Keywords

1 Introduction

As the main stress components of cable-stayed bridges, cable-stayed cables transfer the loads from the main girders to the main towers and play a role in supporting and dispersing the loads, so their integrity plays a crucial role in the safety of the whole cable-stayed bridge. However, in the actual operation of cable-stayed bridges, there are corrosion [1], fatigue [2], fire [3], explosion [4], impact [5] and other factors, and usually a variety of diseases together [6], cable-stayed cables are easy to be damaged, reducing the life of the discovery of the untimely and withstand the extreme conditions, withstand the sudden increase in the load, the cable breakage may occur.

In 2001, the Nanmen Bridge in Yibin City, Sichuan Province, in 2007, the Mezcala Cable-stayed Bridge in Mexico, in 2019, the Nanfang'ao Bridge in Taiwan, and in 2022, a cable-stayed bridge in western India, all suffered from cable breakage, resulting in serious loss of life and property. Therefore, it is important to accurately assess the effects of cable breakage on cable-stayed system bridge structures to ensure the safety of the structures during the design and operation phases.

For the study of broken cables, the American Post-Tensioning Association [7] proposed two simulation methods for broken cables: one is the transient dynamic analysis method; the other is the dynamic amplification factor method, and suggested that the dynamic amplification factor DAF should be taken as 2.0. The European Committee for Standardisation [8] suggested that the DAF should be taken as 1.5. The Institute of Roads and Highways of France [9] suggested that the DAF should be taken in the range of 1.5–2.0. Wolff [10] pointed out that the DAF for bending moment of cable-stayed bridge is generally less than 2.0, and that the DAF for tie force and tower bending moment is higher than 2.0. However, most of them have not considered the vehicle load on the bridge deck, and the force situation of different bridges is different, and the impact of cable breakage is also different. Therefore, the discussion of cable breakage for various bridge types should be subdivided and not generalized. This article studies the cable breakage of a single tower cable-stayed bridge with specific tower beam consolidation, and considers the vehicle loads that are likely to exist in reality.

2 Overview of Cable-Stayed Bridge Project and Finite Element Modelling

The example bridge is a single tower asymmetric cable-stayed bridge with tower beam consolidation, with a main span of 196 m and a side span of 101 + 62 = 163 m. The bridge deck is 29.5 m wide, with a basic cable spacing of 7 m on the main beam, a side span tail cable area of 4.5 m, and a cable spacing of 2.0 m on the tower. A total of 2 × 52 diagonal cables are arranged in a fan-shaped layout, with a steel wire diameter of 7 mm and a standard strength of 1770 Mpa. The cable numbers and arrangements are shown in Fig. 1. The overall layout of the entire bridge is shown in Fig. 2. For convenience, each cross is named as shown in Fig. 2. There are a total of 52 key nodes in the entire bridge, corresponding to the positions of the main beams at 52 pairs of diagonal cables, numbered from left to right as 1–52.

Fig. 1.
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Rope layout

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Fig. 2.
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General layout of the whole bridge

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A three-dimensional fishbone beam finite element model of a cable-stayed bridge was established using ANSYS. The main tower was made of C40 concrete, and the main beam was made of C50 concrete. The three-dimensional solid beam element Beam188 was used for both, and the cables were subjected to axial tension or compression using Link10 rod elements. The main beam has a single box three chamber section, and the bridge tower has a rectangular hollow section, both of which use the actual section input form. The section characteristic values are automatically calculated by the software. The bridge tower and foundation are consolidated, the tower beams are consolidated, and both the left and right supports are constrained in vertical and horizontal displacement. The auxiliary piers are constrained in vertical displacement. A finite element model for the calculation of this cable-stayed bridge is established, as shown in Fig. 3.

Fig. 3.
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Finite element model of cable-stayed bridge

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3 Diagonal Cable Fracture Vibration Analysis

To study the dynamic response of the bridge structure of cable-stayed bridges under broken cables, firstly, the deflections of the structure under self-weight, phase II and tension cables are calculated considering the geometric nonlinearity as the initial state of the structure before breaking the cables, and then the dynamic response of the structure after breaking the cables is analysed by removing the tension cables unit in the dynamic time-course calculations.

According to the General Specification for Highway Bridge and Culvert Design (JTG D60-2015) and Design Specification for Highway Cable-stayed Bridges (JTG/T3365-01-2020), the deflection of the main girder of cable-stayed bridge is calculated under the limit state of normal use, and the deformation of the main girder of ANSYS cable-stayed bridge model is shown in Fig. 4, and the maximum value of the main girder downward deflection occurs at the key point 8, with the maximum value of 163 mm, the deflection-to-span ratio is less than L/500 (L is the calculated span diameter) required by the design specification, which meets the specification requirements.

Fig. 4.
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Displacement of main beam under static load

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3.1 Selection of Single Rope Breaking Condition

The power response method is used to analyse the vibration impact of different locations of the cable breakage on the whole bridge, the example bridge is an asymmetric structure, the left and right sides of the bridge tower span of the diagonal cable breakage situation is different, it should be considered separately, and choose a total of 10 kinds of single diagonal cable breakage conditions, see Table 1.

Table 1. Single cable breakage conditions
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3.2 Heavy-Duty Arrangement of Single Broken Ropes

According to the “General Code for Design of Highway Bridges and Culverts” (JTG D60-2015), the standard value of uniformly distributed load on Class I lanes of highways is fully distributed on the same influence line that causes the most adverse effect on the structure. The standard value of concentrated load only acts on the peak value of one influence line in the corresponding influence line. Each working condition applies a uniformly distributed load qk = 10.5 kN/m on the span where the cable is broken, and a concentrated load pk = 360 KN is applied at the critical point of the cable break, which serves as a rough simulation of the vehicle's heavy load effect. The heavy load arrangement for each rope break condition is shown in Table 2.

Table 2. Heavy duty arrangement
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3.3 Effect of a Single Broken Cable on the Main Girder Line Shape

The maximum displacement generated by the broken cable under heavy load, need to break the cable node vibration time course analysis, the main beam 52 on the diagonal cable, to the cable MC19 and the main beam anchorage at the key point 8, for example, its vibration time course analysis results in working condition 3 in Fig. 5.

Fig. 5.
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Vibration time histories at critical point 8 under operating condition 3

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From Fig. 5, it can be seen that in the vibration of a single broken cable under heavy load, the maximum displacement after MC19 breaks the cable is 269 mm, and the stable displacement after vibration stops is 212 mm. The entire vibration process is in a downward bending state, with the maximum value of 269 mm representing the deflection value of key point 8 after breaking the cable under condition 3. Similarly, for other key points, the peak displacement of the vibration after breaking the cable is taken as the displacement value of the key point after breaking the cable. The displacement values of each key point of the main beam under condition 1–10 of cable breaking are obtained as shown in Fig. 6.

Fig. 6.
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Displacement of key points under a single broken cable

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From the upper displacement Fig. 6, it can be concluded that the main girder displacement is the largest in Case 3, and the maximum displacement value occurs at critical point 8, reaching 269 mm. The larger the value of the cable force, the larger the displacement produced by the cable breakage. Therefore, when considering the most unfavourable effect of multi-cord breakage, it is only necessary to increase the broken cords near the maximum displacement of the single-cord breakage result to find out the most unfavourable result of the broken cords.

3.4 Selection of Working Conditions for Multiple Break Ropes

CHEN [11] pointed out that in the case of double-rope fracture, compared with the origin symmetric fracture and transverse symmetric fracture, the longitudinal symmetric fracture has the greatest impact on the line shape, combined with the results of Fig. 5, the location of MC19 (key point 8) was selected for the simulation of multiple-rope fracture, with a total of two types of double-rope fracture conditions and two types of triple-rope fracture conditions, and the specific conditions of rope breakage are shown in Table 3.

Table 3. Multiple tie breakage conditions
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The heavy load arrangement for the multi-cord break is the same as for the single cable breakage condition 3.

3.5 Effect of Multiple Broken Cables on the Main Girder Line Shape

The displacement values of each key point of the main girder under the broken rope condition 11–14 are shown in Fig. 7.

Fig. 7.
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Displacement map of each key point under multiple broken cables

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As can be seen from the resultant Fig. 7, in the case of multi-cord breakage, the same-side broken cords produce larger displacements than the two-side broken cords, which have a greater impact on the main beam line shape. The maximum displacements of the main girder all occurred near the cable MC19 (key point 8), and the maximum value was 366 mm in case 11, the maximum value of the main girder displacement was 323 mm in case 12, the maximum value of the main girder displacement was 481 mm in case 13, and the maximum value of the main girder displacement was 425 mm in case 14. The displacements generated by the 2 conditions of the three cable breakage, 481 mm and 425 mm, have already exceeded the L/500, i.e. 392 mm.

4 Conclusion

In current research, more attention has been paid to the study of the dynamic amplification factor of broken cables, and insufficient attention has been paid to the actual vibration process of broken cables and vehicle loads. Sometimes, the conclusion of a certain bridge is used for all cable-stayed bridges and even cable load-bearing bridges, which is biased towards one sidedness. Different bridge types should be analyzed separately.

This article presents the finite element analysis results of a representative single broken cable, multiple broken cables, the combined action of heavy load and single broken cables, as well as the combined action of heavy load and multiple broken cables in a tower beam consolidated single tower cable-stayed system. The influence of each broken cable on the main beam line shape is obtained, and the conclusions are as follows:

  1. (1)

    The influence of broken cables on the main girder line shape is different in different locations, the smaller the tie force is and the smaller the bending moment is, the smaller the influence of broken cables on the main girder line shape is, and the larger the tie force is and the larger the bending moment is, the larger the influence of broken cables on the main girder line shape is. In the case of multiple breakage of tie ropes, the displacement produced by the same side breakage is larger than that produced by the two sides breakage, which indicates that the most unfavourable result is produced by the nearest tie ropes breaking together on the same side in the case of multiple breakage of tie ropes.

  2. (2)

    Considering the cable breakage after heavy load, the maximum displacement caused by a single cable breakage is 269 mm, the maximum displacement caused by two cable breakage is 366 mm, and the maximum displacement caused by three cable breakage is 481 mm, which is 1.65 times, 2.25 times, and 2.95 times that of the original bridge 163 mm, respectively. When three cables break, the displacement exceeds the allowable value in the specifications, indicating that a single tower cable-stayed bridge with tower beam consolidation has a high stiffness and is less affected by broken cables than a general cable-stayed bridge. For the possible combined effects of heavy loads and broken cables, a single tower cable-stayed bridge with tower beam consolidation can also have a greater safety reserve for broken cables

This article only studied the displacement variation law of the main beam of a single tower cable-stayed bridge after cable breakage, and the conclusion drawn is slightly different from that of a general cable-stayed bridge after cable breakage. It is hoped that in future research, scholars will pay attention to the cable breakage process and vehicle load, and consider the impact effect of vehicle load to participate in time history analysis, so as to continue to discover the variation law of internal forces, cable forces, etc. of other types of cable-stayed bridges, suspension bridges, and suspension arch bridges after cable breakage, Subdivide and summarize the broken rope module.