F-C-3 Three flexible central buckles *5.2. Evaluation Process*

F-C-3 Three flexible central buckles *5.2. Evaluation Process*  The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, F-C-3 Three flexible central buckles *5.2. Evaluation Process*  The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calcu-*5.2. Evaluation Process*  The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calculated on the basis of the hourly occupancy rate of different traffic flows in one year, and the change law of crack depth and crack development rate with time is obtained. The *5.2. Evaluation Process*  The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calculated on the basis of the hourly occupancy rate of different traffic flows in one year, and the change law of crack depth and crack development rate with time is obtained. The process includes the following steps: (1) analyzing the characteristics of traffic flow parameters and generating random traffic flow samples on the basis of the WIM data; (2) The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calculated on the basis of the hourly occupancy rate of different traffic flows in one year, and the change law of crack depth and crack development rate with time is obtained. The process includes the following steps: (1) analyzing the characteristics of traffic flow parameters and generating random traffic flow samples on the basis of the WIM data; (2) taking traffic flow

tion by integrating vehicle flow effects of different levels until failure.

tion by integrating vehicle flow effects of different levels until failure.

tion by integrating vehicle flow effects of different levels until failure.

tion by integrating vehicle flow effects of different levels until failure.

and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calculated on the basis of the hourly occupancy rate of different traffic flows in one year, and

lated on the basis of the hourly occupancy rate of different traffic flows in one year, and the change law of crack depth and crack development rate with time is obtained. The

taking traffic flow into the vehicle bridge coupling analysis system and obtaining the time history results of suspender stress; (3) simulating the uniform corrosion process of steel

process includes the following steps: (1) analyzing the characteristics of traffic flow parameters and generating random traffic flow samples on the basis of the WIM data; (2)

process includes the following steps: (1) analyzing the characteristics of traffic flow parameters and generating random traffic flow samples on the basis of the WIM data; (2) taking traffic flow into the vehicle bridge coupling analysis system and obtaining the time history results of suspender stress; (3) simulating the uniform corrosion process of steel wire and generating random samples of pitting corrosion; (4) calculating crack propaga-

wire and generating random samples of pitting corrosion; (4) calculating crack propaga-

taking traffic flow into the vehicle bridge coupling analysis system and obtaining the time history results of suspender stress; (3) simulating the uniform corrosion process of steel wire and generating random samples of pitting corrosion; (4) calculating crack propagainto the vehicle bridge coupling analysis system and obtaining the time history results of suspender stress; (3) simulating the uniform corrosion process of steel wire and generating random samples of pitting corrosion; (4) calculating crack propagation by integrating vehicle flow effects of different levels until failure. taking traffic flow into the vehicle bridge coupling analysis system and obtaining the time history results of suspender stress; (3) simulating the uniform corrosion process of steel wire and generating random samples of pitting corrosion; (4) calculating crack propagation by integrating vehicle flow effects of different levels until failure.

The fatigue life of suspender wires can be predicted by the corrosion fatigue theory, and the detailed prediction process is shown in Figure 11. Considering the traffic density variation, the contribution of the corresponding stress cycle times to crack depth is calculated on the basis of the hourly occupancy rate of different traffic flows in one year, and the change law of crack depth and crack development rate with time is obtained. The process includes the following steps: (1) analyzing the characteristics of traffic flow parameters and generating random traffic flow samples on the basis of the WIM data; (2)

*Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 18

**Table 3.** Analysis conditions of the FE model.

N-C No central buckle

F-C-1 Single flexible central buckle

F-C-2 Double flexible central buckles

F-C-3 Three flexible central buckles

*5.2. Evaluation Process*

**Condition Description Schematic**

**Figure 11.** Simulation process of wire life. **Figure 11.** Simulation process of wire life.

#### *5.3. Result Discussion 5.3. Result Discussion*

The dynamic test under truck load in Figure 6 is used to analyze the dynamic response of the bridge structure with or without the central buckle. First, the suspender The dynamic test under truck load in Figure 6 is used to analyze the dynamic response of the bridge structure with or without the central buckle. First, the suspender response of the running test is shown in Figure 12 to analyze the difference between suspenders. The suspender bears axial stress and bending stress due to the relative movement between the main cable and the stiffening beam. The bending stress of the suspender cannot be directly obtained by the LINK10 element. Thus, the Wyatt theoretical formula is introduced to calculate bending stress according to computed axial stress and the angle caused by relative movement between the main cable and the stiffening girder [36]. The Wyatt theoretical formula can only be applied to an object that is a round wire, which is not suitable for a set of strands such as the prototype bridge. Kondoh proposed that the bending stress in this kind of suspender at the joint was assumed to be 60% of the theoretical Wyatt formula based on the experimental results as Equation (11) [37]

$$
\sigma\_b = 1.2 \tan \theta \cdot \sqrt{\sigma\_d E} \tag{11}
$$

where *σ<sup>a</sup>* is axial stress, *E* is the elasticity modulus of steel wire, and *θ* is the angle caused by the relative movement between the main cable and the stiffening girder.

40

50

60

**Figure 12.** Comparison of suspender stress. (**a**) Bending stress; (**b**) axial stress. **Figure 12.** Comparison of suspender stress. (**a**) Bending stress; (**b**) axial stress.

The traffic load is divided into different levels according to traffic density and then used for loading to calculate the structural dynamic response under traffic conditions. Figure 13 shows the suspender stress under the traffic flow at level 5. The time history of axial stress is consistent for different conditions, and the bending stress presents a remarkable difference in that the peak values are greatly reduced. N-C F-C-1 Suspender:No.26 150 Only the bending stress and axial stress of partial suspenders are shown due to layoutconstraints. The variation of the axial stress of short suspenders is small, whereas the lengthof short suspenders near the midspan is too small to release stress; the bending stress is greater than long suspenders. The influence of bending stress cannot be neglected in the analysis of suspender degradation. The settlement of flexible buckles considerably reduces the bending stress of suspenders but has minimal effect on the axial stress. The bending stress of short suspenders near suspender no. 26 (midspan) slightly decreases, whereas the long suspenders are almost unaffected. Thus, short suspenders nos. 21–26 are selected as analysis objects.

 F-C-2 F-C-3 0 50 100 Bending stress (MPa) N-C F-C-1 F-C-2 F-C-3 Suspender:No.26 The traffic load is divided into different levels according to traffic density and then used for loading to calculate the structural dynamic response under traffic conditions. Figure 13 shows the suspender stress under the traffic flow at level 5. The time history of axial stress is consistent for different conditions, and the bending stress presents a remarkable difference in that the peak values are greatly reduced. *Materials* **2022**, *15*, x FOR PEER REVIEW 13 of 18

der different conditions. F-C-1 has an improvement effect on suspender no. 26, whereas **Figure 13.** Time history of suspender stress (level 5). (**a**) Axial stress; (**b**) bending stress. **Figure 13.** Time history of suspender stress (level 5). (**a**) Axial stress; (**b**) bending stress.

F-C-2 has an improvement effect on suspender no. 25; these results are related to the position of buckles. The settlement of buckles shares the axial stress of the suspender between inclined cables. In terms of bending stress, the flexible central buckle can remarkably reduce the peak value of short suspenders, but the weakening effect is not significantly improved with the increase in the number of buckles. Figure 14 shows the peak values of bending stress and axial stress. The suspender stress response under different traffic flows is different, and the axial stress is slightly influenced by different traffic flows, whereas the bending stress is greatly reduced by flexible central buckles. The settlement of flexible central buckles has a certain influence on the axial stress and bending stress of short suspenders. The axial stress peak values vary under different conditions. F-C-1 has an improvement effect on suspender no. 26, whereas Figure 14 shows the peak values of bending stress and axial stress. The suspender stress response under different traffic flows is different, and the axial stress is slightly influenced by different traffic flows, whereas the bending stress is greatly reduced by flexible central buckles. The settlement of flexible central buckles has a certain influence on the axial stress and bending stress of short suspenders. The axial stress peak values vary under different conditions. F-C-1 has an improvement effect on suspender no. 26, whereas F-C-2

F-C-2 has an improvement effect on suspender no. 25; these results are related to the position of buckles. The settlement of buckles shares the axial stress of the suspender be-

(**a**) (**b**)

N-C F-C-1 F-C-2 F-C-3

Analysis condition

 No.21 No.22 No.23 No.24 No.25

Level-5

Suspender:

N-C F-C-1 F-C-2 F-C-3

 5 4 3 2 1 Suspender: No.26

Level:

Analysis condition

350

390

430

Maximum axial stress (Mpa)

470

510

550

**Figure 14.** Comparison of peak values of stress. (**a**) Axial stress; (**b**) bending stress.

0

60

120

Maximum bending stress (Mpa)

180

240

300

The generation of pitting corrosion is a random process, and the maximum pitting depth directly affects the generation of cracks and fatigue life. In order to reflect the difference in steel wire life, the corrosion fatigue degradation of steel wire under different working conditions was simulated. A total of 150 samples for each analysis condition were sampled based on randomly generated maximum pitting coefficients, and then the transition from pitting corrosion to cracking and the crack development were calculated on the basis of the proposed predicting process. Figure 15 shows the crack development of the steel wire samples of suspender nos. 21–26. Under the N-C condition, the average

N-C F-C-1 F-C-2 F-C-3

 5 4 3 2 1 Suspender: No.26

Level: No.21

Suspender:

Level-5

N-C F-C-1 F-C-2 F-C-3 Analysis condition

 No.22 No.23 No.24 No.25

Analysis condition

improved with the increase in the number of buckles.

10

20

30

Axial stress (MPa)

40

50

60

has an improvement effect on suspender no. 25; these results are related to the position of buckles. The settlement of buckles shares the axial stress of the suspender between inclined cables. In terms of bending stress, the flexible central buckle can remarkably reduce the peak value of short suspenders, but the weakening effect is not significantly improved with the increase in the number of buckles. F-C-2 has an improvement effect on suspender no. 25; these results are related to the position of buckles. The settlement of buckles shares the axial stress of the suspender between inclined cables. In terms of bending stress, the flexible central buckle can remarkably reduce the peak value of short suspenders, but the weakening effect is not significantly improved with the increase in the number of buckles.

Figure 14 shows the peak values of bending stress and axial stress. The suspender stress response under different traffic flows is different, and the axial stress is slightly influenced by different traffic flows, whereas the bending stress is greatly reduced by flexible central buckles. The settlement of flexible central buckles has a certain influence on the axial stress and bending stress of short suspenders. The axial stress peak values vary under different conditions. F-C-1 has an improvement effect on suspender no. 26, whereas

 N-C F-C-1 F-C-2 F-C-3

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Suspender:No.26

Time (min)

*Materials* **2022**, *15*, x FOR PEER REVIEW 13 of 18

Suspender:No.26

(**a**) (**b**)


−150


−100−50


0

Bending stress (MPa)

50

100

150

0.0 2.5 5.0 7.5 10.0 12.5 15.0

 N-C F-C-1 F-C-2 F-C-3

Time (min)

**Figure 13.** Time history of suspender stress (level 5). (**a**) Axial stress; (**b**) bending stress.

**Figure 14.** Comparison of peak values of stress. (**a**) Axial stress; (**b**) bending stress. **Figure 14.** Comparison of peak values of stress. (**a**) Axial stress; (**b**) bending stress.

The generation of pitting corrosion is a random process, and the maximum pitting depth directly affects the generation of cracks and fatigue life. In order to reflect the difference in steel wire life, the corrosion fatigue degradation of steel wire under different working conditions was simulated. A total of 150 samples for each analysis condition were sampled based on randomly generated maximum pitting coefficients, and then the transition from pitting corrosion to cracking and the crack development were calculated on the basis of the proposed predicting process. Figure 15 shows the crack development of The generation of pitting corrosion is a random process, and the maximum pittingdepth directly affects the generation of cracks and fatigue life. In order to reflect the difference in steel wire life, the corrosion fatigue degradation of steel wire under different working conditions was simulated. A total of 150 samples for each analysis condition were sampled based on randomly generated maximum pitting coefficients, and then the transition from pitting corrosion to cracking and the crack development were calculated on the basis of the proposed predicting process. Figure 15 shows the crack development of the steel wire samples of suspender nos. 21–26. Under the N-C condition, the average crack development in the steel wire samples of suspender nos. 24–26 is remarkably faster than that of suspender nos. 21–23, satisfying the service requirements. Although the corrosion resistance of the aluminum alloy steel wire is better than that of the galvanized steel wire, once the steel wire coating is consumed, the crack growth rate caused by the substrate pitting pit mainly depends on the stress response of the suspender. The bending stress of the short suspender under the N-C condition is larger, but the steel wire fatigue life remains lower than the design life of the bridge. Figure 16 shows the crack speed of the wires of suspender no. 26; the settlement of flexible central buckles substantially improves their fatigue life, and the improvement effect is similar to that of the rigid central buckle [17]. However, the increase in the number of buckles does not considerably weaken the crack growth rate. The fatigue lives were fitted, and the results are shown in Figure 17; all of them obey a normal distribution. The mean values of the fatigue lives of suspender nos. 24, 25, and 26 have small differences because the length of these suspenders is close. The 5% fractiles are taken as the characteristic service life, in which the service life of the steel wire has a 95% assurance rate. The service life of the short suspender is 20.04, 21.02, and 24.18, respectively. All of them are lower than expected, but the service life can be significantly improved by increasing the length of the steel wire; that is, reducing the bending stress.

**Suspender Number (Length)** 

> 24 (2.75 m)

> 25 (2.45 m)

> 26 (2.41 m)

**Analysis Condition** 

**Maximum Bending Stress (MPa)** 

**6. Conclusions** 

the steel wire samples of suspender nos. 21–26. Under the N-C condition, the average crack development in the steel wire samples of suspender nos. 24–26 is remarkably faster than that of suspender nos. 21–23, satisfying the service requirements. Although the corrosion resistance of the aluminum alloy steel wire is better than that of the galvanized steel wire, once the steel wire coating is consumed, the crack growth rate caused by the substrate pitting pit mainly depends on the stress response of the suspender. The bending stress of the short suspender under the N-C condition is larger, but the steel wire fatigue life remains lower than the design life of the bridge. Figure 16 shows the crack speed of the wires of suspender no. 26; the settlement of flexible central buckles substantially improves their fatigue life, and the improvement effect is similar to that of the rigid central buckle [17]. However, the increase in the number of buckles does not considerably weaken the crack growth rate. The fatigue lives were fitted, and the results are shown in Figure 17; all of them obey a normal distribution. The mean values of the fatigue lives of suspender nos. 24, 25, and 26 have small differences because the length of these suspenders is close. The 5% fractiles are taken as the characteristic service life, in which the service life of the steel wire has a 95% assurance rate. The service life of the short suspender is 20.04, 21.02, and 24.18, respectively. All of them are lower than expected, but the service life can be significantly improved by increasing the length of the steel wire; that is, reducing the

the steel wire samples of suspender nos. 21–26. Under the N-C condition, the average crack development in the steel wire samples of suspender nos. 24–26 is remarkably faster than that of suspender nos. 21–23, satisfying the service requirements. Although the corrosion resistance of the aluminum alloy steel wire is better than that of the galvanized steel wire, once the steel wire coating is consumed, the crack growth rate caused by the substrate pitting pit mainly depends on the stress response of the suspender. The bending stress of the short suspender under the N-C condition is larger, but the steel wire fatigue life remains lower than the design life of the bridge. Figure 16 shows the crack speed of the wires of suspender no. 26; the settlement of flexible central buckles substantially improves their fatigue life, and the improvement effect is similar to that of the rigid central buckle [17]. However, the increase in the number of buckles does not considerably weaken the crack growth rate. The fatigue lives were fitted, and the results are shown in Figure 17; all of them obey a normal distribution. The mean values of the fatigue lives of suspender nos. 24, 25, and 26 have small differences because the length of these suspenders is close. The 5% fractiles are taken as the characteristic service life, in which the service life of the steel wire has a 95% assurance rate. The service life of the short suspender is 20.04, 21.02, and 24.18, respectively. All of them are lower than expected, but the service life can be significantly improved by increasing the length of the steel wire; that is, reducing the

*Materials* **2022**, *15*, x FOR PEER REVIEW 14 of 18

**Figure 15.** Development of average crack depth. **Figure 15.** Development of average crack depth. **Figure 15.** Development of average crack depth.

bending stress.

bending stress.

**Figure 16.** Crack speed of wire samples (suspender No. 26). **Figure 16.** Crack speed of wire samples (suspender No. 26). **Figure 16.** Crack speed of wire samples (suspender No. 26).

**Figure 17.** Distribution of steel wire life under the N-C condition. (**a**) Suspender no. 24; (**b**) suspender no. 25; (**c**) suspender no. 26. **Figure 17.** Distribution of steel wire life under the N-C condition. (**a**) Suspender no. 24; (**b**) suspender no. 25; (**c**) suspender no. 26.

Table 4 shows the comparison of the equivalent stress amplitude and fatigue life of the steel wires of suspender nos. 24–26. With consideration of the responses under traffic flow of five levels, the fatigue life of the steel wire after setting buckles meets the service requirements, but the increase in the number of buckles has hardly improved the life of the suspender steel wire. The 95% confidence interval results of fatigue life are shown in Table 4. When the number of samples is sufficient, the confidence interval length is small. The length of the confidence intervals of N-C steel wire is less than 1.5 years, and those of F-C are all less than 8 years. The average life of steel wire tends to be stable, thus the sampling results are proven to be reliable. The setting of buckles can considerably improve the extreme value of bending stress and the equivalent stress replication. When two buckles are settled, the extreme value and the equivalent stress amplitude of the steel wire tend to be stable, and increasing the number of buckles is unnecessary. The equivalent Table 4 shows the comparison of the equivalent stress amplitude and fatigue life of the steel wires of suspender nos. 24–26. With consideration of the responses under traffic flow of five levels, the fatigue life of the steel wire after setting buckles meets the service requirements, but the increase in the number of buckles has hardly improved the life of the suspender steel wire. The 95% confidence interval results of fatigue life are shown in Table 4. When the number of samples is sufficient, the confidence interval length is small. The length of the confidence intervals of N-C steel wire is less than 1.5 years, and those of F-C are all less than 8 years. The average life of steel wire tends to be stable, thus the sampling results are proven to be reliable. The setting of buckles can considerably improve the extreme value of bending stress and the equivalent stress replication. When two buckles are settled, the extreme value and the equivalent stress amplitude of the steel wire tend to be stable, and increasing the number of buckles is unnecessary. The equivalent stress

stress amplitude of axial stress is unaffected, and the suspender stress of long-span sus-

**Fatigue Life (Year)** 

**Confidence Intervals (95% CI)**

*μ σ*

**Equivalent Axial Stress (MPa)** 

**Table 4.** Fatigue life of suspender wire under different analysis conditions.

N-C 208.31 37.30 10.23 35.2 5.51 (34.3,36.1) F-C-1 70.52 14.19 9.60 179.3 19.31 (176.2,182.4) F-C-2 54.14 11.38 9.51 178.5 19.54 (175.4,181.6) F-C-3 54.67 12.46 9.45 181.3 19.59 (178.2,184.4)

N-C 238.91 44.32 9.63 27.6 3.29 (27.1,28.1) F-C-1 77.16 15.68 10.12 177.8 19.32 (174.7,180.9) F-C-2 50.81 11.62 10.33 177.9 20.39 (174.6,181.2) F-C-3 50.79 11.81 9.38 181.1 20.10 (177.9,184.3)

N-C 274.10 46.66 10.03 27.2 3.58 (26.6,27.8) F-C-1 72.17 14.38 10.17 174.2 19.49 (171.1,177.3) F-C-2 50.68 11.93 8.98 178.1 21.35 (174.7,181.5) F-C-3 46.85 12.07 9.30 179.4 20.60 (176.1,182.7)

> The influence of a flexible central buckle on suspension bridge vibration was remarkable, but the control effect on short suspenders is still unknown. This study established the corrosion fatigue degradation model of high-strength steel wire based on traffic composition and explored the influence of flexible central buckles on the corrosion fatigue life of suspenders under traffic flow. To improve the consideration of traffic flow, the WIM data were processed according to traffic density and used to analyze the suspender

pension bridges is determined by the dead load.

**Equivalent Bending Stress (MPa)**  amplitude of axial stress is unaffected, and the suspender stress of long-span suspension bridges is determined by the dead load.


**Table 4.** Fatigue life of suspender wire under different analysis conditions.

## **6. Conclusions**

The influence of a flexible central buckle on suspension bridge vibration was remarkable, but the control effect on short suspenders is still unknown. This study established the corrosion fatigue degradation model of high-strength steel wire based on traffic composition and explored the influence of flexible central buckles on the corrosion fatigue life of suspenders under traffic flow. To improve the consideration of traffic flow, the WIM data were processed according to traffic density and used to analyze the suspender response under traffic flow of different densities. The fatigue life of short suspenders without buckles and with different numbers of buckles was analyzed based on monitoring traffic data. The following conclusions were drawn:


The dynamic motion of the bridges is complex for diverse loads. Moreover, the fatigue behavior of short suspenders and the vibration control effect are influenced by other loads, such as wind, earthquakes, and other special conditions. The optimal design of flexible central buckles should be studied further.

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

**Funding:** This research was funded by the Natural Science Basic Research Program of Shaanxi (Program No. 2022JQ-336); the open fund of Shaanxi Provincial Key Laboratory (Chang'an University) of Highway Bridges and Tunnels (Program No. 300102212509); and the Foundation of Xi'an University of Technology (Grant no. 256082109).

**Data Availability Statement:** Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
