# Fatigue Life Assessment of Key Fatigue Details of the Corroded Weathering-Steel Anchor Boxes of a Cable-Stayed Bridge

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## Abstract

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## 1. Introduction

## 2. Project Background

## 3. Identification of the Key Fatigue Detail of the Weathering-Steel Anchor Box

_{D}is the constant amplitude fatigue limit and Δσ

_{L}is the fatigue stress cutoff limit.

_{D}is the cable force corresponding to the constant amplitude fatigue limit; ΔP

_{L}is the cable force corresponding to the fatigue stress cutoff limit; and ΔP

_{i}and ΔP

_{j}are cable forces corresponding to Δσ

_{I}and Δσ

_{j}in Equation (1).

_{L}. However, to be on the safe side, the fatigue stress cutoff limit is not considered in this paper, and hence it is deemed that the fatigue-load calculation model II two-car model caused fatigue damage. According to the design S-N curve of the fatigue details and the Palmgren–Miner linear cumulative damage principle, Equation (3) [30] is used to calculate the fatigue damage caused by a single pass by the fatigue-load calculation model II two-car model. The calculated fatigue damage is listed in Table 1.

## 4. Determination of Initial Crack Size and Critical Crack Size

_{0}are selected as 1.00 mm, 2.00 mm, and 3.00 mm to study the influence of the corrosion pits and the initial pit size on the fatigue life of the weathering-steel anchor box.

_{max}or be determined according to the fracture mechanics criterion, usually the commonly used K criterion, to obtain a

_{f}, and then the smaller of the values of a

_{max}and a

_{f}is taken [32].

_{max}is taken as the N6′ plate thickness of 24 mm.

_{C}is fracture toughness, in MPa·m

^{1/2}; C

_{v}is the impact energy at a specific temperature, in J; B is the thickness of the material for which the fracture toughness needs to be evaluated, in mm.

^{1/2}. According to the commonly used K criterion, the critical crack length a

_{f}can be obtained according to Equation (5) as [35]:

_{f}is the critical crack length; Y is the geometric correction factor; σ

_{max}is the maximum value of the sum of constant and live load stress; and Q is the shape factor of the semi-elliptical crack, which can be calculated by Equation (6) [35]:

_{f}. From the fracture analysis model for the bearing structure at the end of cable 28 of the steel anchor box, the fitting curve of the geometric correction coefficient at the deepest point of the crack is obtained after regression analysis, as shown in Figure 8, and the geometric correction factor Y is regressed as:

_{f}is taken as 22.9 mm. Since a

_{f}< a

_{max}, the critical crack length is safely taken as 22.9 mm.

_{0}/c

_{0}of the crack is assumed to be 1/1. As mentioned above, this paper safely ignores the fatigue-stress cutoff limit; therefore, in the fatigue-crack growth analysis, the fatigue-crack growth threshold value can be regarded as 0.

## 5. Fatigue Life Assessment of the Key Fatigue Detail

_{0}at the location of the stress concentration of the load-bearing structure is 1 mm, 2 mm, and 3 mm. First, the ratios of the stress-intensity factor ranges ΔK

_{II}and ΔK

_{III}to ΔK

_{I}for the fatigue crack tips in the process of fatigue-crack growth are calculated, respectively, as shown in Figure 9. It can be seen from Figure 9 that the ratios of ΔK

_{II}and of ΔK

_{III}to ΔK

_{I}fluctuate within 3%; therefore, the fatigue crack of this particular fatigue detail is a mixed-mode fatigue crack, which is in line with the assumption; however, the ratios are rather insignificant as the crack is mainly subjected to unidirectional axial force, and hence the fatigue-crack growth is still dominated by mode I opening-mode cracks.

_{I}, K

_{II}, and K

_{III}are the effective stress-intensity factors corresponding to mode I, mode II, and mode III cracks at the crack tip, respectively, and ν is Poisson’s ratio.

_{0}is the initial crack size; a

_{i}is the length of the crack growth; a

_{cr}is the critical crack size; and C and m are the fatigue-crack growth rate parameters measured by the test.

^{11}times; when the initial crack size is 2 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure of the fatigue details is 3.62 × 10

^{11}times; and when the initial crack size is 3 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure of the fatigue details is 2.84 × 10

^{11}times. As is known from Section 2 of this paper, the fatigue damage caused by the fatigue-load calculation model II two-car model through a single pass of the uncorroded fatigue detail is 4.04 × 10

^{−13}, according to the Palmgren–Miner linear cumulative damage principle, the fatigue life of the detail corresponds to 2.48 × 10

^{12}passes. The relationship between the fatigue life and the initial crack size is plotted in Figure 12. It can be illustrated that when the initial crack size increases from 0 to 1 mm, the fatigue life is greatly reduced; when the initial crack size is 1 mm, the fatigue life is 20.3% of that when the initial crack size is 0 mm. When the initial crack size increases from 1 mm to 3 mm, the fatigue life still shows a decreasing trend, but the rate of decrease slows down, and the fatigue life when the initial crack size is 2 mm is 12.7% of that when the initial crack size is 0 mm, and the fatigue life when the initial crack size is 3 mm is 11.5% of that when the initial crack size is 0 mm. The reason for this phenomenon is that when the initial crack is small, the stress-intensity factor range for the tip of the fatigue crack is also small, so the fatigue-crack growth rate is small, and fatigue crack propagation requires more loading times for growing the same distance; thus, even an insignificant initial crack size also greatly reduces the fatigue life of the structure. Therefore, corrosion pits can dramatically reduce the fatigue life of weathering-steel anchor boxes.

## 6. Conclusions

- (1)
- The key fatigue detail of a typical weathering-steel anchor box of a cable-stayed bridge is identified by using the nominal stress method and the finite element analysis; a multi-scale refined finite-element model for the weathering-steel anchor box is established; and the fatigue-life assessment method of corroded weathering steel is extended to the multi-scale model to predict the remaining fatigue life of the corroded fatigue detail;
- (2)
- The remaining fatigue life of the key fatigue detail is calculated when the initial crack size is 1 mm, 2 mm, and 3 mm, with results of 20.3%, 12.7%, and 11.5% of the fatigue life when the initial crack size is 0 mm, respectively. The corrosion pits are equivalent to the introduction of the initial cracks, and even if the initial crack size is small, the fatigue life of the structure is greatly reduced;
- (3)
- The fatigue life of the key fatigue detail of the weathering-steel anchor box calculated by numerical simulation is more than 100 billion times greater, which far exceeds the millions of times that are of concern in engineering practice. The weathering-steel anchor box has excellent fatigue resistance;
- (4)
- A fatigue-life assessment method for corroded weathering-steel structures based on the finite element method and fracture mechanics is proposed, which makes up for the lack of a fatigue-life assessment method for corroded weathering-steel bridges which are in service;
- (5)
- Since the investigated bridge in this paper has been in service for a short period of time, it lacks data on corrosion pits, and the shape and size of the initial cracks are all assumed, for scientific research purposes, to demonstrate the feasibility of the proposed method. The status of the investigated bridge will be monitored and the proposed method will be improved accordingly in future studies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Lu, J.X.; Li, A.B.; Li, Z.G.; Wen, D.H. Development of atmosphere corrosion resistant steel product in baosteel: Review and prospect. China Metall.
**2004**, 12, 23–28. (In Chinese) [Google Scholar] - American Iron and Steel Institute (AISI). Performance of Weathering Steel in Highway Bridges; A Third Phase Report; AISI: Washington, DC, USA, 1995. [Google Scholar]
- Wang, B.Z. Present situation, design and construction of uncoated weathering steel bridges. World Bridges
**1988**, 3, 26–54. (In Chinese) [Google Scholar] - Albrecht, P.; Coburn, S.K.; Wattar, F.M.; Tinklenberg, G.L.; Gallagher, W.P. Guidelines for the Use of Weathering Steel in Bridges; NCHRP: Washington, DC, USA, 1989. [Google Scholar]
- Barsom, J.M. Fatigue Behavior of Weathered Steel Components; Transportation Research Board: Washington, DC, USA, 1984. [Google Scholar]
- Albrecht, P. Fatigue Behavior of Weathered Steel Bridge Components; University of Maryland: College Park, MD, USA, 1982. [Google Scholar]
- Friedland, I.M.; Albrecht, P.; Irwin, G.R. Fatigue of two year weathered A588 stiffeners and attachments. J. Struct. Div.
**1982**, 108, 125–144. [Google Scholar] [CrossRef] - Albrecht, P.; Cheng, J. Fatigue tests of 8-yr weathered A588 steel weldment. J. Struct. Eng. ASCE
**1983**, 109, 2048–2065. [Google Scholar] [CrossRef] - Albrecht, P.; Friedland, I.M. Fatigue tests of 3-yr weathered A588 steel weldment. J. Struct. Div.
**1980**, 106, 991–1003. [Google Scholar] [CrossRef] - Kunz, L.; Luká, P.; Klusák, J. Fatigue strength of weathering steel. Mater. Sci.
**2012**, 18, 18–22. [Google Scholar] [CrossRef] [Green Version] - Xu, S.H.; Qiu, B. Experimental study on fatigue behavior of corroded steel. Mater. Sci. Eng. A
**2013**, 584, 163–169. [Google Scholar] [CrossRef] - Xu, S.H.; Qin, G.C.; Ji, L.X.; Wang, Y.D. Fatigue notch factor of corrosion steel plate considering surface topography. J. Harbin Inst. Technol.
**2016**, 48, 153–157. (In Chinese) [Google Scholar] - Xu, S.H. Estimating the effects of corrosion pits on the fatigue life of steel plate based on the 3D profile. Int. J. Fatigue
**2015**, 72, 27–41. [Google Scholar] [CrossRef] - Xu, S.H.; Ren, S.B.; Wang, Y.D. Effects of Pitting Corrosion on the Fatigue Behavior of Q235 Steel. J. Harbin Inst. Technol. (New Ser.)
**2017**, 24, 81–90. [Google Scholar] - GGarbatov, Y.; Soares, C.G.; Parunov, J.; Kodvanj, J. Tensile strength assessment of corroded small scale specimens. Corros. Sci.
**2014**, 85, 29303. [Google Scholar] [CrossRef] - Saad-Eldeen, S.; Garbatov, Y.; Soares, C.G. Effect of corrosion degradation on ultimate strength of steel box girders. Corros. Eng. Sci. Technol.
**2012**, 47, 272–283. [Google Scholar] [CrossRef] - Saad-Eldeen, S.; Garbatov, Y.; Soares, C.G. Strength assessment of a severely corroded box girder subjected to bending moment. J. Constr. Steel Res.
**2014**, 92, 90–102. [Google Scholar] [CrossRef] - Saad-Eldeen, S.; Garbatov, Y.; Soares, C.G. Effect of corrosion severity on the ultimate strength of a steel box girder. Eng. Struct.
**2013**, 49, 560–571. [Google Scholar] [CrossRef] - Garbatov, Y.; Soares, C.G.; Parunov, J. Fatigue strength experiments of corroded small scale steel specimens. Int. J. Fatigue
**2014**, 59, 137–144. [Google Scholar] [CrossRef] - Guo, X.-Y.; Kang, J.-F.; Zhu, J.-S.; Duan, M.-H. Corrosion Behavior and Mechanical Property Degradation of Weathering Steel in Marine Atmosphere. J. Mater. Civ. Eng.
**2019**, 31, 04019181. [Google Scholar] [CrossRef] - Zong, L.; Shi, G.; Wang, Y.-Q.; Zhou, H. Fatigue assessment on butt welded splices in plates of different thicknesses. J. Constr. Steel Res.
**2017**, 129, 93–100. [Google Scholar] [CrossRef] - Lehner, P.; Krejsa, M.; Pařenica, P.; Křivý, V.; Brozovsky, J. Fatigue damage analysis of a riveted steel overhead crane support truss. Int. J. Fatigue
**2019**, 128, 105190. [Google Scholar] [CrossRef] - Wu, W.; He, X.; He, L.; Wu, C.; He, J.; Zhu, A. Joints Fatigue Damage Prediction for a Steel Truss Suspension Bridge Considering Corrosion Environment. Arab. J. Sci. Eng.
**2022**, 47, 4879–4892. [Google Scholar] [CrossRef] - Su, H.; Wang, J.; Du, J.S. Fatigue behavior of uncorroded non-load-carrying bridge weathering steel Q345qDNH fillet welded joints. J. Constr. Steel Res.
**2020**, 164, 105789. [Google Scholar] [CrossRef] - Su, H.; Wang, J.; Du, J.S. Fatigue behavior of corroded non-load-carrying bridge weathering steel Q345qDNH fillet welded joints. Structures
**2020**, 26, 859–869. [Google Scholar] [CrossRef] - Guo, S.L.; Lei, J.Q.; Huang, Z.W. Study on initial crack characteristics of inside weld in both-side welded U-rib. J. Cent. South Univ. (Sci. Technol.)
**2021**, 52, 3581–3594. (In Chinese) [Google Scholar] - Yosri, A.; Leheta, H.; Saad-Eldeen, S.; Zayed, A. Accumulated fatigue damage assessment of side structural details in a double hull tanker based on spectral fatigue analysis approach. Ocean. Eng.
**2022**, 251, 111069. [Google Scholar] [CrossRef] - Biswal, R.; Mehmanparast, A. Fatigue damage analysis of offshore wind turbine monopile weldments. Procedia Struct. Integr.
**2019**, 17, 643–650. [Google Scholar] [CrossRef] - Ministry of Transport of the People’s Republic of China. JTG D60-2015; General Specifications for Design of Highway Bridge and Culverts. China Communications Press: Beijing, China, 2015. (In Chinese)
- Ministry of Transport of the People’s Republic of China. JTG D64-2015; Specifications for Design of Highway Steel Bridge. China Communications Press: Beijing, China, 2015. (In Chinese)
- General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, Standardization Administration of the People’s Republic of China. GB/T 18590-2001; Corrosion of Metals and Alloys-Evaluation of Pitting Corrosion. Standards Press of China: Beijing, China, 2001. (In Chinese)
- Wang, C.S. Assessment of Remaining Fatigue Life and Service Safety for Riveted Steel Bridges; Tongji University: Shanghai, China, 2003. (In Chinese) [Google Scholar]
- General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, Standardization Administration of the People’s Republic of China. GB/T 4171-2008; Atmospheric Corrosion Resisting Structural Steel. Standards Press of China: Beijing, China, 2008. (In Chinese)
- British Standards Institute. BSI 7910; Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures. British Standards Institute: London, UK, 2005.
- Yang, X.H.; Chen, C.Y. Fatigue and Fracture, 2nd ed.; Huazhong University of Science & Technology Press: Wuhan, China, 2018. (In Chinese) [Google Scholar]

**Figure 6.**The distribution of maximum tensile and compressive stress of the steel anchor box. (

**a**) Maximum tensile stress of the bearing structure at the end of cable 1. (

**b**) Maximum compressive stress of the bearing structure at the end of cable 1. (

**c**) Maximum tensile stress of the bearing structure at the end of cable 28. (

**d**) Maximum compressive stress of the bearing structure at the end of cable 28.

Fatigue Detail | Location of Stress Concentration at the Bearing Structure of Cable 1 | Location of Stress Concentration at the Bearing Structure of Cable 28 |
---|---|---|

Fatigue damage | 7.69 × 10^{−15} | 4.04 × 10^{−13} |

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**MDPI and ACS Style**

Su, H.; Wang, J.; Du, J.
Fatigue Life Assessment of Key Fatigue Details of the Corroded Weathering-Steel Anchor Boxes of a Cable-Stayed Bridge. *Appl. Sci.* **2022**, *12*, 5379.
https://doi.org/10.3390/app12115379

**AMA Style**

Su H, Wang J, Du J.
Fatigue Life Assessment of Key Fatigue Details of the Corroded Weathering-Steel Anchor Boxes of a Cable-Stayed Bridge. *Applied Sciences*. 2022; 12(11):5379.
https://doi.org/10.3390/app12115379

**Chicago/Turabian Style**

Su, Han, Jian Wang, and Jinsheng Du.
2022. "Fatigue Life Assessment of Key Fatigue Details of the Corroded Weathering-Steel Anchor Boxes of a Cable-Stayed Bridge" *Applied Sciences* 12, no. 11: 5379.
https://doi.org/10.3390/app12115379