seismic response of a cable-stayed bridge: active and passive control systems (benchmark problem)

JOURNAL OF STRUCTURAL CONTROLJ. Struct. Control 2003; 10:169–185 (DOI: 10.1002/stc.24)

Seismic response of a cable-stayed bridge: active and passivecontrol systems (Benchmark Problem)

Franco Bontempi1, Fabio Casciati2 and Massimo Giudici2,*

1DISEG, University of Rome ‘La Sapienza’, Via Eudossiana 18, 00184 Rome, Italy2Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100Pavia, Italy

SUMMARY

This paper describes the Benchmark Problem for controlled cable-stayed bridges. The benchmark inquestion is the first to be related directly to bridges. It follows on from past experience andexperimentation, devoted to buildings, developed at an international level. These past experiences focusedupon buildings subjected to wind and earthquake excitation. In this paper seismic excitation is explored inrelation to bridges.Three different schemes of active control are compared with each other. Their performance is also

compared with the two most widely used passive control systems which summarize present energydissipation practice. Copyright # 2003 John Wiley & Sons, Ltd.

KEYWORDS: bridges; earthquake; active control; passive control; elastoplastic devices; viscoelastic devices

1. INTRODUCTION

A long-span cable-stayed bridge submitted to seismic excitation is studied. The particular aim ofthe benchmark [1] is to conceive a competitive control system. Several systems of control will beinvestigated, having either an active or a passive nature. A critical comparison is pursued. Sucha comparison will be carried out by comparing some response variables, such as the shears andmoments, at the base of the central towers and of the lateral piers, and the horizontaldisplacements of the deck.

For active control systems, the linear quadratic Gaussian (LQG) algorithm is adopted. It isnot intended to vary the algorithm from one system to another. The aim is to perform asensitivity analysis of the dynamic behavior of the bridge to changes in the position and thenumber of the control devices.

The dynamic analyses are carried out with reference to the Cape Girardeau Bridge, Missouri,USA. It is a cable-stayed bridge with a central span of 350.6m and lateral spans of 142.7m. It is

Received 15 November 2002Revised 25 April 2003Copyright # 2003 John Wiley & Sons, Ltd.

*Correspondence to: Massimo Giudici, Department of Structural Mechanics, University of Pavia, Via Ferrata 1,27100Pavia, Italy.

yE-mail: [email protected]

Contract/grant sponsor: Italian Ministry of University and Scientific and Technological ResearchContract/grant sponsor: COFIN ’01

currently under construction. Details of the bridge and the principal calculus algorithms can befound elsewhere [1].

All the algorithms are developed in MATLAB [2], in particular the numerical simulations areexecuted using the program SYMULINK.

2. GOVERNING RELATIONS

2.1. Active control

For a generic structure actively controlled, the following dynamic system of equations can bewritten

M .UU þ C ’UU þ KU ¼ �MG .xxg þNf ð1Þ

in which U is the relative displacement vector, M the mass matrix, C the damping matrix, K isthe stiffness matrix, .xxg is the earthquake ground acceleration, f the vector of the control forces,G and N the matrices of assignment that refer the seismic and control forces to the associateddegree of freedom.

Adopting a state variable description, the previous system can be written in the following way

’xx ¼ Axþ Bxuþ E .xxg ð2aÞ

z ¼ CzxþDzuþ Fz .xxg ð2bÞ

ym ¼ CyxþDyuþ Fy .xxg ð2cÞ

in which x ¼ U’UU

� �is the state vector, z is the output vector, ym is the vector of the measured data,

and E, Ci, Di, Fi are state matrices. The vector u denotes the output of the controller whichdrives the control forces f .

The block diagram of Figure 1 represents the active control system with reference to thebenchmark. The block diagram of Figure 2 shows how it simplifies for a passively controlled system.

Figure 1 distinguishes in vector z the vector of external output ye from the vector of the dataentering the sensors ym. The latter is also regarded as different from the output vector from thesensors ys. The vector control u which reaches the devices is a further variable generated by thecontrol algorithm.

In particular, the sensors are both accelerometers (with sensitivity equal to 7V/g) anddisplacements sensors (with sensitivity equal to 30V/m). For the sensors, a noise level of 0.003V(root mean square) [1] is assumed.

Integration of Equations of motion

EarthquakeAccelerograms

SensorsControl

AlgorithmControlDevices

ye

ym

ysu

f

gx

Figure 1. Block diagram of the active control system.

Copyright # 2003 John Wiley & Sons, Ltd. J. Struct. Control 2003; 10:169–185

F. BONTEMPI, F. CASCIATI AND M. GIUDICI170

The control devices are electrohydraulic actuators with a capacity varying from 1000 to7000 kN. The force that they supply is linked to the control vector u by the relationship

f ¼ K fu ð3Þ

In particular, it is assumed that Kf is a diagonal matrix, collecting the gain factors Gd of thedevices.

In the current analysis the dynamics of the actuators is not considered because the intention isto study the general behavior of the bridge with different control systems. Such behavior isinfluenced largely by the modes of vibration with very low frequencies that should not beinfluenced by the high modes which characterize the devices used [3].

The control algorithm used is the classic linear quadratic Gaussian algorithm. This algorithmconsiders .xxg as stationary white noise and minimizes the integral of evaluation:

J ¼ limt!1

Z t

0

CzxþDzuð ÞTQ CzxþDzuð Þ þ uTRu� �

dt ð4Þ

in which R is the identity matrix and Q is a weight matrix applied to the measurements chosenfor the evaluation. Moreover, for the measured data, the ratio g between the power of the noiseon the ground acceleration power spectrum S .xxg .xxg ; and that on the data measurements, Svivi ; isassumed equal to 25 [1] .

Adopting the separation principle [4,5], it is accepted that the control vector is determined bythe following expression

u ¼ �Ku #xx ð5Þ

in which #xx is the estimate of the state vector carried out using the Kalman filter and Ku the gainfeedback matrix.

2.2. Passive control

For a generic structure passively controlled, again an equation analogous to Equation (1) can bewritten

M .UU þ C ’UU þ KU ¼ �MG .xxg þNf p ð6Þ

in which fp is the vector of the forces generated by the passive devices.Such a vector can be expressed in the following form

f p ¼ Kd Udð ÞUd þ Cd’UUad ð7Þ

in which Kd and Cd are respectively a stiffness matrix and a damping matrix referred to thedevices, whereas Ud and ’UUd are the displacements and the velocity of the couples of nodes

Integration ofEquations of motion

EarthquakeAccelerograms

ControlDevices

ye

ymf

gx

Figure 2. Block diagram of the passive control system.


ACTIVE AND PASSIVE CONTROL: CABLE-STAYED BRIDGE 171

linked by the devices. The simplified block diagram of Figure 2 is for a passively controlledsystem.

The passive devices adopted are of three types:(a) viscous dampers;(b) viscoelastic dampers;(c) hysteretic dampers (elastoplastic).In particular, the ith viscous damper responds according to the following law

fi ¼ Cd;iD ’UUad;i ¼ Cd;i ’UUd;i;2 � ’UUd;i;1

� �að8Þ

in which fi is the force supplied by the device, Cd,i the coefficient of viscosity, D ’UUd;i the differencebetween the velocities ’UUd;i;2 and ’UUd;i;1 of the two nodes linked by the damper, a a variablecoefficient usually ranging between 0.2 and 1.2.

The ith viscoelastic damper responds according to the following law

fi ¼ Kd;iDUd;i þ Cd;iD ’UUad;i ¼ Kd;i � Ud;i;2 � Ud;i;1

� �þ Cd;i ’UUd;i;2 � ’UUd;i;1

� �að9Þ

in which fi is the force supplied by the device and, in addition to the terms present in the viscousdamper, Kd;i is the linear stiffness coefficient, DUd;i the difference between the displacementsUd;i;2 and Ud;i;1 of the two nodes linked to the damper. Such a device can be obtained bymounting the previously presented viscous damper in parallel with a spring device supplying anelastic reaction. However, commercial devices are on the market: they consist of metallic andelastomeric components which supply viscoelastic response.

The generic elastoplastic damper i responds according to the following law

fi ¼ Kd;i DUd;i� �

DUd;i ¼ Kd;i Ud;i;2 � Ud;i;1� �

� Ud;i;2 � Ud;i;1� �

ð10Þ

in which fi is the force supplied by the device, Kd;i DUd;i� �

the stiffness coefficient, DUd;i thedifference between the displacements Ud;i;2 and Ud;i;1 of the two nodes linked by the damper.

The response of such devices is represented adopting the bilinear law shown in Figure 3(c), inwhich k1 represents the stiffness before yielding, xy the relative displacement reached at the onsetof yielding, Fy the force supplied at the onset of yielding, k2 the stiffness of the plastic section,xmax and Fmax the maximum displacement and force that the device can supply.

Fmax

xmax

Fmax

xmax

Fy Fmax

xmaxxy

k1

k2(a) (b) (c)

Figure 3. (a) Viscous law; (b) viscoelastic law; (c) bilinear elastoplasic law.



3. RESPONSE OF THE BRIDGE WITHOUT CONTROL

For a cable-stayed bridge subject to an earthquake in the longitudinal direction of the deck,there are three response variables of interest:

1. the actions on the towers;2. the displacement of the deck;3. the variations of force in the stays, which should be confined in the range 0.2 Tf–0.7 Tf,

(with Tf denoting the failure tension) [1].

The bridge without control can assume two distinct configurations: (a) a configuration inwhich the deck is restrained longitudinally to the main piers; (b) a configuration in which thedeck is not restrained longitudinally to the piers and the tie in this direction is supplied only bythe stays.

The analysis will be carried out with reference to three accelerograms, recorded during theearthquakes of El Centro (1940), Mexico City (1985) and Gebze (1999).

Figure 4 displays the graphs of the deck displacement and of the shear at the base of thetowers for the three records, in the two different configurations. The maximum values of suchtrends are displayed in Table I. The value of the maximum moment at the base of the towers, themaximum and minimum values of the tension of the cables, as fraction of Tf and their maximumabsolute variation are also reported in the table.

In configuration (a) the bridge shows limited displacements, but a high shear at the base of thetowers as well as unacceptable variations of tension in the cables. In particular these are foundin the cables anchored in the highest positions on the towers; such cables are those with thegreatest tensions.

In configuration (b), even though there are maximum values of shear and momentrespectively equal to 45.6 and 58.7% of those of configuration (a), one sees an unacceptablesliding of the deck, with a maximum displacement equal to 0.77m. In the subsequent sectionsthe behavior of the bridge coupled with various control systems will be studied. The results willbe compared analyzing the response variables discussed above.

4. RESPONSE OF THE BRIDGE WITH THREE DIFFERENTACTIVE CONTROL SYSTEMS

Three different control systems were designed. They are defined by: location and type of sensors,location and type of actuators and control algorithm. Actually all the control devices areelectrohydraulic actuators whereas the control algorithm is always the Gaussian linearquadratic scheme.

The earthquake motion is supposed to occur along the longitudinal direction. In the responseof the cable-stayed bridge, the cables play a decisive role and, hence, the considered controlsystems involve both the deck and the stays of the bridge.

In particular, the actuators will be located in two different positions (Figure 5): (a) to providea link between the deck of the bridge and the piers and towers; (b) at the lower anchorage of thestays to the deck: in this way the stays act as active tendons [6,7].

In case (b), only the two couples of external stays, which are also the longest ones will beequipped. Indeed these stays are the ones that most characterize the behavior of the bridge.



Deck Displacement (m)

0 20 40 60 80 100-0.1

-0.05

0

0.05

0.1

tempo (s)0 20 40 60 80 100

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

tempo (s)0 20 40 60 80 100

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

tempo (s)

‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake

Maximum Shear at the tower bottom (kN)

0 20 40 60 80 100-5

-2.5

0

2.5

5x 10

4

tempo (s)0 20 40 60 80 100

-1

-0.5

0

0.5

1

1.5x 10

4

tempo (s)0 20 40 60 80 100

-4

-2

0

2x 10

4

tempo (s)‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake (a)


0 50 100 150 200-0.4

-0.2

0

0.2

0.4

tempo (s)0 50 100 150 200

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

tempo (s)0 50 100 150 200

-1

-0.5

0

0.5

1

tempo (s)



0 50 100 150 200-3

-2

-1

0

1

2x 10

4

tempo (s)0 50 100 150 200

-8000

-6000

-4000

-2000

0

2000

4000

6000

tempo (s)0 50 100 150 200

-1.5

-1

-0.5

0

0.5

1x 10

4

tempo (s)

‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake (b)

Figure 4. (a) Base shear and deck displacement for the uncontrolled bridge in configuration (a); (b) forconfiguration (b).



For all the control systems, the sensors (Figure 5) can be either

* accelerometers that detect the acceleration in the horizontal direction (longitudinal): fourof them are located at the top of the towers and one on the middle of the deck;

* displacements sensors that measure the relative displacement between the towers and thedeck : two for each tower.

The active control systems adopted are now described with reference to Figure 6.

Table I. Evaluation parameters for the uncontrolled bridge.

Configuration (a): longitudinally costrained

Earthquake El Centro Mexico GebzeMax Displ. (m) 0.09758 0.02432 0.07192Max Shear (kN) 48782.3 11181.0 30847.7Max M. (kNm) 1027060 198235 697787Tmax cables/ Tf 0.73336 0.47715 0.57698Tmin cables/ Tf 0.07029 0.27899 0.22753∆∆T cables (kN) 1980.97 438.243 945.422

Configuration (b): longitudinally uncostrained

Earthquake El Centro Mexico GebzeMax Displ. (m) 0.36263 0.18410 0.77342Max Shear (kN) 22242.7 622.094 12480.0Max M. (kNm) 398342 173582 603021Tmax cables/ Tf 0.50587 0.45041 0.50092Tmin cables/ Tf 0.23148 0.28424 0.23535∆T cables (kN) 883.202 443.045 1603.71


0.0E+00

1.5E-01

3.0E-01

4.5E-01

6.0E-01

7.5E-01

9.0E-01

Conf. a

Conf. b

GebzeMexicoEl Centro

d0,max

Base Shear (kN)

0.0E+00

1.0E+04

2.0E+04

3.0E+04

4.0E+04

5.0E+04

6.0E+04

Conf. a

Conf. b


S0,max

Base Moment (kNm)

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

Conf. a

Conf. b


M0,max

Figure 5. Positions of the actuators (‘) and of the sensors ( ).



4.1. Active system 1 (Figure 6a)

Two 4000 kN actuators link the deck with the lateral piers. Four 4000 kN actuators are placedto link the deck with the central towers. This is because the central piers are burdened with thegreatest mass and moreover, having dimensions larger than the lateral supports, are able toeasily sustain the reaction force deriving from the actuators.

4.2. Active system 2 (Figure 6b)

Eight actuators of 7000 kN are applied to the eight longest stays on the central and lateral span.

4.3. Active system 3 (Figure 6c)

Sixteen actuators are adopted: the eight devices of system 1 and the eight of system 2. Thefollowing qualities are chosen in Equation (4) for the control algorithm:

* Q=1000 I;* R = I where I is the identity matrixes;* Cz and Dz assembled to represent the horizontal (longitudinal) displacements of the deck

(four degrees of freedom) and of the top of the towers (four degrees of freedom). They arereconstructed by the Kalman filter from the accelerometers measures.

The analyses are carried out with different gain factors Gd from the actuators. In particular, insystem 3 different factors are employed for the devices on the deck Gd,deck and for those on thestays Gd,stays.

Table II displays the most significant results. Figure 7 shows the maximum shear on the piersand of the displacement of the deck for system (3) when Gd,deck=500kN/V, Gd,stays=900kN/V.

It is worth noting that, in all cases, the tension in the cables is reported within acceptablelimits. The smallest stresses are obtained with system 3 which acts on both the stays and thedeck. In such a system, the maximum value of the shear Smax on the piers is 13854.8 kN, which is28.4% of the maximum shear on the uncontrolled bridge in configuration (a) and the 62.3% of

(a)

(b)

(c)

Figure 6. Positions of the actuators for the active systems 1 (a); 2 (b); and 3 (c).



the maximum shear in configuration (b). The maximum moment Mmax obtained is 270483 kNm, equal to 26.3% of the moment in configuration (a) and to 44.9% of the moment inconfiguration (b). The maximum displacement obtained is 0.12395m.

Good results were also obtained by adopting system 1, which supplied values of Smax andMmax equal to 31.6 and 27.8% of the values in the uncontrolled bridge in configuration (a), 69.3and 47.4% of the values in the uncontrolled bridge in configuration (b) and 11.3 and 5.6%larger than the results of system 3. In system 1, however, the overall construction is simple, inthat the devices which link the deck to the piers do not have to supply a reaction when they arenot activated by the control algorithm. In contrast, the devices on the stays must constantlysupply such a reaction.

System 2 gave the worst results, with values of Smax and Mmax of 55.2 and 114% respectively,greater than those of system 3. Also the displacement of the deck resulted as very large

Table II. Evaluation parameters for the bridge with control active systems.

System n. 1) Gd=300 kN/VEarthquake El Centro Mexico GebzeMax Displ. (m) 0.09137 0.04694 0.16403Max Shear (kN) 15424.1 5644.35 13286.0Max M. (kNm) 253079 109535 285753Fmax devices (kN) 2371.48 1002.07 2483.88Tmax cables/ Tf 0.47079 0.43905 0.46267Tmin cables/ Tf 0.26224 0.29288 0.28176∆∆T cables (kN) 582.173 248.608 443.249

System n. 2) Gd=200 kN/VEarthquake El Centro Mexico Gebze Max Displ. (m) 0.17054 0.14655 0.65819Max Shear (kN) 21497.3 11355.8 14019.8Max M. (kNm) 383356 184132 579056Fmax devices (kN) 1844.62 1263.42 2269.61Tmax cables/ Tf 0.49614 0.44333 0.54561Tmin cables/ Tf 0.24384 0.28947 0.21369∆T cables (kN) 866.433 458.039 2029.17

System n. 3) Gd,deck = 500 kN/VGd, stays = 900 kN/V

Earthquake El Centro Mexico Gebze Max Displ. (m) 0.08640 0.03764 0.12395Max Shear (kN) 13854.8 5851.19 12246.7Max M. (kNm) 270483 99111.8 246375Fmax d.deck (kN) 3231.84 1237.98 3141.68Fmax d.stays (kN) 1357.20 599.649 1008.59Tmax cables/ Tf 0.47988 0.43598 0.45581Tmin cables/ Tf 0.25964 0.29449 0.28664


0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

5.0E-01

6.0E-01

7.0E-01

Syst. 1

Syst. 2

Syst. 3


0.16d0,max

Base Shear (kN)

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

Syst. 1

Syst. 2

Syst. 3


0.28S0,max

Base Moment (kNm)

0.0E+00

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

6.0E+05

7.0E+05

Syst. 1

Syst. 2

Syst. 3


0.26M0,max

∆T cables (kN) 398.370 171.448 403.134



(0.65819m). The devices applied to the stays should be able to resist the maximum tensionpresent in the cable, which in the case of system 2 is 6624.58 kN.

A similar argument holds for system 3. The table displays the values of Fmax referring to thedevices on the deck (those that develop the greatest control force). The devices on the staysshould resist a maximum tension of 6295.50 kN.

5. RESPONSE OF THE BRIDGE WITH THREE PASSIVE CONTROL SYSTEMS OFDIFFERENT NATURE

As previously mentioned, three types of passive damper [8–10] are adopted: viscous, viscoelasticand elastoplastic. In all cases, the layout of the devices is analogous. In particular, four devices areplaced on the central piers and two on the lateral ones: these devices link the deck to the piers.

Again the devices are mainly located on the central piers because it is desired to transfer alarge part of the load to them. Indeed, they show a considerably larger section compared withthe lateral supports.

5.1. Viscous and viscoelastic dampers

The viscous dampers follow the law in Equation (8); the viscoelastic devices that in Equation (9).The analyses have been carried out for various values for the coefficient of viscosity C, thestiffness coefficient K and the velocity exponent a.


0 20 40 60 80 100-0.1

-0.05

0

0.05

0.1

tempo (s)0 20 40 60 80 100

-0.04

-0.02

0

0.02

0.04

tempo (s)0 20 40 60 80 100

-0.15

-0.1

-0.05

0

0.05

0.1

tempo (s)

‘El Centro’ Earthquake ‘Mexico City’Earthquake ‘Gebze’ Earthquake


0 20 40 60 80 100-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

tempo (s)0 20 40 60 80 100

-6000

-4000

-2000

0

2000

4000

6000

tempo (s)0 20 40 60 80 100

-1.5

-1

-0.5

0

0.5

1x 10

4

tempo (s)

‘El Centro’ Earthquake ‘Mexico City’Earthquake ‘Gebze’ Earthquake

Figure 7. Base shear and deck displacement for the bridge with active system 3.



The most significant results are displayed in Table III. The maximum shear on the piers andof the displacement of the deck are represented in Figure 8 for K=50000 kN/m, C=1000 anda=0.2. In all cases the tension in the cables is reported within acceptable limits, as well as themaximum displacement of the deck (0.08675m).

The maximum reduction of moment and shear is obtained by adopting viscoelastic damperswith K=50000 kN/m, C=1000 and a=0.2. In particular, the maximum moment obtained is257 325 kNm, namely 25.1% of the moment in configuration (a) and 42.7% of the moment inconfiguration (b). The maximum shear is equal to 13 673.3 kN, namely 28.0% of the shear inconfiguration (a) and the 61.5% of the shear in configuration (b).

Table III. Evaluation parameters for the bridge with viscous and viscoelastic dampers.

Device’s law(Visc.1) f = 2000 0.2

Earthquake El Centro

Mexico Gebze

Max Displ. (m) 0.10440 0.02751 0.14371Max Shear (kN) 17286.5 6800.38 11825.9Max M. (kNm) 273481 111200 271380Fmax devices (kN) 1438.99 1096.52 1422.38Tmax cables/ Tf 0.48463 0.43672 0.46744Tmin cables/ Tf 0.24797 0.29252 0.27649∆∆

∆

T cables (kN) 584.324 211.556 489.582

Device’s law(Visc.2) f = 3000 0.2

EarthquakeEl

Centro Mexico Gebze

Max Displ. (m) 0.10098 0.02726 0.09838Max Shear (kN) 15413.0 6375.61 11003.6Max M. (kNm) 289351 108848 252939Fmax devices (kN) 2105.89 1443.52 2007.53Tmax cables/ Tf 0.47454 0.43778 0.45783Tmin cables/ Tf 0.25155 0.29193 0.28113∆T cables (kN) 543.642 204.914 401.085

Device’s law(Visc.el) f = 50000 + 1000 0.2

Earthquake El Centro

Mexico Gebze

Max Displ. (m) 0. 08675 0.02266 0.07217Max Shear (kN) 13673.3 7107.88 12907.3Max M. (kNm) 257325 103372 243474Fmax devices (kN) 4871.82 1813.20 4141.61Tmax cables/ Tf 0.46967 0.44087 0.45156Tmin cables/ Tf 0.26688 0.29199 0.28417


0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

1.4E-01

1.6E-01

Visc.1

Visc.2

Visc.el.


0.11d0,max

Base Shear (kN)

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

Visc.1

Visc.2

Visc.el


0.28S0,max

Base Moment (kNm)

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

Visc.1

Visc.2

Visc.el


0.25M0,max

∆T cables (kN) 438.248 188.956 383.161

∆

∆ ∆



Figure 9 displays, for the values of K, C and a above, the reaction supplied by the devices as afunction of the displacement between the points of application. It represents the typical behaviorof a viscoelastic damper.

Deck Displacement(m)

0 20 40 60 80 100-0.1

-0.05

0

0.05

0.1

tempo (s)0 20 40 60 80 100

-0.03

-0.02

-0.01

0

0.01

0.02

tempo (s)0 20 40 60 80 100

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

tempo (s)



0 20 40 60 80 100-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

tempo (s)0 20 40 60 80 100

-1

-0.5

0

0.5

1x 10

4

tempo (s)0 20 40 60 80 100

-1.5

-1

-0.5

0

0.5

1x 10

4

tempo (s)


Figure 8. Base shear and deck displacement for the bridge with viscoelastic dampers.

-0.1 -0.05 0 0.05 0.1-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

Displacement (m)

For

ce (

kN)

Figure 9. Response of a viscoelastic device placed at the towers (K=50000 kN/m, C=1000, a=0.2).



5.2. Elastoplastic dampers

The bilinear idealization of Figure 3 is adopted for the elastoplastic dampers of Equation (18).The analyses were carried out for various values of the stiffness coefficient k1, k2 and the forcesupplied at the onset of yielding Fy, Moreover, the dampers should be able to supply amaximum displacement equal to at least � 20 cm.

The values of the coefficients used are within the range of production for these devices.Table IV displays the most significant results . The maximum shear on the piers and of thedisplacement of the deck are shown in Figure 10, for Fy=2000, k1=50 000 and k2=10. Again,in all cases, the tension in the cables is reported within acceptable limits, as well as the maximumdisplacement of the deck (0.17819m).

The maximum reduction of moment is obtained by chosing Fy=2000, k1=50 000and k2=10, which supply a maximum moment equal to 276 699 kNm, namely 26.9% of themoment in configuration (a) and the 45.9% of the moment in configuration (b).

The maximum reduction of shear is obtained by choosing Fy=1000, k1=80000 and k2=10,which supply a maximum shear equal to 13 799.8 kNm, namely 28.3% of the shear inconfiguration (a) and 62.0% of the shear in configuration (b).

Table IV. Evaluation parameters for the bridge with elasto-plastic dampers.

Dev. parameters (1)Fy(kN), ki(kN/m) Fy=2000 k1=50000 k2=10

Earthquake El Centro Mexico Gebze Max Displ. (m) 0.17695 0.05685 0.09633Max Shear (kN) 13799.8 7931.10 13246.4Max M. (kNm) 301567 174324 288286Fmax devices (kN) 2011.37 2010.17 2010.51Tmax cables/ Tf 0.47113 0.43645 0.46874Tmin cables/ Tf 0.26333 0.29517 0.27333∆∆T cables (kN) 517.144 179.303 383.171

Dev. parameters (2)Fy(kN), ki(kN/m)

Fy=1000 k1=80000 k2=10

Earthquake El Centro Mexico GebzeMax Displ. (m) 0.13165 0.03920 0.17819Max Shear (kN) 16864.7 6017.17 11270.7Max M. (kNm) 276699 114959 271710Fmax devices (kN) 1011.19 1010.27 1011.66Tmax cables/ Tf 0.48481 0.43735 0.47256Tmin cables/ Tf 0.24298 0.29130 0.27745∆T cables (kN) 615.002 178.380 530.502


0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

Syst. 1

Syst. 2


0.23d0,max

Base Shear (kN)

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

Syst. 1

Syst. 2


0.28S0,max

Base Moment (kNm)

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

Syst. 1

Syst. 2


0.27M0,max



Figure 11 displays the reaction supplied by the devices versus the displacement between thepoints of application. This figure emphasizes the hysteretic cycles that the device producesduring the analysis.

It is interesting to note how the use of these dampers, produces a response of significantamplitude for a considerably long time. This is because, once the amplitude of the displacement


0 50 100 150-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

tempo (s)0 50 100 150

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

tempo (s)0 50 100 150

-0.1

-0.05

0

0.05

0.1

tempo (s)



0 50 100 150-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

tempo (s)0 50 100 150

-1

-0.5

0

0.5

1x 10

4

tempo (s)0 50 100 150

-1.5

-1

-0.5

0

0.5

1x 10

4

tempo (s)


Figure 10. Base shear and deck displacement for the bridge with elastoplastic dampers.

-0.05 0 0.05 0.1 0.15-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

Displacement (m)

For

ce (

kN)

Figure 11. Response of an elastoplastic device placed at the towers (Fy=2000, k1=50000, k2=10).



TableV.Dim

ensionless

parametersforevaluationofthecontrolsystem

sasproposedfortheBenchmark

controlproblem

[1].

Activesystem

3Elasto-viscoussystem

F=

50000D

’ UU+

1000D

’ UU0.2

Elasto-plastic

system

Fy=

1000k1=

80000k2=

10

ElCentro

Mexico

Gebze

ElCentro

Mexico

Gebze

ElCentro

Mexico

Gebze

J1

2.84�10�1

5.23�10�1

3.97�10�1

2.80�10�1

6.36�10�1

4.18�10�1

3.46�10�1

5.38�10�1

3.65�10�1

J2

9.32�10�1

1.1

8.49�10�1

8.90�10�1

9.65�10�1

9.47�10�1

1.08

1.1

1.13

J3

2.63�10�1

5.00�10�1

3.53�10�1

2.51�10�1

5.21�10�1

3.49�10�1

2.69�10�1

5.80�10�1

3.89�10�1

J4

4.1�10�1

3.81�10�1

5.93�10�1

3.03�10�1

3.48�10�1

4.63�10�1

6.64�10�1

3.60�10�1

9.49�10�1

J5

1.52�10�1

5.44�10�1

8.84�10�2

1.39�10�1

5.42�10�2

7.76�10�2

2.16�10�1

4.79�10�2

1.30�10�1

J6

8.85�10�1

1.55

1.72

8.89�10�1

9.32�10�1

1.00

1.35

1.61

2.48

J7

4.73�10�1

8.61�10�1

6.85�10�1

6.78�10�1

9.18�10�1

8.75�10�1

4.93�10�1

1.07

6.77�10�1

J8

1.38

1.64

1.80

2.09

1.75

2.48

1.97

1.81

3.21

J9

4.62�10�1

8.43�10�1

7.67�10�1

7.34�10�1

8.27�10�1

9.27�10�1

5.22�10�1

1.14

9.90�10�1

J10

7.1�10�1

1.17

1.20

1.1

9.52�10�1

1.26

1.31

1.00

4.30

J11

3.18�10�2

1.29�10�2

1.65�10�2

3.46�10�2

1.39�10�2

1.75�10�2

3.43�10�2

1.14�10�2

2.76�10�2

J12

1.23E�10�2

1.09�10�2

6.16�10�3

9.55�10�3

3.56�10�3

8.12�10�3

1.98�10�3

1.98�10�3

1.98�10�3

J13

5.81�10�1

7.79�10�1

9.42�10�1

5.84�10�1

4.69�10�1

5.50�10�1

8.86�10�1

8.12�10�1

1.36

J14

2.14�10�2

2.16�10�2

2.1�10�2

1.33�10�2

4.46�10�3

1.92�10�2

5.52�10�3

6.57E-03

9.98�10�3

J15

1.70�10�3

1.44�10�3

9.49�10�4

1.06�10�4

2.97�10�5

9.04�10�5

2.19�10�4

2.19E-04

2.35�10�4

J16

20

20

20

12

12

12

12

12

12

J17

99

9}

}}

}}

}J18

30

30

30

}}

}}

}}



of the deck is below the yield threshold of the devices, the devices respond according to thelinear elastic law, without dissipating more energy.

6. CONCLUSIONS

In this research various control systems applied to a long-span cable-stayed bridge subject tolongitudinal seismic activity have been proposed and compared.

System 3, with devices situated on both the deck and the stays, resulted the active systemwhich supplied the best response in terms of stress. Such a system supplied values of momentand maximum shear on the piers equal respectively to 26.3 and 28.4% of the values obtained fora uncontrolled bridge with the deck fixed to the towers, which is the worst configuration (a).These internal actions are equal respectively to 44.9% of the maximum moment and to 62.3% ofthe maximum shear of the bridge in configuration (b). The maximum displacement of the deck is0.124m, equal to 16.0% of the maximum deck displacement of the bridge in configuration (b).

The passive control system which supplied the best results was the one with viscoelasticdampers. This system, coupling the ‘isolating’ contribution of the elastic reaction with thedissipating contribution of the viscous reaction, produced values of maximum moment andshear on the piers equal respectively to 25.1 and 28.0% of the values obtained for a uncontrolledbridge in configuration (a) and 42.7 and 61.5% of the values obtained for an uncontrolledbridge in configuration (b). The maximum displacement of the deck is 0.087m, equal to 11.2%of the maximum deck displacement of the bridge in configuration (b).

Such values are comparable to those obtained when an active system is adopted. Infact, the passive system supplied values of moment and shear lower by 4.9 and 1.3%,respectively.

As a consequence the passive system seems to be the most convenient among the solutionsinvestigated. This system supplies values of internal action similar to the active system and,moreover, its realization is easier. In particular, the availability of electric power supplies is notnecessary, and the use of electric power is not required during the phase of control. This latteraspect also implies that the supply of electric power to the system is ensured, even during anearthquake.

It is worth noting that for a cable-stayed bridge subjected to horizontal action, the behaviorof the towers has a large influence on the overall behavior of the bridge. These towers can besimply schematized as two cantilever beams fixed to the ground and subject to the twoconcentrated forces coming from the deck and the cables. Therefore it is evident that, by varyingthe distribution of the forces on these beams, extremely different responses of the bridge can beobtained.

Given the relative simplicity of the structure here considered, it is easy to understand whypassive control devices are more suitable. Nevertheless, one expects that a more complex designapproach, including transverse motion and an adequate hazard analysis would lead to a greatercomplexity, that a simple passive system might be unable to deal with.

APPENDIX.

Table V shows the evaluation parameters in the form proposed for the Benchmark ControlProblem [1].



ACKNOWLEDGEMENTS

Massimo Giudici acknowledges the Ministry of Universities and Scientific and Technological Research,who through a Young Researchers grant made this research possible. Fabio Casciati’s contribution to thispaper was supported by a COFIN ‘01 grant, within a national research program, with Professor F. Dav"ıı ofthe University of Ancona acting as National coordinator.

REFERENCES

1. Dyke SJ, Turan G, Caceido JM, Bergman LA, Hauge S. Benchmark Control Problem for Seismic Response of Cable-Stayed Bridges, Washington University in St. Louis, 2000.

2. MATLAB. The Math Works, Inc. Natick, Massachussets, 1999.3. Casciati F. CISM Lecture Notes, Advanced Dynamics and Control of Structures and Machines, 20024. Stengel RF. Stochastic Optimal Control. Wiley: New York, 1986.5. Skelton RE. Dynamic Systems Control: Linear Systems Analysis and Synthesis. Wiley: New York, 1988.6. Soong TT. Active Structural Control: Theory and Practice. Longman: Essex, 1990.7. Yang JN Giannopolous F. Active control and stability of cable stayed bridge. Journal of the Engineering Mechanics

Division (ASCE) 1979; 105:677–694.8. Soong TT, Dargush GF. Passive Energy Disspiation Systems in Structural Engineering. Wiley: New York, 1997.9. Casciati F, Faravelli L. Standalone controller for a bridge semi-active damper, Smart Systems for Bridge, Structures,

and Highways. Proceedings of SPIE 2001, Liu SC (ed.), Vol. 4330; 399–404.10. Casciati F, Faravelli L, Battaini M. Ultimate vs. serviceability limit state in designing bridge energy. Earthquake

Engineering Frontiers in the New Millennium 2001, Proceedings of the China–U.S. Millennium Symposium onEarthquake Engineering, Beijing, 8–11 November 2000, Spencer Jr BF, Hu YX (eds), Swets & Zeitlinger, 293–297.

