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Article

Experimental Study on Fatigue Characteristics and Life Prediction of Rotating Restricted Short Suspender in Suspension Bridge

by
Lei Zhao
1,
Zhili Yang
2,
Xianneng Tong
2,
Yang Zhang
3 and
Ruifeng Nie
4,*
1
CCCC Construction Group Beijing Testing Technology Co., Ltd., Beijing 101104, China
2
School of Architectural and Civil Engineering, Qingdao Institute of Technology, Qingdao 266300, China
3
Shandong Gold Mining (Yinan) Co., Ltd., Yinan 276300, China
4
College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao 266590, China
*
Author to whom correspondence should be addressed.
Buildings 2025, 15(2), 254; https://doi.org/10.3390/buildings15020254
Submission received: 11 December 2024 / Revised: 7 January 2025 / Accepted: 15 January 2025 / Published: 16 January 2025
(This article belongs to the Special Issue Research on Emerging Technologies for Structural Design, Inspection, and Maintenance)
Browse Figures
Figure 1
<p>Ma’anshan Yangtze River Bridge and its suspender: (<b>a</b>) General layout of the bridge; (<b>b</b>) Cross-section of the parallel wires; (<b>c</b>) General layout of the suspender of the parallel wires.</p> "> Figure 1 Cont.
<p>Ma’anshan Yangtze River Bridge and its suspender: (<b>a</b>) General layout of the bridge; (<b>b</b>) Cross-section of the parallel wires; (<b>c</b>) General layout of the suspender of the parallel wires.</p> "> Figure 2
<p>Technical condition inspection of the short suspender: (<b>a</b>) NY-29-1; (<b>b</b>) NZ-30-1; (<b>c</b>) NZ-39-1; (<b>d</b>) NZ-94-2; (<b>e</b>) NZ-97-1; (<b>f</b>) NZ-97-2.</p> "> Figure 3
<p>Test setup for the degradation assessment: (<b>a</b>) Actual scene; (<b>b</b>) Position indication of the displacement gauges.</p> "> Figure 4
<p>Deformation increments of each specimen: (<b>a</b>) NY-029-1; (<b>b</b>) NZ-030-1; (<b>c</b>) NZ-039-1; (<b>d</b>) NZ-094-2.</p> "> Figure 4 Cont.
<p>Deformation increments of each specimen: (<b>a</b>) NY-029-1; (<b>b</b>) NZ-030-1; (<b>c</b>) NZ-039-1; (<b>d</b>) NZ-094-2.</p> "> Figure 5
<p>Comparison of the elastic modulus test results and fitting results.</p> "> Figure 6
<p>Comparison between the effective area ratios and the measured fracture gap size.</p> "> Figure 7
<p>Setup of fatigue performance test: (<b>a</b>) Loading device; (<b>b</b>) Arrangement of the fatigue specimen; (<b>c</b>) Description of finite rotation; (<b>d</b>) Relationship between the specimen and horizontal line.</p> "> Figure 7 Cont.
<p>Setup of fatigue performance test: (<b>a</b>) Loading device; (<b>b</b>) Arrangement of the fatigue specimen; (<b>c</b>) Description of finite rotation; (<b>d</b>) Relationship between the specimen and horizontal line.</p> "> Figure 8
<p>Damage condition of the specimen at different loading stages: (<b>a</b>) After the 1st loading stage; (<b>b</b>) After the 2nd loading stage; (<b>c</b>) After anatomy; (<b>d</b>) Cross-section of the broken parallel wires.</p> "> Figure 9
<p>Calculation principles of the simplified simulation method.</p> "> Figure 10
<p>Annual average traffic volume change and vehicle proportion.</p> "> Figure 11
<p>Traffic flow characteristics of different lanes: (<b>a</b>) Daily variation in the average traffic volume of heavy vehicles; (<b>b</b>) Probability of different lanes being selected by various heavy vehicles.</p> "> Figure 12
<p>Characteristics of individual vehicles: (<b>a</b>) Normal distribution fitting of the vehicle speed; (<b>b</b>) Ratio of the axle load to the vehicle weight; (<b>c</b>) Multi-peak fitting of the vehicle weight.</p> "> Figure 13
<p>Simulation results based on random traffic theory: (<b>a</b>) Lane distribution; (<b>b</b>) Stress response of the suspenders caused by the vehicles on each lane.</p> "> Figure 14
<p>Relationship between the amplitude and the frequency of fatigue stress in each year: (<b>a</b>) 2014; (<b>b</b>) 2015; (<b>c</b>) 2016; (<b>s</b>) 2017; (<b>e</b>) 2018; (<b>f</b>) 2019; (<b>g</b>) 2020.</p> "> Figure 15
<p>Inherent life of the short suspender under different stress levels.</p> "> Figure 16
<p>The converted fatigue cycle curve and its fitting curve.</p> ">
Versions Notes

Abstract

:
The corrosion of the rotating axis pins of the short suspender will lead to the rotating restriction of its end, which will lead to the corrosion of the parallel wires and affect the performance of the short suspender. In this study, the technical condition of the rotating restricted short suspender unfixed from the suspension bridge was carefully detected. An axial tensile performance test was carried out on these short suspenders, and the subsequent availability of the rotating restricted suspender was evaluated based on the size of the fracture gap. The rotationally limited working conditions of these short suspenders were skillfully simulated by the specially designed tooling, and the fatigue performance test of the rotating restricted short suspender was carried out. A simplified simulation method was proposed based on the random traffic theory. By introducing traffic data obtained from the WIM system, the stress response of the short suspenders caused by vehicles on each lane was simulated, and the simulation results were converted by the rain flow counting method. The residual life of the rotating restricted short suspender was predicted by the comparison between the fatigue test results and the fitting curve of the simulation results. From this study, several of the following conclusions can be summarized: The measured fracture gap size is negatively correlated with the effective area of the suspender, and the gap size of 8mm is a key value. When the fatigue load cycle reaches 345,000 times, the suspender is already in a dangerous state. Additionally, the fractured gap size is considered as the judgment basis for the usability of rotating restricted short suspenders. When the gap size is less than 8 mm, the suspender can be continually used after maintenance and should be updated after 6 years. Otherwise, the suspender needs to be replaced immediately.

1. Introduction

Long-span bridges are subjected to the coupling effect of various loads and environmental erosion in their whole life cycle [1,2]. The suspension bridge with a cable structure as the main bearing structure has a relatively small total stiffness, and some components are also prone to fatigue damage. As a key force transmission component between the main girder and the main cable, the suspenders are prone to rotation restrictions due to the corrosion of its end-axis pins [3]. Compared with the length of the anchorage section at the end of the suspension, the length of the parallel wire section of the short suspension is relatively short [4]. Therefore, the stiffness of the short suspender is obviously larger than that of the long suspender. In the case of limited rotation at the end of the short suspension, the deformation of the suspension is difficult to release through the parallel wire section [5,6]. Under the cyclic loading of the vehicle, corrosion and fracture are prone to occur between the end anchorage section and the parallel wire section, which directly affects the safety of the suspenders and even the suspension bridge [7,8].
The fatigue characteristics of suspenders have been experimentally investigated by some scholars [9,10,11]. Xue et al. [12] focused on the suspenders of parallel wires with different corrosion degrees. The single high-strength steel wire in the suspender was selected as the main research object, and a fatigue performance test was carried out. This study reveals the fatigue properties of steel wires with different degrees of corrosion. On this basis, Xue et al. [13] continued to carry out further research based on the engineering case of the Jinsha River Bridge. The cause of suspender failure was analyzed with a fatigue performance test, and the failure process caused by the wear between steel wires was simulated from the micro-level. In addition, Liu et al. [14] adopted the testing method of field damage inspection and short-term monitoring to the short suspenders of the Jiangyin Yangtze River Bridge and clarified the typical damage forms of short suspenders. It can be judged from the monitoring data of 100 s that the traffic load may be one of the most direct causes of high-stress fatigue damage on the short suspender. Similar to the research of Xue et al. [12,13], a refined simulation model was proposed to explore the failure process of the corroded suspenders of parallel wires.
Through a series of experimental studies, scholars have conducted in-depth analyses from different levels and explored the causes of suspender damage to a certain extent. However, most of these studies focused on the mechanical properties and failure modes of the steel wire bundle or even the single steel wire of the suspender. They failed to reflect on the failure reason of the short suspender from the overall structure of the bridge, and failed to provide some prediction methods or suggestions on whether the suspenders can continue to be used.
Therefore, some research has been conducted on these aspects, as follows. Yuan et al. [15] analyzed the corrosion fatigue of suspenders, considering the coupling effect of wind and traffic. The monitoring data were introduced to simulate the stress response of the suspender, and the corrosion fatigue of the suspenders were evaluated. In this study, compared with the wind load, traffic flow was concluded to be the main cause of the corrosion fatigue of the suspender. Additionally, both He et al. [16] and Deng et al. [17] analyzed the fatigue performance of suspenders and predicted their residual life. Among them, the digital twin Bayesian entropy framework was introduced to the study of He et al., and the 8-month on-site WIM data were adopted to the study of Deng et al. However, Deng et al. only revealed the correlation between the fatigue performance of the suspender and traffic loads, and did not provide a method for evaluating the fatigue performance using traffic loads. Although He et al. provided a fatigue life prediction method for the corrosion suspender, the simulation results still lack a combination with actual tests of the suspender.
In this paper, the damage of the rotating restricted short suspender in the mid-span is studied based on the engineering background of the Ma’anshan Bridge. The performance of the suspender is tested and evaluated from the aspects of the axial tension static performance and fatigue performance, and the residual life is predicted by simulation. Firstly, the deformation of different sections and the maximum axial tension force of the short suspender are obtained by the axial tension performance test. The residual effective area of the rotating restricted short suspender is calculated based on the test data, and the update condition of such a suspender is clarified by comparing it with the test data of the facture gap size. Secondly, a specially designed tooling that can simulate the limited rotation condition of the suspender is designed. The fatigue performance test of the rotating restricted short suspender is carried out, and the number of fatigue cycles is obtained. Finally, a simplified simulation method based on random traffic theory is proposed and the measured WIM data are introduced to simulate the fatigue stress cycle of the suspender from 2014 to 2020. Combined with the simulation results and the fatigue performance test results, the residual life of such rotating restricted short suspenders is fitted and predicted.
Through the research performed in this paper, some novelties have been achieved. Firstly, the evaluation principle of the continuous use of the rotating restricted short suspender and the prediction method of the residual life are innovatively proposed. Two key parameters, the effective area ratios and the measured fracture gap size, are specially selected to characterize the damage law of the short suspender. The qualitative relationship between the two key parameters is compared, and the updating principle of the rotating restricted short suspender is proposed. This avoids updating the remaining useful suspenders and can reduce the maintenance cost of the Ma’anshan Bridge. Additionally, the specially designed tooling can skillfully simulate the rotationally limited working conditions of the short suspender, so that the key test data of the short suspender under fatigue load can be obtained. On this basis, the random traffic theory is innovatively introduced to simulate the fatigue load response of short suspenders over the years. Combined with the test data, the residual life of the rotating restricted short suspender can be reasonably predicted, which effectively guarantees the safety of the suspension bridge cable system.

2. Brief Introduction of Short Suspender and Its Technical Condition

2.1. Brief Introduction of Short Suspender Adopted

The Ma’anshan Yangtze River Bridge is a cross-river channel connecting Chaohu City and Ma’anshan City. The total length of this bridge is 36.274 km, and the main bridge can be divided into the left branch and the right branch. The left branch is a three-tower two-span suspension bridge with a length of 360 + 1080 + 1080 + 360 = 2880 m.
The suspender of parallel wires is adopted in this bridge and the cross-section of the suspender is shown in Figure 1. The diameter of the parallel wires is 5.0 mm, and its standard tensile strength is required to be not less than 1670 MPa. The suspender can be divided into the ordinary one and the special one. Among them, the ordinary type is composed of 109 steel wires, and the special type is composed of 121 or 163 steel wires. In addition, a double-layer HDPE protective layer with a thickness of 8.0 mm is applied to the outer surface of the parallel wires. The suspender anchorages adopted at the two ends are the forked-type hot-cast anchorage.

2.2. Technical Condition Inspection of Short Suspender

After the rotating restricted short suspenders were unfixed from the Ma’anshan Bridge, corresponding tests were conducted to assess the short suspenders. The short suspenders of NY-29-1, NZ-30-1, NZ-39-1, NZ-94-2, NZ-97-1, and NZ-97-2 were selected to destructively inspect after several kinds of loading test were completed. The typical damage status can be observed as presented in Figure 2.
It can be concluded from the inspection that the fracture modes of the short suspender can be divided into the following two types:
  • Type A: damage after partial rotation restriction.
Typically for the suspenders NY-29-1, NZ-30-2, NZ-39-1, and NZ-94-2, the fracture process can be summarized in the following steps:
  • The sealing ring between the parallel wires and connecting sleeve fails first. Moisture enters the threaded connection area of the connecting sleeve, and the threaded connection region begins to rust.
  • The stress at the threaded connection region is complex, and it is easy to form stress concentrations in the connecting sleeve. Due to the coupling effect of corrosion, cracks appear on the connecting sleeve and eventually break off.
  • Moisture enters the parallel wires, and the parallel steel wires rust. Part of the steel wire is destroyed under the action of tensile stress, and this eventually leads to the fracture of the suspender.
  • Type B: damage after complete rotation restriction.
Typically, for the suspenders NZ-97-1 and NZ-97-2, the fracture process can be summarized in the following steps:
  • The lubrication between the forked-type hot-cast anchorage and axis pin, and the friction between the two, increases. The rotation performance of the suspender is weakened.
  • Due to the weakening of the rotation performance of the suspender, the deformation of the suspender body decreases, resulting in an increase in the force of the connecting sleeve.
The subsequent steps will continue to repeat all of the steps in fracture type A.

3. Damage Assessment of Rotating Restricted Short Suspender

In order to assess the damage of the short suspender, it is necessary to test its mechanical characteristics. The short suspenders damaged after partial rotation restriction are selected due to the limited number of short suspenders unfixed from the Ma’anshan Bridge.

3.1. Setup of Axial Tension Performance Test

The setup of the axial tension performance test is presented in Figure 3.
The base and the actuator are adopted and fixed on the floor. The two ends of the tested suspender are fixed on the test equipment with axis pins. Six displacement gauges are arranged at the appropriate position of the suspender to measure the displacement in different regions. The gauges of W1 and E1 are arranged at the position of the axis pin, so the length change in suspender (L1) can be measured; the gauges of W2 and E2 are arranged at the ends of the two sockets, so the length change in the parallel wires (L2) can be measured; the gauges of W2 and W3, and E2 and E3 are, respectively, are arranged at the connection position between the suspender anchorage and the connecting sleeve, so the change in the detached gap between them (L3 and L4) can be measured.
The short suspenders of NY-029-1, NZ-030-1, NZ-039-1, and NZ-094-2 were selected for the tests. The suspenders were tensioned in levels, and each loading level was 10 tons. The maximum tension force, displacement, and failure mode were obtained in this process. The condition for terminating the test was set as the displacement of the actuator reaching 60 cm.

3.2. Result Analysis of Axial Tension Performance Test

Table 1 lists the maximum loading force and damage situation when the test termination condition is reached.
It can be learned from the test that, although the maximum loading force of each specimen can reach more than 3000 kN, the results can meet the requirements of the axial tension test by comparing them with the exiting studies [18,19]. Some differences in the damage situation can be observed. Combined with the inspection results of the technical condition of the short suspender, it can be seen that the region between the suspender anchorage and the connecting sleeve is damaged to varying degrees. The effective capacity of the different sections of the suspender may be different. Therefore, it is necessary to check the actual effective elastic modulus of the suspender.
The variation in the displacement measurement results of the adjacent load levels was calculated; as a result, the displacement change relationship of the suspender could be analyzed more intuitively. The deformation increment of each specimen is presented in Figure 4.
It can be learned from the calculation results that the deformation increment of each measuring point is relatively stable when the load is less than 2000 kN. In addition, some sudden drops or rises can be observed in these curves, which may be caused by the following different reasons.
For the specimen NY-029-1, a sudden drop at the load of 900 kN is mainly caused by the debonding between the suspender anchorage and the connecting sleeve. In particular, for specimens NY-029-1 and NZ-094-2, the sudden drop of ΔL1 and ΔL2 to zero may also be caused by this reason. Continuous loading causes a weak fracture between the suspender anchorage and the connecting sleeve, but it does not have too much of an impact on the parallel wire itself in the segment of L2.
For the specimens NZ-030-1 and NZ-039-1, the rapid increase in ΔL1 and ΔL2 after 2700 kN may be caused by the yield of some parallel wires in segment L2. When some of the parallel wires reach the yield point limit, the displacement increment of each loading stage becomes larger. Subsequently, the force on these yielding wires is redistributed to other wires, and the deformation increment ΔL2 gradually returns to the original level.
For the specimen NZ-094-2, a significant increase can be observed in ΔL1, ΔL2, and ΔL3, which is evidently different from the other specimens. It can be clearly considered that the main section of the steel wire yield is located between the suspender anchorage and the connecting sleeve.

3.3. Damage Assessment Based on Fracture Gap Size

According to the definition of the elastic modulus, the actual elastic modulus of the suspender can be tested and calculated by the deformation of the suspender in a certain micro-interval [20]. The corresponding equation is defined as follows:
E ′ = F n − F 0 A ⋅ L Δ l
In which F n and F 0 are final load and initial load, respectively, A denotes the theoretical cross-sectional area of the suspender, and L and Δ l are the initial length and deformation of the suspender in a certain micro-interval.
For the interval between the displacement gauges W3 and E3, the parallel wire is almost not corroded due to the protection of the external PE protective layer. Therefore, it is reasonable to adopt the calculation results in this interval to characterize the actual elastic modulus of the suspender tested.
The elastic modulus calculation results of each specimen under different load levels are compared in Figure 5.
It can be seen from the comparison that the calculated elastic modulus is relatively large at the early stage of loading (loading force < 1000 kN), which is mainly due to the fact that there may still be a certain gap between the parallel wires. When the loading force is in the range of 1000 kN to 2000 kN, the calculation results tend to be stable. When the loading force is larger than 2000 kN, the calculated elastic modulus tends to decrease, which may be caused by the fracture of parallel wires in other regions. Combined with the calculated deformation increments of each specimen, presented in Figure 4, the calculation results of the elastic modulus in the range of 1000 kN to 2000 kN were selected for the fitting. The fitting results of the elastic modulus of specimens NY-029-1, NZ-030-1, NZ-039-1, and NZ-094-2 are 229.6 MPa, 230.4 MPa, 251.2 MPa, and 253.9 MPa, respectively.
The fitting results of the elastic modulus can be applied to calculate the effective area A ′ of the tested suspender. The effective area ratio of the tested suspender can be calculated by comparing it with the design value of the cross-sectional area [21], as shown in Equation (2):
A e f f = A ′ A 0 = F ′ L E ′ Δ l A 0 × 100 %
In which A e f f denotes the effective area ratio, A ′ and A 0 are the effective area and the design area of the tested suspender, E ′ is the fitting result of the elastic modulus, and F ′ is the loading force.
The effective area ratios are compared with the measured fracture gap size between the suspender anchorage and the connecting sleeve, as shown in Figure 6.
From the comparison, it can be learned that the effective area of the suspender shows a significant decrease with the increase in the measured fracture gap size between the suspender anchorage and the connecting sleeve. This indicates that the fracture gap will lead to the corrosion of the parallel wires, which eventually leads to a decrease in the bearing capacity of the suspender. In addition, when the measured fracture gap size reaches a certain degree, about 8mm, the effective area ratio of the suspender changes little. It can be considered that the connecting sleeve has lost its protective effect at this time, and the parallel wires of the suspender will be directly affected by environmental erosion.
Therefore, the updating criterion of the suspender is proposed according to the analysis of the axial tension performance test. When the measured fracture gap size is small (less than 8 mm), the fracture gap should be filled and closed in time, and the corresponding information and maintenance records should be saved.

4. Fatigue Performance Test of Rotating Restricted Short Suspender

The rotation ability of the suspender is often restricted after corrosion at the position of the axis pins during the actual use of the suspension bridge [22]. At this time, the suspender is in a rotating restricted state and will continue to bear the tensile fatigue load, which places the suspender in an abnormal state [23]. Therefore, it is necessary to carry out experimental research on the tensile fatigue performance of the rotating restricted short suspender. The NZ-97-2 specimen with complete rotation restriction was selected for the fatigue performance test.

4.1. Setup of Fatigue Performance Test

The frame-type loading device was adopted for the fatigue performance loading of the suspender, as shown in Figure 7a,b. For the loading device, the one-shaped beam is the fixed end, and the rectangle-shaped beam is the loading end. The one-shaped beam can be adjusted along with the frame loading device according to the length of the suspender being tested. Both the one-shaped beam and the rectangle-shaped beam are equipped with coaxial circular holes. The suspender specimen can pass through the circular hole. The 250t actuator arranged at the loading end was used for the fatigue loading.
Through the specially designed tooling, the suspender can be fixed on the loading device and the deflection angle can be set at the same time. Additionally, the limited rotation of the suspender axis pins in the actual bridge can be simulated. For example, when the angle of the deflection tooling is set to 0.58°, the angle between the axis of the suspender anchorage and the axis of the suspender parallel wire is 2°. At this time, the angle between the axis of the suspender parallel steel wire and the loading axis is 1.42°, as shown in Figure 7c,d.
According to the General Specification for Design of Highway Bridges and Culverts (JTG D60-2015) [24], the fatigue load calculation model was applied to obtain the deflection angle of the suspender under various load combinations. The truss simulation model of the Ma’anshan Bridge was established. The deflection angle of 12 suspenders (3 suspenders on each side of the mid-span section of the two bridge spans) could then be calculated, and the deflection angle is in the range of 2.0798° to 3.5857°. Therefore, it is more unfavorable to select 2° as the deflection angle adopted in this test.
The NY-097-2 suspender, unfixed from the bridge, was selected for the test. This suspender is composed of 109 parallel wires of Grade 1670 MPa. The length of the suspender is 1.694 m and the diameter of the parallel wires is 5 mm, and the theoretical breaking force Pb is equal to 3574 kN. According to the specification requirements, the upper limit of fatigue stress is 0.35 σ b and the stress amplitude is 150 MPa [25]. That is, the upper limit of the fatigue test load is 1251 kN (0.35 Pb) and the lower limit of the load is 930 kN.
The main mechanical parameters adopted in the fatigue performance test are listed in Table 2.
In particular, when the fatigue cycle is loaded to 345,000 times, the loading process is suspended due to the obvious steel wire fracture, which leads to the deformation limit of the sling exceeding the preset protection value of the test device. A preliminary visual inspection was carried out to observe the fracture of the parallel wire between the parallel wires and the connecting sleeves. About 14 broken steel wires in the outer layer of the suspender could be observed, and the rest of the situation was uncertain. Considering that the broken rate of the wires that can be observed is only 12.8%, which is much less than the termination condition of a 30% broken rate, we can safely assume that the breaking rate of the parallel wires has reached 30%, that is, the remaining 76 unbroken steel wires continue to be loaded. The upper and lower limits of the fatigue load are also adjusted accordingly, but the stress amplitude remains unchanged at 150 MPa, as shown in the second loading stage in Table 1.
When the fatigue cycle is accumulated to 583,850 times, a large number of parallel wires are damaged in the NY-097-2 specimen, in which the failure rate exceeds 30%, so the fatigue test is terminated.

4.2. Reliability Evaluation of the Short Suspender Based on the Fatigue Test Results

The failure process of the suspender specimen NY-097-2 can be described as follows:
Before the fatigue test, the fatigue loading device was calibrated to ensure accuracy.
After the first loading stage: when the fatigue cycle reaches 345,000 times, about 14 parallel wires in the outer layer of the suspender are broken, as presented in Figure 8a. The fracture section of the parallel wire is mainly located between the suspender anchorage and the connecting sleeve.
After the second loading stage: when the fatigue cycle reaches 583,850 times, about 74 parallel wires are broken, exceeding the specified limit of 30% (i.e., 33 wires) [25], as presented in Figure 8b. The position between the suspender anchorage and the connecting sleeve is still the place where the parallel wire breaks at this loading stage.
After the test, the connecting sleeve of the suspender and the PE protective layer near it were gouged. It can be observed that the connecting sleeve and suspender anchorage have been completely corroded and disconnected, and the internal parallel wires are also seriously corroded, as shown in Figure 8c,d.
In practice, the data of the fatigue test can be processed according to the statistical theory of probability [26]. Many fatigue test results show that the fatigue life under a given stress level can be characterized by the Weibull distribution, which is more in line with the law of fatigue failure [27]. The probability that the structure will not be destroyed in time (0, N) can be characterized by the following equations:
L N = e x p − N k e − b S − b C e x p − l n l n 2 / k k × 100 %
k e = 1 / 1 − S / S b
k = V N − 1.08
In which N is the ultimate fatigue life of the suspender, expressed by the number of fatigue cycles; C is a variable related to the characteristic lifetime c, where C is equal to 1013.84. When considering the Goodman correction, the coefficient b is recommended to be 3.5 [28]. S is the stress amplitude adopted in the fatigue test, and Sb is the ultimate strength of the parallel wires; k is the shape factor, and k = V N − 1.08 , where the mean value of VN is recommended to be 0.53.
The probability of suspender specimen NY-097-2 can be calculated by Equation (3). When the fatigue loading cycle is 345,600 times, L(N) is equal to 99.9994967266368%. The corresponding reliability index is 4.4158, which is less than the level that is allowed in the reliability index (equal to 4.6). Therefore, it can be considered that the suspender is already in a dangerous state when the fatigue loading cycle is 345,600 times.

5. Residual Life Prediction of Short Suspender Based on Random Traffic Theory

A simplified simulation method should be established based on random traffic flow theory; as a result, the efficiency of the fatigue performance evaluation of the short suspenders will be improved and the residual life of the short suspenders can be reasonably predicted.

5.1. Fatigue Characteristics Simultion Based on Random Traffic Theory

The system of Weighting in Moving (WIM) was arranged on a section of the Ma’anshan Bridge. The traffic data of two directions from 2014 to 2020 were obtained. The data on the overall traffic flow and individual vehicles were statistically analyzed to facilitate the simulation of the random traffic flow.

5.1.1. Simplified Simulation Method

The Cellular Automata (CA) simulation model of the random traffic flow was established based on MATLAB 2017 [29,30,31]. The actual bridge lane can be discretized into a continuous cell space, and the vehicle position can be updated in real-time in this analysis model by using the determined characteristic parameters of the overall traffic flow and individual vehicles. By updating the position of random vehicles, the axles of each vehicle can be moved on the bridge, while the load applied on the bridge structure is also constantly changing. In general, the structural response of a bridge under the moving load can be expressed by the influence line. In the case of loading the determined load on the bridge, the load effect can be obtained by the relevant structural mechanics calculation principle.
Therefore, the CA simulation model of the random traffic flow can mainly be divided into two parts, as presented in Figure 9. Firstly, the parameters of the overall traffic flow and individual vehicles can realize the simulation of the random traffic flow according to the actual statistical results. Secondly, the influence line can realize the solution of the structural effect. In addition, the cell size will control the calculation accuracy of the whole model.

5.1.2. Characteristics of Overall Traffic Flow

The annual average daily traffic volumes (AADT) in two directions of the target bridge were tested by the WIM system. The tested data were compared, and the proportion of light and heavy vehicles in each direction were also calculated, as depicted in Figure 10.
It can be learned from the comparison that the proportion of light and heavy vehicles in the Hefei direction is slightly higher than that in the Ma’anshan direction. Therefore, the number of heavy vehicles in the Ma’anshan direction is larger, which will lead to more traffic loads on the side of the bridge, and the load response of the suspender may also be more obvious. Thus, this side of the bridge was selected for the simulation.
According to the collected WIM data from 2014 as an example, the variation in the traffic flow in one direction is presented in Figure 11a. Due to the adoption of lane-by-lane driving regulations, the probability of selecting different lanes for different axle-type vehicles varies greatly, and the probability of different lanes being selected by various heavy vehicles is significantly different, as shown in Figure 11b. Among them, it should be noted that, due to the weak response of the bridge structure caused by light vehicles, only the relevant data of heavy vehicles are counted.
It can be concluded from Figure 11 that the average daily traffic flow changes over time, and there is a peak traffic flow between 10 a.m. and 6 p.m. Heavy vehicles are mainly concentrated on the outer two lanes. The average daily traffic volume of the inner lane is 178 veh/d, and that of the middle lane and the outer lane is 2953 veh/d and 2789 veh/d, respectively. In addition, among these heavy vehicles, two-axle vehicles are mostly concentrated in the middle lane, but other-axle vehicles are mostly concentrated in the outer lane.

5.1.3. Characteristics of Individual Vehicles

The distribution law of the speed of heavy vehicles with different axle types is analyzed as shown in Figure 12a.
The speed distribution of heavy vehicles can be fitted by a normal function. The average speed of the two-axle vehicles is 75 km/h, the average speed of the three-axle vehicles is 65 km/h, and the speed of the vehicles that exceed four axles is about 50 km/h. The average speed does not exceed the speed limit of the bridge. Through the data analysis, it can be seen that the distribution of the axle load has a significant relationship with the number of axles. In the case of uniform loading, the ratio of the axle load to the vehicle weight is basically the same for a heavy vehicle with a given axle number, which can be seen in Figure 12b.
However, for the heavy vehicles with different axle numbers, the vehicle weight distribution can also be fitted by a multi-peak Gaussian distribution, as shown in Figure 12c. It can be learned from the fitting results that the peak weights of two-axis to six-axis vehicles are 16 t, 28 t, 30 t, 37 t, and 45 t, respectively. This slightly exceeds the limit of the vehicle weight specified in the Chinese standard GB 1589-2016 [32]. This means that there is a phenomenon of vehicle overload on the Ma’anshan Bridge, which may cause safety hazards to the cable system of the suspension bridge. Similarly, for the distribution law of the axle distance, it presents the characteristics of a multi-peak distribution, which can be fitted by a multi-peak Gaussian distribution.

5.2. Residual Life Prediction of Rotating Restricted Short Suspender

5.2.1. Simulation Result of Short Suspender Stress

The short suspenders on the side of the Hefei direction were selected, which may be adversely affected by the vehicle load. The structural response of the suspender subjected to random traffic flow from 2014 to 2020 was simulated. The maximum stress response of each day from the time history simulation results of the suspender stress caused by vehicle loads from different lanes is shown in Figure 13.
It can be learned from Figure 13 that the change in the stress history curve is irregular. Thus, it is necessary to perform a statistical analysis to obtain the stress amplitude spectrum so that the stress response of the suspenders can be evaluated.

5.2.2. Conversion Calculation of Fatigue Damage

The cycle counting method is usually adopted to transform the irregular stress time history into a series of amplitudes corresponding to the fixed stress level, among which the rain flow counting method is commonly used [33,34]. The rain flow counting method counts the number by considering the hysteresis loop between the stress and strain, which can better reflect the whole process of random traffic loading. Therefore, this method was adopted to statistically analyze the stress history of the suspenders.
Based on the theory of the rain flow counting method, a self-compiled program was applied to calculate the relationship between the amplitude and the frequency of the fatigue stress of the suspenders in each year, as shown in Figure 14. It can be seen from the calculation results that the suspender stress with the highest frequency is about 40 MPa.
According to the classical S-N curve expressed by Equation (6) [35,36], the simulation results of the suspender stress can be converted to the stress amplitude of 150 MPa adopted in the fatigue performance test of the short suspender, as follows:
l o g N = l o g C − m l o g Δ S
In which ΔS denotes the fatigue stress amplitude and N is the number of load cycles required for the component to fail under the action of the fatigue stress amplitude ΔS. Additionally, both m and C are the constants related to the materials.
The stress history of the short suspender is presented as a random load spectrum, which is a typical variable amplitude load effect. According to the Palmgren–Miner linear cumulative damage criterion, the cumulative number of cycles of the short suspender in the fatigue cycle can be calculated by Equation (7), as follows:
N = ∑ i = 1 k N i = ∑ i = 1 k n i × N T n i T
In which N i and n i are the inherent life and fatigue cycle number under the stress level Δ σ i , respectively; N T and n i T are the inherent life and fatigue cycle number under the stress amplitude of 150 MPa.
According to the obtained relationship between the amplitude and the frequency of fatigue stress, the annual inherent life of the short suspender under different stress levels can be calculated as depicted in Figure 15.

5.2.3. Residual Life Prediction Based on Simulation Results

The number of fatigue cycles after conversion can be plotted as a curve, as shown in Figure 16. It can be seen from the converted data that the number of fatigue cycles after conversion showed a significant growth trend between 2014 and 2017, but the growth trend was relatively flat between 2018 and 2020. Its development law is basically similar to the annual change in the total traffic volume of the Ma’anshan Bridge.
On this basis, the Boltzmann fitting formula [37] was applied to fit the data shown in Figure 16, and the fitted formula is shown in Equation (8), as follows:
  y = 5.67   − 3.43 1 + e x − 2014.45 1.19
In which x denotes the year to be converted, which should be greater than in 2014, and y is the number of fatigue cycles corresponding to the required conversion year.
According to the previous fatigue performance test results, the fatigue cycle number of 345,600, which can be considered as a dangerous state, was adopted to calculate the residual life of the rotating restricted suspender, and the fatigue residual life is considered as 6.08 years.
Therefore, the updating criteria for the rotating restricted short suspender can be proposed according to the test results and the simulation results, as follows: when the measured fracture gap size between the suspender anchorage and the connecting sleeve is less than 8 mm, the gap size should be continuously monitored, and this type of suspender should be updated within 6 years.

6. Conclusions

In this paper, the damage to the rotating restricted short suspender of a long-span suspension bridge was evaluated by an axial tensile performance test. On this basis, the residual fatigue life of this kind of suspender was analyzed by a fatigue performance test and random traffic fatigue characteristics simulation. The following conclusions can be drawn:
  • When the rotation of the end of the short suspender is limited, a fractured gap size between the suspender anchorage and the connecting sleeve will occur, and the parallel wires will subsequently be corroded.
  • The maximum loading force of each short suspender can reach more than 3000 kN, which can be considered to meet the purpose of the axial tension test. The anchorages are still normal and most of the parallel wires are still not plastically fractured.
  • With the increase in the measured fracture gap size, the effective area of the suspender will decrease to a certain extent. When the measured fracture gap size is larger than 8 mm, the short suspender should be updated.
  • When the fatigue load cycle reaches 345,000 times, although only 14 parallel wires in the outer layer can be observed to break, the calculated reliability index indicates that it is already in a dangerous state.
  • The fatigue characteristics can be simulated by the proposed simulation method based on random traffic theory. The stress amplitude of the suspender increases with the increase in the annual average daily traffic volumes, but the suspender fatigue stress with the highest frequency is about 40 MPa.
  • For the short suspender with a measured fracture gap size of less than 8 mm, it is expected that there will be a fatigue performance residual life of 6.08 years, after which such the suspenders need to be updated.
Due to the high cost of the fatigue performance test, only a limited number of short suspenders were tested in this study; the number of specimens can be expanded in further studies. In addition, the residual life prediction result was simulated by the WIM data over the years, which can be verified by an on-site inspection after several years.

Author Contributions

Conceptualization, L.Z. and R.N.; methodology, Z.Y.; software, X.T.; validation, L.Z. and Z.Y.; formal analysis, Z.Y.; investigation, R.N.; resources, Y.Z.; data curation, R.N.; writing—original draft preparation, L.Z.; writing—review and editing, R.N.; visualization, X.T.; supervision, R.N.; project administration, Z.Y.; funding acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

All of the data are available within the manuscript.

Acknowledgments

Thanks are extended to the anonymous reviewers whose suggestions improved this manuscript. Additionally, kind thanks for the instructive suggestions by Chaoyu Zhu during the manuscript editing.

Conflicts of Interest

Author Lei Zhao was employed by the company CCCC. Construction Group Beijing Testing Technology Co., Ltd. Author Yang Zhang was employed by the company Shandong Gold Mining (Yinan) Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Ma’anshan Yangtze River Bridge and its suspender: (a) General layout of the bridge; (b) Cross-section of the parallel wires; (c) General layout of the suspender of the parallel wires.
Figure 1. Ma’anshan Yangtze River Bridge and its suspender: (a) General layout of the bridge; (b) Cross-section of the parallel wires; (c) General layout of the suspender of the parallel wires.
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Figure 2. Technical condition inspection of the short suspender: (a) NY-29-1; (b) NZ-30-1; (c) NZ-39-1; (d) NZ-94-2; (e) NZ-97-1; (f) NZ-97-2.
Figure 2. Technical condition inspection of the short suspender: (a) NY-29-1; (b) NZ-30-1; (c) NZ-39-1; (d) NZ-94-2; (e) NZ-97-1; (f) NZ-97-2.
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Figure 3. Test setup for the degradation assessment: (a) Actual scene; (b) Position indication of the displacement gauges.
Figure 3. Test setup for the degradation assessment: (a) Actual scene; (b) Position indication of the displacement gauges.
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Figure 4. Deformation increments of each specimen: (a) NY-029-1; (b) NZ-030-1; (c) NZ-039-1; (d) NZ-094-2.
Figure 4. Deformation increments of each specimen: (a) NY-029-1; (b) NZ-030-1; (c) NZ-039-1; (d) NZ-094-2.
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Figure 5. Comparison of the elastic modulus test results and fitting results.
Figure 5. Comparison of the elastic modulus test results and fitting results.
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Figure 6. Comparison between the effective area ratios and the measured fracture gap size.
Figure 6. Comparison between the effective area ratios and the measured fracture gap size.
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Figure 7. Setup of fatigue performance test: (a) Loading device; (b) Arrangement of the fatigue specimen; (c) Description of finite rotation; (d) Relationship between the specimen and horizontal line.
Figure 7. Setup of fatigue performance test: (a) Loading device; (b) Arrangement of the fatigue specimen; (c) Description of finite rotation; (d) Relationship between the specimen and horizontal line.
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Figure 8. Damage condition of the specimen at different loading stages: (a) After the 1st loading stage; (b) After the 2nd loading stage; (c) After anatomy; (d) Cross-section of the broken parallel wires.
Figure 8. Damage condition of the specimen at different loading stages: (a) After the 1st loading stage; (b) After the 2nd loading stage; (c) After anatomy; (d) Cross-section of the broken parallel wires.
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Figure 9. Calculation principles of the simplified simulation method.
Figure 9. Calculation principles of the simplified simulation method.
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Figure 10. Annual average traffic volume change and vehicle proportion.
Figure 10. Annual average traffic volume change and vehicle proportion.
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Figure 11. Traffic flow characteristics of different lanes: (a) Daily variation in the average traffic volume of heavy vehicles; (b) Probability of different lanes being selected by various heavy vehicles.
Figure 11. Traffic flow characteristics of different lanes: (a) Daily variation in the average traffic volume of heavy vehicles; (b) Probability of different lanes being selected by various heavy vehicles.
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Figure 12. Characteristics of individual vehicles: (a) Normal distribution fitting of the vehicle speed; (b) Ratio of the axle load to the vehicle weight; (c) Multi-peak fitting of the vehicle weight.
Figure 12. Characteristics of individual vehicles: (a) Normal distribution fitting of the vehicle speed; (b) Ratio of the axle load to the vehicle weight; (c) Multi-peak fitting of the vehicle weight.
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Figure 13. Simulation results based on random traffic theory: (a) Lane distribution; (b) Stress response of the suspenders caused by the vehicles on each lane.
Figure 13. Simulation results based on random traffic theory: (a) Lane distribution; (b) Stress response of the suspenders caused by the vehicles on each lane.
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Figure 14. Relationship between the amplitude and the frequency of fatigue stress in each year: (a) 2014; (b) 2015; (c) 2016; (s) 2017; (e) 2018; (f) 2019; (g) 2020.
Figure 14. Relationship between the amplitude and the frequency of fatigue stress in each year: (a) 2014; (b) 2015; (c) 2016; (s) 2017; (e) 2018; (f) 2019; (g) 2020.
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Figure 15. Inherent life of the short suspender under different stress levels.
Figure 15. Inherent life of the short suspender under different stress levels.
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Figure 16. The converted fatigue cycle curve and its fitting curve.
Figure 16. The converted fatigue cycle curve and its fitting curve.
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Table 1. Maximum loading force and damage situation of the tested suspender.
Table 1. Maximum loading force and damage situation of the tested suspender.
Specimen NumberMaximum Tension Force/kNDamage Situation
NY-029-13001No wire breaks, and the anchorage is still normal.
NY-030-13139No wire breaks, and the anchorage is still normal.
NZ-039-13049Four wires are broken, but the anchorage is still normal.
NZ-094-23220No wire breaks, and the anchorage is still normal.
Table 2. Main mechanical parameters adopted in the fatigue performance test.
Table 2. Main mechanical parameters adopted in the fatigue performance test.
Parameter1st Loading Stage2nd Loading Stage
Fmax/kN1251648.2
Fmin/kN930872.2
Favg/kN1090.5760.2
ΔF/kN160.5112.0
Frequency/Hz1.01.0
Δσ/MPa150.0150.0
When the fatigue cycle is less than 345,000 times, it is the 1st loading stage. When the fatigue cycle is between 345.6 thousand times and 583,800 times, it is the 2nd loading stage.
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MDPI and ACS Style

Zhao, L.; Yang, Z.; Tong, X.; Zhang, Y.; Nie, R. Experimental Study on Fatigue Characteristics and Life Prediction of Rotating Restricted Short Suspender in Suspension Bridge. Buildings 2025, 15, 254. https://doi.org/10.3390/buildings15020254

AMA Style

Zhao L, Yang Z, Tong X, Zhang Y, Nie R. Experimental Study on Fatigue Characteristics and Life Prediction of Rotating Restricted Short Suspender in Suspension Bridge. Buildings. 2025; 15(2):254. https://doi.org/10.3390/buildings15020254

Chicago/Turabian Style

Zhao, Lei, Zhili Yang, Xianneng Tong, Yang Zhang, and Ruifeng Nie. 2025. "Experimental Study on Fatigue Characteristics and Life Prediction of Rotating Restricted Short Suspender in Suspension Bridge" Buildings 15, no. 2: 254. https://doi.org/10.3390/buildings15020254

APA Style

Zhao, L., Yang, Z., Tong, X., Zhang, Y., & Nie, R. (2025). Experimental Study on Fatigue Characteristics and Life Prediction of Rotating Restricted Short Suspender in Suspension Bridge. Buildings, 15(2), 254. https://doi.org/10.3390/buildings15020254

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