## 1 Introduction  
Due to their light weight, high bearing capacity and short installation time, orthotropic steel decks (OSDs) have been widely used in long-span steel bridges, suspension bridges, and cable-stayed bridges [1; 2]. However, in some cases, some welded joints of OSDs have shown serious fatigue cracks shortly after the bridge was opened to traffic [3; 4]. For example, after 1.5 to 2 years of the Bronx-Whitestone Bridge being in use, cracks were found in the welded joints of the deck ribs by visual inspection, and the cracks continued to expand [3]. Located in the Yangtze River Delta, an area with the heaviest traffic in China, the Jiangyin Yangtze River Bridge experienced fatigue cracks only nine years after its construction. Figure 1 reveals fatigue cracking in the Jiangyin Yangtze River Bridge. As presented in the figure, many cracks initiated from the weld toe between the deck and the U-rib, which caused serious orthotropic steel deck and pavement damage. Therefore, it is important to evaluate the fatigue life of deck-to-rib welded joints in orthotropic steel decks using appropriate approaches. At present, the main approaches used for steel bridge fatigue life evaluation [5; 6; 7] include the nominal stress approach, the extrapolation hot spot stress approach, and equivalent structural stress approach. The nominal stress approach has been applied to some bridge design codes [8; 9], but it adopts different S-N curves for different welded joint types, and it is difficult to determine the location of nominal stress points of complex structure joints accurately with this approach [10]. The extrapolated hot spot stress approach [11; 12] only needs a unified S-N curve for different welded joints. However, it needs a fine mesh to achieve more accurate results due to its mesh sensitivity, and it is difficult to establish a uniform hot spot extrapolation formula for different weld toe types. The equivalent structural stress approach was proposed by [PERSON] et al. [13; 14], and is mesh insensitive. That is, accurate results can be obtained using a coarser mesh, and computational efficiency is improved. Moreover, for different weld joint types, a unified master S-N curve stress can be used for fatigue life evaluation [13]. To calculate the structural stresses, for the shell elements, [PERSON] et al. provided an approach based on the nodal force [14]. That is, the structural stress is calculated using the nodal force and moment at the weld toe. For solid elements, the structural stress calculation based on stress integration is adopted. [PERSON] et al. provided the stress integration formula of two-dimensional (2D) solid elements [15], and [PERSON] et al. further provided the stress integration formula of three-dimensional (3D) solid elements [16; 17]. That is, the stress on the reference surface, which is at a certain distance from the weld toe, is first extracted from FEM and integrated, and then the structural stress at the crack propagation surface is calculated through the equilibrium equation. There are three parameters in the formula of the stress integration approach: The plate thickness, the isolation body width, and the distance between the crack propagation surface and the reference surface. Understanding of the influence of the deck thickness on fatigue life was provided by [PERSON] et al. [13]. However, the influences of the other two parameters have not been given in previous studies. Therefore, in the present study, the stress integration approach of 3D solid elements is further studied, and the influences of calculated  
Figure 1: Fatigue cracks in the Jiangyin Yangtze River Bridge. parameters on the results are investigated. The master S-N curve given by [PERSON] et al. is mainly used for low-cycle fatigue evaluation [18], and the slope of the curve remains unchanged. However, the bridge fatigue life evaluation under actual traffic load belongs to the high cycle fatigue problem, with low stress amplitude [19].