Structural Engineering, Mechanics and Computation. Proceedings of the International Conference on Structural Engineering, Mechanics and Computation 2–4 April 2001, Cape Town, South Africa

1543 Pages·2001·35.6 MB·English

by  A. Zingoni (Eds.)

FOREWORD
by
Professor Patrick Dowling CBE DL FREng FRS
Vice-Chancellor and Chief Executive
University of Surrey, UK
Chairman, Steel Construction Institute, UK
These Proceedings will be regarded in time as seminal in that they record the enormous
progress which has been made in Structural Engineering, Mechanics and Computation in
the later half of the 20th century, and point the way to the future agenda for research in
those areas for the 12 st century.
The two volumes record the collected work of some of the most able researchers working
on the world stage at this moment in time, and who are laying the foundations to some
exciting new developments in the future. In that respect, the Proceedings should prove
essential reading for those newly entering the field.
It is also most appropriate that the SEMC 2001 International Conference be held in South
Africa where there is a New Dawn unfolding, and the spirit of cooperation is infusing all
sectors of society, not least places of learning such as universities and colleges which are
at the very heart of the New Knowledge Economy and Life Long Leaming Agenda.
I salute all of those involved in the organisation, preparation and presentation of the
Conference, as well as the delegates for their contribution to the shaping of these
Proceedings, and assure all potential readers of the high quality and usefulness of its
contents.
PREFACE
The International Conference on Structural Engineering, Mechanics and Computation
was held in Cape Town (South Africa) from 2 to 4 April 2001. Organised by the
University of Cape Town, the conference (SEMC 2001) aimed at bringing together
from around the world academics, researchers and practitioners in the broad area of
structural engineering and allied fields, to review the achievements of the past 50
years in the advancement of structural engineering, structural mechanics and
structural computation, share the latest developments in these areas, and address the
challenges that the future poses.
The Proceedings contain, in two volumes, a total of 180 papers written by Authors
from around 40 countries worldwide. The contributions include 6 Keynote Papers and
21 Special Invited Papers. In line with the aims of the SEMC 2001 International
Conference, and as may be seen from the List of Contents, the papers cover a wide
range of topics under a variety of themes. There is a healthy balance between papers
of a theoretical nature, concerned with various aspects of structural mechanics and
computational issues, and those of a more practical, nature, addressing issues of
design, safety and construction. As the contributions in these Proceedings show, new
and more efficient methods of structural analysis and numerical computation are
being explored all the time, while exciting structural materials such as glass have
recently come onto the scene. Research interest in the repair and rehabilitation of
existing infrastructure continues to grow, particularly in Europe and North America,
while the challenges to protect human life and property against the effects of fire,
earthquakes and other hazards are being addressed through the development of more
appropriate design methods for buildings, bridges and other engineering structures.
I would like to thank all Authors for preparing their work towards this compilation
which, on account of the wealth of information it contains in just two volumes, will
undoubtedly serve as a useful reference to practitioners, researchers, students and
academics in the areas of structural engineering, structural mechanics, computational
mechanics, and allied disciplines. Special thanks are due to Members of the
International Scientific/Technical Advisory Board of SEMC 2001, who gave their
time in advising on the selection of material contained in these Proceedings. The
financial support of the Sponsoring Organisations is gratefully acknowledged. I am
indebted to my colleagues in the Organising Committee, for the hard work they put
into the preparations for the conference. Last but not least, I would like to thank my
wife Lydia for the immense contribution she made towards making the SEMC 2001
International Conference a success.
A. Zingoni
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VIBRATION ANALYSIS OF BRIDGES UNDER MOVING
VEHICLES AND TRAINS
Y. K. Cheung, Y. .S Cheng and .F .T K. Au
Department of Civil Engineering, The University of Hong Kong,
Pokfulam Road, Hong Kong, People's Republic of China
ABSTRACT
The vibration of bridges under moving vehicles and trains is of great theoretical and practical
significance in civil engineering. This paper describes some recent developments in the vibration
analysis of girder and slab bridges under the action of moving vehicles or trains. A bridge-track-
vehicle element is developed for investigating the dynamic interactions among a moving train, and its
supporting railway track structure and bridge structure. The effect of track structure on the dynamic
response of the bridge structure and the effect of bridge structure on that of the track structure are
identified. The proposed bridge-track-vehicle element can be easily degenerated to a vehicle-beam
element, which is employed to study the effects of the random road surface roughness and the long-
term deflection of concrete deck on the dynamic response of a girder bridge. The plate-vehicle strip
for simulating the interaction between a rectangular slab bridge and moving vehicles is also described.
Two kinds of plate finite strips, namely the plate-vehicle strip and the conventional plate strip, are
employed to model a slab bridge. In the analysis, each moving vehicle is idealised as a one-foot
dynamic system with the unsprung mass and sprung mass interconnected by a spring and a dashpot. A
train is modelled as a series of moving vehicles at the axle locations. The efficiency and accuracy of
the proposed methods are demonstrated by numerical examples.
KEYWORDS
Bridge vibration, moving vehicles, moving trains, finite element method, finite strip method
INTRODUCTION
The vibration of bridges under moving vehicles and trains is of great theoretical and practical
significance in civil engineering, and it has attracted much attention during the last three decades. This
is in part due to the rapid increase in the proportion of heavy vehicles and high-speed vehicles in the
highway and railway traffic, and the trend to use high-performance materials and therefore more
slender sections for the bridges. Vehicle-bridge interaction is a complex dynamic phenomenon
depending on many parameters. These parameters include the type of bridge and its natural
frequencies of vibration, vehicle characteristics, vehicle speed and traversing path, the number of
vehicles and their relative positions on the bridge, roadway surface irregularities, the damping
characteristics of bridge and vehicle etc.
The moving force model, moving mass model and moving vehicle model are three essential
computational models used to analyse the dynamic responses of bridges due to moving vehicles and
trains. The moving force model is the simplest model with which the essential dynamic characteristics
of a bridge under the action of moving vehicles can be captured, although the interaction between the
vehicles and bridge is ignored. Where the inertia of the vehicle cannot be regarded as small, a moving
mass model is often adopted instead. However the moving mass model suffers from its inability to
consider the bouncing effect of the moving mass, which is significant in the presence of road surface
irregularities or for vehicles running at high speeds. The advent of high-speed digital computer a few
decades ago made it possible to analyse the interaction problem with more sophisticated bridge and
vehicle models. The vibration of various types of bridges such as girder bridges, slab bridges, cable-
stayed bridges and suspension bridges due to moving vehicles and trains can be studied by using a
moving vehicle model, in which a vehicle is modelled as a single-axle or multi-axle mass-spring-
damper dynamic system.
The analytical methods as described by Fryba (1999) may be used to solve problems involving simple
structures. As these analytical methods are often limited to simple moving load problems, many
researchers have resorted to various numerical methods such as finite element method (FEM), finite
strip method (FSM) and structural impedance method (SIM) to analyse the interaction problems.
Among various numerical methods for the class of problems, FEM is no doubt the most versatile and
powerful. However, the use of refined meshes often gives rise to large matrix equations with
comparatively large bandwidth. In this respect, FSM is particularly suitable for regular slab bridges,
and this is especially true for simply supported rectangular slab bridges. Consequently, where
applicable, FSM is cheaper than FEM for solutions of comparable accuracy. SIM is also efficient for
the analysis of bridge-vehicle interaction problems, as one can treat the modelling of the bridge and the
vehicles separately, and therefore changes in one part do not affect those in the other. In addition,
modal superposition method does enable further reduction of the problem size.
The vibration analyses of girder bridges under moving vehicles or trains have been extensively
investigated. Some of the more recent work includes Olsson (1985), Coussy et al. (1989), Wang et al.
(1991), Huang et al. (1992), Yang & Lin (1995), Fryba (1996), Yang & Yau (1997), Henchi et al.
(1997), Cheung et al. (1999a), Leet al. (1999), Yang et al. (1999) and Yau et al. (1999). It is noted
that, in most of the previous studies on railway bridge vibration, the effects of the track structure have
been completely neglected or only partially accounted for.
It is inadequate to model wide slab bridges using beam models, particularly when the vehicle paths are
not along the centre-line of the bridge. The vibration of slab bridges modelled as isotropic or
orthotropic plates under the action of moving loads has so far received but scant attention. Wu et al.
(1987), Taheri & Ting (1990), Yener and Chompooming (1994), Humar & Kashif (1995) etc.
separately used the FEM to analyse the vibration of plates under moving vehicles. Taheri & Ting
(1989) used SIM to study the dynamic response of guideways by treating the moving loads and the
guideway as components of an integrated structural system. Cheung et al. (1999b) also utilised SIM to
solve a similar class of problem, but FSM was employed to obtain the influence function. By using the
method of modal analysis, Wang & Lin (1996) analysed the vibration of multi-span Mindlin plates
under a moving load.
In this paper, a bridge-track-vehicle element is proposed for investigating the dynamic interactions
among a moving train, and its supporting railway track structure and bridge structure. One of its
degenerated versions, the vehicle-beam element, is employed to investigate the effects of the random
road surface roughness and long-term deflection of concrete deck on the dynamic response of a girder
bridge. A plate-vehicle strip is reported for simulating the interaction between a rectangular slab
bridge and the moving vehicles. The efficiency and accuracy of the proposed methods are
demonstrated by numerical examples.
THEORY AND FORMULATION
The methods described in this paper are based on the Bemoulli-Euler beam theory and Kirchhoff thin-
plate theory where applicable. The bridges under consideration are treated as plane structural systems
while each vehicle is modelled as a one-foot dynamic system, in which the unsprung mass and sprung
mass are interconnected by a spring and a dashpot. A moving train is modelled as a series of moving
vehicles at the axle locations.
The Bridge- Track- Veh&le Element
Figure 1 shows a typical bridge-track-vehicle element with a few vehicles running on it. The upper
and lower beam elements modelling the rail and the bridge deck respectively are interconnected by a
series of springs and dampers, which reflect the properties of the rail bed. It is assumed that there are
vN moving vehicles in direct contact with the upper beam element and Np spring-damper systems
between the upper beam element and the lower beam element. The ith vehicle proceeds with velocity
)t(~v and acceleration at(t) in the longitudinal direction. The stiffness of the spring and the damping
coefficient of the dashpot of the typical ith vehicle are denoted by k~ and c~ respectively, the unsprung
mass is denoted by n~m and the sprung mass is denoted by 2,,,m where i =1,2 ..... .vN The stiffness and
damping coefficients of the typical jth spring-damper system between the upper and lower beam
elements are respectively jpk and jpC where j= 1,2,..., .pN
~ 2 i v m
[
l i v m
x
I;
xc, _
Figure 1: A typical bridge-track-vehicle element
It is assumed that the upward deflections of rail and bridge deck are taken as positive and that they are
measured with reference to their respective vertical static equilibrium positions. Let rc(x) denote the
top surface irregularities of rail that is defined as the vertically upward departure from the mean
horizontal profile. The vector {re} contains the values of the surface irregularities of the rail at the
contact points ~cX (i=1,2 ..... )vN between the rail and the vehicles. The vertical displacements of the
masses nvm and mva are l~y and 2~y respectively, and they are measured vertically upward with
reference to their respective vertical static equilibrium positions before coming onto the bridge.
A matrix ]cN[ of dimension 4xNv is then defined such that it contains the cubic Hermitian interpolation
functions for the beam element evaluated at the contact points ~cx as follows
[N~]=[{N<}, {Nc}, ... {Nc},<] (1)
The vector {Nc}, in the matrix ]cN[ is given in terms of the vector {N(x)} as
1]
,[xJI1 - 3(xlZ) ~ + ~ 2(x/Z)
crA{ }j = {N(x)l ..... = [ 3(xlt) = - 2(x/t) It i=1,2 ..... ,,N (2)
L xTxlO -(xl t) Jlx
"- icX
where l is the length of the beam element.
In the case of discrete spring-damper support systems, a matrix ]pcN[ of dimension 4xNp, similar to the
matrix ,]cN[ is introduced and it contains the cubic Hermitian interpolation functions for the beam
element evaluated at the positions jpcX (j=1,2 .... )pN, of the spring-damper systems between the rail and
the bridge deck as follows
~}pcN{[=]p~N[ 2}p~N{ ... {N~p}N]=[{N(xcp~) } {N(xcp2) } ... {N(XcpN,)} ] (3)
One may establish the equations of motion for the vehicles, the upper beam element and the lower
beam element respectively. These equations are coupled through the unknown contact forces between
the moving vehicles and the upper beam element, and the supporting reactions acting through the
spring-damper systems between the rail and the bridge deck. Using the constraint equations at the
contact points in terms of the displacements, velocities and accelerations at those points, expressing the
supporting reactions in terms of the displacements and velocities of upper and lower beam elements at
the locations of spring-damper systems and eliminating the degrees of freedom (DOFs) of unsprung
masses, one can obtain the equation of motion for a bridge-track-vehicle element with discrete rail
supporting systems as follows (Cheng, 2000) using the conventional notations:
[[il] ]0[ ]0[ ]f{//bi } T]p~N[]pcE]p~N[+]bcEI T]p=N[]%[]p=NE_ ]o[ ]}bU{'[]
]0[ [m.lJ[{j~2 ]o[ [c,,] [c,]J I.{J,~iJ
[kb]+[N<p][kp][N<p] T -[N<p][kp][N<p] T ]O[ ][{Ub}] f {Pb} ]
_[Nw][kp][N,p] T [kr]+[kii]+[N,p][kp][N,p] T [k,,:l]i {u,/i> = i< {p,/+ {/,/i> (4)
]0[ ],2k[ }2y{.lJ]~k[ j .[ {£,} J
llor[ ] -[N~][m~][N~] ,T [c,,] = ~r.]cN[]v[]~vm[]cN[2 +[Nc][C.][Nc] r (5,6)
[C12 ] = _[Nc ][cv ], [c21 ] ._. _[cv ][Nc T] (7,8)
xr.]~N[]a[],~m[]~N[=],,k[ ~.]~N[2]v[],~m[]~N[+ ]cN[]v[]~C[]cN[+ ~,T +[N¢][k.][N~] T (9)
2,k[ = ] -[N c ~k[] ], = ] [k2, -[c, ][v][N c r.] x -[k~] [N c T] (10,11)
{fb = } -[N~]({fw + } [k.] }~r{ +[c.][v]{rc},~ x.}~r{]a[]l~m[+ + l~m[ 2 ][v] cr{ )x~.} (12)
}2~f{ = [cv][vl{rc}.x + [kv]{r~}, {fw} = {(my,, + mv,2)g} (13,14)
[mv,]=diag[m,,,, ], [m,,2]=cfiag[mv,2J,[c,,]=diag[c,,,],[k,,]=diag[k,,~] /--1,2 ..... ,,N (15-18)
[Cp] = V~ag[cpj ] ,[kp] = diag[kpj ] j=l,2 ..... Np (19,20)
2Y{ = } 2,Y{ }, [a] = diag[a, ], [v] = c~ag[v, ] = i 1,2 ..... Nv (21-23)
Matrices [mb], [Co] and [kb] are the mass, damping and stiffness matrices, respectively, of the lower
beam element representing the bridge deck, the vectors {u b }, bif{ } and {//b} are the corresponding
nodal displacement, velocity and acceleration vectors, respectively. Matrices r [m ], [Or] and [k~] are
the mass, damping and stiffness matrices of the upper beam element representing the track structure
respectively, and vectors {dr} , {dr} and {//,} are their nodal displacement, velocity and acceleration
vectors, respectively. The vector {Pb } is a force vector to account for any external load applied to the
bridge deck apart from the reactions through the spring-damper systems between the rail and the
bridge deck. The vector {Pr} is any other external load vector acting on the track structure apart from
the reactions and contact forces.
In the case of a rail supporting system with continuous distributed stiffness pk (x) and damping pC (X),
the terms [Ncp][kp][Ncp] T and [Ncp][Cp][Ncp] T in Eqn. 4 should be replaced by ~{N}kp(x){N} ~ dx
and t[0a {N}cp(x){N}T dx respectively in which the vector {N(x)} has been defined in Eqn. 2.
A bridge-track-vehicle element can be easily degenerated into a bridge-track element that applies to
the rest of the bridge With no moving vehicles on it. It can also be degenerated into special cases of a
beam on viscoelastic foundation or simply a beam, with or without the moving vehicles on it. The
equation of motion for the vehicle-beam element is obtained by deleting sub-matrices associated with
the DOFs of the lower beam element and the rail supporting systems, i.e.
]llm[-~]rm[[ ]llC[~-]rC[[_~}}}r~i{~l]O[ ]llk[-]-]rk[Ir~}r~{~l]2lC[ 42(}}rf{-~-}rP{~._.~}r~{fl]21k[ )
[o] [m~]J[{¢~ [c~,] [c.]][{p~}J [k~,] [k~]Jl{y~}J L {f,,2}
The Plate-Vehicle Strip
A slab bridge is modelled as a rectangular thin plate simply supported at two opposite edges and is
subjected to N moving vehicles proceeding with velocity v(t) in the longitudinal direction. The bridge
is first split up into a number of rectangular finite strips. The conventional plate finite strips can be
employed for those strips not carrying any moving vehicles. However the strips carrying vehicles have
to be separately treated.
Figure 2 shows a typical plate-vehicle strip with width b and length l, where the surface irregularity of
the bridge is also indicated. It is also assumed that there are vN moving vehicles in direct contact with
the strip. Let w(~,rl, t) denote the vertical deflection at time t of the point (~:,q) of a plate strip (upward
positive), which can be expressed as
,~(w =)t,/r £ }N{ T u{ b .} = }N{ v u{} o )52(
m=l
where r is the number of terms in the series used in the longitudinal part of the displacement function
of a strip, and {N},, and m}bU{ are, respectively, vectors containing the shape functions and nodal line
displacement parameters for the mth term of the displacement function. Obviously, {N}m is a function
of the position (~:,r/) on the strip and m}bU{ is a function of time .t The vectors {N}m and m}bU{ era
given in terms of the typical ith element as
,,,}N{ = {N~}, ,bU{=m}bU{ } /=1,2 ..... sxn~ (26,27)
in which s is the number of nodal lines in a strip and en is the number of DOFs associated with each
nodal line. The vectors {N}m and m}bU{ era given by Cheung & Tham (1998) for various plate finite
strips. For a lower order (LO2) mixed plate finite strip, the vectors {N},, and m}bU{ appear
respectively as
mlWI (/)]
[[1- 3(~/b) + 2 (~/b)3 ] m~Y (r/)]
= ~0'm(')~
=~[1-2(~/b)+(~/b)Z]Yl,(rl)[
(28,29)
m}g{ [ 2 [3(~:/b) -2(¢/b)3]yzm(17) [ ' {ub} m ]W2m(t)[
[ ~[(~/b)2-(~/b)]Y2m(O) J [O2m(t)]
where )/r(mjY is the longitudinal displacement function associated with the nodal line j (j= 1,2 for LO2
strip) of the strip for the mth term of the series; )t(eriw and 0jm(t) are the deflection and rotation
parameters at nodal linesj of the strip for the mth term of the series, respectively.
Let ~:~ = tb and rki = )t(~clr denote the transverse and longitudinal co-ordinates, respectively, of the ith
vehicle. One may then define a matrix ]s~N[ of dimension (rsne)xNv containing the displacement
functions at the positions of the vehicles as follows
ii1,, i I
)03(
IN=l= ...
. . . . . . . .o.
The sub-matrix q]scN[ is given in terms of the typical element as
=o],cN[ ,j~]2(N{ jc/r ) i=1,2 ..... r;j=l,2 ..... ~N (31)
The equation of motion for a plate-vehicle strip can be obtained similarly. Following the sign
conventions adopted in the bridge-track-vehicle element and replacing the mass, damping and stiffness
matrices of the beam element as well as the matrix ]crA[ defined by Eqn. 1 with the counterparts of the
plate strip, the equation of motion for a plate-vehicle strip has the same form as that for a vehicle-beam
element given by Eqn. 24. The plate-vehicle strip can therefore be taken as the extension of the
vehicle-beam element to the two-dimensional case.
t ,lr y
O i vehicle
)t(,clr
vN 0
elcihev 2ivm
ivk ivC
O(vNc~l
O vehicle 1 livm
~,X
~b ~
ib
"~I '~" _..~Nb
IS "-'
2lvm ~mv,2~. ~mvNv2
,Z W
L v • v
x,~
Figure 2: A typical plate-vehicle finite strip
NUMERICAL EXAMPLES
A Three-span Continuous Girder Bridge under a Moving Vehicle
The vehicle-bridge element is employed to analyse the dynamic response of a three-span continuous
girder bridge with equal span length l of 30m. The material properties of the bridge are, respectively,
the mass density p=2400kg/m ,3 Young's modulus E=30GPa, cross sectional area A=2.26m 2 and second
a=vn/coll,
moment of area of cross section I=0.667m .4 The velocity ratio a is defined as where v is the
speed of vehicle and loc is the fundamental frequency of the bridge, which is 21.068rad/s. The
damping effect of the bridge is taken into account by assuming Rayleigh damping of 2%. The moving
vehicle is modelled as a one-foot dynamic system for the study, comprising a sprung mass of 31800kg,
supported by a spring of stiffness of 9.12x106N/m and a dashpot of damping coefficient of 8.6x104
Ns/m.
The random road surface roughness and long-term deflection of concrete deck are considered in the
study. The former is described by a zero-mean stationary Gaussian random process while the latter is
represented as a kind of global roadway surface roughness (Cheng, 2000). The Monte Carlo method is
used in the simulation. 20 profiles of random road surface roughness are generated using the cut-off
spatial frequencies toc = 0.01 cycles/m and uoc = 3.0 cycles/m for general good road surface. Statistics
are computed for each set of results obtained for a certain vehicle velocity. They include the mean and
the standard derivation (SD). For simplicity, the maximum long-term deflection xamr of a span of
length ~l is taken as xamr = 0.0016 .sl Four schemes of deck profiles are studied. They include the
01
perfectly straight and smooth deck (Scheme ,)1 the deck with long-term deflection only (Scheme 2),
the deck with random road surface roughness only (Scheme 3) and the deck with both long-term
deflection and random road surface roughness (Scheme 4).
In the analysis, only two kinds of elements are required to simulate the bridge-vehicle system. They
are the beam elements and the vehicle-bridge elements. The problem was solved by FEM with 48
elements of equal lengths and 1500 equal time steps for the range of velocity ratio 0.06<a~_0.20. The
dynamic magnification factors dD for displacement and mD for bending moment at the mid-point of the
bridge are shown in Figures 3(a) and 3(b) against the velocity ratio a.
2.2 1.8
2.0
1.6 .
1.8
~ • 1.6 ~ • 1.4
.~ 1.4 t~
"~ 1.2
~ 1.2
"i 1.0
~ 0.8 ~0.8
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Velocity ratio to Velocity ratio to
)a( )b(
erugiF :3 naps-eerhT suounitnoc girder cimanyd :egdirb noitacifingam srotcaf )a( dD rof ;tnemecalpsid
dna )b( mD rof gnidneb tnemom ta elddim of napS .2 ~ , emehcS "1 [] , emehcS ;2 ~ •
emehcS 3 naeM( + ;)DS X , emehcS 4 naeM( + .)DS
It is observed that the effects of random road surface roughness on impact are significant while those
of long-term deflection of concrete deck are not as significant. In general, the dynamic magnification
factors for displacement are higher than those for bending moment. The effects of the long-term
deflection of concrete deck generally become more obvious in the higher range of velocity ratio.
However the effects may be adverse or beneficial depending on the velocity ratio.
A Railway Single-Span Simply Supported Bridge Moving a under niarT
A single-span simply supported prestressed concrete railway bridge with the two approaches supported
on embankments as shown in Figure 4 is considered to investigate the interaction among the train and
the supporting track and bridge. The railway track is continuous throughout. The top surface
irregularities of rail have been ignored in this example.
The span of the bridge is Lb=20m. For the analysis, the length of track structure taken into account on
each approach embankment is assumed to be twice the bridge span, i.e. .mO4=2eL=leL The flexural
rigidity per rail is 4.3x 106Nm 2 while the mass per unit length of one rail is 51.5kg/m. Both the discrete
model and the continuous model have been investigated in the modelling of the rail bed. In the
discrete model, the spring-damper systems are at a regular spacing of 0.625m. The stiffness and
damping coefficient of each spring-damper system underlying one rail are respectively 41125kN/m
and 20062.5Ns/m. In the continuous model, the stiffness and damping coefficient of the rail bed for
one rail are 65800kN/m 2 and 32100Ns/m 2 respectively. In the analysis, all the rail properties have to
be doubled to account for the presence of both rails.
The material properties of the bridge are, respectively, the mass per unit length ,m/gk88043=Ap
Young's modulus E=29430MPa and second moment of area of cross section I=3.81m .4 A train
consisting of five cars, as shown in Figure 5, runs over the bridge. Each car can be considered as
comprising two vehicles, as shown in Figure 5(b). Each wheel assembly is modelled as an equivalent