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Analysis of capped pile groups subjected to horizontal and vertical loads
Computers and Geotechnics 26 (2000) 1±21 www.elsevier.com/locate/compgeo
Analysis of capped pile groups subjected to horizontal and vertical loads H.H. Zhang*, J.C. Small Department of Civil Engineering, University of Sydney, NSW 2006, Australia Received 16 April 1999; received in revised form 17 September 1999; accepted 20 September 1999
Abstract This paper presents a method of analysis for an o-ground cap supported by piles embedded in a layered soil and subjected to horizontal and vertical loads. The cap is modelled as a thin plate and the piles as elastic beams and the soil is treated as consisting of horizontal layers of dierent materials. Finite element theory is used to analyse the cap and piles while ®nite layer theory is employed to analyse the layered soil. Using program APPRAF (Analysis of Piles and Piled RAft Foundations) to carry out the analysis described above, comparisons of the behaviour of capped pile groups are made and factors aecting the displacements of capped pile group foundations are examined. Finally, an example related to three types of soils where the moduli increase with depth is illustrated. The results show that the present method is a powerful and useful way to evaluate the behaviour of capped pile foundations embedded in dierent types of soils and subjected to both vertical and horizontal loadings. # 2000 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction O-ground pile groups with sti caps are widely used as foundations for structures and oshore platforms (wharves or drilling platforms). Generally, capped pile groups will be subjected to both vertical loads transferred from the structure above and horizontal loads caused by winds, waves, earth pressures or earthquakes. It is therefore necessary to develop a method which can not only analyse capped pile group foundations under vertical loading but can also analyse their behaviour when * Corresponding author. 0266-352X/00/$ - see front matter # 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0266-352X(99)00029-4
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subjected to horizontal loading. Early investigations were generally only related to groups of piles with a rigid cap (assuming the displacement of each pile is identical) or related to a completely ¯exible cap (assuming the loading acting on each pile is identical). Gradually, cap behaviour was included. Hongladaromp et al. [1] have explored a capped pile group with ¯exibility, but pile±pile and pile±soil surface interactions were ignored. Kuwabara [2] used a boundary element method to analyse the behaviour of a capped pile group. Clancy and Randolph [3] and Poulos [4] presented, respectively, an approximate numerical solution mainly based on ®nite element theory and Mindlin's solution. By introducing ®nite layer theory, Ta and Small [5,6] have analysed piled raft systems subjected to vertical loads in layered soil. However all the previous work on the analysis of piled raft systems has only been related to vertical loads. In this paper, o-ground capped pile group systems subjected to both vertical and horizontal loads and embedded in homogenous elastic soils or elastic soils where the modulus increases with depth, have been analysed by combining ®nite elements to model the cap and the piles with ®nite layer theory to model the soil. The method can also be used for on-ground caps, but this is left for further work. 2. Method of analysis For a capped pile group foundation, external loads including vertical and horizontal concentrated or uniform loads and moments in each of the three (orthogonal) axis directions may be applied to the cap and are transferred from the cap to the piles and then through the piles to the soil. Based on this consideration, analysis of a capped pile group system is carried out by separating a capped pile group into three parts: the cap, the group of piles and the layered soil. The cap is assumed to be a thin elastic plate and the ®nite element method developed by Bogner et al. [7] is employed to analyse the cap. Element division of the cap should be such that the pile head ®ts within one element of the cap. Each cap element has four nodes and 24 degrees of freedom. The contact forces applied to the cap elements that are connected to the pile heads are assumed to be uniform loads on the cap and concentrated loads on the piles as shown in Fig. 1. A pile is modelled as a beam and a simple ®nite element method is used to analyse the pile. The part of a pile which is embedded in the layered soil is divided into the same number of elements as the number of layers in the soil while the upper exposed part of the pile may just be taken as one element or more if it is of some length. A series of vertical or horizontal ring loads are assumed to act on the soil interfaces (corresponding to each node along the pile shaft) but a circular load is assumed to act on the soil at the pile base. Point vertical and horizontal forces are assumed to act on the pile at each node. Interface forces transferred from the cap to the pile heads may be simpli®ed into equivalent concentrated loads, and these loads are considered as external forces for the group of piles. Torsional loadings are not considered along the pile shafts, and so the analysis is limited to where torsion is not of major concern.
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Fig. 1. Schematic diagram of separated cap and piles.
A modi®ed ®nite layer method based on that developed by Small and Booker [8] is used to analyse the layered soil. Generally, the layered soil above the base of a pile is divided into 12±15 layers or more based on the soil properties and required numerical accuracy, while the soil beneath the base of the pile may be divided into one or more layers according to the soil properties and strata. 2.1. Analysis of cap The analysis of the cap and the piles is carried out separately and, in the structural analysis, some nodes on the cap must be restrained from undergoing free body rotations and translations. For convenience, in the present paper two corner nodes of the cap were chosen as points of restraint as shown in Fig. 1. At pin 1, the node is completely ®xed in all directions (i.e. six freedoms) and pin 2 is ®xed only in the ydirection to resist the cap from rotating about the z-axis. Based on the above assumption, the actual displacement fr g at the centre of each cap element containing a pile may be expressed as fr g Ir fPr g fagDx fbgDy fcgDz fdgx fegy f z fr0 g 1 where Ir = in¯uence matrix of the pinned cap fPr g = P1rx ; P1ry ; P1rz ; M1rx ; M1ry ; . . . ; Pnrx ; Pnry ; Pnrz ; Mnrx ; Mnry T , is the vector of interface loads and moments on the cap elements (containing a pile), where Pirx is the uniform interface load in the x-direction on element i, etc. 1 1 n n T ; ry ; . . . ; nrx ; nry nrz ; rx ; ry is the vector of displacements at fr g = 1rx ; 1ry ; 1rz ; rx the centre of the cap elements (containing a pile), where irx is the displacei represents the rotation in the ment in the x-direction for element i and rx direction of Mx (see Fig. 1) for element i, etc.
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fr0 g = displacements in the centre of the cap elements (containing a pile) under the applied loads on the pinned cap. The order of unknowns is the same as fr g. Dx = horizontal translation of the cap at the ®rst pinned point in the x-direction Dy = horizontal translation of the cap at the ®rst pinned point in the y-direction Dz = vertical translation of the cap at the ®rst pinned point in the z-direction = the rotation of the cap about the ®rst pinned point in the direction of Mx x = the rotation of the cap about the ®rst pinned point in the direction of My y z = the rotation of the cap about the z-axis fag = 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; . . . . . .T fbg = 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; . . . . . .T fcg = 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; . . . . . .T fdg = 0; 0; x1 ; 1; 0; 0; 0; x2 ; 1; 0; . . . . . . 0; 0; xn ; 1; 0T feg = 0; 0; y1 ; 0; 1; 0; 0; y2 ; 0; 1; . . . . . . 0; 0; yn ; 0; 1T ff g = y1 ; ÿx1 ; 0; 0; 0; y2 ; ÿx2 ; 0; 0; 0; . . . . . . ; yn ; ÿxn ; 0; 0; 0T where xi ; yi = co-ordinates of the central point of cap element i (containing a pile) relative to the ®rst pinned point, n = number of cap elements (containing a pile). The in¯uence matrix of the pinned cap can be generated as follows. A unit uniform horizontal load in the x-direction may be applied to the ®rst cap element that contains a pile and the central displacements (in the x-, y- and z-directions) plus rotations of all elements containing a pile are computed. These displacements and rotations will form the ®rst column of the in¯uence matrix Ir of the pinned cap. Then a unit uniform horizontal load in the y direction may be applied to the same cap element and the central displacements of all cap elements (containing a pile) in the x-, y- and z-directions plus rotations can be calculated. These displacements will form the second column of Ir and so on for a unit load in the z-direction and a unit moment in the x- and y-directions, respectively. 2.2. Analysis of a group of piles embedded in a layered soil Behaviour of the soil is here assumed to be elastic (although the soil may consist of horizontal layers having dierent elastic properties). This is a reasonable assumption since working loads should be kept well below the failure load and in most cases, this is within the region where the load±de¯ection behaviour is linear. Thus, elastic theory can be used directly for settlement prediction. Model test results [9] also show that the load±de¯ection curves of pile groups are roughly linear up to load level of 13 to 12 of the failure load and so in this paper the elastic theory is used to analyse the soil behaviour. For the group of piles, the displacement of the head of pile i due to the loads and moments acting there as well as loads applied to the tops of other piles may be obtained by superposition using the following equation.
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i
5
n X Iij Pj j1
where Iij is the ¯exibility coecient denoting the displacement at the top of pile i due to a unit load (or moment) at the top of pile j; Pj is the pile±cap interaction force acting on the top of pile j; n is the total number of piles. So the displacement at the top of each pile under the loads transferred from the cap can be expressed as 2 sp Isp Psp where Isp = pile±soil in¯uence matrix = loads on the pile heads Psp sp = displacements at the pile heads The primary problem here is to generate the pile±soil in¯uence matrix. In this paper, an interaction method based on the ®nite layer theory developed by Small and Booker [8] is introduced to generate the in¯uence matrix of the layered soil. 2.2.1. Finite layer theory In order to simplify the equations governing the problem of circular loads applied to a layered soil, double Fourier transformations may be applied to all ®eld quantities (displacements and loads). For general vertical and shear loading, this results in two sets of ®nite layer equations that give the relationship between the transformed stresses and transformed displacements at the interface of each horizontal soil layer. The ®rst set of equations may be written in ¯exibility form (see [8]) i i i 3 F P or in the alternate stiness form i i i K P
4a
where i ÿ T i =ÿUzp ; Up ; Uzm ; Um T P = Np ; Tp ; ÿNm ; ÿTm The transforms of the displacements at the layer interface are designated Uz (vertical), U (displacement in the -direction see Fig. 2) and the subscripts p, m designate the upper and lower faces of the soil layer. The transforms of the stresses at the layer interfaces are designated N (for vertical stress) and T for shear stress in the z-direction (p and m again denote upper and lower faces of the soil layer).
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Fig. 2. Uniform shear loading over a circular region showing axis systems.
The second set of equations also results for the soil layer (called the uncoupled terms) Ku i iu Piu 4b where i T ui = ÿ Up ; Um T Pu = Tp ; ÿTm where U is the transform of displacement in the direction and T is the transform of the shear in the z-direction (see Fig. 2). These stiness matrices may be assembled for each soil layer into a global matrix, and solved to give the transformed displacements Uz ; U ; U at the layer interfaces. Upon addition of layer matrices all interface stresses cancel, and so the right-hand side only consists of the transform of the load at the interface where it is applied. 2.2.1.1. Transform of horizontal loadings. The transform of a horizontal shear loading in the x-axis direction and applied uniformly over a circle may be shown to reduce to the Hankel transform of the load Txc
ihc aJ1 a
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where hc is the shear load. The components of the shear load for use in the ®nite layer Eqs. (4a) and (4b) are therefore T
ihc aJ1 a cos "
T
ihc aJ1 a sin "
and the angle " is as shown in Fig. 2. For a horizontal ring load applied in the x-direction, the transform Txr is given by Txr
iHr J0 a 2
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and again T and T can be found in terms of the angle ", where = transform of applied horizontal circular load Txc = transform of applied horizontal ring load Txr = applied horizontal circular load hc Hr = applied horizontal ring load a = radius of loaded circle J0 a = Bessel function of the ®rst kind (order zero) J1 a = Bessel function of the ®rst kind (order one) =p Hankel transform parameter i = ÿ1 to make the solution for the horizontal load the complex part of the solution. 2.2.1.2. Transforms of vertical loadings. The transforms of the vertical ring loads that are applied along the pile shaft or the uniform load that is applied at the pile base can also be found by using a double Fourier transform. The transform reduces to the Hankel transform of the load. For a uniform vertical load q the transform is Qc
qaJ1 a
8
and for a ring load Qr
Pr J0 a 2
where Qc = transform of the applied uniform circular vertical load Qr = transform of the total applied ring vertical load
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q = applied uniform circular vertical load Pr = total applied ring vertical load. The solution to the ®nite layer equations results in the transformed displacements that may be inverted to yield the actual displacements. The real part is the displacement due to the vertical loading and the complex part is the displacement due to the horizontal loading. In the case of horizontal loading, the components of displacement from both sets of Eqs (4a) and (4b) need to be combined to compute the displacement in say the x- or y-direction. The ®nite layer technique can therefore be used to compute the response of the layered soil to ring loads at the soil layer boundaries or uniform shear loads at the pile toe. 2.2.2. Generation of in¯uence matrix of pile±soil system Similar to the formation of the in¯uence matrix of the cap, the pile±soil in¯uence matrix, Isp , may be generated by using the interaction method for pile groups developed by Zhang and Small [10]. A horizontal load which is equal to the total horizontal load applied on the corresponding cap element may be applied to pile 1 in the x-direction and the displacements used to form the elements of the ®rst column of Isp . In a similar way, the remaining columns of Isp can be generated for loads in the y- and z-directions and moments in the x- and y-directions. The advantage of this method is that it can be used to solve capped pile group problems involving a large group of piles. 2.3. Analysis of capped pile groups If the compatibility of displacements and the equilibrium of interaction forces between the pile heads and the cap are taken into account, we get fr g sp 10 fPr g ÿ Psp Combination of Eqs. (1), (2), (10) and (11) leads to ÿ Ir Isp Psp ÿ fagDx ÿ fbgDy ÿ fcgDz ÿ fdgx ÿ fegy ÿ f z fr0 g
11
12
To guarantee force and moment equilibrium of the cap, the following equations must be satis®ed: 0 13 a Psp Px 0 14 b Psp Py 0 15 c Psp Pz
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0 d Psp Mx
16
0 e Psp My
17
0 f Psp Mz
18
where fa0 g fb 0 g fc 0 g fd 0 g fe0 g ff 0 g
= = = = = =
A1 ; 0; 0; 0; 0; A2 ; 0; 0; 0; 0; . . . . . . ; An ; 0; 0; 0; 0 0; A1 ; 0; 0; 0; 0; A2 ; 0; 0; 0; . . . . . . ; 0; An ; 0; 0; 0 0; 0; A1 ; 0; 0; 0; 0; A2 ; 0; 0; . . . . . . ; 0; 0; An ; 0; 0 0; 0; A1 x1 ; A1 ; 0; 0; 0; A2 x2 ; A2 ; 0; . . . . . . ; 0; 0; An xn ; An ; 0 0; 0; A1 y1 ; 0; A1 ; 0; 0; A2 y2 ; 0; A2 ; . . . . . . ; 0; 0; An xn ; 0; An A1 y1 ; ÿA1 x1 ; 0; 0; 0; A2 y2 ; ÿA2 x2 ; 0; 0; 0; . . . . . . ; An yn ; ÿAn xn ; 0; 0; 0
and where Ai = the area of element i in the cap, and Px ; Py ; Pz are the total loads applied to the cap in the x-, y- and z-directions and Mx ; My are the total moments applied to the cap; Mz is the total moment about the z-axis (at pin 1) due to Px and Py . Solving Eqs. (12)±(18), we may obtain interaction forces on the pile heads at the pile±cap interfaces, and the rigid body rotations and translations of the cap relative to the pins. Substituting the results into Eq. (1) it is possible to work out the actual displacements of the cap elements that contain piles under the applied external forces. By substituting the interaction forces into Eq. (2) it is also possible to calculate the displacements of the pile heads. 3. Results For convenience in the analysis of capped pile group foundations subjected to vertical and horizontal loads, some dimensionless parameters can be de®ned as follows Pile spacing ratio Pile slenderness ratio Pile±soil stiness ratio Cap±soil stiness ratio
De®nition S=D Lem =D Kps Ep =Es Krs Er =Es
Practical range 2.58 10100 10010 000 10010 000
where Ep is the pile Young's modulus; Es is the soil Young's modulus; S is spacing between pile centres; D is pile diameter and Lem embedded pile length. In the following analyses, the Poisson's ratio of the cap and soil will be taken as 0.15 and 0.499, respectively, if not stated otherwise.
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Uniform horizontal load qx or vertical load, qz , is generally taken into account separately in the examples below. The normalised displacement for a capped pile group corresponding to horizontal loads or vertical loads is de®ned as: for capped pile groups under horizontal loads Iuxx uxx Es D=qx Br Lr
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for capped pile groups under vertical loads Iuzz uzz Es D=qz Br Lr
20
where uxx is the displacement of one of the pile heads in the x-direction induced by horizontal loads; uzz is the displacement of one of the pile heads in the z-direction induced by vertical loads; Br and Lr are the breadth and length of the cap, respectively; D is pile diameter. 3.1. Veri®cation by comparisons with results for pile groups 3.1.1. Comparisons for capped pile group with two piles An obvious characteristic of a capped pile group with only two piles is that when a symmetric horizontal loading is applied to the cap in the direction perpendicular to the line connecting the two piles, the cap has no enhancing eect on the two piles and the behaviour of the capped pile group is similar to that of a free head pile group. If the load is applied parallel to the line joining the piles, the cap stiness will have a large eect on the displacement. For veri®cation of this, the pile group as shown in the inset to Fig. 3 was analysed, where the ratio of embedded pile length, Lem , to diameter, D, was chosen to be 25. The ratio of the exposed pile length H0 to pile diameter was 2.5 and the pile±soil and cap±soil stiness ratios are taken as 4000 and 4285, respectively. The results of the analysis of the pile group and capped pile group with only two piles subjected to horizontal loading in the two directions are shown in Figs. 3a and b, respectively. As can be seen in Fig. 3a under horizontal load in the x-direction, as expected, the normalised displacement of the capped pile group is very close to that of the pile group with a rigid cap but much lower than that of the pile group with a completely ¯exible cap. Fig. 3a also shows that when pile spacing is less than 6 times the pile diameter, the displacement factor is sensitive to change in spacing. However, with increase of pile spacing, the horizontal displacement is less and is not greatly aected by the spacing. Fig. 3b demonstrates that under a horizontal load in the y-direction, there will be no dierence among a capped pile group and two-pile group with a completely ¯exible or rigid cap with respect to normalised displacements. The results of Fig. 3b indicate that the program is probably working correctly. However, for problems with more piles, further comparisons are needed.
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Fig. 3. Comparison of 2-pile group with capped pile group: (a) Px only; (b) Py only.
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3.1.2. Comparison with pile groups In an example given by Poulos [11], there is a ®xed-head pile group with 16 piles as shown in the inset to Fig. 4. The ratio of pile length to diameter is 25 and the pile ¯exibility factor KR is equal to 10ÿ5 (de®ned as KR Ep Ip =Es L4em , Ip is the second moment of area of the pile). Here the Poisson's ratio for the soil is taken as 0.499 while 0.5 was used by Poulos [11]. Ep =Es was chosen to be 79.6 while Er =Es was made large (85 250) to represent a rigid cap so that the behaviour of the capped pile group would be like that of a ®xed-head pile group although for the rigid cap some head rotations may still occur. The cap is just clear of the ground (i.e. H0 0:0).
Fig. 4. Typical horizontal load distributions in ®xed head pile group and capped pile group.
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The load distributions against pile spacing are plotted in Fig. 4. We can observe that for a pile spacing ratio of less than 4 there is poor agreement of the typical horizontal load distributions between the ®xed-head pile group and the capped pile group. Load on piles 3 and 4 computed using the present method is higher than that given by Poulos [11], while the load on piles 1 and 2 is lower, especially for a capped pile group with close pile spacing. With increase of the pile spacing, the load distributions tend to become more uniform. Because of the lack of agreement, a further comparison was made with the results of El Sharnouby and Novak [12] for the same problem. It may be seen in Fig. 4 that these results are in fairly good agreement with the present results.
Fig. 5. The eect of pile±soil stiness ratio on de¯ection of capped pile groups under horizontal loads.
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3.2. Parametric analysis In order to demonstrate the use of the present method for the analysis of laterally or axially loaded capped pile groups, the most critical parameters were examined to show their eects on the maximum displacement of a capped pile group. The ratios of the embedded pile length to pile diameter (Lem =D) were chosen to be 10, 20, 40, 80. The Poisson's ratios of the soil and the cap were assumed to be 0.35 and 0.15, respectively. The ratio of the exposed pile length H0 to pile diameter was 2.5 and the pile spacing ratio was taken as 5. The thickness of the cap was 0.5 m, while the breadth and length of the cap were chosen to be 18 and 13 m, respectively. 3.2.1. The eect of soil and pile modulus Varying the pile±soil stiness ratio Ep =Es from 101 (¯exible pile) to 106 (rigid pile) and keeping the cap-soil stiness ratio constant (Er =Es 2000), the present program APPRAF was used to calculate displacements at the central point of each pile head. The normalised displacements for pile 5 (maximum displacements generally occur for this pile) are plotted in Figs. 5 and 6 against the pile±soil stiness ratio.
Fig. 6. The eect of pile±soil stiness ratio on de¯ection of capped pile group under vertical loads.
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It may be seen from Fig. 5 that the horizontal displacements of the capped pile group decrease rapidly with the pile±soil stiness ratio. The results corresponding to four pile lengths shows that there is almost no in¯uence of the pile lengths on the displacement of the capped pile group for a pile±soil stiness ratio of less 103 and for pile slenderness ratios in excess of 10. However, when the piles become substantially rigid (Ep =Es up to 106), the shorter piles will obviously produce higher displacement than the longer piles. Fig. 6 shows that for vertical loading and a pile±soil stiness ratio of less than 100, the displacement of the capped pile group increases sharply as the pile±soil stiness ratio reduces. For pile±soil stiness ratios greater than 100, the displacement of the capped pile group will decrease very slowly, especially when the piles are short. Fig. 6 also shows that there are no obvious dierences in the de¯ections of the group with small pile±soil stiness ratios (<100) for dierent pile lengths. However, for pile±soil stiness ratios of over 100, pile lengths have more eect on the de¯ections of the group and the group with longer piles will produce lower vertical displacements.
Fig. 7. The eect of cap±soil stiness ratio on de¯ection of capped pile groups under horizontal loads.
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3.2.2. The eect of cap modulus The pile±soil stiness ratio was chosen to be 2000 and the cap±soil stiness ratio (Er =Es ) was varied from 101 to 106 in this example. Program APPRAF was used to analyse the capped pile group under horizontal loading and vertical loading, respectively. The displacements of the capped pile group under horizontal loading and under vertical loading are shown in Figs. 7 and 8, respectively, against the cap± soil stiness ratio. Fig. 7 shows that if the capped pile group is subjected to horizontal loading only, the increase of the cap±soil stiness ratio will obviously reduce the horizontal displacement of the group. However, once the cap becomes stier (a cap±soil stiness ratio >1000), further increase of the cap±soil stiness will have a negligible in¯uence on the horizontal displacement of the group. Furthermore, the lengths of the piles have almost no in¯uence on the horizontal displacements of the group. On the contrary, when the capped pile group is subjected to vertical loads only as shown in Fig. 8, the cap±soil stiness ratio has only a minor eect on the vertical de¯ection of the group. However, increase in pile length will greatly reduce the vertical displacements of
Fig. 8. The eect of cap±soil stiness ratio on de¯ection of capped pile groups under vertical loads.
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the group. For example, as shown in Fig. 8, the vertical displacement of the group with a pile slenderness ratio of 80 may be reduced to approximately 43±48% of that of a capped pile group with L=D 10.
Fig. 9. Models for soils where the modulus increases with depth.
Fig. 10. The eect of dierent soils on the displacement of capped pile groups under horizontal loading.
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3.3. Capped pile groups embedded in soils where the modulus increases with depth Three types of elastic soils where the modulus increases with depth are chosen in this paper to demonstrate this method. The soil models are shown in Fig. 9. It is assumed that the capped pile group with 12 piles as shown in Fig. 1 is embedded in the dierent soil types. The stinesses of the dierent soils can be expressed as Es z Es0 m z=Lem n If m 0, the above equation represents a homogeneous soil with a constant elastic modulus, i.e. Es0 Esb ; if Es0 0 and n 1, it represents Gibson's soil; if Es0 > 0 and n 1, it is Banerjee's soil; and the general case is a parabolic variation of soil modulus. The soil modulus Esb was taken as 7 MPa at the toe of the pile and the ratio Lem =D was taken as 20. The pile±soil stiness ratio Ep =Esb and the cap±soil ratio
Fig. 11. The eect of dierent soils on the displacement of capped pile groups under vertical loading.
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Er =Esb were 4000 and 4285, respectively. The cap thickness was chosen to be 1.0 m. The breadth and length of the cap were 20 and 15 m, respectively. Other parameters were the same as those used in Section 3.2. Figs. 10 and 11 show that whether the capped pile group is subjected to horizontal loading or it is subjected to vertical loading, the pile spacing has a signi®cant eect on displacements of the group when the pile spacing is less than about 6 times the pile diameter. Furthermore, they show that the homogeneous soil is one of the best soils with respect to resistance of de¯ection of a capped pile group. On the contrary, the Gibson's soil is the poorest one as may be expected. Moreover, beyond a value of pile spacing of S=D 6, the eect of a change in spacing on the displacement of the group examined will become much less. For the capped pile group subjected to horizontal loading, moments in pile 1 corresponding to two pile spacings are plotted in Figs. 12 and 13. With a pile spacing ratio of 2 as shown in Fig. 12, the largest value of the positive moment in pile 1 is much higher than that in pile 1 with a pile spacing ratio of 6 as plotted in Fig. 13. This demonstrates that a relatively small pile spacing ratio (i.e. less than 6) may lead to a large moment in the pile and is therefore a less economical use of the piles.
Fig. 12. Moment distribution in single pile embedded in dierent soils with close pile spacing.
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Fig. 13. Moment distribution in single pile embedded in dierent soils with large pile spacing.
4. Conclusions In this paper, a powerful method for analysing the behaviour of capped pile groups subjected to both lateral and vertical loading has been developed. The analysis of a capped pile group with only two piles, comparisons with solutions for ®xed-head pile groups, and investigations of the eect of soil, pile and cap parameters demonstrate that: 1. The method can be used to analyse the behaviour of a capped pile group foundation subjected to vertical loading or horizontal loading as well as moments in all axis directions. 2. Through comparison of the load distribution on the pile heads between a capped pile group (where the cap is very sti) and a pile group with a rigid cap, it is shown that the present method for analysis of a capped pile group can reproduce previous results. 3. Pile±soil stiness ratio plays an important role in the resistance of a capped pile group to lateral de¯ection for pile±soil stiness ratios of less than 1000. In this
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range, the displacements of the capped pile group will increase sharply with the reduction of the pile±soil stiness ratio. 4. For the example chosen, change in the pile±soil stiness ratio had a large in¯uence on the displacement of the capped pile group under vertical loading when this ratio was less than 100. When the pile±soil stiness ratio is greater than 100, change in the pile±soil stiness ratio had only insigni®cant eect on the displacement. Pile lengths have a relatively large eect on the de¯ections of the capped pile group under vertical loading, but much less eect for horizontal loading. 5. For lateral loading, small pile spacing (i.e. less than 6 times the pile diameter) will not only result in a large de¯ection of a capped pile group under applied load, but it can also cause larger moment in the piles. 6. The method can conveniently solve problems involving capped pile groups embedded in dierent soil types. The results of analysis con®rm, as expected, that a Gibson's soil provides the poorest lateral restraint for capped pile groups subjected to lateral loading. References [1] Hongladaromp T, Chen N, Lee S. Load distribution in rectangular footings on piles. Geotechnical Engineering 1973;4(2):77±90. [2] Kuwabara F. An elastic analysis for piled raft foundations in a homogeneous soil. Soils and Foundations 1989;29(1):82±92. [3] Clancy P, Randolph MF. An approximate analysis procedure for piled raft foundations. International Journal for Numerical and Analytical Methods in Geomechanics 1993;17:849±69. [4] Poulos HG. An approximate numerical analysis of pile±raft interaction. International Journal for Numerical and Analytical Methods in Geomechanics 1994;18:73±92. [5] Ta LD, Small JC. An approximation for analysis of raft and piled raft foundations. Computers and Geotechnics 1997;20(2):105±23. [6] Ta LD, Small JC. Analysis of piled raft systems in layered soils. Int J for Num and Anal Methods in Geomechs 1996;20:57±72. [7] Bogner FK, Fox RL, Schmit LA. The generation of inter-element-compatible stiness and mass matrices by the use of interpolation formulas. In: Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, 1965. p. 397±443. [8] Small JC, Booker JR. Finite layer analysis of layered elastic materials using a ¯exibility approach. Part 2-circular and rectangular loadings. International Journal for Numerical Methods in Engineering 1986;23:959±78. [9] Rao SN, Ramakrishna VGST, Rao MB. In¯uence of rigidity on laterally loaded pile groups in marine clay. Journal of Geotechnical and Geoenvirnmental Engineering, ASCE 1998;124(6):542±9. [10] Zhang HH, Small JC. Analysis of axially and laterally loaded pile groups embedded in layered soils. In: Proceedings of 8th ANZ Conference on Geomechanics, Hobart (Tasmania), February, 1999. [11] Poulos HG. Behaviour of Laterally-Loaded Piles: II Ð Pile Groups. J Soil Mech Found Div (JSMFD), ASCE 1971;97(SM5):733±51. [12] El Sharnouby B, Novak M. Static and low-frequency response of pile groups. Can Geotech J 1985;22:79±94.
Analysis of capped pile groups subjected to horizontal and vertical loads H.H. Zhang*, J.C. Small Department of Civil Engineering, University of Sydney, NSW 2006, Australia Received 16 April 1999; received in revised form 17 September 1999; accepted 20 September 1999
Abstract This paper presents a method of analysis for an o-ground cap supported by piles embedded in a layered soil and subjected to horizontal and vertical loads. The cap is modelled as a thin plate and the piles as elastic beams and the soil is treated as consisting of horizontal layers of dierent materials. Finite element theory is used to analyse the cap and piles while ®nite layer theory is employed to analyse the layered soil. Using program APPRAF (Analysis of Piles and Piled RAft Foundations) to carry out the analysis described above, comparisons of the behaviour of capped pile groups are made and factors aecting the displacements of capped pile group foundations are examined. Finally, an example related to three types of soils where the moduli increase with depth is illustrated. The results show that the present method is a powerful and useful way to evaluate the behaviour of capped pile foundations embedded in dierent types of soils and subjected to both vertical and horizontal loadings. # 2000 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction O-ground pile groups with sti caps are widely used as foundations for structures and oshore platforms (wharves or drilling platforms). Generally, capped pile groups will be subjected to both vertical loads transferred from the structure above and horizontal loads caused by winds, waves, earth pressures or earthquakes. It is therefore necessary to develop a method which can not only analyse capped pile group foundations under vertical loading but can also analyse their behaviour when * Corresponding author. 0266-352X/00/$ - see front matter # 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0266-352X(99)00029-4
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subjected to horizontal loading. Early investigations were generally only related to groups of piles with a rigid cap (assuming the displacement of each pile is identical) or related to a completely ¯exible cap (assuming the loading acting on each pile is identical). Gradually, cap behaviour was included. Hongladaromp et al. [1] have explored a capped pile group with ¯exibility, but pile±pile and pile±soil surface interactions were ignored. Kuwabara [2] used a boundary element method to analyse the behaviour of a capped pile group. Clancy and Randolph [3] and Poulos [4] presented, respectively, an approximate numerical solution mainly based on ®nite element theory and Mindlin's solution. By introducing ®nite layer theory, Ta and Small [5,6] have analysed piled raft systems subjected to vertical loads in layered soil. However all the previous work on the analysis of piled raft systems has only been related to vertical loads. In this paper, o-ground capped pile group systems subjected to both vertical and horizontal loads and embedded in homogenous elastic soils or elastic soils where the modulus increases with depth, have been analysed by combining ®nite elements to model the cap and the piles with ®nite layer theory to model the soil. The method can also be used for on-ground caps, but this is left for further work. 2. Method of analysis For a capped pile group foundation, external loads including vertical and horizontal concentrated or uniform loads and moments in each of the three (orthogonal) axis directions may be applied to the cap and are transferred from the cap to the piles and then through the piles to the soil. Based on this consideration, analysis of a capped pile group system is carried out by separating a capped pile group into three parts: the cap, the group of piles and the layered soil. The cap is assumed to be a thin elastic plate and the ®nite element method developed by Bogner et al. [7] is employed to analyse the cap. Element division of the cap should be such that the pile head ®ts within one element of the cap. Each cap element has four nodes and 24 degrees of freedom. The contact forces applied to the cap elements that are connected to the pile heads are assumed to be uniform loads on the cap and concentrated loads on the piles as shown in Fig. 1. A pile is modelled as a beam and a simple ®nite element method is used to analyse the pile. The part of a pile which is embedded in the layered soil is divided into the same number of elements as the number of layers in the soil while the upper exposed part of the pile may just be taken as one element or more if it is of some length. A series of vertical or horizontal ring loads are assumed to act on the soil interfaces (corresponding to each node along the pile shaft) but a circular load is assumed to act on the soil at the pile base. Point vertical and horizontal forces are assumed to act on the pile at each node. Interface forces transferred from the cap to the pile heads may be simpli®ed into equivalent concentrated loads, and these loads are considered as external forces for the group of piles. Torsional loadings are not considered along the pile shafts, and so the analysis is limited to where torsion is not of major concern.
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3
Fig. 1. Schematic diagram of separated cap and piles.
A modi®ed ®nite layer method based on that developed by Small and Booker [8] is used to analyse the layered soil. Generally, the layered soil above the base of a pile is divided into 12±15 layers or more based on the soil properties and required numerical accuracy, while the soil beneath the base of the pile may be divided into one or more layers according to the soil properties and strata. 2.1. Analysis of cap The analysis of the cap and the piles is carried out separately and, in the structural analysis, some nodes on the cap must be restrained from undergoing free body rotations and translations. For convenience, in the present paper two corner nodes of the cap were chosen as points of restraint as shown in Fig. 1. At pin 1, the node is completely ®xed in all directions (i.e. six freedoms) and pin 2 is ®xed only in the ydirection to resist the cap from rotating about the z-axis. Based on the above assumption, the actual displacement fr g at the centre of each cap element containing a pile may be expressed as fr g Ir fPr g fagDx fbgDy fcgDz fdgx fegy f z fr0 g 1 where Ir = in¯uence matrix of the pinned cap fPr g = P1rx ; P1ry ; P1rz ; M1rx ; M1ry ; . . . ; Pnrx ; Pnry ; Pnrz ; Mnrx ; Mnry T , is the vector of interface loads and moments on the cap elements (containing a pile), where Pirx is the uniform interface load in the x-direction on element i, etc. 1 1 n n T ; ry ; . . . ; nrx ; nry nrz ; rx ; ry is the vector of displacements at fr g = 1rx ; 1ry ; 1rz ; rx the centre of the cap elements (containing a pile), where irx is the displacei represents the rotation in the ment in the x-direction for element i and rx direction of Mx (see Fig. 1) for element i, etc.
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fr0 g = displacements in the centre of the cap elements (containing a pile) under the applied loads on the pinned cap. The order of unknowns is the same as fr g. Dx = horizontal translation of the cap at the ®rst pinned point in the x-direction Dy = horizontal translation of the cap at the ®rst pinned point in the y-direction Dz = vertical translation of the cap at the ®rst pinned point in the z-direction = the rotation of the cap about the ®rst pinned point in the direction of Mx x = the rotation of the cap about the ®rst pinned point in the direction of My y z = the rotation of the cap about the z-axis fag = 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; . . . . . .T fbg = 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; . . . . . .T fcg = 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 1; 0; 0; . . . . . .T fdg = 0; 0; x1 ; 1; 0; 0; 0; x2 ; 1; 0; . . . . . . 0; 0; xn ; 1; 0T feg = 0; 0; y1 ; 0; 1; 0; 0; y2 ; 0; 1; . . . . . . 0; 0; yn ; 0; 1T ff g = y1 ; ÿx1 ; 0; 0; 0; y2 ; ÿx2 ; 0; 0; 0; . . . . . . ; yn ; ÿxn ; 0; 0; 0T where xi ; yi = co-ordinates of the central point of cap element i (containing a pile) relative to the ®rst pinned point, n = number of cap elements (containing a pile). The in¯uence matrix of the pinned cap can be generated as follows. A unit uniform horizontal load in the x-direction may be applied to the ®rst cap element that contains a pile and the central displacements (in the x-, y- and z-directions) plus rotations of all elements containing a pile are computed. These displacements and rotations will form the ®rst column of the in¯uence matrix Ir of the pinned cap. Then a unit uniform horizontal load in the y direction may be applied to the same cap element and the central displacements of all cap elements (containing a pile) in the x-, y- and z-directions plus rotations can be calculated. These displacements will form the second column of Ir and so on for a unit load in the z-direction and a unit moment in the x- and y-directions, respectively. 2.2. Analysis of a group of piles embedded in a layered soil Behaviour of the soil is here assumed to be elastic (although the soil may consist of horizontal layers having dierent elastic properties). This is a reasonable assumption since working loads should be kept well below the failure load and in most cases, this is within the region where the load±de¯ection behaviour is linear. Thus, elastic theory can be used directly for settlement prediction. Model test results [9] also show that the load±de¯ection curves of pile groups are roughly linear up to load level of 13 to 12 of the failure load and so in this paper the elastic theory is used to analyse the soil behaviour. For the group of piles, the displacement of the head of pile i due to the loads and moments acting there as well as loads applied to the tops of other piles may be obtained by superposition using the following equation.
H.H. Zhang, J.C. Small / Computers and Geotechnics 26 (2000) 1±21
i
5
n X Iij Pj j1
where Iij is the ¯exibility coecient denoting the displacement at the top of pile i due to a unit load (or moment) at the top of pile j; Pj is the pile±cap interaction force acting on the top of pile j; n is the total number of piles. So the displacement at the top of each pile under the loads transferred from the cap can be expressed as 2 sp Isp Psp where Isp = pile±soil in¯uence matrix = loads on the pile heads Psp sp = displacements at the pile heads The primary problem here is to generate the pile±soil in¯uence matrix. In this paper, an interaction method based on the ®nite layer theory developed by Small and Booker [8] is introduced to generate the in¯uence matrix of the layered soil. 2.2.1. Finite layer theory In order to simplify the equations governing the problem of circular loads applied to a layered soil, double Fourier transformations may be applied to all ®eld quantities (displacements and loads). For general vertical and shear loading, this results in two sets of ®nite layer equations that give the relationship between the transformed stresses and transformed displacements at the interface of each horizontal soil layer. The ®rst set of equations may be written in ¯exibility form (see [8]) i i i 3 F P or in the alternate stiness form i i i K P
4a
where i ÿ T i =ÿUzp ; Up ; Uzm ; Um T P = Np ; Tp ; ÿNm ; ÿTm The transforms of the displacements at the layer interface are designated Uz (vertical), U (displacement in the -direction see Fig. 2) and the subscripts p, m designate the upper and lower faces of the soil layer. The transforms of the stresses at the layer interfaces are designated N (for vertical stress) and T for shear stress in the z-direction (p and m again denote upper and lower faces of the soil layer).
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Fig. 2. Uniform shear loading over a circular region showing axis systems.
The second set of equations also results for the soil layer (called the uncoupled terms) Ku i iu Piu 4b where i T ui = ÿ Up ; Um T Pu = Tp ; ÿTm where U is the transform of displacement in the direction and T is the transform of the shear in the z-direction (see Fig. 2). These stiness matrices may be assembled for each soil layer into a global matrix, and solved to give the transformed displacements Uz ; U ; U at the layer interfaces. Upon addition of layer matrices all interface stresses cancel, and so the right-hand side only consists of the transform of the load at the interface where it is applied. 2.2.1.1. Transform of horizontal loadings. The transform of a horizontal shear loading in the x-axis direction and applied uniformly over a circle may be shown to reduce to the Hankel transform of the load Txc
ihc aJ1 a
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7
where hc is the shear load. The components of the shear load for use in the ®nite layer Eqs. (4a) and (4b) are therefore T
ihc aJ1 a cos "
T
ihc aJ1 a sin "
and the angle " is as shown in Fig. 2. For a horizontal ring load applied in the x-direction, the transform Txr is given by Txr
iHr J0 a 2
7
and again T and T can be found in terms of the angle ", where = transform of applied horizontal circular load Txc = transform of applied horizontal ring load Txr = applied horizontal circular load hc Hr = applied horizontal ring load a = radius of loaded circle J0 a = Bessel function of the ®rst kind (order zero) J1 a = Bessel function of the ®rst kind (order one) =p Hankel transform parameter i = ÿ1 to make the solution for the horizontal load the complex part of the solution. 2.2.1.2. Transforms of vertical loadings. The transforms of the vertical ring loads that are applied along the pile shaft or the uniform load that is applied at the pile base can also be found by using a double Fourier transform. The transform reduces to the Hankel transform of the load. For a uniform vertical load q the transform is Qc
qaJ1 a
8
and for a ring load Qr
Pr J0 a 2
where Qc = transform of the applied uniform circular vertical load Qr = transform of the total applied ring vertical load
9
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q = applied uniform circular vertical load Pr = total applied ring vertical load. The solution to the ®nite layer equations results in the transformed displacements that may be inverted to yield the actual displacements. The real part is the displacement due to the vertical loading and the complex part is the displacement due to the horizontal loading. In the case of horizontal loading, the components of displacement from both sets of Eqs (4a) and (4b) need to be combined to compute the displacement in say the x- or y-direction. The ®nite layer technique can therefore be used to compute the response of the layered soil to ring loads at the soil layer boundaries or uniform shear loads at the pile toe. 2.2.2. Generation of in¯uence matrix of pile±soil system Similar to the formation of the in¯uence matrix of the cap, the pile±soil in¯uence matrix, Isp , may be generated by using the interaction method for pile groups developed by Zhang and Small [10]. A horizontal load which is equal to the total horizontal load applied on the corresponding cap element may be applied to pile 1 in the x-direction and the displacements used to form the elements of the ®rst column of Isp . In a similar way, the remaining columns of Isp can be generated for loads in the y- and z-directions and moments in the x- and y-directions. The advantage of this method is that it can be used to solve capped pile group problems involving a large group of piles. 2.3. Analysis of capped pile groups If the compatibility of displacements and the equilibrium of interaction forces between the pile heads and the cap are taken into account, we get fr g sp 10 fPr g ÿ Psp Combination of Eqs. (1), (2), (10) and (11) leads to ÿ Ir Isp Psp ÿ fagDx ÿ fbgDy ÿ fcgDz ÿ fdgx ÿ fegy ÿ f z fr0 g
11
12
To guarantee force and moment equilibrium of the cap, the following equations must be satis®ed: 0 13 a Psp Px 0 14 b Psp Py 0 15 c Psp Pz
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0 d Psp Mx
16
0 e Psp My
17
0 f Psp Mz
18
where fa0 g fb 0 g fc 0 g fd 0 g fe0 g ff 0 g
= = = = = =
A1 ; 0; 0; 0; 0; A2 ; 0; 0; 0; 0; . . . . . . ; An ; 0; 0; 0; 0 0; A1 ; 0; 0; 0; 0; A2 ; 0; 0; 0; . . . . . . ; 0; An ; 0; 0; 0 0; 0; A1 ; 0; 0; 0; 0; A2 ; 0; 0; . . . . . . ; 0; 0; An ; 0; 0 0; 0; A1 x1 ; A1 ; 0; 0; 0; A2 x2 ; A2 ; 0; . . . . . . ; 0; 0; An xn ; An ; 0 0; 0; A1 y1 ; 0; A1 ; 0; 0; A2 y2 ; 0; A2 ; . . . . . . ; 0; 0; An xn ; 0; An A1 y1 ; ÿA1 x1 ; 0; 0; 0; A2 y2 ; ÿA2 x2 ; 0; 0; 0; . . . . . . ; An yn ; ÿAn xn ; 0; 0; 0
and where Ai = the area of element i in the cap, and Px ; Py ; Pz are the total loads applied to the cap in the x-, y- and z-directions and Mx ; My are the total moments applied to the cap; Mz is the total moment about the z-axis (at pin 1) due to Px and Py . Solving Eqs. (12)±(18), we may obtain interaction forces on the pile heads at the pile±cap interfaces, and the rigid body rotations and translations of the cap relative to the pins. Substituting the results into Eq. (1) it is possible to work out the actual displacements of the cap elements that contain piles under the applied external forces. By substituting the interaction forces into Eq. (2) it is also possible to calculate the displacements of the pile heads. 3. Results For convenience in the analysis of capped pile group foundations subjected to vertical and horizontal loads, some dimensionless parameters can be de®ned as follows Pile spacing ratio Pile slenderness ratio Pile±soil stiness ratio Cap±soil stiness ratio
De®nition S=D Lem =D Kps Ep =Es Krs Er =Es
Practical range 2.58 10100 10010 000 10010 000
where Ep is the pile Young's modulus; Es is the soil Young's modulus; S is spacing between pile centres; D is pile diameter and Lem embedded pile length. In the following analyses, the Poisson's ratio of the cap and soil will be taken as 0.15 and 0.499, respectively, if not stated otherwise.
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Uniform horizontal load qx or vertical load, qz , is generally taken into account separately in the examples below. The normalised displacement for a capped pile group corresponding to horizontal loads or vertical loads is de®ned as: for capped pile groups under horizontal loads Iuxx uxx Es D=qx Br Lr
19
for capped pile groups under vertical loads Iuzz uzz Es D=qz Br Lr
20
where uxx is the displacement of one of the pile heads in the x-direction induced by horizontal loads; uzz is the displacement of one of the pile heads in the z-direction induced by vertical loads; Br and Lr are the breadth and length of the cap, respectively; D is pile diameter. 3.1. Veri®cation by comparisons with results for pile groups 3.1.1. Comparisons for capped pile group with two piles An obvious characteristic of a capped pile group with only two piles is that when a symmetric horizontal loading is applied to the cap in the direction perpendicular to the line connecting the two piles, the cap has no enhancing eect on the two piles and the behaviour of the capped pile group is similar to that of a free head pile group. If the load is applied parallel to the line joining the piles, the cap stiness will have a large eect on the displacement. For veri®cation of this, the pile group as shown in the inset to Fig. 3 was analysed, where the ratio of embedded pile length, Lem , to diameter, D, was chosen to be 25. The ratio of the exposed pile length H0 to pile diameter was 2.5 and the pile±soil and cap±soil stiness ratios are taken as 4000 and 4285, respectively. The results of the analysis of the pile group and capped pile group with only two piles subjected to horizontal loading in the two directions are shown in Figs. 3a and b, respectively. As can be seen in Fig. 3a under horizontal load in the x-direction, as expected, the normalised displacement of the capped pile group is very close to that of the pile group with a rigid cap but much lower than that of the pile group with a completely ¯exible cap. Fig. 3a also shows that when pile spacing is less than 6 times the pile diameter, the displacement factor is sensitive to change in spacing. However, with increase of pile spacing, the horizontal displacement is less and is not greatly aected by the spacing. Fig. 3b demonstrates that under a horizontal load in the y-direction, there will be no dierence among a capped pile group and two-pile group with a completely ¯exible or rigid cap with respect to normalised displacements. The results of Fig. 3b indicate that the program is probably working correctly. However, for problems with more piles, further comparisons are needed.
H.H. Zhang, J.C. Small / Computers and Geotechnics 26 (2000) 1±21
Fig. 3. Comparison of 2-pile group with capped pile group: (a) Px only; (b) Py only.
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3.1.2. Comparison with pile groups In an example given by Poulos [11], there is a ®xed-head pile group with 16 piles as shown in the inset to Fig. 4. The ratio of pile length to diameter is 25 and the pile ¯exibility factor KR is equal to 10ÿ5 (de®ned as KR Ep Ip =Es L4em , Ip is the second moment of area of the pile). Here the Poisson's ratio for the soil is taken as 0.499 while 0.5 was used by Poulos [11]. Ep =Es was chosen to be 79.6 while Er =Es was made large (85 250) to represent a rigid cap so that the behaviour of the capped pile group would be like that of a ®xed-head pile group although for the rigid cap some head rotations may still occur. The cap is just clear of the ground (i.e. H0 0:0).
Fig. 4. Typical horizontal load distributions in ®xed head pile group and capped pile group.
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The load distributions against pile spacing are plotted in Fig. 4. We can observe that for a pile spacing ratio of less than 4 there is poor agreement of the typical horizontal load distributions between the ®xed-head pile group and the capped pile group. Load on piles 3 and 4 computed using the present method is higher than that given by Poulos [11], while the load on piles 1 and 2 is lower, especially for a capped pile group with close pile spacing. With increase of the pile spacing, the load distributions tend to become more uniform. Because of the lack of agreement, a further comparison was made with the results of El Sharnouby and Novak [12] for the same problem. It may be seen in Fig. 4 that these results are in fairly good agreement with the present results.
Fig. 5. The eect of pile±soil stiness ratio on de¯ection of capped pile groups under horizontal loads.
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3.2. Parametric analysis In order to demonstrate the use of the present method for the analysis of laterally or axially loaded capped pile groups, the most critical parameters were examined to show their eects on the maximum displacement of a capped pile group. The ratios of the embedded pile length to pile diameter (Lem =D) were chosen to be 10, 20, 40, 80. The Poisson's ratios of the soil and the cap were assumed to be 0.35 and 0.15, respectively. The ratio of the exposed pile length H0 to pile diameter was 2.5 and the pile spacing ratio was taken as 5. The thickness of the cap was 0.5 m, while the breadth and length of the cap were chosen to be 18 and 13 m, respectively. 3.2.1. The eect of soil and pile modulus Varying the pile±soil stiness ratio Ep =Es from 101 (¯exible pile) to 106 (rigid pile) and keeping the cap-soil stiness ratio constant (Er =Es 2000), the present program APPRAF was used to calculate displacements at the central point of each pile head. The normalised displacements for pile 5 (maximum displacements generally occur for this pile) are plotted in Figs. 5 and 6 against the pile±soil stiness ratio.
Fig. 6. The eect of pile±soil stiness ratio on de¯ection of capped pile group under vertical loads.
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It may be seen from Fig. 5 that the horizontal displacements of the capped pile group decrease rapidly with the pile±soil stiness ratio. The results corresponding to four pile lengths shows that there is almost no in¯uence of the pile lengths on the displacement of the capped pile group for a pile±soil stiness ratio of less 103 and for pile slenderness ratios in excess of 10. However, when the piles become substantially rigid (Ep =Es up to 106), the shorter piles will obviously produce higher displacement than the longer piles. Fig. 6 shows that for vertical loading and a pile±soil stiness ratio of less than 100, the displacement of the capped pile group increases sharply as the pile±soil stiness ratio reduces. For pile±soil stiness ratios greater than 100, the displacement of the capped pile group will decrease very slowly, especially when the piles are short. Fig. 6 also shows that there are no obvious dierences in the de¯ections of the group with small pile±soil stiness ratios (<100) for dierent pile lengths. However, for pile±soil stiness ratios of over 100, pile lengths have more eect on the de¯ections of the group and the group with longer piles will produce lower vertical displacements.
Fig. 7. The eect of cap±soil stiness ratio on de¯ection of capped pile groups under horizontal loads.
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3.2.2. The eect of cap modulus The pile±soil stiness ratio was chosen to be 2000 and the cap±soil stiness ratio (Er =Es ) was varied from 101 to 106 in this example. Program APPRAF was used to analyse the capped pile group under horizontal loading and vertical loading, respectively. The displacements of the capped pile group under horizontal loading and under vertical loading are shown in Figs. 7 and 8, respectively, against the cap± soil stiness ratio. Fig. 7 shows that if the capped pile group is subjected to horizontal loading only, the increase of the cap±soil stiness ratio will obviously reduce the horizontal displacement of the group. However, once the cap becomes stier (a cap±soil stiness ratio >1000), further increase of the cap±soil stiness will have a negligible in¯uence on the horizontal displacement of the group. Furthermore, the lengths of the piles have almost no in¯uence on the horizontal displacements of the group. On the contrary, when the capped pile group is subjected to vertical loads only as shown in Fig. 8, the cap±soil stiness ratio has only a minor eect on the vertical de¯ection of the group. However, increase in pile length will greatly reduce the vertical displacements of
Fig. 8. The eect of cap±soil stiness ratio on de¯ection of capped pile groups under vertical loads.
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17
the group. For example, as shown in Fig. 8, the vertical displacement of the group with a pile slenderness ratio of 80 may be reduced to approximately 43±48% of that of a capped pile group with L=D 10.
Fig. 9. Models for soils where the modulus increases with depth.
Fig. 10. The eect of dierent soils on the displacement of capped pile groups under horizontal loading.
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3.3. Capped pile groups embedded in soils where the modulus increases with depth Three types of elastic soils where the modulus increases with depth are chosen in this paper to demonstrate this method. The soil models are shown in Fig. 9. It is assumed that the capped pile group with 12 piles as shown in Fig. 1 is embedded in the dierent soil types. The stinesses of the dierent soils can be expressed as Es z Es0 m z=Lem n If m 0, the above equation represents a homogeneous soil with a constant elastic modulus, i.e. Es0 Esb ; if Es0 0 and n 1, it represents Gibson's soil; if Es0 > 0 and n 1, it is Banerjee's soil; and the general case is a parabolic variation of soil modulus. The soil modulus Esb was taken as 7 MPa at the toe of the pile and the ratio Lem =D was taken as 20. The pile±soil stiness ratio Ep =Esb and the cap±soil ratio
Fig. 11. The eect of dierent soils on the displacement of capped pile groups under vertical loading.
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Er =Esb were 4000 and 4285, respectively. The cap thickness was chosen to be 1.0 m. The breadth and length of the cap were 20 and 15 m, respectively. Other parameters were the same as those used in Section 3.2. Figs. 10 and 11 show that whether the capped pile group is subjected to horizontal loading or it is subjected to vertical loading, the pile spacing has a signi®cant eect on displacements of the group when the pile spacing is less than about 6 times the pile diameter. Furthermore, they show that the homogeneous soil is one of the best soils with respect to resistance of de¯ection of a capped pile group. On the contrary, the Gibson's soil is the poorest one as may be expected. Moreover, beyond a value of pile spacing of S=D 6, the eect of a change in spacing on the displacement of the group examined will become much less. For the capped pile group subjected to horizontal loading, moments in pile 1 corresponding to two pile spacings are plotted in Figs. 12 and 13. With a pile spacing ratio of 2 as shown in Fig. 12, the largest value of the positive moment in pile 1 is much higher than that in pile 1 with a pile spacing ratio of 6 as plotted in Fig. 13. This demonstrates that a relatively small pile spacing ratio (i.e. less than 6) may lead to a large moment in the pile and is therefore a less economical use of the piles.
Fig. 12. Moment distribution in single pile embedded in dierent soils with close pile spacing.
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H.H. Zhang, J.C. Small / Computers and Geotechnics 26 (2000) 1±21
Fig. 13. Moment distribution in single pile embedded in dierent soils with large pile spacing.
4. Conclusions In this paper, a powerful method for analysing the behaviour of capped pile groups subjected to both lateral and vertical loading has been developed. The analysis of a capped pile group with only two piles, comparisons with solutions for ®xed-head pile groups, and investigations of the eect of soil, pile and cap parameters demonstrate that: 1. The method can be used to analyse the behaviour of a capped pile group foundation subjected to vertical loading or horizontal loading as well as moments in all axis directions. 2. Through comparison of the load distribution on the pile heads between a capped pile group (where the cap is very sti) and a pile group with a rigid cap, it is shown that the present method for analysis of a capped pile group can reproduce previous results. 3. Pile±soil stiness ratio plays an important role in the resistance of a capped pile group to lateral de¯ection for pile±soil stiness ratios of less than 1000. In this
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range, the displacements of the capped pile group will increase sharply with the reduction of the pile±soil stiness ratio. 4. For the example chosen, change in the pile±soil stiness ratio had a large in¯uence on the displacement of the capped pile group under vertical loading when this ratio was less than 100. When the pile±soil stiness ratio is greater than 100, change in the pile±soil stiness ratio had only insigni®cant eect on the displacement. Pile lengths have a relatively large eect on the de¯ections of the capped pile group under vertical loading, but much less eect for horizontal loading. 5. For lateral loading, small pile spacing (i.e. less than 6 times the pile diameter) will not only result in a large de¯ection of a capped pile group under applied load, but it can also cause larger moment in the piles. 6. The method can conveniently solve problems involving capped pile groups embedded in dierent soil types. The results of analysis con®rm, as expected, that a Gibson's soil provides the poorest lateral restraint for capped pile groups subjected to lateral loading. References [1] Hongladaromp T, Chen N, Lee S. Load distribution in rectangular footings on piles. Geotechnical Engineering 1973;4(2):77±90. [2] Kuwabara F. An elastic analysis for piled raft foundations in a homogeneous soil. Soils and Foundations 1989;29(1):82±92. [3] Clancy P, Randolph MF. An approximate analysis procedure for piled raft foundations. International Journal for Numerical and Analytical Methods in Geomechanics 1993;17:849±69. [4] Poulos HG. An approximate numerical analysis of pile±raft interaction. International Journal for Numerical and Analytical Methods in Geomechanics 1994;18:73±92. [5] Ta LD, Small JC. An approximation for analysis of raft and piled raft foundations. Computers and Geotechnics 1997;20(2):105±23. [6] Ta LD, Small JC. Analysis of piled raft systems in layered soils. Int J for Num and Anal Methods in Geomechs 1996;20:57±72. [7] Bogner FK, Fox RL, Schmit LA. The generation of inter-element-compatible stiness and mass matrices by the use of interpolation formulas. In: Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, 1965. p. 397±443. [8] Small JC, Booker JR. Finite layer analysis of layered elastic materials using a ¯exibility approach. Part 2-circular and rectangular loadings. International Journal for Numerical Methods in Engineering 1986;23:959±78. [9] Rao SN, Ramakrishna VGST, Rao MB. In¯uence of rigidity on laterally loaded pile groups in marine clay. Journal of Geotechnical and Geoenvirnmental Engineering, ASCE 1998;124(6):542±9. [10] Zhang HH, Small JC. Analysis of axially and laterally loaded pile groups embedded in layered soils. In: Proceedings of 8th ANZ Conference on Geomechanics, Hobart (Tasmania), February, 1999. [11] Poulos HG. Behaviour of Laterally-Loaded Piles: II Ð Pile Groups. J Soil Mech Found Div (JSMFD), ASCE 1971;97(SM5):733±51. [12] El Sharnouby B, Novak M. Static and low-frequency response of pile groups. Can Geotech J 1985;22:79±94.