Abstract
In order to address the impact of the perforated parameters on the mechanical properties of the plate, the ultimate strength of hyper-ellipse flanged-perforated plates under uniaxial compression stress is numerically investigated in this article. The four edges of the flanged-perforated plate are only supported in the out-of-plane direction while the plate is exposed to uniaxial compressive loads. The impact of the cutout size, flange height, cutout position, rotation angle, and cutout form on the ultimate bearing capacity of the perforated plate with varied thicknesses is investigated and compared through a series of elasto-plastic buckling analyses using the ANSYS software. The structure’s stress and deformation analysis is then used to explain the results of the ultimate strength test. The flange efficiently raises the maximum bearing strength of the structure with cutouts. For the limit strength of thick plate, the cutout size, elliptical shape, cutout rotation angle, and cutout position have considerably more of an impact than they do on the maximum bearing capacity of thin plate. The findings can assist the structural layout of this sort of perforated plate, and the right cutout parameters should be chosen in accordance with the various performance specifications.
1 Introduction
A sufficient number of cutouts of various shapes and sizes are frequently constructed in the plate structure in the aerospace industry in order to minimize the weight of the structure, facilitate inspection and maintenance, etc. In comparison to a plate without cutouts, the presence of cutouts not only reduces the weight of the structure, but also has a significant impact on the mechanical properties of the structure, such as natural frequency and structural load-bearing performance of the structure. Therefore, analysis of the mechanical properties of the structure with cutouts has been a hot issue in the research [1].
There have been studies of the effects of various cutouts (including square, circular, elliptic, and triangular shapes) on the ultimate strength and buckling resistance of perforated plates. The impact of circular and rectangular cutouts on the linear and elasto-plastic buckling behavior of a simply supported plate under uniaxial and biaxial loads, for instance, was investigated using finite element analysis in ANSYS by El-Sawy et al. [2,3,4]. The ultimate strength of plates with cutouts was investigated under several stress scenarios, including side shear [5], axial compression [6], biaxial compression, and side shear combined loading [7]. Several closed-form empirical formulas for forecasting the ultimate strength of perforated plates are provided, and they are based on the regression analysis of the findings of the finite element analysis. Researchers investigated the effects of two types of loads on the elastic buckling of a perforated rectangular plate: a linearly variable in-plane normal force [8] and a partial edge load [9]. Researchers looked at the buckling behavior of a simply supported rectangular plate with an elliptical cutout when it was subjected to uniaxial compression [10]. The influence of these cutouts on the buckling behavior of simply supported rectangular plates under in-plane compression and bending moment loads led to the proposal of the ideal location and direction of the circular and rectangular cutouts [11]. Moen and Schafer [12] have established an approximation of the elastic buckling stress of a plate with one or more cutouts under compression or bending loads. Prajapat et al. [13] studied the effect of in-plane boundary conditions on the elastic buckling behavior of solid and perforated plates. The optimal position (both location and orientation) of a single circular/elliptical cutout for maximum buckling load of an orthotropic rectangular plate have been studied by Choudhary and Jana [14], employing a MATLAB optimization routine coupled with buckling computation in ANSYS. Buckling and postbuckling responses of hybrid composite plates with different cutouts under uniaxial compression loading have been conducted through experimental and numerical investigations by Vummadisetti and Singh [15]. Ghannadpour and Mehrparva [16] obtained the nonlinear and post-buckling responses of relatively thick functionally graded plates with oblique elliptical cutouts using a new semi-analytical approach. The buckling and vibration of laminated plates with a cutout under various types of non-uniform compressive edge loads was analyzed by Kurpa et al. [17].
As previously indicated, a significant amount of research has been done on the buckling behavior of perforated plates with various cutout forms, aspect ratios, slenderness ratios, and locations, and it has been proposed that cutout results in decreased buckling strength of this type of plates. Adopting the proper cutout-strengthening techniques is important to overcome the buckling and ultimate strength reduction induced by cutout, in order to comply with specific technical requirements, such as normal use and structural safety. As a result, perforated plate reinforcing techniques were researched. Cheng and Zhao studied the buckling properties of stiffened perforated plates using various stiffeners, including annular stiffeners, flat stiffeners, longitudinal stiffeners, and transverse stiffeners [18]. Kim et al. [19] used axial compression and in-plane edge shear loads to assess the effects of various stiffening techniques, such as carling, surface plating, and pressure doubling, on the buckling and ultimate strength of perforated plates. Investigations on the axial buckling of a structure reinforced with band and middle tube resulted in a significant increase in the buckling strength [20]. In some cases, the stiffened plate has buckling load greater than the perfect plate. Jana [21] considered maximization of the buckling load of a simply supported rectangular perforated plate. The effects of planer, longitudinal, and ring type stiffeners on the buckling strength of composite laminates with circular cutout were investigated by Shojaee et al. [22], using experimental and numerical method. Rajanna et al. [23,24,25,26] addressed the effects of reinforced/unreinforced circular/elliptical cutouts and non-uniform in-plane edge loads on the buckling behavior of composite panels by employing the finite element approach. Although there are numerous techniques that aim to increase the buckling and ultimate bearing capacity, investigations of their efficacy are typically incomplete and rely heavily on engineering expertise.
In actual engineering, when cutouts are made in a structure, a particular height of the flange will also be generated simultaneously and is referred to in our research as flanged cutout. Our earlier research reported the effects of flanged cutouts on structural dynamic performance, taking into account the effects of various circular [27], circle-rectangular [28], and hyper-elliptical cutout parameters [29]. The ultimate strength of a rectangular plate with a flanged hyper-ellipse cutout is a key topic in this article. The four edges of the plate are only supported in the out-of-plane direction while the plate is subjected to uniaxial compressive pressures. The impact of the cutout size, flange height, cutout position, rotation angle, and cutout form on the ultimate bearing capacity of the perforated plate with various thicknesses is examined using a series of ANSYS finite element analyses. The research findings in this work have some practical relevance for the structural design.
2 Model of hyper-ellipse flanged-perforated plates
2.1 Description of the cutout boundary
Figure 1 depicts a rectangular plate with an elliptical flanged-cutout. The cutoff border is specified by the hyper-elliptic equation to reduce stress concentration as follows:
where η is the exponential of a hyper-elliptic equation, such as an ellipse (η = 2) or a rectangle (η = 5), and a and b are the ellipse’s semi-axis lengths along the x and y axes, respectively. The cutout border is given in terms of parameters that characterize both the position and rotation as follows:
where (x 0, y 0) is the ellipse centroid’s coordinate and β ∈ (0, 2π) is an auxiliary parameter. The cutout’s area is expressed as follows:
where B(1/η, 1/η) is a beta function, when η = 2, B(1/2, 1/2) = π. Then, the cutoff ratio is therefore defined as follows:
Rectangular plate with ellipse cutout.
2.2 Finite element model and meshing
The model of hyper-ellipse flanged-perforated plate is depicted in Figure 2(a). The size of the flanged-perforated plate is 450 (L) × 300 (W), thickness is 2, 4, and 8 mm, respectively, and the distance between the inner and outer ellipse is 5 mm. They are made of the same material and thickness. The material has Young’s modulus E = 210 GPa, Poisson’s ratio
A plate with hyper-ellipse flanged cutout (a) and its finite element model (b).
Cutout plates are frequently supported by plates next to them in engineering constructions, which places them in a state between rotatable constraint and fixed constraint in the out-of-plane direction. Therefore, this study considers that all edges of the plate are simply supported, this approximation has been proved in previous studies [18]. In addition, considering that no rigid body motion and translation in the plane of the plate are allowed, the in-plane boundary conditions should be applied to the finite element model. In this work, the midpoint of the bottom edge is constrained along the x direction, and the degrees of freedom of the two bottom corners are fixed along the y direction, as illustrated in Figure 2.
In the process of analysis, ANSYS is used as the numerical tool for elasto-plastic buckling analysis. Plate elements (SHELL181) with four nodes and six degrees of freedom for each node were used in the present research, with the influence of transverse shear deformation being considered. Considering the stress sensitivity to the mesh density around the cutout, the meshes are subdivided near the cutout. To verify the applicability of mesh size, a typical hyper-ellipse flanged-perforated plate (the parameters are a = 100 mm, b = 50 mm, h = 6 mm, t = 2 mm, θ = 0, x 0 = 0, y 0 = 0, and η = 2) is employed. Table 1 compares the effects of various mesh sizes on the accuracy of the results of buckling analysis. The findings demonstrate that the elasto-plastic ultimate strengths deviate very little (related to mesh size of 4 mm). Next a mesh size of 4 mm is chosen because it is thought to be accurate enough for the investigation.
Verification of mesh independence
| Size (mm) | σ xu /σ Y | Deviation (%) |
|---|---|---|
| 4 | 0.229706 | 0.00 |
| 2 | 0.229179 | 0.23 |
| 8 | 0.230891 | 0.52 |
2.3 Buckling analysis method
Eigenvalue (or linear) buckling analysis and nonlinear buckling analysis are the two methods available in the ANSYS professional programs for estimating the buckling mode shape and buckling load of a structure. According to the elastic buckling analysis, eigenvalue buckling analysis predicts the theoretical buckling strength (the bifurcation point, Figure 3) of an ideal linear elastic structure. For instance, the conventional Euler solution will match the eigenvalue buckling analysis of a column. The majority of real-world structures, however, are unable to reach their theoretical elastic buckling strength due to flaws and nonlinearities.
Linear (eigenvalue) and nonlinear buckling curves.
The more accurate method, nonlinear buckling analysis, is typically advised for designing or assessing real structures. This method makes use of a nonlinear static analysis with progressively higher loads to find the limit load (Figure 3) at which the structure collapses. In this study, it is assumed that the collapse takes place at the peak of the in-plane load and out-of-plane deflection curves, with the ultimate strength bearing the brunt of the load during the whole loading procedure. The model can contain characteristics related to elasto-plastic buckling analysis, such as initial flaws, plastic behavior, gaps, and large-deflection response, using the nonlinear technique.
The equilibrium equation for the nonlinear buckling solution can be written as follows:
where
The consequences of geometric nonlinearity and material nonlinearity further exacerbate the issue for elasto-plastic buckling analysis. In order to avoid the solution’s instability close to the peak value of the load-deflection curve, the arc-length approach is employed to solve the nonlinear equation system to get the perforated plate’s ultimate strength σ xu .
3 Ultimate strength analysis
3.1 Influence of cutout size
First, the effect of the center flanged ellipse cutout size on the ultimate strength of the plate structure is investigated, while maintaining the values of a = 100, η = 2, θ = 0 and increasing b from 20 to 90 mm by an increment of 10 mm. Table 2 provides the modifications of cutoff ratio. Figure 4 illustrates the ultimate strength of the flanged perforated plates with thicknesses of 2, 4, and 8 mm.
Changes in cutout ratio
| ϕ | 0.047 | 0.070 | 0.093 | 0.116 | 0.140 | 0.163 | 0.186 | 0.209 |
|---|---|---|---|---|---|---|---|---|
| a (mm) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| b (mm) | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
Influence of cutout area on ultimate strength.
The following conclusions can be made in light of Figure 4:
With the increase in the cutout ratio, the ultimate strength of the perforated plate with thickness of 2 mm shows a continuous increase, the ultimate strength of the perforated plate with thickness of 4 mm exhibits a tendency of growing and subsequently decreasing, while the ultimate strength of perforated plates with thicknesses of 8 mm has been reducing as the cutout ratio increased.
With the same plate thickness and cutout ratio, it has been discovered that the higher the flanging height, the greater the ultimate strength,
The ultimate strength of the structure constantly rises as the plate thickness increases under the same cutout ratio (i.e., as the slenderness ratio falls from 225 to 56.25).
The ultimate strength rises with the increase in the cutout ratio for rectangular plate with t = 4 mm, when the cutout ratio is between 0.047 and 0.093. The ultimate bearing capacity of this plate, which has a flange height of h = 6–8 mm, progressively drops when the cutoff ratio reaches 0.093–0.116, but the ultimate strength of the rectangular plate, which has a flange height of h = 9–10 mm, gradually increases. The ultimate strength of this structure, which has various flange heights, steadily declines with an increase in the cutout ratio when it reaches 0.116–0.209.
Here we compared the out-of-plane bending stiffness provided by the flanging and the plate itself, in order to provide some explanations for these variances. For the thin plate with 2 mm thickness, the out-of-plane stiffness provided by the plate itself gradually decreases as the cutout ratio increases, while the out-of-plane stiffness provided by the flanging increases for the whole structure, and the ultimate strength of the structure increases as the opening rate increases as seen in Figure 4(a).
For the plate with a thickness of 8 mm, the plate itself provides out-of-plane bending stiffness has been playing a dominant role due to the large thickness, so the ultimate strength of the structure decreases with the increase in the cutout ratio (Figure 4(b)), the impact of the flanging is very small at this time.
There is a critical point in the out-of-plane bending stiffness between the flanging and the plate itself, which may be a plausible explanation for the phenomenon of peak points in Figure 4(b).
3.2 Influence of ellipse shape
The influence of ellipse shape on ultimate strength is investigated using the assumption that the rectangular plate has an elliptical cutout in the center and the same cutout ratio, namely a = 100 mm, b = 50 mm, and ϕ = 0.116. Although the lengths of the two half-axes, a and b, can vary, they must still meet equation (4). The length variations of the two half axles are shown in Table 3. Figure 5 illustrates how the elliptical shape affects the final strength.
Length changes of the two half axle
| a (mm) | 125 | 100 | 90 | 80 | 70 | 60 | 50 |
| b (mm) | 40 | 50 | 55.6 | 62.5 | 71.4 | 83.3 | 100 |
Influence of elliptical shape on ultimate strength.
The inferences from Figure 5 are as follows:
The ultimate strength curve for the rectangular plate with t = 2 mm exhibits an upward tendency as the half axis b rises. For a rectangular plate with t = 4 mm, the difference in the ultimate bearing capacity progressively diminishes, and the ultimate strength gradually increases when b = 40–50 mm and gradually reduces when b = 50–100 mm.
The ultimate strength curves for rectangular plates with t = 8 mm essentially coincide and exhibit a linear decline with an increase in half axis b, demonstrating that, when the cutout ratio is constant, changing only the shape of the elliptical cutout has little impact on the structure’s bearing capacity.
The ultimate strength increases with the increase in the flange height for t = 2 mm and t = 4 mm plates. The ultimate strength of a rectangular plate gradually increases as plate thickness (reduction in slenderness ratio) increases.
In the case of the same cutout ratio, with the change in the elliptical radius a and b, the flanging gradually approaches the mid-span in the compression direction, increasing the associated out-of-plane stiffness and the ultimate strength of the structure. So when b ≥ 50 mm, the ultimate strength values of the same thickness plates in Figure 5 are all higher than the results shown in Figure 4. The trend of ultimate strength variation for different plate thicknesses is similar to that of Figure 4, which can be explained by the same comparison of out-of-plane bending stiffness.
3.3 Influence of cutout rotation angle
The structure with a central elliptic cutout (a = 100 mm, b = 50 mm, and η = 2) is chosen as the research subject to examine the impact of counterclockwise rotation on ultimate strength. The elliptical cutout’s long axis is rotated counterclockwise by 15° increments, escalating to 90° at the end. Figure 6 shows the change curve for the ultimate buckling strength of a structure with various thicknesses (different slenderness ratios), with the influence of flange height serving as a guide.
Influence of cutout rotation angle on ultimate strength.
The following conclusions can be drawn from Figure 6:
The ultimate strength of the rectangular plate with t = 2 mm and h = 6–9 mm grows gradually as the rotation angle increases from 0° to 75° and drops gradually as the rotation angle increases from 75° to 90°. The ultimate strength curve has an increasing trend at h = 10 mm. The ultimate strength of the rectangular plate with t = 4 mm and t = 8 mm thickness rapidly decreases as the rotation angle increases.
The ultimate strength gap will gradually close as plate thickness rises, and thick plates are essentially unaffected by flange height.
A similar rationale may be used to explain the variations in the bearing capacity of the structure, since changes in the a/b axis of the elliptical cutout with the increase in rotation angle are comparable to the changes in the elliptical form in Section 3.2.
3.4 Influence of cutout position
Figure 7 illustrates the results of a study on the effects of cutout position on the ultimate strength of a rectangular plate, by moving the cutout center position along the positive x-axis while leaving the cutout area unchanged (a = 100 mm, b = 50 mm, and η = 2). With an incremental displacement of 15 mm and a movement range of 0–75 mm, the cutout center moved horizontally to the right.
Influence of cutout location on ultimate strength.
The following conclusions can be reached from Figure 7:
The ultimate strength of rectangular plates with t = 2 mm increases initially before beginning to decline when x > 45 mm; the strength declines most noticeably in rectangular plates with h = 6 mm.
With an increase in x, the ultimate strength of rectangular plates with t = 4 and 8 mm gradually declines, and the ultimate strength decline trend of plates with the same thickness is consistent.
3.5 Influence of cutout shape
Assuming that the rectangular plate’s center has cutouts for hyper-ellipses of various shapes, as shown in Figure 8, and taking into account that η = 1.5–5.0, a = 100 mm, b = 50 mm, where η is the index of the hyper-elliptic equation, Figure 9 depicts the curves for ultimate strength variation for rectangular plates with various hyper-ellipse geometries.
Hyper-ellipse cutout models with different shapes.
Influence of cutout shape on ultimate strength.
The following conclusions can be drawn from Figure 9:
The ultimate strength rises as the equation index rises for the rectangular plate with t = 2 mm. The final strength increases as the cutout progressively becomes a rectangle as the index increases, the flanging gradually travels away from the cutout’s center, and the amount of available out-of-plane bending stiffness gradually rises.
The ultimate strength of the rectangular plate with t = 4 mm rapidly declines with the increase in the equation exponent for flange heights of 6 and 7 mm. The maximum strength of the plate steadily rises before η = 2 and then falls at flange height of h = 8 mm. When the flange height is between 9 and 10 mm, the increase in the flanging height increases the structure’s out-of-plane stiffness, and the ultimate strength of the plate increases as the equation index rises.
With an exponential change in the equation, the ultimate strength of rectangular plates with t = 8 mm does not change at all for flange heights of h = 6 mm and h = 7 mm, indicating that the cutoff from sub-elliptic to hyper-elliptic will modify the ultimate strength of the plate. The ultimate strength rapidly declines with an exponential increase in the equation when the flange height is between 8 and 10 mm.
The ultimate strength of the rectangular plate gradually rises as its slenderness ratio falls.
3.6 Results analysis
There are significant variances in the ultimate strength for various plate thicknesses. In order to analyze the reasons for these results, as an example, Figures 10 and 11 provide an illustration of the analogous von Misses stress distribution of the structure in the limit state for t = 2 mm and t = 8 mm thickness, respectively, considering the influence difference of rotation angle on the ultimate strength.
Stress distribution of the flanged perforated plates with t = 2 mm in the ultimate state.
Stress distribution of the flanged perforated plates with t = 8 mm in the ultimate state.
As shown in Figures 10 and 11, the red region signifies that plastic deformation has occurred and the stress value has reached the yield limit of the material. For the flanged perforated plate with t = 2 mm as shown in Figure 10, the plastic regions are distributed on the two compressed sides of the flanging perforated plate and the two tips of the elliptical cutout. As θ increases, it can be found that the area of the red area at the tip of the elliptical cutout has a slight decrease in the rotation angle θ = 45–75°, and when the rotation angle is θ = 75–90°, there is a slight increase. The change trend is inconsistent with the change trend of the ultimate stress, as shown in Figure 6. However, as shown in Figure 11, for the flanged perforated plate with t = 8 mm, the red region (plastic region) decreases with the increase in θ, which is consistent with the change trend of the ultimate stress in Figure 6, and the plastic region is always maintained at the two tips of the elliptical cutout.
As can be seen, the stress distribution alone is insufficient to determine how the rotation angle will affect the final strength of the structure. The limit state deformation of the perforated plates is further studied.
The deformations in the ultimate state at various rotation angles for the perforated plate with t = 2 mm are depicted in Figure 12 and are similar to the perforated plate’s first-order buckling mode, which is depicted in Figure 13. In this context, the term “sum displacement” refers to the total displacements in the three ANSYS directions of x, y, and z.
Displacement distribution of the perforated plates with t = 2 mm in the ultimate state.
First-order buckling mode of the perforated plates with t = 2 mm.
The findings demonstrate that elastic buckling occurs in the perforated plate’s limit state with t = 2 mm, and that the substantial deformation of the structure that follows buckling results in substantial plastic yield. According to Figure 10, the greater the plastic zone, the more materials must be loaded, and as a result, the structure’s ultimate strength diminishes. It is consistent with the change trend depicted in Figure 6, which shows that the ultimate strength has a minor increase trend with an increase when the rotation angle is between 45° and 75°, and a slight decrease trend when the rotation angle is between 75° and 90°.
The structure transformation in the limit state for the plate with t = 8 mm elliptical flange cutout in Figure 14 is manifestly incongruous with the overall instability mode in Figure 15, suggesting that the structure has undergone plastic instability and the material has reached its yield limit. The majority of the plate’s surface is covered by the plastic zone, which shows that the majority of the perforated plate’s components have been fully utilized. The red area in Figure 11 is reducing, and the ultimate strength is also decreasing, which is compatible with Figure 6. The greater the red area, the higher the bearing capacity will be.
Displacement distribution of the perforated plates with t = 8 mm in the ultimate state.
First-order buckling mode of the perforated plates with t = 8 mm.
4 Conclusion
This study investigates the ultimate strength of a hyper-ellipse flanged-perforated plate and employs ANSYS software to assess the effects of cutout size, flange height, cutout position, rotation angle, and cutout form on the ultimate strength. The findings of the study can support the structural design of this type of perforated plate in real-world engineering applications. The following are the primary conclusions:
Flanging effectively increases the ultimate strength of the perforated structure.
The cutout area, ellipse shape, cutout rotation, and cutout location have little effect on the ultimate strength of thin plate, but have relatively great effect on the ultimate bearing capacity of thick plate.
For the increase in the elastic-plastic ultimate strength of the perforated plate, a larger elliptic cutout area, a smaller a/b (elliptic shape), a larger cutout rotation, a larger η, and the cutout location near the center are the best choices for the thin plate. A smaller elliptic cutout area, a larger a/b (elliptic shape), a smaller cutout rotation, the cutout location near the center, and a small η when flanging height is large are the best choices for thick plates. However, the optimal strengthening effect of these parameters on the elastic-plastic ultimate strength needs more analysis that is detailed.
In actual engineering structure design, appropriate cutout parameters should be selected according to different performance requirements.
Acknowledgements
This research was supported by the Natural Science Foundation of China (Grant Nos. 11402077 and 51975187), State Key Laboratory of Structural Analysis for Industrial Equipment (GZ18114).
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Funding information: Authors state no funding involved.
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Author contributions: Wang Wensheng: data curation, formal analysis, writing – original draft, review, and editing; Ning Huijun: methodology, software; Shang Xin: investigation and visualization.
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Conflict of interest: The authors declared that they have no conflicts of interest in this work.
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Ethical approval: The conducted research is not related to either human or animal use.
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Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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