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1 PhD researcher, Dept. of Civil Engineering, Aristotle University of Thessaloniki, Greece (email:
lesgidis@civil.auth.gr)
2Assoc. Professor, Dept. of Civil Engineering, Aristotle University of Thessaloniki and University of Bristol, UK
3Assoc. Professor, Dept. of Civil Engineering, University of Toronto, Canada
Eleventh U.S. National Conference on Earthquake Engineering
Integrating Science, Engineering & Policy
June 25-29, 2018
Los Angeles, California
A FREQUENCY- AND INTENSITY-
DEPENDENT, LUMPED PARAMETER
MODEL FOR CONSIDERING
SSI EFFECTS ON SEISMIC BRIDGE
DESIGN AND ASSESSMENT
N. Lesgidis1, A. Sextos2 and O.-S. Kwon3
ABSTRACT
The computational demand of soil-structure interaction analysis in seismic risk assessment of a
structure has led the civil engineering community to the development of a variety of methods
towards the model order reduction of the soil-structure dynamic system. Different approaches
have been proposed in the past as computationally viable alternatives to the FEM simulation of
the complete structure-soil domain, such as the nonlinear lumped spring, the macro-element
method and the substructure partition method. Yet, with few exceptions, no approach was
capable of capturing simultaneously the frequency-dependent dynamic properties along with the
nonlinear behavior of the condensed segment of the overall soil-structure system, an essential
step for the accurate assessment of a structure’s performance under a seismic hazard. To this end,
a dual frequency- and intensity-dependent expansion of the lumped parameter modeling method
is proposed in the current paper, materialized through a multi-objective algorithm, capable of
fully capturing the behavior of the nonlinear dynamic system of the condensed segment. The
efficiency of the proposed approach is validated for the case of an existing bridge, wherein the
seismic response is comparatively assessed for both the proposed method and the detailed finite
element model.
Eleventh U.S. National Conference on Earthquake Engineering
Integrating Science, Engineering & Policy
June 25-29, 2018
Los Angeles, California
A frequency- and intensity-dependent, Lumped Parameter Model for
considering SSI effects on seismic bridge design and assessment
N. Lesgidis, A. Sextos and O.-S. Kwon
ABSTRACT
The computational demand of soil-structure interaction analysis in seismic risk assessment of a
structure has led the civil engineering community to the development of a variety of methods
towards the model order reduction of the soil-structure dynamic system. Different approaches
have been proposed in the past as computationally viable alternatives to the FEM simulation of the
complete structure-soil domain, such as the nonlinear lumped spring, the macro-element method
and the substructure partition method. Yet, with few exceptions, no approach was capable of
capturing simultaneously the frequency-dependent dynamic properties along with the nonlinear
behavior of the condensed segment of the overall soil-structure system, an essential step for the
accurate assessment of a structure’s performance under a seismic hazard. To this end, a dual
frequency- and intensity-dependent expansion of the lumped parameter modeling method is
proposed in the current paper, materialized through a multi-objective algorithm, capable of fully
capturing the behavior of the nonlinear dynamic system of the condensed segment. The efficiency
of the proposed approach is validated for the case of an existing bridge, wherein the seismic
response is comparatively assessed for both the proposed method and the detailed finite element
model.
Introduction
Simulation of soil-structure interaction can be a computationally demanding approach.
As a result, a number of order reduction methods have been developed in the past capable of
significantly reducing the SSI problem computational cost. The macroelement approach, a
concept initially introduced by Montrasio et al [1] and further developed by a number of
different research groups [2–4], has successfully provided with an accurate, yet low in
computational effort, representation of the inelastic behavior of the soil-foundation domain.
Although the macroelement approach emulates with great detail the different mechanisms
triggered during a quasi-static excited simulation, it is common in the literature that it either
completely neglects or oversimplifies the dynamic traits of the foundation-soil domain through
the use of complementary Kelvin –Voigt components. As such components are only limited to
accuracy along a specific frequency, their use can lead to significant error in predicting the
behavior of the structure under dynamic excitation as illustrated in [5]. On the other hand, the
lumped parameter (LP) modeling method has provided with an effective order reduction
approach targeting the viscoelastic dynamic properties of the soil foundation domain along a
broad frequency as reported in the literature [6–9]. Despite the above advantage, the LP
modeling method is limited to the reduction of only viscous elastic dynamic systems.
In order to eliminate the drawbacks of the aforementioned approaches, a frequency-
dependent macroelement is presented herein, capable of significantly reducing the order of
inelastic dynamic SSI systems without compromising the accuracy of nonlinear dynamic SSI
analysis.
Inelastic Lumped Parameter Model
Two individual methods are developed for the frequency- and intensity-dependent
reduction of the SSI problem as observed in the model assembly of Figure 1. In both methods the
inelastic behavior of the system is emulated by the selection of an appropriate macroelement as
the base component. On the proposed method (M.1), externally controlled springs are
incorporated and the overall complementary mass, spring and dashpot components of the
assembly are calibrated targeting the impedance functions of the system for different intensity
levels. On the simplified approach (M.2) the conventional complementary components are
calibrated targeting the elastic impedance function of the system. The calibration approach is
accomplished through the reformulation of a multi-objective optimization problem solved by a
trust region nonlinear programming algorithm.
.
Figure 1. Physical representation of the proposed frequency- and intensity-dependent
Lumped Parameter Model.
Complementary Component:
a)M.1:Externally controlled Spring
b)M.2 Conventional Spring
Base Component:
Macroelement
Complementary Component:
Conventional Mass
Complementary Component:
Conventional Dashpot
Numerical Verification of the proposed Method
The methods proposed follow a verification procedure for the case study of a bridge pier
system excited from an earthquake ground motion. The proposed methods M.1 and M.2 along
with the targeted FEM model and the static macroelement are all subjected to the acceleration
time history recorded in the Earthquake of the Imperial Valley at the El Centro Array station
rescaled to a PGA=0.1g. The time history response of the superstructure and the foundation
interface DOFs are compared for the complete FEM simulation and the proposed inelastic LP
method. The vertical component of the earthquake motion is neglected. The time history results
are depicted in Figures 2 and 3 for the two proposed methods M.1 and M.2, the holistic FEM
solution and the conventional static macroelement approach.
Figure 2. Time history results of the (a) foundation rotation (b) foundation horizontal displacement for
the FEM and the reduced model approaches.
Figure 3. (a) foundation rotation to moment relation (b) foundation horizontal displacement to force
relation for the FEM and the reduced model approaches.
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 1 2 3 4 5 6 7 8
Foundation Rotation
(mrad)
Time (Sec)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 1 2 3 4 5 6 7 8
Foundation Hor.
displacement (mm)
Time (Sec)
FEM solution
Complete LP model (M.1)
Simp. LP model (M.2)
NL Macroelement
-450.0
-350.0
-250.0
-150.0
-50.0
50.0
150.0
250.0
350.0
450.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Moment at Foundation (KNm)
Foundation Rotation (mrad)
-80.0
-60.0
-40.0
-20.0
0.0
20.0
40.0
60.0
80.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Horizontal Force at Foundation (KN)
Foundation Displacement (mm)
FEM solution
Complete LP model (M.1)
Simp. LP model (M.2)
NL Macroelement
From the comparison of the foundation response on the three different approaches (i.e.,
FEM, M.1 and M.2), it can be observed that both proposed methods M.1 and M.2 are adequately
emulating the targeted system behavior. The efficiency of the M.2 method can be explained by
the fact that the higher frequency content of the earthquake excitation is filtered out by the
superstructure’s transfer function. This also practically implies that the simplified method M.2 is
adequately accurate mainly for excitations with low predominant frequencies. The static
macroelement on the other hand, is observed to misrepresent the foundation behavior on both
rotation and horizontal displacement.
Conclusions
A lumped parameter modelling method has been proposed in the current paper, capable
of accurately emulating the dynamic behaviour of the soil foundation system. In contrast to
methods presented in the past literature, the proposed method copes with the frequency depended
properties of the soil-foundation system in different intensity regions through the expansion of
the lumped parameter model framework to inelastic dynamic systems. The procedure is
numerically verified through the comparison with the FE model, the complete and the simplified
version of the proposed approach. The results are limited to three interface DOF representations
of simplified bridge model-soil foundation systems and thus future work is essential to the
expansion of the proposed method to more realistic soil foundation systems.
References
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Dynamics and Earthquake Engineering 2009; 29(5): 765–781. DOI:
10.1016/j.soildyn.2008.08.009.
3. Salciarini D, Tamagnini C. A hypoplastic macroelement model for shallow foundations under
monotonic and cyclic loads. Acta Geotechnica 2009; 4(3): 163–176. DOI: 10.1007/s11440-009-
0087-2.
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5. Lesgidis N, Sextos A, Kwon O. Influence of frequency-dependent soil-structure interaction on the
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9. Lesgidis N, Kwon O, Sextos A. A time-domain seismic SSI analysis method for inelastic bridge
structures through the use of a frequency-dependent lumped parameter model. Earthquake
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