Approximate three-dimensional analysis of rectangular barrette–soil–cap interaction

Approximate three-dimensional analysis ofrectangular barrette–soil–cap interaction

G.H. Lei, X. Hong, and J.Y. Shi

Abstract: Rectangular barrettes are increasingly being used to support large-size and heavy-duty structures, but the in-teraction among barrettes, soil, and cap has rarely been studied theoretically. This paper presents an approximate three-dimensional semi-analytical method for the analysis of load–displacement behaviour of single barrettes, barrette groups,and barrette–soil–cap interaction systems. A unique feature of a barrette, which distinguishes it from a circular pile, isits nonaxisymmetrical mechanical behaviour. To take into account this feature, both the barrette–soil and the cap–soilinterfaces are discretized. Mindlin’s solution is adopted to define the load–displacement relationship of the soils next tothe barrette and the cap. By assuming the deformation compatibility at the barrette–soil and cap–soil interfaces, theload–displacement relationship of the soils is incorporated into the static force equilibrium conditions in the interior ofthe barrette and cap structures. In this way, governing equations in finite difference form are derived for obtaining theload–displacement response of the barrette–soil–cap system. The proposed method is verified by comparing the calcu-lated results for a group of square piles using other existing methods. In addition, some factors such as barrette shape,barrette spacing, and barrette group layout and finite-layer depth, which influence the response of the barrette–soil–capsystem, are investigated.

Key words: elasticity, foundations, numerical modelling, piles, theoretical analysis.

Résumé : Des barrettes rectangulaires sont de plus en plus utilisées pour supporter des structures de grandes dimen-sions ou très lourdes, mais l’interaction entre les barrettes, le sol et les casques des pieux a été rarement étudiéethéoriquement. Cet article présente une méthode semi-analytique tridimensionnelle approximative pour l’analyse ducomportement du déplacement sous charge de barrettes uniques, de groupes de barrettes et de l’interaction dans dessystèmes barrette–sol–casque. Une caractéristique unique d’une barrette, qui la distingue d’un pieu circulaire, est soncomportement mécanique non axisymétrique. Pour prendre en compte cette caractéristique, les interfaces barrette–solde même que casque–sol doivent être discrétisées. On a adopté la solution de Mindlin pour définir la relation déplacement–charge des sols avoisinant la barrette et le casque. En supposant qu’il y a compatibilité de déformation aux interfacesbarrette–sol et casque–sol, la relation charge–déplacemment des sols est incorporée dans les conditions d’équilibre desforces statiques à l’intérieur des structures de barrettes et de casques. De cette façon, les équations maîtresses sousforme de différences finies sont dérivées pour obtenir la réaction charge–déplacement du système barrette–sol–casque.On vérifie la méthode proposée en comparant les résultats calculés pour un groupe de pieux carrés au moyen d’autresméthodes existantes. De plus, on étudie des facteurs tels que la forme d’une barrette, l’espacement des barrettes, et ladisposition d’un groupe de barrettes et la profondeur d’une couche-finie, qui influencent la réaction du système barrette–sol–casque.

Mots-clés : élasticité, fondations, modélisation numérique, pieux, analyse théorique.

[Traduit par la Rédaction] Lei et al. 796

Introduction

In the past four decades, rectangular barrettes have beenused as the deep foundations for many high-rise buildings andinfrastructure engineering projects. A collection of informa-tion of the applications and loading tests of barrettes reported

in the literature is presented in Lei (2001). Most of the casehistories of barrettes were not comprehensively documented.Nevertheless, some loading test results provided more detailedand useful information on the normal stress changes at thebarrette–soil interface, the mobilization of barrette shaft resis-tance, the influence of loading direction on the lateral responseof barrettes, and the construction effects on barrette perfor-mance (Fellenius et al. 1999; Ng et al. 2000; Lei 2001; Ngand Lei 2003; Zhang 2003).

There are some typical methods of analysing the behaviourof pile–soil–cap interaction for circular piles, such as theboundary element method (Butterfield and Banerjee 1971;Kuwabara 1989), the finite element method (Ottaviani 1975),the load-transfer function method (Randolph and Wroth 1979),the hybrid method (Chow and Teh 1991), and the variationalmethod (Shen et al. 2000). For barrettes, however, only verylimited theoretical investigations have been carried out be-

Can. Geotech. J. 44: 781–796 (2007) doi:10.1139/T07-017 © 2007 NRC Canada

781

Received 21 March 2006. Accepted 29 January 2007.Published on the NRC Research Press Web site at cgj.nrc.caon 2 August 2007.

G.H. Lei1 and J.Y. Shi. Department of Civil Engineering,Hohai University, 1 Xikang Road, Nanjing, Jiangsu, 210098,P.R. China.X. Hong. Kunshan Construction Engineering Quality TestingCentre, Kunshan, Jiangsu, 215337, P.R. China.

1Corresponding author (e-mail: [email protected]).

cause of their brief history (only about four decades) and thecomplexities and difficulties in analysing the nonaxisym-metrical mechanical behaviour. Without design parametersfrom costly and time-consuming full-scale loading tests, bar-rettes have been commonly designed as drilled shafts or boredpiles of equal cross-sectional area by ignoring the effect ofthe geometrical shape on the load-carrying performance.

To improve the understanding of the load-carrying behav-iour of rectangular barrette–soil–cap interaction, this paperpresents a boundary element method for the analysis of thestresses and displacements at the barrette–soil and cap–soilinterfaces. The nonaxisymmetrical mechanical feature of thebarrette is taken into account by discretizing the interfacesthree-dimensionally. Mindlin’s displacement solution (Mindlin1936) is adopted to define the load–displacement relation-ships of the discretized interface elements. Combining theserelationships with the static force equilibrium conditions ofthe barrette and cap structures, governing equations are derivedin finite difference form for obtaining the load–displacementrelationship of the barrette–soil–cap system, together withthe stresses and displacements at the barrette–soil and cap–soil interfaces. Firstly the governing equations for a singlebarrette are derived, followed by those for a freestandingbarrette group and a barrette–soil–cap system. Then the pro-posed method is verified by comparing with other existingmethods for a square pile–soil–cap system. Finally theeffects of barrette shape, barrette spacing, barrette arrange-ment, and finite-layer depth on the response of the barrette–soil–cap system are investigated.

Load–displacement relationship at thebarrette–soil interface

For a barrette embedded in an isotropic homogeneous soil,the barrette–soil interface can be discretized three-dimensionallyas illustrated in Fig. 1. The length, width, and depth of thebarrette are designated by l, w, and d, respectively. The depthof the barrette is equally divided into M layers. For eachlayer, the longer and shorter sides of the barrette shaft areequally divided into ml and mw number of elements, re-spectively. Therefore, the total number of the elements aren = [2 × (ml + mw)] for each layer of the barrette shaft,m = (ml × mw) at the barrette base, and MP = (M × n + m) atthe whole barrette–soil interface. A shaft element located atthe Ith layer is denoted as Ij, where j represents the number

of elements counted consistently from a corner and a longerside of the barrette, j = 1, …n and I = 1, …M. A base elementis denoted as bi, where i represents the number of elements,i = 1, …m. The vertical load applied at the top of the barretteis P0. The shaft resistance (i.e., the side shear stress) on anarbitrary shaft element Lk at the Lth layer and the end-bearingresistance on a base element bk are designated by pLk andpbk, respectively.

According to Mindlin’s solution (Mindlin 1936) and theprinciple of superposition, the vertical displacements, w Ij

J

and w biJ , at the corresponding centrepoints of shaft element Ij

and base element bi of an arbitrary Jth barrette of a group ofN barrettes may be calculated as follows:

[1] wE

Is p Is pIjJ

IjJ LkK LkK

k

n

IjJ bkK bkK

k

m

L

M

= += ==∑ ∑∑1

s 1 11, ,

=∑K

N

1

[2] wE

Is p Is pbiJ

biJ LkK LkK

k

n

biJ bkK bkK

k

m

L

M

= += ==∑ ∑∑1

s 1 11, ,

=∑K

N

1

where Es is the Young’s modulus of the soils next to the bar-rette; IsIjJ,LkK and IsbiJ,LkK, and IsIjJ,bkK and IsbiJ,bkK are the soildisplacement influence factors, which represent the verticaldisplacements at the centrepoints of elements Ij and bi of theJth barrette induced by a unit shaft resistance, pLk

K , on ele-ment Lk and a unit end-bearing resistance, pbk

K , on elementbk of the Kth barrette, respectively; these influence factorscan be obtained by double integration of Mindlin’s equationas given in Appendix A.

Therefore, the vertical displacements at the centrepointsof all the elements at the barrette–soil interface of the group,{ws}G, can be written in matrix form as

[3] { } [ ] { }wE

Is ps Gs

G s G= 1

in which

[4] { } {w w w w w w wn M Mn b bms G 111

11

11 1

11 1= � � � �

� � � � �w w w w w wNn

NMN

MnN

bN

bmN T

11 1 1 1 }

[5] [ ]

[ ] [ ]

[ ] [ ]

, ,

, ,

Is

Is Is

Is Is

N

N N NG

1 1 1

1

=

�

� � �

�

[6] [ ] ,

, , , ,

Is

Is Is Is Is Is

J K

J K J nK J M K J MnK

=

11 11 11 1 11 1 11 1� � � 1 1 11

1 11 1 1 1

J b K J bmK

nJ K nJ nK n

Is

Is Is Is

, ,

, ,

�

� � � � � � � � � �

� � J M K nJ MnK nJ b K nJ bmK

M J

Is Is Is

Is

, , , ,

,

1 1 1 1 1

1

� �

� � � � � � � � � �

11 1 1 1 1 1 1 1 1K M J nK M J M K M J MnK M J b K M J bIs Is Is Is Is� � � �, , , , , mK

MnJ K MnJ nK MnJ M K MnJ MnIs Is Is Is

� � � � � � � � � �

� � �, , , ,11 1 1 K MnJ b K MnJ bmK

b J K b J nK b J M K

Is Is

Is Is Is, ,

, , ,

1

1 11 1 1 1 1

�

� � � Is Is Is

Is Is

b J MnK b J b K b J bmK

bmJ K

1 1 1 1

11

, , ,

,

�

� � � � � � � � � �

� bmJ nK bmJ M K bmJ MnK bmJ b K bmJ bmKIs Is Is Is, , , , ,1 1 1� � �

© 2007 NRC Canada

782 Can. Geotech. J. Vol. 44, 2007

[7] { } {p p p p p p pn M Mn b bms G 111

11

11 1

11 1= � � � �

� � � � �p p p p p pNn

NMN

MnN

bN

bmN T

11 1 1 1 }

where {ps}G is a vector of the shaft or end-bearing resistanceon the interface elements; the superscript of each coefficientof vectors {ws}G and {ps}G denotes the identity number of agiven barrette in the group; a vector superscripted by T repre-sents the transpose of the vector; [Is]G is a matrix of soil dis-placement influence factors of the group, whose submatrix[Is]J,K represents the displacement of the soil next to the Jthbarrette induced by a unit shaft or end-bearing resistance atthe Kth barrette–soil interface; [Is]J,K is of the order of MPby MP; {ws}G and {ps}G are column vectors with MG rows;[Is]G is of the order of MG by MG; and MG = (MP × N) is thetotal number of elements at the barrette–soil interface in thegroup.

A special case of eq. [3], the load–displacement rela-tionships at the barrette–soil interface for a single barrette(namely, N = 1) can be expressed as

[8] { } [ ]{ }wE

Is pss

s= 1

A single barrette

Force equilibrium equations for a single barretteThe free body force diagram for an arbitrary Ith layer of

the barrette is shown in Fig. 2. The vertical force equilib-rium and the stress–strain relationship may be expressed as

[9] lwz

p pz

Il Ij

j

m

w Ijj m

m m

l

l

l

l w∂σ∂

δ δ δ

= − + +

= = +

+

∑ ∑1 1

pIjj m m

m m

l w

l w

= + +

+

∑

1

2

+

= −

= + +

+

∑δw Ijj m m

m m

Ip pl w

l w

2

2

1

( )

[10]∂∂

σw

z Ez

I

I

= −

P

where the subscript I of a bracketed partial derivative repre-sents the value of the derivative at the Ith layer; σz and wzrepresent the normal stress and vertical displacement alongthe depth of the barrette, respectively; σI and pI represent thenormal stress and shaft resistance at the Ith layer, respectively;δl and δw are the lengths of an element on the longer andshorter sides of the barrette, respectively, δl = l/ml, δw = w/mw;and Ep is the Young’s modulus of the barrette.

Substituting eq. [10] into eq. [9] yields

[11] p E lww

zI

z

I

=

p

2

2

∂

∂

This second-order partial differential equation may besolved by using a finite difference approximation.

For the Ith layer (2 ≤ I ≤ M – 1) of the barrette, the finitedifference formula of eq. [11] may be expressed as

[12] pE lw

w w wIz

I I I= − +− +p

2 1δ

( )2 1

where wI is the vertical displacement at the midpoint of theIth layer, and δz is the thickness of that layer (i.e., δz = d/M).

At the first layer of the barrette (i.e., I = 1) the normalstress is

[13] σ10=

P

lw

© 2007 NRC Canada

Lei et al. 783

Fig. 1. Three-dimensional discretization of a barrette–soil interface at: (a) barrette shaft, and (b) barrette base.

Fig. 2. Stresses acting on a layer of barrette shaft.

To obtain the finite difference formula for the first layer,an imaginary layer of thickness δz, which has a midpoint dis-placement w0, is considered above the first real layer. There-fore, from eqs. [10] and [13], the midpoint displacement ofthe imaginary layer may be related to that of the first reallayer, w1, as

[14] w wP

E lwz

0 10

p

= +δ

By substituting eq. [14] into eq. [12] for I = 1, the shaftresistance at the first layer may be derived as follows:

[15] pE lw

w wP

z z1

p

2 2 10= − +

δ δ( )

For the bottom layer of the barrette (i.e., I = M) the shaftresistance may be related to the vertical displacements of themidpoints of the (M–2)th, (M–1)th, Mth layers and the baseof the barrette. Using a finite difference formula for pointswith nonuniform spacing (Mattes and Poulos 1969), eq. [11]becomes

[16] pE lw

w w w wMz

M M M= − + − +− −p

2 2 1 b0 2δ

( . . )2 5 3 2

where wb is the vertical displacement of the barrette base.To obtain the final equation required for the solution of

the problem, eq. [10] may be applied to the barrette base, us-ing again a finite difference expression for an uneven inter-val spacing (Mattes and Poulos 1969), as given by

[17] pE lw

fw fw fwz

M Mbp

2 b= − + −−δ

( )1.33 12 10.671

where pb is the end-bearing resistance at the barrette base,and f = δz/(4lw).

Equations [12] and [15]–[17] can be written in matrix form as

[18] { } [ ]{ } { }pE lw

Ip w Yz

pp

2 p= +δ

in which

[19] { } { }p p p p pMT

p 1 2 b= �

[20] [ ]Ip =

−−

−

1 1 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0

�

�

�

� � � � � � � � �

2

2

� �

�

�

�

0 0 0 0 0 0 1 2 1 0

0 0 0 0 0 0 0 2 2 5 3 2

0 0 0 0 0 0 0 1 3 12 1

−− −

− −. .

. 3 f f 0 67. f

[21] { } { }w w w w wMT

p 1 2 b= �

[22] { }YP

z

T

=

0

δ0 0�

where {pp} is a vector of the shaft resistance at all the layers of the barrette and the end-bearing resistance at the base; {wp} isa vector of the midpoint displacements of the layers and the base; [Ip] is a matrix of the displacement influence factors for thebarrette; {Y} is a coefficient vector; {pp}, {wp}, and {Y} are column vectors of the order of (M + 1); and [Ip] is a matrix of theorder of (M + 1) by (M + 1).

Load–displacement governing equation of a single barretteBy assuming no slippage at the barrette–soil interface, that is, the displacements are compatible between the barrette and

the soils next to it, then

[23] { } [ ]{ }w A ws p=

in which

[24] [ ]A

n

n

=

1 1 0 0 0 0 0 0

0 0 1 1 0 0 0 0

��

� � � �

� ��

� � �

� � � � � � � � � � � � �

� � � ��

�

� � � � ��

0 0 0 0 1 1 0 0

0 0 0 0 0 0 1 1n

m

T

© 2007 NRC Canada


where [A] is a coefficient matrix of the order of MP by (M + 1); the symbol under a flat-lying brace represents the number of termsinvolved in the coefficients over the brace; and a matrix superscripted by T represents the transpose of the matrix.

Equation [9] can be written in matrix form as

[25] { } [ { }p B pp s= ]

in which

[26] [ ]B

l

m

w

m

l

m

w

ml w l w

=

δ δ δ δ��

��

��

��

� � � � � �0 0 0 0 0 0 0 0

� � � � � � � � � �

� � � � � ��

��

��

�0 0 0 0 0 0 δ δ δ δl

m

w

m

l

m

w

ml w l w

��

� � � � � � � � � ��

0 0

0 0 0 0 0 0 0 0 0 01 1

m mm

where [B] is a coefficient matrix of the order of (M + 1) byMP.

In addition, eq. [18] can be rearranged as

[27] { } [ ] ({ } { })wE lw

Ip p Yzp

pp= −−δ2

1

where the matrix superscripted by “–1” represents the in-verse of the matrix.

Therefore, from eqs. [8], [23], [25], and [27], it can readilybe shown that

[28] { } [ ][ ] [ ] [ ] [ ][ ]p A Ip BE lw

EIs A Ip

zs

p

s

= −

−−

−12

1

1

δ{ }Y

The displacement of the barrette head, S0, can be ex-pressed as a summation of the displacement of the midpointof the first layer and the compression of the barrette segmentabove this midpoint and shown as follows:

[29] S wP p

E lwz z

0 10 1

p

= +−4

8

2δ δ

A barrette group

Force equilibrium equations of a barrette group with arigid cap not in contact with soil

For a barrette group with a rigid cap not in contact withsoil, the displacements at the top of all of the barrettes areequal to the displacement of the cap, SG, and the load carriedby the cap is a summation of the loads carried by all the bar-rettes, PG. These relationships can be expressed as

[30] P P P P PK NG 0

102

0 0= + + + + +� �

[31] S S S S SK NG 0

102

0 0= = = = = =� �

where P K0 and S K

0 represent the load and displacement atthe top of the Kth (K = 1, 2, …, N) barrette, respectively.

Referring to eq. [18], the force equilibrium equation foran arbitrary Kth barrette may be expressed as

[32] { } [ ] { } { }pE lw

Ip w YK

z

K K Kp

p

2 p= +δ

Similarly, referring to eqs. [15] and [29], the shaft resis-tance pK

1 at the first layer and the displacement S K0 at the

top of the Kth barrette may be expressed as the followingtwo equations:

[33] pE lw

w wPK

z

K KK

1 2 2 10= − +p

δ δ( )

z

[34] S wP p

E lwK K

Kz

Kz

0p

= +−

10 1

24

8

δ δ

where w K1 and w K

2 represent the midpoint displacements ofthe first and second layers of the Kth barrette, respectively.

From eqs. [31] and [34], the following formula can be de-rived by a summation of the displacements at the top of allof the barrettes in the group:

[35] SN

wE lw

P pJ

J

Nz

zJ

J

N

Gp

G= + −

= =∑ ∑1

841

11

1

δδ

From eq. [31], S K0 in eq. [34] equals SG in eq. [35], and

hence

[36] PP

N

E lw

Nw wK

z

J K

J

N

0G p= + −

=∑

2 11 1

1δ

− −

=∑δ z J

J

NK

Np p

4

11

11

Substituting eq. [36] into eq. [33] yields

[37]1

4

3

4

231

11 2 2 1

11

Np p

E lww

Nw wJ

J

NK

z

K J

J

NK+ = + −

= =∑ ∑p

δ

+P

N z

G

δ

Equation [37] is the finite difference formula for the forceequilibrium equation of the first layer of the Kth barrette.

© 2007 NRC Canada

Lei et al. 785

Therefore, eq. [37], instead of eq. [15], should be used as thefirst row of the matrix in eq. [32]. For other layers, their fi-nite difference formulae are the same as eqs. [12], [16], and[17]. Based on eqs. [12], [16], [17], [32], and [37], the forceequilibrium equation of each layer of all of the barrettes inthe group can be expressed in matrix form as follows:

[38] [ ] { } [ ] { } { }U pE lw

Ip w Yz

G p Gp

G p G G= +δ2

in which

[39] [ ]

[ ] [ ]

[ ] [ ]

, ,

, ,

U

U U

U U

N

N N NG =

1 1 1

1

�

� � �

�

[40] [ ] ,U

N

J K =

+

1

4

3

40 0

0 1 0

0 0 1

�

�

� � � �

�

when J = K

1

40 0

0 1 0

0 0 1

N�

�

� � � �

�

≠

when J K

[41] { } { }p p p p p p pM bN

MN

bN T

p G = 11 1 1

1� � �

[42] [ ]

[ ] [ ]

[ ] [ ]

, ,

, ,

Ip

Ip Ip

Ip Ip

N

N N NG =

1 1 1

1

�

� � �

�

[43] [ ] ,Ip

N

J K =

−

−−

23 1 0 0 0 0 0 0

1 2 1 0 0 0 0 0

0 1 2 1 0 0 0 0

0

�

�

�

� � � � � � � � �

0 0 0 1 2 1 0

0 0 0 0 2 5

04

312

32

3

�

�

�

−− −

− −

0.2 3.

0 0 0 0

2

f f f

when J = K

20 0 0 0 0 0 0

0 0 0 0 0 0 0 0N

�

�

� � � � � � � � �

�

�

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

≠

when J K

[44] { } { }w w w w w w wM bN

MN

bN T

p G = 11 1 1

1� � �

[45] { }YP

N

P

Nz

M

z

M

GG

1

G

1

=

+ +

δ δ0 0 0 0�

� ��

� �

� ��

T

where {pp}G is a vector of the shaft resistance at all ofthe layers and the end-bearing resistance at all of thebases of the barrettes in the group; {wp}G is a vector ofthe midpoint displacements of the layers and the dis-placements of the bases; the superscript of each coeffi-cient of vectors {pp}G and {wp}G denotes the identitynumber of a given barrette; [Ip]G is a matrix of the

displacement influence factors for the barrettes, whosesubmatrix [Ip]J,K represents the displacement of the Jthbarrette induced by a unit shaft or end-bearing resistanceat the Kth barrette–soil interface; {U}G and {Y}G are co-efficient matrices; {pp}G, {wp}G, and {Y}G are columnvectors of the order of [(M + 1) × N]; the submatrices[Ip]J,K and [U]J,K are of the order of (M + 1) by (M + 1); and[Ip]G and [U]G are matrices of the order of [(M + 1) × N] by[(M + 1) × N].

Force equilibrium equations of a freestanding barrettegroup

A freestanding barrette group refers to a group of barretteswithout the cap. In this case, the load at the top of each bar-rette is known. Therefore, for an arbitrary Kth barrette, theforce equilibrium equations are the same as those derived

© 2007 NRC Canada


from a single barrette, namely eqs. [18]–[22]. By combiningeqs. [18]–[22] for each barrette, the force equilibrium equationsof a freestanding barrette group can be expressed as equationsin the same form as eqs. [38]–[45], except that [U]G is anidentity matrix of the order of [(M + 1) × N] by [(M + 1) × N]and eqs. [42] and [45] shall be replaced, respectively, by thefollowing two equations:

[46] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

Ip

Ip

Ip

Ip

G =

0 0

0 0

0 0

�

�

� � � �

�

[47] { }YP P

z

M

N

z

M

G01

0=

+ +

δ δ0 0 0 0

1 1

�

� ��

� � �

� ��

T

Load–displacement governing equation of a barrettegroup

Provided that displacement is compatible at the barrette–soil interface, the following equation can be derived for abarrette group with no cap connected or with a rigid cap notin contact with soil:

[48] { } [ ] { }w A ws G G p G=

in which

[49] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

A

A

A

A

G =

0 0

0 0

0 0

�

�

� � � �

�

where [A]G is a coefficient matrix of the order of MG by[(M + 1) × N].

In addition, the shaft and end-bearing resistance on eachlayer of the barrettes can also be related to those on each el-ement at the barrette–soil interface as follows:

[50] { } [ ] { }p B pp G G s G=

in which

[51] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

B

B

B

B

G =

0 0

0 0

0 0

�

�

� � � �

�

where [B]G is a coefficient matrix of the order of [(M + 1) × N]by MG.

From eqs. [3], [38], [48], and [50], it can be shown readilythat

[52] { } [ ] [ ] [ ] [ ] [ ]p A Ip U BE lw

EIs

zs G G G G G

p

sG= −

−−

12

1

δ

× −[ ] [ ] { }A Ip YG G G1

A barrette–soil–cap interaction system

For a barrette group with a rigid cap in contact with soil,the cap–soil interface can be discretized as illustrated inFig. 3, where the Arabic numerals represent the identities ofthe barrettes. The lengths of the overhang of the cap in the xand y directions are designated by ox and oy, respectively.The side-to-side spacing among barrettes in the x and y di-rections is designated by sx and sy, respectively. The totalnumber of the discretized cap–soil interface elements is MC.Since the cap is rigid, the vertical displacement at the top ofthe cap, S, will be the same as that at the top of each barrettein the group. The total load carried by the barrette–soil–capsystem, P, is the summation of the contact pressure at thecap–soil interface, PC, and the load carried by the barrettes,PG. These relationships can be expressed as

[53] P P P= +C G

[54] S S S S SK N= = = = = =01

02

0 0� �

Load–displacement relationship at the barrette–soil andcap–soil interfaces

Similar to the derivation procedures for a single barretteand a barrette group, the load–displacement relationship atthe barrette–soil and cap–soil interfaces may be expressed as

[55]{ }

{ }

[ ] [ ]

[ ] [ ],

,

w

w E

Is Is

Is Iss C

s G s

C C G

G C G

=

1 { }

{ }

p

ps C

s G

where {ws}C is a vector of the vertical displacements at thecentrepoints of the elements at the cap–soil interface; {ps}Cis a vector of the contact pressure on those elements; [Is]Cand [Is]G,C are matrices of displacement influence factorsfor soils next to the cap and the barrettes induced by a unitcontact pressure at the cap–soil interface, respectively;[Is]C,G and [Is]G are matrices of displacement influence fac-tors for soils next to the cap and the barrettes induced by aunit shaft or end-bearing resistance at the barrette–soil in-terface, respectively; {ws}C and {ps}C are column vectorswith MC rows; [Is]C is of the order of MC by MC, [Is]C,G isof the order of MC by MG, and [Is]G,C is of the order of MGby MC.

Force equilibrium equations of barrettes and capSince the cap is rigid, the displacements at the centre-

points of the elements at the cap–soil interface are equal to

© 2007 NRC Canada

Lei et al. 787

Fig. 3. Discretization of a cap–soil interface.

S; hence the first row of the matrix in eq. [55] can be rear-ranged as

[56] { } [ ] { } [ ] [ ] { },p E S Is V Is Is ps C s C1

a C1

C G s G= −− −

where {Va} is a column vector with all coefficients equal to1; and it is of the order of MC.

By multiplying eq. [56] by a vector of the areas of thecorresponding elements at the cap–soil interface, {C}, theload carried only by the cap can be derived as

[57] P E S C Is V C Is Is pC s C1

a C1

C G s G= −− −{ }[ ] { } { }[ ] [ ] { },

where {C} is of the order of MC.Substituting eq. [57] into eq. [53] yields the load carried

by the barrettes, PG. By substituting it into eq. [35], the dis-

placement at the top of each barrette can be derived as fol-lows:

[58] SN

wE lw

P pJ

J

Nz

zJ

J

N

= +

−

= =∑ ∑1

84

11

10 p1

δδ

+

− ∑4{ }[ ] [ ] { },C Is Is pC1

C G s G

in which

[59] N NE

E lwC Is Vz

0s

pC1

a= + −δ2

{ }[ ] { }

For the Kth barrette, the following equation can be derivedbased on eqs. [29] and [58]:

[60]1

84 41

11Nw

E lwP p C IsJ z

zJ

J

N

J

N

01

pC+ −

+==

∑∑ δδ { }[ ]− ∑

= +−1

C G s G

2

p

[ ] { },Is p wP p

E lwK

Kz

Kz

10 14

8

δ δ

Rearranging eq. [60] yields the load at the top of the Kth barrette as

[61] PP

N

E lw

Nw w

Np pK

z

J

J

NK z J

01

1

2 1

4

1= + −

− −

=∑

0

p

01 1

0δδ

11

1K

J

N

NC Is Is p

=

−∑

+

0C1

C G s G{ }[ ] [ ] { },

Substituting eq. [61] into eq. [33] yields the shaft resistance at the first layer of the Kth barrette as

[62]1

4

3

4

11

11

Np p

NC Is Is p

E lJ

J

NK

z0 0C1

C G s Gp

=

−∑ + − =δ

{ }[ ] [ ] { },

ww

Nw w

P

Nz

K J

J

NK

zδ δ2 11

232

01

0

+ −

+

=∑

Similar to the derivation procedures for eq. [38], the forceequilibrium equations for the barrettes in the barrette–soil–capinteraction system can be expressed in matrix form as follows:

[63] [ ]{ } [ ] { } { }V pE lw

Ip w Yz

1 s Gp

C p G C= +δ2

in which

[64] [ ] [ ] [ ] { }{ }[ ] [ ] ,V U BN

V C Is Isz

1 C G0

b C1

C G= − −1

δ

[65] { }VM M

T

b

1 1

=

+ +

1 0 0 1 0 0��

� ��

where [U]C, [Ip]C, and {Y}C are matrices in the same formand of the same order as [U]G, [Ip]G, and {Y}G in eqs. [39]and [40], [42] and [43], and [45], respectively, except that Nin these equations should be replaced with N0; [V1] is a coef-ficient matrix of the order of [(M + 1) × N] by MG; {Vb} is acoefficient vector of the order of [(M + 1) × N].

Load–displacement governing equation of a barrette–soil–cap interaction

Substituting eq. [56] into the second row of the matrix ineq. [55] yields

[66] { } [ ] [ ] { } ([ ],w S Is Is VE

Iss G G C C1

as

G= +− 1

− −[ ] [ ] [ ] ){ }, ,Is Is Is pG C C1

C G s G

In addition, eq. [58] can be rewritten in matrix form as

[67] SN

V wE lwN

P V pzz= + −1

84

00c s G

pd s G{ }{ } ( { }{ }

δδ

+ −4{ }[ ] [ ] { } ),C Is Is pC1

C G s G

in which

[68] { }VM n m M n m

c =

× + × +

1 0 0 1 0 0��

� � ��

© 2007 NRC Canada


[69] { }V l

m

w

m

l

m

w

m

M n m

l w l w

d =

× +

δ δ δ δ��

��

��

��

�

� ��

0 0

��

� � ��

��

��

��

�δ δ δ δl

m

w

m

l

m

w

ml w l w

0 0

M n m× +

� ��

where both {Vc} and {Vd} are vectors of the order of MG.Substituting eq. [67] into eq. [66] yields

[70] [ ]{ } { } [ ]{ }V pP

E lwNV V wz

2 s Gp 0

3 4 s G+ =δ

2

in which

[71] [ ] { }( { }[ ] [ ] { }),VE lwN

V Is Is Vzz2

p 03 C

1C G dC= −−δ

δ8

4

+ − −1

EIs Is Is Is

sG G C C

1C G([ ] [ ] [ ] [ ] ), ,

[72] { } [ ] [ ] { },V Is Is V3 G C C1

a= −

[73] [ ] [ ] { }{ }V IN

V V4 3 c= − 1

0

where [V2] and [V4] are matrices of the order of MG by MG;{V3} is a column vector of the order of MG; and [I] is anidentity matrix of the order of MG by MG.

Equations [63] and [70] define the relationships of {ps}Gversus {wp}G and {ps}G versus {ws}G, respectively. There-fore based on the relationship between {ws}G and {wp}G, de-fined by eq. [48], the following equation can be derived:

[74] { } [ ] [ ] [ ] [ ] [ ]pE lw E lw

A Ip V V Vs Gp p

G C1

11= −

− −δ δz z2

4 2

−1

× +

− −δz A Ip YP

NV V[ ] [ ] { } [ ] { }G C

1C

0

13

24

Numerical implementation and modification

Equations [28], [52], and [74] are the load–displacementgoverning equations for a single barrette, a barrette groupwith no cap connected or with a rigid cap not in contact withsoil, and a barrette–soil–cap interaction system, respectively.These explicit equations can be solved straightforwardly toobtain the shaft or end-bearing resistance of all of the ele-ments at the barrette–soil interface, that is, {ps} and {ps}G.For a single barrette and a barrette group, substituting thesolved {ps} and {ps}G into corresponding eqs. [25] and [50]yields the shaft resistance of all of the layers of the barrettesand the end-bearing resistance at the bases, namely {pp} and{pp}G. Subsequently, the vertical displacements at the mid-points of the layers of the barrettes, {wp} and {wp}G, can beobtained by substituting the derived {pp} and {pp}G into cor-responding eqs. [27] and [38]. For a barrette group with arigid cap not in contact with soil, the load at the top of eachbarrette can be calculated using eq. [36]. For a freestandingbarrette group, the displacement at the top of each barrette

can be calculated using eq. [34]. For a barrette–soil–cap in-teraction system, substituting the solved {ps}G into eq. [63]yields the displacements at the midpoints of all the layers ofthe barrettes, namely {wp}G and S. Subsequently, the contactpressure on the elements at the cap–soil interface can be ob-tained by substituting the derived {ps}G and S into eq. [56].Following these solution procedures, a FORTRAN programwas developed to perform the numerical calculation. Withthe proposed method, not only the load–displacement rela-tionship but also the shaft and end-bearing resistance canbe calculated for a single barrette, a barrette group, and abarrette–soil–cap system.

The problem of finite-layer depth, df, which exists belowthe barrette base can be treated approximately using themethod proposed by Poulos and Davis (1968), that is, modi-fying the displacement influence factors for a semi-infinitemass to those for a finite-layer depth. In addition, the pre-ceding analysis can be modified to take into account localyielding at the barrette–soil interface in a similar manner tothat described by Mattes and Poulos (1969) and Poulos andDavis (1980). When local yielding occurs at an interface ele-ment, the compatibility of the barrette and soil displace-ments for that element will no longer hold. Any increase inload will cause a redistribution of stress among the remain-ing elastic elements, and the new stress distribution alongthe barrette can be analysed by setting the known ultimatevalues on the yielded elements and considering the compati-bility of the barrette and soil displacements at the elastic ele-ments. In numerical implementation, when the calculatedshaft or end-bearing resistance on an element reaches itspredefined ultimate value, the displacements at the barrette–soil interface, {ws}G, are given by the original elastic dis-placement matrix, eq. [3], by setting the ultimate value onthat element. The displacements of the layers of the barrettes,{wp}G, are calculated in a similar manner to the fully elasticcase, except that eq. [10], which is applied to the first andbottom layers in the elastic analysis, must now be applied tothe layers nearest the top and bottom layers that have not al-ready yielded. By equating the interface and layer displace-ments at the elements that have not yielded, equations similarto eqs. [28], [52], and [74] are obtained, and these can besolved to determine a new stress distribution. This procedurecan be repeated until no additional elements have yielded.

Apart from the above considerations, it may also be modi-fied to take into account vertically nonhomogeneous soils ormultilayered soils using the method proposed by Lee andPoulos (1990), but this is beyond the scope of this paper.

Without exception, the calculated results from the proposedfinite difference method are sensitive to the number ofdiscretized layers of the barrettes and elements of the barrette–soil and cap–soil interfaces. In this study, the trial and errormethod is used to find out the satisfied number of layers andelements for achieving convergent results (Hong 2004).

© 2007 NRC Canada

Lei et al. 789

Verification of method

To verify the validity of the proposed method, a 3 × 3square pile–soil–cap interaction system was analysed, andthe results from the analysis were compared with those ob-tained using a truly three-dimensional finite-element methodby Ottaviani (1975), a boundary element method byKuwabara (1989), and a variational method by Shen et al.(2000) (the last two being derived from circular piles ofequal cross-sectional area). In this study, the pile cross-section is a square with side length l or w of 1 m; the piledepth d below the ground surface is 40 m; the spacing sx orsy is 3 m; the cap is in contact with soil at a depth, dc, of3 m; the cap overhang ox or oy is 0.5 m; the finite-layerdepth, df, is 60 m; the Young’s modulus of the pile, Ep, is20 GPa; and the Poisson’s ratio of the soil, νs, is 0.45. Thesquare pile was discretized into M = 15 layers and ml = 5 el-ements and mw = 5 elements, and the cap–soil interface wasdiscretized into MC = 108 elements. The loads taken by thecap and individual piles calculated by the proposed methodare summarized in Table 1, where piles 1, 2 and 3, and 4represent the corner, midside, and centre piles, respectively,as shown in Fig. 3. It can be seen that the uniformly distrib-uted load sharing by the piles in the group for the case ofEp/Es = 200 by Ottaviani (1975) is questionable. The resultscalculated using the proposed method are closer to those ob-tained by Kuwabara (1989) and Shen et al. (2000) than tothose obtained by Ottaviani (1975). This is likely becausethe former two methods were also derived on the basis ofMindlin’s solution. The little differences seen here aremainly due to the inherent assumptions of the differentmethods involved, for example, the square piles weretreated as circular piles of equal cross-sectional area byKuwabara (1989) and Shen et al. (2000). Generally reason-able agreement is obtained in the results between these dif-ferent methods.

Results of analyses

Many analytical methods have been applied to investigatethe load-carrying behaviour of circular pile–soil–cap interac-tion and its influencing factors, such as the pile length, pilespacing, pile-to-soil stiffness ratio, etc. (Poulos and Davis1980; Kuwabara 1989; Shen et al. 2000). By using themethod proposed in this paper, these factors were also inves-tigated, and similar results were obtained and are presentedin detail in Hong (2004). In the following, some unique re-sponses of a barrette–soil–cap system related to thenonaxisymmetrical mechanical feature of the barrette are in-vestigated. The length, width, and depth of the barrettes ana-lysed are 2.8, 0.8, and 40 m, respectively. The Young’smoduli of the barrette and soil are 20 GPa and 20 MPa, re-spectively. It has been found that the Poisson’s ratio of thesoil, νs, has a relatively small influence on the calculated re-sults (Hong 2004). The value of νs was chosen as 0.5, so thatthe analysis is particularly applicable to barrettes in un-drained soil where the load transfer takes place mainly byinterface adhesion. The barrette was discretized into M = 15layers and ml = 25 elements and mw = 7 elements.

Nonuniform shaft resistance distribution across a singlebarrette

Using the proposed method, the shaft resistance distribu-tion across a single barrette was investigated. For simplicity,the ultimate shaft resistance is limited to 45 kPa along theentire depth of the barrette, and the ultimate end-bearing re-sistance is limited to 800 kPa.

Figure 4 shows the calculated shaft resistance distributionacross one quarter of the barrette at its mid-depth. At otherdepths throughout the barrette, similar shaft resistance distri-butions were also observed. When the load applied to thebarrette head is less than 8000 kN, the shaft resistance is notuniformly distributed along the perimeter of the barrette.The shaft resistance at the corner is larger than that mobi-lized around the centre of the longer and shorter sides. Theshaft resistance at the centre of the shorter side is abouttwice the shaft resistance as that of the longer side. For theloads less than 8000 kN, the shaft resistance distributions arethe results of the elastic response of the barrette–soil inter-face. After the load is greater than 8000 kN, the shaft resis-tance starts to become increasingly uniformly distributedacross the barrette with increasing load, as a result of localyielding at the barrette–soil interface. Such phenomena areconsistent with the results from a three-dimensional finitedifference analysis (Lei 2001) using FLAC3D (Itasca Con-sulting Group, Inc. 1997).

From the above analyses, it can be concluded that a uni-formly distributed shaft resistance is not mobilized acrossthe barrette until failure of the barrette–soil interface hasbeen reached. At the elastic stage, the shaft resistance mobi-lized at the corner of the barrette is greater than that aroundits shorter side, which in turn is greater than that around itslonger side. The reason for this may be approximately ex-plained by the analytical solution for a rigid rectangularfoundation, which shows similar distribution characteristics(of contact pressure) (Milovi� 1992).

© 2007 NRC Canada


Cap Pile 1Pile 2or 3 Pile 4

Ep/Es = 200

Proposed method 22.7 10.0 7.9 5.7

Ottaviani (1975) 22.0 8.0 8.0 8.0

Ep/Es = 400


Ottaviani (1975) 15.0 10.0 9.0 7.0

Shen et al. (2000) 17.3 11.2 8.2 5.0

Ep/Es = 800


Ottaviani (1975) 11.0 11.0 9.0 7.0

Kuwabara (1989) 11.0 12.0 9.0 5.0

Ep/Es = 2000


Ottaviani (1975) 9.0 12.0 9.0 6.0

Shen et al. (2000) 11.8 13.0 8.2 3.3

Table 1. Load sharing (%) by cap and individual piles for a3 × 3 pile group.

Effect of the arrangement of a barrette groupTo study the effect of barrette arrangement on the load-

carrying behaviour of a barrette group, a 1 × 2 group and a2 × 1 group were analysed using the proposed method. Eachgroup has two barrettes with various side-to-side spacing of0.1, 1.0, 5.0, and 20.0 m. The barrettes in the two groupsare, respectively, in longitudinal and transversal arrays. Eachbarrette is subjected to a vertical load of 2000 kN.

Figure 5 shows the calculated shaft resistance distributionacross one half of a given barrette at its mid-depth in thegroups. It can be seen that the shaft resistance at thebarrette–soil interface is much smaller on the opposite sidesbetween the barrettes than on the other sides because of thegroup effect. With increasing barrette spacing, the shaft re-sistance on the sides inside and outside the group graduallyincreases and decreases, respectively. When the barrettespacing is far enough to eliminate the group effect, the dif-ference between them becomes negligible, and the shaft re-sistance distribution approaches that for a single barrette.These results are consistent with those obtained by Ottaviani(1975).

Figure 6 shows the calculated settlements of differentgroups, SG, normalized by the corresponding settlement ofa single barrette, S0. The barrettes in these groups are ofequal cross-sectional area of 2.24 m2 but with different as-pect ratios of l/w = 1.0, 3.5, and 10.0. The calculated valuesof SG/S0 are plotted against the barrette spacing on a loga-rithmic scale. The magnitude of SG/S0 reflects the group ef-fect. The higher the ratio of SG/S0, the more significant thegroup effect will be, and vice versa. It can be seen fromFig. 6 that at a given barrette spacing, the group effect forbarrettes in transversal array (full curves) increases with in-creasing aspect ratio; but the group effect for barrettes inlongitudinal array (broken curves) decreases with increas-ing aspect ratio. This is because on an equal cross-sectionalarea basis, the higher the aspect ratio, the longer the longer

sides and the shorter the shorter sides of the barrettes willbe. When the barrettes are in transversal array, the oppositesides between the barrettes are the longer sides of the bar-rettes, and hence the group effect increases with increasingaspect ratio. When the barrettes are in longitudinal array,the opposite sides between the barrettes are the shortersides of the barrettes, and hence the group effect decreaseswith increasing aspect ratio. From the analyses of the groupeffect it can be inferred that at any given displacement themaximum load applied to the barrette group increases withdecreasing group effect.

It can also be seen from Fig. 6 that with increasing bar-rette spacing, the ratio of SG/S0 decreases, which means thatthe group effect decreases. For the barrettes considered, thegroup effect starts to attenuate appreciably after the spacingis greater than 2.0 m.

Load-carrying behaviour of a barrette–soil–capinteraction

Using the proposed method, a 3 × 3 barrette–soil–cap sys-tem was analysed to investigate its load-carrying behaviour.The overhang ox or oy and embedment depth dc of the capare 0.5 m and 0 m, respectively. Table 2 shows the calcu-lated loads taken by the cap and individual barrettes of vari-ous side-to-side spacing sx and sy of 3, 5, 10, and 20 m. Itcan be seen that the corner and centre barrettes take themaximum and minimum loads, respectively. This is consis-tent with the results from the elastic analysis using othermethods (Kuwabara 1989). With increasing barrette spacing,namely increasing area of the cap–soil interface, the loadtaken by the cap increases, as expected; the loads taken bythe corner and midside barrettes (c.f., Fig. 3) decrease ac-cordingly; and the load taken by the centre barrette de-creases very slightly. For the barrette group with cap, thedifference between the loads taken by the corner, midside,and centre barrettes is relatively larger than that for the bar-

© 2007 NRC Canada

Lei et al. 791

Fig. 4. Shaft resistance distribution across a barrette.

rette group without cap. In other words, with the presence ofthe cap, the degree of nonuniformity of the load distributionon the barrettes of the group is higher than that without thecap. This indicates that besides minimizing the differentialsettlement of the pile group with a flexible cap (Randolph2003), an optimized design would also be necessary to de-termine the layout and geometry of the barrettes by mini-mizing the differential load sharing of a barrette group witha rigid cap.

Effect of finite-layer depthFigures 7 and 8 show the calculated percentage of the loads

taken by the cap and individual barrettes and the calculatedsettlement of the cap in the barrette–soil–cap interaction sys-tem with finite-layer depths df varying from 1.1d to infinity. Itcan be seen that when the finite-layer depth is greater thantwice the barrette depth d, the finite-layer depth has little in-fluence on the load sharing and the settlement of the barrette–soil–cap system. Therefore, for practical problems, it appears

© 2007 NRC Canada


Fig. 5. Shaft resistance distribution across barrettes: (a) in transversal array, and (b) in longitudinal array.

sufficiently accurate to derive the load sharing and the settle-ment in a semi-infinite space when hard stratum underlyingthe barrette base is encountered at depths greater than twicethe barrette depth. For smaller finite-layer depths, the curvesshown in Figs. 7 and 8 provide some basis for correcting thecalculated values applying to a semi-infinite space. Apart fromthis, they also have implications in estimating the boundary ef-fect in the numerical analysis and model testing of the barrette–soil–cap interaction system.

ConclusionsA boundary element method for the analysis of rectangular

barrette–soil–cap interaction is presented. The method takesinto account the nonaxisymmetrical mechanical feature of thebarrette by discretizing three-dimensionally the barrette–soiland cap–soil interfaces. Using the method, the load-carryingbehaviour of a single barrette, a barrette group, and a barrette–soil–cap interaction system are analysed, and the followingconclusions are drawn:(1) A uniformly distributed shaft resistance is not mobilized

along the cross-section of a barrette until the failure ofthe barrette–soil interface has been reached.

(2) The group effect for barrettes arranged along their shortersides is more significant than that for those arrangedalong their longer sides.

(3) The group effect decreases with increasing side-to-sidespacing of the barrettes, and it starts to attenuate appre-ciably after the spacing is greater than 2.0 m for the bar-rettes considered.

(4) With the presence of a cap, the degree of nonuniformityof the load distribution on the barrettes of the group ishigher than that without the cap.

(5) When the finite-layer depth is greater than twice thebarrette depth, its influence on the load-carrying behav-iour of a barrette–soil–cap system is insignificant.

Acknowledgements

This research project is supported by the Scientific ResearchFoundation for the Returned Overseas Chinese Scholar fromthe State Education Ministry of China. The authors thank theanonymous reviewers for their critical comments and sug-gestions, which greatly assisted in revising the manuscript.

© 2007 NRC Canada

Lei et al. 793

Fig. 6. Influence of barrette spacing on the group effect. Fig. 7. Effect of finite-layer depth on load sharing.

Fig. 8. Effect of finite-layer depth on cap settlement.

sx and sy

(m)Withcap Cap

Barrette1

Barrette2

Barrette3

Barrette4

3 Yes 16.4 12.7 7.8 7.3 2.83 No 14.6 9.5 9.1 4.35 Yes 19.6 12.1 7.5 7.2 2.75 No 14.3 9.6 9.4 4.8

10 Yes 27.6 10.7 6.9 6.6 2.610 No 13.6 9.9 9.9 5.920 Yes 44.6 8.1 5.3 5.0 2.120 No 12.8 10.3 10.3 7.5

Table 2. Load sharing (%) by cap and individual barrettes for a3 × 3 barrette group.

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List of symbols

[ ]A coefficient matrix related to a single barrette[A]G coefficient matrix related to a barrette group

[B] coefficient matrix related to a single barrette[ ]B G coefficient matrix related to a barrette group{C} vector of areas of all of the elements at the cap–soil in-

terfaced depth of barrette base

dc depth of cap basedf depth of finite layerδl length of each element along the longer sides of a barretteδw length of each element along the shorter sides of a barretteδz thickness of each layer of a barretteEp Young’s modulus of barretteEs Young’s modulus of soil

f temporary variable[I] identity matrix

[ ]Ip displacement influence factor matrix for a single barrette[ ]Ip C displacement influence factor matrix for barrettes in

barrette–soil–cap system[ ]Ip G displacement influence factor matrix for a barrette group

[ ]Ip J K, displacement influence factor matrix for Jth barrette dueto Kth barrette

[ ]Ip K displacement influence factor matrix for Kth barretteIsbiJ bkK, centrepoint displacement of base element bi of Jth bar-

rette induced by unit end-bearing resistance of base ele-ment bk of Kth barrette

IsbiJ LkK, centrepoint displacement of base element bi of Jth bar-rette induced by unit shaft resistance of shaft element Lkof Kth barrette

IsIjJ,bkK centrepoint displacement of shaft element Ij of Jth bar-rette induced by unit end-bearing resistance of base ele-ment bk of Kth barrette

IsIjJ LkK, centrepoint displacement of shaft element Ij of Jth bar-rette induced by unit shaft resistance of shaft element Lkof Kth barrette

[ ]Is displacement influence factor matrix for soil next to asingle barrette

[ ]Is C displacement influence factor matrix for soil next to capdue to cap

[ ] ,Is C G displacement influence factor matrix for soil next to capdue to barrettes

[ ]Is G displacement influence factor matrix for soil next to bar-rettes due to barrettes

[ ]Is G,C displacement influence factor matrix for soil next to bar-rettes due to cap

[ ] ,Is J K displacement influence factor matrix for soil next to Jthbarrette due to Kth barrette

Iθ temporary variablel length of barrette

m number of base elements of a barretteM number of layers of a barrette

MC number of elements at cap–soil interfaceMP number of elements at a single barrette–soil interfaceMG number of elements at barrette–soil interface in a groupml number of elements divided along each longer side of a

barrettemw number of elements divided along each shorter side of a

barretten number of elements of each layer of a barretteN number of barrettes in a group

N0 temporary variable related to barrette–soil–cap interactionνs Poisson’s ratio of the soil

© 2007 NRC Canada


ox, oy cap overhangs in the x and y directionspb end-bearing resistance at barrette base

p kb end-bearing resistance of the kth base elementpI shaft resistance at the Ith layer of a barrettepIj shaft resistance of the jth element at the Ith layer of a

barrettepI

K shaft resistance at the Ith layer of the Kth barrettepbk

K end-bearing resistance of element bk of the Kth barrettepIj

K shaft resistance of element Ij of the Kth barrettepLk shaft resistance of the kth element at the Lth layer of a

barretteP load at the top of a cap

P0 load at the top of a barrettePK

0 load at the top of the Kth barrettePC load carried by the cap–soil interfacePG load carried by a barrette group

{ }pp vector of shaft or end-bearing resistance of the layers ofa barrette

{ }pp G vector of shaft or end-bearing resistance of the layers ofbarrettes

{ }p Kp vector of shaft or end-bearing resistance of the layers of

Kth barrette{ }ps vector of shaft or end-bearing resistance of the elements

at the barrette–soil interface{ }ps C vector of contact pressure of the elements at the cap–

soil interface{ }ps G vector of shaft or end-bearing resistance of the elements

at the barrette–soil interface in a groupR1, R2 temporary variables

S displacement at the top of a capS0 displacement at the top of a barrette

S K0 displacement at the top of Kth barrette

SG displacement at the top of a barrette group with cap notin contact with soil

sx, sy side-to-side spacing among barrettes in the x and y di-rections

σ1 normal stress at barrette headσI normal stress at the Ith layer of a barretteσz normal stress along the depth of a barrette

(u,v,c) co-ordinates of the centrepoint of element Lk[ ]U C coefficient matrix related to a barrette–soil–cap interaction[ ]U G coefficient matrix related to a barrette group

[ ]U J K, submatrix of [U]G[ ]V1 coefficient matrix[ ]V2 coefficient matrix{ }V3 coefficient vector[ ]V4 coefficient matrix{ }Va coefficient vector{ }Vb coefficient vector{ }Vc coefficient vector{ }Vd coefficient vector

w width of barrettewb displacement of barrette basew ib centrepoint displacement of ith base elementwI midpoint displacement at the Ith layer of a barrettewIj centrepoint displacement of the jth element at the Ith

layer of a barrettewz vertical displacement along the depth of a barrette

wIK midpoint displacement at the Ith layer of Kth barrette

wbkJ centrepoint displacement of base element bk of Jth barrette

wIjJ centrepoint displacement of shaft element Ij of Jth barrette

{ }wp vector of displacements of the layers of a barrette{ }wp G vector of displacements of the layers of barrettes{ }w K

p vector of displacements of the layers of Kth barrette{ }ws vector of displacements of the elements at the barrette–

soil interface{ }ws C vector of displacements of the elements at the cap–soil

interface{ }ws G vector of displacements of the elements at the barrette–

soil interface in a group(x,y,z) co-ordinates of the centrepoint of element Ij

{ }Y coefficient vector related to a single barrette{ }Y C coefficient vector related to barrette–soil–cap interaction{ }Y G coefficient vector related to a barrette group{ }Y K coefficient vector related to the Kth barrette

Z1, Z2 temporary variables

Appendix A. Expression for the soildisplacement influence factor

Let (x, y, z) and (u, v, c) represent the co-ordinates of thecentrepoints of elements Ij and Lk at the barrette shaft, re-spectively, as shown in Fig. A1. According to the Mindlinsolution (Mindlin 1936), the vertical displacement dw Ij

J , ofthe centrepoint of element Ij of the Jth barrette induced by adifferential load, dpLk

K , acting on element Lk of the Kth bar-rette may be expressed as (Vaziri et al. 1982):

[A1] dwdp

EIIj

J LkK

=s

θ

in which

[A2] Iv

vv

R

Z

R

cz

Rθ π

=+

−− +

− +

( )

( )( )

1

8 13 4

1 2s

ss

1

22

23

23

Z

R12

13

+ +− − −

6 8 1 3 4czZ

R

v v

R22

25

s2

s

2

( ) ( )

[A3] R x u y v Z ii i= − + − + =( ) ( ) ( , )2 2 2 1 2

where Z1 = z – c; Z2 = z + c; and νs is Poisson’s ratio of thesoil.

If the plane of element Lk is parallel to the yz plane inFig. A1, then

[A4] IsE

I v cIjJ LkK

v

v

c

c

, = ∫∫1

1

2

1

2

s

d dθ

If the plane of element Lk is parallel to the xz plane, then xand y in eq. [A3] should be interchanged (Vaziri et al. 1982).In addition, the displacement influence factor, IsbiJ,LkK, for thesoil next to element bi at the base of the Jth barrette inducedby a unit shaft resistance of element Lk of the Kth barrette canalso be calculated by substituting z = d into eqs. [A2]–[A4].

Similarly, the displacement influence factor for soil nextto element Ij of the Jth barrette induced by a unit end-bearing resistance on element bk at the base of the Kth bar-rette may be calculated by

© 2007 NRC Canada

Lei et al. 795

[A5] IsE

I v uIjJ bkK

v

v

u

u

, = ∫∫1

1

2

1

2

s

d dθ

The results of integrating eqs. [A4] and [A5] can be ob-tained from Vaziri et al. (1982).

References

Mindlin, R.D. 1936. Force at a point in the interior of a semi-infinite solid. Physics, 7: 195–202.

Vaziri, H., Simpson, B., Pappin, J.W., and Simpson, L. 1982. Inte-grated forms of Mindlin equation. Géotechnique, 32(3): 275–278.

© 2007 NRC Canada


Fig. A1. Three-dimensional discretization of a barrette–soil interface at: (a) barrette shaft, and (b) barrette base.

Documents

Approximate three-dimensional analysis of rectangular barrette–soil–cap interaction