A boundary element technique for the analysis of raft and piled raft foundation under static vertical loading and supported on an elastic infinite or finite stratum

A. V. Mendonça, V. S. Almeida & J. B. de Paiva Department of Structural Engineering.

Sao Paulo University at Sao Carlos, Brazil. Abstract In this article, static problems involving axially-loaded piled raft foundations are analysed by a boundary element formulation. In this approach, the raft is assumed to have linear elastic properties and is represented by integral equations based on Kirchhoff’s plate model. The soil is supposed to be an infinite layered medium or finite stratum solid. Numerical analyses are presented for different foundation configurations and the results of those analyses are investigated. 1 Introduction In recent decades, soil-foundation interaction problems have been investigated by many researchers in which the supporting medium is supposed to have elastic behaviour. The models for elastic responses of soil have been developed along two distinct lines. In the first, the soil is represented as a mutually-independent spring system (Winkler’s model), in which the disturbances caused in a certain region are not transmitted to another location outside it. In the second approach, the soil is assumed to be a continuous elastic medium, so that the effects of loading can also be observed away from the loading region. Many models have been developed taking into account the geometrical and mechanical properties of the continuum, such as the well-known Mindlin’s solution to the problem of a uniform, homogeneous, semi-infinite, isotropic, linear elastic solid under static concentrated loading. Some researchers have presented formulations for the soil-foundation interactions that occur when the supporting medium is assumed to be a Mindlin half space and these analyses can be divided into three groups: raft-soil , pile-soil and raft-pile-soil interaction problems. In the first group there are analytical solutions to the complete raft-soil problem, based on variational methods for particular geometry and loading of the raft [1,2]. Other authors have proposed numerical solutions using a single technique (either the finite element method (FEM) [3,4] or the boundary element method (BEM) [5] ), or a mixed FEM/BEM model [6]. In solutions to the second type of problem, exploring the interactions between pile groups and the soil, the elastic contribution of each pile to the global performance has been analysed

classically, by using load-transfer curves (LTC) and a direct continuum approach. In the LTC model [7-8], the interactions of two piles were initially obtained and then applied to the study of pile groups by superposing the results for all piles in the group [7-9]. The elastic-continuum model described in Mindlin [10] was employed to represent the soil-pile interaction in many cases such as in Shen et al. [11], in which the tractions of soil-pile interaction are approximated by a finite series, and in Xu and Poulos [12], where each pile is considered as a 3D element for the interface tractions and a linear element beam for its internal tractions. The load-transfer curves have been extended to model the third type problem (raft-pile-soil), both for the rigid rafts [13-16] and the flexible caps [17-20], by invoking Kirchhoff’s plate model and numerical algorithms, based respectively on FEM and BEM. In Fatemi-Ardakani[18] and Poulos[20], the pile contributions are represented by springs and a computer program is used to evaluate their stiffness, but the raft-soil interaction forces are not taken into account to obtain the final value of the elastic constant of spring. Besides the LTC models, the direct application of Mindlin’s solution to raft-pile-soil interaction problems has also been reported by some authors. In Butterfield and Banerjee[14], Shen et al.[16], the interactions between rigid cap and pile groups were modelled using two distinct approaches. In the first, the piles were divided into smaller parts and the problem was solved by the Finite Difference Method (FDM) while in the second, each pile was represented by a finite series and variational methods were used to compute all contribution from soil, pile and rigid raft. Recently, in Mendonça and Paiva [21], a formulation has been given in which the soil, the piles and the raft are assumed to be flexible and represented by a single BEM algorithm; moreover, all interactions among the structural components of foundation and soil are simultaneously taken into account. In all articles discussed so far, the soil is assumed to be a homogeneous, isotropic, linear elastic solid which has an infinite stratum. However, in some cases the infinite thickness of a compressible medium can lead to overestimated values for the elastic displacements in soil-structure problems. In the literature, techniques proposed to model finite layer media fall basically into four distinct types. In the first, the contribution of each layer is substituted by an equivalent spring system [22-23]. However, these approaches are strongly dependent on the model assumptions to evaluate the equivalent stiffness coefficient of the spring system. The second line of research has been developed from Burmister’s model [24-28], which was initially proposed for a layered elastic solid under a concentrated vertical load acting on the free surface. In Chan et al.[29], Pan[30], the Burmister approach was extended to the concentrated forces acting within a layered half space. However, the governing partial differential equations (PDE) for these models lead to a cumbersome procedure to get approximated numerical solutions and, to the authors’ knowledge, there are no closed-form solutions to this fundamental elastic problem. The third approach to a finite elastic stratum is the so-called finite layer technique reported in Small and Booker [31] and in Southcott and Small [32]. In general, this provides no representation of the shear stresses along the shaft at points within the layer domain, so that these stresses are adopted as equivalent concentrated forces acting at the nodal points located on the layer interfaces, and Bessel’s functions are employed to evaluate the soil stiffness matrix. However, solutions to these hypergeometric functions are available only for particular distributions of loading. In some special cases such as the finite stratum, a further line of modelling has been reported which starts from the infinite Mindlin model. Steinbrenner [33] has presented a formulation for the evaluation of displacements at interior points of the elastic stratum by superposition of Boussinesq solutions for load and field points along the same vertical, in which the first is to the desired point and the vertical coordinate of the second point is the depth

of the plane assumed as rigid. Another application of the infinite Mindlin model to elastic layered solids involved discretization of the rigid plane using FEM or BEM, and imposing vertical constraints for displacements at nodal points on the rigid supporting medium. In Banerjee [34], a raft-pile-soil interaction problem is modelled using this treatment for the soil, but the raft is assumed to be rigid. In the present article, a BEM formulation for raft-pile-soil is described, in which all the structural systems and soil are assumed to be flexible. In addition, this formulation has been developed for static loading and full contact at surfaces between the structural components and the supporting medium. The first part of this paper is an extended review of Mendonça and Paiva [21], in which the integral and algebraic representations of raft-pile-soil interaction (with the elastic medium assumed to be an infinite stratum) are described. In the second part, the previous formulation is expanded to analyse piled-raft foundations supported on a finite stratum, using both the Steinbrenner model and direct discretization of the rigid plane by invoking the Mindlin Model. 2 Bending Plate Representation In this section, the integral representations are described for a raft subjected to external loading as reactive forces developed by raft-soil and raft-pile interactions. The raft is assumed to be a thin elastic plate and it is represented by Kirchhoff’s plate model. 2.1 Integral Equations In figure 1 are depicted the forces acting on the median plane of the raft and the efforts on the plate boundary. By performing the weighted residual technique or Betti’s reciprocity theorem, the domain forces and boundary efforts of a real bending plate can be linked to their respective dual quantities of the fundamental problem so that the transverse displacement integral representation of the raft is written as

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )+⋅=⋅+Γ⋅−⋅+ ∫Γ cpwsRcpwsRdspmsspqswpkw ccccnpn ,,,, **** θ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +Ω⋅+Ω⋅+Γ

⋅−⋅ ∑∫∫∫

=ΩΩΓ

Ncel

d

dcelpnn

d

dSpwSpdSpwSgdspsmspwsV1

***.

* ,, ,, θ

( ) ( )∑∫=

ΩΩ⋅

Np

d

dTT

d

dSpwSp1

* , ; Ncc ,...,2,1= (1)

where p refers to the source point; s and S are the field points associated with the boundary and the domain of the plate, respectively; )(smn and )(sVn are the boundary efforts related to the bending moment and Kirchhoff’s shear force, respectively; )(sRc is the corner reaction;

)( ),( p ssw θ are boundary quantities referring to the transverse displacements and normal

slopes, respectively; )(Sg is external loading distributed over the plate domain; )(Sp is the soil-raft contact force; N , FCcN are the total numbers of the corners and concentrated forces, respectively; T and , ΩΩΓ cell are regions of the boundary plate, the raft-soil and raft-pile interaction domains, respectively.

To provide sufficient equations for the boundary unknowns, in this formulation an additional integral equation is obtained by differentiation of the transverse displacements eqn (1), resulting in

( ) ( )++ pwkpwk um ,2,1 ( ) ( ) ( ) ( )[ ] ( ) ( ) =+Γθ−∫Γ

sws,pRd s,pmss,pqsw c*

m,c*

k,np*

m,n

( ) ( ) ( ) ( )[ ] ( ) ( )++Γθ−∫Γ

c,pwsRd s,psms,pwsV *m,cc

*m,pn

*m,n ( ) ( ) +Ω∫Ω dSpwSg m ,*

,

.

( ) ( ) +Ω∑∫=

Ω

Ncel

i

icelm

i

dSpwSp1

*, , ( ) ( )∑∫

=Ω

ΩNp

i

iTm

i

dSpwSp1

*, , , Nc,...,2,1c = , 2,1k = . (2)

For eqns (1) and (2), the free integral terms ( 21 k and k ,k ) and the kernels, marked by ‘*’, are given in Appendix I.

1x

T

m

Tp

g

u

tsn

r

x2

g

x3

Vnmn

Rcp

cell

Figure 1: Boundary Efforts and Domain Forces acting on the Raft. 2.2 Discretization General analytical solution to the integral representations eqn (1) and eqn (2) is not available, so that an alternative strategy in achieving the solution is to employ numerical techniques, and discretization is one of the earliest stages in this. The BEM discretization consists in dividing the problem boundary into smaller regions called boundary elements, while the domain is divided into smaller areas called cells. Nodal points are defined for each boundary element and the problem quantities are approximated by employing their nodal values and interpolation functions. For the domain forces an analogous technique is performed on the cells. In the present work, the raft boundary is divided into linear elements and a linear interpolation is adopted for all continuously distributed boundary quantities of the raft, i.e., transverse displacement w , normal slope pθ , bending moment nm and Kirchhoff’s force nV . The linear interpolations for these quantities can be written

θ

ϕ

ϕ+

θ

ϕ

ϕ=

θ 2

p

2

2

21p

1

1

1

p

w0

0w0

0w, (3)

ϕ

ϕ+

ϕ

ϕ=

2n

2n

2

21n

1n

1

1

n

n

mV

00

mV

00

mV

, (4)

where k

nkn

kp and V , , mwk θ are the nodal values for the transverse displacement, normal slope,

Kirchhoff’s shear force and bending moment, respectively, where the superscript k is assigned the number 1 for the first element node and 2 for the second. iϕ is the linear interpolation function which can be given in dimensionless form as ( ) 2/11 ςϕ −= and ( ) 2/12 ςϕ += , with the coordinate ς defined within range 11 ≤≤− ς . In addition, nodal points for the corner reactions and displacements are located at the non-smooth regions of the plate.

j

ki

k

j

i

kΩΓk

x1

x2x2

x3

1x

Figure 2: Raft Discretization.

Each raft-pile interaction force )(spT is supposed to have a constant distribution over

circular cell dT Ω and a single nodal point is defined to it. The raft domain is divided into

triangular cells and raft-soil interaction force )(sp is assumed to have a linear distribution over

each cell celldΩ , so that nodal points are defined at the triangle vertices, (see figure 3). The

linear interpolation for the raft-soil interaction force over a single cell domain is given in the form

33

22

11)( pppsp ⋅+⋅+⋅= εεε , (5)

where 321 and , ppp are the nodal values of the raft-soil reaction, (see figure 3). iε is the homogeneous coordinate and its values are given by

( ) ( ) ( ) Dxxxx+x xx x xx= t21 2j1k2kjj1k2k2j1/

12

−−+−ε , (5-a)

( ) ( ) ( )[ ] t21 Dxxxx +xxx +xxx =

2k1i2i1k2k1i2i2k/2 −−−ε , (5-b)

( ) ( ) ( )[ ] t21 Dxxxx+xxx +xxx =

2ijj1i1i1j2ji23/

12−−−ε , (5-c)

( ) ( ) ( )xxx+xx x xx x=D 2j2i1kik1j2k2j1it −−+−22

, (5-d)

where αβx ( )kj,i, ;2,1 == βα is the coordinate of the cell nodal points .

2

3p

p1

x3

p1

2

x3 3g

gg

Figure 3: Raft-Soil Force Nodal Values in a Single Cell. The integrals involving external loading or the reactive subgrade forces defined over the

thd cell domain dΩ can be transformed into equivalent integrals defined only along the cell contour dcellΓ . The transformation integral scheme is shown for arbitrary plan form of the cell domain. Starting from eqns (1) and (2), their domain integrals can be reduced to a single form as

( ) ( )SdS kh d

d

*

~d~Ωλ= ∫

Ω

, (6)

where λ is the domain quantity (external loading or subgrade reaction) and

~

*k is the kernel

vector given by

( )( )

=

=S,pwS,pw

KKK *

m,

*

*2

*1

~

* . (7)

If a linear interpolation is adopted for the domain quantity, its geometrical representation results in

( ) ( ) ( ) )(32211 SASxASxAS ss ++=λ (8) where )( (S) 2s1 Sxandx s are the coordinates of point S in the local system )x ,x( 21 . By translating the coordinate system, the coordinates of S in ( )21 x ,x can be written in a

new system )x ,( 21x and then, by transforming the latter system to the polar coordinates ( )θ,r , the domain quantities can be given as follows

( ) )( cos 321 pAsinrArAS ++= θθλ , 2,1i = (9) with

( ) ( ) ( ) )(322113 SApxApxApA ss ++= . (10) The cartesian-polar Jacobian is

θddrrd d =Ω . (11) Substituting eqns (7)-(11) into eqn (6), the domain integral can be re-written as follows

( ) θθθθ

d dr r A+ i r A+ r A K =hR

d 321~

*

~nscos∫ ∫ ; 2,1=i . (12)

From the geometrical relation shown in figure 4, it follows that

dcell d =d Γρϕθ cos ; 2,1=i (13)

where ϕ is the angle between the unit radius vector rr and the unit vector γ

r normal to the cell

contour dcellΓ at the field point s ; ρ is the distance between the field and load points represented respectively by sand p (see figure 4). Substituting eqn (13) into eqn (12), the boundary of the cell is written

( ) Γ∫Γ

dcelld d K =hd

ρϕρ cos

~~, 2,1i = (14)

where the vector components ( ) ( ) ( ) TKKK ρρρ *2

*1

*

~= for transverse displacement and the

slope integral equations of the raft are respectively given by

( ) ( ) ( ) ( )3ln4128

7ln10 400

33

2,21,1

4*1 −+−+= ρ

πρρρρ

πρρ

DAAA

DK , 2,1i = , (15)

( ) ( ) ( ) ( )1ln336

1ln464 ,

23

2,21,1,

3*2 −−−+−= ρρ

πρρρρρ

πρρ jjjj m

DAAAm

DK , 2,1j,i = , (16)

where D is the flexural rigidity of the raft, i,ρ is the direction cosine of the radial vector →ρ

and jj m,,ρ is the cosine between the directions →ρ and

→m (see figure 4).

For the evaluation of the raft-pile domain integrals, a similar procedure can be performed in the circular cell and the coefficients A1 and A2 have zero values because the raft-pile interaction force is assumed to have constant distribution over TΩ , (see figure 1). It should be noted that when the load point is located at the centre of the circular cell, obviously 0=ϕ and therefore

θρ dd cell =Γ . Thus, analytical integration can easily be performed on the integrands of the eqns (15) and (16), resulting in the following forms

( ) ( )3ln464

43*

1 −= ρρρD

AK , (17)

( ) ( )1ln336 ,

33*

2 −−= ρρρρ jjmDAK , 2,1j,i = . (18)

ρ

m

m

s

s

pd

b

c

adcelΓd

dρ

dd

b

ca

d

dcelΓ

ΩΓ

x2

x1

r

dp

d

drdr

Γdcel

x1

_

_x2

ρ r =ρ

Figure 4: Cell Contour Integration Scheme. The integration of kernels over each boundary element or cell contour can be performed by analytical or numerical schemes. The latter technique is more popular and there are many works in which more elaborate treatments for this problem are discussed; see [35-37]. In general, the numerical integration can be represented in compact form as

( ) ( ) ( ) ( ) ( ) ( )~

kkT

~

qN

1kk

*

~

T

~

1

1

*d

d

* WJKdJKdSS,pK λ

εϕε=λ

εεϕε=Γλ ∑∫∫=−Γ

(19)

where J is a Jacobian operator; kε and kW are the coordinate and weight quadrature for point

in the thd boundary element or cell contour; qN is the number of points used to perform the integration. 2.3 Algebraic System After evaluating the integrals of the discretized raft equations for all boundary elements and cells, an algebraic system can be assembled as follows

+

+

=

Γ

Γ

~

2~

1~

0

~T

~cell

~

2

~

2~

1

~

1~

0

~

0

~

~

2~

1~

0

~T

~cell

~

~~~

2~~~

1~~~

0

G

G

G

P

P

SQ

SQ

SQ

V

C

C

C

W

W

W

I0H

0IH

00H

(20)

where

~T

~cell

~W and W,ΓW are the nodal vectors referring respectively to the transverse

displacements at the points along the raft boundary, at the cell vertices and at the pile tops. ΓV is the nodal vector for the boundary efforts and corner reactions of the raft.

~T

~P and cellP are

the nodal quantities respectively associated with the interaction forces of the raft-soil and the

raft-pile problems. 2

~

1

~

0

~, G and GG are the independent vectors related to the external

loading. The influence matrices and vectors marked with superscripts '2' and '1' ,'0' are respectively associated with the raft boundary, the cells and the pile tops.

~I and

~0 are the

identity and null matrices, respectively. 3 Pile-Soil Representation In this section, the integral and algebraic representations of the pile and the soil are presented. The soil is assumed to have either an infinite layered medium or a finite stratum, so that two approaches are described for these geometrical configurations. 3.1 Infinite Layer The soil integral and algebraic representations for an infinitely thick medium are shown first. In the following section, the integral equations are adapted to a finite stratum. 3.1.1 Integral Equations In figure 5 are depicted the interaction forces acting on the surface or within the infinite elastic half space assumed to be the real problem of the soil. By applying a mathematical technique such as the weighted residual method, built from residual functions associated with the equilibrium equations of the real half space and weighted functions associated with Mindlin’s fundamental half space solutions, the following integral equation for vertical displacement can be obtained

( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫∫= Γ Γ

∗∗

Γ

Γ+Γ+Γ=np

ibicpis

pi bis

dSSpudSSpudspspupu1

3333*33 ,, , στ (21)

where ( )Sτ and ( )Scσ are respectively the shear stress on the shaft and the normal stress at the

tip of the pile; np is the number of piles and ( )spu ,*33 is Mindlin’s fundamental solution for

the 3x direction at load and field points, given by

( )

++++

−=∗

32

5

2

452

3

1

231

133 116

1),(Ra

Ra

Ra

Ra

Ra

Gspu

ss νπ (22)

where; )()( sxpxr iii −= ; iirrr = and the index 3,2,1=i ; 0)(3 >= pxc ; 0)(3 >= sxz ,

( )pxi and ( )sxi are the coordinates of the load and field points, respectively, 's is the

reflection of point s in the half space surface plane ( )0x3 = , 221 )( czrR −+= ,

222 )( czrR ++= and coefficient ia is given by

( ) czczaa

czczaacza

sss

s

2))(43( );43(18

;)(6 ;43 ;)(2

52

4

232

21

−+−=−−−=

+=−=−=

ννν

ν. (22-a)

In eqn (22), G and ssν are the Poisson ratio and shear modulus for the soil. bipi and ΓΓ

are the boundaries of the shaft and the pile base. sΓ is the raft-soil contact surface and p is the raft-soil interaction force.

Tpp

Soil surface discretization

1

2x

x

(b)

pj

L/3

L/3

L/3

n

m

l

ττ n

τm

lτ

T

cσσσ

pp

Soil

ττ

cσ

x3

Pile

x1

p

Figure 5: Interaction forces in the pile-soil problem.

3.1.2 Discretization In the previous section, the linear interpolation to approximate the soil-raft interaction forces was described and this treatment is maintained for the pile-soil problem, so that the free traction surface of Mindlin’s half space is divided into triangular boundary elements whose nodal points coincide with the cell nodes of the raft. Each pile is geometrically represented by one linear element and the shear stress along the shaft is interpolated with a quadratic function, (see figure 5), so that the shaft shear stress is represented mathematically by

33

22

11)( τφτφτφτ ++=S (23)

where the interpolation functions are given by ) 2+9 9 (21=)( 2 ξξξφ −1 ,

ξξ−ξφ 6+ 9 =)( 22 and ) 3 9 (

21=)( 2

3 ξ−ξξφ ,with L(s)x= 3ξ and its range 10 ≤ξ≤ ,

and iτ is the shear stress nodal vector. In addition, at the tip of the pile the normal stress is assumed to have a constant distribution and a single nodal point is located there. The discretized integral representation from eqn (21) for all nodal points of the thi pile can be written

( )( )( )( )

[ ]( )( )( )( )

( )∑ ∫= Ω

+

Ω

=

p

k

N

k

kck

k

k

k

k

k

k

k

k

k

k

k

k

i

i

i

i

sA

siusiusiusiu

d

siusiusiusiu

uuuu

1

4*33

3*33

2*33

1*33

3

2

1

321

4*33

3*33

2*33

1*33

4

3

2

1

,,,,

,,,,

στττ

φφφ

( )( )( )( )

[ ]∑ ∫= Ω

Ω

+cell

k

N

cell k

k

k

cell

k

k

k

k

ppp

d

siusiusiusiu

13

2

1

321

4*33

3*33

2*33

1*33

,,,,

εεε . (24)

The last term in eqn (24) refers to the contact between the pile cap and half space surface, so that a further integral equation incorporating the cap-soil interaction forces is needed in order to get a solvable algebraic system. One equation which satisfies this purpose can be obtained by writing an integral representation of the vertical displacement at nodes located on the cap-soil interaction surface, that is

( )[ ] ( ) ( )∑ ∫= Ω

+

Ω=p

k

N

k

kckk

k

k

k

kkcell sAswudswuw1

*33

3

2

1

321*33 ,, σ

τττ

φφφ

( )[ ]∑ ∫= Ω

Ω+cell

k

N

cell k

k

k

cellk

ppp

dswu1

3

2

1

321*33 , εεε . (25)

An analogous procedure for the integral evaluations shown in section 2.2 can be performed with the discretized integral equations from the pile-soil interaction problem. Thus the integration scheme is briefly discussed and is divided into two types. The first refers to a load point located within or on the surface of the half space and a field point located in the discretized cell domain at ground level. The second integration type is associated with a singular configuration of the kernels. If a flat surface is adopted for the cell domain, the radius vector can be decomposed into directions normal and tangential to the cell plane. It should be noted that the vector normal to

the cell plane is invariant (see figure 6) and thus Mindlin’s fundamental solution *33u can be

rewritten as the projection of radius vector rr on the plane of the cell domain.

( ) ( ) [ ] [ ]

+

+−

+

+−−

=∗

21

22

2

23

22

133

)()(116

1,ρρ

νπcz

a

cz

aG

Spuss

[ ] [ ]2122

4

25

22

3

)zc(

a

)zc(

a

ρ++

+

ρ++ ( )[ ]

ρ++

+23

22

5

zc

a . (26)

Substituting eqn (26) into eqn (12) gives the following relation

( ) ( ) ( ) cell

cell

cell

cell

*33 dcosfdSpS,pu Γ

ρϕ

ρ=Ω ∫∫ΓΩ

(27)

where the function ( )ρf is given by

( ) ( ) ( ) ( ) ( ) ( )[ ][ ]

( ) ( ) ( ) ( ) ( )[ ]ββααα

ρρββαααρ

25143322113

2,21,12514332211

hahahahahaA

AAgagagagagaf

++++

++++++= (28)

with the supporting functions ( ) ( )yh and yg ii given in Appendix III; their arguments are

czy −=α= and czy +=β= . When the load and field points are located along distinct piles, in this analysis, to evaluate the integral the shaft shear stresses are assumed to act along the longitudinal axis of the pile. In other words, the load point ‘sees’ all field points located at the circumference within the pile as a single point positioned at the centre of this circle. In general, for this case the numerical evaluation of the integrals can be successfully performed, using popular polynomial quadratures. If the load and field points belong to the same pile, the real location of the shear stress on the shaft surface is adopted. Hence, the kernels tend to be quasi-singular rather than singular. If numerical evaluation is chosen to perform the resulting quasi-singular integrals, special techniques are required to achieve success, such as schemes described in references given in section 2.2. An alternative approach is to employ an analytical treatment of integral evaluation described below. The quadratic interpolation function iφ in eqn (23) has the general form

i3i22

i1i bbb +ξ+ξ=φ and an explicit representation of the pile domain integral (23) is

( ) ( ) ( )∑∫=Γ

++=Γ3

1332211

*33 ,

m

mmmmp HbHbHbdSSpu

p

ττ (29)

where ( ) pdSpuH

p

Ω= ∫Ω

2*331 , ξ , ( ) pdSpuH

p

Ω= ∫Ω

ξ,*332 and ( ) pdSpuH

p

Ω= ∫Ω

,*333 are in Appendix IV.

Source point plane

Cell plane

'

p

s

p

h r

ρ

γρ

ϕ

Γcell

Figure 6: Two-Dimensional Integration Scheme in 3D Space. 3.1.3 Algebraic System After evaluating the integrals shown in eqns (24) and (25) for all cell and pile nodes, the explicit form of the rows of the algebraic representation, related to the thi pile and to thj cell, can be written as follows

=

M

M

M

MMM

LLL

MMM

LLL

MMM

i

i

i

i

mi

uuuu

wI

4

3

2

1

~

~

~

~

~

~~~~~

10000010000010000010

0000

M

M

M

MMM

LLL

MMM

LLL

MMM

ic

i

i

i

mj

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

imimimimmj

ttt

p

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

σ3

2

1

~

4,43,42,41,4~

,4

4,33,32,31,3~

,3

4,23,22,21,2~

,2

4,13,12,11,1~

,1

~4,

~3,

~2,

~1,

~

(30)

or in compact form

=

~

~

~~

~~

~

~

~~

~~

p

cell

pppc

cpcc

p

cell

pppc

cpcc

T

P

AA

AA

U

W

QQ

QQ (31)

where the vectors

~cellW and Up are the displacements of the cell and pile nodes;

~p

~cell T ,P are

their respective nodal force vectors. The displacement vector for cells and piles can be obtained by operating the matrices in (31)

=

~

~

~~

~~

~

~

p

cell

pppc

cpcc

p

cell

T

P

BB

BB

U

W, with

~~

1

~AQB −= . (32)

In eqn (30),

~~IQ = leads to redundant (or unnecessary) operations in 32. However, these

have been shown in this section to give uniformity of presentation with section 3.2.2, in which cases with

~~IQ ≠ occur.

The explicit form of eqn (32), related to the thi pile and to thj cell, can be given by

=

M

M

M

MMM

LLL

MMM

LLL

MMM

M

M

M

ic

i

i

i

jm

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

imimimimmj

iT

i

i

i

mj

ttt

p

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

uuuu

w

σ3

2

1

~

4,43,42,41,4~

,4

4,33,32,31,3~

,3

4,23,22,21,2~

,2

4,13,12,11,1~

,1

~4,

~3,

~2,

~1,

~

3

2

1

~

. (33)

If the radial strain in the pile is neglected, the elongation at internal point z of the thi pile is

( ) ( )dzdzzNAE

uzuuz z

Li

ii

ip

ip ∫∫−=−=∆

0

1 (34)

where ( )zNi is the compression force acting at point z of the thi pile

( ) [ ] ic

iiiici

i

i

iz

Li ztztztzA

ttt

dzzN σσφφφ 4332211

3

2

1

321 +++=+

= ∫ . (35)

If 1iu is taken as reference displacement, the elongations of the points at the top, on the

shaft and at the thi pile tip, i.e., i43i21 u and ,u , ii uu can be written by substituting their

respective depths iii L and /32L /3,L z ,0= into eqn (34), resulting in:

ic

i

i

i

i

i

ii

i

i

i

i

i

EL

ttt

REL

uuuu

σ

+

−−−

=

∆∆∆∆

13/23/1

0

8/54/18/12/127/4108/10216/5536/1216/11

0002

3

2

12

4

3

2

1

(36)

where ,iL iE and iR are, respectively, length, Young’s modulus and radius of the thi pile. Substituting eqn (36) and eqn (35) into eqn (33) results in the following representation

=

M

M

M

MMM

LLL

MMM

LLL

MMM

M

M

M

ic

i

i

i

mj

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

iiiiiiiiji

imimimimmj

i

i

i

i

mj

ttt

p

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

uuuu

w

σ3

2

1

~

4,43,42,41,4~

,4

4,33,32,31,3~

,3

4,23,22,21,2~

,2

4,13,12,11,1~

,1

~4,

~3,

~2,

~1,

~

4

3

2

1

~

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

. (37)

If the set of eqns (37) is written inversely, the rows of algebraic representation referring to the thi pile are given by

( )( )( )( )

( )( )( )( )

∑−

=

−−−

−−−

−−−

−−−

++++++++++++

+

++++++++++++

=

1

1

4

3

2

1

44,4144,4244,4344,4

34,3134,3234,3334,3

24,2124,2224,2324,2

14,1114,1214,1314,1

4

3

2

1

4,43,42,41,4

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

3

2

1pN

jj

j

j

j

jijijiji

jijijiji

jijijiji

jijijiji

i

i

i

i

iiiiiiii

iiiiiiii

iiiiiiii

iiiiiiii

ic

i

i

i

uuuu

bbbbbbbbbbbb

bbbb

uuuu

bbbbbbbbbbbbbbbb

ttt

σ

∑=

−−

−−

−−

−−−

+elN

k k

k

k

kikiki

kikiki

kikiki

kikiki

ppp

13

2

1

3,413,423,4

3,313,323,3

3,213,223,2

23,123,123,1

ββββββββββββ

. (38)

The normal stress on the top of the the thi pile i

TP can be calculated by dividing the total

transferred load iTN , eqn (35) with 0z = , by the cross-sectional area of pile i

pA :

ic

iiiip

iTi

T CCCCANP στττ 4332211 +++== (39)

where ip

i

RLC

21 = ; 0C2 = ; ip

i

RLC

23

3 = and 1C4 = .

If each row i in eqn (38) is multiplied by the constants 4321 C and C ,C ,C and then the four rows are combined linearly, a new algebraic system can be written

( )( )( )( )

( )( )( )( )

∑

∑

=

−−

−−

−−

−−−

−

=

−−−

−−−

−−−

−−−

+

++++++++++++

+

++++++++++++

=

el

p

N

k k

k

k

kikiki

kikiki

kikiki

kikikiT

N

j

T

jijijiji

jijijiji

jijijiji

jijijiji

j

T

iiiiiiii

iiiiiiii

iiiiiiii

iiiiiiii

iiT

ppp

bbbbbbbbb

bbb

CCCC

CCCC

bbbbbbbbbbbbbbbb

u

CCCC

bbbbbbbbbbbbbbbb

uP

13

2

1

3,413,423,4

3,313,323,3

3,213,223,2

23,123,123,1

4

3

2

1

1

1

4

3

2

1

44,4144,4244,4344,4

34,3134,3234,3334,3

24,2124,2224,2324,2

14,1114,1214,1314,1

1

4

3

2

1

4,43,42,41,4

4,33,32,31,3

4,23,22,21,2

4,13,12,11,1

1

(40)

When the rows for the remaining piles are treated by an analogous procedure, the final system of equations can be written in compact form as follows

=

~T

~cell

~22

~21

~12

~11

~T

~cell

W

W

bb

bb

P

P (41)

where

~T

~cell W and W are the vectors referring to the nodal values of the cell and the pile tops.

3.2 Finite Layer The integral equation for the pile-soil problem in which the elastic medium is assumed to be a finite stratum will be discussed first in terms of the Steinbrenner model [33]. An alternative integral formulation will then be obtained from Mindlin’s model. 3.2.1 Steinbrenner’s Model The integral equation for displacements in the pile-soil system, based on Steinbrenner’s scheme is used as fundamental problem, so that the vertical displacement u at a generic point p within the finite single layer medium is given by

( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫∫= Γ Γ

∗∗

Γ

Γ+Γ+Γ=np

ibic

Spi

Ss

S

pi bis

dSSpudSSpudspspuu1

3333*33 ,, , στ (42)

where ( )Sτ and ( )Scσ are defined as in eqn (21) and ( )s,pu s*

33 is Steinbrenner’s displacement kernel for the load point p and the field point s in the vertical direction ( 3x ). In 1934, Steinbrenner proposed an approximate solution for a concentrated vertical force acting on the surface of an elastic finite stratum, using a superposition of Boussinesq´s solutions. Other researches have modified the original work of Steinbrenner to model elastic problems under concentrated loads acting within a finite layered solid, using a superposition of the Mindlin solution. The modified Steinbrenner approach states that the displacements

( )s,pu s*33 can be obtained by taking the difference between Mindlin’s kernel for a load point at

p and another for a collocation point at A, located vertically below point p on the rigid plane assumed to constrain vertical displacements at depth ( hx3 = ), as in figure 7

( ) ( ) ( )s,Aus,pus,pu *33

*33

S*33 −= (43)

where ( )s,pu*

33 is Mindlin’s fundamental solution given in eqn (22).

*Γ

^

*

s

Γ

Γ

pp

r

Rigid base

S

f

p

h

pT

pT

Soil

ττ

bσ

Γx3

Pile

x1

A

σb

Figure 7: Steinbrenner Model.

To perform the transformation of integral equations into an algebraic representation for the pile-soil problem using Steinbrenner’s scheme, a few simple adaptations must be made in the infinite-layer procedures described in the previous section. The original influence matrix related to the infinite layer must be corrected by the algebraic system which is obtained when the projections of the original load points are located on the vertically rigid plane. In other words, all contributions of integral terms in the infinite layer model related to Mindlin’s kernel

*33u must be corrected using the alternative kernel given in eqn (43). Thus, the pile-soil

algebraic representation for this Steinbrenner model can be written similarly to eqn (23)

−

−

−

−

=

~T

~cell

~22

~22

~21

~21

~12

~12

~11

~11

~T

~cell

W

W

bbbb

bbbb

P

P. (44)

3.2.2 Direct Mindlin Scheme In the previous section, the interaction between the finite layer medium and pile groups was discussed using Steinbrenner’s model to perform the pile-soil analysis. In the current section, this problem is going to be investigated using Mindlin’s fundamental solutions to represent the finite stratum elastic responses.

In figure 7 is shown a geometrical and loading configuration of a pile group embedded in a single finite-layer elastic continuum and the displacement integral equation for the pile-soil system can be written

( ) ( ) ( ) ( ) ( ) +Γ=Γ+⋅ ∫∫ΓΓ

dspspudsuspppu ,, *33

*33α ( ) ( ) ( ) ( )∑ ∫ ∫

= Γ Γ

∗∗

Γ+ΓNp

ibicpip

pi bi

dsspudsspu1

3333 ,, στ (45)

where the medium boundary Γ can be divided into three surfaces, in which the first sΓ is the

traction-free half space surface, where ( ) 0,*33 =spp ; the second surface *Γ is the boundary of

Mindlin’s problem, off the surface on which ( ) 0,*33 =spp , and the last surface Γ is the

vertically rigid plane, see figure 7. The traction fundamental solution ( )s,pp*33 - on the

horizontal plane Γ with director cosines ( )1 ,0 ,0 - is obtained by substituting Mindlin’s solutions for stress into the Cauchy formula

( ) ==++= *3333

*3332

*3321

*331

*33 , σσσσ nnnSpp ( )

+−−

51

231

1

181

Rb

Rb

sνπ

+++72

552

432

3

Rb

Rb

Rb

(46)

where coeficient ib in eqn (46) is given by ( )( )czb s −−= ν211 , ( )32 3 czb −= , 13 bb −= ,

( ) ( ) ( )( )czczcczzb s −+−+−= 53413 24 ν , ( )czczb += 305 . In addition, the free term α in

eqn (45) is assigned the following values

Ω∉Γ∈

Ω∈Γ∈=

p if ,0 ˆ p if , 0.5 por p if , 1 s

α

The discretization of integral eqn (45) is performed by using the same interpolations for the geometry and for the interaction forces of the pile discussed in the previous section. To obtain the algebraic representation from integral eqn (45), the evaluation of the integrals of ( )spu ,*

33

and ( )spp ,*33 is required. The integration of the first kernel can be carried out using eqn (27),

and for the second kernel given in (45), an analogous treatment of integrals of ( )spu ,*33 can be

employed, leading to the following expression

( ) ( ) ( ) cellcell dtdSuSppcellcell

Γ=Ω ∫∫ΓΩ

ρϕ

ρcos,*

33 (47)

where the function ( )ρt is given by ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ][ ] +++++++−= 2,21,1463514433211 ρρβββαααρ AAgbgbgbgbgbgbt

( ) ( ) ( ) ( ) ( )[ ] ( )]18/[35144332113 shahahahbhbA νπββααα −++++ (48)

After transforming all the integrals of eqn (45), the rows of the algebraic representation related to the thi pile can be written similarly to eqn (31)

=

M

M

M

M

MMM

LLLL

L

LLL

MMM

LLLL

MMMM

LLLL

MMMM

4

3

2

1

~~,4

~~,3

~~,2

~~,1

~~~~~~

~~~~~~

10000

01000

00100

00010

0000

00000

i

i

i

i

mcell

jbott

ji

ji

ji

ji

mj

kj

uuuu

W

U

a

a

a

q

Iq

q

M

M

M

M

MMM

LLLL

L

LLL

MMM

LLLL

MMMM

LLLL

MMMM

ic

i

i

i

mcell

jbott

iiiiiiiimiji

iiiiiiiimiji

iiiiiiiimiji

iiiiiiiimiji

imimimimmmmj

ikikikikkmkj

ttt

P

F

aaaaaa

aaaaaa

aaaaaa

aaaaaa

aaaaaa

aaaaaa

σ3

2

1

4,43,42,41,4~,4

~,4

4,33,32,31,3~,3

~,3

4,23,22,21,2~,2

~,2

4,13,12,11,1~,1

~,1

~1,

~1,

~1,

~1,

~~

~1,

~1,

~1,

~1,

~~

(49)

Applying a similar transformation to eqn (32) in eqn (49), the vector of displacements can be obtained and then the stiffness of the piles can be inserted using analogous steps described from eqn (34) to eqn (36), resulting in

=

M

M

M

M

MMM

LLLL

L

LLL

MMM

LLLL

MMMM

LLLL

MMMM

M

M

M

M

ic

i

i

i

mcell

jbott

iiiiiiiimiji

iiiiiiiimiji

iiiiiiiimiji

iiiiiiiimiji

imimimimmmmj

ikikikikkmkj

i

i

i

i

mcell

jbott

ttt

P

F

bbbbbb

bbbbbb

bbbbbb

bbbbb

bbbbbb

bbbbbb

uuuu

W

U

σ3

2

1

4,43,42,41,4~,4

~,4

4,33,32,31,3~,3

~,3

4,23,22,21,2~,2

~,2

4,13,12,11,1~,1

~,1

~1,

~1,

~1,

~1,

~~

~1,

~1,

~1,

~1,

~~

1

1

1

1

~

~

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆ. (50)

If algebraic representation (50) is written inversely and a linear combination is performed between the four rows of the thi pile from this resulting system and the constants

4321 C ,C , andCC , an equation equivalent to eqn (40) is obtained

∑∑∑∑∑= == ==

+++=inf

1

3

1

sup

1

3

1211

1111

N

k m

kmmk

N

k m

kmmk

N

j

jpjp

iT Fhpgudud

p

σ . (51)

When this process is repeated for all piles and cells, the final algebraic system can be represented in compact form as follows

=

~

~

~

~33

~32

~31

~23

~22

~21

~13

~12

~11

~

~

~

cell

T

bott

cell

T

bott

W

W

U

ddd

ddd

ddd

P

P

F

(52)

where Tcell W and W are the vectors referring to the nodal values of the cell and the pile tops. In eqn (52), the nodal points at the bottom of the finite layer have their degrees of freedom with respect to displacements restricted in accordance with the rigid stratum supporting it,

~~0=bottu in (52).

The resultant system is uncoupled, so that all remaining unknowns belong only to the raft plane. Hence, eqn (52) can be simplified to

=

~

~

~33

~32

~23

~22

~

~

cell

T

cell

T

W

W

dd

dd

P

P. (53)

4 Raft-pile-soil system coupling To assemble the final system of equations for the raft-pile-soil interaction problem it is necessary to couple the raft algebraic representation and pile-soil contributions from the respective stratum model. In the case where the soil is modelled by the infinite-layer Mindlin model, the algebraic system eqn (41) should be inserted into eqn (6), giving the following relation

+

=

Γ

Γ

~

2~

1~

0

~

~

2~

1~

0

~T

~cell

~

~33

~32

~31

~23

~22

~21

~13

~12

~11

G

G

G

V

C

C

C

W

W

W

HHH

HHH

HHH

. (54)

If the finite stratum Steinbrenner scheme, eqn (44), or direct Mindlin model, eqn(54), is inserted in eqn (6), the final system will give a relation analogous to eqn (54). 5 Evaluation of Domain Quantities After the solution of the final system of equations, the unknowns vector (the boundary displacement and slopes of the raft, the displacements at the node cell , the pile-top deflections and the forces acting on the rigid plane) is determined. If the values of this vector are substituted into eqn (41), eqn (44) or eqn (53), the raft-soil and pile-raft interaction forces can be evaluated and then if these forces and the node cell displacements are inserted into eqn (37) or eqn (50), the interaction shear stresses on the shaft and normal stress at the tip of the pile can also be determined. In the case of direct Mindlin model for layered medium, the forces acting on the rigid base can be calculated using the expression taken from eqn (52), that is

~~

13~12

~cellTbott wdwdF += . (55)

Besides the quantities at the boundary of the raft and interaction forces of the problem, there are further fields which can be required, for instance the domain efforts of the raft. An expression for these efforts can be obtained using the relation taken from classical plate theory. For the bending moments, the constitutive relations can be written

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]pqptpwpqptpwDpmpqptpm jiijiikkijjitq ,, 1 νν −+−== ; 2 1,k j,, =i . (56) On the other hand, the integral representations for the curvatures of the plate can be written by performing double differentiation on the displacement integral eqn (1)

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) =+Γ−+ ∫Γ

swspRptpqdspmsspqswptpqpw cijcjiijnpijnjitq , ,, *,

*,

*,, θ

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) +Ω+Γ− ∫∫ ΩΓ

dSpwSgptpqdspsmspwsVptpq ijjiijpnijnji , ,, *,

*,

*, θ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫=

Ω+Ω+

Ncel

h

hcelijjiijccji

h

dSpSpwptpqSpwSRptpq1

*,

*, ,,

( ) ( ) ( ) ( )∑ ∫=

ΩΩ

Np

h

hTijji

h

dSpSpwptpq1

*, , ; Nc,...,2,1c = ; 2,1j,i = (57)

where jj q nda t are direction cosines referring to the directions of the differentiation at the load point; the kernels in eqn (57) are shown in Appendix II. The discretized integral representation and algebraic equations for curvature can be obtained by applying similar techniques to the displacements and slopes as described throughout the earlier sections. To transform the cell domain integral to an equivalent one defined along the cell boundary, the following equation can be used

( ) ( ) ( ) ( ) ( ) ( ) ( ) celljihcellijji dkptpqdSpSpwptpq

hcell

cell

Γ=Ω∫ ∫ΩΓ

ρϕρχ

cos, **, (58)

where

( ) ( )( )[ ] ( )[ ]jiijjiij DAAA

Dk ,,

32

,,2,21,1

3* 21ln2

1631ln3

36ρρδρ

πρρρδρρρ

πρρχ +−−+−+−= . (59)

6 Numerical analyses In this section are presented some configurations for raft-soil, pile-soil and piled raft interaction problems and for direct loading of the medium.

6.1 Loading on a rectangular area of a rough rigid base In this example, the surface of stratum is subjected to a external loading distributed over a square region (see figure 8, case a). The resulting displacements for the Steinbrenner, the direct finite Mindlin and the Burmister models for the elastic medium are indicated in table 1 for some depths of the rigid base.

Table 1: Displacement of the point A (W*10-3m).

Lh=η Burmister [38] This work

(Direct F. Mindlin) This work

(Steinbrenner) 1 4.3956 4.5023 4.0227 2 7.2214 7.2458 7.4117 4 9.0005 9.2511 9.2726 ∝ 11.2308 11.1691 11.3050

6.2 Raft supported on finite stratum Consider a square plate submitted to a uniform transverse load 201.0 mkNg = and resting on a linear elastic half space, (see figure 8, case b). The length of the side of the plate is mL 10= , thickness mt 26.0= and its elastic properties are Young’s modulus MPaER 21000= and Poisson’s ratio 15.0=Rν . The half space has Young’s modulus MPaEs 1.9= , Poisson’s

ratio 3.0=sν and thickness mh 10= . The displacements and bending moments found at the points A, B and C are shown in table 2.

LηA

CB

g

h=

Rigid base

10 m

10 m

Limits of the half space surface

L=10 m

r=∝

t=0.26 mg

10 m

(a) (b)

Plate

Figure 8: Finite stratum under uniform loading .

Table 2: Some points in the thick plate.

Point of plate [37] This work (Direct. F. Mindlin)

This work (Steinbrenner)

A [Deflections w * 10-3 (m)] 7.30 6.69 3.74 B [ “ ” ] 2.80 2.54 2.43 C [ “ ” ] 4.50 4.63 1.83 A [Moments M11 (kNm/m)] 6.20 7.20 6.00

6.3 Long strip on two piles In this example, a raft on two piles is analysed for two cases (see figure 10). In the first there is no contact between the rigid raft and the supporting medium. The rigid base is located at infinite depth ( ∞=Lh / ). The results are indicated in figure 9 and expressed in terms of two dimensionless parameters. The first is the ratio between the Young’s moduli of the pile and

soil, sp EE , and the second parameter is given by expression dEw

P

st, where P is the total

load acting on the raft, tw is the vertical displacement of the top of the pile and d is the pile diameter. The spacing between piles is d5.2S = , the raft width is d25.1B = , the Poisson’s

ratio is 0.5 and sG is the shear modulus given by )1(2 ss

sEG ν+= .

0 20 40 60 80 1000

20

40

60

80

100

120

140

h/L=∝

Ep/Es=∝

Ep/Es=18000

This work

[14]

This work

[14]

P/(G

s wt d

)

L/d Figure 9: Stiffness of two capped-pile group.

S/2S

g

L=20d

S/2

d

Rigid base

Figure 10: Geometrical configuration of raft-pile-soil problem.

In the second case, the piled raft is resting on a finite stratum, below which the rigid plane is situated at depth ( 2Lh = ). The results of this analysis are indicated in table 3 for several values of Sp EE .

Table 3: Vertical displacements of the piled raft for several Sp EE ratios.

dEwP

st

S

P

EE

[39] This work (Direct. F. Mindlin)

This work (Steinbrenner)

333 14.0 15.1 16.2 3333 16.7 17.8 19.3 ∞ 17.3 18.2 19.8

6.4 Rectangular raft A rectangular plate (with dimensions m 0.25 x m 15.0 x 0.30 m , Young’s modulus

27R kN/m 10 E 0.2= , Poisson’s ratio 2.0=Rν and under uniform loading 2kN/m 0.30=g ) is

resting on infinitely-thick medium and analysed with or without stiffening of pile-groups (see figure 11). In the piled raft cases, the groups of pile have four sets of configurations. The first consists of corner piles 1P , the second is composed of 1P added to 2P piles, the third is represented by the piles located along the boundary of the raft and the last set is all nine piles. The piles are assumed to be incompressible and their length and diameter are m 15=pL and

m 3.0=pD , respectively. The mechanical properties of the soil are Young’s modulus 25 kN/m 10 0.2=sE , Poisson’s ratio 5.0=sν and the raft-soil interface is shown in figure 12.

The results for displacements (at points along the line m 75.31 =x ) are in figure13 and the bending moments along the line m 5.71 =x are in figure14.

7.5 m

15.00 m3

1

P

P

1P

7.5 m

0.30 m

30.00 kN/m

0.25 m

2

P

P

P

7.5 m

P2

P2

4P

7.5m

3.75 m

3.75 m

3.75 m

3.75 m

3

1

1

Figure 11: Raft-pile-soil layout.

1P 2P 1P

3P 4P3P

1P 2P 1P

x1

2

Figure 12: Raft-soil discretization.

0 2 4 6 8 10 12 14 16

2x10-1

2x10-1

2x10-1

2x10-1

2x10-1

3x10-1

0 pile 4 piles 6 piles 8 piles 9 piles

w (m

m)

x1 (m) Figure 13: Displacements along line ( mx 75.32 = ).

0 2 4 6 8 10 12 14 16

-25

-20

-15

-10

-5

0

5

0 pile 4 piles 6 piles 8 piles 9 pilesM

x 1(kN

m/m

)

x1(m) Figure 14: Bending moments

1xm along line ( mx 50.72 = ).

Table 4: Forces on the piles and on the raft.

Number of Piles

P1 P2 P3 P4 Subgrade reaction

% load

4 130.7 12977 4.03 6 130.3 128.9 12721 6.12 8 125.7 128.5 124.5 12491 8.07 9 125.4 122.5 123.9 129.4 12376 9.08

7 Conclusion A boundary element formulation has been described for elastic analysis of the raft and piled-raft foundation submitted to vertical static loading and supported on an infinite layered medium or on a finite stratum. All the interaction forces of the raft-pile-soil system are simultaneously taken into account. The numerical analyses were performed for the direct loading of the medium and raft-soil, pile-soil, piled-raft-interaction problem. In the case of the direct loading of the medium, the displacements are more distorted by the Steinbrenner model than the direct layered Mindlin approach when the responses of Burmister’s scheme are taken as reference values. Although the Steinbrenner model is an attractive technique due to the simplicity of its mathematical representation, it leads to poorer values when the rigid base tends to the free surface of the medium, because of the superposition technique required by Steinbrenner model that not taken into account the influence of the forces mobilized by constraints imposed by the rigid base surface. This fact also applies to the raft supported on a finite stratum, in which the results for displacements in Steinbrenner’s model are lower than those obtained by the direct Mindlin model and by Fraser and Wardle [37]. In the pile-soil interaction problems, the influence of the relative rigidity between the pile and the soil are investigated. The problems consist in analysing a rigid raft on two piles for two cases. In the first, the infinitely-thick medium is not in contact with the raft and the analysis is performed for different ratios (length/diameter) of the piles. In figure 9 is shown a good agreement of response of the present formulation compared to the Butterfied and Banerjee approach [14]. In the second case, all parts are in contact and the medium is considered as a finite-layered solid. The results of the present formulation invoking the direct Mindlin model is closer to that presented by Banerjee [34], due to the similarity of the two approaches for the finite layered medium. The last problem analysed consists of a rectangular raft stiffened or not by pile groups. In figure 14, it may be noted that the concentration of bending moments, along line mx 5.72 = , are restricted to the vicinity of the piles due to the high values of raft-pile reactions distributed within the cross-section of the piles. In addition, the results for the displacements along line

mx 75.32 = (see figure 13), show that these values are effectively not modified when the number of piles in the groups is increased. Similar behaviour can be observed from table 4 for the load transferred to the piles and the raft. For this specific example, it can be seen that the raft rigidity has a crucial influence on the behaviour of the piled-raft system. Acknowledgments This work has been financial supported by FAPESP (Brazil).

References [1] Zaman, M.M., Issa, A. & Kukreti, A.R. Analysis of circular plate-elastic half-space

interaction using an energy approach. Applied Math.l Modelling, 12, pp. 285-292, 1998. [2] Kukreti, A.R., Zaman, M.M. & Issa, A. Analysis of fluid storage tanks including

foundation-superstructure interaction. Applied Mathematical Modelling, 17, pp.618-631, 1993.

[3] Hemsley J.A. Elastic solutions for large matrix problems in foundation interaction analysis. In: Proceedings of the Institute Civil Engineers, Part2,Vol 89, 471-494, 1993.

[4] Hemsley J.A. Application of large matrix interaction analysis to raft foundations. Proceedings of the Institute Civil Engineers, Part2, Vol 89, pp. 495-526, 1993.

[5] Paiva, J.B. Formulação do método dos elementos de contorno para análise da interação solo-estrutura, Associate Professor Qualifying Text (in portuguese), São Carlos Engineering School, São Paulo University, Brazil, 1993.

[6] Messafer T., Coates L.E. An application of FEM/BEM coupling to foundation analysis. Advances in Boundary Methods. eds. C.A. Brebbia & Connor, Computational Mechanics Publications: Southampton and Boston, pp. 211-221, 1993.

[7] Poulos, H.G. & Davis, E.H. The settlement Behaviour of simple axially-loaded incompresible piles and piers. Géotechnique, 18, pp. 351-371, 1968.

[8] Poulos, H.G. & Mates, N.S. Settlement and load distributions analysis of pile groups. Australian Geomechanics Journal, G1(1), pp. 18-28, 1971.

[9] Shen, W.Y., Chow,Y.K. & Yong, K.Y. A variational approach for vertical deformation analysis pile groups. International Journal for Numerical and Analytical Methods in Geomechanics; 21 (11), pp.741-752, 1997.

[10]Mindlin, R.D. A force at the interior point of a semi-infinite solid. Physics, 7, pp. 195-202, 1936.

[11]Shen WY, Chow Y.K & Yong K.Y. A variational approach for vertically loaded pile groups in an elastic half-space. Géotechnique, 49(2), pp.199-213, 1999.

[12]Xu, K.J. & Poulos, H.G. General elastic analysis of piles and pile groups. International Journal for Numerical and Analytical Methods in Geomechanics,24, pp. 1109-1138, 2000.

[13]Butterfield, R. & Banerjee, P.K. The elastic analysis of compressible piles and piles groups. Géotechnique; 21(1), pp. 43-60, 1971.

[14]Butterfield, R. & Banerjee, P.K. The problem of pile group-pile cap interaction. Géotechnique; 21(2), pp.135-142, 1971.

[15]Kuwabara, E. An elastic analysis for piled raft foundations in a homogeneous soil. Soils and Foundations; 29(1), pp. 82-92, 1989.

[16]Shen, W.Y., Chow, Y.K. & Yong, K.Y. A variational approach for the analysis of pile group- pile cap interaction. Géotechnique; 50(4), pp. 349-357, 2000.

[17]Brown, P.T. & Weisner, T.J. The behaviour of uniformly loaded piled strip. Footings. Soil and foundations, 5(4), pp. 13-21, 1975.

[18]Fatemi-Ardakani B. A contribution to the analysis of pile-supported raft foundations. Ph.D Thesis, University of Southampton, Southampton, U.K., 1987.

[19]Hain, S.J. & Lee, I.K. The analysis of flexible pile-raft systems. Géotechnique, 28(1), pp. 65-83, 1978.

[20]Poulos, H.G. An approximate numerical analysis of pile-raft interactions. International Journal for Numerical and Analytical Methods in Geomechanics; 18, pp.73-94, 1994.

[21]Mendonça, A.V. & Paiva, J.B. A boundary element for the static analysis of raft foundations on piles. Engineering Analysis with Boundary Elements; 20, pp. 237-247, 2000.

[22]Randoph, M.F. & Wroth, C.P. Analysis of vertical deformation of pile groups. Géotechnique, 29(4), pp. 423-439, 1979.

[23]Mylonakis, G. & Gazetas, G. Settlement and additional internal forces of grouped piles in layered soil. Géotechnique, 48(1), pp.55-72, 1998.

[24]Burmister, D.M. Theory of stresses and displacements and applications to the design of airport runways. 23rd proc. Highway Research Board, pp.127-248, 1943.

[25]Burmister, D.M. The general theory of stresses and displacements in layered soil system I. Journal of Applied Physics, 16(2), pp.89-96, 1945.

[26]Burmister, D.M. The general theory of stresses and displacements in layered soil system II. Journal of Applied Physics, 16(3), pp.126-127, 1945.

[27]Burmister, D.M. The general theory of stresses and displacements in layered soil system III. Journal of Applied Physics, 16(5), pp.296-302, 1945.

[28]Burmister, D.M. Stresses and displacements characteristics of two layer rigid base soil system. 35th Proc. Highway Research Board, eds. F. Burggraf, E. M. Ward, NAS-NRC Publications: Washington, pp. 773-814, 1956.

[29]Chan, K.S., Karasushi, P. & Lee, S. L. Force at a point in the interior of a layered elastic half space. International Journal of Solids and Structures, 10, pp.1179-1199, 1974.

[30]Pan, E. Static green’s functions in multilayered half spaces. Applied Mathematical Modelling, 21, pp. 509-521, 1997.

[31]Small, J.C. & Booker, J.R. Finite layer analysis of layered elastic materials using a flexibility approach, part i-strip loadings. International Journal of Numerical Methods in Engineering, 20, pp.1025-1037, 1984.

[32]Southcott, P.H. & Small, J.C. Finite layer analysis of vertically loaded piles and pile groups. Computer and Geotechnics, 18(1), pp. 47-63, 1996.

[33]Steinbrenner, W. Tafeln zur Setzungberechnung. Die Strasse, 1, pp.221, 1934. [34]Banerjee, P.K. Effects of the pile cap on the load displacement behaviour of pile groups

when subjected to eccentric loading. Proc. 2nd Australia-New Zealand Conf. Geomech., pp. 179-184, 1975.

[35]Sladek, V. & Sladek, J. Singular Integrals in Boundary Element Method, eds. C.A. Brebbia & Aliabadi, Computational Mechanics Publications: Southampton and Boston, 1998.

[36]Hayama, K. & Brebbia, C.A. A new coordinate transformation method for singular and quasi-singular integrals over general curved boundary elements. In: BEM IX, Vol.1, Springer-Verlag: Berlin, 1987.

[37]Telles, J.C.F. A self-adaptative coordinate transformation for efficient numerical evaluations of general boundary element integrals. International Journal of Numerical Methods in Engineering, 24, pp. 959-973, 1987.

[38]Fraser, R.A. & Wardle, L.J. Numerical analysis of rectangular rafts on layered foundations. Géotechnique, 46(4), pp. 613-630, 1996.

[39]Poulos, H.G. & Davis, E.H. Elastic solutions in soil and rock mass, John Waley & Sons: New york, pp. 113-118, 1974.

Appendix I The free terms in integral eqns (1) and (2) are given by

πβ 2 =k , ( )[ ]βγγπν

πβ

+−+= 22421 sinsinK , ( )[ ]βγγ

πν

+−= 2cos2cos42K

where β is the internal angle of the raft; γ is the angle between ( )sn, and ( )um, coordinates; ν is the raft Poisson ratio. The kernels referred to eqn (1) and its derivative (2) are given by

( )

−=21ln

81, 2* rr

Dspw

π, ( ) rr

Dnr

sp iip ln

4

, ,*π

θ = ; 2,1=i .

( ) ( ) ( )( )[ ]νννπ

+−++−= 2,

* 1ln141, iin rnrspm ; 2,1=i .

( ) ( )( )jjiint rnrtspm ,,*

41,πν−

−= ; 2,1, =ji .

( ) ( )( )[ ] ( )( )jjiiiij

n rnrtR

rtr

rnspV ,,

2,

j ,*21312

4,

πννν

π−

++−−−= ; 2,1, =ji .

( ) rrDmr

spw iim ln

4

, ,*, π

−= ; 2,1=i .

( ) ( )( ) ( )[ ]rnmrnrmD

sp iijjiimp ln4

1, , , *

, +=π

θ ; 2,1, =ji .

( ) ( )( ) ( )( )( ) ( )( )[ ] k ,j ,i ,i ,*

, 12141, rnrmnmrnrm

rspm kjjjiimn −−++= νν

π; 2,1 ,, =kji .

( ) ( )( ) ( )( ) ( )( )( ) k ,j ,i ,k ,j ,*

, 24

1, rtrnrmrntmrtnmr

spm kjikiijiimnt −+−

=πν ; 2,1 , , =kji .

( ) ( )( ) ( )( )( ) ( )( ) ( )( )[ ] k ,j ,k ,j ,i ,q ,2*, 2412

4

1, rtnmrntmrtrnrmrtr

spV kiijiikjiqmn −−−= νπ

( ) ( ) ( )( )[ ] ( )( ) ( )( )[ ]j ,q ,i ,k ,j ,i , 2123 rtrmrmrtRr

rnrmmn jqikjijj −−

+−−+πνν ; 2,1qk, , , =ji .

where D is the flexural rigidity of the raft; r is the modulus of radius vector r

rand i ,r is its

director cosine in direction i ; ii t ,n are the unit vectors normal and tangential to the boundary of the plate in s, ijδ is the Kronecker delta and R

1 is the curvature of the boundary of the raft. Appendix II: The kernels referred to in the raft curvature integral equation eqn (57) are given by

( ) ( ) ( ) ( )[ ( ) +++−−−−+−= kkijjijijiijijn nrnrnrnnrrr

spm ,,,,,2*, 21221

4

1, νδνπ

( )( ) ] 4 2,,, ααδ nrrr jiij − ; 2,1,, =kji .

( ) ( )[ ( )++++−

= ijjikkijjikkijns trtrnrnrnrtrr

spm ,,,,,,2*

, 241,πν ( )]−− jiijkk rrtrnr ,,,, 4δαα ( )ijji tntn + ;

2,1,,, =αkji .

( ) rrrD

spw ijjii ln4

1, ,,*

j , δπ

+= ; 2,1, =ji ,

( ) [ ] jiijkkijjijp nrnrnrrrrD

spi ,,,,,

*, 2

41, −−−= δπ

θ ; 2,1,, =kji ,

( ) ( ) ( ) ( )[ ] ( ) +−+++−−= ijkkijijjijikkijn ttnrnrnrnrnrrrtrr

spV ,,,,,,,2

,3*, 14412

41, νδνπ αααα

( ) ( ) ( )[ ]++−+− ijjiijjikk trtrtrtntntr ,,,, 8212 ααν ( ) ( ) ( )[ ]+++−− ijjikkjiij nrnrnrrr ,,,,, 2432 δν

)(1,,,,2 ααπ

ν rtrtrrtRr iijkk −− ; 2,1,,,, =αkji ,

where the symbols have the same meaning as in Appendix I.

Appendix III: The supporting functions ( ) ( )yygi ih , for ( )ρf , eqn (28), and ( )ρt , eqn (48), are written as

( ) ( )

++++

−= yyRRyR

Ryg ][ln 2222

1 , ( ) ( )

++−

++= yyRRyyRRRyg ][ln21

222222

2 ,

( ) ( ) 23

2223 (

3−− +⋅= yRyRyg , ( ) ( ) ][]25[

1525

224223

4−− +⋅+= yRyRyRyg ,

( ) ( )y

yRyh 121

221 ++−=

− , ( ) ( ) yyRyh −+= 222 ,

( ) ( ) 323

223 3

131 −−

⋅++⋅−= yyRyh , ( ) ( ) 525

224 5

151 −−

⋅++⋅−= yyRyh .

Appendix IV The expressions resulting from the analytical evaluation of integrals in eqn (29) are

( )( ) ( )( ) +−+⋅−−⋅=Γ=

−

−

Γ∫ ]][ln[

22

2322

3

2 13142

231

321

2*331

orzo

p

krz

p RRRRRRk

zRkduH ξ

( ) ( ) ( )

( ) +⋅−⋅+−⋅+⋅⋅+−

+⋅⋅−⋅++⋅++−+−⋅

−−

−

])(23[2

)(])3([

])(23[2

]ln)2(3[2

224

23

20

23

12

20

20

203

4

214

2202

142

20

22

2

RRzRrzcRkRrzrrczRz

RRzrRzcRRRrzRzck

( ) ( ) ( )[ ][ ]+⋅+−++−⋅++++ −122

40

220

244

20

202

13 62928825263

2RrzzczrzzRrzrRzc

Rk

( ) ( )[ ] ( )42213

240

20

354

40

20 ln36][12725942 RRzRrzczrzczzRrzr +−⋅+−+++−⋅ − .

( )( )[ ][ ] ( ) ( )[ ]4222314231131

*332 lnlnln RRzRkRRzRRRRzRkkduH

p

p +−+−++−+=Ω= ∫Ω

ξ

[ ] +⋅+⋅−⋅+++⋅+⋅−++⋅+ −−− 12

203

24

20

224

2023

213

201 ][2])(1[])(1[ RrRzRzrRzczRrRkRzczRrR

( ) ( )[ ]( ) ( )[ ] ( )42

12

40

40

220

24

132

20

20

220

44

ln4][141382

][)513(42

RRzRrzcrzrzRz

RrzcrzczrzRz

+−⋅+−++⋅

+⋅−++−⋅−

−

( ) ( ) ( ) ( ) +−++++−+=Ω= ∫

Ω2

43

1

342323131

*333 lnln1

RR

kRR

RRkkRRkkduH

p

p

where

)1(161

1ssG

kνπ −

= , ( ) k ss )43(18 22 νν −−−= , sk ν433 −= ,

czR −=3 , czR +=4 , )(3 sxz = , )(3 pxc = , 2,1;0 == irrr ii ; 3,2,1),()( =−= isxpxr iii .

( ) ( ) ])()4()()23[2][]1[2 122

20

2204

142

204

2042

12

20

20

−−− ⋅+⋅+⋅+−⋅−⋅++⋅ RrzrRRrzRrRzzRRrczrz