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

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

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

Franco Bontempi1, Fabio Casciati2 and Massimo Giudici2,*
1
DISEG, University of Rome ‘La Sapienza’, Via Eudossiana 18, 00184 Rome, Italy
2
Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

SUMMARY
This paper describes the Benchmark Problem for controlled cable-stayed bridges. The benchmark in
question is the first to be related directly to bridges. It follows on from past experience and
experimentation, devoted to buildings, developed at an international level. These past experiences focused
upon buildings subjected to wind and earthquake excitation. In this paper seismic excitation is explored in
relation 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 energy
dissipation practice. Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS: 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 of
the benchmark [1] is to conceive a competitive control system. Several systems of control will be
investigated, having either an active or a passive nature. A critical comparison is pursued. Such
a comparison will be carried out by comparing some response variables, such as the shears and
moments, at the base of the central towers and of the lateral piers, and the horizontal
displacements of the deck.
For active control systems, the linear quadratic Gaussian (LQG) algorithm is adopted. It is
not intended to vary the algorithm from one system to another. The aim is to perform a
sensitivity analysis of the dynamic behavior of the bridge to changes in the position and the
number 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 350.6 m and lateral spans of 142.7 m. It is

*Correspondence to: Massimo Giudici, Department of Structural Mechanics, University of Pavia, Via Ferrata 1,
27100 Pavia, Italy.
y
E-mail: oogiud@tin.it

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

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

170 F. BONTEMPI, F. CASCIATI AND M. GIUDICI

currently under construction. Details of the bridge and the principal calculus algorithms can be
found elsewhere [1].
All the algorithms are developed in MATLAB [2], in particular the numerical simulations are
executed using the program SYMULINK.

2. GOVERNING RELATIONS

2.1. Active control
For a generic structure actively controlled, the following dynamic system of equations can be
written
. ’
M U þ C U þ KU ¼ ÀMG xg þ Nf. ð1Þ
in which U is the relative displacement vector, M the mass matrix, C the damping matrix, K is
.
the stiffness matrix, xg 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 associated
degree of freedom.
Adopting a state variable description, the previous system can be written in the following way
’ .
x ¼ Ax þ Bx u þ E xg ð2aÞ

.
z ¼ C z x þ Dz u þ F z xg ð2bÞ

.
ym ¼ C y x þ Dy u þ F y xg ð2cÞ
ÂU Ã
in which x ¼ U 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 which
drives the control forces f .
The block diagram of Figure 1 represents the active control system with reference to the
benchmark. 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 data
entering the sensors ym. The latter is also regarded as different from the output vector from the
sensors ys. The vector control u which reaches the devices is a further variable generated by the
control algorithm.
In particular, the sensors are both accelerometers (with sensitivity equal to 7 V/g) and
displacements sensors (with sensitivity equal to 30 V/m). For the sensors, a noise level of 0.003 V
(root mean square) [1] is assumed.

xg ye
Earthquake
Accelerograms Integration of
f Equations of motion ym

Control u Control ys
Sensors
Devices Algorithm

Figure 1. Block diagram of the active control system.

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

ACTIVE AND PASSIVE CONTROL: CABLE-STAYED BRIDGE 171

xg ye
Earthquake
Accelerograms Integration of
Equations of motion

f Control ym
Devices

Figure 2. Block diagram of the passive control system.

The control devices are electrohydraulic actuators with a capacity varying from 1000 to
7000 kN. The force that they supply is linked to the control vector u by the relationship
f ¼ Kfu ð3Þ
In particular, it is assumed that Kf is a diagonal matrix, collecting the gain factors Gd of the
devices.
In the current analysis the dynamics of the actuators is not considered because the intention is
to study the general behavior of the bridge with different control systems. Such behavior is
influenced largely by the modes of vibration with very low frequencies that should not be
influenced by the high modes which characterize the devices used [3].
The control algorithm used is the classic linear quadratic Gaussian algorithm. This algorithm
.
considers xg as stationary white noise and minimizes the integral of evaluation:
Z t
Â Ã
J ¼ lim ðC z x þ Dz uÞT QðC z x þ Dz uÞ þ uT Ru dt ð4Þ
t!1 0

in which R is the identity matrix and Q is a weight matrix applied to the measurements chosen
for the evaluation. Moreover, for the measured data, the ratio g between the power of the noise
on the ground acceleration power spectrum Sxg xg ; and that on the data measurements, Svi vi ; is
. .
assumed equal to 25 [1] .
Adopting the separation principle [4,5], it is accepted that the control vector is determined by
the following expression
#
u ¼ ÀK u x ð5Þ
#
in which x is the estimate of the state vector carried out using the Kalman filter and Ku the gain
feedback matrix.

2.2. Passive control
For a generic structure passively controlled, again an equation analogous to Equation (1) can be
written
. ’ .
M U þ C U þ KU ¼ ÀMG xg þ Nf 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 K d ðU d ÞU d þ C d U’a ð7Þ
d

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



linked by the devices. The simplified block diagram of Figure 2 is for a passively controlled
system.
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
À Áa
’ ’ ’
fi ¼ Cd;i DU a ¼ Cd;i U d;i;2 À U d;i;1 ð8Þ
d;i

’
in which fi is the force supplied by the device, Cd,i the coefficient of viscosity, DU d;i the difference
between the velocities U ’ d;i;2 and U d;i;1 of the two nodes linked by the damper, a a variable
’
coefficient usually ranging between 0.2 and 1.2.
The ith viscoelastic damper responds according to the following law
À Á À Áa
’ ’ ’
fi ¼ Kd;i DUd;i þ Cd;i DU a ¼ Kd;i Á Ud;i;2 À Ud;i;1 þ Cd;i U d;i;2 À U d;i;1 ð9Þ
d;i

in which fi is the force supplied by the device and, in addition to the terms present in the viscous
damper, Kd;i is the linear stiffness coefficient, DUd;i the difference between the displacements
Ud;i;2 and Ud;i;1 of the two nodes linked to the damper. Such a device can be obtained by
mounting the previously presented viscous damper in parallel with a spring device supplying an
elastic reaction. However, commercial devices are on the market: they consist of metallic and
elastomeric components which supply viscoelastic response.
The generic elastoplastic damper i responds according to the following law
À Á À Á À Á
fi ¼ Kd;i DUd;i DUd;i ¼ Kd;i Ud;i;2 À Ud;i;1 Á Ud;i;2 À Ud;i;1 ð10Þ
À Á
in which fi is the force supplied by the device, Kd;i DUd;i the stiffness coefficient, DUd;i the
difference 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), in
which k1 represents the stiffness before yielding, xy the relative displacement reached at the onset
of 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 Fmax Fy Fmax

k1
xmax xmax xy xmax

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 in
which the deck is restrained longitudinally to the main piers; (b) a configuration in which the
deck is not restrained longitudinally to the piers and the tie in this direction is supplied only by
the stays.
The analysis will be carried out with reference to three accelerograms, recorded during the
earthquakes 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 the
towers for the three records, in the two different configurations. The maximum values of such
trends are displayed in Table I. The value of the maximum moment at the base of the towers, the
maximum and minimum values of the tension of the cables, as fraction of Tf and their maximum
absolute variation are also reported in the table.
In configuration (a) the bridge shows limited displacements, but a high shear at the base of the
towers as well as unacceptable variations of tension in the cables. In particular these are found
in the cables anchored in the highest positions on the towers; such cables are those with the
greatest tensions.
In configuration (b), even though there are maximum values of shear and moment
respectively equal to 45.6 and 58.7% of those of configuration (a), one sees an unacceptable
sliding of the deck, with a maximum displacement equal to 0.77 m. In the subsequent sections
the behavior of the bridge coupled with various control systems will be studied. The results will
be compared analyzing the response variables discussed above.

4. RESPONSE OF THE BRIDGE WITH THREE DIFFERENT
ACTIVE 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 are
electrohydraulic actuators whereas the control algorithm is always the Gaussian linear
quadratic scheme.
The earthquake motion is supposed to occur along the longitudinal direction. In the response
of the cable-stayed bridge, the cables play a decisive role and, hence, the considered control
systems 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 provide
a link between the deck of the bridge and the piers and towers; (b) at the lower anchorage of the
stays 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 be
equipped. Indeed these stays are the ones that most characterize the behavior of the bridge.



Deck Displacement (m)
0.1 0.03 0.06

0.02 0.04
0.05
0.02
0.01
0
0 0
-0.02
-0.01
-0.04
-0.05
-0.02 -0.06

-0.1 -0.03 -0.08
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
tempo (s) tempo (s) tempo (s)
‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake

Maximum Shear at the tower bottom (kN)
4 4 4
x 10 x 10 x 10
5 1.5 2

1
2.5
0
0.5
0
0
-2
-2.5
-0.5

-5 -1 -4
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
(a) ‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake

0.4 0.15 1

0.1
0.2 0.5
0.05

0
0 0
-0.05

-0.1
-0.2 -0.5
-0.15

-0.4 -0.2 -1
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200

4
Maximum Shear at the tower bottom (kN) 4
x 10 6000 x 10
2 1
4000
1 0.5
2000

0 0 0

-2000
-1 -0.5
-4000
-2 -1
-6000

-3 -8000 -1.5
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
(b) ‘El Centro’ Earthquake ‘Mexico City’ Earthquake ‘Gebze’ Earthquake

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



Table I. Evaluation parameters for the uncontrolled bridge.
9.0E-01

7.5E-01 d0,max
6.0E-01
Conf. a
Configuration (a): longitudinally
4.5E-01
Conf. b costrained
3.0E-01

1.5E-01
Earthquake El Centro Mexico Gebze
0.0E+00
Max Displ. (m) 0.09758 0.02432 0.07192
El Centro Mexico Gebze Max Shear (kN) 48782.3 11181.0 30847.7
Max M. (kNm) 1027060 198235 697787
Base Shear (kN)
6.0E+04 Tmax cables/ Tf 0.73336 0.47715 0.57698
5.0E+04 S0,max Tmin cables/ Tf 0.07029 0.27899 0.22753
4.0E+04 ∆T cables (kN) 1980.97 438.243 945.422
Conf. a
3.0E+04
Conf. b
2.0E+04
1.0E+04 Configuration (b): longitudinally
0.0E+00
El Centro Mexico Gebze
uncostrained
Base Moment (kNm) Max Displ. (m) 0.36263 0.18410 0.77342
1.2E+06
Max Shear (kN) 22242.7 622.094 12480.0
1.0E+06 M0,max
Max M. (kNm) 398342 173582 603021
8.0E+05
Conf. a Tmax cables/ Tf 0.50587 0.45041 0.50092
6.0E+05
Conf. b Tmin cables/ Tf 0.23148 0.28424 0.23535
4.0E+05
2.0E+05
∆T cables (kN) 883.202 443.045 1603.71
0.0E+00

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

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

* accelerometers that detect the acceleration in the horizontal direction (longitudinal): four
of 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 the
deck : two for each tower.

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



(a)

(b)

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

4.1. Active system 1 (Figure 6a)
Two 4000 kN actuators link the deck with the lateral piers. Four 4000 kN actuators are placed
to link the deck with the central towers. This is because the central piers are burdened with the
greatest mass and moreover, having dimensions larger than the lateral supports, are able to
easily 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. The
following 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 are
reconstructed by the Kalman filter from the accelerometers measures.
The analyses are carried out with different gain factors Gd from the actuators. In particular, in
system 3 different factors are employed for the devices on the deck Gd,deck and for those on the
stays Gd,stays.
Table II displays the most significant results. Figure 7 shows the maximum shear on the piers
and of the displacement of the deck for system (3) when Gd,deck = 500 kN/V, Gd,stays = 900 kN/V.
It is worth noting that, in all cases, the tension in the cables is reported within acceptable
limits. The smallest stresses are obtained with system 3 which acts on both the stays and the
deck. In such a system, the maximum value of the shear Smax on the piers is 13854.8 kN, which is
28.4% of the maximum shear on the uncontrolled bridge in configuration (a) and the 62.3% of



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

System n. 1) Gd=300 kN/V
Max Displ. (m) 0.09137 0.04694 0.16403
7.0E-01 Max Shear (kN) 15424.1 5644.35 13286.0
6.0E-01
Max M. (kNm) 253079 109535 285753
5.0E-01
4.0E-01
Syst. 1 Fmax devices (kN) 2371.48 1002.07 2483.88
3.0E-01
Syst. 2 Tmax cables/ Tf 0.47079 0.43905 0.46267
Syst. 3
2.0E-01 Tmin cables/ Tf 0.26224 0.29288 0.28176
0.16d0,max
1.0E-01
∆T cables (kN) 582.173 248.608 443.249
0.0E+00
System n. 2) Gd=200 kN/V
Base Shear (kN) Earthquake El Centro Mexico Gebze
2.5E+04
Max Displ. (m) 0.17054 0.14655 0.65819
2.0E+04 Max Shear (kN) 21497.3 11355.8 14019.8
1.5E+04 0.28S0,max Syst. 1 Max M. (kNm) 383356 184132 579056
Syst. 2
1.0E+04 Syst. 3
Fmax devices (kN) 1844.62 1263.42 2269.61
5.0E+03 Tmax cables/ Tf 0.49614 0.44333 0.54561
0.0E+00
Tmin cables/ Tf 0.24384 0.28947 0.21369
El Centro Mexico Gebze ∆T cables (kN) 866.433 458.039 2029.17

Base Moment (kNm) Gd,deck = 500 kN/V
7.0E+05 System n. 3)
6.0E+05 G d, stays = 900 kN/V
5.0E+05 Earthquake El Centro Mexico Gebze
Syst. 1
4.0E+05
Syst. 2
Max Displ. (m) 0.08640 0.03764 0.12395
3.0E+05 0.26M0,max
Syst. 3 Max Shear (kN) 13854.8 5851.19 12246.7
2.0E+05
1.0E+05 Max M. (kNm) 270483 99111.8 246375
0.0E+00 Fmax d.deck (kN) 3231.84 1237.98 3141.68
El Centro Mexico Gebze Fmax d.stays (kN) 1357.20 599.649 1008.59
Tmax cables/ Tf 0.47988 0.43598 0.45581
Tmin cables/ Tf 0.25964 0.29449 0.28664
∆T cables (kN) 398.370 171.448 403.134

the maximum shear in configuration (b). The maximum moment Mmax obtained is 270
483 kN m, equal to 26.3% of the moment in configuration (a) and to 44.9% of the moment in
configuration (b). The maximum displacement obtained is 0.12395 m.
Good results were also obtained by adopting system 1, which supplied values of Smax and
Mmax equal to 31.6 and 27.8% of the values in the uncontrolled bridge in configuration (a), 69.3
and 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, in
that the devices which link the deck to the piers do not have to supply a reaction when they are
not activated by the control algorithm. In contrast, the devices on the stays must constantly
supply 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



0.1 0.04 0.1

0.0 5
0.05 0.02

0
0 0
-0.0 5

-0.05 -0.02
-0.1

-0.1 -0.04 -0.1 5
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
‘El Centro’ Earthquake ‘Mexico City’Earthquake ‘Gebze’ Earthquake

4
Maximum Shear at the tower bottom (kN) 4
x 10 6000 x 10
1.5 1

1 4000
0.5
0.5 2000
0
0 0
-0.5
-0.5 -2000

-1 -4000 -1

-1.5 -6000 -1.5
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
‘El Centro’ Earthquake ‘Mexico City’Earthquake ‘Gebze’ Earthquake
Figure 7. Base shear and deck displacement for the bridge with active system 3.

(0.65819 m). The devices applied to the stays should be able to resist the maximum tension
present 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 the
devices on the deck (those that develop the greatest control force). The devices on the stays
should resist a maximum tension of 6295.50 kN.

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

As previously mentioned, three types of passive damper [8–10] are adopted: viscous, viscoelastic
and elastoplastic. In all cases, the layout of the devices is analogous. In particular, four devices are
placed 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 a
large part of the load to them. Indeed, they show a considerably larger section compared with
the 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, the
stiffness coefficient K and the velocity exponent a.



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

Device’s law(Visc.1) f = 2000 ∆ 0.2

El
Earthquake Mexico Gebze
Deck Displacement (m) Centro
1.6E-01 Max Displ. (m) 0.10440 0.02751 0.14371
1.4E-01
1.2E-01 Max Shear (kN) 17286.5 6800.38 11825.9
1.0E-01 0.11d0,max Visc.1 Max M. (kNm) 273481 111200 271380
8.0E-02 Visc.2
6.0E-02
Fmax devices (kN) 1438.99 1096.52 1422.38
Visc.el.
4.0E-02 Tmax cables/ Tf 0.48463 0.43672 0.46744
2.0E-02 Tmin cables/ Tf 0.24797 0.29252 0.27649
0.0E+00
El Centro Mexico Gebze ∆T cables (kN) 584.324 211.556 489.582

2.0E+04
Base Shear (kN) Device’s law(Visc.2) f = 3000 ∆ 0.2

El
1.5E+04 0.28S0,max Earthquake Mexico Gebze
Visc.1
Centro
1.0E+04 Visc.2 Max Displ. (m) 0.10098 0.02726 0.09838
5.0E+03
Visc.el
Max Shear (kN) 15413.0 6375.61 11003.6
Max M. (kNm) 289351 108848 252939
0.0E+00
Fmax devices (kN) 2105.89 1443.52 2007.53
Tmax cables/ Tf 0.47454 0.43778 0.45783
Base Moment (kNm) Tmin cables/ Tf 0.25155 0.29193 0.28113
∆T cables (kN)
3.5E+05
3.0E+05
543.642 204.914 401.085
0.25M0,max
2.5E+05
2.0E+05
Visc.1 Device’s law(Visc.el) f = 50000 ∆ + 1000 ∆ 0.2
Visc.2
1.5E+05
Visc.el El
1.0E+05 Earthquake Mexico Gebze
Centro
5.0E+04
0.0E+00 Max Displ. (m) 0. 08675 0.02266 0.07217
El Centro Mexico Gebze Max Shear (kN) 13673.3 7107.88 12907.3
Max M. (kNm) 257325 103372 243474
Fmax devices (kN) 4871.82 1813.20 4141.61
Tmax cables/ Tf 0.46967 0.44087 0.45156
Tmin cables/ Tf 0.26688 0.29199 0.28417
∆T cables (kN) 438.248 188.956 383.161

The most significant results are displayed in Table III. The maximum shear on the piers and
of the displacement of the deck are represented in Figure 8 for K = 50000 kN/m, C = 1000 and
a = 0.2. In all cases the tension in the cables is reported within acceptable limits, as well as the
maximum displacement of the deck (0.08675 m).
The maximum reduction of moment and shear is obtained by adopting viscoelastic dampers
with K = 50000 kN/m, C = 1000 and a = 0.2. In particular, the maximum moment obtained is
257 325 kN m, namely 25.1% of the moment in configuration (a) and 42.7% of the moment in
configuration (b). The maximum shear is equal to 13 673.3 kN, namely 28.0% of the shear in
configuration (a) and the 61.5% of the shear in configuration (b).



Deck Displacement(m)
0.1 0.02 0.08

0.06
0.01
0.05
0.04
0 0.02
0
-0.01 0

-0.02
-0.05
-0.02
-0.04

-0.1 -0.03 -0.06
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100

4 4 4
x 10 x 10 x 10
1.5 1 1

1
0.5
0.5
0.5
0
0 0
-0.5
-0.5
-0.5
-1
-1

-1.5 -1 -1.5
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Figure 8. Base shear and deck displacement for the bridge with viscoelastic dampers.

4000

3000

2000

1000
Force (kN)

0

-1000

-2000

-3000

-4000

-5000
-0.1 -0.05 0 0.05 0.1
Displacement (m)
Figure 9. Response of a viscoelastic device placed at the towers (K = 50000 kN/m, C = 1000, a = 0.2).

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



Table IV. Evaluation parameters for the bridge with elasto-plastic dampers.
2.0E-01
0.23d0,max
1.5E-01
Dev. parameters (1)
Fy=2000 k1=50000 k2=10
1.0E-01
Syst. 1 Fy(kN), ki(kN/m)
Syst. 2
5.0E-02
Max Displ. (m) 0.17695 0.05685 0.09633
0.0E+00 Max Shear (kN) 13799.8 7931.10 13246.4
Max M. (kNm) 301567 174324 288286
Base Shear (kN) Fmax devices (kN) 2011.37 2010.17 2010.51
2.0E+04 Tmax cables/ Tf 0.47113 0.43645 0.46874
0.28S0,max Tmin cables/ Tf 0.26333 0.29517 0.27333
1.5E+04

Syst. 1
∆T cables (kN) 517.144 179.303 383.171
1.0E+04
Syst. 2

5.0E+03
Dev. parameters (2)
Fy=1000 k1=80000 k2=10
0.0E+00 Fy(kN), ki(kN/m)
El Centro Mexico Gebze Earthquake El Centro Mexico Gebze
Max Displ. (m) 0.13165 0.03920 0.17819
Base Moment (kNm)
3.5E+05 Max Shear (kN) 16864.7 6017.17 11270.7
3.0E+05 0.27M0,max Max M. (kNm) 276699 114959 271710
2.5E+05
Fmax devices (kN) 1011.19 1010.27 1011.66
2.0E+05 Syst. 1
1.5E+05 Syst. 2
Tmax cables/ Tf 0.48481 0.43735 0.47256
1.0E+05 Tmin cables/ Tf 0.24298 0.29130 0.27745
5.0E+04 ∆T cables (kN) 615.002 178.380 530.502
0.0E+00

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 force
supplied at the onset of yielding Fy, Moreover, the dampers should be able to supply a
maximum displacement equal to at least Æ 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 the
displacement 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 maximum
displacement of the deck (0.17819 m).
The maximum reduction of moment is obtained by chosing Fy = 2000, k1 = 50 000
and k2 = 10, which supply a maximum moment equal to 276 699 kN m, namely 26.9% of the
moment 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 kN m, namely 28.3% of the shear in
configuration (a) and 62.0% of the shear in configuration (b).



0.25 0.06 0.1

0.2 0.04
0.05
0.15
0.02
0.1
0 0
0.05
-0.02
0
-0.05
-0.05 -0.04

-0.1 -0.06 -0.1
0 50 100 150 0 50 100 150 0 50 100 150

4 4 4
x 10 x 10 x 10
1.5 1 1

1
0.5
0.5
0.5
0
0 0
-0.5
-0.5
-0.5
-1
-1

-1.5 -1 -1.5
0 50 100 150 0 50 100 150 0 50 100 150
Figure 10. Base shear and deck displacement for the bridge with elastoplastic dampers.

2500

2000

1500

1000
Force (kN)

500

0

-500

-1000

-1500

-2000

-2500
-0.05 0 0.05 0.1 0.15
Displacement (m)
Figure 11. Response of an elastoplastic device placed at the towers (Fy = 2000, k1=50000, k2=10).

Figure 11 displays the reaction supplied by the devices versus the displacement between the
points of application. This ﬁgure emphasizes the hysteretic cycles that the device produces
during the analysis.
It is interesting to note how the use of these dampers, produces a response of signiﬁcant
amplitude for a considerably long time. This is because, once the amplitude of the displacement


Table V. Dimensionless parameters for evaluation of the control systems as proposed for the Benchmark control problem [1].
Active system 3 Elasto-viscous system Elasto-plastic system
’ ’
F = 50000DU +1000DU 0.2 Fy = 1000 k1 = 80000 k2 = 10
El Centro Mexico Gebze El Centro Mexico Gebze El Centro Mexico Gebze
À1 À1 À1 À1 À1 À1 À1 À1
J1 2.84 Â 10 5.23 Â 10 3.97 Â 10 2.80 Â 10 6.36 Â 10 4.18 Â 10 3.46 Â 10 5.38 Â 10 3.65 Â 10À1

Copyright # 2003 John Wiley & Sons, Ltd.
J2 9.32 Â 10À1 1.1 8.49 Â 10À1 8.90 Â 10À1 9.65 Â 10À1 9.47 Â 10À1 1.08 1.1 1.13
J3 2.63 Â 10À1 5.00 Â 10À1 3.53 Â 10À1 2.51 Â 10À1 5.21 Â 10À1 3.49 Â 10À1 2.69 Â 10À1 5.80 Â 10À1 3.89 Â 10À1
J6 8.85 Â 10À1 1.55 1.72 8.89 Â 10À1 9.32 Â 10À1 1.00 1.35 1.61 2.48
J7 4.73 Â 10À1 8.61 Â 10À1 6.85 Â 10À1 6.78 Â 10À1 9.18 Â 10À1 8.75 Â 10À1 4.93 Â 10À1 1.07 6.77 Â 10À1
J8 1.38 1.64 1.80 2.09 1.75 2.48 1.97 1.81 3.21
J9 4.62 Â 10À1 8.43 Â 10À1 7.67 Â 10À1 7.34 Â 10À1 8.27 Â 10À1 9.27 Â 10À1 5.22 Â 10À1 1.14 9.90 Â 10À1
J10 7.1 Â 10À1 1.17 1.20 1.1 9.52 Â 10À1 1.26 1.31 1.00 4.30
J12 1.23E Â 10À2 1.09 Â 10À2 6.16 Â 10À3 9.55 Â 10À3 3.56 Â 10À3 8.12 Â 10À3 1.98 Â 10À3 1.98 Â 10À3 1.98 Â 10À3
J13 5.81 Â 10À1 7.79 Â 10À1 9.42 Â 10À1 5.84 Â 10À1 4.69 Â 10À1 5.50 Â 10À1 8.86 Â 10À1 8.12 Â 10À1 1.36
J14 2.14 Â 10À2 2.16 Â 10À2 2.1 Â 10À2 1.33 Â 10À2 4.46 Â 10À3 1.92 Â 10À2 5.52 Â 10À3 6.57E-03 9.98 Â 10À3
J15 1.70 Â 10À3 1.44 Â 10À3 9.49 Â 10À4 1.06 Â 10À4 2.97 Â 10À5 9.04 Â 10À5 2.19 Â 10À4 2.19E-04 2.35 Â 10À4
J16 20 20 20 12 12 12 12 12 12
J17 9 9 9 } } } } } }
J18 30 30 30 } } } } } }
ACTIVE AND PASSIVE CONTROL: CABLE-STAYED BRIDGE
183

J. Struct. Control 2003; 10:169–185


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

6. CONCLUSIONS

In this research various control systems applied to a long-span cable-stayed bridge subject to
longitudinal seismic activity have been proposed and compared.
System 3, with devices situated on both the deck and the stays, resulted the active system
which supplied the best response in terms of stress. Such a system supplied values of moment
and maximum shear on the piers equal respectively to 26.3 and 28.4% of the values obtained for
a 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% of
the maximum shear of the bridge in configuration (b). The maximum displacement of the deck is
0.124 m, 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 viscoelastic
dampers. This system, coupling the ‘isolating’ contribution of the elastic reaction with the
dissipating contribution of the viscous reaction, produced values of maximum moment and
shear on the piers equal respectively to 25.1 and 28.0% of the values obtained for a uncontrolled
bridge in configuration (a) and 42.7 and 61.5% of the values obtained for an uncontrolled
bridge in configuration (b). The maximum displacement of the deck is 0.087 m, 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. In
fact, 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 solutions
investigated. 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 not
necessary, and the use of electric power is not required during the phase of control. This latter
aspect also implies that the supply of electric power to the system is ensured, even during an
earthquake.
It is worth noting that for a cable-stayed bridge subjected to horizontal action, the behavior
of the towers has a large influence on the overall behavior of the bridge. These towers can be
simply schematized as two cantilever beams fixed to the ground and subject to the two
concentrated forces coming from the deck and the cables. Therefore it is evident that, by varying
the distribution of the forces on these beams, extremely different responses of the bridge can be
obtained.
Given the relative simplicity of the structure here considered, it is easy to understand why
passive control devices are more suitable. Nevertheless, one expects that a more complex design
approach, including transverse motion and an adequate hazard analysis would lead to a greater
complexity, 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 Control
Problem [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 this
paper was supported by a COFIN ‘01 grant, within a national research program, with Professor F. Dav" of
ı
the 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, 2002
4. 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 on
Earthquake Engineering, Beijing, 8–11 November 2000, Spencer Jr BF, Hu YX (eds), Swets & Zeitlinger, 293–297.


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

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