Analysis of Longitudinal Force of Beam and Rail of Continuously Welded Rails (CWR) on Railway Bridge

Abstract

To study the longitudinal force of CWR on viaduct, a track-bridge-pier finite element model is established. Taking a multi-span simply supported beam with a maximum span of 32.7m of an elevated CWR as an example, the additional expansion and contraction forces, displacement between rail and beam and the force of pier are calculated, and whether the rail stress meets the requirements when setting constant resistance fasteners is checked. The results show that: (1) For the left and right lines, the maximum additional expansion forces of single strand rail are both 211.13kN, and the maximum relative displacements between beam and rail are both 6.572mm. (2) The maximum value of the additional expansion and contraction forces and the relative displacement between beam and rail of the same line occur at the same position. The left line is at ZFZ29 pier and the right line is at ZFS31 pier. (3) The maximum force of pier in this section is 500.80kN, and the pier numbers are ZFZ27 and ZFS29. (4) The rail stress is less than the allowable stress of 352MPa, and the rail strength meets the requirements.

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Journal of Physics: Conference Series
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Analysis of Longitudinal Force of Beam and Rail of Continuously Welded
Rails (CWR) on Railway Bridge
To cite this article: Zhiping Zeng et al 2022 J. Phys.: Conf. Ser. 2148 012065
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ICPEM 2021
Journal of Physics: Conference Series 2148 (2022) 012065
IOP Publishing
doi:10.1088/1742-6596/2148/1/012065
1
Analysis of Longitudinal Force of Beam and Rail of
Continuously Welded Rails (CWR) on Railway Bridge
Zhiping Zeng1, 2, Ji Hu1, Qiang Zeng 1, Zhibin Huang1, 3*,Huatuo Yin4 and Hui
Huang4
1 School of Civil Engineering, Central South University, Changsha, Hunan 410075,
China
2 MOE Key Laboratory of Engineering Structure of Heavy Haul Railway, Central
South University, Changsha, Hunan 410075, China
3 Southeast Coastal Railway (Fujian) Co., Ltd, Fuzhou, Fujian, 350001, China
4 Guangzhou Metro Design & Research Institute Co., Ltd., Guangzhou, Guangdong
510010, China
Email: 1146401456@qq.com
Abstract. To study the longitudinal force of CWR on viaduct, a track-bridge-pier finite
element model is established. Taking a multi-span simply supported beam with a maximum
span of 32.7m of an elevated CWR as an example, the additional expansion and contraction
forces, displacement between rail and beam and the force of pier are calculated, and whether
the rail stress meets the requirements when setting constant resistance fasteners is checked. The
results show that: (1) For the left and right lines, the maximum additional expansion forces of
single strand rail are both 211.13kN, and the maximum relative displacements between beam
and rail are both 6.572mm. (2) The maximum value of the additional expansion and
contraction forces and the relative displacement between beam and rail of the same line occur
at the same position. The left line is at ZFZ29 pier and the right line is at ZFS31 pier. (3) The
maximum force of pier in this section is 500.80kN, and the pier numbers are ZFZ27 and
ZFS29. (4) The rail stress is less than the allowable stress of 352MPa, and the rail strength
meets the requirements.
1. Introduction
Under the influence of train load, temperature change and other factors, the rail and beam will have
longitudinal interaction, produce relative displacement and additional internal force [1-3].
The researches on Continuously Welded Rails (CWR) on bridge mainly study the longitudinal
interaction mechanism between track and bridge and the structural design of track and bridge to ensure
that the track and bridge structure meet the requirements of strength and stability under temperature
and train load [4-5].
In this paper, taking the CWR on a viaduct as an example, the spatial structure model is established
by using the finite element software ANSYS [6-8], and the longitudinal mechanical properties are
calculated and analyzed, which could provide a reference basis for the design of laying CWR on the
bridge.

ICPEM 2021
Journal of Physics: Conference Series 2148 (2022) 012065
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doi:10.1088/1742-6596/2148/1/012065
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2. Track-Bridge-Pier Finite Element Calculation Model
2.1. Model Assumptions and Treatment
The finite element model of beam-rail interaction under temperature change is established. There is no
regulator in the design scheme, and the longitudinal force of CWR is calculated according to the
concrete simply supported beam. In order to eliminate the influence of boundary conditions, 180 m
subgrade sections are set at both ends. The rail and bridge are simulated by beam element respectively,
the vertical characteristic of the fastener is simulated by linear spring element, and the longitudinal
characteristic of the fastener is simulated by nonlinear spring. The spring action direction is to prevent
the displacement of rail relative to sleeper.
2.2. Model Parameters
2.2.1. Bridge Parameters. The finite element model of track-bridge structure is established based on
ANSYS. The ballastless track structure is adopted. The bridge is a simply supported beam bridge, and
the bearings are arranged in the form of ‘fixed-movable', the basic parameters of the bridge are shown
in table 1.
Table 1. Bridge parameters.
line
Pier number
Span(m)
Stiffness(kN/cm)
Bearing
Left line
ZFZ9
24.7
26149
Fixed- Movable
ZFZ10
24.7
2178
Fixed- Movable
ZFZ11
24.7
1918
Fixed- Movable
ZFZ12
24.7
2056
Fixed- Movable
ZFZ13
24.7
2178
Fixed- Movable
ZFZ14
/
28149
/
ZFZ27
32.7
28149
Fixed- Movable
ZFZ28
32.7
1667
Fixed- Movable
ZFZ29
/
28149
/
Right line
ZFS9
24.7
28149
Fixed- Movable
ZFS10
24.7
2878
Fixed- Movable
ZFS11
24.7
2358
Fixed- Movable
ZFS12
24.7
2167
Fixed- Movable
ZFS13
24.7
5649
Fixed- Movable
ZFS14
/
28149
/
ZFS29
32.7
28149
Fixed- Movable
ZFS30
32.7
1667
Fixed- Movable
ZFS31
/
28149
/
2.2.2. Track Structure Parameters
(1) Standard 60kg/m rail is adopted, and the elastic modulus of rail is 2.06×105Mpa, the cross-
sectional area of a single rail is 77.45cm2; The linear expansion coefficient is taken as 1.18×10-5/℃.
(2) SFC type constant resistance fastener is adopted, the fastener spacing is 0.6m, the longitudinal
resistance of fastener is taken as 9kN for each group, and the slip value is 2mm.
(3) The elastic modulus of concrete bridge is 35.5GPa, and the linear expansion coefficient of
concrete beam is 1×10-5 /℃ calculation.

ICPEM 2021
Journal of Physics: Conference Series 2148 (2022) 012065
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(4)According to Code for Design of Railway Continuous Welded Rail (TB 10015-2012) [9], the
maximum rail temperature in this area is 61.7 ℃ and the minimum rail temperature is - 1.7 ℃. The
daily temperature difference of concrete beam is calculated as 30 ℃, the locking rail temperature is
selected as 30 ± 5 ℃, and the maximum temperature change range is 36.7 ℃.
2.2.3. Model Establishment. The overall beam and rail interaction models are shown in figure 2.
(a) Left line ZFZ9~ZFZ14
(b) Left lineZFZ27~ZFZ29
(c) Right lineZFS9~ZFS14
(d) Right lineZFS29~ZFS31
Figure 2. Finite element model of expansion force and temperature force of beam rail interaction.
3. Calculation and Analysis
3.1. Calculation of Additional Expansion and Contraction Force of Rail, Relative Displacement
Between Beam and Rail And The Force of Pier
When the concrete bridge temperature rises 30 ℃, the additional expansion and contraction forces of
the rails and relative displacements between the beam and rail are shown in figure 3 to 10. The force
of each pier is shown in table 2.
0100 200 300 400 500
-200
-150
-100
-50
0
50
100
150
200
Force (kN)
Rail coordinate (m)
0 100 200 300 400 500
-6
-4
-2
0
2
4
Displacement (mm)
Rail coordinate (m)
Figure 3. Additional expansion and
contraction force of left rail (ZFZ9~ZFZ14).
Figure 4. Relative displacement of left track
between the beam and rail (ZFZ9~ZFZ14).

ICPEM 2021
Journal of Physics: Conference Series 2148 (2022) 012065
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doi:10.1088/1742-6596/2148/1/012065
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0 100 200 300 400
-200
-100
0
100
200
Force (kN)
Rail coordinate (m)
0 100 200 300 400
-8
-6
-4
-2
0
2
4
Displacement (mm)
Rail coordinate (m)
Figure 5. Additional expansion and
contraction force of left rail (ZFZ27~ZFZ29).
Figure 6. Relative displacement of left track
between the beam and rail (ZFZ9~ZFZ14).
0100 200 300 400 500
-200
-150
-100
-50
0
50
100
150
200
Force (kN)
Rail coordinate (m)
0 100 200 300 400 500
-6
-4
-2
0
2
4
Displacement (mm)
Rail coordinate (m)
Figure 7. Additional expansion and
contraction force of left rail (ZFS9~ZFS14).
Figure 8. Relative displacement of left track
between the beam and rail (ZFS9~ZFS14).
0 100 200 300 400
-200
-100
0
100
200
Force (kN)
Rail coordinate (m)
0 100 200 300 400
-8
-6
-4
-2
0
2
4
Displacement (mm)
Rail coordinate (m)
Figure 9. Additional expansion and
contraction force of left rail (ZFS29~ZFS31).
Figure 10. Relative displacement of left track
between the beam and rail (ZFS29~ZFS31).
According to figure 2 to 9, for the left line, the maximum additional expansion force on the single
strand rail is 211.13kN, which occurs at the pier numbered ZFZ29; The maximum relative
displacement between beam and rail also occurs at this position, which is 6.572mm. For the right line,
the maximum additional expansion force on the single strand rail is 211.13kN, which occurs at the pier

ICPEM 2021
Journal of Physics: Conference Series 2148 (2022) 012065
IOP Publishing
doi:10.1088/1742-6596/2148/1/012065
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numbered ZFS31; The maximum relative displacement between beam and rail also occurs at this
position, which is 6.572mm. Therefore, for the left and right lines, the values of the maximum
additional expansion force of single strand rail and the values of the maximum relative displacement
between beam and rail are the same. The maximum value of the additional expansion and contraction
forces and the relative displacement between beam and rail for each line occur at the same position.
As is shown in table 2, the maximum force of pier in the section is 500.80kN, and the pier numbers
are ZFZ27 and ZFS29.
Table 2. Longitudinal force of pier (kN / rail).
Pier number
Force
Pier number
Force
ZFZ9
165.08
ZFS9
164.12
ZFZ10
34.50
ZFS10
36.14
ZFZ11
22.10
ZFS11
21.00
ZFZ12
30.76
ZFS12
26.86
ZFZ13
63.52
ZFS13
78.10
ZFZ14
0
ZFS14
0
ZFZ27
250.40
ZFS29
250.40
ZFZ28
95.10
ZFS30
95.10
ZFZ29
0
ZFS31
0
Therefore, in the design, construction and maintenance of CWR on the bridge, more attention
should be paid to the bridge-track structure at these piers, ZFZ29, ZFS31, ZFZ27 and ZFS29.
3.2. Checking and Calculation of Rail Strength
Each rail shall have sufficient strength to ensure that it will not fail under the joint action of dynamic
bending stress, temperature stress, expansion additional stress, train starting / braking stress and other
additional stresses. At this time, the sum of various stresses borne by the rail shall not exceed the
specified allowable value [], and the calculation formula is shown in formula (1):
d＋t＋z＋f ≤ [] (1)
Where:
d  Maximum dynamic bending tensile stress of rail (MPa);
t  Temperature stress (MPa);
f  Maximum additional stress on rail (MPa);
z  Traction (braking) stress on rail (MPa);
[]  the allowable stress of rail (MPa), according to Code for Design of Railway Continuous
Welded Rail (TB 10015-2012) [9]. The yield strength of U71Mn rail is 457MPa, and the safety factor
K=1.3, so []=352MPa.
The checking calculation of rail strength is shown in table 3.
Table 3. Statistics of rail stress (MPa).
indexes
Dynamic
bending stress
Temperature
stress
Additional
expansion stress
Additional
braking stress
Total
stress
Allowable
stress
Tension
119.99
91.02
27.26
11.80
250.07
352
compression
154.63
91.02
27.26
11.80
284.71
352

ICPEM 2021
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doi:10.1088/1742-6596/2148/1/012065
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It can be seen from the table that the sum of tensile stress and compressive stress borne by the rail
is 250.07MPa and 284.71MPa, both less than the allowable stress 352MPa, and the rail strength meets
the requirements.
4. Conclusion
Taking the CWR on the multi-span simply supported beam bridge as an example, the following
conclusions can be drawn after analyzing the longitudinal mechanical properties of the beam-rail
interaction model:
(1) For the left line, the maximum additional expansion force on the single strand rail is 211.13kN,
which occurs at the pier numbered ZFZ29; The maximum relative displacement between beam and
rail also occurs at this position, which is 6.572mm. For the right line, the maximum additional
expansion force on the single strand rail is 211.13kN, which occurs at the pier numbered ZFS31; The
maximum relative displacement between beam and rail also occurs at this position, which is 6.572mm.
(2) For the left and right lines, the values of the maximum additional expansion force of single
strand rail and the values of the maximum relative displacement between beam and rail are the same.
(3) The maximum value of the additional expansion and contraction forces and the relative
displacement between beam and rail for each line occur at the same position, and both are located at
the pier.
(4) The maximum stress of pier in the section is 500.80kN, and the pier numbers are ZFZ27 and
ZFS29.
(5) The total tensile stress borne by the rail is 250.07MPa, and the total compressive stress borne by
the rail is 284.71MPa, both less than the allowable stress 352MPa, so the rail strength meets the
requirements.
Acknowledgments
The work described here has been supported by the Project of Science and Technology Research and
Development Plan of China Railway Nanchang Bureau Group Co., Ltd (Grant 202007). To smoothly
complete this paper, I would like to sincerely thank my teacher for his guidance, my senior fellow
disciples for answering some questions and solving some doubts for me, and my friends for their help
and encouragement when I felt depressed. I really feel very lucky to meet them on my growth path!
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