**1. Introduction**

With the rapid development of highway transportation, long-span suspension bridges are constructed across mountains, valleys, and rivers for their good mechanical characteristics and excellent spanning performance. The construction of early large-span suspension bridges was limited by experience and technology, and structural vibration control measures were relatively lacking, leading to obvious vibration responses under external load. The longitudinal vibration displacement of structures caused by external load may lead to the fatigue of expansion joints and other ancillary components. Traffic load has been proven to be one of the main reasons causing the longitudinal vibration displacement of structures. Such vibration may cause the fatigue of expansion joints and other ancillary components [1,2]. Suspenders are key load-bearing components of suspension bridges, of which the degradation is the result of the comprehensive action of corrosion and fatigue, and the corrosion accelerates the generation of fatigue cracks. The corrosion degradation of components seriously affects the reliability of bridge operation [3]. The propagation

**Citation:** Zhao, Y.; Guo, X.; Su, B.; Sun, Y.; Li, X. Evaluation of Flexible Central Buckles on Short Suspenders' Corrosion Fatigue Degradation on a Suspension Bridge under Traffic Load. *Materials* **2023**, *16*, 290. https://doi.org/10.3390/ma16010290

Academic Editor: Costica Bejinariu

Received: 22 November 2022 Revised: 22 December 2022 Accepted: 23 December 2022 Published: 28 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of fatigue cracks is easily affected by vibrations under traffic load; thus, it has been an important research goal to evaluate the stress and service life of short suspenders. Flexible central buckles were set up in the midspan for long-span suspension bridges recently to enhance wind resistance performance with low cost and construction convenience, but the studies on them were limited and mainly focused on wind resistance and the vibration characteristics of the structure itself. Actually, flexible central buckles also play a contributing role to the suspenders' response under traffic flow that may reduce the fatigue degradation of suspenders, but the influence mechanism on the structural vibration under traffic load remains unclear. Thus, the control effect of flexible central cables should be investigated to optimize the designation of the flexible central buckle.

Corrosion fatigue is the phenomenon of crack formation and propagation under the interaction of alternating load and a corrosive medium that leads to a reduction in fatigue resistance [4]. Scholars have studied the corrosion fatigue degradation process of highstrength steel wires in bridge engineering. The surface of the suspender steel wire is provided with a coating to enhance the corrosion resistance. The damage to the coating's passive film is accompanied by pitting corrosion. Roffey indicates that the pit corrosion of the steel wire develops into vertical cracks inside, resulting in the decline of the bearing capacity of the steel wire based on the inspection results of the Fourth Highway Bridge in Scotland [5]. Qiao Yan divided the corrosion fatigue process of steel wire into three stages—coating corrosion, corrosion pit development, and crack development—and gave a calculation method for the development time of each stage [6]. Valor proposed a random model for pitting distribution simulation which uses a non-uniform Poisson process to simulate the generation of pits and verified it with experiments [7]. Nakamura investigated the corrosion of steel wires in different environments for fatigue loading. The results show that the fatigue life of steel wire in a corrosive environment decreases significantly [8]. Suzumura studied the effects of reagent concentration, ambient temperature, and humidity on the corrosion rate of galvanized steel wire through experiments, and gave the loss rate of zinc coating on galvanized steel wire [9]. Although the durability of galvanized steel wire has been significantly improved, it still does not meet the engineering requirements. The corrosion resistance of the coating can be achieved by improving the properties of the coating, such as improving the adhesion and porosity and adding elements that can form a passive film; it is proven that the corrosion resistance can be improved by the oxidation of the Al element [10]. In recent years, Galfan steel wires have been gradually widely used. The evaluation method of steel wire has been well developed, but the existing models mainly use the Paris criterion to calculate the crack growth rate; the influence of the average load factor and the difference in crack growth rate caused by the change in traffic flow intensity are not considered. There is a deviation when using the parameters under the same stress ratio to calculate the crack growth life.

Furthermore, the axial stress and bending stress fluctuations caused by relative displacements between the girder and cables easily damage the short suspenders along with fatigue degradation [11,12]. To reduce the fatigue damage of short suspenders, appropriate vibration control facilities are utilized to control bridge vibration [13,14]. The central buckle is a vibration control measure for long-span suspension bridges, which includes a flexible central buckle and a rigid central buckle. Previous research focuses on the influence of rigid central buckles on the dynamic characteristics of bridges [15]. Wang analyzed the influence of rigid central buckles on the wind-induced buffeting response of long-span suspension bridges and pointed out that rigid central buckles can suppress buffeting vibration [16]. Wang investigated the working and mechanical characteristics of the rigid central buckle of the Runyang Yangtze River Bridge under vehicle load based on measured results and finite element modeling [17]. Liu investigated the effects of central clamps in the midspan (i.e., rigid central buckle) on the fatigue life of short suspenders, and the results revealed that short suspenders were more prone to fatigue than others because of large bending stress, and central clamps can effectively improve their lifespan [18]. In addition to a rigid central buckle, a flexible central buckle cable was set up in the midspan to enhance wind resistance

performance. Wang studied the influence of flexible central buckles on the displacement of stiffening girders. The results showed that the flexible central buckle remarkably reduces the longitudinal amplitude of the stiffening girder and increases its vibration frequency [19]. The influence of flexible buckles on structural vibration under random traffic flow remains unclear. The control effect of flexible buckles under random traffic flow should be studied. [18]. In addition to a rigid central buckle, a flexible central buckle cable was set up in the midspan to enhance wind resistance performance. Wang studied the influence of flexible central buckles on the displacement of stiffening girders. The results showed that the flexible central buckle remarkably reduces the longitudinal amplitude of the stiffening girder and increases its vibration frequency [19]. The influence of flexible buckles on structural vibration under random traffic flow remains unclear. The control effect of flexible buckles

and the results revealed that short suspenders were more prone to fatigue than others because of large bending stress, and central clamps can effectively improve their lifespan

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Traffic flow is an important vibration source in the suspender stress response. Suspension bridges are a flexible system and structural deformation is evident under the action of traffic flow, which varies with traffic density. Characteristics such as traffic flow parameters, vehicle type, and vehicle weight generally have random distribution [20,21]. The load effect of traffic flow can be well considered by a macro traffic flow simulation method [22,23]. Thus, in this study, the influence of flexible central buckles on the stress response and corrosion fatigue life of suspenders under traffic flow were analyzed by numerical modeling. First, the corrosion fatigue of high-strength steel wire based on the Forman criterion was established. Then, the response of suspenders with flexible central buckles was calculated with consideration of the load effect of traffic flow at different levels. Finally, the fatigue life of suspender steel wires and the influence of flexible central buckles were evaluated. This research can provide a reference for the design and maintenance of long-span suspension bridges. under random traffic flow should be studied. Traffic flow is an important vibration source in the suspender stress response. Suspension bridges are a flexible system and structural deformation is evident under the action of traffic flow, which varies with traffic density. Characteristics such as traffic flow parameters, vehicle type, and vehicle weight generally have random distribution [20,21]. The load effect of traffic flow can be well considered by a macro traffic flow simulation method [22,23]. Thus, in this study, the influence of flexible central buckles on the stress response and corrosion fatigue life of suspenders under traffic flow were analyzed by numerical modeling. First, the corrosion fatigue of high-strength steel wire based on the Forman criterion was established. Then, the response of suspenders with flexible central buckles was calculated with consideration of the load effect of traffic flow at different levels. Finally, the fatigue life of suspender steel wires and the influence of flexible central buckles were evaluated. This research can provide a reference for the design and maintenance of long-span suspension bridges.

#### **2. Prototype Bridge 2. Prototype Bridge**

#### *2.1. Bridge Information 2.1. Bridge Information*

This study takes the Zhixi Yangtze River Bridge as the research object. The bridge is a single-span steel–concrete composite girder suspension bridge. The section layout is shown in Figure 1. The span of the main cable is arranged as 250 + 838 + 215 m, and the sagittal span ratio of the midspan main cable is 1/10. The standard distance between the adjacent lifting points of stiffening girders is 16 m, and the suspender adopts a ϕ5.0 mm galvanized aluminum alloy (i.e., Galfan coating) high-strength steel wire. To improve the vibration resistance of the bridge, two flexible central buckles are set near both sides of the middle span of each main cable to form a cable–beam connection. The entire bridge has a total of eight central buckles. The stiffening girder adopts a steel–concrete composite structure in which the steel beam is combined with the concrete deck through shear nails. The half section of the stiffening girder is shown in Figure 2. The full width of the stiffening girder is 33.2 m, the center height is 2.8 m, and the central transverse spacing of the two main cables in the midspan is 26.0 m. The small longitudinal beams are arranged longitudinally at the center line of the girder and the top surface is flush with the top surface of the steel beam. The bridge deck is reinforced concrete with a full width of 25.0 m and a thickness of 0.22 m. This study takes the Zhixi Yangtze River Bridge as the research object. The bridge is a single-span steel–concrete composite girder suspension bridge. The section layout is shown in Figure 1. The span of the main cable is arranged as 250 + 838 + 215 m, and the sagittal span ratio of the midspan main cable is 1/10. The standard distance between the adjacent lifting points of stiffening girders is 16 m, and the suspender adopts a φ5.0 mm galvanized aluminum alloy (i.e., Galfan coating) high-strength steel wire. To improve the vibration resistance of the bridge, two flexible central buckles are set near both sides of the middle span of each main cable to form a cable–beam connection. The entire bridge has a total of eight central buckles. The stiffening girder adopts a steel–concrete composite structure in which the steel beam is combined with the concrete deck through shear nails. The half section of the stiffening girder is shown in Figure 2. The full width of the stiffening girder is 33.2 m, the center height is 2.8 m, and the central transverse spacing of the two main cables in the midspan is 26.0 m. The small longitudinal beams are arranged longitudinally at the center line of the girder and the top surface is flush with the top surface of the steel beam. The bridge deck is reinforced concrete with a full width of 25.0 m and a thickness of 0.22 m.

**Figure 1.** Layout of the prototype bridge. **Figure 1.** Layout of the prototype bridge.

**Figure 2.** Half-section of the stiffening girder (mm). **Figure 2.** Half-section of the stiffening girder (mm).

Cable clamp Main cable

ture.

Flexible central buckle

**Figure 3.** Flexible central buckle.

*2.2. Finite Element (FE) Model* 

measured cable force.

Suspender

As a common vibration control measure, central buckles are used to improve the vi-

To simulate the structural characteristics, a three-girder model of a prototype bridge was established using ANSYS 18.0. The FE model is shown in Figure 4. The stiffening girder of the bridge is a steel-composite girder with an open section, and the longitudinal beams on both sides are the main bearing structures of the stiffening girder. Thus, the BEAM4 element was used to simulate the main stringer, small stringer, steel beam, and main tower. The LINK10 element was used to simulate the cable components. A total of 1836 BEAM4 elements for the girder, 82 BEAM4 elements for the pylon, and 279 LINK10 elements for the main cable and suspender were found. The bridge deck pavement contributes minimally to the stiffness of the stiffening girder; thus, only its mass was considered, and the stress stiffening of the LINK element was conducted in accordance with the

dle span; examples include the Runyang Yangtze River Bridge and the Sidu River Bridge, in which the rigid central buckle is installed in the middle span. Existing research indicates that the rigid central buckle can improve the structure frequency and reduce the longitudinal displacement response of the girder[24]. In the Zhixi Yangtze River Bridge, flexible central buckles that differ from traditional rigid central buckles are set in the middle of the main span of each main cable to coordinate with short suspenders, as shown in Figure 3. The flexible central buckle is composed of an inclined cable connected to a short suspender, forming a cable–girder connection to control the vibration response of the struc-

As a common vibration control measure, central buckles are used to improve the vibration response of suspension bridges. These buckles are generally installed in the middle span; examples include the Runyang Yangtze River Bridge and the Sidu River Bridge, in which the rigid central buckle is installed in the middle span. Existing research indicates that the rigid central buckle can improve the structure frequency and reduce the longitudinal displacement response of the girder [24]. In the Zhixi Yangtze River Bridge, flexible central buckles that differ from traditional rigid central buckles are set in the middle of the main span of each main cable to coordinate with short suspenders, as shown in Figure 3. The flexible central buckle is composed of an inclined cable connected to a short suspender, forming a cable–girder connection to control the vibration response of the structure. As a common vibration control measure, central buckles are used to improve the vibration response of suspension bridges. These buckles are generally installed in the middle span; examples include the Runyang Yangtze River Bridge and the Sidu River Bridge, in which the rigid central buckle is installed in the middle span. Existing research indicates that the rigid central buckle can improve the structure frequency and reduce the longitudinal displacement response of the girder[24]. In the Zhixi Yangtze River Bridge, flexible central buckles that differ from traditional rigid central buckles are set in the middle of the main span of each main cable to coordinate with short suspenders, as shown in Figure 3. The flexible central buckle is composed of an inclined cable connected to a short suspender, forming a cable–girder connection to control the vibration response of the structure.

Centerline

2,800

**Figure 2.** Half-section of the stiffening girder (mm).

1,950 29,300/2=14,650

Concrete deck 2% Main cable

3,600 26,000/2=13,000

2%

941.6

1,590

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25,000/2=12,500

**Figure 3.** Flexible central buckle. **Figure 3.** Flexible central buckle.

#### *2.2. Finite Element (FE) Model 2.2. Finite Element (FE) Model*

To simulate the structural characteristics, a three-girder model of a prototype bridge was established using ANSYS 18.0. The FE model is shown in Figure 4. The stiffening girder of the bridge is a steel-composite girder with an open section, and the longitudinal beams on both sides are the main bearing structures of the stiffening girder. Thus, the BEAM4 element was used to simulate the main stringer, small stringer, steel beam, and main tower. The LINK10 element was used to simulate the cable components. A total of 1836 BEAM4 elements for the girder, 82 BEAM4 elements for the pylon, and 279 LINK10 elements for the main cable and suspender were found. The bridge deck pavement contributes minimally to the stiffness of the stiffening girder; thus, only its mass was considered, and the stress stiffening of the LINK element was conducted in accordance with the To simulate the structural characteristics, a three-girder model of a prototype bridge was established using ANSYS 18.0. The FE model is shown in Figure 4. The stiffening girder of the bridge is a steel-composite girder with an open section, and the longitudinal beams on both sides are the main bearing structures of the stiffening girder. Thus, the BEAM4 element was used to simulate the main stringer, small stringer, steel beam, and main tower. The LINK10 element was used to simulate the cable components. A total of 1836 BEAM4 elements for the girder, 82 BEAM4 elements for the pylon, and 279 LINK10 elements for the main cable and suspender were found. The bridge deck pavement contributes minimally to the stiffness of the stiffening girder; thus, only its mass was considered, and the stress stiffening of the LINK element was conducted in accordance with the measured cable force. *Materials* **2022**, *15*, x FOR PEER REVIEW 5 of 18

**Figure 4.** Bridge FE model. **Figure 4.** Bridge FE model.

The theoretical material properties and cable force vary from the actual state of the structure; thus, the FE model was modified according to the measured material properties and cable force in construction. Then, the structure frequency was calculated by the modal analysis module of ANSYS software, and the Block Lanczos feature solver based on the Lanczos algorithm was used in modal analysis. When calculating the natural frequencies of a certain range contained in the eigenvalue spectrum of a system, the Block Lanczos method is particularly effective for extracting modes. The frequencies of the FE model were compared with measured structure frequency to validate the FE model. The research The theoretical material properties and cable force vary from the actual state of the structure; thus, the FE model was modified according to the measured material properties and cable force in construction. Then, the structure frequency was calculated by the modal analysis module of ANSYS software, and the Block Lanczos feature solver based on the Lanczos algorithm was used in modal analysis. When calculating the natural frequencies of a certain range contained in the eigenvalue spectrum of a system, the Block Lanczos method is particularly effective for extracting modes. The frequencies of the FE model were compared with measured structure frequency to validate the FE model. The research team

team undertook the monitoring of the structural state during the bridge's construction. After construction, the actual vibration mode and frequency of the bridge were measured

quency L1 is 0.113 hz and smaller than other bridge types, which is determined by the flexibility characteristics of the suspension bridge. The error of L1 is 2.7%, and the maximum error is 5.2% in L2. The errors are within the acceptable range, which preliminarily

> Measured value FE model

**Figure 5.** Comparison of structure frequency (L is lateral mode; DV is dissymmetry vertical mode;

Besides the modal test, the vehicle loading experiment was conducted to test its deformation performance under external load. Figure 6 shows the layout of the static and running tests of the bridge. The loading vehicle is a 35 t three-axle truck. The static test has four loading trucks in each row and eight rows in total. A comparison of the maximum girder vertical deflection of the static test is given in Table 1. The computing value and measured value are close, and the error is within 3%. The running test condition is that two 35 t loading vehicles drove through the bridge at a constant speed of 60 km/h, and the vertical dynamic deflection of the main girder in 1/2 L is measured. A comparison

proves the simulation effect of the FE model.

SV is symmetrical vertical mode; T is torsional mode).

0.00

0.15

Error:2.7%

Error:5.2%

0.30

Frequency

0.45

0.60

L1 L2 DV1 DV2 SV1 SV2 SV3 T1 T2 T3

Modality

undertook the monitoring of the structural state during the bridge's construction. After construction, the actual vibration mode and frequency of the bridge were measured through the modal test analysis system. Figure 5 shows the modal analysis results and the test results; the modes of vibration are consistent with the test results. The first-order frequency L1 is 0.113 hz and smaller than other bridge types, which is determined by the flexibility characteristics of the suspension bridge. The error of L1 is 2.7%, and the maximum error is 5.2% in L2. The errors are within the acceptable range, which preliminarily proves the simulation effect of the FE model. team undertook the monitoring of the structural state during the bridge's construction. After construction, the actual vibration mode and frequency of the bridge were measured through the modal test analysis system. Figure 5 shows the modal analysis results and the test results; the modes of vibration are consistent with the test results. The first-order frequency L1 is 0.113 hz and smaller than other bridge types, which is determined by the flexibility characteristics of the suspension bridge. The error of L1 is 2.7%, and the maximum error is 5.2% in L2. The errors are within the acceptable range, which preliminarily proves the simulation effect of the FE model.

The theoretical material properties and cable force vary from the actual state of the structure; thus, the FE model was modified according to the measured material properties and cable force in construction. Then, the structure frequency was calculated by the modal analysis module of ANSYS software, and the Block Lanczos feature solver based on the Lanczos algorithm was used in modal analysis. When calculating the natural frequencies of a certain range contained in the eigenvalue spectrum of a system, the Block Lanczos method is particularly effective for extracting modes. The frequencies of the FE model were compared with measured structure frequency to validate the FE model. The research

*Materials* **2022**, *15*, x FOR PEER REVIEW 5 of 18

**Figure 4.** Bridge FE model.

Besides the modal test, the vehicle loading experiment was conducted to test its deformation performance under external load. Figure 6 shows the layout of the static and running tests of the bridge. The loading vehicle is a 35 t three-axle truck. The static test has four loading trucks in each row and eight rows in total. A comparison of the maximum girder vertical deflection of the static test is given in Table 1. The computing value and measured value are close, and the error is within 3%. The running test condition is that two 35 t loading vehicles drove through the bridge at a constant speed of 60 km/h, and the vertical dynamic deflection of the main girder in 1/2 L is measured. A comparison Besides the modal test, the vehicle loading experiment was conducted to test its deformation performance under external load. Figure 6 shows the layout of the static and running tests of the bridge. The loading vehicle is a 35 t three-axle truck. The static test has four loading trucks in each row and eight rows in total. A comparison of the maximum girder vertical deflection of the static test is given in Table 1. The computing value and measured value are close, and the error is within 3%. The running test condition is that two 35 t loading vehicles drove through the bridge at a constant speed of 60 km/h, and the vertical dynamic deflection of the main girder in 1/2 L is measured. A comparison between the measured results and the FE model is shown in Figure 7; the results are in good agreement. The model can reflect the dynamic response of the bridge and satisfy the requirements of subsequent analysis under traffic flow. **Table 1.** Comparison of stiffening girder vertical deflection under static test. **L/8 3L/8 L/2 5L/8 7L/8**  Measured Value (mm) 175 606 1093 615 180 FE model (mm) 181 624 1120 630 185 Error (%) 0.03 0.03 −2.02 0.02 0.03

**Figure 6.** Load case of bridge static and running tests. (unit: m) −50**Figure 6.** Load case of bridge static and running tests. (unit: m).

Measured

Calcluated

**Figure 7.** Comparison of midspan dynamic strain under running test.

0 10 20 30 40 50 60

Time (s)

The stress of long-span bridge suspenders is caused mostly by dead load; thus, the

amplitude of stress change caused by vehicle load and other live loads is relatively small and is far lower than the fatigue limit of steel wire. Therefore, the degradation process is a typical corrosion fatigue process; that is, the corrosion defects on the steel wire surface develop into initial crack damage. The entire steel wire degradation process can be divided into stages of the development of corrosion and crack propagation, as shown in Figure 8. The tiny corrosion defects on the steel wire surface become the crack initiation site. When the corrosion defects transform into cracks, they continue to develop until

**3. Corrosion Fatigue of Suspender Steel Wire** 

*3.1. Corrosion Fatigue Mechanism* 

Vehicle speed: 60km/h


0

50

100

Vertical displacement(mm)

150

200

250


between the measured results and the FE model is shown in Figure 7; the results are in good agreement. The model can reflect the dynamic response of the bridge and satisfy the

Value (mm) 175 606 1093 615 180

② Arrangement of running test

(mm) 181 624 1120 630 185 Error (%) 0.03 0.03 −2.02 0.02 0.03

1.8 1.921.8

**L/8 3L/8 L/2 5L/8 7L/8** 

**Table 1.** Comparison of stiffening girder vertical deflection under static test.

Mid span ③ Arrangement of static test

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requirements of subsequent analysis under traffic flow.

Measured

FE model

1.4 3.6

14.5t 14.5t 6t

①Truck weight:35t Axle weight

**Table 1.** Comparison of stiffening girder vertical deflection under static test.

**Figure 7.** Comparison of midspan dynamic strain under running test. **Figure 7.** Comparison of midspan dynamic strain under running test.

#### **3. Corrosion Fatigue of Suspender Steel Wire 3. Corrosion Fatigue of Suspender Steel Wire**

#### *3.1. Corrosion Fatigue Mechanism 3.1. Corrosion Fatigue Mechanism*

The stress of long-span bridge suspenders is caused mostly by dead load; thus, the amplitude of stress change caused by vehicle load and other live loads is relatively small and is far lower than the fatigue limit of steel wire. Therefore, the degradation process is a typical corrosion fatigue process; that is, the corrosion defects on the steel wire surface develop into initial crack damage. The entire steel wire degradation process can be divided into stages of the development of corrosion and crack propagation, as shown in Figure 8. The tiny corrosion defects on the steel wire surface become the crack initiation site. When the corrosion defects transform into cracks, they continue to develop until The stress of long-span bridge suspenders is caused mostly by dead load; thus, the amplitude of stress change caused by vehicle load and other live loads is relatively small and is far lower than the fatigue limit of steel wire. Therefore, the degradation process is a typical corrosion fatigue process; that is, the corrosion defects on the steel wire surface develop into initial crack damage. The entire steel wire degradation process can be divided into stages of the development of corrosion and crack propagation, as shown in Figure 8. The tiny corrosion defects on the steel wire surface become the crack initiation site. When the corrosion defects transform into cracks, they continue to develop until destroyed under the action of load cycles. This degradation process can be simulated by the corrosion fatigue theory. *Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 18 destroyed under the action of load cycles. This degradation process can be simulated by the corrosion fatigue theory.

**Figure 8.** Degradation of suspender steel wires. **Figure 8.** Degradation of suspender steel wires.

#### *3.2. Uniform Corrosion and Pitting Corrosion*

**Nominal Diameter (mm)** 

**Tensile Strength (MPa)** 

corrosion rate.

where *t* is corrosion time.

*3.2. Uniform Corrosion and Pitting Corrosion*  The corrosion of steel wire includes uniform corrosion and pitting corrosion. Uniform corrosion describes the degree of average corrosion of the steel wire surface, which directly causes the reduction in the diameter of the steel wire, and the extent of diameter reduction is assumed to stay unchanged along the steel wire length [25]. The steel wire parameters adopted in this study are shown in Table 2. The surface of high-strength steel wire is usually protected by a coating for corrosion resistance. In the prototype bridge, the suspender consists of Galfan-coated steel wires. The Galfan coating should not be less than 300 g/m2 due to the specification of bridge designation [26]. According to the survey of relevant cable manufacturers, the coating quality is usually controlled within 350 g/m2. Thus, the depth of a Zn-Al alloy coating can be calculated according to its density (6.58 g/cm3) and ranges from about 29 μm to 34 μm. The corrosion of steel wire includes uniform corrosion and pitting corrosion. Uniform corrosion describes the degree of average corrosion of the steel wire surface, which directly causes the reduction in the diameter of the steel wire, and the extent of diameter reduction is assumed to stay unchanged along the steel wire length [25]. The steel wire parameters adopted in this study are shown in Table 2. The surface of high-strength steel wire is usually protected by a coating for corrosion resistance. In the prototype bridge, the suspender consists of Galfan-coated steel wires. The Galfan coating should not be less than 300 g/m<sup>2</sup> due to the specification of bridge designation [26]. According to the survey of relevant cable manufacturers, the coating quality is usually controlled within 350 g/m<sup>2</sup> . Thus, the depth of a Zn-Al alloy coating can be calculated according to its density (6.58 g/cm<sup>3</sup> ) and ranges from about 29 µm to 34 µm.

> **Modulus of Elasticity (GPa)**

௨() = ൜ () ≤

The uniform corrosion of high-strength steel wire undergoes a two-stage corrosion process; that is, the corrosion of the coating and the corrosion of the steel wire substrate.

௦() + () ≥

According to the preliminary work of the research team, the corrosion process of Galfan steel wire is measured by an accelerated corrosion test and can be simulated by parabola distribution as Equation (2) [25]. The corrosion rate decreases gradually. The oxidation products of aluminum in the coating form a passive film, which slows down the

In service conditions, the corrosion rate of Galfan coating is significantly different because of the exposure environment. As is well known, field exposure tests are difficult to conduct due to the high cost of time, and it is also difficult to find exactly matched field exposure test results. Thus, the time conversion scale was determined by the field exposure test of Galfan coating by Aoki and Katayama, in which a hot-dipped Galfan-coated steel plate with a 25 μm coating was investigated [27,28]. Assuming that the influence of coating thickness and surface shape on the corrosion rate is negligible, and the test results

௨() = 0.04431 − 0.000014<sup>ଶ</sup> (2)

where ௨() is the depth of uniform corrosion, *d*c is the corrosion depth of the zinc–aluminum alloy coating, *d*s is the corrosion depth of the steel wire, *t* is corrosion time, and *t*<sup>c</sup>

**Coating Quality (g/cm2)** 

**Coating Depth (μm)** 

(1)

**Yield Strength (MPa)** 

Galfan steel wire 5.25 1926 1775 2.08 × 10ହ 337 31.05

The corrosion rate can be described as Equation (1).

is the time when the coating is totally corroded.

**Nominal Diameter (mm) Tensile Strength (MPa) Yield Strength (MPa) Modulus of Elasticity (GPa) Coating Quality (g/cm<sup>2</sup> ) Coating Depth (**µ**m)** Galfan steel wire 5.25 1926 1775 2.08×10<sup>5</sup> 337 31.05

**Table 2.** Parameters of steel wire samples.

The uniform corrosion of high-strength steel wire undergoes a two-stage corrosion process; that is, the corrosion of the coating and the corrosion of the steel wire substrate. The corrosion rate can be described as Equation (1).

$$a\_{\iota}(t) = \begin{cases} \, \_{d\_{\mathcal{C}}}(t) & t \le t\_{\mathcal{C}} \\ d\_{\mathcal{S}}(t) + d\_{\mathcal{C}}(t) & t \ge t\_{\mathcal{C}} \end{cases} \tag{1}$$

where *au*(*t*) is the depth of uniform corrosion, *d*<sup>c</sup> is the corrosion depth of the zinc– aluminum alloy coating, *d*<sup>s</sup> is the corrosion depth of the steel wire, *t* is corrosion time, and *t*<sup>c</sup> is the time when the coating is totally corroded.

According to the preliminary work of the research team, the corrosion process of Galfan steel wire is measured by an accelerated corrosion test and can be simulated by parabola distribution as Equation (2) [25]. The corrosion rate decreases gradually. The oxidation products of aluminum in the coating form a passive film, which slows down the corrosion rate.

$$a\_{\mu}(t) = 0.04431t - 0.000014t^2 \tag{2}$$

where *t* is corrosion time.

In service conditions, the corrosion rate of Galfan coating is significantly different because of the exposure environment. As is well known, field exposure tests are difficult to conduct due to the high cost of time, and it is also difficult to find exactly matched field exposure test results. Thus, the time conversion scale was determined by the field exposure test of Galfan coating by Aoki and Katayama, in which a hot-dipped Galfan-coated steel plate with a 25 µm coating was investigated [27,28]. Assuming that the influence of coating thickness and surface shape on the corrosion rate is negligible, and the test results are applicable to the conversion time scale, it is suggested that 1 h of the accelerated corrosion test corresponds to 0.033~0.052 years in a rural environment, 0.018~0.024 years in an industrial environment, 0.019~0.028 years in a marine environment, and 0.014~0.022 years in a severe marine environment.

When the metal material surface has a passive or protective film, the pitting pit on the substrate surface appears after the protective layer is consumed, greatly affecting the characteristics of the steel wire. Pitting corrosion occurs randomly, accompanied by uniform corrosion [29]. Given the stress concentration effect, pitting corrosion is the site where steel wire fatigue fracture may occur. The pitting pit with the largest depth determines the working state of the steel wire; thus, the pitting pit with the largest depth is the key analysis point in corrosion fatigue analysis. Pitting pit depth can be calculated by uniform corrosion depth and pitting coefficient.

$$a \wedge (t) = a\_p(t) / a\_u(t) \tag{3}$$

where *ap*(*t*) and *au*(*t*) denote the depth of pitting corrosion and uniform corrosion.

The distribution of the maximum pitting coefficient conforms ∧(*t*) to the Gumbel distribution [7], which can be expressed as

$$F(\wedge(t)) = \exp\{-\exp[-\frac{(\wedge(t) - \beta\_0)}{\mathfrak{a}\_0}]\} \tag{4}$$

where *F*(∧(*t*)) is the cumulative probability density function; ∧(*t*) is the maximum pitting coefficient; and *α*<sup>0</sup> and *β*<sup>0</sup> are the distribution parameters.

Then, the distribution parameter of any wires with different lengths and diameters can be calculated by Equation (5):

$$
\beta\_k = \beta\_0 + \frac{1}{\alpha\_0} \ln(\frac{A\_k}{A\_0}), \alpha\_k = \alpha\_0 \tag{5}
$$

where *A<sup>k</sup>* is the surface area of the analysis target, and *A*<sup>0</sup> is the surface area of the wire with a 125 mm length and 8 mm diameter.

#### *3.3. Corrosion Fatigue Crack*

Stress concentration happens due to the shape characteristics of the corrosion pit. As the depth of the corrosion pit increases, a crack will occur when the stress intensity reaches a critical value. The transition process from pitting to cracking can be determined by two methods: (1) the growth rate of the fatigue crack exceeding that of the corrosion pit and (2) the stress intensity factor of the corrosion pit reaching the critical threshold of fatigue crack propagation. This study adopts the former method. The steel wire crack dominates when the development speed of the pitting pit depth exceeds that of the crack.

Corrosion cracks expand until failure under the stress cycle caused by an operating live load. The Forman formula is used to analyze the growth rate of a metal corrosion fatigue crack, as shown in Equation (6).

$$\frac{d\_a}{d\_N} = \mathcal{C} (\Delta \mathcal{K})^m / [\mathcal{K}\_c (1 - \mathcal{R}) - \Delta \mathcal{K}] \tag{6}$$

where *<sup>d</sup><sup>a</sup> d<sup>N</sup>* is the growth rate of the crack, *a* is the depth of the crack, *C* and *m* are the parameters of the Paris criterion [30], *K<sup>c</sup>* is the fracture toughness of the material, ∆*K* is the stress intensity factor range, and *R* is the stress ratio of alternating load.

The stress intensity factor ∆*K* is given by Forman, as follows:

$$
\Delta K = F\_a \left(\frac{a}{b}\right) \Delta \sigma\_a \sqrt{\pi a} + F\_b \left(\frac{a}{b}\right) \Delta \sigma\_b \sqrt{\pi a} \tag{7}
$$

where *F<sup>a</sup> a b* denotes a coefficient related to axial stress, *F<sup>b</sup> a b* denotes a coefficient related to bending stress, *a* is the crack depth, *b* is the diameter of the steel wire, ∆*σ<sup>a</sup>* is the equivalent axial stress amplitude, and ∆*σ<sup>b</sup>* is the equivalent axial stress amplitude.

*F a b* is calculated by Equation (8) [31].

$$\begin{cases} F\_{a}\left(\frac{a}{b}\right) = 0.92 \cdot \frac{2}{\pi} \cdot \sqrt{\frac{2b}{\pi a} \cdot \tan\frac{\pi a}{2b}} \cdot \frac{0.752 + 1.286\left(\frac{a}{b}\right) + 0.37(1 - \sin\frac{\pi a}{2b})^3}{\cos\frac{\pi a}{2b}} \\\\ F\_{b}\left(\frac{a}{b}\right) = 0.92 \cdot \frac{2}{\pi} \cdot \sqrt{\frac{2b}{\pi a} \cdot \tan\frac{\pi a}{2b}} \cdot \frac{0.923 + 0.199\left(1 - \sin\frac{\pi a}{2b}\right)^4}{\cos\frac{\pi a}{2b}} \end{cases} \tag{8}$$

where *a* is the crack depth, and *b* is the diameter of the steel wire.

To consider the effect of daily traffic flow on the structure comprehensively, a crack depth development model is established on the basis of daily traffic flow operation according to the traffic load investigation, as shown in Equation (9).

$$\begin{cases} a\_i = \Delta a + a\_{i-1} \\ \Delta a = C \sum e\_j N\_j \left(\Delta K\_j\right)^m / \left[K\_c \left(1 - R\right) - \Delta K\right] \end{cases} \tag{9}$$

where *a<sup>i</sup>* is the depth of the crack at time *i*; ∆*a* is the increment of the crack; *e<sup>j</sup>* is the operating time of traffic flow with different intensities; ∑ *e<sup>j</sup>* = 24 h; and ∆*K<sup>j</sup>* and *N<sup>j</sup>* are the stress intensity factor range and the number of cycles, respectively. Mayrbaurl pointed out that the critical relative crack depth conforms to the lognormal distribution with an average value of 0.390 and a coefficient of variation of 0.414. Based on the test, the maximum critical relative depth is 0.5, which is used as the judgment standard for steel wire failure.

#### **4. Traffic-Induced Stress Responses of Suspenders 4. Traffic-Induced Stress Responses of Suspenders**

*Materials* **2022**, *15*, x FOR PEER REVIEW 9 of 18

2 <sup>∙</sup> <sup>ඨ</sup>2 ∙ 2 ∙

where *a* is the crack depth, and *b* is the diameter of the steel wire.

cording to the traffic load investigation, as shown in Equation (9).

#### *4.1. Vehicle Bridge System 4.1. Vehicle Bridge System*

failure.

⎩ ⎪⎪ ⎨

⎪⎪ ⎧ ቀ 

ቁ = 0.92 ∙

 ቀ 

2 <sup>∙</sup> <sup>ඨ</sup>2 ∙ 2 ∙

ቁ = 0.92 ∙

൝

Vehicles can be classified into different types according to axle distance, axle number, vehicle load, etc. Vehicle subsystems are commonly simplified as a car body, wheels, a shock mitigation system, and a damping system. The corresponding dynamic models are established on the basis of the hypothesis that the mass of the damper and spring components are ignored. For example, a three-axle vehicle is shown in Figure 9 [32]. The longitudinal vibration of the vehicle is neglected for its few effects on the bridge; thus, the longitudinal degree of freedom is ignored in the analysis [33]. Thus, five degrees of freedom (vertical, horizontal, head nodding, side rolling, and head shaking) are considered for the integral vehicle. The vehicle dynamic models are also classified into five types, and the corresponding dynamic models are constructed. Vehicles can be classified into different types according to axle distance, axle number, vehicle load, etc. Vehicle subsystems are commonly simplified as a car body, wheels, a shock mitigation system, and a damping system. The corresponding dynamic models are established on the basis of the hypothesis that the mass of the damper and spring components are ignored. For example, a three-axle vehicle is shown in Figure 9 [32]. The longitudinal vibration of the vehicle is neglected for its few effects on the bridge; thus, the longitudinal degree of freedom is ignored in the analysis [33]. Thus, five degrees of freedom (vertical, horizontal, head nodding, side rolling, and head shaking) are considered for the integral vehicle. The vehicle dynamic models are also classified into five types, and the corresponding dynamic models are constructed.

0.752 + 1.286(

To consider the effect of daily traffic flow on the structure comprehensively, a crack depth development model is established on the basis of daily traffic flow operation ac-

= Δ+ିଵ

where is the depth of the crack at time *i*; Δ*a* is the increment of the crack; *<sup>j</sup> e* is the operating time of traffic flow with different intensities; ∑ = 24 h; and Δ and are the stress intensity factor range and the number of cycles, respectively. Mayrbaurl pointed out that the critical relative crack depth conforms to the lognormal distribution with an average value of 0.390 and a coefficient of variation of 0.414. Based on the test, the maximum critical relative depth is 0.5, which is used as the judgment standard for steel wire

) + 0.37(1 −

2)ସ

 2

0.923 + 0.199(1 −

 2

<sup>Δ</sup> = (Δ) /[(1−) <sup>−</sup> <sup>Δ</sup>] (9)

2)ଷ

(8)

**Figure 9.** Dynamic model of a three-axle vehicle. **Figure 9.** Dynamic model of a three-axle vehicle.

Vehicle wheels always keep contact with the deck; the bridge deformation caused by an external load leads to the vibration response of the vehicle and bridge subsystems; the dynamic response is influenced by the overall total mass matrix, damping matrix, and the overall stiffness matrix of the subsystem; and the road surface roughness is the main excitation source. Therefore, the interaction force between the vehicle and bridge system is Vehicle wheels always keep contact with the deck; the bridge deformation caused by an external load leads to the vibration response of the vehicle and bridge subsystems; the dynamic response is influenced by the overall total mass matrix, damping matrix, and the overall stiffness matrix of the subsystem; and the road surface roughness is the main excitation source. Therefore, the interaction force between the vehicle and bridge system is a function of the vehicle–bridge system's motion state and road roughness, which can be analyzed in the established vehicle–bridge analysis system [34]. The road surface roughness is described by a power spectral density function, which can be generated through Fourier inversion [35]. . .. . ..

$$\begin{cases} \mathbf{F}\_{\upsilon} = \mathbf{F}\_{\upsilon i} \Big( \mathbf{Z}\_{\upsilon \nu} \dot{\mathbf{Z}}\_{\upsilon \nu} \ddot{\mathbf{Z}}\_{\upsilon \nu} \mathbf{Z}\_{\flat \nu} \dot{\mathbf{Z}}\_{\flat \nu} \ddot{\mathbf{Z}}\_{\flat \nu} \mathbf{i} \Big) \\\ \mathbf{F}\_{\mathsf{b}} = \mathbf{F}\_{\mathsf{b} i} \Big( \mathbf{Z}\_{\upsilon \nu} \dot{\mathbf{Z}}\_{\upsilon \nu} \ddot{\mathbf{Z}}\_{\upsilon \nu} \mathbf{Z}\_{\flat \nu} \dot{\mathbf{Z}}\_{\flat \nu} \ddot{\mathbf{Z}}\_{\flat \nu} \ddot{\mathbf{z}} \Big) \end{cases} \tag{10}$$

where *Z<sup>v</sup>* denotes vehicle displacement, *Z<sup>b</sup>* denotes bridge displacement, and *i* denotes road surface roughness.

#### *4.2. Traffic Load Simulation*

The vehicle load data monitored in a region is used to further evaluate the degradation process of suspenders under traffic load. The data were collected from the traffic load of a long-span bridge for one month by a weigh-in-motion (WIM) system. Figure 10 shows the hourly traffic volume results of the traffic flow, which are divided into five levels based on the range of traffic volume, including level 1 (<300 passenger car unit (pcu)/h), level 2 (300~600 pcu/h), level 3 (600~900 pcu/h), level 4 (900~1050 pcu/h) and level 5 (>1050 pcu/h). The error bar of the hourly traffic volume proves that the traffic volume is relatively stable. Although the standard deviation of peak traffic volume is larger than the trough period, the overall distribution is consistent, which does not affect the division of traffic intensity. The time proportions are 0.25, 0.21, 0.165, 0.21, and 0.165, respectively. The

established random traffic flow simulation method is used to generate traffic flow loads of different strengths for loading [34], so as to obtain the impact of traffic flow level on the stress response of the suspenders. The established random traffic flow simulation method is used to generate traffic flow loads of different strengths for loading [34], so as to obtain the impact of traffic flow level on the stress response of the suspenders.

*Materials* **2022**, *15*, x FOR PEER REVIEW 10 of 18

ቊ

௩ = ௩(௩, ሶ

= (௩, ሶ

Fourier inversion [35].

road surface roughness.

*4.2. Traffic Load Simulation* 

a function of the vehicle–bridge system's motion state and road roughness, which can be analyzed in the established vehicle–bridge analysis system [34]. The road surface roughness is described by a power spectral density function, which can be generated through

where ௩ denotes vehicle displacement, denotes bridge displacement, and *i* denotes

The vehicle load data monitored in a region is used to further evaluate the degradation process of suspenders under traffic load. The data were collected from the traffic load of a long-span bridge for one month by a weigh-in-motion (WIM) system. Figure 10 shows the hourly traffic volume results of the traffic flow, which are divided into five levels based on the range of traffic volume, including level 1 (<300 passenger car unit (pcu)/h), level 2 (300~600 pcu/h), level 3 (600~900 pcu/h), level 4 (900~1050 pcu/h) and level 5 (>1050 pcu/h). The error bar of the hourly traffic volume proves that the traffic volume is relatively stable. Although the standard deviation of peak traffic volume is larger than the trough period, the overall distribution is consistent, which does not affect the division of traffic intensity. The time proportions are 0.25, 0.21, 0.165, 0.21, and 0.165, respectively.

, ሷ , )

, ሷ

, ) (10)

௩, ሷ ௩, , ሶ

௩, ሷ ௩, , ሶ

**Figure 10.** Average traffic volume from WIM data. **Figure 10.** Average traffic volume from WIM data.

#### **5. Numerical Analysis 5. Numerical Analysis**

#### *5.1. Analysis Conditions 5.1. Analysis Conditions*

As a tensioned component, flexible central buckles cannot support the vibration response of the midspan main beam or cable as the rigid central buckles, but they can still affect the overall response of the structure by changing the fastening force. The connection system formed by the central cable and the suspenders changes the distribution of the force of the suspender near the midspan under traffic load. Traffic load is the main inducement of the bridge vibration response. To study the improvement effect of central buckles on bridge vibration, this study analyzes the response of bridges under conditions such as no central buckle and settled flexible central buckles. The detailed analysis conditions are shown in Table 3. As a tensioned component, flexible central buckles cannot support the vibration response of the midspan main beam or cable as the rigid central buckles, but they can still affect the overall response of the structure by changing the fastening force. The connection system formed by the central cable and the suspenders changes the distribution of the force of the suspender near the midspan under traffic load. Traffic load is the main inducement of the bridge vibration response. To study the improvement effect of central buckles on bridge vibration, this study analyzes the response of bridges under conditions such as no central buckle and settled flexible central buckles. The detailed analysis conditions are shown in Table 3. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 18 *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 18 **Table 3.** Analysis conditions of the FE model. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 18 **Table 3.** Analysis conditions of the FE model. **Condition Description Schematic**  *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 18 **Table 3.** Analysis conditions of the FE model. **Condition Description Schematic**  N-C No central buckle

**Table 3.** Analysis conditions of the FE model. **Table 3.** Analysis conditions of the FE model. **Condition Description Schematic**  N-C No central buckle

