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Steel Structures 8 (2008) 285-294 www.ijoss.org
Seismic Behavior Analysis of a Plate-Girder Bridge
Considering Abutment-Soil Interaction
Jeong-Hun Won1, Jia Xu Wu1, Ariunzul Davaadorj1, Sang-Hyo Kim1, and Ho-Seong Mha2*
1School of Civil & Environmental Engineering, Yonsei University, 134 Sinchon-dong, Seodaemun-gu, Seoul, 120-749, Korea2Department of Civil Engineering, Hoseo University, 165 Sechul-ri, Baebang, Asan, Chungnam, 336-795, Korea
Abstract
Longitudinal dynamic behaviors of a multi-span plate-girder bridge under seismic excitations are examined to see the effectof the abutment-soil interaction. The stiffness degradation due to the abutment-soil interaction is considered in the systemmodel, which may play the major role upon the global dynamic characteristics of the whole bridge system. An idealizedmechanical model is proposed, which is capable of considering pounding phenomena, friction at the movable supports, and thecorresponding simulation method is developed. The abutment-soil interaction is modeled as the one degree-of-freedom systemwith nonlinear spring and damper. Using the idealized mechanical model for the bridge system, the longitudinal responses ofstiffness degradation model are compared with those based on the linear system, which excludes the stiffness degradation.Results show that the stiffness degradation of the abutment-backfill system takes an important influence upon the global bridgemotions and the seismic responses may be underestimated in the system only with the constant stiffness considered. Hence,it is concluded that the stiffness degradation should be taken into account in the seismic analysis of the bridge system.
Keywords: Dynamic behavior, Seismic excitation, Stiffness degradation, Abutment-soil interaction, Nonlinear spring
1. Introduction
Recently, there are more earthquakes with bigger
magnitudes occurring world widely, and the report of the
corresponding damages are telling us that the earthquake
is truly shaking our lives as it is still remaining the one of
the biggest threats to human society. For those reasons,
earthquake related disasters have been asking us for the
better seismic concepts to prevent the worst scenario of
the damages. Correct predictions of the dynamic behaviors
of the structures under seismic excitations are more
desired for the better design of the systems.
Bridge systems consisting of several simple spans can
be described as the combination of multiple vibration
units, and the dynamic characteristics of the most units
are similar. However, the global dynamic behaviors of
such a bridge system become complicated to be predicted
due to many factors, which can be abutment-soil interactions,
poundings between girders, frictions at movable supports,
inelastic behaviors of RC piers, foundation motions
(rotation and translation), and so forth. It should be
noticed that the interactions between the adjacent
oscillating units have drawn the interest, since it may play
the major role of the span collapses. In order to prevent
span collapse, it is need to install special device such as
cable restrainers and energy dissipation system (Cho et
al., 2008; Won et al., 2008). Among the interactions
between vibrating units, the abutment-soil interaction
may be the most important component affecting the
global motions of the bridge particularly for bridge
systems with similar vibrating units (Shamsabadi et al,
2007).
It is highly challenging to verify the effect of the
abutment-soil interaction because of the uncertainties
included to the abutment-backfill system in addition to
the soil. The abutment-backfill system is often ignored or
at most modeled as either a linear spring system.
However, the abutment-soil interaction should be considered
in the analysis of bridge motions. An appropriate nonlinear
model corresponding to the abutment-soil interactions
should be adopted for the proper simulations.
A simple and economical way to represent the
abutment-backfill system in seismic behavior analysis is
to use a translational nonlinear spring. Siddharthan et al.
(1997) suggested the nonlinear spring which has
degradation of stiffness with the increment of abutment
movements. In this model, many important factors such
as nonlinear soil behavior, abutment dimensions,
superstructure loads, difference in soil conditions, and so
forth, are considered to determine the nonlinear spring
stiffness. Saadeghvaziri and Yazdani-Motlagh (2008)
Note.-Discussion open until May 1, 2009. This manuscript for thispaper was submitted for review and possible publication on Septem-ber 16, 2008; approved on November 27, 2008
*Corresponding authorTel: +82-41-540-5792; Fax: +82-41-540-5798E-mail: mhah@hoseo.edu
286 Jeong-Hun Won et al.
used the bilinear spring to consider effects of the
abutment on the global bridge response.
In this study, the idealized mechanical model for the
multi-simple span bridge system is proposed, which can
consider the abutment-soil interactions as well as the
pounding and friction and other factors. The abutment-
backfill system is modeled as a one-degree-of-freedom
system with a nonlinear spring and a nonlinear damper to
consider the stiffness degradation.
2. Modeling of Systems
2.1. Bridge model
The bridge considered is a three-span simple plate
girder bridge with 35 m span length as shown in Fig. 1.
Bent type piers, shallow foundations, and seat-type
abutments are used. The pier height is 12 m and the
diameter of a column is 1.7 m. The longitudinal and
transverse widths of pier foundations are 6 m and 14 m.
The height of foundations is 2 m. The abutment height is
6.5 m. The longitudinal and transverse widths of abutments
are 4.6 m and 17 m.
In this study, only the longitudinal motions are of
concern, so the total system can be divided into four
individual vibrating units shown as in the Fig. 1 (a); A1
unit, P1 unit, P2 unit, and A2 unit.
For better efficiency, a simplified mechanical model is
proposed using the lumped mass system, which is
depicted in Fig. 2. In the figure, m1, m5, m9 are the masses
of superstructures, m2, m6 are the masses of piers, m3, m7
are the masses of foundations, m4, m8 are the rotational
mass moments of inertia of foundations, and mA1, mA2 are
masses of abutments. K2, K6 and C2, C6 are the stiffness
and damping constants of the piers, K3, K7 and C3, C7 are
the translational stiffness and damping constants of the
foundations, and K4, K8 and C4, C8 are the rotational
stiffness and damping constants of the foundations,
respectively. KA1, KA2 and CA1, CA2 are the stiffness and
damping constants of the abutments. KA1,1, K2,5 and K6,9
are the stiffness of the fixed supports at individual
vibration units. F1,2, F5,6, and F9,A are the friction forces at
the movable supports. S1,5, S5,9, S9,A2 and C1,5, C5,9, C9,A2
are the stiffness and damping constants of the impact
elements, and L is height of pier. d1,5, d5,9, d9,A2 are the gap
distances between adjacent vibration units, and ug is the
ground displacement.
2.2. Abutment-backfill model
A simplest approach to represent the nonlinear
behavior of the abutment due to abutment-soil interaction
is to use translational nonlinear spring. The computational
procedure of this methodology is relatively simple while
the output shows great coincidence with the realistic
behavior of the abutment, as been verified through a field
test (Siddharthan et al., 1997; Goel & Chopra, 1997).
The abutment-backfill system is modeled in this study
as one-degree-of-freedom system with nonlinear spring
and nonlinear damper to consider the abutment stiffness
degradation as shown in Fig. 3. Since the abutment is
synchronized with the backfill (Siddharthan et al., 1994),
the mass of the abutment-backfill system is assumed to be
the summation of the mass of the abutment and the
backfill. In the Figure, mA is the mass of the abutment-
backfill system, ug(A) is the ground displacement, uA is the
displacement of mass mA, and KA(uA) and CA(uA) are the
translational nonlinear stiffness and damping coefficients
Figure 1. Sample bridge.
Seismic Behavior Analysis of a Plate-Girder Bridge Considering Abutment-Soil Interaction 287
of the soil surrounding the abutment, respectively.
Since the abutment undergoes rigid body movement
(Al-Homoud and Whitmann, 1999; Maroney, 1994), the
abutment displacement at any point is evaluated in terms
of δL and θ as shown in Fig. 3. The nonlinear spring
stiffness is obtained from estimation of the force, PL, for
a given displacement, δL, such that force equilibrium
equation is satisfied.
2.3. Poundings between girders
Two adjacent vibration units may produce poundings
upon the applied seismic excitations with various intensities,
and these poundings are governed by the relative
displacements between the oscillating systems. The
pounding is described in this study by placing spring-
damper elements (impact elements) between the masses
as shown in Fig. 4. The pounding condition and force are
defined as follows.
(1)
(2)
where ui, ui+4 are the displacement of mass mi and mi+4,
ugi, ug(i+1) are the ground displacement, and di,i+4 is the gap
distance between mi and mi+4. In addition, Si,i+4 and Ci,i+4
are denote the spring stiffness and damping constant of
impact element, respectively.
The stiffness of spring, which is typically large and
highly uncertain, and the damping constant, which determines
the amount of energy dissipated, can be obtained by
references (Anagnostopoulos, 1988; Kim et al., 1999).
2.4. Friction of movable supports
The frictions between the superstructures and the
δi ui ui 4+– ugi ug i 1+( )– di i 4+,–+ 0≥=
Fi i 4+, Si i 4+, δi Ci i 4+, δi'+= for δi 0>
Fi i 4+, 0= for δi 0>
Figure 2. Simplified mechanical model of the bridge.
Figure 3. Abutment-backfill model.
Figure 4. Idealization of pounding.
288 Jeong-Hun Won et al.
movable supports are usually neglected in the bridge
dynamic analysis. However, this may not yield the
appropriate results since it ignores the energy dissipation
due to friction. In this study, a modified bilinear coulomb
friction model is utilized (Kim et al., 2000). The
relationship between the friction force and relative
velocity between the adjacent oscillators can be depicted
as shown in Fig. 5. In stick condition, the friction force
increases up to a given value, ε of the relative velocity
and then sustains a constant friction force multiplying
vertical force with friction coefficient (µ). The friction
forces, Fi,i+1 of the stick and sliding conditions are
expressed as follows.
(3)
where ∆i is the relative distance between vibration units
which are connected by movable supports.
2.5. Pier and foundation motions
The nonlinear pier motion is simulated by adopting the
hysteresis loop function (Kim et al., 2000). The foundation
motions are modeled as a two DOF system with
translational and rotational springs and dampers. The
values of springs are determined by the guideline of
FHWA (1986).
3. Longitudinal Abutment Stiffness Evaluation
The longitudinal abutment stiffness varies with types
and conditions of soil surrounding the abutment. The
rational evaluation of longitudinal stiffness degradation
due to abutment-soil interaction is required. This study
evaluates the nonlinear stiffness of abutment following
the procedure suggested by Siddharthan (1997).
For the considered abutment with 6.5 m height, the
longitudinal stiffness (passive state) of abutments can be
obtained as
(4)
(5)
where SL is the secant stiffness (kN ⋅ m−1 ⋅ m−1) per width
of an abutment wall, DL is the dimensionless stiffness
coefficient, xL is the abutment displacement (mm), H is
the abutment height (m), γ is the unit weight of soil (kN/
m3), and B is the abutment width in longitudinal direction
(m).
Considering the surrounding soil condition and the
dimension of abutments (γ=20 kN/m3, H=6.5 m, B=3 m,
and DL=7000), the nonlinear stiffness curve is plotted in
Fig. 6 (Nonlinear model in the figure), where the width of
the abutment in transverse direction (17 m) is included. It
is assumed that the stiffness is constant to the movement
of 1 mm. Then, the longitudinal abutment stiffness
decreases rapidly as the displacement increases when the
displacement is small.
In order to investigate the effects of stiffness degradation
on the global bridge motion, the linear stiffness of the
abutment is also considered. First, the linear model with
the initial constant stiffness of nonlinear model, which is
the stiffness at 1 mm movement of the abutment, is
considered as a linear model (Linear model-Case 1 in Fig.
6).
In addition, the linear model with the initial stiffness
suggested by Caltrans (2001) is considered (Linear
model-Case2 in the Fig. 6) since this method can easily
simulate the longitudinal abutment response. The
constant stiffness is
(6)
(7)
where w is the abutment width in transverse direction and
H/1.7 is the height proportionality factor.
Considering the abutment width in transverse direction
(17 m) and the abutment height (6.5 m), the constant
stiffness of linear model-Case2 is evaluated as 747,500
Fi i 1+,
1
2---µm
ig1
ε---∆
i= for ∆
iε<
Fi i 1+,
1
2---= µm
ig for ∆
iε≥
SL
DLE
L
xL
H-----⎝ ⎠⎛ ⎞
0.96–
=
EL
γB3
H2
--------=
Kabut
SL
Kiw
H
1.7-------⎝ ⎠⎛ ⎞
= =
Ki
11.5 103kN m⁄m
-------------×=
Figure 5. Friction force-relative displacement relationship.
Figure 6. Applied abutment stiffness (passive stiffness).
Seismic Behavior Analysis of a Plate-Girder Bridge Considering Abutment-Soil Interaction 289
kN/m. This value is same to the stiffness of nonlinear
model at the abutment movement of 13.6 mm.
It should be noted that the stiffness in Fig. 6 are passive
stiffness. Since the active condition in a soil deposit can
be mobilized with a much lower abutment movement
than the passive condition (Clough and Duncan, 1991;
Barker et al., 1991), the active stiffness of the abutment
is assumed to be ten times smaller than the passive
stiffness. It is also assumed that once the degradation is
started, the degraded stiffness cannot be recovered and
that the stiffness after the critical displacement is reached
remains constant afterwards regardless of the displacement.
4. Results and Observation
Dynamic responses of the bridge under earthquakes
with various magnitudes of peak ground accelerations are
evaluated to see the effects of the longitudinal stiffness
degradation due to abutment-soil interaction. Three types
of analysis models according to the abutment models are
selected to examine the dynamic responses of the bridge
under earthquakes to see the effect of the stiffness
degradation due to the abutment-soil interaction. Two
linear models are employed, which consist of the constant
abutment stiffness (linear systems) and one model consist
of the nonlinear stiffness considering the abutment
stiffness degradation (nonlinear system). The simulated
bridge models are described in Table 1.
For the input ground motions, artificial seismic excitations
are generated by using the well-known SIMQKE code
(Gasparini, 1976), which is compatible to the design
response spectra specified in the Korean code (MOCT,
2005). An example of the simulated seismic excitation is
shown in Fig. 7.
Since seismic excitations can be indicative by stochastic
processes, the responses of a bridge system exhibit
probabilistic characteristics. Therefore, in this research to
evaluate the fluctuation of response by artificial seismic
excitations, the probabilistic characteristics of maximum
responses are estimated by the mean value and Gumbel
Type-I probability distribution (for each acceleration, the
number of samples is 10).
The equations of motions corresponding to the simplified
mechanical model are derived using the Lagrange equation.
The dynamic responses of a bridge system are then
simulated by adopting the direct integral method (4th
order Runge-Kutta method). The time step size is 2×10−5.
The exact time of pounding is obtained by the Newton
method. The 5 cm-gap distance between adjacent vibration
units is selected, and the friction coefficient for movable
support is assumed to be 0.05.
4.1. Displacements of each vibration unit
Relative displacements of each vibration unit to the
ground motion are evaluated in order to see the effects of
the stiffness degradation. For the seismic excitations with
various magnitudes of peak ground acceleration (PGA)
from 0.1 g to 0.6 g, the maximum displacements are
investigated. Table 2 shows the mean values and 90%
extreme values of Gumbel type-I for maximum displacements.
Fig. 8 represents the comparison results of the extreme
values according to the abutment models.
First, the relative displacements of A2 unit, which is
consisting of the abutment only, are observed. Under
earthquakes with PGA less than 0.3 g, the displacement is
quiet small. Consequently, the results from the linear
model (linear case 1: L-C1) and the nonlinear model are
showing the similar trends, while the other linear model
with smaller stiffness (linear case 2: L-C2) showing the
biggest responses. In the case with the PGA above 0.3 g,
the nonlinear model begins to show the bigger responses
than the other linear models, since the relative displacements
become large enough for the stiffness to dramatically
degrade.
The effect of the stiffness degradation can be seen more
clearly in the results from the A1 abutment unit, which
consists not only of the abutment itself, but also of the
superstructure. The difference between the results from
Table 1. Considered abutment models
Abutment model Description
Nonlinear model- Nonlinear spring according to the Eq. (4)- Nonlinear damper (damping ratio of 5%)
Linear model-Case 1- Linear spring with the stiffness at 1 mm movement of the nonlinear model - Linear damper (damping ratio of 5%)
Linear model-Case 2- Linear spring with the stiffness according to the Caltrans- Linear damper (damping ratio of 5%)
Figure 7. Example of simulated seismic excitation.
290 Jeong-Hun Won et al.
linear and nonlinear system becomes larger.
Secondly, the relative displacements of P1 and P2 units,
which consist of piers and superstructures, are observed.
Under weak earthquakes (PGA<0.3 g), the similar results
are obtained regardless of the types of the abutment
models, since the displacements are quite small, and the
effect of the interactions between the adjacent vibration
units is not big enough to initiate the stiffness degradation.
Under stronger earthquakes (PGA>0.3 g), the pier units
starts to show the influence from the abutment units, and
the largest displacement naturally occur from the
nonlinear system due to the stiffness degradation.
From results, it can be said that the effects of stiffness
degradation upon the displacements relative to the ground
are relatively larger under moderate and strong excitations
with PGA>0.3 g. It should be noticed that the results
using the stiffness obtained by the method suggested by
Caltrans (linear case 2: L-C2) can give the similar results
under PGA below 0.3 g, but that under PGA over 0.3 g,
nonlinear model should be employed for the better
prediction of the bridge motions.
Typical time histories of these displacements are
depicted in Fig. 9-12 for both cases with linear and
nonlinear abutment models. At the level of the peak
ground acceleration (PGA) of 0.3 g, A1 unit shows the
increment of displacement in nonlinear model, while A2
units shows the bigger displacements in linear model-
case 2. In addition, P1 and P2 unit shows the similar time
histories in all considered abutment models. With regard
to the PGA of 0.6 g, the nonlinear model clearly shows
the increased time histories for all vibrating units.
Figure 8. Comparison of displacements for each vibration unit (Gumbel type-90%).
Table 2. Maximum displacements of each vibration unit (unit: cm)
PGAA1 vibration unit P1 vibration unit P2 vibration unit A2 vibration unit
L-C1c L-C2d N-Le L-C1 L-C2 N-L L-C1 L-C2 N-L L-C1 L-C2 N-L
0.1 gMeana 0.04 0.43 0.04 2.62 2.76 2.62 2.40 2.56 2.40 0.03 0.24 0.03
Gumbelb 0.05 0.50 0.05 3.00 3.22 3.00 2.78 2.96 2.79 0.05 0.27 0.05
0.2 gMean 0.18 0.94 0.55 6.45 6.67 6.68 6.17 6.32 6.20 0.13 0.54 0.17
Gumbel 0.26 1.10 0.90 7.88 7.75 8.01 7.02 7.00 6.99 0.26 0.75 0.40
0.3 gMean 0.38 1.51 2.70 8.82 9.88 9.65 8.15 8.72 8.69 0.25 1.00 0.67
Gumbel 0.53 1.75 4.00 10.57 12.30 12.23 9.99 10.33 10.89 0.35 1.36 1.03
0.4 gMean 0.58 2.22 6.13 10.42 11.54 11.80 9.02 9.62 10.13 0.35 1.33 2.38
Gumbel 0.72 2.77 8.61 13.47 13.34 14.38 10.82 11.57 12.21 0.46 1.77 3.13
0.5 gMean 0.82 3.00 9.99 12.79 13.87 14.25 10.31 10.74 12.30 0.46 1.68 4.67
Gumbel 1.09 3.86 12.80 16.72 17.34 19.28 12.90 13.42 15.56 0.69 2.22 6.44
0.6 gMean 0.98 3.82 12.10 14.54 15.75 16.07 10.95 11.70 13.43 0.59 2.10 6.94
Gumbel 1.36 5.06 15.35 19.24 21.18 20.54 14.33 14.09 16.56 0.87 2.85 8.79
a)Mean value of relative displacements of each vibration unitb)90% extreme value of Gumbel Type-Ic)Abutment model with linear spring and dashpot (case 1)d)Abutment model with linear spring and dashpot (case 2)e)Abutment model with nonlinear spring and dashpot
Seismic Behavior Analysis of a Plate-Girder Bridge Considering Abutment-Soil Interaction 291
4.2. Relative distances between vibration units
The relative distance between the adjacent vibration
units are investigated in this section to see the effects of
the stiffness degradation more clearly under the seismic
excitations with various magnitudes of PGAs from 0.1 g
to 0.6 g. The mean values and 90% extreme values of
Figure 9. Time histories of displacement relative to ground (A1 unit).
Figure 10. Time histories of displacement relative to ground (A2 unit).
Figure 11. Time histories of displacement relative to ground (P1 unit).
Figure 12. Time histories of displacement relative to ground (P2 unit).
292 Jeong-Hun Won et al.
maximum relative distances are obtained. The results are
tabulated in Tables 3. The comparison of the maximum
relative distances is represented in Fig. 13 according to
the abutment model types.
The maximum relative distances are observed to occur
between the abutment and pier units. At the location
between A1 and P1 units, where the largest maximum
relative distances occur, the nonlinear case shows the
largest relative distances among three models due to the
stiffness degradation. As PGA increases, this trend
becomes clearer. The relative distances between P2 and
A2 units are similar to each other in the most PGA, but
the nonlinear case shows a little larger distances than the
linear cases under PGA over 0.3 g.
Between pier units, the nonlinear case shows the largest
relative distances when the PGA is over 0.3 g. The
biggest difference in the relative distances obtained from
the linear and nonlinear cases can be found at the location
between pier units.
Time histories of relative distances between the
abutment unit and the pier unit are plotted in Fig. 14 and
Fig. 15. The positive value means that two units are
facing, while the negative values represent that two units
are retreating. The initial distance gap between adjacent
units is 5 cm. Thus, the positive value of the relative
distance cannot exceed this value due to the pounding.
For the time histories between A1 unit and P1 unit (Fig.
14), it is clearly shown that nonlinear model gives the
increased relative distances, which can cause the span
collapse of the simply supported bridges. With regard to
the relative distances between P2 unit and A2 unit (Fig.
15), the small increment of relative distances is shown in
the nonlinear model.
Summarizing the above results, the stiffness degradation
due to the abutment-soil interaction plays the important
role upon the global dynamic characteristics of the whole
Figure 13. Comparison of relative distance between vibration units (90% extreme value of Gumbel type-I).
Table 3. Maximum relative distances between vibration units (unit: cm)
PGAA1-P1 P1-P2 P2-A2
L-C1c L-C2d N-Le L-C1c L-C2d N-Le L-C1c L-C2d N-Le
0.1 gMeana 3.00 3.20 3.00 0.10 0.15 0.10 2.40 2.53 2.40
Gumbelb 3.32 3.81 3.32 0.17 0.27 0.17 2.82 2.97 2.82
0.2 gMeana 7.07 7.40 7.27 3.97 3.78 3.61 5.91 6.06 5.96
Gumbelb 8.55 8.62 8.89 5.55 5.82 5.65 7.24 7.43 7.29
0.3 gMeana 9.62 11.04 11.32 5.29 5.26 5.32 8.08 8.49 8.58
Gumbelb 11.99 13.73 14.08 5.53 5.39 5.49 10.44 10.61 11.01
0.4 gMeana 11.83 13.42 15.02 5.91 5.54 8.12 9.32 9.85 10.25
Gumbelb 15.92 16.23 19.86 7.21 6.16 11.44 11.68 11.99 12.56
0.5 gMeana 14.58 16.28 17.65 6.53 5.56 9.46 10.86 11.21 13.03
Gumbelb 19.91 21.86 25.03 8.83 5.70 12.62 14.48 14.67 15.18
0.6 gMeana 16.96 19.35 20.72 6.37 5.93 11.37 11.84 12.31 14.04
Gumbelb 24.09 26.59 29.85 8.35 6.71 15.33 15.96 16.26 16.32
a)Mean value of relative displacements of each vibration unitb)90% extreme value of Gumbel Type-Ic)Abutment model with linear spring and dashpot (case 1)d)Abutment model with linear spring and dashpot (case 2)e)Abutment model with nonlinear spring and dashpot
Seismic Behavior Analysis of a Plate-Girder Bridge Considering Abutment-Soil Interaction 293
bridge system. Thus, the seismic responses may be
underestimated in the system only with the constant
stiffness considered. Therefore, the stiffness degradation
should be accommodated in the analysis of seismic
responses of the bridges.
5. Conclusions
The effect of the stiffness degradation due to the
abutment-soil interactions upon the dynamic behaviors of
a multi-span plate-girder bridge under seismic excitations
are investigated by observing the relative displacements
to ground and relative distances between the adjacent
vibration units. The abutment-soil interaction is modeled
as the one degree-of-freedom system with nonlinear spring
and nonlinear damper. By comparing results obtained
from those systems with linearly modeled abutment-soil
interaction, the following trends are observed.
(1) The longitudinal abutment stiffness is found to
dramatically decrease at the onset of seismic
excitations. The relative motions to both grounds
and adjacent units are found to be larger in the case
with consideration of the stiffness degradation of
the abutment-backfill system than in the case
without consideration of the stiffness degradation
under moderate and strong seismic excitations with
PGA over 0.3 g.
(2) The effects of the stiffness degradation of the
abutment-backfill system become much larger
when relative distances between pier units are
compared. As PGA increases, this trend becomes
more noticeable. The stiffness degradation of the
abutment-backfill system is found to take an
important influence upon the global bridge motions.
The response motions may be underestimated if
only the linear system is considered. Hence, it is
concluded that the stiffness degradation should be
taken into account in the seismic analysis of the
bridge system especially in the case of strong
seismicity.
Acknowledgment
This work has been supported by Yonsei University,
Center for Future Infrastructure System, a Brain Korea 21
program, Korea.
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294 Jeong-Hun Won et al.
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