Performance Evaluation of KHSR Bridge using Two-Dimensional

Train/Track/Bridge Interaction Analysis Method

Man-Cheol Kim � Woo-Jin Chung � Kee-Dong Kang

Korea Railroad Research Institute

374-1, Woulam-Dong, Uiwang-City, Kyonggi-Do, 437-050, Korea

Telephone: +82-31-461-8531(Ext. 212)

E-mail: kimmc@krri.re.kr

Abstract

Railway bridges are subject to dynamic loads caused by interactions between moving

vehicles and the bridge structures. These dynamic loads result in response fluctuation in

bridge members. Therefore, to investigate the real dynamic behaviors of the bridge, a number

of analytical and experimental investigations should be carried out.

Nowadays, the Korea High Speed Rail Construction Project is under way. This paper

represents the results carried out to investigate the dynamic response characteristics of KHSR

bridge. The dynamic behaviors of the KHSR bridge subjected to the moving train loading are

simulated through the developed two-dimensional train/track/bridge interaction analysis

program. In the developed program, the eccentricity of axle loads and the effect of the

torsional forces acting on the bridge are included for more accurate train/track/bridge

interaction analysis. As railroads are constructed mostly by double track, the eccentricity

between vehicle axle and neutral axis of cross-section of bridges is important factor in

evaluating the dynamic behavior of bridges.

The results of the analyses are compared with the field test data for the typical PSC box

bridge to verify the performance of the developed train/track/bridge interaction analysis

method.

1. Introduction

The Korea High Speed Rail Construction Project for the purpose of commercial operation

speed up to 300km/h in the year of 2004 is under way and also, the testing operations in the

test track section(57.2km) are being conducted. Total length of the Korea High Speed Rail is

412km, of which 112km (27%) consists of elevated viaduct and station structures, and

bridges.

The railroad bridges are subjected to dynamic loads caused by the interactions between the

moving train and the bridges. These dynamic loads lead to impacts and fatigue to the bridge

members. Specially, when the heavy train is running at the high speed on the bridge, the

excessive vibrations are induced. Therefore, it is very important to exactly analyze the

dynamic characteristics of the bridge resulted from the operation of high speed train with

considering the interaction of train/track/bridge in order to obtain the structural safety, train

operation safety and passenger comfort.

In this paper, the analyzing program of considering the interaction of train/track/bridge is

developed which makes it available to evaluate the performance of bridge for high-speed train

in terms of structural safety, train operation safety and passenger comfort. And also, the

numerical analyses on the representative bridge in Korea High Speed Rail are performed by

making use of the developed program and the results are compared with field test data.

2. Train/Track/Bridge Interaction Analysis Program

2.1 Modeling of KTX

KTX(Korea eXpress Train) consists of two power cars, two motorized trailers and sixteen

trailer cars. Each car consists of car body, secondary suspension, bogie, primary suspension

and axle. The most distinctive characteristic of KTX is that each passenger car is linked to the

adjacent car with a connecting structure, an "articulated bogie". Since the whole train is

connected organically by making use of the articulated bogie, the independent movement of

each car is prevented. Therefore, the affection of refraining from vibration phenomenon

typically occurred in the conventional train is acquired.

In this paper, vertical displacement and pitching motion of the car body and the bogie and

vertical displacement at the axle are considered as general coordinates (Figure 1). And also,

the car body and the bogie are assumed as the rigid body with the mass, the train runs at a

constant speed.

L1 L2 L3 L20L19Li

l1 l2 l3 l4 li+1 li+2 l21 l22 l23

Mi,Iiksi,csi

kpj,cpj

kuk

mj,ij

mukAxle

Car BodySecondary Suspension

Primary SuspensionBogie

Hertzian Spring

v1

u1,èi

w1 ,w2

L1 L2 L3 L20L19Li

l1 l2 l3 l4 li+1 li+2 l21 l22 l23

Mi,Iiksi,csi

kpj,cpj

kuk

mj,ij

mukAxle

Car BodySecondary Suspension

Primary SuspensionBogie

Hertzian Spring

v1

u1,èi

w1 ,w2

Figure 1. Material properties and degrees of freedom for KTX

Figure 2. Contact stiffness model between axle and rail

kh : Hertzian spring

w : wheel displacement

yw/r : rail irregularity

v : rail displacement

Pi : dynamic contact force

Pi : dynamic contact force

w : wheel displacement

yw/r : rail irregularity

v : rail displacement

Pi : dynamic contact force

Pi : dynamic contact force

During vehicle/track interaction the forces are transmitted via the wheel/rail contact area. In

this paper, this mechanism is modeled as the Hertzian contact spring( hk ). On account of the

geometry of the contact area between the round wheel and the rail, the relationship between

force and compression, represented by the Hertzian contact spring, is not linear. But it is

assumed as the linear for the simplification in this study.

To derive the equation of motion of KTX, Lagrange’s equation is used.

iiiii

qgD

gV

gT

gT

dtd =

∂∂+

∂∂+

∂∂−

∂∂

&&(1)

where T is the total kinetic energy of the system, V and D are the potential energy and

dissipation energy. ig is the generalized coordinate and iq represents the external force

corresponding to the generalized coordinate ig .

The kinetic, potential and dissipation energies of the front power car can be written as

follows ;

( )

+++

−+

+= ∑∑

==

4

1

22

1

22

2

1

211

2

211 2

122

1

iiui

iiiii wmium

Lvv

Ivv

MT &&&&&&& θ

(2)

∑∑∑===

++=4

1

24

1

22

1

2

21

21

21

ihihi

ipipi

isisi ykykykV

(3)

∑∑==

+=4

1

22

1

2

21

21

ipipi

isisi ycycD &&

(4)

The kinetic, potential and dissipation energies of the motorized trailer car and the trailer car

are given by

( )

+++

−+

+= ∑ ∑∑

= ==

++++19

2

42

5

221

3

22

2

21

2

21

221

i iiui

iiiii

i

iii

iii wmium

L

vvI

vvMT &&&

&&&&θ (5)

∑∑∑===

++=42

5

242

5

221

3

2

21

21

21

ihihi

ipipi

isisi ykykykV (6)

∑∑==

+=42

5

221

3

2

21

21

ipipi

isisi ycycD && (7)

The kinetic, potential and dissipation energies of the rear power car are as following

formula;

( )

+++

−+

+= ∑∑

==

46

43

223

22

22

2

20

232220

2

232220 2

122

1

iiui

iiiii wmium

Lvv

Ivv

MT &&&&&&& θ (8)

∑∑∑===

++=46

43

246

43

223

22

2

21

21

21

ihihi

ipipi

isisi ykykykV (9)

∑∑==

+=46

43

223

22

2

21

21

ipipi

isisi ycycD && (10)

Herewith, the relative displacements of the suspension and the Hertzian contact spring can

be represented as follows;

iisi uvy −= (11)

( ) iiii

ipi wluy −−+= + θ11 (12)

( ) ( )irwirihi xyxvwy /+−= (13)

( )ir xv is the displacement of the rail at the location ix of i th axle and ( )irw xy / represents the

rail irregularity at the corresponding point.

By applying equations (2)_(13) to Lagrange’s equation (1), the following equation of

motion of KTX for the generalized coordinates T

jjjjjt wwuvg 212 −= θ ,

23,,1 KK=j can be obtained;

ttttttt qgKgCgM =++ &&& (14)

where tM , tC , tK and tq represent the mass, damping, stiffness matrices and load vector

applied to the KTX respectively. In equation (14), the load vector applied to the KTX,

T

tjttt qqqq )23()()1( KK= , can be written as follows;

( ) ( ){ }( ) ( ){ }

−+−+

=

−−−−

jrwjrjhju

jrwjrjhju

j

i

jt

xyxvkgm

xyxvkgm

gm

gM

q

2/2)2()2(

12/12)12()12(

)( 0

2

,

23,2220

2119

32

2,11

====

====

jfori

jfori

jfori

jfori

(15)

( ) ( ){ }( ) ( ){ }

−+−+

+

=

−−−−

−−

jrwjrjhju

jrwjrjhju

j

jj

jt

xyxvkgm

xyxvkgm

gm

gMM

q

2/2)2()2(

12/12)12()12(

12

)( 0

2

, 20,,4 KK=j

where g is acceleration due to gravity.

2.2 Track Modeling

The track consists of rail, sleeper and ballast. The rail is modeling into two-dimensional

beam element. The sleeper and ballast are modeling by means of a winkler element. A

winkler element consists of four nodes, each node has vertical and rotational degrees of

freedom. When assuming that the stiffness of ballast track are uniformly distributed in local

coordinate, the stiffness matrix of a winkler element in local coordinate can be obtained as

follows; the displacements ( )ξrv and ( )ξdv of rail and the deck of bridge at the position ξ

can be represented as following formula by means of Hermitian shape function,

TNNNNN 4321= used when forming beam element, and nodal displacements

T

rrrrr vvv 2211 θθ= and T

ddddd vvv 2211 θθ= .

rT

r vNv =)(ξ , dT

d vNv =)(ξ (16)

By the equation (16), the stiffness matrix of a winkler element can be obtained as follows;

−−−−

=

= ∫

2

22

4

22156

3134

135422156

420

Lk

Lkksym

LkLkLk

LkkLkk

L

NdlkNk

v

vv

vvv

vvvv

L vT

w

(17)

The dynamic load )(tPi caused by the interaction between wheel and rail at i th axle is

applied to the rail, which can be represented as follows;

( ) ( ){ }irwirihit xyxvwktP /)( +−= (18)

Interaction force )(tPi affected into rail by train running can be represented as the nodal force

of rail element by the following equation.

( )NtPmfmf i

T =2211 (19)

And also, by making use of these formulas, displacement at node can be represented as the

one at axle as follows;

( ) rT

ir vNxv = (20)

2.3 Modeling of Bridge

Bridge is modeling by means of two-dimensional beam element. In the existing two-

dimensional interaction analysis program, the axle load is assumed to acts through the neutral

axis of section. However, Since the double track on the bridges for high-speed train is

generally constructed, the analysis of considering the eccentricity of axle load caused by the

train running is more reasonable. Therefore, beam element makes it available to consider the

affection with eccentricity of axle load by including the torsional degree of freedom ( )kb xϕ in

addition to vertical ( )kb xv and rotational displacements ( )kb xθ . The displacement of deck on

the bridge ( )kd xv can be represented in terms of torsional displacement ( )kb xϕ and vertical

displacement ( )kb xv at the neutral axis of bridge by the following geometric shape.

P

a vb

¥õb

vb

lo ¥õo

¥õblocos¥õo

vd=vb+¥õblocos¥õo

P

a vb

¥õb

vb

lo ¥õo

¥õblocos¥õo

vd=vb+¥õblocos¥õo

Figure 3. Axis load eccentric modeling

( ) ( ) ( ) ookbkbkd lxxvxv ϕϕ cos+= (21)

Therefore, the relationship of the displacements of node at the neutral axis of section and

vertical displacement of node at the deck of bridge can be expressed by making use of the

following constraint equation.

=

b

b

boo

b

b

b

d vl

v

v

θϕ

ϕ

θϕ

100

010

001

0cos1

(22)

Since in case of PSC box bridge, the affection of shear deformation to the total girder

deformation is tiny, the shear deformation is disregarded in this study.

The damping matrix of bridge is obtained by making use of Rayleigh damping.

bbb KMC βα += (23)

Here, bC , bM and bK mean damping, mass and stiffness matrix of bridge respectively, α

and β are Rayleigh damping coefficients.

3. Evaluation of Dynamic Performance of KHSR Bridge

Based on extensive theoretical studies, field measurements, and experience obtained in

operating its TGV lines, the French National Railways(SNCF) has defined a set of dynamic

performance requirements for train-operation safety and passenger comfort for bridges

supporting high-speed trains(Table 1). We have used the maximum vertical acceleration at the

level of deck and vertical deflection of dynamic performance criteria given in Table 1 as the

basis for assessing the acceptability of the dynamic performances of the KHSR bridges in this

study.

Table 1. Dynamic performance criteria for bridges supporting high-speed trains

Maximum vertical acceleration at deck

levelga 35.0max ≤

Maximum relative rotation of deck

across a jointradina1050 5

max−×≤θ

Criteria for

train-operation

safety

Maximum twist per unit length of deck (Long)3/(Tra)/4.0max mmmm≤ϕ

Maximum vertical-deflection to span-

length ratio700,1/1/max ≤∆ LCriteria for

passenger

comfortMaximum vertical acceleration

experienced by train passengersga 05.0max ≤

The material properties of KTX is detailed on Table 2. The bridge consists of two spans,

length of each span is 40m and total length of the bridge is 80m. The Section type of the

bridge is the PSC box with one cell which is most typical in KHSR. The material properties of

the bridge are summarized on Table 3~5.

Track is modeled as stiffness and damping by making use of a winkler element. And the

inertia by the mass of track is considered as equivalent mass density, which is converted into

the cross-sectional area of each bridge section corresponding to the mass per unit length of

each bridge section and mass per unit length of track.

Table 2. Material properties of KTX(unit: tonf, m)

PC MTC TC

Mass(m ) 5.60 4.36 2.69Car body

Moment of inertia( 33I ) 115.50 167.72 100.14

Mass(m ) 0.25 0.31 0.31Bogie

Moment of inertia( 33I ) 0.265 0.333 0.327

Axle Mass(m ) 0.21 0.21 0.21

Stiffness( vk ) 259.78 75.51 61.22Secondary

suspension Damping( vc ) 4.082 4.082 4.082

Stiffness( vk ) 250 250 168.37Primary

suspension Damping( vc ) 4.08 4.08 4.08

Contact stiffness Stiffness( vk ) 285,714.29 285,714.29 285,714.29

Table 3. Material properties of KHSR bridge(unit: tonf, m)

Young’s modulus( E ) 2.8× 10+6

Poisson’s ratio(ν ) 0.2

Section Area( A ) Moment of inertia( 33I ) Mass density( ρ )

1 12.262 20.397 0.342

2 13.33 21.226 0.335

3 15.991 26.263 0.322

4 15.73 24.377 0.323

5 24.099 28.764 0.299

6 13.996 22.387 0.331

7 14.53 22.802 0.329

Table 4. Material properties of track(unit: tonf, m)

Stiffness( vk ) 8,163.27

Damping( vc ) 24.49

Table 5. Material properties of UIC60 rail(unit: tonf, m)

Young’s modulus( E ) 21,428,571

Mass density( ρ ) 0.806

Poisson’s ratio(ν ) 0.3

Area( A ) 15.388× 10 -3

Moment of inertia( 33I ) 61.1× 10-6

On-site measurements on the above bridge to evaluate the accuracy of results of numerical

analysis are carried out at each running speeds of train of 130, 175, 200, 250, 275 and

300km/h. The measurements of vertical deflection and vertical acceleration are performed in

the middle of the second span for the direction of Pusan. The sampling interval ∆ is 0.005

second and location of sensor is as shown in Figure 4.

A

A

40m 20m

2@40m=80m

Seoul Pusan

20m

Accelerometer

Displacement Transducer

Figure 4. Sensor and location of sensor

Maximum vertical displacements and vertical accelerations at the mid-point of each span

acquired through the numerical analysis and field measurement are detailed in Figure 5_8.

The error in numerical analysis for vertical deflection is 0.01% to 15% which means the

developed program is capable of predicting the dynamic behaviors of bridge caused by the

running of the train at the creditable level. But, the error for acceleration is relatively bigger

than the one for displacement. The relatively bigger error for acceleration as compared with

the one of displacement is because the measured acceleration is a tiny value. The fact that

KHSR Bridges under currently construction meet the requirement of dynamic efficiency

standard of SNCF is proved through this study.

120 140 160 180 200 220 240 260 280 300 3200.4

0.8

1.2

1.6

2.0

2.4

Mid-Point of First Span Measured Data Simulated Data

Dis

plac

emen

t(m

m)

Velocity(km/h) 120 140 160 180 200 220 240 260 280 300 320

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36

Mid-Point of First Span Measured Data Simulated Data

Acc

eler

atio

n(g)

Velocity(km/h)

Figure 5. Vertical deflection of mid-point Figure 6. Vertical acceleration of mid-point

of first span of first span

120 140 160 180 200 220 240 260 280 300 3200.4

0.8

1.2

1.6

2.0

2.4

Mid-Point of Second Span Measured Data Simulated Data

Dis

plac

emen

t(m

m)

Velocity(km/h) 120 140 160 180 200 220 240 260 280 300 320

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36

Mid-Point of Second Span Measured Data Simulated Data

Acc

eler

atio

n(g)

Velocity(km/h)

Figure 7. Vertical deflection of mid-point Figure 8. Vertical acceleration of mid-point

of second span of second span

4. Conclusion

When the heavy train is running at the high speed of 300km/h on the bridges, the

interaction between train and bridge causes the excessive dynamic behaviors to the bridge.

Therefore, in order to acquire the train operation safety, passenger comfort on the courses of

bridge, to analyze the precise dynamic characteristics of bridge by fully analyzing the

interaction of train/track/bridge is required. This study is developing the two dimensional

train/track/bridge interaction analysis program of being capable of evaluating the dynamic

characteristic of bridge resulted from the operation of KTX Trains. In order to evaluate the

efficiency of this program, when comparing the KHSR Bridges with the field measured

results and numerical analysis making use of development program, the results of numerical

analysis is shown a little high value, at the conservative level. This means the results are

analyzed as safety when evaluating the dynamic performance of bridge making use of this

development program and also, when considering the dynamic characteristic of KHSR Bridge

under currently construction in terms of operation safety and passenger comfort, it is analyzed

as excellent.

References

1. S.K. Chauduri (1975), "Dynamic Response of Horizontally Curved I-Girder Highway

Bridges Due to a Moving Vehicle," Dissertation, The University of Pennsylvania.

2. K.H. Chu, and V.K. Garg(1986), "Impact in Railway Prestressed Concrete Bridges,"

Journal of Structural Engineering(ASCE), Vol.112, No.5, pp.1036-1051.

3. Y.B. Yang, J.D. Yau, and L.C. Hsu(1997), "Vibration of Simple Beams due to Trains

Moving at High Speeds," Engineering Structures, Vol.19, No.11, pp.936-944.