# Fatigue Durability Analysis for Suspenders of Arch Bridge Subjected to Moving Vehicles in Southwest China

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establishment of a Standard Fatigue Car Model

#### 2.1. Vehicle Statistics

- Closure of lanes in a reasonable manner. The Xijiang Bridge has four lanes, and there are two lanes in the same direction. Therefore, in the same direction, the roadside lane is closed first, and the other lane is opened to traffic.
- Draw lines and cut grooves in the closed driveway.
- Cleaning and blow-drying the cut groove.
- Install the sensor in the groove, see Figure 1. The circuit leads to the roadside cabinet.
- Fill the groove with caulking glue, and it takes 3–8 h for the caulking glue to cure according to the weather conditions.
- Grind the joint sealant and the road surface.

_{i}, and weight W. Calculate the average and standard deviation of the wheelbase and weight after classification. Second, we combine vehicles with similar average weights and a wheelbase in the first classification, which reduces the number of categories and the calculation amount. The vehicle statistics are listed in Table 1.

#### 2.2. Establish a Standard Fatigue Car Model

_{i}is the i-th wheelbase of a typical fatigue car, D

_{ij}is the i-th wheelbase of the j-th vehicle in a typical vehicle assembly, and n is the total number of vehicles in a typical fatigue set.

_{i}is the axle load of the i-th axle in a typical fatigue car model, f

_{j}is the frequency of the j-th vehicle in a typical vehicle assembly, A

_{ij}is the axle load of the i-th axle in the j-th vehicle in a typical vehicle assembly, and μ

_{i}is the proportion of the axle load of the i-th axles to the weight of the whole car in a typical fatigue car mode.

_{a}, σ

_{a}= (σ

_{max}− σ

_{min})/2. V6 σ

_{a}= 40.46 MPa, V9 σ

_{a}= 47.77 MPa. From the calculation results, it can be seen that V9 is the vehicle that causes the maximum stress amplitude. Therefore, it is confirmed that V9 is used as the blueprint to establish the standard fatigue vehicle model. Model V9 is a six-axle car, and its wheelbase and axle weight distribution coefficient are known. To calculate the equivalent vehicle weight of V9, the axle weights of each axle are converted using fatigue damage equivalence. The calculation formula is Equation (4):

_{i}is the equivalent mass of the i-th typical fatigue car, that is the sum of the above A

_{i}; and f

_{i}is the i-th typical fatigue car frequency.

## 3. Calculation Method of the Fatigue Life of Suspenders

#### Concrete Calculation Steps of the Fatigue Life of Suspenders

- Find or calculate the local standard fatigue car model.
- Traffic flow statistics on the bridge, which can be made by the health monitoring system.
- Establishment of the bridge MIDAS model.
- Simulation of random traffic flow based on the Monte Carlo method.

- 5.
- Calculation of suspenders’ stress spectrum.

_{i}, and in lane no. 2, the influence line for the i-th suspender becomes phasor B

_{i}, and so on.

^{2}), I

_{c}is the section moment of inertia of the midspan section of the structure (m

^{4}), and m

_{c}is the mass per unit length in the middle of the structure (kg/m).

- 6.
- Calculation of the fatigue life of suspenders.

- (a)
- The S–N curve of the material is modified by the mean stress [37].

_{a}is the stress amplitude, and N is the number of cycles.

_{a}′ is the ultimate fatigue stress amplitude, σ

_{ar}is the fatigue limit under symmetrical cycle, σ

_{b}is the tensile strength, and σ

_{m}is the mean value of stress.

^{7}cycles is considered the fatigue limit, so N = 10

^{7}cycles are brought into Equation (7) to obtain the ultimate fatigue stress amplitude σ

_{a}′. Then, σ

_{a}′ is brought into Equation (8) to obtain σ

_{ar}corresponding to an average stress of 1050 MPa. Thus, the ultimate fatigue amplitude is deduced as Equation (9).

_{a}and bring in N = 10

^{7}, σ

_{a}= σ

_{a}′. The formula for obtaining a is as in Equation (10).

- (b)
- Calculation of equivalent stress amplitude σ
_{ae}.

_{ae}

^{3.5}= Σσ

_{a}

^{3.5}, where n is the total number of cycles. The σ

_{ae}can be calculated by Equation (14).

- (c)
- Calculation of the fatigue life according to the fatigue damage degree.

_{ae}into Equation (11) for σ

_{a}to obtain the total number of cycles N corresponding to σ

_{ae}. The daily damage degree is equal to the number of calculation hours multiplied by the fatigue damage in 1 h (i.e., fatigue damage in 1 h under daytime traffic condition × 14 h + fatigue damage in 1 h under night traffic condition × 10 h).

## 4. Engineering Verification and Application

#### 4.1. Brief Introduction of the Project

_{pk}= 1860 MPa, the diameter of each steel strand is 15.20 mm, and the outer diameter of the suspenders is 126 mm. A cross-section view of the suspender is shown in Figure 9, and the Dafeng River Bridge structural layout is shown in Figure 10.

#### 4.2. Finite Element Verification

^{3}, Young’s modulus is 2.95 × 10

^{11}Pa, Poisson’s ratio is 0.25, the tensile strength is f

_{pk}= 1860 MPa, and the inputted S–N curve is as in Figure 11. As shown in Figure 12, the grid is divided by sweeping. One end of the suspender is fixed and restrained, and the other end exerts a pressure of 207.422 MPa.

^{6}cycles. When the stress value is 207.422 MPa in actual engineering, it occurs about once per hour, so the cycle is about 8760 times per year. Thus, the calculated fatigue life unit of the no. 7 suspender is changed from cycles to years as in Equation (15).

#### 4.3. Engineering Application

#### 4.3.1. Traffic Flow Statistics

#### 4.3.2. Establishment of the Bridge MIDAS Model

#### 4.3.3. Simulation of Random Traffic Flow Based on Monte Carlo Method

#### 4.3.4. Simulated Traffic Load and Stress Spectrum Calculation of Suspenders

#### 4.3.5. Fatigue Life Calculation of Suspenders

_{m}= 98.721 MPa, and the ultimate tensile stress of the steel wire of the suspender is σ

_{b}= 1860 MPa.

_{ae}are N = 1,158,594,655 cycles, and the number of actions is 280 cycles. Therefore, during the daytime, the hourly fatigue damage of the no. 7 suspender can be determined.

## 5. Conclusions

- According to the traffic flow statistics of typical road sections, a standard fatigue vehicle model in Southwest China is established. Comparing our model with the standard fatigue vehicle in the Chinese General Code for Design JGD60-2015, we established that the model’s weight is lower. It is proved that even near the port, the number of heavy vehicles and their weight are not necessarily too high. There are differences in different regions, so it is inevitable to establish standard fatigue vehicles in Southwest China. Subsequently, this model can be used for vehicle fatigue simulation of similar road sections in Southwest China and other areas. It is not essential to obtain a large number of vehicle statistics and calculations again, thus improving the efficiency of further vehicle fatigue simulation.
- A set of calculation methods for the fatigue life of suspenders under the vehicle load is put forward. Compared with finite element calculation, this method has an error rate of less than 5% and can be effectively applied to practical projects, which proves its accuracy and feasibility. In the future, this method can be applied to the bridge health monitoring software system. According to the established standard fatigue vehicle model and the road traffic volume counted, the fatigue damage of bridge suspenders in Southwest China can be monitored in real time. Therefore, the fatigue life of bridges in Southwest China can be evaluated, which provides a reference for replacing suspenders.
- In practical engineering, the life of Dafeng River Bridge suspenders is calculated. It is found that the life of nos. 1–7 suspenders is on the rise. The life of the no. 1 suspender is the shortest, even less than one-third of that of the no. 7 suspender, mainly because the no. 1 suspender is short in length and close to the arch foot. Its stress amplitude is large under the action of vehicle load, so its fatigue damage is large and its fatigue life is low. Therefore, more attention should be paid to the health status of short suspenders near the arch foot in practical engineering.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 15.**Scatter diagram of the data distance of each group. (

**a**) Scatter plot of the daytime lane no. 2 data group. (

**b**) Scatter plot of the daytime lane no. 3 data group. (

**c**) Scatter plot of the night lane no. 2 data group. (

**d**) Scatter plot of the night lane no. 3 data group.

**Figure 16.**Stress influence line of the suspender, (

**a**) stress influence line of the no. 1 suspender, (

**b**) stress influence line of the no. 2 suspender, (

**c**) stress influence line of the no. 3 suspender, (

**d**) stress influence line of the no. 4 suspender, (

**e**) stress influence line of the no. 5 suspender, (

**f**) stress influence line of the no. 6 suspender, (

**g**) stress influence line of the no. 7 suspender.

Vehicles | Quantity | D_{1} | d_{1} | D_{2} | d_{2} | D_{3} | d_{3} | D_{4} | d_{4} | D_{5} | d_{5} | W | w | Classification |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 axle-1 | 92 | 1.50 | 0.08 | 0.467 | 0.2 | V1 | ||||||||

2 axle-2 | 11,672 | 2.66 | 0.10 | 1.40 | 0.47 | |||||||||

2 axle-3 | 1754 | 3.20 | 0.16 | 2.60 | 2.17 | |||||||||

2 axle-4 | 134 | 4.20 | 0.25 | 7.9 | 3.08 | V2 | ||||||||

2 axle-5 | 251 | 5.20 | 0.26 | 11.4 | 2.86 | V3 | ||||||||

3 axle-1 | 36 | 1.85 | 0.12 | 3.9 | 1.15 | 13.9 | 7.1 | V4 | ||||||

3 axle-2 | 123 | 3.60 | 0.20 | 1.3 | 0.51 | 20.3 | 12.5 | V5 | ||||||

3 axle-3 | 138 | 4.20 | 0.45 | 1.4 | 0.31 | 23.0 | 13.3 | |||||||

4 axle-1 | 437 | 1.90 | 0.10 | 4.1 | 0.76 | 1.35 | 0.09 | 31.9 | 20.0 | V6 | ||||

4 axle-2 | 19 | 2.90 | 0.66 | 7.4 | 2.10 | 2.50 | 0.74 | 8.20 | 11.50 | V7 | ||||

4 axle-3 | 66 | 2.70 | 0.20 | 7.8 | 1.45 | 2.70 | 0.18 | 3.25 | 0.73 | |||||

5 axle-1 | 4 | 2.00 | 0.17 | 3.0 | 2.07 | 3.50 | 0.57 | 1.3 | 0.26 | 28.4 | 11.3 | V8 | ||

5 axle-2 | 10 | 1.90 | 0.12 | 2.7 | 0.86 | 4.00 | 0.66 | 1.3 | 0.12 | 29.1 | 10.8 | |||

5 axle-3 | 22 | 3.40 | 0.14 | 1.4 | 0.41 | 3.70 | 0.37 | 1.3 | 0.10 | 29.2 | 12.0 | |||

5 axle-4 | 6 | 3.50 | 0.79 | 1.36 | 0.18 | 5.65 | 2.14 | 1.4 | 0.41 | 26.56 | 12.8 | |||

5 axle-5 | 2 | 3.60 | 0.37 | 6.8 | 1.21 | 2.20 | 2.58 | 1.5 | 0.53 | 30.6 | 15.1 | |||

6 axle-1 | 16 | 1.80 | 0.25 | 3.0 | 1.36 | 3.10 | 0.87 | 1.8 | 2.10 | 1.4 | 0.38 | 26.15 | 10.1 | V9 |

6 axle-2 | 96 | 3.40 | 0.15 | 1.36 | 0.31 | 3.60 | 0.67 | 1.3 | 0.27 | 1.3 | 0.06 | 33.5 | 19.9 | |

6 axle-3 | 198 | 3.37 | 0.10 | 1.35 | 0.07 | 5.88 | 0.39 | 1.3 | 0.05 | 1.3 | 0.03 | 43.6 | 23.1 |

_{i}, d

_{i}, D is the mean wheelbase; d is the standard deviation, and the unit is m; i is the i-th wheelbase; W is the average vehicle mass, w is the standard deviation, and the unit is t.

Parameter | V2 | V3 | V4 | V5 | V6 | V7 | V8 | V9 |
---|---|---|---|---|---|---|---|---|

axles | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 |

D_{1} (m) | 4.2 | 5.2 | 1.85 | 3.90 | 1.90 | 2.70 | 2.95 | 3.3 |

D_{2} (m) | 3.90 | 1.35 | 4.10 | 7.70 | 2.00 | 1.4 | ||

D_{3} (m) | 1.35 | 2.65 | 3.95 | 5.0 | ||||

D_{4} (m) | 1.30 | 1.3 | ||||||

D_{5} (m) | 1.3 |

Parameter | V2 | V3 | V4 | V5 | V6 | V7 | V8 | V9 |
---|---|---|---|---|---|---|---|---|

axles | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 |

A_{1}(μ _{1}) | 27 (0.34) | 36 (0.32) | 36 (0.26) | 52 (0.24) | 57 (0.18) | 11 (0.21) | 45 (0.15) | 52 (0.13) |

A_{2}(μ _{2}) | 52 (0.66) | 78 (0.68) | 30 (0.22) | 83 (0.38) | 57 (0.18) | 13 (0.25) | 47 (0.16) | 68 (0.17) |

A_{3}(μ _{3}) | 73 (0.52) | 84 (0.38) | 101 (0.32) | 14 (0.27) | 48 (0.20) | 70 (0.17) | ||

A_{4}(μ _{4}) | 104 (0.18) | 14 (0.27) | 71 (0.24) | 66 (0.16) | ||||

A_{5}(μ _{5}) | 74 (0.25) | 69 (0.17) | ||||||

A_{6}(μ _{6}) | 79 (0.20) | |||||||

G (kN) | 79 | 114 | 139 | 219 | 319 | 52 | 295 | 404 |

Classification | Axles | G(kN) | Model Legend | Quantity | Frequency (%) |
---|---|---|---|---|---|

V2 | 2 | 79 | 134 | 8.61 | |

V3 | 2 | 114 | 251 | 16.11 | |

V4 | 3 | 139 | 36 | 2.31 | |

V5 | 3 | 219 | 261 | 16.75 | |

V6 | 4 | 319 | 437 | 28.05 | |

V7 | 4 | 52 | 85 | 5.45 | |

V8 | 5 | 295 | 44 | 2.82 | |

V9 | 6 | 404 | 310 | 19.9 |

Suspender | Fatigue Life Calculated by ANSYS (Years) | Theoretical Fatigue Life (Years) | Error |
---|---|---|---|

No. 7 | 677.8 | 670 | 1.16% |

No. 6 | 583.4 | 575 | 1.46% |

No. 1 | 185.2 | 178 | 4.04% |

Lane | Periods | The Daily Traffic Flow | The Hourly Traffic Flow |
---|---|---|---|

No. 3 | daytime (6:00–20:00) | 831 | 60 |

night (20:00–6:00) | 208 | 21 | |

No. 2 | daytime (6:00–20:00) | 415 | 30 |

night (20:00–6:00) | 104 | 11 |

Periods | Lane | Average Vehicle Distance (m) |
---|---|---|

daytime | No. 2 | 2758.6 |

No. 3 | 1355.9 | |

night | No. 2 | 8000 |

No. 3 | 4000 |

Data Set | Capacity | U (m) | u (m) | D (m) | d (m) | Error |
---|---|---|---|---|---|---|

daytime lane no. 2 | 29 | 2758.6 | 2759 | 40 | 40.62 | 1.55% |

daytime lane no. 3 | 59 | 1355.9 | 1356 | 20 | 20.11 | 0.55% |

night lane no. 2 | 10 | 8000 | 8000 | 120 | 119.41 | 0.494% |

night lane no. 3 | 20 | 4000 | 4000.1 | 60 | 60.83 | 1.33% |

Lane | Number of Cars | Distance from the Front Car (m) | Position (m) |
---|---|---|---|

No. 2 | 1 | 0 | |

2 | 2770.55 | 2770.55 | |

3 | 2752.07 | 5522.62 | |

4 | 2782.89 | 8305.51 | |

$\vdots $ | $\vdots $ | $\vdots $ | |

29 | 2784.16 | 77,226.51 | |

30 | 2773.49 | 80,000 | |

No. 3 | 1 | 0 | |

2 | 1337.79 | 1337.79 | |

3 | 1358.82 | 2696.61 | |

4 | 1383.86 | 4080.47 | |

$\vdots $ | $\vdots $ | $\vdots $ | |

59 | 1409.4 | 78,644.22 | |

60 | 1355.78 | 80,000 |

Lane | Number of Cars | Distance from the Front Car (m) | Position (m) |
---|---|---|---|

No. 2 | 1 | 0 | |

2 | 7977.34 | 7977.34 | |

3 | 8017.79 | 15,995.13 | |

4 | 7785.93 | 23,781.06 | |

$\vdots $ | $\vdots $ | $\vdots $ | |

10 | 8194.95 | 71,896.35 | |

11 | 8103.65 | 80,000 | |

No. 3 | 1 | 0 | |

2 | 3974.09 | 3974.09 | |

3 | 3949.9 | 7923.99 | |

4 | 4010.78 | 11,934.77 | |

$\vdots $ | $\vdots $ | $\vdots $ | |

20 | 3988.2 | 76,053.54 | |

21 | 3946.46 | 80,000 |

Suspenders | Dead Load (kN) | Dead Load Stress (MPa) |
---|---|---|

No. 1 | 996.8 | 79.942 |

No. 2 | 1250.6 | 100.297 |

No. 3 | 1233.3 | 98.909 |

No. 4 | 1154.6 | 92.598 |

No. 5 | 1186.0 | 95.116 |

No. 6 | 1172.4 | 94.025 |

No. 7 | 1209.5 | 97.001 |

Stress Amplitude σ_{a} (MPa) | Number of Cycles | Stress Amplitude σ_{a} (MPa) | Number of Cycles |
---|---|---|---|

0–1 | 167 | 50–70 | 49 |

1–25 | 3 | 70–95 | 59 |

25–50 | 2 |

Suspender Number | Periods | Equivalent Stress Amplitude σ_{ae} (MPa) | Fatigue Loading Cycles | Daily Cycles | Daily Damage Degree (10^{−6}) | Fatigue Life (Years) |
---|---|---|---|---|---|---|

No. 7 | daytime | 50.22 | 1,158,594,655 | 3920 | 3.38 | 670 |

night | 49.103 | 1,256,558,780 | 890 | 0.71 | ||

No. 6 | daytime | 52.48 | 999,296,804 | 3892 | 3.90 | 575 |

night | 51.95 | 1,037,712,186 | 890 | 0.86 | ||

No. 5 | daytime | 53.15 | 953,834,246 | 3864 | 4.05 | 561 |

night | 51.967 | 1,034,303,102 | 860 | 0.83 | ||

No. 4 | daytime | 54.21 | 894,520,612 | 3920 | 4.38 | 513 |

night | 53.699 | 926,746,173 | 880 | 0.95 | ||

No. 3 | daytime | 55.55 | 810,817,727 | 3612 | 4.46 | 469 |

night | 59.66 | 633,102,176 | 870 | 1.30 | ||

No. 2 | daytime | 59.419 | 638,575,289 | 3976 | 6.23 | 359 |

night | 59.35 | 642,890,010 | 900 | 1.40 | ||

No. 1 | daytime | 82.72 | 209,448,213 | 2702 | 13.00 | 178 |

night | 79.47 | 241,175,283 | 580 | 2.40 |

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**MDPI and ACS Style**

Zhang, Z.; Wang, H.; Yang, T.; Wang, L.; Wang, X.
Fatigue Durability Analysis for Suspenders of Arch Bridge Subjected to Moving Vehicles in Southwest China. *Sustainability* **2022**, *14*, 10008.
https://doi.org/10.3390/su141610008

**AMA Style**

Zhang Z, Wang H, Yang T, Wang L, Wang X.
Fatigue Durability Analysis for Suspenders of Arch Bridge Subjected to Moving Vehicles in Southwest China. *Sustainability*. 2022; 14(16):10008.
https://doi.org/10.3390/su141610008

**Chicago/Turabian Style**

Zhang, Zimo, Hua Wang, Tao Yang, Longlin Wang, and Xirui Wang.
2022. "Fatigue Durability Analysis for Suspenders of Arch Bridge Subjected to Moving Vehicles in Southwest China" *Sustainability* 14, no. 16: 10008.
https://doi.org/10.3390/su141610008