FE-Analysis of piled and piled raft foundations. Jean-Sébastien LEBEAU

Transcription

1 FE-Analysis of piled and piled raft foundations Jean-Sébastien LEBEAU April - August 2008

2 Abstract In the last few years the number of piled raft foundations especially those with few piles, has increased. Unlike the conventional piled foundation design in which the piles are designed to carry the majority of the load, the design of a piled raft foundation allows the load to be shared between the raft and piles and it is necessary to take the complex soil-struture interaction eects into account. The aim of this paper is to describe a nite element analysis of deep foundations: piled and mainly piled raft foundations. A basic parametric study is rstly presented to determine the inuence of mesh discretisation, of materials - loose or dense sand -, of dilatancy and interface elements. Then the behavior of piled raft foundations is analysed in more details using partial axisymmetric models of one pile-raft. We continue by preparing a more sophisticated 3D study to take into account the complex pilepile interaction which occured when the pile spacing is small. So the possibilies of employing the embedded pile concept as implemented into Plaxis 3D foundations is investigated. Finally, some clues about the group eect are indicated. Key words: Piled raft foundation, piles, embedded pile, volume pile, hardening soil model 1

3 Acknowledgements First of all I would like to express my gratefulness to Professor Helmut F. Schweiger for giving me the opportunity to work on geotechnical issues at the Institute for Soil Mechanics and Foundation Engineering of Graz University of Technology. This paper was made possible by the great contribution of my supervisor Dipl.-Ing Franz Tschuchnigg. I am indebted to him for his friendly supervision and guidance throughout the period of my traineeship. I deeply thank him because he conveyed me a better understanding of nite element modeling and analyses. I also would like to thank my French professor, Yvon Riou for getting me in touch with the Institute. Finally, I would like to express my appreciation to all the people I met here who made my ve months stay in Austria very enjoyable. 2

4 Contents 1 Introduction 6 2 Preliminary studies Single pile Presentation of calculations Geometry Boundaries conditions Material properties Meshes Load control and calculation steps Results Mesh dependency Comparison between distributed loads and prescribed displacement Inuence of the interface coecient R inter Inuence of the dilatancy Pile-raft Presentation of calculations Geometry Boundaries conditions Materials properties Meshes Load control and calculation steps Results

5 CONTENTS CONTENTS Mesh dependency Inuence of the interface coecient R inter Inuence of the dilatancy Analysis of 2D models Single-pile Pile-Raft Load-displacement curve Variations of Skin friction and Normal Stresses along the pile Analysis of the α Kpp factor Denition of α Kpp Methodology to calculate α Kpp Comparison and evolution of α Kpp for dierent geometries: Evolution of α Kpp for dierent materials and dilatancy Evolution of α Kpp for dierent values of R inter Eciency of a piled-raft foundation in comparison with a raft foundation Analysis of the pile behavior Base resistance Skin resistance Conclusions Preliminary studies of 3D models Volume pile Finite element models Results Load-displacement curves Variations of skin friction Some remarks about parameters Embedded pile Embedded pile-raft Finite element models

6 CONTENTS CONTENTS Embedded pile with linear skin friction distribution Embedded pile with multilinear skin friction distribution Embedded pile with layer dependent skin friction distribution Comparison of the three options: Linear, multilinear and layer dependent Group eect Presentation of calculations Geometry Finite element model Results Vocabulary details Load-displacement curves Displacement proles More precise analysis of group Conclusion Conclusion 98 5

7 Chapter 1 Introduction In traditional foundation design, it is customary to consider rst the use of shallow foundation such as a raft (possibly after some ground-improvement methodology performed). If it is not adequate, deep foundation such as a fully piled foundation is used instead. In the last few decade, an alternative solution has been designed: piled raft foundation. Unlike the conventional piled foundation design in which the piles are designed to carry the majority of the load, the design of a piled raft foundation allows the load to be shared between the raft and piles and it is necessary to take the complex soil-struture interaction eects into account. The concept of piled raft foundation was rstly proposed by Davis and Poulos in 1972 and is now used extensively in Europe, particularly for supporting the load of high buildings or towers. The favorable application of piled raft occurs when the raft has adequate loading capacities, but the settlement or dierential settlement exceed allowable values. In this case, the primary purpose of the pile is to act as settlement reducer. The aim of this paper is to describe a nite element analysis of deep foundations: piled and mainly piled raft foundations. A basic parametric study is rstly presented to determine the inuence of mesh discretisation, of materials - loose or dense sand -, of dilatancy and interface elements. Then the behavior of piled raft foundations is analysed in more details using partial axisymmetric models of one pile-raft. We continue by preparing a more sophisticated 3D study to take into account the complex pilepile interaction which occured when the pile spacing is small. So the possibilies of employing the embedded pile concept as implemented into Plaxis 3D foundations is investigated. Finally, some clues about the group eect are indicated. 6

8 Chapter 2 Preliminary studies - 2D axisymmetric models - In order to prepare a more sophisticated analysis a large number of calculations have been performed in axisymmetric conditions. This approach oered the possibility to study with reasonable calculation times the inuence of mesh discretisation, dilatancy and interface elements for a single pile and a pile-raft. The dierent models and conclusions are presented in this part. 2.1 Single pile Presentation of calculations Geometry In order to analyze the behavior of the single pile, a model has been made in PLAXIS V8 using an axisymmetric model. A working area of 20 m width and 40 m depth has been used. At the axis of symmetry the pile has been modeled with a length of 15 m and a diameter of 0,8 m. The soil is modeled as a single layer of sand with properties are described in ). The ground water is located at 40 m below the soil surface. In this way we did not take into account the water inuence. Along the length of the pile an interface has been modeled. We extended this interface to 0,5 m below the pile inside the soil body to prevent stress oscillation in this sti corner area. 1 We added two clusters close to the pile to enrich easily the mesh in this more moving area. 1 This longer interface will enhance the exibility of the nite element mesh in this area and will thus prevent non-physical stress results. However, these elements should not introduce an unrealistic weakness in the soil according to PLAXIS V8 manual. 7

9 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Boundaries conditions We used the standard xities PLAXIS tool to dene the boundaries conditions. Thus these boundaries conditions are generated according to the following rules: ˆ Vertical geometry lines for which the x-coordinate is equal to the lowest or highest x-coordinate in the model obtain a horizontal xity (u x = 0). ˆ Horizontal geometry lines for which the y-coordinate is equal to the lowest y-coordinate in the model obtain a full xity (u x = u y = 0). Figure 2.1: Global geometry of the axisymmetric model of the single pile Material properties The constitutive model used for the soil - sand - is the Hardening soil model. The main advantage of this constitutive law is its ability to consider the stress path and its eect on the soil stiness and its behavior. We used two dierent types of sand: one loose and the other dense. We also varied the dilatancy value. 8

10 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES For the concrete pile, a linear elastic material set was applied. The parameters of all this materials are summarized in the following table: Parameter Symbol Loose sand Dense sand Concrete (pile) Unit Material model Model Hardening Soil Hardening Soil Linear Elastic - Unsaturated weigth γ unsat kn/m 3 Saturated weigth γ sat kn/m 3 Permeability k m/day E ref kn/m 3 Stiness Eur ref 1E5 1,8E5 kn/m 3 Power m 0,65 0,55 Poisson ratio ν ur 0,2 0,2 0,2 - Dilatancy y 2/0 8/0 Friction angle f Cohesion c ref 0,1 0,1 kn/m 2 Lateral pressure coe. K 0 1-sinf 1-sinf - Failure ratio Rf 0,9 0,9 - E ref oed E7 kn/m 3 Table 2.1: Materials parameters Meshes To study the mesh dependency 3 analyses were performed: one with a coarse, one with a medium and one with a very ne mesh. For each one we considered 6 models varying the interface elements. Thus we played around the R inter coecient 2 from 0,1 to 1. 2 This factor relates the interface strength (wall friction and adhesion) to the soil strength (friction angle and cohesion) 9

11 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Figure 2.2: A very ne mesh for calculations with interface elements Coarse Medium Very ne Number of elements Number of nodes Elements 15-node Table 2.2: Information on the generated meshes Load control and calculation steps To assign a load at the top of the pile we considered two approaches: one with prescribed displacement, one with distributed loads. With prescribed displacement we impose a certain displacement at the top of the pile whereas with distributed loads we impose a force; results should be the same Results Remark: All the following curves are plotted for the node point located at the top right side of the pile. 10

12 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Figure 2.3: Node point selected for load-displacement curves Mesh dependency By analysing all the calculations made, we can conclude that for each material - loose or dense sand - the curves have the same shapes for calculations performed with coarse, medium and very ne mesh. Nevertheless, we can observe that with ner meshes, we have unphysical premature soil body collapsing. The following gure illustrates this conclusion with some examples. 11

13 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Figure 2.4: Mesh dependency for the loose sand - ψ=2 - and dierent values for R inter les : Geo2Load_Mesh 1/2/3_Rinter0,1/0,7/1_Psi2_HS.plx To avoid this premature failure we decided to restart the medium and very ne calculations switching o the arc length control procedure. But we now observed convergence problems with more or less important oscillations. These oscillations occurred for important displacements (from 20 cm) whatever the material, mesh or R inter value. However, the global shape of the load-displacement curve seems to stay realistic even if there are these stairs. 12

14 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Figure 2.5: Mesh dependency for the loose sand - ψ=2 - and dierent values for R inter les : Geo2Load_Mesh 2_Loose_Rinter0,1/0,7_Psi2_HS(_alc=OFF).plx Figure 2.6: Inuence of arc length control for the loose sand - ψ=2 - and R inter =0,7/0,1 parameters: mesh1= coarse, arc length=on; mesh2= medium, arc length=off; mesh3= Very ne, arc length=off; 13

15 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES The gure 2.6 enables us to conrm that the mesh dependency is negligible for this model Comparison between distributed loads and prescribed displacement The previous paragraph was based on les with the load approach. We did the same calculations with the displacement approach. By comparing these two approaches we can conclude that the shape of the load-settlement curves is exactly the same in each case. Moreover, there are less oscillations with prescribed displacement than with distributed loads. There are no stairs'' even with an important displacement. So because it limits this problem of big oscillations, prescribed displacement seems to be better to study a single pile. The following picture illustrates these conclusions. Figure 2.7: Distributed loads and prescribed displacement, comparison for the loose sand - ψ=0 - and R inter =0,1/0,4/0,7 les : Geo2Disp/Load_Mesh 2_Rinter0,1/0,4/0,7_Psi0_HS(_alc=OFF).plx 14

16 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Remark: The gure 2.7 shows us the settlement with the load [kn] whereas in the previous section we plotted the settlement with the distributed load [kn/m 2 ]. To compare the two approaches we have to: ˆ ˆ Distributed load: Multiply the distributed load [kn/m 2 ] by the area of the pile to get the total force (R tot ) [kn]. Prescribed displacement: Read out the force value in Plaxis output [kn/rad] and multiply it by 2 π to get the total force (R tot ) [kn] Now we can try to interpret the stairs of the load approach by comparing with the same calculations done with the prescribed displacements. Figure 2.8: Comparison Displ and Load approaches for the loose sand ψ=0 medium mesh - No interface les : Geo2Disp/Load_Mesh 2_Loose_RinterNo_Psi=0_HS(_alc=OFF).plx 15

17 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES Figure 2.9: Comparison Displ and Load approaches for the loose sand ψ=0 - R=1 medium mesh les : Geo2Disp/Load_Loose_Mesh 2_Rinter1_Psi=0_HS(_alc=OFF).plx As we can see on gures 2.8 and 2.9, it is impossible to deduce correctly the normal shape from the distributed load curves by interpreting the oscillations. Sometimes, the prescribed displacement curve is under the distributed load one, sometimes it is in the middle. So we have to interpret with caution the shape of the stairs part of the distributed load curves Inuence of the interface coecient R inter We varied the way to model the pile to sand interface by changing the R inter value and doing a model without interface. We also performed one calculation by drawing an interface in Plaxis input and unselected it in Plaxis calculation. We can conclude that the choice of the value for R inter is not negligible when you model a single pile. As we can see in the table, for the same load, the settlements increase by more than 40 % between R=0,4 and 0,1, 80% between R=0,7 and 0,4 and 600 % between R=1 and 0,7 for loose sand. Load=2000 kn Settlement [cm] R inter =0,1 22,5 R inter =0,4 15,7 R inter =0,7 8,6 R inter =1 1,2 Table 2.3: Settlements of the single pile for loose sand, ψ=2 and R tot =2000 kn 16

18 2.1. SINGLE PILE CHAPTER 2. PRELIMINARY STUDIES As we could expect the load-displacement curves have almost the same shape for models with R=1 and without interface. We also noticed that unselecting the interface lead to false results with premature failure (see red following curve) or an unrealistic behavior. So the interface drawn in the Plaxis input must be selected in the calculations steps. The following curves sum up all these conclusions. Figure 2.10: Load-settlement curves for Loose sand, ψ=2 and coarse mesh les : Geo2Load_Mesh1_Loose_Rinter0,1/0,4/0,7/1/Unselected/No_Psi=2_HS.plx Inuence of the dilatancy We tested two values of dilatancy ψ for both materials: (ϕ 30) and 0. As expected, we have less displacement with a high ψ value than without dilatancy. 17

19 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Figure 2.11: Inuence of dilatancy for dense sand, mesh medium les : Geo2Load_Mesh2_Dense_Rinter0,4/0,7_Psi8/0_HS_ALCo.plx 2.2 Pile-raft Presentation of calculations Geometry We performed the same calculations as we have done with the single pile model using an axisymmetric model of a pile-raft foundation. As we did for the single pile, the pile has been modeled with a length of 15 m and a diameter of 0,8 m at the axis of symmetry. We added a slab in concrete with a thickness of 0,5 m. The soil is also modeled as a single layer of sand with the same properties as the single pile. The ground water is located at 40 m below the soil surface. In this way we did not take into account the water inuence. Along the length of the pile an interface has been modeled. We extended this interface to 0,5 m below the pile inside the soil body to prevent stress oscillation in this sti corner area. 18

20 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Boundaries conditions We also used for this study the standard xities PLAXIS tool (see ) Materials properties The parameters of all the materials are recalled in the following table: Parameter Symbol Loose sand Dense sand Concrete Unit Material model Model Hardening Soil Hardening Soil Linear Elastic - Unsaturated weigth γ unsat kn/m 3 Saturated weigth γ sat kn/m 3 Permeability k m/day E ref kn/m 3 Stiness Eur ref 1E5 1,8E5 kn/m 3 Power m 0,65 0,55 Poisson ratio ν ur 0,2 0,2 0,2 - Dilatancy y 2/0 8/0 Friction angle f Cohesion c ref 0,1 0,1 kn/m 2 Lateral pressure coe. K 0 1-sinf 1-sinf - Failure ratio Rf 0,9 0,9 - E ref oed E7 kn/m 3 Table 2.4: Materials parameters Meshes To study the mesh dependency 3 analysis were also performed: one with a coarse, one with a medium and one with a very ne mesh. For each one we considered 6 models varying the interface elements. Thus we varied the R inter coecient from 0,1 to 1. We also performed one batch of calculation with 6-nodes instead of 15 nodes. Coarse Medium Very ne Coarse Number of elements Number of nodes Elements 15-node 6-node Table 2.5: Information on the generated meshes 19

21 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Load control and calculation steps To assign a load at the top of the slab we considered in this case only a distributed load Figure 2.12: Details about a pile-raft geometry with the axisymmetric model, Very ne mesh Results Remark: All the following curves are plotted for the node point A, situated at the top right side of the pile, under the slab (see gure 2.12) Mesh dependency By analysing all the calculations made, we can conclude that for each material - loose or dense sand - the curves have exactly the same shapes for calculations performed with coarse, medium and very ne mesh. 20

22 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Figure 2.13: Example, Mesh dependency for the pile raft model with loose sand, ψ=2, R inter =0,7 les : Geo1_Mesh1/2/3_loose_Rinter0,7_Psi2_HS.plx Inuence of the interface coecient R inter We varied the way to model the pile to sand interface by changing the R inter value and doing a model without interface. As we can see in the table and on the following curve the way you model the interface has a negligible inuence on the settlements. Sand Mesh ψ R inter =0,4 R inter =0,7 R inter =1 Loose Coarse 2-34,3 cm -33,0 cm -32,4 cm Dense Coarse 8-13,6 cm -13,2 cm -13,0 cm Table 2.6: Settlements with dierent values of R inter for load=1000 kn/m 2 21

23 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Table 2.7: Load-settlement curves for Loose sand, ψ=2 and coarse mesh les : Geo1_Mesh1_loose_Rinter0,1/0,4/0,7/1/NO_Psi2_HS.plx Inuence of the dilatancy We tested two values of dilatancy (ψ) for both materials: 2 and 0 for the loose sand, 8 and 0 for the dense sand. We can conclude that the inuence of the dilatancy is negligible for this model even for the dense sand. Sand Mesh ψ R inter =0,4 R inter =0,7 R inter =1 Loose Coarse 2-34,3 cm -33,0 cm -32,4 cm Loose Coarse 0-34,4 cm -33,0 cm -32,4 cm Dense Coarse 8-13,6-13,2-13,0 Dense Coarse 0-13,7-13,2-13,1 Table 2.8: Settlements with dierent values of R inter for load=1000 kn/m 2 22

24 2.2. PILE-RAFT CHAPTER 2. PRELIMINARY STUDIES Figure 2.14: Inuence of dilatancy for dense sand, mesh coarse les : Geo1_Mesh1_dense_Rinter0,4/0,7_Psi8/0_HS.plx 23

25 Chapter 3 Analysis of 2D models - Behavior of a pile and a pile-raft - In chapter 2 we made conclusions about how to dene eciently and correctly an axisymmetric model of a single pile and a pile-raft. Now we present other calculations performed by taking these preliminary practical conclusions into account. In design of piled rafts, design engineers have to understand the mechanism of load transfer from the raft to the piles and to the soil. It requires to take complex interactions into account such as: pile-soil interaction, raft-soil interaction, pile-raft interaction and pile-pile interaction. The aim of this chapter is to have a better understanding of the pile and raft behavior and to check the ability of the software to model such complex interactions. In this part, we only modeled a single pile with a raft so we did not take into account the pile-pile interaction. 3.1 Single-pile In the previous calculations we simulated an axial load test on a bored pile. We get the following load-displacement curve: 24

26 3.1. SINGLE-PILE CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.1: Axial load curve for a single-pile le : Geo2Disp_Mesh2_loose_R=0,7_Psi2_HS.plx Now we observe the mobilisation of the skin friction (q s ) with dierent loads. Figure 3.2: Evolution of the Skin friction with the load (Rtot) le : Geo2Disp_Mesh2_loose_R=0,7_Psi2_HS.plx 25

27 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS s = 1cm s = 8cm s = 15cm R b [kn] R s [kn] R s R b 7,5 1,15 0,75 Table 3.1: Evolution of the skin and base resistance with settlements le : Geo2Disp_Mesh2_loose_R=0,7_Psi2_HS.plx That shows that the maximum skin friction is already reacted when 1,0 cm settlements occur (see gure 3.1). Further, the skin resistance stays the same. 3.2 Pile-Raft Key questions that arise in the design of piled rafts concern the relative proportion of load carried by raft and piles. It depends on the geometric parameters of the pile and of the raft. We performed four new models based on the rst geometry described in chapter 2 to interpret the raft and pile inuence 1. Figure 3.3: Some geometric parameters 1 The Pile-Raft I is the geometry described in details in the chapter 2 26

28 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Paramater Symbol Pile-Raft I Pile-Raft II Diameter of the pile d pile 0,8 m 0,8 m Length of the pile L pile 15 m 15 m Width of the raft L raft 2 m 5 m Depth of the model H model 40 m 40 m Thickness of the slab t raft 0,5 m 0,5 m L raft d pile 2,5 6,25 Table 3.2: Parameters of the rst set of calculations Figure 3.4: Details of Pile-Raft I and II 27

29 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Paramater Symbol Pile-Raft V Pile-Raft III Pile-Raft IV Diameter of the pile d pile 1,5 m 1,5 m 1,5 m Length of the pile L pile 30 m 30 m 30 m Width of the raft L raft 4,5 m 9 m 18 m Depth of the model H model 60 m 60 m 60 m Thickness of the slab t raft 1 m 1 m 1 m L raft d pile Table 3.3: Parameters of the second set of calculations Figure 3.5: Details of Pile-Raft V, III and IV We tested all these geometries with the materials loose and dense sand 2, with and without dilatancy and varying the value of R inter. The outcome was that the inuence of dilatancy and of R inter is very limited. We also performed these calculations with 3 dierent meshes to conrm that there is no mesh dependency. We tryed to have next the pile the same mesh coarseness in 2 See table n

30 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS each model in order to compare precisely the dierent models. The load is a distributed load applied on the slab and the boundaries conditions are those described in chapter 2. In this study we did not take into account the ground water. Remark: As we did in chapter 2, all the load-displacement curves are plotted for the node point A, situated at the top right side of the pile, under the slab (see gure 2.12) Load-displacement curve As we see on the following gure, the load-displacement curve for a pile-raft and a single pile is completely dierent. Figure 3.6: Load settlement curve for pile and pile-raft foundation les : Geo1/1Bis/2load_Mesh1_loose_R=0,7_Psi2_HS.plx Variations of Skin friction and Normal Stresses along the pile For each model we plotted the skin friction and the normal stresses along the pile. This procedure gave us the possibility to illustrate how the load transfer works when the load increases. All the following curves concern dense sand with ψ=8 and R inter = 0, According to Plaxis manual this Rinter value is the most common to model standard situations 29

31 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Remark: All these gures are plotted by selecting the interface in the Plaxis output. In order to get something comparable from one model to an other, we subtracted the rst phase with the pileactivation for each load steps plotted. Thus the Skin friction or Normal Stresses that we present here are only due to the load and the weight of slab. Figure 3.7: Evolution of the skin friction with the load for the Pile-raft I ( d pile l raft = 2, 5) le : Geo1_Mesh1_Dense_R=0,7_Psi2_HS.plx 30

32 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.8: Evolution of the skin friction with the load for the Pile-raft II ( d pile l raft = 6, 25) le : Geo1Bis_Mesh1_Dense_R=0,7_Psi2_HS.plx On the previous gures we can easily see that the mobilization of skin friction of a pile in a piled-raft foundation is completely dierent from the one of with a single pile. For the model Pile-Raft II with a big spacing ( d pile l raft = 6, 25), the slab has a strong inuence on the shear stress distribution along the pile. We notice an increase of shear stresses at the top of the pile, just under the slab. In this case, the slab increases locally the normal stress, so the shear stresses increase in this area provoking this peak in the top area of the pile. For the model Pile-Raft I, the slab does not participate to the load transmission because we do not see such a peak in the distribution: The spacing ( d pile l raft = 2, 5) is too small and almost all the load goes to the pile. Nevertheless, the slab has an inuence too because the distribution is dierent from the one for the single pile. There is an important mobilization of skin friction in the lower part of the pile and no mobilization in the top part. As we can see on the following curves the shape of the normal stresses is in compliance with these observations about the shear stresses. 31

33 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.9: Evolution of the normal stresses with the load for the Pile-raft I ( d pile l raft = 2, 5) le : Geo1_Mesh1_Dense_R=0,7_Psi2_HS.plx Figure 3.10: Evolution of the normal stresses with the load for the Pile-raft II ( d pile l raft = 6, 25) le : Geo1Bis_Mesh1_Dense_R=0,7_Psi2_HS.plx 32

34 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS We now plotted the Skin friction for the second set of calculation. These curves plotted for the geometries with a 30 m length pile and a 1,5 m diameter conrme theses comments. Figure 3.11: Evolution of the skin friction with the load for the Pile-raft V ( d pile l raft = 3) le : Geo1Cinq_Mesh1_Dense_R=0,7_Psi2_HS.plx 33

35 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.12: Evolution of the skin friction with the load for the Pile-raft III ( d pile l raft = 6) le : Geo1Ter_Mesh1_Dense_R=0,7_Psi2_HS.plx Figure 3.13: Evolution of the skin friction with the load for the Pile-raft IV ( d pile l raft = 12) le : Geo1Quater_Mesh1_Dense_R=0,7_Psi2_HS.plx 34

36 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS For the model Pile-raft IV - biggest spacing - with a 1000 kn/m2 loading (gure 3-12), there is positive shear stresses on some centimeters in the top part of the pile. This eect should be studied in further research. By plotting the same curves for the dierent materials -loose and dense sand- and dierent values for ψ we can conclude both dilatancy and materials have very few inuence on the normal stresses and the skin friction distribution. Figure 3.14: Evolution of the skin friction with dilatancy les : Geo1Bis_Mesh1_Dense_R=0,7_Psi0/8_HS.plx 35

37 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.15: Evolution of the skin friction with dense or loose sand les : Geo1Bis_Mesh1_Dense/Loose_R=0,7_Psi0_HS.plx Analysis of the α Kpp factor The previous curves in the last section let us understood some aspects of the behaviour of a piledraft foundation. We easily saw that the bigger the spacing is the more the raft acts in the load transmission. We are now going to describe these observations in a more precise way by calculating the pile/raft stress repartition. In Austria and Germany a common approach consists in calculating the α 4 Kpp factor Denition of α Kpp The α Kpp factor is the ratio between the load carried by the pile and the total load applied on the piled raft foundation.thus it gives us a precise idea of the proportion of load carried by the pile and by the raft. with: α Kpp = R pile R tot 4 In English, Kpp (Kombinierte-Pfahl-Plattengründung) means piled-raft-foundation 36

38 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS ˆ R pile = R b + R s = Load carried by the pile 5 [kn] ˆ R tot =Total load =Distributed load on the slab + weigth of the slab = R raft + R pile 6 [kn] So it means that: ˆ If α Kpp = 1, all the load is carried by the pile ˆ If α Kpp = 0, all the load is carried by the raft We will also use the (1-α Kpp ) coecient which represents the proportion of load carried by the raft. (1-α Kpp )= R raft R tot Remark: Again the weight of the pile is not taken into account Methodology to calculate α Kpp The simplest way to calculate α Kpp with Plaxis 2D consists in realizing a cross section under the slab and reading out the normal stresses on this cross section. Then we just have to sort the normal stresses which are into the pile and into the soil. Remarks: In order to get an accurate value for α Kpp we need to take care of: ˆ Making a cross section which crosses as much stress points as possible because the value is obtained from extrapolation. ˆ Making a cross section not too close to the slab because the junction Slab/pile is a high stress variation area and singularities could occur (take 10 cm to 20 cm under the slab usually leads to accurate values). The following example explains in detail this methodology. 5 R b =Base resistance of the pile [kn]; R s =Skin resistance of the pile [kn] 6 R raft =Load carried by the raft [kn] 37

39 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Example: calculation of α Kpp for Pile-Raft I, Load=1000 kn/m 2, Dense sand, mesh medium,ψ=8 : In this case, we have R tot = 1000.area + weigth of the slab = 3180 kn/m 2 Figure 3.16: Cross sections for Pile-raft I We rst made the cross section n 1 just under the slab. We get the normal stresses as we can see on the prole normal stresses for cross section n 1. Figure 3.17: Normal stresses for cross section n 1 38

40 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS From the values of this prole we calculated R pile and R tot. We found: R pile = 3687 kn and R tot = 3704kN, thus there is an error of 16 % for R tot. In this way we overestimate R pile and R tot because of the unrealistic high normal stress value at the interface. So we started again with the cross section n 2. This one is not directly under the slab, thus we avoid the singular area. Moreover the soil weigth added is negligible in comparison with the load. Now we have the following distribution: Figure 3.18: Normal stresses for cross section n 2 Here we calculate: R pile = 3127 kn and R tot = 3151kN. There is an error of only 1 % for R tot. Thus crossing the section in this way is more accurate. We nally nd for this example α Kpp = 0, Comparison and evolution of α Kpp for dierent geometries: With a small spacing ( W idth raft Diameter pile =2,5 or 3) it seems that the raft takes a small part of the load. In these cases, we calculated an α Kpp equal to 0,99 for all load. With a bigger spacing ( W idth raft Diameter pile =6; 6,25 or 12), we can notice that: ˆ The stress repartition between the raft and the pile evolves with the loading. The higher the loading is, the more the stress is shared. With a load between 0 and 200 kn/m 2 everything goes mostly to the pile (1 <α Kpp < 0,8). From 200 kn/m 2 the raft has a stronger inuence. 39

41 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS ˆ The bigger the spacing is, the more load the raft takes. ˆ In each case the curves converge to an equilibrium state, around α Kpp =0,65 for Pile-Raft III, α Kpp = 0,5 for Pile-Raft II and α Kpp = 0,2 for Pile-Raft V. ˆ The pile obviously carries more load by increasing the length of the pile (compare the geometries Pile-Raft II and III). Figure 3.19: Inuence of geometry on α Kpp for loose sand, ψ=2, R=0,7, mesh medium. W idth Name Length pile Diameter pile Width raft raft Diameter pile Pile-Raft I 15 m 0,8 m 2 m 2,5 Pile-Raft II 15 m 0,8 m 5 m 6,25 Pile-Raft V 30 m 1,5 m 4,5 m 3 Pile-Raft III 30 m 1,5 m 9 m 6 Pile-Raft IV 30 m 1,5 m 18 m 12 Table 3.4: Reminder, basic parameters of each geometry 40

42 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS R tot [kn/m 2 ] 25 + Slab Slab Slab Slab Pile-Raft I 0,99 0,99 0,99 0,99 Pile-Raft II 0,95 0,65 0,57 0,52 Pile-Raft V 0,99 0,99 0,99 0,99 Pile-Raft III 0,96 0,85 0,71 0,62 Pile-Raft IV 0,66 0,29 0,22 0,19 Table 3.5: Few values of α Kpp for loose sand, ψ=2, R=0, Evolution of α Kpp for dierent materials and dilatancy As we can see on the following curves, the material - loose or dense dand - and the dilatancy have a negligible inuence on the stress repartition in the piled-raft foundation. Figure 3.20: Inuence of material on α Kpp, Pile-raft II, R=0,7 41

43 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.21: Inuence of the dilatancy on α Kpp, Pile-raft II, R=0, Evolution of α Kpp for dierent values of R inter In this sub-section we present the evolution of α Kpp with the load (gure 3.19) and the displacement (gure 3.20) for dierent R inter values. In both cases, the tendency is exactly the same. We only added with the displacement because it is also a common presentation in the literature. Concerning the inuence of R inter on α Kpp we can conclude that the part of the load carried by the pile decreases when we reduce the interface strength factor. It is an expected behavior because by reducing the R inter value we decrease the maximum amount of mobilization of the skin friction along the pile 7. 7 On the interface, τ Rinter.(σ n.tanφ soil +c soil ) 42

44 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.22: Inuence of R inter on the evolution of α Kpp with the load, Pile-raft II, R=0,7 Figure 3.23: Inuence of R inter on the evolution of α Kpp with the displacement, Pile-raft II, R=0,7 43

45 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Eciency of a piled-raft foundation in comparison with a raft foundation To evaluate the eciency of a piled-raft foundation in comparison with a raft foundation it is interresting to compare the settlements with and without a pile. So, we performed one new calculation for each geometry putting just the raft without the pile. Then we calculated the β coecient. Denition β is the ratio between the settlements which occured without pile (U raft ) and with the settlements which occured with a pile (U pile+raft ): β= U raft U pile+raft Thus, we necessarily have β 1 and if β 1 the pile is useless. As expected, the evolution of β with the load has the same tendency as α Kpp. When we have a high value for α Kpp the pile carries most of the load and thus acts a lot against displacements. So the value of β is high. On the contrary, when the raft carries a big part of the load - for example with Pile-Raft IV - the settlements are very close to those observed with a raft only. For the model Pile-Raft II in which we have a good sharing of the load, we have a β value from 1,15 to 2,1. Figure 3.24: Evolution of β with the load, Loose sand Ψ=2 - Mesh Medium R inter =0,7 44

46 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS R tot [kn/m 2 ] Pile-Raft I 2,4 2,15 2,0 1,8 Pile-Raft II 2,1 1,3 1,2 1,15 Pile-Raft V 3,1 2,5 2,5 2,2 Pile-Raft III 2,6 1,65 1,4 1,3 Pile-Raft IV 1,5 1,15 1,1 1,0 Table 3.6: Few values of β for loose sand, ψ=2, R=0,7 Figure 3.25: Comparison between the evolution of β and α for Pile-Raft II, loose, ψ=2, R=0, Analysis of the pile behavior The bearing capacity of a pile consists of the base resistance (R b ) and the skin resistance (R s ). Now we study in detail these two forces in order to have a better idea of the pile behavior for dierent geometries Base resistance The method to calculate R b is the same as for R tot. We made a cross section under the pile. In this part, we considered only the contribution of the distributed load by subtracting the two rst phase with the pile and raft activation. On the next gure, we can see the evolution of the base resistance with the load for the model Pile-Raft I and II. The curves are both approximatively linear. It means that in our cases the part of the total load (R tot ) carried by the base of the pile is approximatly constant. 45

47 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.26: Evolution of R b with the load for Pile-Raft I and II, dense sand, ψ=8 Now we compare the R base for the pile-raft I and the single-pile. Figure 3.27: Comparison of Pile-Raft I and Single Pile, evolution of R b with the load for Pile-Raft I and II, dense sand, ψ= Skin resistance The skin friction proles presented previously give us the possibility to work out the skin resistance R s. As we did for R b we only considered in this section the contribution of the distributed load by 46

48 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS subtracting the constribution of the pile and of the raft. Figure 3.28: Evolution of R s with the load for Pile-Raft I and II, dense sand, ψ= Conclusions The following curves sum up the R base, R skin and R raft proportions for various models. Figure 3.29: Repartition of the forces into the single-pile, dense sand, ψ=8 47

49 3.2. PILE-RAFT CHAPTER 3. ANALYSIS OF 2D MODELS Figure 3.30: Repartition of the forces into pile for Pile-Raft I, dense sand, ψ=8 Figure 3.31: Repartition of the forces into pile for Pile-Raft II, dense sand, ψ=8 48

50 Chapter 4 Preliminary studies of 3D models - From 2D axisymmetric models to 3D models - We previously studied the behavior of one pile-raft foundation. Nevertheless the load settlement behavior of piles in a pile group is usually observed to be totally dierent from the behavior of a corresponding single pile. This group eect cannot be studied with axisymmetric models and consequently it requires performing calculations with Plaxis 3D foundation. In order to prepare the group eect analysis, we rstly tested the dierent Plaxis 3D foundation tools to model a pile: the volume pile and a new feature, the embedded pile. These comparisons are presented in this chapter. Remark about the mesh dependency: The previous calculations with axisymmetric models showed a negligible mesh dependency. We also checked that 6-node coarse meshes lead to the same load-displacement behavior as 15-node ne meshes. Due to the bigger size of working areas in 3D models we cannot use eciently ne meshes. Thus, we will perform calculations from coarse to medium meshes. The results should be realistic because of the low sensitivity of the mesh renement observed in 2D. Remark about the mesh generation: To create a mesh with Plaxis 3D foundation we rstly generate a 2D mesh on a horizontal work plane. When the 2D mesh is satisfactory, the 3D mesh is generated from the 2D mesh. Since there is no vertical renement option, badly shaped elements with a higher vertical than horizontal dimension could occur. To get a satisfactory vertical renement, we added multiple work planes in the input, then when the 3D mesh is generated from the 2D one, these additional planes are taken into account and the vertical size of the elements is adapted from their spacing. In this way we get a good medium 3D mesh with a local 3D renement under the slab and at the pile bottom (see gure 4.1). 49

51 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS 4.1 Volume pile The volume pile is a common Plaxis 3D foundation option to model a pile Finite element models To start this study, all the previous geometries (Pile-Raft I, II, III, IV, V) were modeled using Plaxis 3D foundation. The working area was adapted in each case to have the same raft area with 3D and with axisymmetric models. Actually the raft area with axisymmetric models is circular whereas it is a square raft in 3D. Thus in 2D, the raft area is given by the following formula: A raft2d = π ( L raft 2D 2 ) 2 [m 2 ] The 3D width raft is obtained by taking the square root of 2D area raft as followed: L raft3d = A raft2d = π ( L raft 2D 2 ) 2 [m] In this way, the area of the 3D raft is equal to the one in 2D: A raft3d = A raft2d [m 2 ] Figure 4.1: Comparison between the axisymmetric and 3D raft shapes The pile is modeled as a volume pile and we selected the massive circular pile type. Interfaces are modeled along the pile with a R inter = 0, 7. The soil consists of a single layer of dense sand with the same properties as the sand we used previously. The load is modeled as a distributed load on the slab. Two dierent meshes with dierent levels of renement were applied to the rst two geometries. Only a medium one was used for the remaining geometries. The following tables and gures sum up the most important parameters used. W idth Name Thickness slab Depth model Length pile Diameter pile Width raft raft Diameter pile Pile-Raft I 0,5 m 40 m 15 m 0,8 m 1,8 m 2,25 Pile-Raft II 0,5 m 40 m 15 m 0,8 m 4,4 m 5,5 Pile-Raft V 1 m 60 m 30 m 1,5 m 4 m 2,7 Pile-Raft III 1 m 60 m 30 m 1,5 m 8 m 5,3 Pile-Raft IV 1 m 60 m 30 m 1,5 m 16 m 10,6 Table 4.1: Basic parameters of each geometry 50

52 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Parameter Symbol Dense sand Concrete Unit Material model Model Hardening Soil Linear Elastic - Unsaturated weigth γ unsat kn/m 3 Saturated weigth γ sat kn/m 3 Permeability k 1 0 m/day E ref kn/m 3 Stiness Eur ref 1,8E5 kn/m 3 Power m 0,55 Poisson ratio ν ur 0,2 0,2 - Dilatancy y 8 Friction angle f 38 Cohesion c ref 0,1 kn/m 2 Lateral pressure coe. K 0 1-sinf - Failure ratio Rf 0,9 - E ref oed E7 kn/m 3 Table 4.2: Materials parameters Figure 4.2: Details about a pile-raft geometry in 3D, medium mesh (Pile-raft IV) 51

53 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Number of 15-noded elements Medium Fine Pile-Raft I Pile-Raft II Pile-Raft V / Pile-Raft III / Pile-Raft IV / Table 4.3: Information on the generated meshes Results Remark: As we did for axisymmetric models all the following load-settlement curves are plotted for the node point located at the top right side of the pile, under the slab. Figure 4.3: Position of the node point A Load-displacement curves We plotted the load-displacement curve for each geometry. Then we compared these curves with the associated axisymmetric curves. In each case, we noticed a good match with the 3D volume pile-raft and the associated axisymmetric models. Moreover the gures 4.3, 4.4 and 4.5 conrm that the mesh dependency is negligible. 52

54 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.4: Load-displacement curves comparison for Pile-Raft I, dense sand, ψ=8 Figure 4.5: Load-displacement curves comparison for Pile-Raft II, dense sand, ψ=8 53

55 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.6: Load-displacement curves comparison for Pile-Raft IV, dense sand, ψ= Variations of skin friction Remarks: ˆ All the following gures are plotted by selecting the interface in Plaxis output. For Pile-Raft I and II (respectively for Pile-Raft V and IV) we plotted the interface along the line - X = 0, 4 (resp. 0, 75); Y [ 15;0] (respec. [ 30; 0]); Z = 0 (resp. Z = 0 ) -. In order to get something comparable from one model to an other, we subtracted the rst phase from the pile for each load steps plotted. Thus the skin friction presented in this part are only due to the load and the weight of the slab. ˆ Then, we compared the 3D volume pile proles with the axisymmetric ones. They are not strictly comparable because the shape of the raft area is not the same. Nevertheless a comparison stays relevant as we choose the same area for every models. Results of 3D volume pile models are in a very good aggreement with those we got with axisymmetric calculations. We observed almost the same shape of skin friction for each Pile-Raft le. 54

56 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.7: Axisymmetric and 3D volume pile skin friction curves, Pile-Raft I, Dense, ψ = 8, R inter = 0, 7 Figure 4.8: Axisymmetric and 3D volume pile skin friction curves, Pile-Raft II, Dense, ψ = 8, R inter = 0, 7 55

57 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.9: Axisymmetric and 3D volume pile skin friction curves, Pile-Raft V, Dense, ψ = 8, R inter = 0, 7 Figure 4.10: Axisymmetric and 3D volume pile skin friction curves, Pile-Raft III, Dense, ψ = 8, R inter = 0, 7 We can also notice that there are more oscillations in the lowest part of pile with the 3D volume pile models than with 2D axisymmetric models. These non-physical stress oscillations are due to 56

58 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS the high peaks in stresses at the bottom of the pile. As we can see on gure 4.11, we can reduce these numerical inaccuracies by lengthening the interface at the bottom of the pile (+0,5 m). Figure 4.11: Reduction of oscillations by lengthening the interface, Pile-raft III, Dense, ψ = 8, R inter = 0, 7 By analysing in more details the load repartion for each model we can conclude that not only the skin friction but also the base resistance ts well. Axisymmetric Pile-Raft II 250 kn/m kn/m kn/m 2 R skin [kn] R base [kn] R skin R base 1,36 1,13 1,05 Volume Pile-Raft II 250 kn/m kn/m kn/m 2 R skin [kn] R base [kn] R skin R base 1,1 0,94 0,88 Table 4.4: Comparison between volume and axisymmetric pile raft II 57

59 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Remarks: The following values have been calculated without subtracting the weight of the pile and of the raft. For the volume pile, we estimated R base and R skin by reading in Plaxis output the normal force values N at the top and at the bottom of the pile. Then we considered that: R base = N bottom and R skin = N top -N bottom Some remarks about parameters We also varied the value of R inter and ψ with some 3D volume pile models. We can conclude that both dilatancy and R inter have little inuence on results. Figure 4.12: Load-displacement curves for Pile-Raft I for dierent values of R inter, dense sand, ψ=8 58

60 4.1. VOLUME PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.13: Load-displacement curves for Pile-Raft III for dierent values of ψ, dense sand, R inter = 0, 7 Figure 4.14: Skin friction with the load for ψ = 8 and 0, Pile-Raft III, Dense sand, R inter = 0, 7 59

61 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS 4.2 Embedded pile An embedded pile is a pile composed of beam elements that can be placed in arbitrary direction in the sub-soil (irrespective from the alignment of soil volume elements) and that interacts with the sub-soil by means of special interface elements. The interaction may involve a skin resistance as well as a foot resistance. Although an embedded pile does not occupy volume, a particular volume around the pile (elastic zone) is assumed in which plastic soil behaviour is excluded. The size of this zone is based on the (equivalent) pile diameter according to the corresponding embedded pile material data set. This makes the pile almost behave like a volume pile. Nevertheless, when creating embedded piles no corresponding geometry points are created. Thus, contrary to volume pile, embedded piles do not inuence the nite element mesh as generated from the geometry model. So the mesh renement is lower and we save calculation time. 1 In contrast to what is common in the Finite Element Method, the bearing capacity of an embedded pile is considered to be an input parameter rather than the result of the nite element calculation. Plaxis gives us the possibility to enter the skin resistance prole in three ways: ˆ Linear: The user enters the skin resistance at the pile top and the skin resistance at the pile bottom. The skin resistance is dened as linear along the pile. This way of dening the pile skin resistance is mostly applicable to piles in a homogeneous soil layer. ˆ Multi-linear: The skin resistance is dened in a table at dierent positions along the pile. Multi-linear can be used to take into account inhomogeneous or multiple soil layers with dierent properties and, as a result, dierent resistances. ˆ Layer dependent, can be used to relate the local skin resistance to the strength properties of the soil layer in which the pile is located, and the interface strength reduction factor R inter, as dened in the material data set on the corresponding soil layer. Using this approach the pile bearing capacity is based on the stress state in the soil, and thus unknown at the start of a calculation. Nevertheless an overall maximum resistance can be specied before to avoid an undesired too high value at the end. We performed another set of calculations by modeling the previous geometries using embedded piles. This study gave us the possibility to test the reliability of this new feature to model pile-raft structures Embedded pile-raft For this study, we focused our calculations on the two rst geometries named Pile-Raft I and Pile- Raft II. We took exactly the same geometries by using embedded piles instead of volume piles. We 1 See Plaxis manual for more details about embedded piles 60

62 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS considered the pile to raft connection as rigid. As mentioned previously the capacity of the pile is an input parameter for an embedded pile so we had to dene the most relevant skin friction distribution and base resistance. This is the reason why we tested each possibility oered by Plaxis to try to congure in a proper way this new tool Finite element models The parameters we examined for the embedded pile are the same as those desribed previously for the volume pile. We performed calculations for only one material, the dense sand with R inter = 0, 7. The load is modeled as a distributed load on the slab. Two dierent meshes with dierent levels of renement had been used. The following tables sum up the most important parameters. W idth Name Thickness slab Depth model Length pile Diameter pile Width raft raft Diameter pile Pile-Raft I 0,5 m 40 m 15 m 0,8 m 1,8 m 2,25 Pile-Raft II 0,5 m 40 m 15 m 0,8 m 4,4 m 5,5 Table 4.5: Basic parameters of each geometry Parameter Symbol Dense sand Concrete (slab) Unit Material model Model Hardening Soil Linear Elastic - Unsaturated weigth γ unsat kn/m 3 Saturated weigth γ sat kn/m 3 Permeability k 1 0 m/day E ref kn/m 3 Stiness Eur ref 1,8E5 kn/m 3 Power m 0,55 Poisson ratio ν ur 0,2 0,2 - Dilatancy y 8 Friction angle f 38 Cohesion c ref 0,1 kn/m 2 Lateral pressure coe. K 0 1-sinf - Failure ratio Rf 0,9 - E ref oed E7 kn/m 3 Table 4.6: Soil parameters 61

63 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Parameter Name Value Unit Young s modulus E kn/m 3 Weight γ 5 kn/m 3 Properties type Type Massive circular pile - Diameter d pile 0,8 m Length L pile 15 m Table 4.7: Material properties of the embedded pile Number of 15-noded elements Medium Fine Pile-Raft I Pile-Raft II Table 4.8: Information on generated meshes Figure 4.15: Details about an embedded pile-raft geometry in 3D, ne mesh, pile-raft II 62

64 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Remark: ˆ As we did previously all the following load-settlement curves are plotted for the same node point A located at the top right side of the pile, under the slab. ˆ Concerning the skin friction proles, we read out the T skin2 value [kn/m] by selecting the embedded pile in Plaxis output. Then we divided T skin by the perimeter of the pile to get the skin friction q s [kn/m 2 ]. In order to get something comparable from one model to an other, we subtracted the rst phase from the pile for each load step plotted. Thus the skin friction that we present here are only due to the load and the weight of the slab. ˆ We also read out the pile foot force F foot3 [kn] by selecting the embedded pile in the Plaxis output. Thus we compared this value with the base resistance values found with axisymmetric models Embedded pile with linear skin friction distribution For this rst set of calculations we dened linear skin friction distribution using unrealistic high values (see gures bellow). Skin friction distribution linear [-] T top,max 2000 [kn/m] T bot,max 2000 [kn/m] F max [kn] Table 4.9: Linear skin friction distribution n 1 for Pile-Raft I and II 2 The Skin force Tskin, expressed in the unit of force per unit of pile length, is the force related to the relative displacement in the pile s rst direction (axial direction) 3 The pile foot force Ffoot, expressed in the unit of force, is obtained from the relative displacement in the axial pile direction between the foot of the pile and the surrounding soil. 63

65 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Thus we got the following results: Figure 4.16: Load-displacement curves for Pile-raft I, dense sand, ψ=8, R inter = 0, 7 Figure 4.17: Load-displacement curves for Pile-raft II, dense sand, ψ=8, R inter = 0, 7 64

66 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Load=1000 kn/m 2 Settlement [cm] axisymmetric Pile-Raft I -13,2 Embedded Pile-Raft I_Medium -14,8 Embedded Pile-Raft I_Fine -15,8 axisymmetric Pile-Raft II -19,1 Embedded Pile-Raft II_Medium -19,6 Embedded Pile-Raft II_Fine -19,9 Table 4.10: Settlements for the dierent models for 1000 kn/m 2, Dense sand, ψ=8, R inter = 0, 7 We can conclude that the mesh inuence seems to be still quite negligible. We also notice that axisymmetric curves are not in a very good agreement with embedded pile curves. Thus, we note a dierence of around 15% in the settlements for axisymmetric and embedded ne Pile-Raft I (with Load=1000 kn/m 2 ). Now we observe the skin friction prole for these models: Figure 4.18: Skin friction with the load for ψ = 8, Pile-Raft I, Dense sand, R inter = 0, 7 65

67 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.19: Skin friction with the load for ψ = 8, Pile-Raft II, Dense sand, R inter = 0, 7 When we compared these embedded pile skin friction proles with the axisymetric ones we notice that they are very dierent. We cannot observe the increase under the slab we described previously in the 2D analysis. Thus the mobilization of such a dened embedded pile is dierent. So we decided to change our linear skin friction distribution using more realistic values. We dened these values from the axisymetric skin friction proles. Thus we performed new calculations by specifying this new input information: 66

68 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Skin friction distribution linear [-] T top,max 620 [kn/m] T bot,max 620 [kn/m] F max 2260 [kn] Table 4.11: Linear skin friction distribution n 2 for Pile-Raft I Skin friction distribution linear [-] T top,max 1110 [kn/m] T bot,max 1110 [kn/m] F max 8300 [kn] Table 4.12: Linear skin friction distribution n 2 for Pile-Raft II 67

69 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS The load-displacement curves we got with these new parameters are almost strictly the same as those we had with the linear skin friction n 1. However, we can observe some dierences in the skin friction proles: Figure 4.20: Skin friction with the load for Pile-Raft I, Dense sand,ψ = 8, R inter = 0, 7 Figure 4.21: Skin friction with the load for Pile-Raft II, Dense sand,ψ = 8, R inter = 0, 7 68

70 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS For the lowest load, the proles are exactly the same for the distribution -n 1 or n 2-. Nevertheless, with the highest load and the input linear skin friction distribution n 2, the skin friction reaches the input value and stops growing. To conclude we can say that neither linear skin friction distribution n 2 nor linear skin friction distribution n 1 leads to a skin friction prole in aggrement with the realistic one Embedded pile with multilinear skin friction distribution For this second set of calculations we tested three dierent multilinear skin friction distributions. Input: The multilinear skin friction distribution n 1 is a quite simple but realistic multilinear distribution: Skin friction distribution : Multilinear Depth [m] T max [kn/m] , F max [kn ] 2260 Skin friction distribution : Multilinear Depth [m] T max [kn/m] F max [kn ] 8300 Table 4.13: Multilinear skin friction distribution n 1 for Pile-Raft I (left) and II (right) 69

71 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS The multilinear skin friction distribution n 2 is the same as multilinear skin friction distribution n 1 with T max =0 kn/m instead of 1 in the depth equal to 15m. Finally the multilinear skin friction distribution n 3 is a more complex multilinear distribution designed from the axisymetric skin friction prole as followed: Skin friction distribution : Multilinear Depth [m] T max [kn/m] , , , F max [kn ] 8300 Skin friction distribution : Multilinear Depth [m] T max [kn/m] 0 0-1, , , , F max [kn ] 8300 Table 4.14: Multilinear skin friction distribution n 3 for Pile-Raft I (left) and II (right) Output: We noticed that the behaviors observed with distributions n 1 and 2 are exactly the sames. More precisely, the load-displacement curves and the shear stresses distributions we got with n 1 and n 2 are completly equal. 70

72 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS When we compare the n 1&2 load-displacement curve with the axisymmetric one, we see that they do not t very well. There is a dierence of 12,5% (Pile-Raft I) and 7% (Pile-Raft II) in settlements. Figure 4.22: Load-displacement curves for multilinear n 1/2 embedded and axisymmetric pile-raft I Figure 4.23: Load-displacement curves for multilinear n 1/2 embedded and axisymmetric pile-raft II Concerning distribution n 3, the load-settlement curve is almost the same as for distribution n 1/2. 71

73 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Load=25 kn/m 2 Load=500 kn/m 2 Load=1000 kn/m 2 Axisymmetric Pile-Raft II -5,4 mm -110 mm -191 mm Multilinear n 1/2 Emb Pile-Raft II -5,6 mm -112 mm -204 mm Multilinear n 3 Emb Pile-Raft II -5,6 mm -115 mm -210 mm Table 4.15: Comparison Load/Settlements for dierent Pile-Raft II inputs Finally by plotting the shear stresses distributions for each case, no multilinear skin resistance input yields to the realistic skin friction mobilization we calculated with the axisymmetric models. 72

74 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.24: Evolution of skin friction for dierent models and loadings Embedded pile with layer dependent skin friction distribution For this third set of calculations we tested the layer dependent option. According to a recent update on plaxis website, when using the layer dependant skin resistance for the embedded piles, while leaving the linear skin resistance values to their defaults, the calculation kernel will show a "severe divergence" error message. This severe divergence is caused by the zero values for the linear skin resistance, though they do not have any inuence on the layer dependant skin resistance. To overcome this error, users are advised to set the linear skin resistance values to some values not equal to zero, and then activate the layer dependant option. These linear skin resistance values will not have an inuence on the layer dependant values for the skin resistance. 4 We perfomed some tries. Input: For the layer dependent distribution n 1 we let the default values suggested by Plaxis. Skin friction distribution: Layer dependent T max [kn/m] F max [kn] Table 4.16: Layer dependent distribution n 1, parameters 4 Plaxis website, Known issues 3D Foundation 2.1,

75 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS As we explained in the introduction of this section, we did not let the default values for the linear skin resistance. We input 1. For the layer dependent distribution n 1bis we used the values as described in the previous table, but we input 2000 for the linear skin resistance. We tested with dense sand, R inter = 0, 7. Output: The layer dependent distribution n 1 and the layer dependent distribution n 1bis perfectly match. It conrmed that these linear skin resistance values do not have an inuence on the layer dependent results. We just need to write a value not equal to zero in linear to use correctly the layer dependant option. Figure 4.25: Comparison of load-displacement curves Moreover, the skin distribution prole is in a perfect agreement for the layer dependent distribution n 1 and n 1Bis. We also have a quite good match with the axisymmetric prole. If we calculate the dierence of skin friction at half a pile between the axisymmetric and layer dependent results : 7,5m = q s2d ( 7, 5m) - q s3d ( 7, 5m) 74

76 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS We get for a load equal to 500 kn/m 2, 7,5m 25kN/m 2. Figure 4.26: Evolution of skin friction, Pile-raft II We performed another calculation with the parameters of the so called layer dependent distribution n 1. We just changed the R inter value from 0,7 to 1. We compare the load-displacement behavior on the following gures. 75

77 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.27: Comparison of load-displacement curves The load-displacement curves we got for the layer dependent distribution n 1 with R inter = 1 is close to one with R inter = 0, 7 but not exactly equal. In each case, they are not in good aggrement with the axisymmetric results. We have a dierence of around 12,5 % (Pile-Raft II) in the settlements for a distributed load equal to 1000 kn/m 2. Load=25 kn/m 2 Load=500 kn/m 2 Load=1000 kn/m 2 Axisymmetric Pile-Raft II -5,4 mm -110 mm -191 mm Layer dependent 1 - R inter = 0, 7-5,6 mm -126 mm -215 mm Layer dependent 2 - R inter = 1-5,6 mm -123 mm -213 mm Table 4.17: Comparison Load/Settlements for dierent Pile-Raft II inputs For each input distributions we also plotted the skin friction distributions. In each case, the skin distribution prole is in quite good agreement with the axisymmetric prole, particularly with the layer dependent distribution n 1 with R inter = 1. 76

78 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.28: Evolution of skin friction for dierent inputs, Pile-raft II We get for a load equal to 1000 kn/m 2, 7,5m 77kN/m 2 with R inter = 0, 7 and 7,5m 16kN/m 2 with R inter = 1. The layer dependent distribution option seems to be the best way for embedded piles to get skin friction distributions with realistic shapes. Nevertheless further tests must be done to really determine the inuence of the virtual value we need to input in linear skin resistance when we use the layer dependant option Comparison of the three options: Linear, multilinear and layer dependent We now compare the three approaches in order to determine which approach is the best for analysing piled raft foundations. Load-displacement behavior: Concerning the load-displacement behavior, the linear embedded pile raft is the closest to the axisymmetric model. However for common geotechnical displacements (max. 10 cm), whatever the input option for the embedded pile is, the load displacement curve stays quite reasonably close to the axisymmetric one with a dierence of around more or less 1 cm. 77

79 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Load=25 kn/m 2 Load=500 kn/m 2 Load=1000 kn/m 2 Axisymmetric -5,4 mm -110 mm -191 mm Linear embedded +3,7 % +0,9% +3,7% Multilinear embedded +3,7 % +4,5% +12% Layer dependent embedded +3,7 % +14,5% +12,6% Table 4.18: Displacement with the load, for Pile-Raft II Figure 4.29: Load-displacement curves for Pile-Raft II Pile-raft behavior: According to the skin friction distributions presented previously, we saw that the mobilization of the embedded pile-raft foundation and the axisymetric pile-raft foundation is dierent. So we analysed in more details the load repartion for each model. The following values have been calculated by subtracting the weight of the pile and of the raft. 78

80 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Axisymmetric Pile-Raft II 25 kn/m kn/m kn/m kn/m 2 R skin [kn] R base [kn] α Kpp 0,95 0,72 0,63 0,57 R skin R base 4,73 1,38 1,13 1,05 Linear n 2 Emb. Pile-Raft II 25 kn/m kn/m kn/m kn/m 2 R skin [kn] R base [kn] 12,7 95, α Kpp 0,9 0,76 0,66 0,55 R skin R base 33,9 38, Multilinear n 1 Emb. Pile-Raft II 25 kn/m kn/m kn/m kn/m 2 R skin [kn] R base [kn] α Kpp 0,88 0,75 0,66 0,48 R skin R base 18,1 16, ,1 Layer dependent n 2 R inter = 0, 7 Emb. Pile-Raft II 25 kn/m 2 / 500 kn/m kn/m 2 R skin [kn] 438,3 / 2760,9 3756,7 R base [kn] 16,9 / 570,4 967,6 α Kpp 0,99 / 0,34 0,24 R skin R base 25,9 / 4,84 3,88 Layer dependent n 2 R inter = 1 Emb. Pile-Raft II / 500 kn/m kn/m 2 R skin [kn] / 3930,8 5729,2 R base [kn] / 526,6 935,8 α Kpp / 0,45 0,33 R skin R base / 7,46 6,12 Table 4.19: R raft,r skin,and R base repartition for dierent models of Pile-Raft II (Dense sand, ψ=8 ) 79

81 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Figure 4.30: Evolution of skin friction, Load=500 kn/m2 Figure 4.31: Evolution of skin friction, Load=1000 kn/m2 Remark For these previous gures we substracted the weight of the pile and of the raft. 80

82 4.2. EMBEDDED PILE CHAPTER 4. PRELIMINARY STUDIES OF 3D MODELS Axisymmetric Pile-Raft I 25 kn/m kn/m kn/m kn/m 2 R skin [kn] 50,3 451,5 812, R base [kn] 35,8 335, α Kpp 0,99 0,99 0,99 0,99 R skin R base 1,41 1,35 1,04 0,81 Linear n 2 Emb. Pile-Raft I 25 kn/m kn/m kn/m kn/m 2 R skin [kn] R base [kn] 4, ,1 203 α Kpp 1 1 0,99 0,98 R skin R base 16 21,4 17,2 14,1 Multilinear n 1 Emb. Pile-Raft I 25 kn/m kn/m kn/m kn/m 2 R skin [kn] R base [kn] 9 65,2 123,5 252 α Kpp 0,97 0,97 0,97 0,97 R skin R base 7,6 10,7 11,3 11,1 Table 4.20: R raft,r skin,and R base repartition for dierent models of Pile-Raft I (Dense sand, ψ=8 ) The linear and multilinear embedded pile models seem to lead to realistic values of α Kpp. Actually we calculated values of α Kpp very close to the axisymmetric ones. The main problem remains the mobilization of the base resistance because whatever the input is, we underestimate R base with embedded piles in comparison with 2D models. Moreover, the skin friction of embedded piles seems to be overestimated in comparison with chapter 2 except the calculations with the layer dependent skin resistance. But with this approach we always got too much settlements. To conclude we can say that with embedded pile option we did not manage to calculate the pile raft behavior we observed with volume piles or axisymmetric models. 81

83 Chapter 5 Group eect - Analysis of the group eects in piled raft foundations - The previous models gave us a rst idea of a piled raft foundation behavior. These models took into account the pile-soil interaction, the raft-soil interaction and the pile-raft interaction but not the pile-pile interaction. Yet when the piles spacing is small, a partial geometry with a single pile and one section of the raft is not accurate enough. We must consider the pile-pile interaction and design more complex models with a group of piles. In this chapter, we present some observations about the group eect in a piled raft foundation. 5.1 Presentation of calculations We studied a piled raft foundations with 6 6 piles. Thus by using symetries we modeled only 9 piles. To study the inuence of pile length and diameter as well as spacing between piles we designed ve dierent geometries. 82

84 5.1. PRESENTATION OF CALCULATIONS CHAPTER 5. GROUP EFFECT Figure 5.1: 6 6 piled raft foundation, we model the red delimited quarter only Geometry The following pictures present the global geometry of our models. 83

85 5.1. PRESENTATION OF CALCULATIONS CHAPTER 5. GROUP EFFECT Figure 5.2: Model of one quarter of one piled raft foundation (6 6 piles) 84

86 5.1. PRESENTATION OF CALCULATIONS CHAPTER 5. GROUP EFFECT The main geometric paramaters of this study are presented in this gure: Figure 5.3: Some geometric parameters From this scheme, we designed ve dierent geometries: Paramater Symbol Group 1 Group 2 Group 3 Diameter of the pile d pile 0,8 m 0,8 m 0,8 m Length of the pile L pile 15 m 15 m 30 m L - pile d pile 18,75 18,75 37,5 Length of the pile group L g 8,75 m 16,8 m 16,8 m Depth of the model H model 50 m 50 m 80 m Thickness of the slab t raft 1,5 m 1,5 m 1,5 m Spacing between piles a 2,5 m ( 3 d pile ) 4,8 m (6 d pile ) 4,8 m (6 d pile ) Paramater Symbol Group 4 Group 5 Diameter of the pile d pile 1,5 m 1,5 m Length of the pile L pile 30 m 30 m L - pile d pile Length of the pile group L g 15,75 m 31,5 m Depth of the model H model 80 m 80 m Thickness of the slab t raft 1,5 m 1,5 m Spacing between piles a 4,5 m (3 d pile ) 9 m (6 d pile ) Table 5.1: Geometric parameters for each model 85

87 5.2. RESULTS CHAPTER 5. GROUP EFFECT For each group we also performed the same calculations without the piles in order to evaluate the piled raft eciency in comparison with a raft foundation (β-factor) Finite element model We modeled the piles with the volume pile option. Moreover, we generated for each model a medium mesh with around elements (see gure 5.2). Finally, the load is a distributed load applied on the slab. We tested only the dense sand material, with R inter =0,7 and ψ=8. Parameter Symbol Dense sand Concrete Unit Material model Model Hardening Soil Linear Elastic - Unsaturated weigth γ unsat kn/m 3 Saturated weigth γ sat kn/m 3 Permeability k 1 0 m/day E ref kn/m 3 Stiness Eur ref 1,8E5 kn/m 3 Power m 0,55 Poisson ratio ν ur 0,2 0,2 - Dilatancy y 8 Friction angle f 38 Cohesion c ref 0,1 kn/m 2 Lateral pressure coe. K 0 1-sinf - Failure ratio Rf 0,9 - E ref oed E7 kn/m 3 Table 5.2: Material properties 5.2 Results Vocabulary details Please note that in the following sections we will distinguish four types of pile depending on their position in the group: ˆ the center pile is pile A ˆ the middle piles are piles B, D, E 86

88 5.2. RESULTS CHAPTER 5. GROUP EFFECT ˆ the corner pile is pile I ˆ the edge piles are piles C and G Figure 5.4: Name of each pile Remark: To plot the load displacement curves we selected the node point located in the center of the upper face of each pile Load-displacement curves For the so called center pile, edge piles and corner pile of the group 1 and 2 we read out the R pile value and the corresponding pile settlements for each load steps. We got the following curves: Figure 5.5: Displacement of the pile with R pile, group 1 87

89 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.6: Displacement of the pile with R pile, group 2 These curves are in agreement with the shape described in books. For the other models, we plotted the load-settlement curves for the raft. We also compared the behavior with the closest single pile-raft model. As these partial geometries have not exactly the same geometric parameters (see previous chapter) they are not entirely comparable but give us some clues to interpret the group behavior. Figure 5.7: Load-displacement curves for Group 4 88

90 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.8: Load-displacement curves for Group 5 Whatever group we examine the settlement behavior of each pile is not really dierent from one pile to another. The whole behavior is very sti and we could propably improve our models by varying the parameters of the raft in order to have a less sti behavior Displacement proles A cross section has been performed as described in the following gure. It gives us the possibility to observe vertical displacements (u y ) around center, middle and edge piles. All these cross sections are performed for a distributed load equal to 1000 kn/m 2. 89

91 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.9: A, B, C piles cross section Figure 5.10: Group 1 (left) and Group 2 (right) cross sections 90

92 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.11: Single pile-raft II cross section Figure 5.12: Group 4 (left) and Group 5 (right) cross sections 91

93 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.13: Single pile-raft III We notice that the displacement behavior around middle and center piles for groups with a large spacing (group 2 and 5) is quite comparable and in good agreement with the associated single pileraft model. On the other hand, with the small spacing groups (group 1 and 4) the settlements are dierent from one pile to another. In each case the behavior of the edge pile cannot be compared with the one of the associated single pile-raft model because the raft is larger in the area of the edge pile More precise analysis of group 5 We estimated R base and R skin by reading out in Plaxis output the normal force values N at the top and at the bottom of the pile. Then we considered that: R base = N bottom and R skin = N top -N bottom. The value from N bottom are not the highest value of the normal force along the pile but this value normally occurs a bit above the pile toe (around -29,5 m). These values give us an idea of the real base and skin resistance. 92

94 5.2. RESULTS CHAPTER 5. GROUP EFFECT Group 5 Center pile Middle piles edge piles corner pile Single pile-raft III R skin [kn] R base [kn] R skin + R base R skin R base 0,52 0,57 0,56 0,59 1,12 Table 5.3: Base and skin resistance for dierent piles Remark: Paramater Symbol Group 5 Pile-Raft III Diameter of the pile d pile 1,5 m 1,5 m Length of the pile L pile 30 m 30 m L - pile d pile Thickness of the slab t raft 1,5 m 1 m Spacing between piles a 9 m (6 d pile ) 8 m (5,3 d pile ) Table 5.4: Main geometric parameters We noticed that each pile of the group seems to be mobilizated in the same way (around the same values of R skin and R base ). But the mobilization of skin resistance appears to be lower than for a single pile-raft. The skin friction proles we see in the following pictures conrm these observations. This low skin friction mobilization is due to pile-pile interaction. We represented the skin friction prole in four directions for the center, middle and corner piles of group 5. We added the skin friction proles for the single pile-raft III (in blue). 93

95 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.14: Skin friction prole for the center pile, Load=1000 kn/m 2 94

96 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.15: Skin friction prole for a middle pile, Load=1000 kn/m 2 95

97 5.2. RESULTS CHAPTER 5. GROUP EFFECT Figure 5.16: Skin friction prole for the corner pile, Load=1000 kn/m 2 96