Fatigue crack growth behavior of rib-to-deck double-sided welded joints of orthotropic steel decks

Abstract

Innovative double-sided welding is expected to improve the fatigue resistance of rib-to-deck welded joints of orthotropic steel decks (OSDs). Welding crack-like defects are the crucial issue affecting the fatigue performance of rib-to-deck double-sided welded joints. This study presents a numerical simulation of three-dimensional (3D) mixed mode fatigue crack growth behavior of rib-to-deck double-sided welded joints of OSDs. Maximum tensile stress theory and equivalent stress intensity factor (SIF) were used to simulate mixed mode fatigue cracks growth. The Paris law model was employed to predict the fatigue life. Fatigue cracks of rib-to-deck double-sided welded joints were characterized by the presence of mixed mode cracks of modes I (open), mode II (shear), and mode III (tear), which was dominated by mode I. The equivalent SIF was found to be complex at the growth stage with the maximum value at the two ends of the crack front and the minimum value at the midpoint of the crack front. The crack shape became flatter in the later phase of the crack growth. The fatigue crack surface underwent deflections during crack growth, making the final crack shape exhibiting the characteristic of a spatial curved surface. The initial crack geometry showed a significant impact on the fatigue life.

Keywords

orthotropic steel deck rib-to-deck double-sided mixed mode crack fatigue crack growth

Introduction

Orthotropic steel decks (OSDs) are commonly used in cable-stayed, suspension, and other bridge structures which require high strength combined with lightweight (Lu et al., 2017a; Mao et al., 2019). However, the fatigue durability of OSDs encounters a challenge due to the aging of the structural materials combined with increasing traffic load (Lu et al., 2019; Zhang et al., 2020). Several authors have mentioned that fatigue is one of the main causes of structural failures (Deng et al., 2016, Lu et al., 2017b; Song et al., 2016), and it should be reflected as the primary consideration to extend the service life of bridge. Based on a review on fatigue cracks occurring in OSDs, different causes of fatigue cracking were distinguished (Chitoshi 2006; Kolstein 2007), and the authors generally agree with the following categories: (a) Cracking induced by welding crack-like defects: fatigue cracks initiate from welding defects introduced during manufacturing; (b) Load-induced cracking: fatigue cracking due to low fatigue strength of the connection details; and (c) Distortion-induced cracking: fatigue cracks initiation or growth due to out-of-plane stresses and deformations under the wheel loads.

In recent years, the rib-to-deck welded joints of OSDs have received significant attention due to fatigue cracking problems. Fatigue cracks initiating at the weld root of rib-to-deck welded joints are one of the most common types of cracks observed (Kainuma et al., 2016). Recently, even the long through-roof type fatigue cracks have been detected, and the frequency of occurrence of such cracks increases gradually (Kainuma et al., 2016; Ya et al., 2011). These fatigue cracks are the most dangerous, as their identification is difficult during inspection due to the hidden locations, unless the resulting damage of the wear layer becomes visible (Wang et al., 2019b). Several authors have mentioned that weld penetration is the one of the main influencing factors affecting the fatigue performance of rib-to-deck single-sided welded joints. Some studies have shown that increasing the weld penetration has a positive impact on improving the fatigue performance of rib-to-deck single-sided welded joints (Dung et al., 2014; Fu et al., 2017), while other studies have shown that a shallower weld penetration has a positive effect on improving the fatigue performance of rib-to-deck single-sided welded joints (Kainuma et al., 2016; Ya et al., 2011). The effect of weld penetration on fatigue performance of rib-to-deck single-sided welded joints of OSDs is still inconclusive. Therefore, more careful consideration is needed for the design of rib-to-deck welded joints to overcome the disadvantages of single-sided welding.

More recently, various innovative rib-to-deck details have been proposed, which are expected to improve the anti-fatigue performance of rib-to-deck welded joints. Masahiro et al. (2014) proposed the use of double-sided welding for the first time to attach the closed ribs to the deck plates in OSDs, and conducted tests to investigate the effect of the inside fillet weld on fatigue durability of rib-to-deck welded joints. You et al. (2018) conducted a series of comparative tests on the fatigue performance of double-sided and single-sided welding of rib-to-deck welded joints. The results indicated that double-sided welds could significantly improve the fatigue life of rib-to-deck welded joints. Liu et al. (2019) studied the parameter analysis of the influence of weld penetration on the fatigue performance of rib-to-deck double-sided welded joints. The results indicated that the penetration exhibited very little effect on the fatigue performance of the rib-to-deck double-sided welded joints. However, currently the research on fatigue characteristics of rib-to-deck double-sided welded joints is still lacking.

The linear elastic fracture mechanics (LEFM) has been used to understand the fatigue behavior of rib-to-deck welded joints of OSDs. Wu et al. (2019) analyzed the fatigue crack propagation of rib-to-deck single-sided welded joints in OSDs by LEFM of multi-point extrapolation. Wang et al. (2019a) employed a finite element method (FEM) to simulate the crack propagation of rib-to-deck single-sided welded joints, and validated it against fatigue experiments. Wang et al. (2016) employed the extended FEM to simulate the fatigue crack growth of rib-to-deck welded joints. Previous studies mainly dealt with two-dimensional (2D) problems to simulate crack growth of rib-to-deck welded joints. In fact, the defects of rib-to-deck welded joints are subjected to mixed mode loading under the vehicle load. Therefore, better understanding of the mixed mode fatigue crack propagation behavior of the rib-to-deck double-sided welded joints of OSDs is highly desirable.

This study investigated the mixed mode fatigue crack propagation behavior of the rib-to-deck double-sided welded joints of OSDs by LEFM. The 3D FE models of rib-to-deck double-sided welded joints were established to determine the SIF related to the three modes of fracture. Mixed mode crack growth behaviors were analyzed in terms of the SIF variation along the crack front, crack growth paths, and crack shape variation. In this end, the fatigue life of the weld toe of rib-to-deck welded joints was predicted. Finally, the influences of initial crack size and aspect ratio ( $a / c)$ on fatigue life were analyzed.

Details of double-sided welded joints

The more advanced welding technology, namely, U-rib internal welding technology, was adopted to add fillet welds inside the closed longitudinal ribs to form rib-to-deck double-sided welded joints. Compared to the traditional single-sided welded joints, the innovative double-sided welded joints are expected to improve the fatigue performance of rib-to-deck. Recently, the rib-to-deck double-sided welded joints have been successfully applied in construction of actual bridge in China, such as Zhuankou Yangtze River Bridge and Jiayu Yangtze River Bridge. The details of rib-to-deck double-sided welded joints are shown in Figure 1.

Figure 1.

Details of rib-to-deck double-sided welded joints.

Fatigue crack growth simulation and life prediction method

M-Integral for SIF determination

The M-Integral, which is a numerical method for the calculation of SIF related to the three modes of fracture, was first proposed by Yau et al. (1980) from J-Integral (Rice, 1968). The J-Integral is defined in terms of equation (1) as follows:

J = \int_{Γ} (σ_{ij} \frac{\partial u_{i}}{\partial x_{1}} - W δ_{1 j}) \frac{\partial q}{\partial x_{j}} ds

(1)

where $Γ$ is the contour integration around the crack tip, $u_{i}$ is the displacement vector, ds is the increment of arc length along $Γ$ , $x_{1}$ is the position vector of the integration point, $δ_{1 j}$ is the crack tip opening displacement, and q is a function that is 1 at the crack tip and 0 on the boundary of the integration domain; the strain energy density W is defined as $W = \frac{1}{2} σ_{ij} ε_{ij}$ , $σ_{ij}$ is the stress tensor, and $ε_{ij}$ is the strain tensor.

For linear analysis, the field variables associated with the two solutions are assumed and superimposed (Yau et al., 1980), and defined as follows:

\begin{matrix} σ_{ij} = σ_{ij}^{(1)} + σ_{ij}^{(2)} \\ ε_{ij} = ε_{ij}^{(1)} + ε_{ij}^{(2)} \\ u_{ij} = u_{ij}^{(1)} + u_{ij}^{(2)} \end{matrix}

(2)

The J-integral superposed state can be defined (Yau et al., 1980) as:

J = J^{(1)} + J^{(2)} + M^{(1, 2)}

(3)

where

J^{(1)} = \int_{Γ} (σ_{ij}^{(1)} \frac{\partial u_{i}^{(1)}}{\partial x_{1}} - W^{(1)} δ_{1 j}) \frac{\partial q}{\partial x_{j}} ds

(4)

J^{(2)} = \int_{Γ} (σ_{ij}^{(2)} \frac{\partial u_{i}^{(2)}}{\partial x_{1}} - W^{(2)} δ_{1 j}) \frac{\partial q}{\partial x_{j}} ds

(5)

M^{(1, 2)} = \int_{Γ} (σ_{ij}^{(1)} \frac{\partial u_{i}^{(2)}}{\partial x_{1}} + σ_{ij}^{(2)} \frac{\partial u_{i}^{(1)}}{\partial x_{1}} - W^{(1, 2)} δ_{1 j}) \frac{\partial q}{\partial x_{j}} ds / A_{q}

(6)

where $A_{q} = \int_{L} q_{t} ds$ , $q_{t}$ is the value of the q function along the crack front and $W^{(1, 2)}$ is the interaction strain energy density, defined by

W^{(1, 2)} = σ_{ij}^{(1)} ε_{ij}^{(2)} = σ_{ij}^{(2)} ε_{ij}^{(1)}

(7)

and $M^{(1, 2)}$ . is the M-Integral.

For small scale yielding, the energy release rate G is equal to the J-Integral (Lv et al., 2018).

G = J = \frac{1 - ν^{2}}{E} K_{I}^{2} + \frac{1 - ν^{2}}{E} K_{II}^{2} + \frac{1 + ν}{E} K_{III}^{2}

(8)

where v is Poisson’s ratio, E is Young’s modulus, and $K_{i}$ is SIF, $K_{i} = K_{i}^{(1)} + K_{i}^{(2)}, i = I, II, III$ .

The relationship between M-Integral in terms of material properties and the SIF is shown as equation (9) (Lv et al., 2018).

\begin{matrix} M^{(1, 2)} = 2 \\ \times [\frac{1 - ν^{2}}{E} K_{I}^{(1)} K_{I}^{(2)} + \frac{1 - ν^{2}}{E} K_{II}^{(1)} K_{II}^{(2)} + \frac{1 + ν}{E} K_{III}^{(1)} K_{III}^{(2)}] \end{matrix}

(9)

Moreover, the two definitions for the M-Integral are equal.

\begin{matrix} \int_{Γ} (σ_{ij}^{(1)} \frac{\partial u_{i}^{(2)}}{\partial x_{1}} + σ_{ij}^{(2)} \frac{\partial u_{i}^{(1)}}{\partial x_{1}} - W^{(1, 2)} δ_{1 j}) \frac{\partial q}{\partial x_{j}} ds / A_{q} = \\ 2 \times [\frac{1 - ν^{2}}{E} K_{I}^{(1)} K_{I}^{(2)} + \frac{1 - ν^{2}}{E} K_{II}^{(1)} K_{II}^{(2)} + \frac{1 + ν}{E} K_{III}^{(1)} K_{III}^{(2)}] \end{matrix}

(10)

$K_{I}$ , $K_{II}$ , and $K_{III}$ , can be calculated by using equation (10) in FE analysis.

Mixed-mode SIF range

In most engineering cases there is a presence of the mixed mode of modes I, II, and III of fracture. The equivalent SIF $K_{eq}$ of the mixed mode can be calculated by using equation (11) from BS 7910 (2005). The equivalent SIF range $Δ K_{eq}$ is calculated by using equation (12).

K_{eq} = \sqrt{K_{I}^{2} + K_{II}^{2} + \frac{K_{III}^{2}}{1 - v}}

(11)

Δ K_{eq} = K_{eq, \max} - \max (K_{eq, \min}, 0)

(12)

where v is Poisson’s ratio; $K_{I}$ , $K_{II}$ , and $K_{III}$ are separate SIF for all three modes of fracture, respectively; and $K_{eq, \max}$ and $K_{eq, \min}$ are the maximum and minimum values of $K_{eq}$ , respectively.

Kink angle model

The direction of crack propagation is determined by the kink angle. The maximum tensile stress (MTS) theory proposed by Erdogan and Sih (1963) is one of the most popular theories. It assumes that the crack extension direction $θ_{c}$ is consistent with the maximum value of the hoop stress. For materials with isotropic properties, the crack extension direction $θ_{c}$ is given by the equation (13).

θ_{c} = 2 \arctan [\frac{- 2 (K_{II} / K_{I})}{1 + \sqrt{1 + 8 {(K_{II} / K_{I})}^{2}}}]

(13)

Crack extension type

The specified median crack front extension is used to grow and extend the crack, which specifies the median crack growth increments over each growth step. The crack extension at the crack front point i, $Δ a_{i}$ , is computed by using equation (14) (Liu et al., 2016). The specified crack extension is illustrated in Figure 2.

Δ a_{i} = Δ a_{m} {(\frac{Δ K_{eq, i}}{Δ K_{eq, m}})}^{n}

(14)

where $Δ a_{m}$ is the crack extension at crack front point, $Δ K_{eq, i}$ is equivalent SIF range at crack front point i, $Δ K_{eq, m}$ is the median equivalent SIF range, and n is the material parameter.

Figure 2.

Specified median extension.

Fatigue life prediction

The Paris law (1963) was modified to predict the crack growth rate of the mixed mode, which can be expressed as equation (15):

\frac{da}{dN} = C (Δ K_{eq})^{n}

(15)

N = \int_{a_{0}}^{a_{f}} \frac{da}{C {(Δ K_{eq})}^{n}}

(16)

where $a$ is crack length and N is the number of cycles due to which a crack propagates from its initial crack length, $a_{0}$ , to a final crack length, and $a_{f}$ , can be expressed as equation (16). C and $n$ are empirical coefficients, which are specified along with the values for the SIF threshold range, $Δ K_{th}$ .

Step-wise procedure

In this study, FRANC3D (2016) and ABAQUS (2019) were used to simulate mixed mode fatigue crack growth. A typical flowchart is shown in Figure 3.

Figure 3.

Flowchart for simulation of crack growth.

Numerical examples and verification

A 3D mixed mode fatigue crack growth model was used to simulate the surface crack in the T-joint under constant amplitude loading, providing comparative analysis of the calculated number of load cycles to the experimental results provided by Nikfam et al. (2019). ASTM-572 steel welded T-joints specimens and initial crack geometry are shown in Figure 4. The values of the Young’s modulus, ultimate tensile stress, yield stress, and Poisson’s ratio are 207 GPa, 572 MPa, 406 MPa, and 0.3, respectively. Load amplitude of P = 30 kN was used for loading. FRANC3D and ABAQUS were used to simulate the mixed mode fatigue crack growth of T-joint. The T-joint was modeled by using 8-nodes solid elements (C3D8R). The mesh size of the global model is 10 mm, and that of the cracked sub-model is 0.02 mm (Figure 4). The Paris law constants for ASTM-572 steel were taken as n = 2.88 and C = 6.77 × 10⁻¹³ (MPa and mm) from BS 7910 (2005).

Figure 4.

Surface crack in plate under tension.

Figure 5 presents the comparison between the simulation results and the two experimental results by Nikfam et al. (2019). The comparative analysis indicates that the numerical simulation results are in good agreement with the experimental data. The number of cycles of numerical simulation is very close to the mean number of cycles of two experiments, with a difference of 2.1%. Noteworthy, the error in the crack growth simulation may be caused by the initial crack geometry. In the experiments, the initial crack geometry is not a perfect semi-elliptical shape. Therefore, it is feasible to simulate mixed mode crack growth of 3D problem by using FRANC3D and the results are reasonably accurate.

Figure 5.

Crack length versus number of cycles.

Background of project

Jiayu Yangtze River Bridge with the main span of 920 m, which is located in Hubei, China, was considered as the research object. It is an unsymmetrical composite girder cable-stayed bridge with the longest span in the world. The cross section of the main girder is illustrated in Figure 6. The width of the main girder is 33.5 m and the standard height is 3.0 m. The deck plates are designed to be 16 to 24 mm thick and the longitudinal closed U-ribs are 8 mm thick. The ribs are 300 mm high and 300 mm wide on top and have a width of 180 mm at the lower soffit. The distance between the longitudinal ribs is equal to 300 mm. The U-rib and deck plate were connected by double-sided welding. First, a small intelligent welding robot was used to weld the fillet weld inside the joint, and then the conventional welding technology was employed to weld the groove fillet weld with 80% penetration outside the joint. The clearance between the U-rib and the deck plate assembly was no more than 0.5 mm.

Figure 6.

Cross-section of steel box girder (unit: mm).

Finite element model

To study the mixed mode fatigue crack propagation behavior of rib-to-deck double-side welded joints of OSDs, two different 3D FE models were established. The first FE model was used in stress analysis to identify the most adverse situation for loading, which was named as no-crack FE model. The second FE model was used in fracture analysis, and it was named as cracked FE model. The no-crack FE model, which consists of five longitudinal U-ribs and three diaphragms (Figure 7), was established by using ABAQUS with C3D8R. Young’s modulus of 206 GPa and Poisson’s ratio of 0.3 were used in the FE model, which represent material properties for steel Q345qD. The following boundary conditions were preassigned in ABAQUS: (a) The vertical translation (Y direction) and rotations (X and Z directions) are constrained for all nodes of lower edge of diaphragm to simulate the support of the diaphragm; (b) The longitudinal translation (Z direction) and rotations (X and Y directions) are constrained for all nodes of two ends of model to simulate the boundary conditions at the interior rib and deck plate; and (c) The transverse translation (X direction) and rotations (Y and Z directions) are constrained for all nodes of two sides of model to simulate the boundary condition at the interior diaphragm and deck plate. The cracked model was established by using FRANC3D, by inserting fatigue cracks into the no-crack model. In order to optimize the computational cost associated with the process, the uncracked sub-model, shown in Figure 7 as local, was selected and imported into FRANC3D. A semi-elliptic initial crack was inserted into the uncracked sub-model and then remeshing was carried out by using FRANC3D to establish a cracked sub-model, as shown in Figure 7. The cracked sub-model was connected to the global model by merging nodes. In this way, it could be subdivided to avoid remeshing of the global model at each step of crack growth and to favor the transition of the finer mesh (i.e. cracked sub-model mesh) to the coarser mesh (i.e. global mesh). The mesh size of the global model was 10 mm, and that of the cracked sub-model was 0.02 mm.

Figure 7.

FE model of rib-to-deck welded joints (Unit: mm).

Geometry and location of initial flaws

The initial crack geometry is an important parameter that directly affects the fatigue crack propagation. In general, fatigue cracks grow irregularly in a non-uniform stress field (Gadallah et al., 2017). For the sake of simplicity, a semi-elliptical surface initial crack with depth of 0.2 mm and full surface length of 1 mm was inserted into the no-crack sub-model (Figure 7). Moreover, four types of cracks, namely, Crack I, Crack II, Crack III, and Crack IV as shown in Figure 8, were assumed for fatigue analysis.

Crack I: the outside toe-deck, which initiates at the outside weld toe and propagates into the deck.

Crack II: the outside root-deck, which initiates at the outside weld root and propagates into the deck.

Crack III: the inside root-deck, which initiates at the inside weld root and propagates into the deck.

Crack IV: the inside toe-deck, which initiates at the inside weld toe and propagates into the deck.

Figure 8.

Schematic illustration of different forms of crack.

Fatigue load

The fatigue load model III in EN 1991-2 (2003) was used for simulation. The geometry of mode III is shown in Figure 9. The stress influence line in the transverse direction of rib-to-deck welded joints of OSD was short under the wheel loading, it was approximately three times the width of the U-rib opening (Xiao et al., 2008), while the wheel spacing of single axles in mode III was 2 m. Therefore, the stress interference between neighboring wheels of single axle could be neglected. Moreover, the spacing between the second axle and the third axle was 6 m, which was much larger than the stress influence line in the longitudinal direction. For the sake of simplicity, a fatigue load mode consisting of two axles, each of them having one wheel and each wheel with a contact surface of 400 mm by 400 mm was used in the FE model. The geometry of computational fatigue load is shown in Figure 9.

Figure 9.

Fatigue load mode III and computational wheel load.

Stress analysis for no-crack Fe model

The double-axles fatigue load was used to simulate the moving vehicle load (Figure 9). The DLOAD of the ABAQUS user subroutine was employed to simulate the fatigue load movement on the bridge. The increment of the moving step in the transverse direction was 50 mm, and that in the longitudinal direction was 100 mm, forming an array of load cases of 73 × 37 to simulate the moving of fatigue load. Loading cases are shown in Figure 10, where TC is defined as transverse movement and LC is defined as longitudinal movement.

Figure 10.

Loading cases: (a) transversal loading conditions and (b) longitudinal loading conditions.

Fatigue cracks are more sensitive to the transverse tensile stress, which is the driving force of crack propagation (Shi et al., 2013). The longitudinal influence line of transverse stress at the transverse locations ( $X = - 100 mm$ ) of wheel center is presented in Figure 11(a). It is observed that the transverse stress reaches the maximum value (tensile stress) when the double-axle center of fatigue load is in the location of mid-span ( $Z = 0 mm$ ), which is the most adverse situation in the longitudinal direction. Furthermore, when the double-axle center of fatigue load is located in the mid-span ( $Z = 0 mm$ ), the transverse influence line of transverse stress is obtained and presented in Figure 11(b). Clearly, when the transverse position of vehicle load is ( $X = - 100 mm$ ), the maximum transverse stress (tensile stress) of the details can be obtained. Therefore, when the wheel center is located at transverse position of ( $X = - 100 mm$ ), it is the most adverse situation in the transverse direction.

Figure 11.

Stress-time curves: (a) longitudinal loading and (b) transverse loading.

Static crack SIF analysis

The heavy vehicle load is one of the main causes of fatigue of OSD (Lu et al., 2019). In this study, considering the influence of overloading of the vehicle, the weight of each axle of 240 kN was used to perform fatigue analysis at the most adverse load position of vehicle load ( $X = - 100 mm$ , $Z = 0 mm$ ). Furthermore, the SIFs related to the three modes of fracture for the four types of cracks of the rib-to-deck double-sided welded joints in OSDs were determined. Under the most unfavorable loading conditions, the SIF values ( $K_{I}$ , $K_{II}$ , $K_{III}$ ) along the crack front were identified by the non-dimensional curvilinear coordinate system, as shown in Figure 12. It is observed that the fatigue cracks at the rib-to-deck double-sided welded joints was the I–II–III mixed mode, which is dominated by mode I. For the initial crack of the weld toe, the values of $K_{I}$ at the mid-point of the crack front are lower than the value of $K_{I}$ at the two ends of the crack front. However, for the initial crack of the weld root, the values of $K_{I}$ of the mid-point of the crack at the weld root are higher than the value of $K_{I}$ at the two ends of the crack front. This is attributed to the presence of the fatigue cracks in the non-uniform stress field.

Figure 12.

Static SIF of the initial crack: (a) crack I, (b) crack II, (c) crack III, and (d) crack IV.

Figure 13 exhibits the values of $Δ K_{eq}$ of the crack front of the four types of cracks. The $Δ K_{eq}$ of the crack front at the outside weld toe is not much different from that at the inside weld toe. However, the $Δ K_{eq}$ of the crack front at the weld toe is much greater than that at the weld root. It indicates that the fatigue failure of rib-to-deck double-sided welded joints is mainly determined by the fatigue crack of the weld toe. Notably, the maximum value of $Δ K_{eq}$ of the crack front at welded root is 34.12 MPa·mm^1/2, which is far less than the threshold value of the SIF ( $Δ K_{th}$ = 63 MPa·mm^1/2) in the International Institute of Welding (IIW) (Hobbacher, 2008). Therefore, considering only the effect of vehicle load, the weld root of the rib-to-deck double-sided welded joints does not lead to crack growth. In this study, only the crack growth behavior of the weld toe of rib-to-deck double-sided welded joints was simulated.

Figure 13.

Equivalent SIF range of the initial crack.

Crack growth analysis and fatigue life predictions

Crack growth behavior

The identification of crack propagation paths is essential for the comprehensive understanding of the fatigue crack growth behavior. The simulation results in a detailed visualization of the crack growth behavior of the welded toe of the rib-to-deck double-sided welded joints are shown in Figure 14, where various curves on the fatigue crack surface represent the crack propagation at each step. The results indicate that the crack growth behaviors of the outside welded toe and the inside welded toe of rib-to-deck double-sided joints are basically similar. During crack propagation, the surface fatigue crack at the weld toe of rib-to-deck double-sided joints does not remain flat, but slightly deflected. This can be represented by the values of the $K_{IImax} / K_{Imax}$ and $K_{IIImax} / K_{Imax}$ , which quantify the contribution of mode II and mode III to the crack propagation, as shown in Figure 15. Moreover, it is also observed that the overall trend of the contribution of the mode II and mode III increases after a certain number of crack growth steps. Therefore, the contribution of the mode II and mode III cannot be neglected during the crack propagation.

Figure 14.

Spatial shape of crack propagation: (a) outside weld toe and (b) inside weld toe.

Figure 15.

$K_{IImax} / K_{Imax}$ and $K_{IIImax} / K_{Imax}$ variations: (a) outside weld toe and (b) inside weld toe.

Variation of equivalent SIF range

The variations of $Δ K_{eq}$ directly affect the crack growth rate. The SIFs along the crack fronts of the welded toe of rib-to-deck double-sided joints at each stage of crack growth were determined. The variation of $Δ K_{eq}$ with the crack length shown in the graphs refers to five propagation paths along the crack front (Figure 16), namely 0.1, 0.3, 0.5, 0.7, and 0.9, identified by using normalized coordinates. It is observed that the variation trends of $Δ K_{eq}$ of the crack front of the outside weld toe and the inside weld toe are basically consistent during crack propagation. The $Δ K_{eq}$ at the midpoint of the crack front along the propagation path (0.5) is always smaller than that of the crack front along the other propagation paths. Noteworthy, the $Δ K_{eq}$ is along the propagation path (0.5), which gradually increases and then gradually decreases. Moreover, the results shown in Figure 17 illustrate the variation of fatigue crack aspect ratio ( $a / c$ ) during crack propagation. It is observed that the $a / c$ first increases and then decreases. This behavior is mainly due to the different crack growth rate along the front, that is, the crack growth rate at two ends of the crack front is higher than that at midpoint of the crack front after a certain number of crack growth steps.

Figure 16.

Variation curves of equivalent SIF range: (a) outside weld toe and (b) inside weld toe.

Figure 17.

Variation curves of crack aspect ratio.

Fatigue life predictions

The $Δ K_{eq}$ . value and the corresponding crack length estimated at a specific path along the crack front were determined. By using equation (16), the number of load cycles of the rib-to-deck double-sided welded joints was calculated at each step, which is an average of the values computed for all mid-side nodes along the crack front when growing to the crack from one to the next. According to the recommended value of BS 7910 (2005), the Paris laws constants for Q345qD steel were taken as n = 3 and C = 5.21 × 10⁻¹³ (MPa and mm). According to the IIW (2008), it is assumed that the structure fatigue failure occurs when the crack front is close to 50% of thickness of the deck plate. The curves of the fatigue life and the corresponding crack depth in the thickness direction of the deck plate are shown in Figure 18. The results reveal the value of the crack depth (a = 8 mm) of the growth phase related to the crack with outside weld toe is equal to N = 4400093 cycles, and the value in the case with inside weld toe is equal to N = 4367796 cycles. Under the same fatigue load, the fatigue life of the inside welded toe is lower than that of the outside welded toe of rib-to-deck double welded joints.

Figure 18.

Fatigue crack depth versus number of cycles.

Discussion

Effect of initial crack on fatigue life

As discussed previously, a surface semi-elliptical crack was assumed to be an initial defect to simulate crack growth. To investigate the impact of initial crack with a fixed aspect ratio ( $a_{0} / c_{0}$ = 0.4) on fatigue life, the initial crack geometry was assumed to be a variable with three cases, including case 1 in which $a_{0}$ = 0.2 mm, $c_{0}$ = 0.5 mm; case 2 in which $a_{0}$ = 0.4 mm, $c_{0}$ = 0.1 mm; and case 3 in which $a_{0}$ = 1.0 mm, $c_{0}$ = 2.5 mm. The curves of crack depth versus the number of cycles were determined for three cases of outside welded toe as shown in Figure 19. The curves demonstrate that the initial crack geometry significantly affects the fatigue life.

Figure 19.

Crack depth versus number of cycles.

Furthermore, in order to investigate the impact of initial crack aspect ratio ( $a_{0} / c_{0}$ ) on fatigue life, the initial crack depth was assumed to be a constant $a_{0}$ = 0.2 mm, and the surface half-length $c_{0}$ was a variable changing between 0.2, 0.5, and 0.8 mm. The curves of crack depth versus the number of cycles were determined for three cases of outside welded toe as shown in Figure 20. Clearly, $a_{0} / c_{0}$ has a significant effect on fatigue life, and the fatigue life decreases significantly with the decrease of aspect ratio.

Figure 20.

Crack depth versus number of cycles.

Conclusion

This study presented numerical simulation of 3D mixed mode fatigue crack growth behavior of rib-to-deck double-sided welded joints of OSD. Fatigue life of the rib-to-deck double-sided welded joints was predicted by using the mixed mode Paris law model. The influence of initial crack on the fatigue life of the outside weld toe was discussed. The major conclusions are as follows:

Under the vehicle load, the equivalent SIF range values of the crack fronts at the weld root of the rib-to-deck double-sided welded joints were found to be far less than the SIF threshold, and the fatigue crack did not grow.

The mixed mode fatigue crack growth of the weld toe was characterized by the presence of the three modes of fracture which was dominated by mode I. The mode II interacts with the mode III and causes deflection of the crack surface, resulting in the final crack shape exhibiting the characteristic of a spatial curved surface.

The two ends of the crack fronts at the weld toe grow faster than the midpoint of the crack front due to the fact that the two ends of the crack fronts were under the action of higher values of $K_{eq}$ . This resulted in the progressive decrease in the aspect ratio ( $a / c)$ of fatigue cracks in the later phase of the crack growth.

Under the same fatigue load, the fatigue crack at the outside weld toe exhibited a longer fatigue life than that at the inside weld toe.

The initial crack exhibited a significant effect on the fatigue life of outside weld toe of rib-to-deck double-sided welded joints of OSDs, and the flatter initial crack resulted in shortening of the fatigue life.

Highlights

A 3D FE model was established to simulate the mixed mode fatigue crack growth behavior of rib-to-deck double-sided welded joints of orthotropic steel decks.

Mixed mode crack growth behaviors were analyzed in terms of the SIF variation along the crack front, crack growth paths, and crack shape variation.

The fatigue crack growth process was affected by the presence of mixed mode cracks of modes I (open), mode II (shear) and mode III (tear), in which mode I is dominant.

The geometry of the initial fatigue crack has a significant impact on the fatigue life.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Key Fundamental Research Development Program (Program 973) (Grant No. 2015CB055701); National Natural Science Foundation of China (Grant Nos. 51378081, 51308073, 51878072, 51908068); Industry Key Laboratory of Traffic Infrastructure Security Risk Management (CSUST) (Grant No. 18KF04); Graduate Student Research Innovation Project of Hunan Province (CSUST) (Grant No. CX2018B533); and the Natural Science Foundation of Hunan Province (Grant Nos. 2018JJ3540, 2020JJ5140, 2020JJ5143).

ORCID iD

Fanghuai Chen

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