FEM model for dynamic analysis of structures on piles and pile groups in viscoelastic and poroelastic soils

ARTICLE IN PRESS Engineering Analysis with Boundary Elements 33 (2009) 25– 34

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Simplified BEM/FEM model for dynamic analysis of structures on piles and pile groups in viscoelastic and poroelastic soils M.A. Milla´n , J. Domı´nguez Department Estructuras, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, 41092 Sevilla, Spain

a r t i c l e in fo

abstract

Article history: Received 25 August 2007 Accepted 17 April 2008 Available online 16 June 2008

A simplified model for the analysis of the dynamic response of structures on piles and pile groups under time harmonic excitation is presented in this paper. It is a coupled boundary element–finite element model able to take into account dynamic pile–soil–pile interaction in a rigorous manner. Piles and pile groups in viscoelastic or poroelastic soils are considered. Two-node cylindrical boundary elements are used to represent the interface between soil and pile. These elements are connected to beam-type finite elements representing the concrete pile which can be connected to a pile cap and to any superstructure modeled by beam elements. The model is rather simple: two-node beam elements along the pile are directly connected to the BE nodes along the soil hole, and the uppermost node to the soil surface and to the FE nodes of any superstructure. Thus, large structures founded on piles in viscoelastic or poroelastic soils can be represented using a reasonable number of unknowns. In order to validate the procedure, single piles and pile groups in viscoelastic and poroelastic soils are analyzed. The obtained results are compared with those obtained by other authors using more complex or less general approaches. There is a good agreement between the present results and those reported in the literature. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Pile foundations Dynamics Boundary elements Frequency domain Soil–structure interaction Soil–pile interaction

1. Introduction Dynamic analysis of piles and pile groups in viscoelastic soils have received considerable attention in recent years. Several procedures to represent the pile–soil–pile system have been used in this field. Most of the existing approaches contribute to the understanding of pile systems behavior and provide results for one pile or a pile group dynamic stiffness or their transfer functions for incident seismic waves. Nevertheless, dynamic soil–structure interaction is difficult to be taken into account for large structures supported on piles using the existing methods. The model presented in this paper can be used to study pile groups coupled to large structures (like bridges or buildings) in a rigorous manner using a reasonable number of degrees of freedom. Initial models (i.e. Penzien et al. [1]) represented the soil using a Winkler model including dampers and concentrated masses to simulate wave radiation towards infinity and soil damping. Those models are not able to represent pile–soil–pile interaction and, consequently, the behavior of a pile group. In a different way, many researchers (for example Wolf and von Arx [2] and Kuhlemeyer [3,4]) used the finite element method

 Corresponding author. Tel.: +34 954 487293; fax: +34 954 487295.

E-mail addresses: [email protected] (M.A. Milla´n), [email protected] (J. Domı´nguez). 0955-7997/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2008.04.003

(FEM) to analyze the problem. They used axisymmetric elements combined with transmitting boundaries to eliminate the wave reflection at the discretization boundaries. Waas and Hartmann [5] used beam elements connected to the soil FE mesh to represent the piles. The boundary element method (BEM) has been shown to be powerful to solve the problem since it can properly represent the soil half-space. Kaynia and Kausel [6] used Green’s function for the half-space and later [7] for the stratified soil. In order to solve the numerical instabilities of the method, Kobayashi and Nishimura [8] obtained a new fundamental solution and Sen et al. [9] applied it efficiently. Mamoon and Banerjee [10] and Mamoon and Ahmad [11] used a hybrid formulation coupling a Green’s function to the beam differential equation using compatibility and equilibrium conditions. The work by Ferro and Venturini [12] introduced a new procedure to couple BEM and FEM for pile analysis. Later, Coda et al. [13–15] generalized the BEM–FEM coupling introducing the idea of a cylindrical element to model piles in elastic soils. Dynamic behavior of poroelastic soils has also received attention in recent times with important contributions to a proper understanding of their influence on some soil–structure interaction problems. However, very few studies have been presented where the dynamic behavior of piles and pile groups in poroelastic soils is studied. Initial researches were presented by

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Zeng and Rajapakse [16] and Jin et al. [17] for a single pile, and by Wang et al. [18] for pile groups. A more complete analysis of single piles and pile groups has been presented by Maeso et al. [19] using a full model which includes discretization of the pile–soil interface into quadratic boundary elements to represent the concrete pile and the soil. The model presented in this paper is able to represent the dynamic behavior of pile foundations and of structures founded on piles for fluid saturated porous soils as well as for viscoelastic soils. This approach is related to that presented by Maeso et al. [19] and represents a step forward as compared to that work in two significant aspects: (1) it includes a BE–FE coupling that allows for the representation of the whole structural system, i.e., not only the piles but any superstructure founded on them also and (2) the simplified elements used in this paper lead to a very important reduction of the number of unknowns of the model without lost of accuracy. The present model includes a simplified boundary element–finite element representation of piles. It is an extension to time harmonic problems and to viscoelastic and poroelastic soils of a similar model presented in Refs. [13–15] for static and time domain analysis of piles in purely elastic soils.

2. Simplified boundary element–finite element model A three-dimensional coupled boundary element–finite element (BEM–FEM) model for the dynamic analysis of piles is presented next. The model combines three-dimensional-beam finite elements for the piles, and cylindrical boundary elements for the soil. The use of finite elements allows for an easy representation of the behavior of piles as Timoshenko-type beams with 6 degree of freedom per node. The use of BEM elements for the soil has the advantage of requiring only the discretization of the pile surface and, since a full-space fundamental solution is used, of the half-space surface near to the pile or group of piles. Cylindrical boundary elements are used to represent the interface between concrete and soil. Nodes of these cylindrical elements are located at their axis. Nodal variables represent displacement and traction values over the cylindrical surface. They are assumed to have a linear variation along the z-axis and to be constant over the circular coordinate y. This traction distribution is represented in Fig. 1. The pile–soil interface is represented by a row of cylindrical elements, adding a circular element at the bottom for the lower contact surface. This lower surface for the pile allows the model to account for the traction developed at this point. Those tractions may be important for piles having the bottom area in a stiffer soil than the overlying soil. At the top, the circular element does not

Fig. 2. Pile mesh without simplification of the pile–ground surface connection.

exist since the cylindrical boundary elements represent the hollow pile–soil interface and not the surface of the concrete pile. However, connection between cylindrical element at the top of the pile and the soil surface elements require some attention. A BE representation of the cylindrical hole in the soil is shown in Fig. 2. The hollow surface represents the pile–soil interface. A beam is introduced at the axis to represent the concrete pile. In order to properly model the geometry of the connection between the hole cylindrical surface and the soil surface, several elements should be used, increasing significantly the model size (Fig. 2). To reduce this size, a simplification is introduced as presented in Fig. 3. The mesh corresponding to the soil surface is maintained continuous. The desired ‘‘hole’’ in the soil surface (which is a region in the surface where no integration is done) is obtained defining an upper cap element for the pile which is ‘‘subtracted’’ to the surface element of the soil defined at the same position (the integration over the soil surface element minus the integration over the upper cap element is equivalent to the integration over the real surface with the ‘‘hole’’) . Those elements are of different type (upper cap: constant, soil surface: quadratic) and an error is introduced using this assumption. However, this error is small whereas the model size is reduced significantly and the mesh definition is considerably simplified, as shown in Fig. 4. The collocation points in the BEM formulation are the element nodes as usual. Since they are placed along the element axis, they are external nodes to the cylindrical surface and, subsequently, the corresponding free term value is null. For the case of collocation at the node in the lower or upper circular element, the free term value is 0.5. The normal direction of the cylindrical elements points towards the interior of the pile. The connection conditions between boundary elements and finite elements along the pile are defined as follows: displacements and tractions are identical at nodes of the boundary elements and the finite elements. Since linear shape functions are used for these variables in both kinds of elements, equal displacements and tractions are enforced along the pile. Rotations at the finite element nodes are left free.

3. General BEM equation

Fig. 1. Traction distribution over cylindrical element.

Assuming no body forces, and separating the pile boundaries Gp from the rest of the boundaries, the BEM equation can be

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27

ground surface element

ground surface upper cap cylindrical elements pile

beam elements

lower cap Fig. 3. Simplified pile–ground surface connection. The upper cap is subtracted to obtain the actual ground surface geometry.

where Z ^ ij ¼ H t  F dG;

Gij ¼

Gj

Z

u F dG

^ ij for iaj; Hij ¼ H ^ ij þ ci for iaj Calling Hij ¼ H of equations NE X j¼1

Z ¼ Gp

Gp

t i;j uj dGp þ

ui;j t j dGp þ

Z

NE Z X j¼1

Z G

ui;j t j dG

(1)

t  F dGuj ¼

NE Z X

Gj

j¼1

u F dGt j

(2)

Gj

where F is the shape function matrix and ui , t i are vectors containing nodal displacements and tractions, respectively. This equation can be written in a more compact form as c i ui þ

NE X j¼1

(5)

4.1. Viscoelastic case

t i;j uj dG

G

where cki;j are the elements of the free-term matrix for collocation point k. ui;j and t i;j are displacement and traction components, respectively, of the fundamental solution matrix, and uj , t j are, respectively, displacement and traction components at boundary points. After discretization, Eq. (1) can be expressed as c i ui þ

Gij t j

j¼1

4. Element integration

written as Z

NE X

the usual system

is obtained. Eqs. (1)–(5) are valid for viscoelastic and poroelastic soils the only difference being that in the second case displacement and traction vectors are augmented to include as additional component, the fluid equivalent stress and the fluid normal displacement, respectively. Fundamental solution matrices are 3  3 in the viscoelastic case and 4  4 in the poroelastic case. Details of the BE formulation for dynamic poroelastic problems can be found in Dominguez [20–22], Maeso et al. [19] and Azna´rez et al. [23].

Fig. 4. BE mesh with simplified pile–ground surface connection.

cki;j uki þ

Hij uj ¼

(4)

Gj

^ ij uj ¼ H

NE X j¼1

Gij t j

(3)

In general, kernels of the fundamental solution integrals over a boundary element have a singular part and a non-singular part when the collocation point belongs to the integration element. In the case of the cylindrical elements used in this paper, collocation is always done at the pile axis and there is no singularity over the integration cylindrical surface in any case. Singularities only appear when integration is done over the circular elements at the lower and upper surface of the pile and collocation is done at its center point. Let us analyze in detail the different cases, considering only the integration over the elements around the pile and separating the cylindrical boundaries Gperim from the upper and lower circular boundaries (pile cap) Gcap : (1) Collocation node does not belong to the pile cap. There is no singularity in this case. Seven Gauss point have been chosen for the integration over the circular cap. For the integration over the cylindrical surface, an uniform distribution of the Gauss points along the circular coordinate y and a standard distribution along the longitudinal direction are assumed. The number of Gauss point is chosen depending on the distance from the integration

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28

element to the collocation point. Ten Gauss points are used along the circular coordinate and 20 along the longitudinal direction when the collocation point is closer than four times the pile radius to the integration element. Six and 10 Gauss points, respectively, are used otherwise. Z Z ^ ij ¼ H t  dGcap þ t  F dGperim Gcap

¼ pR2

NPG;cap X

Z

NPG;perim X

t  oi þ

t  oj FjJj

(6)

j¼1

u dGcap þ

Z

Gcap

¼ pR2

u F dGperim

Gperim

NPG;cap X

u  oi þ

i¼1

NPG;perim X

u oj FjJj

(7)

j¼1

^ and G are the BEM matrices, u and t  are displacement where H and traction fundamental solution matrices, respectively, R is the pile radius, jJj is the Jacobian of the coordinate transformation, F is the shape function matrix and, NPG and oi are the number of integration points and the weighting values of the Gauss numerical integration, respectively. (2) Collocation at circular pile cap element node. Consider the expressions for the fundamental solution displacement and traction 1 ¼  ½c di; j  w r ;i r ; j  4p m c22

(8)

    1 qr qr  A dij r ;i nj þ r ;i r ; j B þ r ;i nj C ¼ 4p qn qn

(9)

ui; j

t i; j

c22 c21

where the singularity has been removed by analytical integration. 4.2. Poroelastic case

1 2r

w¼

c2 1 1  22 2r c1

1þ

Expressions for the elements of those matrices in terms of auxiliary functions can be written as 1 ~ di; j  w~ r ;i r ; j   ½c 4pm

ioZ ~  Cr ;l 4p

tl ¼

(18)

(19)

þ Oc ðr 0 Þ

(10)

u0k ¼

þ Ow ðr 0 Þ

(11)

t0 ¼

(22)

(12)

  ~ ¼ 1 1þ m c þ Oc~ ðr 0 Þ 2r l þ 2m w~ ¼

  1 m 1 þ Ow~ ðr 0 Þ 2r l þ 2m

(23)

~ ¼ j

  1 m þ Oj~ ðr 0 Þ 2 l þ 2m

(24)

k~ ¼

1 þ Ok~ ðr 0 Þ r

(25)

!

dC w  dr r

  w dw B¼2 2  r dr

(13)



 l dC dw w w C¼  2 2 m dr dr r r

(14)

where m is the soil shear modulus and, c1 and c2 are the p and s wave velocity, respectively. dij is the Kronecker’s delta, Oc and Ow are non-singular functions [22,24]. Taking into account the symmetry relations existing over the pile cap circular surface when collocation point is its center point, it can be easily shown ^ terms are null and only the diagonal elements of the that matrix H G matrix are non-zero. Using a series expansion for the fundamental solution and some algebra, the diagonal elements of the G matrix become (" ! !# c2 c2 1 R Gð1; 1Þ ¼ Gð2; 2Þ ¼  p2 R 1 þ 22  1  22 4pm 2 c1 c1 " NPG;cap X Oc ðr 0 Þoi þ p2 R2 i¼1

p2 R2

As mentioned above, BE equations have the same form as in the viscoelastic case the only difference being that displacement and traction vectors as well as the fundamental solution matrices are augmented by a fourth dimension. Details can be found in [19–23]. The fundamental solution matrices are 0  1 u11 u12 u13 t1 B  C B u21 u22 u23 t2 C B C u ¼ B  C, B u31 u32 u33 t3 C @ A u01 u02 u03 t0 0  1 t 11 t 12 t 13 U n1 B  C B t 21 t 22 t 23 U n2 C B C p ¼ B  (17) C B t 31 t 32 t 33 U n3 C @ A t 01 t 02 t 03 U n0

ui; j ¼

!

c¼

A¼

(16)

Gperim

i¼1

Gij ¼

" ! # NPG;cap X c22 1 2 0 Gð3; 3Þ ¼ pR 1 þ 2 þ pR Oc ðr Þoi 4pm c1 i¼1

NPG;cap X i¼1

#) rOw ðr 0 Þoi

(15)

g ~  Cr ;k 4p

1  k~ 4p

(20)

(21)

and t i; j ¼

    1 qr qr ~ ~ n þ Cr ~ nÞ  A~ dij r ;i nj þ r ;i r ; j B þ ðAr ;i j ;i j 4p qn qn

(26)

t 0k ¼

  g qr ~ ~ Fr ;k þ Gn k 4p qn

(27)

U nl ¼

  1 qr ~ ~ l Dr ;l þ En 4p qn

(28)

^  ¼ U   JX 0 nl ¼ 1 qr H ~ U n0 n0 l 4p qn

(29)

  1 m þ OA~ ðr 0 Þ A~ ¼  2 r l þ 2m

(30)

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  3 m þ OB~ ðr 0 Þ B~ ¼  2 1  l þ 2m r

(31)

  1 m þ OC~ ðr 0 Þ C~ ¼ 2 1  l þ 2m r

(32)

     1 l þ 2m ~ ¼ 1 Z 1 þ ioZJ þ OD~ ðr 0 Þ D 2r l þ 2m m

29

The G matrix has, in this case, all its non-diagonal terms equal to zero. The diagonal terms have the following expressions: (" ! !# c2 c2 1 R Gð1; 1Þ ¼ Gð2; 2Þ ¼  p2 R 1 þ 22  1  22 4pm 2 c1 c1 " NPG;cap X þ p2 R2 Oc ðr 0 Þoi i¼1

(33)

p2 R2

NPG;cap X

#)

rOw ðr 0 Þoi

(41)

i¼1

     1 1 l þ 2m Z 1þ þ ioZJ þ OE~ ðr 0 Þ E~ ¼  2r l þ 2m m

(34)

  1 m þ OF~ ðr 0 Þ F~ ¼  r l þ 2m

(35)

     m l Q Q ~ ¼1 1þ þ 2 þ 1 þ OG~ ðr 0 Þ G r l þ 2m m Rg Rg

(36)

~ ¼ J 1 þ O ~ ðr 0 Þ H H r2

(37)

where g is the fluid unit weight, X 0l is the fluid body force along the l direction, Z and J depend on the material properties and OA and others are non-singular functions [24]. Due to the geometry of the problem at hand, except for a few cases, the integration element does not include the collocation point and, therefore, there are not singularities in the fundamental solution kernels. Numerical integration is done using the same scheme as for the viscoelastic case. A simple Gauss numerical quadrature is applied to obtain expressions for the G and H matrices terms as shown in Eqs. (6), (7). Only in the case when the integration element is the circular element on the pile upper or lower surface and collocation point is its center point, there are singularities in the kernels. In this case, ^ matrix has all its terms, because of the symmetry relations, the H except one, equal to zero. It can be written as 0

0 B0 ^ ¼B H B @0

0

0

0

0

0

0

0

0

0

0

1

0 C C ^ 34 C A H

(38)

0

Z

g ~ 3 dGcap ^  Gn Hð3; 4Þ ¼ 4p Gcap Z R   g 2l  2m Q 1 þ 2pr dr ¼ 4p 0 2ðl þ 2mÞ Rg r  Z ~ dGcap þ OðGÞ cap Z gRðl  mÞ ~ dGtapa þ OðGÞ ¼ 2ðl þ 2mÞ cap

(39)

The free term matrix is diagonal and has the following form: 0:5

B 0 B B ci ¼ B 0 B @ 0

0

0

0

0:5

0

0

0

0:5

0

0

1 1  2 iob  o2 r22

0

" ! # NPG;cap X c2 1 pR 1 þ 22 þ pR2 Oc ðr 0 Þoi 4pm c1 i¼1

Z 1  k~ dGcap 4p Gcap Z R  Z 1 1 2pr dr þ ¼ Ok ðr 0 Þ dGcap 4p 0 r cap Z R 0 Ok ðr Þ dGcap ¼ þ 2 cap

(42)

Gð4; 4Þ ¼

(43)

5. Coupled system formulation 5.1. Finite element equations A structure (including piles) is discretized into three-dimensional Timoshenko beam-type finite elements with distributed mass. The material is linear elastic with internal damping of hysteretic type. Dynamics equations of the motion are formulated in the frequency domain. The equilibrium system of equation can be written as 0 10 1 0 1 Sss Ssb Spb us Ps BS CB C B C (44) @ bs Sbb Sbp A@ ub A ¼ @ Pb A Sps Spb Spp up Pp where s denotes non-connected superstructure nodes, b connected nodes (pile–structure interface), and p denotes nodes along the pile. Notation for Eq. (44) and the following can be seen in Fig. 5. 5.2. Boundary element equations

where

0

Gð3; 3Þ ¼

1 C C C C C A

(40)

The BE model explained in Section 2 is used to represent the soil boundary, i.e., the soil surface and the pile–soil interface. Three-dimensional quadratic boundary elements are used to model the soil surface and the simplified two-node cylindrical boundary element for the soil surface in touch with the piles. Nodes of these cylindrical elements are located at their axis. Nodal variables represent displacement and traction values over the cylindrical surface. They have a linear variation along the z-axis and are constant along the circular coordinate y. The classical BE system of equations can be written as 0 1 0 1 ucp t cp Bu C Bt C B p C B p C C B C ðHcp Hp Hg Hgp ÞB (45) B ug C ¼ ðGcp Gp Gg Ggp ÞB t g C @ A @ A ugp t gp where u and t are displacement and traction values, respectively, and H and G the BEM coefficient matrices. The subscripts g and gp denote the ground surface nodes non-connected and connected to

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30

Non-connected nodes of the structure: us

Connected nodes at pile cap: • base of colum ub • head of piles ucp • Coincident surface nodes ucp

Non-connected nodes of BEM surface: ug

Reference node of pile cap uRef Pile nodes up: • BEM nodes • FEM nodes

Fig. 5. Schematic representation of nodes included in the coupled structure–pile–soil model.

z θz

θy x θx

y L

d (diametro)

s

Fig. 6. Schematic representation of a 2  2 pile group. Fig. 7. Boundary element model of the soil and the 2  2 pile group using XZ and YZ symmetries.

the pile head, respectively, cp and p denote the pile head nodes and the nodes along the pile, respectively. Using compatibility conditions at the pile cap, and adding rows of cp and gp displacements, the boundary element system of equations becomes: 0

ððHcp þ Hgp Þ  D Hp Hg  Gcp

Ggp t gp ¼

Gg t g

1 uref B C B up C B C B C u C  Gp ÞB B g C B C B t cp C @ A tp

It consists of unit matrices when each pile is connected to the structure independently.

5.3. Coupled system of equations

! (46)

where matrix D relates head pile head displacements with displacements at the center of stiffness of a pile group (Fig. 5).

The BEM–FEM coupling is carried out by applying compatibility and equilibrium conditions at the nodes representing the soil–pile interface. The total load amplitudes acting on the nodes, PT , correspond to the summation of the interaction forces between soil and pile, P, and the directly applied forces on nodes of the structure ðPs Þ or pile ðP p0 Þ. The interaction forces between soil and pile along an impervious viscoelastic–poroelastic (pile–soil) interface are P ¼ MFpil:ðt þ tÞ ¼ MFpil:ðt þ f:pÞ

(47)

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31

Single pile - Horizontal stiffness 2.4 simplified model

1.6

Cxx/Kxx (a0 = 0)

Kxx/Kxx (a0 = 0)

2.0

Kxx Kaynia 1.2 0.8 0.4

2.0 1.6 1.2 0.8

simplified model Cxx Kaynia

0.4

0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

a0

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

a0 Single pile - Vertical stiffness 4.5

2.0

Cxx/Kxx (a0 = 0)

Kxx/Kxx (a0 = 0)

4.0 1.6 1.2 0.8 simplified model 0.4 Kzz Kaynia

3.5 3.0 2.5 2.0 1.5

simplified model Czz Kaynia

1.0 0.5

0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

a0

a0 Fig. 8. Single pile stiffnesses in viscoelastic soil.

2x2 pile group - Horizontal stiffness 3.2

1.6

Cxx/4Kxx (a0 = 0)

Kxx/4Kxx (a0 = 0)

2.0

1.2 0.8 0.4

simplified model Kxx Kaynia

2.8 2.4 2.0 1.6 1.2 0.8

simplified model Cxx Kaynia

0.4

0.0

0.0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4 a0

a0 2x2 pile group - Vertical stiffness 10.0 simplified model Kzz Kaynia

4.0

Cxx/4Kxx (a0 = 0)

Kxx/4Kxx (a0 = 0)

5.0

3.0 2.0 1.0 0.0

simplified model Czz Kaynia

8.0 6.0 4.0 2.0 0.0

0

0.2

0.4

0.6

0.8

1

0

0.2

a0

0.4 a0

Fig. 9. The 2  2 pile group stiffnesses in viscoelastic soil.

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Single pile - Horizontal stiffness 2.4

1.6

Cxx/Kxx(a0=0)

Kxx/Kxx (a0 = 0)

2.0

1.2 0.8 modelo simplif Kxx Maeso

0.4

2.0 1.6 1.2 0.8

modelo simplif Cxx Maeso

0.4

0.0

0.0 0

0.2

0.6

0.4

1

0.8

0

0.2

0.6

0.4

0.8

1

a0

a0

5.0

1.6

4.0

Cxx/Kxx(a0=0)

Kxx/Kxx (a0 = 0)

Single pile - Vertical stiffness 2.0

1.2 0.8 simplified model Kzz Maeso

0.4

3.0 2.0 simplified model Czz Maeso

1.0

0.0

0.0 0

0.2

0.6

0.4

1

0.8

0

0.2

0.4

0.6

0.8

1

0.8

1

a0

a0 Fig. 10. Single pile stiffnesses in poroelastic soil.

2x2 pile group - Horizontal stiffness 3.0 simplified model Kxx Maeso

1.6

Cxx/4Kxx (a0 = 0)

Kxx/4Kxx (a0 = 0)

2.0

1.2 0.8 0.4

2.5 2.0 1.5 1.0 simplified model Cxx Maeso

0.5

0.0

0.0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

a0

0.6 a0

2x2 pile group - Vertical stiffness 8.0 simplified model Kzz Maeso

3.0

Cxx/4Kxx (a0 = 0)

Kxx/4Kxx (a0 = 0)

4.0

2.0 1.0

7.0 6.0 5.0 4.0 3.0 2.0

simplified model Czz Maeso

1.0 0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

a0

0.4

0.6 a0

Fig. 11. The 2  2 pile group stiffnesses in poroelastic soil.

0.8

1

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where the term ðt þ tÞ corresponds to total traction on the poroelastic medium, t denotes traction on the solid boundary and t traction on the fluid boundary, p denotes pore pressure and f porosity, and MFpil the transformation matrix between forces and tractions. Considering the nodes at the pile cap and the intermediate nodes of the pile separately, a global relationship between FEM forces and BEM tractions at the soil interface can be written as

P ¼ MFpil:t ¼

!

MFpilcp

0

0

MFpilp

t cp

! (48)

tp

33

6. Numerical studies 6.1. Dynamic stiffnesses of piles and pile groups in viscoelastic soil In order to validate the proposed model, the dynamic stiffness coefficient of single piles and a 2  2 group of piles are calculated and compared with results previously published by Kaynia and Kausel [6]. The model definition of the 2  2 pile group and the corresponding BEM mesh is presented in Figs. 6 and 7, respectively. Piles are 6 m long and they are assumed to be linearly elastic cylindrical members with elasticity modulus Ep , mass

where P denotes the force on the nodes of the pile and t cp and t p denote the tractions over the head and along the pile, respectively. In the poroelastic case, the vector in Eq. (48) stands for total tractions. Using the previous relations, a set of coupled finite and boundary element equations can be written as follows: 0

Sss Ssb D Spb B T T B D Sbs D Sbb D DT Sbp B B B Sps Spb D Spp B @ 0 ðHcp þ Hgp Þ  D Horl p 0 1 us 0 1 Ps B C B uref C B T B C BD P C b0 C B C C B up C B C B C B B P B t C ¼ B p0 C C B ug C B B C BG t C B C @ gp gp C A B t ref C @ A Gg t g tp

0

0

0

DT MFpilb

0

0

Hg

Gcp

1 C C C C MFpilp C C A Gp 0

(49)

where matrix Hp have been completed to Horl p with columns of zeros corresponding to rotation degrees of freedom in uref .

z

d1, d2, d3

y

x

L

s

d (diameter)

Fig. 12. Schematic representation of two 2  2 pile groups.

Fig. 13. Vertical displacement ðuz Þ, longitudinal displacement along the line connecting pile group centers ðuy Þ and rocking ðwz Þ at the center of the 1 and 2 groups when a vertical load is applied at the center of pile group 1.

ARTICLE IN PRESS 34

´n, J. Domı´nguez / Engineering Analysis with Boundary Elements 33 (2009) 25–34 M.A. Milla

density rp, Poisson’s ratio np , diameter d, and length L. The soil medium is a uniform viscoelastic half-space characterized by its elasticity modulus Es , mass density rs, Poisson’s ratio ns , and hysteretic damping ratio bs . Parameters of the problem are rp =rs ¼ 3=2, np ¼ 0:25, ns ¼ 0:33, Ep =Es ¼ 1000, L=d ¼ 20, and b ¼ 0:05. In the case of a 2  2 pile group, the separation between piles is s ¼ 5d. Results for the single pile and for the pile group are shown in Figs. 8 and 9, respectively. The stiffnes components are normalized by the corresponding single-pile static stiffness (four times the single-pile stiffness in the 2  2 pile group case). Results are presented versus excitationpfrequency a0 ¼ od=cs, where the ffiffiffiffiffiffiffiffi shear wave velocity is cs ¼ m=r. A very good agreement with the results published by Kaynia and Kausel [6,7] can be observed for all cases. 6.2. Dynamic stiffnesses of piles and pile groups in poroelastic soil For the case of piles in a poroelastic half-space, the properties considered for the soil are m ¼ 3:2175e7 N=m2 , ns ¼ 0:25, rs ¼ 1425 kg=m3 , rf ¼ 1000 kg=m3 , ra ¼ 0, f ¼ 0:35, b ¼ 3:0e7 N=m2 , Q ¼ 4:61e8 N=m2 , R ¼ 2:4823e8 N=m2 and for the piles Ep =Edrained ¼ 343, Rp =rbulkmaterial ¼ 1:94, np ¼ 0:20. Impervious contact between piles and soil has been considered. The stiffness components are normalized by the corresponding static stiffness component for an equivalent drained viscoelastic medium obtained considering the same m and r values as in the poroelastic case (a viscoelastic medium with the same shear wave velocity cs ). Results for single pile and for the pile group are shown in Figs. 10 and 11, respectively. They are compared with those presented by Maeso et al. [19], using a BE model with many more degrees of freedom. Good agreement is observed between both sets of results. 6.3. Dynamic interaction between pile groups The dynamic interaction between two close pile groups under harmonic vertical load acting on one of the groups is analyzed. A schematic representation of the problem is shown in Fig. 12. Three different distances between centers of pile groups (d1 ¼ 4 m, d2 ¼ 6 m, and d3 ¼ 8 m) have been considered in order to analyze its influence in the results. The same geometrical data and mechanical properties as in previous viscoelastic analysis have been chosen. Results for vertical displacement ðuz Þ, longitudinal displacement along the line connecting pile group centers ðuy Þ and rocking ðwz Þ at the center of the 1 and 2 groups are shown in Fig. 13. The vertical displacement at the 2nd group is around 10–20% of the vertical displacement at the 1st group, decreasing as distance increases. The longitudinal displacement due to the 1st group excitation produces in the 2nd group displacement of the same order of magnitude as vertical displacement in the 1st group, showing the importance of the interaction. Similar result are also obtained for the rocking rotation of the 2nd group pile cap.

7. Conclusions In the present paper a simplified BEM/FEM formulation for the dynamic analysis of structures based on piles and pile groups is presented. One of the goals of this work is the formulation of a cylindrical boundary element to represent the interface between pile and soil for viscoelastic and poroelastic soils. Elements of this type are connected to beam-type finite elements representing the

pile. This coupled FEM–BEM model allows to properly represent the dynamic behavior of piles and groups of piles in a viscoelastic or poroelastic half-space. Since the formulation is general, this pile foundation can be connected to any superstructure (like buildings of bridges) in order to analyze its dynamic behavior accounting for the soil–structure interaction. The obtained dynamic stiffness coefficients are compared with existing results presenting a very good agreement and showing the ability of the model to properly represent the dynamic behavior of structures supported on piles with a relatively low computational cost.

Acknowledgment This work is partially supported by the Comisio´n Interministerial de Ciencia y Tecnologı´a of Spain under research project DPI 2001-2377-CO2-01. References [1] Penzien J, Scheffy C, Parmelee R. Seismic analysis of bridges on long piles. J Eng Mech 1968;90:223–54. [2] Wolf JP, von Arx GA. Impedance function of a group of vertical piles. In: Proceedings of the Specialty Conference on Soil Dynamic Earthquake Engineering, ASCE, Pasadena, CA; 1978. p. 1024–41. [3] Kuhlemeyer RL. Vertical vibration of piles. ASCE J Geotech Eng ASCE 1979;105(2):273–87. [4] Kuhlemeyer RL. Static and dynamic laterally loaded floating piles. ASCE J Geotech Eng ASCE 1979;105(2):289–304. [5] Waas G, Hartmann HG. Analysis of pile foundations under dynamic loads. In: Proceedings of structural mechanics in reactor technology conference, Paris, France; 1981. [6] Kaynia AM, Kausel E. Dynamic stiffness and seismic response of pile groups. Research report R83-03, Massachusetts Institute of Technology Cambridge, MA; 1982. [7] Kaynia AM, Kausel E. Dynamic of piles and pile groups in layered soil media. Soil Dyn Earthquake Eng 1991;10(8):386–401. [8] Kobayashi S, Nishimura N. Green’s tensors for elastic half-spaces em dash an application of boundary integral equation method. Mem Fac Eng Kyoto Univ 1980;42(2):228–41. [9] Sen R, Davies TG, Banerjee PK. Dynamic analysis of piles and pile groups in homogeneus soil. Earthquake Eng Struct Dyn 1985;13:53–65. [10] Mamoon SM, Banerjee PK. Response of piles and pile groups to travelling SH-waves. Earthquake Eng Struct Dyn 1990;19:597–610. [11] Mamoon SM, Ahmad S. Seismic response of piles to obliquely incident SH, SV, and P waves. J Geotech Eng ASCE 1990;116(2):186–204. [12] Ferro NCP, Venturini WS. BEM–FEM coupling for building structure analysis. In: Brebbia CA, Domı´nguez J, Parı´s F, editors. Boundary elements XIV, vol. 2; 1992. p. 451–65. [13] Coda HB, Venturini WS. Three dimensional transient BEM analysis. Comput Struct 1995;56(5):751–68. [14] Coda HB, Venturini WS, Aliabadi MH. A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis. Int J Numer Methods Eng 1999;46:695–712. [15] Coda HB, Venturini WS. On the coupling of 3D BEM and FEM frame model applied to elastodynamic analysis. Int J Solids Struct 1999;36:4789–804. [16] Zeng X, Rajapakse RKND. Dynamic axial load transfer to a half-space from a partially embedded axially loaded rod. J Eng Mech ASCE 1995;125(9): 1048–55. [17] Jin B, Zhou D, Zhong Z. Lateral dynamic compliance of pile embedded in poroelastic half space. Soil Dyn Earthquake Eng 2001;21:519–25. [18] Wang JH, Zhou XL, Lu JF. Dynamic response of pile groups embedded in a poroelastic half-space. Soil Dyn Earthquake Eng 2003;6(3):235–42. [19] Maeso O, Azna´rez JJ, Garcı´a F. Dynamics impedances of piles and groups of piles on saturated soils. Comput Struct 2005;83(10–11):769–82. [20] Dominguez J. An integral formulation for dynamic poroelasticity. J Appl Mech ASME 1991;58:588–91. [21] Dominguez J. Boundary element approach for dynamic poroelasticity. Int J Numer Methods Eng 1992;35:307–24. [22] Dominguez J. Boundary elements in dynamics. Southampton, London: Computational Mechanics Publications, Elsevier Applied Science; 1993. [23] Azna´rez JJ, Maeso O, Dominguez J. BE analysis of bottom sediments in dynamic fluid–structure interaction problems. Eng Anal Boundary Elem 2006;30:124–36. [24] Azna´rez JJ. Efecto de los feno´menos de interaccio´n incluyendo factores espaciales y sedimentos de fondo en la respuesta sı´smica de presas bo´veda. PhD thesis, Universidad de Las Palmas de Gran Canaria, Spain; 2002 [in Spanish].