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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 groups under vertical and horizontal loading has received much attention

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 moments rather 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

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 the layered soil can then easily be computed, as the solution for a layered

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 =Xm

j=1

wherem 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 directionsx, y, z at each location δ i

s are the

displacements at the unit load locationsi, I ij are the influence factors for the displacement at

locationi due to a load at location j, and P j are the loads at locationj This can be written in

matrix form for all displacements

The influence matrix can then be inverted to obtain the stiffness matrix for the soil continuum

e g.,

whereK s = [I s]−1.

FIGURE 1 A raft and a pile group subjected to external forces and interface forces in all directions (they direction is

not shown).

For the piles, a stiffness relationship may be written

whereK 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 pis the load vector for the piles due to shaft loads, andQ 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

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= ∗sandP p = −P s This gives the final stiffness relationship for the pile-soil continuum

whereR is the load vector consisting of loads at the pile heads Q or any loads not along the pile

shaftS 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 surface is loaded, this can be used to determine the behavior of the raft 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)

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

P x , P y , P z are the total external loads applied to the raft in thex, y and z directions,

respectively

M x , M y , M zare the total external moments applied to the raft about thex, y and z axes,

respectively

{a} to {f } and {a0} to {f0} and are auxiliary vectors

D x , D y , D z , θ x , θ yandθ zare 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

In order to determine the accuracy of the finite layer method described above, a 9-pile raft with

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 lengthD of the square pile The piled raft as

shown in Figure 2 was constructed in a layered soil 15 m in depth The pile slenderness ratioL/D

was chosen to be 20 and pile spacing ratiosS/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)

Pile side length 0.5 m (square pile) Equivalent pile diameter 0.564 m (circular pile)

Raft width Lr S/D = 3; 4.5 m S/D = 5; 6.5 m

Raft breadth Br S/D = 3; 4.5 m S/D = 5; 6.5 m

Overhang of raft 0.5 m Raft thickness 0.25 m

Soil Poisson’s ratio 0.3

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

FIGURE 2 Layout of a 9-pile raft embedded in a soil (whereD0is 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 forS/D = 3 and 169 identical elements

forS/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 thex- 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,

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 thex-direction (the

direction of loading) and 24 m in they-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 /PtotalwhereP i is

the axial force in the pile andPtotalis the total load on the raft) in pile 1 (corner pile) and pile 5

(central pile) against pile length It may 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 ForS/D = 3, the central pile carries less load than the corner pile, but

forS/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 thex direction (M xx) It can be seen that the moments in the raft provided by both methods

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 ofS/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 loadingM 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

FIGURE 5 Comparisons of moments in raft along section A-B.

FIGURE 6 Comparisons of deflection of raft along section A-B.

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