The Fatigue Life Prediction of Welded Joints in Orthotropic Steel Bridge Decks Considering Weld-Induced Residual Stress and Its Relaxation Under Vehicle Loads

Abstract

The welded joints in steel bridges have a complicated structure, and their fatigue life is mainly determined by the real stress under the coupling effect of vehicle load stress, as well as weld-induced residual stress and its relaxation. Traditional fatigue analysis methods are inadequate for effectively accounting for weld-induced residual stress and its relaxation, resulting in a significant discrepancy between the predicted fatigue life and the actual fatigue cracking time. A fatigue damage assessment model of welded joints was developed in this study, considering weld-induced residual stress and its relaxation under vehicle load stress. A multi-scale finite element model (FEM) for vehicle-induced coupled analysis was established to investigate the weld-induced initial residual stress and its relaxation effect associated with cyclic bend fatigue due to vehicles. The fatigue damage assessment, considering the welding residual stress and its relaxation, was performed based on the S–N curve model from metal fatigue theory and Miner’s linear damage theory. Based on this, the impact of variations in traffic load on fatigue life was forecasted. The results show that (1) the state of tension or compression in vehicle load stress notably impacts the residual stress relaxation effect observed in welded joints, of which the relaxation magnitude of the von Mises stress amounts to 81.2% of the average vehicle load stress value under tensile stress working conditions; (2) the predicted life of deck-to-rib welded joints is 28.26 years, based on traffic data from Jiangyin Bridge, which is closer to the monitored fatigue cracking life when compared with the Eurocode 3 and AASHTO LRFD standards; and (3) when vehicle weight and traffic volume increase by 30%, the fatigue life significantly drops to just 9.25 and 12.13 years, receptively.

1. Introduction

Orthotropic steel bridge decks (OSBDs) are extensively utilized in bridges worldwide due to their advantages, including their light weight, exceptional load-bearing capacity, and ease of fabrication and construction [1,2]. However, the bridge engineering community has been consistently challenged by fatigue cracking issues, which are often attributed to the structural features and operating environments of bridge decks. Since the discovery of fatigue cracks in steel bridges such as the Severn Bridge in 1971, numerous instances of fatigue cracks in orthotropic decks have surfaced globally. For example, the Humen Bridge—China’s inaugural long-span suspension bridge with a steel box girder constructed in 1998—presented severe fatigue cracks in its steel deck merely five years after its completion and opening to traffic. Similarly, the Jiangyin Yangtze River Bridge (Jiangyin Bridge), located in the bustling Yangtze River Delta region and built in 1999, developed fatigue cracks just nine years post-completion, necessitating multiple closures for repairs. The Fatigue crack patterns in steel bridge deck of Jiangyin Bridge are shown in Figure 1. The second Nanjing Yangtze River Bridge, which was completed in 2001, also exhibited signs of fatigue cracks in 2008 [3,4,5]. These cases collectively highlight the widespread, early-onset, recurrent, and reproducible nature of fatigue cracking in orthotropic decks. This issue has emerged as a significant technical hurdle impeding the sustainable progress of steel bridge structures in China.

The fatigue damage evolution mechanism of OSBDs, which are typically sensitive to high cycle fatigue, is considerably influenced by the welding residual stress field and its subsequent relaxation during service [6,7]. Experimental studies have demonstrated that fatigue cycling, with local plasticity in the hot-spot region, relaxes welding-induced residual stresses. The phenomenon of stress redistribution generates notable changes in the local average stress, thereby influencing the precision of evaluations of fatigue crack initiation and propagation life [8,9]. Traditional evaluation systems primarily encounter the following issues: First, the use of an evaluation criterion solely relying on a single parameter—namely, stress amplitude—disregards the regulatory impact of mean stress. Additionally, the evaluation criterion frequently substitutes the local stress encountered in complex welds with stress distant from the welded area, simplifying the complex multiaxial stress state into an equivalent uniaxial stress; however, this simplification poses a challenge in terms of accurately representing the true stress state [10,11,12]. Furthermore, the fatigue strength S–N curve solely incorporates welding residual stress and its relaxation effect as empirical values, while a dynamic relaxation model remains unestablished, resulting in a dearth of quantitative analysis regarding its impact. As a result, the precise impact on fatigue performance cannot be accurately quantified, leading to substantial errors in fatigue life calculations for complex structures such as OSBDs.

Recent advancements in metal elastic–plastic fatigue theory and multi-field coupling analysis technology for welded structures have enabled the precise determination of the impact of weld-induced residual stress and its relaxation on the fatigue life of such structures [13,14]. Syahroni et al. employed the strip-cutting method to assess the initial residual stress in welded joints and examined the S–N curve while considering the impact of residual stress. Their findings revealed that welding residual stress, acting as an initial stress, does not alter the magnitude of the stress amplitude. Instead, it predominantly influences the fatigue failure behavior of the structure by modifying the average stress beneath the external cyclic load [15]. However, based on the metal fatigue theory, it is widely accepted that the influence of average stress on fatigue life cannot be overlooked when exposed to a low stress amplitude [16]. Therefore, we can ascertain the real stress experienced by a welded joint during its operation under vehicle loading by precisely computing the welding residual stress field and considering the stress relaxation effect that arises from the concurrent influences of vehicle-induced stress and residual stress. Thus, fatigue life analysis of the welded structure can be performed with reference to the steel fatigue life curve obtained through metal fatigue analysis.

Studies on the impacts of residual stress on the life of various welded structures have made significant achievements through the incorporation of experimental and numerical methods [17,18]. Based on a digital twin-driven framework, a physical model for calculating the fatigue crack growth life of welded structures in the welding residual stress field was studied using the weight function method [19]. The corrosion fatigue life prediction and calibration of bridge suspenders with a digital twin Bayesian entropy framework was investigated [20]. However, the residual stress relaxation effect on fatigue life has only been examined in a limited number of studies. The experimental and numerical investigation of the effects of residual stress and its release on the fatigue strength of a typical FPSO unit welded joint has been studied [21]. The residual stress release and its effects on the fatigue strength of typical welded joints in cone–cylinder pressure structures have been studied via non-destructive testing and numerical simulations [22]. The results show that the residual stress near the welding seam would be greatly released under an external load. Moreover, the fatigue life of the structures decreases while considering the welding residual stress.

In this paper, a fatigue damage calculation method for OSD joints that considers weld-induced residual stress and its relaxation is provided. First, a multi-scale finite element model (FEM) for vehicle-induced coupled analysis is established, based on the element birth and death technique and thermal–structural sequential coupling analysis in the ANSYS 2020 R1 finite element software. Then, numerical simulations investigating the coupling effects between standard fatigue vehicle loads and welding residual stresses are conducted, through which the real stress time–history curves at the U-rib-to-deck weld toe under typical tensile and compressive stress cycles during vehicular loading conditions are extracted based on transient dynamic analysis, enabling subsequent analysis of fatigue parameters. Moreover, utilizing the S–N curve model from metal fatigue theory and Miner’s linear damage theory as a foundation, the fatigue damage calculation method considering weld-induced residual stress and its relaxation is proposed by employing the Goodman equation to examine the influence of residual stress and its relaxation on mean stress. Finally, the method seamlessly incorporates actual bridge traffic flow monitoring data, and its computational outcomes are rigorously compared with the Eurocode 3 and AASHTO LRFD standards, thereby validating its precision. Additionally, a comprehensive analysis is undertaken to examine the influence of increases in vehicle weight and traffic flow on the fatigue durability of welded joints in OSDs. The insights gained from this study empower us to quantitatively assess the implications of these factors, ultimately yielding invaluable data to bolster the accuracy of fatigue life assessments and guide informed decision making in the management and maintenance of steel bridge decks.

2. Fatigue Life Assessment Method Considering Impact of Welding Residual Stress Relaxation

It is imperative to initially comprehend the welding residual stress field at welded joints to uncover the mechanism through which welding residual stress impacts the fatigue life of these joints. Furthermore, acquiring the authentic stress field of the welded joints subjected to cyclic external loads is essential. This approach enables the direct utilization of the traditional fatigue S–N curve of metal materials to assess the fatigue life of steel bridge welded joints.

2.1. Multi-Scale Finite Element Model (FEM) for Vehicle-Induced Coupled Analysis

A multi-scale coupling analysis model for steel bridge deck welding residual stress and vehicle stress is constructed in this section, using the Jiangyin Bridge as the engineering background. The process primarily comprises the following four steps:

Step 1. Establish an overall steel box girder model utilizing shell elements. This model is created using the shell63 elastic shell element, including various components such as top decks, U-ribs, bottom decks, the longitudinal rib of the bottom decks, diaphragms, and web plates. The element size is 150 mm, and the total number is 910,090. The steel box girder’s length along the bridge corresponds to the spacing of five diaphragms, specifically 38.7 m and 18.75 m. Additionally, the top deck has a thickness of 12 mm, while the U-rib plate is 6 mm thick.

Step 2. Establish a refined model of a localized rib-to-deck welded joint utilizing solid elements. The deck, rib, and weld seam within the range of 1.5 U-ribs are simulated using the Solid70 element, an eight-node hexahedral solid element; its corresponding structural element is a Solid185 element provided by ANSYS 2020 R1. Based on a series of mesh convergence studies, the smallest element size for the weld seam and heat-affected zone is set as 2 mm, while the largest element size for the part away from the weld is set as 8 mm (Figure 2).

Step 3. Establish the multi-scale finite element model for vehicle-induced coupled analysis by embedding the localized rib-to-deck welded joint refined model (step 2) into the overall steel box girder model (step 1). In this model, the connection method between the shell element (shell63) and the solid element (solid185) is a rigid connection. In the contact connection zone, the surface nodes of the solid element and the nodes of the shell element define the Targe170 target element and the Conta175 contact element, respectively. In addition, the contact algorithm is a multi-point constraint algorithm.

Step 4. Obtain the real stress time–course history curves under fatigue vehicle load. The thermal–structural coupling method is used to simulate the temperature field and stress field of the welding process of the multi-scale FEM. At this point, only the local solid FEM (Figure 3) is activated, the surrounding shell elements are killed, and the boundary constraints of the activated solid element model are applied. When the welding simulation is finished, the shell elements are activated, and the two-end simple support constraints of the multi-scale FEM and then the fatigue vehicle load are applied. The cyclic plasticity constitutive model of Q345 steel [23,24] is adopted in this FEM of weld-induced stress relaxation under vehicle load. The plasticity model of the steel under cyclic vehicle load is defined using the initial yielding condition, plastic flow rule, and hardening rule. Finally, the real stress time–course history curves are extracted which consider the coupling effect of welding residual stress and vehicle load.

2.2. Weld-Induced Residual Stress Distribution of Local Solid FEM

The ‘element birth and death’ technique [25,26] is incorporated to simulate the molten pool dynamic evolution. Figure 4 and Figure 5 depict the temperature and von Mises stress distribution during welding thermal cycles. Figure 6 shows the distribution of longitudinal welding residual stress. The weld-induced residual stress is at a high level in the weld area and rapidly diminishes in regions further away from the weld. The maximum longitudinal welding residual stress reaches the material’s yield stress (the yield strength of Q345qD steel is 345 MPa). In addition, the numerical simulation results of longitudinal residual stress and the test data [27,28] are compared. Despite the difference in size between the model presented in this paper and the experimental model, the variation rules of the numerical model results and the experimental results are approximately similar. The measured data points closely surround the simulated values, which verifies the authenticity and reliability of the numerical simulation results.

2.3. Fatigue Strength S–N Curves Considering Weld-Induced Residual Stress and Its Relaxation

Utilizing the metal fatigue theory as a foundation, the full-range fatigue strength S–N curves for steel, including the low-cycle fatigue region, high-cycle fatigue region, and ultra-high-cycle fatigue region, and the Goodman equation are adopted to infer the fatigue strength S–N curves. The detailed steps are as follows:

Step 1. The full-range fatigue strength S–N curve for a steel standard specimen subjected to symmetric axial cyclic tensile stress (where $S_m$ = 0) can be accurately represented using the Palmgren equation [14,29]:

$$S(N)=a{(N+B)}^b+c$$

(1)

where $S(N)$ represents the material’s fatigue strength, N denotes the fatigue life, and b is a material parameter specific to the material being considered, with a value of −0.2 [14] for steel. Additionally, $a$, C, and B are equation parameters, and their respective values are detailed as follows:

$$\left\{\begin{array}{l}a={\displaystyle \frac{{\sigma }_{GCF}-{\sigma }_k}{N_{GCF}^b-N_k^b}}\hfill \\ c={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{\left({\sigma }_{GCF}+{\sigma }_k\right)\left(N_{GCF}^b-N_k^b\right)-\left({\sigma }_{GCF}-{\sigma }_k\right)\left(N_{GCF}^b+N_k^b\right)}{N_{GCF}^b-N_k^b}}\right]\hfill \\ B={\left({\displaystyle \frac{{\sigma }_u-c}{a}}\right)}^{1/b}\hfill \end{array}\right.$$

(2)

The stress amplitude, denoted by ‘${\sigma }_k$’, is calculated from Equation (3) where $N$ is equal to $N_k$; $N_{GCF}$ represents the number of stress cycles in the superelevation cycle (for instance, when $N\ge {10}^{10}$); ${\sigma }_{GCF}$ denotes the stress amplitude at $N=N_{GCF}$, which is determined using Equation (4); ${\sigma }_u$ represents the ultimate tensile strength; and $H_{\nu }$ denotes the Vickers hardness [30].

$${\sigma }_k=0.5{\sigma }_u={\displaystyle \frac{{\sigma }_u^{1/3}}{1000}}\cdot \left(H_v+120\right)\cdot \left(155-7\cdot {\log }_{10}{N}_k\right)$$

(3)

$${\sigma }_{GCF}={\displaystyle \frac{{\sigma }_u^{1/3}}{1000}}\cdot \left(H_v+120\right)\cdot \left(155-7\cdot {\log }_{10}{N}_{GCF}\right)$$

(4)

Step 2. The full-range fatigue strength S–N curve of Q345 steel in a symmetric cycle (R = −1; S_m = 0) can be determined as Equation (6) using the Q345 material’s parameters, as shown in Equation (5), which is illustrated in Figure 5.

Regarding Q345 steel [31],

$$\left\{\begin{array}{c}{\sigma }_u=518\mathrm{MPa}\\ H_{\nu }=142\mathrm{kgf}/{\mathrm{mm}}^2\\ N_{GCF}={10}^{10}\end{array}\right..$$

(5)

Substituting Equation (5) into Equation (1), the full-range fatigue strength S–N curve of Q345 steel in a symmetric cycle is obtained and depicted in Figure 7 [30]:

$$S(N)=S_{a(R=-1)}=712\times {(N+41)}^{-0.2}+179$$

(6)

Equation (7) can be obtained by taking the logarithm of Equation (6).

$$\lg \left[\Delta {\sigma }_{(R=-1)}-358\right]=-0.2\lg (N+41)+3.15$$

(7)

In the formula, ‘${\mathrm{S}}_{a(R=-1)}$’ and ‘$\Delta {\sigma }_{(R=-1)}$’ denote the stress amplitude and stress range in a symmetrical cycle (R = −1; S_m = 0), respectively. Equation (7) indicates that a welding detail has an infinite fatigue life if the stress range remains below 358 Mpa. The formula is exclusively applicable to symmetric cyclic fatigue loads characterized by zero mean stress; however, most loads encountered in practical projects do not constitute symmetric stress cycles. The initial welding residual stress and the stress relaxation effect upon unloading primarily influence the fatigue failure behavior of the structure by altering the mean stress of the external cyclic load while leaving the magnitude of the equivalent stress amplitude relatively unaffected.

Step 3. The fatigue strength S–N curves considering welding residual stress relaxation can be obtained based on the Goodman equation. For ductile materials, the actual stress state at the weld seam, considering residual stress relaxation, can be converted into the stress level in a symmetrical cycle (R = −1; S_m = 0) using the Goodman equation [31,32], as demonstrated in Equation (8):

$${\displaystyle \frac{\Delta {\sigma }_{real}}{\Delta {\sigma }_{(R=-1)}}}+{\displaystyle \frac{{\sigma }_{m+rex}}{f_y}}=1$$

(8)

where $\Delta {\sigma }_{real}$ and ${\sigma }_{m+res}$ are the true stress range and the mean stress, considering weld-induced residual stress and its relaxation, respectively; $\Delta {\sigma }_{(R=-1)}$ signifies the stress amplitude in a symmetric cycle (R = −1; S_m = 0); and $f_y$ stands for the yield strength of the material, specifically 345 MPa for Q345 steel. The fatigue strength S–N curves considering welding residual stress relaxation can be derived by substituting Equation (8) into Equation (7):

$$\lg \left[{\displaystyle \frac{\Delta {\sigma }_{real}\cdot f_y}{f_y-{\sigma }_{m+rex}}}-358\right]=-0.2\lg (N+41)+3.15$$

(9)

Equation (9) is solved for N to derive Equation (10). Furthermore, the S–N curve serves as a reliable tool to predict fatigue life considering residual stress relaxation based on Equation (10). This prediction is feasible as long as the true stress time–history curve of significant welded joints subjected to external loads can be accurately extracted. The $\Delta {\sigma }_{real}$ and ${\sigma }_{m+res}$ can be determined using the rain flow counting method [32,33].

$$N=5.86\times {10}^{15}\times {\left({\displaystyle \frac{\Delta {\sigma }_{real}\cdot f_y}{f_y-{\sigma }_{m+rex}}}-358\right)}^{-5}-41$$

(10)

3. Fatigue Damage Considering Weld-Induced Residual Stress and Its Relaxation Under Typical Vehicle Loadings

3.1. The Welding Residual Stress Relaxation Effect of Welded Joints Under Typical Vehicle Loadings

In this section, the stress state of the weld position, considering only vehicle loads, is assessed. Concentrated loads from vehicles are applied at various transverse positions in the multi-scale finite element model to determine the most adverse loading condition of the steel bridge deck (Figure 3).

3.1.1. Fatigue Vehicle Load

The fatigue load calculation employs the fatigue vehicle model [33], which is recommended in the guide for fatigue design and maintenance of orthotropic decks. The axle load and distribution regulations of the model are shown in Figure 8. Figure 8a–c show the distribution of the vehicle load of the no. 1 axle and the no. 2/3/4 axle in the transverse and longitudinal positions, respectively. The axle spacing is 3.5 m, 7.0 m, and 1.3 m from the beginning to the middle to the end, respectively. Z1 in Figure 8a and Z2/Z3/Z4 in Figure 8b stand for the axle loads in Figure 8c. The fatigue vehicle load model used for the FEM analysis is simplified as a series of moving concentrated forces, which is directly applied on the surface of the orthotropic steel deck (Figure 9). The dynamic impact coefficient of the vehicle load, which is utilized for fatigue assessment, is 1.15 [34].

Considering vehicle driving randomness, the multi-scale FEM is applied under various vehicle loading conditions with a lateral distance of 150 mm. As shown in Figure 9, the calculation reveals that the fatigue stress under the vehicle, directly impacting the weld seam position, exhibits a tensile stress state (known as TC). Conversely, the maximum compressive stress is observed in the condition under the vehicle load acting at a distance of 150 mm from the weld (known as CC). Taking the most adverse compressive stress working conditions, CC, and the most unfavorable tensile stress working conditions, TC, as representative examples, the loading positions of the multi-scale FEM are presented.

Figure 10 illustrates the transverse stress and von Mises stress time–history curve for the weld position, excluding the welding residual stress, under both CC (the vehicle load acting at a distance of 150 mm from the weld) and TC (the vehicle load acting directly at the weld position). Under two distinct working conditions, where the stress ratio R is −∞ and 0, the peak transverse stress values are recorded as −17.46 → −34.91 → −31.42 → −31.42 Mpa and 6.84 → 13.69 → 12.32 → 12.32 MPa. The corresponding peak von Mises stress values are 10.06 → 18.95 → 16.71 → 16.71 MPa and 10.06 → 18.95 → 16.71 → 16.71 MPa, respectively.

3.1.2. The True Stress Time–History Curve Under Tension/Compressive Stress Working Conditions

Figure 11 illustrates the von Mises stress time–history curve for the weld position, considering the welding residual stress and its relaxation effect. This curve is plotted under the most adverse compressive stress working conditions, CC, and the most unfavorable tensile stress working conditions, TC, of vehicle load. Here, ‘n’ denotes the number of loading cycles applied to the vehicle model (for instance, n = 1 signifies the initial loading). Then, the stress response curves depicted in Figure 10 and Figure 11 are categorized into the fatigue parameters listed in

Table 1. Comparison of fatigue parameters under CC and TC working conditions.

Working Condition			CC	TC
Stress	True stress amplitude σreal/Mpa	Only considering vehicle loadings	73.86	20.52
		Considering welding residual stress and its relaxation effect	67.68	14.92
	Mean stress σm/Mpa	Only considering vehicle loadings	−41.53	11.55
		Considering welding residual stress and its relaxation effect	172.8	249.51
	Equivalent stress amplitude Sa(R = −1)/Mpa	Only considering vehicle loadings	65.92	21.23
		Considering welding residual stress and its relaxation effect	135.59	53.91
	Stress ratio R	Only considering vehicle loadings	−∞	0
		Considering welding residual stress and its relaxation effect	0.62~0.81	0.89~0.96

using Equation (8).

Based on Figure 11a, the initial von Mises stress at the weld position, after the welding and cooling processes, is 340.12 MPa, which is nearly equivalent to the material’s yield strength. After applying the cyclic vehicle loading compressive von Mises stress 36.12 → 72.74 → 65.02 → 65.02 Mpa, the stress decreases by merely 8.91 → 15.53 → 14.02 → 14.02 Mpa. The final coupling stress stabilizes at 339.43 Mpa, with a mere decrease of 0.69 Mpa. This analysis demonstrates that upon superimposing high-amplitude residual tensile stress onto the compressive stress dominating the stress mounted on the vehicle, the coupling stress at the weld toe experiences a minimal decrease, indicating an insignificant residual stress relaxation effect.

Furthermore, based on the Code for Design on Steel Structures of Railway Bridge [TB10091-2017] [35], in welded structures where R ≥ −1, the fatigue stress state is primarily a tension–compression stress state (abbreviated as a tension stress state), where tensile stress prevails. Under this condition, it is imperative to conduct fatigue-checking calculations. When R < −1, the fatigue stress state becomes a tension–compression stress state that is predominantly influenced by compressive stress (abbreviated as a compressive stress state). Under this condition, the impact of compressive stress on cumulative fatigue damage is disregarded. Based on Table 1, when considering the welding residual stress and its relaxation effect under CC working conditions, the true stress amplitude (σ_real) decreases marginally from 73.86 Mpa to 67.68 Mpa. However, the equivalent stress amplitude S_a (R = −1) significantly increases from 65.92 Mpa to 135.59 Mpa. This surge in the stress amplitude will likely have a direct impact on reducing the fatigue life. In addition, the stress ratio R transitions from −∞ to a positive value, significantly altering the fatigue stress condition at the weld toe. Specifically, it shifts from a compressive stress state, which obviates the need for fatigue assessment, to a tensile stress state necessitating fatigue evaluation.

Based on Figure 11b, the initial von Mises stress at the weld position is 339.07 Mpa. After applying the cyclic vehicle loading tensile von Mises stress 10.06 → 18.95 → 16.71 → 16.71 Mpa, the stress increases by 1.93 → 8.84 → 15.82 → 15.82 Mpa. The final coupling stress stabilizes at 326.40 Mpa, with a decrease of 12.67 Mpa. This analysis demonstrates that upon superimposing high-amplitude residual tensile stress onto the tensile stress dominating the stress mounted on the vehicle, the local superimposed stress state surpasses the elastic limit during this period. The result is a notable relaxation effect, demonstrated by the von Mises stress relaxation value being 81.18% of the mean vehicle stress. Furthermore, the stress relaxation effect predominantly manifests during the initial two loading stages of the primary stress cycle, subsequently leading to the stabilization of stress levels. As shown in Table 1, when considering the welding residual stress and its relaxation effect under the TC condition, the stress ratio R rises from 0 to approximately 1 while maintaining a tensile stress state for fatigue stress. The true stress amplitude (σ_real) decreases from 20.52 Mpa to 14.92 Mpa, and this alteration is not particularly evident. However, the mean stress significantly varies, ranging from 11.55 Mpa to 249.51 Mpa. Based on

Table 2. Calculation of fatigue damage for CC and TC using different S–N curves.

Condition	Fatigue Design Curve	Stress Time-Course Curves	Fatigue Parameters			Number of Cycles	Fatigue Damage
			$\mathbf{\Delta }{\mathit{\sigma }}_{\mathit{i}}$	${\mathit{\sigma }}_{\mathit{m}}$	${\mathit{n}}_{\mathit{i}}$	${\mathbf{N}}_{\mathit{i}}$	$\mathbf{D}$
CC	Eurocode 3 specification S–N curve	Figure 10a	35.45	−17.73	0.5	1.086 × 109	4.0871 × 10−8
			35.45	−17.73	0.5	1.086 × 109
			67.45	−33.73	0.5	4.356 × 107
			61.31	−30.65	1	7.024 × 107
			61.31	−30.65	1	7.024 × 107
			67.45	−33.73	0.5	4.356 × 107
	AASHTO specification S–N curve	Figure 10a	35.45	−17.73	0.5	2.585 × 109	5.7193 × 10−8
			35.45	−17.73	0.5	2.585 × 109
			67.45	−33.73	0.5	3.753 × 107
			61.31	−30.65	1	4.999 × 107
			61.31	−30.65	1	4.999 × 107
	S–N curves considering weld-induced residual stress and its relaxation	Figure 11a	4.39	335.59	1	∞	8.002 × 10−8
			7.26	331.95	1	4.341 × 109
			7.71	332.30	0.5	6.888 × 109
			7.36	331.86	0.5	2.915 × 107
			7.36	331.86	0.5	2.916 × 107
			7.36	331.86	0.5	2.934 × 107
TC	Eurocode 3 specification S–N curve	Figure 10a	9.53	4.77	0.5	∞	0
			9.53	4.77	0.5	∞
			18.16	9.09	0.5	∞
			16.24	8.12	1	∞
			16.24	8.12	1	∞
			18.16	9.09	0.5	∞
	AASHTO specification S–N curves	Figure 10a	9.53	4.77	0.5	∞	0
			9.53	4.77	0.5	∞
			18.16	9.09	0.5	∞
			16.24	8.12	1	∞
			16.24	8.12	1	∞
			18.16	9.09	0.5	∞
	S–N curves considering weld-induced residual stress and its relaxation	Figure 11b	3.92	336.08	0.5	∞	1.316 × 10−8
			3.92	336.08	0.5	∞
			6.80	333.20	0.5	1.008 × 108
			6.80	333.20	0.5	1.008 × 108
			7.25	332.85	0.5	4.633 × 108
			7.25	332.85	0.5	4.633 × 108
			7.25	332.85	0.5	4.633 × 108

in Section 3.2, an increase in mean stress leads to a decrease in fatigue life, ultimately causing a transition from infinite to finite fatigue durability.

Figure 10 and Figure 11, along with Table 1, show that the state of tension or compression in vehicle load stress has a notable impact on the residual stress relaxation effect observed in welded joints. The initial residual stress is already close to the material’s yield strength when subjected to compressive stress working conditions (CC), and the relaxation phenomenon at the weld position is minimal. However, when exposed to tensile stress working conditions (TC), the relaxation magnitude of the von Mises stress amounts to 81.2% of the average vehicle load stress value, indicating a notable relaxation impact. In addition, the welding residual stress and its relaxation effect alter the mean stress when subjected to vehicle load stress, thereby influencing the fatigue life of welded joints. For instance, the transverse stress ratio R transitions from negative infinity to positive under CC conditions, resulting in a shift from a compressive to a tensile fatigue stress state. Conversely, the mean stress (σ_m) at the weld position significantly rises under TC conditions, leading to non-zero fatigue damage and a reduction in fatigue life from infinite to finite.

3.2. The Fatigue Damage Results of Welded Joints Considering Weld-Induced Residual Stress and Its Relaxation

The S–N curves of nominal stress methods recommended by the widely used Eurocode 3 [10] and AASHTO [11] specifications in practical engineering are utilized to calculate the fatigue damage, considering typical compressive stress (CC) and tension stress (TC) working conditions, and investigate the influence of welding residual stress relaxation on the fatigue damage of orthotropic decks. In addition, the nominal stress point is determined to be 1.5t away from the weld toe, where t is the thickness of the top deck; that is, the stress value is at 18 mm in this paper. Furthermore, the results are compared with those obtained using the S–N curves considering welding residual stress relaxation.

The calculation formula based on the fatigue design curve recommended by both the Eurocode 3 and AASHTO specifications is given as follows:

$$\left\{\begin{array}{l}\Delta {{\sigma }_n}^{m_1}\cdot N=2\times {10}^6\cdot \Delta {{\sigma }_C}^{m_1}(m_1=3,N\le 5\times {10}^6)\\ \Delta {{\sigma }_n}^{m_2}\cdot N=5\times {10}^6\cdot \Delta {{\sigma }_D}^{m_2}(m_2=5,5\times {10}^6<N\le {10}^8)\\ \Delta {\sigma }_{nL}=0.549\cdot \Delta {\sigma }_D\end{array}\right.$$

(11)

Equation (11) is the Eurocode 3 specification S–N curve, where $\Delta {\sigma }_{str}$ is the nominal stress range, $N$ is the number of stress cycles, and $\Delta {\sigma }_c$, $\Delta {\sigma }_D$, and $\Delta {\sigma }_L$ are the fatigue detail classification, constant-amplitude fatigue limit, and fatigue cut-off limit, respectively. m₁ and m₂ represent the slopes of the first and second segments of the curve, respectively. For the deck-to-rib welded joints, $\Delta {\sigma }_C=71 \mathrm{Mpa}$, $\Delta {\sigma }_D=52 \mathrm{Mpa}$, $\Delta {\sigma }_L=29 \mathrm{Mpa}$.

$$(\Delta F{)}_{\mathrm{n}}={({\displaystyle \frac{A}{N}})}^{1/3}\ge {\displaystyle \frac{1}{2}}(\Delta F{)}_{TH}$$

(12)

Equation (12) is the AASHTO specification S–N curve, where A is a constant, $(\Delta F{)}_{\mathrm{n}}$ represents the fatigue resistance, and $(\Delta F{)}_{TH}$ indicates the constant-amplitude fatigue threshold. Regarding the specifics of the deck-to-rib welded joints, A = 14.4 × 10¹¹ and $(\Delta F{)}_{TH}=69 \mathrm{Mpa}$.

Table 2 utilizes S–N curves with varying fatigue strengths to compute the fatigue damage of deck-to-rib welded joints in OSDs under both CC and TC conditions. The symbol ‘∞’ in Table 2 represents infinity. The results reveal that under the most adverse compressive stress working conditions, CC, the fatigue damage determined using S–N curves from the Eurocode 3 and AASHTO specifications, as well as S–N curves that account for the influence of residual stress relaxation, are all within a similar order of magnitude. Nevertheless, the S–N curves considering residual stress relaxation have more precise calculation results than the first two methods. This finding underscores the inadequacy of China’s specifications for the design of steel structures in railway bridges [35], which neglects the impact of compressive stress on fatigue damage. When tensile stress working conditions, TC, are applied, the S–N curve calculation results from both the Eurocode 3 and AASHTO specifications indicate an infinite lifespan. However, the calculation results based on the S–N curve considering residual stress relaxation proposed in this paper align with the compressive stress working conditions, CC, in terms of order of magnitude.

According to the monitoring results of traffic volume on Jiangyin Bridge, the average daily traffic flow in 2022, with a single-lane total weight exceeding 30 kN, amounts to approximately 1254 vehicles, representing approximately 37% of the total traffic [36,37,38]. As demonstrated in Figure 12, the distribution frequencies $f_k$ (k representing the condition number) for CC and TT are 59% and 18.75%, respectively. Figure 12a illustrates the frequency of the transverse distribution based on a width of 0.1m [25]. Given the inherent randomness of vehicle driving patterns, the transverse distance between the CC and TT working conditions outlined in this paper is set to 0.15 m. Consequently, the transverse distribution of the standard axle centerline is adjusted to align with this 0.15 m width, as illustrated in Figure 12b.

Next, the fatigue damage $D_k$ incurred under CC and TT conditions can be determined by utilizing Formula (13) [39]:

$$D_k={\displaystyle \sum _i{D}_{(i,k)}}={\displaystyle \sum _i\frac{365\cdot N_{v,d}\cdot n_{(i,k)}\cdot f_k}{N_{(i,k)}}}$$

(13)

where $N_{v,d}$ is the average daily traffic flow, approximately 1254.

Based on the calculations, approximately 28% of the total fatigue damage occurs under TC conditions. Consequently, the traditional fatigue life calculation method for welded joints under tensile stress, which relies on the fatigue strength S–N curve without considering a low stress amplitude, cannot be trusted.

4. Fatigue Life Prediction Considering Variations in Traffic Loads

The number of overweight and overloaded highway bridge vehicles has significantly increased, coupled with a steep rise in traffic flow due to the rapid development of the national economy. These factors have greatly accelerated the rate of fatigue damage to steel bridge decks, thereby considerably reducing the service life of bridge engineering [40]. Over the past 20 years, since its opening to traffic in 1999, Jiangyin Bridge has experienced a steady increase in both traffic flow and vehicle weight. With a cumulative traffic flow exceeding 300 million and an average daily traffic flow of 92,000, the bridge has significantly surpassed its original design traffic flow. Traffic loads primarily affect the fatigue life of welded joints in OSDs via vehicle weight and traffic flow.

On the one hand, an increase in vehicle weight will impact the stress amplitude, ultimately leading to a decrease in N in the S–N curve; on the other hand, the increase in traffic flow will affect the number of cycles of fatigue load, resulting in an increase in n in the S–N curve. Generally, both factors will contribute to an increase in fatigue damage, thereby decreasing the ultimate fatigue life. This section employs three fatigue strength S–N curves to predict the fatigue life of the welding detail in deck-to-rib joints and precisely assess and forecast the influence of traffic load variations on the fatigue lifespan of the bridge deck. These three curves represent the S–N curve model recommended by the Eurocode 3 specification in Equation (11), the AASHTO specification in Equation (12), and that deduced in this paper in Equation (10). Moreover, the calculation expression and calculation process have been described in Section 3.2. The fatigue damage $D_k$ under various working conditions is calculated according to Equation (13). Then, the total fatigue cumulative damage $D_{\mathrm{T}}$ can be obtained using Equation (14). Lastly, the predicted fatigue life N_a of Jiangyin Bridge under the standard vehicle weight is calculated.

$${\mathrm{N}}_a={\displaystyle \frac{1}{D_T}}={\displaystyle \frac{1}{{\displaystyle \sum D_k}}}$$

(14)

4.1. Influence of Vehicle Weight Increase on Fatigue Life

Figure 13 illustrates three types of fatigue strength S–N curves which are employed to depict the fluctuating curve representing the fatigue life of the welded details of deck-to-rib joints in Jiangyin Bridge as vehicle traffic increases. These curves represent the S-N curve model that is recommended by the Eurocode3 specification in Equation (11), the AASHTO specification in Equation (12), and that deduced in this paper in Equation (10). Moreover, the calculation expression and calculation process have been described in Section 3.2. Based on the structural health monitoring (SHM) system and site-specific weigh-in-motion (WIM) system of Jiangyin Bridge, as well as the growth rate of vehicle weight and traffic flow observed over the past decade, the predicted growth rate for both vehicle weight and traffic flow has been established at 30% [41,42].

A vehicle weight growth rate of 30% signifies that the axle weight of each axle in the four-axle fatigue vehicle load model depicted in Figure 5 increases uniformly by 30%. The figure reveals that (1) as the vehicle weight increases, the fatigue life decreases. The decline trend of fatigue life calculated using the AASHTO specification S–N curve is roughly similar to that of the S–N curves considering welding residual stress relaxation. However, the fatigue life calculated using the Eurocode 3 specification S–N curve experiences the fastest decline. Additionally, as vehicle weight increases, the fatigue life curve gradually flattens. This is primarily attributed to the fact that in the high-cycle fatigue region where m = 5, even minor stress variations can lead to notable alterations in the number of stress cycles. (2) When utilizing the recommended standard fatigue vehicles outlined in the guide for fatigue design and maintenance of orthotropic decks, the calculated fatigue life varies significantly based on the application of different fatigue strength S–N curves. Among the options considered, the fatigue life determined using the Eurocode 3 specification S–N curve yields the highest value, approximately 92.91 years. The fatigue life determined by the AASHTO specification S–N curve follows closely, with an estimated fatigue life of around 64.75 years. However, when accounting for the influence of welding residual stress and its relaxation effect, the calculated fatigue life significantly drops to just 28.26 years. (3) When the vehicle exceeds its weight limit by 30%, the fatigue life calculated using the AASHTO specification S–N curve reaches its peak at approximately 29.37 years. However, when considering the influence of welding residual stress and the stress relaxation effect, the fatigue life significantly decreases to only about approximately 9.25 years. Hence, being overweight considerably impacts the fatigue life of bridges, and vehicle overload should be strictly controlled.

4.2. Influence of Traffic Flow Growth on Fatigue Life

Three types of fatigue strength S–N curves are employed to ascertain the variation in fatigue life of the deck-to-rib welded details of Jiangyin Bridge as traffic flow increases, as depicted in Figure 14. The figure shows the following: (1) The fatigue life progressively diminishes as the traffic flow increases. Moreover, the general downward trend in the fatigue life, as calculated by the three types of fatigue strength S–N curves, is largely consistent. (2) Upon a 30% increase in the traffic flow, the fatigue life determined using the Eurocode 3 specification S–N curve yields the highest value, approximately 46.91 years. The fatigue life calculated using the AASHTO specification S–N curve follows closely, estimated at approximately 36.52 years. However, when considering the influence of the welding residual stress and relaxation effect, the calculated fatigue life significantly drops to only 12.13 years. This analysis demonstrates that the current specifications for deck-to-rib welded joints are confined to a few standard structural designs. Additionally, the fatigue life calculated using the traditional S–N curve of fatigue strength, which ignores the influence of the welding residual stress and its relaxation effect, yields overly optimistic results. In reality, fatigue cracks appeared on the deck of Jiangyin Bridge merely nine years after it opened to traffic in 1999. Conversely, fatigue life prediction based on the comprehensive S–N curve, which incorporates the effects of welding residual stress and relaxation, aligns more closely with actual engineering observations.

5. Conclusions

This study investigated the influence of weld-induced residual stress and its relaxation on the fatigue life of welded joints in OSBDs via theoretical analysis and numerical simulation. The following major conclusions can be drawn:

(1)

The state of tension or compression in vehicle load stress notably impacts the residual stress relaxation effect observed in welded joints. When subjected to compressive stress working conditions (CC), the relaxation phenomenon at the weld position is minimal. However, when exposed to tensile stress working conditions (TC), the relaxation magnitude of the von Mises stress amounts to 81.2% of the average vehicle load stress value, indicating the notable impact of relaxation.

(2)

The welding residual stress and its relaxation effect alter the stress ratio (R) when subjected to vehicle load stress, thereby influencing the fatigue life of welded joints. For instance, the transverse stress ratio R transitions from negative infinity to positive under CC conditions, resulting in a shift from a compressive to a tensile fatigue stress state. Conversely, the mean stress (σ_m) at the weld position significantly rises under TC conditions, leading to non-zero fatigue damage and a reduction in fatigue life from infinite to finite.

(3)

Under TC conditions, the fatigue damage determined using the S–N curves from the Eurocode 3 and AASHTO specifications indicates an infinite lifespan. However, when calculated using the S–N curves considering weld-induced residual stress and its relaxation, the results align with the order of magnitude observed under CC conditions.

(4)

Based on traffic data from Jiangyin Bridge, the fatigue life of deck-to-rib welding details is 28.26 years, calculated using the method considering the impact of welding residual stress relaxation proposed in this paper. Moreover, when the vehicle weight increases by 30%, the fatigue life significantly drops to just 9.25 years. Similarly, a 30% increase in the traffic volume leads to a fatigue life of 12.13 years. These assessment results demonstrate the accuracy of the fatigue assessment method described in this paper, which accurately determined a value close to the monitored fatigue cracking life of Jiangyin Bridge, i.e., approximately 9 years.