Investigation of Stress Intensity Factors in Welds of Steel Girders Within Steel–Concrete Composite Structures

Abstract

Fatigue damage in steel–concrete composite structures frequently initiates at welded joints due to stress concentrations and inherent defects. This study investigates the stress intensity factors (SIFs) associated with fatigue cracks in the welds of steel longitudinal beams, employing the FRANC3D–ABAQUS interactive technique. A finite element model was developed and validated against experimental data, followed by the insertion of cracks at both the weld root and weld toe. The influences of stud spacing, initial crack size, crack shape, and lack-of-penetration defects on Mode I SIFs were systematically analyzed. Results show that both weld root and weld toe cracks are predominantly Mode I in nature, with the toe cracks exhibiting higher SIF values. Increasing the stud spacing, crack depth, or crack aspect ratio significantly raises the SIFs. Lack of penetration defects further amplifies the SIFs, especially at the weld root. Based on the computed SIFs, fatigue life predictions were conducted using a crack propagation approach. These findings highlight the critical roles of crack geometry and welding quality in fatigue performance, providing a numerical foundation for optimizing welded joint design in composite structures.

1. Introduction

Steel–concrete composite structures have gradually gained favor among designers due to their ability to combine the advantages of both steel and concrete materials [1]. With their widespread use in engineering applications, the fatigue performance of composite girders has increasingly attracted research attention. In such structures, fatigue-critical welded details mainly include the welds between shear studs and steel girders, as well as the welds within the steel girders themselves. Under the combined effects of cyclic traffic loading [2], wind loads [3], and temperature variations [4,5], initial imperfections [6] in these welded joints can gradually evolve into fatigue cracks, which, if further propagated, pose serious threats to the safety and durability of bridges and buildings, potentially resulting in significant economic losses [7].

Fatigue performance studies of steel–concrete composite structures typically focus on shear connectors and steel girder welds [8], employing experimental investigations, numerical simulations, and, more recently, machine learning techniques as predictive tools [9]. FU, et al. [10] used the beach marking method to test two welded H-shaped steel beams and found that cracks typically initiated at the fillet weld between the tensile flange and the web, propagating along both directions. Tong et al. [11] conducted fatigue tests on stud-connected and purely welded H-beams, revealing that cracks in the former initiated at the weld toe of shear studs, while in the latter, they originated from the fillet weld between the lower flange and the web. Zong, et al. [12] performed fatigue tests on cross-shaped fillet welds and identified two typical failure modes: crack propagation from the weld root and from the weld toe, leading to root and toe failures, respectively. Shear studs have been extensively studied since their failure can compromise composite action, thereby increasing the stress burden on the tension flange of steel girders. Previous research has examined their fatigue behavior under adverse conditions such as corrosion [13,14,15], extremely high/low temperatures [16], and initial defects [6]. In contrast, fatigue cracks in steel girders tend to propagate rapidly, often leading to abrupt structural failure. Within fatigue analysis methodologies, linear elastic fracture mechanics (LEFM) has proven especially effective, emphasizing the initiation and growth of fatigue cracks as key indicators for assessing structural functionality and safety [17,18]. For example, Lu et al. [19] studied steel bridge deck top plates and used crack interaction factors to evaluate how various parameters influence the stress intensity factor (SIF) of a reference crack, extending their investigation to multiple crack coalescence near welds between top plates and U-ribs. Their findings revealed that the interaction factor decreases with increasing crack spacing and increases with the depth and length of the interfering crack. Fatigue life was inversely proportional to initial crack size and positively related to crack spacing—results validated by multi-crack co-propagation tests. Tong et al. [11] conducted fatigue tests on H-shaped welded beams with various welding details, identifying crack initiation sites and propagation paths, and used the test data to develop a fatigue crack growth model validated through numerical simulations. Wang et al. [20,21] focused on studs in composite beams, analyzing SIFs variation by changing stud diameter and height, finding a positive correlation with diameter but minimal influence from height. Chen et al. [18] proposed a numerical framework to evaluate the influence of welding residual stress (WRS) relaxation on fatigue crack propagation in double-sided U-rib welds of orthotropic steel decks, confirming the significant impact of WRS on both SIF and fatigue life prediction accuracy. Overall, existing research on the fatigue performance of steel–concrete composite structures has predominantly focused on shear stud connections. However, fatigue failure in steel girders can result in the loss of structural load-bearing capacity, making it equally important to investigate their fatigue behavior. While LEFM has been successfully applied in the fatigue assessment of steel structures and bridges, welded details in steel–concrete composite structures not only involve the shear stud–girder interface, but also the internal welds within the steel girders. The characteristics of SIFs at internal welding defects in steel girders remain insufficiently explored.

To address these gaps, this study develops a refined finite element model to conduct a parametric analysis of SIFs at initial defects in the steel longitudinal girders of steel–concrete composite beams. The analysis comprehensively considers the effects of shear stud longitudinal spacing, initial crack shape, weld penetration rate, and defect location. In addition, fatigue life analyses based on crack propagation were carried out to evaluate the influence of initial defect characteristics on the fatigue durability of welded joints. The findings aim to provide valuable insights for improving the fatigue assessment and optimization of welded details in steel–concrete composite structures.

2. Methodology

SIFs of fatigue cracks can be obtained through analytical, numerical, and experimental approaches. While analytical techniques are effective for idealized or simplified scenarios, they often fall short when considering the intricacies of real-world structural components. In contrast, numerical strategies—particularly the finite element method—are extensively utilized for modeling fatigue cracks in complex geometries [22,23]. The M-integral stands out for its capacity to evaluate all three fracture modes based on fracture mechanics principles. The M-integral evolved from the path-independent J-integral and enables precise evaluation of stress and strain distributions surrounding a crack tip using the equivalent domain integral technique. This method constructs two distinct, self-equilibrated stress states—one corresponding to the actual physical field (denoted by superscript 1) and the other to an auxiliary field (superscript 2). By superimposing these two fields, the resulting interaction integral, i.e., the M-integral, can be formulated as follows [24]:

$$M^{(1,2)}={\int }_{\Gamma }\left({\sigma }_{ij}^{\left(1\right)}{\displaystyle \frac{𝜕u_i^{\left(2\right)}}{𝜕x_1}}+{\sigma }_{ij}^{\left(2\right)}{\displaystyle \frac{𝜕u_i^{\left(1\right)}}{𝜕x_1}}-W^{(1,2)}{\delta }_{1j}\right){\displaystyle \frac{𝜕q}{𝜕x_j}}ds/A_q$$

(1)

where A_q = ∫Lq_tds, in which q_t represents the crack front weighting function, and q is a smooth function that takes the value 1 at the crack tip and gradually decreases to 0 at the boundary of the integration region. The symbol 𝜕u_i represents the displacement component associated with the ith degree of freedom, and Γ refers to the contour path around the crack tip. The parameter δ_1j acts as an indicator function to select the appropriate displacement direction in the evaluation. ^(1,2) is given by:

$$W^{\left(\mathrm{1,2}\right)}={\sigma }_{ij}^{\left(1\right)}{\epsilon }_{ij}^{\left(2\right)}={\sigma }_{ij}^{\left(2\right)}{\epsilon }_{ij}^{\left(1\right)}$$

(2)

where σ_ij and ε_ij represent the components of the stress and strain tensors, respectively.

The SIF can be described through the following formulation:

$$M^{\left(\mathrm{1,2}\right)}=2\times \left[{\displaystyle \frac{1-{\nu }^2}{E}}{K}_{\mathrm{I}}^{\left(1\right)}{K}_{\mathrm{I}}^{\left(2\right)}+{\displaystyle \frac{1-{\nu }^2}{E}}{K}_{\mathrm{I}\mathrm{I}}^{\left(1\right)}{K}_{\mathrm{I}\mathrm{I}}^{\left(2\right)}+{\displaystyle \frac{1+{\nu }^2}{E}}{K}_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\left(1\right)}{K}_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\left(2\right)}\right]$$

(3)

where v denotes Poisson’s ratio and E is the elastic modulus of the material. The SIF for mode im (m = I, II, III) is given by K_m = K_m⁽¹⁾ + K_m⁽²⁾, representing the combined contributions from two loading conditions.

The two aforementioned forms of the M-integral are functionally equivalent. Therefore, Equations (1) and (3) can be jointly used in finite element analysis to extract all three modes of SIFs for fatigue crack assessment. In the authors’ previous studies [5,20], FRANC3D’s integrated M-integral function was used to compute the individual mode SIFs under mixed-mode conditions, and the results were compared with the reference solutions provided by the Raju–Newman criterion. The comparison showed that the discrepancies were within 3%, demonstrating that the coupled FRANC3D–ABAQUS approach can provide high accuracy in fatigue fracture simulations. Considering the length of this report, the process of fatigue crack propagation and life estimation refers to existing research [4,18]. The software versions used in this study are FRANC3D 7.0 and ABAQUS 2020.

3. Model Formulation and Numerical Validation

3.1. Experimental Program

The flexural behavior of the composite beam was investigated by referencing experimental data from previous studies, particularly the reported load–displacement curves [25]. In the test, the composite beam consisted of a 60 mm thick concrete slab with dimensions of 2000 mm in length and 600 mm in width, using C20 concrete. The steel longitudinal beam was fabricated by welding Q235-grade steel plates into an I-beam section, with a web length of 2000 mm, width of 104 mm, and thickness of 5 mm. Both the top and bottom flange plates had dimensions of 100 mm (width) × 8 mm (thickness). The top surface of the upper flange was in full contact with the concrete slab. Stud connectors were arranged in double rows on the steel longitudinal beam, with longitudinal spacing of 60 mm and 200 mm, and transverse spacing of 40 mm. A total stud length of 50 mm and diameter of 10 mm was used. Transverse reinforcement consisted of Q335-grade steel bars with a diameter of 18 mm. Material properties of the steel components are listed in

Table 1. Material properties of steel components.

Materials	Component	Elastic Modulus/MPa	Yield Strength/MPa	Poisson’s Ratio
Q235	Steel plate	210,000	235	0.3
ML15	Stud	210,000	345	0.3

. The specific dimensions of the test beam are shown in Figure 1.

Prior to the main static loading test, a preloading procedure was implemented to minimize the effects of residual adhesion and other initial disturbances. During this phase, the applied load was gradually increased from 20% to 40% of the beam’s elastic limit load, followed by a complete unloading to zero. In the formal loading stage, a stepwise loading approach was employed, with a load increment of 20,000 N per step and a holding time of 2 min at each level. Once the elastic limit was reached, the control mode transitioned to displacement control, applied at a constant rate of 0.25 mm/min, continuing until failure occurred. At every loading step, measurements were taken after a 2 min hold period, including applied load, deflection at selected points, and strain readings. The experimental loading configuration is illustrated in Figure 2 [25].

3.2. Finite Element Model Development

To achieve close agreement between the numerical simulation and experimental results, this study employed the Concrete Damaged Plasticity (CDP) model available in ABAQUS. This model was used to characterize the nonlinear behavior of concrete by incorporating tensile and compressive damage parameters into the material definition. These parameters account for the two primary failure modes of concrete: tensile cracking and compressive crushing.

Concrete of grade C20, consistent with that used in the referenced experimental tests, was utilized in the simulation. The essential material properties were specified as follows: density of 2400 kg/m³, elastic modulus of 255,000 MPa, and Poisson’s ratio of 0.2. The plasticity-related parameters of the CDP model in ABAQUS were specified as follows [26,27]: the dilation angle φ was selected within the typical range of 30° to 38°; the flow potential eccentricity ε was taken as 0.1; the ratio of biaxial to uniaxial compressive strength f_b0/f_c0 was set to 1.16; the shape factor of the yield surface K_c, generally ranging from 0.5 to 1.0, was assigned a value of 0.667; and the viscosity parameter μ was defined as 0.001. ML15 shear connectors were employed, which were fabricated by welding Q235-grade steel plates into I-shaped sections, as illustrated in Figure 3. The commonly used trilinear elastic–plastic constitutive model was adopted for both materials. For the transverse shear reinforcement, Q335-grade steel bars were used, and the commonly applied bilinear constitutive model was selected.

The constitutive relationship of steel is expressed as shown in Equation (4).

$$\sigma =\left\{\begin{array}{ccc}{E}_{\mathrm{s}}\epsilon & \left(\epsilon \le {\epsilon }_y\right)& \\ {\sigma }_y+0.01E_{\mathrm{s}}\left(\epsilon -{\epsilon }_y\right)& \left({\epsilon }_y<\epsilon \le {\epsilon }_u\right)& \\ {\sigma }_u& \left(\epsilon >{\epsilon }_u\right)& \end{array}\right.$$

(4)

In the equation, E_s denotes the elastic modulus of Q235-grade steel, taken as 210 GPa, and σ_y represents the yield strength of Q235 steel.

The constitutive relationship of reinforcement is expressed as shown in Equation (5).

$$\sigma =\left\{\begin{array}{c}{E}_s\left(\epsilon \le {\epsilon }_y\right)\\ {\sigma }_y\left(\epsilon \ge {\epsilon }_y\right)\end{array}\right.$$

(5)

In the equation, E_s denotes the elastic modulus of Q335-grade reinforcement, taken as 200 GPa, and σ_y represents its yield strength.

As the primary focus of this study lies in the stress and fatigue behavior of the steel longitudinal beam, the model was appropriately simplified. Reference points were defined at the centers of the two supports and were kinematically coupled to the bottom flange of the I-shaped steel beam to facilitate force transmission between the supports and the structure.

The interaction between the concrete slab, bearing blocks, and I-beam was modeled using a penalty contact approach for tangential behavior to simulate friction, and a hard contact formulation in the normal direction. The friction coefficient was set to 0.3, as referenced in [27]. A tie constraint was applied to connect the I-beam and shear studs. To simulate the composite action, both the transverse reinforcement and shear connectors were embedded into the concrete slab using the embedded region method. Boundary conditions were defined by assigning a fixed-hinged support at one end of the steel beam and a roller-hinged support at the opposite end. The external load was applied through a reference point placed on top of the bearing block, representing the actual loading setup.

The finite element model of the research subjects consisted of five main components: the steel longitudinal beam, shear studs, concrete slab, reinforcement, and bearing blocks. Except for the transverse shear reinforcement, which was modeled using T3D2 two-node linear 3D truss elements, all other components—including the steel beam, shear studs, concrete slab, and bearing blocks—were simulated using C3D8R eight-node linear hexahedral elements with reduced integration.

In accordance with the experimental configuration, the steel longitudinal beam was fabricated by welding Q235-grade steel plates into an I-shaped section. To ensure the safety of the welds, it was assumed that only the steel beam bears bending moments, and that the steel reaches its shear strength at the location of maximum shear force. The welds were required to remain intact, and the minimum weld size was calculated accordingly. Based on structural detailing requirements, the weld size h_f was determined to be 5 mm. To ensure that the numerical simulation accurately reflects the experimental conditions, welds with a size of 5 mm were modeled at the junctions between the web and flange plates of the steel beam. The welds were simulated using C3D8R elements, and the mesh configuration in the weld region is illustrated in Figure 4.

The mesh size of a finite element model has a significant impact on the accuracy of simulation results. In simulations involving large material deformations, strain localization may occur, and the calculated strain results can vary depending on the mesh size [28]. If the mesh is too coarse, the computational accuracy of the finite element model may be compromised, thereby reducing the reliability of the simulation results. Conversely, an overly refined mesh may generate a large number of elements, increasing computational cost and time, while also introducing distorted elements that could lead to non-convergence in the analysis.

Structured meshing was employed for all components in the finite element model. The shear connectors were discretized using the “Create Solid Sweep” technique, with an element size of 2.5 mm. For the bearing blocks and concrete slab, mesh sizes were defined as 25 mm and 15 mm, respectively. To evaluate mesh sensitivity, the mesh in the vicinity of the weld between the web and flange plates of the steel beam—especially near the potential crack propagation zone—was refined to minimum element sizes of 0.5 mm, 1.0 mm, and 1.5 mm. The corresponding hot-spot stresses at the mid-span of the steel beam under different mesh densities are illustrated in Figure 5.

As shown in Figure 5, when a mesh size of 1.5 mm was used in the region near the weld between the web and flange plates, the discrepancy in calculated hot-spot stresses compared to the 1.0 mm and 0.5 mm mesh models ranged from 0.5% to 1.7% and 1.3% to 5.1%, respectively. Therefore, to ensure sufficient accuracy in stress calculations while improving computational efficiency, a minimum mesh size of 1.5 mm was adopted in the weld region near the crack surface during the fatigue performance analysis of the steel beam [29]. Meshing of the finite element model is shown in Figure 6.

In steel–concrete composite beam structures, shear connectors play a critical role. They not only enable the effective transfer of shear forces between the steel beam and the concrete flange, but also prevent vertical separation between the concrete and the steel components. The degree of shear connection, denoted as γ, is a key parameter influencing the mechanical performance of composite beams. It is affected by factors such as the diameter, spacing, and distribution of the shear connectors.

By definition, γ is the ratio of the total shear resistance V_r provided by the shear connectors within the shear span to the longitudinal horizontal shear force V_s at the interface induced by the ultimate bending moment. The expression is given in Equation (6).

$$\gamma ={\displaystyle \frac{V_r}{V_s}}$$

(6)

In this section, the influence of the degree of shear connection on the mechanical performance of steel–concrete composite beams is investigated. ABAQUS-based finite element models were developed by varying the longitudinal spacing of shear connectors to 60 mm, 100 mm, 150 mm, and 200 mm, respectively. The corresponding degrees of shear connection γ for each model are listed in

Table 2. Finite element model number and the corresponding nominal degree of shear connection.

Number	Spacing (mm)/Number of Rows	Number of Stud Connectors	γ
LK-1	60/2	64	1.31
LK-2	100/2	40	0.82
LK-3	150/2	26	0.53
LK-4	200/2	20	0.41

. Finite element models LK-1 and LK-2, corresponding to the reference test specimens with longitudinal connector spacings of 60 mm and 200 mm, respectively, are shown in Figure 7.

3.3. Verification of the Finite Element Model

In ABAQUS, displacement-controlled loading was applied to the finite element models to obtain the load–deflection curves of steel–concrete composite beams with shear connector longitudinal spacings of 60 mm and 200 mm. A comparison between the finite element simulation results and the experimental load–deflection curves is presented in Figure 8. As illustrated in figure, the finite element simulation demonstrates good correlation with the experimental results in terms of the load–deflection response. To quantitatively evaluate the consistency between the two, interpolation and statistical error analysis were conducted. For specimen LK-1, the root mean square error (RMSE) and coefficient of determination (R²) were 1278.02 and 0.99, respectively. Similarly, for specimen LK-4, the RMSE and R² were 1306.36 and 0.99, respectively. These results indicate that the stiffness and stress state of the FE model are consistent with those of the tested composite beams, demonstrating comparable static performance.

Model LK-1 represents the composite beam with a shear connector spacing of 60 mm, while LK-4 corresponds to the spacing of 200 mm. The deformation process of the composite beams generally comprised two stages: elastic and plastic. A sudden increase in deflection was observed in all beams at approximately 40~45% of the ultimate load. As the load increased further, the composite beams gradually entered the plastic stage, during which a pronounced plateau appeared in the deformation curves.

3.4. Fracture Mechanics Analysis Model

3.4.1. Determination of Key Parameters

The SIF K is influenced by the local stress at the crack location, crack size, structural geometry, and material properties. Prior to performing fatigue performance analysis of the steel longitudinal beam using the FRANC3D–ABAQUS interactive technique, an initial crack must be predefined.

Regarding the location of the initial crack, relevant studies have been conducted by both domestic and international researchers. Based on extensive fatigue testing of steel beams, Fisher, et al. [30] observed that longitudinal fillet weld roots between the flange and web of welded steel beams often contain defects such as voids or inclusions at tack welds or repair welds. These defects serve as critical points for fatigue failure in steel beams. The typical fatigue crack propagation pattern is illustrated in Figure 9, which shows that fatigue cracks generally propagate radially outward from a circular region centered on the weld root.

Based on the research findings summarized in the introduction regarding fatigue-critical locations in steel beams, cracks are generally initiated at locations of stress concentration and initial defects. In welded steel structures, fatigue cracks typically originate at the weld toe or within the weld itself, and subsequently propagate in a direction perpendicular to the normal stress or principal stress. Accordingly, in the numerical simulation of the fatigue performance of the steel longitudinal beam conducted in this study, the crack plane was assumed to be located at the mid-span section of the beam. Two initial crack positions were considered: one with the crack center located at the weld root between the tensile flange and the web, and the other at the weld toe. A comparative analysis was carried out to evaluate the influence of different initial crack locations on the fatigue performance of the steel longitudinal beam.

Regarding the initial crack size, Hirt and Fisher [31] found through fatigue tests on welded H-shaped steel beams that fatigue cracks tend to initiate at the weld root of the fillet weld between the tensile flange and the web. Based on the back-calculation from fatigue life data, the initial crack size a₀ was estimated to be approximately 1 mm, with a circular shape. Ibsø and Agerskov [32] further suggested that welding defects can be modeled as semi-elliptical surface cracks, with a typical aspect ratio of c₀/a₀ = 4, and the initial crack depth ranging from 0.075 mm to 0.4 mm. Based on the findings of these studies, a circular initial crack with a radius of 1 mm was adopted in the numerical simulation of the fatigue performance of the steel longitudinal beam in this study. Furthermore, in the subsequent parametric analysis, initial crack sizes of 0.2 mm to 0.8 mm were also considered and compared with the 1 mm reference case, in order to investigate the influence of different initial crack sizes on the fatigue performance of the longitudinal fillet weld in the steel beam.

3.4.2. Numerical Modeling for SIF Analysis

A submodel with dimensions of 100 mm × 50 mm × 50 mm was extracted from the high-risk region for fatigue crack initiation identified in the global ABAQUS model, focusing on the connection between the tensile flange and the web. Using the fracture mechanics analysis software FRANC3D, an initial crack was introduced into the submodel to analyze the propagation behavior of fatigue cracks at the weld between the tensile flange and the web at the mid-span of the steel longitudinal beam. The procedure for establishing the three-dimensional finite element model based on the FRANC3D–ABAQUS interactive technique is illustrated in Figure 10. The software versions used in this study are FRANC3D 7.0 and ABAQUS 2020.

To investigate the fatigue performance of the steel longitudinal beam in a steel–concrete composite beam under cyclic loading, the interaction between the fracture mechanics analysis software FRANC3D and the finite element software ABAQUS was utilized. Based on the previously established global finite element model of the composite beam, and with reference to relevant studies by other researchers and recommendations from design codes, the initial crack was defined to originate at the weld root and weld toe of the fillet weld between the tensile flange and the web of the steel beam. The initial crack was assumed to be circular in shape, with an aspect ratio of c₀/a₀ = 1, and the crack plane was oriented perpendicular to the longitudinal axis of the beam. This configuration was used to simulate the evolution of SIFs at the weld under fatigue loading.

4. Analysis of Numerical Results

4.1. Evaluation of Simulation Results for the Model Without Cracks

Before analyzing the fatigue performance of the structure using FRANC3D, the stress results of the ABAQUS finite element model must be obtained. Therefore, prior to conducting the fatigue analysis of the steel longitudinal beam in the steel–concrete composite beam, the mechanical response of the composite beam under static loading was first calculated using ABAQUS. Based on the referenced experiment, a concentrated load was applied at the top of the bearing block located at mid-span of the beam. Four ABAQUS numerical models of steel–concrete composite beams with different longitudinal spacings of shear connectors were analyzed under this loading condition. The von Mises stress distributions under static loading are shown in Figure 11. The corresponding calculated von Mises stresses for the four models were 61.51 MPa, 62.07 MPa, 62.94 MPa, and 63.26 MPa. As shown in Figure 11, a significant stress concentration is observed at the connection between the tensile flange and the web near the mid-span region of the steel longitudinal beam. The maximum stress occurs at the bottom flange at mid-span, with stress magnitude increasing as the location approaches mid-span and decreasing with increasing distance from it.

When the longitudinal spacing of the shear connectors is 60 mm, the stress in the steel beam is the lowest, approximately 61.51 MPa. In contrast, when the spacing increases to 200 mm, the stress reaches the highest value, approximately 63.26 MPa. This indicates that as the longitudinal spacing of the shear connectors increases, the load-carrying capacity of the steel–concrete composite beam decreases, resulting in higher stress levels in the steel longitudinal beam. According to the stress analysis results of the steel–concrete composite beam, the maximum stress occurs at the bottom flange of the steel longitudinal beam at mid-span. The fillet weld connecting the bottom flange and the web—where high tensile stress is concentrated—is prone to fatigue crack initiation.

J.W. Fisher et al. [30] reported through extensive fatigue testing of steel beams that typical defects such as lack of penetration, lack of fusion, and pre-existing cracks are often found at the weld root of the longitudinal fillet weld between the flange and web of welded steel beams, especially at tack welds or repair welds. These welding defects tend to induce localized stress concentrations under loading [33], thereby reducing the overall fatigue life of steel structures. Among these, lack of penetration is one of the most common welding defects. It not only reduces the effective cross-sectional area of the weld and weakens the strength of the welded joint but also causes stress concentration at the defect location, accelerating the initiation and propagation of fatigue cracks. This significantly reduces the fatigue strength of the weld and may pose a greater threat to the structural integrity. Based on the aforementioned studies, lack-of-penetration defects were introduced into the welded detail of the steel longitudinal beam in the finite element model of the steel–concrete composite beam. The FRANC3D–ABAQUS interactive technique was employed to analyze the variation in the SIFs in the presence of lack-of-penetration defects, thereby evaluating their impact on the fatigue performance of the steel longitudinal beam. A rectangular void with mesh dimensions of 2.5 mm × 0.5 mm was embedded at the fillet weld between the bottom flange and the web of the steel beam in the ABAQUS model, simulating the lack-of-penetration defect at this location (Case 2). The case without considering lack-of-penetration defects is defined as Case 1.

Stress analysis was conducted on the finite element model of the steel–concrete composite beam with an introduced lack-of-penetration defect. It was assumed that the location of maximum stress in the model with the defect remained the same as that in the model without the defect, occurring at the bottom flange of the steel longitudinal beam at mid-span. At this location, the maximum stresses in the steel beam for longitudinal shear connector spacings of 60 mm, 100 mm, 150 mm, and 200 mm were calculated to be 61.56 MPa, 62.12 MPa, 62.98 MPa, and 63.30 MPa, respectively. The minimum stress occurred when the connector spacing was 60 mm, and the maximum stress occurred at a spacing of 200 mm. Comparative analysis shows that the variation in maximum stress values of the steel longitudinal beam under different shear connector spacings is relatively small, with a difference of approximately 2.8%. However, the maximum stress value only reflects the overall stress level in the steel beam. For the subsequent analysis of the fatigue performance of welded details, it is necessary to focus on the local stress conditions at specific locations.

As shown in Figure 12, a comparison was made between the von Mises stress results of the finite element models with and without the lack-of-penetration defect. It was found that the maximum stress values before and after introducing the defect were nearly identical. The largest difference was observed in the model with a shear connector spacing of 60 mm, with a discrepancy of approximately 0.08%. The overall stress distribution pattern in the steel–concrete composite beam also appeared to be relatively similar between the two models.

As shown in Figure 13, by focusing on the region near the lack-of-penetration defect, it can be observed that the stress distribution and magnitude in the finite element model with the defect differ significantly from those in the original model. Notably, the nodes along the upper surface of the bottom flange at mid-span are located very close to the defect. As illustrated in Figure 14, the position along the upper surface of the bottom flange was normalized, with 0 and 1 representing the two ends, and 0.5 indicating the mid-span node. A comparison of the nodal stresses between the two models shows that the lack-of-penetration defect induces localized stress concentration in the surrounding area. Since the mid-span node is closest to the defect, it exhibits the most significant stress variation. Compared to the model without the defect, the stress at this location increased by approximately 1.3%, which may influence parameters used to assess the fatigue performance of the steel beam.

4.2. Analysis of SIFs

4.2.1. Initial Crack Location

To investigate the influence of different parameters on the SIFs of fatigue cracks at welded joints in the steel longitudinal beam of a steel–concrete composite beam, three parameters were selected for analysis: the longitudinal spacing of shear studs, the initial crack size, and the initial crack shape. The variation in SIF under these parameters was further examined. All SIF calculations were performed under a constant applied load of 20 kN.

Based on the FRANC3D–ABAQUS interactive technique, a deeply embedded crack was inserted at the weld root, and a surface crack was inserted at the weld toe of the fillet weld between the tensile flange and the web of the steel beam, for the case where the longitudinal spacing of shear connectors was 60 mm. Both initial cracks were assumed to be circular with a size of a₀ = 0.2 mm, and their crack planes were oriented perpendicular to the longitudinal axis of the beam. Fifteen-node wedge-shaped singular elements were used along the crack front. The SIFs at the weld details of the steel beam under cyclic loading were analyzed using the built-in M-integral method in FRANC3D. The crack front was normalized, and the resulting distribution of SIFs is shown in Figure 15.

As shown in Figure 15, both the embedded crack at the weld root and the surface crack at the weld toe of the steel beam weld detail are primarily Mode I (opening mode) dominant mixed-mode cracks. The SIFs K_I along the crack front of the embedded crack at the weld root are nearly uniform, whereas the K_I values along the crack front of the surface crack at the weld toe exhibit a concave distribution—decreasing initially and then increasing—with the maximum values located at the two ends of the crack front and the minimum value at the deepest point. Moreover, the overall K_I values of the crack at the weld toe are greater than those at the weld root. The SIFs for Mode II and Mode III (K_II, K_III) are nearly zero, indicating that the fatigue crack propagation is dominated by Mode I (opening mode). Therefore, in the fatigue performance analysis of the steel longitudinal beam, only the Mode I SIF K_I is considered, while the influence of Mode II (sliding mode) and Mode III (tearing mode) on crack propagation can be neglected. The local stress contour plots after crack insertion at the weld root and weld toe are shown in Figure 16. The initial crack at the weld root shown in the figure is a complete circular internal crack, while the crack at the weld toe is a surface crack. The partial contour of the inserted crack is indicated by red dashed lines, and points A and B represent the two ends of the initial crack.

4.2.2. Longitudinal Spacing of Studs

Based on the previously developed ABAQUS numerical models of steel–concrete composite beams with four different longitudinal spacings of shear studs, initial cracks were inserted at both the weld root and weld toe of the fillet weld between the tensile flange and the web of the steel beam in each model. The initial crack depth a₀ was set to 1 mm, with a circular shape (c₀/a₀ = 1), and the crack plane was oriented perpendicular to the longitudinal axis of the beam. The influence of different shear stud spacings on the SIF was analyzed. The crack front was normalized, and the computed SIF values at each node along the crack front are shown in Figure 17. The SIF values at the deepest point of the crack front are presented in Figure 18.

As shown in Figure 17, the K_I of fatigue cracks at both the weld root and weld toe increases gradually with the increase in longitudinal spacing of the shear studs. Under the same spacing condition, the embedded cracks at the weld root consistently exhibit lower K_I values compared to the surface cracks at the weld toe. As illustrated in Figure 18, the minimum K_I value at the deepest point of the fatigue crack occurs when the shear stud spacing is 60 mm, whereas the maximum K_I value is observed when the spacing reaches 200 mm. Specifically, the Mode I SIF for the embedded crack at the weld root ranges from a minimum of 68.11 MPa·mm^1/2 to a maximum of 69.86 MPa·mm^1/2, reflecting an increase of 2.6%. For the surface crack at the weld toe, K_I increases from 69.01 MPa·mm^1/2 to 70.71 MPa·mm^1/2, an increase of 2.5%. Therefore, reducing the longitudinal spacing of shear studs leads to a decrease in the K_I of fatigue cracks. This is beneficial for improving the fatigue performance of welds in the steel longitudinal beam.

4.2.3. Initial Crack Size

In the previous calculation of the K_I for fatigue cracks at welds in the steel longitudinal beam, an initial crack depth of a₀ = 1 mm was selected. To investigate the influence of the initial crack size on the fatigue performance of the structure, this section maintains the longitudinal spacing of shear studs (60 mm) and the initial crack shape (c₀/a₀ = 1) unchanged, and analyzes the SIFs for fatigue cracks with initial depths of a₀ = 0.2 mm, 0.4 mm, 0.6 mm, and 0.8 mm. The crack front was normalized, and the calculated K_I values at each node are shown in Figure 19 and Figure 20.

As illustrated in Figure 19 and Figure 20, when other parameters are held constant, the K_I for fatigue cracks at both the weld root and weld toe increases progressively with the growth of initial crack depth. Under the same initial crack size, the K_I values at the weld root are consistently lower than those at the weld toe. When a₀ = 0.2 mm, the minimum K_I at the deepest point of the fatigue crack is observed; when a₀ = 1.0 mm, the maximum value is reached. Specifically, at the weld root, K_I increases from a minimum of 30.29 MPa·mm^1/2 to a maximum of 68.11 MPa·mm^1/2, an increase of 124.9%. At the weld toe, K_I rises from 30.55 MPa·mm^1/2 to 69.01 MPa·mm^1/2, an increase of 125.9%. These results indicate that the initial crack size has a significant influence on the fatigue performance of welds in the steel longitudinal beam. Controlling the initial crack size is therefore an effective measure to improve fatigue resistance.

4.2.4. Initial Crack Shape

To investigate the influence of the initial crack aspect ratio on the K_I of fatigue cracks, the previously modeled initial crack in the weld of the steel longitudinal beam was assumed to be circular (c₀/a₀ = 1). In this section, the longitudinal spacing of the shear studs (60 mm) and the initial crack depth (a₀ = 0.2 mm) are kept constant, while different initial crack aspect ratios c₀/a₀ = 2, 3, 4, and 5 are considered. The corresponding SIFs are calculated to evaluate the influence of initial crack shape on K_I. The crack front was normalized, and the computed K_I values at each node along the front are presented in Figure 21. The SIFs at the deepest point and at points A and B at the two ends of the crack front are shown in Figure 22.

As shown in Figure 21a, for the embedded crack at the weld root, under different initial crack shapes, when the aspect ratio c₀/a₀ = 1, the crack is circular, and K_I along the crack front is nearly uniform. This indicates that crack propagation in this case is relatively stable and tends to maintain its circular shape as it grows. When c₀/a₀ > 1, the K_I distribution at the weld root exhibits a bimodal wave-like pattern along the crack front. From the two ends of the crack front toward the deepest point, K_I initially increases and then decreases. The maximum values appear at the upper and lower ends of the crack front, while the minimum values are found at the lateral ends. The K_I value at the deepest point is also relatively low, suggesting that the crack is more likely to propagate along the upper and lower surfaces. As illustrated in Figure 22, the K_I values at points A, B, and the deepest point of the weld root crack front are approximately equal and all decrease as c₀/a₀ increases. However, for most other nodes along the crack front, K_I tends to increase with increasing c₀/a₀.

As shown in Figure 21b, for the surface crack at the weld toe, under different initial crack shapes, when the aspect ratio c₀/a₀ = 1, the SIFs K_I at the two end points of the crack front (points A and B) are nearly equal and reach the maximum values, while the minimum K_I occurs at the deepest point of the crack. When c₀/a₀ > 1, the distribution of K_I along the weld toe crack front exhibits an undulating wave-like pattern. In this case, both the locations of the maximum and minimum K_I values shift, and a significant difference arises between the K_I values at points A and B. As c₀/a₀ increases, the fluctuation range of K_I becomes larger. In particular, when c₀/a₀ = 5, the difference between the maximum and minimum values of K_I reaches its greatest extent.

As shown in Figure 22, due to the varying load effects along different positions of the crack front, stresses of different magnitudes are generated, resulting in distinct trends in K_I at different locations along the weld toe crack front as the initial crack aspect ratio c₀/a₀ increases. K_I at point A decreases with increasing c₀/a₀, whereas the K_I values at point B and the deepest point increase with increasing c₀/a₀. Taking point B and the deepest point as examples, when c₀/a₀ = 5, the maximum K_I values are observed, reaching 76.29 MPa·mm^1/2 and 47.37 MPa·mm^1/2, respectively. When c₀/a₀ = 1, the minimum K_I values at the same locations are 35.87 MPa·mm^1/2 and 30.55 MPa·mm^1/2, respectively. In comparison, the increase in c₀/a₀ results in a 112.7% and 55.1% increase in K_I at point B and the deepest point, respectively, along the weld toe fatigue crack front.

Overall, for both the embedded crack at the weld root and the surface crack at the weld toe, K_I at most nodes along the crack front increases with the rise in the aspect ratio c₀/a₀. When c₀/a₀ > 1, the K_I distribution at the weld root exhibits a bimodal wave-like pattern, while the distribution at the weld toe presents a vertical wave-like pattern. Under these conditions, it becomes unsuitable to compare at the deepest point of the crack front across different aspect ratios. Therefore, in this study, the peak values of the wave-like distributions at the weld root and weld toe are selected to analyze the variation in K_I with respect to different crack aspect ratios.

As shown in Figure 23, when c₀/a₀ = 1, the minimum K_I along the fatigue crack front is observed, while the maximum occurs at c₀/a₀ = 5. Specifically, the K_I value at the weld root increases from 30.42 MPa·mm^1/2 to 48.79 MPa·mm^1/2, an increase of 60.4%; at the weld toe, it increases from 30.65 MPa·mm^1/2 to 51.41 MPa·mm^1/2, representing a 67.7% increase. The change in the initial crack shape not only alters the magnitude of K_I for surface cracks at the weld toe, but also affects the variation trend in K_I at different points along the crack front. Based on this analysis, it can be concluded that the initial shape of the fatigue crack has a significant influence on the fatigue performance of welds in the steel longitudinal beam. Controlling the initial crack shape is therefore an effective approach to improving the fatigue performance of welded joints.

4.2.5. Weld Penetration Rate

The crack front was normalized, and the distribution of SIFs under the influence of the lack-of-penetration defect is shown in Figure 24. In this case, the longitudinal spacing of the shear connectors in the composite beam finite element model was 60 mm. An embedded crack was introduced at the weld root, and a surface crack was introduced at the weld toe of the weld between the tensile flange and the web of the steel beam. The initial crack shape was circular (c₀/a₀ = 1), the initial crack depth was a₀ = 0.2 mm, and the crack plane was oriented perpendicular to the longitudinal axis of the beam.

As shown in Figure 24, under the influence of the lack-of-penetration defect, the SIF distribution along the surface crack front at the weld toe remains nearly the same as that in the model without the defect, exhibiting a concave (U-shaped) distribution. The difference in K_I magnitude is minimal. In contrast, the distribution of the Mode I SIF along the embedded crack front at the weld root differs significantly from the defect-free model. In this case, K_I first decreases and then gradually increases, with the maximum values occurring at the crack front nodes closest to the lack-of-penetration defect, and the minimum values occurring at the node farthest from the defect. This trend is attributed to the significant stress concentration that appears at the defect location when the composite beam finite element model with the lack-of-penetration defect is subjected to loading.

To further investigate the influence of lack-of-penetration defects on the fatigue performance of welds in the steel longitudinal beam, these defects were introduced into composite beam finite element models with different longitudinal spacing of shear connectors. The SIFs obtained from models with and without the lack-of-penetration defect were compared. The crack front was normalized, and the calculated K_I values at each node along the crack front are shown in Figure 25. For both the embedded cracks at the weld root and the surface cracks at the weld toe, K_I increases with increasing longitudinal spacing of shear connectors. This trend is consistent with that observed in the models without the lack-of-penetration defect.

The calculated K_I values at the deepest point of the crack front were compared between the models with and without the defect. As shown in Figure 26, K_I at the deepest crack front node increases with increasing shear stud spacing. The minimum value occurs when the spacing is 60 mm, and the maximum when it is 200 mm. Specifically, when the longitudinal spacing is 150 mm, the influence of the lack-of-penetration defect on the embedded crack at the weld root is the most significant. In this case, K_I increases from 69.57 MPa·mm^1/2 (without the defect) to 70.06 MPa·mm^1/2 (with the defect), an increase of 0.7%. Similarly, for the surface crack at the weld toe, the maximum influence occurs at the same spacing, with K_I increasing from 70.41 MPa·mm^1/2 to 70.46 MPa·mm^1/2, an increase of 0.08%.

In summary, by introducing lack-of-penetration defects into composite beam finite element models with varying longitudinal spacings of shear connectors and comparing the calculated SIFs, it was found that K_I along the crack fronts of both embedded cracks at the weld root and surface cracks at the weld toe increased with increasing shear stud spacing. This further confirms that reducing the longitudinal spacing of shear connectors can effectively lower the SIF of fatigue cracks and thereby improve the fatigue performance of welds in the steel longitudinal beam of steel–concrete composite beams. Analysis of the variation trends in K_I at the deepest point of the crack front indicates that the introduction of lack-of-penetration defects increases the SIF of cracks at different weld locations. The defect has a noticeable effect on both the magnitude and distribution of K_I for embedded cracks at the weld root, while its influence on surface cracks at the weld toe is minimal and can be considered negligible.

4.3. Fatigue Crack Propagation and Life Estimation

To further analyze the influence of various parameters on fatigue crack propagation and life of the steel longitudinal beam, this study considers the initial crack location and aspect ratio as key variables. In practical engineering, fatigue cracks may also develop in shear connectors under repeated loading. However, to isolate the effects of other parameters, the following assumptions were made: Fatigue cracking of shear connectors and degradation of their mechanical properties during fatigue loading are not considered. The longitudinal spacing of shear connectors is fixed at 60 mm, and the initial crack is assumed to be circular, with an aspect ratio of c₀/a₀ = 1. Initial crack depths of 0.2 mm, 0.4 mm, 0.6 mm, 0.8 mm, and 1.0 mm were selected to investigate their effect on the fatigue life of welds in the steel longitudinal beam. The termination criterion for crack growth is set as half of the bottom plate thickness [18]. Lastly, the parameters used for fatigue crack propagation were set as follows: C = 2.74 × 10⁻¹³, m = 2.941, and the stress ratio R = 0.1 [10]. The results are shown in the Figure 27.

According to the calculation results, once the crack reaches the termination condition, the a–N curves (crack size vs. number of cycles) for both the weld root and weld toe exhibit similar trends. As crack propagation progresses, the growth rate increases gradually, and most of the fatigue life is consumed during the early stage when the crack is small. When the crack reaches a depth of 4 mm, the fatigue life at the weld root is 1.725 million cycles, while at the weld toe it is 1.625 million cycles, showing a difference of 6.1%. This comparison indicates that an initial defect at the weld toe is more detrimental than one at the weld root. As the initial crack size increases, the fatigue life decreases significantly. For the weld root, the fatigue life is 5.351 million cycles at a₀ = 0.2 mm and decreases to 1.725 million cycles at a₀ = 1.0 mm, representing a 67.8% reduction. For the weld toe, the fatigue life drops from 5.097 million cycles to 1.625 million cycles, a reduction of 68.1%.

Moreover, the most pronounced decrease in fatigue life occurs as the initial crack depth increases from 0.2 mm to 0.4 mm, while the reduction rate slows when the depth exceeds 0.6 mm. Comparatively, the fatigue life at the weld toe is more sensitive to the longitudinal spacing of shear connectors. In conclusion, the initial crack size has a significant impact on the fatigue performance of welds in steel longitudinal beams. Controlling the initial crack size can effectively improve fatigue resistance and extend the fatigue life of such welded structures.

5. Conclusions

In this study, a fully three-dimensional finite element model incorporating cracks was developed based on the FRANC3D–ABAQUS interactive technique to investigate the SIFs of fatigue cracks in the welds of steel longitudinal beams in steel–concrete composite beams. The main conclusions are as follows:

(1)

Both the embedded crack at the weld root and the surface crack at the weld toe are predominantly Mode I-dominated mixed-mode cracks. Due to differences in crack type and stress conditions, the Mode I SIF along the front of the embedded crack at the weld root remains nearly uniform, while that of the surface crack at the weld toe first decreases and then increases, forming a concave distribution. The overall K_I values for the weld toe crack are greater than those for the weld root crack.

(2)

The magnitude of the Mode I SIF is positively correlated with the shear stud longitudinal spacing, initial crack size, and crack aspect ratio c₀/a₀. Among these, the initial crack size and shape have a more significant impact on K_I. Changes in the aspect ratio c₀/a₀ also affect the distribution pattern of K_I along the crack front. With increasing aspect ratio, the K_I at the weld root exhibits a bimodal wave-like distribution, while at the weld toe it develops into a vertical wave-like pattern.

(3)

The introduction of a lack-of-penetration defect leads to an increase in the SIF of weld cracks in the steel longitudinal beam. The effect on the surface crack at the weld toe is minor, while its influence on the embedded crack at the weld root is more pronounced, significantly affecting both the magnitude and the distribution of K_I along the crack front.

(4)

Among the influencing factors considered in this study, the initial crack size has the greatest impact on the SIFs. When the crack depth increases from 0.2 mm to 1.0 mm, K_I at the weld toe increases by 125.9%. Correspondingly, the fatigue life at the weld toe decreases significantly, dropping by approximately 68.1%. Controlling the initial crack size is critical for improving the fatigue performance and service life of steel longitudinal beam welds.

(5)

This study only considered the longitudinal spacing of shear connectors, without further analysis of other critical parameters such as connector diameter, welding method, and stiffness, which presents certain limitations. Future work can build upon this study to conduct multi-parameter coupled analyses, aiming to more comprehensively reveal the effects of connector parameters on the fatigue performance of steel girders and the overall structural response.