Vertical vibration control of suspension bridges …scientiairanica.sharif.edu/article_2408_358b2b497a20908...Scientia Iranica A (2017) 24(2), 439{451 Sharif University of Technology

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Scientia Iranica A (2017) 24(2), 439{451

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Vertical vibration control of suspension bridgessubjected to earthquake by semi-active MR dampers

S. Pourzeynali�, A. Bahar and S. Pourzeynali

Department of Civil Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran.

Received 17 November 2014; received in revised form 4 September 2015; accepted 13 February 2016

KEYWORDSRigid truss;Semi-active control;Suspension bridge;MR damper;Vertical vibration;Earthquakeexcitation.

Abstract. In this paper, a semi-active control technique is presented to mitigate theseismic vertical response of suspension bridges using Magneto-Rheological (MR) dampers.MR dampers, as semi-active control devices, have received signi�cant attention in recentyears because of their adaptability of working as active control devices without requiringlarge power of sources. The most challenging problem in this study is how to install thedampers at the degrees along the bridge span far from the towers. In the present paper, tosolve this problem, a \rigid truss", which is attached to the bottom of the bridge deck, isproposed. The MR damper can be installed between the bridge deck and free-end of thetruss. For numerical study, the Vincent-Thomas suspension bridge is chosen and di�erentschemes, in point of number and location of the dampers view, are proposed to get optimalperformance of the dampers in reducing the bridge vertical responses. To reach this goal, allof the schemes are optimized, and the controlled and uncontrolled responses of the bridgeare calculated under application of 15 major world-wide earthquake accelerogrames. Theresults indicate that the proposed models can e�ectively reduce the vertical responses ofthe example bridge.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Cable-supported bridges have received great attentionin recent years because of their advantages such aselegant appearance as well as convenience in bridginglong spans. These structures are highly vulnerable todynamic loads, such as earthquakes and strong winds,due to their high exibility and low structural damping.Therefore, the vibration control of these bridges is animportant issue to keep them safe and serviceable. Inthe present study, suspension bridges are focused oncontrolling their vertical responses against earthquakeexcitations.

*. Corresponding author. Fax: +98 13 33690271Email addresses: [email protected] (SaeidPourzeynali); [email protected] (A. bahar);[email protected] (Solmaz Pourzeynali)

There are generally two main strategies to re-duce seismic responses of suspension bridges includingstrengthening by use of traditional methods to increasethe capacity of bridge to resist the seismic demandand using the control devices among which the energydissipative devices, such as MR dampers, are knownas the most important ones to minimize the damagecaused by the ground motion of strong earthquakes.Both approaches can also be used to aim for an optimalsolution [1].

In recent decades, many control systems, such asactive dampers [2,3], tuned mass damper [4,5], viscousdampers [6], and tuned liquid column dampers [7],have been studied for seismic control of long spanbridges and tall buildings, but there have always beensome challenges such as cost and maintenance, relianceon external power, etc. To overcome these challenges,semi-active control devices have been proposed. These

440 S. Pourzeynali et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 439{451

systems combine the best features of passive and activecontrol devices, and they do not need a large powersource which is so critical during seismic events whenpower failing may occur [8-10].

Various types of semi-active control devices withdi�erent control algorithms have been studied to con-trol the response of bridges, which most of them havebeen performed to control the longitudinal responses.To direct comparison among these control studies, thebenchmark Bill Emerson Memorial Bridge crossing theMississippi River was developed [11]. Performance ofdi�erent semi-active control devices, such as ResettingSemi-Active Sti�ness Dampers, switching Semi-ActiveSti�ness Dampers, Magneto-rheological dampers, andSemi-Active Friction Dampers, have been investigatedfor this bridge [12].

Agrawal et al. [13] proposed Semi-Active Sti�nessDamper (SASD) which can operate in the resettingmode (RSASD) or switching mode (SSASD). Theyanalyzed performances of both and concluded thatalthough SSASD reduces the responses compared touncontrolled ones, the reduction ratio is less than thatof the RSASD. In addition, they concluded that theRSASD reduction level was comparable to that of thesample active controller. They also investigated theperformance of Semi-Active Friction Damper (SAFD).They noted that the friction damper should be com-bined with linear elastic spring to be able to add thesti�ness after that sliding starts in the damper.

Jung et al. [14] veri�ed the performance of smartdampers. The results demonstrated that smart damp-ing systems have generally the same performance asthe active control system.

Moon et al. [15] investigated the performance ofsliding mode control in lieu of LQG formulation usingMR dampers. Comparison of the results indicated thatMR dampers are more e�ective than the active controlsystems.

Magneto-rheological dampers have received sig-ni�cant attention as a seismic retro�t control systemdue to their great advantages such as low powerconsumption, force controllability, large force capacity,rapid response, and safe manner operation in case ofpower fail [9,16,17]. MR dampers have been mainlyapplied to vibration control of highway bridges, cablesupported bridges, and tall buildings [18-22]. Manymathematical models have been proposed to modelthe intrinsic non-linear behavior of MR dampers suchas Bingham model, viscoelastic-plastic model, Bouc-Wen model, phenomenological model, improved Bouc-Wen model, modi�ed Dahl model, normalized Bouc-wen model, and many others. Some of these models,such as the Bingham model, cannot capture well theforce-velocity non-linear behavior of the damper. Itis very important for a model to be able to portraythe non-linear behavior of the damper and easily

identify the system parameters [9,23-26]. Jung etal. [18] examined the application of MR dampers witha comprehensive study of the adequacy of di�erenttypes of dynamic models for MR uid dampers, such asBingham model, Bouc-Wen model, and modi�ed Bouc-Wen model. The results indicated that the applicationof MR control system is generally similar to that of theactive control system; the Bouc-Wen model and themodi�ed Bouc-Wen model were also shown to be morecomputationally tractable than the others.

Bing et al. [27] investigated the application ofMR dampers in reducing structural responses of cable-stayed bridges under multi-support excitations. Thenumerical results of that study showed that the perfor-mance of the MR damper is nearly the same as that ofthe active control system, and in some cases, it is betterthan the active system while requiring signi�cantly lesspower than the active one.

Alapati et al. [12] conducted a comparison be-tween di�erent semi-active strategies investigated forthe benchmark bridge. They concluded that amongsemi-active control systems studied, the modi�ed Bouc-Wen model of MR dampers gave overall robust perfor-mance in point of response reduction view.

Also, there was a study by Yang et al. [21] onPingsheng Bridge. A phenomenological model of MRdamper was used in this study. The e�ect of variablecurrent and number of dampers on seismic responsecontrol was investigated. The results showed that thelongitudinal displacement of the tower top and bridgegirder decrease with the increase in input current andnumber of MR dampers attached.

There are six long bridges in the Golden Statethat the California Department of Transportation(Caltrans) decided to upgrade them seismically aftertwo major earthquakes (Loma Prieta earthquake ofOctober 1989, and Northridge earthquake of January1994). The Vincent Thomas Suspension Bridge is oneof them which was decided to be upgraded seismicallyin 1995. Several retro�t schemes were developedduring this study on the Vincent-Thomas SuspensionBridge to reduce its longitudinal responses subjectedto earthquake. Nazmy et al. [1] evaluated some ofthe proposed retro�t schemes in order to examinetheir e�ectiveness. Based on that paper, longitudinalviscous dampers were one of the major seismic retro�tmeasures proposed for this bridge. The other retro�tmeasure was to allow the formation of plastic hinges atthe base of the tower shafts during severe seismic eventsin order to limit the longitudinal bending moment atthe tower base to its plastic moment capacity. Itwas concluded that the vulnerabilities associated withlongitudinal and vertical vibrations of the bridge deckcould be reduced by installing longitudinal dampersand allowing plastic hinge formation at the tower base.Also, it has been concluded that installation of dampers

S. Pourzeynali et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 439{451 441

in the side spans of the bridge was a very e�cientscheme in reducing the seismic demand on the trussmembers in these spans.

All the above reviewed studies have consideredthe response reduction of the bridges in longitudinaldirection, while the main objective of the present paperis to apply MR dampers to seismic control of suspen-sion bridges in vertical direction. In the present paper,the Vincent-Thomas Suspension Bridge is chosen as acase study, and the mechanical behavior of the MRdamper is presented by Normalized Bouc-Wen modelwith 2200 kN capacity [28,29]. The damping forceproduced by an MR damper is mainly dependent onthe input voltage to the damper which is regulated hereusing fuzzy logic controller. The analysis is performedin time domain.

On the other hand, the most challenging problemin this research was how to install the dampers onnodes of the bridge deck along the spans far fromthe towers. To overcome this, use of a \rigid truss"is proposed. In this novel method, a \rigid truss" isattached to the bottom of the bridge deck at any nodeof the bridge deck far from the towers, and then theMR damper can be installed between the bridge deckand free-end of the rigid truss. In order to evaluate theoptimal performance of the MR dampers in reducingthe vertical vibrations of the example bridge, di�erentmodels have been proposed and optimized by trial anderror method.

2. Equation of motion of the bridge

The detailed procedure to write the motion equation

of suspension bridges is a complicated problem whichis out of the scope of this paper. For more informationon dynamic behavior of the suspension bridges, readersare referred to the studies performed by Abdel-Gha�arand his co-authors [30]. The motion equation of thesebridges can be written using �nite element methodand Hamilton's energy principles [28,30]. The mainobjective of the present study is to control the verticalvibration response of suspension bridges using MRdampers, and to �nd the optimal solutions for dampers'numbers and locations. To study the performanceof the MR dampers in reducing the vertical responseof these bridges, it is assumed that the bridge issubjected to vertical component of the earthquakeacceleration transmitted to the bridge deck by thepiers/ abutments. For numerical simulation of thebridge responses, 15 world-wide ground motion verticalaccelerogrames, shown in Table 1, are used as the inputexcitations. The accelerogrames are chosen, such thata variety of Peak Ground Accelerations (PGAs), fre-quency content levels, and distance to fault rupture canbe taken into account in the study. Static condensation[29] of the sti�ness matrix is performed to eliminatethe bending rotational degrees of freedom, and thenthe mass of each element is equally concentrated onits end nodes, resulting in a diagonal n � n lumpedmass matrix where n is the total number of verticaldegrees of freedom [4,28]. Finally, the equation ofmotion of the uncontrolled bridge can be written asfollows [29]:

[M ]f�u(t)g+[C]f _u(t)g+[K]fu(t)g=�[M ]frg�ug(t);(1)

Table 1. Earthquake accelerogrames considered in this study.

No. Earthquake Station Data Ms Distance tofault (km)

PGA(g)

Datareference

1 Bam-Iran Golbaf 1382/10/5 5.8 20 0.139 NEIC2 Manjil-Iran Roudbar 1369/03/31 7.4 13 0.217 NEIC3 Ardakul-Iran Torbate Heidarieh 1389/05/8 5.8 - 0.186 NEIC4 Firoozabad-Iran Abbar 1383/03/8 6.3 - 0.237 EMSC5 Sarein-Iran Astara 1375/12/05 6.1 55 0.191 NEIC6 Gheshm-Iran Bandar Abbas 1384/09/6 6.2 56 0.295 IIEES7 Chi-Chi, Taiwan CHY010 1999/09/20 7.6 19.96 0.125 CWB8 Loma Prieta Hollister City Hall 1989/10/18 7.1 - 0.216 USGS9 Imperial Valley Centro Array #1 1979/10/15 6.9 15.5 0.056 USGS10 Kocaeli, Turkey Fatih 1999/08/17 7.8 64.5 0.128 KOERI11 Erzincan, Turkey Erzincan 1992/03/13 6.9 2.0 0.248 -12 San Fernando Hollywood Stor Lot 1971/02/09 6.6 21.2 0.136 USGS13 Tabas, Iran Tabas 1978/09/16 7.7 - 0.688 -14 Northridge-Rinaldi Pacoima Dam 1994/01/17 6.7 8.0 1.229 CDMG15 Kobe Takarazuka 1995/01/16 6.9 1.2 0.433 CUE


where [M ], [C], and [K] are the n � n mass, damp-ing, and condensed sti�ness matrices of the bridge,respectively; frg is the location vector of groundacceleration, �ug(t); the vectors f�u(t)g, f _u(t)g, andfu(t)g are the acceleration, velocity, and displacementof the bridge vertical degrees of freedom, respec-tively, with respect to the ground [29]. It should benoted that in vertical vibration of suspension bridges,occurrence of plastic hinges in bridge deck/towersis not a matter of concern; therefore, no materialnon-linearity is considered in Eq. (1); on the otherhand, due to the fact that the problem under in-vestigation is too complicated, the probable e�ectscannot be judged. However, dynamic behaviors ofthe main cables exhibit a high geometric non-linearitywhich has been considered in Eq. (1) to some ex-tent [30].

Damping matrix of the bridge is also obtained byRayleigh method using mass and sti�ness matrices asfollows [29]:

[C] = a0[M ] + a1[K]; (2)

in which the proportionality coe�cients a0 and a1 arecalculated from the following equation:

12

"1!i !i1!j !j

#�a0a1

�=��i�j

�; (3)

where !i and !j are the natural frequencies of the ithand jth modes of the bridge. In this study, i is chosenas the �rst and j as the 24th vibrational mode of thebridge [4,28]; � is the ratio of structural damping to itscritical value, which is assumed to be the same for bothmodes and about 1%.

In this paper, the equation of motion of thebridge is transformed to state-space format [31] andsolved utilizing the Simulink Toolbox of the MATLABsoftware; and in order to reduce the complexity ofthe problem and its mathematical expressions, thegeneralized modal coordinates are used as the statecoordinates. Moreover, for the example bridge chosenin this study, using trial and error method, it is shownthat the �rst 7 modes of vibration are su�cient toobtain accurate results [32].

The equation of motion of a bridge controlled byMR dampers, in general, can be written as follows [33]:

[M�]f�u(t)g+ [C�]f _u(t)g+ [K�]fu(t)g = [D]ffg� [M�]frg�ug(t); (4)

where [M�], [C], and [K] are the mass, damping, andcondensed sti�ness matrices of the controlled bridge,respectively; [D] is the matrix of the locations on whichthe MR dampers are installed; and ffg is the vectorof control force imposed by the MR dampers to the

Figure 1. Schematic diagram of semi-active controlsystem.

bridge deck. Figure 1 shows the schematic diagramof semi-active control system in a real structure. Inthe current study, the sensors' input data are obtainedfrom the simulations results.

3. Description of MR damper system

The mechanical model of the MR damper must bechosen, such that it can adequately characterize theintrinsic non-linear behavior of the damper. In thisregard, a number of mechanical models have beenrecently proposed by several researchers [9,23-26,34].In the present study, a large-scale MR damper with2200 kN capacity based on normalized Bouc-Wenmodel is used for which the parameter identi�cation hasbeen conducted by Bahar et al. [23]. The mechanicalmodel of this damper is shown in Figure 2. Theequations governing the force f(0) exerted by thismodel of damper to the bridge are as follows [23]:

f (x(t); _x(t); w(t)) = kx(v)x(t) + k _x(v) _x(t) + kww(t);(5)

_w(t) =�(v)�

_x(t)� �(v)j _x(t)jjw(t)j�(v)�1w(t)

+ (�(v)� 1) _x(t)jw(t)j�(v); (6)

kx(v) = kx; (7)

k _x(v) = k _xa + k _xbv; (8)

Figure 2. Mechanical model of MR damper used in thisstudy.


�(v) = �a + �bexp(�13v); (9)

�(v) = �a + �bexp(�14v); (10)

�(v) = �a + �bexp(�14v); (11)

Kw(v) =

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

Kw1 +Kw2v1:15; v � 0:3

Kw3 +Kw4 sin��(v�0:3)

0:8

�+Kw5 sin

�3�(v�0:3)

0:8

�;

0:3 � v � 0:7

Kw6 +Kw7v +Kw8v3

+Kw9v5; 0:7 � v

(12)

where the term kx(v)x(t) represents a linear elasticforce; f(0) is the output force of the MR damper;x(t), _x(t), and v are displacement, velocity, and voltageinputs, respectively. The state variable w(t) has nophysical meaning; so, it should be calculated mathe-matically from Eq. (6). The other parameters includingkx; k _xa; k _xb; �a; �b; �a; �b; �a; �b; kw1; kw2; :::; kw9 are tobe determined by the experimental test for which thenumerical values used in this study are given in Table 2.

4. Numerical study

In this paper, semi-active control system is used to mit-igate the seismic vertical response of suspension bridges

Table 2. The parameters of the large-scale normalizedMR damper used in this study.

Parameter Value

Kx 207K _x;a 89.64K _x;b 292�a 648.95�b -3.86�a 1.44�b 0.02�a 0.76�b 0.009kw1 55.38kw2 2270kw3 619.85kw4 387.34kw5 18.42kw6 -87.52kw7 2665kw8 -3054.7kw9 1545.5

Figure 3. General view of vincent-Thomas suspensionbridge.

using MR dampers. The Vincent-Thomas suspensionbridge, shown in Figure 3, is chosen for numericalstudy. This bridge was constructed to connect SanFernando to the Terminal Island at the port of LongBeach to the south of Los Angeles, California. Itssuperstructure consists of a 460 m center span and two155 m side spans. There are two towers with the heightof 102 m, and sag of the main cable in the center of themain span is 45.72 m [1,30]. The weight of the deckand the two main cables are about 52438 N/m and12390 N/m, respectively.

The cross area of each main cable is almost780 cm2 and the initial tension due to dead loads ineach cable is 30038 � 103 N [1,30]. The bridge hasbeen simulated by many researchers in the past, andthe sti�ness and mass matrices are available in [28].The �nite element model of the bridge is shown inFigure 4. According to this �gure, for �nite elementanalysis of the bridge, 11 elements in each side span and28 elements in the center span are considered [1,30].The bridge deck is considered as a simply supportedbeam between the towers, connected by several hangersto the main cables. The end connections of thebridge girders to the towers are assumed to be hinged.In �nite element model of the bridge, two noddedbeam elements with two degrees of freedom at eachnode, vertical displacement, and bending rotation areconsidered (Figure 4(b)) [1,4,28,30]. The �rst 6 modalfrequencies and mode shapes of the bridge are emergedin Figure 5 [28]. A more detailed description of thebridge can be found in Abdel-Gha�ar [30].

As illustrated above, in this study, semi-activecontrol system utilizing MR dampers is used to reducethe seismic vertical response of suspension bridges.For this purpose, di�erent control models, in pointof MR dampers installation view, are investigated toachieve the optimal model of the control system to getmaximum reduction in bridge responses. The main aimof these models is to optimize the number and locationof the dampers. The �rst model that comes to mind isto install MR dampers between the sti�ening truss ofbridge deck and towers, but due to low displacementthat is expected to happen near the towers, this optionsounds not to be more appropriate. Therefore, theproblem of how to install the dampers along the bridgespan at any node far from the towers should be solved.For this purpose, �rstly, di�erent models have been


Figure 4. (a) Elements and node numbers in each span. (b) One �nite element of the bridge, showing the degrees offreedom.

Figure 5. Vincent-Thomas suspension bridge's �rst 6mode shapes and frequencies.

proposed to install the dampers at the nodes far fromtowers, and then the bridge responses are calculatedunder the application of the earthquake accelerogramesgiven in Table 1 to evaluate the performance of thesemodels. If a proposed model has an appropriateoperation, then its performance is optimized for thenumber and location of the dampers.

Herein, it should be mentioned that in this re-search, semi-active control of the suspension bridgesis studied through fuzzy logic, and in particular,two schemes of passive o� and passive on have beeninvestigated. There are a vast number of parameters,including input and output variables of fuzzy logic,membership functions, fuzzy rules, and the numberand location of dampers, which make the investigationsophisticated. To reduce the complexity, �rst, thenumber and location of dampers have been optimizedthrough passive o� and passive on cases. After that,the optimized models have been studied for semi-activecontrol scheme. It is found that although semi-activecontrol performs in an excellent way to reduce thevertical vibrations of the system; in most of the times,the results are the same as the passive on scheme,and fuzzy controller seems unable to perform betterthan the passive on scheme despite a lot of striving bytrial and error method. In spite of the fact that fuzzylogic is overwhelmingly e�cient in reducing the powersource, it should be noted that the power source is notan important matter when it comes to MR dampers.


Therefore, due to the massive details and results relatedto the fuzzy logic controller and in order to keep thepaper length short, just the displacement results ofsome particular cases of passive o� and passive onschemes are discussed in the present paper.

Finally, among the di�erent models studied in thisinvestigation, 3 models are chosen for optimization andpresented in the following.

Model 1: Installation of MR dampers along thebridge span using a TMD system.

In the �rst proposed model, to install MRdampers, as shown in Figure 6, a Tuned Mass Damper(TMD) system is used to install the dampers to thebottom of the bridge deck far from the towers. Theresults obtained from this model indicate that in spiteof high complexity of this method, it is not appropriate.This model is not only unable to decrease the bridgeresponses, but also it increases them. The complexityof this model is due to the large number of its unknownsincluding the amount of mass of the TMD and sti�nessof the spring, in addition to the number of MR dampersand their locations.

Some of di�erent schemes of this model (Case 1to Case 5), in which only the location and numberof dampers vary along the bridge span, are given inFigure 7. The controlled and uncontrolled averagedmaximum displacement responses of the bridge evalu-ated for the Case 1 of Figure 7 are presented in Figure 8.

It can be seen from the Figure 8(b) that thebridge response at some node points is highly increased,showing that use of the TMD for installing the MRdamper along the bridge span does not provide accept-able reduction in bridge responses. In this case, theTMD mass and its spring sti�ness are chosen about106 kg and 3 � 107 N/m, respectively. The other casesof this model, shown in Figure 7, also show the sametrend in reducing the bridge responses. Therefore, it is�nally concluded that this model is not appropriate forthis example bridge.

It should be noted that many other cases (notshown here) have been studied in this model bychanging the values of mass and sti�ness of the TMDand its location, but none of them showed appropriateresults. Moreover, it is noted that in this study, semi-active performance of the MR dampers is studied, but

Figure 6. Installation of MR dampers at nodes far fromtowers using TMD.

Figure 7. Some di�erent schemes investigated in Model 1by trial and error method.

Figure 8. Comparison of the uncontrolled and controlledresponses in Case 1 of Model 1 for voltage of zero (passiveo�) and maximum voltage (passive on): (a) Averagedmaximum displacements, and (b) response reductionratios.


due to its improper operation, only the extreme results(passive on and passive o�) are presented here.

Model 2: Installation of MR dampers near thetowers.

After investigating the performance of the Model1 and concluding its inability in reducing the bridgeresponses, it is decided to install the dampers betweenthe towers and bridge deck as shown in Figure 9. In thismodel, there is no need for the TMD system; therefore,the number of the dampers is limited to what can beinstalled near the towers and piers (or abutments). Itshould be highlighted that increasing the number ofdampers may not always lead to more response reduc-tion. It is due to high non-linearity and high exibilityof the bridge. As a result, some of the dampers in thisstudy are used with the capacity of 1100 kN, calledscaled dampers, discussed by Rodriguez [25].

In this model, dynamic characteristics of thecontrolled bridge and the uncontrolled one are thesame in the equation of motion. Also, as shown inFigure 10, di�erent cases here (Cases 1 to 6), whichdi�er in number, location, and the capacity of thedampers (2200 kN or 1100 kN dampers), have beeninvestigated by trial and error method to �nd theoptimal case. After comparing the reduction ratioson bridge responses, the optimal case is evaluatedfor which the con�guration of the dampers is shownin Figure 11. In this optimal case, total number of4 dampers are installed between the deck and maintowers as follows: two dampers with the capacity of2200 kN between the towers and nodes number 10and 38; two dampers with the capacity of 1100 kNbetween the towers and nodes number 11 and 37 of thebridge, as shown in Figure 11 (nodes number are givenin Figure 4).

In the next step, the bridge is analyzed for the op-timal case of damper installation, shown in Figure 11,under the 15 earthquake accelerogrames of Table 1, andthen the ensemble average of maximum displacementresponses of the bridge for these 15 accelerogrames arecalculated and compared in Figure 12. As it can beseen from the �gure, the reduction ratio of the bridgeaverage maximum displacements is between 5-15% inpassive o� control (voltage is zero), and it is between10-62% in passive on control (voltage is maximum).The minimum reduction happens in the middle of the

center span, and the maximum reduction happens atthe nodes on which MR dampers are installed. Themost important problem of this model is its inability inreducing the displacements of the middle of the centerspan, which is solved in Model 3 and explained in thenext section.

Model 3: Installation of MR dampers at any de-sired node using a rigid truss (as a novel method).

One of the most challenging problems in thisresearch, as mentioned before, is how to install the MR

Figure 10. Some cases of di�erent schemes investigatedin Model 2 by trial and error method.

Figure 11. Optimal installation con�guration of the MRdampers in Model 2.

Figure 9. Installation of MR dampers on the bridge deck in Model 2.


Figure 12. Comparison of the bridge maximumuncontrolled displacements with those controlled bypassive o� and passive on systems: (a) The ensembleaverage of maximum displacements for 15 accelerogrames,and (b) reduction ratio of the averaged maximumdisplacements.

dampers at a node of bridge deck far from the towers.The �rst idea o�ered for this purpose (use of the TMDas explained in Model 1) was not appropriate, and thesecond one operated well, although it was not expected.As it is seen in the previous section, the Model 2 is notable to reduce the bridge responses at the middle ofthe center span. To overcome this drawback, Model 3is proposed in which, as shown in Figure 13, the MRdampers are installed either between the bridge deckand free-end of a rigid truss at any node far from thetowers or between the bridge deck and towers describedin Model 2. According to the Figure 13-Detail b, theforces produced by the MR damper, installed betweenthe bridge deck and the rigid truss, will be transferredto the bridge deck by elements of truss (forces F1, F2,and F3 in the �gure). It should be noted that, as itcan be seen from the �gure, the rigid truss is attachedto the bridge deck at points A, B, and C, amongwhich only the point B is considered as a node in the�nite element model of the bridge deck (see Figure 4).Therefore, in the present study, the forces F1 and F3cannot be directly inserted into the bridge equation ofmotion, and the resultant of these forces is consideredin the model equation. Thus, some moments areprobably produced during transformation of the MRdampers forces to the deck by the truss. If the dampersare installed symmetrically, these moments will beneutralized; otherwise, they should be considered inthe simulation. In this study, because of the limitationin the �nite element model, this e�ect is neglected.From the results of the bridge controlled responses, itis found that without considering these extra moments,asymmetric results are a few better than those of thesymmetric ones. In order to account for the e�ects ofthese extra moments, there are some strategies amongwhich the �rst one is increasing the number of nodes inthe �nite element model of the bridge, such that all the

Figure 13. Installation details of the MR dampers in Model 3.


Figure 14. Optimal installation con�guration of the MRdampers in Model 3 (Scheme 3-1 with 6 dampers).

Figure 15. Optimal installation con�guration of the MRdampers in Model 3 (Scheme 3-2 with 8 dampers).

forces F1, F2, and F3 could be directly considered inthe equation of motion. The second one is neglectingthe extra moments (used in this study); the last oneis symmetric installation of the dampers instead ofasymmetric without considering the economic issues.In this paper, the results of the asymmetric model alsohave been �gured. Here also, di�erent schemes havebeen also investigated to achieve the optimal number,location, and capacity of dampers using trial and errormethod. Finally, 2 schemes (Schemes 3-1 and 3-2),shown in Figures 14 and 15, are suggested as theoptimal ones.

Figures 16 and 17 show the comparison betweenaveraged values of the controlled and uncontrolled ver-tical maximum displacement response of the examplebridge for Schemes 3-1 and 3-2, respectively. Figures16(a) and 17(a) show the ensemble average of themaximum displacement of the bridge subjected to 15earthquake accelerogrames of Table 1; Figure 16(b) and17(b) show the reduction ratio of this response quan-tity. It is clear from the �gures that the performanceof Scheme 3-1, in which 6 dampers are used, is betterthan Scheme 3-2 in the middle of the center span, whilein the side spans, Scheme 3-2 works better. Moreover,performance of both schemes in Model 3 is much betterthan Model 2.

In passive o� (voltage is zero), the reduction ratioof the average maximum displacement is about 10-20%,while in passive on (voltage is maximum), this value isabout 30-70%. The minimum reduction happens in themiddle of the center span, and the maximum reductionhappens at the nodes at which the MR dampers areinstalled.

The results of Scheme 3-2, in which 8 dampershave been used, are shown in Figure 17. The reductionratio of the average maximum displacement is about10-20% in passive o� (voltage is zero), about 25-70% inpassive on (voltage is maximum), and almost the sameas Scheme 3-1 in passive o� control. Furthermore, the

Figure 16. Comparison of the uncontrolled andcontrolled responses in Model 3 with 6 dampers for voltageof zero (passive o�) and maximum voltage (passive on) forScheme 3-1: (a) Ensemble average of the maximumdisplacements, and (b) response reduction ratios.

reduction in passive on is more than Scheme 3-1 for thesuspended side spans, but it performs between Model2 and Scheme 3-1 for the center spans.

5. Conclusions

Seismic vibration control of suspension bridges inhorizontal direction is well studied and reported in theliterature, but no research is reported on this issue invertical direction. In this paper, the performance ofsemi-active control of vertical vibration of suspensionbridges subjected to earthquake using MR dampers isdiscussed. The Vincent-Thomas suspension bridge lo-cated in Los Angeles, U.S.A., is chosen as a case study,and 15 worldwide major earthquake accelerogrames areused in the numerical analyses. The main challengingproblem in this study was how to install the MRdamper at a node of bridge deck far from the towers.To solve it, di�erent models have been suggested andoptimized by trial and error method, and the optimallocation and number of the dampers are evaluated. Fi-nally, 3 models have been proposed as the appropriateones. In the �rst model, Model 1, the MR dampers areinstalled along the bridge span by use of a TMD sys-tem. In the second one, Model 2, the MR dampers are


Figure 17. Comparison of the uncontrolled andcontrolled responses in Model 3 with 8 dampers and forvoltage of zero (passive o�) and maximum voltage (passiveon) for Scheme 3-2: (a) Ensemble average of the maximumdisplacements, and (b) response reduction ratios.

installed only between the towers and the bridge deck.In the third one, Model 3, a novel rigid truss is used toinstall the MR dampers between free-end of the trussand any desired node of the bridge deck, in additionto the dampers installed between the towers/or abut-ments and bridge deck. In Model 2, 4 dampers are in-stalled between the towers and bridge deck. In Model 3,two schemes (Schemes 3-1, and 3-2) are proposed as theoptimal ones which di�er in the location and number ofthe dampers. The novel method of using rigid truss isproposed in Model 3 to install the MR dampers at thenodes of bridge deck far from the towers; the optimalnumbers of the MR dampers have been evaluated as6 and 8 dampers in Schemes 3-1 and 3-2, respectively.From the results of this numerical study, it is found thatthe use of a TMD system in such problems is useless;however, the other proposed models, Models 2 and 3(Schemes 3-1 and 3-2), are successful in reducing thevertical vibration of the example bridge. The most re-duction ratio of the center span relates to Scheme 3-1 ofModel 3, and those of the side spans belong to Scheme3-2 of this model. Finally, it should be highlightedthat although the reduction ratio has its least value inModel 2, it is introduced as an appropriate model dueto the fact that it has the least number of dampers. In

both of Models 2 and 3, the maximum reduction of thebridge vertical displacement is obtained at the nodesat which the MR dampers are installed.

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Biographies

Saeid Pourzeynali received his MS degree in Struc-tural Engineering, in 1989, from Tehran University,Iran, and joined the University of Guilan, Rasht, Iran,as Lecturer in the Department of Civil Engineering.In 1991, Dr. Pourzeynali obtained his PhD degreein Structural Engineering from IIT Delhi, India, andis currently at the University of Guilan, Iran. Hisresearch interests span a wide range of specialized�elds, including structural dynamics, earthquake engi-neering, random vibrations, reliability analysis, struc-tural control and optimization, and dynamics of cablesupported bridges. Dr. Pourzeynali has more than24 years experience in teaching and has supervisedmore than 30 MS degree students. He has publishedmore than 90 international journal and conferencepapers, and has received various awards, including\best research study on cable supported bridges" in2004 from the Vice-Chancellor for Research at theUniversity of Guilan.

Arash Bahar was born in 1967 in Rasht, Iran. Aftergraduation from the Civil Engineering Departmentat the Isfahan University, Isfahan, Iran, in 1987, hecontinued his studies toward MSc in Structural Engi-neering at the School of Engineering, Shiraz University,Shiraz, Iran. He then joined Guilan University in 1993.He received his PhD degree in Earthquake Engineeringfrom the \Universitat Polit�ecnica de Catalu~na", in 2009in Barcelona, Spain. He is presently Assistant Professorof Structural Engineering division of the Faculty ofEngineering in the University of Guilan. His researchinterests include: structural control, identi�cation,


linear/non-linear dynamic analysis, MR damper/semi-active devices, earthquake engineering, and steel struc-tural design.

Solmaz Pourzeynali was born in 1988 in Tehran,Iran. She obtained her BSc degree in Civil Engineering

from University of Guilan, Iran, in 2010, and herMSc degree in Structural Civil Engineering at thesame university in 2013. Her research interests in-clude structural dynamics, earthquake engineering andbridge engineering. She, also, has several publicationsin suspension bridges.

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