Study on Dynamic Characteristics of Long-Span Highway-Rail Double-Tower Cable-Stayed Bridge

Abstract

The long-span dual-purpose highway-rail double-tower cable-stayed bridge has the characteristics of a large span and large load-bearing capacity. Compared with the traditional cable-stayed bridge, its wind resistance and seismic resistance are weaker, and the dynamic characteristics of the bridge are closely related to the wind resistance and seismic bearing capacity of the bridge. This study investigated the influence of the variations of bridge member parameters on the dynamic characteristics of the bridge and then improved the dynamic characteristics of the bridge. To provide the necessary experimental theory for the research work of the long-span dual-purpose highway-rail double-tower cable-stayed bridges, this paper takes the world’s longest span of the dual-purpose highway-rail double-tower cable-stayed bridge as the background, using the finite element analysis software Midas Civil 2022 v1.2 to establish a three-dimensional model of the whole bridge by changing the steel truss beam stiffness, cable stiffness, pylon stiffness, and auxiliary pier position, as well as study the influence of parameter changes on the dynamic characteristics of the bridge. The results show that the dynamic characteristics of the bridge can be enhanced by increasing the stiffness of the steel truss beam, the cable, and the tower. The stiffness of the steel truss beam mainly affects the transverse bending stiffness and flexural coupling stiffness of the bridge. The influence of cable stiffness is weak. The tower stiffness can comprehensively affect the flexural stiffness and torsional stiffness of the bridge. The position of auxiliary piers should be determined comprehensively according to the site conditions. In practical engineering, the stiffness of components can be enhanced according to the weak links of bridges to improve the dynamic characteristics of bridges and save costs.

1. Introduction

To better fit the development of urbanization, nowadays, the dual-purpose bridge is a widely used bridge type, which needs both a road and a track. Long-span dual-purpose highway-rail cable-stayed bridges are often more complex in structural form and have a larger cross-section due to the need to accommodate both highway and railroad traffic, which may result in a correspondingly larger wind area. Due to the structural characteristics of long-span dual-purpose highway-rail cable-stayed bridges, the lateral forces exerted on them by wind forces may be greater. This may lead to greater deformation and vibration of the bridge structure, thus affecting its wind stability. Long-span dual-purpose highway-rail cable-stayed bridges with large spans and complex structures may have longer self-oscillation periods. Under the action of an earthquake, the long period of ground vibration may lead to larger displacement and vibration of the bridge structure, thus affecting its seismic performance. This suggests that long-span dual-purpose highway-rail cable-stayed bridges for both public and railroads may have relatively weaker wind and seismic capacities compared to conventional cable-stayed bridges. The dual-use characteristics of long-span highway-rail dual-use cable-stayed bridges [1,2] can effectively cut construction times and costs while also resolving the issue of limited land use. When compared to standard cable-stayed bridges, the steel truss girder of the dual-purpose cable-stayed bridges are mostly steel structures, which lowers the dead weight and allows for a wider span. The appearance of an orthotropic steel bridge panel replaces the concrete bridge panel and further reduces the dead weight. Therefore, the cable-stayed bridge has a good economy, environmental protection, and sustainable development [3,4]. Long-span dual-purpose highway-rail bridges have been the subject of extensive domestic and international research [5,67,8,9]. These include changes in the dynamic characteristics and static responses of sea-crossing bridges, highway bridges, railroad bridges, cable-stayed bridges, etc., caused by wind, traffic, and other loads, mainly focusing on static characteristics, seismic design, and reasonable system design. The study of dynamic characteristics of bridges is quite rich in application examples in bridge engineering [1011,12,13,14,15]. By measuring the vibration data of bridges in real-time and combining it with dynamic analysis, the operational status of bridges can be monitored, and possible damages or problems can be detected in time, which can provide a scientific basis for the maintenance and management of bridges. It also plays an important role in the design and optimization of new bridge structures. For example, damage detection based on modal shapes, which utilizes an EFT-based approach to identify constant model deviations, was validated for its applicability to infrastructure [16]. Advanced technologies are also important for health monitoring. For example, DBSCAN-based automated operational modal analysis algorithms monitor structural integrity by estimating modal parameters of vibration [17]. Studying the dynamic properties of long-span dual-purpose highway-rail bridges is crucial because there is currently a dearth of research and data on this sort of dual-purpose long-span bridge.

The upper and lower two-story structures, or the left and right parallel structures, are more sensitive to the effects of longitudinal wind load, temperature load, and live load, and there are obvious differences in the resistance ability of the dual-purpose bridges of different systems. At present, there are few studies on the dynamic characteristics of dual-purpose bridges and most of them stay in the stage of theoretical research. Mutashar et al. [18] focused on studying the influence of wind load, rain load, and static load on long-span dual-purpose highway-rail bridges and found that rainfall would increase the wind load resistance of the bridge deck and cause changes in the wind field, thereby reducing the resistance of the bridge structure to the wind load. Siwowski et al. [19] found that a new type of glass fiber polymer (GFRP) can be used for bridge steel truss beams. The new type of GFRP beam is composed of a trapezoidal steel truss beam and bridge panel, which has strong stiffness, strength, and ultimate bearing capacity. It can be used to replace the steel truss beam of the dual-purpose bridge to increase the span and has a high application value. Based on the results of the static load test, Ademovic [20] discovered that the temperature had a significant impact on the correlation that existed between the structure’s dynamic properties and stiffness. In a comprehensive investigation of the static and dynamic properties of steel arch beams, Gara et al. [21] compared the numerical parameters derived from the finite element model of bridge structures with the experimental modal parameters obtained at various loading stages. They discovered that the loading mode and size had a greater impact on the transverse stiffness. However, little research has been done on the effects of the changes in bridge member parameters on the dynamic characteristics of bridges, and this research is necessary to improve the dynamic characteristics of bridges and thus save costs, which is conducive to the sustainable development of bridges.

Numerous factors affect the dynamic characteristics of bridges, such as environmental factors, material selection, loads, and structural forms. Environmental factors such as temperature, humidity, wind speed, etc., will affect the dynamic characteristics of the bridge. The influence of material selection on the dynamic characteristics of the bridge is mainly reflected in the elastic modulus, Poisson’s ratio, damping ratio, and other parameters of the material. The changes in these parameters will directly affect the stiffness, damping, and other characteristics of the bridge structure. Studies on long-span dual-purpose highway-rail double-tower cable-stayed bridges of similar type are informative. For example, nonlinear time course analysis and nonuniform seismic analysis are used to study the influence of structural parameter changes and traveling wave effects on the seismic response of steel truss cable-stayed bridges with a single tower and a single-cable plane. It was concluded that the influence of the variations of steel truss beam stiffness parameters on the structural internal force of single-tower, single-surface steel truss girder cable-stayed bridges is not prevalent [22]. A double-pylon cable-stayed bridge with a single-cable plane and steel truss girder with a similar structure to the research object of this paper was investigated regarding the influence of a stayed cable surface arrangement, the concentration of the dead load and structural system on the dynamic characteristics of a double-pylon cable-stayed bridge with a single-cable plane and steel truss girder through the dynamic load test [23]. This study employs a long-span cable-stayed bridge in Chongqing—a dual-purpose public railway bridge—as an example to examine the dynamic properties of the bridge produced by parameter changes. The bridge model was modeled using the finite element analysis software Midas Civil. The impact of varying the stiffness of the steel truss beam, cable, pylon, and auxiliary pier on the transverse bending resistance, vertical bending resistance, and flexural coupling performance of the bridge was examined, and the regularity of its influence on the bridge’s dynamic characteristics was summarized. The influence of the variations of bridge member parameters on the dynamic characteristics of the bridge obtained in this paper provides a valuable experimental basis for the current research on the dynamic characteristics of long-span dual-purpose highway-rail cable-stayed bridges. The discovery of this influence law, to a certain extent, can improve the dynamic characteristics of the bridge in the actual project, save production costs, and realize the sustainable development of bridges.

2. Bridge Background

The longest dual-purpose highway-rail bridge in the world is a long-span steel truss cable-stayed bridge located in Chongqing, China. As illustrated in Figure 1a, the bridge has a total length of 1622 m, with a main span of 660 m, a side span of 362 m, and an approach length of 865 m. The steel truss beam configuration is double-layer, with two-way eight-lane lanes in the upper layer that are developed by the urban expressway’s 60 km/h speed standard. There is two-way rail activity on the lower level. The main tower is a drop-shaped concrete pylon and is made of C55 concrete with a compressive strength of 55 MPa, which is a high-strength concrete with excellent compressive properties and durability; thus, it can withstand greater loads, is more resistant to earthquakes and winds, and is less susceptible to environmental factors during long-term use. The main tower foundation adopts an integral cap to connect with the pile group foundation. The total height of the main tower is 236 m, and 20 pairs of stay cables are set at a distance of 15.0 m. The local fundamental earthquake intensity is VI degrees, the bridge seismic fortification intensity is VII degrees, the peak acceleration of ground motion is 0.05 g, and the bridge fortification grade is class A. Following its completion, the bridge will stand as the longest rail transportation bridge in the world. Its features include the following:

• It is the longest dual-purpose bridge in the world, with a total length of 1622 m and a main span of 660 m.
• The main girder of the bridge is a steel truss structure with a width of 38 m.
• The pylon is an aesthetical and unique structure of a “water drop”.

3. Method Introduction

This paper takes a large-span cable-stayed bridge in Chongqing as an example, and the finite element model of the bridge is established by using the finite element software Midas Civil while maintaining the consistency between the model and the actual structure. After the model is established, the dynamic characteristics of ten modes of the bridge are analyzed, and then the method of controlling variables is used to change the steel truss beam stiffness, cables stiffness, bridge pylon stiffness, and position of auxiliary piers, respectively, to observe the effect of the changes in the above parameters on the dynamic characteristics of the bridge, to conclude.

Of course, there are some limitations to the existing methods.

Due to the immaturity of the conditions, this paper lacks relevant research on the change rule of the dynamic characteristics of the bridge caused by the parameter changes in each component of the bridge in the real environment, and it is necessary to carry out more research on the law of its influence if it is necessary to popularize the application of this discovery in the actual project.

This paper temporarily considered changing the steel truss beam stiffness, cable stiffness, pylon stiffness, and auxiliary pier position. These four parameter changes have certain limitations, and other component parameter changes will also have an impact on the dynamic characteristics of the bridge; however, it will allow for a more systematic study of the impact of other parameter changes on the dynamic characteristics of the bridge.

In this paper, the four parameters are changed to explore the changes in the dynamic characteristics of the bridge, and the laws found are used to improve the dynamic characteristics of the bridge. It is expected that more and better methods can be found to improve the dynamic properties of bridges and save construction costs, which is still to be conducted.

4. Computational Theory

The undamped free vibration equation for bridges is commonly used to describe the vibration behavior of bridge structures in the absence of external damping forces. This vibration is usually due to the balance between the bridge’s elastic restoring forces and inertial forces. In the undamped case, the vibration of a bridge can be simplified to a single-degree-of-freedom system, and the undamped free vibration equation of a bridge can be derived from the structural dynamics as follows:

$$[K^{*}-{\omega }^2{M}^{*}]\left\{a\right\}=0$$

(1)

where K is the generalized stiffness matrix, M is the generalized mass matrix, ω represents the frequency of the free vibration circle, and a stands for the eigenvector. This equation is the derived equation of the Cramer rule. The original equation of the Cramer rule is

$$\left|K^{*}-{\omega }^2{M}^{*}\right|=0$$

(2)

5. Finite Element Method (FEM) Modeling of the Bridge

Midas Civil has highly developed modeling and analysis capabilities that allow for accurate modeling to visualize the effects of parameter changes on properties. Therefore, in this study, the finite element model of the bridge was developed using the finite element software MidasCivil 2022 v1.2. To ensure coherence between the model and the actual construction, we used space truss elements to replicate the steel truss beams. Due to its unique shape, we simulated the pylon using a multi-variable cross-section group. To achieve a more uniform and symmetrical distribution of forces on the tension cables that connect the two pylons, we chose truss elements instead of pure tensile elements. Ernst’s formula was used to adjust the modulus of elasticity of the cables, aligning them with their actual nonlinear characteristics. The materials used in the finite element model are consistent with the actual bridge conditions to ensure the accuracy and reliability of the structural analysis. The plate element represents the orthotropic bridge panel in the simulation. The dead weight is determined by the quantities of steel and concrete used in the actual construction, considering the impact of the bridge’s phase II load. Elastic connections are established between the pylon and steel truss beam, as well as between the auxiliary pier and steel truss beam, depending on the engineering requirements. The right pylon’s bottom is fixed, while the other piers have hinged connections. A comprehensive analysis of the bridge’s dynamic properties was conducted after constructing the model.

Table 1. Material parameters of the bridge.

Component Names	Poisson Ratio	Elastic Modulus (GPa)	Mass Density (kg/m3)	Linear Expansion Coefficient (10−5 °C)
Pylon (C55 concrete)	0.2	35.5	2.71	1.0
Stayed cables	0.3	195	8.01	1.2
Q370 Steel	0.3	206	7.85	1.2
Longitudinal connection of beams	0.3	54	7.85	1.2

displays the bridge’s material parameters, and Figure 2 depicts the Midas Civil model.

6. Parameter Analysis

6.1. Analysis of Dynamic Characteristics

The frequency spectrum of the cable-stayed bridge is denser compared to a conventional cable-stayed bridge due to its extensive main span. To enable the load to have more vibration modes within the frequency range, the dynamic properties of this cable-stayed bridge were analyzed using the tenth-order mode. The characteristics of the bridge structure are typically assessed using the undamped free vibration equation, which takes into account the initial stress effects caused by dead weight. In a vertical plane, the deadweight of each location equals the mass of the overlying unit area. To compute the frequency and mode shapes, the subspace iteration method [24,25,26,27] is utilized.

Table 2. Frequency and shape characteristics of the first 10 order natural vibrations of the bridge.

Orders	Natural Frequency (Hz)	Period (s)	Modal Characteristics
1	0.22	4.57	The transverse bending of the first order occurs in the main span of the steel truss.
2	0.25	3.98	Steel truss undergoes first-order symmetric vertical bending.
3	0.34	2.91	Bending–torsion coupling between steel truss and girder on left side
4	0.35	2.87	Symmetrical torsion of steel truss girder
5	0.47	2.12	Steel truss undergoes second-order symmetric vertical bending.
6	0.62	1.61	Bending–torsion coupling of steel truss girders of the first order
7	0.63	1.58	Steel truss undergoes third-order symmetric vertical bending.
8	0.82	1.23	Fourth-order symmetrical vertical bending of steel truss girder
9	0.93	1.08	Second-order bending–torsion coupling of steel truss girder
10	0.97	1.03	Third-order bending–torsion coupling of steel truss girder

shows the characteristics of the natural frequency, the period, and the mode shapes of the first 10 modes of the bridge, while Figure 3 illustrates some of the mode shapes.

Table 2 displays the vibrational modes of the dual cable plane and dual rail cable-stayed bridge, which have the following basic characteristics:

• The mode shapes are transverse vibration, vertical vibration, torsional, and bending–torsion coupling. The torsion occurs in the fourth-order mode, and the fundamental reason for the torsion is that the bridge width is too large, and the cable surface is subjected to too large of a torsional force, so the torsional stiffness and frequency are reduced. When the bridge is subjected to a transverse load such as wind load and earthquake load, the resistance ability of the bridge structure should be considered. Because the stiffness of bridge structures, such as steel truss beams, pylons, cable stay, and pier, is different, the vibration patterns of bridge beams are mainly affected by the stiffness of components, and the first ten steps of this bridge are all the vibration patterns of steel truss beams.
• The long-span dual-purpose highway-rail cable-stayed bridge, with its extensive span and numerous degrees of freedom, exhibits a longer natural vibration period compared to traditional cable-stayed bridges. The transverse vibration of the steel truss beam, characterized as the bridge’s fundamental mode, arises due to the relatively lower stiffness of its transverse components compared to the longitudinal ones. Across all tenth-order mode shapes of the bridge, the maximum frequency remains below 1 Hz. This is attributed to the high stiffness of each bridge component, effectively limiting the vibration of the entire bridge structure.
• Table 2 shows that the mode distribution of the long-span dual-purpose highway-rail cable-stayed bridge is highly proximate. Specifically, within the frequency range of 0.22 Hz to 0.97 Hz, 10 order modes are densely clustered. Consequently, to ensure precision in analysis, when evaluating the dynamic performance of large-span dual-purpose highway-rail cable-stayed bridges, it is recommended to use multi-mode analysis.

6.2. Sensitivity Analysis of the FEM Parameters

Changes in the frequency parameters of a bridge structure can directly affect the overall structural integrity of the bridge. Such changes may further affect the force state and force transmission path of the bridge and may even lead to structural instability or damage. Secondly, changes in the frequency parameters of bridge structures have a significant effect on the dynamic responses of bridges. Changes in frequency parameters will change the vibration characteristics and response modes of the bridge. This may not only affect the usability and comfort of the bridge but also pose a threat to the structural safety of the bridge. To analyze the influence of the changes in structural parameters on the dynamical properties of the bridge, the rigidity of the steel truss, the rigidity of the cable, the stiffness of the pylon, and the position of the auxiliary pier have changed now [28,29,30,31,32,33,34,35,36,37,38]. For the convenience of illustration, the first 10 order modes of the bridge are numbered as M1~M10. It is important to note that the investigation of the pattern of change in the dynamic properties of the bridge by varying the stiffness of the steel truss beams, cables, and bridge pylons is conducted with the known frequency content of the seismic and wind dynamic inputs expected at the site, and the resulting new natural frequencies should be compared to the known excitation frequencies.

6.2.1. Stiffness of Steel Truss Beam

Other parameters of the bridge structure remain unchanged, and the stiffness of the steel truss beam is changed to 0.6, 0.8, 1.2, 1.4, and 1.6 times. The effects of the rigidity change in the structural framework on the vibration mode, the flexural and torsional mode, and the coupling mode of the cable stay bridge are analyzed. The influence of the variations in the stiffness of the steel truss on the bending stiffness and the bending coupling stiffness of the bridge structure is shown in Figure 4.

In the figure, it can be seen that when the stiffness of the steel truss beam increases, the frequency of the first 10 orders of the vibration pattern also tends to increase. However, the amplitude of the increase is different. As shown in the figure, M1, M2, M5, M7, and M8 have a slightly smaller increase, i.e., the increased amplitude of the bending mode is smaller, and M10 has a larger increase, i.e., the increased amplitude of the bending–torsion coupling mode is slightly larger. The M3 and M4 amplitudes are flat, and the M6 and M9 amplitudes also rise slightly. When the rigidity of the bridge steel girder is changed from 0.6 to 1.4, the first-order mode (M1), which is the fundamental frequency, has a frequency increase of 19.09% in the bending mode, while the smallest second-order mode (M2) has a frequency increase of only 4.35%. Although the frequency of the first mode only increased by 0.04 Hz, its frequency only increased by 0.20 Hz, and the frequency increased more frequently, indicating that with the increased stiffness of the steel girder, the transverse bending resistance of the bridge improved more than the vertical bending resistance of the bridge. By increasing the rigidity of the steel truss beam, the frequency of the bridge structure increases, which indicates that when the bending capacity of the large-span dual-purpose highway-rail cable stay bridge is insufficient, it may be improved by increasing the rigidity of the steel truss beam. However, in the bending–torsional coupling mode, with the increased stiffness of the steel truss beam, the frequency increase rate of the tenth-order mode is 15.06%, and its frequency increase value is as high as 0.14 Hz, which is 3.5 times the frequency increase value of the first-order mode (M1). The growth rate of the ninth-order mode (M9) is only 7.69%, but due to its high-frequency value, its frequency growth value also reaches 0.07 Hz, which is higher than the frequency growth value of the first-order mode (M1). Obviously, with the stiffness increase in the steel truss, the increase in the bending–torsional coupling ability of the bridge is greater than that of the simple bending resistance of the bridge, which indicates that the bending–torsional coupling ability of the bridge structure is more sensitive to the change in stiffness of the iron truss. As the frequency of use of the bridge structure increases, its dynamic characteristics improve. For long-span highway/rail/common-rail cable-stayed bridges, the steel truss girders primarily support vehicle and train loads. These loads are transferred to the bridge pylon via the cable stays. As a result, the rigidity of the steel truss is enhanced, reducing the load on the cables and pylon. This improvement improves the overall stiffness and load-carrying capacity of the bridge construction. The improved dynamical properties suggest that the bridge can withstand greater lateral loads, such as wind and seismic forces. To further increase the rigidity of the steel truss, one may consider upgrading the steel grade, densifying the steel truss, or strengthening the bolt anchorage.

6.2.2. Stiffness of Cables

Other parameters of the structure of the bridge remain unaltered, and the rigidity of the rope is changed to 0.6, 0.8, 1.2, 1.4, and 1.6 times. The effect of the variations in the cable rigidity on the vibration mode, the flexural and torsional mode, and the coupling mode of the cable-stayed bridges are analyzed. The influence of the variations in cable rigidity on the bending stiffness and the bending coupling stiffness of the bridge is shown in Figure 5.

As can be observed in the figure, the bridge frequency shows an upward trend as the cable rigidity increases, but it is worth noting that the frequency change in each mode is relatively small. Among the bending modes, the fifth order (M5) has the largest frequency increase rate, which is 14.29%, but the frequency appreciation is very small and is only 0.06 Hz. The revaluation of the eighth-order mode (M8) is the largest at the frequency of 0.06 Hz, which is only 0.01 Hz higher than that of the fifth-order mode. The frequency increase rates of the first-order mode (M1) and the second-order mode (M2) are 1.21% and 0.66%, respectively, which means that the feedback obtained by improving the cable stiffness is small for both the transverse and vertical bending resistance of the bridge. In the bending–torsional coupling mode, the frequency increase rate of the tenth-order mode (M10) is 9.59%, and the frequency appreciation reaches 0.09 Hz. In general, with the improvement in cable rigidity, the frequency values of the whole tenth-order mode increase slightly, and the average growth rate is only 5%, which means that with improvement in cable rigidity, improvement in the bending resistance and bending coupling performance of the bridge structure is weak. Therefore, in the process of bridge design and structure, if the dynamic characteristics of bridges need to be improved, the priority of improving cable stiffness is low.

6.2.3. Stiffness of Bridge Pylon

Other parameters of the bridge structure remain unchanged, and the stiffness of the pylon is changed to 0.6, 0.8, 1.2, 1.4, and 1.6 times. The influence of the change in pylon stiffness on the vibration mode, bending and torsion mode, and coupling mode of the cable-stayed bridge are analyzed. The influence of tower stiffness variations on the flexural stiffness and flexural coupling stiffness of the bridge structure is shown in Figure 6.

As can be observed in the figure, as the rigidity of the pylon increases, the frequency of the entire tenth-order mode exhibits an upward trend. When the rigidity coefficient of the pylon is 0.6, the frequency values of the ninth-order mode (M9) and the tenth-order mode (M10) are 0.81 Hz and 0.92 Hz, respectively, with a difference of 0.11. When the stiffness coefficient of the pylon is 1.6, the frequency difference between the two is only 0.01 Hz. This shows that when the bridge encounters a small flexural and torsional coupling force, increasing the tower stiffness can significantly improve the flexural and torsional coupling abilities of the bridge structure. However, when the bending–torsion coupling force is large, improvement in pylon stiffness is not good for improvement in the flexural and torsional coupling resistance of the bridge. In the flexural mode of the structure, the overall frequency increase rate is small, the growth rate of the third-order mode (M3) reaches 33.45%, and the frequency increase value is 0.24 Hz. The lowest growth rate is the first-order mode (M1), with a growth rate of only 8.85%, which shows that the anti-vertical flexural capacity of the bridge structure is more sensitive to the change in tower stiffness than the anti-transverse flexural capacity. Although a mere increase in the stiffness coefficient from 1.0 to 1.2 results in a minor lifting rate of 0.11% for the second-order mode, a decrease in the pylon’s stiffness coefficient from 1.0 to 0.8 leads to a significant reduction of 9% in the bridge’s natural vibration frequency. This reduction indicates a decline in the bridge structure’s resistance to vertical bending, indicating that as pylon stiffness increases, the anti-vertical bending capacity of the bridge structure decreases. In terms of the bridge’s coupled bending–torsional mode, enhancing the tower stiffness notably boosts the bridge’s overall frequency. Although the frequency growth rate for the ninth-order mode is relatively low at 31.92%, the actual frequency increase amounts to 0.25 Hz. Even the tenth-order mode, which experiences the slowest growth rate, still registers a frequency increase of 0.18 Hz. However, when the pylon rigidity factor increases from 1 to 1.6, the increase in the frequency of the tenth-order mode is only 0.08 Hz. Conversely, when the pylon rigidity factor decreases from 1 to 0.6, the frequency of the tenth-order mode decreases by 0.11 Hz, which is 1.45 times the growth rate. These observations highlight the sensitivity of the flexural and torsional coupling capacities of the bridge structure to reductions in pylon stiffness. Therefore, during bridge construction and design, it is crucial to avoid compromising the pylon’s stiffness, as this could significantly weaken the bridge’s seismic performance and wind resistance.

6.2.4. Position of Auxiliary Piers

With all other parameters of the bridge structure remaining constant, the auxiliary pier is shifted towards the pylon and abutment by one or two segments, respectively. We evaluated the impact of such positional adjustments on the cable-stayed bridge’s vibration mode, bending and torsion mode, and coupling mode. Specifically, Figure 7 illustrates how changes in the auxiliary pier’s location influence the bridge structure’s flexural stiffness and flexural coupled stiffness.

As can be seen in the figure, the auxiliary piers are already in their optimal positions, and any displacement in either direction will result in a decrease in the overall frequency value of the bridge. Specifically, regarding the bending mode, the positional adjustment of the auxiliary pier has minimal impact on the frequency value. The slight variance observed in the first-order mode (M1) is merely 0.003 Hz, resulting in a frequency increase rate of only 1.34%. This indicates that the placement of the auxiliary pier has a negligible effect on the transverse bending resistance of the bridge. The change difference in the eighth-order mode (M8) was 0.11 Hz, and the frequency increase rate was 15.7%. However, the change difference in the second-order mode (M2) was 0.003 Hz, and the change rate was only 1.3%, which means that the bridge structure can obtain better vertical bending resistance by adjusting the position of the auxiliary pier when the bridge is under large vertical bending force. In the bending–torsion coupled mode, the change in the auxiliary pillar position causes a large change in the bridge’s frequency value. In the tenth-order mode (M10), after the auxiliary pier position moves two segments from the original bridge arrangement to the abutment direction, the frequency decrease rate reaches 8.80%, and the frequency difference is 0.08 Hz, which is 27 times the first mode. It means that the bending and torsional capacities of the bridge are sensitive to the position of the auxiliary pier, and even a small deviation of the auxiliary pier can significantly reduce the bending and torsional capacities of the bridge. Therefore, in bridge construction and design, if it is found that the flexural and torsional coupling forces of bridges are too large, a reasonable auxiliary pier location can be found to enhance the total capacity of the bridge.

By improving the dynamic characteristics through the above parameter changes, the response of the bridge structure to vibration can be adjusted to reduce the occurrence of resonance phenomena and reduce the amplitude of vibration. At the same time, it can also increase the bridge structure’s ability to dissipate and absorb vibration energy and significantly improve the bridge’s load-carrying capacity and vibration resistance.

Combining the above conclusions, the stiffness of the steel truss beam, the stiffness of cables, the stiffness of the bridge pylon, and the position of auxiliary piers on the mode vibration frequencies are shown in

Table 3. Influence of the variations of each parameter on the frequency of the vibration.

Variation of Parameters	Variations in Vibration Frequency by Order
	1	2	3	4	5	6	7	8	9	10
Stiffness of steel truss beam (rise)	Slight increase	Tiny increase	Tiny increase	Tiny increase	Slight increase	Slight increase	Slight increase	Slight increase	Slight increase	Increase significantly
Stiffness of cables (rise)	Tiny increase	Tiny increase	Tiny increase	Tiny increase	Slight increase	Tiny increase	Slight increase	Slight increase	Slight increase	Increase significantly
Stiffness of bridge pylon (rise)	Tiny increase	Slight increase	Slight increase	Slight increase	Tiny increase	Tiny increase	Slight increase	Increase significantly	Increase significantly	Increase significantly
Position of auxiliary piers (prime location)	Tiny increase	Tiny increase	Tiny increase	Tiny increase	Increase significantly	Increase significantly	Slight increase	Increase significantly	Increase significantly	Increase significantly

.

7. Conclusions

The results indicate that the dynamic characteristics of the bridge can be enhanced by increasing the steel truss beam stiffness, cable stiffness, and pylon stiffness. The detailed results of the impact of varying the parameters of each bridge member on the dynamic characteristics are presented below as follows:

(1)

As the steel truss beam stiffness increases, the dynamic characteristics of the bridge are also enhanced. For large-span dual-purpose cable-stayed bridges, increasing the steel truss beam stiffness can significantly improve the wind and seismic resistance of the bridge due to the large span. However, since cables are force-transmitting structures, increasing the stiffness of cables has a limited effect on improving the dynamic characteristics of bridges. In engineering, the selection of lower-stiffness materials for cables can reduce wind and seismic resistance and save costs.

(2)

Increasing the stiffness of pylons has a greater impact on enhancing the dynamic characteristics of bridges. Increasing the stiffness of the pylons can significantly enhance the vertical bending and bending coupling resistance of the bridge and effectively enhance the load-carrying capacity of the bridge.

(3)

Regardless of whether the auxiliary pier moves toward the abutment or the tower, the inherent vibration frequency of the entire bridge will be reduced. The location and number of auxiliary piers should be carefully considered regarding the specific circumstances.

Studying the law of changes in dynamic characteristics due to changes in the parameters of bridge components is crucial for guiding the future construction of dual-purpose bridges for both public and railroad use in the future. It also provides a substantial theoretical basis for improving the wind and seismic resistance of dual-purpose bridges.