Research ArticleSeismic Response Analysis of Continuous Multispan Bridgeswith Partial Isolation

E Tubaldi1 A DallrsquoAsta2 and L Dezi1

1Department of Construction Civil Engineering and Architecture (DICEA) Polytechnic University of MarcheVia Brecce Bianche 60131 Ancona Italy2School of Architecture and Design (SAD) University of Camerino Viale della Rimembranza 63100 Ascoli Piceno Italy

Correspondence should be addressed to E Tubaldi etubaldigmailcom

Received 17 December 2014 Accepted 23 March 2015

Academic Editor Laurent Mevel

Copyright copy 2015 E Tubaldi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Partially isolated bridges are a particular class of bridges in which isolation bearings are placed only between the piers top andthe deck whereas seismic stoppers restrain the transverse motion of the deck at the abutments This paper proposes an analyticalformulation for the seismic analysis of these bridges modelled as beams with intermediate viscoelastic restraints whose propertiesdescribe the pier-isolator behaviourDifferent techniques are developed for solving the seismic problemThefirst technique employsthe complex mode superposition method and provides an exact benchmark solution to the problem at hand The two othersimplified techniques are based on an approximation of the displacement field and are useful for preliminary assessment anddesign purposes A realistic bridge is considered as case study and its seismic response under a set of ground motion records isanalyzed First the complex mode superposition method is applied to study the characteristic features of the dynamic and seismicresponse of the system A parametric analysis is carried out to evaluate the influence of support stiffness and damping on theseismic performance Then a comparison is made between the exact solution and the approximate solutions in order to evaluatethe accuracy and suitability of the simplified analysis techniques for evaluating the seismic response of partially isolated bridges

1 Introduction

Partially restrained seismically isolated bridges (PRSI) are aparticular class of bridges isolated at intermediate supportsand transversally restrained at abutmentsThis type of partialisolation is quite common in many bridges all over theworld (see eg [1ndash4]) The growing interest on the dynamicbehaviour of bridges with partial restrain is demonstrated bynumerous recent experimental works [3 4] and numericalstudies [5ndash11] discussing the advantages and drawbacks withrespect to full isolation

An analytical model commonly employed for the analysisof the transverse behaviour of PRSI bridges consists in acontinuous simply supported beam resting on discrete inter-mediate supports with viscoelastic behaviour representingthe pier-bearing systems (see eg [6 8 9]) The damping ispromoted by two different mechanisms the bearings usuallycharacterized by high dissipation capacity and the deckcharacterized by a lower but widespread dissipation capacity

The strongly inhomogeneous distribution of the dissipationproperties along the bridge results in nonclassical dampingand this makes the rigorous analysis of the analytical PRSIbridge model very demanding

In general the exact solution of the seismic problemfor a nonclassically damped system requires resorting to thedirect integration of the equations of motion or to modalanalysis [12] based on complex vibration modes The firstanalysis approach is conceptually simple though computa-tionally costlywhen large-scale systems are analyzedwhereasthe second approach is computationally efficient but oftennot appealing for practical engineering applications sinceit involves complex-valued functions and it is difficult toimplement in commercial finite element codes For thisreason many studies have been devoted to the definition ofapproximate techniques of analysis and to the assessment oftheir accuracy [13ndash17] With reference to the classical caseof fully isolated bridges the studies of Hwang et al [18]and Lee et al [19] have shown that acceptable estimates

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 183756 15 pageshttpdxdoiorg1011552015183756

2 Shock and Vibration

of the modal damping ratios are obtained by consideringthe real undamped modes and the diagonal terms of themodal damping matrix Franchin et al [20] have analyzedsome realistic bridge models by showing that this decouplingapproximation gives also quite accurate estimates of theresponse to a seismic input as compared to the rigorousresponse estimates obtained through complex modal anal-ysis With reference to the specific case of PRSI bridgescharacterized by deformation shapes and dynamic propertiesvery different from those of fully isolated bridges the workof Tubaldi and DallrsquoAsta [9] has addressed the issue ofnonclassical damping within the context of the free-vibrationresponseThe authors have observed that nonclassical damp-ing influences differently the various response parametersrelevant for the performance assessment (ie the transversedisplacement shape is less affected than the bending momentdemand by the damping nonproportionality) However theeffects of the decoupling approximation on the evaluation ofthe seismic response of the proposed PRSI bridgemodel havenot been investigated yet This issue becomes of particularrelevance in consequence of the numerous studies on PRSIbridges that completely disregard nonclassical damping [6ndash8] Thus a closer examination is still required to ensurewhether the use of proportionally damped models providesacceptable estimates of the seismic response of these systems

Another approximation often introduced in the analysisof PRSI bridges concerns the transverse deformation shapeIn this regard many studies are based on the assumption of aprefixed sinusoidal vibration shape [6 8 10]This assumptionpermits deriving analytically the properties of a generalizedSDOF system equivalent to the bridge and estimating thesystem response by expressing the seismic demand in termsof a response spectrum reduced to account for the systemcomposite damping ratio In Tubaldi and DallrsquoAsta [8 9] itis shown that the sinusoidal approximation of the transversedisplacements may be accurate for PRSI bridges if thefollowing conditions are met (a) the superstructure stiffnessis significantly higher than the pier stiffness (b) the variationsof mass and stiffness of the deck and of the supports arenot significant (c) the span number is high and (d) thedisplacement field is dominated by the first vibration modeHowever even if the displacements are well described by asinusoidal shape other response parameters of interest forthe performance assessment such as the transverse bendingmoments may not exhibit a sinusoidal shape Thus furtherinvestigations are required to estimate the error arising dueto this approximation

The aim of this study is to develop an analytical formu-lation of the seismic problem of PRSI bridges modelled asnonclassically damped continuous systems and a rigoroussolution technique based on the complexmode superposition(CMS) method [12 21ndash24] The application of this methodrequires the derivation of the expression of the modalorthogonality conditions and of the impulsive responsespecific to the problem at hand It permits describing theseismic response in terms of superposition of the complexvibration modes which is particularly useful for this case inconsequence of the relevant contribution of higher modes ofvibration to the response of PRSI bridges [8 10] Moreover

kcr ccr

xr

L

m(x) b(x) c(x)

Figure 1 Analytical model for PRSI bridges

the proposed technique permits testing the accuracy oftwo simplified analysis techniques introduced in this studyand commonly employed for the PRSI bridge analysis Thefirst approximate technique describes the transverse motionthrough a series expansion in terms of the classic modesof vibration obtained by neglecting the damping of theintermediate restraints whereas the second technique isbased on a series expansion in terms of sinusoidal func-tions corresponding to the vibration modes of the systemwithout the intermediate restraintsThe introduction of theseapproximations of the displacement field results in a couplingof the equation of motions projected in the space of theapproximating functions which is neglected in the solutionto simplify the response assessment

A realistic case study is considered and its response to a setof groundmotion records is examined first through the CMSmethod with these two objectives to unveil the characteristicfeatures of the performance of PRSI bridges and to assess theinfluence of higher vibration modes and of the intermediatesupport stiffness and damping on the response of the resistingcomponents Successively the seismic response estimatesaccording to the CMS method are compared with thecorresponding estimates obtained by applying the proposedsimplified techniques in order to evaluate their accuracy andreliability

2 Dynamic Behavior of PRSI Bridges

The PRSI bridge model (Figure 1) consists of beam pinned atthe abutments and resting on discrete viscoelastic supportsrepresenting the pier-bearing systems

Let 1198672[Ω] be the space of functions with square inte-

grable second derivatives in the spatial interval Ω = [0 119871] 119881

the space of transverse displacement functions satisfying thekinematic boundary conditions (eg 119881 = V(119909) isin 119867

2[Ω]

V(0) = V(119871) = 0) and 119906(119909 119905) isin 119880 sube 1198622(119881 [1199050 1199051])

the motion defined in the time interval considered [1199050 1199051]

belonging to the space of continuous functions1198622 and known

at the initial instant together with its time derivative (initialconditions)The differential dynamic problem can be derivedfrom the DrsquoAlembert principle [25] and expressed in thefollowing form

int

119871

0

119898 (119909) (119909 119905) 120578 (119909) 119889119909 + int

119871

0

119888 (119909) (119909 119905) 120578 (119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

(119909119903 119905) 120578 (119909

119903) + int

119871

0

119887 (119909) 11990610158401015840

(119909 119905) 12057810158401015840

(119909) 119889119909

Shock and Vibration 3

+

119873119888

sum

119903=1

119896119888119903

119906 (119909119903 119905) 120578 (119909

119903) = minus int

119871

0

119898 (119909) 119892

(119905) 120578 (119909) 119889119909

forall120578 isin 119881 forall119905 isin [1199050 1199051]

(1)

where 120578 isin 119881 denotes a virtual displacement consistentwith the geometric restrains 119873

119888denotes the number of

intermediate supports prime denotes differentiation withrespect to 119909 and dot differentiation with respect to time 119905

The piecewise continuous functions 119898(119909) 119887(119909) and 119888(119909)

denote the mass per unit length the transverse bendingstiffness per unit length and the deck distributed dampingconstant The constants 119896

119888119903and 119888

119888119903are the stiffness and

damping constant of the viscoelastic support located atthe 119903th position = 119909

119903 while

119892(119905) denotes the ground

accelerationThe local form of the problem is obtained by integrating

by parts (1) and can be formally written as

119872 (119909 119905) + 119862 (119909 119905) + 119870119906 (119909 119905) = minus119872119892

(119905)

119906101584010158401015840

(119909 119905) 12057810038161003816100381610038161003816

119871

0= 0 119906

10158401015840(119909 119905) 120578

101584010038161003816100381610038161003816

119871

0= 0

(2)

where 119872 119862 and 119870 denote respectively the mass dampingand stiffness operator They are expressed as (120575 is the Diracrsquosdelta function)

119872 = 119898 (119909)

119870 = 119887 (119909)1205974

1205971199094+

119873119888

sum

119903=1

119896119888119903

120575 (119909 minus 119909119903)

119862 = 119888 (119909) +

119873119888

sum

119903=1

119888119888119903

120575 (119909 minus 119909119903)

(3)

3 Eigenvalue Problem for PRSI Bridges

The free-vibrations problem of the beam is obtained byposing

119892= 0 in (2) The corresponding differential

boundary problem is then reduced to an eigenvalue problemsolvable by expressing the transverse displacement 119906(119909 119905) asthe product of a spatial function 120595(119909) and a time-dependentfunction 119885(119905) = 119885

0119890120582119905

119906 (119909 119905) = 120595 (119909) 119885 (119905) (4)

After substituting (3) and (4) into (2) for 119892

= 0 the followingtranscendental equation is obtained

[119898 (119909) 1205822

+ 119888 (119909) 120582] 120595 (119909) + 119887 (119909) 120595119868119881

(119909)

+

119873119888

sum

119903=1

(119896119888119903

+ 119888119888119903

120582) 120575 (119909 minus 119909119903) 120595 (119909) = 0

(5)

Equation (5) is satisfied by an infinite number of eigenvaluesand eigenvectors that occur in complex conjugate pairs [26]The 119894th eigenvalue 120582

119894contains information about the system

vibration frequency and damping while the 119894th eigenvector120595119894(119909) is the 119894th vibration shapeThe solution of the eigenvalue

problem for constant deck properties is briefly recalled inthe appendix The orthogonality conditions for the complexmodes are [9]

(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909 +

119873119888

sum

119903=1

119888119888119903

120595119894(119909119903) 120595119895(119909119903) = 0

(6)

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 minus 120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(7)

4 Seismic Response of PRSI BridgesBased on CMS Method

41 Series Expansion of the Response In the CMS methodthe displacement of the beam is expanded as a series of thecomplex vibration modes as

119906 (119909 119905) =

infin

sum

119894=1

120595119894(119909) 119902119894(119905) (8)

where 120595119894(119909) = 119894th complex modal shape and 119902

119894(119905) = 119894th

complex generalized coordinate It is noteworthy that inpractical applications the series is truncated at the term 119873

119898

Substituting (8) into (1) written for 120578(119909) = 120595119895(119909) one

obtains

infin

sum

119894=1

[int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909] 119902

119894(119905)

+

infin

sum

119894=1

[int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

+

infin

sum

119894=1

[int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(9)

4 Shock and Vibration

Upon substitution of (7) into (9) the following expression isobtained

infin

sum

119894=1

119902119894(119905) [int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

+

infin

sum

119894=1

119902119894(119905) [int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

+

infin

sum

119894=1

119902119894(119905) [120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(10)

42 Response to Impulsive Loading For 119892(119905) = 120575(119905) the

generalized coordinate assumes the form [22]

119902119894(119905) = 119861

119894119890120582119894119905 (11)

with 119902119894(119905) = 120582

119894119902119894(119905) and 119902

119894(119905) = 120582

119894

2119902119894(119905)

After substituting into (10) one obtains

infin

sum

119894=1

119902119894(119905)

120582119894

[(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

= minus120575 (119905) int

119871

0

119898 (119909) 120595119895(119909) 119889119909

(12)

It can be noted that the right-hand side of (12) vanishesfor 120582119894

= 120582119895by virtue of (6) Thus the problem can be

diagonalized and the 119894th decoupled equation reads as follows

2119894

119902119894(119905) + 119862

119894119902119894(119905) = minus

119894120575 (119905) (13)

where 119894

= int119871

0119898(119909)120595

119894

2(119909)119889119909 119862

119894= int119871

0119888(119909)120595

119894

2(119909)119889119909 +

sum119873119888

119903=1119888119888119903

120595119894

2(119909119888119903

) and 119894= int119871

0119898(119909)120595

119894(119909)119889119909

By assuming that the system is at rest for 119905 lt 0 thefollowing equation holds at 119905 = 0

+

2119894

119902119894(0+) + 119862119894119902119894(0+) = minus

119894 (14)

Thus since 119902119894(0+) = 119861

119894and 119902

119894(0+) = 120582119861

119894 one can finally

express 119861119894as

119861119894=

119894

2119894120582119894+ 119862119894

= minus (int

119871

0

119898 (119909) 120595119894(119909) 119889119909)

sdot (2120582119894int

119871

0

119898 (119909) 120595119894

2(119909) 119889119909 + int

119871

0

119888 (119909) 120595119894

2(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894

2(119909119888119903

))

minus1

(15)

The corresponding expression of the complex modal impulseresponse function is ℎ

119894

119888(119909 119905) = 119861

119894120595119894(119909)119890120582119894119905 and the sum

of the contribution to the complex modal impulse responsefunction of the 119894th mode and of its complex conjugate yieldsa real function ℎ

119894(119909 119905) which may be expressed as

ℎ119894(119909 119905) = 119861

119894120595119894(119909) 119890120582119894119905+ 119861119894120595119894(119909) 119890120582119894119905

= 120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 ℎ119894(119905) + 120573

119894(119909) ℎ119894(119905)

(16)

where 120572119894(119909) = 120585

119894120573119894(119909) minus radic1 minus 120585

119894

2120574119894(119909) 120573

119894(119909) = 2Re[119861

119894120595119894]

120574119894(119909) = 2 Im[119861

119894120595119894] and ℎ

119894(119905) denotes the impulse response

function of a SDOF system with natural frequency 1205960119894

= |120582119894|

damping ratio 120585119894

= minusRe(120582119894)|120582119894| and damped frequency

120596119889119894

= 1205960119894

radic1 minus 120585119894

2 whose expression is

ℎ119894(119905) =

1

120596119889119894

119890minus1205851198941205960119894119905 sin (120596

119889119894119905) (17)

43 CMS Method for Seismic Response Assessment By tak-ing advantage of the derived closed-form expression ofthe impulse response function the seismic input

119892(119905) is

expressed as a sum of Delta Dirac functions as follows

119892

(119905) = int

119905

0

119892

(120591) 120575 (119905 minus 120591) 119889120591 (18)

The seismic displacement response is then expressed in termsof superposition of modal impulse responses

119906 (119909 119905) =

infin

sum

119894=1

int

119905

0

119892

(120591) ℎ119894(119909 119905 minus 120591) 119889120591

=

119873119898

sum

119894=1

120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 119863119894(119905) + 120573

119894(119909) 119894(119905)

(19)

where 119863119894(119905) and

119894(119905) denote the response of the oscillator

with natural frequency 1205960119894and damping ratio 120585

119894 subjected to

the seismic input 119892(119905)

It is noteworthy that in the case of 119888119888119903

= 0 correspond-ing to intermediate supports with no damping the system

Shock and Vibration 5

becomes classically damped and one obtains 120582119894

= minus1205851198941205960119894

+

1198941205960119894

radic1 minus 120585119894

2 120585119894

= 119888119889(21205960119894

119898119889) 119861119894

= minus120588119894119894(21205960119894

radic1 minus 120585119894

2)

120572119894

= 1205881198941205960119894 120573119894

= 0 and 120574119894

= minus120588119894(1205960119894

radic1 minus 120585119894

2) where 120588

119894=

119894th is the real mode participation factor Thus (19) reduces tothe well-known expression

119906 (119909 119905) =

infin

sum

119894=1

120588119894119863119894(119905) (20)

The expression of the other quantities that can be of interestfor the seismic performance assessment of PRSI bridges canbe derived by differentiating (19) In particular the abutmentreactions are obtained as 119877

119886119887(0 119905) = minus119887(0)119906

101584010158401015840(0 119905) and

119877119886119887

(119871 119905) = minus119887(119871)119906101584010158401015840

(119871 119905) the 119903th pier reaction is obtainedas 119877119901119903

= 119896119888119903

119906(119909119903 119905) + 119888

119888119903(119909119903 119905) for 119903 = 1 2 119873

119888 and

the transverse bending moments are obtained as 119872(119909 119905) =

minus119887(119909)11990610158401015840

(119909 119905)

5 Simplified Methods forSeismic Response Assessment

In this paragraph two simplified approaches often employedfor the seismic analysis of structural systems are intro-duced and discussed These approaches are both based onthe assumed modes method [27 28] and entail using in(8) a complete set of approximating real-valued functions(denoted as 120578

119894(119909) for 119894 = 1 2 infin) instead of the exact

complex vibration modes for describing the displacementfieldThe two analysis techniques differ for the approximatingfunction employed The use of these functions leads to asystem of coupled Galerkin equations [27] and the generic119894th equation reads as follows

infin

sum

119895=1

119872119894119895

119902119895(119905) +

infin

sum

119895=1

(119862119888119894119895

+ 119862119889119894119895

) 119902119895(119905)

+

infin

sum

119895=1

(119870119888119894119895

+ 119870119889119894119895

) 119902119895(119905) = minus119872

119901119894119892

(119905)

(21)

where

119872119894119895

= int

119871

0

119898 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119872119901119894

= int

119871

0

119898 (119909) 120578119894(119909) 119889119909

119870119888119894119895

=

119873119888

sum

119903=1

119896119888119903

120578119894(119909119903) 120578119895(119909119903)

119870119889119894119895

= int

119871

0

119887 (119909) 12057810158401015840

119894(119909) 12057810158401015840

119895(119909) 119889119909

119862119888119894119895

=

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903)

119862119889119894119895

= int

119871

0

119888 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119894 119895 = 1 2 infin

(22)

The coupling between the generalized responses 119902119894(119909) and

119902119895(119909) may be due to nonzero values of the ldquonondiagonalrdquo

damping and stiffness terms 119862119888119894119895

and 119870119888119894119895 for 119894 = 119895 In order

to evaluate the extent of coupling due to damping the index120572119894119895is introduced whose definition is [13]

120572119894119895

=

(119862119888119894119895

+ 119862119889119894119895

)2

[119862119888119894119894

+ 119862119889119894119894

] sdot [119862119888119895119895

+ 119862119889119895119895

]

= (

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903) + int

119871

0

119888119889

(119909) 120578119894(119909) 120578119895(119909) 119889119909)

2

sdot ((

119873119888

sum

119903=1

119888119888119903

120578119894

2(119909119903) + int

119871

0

119888 (119909) 120578119894

2(119909) 119889119909)

sdot (

119873119888

sum

119903=1

119888119888119903

120578119895

2(119909119903) + int

119871

0

119888 (119909) 120578119895

2(119909) 119889119909))

minus1

(23)

This index assumes high values for intermediate supportswith high dissipation capacity while in the case of deck andsupports with homogeneous properties it assumes lower andlower values for increasing number of supports since thebehaviour tends to that of a beam on continuous viscoelasticrestraints [9]

A second coupling index is defined for the stiffness termwhose expression is

120573119894119895

=

(119870119888119894119895

+ 119870119889119894119895

)2

[119870119888119894119894

+ 119870119889119894119894

] sdot [119870119888119895119895

+ 119870119889119895119895

]

= (

119873119888

sum

119903=1

119896119888119903

120593119894(119909119903) 120593119895(119909119903) + int

119871

0

119887 (119909) 12059310158401015840

119894(119909) 12059310158401015840

119895(119909) 119889119909)

2

sdot ((int

119871

0

119887 (119909) [12059310158401015840

119894(119909)]2

119889119909 +

119873

sum

119903=1

119896119888119903

120593119894

2(119909119903))

sdot (

119873119888

sum

119903=1

119896119888119903

120593119895

2(119909119903) + int

119871

0

119887 (119909) [12059310158401015840

119895(119909)]2

119889119909))

minus1

(24)

It is noteworthy that 120573119894119895assumes high values in the case of

intermediate supports with a relatively high stiffness com-pared to the deck stiffness Conversely it assumes lower andlower values for homogenous deck and support propertiesand increasing number of spans since the behaviour tendsto that of a beam on continuous elastic restraints for which120573119894119895

= 0An approximation often introduced for practical pur-

poses [13 16] is to disregard the off-diagonal coupling terms

6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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VLSI Design

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Shock and Vibration

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

of the modal damping ratios are obtained by consideringthe real undamped modes and the diagonal terms of themodal damping matrix Franchin et al [20] have analyzedsome realistic bridge models by showing that this decouplingapproximation gives also quite accurate estimates of theresponse to a seismic input as compared to the rigorousresponse estimates obtained through complex modal anal-ysis With reference to the specific case of PRSI bridgescharacterized by deformation shapes and dynamic propertiesvery different from those of fully isolated bridges the workof Tubaldi and DallrsquoAsta [9] has addressed the issue ofnonclassical damping within the context of the free-vibrationresponseThe authors have observed that nonclassical damp-ing influences differently the various response parametersrelevant for the performance assessment (ie the transversedisplacement shape is less affected than the bending momentdemand by the damping nonproportionality) However theeffects of the decoupling approximation on the evaluation ofthe seismic response of the proposed PRSI bridgemodel havenot been investigated yet This issue becomes of particularrelevance in consequence of the numerous studies on PRSIbridges that completely disregard nonclassical damping [6ndash8] Thus a closer examination is still required to ensurewhether the use of proportionally damped models providesacceptable estimates of the seismic response of these systems

Another approximation often introduced in the analysisof PRSI bridges concerns the transverse deformation shapeIn this regard many studies are based on the assumption of aprefixed sinusoidal vibration shape [6 8 10]This assumptionpermits deriving analytically the properties of a generalizedSDOF system equivalent to the bridge and estimating thesystem response by expressing the seismic demand in termsof a response spectrum reduced to account for the systemcomposite damping ratio In Tubaldi and DallrsquoAsta [8 9] itis shown that the sinusoidal approximation of the transversedisplacements may be accurate for PRSI bridges if thefollowing conditions are met (a) the superstructure stiffnessis significantly higher than the pier stiffness (b) the variationsof mass and stiffness of the deck and of the supports arenot significant (c) the span number is high and (d) thedisplacement field is dominated by the first vibration modeHowever even if the displacements are well described by asinusoidal shape other response parameters of interest forthe performance assessment such as the transverse bendingmoments may not exhibit a sinusoidal shape Thus furtherinvestigations are required to estimate the error arising dueto this approximation

The aim of this study is to develop an analytical formu-lation of the seismic problem of PRSI bridges modelled asnonclassically damped continuous systems and a rigoroussolution technique based on the complexmode superposition(CMS) method [12 21ndash24] The application of this methodrequires the derivation of the expression of the modalorthogonality conditions and of the impulsive responsespecific to the problem at hand It permits describing theseismic response in terms of superposition of the complexvibration modes which is particularly useful for this case inconsequence of the relevant contribution of higher modes ofvibration to the response of PRSI bridges [8 10] Moreover

kcr ccr

xr

L

m(x) b(x) c(x)

Figure 1 Analytical model for PRSI bridges

the proposed technique permits testing the accuracy oftwo simplified analysis techniques introduced in this studyand commonly employed for the PRSI bridge analysis Thefirst approximate technique describes the transverse motionthrough a series expansion in terms of the classic modesof vibration obtained by neglecting the damping of theintermediate restraints whereas the second technique isbased on a series expansion in terms of sinusoidal func-tions corresponding to the vibration modes of the systemwithout the intermediate restraintsThe introduction of theseapproximations of the displacement field results in a couplingof the equation of motions projected in the space of theapproximating functions which is neglected in the solutionto simplify the response assessment

A realistic case study is considered and its response to a setof groundmotion records is examined first through the CMSmethod with these two objectives to unveil the characteristicfeatures of the performance of PRSI bridges and to assess theinfluence of higher vibration modes and of the intermediatesupport stiffness and damping on the response of the resistingcomponents Successively the seismic response estimatesaccording to the CMS method are compared with thecorresponding estimates obtained by applying the proposedsimplified techniques in order to evaluate their accuracy andreliability

2 Dynamic Behavior of PRSI Bridges

The PRSI bridge model (Figure 1) consists of beam pinned atthe abutments and resting on discrete viscoelastic supportsrepresenting the pier-bearing systems

Let 1198672[Ω] be the space of functions with square inte-

grable second derivatives in the spatial interval Ω = [0 119871] 119881

the space of transverse displacement functions satisfying thekinematic boundary conditions (eg 119881 = V(119909) isin 119867

2[Ω]

V(0) = V(119871) = 0) and 119906(119909 119905) isin 119880 sube 1198622(119881 [1199050 1199051])

the motion defined in the time interval considered [1199050 1199051]

belonging to the space of continuous functions1198622 and known

at the initial instant together with its time derivative (initialconditions)The differential dynamic problem can be derivedfrom the DrsquoAlembert principle [25] and expressed in thefollowing form

int

119871

0

119898 (119909) (119909 119905) 120578 (119909) 119889119909 + int

119871

0

119888 (119909) (119909 119905) 120578 (119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

(119909119903 119905) 120578 (119909

119903) + int

119871

0

119887 (119909) 11990610158401015840

(119909 119905) 12057810158401015840

(119909) 119889119909

Shock and Vibration 3

+

119873119888

sum

119903=1

119896119888119903

119906 (119909119903 119905) 120578 (119909

119903) = minus int

119871

0

119898 (119909) 119892

(119905) 120578 (119909) 119889119909

forall120578 isin 119881 forall119905 isin [1199050 1199051]

(1)

where 120578 isin 119881 denotes a virtual displacement consistentwith the geometric restrains 119873

119888denotes the number of

intermediate supports prime denotes differentiation withrespect to 119909 and dot differentiation with respect to time 119905

The piecewise continuous functions 119898(119909) 119887(119909) and 119888(119909)

denote the mass per unit length the transverse bendingstiffness per unit length and the deck distributed dampingconstant The constants 119896

119888119903and 119888

119888119903are the stiffness and

damping constant of the viscoelastic support located atthe 119903th position = 119909

119903 while

119892(119905) denotes the ground

accelerationThe local form of the problem is obtained by integrating

by parts (1) and can be formally written as

119872 (119909 119905) + 119862 (119909 119905) + 119870119906 (119909 119905) = minus119872119892

(119905)

119906101584010158401015840

(119909 119905) 12057810038161003816100381610038161003816

119871

0= 0 119906

10158401015840(119909 119905) 120578

101584010038161003816100381610038161003816

119871

0= 0

(2)

where 119872 119862 and 119870 denote respectively the mass dampingand stiffness operator They are expressed as (120575 is the Diracrsquosdelta function)

119872 = 119898 (119909)

119870 = 119887 (119909)1205974

1205971199094+

119873119888

sum

119903=1

119896119888119903

120575 (119909 minus 119909119903)

119862 = 119888 (119909) +

119873119888

sum

119903=1

119888119888119903

120575 (119909 minus 119909119903)

(3)

3 Eigenvalue Problem for PRSI Bridges

The free-vibrations problem of the beam is obtained byposing

119892= 0 in (2) The corresponding differential

boundary problem is then reduced to an eigenvalue problemsolvable by expressing the transverse displacement 119906(119909 119905) asthe product of a spatial function 120595(119909) and a time-dependentfunction 119885(119905) = 119885

0119890120582119905

119906 (119909 119905) = 120595 (119909) 119885 (119905) (4)

After substituting (3) and (4) into (2) for 119892

= 0 the followingtranscendental equation is obtained

[119898 (119909) 1205822

+ 119888 (119909) 120582] 120595 (119909) + 119887 (119909) 120595119868119881

(119909)

+

119873119888

sum

119903=1

(119896119888119903

+ 119888119888119903

120582) 120575 (119909 minus 119909119903) 120595 (119909) = 0

(5)

Equation (5) is satisfied by an infinite number of eigenvaluesand eigenvectors that occur in complex conjugate pairs [26]The 119894th eigenvalue 120582

119894contains information about the system

vibration frequency and damping while the 119894th eigenvector120595119894(119909) is the 119894th vibration shapeThe solution of the eigenvalue

problem for constant deck properties is briefly recalled inthe appendix The orthogonality conditions for the complexmodes are [9]

(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909 +

119873119888

sum

119903=1

119888119888119903

120595119894(119909119903) 120595119895(119909119903) = 0

(6)

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 minus 120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(7)

4 Seismic Response of PRSI BridgesBased on CMS Method

41 Series Expansion of the Response In the CMS methodthe displacement of the beam is expanded as a series of thecomplex vibration modes as

119906 (119909 119905) =

infin

sum

119894=1

120595119894(119909) 119902119894(119905) (8)

where 120595119894(119909) = 119894th complex modal shape and 119902

119894(119905) = 119894th

complex generalized coordinate It is noteworthy that inpractical applications the series is truncated at the term 119873

119898

Substituting (8) into (1) written for 120578(119909) = 120595119895(119909) one

obtains

infin

sum

119894=1

[int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909] 119902

119894(119905)

+

infin

sum

119894=1

[int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

+

infin

sum

119894=1

[int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(9)

4 Shock and Vibration

Upon substitution of (7) into (9) the following expression isobtained

infin

sum

119894=1

119902119894(119905) [int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

+

infin

sum

119894=1

119902119894(119905) [int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

+

infin

sum

119894=1

119902119894(119905) [120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(10)

42 Response to Impulsive Loading For 119892(119905) = 120575(119905) the

generalized coordinate assumes the form [22]

119902119894(119905) = 119861

119894119890120582119894119905 (11)

with 119902119894(119905) = 120582

119894119902119894(119905) and 119902

119894(119905) = 120582

119894

2119902119894(119905)

After substituting into (10) one obtains

infin

sum

119894=1

119902119894(119905)

120582119894

[(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

= minus120575 (119905) int

119871

0

119898 (119909) 120595119895(119909) 119889119909

(12)

It can be noted that the right-hand side of (12) vanishesfor 120582119894

= 120582119895by virtue of (6) Thus the problem can be

diagonalized and the 119894th decoupled equation reads as follows

2119894

119902119894(119905) + 119862

119894119902119894(119905) = minus

119894120575 (119905) (13)

where 119894

= int119871

0119898(119909)120595

119894

2(119909)119889119909 119862

119894= int119871

0119888(119909)120595

119894

2(119909)119889119909 +

sum119873119888

119903=1119888119888119903

120595119894

2(119909119888119903

) and 119894= int119871

0119898(119909)120595

119894(119909)119889119909

By assuming that the system is at rest for 119905 lt 0 thefollowing equation holds at 119905 = 0

+

2119894

119902119894(0+) + 119862119894119902119894(0+) = minus

119894 (14)

Thus since 119902119894(0+) = 119861

119894and 119902

119894(0+) = 120582119861

119894 one can finally

express 119861119894as

119861119894=

119894

2119894120582119894+ 119862119894

= minus (int

119871

0

119898 (119909) 120595119894(119909) 119889119909)

sdot (2120582119894int

119871

0

119898 (119909) 120595119894

2(119909) 119889119909 + int

119871

0

119888 (119909) 120595119894

2(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894

2(119909119888119903

))

minus1

(15)

The corresponding expression of the complex modal impulseresponse function is ℎ

119894

119888(119909 119905) = 119861

119894120595119894(119909)119890120582119894119905 and the sum

of the contribution to the complex modal impulse responsefunction of the 119894th mode and of its complex conjugate yieldsa real function ℎ

119894(119909 119905) which may be expressed as

ℎ119894(119909 119905) = 119861

119894120595119894(119909) 119890120582119894119905+ 119861119894120595119894(119909) 119890120582119894119905

= 120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 ℎ119894(119905) + 120573

119894(119909) ℎ119894(119905)

(16)

where 120572119894(119909) = 120585

119894120573119894(119909) minus radic1 minus 120585

119894

2120574119894(119909) 120573

119894(119909) = 2Re[119861

119894120595119894]

120574119894(119909) = 2 Im[119861

119894120595119894] and ℎ

119894(119905) denotes the impulse response

function of a SDOF system with natural frequency 1205960119894

= |120582119894|

damping ratio 120585119894

= minusRe(120582119894)|120582119894| and damped frequency

120596119889119894

= 1205960119894

radic1 minus 120585119894

2 whose expression is

ℎ119894(119905) =

1

120596119889119894

119890minus1205851198941205960119894119905 sin (120596

119889119894119905) (17)

43 CMS Method for Seismic Response Assessment By tak-ing advantage of the derived closed-form expression ofthe impulse response function the seismic input

119892(119905) is

expressed as a sum of Delta Dirac functions as follows

119892

(119905) = int

119905

0

119892

(120591) 120575 (119905 minus 120591) 119889120591 (18)

The seismic displacement response is then expressed in termsof superposition of modal impulse responses

119906 (119909 119905) =

infin

sum

119894=1

int

119905

0

119892

(120591) ℎ119894(119909 119905 minus 120591) 119889120591

=

119873119898

sum

119894=1

120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 119863119894(119905) + 120573

119894(119909) 119894(119905)

(19)

where 119863119894(119905) and

119894(119905) denote the response of the oscillator

with natural frequency 1205960119894and damping ratio 120585

119894 subjected to

the seismic input 119892(119905)

It is noteworthy that in the case of 119888119888119903

= 0 correspond-ing to intermediate supports with no damping the system

Shock and Vibration 5

becomes classically damped and one obtains 120582119894

= minus1205851198941205960119894

+

1198941205960119894

radic1 minus 120585119894

2 120585119894

= 119888119889(21205960119894

119898119889) 119861119894

= minus120588119894119894(21205960119894

radic1 minus 120585119894

2)

120572119894

= 1205881198941205960119894 120573119894

= 0 and 120574119894

= minus120588119894(1205960119894

radic1 minus 120585119894

2) where 120588

119894=

119894th is the real mode participation factor Thus (19) reduces tothe well-known expression

119906 (119909 119905) =

infin

sum

119894=1

120588119894119863119894(119905) (20)

The expression of the other quantities that can be of interestfor the seismic performance assessment of PRSI bridges canbe derived by differentiating (19) In particular the abutmentreactions are obtained as 119877

119886119887(0 119905) = minus119887(0)119906

101584010158401015840(0 119905) and

119877119886119887

(119871 119905) = minus119887(119871)119906101584010158401015840

(119871 119905) the 119903th pier reaction is obtainedas 119877119901119903

= 119896119888119903

119906(119909119903 119905) + 119888

119888119903(119909119903 119905) for 119903 = 1 2 119873

119888 and

the transverse bending moments are obtained as 119872(119909 119905) =

minus119887(119909)11990610158401015840

(119909 119905)

5 Simplified Methods forSeismic Response Assessment

In this paragraph two simplified approaches often employedfor the seismic analysis of structural systems are intro-duced and discussed These approaches are both based onthe assumed modes method [27 28] and entail using in(8) a complete set of approximating real-valued functions(denoted as 120578

119894(119909) for 119894 = 1 2 infin) instead of the exact

complex vibration modes for describing the displacementfieldThe two analysis techniques differ for the approximatingfunction employed The use of these functions leads to asystem of coupled Galerkin equations [27] and the generic119894th equation reads as follows

infin

sum

119895=1

119872119894119895

119902119895(119905) +

infin

sum

119895=1

(119862119888119894119895

+ 119862119889119894119895

) 119902119895(119905)

+

infin

sum

119895=1

(119870119888119894119895

+ 119870119889119894119895

) 119902119895(119905) = minus119872

119901119894119892

(119905)

(21)

where

119872119894119895

= int

119871

0

119898 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119872119901119894

= int

119871

0

119898 (119909) 120578119894(119909) 119889119909

119870119888119894119895

=

119873119888

sum

119903=1

119896119888119903

120578119894(119909119903) 120578119895(119909119903)

119870119889119894119895

= int

119871

0

119887 (119909) 12057810158401015840

119894(119909) 12057810158401015840

119895(119909) 119889119909

119862119888119894119895

=

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903)

119862119889119894119895

= int

119871

0

119888 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119894 119895 = 1 2 infin

(22)

The coupling between the generalized responses 119902119894(119909) and

119902119895(119909) may be due to nonzero values of the ldquonondiagonalrdquo

damping and stiffness terms 119862119888119894119895

and 119870119888119894119895 for 119894 = 119895 In order

to evaluate the extent of coupling due to damping the index120572119894119895is introduced whose definition is [13]

120572119894119895

=

(119862119888119894119895

+ 119862119889119894119895

)2

[119862119888119894119894

+ 119862119889119894119894

] sdot [119862119888119895119895

+ 119862119889119895119895

]

= (

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903) + int

119871

0

119888119889

(119909) 120578119894(119909) 120578119895(119909) 119889119909)

2

sdot ((

119873119888

sum

119903=1

119888119888119903

120578119894

2(119909119903) + int

119871

0

119888 (119909) 120578119894

2(119909) 119889119909)

sdot (

119873119888

sum

119903=1

119888119888119903

120578119895

2(119909119903) + int

119871

0

119888 (119909) 120578119895

2(119909) 119889119909))

minus1

(23)

This index assumes high values for intermediate supportswith high dissipation capacity while in the case of deck andsupports with homogeneous properties it assumes lower andlower values for increasing number of supports since thebehaviour tends to that of a beam on continuous viscoelasticrestraints [9]

A second coupling index is defined for the stiffness termwhose expression is

120573119894119895

=

(119870119888119894119895

+ 119870119889119894119895

)2

[119870119888119894119894

+ 119870119889119894119894

] sdot [119870119888119895119895

+ 119870119889119895119895

]

= (

119873119888

sum

119903=1

119896119888119903

120593119894(119909119903) 120593119895(119909119903) + int

119871

0

119887 (119909) 12059310158401015840

119894(119909) 12059310158401015840

119895(119909) 119889119909)

2

sdot ((int

119871

0

119887 (119909) [12059310158401015840

119894(119909)]2

119889119909 +

119873

sum

119903=1

119896119888119903

120593119894

2(119909119903))

sdot (

119873119888

sum

119903=1

119896119888119903

120593119895

2(119909119903) + int

119871

0

119887 (119909) [12059310158401015840

119895(119909)]2

119889119909))

minus1

(24)

It is noteworthy that 120573119894119895assumes high values in the case of

intermediate supports with a relatively high stiffness com-pared to the deck stiffness Conversely it assumes lower andlower values for homogenous deck and support propertiesand increasing number of spans since the behaviour tendsto that of a beam on continuous elastic restraints for which120573119894119895

= 0An approximation often introduced for practical pur-

poses [13 16] is to disregard the off-diagonal coupling terms

6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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Shock and Vibration 3

+

119873119888

sum

119903=1

119896119888119903

119906 (119909119903 119905) 120578 (119909

119903) = minus int

119871

0

119898 (119909) 119892

(119905) 120578 (119909) 119889119909

forall120578 isin 119881 forall119905 isin [1199050 1199051]

(1)

where 120578 isin 119881 denotes a virtual displacement consistentwith the geometric restrains 119873

119888denotes the number of

intermediate supports prime denotes differentiation withrespect to 119909 and dot differentiation with respect to time 119905

The piecewise continuous functions 119898(119909) 119887(119909) and 119888(119909)

denote the mass per unit length the transverse bendingstiffness per unit length and the deck distributed dampingconstant The constants 119896

119888119903and 119888

119888119903are the stiffness and

damping constant of the viscoelastic support located atthe 119903th position = 119909

119903 while

119892(119905) denotes the ground

accelerationThe local form of the problem is obtained by integrating

by parts (1) and can be formally written as

119872 (119909 119905) + 119862 (119909 119905) + 119870119906 (119909 119905) = minus119872119892

(119905)

119906101584010158401015840

(119909 119905) 12057810038161003816100381610038161003816

119871

0= 0 119906

10158401015840(119909 119905) 120578

101584010038161003816100381610038161003816

119871

0= 0

(2)

where 119872 119862 and 119870 denote respectively the mass dampingand stiffness operator They are expressed as (120575 is the Diracrsquosdelta function)

119872 = 119898 (119909)

119870 = 119887 (119909)1205974

1205971199094+

119873119888

sum

119903=1

119896119888119903

120575 (119909 minus 119909119903)

119862 = 119888 (119909) +

119873119888

sum

119903=1

119888119888119903

120575 (119909 minus 119909119903)

(3)

3 Eigenvalue Problem for PRSI Bridges

The free-vibrations problem of the beam is obtained byposing

119892= 0 in (2) The corresponding differential

boundary problem is then reduced to an eigenvalue problemsolvable by expressing the transverse displacement 119906(119909 119905) asthe product of a spatial function 120595(119909) and a time-dependentfunction 119885(119905) = 119885

0119890120582119905

119906 (119909 119905) = 120595 (119909) 119885 (119905) (4)

After substituting (3) and (4) into (2) for 119892

= 0 the followingtranscendental equation is obtained

[119898 (119909) 1205822

+ 119888 (119909) 120582] 120595 (119909) + 119887 (119909) 120595119868119881

(119909)

+

119873119888

sum

119903=1

(119896119888119903

+ 119888119888119903

120582) 120575 (119909 minus 119909119903) 120595 (119909) = 0

(5)

Equation (5) is satisfied by an infinite number of eigenvaluesand eigenvectors that occur in complex conjugate pairs [26]The 119894th eigenvalue 120582

119894contains information about the system

vibration frequency and damping while the 119894th eigenvector120595119894(119909) is the 119894th vibration shapeThe solution of the eigenvalue

problem for constant deck properties is briefly recalled inthe appendix The orthogonality conditions for the complexmodes are [9]

(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909 +

119873119888

sum

119903=1

119888119888119903

120595119894(119909119903) 120595119895(119909119903) = 0

(6)

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 minus 120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(7)

4 Seismic Response of PRSI BridgesBased on CMS Method

41 Series Expansion of the Response In the CMS methodthe displacement of the beam is expanded as a series of thecomplex vibration modes as

119906 (119909 119905) =

infin

sum

119894=1

120595119894(119909) 119902119894(119905) (8)

where 120595119894(119909) = 119894th complex modal shape and 119902

119894(119905) = 119894th

complex generalized coordinate It is noteworthy that inpractical applications the series is truncated at the term 119873

119898

Substituting (8) into (1) written for 120578(119909) = 120595119895(119909) one

obtains

infin

sum

119894=1

[int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909] 119902

119894(119905)

+

infin

sum

119894=1

[int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

+

infin

sum

119894=1

[int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909

+

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)] 119902119894(119905)

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(9)

4 Shock and Vibration

Upon substitution of (7) into (9) the following expression isobtained

infin

sum

119894=1

119902119894(119905) [int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

+

infin

sum

119894=1

119902119894(119905) [int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

+

infin

sum

119894=1

119902119894(119905) [120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(10)

42 Response to Impulsive Loading For 119892(119905) = 120575(119905) the

generalized coordinate assumes the form [22]

119902119894(119905) = 119861

119894119890120582119894119905 (11)

with 119902119894(119905) = 120582

119894119902119894(119905) and 119902

119894(119905) = 120582

119894

2119902119894(119905)

After substituting into (10) one obtains

infin

sum

119894=1

119902119894(119905)

120582119894

[(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

= minus120575 (119905) int

119871

0

119898 (119909) 120595119895(119909) 119889119909

(12)

It can be noted that the right-hand side of (12) vanishesfor 120582119894

= 120582119895by virtue of (6) Thus the problem can be

diagonalized and the 119894th decoupled equation reads as follows

2119894

119902119894(119905) + 119862

119894119902119894(119905) = minus

119894120575 (119905) (13)

where 119894

= int119871

0119898(119909)120595

119894

2(119909)119889119909 119862

119894= int119871

0119888(119909)120595

119894

2(119909)119889119909 +

sum119873119888

119903=1119888119888119903

120595119894

2(119909119888119903

) and 119894= int119871

0119898(119909)120595

119894(119909)119889119909

By assuming that the system is at rest for 119905 lt 0 thefollowing equation holds at 119905 = 0

+

2119894

119902119894(0+) + 119862119894119902119894(0+) = minus

119894 (14)

Thus since 119902119894(0+) = 119861

119894and 119902

119894(0+) = 120582119861

119894 one can finally

express 119861119894as

119861119894=

119894

2119894120582119894+ 119862119894

= minus (int

119871

0

119898 (119909) 120595119894(119909) 119889119909)

sdot (2120582119894int

119871

0

119898 (119909) 120595119894

2(119909) 119889119909 + int

119871

0

119888 (119909) 120595119894

2(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894

2(119909119888119903

))

minus1

(15)

The corresponding expression of the complex modal impulseresponse function is ℎ

119894

119888(119909 119905) = 119861

119894120595119894(119909)119890120582119894119905 and the sum

of the contribution to the complex modal impulse responsefunction of the 119894th mode and of its complex conjugate yieldsa real function ℎ

119894(119909 119905) which may be expressed as

ℎ119894(119909 119905) = 119861

119894120595119894(119909) 119890120582119894119905+ 119861119894120595119894(119909) 119890120582119894119905

= 120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 ℎ119894(119905) + 120573

119894(119909) ℎ119894(119905)

(16)

where 120572119894(119909) = 120585

119894120573119894(119909) minus radic1 minus 120585

119894

2120574119894(119909) 120573

119894(119909) = 2Re[119861

119894120595119894]

120574119894(119909) = 2 Im[119861

119894120595119894] and ℎ

119894(119905) denotes the impulse response

function of a SDOF system with natural frequency 1205960119894

= |120582119894|

damping ratio 120585119894

= minusRe(120582119894)|120582119894| and damped frequency

120596119889119894

= 1205960119894

radic1 minus 120585119894

2 whose expression is

ℎ119894(119905) =

1

120596119889119894

119890minus1205851198941205960119894119905 sin (120596

119889119894119905) (17)

43 CMS Method for Seismic Response Assessment By tak-ing advantage of the derived closed-form expression ofthe impulse response function the seismic input

119892(119905) is

expressed as a sum of Delta Dirac functions as follows

119892

(119905) = int

119905

0

119892

(120591) 120575 (119905 minus 120591) 119889120591 (18)

The seismic displacement response is then expressed in termsof superposition of modal impulse responses

119906 (119909 119905) =

infin

sum

119894=1

int

119905

0

119892

(120591) ℎ119894(119909 119905 minus 120591) 119889120591

=

119873119898

sum

119894=1

120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 119863119894(119905) + 120573

119894(119909) 119894(119905)

(19)

where 119863119894(119905) and

119894(119905) denote the response of the oscillator

with natural frequency 1205960119894and damping ratio 120585

119894 subjected to

the seismic input 119892(119905)

It is noteworthy that in the case of 119888119888119903

= 0 correspond-ing to intermediate supports with no damping the system

Shock and Vibration 5

becomes classically damped and one obtains 120582119894

= minus1205851198941205960119894

+

1198941205960119894

radic1 minus 120585119894

2 120585119894

= 119888119889(21205960119894

119898119889) 119861119894

= minus120588119894119894(21205960119894

radic1 minus 120585119894

2)

120572119894

= 1205881198941205960119894 120573119894

= 0 and 120574119894

= minus120588119894(1205960119894

radic1 minus 120585119894

2) where 120588

119894=

119894th is the real mode participation factor Thus (19) reduces tothe well-known expression

119906 (119909 119905) =

infin

sum

119894=1

120588119894119863119894(119905) (20)

The expression of the other quantities that can be of interestfor the seismic performance assessment of PRSI bridges canbe derived by differentiating (19) In particular the abutmentreactions are obtained as 119877

119886119887(0 119905) = minus119887(0)119906

101584010158401015840(0 119905) and

119877119886119887

(119871 119905) = minus119887(119871)119906101584010158401015840

(119871 119905) the 119903th pier reaction is obtainedas 119877119901119903

= 119896119888119903

119906(119909119903 119905) + 119888

119888119903(119909119903 119905) for 119903 = 1 2 119873

119888 and

the transverse bending moments are obtained as 119872(119909 119905) =

minus119887(119909)11990610158401015840

(119909 119905)

5 Simplified Methods forSeismic Response Assessment

In this paragraph two simplified approaches often employedfor the seismic analysis of structural systems are intro-duced and discussed These approaches are both based onthe assumed modes method [27 28] and entail using in(8) a complete set of approximating real-valued functions(denoted as 120578

119894(119909) for 119894 = 1 2 infin) instead of the exact

complex vibration modes for describing the displacementfieldThe two analysis techniques differ for the approximatingfunction employed The use of these functions leads to asystem of coupled Galerkin equations [27] and the generic119894th equation reads as follows

infin

sum

119895=1

119872119894119895

119902119895(119905) +

infin

sum

119895=1

(119862119888119894119895

+ 119862119889119894119895

) 119902119895(119905)

+

infin

sum

119895=1

(119870119888119894119895

+ 119870119889119894119895

) 119902119895(119905) = minus119872

119901119894119892

(119905)

(21)

where

119872119894119895

= int

119871

0

119898 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119872119901119894

= int

119871

0

119898 (119909) 120578119894(119909) 119889119909

119870119888119894119895

=

119873119888

sum

119903=1

119896119888119903

120578119894(119909119903) 120578119895(119909119903)

119870119889119894119895

= int

119871

0

119887 (119909) 12057810158401015840

119894(119909) 12057810158401015840

119895(119909) 119889119909

119862119888119894119895

=

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903)

119862119889119894119895

= int

119871

0

119888 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119894 119895 = 1 2 infin

(22)

The coupling between the generalized responses 119902119894(119909) and

119902119895(119909) may be due to nonzero values of the ldquonondiagonalrdquo

damping and stiffness terms 119862119888119894119895

and 119870119888119894119895 for 119894 = 119895 In order

to evaluate the extent of coupling due to damping the index120572119894119895is introduced whose definition is [13]

120572119894119895

=

(119862119888119894119895

+ 119862119889119894119895

)2

[119862119888119894119894

+ 119862119889119894119894

] sdot [119862119888119895119895

+ 119862119889119895119895

]

= (

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903) + int

119871

0

119888119889

(119909) 120578119894(119909) 120578119895(119909) 119889119909)

2

sdot ((

119873119888

sum

119903=1

119888119888119903

120578119894

2(119909119903) + int

119871

0

119888 (119909) 120578119894

2(119909) 119889119909)

sdot (

119873119888

sum

119903=1

119888119888119903

120578119895

2(119909119903) + int

119871

0

119888 (119909) 120578119895

2(119909) 119889119909))

minus1

(23)

This index assumes high values for intermediate supportswith high dissipation capacity while in the case of deck andsupports with homogeneous properties it assumes lower andlower values for increasing number of supports since thebehaviour tends to that of a beam on continuous viscoelasticrestraints [9]

A second coupling index is defined for the stiffness termwhose expression is

120573119894119895

=

(119870119888119894119895

+ 119870119889119894119895

)2

[119870119888119894119894

+ 119870119889119894119894

] sdot [119870119888119895119895

+ 119870119889119895119895

]

= (

119873119888

sum

119903=1

119896119888119903

120593119894(119909119903) 120593119895(119909119903) + int

119871

0

119887 (119909) 12059310158401015840

119894(119909) 12059310158401015840

119895(119909) 119889119909)

2

sdot ((int

119871

0

119887 (119909) [12059310158401015840

119894(119909)]2

119889119909 +

119873

sum

119903=1

119896119888119903

120593119894

2(119909119903))

sdot (

119873119888

sum

119903=1

119896119888119903

120593119895

2(119909119903) + int

119871

0

119887 (119909) [12059310158401015840

119895(119909)]2

119889119909))

minus1

(24)

It is noteworthy that 120573119894119895assumes high values in the case of

intermediate supports with a relatively high stiffness com-pared to the deck stiffness Conversely it assumes lower andlower values for homogenous deck and support propertiesand increasing number of spans since the behaviour tendsto that of a beam on continuous elastic restraints for which120573119894119895

= 0An approximation often introduced for practical pur-

poses [13 16] is to disregard the off-diagonal coupling terms

6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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Shock and Vibration

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International Journal of

4 Shock and Vibration

Upon substitution of (7) into (9) the following expression isobtained

infin

sum

119894=1

119902119894(119905) [int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

+

infin

sum

119894=1

119902119894(119905) [int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

+

infin

sum

119894=1

119902119894(119905) [120582

119894120582119895int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909]

= minus int

119871

0

119898 (119909) 120595119895(119909) 119892

(119905) 119889119909

(10)

42 Response to Impulsive Loading For 119892(119905) = 120575(119905) the

generalized coordinate assumes the form [22]

119902119894(119905) = 119861

119894119890120582119894119905 (11)

with 119902119894(119905) = 120582

119894119902119894(119905) and 119902

119894(119905) = 120582

119894

2119902119894(119905)

After substituting into (10) one obtains

infin

sum

119894=1

119902119894(119905)

120582119894

[(120582119894+ 120582119895) int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+ int

119871

0

119888 (119909) 120595119894(119909) 120595119895(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

)]

= minus120575 (119905) int

119871

0

119898 (119909) 120595119895(119909) 119889119909

(12)

It can be noted that the right-hand side of (12) vanishesfor 120582119894

= 120582119895by virtue of (6) Thus the problem can be

diagonalized and the 119894th decoupled equation reads as follows

2119894

119902119894(119905) + 119862

119894119902119894(119905) = minus

119894120575 (119905) (13)

where 119894

= int119871

0119898(119909)120595

119894

2(119909)119889119909 119862

119894= int119871

0119888(119909)120595

119894

2(119909)119889119909 +

sum119873119888

119903=1119888119888119903

120595119894

2(119909119888119903

) and 119894= int119871

0119898(119909)120595

119894(119909)119889119909

By assuming that the system is at rest for 119905 lt 0 thefollowing equation holds at 119905 = 0

+

2119894

119902119894(0+) + 119862119894119902119894(0+) = minus

119894 (14)

Thus since 119902119894(0+) = 119861

119894and 119902

119894(0+) = 120582119861

119894 one can finally

express 119861119894as

119861119894=

119894

2119894120582119894+ 119862119894

= minus (int

119871

0

119898 (119909) 120595119894(119909) 119889119909)

sdot (2120582119894int

119871

0

119898 (119909) 120595119894

2(119909) 119889119909 + int

119871

0

119888 (119909) 120595119894

2(119909) 119889119909

+

119873119888

sum

119903=1

119888119888119903

120595119894

2(119909119888119903

))

minus1

(15)

The corresponding expression of the complex modal impulseresponse function is ℎ

119894

119888(119909 119905) = 119861

119894120595119894(119909)119890120582119894119905 and the sum

of the contribution to the complex modal impulse responsefunction of the 119894th mode and of its complex conjugate yieldsa real function ℎ

119894(119909 119905) which may be expressed as

ℎ119894(119909 119905) = 119861

119894120595119894(119909) 119890120582119894119905+ 119861119894120595119894(119909) 119890120582119894119905

= 120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 ℎ119894(119905) + 120573

119894(119909) ℎ119894(119905)

(16)

where 120572119894(119909) = 120585

119894120573119894(119909) minus radic1 minus 120585

119894

2120574119894(119909) 120573

119894(119909) = 2Re[119861

119894120595119894]

120574119894(119909) = 2 Im[119861

119894120595119894] and ℎ

119894(119905) denotes the impulse response

function of a SDOF system with natural frequency 1205960119894

= |120582119894|

damping ratio 120585119894

= minusRe(120582119894)|120582119894| and damped frequency

120596119889119894

= 1205960119894

radic1 minus 120585119894

2 whose expression is

ℎ119894(119905) =

1

120596119889119894

119890minus1205851198941205960119894119905 sin (120596

119889119894119905) (17)

43 CMS Method for Seismic Response Assessment By tak-ing advantage of the derived closed-form expression ofthe impulse response function the seismic input

119892(119905) is

expressed as a sum of Delta Dirac functions as follows

119892

(119905) = int

119905

0

119892

(120591) 120575 (119905 minus 120591) 119889120591 (18)

The seismic displacement response is then expressed in termsof superposition of modal impulse responses

119906 (119909 119905) =

infin

sum

119894=1

int

119905

0

119892

(120591) ℎ119894(119909 119905 minus 120591) 119889120591

=

119873119898

sum

119894=1

120572119894(119909)

10038161003816100381610038161205821198941003816100381610038161003816 119863119894(119905) + 120573

119894(119909) 119894(119905)

(19)

where 119863119894(119905) and

119894(119905) denote the response of the oscillator

with natural frequency 1205960119894and damping ratio 120585

119894 subjected to

the seismic input 119892(119905)

It is noteworthy that in the case of 119888119888119903

= 0 correspond-ing to intermediate supports with no damping the system

Shock and Vibration 5

becomes classically damped and one obtains 120582119894

= minus1205851198941205960119894

+

1198941205960119894

radic1 minus 120585119894

2 120585119894

= 119888119889(21205960119894

119898119889) 119861119894

= minus120588119894119894(21205960119894

radic1 minus 120585119894

2)

120572119894

= 1205881198941205960119894 120573119894

= 0 and 120574119894

= minus120588119894(1205960119894

radic1 minus 120585119894

2) where 120588

119894=

119894th is the real mode participation factor Thus (19) reduces tothe well-known expression

119906 (119909 119905) =

infin

sum

119894=1

120588119894119863119894(119905) (20)

The expression of the other quantities that can be of interestfor the seismic performance assessment of PRSI bridges canbe derived by differentiating (19) In particular the abutmentreactions are obtained as 119877

119886119887(0 119905) = minus119887(0)119906

101584010158401015840(0 119905) and

119877119886119887

(119871 119905) = minus119887(119871)119906101584010158401015840

(119871 119905) the 119903th pier reaction is obtainedas 119877119901119903

= 119896119888119903

119906(119909119903 119905) + 119888

119888119903(119909119903 119905) for 119903 = 1 2 119873

119888 and

the transverse bending moments are obtained as 119872(119909 119905) =

minus119887(119909)11990610158401015840

(119909 119905)

5 Simplified Methods forSeismic Response Assessment

In this paragraph two simplified approaches often employedfor the seismic analysis of structural systems are intro-duced and discussed These approaches are both based onthe assumed modes method [27 28] and entail using in(8) a complete set of approximating real-valued functions(denoted as 120578

119894(119909) for 119894 = 1 2 infin) instead of the exact

complex vibration modes for describing the displacementfieldThe two analysis techniques differ for the approximatingfunction employed The use of these functions leads to asystem of coupled Galerkin equations [27] and the generic119894th equation reads as follows

infin

sum

119895=1

119872119894119895

119902119895(119905) +

infin

sum

119895=1

(119862119888119894119895

+ 119862119889119894119895

) 119902119895(119905)

+

infin

sum

119895=1

(119870119888119894119895

+ 119870119889119894119895

) 119902119895(119905) = minus119872

119901119894119892

(119905)

(21)

where

119872119894119895

= int

119871

0

119898 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119872119901119894

= int

119871

0

119898 (119909) 120578119894(119909) 119889119909

119870119888119894119895

=

119873119888

sum

119903=1

119896119888119903

120578119894(119909119903) 120578119895(119909119903)

119870119889119894119895

= int

119871

0

119887 (119909) 12057810158401015840

119894(119909) 12057810158401015840

119895(119909) 119889119909

119862119888119894119895

=

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903)

119862119889119894119895

= int

119871

0

119888 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119894 119895 = 1 2 infin

(22)

The coupling between the generalized responses 119902119894(119909) and

119902119895(119909) may be due to nonzero values of the ldquonondiagonalrdquo

damping and stiffness terms 119862119888119894119895

and 119870119888119894119895 for 119894 = 119895 In order

to evaluate the extent of coupling due to damping the index120572119894119895is introduced whose definition is [13]

120572119894119895

=

(119862119888119894119895

+ 119862119889119894119895

)2

[119862119888119894119894

+ 119862119889119894119894

] sdot [119862119888119895119895

+ 119862119889119895119895

]

= (

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903) + int

119871

0

119888119889

(119909) 120578119894(119909) 120578119895(119909) 119889119909)

2

sdot ((

119873119888

sum

119903=1

119888119888119903

120578119894

2(119909119903) + int

119871

0

119888 (119909) 120578119894

2(119909) 119889119909)

sdot (

119873119888

sum

119903=1

119888119888119903

120578119895

2(119909119903) + int

119871

0

119888 (119909) 120578119895

2(119909) 119889119909))

minus1

(23)

This index assumes high values for intermediate supportswith high dissipation capacity while in the case of deck andsupports with homogeneous properties it assumes lower andlower values for increasing number of supports since thebehaviour tends to that of a beam on continuous viscoelasticrestraints [9]

A second coupling index is defined for the stiffness termwhose expression is

120573119894119895

=

(119870119888119894119895

+ 119870119889119894119895

)2

[119870119888119894119894

+ 119870119889119894119894

] sdot [119870119888119895119895

+ 119870119889119895119895

]

= (

119873119888

sum

119903=1

119896119888119903

120593119894(119909119903) 120593119895(119909119903) + int

119871

0

119887 (119909) 12059310158401015840

119894(119909) 12059310158401015840

119895(119909) 119889119909)

2

sdot ((int

119871

0

119887 (119909) [12059310158401015840

119894(119909)]2

119889119909 +

119873

sum

119903=1

119896119888119903

120593119894

2(119909119903))

sdot (

119873119888

sum

119903=1

119896119888119903

120593119895

2(119909119903) + int

119871

0

119887 (119909) [12059310158401015840

119895(119909)]2

119889119909))

minus1

(24)

It is noteworthy that 120573119894119895assumes high values in the case of

intermediate supports with a relatively high stiffness com-pared to the deck stiffness Conversely it assumes lower andlower values for homogenous deck and support propertiesand increasing number of spans since the behaviour tendsto that of a beam on continuous elastic restraints for which120573119894119895

= 0An approximation often introduced for practical pur-

poses [13 16] is to disregard the off-diagonal coupling terms

6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

becomes classically damped and one obtains 120582119894

= minus1205851198941205960119894

+

1198941205960119894

radic1 minus 120585119894

2 120585119894

= 119888119889(21205960119894

119898119889) 119861119894

= minus120588119894119894(21205960119894

radic1 minus 120585119894

2)

120572119894

= 1205881198941205960119894 120573119894

= 0 and 120574119894

= minus120588119894(1205960119894

radic1 minus 120585119894

2) where 120588

119894=

119894th is the real mode participation factor Thus (19) reduces tothe well-known expression

119906 (119909 119905) =

infin

sum

119894=1

120588119894119863119894(119905) (20)

The expression of the other quantities that can be of interestfor the seismic performance assessment of PRSI bridges canbe derived by differentiating (19) In particular the abutmentreactions are obtained as 119877

119886119887(0 119905) = minus119887(0)119906

101584010158401015840(0 119905) and

119877119886119887

(119871 119905) = minus119887(119871)119906101584010158401015840

(119871 119905) the 119903th pier reaction is obtainedas 119877119901119903

= 119896119888119903

119906(119909119903 119905) + 119888

119888119903(119909119903 119905) for 119903 = 1 2 119873

119888 and

the transverse bending moments are obtained as 119872(119909 119905) =

minus119887(119909)11990610158401015840

(119909 119905)

5 Simplified Methods forSeismic Response Assessment

In this paragraph two simplified approaches often employedfor the seismic analysis of structural systems are intro-duced and discussed These approaches are both based onthe assumed modes method [27 28] and entail using in(8) a complete set of approximating real-valued functions(denoted as 120578

119894(119909) for 119894 = 1 2 infin) instead of the exact

complex vibration modes for describing the displacementfieldThe two analysis techniques differ for the approximatingfunction employed The use of these functions leads to asystem of coupled Galerkin equations [27] and the generic119894th equation reads as follows

infin

sum

119895=1

119872119894119895

119902119895(119905) +

infin

sum

119895=1

(119862119888119894119895

+ 119862119889119894119895

) 119902119895(119905)

+

infin

sum

119895=1

(119870119888119894119895

+ 119870119889119894119895

) 119902119895(119905) = minus119872

119901119894119892

(119905)

(21)

where

119872119894119895

= int

119871

0

119898 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119872119901119894

= int

119871

0

119898 (119909) 120578119894(119909) 119889119909

119870119888119894119895

=

119873119888

sum

119903=1

119896119888119903

120578119894(119909119903) 120578119895(119909119903)

119870119889119894119895

= int

119871

0

119887 (119909) 12057810158401015840

119894(119909) 12057810158401015840

119895(119909) 119889119909

119862119888119894119895

=

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903)

119862119889119894119895

= int

119871

0

119888 (119909) 120578119894(119909) 120578119895(119909) 119889119909

119894 119895 = 1 2 infin

(22)

The coupling between the generalized responses 119902119894(119909) and

119902119895(119909) may be due to nonzero values of the ldquonondiagonalrdquo

damping and stiffness terms 119862119888119894119895

and 119870119888119894119895 for 119894 = 119895 In order

to evaluate the extent of coupling due to damping the index120572119894119895is introduced whose definition is [13]

120572119894119895

=

(119862119888119894119895

+ 119862119889119894119895

)2

[119862119888119894119894

+ 119862119889119894119894

] sdot [119862119888119895119895

+ 119862119889119895119895

]

= (

119873119888

sum

119903=1

119888119888119903

120578119894(119909119903) 120578119895(119909119903) + int

119871

0

119888119889

(119909) 120578119894(119909) 120578119895(119909) 119889119909)

2

sdot ((

119873119888

sum

119903=1

119888119888119903

120578119894

2(119909119903) + int

119871

0

119888 (119909) 120578119894

2(119909) 119889119909)

sdot (

119873119888

sum

119903=1

119888119888119903

120578119895

2(119909119903) + int

119871

0

119888 (119909) 120578119895

2(119909) 119889119909))

minus1

(23)

This index assumes high values for intermediate supportswith high dissipation capacity while in the case of deck andsupports with homogeneous properties it assumes lower andlower values for increasing number of supports since thebehaviour tends to that of a beam on continuous viscoelasticrestraints [9]

A second coupling index is defined for the stiffness termwhose expression is

120573119894119895

=

(119870119888119894119895

+ 119870119889119894119895

)2

[119870119888119894119894

+ 119870119889119894119894

] sdot [119870119888119895119895

+ 119870119889119895119895

]

= (

119873119888

sum

119903=1

119896119888119903

120593119894(119909119903) 120593119895(119909119903) + int

119871

0

119887 (119909) 12059310158401015840

119894(119909) 12059310158401015840

119895(119909) 119889119909)

2

sdot ((int

119871

0

119887 (119909) [12059310158401015840

119894(119909)]2

119889119909 +

119873

sum

119903=1

119896119888119903

120593119894

2(119909119903))

sdot (

119873119888

sum

119903=1

119896119888119903

120593119895

2(119909119903) + int

119871

0

119887 (119909) [12059310158401015840

119895(119909)]2

119889119909))

minus1

(24)

It is noteworthy that 120573119894119895assumes high values in the case of

intermediate supports with a relatively high stiffness com-pared to the deck stiffness Conversely it assumes lower andlower values for homogenous deck and support propertiesand increasing number of spans since the behaviour tendsto that of a beam on continuous elastic restraints for which120573119894119895

= 0An approximation often introduced for practical pur-

poses [13 16] is to disregard the off-diagonal coupling terms

6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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6 Shock and Vibration

L1 L2 L2 L1

Figure 2 Bridge longitudinal profile

to obtain a set of uncoupled equationsThe resulting 119894th equa-tion describes the motion of a SDOF system with mass 119872

119894119894

stiffness 119870119888119894119894

+ 119870119889119894119894

and damping constant 119862119888119894119894

+ 119862119889119894119894

Thustraditional analysis tools available for the seismic analysisof SDOF systems can be employed to compute efficientlythe generalized displacements 119902

119894(119905) and the response 119906(119909 119905)

under the given ground motion excitation

51 RMS Method for Seismic Response Assessment Thismethod referred to as real modes superposition (RMS)method is based on a series expansion of the displacementfield in terms of the real modes of vibration 120601

119894(119909) for

119894 = 1 2 119873119898of the undamped (or classically damped)

structure The calculation of these classic modes of vibrationinvolves solving an eigenvalue problem less computationallydemanding than that required for computing the complexmodes

It is noteworthy that the real modes of vibration areretrieved from the PRSI bridge model of Figure 1 by neglect-ing the damping of the intermediate supports and that theorthogonality conditions for these modes can be obtainedfrom (6) and (7) by posing 119888

119889(119909) = 0 and 119888

119888119903= 0 for

119903 = 1 2 119873119888 leading to

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 +

119873119888

sum

119903=1

119896119888119903

120595119894(119909119888119903

) 120595119895(119909119888119903

) = 0

(25)

The same expressions are obtained also in the case of massproportional damping for the deck that is for 119888

119889(119909) = 120581119898(119909)

where 120581 is a proportionality constantThe orthogonality conditions of (25) result in nonzero

values of 119862119888119894119895

and zero values of 119870119888119894119895 These terms are

discarded to obtain a diagonal problem

52 FTS Method for Seismic Response Assessment In thesecond method referred to as Fourier terms superposition(FTS) method the Fourier sine-only series terms 120593

119894(119909) =

sin(119894120587119909119871) for 119894 = 1 2 119873119898

are employed to describethe motion This method is employed in [8 10] for thedesign of PRSI bridges and is characterized by a very reducedcomputational cost since it avoids recourse to any eigen-value analysis It is noteworthy that the terms of this seriescorrespond to the vibration modes of a simply supportedbeam which coincides with the PRSI bridge model withoutany intermediate support The orthogonality conditions for

these terms derived from (6) and (7) by posing respectively119888119888119903

= 119896119888119903

= 0 for 119903 = 1 2 119873119888 are

int

119871

0

119898 (119909) 120595119894(119909) 120595119895(119909) 119889119909 = 0

int

119871

0

119887 (119909) 12059510158401015840

119894(119909) 12059510158401015840

119895(119909) 119889119909 = 0

(26)

The application of these orthogonality conditions leads to aset of coupled Galerkin equations due to the nonzero terms119862119888119894119894

and 119870119888119894119895

for 119894 = 119895 These out of diagonal terms arediscarded to obtain a diagonal problem

6 Case Study

In this section a realistic PRSI bridge [8] is considered ascase study First the CMS method is applied to the bridgemodel by analyzing its modal properties and the relatedresponse to an impulsive input Successively the seismicresponse is studied and a parametric analysis is carriedout to investigate the influence of the intermediate supportstiffness and damping on the seismic performance of thebridge components Finally the accuracy of the simplifiedanalysis techniques is evaluated by comparing the results ofthe seismic analyses obtained with the CMS method and theRMS and FTS methods

61 Case Study and Seismic Input Description The PRSIbridge (Figure 2) whose properties are taken from [8] con-sists of a four-span continuous steel-concrete superstructure(span lengths 119871

1= 40m and 119871

2= 60m for a total

length 119871 = 200m) and of three isolated reinforced concretepiers The value of the deck transverse stiffness is 119864119868

119889=

11119864 + 09 kNm2 which is an average over the values of119864119868119889that slightly vary along the bridge length due to the

variation of the web and flange thickness The deck massper unit length is equal to 119898

119889= 1624 tonm The circular

frequency corresponding to the first mode of vibration of thesuperstructure vibrating alone with no intermediate supportsis 120596119889

= 203 rads The deck damping constant 119888119889is such that

the firstmode damping factor of the deck vibrating alonewithno intermediate supports is equal to 120585

119889= 002The combined

pier and isolator properties are described by Kelvin modelswhose stiffness and damping constant are respectively 119896

1198882=

205761 kNm and 1198881198882

= 20633 kNm for the central supportand 1198961198881

= 1198961198883

= 350062 kNm and 1198881198881

= 1198881198883

= 32269 kNmfor the other supportsThese support properties are the resultof the design of isolation bearings ensuring the same sheardemand at the base of the piers under the prefixed earthquakeinput

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

0

02

04

06

08

1

12

14

16

18

2

Mean spectrumCode spectrum

S a(T

5

) (g)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0 1 2 3 4

T (s)

(a)

006263 ya004673 ya000535 ya

006328 ya000199 ya006334 ya000594 ya

0

01

02

03

04

05

06

07

Mean spectrum

S d(T

5

) (m

)

0 1 2 3 4

T (s)

(b)

Figure 3 Variation with period 119879 of the pseudo-acceleration response spectrum 119878119886(119879 5) (a) and of the displacement response spectrum

119878119889(119879 5) (b) of natural records

Table 1 Modal properties complex eigenvalues 120582119894 undamped

modal frequencies 1205960119894 and damping ratios 120585

119894

Mode 120582119894[ndash] 120596

0119894[rads] 120585

119894[ndash]

1 minus01723 minus 26165119894 262 006572 minus02196 minus 83574119894 836 002633 minus02834 minus 184170119894 1842 001544 minus01097 minus 325180119894 3252 000345 minus01042 minus 507860119894 5079 00021

The seismic input is described by a set of seven recordscompatible with the Eurocode 8-1 design spectrum corre-sponding to a site with a peak ground acceleration (PGA) of035119878119892where 119878 is the soil factor assumed equal to 115 (groundtype C) and 119892 is the gravity acceleration They have beenselected from the European strong motion database [29] andfulfil the requirements of Eurocode 8 [30] Figure 3(a) showsthe pseudo-acceleration spectrum of the records the meanspectrum and the code spectrum whereas Figure 3(b) showsthe displacement response spectrum of the records and thecorresponding mean spectrum

62 Modal Properties and Impulsive Response Table 1 reportsthe first 5 eigenvalues 120582

119894of the system the corresponding

vibration periods 1198790119894

= 2120587120596119894 and damping ratios 120585

119894 These

modal properties are determined by solving the eigenvalueproblem corresponding to (5) It is noteworthy that the evenmodes of vibration are characterized by an antisymmetricshape and a participating factor equal to zero and thusthey do not affect the seismic response of the consideredconfigurations (uniform support excitation is assumed)

In general the effect of the intermediate restraints onthe dynamic behaviour of the PRSI bridge is to increase thevibration frequency and damping ratio with respect to thecase of the deck vibrating alone without any restraint In factthe fundamental vibration frequency shifts from the value120596119889

= 203 rads to 12059601

= 262 rads corresponding to afirst mode vibration period of about 240 s The first modedamping ratio increases from 120585

119889= 002 to 120585

1= 00657 It

should be observed that the value of 1205851is significantly lower

than the value of the damping ratio of the internal and exter-nal intermediate viscoelastic supports at the same vibrationfrequency 120596

01 equal to respectively 013 and 012 This is the

result of the low dissipation capacity of the deck and of thedual load path behaviour of PRSI bridge whose stiffness anddamping capacity are the result of the contribution of boththe deck and the intermediate supports [8 9] The dampingratio of the higher modes is in general very low and tends todecrease significantly with the increasing mode order

Figure 4 shows the response of the midspan transversedisplacement (Figure 4(a)) and of the abutment reactions(Figure 4(b)) for a unit impulse ground motion

119892(119905) =

120575(119905) The analytical exact expression of the modal impulseresponse is reported in (16) The different response functionsplotted in Figure 4 are obtained by considering (1) thecontribution of the first mode only (2) the contribution ofthe first and third modes and (3) the contribution of thefirst third and fifth modes Modes higher than the 5th havea negligible influence on the response In fact the massparticipation factors of the 1st 3rd and 5th modes evaluatedthrough the RMS method are 81 92 and 325 Whilethe midspan displacement response is dominated by thefirst mode only higher modes strongly affect the abutment

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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Shock and Vibration

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International Journal of

8 Shock and Vibration

0

01

02

03

04

0 2 4 6 8 10

minus04

minus03

minus02

minus01 t (s)

h(m

)

1st1st + 3rd1st + 3rd + 5th

(a)

0

04

0 2 4 6 8 10

06

02

minus06

minus04

minus02t (s)

1st1st + 3rd1st + 3rd + 5th

h(m

minus2)

times10minus3EId

(b)

Figure 4 Response to impulsive loading (ℎ) versus time (119905) in terms of (a) midspan transverse displacement and (b) abutment reactions

reactions The impulse response in terms of the transversebending moments not reported here due to space con-straints is also influenced by the higher modes contributionbut at a less extent than the abutment reactions

63 Seismic Response In this section the characteristicaspects of the seismic performance of PRSI bridges are high-lighted by starting from the seismic analysis of the case studyand by evaluating successively the response variations dueto changes of the most important characteristic parametersof the bridge that is the deck-to-support stiffness ratio andthe support damping The bridge seismic performance isevaluated by monitoring the following response parametersthe midspan transverse displacement the transverse bendingmoments and the abutment shear The CMS method isapplied to compute the maximum overtime values of theseresponse parameters by averaging the results obtained for theseven different records considered Only the contribution ofthe first three symmetric vibration modes is considered dueto the negligible influence of the other modes

Figure 5 reports the mean of the envelopes of the trans-verse displacements and bending moments obtained for thedifferent records The displacement field coincides with asinusoidal shape This can be explained by observing thatthe isolated piers have a very low stiffness compared to thedeck stiffness and thus their restraint action on the deckis negligible Furthermore the displacement response is ingeneral dominated by the first mode of vibration The trans-verse bendingmoments exhibit a different shape because theyare significantly influenced by the higher modes of vibrationThe support restraint action on the deck moments is almostnegligible It is noteworthy that differently from the case offully isolated bridges the bending moment demand in thedeck may be very significant and attain the critical valuecorresponding to deck yielding a condition that should beavoided according to EC8-2 [30]

In order to evaluate how the support stiffness and damp-ing influence the seismic response of the bridge componentsan extensive parametric study is carried out by considering aset of bridge models with the same superstructure but withintermediate supports having different properties Following[9] the intermediate supports properties can be syntheticallydescribed by the following nondimensional parameters

1205722

=sum119873119888

119894=1119896119888119894

sdot 1198713

1205874119864119868119889

=sum119873119888

119894=1119896119888119894

1205962

119889119871119898119889

120574119888

=sum119873119888

119894=1119888119888119894

21205722120596119889119898119889119871

=sum119873119888

119894=1119888119888119894

120596119889

2 sum119873119888

119894=1119896119888119894

(27)

The parameter 1205722 denotes the relative support to deck

stiffness Low values of 1205722 correspond to a stiff deck relative

to the springs while high values correspond to a more flexibledeck relative to the springs Limit case 120572

2= 0 corresponds to

the simply supported beam with no intermediate restraintsThe parameter 120574

119888describes the energy dissipation of the

supports and it is equal to the ratio between the energydissipated by the dampers and the maximum strain energyin the springs for a uniform transverse harmonic motion ofthe deckwith frequency120596

119889These two parameters assume the

values 1205722

= 0676 and 120574119888

= 0095 for the bridge consideredpreviously

In the following parametric study the values of theseparameters are varied by keeping the same distribution ofstiffness and of damping of the original bridge that is byusing the same scale factor for all the values of 119896

119888119894and the

same scaling factor for all the values of 119888119888119894 In particular

the values of 1205722 are assumed to vary in the range between

0 (no intermediate supports) and 2 (stiff supports relative tothe superstructure) whereas the values of 120574

119888are assumed to

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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International Journal of

Shock and Vibration 9

0

005

01

015

02

025

03

035

04

0 50 100 150 200

x (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

14

0 50 100 150 200

x (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 5 Average peak transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck

Mode 1

05

06

07

08

09Mode 5Mode 3

1

0 02 04 06 08 1 12 14 16 18 2

T(120572

2)T(120572

2=0)

(mdash)

1205722 (mdash)

(a)

01

03

0

02

04

06

05

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d(m

)

d1

d2

(b)

Figure 6 (a) Variation with 1205722 of normalized transverse period 119879(120572

2)119879(120572

2= 0) (b) variation with 120572

2 of displacement demand of the deckin correspondence with the external piers (119889

1) and the central pier (119889

2)

vary in the range between 0 (intermediate supports with nodissipation capacity) and 03 (very high dissipation capacity)

Figure 6(a) shows the variation with 1205722 of the periods

of the first three modes of vibration that participate in theseismic response that is modes 1 3 and 5 normalizedwith respect to the values observed for 120572

2= 0 It can be

seen that only the fundamental vibration period is affectedsignificantly by the support stiffness whereas the highermodes of vibration exhibit a constant value of the vibrationperiod for the different 120572

2 values Figure 6(b) provides someinformation on the seismic response and it shows the influ-ence of120572

2 on the averagemaximum transverse displacementsin correspondence with the outer intermediate supports andat the centre of the deck denoted respectively as 119889

1and 119889

2

These response quantities do not vary significantly with 1205722

This occurs because the displacement demand is controlledby the first mode of vibration whose period for the different1205722 values falls within the region corresponding to the flat

displacement response spectrum

Figure 7(a) shows the variation of the bending momentdemand in correspondence with the piers 119872

1198891 and at the

centre of the deck 1198721198892 These response parameters follow

a trend similar to that of the displacement demand that isthey do not vary significantly with 120572

2 Figure 7(b) shows thevariation with 120572

2 of the sum of the pier reaction forces (119877119901)

and the sum of the abutment reactions (119877119886119887) In the same

figure the contribution of the first mode only to these quan-tities (119877

1199011 1198771198861198871

) is also reported It can be observed that thetotal base shear of the system increases for increasing valuesof 1205722 consistently with the increase of spectral acceleration

at the fundamental vibration period Furthermore while thevalues of 119877

119901tend to increase for increasing values of 120572

2 thevalues of 119877

119886119887remain almost constant This can be explained

by observing that the abutment reactions are significantlyaffected by the contribution of higher modes Thus in thecase of intermediate support with no dissipation capacityincreasing the intermediate support stiffness does not reducethe shear forces transmitted to the abutments

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

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VLSI Design

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Shock and Vibration

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DistributedSensor Networks

International Journal of

10 Shock and Vibration

0

02

04

06

08

1

12

14

16

18

2

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Md

(kN

m)

times105

Md1

Md2

(a)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

Rp

Rab

Rp1

Rab1

(b)

Figure 7 (a) Variation with 1205722 of the peak average bending moments in correspondence with the external piers (119872

1198891) and the central pier

(1198721198892) (b) variation with 120572

2 of the peak average total pier reactions (119877119901) and abutment reactions (119877

119886119887) and of the corresponding first mode

contribution (1198771199011 1198771198861198871

)

In the following the joint influence of support stiffnessand damping on the bridge seismic response is analysedFigure 8(a) shows the variation of the transverse displace-ment 119889

2versus 120572

2 for different values of 120574119888 The values of

1198892are normalized for each value of 120572

2 by dividing themby the value 119889

20corresponding to 120574

119888= 0 As expected the

displacement demand decreases by increasing the dissipa-tion capacity of the supports Furthermore the differencesbetween the displacement demands for the different 120574

119888values

reduce for 1205722 tending to zero

Figure 8(b) shows the total pier (119877119901) and abutment (119877

119886119887)

shear demand versus 1205722 for different values of 120574

119888 These

values are normalized for a given value of 1205722 by dividing

respectively by the values 1198771199010

and 1198771198861198870

corresponding to120574119888

= 0 It can be noted that both the pier and abutmentreactions decrease by increasing 120574

119888 Furthermore differently

from the case corresponding to 120574119888

= 0 the increase of theintermediate support stiffness results in a reduction of theabutment reactions However the maximum decrease of thevalues of the abutment reactions achieved for high 120572

2 levelsis inferior to the maximum decrease of the values of pierreactionsThis is a consequence of the combined effects of (a)the contribution of highermodes which is significant only forthe abutment reactions and negligible for the pier reactionsand (b) the reduced efficiency of the intermediate dampers indamping the higher modes of vibration [9]

Finally Figure 9 shows the total base shear versus 1205722 for

different values of 120574119888 By increasing 120572

2 the total base shearincreases for low support damping values (120574

119888= 00 005)

while for very high 120574119888values (120574

119888= 010 015 020) it first

decreases and then it slightly increases Thus for values of

120574119888higher than 005 the total base shear is minimized when

1205722 is equal to about 05 This result could be useful for

the preliminary design of the isolator properties ensuringthe minimization of the total base shear as performanceobjective

64 Accuracy of Simplified Analysis Techniques In this sec-tion the accuracy of the simplified analysis techniques isevaluated by comparing the exact estimates of the responseobtained by applying the CMS method with the correspond-ing estimates obtained by employing the RMS and FTSmethods In the application of all these methods the firstthree symmetric terms of the set of approximating functionsare considered and the ground motion records are thosealready employed in the previous section

Figure 10 reports the results of the comparison for thecase study described at the beginning of this section Itcan be observed that the RMS and FTS method providevery close estimates of the mean transverse displacements(Figure 10(a)) whose shape is sinusoidal This result wasexpected since vibration modes higher than the first modehave a negligible influence on the displacements as alreadypointed out in Figure 4 With reference to the transversebending moment envelope (Figure 10(b)) the mean shapeaccording to RMS and FTS agrees well with the exact shapethe highest difference being in correspondence with theintermediate restraints The influence of the third mode ofvibration on the bending moments shape is well described byboth simplified techniques

Table 2 reports and compares the peak values of themidspan displacement 119889

2 the transverse abutment reaction

119877119886119887 the midspan transverse moments 119872

119889 and the support

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

d2d

20

(mdash)

120574c = 000

120574c = 005

120574c = 010

120574c = 020

120574c = 015

(a)

0

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

Rp

Rab

120574c = 000120574c = 005

120574c = 010

120574c = 020

120574c = 015

RR

0(mdash

)(b)

Figure 8 Variation with 1205722 and for different values of 120574

119888of normalized deck centre transverse displacement 119889

211988920

and of the normalizedtotal pier and abutment reactions (119877

1199011198771199010

and 119877119886119887

1198771198861198870

)

0 02 04 06 08 1 12 14 16 18 2

1205722 (mdash)

R(k

N)

120574c = 000

120574c = 005

120574c = 010

120574c = 020120574c = 015

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Figure 9 Variation with 1205722 of total base shear 119877 for different values of 120574

119888

Table 2 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01828 01827 01824 21056 21032 21198 71788 71779 72022 3698 3702 3723 3761 3760 37522 05496 05496 05485 41144 40938 41845 188809 188837 189684 11265 11256 11313 11309 11309 112853 02681 02681 02676 33205 33173 33157 103350 103412 103921 5208 5199 5230 5516 5517 55064 04473 04474 04468 31859 31803 32050 143742 143659 144171 9477 9474 9535 9205 9206 91925 06227 06227 06215 36502 36608 37828 194330 194336 195329 12199 12202 12293 12813 12813 127886 02417 02418 02416 33354 33283 33456 116320 116249 116587 5087 5083 5116 4974 4975 49717 01680 01680 01678 16488 16628 16830 65313 65334 65723 3363 3358 3381 3456 3457 3453

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

CMS RMSFTS

0

005

01

015

02

025

03

035

04

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

2

4

6

8

10

12

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

Nm

)

14times104

(b)

Figure 10 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 3 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 03543 0519 03543 0519 03537 0519

119877ab [kN] 305155 0284 304947 0282 309092 0287119872119889[kNm] 12623575 0411 12624935 0411 12677866 0411

1198771199011

[kN] 71852 0515 71822 0515 72273 05151198771199012

[kN] 72905 0519 72909 0519 72783 0519

reactions 1198771199011

and 1198771199012

according to the three analysis tech-niques for the 7 records employed to describe the record-to-record variability effects These records are characterized bydifferent characteristics in terms of duration and frequencycontent (Figure 3) Thus the values of the response param-eters of interest exhibit significant variation from recordto record The level of accuracy of the simplified analysistechniques is in general very high and it is only slightlyinfluenced by the record variabilityThe highest relative errorobserved for the estimates of the bending moment demandis about 084 for the RMS method and 36 for the FTSmethod The coupling indexes for the damping matrix are12057213

= 00454 12057215

= 02259 and 12057235

= 01358 for the RMSmethod and 120572

13= 00461 120572

15= 02242 and 120572

35= 01365 for

the FTSmethodThe coupling indexes for the stiffnessmatrixare 12057313

= 559119890minus4 12057315

= 904119890minus5 and 12057335

= 183119890minus6 for theFTSmethodThus the extent of coupling in both the stiffnessand damping terms is very low and this explains why theapproximate techniques provide accurate response estimates

Table 3 reports themean value and coefficient of variationof the monitored response parameters With reference tothis application it can be concluded that both the simplifiedanalysis techniques provide very accurate estimates of theresponse statistics despite the approximations involved intheir derivationThe highest relative error between the resultsof the simplified analysis techniques and those of the CMS

method is observed for the estimates of the mean bendingmoment demand and it is about 001 for the RMS methodand 043 for the FTS method

The accuracy of the approximate techniques is also evalu-ated by considering another set of characteristic parameterscorresponding to 120572

2= 2 and 120574

119888= 02 and representing

intermediate supports with high stiffness and dissipationcapacity

The displacement shape of the bridge reported inFigure 11(a) is sinusoidal despite the high stiffness anddamping capacity of the supports However the bendingmoment shape reported in Figure 11(b) is significantly influ-enced by the higher modes of vibration and by the restrainaction of the supports The RMS method provides veryaccurate estimates of this shape whereas the FTS method isnot able to describe the support restraint effect

Table 4 reports the peak values of the response parame-ters of interest for each of the 7 natural records consideredaccording to the different analysis techniques The highestrelative error observed for the estimates of the bendingmoment demand is less than 36 for the RMS method and92 for the FTS method

The mean and coefficient of variation of the responseparameters of interest are reported in Table 5 The highestrelative error with respect to the CMS estimate is observedfor the mean bending moment demand and it is less than

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 13

0

005

010

015

02

0175

0125

0075

0025

CMS RMSFTS

0 50 100 150 200

y (m)

um

ax(m

)

(a)

0

1

2

3

4

5

6

7

CMS RMSFTS

0 50 100 150 200

y (m)

Mdm

ax(k

N m

)

times104

(b)

Figure 11 Transverse displacements 119906max (a) and bending moments 119872119889max (b) along the deck according to the exact and the simplified

analysis techniques

Table 4 Comparison of results according to the different analysis techniques for each of the 7 ground motion records considered

Record 1198892[m] 119877ab [kN] 119872

119889[MNm] 119877

1199011[kN] 119877

1199012[kN]

CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS CMS RMS FTS1 01556 01557 01548 14682 15129 14352 65801 65840 65878 8460 8325 8460 9482 9478 94242 02099 02096 02077 21149 21750 21756 74322 73745 72942 11978 11876 12009 12796 12757 126423 01504 01505 01490 18543 18719 19050 60887 59777 59856 8907 8838 8905 9169 9160 90704 02164 02167 02144 21698 21986 21989 78089 78244 77744 11832 11712 11907 13190 13186 130505 03471 03475 03439 20888 20168 22395 105322 104517 103797 21072 20992 21280 21157 21148 209336 01971 01966 01954 19480 19550 17832 83645 82237 82389 10888 11029 11268 12014 11968 118917 00799 00799 00790 9682 9585 9329 30202 29984 30055 4640 4594 4646 4868 4863 4806

Table 5 Comparison of results (mean values and CoV of the response) according to the different analysis techniques

Response parameter CMS RMS FTSMean CoV Mean CoV Mean CoV

1198892[m] 01938 0425 01938 0425 01920 0424

119877ab [kN] 18017 0243 18126 0243 18100 0266119872119889[kNm] 711812 0324 706206 0324 703800 0322

1198771199011

[kN] 11111 0456 11052 0458 11211 04581198771199012

[kN] 11811 0425 11794 0425 11688 0424

079 for the RMS method and 112 for the FTS methodThe coupling indexes for the damping matrix are 120572

13=

00625 12057215

= 04244 and 12057235

= 02266 for the RMSmethod and 120572

13= 00652 120572

15= 04151 and 120572

35=

02305 for the FTS method The coupling indexes for thestiffness matrix are 120573

13= 00027 120573

15= 443119890 minus 4 and

12057335

= 156119890 minus 5 In general these coefficients assume highervalues compared to the values observed in the other bridge(Tables 2 and 4) In any case the dispersion of the responsedue to the uncertainties in the values of the parametersdescribing the pier and the bearings might be more impor-tant than the error due to the use of simplified analysismethods

7 Conclusions

This paper proposes a formulation and different analysistechnique for the seismic assessment of partially restrainedseismically isolated (PRSI) bridges The formulation is basedon a description of the bridges by means of a simply sup-ported continuous beam resting on intermediate viscoelasticsupports whose properties are calibrated to represent thepierbearing systems The first analysis technique developedin this paper is based on the complex modes superposition(CMS) method and provides an exact solution to the seis-mic problem by accounting for nonclassical damping Thistechnique can be employed to evaluate the influence of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

the complex vibration modes on the response of the bridgecomponents and also provides a reference solution againstwhich simplified analysis approaches can be testedThe othertwo proposed techniques are based on the assumed modesmethod and on a simplified description of the dampingproperties The real mode superposition (RMS) methodemploys a series expansion of the displacement field in termsof the classic modes of vibration whereas the Fourier termssuperposition (FTS) method employs a series expansionbased on the Fourier sine-only series terms

A realistic PRSI bridge model is considered and itsseismic response under a set of ground motion recordsis analyzed by employing the CMS method This methodreveals that the bridge components are differently affectedby higher modes of vibration In particular while the dis-placement demand is mainly influenced by the first modeother parameters such as the bendingmoments and abutmentreactions are significantly affected by the contribution ofhigher modes A parametric study is then carried out toevaluate the influence of intermediate support stiffness anddamping on the seismic response of the main components ofthe PRSI bridge These support properties are controlled bytwo nondimensional parameters whose values are varied inthe parametric analysis Based on the results of the study itcan be concluded that (1) in the case of intermediate supportwith low damping capacity the abutment reactions do notdecrease significantly by increasing the support stiffness(2) increasing the support damping results in a decreasein the abutment reactions which is less significant than thedecrease in the pier reactions (3) for moderate to highdissipation capacity of the intermediate supports there existsan optimal value of the damping that minimizes the totalshear transmitted to the foundations

Finally the accuracy of the simplified analyses techniquesis evaluated by comparing the results of the seismic analysescarried out by employing the CMS method with the corre-sponding results obtained by employing the two simplifiedtechniques Based on the results presented in this paper it isconcluded that for the class of bridges analyzed (1) the extentof nonclassical damping is usually limited and (2) satisfactoryestimates of the seismic response are obtained by consideringthe simplified analysis approaches

Appendix

The analytical solution of the eigenvalue problem is obtainedunder the assumption that 119898(119909) 119887(119909) and 119888(119909) are constantalong the beam length and equal respectively to 119898

119889 119864119868119889

and 119888119889 The beam is divided into a set of 119873

119904segments

each bounded by two consecutive restraints (external orintermediate)The function 119906

119904(119911119904 119905) describing themotion of

the 119904th segment of length 119871119904is decomposed into the product

of a spatial function 120595119904(119911119904) and of a time-dependent function

119885(119905) = 1198850119890120582119905 The function 120595

119904(119911119904) must satisfy the following

equation

120595119904

119868119881(119911119904 119905) = Ω

4120595119904(119911119904 119905) (A1)

where Ω4

= minus(1205822119898119889

+ 120582119888119889)119864119868119889

The solution to (A1) can be expressed as

120595119904(119911119904) = 1198624119904minus3

sin (Ω119911119904) + 1198624119904minus2

cos (Ω119911119904)

+ 1198624119904minus1

sinh (Ω119911119904) + 1198624119904cosh (Ω119911

119904)

(A2)

with 1198624119904minus3

1198624119904minus2

1198624119904minus1

and 1198624119904

to be determined basedon the boundary conditions at the external supports andthe continuity conditions at the intermediate restraints Thisinvolves the calculation of higher order derivatives up to thethird order

In total a set of 4119873119904conditions is required to determine

the vibration shape along the whole beam At the first spanthe conditions 120595

1(0) = 120595

10158401015840

1(0) = 0 apply while at the last span

the support conditions are 120595119873119904

(119871119873s

) = 12059510158401015840

119873119904

(119871119873119904

) = 0 Theboundary conditions at the intermediate spring locations are

120595119904minus1

(119871119904minus1

) = 120595119904(0)

1205951015840

119904minus1(119871119904minus1

) = 1205951015840

119904(0)

12059510158401015840

119904minus1(119871119904minus1

) = 12059510158401015840

119904(0)

119864119868119889

[120595101584010158401015840

119904minus1(119871119904minus1

) minus 120595101584010158401015840

119904(0)] minus (119896

119888119904+ 120582119888119888119904

) 120595119904(0) = 0

(A3)

By substituting (A2) into the boundary (supports and conti-nuity) conditions a system of 4119873

119904homogeneous equations

in the constants 1198621 119862

4119873119904

is obtained Since the systemis homogeneous the determinant of coefficients must beequal to zero for the existence of a nontrivial solution Thisprocedure yields the following frequency equation in theunknown 120582

119866 (119898119889 119864119868119889 119888119889 1198711 119871

119873119904

1198961198881

119896119888119873119888

1198881198881

119888119888119873119888

120582)

= 0

(A4)

In the general case of nonzero damping the solution of theequation must be sought in the complex domain It is note-worthy that since the system is continuous an infinite set ofeigenvalues 120582 are obtained which satisfy (A4) However onlyselected values of 120582 are significant because they correspondto the first vibration modes which are usually characterizedby the highest participation factors It is also noteworthythat the eigensolutions appear in complex pairs Thus if 120582

119894

satisfies (A4) then also its complex conjugate 120582119894satisfies

(A4) The eigenvectors corresponding to these eigenvaluesare complex conjugate too Finally it is observed that theundamped circular vibration frequency120596

119894and damping ratio

120585119894corresponding to the 119894th mode can be obtained by recalling

that 120582119894120582119894= 120596119894

2 and 120582119894+120582119894= minus2120585

119894120596119894 For zero damping 120582 = 119894120596

and the vibration modes are real

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

Acknowledgment

The reported research was supported by the Italian Depart-ment of Civil Protectionwithin the Reluis-DPCProjects 2014The authors gratefully acknowledge this financial support

References

[1] M T A Chaudhary M Abe and Y Fujino ldquoPerformanceevaluation of base-isolated Yama-age bridge with high dampingrubber bearings using recorded seismic datardquo EngineeringStructures vol 23 no 8 pp 902ndash910 2001

[2] G C Lee Y Kitane and I G Buckle ldquoLiterature reviewof the observed performance of seismically isolated bridgesrdquoResearch Progress and Accomplishments 2000ndash2001 MCEERState University of New York Buffalo NY USA 2001

[3] R L Boroschek M O Moroni and M Sarrazin ldquoDynamiccharacteristics of a long span seismic isolated bridgerdquo Engineer-ing Structures vol 25 no 12 pp 1479ndash1490 2003

[4] J Shen M H Tsai K C Chang and G C Lee ldquoPerformanceof a seismically isolated bridge under near-fault earthquakeground motionsrdquo Journal of Structural Engineering vol 130 no6 pp 861ndash868 2004

[5] K L Ryan and W Hu ldquoEffectiveness of partial isolation ofbridges for improving column performancerdquo in Proceedings ofthe Structures Congress pp 1ndash10 2009

[6] M-H Tsai ldquoTransverse earthquake response analysis of aseismically isolated regular bridge with partial restraintrdquo Engi-neering Structures vol 30 no 2 pp 393ndash403 2008

[7] N Makris G Kampas and D Angelopoulou ldquoThe eigenvaluesof isolated bridges with transverse restraints at the end abut-mentsrdquo Earthquake Engineering and Structural Dynamics vol39 no 8 pp 869ndash886 2010

[8] E Tubaldi and A DallrsquoAsta ldquoA design method for seismicallyisolated bridges with abutment restraintrdquo Engineering Struc-tures vol 33 no 3 pp 786ndash795 2011

[9] E Tubaldi and A DallrsquoAsta ldquoTransverse free vibrations ofcontinuous bridges with abutment restraintrdquo Earthquake Engi-neering and Structural Dynamics vol 41 no 9 pp 1319ndash13402012

[10] GDellaCorte R deRisi andL di Sarno ldquoApproximatemethodfor transverse response analysis of partially isolated bridgesrdquoJournal of Bridge Engineering vol 18 no 11 pp 1121ndash1130 2013

[11] M N Fardis and G Tsionis ldquoEigenvalues and modes ofdistributed-mass symmetric multispan bridges with restrainedends for seismic response analysisrdquo Engineering Structures vol51 pp 141ndash149 2013

[12] A S Veletsos and C E Ventura ldquoModal analysis of non-classically damped linear systemsrdquo Earthquake Engineering ampStructural Dynamics vol 14 no 2 pp 217ndash243 1986

[13] A M Claret and F Venancio-Filho ldquoModal superpositionpseudo-forcemethod for dynamic analysis of structural systemswith non-proportional dampingrdquo Earthquake Engineering andStructural Dynamics vol 20 no 4 pp 303ndash315 1991

[14] F Venancio-Filho Y K Wang F B Lin A M Claret and WG Ferreira ldquoDynamic analysis of nonproportional dampingstructural systems time and frequency domain methodsrdquo inProceedings of the 16th International Conference on StructuralMechanics in Reactor Technology (SMIRT rsquo01) Washington DCUSA 2001

[15] S Kim ldquoOn the evaluation of coupling effect in nonclassicallydamped linear systemsrdquo Journal of Mechanical Science andTechnology vol 9 no 3 pp 336ndash343 1995

[16] G Prater Jr and R Singh ldquoQuantification of the extent of non-proportional viscous damping in discrete vibratory systemsrdquoJournal of Sound and Vibration vol 104 no 1 pp 109ndash125 1986

[17] M Morzfeld F Ma and N Ajavakom ldquoOn the decouplingapproximation in damped linear systemsrdquo Journal of Vibrationand Control vol 14 no 12 pp 1869ndash1884 2008

[18] J S Hwang K C Chang andM H Tsai ldquoComposite dampingratio of seismically isolated regular bridgesrdquo Engineering Struc-tures vol 19 no 1 pp 55ndash62 1997

[19] S C Lee M Q Feng S-J Kwon and S-H Hong ldquoEquivalentmodal damping of short-span bridges subjected to strongmotionrdquo Journal of Bridge Engineering vol 16 no 2 pp 316ndash323 2011

[20] P Franchin G Monti and P E Pinto ldquoOn the accuracyof simplified methods for the analysis of isolated bridgesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 380ndash382 2001

[21] K A Foss ldquoCoordinates which uncouple the equations ofmotion of damped linear dynamic systemsrdquo Journal of AppliedMechanics vol 25 no 3 pp 361ndash364 1958

[22] G Oliveto A Santini and E Tripodi ldquoComplex modal analysisof a flexural vibrating beam with viscous end conditionsrdquoJournal of Sound andVibration vol 200 no 3 pp 327ndash345 1997

[23] M Gurgoze and H Erol ldquoDynamic response of a viscouslydamped cantilever with a viscous end conditionrdquo Journal ofSound and Vibration vol 298 no 1-2 pp 132ndash153 2006

[24] E Tubaldi ldquoDynamic behavior of adjacent buildings connectedby linear viscousviscoelastic dampersrdquo Structural Control andHealth Monitoring 2015

[25] C Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik Band IIIl Springer Berlin Germany1960

[26] G Prater Jr and R Singh ldquoEigenproblem formulation solutionand interpretation for nonproportionally damped continuousbeamsrdquo Journal of Sound and Vibration vol 143 no 1 pp 125ndash142 1990

[27] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained uniform beam carrying an intermediate lumpedmass and a rotary inertiardquo Journal of Sound and Vibration vol150 no 2 pp 203ndash216 1991

[28] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

[29] N Ambraseys P Smith R Bernardi D Rinaldis F Cottonand C Berge-ThierryDissemination of European Strong-MotionData CD-ROM Collection European Council Environmentand Climate Research Programme 2000

[30] European Committee for Standardization (ECS) ldquoEurocode8mdashdesign of structures for earthquake resistancerdquo EN 1998European Committee for Standardization Brussels Belgium2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of