1. Introduction
Steel structures have the following advantages: they are very strong, convenient to construct, lightweight, and have good mechanical properties, which make them the primary structural forms in large infrastructures. The orthotropic steel deck (OSD) is the main type of deck for long-span steel bridges, especially for cable-supported bridges. However, fatigue cracking is a key problem for the service safety and the serviceability of OSD structures. This is a result of low local stiffness, welding defects and heavy traffic loads [
1,
2]. Fatigue cracking is a problem in many famous long-span bridges, such as the Humen Bridge, the Jiangyin Bridge, and the Junshan Bridge [
3,
4]. Of all the various types of fatigue cracks, the most critical and common cracking type is in the welding roots of the rib-to-deck connections. In addition, the cracks in welding roots are usually under the pavement, making them difficult to detect [
5]. The fatigue performance of welded joints is affected by multiple uncertainties, such as base metal defects, penetration rate, residual stresses, and the random distribution of traffic load. These uncertainties result in a high degree of randomness in fatigue cracking. Because of these uncertain factors, the crack propagation path randomly changes [
6,
7]. However, the traditional deterministic crack growth analysis method is cannot to describe the random crack propagation characteristics effectively. In addition, the research results of this study into the random propagation behavior of fatigue cracks on steel bridge decks could lay a theoretical foundation for their fatigue reliability evaluations and optimal designs.
Many theoretical and experimental studies have been carried out to investigate the defect initiation, crack propagation, and fatigue fractures in rib-to-deck welded joints. In the numerical simulation of fatigue cracks, typical two-dimensional, semi-elliptic, type-I crack, with fixed proportions for the long and short axes, were widely utilized to study crack propagation behavior. Wang et al. [
8] observed the three-dimensional propagation behavior of fatigue cracks in steel bridge decks due to the high strain energy of the type-II and type-III characteristics in the later period. Di et al. [
9] developed a fatigue evaluation approach for in-service steel bridge decks, based on the strain monitoring data. Wang et al. [
10] evaluated the distortion-induced fatigue crack growth rate using the extended finite element method.
Since the mechanical behavior of fatigue cracks in OSDs is determined by three different crack types, the maximum circumferential stress and minimum strain energy density are commonly utilized to judge the spatial crack propagation behavior. Zhang et al. [
11] proposed a simulation method for three-dimensional, semi-elliptic crack propagation at rib-to-deck welded joints, which was verified by an experimental test. Rodenburg et al. [
12] developed an approach for evaluating the growth rate of both penetrating and non-penetrating cracks. Mahmood et al. [
13] investigated the propagation behavior of penetrating fatigue cracks in the I-beams of a steel bridge, based on an advanced equivalent stress intensity approach. Cheng et al. [
14] investigated the crack propagation behavior of rib-to-floor, beam-welded connections in ultra-high performance, concrete-reinforced OSDs, which were subjected to longitudinal flexural loads. Huang et al. [
15] investigated the propagation characteristics and fatigue life of rib-to-diaphragm welded joints under a constant amplitude load. Fang et al. [
16] investigated the fatigue failure mechanism and optimization of double-sided welds in OSDs. Cui et al. [
17] developed fatigue mechanics for a new compression-compression zone for welded joints in OSDs. Xu et al. [
18] investigated a three-dimensional weight function method for the rapid calculation of crack stress intensity factors.
In this study, we developed a computational framework for predicting fatigue crack propagation in the welded joints of OSDs under stochastic traffic loading. Stochastic traffic load models were established based on site-specific traffic data, for the purpose of simulating the fatigue stress spectra of welded joints. The influence of the transverse loading position of trucks’ wheels on the stress intensity factor of the crack tip was also investigated. The random propagation paths of the crack under stochastic traffic loading were evaluated. Both ascending and descending load spectra were considered in the traffic loading pattern. The research results can provide a theoretical basis for the fatigue reliability of existing steel bridge decks, based on practical traffic loading.
5. Random Propagation Characteristics of Fatigue Cracks at Rib-to-Deck Welded Joints under Realistic Traffic Loading
This section combines the stochastic traffic loading and the fatigue stress analysis method to conduct the stochastic propagation investigation. The adverse transverse loading position and the stress reduction factor was considered. The random propagation behaviors of fatigue cracks at the rib-to-deck weld root of steel bridge decks under stochastic traffic loading are also discussed here.
As mentioned, the crack depth,
a, and the crack shape ratio,
a/
c, are two important indexes that affect the crack propagation behavior in the LEFM. The initial, semi-elliptic surface crack shape parameters were as follows: crack depth,
a = 4 mm; surface length, 2
c = 16 mm; and crack morphology ratio,
a/
c = 0.5. A crack-free sub-model of 300 mm × 300 mm × 280 mm, was cut from the midspan of the segment model. The crack and the sub-model were combined in FRANC3D, and the model was re-meshed with a scale characteristic of 0.4 mm. The re-meshed sub-model is shown in
Figure 15.
The analysis of the stress intensity of the crack tip at the rib-to-deck weld root, under an adverse loading position, longitudinally, is shown in
Figure 16. The stress intensity factors of type-I, type-II, and type-III cracks are noted as
KI,
KII, and
KIII, respectively.
As can be observed from
Figure 16,
KI of the fatigue crack at the rib-to-deck welded root was significantly larger than
KII and
KIII. Therefore, the composite crack was mostly affected by the type-I crack. Note that
KI will decrease with the transverse moving from the adverse location. In addition,
KII is negative to all transverse distributions, which restrains the sliding fatigue crack obviously. Considering the transverse distribution of wheel tracks,
KII at the crack front was either positive or negative, and
KI at the deepest crack point was insignificant.
According to these results, type-I cracks cause fatigue cracks at the weld root. A comparative analysis was conducted to estimate the maximum stress intensity factor,
KI, of the fatigue crack at the weld root under the transverse wheel loads.
Figure 17 shows the stress intensity factor that was affected by the transverse loading position.
It can be observed from
Figure 17 that the maximum value of
KI was 568.18 (MPa·mm
1/2) under the adverse transverse loading position. If the wheel tracks moved 450 mm to the right of the adverse loading position, the maximum
KI decreases to 190.63 (MPa·mm
1/2) with a reduction rate of 66.4%. The influence of the transverse distribution of the wheel tracks on the direction of the crack propagation is useful to understand the random crack propagation behavior of OSD cracks. The crack torsion angle at the weld root is shown in
Figure 18 under the adverse loading condition.
It can be observed from
Figure 18 that the propagation angle of the crack tip increased from 0.24° to 0.34°, which is an increase of 42%, under the condition of moving 450 mm transversally. Since the variation of the torsion angle in the middle part was negligible, the stochastic propagation behavior can be ignored. However, there were some differences in the propagation direction of the parts near the two ends. Under the effect of the transverse distribution of the wheel tracks, the torsional angle increased when the wheels moved to the right, but the torsional angle decreased when the wheels moved to the left.
Through the transformation of the wheels’ transverse loads, the transverse deviation value,
e, of the wheel tracks was used as a random variable to analyze the random crack propagation path at the rib-to-deck weld root. The transverse distribution frequency is shown in
Figure 19. MATLAB was used to simulate the transverse random wheel tracks to generate the stochastic traffic loading.
The random load spectra resulting from the stochastic traffic load was simplified in ascending and descending order to reflect the fatigue crack propagation behavior of the weld root of rib-to-deck welded joints more realistically and intuitively. The load spectra are shown in
Figure 19. The random propagation paths under three load spectrum conditions were compared with that under the most adverse loading condition.
The propagation path of the fatigue crack was simulated based on the segment model established by FRANC3D-ABAQUS interaction technology. The basic parameters in the process of crack propagation were set. The parameters in Equation (1) are c = 5.21 × 10
−13 and
n = 3. The model was then submitted to ABAQUS for analysis. The propagation step, Δ
a, of the midpoint of the front end of the fatigue crack was 0.6 mm. The propagation paths of the two endpoints, A and B, of the long axis of the semi-elliptic crack is discussed below. The location of the two endpoints is shown in
Figure 20. The chosen number of propagation steps was 18 to clearly reflect the crack propagation paths and to ensure the efficiency and accuracy of the calculation.
Figure 21 shows the propagation paths of the two endpoints.
In the random propagation paths at both ends of the semi-elliptic crack, the four paths had a small migration and almost coincided with each other within 10 mm of the propagation distance in the x axis direction. When the propagation distance in the x axis direction exceeded 10 mm, different degrees of migration began to appear, and the migration of the crack propagation path under the influence of a descending load spectrum, was more significant in the y axis. In addition, the propagation path under a random load spectrum was between the ascending load spectrum and the descending load spectrum.
6. Conclusions
A computational framework for the fatigue crack propagation of OSDs under stochastic traffic loads, based on the extended finite element model, have been presented in this study. The stochastic traffic load models were established based on site-specific weigh-in-motion measurements to simulate the fatigue stress spectra of welded joints. The influence of the transverse loading position of the wheel tracks on the stress intensity factor of the crack tip was also investigated. Furthermore, the random propagation paths of the crack under stochastic traffic loads were evaluated. Moreover, the influence of both the ascending and descending traffic loading patterns on the fatigue cracking behavior were also investigated.
The numerical results indicated that the equivalent stress amplitudes of the weld root and weld toe of rib-to-deck welded joints under the random transversal wheel loading model were 0.80 and 0.78 times that of the traditional loading model, respectively. The maximum value of KI is 568.18 (MPa·mm1/2) under the most critical transversal condition of the wheel load. However, the maximum KI is 190.63 (MPa·mm1/2), a decrease of 66.4%, under the condition of transversal moving by 450 mm. The transverse distribution of the wheel tracks had a significant influence on the torsion angle of the crack. If the wheel was shifted 450 mm laterally from the most unfavorable position, the propagation angle of the crack tip would increase from 0.24° to 0.34° with an increase ratio of 42%. Under the three stochastic loads spectra and the transverse distribution of the wheel tracks, the propagation paths of the fatigue cracks almost coincided, where the value is within 10 mm of the propagation distance in the direction of the x axis. This phenomenon demonstrates that the migration occurred with different degrees. The migration effect is most significant under the descending load spectrum.
In addition to the fatigue cracks at the rib-to-deck weld root, the influence of the other typical fatigue factors of OSD needs to be further studied. The random fatigue crack propagation characteristics of steel bridge decks, under the combined action of weld defects and the transverse distribution of the wheel tracks, will be the focus of future research. In addition, the combination of the stochastic traffic model and the fatigue crack propagation behavior can be further developed.