analysis of piled raft foundations using a variational approach

Analysis of Piled Raft Foundations 129

The International Journal of Geomechanics Volume 1, Number 2, 129–147 (2001)

© 2003 ASCE DOI: 10.1061/(ASCE)1532-3641(2001)1:2(129)ISSN 15323641

Analysis of Piled Raft FoundationsUsing a Variational Approach

Y. K. Chow, K. Y. Yong,* and W. Y. Shen

Center for Soft Ground Engineering, Department of Civil Engineering,National University of Singapore, Singapore

ABSTRACT: A variational approach for the analysis of piled raft foundations is presented. The raft and piles are bothanalyzed by the use of the principle of minimum potential energy. By representing the deformation of the piles and raftusing finite series, the method is very efficient for the analysis of a piled raft with a large number of piles. Comparisonswith other numerical methods and field measurements have shown reasonable agreement.

INTRODUCTION

Piled raft foundations are widely used in practice to support high-rise buildings. However,only a few numerical methods that can be used to analyze such foundations have been reporteddue to the complexity of the problem. Hain and Lee1 reported an analysis of piled rafts in whicha surface settlement influence factor method was used to approximately describe the interactionbetween the pile group-soil system and the plate finite elements used to model the flexible raft.Griffiths, Clancy and Randolph2 presented a numerical solution in which the hybrid methodpresented by Chow3 is introduced to describe the response of the pile group and soil, and platefinite elements were used for the raft. Poulos4 proposed an approximate numerical method forpile-raft interaction, where the raft was treated using the finite difference method and the pilesmodeled using a boundary element method. The limiting pressure below the raft and bearingcapacity of the piles can be considered in the analysis. Ta and Small5 reported an analysis of piledrafts in layered soils. In their method, a finite layer method and finite element method are usedfor the analysis of the pile group and raft, respectively. Ta and Small6 also reported an analysisof performance of piled raft foundations on layered soils. It is noted that the methods mentionedabove require the discretisation of the piles in a group, which will affect their efficiency whenperforming analysis of large pile groups commonly used in practice.

In this article, a variational approach for the analysis of phm›d raft foundations is described.The present method is an extension of the variational solutions for the bending analysis of raftsby Shen, Chow, and Yong7 and the analysis of pile group-pile cap interaction by Shen, Chow,and Yong.8 The raft is assumed to rest on an elastic half-space reinforced by a pile group. Thepile deformations and raft deflections are each represented by a finite series. The response of thepiled raft and pile group is determined by the use of the principle of minimum potential energy.The discretisation of the pile shafts and raft itself is no longer required, and this is very efficientto large pile groups. A surface stiffness that relates the load-settlement relationship at the pilehead-raft-soil interface is incorporated in the raft analysis, making available an efficient andcomplete solution of a piled raft. Results of the present method are compared with those obtainedusing other numerical methods. Case studies are also carried out to show the application of thetheoretical analysis to actual field cases.

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130 Chow et al.

METHOD OF ANALYSIS

For a piled raft, the forces imposed by the raft and by the piles and soil at the pile head-soil-raft interface, are schematically shown in Figure 1. There are two main procedures involved inthe present analysis. First, a stiffness matrix of the pile group-soil system at raft-pile group-soilinterface is determined. Second, this stiffness is incorporated in the raft analysis and a completesolution of the piled raft is then achieved.

A. Stiffness of pile group-soil system at raft-pile group-soil interfaceThe stiffness of a pile group-soil system at the raft-pile group-soil interface that gives the

load-settlement relationship at the pile head and ground surface only can be obtained based ona variational approach (Shen, Chow, and Yong8). The main procedures are highlighted herein forcompleteness. The potential energy of the pile group-soil system can, in general, be expressedas

π τ σp p z

S

T

z b

A

T

bU w ds w dA= + { } { } + { } { }∫∫ ∫∫12

12

+ −{ } { }∫∫12 p w dB w Ps s t t

B

Τ(1)

The first term on the right side of the above equation is the elastic strain energy of the pilesin the group, the second and third terms are the work done by shear stresses

τ τ τ τz z z znp

T{ } = { }1 2, , ..., along the pile shafts and normal stresses σ σ σ σb b b bnp

T{ } = { }1 2, , ...,

at the pile bases, respectively, where np is the number of piles in the group. The fourth term isthe work done by pressure PS acting on the ground, and the last term is the work done by forces

P P P Pt t t tnp

T{ } = { }1 2, , ..., acting at the pile heads. S, A, and B in Equation (1) are the surface area,

cross-section area of the piles and ground area loaded by the force PS, respectively,

w w w wz z z znp

T{ } = { }1 2, , ..., , w w w wb b b bnp

T{ } = { }1 2, , ..., and are the vectors of settlements of

the piles at the pile shafts, pile bases and pile heads, respectively, and is the settlement of theground surface.

With the pile head-surface soil in contact with the raft discretized into a number of elementsas shown in Figure 2 and Gauss numerical integration employed to conduct the integral with respectto the pile length in the second term of Equation (1), the potential energy can be written as

πp p g g bT

b sT

s tT

tU w P w P w P w P= + { } { } + { } { } + { } { } − { } { }12

12

12

Τ(2)

The vector {wg} in Equation (2) is composed of the settlements of the piles at the Gauss pointsalong the pile shafts, and {Pg} is a vector of the equivalent forces acting at the Gauss points dueto the shear stress {τz}. The vector {Pb} is composed of the forces at the pile bases due to normalstress {σb}. The vector {PS} is composed of the forces acting on the soil elements due to at theground surface, and is composed of the settlements of the soil elements at the ground surface. Thesecond and third terms on the right side of Equation (2) can be assembled to give the followingexpression of the potential energy:

(3)πp p p p sT

s tT

tU w P w P w P= + { } { } + { } { } − { } { }12

12

Τ



FIGURE 1. Forces on raft, pile head and soil.


132 Chow et al.

in which {PP} and {wP} are the forces and settlements of the piles at the Gauss points and at thepile bases.

The relationships between {PP} and {wP}, and {PS} and {wS} in Equation (3) can beexpressed as

P

P

k k

k k

w

wp

s

spp sps

ssp sss

p

s

=

(4)

where [kspp], [ksps], [kssp], and [ksss] are the soil stiffness matrices and can be obtained for asoil modeled as an uniform elastic half-space or with stiffness increasing linearly with depth

(Shen, Chow and Yong8). By making use of Equation (4) and noting that the matrix k k

k kspp sps

ssp sss

is symmetrical, the expression for the potential energy can be obtained as

πp p p spp p p

T

sps sU w k w w k w= + { } [ ]{ } + { } [ ]{ }12

Τ

+ { } [ ]{ } − { } { }12 w k w w Ps

Tsss s t

Tt (5)

The displacement of each pile in the group can be accurately represented by a finite seriesgiven by (Shen, Chow, and Yong9)

w zz

li iji

j

k j

( ) = −

=

−

∑β 11

11

(i = 1,2... np) (6)

where z is the depth coordinate, l is the pile length, and βij are undetermined coefficients. Thus,the strain energy Up, the displacements {wP} and {wt} in Equation (5) can all be related to theundetermined coefficient βij, and principle of minimum potential energy then requires equation

(5) to be ∂π∂β

p

ij

= 0. Eventually, the following matrix can be deduced

k k w Ppp ps s T[ ]{ } + [ ]{ } = { }β (7)

FIGURE 2. Discretization of pile head and soil in contact with raft.



in which [kpp], [kps] are the matrices reflecting pile-pile and pile-soil interaction, {β} is a vectormade up by the undetermined coefficients, and {PT} is the vector reflecting the loads acting onthe pile heads. On the other hand, based on Equations (4) and (6), the following relationship canalso be established

k k w Psp ss s s[ ]{ } + [ ]{ } = { }β (8)

Note that Equations (7) and (8) can be assembled and then inverted to give

βw

k k

k k

P

Ps

pp ps

sp ss

T

s

=

−1

(9)

The above matrix Equation can be transformed into a matrix equation relating the load-settlementrelationship at the pile head and the soil in contact with the raft only through proper addition of

the rows and columns in the matrix k k

k kpp ps

sp ss

−1

corresponding to the vectors {PT} and {β}. This

resulting matrix equation can be expressed as

w

w

F F

F F

P

Pt

s

tpp tps

tsp tss

t

s

=

(10)

By inverting the matrix in Equation (10), the stiffness matrix equation of the pile group-soilsystem at the pile head-soil-raft interface only is obtained as

P

P

F F

F F

w

wt

s

tpp tps

tsp tss

t

s

=

−1

(11)

This equation can be rewritten in the form

P k wt{ } = [ ]{ } (12)

Thus, the stiffness matrix [kt] for a pile group-soil system that considers the reinforcing effectof the pile group is achieved. This matrix can be easily incorporated in the raft analysis to givea complete solution of a piled raft.

The loads taken by the piles and ground soil in contact with the raft may exceed the bearingcapacity of the pile and ground soil in the elastic analysis as described above. This problem istaken into account in the analysis through the use of an “initial stress” technique. The responseof the piles at the pile heads and the soil reaction pressure are assumed to be elastic-perfectlyplastic. The excess loads on the piles above the specified pile bearing capacity and the excess soilreaction pressures of the soil elements above the specified soil bearing capacity are redistributedto other piles and soil elements. This process is repeated until the computed load on the piles andreaction pressures on the soil do not exceed the specified limiting values.

Raft analysisThe potential energy of a raft can, in general, be expressed as


134 Chow et al.

π σr r r

A

r r r

A

U w dA w p dA= + −∫∫ ∫∫12 (13)

in which Ur is the strain energy stored in the raft, wr is the deflection of the raft, and A is the areaof the raft. The second term is the work done by soil reaction pressure σr on the bottom of theraft, and the third term is the work done by external load pr on the upper surface of the raft. Basedon elastic thin plate theory, the strain energy for the raft can be related to its deflection andexpressed as

UD w

x

w

yv

w

x

w

y

w

x ydAr

rr

A

= +

− −( ) −

∫∫2

2 12

2

2

2

2 2

2

2

2

2 2∂∂

∂∂

∂∂

∂∂

∂∂ ∂

(14)

in which DE t

vrr

r

=−

3

212 1( ) is the flexural rigidity of the raft. vr, Er and t are the Poisson’s ratio,

the Young’s modulus and the thickness of the raft, respectively. x and y are the coordinates.

For a raft resting on a pile group-soil system where the pile head-ground surface in contactwith the raft is discretized into a number of elements as shown in Figure 2, the potential energyof the raft can be written in the following form (Shen, Chow, and Yong7):

πr r rT

rT

rU w R W p= + { } { } − { } { }12 (15)

where {wr} and {R} are the deflections and reaction forces of the soil and pile elements. {pr} arethe external pressure acting on the raft corresponding to each soil and pile element, and {Wr} arethe integration of the raft deflections over the pile and soil element areas. At the pile group-soiland raft interface, the compatibility and equilibrium conditions require

w wr{ } = { } (16)

R P{ } = { } (17)

Therefore, by making use of the stiffness matrix in Equation (12), the relationship betweenthe reaction forces {R} and the deflections {wr} can be obtained as

R k wt r{ } = [ ]{ } (18)

The potential energy expression in Equation (15) then becomes

πr r rT

t r rT

rU w k w W p= + { } [ ]{ } − { } { }12 (19)

The deflection of the raft can be represented by a finite series as (Shen, Chow, and Yong7):

w w x y Am x

a

n y

bx y mn

n

k

m

k

= + + +==∑∑0

11

θ θ π πsin sin

+ += =∑ ∑B

m x

aC

n y

bm

m

k

n

n

k

1 1

sin sinπ π

(20)



where a and b are the length and width of the raft, w0 , θx , and θy are the rigid movement androtations in x and y directions of the raft, respectively, and Amn, Bm, and Cn are undeterminedcoefficients. Based on this finite series, the strain energy Ur, the vectors {wr} and {Wr} inEquation (19) all can be related to these coefficients. The principle of minimum potential energy

then requires Equation (19) to be an extremum with respect to these coefficients, i.e., ∂π∂δ

r = 0 ,

where the parameter δ denotes the representation of the coefficients w0 , θx , θy , Amn, Bm, and Cn

and this leads to the following relationship

∂∂δ

∂∂δ

∂∂δ

U wk w

Wpr r

T

t rr

T

r+

[ ]{ } =

{ } (21)

The above expression can finally be written in matrix form as

h Pr[ ]{ } = { }ξ (22)

where [h] = [kr] + [Zr]T [kt][Zr], and{Pr} = [Zr]T {pr} . The matrix [kr] in the matrix [h] reflectsthe stiffness of the raft, [Zr] is a matrix related only to the coordinates, and {ξ} is a vector madeup by the coefficients in the finite series in equation (20).

Equation (22) can then be solved for the unknown constants, i.e., the vector under the knownexternal loads {pr} applied on the raft. The deflection and bending moment of the raft at anylocation can be determined analytically based on Equation (20) and thin plate theory. Thereaction forces, i.e., the forces at the pile heads and at the soil elements in contact with the raft,can be determined using Equation (18).

III. RESULTS OF ANALYSIS

A. Comparison of ResultsComparison with published results of piled raft foundations is carried out. All the present

results are obtained using a computer code named VAFPR (Variational Approach For PiledRafts) that is developed based on the variational approach described above.

Clancy and Randolph10 reported a hybrid approach for the analysis of piled rafts, in whichthe analysis of piles and soil was based on a load transfer treatment of individual piles with elasticinteraction between piles and raft treated using Mindlin’s solution, and analysis of raft based onplate bending finite elements. A uniformly distributed unit load was applied on the raft. Theproperties for a three-by-three and a nine-by-nine piled rafts analyzed are shown in Table 1,where Es, vs, are the Young’s modulus and Poisson’s ratio of the soil, respectively, Ep, and r0, isthe Young’s modulus and radius of the piles, respectively, and s is the pile spacing. Thecorresponding solutions are compared to the present results in Table 2. The average settlementscomputed by the two different methods are in good agreement, but the present results give a lowerpercentage of loads taken by the piles, especially for the larger pile group with nine-by-nine piles.A comparison of the differential settlement for the nine-by-nine pile group is shown in Figure 3.The normalized coordinate is defined with a value of 0.0 at the corner, 0.5 at the center-edge and1.0 at the center of the raft (Clancy and Randolph10). The normalized displacement is defined as

ww w

w wnorav= −

−max min

in which is the actual displacement at a given point, wav , wmax , and wmin are the average, maximumand minimum displacement of a piled raft, respectively. Good agreement is observed between thetwo methods.


136 Chow et al.

TABLE 1

Properties for 3 × 3 and 9 × 9 piled rafts

TABLE 2

Results for 3 × 3 and 9 × 9 piled rafts



FIGURE 3. Normalized differential displacement for 9 × 9 piled raft foundation.

Poulos11 presented a comparison of some methods for the analysis of piled rafts. A hypotheti-cal example consisting of 15 piles with concentrated loads acting on the raft was analyzed byvarious methods. The plan of the foundation together with the concentrated loads termed P1 andP2 acting at the location of some piles are shown in Figure 4. The properties and the two casesanalyzed for the piled rafts are presented in Table 3. The difference in these two cases is in themagnitude of the applied loads. In the present analysis, the concentrated loads applied on the raftare transformed into the distributed loads over the cross-sectional area of each pile. The resultsobtained using different methods for the key responses: average settlement, maximum differen-tial settlement, maximum bending moment, and the proportion of load taken by the piles arepresented in Figure 5. The results of Poulos4 in Figure 5 are based on an approximate springapproach. The results of Ta and Small5 are based on a combined finite element and finite layeredmethod, and the results of Sinha12 are based on a combined finite element and boundary elementmethod. It can be seen that, apart from the method of Sinha,12 the rest of the methods givereasonably similar value of the average settlement. All the methods give reasonably similar

FIGURE 4. Plan of a piled raft with concentrated loadings.


138 Chow et al.

results for the differential settlement between the center and the center-edge, with the method ofPoulos4 giving a slightly higher value. The maximum bending moment computed by Poulos4 isconsiderably larger than the rest of the methods. The result of Sinha12 gives the lowest percentageof load taken by the piles; the rest of the methods for load on the piles are in reasonably goodagreement. To sum up, the present solutions are in close agreement in these key responses withthe results of Ta and Small,5 which involve less assumptions and approximations than themethods of Poulos4 and Sinha.12 It is noted that, in case 2, where the overall applied load exceedsthe ultimate capacity of the piles alone, some nonlinear behavior of the foundation can beexpected. The trend of the results for all the methods is similar. The average settlement, thedifferential settlement and the maximum bending moment increases and the proportion of loadtaken by the piles decreases.

Effect of Pile Spacing and Number of Piles on the Response of Piled RaftsParametric studies are carried out to investigate the effect of pile spacing and number of piles

on the response of piled rafts supported by small pile groups and large pile groups. The keyresponse, i.e., the average settlement, maximum differential settlement, and maximum bendingmoment of the rafts, and the proportion of load taken by the piles are presented. Two square piledrafts subjected to uniformly distributed loading are selected as examples. One of them has an areaa b r r× = ×30 300 0 and is supported by small pile groups and the other has an area

a b r r× = ×90 900 0 and is supported by large pile groups. The pile groups used to support these two

rafts are arranged in a square grid. For the raft with the smaller area 30r0 × 30r0, the small pile groupsused are 3 × 3, 4 × 4, and 5 × 5 groups. The corresponding pile spacings adopted for these pile groupsare s/r0 = 12, 8, and 6, which are the spacings commonly used in practice. For the raft with the largerarea 90r0 × 90r0, the relatively larger pile groups used are 8 × 8, 11 × 11, and 15 × 15 groups. Thecorresponding pile spacing for these pile groups are s/r0 = 12, 8.4, and 6.

The results obtained are presented in Tables 4 and 5 for the critical response of the pile-raftsystem, i.e., the average settlement of the rafts, the differential settlement of the rafts, the load takenby the piles and the bending moment of the rafts. For a given raft area, the average settlement, thedifferential settlement, and the maximum bending moment of the raft increase as the pile spacing

TABLE 3

Properties and cases analyzed for a piled raft with concentrated loadings



Figure 5. Comparison of various solutions (Case 1: ______ Case 2: – – – – – ).


140 Chow et al.

TABLE 4

The response of a piled raft with an area 30r0 × 30r0



TABLE 5

The response of a piled raft with an area 90r0 × 90r0


142 Chow et al.

gets larger and the number of the piles gets less, while the load taken by the pile decreases. However,as can be seen in the two tables, the degree of this effect is quite different between the small pilegroups and the larger pile groups. For the small pile groups, as the pile spacing gets larger and thenumber of the piles gets less, the differential settlement and the maximum bending moment of theraft increase very significantly, while the average settlement of the raft and the load taken by thepiles only increase moderately. However, these quantities increase very little for the larger pilegroups. These observations mean that for larger pile groups with the range of practical pile spacing,i.e., s/r0 = 6–12, the response of the system is not sensitive to the change in the pile spacing and thenumber of the piles. In other words, taking larger pile spacing and reducing a certain number of pilesdo not result in the great increase in the settlement, differential settlement and bending moment ofrafts. This is due to the fact that for larger pile groups, the smaller spacing and larger number of thepiles increase the degree of the interaction effect of the piles and soil significantly, leading to littleincrease in the overall stiffness of the system. These observations indicate that too small a pilespacing, say s/r0 = 6, has very little contribution to the stiffness of a piled-raft system containing alarge number of piles. In other words, adopting a larger pile spacing appears to be acceptable forpractical large pile groups taking into consideration the settlement, bending moment, and loadsharing between the piles and raft.

IV. CASE STUDIES

Two reported case histories of piled raft foundations are analyzed. The theoretical results forthe settlement, percentage of load taken by piles as well as load on individual piles are comparedwith the field measurements.

A. Dashwood HouseThe building was a 15-storey building supported by a piled raft foundation (Green and

Hight13). The pile group consisted of 462 bored piles with diameter of 0.485 m and length of 15m, and was capped by a rectangular raft with a thickness of 1.5 m and a plan of 33.8 m × 32.6m. The center-to-center spacing of the piles was 1.5 m. A part of the foundation plan showingthe positions of piles with load cell is shown in Figure 6.

The shear modulus profile was given by (Shen, Chow, and Yong8)

G = 30 + 1.33z MN / m2

The Poisson’s ratio of the clay was taken to be 0.5. The overall load on the foundation was279 MN, and this load is applied on the raft as a uniformly distributed load in the analysis. Thecomputed average settlement and percentage of load taken by the raft are compared withmeasured results obtained at the end of construction in Table 6. There is reasonable agreementbetween the computed and measured values, although the computed percentage of load taken bythe raft is moderately larger than that measured. The computed loads on the individual piles alongseveral sections are compared with the field measurements in Figure 7. It seems that reasonableagreement is observed, but the computed load on the pile located inside the core area issignificantly less than the measured result. This may be due to the heavy loading condition withinthe core area that is not reflected in the analysis.

B. A Tall Building on Frankfurt ClayThe field measurements of the performance of the raft of the tall building was reported by

Sommer et al.14 The 30-storey tall building was a 130-m-high structure supported by two identicalseparate piled rafts, which had a thickness of 2.5 m and a plan of 17.5 × 25.0 m. The pile groupbeneath each raft comprised of 42 bored piles of diameter 90 cm and a length of 20 m. The pilespacing varied from 6r0 to 7r0. The foundation plan for one raft with pile load cells installed isshown in Figure 8.



FIGURE 6. Layout of piles with load cell for Dashwood House.

TABLE 6

Settlement and load taken by raft


144 Chow et al.

FIGURE 7. Load taken by piles.



The subsoil below the raft was underlain by layers of Frankfurt clay extending to greatdepth. Within the clay, thin calciterous sand, silt inclusions, and isolated floating limestone layerswere embedded. According to laboratory test results, the undrained shear strength of the clay Cu

varied from 100 to 200 kN / m2 increasing with depth.

In the present analysis, the clay undrained shear strength is assumed to increase linearly withdepth from 100 kN / m2 at the foundation level to 200 kN / m2 at the pile toe, and the shear modulusof the clay is taken as G = 200Cu. Thus, the shear modulus profile can be determined by

G = 20 + 1.0z MN / m2

The Poisson’s ratio of the clay is taken to be 0.5. As reported, when the measurements werecarried out, only approximately 75% of the structural load 241 MN was acting on a single raft.This load is uniformly distributed over the raft area in the present analysis.

Comparisons with measured maximum settlement, and the estimated percentage of loadtaken by the raft are shown in Table 6. A comparison of the loads on individual piles is shownin Figure 9. The agreement with the measured values in the settlement and the percentage of theload on the raft is reasonable. Note that the computed loads on the individual piles are larger thanthe measured results. This may be due to the approximate structural load used in the presentanalysis, which may overestimate the real structural load at the time of the measurements andresults in the larger computed loads on the individual piles.

CONCLUSIONS

A variational approach for the analysis of piled raft foundations is described. The raft andpile group-soil system can both be analyzed by the use of the principle of minimum potentialenergy. By representing the deformation of the piles and raft using finite series, the method isvery efficient for the analysis of a piled raft with a large number of piles. The present solutionshave shown good agreement with other numerical methods and can reasonably predict the

FIGURE 8. Foundation plan.


146 Chow et al.

FIGURE 9. Load taken by piles.



response of piled raft foundations in the field. Parametric studies have shown that for a givenpiled raft supported by a larger pile group, increasing pile spacing from s/r0 = 6 to 12, andreducing the number of piles does not result in a great increase in the settlement, differentialsettlement, and bending moment of the raft.

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