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Dynamic impedances of piles and groups of piles in saturated soils
Computers and Structures 83 (2005) 769–782 www.elsevier.com/locate/compstruc
Dynamic impedances of piles and groups of piles in saturated soils Orlando Maeso *, Juan J. Azna´rez, Fidel Garcı´a Instituto Universitario de Sistemas Inteligentes y Aplicaciones Nume´ricas en Ingenierı´a (IUSIANI), Universidad de Las Palmas de Gran Canaria, Edificio Central del Parque Cientı´fico y Tecnolo´gico del Campus Universitario de Tafira, 35017 Las Palmas de Gran Canaria, Spain Accepted 21 October 2004 Available online 5 January 2005
Abstract A three-dimensional boundary element model is presented for the computation of time-harmonic dynamic stiffness coefficients of piles and pile groups embedded in two-phase poroelastic soils. Piles are modelled as continuum elastic solids and soil as a fluid-filled poroelastic half-space governed by BiotÕs theory. Piles are bonded to the surrounding medium along the contact surface where rigorous pile–soil interaction conditions are imposed. The technique has been applied to the calculation of vertical and horizontal dynamic impedance functions of piles and 2 · 2 pile groups. Selected numerical results are presented to portray the influence of excitation frequency, pile stiffness, porous medium properties and kind of pile–soil contact conditions. 2004 Elsevier Ltd. All rights reserved. Keywords: Poroelastic soil; Piles; Pile-groups; Pile–soil interaction; Dynamic impedance; Boundary Elements
1. Introduction The load–displacement response of piles and pile groups embedded in a uniform or layered half-space and subjected to harmonic loads has received considerable attention during the past few decades. Quite a number of papers have appeared that address the problem using both computational [1–6], rigorous [7–9] and simplified analytical [10,11] techniques. A good compilation of used techniques is the one presented by Novak [12]. In all these cases the soil has been assumed to be an elastic or viscoelastic solid medium.
* Corresponding author. Tel.: +34 928 451920; fax: +34 928 451879. E-mail address: [email protected] (O. Maeso).
Little attention has been paid to the analysis of piles in soils saturated by ground water and such problems have been usually studied by treating the soil as an equivalent single-phase solid having a PoissonÕs ratio close to 0.5. Very few studies exist whereby the twophase character of these soils has been taken into account according to BiotÕs theory [13,14] (including the compressibility of both phases, the dissipation due to the fluid viscosity and the coupling between the solid and the fluid deformation and stresses) and most of them contain simplifications that limit their applicability. Zeng and Rajapakse [15] presented the first dynamic studies of a pile in a poroelastic medium. These authors extended the classical elastostatic Muki and Sternberg formulation [16] to analyze steady-state dynamic response of an axially loaded cylindrical elastic bar partially embedded in a homogeneous poroelastic
0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.10.015
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half-space. The pile is treated as a one-dimensional structure and the interaction between the pile and the poroelastic medium is formulated in terms of a Fredholm integral equation of the second kind. In a similar way Jin et al. [17] extended the method proposed by Pak and Jennings [7] to study the dynamic lateral response of a single pile. According to the writersÕ knowledge, Wang et al. [18] have presented recently the only studies corresponding to pile groups embedded in a poroelastic continuum reported in the literatures. These authors analysed the response of pile groups under vertical loading, using Muki and Sternberg method for a single pile, and introducing a simplified complex interaction factor approach and a superposition method to take into account approximately the pile–soil–pile interaction effects. Boundary Element Method (BEM) has shown to be a good strategy for dynamic soil–structure interaction problems. A complete review of boundary element (BE) applications for dynamic problems can be found in Beskos [19]. In the present paper, a three-dimensional boundary element (BE) approach is used to obtain dynamic impedances of piles and pile groups embedded in two-phase poroelastic soils. The technique is based on the BE formulation for poroelastic media obtained by Domı´nguez [20,21] and Cheng et al. [22] after developing an integral equation formulation from BiotÕs differential equations. Piles are modelled as continuum elastic or viscoelastic solids and surrounding medium as a fluid-filled poroelastic one. BEM is used both for piles and soil. This technique is an extension of the two-dimensional BE model presented by Japo´n et al. [23] for the computation of dynamic stiffness of strip foundations resting in poroelastic soils. In comparison with the existing procedures already commented [15,17,18] the technique proposed here requires a bigger computational effort but this is compensated by an invariably superior versatility, generality and accuracy. Any general foundation geometry can be reproduced, including unequal or inclined piles with their actual cross-section. The contact condition between pile and soil may be pervious or impervious. Although the results presented are restricted to a half-space, the technique is very versatile and more complicated underground geometries (including poroelastic and viscoelastic zones) only require a different BE mesh for the soil region. Selected numerical results for vertical and horizontal impedances are presented. The influences of the frequency of excitation, bar flexibility and poroelastic material properties on the response are examined.
2. Governing equations and BE-model The model used for the dynamic analysis of piles and pile groups should be able to represent the dynamic
behaviour of elastic or viscoelastic regions (piles), poroelastic regions (soil) and the interaction between them at soil–pile interface. The model should accommodate unbounded regions using a reasonable number of unknowns and represent radiation damping properly. The soil is assumed to be an isotropic homogeneous fluid-filled poroelastic material governed by BiotÕs theory [13]. The constitutive equations are: Q2 sij ¼ k þ edij þ 2leij þ Qedij ð1aÞ R s ¼ Qe þ Re
ð1bÞ
where sij are the solid skeleton stress components; s is the fluid equivalent stress = /p (p = pore pressure); / the porosity; eij are solid skeleton strain components=1/2(ui,j + uj,i); dij is the Kronecker delta function; e = $u and e = $U are the solid and fluid dilatation, respectively; u is the displacement of the solid; U is the displacement of the pore fluid; k, l are Lame constants for the drained solid skeleton; and Q, R are Biot constants. The governing equations for a time harmonic excitation of the type eixt (x = angular frequency) are obtained from the equilibrium equations, which can be written [20] in terms of four variables, namely the solid displacement components and the fluid stress, as: ^ Q q lr2 u þ ðk þ lÞre þ 12 rs ^22 R q ^212 ^11 q ^22 þ q ^ q q x2 u þ X 12 X0 ¼ 0 ð2aÞ þ ^22 ^22 q q r2 s þ x 2
^22 Q q ^12 q ^22 e þ rX0 ¼ 0 s þ x2 q R R
ð2bÞ
where the dissipation constants have been included as part of complex valued densities in order to simplify the equations ^ 11 ¼ q11 i q
b ; x
^22 ¼ q22 i q
b ; x
^12 ¼ q12 þ i q
b x ð3Þ
0
X and X are body forces in the solid and fluid phase, respectively; q11 = (1 /)qs + qa; q22 = /qf + qa; q12 = qa; qs and qf are solid and fluid phase2 densities, respectively; qa is the added density; b ¼ cf k/ is the dissipation constant; where k (m/s) is the permeability of the poroelastic medium (depending on the fluid viscosity and the intrinsic permeability of skeleton) and cf the specific weight of fluid phase. Internal damping of the solid skeleton can be introduced using complex valued Lame constants of the type l = Re[l](1 + 2in); where n is the damping coefficient. By substitution of plane wave expressions for u and s into equation 2 for zero body forces, a characteristic equation for the wave numbers is obtained. Three
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
solutions of that equation exist corresponding to three kinds of time harmonic plane waves. One is a shear wave transmitted through the solid skeleton. The other two are dilatational waves (P1 and P2). All wave velocities are complex and frequency dependent; i.e. they are dissipative and dispersive. The solid and the fluid dilatation are in phase for the long longitudinal waves (P1) and they are in opposite phase for the short waves (P2), which damps out at short distances from the perturbation. An integral formulation of governing equations (2) can be obtained. Weighting equations (2a) in each direction by a displacement function ui and (2b) by a stress function s*, adding the four resultant equations and integrating in the domain X we can write in a condensed form: Z ½ðGi þ F i Þui þ ðH þ X 0i;i Þs dX ¼ 0 ð4Þ X
where F i ¼ X i qq^^1222 X 0i . An integral reciprocal relation including solid and fluid variables of two states defined in X with boundary C, can be obtained from (4): Z Z ti ui dC þ sðU n þ JX 0 i ni Þ dC C C Z þ ðF i ui þ JX 0i,i s Þ dX Z ZX ti ui dC þ s ðU n þ JX 0i ni Þ dC ¼ C C Z ð5Þ þ ðF i ui þ JX 0 i,i sÞ dX X
where ti = sijnj are the traction components on the solid phase, Un = Uini is the normal displacement of the fluid, and J ¼ x21q^22 . By using two fundamental solutions, one corresponding to a unit point load in the solid phase and the other to a unit point source in the fluid: X i ¼ dðx xk Þdji ; F i ¼ 0 X 0 X 0 i ¼ 0; i;i ¼ dðx xk Þ
ð6Þ
the integral representations of solid displacements and fluid stress for an internal point xk can be written [20] as: Z Z ukj þ tji ui dC U nj s dC ZC ZC ¼ uji ti dC sj U n dC ð7aÞ C
C
Z
Z J sk þ t0i ui dC ðU n0 þ JX 0 i ni Þs dC Z C Z C u0i ti dC s0 U n dC ¼ C
ð7bÞ
C
where uji and tji (i, j = 1, 2, 3) are solid skeleton displacements and stresses in direction i due to unit point load in the solid in direction j. For the same load, sj and U nj are
771
fluid equivalent stress and normal displacement of the fluid, respectively. The terms u0i , t0i , s0 and U n are the responses in both phases to a unit point source in the fluid phase. When xk 2 C Eqs. (7) become boundary integral equations, to be solved numerically by BEM. In a condensed form they are usually written as: Z Z ck uk þ p u dC ¼ u p dC ð8Þ C
C
where u and p are boundary variables vectors: 8 9 8 9 u1 > t1 > > > > > > > > > > = 2 2 u¼ p¼ > > t3 > u3 > > > > > > > > : > : ; ; s Un
ð9Þ
u* and p* are fundamental solution tensors: 2
u11
6 u 6 u ¼ 6 21 4 u31 u01
u12
u13
u22 u32
u23 u33
u02
u03
3
2
t11 6 7 s2 7 6 t21 7 p ¼6 6 t s3 5 4 31 s1
s0
t01
t12 t22
t13 t23
t32
t33
t02
t03
3 U n1 7 U n2 7 7 U n3 7 5 b U n0
ð10Þ
and ck is the local free the form: 2 e c11 ce12 ce13 6 ce ce ce 6 22 23 ck ¼ 6 21 4 ce31 ce32 ce33 0
0
0
term at collocation point xk with 0 0 0
3k 7 7 7 5
ð11Þ
Jcp
where ceij are the same as the free terms for the elastic static equations at collocation point xk, and cp is the same as the scalar static term at xk. The fundamental solution terms (obtained using the thermoelastic analogy and KupradzeÕs et al. [24] solution for that kind of problems) are given in the Appendix A. Different to other expressions in previous works, they are written in a similar way as that used for the time harmonic elastic and scalar fundamental solutions. Thus, the singularities at collocation point are clearly identified. All the singular terms of the fundamental solution tensor components are contained in their limit value for xr ! 0. These limit values can also be written as the corresponding elastic or scalar static fundamental solution plus some regular terms. This fact is important for boundary element implementation since all the algorithms for integration of the singular terms are the same as for the elastic and scalar static problems. All the additional regular terms are evaluated using standard quadratures. A boundary element discretization of Eq. (8) leads to a system of 4N equations: Hu ¼ Gp
ð12Þ
772
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where N is the number of nodes on the boundary; u is a vector containing solid displacement and fluid stress at boundary nodes; p is a vector containing solid traction and fluid normal displacement at boundary nodes; and H, G are 4N · 4N system matrices obtained by integration of the 3-D time harmonic poroelastic fundamental solution times the shape functions, over the boundary elements. Piles are modelled as a continuum isotropic homogeneous linear viscoelastic medium with their actual geometry [25]. The governing equations in the frequency domain are: lr2 u þ ðk þ lÞrru þ qx2 u þ X ¼ 0
ð13Þ
where u is the displacement vector amplitude, k and l are the Lame elastic constants, q is the mass density of the pile, and X is the body forces vector amplitude. The boundary integral representation of u is an equation analogous to (8), where in this case u and p are displacement and traction vectors, respectively, over the boundary of pile, and u* and p* are the fundamental solution displacement and traction tensors, respectively. A BE equation for each pile leads to a system of 3Np equations: Hp up ¼ Gp pp
Compatibility: upn ¼ un ¼ U n
ð16aÞ
upt ¼ ut
ð16bÞ
If the solid material is pervious, the condition on the normal displacement of the pore fluid must be substituted by a condition on the pore pressure: (15a) becomes tpn þ tn ¼ 0 and s = 0; (16a) becomes upn ¼ un ; and (15b) and (16b) do not change. In this study soil free surface is considered drained. For a single impervious pile embedded in poroelastic half-space the boundary element equation for each region (pile and soil) in partitioned form according with boundaries in Fig. 1 are: Pile Hp1 up1 þ Hp2 up2 ¼ Gp1 tp1 þ Gp2 tp2
ð17aÞ
Poroelastic soil ss ss H3 Hsw H2 Hsw u2 u3 3 2 þ ws ws ww ww H2 H 2 H3 H 3 s2 s3 ss ss sw G2 Gsw G G t t3 2 2 3 3 ¼ þ ws ws ww ww G2 G 2 G3 G 3 U n2 U n3 ð17bÞ
ð14Þ
obtained for each pile (denoted by super-index p), where Np is the number of nodes on the boundary; up and pp are vectors containing displacement and traction, respectively, at boundary nodes; and Hp, Gp are 3Np · 3Np system matrices obtained by integration of the 3-D time harmonic elastic fundamental solution times the shape functions, over the boundary elements. Eq. (12) is written for the poroelastic soil (nodes over free surface and piles boundaries) and Eq. (14) is written for each pile. The resulting equations are coupled together through equilibrium and compatibility conditions along piles–soil interfaces. The pile is assumed to be fully bonded, and the hydraulic boundary condition along the contact surface corresponds to an intermediate situation between two limiting cases: full drainage (pervious pile) and impermeable (impervious pile). The equilibrium and compatibility conditions between a two-phase poroelastic region and an impervious viscoelastic pile (denoted by super-index p) are the following:
where sub indexes 1–3 correspond to the boundary in contact with the pile cap where displacement nodal values are known (C1), soil–pile interface (C2), and soil free traction surface (C3), respectively. Substituting in Eq. (17) the compatibility and equilibrium conditions along the pile–soil interface (Eqs. (15) and (16)) and the
( u3 , Un3 ) t3 = 0 , τ3 = 0 ( 3) free boundary (Γ
u1 = u ( t1 ) pile cap (Γ ( 1)
poroelastic soil elastic pile
( u2 ,τ , 2 , t2 ) soil - pile interface
( 2) (Γ
Equilibrium: tpn þ ðtn þ sÞ ¼ 0
ð15aÞ
tpt þ tt ¼ 0
ð15bÞ
where n and t sub-indices stand for one normal and two tangential components, respectively.
Fig. 1. Single impervious pile embedded in poroelastic halfspace. Boundary conditions and unknown values at coupled equations of problem.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
external boundary conditions, the combined equations for the coupled dynamic problem in this case can be written as follows: 2
Gp1 6 4 0 0
Hss 2 Hws 2
Hp2 þ Gsw 2 n2
Gp2 n2 Hsw 2
Gp2 Gss 2
0 Hss 3
þ Gww 2 n2
H ww 2
Gws 2
Hws 3
8 9 p u1 > > < H1 = ¼ 0 > > : ; 0
8 > > > > 3> > 0 > > < sw 7 G3 5 > > > Gww > 3 > > > > :
773
Kzz Kθθ
Kxx
9 > > > > u2 > > > > s2 = > t2 > > > > u3 > > > ; U n3 tp1
L
y
x d s
ð18Þ
Fig. 2. 2 · 2 pile group embedded in a half space. Problem geometry definition.
where u2 (solid skeleton/pile displacement vectors), s2 (stress in fluid of poroelastic soil) and t2 (traction vectors in solid skeleton of poroelastic soil) are the unknown nodal values adopted at the pile–soil interface (C2) respectively, and n2 is the unit normal to the pile on this boundary. The equations for a pervious contact case are not shown for the sake of brevity. An advanced 3-D frequency domain BEM computer code in Fortran 90 has been developed. All the boundaries (piles and soil free surface) are discretized into a finite number of quadratic nine-node and six-node boundary elements. The code allows for choosing collocation points shifted to the interior of elements, where displacement and traction are expressed in terms of nodal and shape function values at collocation point. This partially ÔdiscontinuousÕ elements are used to accommodate edges, corners or discontinuities in the boundary conditions. Integrals containing singularities O(1/r) and O(1/r2) are evaluated in a direct form. No enclosing elements are needed in unbounded domains to carry out the so-called rigid body motion procedure. The code includes symmetry and antisymmetry capabilities, in order to reduce the problem size. All computations were run on a Linux PC computer equipped with one 1.33 GHz Athlone AMD processor and 1 GB of RAM memory. The dynamic pile–soil and pile–soil–pile interaction problems include soil as a semi-infinite region, where the radiation damping plays an important role. The semi-infinite soil and radiation damping are easily represented by BE-technique because the fundamental solutions satisfy radiation conditions [26], so the boundaries of soil at free surface are left open at a certain distance from the zone of interest.
the pile head and the resulting vector of displacements (and rotations) at the same point. For a group of piles, it is assumed that the pile heads are constrained by a rigid pile-cap, and the foundation stiffness is the addition of the contributions of each pile. Fig. 2 illustrates the approached problem for a usual configuration. A more complicated foundation, that may include unequal piles, inclined or variable section does not require any modification of the proposed technique and it only demands to modify the boundary element mesh. The dynamic stiffness terms for a time harmonic excitation are functions of frequency x and they are usually written as:
3. Validation of the model In order to validate the boundary element model, in this section, results of impedances of piles and groups of piles are contrasted with other reference values taken from the literature. The dynamic stiffness matrix Kij of a pile relates the vector of forces (and moments) applied at
K ij ¼ k ij þ ia0 cij
ð19Þ
where kij and cij are the frequency dependent dynamic stiffness and damping coefficients, respectively, a0 is the dimensionless frequency a0 ¼
xd cs
ð20Þ
and cs is the soil shear-wave velocity. Fig. 3a and b show the discretizations in boundary elements used to calculate the stiffness of a pile and a group of four piles, respectively, embedded in a viscoelastic or poroelastic half-space. The developed software incorporates symmetry properties. In the cases shown in Fig. 3, it is only necessary to discretize a quarter of the total geometry of the problem. Discretization criteria (size and shape of elements and length of free surface) have been studied in a previous work, Vinciprova et al. [27]. The terms of the dynamic stiffness matrix Kij are obtained by computing the resulting force and moment at the pile heads by integration of the tractions over piles, when unit displacements (horizontal, vertical or rotation) are imposed. It is assumed that the piles heads are situated at the free surface level. The sets of results selected in this section for validation purposes, include horizontal impedances of single piles and 2 · 2 pile groups embedded in a viscoelastic soil, and vertical impedances of single piles embedded in a poroelastic soil.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
Fig. 3. Boundary discretizations for single pile and 2 · 2 pile group in a half space.
3.1. Piles embedded in a viscoelastic half-space The lateral impedances (real and imaginary parts) of a single pile and a 2 · 2 pile group embedded in a homogeneous isotropic viscoelastic half-space obtained by proposed technique are shown in Fig. 4, and compared with those obtained by Kaynia and Kausel [2]. These authors carried out an analysis of single piles and pile groups considering piles as linear elastic prismatic members and soil as semi-infinite viscoelastic media by constructing the requisite GreenÕs function using a discrete layer matrix approach. The following properties are taken from Kaynia and Kausel [2]: piles (in the sequel denoted by sub-index p) are assumed to be uniform elastic continuum and surrounding soil a uniform viscoelas-
(a)
tic media with internal damping coefficient b = 0.05; the ratio between the material modulae is Ep/E = 103; ratio between densities q/qp = 0.7; and PoissonÕs ratios m = 0.4 (soil) and mp = 0.25 (pile). The piles aspect ratio is L/d = 15. The impedance functions for several spacing to diameter ratios s/d have been normalized with respect to the respective single pile static stiffness (Kxxo) times the number of piles in the group (N). All results are plotted against the dimensionless frequency parameter defined by Eq. (20). Fig. 4 shows that the computed values are in very good agreement with those presented in [2] for single piles and 2 · 2 pile groups for s/d = 5 and 10. For s/d = 2 the proposed method presents a slightly superior real part as the frequency increases. This discrepancy with [2] is also observed by Sen et al. [5] for
(b) 5
s/d = 10
5 single pile
2
2
Fig. 4. Horizontal impedances of single pile and 2 · 2 pile groups in a viscoelastic soil. Comparison with KayniaÕs solution.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
a group of 3 · 3 piles for the same distance between piles. It can be noted that the dynamic stiffness of a group of piles cannot be computed by simply adding the stiffnesses of the individual piles, since each one is affected not only by its own load, but also by the load and deflection of its neighbouring piles. This effect (known in the literatures as pile–soil–pile interaction effect) is strongly frequency-dependent, resulting from waves that are emitted from the periphery of each pile and propagated through the soil to the neighbouring piles.
3.2. Piles embedded in a fluid-filled poroelastic half-space The selected validation results correspond with the ones obtained by Zeng and Rajapakse [15] for the vertical impedance of a single pile embedded in a uniform poroelastic half-space with the procedure described before. The properties of two-phase poroelastic soil have been taken from [15]. They are detailed in terms of the non-dimensional parameters whose definition is shown in Appendix B: k* = 1.5, Q* = 2.87, R* = 2.83, qs ¼ 1:44, qf ¼ 0:53, qa ¼ 0 and two different values of the permeability that correspond to the values of the non-dimensional dissipation constant b* = 232.15 and b* = 0. The porosity is / = 0.482. The pile is an impervious elastic medium whose properties are defined by: flexibility ratio Ep/E = 103 (being E = drained YoungÕs modulus of poroelastic medium), mass density ratio qp/q = 1.2 (being q = density of bulk material) and PoissonÕs ratio mp = 0.3. The pile aspect ratio is L/d = 10. Fig. 5a and b show, respectively, the results obtained from the real and imaginary parts of the vertical stiffness, normalized with respect to the static stiffness value Kzzo for a drained elastic soil. The impedance is represented here in a way different from [15]. The non-dimensional fre-
(a)
775
quency a0 is defined by the Eq. (20) where cs = shear wave velocity of bulk material. It is noted a very good agreement with the results taken from [15]. It is also noticed that the stiffness (real part) and damping values (imaginary part) are bigger for the biggest b* (less pervious medium). This effect of the permeability of the soil on the dynamic impedance will be studied afterwards with greater depth.
4. Numerical analysis and results discussion In what follows, it is going to be presented results of dynamic impedances of piles and 2 · 2 pile groups embedded in a saturated poroelastic half-space whose properties have been taken from Kassir and Xu [28]. The normalized values of properties of this medium are: k* = 1, Q* = 14.33, R* = 7.72, qs ¼ 1:12, qf ¼ 0:78, qa ¼ 0 and b* = 59.30. The porosity is / = 0.35 and the internal damping coefficient of the skeleton is b = 0.05. For the piles, it is assumed viscoelastic behaviour with properties similar to concrete: flexibility ratio Ep/E = 343 (E=drained YoungÕs modulus of poroelastic medium), mass density ratio qp/q = 1.94 (q = density of bulk material), PoissonÕs ratio mp = 0.2 and zero internal damping coefficient. The piles aspect ratio is L/d = 15. Except when some other thing is explicit, impervious contact condition between piles and soil are considered. The influence of the frequency of excitation, pile flexibility, as well as the poroelastic effects relative to the medium permeability and to the pile–soil contact condition, on the dynamic impedances, is evaluated. 4.1. Influence of piles flexibility (single pile) The influence of pile flexibility ratio Ep/E on kxx and cxx is presented in terms of a0 in Fig. 6a and b,
(b)
Fig. 5. Vertical impedance of single pile in a poroelastic half space. Comparison with ZengÕs solution.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 6. Influence of piles flexibility on horizontal impedance of single piles.
respectively. Four values of pile flexibility ratio Ep/ E = 100, 200, 500, 1000 and two poroelastic soils with b* = 0 (permeability k ! 1) and 5.93 (10 times greater than in [28]) are used in this analysis. Some aspects of the problem should be pointed out in the light of Fig. 6: in the whole frequency range, growing stiffness and damping values are obtained as Ep/E increases. For b* = 5.93 the dynamic stiffness results (kxx ) show very little dependence with regard to the frequency of excitation a0 in all cases. Only at very low frequencies it is observed a sudden increase of the stiffness from the static value for media with high values of b*. In these problems the static stiffness does not depend on b* and it coincides with the one obtained for the completely drained soil. This effect will be observable in all the results that will be presented in this paper. For the limit value b* = 0 it is noted an important reduction of the impedance. It is observed a significative and progressive reduction of stiffness as the soil permeability and the frequency of excitation increase. This effect is more important as Ep/E increases. In the next section, where the influence of soil permeability is studied, this topic will mentioned again. With the previous four values of Ep/E, additional results of dynamic impedances have been obtained for L/ d = 5, 10 and 20. In the studied range and for all the values of Ep/E the piles with L/d = 10, 15 and 20 present almost identical horizontal dynamic response. It is only noticed a different behaviour for L/d = 5 and values of Ep/E = 100, 200 and 500. This insensitivity of the horizontal response above a certain slenderness value of the pile (La/d, La = active length of the pile) is an effect known in the bibliography. The active length depends on the pile–soil relative stiffness Ep/E and physically represents that, for piles with L La, the imposed deformation on the pile head is not felt in all its length but simply at a depth smaller than La.
4.2. Influence of soil permeability (single piles and pile groups) It has been mentioned above that the dissipation constant b (inversely proportional to permeability k) affects in a significative way the dynamic response: high values of b (clays) imply a greater difficulty in the fluid transit through the solid skeleton as compared to low values of b (loose sands). To study the influence of medium permeability on the dynamic stiffness, a brief analysis of its effects on the characteristics of the waves in the poroelastic soil is done first. Fig. 7 shows, for three values of non-dimensional frequency (a0 = 0.25, 0.5 and 0.75), the variation of the amplitude of S- and P2-wave propagation velocities in a wide range of the non-dimensional
Fig. 7. Wave propagation velocity amplitudes in the poroelastic soil vs. dissipation constant.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
constant b*. The modulus of the velocities has been normalized with the shear wave velocity of the elastic undrained medium cs In the same range of b* the P1-wave velocity (jcp1/csj 8) only experiences a small reduction of the order of 1% (not shown). The S-wave velocity presents a variation of the order of 20%. The short wave velocity (P2) presents the most important variation with b*. It can be seen in the figure that this velocity grows very fast for b* between 101 and 102; P2-wave velocity being bigger than S-wave velocity for values smaller (approximately) than 101. Little influence of the medium permeability can be expected for values of b* bigger than 59.3, which is the value in the considered soil. This study focuses on analyzing the influence of b* in the range of its more significant values. Three values for b* have been adopted: b* = 59.3, 59.3 · 102 and 59.3 · 104. Figs. 8 and 9 show, respectively, the horizontal and vertical impedances of a single pile for the three values of b*. Again, it is observed a reduction of the real (stiffness) and imaginary (damping) parts as b* decreases, that is, for progressively more pervious soils. The figures also show the impedances corresponding to the two ideal limit single-phase media: drained elastic soil and un-drained elastic soil. High values of b* bring the behavior of the two-phase medium near the undrained ideal elastic soil, except, as can be expected, at low frequencies. For the case of vertical impedance, the drained elastic medium marks the limit value towards which the two-phase medium tends as its permeability increases. The real part of the horizontal stiffness (Fig. 8a) falls below the value corresponding to the drained elastic medium when the soil permeability is very high, and this effect is growing with a0. This fact can be explained. For the case of vertical impedance, and for a very pervious soil, the dynamic axial load transfer from pile to poroelastic soil is carried out basically along the contact surface by means of shear tractions in the solid skeleton: in these conditions the pore
(a)
777
fluid hardly provides stiffness and the medium behavior is similar to that of a drained soil. However, in the case of horizontal impedance, the lateral deflection of the pile loads on the two phases on the ground, causing pressures and suctions zones of the pore pressure. This modifies the behavior as opposed to the drained soil. Figs. 10 and 11 show, respectively, the horizontal and vertical impedances of a group of 2 · 2 piles with separation s/d = 5. For s/d = 10 the results are presented in Figs. 12 and 13. It is observed that, for a pile group, the dynamic response is something more complex: the general tendencies of increasing values for the impedance with b* are maintained, but in this case wave reflection phenomena take place between piles in terms of their separation and the medium properties. This causes increases and decreases of the stiffness in certain frequency ranges. Because of that, higher impedance values may be found in media with smaller values of b*. Thus, one can see in Figs. 10a, 11a and 13a, higher stiffness maxima for b* = 59.3 · 104 than for b* = 59.3 · 102. This effect is combined with curves displacement towards the right side for increasing permeabilities. It is made possible by the increasing of the propagation velocities in the porous medium with permeability, and by the fact that in all cases the velocity used to normalize the frequency a0 in (20) is the one of the S-wave in a undrained medium (which is its lower limit). Finally, as it happens for a single pile, the real part of the horizontal stiffness of a very pervious poroelastic medium can be lower than the one of an elastic drained soil, as it is noticed in Fig. 12a. 4.3. Influence of hydraulic boundary condition along pile–soil interface (pile groups) In order to study this effect, the difference that experiences the impedance of a 2 · 2 piles group with separation s/d = 10, in two limit cases: completely impervious contact (same normal displacement in both medium
(b)
Fig. 8. Influence of soil permeability. Horizontal impedance. Single pile.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 9. Influence of soil permeability. Vertical impedance. Single pile.
(a)
(b)
Fig. 10. Influence of soil permeability. Horizontal impedance. 2 · 2 pile group; s/d = 5.
(a)
(b)
Fig. 11. Influence of soil permeability. Vertical impedance. 2 · 2 pile group; s/d = 5.
phases) and completely drained contact (s = 0), are analyzed. According to the conclusions of the previous subsection, it can be expected that the influence of the
hydraulic contact condition becomes more important as b* decreases. This fact is confirmed by the results. Fig. 14 shows the horizontal impedances obtained for
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
779
(b)
Fig. 12. Influence of soil permeability. Horizontal impedance. 2 · 2 pile group; s/d = 10.
(a)
(b)
Fig. 13. Influence of soil permeability. Vertical impedance. 2 · 2 pile group; s/d = 10.
(a)
(b)
Fig. 14. Influence of hydraulic boundary condition along pile–soil interface. Horizontal impedance of 2 · 2 pile group; s/d = 10; b* = 59.3.
the medium under study (b* = 59.3) for both boundary conditions. The obtained differences are negligible. The
influence is noticed when the soil is very pervious (b* = 59.3 · 104) as it is shown in Fig. 15. The real part
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 15. Influence of hydraulic boundary condition along pile–soil interface. Horizontal impedance of 2 · 2 pile group; s/d = 10; b* = 59.3 · 104.
of the impedance shows increasing differences with a0 from a0 = 0.4. It is also appreciated that the stiffness corresponding to the pervious contact is near the values of an ideal elastic drained soil. The explanation proposed before for the effect of loss of stiffness in the pervious saturated soil with respect to the elastic drained soil, caused by the presence of the fluid phase, remains reinforced in the light of these results. The damping of the foundation is insensible to the type of contact condition.
5. Conclusions In the present paper, a three-dimensional BE approach for the computation of time-harmonic dynamic stiffness coefficients of piles and pile groups embedded in two-phase poroelastic soils has been presented. Piles are modelled as continuum elastic or viscoelastic solids and surrounding medium as a fluid-filled poroelastic half-space. The study of more complicated geometries of the soil, including poroelastic and viscoelastic zones, can be conducted without difficulty. The technique has been applied to the calculation of vertical and horizontal dynamic impedance functions of cylindrical single piles and 2 · 2 pile groups. Piles are assumed to be fully bonded and pile–soil interaction effects are taken into account through equilibrium and compatibility conditions at interfaces. In the cases in which these exist, the model has been validated comparing with results from the bibliography. Impedance results have been presented and it has been studied the influence of aspects such as: frequency of excitation, pile stiffness, effects of pile–soil–pile interaction, permeability of the saturated soil and hydraulic condition of pile–soil contact. From the study carried out, we can extract the following conclusions:
In the saturated porous soil the foundation presents an increment of the stiffness from very low frequency values with respect to the elastic drained soil. This effect is less appreciated as constant b decreases in the soil. In the studied frequency range, the increase of pile/ soil relative stiffness implies an increase of the dynamic impedance, independently from the soil permeability. The influence on the dynamic behaviour of the dissipation constant b (dependent on the medium viscosity and the intrinsic permeability of the skeleton) is big because it affects noticeably the soil wave velocities. In general, increasing values of the dynamic impedances with b are obtained, tending towards the values corresponding to an ideal elastic un-drained soil. Very pervious porous soils can present horizontal stiffnesses lower than the ones of the ideal elastic drained soil. This effect is not noticed in the case of vertical stiffness. The dynamic impedance of a group of piles is more dependent on the frequency than the one of a single pile due to dynamic pile–soil–pile interaction effects. These effects depend on separation between piles and soil properties. The degree of permeability of the pile–soil contact condition only has an appreciable influence for very low values of b, and for the horizontal stiffness case. In conclusion, simulating the dynamic behaviour through a drained or un-drained single-phase model can lead, depending on the medium properties and the geometric configuration of the foundation, to unrealistic results. Any model to be used for the dynamic analysis of piles and groups of piles in poroelastic soils must include all the parameters of the material. The technique proposed in this paper does that, and permits a representation more general and versatile than other existent techniques.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
Acknowledgment This work was supported by the Ministerio de Ciencia y Tecnologı´a of Spain (DPI2001-2377-C02-02). The financial support is gratefully acknowledged.
Appendix A Fundamental solution for fluid-filled poroelastic material under dynamic loading Point load in solid skeleton in j direction:
781
with r = jx xkj, zm = ikm (m = 1, 2, 3) (km: wave number), z21 ¼ z22 z21 and Em ¼ 1r ezm r . The functions A, B, C, D, E, F, G, y H in terms of w, v, u, y j are: dw v dr r v dv B¼2 2 r dr
ðA:4aÞ
A¼
ðA:4bÞ
k dw dv v v Q 2 2 þ ixgu l dr dr r r R
C¼
ðA:4cÞ
uji
1 ðwdji vr;j r;i Þ ¼ 4pl
ðA:1aÞ
du u Z þ v D ¼ ixgJ dr r l
tji
1 or Adji þ Br;j r;i þ Ar;i nj þ Cr;j ni ¼ 4p on
ðA:1bÞ
E ¼ ixgJ
ðA:1cÞ
du u F ¼ 2l dr r
ðA:4fÞ
ðA:1dÞ
du u u Q þ2 2l þ j G¼k dr r r Rc
ðA:4gÞ
ixg ¼ ur;j 4p c or Dr;j þ Enj U nj ¼ 4p on sj
Point source in pore fluid: c ur;i 4p c or Fr;i þ Gni t0i ¼ 4p on u0i ¼
H ¼ J ðA:2aÞ
1 ¼ j 4p
ðA:2bÞ
ðA:2cÞ
b ¼ U þ JX 0i ni ¼ 1 or H U n0 n0 4p on
ðA:2dÞ
3 X ð1Þm
l ix z2m ðdm1 þ dm2 Þ þ dm3 ðk þ 2lÞz21 K m¼1 1 1 ðA:3aÞ þ dm3 Em 2 2 zm r zm r 3 X
l ix 2 v¼ ð1Þ zm ðdm1 þ dm2 Þ þ dm3 ðk þ 2lÞz21 K m¼1 3 3 þ 1 Em 2 2 ðA:3bÞ zm r zm r u¼
ðA:4eÞ
dj Zcu dr
ðA:4hÞ
^212 ^11 q ^22 q q ^22 q
ðA:5aÞ
Q þZ R Q ^12 q ^22 g ¼ ix q R
ðA:5bÞ
c¼
where: w¼
u Z w r l
In all these expressions (thermo-poroelastic analogy, see e.g. [26]): q¼
s0
ðA:4dÞ
K¼
ðA:5cÞ
R ix^ q22
ðA:5dÞ
and Z ¼ qq^^1222 .
m
2 X ð1Þmþ1 1 zm 1 Em ðk þ 2lÞz21 zm r m¼1
2 X ð1Þmþ1 l 2 2 z zm E m j¼ k þ 2l 3 z21 m¼1
ðA:3cÞ
ðA:3dÞ
Appendix B The material properties of the poroelastic soil are nondimensionalized as [15]: k ¼
k l
Q ¼
Q l
qs ¼
qs q
R ¼
R l
qf ¼
qf q
qa ¼
bd b ¼ pffiffiffiffiffiffi lq
qa q
ðB:1a–dÞ
ðB:1e–gÞ
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
where q = (1 /)qs + /qf is the density of bulk material, l the shear modulus of drained solid skeleton and d the pile diameter.
References [1] Wolf JP, Arx GA. Impedance functions of a group of vertical piles. Proc ASCE Specialty Conf Earthquake Eng Soil Dyn, Pasadena, CA 1978;II:1024–41. [2] Kaynia AM, Kausel E. Dynamic stiffness and seismic response of pile groups. Research Report R83-03. Massachusetts Institute of Technology Cambridge, Mass. 1982. [3] Velez A, Gazetas G, Krishnan R. Lateral dynamic response of constrained-head piles. J Geotech Eng ASCE 1983;109(8):1063–81. [4] Roesset JM. Dynamic stiffness of pile groups. In: Meyer JR, editor. Analysis and design of pile foundations. ASCE; 1984. p. 263–86. [5] Sen R, Davies TG, Banerjee PK. Dynamic analysis of piles and pile groups embedded in homogeneous soils. Earthquake Eng Struct Dyn 1985;13:53–65. [6] Mamoon SM, Kaynia AM, Banerjee PK. Frequency domain analysis of piles and pile groups. J Eng Mech ASCE 1990;116(10):2237–57. [7] Pak RSY, Jennings PC. Elastodynamic response of the pile under transverse excitation. J Eng Mech ASCE 1987;113(7):1101–16. [8] Rajapakse RKND, Shah AH. On the longitudinal harmonic motion of an elastic bar embedded in an elastic halfspace. Int J Solids Struct 1987;23(2):267–85. [9] Rajapakse RKND, Shah AH. On the lateral harmonic motion of an elastic bar embedded in an elastic half-space. Int J Solids Struct 1987;23(2):287–303. [10] Gazetas G, Makris N. Dynamic pile–soil–pile interaction. Part I: Analysis of axial vibration. Earthquake Eng Struct Dyn 1991;20:115–32. [11] Makris N, Gazetas G. Dynamic pile–soil–pile interaction. Earthquake Eng Struct Dyn 1992;21:145–62. [12] Novak M. Piles under dynamic loads. 2nd Int Conf Recent Adv Geotech Earthquake Eng Soil Dyn, St Louis, Missouri 1991;22(3):250–73. [13] Biot MA. Theory of propagation of elastic waves in a fluidsaturated porous solid. I: Low frequency range. J Acoust Soc Am 1956;28(2):168–78.
[14] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 1962;33:1482–98. [15] Zeng X, Rajapakse RKND. Dynamic axial load transfer from elastic bar to poroelastic medium. J Eng Mech ASCE 1999;125(9):1048–55. [16] Muki R, Sternberg E. Elastostatic load transfer to a halfspace from a partially embedded axially loaded rod. Int J Solids Struct 1970;6:69–90. [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 medium. Soil Dyn Earthquake Eng 2003;23:235–42. [19] Beskos DE. Boundary element methods in dynamic analysis: Part II (1986–1996). Appl Mech Rev 1997;50:149–97. [20] Domı´nguez J. An integral formulation for dynamic poroelasticity. J Appl Mech ASME 1991;58(2):588–91. [21] Domı´nguez J. Boundary element approach for dynamic poroelastic problems. Int J Num Meth Eng 1992;35(2): 307–24. [22] Cheng AHD, Badmus T, Beskos DE. Integral equation for dymanic poroelasticity in frequency domain with BEM solution. J Eng Mech ASCE 1991;117(5):1136–57. [23] Japo´n BR, Gallego R, Domı´nguez J. Dynamic stiffness of foundations on saturated poroelastic soils. J Eng Mech 1997;123(11):1121–9. [24] Kupradze VD, Gegelia TG, Basheleishvili MO, Burchuladze TV. Three-dimensional problems of the mathematical theory of elasticity and thermo- elasticity. Amsterdam: NorthHolland; 1979. [25] Kattis SE, Polyzos D, Beskos DE. Vibration isolation by a row of piles using a 3-D frequency domain BEM. Int J Num Meth Eng 1999;46:713–28. [26] Domı´nguez J. Boundary elements in dynamics. Southampton, New York: Computational Mechanics Publications & Elsevier Applied Science; 1993. [27] Vinciprova F, Maeso O, Azna´rez JJ, Oliveto G. Interaction of BEM analysis and experimental testing on pile–soil systems. In: Davini C, Viola E, editors. Problems in structural identification and diagnostics: General aspects and applications. Wien, New York: Springer-Verlag; 2003. p. 195–227. [28] Kassir M, Xu J. Interaction functions of a rigid strip bounded to saturated elastic half-space. Int J Solids Struct 1988;24(9):915–36.
Dynamic impedances of piles and groups of piles in saturated soils Orlando Maeso *, Juan J. Azna´rez, Fidel Garcı´a Instituto Universitario de Sistemas Inteligentes y Aplicaciones Nume´ricas en Ingenierı´a (IUSIANI), Universidad de Las Palmas de Gran Canaria, Edificio Central del Parque Cientı´fico y Tecnolo´gico del Campus Universitario de Tafira, 35017 Las Palmas de Gran Canaria, Spain Accepted 21 October 2004 Available online 5 January 2005
Abstract A three-dimensional boundary element model is presented for the computation of time-harmonic dynamic stiffness coefficients of piles and pile groups embedded in two-phase poroelastic soils. Piles are modelled as continuum elastic solids and soil as a fluid-filled poroelastic half-space governed by BiotÕs theory. Piles are bonded to the surrounding medium along the contact surface where rigorous pile–soil interaction conditions are imposed. The technique has been applied to the calculation of vertical and horizontal dynamic impedance functions of piles and 2 · 2 pile groups. Selected numerical results are presented to portray the influence of excitation frequency, pile stiffness, porous medium properties and kind of pile–soil contact conditions. 2004 Elsevier Ltd. All rights reserved. Keywords: Poroelastic soil; Piles; Pile-groups; Pile–soil interaction; Dynamic impedance; Boundary Elements
1. Introduction The load–displacement response of piles and pile groups embedded in a uniform or layered half-space and subjected to harmonic loads has received considerable attention during the past few decades. Quite a number of papers have appeared that address the problem using both computational [1–6], rigorous [7–9] and simplified analytical [10,11] techniques. A good compilation of used techniques is the one presented by Novak [12]. In all these cases the soil has been assumed to be an elastic or viscoelastic solid medium.
* Corresponding author. Tel.: +34 928 451920; fax: +34 928 451879. E-mail address: [email protected] (O. Maeso).
Little attention has been paid to the analysis of piles in soils saturated by ground water and such problems have been usually studied by treating the soil as an equivalent single-phase solid having a PoissonÕs ratio close to 0.5. Very few studies exist whereby the twophase character of these soils has been taken into account according to BiotÕs theory [13,14] (including the compressibility of both phases, the dissipation due to the fluid viscosity and the coupling between the solid and the fluid deformation and stresses) and most of them contain simplifications that limit their applicability. Zeng and Rajapakse [15] presented the first dynamic studies of a pile in a poroelastic medium. These authors extended the classical elastostatic Muki and Sternberg formulation [16] to analyze steady-state dynamic response of an axially loaded cylindrical elastic bar partially embedded in a homogeneous poroelastic
0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.10.015
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
half-space. The pile is treated as a one-dimensional structure and the interaction between the pile and the poroelastic medium is formulated in terms of a Fredholm integral equation of the second kind. In a similar way Jin et al. [17] extended the method proposed by Pak and Jennings [7] to study the dynamic lateral response of a single pile. According to the writersÕ knowledge, Wang et al. [18] have presented recently the only studies corresponding to pile groups embedded in a poroelastic continuum reported in the literatures. These authors analysed the response of pile groups under vertical loading, using Muki and Sternberg method for a single pile, and introducing a simplified complex interaction factor approach and a superposition method to take into account approximately the pile–soil–pile interaction effects. Boundary Element Method (BEM) has shown to be a good strategy for dynamic soil–structure interaction problems. A complete review of boundary element (BE) applications for dynamic problems can be found in Beskos [19]. In the present paper, a three-dimensional boundary element (BE) approach is used to obtain dynamic impedances of piles and pile groups embedded in two-phase poroelastic soils. The technique is based on the BE formulation for poroelastic media obtained by Domı´nguez [20,21] and Cheng et al. [22] after developing an integral equation formulation from BiotÕs differential equations. Piles are modelled as continuum elastic or viscoelastic solids and surrounding medium as a fluid-filled poroelastic one. BEM is used both for piles and soil. This technique is an extension of the two-dimensional BE model presented by Japo´n et al. [23] for the computation of dynamic stiffness of strip foundations resting in poroelastic soils. In comparison with the existing procedures already commented [15,17,18] the technique proposed here requires a bigger computational effort but this is compensated by an invariably superior versatility, generality and accuracy. Any general foundation geometry can be reproduced, including unequal or inclined piles with their actual cross-section. The contact condition between pile and soil may be pervious or impervious. Although the results presented are restricted to a half-space, the technique is very versatile and more complicated underground geometries (including poroelastic and viscoelastic zones) only require a different BE mesh for the soil region. Selected numerical results for vertical and horizontal impedances are presented. The influences of the frequency of excitation, bar flexibility and poroelastic material properties on the response are examined.
2. Governing equations and BE-model The model used for the dynamic analysis of piles and pile groups should be able to represent the dynamic
behaviour of elastic or viscoelastic regions (piles), poroelastic regions (soil) and the interaction between them at soil–pile interface. The model should accommodate unbounded regions using a reasonable number of unknowns and represent radiation damping properly. The soil is assumed to be an isotropic homogeneous fluid-filled poroelastic material governed by BiotÕs theory [13]. The constitutive equations are: Q2 sij ¼ k þ edij þ 2leij þ Qedij ð1aÞ R s ¼ Qe þ Re
ð1bÞ
where sij are the solid skeleton stress components; s is the fluid equivalent stress = /p (p = pore pressure); / the porosity; eij are solid skeleton strain components=1/2(ui,j + uj,i); dij is the Kronecker delta function; e = $u and e = $U are the solid and fluid dilatation, respectively; u is the displacement of the solid; U is the displacement of the pore fluid; k, l are Lame constants for the drained solid skeleton; and Q, R are Biot constants. The governing equations for a time harmonic excitation of the type eixt (x = angular frequency) are obtained from the equilibrium equations, which can be written [20] in terms of four variables, namely the solid displacement components and the fluid stress, as: ^ Q q lr2 u þ ðk þ lÞre þ 12 rs ^22 R q ^212 ^11 q ^22 þ q ^ q q x2 u þ X 12 X0 ¼ 0 ð2aÞ þ ^22 ^22 q q r2 s þ x 2
^22 Q q ^12 q ^22 e þ rX0 ¼ 0 s þ x2 q R R
ð2bÞ
where the dissipation constants have been included as part of complex valued densities in order to simplify the equations ^ 11 ¼ q11 i q
b ; x
^22 ¼ q22 i q
b ; x
^12 ¼ q12 þ i q
b x ð3Þ
0
X and X are body forces in the solid and fluid phase, respectively; q11 = (1 /)qs + qa; q22 = /qf + qa; q12 = qa; qs and qf are solid and fluid phase2 densities, respectively; qa is the added density; b ¼ cf k/ is the dissipation constant; where k (m/s) is the permeability of the poroelastic medium (depending on the fluid viscosity and the intrinsic permeability of skeleton) and cf the specific weight of fluid phase. Internal damping of the solid skeleton can be introduced using complex valued Lame constants of the type l = Re[l](1 + 2in); where n is the damping coefficient. By substitution of plane wave expressions for u and s into equation 2 for zero body forces, a characteristic equation for the wave numbers is obtained. Three
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
solutions of that equation exist corresponding to three kinds of time harmonic plane waves. One is a shear wave transmitted through the solid skeleton. The other two are dilatational waves (P1 and P2). All wave velocities are complex and frequency dependent; i.e. they are dissipative and dispersive. The solid and the fluid dilatation are in phase for the long longitudinal waves (P1) and they are in opposite phase for the short waves (P2), which damps out at short distances from the perturbation. An integral formulation of governing equations (2) can be obtained. Weighting equations (2a) in each direction by a displacement function ui and (2b) by a stress function s*, adding the four resultant equations and integrating in the domain X we can write in a condensed form: Z ½ðGi þ F i Þui þ ðH þ X 0i;i Þs dX ¼ 0 ð4Þ X
where F i ¼ X i qq^^1222 X 0i . An integral reciprocal relation including solid and fluid variables of two states defined in X with boundary C, can be obtained from (4): Z Z ti ui dC þ sðU n þ JX 0 i ni Þ dC C C Z þ ðF i ui þ JX 0i,i s Þ dX Z ZX ti ui dC þ s ðU n þ JX 0i ni Þ dC ¼ C C Z ð5Þ þ ðF i ui þ JX 0 i,i sÞ dX X
where ti = sijnj are the traction components on the solid phase, Un = Uini is the normal displacement of the fluid, and J ¼ x21q^22 . By using two fundamental solutions, one corresponding to a unit point load in the solid phase and the other to a unit point source in the fluid: X i ¼ dðx xk Þdji ; F i ¼ 0 X 0 X 0 i ¼ 0; i;i ¼ dðx xk Þ
ð6Þ
the integral representations of solid displacements and fluid stress for an internal point xk can be written [20] as: Z Z ukj þ tji ui dC U nj s dC ZC ZC ¼ uji ti dC sj U n dC ð7aÞ C
C
Z
Z J sk þ t0i ui dC ðU n0 þ JX 0 i ni Þs dC Z C Z C u0i ti dC s0 U n dC ¼ C
ð7bÞ
C
where uji and tji (i, j = 1, 2, 3) are solid skeleton displacements and stresses in direction i due to unit point load in the solid in direction j. For the same load, sj and U nj are
771
fluid equivalent stress and normal displacement of the fluid, respectively. The terms u0i , t0i , s0 and U n are the responses in both phases to a unit point source in the fluid phase. When xk 2 C Eqs. (7) become boundary integral equations, to be solved numerically by BEM. In a condensed form they are usually written as: Z Z ck uk þ p u dC ¼ u p dC ð8Þ C
C
where u and p are boundary variables vectors: 8 9 8 9 u1 > t1 > > > > > > > > > > = 2 2 u¼ p¼ > > t3 > u3 > > > > > > > > : > : ; ; s Un
ð9Þ
u* and p* are fundamental solution tensors: 2
u11
6 u 6 u ¼ 6 21 4 u31 u01
u12
u13
u22 u32
u23 u33
u02
u03
3
2
t11 6 7 s2 7 6 t21 7 p ¼6 6 t s3 5 4 31 s1
s0
t01
t12 t22
t13 t23
t32
t33
t02
t03
3 U n1 7 U n2 7 7 U n3 7 5 b U n0
ð10Þ
and ck is the local free the form: 2 e c11 ce12 ce13 6 ce ce ce 6 22 23 ck ¼ 6 21 4 ce31 ce32 ce33 0
0
0
term at collocation point xk with 0 0 0
3k 7 7 7 5
ð11Þ
Jcp
where ceij are the same as the free terms for the elastic static equations at collocation point xk, and cp is the same as the scalar static term at xk. The fundamental solution terms (obtained using the thermoelastic analogy and KupradzeÕs et al. [24] solution for that kind of problems) are given in the Appendix A. Different to other expressions in previous works, they are written in a similar way as that used for the time harmonic elastic and scalar fundamental solutions. Thus, the singularities at collocation point are clearly identified. All the singular terms of the fundamental solution tensor components are contained in their limit value for xr ! 0. These limit values can also be written as the corresponding elastic or scalar static fundamental solution plus some regular terms. This fact is important for boundary element implementation since all the algorithms for integration of the singular terms are the same as for the elastic and scalar static problems. All the additional regular terms are evaluated using standard quadratures. A boundary element discretization of Eq. (8) leads to a system of 4N equations: Hu ¼ Gp
ð12Þ
772
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
where N is the number of nodes on the boundary; u is a vector containing solid displacement and fluid stress at boundary nodes; p is a vector containing solid traction and fluid normal displacement at boundary nodes; and H, G are 4N · 4N system matrices obtained by integration of the 3-D time harmonic poroelastic fundamental solution times the shape functions, over the boundary elements. Piles are modelled as a continuum isotropic homogeneous linear viscoelastic medium with their actual geometry [25]. The governing equations in the frequency domain are: lr2 u þ ðk þ lÞrru þ qx2 u þ X ¼ 0
ð13Þ
where u is the displacement vector amplitude, k and l are the Lame elastic constants, q is the mass density of the pile, and X is the body forces vector amplitude. The boundary integral representation of u is an equation analogous to (8), where in this case u and p are displacement and traction vectors, respectively, over the boundary of pile, and u* and p* are the fundamental solution displacement and traction tensors, respectively. A BE equation for each pile leads to a system of 3Np equations: Hp up ¼ Gp pp
Compatibility: upn ¼ un ¼ U n
ð16aÞ
upt ¼ ut
ð16bÞ
If the solid material is pervious, the condition on the normal displacement of the pore fluid must be substituted by a condition on the pore pressure: (15a) becomes tpn þ tn ¼ 0 and s = 0; (16a) becomes upn ¼ un ; and (15b) and (16b) do not change. In this study soil free surface is considered drained. For a single impervious pile embedded in poroelastic half-space the boundary element equation for each region (pile and soil) in partitioned form according with boundaries in Fig. 1 are: Pile Hp1 up1 þ Hp2 up2 ¼ Gp1 tp1 þ Gp2 tp2
ð17aÞ
Poroelastic soil ss ss H3 Hsw H2 Hsw u2 u3 3 2 þ ws ws ww ww H2 H 2 H3 H 3 s2 s3 ss ss sw G2 Gsw G G t t3 2 2 3 3 ¼ þ ws ws ww ww G2 G 2 G3 G 3 U n2 U n3 ð17bÞ
ð14Þ
obtained for each pile (denoted by super-index p), where Np is the number of nodes on the boundary; up and pp are vectors containing displacement and traction, respectively, at boundary nodes; and Hp, Gp are 3Np · 3Np system matrices obtained by integration of the 3-D time harmonic elastic fundamental solution times the shape functions, over the boundary elements. Eq. (12) is written for the poroelastic soil (nodes over free surface and piles boundaries) and Eq. (14) is written for each pile. The resulting equations are coupled together through equilibrium and compatibility conditions along piles–soil interfaces. The pile is assumed to be fully bonded, and the hydraulic boundary condition along the contact surface corresponds to an intermediate situation between two limiting cases: full drainage (pervious pile) and impermeable (impervious pile). The equilibrium and compatibility conditions between a two-phase poroelastic region and an impervious viscoelastic pile (denoted by super-index p) are the following:
where sub indexes 1–3 correspond to the boundary in contact with the pile cap where displacement nodal values are known (C1), soil–pile interface (C2), and soil free traction surface (C3), respectively. Substituting in Eq. (17) the compatibility and equilibrium conditions along the pile–soil interface (Eqs. (15) and (16)) and the
( u3 , Un3 ) t3 = 0 , τ3 = 0 ( 3) free boundary (Γ
u1 = u ( t1 ) pile cap (Γ ( 1)
poroelastic soil elastic pile
( u2 ,τ , 2 , t2 ) soil - pile interface
( 2) (Γ
Equilibrium: tpn þ ðtn þ sÞ ¼ 0
ð15aÞ
tpt þ tt ¼ 0
ð15bÞ
where n and t sub-indices stand for one normal and two tangential components, respectively.
Fig. 1. Single impervious pile embedded in poroelastic halfspace. Boundary conditions and unknown values at coupled equations of problem.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
external boundary conditions, the combined equations for the coupled dynamic problem in this case can be written as follows: 2
Gp1 6 4 0 0
Hss 2 Hws 2
Hp2 þ Gsw 2 n2
Gp2 n2 Hsw 2
Gp2 Gss 2
0 Hss 3
þ Gww 2 n2
H ww 2
Gws 2
Hws 3
8 9 p u1 > > < H1 = ¼ 0 > > : ; 0
8 > > > > 3> > 0 > > < sw 7 G3 5 > > > Gww > 3 > > > > :
773
Kzz Kθθ
Kxx
9 > > > > u2 > > > > s2 = > t2 > > > > u3 > > > ; U n3 tp1
L
y
x d s
ð18Þ
Fig. 2. 2 · 2 pile group embedded in a half space. Problem geometry definition.
where u2 (solid skeleton/pile displacement vectors), s2 (stress in fluid of poroelastic soil) and t2 (traction vectors in solid skeleton of poroelastic soil) are the unknown nodal values adopted at the pile–soil interface (C2) respectively, and n2 is the unit normal to the pile on this boundary. The equations for a pervious contact case are not shown for the sake of brevity. An advanced 3-D frequency domain BEM computer code in Fortran 90 has been developed. All the boundaries (piles and soil free surface) are discretized into a finite number of quadratic nine-node and six-node boundary elements. The code allows for choosing collocation points shifted to the interior of elements, where displacement and traction are expressed in terms of nodal and shape function values at collocation point. This partially ÔdiscontinuousÕ elements are used to accommodate edges, corners or discontinuities in the boundary conditions. Integrals containing singularities O(1/r) and O(1/r2) are evaluated in a direct form. No enclosing elements are needed in unbounded domains to carry out the so-called rigid body motion procedure. The code includes symmetry and antisymmetry capabilities, in order to reduce the problem size. All computations were run on a Linux PC computer equipped with one 1.33 GHz Athlone AMD processor and 1 GB of RAM memory. The dynamic pile–soil and pile–soil–pile interaction problems include soil as a semi-infinite region, where the radiation damping plays an important role. The semi-infinite soil and radiation damping are easily represented by BE-technique because the fundamental solutions satisfy radiation conditions [26], so the boundaries of soil at free surface are left open at a certain distance from the zone of interest.
the pile head and the resulting vector of displacements (and rotations) at the same point. For a group of piles, it is assumed that the pile heads are constrained by a rigid pile-cap, and the foundation stiffness is the addition of the contributions of each pile. Fig. 2 illustrates the approached problem for a usual configuration. A more complicated foundation, that may include unequal piles, inclined or variable section does not require any modification of the proposed technique and it only demands to modify the boundary element mesh. The dynamic stiffness terms for a time harmonic excitation are functions of frequency x and they are usually written as:
3. Validation of the model In order to validate the boundary element model, in this section, results of impedances of piles and groups of piles are contrasted with other reference values taken from the literature. The dynamic stiffness matrix Kij of a pile relates the vector of forces (and moments) applied at
K ij ¼ k ij þ ia0 cij
ð19Þ
where kij and cij are the frequency dependent dynamic stiffness and damping coefficients, respectively, a0 is the dimensionless frequency a0 ¼
xd cs
ð20Þ
and cs is the soil shear-wave velocity. Fig. 3a and b show the discretizations in boundary elements used to calculate the stiffness of a pile and a group of four piles, respectively, embedded in a viscoelastic or poroelastic half-space. The developed software incorporates symmetry properties. In the cases shown in Fig. 3, it is only necessary to discretize a quarter of the total geometry of the problem. Discretization criteria (size and shape of elements and length of free surface) have been studied in a previous work, Vinciprova et al. [27]. The terms of the dynamic stiffness matrix Kij are obtained by computing the resulting force and moment at the pile heads by integration of the tractions over piles, when unit displacements (horizontal, vertical or rotation) are imposed. It is assumed that the piles heads are situated at the free surface level. The sets of results selected in this section for validation purposes, include horizontal impedances of single piles and 2 · 2 pile groups embedded in a viscoelastic soil, and vertical impedances of single piles embedded in a poroelastic soil.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
Fig. 3. Boundary discretizations for single pile and 2 · 2 pile group in a half space.
3.1. Piles embedded in a viscoelastic half-space The lateral impedances (real and imaginary parts) of a single pile and a 2 · 2 pile group embedded in a homogeneous isotropic viscoelastic half-space obtained by proposed technique are shown in Fig. 4, and compared with those obtained by Kaynia and Kausel [2]. These authors carried out an analysis of single piles and pile groups considering piles as linear elastic prismatic members and soil as semi-infinite viscoelastic media by constructing the requisite GreenÕs function using a discrete layer matrix approach. The following properties are taken from Kaynia and Kausel [2]: piles (in the sequel denoted by sub-index p) are assumed to be uniform elastic continuum and surrounding soil a uniform viscoelas-
(a)
tic media with internal damping coefficient b = 0.05; the ratio between the material modulae is Ep/E = 103; ratio between densities q/qp = 0.7; and PoissonÕs ratios m = 0.4 (soil) and mp = 0.25 (pile). The piles aspect ratio is L/d = 15. The impedance functions for several spacing to diameter ratios s/d have been normalized with respect to the respective single pile static stiffness (Kxxo) times the number of piles in the group (N). All results are plotted against the dimensionless frequency parameter defined by Eq. (20). Fig. 4 shows that the computed values are in very good agreement with those presented in [2] for single piles and 2 · 2 pile groups for s/d = 5 and 10. For s/d = 2 the proposed method presents a slightly superior real part as the frequency increases. This discrepancy with [2] is also observed by Sen et al. [5] for
(b) 5
s/d = 10
5 single pile
2
2
Fig. 4. Horizontal impedances of single pile and 2 · 2 pile groups in a viscoelastic soil. Comparison with KayniaÕs solution.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
a group of 3 · 3 piles for the same distance between piles. It can be noted that the dynamic stiffness of a group of piles cannot be computed by simply adding the stiffnesses of the individual piles, since each one is affected not only by its own load, but also by the load and deflection of its neighbouring piles. This effect (known in the literatures as pile–soil–pile interaction effect) is strongly frequency-dependent, resulting from waves that are emitted from the periphery of each pile and propagated through the soil to the neighbouring piles.
3.2. Piles embedded in a fluid-filled poroelastic half-space The selected validation results correspond with the ones obtained by Zeng and Rajapakse [15] for the vertical impedance of a single pile embedded in a uniform poroelastic half-space with the procedure described before. The properties of two-phase poroelastic soil have been taken from [15]. They are detailed in terms of the non-dimensional parameters whose definition is shown in Appendix B: k* = 1.5, Q* = 2.87, R* = 2.83, qs ¼ 1:44, qf ¼ 0:53, qa ¼ 0 and two different values of the permeability that correspond to the values of the non-dimensional dissipation constant b* = 232.15 and b* = 0. The porosity is / = 0.482. The pile is an impervious elastic medium whose properties are defined by: flexibility ratio Ep/E = 103 (being E = drained YoungÕs modulus of poroelastic medium), mass density ratio qp/q = 1.2 (being q = density of bulk material) and PoissonÕs ratio mp = 0.3. The pile aspect ratio is L/d = 10. Fig. 5a and b show, respectively, the results obtained from the real and imaginary parts of the vertical stiffness, normalized with respect to the static stiffness value Kzzo for a drained elastic soil. The impedance is represented here in a way different from [15]. The non-dimensional fre-
(a)
775
quency a0 is defined by the Eq. (20) where cs = shear wave velocity of bulk material. It is noted a very good agreement with the results taken from [15]. It is also noticed that the stiffness (real part) and damping values (imaginary part) are bigger for the biggest b* (less pervious medium). This effect of the permeability of the soil on the dynamic impedance will be studied afterwards with greater depth.
4. Numerical analysis and results discussion In what follows, it is going to be presented results of dynamic impedances of piles and 2 · 2 pile groups embedded in a saturated poroelastic half-space whose properties have been taken from Kassir and Xu [28]. The normalized values of properties of this medium are: k* = 1, Q* = 14.33, R* = 7.72, qs ¼ 1:12, qf ¼ 0:78, qa ¼ 0 and b* = 59.30. The porosity is / = 0.35 and the internal damping coefficient of the skeleton is b = 0.05. For the piles, it is assumed viscoelastic behaviour with properties similar to concrete: flexibility ratio Ep/E = 343 (E=drained YoungÕs modulus of poroelastic medium), mass density ratio qp/q = 1.94 (q = density of bulk material), PoissonÕs ratio mp = 0.2 and zero internal damping coefficient. The piles aspect ratio is L/d = 15. Except when some other thing is explicit, impervious contact condition between piles and soil are considered. The influence of the frequency of excitation, pile flexibility, as well as the poroelastic effects relative to the medium permeability and to the pile–soil contact condition, on the dynamic impedances, is evaluated. 4.1. Influence of piles flexibility (single pile) The influence of pile flexibility ratio Ep/E on kxx and cxx is presented in terms of a0 in Fig. 6a and b,
(b)
Fig. 5. Vertical impedance of single pile in a poroelastic half space. Comparison with ZengÕs solution.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 6. Influence of piles flexibility on horizontal impedance of single piles.
respectively. Four values of pile flexibility ratio Ep/ E = 100, 200, 500, 1000 and two poroelastic soils with b* = 0 (permeability k ! 1) and 5.93 (10 times greater than in [28]) are used in this analysis. Some aspects of the problem should be pointed out in the light of Fig. 6: in the whole frequency range, growing stiffness and damping values are obtained as Ep/E increases. For b* = 5.93 the dynamic stiffness results (kxx ) show very little dependence with regard to the frequency of excitation a0 in all cases. Only at very low frequencies it is observed a sudden increase of the stiffness from the static value for media with high values of b*. In these problems the static stiffness does not depend on b* and it coincides with the one obtained for the completely drained soil. This effect will be observable in all the results that will be presented in this paper. For the limit value b* = 0 it is noted an important reduction of the impedance. It is observed a significative and progressive reduction of stiffness as the soil permeability and the frequency of excitation increase. This effect is more important as Ep/E increases. In the next section, where the influence of soil permeability is studied, this topic will mentioned again. With the previous four values of Ep/E, additional results of dynamic impedances have been obtained for L/ d = 5, 10 and 20. In the studied range and for all the values of Ep/E the piles with L/d = 10, 15 and 20 present almost identical horizontal dynamic response. It is only noticed a different behaviour for L/d = 5 and values of Ep/E = 100, 200 and 500. This insensitivity of the horizontal response above a certain slenderness value of the pile (La/d, La = active length of the pile) is an effect known in the bibliography. The active length depends on the pile–soil relative stiffness Ep/E and physically represents that, for piles with L La, the imposed deformation on the pile head is not felt in all its length but simply at a depth smaller than La.
4.2. Influence of soil permeability (single piles and pile groups) It has been mentioned above that the dissipation constant b (inversely proportional to permeability k) affects in a significative way the dynamic response: high values of b (clays) imply a greater difficulty in the fluid transit through the solid skeleton as compared to low values of b (loose sands). To study the influence of medium permeability on the dynamic stiffness, a brief analysis of its effects on the characteristics of the waves in the poroelastic soil is done first. Fig. 7 shows, for three values of non-dimensional frequency (a0 = 0.25, 0.5 and 0.75), the variation of the amplitude of S- and P2-wave propagation velocities in a wide range of the non-dimensional
Fig. 7. Wave propagation velocity amplitudes in the poroelastic soil vs. dissipation constant.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
constant b*. The modulus of the velocities has been normalized with the shear wave velocity of the elastic undrained medium cs In the same range of b* the P1-wave velocity (jcp1/csj 8) only experiences a small reduction of the order of 1% (not shown). The S-wave velocity presents a variation of the order of 20%. The short wave velocity (P2) presents the most important variation with b*. It can be seen in the figure that this velocity grows very fast for b* between 101 and 102; P2-wave velocity being bigger than S-wave velocity for values smaller (approximately) than 101. Little influence of the medium permeability can be expected for values of b* bigger than 59.3, which is the value in the considered soil. This study focuses on analyzing the influence of b* in the range of its more significant values. Three values for b* have been adopted: b* = 59.3, 59.3 · 102 and 59.3 · 104. Figs. 8 and 9 show, respectively, the horizontal and vertical impedances of a single pile for the three values of b*. Again, it is observed a reduction of the real (stiffness) and imaginary (damping) parts as b* decreases, that is, for progressively more pervious soils. The figures also show the impedances corresponding to the two ideal limit single-phase media: drained elastic soil and un-drained elastic soil. High values of b* bring the behavior of the two-phase medium near the undrained ideal elastic soil, except, as can be expected, at low frequencies. For the case of vertical impedance, the drained elastic medium marks the limit value towards which the two-phase medium tends as its permeability increases. The real part of the horizontal stiffness (Fig. 8a) falls below the value corresponding to the drained elastic medium when the soil permeability is very high, and this effect is growing with a0. This fact can be explained. For the case of vertical impedance, and for a very pervious soil, the dynamic axial load transfer from pile to poroelastic soil is carried out basically along the contact surface by means of shear tractions in the solid skeleton: in these conditions the pore
(a)
777
fluid hardly provides stiffness and the medium behavior is similar to that of a drained soil. However, in the case of horizontal impedance, the lateral deflection of the pile loads on the two phases on the ground, causing pressures and suctions zones of the pore pressure. This modifies the behavior as opposed to the drained soil. Figs. 10 and 11 show, respectively, the horizontal and vertical impedances of a group of 2 · 2 piles with separation s/d = 5. For s/d = 10 the results are presented in Figs. 12 and 13. It is observed that, for a pile group, the dynamic response is something more complex: the general tendencies of increasing values for the impedance with b* are maintained, but in this case wave reflection phenomena take place between piles in terms of their separation and the medium properties. This causes increases and decreases of the stiffness in certain frequency ranges. Because of that, higher impedance values may be found in media with smaller values of b*. Thus, one can see in Figs. 10a, 11a and 13a, higher stiffness maxima for b* = 59.3 · 104 than for b* = 59.3 · 102. This effect is combined with curves displacement towards the right side for increasing permeabilities. It is made possible by the increasing of the propagation velocities in the porous medium with permeability, and by the fact that in all cases the velocity used to normalize the frequency a0 in (20) is the one of the S-wave in a undrained medium (which is its lower limit). Finally, as it happens for a single pile, the real part of the horizontal stiffness of a very pervious poroelastic medium can be lower than the one of an elastic drained soil, as it is noticed in Fig. 12a. 4.3. Influence of hydraulic boundary condition along pile–soil interface (pile groups) In order to study this effect, the difference that experiences the impedance of a 2 · 2 piles group with separation s/d = 10, in two limit cases: completely impervious contact (same normal displacement in both medium
(b)
Fig. 8. Influence of soil permeability. Horizontal impedance. Single pile.
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O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 9. Influence of soil permeability. Vertical impedance. Single pile.
(a)
(b)
Fig. 10. Influence of soil permeability. Horizontal impedance. 2 · 2 pile group; s/d = 5.
(a)
(b)
Fig. 11. Influence of soil permeability. Vertical impedance. 2 · 2 pile group; s/d = 5.
phases) and completely drained contact (s = 0), are analyzed. According to the conclusions of the previous subsection, it can be expected that the influence of the
hydraulic contact condition becomes more important as b* decreases. This fact is confirmed by the results. Fig. 14 shows the horizontal impedances obtained for
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
779
(b)
Fig. 12. Influence of soil permeability. Horizontal impedance. 2 · 2 pile group; s/d = 10.
(a)
(b)
Fig. 13. Influence of soil permeability. Vertical impedance. 2 · 2 pile group; s/d = 10.
(a)
(b)
Fig. 14. Influence of hydraulic boundary condition along pile–soil interface. Horizontal impedance of 2 · 2 pile group; s/d = 10; b* = 59.3.
the medium under study (b* = 59.3) for both boundary conditions. The obtained differences are negligible. The
influence is noticed when the soil is very pervious (b* = 59.3 · 104) as it is shown in Fig. 15. The real part
780
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
(a)
(b)
Fig. 15. Influence of hydraulic boundary condition along pile–soil interface. Horizontal impedance of 2 · 2 pile group; s/d = 10; b* = 59.3 · 104.
of the impedance shows increasing differences with a0 from a0 = 0.4. It is also appreciated that the stiffness corresponding to the pervious contact is near the values of an ideal elastic drained soil. The explanation proposed before for the effect of loss of stiffness in the pervious saturated soil with respect to the elastic drained soil, caused by the presence of the fluid phase, remains reinforced in the light of these results. The damping of the foundation is insensible to the type of contact condition.
5. Conclusions In the present paper, a three-dimensional BE approach for the computation of time-harmonic dynamic stiffness coefficients of piles and pile groups embedded in two-phase poroelastic soils has been presented. Piles are modelled as continuum elastic or viscoelastic solids and surrounding medium as a fluid-filled poroelastic half-space. The study of more complicated geometries of the soil, including poroelastic and viscoelastic zones, can be conducted without difficulty. The technique has been applied to the calculation of vertical and horizontal dynamic impedance functions of cylindrical single piles and 2 · 2 pile groups. Piles are assumed to be fully bonded and pile–soil interaction effects are taken into account through equilibrium and compatibility conditions at interfaces. In the cases in which these exist, the model has been validated comparing with results from the bibliography. Impedance results have been presented and it has been studied the influence of aspects such as: frequency of excitation, pile stiffness, effects of pile–soil–pile interaction, permeability of the saturated soil and hydraulic condition of pile–soil contact. From the study carried out, we can extract the following conclusions:
In the saturated porous soil the foundation presents an increment of the stiffness from very low frequency values with respect to the elastic drained soil. This effect is less appreciated as constant b decreases in the soil. In the studied frequency range, the increase of pile/ soil relative stiffness implies an increase of the dynamic impedance, independently from the soil permeability. The influence on the dynamic behaviour of the dissipation constant b (dependent on the medium viscosity and the intrinsic permeability of the skeleton) is big because it affects noticeably the soil wave velocities. In general, increasing values of the dynamic impedances with b are obtained, tending towards the values corresponding to an ideal elastic un-drained soil. Very pervious porous soils can present horizontal stiffnesses lower than the ones of the ideal elastic drained soil. This effect is not noticed in the case of vertical stiffness. The dynamic impedance of a group of piles is more dependent on the frequency than the one of a single pile due to dynamic pile–soil–pile interaction effects. These effects depend on separation between piles and soil properties. The degree of permeability of the pile–soil contact condition only has an appreciable influence for very low values of b, and for the horizontal stiffness case. In conclusion, simulating the dynamic behaviour through a drained or un-drained single-phase model can lead, depending on the medium properties and the geometric configuration of the foundation, to unrealistic results. Any model to be used for the dynamic analysis of piles and groups of piles in poroelastic soils must include all the parameters of the material. The technique proposed in this paper does that, and permits a representation more general and versatile than other existent techniques.
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
Acknowledgment This work was supported by the Ministerio de Ciencia y Tecnologı´a of Spain (DPI2001-2377-C02-02). The financial support is gratefully acknowledged.
Appendix A Fundamental solution for fluid-filled poroelastic material under dynamic loading Point load in solid skeleton in j direction:
781
with r = jx xkj, zm = ikm (m = 1, 2, 3) (km: wave number), z21 ¼ z22 z21 and Em ¼ 1r ezm r . The functions A, B, C, D, E, F, G, y H in terms of w, v, u, y j are: dw v dr r v dv B¼2 2 r dr
ðA:4aÞ
A¼
ðA:4bÞ
k dw dv v v Q 2 2 þ ixgu l dr dr r r R
C¼
ðA:4cÞ
uji
1 ðwdji vr;j r;i Þ ¼ 4pl
ðA:1aÞ
du u Z þ v D ¼ ixgJ dr r l
tji
1 or Adji þ Br;j r;i þ Ar;i nj þ Cr;j ni ¼ 4p on
ðA:1bÞ
E ¼ ixgJ
ðA:1cÞ
du u F ¼ 2l dr r
ðA:4fÞ
ðA:1dÞ
du u u Q þ2 2l þ j G¼k dr r r Rc
ðA:4gÞ
ixg ¼ ur;j 4p c or Dr;j þ Enj U nj ¼ 4p on sj
Point source in pore fluid: c ur;i 4p c or Fr;i þ Gni t0i ¼ 4p on u0i ¼
H ¼ J ðA:2aÞ
1 ¼ j 4p
ðA:2bÞ
ðA:2cÞ
b ¼ U þ JX 0i ni ¼ 1 or H U n0 n0 4p on
ðA:2dÞ
3 X ð1Þm
l ix z2m ðdm1 þ dm2 Þ þ dm3 ðk þ 2lÞz21 K m¼1 1 1 ðA:3aÞ þ dm3 Em 2 2 zm r zm r 3 X
l ix 2 v¼ ð1Þ zm ðdm1 þ dm2 Þ þ dm3 ðk þ 2lÞz21 K m¼1 3 3 þ 1 Em 2 2 ðA:3bÞ zm r zm r u¼
ðA:4eÞ
dj Zcu dr
ðA:4hÞ
^212 ^11 q ^22 q q ^22 q
ðA:5aÞ
Q þZ R Q ^12 q ^22 g ¼ ix q R
ðA:5bÞ
c¼
where: w¼
u Z w r l
In all these expressions (thermo-poroelastic analogy, see e.g. [26]): q¼
s0
ðA:4dÞ
K¼
ðA:5cÞ
R ix^ q22
ðA:5dÞ
and Z ¼ qq^^1222 .
m
2 X ð1Þmþ1 1 zm 1 Em ðk þ 2lÞz21 zm r m¼1
2 X ð1Þmþ1 l 2 2 z zm E m j¼ k þ 2l 3 z21 m¼1
ðA:3cÞ
ðA:3dÞ
Appendix B The material properties of the poroelastic soil are nondimensionalized as [15]: k ¼
k l
Q ¼
Q l
qs ¼
qs q
R ¼
R l
qf ¼
qf q
qa ¼
bd b ¼ pffiffiffiffiffiffi lq
qa q
ðB:1a–dÞ
ðB:1e–gÞ
782
O. Maeso et al. / Computers and Structures 83 (2005) 769–782
where q = (1 /)qs + /qf is the density of bulk material, l the shear modulus of drained solid skeleton and d the pile diameter.
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