Proceedings of the Institution of Mechanical Engineers, Part F- Journal of Rail and Rapid Transit-2004-Koro-159-72

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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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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

K KORO, K ABE, M ISHIDA AND T SUZUKI160

Proc. Instn Mech. Engrs Vol. 218 Part F: J. Rail and Rapid Transit F00204 # IMechE 2004



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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





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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)





7/15

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





8/15

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)





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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)





10/15

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





11/15

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





12/15

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





13/15

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.

REFERENCES

1 Knothe, K. and Grassie, S. L. Modelling of railway track

and vehicle/track interaction at high frequencies. Veh.

System Dynamics, 1993, 22, 209262.

2 Knothe, K., Strzyzakowski, Z. and Willer, K. Rail vibra-

tions in the high frequency range. J. Sound Vibr., 1994,

169(1), 111123.3 Lunde n, R. and A kesson, B. Damped second-order

RayleighTimoshenko beam vibration in spacean exact

complex dynamic member stiffness matrix. Int. J. Numer.

Meth. Engng, 1983, 19, 431449.

4 Nielsen, J. C. O. and Igeland, A. Vertical dynamic

interaction between train and trackinfluence of wheel

and track imperfections.J. Sound Vibr., 1995,187(5), 825

839.

5 Sa llstro m, J. H. Fluid-conveying damped Rayleigh

Timoshenko beams in transverse vibration analyzed by

use of an exact finite element. Part II: applications. J.

Fluids Structs, 1990, 4, 573582.

6 Thomas, J. and Abbas, B. A. H. Finite element model for

dynamic analysis of Timoshenko beam. J. Sound Vibr.,1975, 41(3), 291299.

7 Dong, R. G., Sankar, S. and Dukkipati, R. V. A finite

element model of railway track and its application to the

wheel flat problem.Proc. Instn Mech. Engrs, Part F: J. Rail

and Rapid Transit, 1994, 208(F1), 6172.

8 Luo, Y., Yin, H. and Hua, C. The dynamic response of

railway ballast to the action of trains moving at different

speeds.Proc. Instn Mech. Engrs, Part F: J. Rail and Rapid

Transit, 1996, 210(F2), 95101.

9 Nickel, R. E.and Secor, G. A.Convergence of consistently

derived Timoshenko beam finite elements. Int. J. Numer.

Meth. Engng, 1972, 5, 243253.

10 Vermeulen, A. H. and Heppler, G. R. Predicting and

avoiding shear locking in beam vibration problems using

the B-spline field approximation method. Computer Meth.

Appl. Mechanics Engng, 1998, 158, 311327.

11 Kataoka, H., Abe, N., Wakatsuki, O. and Oikawa, Y. A

dynamic stress analysis of joint rails using finite beam

element model (in Japanese). In Proceedings of the 57th

JSCE Annual Conference, 2002, pp. 283284.

12 Abe, K. Application of time integration scheme based on

integral equation to elastodynamic FE/BE coupling

analysis (in Japanese). In Proceedings of the 14th Japan

National Symposium onBoundary Element Methods, 1997,

pp. 9398.

13 Andersson, C. and Dahlberg, T. Wheel/rail impacts at a

railway turnout crossing. Proc. Instn Mech. Engrs, Part F:J. Rail and Rapid Transit, 1998, 212, 123134.

14 Abe, K., Morioka, T. and Furuta, M. A numerical model

for track vibration analysis using Timoshenko beam (in

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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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Proceedings of the Institution of Mechanical Engineers, Part F- Journal of Rail and Rapid Transit-2004-Koro-159-72