Abstract
To explore the difference in the impact of transverse bracing on the seismic effect of through concretefilled steel tube arch bridges with nonisolated and earthquakeisolated, nine nonisolated and earthquakeisolated structural models under different crossbracing arrangements were established, and Elcentro seismic waves were selected. The internal force, displacement, velocity, absolute acceleration, relative acceleration, and separation of arch ribs of each model were compared and analyzed under uniform excitation along the bridge, transverse and vertical directions, multidimensional combined excitation, and multipoint excitation considering the traveling wave effect. Based on the shear force and displacement of the earthquake support, it is concluded that the internal force response of different excitations of various models is more complicated. The installation of transverse bracing on the upper part of the arch rib can reduce the vertical displacement of the arch rib of the nonseismic structure. The “X”shaped cross brace at the top of the arch rib and the “K”shaped cross brace at the lower part help to reduce the transverse acceleration of the arch rib. The absolute acceleration and relative acceleration of the seismic structure arch ribs are significantly reduced.
Highlights
 Timehistory response of arch foot axial force
 Timehistory response of vault displacement under various conditions of nonseismic isolation
 Timehistory response of vault lateral velocity in each case of seismic isolation
 Timehistory response of lateral acceleration of nine vaults under working conditions
 Hysteresis curve
1. Introduction
In recent years, there have been frequent earthquakes around the world. As a lifeline project for postdisaster reconstruction and disaster relief, bridges have always received extensive attention. A large number of concretefilled steel tube arch bridges have been built, and research on related crossbracing arrangements is also ongoing.
Dong Rui et al. [1] studied the effectiveness of new Lshaped crossbraces in the stability of longspan concretefilled steel tube truss arch bridges. Hejiang Third Bridge was taken as the engineering background, using a combination of numerical calculation and theoretical analysis to compare and analyze its mechanical performance and stability, and use orthogonal experiment and variance analysis methods to evaluate the significance of Lshaped cross braces in the stability of longspan CFST truss arch bridges Zhang Sumei and Yundi [2] analyzed and compared the possible layout schemes of cross braces and X braces for a 360meterspan halfthrough concretefilled steel tube arch bridge, and proposed the rationality of X braces and cross braces accordingly. According to the principle of equal bracing area and similar material consumption of transverse bracing system, four bracing schemes were proposed and analyzed for ultimate bearing capacity respectively; Wan Peng et al. [3] designed the Guangzhou Xinguang Bridge with a main span of 428 meters in plan, the largescale finite element software ANSYS was used to establish a threedimensional finite element model of the full bridge, and the influence of the number and position of the transverse braces on the elastic stability and the ultimate bearing capacity of the plane was analyzed. Jin Bo et al. [4] used the finite element method to analyze the influence of transverse bracing on the overall stability of a cablestayed concretefilled steel tube arch bridge; Chen Baochun et al. [5] found arch and archgirder composite bridges are the main ones; Liu Zhao et al. [6] derived the analytical calculation formula for the lateral elastic stability bearing capacity of arch bridges with transverse braces based on the energy principle, and verified the proposed finite element numerical solution through a numerical example. The correctness of the analytical formula and finally discussed the influence of structural parameters on the stability of bearing capacity; Wu Meirong et al. [7] stepped into the nonthrust halfthrough concretefilled steel tube arch bridge in terms of risespan ratio, widthspan ratio, main arch rib stiffness, transverse bracing Changes in the dynamic characteristics of the bridge structure when the layout mode, suspender failure, and support layout are changed; Kong Dandan et al.[8] took a steel truss arch bridge in a certain city as the research object and showed that increasing the number of wind bracing structures can significantly improve the structure’s performance stability; but when the number of wind bracing is sufficient, the continue to increase the number of wind bracing structures, the stability of the structure cannot be greatly improved, and the setting of diagonal braces has a great influence on the overall stability, especially “K” and “X” diagonal braces have a significant impact on the structural stability; Li Xiayuan et al. [9] relying on a certain throughtype steel tube concrete arch bridge, based on the original bridge wind bracing form, using the MIDAS Civil finite element analysis software to establish the “” The calculation model for the throughtype steel tube concrete arch bridge with “X”shaped wind bracing, “K”shaped wind bracing, “m”shaped wind bracing, and “X”shaped wind bracing, extracts the first 20order natural frequency and The vibration mode types of the first 6 steps were compared and analyzed with the original bridge; Zheng Xiaoyan et al. [10] studied the stability of the tied arch bridge during the construction phase and the influence of temporary transverse bracing on the structural stability.
In this paper, nine nonisolated and earthquakeisolated structural models under different crossbracing arrangements were established, and Elcentro seismic waves were selected. The internal force, displacement, velocity, absolute acceleration, relative acceleration, and separation of arch ribs of each model were compared and analyzed under uniform excitation along the bridge, transverse and vertical directions, multidimensional combined excitation, and multipoint excitation considering the traveling wave effect.
The layout position and layout of the transverse bracing have different effects on the throughtype concretefilled steel tube seismic arch bridge and the seismic isolation arch bridge. The article will conduct comparative analysis and research to provide the necessary references for the design and construction of similar arch bridges.
2. Principles of time history analysis
The vibration equation for dynamic time history analysis is:
where ${M}_{S}$, ${C}_{S}$, ${K}_{S}$ denote the mass matrix, damping matrix and stiffness matrix of the corresponding structural nonsupporting position, respectively, use ${M}_{b}$, ${C}_{b}$, ${K}_{b}$ to denote the mass matrix, damping matrix and stiffness matrix of the corresponding structural support position, respectively, and use ${\ddot{y}}_{s}$, ${\dot{y}}_{s}$, ${y}_{s}$ to denote the structural nonsupporting position under earthquake action, the acceleration, velocity and absolute displacement of the support, with ${\ddot{y}}_{b}$, ${\dot{y}}_{b}$, ${y}_{b}$, respectively represent the acceleration, velocity and absolute displacement vector of the structural support position under the action of an earthquake. ${F}_{b}$ is the reaction force of the support under the action of an earthquake. Then the vibration equation can be expressed in the following form:
3. Finite element model
Taking an actual through arch bridge as the background, nine nonseismic and seismic finite element models of different transverse bracing arrangements are established. The transverse bracing arrangement and finite element model are shown in Table 1 and Fig. 1. The seismic isolation model is equipped with leadcore rubber seismic isolation bearings, and the bearing parameters are shown in Table 2. The bridge has a main span of 127 m and a bridge deck width of 31 m. The arch rib crosssection is dumbbellshaped. The diameter of the upper and lower arch ribs is 1.2 m, and the diameter of the cross brace is 1.3 m.
Table 1The layout of transverse bracing in various working conditions
Working condition 1  Working condition 2  Working condition 3  Working condition 4  Working condition 5  Working condition 6  Working condition 7  Working condition 8  Working condition 9 
A cross brace in the shape of “” on the vault  Three “”shaped cross braces on the vault and the middle and upper parts  Three “” shaped cross braces on the vault and the middle and lower parts  Fiveway "" cross brace  One “” shaped cross brace on the vault, and two “K” cross braces in the middle and upper part  One “” shaped cross brace on the vault, two “K” cross braces in the middle and lower part  The vault has one “” shaped cross brace and four “K”shaped cross braces  One “” shaped cross brace on the vault and four “X” cross braces  Five “X” shaped cross braces 
Table 2Parameter table of lead rubber bearing
Support plane size (mm×mm)  Lead core yield (kN)  Rigidity before yielding (kN/mm)  Rigidity after yielding (kN/mm)  Horizontal equivalent stiffness (kN/mm) 
1320×1320  964  25.6  3.9  6.4 
Fig. 1Finite element model diagram
a) Working condition 1
b) Working condition 2
c) Working condition 3
d) Working condition 4
e) Working condition 5
f) Working condition 6
g) Working condition 7
k) Working condition 8
l) Working condition 9
4. Analysis of dynamic characteristics
Through the finite element software analysis of the dynamic characteristics, the frequency and mode shape of the nonisolated and isolated models under nine working conditions are obtained. The first three orders are shown in Table 3, and the frequency comparison is shown in Fig. 2. It can be seen that the firstorder modes of the two models under nine working conditions are all arch rib lateral inclination, and the firstorder frequencies of working conditions 1, 2, 3, and 4 have little difference, while the firstorder frequencies of working conditions 8 and 9 are relatively different. Large “K”shaped cross braces and “X” cross braces can increase the fundamental frequency, and the effect of being close to the lower part of the arch rib is obvious. The “X” cross brace on the dome actually reduces the fundamental frequency. The second and third order frequencies and modes of the two models are quite different, and the influence of the cross bracing of the nonisolated model is more obvious than that of the isolated model.
Table 3The first thirdorder frequency and mode shape of each working condition
Working condition  Types  Frequency and mode shape  First order  Second order  Third order 
Working condition 1  Nonisolated  Mode shape  
Frequency  0.167  0.520  0.521  
isolation  Mode shape  
Frequency  0.160  0.253  0.283  
Working condition 2  Nonisolated  Mode shape  
Frequency  0.167  0.517  0.647  
isolation  Mode shape  
Frequency  0.160  0.252  0.284  
Working condition 3  Nonisolated  Mode shape  
Frequency  0.168  0.518  0.647  
isolation  Mode shape  
Frequency  0.161  0.252  0.284  
Working condition 4  Nonisolated  Mode shape  
0.168  0.515  0.645  
isolation  Mode shape  
Frequency  0.161  0.252  0.284  
Working condition 5  Nonisolated  Mode shape  
Frequency  0.188  0.522  0.645  
isolation  Mode shape  
Frequency  0.177  0.252  0.290  
Working condition 6  Nonisolated  Mode shape  
Frequency  0.213  0.546  0.645  
isolation  Mode shape  
Frequency  0.195  0.252  0.297  
Working condition 7  Nonisolated  Mode shape  
Frequency  0.231  0.548  0.642  
isolation  Mode shape  
Frequency  0.205  0.252  0.306  
Working condition 8  Nonisolated  Mode shape  
Frequency  0.332  0.619  0.640  
isolation  Mode shape  
Frequency  0.233  0.251  0.363  
Working condition 9  Nonisolated  Mode shape  
Frequency  0.330  0.640  0.691  
isolation  Mode shape  
Frequency  0.232  0.251  0.365 
5. Selection of seismic wave and apparent wave speed
The seismic fortification intensity of the area where the bridge is located is 8 degrees (0.2 g), and the site category is Type II. The El Centro seismic wave is selected, and the peak acceleration value of the seismic wave is multiplied by a coefficient of 0.339 for adjustment. The adjusted seismic wave is shown in Fig. 3, and the action time is taken as 20 s, the excitation direction is uniform excitation along the bridge direction, uniform excitation across the bridge direction, uniform excitation vertical direction, multidimensional combination one (long bridge direction + 0.3 horizontal bridge direction + 0.3 vertical) excitation, multidimensional combination two (0.3 forward bridge direction + Transverse bridge direction + 0.3 vertical direction) excitation, multidimensional combination three (0.3 along bridge direction + 0.3 transverse bridge direction + vertical direction) excitation and the apparent wave speed is 100 m/s, 200 m/s, 300 m/s, 400 m/s, Multipoint excitation of 500 m/s, 1000 m/s, 1500 m/s, 2000 m/s.
Fig. 2Frequency comparison
a) First order
b) Second order
c) Third order
Fig. 3Elcentro seismic wave adjusted
6. Earthquake response analysis
6.1. Internal force of arch rib
See Table A1 for the maximum internal force and damping rate of arch ribs in different models under uniform excitation. See Table A2 for the maximum internal force and damping rate of arch ribs in different models under multidimensional combined excitation. Under multipoint excitation considering traveling wave effect, the maximum internal force and shock absorption rate of arch ribs in different models under various working conditions are shown in Table A3. The timehistory response of partial arch foot axial force is shown in Fig. 4.
Through the comparison of Table A1 to Table A3 and Fig. 4, we can get:
(1) Under the action of seismic waves with different wave speeds in the bridge direction, transverse bridge direction, combination 1 and bridge direction, the main internal force of the seismic isolation structure arch rib in each working condition is significantly reduced;
(2) Under the action of vertical earthquake, the main internal forces of the seismic isolation structure arch ribs in various working conditions increased, the shear force ${F}_{Z}$ increased by more than twice, and the bending moment ${M}_{y}$ increased by more than three times;
(3) Under the action of the second combination earthquake, the arch rib axial force of each working condition of the seismic isolation structure decreases, the shear force ${F}_{z}$ increases, the bending moment ${M}_{z}$ in working condition 8 and 9 increase, and the rest decrease. Under the action of the combination three earthquakes. The main internal force of the arch rib of the seismic isolation structure in the working condition increased, the shear force ${F}_{Z}$ increased more than doubled, and the bending moment ${M}_{y}$ increased more than doubled;
(4) Under the effects of lateral earthquake and combination, the main internal force of the seismic isolation structure arch ribs in working conditions 8 and 9 increase significantly.
Fig. 4Time history response of arch foot axial force
a) Working condition1 along the bridge
b) Condition 7 along the bridge
c) Condition 1 Crossbridge direction
d) Working condition 5 Combination one
6.2. Arch rib displacement
The maximum displacement of the arch rib under transverse excitation is shown in Table 4, and the timehistory response of the DY time history of the vault displacement under nonseismic conditions is shown in Fig. 5.
Fig. 5DY timehistory response of vault displacement under various conditions of nonseismic isolation
Through the comparative analysis of Table 4 and Fig. 5, we can get:
(1) Under the action of transverse bridge seismic wave, the arch ribs of nonseismic and isolation models mainly undergo lateral displacement. The lateral displacements of working conditions 1, 2, 3, and 4 are not much different. The lateral displacements of working conditions 5, 6, and 7 are more than other, the working condition is small, and it is concluded that the “K”shaped cross brace is better than the “” cross brace and the “meter” cross brace in reducing the lateral displacement of the arch rib;
(2) Comparing various working conditions, it can be concluded that setting up transverse bracing on the upper part can reduce the vertical displacement of the arch rib of the nonseismic model.
Table 4Arch rib displacement (unit: cm)
Incentive direction  Displacement direction  Model  Working condition  
1  2  3  4  5  6  7  8  9  
Cross bridge  Along the bridge  Nonisolated  0.220588  0.217772  0.218886  0.216088  0.19172  0.215155  0.190403  0.234662  0.238253 
Vertical Horizontal Vertical Isolated  0.095618  0.095142  0.095793  0.095229  0.08672  0.090369  0.097304  0.14055  0.138715  
Cross bridge  Nonisolated  12.178108  12.190638  12.160166  12.177714  10.963965  9.490335  8.852437  11.574302  11.437827  
Vertical Horizontal Vertical Isolated  13.456979  13.433771  13.436001  13.414679  12.387384  11.74146  10.855834  17.104378  16.846004  
Vertical  Nonisolated  0.575183  0.569438  0.572487  0.566709  0.505177  0.567788  0.505335  0.55977  0.564412  
Isolated  0.225666  0.225264  0.225162  0.224671  0.19714  0.210035  0.216076  0.3417  0.344369 
6.3. Arch rib speed
The maximum speed of arch ribs under transverse excitation is shown in Table 5. The timehistory response of the transverse velocity of the vault under each condition of seismic isolation is shown in Fig. 6. Through the comparative analysis of Table 5 and Fig. 6, we can get:
(1) Under the action of transverse bridge seismic waves, the lateral velocity of arch ribs in nonseismic and seismic isolation models basically increases in working conditions 1 to 8, while working condition 9 decreases slightly;
(2) Under the action of transverse bridge seismic waves, the longitudinal and vertical speeds of arch ribs in nonseismic and seismic models are relatively small in condition five;
(3) The speed of the arch ribs of the seismic isolation structure in each working condition is reduced.
Table 5Arch rib speed (unit: cm/s)
Incentive direction  Speed direction  Model  Working condition  
1  2  3  4  5  6  7  8  9  
Cross bridge  Along the bridge  Nonisolated  1.572553  1.573233  1.55233  1.555467  1.481314  1.592136  1.495981  1.442885  1.443201 
Isolated  0.8401  0.842009  0.831214  0.836434  0.801512  0.875925  0.845275  0.88414  0.885579  
Cross bridge  Nonisolated  25.172737  25.105125  25.468387  25.328813  27.179484  29.70583  31.043626  39.780206  38.68521  
Isolated  19.793487  19.782502  19.837443  19.837609  18.940581  23.446982  26.028057  39.182534  38.69969  
Vertical  Nonisolated  3.808268  3.734763  3.786084  3.709999  3.400459  3.806481  3.455232  3.459467  3.521407  
Isolated  1.711642  1.729262  1.698642  1.715102  1.625263  1.767564  1.702519  1.688422  1.690779 
Fig. 6Timehistory response of vault lateral velocity in each case of seismic isolation
6.4. Absolute acceleration of arch rib
The maximum absolute acceleration of the arch rib under transverse excitation is shown in Table 6, and the timehistory response of the lateral acceleration of the nine vaults under working conditions is shown in Fig. 7. Through the comparative analysis of Table 6 and Fig. 7, it can be obtained:
(1) Under the action of the transverse bridge seismic wave, the nonisolated and isolated model arch rib lateral acceleration, the nonseismic structure working condition 5 and working condition 7 are smaller, the seismic isolation structure working condition 7 is relatively small, and the working condition 9 is relatively small. Working condition 8 is reduced, it can be inferred that the “米”shaped cross brace at the top of the arch rib, and the “K”shaped cross brace at the lower part will help reduce the absolute acceleration of the arch rib.
(2) The absolute acceleration of the arch rib of the seismic isolation structure in each working condition is significantly reduced.
Table 6Absolute acceleration of arch ribs (unit: cm/s2)
Incentive direction  Absolute acceleration direction  Model  Working condition  
1  2  3  4  5  6  7  8  9  
Cross bridge  Cross bridge  Nonisolated  302.069666  298.588074  314.635446  306.619751  291.123396  315.517481  291.100165  343.150259  338.930472 
Isolated  83.273905  82.935068  82.92821  83.290524  82.154905  81.068172  79.804811  91.825414  88.816718 
Fig. 7Timehistory response of lateral acceleration of nine vaults under working conditions
6.5. Relative acceleration of arch rib
The maximum relative acceleration of arch ribs under transverse excitation is shown in Table 7.
Table 7Relative acceleration of the arch ribs (unit: cm/s2)
Incentive direction  Relative acceleration direction  Model  Working condition  
1  2  3  4  5  6  7  8  9  
Cross bridge  Cross bridge  Nonisolated  350.799422  353.664181  349.011316  349.716454  351.965781  330.543943  325.231323  348.104201  346.062742 
Isolated  157.707118  157.621842  157.721081  157.612437  156.244506  156.104435  154.734779  150.697906  151.129868 
Through the comparative analysis of Table 7, we can get:
(1) Under the action of the transverse bridge seismic wave, the relative acceleration of the arch ribs of the nonseismic and isolation models is relatively small for the nonseismic structure working conditions 6 and 7, and the seismic isolation structure working conditions 1 to 7 basically show a decreasing trend;
(2) The relative acceleration of the arch rib of the seismic isolation structure in each working condition is significantly reduced.
6.6. Shear force and displacement of seismic isolation support
See Table 8 for the maximum shear force and displacement of the seismic isolation support. See Fig. 8 for the shear force comparison of some supports. See Fig. 9 for the displacement comparison of some supports.
Table 8Maximum shear force and displacement of seismic isolation support
Incentive direction  Working condition  1  2  3  4  5  6  7  8  9 
Along the bridge  Shear force / kN  940.21  939.78  939.62  940.56  940.73  940.49  939.69  938.43  938.22 
Displacement / cm  4.87  4.87  4.86  4.90  4.90  4.90  4.89  4.84  4.84  
Cross bridge  Shear force / kN  873.87  873.54  873.76  873.44  861.16  844.86  843.41  762.70  753.41 
Displacement / cm  4.34  4.33  4.34  4.33  4.23  4.09  3.95  3.40  3.35  
Vertical  Shear force / kN  190.50  190.67  190.48  190.66  190.81  190.56  190.93  191.25  191.63 
Displacement / cm  0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.76  
Combination one  Shear force / kN  934.84  937.98  937.77  936.80  934.97  936.03  937.64  938.89  938.63 
Displacement/cm  4.97  4.99  5.00  5.01  4.98  4.98  4.96  4.94  4.91  
Combination two  Shear force / kN  852.10  851.64  855.63  852.19  837.90  822.17  815.99  745.03  741.08 
Displacement / cm  4.37  4.38  4.41  4.37  4.22  4.18  3.91  3.45  3.34  
Combination three  Shear force / kN  448.11  450.55  448.51  452.70  440.66  442.84  446.90  442.42  442.31 
Displacement / cm  1.88  1.84  1.84  1.81  1.88  1.85  1.85  1.86  1.86 
Fig. 8Comparison of bearing shear force
a) Along the bridge and combination one
b) Crossbridge direction and combination two
Fig. 9Comparison of bearing displacement
a) Along the bridge and combination one
b) Crossbridge direction and combination two
See Figure 10 for the shear response time history of some supports. Hysteresis curve for some supports. See Fig. 11. From the analysis of Table 8 and Fig. 8 to Fig. 11, we can get:
(1) The maximum shear force of the seismic isolation support under the excitation of working conditions 1 to 7 is greater than that of combination 1, and the maximum shear force of the seismic isolation support under the excitation of working conditions 8 and 9 is greater than the excitation of the forward bridge;
(2) Under the action of the transverse bridge direction and combination two, the maximum shear force of the seismic isolation support of working conditions 1 to 9 shows a decreasing trend, and the linear direction is basically similar, and the transverse bridge excitation of each working condition is greater than the combination two excitation;
(3) The maximum displacement of each working condition is that the excitation of combination one is greater than the excitation along the bridge direction, and the crossbridge direction and combination two are basically the same, and there is a decreasing trend from working condition 1 to working condition 9;
Under the vertical excitation, the shear force and displacement of all working conditions are basically the same.
Fig. 10Time history response of bearing shear force
a) Working condition 5, along the bridge direction excitation
b) Working condition 1 crossbridge excitation
Fig. 11Hysteresis curve
a) Condition 1 along the bridge direction excitation
b) Working condition 9 combination one incentive
7. Conclusions
Nine nonisolated and earthquakeisolated structural models under different crossbracing arrangements were established, and Elcentro seismic waves were selected. The internal force, displacement, velocity, absolute acceleration, relative acceleration, and separation of arch ribs of each model were compared and analyzed under uniform excitation along the bridge, transverse and vertical directions, multidimensional combined excitation, and multipoint excitation considering the traveling wave effect.
Through the above comparative analysis, we can get:
1) The main internal force of the arch ribs of the seismic isolation structure in each working condition decreases significantly under the action of the bridge direction, the horizontal bridge direction, the combination one, and the seismic waves with different wave speeds. Under the vertical earthquake action, the arch of the seismic isolation structure, the main internal force of the rib increases. Under the action of the second combination earthquake, the axial force of the arch rib in each working condition of the seismic isolation structure decreases, the shear force ${F}_{z}$ increases, the bending moment ${M}_{z}$ working conditions eight and nine increase, and the rest decrease, and the combination three under the action of an earthquake, the main internal forces of the seismic isolation structure arch ribs in various working conditions have increased;
2) Under the action of transverse bridge seismic waves, the arch ribs of nonseismic and isolation models mainly undergo lateral displacement. The “K”shaped cross brace is better than the “” cross brace and the “meter” shape in reducing the lateral displacement of the arch rib. Transverse bracing, setting transverse bracing on the upper part of the arch rib can reduce the vertical displacement of the arch rib of the nonseismic model;
3) Under the action of transverse seismic waves, the lateral velocity of the arch ribs of the nonseismic and isolation models basically increased, and the velocity of the arch ribs of the seismic isolation structure under various working conditions decreased;
4) The “meter”shaped cross brace at the top of the arch rib and the “K”shaped cross brace at the lower part help reduce the lateral acceleration of the arch rib. The absolute acceleration and relative acceleration of the arch rib of the seismic isolation structure under various working conditions are significantly reduced;
5) Under the action of the maximum shear force of the seismic isolation support in the transverse direction and the combination two, working conditions 1 to 9 show a decreasing trend, and the linear directions are basically similar. In all conditions, the excitation of combination one is greater than the excitation along the bridge direction, and the crossbridge direction and combination two are basically the same, and there is a decreasing trend from working condition one to working condition nine.
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About this article
This research was funded by the Fundamental Research Funds for the Central Universities (31920210078), the National Natural Science Foundation of China (51868067).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that they have no conflict of interest.