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1
Optimal Seismic Isolation Design for A Highway Bridge
with Nonlinear Base Isolator Modeling
Yili Huo and Bulent N. Alemdar1
ABSTRACT
Time history analysis is the recommended analysis method for seismic isolation of
bridges in many design codes. This type of analysis imposes much more work to structural
engineers for isolation design than traditional non-isolation design. Optimal choices of seismic
isolation design parameters play a crucial role in mitigating bridge superstructure damages. This
study utilizes fragility method and generic algorithm search to address optimal seismic isolation
parameters. These methods can be very beneficial for bridge engineers to expedite and provide
supplemental information during design process. To demonstrate this, a two-span concrete
bridge is chosen in this study and an elastomeric rubber bearing isolator is placed between the
pier column top and girder. A series of numerical studies is conducted with the aforementioned
methods to find optimal design parameters of the rubber isolator in regards with minimizing
damages in the bridge. Performance-Based Earthquake Engineering framework by PEER is
followed during the numerical study to evaluate the bridge damages equipped with the isolation
bearing. It is found that the characteristic yielding strength and the post-yielding stiffness of the
bearings are crucial for seismic damage mitigation, but the pre-yielding stiffness and several
other hysteretic controlling constants are not influential. The identified design parameters could
be adopted for isolation design of bridges with similar properties. The proposed methods for
searching optimal isolation design could be useful in bridge engineering practice.
Yili Huo, Software Research Engineer I, Bentley Systems Inc., 2744 Loker Ave. West, Carlsbad, CA 92010 Bulent N. Alemdar, Senior Software Product Research Engineer, Bentley Systems Inc., 2744 Loker Ave. West, Carlsbad, CA 92010
2
INTRODUCTION
Various kinds of optimization tasks prevail in many structural engineering tasks. Most
common optimization objectives include the structural performance under certain types of
loadings, the material consumption or cost, the building geometries, etc. The variables to be
optimized are typically structural geometries, structural configurations, element sizes and
properties or material properties. Among so many kinds of optimization problems, seismic
related optimization could be more complicated than most other problems, due to the
probabilistic nature of the seismic loads.
This study attempts to illustrate the feasibility and details of using fragility method and
generic algorithm to optimize structural element properties in order to approach best seismic
performance. An optimization problem is created and solved in such a way that a seismic isolator
in a highway bridge is to be designed to mitigate bridge damages under earthquakes most
effectively. To this end, the main objective is to use the aforementioned methods to find optimal
design parameters for the seismic isolator to minimize superstructure damages in the bridge.
Several previous studies have focused on similar isolator optimization problems with
varied methodologies. For example, Jangid 2005, 2007 enumerated possible isolator
configurations and found the optimized property values with 6 ground motions. Ghobarah and
Ali 1998 also identified the optimized properties with 3 ground motions. Another similar study is
done by Kunde and Jangid 2006 also with 3 ground motions. One critical question in these
studies is that whether the results with chosen motions are validated for other ground motions.
Therefore, to deal with the uncertainty associated with ground motions, several other studies
applied probability methods, mostly fragility method, to statistically answer the question. Such
researches include the ones by Nielson and DesRoches 2007, Padgett and DesRoches, 2008 and
Zhang and Huo 2009. As a price of incorporating the uncertainties, the computation demand
caused by probabilistic method is very high. And, because enumeration is also conducted to find
the optimum, the total computation work is dramatically huge, which is almost impossible for
practical use.
Given the advantages and disadvantages of above reviewed methods, this study attempts
to solve the proposed optimization example in three solutions and compares the findings and
efficiencies from them. First, the Probabilistic Seismic Demand Analysis (PSDA, the fragility
method) is used to solve the problem with enumeration over isolator property values. Second, a
Generic Algorithm Search (GAS) is employed to search the optimal isolator properties for given
ground motion histories. Finally, the PSDA and GAS are combined for searching optimal design
values.
OPTIMIZATION PROBLEM STATEMENT
Modeling
A two-span concrete bridge model is built for this study. The modeling prototype is a
combination of two California bridges, the Overcrossing of I91/5 Highway and the Mendocino
Bridge. The bridge model geometry and cross-section properties of pier column are defined in
Fig 1. The material values are the same as that in the study of Zhang and Huo 2009. The
3
calculated mass for deck is 340.0 tons for each span, and for column is 44.0 tons. An isolator is
placed between the pier column top and girder.
(a) Bridge model side view sketch
(b) Bent and deck sketch (c) Column cross section
Figure 1. Model sketches and geometry.
To simulate the column’ yielding behavior, a rotational spring is inserted at the bottom of
the column. The spring represents a rigid-plastic hinge behavior (i.e., it has an infinite initial
stiffness before yielding). A pushover analysis with a fiber cross-section idealization is carried
with OpenSEES (http://opensees.berkeley.edu) to obtain nonlinear characteristics of the plastic
hinge. The followings are obtained: My,hinge = 7800.0 kN·m, θy,hinge=0.00427 rad, and
Ky,hinge=4.7×104
kN·m/rad.
The isolator element is formulated based on an evolution equation given by Park et al.,
(1986). In this equation, the force-displacement relationship of the two transverse directions is
coupled:
(1a)
(1b)
where , , , and are the shearing force, deformation, initial stiffness, yielding force
and post-yielding stiffness ratio, respectively and . The terms and are referred to
as evolutionary variables and they represent hysteretic components of the restoring forces. These
terms are dimensionless and defined in the following ordinary different equations (ODE):
(2)
in which and are the yielding displacements, respectively. The constants , and are
controlling constants and they define the shape of hysteresis loop. In this study, these constants
are , and . The other property parameter values are referred to as
optimization parameters and they are discussed in the following sections.
4
Time History Analysis and Ground Motions
Nonlinear time history analyses are carried out to evaluate the seismic responses of the
model. The seismic loading is applied only in the longitudinal direction, i.e. in the plane of
Figure 1a. It is acknowledged by the authors that the seismic response in transverse direction is
more crucial for isolated bridge and the coupled effect between two directions is important, but
this study only investigates the longitudinal response because it is mainly aimed in the paper to
demonstrate the use of the of the proposed methods. A total of 50 ground motions are selected
from PEER Strong Motion Database (http://peer.berkeley.edu/smcat/). The selection intends to
uniformly distribute peak ground acceleration (PGA) from 0.05g to 1.5g.
Optimization Parameters and Objectives
For the isolator element, the following three parameters dominantly govern its response:
initial stiffness (K0), yielding strength (Fy) and post-yielding stiffness ratio ( ). For a certain type
isolator, the post-stiffness ratio is usually fixed in a certain range. For example, it is typically
chosen between 1/5 - 1/15 for an elastomeric rubber bearing (ERB), and 1/15 - 1/30 for a lead-
plug rubber bearing (LRB), and 1/50 - 1/100 for a friction pendulum system (FPS). In the current
study, a constant value of 1/20 is chosen for , which is a typical choice for aLRB type isolator.
To achieve the best isolator design, it is essential to find optimal values of K0 and Fy,
which are the chosen optimization parameters in the present study. Based on the column stiffness
and the hinge properties, these parameters are searched in the following ranges:
(3a)
(3b)
in which is the elastic stiffness of the pier column under cantilever
boundary condition and is the column height.
Damage measures (DMs) are defined according to the deformations measured in the
isolator element and at the column hinge. The damage measure in the isolator is defined as
follows:
(4)
where is the deformation of the isolator. The damage measure for the column hinge is
defined according to FEMA356 (FEMA, 2000):
(5)
where is the column rotation measured in the column hinge.
A global level DM is then defined as a proportional summation of the two component
DMs:
(6)
The weight ratios are chosen based on the consequences of the corresponding component
damage. A larger weight value is assigned to DMisolator and this is because a large deformation in
the isolator also means extensive deck movement, which could induce other damages such as
span collapse, pounding at joints and foundation, and abutment failures.
The optimization goal is to minimize the damage in the bridge system by choosing
appropriate isolator properties. It can be mathematically expressed as:
(7a)
while subjected to:
5
(7b)
With above expression, the problem is not fully defined. In other words, the optimization could
be done for a certain earthquake, or for a group of earthquakes, or for a certain ground motion
intensity measure (IM), or at a certain damage level. This ambiguity will be discussed more
during running the optimization.
OPTIMIZATION
Solution I: Enumeration with PSDA (Fragility Method)
Ten uniformly distributed values for each of K0 and Fy are selected from the search
ranges defined in Eq. 7b. Hence, a total of hundred (i.e. ten by ten) isolator configuration is
addressed. Nonlinear time history analysis is performed for each configuration subjected to the
selected 50 ground motions (i.e., a total of 5000 analyses run carried out). The responses for each
analysis are calculated and interpreted with probabilistic seismic demand analysis (PSDA)
method. And the optimal configuration is identified via direct comparison of the PSDA results.
Figure 2 shows analysis results of 50 ground motion histories for a configuration of
and . The results are portrayed graphically as bridge
global level DM against ground motion IM. The figure also includes a regression analysis result,
which is further explained below. In a similar way, the same exercise is repeated for all 100
configurations.
The regression analysis for each configuration is carried out as follows:
(8)
where a and b are the two regressed constants. The peak ground velocity (PGV) is selected as the
IM for this regression. Although structural engineers are much more familiar with peak ground
acceleration (PGA) and it is also usually used for such regression analyses, the current study
adopts PGV because it is believed to be a better indicator for representing ground motion
intensities and therefore leads to better data fitting in the regression analysis (i.e. smaller
standard deviation). Further discussion on this issue is not preceded here due to paper length
limitation.
Figure 2. DM-IM data and regression for a model with an isolator configuration of
and .
0.008
0.022
0.058
0.157
0.425
1.148
3.099
0.01 0.03 0.07 0.20 0.53 1.43
Gla
ba
l Da
ma
ge
Mea
sure
Intensity Measure, PGV (m/s)
Data
Regression
At DM=1,2,3,4
At PGV=0.25,0.5,0.75,1.0
6
Corresponding to the 100 analyzed isolator configurations, there are totally 100 regressed
lines. Apparently, the “lowest” line among them represents the best performance, because it
provides the smallest DI under same IM, and sustains the largest IM to reach the same DI.
However, it is not possible to find a single line be lower than all other lines through all the IM or
DM range, because the lines cross-over each other. Therefore, an alternative way to harness the
data is proposed so that the data is compared at certain IM or DM values. For example, the DMs
at four PGV levels (0.25m/s, 0.5m/s, 0.75m/s and 1.0m/s) are interpolated in Fig 2.
Figure 3 compares the interpolated DM values for the 100 models with different isolator
configurations. The lowest point of the surface corresponds to the optimal isolator configuration
leading to the best seismic mitigation. Except Fig. 3a, these plots show almost the same trend:
the DM values are much more sensitive to the yielding strength Fy than to the elastic stiffness K0.
The optimal range for Fy is about 0.6 to 0.9Mhinge/Hcolumn. The surface in Figure 3a demonstrates
a different trend than the other three plots. That is because the damage scenarios under small
ground motions are different from relatively large earthquakes. The pier column remains elastic
and only the isolator contributes to damage measure.
(a) At PGV=0.25m/s (b) At PGV=0.5m/s
(c) At PGV=0.75m/s (d) At PGV=1.0m/s
Figure 3. DM of models with different isolator configurations.
Figure 4 provide further information deducted from the results shown in Figure 3. In this
figure, the first top five optimal parameters are plotted against PGV. It is seen that optimal Fy is
decreasing with increasing PGV, and optimal K0 decreases along PGV. These finding are
0.3
0.5
0.7
0.9
1.1
0.2
0.25
0.3
0.35
0.4
0.4 0.5 0.6 0.7 0.8 0.9 11.1
1.21.3
K0/K
co
lum
n
Da
ma
ge
Mea
sure
Fy/(Mhinge/Hcolumn)
0.3
0.5
0.7
0.9
1.1
0.60.650.7
0.750.8
0.850.9
0.951
1.051.1
0.4 0.5 0.6 0.7 0.8 0.9 11.1
1.21.3
K0/K
co
lum
n
Da
ma
ge
Mea
sure
Fy/(Mhinge/Hcolumn)
0.3
0.5
0.7
0.9
1.1
11.1
1.21.3
1.41.5
1.6
1.7
1.8
0.4 0.5 0.6 0.7 0.8 0.9 11.1
1.21.3
K0/K
co
lum
n
Dam
age
Mea
sure
Fy/(Mhinge/Hcolumn)
0.3
0.5
0.7
0.9
1.1
1.61.71.81.9
22.12.22.32.4
2.52.6
0.4 0.5 0.6 0.7 0.8 0.9 11.1
1.21.3
K0/K
co
lum
n
Da
ma
ge
Mea
sure
Fy/(Mhinge/Hcolumn)
7
expected because DM is mostly contributed from the isolator component for smaller
earthquakes.
Figure 4 provide further information deducted from the results shown in Figure 3. In this
figure, the first top five optimal parameters are plotted against PGV. It is seen that optimal K0
decreases with increasing PGV, and optimal Fy is increasing along PGV. These finding are
expected. For smaller earthquakes, DM is mostly contributed from the isolator component.
Therefore, a smaller Fy could prevent column hinge yielding, and meanwhile a bigger K0 can
also reduce the deformation in isolators. For larger motions, the column yielding is inevitable. A
bigger Fy limits the severe deformation in bearings. And at the same time, a deducted K0 could
soften the system and reduce system resonance.
(a) Optimal K0 against PGV
(b) Optimal Fy against PGV
Figure 4. Optimal isolator configurations.
Solution II: Generic Algorithm Search (GAS)
A different approach is followed in this solution in such a way that optimal values of K0
and Fy are searched for each ground motion history. An optimum solution is targeted solution at
which minimum damage is measured with optimal values of K0 and Fy. In this case, it is not
required to run all possible scenarios as carried out in the previous solution. Instead, an
optimization problem, as defined in Eq. (7), is solved for each ground motion and hence, number
of runs to attain optimum values is significantly reduced.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
K0/K
co
lum
n
PGV (m/s)
1st optimal
2nd optimal
3rd optimal
4th optimal
5th optimal
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Fy/(
Mh
ing
e/H
co
lum
n)
PGV (m/s)
1st optimal
2nd optimal
3rd optimal
4th optimal
5th optimal
8
To fulfill this objective, Darwin optimization software package (Wu Z.Y., et al. 2011) is
employed in this approach. This optimization package is designed and developed as a general
optimization toolkit to address single and multiple objective optimization problems. Optimal
seismic isolator parameters (i.e., K0 and Fy) for each ground motion is searched with the help of
the optimization software. More specifically, a series of nonlinear time history analysis runs is
performed and at each run, a new set of (K0 , Fy) predicted by the optimization software is
executed. This process is repeated until optimum values of (K0 , Fy) are obtained.
The aforementioned procedure is carried out separately for each ground motion history
and for each case, the optimum values are searched. It is noted that almost 500 runs suffice to
obtain the optimal values for all ground motions (i.e., 10 iteration for each ground motion
observed). The results are portrayed graphically in Figs. 5-7. Figure 5 shows calculated global
level damage measures ( ) obtained with optimum values of (K0 , Fy) for each ground
motion. Corresponding optimal values of K0 and Fy are given in Fig. 6 and 7, respectively. It is
interesting to note from these figures that yielding strength (i.e., Fy ) is more crucial for seismic
damage mitigation than initial stiffness of the isolator. The same observation is also drawn with
the previous solution.
Figure 5. Damage measures with respect to optimal values of K0 and Fy.
Figure 6. Normalized optimal values of K0 with minimum .
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Dam
age
Me
asu
re (D
Mgl
obal
)
Ground Motions
0.0
0.2
0.4
0.6
0.8
1.0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
K0
/ K
colu
mn
Ground Motions
9
Figure 7. Normalized optimal values of Fy with minimum .
Figure 8 shows with respect to ground motion intensities. A polynomial curve
that best fits the data is also shown in the figure. Each mark in the plot indicates a
value calculated with respect to a specific ground motion and with the optimal values of K0 and
Fy. In other words, the region above the curve contains all possible solutions while the curve
itself represents a solution set of optimal values of K0 and Fy under different ground motion
intensities. A similar exercise is carried out for optimal values of K0 and Fy, as shown in Fig. 9. It
is noted from the figure that the optimal ranges for K0 and Fy are about 0.6Kcolumn - 1.0Kcolumn and
0.6Mhinge/Hcolumn - 1.0Mhinge/Hcolumn, respectively. The only exception is for the cases recorded
with low ground motion intensities.
Figure 8. Optimal values for different ground motion intensities.
0.0
0.2
0.4
0.6
0.8
1.0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
F y/
(Mhi
nge/H
colu
mn)
Ground Motions
0
1
2
3
4
5
6
0 0.4 0.8 1.2 1.6 2
Dam
age
Me
asu
re (D
Mgl
ob
al)
Intensity Measure (PGV (m/s))
10
Figure 9. Optimal K0 and Fy values for different ground motion intensities.
Solution III: GAS and PSDA
This last solution combines the fragility method and generic search algorithm. The
intention is to utilize fragility method to incorporate the uncertainties in ground motions and to
use generic search algorithm to search for optimal solutions. With this in mind, the procedure is
applied as follows: with a given isolator configuration (Fy and K0), the model is analyzed with 50
ground motions and then, the PSDA formula is used to find DM-IM relationship by the help of
regression analysis. The DM value is calculated for a targeted IM value and this value is used to
predict the next trial values of (Fy and K0) by the optimization package. Then, the analysis is
repeated with 50 ground motions and with the new values of (Fy and K0). This execution is
reiterated until optimal solutions of (Fy and K0) is found (i.e., ) for the targeted
IM. Note that this exercise is performed separately for each of the following targeted IM values:
PGV=0.25m/s, 0.50m/s, 0.75m/s and 1.0m/s
Table I summarizes the optimal configurations predicted by the three different solutions
addressed in the paper. For Solution 1 and 2, the optimal values are read directly from the fitting
curves in Figure 4 and 9, respectively.
TABLE I. OPTIMAL ISOLATOR CONFIGURATIONS FROM DIFFERENT SOLUTIONS
At
PGV=0.25m/s
At
PGV=0.50m/s
At
PGV=0.75m/s
At
PGV=1.0 m/s
Solution I: PSDA
Enumeration
K0/Kcolumn 1.17 0.99 0.84 0.70
Fy/(Mhinge/Hcolumn) 0.62 0.66 0.69 0.73
Solution II: Generic
Algorithm Search
K0/Kcolumn 0.86 0.82 0.78 0.77
Fy/(Mhinge/Hcolumn) 0.67 0.64 0.64 0.66
Solution III: GAS
and PSDA
K0/Kcolumn 1.10 1.10 0.90 0.90
Fy/(Mhinge/Hcolumn) 0.60 0.60 0.70 0.70
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.4 0.8 1.2 1.6 2
K0
/ K
colu
mn
Intensity Measure (PGV (m/s))
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.4 0.8 1.2 1.6 2
F y/
(Mh
inge
/Hco
lum
n)
Intensity Measure (PGV (m/s))
11
CONCLUSIONS
This study proposes a solution framework to find optimal design parameters of a
seismically isolated highway bridge. With the proposed solutions, the following observations are
concluded:
The yielding strength of the isolator (Fy) is more influential in controlling damage
than the initial stiffness of the isolator (K0). It is observed that all three solutions
approximately predict Fy = 0.65Mhinge/Hcolumn as an optimal choice for all ground
motion histories.
The generic algorithm search solution can be a highly efficient and effective tool
in various stages of a design process. For most cases, only a few analyses run is
sufficient to obtain optimal values of governing design parameters. Because a
typical design process only concerns one single set of design configuration rather
than revealing influential mechanism and secondary optimal configurations, it can
be effectively fit in design process.
The fragility method with enumerating all possible values is good for an academic
study point of view. With further data interpreting, the approach can reveal
important information about impact of different properties in governing
responses. However, the computation demand can be overwhelmingly huge even
for a small problem and hence, it can be almost unaffordable for practical usage.
If “N” number of ground motion is selected and if “m” number of optimization
parameter is targeted, the expected computational demand is approximately in the
order of , , and for Solution I, Solution II and
Solution III, respectively. Clearly, Solution 2 and Solution 3 have an
overwhelming computational demand advantage over the Solution 1 if “m” is big.
The Solution 3 serves as a compromise between other two solutions. By using
fragility method, it includes the uncertainties from the ground motions and by
using a generic algorithm search it eliminates the need of enumerating all possible
solutions.
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Kunde, M. C. and Jangid, R. S., (2006). “Effects of Pier and Deck Flexibility on the Seismic Response of the
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12
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