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STRUCTURAL CONTROL AND HEALTH MONITORINGStruct. Control Health Monit. 2007; 14:109143Published online 11 July 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/stc.87

MATLAB-based seismic response control of a cable-stayed

bridge: cable vibration

Chin-Hsiung Loh*,y,1 and Chia-Ming Chang2

Department of Civil Engineering, National Taiwan University, Taiwan

SUMMARY

This paper presents the cable vibration problem of structural control for a cable-stayed bridge. The goal ofthis study is not only to provide a test for development of strategies for control of cable vibration, but alsoto examine the effect of cable vibration on control of the structure itself. Based on the detailed drawings of

the Gi-Lu bridge, a three-dimensional numerical model has been developed to represent complex behaviorof the full-scale bridge. Differential motion at multiple supports and cable flexibility are included in theanalysis. A MATLAB-based structural analysis tool has been developed that considers of geometricallynonlinear behavior of beam elements that represent the cables. The dynamic characteristics of cables areverified using field experiments, a commercial finite element code, and the MATLAB program. Evaluationcriteria are presented for the design problems that are consistent with the goal of control of seismicresponse control of a cable-stayed bridge. Control devices are assumed to be installed either between thedeck and the end-abutment and/or between the deck and cables. Passive and active devices are used tostudy behavior of the model. A comparison of the control responses using truss elements or usinggeometrically nonlinear beam elements to represent the cable is also presented. Copyright # 2005 JohnWiley & Sons, Ltd.

KEY WORDS: cable-stayed bridge; active control algorithms; stayed cable; vibration

INTRODUCTION

In the past few decades, cable-stayed bridges have found wide application throughout the world.

The main spans of these bridges have reached a length of 900 m, leading to very long stayed

cables. The control of flexible bridge structures, such as cable-stayed bridges, is viewed as a

unique and challenging problem with many complexities in modeling, control design, and

implementation. Long-span cables are especially susceptible to vibration with large amplitude

under wind/rain loading or support excitations due to their high flexibility, relatively small mass

Received 5 January 2005Revised 16 March 2005

Accepted 2 May 2005Copyright# 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected].

*Correspondence to: Chin-Hsiung Loh, Department of Civil Engineering, National Taiwan University, Taiwan.

2Research Assistant.

Contract/grant Sponsor: Taiwan National Science Council; Contract Grant Number: NSC 93-2211-E-002-005

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and very low inherent damping [1,2]. Therefore, cable-stayed bridges might be vulnerable to

dynamic loading such as earthquakes and strong wind loads. In 1993 Warnitchai et al. [3]

experimentally and analytically studied active tendon control of cable-stayed bridges, subjected

to a vertical sinusoidal force. They utilized a simple cable-supported cantilever beam as a model.

Up to now, many research efforts have been focused on the interaction of cables with the deckand attenuation of the cable movement [46]. Other research efforts have been aimed at

understanding the overall dynamic behavior of cable-stayed bridges and developing finite

element models [7,8]. The working group on bridge control within the ASCE Committee on

Structural Control recently posted a first-generation benchmark structural control problem

based on the Cape Girardeau Bridge [7,8]. This problem focus on one-dimensional ground

acceleration applied in the longitudinal direction that is uniformly and simultaneously applied

at all supports. In the work of Moon et al. [9,10], a semi-active system for the benchmark bridge

employing MR dampers in conjunction with a LDG/clipped optimal control (LQG/MR) and a

sliding mode semi-active control system (SMC/MR) was studied. In addition, a hybrid base

isolation system employing semi-active control devices (using MR-dampers), often termed

smart dampers, has been suggested to control the damping force of bridge structure [11]. In the

design of modern feedback control systems, the question of uncertainties in the models and theexcitations, within the context of robust control, is becoming an important issue. For example,

an active scheme that uses active tendons via feedback of the states of the system for vibration

control of a cable-stayed bridge under seismic loads has also been proposed by Rodellar et al.

[12]. While different approaches have been adopted for robust control of civil engineering

structures, such as H1 and related tools [13], neural networks and fussy logic [14], and sliding

mode control [15,16], the use of these approaches to control of cable-stayed bridges has been

limited so far.

In the ASCE phase II benchmark control problem, a three-dimensional evaluation model has

been developed to represent complex behavior of the multi-support and transverse excitations of

the Cape Girardeau Bridge [17]. The benchmark problem and a sample control design have been

made available in the form of a set of MATLAB equations.

In this research the control of cable-stayed bridge is studied. Based on the detailed informationof the Gi-Lu cable-stayed bridge in Taichung County, Taiwan, a three-dimensional numerical

model has been developed to represent the complex behavior of the full-scale bridge. Dynamic

behavior of the cable including sag geometry, pre-stress tension force, and nonlinear response are

considered in this analysis. The formulation of the cable stiffness is carried out both in ABAQUS

and in MATLAB equations for consistency. This paper is focused on the following issues:

1. A three-dimensional evaluation model is developed in a realistic way to represent complex

behavior of a full-scale cable-stayed bridge. Nonlinear beam elements are used to represent

each cable. A comparison of the dynamic response of the bridge is made by using a truss

model or nonlinear beam element model for the cables.

2. A comparison of the control effectiveness is discussed for: (a) using active control strategy

by employing actuators between the bridge deck and the end-abutments; or (b) using bothviscous dampers between the cables and the bridge deck and actuators between the deck

and the end-abutments.

3. Based on the different evaluation criteria the control effectiveness between two different

control algorithms, active and hybrid, is examined.

4. A MATLAB-based model for control of a cable-stayed bridge is developed.

Copyright # 2005 John Wiley & Sons, Ltd. Struct. Control Health Monit. 2007; 14:109143

C.-H. LOH AND C.-M. CHANG110

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MODELING OF CABLE-STAYED BRIDGE

The cable-stayed bridge used for this study is the Gi-Lu bridge, located at Nantou County,

Taiwan. It is a modern, pre-stressed concrete cable-stayed bridge, which crosses the

Juosheui River in Taiwan. As shown in Figure 1(a) the bridge has a single pylon constructedwith reinforced concrete (58 m above the deck), two rows of harped cables (68 cables in total),

Figure 1. (a) Top view and side view of Gi-Lu bridge; and (b) cross-section of Gi-Lu bridge.


MATLAB-BASED SEISMIC RESPONSE CONTROL 111

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and a streamline-shape single box girder, as shown in Figure 1(b). With depth 2.75 m andwidth 24 m, the box girder rigidly connects with the pylon and spans 120 m to each side span.

17 pairs of near parallel pre-stressed steel cables extending from each side of the pylon

were developed. The material properties of the bridge are shown in Table I. The sag of each

cable is shown in Table II. Cables 1 and 2 indicate the shortest cables and cables 33 and 34 are

the longest cables. R and L indicate the left-hand and right-hand sides of the pylon. The

vibration frequency of each cable was also estimated from vibration test (Table III). The density

of the cable ranges from 0.410 to 0.613 kg/cm and the design cable force ranges from 220 to

290 ton.

Dynamic characteristics of the cable

A three-dimensional finite element model of the Gi-Lu bridge was developed in MATLAB.

A linear evaluation model is used in this cable-stayed bridge model. The finite element

model employs beam elements and rigid links. Due to large deformation of the cables

and their sag, geometric nonlinearity of beam elements needs to be considered in this

analysis. Analytical solutions of geometrically nonlinear beam element are available for

several special cases; for instance, a fixed beam subjected to concentrated loads.

Generally, conventional linear two-dimensional beam elements cannot consider the effect

of axial stresses due to large bending deformation. It is necessary to represent an

appropriate nonlinear behavior of beam. In this study, by using an energy method a

geometrically nonlinear beam element was generated that includes the nonlinear terms

plus the terms from the conventional linear beam element. Therefore, a geometric stiffness

matrix can be derived based on energy methods from Euler-Bernoulli equation as shown

follows [18]:Fext Klinear Knonlinearu 1

Table I. Material properties of Gi-Lu bridge.

Element Youngsmodulus (N/m2)

Area (m2) Moment ofinertia (m4)

Poissonsratio

Density(kg/m3)

Main girder (deck) 2:77 1010 12.27 9.77, 334.55 0.2 3202

Plate (deck) 2:77 1010 4 334.55, 334.55 0.2 0.001Side span 2:77 1010 8.121 6.66, 84.29 0.2 2400Pier 2:77 1010 8.121 6.66, 84.29 0.2 2400

Cable (1) 2:04 1011 0.005163 0.515 105, 0.515 105 0 8000Cable (2) 2:04 1011 0.006 0.45 105, 0.45 105 0 8000Cable (3) 2:04 1011 0.00756 0.67 105, 0.67 105 0 8000Pylon 2:77 1010 8.670.07 6.8563.63, 14.1869.02 0.2 2400

Varying cross-section.



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Table II. Left-hand side four columns indicate the sag of each cable (middle node) and right-hand sidefour columns indicate the 1st natural frequency of each cable (R** means west-side cables, L** means east-

side cables).

Sag (maximum) (m) 1st natural frequency (Hz)

0.51839 R33 L33 0.51839 0.75823 R33 L33 0.758230.52236 R31 L31 0.52236 0.75513 R31 L31 0.75513

0.4723 R29 L29 0.4723 0.79146 R29 L29 0.79146

0.29116 R27 L27 0.29116 0.99956 R27 L27 0.99956

0.23382 R25 L25 0.23382 1.111 R25 L25 1.111

0.23022 R23 L23 0.23022 1.1161 R23 L23 1.1161

0.16688 R21 L21 0.16688 1.3042 R21 L21 1.3042

0.13945 R19 L19 0.13945 1.4196 R19 L19 1.4196

0.091567 R17 L17 0.091567 1.7411 R17 L17 1.7411

0.072009 R15 L15 0.072009 1.9493 R15 L15 1.9493

0.070109 R13 L13 0.070109 1.9581 R13 L13 1.9581

0.058809 R11 L11 0.058809 2.1138 R11 L11 2.11380.049284 R9 L9 0.049284 2.2756 R9 L9 2.2756

0.033323 R7 L7 0.033323 2.7106 R7 L7 2.7106

0.022885 R5 L5 0.022885 3.1737 R5 L5 3.1737

0.018332 R3 L3 0.018332 3.3889 R3 L3 3.3889

0.011206 R1 L1 0.011206 4.0191 R1 L1 4.0191Pylon &

Pier No.2

Abutment

(Pier No.1)(Lu-Ku side)

Abutment

Pier No.3

(Gi-Lu side)

120 m

120 m

0.010133 R2 L2 0.010133 4.0193 R2 L2 4.0193

0.016358 R4 L4 0.016358 3.3891 R4 L4 3.3891

0.020243 R6 L6 0.020243 3.1738 R6 L6 3.1738

0.029262 R8 L8 0.029262 2.7108 R8 L8 2.7108

0.043072 R10 L10 0.043072 2.2758 R10 L10 2.2758

0.051267 R12 L12 0.051267 2.114 R12 L12 2.114

0.060964 R14 L14 0.060964 1.9582 R14 L14 1.9582

0.062502 R16 L16 0.062502 1.9493 R16 L16 1.9493

0.07934 R18 L18 0.07934 1.741 R18 L18 1.741

0.12064 R20 L20 0.12064 1.4193 R20 L20 1.4193

0.14422 R22 L22 0.14422 1.3037 R22 L22 1.3037

0.19876 R24 L24 0.19876 1.1149 R24 L24 1.1149

0.20176 R26 L26 0.20176 1.11 R26 L26 1.11

0.25108 R28 L28 0.25108 0.99808 R28 L28 0.99808

0.40785 R30 L30 0.40785 0.78728 R30 L30 0.78728

0.45105 R32 L32 0.45105 0.75068 R32 L32 0.75068

0.44722 R34 L34 0.44722 0.75464 R34 L34 0.75464



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TableIII.Comparisonontheestimatedcablevibrationfrequenciesfromvibrationmeas

urement,fromABAQUSandfromBcodeusing

nonlinearbeamelements.

Analyzenaturalfrequencyofcables(R1,R17,R34)

R1(shortest)

R34(longest)

Elements

10

00

15

50

10

0

200

Elements

1000

15

50

100

Method

ABA

QUS

MATLAB

MATLAB

MAT

LAB

MATLAB

Method

MEAS

URE

ABAQUS

MATLAB

MATLAB

MATLAB

Mode

H

z

Hz

Hz

H

z

Hz

Mode

Hz

Hz

Hz

Hz

Hz

1

4.12

4.1301

4.1238

4.1

237

4.1237

1

0.7

69

0.769

0.75119

0.759

74

0.75956

2

8.43

8.1687

8.1566

8.1

568

8.1569

2

1.4

99

1.496

1.4885

1.506

2

1.5054

3

12.97

12.027

12.023

12.0

25

12.026

3

2.2

48

2.247

2.2266

2.257

6

2.2565

4

17.93

15.622

15.671

15.6

82

15.685

4

2.9

97

2.999

2.9552

3.006

7

3.0052

5

23.26

18.884

19.078

19.1

09

19.117

5

3.7

51

3.755

3.6689

3.753

3.7512

21.763

22.245

22.3

1

22.327

6

4.5

11

4.514

4.3602

4.495

7

4.4938

24.236

25.189

25.3

07

25.338

7

5.2

79

5.278

5.0201

5.234

2

5.2321

R17(middle)

8

6.0

44

6.047

5.6376

5.967

7

5.9657

Elements

1000

15

50

100

9

6.8

33

6.822

6.2036

6.695

6

6.694

Method

MEA

SURE

ABAQUS

MATLAB

MAT

LAB

MATLAB

10

7.5

91

7.604

6.7041

7.417

1

7.4162

Mode

H

z

Hz

Hz

H

z

Hz

11

8.3

86

8.393

7.132

8.131

6

8.132

1

1.7327

1.7325

1.7318

1.7

208

1.7202

12

9.1

76

9.19

7.4739

8.838

4

8.8407

2

3.4628

3.4629

3.454

3.4

342

3.433

13

9.9

78

9.996

7.7248

9.536

8

9.5421

3

5.1923

5.2001

5.1587

5.1

349

5.1331

14

10.8

02

10.81

7.881

10.226

10.235

4

6.9496

6.9435

6.8349

6.8

16

6.8139

15

11.6

31

11.63

15.761

10.906

10.921

5

8.7043

8.696

8.4706

8.4

711

8.4695

16

12.4

55

12.47

15.766

11.575

11.597

6

10.476

10.459

10.052

10.0

94

10.095

17

13.3

03

13.32

15.829

12.233

12.264

7

12.2541

12.236

11.564

11.6

79

11.685

18

14.1

67

14.18

15.846

12.88

12.922

8

14.0303

14.028

12.99

13.2

2

13.235

19

15.0

21

15.05

15.934

13.514

13.571

9

15.8487

15.837

14.312

14.7

13

14.743

20

15.9

07

15.93

15.967

14.136

14.21

10

17.6494

17.665

15.509

16.1

52

16.206

21

16.8

21

16.83

14.744

14.839

11

19.5359

19.515

16.557

17.5

34

17.621

22

17.7

33

17.74

15.338

15.458

12

21.3985

21.387

17.428

18.8

56

18.987

23

18.6

35

18.67

15.916

16.067

Note:(a)R1,R17,

andR34indicatetheshortest,middle,andlongestcables,respectively;(b)translatio

nalandtransversedegrees-of-freedomof

eachcablenode

arelocked.



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or

Fx1

Fy1

Mz1

Fx2

Fy2

Mz2

2666666666664

3777777777775

EA

L

0

12EI

L3

06EI

L24EI

L

EA

L0 0

EA

L

0 12EI

L3

6EI

L20

12EI

L3

06EI

L22EI

L0

6EI

L24EI

L

2

666666666666666666664

3

777777777777777777775

N

0

06

5L

01

10

2L

15

0 0 0 0

0 6

5L

1

100

6

5L

01

10

L

300

1

10

2L

15

26666666666666666664

37777777777777777775

x1

y1

f1

x2

y2

f2

2666666666664

3777777777775

1a

where E is the Youngs modulus, A is the cross-sectional area, I is the moment inertia, L is the

length of the element, Fis the force, Mis the moment, x and y are the translational deformation

of both ends of the element, and j is the rotational deformation of both ends of the element.

Subscripts (1 and 2) indicate the two ends of the beam, the first term of the stiffness matrix is the

linear stiffness matrix, the second term is due to nonlinearity, and Nis related to the axial force

due to large deformation:

N EA

LL0 L 2

in which L0 is the actual length along the center line of the beam. However, to deal with the

complex behavior of cables, the computational procedures must be modified. In development of

the initial cable stiffness is important to obtain the actual pre-tensioned values (not designed

values) and to include correct element properties (modulus of elasticity, moment inertia of cross-

sectional area, length of taut cables, and Poissons ratio). The procedures for developing the

initial cable stiffness are shown below:

1. Assume that Poissons ratio equals zero.

2. Choose how many elements to use in the formulation, and transform Equation (1) into a

three dimensional nonlinear beam element.



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3. Use the initial length to generate the stiffness matrix where N equals the pre-tensioned

value.

4. Divide the external force (self-weight of cables) into an appropriate number of equal parts

n.

5. Use NewtonRaphson (see Figure 1) iterations to solve KfDg fFextg by appropriateconditions of convergence.

6. Calculate the present length due to large lateral deformation and re-formulate the cable

stiffness matrix where N is still the pre-tensioned value.

7. Repeat Step 5 until the total steps equal to n (in Step 4).

From these procedures, stiffness matrices of the cables can satisfy effects of sag and measured

natural frequencies.

In order to evaluate the dynamic characteristics of the stayed-cable in this model, the

vibration frequencies of each stayed-cable in the Gi-Lu bridge were examined from three

different approaches: (1) experiments; (2) ABAQUS; and (3) MATLAB equations. For example,

for the longest cables (33 and 34), it is described as: (i) angle of elevation 26 8; (ii) designed length

126.42 m; (iii) length density 47.9 kg/m; (iv) Youngs modulus 1:83 1011 N=m2; (v) crosssectional area 0.0060 m2; and (vi) gravity 9.81m/sec2. The FEM model developed using

ABAQUS has 1000 beam elements and considers the cable structure to be subjected to gravity

and to have fixed-end boundary conditions. The cable force and moment of inertia were

identified by matching the 21 vibration frequencies of the cables to those obtained from the field

experiment. Through an optimally iterative process, the identified cable force is 1 :66 106 N;and the identified moment of inertia is 5:0 106 m4: This information is used to formulateequations in MATLAB for the stiffness matrix of the cable system. The vibration frequency of

each cable predicted by MATLAB is verified using field experiments. The configuration of

Cables R1, R17 and R34 is shown in Figure 2, and the identified cable force is only 91.4%

(1:66 106=1:81 106 N) of the force estimated by string vibration theory. Comparisonbetween the identified vibration frequencies of the cable R1, R17 and R34 using MATLAB and

ABAQUS is shown in Table III. The vibration modes of cable R34 (longest) and cable L17 areshown in Figure 3. To catch the dynamic characteristics of the cable 1000 beam elements are

used in ABAQUS to analyze the detail of the dynamic characteristics of a single cable, but on

Figure 2. Configuration of Cable-R34 and the identified.



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the contrary, only 9 nodes were used to form a single cable which is sufficient to involve the low-

frequency dynamic behavior (0.74 Hz) as in relating to the whole bridge system.

DESCRIPTION OF FINITE ELEMENT MODEL

The finite element model of the cable-stayed bridge, as shown in Figure 4, has a total of 1009nodes. The pylon is modeled by 90 nodes with 540 degrees of freedom (DOF) (25 nodes are

above the deck, 5 nodes are below the deck, and 60 nodes are near the anchors). The deck is

modeled by 729 nodes with 4374 DOFs. Pier 2 (under the pylon) is modeled by 5 nodes

(including the node attached to the ground). Pier 1 (North side) and pier 3 (South side) have 4

nodes for each (including the node attached to the ground) with a total of 56 DOFs. The cable is

Figure 3. The first six mode shapes: (a) cable R34; and (b) cable R17.



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modeled with 9 nodes for each cable and total of 612 nodes for 64 cables with 3672 DOFs. The

mass matrix is formed by a lumped mass approach. The proportional damping formulation is

used in each element and then to form the full damping matrix is formed with the sameprocedure as is used to form the stiffness matrix. The damping ratio for deck, pylon and piers

(including all supports) is assumed to be 5% and for cables is assumed to be 1%. Constraints are

applied to restrain the both ends of the deck (boundary conditions). All DOFs at the bottom of

both piers are fixed. As for the boundary condition of the bridge structure, all translational

DOFs and torsion DOF at the side spans connected to the embankment are fixed. The

transverse and vertical degrees of freedom at Piers 1 and 3, and at the side spans of the bridge

deck deform consistently (i.e. y- and z-directions are constraint, and x-, fx-, fy-, fz}directions

are free to move).

EVALUATION MODEL

The control system of the cable-stayed bridge includes: evaluation model, sensor processor,

controller and control devices. Based on the above-mentioned finite element model the

formulation of the model is described first.

Problem formulation

Since a precise mathematical model for analyzing the dynamic behavior of cable-stayed bridges

is very complicated, appropriately reduced methods were used to formulate the equation of

motion for the Gi-Lu bridge. The reduced methods use static condensation and quasi-static

reduction. Consider the general equation of motion for a structural system subjected to seismic

loads

Mtotal .U Ctotal U KtotalU Pext 3

where system matrices can be written as

Mtotal Mcdp Mcoup

MTcoup Msp

" #Ctotal

Ccdp Ccoup

CTcoup Csp

" #Ktotal

Kcdp Kcoup

KTcoup Ksp

" #3a

Figure 4. Model for cable-stayed bridge (Gi-Lu bridge) with the span length of 318.9m, the pylon heightof 58 m from deck, and the deck width of 24 m.



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where subscript cdp indicates DOFs of cables, deck and pylon, subscript sp indicates the DOFs

for support from piers and side spans and subscript coup indicates coupled terms between

supports and others; U is the displacement response vector; Mtotal, Ctotal, Ktotal are the mass,

damping and stiffness matrices respectively of the whole structural system, and Pext is the vector

of external forces.

Model reduction (static condensation and quasi-static reduction)

The finite element formulation of the bridge model has a large number of degrees of freedom.

Static condensation is applied to reduce the redundant DOFs and retain the main DOFs. The

main DOFs is taken here to include the nodes of the main girder, pylon, piers, side span, and

cables including the nodes connecting the cables to the deck and pylon. The redundant DOFs

refers to the other nodes of the deck since these nodes are used to analyze static behavior. It is

assumed that the total displacement can be divided into two parts: active DOFs (main nodes)

and dependent DOFs (redundant nodes). Therefore, the equation for the static condensation

can be formed as

Kaa Kad

KTad Kdd

" #Ua

Ud

" # P

a

0

" #4

where the subscript a denotes the active DOFs and d denotes dependent DOFs. Then the total

displacement vector can be transformed into

Ua

Ud

" #

I

K1dd KTad

" #Ua or

Ua

Ud

TRUa 5

where [TR] is the transformation matrix of the static condensation. Consequently, the system

total mass, damping, and stiffness matrices can be changed as follows:

M

_

total TRT

MtotalTR; C

_

total TRT

CtotalTR; K

_

total TRT

KtotalTR 6

To consider the seismic excitation of the multiple-supported system both quasi-static and

dynamic analyses must be employed. By using the transformation matrix of the static

condensation Equation (1) can be re-arranged as follows:

M_

total.Ua C

_

totalUa K

_

totalUa TRTPext 7

where the active DOFs of Ua represent two components: Ua,cdp, the DOFs from cables, deck

and pylon, and Ua,sp, the DOFs from piers and side spans (supports). The total displacement

Ua,cdp can also be separated into the displacement Ua,cdps

due to static application of the groundmotion, and the dynamic displacement Ua,cdp

d relative to the quasi-static displacement. The

relationship between these displacement components is

Ua;cdp

Ua;sp

" #

Usa;cdp

Ua;sp

" #

Uda;cdp

0

" #8



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By neglecting the dynamic terms in Equation (7), the quasi-static solution can be obtained

from the following equation

Ka;cdp Ka;coup

KT

a;coupKa;sp" #

Usa;cdp

Ua;sp

" # 0

Ps

ext" # 9

Solving Equation (9) for quasi-static displacement leads to

Usa;cdp K1a;cdpKa;coupUa;sp

or

Usa;cdp RsUa;sp 10

Substituting Equations (8) and (10) into (7), one can obtain the dynamic equation of motion

of the structure as follows:

M_

a;cdp.Uda;cdp C

_

a;cdpUda;cdp K

_

a;cdpUda;cdp

M_

a;cdpRs M_

a;sp .Ua;sp C_

a;cdpRs C_

a;sp Ua;sp 11

To solve for the dynamic response of a cable-stayed bridge the state-space formulation is

used. A condition for convergence of the discrete state space calculation is that the following

criteria must be satisfied:

DT4p

o12

where o is the highest modal frequency (rad/sec) that is considered. In this study the first 400

modes of vibration are used for the analysis. Since the sampling time for the input ground

motion is 0.005 s, the convergence condition of Equation (12) is satisfied.

After the model has been reduced the equation of motion for the damped structural systemresults from Equation (11):

%Ma;cdp .Y %Ca;cdp Y %Ka;cdpY FTM

_

a;cdpRs M_

a;sp .Ua;sp

FTC_

a;spRs C_

a;sp Ua;sp 13

where

Y FUda;cdp; %Ma;cdp FT M

_

a;cdpF; %Ca;cdp FT C

_

a;cdpF; %Ka;cdp FT K

_

a;cdpF

and F is the modal matrix. A state space form for the cables, deck and pylon is as follows:

x

Y

.Y

" # Ax Euext

0 I

%M1a;cdp %Ka;cdp %M1a;cdp

%Ca;cdp

" # YY

" #

0 0

%M1a;cdpFTM

_a;cdpRs M

_a;sp %M

1a;cdpF

TC_

a;cdpRs C_

a;sp

24

35 .Ua;sp

Ua;sp

" #14



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Figure 5. (a) The first six vibration modes of the cable-stayed bridge. Its corresponding modal frequenciesare 0.51487, 0.73124, 0.75022, 0.75069, 0.75393, and 0.75464 Hz, respectively; (b) the 31st (f 1:506 Hz),64th (f 1:4456 Hz), 97th (f 1:756 Hz), 102nd (f 1:8941 Hz) and 115th (f 2:0378 Hz) vibrationmodes of the cable stayed bridge; and (c) first six mode shapes of cable-stayed bridge using truss elementsto simulate the cable behavior (the vibration frequencies of the first six modes are: 0.42152, 0.44952,

0.80013, 1.3695, 1.4455, and 1.5871 Hz).



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Through static condensation the final reduced model for the pylon and the deck reduce from

90 to 85 nodes and from729 to 243 nodes, respectively; the number of nodes for the pier and

cable remain the same.

Based on this formulation the mode shapes of the cable-stayed bridge are calculated. Figure 5(a)shows the first six vibration modes of the cable-stayed bridge. The corresponding modal

frequencies are 0.51487, 0.73124, 0.75022, 0.75069, 0.75393, and 0.75464 Hz, respectively. It is

found that the first fundamental mode is mainly dominated by the vibration of deck, and that

the following five modes are dominated by vibration of the cables. In addition, the 31st

(f 1:506 Hz), 64th (f 1:4456 Hz), 97th (f 1:756 Hz), 102th (f 1:8941 Hz) and 115th(f 2:0378 Hz) vibration modes of the cable stayed bridge are also shown in Figure 5(b),contributions from both cable and deck vibration to these higher modes are observed. Since the

implementation of geometric stiffness matrix to simulate the cable using beam element, then the

sag behavior of cable can be truly reflected. Besides, the cable vibration in transverse direction

and the pre-stressed force in the cable can also be used. Figure 5(c) shows the calculated lowest

six vibration modes of the cable-stayed bridge that are calculated using truss element to simulate

the cable (the vibration frequencies of the first six modes are: 0.42152, 0.44952, 0.80013, 1.3695,1.4455, and 1.5871 Hz). Therefore, a significant difference between the bridge vibration modes is

obtained if the cable is modeled using truss elements instead of dynamics of geometrically

nonlinear beam elements. A comparison of the acceleration and displacement response in the

longitudinal, transverse and vertical directions at the top of the pylon is shown in Figure 6. It

can be observed that larger displacement and acceleration responses occur in the transverse

Figure 5. Continued.



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Figure 6. Comparison of estimated acceleration and displacement response at the top of pylon using trusselements for cables and nonlinear beam elements for cables.



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direction at the top of pylon when the truss model is used for the cable; by contrast, the

longitudinal direction at the top of the pylon has the most significant responses when using the

nonlinear beam elements. Figure 7 compares the acceleration and displacement response at

the northern end of the bridge deck when using truss elements for the cables (blue line) and

nonlinear beam elements for the cables (red line). It is found that use of the truss model for thecables induces larger acceleration response at both ends of the bridge deck. For estimation of

transverse direction at both ends of the bridge, a larger displacement is observed by using the

nonlinear beam element model for the cables. It is believed that use of nonlinear beam elements

to model for the cables is a reasonable and necessary approach for predicting behavior of the

stayed cables. Therefore, in what follows only the modeling of the bridge cables using the

nonlinear beam elements for the cables is used to evaluate the control effectiveness. It is believed

that using truss elements to model the cable for the cable-stayed bridge may induce significant

bias on the response calculation.

CONTROL SYSTEM DESIGN

In this study a controller is designed to serve as an active control system for the equivalent linear

cable-stayed bridge model. Readings from acceleration and displacement transducers are fed

back to the control algorithm. The sample control system employs a total of 12 hydraulic

actuators located at both ends of the bridge between the deck and the top of each pier (6

actuators for each pier); actuators are oriented to apply forces longitudinally (X-direction). It is

assumed that each actuator can provide up to 1500 kN.

A total of 19 sensors are used to collect the response of the bridge and are used for control

purposes. The locations of these sensors are specified as follows:

(a) four sensors: acceleration and displacement sensors at the northern end of the deck (in

both longitudinal (x) and transverse (z) directions;

(b) four sensors: acceleration and displacement sensors at the southern end of the deck (inboth longitudinal (x) and transverse (z) directions;

(c) four sensors: acceleration and displacement sensors at the middle node of the deck

(in both longitudinal (x) and transverse (z) directions;

(d) four sensors: acceleration and displacement sensors at the top of the pylon (in both

longitudinal (x) and transverse (z) directions;

(e) three sensors: displacement sensors located at the top of all three piers in the longitudinal

direction.

Both sensors from bridge and cables are measured. In the present study the cable sensors

sometimes are and sometimes are not connected to the controller. This is done to study the effect

of the control on the bridge as well as on the vibration of the cables.

CONTROL DESIGN PROBLEM

A test bed for development of effective control strategies for a cable-stayed bridge has also been

developed with emphasis on nonlinear behavior of the stayed-cable. This sample active control

design study defines the excitation, evaluated model, devices, sensors, and control algorithms. A



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Figure 7. Comparison of estimated acceleration and displacement response at the northern end of thebridge deck using truss elements for cables and nonlinear beam elements for cables.



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SIMULINK block diagram for this active control study is shown in Figure 8. It is patterned for

SIMULINK block diagram provided by Dyke et al. [7] and Caicedo et al. [17]. A detaileddescription of the SIMULINK block diagram is now given.

Excitation

The earthquake record from the Chi-Chi earthquake can be selected as the excitation to this

cable-stayed bridge. Each earthquake record contains three-dimensional acceleration and

velocity data (from integration of acceleration data). To consider the spatial variation of

earthquake excitation, uniform input with a phase delay was specified (4 km/s apparent wave

velocity was assumed).

Evaluation model

The evaluation model in SIMULINK contains two input ports and six output ports (Figure 9).The excitation input port includes nine acceleration values and nine velocity values to describe

the multi-support problem in three-dimensional and the time-delayed excitation. The input port

of the control force is developed from the design of actuators or dampers. The evaluated output

serves to determine the control efficiency of various control criteria. The output port of sensors

gives the predicted responses of the deck or the pylon (or piers). At the same time, the output of

Figure 8. Modified SIMULINK block diagram with active control.



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sensors is used to estimate the control forces which include a Kalman filter estimator for

feedback control. The output port of cable sensors gives the responses of the cables. If the

control algorithm focuses on the cables, these cable sensors can also be used to determine the

feedback control. The device sensors provide the responses of actuators or dampers between

connections. The output port of tension estimates the variation of cable tensile force using the

deformation of cables. The cable deformations are selected to consider the dimensionless

responses at the middle node of each cable.

Sensor processor

The sensor processor converts the responses of the bridge to the voltage signal. The converter

for each sensor has a range of 10 V: Each of the measured responses contains a noise levelwith an rms value of 0.01 V. Measurement noises are modeled with Gaussian rectangular pulses

that have a pulse width equal to the integration step. Figure 10 shows the SIMULINK block of

the sensor processor.

Controller

The controller contains a signal converter and a force estimator. The signal converter mainly

transforms the sensors signal into real responses with a constraint. The constraint within the

signal converter has an upper bound (10 V) and a lower bound (10 V). In regard to the force

estimator, it follows with H2 control algorithm and Kalman estimator for use of actuators.First, an appropriate design model must be developed. The design model is formed from the

evaluation model by choosing suitable modes of the system. In this case, in order to use

actuators to control the responses between the deck and piers (end-abutments), the piers (1 and 3)

or abutments are selected to apply the control devices. The control design model contains 44

modes that significantly affect responses of the deck and piers. Originally, the first 100 modes of

Figure 9. SIMULINK block: evaluation model.



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the whole bridge system were used to obtain the control gain using H2 control theory. In real

applications, it may waste much time to calculate the optimal control forces if the matrix of thecontrol gain has a large dimension. However, there are some modes from the first 100 modes

which do not directly and strongly contribute to control the system. Therefore, these modes with

minor contributions in control gain are eliminated and remain the 44 modes which can be used

to generate the optimal control force.

The measured outputs focus on the sensor outputs (responses of the deck and the pylon). The

state-space (discrete form) system of the control design model is represented as follows

xn 1 Adconxn Bdconun Edcon.xgn

xgn

" #15

ysn Cdconxn Ddcon.xgnxgn

" # Fdconun 16

where x is the state vector, and y is a vector of sensor outputs. The controller employs an H2control algorithm to estimate the appropriate control force. To obtain the optimal control force,

the external disturbances, such as the ground excitations, are assumed to be independent with

respect to the control force. First, the objective function is defined as

J2 Xk!1kk0

xTkQxk uTkRuk 17

where R is a weighting matrix related to the optimal control force, Q is a weighting matrix

related to the system of the design model, and k0 is the initial time. Through the computation of

the variation method, the optimal control force can be obtained as

ATdconPkAdcon ATdconPkBdcon2R B

TdconPkBdcon

1BTdconPkAdcon 2Qjk!1 0 18

uk 2R BTdconPBdcon1BTdconPAdconxk Gxk 19

Figure 10. SIMULINK block: sensor processor.



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where Equation (18) is the Riccati equation, P in Equation (19) is the solution of Equation (18),

and G is the optimal control gain. Although the control gain can be obtained from Equation

(19), the optimal control force needs the full state vector of the design model. Therefore, the full

state vector must be computed from the Kalman estimator, given by

x_

k 1 Adcon x_

k Bdconuk Lysk Cdcon x_

k Fdconuk 20

where L is the Kalman gain solved from the Ricatti equation. The optimal control force

combined with the estimator (Equation 20) can be obtained as

uk G x_

k 21

and the completed controller can be expressed as

x_

k 1 Adcon BdconG LCdcon LDdconG x_

k Lysk 22

uk 1 G x_

k 1 23

A SIMULINK block form of the controller is shown in Figure 11.

Control devices

The block of control devices, as shown in Figure 12, contains a converter for the actuators

(forces to voltages) and a real output force. The converter for the actuators transforms

command forces into corresponding voltages for single actuators. The real output force is

indicated by the force from single actuator multiplied by number of control devices. The

connection between the command forces and the output forces is limited by bounds on the

capacity of the device, which is 1500 kN per actuator and the minimum force is 1500 kN per

actuator.

Two kinds of control devices are employed for the vibration Gi-Lu Bridge. The first type of

control device is the actuator, and the other kind of control device is the viscous damper. Twostrategies are used to arrange the locations of the control devices:

Strategy 1: actuators are placed at the top of the Piers 1 and 3 in connection with the deck. Six

actuators are placed between the deck and piers at each location. The voltage command for each

actuator with respect to the output control force is 10 V per 1500 kN. The limitation of actuators

between the control force and the desired force is bounded from 1500 to 1500 kN.

Figure 11. SIMULINK block: controller.



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Strategy 2: besides using the Strategy 1 dampers are also implemented between cables and the

bridge deck (which is placed between cables and the deck with 1 piece at each cable). Each

damper is connected at the lowest nodal point of the cable relative to the deck. Based on the

uncontrolled responses of the Gi-Lu Bridge to six specified excitations, four types of viscous

dampers are selected. The damping coefficient of all types of dampers is: 120 000 (N s/m) for

cable 1 through 10, 42 857 (N s/m) for cable 11 and 12, 30 000 (N s/m) for cable 13 and 14, and7500 (N s/m) for cable 15 through 17, respectively. The input used to drive the dampers is the

relative velocity response between the connecting cables and the deck.

Figure 13 shows the SIMULINK block of the strategy 2 control devices. Figure 14 shows the

schematic diagram of the locations of actuators and viscous dampers for control strategies 1 and

2. On the practical implementation of the dampers in stayed cable will discuss later.

Figure 13. SIMULINK block: control devices (case B).

Figure 12. SIMULINK block: control devices (Strategy 1).



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Evaluation criteria

To evaluate the performance of the control algorithm used in this study, the shear forces and

moments in the pylon and piers induced by earthquake excitation must be considered. Thus,

these evaluation criteria must consider the ability of the controller to reduce the peak responses,

the normalized responses over the entire time record, and the control requirements. For this

purpose, a set of 45 criteria have been employed to evaluate the ability of each control strategy.

Shear forces and moments at each pier, including the base of the pylon, are considered in the

evaluation criteria. Other criteria contain the displacements at the top node of the pylon, the

middle node on the deck, and the north and south ends of the deck. Additionally, the behavior

of the cable tension force and the deformation of the middle node of each cable are important to

observe. Because the earthquake is assumed to have three-dimensional components at a

specified incidence angle, several criteria are evaluated in both the X (longitudinal) and Z(transverse) directions. These evaluation criteria are listed and shown in Table V.

CASE STUDY OF CONTROL EVALUATION

The basic problem for control of the Gi-Lu cable-stayed bridge in this study focuses on

modeling of the stayed cables. Both truss and nonlinear beam elements are selected to model the

cable. The effect of cable modeling on the seismic response of cable-stayed bridge is examined

first. Earthquake ground motion data from the Chi-Chi earthquake from recorded station

TCU089 is selected as the excitation (PGA in the EW direction is 244 gal, in the NS direction

324 gal and in vertical direction 190 gal). In this study a uniform input motion with three

components (two horizontal and one vertical component) is used. Figure 15 shows theacceleration response spectrum of the recorded data. Before the evaluation of control

effectiveness of this bridge is considered response of uncontrolled case using different models

is examined first.

As discussed earlier there are two cases of control devices in this study, control strategy 1:

using actuators only and control strategy 2: using actuators at both ends of the deck and

Figure 14. (a) Location of actuators between deck and abutment; and (b) location of viscous damperbetween deck and cable.



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Table IV. Comparison of the control effectiveness using two different control systems: Case A: puttingactuators between the abutments (both ends) and the bridge deck; Case B: besides the actuators (in Case A)

viscous dampers were also added on the cables.

Case A Case B

J1 1.044 0.848J2 0.937 0.921J3 0.926 0.825J4 1.0 0.999J5 0.999 0.999J6 1.0 0.999J7 1.067 0.882J8 1.0 0.998J9 1.0 0.999J10 1.0 0.999J11 1.0 0.999J12 0.921 0.879J13 0.922 0.868J14 0.911 0.694

J15 1.0 0.999J16 1.010 0.846J17 0.956 0.755J18 1.011 0.844J19 0.950 0.750J20 0.999 0.999J21 1.0 1.001J22 0.999 0.999J23 0.995 0.789J24 0.999 1.030J25 1.0 0.999J26 1.0 1.004J27 1.0 0.999J28 0.721 0.443

J29 0.970 0.756J30 0.721 0.444J31 0.999 1.030J32 0.995 0.789J33 0.802 0.807J34 0.814 0.750J35 1.0 0.999J36 0.878 0.894J37 0.893 0.739J38 0.999 1.001J39 0.878 0.773J40 0.872 0.700J41 1.0 0.999J42 0.961 0.811J43 0.888 0.783J44 0.999 1.001J45 1.0 1.0



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attaching viscous dampers to the cables. Figure 16 compares the displacement and acceleration

responses at the north end of the deck for the uncontrolled case and the case of using control

strategy 1. The proposed control devices are shown to effectively reduce the longitudinal

displacement response of the deck. If control strategy 2 is used, an improved level of control can

be observed, as shown in Figure 17. This is the case because viscous dampers are added betweendeck and cables in control strategy 2, which affords a significant reduction in the response of the

cables as is clearly shown in Figure 18.

Based on the proposed 45 evaluation criteria the performance indices are evaluated for the

controlled and uncontrolled cases. On the left of Figure 19(a) shows the evaluation criteria with

respect to different indices for control strategy 1. The majority of the evaluation criteria have

values less than one, except for indices J1 (longitudinal shear force at the first pier) and J7(longitudinal shear force at the base of the pylon). This means that by putting actuators at both

ends of the deck, the pylon base shear force is larger than for uncontrolled case. The right hand

side of Figure 19(a) shows the estimated value of the normalized displacement at the midpoint

of each cable for control case A. For control strategy 2 the evaluation criteria and the estimated

normalized displacement at the midpoint of each cable is shown in Figure 19(b). It is observed

that control strategy 2 provides an improved level of control effectiveness not only for theresponse of the bridge itself but also for the stayed cables. Most of the displacement at the

midpoint of the cable is reduced after control. Table IV compares the normalized displacement

(with respect to the uncontrolled case) of the evaluation criteria for both control cases A and B.

Figure 20 shows the acceptable cable tension between the provided ranges after the control (for

control strategy 2).

Figure 15. Plot of acceleration response spectrum of ground motion data collected from Station TCU089,Chi-Chi earthquake.



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DISCUSSIONS ON CONTROL STRATEGY

In control strategy 2 dampers were attached between cable and the bridge deck. Because only

eight nonlinear beam elements were modeled for cable and the lowest nodal point was used to

connect the damper to the bridge deck, considering the height of the lowest nodal point of the

cable it is impractical to implement the damper for such a location (almost 6.9 m above the deck

level) under control strategy 2. Therefore an alternative control strategy, control strategy 3, was

used to implement the dampers for the cable. Instead of installing the damper between the cable

and the bridge deck, forty dampers are placed between the deck and cables through cable 1 to

cable 10 at each side. Remaining dampers are placed at the middle of the cable connecting twoneighboring cables through cable 10 to cable 17 at each side, as shown in Figure 21. In this case,

three types of MR dampers are selected to control cable vibration. Type I of MR dampers is

used to position between cable 4 to cable 9 and the deck. Type II MR dampers are used between

cables 13 and the deck, between cable 10 and the deck, and between cable 1012. Type III MR

dampers are positioned between cables 1317.

Figure 16. Comparison of the estimated displacement and acceleration responses between uncontrolledand controlled case using strategy 1: (a) longitudinal responses at northern end of the deck; (b) longitudinal

responses at southern end of the deck; and (c) longitudinal responses at the top of pylon.



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The model of the MR damper for use in structural control of this study combines the

modified bi-viscous model and the bilinear model [19]. The command voltage is sent to the

modified bi-viscous model with the states as input and then generates the corresponding force.

The voltage command ranged from 0 to 1.2 V. A constraint function of voltage with a second-

order polynomial function is used to obtain the lower bound of the force. If the corresponding

force is greater than the lower bound, the output force is still the force generated by the modified

bi-viscous model. On the contrary, the output force is changed to obtain the bilinear model.

There are three types of MR dampers including: (1) 120 mm stroke, 350 mm/s maximum

velocity, and 30 kN force capacity; (2) 120 mm stroke, 1050 mm/s maximum velocity, and30 kN force capacity; and (3) 240 mm stroke, 3500 mm/s maximum velocity, and 30 kN force

capacity.

Figure 22 shows the comparison on the 45 evaluation criteria of three different control

systems. It is found that J1, J2, J3, and J7 are larger for strategy 3 using the input motion from

TCU089. But the other indices are all smallest for using strategy 3 and the control of cable

Figure 17. Comparison on the estimated displacement and acceleration responses between uncontrolled

and controlled case of strategy 2: (a) longitudinal responses at northern end of the deck; (b) longitudinalresponses at southern end of the deck; and (c) longitudinal responses at the top of pylon.



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Figure 18. Responses of cable (EN-1): (a) longitudinal displacement; (b) vertical displacement; (c)transverse displacement; (d) longitudinal acceleration; (e) vertical acceleration; and (f) transverse

acceleration.



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vibration the strategy 3 provides much better results than others. Figure 23 shows the

comparison on the cable vibration. It is also found that with the control strategy 3 the vibration

of cable was significantly reduced.

Figure 19. Plot of evaluation criteria with respect to different index, and the normalized displacement ofmiddle point of all cables: (a) for case A control; and (b) for case B control. (Note: The number of Dindices for 117: R1, R3,. . ., R31, R33; for 1834: R2, R4,. . ., R32, R34; for 3551: L1, L3,. . ., L31, L33;

for 5268: L2, L2,. . ., L32, L34).



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Tab

leV.Summaryofevaluationcriteria.

Jm

maxm;

tjFpm

xtj

F0x;mmax

;

m

13

Fpmx

indicatesthelongitudinal(x-direction)shea

rforceatmth

pier;F0x;mmax

isthemaximumuncontrolledshearfor

ceatmthpier;

m

1(north

pier),2(pierbelowpylon),3(south

pier).

Jn

maxm;

tjFpmz

tj

F0z;mmax

;

n

426

Fpmzindicates

thetransverse(z-direction)shearforceatmthpier;

F0z;mmax

isthem

aximumuncontrolledshearforceat

mthpier.

J7

maxtjFdxtj

F0dxmax

Fdxisthelong

itudinal(x-direction)shearforceatthedecklevelof

thepylon;F0dx

m

axindicatesthemaximumuncontrolledshearforce

atthedecklevelofthepylon.

J8

maxtjFdztj

F0dzmax

Fdzisthetran

sverse(z-direction)shearforceatth

edecklevelof

thepylon;F0dz

m

axindicatesthemaximumuncontrolledshearforce

atthedecklevelofthepylon.

Jl

maxm;

tMpm

xt

M0x;mmax

;

l911

Mpmxindicate

sthelongitudinal(x-direction)momentatmthpier;

M0x;mmax

isthem

aximumuncontrolledmomentatmthpier.

Jk

maxm;

tMpm

zt

M0z;mmax

;

k

12214

Mpmzindicate

sthetransverse(z-direction)momentatmthpier;

M0x;mmax

isthem

aximumuncontrolledmomentatmthpier.

J15

maxtMdxt

M0dxmax

Mdx

indicates

thelongitudinal(x-direction)momentatthedeck

levelofthepy

lon;M0dxmaxisthemaximumuncontrolledmomentat

thesameloca

tion.

J16

maxtMdzt

M0dzmax

Mdzindicates

thetransverse(z-direction)momenta

tthedecklevel

ofthepylon;

M0dzmaxisthemaximumuncontrolled

momentatthe

samelocation

.

Ji

normm;

tjFpmx

tj

F0x;mnorm

;

i17219

F0x;mnorm

isthe

normed

valueofuncontrolledsh

earforcesin

the

longitudinal(x-direction)direction

atmth

pier;where

norm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

tftt0

t2

q

:

Jj

normm;

tjFpmz

tj

F0z;mnorm

;

J

20222

F0z;mnorm

isthenormedvalueofuncontrolledshearforcesinthe

transverse(z-direction)directionatmthpier.

J23

normtjFdx

tj

F0dx

norm

F0dx

norm

isthenormedvalueofuncontrolledshearforcesofthe

pyloninthelongitudinal(x-direction)directionat

thedecklevel.



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J24

normtjFdz

tj

F0dz

norm

F0dz

norm

isthenormedvalueofuncontrolledshearforcesofthe

pyloninthet

ransverse(z-direction)directionatthedecklevel.

Jw

normm;

tjM

pmx

tj

M0x

;m

norm

;

w

25227

M0x;mnorm

isthe

normedvalueofuncontrolledmomentsinthe

longitudinal(

x-direction)directionatmthpier.

Js

normm;

tjM

pmz

tj

M0z

;mnorm

;

s

28230

M0z;mnorm

isthe

normedvalueofuncontrolledmomentsinthe

transverse(z-direction)directionatmthpier.

J31

normtjMdxtj

M0dxnorm

M0dxno

rmisthen

ormedvalueofuncontrolledmomen

tsofthepylon

inthelongitu

dinal(x-direction)directionatthedecklevel.

J32

normtjMdztj

M0dznorm

M0dzno

rmisthenormedvalueofuncontrolledmoment

ofthepylonin

thetransverse

(z-direction)directionatthedecklevel.

J33

maxtjxp1x

tj

x0x;1

max

;

J34

maxtjxp1y

tj

x0y;1

max

;

J35

maxtjxp1z

tj

x0z;1max

x:longitudinal,y:vertical,z:transverse

Subscript(1)indicatesthenorthendofthedeckdisplacementsat

thenorthend

ofthedecklevel(atthetopofPier

1),andx0;1maxis

themaximum

displacementoftheuncontrolledre

sponse.

J36

maxtjxp2x

tj

x0x;2

max

;

J37

maxtjxp2y

tj

x0y;2

max

;

J38

maxtjxp2z

tj

x0z;2max

Subscript(2)indicatesthemiddlenodeofthedeck

displacements

atthemiddle

nodeofthedecklevel(atthebottom

ofthepylon).

J39

maxtjxp3x

tj

x0x;3

max

;

J40

maxtjxp3y

tj

x0y;3

max

;

J41

maxtjxp3z

tj

x0z;3max

Subscript(3)indicatesthesouthernendofthedeck

displacements

atthesouthernendofthedecklevel(atthetopo

fpier3).

J42

maxtjxtpx

tj

x0x;tpmax

;

J43

maxtjxtpy

tj

x0y;tp

max

;

J44

maxtjxtpz

tj

x0z;tp

max

Subscript(tp)

indicatesthetopofthepylondispla

cementsatthe

topofthepylon.

J45

maxt;v

Fcon;v

t

Fcon;capa

city

;

Fcon,capacity=

thecapacityo

fcontroldevices

Themaximum

controlforcenormalizedtothecapacityofcontrol

devices(capacityofcontroldevicesisassumedto

be1500kN).



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CONCLUSIONS

This paper presents a detailed study of structural control for a seismically excited cable-

stayed bridge. The goal of this paper is to develop an evaluation model for control that portrays

salient features of the structural system, particularly the sag cables and their pre-tension

forces. To this end, a MATLAB-based computer program has been developed to simulate athree-dimensional cable-stayed bridge; the code includes the nonlinear beam elements that

represent the sag cables. Both active control and hybrid-control algorithms are given to

illustrate some of the design challenges of the problem. Evaluation criteria are presented for the

design problems that are consistent with the goals of seismic response control of a cable-stayed

bridge.

Figure 20. Variation of cable tension force for un-controlled case and control case B.

MR Damper withconstant Voltage

VE Dampers

Figure 21. Schematic diagram of the location of dampers in the middle point of cables (Line 1017) andbetween deck and cable (Line 110).



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Through this study the following conclusions are made:

(a) A comparison is made between simulation results using either truss elements or nonlinear

beam elements to represent dynamic behavior of the stayed cable of the Gi-Lu bridge.

Significant differences in the dynamic characteristics of cable-stayed bridge are estimated

using different models. To form the stiffness matrix of the cable the truss element can only

consider the linear system with the basic properties (Youngs modulus, cross-sections and

length of elements) for formulation. On the contrary, the nonlinear beam elements not onlyuse the conventional linear stiffness system (partially same as truss elements), but also

include the geometrically nonlinear beam elements to form the stiffness matrix and the self-

weight of the cable and the pre-tension force in the cable can be implemented. Using data

from field tests it is believed that the nonlinear beam elements to simulate the stayed cable

in a realistic manner.

(b) Comparison on the responses of the cable-stayed bridge, as shown in Figures 6 and 7,

cannot prove that using the nonlinear beam element to represent the cable is more

reasonable than using the truss element. But there are some defects associated with using

the truss element for cable, e.g., using truss elements cannot obtain the bridge out-plane

responses and underestimates the cable tensile force.

(c) The simulation results show that use of the actuators as control devices between the bridge

deck and both end-abutments is not very effective for controlling the vibration of the bridgedeck, pylon, and the stayed cables.

(d) Instead of using an active control method, hybrid control devices are applied to the

structure to control response due to seismic excitation. Because nonlinear beam elements

are used to simulate the stayed cable, it is a relatively simple matter to add dampers between

cables and the deck. It is shown that adding viscous dampers to each cable can provide

Figure 22. Comparison of evaluation criteria of three different control systems.



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good control effectiveness, that not only the reduces the deck and pylon response, but also

the response of the cables.

(e) With implementation of extra viscous dampers between the deck and the stayed cables, the

displacement at the top of the pylon (J42 and J43) in both longitudinal and transverse

directions and shear force at the bottom of pylon (J7) and at the top of the pier 3 (J3) in the

longitudinal direction are significantly reduced. For practical consideration the control

strategy 3 can also provide good control effectiveness in most of the evaluation criteria.

(f) In this study the inherent damping ratio for cable is assumed 1% and with thesupplementation of damping ratio from the viscous damper in the cable (1.320% from

damper) the damping ratio for cable is ranging from 2.3% (longest cable) to 21% (shortest

cable).

(g) A formulation of the structural model and a simple control design has been made in the

form of a set of MATLAB equations. This code can provide another type of benchmark

Figure 23. Comparison of the normalized displacement at the middle point of all cables: (a) North-West;(b) South-West; (c) North-East; and (d) South-East sides.



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structural control problem for cable-stayed bridges. The numerical results from use of a

hybrid control system look promising; however, a number of practical aspects not

considered here need to be tackled in further studies in order to gain a deeper evaluation of

the potential effectiveness for mitigation of vibration.

ACKNOWLEDGEMENT

The authors wish to express their thanks to Dr Z. K. Lee (Associate Research Fellow of NCREE) toprovide experimental data of Gi-Lu bridge for this study and the support from National Science Councilunder Grant NSC93-2211-E-002-005 is also acknowledged.

REFERENCES

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2. Pacheco BM, Fujino Y, Sulekh A. Estimation curves for modal damping in stay cables with viscous dampers. ASCEJournal of Structural Engineering 119(6);19611979.

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4. Fujino Y, Warnitchai P, Pacheco BM. Active stiffness control of cable vibration. ASCE Journal of AppliedMechanics 1993; 60:948953.

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7. Dyke JS, Caicedo JM, Turan G, Bergman LA, Hague D. Benchmark control problem for seismic response of cable-stayed bridge. http://wusceel.cive.wustl.edu/quake/benchmark/2000

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10. Moon SJ, Bergman LA, Voulgaris PG. Sliding mode control of cable-stayed bridge subjected to seismic excitation.ASCE Journal of Engineering Mechanics 2003; 129:7177.

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14. Kim DH, Lee IW. Neuro-control of seismically excited steel structures through sensitivity evaluation scheme.Earthquake Engineering and Structural Dynamics 2001; 30:13611377.

15. Luo N, Rodeller J, de la Sen M. Composite robust active control of seismically excited structures with actuatordynamics. Earthquake Engineering and Structural Dynamics 27:301311.

16. Yang JN, Wu JC. Sliding mode control for nonlinear and hysteretic structures. ASCE Journal of EngineeringMechanics 1995; 121:13301339.

17. Caicedo JM, Dyke SJ, Moon SJ, Bergman LA, Turan G, Hague S. Phase II benchmark control problem for seismicresponse of cable-stayed bridges. Journal of Structural Control 2003; 10:137168.18. Przemieniecki SP. Theory of Matrix Structural Analysis. McGraw-Hill: New York, 1968.19. Loh CH, Chang CM. Vibration control assessment of ASCE benchmark model of cable-stayed bridge. Journal of

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