
Que o Deus da esperança vos mantenha felizes e cheios da paz que nasce pela fé, para que abundeis na esperança pelo poder do Espírito Santo. Romanos 15:13
ABSTRACT The purpose of this thesis is to analyse the behaviour of single piles under axial loading, as far as settlement and load transfer mechanisms are concerned. It includes a literary review of two elastic theory-based methods, the Poulos and Davis method and the Randolph and Wroth method, as well as axisymmetric elastic modelling using the finite element-based program Plaxis. The results given by each method are organized in dimensionless charts of loadsettlement ratio and proportion of load transferred to the pile base in terms of the pile slenderness ratio and the soil inhomogeneity and compared amongst each other. This thesis also includes two comparison studies which involve axisymmetric elastoplastic modelling in Plaxis, considering the Mohr-Coulomb failure criterion. The first one is a comparison with two previous finite element simulations subject to very similar conditions: a GEFdyn 3D simulation and a CESARLCPC 2D axisymmetric simulation. It showed that in the simulation performed by Plaxis the value of load transferred to the pile base was lower than the others, although the total load-settlement curve was very similar in all three cases. The second one is a case study of the simulation of a static load test performed on a test pile. It includes the geological description of the site and the justification of the choice of parameters introduced in the model. Although limited information is available regarding the geological and geotechnical conditions of the site, the overall results were quite satisfactory. Key words: axially loaded pile, numerical modelling, soil-pile interaction, Poulos and Davis, Randolph
and Wroth
RESUMO O objectivo desta tese é analisar o comportamento de estacas isoladas sob carga axial, em termos de assentamento e mecanismos de transferência de carga. Inclui uma revisão literária de dois métodos baseados na teoria da elasticidade, o método de Poulos e Davis e o método de Randolph e Wroth, bem como a modelação elástica em estado axissimétrico utilizando o programa de elementos finitos Plaxis. Os resultados obtidos a partir destes métodos estão organizados em ábacos da relação carga-assentamento e carga da ponta-carga total em termos do factor de esbelteza da estaca e da variação de rigidez do solo, e são comparados entre si. Esta tese inclui também dois estudos de comparação que incluem modelação elastoplástica em estado axissimétrico utilizando Plaxis, considerando o critério de cedência de Mohr-Coulomb. O primeiro estudo trata-se de uma comparação com duas simulações numéricas realizadas anteriormente em condições idênticas: uma simulação 3D utilizando GEDdyn e uma simulação em estado axissimétrico utilizando CESAR-LCPC. Este estudo mostra que na simulação em Plaxis o valor da carga transferida para a base era inferior ao das outras simulações, apesar de a curva de carga no topo-assentamento no topo ser semelhante nos três casos. O segundo é um caso de estudo que consiste na simulação de um ensaio de carga estático executado numa estaca experimental. Inclui uma descrição geológica do local e a justificação da escolha de parâmetros introduzidos no modelo. Apesar da pouca informação disponível relativa às condições geológicas e geotécnicas do local, os resultados foram bastante satisfatórios. Palavras-chave: estacas sob carga axial, modelação numérica, interacção solo-estaca, Poulos e
Davis, Randolph e Wroth
ACKNOWLEDGEMENTS Firstly, I want to thank Professor Jaime Alberto dos Santos, for proposing this theme and guiding me through my research. I appreciate the confidence placed in me by permitting that I develop each chapter according to my will, allowing me to learn for myself. I also acknowledge the transmission of the value of discipline, organization and patience at work – I will not forget the importance of finishing a task before beginning the next one. Secondly, I want to thank João Camões, for answering my every question so promptly and thoroughly. His help with working with finite elements was priceless, teaching me how to find solutions methodically and observing details critically. He explained to me how useful a tool like Plaxis can be, and also how easily it can become a “black hole”. I will never forget that “no model works well the first time you run it”. I wish him all the luck with his career, and I know he will excel at whatever he
attempts. The gratitude I feel towards my parents cannot be expressed. I thank my father for encouraging me to pursue this career and setting up a fine professional example; he taught me that “to be an en gineer is not a job, it is a way of life”. I thank my mother for listening with infinite patience and being an endless
source of care and support. I thank both for providing me with the best conditions to study and learn and never pressing or demanding anything in return. There is nothing more a daughter can ask. I also thank my grandparents, my brothers and the rest of my family, for the good environment I grew up in, and for allowing me to discharge whenever I go home. I particularly thank my uncle António Carlos, for clarifying my doubts whenever I needed; I hope to have inherited at least some of his talent for engineering. I thank Marta for being the sister I never had, my most trustworthy friend. I thank her for always being there, and for so often disregarding her own work to help me with mine. We shared almost everything the last few years, the very best and the very worst moments. There is no one I would rather have spent this time with. I thank all my friends and colleagues who have in one way or another helped during university years, especially Francisco, Caldinhas, António, Miguel Melo, Ana Bento, Tiago Schiappa, Margarida and Guilherme. Because of them, Lisbon became a home to me; they made studying and working in projects much easier and more agreeable. I am proud to belong to such a group of civil engineers, as I am sure they will all be excellent professionals.
CONTENTS 1
2
3
Introduction ...................................................................................................................................... 1 1.1
Context and Motivation ............................................................................................................ 1
1.2
Objectives and Methodology ................................................................................................... 2
Elastic Theory-Based Methods for Analysis of Single Axially Loaded Piles ................................... 3 2.1
Introduction .............................................................................................................................. 3
2.2
Poulos and Davis Method ........................................................................................................ 6
2.3
Randolph and Wroth Method................................................................................................. 16
Comparison between Elastic Theory-Based Methods and the Finite Element Method in the
Analysis of Single Axially Loaded Piles ................................................................................................. 33
4
3.1
Introduction ............................................................................................................................ 33
3.2
The Model .............................................................................................................................. 34
3.2.1
Geometry ....................................................................................................................... 34
3.2.2
Loading .......................................................................................................................... 36
3.2.3
Materials and Interfaces ................................................................................................ 37
3.2.4
The Mesh ....................................................................................................................... 38
3.3
Initial Conditions .................................................................................................................... 38
3.4
Calculations and Results ....................................................................................................... 38
Numerical Validation of Elastoplastic Modelling of a Single Axially Loaded Pile .......................... 47 4.1
Introduction ............................................................................................................................ 47
4.2
The Model .............................................................................................................................. 48
4.2.1
Geometry ....................................................................................................................... 48
4.2.2
Loading .......................................................................................................................... 48
4.2.3
Materials and Interfaces ................................................................................................ 49
4.2.4
The Mesh ....................................................................................................................... 52
4.3
4.3.1
Water pressure generation ............................................................................................ 52
4.3.2
Initial stress field generation .......................................................................................... 53
4.3.3
Introduction of the Pile ................................................................................................... 55
4.4
5
Initial Conditions .................................................................................................................... 52
Calculation and Results ......................................................................................................... 56
4.4.1
Sensitivity Analysis of the Interface ............................................................................... 56
4.4.2
Plaxis Results Compared to CESAR-LCPC and GEFdyn Results ............................... 57
4.4.3
Analysis Including Soil Dilatancy ................................................................................... 59
Case Study of a Static Load Test Performed on a Test Pile ......................................................... 61 5.1
Introduction ............................................................................................................................ 61
5.2
Geological Characterization .................................................................................................. 62
5.3
The Static Load Test (SLT) ................................................................................................... 63 i
5.3.1
Preparation .................................................................................................................... 63
5.3.2
Procedure ...................................................................................................................... 64
5.3.3
Instrumentation .............................................................................................................. 67
5.3.4
Results ........................................................................................................................... 68
5.4
6
Numerical Analysis by Plaxis................................................................................................. 72
5.4.1
The Model ...................................................................................................................... 72
5.4.2
Initial Conditions ............................................................................................................ 79
5.4.3
Calculations and Results ............................................................................................... 83
Concluding Remarks ..................................................................................................................... 91 6.1
Conclusions ........................................................................................................................... 91
6.2
Further Research ................................................................................................................... 93
Bibliography ........................................................................................................................................... 95 References ........................................................................................................................................ 95 Consulted Bibliography ...................................................................................................................... 96 Appendixes ............................................................................................................................................ 97 Appendix A – Dimension of the Model Used in the Elastic Simulations ........................................... 97 Appendix B1 – Load Settlement Ratio in Terms of the Pile Slenderness Ratio ................................ 98 Appendix B2 – Proportion of Load Transferred to the Pile Base in Terms of the Pile Slenderness Ratio ................................................................................................................................................ 107 Appendix C – Load Settlement Curves Determined in the Elastoplastic Modelling for Numerical Validation ......................................................................................................................................... 116 Appendix D1 – Load Settlement Curve Determined in the Elastoplastic Modelling for Comparison with the Static Load Test ................................................................................................................. 120 Appendix D2 – Normal Stress along the Pile in the Elastoplastic Modelling for Comparison with the Static Load Test ............................................................................................................................... 121 Appendix D3 – Shaft Load Curves Determined in the Elastoplastic Modelling for Comparison with the Static Load Test ......................................................................................................................... 122 Appendix D4 – Load Settlement Curves Determined in the Elastoplastic Modelling for Comparison with the Static Load Test ................................................................................................................. 123
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LIST OF FIGURES Figure 2.1: (a) Stresses acting in the soil adjacent to the pile; (b) Stresses acting on the pile; (c) Stresses acting on a division of the pile. Adapted from (Poulos & Davis, 1980), p. 75. ......................... 4 Figure 2.2: Pile under axial loading – relevant parameters ..................................................................... 5 Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I 0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis, 1980), p.89. .................................................................................................................. 8 Figure 2.4: Correction factor for the pile compressibility, R k: in terms of the relation between the pile’s and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89.
................................................................................................................................................................. 9 Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, R h, in terms of the relation between the total depth of the soil layer and the length of the pile, h/L, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89. .................................................................................................. 9 Figure 2.6: Correction factor for the Poisson’s ratio of the so il, Rν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos &
Davis, 1980), p.89. ................................................................................................................................ 10 Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis, 1980), p.86. ................................................................................................................................ 10 Figure 2.8: Correction factor for pile compressibility, Ck, in terms of relation between the pile and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.86. ........ 11
Figure 2.9: Correction factor for pile compressibility, C ν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.86.
............................................................................................................................................................... 11 Figure 2.10: Load settlement ratio in terms of the pile slenderness ratio, for different inhomogeneity factors and λ=975, according to the Poulos and Davis method. ........................................................... 14 Figure 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for different inhomogeneity factors and λ=975, according to the Poulos and Davis method. .................... 15 Figure 2.12: (a) Upper and lower soil layers; (b) Separate deformation patters of the upper and lower soil layers. Adapted from (Randolph & Wroth, 1978), p.1469. .............................................................. 16 Figure 2.13: Hypothetical variation of the radius of influence of the pile, r m. Adapted from (Randolph & Wroth, 1978), p. 1471. ........................................................................................................................... 18 Figure 2.14: Load settlement ratio in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, according to the Randolph and Wroth method. ............................................... 25 Figure 2.15: Load settlement ratio in terms of the pile slenderness ratio for compressible piles, for different inhomogeneity factors and λ=975, according to the Randolph and Wroth Method. ............... 26
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Figure 2.16: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles (λ=975), for different inhomogeneity factors, according to the Randolph and Wroth method. ..... 27 Figure 2.17: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles, for different inhomogeneity factors and soil-pile stiffness ratios, according to the Randolph and Wroth method. ....................................................................................................................................... 28 Figure 2.18: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, according to the Randolph and W roth method. .................. 29 Figure 2.19: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for compressible piles, for different inhomogeneity factors and λ=975, according to the Randolph and Wroth method. ....................................................................................................................................... 30 Figure 2.20: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid and compressible piles, for different inhomogeneity factors and λ=975, according to the Randolph and Wroth method. ....................................................................................................................................... 31 Figure 2.21: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors and soil-pile stiffness ratios, according to the Randolph and Wroth method. ....................................................................................................................................... 32 Figure 3.1: Geometry of the model. ....................................................................................................... 35 Figure 3.2: Geometry of the model, L=20m. ......................................................................................... 35 Figure 3.3: (a) Point load at the centre; (b) Point load on the side; (c) Distributed load; (d) Prescribed displacements. ....................................................................................................................................... 36 Figure 3.4: Vertical displacement field, for 1m prescribed displacement at the pile top, L=20m. ......... 38 Figure 3.5: Vertical stress field, for 1m prescribed displacement at the pile top, L=20m...................... 39 Figure 3.6: Vertical stress field near the pile base, for 1m prescribed displacement at the pile top, L=20m. ................................................................................................................................................... 39 Figure 3.7: Normal stress diagram at the pile top, for 1m prescribed displacement at the pile top, L=20m. ................................................................................................................................................... 40 Figure 3.8: Normal stress diagram at the pile base, for 1m prescribed displacement at the pile top, L=20m – both at the pile and at the soil side of the interface. ............................................................... 40 Figure 3.9: Load settlement ratio in terms of the pile sl enderness ratio for homogeneous soils (ρ=1) and λ=975. ............................................................................................................................................. 41 Figure 3.10: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,75 and λ=975. ................................................................................................................................. 42 Figure 3.11: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,5
and λ=975. ............................................................................................................................................. 43 Figure 3.12: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for homogeneous soils (ρ=1) and λ =975. ................................................................................................... 44
Figure 3.13: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,75 and λ =975. .............................................................................................. 45
Figure 3.14: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for inhomogeneity factor ρ=0,5 and λ=975. ................................................................................................ 46
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Figure 4.1: Geometry of the model. ....................................................................................................... 48 Figure 4.2: Distribution of nodes in interface elements and respective connection to 15-node triangular elements. Adapted from Plaxis Manual (Brinkgreve, 2006). ................................................................. 49 Figure 4.3: Mesh. ................................................................................................................................... 52 Figure 4.4: Diagram of pore pressure. .................................................................................................. 53 Figure 4.5: Initial vertical effective stress field, not including the pile. ................................................... 54 Figure 4.6: Initial horizontal effective stress field, not including the pile. .............................................. 55 Figure 4.7: Total load, load transmitted to the pile shaft and load transmitted to the pile base in terms of the settlement at the pile top, for smooth and rough interfaces. ....................................................... 56 Figure 4.8: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top, obtained through Plaxis and GEFdyn. ................................................... 57 Figure 4.9: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top, obtained through Plaxis (for smooth and rough interfaces) and GEFdyn. ............................................................................................................................................................... 58 Figure 4.10: Plastic points due to the Mohr-Coulomb criterion at the pile base for s=8mm. ................ 58 Figure 4.11: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top, obtained through Plaxis (including soil dilatancy at the base) and GEFdyn.................................................................................................................................................. 59 Figure 5.1: Geological profile where the static load test was performed. ............................................. 62 Figure 5.2: Pile and soil layers. Adapted from (Santos, 2005). ............................................................. 63 Figure 5.3: (a) Driving of the temporary casing; (b) Welding of the casing; (c) Introduction of the reinforcing cage. (Viaponte, 2003). ....................................................................................................... 64 Figure 5.4: Reaction system (Viaponte, 2003). ..................................................................................... 64 Figure 5.5: Loading plan. Adapted from (Santos, 2005). ...................................................................... 65 Figure 5.6: Vibrating wire extensom eter welded to the reinforcing cage (Viaponte, 2003). ................. 67 Figure 5.7: Depth of each level of extensometers. Adapted from (Santos, 2005). ............................... 68 Figure 5.8: Load at the pile top, measured by pressure gauges, in terms of time in minutes. Adapted from (Santos, 2005). .............................................................................................................................. 69 Figure 5.9: Measured load at the pile top in terms of the measured settlement of the pile top. Adapted from (Santos, 2005). .............................................................................................................................. 69 Figure 5.10: Normal stress at different levels along the pile (Santos, 200 5). ....................................... 70 Figure 5.11: Normal stress along the pile shaft measured for load steps 4 and 19. Adapted from (Santos, 2005). ...................................................................................................................................... 70 Figure 5.12: Lateral stress between different levels along the pile (Santos, 2005). ............................. 71 Figure 5.13: Model geometry. ............................................................................................................... 72 Figure 5.14: Mesh. ................................................................................................................................. 79 Figure 5.15: Diagram of pore pressure. ................................................................................................ 80 Figure 5.16: Initial vertical effective stress field, including removed soil and not including the pile. ..... 80 Figure 5.17: Initial horizontal effective stress field, including removed soil but not the pile. ................. 81 Figure 5.18: Total displacements after the removal of soil at the top. .................................................. 82
v
Figure 5.19: Load at the pile top in terms of the total settlement at the pile top , given by Plaxis. ........ 83 Figure 5.20: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and Plaxis, for the first loading cycle. ........................................................................................................... 85 Figure 5.21: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and Plaxis, for the second loading cycle. ..................................................................................................... 86 Figure 5.22: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and Plaxis, for both loading cycles. .............................................................................................................. 86 Figure 5.23: Normal stress along the pile shaft at the load peaks (steps 4 and 19), given by the SLT and Plaxis. ............................................................................................................................................. 87 Figure 5.24: Shaft load between different levels along the pile and total applied load, obtained by Plaxis. .................................................................................................................................................... 88 Figure 5.25 Shaft load by layer of soil and total applied load, obtained by Plaxis. ............................... 88 Figure 5.26: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top for the second loading cycle, obtained through Plaxis. ...................... 90
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LIST OF TABLES Table 2.1: Limit pile slenderness ratio between rigid and compressible piles, for different values of the soil-pile stiffness factor, according to (Fleming, 1992). ......................................................................... 28 Table 3.1: Material Properties. .............................................................................................................. 37 Table 3.2: Young’s Modulus of the soil, E. ............................................................................................ 37
Table 4.1: Material properties (Neves, 2001a). ..................................................................................... 49 Table 4.2: Interface properties used in the CÉSAR-LCPC and in the GEFdyn simulations. ................ 50 Table 4.3: Interface properties used in the Plaxis simulation. ............................................................... 51 Table 5.1: Loading plan. Adapted from (Santos, 2005). ....................................................................... 66 Table 5.2: Depth of each level of extensometers. ................................................................................. 67 Table 5.3: Mobilized shaft load for load steps 4 and 19. Adapted from (Santos, 2005). ...................... 71 Table 5.4: Pile properties....................................................................................................................... 73 Table 5.5: Soil properties....................................................................................................................... 73 Table 5.6: Pile properties (2). ................................................................................................................ 74 Table 5.7: Soil properties (2). ................................................................................................................ 77 Table 5.8: Analytical base resistance, R b. ............................................................................................. 77 Table 5.9: Analytical shaft resistance, R s. ............................................................................................. 78 Table 5.10: Shaft load for load step 4. .................................................................................................. 89 Table 5.11: Shaft load for load step 19. ................................................................................................ 89 Table 5.12: Poulos and Davis estimation. ............................................................................................. 84 Table 5.13: Randolph and Wroth (compressible piles) estimation. ....................................................... 84
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viii
S YMBOLS
Latin alphabet
A: area of the cross section of the pile Ab: area of the pile base As: area of the pile shaft CK: correction factor for pile compressibility Cν: correction factor for Poisson’s ratio of soil c: cohesion of the soil cu: undrained strength of the soil
: average undrained resistance along the pile shaft
d: diameter of the pile shaft db: diameter of the pile base E: Young’s modulus of the soil
Einterface: Young’s modulus of the soil-pile interface EL/2: Young’s modulus of the soil at the middle of the pile EL: Young’s modulus of the soil at the pile base Ep: Young’s modulus of the pile Eoed: Young’s modulus of the soil for oedometer loading conditions Eoed, interface : Young’s modulus of the soil-pile interface for oedometer loading conditions G: shear modulus of the soil Ginterface: shear modulus of the soil-pile interface GL/2: shear modulus of the soil at the middle of the pile GL: shear modulus of the soil at the pile base h: total depth of the soil layer, i. e. distance between the soil surface and the rigid layer I: coefficient used in the calculation of the total settlement of the pile I0: settlement-influence factor for a rigid pile in a semi-infi nite incompressible soil (ν=0 .5) Ib: vertical displacement factor for a pile element due to the normal stress at the pile base Is: vertical displacement factor for a pile element due to the shear stress at the pile shaft K: relation between the pile’s and the soil’s Young’s modulus
Ks: earth pressure coefficient L: length of the pile Nc: end-bearing capacity factor Pb: load transferred to the pile base Ps: load transferred to the pile shaft Pt: total (applied) load r: horizontal distance to the pile axis ix
r 0: radius of the pile shaft r m: radius of influence of the pile, i.e. maximum distance past which shear stress is negligible Rb: resistance of the pile base Rc: total resistance of a pile under compression Rh: correction factor for finite depth of layer on a rigid base Rk: correction factor for pile compressibility Rs: resistance of the pile shaft Rν: correction factor for the Poisson’s ratio of the soil (when ν<0 .5) Rinter : interface resistance factor tinterface: thickness of the soil-pile interface u: radial displacement of the soil w: vertical displacement of the soil wb: vertical displacement of the pile base ws: settlement of the pile shaft wt: total vertical displacement (settlement) of the pile x: horizontal coordinate z: vertical coordinate (depth)
Greek alphabet
α: adhesion factor of the pile shaft β0: tip-load proportion for incompressible pile in uniform half- space (ν=0.5) γ: shear strain γ: specific weight δ: friction angle of the soil -pile interface ζ: relation between the radius of influence of the pile and t he radius of the pile shaft η: depth factor of interaction between layers λ: soil-pile stiffness ratio ν: Poisson’s ratio of the soil ρ: inhomogeneity factor of the soil σ: normal stress σ0: normal stress at the pile base σz: vertical stress
̅
: average effective vertical stress along the pile shaft
τ: shear stress τ0: shear stress at the pile shaft φ’: effective friction angle of the soil
x
ACRONYMS CPTU: Piezocone Test FVT: Field Vane Test GEFdyn : Géo Mécanique Eléments Finis Dynamique P&D: Poulos and Davis R&W: Randolph and Wroth SCPTU: Seismic Cone Penetration Test SLT: Static Load Test SPT: Standard Penetration Test
xi
xii
1 INTRODUCTION
1.1 Context and Motivation Piles are deep foundations, necessary when the bearing capacity of shallow foundations is not enough to ensure the support of the superstructure. This superstructure results in vertical forces, due to its weight as well as additional loads, which are axially transferred to the pile and, through its shaft and base, to the soil, possibly reaching a stiffer layer. The analysis of the load transfer mechanism in single piles under axial loading is therefore an essential basis for deep foundation design. It is very important that the physical interaction between pile and soil is carefully studied. The settlement analysis is also fundamental, for the maximum allowable settlement of a foundation is often the most important criterion in its design. Thus, it should be estimated accurately. The behaviour of single piles under axial loading, as far as load distribution and settlement along the pile are concerned, have been analysed through numerous methods. They can be divided into three main categories, according to (Poulos & Davis, 1980): 1) Load-transfer methods, which involve a comparison between the pile resistance and the pile movement in several points along its length; 2) Elastic theory-based methods, which employ the equations described in (Mindlin, 1936) for surface loading within a semi-infinite mass (such as the Poulos and Davis method), or other analytical formulations that impose compatibility between the displacements of the pile and of the adjacent soil for each element of the pile (such as the Randolph and Wroth method); 3) Numerical methods, such as the finite element method. Elastic theory based methods, such as the ones presented in this work, do not explain the behaviour of the pile near failure. In this thesis, their results are used in comparison with the results of a finite element method program, Plaxis 2D version 8. Numerical methods are powerful and very useful tools when used carefully and calibrated with the appropriate tests. They also constitute a valuable way of performing a sensitivity analysis of the soil parameters.
1
1.2 Objectives and Methodology This thesis has three main objectives: 1) To describe two elastic theory-based methods of analysis of single piles under axial loading; 2) To compare solutions given by finite element 2D elastic modelling of piles with the results of the elastic theory-based methods; 3)
To perform finite element 2D elastoplastic modelling of single piles under axial loading, validating its results with former numerical simulations and finally analysing a real case study.
This thesis consists of six chapters, the first being the introduction. Chapter 2 describes two elastic theory-based methods that analyse the behaviour of single piles under axial loading: the Poulos and Davis method and the Randolph and Wroth method. This review focuses in settlement and load transfer mechanisms. Results are organizes in dimensionless charts. Chapter 3 presents the results of 2D axisymmetric elastic modelling of single piles under axial loading, also organized in dimensionless charts. These are compared to the ones obtained from the analytical methods. Chapter 4 compares the results of 2D axisymmetric elastoplastic modelling of a single pile under vertical loading with the ones obtained by other authors under similar conditions. The objective is to validate the used finite element method-based program with other similar ones. Chapter 5 presents a 2D axisymmetric elastoplastic modelling of a single pile in a real case study, simulating a static load test performed on a test pile. It includes a geological description of the site and subsequent choice of the attributed soil parameters that calibrate the numerical model. Chapter 6 list the concluding remarks derived from this thesis, as well as indications on further research.
2
2 ELASTIC THEORY-BASED METHODS ANALYSIS
OF
FOR
SINGLE AXIALLY LOADED
PILES
2.1 Introduction In this thesis, two elastic theory-based methods are analysed: the Poulos and Davis method, introduced by (Poulos & Davis, 1968) and the Randolph and Wroth method, firstly described in (Randolph & Wroth, 1978). Elastic theory-based methods usually consist of dividing the pile into uniformly-loaded elements, as shown in Figure 2.1(a). 2.1(a). Shear stress, τ, acts along the shaft, whereas normal stress , σ, acts on the base of the pile, as represented in Figure in Figure 2.1(b). 2.1(b). These are assumed to be uniform in each division (as shown in in Figure 2.1(c)), 2.1(c)), and the resultant is equal to the total applied load, P t. Equilibrium and compatibility between the displacements of the pile and of the soil adjacent to it are imposed for each element. Usually, there is no imposition of radial compatibility of displacements between pile and soil, since it is assumed that there is no horizontal movement (du/dz=0).
3
(a)
(b)
(c)
Figure 2.1: (a) Stresses acting in the soil adjacent to the pile; (b) Stresses acting on the p ile; (c) Stresses acting on a division of the pile . Adapted from (Poulos & Davis, 1980), p. 75.
The methods are distinct because of the different assumptions made on the distribution of shear stress along the pile. It may be represented as a single point load acting on the axis of each element or as a uniformly-loaded circular area at the centre of each element, for instance. In the Poulos and Davis method, shear stress is considered to be uniformly distributed around the circumference of the pile, which has proved to be an acceptable assumption, especially for shorter piles, according to (Poulos & Davis, 1980). For both elastic theory-based methods, a cylindrical pile is considered, of length L and diameter of the shaft d. Although the possibility of the shaft diameter, d, and the diameter at the pile base, d b, assuming different values is considered in both methods, that case is not analysed in this thesis. Therefore, in the rest of this document, the diameter is considered to be constant along the entire pile. The pile radius is represented by r 0, and the cross section area by A. As a general rule, the index “s” refers to the pile shaft, and the index “b” to the pile base.
The soil is considered to be an ideal isotropic elastic mass, being the Young’s modulus, E, and the Poisson’s ratio, ν, its linear elastic parameters that are not influenced by the presence of the pile. The
total depth of the soil layer, i.e. the distance between the soil surface and the rigid layer, is represented by h. The Young’s modulus of the pile, E p, is assumed to be constant. The Poisson’s ratio of the pile is generally not taken into consideration, as it has negligible effect in the overall behaviour. In fact, even the Poisson’s ratio of the soil, ν, has little ef fect in the end results.
It is assumed that both the pile and the soil are initially stress-free, and there is no residual stress, i.e. effects of installation are not taken into consideration in any way. It may be argued that this is an oversimplification; actually, there are quick ways of simulating the installation, such as employing adjusted shear factor values, as stated in (Poulos & Davis, 1980). However, that is beyond the scope of this thesis. There is an applied axial load Pt in the pile head. The settlement and load results refer to the applied load only: the difference between specific weights, γ, of the pile and of the soil is not considered. Since the conditions of this analysis are purely elastic, the interface between the soil and the pile is considered to be rigid – there is no relative movement between them.
4
The shear modulus of t he soil, G, is used instead of the Young’s modulus, E, in (Randolph & Wroth, 1978) because the soil deforms primarily in shear and also because G is assumed to be unaffected by whether the load is drained or undrained. The shear modulus of the soil may be obtained from the Young’s Modulus, through eq. (2.1), which results from Hooke’s law of isotropic linear elasticity :
(2.1)
The relevant parameters that influence the vertical displacement, w, of a floating pile under axial loading are stated in eq. (2.2):
(2.2)
These parameters are illustrated in Figure in Figure 2.2. It is useful to have dimensionless solutions for the pile behaviour, so as to simplify and quicken their employment. Results from the methods subsequently presented are therefore arranged in dimensionless units, such as the pile slenderness ratio, L/r 0, the proportion of load transferred to the pile base, Pb/Pt, and the load settlement ratio, Pt/(wtr 0GL).
Figure 2.2: Pile under axial loading – relevant parameters
5
2.2 Poulos and Davis Method This method, firstly presented in (Poulos & Davis, 1968), allows a quick estimation of both the proportion of load which reaches the pile base and the total settlement of a pile in the conditions described in the last section. It is necessary to determine the values of the stress acting on the pile (see Figure 2.1) that satisfy the condition of displacement compatibility. As previously stated, only vertical displacement are considered. In order to obtain the values of shear stress, τ, normal stress, σ, and displacement of the
pile top, i.e. total displacement of the pile, wt, expressions that relate vertical displacement with unknown stresses must be determined, imposing the compatibility conditions and solving the resulting equations. The vertical displacement of the soil adjacent to a pile element due to shear stress at the pile shaft is given by eq. (2.3):
(2.3)
Where: d: diameter of the pile shaft Is: vertical displacement factor for the pile element due to the shear stress at the pile shaft τ0: shear stress acting at the pile shaft E: Young’s modulus of the soil
Considering all n pile elements, the resulting vertical displacement of the soil, w t, is provided by eq. (2.4):
∑
(2.4)
Where: Ib: vertical displacement factor for the pile element due to the normal stress at the pile base σ0: normal stress acting at the pile base
This expression is valid for piles with constant diameter. The mentioned factors are determined using the integration of the (Mindlin, 1936) equations for the displacement caused by a point load within a semi-infinite mass. If the presence of a rigid layer at a certain depth is to be accounted for, then the factor Is is to be altered accordingly.
6
For calculating the displacement of the pile elements, only axial compression of the pile is considered. The vertical equilibrium of a cylindrical pile is provided by eq. (2.5):
(2.5)
Where: σ: normal stress acting on the pile (average over the cross section)
r 0: radius of the pile shaft Eq. (2.5) may be applied to the pile top, resulting in eq. (2.6):
(2.6)
Where: Pt: total applied load A: area of the pile cross section Eq. (2.5) may also be applied to the pile base, resulting in eq. (2.7):
(2.7)
The displacement compatibility condition is satisfied by imposing the same displacement for the pile and the soil in each element (rigid interface). The load settlement ratio is expressed in terms of a coefficient, I, as shown in eq. (2.8):
(2.8)
However, it has been mentioned tha t the shear modulus, G, is used instead of the Young’s modulus, E. Thus, eq. (2.8) may be rewritten as eq. (2.9):
(2.9)
7
The difference between the general shear modulus, G, and the shear modulus at the pile base, G L, is clarified as the soil inhomogeneity is taken into consideration, further in this chapter. The load settlement ratio is from now on represented as in eq. (2.9): Pt/(wtr 0GL). This coefficient I is obtained by multiplying other coefficients, as shown in eq. (2.10):
(2.10)
Where: I0: settlement-influence factor for a rigid pile in a semi-infinite incompre ssible soil (ν=0 .5) Rk: correction factor for pile compressibility Rh: correction factor for finite depth of layer on a rigid base Rν: correction factor for the Poisson’s ratio of the soil (when ν<0 .5) The values of I0, Rk, Rh and Rν are plotted in Figure 2.3, Figure 2.4, Figure 2.5 and Figure 2.6, respectively.
=
L Figure 2.3: Settlement-influence factor for a rigid pile in a semi-infinite incompressible soil, I0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis, 1980), p.89.
8
L
Figure 2.4: Correction factor for the pile compressibility, Rk: in terms of the relation between the pile’s and the soil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89.
The correction factor for the pile compressibility, Rk, is function of the relation between the pile’s Young’s modulus, Ep, and the soil’s, E. This relation is represented by K, as shown in eq. (2.11):
(2.11)
The more relatively compressible the pile, the smaller the value of K.
L
Figure 2.5: Correction factor for the finite depth of the layer on a rigid base, Rh, in terms of the relation between the total depth of the soil la yer and the length of the pile, h/L, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.89.
9
ν
Figure 2.6: Correction factor for the Poisson’s ratio of the soil, R ν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.89.
Figure 2.6 confirms the previous statement that the Poisson’s ratio of the soil, ν, has no great influence in the total settlement of the pile, since the correction factor R ν varies between 0.8 and 1.0, for normal cases. The proportion of load transferred to the pile base, Pb/Pt, for a floating pile may be calculated by eq. (2.12), first presented in (Poulos, 1972):
(2.12)
Where: β0: tip-load proportion for incompressible pile in uniform half-space ( ν=0.5)
CK: correction factor for the pile compressibility Cν: correction factor for the Poisson’s ratio of the the soil The values of β0, CK and Cν are plotted in Figure 2.7, Figure 2.8 and Figure 2.9, respectively.
L
Figure 2.7: Tip-load proportion for incompressible pile in uniform half-space, β0, in terms of the pile slenderness ratio, L/d, and of the relation between the base and the shaft diameters, db/d. (Poulos & Davis, 1980), p.86.
10
L
Figure 2.8: Correction factor for pile compressibility, Ck, in terms of relation between the pile and the s oil’s Young’s modulus, K, and of the pile slenderness ratio, L/d. (Poulos & Davis, 1980), p.86.
The pile’s compressibility has the effect of decreasing the load transferred to the tip. On the other hand, the load transferred to the tip tends to increase with the relative stiffness of this stratum, and this is more pronounced for slender piles.
ν
Figure 2.9: Correction factor for pile compressibility, Cν, in terms of the soil’s Poisson’s coefficient, ν, and of the relation between the pile’s and the soil’s Young’s modulus, K. (Poulos & Davis, 1980), p.86.
The distance to a rigid layer, h, is not present nor has any influence on any term of eq. (2.12). In fact, the proportion of load which reaches the pile base is not greatly affected by it, when its value is higher than 2L, according to (Poulos & Davis, 1980). This must be taken into consideration when comparing results provided by different methods – if the depth h is not to be accounted for, then the limit of 2L must be respected. There are obviously other factors that may have influence on the proportion of load that reaches the pile tip, such as the presence of a pile cap resting on the soil surface, or of enlarged bulbs along the pile, but they are beyond the scope of this study.
11
A layered or a vertically non-homogeneous soil may also be analysed by using the equations in (Mindlin, 1936) for a uniform mass, if approximate values of Young’s modulus and Poisson’s ratio at various points along the pile are employed. Thus, the stress distribution is assumed to be unaltered, as if the soil was homogeneous, but the soil displacement at a point adjacent to the pile is function of the soil’s Young’s modulus at that point. The
result of the soil-displacement equation changes (see eqs. (2.3) and (2.4)), but not the piledisplacement’s one. An average Young’s modulus may be calculated using eq. (2.13):
∑
(2.13)
Where: Ei: Young’s modulus of layer i hi: thickness of layer i n: number of layers/divisions of the soil This may be used when the soil is divided into different layers but the Young’s modulus does not vary much. In those cases, the solution may be calculated with this new value of the Young’s modulus and is very close to the one provided by the finite element method (errors inferior to 15%), according to (Poulos & Davis, 1980). This approach is an approximation, but its solution is considered to be accurate enough for practical purposes. It must not be forgotten, however, that this is an approximation and that it does not provide an accurate solution of the load or settlement distribution along the pile. Only the total values are considered relevant. The variations in the Poisson’s ratio along the depth may be ignored, since, as discussed before, this parameter has little influence in the total settlement of the pile. A relevant form of soil non-homogeneity is one in which the shear modulus varies linearly with depth. A measure of this variation is the inhomogeneity factor, ρ; it is calculated through eq. (2.14):
(2.14)
In the extreme case of ρ=0 .5, the shear modulus at the surface must be null – this is called a “Gibson soil”.
The factor of inhomogeneity enables a comparison between different types of soil, with more or less vertical inhomogeneity. Since eq. (2.13) is also applicable in this case, it is us ed in the calculation of ρ , through eq. (2.14), eq. (2.14), using using the relation described in eq. (2.1).
12
In Figure 2.10, results given by eq. (2.9) are plotted, for different values of the soil inhomogeneity factor , ρ. The value of Poisson’s coefficient of the soil, ν, is 0.3. The radius of the pile, r 0, is equal to the unity in every case, for simplification reasons. The length of the pile, L, varies between 4m and 100m. The rigid layer is assumed to be at a distance of 2.5L of the surface (h). The Young’s modulus at the pile base, EL, is equal to 80×103 kPa in every case – its value varying in the rest of the soil according to ρ.
This has very little influence in the overall results, since the charts are normalized for the soil rigidity; thus, it only affects the value of K. K assumes the values of 375, 500 and 750 for ρ=1, ρ=0 .75 and ρ=0.5, respectively. The reason why it is not given a constant value is that it is not possible, if the said
values of soil inhomogeneity are to be tested and simultaneously the shear modulus at the pile base, GL, is to be the same in all cases, for K is a relation between the pile’s Young’s modulus and the average shear modulus along the shaft. Nevertheless, K has not great influence either over the load settlement ratio or over the proportion of load that is transferred to the pile base for normal values of Ep and GL. Besides, in (Randolph & Wroth, 1978) a similar relation is presented, the soil-pile stiffness ratio, λ, calculated through eq. (2.15):
(2.15)
Since both the Young’s modulus of the pile, E p, and the shear modulus of the soil at the pile base, GL,
are the same for every case, the soil-pile stiffness ratio is constant and λ=975. Below the pile, the Young’s modulus is constant and has the same value as at the pile base. The Young’s modulus of the soil, E, used in its calculation is the one at the middle of the pile. The Young’s
modulus of the pile is 30×106 kPa. The chart in Figure 2.10 was built according to non-linear functions created from the few exact points given by Figure 2.3 to Figure 2.6, since the intermediate values cannot be interpolated linearly. Therefore, the load settlement ratio was calculated for each natural number of pile slenderness between 4 and 100. All those values are grouped in tables in Appendix B1.
13
140
120
100 ) L G 80 × 0 r × t w ( 60 / t P
ρ=1 ρ=0,75 ρ=0,5
40
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.10: Load settlement ratio in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Poulos and Davis method.
The chart shows that the load settlement ratio increases with the pile slenderness ratio. This is expected, since, when subject to the same conditions, longer piles settle less. The shapes of the three curves are very similar. In fact, the inhomogeneity factor, ρ, influences the pile compressibility, K, and the expression of the load settlement ratio, eq. (2.9), only; the distance between the curves increases slightly with the slenderness ratio, L/r 0. The pile settlement ratio increases with the inhomogeneity factor, ρ. Since the same shear modulus at the pile base is considered for the three cases, the one with the smallest ρ is the one in which the
shear modulus at the middle of the pile (G, in eq. (2.9)) is the smallest, i .e. the average Young’s modulus Eav is the lowest. Thus, according to eq. (2.9) and Figure 2.4 and Figure 2.6, the load settlement ratio will also be the lowest. It is expected that, the lower the ρ, the worst the results, since eq. (2.13) obviously provides a very gross approximation, the grosser the less homogeneous the soil. Since there were few exact points to be extracted from the original charts, the error associated with these charts is considerable. The non-linearity of these functions is the cause of irregularity of the resulting curves. In Figure 2.11, results given by eq. (2.12) are plotted, for different values of the soil inhomogeneity factor , ρ. The conditions of the pile and the soil are identical to the ones described for Figure 2.10. Once again, it is built from non-linear functions created from the few exact points given by Figure 2.7 to Figure 2.9. Therefore, the proportion of load transferred to the pile tip was calculated for each natural number of pile slenderness between 4 and 100. All those values are grouped in tables in Appendix B2.
14
0,40 0,35 0,30 0,25 t
ρ=1
P / 0,20 b P
ρ=0,75
0,15
ρ=0,5
0,10 0,05 0,00 0
10
20
30
40
50 L/r0
60
70
80
90
100
Figure 2.11: Proportion of the load taken by the pile base in terms of the pile slenderness ratio, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Poulos and Davis method.
This chart shows that the proportion of load transferred the pile base decreases non-linearly with the slenderness ratio: its value reduces quickly for low values of L/r 0, but tends to stabilize. In longer piles, the shaft plays a more important role in the load transfer mechanism, given its dimension. Thus, less load reaches the pile base. In fact, some studies consider it to be null, for values of the pile slenderness ratio higher than a certain limit. Besides, the values of the load settlement ratio increase inversely with ρ , although very slightly. In fact, the difference in the shear modulus of the soil distribution only affect C k (Figure 2.8) and Cν (Figure 2.9), and in this last case the change is negligible. However, it is natural that, in piles with a higher value of the relation K, more load is transferred to the tip. Once again, it should not be forgotten that the information used to build this chart has come from Figure 2.7 to Figure 2.9, and so there is a significant error associated with it. Although some of these parameters are taken as independent from each other (as the measure of pile compressibility, K, and the total depth of the soil layer, h, used in the calculation of the load settlement ratio), and other factors are not considered, this method is very convenient and adequate for practical purposes. The Poulos and Davis method has proved to provide relatively good solutions, considering its simplicity. Its results for the load settlement ratio are usually slightly higher than the ones given by numerical methods, i.e. settlement values are lower, according to (Poulos & Davis, 1980). In Chapter 3, the pertinence of this statement is tested.
15
2.3 Randolph and Wroth Method This method, firstly introduced in (Randolph & Wroth, 1978), has been developed in order to explain the axial load transfer process between pile and soil. It is particularly useful in cases where the soil is non-homogenous, since the previously developed methods, the Poulos and Davis method amongst them, had great limitations in that aspect. Initially, the shaft and base behaviours are studied separately. An imaginary horizontal plane AB at the depth of the pile base separates base and shaft, as represented in Figure 2.12(a). Thus, it is considered that above that plane the soil deforms due to the pile shaft only, and that below the plane the soil deforms due to the pile base only, as shown in Figure 2.12(b). The deformation above and below the plane is not compatible and that allows for interaction between the upper and lower layers of soil. This is a simplification which will obviously not provide the exact solution, but that has proved to be satisfactory.
(a)
(b)
Figure 2.12: (a) Upper and lower soil layers; (b) Separate deformation patters of the upper and lower soil layers. Adapted from (Randolph & Wroth, 1978), p.1469.
The soil is considered to be linear elastic. Thus, the effects of installation (residual stresses) are ignored. As explained before, it is also assumed that the parameters of the soil are not affected by the installation of the pile. The deformation of the soil surrounding the pile is similar to shearing of concentric cylinders. The vertical equilibrium of an element of soil is given by eq. (2.16):
(2.16)
Where: r: horizontal distance to the pile axis τ: shear stress
x: horizontal coordinate σz: vertical stress
z: depth
16
Since the shear stress caused by pile loading is much greater than the vertical stress, the second summand is insignificant. Thus, the integration of the previous equation gives the shear stress in the soil surrounding the pile, according to eq. (2.17):
(2.17)
Where: r 0: radius of the pile shaft τ0: shear stress at the pile shaft
The shear strain of the soil is calculated through eq. (2.18):
(2.18)
As mentioned before, the displacement is considered to be mainly vertical (the radial component is negligible), so eqs. (2.16) to (2.18) may be rewritten as eq. (2.19):
∫ ∫
(2.19)
When including r m in the equation, an upper boundary of the radius of influence of the place, i.e. the distance past which shear stress becomes negligible, eq. (2.19) may be rewritten as eq. (2.20):
(2.20)
The factor ζ is as a relation between the radius of influence of the pile and the radius of the pile shaft,
as shown in eq. (2.21):
(2.21)
The pile acts on the layer below as a rigid punch; its effect is more significant, i.e. the lower layer suffers more deformation, nearer the pile. The lower layer restrains the deformation of the upper layer. The result is the generation of positive vertical stresses (σ z>0, compression), which indicates that, by
the equation of soil equilibrium, shear stresses will decrease more rapidly than linearly, contrary to what eq. (2.17) implies.
17
The consequence is that the magnitude of ∂/∂r(rτ) (the variation of shear stress with distance do the pile) decreases with depth along the pile and, consequently, so does the value of r m, as represented in Figure 2.13. .
Figure 2.13: Hypothetical variation of the radius of influence of th e pile, r m. Adapted from (Randolph & Wroth, 1978), p. 1471.
Rigid Piles in Homogeneous Soil
For rigid piles, the shaft settlement, ws, is independent of the depth, since there is no shortening (the pile is considered as non-compressible). Thus, the shear stress at the pile shaft, τ 0, must also vary, increasing with depth, so that the shaft displacement given by eq. (2.20) is constant. Thus, the higher the shear stress, the smaller the distance at which its value is significant. The variation of r m with depth is generally disregarded, and its value is taken, in the case of homogeneous soil, as an average, given by eq. (2.22):
(2.22)
Generally, ζ varies between 3 and 5.
The shear stress at the pile shaft may be considered as described in eq. (2.23):
(2.23)
Assuming both this and the radius of influence of the pile, r m, constant with depth, the settlement relative to the pile shaft is given by eq. (2.24):
(2.24)
18
Since the behaviour of the pile base resembles a rigid punch, the resulting displacement is obtained by eq. (2.25):
(2.25)
η is the factor of interaction between the upper and lower layers of soil.
The factor η refers to the stiffening effect of the soil above the loaded area. The value to attribute to this factor is subject to discussion, but it is generally accepted that to adopt the unity is adequate, according to (Randolph & Wroth, 1978). In (Fleming, 1992), this factor is considered as the relation between the base and shaft diameters, as shown in eq. (2.26):
(2.26)
This factor is included in the following equations, but is considered equal to the unity whenever calculations are performed. In a rigid pile, there is by definition no relative displacement within, and so eq. (2.27) is applicable:
(2.27)
Therefore, every point in the pile has the same settlement. Besides, the total load is given by eq. (2.28):
(2.28)
For rigid piles in a homogeneous soil, the load settlement ratio may be calculated through eq. (2.29):
(2.29)
Also, the fraction of the load that is taken by the base may be calculated by eq. (2.30):
(2.30)
19
Compressible Piles in Homogeneous Soil
Not always may the pile be considered as rigid. For piles with certain properties, relative displacement within the pile must be taken into consideration. Thus, for compressible piles, the shaft settlement varies with depth, since shear stress can no longer be assumed as constant along the pile, as in eq. (2.31):
(2.31)
The pile is considered as elastic, and its compressive strain in depth may be expressed by terms of the transmitted load, as in eq. (2.32):
(2.32)
The soil-pile stiffness ratio, λ, is calculated by eq. (2.15). It is relevant to point out that the Young ’s modulus of the pile, Ep, is taken into account in the calculations relative to compressible piles only, since for rigid piles it is considered to be infinite. The transmitted load is related to the variable shear stress on the shaft surface, as in eq. (2.33):
(2.33)
Differentiating and combining the last equations will result in the governing differential equation, as in eq. (2.34):
(2.34)
(2.35)
And its solution is described in eq. (2.35):
Taking the following measure of pile compressibility, described in eq. (2.36):
(2.36)
20
The constants A and B are found by using the boundary conditions at the base of the pile, and equation (2.35) may be rewritten as eq. (2.37):
[ ] [ ]
(2.37)
The term (πr 0λμ)-1 is very small (inferior to 0.02 in normal cases) and, through eq. (2.25), eq. (2.37) may be simplified to eq. (2.38):
() ()
(2.38)
It is possible to particularize eq. (2.38), expressing the total settlement of the pile in terms of the settlement of the base, as in eq. (2.39):
()
(2.39)
The mentioned simplification may also be used, as in eq. (2.40):
(2.40)
The integration of eq. (2.33) and substitution of the latest equations provides the expression of the total load supported by the pile in terms of depth, as in eq. (2.41):
() ()
(2.41)
Thus, the expression of the load settlement ratio is provided by eq. (2.42):
(2.42)
This allows for a comparison between the pile compressibility and the load deformation behaviour. Also, the fraction of the load that is taken by the base may be calculated by eq. (2.43):
(2.43)
21
It is also useful to adapt the former equations to the case of a non-homogenous soil. Only vertical nonhomogeneity is considered – the concept of radial inhomogeneity not being developed, given its lack of relevance for the present study. As in the chapter referent to the Poulos and Davis method, two types of non-homogeneous soils may be considered: a layered soil (in which every layer has a constant shear modulus) and a soil in which the shear modulus varies linearly. For the case of the layered soil, the shear strain γ dist ribution is unaltered (see eq. (2.18)), the shear stress τ0 distribution being obtained by multiplying by an appropriate shear modulus.
Once again, the second type of soil inhomogeneity, in which the shear modulus varies linearly, is more carefully analysed. Rigid Piles in Non-Homogeneous Soil
The behaviour of a rigid pile in this type of soil is considered. The shear stress, which would be assumed constant in a homogenous soil, increases approximately linearly with depth in this case. The distance at which the shear stresses become negligible, r m, will also decrease. For a pile in an infinite half space, r m may be calculated by eq. (2.44):
(2.44)
And, for a pile in a space where there is a rigid layer at the depth of 2.5L, which is more commonly used since it allows a comparison with finite element method program s’ results, r m may be calculated by eq. (2.45):
(2.45)
The inhomogeneity factor, ρ, is given by eq. (2.14).
Below the pile base, the shear modulus is considered to be constant. However, analyses using the finite element method have proved that the difference between this and the case where the shear modulus continues to increase is negligible (inferior to 5% in the total settlement value). The value of shear stress may be written as in eq. (2.46):
(2.46)
And thus, the shaft settlement, ws, may be calculated through eq. (2.47):
(2.47)
22
The total load taken by the pile shaft may be written as in eq. (2.48):
∫
(2.48)
Eq. (2.48) may be rewritten as eq. (2.49):
(2.49)
And the load settlement ratio may be written as in eq. (2.50):
(2.50)
The fraction of the load that is taken by the base may be calculated by eq. (2.51):
(2.51)
Compressible Piles in Non-Homogeneous Soil
For compressible piles, eq. (2.42), used for homogeneous soils, may be used for non-homogeneous soils when modified by the introduction of the inhomogeneity factor, as in eq. (2.52):
(2.52)
Also, the fraction of the load that is taken by the base may be calculated by eq. (2.53):
(2.53)
23
A relevant statement is that a pile with L/r 0=100 could hardly be considered “rigid” – this would be the case of a pile with L=100m and d=2m, or L=8m and d=16cm. The authors of this method have not provided any means of distinction between rigid and compressible piles. According to (Fleming, 1992), eq. (2.54) can be used as a general rule to determine if a pile is may be considered as rigid:
√
(2.54)
In this case, for the conditions described, eq. (2.54) is calculated as shown in eq. (2.55):
(2.55)
In the charts that display results from the Randolph and Wroth method presented from now on, a black line represents this limit, and the charts are divided in “Rigid” and “Compressible” areas.
In Figure 2.14, results given by eq. (2.50) are plotted, for different values of the soil inhomogeneity factor , ρ. The factor of interaction between layers, η, is given the value of the unit y, and so is the pile radius, r 0, in every case. The value of Poisson’s coefficient, ν , is 0.3. The relation between the radius of influence of the pile and the radius of the pile shaft, ζ, is given by equation (2.21), using values of the radius of influence of the pile, r m, given by equations (2.22) and (2.45), for homogeneous (ρ=1) and inhomogeneous (ρ≠1) soils respectively; thus, the rigid layer is assumed to be at the distance of
2.5L from the surface. The Young’s modulus at the pile base, E L, is equal to 80×103 kPa in every case – its value varying in the rest of the soil according to ρ. Below the pile, the Young’s modulus is
constant and has the same value as at the pile base. These conditions are the closest possible to the ones used to analyse the Poulos and Davis method (see Figure 2.10).
24
140
120
100 ) L G × 80 0 r × t w ( / t P 60
ρ=1 ρ=0,75 ρ=0,5
40
20
0 0
10
20
30
40
50 L/r0
60
70
80
90
100
Figure 2.14: Load settlement ratio in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
This chart shows that, for rigid piles, the load settlement ratio increases approximately linearly with the pile slenderness ratio. The equations that originate these curves show very clearly that the contribution of the pile base is constant, and that the shaft contribution varies linearly with the pile slenderness. Besides, the higher the inhomogeneity factor, ρ, the higher the load settlement ratio. As expected, as ρ decreases, so does the slope of the line that represents the function.
The values that originate these curves are grouped in tables in Appendix B1.
25
In Figure 2.15, results given by eqs. (2.42) and (2.52) are plotted, for different values of the soil inhomogeneity factor. The Young’s modulus of the pile is 30×10 6 kPa. The conditions are similar to the ones described for Figure 2.14.
140
120
100 ) L G 80 × 0 r × t w ( / t 60 P
ρ=1 ρ=0,75 ρ=0,5
40
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.15: Load settlement ratio in terms of the pil e slenderness ratio for compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth Method.
This chart shows that, for compressible piles, the load settlement ratio increases non-linearly with the pile slenderness ratio. This suggests that the total settlement does not reduce infinitely as the pile slenderness increases, but instead that it stabilizes, indicating the existence of an asymptote. This did not happen for rigid piles, as in that case the contribution of the shaft for the load settlement ratio increased infinitely with the pile slenderness. For compressible piles, the contribution of the base remains the same. However, with the introduction of the measure of the pile compressibility, µL, the proportion of the load taken by the shaft is reduced. Thus, the load settlement ratio for a given value is lower if the pile is considered compressible than if it were considered rigid, and the difference increases with the pile slenderness (see fig. Figure 2.16). This is an overall more realistic approach for high pile slenderness values. Once again, the load settlement ratio also becomes higher as the inhomogeneity factor ρ approaches the unity. The values that originate these curves are grouped in tables in Appendix B1.
26
Figure 2.16 allows a clear comparison between curves relative to rigid and to compressible piles, as well as the mentioned division. 140
Rigid
Compressible
120
ρ=1 compressible
100
ρ=0,75 compressible
) L G 80 × 0 r × t w ( 60 / t P
ρ=0,5 compressible ρ=1 rigid ρ=0,75 rigid
40
ρ=0,5 rigid
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.16: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles ( λ=975), for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
It is now evident how different is the behaviour of each type of pile, rigid or compressible. The domain of rigid piles is clearly very limited, accordi ng to the division criterion used. This analysis would not be complete, however, if a soil-pile stiffness ratio sensitivity analysis was not performed, since the presented charts are not normalized by it. Figure 2.17 shows the load settlement ratio in terms of the pile slenderness ratio for three different values of soil inhomogeneity factor (ρ=1, ρ=0.75 and ρ=0.5) and for three different values of the soil- pile stiffness ratio (λ=3 ,000, λ=975 and λ=300). This sensitivity analysis is only applied to compressible piles, since the soil-pile stiffness factor
is assumed to be infinite in rigid piles.
27
140
120 ρ=1; λ=3000
100
ρ=0,75; λ=3000 ρ=0,5; λ=3000
) L G 80 × 0 r × t w ( / t 60 P
ρ=1; λ=975 ρ=0,75; λ=975 ρ=0,5; λ=975 ρ=1; λ=300 ρ=0,75; λ=300
40
ρ=0,5; λ=300
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.17: Load settlement ratio in terms of the pile slenderness ratio for rigid and compressible piles, for different inhomogeneity factors and soil-pile stiffness ratios, ν=0.3 and h=2.5L, according to the Randolph and Wroth method.
This chart shows that the load settlement ratio is quite sensitive to the soil-pile stiffness factor, for any given value of the soil inhomogeneity factor. However, it should be borne in mind that, for a constant Young’s modulus of the pile of 30GPa, a soil -pile stiffness factor of 300 requires the shear modulus of
the soil at the pile base to be equal to 100MPa, whereas a soil-pile stiffness factor of 3,000 requires the shear modulus of the soil at the pile base to be equal to 10MPa, which is quite a wide range. The curves relative to λ=3,000 are obviously closer to the rigid pile solution (see Figure 2.14). Application of eq. (2.54) provides the boundary values between rigid and compressible piles, for the given values of soil-pile stiffness factor, shown in Table 2.1. Table 2.1: Limit pile slenderness ratio between rigid and compressible piles, for different values of the soil-pile stiffness factor, according to (Fleming, 1992). λ
L/r 0
300
8.66
975
15.61
3,000
27.39
This table shows that the boundary between rigid and compressible piles is also quite sensitive to the soil-pile stiffness factor (as is, in fact, clear in eq. (2.54)). This gives an idea of the utility of the rigid solution (up to which value of the pile slenderness ratio the rigid solution is applicable). 28
In Figure 2.18, results given by eqs. (2.30) and (2.51) are plotted, for different values of the soil inhomogeneity factor. Once again, the conditions are the same as the ones described for Figure 2.14.
0,40 0,35 0,30 0,25 t
ρ=1
P / b 0,20 P
ρ=0,75 ρ=0,5
0,15 0,10 0,05 0,00 0
10
20
30
40
50 L/r0
60
70
80
90
100
Figure 2.18: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for rigid piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
This chart shows that, for rigid piles, the proportion of load that reaches the pile base decreases nonlinearly with the slenderness ratio: its value reduces quickly for low values of L/r 0, but tends to stabilize. Besides, the values increase inversely with ρ ; this is expectable, since the contribution of the base to the load settlement ratio is not affected by ρ and the contribution of the shaft is proportional to ρ (see
eq. (2.50)), as it increases with the shear modulus along the pile (see eq. (2.24)). The values that originate these curves are grouped in tables in Appendix B2.
29
In Figure 2.19, results given by equations (2.43) and (2.53) are plotted, for different values of the soil inhomogeneity factor.
0,40 0,35 0,30 0,25 ρ=1
t
P / b 0,20 P
ρ=0,75 ρ=0,5
0,15 0,10 0,05 0,00 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.19: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
This chart shows that, in terms of the proportion of load taken by the pile base, piles regarded as compressible behave very similarly to the ones considered to be rigid. The shapes of the curves are almost identical, the only difference being that values relative to the compressible piles are slightly lower. This is due to the fact that, in this case, the calculation of the contribution of the shaft includes the variation of shear stress with depth, and its results are higher. Besides, the longer the pile, the more load is transferred to the shaft, resulting in less load reaching the pile base. It is interesting that the difference between curves ρ=1 and ρ=0 .75 is lower than the difference between curves ρ=0.75 and ρ=0.5.
The values that originate these curves are grouped in tables in Appendix B2.
30
Figure 2.20 allows a clear comparison between curves relative to rigid and to compressible piles, as well as the mentioned division.
0,40
Rigid
Compressible
0,35
0,30 ρ=1 compressible
0,25
ρ=0,75 compressible
t
ρ=0,5 compressible
P / b0,20 P
ρ=1 rigid ρ=0,75 rigid
0,15
ρ=1 rigid
0,10
0,05
0,00 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 2.20: Proportion of the load taken by the pile base in terms of the pile slenderness ra tio for rigid and compressible piles, for different inhomogeneity factors, ν=0.3, h=2.5L and λ=975, according to the Randolph and Wroth method.
Once again, a soil-pile stiffness ratio sensitivity analysis is performed. Figure 2.21 shows the proportion of load transferred to the pile base in terms of the pile slenderness ratio for three different values of soil inhomogeneity factor (ρ=1, ρ=0 .75 and ρ=0.5) and for three different values of the soilpile stiffness ratio (λ=3 ,000, λ=975 and λ=300).
31
0,40
0,35
0,30
0,25
ρ=1; λ=975 ρ=0,75; λ=975
t
P 0,20 / b P
ρ=0,5; λ=975 ρ=1; λ=3000 ρ=0,75; λ=3000
0,15
ρ=0,5; λ=3000 ρ=1; λ=300
0,10
ρ=0,75; λ=300 ρ=0,5; λ=3000
0,05
0,00 0
10
20
30
40
50 L/r0
60
70
80
90
100
Figure 2.21: Proportion of the load taken by the pile bas e in terms of the pile slenderness ratio for rigid pi les, for different inhomogeneity factors and soil-pile stiffness ratios, ν=0.3 and h=2.5L, according to the Randolph and Wroth method.
When compared to the case of load settlement ratio, the proportion of load transferred to the pile base is less sensitive to the soil-pile stiffness ratio, as it is to the soil inhomogeneity factor. Besides, the difference between rigid and compressible solutions of the proportion of load transferred to the pile base is also not as relevant as the one of the load settlement ratio. Thus, it is possible to conclude that the soil-pile stiffness ratio has more influence over the load settlement ratio than over the proportion of load transferred to the pile base. The authors of this method state that comparisons made between the results given by this method and by the finite element method show great similarity. They also declare, however, that this method proves to be unstable for very long compressible piles and is considered unsuitable when (L/r0) 2/λ>20; it provides unreliable results for values of the pile slenderness ratio higher than 80. In Chapter 3, tests will be made to confirm these statements. The list of this method’s advantages includes its simplicity, which allows a good estimation of the total
settlement of a pile with no need of a computer. Also, the non-homogeneity of the soil are more thoroughly taken into account than in the previous elastic theory-based methods. In the next chapter, charts provided by the Poulos and Davis method and by the Randolph and Wroth method are compared to each other and to others, given by the finite element method-based program Plaxis. 32
3 COMPARISON BETWEEN ELASTIC THEORYBASED
METHODS
ELEMENT METHOD
AND IN THE
THE
FINITE
ANALYSIS
OF
SINGLE AXIALLY LOADED PILES
3.1 Introduction Nowadays, the use of finite element method-based programs in structural and geotechnical analyses is generalised. These are particularly useful in study of inelastic behaviour, when other methods are no longer applicable. In this chapter, the finite element method program Plaxis 2D version 8 is used to model single piles ’ behaviour under axial loading. Its results are compared to the results given by the two described elastic theory-based methods, the Poulos and Davis method and the Randolph and Wroth method. The purpose is to obtain curves of the load settlement ratio and of the proportion of load taken by the pile base in terms of the pile slenderness, traced from the results provided by the finite element program.
33
3.2 The Model The analysed piles are cylindrical and isolated and the loading is performed axially. These conditions allow for the use of symmetry; thus, the single piles behaviour is modelled in two dimensions, using axisymmetry. 15-node triangular elements are used to create the mesh.
3.2.1 Geometry The model consists of two materials: the pile and the soil. Since axisymmetry is used, only half of the problem is represented, as shown in Figure 3.2. The length of the pile, L, defines the rest of the geometry; its value varies between 4m 100m. The pile radius, r 0, is given the value of the unity. The distance between the surface and the rigid layer, h, is 2.5L. This limit will be used in every analysed case, and has special importance because the proximity of the rigid layer has great influence in the pile settlement results. In order to compare Plaxis’ results with Poulos and Davis’, it is required that this distance must be greater than 2L, so it does not influence the proportion of load that reaches the pile base. However, the Randolph and Wroth’s expression of the radius of influence of the pile, r m, which is empirical and subject to discussion, given by equations (2.22) and (2.45), has proved to be adequate for that particular distance, and there is no proof of its validity for others. In Randolph & Wroth (1978), it is recommended that the distance between the symmetry axis and the vertical outer boundary, l, should be equal or higher than 2L, since in pile settlement analyses performed with closer limits, results have been affected by it. The choice of these properties is determined by the two mentioned elastic theory-based methods: since the purpose of this simulation is to enable a comparison with results provided by them, the conditions must be as similar as possible. Appendix A shows the dimensions of the model used for each pile slenderness value, L/r 0, which acquires the same value as the pile length, L. Figure 3.1 shows the general geometry of the model and Figure 3.2 shows the geometry of the model, in the case when L=20m.
34
Figure 3.1: Geometry of the model.
Free boundary
Horizontal
Horizontal
displacements
displacements
not allowed
not allowed
Vertical and horizontal displacements not allowed Figure 3.2: Geometry of the model, L=20m.
35
3.2.2 Loading Plaxis provides several ways of introducing data into the model. It is important, however, to determine which is the best way to do so, i.e. which way provides results closer to reality. As far as loading is concerned, there are three possible means of introducing it: using a point load, a distributed load or a prescribed displacement. The analysis of the diagram of normal stresses in the top of the pile provides a good evaluation of the quality of the results given by each one of these cases. The diagram is expected to be uniform with an increase near the soil-pile interface, resulting from the interaction between the two materials. Figure 3.3 shows the qualitative diagram of normal stresses for each of the mentioned cases, for a generic geometry of the pile and load.
Figure 3.3: (a) Point load at the centre; (b) Point load on the side; (c) Distributed load; (d) Prescribed displacements.
The observation of the diagrams which result from point loads enables an immediate discard of the possibility of using this type of loading. Although the diagram referent to distributed load is considerably better, it is clear that results closer to reality are provided by prescribed displacements. Therefore, this is the method used in the model: a 1m displacement is applied at the pile top, and the resulting stress is used in the pile-settlement ratio and the proportion of load transferred to the pile base charts. This may seem an unrealistic approach, but since Plaxis’ results showed no sign of numerical error for w=1m and all the materials are linear elastic, any value of the displacement may be used as an input. For the boundaries, standard fixities are used: the bottom boundary’s vertical and horizontal displacements are null (representing the rigid layer), the left and right boundaries’ horizontal
displacements are null and the upper boundary is free, as represented in Figure 3.2. 36
3.2.3 Materials and Interfaces Two different materials are used in this analysis: reinforced concrete (referred to as “pile”) and soil. The simulation is performed under drained conditions. This is an elastic analysis, and therefore it is considered that the materials do not yield. Since the difference between the specific weight, γ, of each material is not accounted for, and that the initial stress state is not taken into consideration – being all the produced stress a consequence of loading only – in the elastic theory-based methods, the specific weight introduced in the program is null for both materials. The soil-pile interface is rigid (parameter Rinter =1), i.e. there is no slip or gap (relative displacement) between the two materials, when the load is applied. The value of the P oisson’s ratio, ν, is commonly used. The value of the Young’s modulus of the pile, Ep, is widely used for reinforced concrete, and the one of the soil, E, is variable. Table 3.1 summarizes the described material properties, and Table 3.2 shows the values of the soil’s Young’s modulus, E, used for each case (varying according to the inhomogeneity factor, ρ) . Since the charts are normalized for the soil rigidity, the results would be the same for other Young’s modulus values, if
the soil homogeneity factor remained the same.
Table 3.1: Material Properties.
Material Model
Material Type
ν
E [kN/m ]
Rinter
Pile
Linear Elastic
Drained
0.15
30×106
1
Soil
Linear Elastic
Drained
0.30
Variable
1
Table 3.2: Young’s Modulus of the soil, E. ρ=1
ρ=0.75
ρ=0.5
Ei [kN/m ]
80×10
40×10
0
EL/2 [kN/m ]
80×10
60×10
40×10
EL [kN/m ]
80×10
80×10
80×10
The soil-pile stiffness ratio, λ, calculated by eq. (2.15), will consequently have the value of 975 in every case.
37
3.2.4 The Mesh Since geometry changes in every case, it is not possible to define a fixed mesh. The dimension of the problem varies significantly with the simulation (see Appendix A) and the number of elements alters accordingly. However, care is taken to ensure that the element size is adequate, the mesh being refined where the highest gradients of stress occur. In most cases, it is proved to be an efficient procedure to use transition areas: clusters which have the sole purpose of easing the change between fine and coarse regions of the mesh. All elements are triangular with 15 nodes.
3.3 Initial Conditions No water table is taken into account. Besides, as previously mentioned, the specific weights of the soil and of the pile are considered to be equal to zero. Therefore, both the pore pressure and the stress fields are null.
3.4 Calculations and Results As previously mentioned, the initial stress field is null. Therefore, the calculation can be performed in one phase only, which corresponds to the loading. In order to compare results provided by Plaxis with the ones presented in former chapters, total displacement and load at the pile base must be computed and analysed. The maximum displacement is measured at the pile top, and the load at the pile base is calculated from the diagram of normal stress at the pile base. As previously mentioned, the load is applied in the form of a prescribed displacement of 1m. The resulting displacement field within the model boundaries is displayed in Figure 3.4.
Figure 3.4: Vertical displacement field, for 1m prescribed displacement at the pile top, L=20m.
38
This displacement field implies that the model behaves as expected, as far as the settlement is concerned; the maximum value occurs at the pile top, and is also quite high along the pile, gradually decreasing with the distance to the pile. The vertical stress field is displayed in Figure 3.5 and Figure 3.6.
Figure 3.5: Vertical stress field, for 1m prescribed displacement at the pile top, L=20m.
Figure 3.6: Vertical stress field near the pile base, for 1m prescribed displacement at the pile top, L=20m .
39
As expected, high gradients of stress occur at the pile base, while the rest of the soil mass is less disturbed. The existence of a stress bulb is evident. The maximum and minimum stress points result from the soil-pile interaction. Figure 3.7 and Figure 3.8 represent the normal stress diagram at the pile top and at the pile base, respectively.
Figure 3.7: Normal stress diagram at the pile top, fo r 1m prescribed displacement at the pile top, L=20m.
Figure 3.8: Normal stress diagram at the pile base, for 1m prescribed displacement at the pile top, L=20m – both at the pile and at the soil side of the interface.
These diagrams attest for the good behaviour of the model, as far as normal stress is concerned; their shapes are as expected, showing that the load is transmitted to the pile and also how the pile and the soil interact. Their smoothness shows that the mesh is adequate. 40
The obtained results for every different model geometry are organized in the following charts. The curves of the load settlement ratio in terms of the slenderness ratio of the pile are presented in Figure 3.9 (for homogeneous soils, ρ=1), in Figure 3.10 (for non-homogeneous soils, ρ=0.75) and in Figure 3.11 (for non-homogeneous soils, ρ=0 .5).
140
Rigid
Compressible
120
100 ) L G 80 × 0 r × t w ( / t 60 P
P&D R&W (rigid) R&W (compressible) Plaxis
40
20
0 0
10
20
30
40
50 L/r0
60
70
80
90
100
Figure 3.9: Load settlement ratio in terms of the pile slenderness ratio for homogeneous soils (ρ=1) , ν=0.3, h=2.5L and λ=975.
In the chart above, which refers to homogeneous soils, there is an obvious similarity between the results given by Plaxis and by the Randolph and Wroth method for compressible piles. Both indicate the existence of an asymptote. The difference between these results increases with the slenderness ratio, and shows that, for the same applied load, vertical displacements calculated through Plaxis are higher than those calculated by the other methods. Values referent to the Poulos and Davis method are between the Randolph and Wroth method ’s referent to rigid and compressible piles. This is expected, since the Poulos and Davis method refers to rigid piles, allowing for the introduction of a correction factor for pile compressibility, R k; this factor should have a lower (thus, more significant) value in the calculation of the load settlement ratio of the pile, if results provided by Plaxis are assumed to be more correct solutions. The settlement provided by the Poulos and Davis method is indeed lower than the one provided by the finite element program, as stated in (Poulos & Davis, 1980). As stated in (Randolph & Wroth, 1978), when L/r 0>80, the difference between the method’s and numerical results is significant.
41
140
Rigid
Compressible
120
100 P&D
) L G 80 × 0 r × t w ( 60 / t P
R&W (rigid) R&W (compressible) Plaxis
40
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 3.10: Load settlement ratio in terms of the pile slenderness ratio for inhomogeneity fact or ρ=0 .75, ν=0.3, h=2.5L and λ=975.
Similar conclusions may be drawn from this chart, as far as the relation between each method’s results is concerned. As mentioned before, all elastic theory-based methods, as well as the numerical analysis, provide lower values of the load settlement ratio in this case, comparing to the homogeneous soil. For ρ=0.75, the existence of an asymptote is even clearer. The shaft contribution to the load
settlement ratio decreases, i.e. the settlement due to the shaft is higher, since the soil does not develop as much shaft load as in homogeneous soils, given the same shear modulus at the pile base.
42
140
Compressible
Rigid
120
100 P&D ) L G 80 × 0 r × t w ( 60 / t P
R&W (rigid) R&W (compressible) Plaxis
40
20
0 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 3.11: Load settlement ratio in terms of the pile slenderness ratio fo r inhomogeneity factor ρ=0 .5, ν=0.3, h=2.5L and λ=975.
As previously mentioned, the Poulos and Davis procedure to account for the soil inhomogeneity is to assume constant stiffness and use the Young’s modulus at the depth corresponding to L/2. This might be an oversimplification. The graph of Plaxis shows that, for values of the pile slenderness higher than 70, the load settlement ratio starts decreasing, although very slightly. This cannot correspond to the real behaviour, because the total settlement of the pile may not increase with the pile length, if all the other factors remain constant. Therefore, this decrease may be explained by the insufficient refinement of the mesh and numerical inaccuracies; however, these differences may be considered negligible. The curves of the proportion of load taken by the pile base in terms of the slenderness ratio of the pile are presented in Figure 3.12 (for homogeneous soils, ρ=1), in Figure 3.13 (for non-homogeneous soils, ρ=0.75) and in Figure 3.14 (for non-homogeneous soils, ρ=0 .5).
43
0,40
Rigid
0,35
Compressible
0,30 P&D
0,25
R&W (rigid)
t
P / b 0,20 P
R&W (compressible)
0,15
Plaxis
0,10
0,05
0,00 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 3.12: Proportion of the load taken by the pile base in terms of the pile slenderness ratio for homogeneous soils (ρ=1), ν=0.3, h=2.5L and λ=975.
Results delivered by Plaxis are very consistent with the ones provided by the Randolph and Wroth method for compressible piles. They are also extremely close to the ones given by the Poulos and Davis. This proximity is expected to diminish as the inhomogeneity factor reduces, not only because the results of the finite-element program are subject to more error, but also due to the simplification of the Poulos and Davis method for non-homogeneous soils.
44
0,40
Rigid
0,35
Compressible
0,30 P&D
0,25 R&W (rigid) t
P / b0,20 P
R&W (compressible)
0,15
Plaxis
0,10
0,05
0,00 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 3.13: Proportion of the load taken by the pile bas e in terms of the pile slenderness ratio for inhomogeneity factor ρ=0.75, ν=0.3, h=2.5L and λ=975.
This chart indicates that the finite element program is more sensitive to the change in the inhomogeneity factor than the Randolph and Wroth method for compressible piles: the vertical shift of its curve is more evident. In fact, Plaxis’ results are the ones which vary the most with the increase of the inhomogeneity of the soil.
45
0,40
Compressible
R
0,35
0,30
P&D
0,25
R&W (rigid)
t
P 0,20 / b P
R&W (compressible)
0,15
Plaxis
0,10
0,05
0,00 0
10
20
30
40
50
60
70
80
90
100
L/r0
Figure 3.14: Proportion of the load taken by the pile bas e in terms of the pile slenderness ratio for inhomogeneity factor ρ=0.5, ν=0.3, h=2.5L and λ=975.
This chart shows the same tendency as the last one: once again, results provided by Plaxis are the ones with the highest relative increase due to change in the soil inhomogeneity. For L/r 0→0, it is expected that P b/Pt→1. The chart shows that the Randolph and Wroth method’s solution presents a slight inflexion for L/r 0=4. This means the method cannot be applied for small values of the pile slenderness ratio (when it tends to become a shallow foundation). Nevertheless, the curves obtained by the different methods correspond well. The Randolph and Wroth solution for compressible piles proves to be adequate for both load distribution and settlement estimation. Its solutions for vertically non-homogeneous soils are accurate, and are useful for expeditious calculations.
46
4 NUMERICAL
VALIDATION
ELASTOPLASTIC MODELLING
OF A
OF
SINGLE
AXIALLY LOADED PILE
4.1 Introduction In this chapter, a calculation performed through the finite element method-based program Plaxis will be validated with a set of published results provided by other programs, under the same conditions. This case was first analysed in (Neves, 2001b). In this document, results provided by the software CÉSAR-LCPC are displayed – a two-dimensional axisymmetric simulation, as the one performed by Plaxis. The parameters of the different layers of soil were determined in (Neves, 2001a). Later, the case was analysed again in (D'aguiar, 2008), using the software GEFdyn, in a three-dimensional calculation. The results from these two different simulations correspond very well. Results provided by Plaxis will be compared with the ones provided by GEFdyn. This will validate the model used in the Plaxis simulation, and particular attention is given to the behaviour of the soil-pile interface.
47
4.2 The Model The analysed pile is cylindrical and isolated, subject to axial loading. Therefore, it is once again possible to perform a two-dimensional axisymmetric simulation. 15-node triangular elements are used.
4.2.1 Geometry The model consists of five different materials: the pile (made of reinforced concrete) and four layers of soil. The pile is 10m long, with a 0.4m diameter. Its shaft crosses three different layers of granular soil, whose thicknesses are 6.3m, 2m and 2.8m, respectively. There is yet another layer of soil underneath the pile, with thickness of 8.9m, at the bottom of which a rigid layer is assumed to exist. The water table is located 0.4m above the pile base. Since axisymmetry is used, only half of the problem is represented. The total dimensions of the model are 10×20m2, as used in the previous simulations. Figure 4.1 shows the geometry of the model.
Figure 4.1: Geometry of the model.
4.2.2 Loading In the last chapter, it was proved that the best way of introducing load into the model is to use prescribed displacement. Since the general conditions are the same in this chapter (cylindrical pile, axial loading at the pile top, etc), the same type of loading will be applied.
48
4.2.3 Materials and Interfaces Five materials were used in this analysis: reinforced concrete (referred to as “pile”) and four different layers of granular soil. In one of the layers, there is a distinction between saturated and unsaturated, since it is crossed by the water table. The simulation is performed under drained conditions. The pile is modelled as linear elastic, and the soil as elastoplastic, using the Mohr-Coulomb criterion. In Table 4.1, the material properties introduced in the program are listed. These are exactly the ones used in the previous simulations. Table 4.1: Material properties (Neves, 2001a).
Reinforced Concrete
Pile Layer 1 Layer
Soil
2 Layer 3 Layer 4
L
Material
Material
(m)
Model
Type
10.0 6.3 2.0 2.8 8.9
Linear Elastic MohrCoulomb MohrCoulomb MohrCoulomb MohrCoulomb
φ'
ψ'
[kN/m3] [kN/m2] (º)
(º)
γ
c'
ν
k0 -
E [kN/m2] 29×106
Drained
24.0
-
-
0
0.3
Drained
16.7
13
26
0
0.12 0.562
9150
Drained
18.8
12
23
0
0.12 0.609
13510
Drained
19.8
14
23
0
0.07 0.609
13570
Drained
20.0
17
23
0
0.05 0.609
19300
The soil-pile interaction is modelled through an interface. This is, in the case of 15-node triangular elements, five pairs of nodes that allow for relative displacement between the two materials that are linked through it. Figure 4.2 represents the geometry of an interface.
Figure 4.2: Distribution of nodes in interface elements and respective connection to 15-node triangular elements. Adapted from Plaxis Manual (Brinkgreve, 2006).
In Table 4.2, the interface properties of the layers of soil crossed by the pile shaft used in the previous analyses are listed. The interface model, cohesion and friction angle were used in both the CÉSARLCPC and the GEFdyn calculations; however, the information in the last three columns of Table 4.2 refers to the latter simulation only, since these parameters were not mentioned in the former.
49
Table 4.2: Interface properties used in the CÉSAR-LCPC and in the GEFdyn simulations.
L
Material
Interface
c'interface
φ' interface
(m)
Model
Model
[kN/m2]
(º)
Mohr-
Mohr-
Coulomb
Coulomb
6
23
Mohr-
Mohr-
Coulomb
Coulomb
5
Mohr-
Mohr-
Coulomb
Coulomb
7
Layer 1
6.3
Layer 2
2.0
Layer 3
2.8
Ginterface
Einterface
[kN/m2]
[kN/m2]
0.562
4×104
9×1017
20
0.609
4×104
9×1017
20
0.609
4×104
9×1017
k0
As far as the interface is concerned, the two-dimensional analysis performed by Plaxis provides the user with limited control of the parameters. Actually, there are only two interface properties that may be altered directly: the interface strength (R inter ) and the virtual thickness factor. The interface is always elastoplastic, by the Mohr-Coulomb criterion. The interface strength, Rinter , may vary between 0.01 and 1.00. The upper limit indicates that the interface is assumed to be weaker and more flexible than the correspondent soil, since the factor R inter relates the interface parameters with the soil parameters, according to eq. (4.1) to eq. (4.3):
(4.1)
(4.2)
(4.3)
It is evident that there is no value of R inter that fulfils these three conditions simultaneously, when a set of material properties as in Table 4.1 and a set of interface properties as in Table 4.2 are used. For instance, for the first layer of soil, eq. (4.1) results in R inter =0.462, eq. (4.2) results in Rinter =0.870 and eq. (4.3) results in R inter =3.129 (!) – this is due to the fact that, in the previous simulations, the interface was assumed to be less flexible than the adjacent soil, which is simply not possible in this calculation. Since the friction angle is more important to the interface behaviour than its cohesion, the results from eq. (4.2) are used directly, as a first iteration. The Poisson’s ratio of the interface, for R inter <1, are fixed and equal to 0.45. Table 4.3 lists the interface strength used in each layer of soil, as well as the resulting interface parameters. The different between the interface cohesion, c’, shear modulus, G interface, and Young’s
modulus, Einterface, in Table 4.2 and Table 4.3 is evident. 50
Table 4.3: Interface properties used in the Plaxis simulation.
L (m) Layer 1
6.3
Layer 2
2.0
Layer 3
2.8
Material
Interface
Model
Model
Mohr-
Mohr-
Coulomb
Coulomb
Mohr-
Mohr-
Coulomb
Coulomb
Mohr-
Mohr-
Coulomb
Coulomb
Rinter
c'interface [kN/m2]
φ' (º)
νinterface
Ginterface
Einterface
[kN/m2]
[kN/m2]
0.870
11
23
0.45
3092
8966
0.857
10
20
0.45
4430
12846
0.857
12
20
0.45
4657
13506
The other interface-related option than may be changed by the user in Plaxis calculations is the virtual thickness factor. This factor, when multiplied by the average element size, results in the virtual thickness of the interface, tinterface. The virtual thickness influences the interface behaviour because it affects the relative displacements between the pile and the soil, as described in eq. (4.4) and (4.5).
(4.4)
(4.5)
Eoed, interface is the Oedometer modulus of the interface, and may be calculated through eq. (4.6):
(4.6)
Since the values of both the shear modulus and the Young’s modulus of the interface used in the Plaxis calculation are significantly lower than the ones used in the two previous simulations, the virtual thickness of the interface is reduced, in an attempt to produce elastic slips closer to the ones obtained before – i.e. lower values of relative displacement between soil and pile. It is important to notice that these are not calculated in the same way as in the other softwares, and therefore there is no exact value of the interface virtual thickness to introduce that may be considered “correct” . The default value of the interface virtual thickness factor is 0.1. This was changed to 0.01, which is the lowest possible value – resulting in a ten times lower virtual thickness, whereas the shear modulus and the Young’s modulus used in the former simulations are over ten times higher than the ones used in the present simulation. Therefore, only a small improvement is to be expected. This alteration will provide a stiff interface, allowing for less movement. A few tests were run using this approach, in order to ensure no num erical error derived from it. Indeed, the results showed no sign of error, and the difference to the ones using the default interface thickness was almost insignificant.
51
4.2.4 The Mesh The mesh is constituted by 15-node triangular elements, in a total of 11559 elements and 95613 nodes, with an average element size of 0.131m. The mesh is much more refined around the pile. To achieve this, two transition areas are drawn, so as to obtain a good connection between highly refined and coarse parts of the mesh. This mesh is considerably refined, as may be observed in Figure 4.3.
Figure 4.3: Mesh.
4.3 Initial Conditions
4.3.1 Water pressure generation The initial conditions of the model refer to both the water pressure and the stress field, before any calculation is performed. The pore pressure field due to the water table is generated in the first phase, using the specific weight 3
of the water, γ=10kN/m . The resulting pore pressure diagram, which is constant along the model, is
displayed in Figure 4.4.
52
Figure 4.4: Diagram of pore pressure.
4.3.2 Initial stress field generation generation The initial stress field generation is not as immediate as the pore pressure field. In the two previous analyses, the simulation was carried out with no generation of shaft load due to the pile installation. Thus, the weight of the pile was completely transmitted to the base, without shear stress being developed along the shaft. In order to replicate these conditions, the ideal procedure would be to generate the initial stress field with the different soil layers as well as the pile, keeping the interface “off”, so there would be no relative movement between the pile and the soil. However, the normal stress generation method (the “k0-procedure”) carried out by Plaxis provides vertical equilibrium (with the self-weight of the soil) only,
horizontal equilibrium not being ensured, since horizontal stresses are generated from the introduced value of the coefficient of lateral earth pressure, k 0. Thus, if two different materials are introduced in the model at the same depth, the k0-procedure does not ensure horizontal equilibrium, according to (Brinkgreve, 2006), the Plaxis Manual. Given that neither the specific weight nor the k0 coefficient of the pile are the same as the soil’s, and that the geometry of t he problem implies the existence of pile material and soil material at the same depth – i.e., horizontal equilibrium cannot be achieved – this apparently ideal way of generating initial stress does not provide adequate results. Another method of stress generation is available in Plaxis – “gravity loading”. This has the advantage of granting equilibrium. However, in this type of stress generation, the coefficient of lateral earth pressure, k0, is calculated through the Poisson’s ratio, ν, as in eq. (4.7), and not through the friction angle, φ’, as in eq. (4.8):
(4.7)
(4.8)
53
In the case of layer 1, for instance, eq. (4.7) results in k0=0.136 (which is not very realistic) and eq. (4.8) results in k 0=0.562. These are indeed very different values. That is the reason why Plaxis manuals recommend that the Poisson’s c oefficient be adjusted so that eq. (4.7) provides the desired
value of k0. Since the objective of this work is to simulate the conditions of the previous analyses, attributing a different Poisson’s coefficient to the soil would not be appropriate, and thus this type of stress generation should not be used. For these reasons, and since in this simulation the effects of the installation in the soil cannot be considered, the initial stress field must be generated from the different layers of soil only, using the k 0procedure. Using the information contained in Table in Table 4.1, the 4.1, the initial stresses were then generated, and the resulting vertical and horizontal effective stress fields are shown in Figure 4.5 and Figure 4.6, respectively.
Figure 4.5: Initial vertical effective stress field, not including the pile.
54
Figure 4.6: Initial horizontal effective stress field, not including the pile.
These two figures show satisfactory stress fields, with increasing stress values along the depth, and constant along the radius. The values correspond to the ones calculated manually, with an error in the order of 0.01kN/m2. Thus, this initial stress field is considered to be adequate and is used in the following calculations.
4.3.3 Introduction Introduction of the Pile The question of how to include the pile in the model is yet to be addressed. The assumption that the adjacent soil remains undisturbed by the pile installation implies that no shear stress in the soil-pile interface occurs before the first loading, and consequently that the totality of the pile weight reaches the base. In order to achieve this, the soil-pile interface should be considered as completely smooth, in the first step of the calculation, which corresponds to the introduction of the pile with no loading. In the program, this is achieved by replacing the volume of soil material which occupied the place of the pile during the initial stress generation by pile material, while maintaining the interface strength R inter set to the minimum value – 0.01, since the introduction of 0.00 is not allowed. This is an artificial step, and the proper interface strength Rinter value is restored when loading is applied. Immediately after it, the incremental loading starts, with a null displacement field that is achievable by selecting the option “reset displacements”.
Although in the program this step is already part of the th e calculation, and not of the initial conditions, it is included in this section because, along with the initial stress generation, it is common to all the calculations performed subsequently.
55
4.4 Calculation and Results The following calculations are performed with incremental loading, the size of each step increasing gradually. This is a way to avoid numerical errors that might result from applying all the load at once. The result of each increment is displayed as a mark in the correspondent curve. All the points that constitute the charts are shown in Appendix C.
4.4.1 Sensitivity Analysis of the Interface In a first observation, it seems clear that the most evident variation of the model used in Plaxis from the ones used in the two previous calculations (by CESAR-LCPC and GEFdyn) is the soil-pile interface. Therefore, a sensitivity analysis of the interface is performed. Two interfaces with extreme values of interface strength are tested: Rinter =0.01 and Rinter =1.00 (rigid interface). All the other parameters used are the ones described in the previous section. In Figure 4.7, the total load Pt (sum of the applied load and the pile total weight), shaft load P s and base load Pb in terms of the displacement at the pile top wt are plotted, for the cases of smooth and rigid interfaces.
700 650 600 550 500 450
Pt Rinter=0,01
400
Ps Rinter=0,01
) N k ( 350 P
Pb Rinter=0,01 Pt Rinter=1
300
Ps Rinter=1
250
Pb Rinter=1
200 150 100 50 0 0
5
10
15
20
25
30
wt (mm)
Figure 4.7: Total load, load transmitted to the pile shaft and load transmitted to the pile base in terms of the settlement at the pile top, for smooth and rough interfaces.
This chart shows that the applied load necessary to cause a certain displacement is much higher when the interface is rigid. The curves of the load transferred to the pile base are very close, indicating 56
that, as expected, the interface strength has a much greater influence over the shaft load than over the base load. The shaft load is, as expected, very small in the case of smooth interface. Since the calculated interface strengths are Rinter =0.87 and Rinter =0.857, the correspondent curves are expected to be closer to the rigid interface ones.
4.4.2 Plaxis Results Compared to CESAR-LCPC and GEFdyn Results The calculation with the interface values shown in Table 4.3 is now performed. In Figure 4.8, the curves of total load, load transmitted to the pile shaft and load transmitted to the pile base obtained by Plaxis are plotted, along with the ones obtained by CESAR-LCPC and GEFdyn.
500 450 400 350 Pt CESAR-LCPC 300
Ps CESAR-LCPC
) N k ( 250 P
Pb CESAR-LCPC Pt GEFdyn Ps GEFdyn
200
Pb GEFdyn Pt Plaxis
150
Ps Plaxis Pb Plaxis
100 50 0 0
5
10
15 wt (mm)
20
25
30
Figure 4.8: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top, obtained through Plaxis and GEFdyn.
The curves of total load show correspond very well. However, there is an evident discrepancy in the other two curves – results provided by Plaxis indicate a lower base load, i. e. the base resistance is not as mobilized as in the other models. As proved in Figure 4.7, changing the value of the interface strength, Rinter , barely alters these results, since it affects the load that reaches the pile base very little. In fact, the curve that corresponds to the maximum possible base load (smooth interface, R inter =0.01) is still below the one given by GEFdyn, as shown in Figure 4.9. Therefore, the calculated value of the interface strength may be considered acceptable, and the inconsistency of results must be explained by something else.
57
700 650 600 550 500 450
Pt Plaxis Rinter=0,01
400
) N k ( 350 P
Pb Plaxis Rinter=0,01 Pt Plaxis Rinter=1
300
Pb Plaxis Rinter=1
250
Pt GEFdyn
200
Pb GEFdyn
150 100 50 0 0
5
10
15 wt (mm)
20
25
30
Figure 4.9: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top, obtained through Plaxis (for smooth and rough interfaces) and GEFdyn.
In order to understand the difference of the load transfer behaviour between the GEFdyn model and the Plaxis model, the failure mechanism is analysed. The plastic points due to the Mohr-Coulomb failure envelope are displayed in Figure 4.10, for a prescribed displacement of s=8mm (when the interface along the entire pile is in the plastic state).
Figure 4.10: Plastic points due to the Mohr-Coulomb criterion at the pile base for s=8mm.
This figure shows that the developed failure mechanism includes a wedge. The distribution of the plastic points does completely resemble the failure mechanism normally attributed to piles under compression. This indicates that the discrepancies shown in Figure 4.8 might be a consequence of the difference between the finite element algorithms. 58
4.4.3 Analysis Including Soil Dilatancy As an attempt to improve these results, a calculation including dilatancy of the soil beneath the pile base is performed. It is expected that if the soil at the pile base has a positive dilatancy angle, the base resistance will be more easily mobilized as the load is applied, since the soil will expand. The introduced value for the angle of dilatancy, ψ, is the same as the friction angle, φ (ψ=φ=23º).
Figure 4.11 shows the results of this calculation, compared to the ones given by GEFdyn.
500 450 400 350 Pt GEFdyn
300 ) N k ( 250 P
Ps GEFdyn Pb GEFdyn
200
Pt Plaxis
150
Ps Plaxis Pb Plaxis
100 50 0 0
5
10
15
20
25
30
wt (mm)
Figure 4.11: Total load, load transferred to the pile shaft and load transferred to the pile base i n terms of the settlement at the pile top, obtained through Plaxis (including soil dilatancy at the base) and GEFdyn.
This chart shows that introducing dilatancy as a soil property does indeed have the effect of mobilizing the base. In this case, an angle of dilatancy equal to the friction angle is still not enough to obtain results identical to the ones given by GEFdyn; however, it does approximate the base load curves, and the difference seems to decrease with the loading. The simulation performed in this chapter led to the conclusion that there are difficulties in modelling the behaviour of the pile base in Plaxis, since in this program the base resistance is not as mobilized as in CESAR-LCPC and GEFdyn.
59
60
5 CASE STUDY
OF A
STATIC LOAD TEST
PERFORMED ON A TEST PILE
5.1 Introduction In this chapter, a calculation performed by the finite element method-based program Plaxis is compared to the results of a static load test performed on a test pile. The static load test is part of a planned set of tests that support the design of the project “Modernização da Linha do Norte, Sub Troço 14 – Azambuja/Vale de Santarém, Viaduto de Santana do Cartaxo”, the construction of a viaduct. It was performed by Mota-Engil, Engenharia e Construção,
S. A. and the results were analysed by Instituto de Engenharia de Estruturas, Território e Construção do Instituto Superior Técnico (ICIST). The objective of a static load test is to analyse the load transfer process between the pile and the soil. The results are therefore carefully analysed and conclusions on the soil behaviour are drawn.
61
5.2 Geological Characterization The geological prospection included the excavation of 37 boreholes that enabled the execution of SPT, CPTU, SCPTU and FVT tests, as well as laboratory tests, which included edometric and triaxial tests. The boreholes were executed until a hard layer was reached. These tests allowed the distinction of several soil layers, which constitute the lithological profile of the area. According to the report by (Viaponte, 2003), four different lithostratigraphic units are identified, following in the order of crescent depth: - Superficial deposits: dark brown soil, formed by silty sand and sandy organic clay; - Recent alluvial deposits: constituted by soft grey organic clay (0
Figure 5.1: Geological profile where the static load test was performed.
62
5.3 The Static Load Test (SLT) 5.3.1 Preparation The static load test (SLT) was performed on a cylindrical concrete pile with a total length of 40.6m and diameter of 800mm. The pile cap was built 2.1m below the soil surface, and the shaft was extended to the hard layer. This geometry is displayed in Figure 5.2.
Figure 5.2: Pile and soil layers. Adapted from (Santos, 2005).
A retrievable casing tube was necessary due to the position of the water table and to the properties of the first layers of soil, which are very soft. This temporary casing extended to the depth of 32.0m (or 29.9m below the pile top) as shown in Figure 5.2, since the soil was considered to be consistent enough for it to be unnecessary from that point on. As shown in Figure 5.3(a), the casing was introduced into the soil using static pressure and small rotations, in a gradual way, by welding several pieces until the necessary length had been achieved (Figure 5.3(b)). The soil inside the casing was constantly being removed. When the casing reached the depth of 32.0m, the reinforcing cage was introduced as displayed in Figure 5.3(c).
63
(a)
(b)
(c)
Figure 5.3: (a) Driving of the temporary casing; (b) Welding of the casing; (c) Introduction of the reinforcing cage. (Viaponte, 2003).
The casing was removed as the pile was being concreted. The concrete had a resistance class of C35/45 and a Young’s modulus of E p=33.5GPa. When the concreting process ended, the pile top was
cleaned and the cap was built.
5.3.2 Procedure The execution of the test began on January 20th 2005 at 11.10 a.m. Care was taken to ensure that enough time had passed since the building of the pile to let the vertical displacements and the properties of the soil to stabilize. Two hydraulic jacks were used; they exerted pressure against the reaction system, which was constituted by a metallic truss, since these structures are rigid enough not to deform excessively as the load is applied, thus being very efficient. The foundations of this reaction system are constituted by a total of four piles, built by following the same procedure as for the tested pile, described above. Figure 5.4 shows all these elements.
Figure 5.4: Reaction system (Viaponte, 2003).
64
The usual procedure for static load tests is to apply a maximum load of twice the calculated service load, Pser ; in this case, the correspondent value would be 2×P ser =2×2,800kN=5,600kN. However, since the maximum load that could be applied to the reaction system was 8,000kN (its maximum resistance), it was decided to take advantage of that limit. The test was performed incrementally, the loading and subsequent unloading being executed twice; the first loading lasted until service load (2,800kN) was reached, and the second until the reaction system limit (8,000kN) was reached. Before proceeding to the next load step, stabilization criterions were applied. In the loading phases, it was ensured that either the settlement rate was below 0.3mm/h or the difference between two consecutive displacement results was under 5% of the total settlement, before continuing to the next increment. Besides these two criterions, time limits were also established, the most important being that the step that corresponded to the first peak had to last at least 2h and the one that corresponded to the second peak had to last at least 12h. During the unloading, the only criterion was that each step should last 15min, except for the last step of each loading cycle, which in the first unloading had to last at least 2h and in the second unloading had to last at least 4h. Table 5.1 shows the applied load in each load step and its duration. Figure 5.5 traduces this information in a chart, considering the minimum possible duration for each load step.
8000 7000 6000 5000
) N k ( d 4000 a o L
3000 2000 1000 0 0
5
10
15
20
25
30
Time (h)
Figure 5.5: Loading plan. Adapted from (Santos, 2005).
65
Table 5.1: Loading plan. Adapted from (Santos, 2005). Duration Load Step
Applied Load Pt
Proportion of Applied Load Pt/Pser
Minimum
Maximum
1
700
0.25
30min
4h
2
1,400
0.50
30min
4h
3
2,100
0.75
30min
4h
4
2,800
1.00
2h
4h
5
2,100
0.75
15min
6
1,400
0.50
15min
7
700
0.25
15min
8
0
0.00
2h
4h
9
700
0.25
30min
4h
10
1,400
0.50
30min
4h
11
2,100
0.75
30min
4h
12
2,800
1.00
30min
4h
13
3,500
1.25
30min
4h
14
4,200
1.50
30min
4h
15
4,900
1.75
30min
4h
16
5,600
2.00
30min
4h
17
6,300
2.25
30min
4h
18
7,000
2.50
30min
4h
19
8,000
2.86
12h
-
20
7,000
2.50
15min
21
5,600
2.00
15min
22
4,200
1.50
15min
23
2,800
1.00
15min
24
1,400
0.50
15min
25
0
0.00
4h
-
66
5.3.3 Instrumentation As previously mentioned, the load was applied through two hydraulic jacks. The capacity of the jack should exceed the maximum applied load by a considerable margin; for very high values of maximum load, as in this case, more than one may be necessary. For the load measurement at the pile top, two different methods were used, so there would be two sets of results; this redundancy assesses the quality of the results. The first was a pressure gauge, and the second a load cell. The two sets of results corresponded well and the values provided by the pressure gauge were the ones used in the interpretation of the test results. For the settlement measurement at the pile top, two independent methods were used: the first was a set of dial gauges and transducers and the second was optical levelling. At the pile top, two mechanical dial gauges and two digital transducers were installed. They were placed on a metallic structure, and the topographic measurements showed no relative displacement between this and the reference level during the test. Both the results from the set of dial gauges and transducers and from the optical levelling corresponded very well. For the displacement measure along the pile shaft, and consequent determination of the mobilized resistance, vibrating wire extensometers were used, by welding to the reinforcing cage, as shown in Figure 5.6. Sets of three extensometers were placed at eight different depths, as shown in Table 5.2 and displayed in Figure 5.7.
Figure 5.6: Vibrating wire extensometer welded to the reinforcing cage (Viaponte, 2003).
Table 5.2: Depth of each level of extensometers. Level
1
2
3
4
5
6
7
8
Depth (m)
20.7
25.7
29.7
31.7
32.7
36.7
37.7
40.7
67
Figure 5.7: Depth of each level of extensometers . Adapted from (Santos, 2005).
Due to the fact that the records at levels 1, 3 and 7 were unusual (unexpectedly high or low, or with sudden peaks), only the normal stresses provided by the extensometers at the other levels are considered.
5.3.4 Results There are three sets of results which must be analysed to understand the behaviour of the soil in this static load test: 1)
The first is the set of results of the applied load at the pile top, provided by the pressure
gauges. Due to load losses, the measured load may not be exactly the same as the applied load. Figure 5.8 shows the measured load in terms of the time. When compared to the plan (see Figure 5.5), it becomes evident that the test took longer than the minimum scheduled amount of time, and that the load had to be adjusted several times to achieve the intended value, as in P t=7,000kN.
68
Figure 5.8: Load at the pile top, measured by pressure gauges, in terms of time in minutes. Adapted from (Santos, 2005).
2)
The second set of results provides the vertical displacement at the pile top, measured by the
dial gauges and transducers. These corresponded very well. Figure 5.9 displays the applied load at the pile top measured by the pressure gauges (shown in Figure 5.8) in terms of the average settlement measured by the dial gauges and transducers.
8 000
7 000
6 000
5 000 ) N k ( 4 000 t P
3 000
2 000
1 000
0 0
10
20
30
40
50
60
70
wt (mm)
Figure 5.9: Measured load at the pile top in terms of the measured settlement of the pile top. Adapted from (Santos, 2005).
69
3)
The results of normal stress along the pile shaft, provided by the vibrating wire extensometers.
As previously mentioned, only 5 out of 8 levels of extensometers provided acceptable results. It should be borne in mind that this kind of instrument measures the increase of extension with load along the pile; normal stress is calculated from its results. Figure 5.10 shows the normal stress measured at five different levels and Figure 5.11 shows the normal stress along the depth for service load (2,800kN, load step 4) and for maximum load (8,000kN, load step 19). L2 L4 L5 L6
L8
Figure 5.10: Normal stress at different levels along the pile (Santos, 2005).
N (kN) 0
2000
4000
6000
8000
0 5 10 15 ) m20 ( z
25
Load Step 4
Load Step 19
30 35 40 45
Figure 5.11: Normal stress along the pile shaft measured for load steps 4 and 19. Adapted from (Santos, 2005).
70
Figure 5.12 shows the lateral stress measured between two levels, calculated from the normal stress, and Table 5.3 shows the estimated mobilized shaft resistance measure at load steps 4 and 19.
L45
L24 L56 L68
Figure 5.12: Lateral stress between different levels along the pile (Santos, 2005).
Table 5.3: Mobilized shaft load for load steps 4 and 19. Adapted from (Santos, 2005).
Mobilized Shaft Load (kPa) Between levels
Load Step 4
Load Step 19
2 and 4
15 to 25
60 to 70
4 and 5
100 to 110
180 to 200
5 and 6
15 to 25
100 to 110
6 and 8
20 to 30
≥ 130
The chart in Figure 5.12 appears to indicate that, between levels 6 and 8, the mobilized shaft resistance has not reached its maximum for the maximum applied load. Given that this corresponds to the last layer of soil (a very consistent clay), it makes sense that the maximum resistance has not been reached yet. Also, it is interesting to notice that the mobilized shaft resistance between levels 4 and 5 is so high compared to the mobilized shaft resistance between levels 5 and 6, given that the same soil (very stiff clay) is measured. This is probably due to the effect of the casing (that reached the distance between levels 4 and 5), which may have disturbed the stress state of the soil at its bottom.
71
5.4 Numerical Analysis by Plaxis The static load test is now modelled by the finite element-based program Plaxis, in order to compare its results with the ones provided by the different types of instrumentation described in the previous section. The geological profile in the model is identical to the one given by the static load test report, and the different layers of soil are related to the lithostratigraphic units described above. The analysed pile is cylindrical and isolated, subject to axial loading. Therefore, it is possible to perform a two-dimensional axisymmetric simulation. 15-node triangular elements are used.
5.4.1 The Model 1) Geometry The model consists of nine different materials: the pile (made of reinforced concrete) and eight layers of soil. The pile is 40.6m long, with a 0.8m diameter. The pile top is not located at the soil surface; its cap was built 2.1m below. The pile shaft crosses eight different layers of soil. The water table is located 3.4m below the soil surface, and 1.3m below the pile top; this depth corresponds to 0.4m below the base of the first layer of soil. For simplification reasons, since 0.4m is a very small length compared to the overall size of the model and thus difficult to mesh, the water table is introduced so as to coincide precisely with the base of the first layer of soil. Since axisymmetry is used, only half of the problem is represented. The chosen dimension of the model is 50×55m2, since the horizontal dimension should be at least as long as the pile (50.0m>40.6m) and the distance from the pile base to the lower boundary should be enough not to affect the behaviour (in this case, the distance of 12.3m is considered enough, for the last layer is very resistant). Figure 5.13 shows the geometry of the model.
Figure 5.13: Model geometry.
72
2) Loading The general conditions in this case are the same as in the previous simulations: cylindrical pile, axial loading at the pile top, etc. Therefore, once again, the loading is applied through prescribed displacements. 3) Materials and Interfaces Nine materials are used in this analysis: reinforced concrete (referred to as “pile”) and eight different layers of soil. The soil properties vary considerably from layer to layer, but there is a clear increase in consistency with depth. The first soil layer is unsaturated, and the seven others are saturated. The pile is modelled as linear elastic, and every layer of soil as elastoplastic, according to the Mohr-Coulomb criterion or to the Tresca criterion, depending on if it is drained or undrained, respectively. Table 5.4 and Table 5.5 describe the pile and the soil layers, respectively. Table 5.4: Pile properties.
Material
L (m)
d (m)
Designation
Material Model
Reinforced concrete
40.6
0.8
Pile
Linear elastic
Table 5.5: Soil properties.
Layer
L (m)
Designation
Material Model
1
3.0
Superficial deposit
Mohr-Coulomb
2
12.0
Soft clay
Tresca
3
9.0
Medium clay
Tresca
4
4.0
Firm clay
Tresca
5
2.5
Gravel
Mohr-Coulomb
6
6.0
Very stiff clay
Tresca
7
4.0
Dense sand
Mohr-Coulomb
8
14.5
Hard clay
Tresca
Unlike the case of the last chapter, the information about each material type is very limited. In some cases, (Viaponte, 2003) suggests a value or a range of values; in others, there is no indication. The attributed soil properties for each case is described and discussed below. - Pile (Santos, 2005) indicates for the pile’s Young’s modulus the value of E p=33.5GPa. For the other properties, values that are generally accepted for reinforced concrete piles are used: specific weight equal to γ=25kN/m3 and Poisson’s ratio of ν=0.3. In Table 5.6, the pile properties introduced in the program are listed.
73
Table 5.6: Pile properties (2).
Material
Designation
γ [kN/m ]
ν
E [MN/m ]
Rinter
Reinforced concrete
Pile
25.0
0.3
33500
1.000
- Layer 1: Superficial Deposit There is little information available about this material. It is assumed that it corresponds to the superficial deposit described in the geological characterization. In fact, its properties are not expected to have great influence in the final results, since its thickness is 3m only, and the interface with the pile only 0.9m long; besides, being the top layer, it is probably the least resistant. It is modelled under drained conditions, γ=18kN/m3, E=25MPa, ν=0.3 and friction angle φ=30º. For the interface strength factor, Rinter , it is assumed that the most important parameter is the friction angle (see eq. (4.2)). Generally, the friction angle of the soil-pile interface is taken as eq. (5.1):
(5.1)
Thus, through eq. (4.2), the correspondent interface strength factor is Rinter =0.63. - Layer 2: Soft clay This type of soil is assumed to be a recent alluvial deposit (a1) – grey and very soft to soft organic clay (0
(5.2)
In this case, the variation of the undrained strength with depth follows eq. (5.3):
(5.3)
The initial value of undrained strength (at the layer top), is considered to be c u=8kPa. Although the recommended value of Poisson’s ratio is ν=0 .49, the maximum value that may be introduced in the program is ν=0.35 for undrained conditions. According to (Brinkgreve, 2006), this is due to numerical
issues: for undrained simulations, Plaxis attributes to the pore fluid an implicit Poisson’s ratio of ν=0.495 (≈0.5) and if the effective Poisson’s ratio of the soi l were higher than 0.35, the stiffness of the
74
water compared to the soil’s would not be high enough. Therefore, the drained value of the Poisson’s
ratio is not taken into account into undrained simulations. As for the interface strength factor, Rinter =0.6 is applied. Generally, this factor is supposed to decrease with the soil’ resistance. In this case, since the rule used in drained situations may not be applied, a
value slightly below the used in the previous layer is chosen. - Layer 3: Medium sandy clay This type of soil is assumed to be a recent alluvial deposit (a2) – green and grey fine sand with high percentage of silt and clay (2
(5.4)
The initial value of undrained strength (at the layer top), is considered to be the same as the one in the bottom of the previous layer: cu=26kPa. (Viaponte, 2003) suggests a value for the Poisson’s ratio of ν=0 .44 for this soil. The interface strength value is the same as in the previous layer: Rinter =0.6. - Layer 4: Firm sandy clay This layer of soil is compared to the Pleistocene/Miocene (?) soil. It is described by the static load test report as green and stiff sandy clay. The Pleistocene/Miocene (?) is described as a silty clay with a high percentage of sand, and NSPT>10. (Viaponte, 2003) suggests for this soil γ=20kN/m3 and E=176.1MPa. The soil is modelled under undrained conditions. For the undrained strength, the value at the bottom of the previous layer, cu=44kPa, is used, for it is generally considered an appropriate value for stiff clay; this is supposed to be constant along depth. (Viaponte, 2003) suggests a value for the Poisson’s ratio of ν=0 .48 for this soil. The interface strength value is the same as in the previous layers: Rinter =0.6. - Layer 5: Gravel This layer is assumed to be part of the Miocene, which, according to (Viaponte, 2003), has the properties of γ=21.5kN/m3 and E=745.0MPa. On the other hand, (Viaponte, 2003) also suggests, for gravel in that region, the values of γ=20.5kN/m 3 and E=100.3MPa. It is decided that the specific weight of the gravel, γ=20.5kN/m3, is more appropriate. As for the Young’s modulus, the suggested value for the Miocene is clearly too high, and the value for the gravel
75
too low, since the previous layer’s Young’s modulus is E=176 .1MPa, and it is supposed to increase with depth. Thus, the compromise value of E=180MPa is used. The simulation is performed under drained conditions, and the attributed value of the Poisson’s ratio is ν=0.35. The friction angle is supposed to be very high, so the value o f φ’=45º is chosen. Using eqs.
(4.2) and (5.1), this leads to an interface strength factor of Rinter =0.577. This value is slightly lower than the one of the previous layer (Rinter =0.6), so it is considered appropriate. - Layer 6: Very stiff yellow clay This layer is clearly part of what is assumed to be the Miocene, which, according to (Viaponte, 2003), has the properties of γ=21.5kN/m3 and E=745.0MPa and is formed by calcareous marl and clay. The indication of the specific weight is respected, but not the Young’s modulus’, since it is excessively
high, especially given that the value of the previous layer is E=180MPa. The value of E=200MPa appears to be more appropriate. The indicated Poisson’s ratio is ν=0 .25.
The simulation is performed under undrained conditions. The undrained strength is assumed to be constant and equal to cu=200kPa, since that is a generally accepted value for very stiff clay. Since there is an increase in resistance, it seems adequate to slightly lower the value of the interface strength factor, and Rinter =0.5 is introduced in the model. - Layer 7: Dense yellow sand This layer also belongs to the Miocene; thus, a specific weight of γ=21.5kN/m3 is used. It is considered appropriate to use the same Young’s modulus as the previous layer, E=200MPa, and also the same Poisson’s ratio, ν=0 .25. This layer is modelled under drained conditions. For dense sand, a friction angle equal to φ’=40º is generally considered to be appropriate. If eqs. (4.2) and (5.1) were to be
used, a Rinter =0.599 would be introduced in the program; however, since the previous layer is attributed Rinter =0.5, this value is also used in this one, at least as a first iteration. - Layer 8: Hard yellow clay This layer has similar characteristics as layer 6, but, given its depth, is supposed to be more consistent. Most soil properties introduced in the program are the same: γ=21.5kN/m3 and E=200MPa. It also behaves under undrained conditions, but the attributed value of undrained strength is higher: cu=600kPa (for hard clay: cu>400kPa). The same value of interface strength factor as in the previous layers is introduced: Rinter =0.5. In Table 5.7, the soil properties of all layers introduced in the program are listed.
76
Table 5.7: Soil properties (2).
L
Material
γ
cu0
cu/z
φ'
(m)
Type
[kN/m3]
[kN/m2]
[kN/m2/m]
(º)
1
3.0
Drained
18.0
-
-
30
0.3
2
12.0
Undrained
16.0
8
1.5
-
3
9.0
Undrained
18.0
26
2.0
4
4.0
Undrained
20
44
5
2.5
Drained
20.5
6
6.0
Undrained
7
4.0
8
14.5
Layer
E*
ν'
Rinter
k0
25.0
0.630
0.500
0.35
59.0
0.600
0.550
-
0.35
77.7
0.600
1.000
0
-
0.35
176.1
0.600
1.000
-
-
45
0.35
180.0
0.577
0.293
21.5
200
0
-
0.25
200.0
0.500
1.000
Drained
21.5
-
-
40
0.25
200.0
0.500
0.357
Undrained
21.5
600
0
-
0.25
200.0
0.500
1.000
[MN/m2]
-4
* For γ=10 .
In order to verify the consistency of these parameters, the analytical shaft resistance is calculated and compared to the shaft resistance given by the SLT (see Table 5.3). All the following equations are found in (Santos, 2008b). The equation that expresses pile resistance to compression, Rc, is eq. (5.5):
Where:
(5.5)
Rb: base resistance of the pile Rs: shaft resistance of the pile In undrained conditions, the base resistance, Rb, is calculated through eq. (5.6):
(5.6)
Where: Ab: area of the pile base cu: undrained resistance at the pile base Nc: load capacity factor (Nc=9, according to (Santos, 2008a)) Table 5.8 shows the calculated base resistance, Rb, for this case. Table 5.8: Analytical base resistance, Rb.
Layer
L (m)
Material Type
Ab (m )
cu (kPa)
Nc
r b (kPa)
Rb (kN)
8
3.0
Undrained
0.503
600
9
5400.00
2714.34
77
In undrained conditions, the shaft resistance, Rc, is calculated through eq. (5.7):
(5.7)
Where: As: area of the pile shaft α: adhesion factor (α=Rinter )
: average undrained resistance along the pile shaft
In drained conditions, the shaft resistance, Rs, is calculated through eq. (5.8):
̅
(5.8)
Where: Ks: earth pressure coefficient
̅
: average effective vertical stress along the pile shaft
δ’: friction angle at the soil-pile interface
Table 5.9 shows the calculated shaft resistance, R s, for the different soil layers.
Table 5.9: Analytical shaft resistance, Rs.
k0
0.630
-
0.500
Undrained
0.600
17
9.0
Undrained
0.600
4
4.0
Undrained
5
2.5
6
Material
Layer
L (m)
1
3.0
Drained
2
12.0
3
Rinter
δ’ (º)
r s (kPa)
Rs (kN)
27.00
20.00
4.91
37.05
0.550
90.00
-
10.20
307.62
35
1.000
162.00
-
21.00
475.01
0.600
44
1.000
218.00
-
26.40
265.40
Drained
0.577
-
0.293
251.13
30.00
42.47
266.82
6.0
Undrained
0.500
200
1.000
298.75
-
100.00
1507.96
7
4.0
Drained
0.500
-
0.357
356.25
26.67
63.91
642.50
8
2.2
Undrained
0.500
600
1.000
462.63
-
300.00
1658.76
Type
(kPa)
(kPa)
Total These
tables
indicate
that
this
pile’s
5161.13
analytical
resistance
to
compression
is
Rc=2714.34+5161.13=7875.47kN, which seems an acceptable value, given the SLT’s results.
78
4) The Mesh The mesh is created from 15-node triangular elements, in a total of 9970 elements and 84399 nodes, with an average element size of 0.525m. The mesh is much more refined around the pile. To achieve this, three transition areas are drawn, so as to obtain a good connection between highly refined and coarse parts of the mesh. This mesh is considerably refined, as may be observed in Figure 5.14.
Figure 5.14: Mesh.
5.4.2 Initial Conditions 1) Water Pressure Generation The initial conditions of the model refer to both the water pressure and the stress field, before any calculation is performed. The pore pressure field due to the water table is generated in the first phase, using the specific weight 3
of the water, γ=10kN/m . The resulting pore pressure diagram, which is constant along the model, is
displayed in Figure 5.15.
79
Figure 5.15: Diagram of pore pressure.
2) Initial Stress Field Generation The initial stress field generation must be executed carefully in Plaxis, as discussed in the previous chapter. For the same reasons as in the previous chapter (the K0-procedure does not impose horizontal equilibrium), the pile is not included in this generation. In this case, however, there is another relevant issue, in the form of the volume of superficial soil removed for the execution of the pile. Since it is not advisable to use the normal initial stress generation procedure for non-horizontal surfaces, it is decided to ignore the removal of the soil in the initial stress generation. The values of the coefficient of earth pressure displayed in Table 5.7 lead to the diagram of effective vertical stresses in Figure 5.16 and the diagram of effective horizontal stresses in Figure 5.17.
Figure 5.16: Initial vertical effective stress field, including removed soil and not including the pile.
80
Figure 5.17: Initial horizontal effective stress field, including removed soil but not the pile.
These two figures show satisfactory stress fields, with increasing stress values along the depth, and constant along the radius. The values correspond to the ones calculated manually, with a negligible error in the area of the pile (since the mesh is much more refined there). Thus, this initial stress field is considered to be adequate and is used in the following calculations.
3) Removal of Soil at the Top The next step is to remove the volume of soil at the pile top. According to the profile published by (Santos, 2005), the ICIST report (see Figure 5.2), the excavation is horizontal at the bottom, where the pile cap is located, and its slopes have an unclear inclination value. For modelling purposes, it is important that this excavation does not cause a failure mechanism in the slope; thus, an inclination of 26.57º is used (10m×5m). The total displacements in the model after this step are displayed in Figure 5.18.
81
Figure 5.18: Total displacements after the removal of soil at the top.
It is clear that this inclination is sufficient to avoid a failure mechanism. The vertical displacements that results from this step are also acceptable. The soil removal also causes a slight change in the stress fields; however, there is no available data to compare with the stress field provided by Plaxis.
4) Introduction of the Pile
Unlike the case of the last chapter, which was an academic analysis that pursued a comparison with other simulations, this chapter aims to compare results provided by Plaxis with the ones from a static load test. In the last chapter, the introduction of the pile into the soil was simulated so as not to disturb the soil, i. e. not generating shear stress. In this case, although there is no direct way to determine the degree of disturbance caused by the introduction of the pile into the soil, it seems appropriate to consider it as a non-displacement pile. This is achieved by simply replacing the volume of soil for pile material, using the interface strength factors listed in Table 5.7. The consequence is that the pile introduction generated shear stress along the shaft. As a simplification, the use of the temporary casing is not modelled, although it is probable that its effect is not entirely negligible.
82
5.4.3 Calculations and Results The calculation is performed through steps, each one representing one of increments of the static load test. Since the loading is applied through prescribed displacements, the inversed method is used: the settlements recorded from the test are introduced in the program, and the consequent load along the pile is the output. Since the goal is to compare the results with the ones recorded from the static load test, the “reset displacements” option is activated before the first load is applied, for the measurements refer to
variations on settlement and stress that result from loading. The results provided by Plaxis are presented in comparison to the ones found in the static load test report. Figure 5.19 shows the load at the pile top, Pt, in terms of the total settlement at the pile top, w t, resulting from the Plaxis simulation.
8 000
7 000
6 000
5 000
) 4 000 N k ( t P
3 000
2 000
1 000
0 0
10
20
30
40
50
60
70
wt (mm)
Figure 5.19: Load at the pile top in terms of the total settlement at the pile top, given by Plaxis.
This curve indicates that the overall behaviour of the model is quite acceptable, for it is similar to the results from the static load test, shown in Figure 5.9.
83
As mentioned in chapters 2 and 3, elastic theory-based methods are useful tools for analysing the settlement and load distribution in early stages of loading, when the soil is still in plastic state. Thus, the two described methods are applied to this case. Table 5.10 and Table 5.11 show the load settlement ratio and the proportion of load transferred to the pile base, as well as the relevant parameters for the calculations. Table 5.7 provides the rest of the necessary information. Table 5.10: Poulos and Davis estimation.
Eav (kN/m2)
I0
Rk
Rh
Rν
β0
Ck
Cν
118892
0.045
1.9
0.8
0.98
53.21
0.035
0.45
0.85
0.0134
. Table 5.11: Randolph and Wroth (compressible piles) estimation.
ζ
μL
3.09
4.27
50.12
0.0116
The two methods provide similar solutions, which is a good indicator. The Poulos and Davis method gives slightly higher values, as occurred in the results displayed in chapters 2 and 3. Figure 5.20 shows the load at the pile top, Pt, in terms of the total settlement at the pile top, w t, resulting from the static load test and from the calculation performed by Plaxis for the first loading (load steps 1 to 8). The solution by Randolph and Wroth is also plotted. The points which constitute this chart are listed in Appendix D1.
84
3 000
2 000 ) N k ( t P
SLT Plaxis
1 000
R&W
0 0
1
2
3
4
5
wt (mm)
Figure 5.20: Load at the pile top in terms of the total settlement at the pile top, given by the SLT, Plaxis and the Randolph and Wroth method, for the first loading cycle.
The curve of the first loading produces satisfactory results, which traduces in a good behaviour of the model. In this load step, only the first 4 layers of soil are in plastic state, the 4 in the bottom still being in the elastic state. This is expected, since resistance increases with depth. Thus, the parameters attributed to the soil are considered to be acceptable. Results provided by the Randolph and Wroth method show that the total settlement is underestimated, when compared with the ones from the SLT and Plaxis. Figure 5.21 shows the load at the pile top, Pt, in terms of the total settlement at the pile top, w t, resulting from the static load test and from the calculation performed by Plaxis for the reloading (load steps 8 to 25). Figure 5.22 shows the load at the pile top, Pt, in terms of the total settlement at the pile top, w t, resulting from the static load test and from the calculation performed by Plaxis for both loading cycles.
85
8 000
7 000
6 000
5 000 SLT
) 4 000 N k ( t P
Plaxis
3 000
2 000
1 000
0 0
10
20
30
40
50
60
70
wt (mm)
Figure 5.21: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and Plaxis, for the second loading cycle. 8 000
7 000
6 000
5 000 SLT
)4 000 N k ( t P
Plaxis
3 000
2 000
1 000
0 0
10
20
30
40
50
60
70
wt (mm)
Figure 5.22: Load at the pile top in terms of the total settlement at the pile top, given by the SLT and Plaxis, for both loading cycles.
86
For the last load step, the simulation by Plaxis shows that all the soil layers are in the plastic state. The similarity of shapes between the curves accounts for the good behaviour of the model. Figure 5.23 shows the normal stress along the shaft for both loading peaks (step 4 and 19). The points which originate this chart are listed in Appendix D2.
0
2000
N (kN) 4000
6000
8000
0 5 10 15 ) m ( 20 z
25
SLT - Load Step 4 Plaxis - Load Step 4 SLT - Load Step 19 Plaxis - Load Step 19
30 35 40 45
Figure 5.23: Normal stress along the pile shaft at the load peaks (steps 4 and 19), given by the SLT and Plaxis.
This chart shows that the shaft load is appropriately distributed, for the slope between levels is close to the one given by the SLT. It is interesting to notice that the results are better for load step 19. This accounts for the resistance properties of the soil. These values refer to the load increment; i.e. normal stress at the initial state (after the pile installation) was subtracted to the normal stress at the steps 4 and 19, otherwise a comparison to the vibrating wire extensometers would not make sense. The same procedure was applied to the shaft load. Nevertheless, the values of initial stress are almost negligible when compared to the ones caused by loading. Figure 5.24 shows the mobilized shear stress along the pile for different levels of depth and Figure 5.25 shows the shaft load in every soil layer, as well as the corresponding applied load at the pile top, resulting from the Plaxis calculations.
87
) a P k ( τ
200
8000
150
6000 R24
R45
R56
R68
Pt
100
4000
50
2000
0
P t ( k N )
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Load Step
Figure 5.24: Shaft load between different levels along the pile and total applied load, obtained by Plaxis.
400
8000
350
7000
300 250 ) a P k ( τ
Layer 1
Layer 2
Layer 3
Layer 4
Layer 5
Layer 6
Layer 7
Layer 8
6000 5000
Pt
200
4000
150
3000
100
2000
50
1000
0
P t ( k N )
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Load Step
Figure 5.25 Shaft load by layer of soil and total applied load, obtained by Plaxis.
88
Table 5.12 and Table 5.13 show a comparison between mobilized shear load between levels, measured during the SLT, calculated analytically and determined by Plaxis, for load step 4 and 19. Table 5.12: Shaft load for load step 4. Load Step 4
Mobilized shaft resistance (kPa)
Between levels
SLT
Plaxis
2 and 4
15 to 25
31
4 and 5
100 to 110
32
5 and 6
15 to 25
28
6 and 8
20 to 30
27
Table 5.13: Shaft load for load step 19. Load Step 19
Mobilized Shaft Resistance (kPa)
Between levels
SLT
Analytic
Plaxis
2 and 4
60 to 70
48
51
4 and 5
180 to 200
100
105
5 and 6
100 to 110
98
99
6 and 8
≥ 130
76
70
For load step 4, the mobilized resistance given by Plaxis is slightly superior to the measurements of the vibrating wire extensometers, except between levels 4 and 5. For load step 19, comparison with Figure 5.12 shows that the resistance of the model between levels 2 and 4 and also between levels 5 and 6 is slightly lower than the measured in the SLT. As far as the resistance between levels 4 and 5 is concerned, it is much lower than the measured. Thus, it is possible to conclude that the temporary casing affected the area between levels 4 and 5 by increasing shear resistance, but its influence did not reach the region between levels 5 and 6. Shear resistance between levels 6 and 8 obtained by Plaxis are lower than the ones obtained in the SLT. Figure 5.25 shows the distribution of shaft load by the different soil layers; it accounts for the good behaviour of the model. The layers closer to the surface have lower resistance; the last layer (hard clay) does not reach the maximum resistance. It is interesting that, in the unloading phase, the top layers are unloaded before the bottom layer – layer 8 loses little load between load steps 19 and 24. Figure 5.26 shows the total load, base load and shaft load in terms of the settlement at the pile top, determined by Plaxis, for the second loading cycle. It also shows the measured values for load step 19 of the SLT, since that was the only step of the second loading cycle for which there were results available. The total load and the base load in terms of the settlement at the pile top provided by the Randolph and Wroth method for compressible piles is also displayed. These results concern the displacement caused by the loading alone, disregarding the weight of the pile.
89
8 000 7 000 Pt Plaxis
6 000
Ps Plaxis Pb Plaxis
5 000
Pt SLT
) N k ( 4 000 t P
Pt SLT load step 19 Ps SLT load step 19
3 000 Ps SLT load step 19 Pt R&W
2 000
Pb R&W
1 000 0 0
10
20
30
40
50
60
wt (mm)
Figure 5.26: Total load, load transferred to the pile shaft and load transferred to the pile base in terms of the settlement at the pile top for the s econd loading cycle, obtained through Plaxis.
These curves given by Plaxis are a good indicator, shaft failure occurring for load values between 5,000kN and 6,000kN (the analytic calculation pointed to 5161.13kN). As far as the analytical methods are concerned, they underestimate the total settlement, as happened in the calculations performed in chapter 3. The difference in this case, however, is more evident. That is expected, since the conditions in this case study make it much more complex. One of the most important factors is the relative movement between soil and pile; the analytical methods consider the interface strength factor to be equal to 1 and the values introduced in the program were between 0.5 and 0.6. This causes the shear stress along the pile to develop differently: the shaft generates less load, and therefore the total load curve is formed by lower values. As far as the base load is concerned, the analytical result is more satisfactory, since the slope of the curve is similar to the slope of the Plaxis curve – their values differing due to the fact that the Randolph and Wroth method does not take into account the weight of the pile. All the points that constitute the Plaxis curves in Figure 5.26 are shown in Appendix D4. In this chapter, the calculations performed by Plaxis were quite satisfactory; the static load test was accurately simulated. Given the little available information about the site geologic conditions, the parameters introduced into the model provided good results. It is relevant to state that the goal of this work was not to manipulate parameters indiscriminately in order to obtain the same results as the SLT’s; it was to introduce in the program acceptable values within the commonly used ranges and
taking into account the geological report recommendations, calibrating the model as possible. It is possible to conclude that the effect of the casing was not negligible, and the fact that it was not considered in the model affected the quality of the results. 90
6 CONCLUDING REMARKS
6.1 Conclusions The purpose of this thesis was to understand the load transfer mechanism in single piles under axial loading, as well as to determine the load settlement relation. The first step was to understand the background of elastic analyses, by studying elastic theory-based methods. This was complemented by an elastic numerical analysis using rigid interfaces. The second step was to study the soil behaviour when it was no longer linear elastic. In order to perform inelastic analyses, it was essential to use numerical methods, since it was no longer possible to resort to the elastic theory-based methods. Both the soil and the soil-pile interface were assumed to have elastoplastic behaviour, according to the Mohr-Coulomb criterion or to the Tresca criterion. As far as the first part of this thesis is concerned, in which chapters 2 and 3 are included, the resulting dimensionless charts of load settlement ratio and proportion of load transferred to the pile base in terms of the slenderness ratio and of the soil inhomogeneity factor constitutes a quick way to estimate the total settlement and the base load of a given single pile subject to vertical loading. Thus, it is possible to conclude that:
The load settlement ratio increases with the pile slenderness ratio whereas the proportion of load transferred to the pile base decreases with it.
The soil-pile stiffness factor has more influence over the load settlement ratio than over the proportion of load transferred to the pile base, according to the Randolph and Wroth method. It also influences the limit of applicability between rigid and compressible pile solutions;
91
The Randolph and Wroth method for compressible piles gave solutions which are very close to results provided by Plaxis, indicating its adequacy for the studied cases (homogeneous and vertically non-homogeneous soil with linear stiffness increase with depth). It provides an expeditious method for estimating settlement and load distribution along the pile. However, it is not applicable to very short piles;
The Poulos and Davis method does underestimate total settlement, as affirmed by the authors and confirmed by Plaxis results. This is probably related to the fact that the pile compressibility is not thoroughly taken into account, especially for high values of pile slenderness;
Results given by Plaxis are more sensitive to the soil inhomogeneity than the elastic theorybased methods, as far as load transferred to the pile base is concerned. This may be due to oversimplification of the analytical methods (especially in the Poulos and Davis method), but also to difficulties in the modelling of a high rigidity gradient in the soil;
Chapter 4 concerned the comparison of a simulation by Plaxis with two other finite element programs. The following conclusions were drawn:
The sensitivity analysis of the interface strength factor, Rinter , showed that it is the parameter that affected the total load and the shaft load the most; however, the influence over the base load was not very significant.
Although the total load-settlement curves provided by the different programs correspond well, the discrepancy regarding the proportion of load transferred to the pile base is not negligible. In the simulation performed by Plaxis, the tip was not as mobilized, even if a smooth interface was considered;
This indicates that the algorithm used in Plaxis might have significant differences to the other ones’;
When dilatancy in the soil at the pile base was introduced, the base was indeed a lot more mobilized.
The simulation performed in chapter 5, however, lead to satisfactory results. It is possible to conclude that:
The goal of this work was not to manipulate parameters indiscriminately in order to obtain the same results as the SLT’s; it was to introduce in the program acceptable values within the
commonly used ranges and taking into account the geological report recommendations, calibrating the model as possible. Given the little available information of the site’s geology , the number of layers (and consequently of soil parameters) and the considerable discrepancies between the geological profile of the static load test report and the ones given by the closest boreholes, as well as the differences between the applied and measured load and the normal errors associated with in-situ tests, the results provided by Plaxis correspond very well with the ones from the instrumentation used in the SLT;
Not all the parameters indicated in the geological report were used, since some of them did not seem appropriate; the ones which were used, however, lead to good results. As for the
92
parameters for which there was no available information, values of the generally used ranges were successfully applied;
The normal stress distribution along the pile calculated by Plaxis corresponded well with the SLT results, for the two load steps with available information. As for the shaft load, the distribution by layer provided by the model was consistent with the expectations regarding the increase of soil resistance with depth. The last layer of soil, hard clay, did not reach its maximum resistance; besides, during the unloading phase, the other less resistant layers unloaded first.
It is possible to conclude that the effect of the casing was not negligible, and the fact that it was not considered in the model affected the quality of the results. It is likely that the casing has caused an enlargement of the borehole, increasing stresses in the area;
When applied to this case study, the elastic theory-based methods do not provide satisfactory load-settlement results. This is due to the number of simplifications assumed by the methods, such as admitting an average value for the Young’s modulus of the soil. The most relevant
difference is perhaps the interface strength factor, which is equal to 1 in the analytical methods, whereas the values introduced in the model are between 0.5 and 0.6; therefore, Plaxis provides lower values of generated shear stress, and thus lower total load.
6.2 Further Research The described elastic theory-based methods are applicable to piles with non-circular cross sections. Since 2D axisymmetry was used, it was not possible to validate numerical methods for this case. In the future, it would be interesting to use 3D modelling to study the behaviour of piles with non-circular cross sections. There are non-linear load-transfer methods, which can be applied and compared to the elastic theorybased methods. Although only floating piles were analysed in this work, the Randolph and Wroth method may also be applied cases of end-bearing piles, i.e. where the Young’s modulus of the soil increases abruptly at the pile base. It would be interesting to study the results of numerical analyses in those conditions. Besides, it would be interesting to draw charts of the load settlement ratio and the proportion of load transferred to the pile base in terms of the pile slenderness ratio for different values of the soil-pile stiffness factor obtained by Plaxis, since the simulations made in chapter 3 all refer to the same value. The calibration of numerical methods through static load tests is an interesting and useful work. In this thesis, the soil was modelled using the Mohr-Coulomb criterion and the Tresca criterion. It would be appropriate to apply other material models, such as the hardening soil, to this and other cases, and analyse the differences. The pile-soil interface could also be modelled using another failure criterion.
93
94
BIBLIOGRAPHY
References Brinkgreve, R. B. (2006). Plaxis - 2D Version 8 Manual. The Netherlands: Delft University of Technology and Plaxis b. v. D'aguiar, S. (2008). Numerical Modelling of Soil-Pile Axial Load Transfer Mechanisms in Granular Soils. PhD Thesis. École Centrale des Arts et Manufactures – École Centrale Paris; Universidade Técnica de Lisboa – Instituto Superior Técnico. Fleming, K. W. (1992). Piling Engineering. 3rd Edition. Glasgow: Black Academic and Professional. Mindlin, R. D. (1936). Force at a Point in the Interior of a Semi-Infinite Soil. In Physics, vol. 7. Neves, M. M. (2001a). Étude du Comportement de Pieux Forés. I. Expérimentations in situ et en Laboratoire. Bulletin des Laboratoires des Ponts et Chaussées, 231, pp. 39-54. Neves, M. M. (2001b). Étude du comportement de pieux fores. II. modelisation par elements finis. Bulletin des Laboratoires des Ponts et Chaussées, 231, pp. 55-67. Poulos, H. G. (1972). Load-Settlement Prediction for Piles and Piers. JSMFD ASCE vol. 98 SM9, pp. 879-897. Poulos, H. G., & Davis, E. (1980). Pile Foundation Analysis and Design. New York: W iley and Sons. Poulos, H. G., & Davis, E. H. (1968). The Settlement Behaviour of Single Axially-Loaded Incompressible Piles and Piers. Geotechnique, vol. 18 , pp. 351-371. Randolph, M., & Wroth, C. (1978). Analysis of Deformation of Vertically Loaded Piles. JGED, ASCE 104(GT12), pp. 1465-1488. Santos, J. A. (2005). Relatório do Ensaio de Carga numa Estaca Experimental no Âmbito do Projecto de Modernização da Linha do Norte, Viaduto de Santana do Cartaxo. Instituto de Engenharia de Estruturas, Território e Construção (ICIST), Instituto Superior Técnicno. Santos, J. A. (2008). Fundações por Estacas. Acções Verticais. . Elementos Teóricos da disciplina de Obras Geotécnicas. Instituto Superior Técnico. Santos, J. A. (2008). Fundações por Estacas. Formulário. Elementos Teóricos da disciplina de Obras Geotécnicas. Instituto Superior Técnico. Viaponte. (2003). Memória Descritiva e Justificativa – Geologia e Geotecnia, Doc nº PE 27.5 MDg 2. Vol. 10 – Viaduto de Santana do Cartaxo, Projecto de Modernização da Linha do Norte – Sub-troço 1.4: Azambuja (excl.) – Vale de Santarém (inc.).
95
Consulted Bibliography Bowles, J. E. (1996). Foundation Analysis and Design, 5th Edition. McGraw-Hill. Das, B. M. (2006). Principles of Geotechnical Engineering 7th Edition. Cengage Learning. Fellenius, B. H. (2011). Basics of Foundation Design. Electronic Edition. Frank, R. (1975). Étude Théorique du Comportement des Pieux sous Charge Verticale. Introduction de la Dilatance. Laboratoire Central des Ponts et Chaussées. Pereira, T. (2008). Estacas Sob Acções Verticais, Considerando os Efeitos da Carga Residual. MsC Thesis. Universidade Técnica de Lisboa - Instituto Superior Técnico. Tadjbakhsh, S. a. (1985). Étude par la Méthode des Éléments Finis du Comportement Élastoplastique de Sols Dilatants. Application aux Pieux sous Charge Verticale. Laboratoire Central des Ponts et Chaussées. Timoshenko, S. P. (1936). Theory of Elastic Stability. New York: McGraw-Hill. Velloso, D. A. (2002). Fundações. Vol. 2 - Fundações Profundas. COPPE-UFRK. Oficina de Textos.
96
APPENDIXES
Appendix A – Dimension of the Model Used in the Elastic Simulations
L/r 0
h (m)
l (m)
36
90.0
72.0
69
172.5
138.0
4
10.0
8.0
37
92.5
74.0
70
175.0
140.0
5
12.5
10.0
38
95.0
76.0
71
177.5
142.0
6
15.0
12.0
39
97.5
78.0
72
180.0
144.0
7
17.5
14.0
40
100.0
80.0
73
182.5
146.0
8
20.0
16.0
41
102.5
82.0
74
185.0
148.0
9
22.5
18.0
42
105.0
84.0
75
187.5
150.0
10
25.0
20.0
43
107.5
86.0
76
190.0
152.0
11
27.5
22.0
44
110.0
88.0
77
192.5
154.0
12
30.0
24.0
45
112.5
90.0
78
195.0
156.0
13
32.5
26.0
46
115.0
92.0
79
197.5
158.0
14
35.0
28.0
47
117.5
94.0
80
200.0
160.0
15
37.5
30.0
48
120.0
96.0
81
202.5
162.0
16
40.0
32.0
49
122.5
98.0
82
205.0
164.0
17
42.5
34.0
50
125.0
100.0
83
207.5
166.0
18
45.0
36.0
51
127.5
102.0
84
210.0
168.0
19
47.5
38.0
52
130.0
104.0
85
212.5
170.0
20
50.0
40.0
53
132.5
106.0
86
215.0
172.0
21
52.5
42.0
54
135.0
108.0
87
217.5
174.0
22
55.0
44.0
55
137.5
110.0
88
220.0
176.0
23
57.5
46.0
56
140.0
112.0
89
222.5
178.0
24
60.0
48.0
57
142.5
114.0
90
225.0
180.0
25
62.5
50.0
58
145.0
116.0
91
227.5
182.0
26
65.0
52.0
59
147.5
118.0
92
230.0
184.0
27
67.5
54.0
60
150.0
120.0
93
232.5
186.0
28
70.0
56.0
61
152.5
122.0
94
235.0
188.0
29
72.5
58.0
62
155.0
124.0
95
237.5
190.0
30
75.0
60.0
63
157.5
126.0
96
240.0
192.0
31
77.5
62.0
64
160.0
128.0
97
242.5
194.0
32
80.0
64.0
65
162.5
130.0
98
245.0
196.0
33
82.5
66.0
66
165.0
132.0
99
247.5
198.0
34
85.0
68.0
67
167.5
134.0
100
250.0
200.0
35
87.5
70.0
68
170.0
136.0
97
Appendix B1 – Load Settlement Ratio in Terms of the Pile Slenderness Ratio
ρ=1, λ=975, ν=0.3, h=2.5L
Load Settlement Ratio, Pt/(wtr 0GL)
Pile Slenderness
Randolph and
Ratio,
Poulos and Davis,
Randolph and Wroth,
L/r 0
P&D
R&W (rigid piles)
4
24.96
18.63
18.42
5
26.08
20.20
19.90
6
27.45
21.75
21.34
7
29.03
23.27
22.74
8
30.71
24.76
24.10
9
32.42
26.23
25.41
10
34.06
27.67
26.67
11
35.54
29.08
27.89
12
36.86
30.48
29.08
13
37.98
31.86
30.22
14
38.91
33.21
31.33
15
39.63
34.56
32.41
16
40.19
35.88
33.44
17
40.61
37.20
34.45
18
40.94
38.50
35.42
19
41.25
39.78
36.37
20
41.72
41.06
37.28
21
42.15
42.32
38.16
22
42.70
43.58
39.01
23
43.35
44.82
39.84
24
44.09
46.06
40.64
25
44.92
47.29
41.41
26
45.82
48.51
42.15
27
46.78
49.72
42.87
28
47.80
50.92
43.57
29
48.83
52.12
44.24
30
50.14
53.30
44.89
31
51.25
54.49
45.51
32
52.37
55.66
46.12
33
53.47
56.83
46.70
34
54.56
58.00
47.26
35
55.64
59.16
47.80
Wroth, R&W
Plaxis
(compressible piles) 19.30 22.11 24.86 27.35 29.74
32.93 34.87
37.48 39.05
41.83
44.18
46.62
98
36
56.71
60.31
48.32
37
57.75
61.46
48.82
38
58.78
62.60
49.30
39
59.79
63.74
49.77
40
60.53
64.87
50.21
41
61.41
66.00
50.64
42
62.25
67.12
51.06
43
63.04
68.24
51.46
44
63.79
69.36
51.84
45
64.50
70.47
52.20
46
65.17
71.58
52.56
47
65.80
72.68
52.89
48
66.39
73.78
53.22
49
66.95
74.88
53.53
50
67.62
75.97
53.83
51
68.02
77.06
54.11
52
68.40
78.15
54.39
53
68.77
79.23
54.65
54
69.13
80.31
54.90
55
69.51
81.38
55.14
56
69.89
82.46
55.37
57
70.29
83.53
55.59
58
70.72
84.59
55.80
59
71.18
85.66
56.00
60
71.26
86.72
56.19
61
71.68
87.78
56.37
62
72.12
88.83
56.55
63
72.58
89.89
56.71
64
73.04
90.94
56.87
65
73.50
91.99
57.02
66
73.94
93.03
57.16
67
74.37
94.07
57.30
68
74.75
95.11
57.43
69
75.09
96.15
57.55
70
75.63
97.19
57.67
71
75.93
98.22
57.78
72
76.18
99.26
57.89
73
76.39
100.28
57.98
74
76.56
101.31
58.08
75
76.71
102.34
58.17
76
76.84
103.36
58.25
77
76.97
104.38
58.33
78
77.10
105.40
58.41
48.63
51.28
52.90
53.77
99
79
77.25
106.42
58.48
80
77.57
107.43
58.54
81
77.81
108.45
58.60
82
78.08
109.46
58.66
83
78.37
110.47
58.72
84
78.67
111.47
58.77
85
78.96
112.48
58.81
86
79.24
113.48
58.86
87
79.49
114.49
58.90
88
79.70
115.49
58.94
89
79.85
116.49
58.97
90
80.60
117.48
59.00
91
80.80
118.48
59.03
92
80.96
119.47
59.06
93
81.05
120.47
59.08
94
81.05
121.46
59.11
95
81.05
122.45
59.13
96
81.05
123.43
59.14
97
81.05
124.42
59.16
98
81.15
125.40
59.17
99
81.11
126.39
59.18
100
81.19
127.37
59.19
54.17
54.42
54.06
100
ρ=0.75, λ=975, ν=0.3, h=2.5L
Load Settlement Ratio, Pt/(wtr 0GL)
Pile Slenderness
Randolph and
Randolph and Wroth,
Wroth, R&W
R&W
(rigid piles)
(compressible piles)
18.72
18.85
18.61
5
19.62
19.92
19.60
6
20.73
21.08
20.65
7
21.99
22.25
21.70
8
23.34
23.43
22.74
9
24.72
24.60
23.76
10
26.02
25.76
24.74
11
27.17
26.90
25.70
12
28.18
28.03
26.63
13
29.06
29.15
27.53
14
29.77
30.26
28.40
15
30.34
31.35
29.24
16
30.77
32.44
30.05
17
31.09
33.51
30.83
18
31.35
34.57
31.59
19
31.59
35.63
32.32
20
31.95
36.67
33.03
21
32.32
37.71
33.71
22
32.77
38.73
34.36
23
33.31
39.75
35.00
24
33.91
40.76
35.61
25
34.58
41.77
36.19
26
35.30
42.76
36.76
27
36.08
43.76
37.30
28
36.88
44.74
37.83
29
37.71
45.72
38.33
30
38.74
46.69
38.82
31
39.63
47.66
39.28
32
40.51
48.62
39.73
33
41.38
49.58
40.16
34
42.24
50.53
40.57
35
43.10
51.48
40.97
36
43.93
52.42
41.35
37
44.75
53.36
41.71
38
45.56
54.29
42.06
39
46.35
55.22
42.39
40
46.93
56.15
42.71
41
47.61
57.07
43.02
Ratio,
Poulos and Davis,
L/r 0
P&D
4
Plaxis 18.96 20.95 23.01 24.92 26.76
29.22 30.72
32.73 33.94
36.07
37.85
39.68
41.17
101
42
48.26
57.98
43.31
43
48.87
58.90
43.59
44
49.45
59.81
43.86
45
49.99
60.72
44.12
46
50.50
61.62
44.36
47
50.98
62.52
44.60
48
51.43
63.42
44.82
49
51.85
64.31
45.03
50
52.34
65.20
45.24
51
52.74
66.09
45.43
52
53.12
66.97
45.61
53
53.49
67.86
45.79
54
53.86
68.74
45.96
55
54.23
69.61
46.12
56
54.60
70.49
46.27
57
54.99
71.36
46.41
58
55.39
72.23
46.54
59
55.82
73.09
46.67
60
55.95
73.96
46.80
61
56.34
74.82
46.91
62
56.75
75.68
47.02
63
57.16
76.54
47.12
64
57.58
77.39
47.22
65
57.99
78.24
47.31
66
58.39
79.09
47.40
67
58.77
79.94
47.48
68
59.12
80.79
47.56
69
59.43
81.63
47.63
70
59.89
82.48
47.70
71
60.16
83.32
47.76
72
60.39
84.16
47.82
73
60.59
84.99
47.87
74
60.75
85.83
47.93
75
60.89
86.66
47.97
76
61.02
87.49
48.02
77
61.14
88.32
48.06
78
61.26
89.15
48.10
79
61.39
89.97
48.13
80
61.66
90.80
48.16
81
61.86
91.62
48.19
82
62.08
92.44
48.22
83
62.31
93.26
48.24
84
62.55
94.08
48.26
43.04
44.09
44.53
44.61
102
85
62.79
94.90
48.28
86
63.01
95.71
48.30
87
63.20
96.52
48.31
88
63.36
97.34
48.33
89
63.47
98.15
48.34
90
64.05
98.96
48.35
91
64.10
99.76
48.35
92
64.10
100.57
48.36
93
64.10
101.37
48.36
94
64.10
102.18
48.36
95
64.10
102.98
48.36
96
64.23
103.78
48.36
97
64.23
104.58
48.36
98
64.23
105.38
48.36
99
64.23
106.17
48.35
100
64.28
106.97
48.35
44.55
44.03
103
ρ=0.5, λ=975, ν=0.3, h=2.5L
Load Settlement Ratio, Pt/(wtr 0GL)
Pile Slenderness
Randolph and
Randolph and Wroth,
Wroth, R&W
R&W
(rigid piles)
(compressible piles)
12.48
17.92
17.66
5
13.08
18.25
17.92
6
13.82
18.85
18.43
7
14.66
19.55
19.02
8
15.56
20.30
19.66
9
16.48
21.08
20.30
10
17.37
21.86
20.94
11
18.17
22.64
21.56
12
18.88
23.43
22.18
13
19.50
24.21
22.77
14
20.01
24.98
23.35
15
20.42
25.76
23.91
16
20.74
26.52
24.45
17
21.00
27.28
24.97
18
21.20
28.03
25.48
19
21.40
28.78
25.96
20
21.67
29.52
26.43
21
21.95
30.26
26.88
22
22.29
30.99
27.32
23
22.68
31.72
27.73
24
23.13
32.44
28.13
25
23.61
33.15
28.52
26
24.14
33.87
28.89
27
24.70
34.57
29.24
28
25.28
35.28
29.58
29
25.88
35.98
29.90
30
26.62
36.67
30.22
31
27.26
37.36
30.51
32
27.90
38.05
30.80
33
28.53
38.73
31.07
34
29.16
39.41
31.33
35
29.78
40.09
31.58
36
30.39
40.76
31.82
37
31.00
41.43
32.05
38
31.59
42.10
32.27
39
32.17
42.76
32.47
40
32.60
43.43
32.67
41
33.11
44.08
32.86
Ratio,
Poulos and Davis,
L/r 0
P&D
4
Plaxis 17.49 18.58 19.85 21.12 22.32
23.97 24.97
26.32 27.12
28.52
29.67
30.81
31.69
104
42
33.60
44.74
33.04
43
34.06
45.39
33.21
44
34.50
46.04
33.37
45
34.91
46.69
33.52
46
35.30
47.34
33.67
47
35.67
47.98
33.81
48
36.01
48.62
33.94
49
36.34
49.26
34.06
50
36.73
49.90
34.18
51
37.04
50.53
34.29
52
37.34
51.16
34.40
53
37.64
51.79
34.50
54
37.93
52.42
34.59
55
38.23
53.04
34.68
56
38.53
53.67
34.76
57
38.84
54.29
34.84
58
39.16
54.91
34.91
59
39.50
55.53
34.98
60
39.62
56.15
35.05
61
39.94
56.76
35.11
62
40.26
57.37
35.17
63
40.60
57.98
35.22
64
40.93
58.59
35.27
65
41.26
59.20
35.31
66
41.59
59.81
35.36
67
41.90
60.41
35.40
68
42.19
61.02
35.43
69
42.45
61.62
35.47
70
42.82
62.22
35.50
71
43.06
62.82
35.52
72
43.26
63.42
35.55
73
43.45
64.01
35.57
74
43.61
64.61
35.59
75
43.75
65.20
35.61
76
43.89
65.79
35.63
77
44.02
66.39
35.64
78
44.15
66.97
35.66
79
44.29
67.56
35.67
80
44.53
68.15
35.68
81
44.73
68.74
35.68
82
44.93
69.32
35.69
83
45.15
69.90
35.69
84
45.38
70.49
35.70
32.69
33.07
33.01
32.68
105
85
45.60
71.07
35.70
86
45.81
71.65
35.70
87
46.00
72.23
35.70
88
46.17
72.81
35.70
89
46.43
73.38
35.69
90
46.78
73.96
35.69
91
46.95
74.53
35.68
92
47.08
75.11
35.68
93
47.18
75.68
35.67
94
47.25
76.25
35.66
95
47.29
76.82
35.65
96
47.30
77.39
35.64
97
47.38
77.96
35.63
98
47.45
78.53
35.62
99
47.51
79.09
35.61
100
47.54
79.66
35.60
32.24
31.52
106
Appendix B2 – Proportion of Load Transferred to the Pile Base in Terms of the Pile Slenderness Ratio
ρ=1, ν=0.3, h=2.5L
Proportion of Load taken by the Pile Base, P b/Pt
Pile Slenderness
Randolph and
Randolph and Wroth,
Wroth, R&W
R&W
(rigid piles)
(compressible piles)
0.265
0.307
0.305
5
0.229
0.283
0.281
6
0.202
0.263
0.261
7
0.181
0.246
0.243
8
0.164
0.231
0.228
9
0.151
0.218
0.215
10
0.139
0.207
0.203
11
0.130
0.196
0.193
12
0.121
0.187
0.183
13
0.114
0.179
0.175
14
0.108
0.172
0.167
15
0.102
0.165
0.160
16
0.097
0.159
0.154
17
0.093
0.154
0.148
18
0.089
0.148
0.142
19
0.085
0.144
0.137
20
0.082
0.139
0.133
21
0.079
0.135
0.128
22
0.076
0.131
0.124
23
0.073
0.127
0.120
24
0.071
0.124
0.116
25
0.068
0.121
0.113
26
0.066
0.118
0.109
27
0.064
0.115
0.106
28
0.062
0.112
0.103
29
0.060
0.110
0.100
30
0.059
0.107
0.098
31
0.057
0.105
0.095
32
0.056
0.103
0.093
33
0.054
0.101
0.090
34
0.053
0.099
0.088
35
0.051
0.097
0.086
36
0.050
0.095
0.084
Ratio,
Poulos and Davis,
L/r 0
P&D
4
Plaxis 0.257 0.203 0.167 0.148 0.128
0.110 0.101
0.089 0.083
0.073
0.065
0.056
107
37
0.049
0.093
0.082
38
0.048
0.091
0.080
39
0.047
0.090
0.078
40
0.046
0.088
0.076
41
0.044
0.087
0.074
42
0.043
0.085
0.073
43
0.043
0.084
0.071
44
0.042
0.082
0.069
45
0.041
0.081
0.068
46
0.040
0.080
0.066
47
0.039
0.079
0.065
48
0.038
0.077
0.064
49
0.037
0.076
0.062
50
0.036
0.075
0.061
51
0.036
0.074
0.060
52
0.035
0.073
0.058
53
0.034
0.072
0.057
54
0.034
0.071
0.056
55
0.033
0.070
0.055
56
0.032
0.069
0.054
57
0.032
0.068
0.053
58
0.031
0.068
0.052
59
0.030
0.067
0.051
60
0.030
0.066
0.050
61
0.029
0.065
0.049
62
0.029
0.064
0.048
63
0.028
0.064
0.047
64
0.028
0.063
0.046
65
0.027
0.062
0.045
66
0.027
0.061
0.044
67
0.026
0.061
0.043
68
0.026
0.060
0.043
69
0.025
0.059
0.042
70
0.025
0.059
0.041
71
0.024
0.058
0.040
72
0.024
0.058
0.039
73
0.023
0.057
0.039
74
0.023
0.056
0.038
75
0.022
0.056
0.037
76
0.022
0.055
0.037
77
0.022
0.055
0.036
78
0.021
0.054
0.035
79
0.021
0.054
0.035
0.050
0.040
0.033
0.028
108
80
0.020
0.053
0.034
81
0.020
0.053
0.033
82
0.020
0.052
0.033
83
0.019
0.052
0.032
84
0.019
0.051
0.032
85
0.019
0.051
0.031
86
0.018
0.050
0.030
87
0.018
0.050
0.030
88
0.018
0.049
0.029
89
0.017
0.049
0.029
90
0.017
0.049
0.028
91
0.017
0.048
0.028
92
0.017
0.048
0.027
93
0.016
0.047
0.027
94
0.016
0.047
0.026
95
0.016
0.047
0.026
96
0.015
0.046
0.025
97
0.015
0.046
0.025
98
0.015
0.046
0.025
99
0.015
0.045
0.024
100
0.014
0.045
0.024
0.023
0.020
0.017
109
ρ=0.75, λ=975, ν=0.3, h=2.5L
Proportion of Load taken by the Pile Base, P b/Pt
Pile Slenderness
Randolph and
Randolph and Wroth,
Wroth, R&W
R&W
(rigid piles)
(compressible piles)
0.266
0.303
0.301
5
0.230
0.287
0.285
6
0.202
0.271
0.268
7
0.181
0.257
0.254
8
0.164
0.244
0.240
9
0.151
0.232
0.228
10
0.139
0.222
0.217
11
0.130
0.212
0.207
12
0.122
0.204
0.198
13
0.115
0.196
0.190
14
0.108
0.189
0.183
15
0.103
0.182
0.175
16
0.098
0.176
0.169
17
0.094
0.171
0.163
18
0.090
0.165
0.157
19
0.086
0.160
0.152
20
0.083
0.156
0.147
21
0.080
0.152
0.142
22
0.077
0.148
0.138
23
0.074
0.144
0.134
24
0.072
0.140
0.130
25
0.070
0.137
0.126
26
0.068
0.134
0.122
27
0.065
0.131
0.119
28
0.064
0.128
0.116
29
0.062
0.125
0.113
30
0.060
0.122
0.110
31
0.059
0.120
0.107
32
0.057
0.118
0.104
33
0.056
0.115
0.102
34
0.054
0.113
0.099
35
0.053
0.111
0.097
36
0.052
0.109
0.094
37
0.051
0.107
0.092
38
0.049
0.105
0.090
39
0.048
0.103
0.088
40
0.047
0.102
0.086
Ratio,
Poulos and Davis,
L/r 0
P&D
4
Plaxis 0.314 0.251 0.208 0.185 0.161
0.139 0.128
0.114 0.106
0.094
0.084
0.073
0.066
110
41
0.046
0.100
0.084
42
0.045
0.099
0.082
43
0.044
0.097
0.080
44
0.043
0.096
0.078
45
0.043
0.094
0.077
46
0.042
0.093
0.075
47
0.041
0.091
0.073
48
0.040
0.090
0.072
49
0.039
0.089
0.070
50
0.039
0.088
0.069
51
0.038
0.086
0.067
52
0.037
0.085
0.066
53
0.036
0.084
0.065
54
0.036
0.083
0.063
55
0.035
0.082
0.062
56
0.035
0.081
0.061
57
0.034
0.080
0.060
58
0.033
0.079
0.058
59
0.033
0.078
0.057
60
0.032
0.077
0.056
61
0.032
0.076
0.055
62
0.031
0.076
0.054
63
0.030
0.075
0.053
64
0.030
0.074
0.052
65
0.029
0.073
0.051
66
0.029
0.072
0.050
67
0.028
0.071
0.049
68
0.028
0.071
0.048
69
0.027
0.070
0.047
70
0.027
0.069
0.046
71
0.027
0.069
0.045
72
0.026
0.068
0.044
73
0.026
0.067
0.043
74
0.025
0.067
0.043
75
0.025
0.066
0.042
76
0.024
0.065
0.041
77
0.024
0.065
0.040
78
0.024
0.064
0.039
79
0.023
0.064
0.039
80
0.023
0.063
0.038
81
0.023
0.062
0.037
82
0.022
0.062
0.037
83
0.022
0.061
0.036
0.054
0.046
0.039
0.033
111
84
0.021
0.061
0.035
85
0.021
0.060
0.035
86
0.021
0.060
0.034
87
0.020
0.059
0.033
88
0.020
0.059
0.033
89
0.020
0.058
0.032
90
0.020
0.058
0.032
91
0.019
0.057
0.031
92
0.019
0.057
0.030
93
0.019
0.056
0.030
94
0.018
0.056
0.029
95
0.018
0.055
0.029
96
0.018
0.055
0.028
97
0.018
0.055
0.028
98
0.017
0.054
0.027
99
0.017
0.054
0.027
100
0.017
0.053
0.026
0.029
0.025
112
ρ=0.5, λ=975, ν=0.3, h=2.5L
Proportion of Load taken by the Pile Base, P b/Pt
Pile Slenderness
Randolph and
Randolph and Wroth,
Wroth, R&W
R&W
(rigid piles)
(compressible piles)
0.267
0.319
0.316
5
0.231
0.313
0.310
6
0.204
0.303
0.299
7
0.183
0.292
0.287
8
0.166
0.281
0.276
9
0.153
0.271
0.265
10
0.141
0.261
0.255
11
0.132
0.252
0.245
12
0.124
0.244
0.236
13
0.117
0.236
0.227
14
0.110
0.229
0.219
15
0.105
0.222
0.212
16
0.100
0.215
0.205
17
0.096
0.209
0.198
18
0.092
0.204
0.192
19
0.088
0.199
0.186
20
0.085
0.194
0.180
21
0.082
0.189
0.175
22
0.079
0.184
0.170
23
0.076
0.180
0.165
24
0.074
0.176
0.161
25
0.072
0.172
0.156
26
0.070
0.169
0.152
27
0.068
0.165
0.148
28
0.066
0.162
0.144
29
0.064
0.159
0.141
30
0.063
0.156
0.137
31
0.061
0.153
0.134
32
0.060
0.150
0.130
33
0.058
0.148
0.127
34
0.057
0.145
0.124
35
0.056
0.143
0.121
36
0.055
0.140
0.118
37
0.054
0.138
0.116
38
0.053
0.136
0.113
39
0.051
0.134
0.110
40
0.050
0.132
0.108
Ratio,
Poulos and Davis,
L/r 0
P&D
4
Plaxis 0.391 0.320 0.270 0.242 0.213
0.186 0.173
0.156 0.146
0.131
0.118
0.104
0.096
113
41
0.050
0.130
0.105
42
0.049
0.128
0.103
43
0.048
0.126
0.101
44
0.047
0.124
0.099
45
0.046
0.122
0.096
46
0.045
0.121
0.094
47
0.044
0.119
0.092
48
0.044
0.118
0.090
49
0.043
0.116
0.089
50
0.042
0.115
0.087
51
0.041
0.113
0.085
52
0.041
0.112
0.083
53
0.040
0.110
0.081
54
0.039
0.109
0.080
55
0.039
0.108
0.078
56
0.038
0.106
0.076
57
0.037
0.105
0.075
58
0.037
0.104
0.073
59
0.036
0.103
0.072
60
0.036
0.102
0.070
61
0.035
0.101
0.069
62
0.034
0.100
0.068
63
0.034
0.099
0.066
64
0.033
0.098
0.065
65
0.033
0.097
0.064
66
0.032
0.096
0.062
67
0.032
0.095
0.061
68
0.031
0.094
0.060
69
0.031
0.093
0.059
70
0.030
0.092
0.058
71
0.030
0.091
0.057
72
0.029
0.090
0.056
73
0.029
0.089
0.054
74
0.028
0.088
0.053
75
0.028
0.088
0.052
76
0.028
0.087
0.051
77
0.027
0.086
0.050
78
0.027
0.085
0.049
79
0.026
0.085
0.048
80
0.026
0.084
0.048
81
0.026
0.083
0.047
82
0.025
0.082
0.046
83
0.025
0.082
0.045
0.080
0.069
0.060
0.052
114
84
0.025
0.081
0.044
85
0.024
0.080
0.043
86
0.024
0.080
0.042
87
0.024
0.079
0.042
88
0.023
0.078
0.041
89
0.023
0.078
0.040
90
0.023
0.077
0.039
91
0.022
0.077
0.039
92
0.022
0.076
0.038
93
0.022
0.076
0.037
94
0.022
0.075
0.036
95
0.021
0.074
0.036
96
0.021
0.074
0.035
97
0.021
0.073
0.034
98
0.021
0.073
0.034
99
0.020
0.072
0.033
100
0.020
0.072
0.033
0.047
0.041
115
Appendix C – Load Settlement Curves Determined in the Elastoplastic Modelling for Numerical Validation
Rinter =0.01
Rinter =1.00
s (mm)
Pb (kN)
Ps (kN)
Pt (kN)
Pb (kN)
Ps (kN)
Pt (kN)
0.0
0.00
0.00
0.00
0.00
0.00
0.00
0.0
29.23
0.42
29.66
29.23
0.42
29.66
0.1
30.07
0.49
30.56
29.54
7.79
37.33
0.2
30.88
0.54
31.42
29.84
15.16
45.00
0.3
31.68
0.60
32.28
30.13
22.53
52.67
0.4
32.44
0.65
33.09
30.42
29.91
60.33
0.5
33.18
0.71
33.89
30.70
37.29
68.00
0.6
33.90
0.76
34.66
30.99
44.67
75.66
0.7
34.61
0.81
35.42
31.27
52.06
83.33
0.8
35.31
0.86
36.17
31.55
59.44
90.99
0.9
35.97
0.92
36.89
31.82
66.83
98.65
1.0
36.62
0.98
37.60
32.09
74.22
106.31
1.5
39.58
1.26
40.84
33.40
111.19
144.59
2.0
42.21
1.55
43.76
34.68
148.14
182.81
2.5
44.58
1.80
46.38
35.90
185.03
220.94
3.0
46.71
2.05
48.76
37.09
221.83
258.92
3.5
48.67
2.30
50.97
38.24
258.49
296.72
4.0
50.45
2.54
52.98
39.36
294.95
334.31
4.5
52.10
2.73
54.83
40.46
331.11
371.57
5.0
53.68
2.86
56.54
41.54
366.79
408.33
5.5
55.45
2.90
58.35
42.58
400.88
443.46
6.0
56.79
2.92
59.71
43.62
432.22
475.84
6.1
57.11
2.92
60.02
43.81
438.13
481.94
6.2
57.43
2.91
60.34
44.02
443.30
487.32
6.3
57.67
2.90
60.58
44.26
447.28
491.54
6.4
57.99
2.89
60.88
44.49
451.27
495.76
6.5
58.27
2.89
61.15
44.71
455.19
499.90
6.6
58.54
2.88
61.42
44.93
458.99
503.92
6.7
58.80
2.87
61.68
45.16
462.63
507.79
6.8
59.07
2.87
61.94
45.39
466.14
511.53
6.9
59.33
2.86
62.19
45.61
469.50
515.11
7.0
59.59
2.86
62.45
45.83
472.61
518.44
7.1
59.91
2.85
62.75
46.05
475.50
521.55
7.2
60.12
2.84
62.97
46.28
478.06
524.34
7.5
60.91
2.82
63.73
47.00
480.14
527.14
8.0
62.20
2.79
64.98
48.08
490.54
538.62
116
9.0
64.44
2.75
67.19
50.25
498.45
548.71
10.0
66.80
2.70
69.50
52.29
505.38
557.67
12.0
71.01
2.63
73.64
56.08
521.24
577.32
14.0
74.97
2.56
77.53
59.50
535.56
595.05
16.0
78.70
2.49
81.19
62.67
549.35
612.02
18.0
82.18
2.45
84.62
65.63
561.87
627.51
20.0
85.46
2.40
87.86
68.36
573.37
641.72
22.0
88.59
2.35
90.94
70.90
584.00
654.90
24.0
91.61
2.30
93.91
73.26
594.27
667.53
26.0
94.48
2.25
96.73
75.54
603.58
679.12
28.0
97.26
2.20
99.47
77.67
612.51
690.18
30.0
99.98
2.16
102.14
79.65
621.22
700.86
117
Rinter =0.870 or Rinter =0.857 Ψ=0º
Ψ=23º
s (mm)
Pb (kN)
Ps (kN)
Pt (kN)
Pb (kN)
Ps (kN)
Pt (kN)
0.0
0.00
0.00
0.00
0.00
0.00
0.00
0.0
29.23
0.42
29.66
29.23
0.42
29.66
0.1
29.55
7.76
37.30
29.55
7.75
37.31
0.2
29.85
15.10
44.95
29.87
15.08
44.95
0.3
30.15
22.45
52.60
30.18
22.42
52.60
0.4
30.44
29.80
60.24
30.48
29.76
60.25
0.5
30.73
37.15
67.88
30.79
37.10
67.89
0.6
31.02
44.50
75.52
31.09
44.44
75.54
0.7
31.31
51.86
83.17
31.39
51.79
83.18
0.8
31.59
59.21
90.81
31.69
59.13
90.82
0.9
31.87
66.57
98.44
31.98
66.48
98.46
1.0
32.15
73.93
106.08
32.28
73.83
106.11
1.5
33.48
110.76
144.25
33.72
110.57
144.29
2.0
34.78
147.57
182.35
35.16
147.28
182.44
2.5
36.06
184.27
220.34
36.63
183.86
220.49
3.0
37.41
220.72
258.13
38.22
220.17
258.39
3.5
38.78
256.84
295.62
39.88
256.14
296.03
4.0
40.18
292.45
332.64
41.62
291.63
333.25
4.5
41.63
326.68
368.31
43.45
325.77
369.23
5.0
43.11
357.70
400.82
45.41
356.77
402.18
5.5
44.82
378.61
423.43
47.72
377.99
425.71
6.0
46.60
390.54
437.14
50.22
390.00
440.22
6.1
46.96
391.45
438.41
50.75
391.06
441.81
6.2
47.34
391.14
438.48
51.29
390.99
442.28
6.3
47.71
391.22
438.93
51.83
390.89
442.71
6.4
48.07
391.18
439.26
52.36
390.87
443.23
6.5
48.43
391.23
439.66
52.88
390.87
443.74
6.6
48.78
391.20
439.98
53.39
390.86
444.25
6.7
49.13
391.24
440.37
53.90
390.85
444.75
6.8
49.48
391.20
440.68
54.41
390.84
445.25
6.9
49.82
391.25
441.06
54.91
390.83
445.74
7.0
50.15
391.21
441.37
55.41
390.81
446.23
7.1
50.49
391.26
441.74
55.91
390.80
446.71
7.2
50.82
391.22
442.04
56.40
390.79
447.19
7.5
51.79
391.37
443.15
57.86
390.90
448.76
8.0
53.36
389.68
443.04
60.28
389.01
449.30
9.0
56.24
390.87
447.11
64.79
389.71
454.50
10.0
58.93
391.14
450.07
69.08
389.67
458.76
12.0
63.82
392.08
455.90
77.09
390.75
467.84
118
14.0
68.24
392.07
460.31
84.56
389.96
474.53
16.0
72.30
392.08
464.38
91.60
389.45
481.05
18.0
76.09
392.22
468.31
98.29
389.14
487.43
20.0
79.68
392.29
471.97
104.69
388.94
493.63
22.0
83.07
392.25
475.32
110.85
389.05
499.90
24.0
86.27
392.20
478.47
116.82
388.72
505.53
26.0
89.29
392.28
481.57
122.60
388.15
510.75
28.0
92.18
392.24
484.42
128.23
387.88
516.11
30.0
94.96
392.19
487.15
133.72
387.45
521.17
119
Appendix D1 – Load Settlement Curve Determined in the Elastoplastic Modelling for Comparison with the Static Load Test
s (mm)
P (kN)
0.00
0.15
0.60
426.49
1.55
1049.62
2.65
1668.84
4.25
2474.16
3.80
2153.25
3.25
1762.21
2.35
1131.30
0.80
155.19
1.50
653.12
2.50
1359.82
3.00
1696.23
4.00
2323.66
5.00
2810.61
7.50
3759.55
9.50
4426.71
12.50
5096.28
18.00
6270.01
27.50
6992.60
63.00
7785.68
62.00
7035.90
58.50
4788.97
56.00
3470.83
53.00
2021.50
50.00
812.07
120
Appendix D2 – Normal Stress along the Pile in the Elastoplastic Modelling for Comparison with the Static Load Test
Load Step 4 Level
z (m)
N (kN)
0
2.1
2474.01
2
25.7
1305.82
4
31.7
841.81
5
32.7
765.20
6
36.7
489.26
8
40.7
234.33
Load Step 19 Level
z (m)
N (kN)
0
2.1
7826.85
2
25.7
6434.54
4
31.7
5675.60
5
32.7
5420.72
6
36.7
4424.93
8
40.7
3751.34
121
Appendix D3 – Shaft
Load
Curves
Determined
in
the
Elastoplastic Modelling for Comparison with the Static Load Test
Load
s (mm)
P (kN)
R24
R45
R56
R68
0
0.00
0.15
1.53
1.44
1.35
1.40
1
0.60
426.49
6.08
5.36
4.83
4.85
2
1.55
1049.62
13.34
11.59
10.35
10.32
3
2.65
1668.84
22.31
19.17
17.05
16.88
4
4.25
2474.16
32.46
32.94
28.86
28.09
5
3.80
2153.25
29.00
30.03
26.27
25.52
6
3.25
1762.21
24.83
26.44
23.08
22.36
7
2.35
1131.30
18.01
20.57
17.87
17.19
8
0.80
155.19
7.92
10.35
8.82
8.26
9
1.50
653.12
11.39
14.92
12.87
12.28
10
2.50
1359.82
18.97
21.45
18.67
18.02
11
3.00
1696.23
22.77
24.71
21.56
20.89
12
4.00
2323.66
30.46
31.30
27.40
26.66
13
5.00
2810.61
34.53
41.26
35.66
34.30
14
7.50
3759.55
43.49
75.78
63.91
49.36
15
9.50
4426.71
48.37
103.76
92.11
51.20
16
12.50
5096.28
48.27
103.23
96.86
56.78
17
18.00
6270.01
48.75
103.23
96.94
66.94
18
27.50
6992.60
49.17
103.36
97.12
68.21
19
63.00
7785.68
51.87
105.75
99.56
70.77
20
62.00
7035.90
46.89
103.08
97.08
67.49
21
58.50
4788.97
25.47
81.10
78.10
50.76
22
56.00
3470.83
24.08
62.22
61.48
35.05
23
53.00
2021.50
18.26
34.69
38.80
19.03
24
50.00
812.07
15.02
3.31
10.74
17.65
Step
122
