# Dynamic Characteristics of a Double-Pylon Cable-Stayed Bridge with Steel Truss Girder and Single-Cable Plane

Authors: Yong Zeng; Yutong Zeng; Hao Yu; Yujie Tan; Hongmei Tan; Huijun Zheng
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The dynamic characteristics are closely linked to the seismic stability and wind-resistant of the bridge. But different bridge types have different dynamic characteristics. In order to study the dynamic characteristics of a double-pylon cable-stayed bridge with a single-cable plane and steel truss girder whose main span is the longest in the world, the dynamic load test was done, and the finite element and the subspace iteration methods were used to analyze the vibration mode of the bridge. The influence of different structural parameters on the dynamic characteristics of the bridge was analyzed. The changed structural parameters are cable layout, stiffness of steel truss girder, stiffness of stayed cables, stiffness of pylons, the concentration of dead load, number and location of auxiliary piers, and structural system. The results show that the bending and torsion resistance of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder is weak. The torsional stiffness of the cable-stayed bridge with a double-cable plane is stronger than that of the cable-stayed bridge with a single-cable plane. The seismic stability and wind-resistant of the bridge can be improved by using light dead load, improving the stiffness of pylon and girder, and adding auxiliary piers scientifically. However, the change of cable stiffness has a complex influence on the dynamic characteristics of the bridge. The conclusion can offer references for the construction, maintenance, and design of the same type of bridges.


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## Body

## 1. Introduction

Nowadays, more and more beautiful bridges are being built. A beautifully shaped bridge can become a landmark of a city or even a symbol of urban culture. The double-pylon cable-stayed bridge with single-cable plane and steel truss girder has not only strong performance but also a beautiful appearance. It is a functional artwork integrated with the environment. Artworks are usually kept in museums and hardly suffer from any natural disasters. However, modern bridges, which are functional works of art, need to withstand earthquakes, strong wind, and other natural disasters. The dynamic characteristics of the bridge are closely related to the seismic stability and wind-resistant of the bridge [1, 2]. Moreover, the dynamic characteristics can also reflect the stability of the traffic and detect the health status of the bridge [3, 4]. The dynamic characteristics of each bridge type are unique and need to be studied separately. The dynamic characteristics of the same type of bridge are similar. If the dynamic characteristics of a certain type of bridge can be determined, it can offer references for similar bridges [5]. The structure of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder is complex and its dynamic characteristics are not clear. Therefore, it is necessary to study its dynamic characteristics, which are of great engineering significance.A study of the dynamic characteristics of cable-stayed bridges can reveal how the bridges react under various dynamic excitation [6, 7]. The dynamic characteristics of the bridge can be measured by field tests or by finite element method analysis. Field measurement is usually used to test the real bridge, and the structural parameters cannot be changed. However, the finite element method analysis can change different structural parameters to find a method to optimize the dynamic characteristics of a certain bridge type [8–10]. If the modeling is accurate and the boundary conditions are simulated properly, there is little difference between the finite element method and the field measurement results. Subspace iterative method is one of the methods to analyze the dynamic characteristics of bridges, which has enough accuracy [11–14]. Changing some structural parameters of the bridge may enhance its dynamic characteristics [15, 16]. The dynamic characteristics of a long-span bridge will change with the change of stiffness of each component and the change of constraint between the pylon and girder [17, 18]. The change of boundary conditions also results in the change of dynamic characteristics of the bridge [19]. The influence of dead load concentration on the dynamic characteristics of the bridge cannot be ignored. For a long-span cable-stayed bridge, it is necessary to verify the reliability of the dynamic response of heavy-haul trains resting on the bridge [20]. Auxiliary piers also have a certain influence on the dynamic characteristics of the cable-stayed bridge [21]. Zhou et al. [22] investigated the dynamic characteristics of an extrawide concrete self-anchored suspension bridge. The width of this bridge is very large, and its dynamic characteristics are worth studying. Ventura et al. [23] studied the dynamic characteristics of the Colquitz River Bridge by full-scale testing. Their research was later used by bridge owners to assess the expected response of bridges during earthquakes. Roy and Dash [24] studied the dynamic behavior of the multispan continuous girder bridge. Although the multispan continuous girder bridge is not a new type of bridge, many bridges of this type are built. The study of its dynamic characteristics is still valuable. Xie et al. [25] investigated the dynamic characteristics of a long-span cable-stayed bridge with CFRP cables. It is found that CFRP has excellent mechanical properties and can be used in long-span cable-stayed bridges. Casalegno and Russo [26] also studied the dynamic characteristics of the bridge using new material. The dynamic characteristics of many other bridges have been studied, including ordinary cable-stayed bridges, suspension bridges, and continuous rigid frame bridges [27–37]. These studies mainly focus on the dynamic characteristics of widely used general bridges and new bridges. The studies of the dynamic characteristics of the widely used general bridges are helpful to the formulation of the code. The research on the dynamic characteristics of the new bridges can provide a reference for similar bridges to be built in the future. The double-pylon cable-stayed bridge with steel truss girder and a single-cable plane is a new type of bridge, and its dynamic characteristics are is worth research. However, there are few studies on the influence of different stayed cable surface arrangements, girder stiffness, cable stiffness, pylon stiffness, the concentration of the dead load, number of auxiliary piers, and structural system on the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder.A double-pylon cable-stayed bridge with a single-cable plane and steel truss girder was taken as an example. The dynamic load test was done to verify the FE model, and the finite element method was adopted. The influence of different stayed cable surface arrangements, girder stiffness, cable stiffness, pylon stiffness, the concentration of the dead load, number of auxiliary piers, and structural system on the dynamic characteristics of a double-pylon cable-stayed bridge with single-cable plane and steel truss girder was analyzed. The dynamic characteristics of the same type of long-span bridges are similar, so the conclusions of this paper can be referenced by similar bridges and be of universal significance. This paper is a supplement to the study of the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder, which can offer references for the construction, maintenance, and design of the same type of bridges and offer some new ideas for the study of the dynamic characteristics of other types of bridge.


## 2. Dynamic Analysis Theory Based on Finite Element Method

### 2.1. Characteristic Equations

When a bridge is in free vibration, the vibration equation of the multifree vibrating system under the condition of neglecting damping is as follows:(1)K∗−ω2M∗a=0,where K is generalized stiffness matrix; M is generalized mass matrix; W is free vibration circle frequency; a is a characteristic vector.Equation (1) can be deduced by the Cramer rule. The characteristic equation is as follows:(2)K∗−ω2M∗=0.


### 2.2. Dynamic Characteristic Analysis Process

The subspace iteration method was used to analyze the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder. Subspace iterative method is one of the methods to analyze the dynamic characteristics of bridges, which has enough accuracy. The firstn-order frequencies and mode shapes of the double-pylon cable-stayed bridge with single-cable plane and steel truss girder were analyzed by using the subspace iteration method. The analysis process is as follows:(1)
Based on the finite element method, the whole process from the beginning of construction to the completion of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder was calculated. Based on the bridge state, the total stiffness matrix and the total mass matrix were obtained.(2)
The formula,D=K−1M, was used to obtain the dynamic matrix of the system D.(3)
After the assumption ofn linearly independent normalized modes is selected, they were arranged into an n×n-order matrix A0=ψ1ψ2⋯ψn.(4)
Each array was normalized to obtainψ1=DA0. After further analysis, the generalized stiffness matrix K∗=ψITKψ1 and the generalized mass matrix M∗=ψITMψ1 were obtained.(5)
After the equationK∗a−ω2M∗a=0 was converted to K∗−ω2M∗=0, the first n-order frequency ω1,ω2,…,ωn and the corresponding eigenvector a1,a2,…,an were calculated.(6)
Inequalitya−I<ξ, where I is the unit vector and ξ is the minimum tolerance, needs to be judged. If the inequality is true, the matrix composed of the first n modes is A1=ϕ1Ta1a2…an and the first n frequencies are ω1,ω2,…,ωn. If not, go back to step (4) and continue with the following steps, instead.


## 2.1. Characteristic Equations

When a bridge is in free vibration, the vibration equation of the multifree vibrating system under the condition of neglecting damping is as follows:(1)K∗−ω2M∗a=0,where K is generalized stiffness matrix; M is generalized mass matrix; W is free vibration circle frequency; a is a characteristic vector.Equation (1) can be deduced by the Cramer rule. The characteristic equation is as follows:(2)K∗−ω2M∗=0.


## 2.2. Dynamic Characteristic Analysis Process

The subspace iteration method was used to analyze the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder. Subspace iterative method is one of the methods to analyze the dynamic characteristics of bridges, which has enough accuracy. The firstn-order frequencies and mode shapes of the double-pylon cable-stayed bridge with single-cable plane and steel truss girder were analyzed by using the subspace iteration method. The analysis process is as follows:(1)
Based on the finite element method, the whole process from the beginning of construction to the completion of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder was calculated. Based on the bridge state, the total stiffness matrix and the total mass matrix were obtained.(2)
The formula,D=K−1M, was used to obtain the dynamic matrix of the system D.(3)
After the assumption ofn linearly independent normalized modes is selected, they were arranged into an n×n-order matrix A0=ψ1ψ2⋯ψn.(4)
Each array was normalized to obtainψ1=DA0. After further analysis, the generalized stiffness matrix K∗=ψITKψ1 and the generalized mass matrix M∗=ψITMψ1 were obtained.(5)
After the equationK∗a−ω2M∗a=0 was converted to K∗−ω2M∗=0, the first n-order frequency ω1,ω2,…,ωn and the corresponding eigenvector a1,a2,…,an were calculated.(6)
Inequalitya−I<ξ, where I is the unit vector and ξ is the minimum tolerance, needs to be judged. If the inequality is true, the matrix composed of the first n modes is A1=ϕ1Ta1a2…an and the first n frequencies are ω1,ω2,…,ωn. If not, go back to step (4) and continue with the following steps, instead.


## 3. Bridge Description

A double-pylon cable-stayed bridge with a single-cable plane and steel truss girder is in Chongqing, China, whose span arrangement is 225.5 m + 445 m (main span) + 190.5 m. Its main span is the longest in the world, and the bridge system is semifloating. Compressive bearings are set between girders, pylons, and platforms. The side span’s bearings on both sides are longitudinally free, one pylon is longitudinally restrained, and the other pylon is longitudinally free. The bridge is shown in Figure1.Figure 1
A double-pylon cable-stayed bridge with a single-cable plane and steel truss girder.The height of the pylons on both sides is 162 m and 172 m, and the material is C50 concrete. As shown in Figure2, the main girder is arranged with a double-layer steel truss, whose height span ratio is about 1/38. The lower layer is the railway and the upper layer is the highway. Both the upper and lower decks are orthogonal special-shaped decks with a width from 24.500 to 36.990 m due to different functions. The parallel steel strand is selected as the stayed cable; the material is Strand1860. The main span from 400 to 500 m is the economic span range of a double-pylon cable-stayed bridge with a single-cable plane and steel truss girder. The main spans of most similar bridges in the world are concentrated in 400–500 m. The dynamic characteristics of the same type of long-span bridges are similar, so this bridge is a very suitable choice, and the conclusions obtained have the value of promotion. The physical properties of the bridge materials are listed in Table 1.Figure 2
Control section diagram of main girder/mm. (a) Section of the main girder. (b) Section of main girder cable zone.
(a)(b)Table 1
Physical properties of main materials.
TypeComponentsElastic modulus (E/GPa)Poisson’s ratioLinear expansion coefficient (10–5°C)C50Pylon34.50.21.0Q345Girder206.00.31.2Strand1860Stayed cables195.00.31.2The bridge in this paper was the first bridge in the world to adopt a cable anchor girder fixed structure, and the world records when the bridge was completed are as follows:(1)
The span of the same type of bridge was the longest in the world.(2)
The anchorage form of cable and girder was first used in the world.(3)
The tonnage of the cable pylon anchor was the biggest in the world.(4)
The sparse cable scheme was adopted. There are only 36 stayed cables in the whole bridge. The single-cable is made up of 139 strands of parallel steel strands. The steel strands of stayed cables are about 980 tons. The tonnage of cables is the biggest in the world.(5)
The large tonnage bearing under the main pylon is supported by a corbel, which was the largest tonnage in the world.


## 4. Finite Element Simulation

Several finite element models of bridges with different structural parameters were established. The model with all construction completed and unchanged parameters has a total of 17846 nodes and 30128 elements with sufficient accuracy. The pylon and main girder were simulated by beam element, and the cable was simulated by tension only element. In order to accurately reflect the stress situation of the pylons, the section characteristics of each component were selected according to the actual size of the structure. All construction stages (62 in total) were considered, and the geometric nonlinear effect was considered in the modeling. The subspace iteration method was adopted to realize the analysis and research on the bridge vibration mode. Different structural parameters were changed, and several finite element models were established. Except for the change of cable plane layout, other parameters’ changes do not change the structure appearance. The finite element model is shown in Figure3, and the boundary conditions are listed in Table 2.Figure 3
Finite element model of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder.Table 2
Boundary conditions of FE model.
Structural partsUxUyUzROTxROTyROTzAbutment A0Upstream011000Downstream001000Bottom of pylon P1Upstream111111Downstream111111Connection of pylon P1 and girderUpstream111000Downstream101000Bottom of pylon P2Upstream111111Downstream111111Connection of pylon P2 and girderUpstream011000Downstream001000Pier P3Upstream011000Downstream001000Note. 0 for no; 1 for yes.


## 5. Dynamic Load Test

There are two layers in the steel truss girder, forming a double-layer bridge deck. When secondary dead load had not been laid on the lower deck, the dynamic load test was done on the double-pylon cable-stayed bridge with steel truss girder and single-cable plane. The random vibration method was used for the test and there was no traffic load on the bridge deck during the test. The bridge would vibrate slightly when the water strikes the pier or the wind blows to the bridge. Then, the dynamic characteristics of the bridge could be measured. That is the random vibration method. The advantage of this method is that the experimental process is simple, but the disadvantage is that it can only test the frequencies of the first few modes.


### 5.1. Introduction of the Test

When secondary dead load had not been laid on the lower deck, the dynamic load test was done on the double-pylon cable-stayed bridge with steel truss girder and single-cable plane by the random vibration method. As shown in Figure4, some pick-up instruments were installed at the key parts of the bridge. When the bridge vibrates, the sensor will record the amplitude and phase of each measurement point. The modal frequency can be obtained by comparing the amplitude and phase of each measuring point.Figure 4
The position of the pick-up instruments on the bridge.A total of 28 pick-up instruments were installed on the bridge, 26 on the steel truss girder, and two on the pylons. The type 941-B longitudinal pulse acceleration sensors and 991-B transverse pulse acceleration sensors were selected for the pick-up instruments, and the amplifiers were set at the same time. The amplifier can amplify the collected vibration signal and transmit it to the computer.


### 5.2. Comparison of the Results

During the dynamic load test of the bridge, the secondary dead load had not been laid on the lower deck. So, the dynamic characteristics of the two FE models were calculated. Mode 1 is the FE model without secondary dead load on the lower deck, and mode 2 is the FE bridge model whose secondary dead load has been laid on the lower deck and all construction processes have been completed. The random vibration method was used for the dynamic load test. During the test, there were no traffic loads on the bridge deck and no other irregular vibration sources around the bridge site. Dynamic loads were random wind load, water flow load, and other random loads at the bridge site. Because the random vibration method was used for the dynamic load test, the number of vibration modes is less, only the first five modes. The results are listed in Table3.Table 3
The comparison between measured frequencies and calculated frequencies.
Mode numberMeasured frequencies (Hz)Calculated frequencies 1 (Hz)Calculated frequencies 2 (Hz)Mode shape10.350.320.24The 1st-order cross bending of the main span of the girder20.430.380.31The 1st-order symmetrical vertical bending of the girder30.690.530.47The 2nd-order symmetrical vertical bending of the girder40.730.630.58The bending torsion-coupling of the left span of the girder50.910.730.61The 1st-order symmetrical bending of the girderNote. Calculated frequencies 1, calculated by the FE model 1; calculated frequencies 2, calculated by the FE model 2.During the dynamic load test of the bridge, the secondary dead load had not been laid on the lower deck and the FE model 1 simulates the bridge at that moment. The FE model 2 simulates a fully completed bridge. Because the modeling is accurate and the boundary simulation is correct, the frequencies calculated by the FE model 1 are close to the measured frequencies. For the first five modes, the frequencies calculated by the FE model 1 are slightly higher than the calculated frequencies. During the design of the bridge, the structure would be checked. The structure would not be built until it has been proved safe. The measured frequencies of the bridge are higher than the calculated frequencies, indicating that the stiffness of the bridge is large enough. When secondary dead load laying is completed, the first five modes’ frequencies decrease. It conforms to the changing law of the dynamic characteristics of the bridge. In a word, the comparison between measured and calculated values shows that the FE model modeling is accurate and the boundary simulation is correct. The model can be used for subsequent analysis.


## 5.1. Introduction of the Test

When secondary dead load had not been laid on the lower deck, the dynamic load test was done on the double-pylon cable-stayed bridge with steel truss girder and single-cable plane by the random vibration method. As shown in Figure4, some pick-up instruments were installed at the key parts of the bridge. When the bridge vibrates, the sensor will record the amplitude and phase of each measurement point. The modal frequency can be obtained by comparing the amplitude and phase of each measuring point.Figure 4
The position of the pick-up instruments on the bridge.A total of 28 pick-up instruments were installed on the bridge, 26 on the steel truss girder, and two on the pylons. The type 941-B longitudinal pulse acceleration sensors and 991-B transverse pulse acceleration sensors were selected for the pick-up instruments, and the amplifiers were set at the same time. The amplifier can amplify the collected vibration signal and transmit it to the computer.


## 5.2. Comparison of the Results

During the dynamic load test of the bridge, the secondary dead load had not been laid on the lower deck. So, the dynamic characteristics of the two FE models were calculated. Mode 1 is the FE model without secondary dead load on the lower deck, and mode 2 is the FE bridge model whose secondary dead load has been laid on the lower deck and all construction processes have been completed. The random vibration method was used for the dynamic load test. During the test, there were no traffic loads on the bridge deck and no other irregular vibration sources around the bridge site. Dynamic loads were random wind load, water flow load, and other random loads at the bridge site. Because the random vibration method was used for the dynamic load test, the number of vibration modes is less, only the first five modes. The results are listed in Table3.Table 3
The comparison between measured frequencies and calculated frequencies.
Mode numberMeasured frequencies (Hz)Calculated frequencies 1 (Hz)Calculated frequencies 2 (Hz)Mode shape10.350.320.24The 1st-order cross bending of the main span of the girder20.430.380.31The 1st-order symmetrical vertical bending of the girder30.690.530.47The 2nd-order symmetrical vertical bending of the girder40.730.630.58The bending torsion-coupling of the left span of the girder50.910.730.61The 1st-order symmetrical bending of the girderNote. Calculated frequencies 1, calculated by the FE model 1; calculated frequencies 2, calculated by the FE model 2.During the dynamic load test of the bridge, the secondary dead load had not been laid on the lower deck and the FE model 1 simulates the bridge at that moment. The FE model 2 simulates a fully completed bridge. Because the modeling is accurate and the boundary simulation is correct, the frequencies calculated by the FE model 1 are close to the measured frequencies. For the first five modes, the frequencies calculated by the FE model 1 are slightly higher than the calculated frequencies. During the design of the bridge, the structure would be checked. The structure would not be built until it has been proved safe. The measured frequencies of the bridge are higher than the calculated frequencies, indicating that the stiffness of the bridge is large enough. When secondary dead load laying is completed, the first five modes’ frequencies decrease. It conforms to the changing law of the dynamic characteristics of the bridge. In a word, the comparison between measured and calculated values shows that the FE model modeling is accurate and the boundary simulation is correct. The model can be used for subsequent analysis.


## 6. Analysis of Dynamic Characteristic Analysis Based on Variable Parameters

### 6.1. Influence of Cable Layout on Dynamic Characteristics

When the bridge was in operation, the secondary dead load had been laid on the lower deck. Therefore, all models used in this chapter are models that have completed all construction. Usually, the study of bridge dynamic characteristics mainly focuses on the first 10 modes, because they are representative and easy to occur in reality. As shown in Table4, in order to study the influence of the cable plane layout on the dynamic characteristics of the steel truss cable-stayed bridge, the first 12 modes of the cable-stayed bridge with a single-cable plane and the cable-stayed bridge with a double-cable plane were compared. As shown in Figure 5, some modes’ shapes of the two bridge types with different cable layouts are compared intuitively. In Figure 5, except for the 1st mode, other mode shapes of the two bridge types are different.Table 4
Main vibration mode characteristics.
Mode numberBridge typeFrequencies (Hz)Mode shape1S0.244The 1st-order cross bending of the main span of the girderD0.243The 1st-order cross bending of the main span of the girder2S0.306The 1st-order symmetrical vertical bending of the girderD0.325The 1st-order symmetrical vertical bending of the girder3S0.467The 2nd-order symmetrical vertical bending of the girderD0.472The 2nd-order symmetrical vertical bending of the girder4S0.581The bending torsion-coupling of the left span of the girderD0.584The bending torsion-coupling of the left span of the girder5S0.612The 1st-order symmetrical bending of the girderD0.608The 1st-order symmetrical bending of the girder6S0.656The 3rd-order symmetrical vertical bending of the girderD0.661The 2nd-order symmetrical vertical bending of the girder7S0.671The obvious bending torsion-coupling of the girderD0.673The slight bending torsion-coupling of the girder8S0.716The bending torsion-coupling of the main and right span of the girderD0.718The bending torsion-coupling of the right span of the girder9S0.758The obvious bending torsion-coupling of the girderD0.763The slight bending torsion-coupling of the girder10S0.774The 4th-order symmetrical vertical bending of the girderD0.822The 4th-order symmetrical vertical bending of the girder11S0.830The obvious bending torsion-coupling of the main span of the girderD0.919The obvious bending torsion-coupling of the main span of the girder12S0.946The 5th-order symmetrical vertical bending of the girderD1.006The 5th-order symmetrical vertical bending of the girderNote. S, cable-stayed bridge with a single-cable plane; D, cable-stayed bridge with a double-cable plane.Figure 5
Some vibration modes of two bridges with different cable plane arrangements. (a) The 1st-order mode of the cable-stayed bridge with a single-cable plane. (b) The 1st-order mode of the cable-stayed bridge with a double-cable plane. (c) The 5th-order mode of the cable-stayed bridge with a single-cable plane. (d) The 5th-order mode of the cable-stayed bridge with a double-cable plane. (e) The 6th-order mode of the cable-stayed bridge with a single-cable plane. (f) The 6th-order mode of the cable-stayed bridge with a single double plane. (g) The 7th-order mode of the cable-stayed bridge with a single-cable plane. (h) The 7th-order mode of the cable-stayed bridge with a double-cable plane. (i) The 8th-order mode of the cable-stayed bridge with a single-cable plan. (j) The 8th-order mode of the cable-stayed bridge with a double-cable plane. (k) The 9th-order mode of the cable-stayed bridge with a single-cable plane. (l) The 9th-order mode of the cable-stayed bridge with a double-cable plane.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)From the analysis of Table4 and Figure 5, the conclusions are as follows:(1)
For the first five modes, the vibration frequency and period of the cable-stayed bridge with a single-cable plane and the cable-stayed bridge with a double-cable plane are different, but the mode shapes are consistent. When the mode shape is bending torsion-coupling (the 4th-order mode), the frequency of the cable-stayed bridge with a double-cable plane is higher than that of the cable-stayed bridge with a single-cable plane, because the cables of the cable-stayed bridge with a single-cable plane cannot bear the torsion caused by eccentric load and the double-cable plane can enhance the torsional stiffness of cable-stayed bridge. The fundamental frequency of the original structure is 0.244 Hz, and the shape of the 1st-order mode is the first-order transverse bending of the middle span of the main girder, which is the basic mode of the bridge, indicating that the lateral bending stiffness of the double-pylon cable-stayed bridge with single-cable plane and steel truss girder is small. The shape of the 2nd-order mode is the 1st-order symmetrical vertical bending, and the mode shape appears earlier, which proves that the bridge has low longitudinal flexural rigidity and low transverse and longitudinal flexural rigidity. The wind-resistant of the bridge is weak. The shape of the 4th-order mode is the coupling of bending and torsion of the girder. The bending and torsion capacity of the bridge is relatively balanced and weak, which is unfavorable to the wind-resistant of the bridge.(2)
For the 6th-order mode to the 12th-order mode, the frequencies of the cable-stayed bridge with a double-cable plane are greater than those of the cable-stayed bridge with a single-cable plane. From the 6th-order mode to the 9th-order mode, the mode shapes are different. At the 6th-order mode, the 3rd-order symmetrical vertical bending of the main girder of the cable-stayed bridge with a single-cable plane appears, but the 2nd-order symmetrical vertical bending mode of the main girder of the cable-stayed bridge with a double-cable plane appears. At the 7th-order mode and the 9th-order mode, obvious bending torsion-coupling mode of the main girder of the cable-stayed bridge with single-cable plane appears, but slight bending torsion-coupling mode of the main girder of the cable-stayed bridge with a double-cable plane appears. From the 7th-order mode to the 9th-order mode, bending torsion coupled vibration modes appear. The torsion degree of the cable-stayed bridge with a single-cable plane is larger than that of the cable-stayed bridge with a double-cable plane, which proves that the double-cable plane arrangement can improve the torsional characteristics of steel truss girder cable-stayed bridge. At the 10th-order mode and the 12th-order mode, the girder symmetrical vertical of the main span appears. At the 11th-order mode, obvious torsion of the main span of the girder appears. But the double-cable plane only increases the frequency of the mode, and the mode shape is not changed.(3)
There is no pure torsional mode in the first 12 modes; only the coupled mode of bending and torsion is found, which further shows that the bending and torsion capacity of the bridge is equal and weak. This cable-stayed bridge with a single-cable plane and steel truss girder double-pylon belongs to the semifloating system. The overall bridge is relatively flexible which is conducive to reducing the seismic response. However, from the vibration mode characteristics, the design of a single-cable plane makes the bridge have weak torsional stiffness, which is extremely unfavorable to the wind-resistant. By changing the single-cable plane layout to the double-cable plane layout, the torsional stiffness of the double-pylon cable-stayed bridge with steel truss girder is improved, which can avoid the occurrence of bending and torsion-coupling mode to a certain extent and enhance the wind resistance of the bridge.


### 6.2. Influence of Other Structural Parameters on Dynamic Characteristics

In order to find measures to improve the wind resistance of a double-pylon cable-stayed bridge with a single-cable and steel truss girder and to maintain its good seismic resistance, the influence of different girder stiffness, cable stiffness, pylon stiffness, the concentration of the dead load, number of auxiliary piers, and structural system on the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder was analyzed. In order to study the dynamic characteristics of the bridge more precisely, it is divided into two types: bending mode and bending torsion-coupling mode. The first 12 modes are represented byf1 to f12. The bending mode includes f1, f2, f3, f4, f5, f10, and f12, respectively. The bending torsion-coupling mode includes f6, f7, f8, f9, and f11, respectively.


#### 6.2.1. The Stiffness of Steel Truss Girder

As shown in Figure6, the stiffness of other components is kept unchanged, the stiffness change coefficient of the girder is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the main girder will not affect the mode shape of the bridge.Figure 6
Influence of stiffness change coefficient of the girder on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure6, with the stiffness coefficient of the girder rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. Although they are all rising, it is obvious that the frequency rise of bending mode and bending torsion-coupling mode is different from the rising rate. For bending mode, when the stiffness coefficient increases from 0.6 to 1.6, the maximum rising rate is the fundamental frequency, that is, the frequency of the 1st-order mode (f1) reaches 60.7%, and the frequency of the 3rd-order mode (f3) has the smallest rising rate, which is 23.1%. For the 1st-order mode, when the coefficient increases from 0.6 to 1.6, although it only increases by 0.114 Hz, the rising rate is the largest due to its low frequency. When the stiffness of the girder is increased, the bending mode frequency increases, which indicates that the bending capacity of the bridge can be enhanced by increasing the stiffness of the girder. For bending torsion-coupling mode, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 11th-order mode f11 reaches 60.0%, and the frequency of the 7th-order mode f7 has the smallest rising rate, which is 26.9%. Due to the large cardinality of the 11th-order mode itself, the frequency of the girder increases by 0.391 Hz when the stiffness coefficient increases from 0.6 to 1.6, which is far greater than that of the first mode. The increase of the whole vibration mode frequency shows that the bending and torsional coupling ability of the bridge can be enhanced by improving the stiffness of the girder. The rise of the modal frequency of dynamic analysis means the enhancement of bridge dynamic characteristics. The cable-stayed bridge is an integral structure connecting pylon, stayed cable, and girder. Because the stayed cable pulls the girder and the girder is connected with the pylon, the overall stiffness can be improved after the stiffness of the girder is increased. The improvement of overall stiffness will lead to the improvement of dynamic performance. Improving the stiffness of the main girder can greatly enhance the dynamic characteristics of the bridge, which means that the stiffness of the main girder can improve the wind resistance and seismic resistance of the bridge. There are many measures to improve the stiffness of the main girder, such as replacing the steel with stronger rigidity, increasing the members, and using stronger connection devices.


#### 6.2.2. The Stiffness of Stayed Cables

As shown in Figure7, the stiffness of other components is kept unchanged, the stiffness change coefficient of the stayed cables is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the stayed cables has some influence on the mode shape of the bridge.Figure 7
Influence of stiffness change coefficient of the stayed cables on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure7 that when the cable stiffness coefficient is 0.8, the first 12 main mode frequencies have a sudden drop. The reason is that when the cable stiffness coefficient is 0.8, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode (as is shown in Figure 8), while when the cable stiffness coefficient is 1.0, 1.2, 1.4, and 1.6, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode. Excluding the special case of the cable stiffness coefficient of 0.8, with the stiffness coefficient of the cable rising from 0.6 to 1.6, the rising rates of the bending modes and bending torsion-coupling modes are very low. The lowest rising rate is the frequency of the 7th-order mode, less than 1%. The median rate of increase is less than 5%. From the original design as a benchmark, the change in cable stiffness has a more complicated impact on the dynamic characteristics of the bridge. When the stiffness of the cable drops from the benchmark, the dynamic characteristics of the bridge may change significantly. Therefore, adjusting the stiffness of the cables in the design of the bridge may have a greater impact on the dynamic characteristics of the bridge, thereby affecting the seismic stability and wind-resistant of the bridge.Figure 8
The modal shape of the fundamental frequency with the stiffness coefficient of 0.8.


#### 6.2.3. The Stiffness of Pylon

As shown in Figure9, the stiffness of other members was kept unchanged, the stiffness change coefficient of the pylons was changed to 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of pylon stiffness will not affect the mode shape of the bridge.Figure 9
Influence of stiffness change coefficient of the pylons on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure9 that with the stiffness coefficient of the pylons rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. For bending modes, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 3rd-order mode has the largest rising rate, reaching 23.5%, and the lowest rising rate is 9.8% at the 10th-order mode. However, it should be noted that when the stiffness coefficient of the pylons is increased from 1 to 1.2, 1.4, and 1.6, the frequency rising rate is very low, and the frequency of the 2nd-order mode has the largest rising rate, which is only 4.2%. This shows that the stiffness of C50 concrete pylon is enough, and the dynamic performance will be weakened if it is reduced. The law of bending torsion-coupling mode is different from that of bending torsion-coupling mode. For bending torsion-coupling mode, when the pylon stiffness coefficient is increased from 1 to 1.2, 1.4, and 1.6, the rising rate of modal frequency is not small. The lowest rate of increase was 6.2%, and the highest rate was 10.8%. This shows that when the pylon stiffness increases, although it cannot improve the overall bending strength of the bridge, it is helpful for the torsional strength. Adjusting the stiffness of the pylon will have some impact on the seismic stability and wind-resistant of the bridge. The stiffness of the pylon can be appropriately increased in the design of the bridge to improve the seismic stability and wind-resistant of the bridge.


#### 6.2.4. Concentration of the Dead Load in the Girder

Through the observation of the mode shape of the bridge after the change of the concentration of the dead load, it was found that the change of the concentration of the dead load did not affect the mode shape of the bridge. The change coefficients are set as 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, respectively. As shown in Figure10, the bending modes and bending torsion-coupling modes are compared and studied.Figure 10
Influence of change of the concentration of the dead load on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure10 that with the concentration of the dead load coefficient increasing from 0.6 to 1.6, the frequencies of the first 12 modes of the bridge show a downward trend. For bending modes, the frequency of the 1st-order mode with the highest drop rate is 21.6%, and the lowest one is 14.1% in the 10th-order mode. For bending torsion-coupling modes, the largest descending rate is the 6th-order mode, which reaches 20.6%, and the frequency of the 8th-order mode has the smallest descending rate, reaching 10.4%. The increase of the concentration of the dead load is unfavorable to the bending and torsion resistance of the bridge. When the dead load concentration increases, the bridge needs to bear more loads, which will lead to the decline of dynamic performance. In order to ensure the seismic stability and wind-resistant of the bridge, reasonable light materials should be selected as the second-stage dead load of the bridge.


#### 6.2.5. Number and Location of Auxiliary Piers

As is shown in Figure11, the location and number of auxiliary piers will not affect the basic frequency of the bridge. The number of auxiliary piers is set from 1 to 6. Because the left span of the bridge is larger than the right span, one auxiliary pier is set in the left-side span when the number of auxiliary piers is 1. When the number of auxiliary piers is 2, one auxiliary pier is set in the left-side and right-side spans, respectively. When the number is 3, two auxiliary piers are set in the left-side span and one auxiliary pier is set in the right-side span, and so on. As is shown in Figure 12, the influence of the number of auxiliary piers on the dynamic characteristics of the bridge was studied.Figure 11
The 1st-order mode shape comparison of a cable-stayed bridge with and without auxiliary piers. (a) The shape of 1st-order mode of a cable-stayed bridge without auxiliary piers. (b) The shape of 1st-order mode cable-stayed bridge with auxiliary piers.
(a)(b)Figure 12
Influence of the number of auxiliary piers on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure12 that with the number of auxiliary piers increasing from 0 to 6, the frequencies of the first 12 modes of the bridge show an upward trend. For bending modes, when the auxiliary pier increases from 0 to 2, the frequencies increase obviously, the maximum is the 12th mode, reaching 23.4%. However, when it increases from 2 to 3, 4, 5, and 6, the rising rate is low, and the maximum is the 12th-order mode, which is only 7.4%. The law of bending torsion-coupling mode is similar to that of bending mode. When the number of auxiliary piers is more than 2, the frequency rise is small. Therefore, the installation of auxiliary piers on the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder can improve its bending and torsion resistance, which is beneficial to the seismic stability and wind-resistant of the bridge. In the bridge design, the stability of the bridge can be strengthened by adding auxiliary piers. However, when the number of auxiliary piers is greater than 2, the increase of frequency of each vibration mode is limited.


#### 6.2.6. Structural System of Girder and Pylon

The structural system of a double-pylon cable-stayed bridge usually includes the rigid frame system, semifloating system, pylon and girder consolidation system, and floating system. Several systems of double-pylon cable-stayed bridges are shown in Figure13. The influence of structural system changes on the bending mode and bending torsion-coupling mode is shown in Figure 14.Figure 13
Several systems of the double-pylon cable-stayed bridges. (a) The floating system. (b) The semifloating system. (c) The pylon and girder consolidation system. (d) The rigid frame system.
(a)(b)(c)(d)Figure 14
Influence of the changes of the structural system on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure14 that the first 12 modal frequencies of the four structural systems are not consistent. The frequencies of the semifloating system and the pylon and girder consolidation system are relatively high, and the frequencies of the floating system and the rigid frame system are relatively low. Because there is no connection between the pylon and girder of the floating system, and the whole girder is floating in the air except for the support at both ends, the dynamic characteristics of the floating system are unstable. On the one hand, the pylons and girder of the rigid frame system are consolidated, the pylon and girder form a whole, the number of indeterminate increases, and the uncertainty factors increase at the same time. On the other hand, if the flexibility of the tower is not enough, the dynamic characteristics of the rigid frame system may become unstable. For the pylon and girder consolidation system and semifloating system, there is a certain connection between the pylon and the girder, but not consolidation. The interaction between the pylons and the girder is smaller than that of the rigid frame system but larger than that of the floating system. Therefore, the dynamic characteristics of the semifloating system and pylon and girder consolidation system are relatively stable.


## 6.1. Influence of Cable Layout on Dynamic Characteristics

When the bridge was in operation, the secondary dead load had been laid on the lower deck. Therefore, all models used in this chapter are models that have completed all construction. Usually, the study of bridge dynamic characteristics mainly focuses on the first 10 modes, because they are representative and easy to occur in reality. As shown in Table4, in order to study the influence of the cable plane layout on the dynamic characteristics of the steel truss cable-stayed bridge, the first 12 modes of the cable-stayed bridge with a single-cable plane and the cable-stayed bridge with a double-cable plane were compared. As shown in Figure 5, some modes’ shapes of the two bridge types with different cable layouts are compared intuitively. In Figure 5, except for the 1st mode, other mode shapes of the two bridge types are different.Table 4
Main vibration mode characteristics.
Mode numberBridge typeFrequencies (Hz)Mode shape1S0.244The 1st-order cross bending of the main span of the girderD0.243The 1st-order cross bending of the main span of the girder2S0.306The 1st-order symmetrical vertical bending of the girderD0.325The 1st-order symmetrical vertical bending of the girder3S0.467The 2nd-order symmetrical vertical bending of the girderD0.472The 2nd-order symmetrical vertical bending of the girder4S0.581The bending torsion-coupling of the left span of the girderD0.584The bending torsion-coupling of the left span of the girder5S0.612The 1st-order symmetrical bending of the girderD0.608The 1st-order symmetrical bending of the girder6S0.656The 3rd-order symmetrical vertical bending of the girderD0.661The 2nd-order symmetrical vertical bending of the girder7S0.671The obvious bending torsion-coupling of the girderD0.673The slight bending torsion-coupling of the girder8S0.716The bending torsion-coupling of the main and right span of the girderD0.718The bending torsion-coupling of the right span of the girder9S0.758The obvious bending torsion-coupling of the girderD0.763The slight bending torsion-coupling of the girder10S0.774The 4th-order symmetrical vertical bending of the girderD0.822The 4th-order symmetrical vertical bending of the girder11S0.830The obvious bending torsion-coupling of the main span of the girderD0.919The obvious bending torsion-coupling of the main span of the girder12S0.946The 5th-order symmetrical vertical bending of the girderD1.006The 5th-order symmetrical vertical bending of the girderNote. S, cable-stayed bridge with a single-cable plane; D, cable-stayed bridge with a double-cable plane.Figure 5
Some vibration modes of two bridges with different cable plane arrangements. (a) The 1st-order mode of the cable-stayed bridge with a single-cable plane. (b) The 1st-order mode of the cable-stayed bridge with a double-cable plane. (c) The 5th-order mode of the cable-stayed bridge with a single-cable plane. (d) The 5th-order mode of the cable-stayed bridge with a double-cable plane. (e) The 6th-order mode of the cable-stayed bridge with a single-cable plane. (f) The 6th-order mode of the cable-stayed bridge with a single double plane. (g) The 7th-order mode of the cable-stayed bridge with a single-cable plane. (h) The 7th-order mode of the cable-stayed bridge with a double-cable plane. (i) The 8th-order mode of the cable-stayed bridge with a single-cable plan. (j) The 8th-order mode of the cable-stayed bridge with a double-cable plane. (k) The 9th-order mode of the cable-stayed bridge with a single-cable plane. (l) The 9th-order mode of the cable-stayed bridge with a double-cable plane.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)From the analysis of Table4 and Figure 5, the conclusions are as follows:(1)
For the first five modes, the vibration frequency and period of the cable-stayed bridge with a single-cable plane and the cable-stayed bridge with a double-cable plane are different, but the mode shapes are consistent. When the mode shape is bending torsion-coupling (the 4th-order mode), the frequency of the cable-stayed bridge with a double-cable plane is higher than that of the cable-stayed bridge with a single-cable plane, because the cables of the cable-stayed bridge with a single-cable plane cannot bear the torsion caused by eccentric load and the double-cable plane can enhance the torsional stiffness of cable-stayed bridge. The fundamental frequency of the original structure is 0.244 Hz, and the shape of the 1st-order mode is the first-order transverse bending of the middle span of the main girder, which is the basic mode of the bridge, indicating that the lateral bending stiffness of the double-pylon cable-stayed bridge with single-cable plane and steel truss girder is small. The shape of the 2nd-order mode is the 1st-order symmetrical vertical bending, and the mode shape appears earlier, which proves that the bridge has low longitudinal flexural rigidity and low transverse and longitudinal flexural rigidity. The wind-resistant of the bridge is weak. The shape of the 4th-order mode is the coupling of bending and torsion of the girder. The bending and torsion capacity of the bridge is relatively balanced and weak, which is unfavorable to the wind-resistant of the bridge.(2)
For the 6th-order mode to the 12th-order mode, the frequencies of the cable-stayed bridge with a double-cable plane are greater than those of the cable-stayed bridge with a single-cable plane. From the 6th-order mode to the 9th-order mode, the mode shapes are different. At the 6th-order mode, the 3rd-order symmetrical vertical bending of the main girder of the cable-stayed bridge with a single-cable plane appears, but the 2nd-order symmetrical vertical bending mode of the main girder of the cable-stayed bridge with a double-cable plane appears. At the 7th-order mode and the 9th-order mode, obvious bending torsion-coupling mode of the main girder of the cable-stayed bridge with single-cable plane appears, but slight bending torsion-coupling mode of the main girder of the cable-stayed bridge with a double-cable plane appears. From the 7th-order mode to the 9th-order mode, bending torsion coupled vibration modes appear. The torsion degree of the cable-stayed bridge with a single-cable plane is larger than that of the cable-stayed bridge with a double-cable plane, which proves that the double-cable plane arrangement can improve the torsional characteristics of steel truss girder cable-stayed bridge. At the 10th-order mode and the 12th-order mode, the girder symmetrical vertical of the main span appears. At the 11th-order mode, obvious torsion of the main span of the girder appears. But the double-cable plane only increases the frequency of the mode, and the mode shape is not changed.(3)
There is no pure torsional mode in the first 12 modes; only the coupled mode of bending and torsion is found, which further shows that the bending and torsion capacity of the bridge is equal and weak. This cable-stayed bridge with a single-cable plane and steel truss girder double-pylon belongs to the semifloating system. The overall bridge is relatively flexible which is conducive to reducing the seismic response. However, from the vibration mode characteristics, the design of a single-cable plane makes the bridge have weak torsional stiffness, which is extremely unfavorable to the wind-resistant. By changing the single-cable plane layout to the double-cable plane layout, the torsional stiffness of the double-pylon cable-stayed bridge with steel truss girder is improved, which can avoid the occurrence of bending and torsion-coupling mode to a certain extent and enhance the wind resistance of the bridge.


## 6.2. Influence of Other Structural Parameters on Dynamic Characteristics

In order to find measures to improve the wind resistance of a double-pylon cable-stayed bridge with a single-cable and steel truss girder and to maintain its good seismic resistance, the influence of different girder stiffness, cable stiffness, pylon stiffness, the concentration of the dead load, number of auxiliary piers, and structural system on the dynamic characteristics of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder was analyzed. In order to study the dynamic characteristics of the bridge more precisely, it is divided into two types: bending mode and bending torsion-coupling mode. The first 12 modes are represented byf1 to f12. The bending mode includes f1, f2, f3, f4, f5, f10, and f12, respectively. The bending torsion-coupling mode includes f6, f7, f8, f9, and f11, respectively.


### 6.2.1. The Stiffness of Steel Truss Girder

As shown in Figure6, the stiffness of other components is kept unchanged, the stiffness change coefficient of the girder is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the main girder will not affect the mode shape of the bridge.Figure 6
Influence of stiffness change coefficient of the girder on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure6, with the stiffness coefficient of the girder rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. Although they are all rising, it is obvious that the frequency rise of bending mode and bending torsion-coupling mode is different from the rising rate. For bending mode, when the stiffness coefficient increases from 0.6 to 1.6, the maximum rising rate is the fundamental frequency, that is, the frequency of the 1st-order mode (f1) reaches 60.7%, and the frequency of the 3rd-order mode (f3) has the smallest rising rate, which is 23.1%. For the 1st-order mode, when the coefficient increases from 0.6 to 1.6, although it only increases by 0.114 Hz, the rising rate is the largest due to its low frequency. When the stiffness of the girder is increased, the bending mode frequency increases, which indicates that the bending capacity of the bridge can be enhanced by increasing the stiffness of the girder. For bending torsion-coupling mode, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 11th-order mode f11 reaches 60.0%, and the frequency of the 7th-order mode f7 has the smallest rising rate, which is 26.9%. Due to the large cardinality of the 11th-order mode itself, the frequency of the girder increases by 0.391 Hz when the stiffness coefficient increases from 0.6 to 1.6, which is far greater than that of the first mode. The increase of the whole vibration mode frequency shows that the bending and torsional coupling ability of the bridge can be enhanced by improving the stiffness of the girder. The rise of the modal frequency of dynamic analysis means the enhancement of bridge dynamic characteristics. The cable-stayed bridge is an integral structure connecting pylon, stayed cable, and girder. Because the stayed cable pulls the girder and the girder is connected with the pylon, the overall stiffness can be improved after the stiffness of the girder is increased. The improvement of overall stiffness will lead to the improvement of dynamic performance. Improving the stiffness of the main girder can greatly enhance the dynamic characteristics of the bridge, which means that the stiffness of the main girder can improve the wind resistance and seismic resistance of the bridge. There are many measures to improve the stiffness of the main girder, such as replacing the steel with stronger rigidity, increasing the members, and using stronger connection devices.


### 6.2.2. The Stiffness of Stayed Cables

As shown in Figure7, the stiffness of other components is kept unchanged, the stiffness change coefficient of the stayed cables is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the stayed cables has some influence on the mode shape of the bridge.Figure 7
Influence of stiffness change coefficient of the stayed cables on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure7 that when the cable stiffness coefficient is 0.8, the first 12 main mode frequencies have a sudden drop. The reason is that when the cable stiffness coefficient is 0.8, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode (as is shown in Figure 8), while when the cable stiffness coefficient is 1.0, 1.2, 1.4, and 1.6, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode. Excluding the special case of the cable stiffness coefficient of 0.8, with the stiffness coefficient of the cable rising from 0.6 to 1.6, the rising rates of the bending modes and bending torsion-coupling modes are very low. The lowest rising rate is the frequency of the 7th-order mode, less than 1%. The median rate of increase is less than 5%. From the original design as a benchmark, the change in cable stiffness has a more complicated impact on the dynamic characteristics of the bridge. When the stiffness of the cable drops from the benchmark, the dynamic characteristics of the bridge may change significantly. Therefore, adjusting the stiffness of the cables in the design of the bridge may have a greater impact on the dynamic characteristics of the bridge, thereby affecting the seismic stability and wind-resistant of the bridge.Figure 8
The modal shape of the fundamental frequency with the stiffness coefficient of 0.8.


### 6.2.3. The Stiffness of Pylon

As shown in Figure9, the stiffness of other members was kept unchanged, the stiffness change coefficient of the pylons was changed to 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of pylon stiffness will not affect the mode shape of the bridge.Figure 9
Influence of stiffness change coefficient of the pylons on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure9 that with the stiffness coefficient of the pylons rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. For bending modes, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 3rd-order mode has the largest rising rate, reaching 23.5%, and the lowest rising rate is 9.8% at the 10th-order mode. However, it should be noted that when the stiffness coefficient of the pylons is increased from 1 to 1.2, 1.4, and 1.6, the frequency rising rate is very low, and the frequency of the 2nd-order mode has the largest rising rate, which is only 4.2%. This shows that the stiffness of C50 concrete pylon is enough, and the dynamic performance will be weakened if it is reduced. The law of bending torsion-coupling mode is different from that of bending torsion-coupling mode. For bending torsion-coupling mode, when the pylon stiffness coefficient is increased from 1 to 1.2, 1.4, and 1.6, the rising rate of modal frequency is not small. The lowest rate of increase was 6.2%, and the highest rate was 10.8%. This shows that when the pylon stiffness increases, although it cannot improve the overall bending strength of the bridge, it is helpful for the torsional strength. Adjusting the stiffness of the pylon will have some impact on the seismic stability and wind-resistant of the bridge. The stiffness of the pylon can be appropriately increased in the design of the bridge to improve the seismic stability and wind-resistant of the bridge.


### 6.2.4. Concentration of the Dead Load in the Girder

Through the observation of the mode shape of the bridge after the change of the concentration of the dead load, it was found that the change of the concentration of the dead load did not affect the mode shape of the bridge. The change coefficients are set as 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, respectively. As shown in Figure10, the bending modes and bending torsion-coupling modes are compared and studied.Figure 10
Influence of change of the concentration of the dead load on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure10 that with the concentration of the dead load coefficient increasing from 0.6 to 1.6, the frequencies of the first 12 modes of the bridge show a downward trend. For bending modes, the frequency of the 1st-order mode with the highest drop rate is 21.6%, and the lowest one is 14.1% in the 10th-order mode. For bending torsion-coupling modes, the largest descending rate is the 6th-order mode, which reaches 20.6%, and the frequency of the 8th-order mode has the smallest descending rate, reaching 10.4%. The increase of the concentration of the dead load is unfavorable to the bending and torsion resistance of the bridge. When the dead load concentration increases, the bridge needs to bear more loads, which will lead to the decline of dynamic performance. In order to ensure the seismic stability and wind-resistant of the bridge, reasonable light materials should be selected as the second-stage dead load of the bridge.


### 6.2.5. Number and Location of Auxiliary Piers

As is shown in Figure11, the location and number of auxiliary piers will not affect the basic frequency of the bridge. The number of auxiliary piers is set from 1 to 6. Because the left span of the bridge is larger than the right span, one auxiliary pier is set in the left-side span when the number of auxiliary piers is 1. When the number of auxiliary piers is 2, one auxiliary pier is set in the left-side and right-side spans, respectively. When the number is 3, two auxiliary piers are set in the left-side span and one auxiliary pier is set in the right-side span, and so on. As is shown in Figure 12, the influence of the number of auxiliary piers on the dynamic characteristics of the bridge was studied.Figure 11
The 1st-order mode shape comparison of a cable-stayed bridge with and without auxiliary piers. (a) The shape of 1st-order mode of a cable-stayed bridge without auxiliary piers. (b) The shape of 1st-order mode cable-stayed bridge with auxiliary piers.
(a)(b)Figure 12
Influence of the number of auxiliary piers on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure12 that with the number of auxiliary piers increasing from 0 to 6, the frequencies of the first 12 modes of the bridge show an upward trend. For bending modes, when the auxiliary pier increases from 0 to 2, the frequencies increase obviously, the maximum is the 12th mode, reaching 23.4%. However, when it increases from 2 to 3, 4, 5, and 6, the rising rate is low, and the maximum is the 12th-order mode, which is only 7.4%. The law of bending torsion-coupling mode is similar to that of bending mode. When the number of auxiliary piers is more than 2, the frequency rise is small. Therefore, the installation of auxiliary piers on the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder can improve its bending and torsion resistance, which is beneficial to the seismic stability and wind-resistant of the bridge. In the bridge design, the stability of the bridge can be strengthened by adding auxiliary piers. However, when the number of auxiliary piers is greater than 2, the increase of frequency of each vibration mode is limited.


### 6.2.6. Structural System of Girder and Pylon

The structural system of a double-pylon cable-stayed bridge usually includes the rigid frame system, semifloating system, pylon and girder consolidation system, and floating system. Several systems of double-pylon cable-stayed bridges are shown in Figure13. The influence of structural system changes on the bending mode and bending torsion-coupling mode is shown in Figure 14.Figure 13
Several systems of the double-pylon cable-stayed bridges. (a) The floating system. (b) The semifloating system. (c) The pylon and girder consolidation system. (d) The rigid frame system.
(a)(b)(c)(d)Figure 14
Influence of the changes of the structural system on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure14 that the first 12 modal frequencies of the four structural systems are not consistent. The frequencies of the semifloating system and the pylon and girder consolidation system are relatively high, and the frequencies of the floating system and the rigid frame system are relatively low. Because there is no connection between the pylon and girder of the floating system, and the whole girder is floating in the air except for the support at both ends, the dynamic characteristics of the floating system are unstable. On the one hand, the pylons and girder of the rigid frame system are consolidated, the pylon and girder form a whole, the number of indeterminate increases, and the uncertainty factors increase at the same time. On the other hand, if the flexibility of the tower is not enough, the dynamic characteristics of the rigid frame system may become unstable. For the pylon and girder consolidation system and semifloating system, there is a certain connection between the pylon and the girder, but not consolidation. The interaction between the pylons and the girder is smaller than that of the rigid frame system but larger than that of the floating system. Therefore, the dynamic characteristics of the semifloating system and pylon and girder consolidation system are relatively stable.


## 6.2.1. The Stiffness of Steel Truss Girder

As shown in Figure6, the stiffness of other components is kept unchanged, the stiffness change coefficient of the girder is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the main girder will not affect the mode shape of the bridge.Figure 6
Influence of stiffness change coefficient of the girder on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure6, with the stiffness coefficient of the girder rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. Although they are all rising, it is obvious that the frequency rise of bending mode and bending torsion-coupling mode is different from the rising rate. For bending mode, when the stiffness coefficient increases from 0.6 to 1.6, the maximum rising rate is the fundamental frequency, that is, the frequency of the 1st-order mode (f1) reaches 60.7%, and the frequency of the 3rd-order mode (f3) has the smallest rising rate, which is 23.1%. For the 1st-order mode, when the coefficient increases from 0.6 to 1.6, although it only increases by 0.114 Hz, the rising rate is the largest due to its low frequency. When the stiffness of the girder is increased, the bending mode frequency increases, which indicates that the bending capacity of the bridge can be enhanced by increasing the stiffness of the girder. For bending torsion-coupling mode, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 11th-order mode f11 reaches 60.0%, and the frequency of the 7th-order mode f7 has the smallest rising rate, which is 26.9%. Due to the large cardinality of the 11th-order mode itself, the frequency of the girder increases by 0.391 Hz when the stiffness coefficient increases from 0.6 to 1.6, which is far greater than that of the first mode. The increase of the whole vibration mode frequency shows that the bending and torsional coupling ability of the bridge can be enhanced by improving the stiffness of the girder. The rise of the modal frequency of dynamic analysis means the enhancement of bridge dynamic characteristics. The cable-stayed bridge is an integral structure connecting pylon, stayed cable, and girder. Because the stayed cable pulls the girder and the girder is connected with the pylon, the overall stiffness can be improved after the stiffness of the girder is increased. The improvement of overall stiffness will lead to the improvement of dynamic performance. Improving the stiffness of the main girder can greatly enhance the dynamic characteristics of the bridge, which means that the stiffness of the main girder can improve the wind resistance and seismic resistance of the bridge. There are many measures to improve the stiffness of the main girder, such as replacing the steel with stronger rigidity, increasing the members, and using stronger connection devices.


## 6.2.2. The Stiffness of Stayed Cables

As shown in Figure7, the stiffness of other components is kept unchanged, the stiffness change coefficient of the stayed cables is set to 0.6, 0.8, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of the stiffness of the stayed cables has some influence on the mode shape of the bridge.Figure 7
Influence of stiffness change coefficient of the stayed cables on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure7 that when the cable stiffness coefficient is 0.8, the first 12 main mode frequencies have a sudden drop. The reason is that when the cable stiffness coefficient is 0.8, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode (as is shown in Figure 8), while when the cable stiffness coefficient is 1.0, 1.2, 1.4, and 1.6, the first-order vertical bending of the main span of the girder appears at the fundamental frequency mode. Excluding the special case of the cable stiffness coefficient of 0.8, with the stiffness coefficient of the cable rising from 0.6 to 1.6, the rising rates of the bending modes and bending torsion-coupling modes are very low. The lowest rising rate is the frequency of the 7th-order mode, less than 1%. The median rate of increase is less than 5%. From the original design as a benchmark, the change in cable stiffness has a more complicated impact on the dynamic characteristics of the bridge. When the stiffness of the cable drops from the benchmark, the dynamic characteristics of the bridge may change significantly. Therefore, adjusting the stiffness of the cables in the design of the bridge may have a greater impact on the dynamic characteristics of the bridge, thereby affecting the seismic stability and wind-resistant of the bridge.Figure 8
The modal shape of the fundamental frequency with the stiffness coefficient of 0.8.


## 6.2.3. The Stiffness of Pylon

As shown in Figure9, the stiffness of other members was kept unchanged, the stiffness change coefficient of the pylons was changed to 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, and the bending modes and bending torsion-coupling modes are compared and studied. It is found that the change of pylon stiffness will not affect the mode shape of the bridge.Figure 9
Influence of stiffness change coefficient of the pylons on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure9 that with the stiffness coefficient of the pylons rising from 0.6 to 1.6, the frequency of the first 12 modes of the bridge shows an upward trend. For bending modes, when the stiffness coefficient increases from 0.6 to 1.6, the frequency of the 3rd-order mode has the largest rising rate, reaching 23.5%, and the lowest rising rate is 9.8% at the 10th-order mode. However, it should be noted that when the stiffness coefficient of the pylons is increased from 1 to 1.2, 1.4, and 1.6, the frequency rising rate is very low, and the frequency of the 2nd-order mode has the largest rising rate, which is only 4.2%. This shows that the stiffness of C50 concrete pylon is enough, and the dynamic performance will be weakened if it is reduced. The law of bending torsion-coupling mode is different from that of bending torsion-coupling mode. For bending torsion-coupling mode, when the pylon stiffness coefficient is increased from 1 to 1.2, 1.4, and 1.6, the rising rate of modal frequency is not small. The lowest rate of increase was 6.2%, and the highest rate was 10.8%. This shows that when the pylon stiffness increases, although it cannot improve the overall bending strength of the bridge, it is helpful for the torsional strength. Adjusting the stiffness of the pylon will have some impact on the seismic stability and wind-resistant of the bridge. The stiffness of the pylon can be appropriately increased in the design of the bridge to improve the seismic stability and wind-resistant of the bridge.


## 6.2.4. Concentration of the Dead Load in the Girder

Through the observation of the mode shape of the bridge after the change of the concentration of the dead load, it was found that the change of the concentration of the dead load did not affect the mode shape of the bridge. The change coefficients are set as 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, respectively. As shown in Figure10, the bending modes and bending torsion-coupling modes are compared and studied.Figure 10
Influence of change of the concentration of the dead load on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure10 that with the concentration of the dead load coefficient increasing from 0.6 to 1.6, the frequencies of the first 12 modes of the bridge show a downward trend. For bending modes, the frequency of the 1st-order mode with the highest drop rate is 21.6%, and the lowest one is 14.1% in the 10th-order mode. For bending torsion-coupling modes, the largest descending rate is the 6th-order mode, which reaches 20.6%, and the frequency of the 8th-order mode has the smallest descending rate, reaching 10.4%. The increase of the concentration of the dead load is unfavorable to the bending and torsion resistance of the bridge. When the dead load concentration increases, the bridge needs to bear more loads, which will lead to the decline of dynamic performance. In order to ensure the seismic stability and wind-resistant of the bridge, reasonable light materials should be selected as the second-stage dead load of the bridge.


## 6.2.5. Number and Location of Auxiliary Piers

As is shown in Figure11, the location and number of auxiliary piers will not affect the basic frequency of the bridge. The number of auxiliary piers is set from 1 to 6. Because the left span of the bridge is larger than the right span, one auxiliary pier is set in the left-side span when the number of auxiliary piers is 1. When the number of auxiliary piers is 2, one auxiliary pier is set in the left-side and right-side spans, respectively. When the number is 3, two auxiliary piers are set in the left-side span and one auxiliary pier is set in the right-side span, and so on. As is shown in Figure 12, the influence of the number of auxiliary piers on the dynamic characteristics of the bridge was studied.Figure 11
The 1st-order mode shape comparison of a cable-stayed bridge with and without auxiliary piers. (a) The shape of 1st-order mode of a cable-stayed bridge without auxiliary piers. (b) The shape of 1st-order mode cable-stayed bridge with auxiliary piers.
(a)(b)Figure 12
Influence of the number of auxiliary piers on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure12 that with the number of auxiliary piers increasing from 0 to 6, the frequencies of the first 12 modes of the bridge show an upward trend. For bending modes, when the auxiliary pier increases from 0 to 2, the frequencies increase obviously, the maximum is the 12th mode, reaching 23.4%. However, when it increases from 2 to 3, 4, 5, and 6, the rising rate is low, and the maximum is the 12th-order mode, which is only 7.4%. The law of bending torsion-coupling mode is similar to that of bending mode. When the number of auxiliary piers is more than 2, the frequency rise is small. Therefore, the installation of auxiliary piers on the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder can improve its bending and torsion resistance, which is beneficial to the seismic stability and wind-resistant of the bridge. In the bridge design, the stability of the bridge can be strengthened by adding auxiliary piers. However, when the number of auxiliary piers is greater than 2, the increase of frequency of each vibration mode is limited.


## 6.2.6. Structural System of Girder and Pylon

The structural system of a double-pylon cable-stayed bridge usually includes the rigid frame system, semifloating system, pylon and girder consolidation system, and floating system. Several systems of double-pylon cable-stayed bridges are shown in Figure13. The influence of structural system changes on the bending mode and bending torsion-coupling mode is shown in Figure 14.Figure 13
Several systems of the double-pylon cable-stayed bridges. (a) The floating system. (b) The semifloating system. (c) The pylon and girder consolidation system. (d) The rigid frame system.
(a)(b)(c)(d)Figure 14
Influence of the changes of the structural system on bending modes and bending torsion-coupling modes. (a) Bending modes. (b) Bending torsion-coupling modes.
(a)(b)It can be seen from Figure14 that the first 12 modal frequencies of the four structural systems are not consistent. The frequencies of the semifloating system and the pylon and girder consolidation system are relatively high, and the frequencies of the floating system and the rigid frame system are relatively low. Because there is no connection between the pylon and girder of the floating system, and the whole girder is floating in the air except for the support at both ends, the dynamic characteristics of the floating system are unstable. On the one hand, the pylons and girder of the rigid frame system are consolidated, the pylon and girder form a whole, the number of indeterminate increases, and the uncertainty factors increase at the same time. On the other hand, if the flexibility of the tower is not enough, the dynamic characteristics of the rigid frame system may become unstable. For the pylon and girder consolidation system and semifloating system, there is a certain connection between the pylon and the girder, but not consolidation. The interaction between the pylons and the girder is smaller than that of the rigid frame system but larger than that of the floating system. Therefore, the dynamic characteristics of the semifloating system and pylon and girder consolidation system are relatively stable.


## 7. Conclusions

The double-pylon cable-stayed bridge with single-cable plane and steel truss girder has the feature of novel bridge type, complex structure, and uncertain dynamic characteristics. The dynamic load test was done and the correctness of the FE model was verified. The influence of different stayed cable surface arrangement, girder stiffness, cable stiffness, pylon stiffness, the concentration of the dead load, number of auxiliary piers, and structural system on the dynamic characteristics of the double-pylon cable-stayed bridge with single-cable plane and steel truss girder was analyzed. The conclusion is as follows:(1)
The bending and torsion rigidities of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder are relatively balanced and weak, which is unfavorable to the wind-resistant of the bridge. However, due to its semifloating system, the seismic responsibility is relatively improved. By changing the single-cable plane layout to the double-cable plane layout, the torsional stiffness of the double-pylon cable-stayed bridge with steel truss girder is improved, which can avoid the occurrence of bending and torsion-coupling mode to a certain extent and enhance the wind resistance of the bridge.(2)
The increase of the concentration of the dead load is unfavorable to the bending and torsion resistance of the bridge. In order to enhance the seismic stability and wind-resistant of the bridge, reasonable light materials should be selected as the second-stage dead load of the bridge.(3)
Adjusting the stiffness of the pylon and girder will affect the seismic stability and wind-resistant of the bridge, and improving the stiffness of the pylon and girder can increase the seismic stability and wind-resistant of the bridge. The change of cable stiffness has a complex influence on the dynamic characteristics of the bridge. When the cable stiffness is 0.8 times of the original structure, the mode shape of the bridge changes greatly. Therefore, adjusting the cable stiffness in the bridge design may have a great impact on the seismic stability and wind-resistant of the bridge.(4)
The installation of auxiliary piers on the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder can improve its bending and torsion resistance, which is beneficial to the seismic stability and wind-resistant of the bridge. In the bridge design, the stability of the bridge can be strengthened by adding auxiliary piers.(5)
The dynamic characteristics of the floating system and the rigid frame system are unstable, while the dynamic characteristics of the semifloating system and the pylon and girder consolidation system are better. The semifloating system or the pylon and girder consolidation system should be used in the follow-up similar bridges.



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*Source: 9565730-2021-12-16.xml*