Computers and Geotechnics 37 (2010) 515–528
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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.
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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: 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).
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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 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.
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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.
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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).
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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 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.
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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 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).
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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
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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 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. It was found that the standard
values of the time-stepping coefficients for the Generalised Newmark scheme represent the best compromise to obtain satisfactory
results both in terms of frequency content and peak ground
acceleration.
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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