bridge under moving load

8/7/2019 bridge under moving load

1/18

Engineering Structures 30 (2008) 11601177www.elsevier.com/locate/engstruct

Dynamic impact analysis of long span cable-stayed bridgesunder moving loads

D. Bruno , F. Greco, P. LonettiDepartment of Structural Engineering, University of Calabria, 87030 - Rende (CS), Italy

Received 5 September 2006; received in revised form 19 June 2007; accepted 2 July 2007Available online 21 August 2007

Abstract

The aim of this paper is to investigate the dynamic response of long span cable-stayed bridges subjected to moving loads. The analysis isbased on a continuum model of the bridge, in which the stay spacing is assumed to be small in comparison with the whole bridge length. Asa consequence, the interaction forces between the girder, towers and cable system are described by means of continuous distributed functions.A direct integration method to solve the governing equilibrium equations has been utilized and numerical results, in the dimensionless context,have been proposed to quantify the dynamic impact factors for displacement and stress variables. Moreover, in order to evaluate, numerically, theinuence of coupling effects between bridge deformations and moving loads, the analysis focuses attention on the usually neglected non-standardterms related to both centripetal and Coriolis forces. Finally, results are presented with respect to eccentric loads, which introduce both exuraland torsional deformation modes. Sensitivity analyses have been proposed in terms of dynamic impact factors, emphasizing the effects producedby the external mass of the moving system and the inuence of both A and H shaped tower typologies on the dynamic behaviour of the bridge.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Moving loads; Dynamic impact factors; Cable-stayed bridges; A and H shaped towers

1. Introduction

Cable-stayed systems have been employed, frequently, toovercome long spans, because of their economic and structuraladvantages. Moreover, improvements in the use of lightweightand high strength materials have been proposed in differentapplications, and, consequently, more slender girder crosssections have been adopted. As a result, the external loadshave become comparable with those involved by the bridgeself-weight ones and an accurate description of the effectsof the moving loads is needed to properly evaluate dynamicbridge behaviour. At the same time, new developments inrapid transportation systems make it possible to increase theallowable speed range and trafc load capacity; consequently,the moving system can greatly inuence the dynamic bridgevibration, by means of non-standard excitation modes. To thisend, investigation is needed to quantify the effects produced bythe inertial forces of the moving system on the bridge vibration.

Corresponding author. Tel.: +39 0984 496914; fax: +39 0984 494045.E-mail address: [email protected] (D. Bruno).

The extension of the moving load problem to longspan cable-supported bridges requires a consistent approach,appropriately formulated, in order to fully characterize thebridge kinematics and traingirder interaction. In the literature,several studies have been developed, which analyse dynamicbridge behaviour with respect to different assumptions andframeworks. In particular, Fryba and Timoshenko [ 1,2],provided a comprehensive treatment concerning primarilythe dynamic response of simply supported girder structurestravelled by vehicles, and analytical as well as numerical

solutions for some specic problems have been presented.During the last few decades, with advances in high performancecomputers and computational technologies, more realisticmodelling of the dynamic interaction between a moving systemand bridge vibration has become feasible. In particular, Yanget al. [3] presented a closed-form solution for the dynamicresponse of simple beams subjected to a series of moving loadsat high speeds, in which the phenomena of resonance andcancellation have been identied. Moreover, Lei and Noda [ 4]proposed a dynamic computational model for the vehicle andtrack coupling system including girder prole irregularity by

0141-0296/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2007.07.001
http://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2007.07.001http://dx.doi.org/10.1016/j.engstruct.2007.07.001mailto:[email protected]://www.elsevier.com/locate/engstruct


2/18

D. Bruno et al. / Engineering Structures 30 (2008) 11601177 1161

Nomenclature

Longitudinal stay geometric slope 0 Longitudinal anchor stay geometric slopeAs Stay cross sectional areaAs0 Anchor stay cross sectional areab Half girder cross section width Transverse stay geometric slopec Moving system speed Stay spacing stepe Eccentricity of the moving loads with respect to

the girder geometric axisE Cable modulus of elasticityE I Flexural girder stiffnessE A Axial girder stiffnessE s Stay Dischinger modulusE s0 Anchor stay Dischinger modulusg Girder self-weight per unit length

Stay specic weightG J t Torsional girder stiffnessH Pylon heightI p0 Pylon polar mass momentK p Flexural top pylon stiffnessK p0 Torsional top pylon stiffnessl Lateral bridge spanL Central bridge spanL p Total train length Mass function of the moving system per unit

length 0 Polar mass moment of the moving system with

respect to girder geometric axis per unit lengthM p Lumped top pylon equivalent mass Girder mass per unit length 0 Polar inertial moment of the girder per unit length Girder torsional rotationp Live loads a Allowable stay stress g Stay stress under self-weight loading g0 Anchor stay stress under self-weight loading L ( R) Left (L) and right ( R) top pylon torsional

rotationsu L( R) Left (L) and right ( R) horizontal top pylon

displacement

v Girder vertical displacementw Girder horizontal displacement

the nite element method, whereas additional references tothe inuence of AASHTO live-load deection criteria on thevibration in a railway track under moving vehicles can be foundin [57].

With reference to cable-stayed bridges, in order to evaluatethe amplication effects produced by the moving system,different investigations have been proposed. In particular, Auet al. [8,9] investigated the dynamic impact factors of cable-stayed bridges under railway trafc using various vehicle

models, evaluating the effects produced by random road

surface roughness and long term deection of the concretedeck. An efcient numerical modelling has been developedby Yang and Fonder [ 10] to analyse the dynamic behaviourof cable-stayed bridges subject to railway loads, taking intoaccount nonlinearities involved in the cable system. Dynamicinteraction of cable-stayed bridges with reference to railway

loads has been investigated in [11], in which strategies toreduce the multiple resonant peaks of cable-stayed bridgesthat may be excited by high-speed trains have been proposedfor a small length bridge structure. Finally, a computationalmodel and a parametric study have been proposed in [ 12]to investigate bridge vibration produced by vehicular trafcloads. The literature referred to above investigates dynamicbridge behaviour properly taking into account the effectsof interaction between bridge vibration and the movingsystem. However, only a few studies have concentrated onthe dynamic responses of long span bridges. This paper,therefore, focuses on the dynamic behaviour of long span cable-stayed bridges, evaluating the effects produced by the movingsystem on the dynamic bridge behaviour. In particular, themain aims of this paper are to propose a parametric studyin a dimensionless context, which describes the relationshipbetween dynamic amplication factors and moving loads andbridge characteristics.

The structural model is based on a continuum approach,which has been widely used in the literature to analyselong span bridges [1315 ]. In particular, Meisenholder andWeidlinger [13] have schematized bridge structures as an elasticbeam resting on an elastic foundation, whose stiffness is strictlyconnected to the geometrical and stiffness properties of thestays. Moreover, extended models which generalize the bridge

kinematics have been proposed in [14,15], in which the stayspacing is assumed to be small in comparison with the centralbridge span. As a result, the interaction forcesbetween thecablesystem and the girder can be assumed as continuous functionsdistributed over the whole girder length. The accuracy of thecontinuum approach has been validated in previous worksdeveloped in both static and dynamic frameworks, throughcomparisons with numerical results obtained by using a niteelement model of the discrete cable system bridge [1416 ].

In the present paper, the bridge kinematics and the inertialforces have been considered in a tridimensional context, inwhich both in-plane and out-of-plane deformation modes havebeen accounted for. Cable-stayed bridges based on both Hand A shaped typologies with a double layer of stayshave been considered. However, cable-stayed bridges withone central layer of stays, especially for eccentric railwaybridges, are characterized by high deformability, and difcultiesverifying the design rules on maximum displacements occurfrequently. In particular, the girder torsional stiffness needsto be signicantly improved with respect to those involvedfor H and A shaped typologies, because contributionsarising from the cable system are practically negligible. Asa matter of fact, torsional analysis carried out for typicalconcrete or steel girder cross sections shows that in order tolimit torsional rotation to reasonable values (i.e. below 0.02),

the maximum allowable central length must be approximately


3/18

1162 D. Bruno et al. / Engineering Structures 30 (2008) 11601177

equal to 400 m [16]. The equations of motion for the vehicle-track-bridge element are derived by means of the Hamiltonprinciple. Subsequently, the boundary value problem, due tothe equilibrium equations, was solved, numerically, by meansof a nite difference scheme based on -family methods,in which proper interpolation functions on both spatial and

time domains were adopted to obtain stable and accurateresults. A parametric study in a dimensionless context has beenanalysed by means of numerical results, in terms of typicalkinematic and stress bridge variables for both in-plane andeccentric loading conditions. In particular, results are proposedto investigate the effects of moving the system descriptionwith reference to non-standard forces, usually neglected inconventional dynamic analyses, i.e. Coriolis and centripetalaccelerations. Finally, the inuence on the dynamic bridgebehaviour of pylon typology with reference to both A andH shapes has been analysed,and comparisons in terms of bothmoving loads and tower characteristics have been proposed.

2. Cable-stayed bridge model

To begin with, the bridge geometry is presented withrespect to a fan-shaped self-anchored scheme and both exuraland torsional deformation modes are evaluated for an Hshaped pylon typology. Subsequently, the formulation isadapted for A shaped pylons. This can be easily derived,as explained in the following sections, starting from the Hones and introducing slight modications to the main governingequations. In both formulations, the cable system is arrangedsymmetrically with respect to both zx and yz planes.

Long span bridges based on cable-stayed systems are

frequently analysed by means of a continuum approach, inwhich the stays are assumed to be uniformly distributed alongthe deck. In particular, the stay spacing is quite small incomparison to the central bridge span (i.e. / L 1). Asa result, the self-weight loads produce negligible bendingmoments on the girder with respect to that raised by the movingloads. The initial stress distribution, at the zero conguration,is supposed to be produced by a correct erection process whichyields tension in the stays and compression in both the girderand the pylons. Moreover, under dead loads only, the girder isarranged with an initial straight prole, which is practically freefrom bending moments for reduced values of the stay spacingstep. In particular, the erection procedure is based on the freecantilevered method, which is able to control the initial tensiondistribution in the cable system to a value practically constantin each stay. This assumption has been veried for long spanbridges, in view of the prevailing truss behaviour of the cable-supported structures [ 1518]. Therefore, the moving loadsmodify the initial conguration and, consequently, produceadditional stress and deformation states. It is worth noting thatfor long span bridges, the initial stress state produced by thedead loading needs to be accounted mainly in the cable-stayedsystem, in which the initial tension strongly affects the staysbehaviour due to Dischinger effects [ 17,18].

The geometry and stiffness characteristics of the bridges are

selected with respect to typical ranges suggested by practical

design rules [ 17,18]. In particular, the cross sectional stay areasare designed so that the dead loads ( g) produce constant stressover all the distributed elements, which are assumed equal toa xed design value, namely g . As a result, the geometricarea of the stays varies along the girder, but the safety factorsare practically constant for each element of the cable system.

Moreover, for the anchor stays, the cross sectional geometricarea, As0 , is designed in such a way that the allowable stress isobtained in the static case, for live loads applied to the centralspan only. Therefore, the geometric measurement for the cablesystem can be expressed by the following equations:

As =g

g sin ,

As0 =gl

2 g1 +

lH

2 1/ 2 L2l

2 1 ,

(1)

where is the slope of a generic stay element with respectto the reference system, ( L , l , H ) are representative geometriclengths of the bridge structure, and is the stay spacing step(for more details see Fig. 1). The bridge analysis is based on thefollowing assumptions:

(1) the stress increments in the stays are proportional to the liveloads, p;

(2) a long span fan shaped bridge is characterized by adominant truss behaviour.

In this framework, the tension g and g0 for distributed andanchor stays, respectively, can be expressed by the followingrelations:

g =g

g + p a ,

g0 = a 1 +pg

1 2Ll

2 1 1

.(2)

It is worth noting that the allowable stay stress, a ,represents a known variable of the cable system in terms of which the design tension under dead loading can be determinedbythe use ofEq. (2). Since it is assumed that for dead loads onlythe bridge structure remains in the undeformed conguration,the application of moving loads leads to additional stress anddeformation increments with respect to the self-weight loadingcondition. In particular, as reported in Fig. 1 with respect tothe reference system with the origin xed at the midspan girdercross section, the bridge kinematic, for the H shaped towertypology, is described by following displacement variables:

horizontal and vertical girder displacements [u ( x , t ),v( x , t )],

left and right horizontal pylon top displacements [u L ( t ),u R ( t )],

girder torsional rotation [( x , t )], left and right pylon top torsional rotations [ L ( t ), R ( t )].

In particular, bridge deformations related to exure andtorsion for the girder and pylons and axial deformations for

the girder and stays have been taken into account, whereas


4/18


Fig. 1. H shaped tower moving load problem: bridge kinematics and representative stiffness parameters.

pylon axial deformability has been neglected. Consistentlywith the bridge conguration reported in Fig. 1, the bridgescheme is constrained with respect to both vertical and torsionaldisplacements at boundary cross sections of the bridge and atgirder/pylon connections.

The stays are modelled as bar elements and thenonlinear behaviour is evaluated consistently with theDischinger formulation [ 17], which takes into accountgeometric nonlinearities of the inclined stays introducing actitious elastic modulus for an equivalent straight member, inthis way:

E s =E

1 + 2 l20 E

12 30

1+ 2 2

, with = 0

, (3)

where E s is known as the secant Dischinger modulus, E is theYoungs modulus of the cable material, the specic weight,l0 the horizontal projection of the stay length and 0 and arethe initial and actual tension values of the stay, respectively,i.e. 0 = g for the double layer of stays and 0 = g0 forthe anchor stays. Moreover, the tangent value of the Dischingermodulus can be obtained from Eq. (3) by putting = 1. Asfar as the secant modulus is concerned, its value depends oncable stress statesunder self-weight and live loading conditions.Sufcient accuracy in the actual stress state might even beachieved by assuming as proportional to the ratio betweenlive and self-weight loads (dominant truss behaviour) [1618 ],i.e. = a g

p+ gg for the double layer of distributed stays

and = a g p+ g

g( L / 2l )2

( L/ 2l )2 1for the anchor stays. On the

other hand, numerical investigations have been developed toanalyse the inuence of adopting the secant or the tangentequivalent moduli on the dynamic impact factors prediction.

The results, not presented here for the sake of brevity, show

that this inuence is practically negligible (less than 3%),whereas maximum relative percentage differences, less than10%, are observed for speeds above 140 m / s. In addition, it hasbeen observed that the analysed maximum amplication factorsoccur when the moving system is basically applied on thecentral bridge span. Therefore, the analysis has been developedby assuming the tangent modulus for the double layer of staysacting on the lateral spans, whereas for the double layer of staysacting on the central span and for the anchor stays the secantmodulus has been employed.

The axial deformation increments of a stay generic for theleft ( L) and right ( R) pylons produced by the moving systemdepend on both kinematic and geometric variables, as in thefollowing relationships:

L =1H

[(v b) sin2

(u L L b u ) sin cos ], (4)

R =1H

[(v b) sin2

(u R Rb + u ) sin cos ], (5)

where (+ / ) refers, in Eqs. (4) and (5) and in the followingones, to the right (+ ) and left ( ) distributed stays withrespect to the longitudinal girder geometric axis. Similarly, forthe left and right pylon anchor stays, the incremental axialdeformations are described as

L0 =1H

[(u L L b u ) sin 0 cos 0] , (6)

R0 =1H

[(u R Rb + u ) sin 0 cos 0] , (7)

where 0 is the girder/anchor stay orientation angle ( Fig. 1).The external loads evolve at constant speed from left to right

along the bridge development. A perfect connection between


5/18


the girder and the moving system is assumed. Interactionforces produced by girder prole roughness and friction aresupposed to generate negligible effects with respect to theglobal bridge vibration. This assumption has been veriedin the context of long span cable-supported bridge, whereroughness effects have been considered as negligible [19]. As a

result, the moving system has the same vertical displacementsas the girder. Nevertheless, non-standard contributions arisingfrom Coriolis and centripetal inertial forces, produced by thecoupling behaviour between the moving system and bridgedeformations, have been taken into account. With respect to axed reference system, the velocity and acceleration functionsof the moving system are evaluated consistently with a Euleriandescription of the moving loads as

v =v t

+v x

c, v = 2v t 2

+2v

x t 2c +

2v x2

c2 ,

with c = x t

. (8)

The moving loads are consistent with a train systemtypology, modelled by a sequence of lumped and distributedmasses, representative of both bogie components and vehiclebodies. However, for long span bridges, the internal bogiespacing for an elementary vehicle is, usually, small incomparison with the whole bridge length. Moreover, within thesame approximation level, the locomotive, even if it is muchheavier than the carriage, is distributed on a length, which isassumed, in this context, to be smaller than the whole lengthof the train. As a result, the moving system is supposed to bedescribed by equivalent uniformly distributed loads and massesacting on the girder prole. However, improvements to themoving load distribution can be easily provided just modifyingEqs. (9)(12) and introducing a piecewise constant function todescribe carriages and locomotive loads.

With respect to a moving reference system, x1 , from theleft end of the bridge, the mass and loading functions duringthe external loading advance can be written by the followingexpressions, respectively:

= H x1 + L p ct H (ct x1) , (9)

f = pH x1 + L p ct H (ct x1) , (10)

where x1 = x + ( L / 2 + l ) , (, p) are the vehicle body

mass and loading forces per unit length and H (

) is theHeaviside function. Moreover, the moving system is assumedto be eccentrically located with respect to the bridge half width,and, consequently, distributed moment and rotatory inertialfunctions are introduced to properly describe the external loadsas

0 = 0 H x1 + L p ct H (ct x1) , (11)

m = p eH x1 + L p ct H (ct x1) , (12)

where 0 represents the torsional distributed polar massmoment produced by the external loading and e is theeccentricity of the moving loads with respect to the cross

sectional geometric axes. An energy approach based on the

Hamilton principle is utilized to derive the dynamic equilibriumequations. In particular, it is assumed that the damping energyis practically negligible. This hypotheses is quite veried inthe context of long span bridges, where it has been provedthat the bridge damping effects tend to decrease as spanlength increases [ 20,21]. Detailed results about the inuence of

damping effects on the dynamic amplication factors (DAFs)have been presented in [9], from which it transpires that theassumption of an undamped bridge system leads to greaterDAFs. It is well known that the Hamilton principle can beexpressed for a conservative system and a generic time intervalas

t 2t 1 ( T V ) dt = 0 (13)where T and V are the kinetic and the potential energyof the whole dynamic system, respectively, and t 1 and t 2dene the observation period. Therefore, with respect to these

kinematic elds, the kinetic energy functional of the combinedbridgemoving-load system may now be formulated as

T =12 L L v2 + u 2 dx + 12 L L 02dx+

12

I p0 2L +

2D +

12

M p u2L + u2D

+12 L L v2 + u2 dx + 12 L L 02dx ,

with L = l + L / 2. (14)

In particular, (, 0) and ( , 0) are the mass functions and

polar inertial moments per unit length for both moving loadsand girder, respectively. Moreover, M p is the equivalent lumpedmass which refers to horizontal top pylon displacement, b is thebridge semi-width and I p0 is the pylon polar mass moment forboth left and right pylons, with I p0 = b

2 M p .The total potential energy of the system, using a small

displacements formulation, can be written as

V =12 L L E I v 2 + E Au 2 dx + 12 L L G J t 2dx+

12

0

L

E s As H sin

2L dx +12

L

0

E s As H sin

2Rdx

+12

E s0 As0 2L0 +

2R0

+12

K p0 2L +

2D + K

p u 2L + u2D

L L ( f v + m ) dx+

12 L L v x + L2 + x L2 v2dx (15)

where ( E I , E A, G J t ) are the exural, axial and torsional girderstiffness and E s As , E

s0 As0 are the axial stiffnesses of a

generic stay or the anchor cables. Moreover, K p0 , K p are the


6/18


torsional and exural top pylon stiffness and ( ) is the Diracdelta function. In particular, the last term on the right-hand sideof Eq. (15) denotes a penalty functional, with v representingthe penalty parameter, introduced to penalize girder verticaldisplacements and, consequently, to reproduce the connectionbetween the girder and the pylons correctly.

By integrating by parts the rst variation of the kineticenergy functional of the combined bridge/moving loads systemand assuming that the virtual displacements vanish at both thebeginning and end of the actual varied path, the followingexpression is derived:

t 1t 1 T dt = t 2t 1 L L [ ( vv + uu ) + 0 ] dxdt t 2t 1 I p0 L L + R R dt

t 2

t 1

M p ( u L u L + u Ru R ) dt

t 2t 1 L L 0 ( ) dxdt t 2t 1 L L 0 dxdt t 2t 1 L L (vv + uu ) dxdt t 2t 1 L L ( vv + uu ) dxdt . (16)

It is worth noting that in Eq. (16), the last two terms onthe right-hand side denote the kinetic energy contributionsproduced by the external moving mass, in which thetime derivative for vertical displacement has been assumedconsistently with Eq. (8)). As a consequence, non-standardterms due to both Coriolis and centripetal forces are introducedin the kinetic functional, which are strictly connected with theinteraction behaviour between the bridge deformations and themoving system. Moreover, the time dependence of the movingmass determines additional contributions to the inertial forces,which are, basically, produced by an unsteady distribution of the train/bridge system mass. Taking into account the rstvariation of the total potential energy function, expressed byEq. (15), the following expression is obtained:

t 1t 1 V dt = t 1t 1 L L E I v I V v E Au u G J t dx+ T v M v + N u + M t

L L

+ 0 L (qv L v + qhL u + m L ) dx+ L0 (qv Rv + qh R u + m R ) dx+

0

Lq

u Lu

L+ m

L

Ldx

+ L0 qu R u R + m R R dx+ K p0 ( L L + R R )

+ K p (u L u L + u Ru R )

+ S0L u S0L u L + M

0L L x= L

+ S0Ru + S0Ru R + M

0R R x= L

L L f v dx L L m dx L L [( x + L / 2) + ( x L / 2)] v v v dx dt

(17)

where

qv L( R) =E S ASH

v sin3 u L( R) (+ )u sin2 cos ,

qhL ( R) = E S AS

H ( )v sin2 cos

( ) u L( R) u cos2 sin ,

quL =E S ASH

v sin2 cos + (u L u ) cos2 sin ,

qu R =E S ASH

v sin2 cos + (u R + u ) cos2 sin ,

m L( R) =E S ASb

2

H sin3 L( R) cos sin2 ,

m L( R) =E S ASb

2

H sin2 cos + L( R) cos2 sin .

(18)

S0L ( R) =E s0 As0

H u (+ )u L( R) cos2 0 sin 0,

M 0L ( R) =E s0 As0b

2

H L ( R) cos2 0 sin 0.

(19)

It is worth noting that Eqs. (18) and (19) correspond todistributed internal forcesdue to cable system/girder interactionand concentrated forces and moments applied to the leftand right girder cross section ends due to the anchor stays,respectively (Fig. 2). Moreover, in Eq. (18), the angularparameter , as depicted in Fig. 1, depends on the longitudinal

coordinate x and the geometrical bridge lengths (l , H , L) .Assuming thebridge scheme reported in Fig. 1 , the boundaryconditions at the left and right girder cross section ends requirenull values for vertical displacements, bending moments,torsional rotations and specied horizontal axial forces, as inthe following equations:

= 0, v = 0, v = 0 at x = L ;

E Au L + S0L = 0, E Au L S0R = 0. (20)

The dynamic equilibrium equations can be obtained inexplicit form by means of the variation statement of theHamilton principles. In particular, by substituting Eqs. (16)

and (17) into Eq. (13) and taking into account the boundary


7/18


Fig. 2. Dynamic interaction forces between bridge components.

conditions, the following dynamic equilibrium equations arederived:Girder

v + E I v I V + H (x) qv R + H ( x) qv L + v

+ v + 2cv + c2v

f (x + L / 2) + (x L / 2) v v = 0, (21)

u E Au + H ( x) qh R + H ( x) qhL + u + u = 0, (22) 0 G J t + H (x) m R + H ( x) m L + 0

+ 0 m = 0. (23)

Left top pylon

S0L 0 L quL dx K P u L + M p u L = 0, L x 0,(24)

I p0 L + K p0 L + 0 L m L dx + M 0L = 0, L x 0.

(25)

Right top pylon

S0R + L0 qu R dx + K P u R + M p u R = 0, 0 x L(26)

I p0 R + K p0 R + L0 m Rdx + M 0R = 0, 0 x L .

(27)

A synoptic representation of both internal and externalforces has been proposed in Fig. 2, and, as an alternative tothe variational approach, the governing equations can be easily

derived consistently with a local approach taking into account

the equilibrium conditions for both internal and external forces.In order to develop a generalized formulation, the dynamicequilibrium equations have been proposed in dimensionlessform, introducing the following parameters, related to bothbridge and moving load characteristics:

Girder

F =4 I g

H 3g

1/ 4, =

C t g

Eb2

Hg

1/ 2,

A =A gHg

, =

, 0 = 0

b2.

(28)

Pylon, cable system

P =M P H

, p =K p g

Eg, a =

2 H 2 E 12 3g

,

a = a1 + 2 2

.

(29)

Moving loads

p = p gEg

, e = eb

, = c gEgH

1/ 2,

= t Eg

H g

1/ 2.

(30)

Bridge kinematic

V =vH

, U =uH

, U R =u RH

, U L =u LH

. (31)

In particular, ( F , A, ) are the relative bending, axialand torsional stiffness ratios between the girder and the cablesystem, respectively. Moreover, (a ) identies a bridge size

parameter strictly connected with the deformability of the cable


8/18


system caused by the Dischinger effect and p , P dene theinertial and stiffness properties for both right and left pylons.

The moving load characteristics are reported in Eq. (30),in which ( p , e) describe the dimensionless applied loads andeccentricity with respect to the girder cross sectional geometricaxis, whereas (, 0) correspond to the mass and polar mass

moment ratio between the external loads and the girder. Finally,(,) dene the dimensionless moving load speed and thetime variables. Adopting the following notation for spatial andtime derivatives, f = f X , f =

f , with X =

xH , the

dynamic equilibrium equations are now presented in termsof the normalized kinematic variables, dimensionless bridgeand moving system parameters, by means of the followingexpressions:Girder

4F 4

V I V + V + V

1 H 1 (U L U ) + H 2 (U R + U ) p f 1

+

f 2 V + f 1 V + 2 V + 2V

+ V ( X + L / 2H ) + ( X L / 2H ) V = 0, (32)

AU U H 1 (1V + 2(U U L ))

+ H 2 ( 1V + 2(U + U R))

f 2U f 1U = 0, (33)

0 + 1 H 1 L + H 2 R

+H b

p f 1e 0

f 2 0 f 1 = 0. (34)

Left pylon

0 L/ H [ 1V 2 (U U L )] d X + pU L + P U L ( U U L ) = 0, (35)

p L + p + L + 0 L / H (1 2 L ) dX = 0. (36)Right pylon

L/ H 0 [1V 2 (U + U R)] d X + pU R + P U R+ (U + U R) = 0, (37)

p R + p + R + L/ H 0 (1 2 R ) dX = 0, (38)with H 1 = H ( X ) , H 2 = H ( X ) , =

E S0 A0E

ggH sin 0

cos2 0 , V = v gEg , whereas the functions ( f 1 , f 2, , 1,2) ,

for the sake of brevity, are reported in Appendix A . Moreover,details of the derivation of dynamic equilibrium equations canbe found in Appendix B , in which, for conciseness, only Eq.(32) has been discussed.

The dynamic equilibrium equations introduce an integro-differential boundary value system. Moreover, in Eqs. (35)

(38), the kinematic variables, related to both left and right

pylons, depend only on the time variables. Therefore, in orderto utilize standard numerical methods to solve PDE systemequations, the integro-differential equations, given by Eqs.(32)(38) , are converted into a purely differential form. Theequivalence between the original and the modied systemsis guaranteed by the use of the penalty method, by which

it is possible to convert the equilibrium conditions, fromintegral to local form. In particular, the pylon kinematics(namely U L , U R R , L ) are assumed, ctitiously, to dependonboth time and space variables. However, they are constrainedover the spatial domain by means of penalty functionals,involving two penalty parameters, i.e. K U , K . Moreover,distributed and lumped forces arising from the cable systemand pylons have been taken into account, utilizing Heavisideand delta Dirac functions, respectively. As a result, the integro-differential dynamic equilibrium equations, i.e. Eqs. (35)(38) ,are transformed into the following relationships:

H 1 [ 1V 2 (U U L )] + X +L

2H pU L + P U L + K U U L = 0, (39)

H 1 (1 2 L ) + X +L

2H p L + p L

+ K L = 0, (40)

H 2 [1V 2 (U + U R )] + X L

2H

pU R + P U R + K U U R = 0, (41)

H 2 (1 2 R ) + X L

2H p R p R

+ K = 0. (42)

By comparing the sets of Eqs. (39)(42) and Eqs. (35)(38) ,related to the differential and the integro-differential forms,respectively, it is easy to recognize that the internal forcesreferred to the anchor stays have not yet been introduced. Inparticular, the transformation of equilibrium equations relatedto the pylons, from global into local form, leads to consideringthe concentrated forces and moments arising from anchor stayaxial forces as boundary conditions, by means of the followingequations:

U L

( L / H , ) = 1

K U (U U

L) , U

L( L / H , ) = 0

U R( L / H , ) =1

K U (U + U R ) , U R ( L / H , ) = 0

L ( L / H , ) =1

K , L ( L / H , ) = 0,

R ( L / H , ) =1

K , R ( L / H , ) = 0.

(43)

Finally, the following relationships summarize the remain-ing boundary conditions, i.e. Eq. (20), which impose that thevertical and torsional displacements vanish at the left and rightand bridge cross section ends, natural conditions for girder ax-

ial displacements and homogeneous relationships with respect


9/18


Fig. 3. Moving load problem for A shaped tower typology: bridge kinematics and internal force distribution.

to the time variable:U ( L / H , ) =

A

(U U L ) ,

U ( L / H , ) = A

(U + U R) ,

V ( L / H , ) = 0, V ( L/ H , ) = 0,

V ( L / H , ) = 0, V ( L / H , ) = 0,

( L / H , ) = 0, ( L / H , ) = 0,( X , 0) = 0, ( X , 0) = 0,

U ( X , 0) = 0, U ( X , 0) = 0,

V ( X , 0) = 0, V ( X , 0) = 0,

i ( X , 0) = 0, i ( X , 0) = 0, U i ( X , 0) = 0,

U i ( X , 0) = 0 i = R , L .

(44)

From a numerical point of view the penalty stiffnessparameters, V , K U , K , are assumed to be sufciently highto impose proper constraint conditions, but not, exceedingly,to introduce numerical instabilities in the computation. For anumerical evaluation of the stiffness values, from the Authorsexperience of applying the model over different structures, arange between [10 5106] is suggested.

Long span bridges are frequently designed with A shapedpylon typologies, which are able to efciently reduce bothtorsional and exural deformations produced by eccentric ortransverse loading. In order to analyse such a bridge typology,the previous formulation still applies, but slight modicationsto the main governing equations need to be performed. Asa matter of fact, the dynamic equilibrium equations withrespect to exural deformation in the plane xy are, basically,the same. In contrast, the top pylon kinematic is governedby longitudinal displacements only, because both out-of-plane(xz) and torsional (with respect to y) kinematic parametersare assumed to be meaningless. Moreover, the distributedstays are now inclined with respect to the vertical direction,i.e. the y axis, and determine a coupling behaviour between

out-of-plane exural and torsional deformations. As depicted

in Fig. 3, introducing the misalignment angles, i.e. , ,which describe the orientation of a generic stay element withrespect to vertical and longitudinal directions, respectively,and transverse displacement w regarding the xy plane, theinteraction forces between cable system and girder assume thefollowing expressions:

qv L ( R) =E S ASH

v sin3

u L( R) (+ )u sin2 cos cos ,

qhL ( R) =E S ASH

( )v sin2 cos cos

u L ( R) u cos2 sin cos2 ,

quL ( R) =E S ASH

v sin2 cos cos

+ u L ( R) (+ )u cos2 sin cos2 ,

m L( R) =E S ASb

2

H sin3 +

w x

cos sin2 cos

wb

sin2 cos sin

S0L( R) =E s0 As0

H u (+ )u L ( R) cos2 0 sin 0 cos ,

(45)

qw L( R) =E S ASH

sin cos sin

b sin (+ )w x

b cos cos w cos sin ,

mw L ( R) = (+ )E S ASb

H sin cos cos

b sin (+ )w x

b cos cos (+ )w cos sin ,

(46)

where (qi , m , S0) with ( i = v, h , u ) , are the internal forces

produced by the cable system girder and pylons. Moreover,


10/18


Table 1Comparison of maximum normalized torsional rotation in terms of the relative torsional stiffness ( ) and bridge size parameter ( a ) between simplied (S) andgeneral (G) formulations

a Model = 0.05 = 0.1 = 0.15 = 0.2 = 0.25

0.05 G 115.39 93.21 74.85 61.69 53.08S 115.89 93.57 75.06 62.05 53.50

0.10 G 184.65 139.79 107.17 86.44 77.08S 183.78 137.35 105.84 86.82 77.22

0.15 G 237.77 174.45 133.39 111.17 237.77S 237.84 174.37 133.29 111.14 237.84

0.20 G 272.12 209.57 164.16 135.73 272.12S 271.76 210.71 164.20 135.97 271.76

0.25 G 338.72 250.96 195.65 160.61 338.72S 332.65 246.57 191.63 157.65 332.65

0.30 G 381.34 283.29 218.71 176.48 160.61S 377.06 280.07 217.45 176.61 151.20

Table 2Comparisons of maximum normalized torsional rotation in terms of speed parameter ( ) and relative torsional stiffness ( ) between simplied (S) and general (G)formulations

Model a = 0.1 a = 0.2 = 0.1 = 0.15 = 0.2 = 0.1 = 0.15 = 0.2

0.05 G 117.99 95.31 83.67 191.55 152.52 128.74S 117.31 94.97 83.66 188.85 150.41 129.68

0.07 G 123.33 99.15 84.60 190.29 152.44 131.02S 122.80 98.82 84.49 191.65 153.11 131.01

0.09 G 127.93 101.54 85.06 209.95 164.71 136.03S 127.98 101.82 85.38 208.57 163.62 135.53

0.11 G 139.56 106.54 88.09 223.88 172.00 140.75S 138.49 106.20 87.78 222.20 170.95 139.88

0.13 G 142.69 112.63 92.10 238.69 181.20 144.29S 143.75 113.48 92.64 237.56 181.12 145.11

qw L ( R) and mw L ( R) represent distributed forces and momentsin the (xz) plane (Fig. 3), produced by the coupling behaviourby torsional and exural deformations, arising from the inclinedstays of the A shaped tower typology.

The dynamic equilibrium equations, for A shaped bridgetypology, can be easily derived, starting from Eqs. (21)(26) and making use of Eq. (45) to describe the interactionforces between the cable system and the girder. Moreover, anadditional equilibrium equation is required due to the presence

of the transverse displacement, w . In particular, as reported inFig. 3, the following expression regarding exural deformationin the xy plane is introduced:

w + E I s w I V + H (x) qs R + m s R

+ H ( x) qsL + m sL = 0. (47)

Finally, the dynamic equilibrium equations for the Ashaped tower typology can be summarized by Eqs. (21)(26)and (47). However, in the typical range of both geometrical,mechanical and moving load characteristics, investigationshave been shown that for in-plane loading conditions (loadsapplied in the xy plane), the transverse deformations are

practically negligible and do not inuence dynamic bridge

behaviour. To this end, sensitivity analyses have been proposedin terms of maximum normalized torsional rotation duringmoving load application, i.e. (/ pe ) . In particular, resultsconcerning the actual solution, namely the General Approach(GA), derived from Eqs. (21)(26) and (47), and a simpliedone, namely the Simplied Approach (SA), obtained assuming,a priori, that the transverse displacements are negligible,i.e. w( x , t ) = 0, have been compared. The following bridgeand moving load parameters, typically utilized in practicalapplications, have been assumed constant during the analyses:

e/ b = 0.5, p / g = 1, J w / J = 100, Lp = 750 m,

= 1, 0 = 0.5, p = 0.085. (48)

Variability with respect to dimensionless torsional girderstiffness, , bridge size parameter, a , and moving systemspeed, , have been investigated. The results reported inTables 1 and 2 denote that the dynamic bridge behaviour ispractically unaffected by the transverse exural deformationsderiving from w displacements. The actual solution and thesimplied ones make the same prediction, with an error of less

than 2%.


11/18


The dynamic equilibrium equations for A shaped towersin the dimensionless context are not presented here for the sakeof brevity, but they can be easily derived by introducing thedimensionless parameters previously dened in Eqs. (28)(31)in the governing equations.

3. Numerical procedure

The dynamic equilibrium equation system introduces a PDEsystem from which it is quite difcult to derive an analyticalsolution, because a large variable number and complexities areinvolved in the main equations. The governing equations areconverted to an equivalent differential system of the rst order.In particular, for any dependent variables involved with anorder higher than the rst one, additional functions representingall lower order time derivatives are introduced, by means of supplementary equations, which are appended to the mainsystem. As a result, the reformulated boundary value problemassumes the following form:

a i j ( X , ) y j + bi j ( X , ) y j = f i ( y1, y2 , y3 , . . . , y14 , X ) ,

with i , j = 1, 14, (49)

where y = (V , V , V , V , V , U , U , U , U R , U R , U R , U L ,U L , U L ) is the vector of unknown functions or primaryvariables and a i j , bi j are constants depending on both themoving loads and the bridge properties. Moreover, f i representproper transformation operators, which dene the relationshipbetween primary variables and known quantities, in accordancewith the PDE system given by Eqs. (32)(34) and (39)(42) .In accordance with Eqs. (43) and (44), initial and boundaryconditions with respect to both space and time are introducedby means of the following equations:

B1 yi X , 0 = yi , B2 yi (0, ) = yi ,

with X = L / H , i = 1 . . . 14, (50)

where B1 , B2 are proper transformation matrices, whichguarantee the consistency of the boundary conditions with Eq.(49), and yi , yi represent known quantities related to thetemporal and spatial variables, respectively.

A numerical integration scheme has been utilized by meansof a nite difference method, which uses a second-order centredimplicit scheme for both time and spatial derivatives [22].The method has a truncation error of O (( t )2( x)2) and isunconditionally stable for all time steps. In order to capture therapid changes of the solution during the time integration, thewhole domain has been discretized by means of an accuratemesh point number. Moreover, requested values, those whichdo not lie on a mesh point, have been computed using aLagrange interpolation that uses four space subdivisions andthree time points. The solution process is obtained consistentlywith an error control procedure, which is able to integratethe nonlinear equations with respect to upper bounds errortolerances related to both time and spatial variables. Inparticular, the spatial error estimate is obtained by comparingthe solution to another one computed on a coarser spatial mesh,

but assuming the same time step. This gives an estimate of the

O (h 2) local truncation error of second order in h (where his the mesh spacing for the spatial variable). Alternatively, thetime error estimate is obtained by comparing the solution toanother computed with a larger time step, but the same spatialmesh. This gives an estimate of the O (k 2) local truncationerror of second order in k (where k is the time step). However,

these estimates are local, so they do not account for situationswhere a small perturbation in the solution at a time t = t 1 canlead to a large change in the solution at a later time.

The numerical results are derived providing at rst atrial integration time step, which is subsequently reduced bymeans of a proper adaptive procedure in order to satisfy theconvergence conditions. Contrarily, the spatial discretizationremains xed during the analysis, and consequently, in orderto minimize the integration errors, a proper mesh point numberover the bridge structure has been adopted. In the followingresults, the spatial domain is discretized utilizing more than10 000 subdivisions over the whole bridge length. The initialintegration time step, which is automatically reduced due to thetime adaptation procedure, is assumed as at least 1 / 1000 of theobservation period dened as the time necessary for the movingtrain to cross the bridge. On a Pentium IV processor at 3000 Mzthe CPU time required for performing the time history for eachcase was approximately 3 min.

4. Numerical results and parametric study

The results dene the relationship between the characteris-tics of the bridge and applied moving loads, emphasizing theeffects produced by the external mass on the dynamic bridgevibrations. In particular, a parametric study is proposed, which

describes cable-stayed bridge behaviour in terms of dimension-less variables, strictly related to both the moving loads and thebridge characteristics. Numerical results are presented in termsof dynamic impact factors, in order to quantify the amplicationeffects produced by the moving loads over the static solution(i.e. st ), by means of the following relationship:

X =max X t = 0... T

X st (51)

where T is the observation period and X is the variable underinvestigation. The parametric study has been developed toinvestigate the following variables:

V dynamic amplication factor of the midspan verticaldisplacement, M dynamic amplication factor of the midspan bending

moment, 0 dynamic amplication factor of the axial force in the

anchor stay, dynamic amplication factor of the axial force in the

longest central span stay. dynamic amplication factor of the midspan girder

torsional rotation.

The bridge and moving load dimensioning is selected inaccordance with the values utilized in practical applications

and due mainly to both structural and economical factors.


12/18


Table 3Percentage errors of midspan vertical displacement and bending momentdynamic amplication factors ( V , M ) between the moving force model(MFM), standard acceleration (SA) and proposed results for differentnormalized speed parameters ( )

V M Error % SA Error MFM % Error SA % Error MFM %

0.02 0.37 0.51 0.87 0.210.04 1.16 3.54 1.58 21.93

0.05 5.11 6.40 2.59 33.640.07 9.24 10.60 11.15 33.870.09 2.42 11.25 18.23 39.69

0.11 4.65 20.74 38.32 52.410.13 19.17 31.34 37.58 42.87

The dimensionless parameters related to aspect ratio, pylonstiffness, allowable cable stress and moving load characteristicsare assumed as equal to the following representative values:

L / 2H = 2.5, l / H = 5/ 3,

a / E = 7200 / 2.1 106 , K p / g = 50,p / g = [ 0.5 1], 0 = [ 0 2].

(52)

In order to evaluate the inuence of both mass schematiza-tion and moving system speed, comparisons in terms of thedynamic amplication factors (DAFs) have been proposed inFigs. 47 . In particular, the actual solution has been comparedwith numerical results based on the following assumptions:

(1) Inertial description of the moving system completelyneglected, i.e. p = 0, = 0, namely the moving forcemodel (MFM).

(2) Inertial description of the moving system neglected withrespect to non-standard inertial forces, i.e. p = 0, = 0,f 2 = 0, 2 V = 2V = 0, namely Standard Analysis(SA).

The proposed results do not agree with those arising from theSA, especially at high speed of the moving system, where ithas been shown that non-standard terms in the accelerationfunction provide notable amplications in both kinematicand stress variables. Moreover the comparisons between theproposed formulation and those concerning the MFM are notin agreement in wide ranges of the speed parameter. However,for reduced values of moving system speed, i.e. 0,the results arising from the dynamic and static solutions arepractically coincident and, consequently, the inuence of themass schematization becomes negligible. In order to quantifynumerically the inuence of the inertial effects of the movingsystem, the percentage errors between the SA and the MFM andthe proposed formulation have been reported in Table 3 . Finally,in Figs. 4 and 5, dynamic amplication variability with respectto the speed parameter for different intensity ratios between liveand self-weight loads, are proposed.

In Tables 4 and 5, the inuence of the geometric ratios of the bridge, i.e. L / l and l / H , on the DAFs is investigated atxed speed of the moving system and relative girder stiffness,(i.e. = 0.2, = 0.10. In particular, the bridge geometry is

assumed to verify well-known design rules derived from both

Fig. 4. Midspan vertical displacement dynamic impact factor ( V ) vs normal-ized speed parameter ( ).

Fig. 5. Bending moment dynamic impact factor ( M ) vs normalized speedparameter ( ).

structural and practical conditions, which guarantee stabilityof the anchor stays, avoiding excessive steel quantity amountin the cable system [1618 ]. The DAFs for both bendingmoments and displacements generally grow for increasingratios between the central span and the height of the pylons,because of the most greater deformability of the structuralsystem. However, the impact factors based on the bending

moments are quite dependent from the geometric aspect


13/18


Table 4Dynamic amplication factors for midspan vertical displacement ( V ) vs geometric bridge ratios L/ l and H / L

L/ l a = 0.1 a = 0.2L/ H = 4 L/ H = 5 L/ H = 6 L/ H = 4 L/ H = 5 L/ H = 6

2.75 1.308 1.356 1.390 1.25 1.29 1.353.00 1.282 1.332 1.378 1.20 1.25 1.32

3.25 1.256 1.312 1.371 1.18 1.23 1.313.50 1.235 1.304 1.374 1.16 1.22 1.32

Table 5Dynamic amplication factors for midspan bending moment ( M ) vs geometric bridge ratios L/ l and H / L

L/ l a = 0.1 a = 0.2L/ H = 4 L/ H = 5 L/ H = 6 L/ H = 4 L/ H = 5 L/ H = 6

2.75 1.903 2.394 3.094 1.464 1.855 2.5903.00 1.664 2.280 2.908 1.431 1.727 2.407

3.25 1.674 2.090 2.390 1.722 1.652 2.3613.50 1.563 1.906 2.583 1.588 1.821 2.262

Fig. 6. Anchor stay dynamic impact factor ( 0 ) vs normalized speedparameter ( ).

ratios than the corresponding ones for vertical displacements.Moreover, at xed L/ H , the DAFs for vertical displacementsare quite unaffected by the ratio between the main and centralspans, because the cable system stiffness remains practicallyconstant during the analyses.

The relationship between the DAFs and bridge size isinvestigated in Figs. 8 and 9. In typical allowable rangesof the a parameter, the DAFs, related to cinematic andstress bridge variables, are analysed for an external movingsystem with a constant speed advance and different loadinglengths (namely c = 120 m/ s, Lp1,2 = 500, 1000 m,M p g / K = 2.3). The comparisons are proposed toinvestigate the effect of the external moving mass on the

dynamic bridge vibrations. In particular, the results show a

Fig. 7. Longest centre span stay dynamic impact factor ( ) vs normalizedspeed parameter ( ).

tendency to decrease with an oscillating behaviour and somelocal peaks in curve development. The inertial effects produceconsiderable amplications in both the displacement and stressvariables, especially, for low values of the bridge size parametera . Moreover, results concerning the MFM determine notableunderestimates in both stress and displacement DAFs.

In Figs. 10 and 11, the dynamic behaviour of the bridgeis investigated with respect to the dimensionless parameter F , which denes the normalized stiffness of the girder withrespect to the cable system. In particular, the analysis hasbeen developed at a constant speed of the moving systemand for different values of the bridge size parameter, namely

a = (0.1, 0.2) , emphasizing the inuence of the external


14/18


Fig. 8. Midspan displacement dynamic impact factor ( V ) vs bridge sizeparameter ( a ).

Fig. 9. Bending moment dynamic impact factor ( M ) vs bridge size parameter(a ).

moving mass description on the dynamic behaviour of thebridge. As a matter of fact, the actual solution is comparedto the case in which the inertial effects of the train loadshave not been accounted for. The dynamic bridge behaviourappears to be quite sensitive to the external mass description,and underestimates of the dynamic impact factors are notedif the travelling mass has not been properly evaluated. The

major amplication effects are noted for low ranges of the

Fig. 10. Midspan displacement dynamic impact factor ( V ) vs relative girderstiffness parameter ( F ).

Fig. 11. Bending moment dynamic impact factor ( M ) vs relative girderstiffness parameter ( F ).

F parameter, in which the bridge structure is basically moreexible and, mainly, dominated by the cable-stayed system.In contrast, for high values of F , corresponding to girder-dominated bridge structures, the effects of the inertial forcesof the moving system are notably reduced.

The dynamic bridge behaviour is analysed with respect toeccentric loads, which involve both exural and torsional de-

formations. In particular, in order to evaluate the amplication


15/18


Fig.12. Midspan torsional rotationdynamic impact factors ( ) vs normalizedspeed parameter ( ) for HST.

Fig.13. Midspan torsional rotationdynamic impact factors ( ) vs normalizedspeed parameter ( ) for AST.

effects produced by moving loads for bridge structures basedon both A and H shaped towers (namely AST, HST), a sen-sitivity analysis has been developed. The results are presentedin term of maximum normalized torsional rotation and rela-tive DAF produced by the moving load application, at midspangirder cross section, i.e. X = 0.

In Figs. 12 and 13, the effects of mass distribution of themoving system for both AST and HST is investigated in termsof midspan torsional rotation for different values of the loadingstrip length L p . In particular, in Fig. 12, for the HST typology,dynamic amplication displays a tendency to grow especiallyfor reduced ratios of loading application length and central

bridge span. Moreover, the inertial forces of the moving system

Fig. 14. Midspan torsional rotation dynamic impact factors ( ) vs relativegirder stiffness ( ) parameter for AST-HST.

Fig. 15. Maximum normalized displacement () vs relative girder stiffness( ) parameter for AST-HST.

determine an intensication of the DAFs. In contrast, in Fig. 13,the AST denotes a smaller dependence on the loading striplength and the moving mass schematization. This behaviourcan be explained due to the fact that the AST typologies show,generally, greater stiffness with respect to the HST ones, whichstrongly reduce dynamic amplications, and as a result theeffects of the moving mass becomes negligible.

In Figs. 14 and 15, sensitivity analyses of DAFs andmaximum normalized torsional rotation, i.e. and = / perespectively, with respect to girder torsional stiffness parameter,

, are proposed. The comparisons reported in Fig. 14, denote


16/18


17/18


Appendix A

In order to simplify the presentation of the dynamicequilibrium equations in dimensionless formulation, thefollowing relationships have been utilized:

f 1 = H ( X 1) H X 1 + L pH , (A.1)

f 2 = ( X 1) H X 1 +L pH

X 1 +L pH

H ( X 1) , (A.2)

( X )

=

11 + a X 2

11 + X 2

LH

X 0,

1

1 + a L2H X 2

1

1 + L2H X 2 , 0 X

LH

,(A.3)

1 ( X )

=

11 + a X 2

X 1 + X 2

, LH

X 0,

1

1 + a L2H X 2

L2H X

1 + L2H X 2 , 0 X

LH

,(A.4)

2 ( X )

=

11 + a X 2

X 2

1 + X 2,

LH

X 0,

1

1 + a L2H X 2

L2H X

2

1 + L2H X 2 , 0 X

LH

,(A.5)

with X 1 = x1/ H .

Appendix B

Starting from Eqs. (28)(31) , the following expressions canbe determined:

v t

= H V

t

= H V Eg

H g

1/ 2,

2v t 2

= H 2V 2

Eg H g

,

(B.1)

v x

= H V X

X x

= V , 2v x2

=1H

2V X 2

=1H

V . . .

4v x4

=1

H 34V X 4

=1

H 3V I V

(B.2)

2v t x

= V Eg

H g

1/ 2, c =

EgH g

1/ 2. (B.3)

Moreover, in view of Eqs. (B.1) and (B.3) and Eqs.(28)(31) , the interaction forces between the cable systemand the girder (qv L , qv R) and the mass function of themoving system ( ) can be expressed by the following

relationships:

qv L =E S ASH

v sin3 (u L u ) sin2 cos

=Eg g

(V (U L U ) 1) (B.4)

qv R =E S ASH

v sin3 (u R + u ) sin2 cos

= Eg g

(V (U L + U ) 1) (B.5)

= H x1 + L p ct H (ct x1)

= H X 1 +L pH

H ( X 1) (B.6)

=Eg

H g

1/ 2

X 1 +L pH

H ( X 1)

H X 1 +L pH

( X 1) . (B.7)

By substituting Eqs. (B.4)(B.7) in Eq. (21), and taking intoaccount Eqs. (B.1)(B.3) , the following equation is obtained:

2V 2

g I g H 3

V I V H ( X ) ( V (U L + U ) 1)

H ( X ) ( V (U L U ) 1)

X 1 +

L pH

H ( X 1)

H X 1 +L pH

( X 1) V

H X 1 +

L pH

H ( X 1)

2V 2 + 2 V +

2V + p f 1

+ ( X + L / 2H ) + ( x L / 2H ) v gEg

V = 0,

(B.8)

and taking into account Eqs. (28)(31) and Eqs. (A.1) and(A.2) , Eq. (32) is nally determined.

References

[1] Fryba L. Vibration of solids and structures under moving loads. London:Thomas Telford; 1999.

[2] Timoshenko SP, Young DH. Theory of structures. New York: McGraw-Hill; 1965.

[3] Yang YB, Liao SS, Lin BH. Impact formulas for vehicles moving oversimple and continuous beams. J Struct Eng 1995;121(11):164450.

[4] Lei X, Noda NA. Analyses of dynamic response of vehicle and track coupling system with random irregularity of track vertical prole. J SoundVibration 2002;258(1):14765.

[5] Roeder CW, Barth KE, Bergman A. Effect of live-load deections on steelbridge performance. J Bridge Eng 2004;9(3):25967.

[6] Warburton GB. The dynamical behavior of structures. Oxford: Pergamon;1976.

[7] Wiriyachai A, Chu KH, Garg VK. Bridge impact due to wheel and track irregularities. J Eng Mech Div 1982;108:64865.

[8] Au FTK, Wang JJ, Cheung YK. Impact study of cable-stayed bridge underrailway trafc using various models. J Sound Vibration 2001;240(3):44765.

[9] Au FTK, Wang JJ, Cheung YK. Impact study of cable-stayed railway

bridges with random rail irregularities. Eng Struct 2001;24(5):52941.


18/18


[10] Yang F, Fonder GA. Dynamic response of cable-stayed bridges undermoving loads. J Eng Mech 1998;124(7):7417.

[11] Yau JD, Yang YB. Vibration reduction for cable-stayed traveled by high-speed trains. Finite Element Anal Design 2004;40:34159.

[12] Huang D, Wang TL. Impact analysis of cable stayed bridges. ComputStruct 1992;43(5):897908.

[13] Meisenholder SG, Weidlinger P. Dynamic interaction aspects of cable-stayed guide ways for high speed ground Transportation. J Dyn Syst MeasControl ASME 1974;74-Aut-R:18092.

[14] Chatterjee PK, Datta TK, Surana CS. Vibration of cable-stayed bridges under moving vehicles. Struct Eng Int 1994;4(2):116121.

[15] Bruno D, Leonardi A. Natural periods of long-span cable-stayed bridges.J Bridge Eng 1997;2(3):10515.

[16] De Miranda F, Grimaldi A, Maceri F, Como M. Basic problems in long

span cable stayed bridges. Internal Report no 25. Department of StructuralEngineering, University of Calabria; 1979.

[17] Troitsky MS. Cable stayed bridges. London: Crosby Lockwood Staples;1977.

[18] Gimsing NJ. Cable supported bridges: Concepts and design. John Wiley& Sons Ltd; 1997.

[19] Xia H, Xu YL, Chan THT. Dynamic interaction of long suspensionbridges with running trains. J Sound Vibration 2000;237(2):26380.

[20] Kawashima K, Unjoh S, Tsunomoto M. Estimation of camping ratio of cable-stayed bridges for seismic design. J Struct Eng, ASCE 1993;119(4):101531.

[21] Yamaguchi Hiroki, Ito Manabu. Mode-dependence of structural dampingin cable-stayed bridges. J Wind Eng Industrial Aerodynam 1997;72(13):289300.

[22] MAPLE Maplesoft. Waterloo Maple Inc. 2006.

Documents

bridge under moving load