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

Abstract

In order to study the influence of corrosion on the fatigue life of weathering steel bridges, firstly the nominal stress method is employed to identify the key fatigue details of a typical weathering-steel anchor box of a cable-stayed bridge and then, a multi-scale refined finite-element model for the weathering-steel anchor box is established. The established fatigue-life assessment method for corroded weathering steel is extended to the multi-scale model, the remaining fatigue life for the fatigue details under corrosion are predicted, and the influence of the initial pit size on the fatigue life of weathering steel structures is analyzed. The research results show that corrosion pits can be equated to the introduction of initial cracks in the structure, and the remaining fatigue life for key fatigue details is calculated when the initial crack size is 1 mm, 2 mm, and 3 mm of the fatigue life when the initial crack size is 0 mm, with results of 20.3%, 12.7%, and 11.5%, respectively. Therefore, the research results show that the corrosion pits are equivalent to the introduction of initial cracks in the structure and, even if the initial crack size is small, the fatigue life of the structure is greatly reduced. The fatigue-life assessment method for corroded weathering-steel structures established in this paper is based on the finite element method and fracture mechanics, which offer universal applicability and generalizability and make up for the lack of a fatigue-life assessment method for corroded weathering-steel bridges in service.

1. Introduction

Weathering steel is composed of ordinary carbon steel and small amounts of corrosion-resistant elements, such as copper and nickel. The atmospheric corrosion resistance of weathering steel is 2–8 times that of ordinary carbon steel, and the coating life of coated weathering steel is 1.5–10 times that of ordinary carbon steel [1]. Measured by the cost of the entire life cycle, the cost of uncoated weathering–steel bridges is lower than that of ordinary carbon steel bridges [2,3]; thus, the application of weathering steel in bridges is gradually increasing.

In order to maximize the atmospheric corrosion resistance of weathering steel, weathering steel should be used in its uncoated form. However, pitting corrosion may occur on the surface of weathering steel, which may cause stress concentration. Under the repeated action of traffic loads, fatigue cracks may develop in the pits and cause final fatigue failure. Therefore, there are potential fatigue problems in weathering steel bridges. Albrecht et al. [4] studied the fatigue life of weathering-steel transverse stiffeners, cover plates, and other fatigue details exposed to the atmosphere for many years and found that fatigue cracks were more likely to appear, and developed faster, in a humid environment; the greater the stress concentration of the fatigue details, the less the fatigue life was affected by corrosion. Barsom [5] investigated the life degradation of weathering-steel base materials exposed to the atmosphere for 2 and 4 years, of weathering-steel butt welds, weathering-steel non-load-carrying fillet welds, and of weathering steel covers exposed to the atmosphere for 11 years and found that the corrosion pits caused the increase in surface roughness of the weathering steel and, therefore, the stress concentration, which led to a decrease in fatigue life. Albrecht et al. [6,7] studied the fatigue life of the fatigue details of A588 weathering-steel hand-welded transverse stiffeners exposed to the atmosphere for 2 and 4 years and results showed that corrosion reduced the fatigue life of the specimen by up to 22%. Albrecht et al. [8,9] then conducted an experimental study on the fatigue life of the fatigue details of A588 weathering-steel automatic submerged arc-welding transverse stiffeners exposed to the atmosphere for 3 years and 8 years, and also on that alternately exposed to the atmosphere for 3 years, and their results indicated the fatigue lives of the corroded specimens were 42%, 54%, and 42% lower, respectively, than the uncorroded samples; corrosion led to pitting and stress concentration, which in turn resulted in a decrease in fatigue life. Kunz et al. [10] tested the fatigue life of Atmofix 52 weathering-steel specimens exposed to the atmosphere for 20 years and showed that the fatigue life of corroded steel was much lower than that of uncorroded steel under symmetrical cyclic loading; the occurrence of fatigue cracks was highly related to the surface roughness caused by corrosion pits. Xu et al. [11,12,13,14] experimentally studied the fatigue life of uncorroded and corroded Q235 ordinary carbon-steel specimens, and it was found that the shape of the corrosion pits gradually developed from the narrow and deep type to the wide and shallow type; the fatigue life of the component decreased with the increase in the degree of corrosion, and the fatigue life of the components with a corrosion degree of 15% was only 3% that of the uncorroded components. The characteristic curved surface of the corroded surface was fitted with a PS50 three-dimensional non-contact surface profiler and Geomagic Studio software, and it was then exported to ANSYS to obtain the fatigue notch coefficient; fracture analysis showed that the fatigue cracks in almost all components, at different corrosion levels, were initiated by the corrosion pits.

Garbatov and Saad-Eldeen et al. [15,16,17,18] studied the effect of corrosion on the mechanical properties of ship steel plates with a yield strength of 235 Mpa and found that modulus of elasticity, yield strength, tensile strength, and elongation after the fracture of the steel plate decreased with the increase in the degree of corrosion and corrosion made the steel plate more brittle; a new stress–strain relationship was proposed to consider the influence of corrosion and residual stress on the mechanical properties of the steel plate. Garbatov et al. [19] studied the same batch of corroded ship steel plates and found that the fatigue strength of the uncorroded sample was 86 Mpa and the fatigue strength of the corroded sample was between 50 MPa and 100 Mpa. The increase in the surface roughness of the steel plate caused by corrosion was directly related to the deterioration in the fatigue life. Guo et al. [20] experimentally studied the mechanical behaviors of weathering steels Q420qNH and Q420qNHY in a marine environment and found that local corrosion or pitting can affect the yield strength, ultimate strength, and corresponding strains, especially elongation.

Modeling of the development of fatigue cracks and the numerical prediction of the remaining fatigue life have also been studied extensively. Zong et al. [21] experimentally studied the fatigue behavior of Q345qD ordinary carbon-steel butt-welded splices. As a result, combined from linear elastic fracture mechanics and finite element analysis, a numerical method to predict the remaining fatigue life of steel specimens with initial defects was proposed. Lehner et al. [22] combined the Monte Carlo method, the rainflow counting technique, and the Palmgren–Miner rule and proposed a method to evaluate the residual life of a nearly 100-year-old exterior-riveted crane support truss; the Monte Carlo method is used to generate a time-dependent load based on the operation of the overhead crane, and detailed numerical modeling is based on the rainflow counting technique and the Palmgren–Miner rule and is used to assess the fatigue resistance and estimate the residual life. Wu et al. [23] Proposed a linear corrosion prediction model based on Miner’s linear cumulative damage concept to solve the difficulty of implementing accelerated corrosion tests, and combined with the fatigue test results, a fatigue damage prediction model that considers corrosion factors is established, and the law of fatigue damage and stiffness degradation of the structure under a corrosive environment is determined. Su et al. [24,25] found, through experimental research, that the fatigue properties of corroded and uncorroded Q345qDNH weathering-steel non-load-carrying fillet welded specimens all met the requirements of the Chinese design code JTG D64-2015 “Specifications for Design of Highway Steel Bridge” with a certain safety margin; the fatigue strength of the atmosphere-exposed corroded specimens was reduced by 26.2% compared with the uncorroded specimens; however, the value was still 13.9% larger than the prescribed design fatigue strength. In addition, based on the finite element method and fracture mechanics, and with the help of FEA software Abaqus and fracture mechanics analysis software FRANC3D, it was found that the 3-dimensional initial cracks can be equated to 2-dimensional semi-elliptical initial cracks, and the remaining fatigue life can be reasonably accurately calculated. The influences of initial crack size, initial crack shape, and initial crack location on the remaining fatigue life were studied, and the influence of the initial crack size on fatigue behavior was found to be significant, while the influences of initial crack shape and initial crack location were found to be insignificant. Consequently, a fatigue-life assessment method for corroded weathering steel at the specimen level was established and verified. Guo et al. [26] studied the initial crack propagation characteristics and the initial angle effect of the inside weld in both-side-welded U-rib by using FRANC3D-ANSYS, and a fitting formula for the initial crack-growth effective angle range was derived. The initial crack angle of the inside weld toe of the U-rib was found to mainly affect the fatigue-life distribution of the structure at the crack growth stage, and the cracks with larger initial angles enter unstable propagation relatively quickly. Yosri et al. [27] investigated the fatigue life of structural details in the double side of a tanker ship by employing the spectral fatigue analysis approach, and the stochastic environmental loads were derived based on 3D diffraction theory in the frequency domain and then mapped with a structural finite-element model for the ship hull. The stress response for the considered details was extracted from the finite element analysis and the accumulated damage was estimated by Palmgren–Miner’s rule. The reduction of the hull girder section modulus and the deterioration in fatigue strength due to corrosion were taken into account, and the influences of different wave scatter diagrams, the reduction of hull girder section modulus, the stress concentration factor, the free corrosion S–N curve, wave direction, wave spectrum, and the operational profile on the accumulated fatigue damage were also investigated. Biswal and Mehmanparast [28] performed a realistic fatigue-life estimation of offshore wind turbine (OWT) monopile using the operational service loads recorded by online monitoring systems. Fatigue damage analysis was conducted at the circumferential weld joints using the finite element (FE) method by considering the geometrical and material property discontinuities, and the global–local modeling of the OWT was performed in as-welded condition to capture the local stress range for the weld toe. The S-N fatigue design approach and maximum stress range for the weld toe were used to determine the fatigue-crack-initiation life in monopiles, and the results from the proposed approach show that a realistic life assessment of monopile structures can be performed by accounting for the geometrical effects at the circumferential welds.

Until now, the fatigue life assessment of weathering steel structures is only applicable at the specimen level. Based on the fatigue-life assessment method of corroded weathering steel at the specimen level proposed by Su et al. [24,25], this paper generalized the method to a weathering steel bridge which is in service, and the multi-scale refined finite-element model is established, the key fatigue details of a typical weathering-steel anchor box of a typical cable-stayed bridge is evaluated by the finite element method, the remaining fatigue life after corrosion of the fatigue details is predicted, and the fatigue life evaluation for the corroded weathering-steel bridge is realized. The generalized method makes up for the lack of a fatigue-life assessment method for corroded weathering steel bridges in service. However, since the investigated bridge in this paper has only been in service for a short period of time, it lacks data on the 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 development laws of pitting corrosion will continue to be tracked, information about the corrosion pits measured on site will be incorporated into the proposed method, and the method will continue to be improved accordingly in future studies.

2. Project Background

In this paper, a cross-sea bridge with weathering steel used in the steel anchor boxes is selected as a research case. The bridge is a double-tower steel-box-girder cable-stayed bridge with a span layout of 110 + 236 + 458 + 236 + 110 = 1150 m. The middle and secondary side spans are equipped with stay cables. The cables are arranged in the shape of a fan with double cable planes and anchored on the outside. The tower adopts a horizontal H-shaped frame with a bridge deck width of 33.1 m. The layout of the bridge is shown in Figure 1.

In order to reduce maintenance work during the operational period, the main structure of the steel anchor box, the steel anchor box diaphragm, the steel anchor box segment connecting plate, and the steel anchor box stay-cable sleeve of the cross-sea bridge are made of Q355NHD weathering steel. The steel anchor boxes are uncoated and generate a stable protective rust layer on their own to achieve effective anti-corrosion within the operational period. Steel anchor boxes are divided into 12 sections, and their numbers correspond to the cable numbers. For example, the stay cables S14 and M14 correspond to the No. 14 steel anchor boxes. Steel anchor boxes are divided into types A, B, and C: the type-A steel anchor box is suitable for the No. 14 steel anchor box, which corresponds to stay cables S14 and M14 at the highest anchoring position; the type-B steel anchor box is suitable for the No. 4 to 13 steel anchor boxes, which correspond to stay cables S4–S13 and M4–M13 that are positioned in the middle; the type-C steel anchor box is suitable for the No. 3 steel anchor box corresponding to the stay cables S3 and M3, with the lowest anchoring position. There are 28 steel anchor boxes on the two towers, with 56 pairs of stay cables connected to them.

The steel anchor boxes are all 5.2 m long and 1 m wide. The height of the type-A and -B steel anchor boxes is 2.5 m, and the height of the type-C steel anchor box is 3 m. The steel anchor box is a box-shaped structure composed of side pull plates, end bearing plates, webs, anchor plates, transverse diaphragms, connecting plates, and stiffening ribs. The side pull plate mainly bears the horizontal pulling force of the stay cable; the plate thickness is 40 mm, the two sides are equipped with a vertical manhole, and the outer side is equipped with vertical stiffeners that also serve as connecting plates. The end bearing plate has a thickness of 30 mm and a width of 1300 mm; shear keys are set on its outer side to connect to the concrete of the tower. The web plays an important role in transmitting the cable force to the side pull plate, for which the thickness is 40–60 mm, the height varies with the angle of the stay cable, and stiffeners are provided on both sides. The transverse diaphragm is set horizontally between pull plates; it is a ribbed stiffened-steel plate with a thickness of 16 mm and with a manhole, which is used as a construction platform when the stay cable is stretched. The anchor plate is 40 mm thick, and the anchor pad is 80 mm thick. Since the tension in it is provided by the stay cable varies, there are two specifications for the anchor pads: 740 mm × 740 mm and 640 mm × 640 mm.

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

Firstly, Midas Civil is used to establish the full bridge model, then the car lane load in JTG D60-2015 “General Specifications for Design of Highway Bridges and Culverts” [29] is adopted, and the maximum and minimum cable force enveloping diagrams of the 56 pairs of cables are obtained through the most unfavorable loading of the influence line, as shown in Figure 2. Each steel anchor box is connected to two cables, and the component forces of the two cables in the horizontal direction are opposite and partially offset. Therefore, the type-A steel anchor box (that connecting cable 1 and cable 28) with the largest horizontal force is selected as the research object, and Abaqus is used to establish a refined finite element model.

The bridge deck width is 33.1 m. According to JTG D60-2015, the number of unidirectional design lanes of the bridge is four. According to the finite element model of the full bridge, the cable force influence line of cable 1 for the four car lanes is obtained, as shown in Figure 3. It can be seen that the value of the fourth car lane influence-line is the largest; therefore to be on the safe side, this paper takes the influence line of the fourth car lane to conduct further analysis. The influence lines of cable 1 and cable 28 on the fourth car lane are obtained and shown in Figure 4.

MATLAB’s data fitting tool CFTOOL was employed to fit the influence lines of cable 1 and cable 28 on the fourth car lane, and then the fatigue-load calculation model II two-car model specified in JTG D64-2015 [30] was utilized to calculate the cable force history under fatigue load. The cable force histories for cable 1 and cable 28 are shown in Figure 5.

Steel bridge fatigue belongs to high-cycle fatigue. The Palmgren–Miner linear cumulative damage principle is usually used to calculate the equivalent damage, and the actual variable amplitude load spectrum is converted into equivalent stress amplitude or equivalent cable force amplitude for fatigue damage evaluation, as shown in Equations (1) and (2).

To calculate the equivalent stress amplitude [30]:

$$D=\underset{\Delta {\sigma }_i\ge \Delta {\sigma }_D}{\underbrace{{\displaystyle \sum }_{i=1}^{k_i}\frac{n_i}{5\times {10}^6}{\left(\frac{\Delta {\sigma }_i}{\Delta {\sigma }_D}\right)}^{{\beta }_s}}}+\underset{\Delta {\sigma }_L\le \Delta {\sigma }_i\le \Delta {\sigma }_D}{\underbrace{{\displaystyle \sum }_{j=1}^{k_2}\frac{n_j}{5\times {10}^6}{\left(\frac{\Delta {\sigma }_j}{\Delta {\sigma }_D}\right)}^{{\beta }_s+2}}}={\left(\frac{\Delta {\sigma }_{eq}}{\Delta {\sigma }_D}\right)}^{{\beta }_s}\frac{1}{5\times {10}^6}$$

(1)

where Δσ_D is the constant amplitude fatigue limit and Δσ_L is the fatigue stress cutoff limit.

To calculate the equivalent cable force amplitude:

$$D=\underset{\Delta P_i\ge \Delta P_D}{\underbrace{{\displaystyle \sum }_{i=1}^{k_i}\frac{n_i}{5\times {10}^6}{\left(\frac{\Delta P_i}{\Delta P_D}\right)}^{{\beta }_s}}}+\underset{\Delta P_L\le \Delta P_i\le \Delta P_D}{\underbrace{{\displaystyle \sum }_{j=1}^{k_2}\frac{n_j}{5\times {10}^6}{\left(\frac{\Delta P_j}{\Delta P_D}\right)}^{{\beta }_s+2}}}={\left(\frac{\Delta P_{eq}}{\Delta P_D}\right)}^{{\beta }_s}\frac{1}{5\times {10}^6}$$

(2)

where ΔP_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).

However, the weathering-steel anchor box is connected with two cables, and the stress distribution of the weathering-steel anchor box is affected by the interaction of the two cable forces over time. Therefore, the equivalent cable force cannot be simply used and the actual load spectrum should be used instead.

The S4R shell element is used in ABAQUS to build the finite element model of the steel anchor box. The 45 s stress history in Figure 5 is transformed into 46 load steps, and the stress concentration area of the steel anchor box at each load step is obtained. The maximum tensile stress and maximum compressive stress on both sides of the steel anchor box in the entire stress history appear at the load-bearing structure at the end of the cable, as shown in Figure 6. The maximum tensile stress of the load-bearing structure at the end of cable 1 appears in the 7th load step, and the tensile stress is 805.5 kPa. The maximum tensile stress of the load-bearing structure at the end of cable 28 appears in the 17th load step, and the tensile stress is 424.5 kPa. The maximum compressive stress of the load-bearing structure at the end of cable 1 appears in the 5th load step, and the compressive stress is 908.9 kPa. The maximum compressive stress of the load-bearing structure at the end of cable 28 appears in the 22nd load step, and the compressive stress is 3538.6 kPa. The maximum tensile stress and maximum compressive stress of the two load-bearing structures appear at the same position; therefore, this paper selects these two fatigue details for research.

The load-bearing structures at the ends of cable 1 and cable 28 are composed of plates N6 and N8 and N6′ and N8′, respectively, and the design fatigue strength is 70 MPa. The thickness of the plate N6, where the stress is concentrated at the end of cable 1, is 40 mm, and the thickness of the plate N6′ at the end of the cable 28 is 24 mm. N6 and N6′ are connected to the end bearing plates N8 and N8′, respectively, by double-sided grooved partial-penetration fillet welds, and the strength of the weld is the same as the base material. The height of the weld leg at the end of cable 1, where the structural stress is concentrated, is 19 mm, and the height of the weld foot at the end of cable 1 is 19 mm; the height of the weld foot at the end of cable 28 is 11 mm.

The stress history of the load-bearing structures at the ends of cable 1 and cable 28 extracted from the ABAQUS finite element model is shown in Figure 7:

The rainflow counting method is employed to obtain the equivalent constant amplitude stress-load spectrum of the load bearing structure. A single pass of the fatigue-load calculation model II two-car model brings 7 semi-cyclic loads at the end of cable 1 at the load-bearing structure and 5 semi-cyclic loads and 1 full-cyclic load at the end of cable 28 of the load-bearing structure. The stress amplitudes of the load-bearing structures obtained from the fatigue-load calculation model II two-car model are all less than the fatigue stress cutoff limit Δσ_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. Fatigue damage caused by a single pass by fatigue-load calculation model II.

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

.

$$D=\underset{\Delta {\sigma }_i\ge \Delta {\sigma }_D}{\underbrace{{\displaystyle \sum }_{i=1}^{k_i}\frac{n_i}{5\times {10}^6}{\left(\frac{\Delta {\sigma }_i}{\Delta {\sigma }_D}\right)}^{{\beta }_s}}}+\underset{\Delta {\sigma }_L\le \Delta {\sigma }_i\le \Delta {\sigma }_D}{\underbrace{{\displaystyle \sum }_{j=1}^{k_2}\frac{n_j}{5\times {10}^6}{\left(\frac{\Delta {\sigma }_j}{\Delta {\sigma }_D}\right)}^{{\beta }_s+2}}}$$

(3)

As shown in Table 1, the fatigue-load calculation model II two-car model caused a greater amount of fatigue damage at the end of cable 28 in a single pass than at the end of cable 1; therefore, fatigue failure is more likely to occur at the end of cable 28 and this specific fatigue detail requires additional attention.

4. Determination of Initial Crack Size and Critical Crack Size

Chinese code GB/T 18590-2001 “Corrosion of metals and alloys-Evaluation of pitting corrosion” [31] states that radiography, one of the in situ measurement methods, can be used to measure the depth of the corrosion pits on real bridges. Due to the short service time of the bridge, the actual measurement data are temporarily difficult to obtain. Therefore, from the perspective of scientific research, the assumed initial crack size, initial shape, and other crack information are used. The thickness of the plate N6′, where the structural stress is concentrated at the end of the cable 28, is 24 mm, and the initial crack sizes a₀ 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.

The critical crack size can either be determined according to the applicability principle to obtain a_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].

For the load-bearing structure at the end of cable 28, the applicability principle assumes that once the crack penetrates the steel plate and the critical crack size is reached, the component will fail and the calculation will be terminated. Therefore, a_max is taken as the N6′ plate thickness of 24 mm.

The technical conditions of the weathering steel Q355NHD used in weathering-steel anchor boxes meets the requirements of Chinese code GB/T 4171-2008 “Atmospheric corrosion resisting structural steel” [33], and the impact energy at the temperature of −20 degrees is not less than 34 J. According to the conversion formula of impact energy and fracture toughness recommended in Appendix J of British code BS 7910 [34], the fracture toughness of Q355NHD can be obtained as:

$$K_C=\left[\left(12\sqrt{C_{\nu }}-20\right){\left(\frac{25}{B}\right)}^{0.25}\right]+20$$

(4)

where K_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.

Therefore, the fracture toughness of the Q355NHD steel plate at the service temperature is not less than 70.5 MPa·m^1/2. According to the commonly used K criterion, the critical crack length a_f can be obtained according to Equation (5) as [35]:

$$a_f=\frac{Q{\left(\frac{K_C}{Y{\sigma }_{max}}\right)}^2}{\pi }$$

(5)

where a_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]:

$$Q=1+1.464{\left(\frac{a}{c}\right)}^{1.65} \left(\frac{a}{c}\le 1\right)$$

(6)

The variable amplitude load at the end of cable 28 is mainly the load in the direction of the cable force, while the load amplitude in other directions is much smaller, and therefore, the critical crack obtained by the mode I opening-mode crack is taken as a_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:

$$Y=0.9262-0.23461\left(\frac{a}{t}\right)+1.40921{\left(\frac{a}{t}\right)}^2-1.91653{\left(\frac{a}{t}\right)}^3+0.57421{\left(\frac{a}{t}\right)}^4+0.27246{\left(\frac{a}{t}\right)}^5$$

(7)

where t is the steel plate thickness.

Substituting Equations (6) and (7) into Equation (5), the effective solution for the plate thickness range is 22.9 mm, so a_f is taken as 22.9 mm. Since a_f < a_max, the critical crack length is safely taken as 22.9 mm.

The initial minor to major axis ratio a₀/c₀ 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

Numerical calculations of mixed-mode fatigue-crack growth are carried out for the fatigue life when the initial crack size a₀ 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.

The effective stress-intensity factor range for the tip of the fatigue crack in the process of fatigue-crack growth can be obtained by Equation (8) [24]:

$$\Delta K_{eff}=\sqrt{K_I^2+K_{II}^2+\frac{K_{III}^2}{\left(1-\nu \right)}}$$

(8)

where K_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.

The effective stress-intensity factor ranges of the fatigue crack tip during the fatigue-crack growth process for the three initial crack sizes are calculated. The relationship between the effective stress-intensity factor range and the fatigue crack length is shown in Figure 10. It can be seen from the figure that the effective stress-intensity factor range increases approximately linearly with the increase of the fatigue crack length, and the trends are almost the same for the three cases.

After obtaining the relationship between the effective stress-intensity factor range for the tip of the fatigue crack and the length of the fatigue crack, the number of loadings required for each step of the fatigue-crack growth and the fatigue life of the particular fatigue detail can be obtained by Equation (9) [24]:

$$N={\displaystyle \sum }_1^n{N}_i={\displaystyle \sum }_1^n{{\displaystyle \int }}_{a_0}^{a_i}\frac{da}{C{\left(\Delta K\right)}^m}={{\displaystyle \int }}_{a_0}^{a_{cr}}\frac{da}{C{\left(\Delta K\right)}^m}$$

(9)

where N is the fatigue life; a₀ 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.

The number of loadings required for each step of fatigue-crack growth and the fatigue life of the fatigue detail are calculated, as shown in Figure 11. When the initial crack size is 1 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure for the fatigue details is 5.03 × 10¹¹ 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 × 10¹¹ 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 × 10¹¹ 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 × 10¹² 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.

The above fatigue-crack growth analysis process extends the established fatigue-life evaluation method of corroded weathering steel to a multi-scale refined finite-element model. However, the fatigue life of the particular fatigue detail of the weathering-steel anchor box calculated by numerical simulation is more than 100 billion times greater, far exceeding the concerns of engineering practice, which is normally in the magnitude of millions of times. It can be concluded that the life-assessment method of the corroded weathering-steel structure established in this paper has universal applicability and generalizability, and the weathering-steel anchor box of the research case has excellent fatigue resistance.

6. Conclusions

In this paper, the fatigue-life assessment method of corroded weathering steel is extended to the multi-scale refined finite-element model, and the remaining fatigue life of the fatigue detail of the corroded weathering-steel anchor box is predicted. The main conclusions of this paper are as follows:

(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.