As one of the most fatigue-sensitive parts of an orthotropic steel bridge deck, the weld between the U-rib and the top deck is prone to fatigue cracking under the actions of the stress concentration, welding residual stress, and vehicle load. To investigate the mechanism of fatigue crack propagation and the influence of the welding residual stress on the propagation patterns of fatigue cracks, a multiscale modeling method was proposed, and the static analysis and the dynamic propagation analysis of fatigue crack were carried out in this paper. First, a multiscale finite element model was established, including whole bridge models with a scale feature of 10^{2} m, orthotropic bridge deck models with a scale feature of 10^{0} m, and crack models with a scale feature of 10^{−3} m. Then, a segmental model of the bridge deck was extracted, which is regarded as a critical location of the bridge, and the shell-solid coupling method is adopted in the segmental model in order to further analyze the crack propagation rule. Moreover, based on the extended finite element method (XFEM), the static crack and dynamic crack propagation in this critical position were analyzed. Finally, thermoelastoplastic analysis was carried out on the connection of the U-rib and deck with a length of 500 mm to obtain the residual stress, and then the results of residual stress were introduced into the segmental model to further study its influence on the evolution of fatigue crack propagation. The analysis of the welding process shows that near the weld region of the connection of the U-rib and deck, the peak value of the residual tensile stress can reach the material yield strength. The static analysis of fatigue cracks shows that under the single action of a standard fatigue vehicle load, the fatigue details at the weld toe of the deck cannot reach the tensile stress required for fatigue crack propagation, and only the fatigue details at the weld toe of the U-rib can meet the requirements of fatigue crack propagation. The dynamic analysis of fatigue cracks reveals that the crack in the weld toe of the U-rib is a mixed-mode crack with modes I, II, and III. The propagation of a fatigue crack without a residual stress field will be terminated until the crack length is extended to a certain length. Nevertheless, when the residual stress field was introduced, the growth angle and size of the fatigue crack would increase, and no crack closure occurs. For the crack in the weld toe of the deck, the crack is in the closed state under the standard fatigue vehicle load. When the residual stress field is introduced, the tensile stress of the fatigue details increases. Meanwhile, the fatigue crack will become a mixed-mode crack with modes I, II, and III that will be dominated by mode I and extend toward the weld at a slight deflection angle. The results of various initial crack sizes at the weld toes of the top deck are analyzed, which shows that the initial crack size has a certain effect on the fatigue crack growth rate, especially the initial crack depth.

The orthotropic steel bridge deck is the preferred deck structure for long-span steel bridges and is widely used in modern bridges due to its light weight, construction convenience, performance stability, etc. Nevertheless, as a result of the structural complexities, crowded welds, and stress concentration problems, the deck structure is prone to fatigue failure under the repeated action of a vehicle load. According to the investigation results of the fatigue and fracture subcommittee of the American Society of Civil Engineering (ASCE), 80–90% of steel structure damage is related to fatigue, and the fatigue fracture has become one of the main reasons for steel bridge failure [

In recent years, the fracture mechanics method has been widely used in numerical simulations of macroscopic crack propagation processes and fatigue life analyses of orthotropic steel bridge decks. The extended finite element method (XFEM) is a new finite element method and was first proposed by Belyschko and Black at Northwestern University in 1999. This method is used to solve the problem of describing crack propagation in the FE method by using the idea of independent mesh division. The method does not need remeshing for the internal cracks and other defects in the analysis process and can retain all of the advantages of traditional finite elements. Zhang et al. [

Therefore, based on the XFEM, the fatigue crack propagation mechanism at the welding seam at the connection of the U-rib and top deck is discussed in this paper. First, a multiscale modeling method is established, including the whole bridge with a scale feature of 10^{2} m, box girder components with a scale feature of 10^{0} m, and cracks with a scale feature of 10^{−3} m. A segmental model of the bridge deck at critical locations is extracted, and the shell-solid coupling method is adopted in the segmental model in order to further analyze the crack propagation rule. Afterwards, the fatigue cracks are statically and dynamically analyzed based on the XFEM. Finally, the thermoelastoplastic analysis of the welding seam between the U-rib and top deck is carried out. The life-and-death element technology is adopted to simulate the welding process, and the welding residual stress field is obtained. By introducing the results of the residual stress into the segmental model, the propagation rule of fatigue crack is studied, and the influences of welding residual stress and initial fatigue crack length and depth on fatigue crack propagation rule are further investigated.

The XFEM is a modified finite element method proposed by Belytschko and Black [

In the XFEM, the element grid is divided into three types, as shown in Figure

Element grid of XFEM.

In equation (

In equation (

In addition to introducing the enrichment function, the XFEM also uses the horizontal set function to determine the location of the crack to improve the computational efficiency [

Plane cracks represented by two level set functions

After the displacement function of the XFEM is obtained, the virtual work principle is used to calculate the virtual work equation of the structure with a crack, and then the control equation of the XFEM for the structure with cracks can be obtained by discretizing the virtual work equation.

Similar to the conventional finite element method, the XFEM solves the nodal displacement by the overall stiffness matrix of the element and the load matrix and then obtains the nodal stress. In contrast to the conventional finite element method, the XFEM adopts different integration methods for the different types of elements when solving the element stiffness matrix. Generally, the Gaussian integral method is adopted for conventional elements. Nevertheless, for an element penetrated by a crack and a crack tip element, because their displacement functions are discontinuous, their stiffness matrices are also discontinuous. In this case, these two types of elements are usually divided into several subelements bounded by crack edges, as shown in Figure

Element division for unconventional finite elements. (a) Subelements penetrated by crack. (b) Subelements with crack tip.

In the theory of LEFM, the Paris formula is commonly used to analyze fatigue crack propagation under cyclic loads, which can be expressed as follows:

In the commercial finite element software ABAQUS, the Paris formula is expressed by the crack propagation rate and energy release rate, which can be written as follows:

According to the relationship between the energy release rate

When Δ_{th} but less than _{pl}, fatigue cracks begin to propagate. Here, _{pl} is the maximum energy release rate, which is close to the fracture toughness _{C} of steel. _{th} is the threshold of the energy release rate. In this paper, the threshold of the SIF _{th} can be obtained.

When the maximum energy release rate is greater than its threshold, fatigue cracks initiate and propagate. The crack propagation direction is the direction of the maximum shear stress of the crack tip element, and the fatigue crack propagation process is shown in Figure

Simulation flow of the fatigue crack propagation process.

A refined model of the bridge containing an initial crack of length _{0} was established. Under the ^{th} cyclic load, the maximum strain energy release rate is calculated to determine whether the crack propagates. If the maximum strain energy release rate is greater than the threshold _{th}, the crack propagates; otherwise, the crack does not propagate, and the program ends. It is assumed that the ^{th} element in the crack front along the propagation direction happens to be cracking according to the normal level set function _{i} of the ^{th} element. Virtual crack closure technology (VCCT) [_{i} of ^{th} element, and then the cyclic number increment Δ_{i} of ^{th} element can be obtained by the integral form of the Paris formula. The element with the smallest increment of the cycle number begins to crack and is assumed to be the ^{th} element. The propagation length Δ_{k} and the increment of the cycle number Δ_{k} of the ^{th} element are recorded and the crack length _{j} is updated to be equal to (_{j−1} + Δ_{k}) and the cycle number _{j} is equal to (_{j−1} + Δ_{k}). The next cycle is entered until the maximum number of cycles _{max} set by the program is reached.

A multiscale finite element model of the Runyang Yangtze River Highway cable-stayed bridge in China, as shown in Figure ^{2} m; Figure ^{0} m; and Figures ^{−3} m. Considering that the midspan of the whole bridge may be the critical location [

A multiscale finite element model of Runyang Yangtze River Highway Bridge. (a) Integral bridge model. (b) Orthotropic steel box girder including U-rib. (c) Crack at toe of top deck. (d) Crack at toe of U-rib. (e) Half of steel box girder model.

The standard fatigue vehicle given in [

Standard fatigue load [

Figure

Loading cases. (a) Longitudinal loading cases. (b) Lateral loading cases.

There are four types of fatigue cracks in the connection of the U-rib stiffened plate and top deck, as shown in Figure

Four types of fatigue cracks in the connection of U-rib stiffened plate and top deck.

The maximum principal stress nephograms for crack type. (a) Crack from the welding root of top deck (type a) and (b) crack from the welding toe of top deck (type b).

According to the stress characteristics and propagation paths of the cracks, the fracture modes of the cracks can be divided into three types: opening fracture mode, sliding fracture mode and tearing fracture mode, as shown in Figure

Three fracture modes: (a) opening fracture mode (mode I crack), (b) sliding fracture mode (mode II crack), and (c) tearing fracture mode (mode III crack).

According to the loading method mentioned in Section _{I} is shown in Figure _{0} and the length of the long axis is 2_{0}, and the ratio of _{0} to (2_{0}) is equal to 2.5 mm/10 mm, as shown in Figure

Influence line of _{I} of the crack at the welding toes of the U-rib and top deck. (a) Crack at the welding toe of the top deck. (b) Crack at the welding toe of the U-rib web.

Figure _{I} for the mode I crack at the welding toe of the top deck. The valley value of _{I} is approximately _{I} is approximately _{I}. Therefore, the fatigue crack at the welding toe of the top deck will not propagate.

Figure _{I} for the mode I crack at the welding toe of the U-rib stiffened plate. The peak value of _{I} is _{I} significantly exceeds the threshold value, indicating that the crack initiating at the welding toe of the U-rib stiffened plate has a strong driving force for crack propagation.

During the welding process, the heat at the connection of U-rib and top deck will change rapidly with time and space, which is a typical nonlinear transient heat transfer problem. In this paper, in order to consider the effect of the welding residual stress on the static crack and dynamic crack propagation, thermoelastoplastic analysis is carried out for the joint of the U-rib and top deck. A bilinear isotropic strengthening model is established, and the incremental method is used to gradually solve the temperature fields and stress fields of the U-rib and top deck. The welding residual stress is solved by thermal analysis and order of the structural analysis.

The symmetry is used to simplify the model, and half of the U-rib and top deck is selected. The finite element model and boundary conditions are shown in Figure

The finite element model and boundary conditions for welding analysis.

During the structure analysis, the

The thermodynamical and mechanical parameters of steel in the welding process analysis are given in [

Thermal-dynamical and mechanical parameters of Q345 steel in welding process analysis. (a) Thermodynamical parameters. (b) Mechanical parameters.

The equivalent stress clouds at different times of the welding process are shown in Figure

The equivalent stress clouds at different times of the welding process at the connection of the U-rib and the top deck. (a) The simulation result in this paper. (b) Results in reference [

Figure

The residual stress distribution at 1/2 section of the U-rib and top deck. (a) The longitudinal residual stress of the top deck. (b) The longitudinal residual stress of the U-rib.

According to the results of the static analysis of the fatigue crack in Section _{0} and the length of the long axis is 2_{0}, and the ratio of _{0} to (2_{0}) is equal to 2.5 mm/10 mm. The morphology of crack propagation without residual stress field and with residual stress field is shown in Figure

The morphology of crack propagation at the welding toe of U-rib. (a) Without residual stress field. (b) With residual stress field.

Figures

The cumulative strain energy release rate during the process of crack propagation is shown in Figure _{II} is close to _{III}. The ratio of _{I} to _{II} decreases with increasing cycle number. At the end of the cycle, the ratio of _{II} to _{I} is approximately 0.2. Therefore, for a cycle number of 200, the fatigue crack is a mixed-mode I-II-III crack. The influence of the mode II fracture and mode III fracture on crack propagation cannot be ignored, which will cause crack deflection in the process of crack propagation, and the deflection angle will increase with the increase in the cycle number.

The cumulative strain energy release rate during the process of crack propagation at the welding toe of U-rib.

The static analysis of the fatigue crack shows that the stress at the crack tip of the weld toe of the top deck could not drive the crack propagation when it was subjected to the most unfavorable loading conditions without a residual stress field. Therefore, to study the crack propagation at the weld toe of the top deck, it is necessary to consider the existence of a residual stress field. Here, we still assume that the short axis length of the initial semielliptical crack is _{0}, the length of the long axis is _{0}, and the ratio of _{0} to (2_{0}) is equal to 2.5 mm/10 mm. The morphology of crack propagation of the top deck is shown in Figure

The morphology of crack propagation at the welding toe of top deck: (a) simulation result in this paper; (b) result in reference [

Figure

Figure _{II} to _{I} is 0.017, and that of _{III} to _{I} is 0.035, respectively. The ratios are small but the sliding fracture mode and tearing fracture mode will cause a small deflection of the crack propagation direction.

The cumulative strain energy release rate at the welding toe of top deck. (a) Opening fracture mode (Mode I crack). (b) Sliding and tearing fracture modes (Mode II and III cracks, respectively).

Considering the influence of the residual stress field, different semielliptical initial cracks with ratios of _{0} to (2_{0}) of 2.5 mm/10 mm, 2.5 mm/15 mm, and 4 mm/10 mm are established at the welding toe of the top deck for fatigue crack propagation analysis. Figure

The morphology of crack propagation after the end of the cycle. (a) 2.5 mm/10 mm. (b) 2.5 mm/15 mm. (c) 4 mm/10 mm.

The propagation behavior of the initial crack with the first two sizes is the same, and the crack surface basically maintains a semielliptical shape during the process of propagation. Nevertheless, the crack with a ratio of _{0} to (2_{0}) of 4 mm/10 mm has a different propagation shape along the depth direction. As the welding residual stress has certain stress gradients in the directions of the thickness and length of the plate, and the stress of the mother plate close to the weld area is high, this situation leads to a deflection of the crack in the depth direction and the deflection angle gradually increases and finally causes the crack surface in the depth direction almost to be parallel to the edge of the top deck.

Figure _{0} to (2_{0}) of 4 mm/10 mm is the largest, that is, the fastest crack growth rate, followed by the initial crack with a ratio of _{0} to (2_{0}) of 2.5 mm/10 mm. Although the initial crack with a ratio of _{0} to (2_{0}) of 2.5 mm/15 mm is the longest, under 200 load cycles, the crack has the lowest crack propagation rate.

The relationship between the number of cracking elements and load cycle numbers.

The fatigue cracking problem at the weld toe of the U-rib and top deck is discussed in this paper. The fatigue crack propagation mechanism of the weld toe under a vehicle load was studied based on the XFEM, and the influence of the welding residual stress field on crack propagation was further investigated. The following conclusions were obtained:

The calculated SIF at the crack tip at the weld toe of the top deck under the action of a standard vehicle fatigue load is lower than the SIF threshold of steel; as a result, the crack will not propagate. Residual stress analysis of the welding process shows that the peak of the residual tensile stress at the U-rib and top deck joint can reach the material yield strength. After the residual stress field is introduced, the fatigue crack at the weld toe of the top deck is a mixed-mode I-II-III crack, which is dominated by the opening fracture mode. The sliding fracture mode and tearing fracture mode will lead to a small deflection of the fatigue crack to the weld throat area.

Under the action of a standard vehicle load, the crack at the weld toe of the U-rib will propagate because the stress field of the crack tip has enough driving force to drive its propagation. The crack tip extends to approximately 12.5 mm along the length of the U-rib until the stress in the stress field at the new crack tip is reduced to a new low level, and the fatigue crack stops. After the residual stress is taken into account, the fatigue crack extends in the directions of the crack length and depth, and the angle between the crack and weld length increases with increasing crack propagation.

Under a low cyclic load, the initial crack size at the weld toe of the top deck in the residual stress field will affect the crack propagation rate. Compared to the crack length, the initial crack depth has a larger effect on the rate, and the size of the initial depth will have a direct impact on the rate.

The data used to support the findings of this study are included within the supplementary information files.

The authors declare that they have no conflicts of interest.

The works described in this paper are substantially supported by the grant from the National Natural Science Foundation of China (grant no. 51678135); the Natural Science Foundation of Jiangsu Province (no. BK20171350); the Fundamental Research Funds for the Central Universities (no. 2242016R30009); the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Top-Notch Academic Program Project of Jiangsu Higher Education Institutions (TAPP); and the Six Talent Peak Projects in Jiangsu Province (JNHB-007), which are gratefully acknowledged.

S1: the original data used to plot the influence line of KI of the crack at the welding toe of top deck in Figure