Due to the complex structure and dense weld of the orthotropic steel bridge deck (OSBD), fatigue cracks are prone to occur in the typical welding details. Welding residual stress (WRS) will cause a plastic zone at the crack tip. In this paper, an elastoplastic constitutive model based on the Chaboche kinematic hardening model was introduced, and the extended finite element method (XFEM) was used to study the influence of material elastoplasticity and crack tip plastic zone on the law of fatigue crack propagation. By judging the stress state of the residual stress field at the crack tip and selecting different crack propagation rate models to investigate the crack propagation law when plastic deformation was considered, the propagation path and propagation rate of fatigue crack of the OSBD were obtained. The results show that, whether the residual stress field is considered or not, the plastic deformation at the crack tip will not cause the obvious closure of the fatigue crack at the U-rib toe during the crack propagation process, but will significantly affect the crack propagation path. When material plasticity is considered, the propagation angle of fatigue crack at the U-rib toe basically remains unchanged along the short-axis direction of the initial crack, but is going up along the long-axis direction, and the crack tip plastic zone inhibits the propagation of the crack tip on one side. Compared with linear elastic materials, the crack propagation law considering material plasticity is more consistent with that in actual bridge engineering. In terms of the propagation rate, if the residual stress field is not considered, the fatigue crack propagation rate at U-rib toe with plasticity considered is slightly higher than that without plasticity considered, because plastic deformation will affect the amplitude of energy release rate. When considering the WRS field, the fatigue crack propagation rate at U-rib toe is increased due to the combined actions of plastic deformation and stress ratio

Due to its characteristics of light deadweight and high strength-to-weight ratio, orthotropic steel bridge deck (OSBD) have been widely used in large and medium span bridges [

The results show that the stress intensity factor _{I}_{eff}_{I}

Therefore, for the whole structure of the bridge, this paper focuses on the change of fatigue crack propagation law caused by WRS field and crack tip plastic zone. Considering that welding toe of U-rib at the connection of U-rib and diaphragm was significantly affected by WRS, the fatigue crack propagation law of U-rib-toe in the whole bridge structure was mainly studied in this paper. To this end, a Multiscale finite element model of long-span bridge is firstly established, which includes the cables, bridge towers and equivalent steel box girder models at non-critical locations with the characteristic length of 10^{0}–10^{2} m, refined steel box girder model including the top deck, bottom deck, inclined web, diaphragm and U-shaped stiffener with the characteristic length of 10^{−2}–10^{0} m, and the weld model and the fatigue crack models with the characteristic length of less than 10^{−3} m. For finite element models with different characteristic lengths, different physical equations are used to describe them. Linear elasticity theory is used to describe the whole structure model and the local component model. For the welding area, the elastoplastic theory is adopted. Fracture mechanics and XFEM are used to describe the local crack model. Then, the residual stress field was simulated based on commercial software ABAQUS. Finally, the XFEM is used to study the propagation law of fatigue crack at U-rib toe under the action of vehicle load alone and the residual stress field and vehicle load together when the material near the crack is plastic.

The direct cyclic loading method in ABAQUS software cannot directly output the energy release rate at the beginning of each cycle, that is, when the cyclic load is zero, the energy release rate is not available. When the influence of WRS on fatigue crack propagation is considered, there is an initial WRS in the structure. However, the fatigue load applied to the bridge deck at this time is 0, so ABAQUS cannot output the energy release rate generated by initial WRS. In order to calculate the energy release rate generated by WRS, the interaction integral theory is presented here, in which the stress intensity factor at the beginning and end of each loading cycle is calculated so as to further obtain the energy release rate. The energy release rate induced by fatigue crack propagation can be obtained by subtracted the energy release rate caused by WRS from the energy release rate caused by fatigue load peak. Interaction integral is a new calculation method developed on the basis of

The location of the crack tip should be determined first when the interaction integral is used to solve the stress intensity factor. In the XFEM, the position of the crack surface is traced and defined by the level set function. The normal horizontal set function represents the vertical distance between the element node and the crack face. The extended finite element module in ABAQUS software can directly output the value of the node’s normal level set function, but cannot directly output the coordinates of the crack surface. Instead, the coordinates of the crack tip in the process of crack propagation can only be calculated indirectly according to the node’s normal level set function, and then the location of the crack surface can be determined.

For OSBD, arbitrary hexahedral eight-node elements are adopted in this paper, as shown in

where,

After the coordinate value of the crack tip is obtained,

where

where _{ijkl}

In fracture mechanics theory, the relation between

where

Based on the ^{int} superposed by the real field and the auxiliary field at the crack tip, as shown in

where ^{aus} is the

The auxiliary field of the crack tip is obtained according to Westergaard stress function, and its displacement field is shown in

Among them:

According to

According to

The relationship between interaction integral

Stress intensity factor of the real displacement field can be got by controlling _{(1)} can be got when _{(2)} can be got when _{(3)} can be got when

The discrete integral formula for calculating the

where,

According to _{1} was obtained; when _{2} is obtained; when _{3} is obtained. The _{1}, _{2} and _{3} are obtained by integrating the 8 Gauss integral points of all elements in the integral domain respectively, then the interaction integral _{3} − _{2} − _{1}. By using different virtual displacement fields _{1}, _{2} and _{3} of the three crack types can be obtained, and the stress intensity factors

To be sure, the singularity of the crack tip cannot be simulated in this paper, which is due to the limitations of the extended finite element module in the commercial software ABAQUS. The extended finite element method in ABAQUS omits the third term of the extended finite element formula which can simulate the crack tip singularity, so the final calculation result does not simulate the crack tip singularity in theory.

In the commercial software ABAQUS, fatigue crack initiation can be determined by the increase of energy release rate _{max}_{th}

where, _{1} and _{2} are material constants, and _{max}_{pl}_{pl}_{C}

Under the influence of WRS, the fatigue crack propagation law can be obtained by Walker [

where _{max}_{min}_{r}^{−12},

The influence of stress ratio _{op}

The material constants

There are many methods for the solution of crack opening stress intensity factor _{op}

(1) Nodal displacement method [

where, _{res}

When considering the WRS field of components, especially the welding residual compressive stress at the crack tip, the fatigue crack propagation life can be analyzed based on the Elber crack closure theory, Because Elber’s theory that the plastic deformation of the crack tip generates the residual compressive stress and the crack closure is essentially the same as the external compressive WRS field causes the crack closure [

In the finite element software ABAQUS, the energy release rate is taken as the parameter to analysis the fatigue crack propagation. In the fatigue crack propagation rate model of

where

In this paper, after the stress intensity factor of the crack tip is obtained based on the interaction integration method, the WRS state (tensile stress or compressive stress) of the crack tip is judged according to the positive and negative values of the stress intensity factor, then, different models of crack propagation rate are selected to calculate the number of load cycles, where the number of load cycles is

_{1}/MPa |
_{2}/MPa |
_{3}/MPa |
_{4}/MPa |
||||||
---|---|---|---|---|---|---|---|---|---|

21 | 1.2 | 7993 | 175 | 6773 | 116 | 2854 | 34 | 1450 | 29 |

There is no universal method to analyze the influence of WRS on the mechanical properties of welded structures [

Specifically, the bilinear isotropic hardening model and the thermal elastic-plastic finite element model of typical welding details were firstly established, and the welding heating process was simulated by using the birth and death element technique. The thermal-structural coupling analysis of welding process and weld condensation process was realized, and the temperature field of the welding detail under the double ellipsoid heat source was solved step by step according to Newton iterative method. Then, the temperature field was applied as the temperature load to the structural model of the typical welding detail, and the WRS was obtained by using thermal-structural sequential coupling analysis. Finally, the WRS field was introduced as the initial stress field into the Multiscale finite element model of the bridge, and then the XFEM was used to analyze the fatigue crack propagation law of typical fatigue details of the bridge deck under vehicle loads after the WRS field was introduced.

When the finite element mesh is inconsistent, mesh mapping is needed from welding process analysis to crack propagation analysis. In order to ensure the accuracy of WRS in the crack propagation analysis, the element size of the solid model used in welding analysis and crack propagation analysis are the same, so there is no need for mesh mapping.

A refined solid element model of the connection of the U-rib and diaphragm with welds is established, as shown in

Parameter | The connection of U-rib and diaphragm |
---|---|

Length of the front axle _{1}/mm |
6 |

Length of the front rear _{2}/mm |
14 |

8 | |

5 | |

Welding speed/ |
10 |

Welding voltage |
250 |

Welding current |
25 |

According to References [

In order to study the stress characteristics and crack propagation rule at critical locations under vehicle load, a Multiscale finite element model of bridge was established. Taking the cable-stayed bridge of Runyang Yangtze River Highway Bridge as an example, in the commercial finite element software ABAQUS, the global finite element model of the bridge is first established, as shown in ^{0}–10^{2} m. The concrete counterweight of steel box girder at the end of the bridge is simulated by mass unit. Three translational and three rotational degrees of freedom are constrained at the end of the bridge and at the bottom of the bridge. The bridge tower is simulated by beam element B31, and the cable is simulated by rod element T3D2. In order to improve the calculation efficiency, according to the principles of composite mechanics, mass equivalence and stiffness equivalence, the OSBD with U-ribs is equivalent to the physically equivalent OSBD without the U-rib at non-critical locations [^{−2}–10^{0} m, which are simulated by solid element C3D8 in ABAQUS, as shown in ^{−3} m, as shown in

Therefore, the characteristic lengths of different components are on different scales. Moreover, for finite element models with different characteristic lengths, different physical equations are used to describe them. Linear elasticity theory is used to describe the whole structure model and the local component model. For the welding area, the elastoplastic theory is adopted. Fracture mechanics and extended finite element theory are used to describe the local crack model.

The vehicle load model selected in this paper refers to fatigue load model III in reference [

The welding process of steel box girder of long-span bridge is generally divided into two steps: welding and assembly of each plate of steel box girder in the factory and welding between each section of the girder at the construction site. The former includes the welding of the top deck and the U-rib, the bottom deck and the U-rib, the U-rib and the diaphragm, and the top deck or bottom deck and the diaphragm, and so on. The latter generally refers to the welding of two adjacent sections of steel box girder or the welding of the arm and the box girder section. This paper mainly focuses on the welding process of each plate of OSBD, and considers that the WRS is mainly generated in the process. This process is completed in the factory.

The WRS field can be obtained through thermal-structural sequence coupling, as shown in

Firstly, without considering the influence of WRS field of welding, cyclic vehicle load as shown in

Similarly, when plastic deformation of materials and the influence of WRS field are considered, vehicle loads are applied to the multiscale finite element model of the bridge to obtain the propagation path of fatigue crack at welding toe of U-rib, as shown in

The fatigue crack propagation paths of U-rib toe are compared under the following four conditions: considering WRS and material plasticity, considering WRS and material elasticity, not considering WRS and material plasticity, not considering WRS and material elasticity. The fatigue crack propagation steps at U-rib toe under four conditions were set as 90 steps. The propagation step represents the number of crack propagation. In the extended finite element analysis, ABAQUS software can obtain the value of stress intensity factor corresponding to all elements at the leading edge of the crack each time the load is applied, and then the life of each element

OSBD of Runyang Bridge in China are made of Q345D steel and made of isotropic material. The structural orthotropic nature is only due to the different stiffness in the longitudinal and transverse bridge directions caused by U-shape stiffeners. However, for the individual components of the box girder, such as the U-rib and diaphragm, they are all made of Q345D steel, which is an isotropic material. Therefore, the formulas for calculating stress intensity factor are valid for orthotropic steel bridge decks. Based on the elastic-plastic analysis of fatigue crack in U-rib toe, the crack opening stress intensity factor _{op}_{res}_{op}_{res}

When the WRS field is introduced, the crack tip stress field of the fatigue crack at the U-rib toe reaches the yield stress of the material before the crack propagation. In the process of crack propagation, the crack propagation step with a large plastic zone is selected to obtain the equivalent plastic strain in the crack tip region of the U-rib, as shown in

Considering the influence of the plastic zone, the stress intensity factor corresponding to the WRS field is calculated by executing the interaction integral process in

The relationship between the length of the crack 2

Similarly,

In this paper, an elastoplastic constitutive model based on the Chaboche kinematic hardening model was introduced, and the XFEM was used to study the influence of material elastoplasticity and crack tip plastic zone on the law of fatigue crack propagation with or without WRS field. The interaction integral method was used to obtain the stress intensity factor at the crack tip when the cyclic load dropped to zero during the crack propagation process. The following conclusions are drawn:

(1) For the fatigue crack propagation path, when the plastic deformation is considered, the fatigue crack at the welding toe of U-rib basically remains unchanged in the short-axis direction of the initial half elliptical crack, but is going up along the long-axis direction. Compared with linear elastic materials, the crack propagation path is more consistent with that in actual bridges. In the process of crack propagation, the plastic zone of the crack tip on one side of the diaphragm is larger than that on the other side, which will inhibit the crack propagation, and the inhibition effect will be more significant when the WRS field is considered. The reason may be that residual compressive stress is generated at the crack tip on one side of the plastic zone, resulting in the stress intensity factor here lower than that of the other side of the crack, and thus inhibits the crack propagation.

(2) As the cyclic load drops to zero, if the WRS is not considered, the plastic zone near the fatigue crack tip at the weld toe of U-rib is small, and the stress intensity factor of the residual compressive stress produced by the plastic deformation is less than

(3) The relationship between the crack length 2