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In order to study the coupling vibration between a bridge and a train under the action of crosswind loads, a dynamic interaction model of the wind–train–bridge system is established considering the geometric nonlinear factors of a long-span suspension bridge. A calculation frame is composed, and a corresponding computer program is written. A long-s...
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... of which the main span is 1120 m. The configuration of the suspension bridge is shown in Figure 1. The stiffening girder of the suspension bridge is a Warren-type steel-truss girder with a height of 16 m, the internode length is 14 m, the main girder width is 26 m, and the bridge tower height is 160 m. ...
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... wind and bridge systems are coupled through the iterative calculation of the self-excited wind force. The interaction between the three is shown in Figure 10. The bridge data input files are formed by the bridge's natural vibration frequency and vibration shape calculated by the MIDAS. ...
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... to the formula in reference [42], the í µí±´, í µí±ª, and í µí±² matrix, and the í µí± load vector of the bridge and vehicle are calculated and the vehicle-bridge dynamic equilibrium differential equation of Equation (9) is formed and solved. The specific calculating process is shown in Figure 11. Finally, the bridge calculation program NWVB.FOR is developed by the FORTRAN Language. ...
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... train runs for 10 s under the excitation of track irregularity before boarding. The calculation process of solving the dynamic responses of the train and the bridge in each time step is shown in Figure 12, where cycle j is used to recalculate the structural stiffness based on the bridge displacement. Compared with the constant-value structural stiffness in the linear calculation, the nonlinear calculation is relatively slow, and the integration process is prone to nonconvergence, so the calculated efficiency is quite low. ...
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... order to analyze the influence of the large displacement effect on the vibration characteristics of the system, the wind-train-bridge coupling vibration calculation program WVB.FOR and the nonlinear train-bridge calculation program NWVB.FOR are written. In the calculation, the main point number of the key positions of the bridge is shown in Figure 13. The train running on the bridge is composed of one SS8 locomotive followed by 18 passenger cars; the train speed is 100 km/h; the track irregularity is the American class-5 spectrum; the bridge structure damping ratio is 2 %; the average wind velocity is 10 m/s. ...
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... integral step is 0.005 s, which is determined according to the running speed and program convergence. The maximum displacements of each bridge node obtained by the linear and nonlinear calculation programs are listed in Table 3, and the displacement and acceleration time history curves of some nodes at the main span are drawn in Figures 14 and 15. ...
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... Compared with the change in acceleration time history curves of each node in Fig- ures 13 and 14, the bridge displacement curves are more obviously changed. Since the first mode of the bridge is lateral, the lateral mode appears more frequently in the first 50th vibration modes. ...
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... vibration responses of the bridge are calculated when the vehicle speed is 100 km/h and the average wind velocity increases from 0 to 30 m/s. The time history curves of displacement and acceleration are at points ③, ④, and ⑤, and their changing tendency with wind velocity are shown in Figures 16-19 The corresponding maximum responses of each node are shown in Table 4. ...
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... can be seen from the lateral deformation time history curve of the bridge in Figure 16 that the lateral deformation of the midspan point ③ is much larger than that of the 1/4 midspan point ④, which conforms to the vibration law of bridge natural vibration modes. Based on the data in Table 3, it can be seen that without considering the wind loads, the lateral deformation of point ③ caused by the yaw force of the train is 3.0 cm, and the displacement of point ④ and point ⑤ is close to the maximum of 1.7 cm and 2.1 cm, respectively. ...
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... if the wind velocity increases to 30 m/s, the maximum displacement of point ③ reaches 114.8 cm, and points ④ and ⑤ also reach 73.9 cm and 75.5 cm, respectively, an increase of 38 times, 43 times, and 36 times. The slope of the maximum lateral deformation with wind velocity in Figure 16c is very large, which proves the sensitivity of the long-span suspension bridge lateral deformation to wind loads. • The vertical displacement time history curves of the bridge and the changing tendency of the maximum with the wind velocity are plotted in Figure 17. ...
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... slope of the maximum lateral deformation with wind velocity in Figure 16c is very large, which proves the sensitivity of the long-span suspension bridge lateral deformation to wind loads. • The vertical displacement time history curves of the bridge and the changing tendency of the maximum with the wind velocity are plotted in Figure 17. It can be seen that when the wind velocity is small (≤20 m/s), the vertical displacement of the bridge is not very sensitive to the wind loads. ...
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... can be seen that when the wind velocity is small (≤20 m/s), the vertical displacement of the bridge is not very sensitive to the wind loads. The time history curves of vertical displacement under different wind velocities in Figure 17a,b are close. The maximum vertical displacement of the bridge decreases slightly due to the wind loads. ...
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... when the wind velocity exceeds 20 m/s, the suspension bridge appears buffeted under the effect of the wind loads and the vertical vibration will intensify. The maximum vertical displacement of some nodes will show an increasing tendency, which can be clearly seen in Figure 17c. The lateral and vertical vibration acceleration time history curves of the bridge and the changing tendency of the maximum with the wind velocity are plotted in Figures 18 and 19, respectively. ...
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... maximum vertical displacement of some nodes will show an increasing tendency, which can be clearly seen in Figure 17c. The lateral and vertical vibration acceleration time history curves of the bridge and the changing tendency of the maximum with the wind velocity are plotted in Figures 18 and 19, respectively. It can be seen that when there are no wind loads, the maximum lateral acceleration at point ③ is 3.9 cm/s 2 , and at point ⑤ it is 0.5 cm/s 2 , but when the wind velocity increases to 30 m/s, the maximum values of points ③ and ⑤ are increased to 51.0 cm/s 2 and 43.1 cm/s 2 , an amplification of 13 times and 86 times, respectively; the amplification of the corresponding maximum vertical acceleration is 7 times and 8 times, respectively. ...
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Citations
... It is the key load-bearing component of the bridge, and its failure will cause the instability of bridge structure or even collapse [1,2]. In the service process, the suspension bridge is subjected to the coupling action of the dead load (stiffened beam, deck system, main cable, etc.), the wind load, the live load (automobile load, etc.) and the salt spray environment (NaCl, MgCl 2 , etc.) [3][4][5][6]. These lead to the friction, corrosion, and fatigue behaviors between the parallel wires inside the main cable at the main saddle. ...
The main cable bent around the saddle of the suspension bridge is subjected to the wind, the vehicle, the bridge’s own weight and the corrosive media. The coupling of the three loads and the environments causes the friction, the corrosion, and the fatigue (tribo-corrosion-fatigue) among the wires inside the main cable. In this paper, a wire bending tribo-corrosion-fatigue test rig was designed and developed. The effect of fatigue load on the bending friction behaviors between the cable wires in ultrapure water and 3.5% NaCl solution was explored. The tribological properties and electrochemical corrosion behaviors under different fatigue loading ranges were investigated. The tribo-corrosion-fatigue interaction between the cable wires was quantitatively characterized, and the mechanism of the interaction was analyzed. The results demonstrate that the increasing fatigue load exacerbates the coupling damage of the cable wires attributed to the enhanced interaction. The findings carry theoretical importance when assessing the main cable’s deterioration and the load-bearing safety of a suspension bridge.
... However, it is very complicated for the finite-terms Galerkin discrete model or the direct perturbation method to obtain the solutions for the nonlinear equations. In fact, the most convenient and effective method of solving the mechanical problems of continuous structures is the finite element method [30][31][32][33][34][35][36]. This method is widely used in static and dynamic problems of structures [33][34][35][36]. ...
A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.
... Lateral pretension cables can improve the aerodynamic stability of small-stiffness suspension bridges [18]. The vibrations of suspension bridges on complex loads, such as wind and heavy vehicle loads, still need further research [19]. Recent studies suggested that the vortex-excited vibrations of cylinders are not fully understood. ...
... Equation (19) shows that the static wind load produces a static deformation, f 0 U 2 /ω 2 . This deformation is inversely proportional to the linear stiffness of the bridge. ...
... This also means that the present research ignores the effect of the static wind load on the natural frequency of the bridges. Substituting Equation (19) into Equation (17) obtains ...
A low stiffness makes long-span suspension bridges sensitive to loads, and this sensitivity is particularly significant for wind-induced nonlinear vibrations. In the present paper, nonlinear vibrations of suspension bridges under the combined effects of static and vortex-induced loads are explored using the nonlinear partial differential–integral equation that models the plane bending motion of suspension bridges. First, we discretized the differential–integral equation through the Galerkin method to obtain the nonlinear ordinary differential equation that describes the vortex-induced vibrations of the bridges at the first-order symmetric bending mode. Then, the approximate analytical solution of the ordinary differential equation was obtained using the multiple scales method. Finally, the analytical solution was applied to reveal the relationships between the vibration amplitude and other parameters, such as the static wind load, the frequency of dynamic load, structural stiffness, and damping. The results show that the static wind load slightly impacts the bridge’s vibrations if its influence on the natural frequency of bridges is ignored. However, the bridge’s vibrations are sensitive to the load frequency, structural stiffness, and damping. The vibration amplitude, as a result, may dramatically increase if the three parameters decrease.
In order to prevent skid accidents of main cable of suspension bridge, dynamic contact behaviors between saddle materials and steel wires were explored in the present study. Dynamic contact tests of saddle materials were conducted employing the self-made dynamic contact test rig. Dynamic contact behaviors of the saddle material (transverse slip, longitudinal deformation, contact state, wear mechanism) in a friction cycle were investigated. Effects of transverse and longitudinal locations, friction cycles and saddle materials on the dynamic contact behaviors were presented. The results show that dynamic contact states between saddle specimen and steel wireexhibits the adhesion-gross slip-adhesion-gross slip state in a friction cycle. The transverse slip and longitudinal deformation of saddle specimen both decrease with increasing distance to the contact interface, decrease along the transverse sliding direction, and increase with increasing friction cycles. The ZG270-480 saddle material exhibits the largest average transverse slip and longitudinal deformation, while ZG275-485 saddle material shows the smaller values.
The long-span multi-tower suspension bridge is widely used in the construction of river and sea crossing bridges. The load-bearing safety and anti-sliding safety of its main cable are directly related to the structural safety of a suspension bridge. Failure mechanisms of the main cable of a long-span multi-tower suspension bridge are discussed. Meanwhile, the tribo-corrosion-fatigue of main cable, contact, and slip behaviors of the saddle and service safety assessment of the main cable are reviewed. Finally, research trends in service safety assessment of main cable are proposed. It is of great significance to improve the service safety of the main cable and thereby to ensure the structural safety of long-span multi-tower suspension bridges.
