Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 302706, 20 pageshttp://dx.doi.org/10.1155/2013/302706

Research ArticleDynamic Analysis of Cable-Stayed Bridges Affected byAccidental Failure Mechanisms under Moving Loads

Fabrizio Greco, Paolo Lonetti, and Arturo Pascuzzo

Department of Structural Engineering, University of Calabria, P. Bucci Road, Cubo39-B, Rende, 87030 Cosenza, Italy

Correspondence should be addressed to Paolo Lonetti; lonetti@unical.it

Received 21 August 2012; Accepted 14 November 2012

Academic Editor: Xiaojun Wang

Copyright © 2013 Fabrizio Greco et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Thedynamic behavior of cable-stayed bridges subjected tomoving loads and affected by an accidental failure in the cable suspensionsystem is investigated.Themain aim of the paper is to quantify, numerically, the dynamic amplification factors of typical kinematicand stress design variables, by means of a parametric study developed in terms of the structural characteristics of the bridgecomponents. The bridge formulation is developed by using a geometric nonlinear formulation, in which the effects of localvibrations of the stays and of large displacements in the girder and the pylons are taken into account. Explicit time dependentdamage laws, reproducing the failure mechanism in the cable system, are considered to investigate the influence of the failure modecharacteristics on the dynamic bridge behavior. The analysis focuses attention on the influence of the inertial characteristics of themoving loads, by accounting coupling effects arising from the interaction between girder and moving system. Sensitivity analysesof typical design bridge variables are proposed. In particular, the effects produced by the moving system characteristics, the towertypologies, and the failure mode characteristics involved in the cable system are investigated by means of comparisons betweendamaged and undamaged bridge configurations.

1. Introduction

Cable supported bridges are frequently employed in thecontext of long spans, leading to slender structures, inwhich, typically, the dead loads are comparable with thoseinvolved in the live load configuration. As a consequence, theexternal loads are able to produce high amplifications of themain bridge kinematic and stress design parameters, leadingto nonstandard excitation modes and unexpected damagemechanisms in the structural components of the bridge. Inorder to simulate the actual behavior of the bridge, the struc-turalmodeling should be able to reproduce the source of non-linearities in the cable system, in the girder or in the pylonsdue to cable sag or beam-column large deformation effects,respectively. Moreover, additional complexities arise in theprediction of the bridge behavior in the presence of damagemechanisms, which strongly reduce the structural integrity ofthe bridge. As amatter of fact, cable-supported structures are,typically, affected by degradation effects such as corrosion,abrasion, and fatigue, which may cause a reduction of thestiffness properties or, in extreme cases, the complete failure

of a single or multiple cable elements. In the literature, manyinvestigations have been developed to analyze the influenceof the moving loads on the dynamic behavior of cable-supported bridges, mainly, for undamaged bridge structures.Actually, many models are devoted to predicting dynamicbridge behavior by using refined structural schematizationsas well as accurate descriptions of the moving loads [1–4]. Inthis framework, the bridge behavior is analyzed by means ofanalytical continuum approaches or finite element models, inwhich, the behavior of the cable suspension system is typicallydescribed by means of linear equations expressed in termsof tangent or secant Dischinger elastic moduli [5–8]. Thisassumption is frequently supported by experimental evidencefor static analyses [9, 10]. However, in order to reproducethe dynamic behavior correctly, especially when the bridgeis subjected to extreme loading conditions, local vibrationeffects of the cable elements should be, properly, taken intoaccount [2, 11]. Additional complexities in the prediction ofthe dynamic behavior of long span bridges arise from thedescription of the interaction behavior betweenmoving loadsand bridge vibrations. At this aim, many papers have been

2 Mathematical Problems in Engineering

developed to analyze the influence of the external mass andits motion on the bridge behavior, introducing an accuratedescription of the inertial forces between bridge deformationand moving load kinematic [7, 8, 12]. However, a brief litera-ture review denotes that the behavior of cable-stayed bridgesis mainly analyzed for undamaged structures, whereas theinfluence of cable failure mechanisms produced by loss ofstiffness due to cable degradation or due to an accidentalfailure is rarely analyzed. Typically, many papers are devotedto investigating the effects on the dynamic behavior of asingle cable element of a localized or a distributed timeindependent damage mechanisms [13], without entering indetail on time dependence characteristics of the failure modeor the coupling behavior between damaged and undamagedelements of the cable system. The results proposed in thisframework show how damage phenomena induce tensionloss and sag augmentation of the static undamaged cableprofile.

Since cable elements are mostly utilized in supportedbridges, studies developed on single cables have been gener-alized to evaluating the influence of damage phenomena onrealistic cable structures. In particular, damage behavior ofcable-stayed beam has been analyzed by means of numericalapproaches, in which the effects of diffused inelastic damageconcerning the intensity and location characteristics havebeen investigated [14]. Such results show how the damageeffects produce relevant modifications with respect to theundamaged configuration of the natural frequencies andmode shapes of the structure.Moreover, further investigationhas been developed in the framework of cable-stayed bridge.

It is worth noting that fromadesign point of view, existingcodes on cable-stayed bridges, that is, PTI [15] and SETRA[16], in order to identify the amplification effects providedby the failure mechanism in the cable system, recommend toamplify the results obtained in the framework of quasistaticanalyses by using fictitious amplification factors suggested inthe range between 1.5 and 2.0. In particular, the codes identifythe dynamic characteristics of the failure mode of a genericelement of the cable system, introducing a static loadingconfiguration, in which the cable failure is reproduced bymeans of compression forces to simulate cable release. Thestress distribution arising from such a loading scheme iscombined with the effects of other existing loading schemesby means of proper factored loading combinations. However,recent papers have demonstrated that such a simplifiedapproach becomes unsafe in many cases, leading to dynamicamplification factors higher than those suggested by existingrecommendations [17–19]. In particular, some parametricstudies have been developed for bridge typologies subjectedto accidental cable failure by using a numerical approachbased on classical standard linear dynamic framework [18,19]. Such analyses denote that the results obtained by usingsuch code prescriptions are affected by high underestimationsin the prediction of typical design bridge variables relatedto the girder and pylons. However, in order to correctlyreproduce the bridge behavior, additional developments arerequired to be simulate the effect produced by the nonlin-earities and damage mechanisms involved cable system andby the inertial coupling between girder and moving loads

deformations. Actually, comprehensive analyses that includesuchnon linear dynamic effects are quite rare and thus furtherinvestigations to verify code prescriptions and to quantify theinfluence on the bridge behavior of the dynamic excitationproduced by the failure mode characteristics of the cablesystem are much required.

In the present paper, the dynamic behavior of cable-stayed bridges subjected to cable failure mechanisms isinvestigated. The numerical study consists on a tridimen-sional finite element model, in which both A-shaped andH-shaped pylons are investigated. The aim of the paper isto propose a parametric study to investigate the influenceon the bridge behavior of the dynamic excitation producedby damage mechanisms in the cable system and the transitof moving loads. Differently from existing papers availablein the literature, the proposed modeling reproduces theinertial description of the moving loads by means of arefined schematization of the inertial forces produced bythe moving system and girder bridge interaction. Moreover,local vibrations of the cable elements are taken into accountby reproducing the nonlinearities involved in the cable-sageffect in the cable system as well as large deformations inthe girder and the pylons. Finally, the stay failure mecha-nism is formulated, consistently with a Continuous DamageMechanics approach, introducing time dependent damagefunctions, which control the constitutive behavior of thefailure mechanism and thus the inertial characteristics of theloss of cable stiffness. The outline of the paper is as follows:Section 2 presents the general formulation of the cable andthe girder elements, the damage description of the cablemodel, and the evaluation of the initial configuration; thenumerical implementation is reported in Section 3; paramet-ric studies in terms of bridge andmoving loads characteristicsand failure mode typology in the cable system are reported inSection 4.

2. Bridge Formulation

In this section the governing equations for the bridgeconstituents as well as the main assumptions concerningthe kinematic modeling are discussed. In particular, themain assumptions concerning the kinematic modeling ofthe bridge, the damage formulation adopted to simulatecable failure, and the inertial forces to define the movingloads/girder interaction are discussed. Such governing equa-tions represent the basis for the theoretical formulation ofthe model, whose numerical implementation is presented inSection 3.

2.1. Cable System Formulation and Initial Configuration underDead Loading. The bridge scheme is based on a tridimen-sional modeling, in which both in plane and out of planedeformation modes are considered. The structural model,reported in Figure 1, is consistent with a fan-shaped anda self-anchored cable-stayed bridge scheme. Moreover, thepylons, analyzed by the present paper, refer to A- or H-shaped typologies. Every single cable of the suspensionsystem is simulated by using a sequence of the 𝑛 truss

Mathematical Problems in Engineering 3

l L l

L

b

ct

sc

X

X

KP

H

A

U1(t)

s

(2l+L,t)

KPA0

P

X

X

b

eH

U1

3

2

1

3

1

P

b b

e

c A0c

c Ac

G

EGAG, EGIG2 , EGIG2 , J

Gt

UG2 (X1 ,t)

UG1 (X1 ,t)

−UG3 (X1 ,t)

−UG3 (X1 ,t)

−UC3 (s,t)

ΦG1 (X1 ,t)

UG1 (−1 + 1/2, t)

Sc1Ac

Sc1Ac

α0α0

∆

α(X1)

Cc, Sc1

LS(x)

Figure 1: Cable-stayed bridge scheme: bridge kinematic, pylons, girder, and cable system characteristics.

l L/2

ULP

S1C

S2CS3C S4

C

Sn/2C

UjG

Uj+1G

UG

H

g

Sj+1C

SjC

Figure 2: Cable-stayed configuration under dead loading: internal stresses and kinematic parameters.

elements, which are connected at the cross-section ends ofthe girder profile and at the top cross-section of the tower.The formulation is consistent with a large deformation theorybased on the Green-Lagrange strain measure and the secondPiola-Kirchhoff stress [20], whereas the material behavior isassumed to be linearly elastic. Since the cable behavior ismostly influenced by the preexisting stress and strain status,the initial configuration under the dead loading must beidentified in advance. In particular, with respect to the finalstep of a balanced cantilever construction method, in whichonly the central segment of the deck is left, the geometricshape design of the bridge and the corresponding internalstresses of the cable system are obtained enforcing the deckand the top pylon cross-section kinematic variables to remainin the undeformed configuration, that is, “zero configuration”[10]. In order to calculate the initial stress distribution inthe cable system, an optimization solving problem should bedeveloped. In particular, with reference to the cable-stayed

bridge scheme, reported in Figure 2, with 𝑛 number of stays,the objective functions are represented by the displacementvector

˜𝑈 containing the 𝑛 − 2 vertical displacements, that

is, (𝑈𝐺2, . . . , 𝑈

𝐺

𝑛−3, 𝑈

𝐺

𝑛−2), excluding those points associated

with the anchor stays at the cable/girder connections, andthe horizontal displacements of the top cross-section of thepylons (𝑈𝑃

𝐿, 𝑈

𝑃

𝑅):

˜𝑈𝑇= [𝑈

𝑃

𝐿, 𝑈

𝑃

𝑅, 𝑈

𝐺

2, . . . , 𝑈

𝐺

𝑛−3, 𝑈

𝐺

𝑛−2] . (1)

Moreover, the design variables, which have to be calcu-lated, correspond to the internal stress distribution of thecable system, that is,

˜𝑆𝑇= [𝑆

𝐶

1, 𝑆𝐶

2, . . . , 𝑆

𝐶

𝑛𝑐−1, 𝑆𝐶

𝑛]. Since

the relationships between displacements and cable stressesare essentially nonlinear, a specialized solving procedure tocalculate the initial configuration is required. In particular,starting from an initial trial distribution in the cable system,that is,

˜𝑆0, the vertical displacements under the dead loading

4 Mathematical Problems in Engineering

can be expressed to the first order by the Taylor expansion interms of the incremental cable stress distribution, by meansof the following linearized equations:

˜𝑈(

˜𝑆0+ Δ

˜𝑆, 𝑝) =

˜𝑈 (

˜𝑆0, 𝑝) +

𝑑

˜𝑈

𝑑

˜𝑆

(

˜𝑆0,𝑝)

⋅ Δ

˜𝑆 + 𝑜

Δ

˜𝑆2,

(2)

where˜𝑈(

˜𝑆0, 𝑝) is a vector containing the displacements

in the self weight loading condition and subjected to thestress distribution

˜𝑆0, 𝑝 is the loading parameter associated

to the application of the dead loading, and (𝑑˜𝑈/𝑑

˜𝑆)|

(

˜𝑆0,𝑝)

is the directional derivative of˜𝑈 at

˜𝑆0coinciding with

the flexibility matrix of the structure. In (2), the unknownquantity is represented by the incremental vector related tothe cable stress distribution, namely,Δ

˜𝑆, which is determined

enforcing the displacement vector˜𝑈(

˜𝑆 + Δ

˜𝑆, 𝑝) to be zero

under the action of the dead loading. Since the structureis affected by a nonlinear behavior, an iterative procedurebased on the Newton-Raphson scheme is adopted. Moredetails regarding the solving procedure to calculate the zeroconfiguration are reported in Section 3, which is, essentially,devoted to analyzing the numerical implementation of theproposed modeling.

With reference to the structural bridge scheme reportedin Figure 1, it is assumed that the cable system is composed ofundamaged elements and a fixed number of elements, whichare affected during the moving loads by an internal cablefailure mechanism. In particular, the damage formulation isdeveloped by means of explicit time dependence laws, whichreproduce the phenomenon of the cable release in a shorttime step due to an accidental failure in the cable system,such as the one occurring during a terroristic or vandalismattack and thus is not related to creep damage. On thecontrary, the remaining stays are assumed to be undamagedand thus the damage function is set to zero. The stay failureof the 𝑖th generic stay is simulated, by introducing a damageevolution law in the constitutive relationships. In particular,a time dependent damage function is utilized, which reduces,during the assumed failure time step, the elastic materialproperties of the cable as well as the corresponding initialinternal stresses. Among the several approaches availablefrom the literature, the formulation is assumed to be consis-tent with the ContinuumDamageMechanicsTheory [21, 22].In this context, degradation functions, physically based, areintroduced to reproduce a typical damage phenomenon ofthe cable system. The definition of the damage function isdeveloped introducing a scalar variable 𝐷 representative ofthe failure mechanism, defined by means of the followingscalar expression:

𝐷(𝑠, 𝑡) =

𝑘 (𝑠, 𝑡) − 𝑘𝑑 (𝑠, 𝑡)

𝑘 (𝑠, 𝑡)

with 𝐷 ∈ [0, 1] , (3)

where 𝑠 is the curvilinear coordinate used to describe the arclength of the cable; 𝑘 and 𝑘

𝑑are the actual and residual stiff-

ness properties, for example, area or modulus, of the cable,respectively. In order to reproduce the failure mechanismof the generic cable element, a time dependent damage law

G

P

Linear configurationStatic profile

1X

3X

2X

ij

ij

Dynamic configuration

UG2

UP1UP2

UP3

UG1

UG3

Figure 3: Initial and current configurations of the cable, supportmotion due to girder (𝐺), and pylon (𝑃) deformation.

based on Kachanov or Rabotnov type approach [21, 22] isintroduced. The damage function is expressed through anexplicit evolution law in the failure time domain, definedby the initial (𝑡0) and the final (𝑡𝑓) times of the failuremechanism and the critical value of the damage variable 𝐷𝑐,by means of the following expression:

𝐷(𝑠, 𝑡) = 1 − (1 −

𝑡𝑓− (𝑡 − 𝑡

0)

𝑡𝑓

)

1/(1+𝑚)

, (4)

with

𝑡𝑓 =

1 − (1 − 𝐷𝑐)𝑚+1

𝑚 + 1

(

𝑆

𝐴0

)

−𝑚

, (5)

where 𝑚 and 𝐴0are material damage parameters. More

details, about the mathematical relationships leading to (4)-(5) can be found in the appendix. It is worth noting thatthe critical damage parameter represents the value corre-sponding to the occurrence of the complete failure of thecable element, in which the 𝑖th generic stay does not havethe possibility of transferring any internal stresses fromthe girder and the tower and thus can be considered notto produce any significant effects on the global behaviorof the bridge. Moreover, the above formulation cannot beconsidered in the creep damage framework, since the cablerelease phenomenon is produced in a short time step and it isconcerned to simulate, essentially, an accidental failure in thecable system.

The formulation is presented assuming that the cableelement is deformed in its initial cable configuration underdead loading, that is,Ω

𝑐, and thus the deformed configuration

of the cable due to the application of the live loads can bedescribed by the following additive expression (Figure 3):

˜𝜑 (

˜𝑋, 𝑡) = (𝑋1

+ 𝑈𝐶

1(

˜𝑋, 𝑡))

˜𝑛1+ (𝑋

2+ 𝑈

𝐶

2(

˜𝑋, 𝑡))

˜𝑛2

+ (𝑋3 + 𝑈𝐶

3(

˜𝑋, 𝑡))

˜𝑛3,

(6)

where˜𝑋 with

˜𝑋𝑇= [𝑋

1, 𝑋

2, 𝑋

3] is the positional vector of

the cable cross-sections𝑈𝐶𝑖with 𝑖 = 1, 3 are the displacement

components in the local reference axis 𝑋𝑖described by the

basis 𝑛𝑖of the coordinate system. The main equations for

Mathematical Problems in Engineering 5

X

2c

R(s) R(s+ds)

ds

s

X3

X1

e

X

ΦG1

ΦG2ΦG3

UG2

UG1

UG3

RX3X2

UmX3

RX3

RX3

λ, λ0

λ, λ0

(a) (b)

j

i

Figure 4: Girder kinematic (a) and moving loads description (b).

the generic 𝑖th stay are described by introducing the damageevolution law defined by (4) in the constitutive law definedin terms of the Green-Lagrange strain and the second Piola-Kirchhoff stress as

𝑆𝐶

1(

˜𝑋, 𝑡) = 𝐶

𝐶[1 − 𝛿

𝑖𝑘𝐷 (𝑡)] 𝐸

𝐶

1(

˜𝑋, 𝑡) + 𝑆

𝐶

0,

𝐸𝐶

1(

˜𝑋, 𝑡) = 𝑈

𝐶

1,𝑋1

(

˜𝑋, 𝑡)

+

1

2

[𝑈2

1,𝑋1

(

˜𝑋, 𝑡) + 𝑈

2

2,𝑋1

(

˜𝑋, 𝑡) + 𝑈

2

3,𝑋1

(

˜𝑋, 𝑡)]

𝐶

,

(7)

where 𝐶𝐶 is the elastic modulus, 𝛿𝑖𝑘 is the Kronecker deltafunction, 𝑆𝐶

1is the second Piola-Kirchhoff axial stress, 𝐸𝐶

1is

the Green-Lagrange axial strain, and 𝑆𝐶0is the second Piola-

Kirchhoff axial stress in the zero configuration.Thegoverningequations of the motion of a single cable are expressed bymeans of the following partial differential equations [23]:

𝑑

𝑑𝑋1

[𝑁𝐶

1+ 𝑁

𝐶

1

𝑑𝑈𝐶

1

𝑑𝑋1

] − 𝑏1 − 𝜇𝑐�̈�𝐶

1= 0,

𝑑

𝑑𝑋1

[𝑁𝐶

1

𝑑𝑈𝐶

2

𝑑𝑋1

] − 𝜇𝑐�̈�𝐶

2= 0,

𝑑

𝑑𝑋1

[𝑁𝐶

1

𝑑𝑈𝐶

3

𝑑𝑋1

] − 𝑏2− 𝜇

𝑐�̈�𝐶

3= 0,

with 𝑏1 = 𝜇𝐺𝑔 cos (𝜑1) ,

𝑏2= 𝜇

𝐺𝑔 sin (𝜑

2) ,

(8)

where 𝑁𝐶1is the axial force defined as 𝑁𝐶

1= 𝑆

𝐶

1𝐴𝐶 with

𝐴𝐶 the area of generic 𝑖th cable element, 𝜑

1and 𝜑

2are

the slope angles of the cable along the 𝑋1𝑋2and 𝑋

1𝑋3,

respectively, and 𝑏1and 𝑏

2are the body load projections

in the 𝑋1𝑋2and 𝑋

1𝑋3, respectively. In addition to (8),

boundary conditions on the cable kinematic are needed to

describe the support motion produced by the girder andpylon motion at the corresponding intersection points withthe cable development:

˜𝑈𝐶(

˜𝑋0) =

˜𝑈𝑃, ̇

˜𝑈

𝐶

(

˜𝑋0) = ̇

˜𝑈

𝑃

,

˜𝑈𝐶(

˜𝑋𝑐) =

˜𝑈𝐺, ̇

˜𝑈

𝐶

(

˜𝑋0) = ̇

˜𝑈

𝐺

,

(9)

where˜𝑋0and

˜𝑋𝑐correspond to the initial and final cross-

sections of the cable element; (˜𝑈𝑃, ̇

˜𝑈

𝑃

) and (˜𝑈𝐺, ̇

˜𝑈

𝐺

) are thedisplacement and speed of the pylon (𝑃) and girder (𝐺),respectively, at the corresponding intersections with the cabledevelopment.

2.2. Girder and Moving Loads Interaction. Girder and towersare described by tridimensional geometric nonlinear beamelements bymeans of a formulation based on Euler-Bernoullikinematic assumptions and a Green-Lagrange strain mea-sure.The girder is connected with the cable system at the endpoints of the cable cross-sections and at the bottom cross-section ends of the towers, consistently defined with “H-”or “A-” shaped typologies. With reference to Figure 4(a), thedisplacements of the cross-section for a generic point locatedat the (𝑋1, 𝑋2, 𝑋3) coordinate, that is, (𝑈

𝐺

1, 𝑈

𝐺

2, 𝑈

𝐺

3), are

expressed by the following relationships:

𝑈

𝐺

1(𝑋

1, 𝑋

2, 𝑋

3, 𝑡) = 𝑈

𝐺

1(𝑋

1, 𝑡) + Φ

𝐺

2(𝑋

1, 𝑡) 𝑋

3

− Φ𝐺

3(𝑋

1, 𝑡) 𝑋

2,

𝑈

𝐺

2(𝑋

1, 𝑡) = 𝑈

𝐺

2(𝑋

1, 𝑡) ,

𝑈

𝐺

3(𝑋

1, 𝑋

2, 𝑋

3, 𝑡) = 𝑈

𝐺

3(𝑋

1, 𝑡) + Φ

𝐺

1(𝑋

1, 𝑡) 𝑋

2,

(10)

where (𝑈𝐺1, 𝑈

𝐺

2, 𝑈

𝐺

3) and (Φ𝐺

1, Φ

𝐺

2, Φ

𝐺

3) are the displacement

and rotation fields of the centroid axis of the girder withrespect to the global reference system, respectively.

The external loads are assumed to proceed with constantspeed c from left to right along the bridge development and

6 Mathematical Problems in Engineering

are supposed to be located, eccentrically with respect to thegeometric axis of the girder. The moving system descrip-tion refers to railway vehicle loads, which are reproducedby means equivalent uniformly distributed loads, perfectlyconnected to the girder profile. As a result, the kinematicparameters of the moving system coincide with the onesdefined by the girder, neglecting frictional forces arising fromthe external loads, roughness effects of the girder profile,and local loading distribution produced by railway loadcomponents. However, these assumptions are quite recurrentin the framework of cable supported bridges with longspans, in which, typically, such interaction forces producedby localized dynamic effects are negligible with respect tothe global bridge vibration [24]. Moreover, it is assumed thatthe damping energy is practically negligible. This hypothesisis verified in the context of long span bridges, where it hasbeen proved that the bridge damping effects tend to decreaseas span length increases [25, 26]. Detailed results about theinfluence of damping effects on DAFs have been presentedin [5, 6], from which it transpires that the assumption ofan undamped bridge system leads to greater DAFs. Withreference to the structural scheme reported in Figure 4(b),the infinitesimal reaction forces produced by themoving loadon the girder profile can be expressed as a function of timedependent positional variable 𝑠, with 𝑠(𝑡) = 𝑐𝑡, by means ofthe balance of linear momentum, as follows:

𝑑𝑅𝑋3

= 𝑑𝑋1

{

{

{

𝜆𝑔+

𝑑

𝑑𝑡

[

[

𝜆

𝑑̇𝑈

𝑚

3

𝑑𝑡

(𝑠 (𝑡))]

]

}

}

}

𝑠=𝑋1

= 𝑑𝑋1

{

{

{

𝜆𝑔+

𝑑𝜆

𝑑𝑡

𝑑̇𝑈

𝑚

3

𝑑𝑡

(𝑠 (𝑡))+𝜆

𝑑2 ̇𝑈

𝑚

3

𝑑𝑡2(𝑠 (𝑡))

}

}

}

𝑠=𝑋1

,

𝑑𝑅𝑋2

= 𝑑𝑋1

{

{

{

𝑑

𝑑𝑡

[

[

𝜆

𝑑̇𝑈

𝑚

2

𝑑𝑡

(𝑠 (𝑡))]

]

}

}

}

𝑠=𝑋1

= 𝑑𝑋1

{

{

{

𝑑𝜆

𝑑𝑡

𝑑̇𝑈

𝑚

2

𝑑𝑡

(𝑠 (𝑡)) + 𝜆

𝑑2 ̇𝑈

𝑚

2

𝑑𝑡2(𝑠 (𝑡))

}

}

}

𝑠=𝑋1

,

𝑑𝑅𝑋1

= 𝑑𝑋1

{

{

{

𝑑

𝑑𝑡

[

[

𝜆

𝑑̇𝑈

𝑚

1

𝑑𝑡

(𝑠 (𝑡))]

]

}

}

}

𝑠=𝑋1

= 𝑑𝑋1

{

{

{

𝑑𝜆

𝑑𝑡

𝑑̇𝑈

𝑚

1

𝑑𝑡

(𝑠 (𝑡)) + 𝜆

𝑑2 ̇𝑈

𝑚

1

𝑑𝑡2(𝑠 (𝑡))

}

}

}

𝑠=𝑋1

,

(11)

where 𝑔 is the gravitational acceleration, 𝜆 is the externalmass per unit length, and 𝑈𝑚

𝑖with 𝑖 = 1, 3 are the displace-

ment functions along 𝑋𝑖axis of the moving mass, identified

by the girder kinematic by using (10), as 𝑈𝑚𝑖= 𝑈

𝑖. It

is worth noting that in (11).1, at right hand side, the firstterm represents the dead loading contribution, whereas, thesecond term is produced by the unsteady mass distribution

in the system due to time dependence character of the massfunction arising from the moving loads. Finally, the thirdterm must be calculated taking into account the relativemotion between bridge and the moving mass as follows [27]:

𝑑𝑈

𝑚

𝑖

𝑑𝑡

=

𝜕𝑈

𝑚

𝑖

𝜕𝑡

+

𝜕𝑈

𝑚

𝑖

𝜕𝑠

𝜕𝑠

𝜕𝑡

=

𝜕𝑈

𝑚

𝑖

𝜕𝑡

+

𝜕𝑈

𝑚

𝑖

𝜕𝑠

𝑐,

(12)

𝑑2𝑈

𝑚

𝑖

𝑑𝑡2=

𝑑

𝑑𝑡

[

𝜕𝑈

𝑚

𝑖

𝜕𝑡

+

𝜕𝑈

𝑚

𝑖

𝜕𝑡

𝜕𝑠 (𝑡)

𝜕𝑡

]

=

𝜕2𝑈

𝑚

𝑖

𝜕𝑡2+ 2𝑐

𝜕2𝑈

𝑚

𝑖

𝜕𝑡𝜕𝑠

+ 𝑐2𝜕2𝑈

𝑚

𝑖

𝜕𝑠2.

(13)

It is worth noting that the Eulerian description of themoving system introduces in (13) three terms correspondingto standard, centripetal, and Coriolis acceleration functions,respectively. However, the last two contributions in theacceleration function for the transverse and longitudinaldisplacements, that is, when 𝑖 = 1, 2, are typically negligiblein comparison to the term associated with the standardacceleration and thus they are not considered in the followingcomputations. Substituting (10) into (11) with 𝑈𝑚

𝑖= 𝑈𝑖,

and making the use of (12)-(13), the reaction forces per unitlength produced by the moving system are described by thefollowing expressions:

𝑝𝑋3

=

𝑑𝑅𝑋3

𝑑𝑋1

= 𝜆𝑔 +

𝑑𝜆

𝑑𝑡

[(

𝜕𝑈𝐺

3

𝜕𝑡

+ 𝑒

𝜕Φ𝐺

1

𝜕𝑡

) + 𝑐(

𝜕𝑈𝐺

3

𝜕𝑋1

+ 𝑒

𝜕Φ𝐺

1

𝜕𝑋1

)]

+ 𝜆[

𝜕2𝑈𝐺

3

𝜕𝑡2+ 2𝑐

𝜕2𝑈𝐺

3

𝜕𝑡𝜕𝑋1

+ 𝑐

𝜕2𝑈𝐺

3

𝜕𝑋1

2]

+ 𝜆 ⋅ 𝑒 [

𝜕2Φ𝐺

1

𝜕𝑡2+ 2𝑐

𝜕2Φ𝐺

1

𝜕𝑡𝜕𝑋1

+ 𝑐

𝜕2Φ𝐺

1

𝜕𝑋1

2] ,

𝑝𝑋2

=

𝑑𝑅𝑋2

𝑑𝑋1

=

𝑑𝜆

𝑑𝑡

𝜕𝑈𝐺

2

𝜕𝑡

+ 𝜆

𝜕2𝑈𝐺

2

𝜕𝑡2,

𝑝𝑋1

=

𝑑𝑅𝑋1

𝑑𝑋1

=

𝑑𝜆

𝑑𝑡

[(

𝜕𝑈𝐺

1

𝜕𝑡

− 𝑒

𝜕Φ𝐺

3

𝜕𝑡

)] + 𝜆

𝜕2𝑈𝐺

1

𝜕𝑡2− 𝜆 ⋅ 𝑒

𝜕2Φ𝐺

3

𝜕𝑡2

=

𝑑𝜆

𝑑𝑡

[(

𝜕𝑈𝐺

1

𝜕𝑡

− 𝑒

𝜕2𝑈𝐺

2

𝜕𝑡𝜕𝑋1

)] + 𝜆

𝜕2𝑈𝐺

1

𝜕𝑡2− 𝜆 ⋅ 𝑒

𝜕3𝑈𝐺

3

𝜕𝑡2𝜕𝑋

1

,

(14)

where 𝑒 is the eccentricity of the moving loads with respectto the girder geometric axis. Moreover, in (14), the massfunction during the external load advance can be expressed

Mathematical Problems in Engineering 7

with respect to the global reference system assumed from theleft end of the bridge as

𝜆 (𝑠1, 𝑡) = 𝜆ML𝐻(𝑠1 + 𝐿𝑝 − 𝑐𝑡)𝐻 (𝑐𝑡 − 𝑠1) , (15)

where 𝐻(⋅) is the Heaviside step function, 𝐿𝑝 is the lengthof the moving loads, 𝑠1 is the referential coordinate locatedat the left end of the girder cross-section, that is, 𝑠1 = 𝑙 +𝐿/2 + 𝑋

1, and 𝜆

𝑀is the mass linear density of the moving

system.The kinematic model is consistent with the geomet-

ric nonlinear Euler-Bernoulli theory, in which moderatelylarge rotations are considered [26]. The torsional behaviorowing to eccentric loading is described by means of theclassical De Saint Venant theory. In particular, the strainsare based on the Green-Lagrange measure in which onlythe square of the terms 𝑈2𝐺

𝑖,𝑋1

representing the rotationsof the transverse normal line in the beam is consid-ered. Therefore, starting from (10), the following relation-ships between generalized strain and stress variables areobtained:

𝑁𝐺

1= 𝑁

0(𝐺)

1+ 𝐸𝐴

𝐺𝜀𝐺

1

= 𝑁0(𝐺)

1+ 𝐸

𝐺𝐴𝐺{𝑈

𝐺

1,𝑋1

+

1

2

[𝑈2𝐺

1,𝑋1

+ 𝑈2𝐺

2,𝑋1

+ 𝑈2𝐺

3,𝑋1

]} ,

𝑀𝐺

2= 𝑀

0(𝐺)

2+ 𝐸𝐼

𝐺

2𝜒𝐺

2

= 𝑀0(𝐺)

2+ 𝐸

𝐺𝐼𝐺

2Φ𝐺

2,𝑋1

= 𝑀0(𝐺)

2− 𝐸𝐼

𝐺

2𝑈𝐺

3,𝑋1𝑋1

,

𝑀𝐺

3= 𝑀

0(𝐺)

3+ 𝐸𝐼

𝐺

3𝜒𝐺

3

= 𝑀0(𝐺)

3+ 𝐸

𝐺𝐼𝐺

3Φ𝐺

3,𝑋1

= 𝑀0(𝐺)

3+ 𝐸𝐼

𝐺

3𝑈𝐺

2,𝑋1𝑋1

,

𝑀𝐺

1= 𝐺𝐽

𝐺

𝑡Θ𝐺= 𝐺

𝐺𝐽𝐺

𝑡Φ𝐺

1,𝑋1

,

(16)

where 𝐸𝐴𝐺 and 𝜀𝐺1are the axial stiffness and strain, 𝜒𝐺

2and

𝜒𝐺

3or 𝐸𝐺𝐼𝐺

2and 𝐸𝐺𝐼𝐺

3are the curvatures or the bending

stiffnesses with respect to the 𝑋2 and 𝑋3 axes, respectively,Θ𝐺 and 𝐺𝐺𝐽𝐺

𝑡are the torsional curvature and stiffness,

respectively,𝑁𝐺1is the axial stress resultant,𝑀𝐺

2and𝑀𝐺

3are

the bending moments with respect to the 𝑋2and 𝑋

3axes,

respectively,𝑀𝐺1and 𝐺𝐺𝐽𝐺

𝑡are torsional moment and girder

stiffness, respectively, and (⋅)0 represents the superscript con-cerning the variables associatedwith the “zero configuration.”On the basis of (16), taking into account of (14)-(15) andnotation reported in Figure 4(a), the following governing

equations are derived by means of the local form of dynamicequilibrium equations:

𝑑

𝑑𝑋1

{𝑁𝐺

1(1 +

𝑑𝑈𝐺

1

𝑑𝑋1

)} − 𝜇𝐺�̈�𝐺

1+ 𝑝

𝑋1

= 0,

− 𝐸𝐺𝐼𝐺

2

𝑑4𝑈𝐺

3

𝑑𝑋4

1

−

𝑑

𝑑𝑋1

(𝑁𝐺

1

𝑑𝑈𝐺

3

𝑑𝑋1

)

− 𝜇𝐺�̈�𝐺

3− Φ̈

𝐺

2,𝑋1𝐼𝐺

02− 𝑝

𝑋3

= 0,

𝐸𝐺𝐼𝐺

3

𝑑4𝑈𝐺

2

𝑑𝑋4

1

−

𝑑

𝑑𝑋1

(𝑁𝐺

1

𝑑𝑈𝐺

2

𝑑𝑋1

)

− 𝜇𝐺�̈�𝐺

2− Φ̈

𝐺

3,𝑋1𝐼𝐺

03+ 𝑝

𝑋2

= 0,

𝐺𝐺𝐽𝐺

𝑡

𝑑2Φ𝐺

1

𝑑𝑋2

1

− (𝐼𝐺

01+ 𝜆

0

ML) Φ̈𝐺

1− 𝑝𝑋

3

𝑒 = 0,

(17)

where 𝜆0ML is the per unit length torsional girder mass; 𝜇𝐺is the girder mass per unit length. Additional equationsare required to take into account interelement continuityand initial conditions concerned with solving the dynamicproblem, which can be expressed with reference to the 𝑖thgirder element as follows:

𝑈𝐺

𝑘(0) = 0, �̇�

𝐺

𝑘(0) = 0,

𝑈𝑖(𝐺)

𝑘(0) = 𝑈

𝑖−1(𝐺)

𝑘(𝑙𝑖−1

𝑒) , 𝑈

𝐺

𝑘(𝑙𝑖

𝑒) = 𝑈

𝑖+1

𝑘(0) ,

Φ𝐺

𝑘(0) = 0, Φ̇

𝐺

𝑘(0) = 0,

Φ𝑖(𝐺)

𝑘(0) = Φ

𝑖−1(𝐺)

𝑘(𝑙𝑖−1

𝑒) , Φ

𝐺

𝑘(𝑙𝑖

𝑒) = Φ

𝑖+1

𝑘(0) ,

(18)

where the superscripts 𝑖 + 1 and 𝑖 − 1 indicate the previous orthe next girder elements and the subscript 𝑘, with 𝑘 = 1, 2, 3defines the displacement and rotation directions with respectto the coordinate reference system.

3. Finite Element Implementation

The governing equations reported in the previous sectionintroduce a nonlinear partial differential system, whoseanalytical solution is quite complex to be extracted. Asa consequence, a numerical approach based on the finiteelement formulation is utilized. In particular, starting from(8) and (17), the corresponding weak forms for the 𝑖th finiteelement related to the girder (𝐺) and the cable system (𝐶),respectively, are defined by the following expressions.

Girder

∫

𝑙𝑖

𝑒

𝑁𝐺

1(1 + 𝑈

𝐺

1,𝑋1

)𝑤1,𝑋1

𝑑𝑋1− 𝜇

𝑔 ∫

𝑙𝑖

𝑒

�̈�𝐺

1𝑤1𝑑𝑋

1

− 𝜆ML ∫𝑙𝑖

𝑒

[−𝛿1+ 𝛿

2] �̇�

𝐺

1𝑤2𝑑𝑋

1− 𝜆ML ∫

𝑙𝑖

𝑒

𝐻1𝐻2�̈�𝐺

1𝑤2𝑑𝑋

1

−

2

∑

𝑗=1

𝑁𝐺

1𝑗𝑈𝐺

1𝑗= 0,

8 Mathematical Problems in Engineering

∫

𝑙𝑖

𝑒

{−𝑀𝐺

2𝑤2,𝑋1𝑋1

+ (𝑁𝐺

1𝑈𝐺

3,𝑋1

)𝑤2,𝑋1

} 𝑑𝑋1

− 𝜇𝑔 ∫

𝑙𝑖

𝑒

�̈�𝐺

3𝑤2𝑑𝑋1 + 𝐼

𝐺

02∫

𝑙𝑖

𝑒

�̈�𝐺

3,𝑋1

𝑤2𝑑𝑋1

− 𝜆ML ∫𝑙𝑖

𝑒

[−𝛿1+ 𝛿

2] (�̇�

𝐺

3+ 𝑐𝑈

𝐺

3,𝑋1

)𝑤2𝑑𝑋

1

− 𝜆ML ∫𝑙𝑖

𝑒

𝐻1𝐻2[(�̈�

𝐺

3+ 2𝑐�̇�

𝐺

3,𝑋1

+ 𝑐2𝑈𝐺

3,𝑋1𝑋1

) + 𝑔]𝑤2𝑑𝑋

1

−

2

∑

𝑗=1

𝑇𝐺

3𝑗𝑈𝐺

3𝑗−

2

∑

𝑗=1

𝑀𝐺

2𝑗Φ𝐺

3𝑗= 0,

∫

𝑙𝑖

𝑒

{𝑀𝐺

3𝑤3,𝑋1𝑋1

+ (𝑁𝐺

1𝑈𝐺

2,𝑋1

)𝑤3,𝑋1

} 𝑑𝑋1

− 𝜇𝑔∫

𝑙𝑖

𝑒

�̈�𝐺

2𝑤3𝑑𝑋

1− 𝐼

𝐺

03∫

𝑙𝑖

𝑒

�̈�𝐺

2,𝑋1

𝑤3𝑑𝑋

1

− 𝜆ML ∫𝑙𝑖

𝑒

[−𝛿1 + 𝛿2] �̇�𝐺

2𝑤2𝑑𝑋1

− ∫

𝑙𝑖

𝑒

𝐻1𝐻2[�̈�

𝐺

2+ 𝑔 − 𝑒�̈�

𝐺

2,𝑋1]𝑤

2𝑑𝑋

1

− 𝜆ML𝑒 ∫𝑙𝑖

𝑒

[−𝛿1+ 𝛿

2] �̇�

𝐺

2,𝑋1

𝑤2𝑑𝑋

1

−

2

∑

𝑗=1

𝑇𝐺

2𝑗𝑈𝐺

2𝑗−

2

∑

𝑗=1

𝑀𝐺

3𝑗Φ𝐺

2𝑗= 0,

∫

𝑙𝑖

𝑒

𝑀𝐺

1𝑤4,𝑋1

𝑑𝑋1− 𝐼

01 ∫

𝑙𝑖

𝑒

Φ̈𝐺

1𝑤4𝑑𝑋

1

+ 𝜆ML (𝑒 +𝜆0

ML𝜆ML

)

× ∫

𝑙𝑖

𝑒

𝐻1𝐻2 (Φ̈𝐺

1+ 2𝑐Φ̇

𝐺

1,𝑋1

+ 𝑐2Φ𝐺

1,𝑋1𝑋1

)𝑤4𝑑𝑋1

− 𝜆𝑀𝐿(𝑒 +

𝜆0

ML𝜆ML

)∫

𝑙𝑖

𝑒

[−𝛿1+ 𝛿

2] (Φ̇

𝐺

1+ 𝑐Φ

𝐺

1,𝑋1

)𝑤4𝑑𝑋

1

−

2

∑

𝑗=1

𝑀𝐺

1𝑗Φ𝐺

1𝑗= 0.

(19)

Cable System

∫

𝑙𝑖

𝑒

𝑁𝐶

1(1 + 𝑈

𝐶

1,𝑋1)𝑤

1,𝑋1

𝑑𝑋1− 𝜇

𝑐 ∫

𝑙𝑖

𝑒

�̈�𝐶

1𝑤1𝑑𝑋

1

− ∫

𝑙𝑖

𝑒

𝑏1𝑤1𝑑𝑋1 −

2

∑

𝑗=1

𝑁1𝑗𝑈𝐶

1𝑗= 0,

∫

𝑙𝑖

𝑒

𝑁𝐶

1𝑤2,𝑋1

𝑑𝑋1− 𝜇

𝑐 ∫

𝑙𝑖

𝑒

�̈�𝐶

2𝑤2𝑑𝑋

1−

2

∑

𝑗=1

𝑇𝐶

2𝑗𝑈𝐶

2𝑗= 0,

∫

𝑙𝑖

𝑒

𝑁𝐶

1𝑤3,𝑋

1

𝑑𝑋1 − 𝜇𝑐 ∫

𝑙𝑖

𝑒

�̈�𝐶

3𝑤3𝑑𝑋1

− ∫

𝑙𝑖

𝑒

𝑏3𝑤3𝑑𝑋

1−

2

∑

𝑗=1

𝑇𝐶

3𝑗𝑈𝐶

3𝑗= 0,

(20)

where 𝐻1= 𝐻(𝑠

1+ 𝐿

𝑝− 𝑐𝑡), 𝐻

2= 𝐻(𝑐𝑡 − 𝑠

1), 𝛿

1=

𝛿(𝑠1 + 𝐿𝑝 − 𝑐𝑡), 𝛿2 = 𝛿(𝑐𝑡 − 𝑠1), 𝛿(⋅) represents the delta

Dirac functions, (𝑁1𝑖, 𝑇2𝑖, 𝑇3𝑖,𝑀2𝑖,𝑀3𝑖)𝑘 with 𝑘 = 𝐶, 𝐺 and

𝑖 = 1, 2 represents the internal forces applied at the end node𝑖 of the generic cable (𝐶) or girder (𝐺) element. Moreover,the pylon governing equations can be easily obtained from(19), by removing all the terms related to the moving loadsand changing the relative variables from the superscript (⋅)𝐺

to (⋅)𝑃and the parameters concerning the mechanical andmaterial characteristics. As a consequence, for conciseness,the governing equations concerning the pylon dynamicbehavior are not reported.

Finite element expressions are written starting from theweak forms previously reported, introducing Hermit cubicinterpolation functions (𝜉

𝑖) for the girder and pylon flexures

in the𝑋1𝑋2and𝑋

2𝑋3deformation planes and Lagrange lin-

ear interpolation functions (𝜁𝑖) for the cable system variables

and the remaining variables of the girder and the pylons:

˜𝑈𝐶(

˜𝑟, 𝑡) =

˜𝑁𝐶(

˜𝑟)

˜𝑞𝐶(𝑡) ,

˜𝑈𝐺(

˜𝑟, 𝑡) =

˜𝑁𝐺

˜𝑞𝐺(𝑡) ,

˜𝑈𝑃(

˜𝑟, 𝑡) =

˜𝑁𝑃

˜𝑞𝑃(𝑡) ,

(21)

where˜𝑞𝐶and

˜𝑞𝐺are the vectors collecting the nodal degrees

of freedom of the cable, girder respectively,˜𝑁𝐶,

˜𝑁𝐺, and

˜𝑁𝑃

are the matrixes containing the displacement interpolationfunction for cable element (𝐶), girder (𝐺), and pylons (𝑃),and

˜𝑟 is the local coordinate vector of the 𝑖th finite element.

The discrete equations in the local reference system of the 𝑖thelement are derived substituting (21) into (19)-(20), leading tothe following equations in matrix notation:

(

˜𝑀𝐺

𝑆+

˜𝑀𝐺

NS)̈

˜𝑈

𝐺

+ (

˜𝐶𝐺

𝑆+

˜𝐶𝐺

NS) ˜�̇�

𝐺

+ (

˜𝐾𝐺

𝑆+

˜𝐾𝐺

NS) ˜𝑈𝐺=

˜𝑃𝐺+

˜𝑄𝐺,

˜𝑀𝑃̈

˜𝑈

𝑃

+

˜𝐶𝑃

˜�̇�

𝑃

+

˜𝐾𝑃

˜𝑈𝑃=

˜𝑃𝑃+

˜𝑄𝑃,

˜𝑀𝐶̈

˜𝑈

𝐶

+

˜𝐶𝐶

˜�̇�

𝐶

+

˜𝐾𝐶

˜𝑈𝐶=

˜𝑈𝐶+

˜𝑄𝐶,

(22)

where˜𝑀𝑖is the mass matrix,

˜𝐶𝑖is the damping matrix,

˜𝐾𝑖is

the stiffnessmatrix,˜𝑃𝑖is the load vector produced by the dead

and live loading,˜𝑄𝑖is the unknown force vector collecting

the point sources, and the subscripts (⋅)𝑆or (⋅)NS refer to

standard or nonstandard terms, respectively, introduced inthe discrete equations. Most of the matrixes reported in (22)can be easily recovered from the literature [26]. Contrarily,the matrixes

˜𝑀NS, ˜

𝐶NS, and ˜𝐾NS collect the nonstandard

terms arising from the inertial description of the live loadsand the interaction behavior between moving loads andbridge motion and are defined by the following expressions:

(

˜𝑀𝐺

NS) = ∫𝑙𝑖

𝑒

𝐻1𝐻2(

˜𝑁𝐺(𝑇)

˜Λ𝑀1 ˜𝑁𝐺) 𝑑𝑋

1,

(

˜𝐶𝐺

NS)𝑖𝑗= ∫

𝑙𝑖

𝑒

[(−𝛿1+ 𝛿

2) (

˜𝑁𝐺(𝑇)

˜Λ𝐶1 ˜𝑁𝐺)

+𝐻1𝐻2(

˜𝑁𝐺(𝑇)

,𝑋1 ˜Λ𝐶2 ˜𝑁𝐺)] 𝑑𝑋

1,

Mathematical Problems in Engineering 9

(

˜𝐾𝐺

NS)𝑖𝑗= ∫

𝑙𝑖

𝑒

[(−𝛿1+ 𝛿

2) (𝑁

∼

𝐺(𝑇)

1,𝑋1 ˜Λ𝐾1 ˜𝑁𝐺)

+𝐻1𝐻2(

˜𝑁𝐺(𝑇)

,𝑋1𝑋1 ˜Λ𝐾2 ˜𝑁𝐺)] 𝑑𝑋

1,

(23)

where the matrixes˜Λ𝑀1

,

˜Λ𝐶1,2

, and˜Λ𝐾1,2

, which assemble thecoefficients associated with the inertial contribution arisingfrom the moving loads and girder interaction, are defined as:

Λ𝑀1

= diag [𝜆ML, 𝜆ML, 𝜆ML, 𝜆ML𝑒 + 𝜆0

ML, 0, 0] ,

Λ𝑀1

= diag [0, −𝑒𝜆ML, 0, 0, 0, 0] ,

Λ𝐶1

= diag [𝜆ML, 𝜆ML, 𝜆ML, 𝜆ML𝑒 + 𝜆0

ML, 0, −𝜆ML𝑒] ,

Λ𝐶2

= diag [0, 0, 2𝑐𝜆ML, 2𝑐 (𝜆ML𝑒 + 𝜆0

ML) , 0, 0] ,

Λ𝐾1

= diag [0, 0, 𝑐𝜆ML, 𝑐 (𝜆ML𝑒 + 𝜆0

ML) , 0, 0] ,

Λ𝐾2

= diag [0, 0, 𝑐2𝜆ML, 𝑐2(𝜆ML𝑒 + 𝜆

0

ML) , 0, 0] .

(24)

In order to reproduce the bridge kinematic correctly,additional relationships to define the connections betweengirder, pylon, and cable system are necessary. In particular,the cable system displacements should be equal to those ofthe girder and the pylons at the corresponding intersectionpoints; thus, the bridge kinematic is restricted by means ofthe following constrain equations:

𝑈𝐺

3(

˜𝑋𝐶𝑖

, 𝑡) + Φ𝐺

1(

˜𝑋𝐶𝑖

, 𝑡) 𝑏 = 𝑈𝐶

3(

˜𝑋𝐶𝑖

, 𝑡) ,

𝑈𝐺

1(

˜𝑋𝐶𝑖

, 𝑡) − Φ𝐺

3(

˜𝑋𝐶𝑖

, 𝑡) 𝑏 = 𝑈𝐶

1(

˜𝑋𝐶𝑖

, 𝑡) ,

𝑈𝑃

1(

˜𝑋𝑃, 𝑡) = 𝑈

𝐶

1(

˜𝑋𝑃, 𝑡) ,

𝑈𝑃

2(

˜𝑋𝑃, 𝑡) = 𝑈

𝐶

2(

˜𝑋𝑃, 𝑡) ,

𝑈𝑃

3(

˜𝑋𝑃, 𝑡) = 𝑈

𝐶

3(

˜𝑋𝑃, 𝑡) ,

(25)

where˜𝑋𝐶𝑖

and˜𝑋𝑃represent the vectors containing the inter-

section positions of the 𝑖th cable element and the pylon topcross-section, respectively. Finally, starting from (22), takinginto account (25) as well as the balance of secondary variablesat the interelement boundaries, the resulting equations of thefinite element model are

˜𝑀 ̈

˜𝑄 +

˜𝐶 ̇

˜𝑄 +

˜𝐾

˜𝑄 =

˜𝑃, (26)

where˜𝑄with

˜𝑄 =

˜𝑈𝐵∪

˜𝑈𝐺∪

˜𝑈𝑃is the generalized coordinate

vector containing the kinematic variables associated withthe girder, the pylons, and the cable system,

˜𝑀,

˜𝐶, and

˜𝐾

are the global mass, stiffness, and damping matrixes, and˜𝑃

is the loading vector. Since the structural behavior of eachelement depends on the deformation state of the members,the governing equations defined by (26) will change continu-ously as the structure deforms. Moreover, the external loadsowing to the presence of its own moving mass determinea time dependent mass distribution function on the girder

profile. Consequently, the discrete equations are affected bynonlinearities in the stiffness matrix and time dependence inthe mass matrix.

The governing equations are solved numerically, using auser customized finite element program, that is, COMSOLMultiphysics TM version 4.1 [27]. The analysis is performedby means of two different stages. Initially, a preliminaryanalysis is devoted to calculating the initial stress distributionin the cable system, that is, “zero configuration.” In thiscontext, the shape optimization procedure is developed,consistently with a Newton-Raphson iteration scheme. Sincethe loading condition refers to the application of dead loadingonly, the analysis is developed in the framework of a staticanalysis. In particular, with reference to (2), at the 𝑘thiteration, the incremental value of the cable system stressesΔ

˜𝑆𝑘is determined by neglecting the second order residuum

obtained by the Taylor expansion, that is, 𝑜‖Δ˜𝑆2

𝑘‖ and by

solving the linear equation system enforcing the bridgedeformations to verify the constraint equations given by thezero configuration, namely,

˜𝑈(

˜𝑆𝑘+ Δ

˜𝑆𝑘, 𝜆) = 0, as.

Δ

˜𝑆𝑘= −[

𝑑

˜𝑈

𝑑

˜𝑆

−1

(˜𝑆𝑘,𝜆)

]

𝑘˜𝑈 (

˜𝑆𝑘, 𝜆) . (27)

The iterative values of the cable stresses are determinedfrom (27) as

˜𝑆𝑘+1

=

˜𝑆𝑘+ Δ

˜𝑆𝑘, whereas the final quantities

are obtained by means of an incremental-iterative procedure,which is developed on the basis of a tolerance convergencecriterion. In particular, the residual norm is checked to belower than a fixed prescribed tolerance, that is, ‖

˜𝑈(

˜𝑆𝑘+

Δ

˜𝑆𝑘, 𝜆)‖ ≤ toll; if it is not verified, new values of the stresses

are determined by means of an iterative scheme defined onthe basis of (27). Once the initial configuration is determinedin terms of the initial cable stress and strain distribution, asequential dynamic analysis is developed. Since the governingequations introduce essentially a time dependent nonlinearequation system, an incremental and iterative integrationscheme is required. The algebraic equations are solved bya direct integration method, which is based on an implicittime integration scheme. In particular, an implicit temporaldiscretization of order two using a Backward DifferentiationFormula (BDF-2) with an adaptative time step is utilized.Moreover, a Newton-Raphson scheme in the time stepincrement based on the secant formulation is utilized forthe nonlinearities involved in the governing equations [28].In order to guarantee accuracy in the predicted results,particular attention is devoted to the choice of the timeintegration step, which, assuming small vibrations about thenonlinear equilibrium configuration under dead loads, canbe defined as a function of the periods of those vibrationmode shapes having a relevant participation on the response.However, in the case of moving load excitation, the dynamicsolution strongly depends on the speed of themoving system,since different vibration frequencies are activated for low orlarge transit speeds [27]. In the present analyses, the initialintegration time step, which is automatically reduced due tothe time adaptation procedure, is assumed as at least 1/1000of the observation period defined as the time necessary forthe moving train to cross the bridge. This value turns out to

10 Mathematical Problems in Engineering

be always lower than 1/100 of the 50th natural period of thebridge structure, the first natural period being the largest one.

4. Results

Results are proposed to investigate the behavior of cable-stayed bridge subjected to an accidental failure in the cablesystem. From the design point of view, existing codes on thisargument, that is, PTI [15] and SETRA [16], recommend toreproduce the effects of such loading scheme by performinga quasistatic analysis and taking into account the dynamicamplification effects by means of amplification factors in therange between 1.5–2.0. For this reason, the purpose of thefollowing results is to identify the dynamic amplificationeffects produced by the cable failure by using a refineddynamical formulation in place of a simplified quasi-staticone. In order to quantify the amplification effects producedby themoving loads over the static solution, numerical resultsare proposed in terms of dynamic amplification factors forundamaged (UD) and damaged (𝐷) cable-stayed structuresin terms of the moving loads and the bridge characteristics. Itis worth noting that UD cable-stayed configurations refer toa bridge structures, in which the stays are not affected by anydamage mechanisms. Contrarily, damaged (𝐷) cable-stayedscheme corresponds to bridge configurations, in which oneormore stays are subjected to the explicit damagemechanismdefined on the basis of (3). Moreover, in the followingresults, the cross-section exposed to damage is assumed tobe located at the intersection between the stay and girder.Thedynamic amplification factors (DAFs) for the generic variable𝑋 under investigation related to damaged or undamagedstructural configuration, namely,Φ𝐷

𝑋andΦUD

𝑋are defined by

the following relationships:

Φ𝐷

𝑋=

max (𝑋𝐷, 𝑡 = 0 ⋅ ⋅ ⋅ 𝑇)𝑋𝐷

ST,

ΦUD𝑋=

max (𝑋UD, 𝑡 = 0 ⋅ ⋅ ⋅ 𝑇)𝑋

UDST

,

(28)

where 𝑇 is the observation period and the subscript (⋅)STrefers to the value of the variable determined in a staticanalysis. Moreover, an additional description of the DAFis proposed to quantify the relationship between damagedand undamaged configurations by means of the followingexpression:

Φ𝐷−UD𝑋

=

max (𝑋𝐷, 𝑡 = 0 ⋅ ⋅ ⋅ 𝑇)𝑋

UDST

. (29)

It is worth noting that the formulation of theDAF definedby (30) characterizes the dynamical amplification effects ofthe investigated variable with respect to the static response inthe undamaged structural configuration of the bridge. Thisparameter can be useful for design purpose, since when thiskind of DAF is known in advance the designer is able tocontrol the amplification effects of a generic bridge variabledue to the combined action produced by the failure mecha-nism and the inertial forces, avoiding the analysis suggested

by the existing codes on the argument [15, 16]. The bridgeandmoving load dimensioning is selected in accordance withthe values utilized in practical applications and due, mainly,to both structural and economical reasons. The parametricstudy is developed by using dimensionless parameters, whichare typically utilized to identify the structural andmechanicalproperties of a long-span bridge and moving loads. Inparticular, the cross-sectional stay areas are designed insuch a way that the tensile stresses in the cables are alwaysless than a design maximum allowable value, namely, 𝜎𝑎,adopted on the basis of fatigue requirements. Such value isdefined by the existing codes and it is considered as a knownquantity, depending on the cross-section configuration andmaterial technology [9, 10], which is, typically, provided bythe manufactures of the cables and assumed as a fractionof the rupture stress of the steel [29]. At first, the staydimensioning procedure is defined on the basis of the self-weight configuration. The stresses of a generic stay or theanchor stays are assumed to be equal to fixed working stressvalues, namely, 𝜎

𝑔and 𝜎

𝑔0, respectively, which are defined in

the basis of the ratio between live and self-weight loads, theallowable stay stress, and the geometric characteristics of thebridges, bymeans of the following relationships [7, 10, 29–31]:

𝜎𝑔=

𝑔

𝑔 + 𝑝

𝜎𝑎

𝜎𝑔0= 𝜎

𝑎{1 +

𝑝

𝑔

[1 − (

2𝐿

𝑙

)

2

]

−1

}

−1

.

(30)

It is worth noting that since the stress increments in thestays can be supposed to be proportional to the live loads𝑝, 𝜎

𝑔and 𝜎

𝑔0, defined on the basis of (28), are able to

control the stress variation in the cable system producedby the live load application, avoiding that internal stressesin the stays exceed the allowable stress 𝜎

𝑎. Moreover, since

the present analysis is devoted to investigate the behavior oflong span bridges, it is reasonable to consider the girder aspractically free from bending moments for reduced valuesof the stay spacing step and thus that it is dominated bymeans of a prevailing truss behavior [9, 10, 29, 30]. As aconsequence, it is possible to regard the deck as a continuousstructure supported by a continuous distribution of stays,whose reaction forces can be calculated by using simple equi-librium relationships, able to provide cable dimensioning. Inparticular, the cross-sectional of the 𝑖th stay area, namely,𝐴𝐶

𝑖, is designed in such a way that the dead loads produce

a constant stress level over all the distributed elements equalto 𝜎

𝑔. Similarly, for the anchor stays the cross-sectional

geometric area𝐴𝐶0is designed in such away that the allowable

stress 𝜎𝑔0 is obtained for live loads 𝑝 applied to the centralspan only. Therefore, the geometric measurement for thecables system can be expressed by the following equations[7–10, 30]:

𝐴𝐶

𝑖=

𝑔Δ𝑖

𝜎𝑔sin𝛼

𝑖

,

𝐴𝐶

0=

𝑔𝑙

2𝜎𝑔0

[1 + (

𝑙

𝐻

)

2

]

1/2

[(

𝐿

2𝑙

)

2

− 1] ,

(31)

Mathematical Problems in Engineering 11

where 𝛼𝑖is the slope of a generic stay element with respect

to the reference system, (𝐿, 𝑙,𝐻) are representative geometriclengths of the bridge structure, and Δ is the stay spacing step.Moreover, aspect ratio, pylon stiffness, allowable cable stressand bridge, and moving loads characteristics are assumed asequal to the following representative values [7–10]:

𝐿

2𝐻

= 2.5,

ℓ

𝐻

=

5

3

,

𝐿

Δ

=

1

100

,

𝑎 = (

𝛾2𝐻2𝐸𝐶

12𝜎3

𝑔

) , (bridge geometry)

𝐼𝐺

2

𝐼𝐺

3

=

1

10

,

𝐼𝐺

02

𝐼𝐺

03

=

1

10

,

𝐽𝐺

𝑡

𝐴2= 100,

𝐼𝑃

2

𝐼𝑃

3

=

1

10

,

𝐸𝐺,𝑃

𝐶𝐶= 1 (girder and pylon)

𝜀𝐹= (

4𝐼𝐺

2𝜎𝑔

𝐻3𝑔

)

1/4

,

𝐾𝑃

𝑔

= 50, (girder and pylon)

𝜆0

ML𝜇𝐺𝑏2= 1,

𝜆ML𝜇𝐺

= 1,

𝑝

𝑔

= 1,

𝑒

𝑏

= 0.5, (moving loads)

𝐴𝐶

0𝐸𝐶

𝑔𝐿

= 100, 𝜏0 =

𝑐𝑡0

𝐿

, 𝑚 = 1,

𝜎𝑎

𝐶𝐶=

7.2E82.1E11

(cable system) ,(32)

where 𝐻 is the pylon height, 𝐶𝐶 is the modulus of elasticityof the cable, 𝐾

𝑃is the in-plane flexural top pylon stiffness,

𝑏 is half girder cross-section width, 𝑡0is the initial time

in which the damage mechanism starts the degradationeffects, 𝜏

0is the normalized failure time, and 𝑚 is the

parameter which controls the time evolution of the damagecurve.

At first, the failure condition, located at the girder/cableintersection, is supposed to be produced in one anchorstay, located laterally to the longitudinal axis of the girder.The time of the failure mode is assumed to be consistentwith values typically observed in experimental tests, whoserepresentative value in the computations is assumed to be0.005 sec [16]. It is worth noting that additional analyses, notreported for the sake of brevity, show that the influence ofthe failure time step on the dynamical amplification factorsis practically negligible and within 8% up to very highmoving load speeds, that is, 𝑐 = 160m/s. Moreover, in thispreliminary analysis it is assumed that the failure of the staystarts when the moving load front reaches the midspan, thatis, 𝑡

𝑅= 𝑐/(𝑙 + 𝐿/2). This configuration can be considered as

an average value with respect to the position which assumesthe moving system on the bridge development and will betaken as a reference in the subsequent developments, in

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

1

1.5

2

2.5

3

b b

e

ΘMass

No massΦD

ΦD−UD

ΦUD

εF = 0.1, a = 0.1

ΦU

G 3

Figure 5: Dynamic amplification factors of the midspan verticaldisplacement for a bridge structure with A-shaped tower as afunction of the normalized speed parameter: effect of the failuremechanism and moving load schematization.

which different instants where failure starts are considered.The behavior of the bridge is analyzed to investigate therelationship between dynamic amplification factors (DAFs)and the normalized speed parameter of the moving system,that is, Θ = 𝑐(𝜇

𝐺𝜎𝑔/𝐸𝐺𝑔𝐻)

1/2as a function of the towertopology and the moving mass schematization. Moreover,the dynamic response of the bridge is evaluated by meansof comparisons between damaged (𝐷) or undamaged bridge(UD) structures. The results, reported in Figures 5, 6, 7,and 8, are defined through the relationships between movingsystem normalized speed and dynamic amplification factorsfor the midspan vertical displacement and bending moment.Nevertheless, the DAF evolution curves denote a tendencyto increase with the speeds of the moving system. Theresults show that the DAFs developed for bridge structuresaffected by a failure mechanisms in the cable system are,typically, larger than those obtained assuming undamagedbridge configurations. Moreover, underestimations in theDAF predictions are observed in those cases, in whichthe inertial contributions arising from the external movingmass are completely neglected. The analyses presented abovein terms of the DAFs Φ𝐷−UD

𝑋for both the damaged and

undamaged configurations point out that bridge structureswith A- or H-shaped typologies undergoing damage arecharacterized by large dynamic amplifications with respectto the undamaged case. As a matter of fact, the ranges ofmaximum value of the DAFs increase from [1.47–1.52] in theundamaged configuration to [2.5–3.6] in the damaged onefor the midspan displacement, and similarly from [2.5–4.5]to [5.4–8.3] for the midspan bending moment. It is worth

12 Mathematical Problems in Engineering

0.5

1

1.5

2

2.5

3

3.5

4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Mass

No massΦD

ΦD−UD

ΦUD

b b

e

εF = 0.1, a = 0.1

Θ

ΦU

G 3

Figure 6: Dynamic amplification factors of the midspan verticaldisplacement for a bridge structure with H-shaped tower as afunction of the normalized speed parameter: effect of the failuremechanism and moving load schematization.

noting that the DAFs from the undamaged bridge configu-ration are affected by large amplifications, especially for thevariables concerning the bending moments. This behaviorcan be explained in view of the prevailing truss behaviorof the structure and the nonstandard inertial forces arisingfrom the moving load application, which produce largerbending moments with respect to the ones obtained in thestatic configuration [8]. For all investigated cases, the bridgestructures based on H-shaped tower topology are affectedby larger dynamic amplifications than those structures basedon A-shaped tower. This behavior can be explained in viewof the differences in the cable stress distribution betweenundamaged and damages structures. In particular, the H-shaped tower bridges with respect to the A-shaped ones,owing to the failure of the lateral anchor stay, are affected byan unbalanced distribution of the internal stresses in the cablesystem, which produce larger torsional rotations and verticaldisplacements of the tower and the girder, respectively.To this aim, in Figure 9(a), a comparison of A- and H-shaped towers in terms of the DAFs (Φ

Φ𝐺

1

) and maximumobserved value of the torsional rotation (Φ1) at the midspancross-section is reported. These results show how theH-shaped towers are much more affected by the investigatedfailure condition than the A-shaped towers, since lagertorsional rotations of the tower and the girder are expected.In Figure 9(b), a synoptic representation of this deformationscheme affecting H-shaped tower bridges is reported. Finally,the influence of the failure mode characteristics on the DAFsconcerning the position which the moving system assumeson the bridge development is investigated. In particular, fora fixed value of the moving system speed, that is, 𝜗 = 0.102,

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Mass

No massΦD

ΦD−UD

ΦUD

εF = 0.1, a = 0.1

Θ

b b

e

ΦM

G 2Figure 7: Dynamic amplification factors of the midspan bendingmoment for a bridge structure with A-shaped tower as a functionof the normalized speed parameter: effect of the failure mechanismand moving load schematization.

analyses are developed in terms of normalized failure time𝜏0. The analyzed cases correspond to failure modes in which

the moving load front is located at the entrance, the exitconfigurations, or at specified positions on the bridge devel-opment, that is, 𝑋

1/𝐿 = [𝑙/𝐿, (𝑙 + 𝐿/2)/𝐿, (𝑙 + 𝐿)/𝐿, (2𝑙 +

𝐿)/𝐿]. The results, reported in Tables 1 and 2 in terms ofDAFs of the midspan vertical displacement and bendingmoment, show how the dynamic behavior is quite influencedby 𝜏

0variable, since from a numerical point of view the

values of the DAFs change, significantly, in the investigatedranges. Moreover, the results show that DAFs for damagedstructures, much more for H-shaped tower bridges, aretypically larger than those evaluated for undamaged bridgecases.

Additional results are developed to investigate the effectsof the bridge geometry on the DAFs and on the maximumvalues of typical bridge design variables for both damagedand undamaged bridge configurations. In particular, resultsare proposed in terms of the dimensionless parameter 𝑎,which describes the bridge size characteristics of the struc-ture, and refer to a failure mechanism involving the completefailure of one lateral anchor stay of the cable system. Theanalyzed structures are consistent with a long-span bridgegeometry, whose main span length varies from 500 to 1300mand thus with a total length of the bridge between 900mand 2100m. The results, reported in Figures 10, 11, 12, 13,14, and 15 concerning the undamaged configurations, showa tendency to decrease with increasing values of the bridgesize variable. Contrarily, for damaged structures, the DAFs

Mathematical Problems in Engineering 13

0

1

2

3

4

5

6

7

8

9

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Mass

No massΦD

ΦD−UD

ΦUD

b b

e

εF = 0.1, a = 0.1

Θ

ΦM

G 2

Figure 8: Dynamic amplification factors of the midspan bendingmoment for a bridge structure with H-shaped tower as a functionof the normalized speed parameter: effect of the failure mechanismand moving load schematization.

as well as the maximum values of the investigated kinematicand stress parameters display an oscillating behavior andsome local peaks in curve development. The comparisonsbetween damaged or undamaged bridge structures in termsof the tower typology show, essentially, that H-shaped towerstructures are much more affected by damage failure. Asa matter of fact, the DAFs defined by the ratios betweendynamic damage value and the corresponding undamagedstatic quantity, that is, Φ𝐷−UD

𝑋, show that, for the analyzed

cases regarding H-shaped tower typology, the displacementsor the bending moments are typically greater than the bridgestructures based onA-shaped tower typology. Similar conclu-sions can be drawn in terms of the maximum values of theobserved variables. In particular, the comparisons betweendamaged and undamaged bridge behavior indicate that theA-shaped tower is able to reduce the increments producedby the failure mechanism, since the ratios defined in termsof displacements or bending moments are equal to 1.57and to 1.66, respectively. Contrarily, the results concerningH-shaped tower denote that the undamaged variables areamplified with respect to factors equal to 2.2 and 2.5 fordisplacement and bendingmoments, respectively. A differentbehavior is observed with regard to the axial stresses of theundamaged anchor stay. In particular, the results reported inFigures 14 and 15 show that the H-shaped tower bridges areaffected by higher DAFs but lower maximum stresses withrespect to the bridge structures based on A-shaped towers.

Finally, the behavior of the cable-stayed bridges is ana-lyzed in terms of the failure mode, which affects the bridge

structure. In particular, for both H- or A-shaped towertypologies, different damagemechanisms on the cable systemare supposed on the cable system and the effects on the bridgestructures are examined. The failure conditions, reported inFigure 16, are supposed to produce the complete collapseof the lateral anchor stay (Mode A) or the last three stayson both sides of the central part of the main span (ModeB). Additional modes of failure, concerning the damage toa number of internal stays of the cable system, are alsoinvestigated.However, since they do not produce any relevanteffect on the investigated variables, for the sake of brevity,they are not discussed. The analyses are developed involvinga typical geometry of the bridge structure, which is consistentwith a long-span bridge-based typology.The results, reportedin Figures 17, 18, 19, 20, 21, and 22, are proposed by means ofcomparisons of damaged and undamaged behavior, in termsof time histories and DAFs of the following cinematic andstress quantities:

(i) midspan displacements along 𝑋3 axis of the girder,midspan torsional rotation of the girder;

(ii) bending moment in the𝑋1𝑋3plane of the girder and

undamaged anchor stay axial stress.

The analyses are developed for a fixed transit speed of themoving system, whereas the failure mechanism is supposedto begin when the moving system front reaches the midspanof the bridge, that is, Θ = 0.102 and 𝜏0 = 0.87, respectively.The results show that the larger amplification effects on thevertical midspan displacement are obtained for the case inwhich the mode of failure affects the lateral anchor stay ofthe cable system, that is, Mode A. In this context, the verticaldisplacements from the undamaged bridge configuration aresignificantly amplified with respect to a multiplicative factorequal to 1.48 or to 2.28 for the A- or H-shaped tower topolo-gies, respectively. Moreover, the DAFs for the investigatedconfigurations are equal to 1.45 or 1.94, whereas the DAFsdefined as the ratio between 𝐷 on the UD quantities, thatis, Φ𝐷−UD

𝑈𝐺

3

, are equal to 2.1 or to 3.2, respectively. The resultsconcerning the bending moments indicate that the greatereffects are produced by the damage mechanism affecting thecentral part of the cable system, that is, Mode B. In thiscontext, the increments of the bendingmoments with respectto the undamaged values are equal to 3.2 or 4.1 for bridgeconfigurations based on H- or A-shaped tower topology,respectively. Moreover, the maximumDAFs are equal to 4.63and 8.19 for A- and H-shaped tower typologies, respectively,and both of them refer to a damage condition involvingthe failure of the anchor stay. It is worth noting that theresults concerning the cases of undamaged bridge structuresbased on A- or H-shaped towers denote, essentially, thesame prediction on the investigated variables. Contrarily,when damage mechanisms affect the cable system, notableamplifications of the investigated parameters are observed.In particular, the comparisons developed in terms of towertypologies indicate that H tower-based structures are muchmore affected by the damage mechanisms of the cable systemthan the A-shaped ones. This behavior can be explained bythe fact that the failure of the anchor stay (Mode A) or

14 Mathematical Problems in Engineering

0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5

b b

e

b b

e

H-st A-st

ΦD−UDΦUD

Θ

0

0.2

0.4

0.6

0.8Φ1

ΦΦ

G 1

Φ1

Φ1

(a)

Φ1

(b)

Figure 9: (a) Dynamic amplification factors and maximum value of the midspan torsional rotation as a function of the normalized speedparameter Θ. (b) Schematic deformation produced by the failure mechanism.

Table 1: DAFs of the midspan bending moments as a function of the normalized time of failure on the bridge development comparisons interms of A and H shaped.

𝜏0

H-shaped tower A-shaped towerΦ𝐷

Φ𝐷−UD

ΦUD

Φ𝐷

Φ𝐷−UD

ΦUD

0.00 6.26 9.48 2.81 5.85 6.84 3.240.50 5.89 8.92 2.81 5.49 6.42 3.240.87 7.52 11.39 2.81 6.10 7.13 3.241.25 6.45 9.77 2.81 6.20 7.25 3.241.62 4.82 7.30 2.81 5.84 6.82 3.242.00 5.67 8.58 2.81 5.33 6.23 3.242.50 1.38 2.09 2.81 3.26 3.81 3.24

Table 2: DAFs of the midspan vertical displacements as a function of the normalized time of failure on the bridge development comparisonsin terms of A and H shaped.

𝜏0

H-shaped tower A-shaped towerΦ𝐷

Φ𝐷−UD

ΦUD

Φ𝐷

Φ𝐷−UD

ΦUD

0.00 1.973 3.046 1.451 1.62 2.03 1.600.50 1.890 2.918 1.451 1.58 1.98 1.600.87 2.032 3.137 1.451 1.75 2.20 1.601.25 2.096 3.237 1.451 1.72 2.15 1.601.62 2.264 3.495 1.451 1.61 2.02 1.602.00 2.238 3.456 1.451 1.65 2.07 1.602.50 0.940 1.451 1.451 1.26 1.59 1.60

the stays in the central part (Mode B) produce unbalancedforces in the cable system and thus on the girder leadingto high rotations of the pylons and the girder along vertical𝑋3and 𝑋

3axes, respectively. This behavior is confirmed

by the results reported in Figures 21 and 22, in which timehistories of the torsional rotation and anchor stay axial forcein terms of the damage mechanism and tower typology arereported. The analyses show that bridge structures based

Mathematical Problems in Engineering 15

0.025 0.05 0.075 0.1a

0.125 0.15

1

1.25

1.5

1.75

0.014

0.012

0.012

0.008

0.006

0.004

2

2.25

D UD

b b

e

UG3 /L

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

ΦU

G 3

ΦD

ΦD−UD

ΦUD

UG3 /L

UG3 /L

Figure 10: Dynamic amplification factors of the midspan verticaldisplacement for a bridge structure with A-shaped tower: responseof damaged and undamaged bridge structures as a function of thebridge size parameter 𝑎.

0.025 0.05 0.075 0.1 0.125 0.15

1

1.5

2

2.5

3

a

0

0.005

0.01

0.015

0.02

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

b b

e

D UD

UG3 /L

ΦU

G 3

ΦD

ΦD−UD

ΦUD

UG3 /L

UG3 /L

Figure 11: Dynamic amplification factors of the midspan verticaldisplacement for a bridge structure with H-shaped tower: responseof damaged and undamaged bridge structures as a function of thebridge size parameter 𝑎.

on A-shaped tower are practically unaffected by the failuremodes, since the same prediction on the maximum torsionalrotations is observed. On the contrary, the H-shaped towerbridges owing to the concurrent rotations of the girder andthe pylons are affected by greater deformations and DAFs.Moreover, the results shown in terms of the anchor stay axial

1

1.5

2

2.5

3

3.5

4

4.5

5

0.025 0.05 0.075 0.1a

0.125 0.15

0.005

0.004

0.003

0.002

0.001

0

D UD

ΦD

ΦD−UD

MG22/gL

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

b b

e

ΦM

G 2

MG22/gL

MG22/gL

ΦUD

Figure 12: Dynamic amplification factors of the midspan bendingmoment for a bridge structure with A-shaped tower: response ofdamaged and undamaged bridge structures as a function of thebridge size parameter 𝑎.

1

2

3

4

5

6

7

8

0

0.001

0.002

0.003

0.004

0.005

0.006

0.025 0.05 0.075 0.1a

0.125 0.15

D UD

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

b b

eΦM

G 2

ΦD−UD

ΦUD

ΦD

MG22/gL

MG22/gL

MG22/gL

Figure 13: Dynamic amplification factors of the midspan bendingmoment for a bridge structure with H-shaped tower: response ofdamaged and undamaged bridge structures as a function of thebridge size parameter 𝑎.

stresses, reported in Figure 22, denote that “Mode A” failuremechanism produces high increments in the axial stress,mainly, for the A-shaped tower typology. This behavior canbe explained by the geometric configuration of the A-shapedpylon and by its ability to redistribute the internal stressesfrom the damaged to the undamaged anchor stay.

16 Mathematical Problems in Engineering

0.8

1

1.2

1.4

1.6

1.8

2

0

1

2

3

4

5

6

7

8

0.025 0.05 0.075 0.1

a

0.125 0.15

D UD

σ0L/g

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

b b

e

×106Φ

σ0

ΦD−UD

ΦUD

ΦD

σ0L/g

σ0L/g

Figure 14: Dynamic amplification factors of the undamaged anchorstay axial stress for A-shaped tower bridge configurations: damagedand undamaged bridge response as a function of the bridge sizeparameter 𝑎.

5. Conclusions

Long-span bridges under moving loads have been ana-lyzed by means of tridimensional deformation modes, interms of dynamic impact factors and maximum values oftypical design bridge variables. The proposed model takesinto account nonlinearities involved by large displacementseffects, whereas in those stays affected by internal damage,time dependent damage functions are introduced in theconstitutive relationships. The purpose of this investigationis to analyze the amplification effects of the bridge structureproduced by the moving load application and damage mech-anisms in the cable system. The analyses have shown thatthe presence of damage mechanism in the cable system isable to produce larger DAFs then those obtained for undam-aged bridge configurations. The DAFs strongly depend onthe moving system speeds and the mass schematization.Underestimations in prediction of DAFs and maximumdesign bridge variables are noted for the cases in whichthe inertial description of the moving mass is not properlytaken into account. The results developed in terms of thedamage mechanism configuration, movingmass description,and bridge properties have shown that recommendationsprovided by existing codes, that is, PTI [15] and SETRA [16],become unsafe in many cases. As a matter of fact, resultsdeveloped in terms of the damage mechanism characteristicshave shown that the damage mode which produces theworse effects on the bridge behavior is that associated withthe failure of the anchor stay. In particular, the analyseshave pointed out that damage mechanisms involving thefailure of the lateral anchor stay are able to produce large

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

1

2

3

4

5

6

0.025 0.05 0.075 0.1

a

0.125 0.15

σ0L/g

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87

×106

D UD

b b

e

Φσ

0

σ0L/g

σ0L/g

ΦD

ΦD−UD

ΦUD

Figure 15: Dynamic amplification factors of the undamaged anchorstay axial stress for H-shaped tower bridge configurations: damagedand undamaged bridge response as a function of the bridge sizeparameter 𝑎.

amplifications in the investigated parameters, whose rangeswith respect to the static undamaged value are equal to 2.5–3.5 for vertical midspan displacement, to 5.5–8.5 for midspanbending moment, to 1.3–2.8 for midspan torsional rotation,and to 1.9–2.3 for the anchor stay axial stress. The resultsdeveloped in terms of the damage mechanism characteristicshave shown that the damage mode, which produces theworse effects on the bridge behavior is that associated withthe failure of the anchor stay. Comparisons developed interms of tower topology have shown that the H-shaped towerbridge is much more affected than the A-shaped one, sincethe failure modes produce an unbalanced distribution ofthe internal stresses in the cable system, leading to largertorsional rotations and vertical displacements of the towerand the girder, respectively. It is worth noting that, in thepresent paper, only the effects produced by moving loadsare considered in the results. However, in the frameworkof cable-stayed bridges, another severe loading condition isthe one related to wind effects. As a matter of fact, damagemechanisms in the cable system, among which a typicalexample is the one considered in the present paper, mayamplify the resonance effects related to aeroelastic instabilityphenomena, leading to a premature bridge collapse.However,the study of the coupled aeroelastic and damage effects isbeyond the scope of the present paper and will be an objectof future investigation.

Appendix

The damage description is developed by means of a phe-nomenological approach based on the Continuum Damage

Mathematical Problems in Engineering 17

Failure mode A

MG2UG3

ΦG1

(a)

σ0

Failure mode B

M2UG3

ΦG1

(b)

Figure 16: Synoptic representation of the failure modes: (a) failure in the lateral anchor stay and (b) failure in the central part of the cablesystem.

0 50 100 150 200 250 300−0.0125−0.01

−0.0075−0.005−0.0025

0

0.0025

0.005

0.0075

Failure initiation

StaticDynamic

ct/L

UDD-1D-2

UD D1 D21.45 1.74 1.95

DAFs

b b

e

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87,Θ = 0.102

UG 3/L

Figure 17: Time history of the dimensionlessmidspan displacementand evaluation of the DAFs: comparisons between damaged andundamaged A-shaped bridge configurations.

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0 50 100 150 200 250 300

Failure initiation

StaticDynamic

ct/L

UDD-1D-2

UD D1 D21.44 1.86 1.95

DAF

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87,Θ = 0.102

UG 3/L

b b

e

Figure 18: Time history of the dimensionlessmidspan displacementand evaluation of the DAFs: comparisons between damaged andundamaged H-shaped bridge configurations.

−0.002

−0.001

0

0.001

0.002

0.003

0.004

0.005

0 50 100 150 200 250 300

Failure initiation

StaticDynamic

ct/L

DDD-1D-2

UD D1 D22.19 4.75 4.05

DAF

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87,Θ = 0.102

MG

22/gL

b b

e

Figure 19: Time history of the dimensionless midspan bendingmoment and evaluation of the DAFs: comparisons between dam-aged and undamaged A-shaped bridge configurations.

−0.0045−0.003−0.0015

0

0.0015

0.003

0.0045

0.006

0.0075

0 50 100 150 200 250 300

Failure initiation

StaticDynamic

ct/L

DDD-1D-2

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87,Θ = 0.102

MG

22/gL

b b

e

Figure 20: Time history of the dimensionless midspan bendingmoment and evaluation of the DAFs: comparisons between dam-aged and undamaged H-shaped bridge configurations.

18 Mathematical Problems in Engineering

0 25 50 75 100 125 150 175 200

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

UD-HD1-HD2-H

UD-A D1-A D2-A

b b

e

b b

e

ct/L

e/b = 0.5, a = 0.1, εF = 0.1, τ0 = 0.87,Θ = 0.102

ΦG 1

Failure initiation

Figure 21: Time history of the dimensionless midspan torsional rotation: comparisons in terms of the failure mode and tower typology.

0 25 50 75 100 125 150 175 200

200 175 150 125 100 75 50 25

6.6 × 1076.05 × 1076.5 × 107

4.95 × 1074.4 × 107

3.85 × 1073.3 × 107

2.75 × 1072.2 × 107

1.65 × 1071.1 × 1075.5 × 106

b b

e

b b

e

ct/L ct/L

σ0L/g

UD D1 D21.14 1.24 1.08

DAF

Failure initiation

StaticDynamic

UDD-1D-2

e/b = 0.5, a = 0.1, εF = 0.1,

τ0 = 0.87,Θ = 0.102

Figure 22: Time history of the dimensionless of the undamaged anchor stay axial force: comparisons in terms of the failure mode and towertypology.

Mechanics. The damage evolution is determined by meansof a Kachanov type law based on the following expression[22]:

𝑑𝐷 (𝑠, 𝑡)

𝑑𝑡

= [

𝑆

𝐴0 (1 − 𝐷 (𝑠, 𝑡))

]

𝑚

, (A.1)

where𝐴0and𝑚 are two characteristic damage coefficients for

the material. Introducing the initial condition corresponding

to the undamaged state, that is, 𝐷(0) = 0, and, integrating(A.1), the damage evolution law is obtained as

𝐷(𝑠, 𝑡) = 1 − [−

𝐴𝑚

0𝑆−𝑚

(𝑚 + 1) 𝑡 − 𝑆−𝑚𝐴𝑚

0

]

1/(𝑚+1)

. (A.2)

The time to failure (𝑡𝑓) is obtained from (A.2) and

enforcing the failure condition defined in terms of

Mathematical Problems in Engineering 19

the critical damage value by means of the followingrelationships:

𝐷(𝑠, 𝑡𝑓) = 𝐷

𝑐, 𝑡

𝑓=

1 − (1 − 𝐷𝑐)𝑚+1

𝑚 + 1

(

𝑆

𝐴0

)

−𝑚

. (A.3)

Finally, substituting (A.3) into (A.2), the damage law can beexpressed by the following relationship:

𝐷(𝑠, 𝑡) = 1 − (1 −

𝑡𝑓− (𝑡 − 𝑡

0)

𝑡𝑓

)

1/(1+𝑚)

, (A.4)

Nomenclature

(⋅)𝐺: Stiffening girderproperty

(⋅)𝑃: Pylon property

(⋅)𝐶: Cablesystemproperty

(⋅)𝑚: Movingmassproperty

𝛼𝑖: Longitudinal geometric slop