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The International Journal of Geomechanics Volume 2, Number 1, 29–45 (2002) Behavior of Piled Raft Foundations Under Lateral and Vertical Loading J.C. Small and H.H. Zhang Received June 2, 2001 B.Sc. (Eng), Ph.D., F.I.E.Aust., MASCE Department of Civil Engineering, University of Sydney, NSW 2006, Australia B.E., M.E., Ph.D. Department of Civil Engineering, University of Sydney, NSW 2006, Australia ABSTRACT. This article presents a new method of analysis of piled raft foundations in contact with the soil surface. The soil is divided into multiple horizontal layers depending on the accuracy of solution required and each layer may have different material properties. The raft is modeled as a thin plate and the piles as elastic beams. Finite layer theory is employed to analyze the layered soil while finite element theory is used to analyze the raft and piles. The piled raft can be subjected to both loads and moments in any direction. Comparisons show that the results from the present method agree closely with those from the finite element method. A parametric study for piled raft foundations subjected to either vertical or horizontal loading is also presented. I. Introduction The behavior of pile groupsundervertical and horizontal loadinghasreceivedmuchattention in the past. In early methods developed by Butterfield and Banerjee [1], Davis and Poulos [3], and Kuwabara [5] for the analysis of piled raft foundations subjected to vertical loadings, the raft was considered to be either perfectly flexible or completely rigid. More recently, a variational approach has been developed by Shen et al. [8] for pile groups with a rigid cap. However, these methods cannot deal with a pile group connected at the heads by a flexible raft of any stiffness. The method developed by Hain and Lee [4] considered the interactions of the piles, raft and soil, but the rotations and horizontal movements of a pile head induced by a vertical load applied to an adjacent pile or the soil surface were ignored. Clancy and Randolph [2] and Poulos [6] developed approximate methods for analysis of piled raft foundations subjected to vertical loading or momentsrather than horizontal loads. Based on finite layer theory, Ta and Small[9] developed a method for analysis of a piled raft (with the raft on or off the ground). As for Hain and Lee’s method, the solutions were only for vertical loads. Zhang and Small[11] subsequently developed a method for analysis of piled raft foundations subjected to both vertical and horizontal loadings. In this method, the interactions between raft and piles, raft and soil, piles and piles, piles and © 2003 ASCE DOI: 10.1061/(ASCE)1532-3641(2002)2:1(29) ISSN 1532-3641 Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 30 J.C. Small and H.H. Zhang soil, and soil and soil are fully considered. However, the method can only deal with piled raft foundations clear of the ground. In this article, an extension of the method presented by Zhang and Small [11] has been devel- oped, where the raft can be in contact with the ground surface. The approach uses a combination of the finite layer method for modeling the soil and the finite element method for simulating the raft and piles. The piled raft foundations can be subjected to horizontal and vertical loads as well as moments, and the movements of the piled raft in three directions (x, y,z) and rotations in two directions (x, y) may be computed by the present program APRAF (Analysis of Piled RAft Foundations). Comparisons of the present solutions with those of the finite element method have been made and the effects of parameters (adopted for the soil and raft) on the behavior of piled rafts have been examined. II. Method of analysis As shown in Figure 1, the problem of the piled raft foundation can be solved by assuming that the forces between the piles and layered soil can be treated as a series of ring loads applied to “nodes” along the pile shaft. These loads are both horizontal and vertical, and if enough are used, they well approximate the continuous forces that act along the pile shaft. The contact stresses that act between the raft and the soil can be considered to be made up of uniform rectangular blocks of pressure that approximate the actual stress distribution. These can be considered uniform vertical blocks of pressure or uniform shear stresses. The displacement of thelayeredsoilcantheneasilybecomputed,asthesolutionforalayered soil subjected to ring loads at the layer interfaces may be obtained from finite layer theory [7]. The same theory may be used to determine the deflection of the soil due to vertical and horizontal loads applied over rectangular regions on the soil surface. Firstly, the response of the piles and soil (with no raft) is computed by applying unit surface loads to the rectangular regions on the ground surface or unit ring loads to the soil along the pile shaft, or a unit uniform circular load at the base of the pile. The deflections so computed can be used to form the influence matrix for the soil. The columns of the influence matrix are made up of the deflections at the centers of other loaded areas or at the positions of the ring loads due to application of one of the unit loads. Therefore, we can write δ i s = m j=1 I ij P j (1) where m is the total number of unit loads which will be three times the number of ring load or surface pressure blocks because there are three force directions x,y,z at each location δ i s are the displacements at the unit load locations i, I ij are the influence factors for the displacement at location i due to a load at location j, and P j are the loads at location j. This can be written in matrix form for all displacements δ s = I s P s (2) The influence matrix can then be inverted to obtain the stiffness matrix for the soil continuum e. g., K s δ s = P s (3) where K s =[I s ] −1 . Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical Loading 31 FIGURE 1 A raft and a pile group subjected to external forces and interface forces in all directions (the y direction is not shown). For the piles, a stiffness relationship may be written K p δ p = P p + Q (4) where K p is the stiffness matrix for all the piles in the group, ∗ p are the displacements at the nodes of piles in the group, P p is the load vector for the piles due to shaft loads, and Q is the load vector for the applied load at the pile heads. Three noded linear bending elements were used to model the piles for the work presented in this article. The stiffness matrix of the soil and the stiffness matrix of the piles may now be added, but because the piles have 5 degrees of freedom at each node (3 displacements and 2 rotations) the stiffness matrix of the soil needs to be added to the stiffness matrix of the piles allowing for the Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 32 J.C. Small and H.H. Zhang difference in the numbers of degrees of freedom. By using the fact that the displacements at the pile shaft and base are equal to the displacement of the soil, and the forces are equal and opposite, i. e., ∗ p =∗ s and P p =−P s . This gives the final stiffness relationship for the pile-soil continuum [K s + K p ] δ s = R (5) where R is the load vector consisting of loads at the pile heads Q or any loads not along the pile shaft S such as surface loads. Therefore, deflections of the soil or of the piles can be obtained for loads applied to the pile heads from the above equation. This method is not as efficient computationally as computing the interaction between two piles only (i. e., the interaction method). However it is much more accurate, especially for piles at close spacing because all the piles are considered at once. If the deflection of a pile caused by loading another is carried out using interaction factors (i. e., between two piles only), then the stiffening effect of all the other piles in the group is neglected, and this leads to error. Because the deflection of the piles can be computed when one is loaded at the head, or when the ground surfaceisloaded, thiscanbeusedtodeterminethebehavior oftheraft. The rectangular blocks of uniform pressure that represent the contact pressures are assumed to correspond to each rectangular finite element in the raft. The pressure is applied to the ground surface (either horizontally or vertically) if no pile is present and is applied to a pile head if a pile is present beneath an element of the raft. For a pile, a moment also needs to be applied to the pile head. By applying unit pressures to the ground or unit pressures and moments to the pile heads, an influence matrix for the soil-piles may be obtained. The influence matrix consists of columns that contain the deflections at the centers of each element in the raft due to a unit pressure or moment being applied to the ground surface or pile head. By applying the same unit loads to the raft, the influence matrix for the raft may be obtained. In order to apply loads to the raft, it must be restrained in some way, and this is done by “pinning” one node against rotation and translation. By considering equilibrium of applied forces and moments acting on the piles and raft, and compatibility of displacements of the soil and raft (and of displacement and rotation of the pile head and raft) enough equations may be assembled to obtain the solution under general loading. These equations are given below: ([I r ]+[I sp ]){P sp }−{a}D x −{b}D y −{c}D z −{d}θ x −{e}θ y −{f }θ z ={δ r0 } (6) {a }{P sp }=P x (7) {b }{P sp }=P y (8) {c }{P sp }=P z (9) {d }{P sp }=M x (10) {e }{P sp }=M y (11) {f }{P sp }=M z (12) where [I r ]=influence matrix of the pinned raft [I sp ]=influence matrix of the pile enhanced soil continuum {P sp }=interface load vector between the raft and the pile-enhanced soil δ r0 = displacements at the centers of the raft elements due to applied loads on the pinned raft Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical Loading 33 P x ,P y ,P z are the total external loads applied to the raft in the x,y and z directions, respectively. M x ,M y ,M z are the total external moments applied to the raft about the x,y and z axes, respectively. {a} to {f } and {a } to {f } and are auxiliary vectors D x ,D y ,D z ,θ x ,θ y and θ z are rigid body translations and rotations about a pinned point of the raft. It may be noted that the unknown rotations and translations of the “pinned” raft also form unknowns in the solution [Equation (6)]. III. Comparisons with finite element method Inorderto determinethe accuracyof thefinitelayer methoddescribed above, a 9-pileraftwith two pile spacing ratios has been analyzed by using the present method and the (three-dimensional) finite element method. A square pile was modeled in the finite element method and a circular pile was assumed in the present method. The cross-sectional areas of the square and circular piles were assumed equal, and this makes them equivalent for vertical loading, but because the second moment of area is larger for the square pile, the bending stiffness is 4.7% higher. The “diameter” (D) of a pile referred to in the following comparisons is the edge length of the cross-section of the square pile, whereas the equivalent diameter was used in the analysis of APRAF. It should be noted that the pile spacing ratio in the following comparisons is therefore defined as the ratio of the center-to-center distance of the piles to the side length D of the square pile. The piled raft as shown in Figure 2 was constructed in a layered soil 15 m in depth. Thepile slenderness ratio L/D was chosen to be 20 and pile spacing ratios S/D of either 3 or 5 were used. Both the pile-soil stiffness ratio (E p /E s ) and raft-soil ratio (E r /E s ) were assumed to be 3000. The breadth and length of the raft therefore vary with the pile spacing ratios. All of the properties of the piled raft are listed in Table 1. TABLE 1 Properties of Piled Raft (3x3 group) Quantity Value Pile side length 0.5 m (square pile) Equivalent pile diameter 0.564 m (circular pile) Pile length 10 m Depth of soil 15 m Raft width L r S/D = 3; 4.5 m S/D = 5; 6.5 m Raft breadth B r S/D = 3; 4.5 m S/D = 5; 6.5 m Overhang of raft 0.5 m Raft thickness 0.25 m Soil modulus 10 MPa Soil Poisson’s ratio 0.3 Raft modulus 30,000 MPa Raft Poisson’s ratio 0.3 A uniform vertical loading of 100 kPa or 18 unit concentrated horizontal loadings (applied to pile heads as shown in Figure 2) will be examined, respectively, in the following example. In the Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 34 J.C. Small and H.H. Zhang FIGURE 2 Layout of a 9-pile raft embedded in a soil (where D is equivalent pile diameter used in the present method). analysis, no slip was allowed along the pile-soil interface, and no lift-off of the raft was allowed. These features can be modeled by the present method by limiting forces beneath the raft, but this is not considered in this article. In the APRAF analysis, the pile was divided into 11 sections (elements) along its length and the raft was divided into 81 identical square elements for S/D = 3 and 169 identical elements for S/D = 5. An equivalent diameter D ρ of the circular pile (based on cross-section area) was used in the APRAF analysis. For the finite element analysis, a quarter of the piled raft was meshed by taking advantage of the symmetry of geometry of the piled raft. When the piled raft is under vertical loading, the mesh in the x- and y-directions extends to 32 m from the center of the piled raft and 15 m in the z-direction. Twenty noded solid isoparametric elements were used to model the soil and piles, Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical Loading 35 while 8 noded shell elements were used for the raft. There were 2535 elements in total. When the piled raft is subjected to horizontal loading, the mesh extends to 41 m in the x-direction (the direction of loading) and 24 m in the y-direction involving 2475 elements. The mesh used for the horizontal loading analysis is shown in Figure 3, where the mesh may be seen to be longer in the direction of loading to reduce boundary effects. FIGURE 3 Finite element mesh used for 3 × 3 pile group loaded in the x-direction. Computations showed that it takes about 3 h and 50 m to obtain a solution using a Pentium III processor for the finite element method whereas an equivalent analysis only took about 1 h and 1 m with a Pentium II processor. The piled raft was firstly analyzed for a uniform vertical load by using the finite layer method for two pile spacings (S/D = 3 and 5) and then the same problem was reanalyzed by using the finite element method. Figure 4 shows the plot of normalized axial forces (P i /P total where P i is the axial force in the pile and P total is the total load on the raft) in pile 1 (corner pile) and pile 5 (central pile) against pile length. Itmay be noted that the axial forces in the central pile calculated by the present method are in good agreement with those provided by the finite element method for the different pile spacings. For S/D = 3, the central pile carries less load than the corner pile, but for S/D = 5, the center and corner piles carry almost the same load and have similar axial load distributions along their lengths. For the central pile, the solution from the finite layer method is higher than that of the finite element method, whereas for the corner pile the axial force from the finite element method is higher. Comparisons indicate that the maximum difference between the axial forces computed by the two methods is less than 15%. Figure 5 shows the moment in the raft along section A-B (as shown in Figure 2) for the piled raft under uniform vertical loading. The moment presented is the moment per unit length in the x direction (M xx ). It can be seen that the moments in the raft provided by both methods Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 36 J.C. Small and H.H. Zhang FIGURE 4 Comparisons of axial forces in piles. for different pile spacings are very close. The maximum difference occurs around the heads of the edge piles and is less than 9%. Figure 6 shows the displacement along section A-B where the piled raft is again subjected to uniform vertical loading. The figure shows that the present solutions agree excellently with those of the finite element method. The maximum difference is less than 1%. The comparisons between the present method and the finite element method for the piled raft subjected to concentrated horizontal loads (see Figure 2) are shown in Figures 7 to 9. The case for a pile spacing of S/D = 3 was analyzed. Figures 7 to 9 show the moment in pile 1 and pile 5 against pile length, moment in the raft along section A-B (moment is moment/unit length in the direction of the loading M xx ) and the displacement along section A-B, respectively. It may be observed that the present solutions agree closely with those of the finite element method and Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical Loading 37 FIGURE 5 Comparisons of moments in raft along section A-B. FIGURE 6 Comparisons of deflection of raft along section A-B. Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 38 J.C. Small and H.H. Zhang the maximum difference is less than 12%, in the case of moments in the raft. FIGURE 7 Comparisons of bending moment in pile. IV. Parametric study Example 1. Shown as the inset to Figure 10 is a 9 pile (3 × 3) group driven into a deep uniform soil layer (the ratio of the soil depth to pile length is assumed to be 10). The cap or raft connecting the pile heads is assumed to be constructed in contact with the ground or just clear of the ground, and can Downloaded 27 Sep 2009 to 155.69.4.4. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright [...]...Behavior of Piled Raft Foundations Under Lateral and Vertical Loading FIGURE 8 39 Comparisons of moments in raft along section A-B have a finite flexibility H0 (the distance of the raft from the ground) is 0.0 for an on-ground raft and > 0.0 if the raft is just clear of the ground Two cases of loading have been examined; firstly a horizontal uniform load is applied to the raft... lateral deflection of the raft ux at the central pile is plotted in nondimensional form Iuxx against the pile-soil stiffness ratio Ksp (Ksp = Ep /Es ) For the vertical loading case the vertical displacement of the raft uz at the central pile is plotted in nondimensional form Iuzz also against the pile-soil stiffness ratio Normalized displacements Iuxx and Iuzz may be expressed as Es D ux qx Br Lr Es D =... will Downloaded 27 Sep 2009 to 155.69.4.4 Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright 40 FIGURE 9 J.C Small and H.H Zhang Comparisons of z-displacement of raft along section A-B deflect more if the piles are shorter but only when the piles become stiffer For flexible piles the pile length does not affect the deflection because the piles (at L/D > 25) are longer... the piled raft embedded in the three types of soils as before It should be noted that the modulus of the soil, Es , used in equation 14 is that at the pile tip The results for normalized horizontal displacement of the central pile Iuxx are plotted in Figure 14 against the pile-soil stiffness ratio Ksp = Ep /Es (where again the soil modulus at pile tip level is used) It may be seen that the displacement... Small and H.H Zhang Variation of normalized vertical displacement (central pile) with pile-soil stiffness ratio V Conclusions A method for analyzing the behavior of piled rafts constructed in elastic soils has been presented Comparisons of forces in piles, moments in the raft and piles and displacement in a 9-pile raft subjected to either vertical loading or horizontal loading show that the present... H.H Zhang FIGURE 14 Variation of displacement with pile-soil stiffness ratio for three types of soil Acknowledgments The assistance of Alex Edwards, Alan Millar and Andrew de Ambrosis in obtaining the finite element solutions presented in this article is gratefully acknowledged References [1] R Butterfield, and P.K Banerjee, The problem of pile group - pile cap interaction, Géotechnique, 21(2), pp 135–... G2(1), pp 21– 27, (1972) [4] S.J Hain, and I.K Lee, The analysis of flexible raft-pile systems, Géotechnique, 28(1), pp 65–83, (1978) [5] F Kuwabara, An elastic analysis for piled raft foundations in a homogeneous soil, Soils and Foundations, 29(1), pp 82–92, (1989) [6] H.G Poulos, An approximate numerical analysis of pile-raft interaction, Int Jl for Num and Anal Methods in Geomechanics, 18, pp 73–92,... materials using a flexibility approach, Part 2 — Circular and rectangular loadings, International Journal for Numerical Methods in Engineering, 23, pp 95 9-9 78, (1986) [8] W.Y Shen, Y.K Chow, and K.Y Yong, A variational approach for the analysis of pile Group-Pile cap interaction, Géotechnique, 50(4), pp 349–357, (2000) [9] L.D Ta, and J.C Small, Analysis of piled raft systems in layered soils, Int J for... was assumed to be constant For Banerjee’s soil, the modulus at the ground surface was chosen to be half of that at the pile tip The other main parameters used in this example are as shown in Figure 14 Equation 14 is used Downloaded 27 Sep 2009 to 155.69.4.4 Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical... = Iuzz (13) (14) where Es is the soil modulus; D is the pile diameter; qx and qz are the lateral and vertical loads; Br and Lr are the breadth and length of the raft in plan (in this case the raft is square, so Br = Lr ) Poisson’s ratio of the soil was chosen to be 0.499, the ratio of the thickness of the raft to the pile diameter tr /D = 1.17 and the pile spacing to diameter ratio S/D was 5 (with an . http://pubs.asce.org/copyright Behavior of Piled Raft Foundations Under Lateral and Vertical Loading 45 [2] P.Clancy ,and M.F.Randolph, Anapproximateanalysis procedure forpiledraft foundations, Int. Jl. forNumerical and Analytical. analysis, a quarter of the piled raft was meshed by taking advantage of the symmetry of geometry of the piled raft. When the piled raft is under vertical loading, the mesh in the x- and y-directions. piles and raft, and compatibility of displacements of the soil and raft (and of displacement and rotation of the pile head and raft) enough equations may be assembled to obtain the solution under