Static and Dynamic Performance Analysis of Cable-Stayed Bridges with Cables Damaged Fire

Abstract

Cable-stayed bridges have been widely used in large-span bridge engineering because of their large span capacity and novel structure. The frequent traffic of vehicles transporting flammable and explosive materials has increased the incidence of bridge fires. After being burned, the cable-stayed bridge will suffer from varying degrees of damage, which affects its performance. Therefore, mechanical analysis and evaluation of the fire-damaged cable-stayed bridge are necessary. Due to the development of technology, the structural analysis of cable-stayed bridges has gradually shifted from experimental methods to numerical simulation or artificial intelligence methods, and from local performance research to holistic research. In this paper, a fire accident in the Sifangtai Bridge in Harbin, China, is taken as a case study. Finite element software and damage theory calculation methods were used, and the static and dynamic performances of the bridge under the condition of cable fire damage were analyzed. The results show that the variation of cable force during the movement of vehicle load along the bridge is relatively small, within the range of 7% to 12%. The fusing of the upper cables of the bridge tower has the greatest impact on the deflection of the beam, while the fusing of lower cables has the same impact on the deflection of the entire bridge as the undamaged state. Near the fused cables, cable forces change significantly, increasing by over 20%. As the degree of damage increases, the increase in deflection of the beam becomes more pronounced. The impact of different degrees of cable damage on the dynamic performance of cable-stayed bridges is reflected in quantitative changes. As the degree of cable damage and the amount of fusing increase, the change in structural frequency becomes more pronounced. This paper not only provides technical support and a theoretical basis for the performance analysis of cable-stayed bridges damaged by fire, but it also improves the research content of combining static and dynamic performance, which provides important reference values for similar research in the future.

1. Introduction

Since the beginning of the 21st century, the rapid development of the economy has led to a gradual increase in the number of vehicles transporting flammable and explosive materials. Once a fire occurs on a vehicle transporting flammable and explosive materials, it will cause harm to the bridge where it is located, not only causing damage to people’s lives and property, but also potentially causing traffic paralysis [1,2]. Especially for some large cable-stayed bridges, which constitute traffic bottlenecks, once a fire accident occurs, the losses cannot be estimated.

Unlike small-span bridges, the impact of fires on large-span bridges is rarely observed and studied, so fires on large-span bridges should be given sufficient attention. On 15 July 2009, an oil truck collided with a pillar on a 1.75 km overpass in Michigan, causing a fire which collapsed the elevated bridge onto an interstate road. This accident interrupted traffic for a month, and the elevated bridge was demolished and rebuilt within 65 days [3].

Due to the fact that the stay cables are located above the bridge deck and are usually close to the carriageway, the probability of a fire occurring in the stay cables is self-evident. At present, the cables of cable-stayed bridges are usually wrapped with high-strength steel wires or steel strands contained within high-density polyethylene (HDPE) sheaths. The fusing point of the polyethylene sheaths is low, which leads to a weak point of poor high-temperature performance inside the cable protection sleeve [4,5,6,7]. If a fire occurs and causes the cables to catch fire, it is particularly important to evaluate the cable-stayed bridge in a timely and effective manner [8,9,10].

At present, domestic and foreign scholars have conducted extensive research on the static and dynamic performance of cable-stayed bridges under various influencing factors. When it comes to the static performance of a bridge, several factors come into play. These include the displacement response of cable-stayed bridges under different loads or self-weight conditions, comparison of stress characteristics, influence of linear and nonlinear simulation methods on bridge displacement, application of material reinforcement, and so on [11,12,13,14,15,16]. When it comes to the dynamic performance of a bridge, several factors come into play. These include how the bridge responds to internal forces when subjected to non-linear loads, how it behaves during a progressive collapse, its overall stiffness, and how it performs dynamically when the cables break. These factors have been studied and documented extensively in the literature [17,18,19,20,21].

Li and Zhen [22] derived the calculation formula for the overall potential energy of cable-stayed bridges using the iterative method and solved the nonlinear equation using the Newton-Raphson method, providing a practical method for the nonlinear analysis of cable-stayed bridges. Hidayat and Santoso [23] used taut string theory and beam string theory approaches to evaluate the tension of cables based on dynamic tests. Chen and Qu [24] used the finite element program ANSYS to establish a comprehensive dynamic analysis model of the entire bridge, and calculated the dynamic characteristics of the bridge under four working conditions. Al Washali et al. [25] studied the performance of long-span cable-stayed bridges after cable breakage and conducted a detailed numerical analysis of cable damage accidents in cable-stayed bridges. Zeng et al. [26] analyzed the dynamic characteristics of a cable-stayed bridge with a single pylon and single cable plane. The analysis results show that a cable-stayed bridge with a single pylon and single cable plane has more flexibility and that the lateral rigidity and torsional rigidity are smaller. Yang et al. [27] analyzed the dynamic response of cables under train loads based on the finite element method and a numerical model. Research has shown that the maximum displacement amplitude occurs at the longest cable near the middle of the main span. Hoang et al. [28] utilized an experimental program to simulate cable fracture events, and proposed a method for dynamic response analysis using strain waveforms from the experimental program.

Research on the working performance of cable-stayed bridges is no longer limited to the material properties of the components themselves, and the exploration of various composite damage scenarios has become a mainstream trend [29,30,31]. Kawai et al. [32] investigated in more detail the possibility of brittle shear fracture of the clamp’s pin from the viewpoint of linear fracture mechanics by utilizing the measured Charpy absorbed energy. Kallingal and Singh [33] conducted dynamic analysis research on cable-stayed bridges with different tower column arrangements. The results indicate that as complexity increases, the behavior of the structure changes. Soto et al. [34] proposed a method for optimizing the cable system of cable-stayed bridges, taking into account the accidental fracture of one cable during the design process. Wilson and Liu [35] conducted comprehensive environmental vibration tests on a 542 m cable-stayed bridge. A total of 25 modal frequencies and related modal modes were determined. Experimental data indicates the existence of many closely spaced modal frequencies and spatially complex modal shapes.

With the rapid development of technology, we have seen the potential to use machine learning (ML), artificial intelligence (AI), and artificial neural network (ANN) algorithms to solve practical engineering problems [36,37,38,39]. In terms of convergence speed in simulation calculations, multi-modal motion prediction models for vehicles, and solving numerical models with disturbance suppression, ML and AI have significant advantages [40,41,42]. The numerical simulations or AI methods will also have more applications in the field of geotechnical engineering, such as construction process monitoring, multi-physical field coupling, and reliability analysis, which can effectively promote the development of geotechnical engineering [43,44,45]. For example, Shen et al. [46] proposed a new time series clustering algorithm for long-term settlement analysis of shield tunnels, which alleviates the limitations of low efficiency in long-term settlement analysis. Therefore, from the above examples, it can be seen that these models can provide fast and accurate predictions based on existing theoretical and experimental trends and results.

However, there is not much existing literature on the combined analysis of the static and dynamic performance of cable-stayed bridges experiencing cable fire damage. Especially in the case of damage caused by cable fire, the overall performance impact and evaluation of the bridge are more prominent. Ensuring the reliability and economy of cable repair projects for fire damage, reducing losses, and timely and effective testing and evaluation of the static performance of bridges after a fire, are issues that continue to be addressed in practical engineering. In this paper, a fire accident on the Sifangtai Bridge in Harbin, China, is taken as a case. Finite element software and damage theory calculation methods were used, and the static and dynamic performance of the bridge under the condition of cable fire damage were analyzed. Firstly, based on the damage mechanism and damage mechanics model of cables, and combined with the material characteristics under high temperatures, the simulation method for cable fire damage was determined. Then, a finite element model of the cable-stayed bridge was established, and the effects of vertical loads, cable fusing, and different cable damage on the static performance of the cable-stayed bridge were analyzed. Finally, based on the natural vibration characteristics of cable-stayed bridges, the structural dynamic changes caused by cable fusing and different cable damage were analyzed.

2. Engineering Background

Sifangtai Highway Bridge is located on the Songhua River in Harbin, China. It is an important bridge connecting the south and north transportation systems of Harbin. The total length of the bridge is 696 m, with spans arranged at 44 m (transition span) + 136 m (side span) + 336 m (middle span) + 136 m (side span) + 44 m (transition span). It is a double tower cable-stayed bridge, and the bridge layout is shown in Figure 1a,b. This bridge has a 33.2 m wide deck carrying four traffic lanes. The beam is a steel–concrete composite beam with a prestressed concrete bridge deck on the upper side which has a thickness of 0.25 m and a concrete grade of C60. The steel longitudinal beam on the lower side employs three I-beams arranged symmetrically, with a height of 1.95 m on both sides and 0.55 m in the middle. Longitudinal beams are connected by steel crossbeams, with stiffeners installed every 1.65 m, and pedestrian cantilever steel beams are installed on both sides. The fireproof material used for the steel structure in the steel–concrete composite beam is a thin steel structure fireproof coating, which is brushed on during construction to prevent fire in the steel structure. The bridge cross-section and the steel crossbeam are shown in Figure 1c,d.

The bridge tower is a portal tower, with the south tower reaching a height of 110.80 m and the north tower reaching a height of 106.10 m, with a single chamber hexagonal cross-section. The bridge tower is equipped with two horizontal beams, upper and lower, dividing it into three parts: upper, middle, and lower tower columns. The crossbeam is a hollow box girder, the upper crossbeam is 3.0 m × 3.0 m, and the lower crossbeam is 5.0 m × 3.0 m. The bridge tower is shown in Figure 2.

The cable is a semi-parallel hot extruded polyethylene cable, with a spatial fan-shaped cable surface that tilts towards the outside of the bridge from top to bottom. The vertical spacing between the anchor points of the tower’s inner cable is 2.0 m, and the distance between the top cable anchor point and the top surface of the tower is 4.92 m. The horizontal spacing between the anchor points of the cables inside the beam is 12.0 m, and the distance between the anchor points of the first cable and the center of the tower is 15.2 m. There are a total of 52 pairs of cables for the entire bridge, using φ7 low-relaxation prestressed galvanized high-strength steel wire, wrapped with high-density polyethylene (HDPE) protective material. The fire resistance of the cables mainly relies on the hot extruded HDPE protective cover wrapped around them. Firstly, a black HDPE protective inner cover is wrapped around the galvanized steel wire, and then an outer layer of colored HDPE protective material is wrapped around the cable to provide dual protection. Through the material selection, structural design, and fire prevention management of this bridge, the sustainability and stability of the cable-stayed bridge under fire loads have been effectively improved. The cable design is shown in Figure 3.

In 2011, a traffic accident occurred on the bridge, causing a car to catch fire and burn near cables. The damaged cables were N12 and N13, upstream of the north bank. The PE protective material of two cables was partially burned, and both wires were exposed. The burnt length of N12 was 16.5 m, and the burnt length of N13 was 5 m. After removing the burnt material and exposing the cable steel wire, it was found that the surface color of the steel wire had changed to a certain extent. The damaged condition of the cable is shown in Figure 4.

3. Finite Element Model

3.1. Simulation of the Cable Damage Due to Fire

When a cable is damaged by fire, any fused steel wires are completely out of service. For unfused steel wires, the elastic modulus and ultimate strength are calculated based on the material properties of the steel after high temperatures [7]. The material properties of prestressed steel wire will change after undergoing high temperature effects. Under air cooling, the degradation law of the elastic modulus and ultimate strength of the steel wire at a given temperature conforms to Equations (1) and (2), and the corresponding change curve is shown in Figure 5.

E_sT/E_s = −1.2500 × 10⁻⁹T³ + 1.2703 × 10⁻⁶T² − 3.1131 × 10⁻⁴T + 1.0205

(1)

σ_bT/σ_b = 4.4621 × 10⁻⁹T³ − 6.0530 × 10⁻⁶T² + 1.2195 × 10⁻³T + 0.9257

(2)

where T is the temperature; E_s is the elastic modulus of undamaged steel wire; E_sT is the elastic modulus of the steel wire after fire damage; σ_b is the ultimate strength of undamaged steel wire; σ_bT is the ultimate strength of the steel wire after fire damage.

As shown in Figure 5, the elastic modulus of the steel wire exhibits small fluctuations within the temperature range of 0~600 °C, without a significant increasing or decreasing trend. After 600 °C, the elastic modulus of the steel wire gradually decreases, and the rate of decrease continues to increase. The ultimate strength of steel wire increases significantly in the range of 0~100 °C. In the range of 100~800 °C, ultimate strength decreases continuously with the increase of temperature. However, after 800 °C, ultimate strength is increased and shows an upward trend.

The bearing capacity of a cable is related to its effective bearing area. When calculating bearing capacity, the entire cross-section of the cable’s steel wire strands is considered as a continuous medium. When some steel wires are fused by fire, the void defects formed will reduce the effective bearing area of the cable, leading to a decrease in bearing capacity. The damage rate and bearing capacity of the cable after fire can be calculated using Equations (3) and (4).

$$\zeta =\left(A-\tilde{A}\right)/A=1-\tilde{A}/A$$

(3)

$$F_T={\sigma }_{bT}\times \tilde{A}$$

(4)

where $\zeta$ is the damage rate of the cable; A is the area of undamaged cable steel wire; $\tilde{A}$ is the effective bearing area of the cable; F_T is the bearing capacity of the cable after fire.

When simulating the fire damage, the cables are first segmented and divided into damaged and non-damaged areas. The elastic modulus and ultimate strength of cables in the non-damaged area remain unchanged. The elastic modulus and ultimate strength of the steel wire after high temperature for cables in the damaged area are reduced according to Equations (1) and (2). Finally, the bearing capacity of the cable after fire damage is calculated according to Equations (3) and (4).

Figure 6 is a schematic diagram of a cable damaged by fire. From the figure, it can be seen that before being damaged by fire, the bearing capacity of the cable is determined by the ultimate strength and bearing area of the steel wire. When damaged by fire, the cable undergoes two processes. Firstly, the elastic modulus and ultimate strength of steel wires degrade under high temperature conditions. Secondly, some steel wires may fuse after their performance deteriorates. After being damaged by fire, when calculating the bearing capacity of the cable, it is necessary to comprehensively consider the ultimate strength of the degraded steel wire and the effective bearing area after fusing.

3.2. Establishment of Finite Element Model

Based on the summary of previous research [30], due to the shortcomings of existing analytical models, there are certain opportunities for optimizing numerical models. On the basis of traditional modeling, this article adopts a more reasonable modeling method to improve the accuracy of numerical model calculations. The main assumptions for numerical simulation of cable-stayed bridges are as follows.

(1)

The geometric behavior caused by the sag of the stay cables under their own weight is nonlinear. The equivalent elastic modulus method is used to introduce the nonlinear behavior of the stay cables into the model elements.

(2)

The main beam and bridge tower are spatial beam elements, and the stay cables are spatial rod elements. Ignoring the catenary shape of the stay cables space, they are considered as parabolic shapes.

(3)

Ignoring the interaction between the cable-stayed bridge and the foundation, treating the foundation as a rigid support.

(4)

The safety factor of cable-stayed bridges is considered to be no less than 2.5.

A bridge model was established using the finite element software MIDAS CIVIL/2021 (v1.1), with a total of 1933 nodes and 2272 elements. Verification of different unit division lengths was conducted for the natural frequency and deflection of cable-stayed bridge structures, ensuring the accuracy of the analysis. The overall finite element model of the Sifangtai Bridge is shown in Figure 7a, and the basic parameters of the bridge model are as follows.

(1)

Steel–concrete composite beam

The steel beam is the main load-bearing component, which is connected to the steel crossbeam to form a spatial beam grid structure. The steel specification is Q420E, with a tensile strength of 520 MPa and an elongation of 20%. They are simulated using seven degree of freedom (7-DOF) beam elements and are tetrahedral 3D elements. The prefabricated bridge deck is made of C60 concrete, simulated using beam elements, which are tetrahedral 3D elements, and uniformly arranged on the steel main beam and steel crossbeam. The finite element simulation of the steel–concrete composite beam is shown in Figure 7b.

(2)

Bridge towers and piers

In the bridge towers, the tower columns are made of C50 concrete, and the tower bases are made of C40 concrete. In the bridge piers, the pier cap is made of C40 concrete, and the pier columns and pile cap are made of C30 concrete. The bridge towers and piers are simulated using beam elements, which are tetrahedral 3D elements, and the tower elements are precisely divided. The upper and lower crossbeams are connected to the tower columns through shared nodes, and a “General support” is used to fix all translational and rotational degrees of freedom. The finite element simulation of the bridge tower and pier is shown in Figure 7c,d.

(3)

Cable

The cable is made of φ7 low-relaxation prestressed galvanized high-strength steel wire, with a standard tensile strength of 1570 MPa and an elastic modulus of 2.05 × 10⁵ MPa. The cable is simulated using truss elements, which are 3D elements. The two ends of the cable are connected to the bridge tower and beam through rigid arms, a “Rigid” in “elastic connection” is used. The finite element simulation of the cable is shown in Figure 7e.

Prestressed steel bars are installed in the bridge deck, pier cap, and cable towers, which are made of φj 15.24 low-relaxation high-strength steel strands. The standard tensile strength is 1860 MPa, the controlled tensile strength is 1395 MPa, and the elastic modulus is 195,000 MPa. The prestressed steel bars are simulated by defining the characteristics and shape of the steel strands, embedded in the truss, as a 1D element. The material mechanical performance parameters of the main components are shown in

Table 1. Component material parameters.

Component	Material Type		Design Compressive Strength (MPa)	Design Tensile Strength (MPa)	Elastic Modulus (MPa)	Linear Expansion Coefficient (×10−5/°C)
Steel beam	Steel	Q420E	/	520	2.06 × 105	1.2
Steel crossbeam
Bridge deck	Concrete	C60	26.5	1.96	3.60 × 105	1
Tower column		C50	22.4	1.83	3.45 × 105	1
Tower base		C40	18.4	1.65	3.25 × 105	1
Pier cap
Pier column		C30	13.8	1.39	3.00 × 105	1
Pile cap
Cable	High-strength steel wire		1570	1570	2.00 × 105	1.2
Prestressed steel			1860	1860	1.95 × 105	1.2

.

4. Static Performance Analysis

4.1. The Influence of Vertical Loads

The static performance response of the bridge under vertical loads was used as the basis for static analysis. Vertical loads include the self weight of the beam, secondary loads, and vehicle loads. The static analysis of this bridge is a holistic analysis, which uses lane loads for loading, and designed according to the requirements of Class 20 for automobiles. Each lane consists of one heavy vehicle and several main vehicles, with one heavy vehicle weighing 550 kN and one main vehicle weighing 200 kN. Due to this bridge being a two-way four-lane road, the vehicle load is reduced laterally by 33% [47]. The layout of lane loads is shown in Figure 8.

Figure 9 shows the vertical displacement variation of the beam under loads. The horizontal axis is based on the mid span of the bridge, dividing the bridge into two parts: the south tower and the north tower. The vertical axis represents the vertical displacement of the beam. As can be seen in Figure 9, the vertical displacement of the mid span beam varies greatly, while the vertical displacement of the side span beam remains almost unchanged. This indicates that the support effect of the bridge towers and transition piers on both sides hinders the vertical expansion of the beam to the outside. The maximum vertical displacement of the beam is 1.26 m, which occurs in the middle section of the mid span and gradually decreases towards the bridge towers on both sides.

Figure 10 shows the variation in cable force of typical cables (S13, S11, S6, S1, SC1, SC7, SC13) when the vehicle load moves longitudinally along the beam. The horizontal axis represents the position of the load, and the vertical axis represents the cable force value of typical cables. As can be seen in Figure 10, the cable forces of S13, S11, and SC13 constantly change during the vehicle’s movement. The trend of cable force changes in S13 and S11 is the same, while S13 and SC13 are opposite. This is because S13 and S11 are cables on the same side of the bridge tower and are closer in distance, while SC13 and SC11 are cables on opposite sides and symmetrical to the bridge tower. When the vehicle load acts on the middle section of the mid span, the maximum changes in cable forces for S13, S11, and SC13 are 516 MPa, 440 MPa, and 522 MPa, respectively. This indicates that in the middle section of the mid span, the vehicle load is mainly transmitted to the upper part of the bridge tower through cable SC13. The smaller stiffness of the upper part of the bridge tower produces a larger horizontal displacement, causing several cables to increase their force significantly, and to change continuously in order to prevent deformation. Therefore, the vehicle load has a significant impact on the forces of the upper cable of the bridge tower in the mid span.

The cable forces of S6, S1, SC1, and SC7 reach their maximum values when the vehicle reaches their corresponding positions, which are 456 MPa, 405 MPa, 402 MPa, and 455 MPa, respectively. The fluctuations are relatively small at other positions. This indicates that due to the high stiffness of the middle and lower parts of the bridge tower, the tension of the cables on the beam is similar to that of multi-point elastic support, and the vehicle load only has a significant effect on the support at the position of action. Therefore, for the lower part of the bridge tower, when the vehicle load acts on its corresponding position, the cable force changes significantly.

Table 2. Variation of cable force for typical cables.

Cable	Initial Force (MPa) ①	Maximum Value (MPa) ②	Minimum Value (MPa) ③	Change Rate (%) ④ = (② − ③)/①
S13	489	516	473	8.6
S11	411	440	395	10.9
S6	419	456	419	8.8
S1	375	405	365	10.6
SC1	371	402	361	11.5
SC7	416	455	415	9.6
SC13	492	520	485	7.1

shows the variation of cable force for typical cables. It can be seen from the table that due to the large initial cable force, the variation of cable force during the vehicle load’s movement along the bridge is relatively small, within the range of 7% to 12%.

4.2. The Influence of Vertical Loads

4.2.1. Beam Deflection

(1)

Symmetrical fusing of side-span cables

To consider the fusing of side span cables due to fire, cables with the same number on the bridge tower side span are taken as the research object to analyze their impact on beam deflection. The symmetrical fusing mode of the side span cable is shown in

Table 3. Symmetric fusing mode of side span cable and beam deflection.

Mode	Description	Maximum Deflection of Side Span (mm)			Maximum Deflection of Mid Span (mm)
		Undamaged	Fusing	Deflection Increment	Undamaged	Fusing	Deflection Increment
Undamaged	Undamaged cable	−12.8	/	/	−1269.5	/	/
S13 + N13	Upper cable S13 + N13 fusing	2.2	99.8	97.6	−1269.5	−1455.1	−185.6
S11 + N11	Upper cable S11 + N11 fusing	2.2	57.3	55.1	−1269.5	−1388.7	−119.2
S8 + N8	Middle cable S8 + N8 fusing	2.6	−60.1	−62.7	−1269.5	−1296.2	−26.7
S6 + N6	Middle cable S6 + N6 fusing	−6.2	−75.4	−69.2	−1269.5	−1269.7	−0.2
S1 + N1	Lower cable S1 + N1 fusing	−23.1	−23.1	0	−1269.5	−1269.2	0.3

, the deflection change of the beam is shown in Figure 11a, and the schematic diagram of changes in the bridge under side span cable fusing is shown in Figure 12a.

As can be seen in Figure 11a, the fusing of upper cables S13 + N13 of the bridge tower has the greatest impact on the deflection of the beam, followed by S11 + N11. The maximum deflection of the mid span is 1455.1 mm and 1388.7 mm, respectively, and the maximum deflection of the side span is 99.8 mm and 57.3 mm. This is because after the upper cable fusing of the bridge tower, the cables (SC13 + NC13) at the corresponding position of the mid span are subjected to unbalanced forces. The smaller stiffness in the upper and middle parts of the bridge tower causes the tower to tilt towards the mid span, resulting in a downward deflection of the mid span beam. The upper and middle cables (S12~S9, S8~S5) on the same side are subjected to more redistributed loads, resulting in an increase in cable force and an upward deflection of the side span beam (as shown in Figure 12a ②). The fusing of middle cables S8 + N8 and S6 + N6 of the bridge tower has a relatively small impact on the deflection of the mid span beam, but a significant impact on the deflection of the side span. The maximum deflection of the mid span is 1296.2 mm and 1269.7 mm, respectively, and the maximum deflection of the side span is 60.1 mm and 75.4 mm. This is because after the fusing of the middle cable of the bridge tower, the upper cables (SC13~SC9) of the mid span are less affected and only produce a small displacement towards the mid span. The upper and middle cable forces (S13~S9, S8~S5) on the same side increase, and the smaller stiffness in the upper and middle parts of the bridge tower causes the tower to tilt outward under greater tension, resulting in a significant downward deflection of the beam in the middle section of the side span after losing some forces (as shown in Figure 12a ③). The fusing of lower cables S1 + N1 of the bridge tower has the same impact on the deflection of the entire bridge as the undamaged state. This is because after the fusing of the lower cables of the bridge tower, the cables on the same side share the load, and the lower part of the bridge tower has a large stiffness, which prevents displacement of the bridge tower, meaning there is no significant change in the shape of the beam (as shown in Figure 12a ④).

(2)

Symmetrical fusing of mid-span cables

To consider the fusing of mid-span cables due to fire, cables with the same number on the bridge tower mid span are taken as the research object. The symmetrical fusing mode of the mid span cable is shown in

Table 4. Symmetric fusing mode of mid-span cable and beam deflection.

Mode	Description	Maximum Deflection of Side Span (mm)			Maximum Deflection of Mid Span (mm)
		Undamaged	Fusing	Deflection Increment	Undamaged	Fusing	Deflection Increment
Undamaged	Undamaged cable	−12.8	/	/	−1269.5	/	/
SC13 + NC13	Upper cable SC13 + NC13 fusing	−12.8	−15.2	−2.4	−1269.5	−1631.7	−362.2
SC7 + NC7	Middle cable SC7 + NC7 fusing	−12.8	−12.8	0	−1269.5	−1264.7	4.8
SC1 + NC1	Lower cable SC1 + NC1 fusing	−12.8	−10.1	2.7	−1269.5	−1268.9	0.6

, the deflection change of the beam is shown in Figure 11b, and the schematic diagram of changes in the bridge under mid-span cable fusing is shown in Figure 12b.

As can be seen in Figure 11b, the fusing of upper cables SC13+NC13 of the bridge tower has the greatest impact on the deflection of the mid span beam, with a maximum deflection of 1631.7 mm. The impact on the deflection of the side span beam can be ignored, with a maximum deflection of only 15.2 mm. This is because the self weight of the mid-span beam is borne jointly by the cables on both sides of the bridge tower. After the upper cables on both sides of the mid span fuse simultaneously, the beam in the middle section of the mid span is unsupported by cables, resulting in downward deflection under the self weight. After loss of the cable force, the upper cables forces (SC9~SC12, NC12~NC9) increase, and the smaller stiffness in the upper part of the bridge tower tilts towards the mid span under tension, resulting in additional deflection (as shown in Figure 12b ②). The fusing of middle cables SC7 + NC7 of the bridge tower has a significant impact on the deflection of the beam only at its location, and has no effect at other locations. This is because after the fusing of the middle cable of the bridge tower, the cables on the same side share the beam load, causing an increase in cable force and pulling the middle part of the bridge tower outward. At the same time, under the weight of the beam, there is a downward increase in deflection (as shown in Figure 12b ③). The fusing of lower cables SC1 + NC1 of the bridge tower has the same impact on the deflection of the entire bridge as the undamaged state. This is because after the fusing of the lower cables of the bridge tower, the weight of the beam is borne by other cables and the bridge tower, and the stiffness of the lower part of the bridge tower is relatively high, making it less prone to deformation (as shown in Figure 12b ④).

(3)

Asymmetrical fusing of cables

According to the symmetrical fusing analysis of side-span and mid-span cables, it can be concluded that the fusing of side-span cables S13 and N13, as well as mid-span cables SC13 and NC13, has a significant impact on the deflection of the beam. Below, cables S13, N13, SC13, and NC13 are used to analyze the changes in the deflection of the beam under asymmetric cable fusing. The asymmetric fusing mode of the side-span and mid-span cables is shown in

Table 5. Asymmetric fusing mode of cables and beam deflection.

Mode		Description	Maximum Deflection of Side Span (mm)			Maximum Deflection of Mid Span (mm)
			Undamaged	Fusing	Deflection Increment	Undamaged	Fusing	Deflection Increment
Undamaged		Undamaged cable	−12.8	/	/	−1269.5	/	/
Side-span cable	SU13 + SD13	U represents upstream, D represents downstream. Cable in fused state	2.2	95.2	93	−1269.5	−1368.9	−99.4
	SU13		2.2	43.8	41.6	−1269.5	−1314.0	−44.5
Mid-span cable	SCU13 + SCD13		−12.8	−29.9	−17.1	−1269.5	−1427.2	−157.7
	SCU13		−12.8	−20.5	−7.7	−1269.5	−1340.3	−70.8

, and the deflection of the side-span and mid-span beams is shown in Figure 11c and Figure 11d, respectively.

As can be seen in Figure 11c,d, for the mid-span beam, the impact of mid-span cable fusing is greater than that of side-span cables. The side-span cables SU13 + SD13, and asymmetric fusing of SU13 only, has a significant impact on half of the bridge’s deflection. The peak deflection in the middle section of the mid span will lean towards the side with severe damage. The maximum deflection of the mid span is 1368.9 mm and 1314.0 mm, respectively, and the maximum deflection of the side span is 95.2 mm and 43.8 mm, respectively. The principle of this phenomenon corresponds to Figure 12a ②. The mid span cables SCU13 + SCD13 and asymmetric fusing of SCU13 has a significant impact on the deflection of mid-span beams, and the deflection of mid-span and side-span beams shows a weak antisymmetric trend along the centerline of the bridge span. The maximum deflection of the mid span is 1427.2 mm and 1340.3 mm, and the maximum deflection of the side span is 29.9 mm and 20.5 mm, respectively. The principle of this phenomenon corresponds to Figure 12b ②.

4.2.2. Cable Force

The cable has a relatively high initial stress, and when it is fused by fire, its stress will be redistributed, affecting the forces of other cables. This section is based on the symmetrical fusing mode of the side-span cable in Section 4.2.1, and explores the variation characteristics of other cable forces under the condition of one cable being fused.

Table 6. Asymmetric fusing mode of the upstream cable.

Mode		Description
Undamaged		Undamaged cable
Upper cable	SU13	Cable in fused state. The upstream and downstream cable forces are denoted as U-SU13 and D-SU13, respectively
	SU11	Cable in fused state. The upstream and downstream cable forces are denoted as U-SU11 and D-SU11, respectively
Middle cable	SU8	Cable in fused state. The upstream and downstream cable forces are denoted as U-SU8 and D-SU8, respectively
	SU6	Cable in fused state. The upstream and downstream cable forces are denoted as U-SU6 and D-SU6, respectively
Lower cable	SU1	Cable in fused state. The upstream and downstream cable forces are denoted as U-SU1 and D-SU1, respectively

shows the asymmetric fusing mode of the upstream cable on the south bank. Due to the asymmetry of the cable fusing with respect to the bridge centerline, there may be differences in the cable forces between the upstream and downstream. Therefore, the cable forces for the upstream and downstream are presented separately, and compared with the cable forces in the undamaged state. Figure 13 shows the cable force on both sides of the bridge after typical upstream cables (SU13, SU11, SU8, SU6, SU1) are fused. Figure 14 shows a schematic diagram of changes in cable force under cable fusing.

Based on the data in Figure 13, the impact of the fusing of the upper cables SU13 and SU11 of the bridge tower, and of the middle cables SU8 and SU6 on the two side cables, is as follows: the upstream cable force of the side span is greater than that downstream cable force of the side span, and the upstream cable force of the mid span is smaller than the downstream cable force of the mid span. The upstream cable force of the side span increases, while the upstream cable force of the mid span decreases. Near the fused cable, the cable force changes significantly. When cable SU6 is fused, the force on the adjacent cable SU5 is the highest, reaching 520 MPa. Within the same area of the fused cable, the force of several cables near the downstream of the side span increases, while the force of other cables at the downstream of the side span decreases. The downstream cable force of the middle span also increases. This is because after the fusing of upper cables of the bridge tower, the beam load is redistributed, causing the cables on the same side of the side span and the cables on the opposite side of the side span to bear more load in the same area (as shown in Figure 14a,b). The fusing of lower cable SU1 of the bridge tower only affects the cables within the lower range, and the change in cable force is relatively small. The lower cable force of the side span increases, while the lower cable force of the mid span decreases, and the other cable forces are consistent with the undamaged state. This is because after the fusing of the lower cable of the bridge tower, the lower cable and the bridge tower jointly share the beam load, and more load is transmitted to the bridge tower, resulting in less significant changes in the lower cable force of the bridge tower (as shown in Figure 14c).

The fusing of the upper cables of the bridge tower has a significant impact on other cable forces, and can be considered as the most unfavorable instance of cable fusing. Therefore, taking the upper cable of the bridge tower as an example, the change rate of cable force can more intuitively display the growth of cable force.

Table 7. Fusing mode of the upper cable.

Mode	Description
Undamaged	Undamaged cable
SU13	Cable in fused state. The upstream cable forces are denoted as U-SU13
SU13 + NU13	Cable in fused state. The upstream cable forces are denoted as U-SU13 + NU13
S13	Cable in fused state. The upstream cable forces are denoted as U-S13
S13 + N13	Cable in fused state. The upstream cable forces are denoted as U-S13 + N13

shows the fusing mode of the upper cable, and Figure 15 shows the cable force and change rate of other cables under upper-cable fusing.

As can be seen in Figure 15, when the upper cable of the bridge tower fuses, the side-span cable force generally increases, while the mid-span cable force decreases. Moreover, the change rate of cable force is significant when it is close to the fused cable. The upper cables of the bridge tower are more sensitive, and when the symmetrical cables S13 + N13 fuse, the adjacent cable forces in the upper part increase the most significantly, with a maximum cable force change rate of 39%. The change rate of cable forces in the middle and lower cables of the bridge tower is relatively small, less than 5%. When paired cables (S13, S13 + N13) are fused simultaneously, the force on the three nearby cables increases significantly, and the change rate reaches more than 20%. When cables (SU13, S13) fuse asymmetrically, the cable force in the middle section of the mid span decreases significantly. The principles of the above phenomena correspond to Figure 12a ② and Figure 14a. Due to the design strength of the cable being 1570 MPa, considering a safety factor of 2.5 times, the maximum service strength of the cable cannot exceed 628 MPa. When upper cable S13 of the bridge tower is fused, although the force on adjacent cable S12 has not reached the warning value, the margin is already very small. Therefore, the upper cable should be given special attention in bridge operation.

4.3. The Influence of Different Degrees of Cable Damage

According to the beam deflection and cable force characteristics under the condition of cable fusing in Section 3.2, damage to the upper cables of the bridge tower has a significant impact on the overall performance of the bridge, while other cable damage mainly affects the local performance of the bridge. This section comprehensively considers the most unfavorable situation, selecting side-span cables S13 and S12, and mid-span cables SC13 and SC12, to analyze the impact of different degrees of cable damage on the beam deflection and cable force. According to Equations (1)–(4), damage simulation of the cable force is carried out by reducing the elastic modulus based on the fire temperature and cable damage rate.

Table 8. Elastic modulus of a cable damaged by fire.

Damage Degree	Elastic Modulus Ei (MPa)
Undamaged	E = 1.9500 × 105
25%	E1 = 1.4625 × 105
50%	E2 = 0.9750 × 105
75%	E3 = 0.4875 × 105
100%	E4 = 0

shows the elastic modulus of a cable damaged by fire, and

Table 9. Different degrees of cable damage.

Mode		Deflection Increment (mm)	Change Rate of Deflection (%)
Undamaged		0	/
Upper cable	S13 + S12 − 25%	38.9	3.1
	S13 + S12 − 50%	90.6	7.1
	S13 + S12 − 75%	162.7	12.2
	S13 + S12 − 100%	270.0	21.3
Middle cable	SC13 + SC12 − 25%	58.9	4.6
	SC13 + SC12 − 50%	136.0	10.7
	SC13 + SC12 − 75%	241.5	19.0
	SC13 + SC12 − 100%	394.6	31.1

shows the different degrees of cable damage.

Figure 16 shows the beam deflection under different degrees of cable damage. As can be seen in Figure 16, within the elastic working range, the impact of different degrees of cable damage on the deflection of the beam remains unchanged, manifesting as a quantitative change. As the degree of damage increases, the increase in deflection of the beam gradually becomes apparent. Under the same degree of damage, the impact of mid-span cable damage on the deflection of the beam is greater than that of side-span cable damage. When the cable damage occurs at the side span, it has a significant impact on half of the bridge’s deflection, and the peak deflection in the middle section of the mid span will lean towards the side with severe damage. The case of 100% damage to cable S13 + S12 has a significant impact on the deflection of the beam, with a deflection increment of 270 mm at the mid span and a deflection change rate of 21.3%. Then, as the degree of damage decreases, the deflection of the beam decreases. When cable damage occurs at the mid span, the deflection of the mid-span and side-span beams shows a weak antisymmetric trend along the centerline of the bridge span. The case of 100% damage to cable SC13 + SC12 has the greatest impact on the deflection of the mid-span beam, with a deflection increment of 394.6 mm at the mid span and a deflection change rate of 31.1%. The different effects of damage to the side-span and mid-span cables on the deflection of the beam verify the case of asymmetric damage to the cables in Section 4.2.1.

Figure 17 shows the change rate of cable force under different degrees of cable damage. As can be seen in Figure 17, the upper cable force of the bridge tower is more sensitive, and the change rate of cable force is significant when it is close to the fused cable. Damage to the upper cables of the side span and the upper cables of the mid span have a mutual influence on each other’s cable forces, but compared to the change rate of the adjacent cable force, their impact is relatively small. As the degree of damage increases, the change rate of the cable force gradually becomes more severe. When the degree of damage is above 50%, the change rate of the adjacent three cable forces exceeds 10%. The variation of cable force in the middle and lower parts of the bridge tower is gentle, and is less affected by upper cable damage, with a change rate of less than 5%. When the side-span cables S13 + S12 are damaged by 75%, the increase in force between adjacent cables is the most significant, exceeding 20%, and the maximum change rate of cable force reaches 50%. When the mid-span cables SC13 + SC12 are damaged by 75%, the increase in force between adjacent cables is relatively weakened, exceeding 10%, and the maximum change rate of the cable force is 41%. The principles of the above phenomena correspond to the analysis of cable force in Section 4.2.2 and the schematic diagram of cable force in Figure 14.

5. Dynamic Performance Analysis

5.1. Natural Vibration Characteristics of Cable-Stayed Bridges

The natural vibration characteristics of bridge structures are of great significance for bridge detection, monitoring, and maintenance, reflecting the stiffness indicators of bridges [48]. The important parameters in structural dynamic analysis are the natural frequency and main vibration mode, which can be used to determine structural damage. Due to the presence of spatial cables, the lateral bending and torsion of cable-stayed bridges will overlap, resulting in almost no simple vibration mode, with vibration modes dominated by lateral bending or torsion [24,25]. In the three-dimensional finite element model of the bridge, the Lanczos method was used to solve the natural vibration characteristics of the bridge structure. The 10th order natural vibration characteristics of the Sifangtai bridge in an undamaged state are shown in Figure 18.

As can be seen in Figure 18, the natural frequencies of the first 10 orders of the bridge are within 1.0. As the order of vibration modes increases, the natural frequencies of the bridge continue to increase. The first order has the smallest natural frequency of 0.12664, while the tenth order has the largest natural frequency of 1.00327. The variation in natural frequency from the third to the fifth order is relatively small. The vibration mode of the bridge is the result of a combination of multiple vibration modes, with the beam mainly exhibiting vertical bending and the bridge tower mainly exhibiting lateral bending. Taking the eighth order of vibration mode as an example, the vibration mode of this bridge is composed of the second-order antisymmetric vertical bending of the beam and antisymmetric lateral bending of the tower.

5.2. Dynamic Analysis

5.2.1. Symmetrical Fusing of Cables

Based on the condition of cable fusing at the side span and mid span in Section 3.2, two cables in the upper, middle, and lower parts of the bridge tower are selected to analyze the dynamic impact of symmetrical cable fusing on the bridge structure.

Table 10. Structural frequency under symmetric cable fusing (Hz).

Order		1	2	3	4	5	6	7	8	9	10
Natural frequency		0.127	0.363	0.460	0.472	0.488	0.651	0.733	0.839	0.890	1.003
Side-span cable	S13 + N13	0.126	0.328	0.427	0.472	0.488	0.651	0.727	0.835	0.889	1.003
	S11 + N11	0.125	0.339	0.437	0.472	0.488	0.651	0.730	0.838	0.890	1.003
	S8 + N8	0.126	0.362	0.459	0.472	0.488	0.651	0.726	0.808	0.859	0.991
	S6 + N6	0.127	0.362	0.457	0.472	0.488	0.651	0.712	0.784	0.846	0.992
	S3 + N3	0.127	0.362	0.458	0.472	0.488	0.651	0.728	0.824	0.880	1.003
	S1 + N1	0.127	0.362	0.459	0.472	0.488	0.651	0.733	0.838	0.890	1.003
Mid-span cable	SC13 + NC13	0.127	0.345	0.450	0.472	0.488	0.645	0.651	0.832	0.886	0.997
	SC11 + NC11	0.127	0.352	0.456	0.472	0.488	0.651	0.707	0.805	0.889	0.984
	SC8 + NC8	0.127	0.360	0.450	0.472	0.488	0.651	0.730	0.823	0.861	1.000
	SC6 + NC6	0.127	0.361	0.452	0.472	0.488	0.651	0.713	0.836	0.854	0.937
	SC3 + NC3	0.127	0.362	0.459	0.472	0.488	0.651	0.727	0.837	0.887	0.956
	SC1 + NC1	0.127	0.363	0.459	0.472	0.488	0.651	0.732	0.838	0.890	0.999

shows the structural frequencies under symmetric cable melting, while Figure 19 shows the frequency-order and frequency change rate-order under symmetric cable fusing.

As can be seen in Figure 19a,b, the frequency change of the cable-stayed bridge is relatively small, and the maximum frequency variation does not exceed 10%. Side-span cable fusing has an impact on the vertical bending vibration mode of cable-stayed bridges, but has a relatively small impact on the longitudinal and lateral bending vibration modes. The upper and middle cables of the bridge tower are the main influencing factors. The upper cables of the side span have a significant impact on the low-order frequency of the bridge. Among them, when cables S13 + N13 are fused, this has the greatest impact on the frequency of the first-order symmetric vertical bending and first-order antisymmetric vertical bending modes (second and third order), with a maximum change rate of 9.6%. The middle cables of the side span have a significant impact on the high-order frequency of the bridge. Among them, the fusing of cables S6 + N6 has the greatest impact on the frequency of second-order antisymmetric vertical bending and high-order symmetric vertical bending modes (eighth and ninth order), with a maximum change rate of 6.6%. As can be seen in Figure 19c,d, the maximum frequency change of the cable-stayed bridge exceeds 10%. The fusing of mid-span cables also has a significant impact on the vertical bending vibration mode of the bridge, but has no effect on the longitudinal floating and lateral bending vibration modes. Both the upper and middle cables of the mid span have a significant impact on the high-order frequency of the bridge. Among them, when cables SC13 + NC13 are fused, this has the greatest impact on the frequency of the second-order symmetric vertical bending modes (seventh order), with a maximum change rate of 11.2%. When cables S6 + N6 are fused, this has the greatest impact on the frequency of high-order antisymmetric vertical bending modes (tenth order), with a maximum change rate of 6.6%.

5.2.2. Asymmetrical Fusing of Cables

As can be seen from the previous section, upper-cable fusing is the most important factor affecting the frequency of cable-stayed bridge structures. Therefore, the upper cables of the side span and mid span are selected to analyze the dynamic impact of asymmetric cable fusing on the bridge structure.

Table 11. Structural frequency under asymmetric cable fusing (Hz).

Order		1	2	3	4	5	6	7	8	9	10
Natural frequency		0.127	0.363	0.460	0.472	0.488	0.651	0.733	0.839	0.890	1.003
Side-span cable	SU13	0.127	0.353	0.453	0.472	0.488	0.651	0.731	0.838	0.890	1.003
	SU13 + SD13	0.126	0.342	0.446	0.472	0.488	0.651	0.730	0.837	0.890	1.003
	SU13 + NU13	0.126	0.346	0.444	0.472	0.488	0.651	0.730	0.837	0.889	1.003
	S13 + N13	0.126	0.328	0.427	0.472	0.488	0.651	0.727	0.835	0.889	1.003
Mid-span cable	SCU13	0.127	0.359	0.457	0.472	0.488	0.651	0.712	0.838	0.889	1.002
	SCU13 + SCD133	0.127	0.355	0.454	0.472	0.488	0.651	0.690	0.836	0.888	1.001
	SCU13 + NCU13	0.127	0.356	0.455	0.472	0.488	0.651	0.690	0.835	0.888	1.000
	SC13 + NC13	0.127	0.345	0.450	0.472	0.488	0.645	0.651	0.832	0.886	0.997

shows the structural frequencies under asymmetric cable fusing, while Figure 20 shows the Frequency-order and frequency change rate-order under asymmetric cable fusing.

As can be seen in Figure 20, asymmetric fusing of the upper cables has a significant impact on the vertical bending vibration mode of the cable-stayed bridge, and as the amount of cable fusing increases, the frequency of the bridge structure increases, while it has almost no effect on the frequency of other vibration modes. The impact of asymmetric fusing of the mid-span upper cable on the bridge is greater than that of the side span. The asymmetric fusing of the side-span upper cable has a significant impact on the low-order frequency of the bridge. Among them, when the cable SU13 + SD13 is fused, this has the greatest impact on the frequency of the first-order symmetric vertical bending mode (second order), with a maximum change rate of 5.7%. The impact on high-order frequency is relatively small, and the maximum frequency change rate of the second-order symmetric vertical bending mode (seventh order) is only 0.4%. The asymmetric fusing of the mid-span upper cable has a significant impact on the high-order frequency of the bridge. Among them, when the cable SCU13 + SCD13 is fused, this has the greatest impact on the frequency of the second-order symmetric vertical bending mode (seventh order), with a maximum change rate of 5.9%. The impact on low-order frequency is relatively small, and the maximum frequency change rate of the first-order symmetric vertical bending mode (second order) is 2.0%.

5.2.3. Different Degrees of Cable Damage

As in Section 4.3, using the worst-case scenario as the standard, cables S13, S12, SC13, and SC12 were selected to analyze the impact of different degrees of cable damage on the dynamic performance of cable-stayed bridges.

Table 12. Structural frequency under different degrees of cable damage.

Order		1	2	3	4	5	6	7	8	9	10
Natural frequency		0.127	0.363	0.460	0.472	0.488	0.651	0.733	0.839	0.890	1.003
Side-span cable	S13 + S12 − 25%	0.126	0.354	0.453	0.472	0.488	0.651	0.732	0.838	0.890	1.003
	S13 + S12 − 50%	0.126	0.343	0.447	0.472	0.488	0.651	0.730	0.837	0.890	1.003
	S13 + S12 − 75%	0.126	0.329	0.441	0.472	0.488	0.651	0.728	0.836	0.889	1.003
	S13 + S12 − 100%	0.126	0.309	0.436	0.472	0.488	0.651	0.727	0.835	0.889	1.003
Mid-span cable	SC13 + SC12 − 25%	0.127	0.360	0.458	0.472	0.488	0.651	0.716	0.836	0.890	1.001
	SC13 + SC12 − 50%	0.127	0.356	0.457	0.472	0.488	0.651	0.696	0.834	0.889	0.998
	SC13 + SC12 − 75%	0.127	0.350	0.455	0.472	0.488	0.651	0.672	0.832	0.888	0.995
	SC13 + SC12 − 100%	0.127	0.342	0.453	0.472	0.488	0.644	0.651	0.830	0.888	0.992

shows the structural frequency under different degrees of cable damage, while Figure 21 shows the frequency-order and frequency change rate-order under different degrees of cable damage.

As can be seen in Figure 21, damage to the side-span and mid-span cables has an impact on the low and high-order frequency of the cable-stayed bridge, mainly changing the frequency of the vertical bending mode. Side-span cable damage mainly affects low-order frequency, while mid-span cable damage mainly affects high-order frequency. Although the stages of action are different, the maximum frequency change rate of the structure under side-span cable damage is greater than under mid-span cable damage. As the degree of damage to the cables increases, the amplitude of frequency changes in the structure gradually becomes apparent. When the side-span cable is damaged, the frequency change of the first symmetric vertical bending mode (second order) is the largest, with a maximum change rate of 14.8%, corresponding to the case where the damage degree of cables S13 + S12 reaches 100%, and the frequency of other modes significantly decreases. When the mid-span cable is damaged, the frequency change of the second-order symmetric bending mode (seventh order) is the largest, with a maximum change rate of 11.2%, followed by the frequency change of the first-order symmetric vertical bending mode (second order), with a maximum change rate of 5.5%.

6. Conclusions

Based on the Sifangtai Bridge, finite element software and damage theory calculation methods were used to analyze the static and dynamic performance of cable-stayed bridges with cables damaged by fire. The following conclusions were drawn:

• During the movement of vehicle load along the bridge, the supporting effect of the bridge towers on both sides hinders the expansion of the main beam deflection. Due to the smaller stiffness of the upper part of the bridge tower, the tension of the cables on the upper part of the bridge tower increases in order to prevent its deformation. When the load acts on the corresponding positions of the middle and lower cables of the bridge tower, the cable force increases significantly. The existence of initial cable force makes the amplitude of cable force variation not significant, ranging from 7% to 12%.
• After the upper and middle cables of the bridge tower are fused, the remaining cables are subjected to unbalanced forces, causing the bridge tower to tilt and thus having a significant impact on the deflection of the beam. Due to the high stiffness of the lower part of the bridge tower, the impact of cable fusing on the deflection of the entire bridge is the same as in the undamaged state.
• The cable has a relatively high initial stress, and when it fuses due to fire, its stress will be redistributed, affecting the forces of other cables. The fusing of upper cables of the bridge tower has a global effect on the variation of cable force, while the fusing of other cables has a local effect. After cable fusing, the load on the beam is redistributed, causing the cables on the same side and opposite side to bear more load in the same area. Therefore, the rate of change in cable force is significant in proximity to the fused cables, which requires special attention during bridge operation.
• Within the elastic working range, the impact of different degrees of cable damage on the deflection of the beam remains unchanged, manifesting as a quantitative change. As the degree of damage increases, the effective area of the inner steel wire of the cable decreases, leading to a gradual decrease in its bearing capacity, resulting in a significant increase in beam deflection. Under the same degree of damage, the impact of mid-span cable damage on the deflection of the beam is greater than that of side-span cable damage, and the deflection changes of the side-span and mid-span beams are consistent with the situation of cable fusing due to fire.
• The vibration mode of the bridge is the result of a combination of multiple vibration modes, with the beam mainly exhibiting vertical bending, and the bridge tower mainly exhibiting lateral bending. As the order of vibration modes increases, the higher-order modal changes become increasingly complex, leading to a continuous increase in the natural frequency of cable-stayed bridges.
• In terms of dynamic performance, cable damage or fusing has a greater impact on the vertical bending vibration mode of the entire bridge, but has a smaller impact on the longitudinal floating and lateral bending. The upper cable of the bridge tower is the main influencing factor, the upper cable of the side span mainly affects the low-order frequency, and the upper cable of the mid span mainly affects the high-order frequency. The impact of different degrees of cable damage on the dynamic performance of cable-stayed bridges is reflected in quantitative changes. As the degree of cable damage and the number of fused cables increase, the maximum bearing capacity of the bridge will continue to decrease, leading to more significant changes in structural frequency.

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