EVALUATION OF DYNAMIC LOAD ON RAILTRACK …kknair/pdf/Kumaran.pdf · Dynamic Analysis of Railtrack Sleepers in Track Structure The dynamic wheel loads generated by a moving train

September 10, 2002 8:38 WSPC/165-IJSSD 00059

International Journal of Structural Stability and DynamicsVol. 2, No. 3 (2002) 355–374c© World Scientific Publishing Company

EVALUATION OF DYNAMIC LOAD ON RAILTRACKSLEEPERS BASED ON VEHICLE-TRACK

MODELING AND ANALYSIS

G. KUMARAN∗, DEVDAS MENON† and K. KRISHNAN NAIR‡

Department of Civil Engineering, Indian Institute of Technology Madras,Chennai 600 036, India

∗[email protected]†[email protected]‡[email protected]

Received 22 April 2002Accepted 1 June 2002

The present study reports the results of a rigorous dynamic interaction analysis thataccounts for the vehicle-track characteristics and rail imperfections. In order to performa rigorous dynamic analysis, a model involving all components of the track structureand vehicle parameters are required. A vehicle model (conforming to Indian Railways)with 17 degrees of freedom has been considered. The track model consists of rail, rail-pad, sleeper, ballast, sub-ballast, subsoil and a track length encompassing 12 prestressedconcrete sleepers. The dynamic interactive analysis is carried out between the vehicle andtrack in the time domain using a finite element software. The results of the interactiveanalysis give responses in the form of reaction time histories at the rail-seat locationsduring the passage of vehicle. A parametric study is carried out to assess the influence ofdifferent track parameters on the dynamic load on the railtrack sleepers. Based on thisstudy, suitable load amplification factors are arrived at to facilitate an improved designbasis for an equivalent static analysis in practical designs of sleepers.

Keywords: Railtrack sleepers; dynamic analysis; vehicle model; track model; finiteelement analysis; track imperfections.

1. Introduction

The railtrack structure system consists of the wheels, rails, rail-pads, fastenings,

sleepers, ballast and subgrade, cf. Fig. 1. In this track structure system, the load

transmission occurs, from the axle to the wheels, from wheels to rails, from rails

to sleepers, and finally from sleepers to the supporting ballast and subgrade. For

the purpose of design, railtrack seat loads are modeled as equivalent static loads

(including an “impact factor” for dynamic amplification derived empirically) equi-

librating with the ballast pressures provided by the underlying ballast-subgrade

∗Ph.D Research Scholar.†Associate Professor.‡M.S Research Scholar.

355


356 G. Kumaran, D. Menon & K. K. Nair

Wheel Axle

Ballast

Rail

Fastenings Slope 1 in 20

PSC Sleeper

Fig. 1. Transmission of wheel load in track support system.

medium.1–4 However, such simplified load modeling does not comprehensively

account for dynamic load effects due to the vehicle-track interaction.

Furthermore, the dynamic loads on sleepers are mainly due to the vehicle and

track characteristics.5–9 Although considerable research has been done in the area

of vehicle-rail interaction in the past, attention has been focussed primarily on the

vehicle suspension system and not on the sleeper (which is merely treated as a part

of the substructure system). The analysis of the dynamic loading at the rail-seat

location of the sleeper is a problem that has not been effectively addressed in the

literature.

Hence, there is a need to accurately assess the loading characteristics near the

rail-seat zone of the sleeper in track structure based on detailed vehicle track in-

teraction studies. This will facilitate a more rational and optimal design, having

significant economic implications in view of the large number of sleepers being

manufactured world-wide.

In order to examine the dynamic loading on the railtrack sleepers in a railtrack

structure system, a model involving all components of the track structure and

vehicle parameters is required. In this paper, the dynamic loading on the railtrack

sleepers is investigated, considering the vehicle (axle type), condition of the suspen-

sion system, vehicle speed, wheel and rail defects (irregularities), contact between

sleeper and ballast, and elastic properties of the ballast-subgrade. This detailed

dynamic study requires appropriate modeling of vehicle and track system.10–12

A vehicle model is used to determine the rail-seat load history that is transmitted

to the sleepers during the passage of vehicle with different imperfections such as rail

corrugation, wheel-flats and rail joints. In this model, a track length encompassing

12 sleepers is considered. The interaction between the vehicle and track is carried

out in the time domain using finite element software. The dynamic load on the

sleeper is obtained in the form of reaction time histories at the rail-seat locations. A

parametric study is carried out to assess the influence of different track parameters

on the dynamic load acting on the railtrack sleepers.


Dynamic Load of Railtrack Sleepers 357

The sleeper can be easily designed once the load on the sleeper is established

from the rigorous dynamic analysis, and this will provide a rational basis for

improved design recommendations.

2. Model of Railway Vehicle

Figure 2 illustrates the vehicle model adopted in this study. It consists of a vehicle

body, two bogie frames and four wheel sets. The vehicle is modeled as a rigid body

having a mass mc, and having second moments of area Jcx and Jcy about the

transverse and longitudinal horizontal centroidal axes. Similarly, each bogie frame

is considered as a rigid body with a mass mb (mb1 and mb2) with second moments

of area Jbx and Jby about the transverse and longitudinal horizontal centroidal

axes. Each axle along with the wheel set has a mass mw (for four axles mw1, mw2,

mw3, and mw4). The spring and the shock absorber in the primary suspension

for each axle are characterized by spring stiffness kp and damping coefficient cprespectively. Likewise, the secondary suspension is characterized by spring stiffness

ks and damping coefficient cs respectively.

The various displacements of the vehicle are described with respect to the equi-

librium positions. As the vehicle car body is assumed to be rigid, its motion may

be described by the vertical displacement (bounce or zc) about the horizontal cen-

troidal axis, rotation (pitch or θc) about the about the transverse horizontal cen-

troidal axis and rotation (roll or φc) about the longitudinal horizontal centroidal

axis. Similarly, the movements of the two bogie units are described by three degrees

of freedom zb1, θb1 and φb1 each about their centriods. Each axle set is described

by two degrees of freedom zw1, and φw1 about their centriods. Totally, 17 degrees

of freedom have been considered in this study for the vehicle model,10,12 cf. Fig. 2.

The displacement vector {yt} for the train vehicle can therefore be written in terms

of displacements at the centriod of the vehicle body, bogie and axle sets as

{yt} = {{z}{θ}{φ}} (1)

θθθθc

mc

CAR BODY

BOGIE

WHEEL

φφφφc

ks

kpcp

csθθθθb1 θθθθb2

mb1 mb2

mw1 mw2 mw3 mw4

zb1 zb2

zc

zw1 zw2 zw3 zw4

Fig. 2. Model of railway vehicle.



where

{z} = {zc, zb1, zb2, zw1, zw2, zw3, zw4}; {θ} = {θc, θb1, θb2}

{φ} = {φc, φb1, φb2, φw1, φw2, φw3, φw4} .

The dynamic reactions at the left and right points on the rails are expressed in

terms of the dynamic reactive components WL1(t), WL2(t), WL3(t), WL4(t), and

WR1(t), WR2(t), WR3(t), WR4(t) respectively. Nonlinearities in the behaviour of

suspension the primary and secondary suspension units are not considered in this

present study.

The response is obtained when the vehicle traverses 12 sleeper bays of a model of

a railway track with constant speed. An initially straight track with imperfections

and without imperfections is considered. The vertical dynamic response of railtrack

sleeper excited under moving multi-axle loads on track structure is investigated. The

wheels of the train are assumed to have infinite stiffness. It is also assumed that

there is no loss of contact between the wheels of the vehicle and the rails. A Digital

Power FORTRAN subroutine has been written to perform dynamic analysis of the

vehicle effectively using direct integration technique.13 The equation of motion for

the vehicle model consisting of carbody, bogie units and wheel sets are derived

from rigid body idealization (detailed derivation is given in Appendix A) and can

be expressed as follows:

my + cy + ky = W(t) (2)

where m, c and k are the mass, damping and stiffness matrices of the vehicle

respectively and W(t) and y denote the load and displacement vectors respectively.

3. Model of Track Structure

In this study, a linear 3-D finite element model consisting of the railway track

structure is considered, accounting for the elastic properties of rail-pad, ballast,

different ballast contact between sleeper and ballast along the sleeper length, sub-

ballast and sub-soil stiffness, cf. Fig. 3. The study is carried out with the following

assumptions:

(1) The rail, rail-pad, sleeper ballast and subsoil materials are assumed to be

linearly elastic, homogeneous and isotropic.

(2) In this study, the semi-infinite elastic half space of the subsoil media is replaced

by an equivalent finite elastic boundary. The boundary conditions are imposed

on the finite boundary of the finite element track model along each node.

(3) The sleeper considered in the present study conforms to specifications for broad

gauge tracks of Indian railways. The spacing of the sleepers are assumed to be

0.6 m apart.

(4) The standard rail section conforming to UIC 60 kg/m is considered for this

study.



Ballast

Rail

Boundaryconditions

Sleeper

Wheel loads

Fig. 3. Finite element model of track structure.

The railtrack model used in the present study consists of two rails supported by

rail-pads, sleepers on ballast, sub-ballast and substrata. The rail section is modeled

with solid elements, resting on the rail-pad, which is modeled as a spring element,

with vertical stiffness kp and viscous damping coefficient cp. The spring behaviour

of the rail fastenings at an uplift motion of the rail is not considered in this study.

The sleeper, ballast and sub-ballast are modeled as solid elements, accounting for

their different elastic properties. The substrata are modeled with linear boundary

elements and absorbing boundary conditions. The finite boundaries of such model of

the soil medium should be such that the elastic waves, which propagate through the

medium, do not reflect back from these boundaries to the vibrating medium.14–15

The finite element models are checked for maximum allowable values of aspect

ratio, taper, internal angles and warping in order to avoid the violent distortion of

elements. A finite length of track model including 12 sleeper bays is considered for

dynamic track analysis. The vehicle is assumed to move at constant speed over the

track.

4. Dynamic Analysis of Railtrack Sleepers in Track Structure

The dynamic wheel loads generated by a moving train are mainly due to vari-

ous wheel/track imperfections. These imperfections are considered as the primary

source of dynamic track input to the railroad vehicles.5–9 Normally, the imper-

fections that exist in the railtrack structure are associated with the vertical track

profile, cross-level, rail joint, wheel flatness, wheel/rail surface corrugations and un-

even support of the sleepers. For the present study, the average vertical profile data



on left and right rails of a continuously welded mainline track of the Indian railway

is used.12 The average vertical profile data of a continuously welded mainline track

of the Indian railway has been superimposed on the periodic irregularities (repeated

rail joint along with wheel flatness) to provide realistic track input to the vehicle

model, cf. Fig. 4. The initial excitation force F (t) due to the track irregularities

applied to the vehicle model can be expressed as

F (t) = kpy(t) (3)

where y(t) is the temporal frequency track data rad/s. The initial excitation force

due to the track irregularities are fed to the vehicle system at each primary

suspension unit of wheels with the time lags of t1, t2, t3, and t4 corresponding to the

distance between two consecutive axles, cf. Fig. 5. These force inputs are repeated

between successive time intervals till the complete length of train passes out with

-15

-10

-5

0

5

10

15

1.5 3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5

Time in s

Avg

ver

tical

pro

file

in m

m

Fig. 4. Average vertical profile of the track with all irregularities.

v

F(t1)F(t3)

F(t4)

l1l2l4 l3

F(t2)

Fig. 5. Distances of wheel contact points.



W(t)W(t)

ks

kpcp

cs

Fig. 6. Transmission of wheel load at the wheel-rail contact points.

Response fromvehicle model

Vehicle dynamics

Vertical vehicledynamics model

Track Input for vehicle modelTrack input to vehicle model

R(t) = kpy(t)

Track FEM Model

Checking the compatibilityconditions at the contactnodes between wheel and

track

If satisfied

Combined vehicle and track responses

Upd

ate

d v

alu

e o

f y (t )

Load history near the sleepers P(t)

Sleeper response to various train speeds

Not

Sat

isfie

d

Fig. 7. Flow chart illustrates the interaction between vehicle and track in time domain.



P(t)

Railtrack Sleeper

Rail

Railpad

Fig. 8. Dynamic load on the sleeper.

constant velocity v. The time lags are given by

t1 =l1

v; t2 =

l2

v; t3 =

l3

v; t4 =

l4

v(4)

where l1, l2, l3 and l4 denote the distances between the axles, cf. Fig. 5.

The dynamic interaction of combined systems such as vehicle and track requires

special attention when the fast moving heavy vehicles interact with the track struc-

ture. The objective of the combined vehicle-track interaction analysis is to deter-

mine the time histories of the wheel load reactions transmitted to the sleepers. The

dynamic wheel load on left and right rails at the contact points can be expressed

as follows (detailed derivations are given in Appendix A):

W (t)L = mw zw1 + 2cpzw1 − 2cpzb1 + 2cplbθb1 + 2kpzw1 − 2kpzb1 + 2kplbθb1

+ 2kwzw1 − Jwxφw1 + 2cplgφw1lp + 2cplbφb1lp − 2kwφw1lg2

+ 2kplgφw1lp + 2kplbφb1lp (5)

W (t)R = mw zw1 + 2cpzw1 − 2cpzb1 + 2cplbθb1 + 2kpzw1 − 2kpzb1 + 2kplbθb1

+ 2kwzw1 + Jwxφw1 − 2cplbφw1lp − 2cplbφb1lp + 2kwφw1lg2

− 2kplgφw1lp − 2kplbφb1lp (6)

where WL(t) and WR(t) are the reactions at the left and right rail respectively. The

equation of motion for the entire vehicle track structure is written as[mv 0

0 mT

]y +

[cv 0

0 cT

]y +

[kv 0

0 kT

]y = W(t) (7)

where mv, cv and kv are the mass, damping and stiffness matrices of the vehicle

system respectively and the corresponding parameters with the subscript T refers



to the track system. The external force vector W(t) resulting from the wheel loads

is dependent on the wheel and track (rail) interaction, and hence is time-dependent.

The interaction between vehicle and track is solved numerically using the time

domain approach. In this study, a digital visual FORTRAN code is written to

calculate the vertical response of the vehicle by direct integration technique using

Newmark’s method. The free vibration analysis is carried out for the combined

vehicle and track system using MSC/NASTRAN software. The time steps for direct

integration are calculated from the natural frequency of the combined vehicle and

track system. The minimum time step to carry out the direct integration scheme

is taken as one-tenth of the fundamental time period, in the interest of numerical

stability.16 The dynamic wheel reaction from the vehicle model is fed to the 3-

D finite element track model. The compatibility conditions are satisfied at each

node and at each time step by considering the displacements of wheel and the

displacement of rail. Figure 7 illustrates vehicle-track interaction analysis in the

time domain.

5. Dynamic Load on Railtrack Sleeper

The dynamic wheel load from interaction analysis gets transmitted as reactive forces

on sleeper. The dynamic force on the railtrack sleeper is calculated as the sum of

the reaction forces obtained at the nodes of the finite element model, cf. Fig. 8. The

following relationship is used to determine the dynamic load history P (t) near the

rail-seat location:

P (t) = kp(zs − zr) + cp(zs − zr) (8)

where, kp and cp are the rail-pad stiffness and damping respectively; zr and zs are

the vertical displacements of the rail and the sleeper at the rail-seat locations respec-

tively and zr and zs are the vertical velocities of the rail and the sleeper respectively.

0

10000

20000

30000

40000

50000

60000

70000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time in s

Dyn

amic

rai

l-se

at lo

ad in

N

Fig. 9. Dynamic load history near the left rail-seat area for soft rail-pad.



0

10000

20000

30000

40000

50000

60000

70000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time in s

Rai

lsea

t loa

d hi

stor

y in

N

Fig. 10. Dynamic load history near right rail-seat area for soft rail-pad.

0

10000

20000

30000

40000

50000

60000

70000

0 0.5 1 1.5 2 2.5 3 3.5 4

Time in s

Dyn

amic

rai

l-sea

t loa

d in

N

Fig. 11. Dynamic load history near left rail-seat area for stiff rail-pad.

The load amplification factors can be arrived at by dividing the maximum value of

P (t) by the static wheel load for different parametric conditions.

6. Vehicle Track Parameters

The following vehicle track parameters are considered in this study.



0

10000

20000

30000

40000

50000

60000

70000

0 0.5 1 1.5 2 2.5 3 3.5Time in s

Dyn

amic

rai

lsea

t loa

d in

N

Fig. 12. Dynamic load history near right rail-seat area for stiff rail-pad.

Vehicle parameters :10,11 mass of car body mc = 70 000 kg; mass of bogie unit

mb = 5800 kg; mass of wheelset mw = 1900 kg. Stiffness of primary suspension

springs (per wheel) kp = 1260 N/mm; vertical damping of primary suspension:

cp = 20 Ns/mm; stiffness of secondary suspension springs ks = 900 N/mm and

vertical damping of secondary unit cp = 60 Ns/mm.

Track parameters: mass per unit length (Rail UIC) = 60 kg/m; sleeper Young’s

modulus Ec = 38 450 MPa; sleeper spacing = 600 mm; elastic modulus of ballast

Eb = 100 MPa to 350 MPa.11,17

Rail-pad parameters :18 rail-pad stiffness (kp) and damping (cp) of (a) soft pad:

kp = 100 kN/mm and cp = 43 Ns/mm (b) stiff pad: kp = 600 kN/mm and cp =

93 Ns/mm.

Centre binding between the sleeper and ballast : Two extreme cases are

considered: (i) total loss of contact in the central region (which occurs in the

early life of the track structure) and (ii) full contact at all points (which occurs

subsequently).

Elastic properties of subgrade stiffness (ks) = 200 to 600 kN/mm); density of

subgrade medium (ρ) = 18 × 10−6 N/mm3; Poisson’s ratio (γ) = 0.33 and shear

wave velocity (Vs) = 150 m/s, which is well above the critical speed CT of the train

(in the order of 30 m/s to 40 m/s, adopted in most of the stretches in India.4

7. Results of Parametric Study

The results of the parametric study are depicted in the form of graphs comprising

rail-seat load histories and variation of dynamic amplification factors at the critical

locations, cf. Figs. 9–20.



8. Summary and Conclusions

The dynamic load on the railtrack sleeper is studied based on the comprehensive

vehicle and track models. The vehicle considered in this study is modeled using

rigid body idealization with 17 degrees of freedom and the track is modeled using

finite element software with a track length encompassing 12 sleepers. The common

imperfections, viz. railjoints, wheel flatness and track unevenness have been consid-

ered for the dynamic input to the vehicle model. The dynamic analysis of vehicle

and track structure has been carried out numerically using finite element method

for different speeds and parametric conditions.

The time history plots of load response near the rail-seat, cf. Figs. 9–12, and the

dynamic amplification factors for different parametric conditions are investigated.

It is observed that the effect of different rail-pad stiffness is more significant than

the other track parameters. It is seen that the load amplification factors (β) near the

rail-seat under no rail-pad condition gives the highest value of dynamic amplification

factor (β = 2.55) near the rail-seat locations, cf. Fig. 13.

It has been observed that the frequencies excited by the irregularities on the run-

ning surfaces of wheel and rail are significantly higher when vehicle speeds approach

70 to 150 kmph cf. Figs. 13 and 14. This peak is probably related to the coupled

vehicle-track resonance frequency.

It is observed that when there are no track irregularities, the response is almost

unaffected by increased train speed. cf. Fig. 15.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

1 Eb = 350 MPa2 Eb = 250 MPa3 Eb = 150 MPa

1

3

2

Fig. 13. Amplification factor verses frequency for no rail-pad condition and sub-soil stiffnessks = 600 kN/mm.



0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

No rail pad1 Eb = 350 MPa ; ks = 200 kN/mm2 Eb = 250 MPa ; ks = 200 kN/mm3 Eb = 150 MPa ; ks = 200 kN/mm

1

3

2

Fig. 14. Amplification factor verses frequency for no rail-pad condition and sub-soil stiffnessks = 200 kN/mm.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

321

No irregularity

1 Eb = 350 MPa2 Eb = 250 MPa3 Eb = 150 MPa

Fig. 15. Amplification factor verses frequency for soft rail-pad and sub-soil stiffness ks =600 kN/mm.



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

Soft pad kp =100 kN/mm1 Eb = 350 MPa ; ks = 200 kN/2 Eb = 250 MPa ; ks = 200 kN/3 Eb = 150 MPa ; ks = 200 kN/

13 2

Fig. 16. Amplification factor verses frequency for soft rail-pad and sub-soil stiffness ks =200 kN/mm.

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

Stiff pad kp = 600 kN/mm1 Eb = 350 MPa ; ks = 600 kN/m2 Eb = 250 MPa ; ks = 600 kN/m3 Eb = 150 MPa ; ks = 600 kN/m

1

3

2

Fig. 17. Amplification factor verses frequency for stiff rail-pad and sub-soil stiffness ks =600 kN/mm.



It is also observed that the stiff rail-pad condition gives slightly higher load

amplification factor near the rail-seat locations (β = 2.21), cf. Figs. 17 and 18.

The results of the dynamic analysis gives the load amplification factor for various

frequencies near the rail seat location, cf. Figs. 19 and 20 under non-uniform loss

of contact between sleeper and ballast.

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

Stiff pad kp = 600 kN/mm1 Eb = 350 MPa ; ks = 200 kN/mm2 Eb = 250 MPa ; ks = 200 kN/mm3 Eb = 150 MPa ; ks = 200 kN/mm

1

32

Fig. 18. Amplification factor verses frequency for stiff rail-pad and sub-soil stiffness ks =200 kN/mm.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

Portion of sleeper-ballast loss of contactSoft pad kp=100 kN/mm

L = 750mm ; ks = 200 kN/mm

L

Fig. 19. Amplification factor verses frequency for soft rail-pad and sleeper-ballast loss of contact.



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 50 100 150 200 250 300 350 400 450

Frequency in Hz

Am

plifi

catio

n fa

ctor

Portion of sleeper-ballast loss of contacStiff pad kp = 600 kN/mm

L = 750mm ; ks = 600 kN/mmL

Fig. 20. Amplification factor verses frequency for stiff rail-pad and sleeper-ballast loss of contact.

For a quasi-static analysis (as an alternative to dynamic analysis), the peak val-

ues of amplification factor generated from the present study are applicable. Also,

the corresponding ballast pressure distribution under the sleeper needs to be con-

sidered in order to arrive at the design bending moments in the sleeper. These

studies on the “sleeper in isolation” are reported elsewhere.13

Appendix A

The equation of motion for the vehicle model comprising of carbody, bogie units

and wheel sets are derived from rigid body idealization.

Car body: The equation of motion is written for bounce (zc), pitch (θc) and roll

(φc) degrees of freedom with respect to transverse horizontal centroidal axis

(a) zc bounce of the car body

mczc + 4cpzc − 2cszb1 − 2cszb2 + 4kpzc − 2kszb1 − 2kszb2 = 0 (A.1)

(b) θc pitch of the car body

Jcy θc + 4cslc2θc − 2cszb1lc + 2cszb2lc + 4kslc

2θc − 2kszb1lc

+ 2kszb2lc = 0 (A.2)

(c) φc roll of the car body

Jcφc + 4cszcls + 4csls2φc − 2csφb1ls

2 − 2csφb2ls2 + 4kszcls + 4ksls

2φc

− 2ksφb1ls2 − 2ksφb2ls

2 = 0 . (A.3)



Bogie frames: The equation of motion is written for bounce (zb), pitch (θb) and roll

(φb) degrees of freedom with respect to transverse horizontal centroidal axis

(a) zb1 bounce of the first bogie frame

mbzb1 + 2cszb1 + 2cslcθc − 2cszc + 4cpzb1 − 2cpzw1 − 2cpzw2 + 2kszb1

+ 2kslcθc − 2kszc + 4kpzb1 − 2kpzw1 − 2kpzw2 = 0 . (A.4)

Similarly for other bogie frame zb2

mbzb2 + 2cszb2 + 2cslcθc − 2cszc + 4cpzb2 − 2cpzw3 − 2cpzw4 + 2kszb2

+ 2kslcθc − 2kszc + 4kpzb2 − 2kpzw3 − 2kpzw4 = 0 . (A.5)

(b) θb1 pitch of the first bogie frame

Jby θb1 + 4cslb2θb1 − 2cpzw1lb + 2cpzw2lb + 4kslb

2θb1

− 2kpzw1lb + 2kpzw2lb = 0 . (A.6)

Similarly for other bogie frame θb2

Jby θb2 + 4cslb2θb2 − 2cpzw3lb + 2cpzw4lb + 4kslb

2θb2

− 2kpzw3lb + 2kpzw4lb = 0 (A.7)

(c) φb1 roll of the first bogie frame

Jbφb1 + 2cslsφb1ls + 4cplpφb1lp + 2kslsφb1ls + 4kplpφb1lp = 0 . (A.8)

Similarly for other bogie frame φb2

Jbφb2 + 2cslsφb2ls + 4cplpφb2lp + 2kslsφb2ls + 4kplpφb2lp = 0 . (A.9)

Wheel sets: The equation of motion is written for bounce (zw), pitch (θw) and roll

(φw) degrees of freedom with respect to transverse horizontal centroidal axis,

(a) zw1 bounce of the first wheel sets,

mw zw1 + 2cpzw1 − 2cpzb1 + 2cplbθb1 + 2kpzw1

− 2kpzb1 + 2kplbθb1 + 2kwzw1 = R(t)1 . (A.10)

Similarly for other wheel sets zw2, zw3, zw4

mwzw2 + 2cpzw2 − 2cpzb1 + 2cplbθb1 + 2kpzw2 − 2kpzb1

+ 2kplbθb1 + 2kwzw2 = R(t)2 (A.11)


+ 2kplbθb2 + 2kwzw3 = R(t)3 (A.12)


+ 2kplbθb2 + 2kwzw4 = R(t)4 (A.13)



(b) φw1 roll of the first wheel sets

Jwxφw1 − 2cplgφw1lp − 2cplbφb1lp + 2kwφw1lg2 − 2kplgφw1lp

− 2kplbφb1lp = R(t)5 . (A.14)

Similarly for other wheel sets φw2, φw3, φw4

Jwxφw2 − 2cplgφw2lp − 2cplbφb1lp + 2kwφw2lg2

− 2kplgφw2lp − 2kplbφb1lp = R(t)6 (A.15)


− 2kplgφw3lp − 2kplbφb2lp = R(t)7 (A.16)


− 2kplgφw4lp − 2kplbφb2lp = R(t)8 . (A.17)

The equations of motions of the vehicle can also be written in matrix notations,

my + cy + ky = W(t) . (A.18)

The above equation of motion for the entire vehicle model can be written in terms

of the sub-matrices as

m = diag[mcmb1mb2mw1mw2mw3mw4 Ic Ib1 Ib2 Jc Jb1 Jb1 Jw1 Jw2 Jw3 Jw4]

where m is the mass matrix written in diagonal form yT is the acceleration

vector

yT = {zc zb1 zb2 zw1 zw2 zw3 ww4 θc θb1 θb2 φc φb1 φb2 φw1 φw2 φw3 φw4}

[c]1 and [c]2 are the sub matrices of the damping matrix

[c]1 =

4cs −2cs −2cs 0 0 0 0 0 0 0

−2cs 2cs + 2cp 0 −2cp −2cp 0 0 2cslc 0 0

−2cs 0 2cs + 2cp 0 0 −2cp −2cp −2cslc 0 0

0 −2cp 0 2cp 0 0 0 0 2cplb 0

0 −2cp 0 0 2cp 0 0 0 −2cplb 0

0 0 −2cp 0 0 2cp 0 0 0 2cplb

0 0 −2cp 0 0 0 2cp 0 0 −2cplb

0 2cslc −2cslc 0 0 0 0 2cslc2 0 0

0 0 0 2cplb −2cplb 0 0 0 2cplb2 0

0 0 0 0 0 2cplb −2cplb 0 0 2cplb2



[c]2 =

4cslc12 −2cslc1

2 −2cslc12 0 0 0 0

−2cslc12 2cslc1

2+

4cplb12

0 −2cplb1lg −2cplb1lg 0 0

−2cslcl2 0

2cslc12+

2cplb12

0 0 −2cplb1lg −2cplb1lg

0 −2cplb12 0 2cplb1lg 0 0 0

0 −2cplb12 0 0 2cplb1lg 0 0

0 0 −2cplb12 0 0 2cplb1lg 0

0 0 −2cplb12 0 0 0 2cplb1lg

{y1}T and {y2}T are the velocity vectors

{y1}T = {zc zb1 zb2 zw1 zw2 zw3 zw4 θc θb1 θb2}

{y2}T = {φc φb1 φb1 φw1 φw2 φw3 φw4}[k]1 and [k]2 are the sub matrices of the stiffness matrix

[k]1 =

4ks −2ks −2ks 0 0 0 0 0 0 0

−2ks 2ks + 2kp 0 −2kp −2kp 0 0 2kslc 0 0

−2ks 0 2ks + 4kp 0 0 −2kp −2kp −2kslc 0 0

0 −2kp 0 2kp 0 0 0 0 2kplb 0

0 −2kp 0 0 2kp 0 0 0 −2kplb 0

0 0 −2kp 0 0 2kp 0 0 0 2kplb

0 0 −2kp 0 0 0 2kp 0 0 −2kplb

0 2kslc −2kslc 0 0 0 0 4kslc2 0 0

0 0 0 2kplb −2kplb 0 0 0 4kplb2 0

0 0 0 0 0 2kplb −2kplb 0 0 4kplb2

[k]2 =

4kslc12 −2kslc1

2 −2kslc12 0 0 0 0

−2kslc12 2kslc1

2+

4kplb12

0 −2kplb1lg −2kplb1lg 0 0

−2kslc12 0

4kslc12+

2kplb12

0 0 −2kplb1lg −2kplb1lg

0 −2kplb12 0 2kplb1lg 0 0 0

0 −2kplb12 0 0 2kplb1lg 0 0

0 0 −2kplb12 0 0 2kplb1lg 0

0 0 −2kplb12 0 0 0 2kplb1lg

{y1}T and {y2}T are the displacement vectors

{y1}T = {zc zb1 zb2 zw1 zw2 zw3 zw4 θc θb1 θb2}

{y2}T = {φc φb1 φb1 φw1 φw2 φw3 φw4}

W(t)T = {0 0 0R(t)1R(t)2R(t)3R(t)4 0 0 0 0 0 0R(t)5R(t)6R(t)7R(t)8}where m, c and k are the mass, damping and stiffness matrices of the vehicle respectivelyand W(t) is load vector.



References

1. A. N. Talbot, Stresses in Railroad Track — Reports of the Special Committee onStresses in Railroad Track, Seven Progress Reports, American Railway Engineeringand Maintenance of Way Association, AREMA, 1934.

2. H. P. J. Taylor, “The prestressed concrete railway sleepers — 50 Years of pretensioned,prestressed concrete,” J. Struct. Eng. 71, pp. 281–289, 1993.

3. H. E. Stewart and D. O’Rourke, “Load factor method for dynamic track loading,” J.Transport. Eng. ASCE 114, pp. 21–39, 1998.

4. Standard Specification for Pretensioned Concrete Sleepers, RDSO, Third Revision,Indian Railways, Lucknow, 1996.

5. S. L. Grassie and S. J. Cox, “The dynamic response of railway track with flexiblesleepers to high frequency vertical excitation,” Proc. Inst. Mech. Eng. D198, pp. 117–124, 1983.

6. K. L. Knothe and S. L. Grassie, “Railway track and vehicle/track interaction,”J. Vehicles Syst. Dynam. 22, 217–262, 1993.

7. J. C. O. Nielsen, “Vertical dynamic interaction between train and track — Influenceof wheel and track imperfections,” J. Sound Vib. 87(5), pp. 825–839, 1995.

8. Y. B. Yang and J. D. Yau, “Vehicle-bridge interaction element for dynamic analysis,”J. Struct. Eng. ASCE 123(11), pp. 1512–1518, 1997.

9. Y. S. Wu., Y. B. Yang and J. D. Yau, “Three-dimensional analysis of train-rail-bridgeinteraction problems,” Vehicle Syst. Dynam. J. Vehicle Mech. Mob. 36(1), pp. 1–35,2001.

10. V. Dukkipati and J. R. Amyot, Computer-Aided Simulation in Railway Dynamics,Marcel Dekker Inc., New York, 1988.

11. C. S. Desai and A. M. Siriwardane, “Numerical models for track support structures,”J. Geotech. Eng. Div. ASCE 108, pp. 461–480, 1982.

12. K. V. Gangadharan, “Analytical and experimental studies on dynamics of rail-road vehicles,” PhD Thesis, Department of Applied Mechanics, Indian Institute ofTechnology Madras, 2001.

13. G. Kumaran, D. Menon and K. K. Nair, “Dynamic response of railtrack sleepers tomoving loads,” Proc. EURODYN 2002 (In print).

14. P. Wolf, Foundation Vibration Analysis using Simple Physical Models, Prentice-Hall,Englewood Cliffs, New Jersey, 1994.

15. J. Dinkel and H. Grundmann, “Winkler parameters for railway dynamics derived from3-D half space analysis,” Proc. Struct. Dynam. — EURODYN 99, pp. 831–836, 1999.

16. A. K. Chopra, Dynamics of Structures Theory and Applications to Earthquake Engi-neering Prentice-Hall, Pearson Education Co., New Jersey, 2001.

17. D. Li and T. Ernest, “Wheel/track dynamic interaction — Track substructureperspective,” J. Vehicle Syst. Dynam. 24, pp. 183–196, 1995.

18. B. Ripke and K. Knothe, “Simulation of high frequency vehicle-track interactions,”J. Vehicle Syst. Dynam. 24(Suppl.), pp. 72–85, 1995.

Documents

EVALUATION OF DYNAMIC LOAD ON RAILTRACK …kknair/pdf/Kumaran.pdf · Dynamic Analysis of Railtrack Sleepers in Track Structure The dynamic wheel loads generated by a moving train