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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Dynamic response analysis of a wind-train-bridge system with wind barriers
T. Zhang1, 2, W. W. Guo1, H. Xia1, Q. P. Hou3,Y. Tian1
School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
2
Institute of Road and Bridge Engineering, Dalian Maritime University, Dalian 116026, China
3
Research Institute of National Defense Communication, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
email: [email protected], [email protected], [email protected], [email protected], [email protected]
1
ABSTRACT: An analysis framework for dynamic interaction of train-bridge system with wind barriers under wind load is
proposed based on the theory of bridge wind engineering and structure dynamics. The wind forces acting on bridge, wind
barriers and train vehicles include steady forces induced by mean wind and unsteady forces induced by fluctuating wind. The
detailed calculation formula of unsteady aerodynamic forces on bridge and vehicles are derived according to wind vibration
theory. The bridge is described by modal superposition method based on three-dimensional finite element model, and the
vehicle is modeled by a multi-rigid-body system connected with a series of springs and dampers. The effect of wind barriers on
the dynamic responses of the train-bridge system is analyzed. By taking a continuous beam bridge on a high-speed railway as an
illustrating example, the dynamic responses of the bridge and the train vehicles subjected to strong wind are calculated in the
time domain when a train vehicle passing the bridge with or without wind barriers and the running safety indices of vehicles are
evaluated.
KEY WORDS: Wind-train-bridge coupled system; Wind field; Wind barriers; Dynamic response.
1
INTRODUCTION
Wind disaster is one of the main natural disasters that affect
the operation safety of railways. In the northwest region of
China, railway lines pass through some strong wind areas,
where the train service is often interrupted by the wind, and
the train overturning accidents occasionally occur [1,2], so
effective measures should be taken to ensure the operation of
running trains. The experience of Japan's Shinkansen shows
that wind barriers can obviously decrease the number of train
stops under strong wind [3, 4], so it is necessary and effective
to set wind barriers on bridges in strong wind field to ensure
the running safety and stability of trains.
The high-speed railway line from Lanzhou to Urumqi in
west China under construction passes through the strong wind
area where the instantaneous wind velocity is up to 60 m/s.
When the high-speed train runs on the bridges in this area, the
wind force may induce the train to overturn or derail as well
as the bridge to vibrate intensely. The dynamic response of the
train will decrease under the protection of wind barrier, while
the aerodynamic force on the bridge will increase to
exacerbate the bridge vibration when the wind barrier is
installed on the bridge. Therefore, the running safety of the
train on the bridge under wind load and the influence of the
wind barrier on the bridge and the train become a vital factor
that should be analyzed in the design of railway bridges.
With the new built Lanzhou-Xinjiang railway as a research
background, this paper research the dynamic problem of a
high-speed train running through a continuous beam bridge
with and without wind barrier subjected to turbulent wind. A
rational analysis framework is established to estimate the
vibration response of the coupled vehicle-bridges system in
the wind field, so as to evaluate the dynamic performance of
the bridge and the running safety of high-speed trains. At the
same time, the windbreak structures are studied to improve
the running safety of the train, to reduce the interruption on
bridge operation when strong wind occurs.
2
2.1
NUMERICAL SIMULATION OF WIND LOAD
Wind velocity field
An autoregressive model can be used to simulate the wind
velocity field which is essentially stochastic time series. m
related turbulent wind time histories v(X, Y, Z, t) = [v1(x1,y1,
z1,t) v2(x2, y2, z2, t) …vm (xm , ym, zm, t)] can be generated by
p
v (X,Y,Z,t )   fk v (X,Y,Z,t  k  Dt )  N (t )
(1)
k 1
where X  [x1 x2  xm ]T , Y  [y1 y2  ym ]T , Z  [z1 z2 zm ]T ,
( xi , yi , z i ) is the coordinate the i-th point, i=1,2,3,…,m; p is
the order of AR model; t is the time interval of wind field
simulation; k is the autoregressive coefficient m  m square
matrix of AR model, k=1,2,…, p; and N(t) is a zero-mean
independent stochastic process with given variance.
For convenience, v(X, Y, Z, t) is written as v(t). Based on
the assumption of wind field simulation and the characteristics
of auto-correlation function expressed by
R(  j  Dt )  E[v (t )  v T (t  j  Dt )]
(2)
R (  j  Dt )  R ( j  Dt )
(3)
the relationship between the correlation function R ( j  t )
and the autoregressive coefficient k is given by
p
R( j  t )    k R[( j  k )  t ]  N (t )  v T (t  j  t )
k 1
(j  1,2,, p )
(4)
1147
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
p
R (0)  
  R ( k  t )  R
k
N
(5)
Dse (t ) 
k 1
where RN is the covariance matrix. Using Equation (4), it
finally gets the autoregressive coefficient matrix k, and then
substituting k into Equation (5), RN can be solved.
When the coefficient matrix k and the covariance matrix
RN of AR model are calculated, the fluctuating wind velocity
time series can be easily generated using Equation (1).
2.2
1
Lst =  u 2CL ( ) B
2
1
Dst =  u 2CD ( ) D
2
1 2
M st =  u CM ( ) B 2
2
(6-a)
(6-b)
(6-c)
where the subscript st represents the static forces; u is the
mean wind velocity; CL(α), CD(α) and CM(α) are, respectively,
the non-dimensional lift, drag and moment coefficients, which
are determined by the structure size and wind attack angle α,
whose values can be measured from wind tunnel tests of
section model; B and D are, respectively, the width and height
of the bridge deck segment.
The buffeting forces per unit length are commonly
expressed in terms of quasi-steady model as follows [5]:
1 2  D
u (t ) 
 u B  2 CD ( )
(7-a)
2
u 
 B
u (t )
D
w( t ) 
1

 [CL ( )  CD ( )]
Lbf (t )   u 2 B  2CL ( )
(7-b)
u
B
u 
2

u (t )
w( t ) 
1

 CM ( )
M bf (t )   u 2 B 2  2CM ( )
(7-c)
u
u 
2

Dbf (t ) 
where the subscript bf represents the buffeting
force; CD  dCD d , CL  dCL d , CM  dCM d ; u(t)
and w(t) are the lateral and vertical components of the
fluctuating wind velocity, respectively.
The self-excited forces per unit length, i.e. lift Lse(t), drag
Dse(t), and moment Mse(t) are commonly described utilizing
flutter derivatives in frequency domain as follows [6]:
Lse (t ) 
h
B
1 2 
 u B  KH1* ( K ) b  KH 2* ( K ) b  K 2 H 3* ( K )b
u
u
2

 K 2 H 4* ( K )
1148
 K 2 P4* ( K )
M se (t ) 
hb
p
p 
 KH 5* ( K ) b  K 2 H 6* ( K ) b 
B
u
B
(8-a)
pb
h
h 
 KP5* ( K ) b  K 2 P6* ( K ) b 
B
u
B
(8-b)
h
B
1 2 2 *
 u B  KA1 ( K ) b  KA2* ( K ) b  K 2 A3* ( K )b
u
u
2

+ K 2 A4* ( K )
Wind load acting on the bridge
The wind forces acting on the bridge include the static force
caused by mean wind, the buffeting force caused by
fluctuating wind, and the self-exciting force caused by the
interaction between the wind and bridge motions. Each
component contains forces from three directions of drag force,
lift force and moment. The lift force Lst, drag force Dst and
moment Mst per unit length caused by mean wind can be
calculated according to the classical airfoil theory:
p
B
1 2  *
 u B  KP1 ( K ) b  KP2* ( K ) b  K 2 P3* ( K )b
u
u
2

hb
p
p 
 KA5* ( K ) b  K 2 A6* ( K ) b 
B
u
B
(8-c)
where the subscript se represents the self-excited wind forces;
H i* , Pi * and Ai* (i=1,…,6) are frequency dependent flutter
derivatives from wind tunnel tests; K  B u is reduced
frequency;  is the circular frequency of vibration; hb, pb, and
b are vertical, lateral, and torsional displacement of bridge,
respectively.
2.3
Wind load acting on the train
Similar to the bridge, wind forces acting on a train in crosswind field can be divided into two parts, i.e. steady
aerodynamic forces induced by the mean wind velocity
component of natural wind and unsteady aerodynamic forces
induced by the fluctuating wind velocity component [7, 8].
The wind forces acting on the bogie and wheel-sets of the
vehicle are neglected because of their small windward area,
thus only wind forces acting on the car-body are taken into
account, which mainly refer to side force FS, lift FL and
rolling moment M with respect to the mass center of the car
body. The wind forces acting on the car-body can be given [9]:

w
1
1
2uu
FS =  AVR2 CFS ( )+  AVR2 CFS ( )  2  CFS ( )   (9-a)
VR
u
2
2


w
1
1
2uu
FL =  AVR2 CFL ( )+  AVR2 CFL ( )  2  CFL ( )   (9-b)
VR
u
2
2

M 

w
1
1
2uu
 ( )  
 AVR2 HCMv ( )+  AVR2 H CMv ( )  2  CMv
V
u
2
2


R
(9-c)
where VR2  VT2  u 2 , VR is the wind velocity relative to the
train, VT is the train speed; A and H are the reference area and
height of the vehicle, respectively; Ci ( ) and Ci( )
( i  FS , FL , M v ) are the side force, lift force and moment
coefficients of the vehicle and their first order derivatives at
the wind attack  =0 . In the right side of Equation (9), the
first term represents the steady aerodynamic force, while the
other two terms represent the unsteady, respectively.
It can be seen from Equation (9) that the fluctuating
components of wind velocity field u and w should be obtained
in order to decide the unsteady aerodynamic forces. In
addition, the aerodynamic coefficients of the vehicle and their
first order derivations also should be provided. Of course, the
turbulent wind velocities can be simulated at a series of points
along a longitudinal line passing through the mass center of
the car-body by the method in Section 2.1.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Modeling of train
In this paper, a 4-axle vehicle with two suspension systems is
taken as an example to demonstrate the modeling of the
vehicle. To simplify the analysis but with enough accuracy,
the five assumptions which are described in detail in Ref. [10]
are used in the modeling of the vehicle subsystem.
Both the car-body and each bogie have five degrees-offreedom, including the floating, the lateral movement, the
rolling, the yawing, and the pitching. Each wheel-set has three
degrees-of-freedom, including the floating, the lateral
movement, and the rolling with respect to its mass center. As
a result, the total degrees-of-freedom of the vehicle are 27.
Then the vehicle and the bridge are associated with the
supposed wheel-rail relation and take the track irregularity as
the excitation source.
3.2
Modeling of bridge
The bridge is composed of girders, piers, abutments, deck
system and track system. Due to the complexity of coupled
components, suppose the girder and the track have no relative
deformation and the elastic deformation of the track system is
also neglected. Thus based on the finite element fundamentals,
the bridge is discretized as a three-dimensional finite element
model. By applying the modal comprehension analysis
technique where the generalized coordinates of bridge
vibration modes are solved rather than the motion equations of
the bridge directly, the total number of the degrees-of-freedom
of the system is significantly reduced and the coupled
equations of motion are efficiently solved. Detailed
formulations can be found in Ref. [11].
3.3
Wind-train-bridge system considering wind barriers
Based on the previous model of the vehicle, bridge and wind
field, the wind-train-bridge dynamic interaction analysis
model is derived considering the effect of the wind barrier.
Since the stiffness of wind barrier on is small relative to the
bridge, and only the stand column of the wind barrier is
connected with the bridge, it is not considered that the wind
barrier influence on the bridge modes. So only the effect of
the wind barrier on the wind forces acting on the bridge and
the train is considered to calculate the dynamic response of
the system.
The motion equations of train-bridge coupling system under
wind load can be expressed as follows:
M vv
 0

  Cvv
0   X
v
   + 

M bb   X b  Cbv
K
  vv
 K bv
 
Cvb   X
v
 

Cbb   X b 
K vb   X v  Fv0   Fvst +Fvust 
    

K bb   X b  Fb0  F bst +F bbf +F bse 
acceleration, velocity and displacement vectors of the train
and the generalized coordinate vector of the bridge, its first
order derivation, its second order derivation, respectively; Fv0
and Fb0 are the force vectors due to the train-bridge interaction
through the track and wheels under the moving train,
respectively; F bst , F bbf , and F bse are the modal static wind force
vector, buffeting force vector and the self-excited force vector
of the bridge, respectively; Fvst and Fvust are the steady force
vector and unsteady force vector of the vehicle. Note that
different aerodynamic coefficients are adopted with or without
wind barriers when calculating the aerodynamic forces on the
bridge and the train.
CASE STUDY
4
4.1
System input data
During the design of the Lanzhou-Xinjiang high-speed
railway, the wind-train-bridge calculation program was used
to calculate the dynamic response of the bridge and the
running safety indices of the train traveling on the bridge with
or without wind barriers when the wind is normal to the
motion of the vehicle at the level of the vehicle mass center.
In the wind prone region of the Lanzhou-Xinjiang railway,
the whole bridge is installed with the single-side 4 m height
wind barrier, as shown in Figure 1.
400
3.1
DYNAMIC MODEL OF WIND-TRAIN-BRIDGE
SYSTEM WITH WIND BARRIERS
30
1340/2
1340/2
305
3
Figure 1. Single-side wind barriers on the bridge (Unit: cm)
The bridge concerned is composed of 5-span continuous
PC beams with the length of 40+3×64+40 m, whose finite
element model is shown in Figure 2. The natural vibration
characteristics of the bridge are analyzed to obtain the
frequency and the vibration modes. The range of the first 60
order natural vibration frequency is 0.437 Hz~32.01 Hz.
(10)
where: the subscripts v and b represent the train and bridge,
respectively; Mvv, Cvv and Kvv are the mass, damping and
stiffness matrices of the train, respectively; Mbb, Cbb and Kbb
are the mass, damping and stiffness matrices of the bridge,
respectively; Kvb and Kbv , Cvb and Cbv are the stiffness and
damping matrices due to the interaction between the bridge
 , X
 and X , X
 , X
 are the
and the train; X v , X
b
v
v
b
b
Figure 2. Finite element model of the continuous beam bridge
The train in the case study is the ICE train in Germany,
composed of 2×(3M+1T), where M represents the motor-car
and T the trailer-car. The height and width of the car-body are
3.5 m and 2.7 m. The average static axle loads are 160 kN for
a motor-car and 146 kN for a trailer car. The other parameters
of the ICE train can be found in Ref. [5]. All the vehicles are
the same in a train and the wind is normal to the motion of the
1149
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 1 Aerodynamic coefficient
CM
Bridge
1.09
0.47
0.15
Vehicle
1.37 0.052 0.82
Bridge
2.03 -0.13 0.24 0.057 0.13
Vehicle
0.33 0.094 0.21
4.2
0.74 -0.40
---
2
15
10
5
0
-5
-10
-15
--
Bridge response
Shown in Figure 3 are the maximum displacements of the
bridge with and without wind barriers under different mean
wind velocities when the train speed is 200 km/h.
0
2
acceleration [cm/s ]
0.10
0.05
0.00
5
10
15
20
25
30
35
mean wind velocity [m/s]
5
10
2
acceleration [cm/s ]
vertical displacement [cm]
without wind barrier
with wind barrier
0.25
0.24
0.23
0.22
5
10 15 20 25 30
mean wind velocity [m/s]
35
Figure 3. Maximum displacement of bridge vs. wind velocity
at VT=200 km/h
1150
15
time [s]
20
30
25
30
u=0 m/s
5
0
-5
-10
5
10
0.27
0
25
10
0
0.26
15
20
time [s]
(a) Lateral acceleration
0.15
0
10
u=30 m/s
without wind barrier
with wind barrier
0.20
5
15
10
5
0
-5
-10
-15
0.25
lateral displacement [cm]
u=0 m/s
0
--
2
With 4m wind
barriers
CM
CL
acceleration [cm/s ]
Without wind
barriers
CL
CD
Item
It can be seen that wind force has obvious influence on the
lateral displacements of the bridge, which increase with the
wind velocity significantly, while it has relatively little effect
on the vertical displacements. At the same time, the lateral
displacements of the bridge with wind barrier are larger than
those without wind barrier, while the vertical displacements of
the bridge with wind velocity are the opposite.
For the bridge without wind barrier, Figure 4 shows the
time histories of lateral and vertical accelerations of the bridge
when the train runs at 200 km/h on the third span whether or
not considering the wind forces.
acceleration [cm/s ]
vehicle at the level of the vehicle mass center. The train runs
on a straight track at a constant speed.
The track irregularities are generated by harmonic synthetic
method on the basis of the German PSD functions of rail
irregularities for high-speed railways, whose detailed
expressions can be found in Ref. [5]. The length of the
simulated data is 2000 m with the maximum amplitude being
4.20 mm in the lateral direction, 5.80 mm in the vertical
direction and 0.002 rad in the torsional direction. Then the
wind velocity time series are simulated according to the given
PSD functions in Ref. [12] adopting the method in Section 2.1.
In the addition, the aerodynamic coefficient of the bridge and
the vehicle with respect to wind angle at the zero wind angle
of attack also should be given to calculate the wind forces by
the wind tunnel test, listed in Table 1.
15
10
5
0
-5
-10
-15
15
20
time [s]
25
30
u=30 m/s
0
5
10
15
20
time [s]
25
30
(b) Vertical acceleration
Figure 4. Mid-span acceleration time histories of the third
span of the bridge at VT=200 km/h
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 2 Maximum accelerations of the bridge at the third span
15
20
25
30
without wind barrier
with wind barrier
offload factor
0.40
0.30
0.20
0.10
0.00
0
5
10 15 20 25 30
mean wind velocity [m/s]
35
0.50
without wind barrier
with wind barrier
0.40
0.30
0.20
11.90 12.10 11.90 12.00 12.30
0.10
11.80 12.00 11.90 11.50 12.80
0
5
8.10 8.40 8.62 7.54 9.68
Vehicle response
The maximum car-body accelerations as the train runs at 200
km/h are listed in Table 3.
Table 3 Maximum car-body accelerations of the train
Mean wind velocity
5
10
15
20
25
30
/(m/s)
Without
41.14 43.07 52.32 60.60 82.55 110.39
Lateral/ wind barriers
(cm/s2) With wind
41.22 40.65 41.30 43.03 46.56 48.31
barriers
Without
59.53 59.51 59.57 59.57 58.94 59.24
Vertical/ wind barriers
2
(cm/s ) With wind
59.49 59.45 59.58 59.59 58.34 61.09
barriers
It can be seen from the table that the lateral car-body
accelerations increase obviously with the wind velocity
without installing wind barriers, while increase slowly with
wind barriers. When the mean wind velocity is 30 m/s, the
maximum lateral car-body accelerations are 48.31 cm/s2 and
110.39 cm/s2 with and without wind barriers, respectively.
However, the vertical car-body acceleration has no big change
with the increase of wind velocity.
The running safety indices of the train such as offload
factors, derailment factors, overturning factors and lateral
10 15 20 25 30
mean wind velocity [m/s]
35
2.50
7.08 8.04 8.14 8.02 8.67
overturning factor
4.3
10
0.50
without wind barrier
with wind barrier
2.00
1.50
1.00
0.50
0.00
0
lateral force of wheel-set [kN]
Mean wind velocity
5
/(m/s)
Without
11.60
Lateral/ wind barriers
2
(cm/s ) With wind
11.80
barriers
Without
8.18
Vertical/ wind barriers
2
(cm/s ) With wind
8.19
barriers
forces of the wheel-set are shown in Figure 5, as the mean
wind velocity from 0 m/s to 30 m/s at VT=200 km/h.
derailment factor
The time histories show that the accelerations reach the
maximum value when the train travels on the bridge, while the
wind force has slight effect, because the accelerations are
mainly caused by the track irregularity and train excitation.
The acceleration decreases rapidly after the train leaves the
span without wind action, while it obviously has turbulent
characteristic with wind action. The maximum lateral and
vertical accelerations of the third span are 11.8 cm/s2 and 8.18
cm/s2 without wind, and they are 12.3 cm/s2 and 9.68 cm/s2
with the mean wind velocity 30 m/s, respectively, indicating
that the wind force has no great influence on the bridge
acceleration.
Furthermore, the bridge acceleration is also calculated with
wind barriers. The results of the mid-span maximum
accelerations of the third span of the bridge are listed in Table
2 when the train runs at 200 km/h. It can be found that the
lateral and vertical accelerations of the bridge changed little
with the wind velocity, and the lateral accelerations are larger
than the vertical ones, indicating that the wind barriers have
little effect on the maximum acceleration of the bridge.
5
50
10 15 20 25 30
mean wind velocity [m/s]
35
without wind barrier
with wind barrier
45
40
35
30
0
5
10
15 20
25
30
mean wind velocity [m/s]
35
Figure 5. Running safety indices of the train
1151
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
As can be seen, the safety indices of the train without wind
barriers are much larger than those with wind barriers,
especially when the wind velocity is high. These indices
increase rapidly with the wind velocity, and the greater the
wind velocity is, the faster the increase growth, particularly
for offload factors and overturning factors. The growth
amplitude is larger without wind barriers than that with wind
barriers. Without wind barriers, the maximum overturning
factor is up to 1.25 when the mean wind velocity reaches 25
m/s, while it is 0.32 with wind barriers. Moreover, these
safety indices change slowly with the wind velocity when
wind barriers are installed. Therefore, the wind barrier can
effectively reduce the dynamic response of the train and
improve the running safety.
5
CONCLUSIONS
This paper studies the dynamic response of a multi-span
continuous beam bridge passing through a high-speed train
under turbulent winds. There are several cases, including
different mean wind velocities, the bridge without and with
wind barriers, to obtain the change rule of the dynamic
response and the effect of the wind barriers. Some conclusions
are summarized as follows:
(1) Wind forces have influence on the displacement
responses of the bridge. The lateral displacement is very small
without wind action but significantly increases with the wind
velocity, and the lateral displacement is larger with wind
barriers than that without wind barriers. However, the vertical
displacement has little change with the wind velocity. Wind
force has no great impact on bridge accelerations whether or
not wind barriers are installed.
(2) For bridge without wind barriers, wind has strong
influence on lateral car-body accelerations, and they increase
rapidly with the wind velocity, while for the bridge with wind
barriers, this influence is very slight, which indicates that the
wind barriers can enhance the passengers comfort level. Wind
force does not produce significant impact on vertical car-body
accelerations whether or not to adopt wind barriers.
(3) The offload factor, derailment factor, overturning factor
and lateral wheel-set force of the train under wind action
increase with wind velocity, and the greater the wind velocity,
the faster the increase rate. The increasing amplitude without
wind barriers is much larger than that with wind barriers,
indicating that the running safety of the train on the bridge can
be improved by adopting the proposed wind barriers.
(4) The calculated results show that wind barriers lead to
reduction of vehicle response, but increase of bridge response,
in general, more advantages than disadvantages.
ACKNOWLEDGMENTS
The research described in this paper is supported by the Major
State Basic Research Development Program of China (“973”
Program: 2013CB036203), and the Fundamental Research
Funds for the Central Universities of China (No.2013YJS054),
the Natural Science Foundation of China (51308034).
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