effect of damper on seismic response of cable …...long-span bridge. cable bridges are divided into...

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 9, Issue 2, February 2018, pp. 326–339, Article ID: IJCIET_09_02_032

Available online at http://http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=2

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication Scopus Indexed

EFFECT OF DAMPER ON SEISMIC RESPONSE

OF CABLE-STAYED BRIDGE USING ENERGY-

BASED METHOD

Fereidoun Amini

Professor, Department of Civil Engineering,

Iran University of Science & Technology, Tehran, Iran

Solmaz Sadoughi Shabestari, Mohammad Emamian

M.Sc., Department of Civil Engineering,

Iran University of Science & Technology, Tehran, Iran

ABSTRACT

Bridges are a vital element of transportation networks. As any disorder in the

maintenance or utilization of bridges may cause irrecoverable incidents, the

vulnerability of bridges to earthquakes must be seriously considered. Dampers are

extensively used in the handling of earthquake forces; however, their optimization and

optimal location have not been clarified. Considering the unjustified use of force-

displacement design methods in the seismic design of important structures, such as

bridges, the use of an energy-based approach with nonlinearly behaving materials

and the concept of hysteretic energy are more rational and practical. Therefore, in

this study, the effect of dampers and their location on the seismic response of cable-

stayed bridges is investigated by an energy-based method. For this purpose, a case

study is examined via a numerical study. The results indicate that, to reduce the

bridge’s seismic response to earthquake forces, a damper installed between the deck

and pylon is more effective than a damper installed on the cables.

Key words: Control of structure, Cable-stayed Bridge, Damper, Energy-based

method.

Cite this Article: Fereidoun Amini, Solmaz Sadoughi Shabestari, Mohammad

Emamian, Effect of Damper on Seismic Response of Cable-Stayed Bridge Using

Energy-Based Method. International Journal of Civil Engineering and Technology,

9(2), 2018, pp. 326-339.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=2

1. INTRODUCTION

Typically, long span bridges connect important points within transportation systems, and it is

crucial for that connection to be maintained after an earthquake. Therefore, their seismic

performance should be acceptable. Cable bridges are one of the most common structural

systems for long span bridges. For spans over 200 m, cable bridges are very suitable and

Effect of Damper on Seismic Response of Cable-Stayed Bridge Using Energy-Based Method


should be considered as one of the main options when selecting the structural system of a

long-span bridge. Cable bridges are divided into two types: cable-stayed bridges and

suspension bridges [1].

Figure 1 Cable-Supported Bridges: a) cable-stayed bridge; b) suspension bridge

Designers are always investigating safe bridge designs and the retrofitting of existing

bridges against earthquakes. Thus, various methods have been utilized to achieve these

objectives. The most efficient approach consists of using various structural control systems.

Generally, structural control is performed by using tools or members in the structure to

control its behavior against lateral forces. In recent decades, the seismic control of structures

has drawn considerable attention from engineers.

During strong earthquakes, structures usually exceed their elastic limit. Therefore, the

design methods that have been used in recent decades, including force-based methods, do not

provide complete information on the nonlinear behavior of structures during an earthquake.

These methods usually result in forces that need to be considered. Hence, these forces are not

optimal; therefore, the design of a bridge with consideration to these forces is not optimized

and does not ensure an acceptable structural performance during an earthquake.

Plastic deformation should be considered when estimating earthquake damage or when

determining the required maximum response. Regarding the nature of the earthquake force,

the structure, or any member of the structure, is subjected to several opposite forces and

different time-dependent deformations. This can be investigated by considering the energy

absorbed into the structure.

Thus, the use of energy-based methods by incorporating the nonlinear behavior of the

structure and the hysterical energy concept is a good alternative to methods based on force

and displacement. The objective of developing the design of a bridge by employing energy

based-methods is to minimize the response and hysteretic energy absorbed by the structural

members. Thereby, the seismic response of the structure is improved.

The energy-based method was first proposed by Housner in 1956 [2]. In his method, the

energy absorption capacity of the structure against large earthquakes was introduced as a

structural evaluation parameter, and the equivalent velocity was defined as an indicator of

absorbed energy. Therefore, Housner used this velocity as an approximation to the structure’s

nonlinear behavior demand. Uang and Bertero were among the pioneers of the energy

method. In 1990, they conducted a parametric study on the energy method [3] and obtained

the main parameters and their effectiveness on energy. Finally, they defined two types of

equilibrium equations for energy: one for absolute energy and another for relative energy. As

Fereidoun Amini, Solmaz Sadoughi Shabestari, Mohammad Emamian


𝐸𝐼 = 𝐸𝑘 + 𝐸𝜉 + 𝐸𝑆 +

𝐸𝐻

they pointed out in their report, the absolute energy was more physically meaningful in

comparison to relative energy. Additionally, for structures with low and high periods, the

calculations of the input energy were different in absolute and relative terms. In 1994, Fajfar

and Vidic studied the input energy, the residual energy spectra, and their ratio [4]. In 2001,

Riddell and Garcia, conducted a study on the nonlinear response of single-degree-of-freedom

systems (SDOFs) under earthquake excitation. They concluded that by using the amount of

energy absorbed through nonlinear deformations, the deformation capacity required to carry

out an appropriate design could be achieved [5]. In 2002, Leelataviwat et al. used the concept

of energy balance to calculate the design forces for an SDOF structure. Later they expanded

the application of the concept to a multi-degree-of-freedom structure [6]. In 2003, Chou and

Uang suggested the use of energy as a substitute for force and displacement responses.

Accordingly, they used energy as an indicator of time-dependent structural damage [7]. Since

then, the energy-based method has been extensively applied to seismic design and analysis.

For instance, in 2010, based on the energy-based method and by considering the flexural

rigidity and sag-extensibility of the cable, Cheng et al., investigated the damping properties of

a cable-damper system [8]. In 2013, Habibi et al. proposed a stepwise multi-mode energy-

based design method for seismic retrofitting with passive energy dissipation systems as an

alternative to strength and displacement-based methods [9].

In this study, the effect of the damper and its location on the reduction of the cable-stayed

bridge’s responses to earthquakes was investigated using an energy-based method and

considering the hysterical energy concept. Through a case study, the analysis, modeling, and

structural responses against earthquake were evaluated with and without dampers. The

obtained results and their interpretation are presented in this paper.

2. BRIEF REVIEW OF ENERGY-BASED METHOD EQUATIONS

In the same way as in other methods, the design carried out in this study was based on the

following equation:

(1)

By considering the dynamic equilibrium equation of an SDOF system, and by integrating

it with the displacement occurring during an earthquake, the structural capacity and demand

for SDOFs could be defined within an absolute and relative state.

2.1. Absolute State

In this case, the general energy relationship is expressed follows:

(2)

where is the absolute kinetic energy, is the damping energy, is the absorbed

energy, is the elastic strain energy, is the energy dissipated by the hysteretic or plastic

deformation, and is the absolute input energy. These parameters were obtained from the

following equations:

(3)

∫ = ∫

Demand > Capacity



∫ = +

∫

where c is the damping coefficient, m is the mass of the system, is the absolute

displacement of the system, ν is the relative displacement of the system, is the ground

displacement, and is the force stored in the system. With respect to Equations (1) and (2),

the left side of Equation (2) can be assumed equivalent to the structural demand, while its

right side is equal to the structural capacity.

2.2. Relative State

In this case, the general energy relationship is as follows:

(4)

where is the relative kinetic energy and

is the relative input energy, which can be

obtained as follows:

(5)

∫

In this case, with respect to Equations (1) and (4), the left side of Equation (4) can be

assumed equivalent to the structural demand, while its right side is equal to the structural

capacity.

3. MODELING

In this paper, a case study was carried out to investigate the effect of dampers on the reduction

of structural responses to earthquakes. Thus, the Vasco da Gama cable-stayed bridge in

Lisbon, Portugal, which has been presented by Pedro and Reis in 2010 [10], was modeled. In

2016, Maleki and Shabestari, modeled this bridge and investigated its response modification

factor [11]. In the present work, their model was modified and considered as a case study. The

bridge was modeled using the CSI Bridge 2017 software, which is developed by Computers

and Structures, Inc.

Figure 2 Vasco da Gama cable-stayed bridge in Lisbon, Portugal

𝐸𝐼 = 𝐸𝑘

+ 𝐸𝜉 + 𝐸𝑆 +

𝐸𝐻



Figure 3 Perspective of bridge model

In this study, the Vasco da Gama cable-stayed bridge was modeled with applied loading

under three different conditions, as follows:

Without damper

With dampers on the bridge cables

With dampers between the deck and pylons

3.1. Model A: Cable-Stayed Bridge without Damper

3.1.1. Modeling specifications and assumptions

As can be seen in Fig. 4, the bridge has three spans. The length of the main span is 300 m,

while the length of the two side spans is 120 m.

Figure 4 Longitudinal geometry of bridge

The structural system of the deck consists of a slab and girder system. The width of the

bridge deck is 22 m. The deck was modeled via single frame modeling. Considering that most

of the seismic behavior of cable-stayed bridges is related to pylon behavior rather than to deck

behavior, a deck model with a single frame was considered appropriate. As shown in Fig. 5,

the pylons in this bridge are H-shaped and have a height of 135 m. The pylons were modeled

by using frame elements. At the level of the deck, each pylon is connected to the deck by a

main transverse beam.



Figure 5 Elevation view of H-shaped pylon

In this bridge, a semi-fan cable arrangement system is used, and there are 56 cables

attached to each pylon in the two planes on the left and right sides of the deck, respectively. In

the main span, the spacing of the cables is 10 m, while the side span spacing is 8 m. The

cables were modeled by frame elements, and their section was defined such that it represented

a 30-strand cable. By modeling the cables in this way, their flexural stiffness could be

considered. The geometric nonlinear behavior of the cables was captured by large

displacement analysis. The labels of the cables in the model are presented in Fig. 6.

Figure 6 Cable labels in the model

Additionally, due to the fact of the deck being modeled by a single element, and of the

cables being located on both sides of the deck, rigid elements were used to make cables with

an exact angle. These elements were modeled by link elements and defined as rigid elements.

To capture the nonlinear behavior of the structure, many nonlinear concentrated hinges

were defined as fiber hinges. These hinges consider the nonlinear behavior of elements in the

combination of axial forces and flexural moments simultaneously. To define these hinges, the

sections of the elements were divided into many smaller elements.

𝑏 → 𝑏 4 𝑐 4 → 𝑐 𝑐 → 𝑐 4 𝑏 4 → 𝑏



3.1.2. Loading Specifications

In this model, various loads were applied to the bridge. The most important applied load was

the dead load due to the weight of the constituent elements. The live load was defined based

on the AASHTO specifications and with an impact factor of 1.3. The earthquake loads were

assigned to the bridge according to the AASHTO specifications and with consideration to the

non-urban bridge guidelines [12].

After developing a complete model of the structure and its loading, nonlinear time history

analysis was used with consideration to the P-delta and large displacement effects. For each

time history analysis, three datasets were assigned in each direction. The Northridge, El

Centro, and San Fernando records were used as the time history datasets matching the design

spectrum. Respectively, these three earthquake records were considered as Fema 1-1, Fema 6-

1, and Fema 21-1, as shown in Fig. 7.

a b c

Fem

a

1-1

Fem

a

6-1

Fem

a

21-1

Figure 7 Time history of Fema 1-1, Fema 6-1, and Fema 21-1. a) 1 dir; b) 2 dir; 3) up dir

3.1.3. Cable Forces

In cable-stayed bridges, the vertical displacement of the deck due to the dead load should be

very small, and the bridge should have a predetermined longitudinal profile. Therefore, the

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec) -1

-0.5

0

0.5

1

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.4

-0.2

0

0.2

0.4

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.4

-0.2

0

0.2

0.4

0 20 40rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.6

-0.4

-0.2

0

0.2

0.4

0 20 40

rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.2

-0.1

0

0.1

0.2

0 20 40rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec)

-0.2

-0.1

0

0.1

0.2

0 10 20 30rele

tive

acc

eler

atio

n (

g)

time (sec)



cable tension on these bridges was determined accordingly. Thus, the deck of the bridge was

modeled as a beam, whose supports were rollers aligned with the cables. Then, the dead loads

were applied to the beam and the forces of the cables were obtained by analyzing the beam.

The basic specifications of the cables were considered based on these forces. Subsequently,

by using a trial and error process, the cable forces changed until the deflection profile under

the dead loads was close enough to the predetermined desired state. The results of this trial

and error process are presented in Tables 1 and 2.

Table 1 Force and deck displacement in cables b1 to b14

Table 2 Force and deck displacement in cables c1 to c14

3.2. Model B: Cable-Stayed Bridge with Damper Installed on Cables

Under this condition, a nonlinear damper with a stiffness of 100,000

and a damping factor

of 15.13

was attached to each cable. Each damper was connected to the deck on one

side, and to the cable on the other side. The location of the damper on the cable was at a

distance equal to 0.06 of the cable length before the damper was attached. Time history

analysis was also conducted under this condition.

cable name cable force in deck level (ton) displacement of deck level (mm)

b1 175.38 -2.17

b2 172.4 -3.57

b3 168.86 -4.26

b4 164.8 -4.34

b5 160.25 -3.97

b6 155.31 -3.3

b7 150.1 -2.55

b8 144.77 -1.94

b9 139.54 -1.74

b10 134.79 -2.2

b11 131.12 -3.57

b12 129.42 -5.92

b13 130.65 -8.95

b14 134.57 -11.68

cable name cable force in deck level (ton) displacement of deck level (mm)

c1 180.89 -4.85

c2 180.88 1.24

c3 181.39 5.4

c4 183.12 5.68

c5 112.87 2.9

c6 115.5 0.98

c7 117.17 1.73

c8 117.26 5.12

c9 115.62 9.94

c10 112.58 14.35

c11 109.22 16.4

c12 107.64 14.47

c13 111.18 7.87

c14 122.83 -2.36



3.3. Model C: Cable-Stayed Bridge with Damper between Deck and Pylons

Under this condition, a nonlinear viscous damper, with a stiffness of 500,000

, damping

coefficient of 2,000

, and equal to 0.5, was placed at the transverse beam location

between the deck and pylons. Time history analysis was also conducted under this condition.

4. RESULTS AND FIGURES

The diagrams showing the performance of each model are presented according to the results

obtained by the nonlinear time history analysis of the three described structural models.

First, the damper’s energy was investigated as a parameter in the evaluation of its

performance. The diagrams below show the energy of the utilized damper in the modeling

carried out with regard to each of the three earthquake records. As can be seen in the

diagrams, the energy of the dampers in model B is negligible in comparison to the energy in

model C. This suggests that these dampers are not efficient and that their placement in the

structure does not affect the structure's response. Therefore, the different structural responses

were compared under the conditions of A and C, as shown in Figures 9, 10, and 11.

Figure 8 Damper energy in Models B and C. a) Fema 1-1; b) Fema 6-1; c) Fema 21-1

……

-1.0E+06

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

0 10 20 30 40

dam

per

en

ergy

(J)

time (sec)

Model b

Model c

-1.0E+060.0E+001.0E+062.0E+063.0E+064.0E+065.0E+066.0E+06

0 10 20 30

dam

per

en

ergy

(J)

time (sec)

Model b

Model c

-5.0E+06

0.0E+00

5.0E+06

1.0E+07

1.5E+07

0 10 20 30

dam

per

en

ergy

(J)

time (sec)

Model b

Model c

0.0E+00

2.0E+06

4.0E+06

6.0E+06

8.0E+06

1.0E+07

1.2E+07

1.4E+07

0 20

inp

ut

ener

gy (

J)

time (sec)

Model a

Model c

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

1.4E+06

1.6E+06

0 20

hys

tere

tic

ener

gy (

J)

time (sec)

Model a

Model c

b

a

c

b



Figure 9 Results of nonlinear time history Fema 1-1 analysis for three described structural models. a)

Time history of input energy; b) time history of hysteretic energy; c) time history of kinetic energy; d)

time history of base shear; e) time history of base moment; f) time history longitudinal displacement at

mid-span; g) time history vertical displacement at mid-span; ( Cable-stayed bridge without damper,

Cable-stayed bridge with cable damper, Cable-stayed bridge with damper between deck and pylon)

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

0 10 20 30

kin

etic

en

ergy

(J)

time (sec)

Model a

Model c

c

-1.5E+07

-1.0E+07

-5.0E+06

0.0E+00

5.0E+06

1.0E+07

1.5E+07

0 20

bas

e sh

ear

(N)

time (sec)

Model a

Model c

d

-8.0E+10

-7.5E+10

-7.0E+10

-6.5E+10

-6.0E+10

-5.5E+10

-5.0E+10

-4.5E+10

0 20

bas

e m

om

ent

(N-m

)

time (sec)

Model a

Model c

e

-0.15

-0.1

-0.05

0

0.05

0.1

0 10 20 30Lon

gitu

din

al d

isp

lace

men

t in

th

e

mid

dle

of

the

span

(m

) time (sec)

Model a

Model c

f

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30vert

ical

dis

pla

cem

ent

in t

he

m

idd

le o

f th

e sp

an (

m)

time (sec)

Model a

Model c

g

0.0E+00

2.0E+06

4.0E+06

6.0E+06

8.0E+06

1.0E+07

1.2E+07

0 10 20 30

inp

ut

ener

gy (

J)

time (sec)

Model a

Model c

a

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

0 10 20 30

hys

tere

tic

ener

gy (

J)

time (sec)

Model a

Model c

b



Figure 10 Results of nonlinear time history Fema 6-1 analysis for three described structural models. a)

Time history of input energy; b) time history of hysteretic energy; c) time history of kinetic energy; d)

time history of base shear; e) time history of base moment; f) time history longitudinal displacement at

mid-span; g) time history of vertical displacement at mid-span; ( Cable-stayed bridge without damper,

Cable-stayed bridge with cable damper, Cable-stayed bridge with damper between deck and pylon)

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

0 10 20 30

kin

etic

en

ergy

(J)

time (sec)

Model a

Model c

c

-1.5E+07

-1.0E+07

-5.0E+06

0.0E+00

5.0E+06

1.0E+07

1.5E+07

0 10 20 30bas

e sh

ear

(N)

time (sec)

Model a

Model c

d

-7.5E+10

-7.0E+10

-6.5E+10

-6.0E+10

-5.5E+10

-5.0E+10

0 10 20 30

bas

e m

om

ent

(N-m

)

time (sec)

Model a

Model c

e

-3.0E-01

-2.0E-01

-1.0E-01

0.0E+00

1.0E-01

2.0E-01

3.0E-01

0 10 20 30Lon

gitu

din

al d

isp

lace

men

t in

th

e

mid

dle

of

the

span

(m

) time (sec)

Model a

Model c

f

-0.1

-0.05

0

0.05

0.1

0 10 20 30

vert

ical

dis

pla

cem

ent

in t

he

m

idd

le o

f th

e sp

an (

m)

time (sec)

Model a

Model c

g

0.0E+00

5.0E+06

1.0E+07

1.5E+07

2.0E+07

2.5E+07

0 10 20 30

inp

ut

ener

gy (

J)

time (sec)

Model a

Model c

a

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

6.0E+06

0 10 20 30

hys

tere

tic

ener

gy (

J)

time (sec)

Modela

b



Figure 11 Results of nonlinear time history Fema 21-1 analysis for three described structural models.

a) Time history of input energy; b) time history of hysteretic energy; c) time history of kinetic energy;

d) time history of base shear; e) time history of base moment; f) time history longitudinal displacement

at mid-span; g) time history vertical displacement at mid-span; ( Cable-stayed bridge without

damper, Cable-stayed bridge with cable damper, Cable-stayed bridge with damper between deck

and pylon)

Moreover, in Table 3, the maximum values for each structural response in Models A and

C are presented for Fema 1-1.

Table 3 Maximum structural response values in Models A and C for Fema 1-1

Model A Model C

Input energy (J) 1.19E+07 1.21E+07

Link hysteretic energy (J) 1.49E+06 9.10E+05

Kinematic energy (J) 4.46E+06 4.55E+06

Base shear (N) 1.21E+07 8.73E+06

Base moment (N-m) -5.35E+10 -5.37E+10

Longitudinal displacement (m) 8.21E-02 8.79E-02

Vertical displacement (m) 9.99E-02 5.62E-02

Damper energy (J) 0.00E+00 4.22E+06

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

6.0E+06

0 10 20 30

kin

etic

en

ergy

(J)

time (sec)

Model a

Model c

c

-3.0E+07

-2.0E+07

-1.0E+07

0.0E+00

1.0E+07

2.0E+07

3.0E+07

0 10 20 30

bas

e sh

ear

(N)

time (sec)

Model a

Model c

d

-7.5E+10

-7.0E+10

-6.5E+10

-6.0E+10

-5.5E+10

-5.0E+10

0 10 20 30

bas

e m

om

ent

(N-m

)

time (sec)

Model a

Model c

e

-4.0E-01-3.0E-01-2.0E-01-1.0E-010.0E+001.0E-012.0E-013.0E-014.0E-01

0 20Lon

gitu

din

al d

isp

lace

men

t in

th

e

mid

dle

of

the

span

(m

) time (sec)

Model a

Model c

f

-0.2

-0.1

0

0.1

0.2

0 10 20 30

vert

ical

dis

pla

cem

ent

in t

he

m

idd

le o

f th

e sp

an (

m)

time (sec)

Model a

Model c

g



Based on Equation 2, the input energy was equal to the sum of the strain, kinetic,

damping, and hysteretic energy. Therefore, according to the Energy Conservation Law, as the

energy dissipated by the dampers increased, the structural responses, such as the

displacement, velocity, and acceleration, were reduced as a consequence of the decreasing

strain and kinetic energy. Additionally, at a certain seismic level, the lost hysteresis energy owing to the nonlinear behavior of members, such as the formation of plastic hinges, was

reduced. Therefore, the structural damage caused by the earthquake was also reduced.

In the diagrams presented above, it can be seen that by adding a damper between the

pylon and deck, the maximum hysterical and kinetic energy values of the structure decreased

by approximately 35% and 5%, respectively. The maximum base shear value at the end of the

pylon was reduced by approximately 25%. Moreover, the time until the occurrence of the

maximum values was decreased, and the maximum longitudinal and transverse displacement

values at mid-span were reduced by approximately 7% and 40%, respectively.

5. CONCLUSIONS

In this study, the effects of dampers on the responses of a cable-stayed bridge to earthquake

were investigated using an energy-based method. Specifically, a case study was carried out

regarding the effect exerted by dampers placed on the cables and the effect exerted by

dampers placed between the pylons and deck on the structural responses to the recorded

Northridge, El Centro, and San Fernando earthquakes. The numerical modeling results

indicated that the dampers on the cables did not have any effect on the bridge’s structural

responses to earthquakes. However, the damper between the pylon and deck reduced the

structural responses considerably, including the hysterical energy, base shear, and vertical

displacement responses. It was also shown that a large portion of the seismic force acting on

the bridge was related to the weight of the pylons. For further investigations on the

performance improvement of cable-stayed bridges, it is suggested that consideration be given

to the behavior of the pylons and control systems.

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effect of damper on seismic response of cable …...long-span bridge. cable bridges are divided into...

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