In ABAQUS, the C3D20R element was selected, and the mesh around the hole was locally refined. Under uniaxial tensile conditions, the left end of the sheet steel was fixed, and a uniform load of 200N was applied to the right end (Figure 3 illustrates the example of plate with a circular hole). The stress-strain conditions around the hole, as calculated, were depicted in Figure 4, and the mechanical properties were presented in Table 1.

From Figures 3, 4; Table 1, the results were shown as: 1) The distribution of stress and strain around the hole periphery was expressed a typical “butterfly shape” and stress concentration. The geometric discontinuity caused by the hole opening led to the distortion of stress lines around the hole. The denser the strain lines around the hole, the more pronounced the stress concentration was observed to be. 2) The stress concentration around the diamond-shaped hole was the most pronounced, with the maximum equivalent stress value reaching 46.13 MPa; the stress concentration around the circular hole plate was the lowest, with the maximum equivalent stress value being 21.05 MPa. The shape of the hole directly influenced the distribution of stress.

In accordance with the relevant provisions of the Chinese National Standard GB/T228.1-2010 (Li et al., 2024), specimen preparation and tensile testing were conducted. Tensile tests were conducted using an electronic universal testing machine, with a maximum static load of 300 kN and a loading rate of 2 mm/min. Each type of specimen was tested three times, grouped and numbered as TS1, TS2, and TS3. The load-displacement curves for the unperforated and four perforated specimens are shown in Figures 5, 6, respectively.

The experimental results indicated that: Macroscopically, 1) DC04 unperforated specimens exhibited significant plastic deformation when subjected to load, without immediate fracture. After the specimen fractured, the fracture angle was approximately 60°, which differed from the theoretical value of 45°. The reasons for this discrepancy included the specimen’s good toughness, its thinness, susceptibility to warping deformation during the manufacturing process, and the influence of material elemental characteristics on the stress state at the fracture. 2) Post-yield, the force-displacement curves and the morphologies of the holes began to exhibit noticeable differences. Rectangular and diamond-shaped holes were the first to crack, with significant stress concentration at the two sharp corners perpendicular to the stretching direction, leading to rapid fracture after reaching the maximum tensile stress. In contrast, the stress concentration around circular and oval holes was less pronounced, with smaller cracks around the holes and better ductility.

In summary, numerical simulations and experimental results had demonstrated that the shape of holes significantly influenced stress concentration and crack propagation. Holes with smooth boundaries and gradual geometric transitions, such as circular and oval holes, exhibited less stress concentration and smaller cracks around them, resulting in better ductility. Consequently, the study primarily focused on the stress concentration issues of plates with circular and oval holes in the subsequent research.

A model of the sheet steel with a central circular hole was shown in Figure 7. Its length width and thickness were expressed as A , B and t . A radius of the circular hole was r . Uniform tensile loads q were applied to both sides of the plate. The coordinate origin set at the center of the hole and the axes of the coordinates aligned parallel to the boundaries of sheet steel. Assuming that r , t ≪ A , B , the model was simplified to a plane problem for analysis.

The circumferential normal stress component surrounding the hole was derived using the superposition method, as reported in (Xu, 2013): σ ρ = q 2 1 − r 2 ρ 2 + q 2 cos ⁡ 2 φ 1 − r 2 ρ 2 1 − 3 r 2 ρ 2 where the polar coordinate system, ρ represents the radial distance; φ denotes the angle. Along the direction of axis x and y , when φ = 0 ∘ , φ = 9 0 ∘ , the circumferential normal stress σ φ = 0 ∘   σ φ = 9 0 ∘ were derived as: σ φ = 0 ° = − q 2 r 2 ρ 2 3 r 2 ρ 2 − 1 σ φ = 90 ° = q 1 + 1 2 r 2 ρ 2 + 3 2 r 4 ρ 4

When ρ = r , σ φ = 90 ° max = 3 q , the maximum circumferential normal stress was located around the hole, with a stress value approximately three times that of the applied uniform tensile load, and as the distance from the hole increased, the stress value gradually approached q . Hence, the stress concentration factor in this mechanical model was considered to be: K r = σ φ = 90 ° max q = 3

When B was set to 30 mm, the selection of hole diameter r was referred to Table 2, following the model depicted in Figure 7C. The boundary conditions were consistent with those shown in Figure 3. The aspect ratio, denoted as α = 2 r / B , was considered in the analysis.

The numerical solution K and the analytical solution K r for the complete cross-section were analyzed, and their relative error rate E 1 was computed. A comparative analysis was conducted between the numerical solution k and the finite element solution k r for the net cross-section, and their relative error rate E 2 was calculated. The stress concentration factor k r , which is calculated based on the film stress σ f obtained from finite element analysis (in the case of oval hole plate, it is denoted as k e ), was determined. The relationship curve between α and K , k was depicted in Figure 8.

These results indicated that: As shown in Figure 8A, when α > 0.2 , the relative errors of K and K r gradually increased due to the continuous reduction of the ligament area surrounding the hole. When α ≤ 0.2 , K rapidly approached K r = 3 , E 1 ≤ 4.853 % , indicating that the stress concentration problem around a finite large plate hole could be approximated as an infinitely large plate hole problem, hence simplifying to the ‘small hole problem’. In such cases, K r was used to approximate K .

In the relationship curve of Figure 8B, k and k r exhibited a negative correlation with α . The relative error between k and k r , E 2 < 2 % , and as α increased, the two tended to equalize. When α = 1 was reached, k r would decrease to around 2. Consequently, by utilizing the Levenberg-Marquardt iterative algorithm to fit the k r curve, an empirical formula for the stress concentration factor of the central circular hole sheet steel based on the net section was derived: k r = 1.86 + 1.07 × 0.1 α (3)

The formula was compared with the computational results of Hey Wood (Frankl, 1953) (referred to as k h in the text) and was found to have higher precision, while the trend of the relationship curves between the two was consistent. It was also discovered that as α increased, k r approached its lower limit value of 2, which was a finite value for any given α , while K itself grew larger without an upper limit. The influence of pores present on the material cross-section was considered for k r . Therefore, using k r to represent the stress concentration around the hole is more accurate, and Equation 3 is more in line with practical engineering applications.

The same mechanical model of the sheet steel as in Figure 9A was used and the mechanical model was simplified to a planar problem for analysis (Figure 9B). The hole at the center of the thin plate was designed as an oval hole (with the major axis as a , and the minor axis as b , a > b ), the boundary conditions were set in accordance with the previous specifications.

By referring to the work cited as (Tan, 2015), the problem was addressed using methods from complex function theory, resulting in the formulation of σ φ . σ φ = q 1 − m 2 − 2 m ⁡ cos ⁡ 2 α + 2 ⁡ cos ⁡ 2 φ + α 1 + m 2 − 2 m ⁡ cos ⁡ 2 φ where α was defined as the angle between the x -axis and the direction in which the load was applied; φ was referred to as the angle in the polar coordinate system; m was described as a real constant ( m = a − b a + b ).

When α = 0 ° , the minor axis of the oval hole became parallel to the direction of the tensile force, and the maximum circumferential tensile stress σ φ = 0 , π max = q 1 + 2 a b was observed at the extremities of the major axis. Thus, the maximum stress concentration factor within that mechanical model was identified to be: K e = σ φ = 0 , π max q = 1 + 2 a b

When B was set to 30 mm, the selection of hole diameter r was referred to Table 3. The boundary conditions were set in accordance with the previous specifications. The aperture factor, denoted as β = b / a , was considered in the analysis.

The numerical solution K and the analytical solution K e for the complete cross-section were analyzed, and their relative error rate E 3 was computed. A comparative analysis was conducted between the numerical solution k and the finite element solution k e for the net cross-section, and their relative error rate E 4 was calculated. The relationship curve between β and K , k was depicted in Figure 10.

Research indicated that K , K e and β exhibited a negative correlation. As β increased, the two rapidly approached each other, leading to an extreme scenario in the evolution of the oval hole diameter, namely a = b , where the elliptical hole evolved into a circular hole, at which point K e = K r = 3 occurred (Figure 10A). Within this range, when β ≥ 0.7 , E 3 ≤ 4.98 % was in accordance with the standards of practical engineering, proving that K e could be used to approximate K in solving the stress concentration problem of the oval hole plate; when β ≤ 0.7 , another extreme situation in the evolution of the oval hole diameter occurred, namely a ≫ b , where the oval hole evolved into a slender “crack”. Changes in its shape ratio led to changes in the geometric properties of the structure, especially a sharp decrease in the radius of curvature ( ρ = b 2 / a ) at the ends of the major axis, and this kind of geometric singularity significantly intensified the stress concentration in that area.

In Figure 10B, a negative correlation was observed between k , k e , and β . Furthermore, as β increased, E 4 < 3 % was observed to rapidly converge towards equality with the other variables. Consequently, by employing the Levenberg-Marquardt iterative algorithm to fit the k e curve, an empirical formula for the stress concentration factor of a centrally located oval hole in sheet steel, based on the net section, was obtained. k e = 2.29 + 8.54 × 0.03 β (4)

By maintaining the constant area of the perforation in sheet steel and analyzing the stress concentration around the hole with the evolution of the oval hole aperture, the study of stress concentration in finite large plates containing oval holes had been refined to a certain extent. Equation 4 employed a unique variable to precisely quantify the degree of stress concentration around the hole, with succinctly characterizing the two extreme scenarios of oval hole diameter evolution. This approach enhanced efficiency in structural design and engineering calculations.