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

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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

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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))

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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.

REFERENCES

Jangid, R. S., (2005). “Optimal Friction Pendulum System for Near-Fault Motions.” Engineering Structures, Vol.

27, pp. 349-359.

Jangid, R. S., (2007). “Optimal Lead-Rubber Isolation Bearings for Near-Fault Motions.” Engineering Structures,

Vol. 29, pp. 2503-2513.

Ghobarah, A. and Ali, H. M., (1988). “Seismic Performance of Highway Bridges.” Engineering Structure, Vol. 10,

pp. 157-166.

Kunde, M. C. and Jangid, R. S., (2006). “Effects of Pier and Deck Flexibility on the Seismic Response of the

Isolated Bridges.” Journal of Bridge Engineering ASCE, Vol. 11, pp. 109-121.

Nielson, B. G. and DesRoches, R., (2007). “Seismic Fragility Methodology for Highway Bridges Using a

Component Level Approach.” Earthquake Engineering and Structural Dynamics, Vol. 36, pp. 823-839.

Padgett, J. E. and DesRoches, R., (2008). “Methodology for the Development of Analytical Fragility Curves for

Retrofitted Bridges.” Earthquake Engineering and Structure Dynamics, Vol. 37, No. 8, pp. 1157-1174.

Zhang, J. and Huo, Y., (2009). “Evaluating Effectiveness and Optimum Design of Isolation Devices for Highway

Bridges Using Fragility Function Method.” Engineering Structures, Vol. 31, No. 8, pp. 1648-1660.

Federal Emergency Management Association (FEMA), (2000). FEMA356: Prestandard and Commentary for the

Seismic Rehabilitation of Buildings. Washington, D.C.

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Park, Y. J., Wen, Y. K. and Ang, A. H-S., (1986). “Random Vibration of Hysteretic Systems under Bi-Directional

Ground Motions.” Earthquake Engineering and Structural Dynamics. Vol. 14, pp. 543-557.

Wu, Z. Y., Wang, Q., Butala, S. and Mi, T., (2011). Darwin Optimization Framework User Manual. Bentley

Systems, Incorporated, Watertown, C.T.