[amorosi] parametric study on seismic ground response by finite element modelling

8/11/2019 [AMOROSI] 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 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, performedadopting 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 com-

parable 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 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 performed

using one-dimensional frequency-domain numerical scheme

based on the equivalent visco-elastic approach [13]. This ap-

proach has successfully been adopted in the last 30 years and it

is widely accepted in the engineering practice, although its limita-

tions 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-elas-

tic schemes, disregarding the soilfluid interaction [4] and

possible build-up of excess pore water pressure during seismic

events;

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. [59]. In

contrast, a fully-reversible soil model, with constant visco-elas-

tic soil properties (i.e. shear stiffness and damping ratio) over

the duration of earthquake shaking, is adopted in the traditional

frequency-domain analysis methods;

many engineering problems cannot be scaled down to the one-

dimensional case but require a soilstructure interaction analy-sis 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 solidfluid 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 structurese.g.

[1321].

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 docu-

mented, 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.

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) 515528

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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: vis-

cous damping, generally introduced through the Rayleigh [22]

formulation, and the hysteretic dissipation associated to the

irreversible material response. The amount of hysteretic damp-

ing, 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-dissipa-

tive 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 functione.g. [2426],

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, the

characteristics of the spatial discretisation (i.e. the dimension

and the type of elements) and the appropriate boundary condi-

tions to artificially simulate the far-field medium. While all

these aspects are well understood in the context of static anal-

yses, the literature concerning numerical analyses in dynamic

conditions 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 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 possibledrawbacks they can contain, are taken as target solutions for the

2D 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 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 behav-

iour, provided an appropriate calibration of the parameters is

adopted. The above comparison scheme is obviously no longer va-

lid 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 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-plasticity

assumptions. Realism is introduced in the investigation by consid-

ering a soil deposit characterised by variable stiffness and damping

ratio with depth.

The first part of the paper outlines the geometrical and geotech-

nical 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 em-

ployed for 1D and 2D ground response simulations and summa-

rises the results of equivalent-linear visco-elastic analyses

performed in the frequency-domain. A new procedure for the cal-

ibration 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 dis-

cretisation 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 pro-

file, characterised by the following physical and mechanical

parameters: plasticity index IP = 44%, unit weight of volume of

the saturated soil c= 17kN/m3, overconsolidation ratio in termsof mean effective stress R= 1.5, small-strain shear stiffness

G0= variable with depth, Poissons ratio t0 = 0.25, small-straindamping 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 velocityVS (b).

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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 Viggi-

ani and Atkinson[27]:

G0pr

S p0

pr

nRm 1

wherepris 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 pro-

posed by Viggiani and Atkinson [27] for fine-grained soils) and R

is the overconsolidation ratio in terms of mean effective stress.

The variation ofG0

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 frequencyfp (Hz)

Tarcento NS Friuli (Italy), 1976 0.21 16.85 10.10

Gilroy 2 50 Coyote Lake (USA), 1979 0.20 18.00 5.00

Kalamata X Kalamata (Greece), 1986 0.24 29.75 1.63

Port 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.

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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 tri-

angular finite elements, characterised by a reduced integration for-

mulation 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 character-

ised by the viscous boundaries proposed by Lysmer and Kuhlemey-

er[37], with parametersa = 1.0 andb = 0.25.

All PLAXIS analyses were performed under undrained condi-

tions. 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 SWAN-

DYNE 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 approxi-

mately one-tenth to one-eighth of the wavelength associated with

the maximum frequency componentfmax of the input wave[38]:

Dlnode 6 kmin=810 VS;min=810fmax 4

where VS,min is the lowest wave velocity. An example of the mesh

employed in the PLAXIS analyses is sketched inFig. 6.

It is nowadays well established that the time discretisation can

play a significant role on the accuracy of dynamic finite elementFig. 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).

Fig. 9. Calibration strategies for the Rayleigh coefficients (240-m thick deposit and Tarcento earthquake).

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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 andD profiles were defined

in order to match the ones resulting from the corresponding EERA

analyses. To this aim, the numerical models in PLAXIS and SWAN-

DYNE were subdivided into the same number of layers employed

in EERA and for each layer a value ofGand Dwas selected with ref-

erence to the shear deformation level resulting from the EERA

analyses at the corresponding depth.Fig. 8shows theGandDpro-

files adopted in the FE analysis for the same case ofFig. 7. Rayleigh

damping introduced in the simulations is defined by selecting the

coefficients aRandbR, which depend onD and on the adopted fre-quency intervalfmfn according to Eq.(3). Different possible cali-

bration procedures were proposed in the literature to identify

the interval fmfn. In particular, a well established one e.g.

[24,25] suggests to selectfmas the first natural frequency of the de-

positf1, whilefnis assumed equal to ntimesfm, where nis the clos-

est odd integer larger than the ratiofp/f1between 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 fmandfn, respec-

tively. 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 Tarcen-

to 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 funda-

mental 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 condi-

tion leads to a significant under-damped response of the sys-

tem 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 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.

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In order to obtain a better match between the linear time-do-

main 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 ex-

ited by the earthquake, fm, should be identified by comparing the

EERA amplification function and the Fourier spectrum of the input

motion. In the case ofFig. 9 is evident that the Tarcento earthquake

is 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 asfm.

As regards the second frequency fn, it should be identified consid-

ering 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 inFig. 9,the proposed procedure leads

to a value offn 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 offmandfnis plot-

ted in Fig. 9a (dashed line). The resultingaRandbRprofiles adoptedin the FE analysis are shown inFig. 11a and b, respectively, and

compared to those resulting from the standard calibration ap-

proach. 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 proce-

dure: 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 proce-

dure for the definition of Rayleigh damping coefficient profiles, a

reasonably 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 acceleration

time 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 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 er-

rors 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 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).

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Fig. 15. Comparison between response spectra obtained withEERA and the different investigated FE visco-elasto-plastic analyses at different depths (60-m thickdeposit and

Gilroy 2-050 earthquake).



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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 MohrCoulomb 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-domain

solution 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, with

a magnification factor of 1.2 over the peak base amplitude. The

amount 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 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 offm=f1= 0.54 Hz andfn= 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-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 ob-

tained by the EERA analysis, i.e. assuming as negligible the plas-

ticity-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 adoptedin the previous case (Fig. 14b). As illustrated in Fig. 14c, this im-

plies 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. 15reports the comparison between the results of the three

FE visco-elasto-plastic analyses and the corresponding EERA simu-

lation in terms of response spectra obtained at different depths

along the deposit. All the plasticity-based analyses show a contrac-

tion of the spectra as compared to the EERA one, this being related

to the additional damping supplied by the MohrCoulomb 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).

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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).



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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 fun-

damental 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 hyster-

etic 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 de-

posit. In this last circumstance, plasticity might prevail and bias

the picture traditionally obtained by means of visco-elastic analy-

ses. Under these latter conditions, permanent displacement and

corresponding variation of the effective stress state occur, signifi-

cantly modifying the soilstructure interaction in any geotechnical

contexte.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 Sec-

tions5 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 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. 16shows, 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 be-

tween 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-

dure[31], the displacement (u) and velocity ( _u) vectors in a solid

node at timen + 1 are expressed as:

un1 un _unDt 1

21 b2un

1

2b2un1

Dt2 5

_un1 _un 1 b1un b1un1Dt 6

while the pore pressure (p) vector in a fluid node, at the same time

n+ 1, can be obtained from:

pn1 pn 1 b1_pnb1_pn1

Dt 7

The algorithm is unconditionally stable if the following condi-

tions apply:

b1 P1

2; b2 P

1

2

1

2b1

2; b

1P

1

2 8

The choice ofb1= b2= b

1= 0.5 (corresponding to the higher or-

der accurate trapezoidal scheme) guarantees the stability of the

time-stepping scheme for any value ofDt (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 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 Section3). To as-

sess 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 b1in the range 0.50.9, setting b2according to condition

(8) and assuming b

1= 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 intro-

duced 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 13 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 over-damped 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 ele-

ment 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 vis-

cous 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 anal-

yses performed in the frequency-domain by the code EERA. Those

latter are thus takenas reference in the validation procedure. Using

this approach, the paper shows that the traditionally adopted pro-

cedures for the calibration of the Rayleigh coefficients can lead to

large overestimation of the peak ground acceleration. A novel cal-ibration procedure is here proposed and discussed: in this case the

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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 plastic-

ity 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 ap-

proaches allowed to achieve a good matching between the FE anal-

yses 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 irrevers-

ible 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 simula-

tions was investigated by re-running the numerical analyses with

the finite element code PLAXIS, adopting the Lysmer and Kuhle-

meyer conditions at the lateral boundaries. The match between

the 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, while

satisfactory 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 standard

values of the time-stepping coefficients for the Generalised New-

mark scheme represent the best compromise to obtain satisfactory

results both in terms of frequency content and peak groundacceleration.

Acknowledgements

The Authors gratefully acknowledge the financial support of the

Italian Ministry of Instruction, University and Research (Grants:

PRIN 2007 Seismic response of slopes, excavations and tunnels

and PRIN 2008 Design of underground constructions in seismic

conditions) and the ReLUIS (Italian University Network of Seismic

Engineering Laboratories) network. The employed accelerograms

were extracted from the SISMA (Site of the Italian Strong Motion

Accelerograms) website.

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