Parametric study on seismic ground response by finite element modelling

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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, Italyb University of Bologna, Department of Civil, Environmental and Material Engineering, viale Terracini 28, 40136 Bologna, Italyc Newcastle University, School of Civil Engineering & Geosciences, Drummond Building, NE1 7RU Newcastle upon Tyne, United Kingdom

a r t i c l e i n f o

Article history:Received 22 September 2009Received in revised form 5 February 2010Accepted 10 February 2010Available online 21 March 2010

Keywords:Seismic ground response analysisConstitutive modelsNumerical modellingFinite 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, performedadopting linear visco-elastic and visco-elasto-plastic constitutive models, are presented. The viscous andlinear elastic parameters are selected according to a novel calibration strategy, leading to FE results com-parable to those obtained by 1D equivalent-linear visco-elastic frequency-domain analyses. The influenceof plasticity on the numerical results is also investigated, with particular reference to the relationbetween the hysteretic and viscous damping effects. Finally, different boundary conditions, spatial dis-cretisation and time integration parameters are considered and their role on the FE results discussed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The site response analysis has traditionally been performedusing one-dimensional frequency-domain numerical schemebased on the equivalent visco-elastic approach [1–3]. This ap-proach has successfully been adopted in the last 30 years and itis widely accepted in the engineering practice, although its limita-tions are well-known. In particular, concerning this latter aspect, itis worth remarking the following points:

– soil behaviour is controlled by effective stresses while a totalstress approach is implemented in most equivalent visco-elas-tic schemes, disregarding the soil–fluid interaction [4] andpossible build-up of excess pore water pressure during seismicevents;

– the mechanical behaviour of soil under cyclic loads is character-ised by strong non-linearity, dependence on past stress-history,reduction of shear stiffness with consequent hysteretic dissipa-tion during the cycles, early irreversibility, etc. e.g. [5–9]. Incontrast, a fully-reversible soil model, with constant visco-elas-tic soil properties (i.e. shear stiffness and damping ratio) overthe duration of earthquake shaking, is adopted in the traditionalfrequency-domain analysis methods;

– many engineering problems cannot be scaled down to the one-dimensional case but require a soil–structure interaction analy-sis in two- or three-dimensional conditions [10,11].

Time-domain finite element or finite difference schemes arenowadays available to solve the wave propagation problem in amore realistic way, accounting for the solid–fluid interaction bymeans of a fully coupled effective stress formulation [4,12]. Inthose schemes, the behaviour of the soil can be described usingeither simple or sophisticated non-linear constitutive models ofdifferent level of complexity. These numerical approaches permitto include in one single analysis the evaluation of the site responseand the corresponding interaction with the existing structures e.g.[13–21].

Such approaches are seldom adopted in engineering practice bynon-expert users because both the model calibration proceduresand the code usage protocols are often unclear or poorly docu-mented, leading to unrealistic results and, as such, obscuring thepossible benefits of the time-domain numerical analysis. The maindifficulties can be summarised as follow:

– in linear finite element or finite difference analyses constantvalues of stiffness and viscous properties have to be selectedaccording to a representative level of strain assumed to occurduring the earthquake. Depending on the characteristics ofthe soil deposits these properties can be constant or variablewith depth;

– sophisticated constitutive formulations are not yet availablein most commercial finite element or finite difference codes.

0266-352X/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compgeo.2010.02.005

* Corresponding author. Tel.: +39 080 5963693; fax: +39 080 5963675.E-mail addresses: [email protected] (A. Amorosi), [email protected] (D.

Boldini), [email protected] (G. Elia).1 Tel.: +39 051 2090233; fax: +39 051 2090247.2 Tel.: +44 191 2227934; fax: +44 191 2225322.

Computers and Geotechnics 37 (2010) 515–528

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When implemented, their calibration is not straightforward andrequires non-conventional geotechnical tests, often notincluded in standard geotechnical characterisation;

– in time-domain schemes there are two sources of damping: vis-cous damping, generally introduced through the Rayleigh [22]formulation, and the hysteretic dissipation associated to theirreversible material response. The amount of hysteretic damp-ing, which is frequency independent, is strictly related to theadopted material model. Viscous damping, which in contrastis frequency dependent, is added to the dynamic equations ofmotion to obtain stable numerical solutions and to accountfor the soil damping at small strains, if an hysteretic model isemployed, or for the total amount of damping, if a non-dissipa-tive constitutive formulation is adopted. In this respect, themain issues are associated to the selection of the appropriatetarget viscous damping ratio e.g. [23] and of the frequencyrange 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 discre-tised domain to represent the infinite continuous soil medium.Users are asked to define the extension of the finite domain, thecharacteristics of the spatial discretisation (i.e. the dimensionand the type of elements) and the appropriate boundary condi-tions to artificially simulate the far-field medium. While allthese aspects are well understood in the context of static anal-yses, the literature concerning numerical analyses in dynamicconditions is less exhaustive;

– the integration of the dynamic equations of motion can be per-formed adopting time-stepping schemes characterised by dif-ferent accuracy, stability, algorithmic damping and run-time.

In the paper, some of these features are investigated by compar-ing a set of 1D ground response numerical analyses performed inthe frequency-domain with the corresponding time-domain based2D finite element simulations. 1D frequency-domain analyseswere performed modelling the soil as a single phase visco-elasticequivalent-linear medium. These results, besides the possibledrawbacks they can contain, are taken as target solutions for the2D finite element analyses based on linear visco-elasticity. This lat-ter assumption is underpinned by the following hypothesis: the re-sults of any 1D analysis performed in the frequency-domain andbased 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 behav-iour, provided an appropriate calibration of the parameters isadopted. The above comparison scheme is obviously no longer va-lid once more complex constitutive laws are adopted in the FEanalysis as, for example, when plasticity is included in theformulation.

In order to provide a useful framework for standard finite ele-ment users, the use of advanced constitutive models was avoided.Soil behaviour, in fact, is described either in terms of visco-elastic-ity, with viscous damping accounting for all the dissipative mate-rial behaviour, or by means of simple visco-elasto-plasticityassumptions. Realism is introduced in the investigation by consid-ering a soil deposit characterised by variable stiffness and dampingratio with depth.

The first part of the paper outlines the geometrical and geotech-nical characteristics of the ideal soil deposits under study, the mainfeatures of the adopted seismic motions and the criteria followedfor their selection. It also describes the numerical models em-ployed for 1D and 2D ground response simulations and summa-rises the results of equivalent-linear visco-elastic analysesperformed in the frequency-domain. A new procedure for the cal-ibration of the Rayleigh parameters in FE time-domain analyses isthen proposed and validated.

In the subsequent section, the paper investigates the effect ofthe introduction of plasticity in the soil constitutive assumptionand illustrates the results of different strategies adopted in orderto obtain a good matching between frequency and time-domainanalyses.

Finally, the influence of the boundary conditions, spatial dis-cretisation and the time integration parameters on the results ofthe FE simulations is reviewed.

2. Outline of the idealised problem

An ideal deposit of soft clay is assumed as the reference soil pro-file, characterised by the following physical and mechanicalparameters: plasticity index IP = 44%, unit weight of volume ofthe saturated soil c = 17 kN/m3, overconsolidation ratio in termsof mean effective stress R = 1.5, small-strain shear stiffnessG0 = variable with depth, Poisson’s ratio t0 = 0.25, small-straindamping ratio D0 = 1.0%, coefficient at rest K0 = 0.6, cohesionc0 = 0 and friction angle u0 = 24�. The water table is assumed atthe ground surface.

Fig. 1. Profiles of the small-strain shear stiffness G0 (a) and shear wave velocity VS (b).

516 A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528


Three different thicknesses were considered for this ideal soildeposit, namely 60, 120 and 240 m.

The assumed profile of the small-strain shear stiffness G0 withdepth was calculated adopting the relationship proposed by Viggi-ani and Atkinson [27]:

G0

pr¼ S � p0

pr

� �n

� Rm ð1Þ

where pr is a reference pressure taken equal to 1 kPa, p0 is the meanpressure (resulting from the lithostatic pressure and the applicationof a uniform load at the surface equal to 30 kPa), S, n and m areparameters depending on the plasticity index IP (here set equal to550, 0.82 and 0.36, respectively, according to the correlations pro-posed by Viggiani and Atkinson [27] for fine-grained soils) and Ris the overconsolidation ratio in terms of mean effective stress.The variation of G0 with depth and the corresponding shear wave

Table 1Main characteristics of the adopted input seismic motions.

Station Component Earthquake PGA (g) Duration (s) Dominant frequency fp (Hz)

Tarcento NS Friuli (Italy), 1976 0.21 16.85 10.10Gilroy 2 50 Coyote Lake (USA), 1979 0.20 18.00 5.00Kalamata X Kalamata (Greece), 1986 0.24 29.75 1.63Port Island 90 Kobe (Japan), 1995 0.28 42.00 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.

A. Amorosi et al. / Computers and Geotechnics 37 (2010) 515–528 517


velocity VS are reported in Fig. 1 where the three considered depositthicknesses are also indicated.

In the present study four different acceleration time historieswere considered, namely Kalamata, Gilroy 2, Tarcento and Port Is-land. A summary of ground motion main characteristics is given inTable 1. All input signals were scaled to 0.35 g and were filtered toprevent frequency levels higher than 12 Hz. This latter frequencywas selected in order to limit the minimum element dimensionadopted in the finite element analyses. The selected accelerationtime histories after manipulation are given in Fig. 2 while the cor-responding Fourier amplitude spectra and acceleration responsespectra are shown in Figs. 3 and 4, respectively. It is apparent thatthe seismic signals are characterised by significantly different fre-quency contents, in order to be representative of a wide range ofpossible events.

The input seismic signal was considered applied at the rigidbase of the deposit.

3. Numerical models

The ground response analyses were performed using the equiv-alent-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-linearvisco-elastic soil behaviour. The equivalent-linear model assumesthat the shear modulus G and damping ratio D are function ofthe shear strain amplitude c. The equivalent-linear analysis is re-peated with updated values of G and D until the values of G andD are compatible with the so-called effective shear strain inducedin all the layers of the numerical model. Modulus reduction curve

G/G0 and variation of damping ratio D with shear strain level cwere 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 shownin Fig. 1 were discretised by constant stiffness sub-strata of thick-ness ranging from a maximum of 4.5 m (at the base of the deeper240 m model) to 1.0 m at the surface.

The adopted FE codes allow to perform linear and non-linearanalyses: under static conditions the Newton–Raphson integrationscheme to solve the field equations at the global level is employed,while the Generalised Newmark method [31] is adopted for thetime integration under dynamic conditions. In this latter case the

Fig. 3. Frequency-filtered Fourier amplitude spectra of the four selected acceleration time histories.

Fig. 4. Elastic acceleration response spectra of the four selected acceleration timehistories.



following values of the Newmark parameters were selected in allthe analyses illustrated in this note: b1 = 0.6 and b2 = 0.605 forthe solid phase and b�1 = 0.6 for the fluid phase. Those values ensurethat the algorithm is unconditionally stable, while being dissipa-tive only for the high frequency modes, as discussed in detail inSection 8.

In order to perform a comparative analysis with the EERA re-sults, a linear visco-elastic constitutive model was first consideredin the FE analyses. Plasticity was then introduced by adopting anon-associated visco-elasto-plastic constitutive assumption, witha Mohr–Coulomb yield criterion and a null dilatancy angle. Re-cently a number of authors have introduced plasticity into groundresponse analyses, applying advanced constitutive models and

demonstrating the importance of inelastic deformations and buildof excess pore pressure during and after the seismic loading e.g.[32–35].

Viscous damping is introduced here by means of the Rayleighformulation, whose damping matrix is defined as follows:

½C� ¼ aR½M� þ bR½K� ð2Þ

where [M] and [K] are the mass and the stiffness matrix of the sys-tem, respectively. The coefficients aR and bR are obtained consider-ing the following relationship with the damping ratio D e.g. [36]:

aR

bR

� �¼ 2D

xm þxn

xmxn

1

� �ð3Þ

where xm and xn are the angular frequencies related to the fre-quency interval fm�fn over which the viscous damping is equal toor lower than D.

The mesh employed in SWANDYNE is characterised by a widthequal to 5 m. The domain was discretised with a maximum num-ber of 430 isoparametric quadrilateral finite elements with eightsolid nodes and four fluid nodes. The boundary conditions adoptedfor the static stages of the analyses were the standard ones: nodesat the bottom of the mesh were fixed in both vertical and horizon-tal directions, while those along the lateral sides were only fixed inthe horizontal direction. In the dynamic analyses the bottom of themesh was assumed to be rigid, while the nodes along the verticalsides were characterised by the same displacements (‘‘tied-nodes”boundary conditions). The code SWANDYNE performs fully cou-pled dynamic analysis, solving a unique set of equations for the so-lid and the fluid at each time step (for details see Ref. [12]). Thissolution scheme requires the hydraulic conductivity and void ratioto be defined as input parameters: they where here assumed con-stant with depth and equal to 1.0E-08 m/s and 0.7, respectively.

Fig. 5. Modulus reduction curve G/G0 and variation of damping ratio D with shearstrain c adopted in EERA.

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).



Base and lateral hydraulic boundaries were assumed as imperviouswhile drained condition was imposed at the top of the mesh.Although water flow was thus allowed within the mesh, suchmovement was not large enough to be detected, due to the sub-

stantially undrained condition that characterises the dynamicanalyses in relation to the earthquake duration and the assumedlow hydraulic conductivity.

The PLAXIS domain was discretised by 15-node plane strain tri-angular finite elements, characterised by a reduced integration for-mulation for the pore water pressures. The boundary conditionsadopted for the static stages were the same as the ones used inSWANDYNE, while in the dynamic analyses the bottom of themesh was assumed to be rigid and the lateral sides were character-ised by the viscous boundaries proposed by Lysmer and Kuhlemey-er [37], with parameters a = 1.0 and b = 0.25.

All PLAXIS analyses were performed under undrained condi-tions. This option, selected due to the incapability of the code toperform fully coupled dynamic analyses, made the PLAXIS resultsbeing consistent with those obtained by SWANDYNE, as discussedin the following sections.

The characteristic dimension of the elements h in the SWAN-DYNE analyses and in the central portion of the domain in thePLAXIS analyses always satisfies the condition that the spacing ofthe finite element nodes, Dlnode, must be smaller than approxi-mately one-tenth to one-eighth of the wavelength associated withthe maximum frequency component fmax of the input wave [38]:

Dlnode 6 kmin=ð8�10Þ ¼ VS;min=ð8�10Þfmax ð4Þ

where VS,min is the lowest wave velocity. An example of the meshemployed in the PLAXIS analyses is sketched in Fig. 6.

It is nowadays well established that the time discretisation canplay a significant role on the accuracy of dynamic finite element

Fig. 8. G and D profiles assumed in the FE visco-elastic analyses on the basis ofEERA results (240-m thick deposit and Tarcento earthquake).

Fig. 9. 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 allthe analyses proposed in this work. This value was selected basedon a preliminary parametric study aimed at detecting the optimaltime discretisation to achieve a satisfactory level of accuracy of theanalyses and, at the same time, a reasonable calculation time toperform them.

4. Reference results

The twelve cases considered in the present work, relative tothree different thickness soil deposits (60, 120 and 240 m) for eachselected input motion, were initially analysed using EERA. The re-sults obtained from the equivalent-linear visco-elastic approachwere assumed as reference for the evaluation of the correspondingFE solutions.

As an example, the detailed frequency-domain solution for thecase of the 240 m thick soil deposit exited at the bedrock by theTarcento input motion is illustrated in the following. In particular,the profiles with depth of maximum shear strain (cmax), normal-ized shear modulus (G/G0), damping ratio (D) and maximum accel-eration (amax) obtained at the end of the EERA analysis are reportedin Fig. 7a–d, respectively.

In this case, the predicted peak ground acceleration is equal to0.43 g, with a magnification factor of 1.23 over the peak baseamplitude. The shape of the maximum acceleration profile clearlypoints out that more than one natural mode of the system is in-volved in the propagation process. During the seismic action, shearstrains reach their highest values in the upper part of the deposit,

where the maximum damping ratio and the minimum shear stiff-ness are attained.

Fig. 10. Comparison between Fourier and response spectra obtained with EERA and FE visco-elastic analyses at surface, according to the two investigated calibrationstrategies for the Rayleigh coefficients (240-m thick deposit and Tarcento earthquake).

Fig. 11. aR and bR profiles assumed in the FE visco-elastic analyses according to thetwo investigated calibration strategies for the Rayleigh coefficients (240-m thickdeposit and Tarcento earthquake).



5. Calibration of stiffness and viscous parameters in FE analyses

The simulation of the wave propagation problem through FEanalyses employing a linear visco-elastic model variable withdepth requires the appropriate definition of the elastic and viscousparameters for each sub-layer of the discretised deposit. In fact, itis well-known that the solution strongly depends on the assumedprofile of the stiffness and damping coefficients with depth.

In this paper, a recently developed calibration procedure of thevisco-elastic parameters to be assumed in dynamic FE analyses isadopted [40]. In each FE analysis, G and D profiles were definedin order to match the ones resulting from the corresponding EERAanalyses. To this aim, the numerical models in PLAXIS and SWAN-DYNE were subdivided into the same number of layers employedin EERA and for each layer a value of G and D was selected with ref-erence to the shear deformation level resulting from the EERAanalyses at the corresponding depth. Fig. 8 shows the G and D pro-files adopted in the FE analysis for the same case of Fig. 7. Rayleighdamping introduced in the simulations is defined by selecting thecoefficients aR and bR, which depend on D and on the adopted fre-quency interval fm�fn according to Eq. (3). Different possible cali-bration procedures were proposed in the literature to identifythe interval fm�fn. In particular, a well established one e.g.[24,25] suggests to select fm as the first natural frequency of the de-posit f1, while fn is assumed equal to n times fm, where n is the clos-est odd integer larger than the ratio fp/f1 between the predominantfrequency of the input earthquake motion (fp) and the fundamentalfrequency of the soil deposit (f1). This latter assumption was based

on the evidence that the higher modes of a shear beam are oddmultiples of the fundamental mode of the beam. Recently, Kwoket al. [26] proposed to select, as a first approximation, the firstmode of the site and five times this frequency for fm and fn, respec-tively. More generally, in order to obtain a good matching betweenthe linear time-domain and the frequency-domain solutions, theysuggested to identify the two frequencies through an iterativeprocedure.

For the case of a 240 m thick deposit exited by the Tarcen-to earthquake, Fig. 9a shows the amplification function of thesignal between the bedrock and the surface obtained throughthe frequency-domain analysis, while the Fourier spectrum ofthe input motion is reported in Fig. 9b. Assuming, for example,a target damping ratio of 5%, the standard procedure wouldlead to the selection of fm = f1 = 0.29 Hz (equal to the funda-mental frequency of the deposit, represented by the first peakof the amplification function) and fn = 10.15 Hz, being the ratiofp/f1 equal to 34.83 (and, therefore, n = 35). The correspondingRayleigh damping curve is reported in the same figure witha solid line: it plots well below D = 5% target line. This condi-tion leads to a significant under-damped response of the sys-tem in the frequency range characterised by an amplificationfactor larger than one, i.e. in the frequency interval in whichthe site effects would be more relevant. Fig. 10a illustratesthe above issue by comparing the Fourier and response spectraof the acceleration, as obtained at the surface level by the FEvisco-elastic analysis performed by SWANDYNE, with the corre-sponding 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 forthe Rayleigh coefficients.



In order to obtain a better match between the linear time-do-main and frequency-domain solutions, a new procedure for theselection of the two Rayleigh frequencies is here proposed. The firstnatural frequency of the system which results as significantly ex-ited by the earthquake, fm, should be identified by comparing theEERA amplification function and the Fourier spectrum of the inputmotion. In the case of Fig. 9 is evident that the Tarcento earthquakeis characterised by a very low energy content for the first two nat-ural frequencies of the deposit, such that the third natural fre-quency of the system (equal to 1.12 Hz) should be selected as fm.As regards the second frequency fn, it should be identified consid-ering the range over which the input motion is amplified duringthe propagation process: in particular, fn should be selected equalto the frequency where the amplification function gets lower thanone. For the case described in Fig. 9, the proposed procedure leadsto a value of fn equal to 3.86 Hz, significantly lower than the oneobtained by the standard calibration procedure. The Rayleighdamping curve corresponding to the new values of fm and fn is plot-ted in Fig. 9a (dashed line). The resulting aR and bR profiles adoptedin the FE analysis are shown in Fig. 11a and b, respectively, and

compared to those resulting from the standard calibration ap-proach. Fig. 10b reports the Fourier and response spectra of theacceleration recorded at the surface during the FE visco-elasticanalysis, performed with SWANDYNE employing the new proce-dure: the results are in fair agreement with those obtained forthe same deposit and at the same depth by the frequency-domainbased EERA analysis. More generally, adopting the proposed proce-dure for the definition of Rayleigh damping coefficient profiles, areasonably good matching between the EERA and the FE visco-elastic analysis results was achieved at each depth for all the inves-tigated cases, both in terms of frequency response and accelerationtime histories. A comparison between the peak ground accelera-tions obtained with linear time-domain analyses and those result-ing from the corresponding frequency-domain solution is reportedin Fig. 12, for all the cases analysed in this work. It can be observedthat the use of the standard procedure may lead to significant er-rors for increasing values of the ratio fp/f1 and of the soil depositthickness (Fig. 12a). On the contrary, the difference between theresults obtained adopting the new procedure and the correspond-ing 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 2-050earthquake).



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 andGilroy 2-050 earthquake).



6. Influence of plasticity

In order to investigate the effects of non-linearity on the wavepropagation process, plasticity was added to the FE visco-elasticanalyses through a non-associated visco-elasto-plastic constitutiveassumption, with a Mohr–Coulomb yield criterion and a null dilat-ancy 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 dis-cussed in detail, as considered representative of the entire set of re-sults obtained with the FE analyses. The frequency-domainsolution obtained by the code EERA for the selected case is summa-rised in Fig. 13. The wave propagation from the bedrock to the sur-face leads, in this case, to a peak ground acceleration of 0.42 g, witha magnification factor of 1.2 over the peak base amplitude. Theamount of viscous damping resulting from the iterative equiva-lent-linear procedure attains an average value of about 7.5%.

All the analyses performed in the time-domain were carried outwith the code SWANDYNE. The stiffness and damping profile wereselected according to the calibration procedure discussed in theprevious Section. In particular, the Rayleigh parameters assumein this case the values of fm = f1 = 0.54 Hz and fn = 4.37 Hz.

Adopting the same stiffness profile resulting from the EERAanalysis (Fig. 14a), three different hypotheses concerning the

amount of viscous damping to be introduced in the non-lineartime-domain analyses were explored. In the first simulation(named FE_vep_1), the target damping ratio at each depth ofthe column was selected equal to the corresponding value ob-tained by the EERA analysis, i.e. assuming as negligible the plas-ticity-related hysteretic dissipation provided by the constitutivemodel (Fig. 14b). In the second simulation (FE_vep_2), theamount of viscous damping was set equal to 60% of that adoptedin the previous case (Fig. 14b). As illustrated in Fig. 14c, this im-plies that the reduction of the target damping ratios is, in thiscase, more pronounced in the upper part of the clayey depositas compared to the remaining portion of it. Finally, in the thirdFE analysis (FE_vep_3) the EERA damping profile was reducedat each depth by DD = 3%, resulting in the profile also shown inFig. 14b.

Fig. 15 reports the comparison between the results of the threeFE visco-elasto-plastic analyses and the corresponding EERA simu-lation in terms of response spectra obtained at different depthsalong the deposit. All the plasticity-based analyses show a contrac-tion of the spectra as compared to the EERA one, this being relatedto the additional damping supplied by the Mohr–Coulomb model.This effect is more pronounced in the uppermost portion of the de-posit, between 0 and 15 m depth, where the shear strains attain

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).



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-mthick deposit and Tarcento earthquake).



their maximum values. The frequency range where this effect isprominent is between 3.4 Hz and 5.5 Hz. The non-linearity inducedby the plasticity assumption does not significantly modify the fun-damental modes of vibration of the soil deposit.

None of the three proposed approach for the reduction of theviscous damping is able to balance the introduction of the hyster-etic dissipation, at least when the results are compared to thoseobtained by EERA. Concerning this latter outcome, it is worthremarking that the EERA results might not be the right term ofcomparison when strong motions induce large strain in a soil de-posit. In this last circumstance, plasticity might prevail and biasthe picture traditionally obtained by means of visco-elastic analy-ses. Under these latter conditions, permanent displacement andcorresponding variation of the effective stress state occur, signifi-cantly modifying the soil–structure interaction in any geotechnicalcontext e.g. [41].

7. Influence of boundary conditions and spatial discretisation

The analyses performed with the code SWANDYNE adopting the5-m wide mesh characterised by ‘‘tied-nodes” boundaries (see Sec-tions 5 and 6) are representative of ideal 1D problems. For 2D and3D problems wider meshes should be employed and the hypothe-sis of tied horizontal displacements of the lateral boundaries needsto be abandoned. Therefore questions concerning the appropriatelateral extension of the FE mesh arise.

A numerical investigation regarding this issue was performedwith the code PLAXIS adopting the viscous boundaries proposedby Lysmer and Kuhlemeyer [37] and meshes characterised bydifferent width. The twelve visco-elastic analyses described inSection 5 were re-simulated assuming the standard values ofthe Lysmer and Kuhlemeyer parameters (a = 1.0 and b = 0.25).The horizontal dimension of the mesh, L, was assumed equalto 2, 4 and 8 times the thickness H. In this context, therefore,the results obtained by the code SWANDYNE are assumed asreference.

Fig. 16 shows, as an example, the comparison between Fourierand response spectra at the surface obtained for a 120-m thickdeposit excited by the Kalamata earthquake. The similarity be-tween the results of the PLAXIS analysis characterised by L = 8Hand the reference analysis is clearly recognizable. A satisfactoryagreement between the analyses is already attained for L = 4H.This value can be considered as a good compromise betweenaccuracy and time required to perform the analysis of a 2Dboundary value problem.

The same trend was indeed observed in all the other eleveninvestigated cases. In addition, no significant differences wereidentified in the numerical results when adopting different valuesof the Lysmer and Kuhlemeyer parameters a and b in the range0�1.

8. Influence of time integration parameters

According to the Generalised Newmark time-stepping proce-dure [31], the displacement (u) and velocity ( _u) vectors in a solidnode at time n + 1 are expressed as:

unþ1 ¼ un þ _unDt þ 12ð1� b2Þ€un þ

12

b2 €unþ1

� �Dt2 ð5Þ

_unþ1 ¼ _un þ ½ð1� b1Þ€un þ b1 €unþ1�Dt ð6Þ

while the pore pressure (p) vector in a fluid node, at the same timen + 1, can be obtained from:

pnþ1 ¼ pn þ ð1� b�1Þ _pn þ b�1 _pnþ1�

Dt ð7Þ

The algorithm is unconditionally stable if the following condi-tions apply:

b1 P12

; b2 P12

12þ b1

� �2

; b�1 P12

ð8Þ

The choice of b1 = b2 = b�1 = 0.5 (corresponding to the higher or-der accurate trapezoidal scheme) guarantees the stability of thetime-stepping scheme for any value of Dt (i.e. the algorithm re-mains implicit) and does not provide any numerical (or algorith-mic) damping during the integration of the governing equations.In this case, numerical oscillations may occur during the analysisif no physical (viscous or hysteretic) damping is present [12]. Assuch, some numerical damping is typically introduced adoptingcoefficient values larger than 0.5, consistently with condition (8).

All the time-domain simulations illustrated in this note wereperformed assuming a set of Newmark parameters which leadsto a small amount of algorithmic dissipation (see Section 3). To as-sess the influence of the numerical damping on the FE results, thecase of a 240 m thick deposit exited by the Tarcento earthquakewas studied with the code SWANDYNE, varying the values of theparameter b1 in the range 0.5–0.9, setting b2 according to condition(8) and assuming b�1 = b1.

The comparison between Fourier and response spectra at theground surface obtained with the different Newmark parametersand the corresponding EERA reference results is reported in Fig. 17.

The figure clearly indicates that the numerical dissipation intro-duced by the time-stepping scheme is more pronounced at highfrequencies. The FE analysis performed with an un-damped timeintegration scheme (b1 = 0.5) gives the best agreement with thefrequency-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 inthe analyses discussed in the previous Sections) represents a goodcompromise between a satisfactory agreement with EERA in termsof frequency response and a small under-estimation of the peakground acceleration. Increasing values of b1 induce an over-damped response, especially for the high frequency modes, leadingto significantly reduced peak ground accelerations.

9. Conclusions

This paper describes a set of 2D finite element analyses for thesimulation of the seismic ground response of a clayey deposit.Some of the several factors potentially influencing the numericalresults are highlighted and critically discussed. In particular, thestiffness values and the amount of viscous damping in visco-elasticanalyses, the hysteretic damping when plasticity is added to thesoil model, the spatial and time discretisation and the nature ofboundary 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 ele-ment codes and two different boundary conditions.

Most of the analyses were performed using a linear visco-elasticsoil model characterised by the Rayleigh formulation for the vis-cous damping. The calibration of the Rayleigh coefficients as wellas the selection of the appropriate mobilised stiffness representcritical issues for this kind of simulations. In the note the validationof finite element visco-elastic analyses is performed comparing theresults with those obtained by equivalent-linear visco-elastic anal-yses performed in the frequency-domain by the code EERA. Thoselatter are thus taken as reference in the validation procedure. Usingthis approach, the paper shows that the traditionally adopted pro-cedures for the calibration of the Rayleigh coefficients can lead tolarge overestimation of the peak ground acceleration. A novel cal-ibration procedure is here proposed and discussed: in this case the



results of the FE analyses compare nicely with those obtained bythe frequency-domain approach.

A second set of FE analyses were carried out introducing plastic-ity in the soil constitutive formulation. The appropriate selection ofthe viscous damping to be added in the model was subjected tofurther investigation. Different strategies were attempted in orderto optimise the balance between the hysteretic dissipation and theviscous component of the damping. None of the proposed ap-proaches allowed to achieve a good matching between the FE anal-yses and the corresponding frequency-domain ones. Concerningthis latter outcome, it is worth remarking that the EERA resultsshould not be considered as the right term of comparison whenmodelling strong motion earthquakes, as those selected for thisstudy. In fact, intense shaking results in large and partly irrevers-ible strains associated with modification of the effective stressstate induced by excess pore pressures build-up. Those featurescannot be accounted for by visco-elasticity based constitutive laws,as that adopted in EERA, making the plasticity-based time domainapproach more realistic.

The simulations were performed by the finite element codeSWANDYNE, adopting a 5-m wide mesh characterised by tiednodes at the lateral boundaries, thus limiting the case to the 1Dcondition. The possibility of performing 2D finite element simula-tions was investigated by re-running the numerical analyses withthe finite element code PLAXIS, adopting the Lysmer and Kuhle-meyer conditions at the lateral boundaries. The match betweenthe results of the two different geometrical configurations as-sumed in the two codes were obtained employing 2D meshes char-acterised by a width-height ratio larger than eight, whilesatisfactory results were already achieved for a ratio equal to four.No influence of the values of the Lysmer and Kuhlemeyer coeffi-cients was observed in the 2D analyses.

Finally, accuracy and damping characteristics of the time inte-gration algorithm were analysed. It was found that the standardvalues of the time-stepping coefficients for the Generalised New-mark scheme represent the best compromise to obtain satisfactoryresults both in terms of frequency content and peak groundacceleration.

Acknowledgements

The Authors gratefully acknowledge the financial support of theItalian Ministry of Instruction, University and Research (Grants:PRIN 2007 ‘‘Seismic response of slopes, excavations and tunnels”and PRIN 2008 ‘‘Design of underground constructions in seismicconditions”) and the ReLUIS (Italian University Network of SeismicEngineering Laboratories) network. The employed accelerogramswere extracted from the SISMA (Site of the Italian Strong MotionAccelerograms) website.

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Parametric study on seismic ground response by finite element modelling

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