Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
Analysis and Comparison on Dynamic Characteristics of the Bridge
Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
College of Civil and Architecture Engineering, Northeast Petroleum University, China
Heilongjiang Key Laboratory of Disaster Prevention, Mitigation and Protection Engineering,
Daqing, China, jijing1977@163.com
Abstract
The deformation of bridge subjected to vehicles with different velocities is very complicated, and
some attention has been paid to it in engineering community. On the basis of typical theory on
vibration analysis between bridge and vehicles, finite element model of bridge is established by ANSYS
software. Through the numerical simulation analysis dynamic response characteristics of the bridge
body are acquired when the vehicle passes through the bridge at different speeds and different
frequents, and inner force of bridge is gotten. By comparison the maximum displacement is 1.64 times
than the static displacement, thus it is clear that the dynamic response of the bridge under the moving
load must be considered. These will provide reference for improving the vibration control measures of
bridge under moving loads.
Keywords: Moving Load, Simple Supported Beam, Dynamic Response, ANSYS, Harmonic load
1. Introduction
In general the forced vibration discussed in structural mechanics refers to the vibration problem of
the structure under cyclical disturbing force at fixed location [1-5]. The resonance will occur when the
disturbance frequency is equal to the natural frequency of the system. If forced vibration of bridge
under the load of moving vehicles is studied, resonance condition need also be analyzed. The
difference is that the dynamic characteristics of the bridge-vehicle coupling system change constantly
with the movement of the load location because the load can move and the vehicle is also a vibration
system with the quality. The bridge will vibrate under moving loads, and deformation and stress which
the bridge produces are bigger than those of the bridge under the static load, this kind of dynamic effect
of the moving load can not be ignored. If the resonance conditions are met at the position of the most
disadvantageous static load, a greater dynamic response will happen, and lead to the destruction of
bridge.
When steam locomotives pass through the railway bridge, the hammering force produced by the
imbalance weight of the drive wheels is a moving harmonic force. Vehicles pass through the bridge
with natural vibration frequency after Cars suffer the incentive of the road surface roughness at the
bridge, the inertia force of the vehicles is also a harmonic force [1]. So it is very meaningful for
discussing the forced vibration caused by the moving harmonic force for bridge. On the basis of typical
theory on vibration analysis of bridges and vehicles, by using the ANSYS software, the character of
dynamic response of bridge is analyzed when constant loads and harmonic Loads with different
frequents pass through the bridge, the characteristics of the dynamic response of the bridge body are
acquired through numerical simulation when vehicles pass through the bridge with different speeds,
these research can provide reference for the improvement of vibration control measures of the bridge
under moving load.
2. The classical theory on vibration analysis of the Bridge
2.1. Constant loads
The constant force F moves towards the right with a constant speed on the simple supported bridge
as shown in Figure 1, here the quality of the moving load is ignored. When the time is equal to 0, F is
located in the left supporting place, and when the time is equal to T, F moves to the right supporting
Journal of Convergence Information Technology(JCIT)
Volume7, Number8, May 2012
doi:10.4156/jcit.vol7.issue8.18
159
Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
place, according to the vibration analysis, the expression of vibration equation of the bridge is as
follows:
v
vt
F
m, EI
X
L
Y
Figure 1. The constant force through the simple supported bridge with constant speed
4 y
2 y
 m 2  F ( x, t )
4
x
x
Where EI stands for the bending stiffness of the bridge and m is the quality of unit length. Assume
that the dynamic displacement of the forced vibrations y(x, t) can be expressed as series form of
vibration mode:
EI
N
y ( x, t )   An (t ) n ( x)
N 1
The forced vibration equation can be gotten by putting the formula (2) into the formula (1) and the
orthogonality of vibration mode. Using the standardization of vibration mode, the simplified vibration
equation of bridge under the moving constant force with a constant speed is:
   2 A  2 F sin n vt ( n  1, 2,...N )
A
n
n n
ml
l
Vibration mode of the simple supported beam bridge is expressed for equation (4), so the expression
of the dynamic response is shown as equation (5).
n x
 n ( x)  sin(
)
l
2F N
1

n x
y ( x, t ) 
(sin nt  n sin nt )sin

ml N 1 n2  n2
l
n
Where ωn is each order natural frequency of the simple supported beam and Ωn=mπv/l is the
generalized flexible dynamic frequency of the moving constant force.
2.2. Harmonic forces
When the harmonic force passes through the simple supported beam at a constant speed, the
dynamic response expression is:
n   p


F N 
1
y ( x, t )  1   2
sin nt  
sin( n   p )t 
n
ml N 1  n  ( p   n ) 2 

  n

 
1
n x
sin( p   n )t  p
sin nt   sin
2 
l
  ( p   n ) 
n
 
2
n
Where ωn is each order natural frequency of the simple beam, Ωn is each order generalized
frequency related with the moving speed, and Ωp is disturbance frequency of the harmonic load.
When only the resonance of fundamental vibration mode is considered, resonance will occur at Ωp=
ω1 , and the maximum dynamic response will appear in the time that harmonic force leaves bridge
spans , that is t=l/v, at this time dynamic response is expressed as follows:
2 F1
x
l
sin
sin 1
l
v
1m v
Maximum mid-span deflection will occur in sin(l / v)  1 . At this point the dynamic magnification
y ( x, t ) 
factor is
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu

1
T
2 c
T1
1
Where Tc is the required time of the harmonic force through the whole beam. From equation 2 we
can see, for moving harmonic force, resonance occurs in Ωp= ω1 , the dynamic magnification factor
will depend on the speed, the slower the speed is, the more time is through the beam and the greater the
vibration response is.
3. Modern theory on vibration analysis of bridge and vehicles
Owing to the introduction and extensive application of the computer and the finite element method,
since the 1970s, the main features of modern theory on vibration analysis of vehicles are: considering
vehicle model that can be closer to the truth and idealizing bridge for the finite element or finite item
model with more quality. The main theories include multi-axial vehicle model, the applications of
finite item method and modal analytical method. This paper analyzes the dynamic response of the
bridge by finite element method.
4. Numerical simulations under constant loads
4.1. The finite element model
A car passes through a single-span bridge at a constant speed as shown in Figure 1. Two kinds of
assumptions are given for the load which the car puts on the bridge: one is that moving car can be
simplified for the constant force moving at a constant speed without quality and the other is the singlespan bridge is simplified for simple supported beam. Several parameters of the bridge are shown in
Table 1.
Table 1. Several parameters of simple supported beam
Elastic modulus
(N/m2)
Beam length
(m)
Area of section
(m2)
Height
(m)
Wheel spacing
(m)
2.06×1011
32
0.1
0.1
2.56
The constant force is 2000N and the speed of moving load is selected for 120 km/h, 80 km/h and 40
km/h. The type of element is defined for two-dimension BEAM3 beam element [6].Through meshing
the bridge is divided into 100 elements and 101 nodes. Transient analysis of finite element model is
carried out, and FULL method is selected to solve it.
4.2. Displacement response of the bridge
Displacement(m)
0.04
midspan
0
-0.04
-0.08
-0.12
-0.16
0.0
0.3
0.6
0.9
1.2
Time(s)
Figure2. The displacement-time curves for 120km/h Figure3. The deformation of beam for 120 km/h
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
0.05
Midspan
Displacement(m)
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
0.0
0.3
0.6
0.9 1.2
Time(s)
1.5
1.8
Figure4. The displacement-time curves for 80km/h
Figure5.The deformation of beam for 80 km/h
0.1
Displacement(m)
Midspan
0
-0.1
-0.2
-0.3
-0.4
0.0 0.5
1.0 1.5 2.0 2.5
Time(s)
3.0 3.5
Figure 6. The displacement-time curves for 40km/h Figure 7. The deformation of beam for 40 km/h
Nodal DOF result [7] is chosen when time-history post processing is used for analyzing the results.
Displacement of the 51st node in the midspan is chosen as Y axis, and time is chosen as horizontal axis,
thus displacement-time relationship curves of the node in the midspan can be drawn as shown in Figure
2- Figure 7 when the car passes through the bridge with a speed of 120km/h, 80km/h and 40km/h. The
deformation of simple supported beam is also shown in Figure 2- Figure 7 when the displacement of
the mid-span node reaches the maximum value.
We can be seen from the figures, when the constant force moves at a uniform speed of 120km/h、
80km/h and 40km/h, the maximum dynamic deflection at the midspan of simple supported beam is 0.147m, -0.257m and -0.320m respectively. It shows that the maximum dynamic deflection at the
midspan of beam gradually reduces with the increasing of the speed, and the maximum deflection in
the midspan gradually moves towards the back [2, 8]. When moving speed is larger, the maximum
dynamic deflection at the midspan of beam will happen in the moment which the moving load leaves
from the beam.
4.3. Speed response of bridge
When the constant force moves at the speed of 120km/h、80km/h and 40km/h, the speed-time
curves of the midspan node are shown in Figure8.Along with the increasing of vehicle speed, the first
negative peak on the velocity response of the mid-span node of simple supported beam will be more
and more in advance, the velocity response curves gradually tend to sine curves.
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
0.1
0.1
M idspan
Midspan
0
Speed(m/s)
Speed(m/s)
0
-0.1
-0.2
-0.3
-0.1
-0.2
-0.3
-0.4
-0.4
-0.5
0.0
0.3
0.6
0.9
-0.5
0.0 0.3 0.6 0.9 1.2 1.5 1.8
1.2
Time(s)
Time(s)
(a)120km/h
(b) 80km/h
0.8
M idspan
Speed(m/s)
0.6
0.4
0.2
0
-0.2
-0.4
0.0
0.5
1.0
1.5 2.0
Time(s)
2.5
3.0
3.5
(c) 40km/h
Figure8. The speed-time relationship curves for 120km/h, 80km/h and 40km/h
M idspan
Acceleration(m/s 2)
4
3
2
1
0
-1
-2
-3
-4
0.0
0.3
0.6
0.9
1.2
4
3
2
1
0
-1
-2
-3
-4
0.0
M idspan
0.3
0.6
0.9
1.2
1.5
1.8
Time(s)
Time(s)
(a) 120km/h
(b) 80km/h
2
Acceleration(m/s 2)
Acceleration(m/s 2)
Under the moving load the peak on the speed response of the mid-span node of simple supported
beam occurs in the front moment of the displacement peak, that is the displacement peak at the
midspan node has a lagging effect in comparison with the speed peak. The acceleration-time
relationship curves for 120km/h, 80km/h and 40 km/h are shown in Figure 9.We can see from Figure 9
the acceleration of the bridge becomes smaller in negative direction with the increasing of speed and
there is on regulation in positive direction. Overall acceleration curves show sinusoidal variation.
Midspan
1.5
1
0.5
0
-0.5
-1
0.0
0.7
1.4
2.1
2.8
3.5
Time(s)
(c) 40km/h
Figure9.The acceleration -time relationship curves for 120km/h, 80km/h and 40km/h
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
5. Numerical simulation under harmonic loads
5.1. Finite Element Model
A car passes through a single span bridge at a constant speed as shown in Figure 10. When t is equal
to 0, Fcosωt locates at the left position of bridge. When t is equal to T, Fcosωt moves to the right
support. Single span bridge is simplified for simple supported beam, parameters of bridge are listed in
Table 1.
The function of harmonic load is 1000cosωt, speed of moving load is selected for 120 km/h, 80
km/h and 40 km/h, vibration frequency of harmonic force is taken as 5, 10 and 20. The type of element
is defined for two-dimension BEAM3 beam element [2].Through meshing the bridge is divided into
100 elements and 101 nodes. Transient analysis of finite element model is carried out, and FULL
method is selected to solve it.
vt
Fcosωt
X
L
v
m, EI
Y
Figure10. Simplified model of vehicle and bridge
5.2. Displacement response of the bridge
Displacement （m）
0.1
0.08
0.06
W=5
W=10
W=20
0.04
0.02
0
-0.02
0.0
0.3
0.6
Time（s）
0.9
1.2
Figure 11. The relationship curve of mid-span node
Nodal DOF result [3] is chosen when time-history post processing is used for analyzing the results.
Displacement of 51 node in the midspan is chosen as Y axis, and time is chosen as horizontal axis, thus
displacement-time relationship curves of the node in the midspan can be drawn under different
excitation frequencies when the car passes through the bridge with a speed of 120km/h , as shown in
Figure 11.
As can be seen from Figure 11, under harmonic force with constant speed, with the increasing of the
excitation frequency of harmonic force, the maximum dynamic deflection in the midspan of beam
gradually decreases .when the excitation frequency of harmonic force is lower, the maximum dynamic
deflection in the midspan of beam happens in the moment that harmonic force is about to leave the
beam. Thus it is clear that when the excitation frequency reduces gradually and closes to the natural
frequency of the beam, the resonance will occur and the displacement response in the moment that the
load leaves beam is the largest.
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
0.03
120
Displacement （m）
Displacement （m）
0.15
0.1
0.05
0
120
80
40
-0.05
-0.1
0.0
1.0
2.0
3.0
80
40
0.02
0.01
0
-0.01
-0.02
0.0
4.0
1.0
Time（s ）
(a) Excitation frequency for 5
2.0
Time（s）
3.0
4.0
(b) Excitation frequency for 10
We can see from figure 12, when the excitation frequency is unchanged, with the increasing of the
vehicle speed, it is not obvious for the variable ruler of displacement in the midspan. But the maximum
dynamic deflection in the midspan of beam gradually decreases with excitation frequency of the
harmonic force increases.
0.015
Displacement （m）
0.01
0.005
0
-0.005
120
80
40
-0.01
-0.015
0.0
1.0
2.0
Time（s ）
3.0
4.0
(c) Excitation frequency for 20
Figure12. The comparison of displacement - time relationship curves
5.3. Speed response of bridge
speed （m/s）
0.3
0.2
0.1
W=5
W=10
W=20
0
-0.1
-0.2
0.0
0.3
0.6
Time（s ）
0.9
1.2
Figure13. The speed-time relationship curves with different excitation frequencies
When a car pass through the bridge with a speed of , the speed-time relationship curves of the midspan node under different excitation frequency are got as shown in Figure 13. With the increasing of
the excitation frequency of harmonic force, the peak speed of the mid-span node of the simple
supported beam gradually decreases, and the first peak of velocity response will appear earlier. By
contrasting, it is known that the maximum speed which the harmonic force results in is smaller than
constant force in the mid-span of beam under constant speed.
5.4. Internal force analysis of the bridge
From the dynamic response analysis of moving load, we can get the shear diagram and bending
moment diagram of the bridge in different speed, as shown from Figure 14 to Figure 19. The maximum
shear value and bending moment value of the bridge under different speed are listed in Table 2.
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
Figure14. Beam shear with a speed of 120km/h
Figure15. Beam moment with a speed of 120km/h
Figure16. Beam shear with a speed of 80km/h
Figure17. Beam moment with a speed of 80km/h
Figure18. Beam shear with a speed of 40km/h
Figure19. Beam moment with a speed of 40km/h
As can be seen from figures, with increasing of the speed, the maximum positive shear value
maintains at rearward position. But the largest negative shear values transfers gradually from the end to
the middle part, which shows that the shear value is an increasing process, and can reach maximum
value when the load leaves the beam. We can see that the greatest negative moment value can reach the
maximum value when the load leaves the beam. The maximum positive bending moment value is the
process that gradually moving from the mid-span part to the end, at the same time，the maximum
positive bending moment value of the beam increases with the increasing of speed.
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Analysis and Comparison on Dynamic Characteristics of the Bridge Subjected to Moving Load Based on ANSYS
Jing Ji, Wenfu Zhang, Wenyan Zhao, Chaoqing Yuan, Yang Yu
Speed
(km/h)
Table 2. Shear and bending moment of the bridge at different speeds
120
80
40
Vmax
Vmin
Vmax
Vmin
Vmax
Vmin
Shear (N)
3484
-2950
3058
-1778
2850
-1602
Moment (N. m)
10286
-13979
9585
-14680
8674
-19071
6. The comparison of results between moving load and static load
The vehicle with a weight of 2000N is divided into two parts and two parts is exerted in the 47th
node and the 54th node respectively, the two nodes immediate is on both sides of the midspan node. By
static analysis, the maximum displacement of the midspan is -0.195m. We can know from Figure 2 to
Figure7, the maximum displacements of the midspan under moving constant forces with different
speeds are -0.147m, -0.257m and -0.320m. when the movement speed of vehicle is faster, the biggest
displacement in the midspan will occur at the moment the vehicle is going to leave from the bridge,
and when the movement speed of vehicle is slower, the biggest displacement will occur at the moment
that the vehicle is near the midspan, at that time, the maximum displacement is 1.64 times than the
static displacement, thus it is clear that the dynamic response of the bridge under the moving load must
be considered.
7. Conclusions
The modern theory by means of computer analysis and the finite element method can be more
realistic to simulate the state of the vehicle and the bridge, the speed and interaction between load and
the vehicle - bridge system. The maximum displacement is 1.64 times than the static displacement,
thus it is clear that the dynamic response of the bridge under the moving load must be considered. The
slower the speed of the vehicle is, the greater the vibration displacement response of the bridge is. For
this reason, the vehicle need be controlled at speed aspect when it reaches the bridge. The driver not
only make it quickly pass through the bridge, but also maintain a certain distance between vehicles.
When the excitation frequency reduces and gradually closes to the natural frequency of the beam, the
resonance will occur and the displacement response happened in the moments that the load leaves
beams is the largest. the frequency generated by a moving vehicle should be largely different from the
natural frequency of the bridge，so resonance can be avoided. When vehicles move on the bridge
under the same speed, the maximum displacement and speed of harmonic force is smaller than those of
a constant force.
8. Acknowledgements
The study described in this paper was supported by the Heilongjiang Provincial Department of
Education Science and technology research project (project number: 12511022) and The National
Natural Science Foundation of China (project number: 51178087).These supports are gratefully
acknowledged.
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