COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Effect of Pier and Abutment Non-Uniform Settlement on Train Running
Behavior
Jianzhen Xiong *, Hanbin Yu, Mangmang Gao
China Academy of Railway Science, No.2 daliushu Road, Haidian district, Beijing, 100081 China
Email: xjz@rails.com.cn
Abstract Dynamic behavior of running vehicle on bridge is influenced by the attached irregularity of track caused by
pier and abutment non-uniform settlement. In order to study the relationship between them, train-track-bridge vertical
coupling system dynamic model is established, in which the four-axle car with two suspension systems on independent
bogies is adopted, the bridge is simulated with various finite elements. The vehicle components is treated as rigid
bodies , assembled with linear springs and dampers. There are totally 31 degrees of freedom, with carbody and each
bogie having 5 units of degree of freedom, and each wheel-set having 4 units of degree of freedom. As one part of track
irregularity, the attached deflection is determined by static mechanics. Dynamic analysis uses time domain method,
and nonlinear elastic hertz contact theory is adopted in wheel-rail vertical direction, at the same time the wheel-rail
deviations can be considered. Based on ballast track and prestressed concrete box girder bridges with 40m span, the
homemade Pioneer within 160~220km/h, pier and abutment non-uniform settlement with 5mm, 10mm, 15mm, 20mm,
25mm and 30mm taking into account respectively, the dynamic response of train and bridge corresponding to different
settlement at various running speed are obtained, which can provide a reference to the practice engineering design of
railroad bridge.
Key words: Train; running behavior; pier and abutment; non-uniform settlement; vertical coupling vibration
INTRODUCTION
Regarding pier and abutment settlement of railway bridge, domestic and foreign studies mainly focus on calculating
method, monitoring, reinforcement, etc., while few consider influence of pier and abutment settlement on train running
and bridge vibration, and testing materials in this field is scarce.
Settlement of Pier and abutment, in particular uneven settlement between adjacent piers will cause change of track
geometries, generating a kind of irregularity and affecting directly running performance of train. Therefore, it is
necessary to carry out dynamic analysis on train-track-bridge coupling vibration in bridge that has uneven pier
settlement. Provision concerning uneven settlement between adjacent piers on static structure (simple girder bridge) in
China’s specification [1] is:
Bridge with ballasted bed: Δ=15 mm
This paper takes a 9×40 m single supported bridge with ballasted bed as example, analyzes the dynamic responses of
train and bridge when the train pass the bridge with non-uniform pier settlement of 0~30 mm.
DYNAMIC MODEL OF TRAIN-TRACK-BRIDGE
1. Vehicle dynamic model and motion equation In the paper, vehicle model uses the homemade Pioneer train unit.
locomotive and rolling stock are consisted of carbody, front and rear bogies, wheelset, primary and secondary stage
suspension spring, etc. When establishing vehicle motion equation,, it is assuming that [2]:
(1) Carbody, bogie and wheelset are rigid bodies, which will have vibration of small displacement near the basic
equilibrium position;
(2) Carbody is symmetrical to the center of mass in both left and right as well as front and back and the train makes
uniform motion along tangent track, disregarding the effect of longitudinal dynamic vibration;
⎯ 1215 ⎯
(3) The damping among suspension systems of vehicle are viscous dampings and all springs are linear ones;
(4) Wheel is allowed to jump off the rail in vertical direction and there are normal Hertz nonlinear elastic contact and
tangent nonlinear creep contact between wheel and rail.
In the light of the above assumptions, carbody and bogie have 5 units of degree of freedom respectively, including
bounce, horizontal movement, roll, pitch and yaw. Each wheelset has 4 units of degree of freedom such as bounce,
horizontal movement, roll and yaw. As for 4-axle vehicle, each has 31 units of degree of freedom and for 6-axle one,
each has 35 units. For vehicle computation model, refer to Fig. 1.
Applying D’Alembert principle to the carbody, front and rear bogie and every wheelset, the motion equations of all
components can be obtained, which can be simplified as Eq. (1)
ue = { ye
M eu&&e + FeK2 + FeC2 = 0
ze θ e ϕ e ψ e }
M t u&&t + FtK1 + FtK2 + FtC1 + FtC2 = 0
ut = { yt
zt θ t ϕ t ψ t }
1
1
M wu&&w + FwK
+ FwC
+R+N =0
xw = { y w
zw θ w ψ w }
(1)
M e , M t , M w , xe , ut , uw — generalized mass and generalized displacement of carbody, bogie and wheelset
1
1
1
FtK1 ， FwK
， FtC ， FwC — elastic force and damping force by primary suspension on bogie and wheelset
FeK2 ， FtK2 ， FeC2 ， FtC2 — elastic force and damping force by secondary suspension on bogie and wheelset
y ， z ， θ ， ϕ ，ψ — horizontal movement, vertical movement, roll, pitch and yaw displacement
R ， N — generalized creep force and generalized normal force between wheel and rail
Assuming that the vehicle’s total displacement vector is {X C } , total speed vector X& C , total acceleration vector
{X&& } , then the vehicle’s motion equation can be written in matris form as Eq. (2)
{ }
C
[M C ]{X&&C }+ [CC ]{X& C }+ [KC ]{X C } = {PC }
(2)
Where [M C ] , [CC ] , [K C ] represent separately the vehicle’s mass, damping and stiffness matrix, {PC } represents the
load vector applying to various units of degree of freedom on vehicle.
Yc
ϕc
θc
Zc
Zc
Mc
Mc
k2z
c2z
k1z
c1z
Z ti
k2y
c2y
ϕ ti
ϕ ti
Z ti
k2z
c2z
θti
Y ti
2b2
θwi
Mw
Y wi
k2y,c2y
k1y,c1y
Mt
k1z
Zti
Z wi
c1z
k1y
c1y
2b1
Y ti
ψ ti
Yc
ψc
Y ti
ψti
Figure 1: Vehicle dynamic analysis model
2. Dynamic model and motion equation of track structure During the modeling of track, both left and right rails are
treated as the Euler beam of infinite length on continuous elastic disperse-point-supported foundation, taking into
account the vertical, lateral and torsional vibration of rail, while the sleeper is regarded as rigid body and the linear
spring and viscous damping are used to connect the sleeper with the rail and the sleeper with ballast bed in both vertical
and lateral directions taking into account the Vertical and lateral vibration of sleeper as well as rigid body rotation. The
mass of ballast bed is incorporated into the second stage dead load of bridge for consideration, while the effect on
⎯ 1216 ⎯
sleeper with elastic and damping property of ballast bed is counted into model, see Fig. 2.
mr
-∞←
KSR
EIy
→ +∞
CSR
Ms
KSB
CSB
Mb
KBB
CBB
Figure 2: Dynamic analysis model of track structure
1) Motion Equation of Rail To establish motion equation adopts introducing rail regular vibration mode coordinates
and uses modal analysis method.
Assuming that vibration displacement variable of rail is ur , then its vibration differential equation is:
EI
nw
∂ 4 u r ( x, t )
∂ 2 u r ( x, t ) nw
sr
sr
+
m
+
(
F
+
F
)
δ
(
x
−
x
)
+
( R + N )δ ( x − x wi ) = 0
∑
∑
r
rK
rK
spi
∂x 4
∂x 2
i =1
i =1
ur generalized displacement of rail ur = {zr
(3)
yr θ r }
nsp ， nw — number of sleeper on bridge, number of wheelset on bridge
xspi ， xwi ٛ x coordinate of the i sleeper and the i wheelset
FrKsr ， FrCsr — elastic force and damping force on rail by sleeper
mr — rail mass in unit length
EI — flexural rigidity of rail
When introducing the track regular vibration mode coordinates qK(t) , then the displacement of rail can be written in
the following form:
NM
ur ( x, t ) = ∑ ur ( x ) qk (t )
(4)
k =1
where NM is mode number intercepted, Convergence modular number can be taken according to the computed length
of rail [4].
2) Motion Equation of Sleeper As sleeper is regarded as rigid body and linear spring and viscous damping are used to
connect the sleeper with the rail and the sleeper with the ballast bed in both vertical and lateral directions, thus, the
vertical and lateral motions and rotation of the rigid body are considered for sleeper.The reduced form of motion
equation of sleeper is:
M sU&&s + FsKrs + FsCrs + FsKrB + FsCrB = 0
(5)
M s ， us — sleeper mass, generalized displacement of sleeper us = {zs
ys θ s }
FsKrs ， FsCrs — elastic force and damping force on sleeper by rail
FsKsB ， FsCsB — elastic force and damping force on sleeper by bridge
So far, motion equations of all parts of track─rail and sleeper have been established separately, when we make
⎯ 1217 ⎯
{ }
simultaneous solution, if we assume that total displacement vector of track is {X S } , total speed vector X& S , and total
{ }
&& , then motion equation of track can be written as matrix.
acceleration vector X
S
[M S ]{X&& S }+ [CS ]{X& S }+ [K S ]{X S } = {PS }
(6)
where [M S ] [C S ] and [K S ] represent respectively the mass, damping and stiffness matrix, {PS } represents the load
,
vector on every degree of freedom of track during vibration.
3. Bridge model Bridge uses the member in bending of uniform cross-section beam element as basic elements. The
beam element adopts two-joint space straight beam element with each node including 3 linear displacements and 3
angular displacements and the whole element having 12 units of degree of freedom. If it is assumed that whole
&& ,
displacement vector of bridge structure is as {X B } , whole speed vector as X& B and whole acceleration vector as X
B
then the motion equation of bridge can be written as
{ }
Bs
Bs
M B X&& B + CB X& B + K B X B + FBK
+ FBC
=0
{ }
(7)
M B ， CB ， K B — mass matrix, damping matrix and stiffness matrix of bridge
FsKBs ， FsCBs — elastic force and damping force on bridge by sleeper
X B — generalized displacement vector of bridge
In calculation of train-track-bridge coupling vibration, in order to achieve adequate vibration of bridge, the bridge
model establishes 40m 9-span single supported beam.
WHEEL-RAIL INTERACTION RELATIONSHIP
The coupling relationship between vehicle system and track system is reflected as wheel-rail interaction, whose core is
to determine wheel-rail interaction force by applying wheel-rail rolling contact theory.
The component of wheel-rail normal contact force is determined by famous Hertz nonlinear elastic contact theory [4].
The computation should be conducted for left and right wheel separately and wheel-rail elastic compression volume
should include two parts such as the wheel static compression volume and wheel-rail relative motion displacement. If
the wheel-rail elastic compression volume equals to 0, then it means that wheel jumps off the rail and the normal
contact force is 0.
As for the computation of wheel-rail tangent creep force, the paper determines firstly the creep force by applying
Kalker linear theory and then makes nonlinear modification with Johnson method [4].
VIBRATION SOURCE AND SOLUTION OF TRAIN-TRACK-BRIDGE COUPLING VIBRATION
EQUATION
In accordance with the research achievements of locomotive & rolling stock dynamics, it is held that wheel-rail
excitation is the source for causing vibration of vehicle and track system. As the most important excitation source of
wheel-rail system, track irregularity time domain sample generated from German high speed low disturbance track
spectra is used in calculation. The wavelength range is from 1 to 80 m.
This paper establishes and solves motion equations of train, track and bridge and uses the iteration process to meet the
requirements of geometric compatibility and interaction force equilibrium. The coordination and uniformity for
displacement of train, track and bridge are served as the convergence requirement when conducting the iteration
solution of the vibration differential equation set.
TRAIN-TRACK-BRIDGE COUPLING ANALYSIS UNDER NON-UNIFORM PIER SETTLEMENT
Before vibration analysis, Rail- ballast bed- bridge model are established firstly, and static force method is used to
calculate additional deformation when non-uniform settlement between adjacent abutments occurs, then this additional
deformation will be regarded as a kind of track irregularity, which, together with original track irregularity, functions as
vibration source of train-track-bridge coupling analysis[3].
Fig. 3 shows additional vertical deformation of rail on a 40m single supported bridge when non-uniform pier
settlement are 5 mm, 10 mm, 15 mm, 20 mm, 25 mm and 30 mm respectively.
⎯ 1218 ⎯
This paper conducts calculation on train-track-bridge coupling vibration when Pioneer train passes bridge at 160, 180,
200 and 220km·h-1 separately. Because the pier settlement only influences the vertical dynamic response of train and
bridge in vibration calculation, the lateral response isn’t given in the paper. The result of vertical response listed in
Table 1 and 2.
Table 1 Vertical vibration response of train
Locomotive
Car
Pier
settelnent Train speed
/km·h-1 Rate of wheel Acceleration Sperling Rate of wheel Acceleration
/mm
index load reduction
load reduction
/m/s2
/m/s2
0
5
10
15
20
25
30
Sperling
index
160
0.333
0.906
2.414
0.325
0.806
2.347
180
0.354
0.981
2.507
0.345
0.776
2.391
200
0.369
1.049
2.548
0.366
0.810
2.403
220
0.383
1.130
2.625
0.388
0.804
2.405
160
0.393
0.906
2.406
0.385
0.802
2.364
180
0.414
0.982
2.468
0.405
0.798
2.412
200
0.429
1.055
2.533
0.426
0.824
2.420
220
0.445
1.130
2.629
0.448
0.803
2.423
160
0.433
0.906
2.415
0.445
0.798
2.373
180
0.454
0.983
2.461
0.468
0.820
2.406
200
0.469
1.061
2.561
0.487
0.837
2.426
220
0.486
1.130
2.607
0.504
0.809
2.437
160
0.511
0.951
2.411
0.543
0.829
2.377
180
0.528
0.984
2.490
0.564
0.843
2.425
200
0.546
1.067
2.549
0.583
0.851
2.451
220
0.566
1.130
2.620
0.598
0.908
2.431
160
0.597
0.989
2.456
0.63
0.888
2.393
180
0.615
1.051
2.508
0.651
0.913
2.427
200
0.631
1.097
2.530
0.668
0.943
2.428
220
0.652
1.155
2.611
0.683
1.007
2.426
160
0.717
1.002
2.411
0.753
0.894
2.378
180
0.736
1.099
2.494
0.772
0.901
2.411
200
0.752
1.142
2.541
0.789
0.932
2.428
220
0.773
1.231
2.632
0.802
0.954
2.451
160
0.957
1.063
2.456
0.996
1.007
2.406
180
0.977
1.165
2.519
1
1.070
2.433
200
1
1.211
2.560
1
1.144
2.452
220
1
1.312
2.631
1
1.231
2.461
⎯ 1219 ⎯
\mm
Displacement of rail
0
-5
-10
-15
-20
-25
-30
settlement 5mm
settlement 15mm
settlement 25mm
-35
settlement 10mm
settlement 20mm
settlement 30mm
Figure 3: Displacements of rail with different pier settlement values
Table 2 Vertical displacements and accelerations of mid-span of bridge
Pier
settelnent
Train speed Displacement Acceleration Pier settelnent
/mm
0
10
20
30
/km·h-1
/mm
/m/s2
160
1.0789
0.111
180
1.1724
0.157
200
1.3550
0.283
220
1.1254
160
Train
speed
displacement Acceleration
/mm
/m/s2
160
1.0789
0.111
180
1.1724
0.157
200
1.3550
0.283
0.166
220
1.1237
0.166
1.0789
0.111
160
1.0789
0.114
180
1.1724
0.159
180
1.1725
0.171
200
1.3550
0.283
200
1.3550
0.296
220
1.1220
0.167
220
1.1203
0.178
160
1.0789
0.140
160
1.0790
0.167
180
1.1725
0.183
180
1.1725
0.201
200
1.3550
0.316
200
1.3550
0.335
220
1.1186
0.188
220
1.1177
0.202
160
1.082
0.194
180
1.176
0.223
200
1.359
0.355
220
1.123
0.218
/mm
5
15
25
/km·h-1
CONCLUSIONS
(1) When Pioneer train passes the bridge at 160-220km·h−1 and pier settlement is no more than 15mm, maximum rate
of wheel load reduction less than 0.6 [6], and vertical acceleration of carbody less than 0.13g [5], it meet the
requirements according to relevant code ,thus safety of train running could be guaranteed. When pier settlements
become 20-30mm, maximum rate of wheel load reduction is larger than specified first limit value by 0.65 [6],
maximum vertical acceleration of carbody exceeds by 0.13g, failing to meet safe running requirements.
(2) When Pioneer train passes the bridge at 160-220 km·h−1 the vertical passenger comfort under each pier settlement
reaches the excellent limit (the excellent stability coefficient limit being 2.5) [6], therefore, it is concluded that the
train’s stability coefficient would not be changed when pier settlement is less than 30mm.
⎯ 1220 ⎯
(3) When Pioneer train passes bridge at 160~220 km·h-1, the maximum ratio of deflection-to-span under every
condition is 1/6696, less than the specified limit value [1]. Except condition of 30mm settlement, maximum vertical
bridge vibration acceleration under each settlement is less than specified 0.35g [7], meeting the desired value of “code
for rating existing railway bridges”.
(4) In conclusion, when pier settlement on a 40m single supported bridge varies among 0-15 mm, dynamic response
values of train and bridge meet requirements. When pier settlement is up to 20mm, vibration acceleration of bridge
does not exceed limit value, bridge’s dynamic response meets requirements, however, rate of wheel load reduction
exceeds limit value, for safety of train running could not reach a reliable guarantee. When pier settlement is 30 mm and
especially train’s speed over 180 km·h-1, both vibration acceleration of bridge and rate of wheel load reduction exceed
limit value, at the same time the wheel-rail deviations occur, dynamic response of train and bridge could not meet
requirements.
REFERENCES
1. Ministry-Issued Standard of PR China, 2005. Provisional Specifications for Design of Jinghu Railway. Tie Jian
She [2004] No. 157 (in Chinese).
2. Gao Mangmang. Studies on Train-Track-Bridge Coupling Vibratiion And Runnablility of Train on High-Speed
Railway Bridges. Doctor Thesis of China Academy of Railway Science, 2001,(in Chinese).
3. Zhang Geming. Vehicle-Track-Bridge System Dynamic Analysis Model And Track Irregularities Control on
Quasi & High-speed Railway. Doctor Thesis of China Academy of Railway Science, 2001 (in Chinese).
4. Zhai Wanming. Vehicle-Track Coupling Dynamics. China Railway Publishing House, 2002 (in Chinese).
5. Wu Wangqing, A study on suggested values for track irregularity management criteria on 300 km/h
comprehensive experimental section oOf Qinshen passenger dedicated railway. Railway Standard Design, 2003;
(4): 1-3 (in Chinese).
6. State Standard of PR China,1985. Dynamic Performance Evaluation and Test Identification Specification for
Railway Vehicle GB 5599-85. China Railway Press, Beijing.
7. Ministry-Issued Standard of PR China, 2004. Code for Rating Railway Bridges. Tie Yun Han [2004] No.120,
China Railway Press (in Chinese).
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