Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model

Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model ∗ L.A. Padr´on, J.J. Azn´arez, O.Maeso Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num´ericas en Ingenier´ıa (SIANI) Universidad de Las Palmas de Gran Canaria Edificio Central del Parque Cient´ıfico y Tecnol´ogico Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Spain {lpadron,jjaznarez,omaeso}@siani.es 4 July 2007 Abstract In this paper, a 3D BEM-FEM coupling model is used to study the dynamic behavior of piled foundations in elastic layered soils in presence of a rigid bedrock. Piles are modelled by FEM as beams according to the Bernoulli hypothesis, and every layer of the soil is modelled by BEM as a continuum, semi-infinite, isotropic, homogeneous, linear, viscoelastic medium. First, the main points of the model are set out. Then, several results of vertical, horizontal and rocking impedances for single piles and 2 ×2 pile groups embedded in a stratum resting on a rigid bedrock, are presented. The influence on the dynamic response of stratum depth, soil stiffness and piled foundation configuration is discussed. Finally, the influence of the stratigraphy on the seismic response of a 3 ×3 pile group is analyzed, together with the pile-to-pile kinematic interaction and the wave-scattering phenomena. 1 Introduction The steady-state dynamic response of piled foundations in elastic soils has been the subject of much research, in both the kinematic and the forced vibration analyses (see e.g. [1]). For the study of piles and pile groups embedded in a homogeneous half-space, several frequency domain boundary integral formulations in conjunction with the monodimensional Finite Element Method (FEM) have been used by different authors [2, 3, 4, 5]. Linear analyses for non-homogeneous media or layered soils have also been carried out using the same kind of approach [6, 7, 8, 9]. Subsequently, more versatile and rigorous linear numerical models have been developed using the Boundary Element Method (BEM) for both soil and piles [10, 11, 12, 13], but with the disadvantage of a high computational cost. Although a great deal of this research has been focused on the forced vibration problem, much of it has also dealt with the kinematic response of piled foundations. For instance, parametric studies of the seismic response of single piles and pile groups to Rayleigh waves and to vertically and obliquely incident body waves have been reported in [14, 15, 16, 17]. ∗This is the peer reviewed version of the following article: L.A. Padr´on, J.J. Azn´arez, O.Maeso, Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model, Soil Dynamics and Earthquake Engineering 28 (2008) 333-346, which has been published in final form at doi:10.1016/j.soildyn.2007.07.005. This work is released with a Creative Commons Attribution Non-Commercial No Derivatives License. 1 On the other hand, due to the fact that piles are commonly used when avoiding shallow soil of low bearing capacity and transferring load to deeper soil or rock of high bearing capacity is needed, a particular case of interest in the dynamic analysis of piles is that of piled foundations embedded in a viscoelastic stratum resting on a rigid bedrock, for both floating and hinged piles. However, not many papers have been reported focusing on this topic [18, 19], though some have dealt with it while studying the pile-soil-structure interaction problem [20, 21, 22, 23]. For this reason, the aim of this paper is to present a method for the dynamic analysis of piles and pile groups and, making use of it, contribute to the topic discussed above by investigating, through parametric studies: a) the influence of the presence of a rigid bedrock on the dynamic impedances of piled foundations, and b) the influence of the stratigraphy on the seismic response of hinged pile groups. To this end, a BEM-FEM coupling model previously presented by the authors [24] is used to compute time-harmonic dynamic impedances and the seismic response of piled foundations embedded in viscoelastic zoned-homogeneous layered soils. The model has been validated for both the kinematic and the stiffness problems with several results taken from the literature (e.g. [2, 18, 19, 20]), though these comparisons are not presented here for the sake of brevity. In this approach, and in the line of a previous static model developed in [25, 26, 27], the pile-soil interaction takes place, from the integral equation point of view, through internal forces, as it is assumed that the soil continuity is not altered by the presence of the piles. These are modeled by FEM as beams according to the Bernoulli hypothesis, and every stratum of the soil is modeled by BEM as a continuum, semi-infinite, isotropic, homogeneous, linear, viscoelastic medium. The model not only allows the dynamic analysis of piled foundations embedded in a half-space or in a stratum resting on a rigid bedrock, but also in multilayered soils of generic stratigraphy and topography, including deposits and inclusions. Firstly, the main points of the BEM-FEM coupling model formulation are set out. Secondly, several results of vertical, horizontal and rocking time-harmonic dynamic impedances of single piles and 2 ×2 pile groups embedded in a stratum resting on a rigid bedrock, are presented. Different depths of the stratum and three foundation configurations are studied, and the effects associated to these parameters are discussed. Finally, the influence of the stratigraphy on the seismic response of a 3×3 pile group is analyzed. To this end, displacement transfer functions for vertically-incident plane time-harmonic shear waves and response spectra for a particular configuration, under two different strong ground motions, are presented for several soil profiles. Pile-to-pile kinematic interaction and wave-scattering phenomena, are also investigated. Regarding the impedances study, it is shown that the effect of the presence of the rigid bedrock is vital for the horizontal impedance of single piles and pile groups, and also for the rocking behavior of single piles, while it is almost negligible for the rocking response of pile groups. Regarding the vertical impedances of floating piled foundations, the effect of the bedrock is not important except for frequencies below 1.5 times the fundamental natural frequency of the stratum. As for the performed seismic analysis, it can be seen that the presence of a soft layer at the top of a stratum yields to rapidly decreasing transfer functions and, consequently, to a weakened seismic response at the pile cap. However, taking into account additional layers with shear wave velocity increasing with depth is of minor importance. 2 Pile-soil interaction model The behavior of a pile submitted to harmonically varying loads, considering zero internal damping, can be described by the equation ¯ K up=Fext +Q qp,(1) 2 where ¯ K=K−ω2M, being Kand Mthe stiffness and mass matrices of the pile, ωis the circular frequency of excitation, and upthe vector of nodal translation and rotation amplitudes along the pile. Fext includes the forces at the top Ftop and the axial force at the tip of the pile Fp,qpis the vector of tractions along the pile-soil interface, whereas Qis the matrix that transforms these nodal traction components to equivalent nodal forces. By means of Eq. (1), piles are modelled as vertical Bernoulli beams by three-node FEM elements on which there are defined 13 degrees of freedom: three lateral displacements on each node, and two rotations on each one of the extreme nodes. On the other hand, each stratum of the soil is modelled by BEM as a linear, homogeneous, isotropic, viscoelastic, unbounded region. The boundary integral equation for a time-harmonic elastodynamic state defined in a domain Ωmwith boundary Γmcan be written in a condensed and general form as ckuk+ZΓm p∗udΓ = ZΓm u∗pdΓ + ZΩm u∗XdΩ,(2) where ckis the local free term matrix at collocation point ‘k’, Xare the body forces in the domain Ωm,uand pare the displacement and traction vectors, and u∗and p∗are the elastodynamic fundamental solution tensors for a time-harmonic concentrated load at point ‘k’. A hysteretic damping model is used for the soil through a complex valued shear modulus µof the type µ= Re[µ](1 + 2iξ), being ξthe damping coefficient. Details about BEM formulation can be found in [28]. Generally, body forces Xare considered to be zero in most of elastodynamic problems. Nevertheless, in this approach, from the integral equation point of view, the pile-soil interaction takes place through internal punctual forces placed at the geometric piles tip and through load-lines placed along the piles axis, as it is assumed that the soil continuity is not altered by the presence of the piles. The load-lines within the soil, the tractions along the pile-soil interface acting over the pile and within the soil (qpj=−qsj), and the internal punctual forces Fpjat the tip of the piles, are represented in Fig. 1, where a sketch of the model is shown. Figure 1: Load-lines representation of two piles embedded in a layered soil. Then, Eq. (2) can be written as ckuk+ZΓm p∗udΓ = ZΓm u∗pdΓ + nm ll X j=1 "ZΓm pj u∗qsjdΓpj−δjΥj kFpj#,(3) where Γm pjis the pile-soil interface along the load-line jwithin the domain Ωm;nm ll is the total number of load-lines in the domain Ωm;δjis equal to one if the load-line jcontains the tip of a floating pile and zero otherwise; and Υj kis a three-component vector that represents the contribution of the axial force Fpjat the tip of the jth load-line. 3 Once all boundaries have been discretized, Eq. (3) can be written for each region for all nodes in Γmand Γm pjto obtain two matrix equations of the type Hssus−Gssps− nm ll X j=1 Gspjqsj+ nm ll X j=1 δjΥsjFpj= 0,(4) and upi+Hpisus−Gpisps− nm ll X j=1 Gpipjqsj+ nm ll X j=1 δjΥpijFpj= 0,(5) respectively, where Hand Gare coefficient matrices obtained by integration over the elements of the fundamental solution times the corresponding shape functions, usand psare the vectors of nodal displacements and tractions of boundary elements, and upiis the vector of nodal displacements of the load-line i. A coupled system of equations, of the type A{us,ps,qs,Fp,up}T=B,(6) representing the layered soil-piles system, can be obtained from Eqs. (1), (4) and (5), where equilibrium and compatibility fully-bonded contact conditions over the different interfaces of the problem have been imposed. Aand Bare the square matrix of coefficients and the known vector respectively, both computed by rearranging the equations and prescribing the known boundary conditions. In this work, two different kinds of analysis are carried out, each one of them with a different set of boundary conditions, though, in both cases, piles in a group are considered to be fixedly connected to a rigid massless cap (not in contact with the soil) and zero-traction free surface is assumed. Firstly, dynamic stiffness of piles and pile groups are computed by prescribing forced vibration at the pile cap. Secondly, the seismic response of a piled foundation subject to vertically-incident plane time-harmonic shear waves is analyzed. To this end, the displacement and traction fields in the layered soil are considered as the superposition of two: the uniform viscoelastic layered soil fields uIand pI, whose analytic expression are known; and the scattered wave fields uSand pS, produced by the presence of the piles. Consequently, the total fields are the sum of these two (u=uI+uS,p=pI+pS), while it is only in the scattered wave field that the tractions along the pile-soil interface and the forces at the pile tip exist (q=qS,Fp= (Fp)S). In this case, the above equations are written for the scattered fields in all regions, and then expressed in terms of the total and the incident fields as Hssus−Gssps− nm ll X j=1 Gspjqsj+ nm ll X j=1 δjΥsjFpj=Hssus I−Gssps I(7) and upi+Hpisus−Gpisps− nm ll X j=1 Gpipjqsj+ nm ll X j=1 δjΥpijFpj=upi I+Hpisus I−Gpisps I(8) where the right hand terms are known. Only the main points of the formulation have been presented here. However, more details can be found in Padr´on et al. [24], where the formulation, focused on pile groups embedded in a half-space, has been presented. 4 3 Dynamic stiffness of piled foundations in homogeneous strata The dynamic stiffness matrix Kij of a pile relates the vector of forces (and moments) applied at the pile top to 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, where Land dare used to denote the length and diameter of the piles, s refers to the distance between adjacent piles and Hdenotes the depth of the stratum. Figure 2: 2x2 pile group embedded in a stratum resting on a rigid bedrock. Problem geometry definition. The dynamic stiffness terms for a time-harmonic excitation are functions of frequency ω, and they are usually written as Kij =kij + iaocij,(9) where kij and cij are the frequency dependent dynamic stiffness and damping coefficients, respectively, aois the dimensionless frequency ao=ωd cs (10) and csis the soil shear-wave velocity. Figure 3: 2x2 pile group and stratum BEM-FEM discretization (only a quarter of the geometry). Fig. 3 shows a sketch of the discretizations used to obtain the stiffness of different pile groups embedded in a viscoelastic homogeneous stratum resting on a rigid bedrock. As the developed software incorporates symmetry properties, only a quarter of the total geometry of the problem has to be discretized. 5 In what follows, vertical, horizontal and rocking impedances of different piled foundation configurations are presented. In the case of single piles, the stiffness and damping functions are normalized by the respective static stiffness. As for pile groups, the vertical and horizontal impedance functions are divided by the respective single pile static stiffness (ks zzoand ks xxo) times the number (N) of piles in the group. Finally, the rocking impedances are normalized with respect to the sum of the products of the respective single pile vertical static stiffness (ks zzo) times the square of the distance to the rotation axis (xi). All results are plotted versus the dimensionless frequency defined by Eq. (10), and several ratios H/L have been considered. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 ao czz/(kzzo ) H=L H=1.5L H=2L H=3L H=5L H=10L half−space Figure 4: Vertical impedances of a single pile embedded in a homogeneous stratum. In the first place, vertical impedances of a single pile embedded in a homogeneous stratum of depth Hresting on a rigid bedrock are shown in Fig. 4 for ratios between stratum depth and pile length of H/L = 1 (hinged pile), 1.5, 2, 3, 5 and 10, together with the response for the half-space. It is assumed that the stratum properties are: internal damping coefficient βs= 0.05 and Poisson ratio νs= 0.4. The ratio between densities is ρs/ρp= 0.7, the aspect ratio of the pile is L/d = 15, and the pile/soil modulus ratio is Ep/Es= 103. As can be seen from the figures, the presence of a rigid bedrock below a floating pile has a strong influence on the impedances in the frequency band from the static value to approximately 1.5 times the dimensionless fundamental natural frequency of the stratum in compression-extension mode. Above this frequency, stiffness and damping coefficient are coincident with those of a floating pile in a half-space, which reveals that the main damping mechanism at intermediate and high frequencies is the energy dissipation through surface waves for both the half-space and the stratum. Below the first natural frequency, the damping coefficients are far lower than the ones corresponding to the floating pile in a half-space, because there cannot be surface waves in a stratum at low frequencies and, thus, the energy is confined in it. As expected, the static value of the vertical stiffness of a hinged pile is much higher than the corresponding to a floating pile. More precisely, it is 5.1 times higher than the corresponding to a floating pile embedded in a stratum of depth H= 1.5L. On the other hand, the fundamental natural frequency of the system related to the compression-extension mode is clearly highlighted in each case. Note that the influence of the presence of the rigid bedrock is still noticeable for a stratum depth five times the pile length, while it is hardly significant when the stratum depth is ten times the pile length. Horizontal and rocking impedances of a single pile embedded in a homogeneous stratum of depth Hresting on a rigid bedrock are shown in Figs. 5 and 6 respectively. The characteristics of piles and soils for these and further cases are the ones defined above and, for the sake of clarity, only results for ratios between stratum depth and pile length of H/L = 1 (hinged pile), 1.5 and 2, together with the response for the half-space, are presented. The range of interest in each stiffness figure has been enlarged, and the natural frequencies to which every peak is associated have been labeled for the 6 horizontal cases. In this case, the resonance effects are associated to both the shear and compression-extension modes. The influence of the presence of the rigid bedrock is significant approximately until the second natural frequency related to the shear mode. In this range, the impedances fluctuate about the half-space solution for both the stiffness and the damping coefficients. As for the latter, they are small at low frequencies as discussed above. For higher frequencies, the impedance behavior is similar to the one corresponding to a pile embedded in a half-space. Vertical and horizontal impedances of a 2×2 pile group embedded in a homogeneous stratum of depth Hresting on a rigid bedrock are shown in Figs. 7 and 8, respectively. Ratios between pile separation and diameter of s/d = 2, 5 and 10 are presented. For scale reasons, the peaks associated to the stratum natural frequencies do not appear very clearly in the figures, but their magnitudes are proportional to those of the single pile case, and even increase with the ratio s/d. Besides, the vertical impedances at intermediate and high frequencies are equivalent to those of a floating pile group embedded in a half-space. However, the group effect is predominant over the influence of the presence of the rigid bedrock. On the other hand, for the horizontal case, the influence of the rigid bedrock is relatively much more important, even though the group effect is still predominant. The impedances fluctuate about the half-space solution, and the amplitude of this fluctuation increases with the ratio s/d. Table 1 presents the comparison between the values of dimensionless natural frequencies obtained with the presented coupling model and those analytically computed for an undamped stratum. Only the values obtained for the horizontal impedance of a pile group with a ratio s/d = 10 are presented. As can be seen, the proposed method predicts adequately the actual fundamental frequencies. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 ao cxx/(kxxo ) H=L H=1.5L H=2L half−space Figure 5: Horizontal impedances of a single pile embedded in a homogeneous stratum. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 ao cφφ/(kφφo ) H=L H=1.5L H=2L half−space Figure 6: Rocking impedances of a single pile embedded in a homogeneous stratum. 7 Stratum natural frequencies in shear mode compressionextension mode as o(1)as o(2)ap o(1)H/L Undamped stratum 0.11 0.31 0.26 1 Pile group-soil system 0.12 0.33 0.25 Undamped stratum 0.07 0.21 0.17 1.5 Pile group-soil system 0.09 0.23 0.17 Undamped stratum 0.05 0.16 0.13 2 Pile group-soil system 0.07 0.16 0.12 Table 1: Comparison between natural frequencies for an undamped stratum and for a 2 ×2 pile group-stratum system. s/d = 10. Finally, rocking impedances of a 2×2 pile group embedded in a homogeneous stratum on a rigid bedrock are shown in Fig. 9. Only the ratio s/d =10, for which the influence of the rigid bedrock is stronger, has been displayed in order to illustrate the fact that the rocking impedances of pile groups are little influenced by the presence of the bedrock. 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 ao cG zz/(N ks zzo ) H=L H=1.5L H=2L half−space 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 0 2 4 6 8 ao kG zz/(N ks zzo ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 ao cG zz/(N ks zzo ) H=L H=1.5L H=2L half−space 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 0 2 4 6 8 ao kG zz/(N ks zzo ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 ao cG zz/(N ks zzo ) H=L H=1.5L H=2L half−space s d= 2 s d= 5 s d= 10 Figure 7: Vertical impedances of a 2x2 pile group embedded in a homogeneous stratum. s/d=2, 5 and 10 (from top to bottom). 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T (s) acceleration (g) Free field Half−space 1 stratum 1 soft stratum 2 strata 3 strata 4 strata Figure 16: 5 per cent-damped acceleration response spectra for El Centro (1940) earthquake specified at the free surface. parameters such as densities, damping coefficients and Poisson ratios, conclusions from this study may not be generalized. On the other hand, only the horizontal response has been treated here but, depending on the problem, the rocking behavior could also be an important parameter. 5 Conclusions In this paper, a BEM-FEM coupling model previously presented by the authors has been used to compute dynamic impedances and kinematic response of piled foundations embedded in viscoelastic homogeneous strata resting on a rigid bedrock. This way, the two first steps of the substructuring method are performed. Piles have been modelled by FEM as beams according to the Bernoulli hypothesis, and strata have been modelled by BEM as continuum, semi-infinite, isotropic, homogeneous, linear, viscoelastic media. The model not only allows the dynamic analysis of piled foundations embedded in a half-space or in a stratum resting on a rigid bedrock, but also in multilayered soils of generic stratigraphy, including deposits and inclusions. Any kind of geometry can be modelled by boundary discretizations, thus, more complex stratigraphies and topographies than those reported in this paper can be analyzed. Firstly, several results of vertical, horizontal and rocking impedances for single piles and three different configurations of 2 ×2 pile groups embedded in a stratum have been presented. Several depths of the stratum have also been studied. From the analysis of these results, the following conclusions can be drawn: •The pile-soil system presents peaks associated to the stratum natural frequencies that the numerical model is able to predict adequately. •The vertical impedance functions of a piled foundation embedded in strata of the analyzed depths are equivalent to those embedded in a half-space at frequencies above 1.5 times the fundamental natural frequency of the stratum in the compression-extension mode. Besides, the foundation vertical behavior is only influenced by this natural frequency. •The influence of the stratum depth over the horizontal impedance functions of a piled foundation is significant over a broader band of frequencies. Thus, several peaks associated to both the shear and the compression-extension modes appear in the foundation response. 16 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) Free field 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) 1 stratum 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) 1 soft stratum 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) 2 strata 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) 3 strata 0 2 4 6 8 10 −0.8 −0.4 0 0.4 0.8 t (s) acceleration (g) 4 strata pga=0.778 g pga=0.769 g pga=0.529 g pga=0.532 g pga=0.567 g pga=0.566 g Figure 17: Response acceleration histories for El Centro (1979) earthquake specified at the free surface. •This influence is even more evident in the damping functions, which at low frequencies are much smaller than the corresponding to the half-space. On the contrary, their values are similar at higher frequencies, which reveals that the main damping mechanism at intermediate and high frequencies is the energy dissipation through surface waves for both the half-space and the strata. •The influence of the presence of the rigid bedrock is still noticeable for a stratum depth five times the pile length. •The group effect is predominant over the influence of the presence of the rigid bedrock, especially in vertical and rocking impedances, in which the peaks associated to the natural frequencies of the stratum are of small magnitude. However, the effects of the presence of the bedrock on the horizontal behavior are stronger and become apparent along a broader frequency band. In all cases, the influence of the rigid bedrock increases with the ratio s/d. •The rocking behavior is the less influenced by the presence of a rigid bedrock. Also, additional experiments (not shown here for the sake of brevity) that have been carried out in order to investigate the role of the ratio Ep/Es, show that the frequency band in which the foundation response is influenced by the rigid bedrock broadens with the increase of the soil stiffness. Also, a larger number of natural frequencies of the stratum in both the shear and the compressionextension modes become evident for the horizontal and rocking cases as the soil stiffness increases. At the same time, the group effect becomes more evident as the ratio Ep/Esturns higher. 17 Figure 18: 5 per cent-damped acceleration response spectra for El Centro (1979) earthquake specified at the free surface. Secondly, the seismic response of a square 3 ×3 hinged pile group embedded in different soil profiles and under vertically-incident plane time-harmonic shear waves and two different strong ground motions, have been studied. General conclusions cannot be drawn without more and deeper analysis in the subject but, as for the presented case, it has been shown that: •the presence of a soft layer atop the soil yields to much more rapidly decreasing displacement transfer functions than those corresponding to foundations in the reference homogeneous soil. In fact, two different trends are clearly defined, depending on whether the soft layer is considered or not. •the addition of further intermediate shear wave velocity layers in between is of minor importance. •the incident field at the free surface is perturbed by the presence of the piled foundation, being the magnitude of this perturbation larger when a soft layer exists atop. The maximum value of this perturbation is of similar order in both the direction of motions induced by the S-waves and its perpendicular. •pile-to-pile interaction under seismic excitation is almost nonexistent. •piled foundations embedded in the non-homogeneous soil profiles studied here filter out a great part of the harmonic components of the seismic input, in such a way that the resulting acceleration response spectra are significantly lower than those corresponding to the free field. •the simplification of the soil profile to just one stratum or to a half-space would lead to overor underestimating, depending on the chosen properties, the excitation at the base of a superstructure in a substructuring analysis. •these conclusions are valid for earthquake motions specified at both the free surface or at the bedrock. Acknowledgements The authors would like to thank the reviewers for their valuable comments, that have contributed to improve the paper. This work was supported by the Ministry of Education and Science of Spain through research project BIA2004-03955-C02-02 and co-financed by the European Fund of 18 Figure 19: 5 per cent-damped acceleration response spectra for El Centro (1940) earthquake, specified at the bedrock. Regional Development. L.A. Padr´on is recipient of the FPU research fellowship AP-2004-4858 from the Ministry of Education and Science of Spain. The authors would like to thank for this support. References [1] Beskos DE. Boundary element methods in dynamic analysis: Part II (1986-1996). Appl Mech Rev 1997;50:149-97. [2] Kaynia AM. Dynamic stiffness and seismic response of pile groups. Research Report R83-03, Massachusetts Institute of Technology. Cambridge, MA; 1982. [3] 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. [4] Mamoon SM, Kaynia AM, Banerjee PK. Frequency domain dynamic analysis of piles and pile groups. J Eng Mech ASCE 1990;116(10):2237-57. [5] Mamoon SM, Banerjee PK. Response of piles and pile groups to travelling SH-waves. Earthquake Eng Struct Dyn 1990;19:597-610. [6] Kaynia AM, Kausel E. Dynamics of piles and pile groups in layered soil media. Soil Dyn Earthquake Eng 1991;10:386-401 [7] Gazetas G, Fan K, Kaynia AM, Kausel E. Dynamic interaction factors for floating pile groups. J Geotech Eng ASCE 1991;117:1531-48. 19 [8] Miura K, Kaynia AM, Masuda K, Kitamura E. Seto Y. Dynamic behaviour of pile foundations in homogeneous and non-homogeneous media. Earthquake Eng Struct Dyn 1994;23:183-92. [9] Davies TG, Sen R, Banerjee PK. Dynamic behavior of pile groups in inhomogeneous soil. J Geotech Eng 1985;111(12):1365-79. [10] Kattis SE, Polyzos D, Beskos DE. Vibration isolation by a row of piles using a 3-D Frequency domain BEM. Int J Numer Meth Eng 1999;46:713-28. [11] Kattis SE, Polyzos D, Beskos DE. Modelling of pile wave barriers by effective trenches and their sreening effectiveness. Soil Dyn Earthq Eng 1999;18:1-10. [12] Vinciprova F, Azn´arez JJ, Maeso O, Oliveto G. Interaction of BEM analysis and experimental testing on pile-soil systems. In: Davini C, Viola E, editors. Problems in structural identification and diagnostic: General aspects and applications. Berlin: Springer; 2003. p. 195-227. [13] Maeso O, Azn´arez JJ, Garc´ıa F. Dynamic impedances of piles and groups of piles in saturated soils. Comput Struct 2005;83:769-82. [14] 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. [15] Masayuki H, Shoichi N. A study on pile forces of a pile group in layered soil under seismic loadings. In: Proc. of the II Int Conf on Recent Advances in Geotech Earthquake Eng and Soil Dyn, St Louis, Missouri 1991. [16] Fan K, Gazetas G, Kaynia A, Kausel E, Ahmad S. Kinematic seismic response of single piles and pile groups. J Geotech Eng ASCE 1991;117(12):1860-1879. [17] Kaynia A, Novak M. Response of pile foundations to Rayleigh waves and obliquely incident body waves. Earthquake Eng Struct Dyn 1992;21:303-318. [18] Nogami T. Dynamic group effect of multiple piles under vertical vibration. In: Proc. of the ASCE Engineering Mechanics Division Specialty Conference, Austin, Texas. 1979, pp. 750-754. [19] Velez A, Gazetas G, Krishnan R. Lateral dynamic response of constrained-head piles. J Geotech Eng ASCE 1983;109(8):1063-81. [20] Gazetas G, Fan K, Tazoh T, Shimizu K, Kavvadas M, Makris N. Seismic pile-group-structure interaction. Geotechnical Special Publication NO.34, ASCE 1992;56-93. [21] Mylonakis G, Nikolaou A. Soil-pile-bridge seismic interaction: kinematic and inertial effects. Part I: Soft soil. Earthquake Eng Struct Dyn 1997;26:337-359. [22] Guin J, Banerjee PK. Coupled soil-pile-structure interaction analysis under seismic excitation. J Struct Eng 1998;124(4):434-444. [23] Koo KK, Chau KT, Yang X, Lam SS, Wong YL. Soil-pile-structure interaction under SH wave excitation. Earthquake Eng Struct Dyn 2003;32:395-415. [24] Padr´on LA, Azn´arez JJ, Maeso O. BEM-FEM coupling model for the dynamic analysis of piles and pile groups. Eng Anal Boundary Elem 2007;31:473-484. [25] Mendon¸ca AV, de Paiva JB. A boundary element method for the static analysis of raft foundations on piles. Eng Anal Boundary Elem 2000;24:237-47. 20 [26] Mendon¸ca AV, Paiva JB. An elastostatic FEM/BEM analysis of vertically loaded raft and piled raft foundation. Eng Anal Boundary Elem 2003;27:919-33. [27] Matos Filho R, Mendon¸ca AV, Paiva JB. Static boundary element analysis of piles submitted to horizontal and vertical loads. Eng Anal Boundary Elem 2005;29:195-203. [28] Dom´ınguez J. Boundary elements in dynamics. Southampton, New York: Computational Mechanics Publications & Elsevier Applied Science; 1993. 21