Computers and Geotechnics 37 (2010) 515–528 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo Parametric study on seismic ground response by ﬁnite element modelling Angelo Amorosi a,*, Daniela Boldini b,1, Gaetano Elia c,2 a Technical University of Bari, Department of Civil and Environmental Engineering, via Orabona 4, 70125 Bari, Italy University of Bologna, Department of Civil, Environmental and Material Engineering, viale Terracini 28, 40136 Bologna, Italy c Newcastle University, School of Civil Engineering & Geosciences, Drummond Building, NE1 7RU Newcastle upon Tyne, United Kingdom b a r t i c l e i n f o Article history: Received 22 September 2009 Received in revised form 5 February 2010 Accepted 10 February 2010 Available online 21 March 2010 Keywords: Seismic ground response analysis Constitutive models Numerical modelling Finite element analysis a b s t r a c t In this paper the results of 2D FE analyses of the seismic ground response of a clayey deposit, performed adopting linear visco-elastic and visco-elasto-plastic constitutive models, are presented. The viscous and linear elastic parameters are selected according to a novel calibration strategy, leading to FE results comparable to those obtained by 1D equivalent-linear visco-elastic frequency-domain analyses. The inﬂuence of plasticity on the numerical results is also investigated, with particular reference to the relation between the hysteretic and viscous damping effects. Finally, different boundary conditions, spatial discretisation and time integration parameters are considered and their role on the FE results discussed. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The site response analysis has traditionally been performed using one-dimensional frequency-domain numerical scheme based on the equivalent visco-elastic approach [1–3]. This approach has successfully been adopted in the last 30 years and it is widely accepted in the engineering practice, although its limitations are well-known. In particular, concerning this latter aspect, it is worth remarking the following points: – soil behaviour is controlled by effective stresses while a total stress approach is implemented in most equivalent visco-elastic schemes, disregarding the soil–ﬂuid interaction [4] and possible build-up of excess pore water pressure during seismic events; – the mechanical behaviour of soil under cyclic loads is characterised by strong non-linearity, dependence on past stress-history, reduction of shear stiffness with consequent hysteretic dissipation during the cycles, early irreversibility, etc. e.g. [5–9]. In contrast, a fully-reversible soil model, with constant visco-elastic soil properties (i.e. shear stiffness and damping ratio) over the duration of earthquake shaking, is adopted in the traditional frequency-domain analysis methods; * Corresponding author. Tel.: +39 080 5963693; fax: +39 080 5963675. E-mail addresses: a.amorosi@poliba.it (A. Amorosi), daniela.boldini@unibo.it (D. Boldini), gaetano.elia@ncl.ac.uk (G. Elia). 1 Tel.: +39 051 2090233; fax: +39 051 2090247. 2 Tel.: +44 191 2227934; fax: +44 191 2225322. 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.02.005 – many engineering problems cannot be scaled down to the onedimensional case but require a soil–structure interaction analysis in two- or three-dimensional conditions [10,11]. Time-domain ﬁnite element or ﬁnite difference schemes are nowadays available to solve the wave propagation problem in a more realistic way, accounting for the solid–ﬂuid interaction by means of a fully coupled effective stress formulation [4,12]. In those schemes, the behaviour of the soil can be described using either simple or sophisticated non-linear constitutive models of different level of complexity. These numerical approaches permit to include in one single analysis the evaluation of the site response and the corresponding interaction with the existing structures e.g. [13–21]. Such approaches are seldom adopted in engineering practice by non-expert users because both the model calibration procedures and the code usage protocols are often unclear or poorly documented, leading to unrealistic results and, as such, obscuring the possible beneﬁts of the time-domain numerical analysis. The main difﬁculties can be summarised as follow: – in linear ﬁnite element or ﬁnite difference analyses constant values of stiffness and viscous properties have to be selected according to a representative level of strain assumed to occur during the earthquake. Depending on the characteristics of the soil deposits these properties can be constant or variable with depth; – sophisticated constitutive formulations are not yet available in most commercial ﬁnite element or ﬁnite difference codes. 516 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 When implemented, their calibration is not straightforward and requires non-conventional geotechnical tests, often not included in standard geotechnical characterisation; – in time-domain schemes there are two sources of damping: viscous damping, generally introduced through the Rayleigh [22] formulation, and the hysteretic dissipation associated to the irreversible material response. The amount of hysteretic damping, which is frequency independent, is strictly related to the adopted material model. Viscous damping, which in contrast is frequency dependent, is added to the dynamic equations of motion to obtain stable numerical solutions and to account for the soil damping at small strains, if an hysteretic model is employed, or for the total amount of damping, if a non-dissipative constitutive formulation is adopted. In this respect, the main issues are associated to the selection of the appropriate target viscous damping ratio e.g. [23] and of the frequency range required by the Rayleigh damping function e.g. [24–26], as they can play a crucial role on FE results; – ﬁnite element or ﬁnite difference methods use a ﬁnite discretised domain to represent the inﬁnite continuous soil medium. Users are asked to deﬁne the extension of the ﬁnite domain, the characteristics of the spatial discretisation (i.e. the dimension and the type of elements) and the appropriate boundary conditions to artiﬁcially simulate the far-ﬁeld medium. While all these aspects are well understood in the context of static analyses, the literature concerning numerical analyses in dynamic conditions is less exhaustive; – the integration of the dynamic equations of motion can be performed adopting time-stepping schemes characterised by different accuracy, stability, algorithmic damping and run-time. In the paper, some of these features are investigated by comparing a set of 1D ground response numerical analyses performed in the frequency-domain with the corresponding time-domain based 2D ﬁnite element simulations. 1D frequency-domain analyses were performed modelling the soil as a single phase visco-elastic equivalent-linear medium. These results, besides the possible drawbacks they can contain, are taken as target solutions for the 2D ﬁnite element analyses based on linear visco-elasticity. This latter assumption is underpinned by the following hypothesis: the results of any 1D analysis performed in the frequency-domain and based on (equivalent) linear visco-elasticity should, in principle, coincide with the corresponding 2D ﬁnite element analysis per- formed in the time-domain assuming the same constitutive behaviour, provided an appropriate calibration of the parameters is adopted. The above comparison scheme is obviously no longer valid once more complex constitutive laws are adopted in the FE analysis as, for example, when plasticity is included in the formulation. In order to provide a useful framework for standard ﬁnite element users, the use of advanced constitutive models was avoided. Soil behaviour, in fact, is described either in terms of visco-elasticity, with viscous damping accounting for all the dissipative material behaviour, or by means of simple visco-elasto-plasticity assumptions. Realism is introduced in the investigation by considering a soil deposit characterised by variable stiffness and damping ratio with depth. The ﬁrst part of the paper outlines the geometrical and geotechnical characteristics of the ideal soil deposits under study, the main features of the adopted seismic motions and the criteria followed for their selection. It also describes the numerical models employed for 1D and 2D ground response simulations and summarises the results of equivalent-linear visco-elastic analyses performed in the frequency-domain. A new procedure for the calibration of the Rayleigh parameters in FE time-domain analyses is then proposed and validated. In the subsequent section, the paper investigates the effect of the introduction of plasticity in the soil constitutive assumption and illustrates the results of different strategies adopted in order to obtain a good matching between frequency and time-domain analyses. Finally, the inﬂuence of the boundary conditions, spatial discretisation and the time integration parameters on the results of the FE simulations is reviewed. 2. Outline of the idealised problem An ideal deposit of soft clay is assumed as the reference soil proﬁle, characterised by the following physical and mechanical parameters: plasticity index IP = 44%, unit weight of volume of the saturated soil c = 17 kN/m3, overconsolidation ratio in terms of mean effective stress R = 1.5, small-strain shear stiffness G0 = variable with depth, Poisson’s ratio t0 = 0.25, small-strain damping ratio D0 = 1.0%, coefﬁcient at rest K0 = 0.6, cohesion c0 = 0 and friction angle u0 = 24°. The water table is assumed at the ground surface. Fig. 1. Proﬁles of the small-strain shear stiffness G0 (a) and shear wave velocity VS (b). 517 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 Three different thicknesses were considered for this ideal soil deposit, namely 60, 120 and 240 m. The assumed proﬁle of the small-strain shear stiffness G0 with depth was calculated adopting the relationship proposed by Viggiani and Atkinson [27]: 0 n G0 p ¼S Rm pr pr ð1Þ where pr is a reference pressure taken equal to 1 kPa, p0 is the mean pressure (resulting from the lithostatic pressure and the application of a uniform load at the surface equal to 30 kPa), S, n and m are parameters depending on the plasticity index IP (here set equal to 550, 0.82 and 0.36, respectively, according to the correlations proposed by Viggiani and Atkinson [27] for ﬁne-grained soils) and R is the overconsolidation ratio in terms of mean effective stress. The variation of G0 with depth and the corresponding shear wave Table 1 Main characteristics of the adopted input seismic motions. Station Component Earthquake PGA (g) Duration (s) Dominant frequency fp (Hz) Tarcento Gilroy 2 Kalamata Port Island NS 50 X 90 Friuli (Italy), 1976 Coyote Lake (USA), 1979 Kalamata (Greece), 1986 Kobe (Japan), 1995 0.21 0.20 0.24 0.28 16.85 18.00 29.75 42.00 10.10 5.00 1.63 0.91 Fig. 2. Plot of the four selected acceleration time histories scaled at 0.35 g: (a) Kalamata, (b) Gilroy 2-050, (c) Tarcento and (d) Port Island. 518 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 velocity VS are reported in Fig. 1 where the three considered deposit thicknesses are also indicated. In the present study four different acceleration time histories were considered, namely Kalamata, Gilroy 2, Tarcento and Port Island. A summary of ground motion main characteristics is given in Table 1. All input signals were scaled to 0.35 g and were ﬁltered to prevent frequency levels higher than 12 Hz. This latter frequency was selected in order to limit the minimum element dimension adopted in the ﬁnite element analyses. The selected acceleration time histories after manipulation are given in Fig. 2 while the corresponding Fourier amplitude spectra and acceleration response spectra are shown in Figs. 3 and 4, respectively. It is apparent that the seismic signals are characterised by signiﬁcantly different frequency contents, in order to be representative of a wide range of possible events. The input seismic signal was considered applied at the rigid base of the deposit. G/G0 and variation of damping ratio D with shear strain level c were deﬁned according to typical results reported in the literature [9] as a function of IP (Fig. 5). In all EERA analyses the proﬁles of small strain stiffness shown in Fig. 1 were discretised by constant stiffness sub-strata of thickness ranging from a maximum of 4.5 m (at the base of the deeper 240 m model) to 1.0 m at the surface. The adopted FE codes allow to perform linear and non-linear analyses: under static conditions the Newton–Raphson integration scheme to solve the ﬁeld equations at the global level is employed, while the Generalised Newmark method [31] is adopted for the time integration under dynamic conditions. In this latter case the 3. Numerical models The ground response analyses were performed using the equivalent-linear visco-elastic code EERA [28] and the ﬁnite element (FE) codes SWANDYNE [29] and PLAXIS 2D [30]. The code EERA is based on the assumption of equivalent-linear visco-elastic soil behaviour. The equivalent-linear model assumes that the shear modulus G and damping ratio D are function of the shear strain amplitude c. The equivalent-linear analysis is repeated with updated values of G and D until the values of G and D are compatible with the so-called effective shear strain induced in all the layers of the numerical model. Modulus reduction curve Fig. 4. Elastic acceleration response spectra of the four selected acceleration time histories. Fig. 3. Frequency-ﬁltered Fourier amplitude spectra of the four selected acceleration time histories. 519 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 following values of the Newmark parameters were selected in all the analyses illustrated in this note: b1 = 0.6 and b2 = 0.605 for the solid phase and b1 = 0.6 for the ﬂuid phase. Those values ensure that the algorithm is unconditionally stable, while being dissipative only for the high frequency modes, as discussed in detail in Section 8. In order to perform a comparative analysis with the EERA results, a linear visco-elastic constitutive model was ﬁrst considered in the FE analyses. Plasticity was then introduced by adopting a non-associated visco-elasto-plastic constitutive assumption, with a Mohr–Coulomb yield criterion and a null dilatancy angle. Recently a number of authors have introduced plasticity into ground response analyses, applying advanced constitutive models and Fig. 5. Modulus reduction curve G/G0 and variation of damping ratio D with shear strain c adopted in EERA. demonstrating the importance of inelastic deformations and build of excess pore pressure during and after the seismic loading e.g. [32–35]. Viscous damping is introduced here by means of the Rayleigh formulation, whose damping matrix is deﬁned as follows: ½C ¼ aR ½M þ bR ½K ð2Þ where [M] and [K] are the mass and the stiffness matrix of the system, respectively. The coefﬁcients aR and bR are obtained considering the following relationship with the damping ratio D e.g. [36]: aR bR ¼ 2D xm þ xn xm xn 1 ð3Þ where xm and xn are the angular frequencies related to the frequency interval fmfn over which the viscous damping is equal to or lower than D. The mesh employed in SWANDYNE is characterised by a width equal to 5 m. The domain was discretised with a maximum number of 430 isoparametric quadrilateral ﬁnite elements with eight solid nodes and four ﬂuid nodes. The boundary conditions adopted for the static stages of the analyses were the standard ones: nodes at the bottom of the mesh were ﬁxed in both vertical and horizontal directions, while those along the lateral sides were only ﬁxed in the horizontal direction. In the dynamic analyses the bottom of the mesh was assumed to be rigid, while the nodes along the vertical sides were characterised by the same displacements (‘‘tied-nodes” boundary conditions). The code SWANDYNE performs fully coupled dynamic analysis, solving a unique set of equations for the solid and the ﬂuid at each time step (for details see Ref. [12]). This solution scheme requires the hydraulic conductivity and void ratio to be deﬁned as input parameters: they where here assumed constant with depth and equal to 1.0E-08 m/s and 0.7, respectively. Fig. 6. Example of mesh employed in the FE analyses performed with PLAXIS. Fig. 7. Results of the 1D ground response analysis performed with EERA (240-m thick deposit and Tarcento earthquake). 520 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 Base and lateral hydraulic boundaries were assumed as impervious while drained condition was imposed at the top of the mesh. Although water ﬂow was thus allowed within the mesh, such movement was not large enough to be detected, due to the sub- stantially undrained condition that characterises the dynamic analyses in relation to the earthquake duration and the assumed low hydraulic conductivity. The PLAXIS domain was discretised by 15-node plane strain triangular ﬁnite elements, characterised by a reduced integration formulation for the pore water pressures. The boundary conditions adopted for the static stages were the same as the ones used in SWANDYNE, while in the dynamic analyses the bottom of the mesh was assumed to be rigid and the lateral sides were characterised by the viscous boundaries proposed by Lysmer and Kuhlemeyer [37], with parameters a = 1.0 and b = 0.25. All PLAXIS analyses were performed under undrained conditions. This option, selected due to the incapability of the code to perform fully coupled dynamic analyses, made the PLAXIS results being consistent with those obtained by SWANDYNE, as discussed in the following sections. The characteristic dimension of the elements h in the SWANDYNE analyses and in the central portion of the domain in the PLAXIS analyses always satisﬁes the condition that the spacing of the ﬁnite element nodes, Dlnode, must be smaller than approximately one-tenth to one-eighth of the wavelength associated with the maximum frequency component fmax of the input wave [38]: Dlnode 6 kmin =ð810Þ ¼ V S;min =ð810Þfmax Fig. 8. G and D proﬁles assumed in the FE visco-elastic analyses on the basis of EERA results (240-m thick deposit and Tarcento earthquake). ð4Þ where VS,min is the lowest wave velocity. An example of the mesh employed in the PLAXIS analyses is sketched in Fig. 6. It is nowadays well established that the time discretisation can play a signiﬁcant role on the accuracy of dynamic ﬁnite element Fig. 9. Calibration strategies for the Rayleigh coefﬁcients (240-m thick deposit and Tarcento earthquake). A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 521 Fig. 10. Comparison between Fourier and response spectra obtained with EERA and FE visco-elastic analyses at surface, according to the two investigated calibration strategies for the Rayleigh coefﬁcients (240-m thick deposit and Tarcento earthquake). analyses e.g. [39]. A time step equal to 0.01 s was assumed in all the analyses proposed in this work. This value was selected based on a preliminary parametric study aimed at detecting the optimal time discretisation to achieve a satisfactory level of accuracy of the analyses and, at the same time, a reasonable calculation time to perform them. 4. Reference results The twelve cases considered in the present work, relative to three different thickness soil deposits (60, 120 and 240 m) for each selected input motion, were initially analysed using EERA. The results obtained from the equivalent-linear visco-elastic approach were assumed as reference for the evaluation of the corresponding FE solutions. As an example, the detailed frequency-domain solution for the case of the 240 m thick soil deposit exited at the bedrock by the Tarcento input motion is illustrated in the following. In particular, the proﬁles with depth of maximum shear strain (cmax), normalized shear modulus (G/G0), damping ratio (D) and maximum acceleration (amax) obtained at the end of the EERA analysis are reported in Fig. 7a–d, respectively. In this case, the predicted peak ground acceleration is equal to 0.43 g, with a magniﬁcation factor of 1.23 over the peak base amplitude. The shape of the maximum acceleration proﬁle clearly points out that more than one natural mode of the system is involved in the propagation process. During the seismic action, shear strains reach their highest values in the upper part of the deposit, Fig. 11. aR and bR proﬁles assumed in the FE visco-elastic analyses according to the two investigated calibration strategies for the Rayleigh coefﬁcients (240-m thick deposit and Tarcento earthquake). where the maximum damping ratio and the minimum shear stiffness are attained. 522 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 5. Calibration of stiffness and viscous parameters in FE analyses The simulation of the wave propagation problem through FE analyses employing a linear visco-elastic model variable with depth requires the appropriate deﬁnition of the elastic and viscous parameters for each sub-layer of the discretised deposit. In fact, it is well-known that the solution strongly depends on the assumed proﬁle of the stiffness and damping coefﬁcients with depth. In this paper, a recently developed calibration procedure of the visco-elastic parameters to be assumed in dynamic FE analyses is adopted [40]. In each FE analysis, G and D proﬁles were deﬁned in order to match the ones resulting from the corresponding EERA analyses. To this aim, the numerical models in PLAXIS and SWANDYNE were subdivided into the same number of layers employed in EERA and for each layer a value of G and D was selected with reference to the shear deformation level resulting from the EERA analyses at the corresponding depth. Fig. 8 shows the G and D proﬁles adopted in the FE analysis for the same case of Fig. 7. Rayleigh damping introduced in the simulations is deﬁned by selecting the coefﬁcients aR and bR, which depend on D and on the adopted frequency interval fmfn according to Eq. (3). Different possible calibration procedures were proposed in the literature to identify the interval fmfn. In particular, a well established one e.g. [24,25] suggests to select fm as the ﬁrst natural frequency of the deposit f1, while fn is assumed equal to n times fm, where n is the closest odd integer larger than the ratio fp/f1 between the predominant frequency of the input earthquake motion (fp) and the fundamental frequency of the soil deposit (f1). This latter assumption was based on the evidence that the higher modes of a shear beam are odd multiples of the fundamental mode of the beam. Recently, Kwok et al. [26] proposed to select, as a ﬁrst approximation, the ﬁrst mode of the site and ﬁve times this frequency for fm and fn, respectively. More generally, in order to obtain a good matching between the linear time-domain and the frequency-domain solutions, they suggested to identify the two frequencies through an iterative procedure. For the case of a 240 m thick deposit exited by the Tarcento earthquake, Fig. 9a shows the ampliﬁcation function of the signal between the bedrock and the surface obtained through the frequency-domain analysis, while the Fourier spectrum of the input motion is reported in Fig. 9b. Assuming, for example, a target damping ratio of 5%, the standard procedure would lead to the selection of fm = f1 = 0.29 Hz (equal to the fundamental frequency of the deposit, represented by the ﬁrst peak of the ampliﬁcation function) and fn = 10.15 Hz, being the ratio fp/f1 equal to 34.83 (and, therefore, n = 35). The corresponding Rayleigh damping curve is reported in the same ﬁgure with a solid line: it plots well below D = 5% target line. This condition leads to a signiﬁcant under-damped response of the system in the frequency range characterised by an ampliﬁcation factor larger than one, i.e. in the frequency interval in which the site effects would be more relevant. Fig. 10a illustrates the above issue by comparing the Fourier and response spectra of the acceleration, as obtained at the surface level by the FE visco-elastic analysis performed by SWANDYNE, with the corresponding EERA results. Fig. 12. Comparison between peak ground acceleration values obtained with EERA and FE visco-elastic analyses according to the two investigated calibration strategies for the Rayleigh coefﬁcients. A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 In order to obtain a better match between the linear time-domain and frequency-domain solutions, a new procedure for the selection of the two Rayleigh frequencies is here proposed. The ﬁrst natural frequency of the system which results as signiﬁcantly exited by the earthquake, fm, should be identiﬁed by comparing the EERA ampliﬁcation function and the Fourier spectrum of the input motion. In the case of Fig. 9 is evident that the Tarcento earthquake is characterised by a very low energy content for the ﬁrst two natural frequencies of the deposit, such that the third natural frequency of the system (equal to 1.12 Hz) should be selected as fm. As regards the second frequency fn, it should be identiﬁed considering the range over which the input motion is ampliﬁed during the propagation process: in particular, fn should be selected equal to the frequency where the ampliﬁcation function gets lower than one. For the case described in Fig. 9, the proposed procedure leads to a value of fn equal to 3.86 Hz, signiﬁcantly lower than the one obtained by the standard calibration procedure. The Rayleigh damping curve corresponding to the new values of fm and fn is plotted in Fig. 9a (dashed line). The resulting aR and bR proﬁles adopted in the FE analysis are shown in Fig. 11a and b, respectively, and 523 compared to those resulting from the standard calibration approach. Fig. 10b reports the Fourier and response spectra of the acceleration recorded at the surface during the FE visco-elastic analysis, performed with SWANDYNE employing the new procedure: the results are in fair agreement with those obtained for the same deposit and at the same depth by the frequency-domain based EERA analysis. More generally, adopting the proposed procedure for the deﬁnition of Rayleigh damping coefﬁcient proﬁles, a reasonably good matching between the EERA and the FE viscoelastic analysis results was achieved at each depth for all the investigated cases, both in terms of frequency response and acceleration time histories. A comparison between the peak ground accelerations obtained with linear time-domain analyses and those resulting from the corresponding frequency-domain solution is reported in Fig. 12, for all the cases analysed in this work. It can be observed that the use of the standard procedure may lead to signiﬁcant errors for increasing values of the ratio fp/f1 and of the soil deposit thickness (Fig. 12a). On the contrary, the difference between the results obtained adopting the new procedure and the corresponding EERA solutions is always lower than 10% (Fig. 12b). Fig. 13. Results of the 1D ground response analysis performed with EERA (60-m thick deposit and Gilroy 2-050 earthquake). Fig. 14. Shear modulus (a), damping ratio (b) and damping reduction (c) proﬁles for the different FE visco-elasto-plastic analyses (60-m thick deposit and Gilroy 2-050 earthquake). 524 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 Fig. 15. Comparison between response spectra obtained with EERA and the different investigated FE visco-elasto-plastic analyses at different depths (60-m thick deposit and Gilroy 2-050 earthquake). A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 525 Fig. 16. Comparison between Fourier and response spectra obtained during SWANDYNE and PLAXIS visco-elastic analyses at surface for different extensions of the mesh (120-m thick deposit and Kalamata earthquake). 6. Inﬂuence of plasticity In order to investigate the effects of non-linearity on the wave propagation process, plasticity was added to the FE visco-elastic analyses through a non-associated visco-elasto-plastic constitutive assumption, with a Mohr–Coulomb yield criterion and a null dilatancy angle. In the following, the case of a 60 m thick soil deposit, exited at the bedrock by the Gilroy 2-050 input motion, is discussed in detail, as considered representative of the entire set of results obtained with the FE analyses. The frequency-domain solution obtained by the code EERA for the selected case is summarised in Fig. 13. The wave propagation from the bedrock to the surface leads, in this case, to a peak ground acceleration of 0.42 g, with a magniﬁcation factor of 1.2 over the peak base amplitude. The amount of viscous damping resulting from the iterative equivalent-linear procedure attains an average value of about 7.5%. All the analyses performed in the time-domain were carried out with the code SWANDYNE. The stiffness and damping proﬁle were selected according to the calibration procedure discussed in the previous Section. In particular, the Rayleigh parameters assume in this case the values of fm = f1 = 0.54 Hz and fn = 4.37 Hz. Adopting the same stiffness proﬁle resulting from the EERA analysis (Fig. 14a), three different hypotheses concerning the amount of viscous damping to be introduced in the non-linear time-domain analyses were explored. In the ﬁrst simulation (named FE_vep_1), the target damping ratio at each depth of the column was selected equal to the corresponding value obtained by the EERA analysis, i.e. assuming as negligible the plasticity-related hysteretic dissipation provided by the constitutive model (Fig. 14b). In the second simulation (FE_vep_2), the amount of viscous damping was set equal to 60% of that adopted in the previous case (Fig. 14b). As illustrated in Fig. 14c, this implies that the reduction of the target damping ratios is, in this case, more pronounced in the upper part of the clayey deposit as compared to the remaining portion of it. Finally, in the third FE analysis (FE_vep_3) the EERA damping proﬁle was reduced at each depth by DD = 3%, resulting in the proﬁle also shown in Fig. 14b. Fig. 15 reports the comparison between the results of the three FE visco-elasto-plastic analyses and the corresponding EERA simulation in terms of response spectra obtained at different depths along the deposit. All the plasticity-based analyses show a contraction of the spectra as compared to the EERA one, this being related to the additional damping supplied by the Mohr–Coulomb model. This effect is more pronounced in the uppermost portion of the deposit, between 0 and 15 m depth, where the shear strains attain 526 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 Fig. 17. Comparison between Fourier and response spectra obtained with EERA and FE visco-elastic analyses at surface for different values of Newmark parameters (240-m thick deposit and Tarcento earthquake). 527 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 their maximum values. The frequency range where this effect is prominent is between 3.4 Hz and 5.5 Hz. The non-linearity induced by the plasticity assumption does not signiﬁcantly modify the fundamental modes of vibration of the soil deposit. None of the three proposed approach for the reduction of the viscous damping is able to balance the introduction of the hysteretic dissipation, at least when the results are compared to those obtained by EERA. Concerning this latter outcome, it is worth remarking that the EERA results might not be the right term of comparison when strong motions induce large strain in a soil deposit. In this last circumstance, plasticity might prevail and bias the picture traditionally obtained by means of visco-elastic analyses. Under these latter conditions, permanent displacement and corresponding variation of the effective stress state occur, signiﬁcantly modifying the soil–structure interaction in any geotechnical context e.g. [41]. 7. Inﬂuence of boundary conditions and spatial discretisation The analyses performed with the code SWANDYNE adopting the 5-m wide mesh characterised by ‘‘tied-nodes” boundaries (see Sections 5 and 6) are representative of ideal 1D problems. For 2D and 3D problems wider meshes should be employed and the hypothesis of tied horizontal displacements of the lateral boundaries needs to be abandoned. Therefore questions concerning the appropriate lateral extension of the FE mesh arise. A numerical investigation regarding this issue was performed with the code PLAXIS adopting the viscous boundaries proposed by Lysmer and Kuhlemeyer [37] and meshes characterised by different width. The twelve visco-elastic analyses described in Section 5 were re-simulated assuming the standard values of the Lysmer and Kuhlemeyer parameters (a = 1.0 and b = 0.25). The horizontal dimension of the mesh, L, was assumed equal to 2, 4 and 8 times the thickness H. In this context, therefore, the results obtained by the code SWANDYNE are assumed as reference. Fig. 16 shows, as an example, the comparison between Fourier and response spectra at the surface obtained for a 120-m thick deposit excited by the Kalamata earthquake. The similarity between the results of the PLAXIS analysis characterised by L = 8H and the reference analysis is clearly recognizable. A satisfactory agreement between the analyses is already attained for L = 4H. This value can be considered as a good compromise between accuracy and time required to perform the analysis of a 2D boundary value problem. The same trend was indeed observed in all the other eleven investigated cases. In addition, no signiﬁcant differences were identiﬁed in the numerical results when adopting different values of the Lysmer and Kuhlemeyer parameters a and b in the range 01. 8. Inﬂuence of time integration parameters According to the Generalised Newmark time-stepping proce_ vectors in a solid dure [31], the displacement (u) and velocity (u) node at time n + 1 are expressed as: unþ1 ¼ un þ u_ n Dt þ 1 1 € n þ b2 u € nþ1 Dt 2 ð1 b2 Þu 2 2 € n þ b1 u € nþ1 Dt u_ nþ1 ¼ u_ n þ ½ð1 b1 Þu ð5Þ ð6Þ while the pore pressure (p) vector in a ﬂuid node, at the same time n + 1, can be obtained from: pnþ1 ¼ pn þ ð1 b1 Þp_ n þ b1 p_ nþ1 Dt ð7Þ The algorithm is unconditionally stable if the following conditions apply: b1 P 1 ; 2 b2 P 2 1 1 þ b1 ; 2 2 b1 P 1 2 ð8Þ The choice of b1 = b2 = b1 = 0.5 (corresponding to the higher order accurate trapezoidal scheme) guarantees the stability of the time-stepping scheme for any value of Dt (i.e. the algorithm remains implicit) and does not provide any numerical (or algorithmic) damping during the integration of the governing equations. In this case, numerical oscillations may occur during the analysis if no physical (viscous or hysteretic) damping is present [12]. As such, some numerical damping is typically introduced adopting coefﬁcient values larger than 0.5, consistently with condition (8). All the time-domain simulations illustrated in this note were performed assuming a set of Newmark parameters which leads to a small amount of algorithmic dissipation (see Section 3). To assess the inﬂuence of the numerical damping on the FE results, the case of a 240 m thick deposit exited by the Tarcento earthquake was studied with the code SWANDYNE, varying the values of the parameter b1 in the range 0.5–0.9, setting b2 according to condition (8) and assuming b1 = b1. The comparison between Fourier and response spectra at the ground surface obtained with the different Newmark parameters and the corresponding EERA reference results is reported in Fig. 17. The ﬁgure clearly indicates that the numerical dissipation introduced by the time-stepping scheme is more pronounced at high frequencies. The FE analysis performed with an un-damped time integration scheme (b1 = 0.5) gives the best agreement with the frequency-domain result in terms of peak ground acceleration, but tends to over-predict the energy content in the range 1–3 Hz. The simulation characterised by b1 = 0.6 (the adopted value in the analyses discussed in the previous Sections) represents a good compromise between a satisfactory agreement with EERA in terms of frequency response and a small under-estimation of the peak ground acceleration. Increasing values of b1 induce an overdamped response, especially for the high frequency modes, leading to signiﬁcantly reduced peak ground accelerations. 9. Conclusions This paper describes a set of 2D ﬁnite element analyses for the simulation of the seismic ground response of a clayey deposit. Some of the several factors potentially inﬂuencing the numerical results are highlighted and critically discussed. In particular, the stiffness values and the amount of viscous damping in visco-elastic analyses, the hysteretic damping when plasticity is added to the soil model, the spatial and time discretisation and the nature of boundary conditions are examined. To generalise the investigation, a parametric study was carried out using four earthquake signals, three deposits characterised by different heights, two ﬁnite element codes and two different boundary conditions. Most of the analyses were performed using a linear visco-elastic soil model characterised by the Rayleigh formulation for the viscous damping. The calibration of the Rayleigh coefﬁcients as well as the selection of the appropriate mobilised stiffness represent critical issues for this kind of simulations. In the note the validation of ﬁnite element visco-elastic analyses is performed comparing the results with those obtained by equivalent-linear visco-elastic analyses performed in the frequency-domain by the code EERA. Those latter are thus taken as reference in the validation procedure. Using this approach, the paper shows that the traditionally adopted procedures for the calibration of the Rayleigh coefﬁcients can lead to large overestimation of the peak ground acceleration. A novel calibration procedure is here proposed and discussed: in this case the 528 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 results of the FE analyses compare nicely with those obtained by the frequency-domain approach. A second set of FE analyses were carried out introducing plasticity in the soil constitutive formulation. The appropriate selection of the viscous damping to be added in the model was subjected to further investigation. Different strategies were attempted in order to optimise the balance between the hysteretic dissipation and the viscous component of the damping. None of the proposed approaches allowed to achieve a good matching between the FE analyses and the corresponding frequency-domain ones. Concerning this latter outcome, it is worth remarking that the EERA results should not be considered as the right term of comparison when modelling strong motion earthquakes, as those selected for this study. In fact, intense shaking results in large and partly irreversible strains associated with modiﬁcation of the effective stress state induced by excess pore pressures build-up. Those features cannot be accounted for by visco-elasticity based constitutive laws, as that adopted in EERA, making the plasticity-based time domain approach more realistic. The simulations were performed by the ﬁnite element code SWANDYNE, adopting a 5-m wide mesh characterised by tied nodes at the lateral boundaries, thus limiting the case to the 1D condition. The possibility of performing 2D ﬁnite element simulations was investigated by re-running the numerical analyses with the ﬁnite element code PLAXIS, adopting the Lysmer and Kuhlemeyer conditions at the lateral boundaries. The match between the results of the two different geometrical conﬁgurations assumed in the two codes were obtained employing 2D meshes characterised by a width-height ratio larger than eight, while satisfactory results were already achieved for a ratio equal to four. No inﬂuence of the values of the Lysmer and Kuhlemeyer coefﬁcients was observed in the 2D analyses. Finally, accuracy and damping characteristics of the time integration algorithm were analysed. 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