piled raft foundation

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(1)The University of Sydney School of Civil Engineering. Analysis of Piled-Raft Foundations with Piles of Different Lengths and Diameters. by. Helen Sze Wai Chow, B.E, M.E., M.B.A.. A thesis submitted for the Degree of Doctor of Philosophy in the University of Sydney. August 2007.

(2) SYNOPSIS In recent years, there have been an increasing number of structures using piled rafts as the foundation to reduce the overall and differential settlements. For cases where a piled raft is subjected to a non-uniform loading, the use of piles with different sizes can improve the performance of the foundation. Extensive research work has been performed in the past to examine the behaviour of piled rafts. However, most of the research was focused on piled rafts supported by identical piles, and the use of non-identical piles has not received much attention. In this thesis, the behaviour of piled rafts supported by non-identical piles is examined by the use of a computer program APRILS based on the finite layer and finite element methods. The finite layer method is used for the analysis of the layered soil system. The application of this method to different shapes of loadings is presented and has been shown to be in good agreement with the theoretical solutions. The finite element method is used for the analysis of the raft and piles. Full interaction between raft, piles and soil which is of major importance in the behaviour of piled rafts is considered in the analysis. Among the four different types of interaction present in the piled raft foundation, the interaction between piles plays an important role. Interaction between non-identical piles subjected to both horizontal and vertical loadings is examined. It is found that for a pair of piles under consideration, if the diameter and length of one of the piles is a multiple of the diameter or length of the other pile, the reciprocal theorem holds. Program APRILS can be used for the analysis of piled rafts subjected to horizontal and vertical loadings in which the base of the raft is treated as being rough and slip along the pile-soil interface is allowed in the analysis. By having a rough raft base, there is a significant effect on the vertical movement of the raft due to the lateral resistance of the soil and the piles. For piled rafts embedded in a non-homogeneous layered soil, the. i.

(3) modulus of each layer of soil is used in the computation and accurate solutions are obtained without the use of an averaging technique. The performance of piled rafts is affected by several factors such as the soil conditions, pile dimensions and arrangements. Detailed examinations of these factors have shown that the displacement of the raft and the proportion of load carried by piles are significantly affected by them. For piled rafts subjected to non-uniform vertical or horizontal loads, the use of non-identical piles can improve the performance of the piled rafts. For vertically loaded piled rafts, longer piles are prefered to be used underneath the heavily loaded region while for horizontally loaded piled rafts, larger diameter piles are more preferable. Several field cases are studied in this thesis, and they show that the solutions obtained from program APRILS are in good agreement with field measurements. The use of the insitu test results and back-analysis is used to obtain the correlation between the modulus of the soil and field test results for laterally loaded piles.. ii.

(4) PREFACE This thesis is submitted to the University of Sydney, Australia, for the degree of Doctor of Philosophy. The work described in this thesis was carried out by the candidate during the years 2002-2007 in the School of Civil Engineering at the University of Sydney, under the supervision of Professor John Small. The By-laws of the University of Sydney require a candidature for the degree of Doctor of Philosophy to indicate the originality of the work. Unless otherwise indicated in the text, the candidature submits that the work presented in this thesis is original and it includes the following: ¾ The solution of problems for different types of interaction due to the application of horizontal or vertical loadings. ¾ The numerical method (Method II) for the analysis of piled rafts with piles of different sizes and embedded in layered soils subjected to uniform or non-uniform horizontal and vertical loadings. ¾ The solutions for problems involving rafts, pile groups and piled rafts subjected to uniform or non-uniform horizontal and vertical loadings in homogeneous or nonhomogeneous soils. ¾ The technique for predicting the modulus of soil based on the field test results.. iii.

(5) The following supporting papers have been written based on the work described in this thesis: 1.. Small, J. C., Zhang, H. H. and Chow, H. (2004). “Behaviour of Piled Rafts with Piles of Different Lengths and Diameters.”, Proc. 9th Australia – New Zealand Conference on Geomechanics, 8-11 February, Auckland, New Zealand, Vol. 1, pp. 123-129.. 2.. Chow, H. and Small, J. C. (2005a). “Behaviour of Piled Rafts with Piles of Different Lengths and Diameters under Vertical Loading.” Geotechnical Special Publication No. 130-142, Geo-Frontiers, 2005, Austin, USA, pp. 841-855.. 3.. Chow, H. and Small, J. C. (2005b). “Finite Layer Analysis of Combined Pile-Raft Foundations with Piles of Different Lengths.”, Proc. 11th International Conference on Computer Methods and Advances in Geomechanics (IACMAG), 19-24 June, Torino, Italy, Edited by G. Barla and M. Barla, Vol. 2, pp. 429-436.. 4.. Chow, H. S. W. and Small, J. C. (2006a). “Settlement of a Piled-Raft Foundation considering Lateral Pile Resistance.”, Proc. 10th International Conference on Piling and Deep Foundations, 31 May – 2 June, Amsterdam, The Netherlands, pp. 232-241.. 5.. Chow, H. S. W. and Small, J. C. (2006b). “Analysis of piled raft foundations with piles of different lengths subjected to horizontal and vertical loads.”, Proc. 6th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE06), 6-8 September, Graz, Austria, Edited by H. Schweiger, pp. 583-588.. 6.. Chow, H. and Small, J. C. (2007). “Effect of a thin soft layer on the settlement of piled rafts” (accepted for publication in 10th Australia – New Zealand Conference on Geomechanics).. iv.

(6) ACKNOWLEDGEMENTS This thesis was made possible by the great contribution of my supervisor Professor John Small. I am deeply indebted to him for his outstanding supervision, encouragement and guidance throughout the period of my candidature. I would like to express my appreciation to Dr. Tim Hull and Nigel Balaam for their valuable advice. Thanks are extended to the staff at the Centre for Geotechnical Research, Mr. Ross Barker and Antonio Reyno for their assistance and friendship. I would like to thank the staff and postgraduates in the School of Civil Engineering. Special thanks are due to my fellow research students: Dr. Ezzat William, Dr. Bosco Poon, Ms. Nooshin Jabiri, Mr. Jurgen Becque, Mr. Ryan Chen, Mr. Benoit Gilbert, Mr. David Cao, Mr. Derrick Yap, Mr. Niphan Yaiaroon, Mr. Frank Zhu, Mr. Ali Hanandeh, Mr. Thanh Binh Nguyen, Mr. Cao Hung Pham and Mr. Tayakorn Chandranqsu for their friendship and understanding. Finally, I would like to thank my family, especially to my mother for her understanding, encouragement, support and patience, making it possible for me to pursue the challenging work of my interest.. v.

(7) NOTATION The following are some of the more frequently used symbols in this thesis. Symbols used infrequently are not included in the list.. English Letters A. cross-sectional area of pile. Ai. area of element i in the raft. Ax. axial force along the pile. a. radius of ring or circular load. B or Br. width of raft. C. circumference of pile. ca. pile-soil adhesion. D or d. diameter of pile. E. Young’s modulus. Ep. Young’s modulus of pile. Er. Young’s modulus of raft. Es. Young’s modulus of soil. F. vector of nodal forces. G. shear modulus of soil. h. depth of soil. I. influence factor. [Ip]. influence matrix of pile group. [Ir]. influence matrix of the pinned raft. [Is]. influence matrix of the soil. [Isp]. influence matrix of the pile enhanced soil continuum. [Ipr]. influence matrix of the pinned piled raft. [k]i. stiffness of pile element i. [Kp]. stiffness matrix of pile group. vi.

(8) Krs. raft-soil stiffness matrix. [Ks]. stiffness matrix of the soil. L or l. length of the pile. Lr. length of the raft. Mb. bending moment along the pile. Mx , My. total applied bending moments in the x- and y-directions respectively. Mz. bending moment about the z-axis. [Pp]. vector of nodal forces acting at the nodes along the pile. Ppr, Pr. contact stresses on the raft elements. [Ps]. vector of nodal forces at the soil nodes. [Psp]. vector of interface loads between the raft and the pile enhanced soil. Pu. ultimate load capacity of pile. Px, Py, Pz. total applied loadings in the x-, y- and z-directions respectively. Q. total applied load. Qx, Qy, Qz. total external loads in the x-, y- and z-directions applied to the pile group. q. uniform distributed load. r. distance from the centre of the loaded area. S. shear forces along the pile. s. spacing between piles. su. shear strength of soil. Sz. Hankel transform for vertical loads. Tx, Ty. Hankel transform for horizontal circular and ring loads in the x- and ydirections respectively. tr. thickness of the raft. u. horizontal displacement. Ux, Uy, Uz. transform of displacements in the x-, y-, and z-directions. ux, uy, uz. displacements in the x-, y-, and z-directions. v. vertical displacement. W. width of raft. x, y, z. Cartesian co-ordinates. vii.

(9) Greek Letters αpp. pile-pile interaction factor. αps. pile-soil interaction factor. αsp. soil-pile interaction factor. αss. soil-soil interaction factor. δ. displacement vector. δl. vector of soil displacements due to a unit surface load in the x-, y- and zdirections. δp. vector of pile displacements at the nodes in the x-, y- and z-directions. δpr0. vector of displacements at the centre of each raft element or at the nodes of each pile in the x-, y- and z-directions for the pinned raft. δr. vector of displacements at the centres of the raft elements in the x-, y- and z-directions. δr0. vector of displacements at the centres of each raft element in the x-, y- and z-directions for the pinned raft. δs. vector of soil displacements at the soil nodes or at the centres of soil elements. ∆x, ∆y, ∆z. translation of the pinned raft or piled raft in the x-, y- and z-directions. ε. vector of strains. λ. pile-soil relative stiffness. νr. Poisson’s ratio of the raft. νs. Poisson’s ratio of the soil. θx, θy. (1) rotations of the pile head about the x- and y- axes (2) rotations of the pinned raft about the x- and y- axes (3) rotations of the pinned piled raft about the x- and y- axes. θz. rotations of the pinned raft or pinned piled raft about the z-axis. σ. vector of stresses. viii.

(10) CONTENTS Synopsis. i. Preface. iii. Acknowledgements. v. Notation. vi. Contents. ix. CHAPTER 1. INTRODUCTION. 1. 1.1. BACKGROUND. 1. 1.2. OBJECTIVES. 2. 1.3. THESIS OUTLINE. 3. CHAPTER 2 LITERATURE REVIEW. 6. 2.1. INTRODUCTION. 6. 2.2. RAFT FOUNDATIONS. 6. 2.2.1. Analytical Methods. 7. 2.2.2. Boundary Element Methods. 7. 2.2.3. Finite Element Methods. 9. 2.2.4. Hybrid Approach. 11. PILE GROUP FOUNDATIONS. 12. 2.3. 2.3.1. Simplified Analytical Method. 12. 2.3.2. Hybrid Method. 13. 2.3.3. Boundary Element Method. 14. 2.3.4. Finite Element Method. 19. 2.3.5. Infinite Layer Method. 20. 2.3.6. Finite Layer Method. 21. ix.

(11) 2.4. PILED RAFT FOUNDATIONS. 23. 2.4.1. Approximation Method. 23. 2.4.2. Boundary Element Method. 24. 2.4.3. Finite Element Method. 26. 2.4.4. Combined Boundary Element and Finite Element Method. 29. 2.4.5. Combined Finite Layer and Finite Element Method. 32. 2.4.6. Variational Approach. 32. 2.5. 34. CONCLUSIONS. CHAPTER 3 FINITE LAYER ANALYSIS OF LAYERED SOIL. 38. 3.1. INTRODUCTION. 38. 3.2. ANALYSIS OF LAYERED SOIL. 39. 3.2.1. Circular and Ring Loads. 39. 3.2.1.1. Basic Equations for Elastic Materials. 39. 3.2.1.2. Hankel Transforms. 42. 3.2.2. Rectangular Loads. 46. 3.2.2.1. Basic Equations for Elastic Materials. 47. 3.2.2.2. Fourier Transformations. 48. 3.2.2.3. Transformed Equations in the ξ, η Co-ordinate System. 48. 3.2.3. Stiffness Relationship. 52. 3.2.4. Transform of Loads. 53. 3.2.4.1. Hankel Transform of Vertical and Horizontal Ring Load. 53. 3.2.4.2. Hankel Transform of Vertical and Horizontal Circular Load. 54. 3.2.4.3. Hankel Transform of Vertical and Horizontal Rectangular Load. 55. 3.2.5. Solutions for Displacements. 55. 3.2.5.1. Soil Displacements for Vertical Ring or Circular Loads. 55. 3.2.5.2. Soil Displacements for Horizontal Ring or Circular Loads. 56. 3.2.5.3. Soil Displacements due to Vertical Rectangular Loads. 57. 3.2.5.4. Soil Displacements due to Horizontal Rectangular Loads. 58. 3.3. NUMERICAL INTEGRATION. 59. 3.4. NUMERICAL EXAMPLES. 62 x.

(12) 3.4.1. 62. 3.4.1.1. Ring Loads. 62. 3.4.1.2. Circular Loads. 63. 3.4.1.3. Rectangular Loads. 64. 3.4.2. 3.5. Soil subjected to Vertical Loadings. Soil subjected to Horizontal Loadings. 65. 3.4.2.1. Ring Loads. 65. 3.4.2.2. Circular Loads. 66. 3.4.2.3. Rectangular Loads. 67 68. CONCLUSIONS. CHAPTER 4 Analysis of Rafts and Pile Groups in Layered Soil. 91. 4.1. INTRODUCTION. 91. 4.2. ANALYSIS OF RAFTS. 92. 4.2.1. Raft Subjected to Membrane Action. 92. 4.2.2. Raft Subjected to Bending Action. 94. 4.2.3. Combination of Membrane and Bending Actions. 96. 4.2.4. Rafts on Layered Soils. 96. 4.3. 4.2.4.1. Influence Matrix for the Raft. 96. 4.2.4.2. Influence Matrix for the Soil. 98. 4.2.4.3. Analysis of Foundations. 99. ANALYSIS OF PILE GROUPS. 101. 4.3.1. Stiffness Matrix of Pile Group. 101. 4.3.2. Stiffness Matrix for Layered Soil. 103. 4.3.3. Pile Groups in Layered Soil. 104. 4.4. NUMERICAL EXAMPLES. 105. 4.4.1. Examples for Rafts. 105. 4.4.2. Examples for Pile Groups. 110. 4.5. CONCLUSIONS. 115. xi.

(13) CHAPTER 5 SOIL-STRUCTURE INTERACTION. 140. 5.1. INTRODUCTION. 140. 5.2. EFFECT OF SUPERSTRUCTURE ON FOUNDATIONS. 140. 5.3. INTERACTION MECHANISM OF PILED RAFT. 142. 5.3.1. Pile-Pile Interaction. 142. 5.3.2. Pile-Soil Interaction. 144. 5.3.3. Soil-Pile Interaction. 144. 5.3.4. Soil-Soil Interaction. 146. 5.4. NUMERICAL EXAMPLES. 146. 5.4.1. Pile-Pile Interaction. 146. 5.4.2. Pile-Soil Interaction. 152. 5.4.3. Soil-Pile Interaction. 153. 5.4.4. Soil-Soil Interaction. 153. 5.5. CONCLUSIONS. 154. CHAPTER 6 ANALYSIS OF PILED RAFTS. 171. 6.1. INTRODUCTION. 171. 6.2. METHODS OF ANALYSIS. 172. 6.2.1. Method I – Isolated Raft and Pile Group Embedded in Layered Soil. 173. 6.2.2. Method II – Piled Raft and the Layered Soil. 174. 6.2.2.1. Analysis of the Isolated Piled Raft. 174. 6.2.2.2. Analysis of the Layered Soil. 178. 6.2.2.3. Analysis of a Piled Raft in a Layered Soil. 180. 6.2.3. Comparison of the Two Methods. 181. 6.2.4. Non-Linear Analysis of Piled Raft. 184. 6.2.5. Verification of Results with Other Numerical Solutions. 185. 6.3. 194. CONCLUSIONS. xii.

(14) CHAPTER 7 PERFORMANCE OF PILED RAFTS. 233. 7.1. INTRODUCTION. 233. 7.2. PARAMETRIC STUDIES. 234. 7.2.1. Types of Foundation. 235. 7.2.2. Soil Conditions. 237. 7.2.3. Dimension of Piles. 241. 7.3. CONCLUSIONS. 249. CHAPTER 8 Application of Program to Piled Raft Foundations. 285. 8.1. INTRODUCTION. 285. 8.2. CASE STUDIES. 285. 8.2.1. Case 1: Centrifuge Model Tests. 285. 8.2.2. Case 2: Messe-Torhaus Building, Frankfurt, Germany. 288. 8.2.3. Case 3: Liquid Gas Terminal, Gdansk, Poland. 289. 8.2.4. Case 4: Roosevelt Bridge, Florida. 291. 8.2.5. Case 5: Bridge Foundation, Portugal. 294. 8.3. CONCLUSIONS. 298. CHAPTER 9 Conclusions and Recommendations for Future Research 319 9.1. CONCLUSIONS. 319. 9.2. RECOMMENDATIONS FOR FUTURE RESEARCH. 325. APPENDIX I. Exact Solution for Auxiliary Function φ. APPENDIX II. Flexibility Matrix for a Single Layer of Material. 327. 328. xiii.

(15) APPENDIX III. Boundary Conditions for Circular and Rectangular Loadings Applied to a Layered System. APPENDIX IV. 330. Shape Function for Raft Elements (8 Noded Isoparametric Element). REFERENCES. 331 332. xiv.

(16) Chapter 1 - Introduction. CHAPTER 1 INTRODUCTION 1.1 BACKGROUND For most piled raft foundations, the primary purpose of the piles is to act as settlement reducers. The proportion of load carried by the piles is considered as a secondary issue in the design. Over the past decades, extensive research work has been presented, aimed at improving the accuracy in the prediction of the behaviour of piled rafts. In the design of piled rafts, design engineers have to understand the mechanism of load transfer from the raft to the piles and to the soil to predict (i) the behaviour of the raft which includes the settlements, bending moments and the proportion of load carried by the raft, and (ii) the behaviour of the piles which includes the displacements and load distributions along the piles. Interactions between piles, raft and soil are of major concern in the analysis. The concept of interaction between piles introduced by Poulos (1968) was used in the analysis of pile groups and can be extended to the analysis of piled rafts. Methods that have been used for analysis range from simplified calculations to numerical methods such as the boundary element method (Butterfield and Banerjee, 1971b; Brown and Wiesner, 1975; Kuwabara, 1989; Mendonça and de Paiva, 2000) and the finite element method (Hooper, 1973; Ottaviani, 1975; Chow 1987a; Liu and Novak, 1991; Katzenbach and Reul, 1997; Prakoso and Kulhawy, 2001; Reul and Randolph, 2003). In early years, because of the limited availability of computer memory and processing speed, the use of numerical methods was confined to simple problems. In the last two decades due to the rapid development in computer technologies, numerical methods such as full three-dimensional finite element methods are often used to solve complex problems. A finite layer technique based on the flexibility approach was developed by Small and Booker (1984 and 1986) to determine the soil deflections of horizontally layered soil 1.

(17) Chapter 1 - Introduction subjected to different shaped loadings by the use of Fourier transforms. This technique was used in combination with the finite element technique for the analysis of rafts, pile groups and piled rafts embedded in layered soils (Zhang and Small, 1991, Lee and Small, 1991a and b; Ta and Small, 1995; Southcott and Small, 1996) subjected to vertical loadings and then extended to horizontal loadings (Zhang and Small, 1999 and 2000). In the above analyses, the raft was modelled as a thin elastic plate and analysed by a twodimensional finite element method, the piles were modelled as rod or beam elements and analysed by a one-dimensional finite element method and the horizontally layered soil was analysed by the finite layer method. Full interaction between the raft, piles and soil was taken into account in the analyses. However, there are some limitations with these methods: (1) an identical size for each of the piles and a uniform thickness for the raft was used (2) the piles are contained within the raft element which may not be a good simulation of the loads acting at the pile heads for large piled rafts. (3) analysis is separated into an isolated raft and a pile group embedded in the soil and so there is no moment transfer from the pile head to the raft. (4) applied moments have to be transformed into equivalent forces along the edges of the raft elements. 1.2 OBJECTIVES For cases where the superstructures on the piled rafts apply non-uniform loadings, the use of piles of different dimensions underneath the raft might be preferable to minimize the overall and differential settlements and the tilting of the superstructures. Tilting can be a significant problem for a tall building as a small tilt can mean a large horizontal movement at the top of the building. Most of the recent research has been performed for piled rafts with piles of identical size. The objective of this thesis is to develop a numerical method for the analysis of piled rafts with piles of different lengths and diameters by the use of the finite layer method for the analysis of the layered soil and the finite element method for the analysis of the piles. 2.

(18) Chapter 1 - Introduction and the raft. This new method overcomes the limitations mentioned in (1 – 4) above and has the following features: (1) both horizontal and vertical loadings can be applied to the piled raft by treating the base of the raft as being rough. (2) the piled raft is analysed as a whole structure by attaching the piles to the nodes on the raft (3) piles can have different lengths and diameters and the raft can have different thicknesses and any shape (4) applied moments are directly transferred from the raft to the pile heads (5) slip along the pile-soil interface is allowed in the analysis Effects of the dimensions of piles on different kinds of interaction and on the overall behaviour of the piled raft subjected to horizontal and vertical loadings will be examined by the use of the new method developed in this thesis. Results obtained from this new method are compared with those from existing methods and also with field measurements.. 1.3 THESIS OUTLINE The thesis is presented in the following eight chapters. Chapter 2 presents a brief review of previous research work on rafts, pile groups and piled raft foundations. The review outlines the development of different numerical methods for the analysis of the different types of foundations. Chapter 3 presents the general concept of the finite layer theory and the application of the theory to solve for the displacements in a layered soil subjected to vertical or horizontal circular, ring or rectangular loads at the soil surface or soil interface. The displacements can be obtained at any point in a layered soil system. The accuracy of the solutions depends on the numerical integration schemes.. 3.

(19) Chapter 1 - Introduction Chapter 4 presents the use of the finite layer method in the analysis of rafts resting on the soil surface and pile groups embedded in layered soils. The foundation system is analysed by a combination of the finite layer and finite element methods. The analysis of rafts is performed by the use of a two-dimensional finite element method which models the raft as a thin plate. The raft can be of any shape and is divided into a number of rectangular elements. The raft elements can have different sizes and thicknesses. The soil surface underneath the raft is divided into a number of rectangular elements corresponding to the rectangular elements of the raft and is analysed by the finite layer method as described in Chapter 3 for rectangular loads. The analysis of piles is performed by the use of a onedimensional finite element method in which the piles are modelled as beam elements. The loads acting along the pile shaft are modelled as a series of ring loads and the base load is modelled by a uniform circular load at the pile base. The layered soil is analysed by the finite layer method for circular and ring loads. Results obtained from the present method for the raft and pile groups subjected to horizontal and vertical loadings are presented in this chapter. Comparisons with the results from the other numerical methods are also presented which show that the present method is capable of producing accurate results. Chapter 5 presents the interaction mechanism of piled rafts which is comprised of four different types of interaction. Methods used for the computation of each type of interaction are described. Interaction factors between two identical piles embedded in different soil conditions are presented and compared with other existing methods. Interaction factors between two non-identical piles and the effect of the size (i.e. diameter) and length of piles on different kinds of interactions are also presented in this chapter. Chapter 6 presents a new method (Method II) for the analysis of piled rafts in a layered soil system based on the theories described in Chapters 4 and 5. This new method is compared with Method I which was developed in previous research work based on the same theories. In the new method (Method II), the piled raft is analysed as a whole structure and the piles are attached to the nodes on the raft. The interface between the raft and soil is treated as being rough. Slip along the pile-soil interface is also allowed in the. 4.

(20) Chapter 1 - Introduction new method. Comparison of the results between a smooth and a rough rigid base shows that the raft interface has significant effect on the behaviour of the raft. The examples presented in this chapter show that the number of piles, pile length and raft thickness also have effects on the behaviour of piled rafts with identical piles. Chapter 7 presents comparisons of the performance of different types of foundations. The comparisons show that piled raft is effective in reducing the overall and differential settlement and also the skin friction at the pile heads for vertical loadings. Factors such as the soil conditions, pile spacing and pile dimensions on the performance of piled rafts with identical or non-identical piles with respect to the displacements, bending moments and load distributions for the rafts and piles are carefully examined. Non-uniform or uniform loadings are applied to the piled rafts. Chapter 8 presents several case studies that include centrifuge model tests, commercial buildings in Europe and full scale load tests for bridges. Load tests and back analysis of a load test on a single pile are used to obtain the soil modulus that gives a good fit to measured performance in the analyses and then this modulus is used to determine correlation factors that can be used with insitu test results. Chapter 9 summarizes the main conclusions drawn from the research work and suggestions for future research work in this field.. 5.

(21) Chapter 2 – Literature Review. CHAPTER 2 LITERATURE REVIEW 2.1 INTRODUCTION In foundation design, rafts, pile groups and piled rafts are commonly used to support structures. Extensive research work has been carried out and published in the past decades, and different analysis methods have been developed that can be classified into several categories: empirical, analytical and numerical methods. In this chapter, a brief review of the techniques developed for the analysis of raft, pile group and piled raft foundations are presented.. 2.2 RAFT FOUNDATIONS In the design of raft foundations, the soil can be treated (I) as a series of individual springs – known as a Winkler model (1867) or (II) as a continuum. The Winkler model treats the soil as a series of springs and assumes that the pressure at any point on the surface of the soil is related to the modulus of subgrade reaction (or Winkler spring stiffness) and the deflection of the soil. The spring stiffness depends on the settlement characteristic of the soil and the geometry of the foundation. This model can be easily handled by mathematical equations and can produce reasonably accurate results. However, it neglects the interaction between each individual spring and the supporting soil is therefore not modelled as a continuum. An alternative approach that treats the supporting soil as an elastic continuum can better represent the physical behaviour of the supporting soil. The soil parameters used in this approach depend on the field stress state and have to be carefully evaluated (Hain and Lee, 1974). Different methods ranging from one-dimensional to full three-dimensional models have been developed for the analysis of raft foundations.. 6.

(22) Chapter 2 – Literature Review. 2.2.1 Analytical Methods The use of analytical methods for the analysis of rafts on elastic foundations has been investigated by numerous researchers. However, this approach is limited to simple geometrical shapes of the raft and homogeneous soils. Zhemochkin and Sinitsyn (1962) obtained the analytical solution by assuming that the contact pressures between the raft and soil were uniform blocks of pressure. The deflections of the raft and the soil due to the pressure could be determined by considering the compatibility of the displacements of the raft and the soil at a number of points beneath the raft. Brown (1969a) employed a similar method for the analysis of a circular raft on an elastic foundation of finite depth. The raft was divided into a number of equal width annular elements and the contact stress was assumed to be uniformly distributed over the annular elements. Solutions from the analysis were based on the solution presented by Burmister (1956) for a two layer system subjected to a surface point load. Based on the integral transform techniques presented by Sneddon (1951), Brown (1969b) later presented an improved method which provided greater accuracy and less computation. In this improved method the contact stress was represented by a series of mathematical functions instead of uniform annular pressures. The same method was used by Booker and Small (1983) for analysis of liquid storage tanks resting on homogeneous soils. Kay and Cavagnaro (1983) presented a method for the prediction of settlement for raft foundations by the use of field parameters in which the soil can have numerous sublayers having different properties. The raft was replaced by an equivalent uniformly loaded circular area such that the influence of the raft stiffness was considered in the assessment of differential settlement.. 2.2.2 Boundary Element Methods The boundary element method is a powerful tool that can be applied in engineering applications as only the boundary has to be discritized which reduces the amount of computer memory and the time to solve the problem. Katsikadelis and Armenàkas (1984a and 1984b) and Costa and Brebbia (1985 and 1986) used the boundary integral equation. 7.

(23) Chapter 2 – Literature Review method for the analysis of plates resting on a Winkler type elastic foundation. In this method the boundary of the plate was divided into a finite number of elements with a node defined at the midpoint of each element. Each boundary element was approximated by a curve so that the boundary of the plate can be approximated by straight line or curved line segments. The domain was assumed to be bounded by a continuous curve. In order to reduce the domain integrals, Costa and Brebbia (1985 and 1986) suggested that the domain integrals have to be transformed into boundary integrals. Bezine (1988) developed a new boundary element method for bending of plates on a Winkler foundation which used an original boundary integral equation method. The fundamental solution is for plate flexure problems based on the Kirchhoff’s theory. This method required the discretization of the boundary into a number of straight elements and the domain into a number of rectangular panels. The distribution of contact pressures at the interface was replaced by the equivalent loads applied at each node of the discretised domain. The solution was obtained by Gauss-Legendre integration over the elements on the boundary and the panels on the domain. Analysis of unilaterally supported plates on elastic foundations have been presented by Sapountzakis and Katsikadelis (1992) and Hu and Hartley (1994). Sapountzakis and Katsikadelis (1992) presented a boundary element solution for unilaterally supported plates resting on a homogeneous or non-homogeneous elastic foundation. The soil was modelled by independent springs with the subgrade reaction dependent linearly or nonlinearly on the deflection. The contact between the plate and soil was unbonded and separation contact between the plate and soil had been considered. Discretization along the boundary and within the domain was required and the solution was obtained by an iterative process. Hu and Hartley (1994) carried forward the same work by incorporating an elastic half-space model into the analysis. A direct boundary element method was developed which took into account the interaction between an elastic plate and an elastic half-space. The plate and the elastic foundation were separated into two mathematical models. The boundary of the plate was divided into boundary elements and plate bending was modelled by the thin plate boundary integral formulation. The contact area was. 8.

(24) Chapter 2 – Literature Review divided into a set of contact elements or quadratic elements. The contact pressure within the contact elements was assumed to be constant, while for the contact pressure within a quadratic element was expressed by a second order polynomial. de Paiva and Butterfield (1997) presented a formulation for the analysis of plate-soil interaction where the interface was divided into triangular elements in which the displacements and the subgrade reaction varied linearly. Linear functions were used for the approximation of the displacements and bending moment of the boundary elements and the tractions on the triangular elements on the plate-soil interface. The shear force on the boundary was approximated by reaction forces applied to the nodes of the elements.. 2.2.3 Finite Element Methods The first solution which employed the finite element method for the analysis of foundation structures on an elastic half–space was obtained by Cheung and Zienkiewicz (1965). The behaviour of the raft was obtained by the finite element technique in which the raft was divided into a number of rectangular elements joined at a discrete number of nodal points. The soil was modelled either by the Winkler model in which interactions between springs were not considered or by the elastic continuum model in which separation between the raft and the soil was not allowed when negative reactions existed. The stiffness matrix for the whole system was formed by combining the stiffness of the soil (which was derived by using the Boussinesq equation) with the stiffness matrix of the plate bending elements. Contact stresses were represented by equivalent forces applied at nodal points of the finite element mesh. Cheung and Nag (1968) extended this method to incorporate the shear stresses beneath the raft and examined the effects of uplift between the raft and the soil. Svec and Gladwell (1973) developed an improved method for the analysis of a thin plastic plate on an elastic half space. The plate and the surface of the elastic half-space which was in contact with the plate were divided into a number of 10 noded triangular elements. The continuous contact pressure distribution beneath the plate was represented by a cubic polynomial on each of the triangular regions. The displacements at the surface of the elastic half-space due to the contract pressure were determined from the Boussinesq equation.. 9.

(25) Chapter 2 – Literature Review. Wardle and Fraser (1974) and Fraser and Wardle (1976) extended the approach of Cheung and Zienkiewicz (1965) to a multi-layered soil system with isotropic or crossanistropic properties. The layered soil system was divided into a number of horizontal layers of uniform thickness with infinite lateral extent. The loaded surface of the soil mass was discretized into surface elements corresponding to the raft elements, and the raft was modelled by conventional finite elements. The contact between the raft and the soil was assumed to be smooth. The surface element stiffness matrix of the soil was derived from the surface settlements due to uniformly loaded rectangular areas by using the integral transform techniques of Gerrard and Harrison (1971). With the use of surface elements and integral transform techniques, the required computer storage and computational time were less than those required for the three-dimensional finite element method. The approach of Cheung and Zienkiewicz (1965) was also extended by Wood and Larnach (1974 and 1975) to include layered soils and time-dependent consolidation effects in the analysis. Wood (1977) then extended the method further to include applied moments. The raft can be of irregular shape subjected to non-uniform loadings and resting on a non-homogeneous soil mass. Hain and Lee (1974) suggested that in the analysis of raft foundations, the structure, foundation and supporting soil have to be analysed as a system. The stiffness of the structure can have an influence on the distribution of loads and moments transferred to the raft. The structure-raft-supporting soil system was analysed by the “substructure” method developed by Przemieniecki (1968). The supporting soil was modelled by both the Winkler model and the linear elastic model. Results have revealed that there were significant differences in the behaviour of the raft predicted by the use of different soil models for the supporting soil. Flexibility of the raft has significant effects on the distribution of column loads and moments. Results have shown that the linear elastic continuum model provided a more realistic solution to the behaviour of the raft and is more preferable to use in modelling the supporting soil.. 10.

(26) Chapter 2 – Literature Review. Sharma et al. (1984) used the finite element method for the analysis of rafts of any shape resting on an elastic half space. The raft was modelled by using eight noded isoparametric quadrilateral plate bending elements in which shear deformations were taken into consideration. The global plate bending stiffness matrix was formed by assembling the stiffness matrices for each element. The distribution of contact pressure in an element was represented in terms of shape functions and the vertical deflection at the node due to the contact pressure on an element was obtained by the Boussinesq solution.. 2.2.4 Hybrid Approach Zhang and Small (1991) presented a method for the analysis of soil-raft interaction. This method employed the finite layer technique to determine the behaviour of the soil and the finite element technique for the analysis of the raft. The contact pressure between the raft and the soil was represented by uniform blocks of pressure. The response of the soil due to the contact pressure was obtained by the Fourier transform technique. This method can be used for the analysis of rafts of any shape in plan and subjected to uniform, concentrated or eccentric loads. The elastic soil can be anisotropic or non-homogeneous. Mandal and Ghosh (1999) presented a coupled finite element and boundary element approach for the prediction of elastic settlement of a raft on a semi-infinite elastic continuum. The raft was modelled by isoparametric plate bending finite elements and the raft-soil interface was modelled by boundary elements. The domain of the boundary was divided into a number of isoparametric quadrilateral quadratic elements. The raft was divided into elements corresponding to the boundary elements of the soil and the response of the soil due to the load was obtained from the Mindlin solution for a point load. Rashed (2005) developed a new boundary/domain element method for the analysis of a raft on elastic foundations. Shear deformable plate bending theory was used to model the raft, the soil was modelled by continuous springs following the Winkler model and the. 11.

(27) Chapter 2 – Literature Review raft domain was divided into trapezoidal or general shaped cells. The associate domain integral was replaced by using an equivalent boundary integral along each cell contour.. 2.3 PILE GROUP FOUNDATIONS 2.3.1 Simplified Analytical Method Randolph and Wroth (1978) developed an approximate closed form solution for the analysis of single vertically loaded piles. In this approach, the soil was divided into an upper and a lower layer in which the base of the upper layer corresponded to the level of the base of the pile. The deformation of the upper layer was due to the load acting along the pile shaft, whereas the deformation of the lower layer was due to the load acting on the pile base. For the upper layer, the deformation of the soil around the pile shaft was modelled as shearing of concentric cylinders (Cooke, 1974). For the lower layer, the base of the pile was assumed to act as a rigid punch on the surface of the layer, and this layer was acting as a restraint on the deformation of the upper layer. This approach was then extended to the analysis of pile groups by the same authors (Randolph and Wroth, 1979) by incorporating the interaction between loaded piles. The interaction factors for the pile shaft and base were considered separately. For rigid pile groups, the interaction factors were computed using an approximate closed-form expression, while for compressible pile groups, the interaction factors were obtained by an iterative procedure to establish a relationship which expressed the shaft displacement in terms of the pile head and pile base displacement. The overall displacement of a pile with the presence of adjacent loaded piles was obtained by the principle of superposition. This approach was limited to piles of the same embedded length. Lee (1993) presented an approach which was modified from expressions employed by Randolph and Wroth (1978). The expression for approximating the load-settlement behaviour of a pile was modified for a compressible pile embedded in a soil with the stiffness increasing linearly with depth. The interaction factors were evaluated directly from the approximate closed-form analytical model for compressible piles. The settlement of pile groups could then be obtained by the principle of superposition.. 12.

(28) Chapter 2 – Literature Review. Guo and Randolph (1996 and 1999) developed a program GASGROUP which employed an exact closed form solution for predicting the settlement of pile groups in nonhomogeneous soil by using a load transfer approach. The stiffness of the soil was assumed to increase with some power of depth (Booker et al., 1985). The pile-soil interaction was represented by a series of independent springs along the pile shaft and at the pile base. The load transfer factors take into account the pile slenderness ratio, the soil non-homogeneity factor and the Poisson’s ratio. The interaction factor between the two identical piles was computed by modifying the load transfer factor in the closed form solution for a single pile to account for the presence of a neighbouring pile. The settlement of an individual pile in the group was then computed by the superposition of the interaction factors. Lee and Xiao (2001) presented an approach for non-linear analysis of pile groups in multi-layered soils. The approach employed the solution presented by Randolph and Wroth (1979) to simulate the interaction between two piles and a hyperbolic function as the load transfer function to model the non-linear behaviour between the shear stress and displacement of the pile shaft along the pile-soil interface. The non-linear displacement was approximated as displacement discontinuity to model the non-linear local shear displacement developed at the disturbed soil around the pile shaft. The interaction effect between two identical piles was assumed to be linearly elastic and the shaft and base interaction for individual piles in the group was considered separately. By the principle of superposition, the load transfer function for the pile group was obtained which accounts for the non-linear local shear displacement surrounding the pile shaft and the interactive effect of adjacent piles.. 2.3.2 Hybrid Method A hybrid model was proposed by O’Neill et al. (1977) for the analysis of pile groups. The response of the pile was modelled by the load-transfer method and the pile-soil-pile interaction was determined based on Mindlin’s solution. In this approach, by using the. 13.

(29) Chapter 2 – Literature Review solution for a single pile (i.e. interaction effects were ignored), the response of individual piles in the group was determined. The computed soil reactions were then used to determine the additional soil displacements at the nodes of other piles in the group using Mindlin’s solution. The additional soil displacements were used to adjust the load transfer curve to account for the group effects. The solution was obtained following an iterative procedure. Chow (1986) refined the approach by considering the pile-soil-pile interaction directly in the hybrid model. The response of the individual pile was computed by the load transfer curve presented by Randolph and Wroth (1978) and the non-linear behaviour of the soil was approximated by a hyperbolic function. The interaction between piles based on Mindlin’s solution was computed by replacing the continuous distributed loads at the pile shaft and pile base by point loads at the nodes on the piles. The non-linear response of an individual pile was approximated by introducing a discontinuity in the soil. Since the non-linearity is only confined to a narrow zone adjacent to the pile it was assumed that it would not have any effect on other piles, and therefore, the interaction between piles would remain elastic. Leung and Chow (1987) extended the approach by Chow (1986) to analyse laterally loaded pile groups. For lateral loading, the soil response was modelled by the modulus of subgrade reaction approach. The pile was divided into a number of discrete beam elements and the soil was modelled by non-linear springs at the nodal points on the pile elements. The pile and soil stiffness were determined by the conventional finite element method. The individual pile response was obtained from a load-transfer curve. The interaction between piles was obtained by Mindlin’s solution for a unit horizontal point load in a homogeneous, isotropic elastic half-space. An averaging procedure was used for the analysis of pile groups in non-homogeneous soil.. 2.3.3 Boundary Element Method Poulos and Davis (1968) presented the solution for a single pile by using Mindlin’s equations. Poulos (1968) further extended the method to the analysis of pile groups by introducing an interaction factor, α, to examine the interaction effect between two identical piles. The interaction factor was defined as. 14.

(30) Chapter 2 – Literature Review. additional settlement due to an adjacent pile α = _____________________________________ settlement of a pile under its own load. (2.1). In the analysis, each pile was divided into a number of cylindrical elements. Each element was subjected to a uniform load around the periphery of the element and a uniform circular load at the circular base of the pile as shown in Figure 2.1. The shaft of the pile was assumed to be perfectly rough while the base was assumed to be perfectly smooth such that shear stresses were not developed on the base. The vertical displacement of the soil adjacent to the pile was expressed as. [ρ ] = ([1 I ] + [ 2 I ])[ p ] + ([1 I b ] + [ 2 I b ]) pb. (2.2). where [ρ] = displacement of the soil adjacent to the pile [1I] = vertical displacement influence factors for elements due to a uniform load on each element on pile 1 [2I] = vertical displacement influence factors for elements due to a uniform load on each element on pile 2 [1Ib] = vertical displacement influence factors for the pile base due to a uniform load on the base of pile 1 [2Ib] = vertical displacement influence factors for the pile base due to a uniform load on the base of pile 2 [p] = uniform shear load on pile shaft [pb] = uniform vertical stress on pile base The displacement factors were obtained by integration of the Mindlin equation for vertical displacement due to a point load within a semi-inifinite soil mass. By considering the compatibility of the vertical displacement (i.e. unit displacement, ρ = 1), equation (2.2) can be solved to obtain the distribution of the shear stress along the pile shaft and the vertical stress on the pile base and subsequently the displacement of the pile can be determined. For a group of m piles, the displacement of an individual pile in the group was obtained by superposition j =m. ρ k = ρ1 ∑ Pj α kj j =1. 15. (2.3).

(31) Chapter 2 – Literature Review. αkj = interaction factors for piles k and j. where. Pj = load on pile j. ρ1 = displacement of a single pile under unit load Poulos (1971) further extended the method to laterally loaded pile groups by introducing the interaction factor for rotation, αθ, which is defined as additional rotation due to adjacent pile αθ = _____________________________________ rotation of pile under its own load. (2.4). In the analysis, each pile was assumed to be a vertical strip with a length and breadth of L and d respectively. The pile was divided into (n + 1) elements and each of the elements was subjected to a uniform horizontal stress. The length of the elements at the pile top and base were L/2n, whereas the length of elements along the pile shaft was L/n as shown in Figure 2.2. The lateral displacements at the soil surface can be obtained from the method presented by Poulos (1968) by replacing the vertical loads with horizontal loads. The displacement of the pile can be expressed as ⎛ j =m ⎞ ⎜ ⎟ ρ k = ρ H ⎜ ∑ H j α ρHkj + H k ⎟ ⎜ jj =≠1k ⎟ ⎝ ⎠ where Hj. ρH. (2.5). = load on pile j = unit reference displacement of a single free-head pile under a unit horizontal load. αρHk = interaction factor for pile k and j Butterfield and Banerjee (1971a) presented the analysis of compressible piles and pile groups with a rigid floating pile cap based on the integration of the Mindlin solution. The pile shaft was divided into a number of equal segments and the base into a number of rings. For the analysis of a compressible pile, the pile was assumed to be perfectly bonded to the soil medium. The same method was also used for the analysis of compressible pile groups with the pile cap in contact with the ground (Butterfield and Banerjee, 1971b). The cap-soil and pile-soil interfaces were divided into discrete. 16.

(32) Chapter 2 – Literature Review elements. The displacements of the soil due to the direct stresses acting on the interfaces were obtained from Mindlin’s equations. Results from this method have shown that the load-displacement behaviour of similar pile groups with floating or contacting caps were of little difference while the load distribution for groups with contacting caps differed substantially from those groups with floating caps. Chin et al. (1990) presented a method based on the analytical layered solutions of Chan et al. (1974) for the analysis of an axially loaded vertical pile group embedded in a layered soil with the pile cap not in contact with the ground. The foundation system was decomposed into (i) a pile group subjected to the pile-soil interaction forces along the pile shaft and base and the applied loads (ii) the soil continuum subjected to interface forces. The piles were divided into a number of axial bar elements. The flexibility coefficient which defined as the displacement at a node due to unit interaction force at another node was obtained from the analytical two-layered solution of Chan et al. (1974) while for the nodes on the same pile, the displacement was obtained from the Gaussian quadrature method by applying a uniformly distributed force over the element. Slip at the pile-soil interface was allowed by limiting the pile-soil interaction forces as described by Chow (1986). Pile groups with piles of different radii, lengths and moduli can be analysed by this approach. Lee and Poulos (1990) developed an approach for the analysis of pile groups in nonhomogeneous soil. This method involved the development of soil models which account for the soil modulus of all soil layers and the horizontal non-homogeneity of the soil due to soil disturbance caused by pile installation. The approach was a modification of the methods by Poulos (1979) and Yamashita et al. (1987). Poulos (1979) used some equivalent soil moduli computed from moduli of the influencing and influenced elements. Yamashita et al. (1987) modified the method by computing an equivalent soil modulus from the weighted averages of the soil modulus at every layer. Xu and Poulos (2000) developed a fully coupled load-deformation computer program GEPAN for the analysis of single piles and pile groups subjected to three-dimensional. 17.

(33) Chapter 2 – Literature Review loadings and ground movements. The analysis was based on the principles of the threedimensional boundary element method and incorporated the effects of defective piles, soil movements and on/off pile loadings. The global matrix for the governing equation was derived using the concept of hierarchical structures and a basic influence factor matrix. The piles were assumed to be circular in cross-section and each pile was divided into a series of cylindrical elements on the shafts and ring elements on the bases and discontinuities. The cylindrical elements were then divided into several sub-elements. The soil-pile interface was modelled by soil elements and pile elements which were meshed in partly cylindrical or annular surfaces. In the analysis, the cylindrical or annular boundary elements were transformed into rectangular elements by mathematical transforms. The rectangular elements were then divided into a number of smaller rectangles and interaction between the elements was obtained from the integration of Mindlin’s equation. The program GEPAN can be applied to a variety of pile problems such as (i) the off-line effects of piles which is horizontal pile head movement due to vertical load on the pile group, (ii) interaction factors between piles and pile groups, (iii) pile groups containing defective piles, (iv) non-linear and elastoplastic analysis. Results for the off-line effect of piles have shown that for a highly compressible and closely spaced pile group, loading the piles axially would cause significant horizontal pile movements. Basile (2003) extended the approach proposed by Butterfield and Banerjee (1971a) and developed a computer program PGROUPN which was based on a complete non-linear boundary element formulation. The program PGROUP accounts for the interactions between piles, group stiffening effects, load-deformation coupling and non-linearity of the soil. The pile-soil interface was discretized into a number of cylindrical elements and the base was represented by a circular element. A substructuring technique was employed such that the piles and the surrounding soil were considered separately. For the soil domain, the flexibility matrix was obtained from the integration of the Mindlin functions. The piles were modelled by simple beam-columns and Bernoulli-Euler beam theory was applied to obtain the flexibility matrix. Slip was allowed in the computation by limiting. 18.

(34) Chapter 2 – Literature Review the stresses at the pile-soil interfaces and the pile base. The non-linear soil behaviour was approximated by a hyperbolic stress-strain relationship.. 2.3.4 Finite Element Method Ottaviani (1975) used the three-dimensional finite element method for the analysis of vertically loaded pile groups with or without pile caps. Due to the complexity of the single element stiffness computation and large number of elements, the piles and the soil were assumed as weightless linearly elastic homogeneous media for the examination of the load transfer mechanism. It was found that the presence of a cap would cause a nonuniform distribution of load among the piles of the group. If the cap is in contact with the soil surface, reduction of the shear stress in the soil around the upper portion of the pile was found. Chow (1987a) presented a method based on elasticity theory for the analysis of axially and laterally loaded pile groups embedded in isotropic non-homogeneous soils. The axial and lateral group response was assumed to be uncoupled. The foundation system was decomposed into two systems – (i) a pile group subjected to external loads and pile-soil interaction forces, and (ii) a layered soil continuum subjected to pile-soil interaction forces. The piles were discretized into discrete elements and the soil was discretized into constant elements at the pile-soil interface with uniformly distributed vertical or lateral forces acting over each element. The pile group and the soil continuum were analysed separately. The load deformation relationship of the soil was determined using the flexibility approach in which the soil flexibility coefficients were evaluated using the finite element method with a Fourier series. By applying equilibrium of the pile-soil interaction forces and the compatibility of the pile and soil displacements, the load deformation relationship of the pile was determined and expressed as ([Kp] + [Ks]){wp} = {Q}. (2.6). where [Kp] = stiffness matrix of piles [Ks] = stiffness matrix of the soil obtained by inverting the soil flexibility matrix, [Fs], i.e [Ks] = [Fs]-1 {wp} = vector of deformations at the pile nodes. 19.

(35) Chapter 2 – Literature Review {Q} = vector of external applied loads The reinforcing effect of all piles in the group was considered in the formulation. Nonhomogeneity of the soil was taken into account by incorporating a continuously varying soil stiffness into the numerical integration process during the formulation of the element stiffness matrices. The soil non-linearity was assumed to be elasto-plastic and was analysed by limiting the shear forces or lateral soil pressures to some values and the excess soil forces at the nodes were then redistributed to other nodes. Chow (1989) extended the approach to analyse pile groups in cross-anisotropic soils. This was done by replacing the constitutive isotropic soil model with a cross-anisotropic soil model. Results have shown that the effect of soil anisotropy on small pile groups embedded in homogeneous soils was small, however, the effect on large pile groups in nonhomogeneous soils was significant.. 2.3.5 Infinite Layer Method Guo et al. (1987) presented an inifinite layer method for the analysis of piles. This method was based on the finite layer concept presented by Cheung (1976). The analysis was carried out in the cylindrical co-ordinate system as shown in Figure 2.3. The layered soil mass was divided into a number of horizontal layers, each layer was represented by an infinite layer element. The pile was divided into a number of elements along its length and was treated as a solid bar. Interaction between the soil and pile was defined by the compatibility conditions along the pile-soil interface. Cheung et al. (1988) extended the method to the analysis of pile groups with caps by incorporating an iteration procedure to determine the interaction between two identical piles. The cap was discretized into a number of eight noded isoparametric bending elements and analysed by Mindlin thick plate bending theory. The connection between the cap and piles was assumed to be a sliding ball joint such that only the vertical forces were transmitted from the cap to the pile heads.. 20.

(36) Chapter 2 – Literature Review. 2.3.6 Finite Layer Method The finite layer method developed by Small and Booker (1984 and 1986) was first introduced into the analysis of axially loaded piles embedded in isotropic and crossanisotropic layered soils by Lee and Small (1991a). This method is similar in principle to that of the infinite layer method of Guo et al. (1987). A single pile embedded in a layered soil was treated as two components: (i) the single isolated pile, and (ii) the layered soil. The pile was divided into a series of one-dimensional two noded elastic cylindrical solid elements. The forces acting on the nodes of the elements were treated as point loads and were evaluated from the uniformly distributed interaction forces acting on the circumferential area of each pile shaft element and for the base node from the uniform pressure acting over the area of the pile base. The layered soil component was subjected to the interaction forces along the pile-soil interface which were represented by a series of ring loads on the soil nodes and a uniform load on the base. The displacements at all soil nodes, which form the soil flexibility matrix, were determined by applying a unit load at each soil node in turn. The soil stiffness matrix was obtained by inverting the flexibility matrix. Combining the pile and soil stiffness formed the stiffness matrix for the system and displacements of the pile group were then obtained by solving a set of stiffness equations. Lee and Small (1991b) modified this approach for the analysis of laterally loaded piles by replacing the vertical loads acting along the pile shaft and base with horizontal loads. Southcott and Small (1996) extended the approach of Lee and Small (1991a) to the analysis of vertically loaded pile groups. Instead of using the flexibility approach to generate the flexibility matrix for the soil, the stiffness method was used so that loads could be applied at the layer interfaces. Zhang and Small (2000) proposed two methods based on the finite layer theory to analyse axially and laterally loaded pile groups embedded in homogeneous and nonhomogeneous soils. The principle of the methods is similar to that used by Lee and Small (1991a and b). The finite layer theory was employed for the layered soil and simple beam theory for the piles. The piles were divided into a series of finite elements and the soil 21.

(37) Chapter 2 – Literature Review was divided into corresponding layers. Interaction and stiffness methods were developed to generate the influence matrices for the soil and the pile group. In the interaction method, each pair of piles in the pile group was considered in turn to compute the soil influence and pile influence matrices. The soil influence matrix was formed by applying unit ring loads (to each node along the shaft in turn) or a circular load (to the base node) to compute the displacement at each node of the layered soil. The pile influence matrix was formed by pinning the top of the pile to stop rigid body rotations and translations of the pile and a similar method was used to form the soil influence matrix. For the stiffness method, the soil stiffness matrix was formed by first generating the soil influence matrix. The influence matrix was formed by applying a unit ring load or circular load (base node) to each node on all piles in turn to obtain the displacement of the soil. The soil stiffness matrix was obtained by inverting the soil influence matrix. The pile stiffness matrix was generated by assembling the stiffness matrix of all the elements of the piles. Comparison of results between the two methods has shown that the stiffness method is the most accurate method, however, it is not practicable for large pile groups as it requires a large amount of computer memory. The shortcoming of the interaction method is that the formation of the pile influence matrix only considers the interaction between a pair of piles and ignores the effects of other piles in the pile group. Therefore, the stiffness method is more suitable for small pile groups with any pile spacing while the interaction method is suitable for large pile groups with large pile spacing. Zhang and Small (2000) further extended the method to include the pile cap in the analysis. The analysis was separated into three parts: the cap, the piles and the layered soil. The cap was assumed to be a thin elastic plate and analysed by the finite element method. The element division of the cap was such that the pile head fitted within one element of the cap. In the analysis of the cap, the cap has to be restrained from rigid body rotations and translations by pinning two corner nodes. The influence matrix of the pinned cap was generated by applying a uniform horizontal or vertical load to each of the cap elements in turn to obtain the central displacements of all cap elements. The influence matrices of the piles and the layered soil were obtained from the iteration method presented by Zhang and Small (1999). By considering the equilibrium of interaction forces and compatibility of. 22.

(38) Chapter 2 – Literature Review displacements between the pile heads and the cap, the contact stresses can then be obtained and displacement of the raft can then be computed by solving the equations.. 2.4 PILED RAFT FOUNDATIONS 2.4.1 Approximation Method One approach that treated the raft as a thin plate, the piles as springs and the soil as an elastic continuum, was used by Hongladaromp, Chen and Lee (1973) in which the interaction effects between the piles were ignored. Poulos (1994) developed a program GARP (Geotechnical Analysis of Raft with Piles) which employed a finite difference method for the raft with the consideration of the interaction effects between the piles and raft. Allowances were made for the piles to reach their ultimate capacities and local bearing failure of the raft. Russo (1998) employed a similar method where the piles and soil were modelled by linear or non-linear interacting springs. The soil displacements were calculated using Boussinesq’s solution thus yielding a closed form solution. The non-linear behaviour of the piles was modelled by the assumption of a hyperbolic load-settlement curve for a single pile. This method has the limitation of only allowing for pure vertical interaction between the raft, piles and soil. Randolph (1983) presented a method to compute the interaction between a single pile and a circular raft. A flexibility matrix method was then used to calculate the overall stiffness of the piled raft foundation by combining the individual stiffness of a single pile-raft unit. Clancy and Randolph (1993) employed a hybrid method which combined finite elements and analytical solutions. The raft was modelled by two-dimensional thin plate finite elements, the piles were modelled by one-dimensional rod finite elements and the soil response was calculated by using an analytical solution. The pile was attached to a raft element at a common node, such that the vertical freedoms are common at the connected. 23.

(39) Chapter 2 – Literature Review nodes. Mindlin’s solution was used to compute the interaction between the components. Effects of the pile and raft stiffnesses on displacements and bending moments of the foundation were examined and it was demonstrated that the differential displacements and bending moments were dependent on the raft-soil stiffness ratio which was introduced by Hain and Lee (1978). The load sharing and the average displacement of the raft were dependent on the pile-soil stiffness ratio. This method took into account the non-linearity of pile behaviour and slip was allowed to occur at the pile-soil interface. However, this method is limited to homogeneous soil conditions. Kitiyodom and Matsumoto (2003) presented a similar approach to Hain and Lee (1978), but the piles were modelled by elastic beams and the interactions between structural members were approximated by Mindlin’s solutions. The foundations can be subjected to both axial and lateral loads and embedded in non-homogeneous soil. This approach incorporated both the vertical and lateral resistance of the piles and the base of the raft in the analysis.. 2.4.2 Boundary Element Method In this method, discretization is only required on the boundary of the system under consideration. This technique requires the transformation of the governing partial differential equation into an integral equation. As only the boundaries have to be discretized, the number of sets of equations to be solved is generally smaller than the finite element or finite difference methods. Solutions such as stresses and displacements can be obtained directly by solving the system of equations. Since only the boundaries are discretized, interpolation errors are confined to the boundaries. As this method provides a direct and accurate solution for the analysis, is fast, and requires a moderate amount of computer storage space, it can be used for the analysis of large pile groups. Butterfield and Banerjee (1971b) employed the boundary element method to study the behaviour of a pile group embedded in an ideal elastic half space with a perfectly rigid cap not in contact with the ground. Soil-structure interaction was taken into account in. 24.

(40) Chapter 2 – Literature Review the analysis. Mindlin’s solution was used to describe the soil response and the interaction effects. Brown and Wiesner (1975) used the boundary element method to analyse a strip footing supported by equally spaced identical piles embedded in an isotropic homogenous elastic half space. In this method, the raft and piles were divided into a number of zones in which interface forces or pressures acted on the corresponding zones. Application of Mindlin’s solution was used to determine the interaction relationships due to the interface forces. Kuwabara (1989) described a boundary element analysis based on elastic theory to examine the behaviour of a piled raft foundation in a homogeneous elastic soil mass. In the analysis, the raft was assumed to be rigid but compressibility of the piles was considered. The raft was discretized into a series of rectangular elements and the pile was discretized into a series of shaft and base elements. Poulos (1993) extended the method to incorporate the effect of free-field soil movement, load cutoffs for the pile-soil and raftsoil interfaces to examine the interaction mechanism between the piled raft and a soil subjected to externally imposed vertical movement. The analysis is implemented via a computer program PRAWN (Piled Raft With Negative Friction) Mendonça and de Paiva (2000) presented a boundary element method for the analysis of piled rafts in which full interaction between the raft, piles and the soil is considered. Unlike the other approaches, discretization of the foundation system was not required in this approach. The soil was represented by a Mindlin elastic linear homogeneous half space. The raft was assumed to be a thin plate and was represented by integral equations. The pile was represented by a single element and the shear stresses along it were approximated by a second-degree polynomial. The interaction between the raft and soil was analysed by dividing the interface into triangular elements and the subgrade reaction was assumed to vary linearly across each element.. 25.

(41) Chapter 2 – Literature Review. 2.4.3 Finite Element Method The finite element method is one of the most powerful tools for the analysis of piled rafts. It requires the discretation of both the structural foundation system and the soil. In order to reduce the computational effort, problems are sometimes simplified to an axisymmetric problem or a plane-strain problem. An early example of the analysis of a piled raft (the Hyde Park Barracks) was given by Hooper (1973), in which an axisymmetric model with eight noded isoparametric elements was used. In the analysis, approximation of the equivalent stiffness of the pile group was made such that each concentric row of piles was modelled by a continuous annulus with an overall stiffness that was equivalent to the sum of the stiffnesses of the individual piles. The soil was assumed to be a linear elastic isotropic material with the modulus increasing linearly with depth. In order to incorporate the additional stiffening effect of the superstructure into the analysis, an equivalent raft thickness which had the same bending stiffness as the combined raft and the superstructure was introduced. However, Hooper’s results have shown that the contribution of the stiffening effect of the superstructure on the behaviour of the piled raft was relatively small in the case of the Hyde Park Barracks, although this may not be true in all cases. Ottaviani (1975) applied this method to the analysis of a rigid raft resting on a pile group embedded in a homogeneous medium. The piles and the soil were assumed to be a weightless linearly elastic homogeneous media. Chow and Teh (1991) presented a numerical method to examine the behaviour of a rigid piled raft embedded in a non-homogeneous soil. The raft was discretized into square subelements. The base of the raft was assumed to be perfectly smooth and the interface of the raft and the soil medium was approximated by square subdivisions (Chow 1987a). The soil was assumed to be a linearly elastic, isotropic material and the Young’s modulus assumed to increase linearly with depth. The piles were assumed to have a circular crosssection and were discretized into two noded elements at the pile-soil interface (Chow, 1987b). Interactions between the piles, the raft and the soil were taken into account and the vertical deformation of the soil was determined by the principle of superposition in which equilibrium of the raft-pile-soil system was considered. 26.

(42) Chapter 2 – Literature Review. Liu and Novak (1991) employed the finite element method to examine the behaviour of a raft supported by a single pile at the centre. In the analysis, the cap was assumed to be circular and to contact the soil perfectly. Nine noded isoparametric elements were used to model the cap, the pile and the near-field soil medium. The cap, the pile and the surrounding soil medium were modelled by finite elements. Mapped infinite elements proposed by Danjanic and Owen (1984) are used to model the far-field soil medium to stimulate an unbounded domain. The pile was assumed to be linearly elastic and the soil was assumed to be either elastic or elastic perfectly plastic. A weak zone around the pile with lower strength and modulus was introduced to account for the slip at the pile-soil interface. This method allows for the analysis of a piled raft where the raft is embedded as well as in contact with the ground. The use of inifinite elements to model the far-field soil medium reduced the number of elements required significantly which made the analysis more efficient than the conventional finite element method. Wiesner (1991) presented a method for the analysis of a circular piled raft that was constructed in Cairns. The raft was treated as a thin elastic plate and modelled by rectangular plate bending finite elements. The reaction forces acting on the raft-soil interfaces were assumed to be rectangular blocks of uniform vertical stresses. The piles were represented by elastic cylinders and the soil was assumed to be linearly elastic. The reaction forces on the pile-soil interfaces were treated as uniform vertical shear stresses along the pile shaft and as a uniform vertical stress at the pile base. To take interaction into account, the reciprocal theorem was applied to the pile, and influence factors were calculated based on elastic theory. Non-linear behaviour of piles and soil were considered by comparing the reaction forces with the limiting values during the iterative process. Katzenbach and Reul (1997) described a structural model which employed the finite element method for the geometrical modelling of the continuum, an elastoplastic constitutive model to describe the soil behaviour and a step-by-step analysis for numerical simulation. The piles were modelled by 3-dimensional isoparametric finite elements and the raft was modelled by shell elements. A realistic stress-strain behaviour. 27.

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