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Engineers, Part F: Journal of Rail and RapidProceedings of the Institution of Mechanical
http://pif.sagepub.com/content/218/2/159Theonline version of this article can be foundat:
DOI: 10.1243/0954409041319687
2004 218: 159Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid TransitK Koro, K Abe, M Ishida and T Suzuki
railway tracktrack vibration analysis and its application to jointedTimoshenko beam finite element for vehicle
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7/22/2019 Proceedings of the Institution of Mechanical Engineers, Part F- Journal of Rail and Rapid Transit-2004-Koro-159-72
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Timoshenko beam finite element for vehicletrackvibration analysis and its application to jointed railwaytrack
K Koro1*, K Abe2, M Ishida3 and T Suzuki3
1Graduate School of Science and Technology, Niigata University, Japan2Department of Civil Engineering and Architecture, Niigata University, Japan3Railway Dynamics Division, Railway Technical Research Institute, Tokyo, Japan
Abstract: A Timoshenko beam finite element suitable for vehicletrack vibration analysis is proposed
and is applied to a jointed railway track. In several simulation models, the track vibration excited by a
train running on the rail is formulated as a dynamic problem where a sequence of concentrated loads
moves on the discretely supported Timoshenko beam. The external force is then defined by the
concentrated load. The Timoshenko beam subjected to concentrated loads deforms with the slope
discontinuity at the loading points. This deformation cannot be represented by the usual finite
elements, which causes the fictitious responses of the beam. The present finite element model removes
the undesirable response by completely modelling the slope discontinuity. This is achieved by the TIM7
element with the piecewise-linear hat functions. The jointed track model constructed by this finite
element is employed to predict the impulsive wheeltrack contact force excited by the wheel passage on
rail joints. The rail joints with fishplates are of great concern to track deterioration, the settlement of
ballast track and the failure of track components. In the present paper the effects of train speed and gap
size of the joints on the impact force are assessed from simulation results.
Keywords: Timoshenko beam, moving loads, discontinuity in slope deflection, rail joint, impact load
NOTATION
A cross-sectional area of rail
E Youngs modulus for rail steel
Fbi reaction force transferred from rigid
foundation to the ith sleeper
Fi railith sleeper reaction
G shear modulus for rail steel
I rail second moment of area
kbi stiffness of the ith sleeper support unit
kci Hertzian spring stiffness corresponding tothe ith wheel
ksi railpad stiffness at the ith sleeper
K Timoshenko shear coefficient
mbi ith wheel massmsi ith sleeper mass
Pbi time-invariant load transferred from the
upper component of train to the ith wheelPi railith wheel contact force
u rail deflection
ubi vertical displacement of theith wheel
usi vertical displacement of theith sleeper
uui rail deflection at theith wheel contact point
a,k modification parameters for the Hertzian
contact modelDt time increment
Zbi damping of the ith sleeper support unit
Zsi railpad damping at the ith sleeper
r rail mass densityc rail rotation angle
1 INTRODUCTION
The vertical vibration of a railway track is excited by
trains running on the track. The source of vibration is
the wheelrail contact force. Large contact forces are
induced under the existence of imperfections in vehicle
and track components and affect the cause and progres-sion of damage in the components. The quantitative
The MS was received on 21 January 2004 and was accepted afterrevision for publication on 8 April 2004.
* Corresponding author: Graduate School of Science and Technology,Niigata University, 8050 Igarashi 2-Nocho, Niigata 950-2181, Japan.
159
F00204 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part F: J. Rail and Rapid Transit
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estimation of the dynamic loads is thus essential to
prevent the track from serious deterioration.
The development of a mathematical model and the
simulation technique for vehicletrack vibration pro-
blems will be helpful to achieve improved component
design and maintenance schedules. These models areused to understand the interactions of the track and
vehicle components: the interactions of interest are in the
201500 Hz [1] frequency range. In such mathematical
vibration models, the rail is usually represented by a
uniform beam. The RayleighTimoshenko beam model
is, in particular, available for simulating the dynamic
response in the frequency range up to 2000 Hz [2].
The finite element discretization method is widely
used for RayleighTimoshenko beam models. Nowa-
days many types of finite element for this type of beam
model have been proposed. Lunde n and A kesson [3]
have derived an element from the homogeneous solution
of the modal equation of a corresponding beam. Theinterpolation functions of this element are dependent on
the frequency, and thus both the eigenvalues and the
eigenmodes of the discretized equation are calculated
through non-linear eigenvalue analysis. Nielsen and
Igeland [4] have avoided the non-linear eigenvalue
problem by replacing the finite elements by polyno-
mial-type elements. In reference [4], they have used the
elements defined by the homogeneous solution of the
static equilibrium equation of RayleighTimoshenko
beam [5]. In contrast with the elements that satisfy the
governing equation of the beam, the finite element
proposed by Thomas and Abbas [6] approximates thebeam deflection and rotation by a cubic Hermite inter-
polation. This element has been applied to vehicle
track interaction analysis by Dong et al. [7] and Luo et
al. [8]. Nickel and Secor [9] have developed the TIM7
element with a C1 class cubic interpolation for the
deflection and a C0 class quadratic approximation for
the rotation.
In the finite element models for Rayleigh
Timoshenko beam theory, an inappropriate choice of
the finite elements causes non-physical responses of the
rail. These are excited not only by the widely known
shear locking [10] but also by the incompatibility of
beam deformation in the elements. This incompat-ibility arises from the existence of discontinuity in the
slope deflection at the acting points of concentrated
loads. In many track models, the rail is modelled as a
beam discretely supported by sleepers. Moreover, the
contact force between a wheel and a rail, exciting the
vertical vibration, is calculated using a non-linear
Hertzian contact stiffness model in, for example,
references [4] a n d [7]. The external force acting on
the rail is usually defined as the concentrated load.
Consequently the slope deflection at the loading points
has discontinuity at these loading points. The above
elements [5, 6, 9] are locking free and can represent theslope jump by introducing double nodes. The con-
tinuity reduction by the double nodes can represent
the slope discontinuity at the fixed points such as
support points of rail, while the non-physical responses
concerning moving concentrated loads cannot be
removed even by the double nodes. For this settle-
ment, it is necessary to introduce a slope discontinuitywhich follows the moving loads.
The non-physical responses induced by moving
concentrated loads have been pointed out by Nielsen
and Igeland [4]. These fictitious responses are sufficiently
smaller than those excited by a wheel running on the rail
with irregularity. They have thus concluded that the
effect of the slope discontinuity is negligible. The surface
irregularities are, however, introduced only to reproduce
the impact response concerning the wheel and rail
imperfections. Of course, the rail irregularity is not
considered when investigating a fundamental dynamic
interaction of wheel and track. In this case the non-
physical response stated above may not be negligible.This paper presents a RayleighTimoshenko finite
element that can represent the slope discontinuity
associated with moving concentrated loads. The pro-
posed element consists of the TIM7 elements developed
by Nickel and Secor and the hat functions that are
introduced corresponding to each moving load. The
TIM7 elements contribute to represent the C1 class
deflection component, while the hat functions are used
for the slope discontinuity associated with the moving
loads. The discontinuity at fixed loading points is
defined by the double nodes with respect to the slope.
The introduction of the hat functions forces us to updatea part of the resulting stiffness and mass matrices at
every time step. The rail deflection, slope and rotation
are hence calculated by a time-stepping routine, without
modal decomposition used in reference [4].
In this paper, the present simulation method is applied
to predict the impact responses at a rail joint caused by
the passage of a train. In most simulation models the rail
discontinuity at a rail joint is neglected, and the impact
loads excited by rail-joint passage are usually simulated
by setting a surface irregularity on a continuous rail. A
model representing the rail discontinuity at a rail joint
has been proposed by Kataoka et al. [11]. In the present
paper, a rail-joint model similar to the Kataoka et al.
model is constructed using the proposed finite element.
This model is used to study the effects of train speed and
the gap size of rail joints on the impact load. The
quantitative and qualitative investigations of these
effects are made on the basis of simulation results.
2 MODELLING OF VEHICLE AND TRACK
COMPONENTS
The present simulation model consists of the followingcomponents: wheels, a Hertzian non-linear contact
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spring, a rail, railpads, sleepers and the support units for
sleepers. In modelling a rail joint, a joint bar and spring
units for connecting rails and the joint bar are added in
the track components. The present section describes the
numerical modelling of the vehicle andtrack components.
2.1 Finite element formulation for a rail
A rail is modelled as a single, uniform and straight
RayleighTimoshenko beam. The single beam is dis-
cretely supported by sleepers. The reaction force acting
at the sleeper support is consequently modelled as a
concentrated load. The wheelrail contact force, repre-
sented by the non-linear Hertzian contact model, is
defined as a moving concentrated load, as will be shown
in section 2.2.
Under the present loading condition, the variational
form of vertical motion of the rail is described asL0
EIc0 dc0 dx
L0
GAKcu0dc du0 dx
L0
rAuu duIccdc dx
Xni1
duxi ctPit XNj1
duaj Fjt 1
where x is the longitudinal coordinate and t is the time.
The prime and double dot denote spatial differentia-
tion and temporal differentiation respectively. aj(j 1,2, . . . , N, where N is the number of sleepers) isthe x coordinate at the support point by the jth sleeper.
A sequence of wheels, representing trains, runs on the
rail of lengthLwith a certain constant velocity c. Theith
wheel starts from the points x xi (i 1,2, . . . , n,where n is the number of wheels). I and A are the
moment of inertia and the area of rail cross-section
respectively. E and G are Youngs modulus and the
shear modulus respectively. r is the rail density and Kis
the Timoshenko shear factor. In equation (1), u and c
are the downward deflection and the rotation respec-
tively, and du and dc are their variational components.
Pi denotes the contact force associated with the ithwheel, whileFjrepresents the reaction force acting at the
jth fixed support point. In the present model, the
reaction force Fjwill be defined in equation (16).
The deflectionu and the rotationc on an element are
approximated using the interpolation functionNxandfx of the TIM7 element [9] and the additive hatfunction wx:
ux, t&Nxut wxDu, cx, t&fxwt 2
where u and w are the nodal deflection (including slope
deflection) and rotation vectors. Their components are
defined by u u1, y1, u3, y3 and w c1,c2,c3, asillustrated in Fig. 1a. N and f are the interpolation
functions corresponding to u and w in the TIM7
element. That is,N forms a cubic Hermite interpolation,
and f is the basis function of quadratic Lagrange
interpolation. The detail of these functions will be
shown in the Appendix. The finite element discretization
using the TIM7 elements is carried out in every sleeper
span. The slope discontinuity at the sleepers is modelled
by introducing double nodes associated with the slopedeflection qu=qx.The additive hat functions w fwij i 1,2, . . . , ng
represent the slope discontinuity at the wheelrail
contact points. The function wi i s formed by a
combination of two piecewise linear polynomials and
is arranged on the sleeper span on which the ith wheel
rests (see Fig. 1b). The shape and support ofwiare to be
updated at every time step due to the wheel running. The
deflection components corresponding to the function w
is designated by the vector Du.
Now the variational components du and dc are
defined as du N du w dDu, dc f dw; also, theseexpressions and equation (2) are substituted intoequation (1). The following ordinary differential equa-
tion is consequently obtained by calculating the
integrations on every element and assembling the
stiffness and mass matrices:
M DMTt
DMt lt
" #( UU
Duu
)
K DKTt
DKt jt
" #(U
Du
)
Tt
XPt
fPtg
B
XFt
fFtg 3
where M and K are the mass and stiffness matrices
respectively associated with the interpolation functionsN and f. DM and DK are generated by calculating the
integrals including the functionsN, fandw. l and j are
the submatrices concerning the additive function w.
Fig. 1 Definition of the nodal values of deflection u, slope y qu=qx and rotation w (FE, finite element)
TIMOSHENKO BEAM FE FOR VEHICLETRACK VIBRATION ANALYSIS 161
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The nodal displacement vector U consists of the
vectors u and w. Pt fPit j i 1, . . . , ng andFt fFit j i 1, . . . , Ng are the contact forceand the rail reaction respectively. The matrices T, XP,
B and XFare defined as
Tt fNTxi ct j i 1, . . . , ng
XPt fwTxi ct j i 1, . . . , ng
B fNTai j i 1, . . . , Ng
XFt fwTai j i 1, . . . , Ng
4
The time integration of equation (3) is achieved by
Abes unconditionally stable scheme [12]. This scheme is
based on the weighted residual representation in the
time domain and gives the trapezoidal rule for free
vibration. By implementation of the time integration thealgebraic equation on the rail displacement at the
sequential two time steps M and M 1 is derivedfrom differential equation (3) as
MDt2
4 K DMTM
Dt2
4 DKTM
DMMDt2
4 DKM lM
Dt2
4 jM
2664
3775(UM
DuM
)
Dt2
2
TtM
XPtM
fPMg
Dt2
2
B
XFtM
fFMg
M
Dt2
4 K D
M
T
M
Dt2
4 D
K
T
M
DMMDt2
4 DKM lM
Dt2
4 jM
2664 3775
6UM1
DuM1
( ) Dt
M DMTM
DMM lM
" #
6
_UUM1
D _uuM1
( ) 5
where Dt is the time increment, tMMDt andtM1 M1Dt. The subscript M labelling thevectors in equation (5) implies that they are the nodal
values at t tM. The submatrices with the subscriptMare generated on the basis of the position of the wheels at
the Mth time step.
The velocity vectors _UUM1 and D _uuM1 have to be
given at every step to satisfy the equation
M DMTM
DMM lM
" # _UUM
D _uuM
( )
M DMTM
DMM lM
" # _UUM1
D _uuM1
( )
Dt
2
K DKTM
DKM jM
" # UM UM1
DuM DuM1
( )
DtTtM
XPtM
fPMg DtB
XFtM
fFMg 6
Substituting equation (5) into equation (6), the velocity
components at the Mth temporal step are consequently
calculated as
_UUM
D _uuM( )
_UUM1
D _uuM1( )
2
Dt
UM UM1
DuM DuM1
( ) 7
2.2 Modelling of train and wheelrail contacts
A train with several wheels is represented by an
assembly of masses; the bogie models, used in reference
[13], are not adopted. The interactions of each mass arethus neglected. This is because the wheel motion is
isolated to that of the upper parts of a train in the
frequency range greater than 10 Hz, which is of interest.
The wheels are modelled as a sequence of unsprung
masses, subject to
mbiuubiPbi mbigPi i 1,2, . . . , n 8
wherePbiis the time-invariant load transferred from the
upper components of the train to the ith wheel. mbig is
the weight of the wheel mass mbi, and g is the
acceleration due to gravity. As a simple vehicle model,
an unsprung mass, which is very often defined as awheelset mass basically depending on a bogie structure,
is adopted. However, a wheel mass is used here as a very
brief vehicle model.
The first step for the time integration of equation (8) is
to consider the convolutiont0
fmbiuubit Pbit mbig Pitg
6ubitt dt 0 9
where ubit tHt=mbi and Ht is the Heavisidefunction. In equation (9) the gravity and the external
force are assumed to be constant between sequential twotime steps. After the application of integrations by parts
to equation (9), the vertical displacement uMbi of theith
wheel at the Mth time step is given by
K KORO, K ABE, M ISHIDA AND T SUZUKI162
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uMbi
XMk1
Dt2
mbiM k
1
2
mbigPbi Pi
k 10
where the superscript k denotes the kth time step.The wheelrail contact force P is modelled on the
basis of the non-linear Hertzian contact theory forelastic bodies. The contact force Pi between the ith
wheel and the rail is given by
Pi kcid3=2ci , dciubi uci 11
whereuciis the rail deflection at the contact point of the
ith wheel. kci is the Hertzian spring stiffness. Since
equation (11) is non-linear, the contact force Pi is
calculated by the iterative routine using the following
linearized equation of equation (11):
Pki
kkcidcik kcid3=2cik1
kkcidcik1
kkci :32 kci
ffiffiffiffiffiffiffiffiffiffiffiffiffiffidcik1
p dcik1 > 0
0 dcik1 4 0
(12
In equations (12), the subscript k indicates that thecurrent step for solving the non-linear equation is k.
Applying Abes time-integration scheme [12] with the
weights Ht on the time interval M1 Dt4t4MDt to equation (12), consequently
PM, ki
1
2kkcidciM, k
12
kkciM, k1 1Dt
Dt
0
kci~ddciM, k13=2
dt
~ddciM, k1 : dciM, k1dciM1
Dt tdciM1
13
where ? is the truncated power function of orderzero. This function indicates the non-zero values when
the argument of the function is positive.
2.3 Modelling of sleepers, railpads and support units for
sleepers
Sleepers are modelled as masses, while the railpads and
the support units for the sleepers are represented by
Voigt units with linear springs and dashpots, as shown
in Fig. 2. The equation of motion of the sleepers can be
expressed in a similar way to those of the unsprung
masses, described in the previous section. Since the force
acting on the ith sleeper is the railsleeper reaction Fiand the supporting force Fbi, the vertical displacement
usiof the ith sleeper is calculated as
uMsi
XMk1
Dt2
msiM k
1
2
Fi Fbi
k
i 1,2, . . . , N 14
where msi is the mass of the ith sleeper. Note that the
time integration in conjunction with the equation of
sleeper is dealt with using the same scheme as applied to
the unsprung mass.
The reaction force Fibetween the ith sleeper and the
rail is defined by a Voigt unit with the stiffness ksi and
the damping coefficient Zsi, and hence Fi is given as
Fiksiuui usi Zsi _uuuui _uusi 15
where uui is the rail deflection at the point connected to
the ith sleeper. The time integration of equation (15)
starts with the convolution of the equation and the
weight Ht in the interval M 1Dt4t4MDt. By
Fig. 2 Mathematical model for simulating dynamic track responses (FE, finite element)
TIMOSHENKO BEAM FE FOR VEHICLETRACK VIBRATION ANALYSIS 163
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assuming thatFiis stepwise constant and the behaviours
of uui and usi in a time interval are stepwise linear, the
following equation is derived:
FMi
ksi
2
ZsiDt uu
Mi u
Msi
ksi
2
ZsiDt
uu
M1i u
M1si
16
The reaction forceFbitransferred from rigid foundation
to the ith sleeper can be calculated by a similar scheme
to that for Fi as
FMbi
kbi
2
ZbiDt
u
Msi
kbi
2
ZbiDt
u
M1si 17
where kbi and Zbi are the stiffness and damping
coefficient respectively of the supporting Voigt unit for
the ith sleeper.
3 SUPPRESSION OF THE NON-PHYSICAL
RESPONSE IN THE RAIL
The dynamic responses of the vehicletrack system
shown in Fig. 3 were simulated using the present model.
The beam (rail) ends in Fig. 3 are free, and the physical
properties of the vehicle and track components are
specified in Table 1. A numerical test was carried out to
verify the cause of the non-physical response of the rail
modelled as a RayleighTimoshenko beam and thesuppression of this undesirable response by the present
finite elements. In this numerical experiment, the train
running on the rail is represented by a single-wheel
model. The contact stiffness between wheels and the rail
is represented by a linear spring for simplicity.
The reduction in the fictitious response using the
present finite element is verified on the basis of the
wheelrail contact force Pt. The numerical analysisusing the time-domain integral equation method [14]
was also undertaken to obtain the rail deflection without
non-physical fluctuation. In this method, the rail is
modelled as an infinite beam subject to a periodic
dynamic state.
Figure 4 shows the contact force calculated by thepresent simulation model. In the numerical test a
support span was divided into one or three TIM7
element(s). The combination of the TIM7 elements and
a hat function, proposed in this paper, can completely
represent the slope discontinuity caused by the concen-
trated loads acting on the Timoshenko beam. The non-
physical responses concerning the slope discontinuity
are thus removed, and the contact force calculated by
the present model shows good agreement with the
results of the time-domain integral equation method.
However, these excellent results are not obtained if
the incompatibility on rail deflection remains in theapproximation functions of the finite elements. This fact
can be found from the numerical results shown in Fig. 5;
the contact force depicted in Fig. 5 was obtained by the
model in which the slope discontinuity associated with
the moving load is neglected. In this case the discon-
tinuity of the slope deflection at the support can be
represented, and hence the response of the contact force
at passing above sleepers is simulated accurately. On the
other hand, the contact force fluctuates considerably
when the wheel exists in a location except for the
Fig. 3 Track and vehicle structures. This model is used in the numerical test to verify the reduction in thefictitious response of a rail by the present finite element model
Table 1 Physical properties of the vehicle and trackcomponents
Time increment Dt 1/8000 sNumber of time steps 5000
Wheelrail contact stiffnesskc 2000 MN/m
Rail density r 7880kg/m3
Area of a rail section A 64.056 10 4 m2
Youngs modulus of a rail E 206GPaMoment of inertia on a rail I 19606 10 8 m4
Shear modulus of a rail G 77.3GPa
Shear factor of a rail K 0.34Railpad stiffness ks 110 MN/mDamping coefficients of a railpad Zs 100kN s/m
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support points. The numerical results including the non-
physical fluctuation tend to approach to the contact
force calculated by the time-domain integral equation,
in response to the progress of the number of elements.
Indeed, it needs an extremely fine resolution to remove
the fictitious responses. The analysis with plain TIM7
elements will thus be undesirable.
Figure 6 depicts the contact force calculated by thecubic Hermite finite element model. This element is
widely used in traintrack vibration analysis [7, 8].
Figure 6 indicates that the non-physical response is
also included in the numerical results obtained by the
conventional finite elements. The slope discontinuity at
the acting points of concentrated loads is quite
neglected. The behaviour of the contact force obtained
by the concerned model is clearly different from that by
the integral equation model. In particular the fluctuationcaused by the passage of a wheel on the rail supports
Fig. 4 Wheelrail contact force calculated by the present finite element (FE) method where the finite elementsare defined as a combination of the TIM7 elements and a hat function. The dotted lines indicate thepassing time of the wheel on the support points of the rail (Integ. Eqn., integral equation)
Fig. 5 Wheelrail contact force calculated by the simulation model using the TIM7 elements (FE, finiteelement). The slope discontinuity at the fixed support points is considered, while a hat function is not
used, and hence the discontinuity associated with the contact force is neglected. The dotted linesindicate the passing time of the wheel on the support points of the rail (Integ. Eqn., integral equation)
TIMOSHENKO BEAM FE FOR VEHICLETRACK VIBRATION ANALYSIS 165
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cannot be simulated. This fact spoils the main advantage
of RayleighTimoshenko beam models by which the
pinnedpinned resonance modes can be calculated. The
fictitious fluctuation of rail deflection should be com-
pletely removed by modelling the slope discontinuity,
like the above TIM7 model. The use of the cubic
Hermite element, however, has a disadvantage forcomputational work on the TIM7 model. Although
these two elements are non-locking, the thin-beam limit
solution of the cubic Hermite element reduces to the
TIM7 solution, and the degrees of freedom will then
degenerate by one.
In the numerical tests using beam elements, both the
mass and the stiffness matrices have a band structure.
The band width o depends on the number of unknowns
defined on an beam element. Little difference between
the band widths for the TIM7 or the cubic Hermite
elements exists. The widths are less than 10, which are
sufficiently smaller than the total number N of
unknowns in the finite element equations. The totalcomputational work in every numerical test hence has
little difference.
The numerical solutions were calculated using the LU
factorization in the present numerical tests; the compu-
tational work for the factorization isOo2N. The linearcomplexity of the calculation is not an obstacle to
application of the present finite elements to the vehicle
track vibration analysis. If the hat functions have to
be introduced to represent the slope discontinuities
associated with the moving concentrated loads, the
additive computational work is relatively small com-
pared with the total work for the time-steppingcalculation. This is because the updated coefficients in
the mass and stiffness matrices at every time step are
only the entries associated with the hat functions and the
factorization algorithm enables only the factorization of
the additive coefficients to be selectively carried out.
The non-physical response will also be removed by
using the mesh refinement of the conventional elements.
The mesh width, determined from the running speed andtime increment, has then to be set to an extremely small
value. Such a refined mesh is unnecessary in considera-
tion of the availability of the RayleighTimoshenko
beam for rail vibration analysis. A huge scale problem,
of course, is needed for good accuracy which is
comparable with the proposed method. Therefore, in
the context of computational cost, the advantage of the
present method should be evident.
4 ANALYSIS OF THE IMPACT LOADS CAUSED
BY RAIL-JOINT PASSAGE
4.1 Mathematical modelling of jointed railway track
The present model, representing the slope discontinuity
of a Timoshenko beam in conjunction with concentrated
loads, is applied to predict the impact responses exciting
when a wheel passes on a rail joint. In the traditional
jointed track, adjoining rails are connected using two
joint bars, widely called fishplates by railway engineers,
at the rail ends. Using the joint bars the motion of the
rail end is confined horizontally and vertically, and
hence a smooth running surface can be sustained.
Moreover, the joint bars play an important role incompensating the missing vertical bending stiffness due
Fig. 6 Wheelrail contact force calculated by the simulation model using the cubic Hermite elements (FE,finite element). The slope discontinuity at the points of application of concentrated loads is totallyneglected. The dotted lines indicate the passing time of the wheel on the support points of the rail(Integ. Eqn., integral equation)
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to the rail discontinuity at the joint. The bending stiffness
of two joint bars is, however, much smaller than that of
the rail. For example the widely used rails in Japan have a
moment of inertia about three times larger than those of
the corresponding two joint bars. The resulting rail joint
is thus a weak spot in railway tracks. The large verticaldeflection and impulsive dynamic force are caused by the
passage of wheels on the rail joint.
The impact responses at the rail joint can be simulated
using either the continuous-beam model [13] or the
discontinuous-beam model [11]. In the continuous-beam
model, a fictitious beam element with sufficiently small
bending stiffness is inserted in the gap of adjoining rails
to avoid the numerical problems which are the origin of
the discontinuities in vertical displacement, velocity and
acceleration of rails. A trough between the ends of two
rails, which causes the wheelrail contact force to
fluctuate, is represented by the fictitious surface irregu-
larity. To simulate accurately the above impact responseswithout such an element and irregularities, the simula-
tion model with the joint structure in which two rails are
connected with joint bars [11] has to be employed.
In the present paper, the adjoining rails and the joint
bars are modelled as a single Timoshenko beam as shown
in Fig. 7. Although two joint bars are in general attached
to the rail, these bars are regarded as an equivalent single
Timoshenko beam. This beam is connected to several
springs, which represent the bolts for fastening rails and
joint bars. Wheelrail contact is modelled on the basis of
the Hertzian non-linear contact theory for two elastic
bodies. The contact force is thus calculated by equation(11). In Hertzian theory, the contact force acting on the
interface between two elastic bodies is calculated on the
assumption that the deformation of the contacting
bodies in the vicinity of contact area can be approxi-
mated by that of semi-infinite elastic media. This
important assumption is consistent with the actual
deformation of a wheel and a rail when the wheel runs
at the far position from rail joints. An exceptional case to
this assumption occurs in a situation in which the wheelmakes contact with the rail either in the vicinity of the
joint or at the rail edges. To cope with this situation, a
modified constitutive relation of Hertzian contact model
is introduced as follows [11]:
P kkcdac 18
where P is the wheelrail contact force and k is the
reduction factor of the contact stiffness kc. In reference
[11], the parameters k and a have been determined
through three-dimensional finite element analysis on a
wheelrail contact and using Kalkers algorithm [15].
Note that the ordinary Hertz model has k 1 anda 3=2. Moreover, dc is the relative displacementbetween the barycentre of the wheel and the contact
point on the rail. In the present model, the tread of the
wheel is given as the lateral face of a cylinder. The surface
profile of the rail in the vicinity of the contact point is
approximated by a plane specified by the rail deflection
and the slope at either the point under the barycentre of
the wheel or the rail edge. The surface irregularities, such
as positive or negative step and corrugation, can be easily
taken into account by defining the rail profile on the basis
of both the rail deformations and the irregularities. The
positions of the wheel and the rail in these situations areillustrated in Fig. 8. Note that Figs 8a and b both indicate
the wheelrail position where the wheel makes contact
with the rail surface. Indeed, the contact point on the rail
Fig. 7 Mathematical model of a jointed railway track. The upper and lower rails are modelled as singleTimoshenko beams
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may coincide with a rail edge, when the wheel passes a
rail joint. In the present model, if the geometrical
condition
a ? v > 0 19
is satisfied, then the contact point on a rail is chosen to
the rail edge. In this state the relative displacement dc is
given by
dc R kvk 20
where R is the wheel radius and a is the unit tangential
vector of the rail surface at the rail edge; the detailed
definition is shown in Fig. 8b. v is the vector from the railedge to the barycentre of the wheel.
In the present model, the wheelrail contact is
specified separately for both rails at every time step. It
is assumed that a wheel and either the upper or the lower
rail make contact at a single point respectively. The
contact of a wheel with a rail is determined on the basis
of the geometry of these two bodies in motion. The
single-point contact is thus smoothly shifted to the state
of two-point contact.
4.2 Numerical results
The present simulation model is employed to calculate
the impulsive contact force excited by a wheel passing
on a rail joint. The simulations were undertaken to
investigate the effects of train speed and gap size on the
impact force. In numerical tests the material and
structural parameters of the wheel and the track were
chosen to the values listed in Table 2. The parameters k
anda used in the modified Hertzian contact model were
given by the results shown in reference [11]. The values
of the parameters are depicted in Fig. 9. The origin of
the transverse axis of this figure is set as the rail edge,and the negative abscissae indicate that the wheel exists
Fig. 8 Descriptions of the geometry of the wheel and the upper rail in the vicinity of the rail joint. Thesurface profile of the rail is specified by the deflection, the slope and the irregularity at either (a) thepoint C0x xw or (b) the point E x xend
Table 2 Material and structural parameters of the wheel andthe track in the numerical tests (JIS, JapaneseIndustrial Standard)
Rails and the corresponding joint bars JIS 50 k g NNumber of tie spring between the rail and the
joint bars4
Number of sleepers 21Length of sleeper span 0.58 mMass of a sleeper 80 kgRailpad stiffness 60 MN/mRailpad damping 98 kN s/mStiffness of a sleeper support unit 60 MN/mDamping of a sleeper support unit 42 kN s/m
Time-independent load 56 050 NUnsprung mass 697.5 kgWheel radius 0.43 m
Elastic modulus of the wheel 206 GPaPoissons ratio of the wheel 0.3
Fig. 9 The parametersk anda used in the present simulation.The origin of the transverse axis is set to the rail edge.
The negative abscissaes indicate that a wheel existsover the trough of a rail joint
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over the trough of the rail joint. The original data in
reference [11] are provided in the range of wheel
position from 5 mm to 10 mm. Thus, no contactforce is caused when a wheel exists in the range
?, 10 mm. In order to avoid the sudden unload-
ing associated with the definition of the constitutiverelation on the wheelrail contact, the data on the
parameters k and a were added in the range from 10to 45 mm. In the interval 10 mm, 15mm theparameters were given by the extrapolation using the
known information. The data in the range 15 mm, 45mm were assumed to be piecewise constant withthe level at 15 mm. The addition of these data wasintroduced for simplicity; the better setting of insuffi-
cient data requires a contact analysis to be carried out.
The effects of this treatment on dynamic responses will
have to be investigated by comparison with in situ
measurements.
Figure 10 shows the wheelrail contact force when thewheel transfers from the upper rail to the lower rail in
the vicinity of the rail joint. The present results were
obtained for train speeds of 50 and 150 km/h and rail-
joint gaps of 3, 6 and 14 mm. The impact force is excited
by the contact between the wheel and the lower rail.
The peak of the impact force is observed when the
barycentre of the wheel approaches over the lower rail.
As shown in Figs 10a(ii) and b(ii), the maximum contact
force tends to increase in progression of train speed.
This tendency is clearly found for a smaller size of rail-
joint gap. For a 3 mm gap, the maximum force for
150 km/h running speed is about 1.5 times that for50 km/h. When the wheel passes a larger gap of 14 mm,
the peak of the impact loads is roughly independent of
train speed. The maximum force is 1415 kN, which is
200250 per cent of static loads.
The peak level of the impact force depends not only
on the train speed but also on the size of rail-joint gap.
The train passage on larger rail-joint gap excites a higher
peak impact force, which is found from the results for
both lower and higher speeds. The largest difference
between the maximum forces for 3 and 14 mm gaps is
obtained for a train speed of 50 km/h; the maximum
force of 65kN for a 3mm gap rises to 150kN for a
14 mm gap owing to the increase in the gap size. Thedifference decreases from about 85 kN for 50 km/h to
40 kN for the higher speed of 150 km/h.
Conclusions from the above discussion are as follows:
1. For a lower train running speed the peak level of the
impact force mainly depends on the gap size of the
rail joint.
2. The effect of the train speed on the maximum
impact force is relatively small in comparison with
that of the gap size, at least for a train speed below
150 km/h.
On the other hand, the contact force acting on theupper rail does not include the impact response. This is
because the wheel leaves the upper rail after passing the
rail joint. When the gap size of rail joint is fixed, the
wheelupper rail contact force is unloaded without
fluctuation. This dynamic behaviour is independent of
the train speed, except for a 14 mm gap and a 50 km/h
train running speed. In an exceptional case, theprogression of the unloading is relatively slow in
comparison with the other situations. The slow unload-
ing hardly influences the peak level of the impact force
acting on the lower rail, as a result.
5 CONCLUSIONS
In the present paper a Timoshenko beam finite element
model has been developed for vehicletrack vibration
analysis. The rail was modelled as a discretely supported
beam. The wheelrail contact force was calculated onthe basis of the Hertzian non-linear contact theory. All
the external force acting on the rail was consequently
given as a concentrated load. When a concentrated load
acts on a Timoshenko beam, the slope at the loading
point becomes discontinuous. The widely used finite
elements can represent the slope discontinuity at the
fixed loading points by locating double nodes at these
points. On the other hand, the discontinuous slope in
conjunction with moving concentrated loads cannot be
represented without remeshing. This is why a fictitious
response on the rail deflection is caused. This undesir-
able response has been removed by using a combinationof Nickels TIM7 finite element and an additive hat
function as the finite elements. The effect of this element
on the removal of the non-physical responses has been
verified by numerical tests. As a result, the use of the
present element, where the deflection is approximated
using not only the nodal deflection but also the nodal
value of the slope, was effective for removing the
fictitious rail response.
The present finite element for RayleighTimoshenko
beams has been adopted for simulation of the impact
loads excited by the passage of a wheel on a rail joint.
The rail joint has the structure where two adjoining rails
are fastened to two joint bars by several bolts. Thepresent model has presented the joint structure by
connecting two adjoining beams and an effective beam
modelling the joint bars with linear springs. The wheel
rail contact force is calculated by the Hertzian contact
model. This model is based on the semi-infinite
approximation on the elastic bodies, which is no longer
consistent when a wheel makes contact with the vicinity
of rail edges. The constitutive relation of the wheel/rail
contact around a rail joint has been defined by the
modification of the Hertzian model, as has been
presented in reference [11].
The simulation with this vehicletrack model has beenundertaken to predict the impulsive contact force
TIMOSHENKO BEAM FE FOR VEHICLETRACK VIBRATION ANALYSIS 169
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excited by wheel passage on a rail joint. In particular,
the influence of the train speed and the gap size of the
joint on the impact force have been investigated. The
simulation results show that for lower running speed the
peak level of the impact load acting on the lower rail
mainly depends on the gap size. The effect of the trainspeed on the maximum impact force is relatively small in
comparison with that of the gap size, for train speeds
below 150km/h.
Through the numerical results shown in the present
paper, the impact loads excited when the wheel passes a
rail joint are relatively large in comparison with the
dynamic loads observed for the wheel running on acontinuous rail head. The effects of the modelling of the
Fig. 10 Effects of the gap size of the adjoining rails on the increased wheelrail contact force Dt1=16000s
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slope discontinuity on the impact force may thus be
insignificant. If rail surface irregularities are considered
in the simulation of the impact loads, the peak level of
the impact loads may be mainly influenced by the
longitudinal profile of the irregularities. The present
model is effective for understanding the fundamentalbehaviour of a wheelrail dynamic system without rail
surface irregularities.
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4 Nielsen, J. C. O. and Igeland, A. Vertical dynamic
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APPENDIX
Interpolation functions of TIM7 element and additive hat
function
In the TIM7 element, the beam deflection u and the
rotation c are approximated by
u& N1x N2x N3x N4x
u1
y1u3
y3
8>>>>>:9>>>=>>>;
c& f1x f2x f3x
c1
c2
c3
8>:
9>=>;
21
where the nodal values ui, yi i 1, 3 and ci i1,2,3 are defined as shown in Fig. 1a.
The interpolation functions Nix i 1, 2, 3, 4 and
fix i 1, 2, 3 are defined as
N1x : 1
L32xLxL2
N2x : x
L2Lx2
N3x : x2
L33L2x
N4x : x2
L2xL
22
and
f1x : 1
L22xLxL
f2x : 4
L2xxL
f3x : x
L22xL
23
where 04x4L (L is the length of the element). The
functions Nix i 1, 2, 3, 4 form the cubic Hermite
interpolation, and fix i1, 2, 3 are the Lagrangepolynomials of the second order.
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The additive hat function wx, which has the shapeshown in Fig. 1b, is given by
wx :
x
xc04 x4xc
L0xL0 xcxc < x4L0
8>>>: 24
In equations (24), the coordinate x04x4L0 isdefined in the sleeper span where a corresponding
moving concentrated load exists. xc is the position of
the moving load and L0 is the length of the sleeper
span.
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