Abstract
To explore the different influence of traveling wave effect on the isolated and nonisolated arch bridge, the isolated and nonisolated multispan arch bridges models are established respectively, three measured seismic waves were selected, and under the eight kinds of apparent wave velocities and multipoint consistent excitations, the structural responses of the two models, including the internal force response, the arch rib velocity, the pier’s internal force response, the bridge deck acceleration, and the shear force and displacement of isolation support of different arch ribs position in two models, were compared and analyzed. The results show that the isolation effect of the isolation structure is obvious; the wave effect of the isolated structure is significant, the two structures show different internal force response curves in different positions of the arch ribs under the influence of different seismic waves; the influence of the apparent wave velocity on the force and shock reduction rate of the arch ribs is complex; the vertical acceleration of the arch ribs and bridge deck of the isolated structure is reduced significantly; and the shear force and displacement of the isolation bearing increase with the increase of the apparent wave velocity. The study shows the traveling wave effect of multispan through a concretefilled steel tubular arch bridge with and without isolation, and the results will be used for the seismic design and analysis of structural diseases caused by the wave effect.
Highlights
 Different influence of traveling wave effect on the isolated and nonisolated arch bridge was analyzed.
 Wave effect of the isolated structure is significant compared with the nonisolated.
 Influence of the apparent wave velocity on the force and shock reduction rate of the arch ribs is complex.
1. Introduction
Earthquakes occur frequently, and the seismic energy propagates from the source to the surface in the form of waves. Due to the different characteristics of topography and geological structure at different points on the ground, the time of receiving seismic waves at different points is different, that is, there is phase difference. If the geological conditions of the site where the structure is located are complex, the spatial influence of ground motion is great for longspan structures. Traveling wave effect is due to the fact that the seismic wave propagates at a certain speed, and there is time difference and phase difference when it reaches different points, which will produce traveling wave effect on longspan bridge structure. Concretefilled steel tube (CFST) arch bridge is widely used in bridges, especially in longspan bridges, its seismic response is complex, and the impact of traveling wave effect cause the unsynchronized vibration to the structure elements which led to the lager responses, and cannot be ignored [1, 2]. Most of the previous studies devote to the static behaviors, thermal and creep effects of concrete and steel, forms of structural members, or the construction technology; few researches have been carried out on the analysis of seismic behaviors of the multispan CFST arch bridge [3], especially on its seismic response considering traveling wave effect.
Some researchers focused on seismic response of some kinds of arch bridges, different bridge structure shows different influence degree. Xu Yan et al. [2] discussed a case study on the seismic response of a steel arch bridge under selected near fault ground motions by considering the traveling wave effect with variable apparent velocities. Liu AiRong et al. [4] investigated the spatial variability effects of ground motion on the seismic response of longspan continuous rigidframe arch bridge, in his study, the spatial variability of ground motions between pivots of the bridge was taken into account with multisupport excitation and traveling wave effect by exciting the bridge in both transverse and longitudinal directions of the bridge. Hu Zhiming et al. [5] took the traveling wave effect is into account to establish the model of the whole bridge of a concretefilled steel tube through arch bridge with inclined arch ribs and inclined hangers, and analyzed the relevant natural vibration characteristics and seismic response. Song Bo et al. [6] researched the traveling wave effect on the seismic response of a 260 meters of halfthrough steel arch bridge, and advised the traveling wave effect should be considered in the aseismic design of longspan Bridges. Wang Hao et al. [7] thought due to the apparent wave velocities can be chosen in a wide range for longspan arch bridge, there had not yet been definite conclusions about the influence of traveling wave, and their research result show that the influences are closely correlated with the characteristics of the bridge structure and the seismic wave. In Ref. [8], Using the FE model of a onespan concretefilled steel tubular arch bridge, stochastic seismic analysis of the CFST arch bridge is conducted, by considering the dimensionality, incoherence effect, wavepath effect of ground motions, and local site effects with different and irregular site conditions. Wang Ruolin et al. [9] adopted Taft and ElCentro earthquake waves to analyze the seismic response properties of a longspan steel trussarch railway bridge in consideration of the travellingwave effect of multiple earthquake inputs, and discussed the influence of wave velocity and earthquake property on the structural dynamic responses taking into account travellingwave effect. In order to study, Zhang Yongliang et al. [10]^{}took a decktype railway steel truss arch bridge with the span of 490m as a research object, and established the dynamic calculation model of the whole bridge to analyze the law of seismic response of railway longspan steel truss arch bridges under traveling wave effect. The similar work was given by Wu Yuhua, et al. [11], Lou Menglin, et al. [12], Chen Yanjiang, et al. [13], Liu Zhen, et al. [14], Dai Gonglian, et al. [15], Tang Tang, et al. [16].
In this paper, the response difference of different apparent wave velocities between nonisolated arch bridges and isolated arch bridges under seismic action is explored, the influence of different wave velocities is compared, and the isolation effect of leadcore rubber isolation supports is discussed. It provides a reference for the seismic and isolation analysis of this kind of bridge and a theoretical reference for the extension and application of leadcore rubber isolation bearing in this kind of bridge. It provides some ideas and reference value for the construction and site selection of this kind of bridge.
2. Analysis method of traveling wave effect
Based on D’Alembert principle, the dynamic equilibrium equation of the whole system expressed by absolute displacement is established as:
where subscript $f$ is the base node; subscript $s$ is the structure node; subscript $c$ is the common node. $U$ is the displacement. The displacement of each node can be divided into relative dynamic term and quasistatic term:
Equation of motion of foundation free field:
In the simultaneous Eqs. (1) to (3), the contribution of massless spring damping force to the structure due to the rigid connection between the foundation and the structure is omitted, and Eq. (4) is not consistent in many points:
where ${u}_{f}$ is the foundation displacement caused by the structure movement; ${u}_{c}$ is the common node displacement caused by the structure movement; ${v}_{c}$ is the common node displacement caused by the free field movement; ${v}_{f}$ is the foundation displacement caused by the free field movement.
3. Bridge survey and finite element model
3.1. Overview of bridges
Taking an actual three span through arch bridge as the background, the seismic fortification intensity of the area where the bridge is located is 8 degrees (0.2 g), and the site category is class II. The main span of the bridge is 127 m, the two side spans are 87 m, the deck width is 31 m, the arch rib section is dumbbell shaped, the diameter of the middle arch rib is 1.2 m, the diameter of the side arch rib is 1.0 m, the steel pipe of the arch rib is filled with C50 concrete, and the pier is C40 reinforced concrete with a diameter of 3.5 m. The Elastic modulus of the arch rib steel tube is 2.06×10^{5}^{ }N/mm^{2}, and the Poisson’s ratio is 0.3. The Elastic modulus of C50 concrete is 3.34×10^{4 }N/mm^{2}, and the Poisson’s ratio is 0.2. The Elastic modulus of C40 concrete is 3.25×10^{4 }N/mm^{2}, and the Poisson’s ratio is 0.2.
3.2. Finite element model
The finite element model was built based on a realistic threespan through the arch bridge, in which the arch rib, cross beam, longitudinal beam, cover beam, and bridge pier were simulated by beam element, the bridge deck was simulated by plate element, and the suspender and tie rod were simulated by tensiononly link element. The bottom of the pier is set as consolidation. The bridge deck and beam are arranged as elastic connections. The piers and cover beams in the nonisolated model are connected, however, leadcore rubber isolation supports are arranged at the connection of each pier and cover beam in the isolated model, the bridge bearing parameters is shown in Table 1, and Fig. 1 shows the finite element model with a locally enlarged view of nonisolation and isolation.
Table 1Leadcore rubber support parameters
Support plane dimension (mm×mm)  Lead core yield force (kN)  Preyield stiffness (kN/mm)  Postyield stiffness (kN/mm)  Horizontal equivalent stiffness (kN/mm) 
1320×1320  964  25.6  3.9  6.4 
4. Dynamic characteristic analysis
Using the function of characteristic value analysis in finite element method, the 105order vibration type was analyzed by multiple Ritz vector method, and the results show that basic period of the nonisolated model is 3.72 s and the basic period of the isolated model is 4.55 s. The first ten order vibration type of the isolated model is shown in Fig. 2. The modeling assumes that the seismic wave propagates from left to right, and sets the time for the seismic wave to reach the left pier, the left middle pier, the right middle pier, and the right pier successively, so as to realize the traveling wave effect.
Fig. 1Finite element model with locally enlarged view of nonisolation and isolation
Fig. 2Mode of vibration
a) 1st order vibration type
b) 2nd order vibration type
c) 3rd order vibration type
d) 4th order vibration type
e) 5th order vibration type
f) 6th order vibration type
g) 7th order vibration type
h) 8th order vibration type
i) 9th order vibration type
j) 10th order vibration type
5. Velocity selection of seismic and apparent wave
The seismic fortification intensity of the bridge area is 8 degrees (0.2 g), the site category is class II, Three seismic waves, ElCentro, Taft and San Fernando, were selected, in order to coordinate with the standard reaction spectrum curve and compare the traveling wave effect of three seismic waves, the seismic wave peak acceleration value was adjusted to the same value, the amplitude modulation coefficient is 0.339, 0.784 and 0.387, input three kinds of seismic waves (Fig. 3). The action time is taken as 20 s, and apparent wave velocity of ground motion is selected as 100 m/s, 200 m/s, 300 m/s, 400 m/s, 500 m/s, 1000 m/s, 1500 m/s, 2000 m/s and infinity respectively.
Fig. 3Seismic wave
a) Adjusted ElCentro wave
b) Adjusted Taft wave
c) Adjusted San Fernando wave
6. Seismic response analysis
6.1. Internal force response of arch rib
Compare the maximum internal force value of the four positions of the arch ribs and the corresponding shock absorption rate of the isolation structure, the corresponding position is shown in Table 2. The contrast of the internal forces of the nonisolated structure and the isolated structure, and the shock absorption rate of the isolation structure are shown in Fig. 4 to Fig. 12.
Table 2Analyze the arch rib corresponding to the position
Position 1  Position 2  Position 3  Position 4 
Midarch vault  Midarch 1/4 rib (Left)  Left arch foot of left arch  Left arch foot of right arch 
By comparing Fig. 4 to Fig. 6, it can be obtained: (1) The shaft force and bending moment of middle arch vault position in nonisolated structure decrease with the increase of the apparent wave velocity, and the shaft force increase of the medium arch vault is most obvious when the apparent wave velocity of the Taft wave is 200 m/s; (2) Shaft force and bending moment of left arch foot position of left arch of nonisolated structure, shows basically decreased trend with the increase of apparent wave velocity under the action of the Elcentro wave, the Ushaped under the action of the Taft wave, and basically consistent under the action of the San Fernando wave; (3) The influence of apparent wave velocity on the internal force of the arch ribs of nonisolated structure is complex, and different seismic waves and different internal forces show different change curves; (4) Under the effect of different apparent wave velocity of three seismic waves, the shaft force and bending moment of each arch rib position have a similar curve trend for the nonisolated structure; (5) For nonisolated structure in the arch rib position 1, position 3, position 4, when the apparent wave velocity is small, the shaft force and bending moment is much larger than the infinity excitation, even more than 10 times, if the traveling wave effect is not taken into account, the internal force response of nonisolated structures may be seriously underestimated.
Fig. 4Comparison of nonisolated axial force
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 5Comparison of nonisolated shear force
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 6Comparison of nonisolated bending moment
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 7Comparison of isolation axial force
a) Position 1
b) Position 2
c) Position 3
d) Position 4
A comparison between Fig. 7 to Fig. 9, it can be obtained: (1) The shaft force and bending moment of middle arch vault position in isolated structure decrease with the increase of the apparent wave velocity, and the shaft force increase of the medium arch vault is most obvious when the apparent wave velocity of the Taft wave is 200 m/s; (2) Shaft force, shear force and bending moment of position 3 isolated structure, shows basically decreased trend with the increase of apparent wave velocity under the action of the ElCentro wave, the basically consistent under the action of the Taft wave, and the generally convex shape under the action of the San Fernando wave; (3) The axis force and the bending moment of position 2 and 4 of the isolated structure, and the shear force of position 1 increases with the growth of the apparent wave velocity; (4) Under the effect of different apparent wave velocity of three seismic waves, the shaft force and bending moment of each selected arch rib position have a similar curve trend for the isolated structure.
Fig. 8Comparison of isolation shear force
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 9Comparison of isolation bending moment
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 10Comparison of axial force aseismic ratio
a) Position 1
b) Position 2
c) Position 3
d) Position 4
Fig. 11Comparison of shear force aseismic ratio
a) Position 1
b) Position 2
c) Position 3
d) Position 4
A comparison from Fig. 10 to Fig. 12, is can be obtained: (1) the isolated structure has good isolation effect, except for ElCentro wave and Taft wave in speed of about 400 m/s shock absorption rate for the shaft force and shear force is less than 79.9 % in the 1/4 arch rib (left) position of middle arch rib, the shock absorption rate for the shaft force, shear force and bending moment is more than 79.9 % in the other analysis positions under the action of three kinds of seismic waves; (2) The influence of apparent wave velocity on the shock absorption rate for internal force of the arch ribs of is complex, and different seismic waves and different internal forces show different change curves, but the vast majority of the shock absorption rate curves show basically Ushaped with the wave speed increasing from small; (3) Except for the middle arch vault position, under the action of three seismic waves with different apparent wave velocities, the shock absorption rate of the axial force and the bending moment at the other analyzed positions have a similar curve trend.
Fig. 12Comparison of bending moment aseismic ratio
a) Position 1
b) Position 2
c) Position 3
d) Position 4
The time history diagram for axial force response of arch rib of different apparent wave velocities excitation of the ElCentro seismic wave in the nonisolated model is shown in Fig. 13, to consider the time history response effect of the traveling wave effect.
6.2. Maximum displacement of arch rib
The maximum displacement and rate of change of the middle arch ribs are shown in Table 3.
Fig. 13Axial force timehistories of arch rib in nonisolated model under the action of different apparent wave velocities excitation of ElCentro seismic waves
Table 3Maximum displacement (unit: mm) and rate of change in midarch rib
Seismic wave  Direction  Model and rate of change  Wave velocity  
100 m/s  200 m/s  300 m/s  400 m/s  500 m/s  1000 m/s  1500 m/s  2000 m/s  Infinity  
ElCentro  Direction $X$  Nonisolated  83.08  69.72  71.46  73.1  74.77  79.17  84.53  86.35  87.77 
Isolated  75.19  79.18  84.91  85.39  83.92  87.42  94.64  96.99  100.09  
Rate of change  9.5 %  13.6 %  18.8 %  16.8 %  12.2 %  10.4 %  12.0 %  12.3 %  14.0 %  
Direction $Y$  Nonisolated  0.11  0.15  0.1  0.13  0.11  0.15  0.21  0.24  0.25  
Isolated  0.03  0.04  0.04  0.03  0.03  0.06  0.06  0.06  0.06  
Rate of change  72.7 %  73.3 %  60.0 %  76.9 %  72.7 %  60.0 %  71.4 %  75.0 %  76.0 %  
Direction $Z$  Nonisolated  38.2  30.88  18.96  20.2  20.23  23.43  25.22  26.96  29.16  
Isolated  2.33  3.28  5.02  5.64  5.82  6.46  6.56  6.61  6.72  
Rate of change  93.9 %  89.4 %  73.5 %  72.1 %  71.2 %  72.4 %  74.0 %  75.5 %  77.0 %  
Taft  Direction $X$  Nonisolated  32.99  40.79  40.55  49.26  50.92  53.01  54.17  54.61  56.77 
Isolated  27.97  55.33  67.13  83.2  86.88  79.96  76.49  75.84  78.26  
Rate of change  15.2 %  35.6 %  65.5 %  68.9 %  70.6 %  50.8 %  41.2 %  38.9 %  37.9 %  
Direction $Y$  Nonisolated  0.12  0.14  0.12  0.15  0.12  0.14  0.18  0.21  0.22  
Isolated  0.04  0.02  0.04  0.03  0.02  0.05  0.05  0.05  0.06  
Rate of change  66.7 %  85.7 %  66.7 %  80.0 %  83.3 %  64.3 %  72.2 %  76.2 %  72.7 %  
Direction $Z$  Nonisolated  34.58  33.8  36.38  13.18  18.02  32.48  34.31  33.79  33.11  
Isolated  1.67  4.3  5.92  6.82  7.47  8.43  8.66  8.71  8.78  
Rate of change  95.2 %  87.3 %  83.7 %  48.3 %  58.5 %  74.0 %  74.8 %  74.2 %  73.5 %  
San Fernando  Direction $X$  Nonisolated  18.57  21.28  23.34  23.38  22.15  23  25.49  27.59  31.98 
Isolated  18.16  27.59  32.04  33.4  32.6  34.14  37.09  39.13  43.03  
Rate of change  2.2 %  29.7 %  37.3 %  42.9 %  47.2 %  48.4 %  45.5 %  41.8 %  34.6 %  
Direction $Y$  Nonisolated  0.13  0.13  0.08  0.11  0.1  0.1  0.14  0.16  0.17  
Isolated  0.03  0.12  0.02  0.02  0.02  0.05  0.04  0.03  0.02  
Rate of change  76.9 %  7.7 %  75.0 %  81.8 %  80.0 %  50.0 %  71.4 %  81.3 %  88.2 %  
Direction $Z$  Nonisolated  14.53  16.01  18.35  16.76  15.11  12.59  14.49  15.72  17.64  
Isolated  0.53  1.36  1.44  1.83  2.05  2.34  2.39  2.4  2.4  
Rate of change  96.4 %  91.5 %  92.2 %  89.1 %  86.4 %  81.4 %  83.5 %  84.7 %  86.4 %  
The mean of the three waves  Direction $X$  Average rate of change  8.97 %  26.29 %  40.55 %  42.86 %  43.35 %  36.56 %  32.89 %  31.01 %  28.81 % 
Direction $Y$  Average rate of change  72.11 %  55.58 %  67.22 %  79.58 %  78.69 %  58.10 %  71.69 %  77.48 %  78.99 %  
Direction $Z$  Average rate of change  87.76 %  79.53 %  74.20 %  61.93 %  65.36 %  65.46 %  66.30 %  49.90 %  50.15 % 
According to Table 3, under the multipoint excitation along the bridge considering traveling wave effect, for the arch rib displacement:
(1) It is larger in the longitudinal direction and vertical direction, but smaller in the transverse direction for the nonisolated structure; it is larger in the longitudinal direction, but smaller in the vertical direction and the transverse direction for the isolated structure.
(2) It shows increasing phenomenon in the longitudinal direction, reducing in the transverse direction, and obvious decreasing in the vertical direction.
6.3. Internal force response of the pier
Select the maximum internal force at the bottom of the left pier for comparative analysis.
Fig. 14Comparison of nonisolated internal force
a) Axial force
b) Shear force
c) Bending moment
By contrast from Fig. 14 to Fig. 16, with the increasing of the apparent wave velocity: (1) The maximum shaft force of the nonisolated bridge is roughly decreased and then increased, and the maximum shear force and bending moment are decreasing; (2) The maximum shaft force of the isolated structure bridge is fluctuating, but after considering the traveling wave effect it is greater than the consistent excitation response (infinity), the maximum shear force and the bending moment are decreasing under the action of the ElCentro seismic wave, and the change under the action of Taft wave and San Fernando wave is not obvious; (3) The maximum shaftforce aseismic ratio of the pier is roughly Ushaped, the maximum shear force is decreasing trend, the maximum bending moment is decreasing under the action of the ElCentro seismic wave, the change is not obvious under the action of the Taft wave and the San Fernando wave.
Fig. 15Comparison of isolated internal force
a) Axial force
b) Shear force
c) Bending moment
Fig. 16Comparison of internal forces and aseismic ratio in the isolation structure
a) Axial force
b) Shear force
c) Bending moment
6.4. Deck acceleration
The maximum absolute acceleration and aseismic ratio of the bridge deck are shown in Table 4.
Table 4Deck acceleration (Unit: cm/s2) and aseismic ratio
Seismic wave  Direction  Model and rate of change  Wave velocity  
100 m/s  200 m/s  300 m/s  400 m/s  500 m/s  1000 m/s  1500 m/s  2000 m/s  Infinity  
Elcentro  Direction $X$  Nonisolated  175.32  163.96  148.13  150.17  161.07  135.85  264.69  227.03  281.28 
Isolated  120.25  126.52  124.72  126.66  122.09  122.71  128.86  122.32  122.37  
Rate of change  31.4 %  22.8 %  15.8 %  15.7 %  24.2 %  9.7 %  51.3 %  46.1 %  56.5 %  
Direction $Y$  Nonisolated  12.53  11.93  11.99  12.13  12.49  12.54  12.34  12.6  12.71  
Isolated  12.87  12.12  12.72  12.27  12.45  12.4  12.5  12.98  12.32  
Rate of change  2.7 %  1.6 %  6.1 %  1.2 %  0.3 %  1.1 %  1.3 %  3.0 %  3.1 %  
Direction $Z$  Nonisolated  160.22  203.77  109.82  158.39  159.64  176.54  282.36  344.86  405.39  
Isolated  32.85  25.13  23.35  25.38  26.28  37.67  27.99  32.16  17.01  
Rate of change  79.5 %  87.7 %  78.7 %  84.0 %  83.5 %  78.7 %  90.1 %  90.7 %  95.8 %  
Taft  Direction $X$  Nonisolated  142.24  128.99  143.5  146.26  135.89  129.2  189.09  177.61  163.33 
Isolated  121.62  128.52  128.85  130.39  127.67  129.61  129.77  130.46  128.19  
Rate of change  14.5 %  0.4 %  10.2 %  10.9 %  6.0 %  0.3 %  31.4 %  26.5 %  21.5 %  
Direction $Y$  Nonisolated  13.18  13.22  12.82  13.26  12.58  12.59  12.76  12.56  12.7  
Isolated  13.15  13.18  12.74  13.08  12.53  12.61  12.7  12.43  12.49  
Rate of change  0.2 %  0.3 %  0.6 %  1.4 %  0.4 %  0.2 %  0.5 %  1.0 %  1.7 %  
Direction $Z$  Nonisolated  178  207.31  115.79  132.16  113.48  174.5  260.36  296.36  333.22  
Isolated  15.13  14.98  17.46  17.44  16.47  19.41  16.37  14.96  12.3  
Rate of change  91.5 %  92.8 %  84.9 %  86.8 %  85.5 %  88.9 %  93.7 %  95.0 %  96.3 %  
San Fernando  Direction $X$  Nonisolated  114.87  102.68  136.57  135.55  105.5  105.36  181.72  130.09  143.24 
Isolated  118.79  121.51  121.64  119.97  121.98  122.52  121.8  120.97  119.05  
Rate of change  3.4 %  18.3 %  10.9 %  11.5 %  15.6 %  16.3 %  33.0 %  7.0 %  16.9 %  
Direction $Y$  Nonisolated  7.26  7.49  7.55  7.37  7.05  7.15  7.36  7.31  7.17  
Isolated  7.63  7.41  7.62  7.25  7.26  7.05  7.62  7.3  7.02  
Rate of change  5.1 %  1.1 %  0.9 %  1.6 %  3.0 %  1.4 %  3.5 %  0.1 %  2.1 %  
Direction $Z$  Nonisolated  132.36  200.76  115.88  111.28  124.26  120.58  184.75  222.82  286.87  
Isolated  11.62  14.04  12.45  12.65  12.37  17.52  14.33  14.42  7.31  
Rate of change  91.2 %  93.0 %  89.3 %  88.6 %  90.0 %  85.5 %  92.2 %  93.5 %  97.5 %  
The mean of the three waves  Direction $X$  Average rate of change  14.17 %  1.62 %  12.31 %  12.67 %  4.88 %  2.31 %  38.55 %  26.56 %  31.63 % 
Direction $Y$  Average rate of change  2.53 %  0.07 %  2.13 %  0.61 %  0.75 %  0.79 %  1.45 %  0.61 %  2.27 %  
Direction $Z$  Average rate of change  84.50 %  87.62 %  82.51 %  82.42 %  83.88 %  76.47 %  82.61 %  61.88 %  64.04 % 
Available from Table 4, considering the traveling wave effect along the bridge under multipoint excitation, the absolute acceleration of the bridge deck is as follows: (1) Larger in longitudinal and vertical direction, smaller in transverse direction in nonisolated structure; larger in longitudinal direction, smaller in transverse and vertical direction in isolated structure; (2) For the isolated structure, the impact of the longitudinal wave speed is more complex, about slightly increased in the transverse direction, and decreasing obvious in the vertical direction.
6.5. Shear force and displacement of isolation support
The maximum shear force and displacement of the isolation support are shown in Table 5.
Table 5Maximum shear force and displacement of isolation support
Seismic wave  Wave velocity  100 m/s  200 m/s  300 m/s  400 m/s  500 m/s  1000 m/s  1500 m/s  2000 m/s  Infinity 
Elcentro  Shear force/ kN  120.04  181.72  227.39  269.83  294.97  332.37  340.9  344.07  349.39 
Displacement / mm  18.76  28.39  35.53  42.16  46.09  51.93  53.27  53.76  54.59  
Taft  Shear force / kN  96.73  234.03  324.83  355.61  368.08  406.69  420.27  425.06  431.71 
Displacement / mm  15.11  36.57  50.75  55.56  57.51  63.55  65.67  66.42  67.45  
San Fernando  Shear force / kN  30.29  69.03  81.47  87.59  93.82  103.87  102.25  100.22  102.8 
Displacement / mm  4.73  10.79  12.73  13.69  14.66  16.23  15.98  15.66  16.06 
It can be seen from Table 9 that the shear force and displacement of the isolation support increase with the increase of apparent wave velocity. The maximum shear force and displacement under infinity excitation are approximately 34 times that of the apparent wave velocity of 100 m/s.
7. Conclusions
From the comparative analysis above, it can be concluded that:
1) For this kind of structure, the isolation model has a very good isolation effect, under the action of earthquakes with different apparent wave velocities, the shock absorption rates of axial force, shear force, and bending moment of arch ribs are basically over 79.9 %;
2) The seismic structure should not ignore the effect of the wave effect, and part of the internal force will even increase by ten times if considering the wave effect;
3) The two structural models show similar curve movements with the change of the curve speed of the arch rib shaft force and bending moment;
4) The influence of apparent wave velocity on the internal force and aseismic ratio of arch ribs in isolated structure and nonisolated structure is complex, and different seismic waves show different curves;
5) The shear force and displacement of the isolation bearing increase with the increase of apparent wave velocity, and the maximum shear force and displacement under uniform excitation are approximately 34 times as much as the apparent wave velocity of 100 m/s.
Acknowledgements
This research was funded by the Fundamental Research Funds for the Central Universities (31920210078), the National Natural Science Foundation of China (51868067), the Project of Innovation Fund for Colleges and Universities in Gansu (2020B067), the Scientific and Technological Research Project of Gansu Provincial Construction Department (JK201910) and the Special Fund for Basic Scientific Research Operation Expenses of Central University (31920170068).
References

J. Ma and Y. Li, “Analysis of traveling wave effect on halfthrough CFST arch bridge by large mass method,” Key Engineering Materials, Vol. 540, pp. 21–28, Jan. 2013, https://doi.org/10.4028/www.scientific.net/kem.540.21

Y. Xu and G. C. Lee, “Traveling wave effect on the seismic response of a steel arch bridge subjected to near fault ground motions,” Earthquake Engineering and Engineering Vibration, Vol. 6, No. 3, pp. 245–257, Sep. 2007, https://doi.org/10.1007/s118030070761z

Q. Wu, M. Yoshimura, K. Takahashi, S. Nakamura, and T. Nakamura, “Nonlinear seismic properties of the Second Saikai Bridge,” Engineering Structures, Vol. 28, No. 2, pp. 163–182, Jan. 2006, https://doi.org/10.1016/j.engstruct.2005.05.003

A.R. Liu, Q.C. Yu, and J.P. Zhang, “Seismic response of long span continuous rigidframe arch bridge,” Shenzhen Daxue Xuebao (Ligong Ban), Vol. 24, No. 3, pp. 228–233, 2007.

Z.M. Hu, J.Y. Li, and T.Y. Huang, “Seismic response analysis of concretefilled steel tube arch bridge based on traveling wave effect,” (in Chinese), Bridge Construction, pp. 25–28, 2010.

Song Bo, Yi Hanbin, and Zhou Hongyu, “Seismic response analysis for longspan arch bridges under consideration of traveling wave effect,” (in Chinese), Journal of Beijing University of Technology, Vol. 37, No. 3, pp. 375–380, 2011.

Wang Hao et al., “Influence analysis of seismic traveling wave for longspan CFST arch bridge,” (in Chinese), Journal of Vibration Engineering, Vol. 25, No. 5, pp. 556–563, 2012.

D.Y. Zhang, X. Li, W.M. Yan, W.C. Xie, and M. D. Pandey, “Stochastic seismic analysis of a concretefilled steel tubular (CFST) arch bridge under tridirectional multiple excitations,” Engineering Structures, Vol. 52, pp. 355–371, Jul. 2013, https://doi.org/10.1016/j.engstruct.2013.01.031

R. Wang and L. Xu, “Earthquake response analysis with travellingwave for a longspan steel trussarch railway bridge” Advances in Structural Engineering, Vol. 16, No. 8, pp. 1365–1370, Aug. 2013, https://doi.org/10.1260/13694332.16.8.1365

Zhang Yongliang et al., “Influence of traveling wave effect on seismic response of a longspan decktype railway steel truss arch bridge,” (in Chinese), Zhendong yu Chongji/Journal of Vibration and Shock, Vol. 39, No. 12, pp. 213–220, 2020.

Wu Yuhua, Qi Xingjun, and Guo Jianfei, “Analysis of nonlinear seismic response of longspan CFST arch bridge under traveling wave effect,” Journal of Highway and Transportation Research and Development, Vol. 28, No. 1, pp. 80–85, 2011.

Lou Menglin and Tang Yu, “Consistent response spectrum method for horizontally travelling seismic wave response analysis of long span arch bridges,” (in Chinese), Journal of disaster prevention and mitigation engineering, Vol. 33, pp. 119–124, 2013.

Chen Yanjiang et al., “Stochastic seismic analysis of a CFST arch bridge under spatially varying ground motions,” (in Chinese), Gongcheng Lixue, Vol. 30, No. 12, pp. 99–106, 2013.

Liu Zhen, Han Xiaoyu, and Zhang Zhe, “Seismic response analysis of flyswallow type concretefilled steel tube arch bridge based on traveling wave effect,” (in Chinese), Journal of Sichuan University ( Engineering Science Edition), Vol. 47, No. 6, pp. 54–60, 2015.

Dai Gonglian and Wang Yu, “Seismic response and cushioning research of longspan railway continuous beamarch bridge,” (in Chinese), Journal of Huazhong University of Science and Technology (Natural Science Edition), Vol. 43, No. 7, pp. 19–23, 2015.

Tang Tang and Qian Yongjiu, “Traveling response analysis of largespan reinforced concrete arch bridge,” (in Chinese), Earthquake Engineering and Engineering Dynamics, Vol. 36, No. 2, pp. 111–116, 2016.