Residual Stress and Fatigue Strength Analysis of Stiffener Welds of Steel-Plate Composite Girder Bridge Considering Welding Sequence

Abstract

Welding residual stress will aggravate the fatigue cracking damage of the structure and have an essential impact on the structure’s load-bearing capacity. The welding sequence will directly affect the size and distribution of welding residual stress. To this end, this paper establishes a thermal–mechanical sequential indirect-coupling finite-element analysis model, researches the residual stress of steel-plate composite girder bridges considering the welding sequence, and verifies the analysis results through field tests. Then, a three-span steel-plate composite continuous girder bridge was taken as the research object, and the residual stress of the stiffener welds in seven welding sequences was analyzed. On this basis, the equivalent peak-stress method is used to evaluate and predict the fatigue strength of the weld. The research results show that welding residual stresses change the multiaxial stress state of fatigue details. Although under the same external load cyclic stress, the difference in welding sequence directly leads to a significant difference in the equivalent peak stress of the stiffeners, and this difference results in different fatigue properties of the stiffeners. The research results can provide a basis for the welding process and fatigue analysis of stiffener welds in steel-plate composite girder bridges.

1. Introduction

Steel-plate composite girder bridges have the advantages of being lightweight, having a simple structure, convenient fabrication, and construction, etc., and are widely used in short- and medium-span bridges. It is one of the main bridge types in the United States, Japan, and other countries [1,2,3,4]. Stiffening ribs are an essential structural measure to ensure the stability of the main beam of the steel-plate composite girder bridge. They are vertically connected to the main beam’s top, bottom, and web plates through welding. Such non-load-bearing components can create unexpected fatigue problems due to welding residual stresses. Among them, the weld cracking at the connection between the stiffener and the roof is an atypical cyclic pressure fatigue cracking problem [5,6,7], as shown in Figure 1. This kind of disease is especially prevalent in steel-plate composite girder bridges with a small girder system that has been vigorously promoted by Japan and China [8,9,10]. However, fatigue analysis of this kind of local welded joint is complex and has not been effectively solved at the design level since its discovery. The top and bottom edges of the stiffeners are generally welded manually. There is randomness in the construction process, and different welding sequences will directly affect the distribution and size of welding residual stress [11,12,13,14]. Therefore, it is necessary to start from the fatigue detail cracking mechanism caused by welding, explore the influence of the welding process on the residual stress and fatigue strength of the weld at the upper edge of the stiffener, and put forward the optimization strategy of the welding process in a targeted manner to have the opportunity to solve this fatigue problem fundamentally.

Regarding the influence of the welding process on joint residual stress and fatigue performance, existing research mainly focuses on flat-plate butt joints, T-shaped joints, and round-pipe butt joints. For example, Teng et al. [17] used the finite-element method to study the effects of welding speed, preheating before welding, etc., on the welding residual stress of flat butt joints. They found that the magnitude of residual stress decreased as the welding speed increased, and preheating before welding could effectively reduce the peak residual stress. Xu et al. [18] studied the effects of welding heat input, preheating temperature, etc., on the welding residual stress of steel-pipe butt joints. The results show that the increase in welding heat input increases the peak stress of the steel-pipe joint’s radial, circumferential, and axial residual stress, while preheating before welding reduces the peak stress of the residual stress. Zhao et al. [19] and Akbari et al. [20] studied the influence of welding heat input and other factors on the welding residual stress of dissimilar steel-pipe butt joints. It was found that, as the welding heat input decreases, the residual compressive stress on the stainless-steel side of the steel-pipe joint and the peak residual tensile stress on the outer surface of the steel pipe both decrease. Zhao et al. [21] studied the effect of post-weld heat treatment on the welding residual stress of T-shaped and Y-shaped joints through experiments. The test showed that post-weld heat treatment can significantly reduce the level of welding residual stress. Nassiraei et al. [22,23] conducted a numerical analysis and experimental research on the stress concentration and static strength of T/Y-type pipe joints. It can be seen that different welding processes can dramatically affect the distribution and size of the welding residual stress, thereby affecting the fatigue performance of the joint. However, the current research on the residual stress of welded joints of composite beam stiffeners has not been reported, even though scholars have carried out some direct fatigue tests to explore the fatigue detail strength of the welds at the upper edge of the stiffeners [24,25,26]. However, the research method adopted cannot essentially explain its cyclic pressure cracking mechanism and cannot provide an adequate basis for a quantitative analysis of its fatigue strength.

This paper relies on the actual engineering of steel-plate composite girder bridges by Zhejiang Jiaogong Equipment Engineering Co., Ltd., Zhoushan, China. Based on the established thermal sequential indirect-coupling finite-element analysis method, numerical simulates the welding temperature field and residual stress field of the steel-plate composite girder bridge stiffeners. It verifies the correctness and accuracy of the finite-element analysis method through field-test results. The permutation and combination method was used to numerically simulate the residual stress-distribution characteristics of seven typical stiffener welding sequences. Finally, taking the three-span steel-plate composite continuous girder bridge in the general diagram of the assembled I-beam composite beam steel bridge as the research object, the stiffener welding temperature field and residual stress field were numerically simulated. Analyze the influence of different welding sequences on the welding residual stress field of steel-plate composite girder bridges. In addition, taking the residual stress at the corners of the stiffeners as the research object, the fatigue strength of the welds at the upper edge of the stiffeners of the steel-plate composite girder bridge was evaluated. The research results provide suggestions for the welding process of stiffeners of steel-plate composite girder bridges and provide theoretical basis and technical support for fatigue detail optimization design and anti-fatigue design methods of steel bridges.

The remainder of this paper is organized as follows. Section 2 introduces the basic theory of the welding heat-stress analysis. Section 3 presents the residual stress analysis method and verifies it through field experiments. Based on this, the residual stress distribution of different welding sequences of stiffeners is analyzed. Section 4 describes and presents the fatigue strength analysis of the upper flange of a steel-plate composite beam bridge considering the effect of residual stresses. Finally, conclusions are drawn in Section 5.

2. Basic Theory of Welding Heat-Stress Analysis

2.1. Welding Temperature Field

This paper uses a double ellipsoid heat-source model to represent the heat generated by the welding gun and molten liquid during the welding process. During the welding process, only considering $\rho ,K,c$ as the temperature function [25,26], the three-dimensional transient heat conduction control partial differential equation derived from the thermal balance equation follows.

$$\frac{\partial }{\partial x}\left(K\left(T\right)\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(K\left(T\right)\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(K\left(T\right)\frac{\partial T}{\partial z}\right)+Q=\rho \left(T\right)c\left(T\right)\frac{\partial T}{\partial t}$$

(1)

In the formula, $T$ is the temperature, $K$ is the thermal conductivity, $c$ is the specific heat capacity, $\rho$ is the density, and $Q$ is the heat per unit volume.

According to the characteristics of the double ellipsoid heat source, the heat can be divided into the front and rear parts, as shown in Figure 2.

The expression for the first half of the ellipsoidal heat source is:

$$q_1\left(x,y,z,t\right)=\frac{6\sqrt{3}Q{f}_1}{ab{c}_1{\mathsf{\pi }}^{\frac{3}{2}}}\exp \left(-3\left[\frac{{(x-vt)}^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c_1^2}\right]\right)$$

(2)

The expression for the second half of the ellipsoidal heat source is:

$$q_2\left(x,y,z,t\right)=\frac{6\sqrt{3}Q{f}_2}{ab{c}_2{\mathsf{\pi }}^{\frac{3}{2}}}\exp \left(-3\left[\frac{{(x-vt)}^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c_2^2}\right]\right)$$

(3)

In the formula, $q_1$ and $q_2$ represent the heat flow distribution in the front and rear hemiellipsoids. $(x,y,z)$ indicates the location of the heat source. $Q$ represents the effective heat input, $Q=\eta UI$. Among them, $\eta$ is the thermal efficiency, which is taken as 0.9; $U$ is the arc voltage, and $I$ is the current. $f_1$ and $f_2$ represent the energy-distribution coefficient before and after the double ellipsoid heat source, $f_1+f_2=2$. $a$, $b$, and $c$ represent the three semi-axes of $x$, $y$, and $z$ of the double ellipsoid heat source.

2.2. Welding–Loading Coupling Stress Field

The stress field is solved using the thermoelastoplastic method. During the analysis, only the weak coupling of the temperature field and the stress field is considered. That is, only the influence of the temperature field on the stress field is considered. When the temperature changes, the constitutive relationship of stress and strain is calculated using Formula (4) [21]. After each temperature increase, the stress increment $d\sigma$ on each node is obtained, thereby obtaining the change process of the dynamic stress field during the entire welding process.

$$d\sigma =D_{ij}\left(d{\epsilon }_{ij}^e+d{\epsilon }_{ij}^p+d{\epsilon }_{ij}^T\right)$$

(4)

In the formula, $d\sigma$ is the stress increment. $D_{ij}$ is the tensor coefficient. $d{\epsilon }_{ij}^e$, $d{\epsilon }_{ij}^p$, and $d{\epsilon }_{ij}^T$ are the elastic strain increment, plastic strain increment, and thermal strain increment, respectively.

2.3. Welding Residual Stress–Load Coupling Stress Field

Under the action of welding residual stress and vehicle cyclic loads, continuous loading and unloading cause the structure to produce a ratcheting effect. Therefore, local plastic yielding is prone to occur at the welded joint. When the stress generated by cyclic loading is superimposed on the welding residual stress, and the stress value exceeds the steel yield strength, the steel will harden. The nonlinear kinematic hardening model accurately simulates steel’s reverse cyclic plastic loading conditions [24]. The kinematic hardening rules in this model are as follows.

$$\mathrm{d}a=\frac{2}{3}Cd{\epsilon }^p-\gamma adp$$

(5)

$$\mathrm{d}p=\left|\mathrm{d}{\epsilon }^p\right|={\left[\frac{2}{3}\mathrm{d}{\epsilon }^p·\mathrm{d}{\epsilon }^p\right]}^{\frac{1}{2}}$$

(6)

In the formula, $a$ is the deviatoric backstress tensor. $C$ and $\gamma$ are parameters calibrated by cyclic loading experiments. $dp$ is the magnitude of the plastic strain increment tensor. $d{\epsilon }^p$ is the plastic strain increment tensor.

3. Residual Stress Analysis in Stiffener Welds of Steel-Plate Composite Beams

3.1. Analytical Approach of Residual Stress

ABAQUS 2021 finite-element software was used to analyze the welding residual stress field of the stiffeners of the steel-plate specimen. In order to correctly describe the welding thermal cycle process, a heat-source model with a double ellipsoid function distribution is used, and the subroutine function is used to realize the movement of the heat source. Mobile heat-source simulation includes two essential contents: the application of the heat-source model and the execution of the life and death unit. It is implemented by writing a Fortran-based Dflux subroutine and Python 3.11.3-based script programs.

The temperature-field results at each load step or time point of the welding thermal analysis are used as the thermal load of the mechanical analysis to solve the thermal elastic–plastic problem. In the stress analysis, the calculation model is the same as in the thermal analysis, but the thermal analysis’ solid unit DC3D8 is replaced by the structural analysis’ unit C3D8R. The welding heat-stress sequential coupling process is shown in Figure 3. The calculation process assumes that the material is isotropic, the constitutive relationship adopts the bilinear isotropic strengthening (BSIO) model, and the plastic behavior complies with the Von Mises criterion. In order to ensure the accuracy of numerical simulation, the characteristics of the material’s thermophysical properties and mechanical properties that change with temperature are considered during the calculation process. At the same time, except for the yield strength, it is assumed that the weld metal and the base metal have the same thermophysical and mechanical property parameters.

The material of the calculation model is Q355D steel, and its thermodynamic properties are related to temperature. It is assumed that the material’s physical properties (melting point and above) remain unchanged at high temperatures. The values of the parameters, such as specific heat cp, density ρ, thermal conductivity k, thermal expansion coefficient α, Poisson’s ratio μ, and elastic modulus E of the material at different temperatures, are shown in

Table 1. Mechanical property parameters of steel at room temperature.

T (°C)	k W/(m°C)	ρ (kg·m−3)	cp KJ/(kg°C)	μ	α 105 (°C−1)	E (GPa)	${\mathit{\sigma }}_{\mathit{s}}$ (MPa)
20	50	7820	460	0.28	1.10	205	350
250	47	7700	480	0.29	1.22	187	264
500	40	7610	530	0.31	1.39	150	201
1000	30	7490	670	0.40	1.34	20	12.3
1500	35	7350	660	0.45	1.33	2	1
2000	45	7300	750	0.48	1.32	1.5	1

. The simulated welding current is 235 A. The voltage is 23 V. The welding speed is 10 mm/s. The weld toe hf is 10 mm, and the welding sequence is shown in Figure 4a. The initial temperature is room temperature, which is 20 °C.

The model size is consistent with the size of the steel-plate composite beam specimen. The top plate is 60 × 300 × 28 mm. The base plate is 700 × 300 × 36 mm. The web is 3036 × 300 × 20 mm. The longitudinal stiffener is 2900 × 220 × 16 mm. The transverse stiffener is 3024 × 220 × 16 mm. According to the size of the steel-plate specimen used in the test, the finite-element model shown in Figure 4b was established. A subroutine was written using Fortran to realize the movement of the heat source in the weld area and simulate the welding process. After testing, the current of the simulated weld is 235 A, the voltage is 23 V, the welding speed is 10 mm/s, the welding toe hf is 10 mm, and the welding sequence is shown in Figure 4a. A local model of the test beam was used for research to conduct a refined analysis of the fatigue stress of the steel-plate composite beam specimen welds. The regional model size is 300 × 600 × 500 mm, as shown in Figure 4e. Since the stress near the weld is relatively large, the mesh is densified at the transverse weld position, with a fine mesh size of 1 mm and a global mesh size of 4 mm. The number of units in the local model is 321095. The fatigue load is simulated using a single-vehicle model, and the nodes on the regional model boundary inherit the displacement of the global model.

3.2. Approach Verification Based on Measurements

According to the principle of acoustic elasticity [27,28,29], the residual stress in the material will affect the propagation speed of ultrasonic longitudinal waves. When the direction of the residual stress is consistent with the direction of the longitudinal wave, the tensile stress will slow down the propagation speed of the ultrasonic longitudinal wave or extend the propagation time t, and the compressive stress will speed up the propagation speed of the ultrasonic longitudinal wave or shorten the propagation time t. Therefore, on the premise that the distance between the excitation and receiving transducers (probe spacing) remains unchanged if the ultrasonic propagation time $t_0$ corresponds to zero stress ${\sigma }_0$ and the ultrasonic propagation time t corresponds to the stress $\sigma$ of the tested piece, it is measured. According to the time difference, the absolute value $\sigma$ of the residual stress in the tested specimen can be calculated according to Formula (7) or Formula (8); that is:

$$\sigma -{\sigma }_0=K(t-t_0)$$

(7)

$$c=K\Delta t$$

(8)

In the formula, $\Delta \sigma$ is the change in residual stress (stress difference), $\Delta \sigma =\sigma -{\sigma }_0$. $\Delta t$ is the change in propagation time (acoustic time difference), $\Delta t=t-t_0$. $K$ is the stress coefficient, which is related to the material of the test piece and the distance between the probes. It can be obtained through tensile test calibration.

The test equipment in this article is the Jianwei ultrasonic stress meter. The model is USG-S2-AN-P 1.3. The time domain resolution is 0.37 ns. The sampling accuracy is 12 bit. The center frequency of the measurement probe is 5 MHz. The chip diameter is 6 mm. The transceiver separation method is used, and the incident angle of the matching block is 23°. The stress coefficient is based on the value of “Non-Destructive Testing-Test Method for Measuring Residual Stress Using Ultrasonic Critical Refracted Longitudinal Wave” GB/T 32073-2015 [30]. Five locations of stiffener welds of steel-plate composite beams were selected for residual stress testing, with 23 measuring points tested at each location. The layout of the measuring points is shown in Figure 5. According to the test specification requirements, the vertical and horizontal distances between each measuring point are 2.5 cm. Before the test, the base wave collection and zero-stress calibration work are carried out, as shown in Figure 6. The results of the field test of the residual stress of the specimen are shown in Figure 7, where Area 1 represents the front side of the transverse stiffener steel plate (the direction shown in Figure 5), and Area 2 represents the backside.

Focus on testing and analyzing the Area 1 surface, where the residual stress gradient changes significantly. Figure 8 shows the simulated calculation cloud diagram of the residual stress of the stiffener weld. Path 1 to Path 4 are four stress-distribution paths, respectively. It can be seen that the maximum stress generated by welding is in the weld zone, and away from the weld zone, the residual stress gradually decreases. In the figure, S11 is the stress in the horizontal direction, and S22 is the stress in the vertical direction.

It can be seen from the numerical analysis results in Figure 9 that, on the path parallel to the weld direction, due to the welding effect, there are residual stress changes in both the stiffener area and the weld area. The maximum stress value is at the opening position at the intersection of the top plate and the web plate, and the stress value changes more obviously in the weld area. This phenomenon is consistent with the field-measurement results in Figure 7, especially the residual stress S11 in the horizontal direction. The closer to the center position along the X direction, the greater the tensile stress. In addition, a comparison between the numerical analysis results and the test results shows that, for Path 1, the final residual stress tends to be around 20 Mpa as it gets farther from the center. For Path 2, the final residual stress tends to be around 10 Mpa, which is consistent with the on-site measured residual stress results. Through on-site testing and numerical analysis of the residual stress in the stiffener welds of steel-plate composite beams, it can be seen that the welding process will produce a sizeable residual stress field in the stiffener area. At the same time, the correctness of the numerical simulation method of stiffener weld residual stress in this paper has been verified.

3.3. Residual Stress under Different Welding Sequences

Different welding sequences impact the distribution of the residual stress field and the stress peak value. The effects of different welding sequences on residual stress are discussed based on the verified finite-element analysis model of stiffener welds. According to the possible sequence of the stiffener welding process, seven typical welding sequences (case 1 to case 7) were determined using permutations and combinations, as shown in Figure 10. The arrows and numbers indicate the welding direction and welding sequence, respectively.

Figure 11 shows the distribution of residual stress on the front and back sides of the stiffener along the horizontal direction of the weld toe that corresponds to seven typical welding sequences. Figure 12 is a calculation cloud diagram of seven welding sequences. The calculation results show that the stress peak value of residual stress along the Path 3 direction changes significantly, and the maximum tensile stress peak value and the maximum compressive stress peak value appear at both ends of the weld toe. The residual stress in the middle of the weld is uniformly distributed, manifesting as residual tensile stress with a stress value of 300 MPa.

From the analysis results of cases 1 and 2 and cases 6 and 7, it can be seen that, when the welding sequence is anti-symmetrical, the longitudinal stress-distribution trend is consistent, indicating that the difference in welding direction will not affect the residual stress. The analysis results of cases 1 to 5 and cases 6 and 7 show that, when both ends are welded first, there is residual compressive stress at the outermost corner of the stiffener, with a stress value of 400 MPa. This result favors the location where the stiffeners intersect the roof plate. It can be seen from the results of working cases 1 and 2 that, when the starting welding point is on the outside of the stiffener, the outermost corner point position is in a state of tension on one side and compression on the other side. The peak stress on the tensile side reaches about 170 MPa. It can be seen from the results of working cases 6 and 7 that, when the welding point is on the inside of the stiffener, the outermost corner point appears to be in tension, and the stress peak reaches more than 270 MPa. Under the cyclic compressive stress of the vehicle load, the weld at the upper end of the stiffener will first initiate cracks at the corner point, and then, the crack will expand horizontally along the straight line of the weld toe. This is also the fundamental reason fatigue cracking is prone to occur at weld locations.

Table 2. Stress results at corner position S22 (MPa).

Cases	Front Side	Back Side
1	−97.68	−424.27
2	−421.39	−101.63
3	−441.82	−476.94
4	−472.91	−442.38
5	−359.89	−491.33
6	−104.45	297.47
7	273.86	−110.80

shows the calculation results of the residual stress at the corners of the stiffeners under seven typical welding sequences. For cases 1 to 5, there is a significant residual compressive stress at the corner point of the stiffener. The specific compressive stress magnitude is shown in Table 2. This is beneficial to the structure to avoid cracking. For working cases 1 and 2, one side of the corner point of the stiffener is compressive stress, and the other side is residual tensile stress. For working cases 6 and 7, there are residual tensile stresses on both sides of the stiffeners, which will lead to the occurrence of structural fatigue cracks.

4. Fatigue Strength Assessment of Welds on the Upper Edge of Stiffeners in Steel-Plate Composite Girder Bridges

4.1. Stiffener Local Stress Analysis Method under the Joint Action of Welding and Fatigue Loads

This paper takes the 3 × 40 m two-way four-lane composite I-shaped continuous girder bridge in the “General Drawing of Prefabricated I-Type Composite Beam Bridge” [31] as the research object. The weld’s fatigue strength at the stiffener’s upper edge is evaluated by considering the effect of welding residual stress. The bridge is 26 m wide, and the main beam adopts a combined structure of I-shaped steel beams + concrete bridge decks. The beam height is 2.2 m, and transverse connections are set between each steel beam. The entire bridge’s steel is made of Q370qD. The thickness of the stiffeners at the abutment is 18 mm, and the stiffeners at the piers are 25 mm. The stiffeners at other locations are 12 mm or 16 mm and are welded to the upper and lower flanges of the main beam. The design load is the highway-level I load of the “Specifications for Design of Highway Steel Bridge” (JTG/T D64-01-2015) [32], which provides three load models. For the weld details on the upper edge of the stiffener, the cracking is caused by the deformation of the bridge deck system, so the fatigue model III is used for its fatigue analysis.

ABAQUS software was used to establish the full-bridge finite-element analysis model and the local sub-model, considering the most unfavorable stress position of the welding residual stress effect. In the full-bridge model, the main beam is simulated using S4R shell elements with a mesh size of 50 mm. The steel bars are simulated using T3D2 beam elements, and the concrete slab is simulated using C3D8R hexahedral elements with a mesh size of 100 mm. In the local sub-model, the mesh size in the weld area is 1 mm, and the mesh size away from the weld area is 6 mm. The welding material properties are consistent with Table 1. The whole bridge has a total of 4,854,579 nodes and 4,923,923 elements. Figure 13a,b are, respectively, the elevation and cross-sectional view of the finite-element model of one span. To analyze the fatigue strength of the stiffener weld, fatigue load model III is applied to the entire bridge according to “JTG/T D64-01-2015”. Fifty working conditions are set in the longitudinal bridge direction, and 36 are set in the transverse bridge direction. The iterative calculations are performed sequentially to determine the location of the most unfavorable working conditions, as shown in Figure 13g. The maximum compressive stress calculation result is 39.2 MPa. Taking the unfavorable load position as the research object, a sub-model for fatigue analysis of stiffener welds is established, as shown in Figure 13h. The temperature-field calculation results of the full-bridge model are inherited from the sub-model as a heat source. The calculation results of the sub-model temperature field and vehicle load stress field are shown in Figure 13i,j. The calculation results of the temperature and vehicle load coupled stress fields of the sub-model are shown in Figure 13k.

4.2. Fatigue Assessment of Stiffener Corner Spots Based on Equivalent Peak-Stress Method

The fatigue strength of stiffener corner points is analyzed using the equivalent peak-stress method [33,34]. In the fatigue design framework of welded structures, the equivalent peak-stress method is a method that uses finite elements to estimate the notch stress intensity factor quickly. It requires establishing a mapping mesh containing only solid elements in the seam welding area, which can be achieved through the sub-model technology in ABAQUS. Based on the peak-stress value calculated by the finite-element model, the fatigue effect that the model can withstand is detected. The stiffener weld toe can be simplified into a V-shaped notch with a tip radius of zero. The polar coordinate angle $\theta$ starts from the bisector of the notch angle, and the opening angle is approximately 2α = 135°, as shown in Figure 14. According to reference [33], the finite-element mesh size used in the calculation of equivalent peak stress is calibrated to 1 mm. Therefore, the mesh size of the weld area in the finite-element modeling of this paper is 1 mm. The core principle of this method is to use the critical strain energy density to define the control radius of the stress intensity factor at the singular point. It uses the principle that the stress distribution at the weld toe under polar coordinates is the superposition of the symmetric stress field and anti-symmetric stress field and derives the stress intensity factor of the crack represented by the stress component of the singular point angle bisector, such as open type (I type), slip type (Type II) and tear type (Type III). This article focuses on the plane stress fatigue problem at the corners of stiffeners with complex boundaries, which belongs to the type I + II cracking mode.

The calculation formula of equivalent peak stress is as follows:

$$\Delta {\sigma }_{\mathrm{eq},\mathrm{peak}}=\sqrt{c_{w1}·f_{w1}^2·\Delta {\sigma }_{\theta \theta }^2+c_{w3}·f_{w3}^2·\Delta {\tau }_{\theta z}^2}$$

(9)

In the formula, $\Delta {\sigma }_{\theta \theta }$, $\Delta {\tau }_{\theta z}$ represents the stress component calculated along the notch bisector, which means the normal stress along the $\theta$ direction and the shear stress along the $z$ direction (as shown in Figure 14). According to the plane stress state, its stress components can be obtained. $C_w(R)$ and $f_{wi}$ represent the stress effect coefficient and correction parameter, respectively. The calculation formula is as follows:

$$C_{\mathrm{w}}(R)=\left\{\begin{array}{cc}\frac{1+R^2}{{(1-R)}^2}\hfill & -1\le R\le 0\\ \frac{1-R^2}{{(1-R)}^2}\hfill & 0\le R<1\end{array}\right.$$

(10)

$$f_{wi}=K_{FE}·\sqrt{\frac{2e_i}{1-v^2}}·{\left(\frac{d}{R_0}\right)}^{1-\lambda i}$$

(11)

In the formula, $R$ represents the stress ratio. Considering the influence of residual stress, $R$ is equal to the ratio of the residual stress in the $\theta$ direction to its coupled stress under the vehicle load. $C_w(R)$ can be calculated from this. Reference [33] calibrated $f_{wi}$ at the opening angle 2α = 135°, which is $f_{wi}$ = 1.064 and $f_{wi}$ = 1.877.

First, ${\sigma }_{y,w}$ is the initial stress field formed during welding, as shown in Figure 13i. ${\sigma }_{y,w+v}$ is the coupled stress field formed by the interaction of the initial and moving load stress, as shown in Figure 13k. Calculate ${\sigma }_{\theta \theta }$ and ${\tau }_{\theta z}$ according to Mohr’s stress circle and obtain the equivalent peak stress $\Delta {\sigma }_{\mathrm{eq},\mathrm{peak}}$ through Formula (9). The calculation results of residual stress fields for seven different welding sequences are shown in Figure 15. It can be seen from the calculation results that cases 3 to 5 are generally in a compressed cyclic stress state (* indicates that no cracking occurs at the upper edge of the stiffener under this case). Cases 1, 2, and 6 are in a tensile cyclic stress state due to residual tensile stress. Case 7 is in the tension–compression cyclic stress state. It can be seen that the initial residual stress field in the stiffener will change with the action of fatigue load and become stable as the number of cycles increases. The fatigue strength calculation results of steel-plate composite bridge stiffeners are shown in

Table 3. Calculation results of fatigue strength at corner spot (MPa).

Cases	$\mathbf{\Delta }{\mathit{\sigma }}_{\mathit{y}}$	$\mathbf{\Delta }{\mathit{\tau }}_{\mathit{x}\mathit{z}}$	${\mathit{C}}_{\mathit{w}}(\mathit{R})$	$\mathbf{\Delta }{\mathit{\sigma }}_{\mathit{\theta }\mathit{\theta }}$	$\mathbf{\Delta }{\mathit{\tau }}_{\mathit{\theta }\mathit{z}}$	$\mathbf{\Delta }{\mathit{\sigma }}_{\mathbf{eq},\mathbf{peak}}$	N (Cycles)
1-1	73.57	0.81	4.76	53.19	10.63	133.17	21.96 × 106
2-2	149.17	0.76	1.98	107.98	15.01	173.07	10.01 × 106
* 3-2	147.10	2.51	6.25	109.31	19.98	308.92	-
* 4-2	146.97	2.50	5.64	109.21	19.99	303.09	-
* 5-2	147.21	2.55	4.29	109.40	20.01	263.41	-
6-2	147.18	0.50	2.87	101.28	16.56	189.80	7.59 × 106
7-2	147.24	0.03	0.68	111.65	17.40	126.57	25.58 × 106

Note: * indicates that no cracking occurs at the upper edge of the stiffener under this case.

. It is shown through stress tensor analysis that residual stress changes the multiaxial stress state of fatigue details. Although under the same external load cyclic stress, there are significant differences in the equivalent peak stress calculated in different welding sequences. This results in considerable variations in the fatigue performance of stiffener upper welds.

Reference [35] gives the peak stress S-N curve of a 97.7% survival rate under type I + II loading conditions, $\Delta {\sigma }^{3}N$ = 5.19 × 10¹³. Based on this, the number of cycles for cases 2 and 6 is calculated to be 10.01 million times and 7.59 million times. Assuming that the total traffic flow of the bridge is two million vehicles/year, the crack initiation life of this fatigue detail is about 4 to 6 years, showing that cracking occurs extremely easily at the corner spot welds of the stiffeners. Therefore, the fatigue details focus on the construction sequence of the welding process.

5. Conclusions

The existence of residual stress in the stiffener welds of steel-plate composite girder bridges will directly affect the accuracy, mechanical performance, and fatigue performance of the structure. However, the stiffener welding process is complex, and there are many welding paths. So, residual stress inevitably appears under the influence of multiple uncertain factors. This paper uses numerical analysis and experimental verification methods to study the influence of the welding sequence of steel-plate composite girder bridges on the residual stress of stiffener welds. The main conclusions are as follows:

(1) Based on the heat-source model of double ellipsoid function distribution and using the ABAQUS subroutine function, a thermal sequential indirect-coupling analysis model for the simulation of the residual stress field of the stiffener weld of the steel-plate composite beam was established. The numerical model was calibrated through the test results of field tests to verify the correctness of the finite-element analysis results;

(2) Numerical analysis results show that, when the welding sequence is antisymmetric, the weld’s residual longitudinal stress-distribution trend is the same, and the welding sequence does not change the stress value. When both ends are welded first, residual compressive stress is generated at the most unfavorable position of the corner point of the stiffener, and the structure does not crack;

(3) Welding residual stresses change the multiaxial stress state of fatigue details. Although under the same external load cyclic stress, the difference in welding sequence directly leads to a significant difference in the equivalent peak stress of the stiffeners, and these mechanics lead to considerable variations in the fatigue performance of stiffener upper welds;

(4) The results based on the equivalent peak-stress method show that, for cases 2 and 6, when the stress amplitude is 100 MPa, the cycles are 10.01 million and 7.59 million. Assuming that the traffic flow of the bridge is two million vehicles/year, the crack initiation life of this fatigue detail is about 4 to 6 years, showing that it is straightforward to crack. Therefore, the details of fatigue should focus on the construction sequence of the welding process.