FEM and ANN modeling of stress concentration factors (K_t) of circular plates with various circular holes according to internal and external pressures

1 Introduction

Circular perforated circular plates are used in many engineering applications, especially in aerospace, automotive and civil engineering, for filtration, flow control, heat exchange, sound insulation, and decorative purposes. Perforated structures are widely used to support parts in nuclear power plants, and the reactor coolant flows through the perforated plates (Jeong & Jhung) [1]. These plates are subjected to various types of loads such as pressure, tension, compression, vibration, shock, thermal stresses, corrosion, shear, and bending forces. Correct material selection and appropriate design are important for their efficient and safe operation. The stress concentration factor plays a vital role in the design and safety of circular perforated circular plates. This factor is studied to evaluate and optimize the stress distribution, fracture risk, material durability, structural life, and safety.

2 Review of studies with circular plates with circular holes SCF

In their 2008 study, Achtelik, Gasiak, and Grzelak [2] investigated the strength and stress condition of axially symmetric perforated plates, which are frequently seen in chemical reactors. They used the finite element method (FEM) to simulate the effects of various hole configuration of stress. Their results reveal that the holes placement and form have an important impact on the amount of stress concentrated, and that certain combinations may additonally enhance the plates structural integrity. Atanasiu and Sorohan [3] studied the displacement and stress distributions of circular plates with holes under bending. Stress and displacement analyses were performed for different hole diameters and configurations using the Finite Element Method (FEM) and compared with analytical solutions. The results showed that the maximum stress increased up to 25 % when the ratio of hole diameter to plate diameter exceeded 0.3. Furthermore, the displacement around the hole was found to be approximately 12 % higher compared to the solid plate. Bhaskar and Varadan [4] examined the mechanical behavior of plates and discussed different plate theories and their engineering applications. In the study, classical plate theory (CPT), First-Order Shear Deformation Theory (FSDT), and higher-order plate theories were compared, and the plates’ stress and deformation responses were analyzed by analytical and numerical methods. Example calculations show that FSDT gives 10–15 % more accurate results than CPT for thick plates. Buivol and Hulbert [5], [6], [7] investigated the effect of the ratios RˑR₀ ⁻¹ and R₀ ⁻¹ on the maximum Ktg (stress concentration factor) in circular elements with a central hole and a six-hole ring. In the first design, Ktg = 4.745 was found for RˑR₀ ⁻¹ = 0.65 and aˑR₀ ⁻¹ = 0.2, while Ktg increased to 5.754 when RˑR₀ ⁻¹ = 0.7 and aˑR₀ ⁻¹ = 0.25. In the second design, Ktg = 7.459 for RˑR₀ ⁻¹ = 0.6 and aˑR₀ ⁻¹ = 0.2, while Ktg = 9.909 for RˑR₀ ⁻¹ = 0.65 and aˑR₀ ⁻¹ = 0.2. These results show that the positioning of the holes and the geometric proportions significantly increase the stress concentration. The second pattern resulted in approximately 2 times higher stress concentration compared to the first one. Deogade and Bhope [8] and Deogade [9] studied the stress distributions created by off-center circular holes in rotating disks. The effects of different hole locations on stress concentration were analyzed using Finite Element Analysis (FEA). The results showed that the maximum stress increased significantly as the hole moved away from the center and the stress concentration reached the highest level at the hole edges. For example, an increase of about 18–20 % in the maximum stress was observed when the hole was shifted a certain distance from the center. It was also emphasized that the stress distribution could be optimized by changing the hole location. Green, Hooper, & Hetherington [10] studied the stress distribution in rotating disks with off-center holes. Using analytical methods and experimental tests, they analyzed the effects of the location of the holes and the rotational speed of the disk on the stress distribution. The results show that the maximum stress increases significantly as the hole is moved away from the center, and this effect becomes more pronounced at high rotational speeds. For example, it has been reported that under certain conditions, a hole positioned off-center can increase the maximum stress by up to 30 %. Hulbert and Niedenfuhr [11] developed numerical methods to calculate the stress distribution on multihole plates accurately. In the study, stress analyses were performed under different geometry and loading conditions using finite difference and finite element methods. The results obtained were compared with analytical solutions, and it was shown that the error rate in the maximum stress values was below 2 %. Hulbert [12] conducted a study using the theory of elasticity to solve two-dimensional problems by numerical methods. Using the finite difference method, stress and displacement distributions were calculated under different boundary conditions and loading conditions. The maximum stress value in a given plate was calculated with an error of 1 % to the analytical solution. Konieczny, Gasiak, & Achtelik [13] used ANSYS and Photoelasticity techniques to study the stress analysis of circular plate with holes positioned to the center. The area with the outer circle (R1 = 22.5 mm) had the maximum stress concentraiton in the plate with holes of varying diameters (σ _max = 416.79 MPa). The results of the finite element analysis (FEA) matched with the experimental data. In addition, in another study, Konieczny, Gasiak, & Achtelik [14] experimentally investigated the stress state in a steel-titanium perforated plate loaded with a central force in a different investigation. It was examined with resistive strain gauges and specialied test bench. Maximum stress zones were identifed and radial, circumferential, and von Mises streses were measured. The experimental findings were in good accord with the ANSYS numerical calculations. Konieczny, Achtelik, & Gasiak [15] and Konieczny, Gasiak, & Achtelik [16] investigated the effect of the coating applied on bimetallic perforated plates on the stress state. In the study, the mechanical behavior of the plate was investigated under two different loading conditions, and analyses were made with the finite element method (FEM). In the results, it was determined that the maximum stress decreased by 20–30 %, especially with the increase in coating thickness. Kraus [17] investigated the bending behavior of circular plates with holes arranged in a ring shape. In this study, analytical solutions were developed using the theory of elasticity, and these solutions were verified by numerical methods. The results obtained show that the number and location of the holes have a significant effect on the maximum bending stresses of the plate; in certain cases, the stress concentration factor was found to vary between 3 and 6. Kraus [18] investigated the compressive stresses in objects with multiple holes. In the study, the effects of different hole arrangements on the stress distribution in materials subjected to internal pressure were analyzed using analytical solutions based on the theory of elasticity and numerical methods. The results show that the maximum stress concentration factor around the holes can vary between 2 and 5 depending on the hole diameter and the distance between them. Kraus [19] investigated the stress concentration factors for objects in the geometry of rings with holes loaded in the plane. In the study, the effects of different hole diameters and locations on the stress distribution were investigated using analytical and numerical methods. The results showed that the stress concentration factor increases significantly with increasing hole diameter and that this factor can vary between 3 and 6 in certain configurations. Kraus [20] updated the stress distribution of multihole plates by Hulbert and Niedenfuhr. The accuracy and applicability of the numerical methods were emphasized and the sensitivity of the proposed calculation methods were examined. Kraus stated that the results presented by the authors were generally correct, but that there could be small margins of error under certain boundary conditions. In particular, differences ranging from 5 to 10 % were observed in the calculation of stress concentrations due to the interaction of holes. Kraus, Rotondo, and Haddon [21] investigated the mechanical behavior of radially deformed holed flanges. In the study, analytical solutions were obtained using the theory of elasticity, and these results were confirmed by experimental data. The findings show that the hole diameter and distribution have a significant effect on the stress distribution and rigidity of the flange. In particular, the presence of holes causes an increase of 15–30 % in the maximum stress regions, while it has been found that this effect can be reduced by certain geometric optimizations. Kubair [22] investigated the stress concentration factor (SCF) in the structures of functionally graded (FGM) plates containing circular holes. In the study, SCF values were calculated under different material gradients and loading conditions using analytical and numerical methods (finite element analysis). The results show that the material gradient has a significant effect on the SCF, especially the change in stiffness can reduce the stress concentration by up to 20 %. It was also found that the hole size and location play a decisive role in the SCF values. Mateusz, Henryk, & Grzegorz [23], investigated the maximum stress regions in circular plates under the effect of the resultant force. In the study, the stress distribution under different loading and support conditions was analyzed by the finite element method (FEM). The results showed that resultant loads applied to the center of the plate can increase the maximum stress by up to 40 %, and the largest stress concentration occurs in the regions close to the load application point. It was also determined that the presence of holes significantly changes the stress distribution and affects the structural strength. Mínguez and Vogwell [24] investigated the mechanical behavior of perforated plates subjected to lateral pressure. The study analyzed the effects of different hole configurations on stress distribution and deformation using the finite element method (FEM) and experimental tests. The results showed that the maximum stress increased by approximately 35 % with the increase in hole diameter, and the holes significantly reduced the load-carrying capacity of the plate. It was also emphasized that the positioning of the holes could increase structural strength by reducing stress concentration in critical areas. Ozkan and Erdemir [25] used finite element analysis (FEA) and artificial neural network (ANN) models to study the stress concentration factors (SCF) under various conditions. These models predicted the SCF behavior with high accuracy. Ozkan and Toktaş [26] compared different methods for SCF calculation in rectangular plates and proved that ANN models provide the highest accuracy (R² = 0.999999788, RMSE = 0.000934125, MEP = 0.01 %). Saracoğlu, Uslu, and Albayrak [27] discussed the stress and displacement analysis of circular plates with holes. In the study, stress distribution and displacement results were obtained for different hole diameters and positions using the finite element method (FEM). The analyses showed that the maximum stress increased significantly and the displacement also increased with the increase in hole diameter. Numerically, it was found that the maximum stress was approximately 40 % higher than the reference model when the hole diameter to plate ratio was 0.3. Timoshenko [28] studied the distribution of stresses caused by forces acting on circular plates. In the study, the theory of elasticity was used to determine the stress distribution under forces acting in two opposite directions on the ring. As a result, it was shown that the maximum stresses were concentrated at the load application points and in the inner part of the ring, and this distribution was detailed with analytical solutions.

Atalay and Toktaş [29] explored how stresses impact design and lead to changes in the geometry of parts, especially in spots with sharp corners, holes, or notches where the cross section shifts suddenly. In their study, they considered the estimated notch factor for a cylinder subjected to internal pressure. To determine stress concentration factors, they used four different approaches: finite element analysis, neural networks, analytics, and regression. They found that the artificial neural network model delivered the most accurate results compared to the other methods.

Mayr and Rother [30] introduced enhanced stress concentration factors (SCFs) for circular shafts subjected to uniaxial and combined loading conditions. Their research, which drew on both experimental data and numerical analyses, showed that these updated SCF values offer more precise predictions than the conventional theoretical models. Ekşi [31], conducted a study in which the author took examined the numerical analysis of stress concentration factors (SCFs) for subsurface and undercutting pits. The research demonstrates how geometricical parameters affect the distribution of stress. Both numerical and artificial neural network (ANN) in this study may be validated against these insights, which are useful for predicting SCFs in perforated structures.

In recent years, studies on the determination of the stress concentration factor (Kt) in perforated composite structures have made significant contributions to the development of numerical and experimental approaches in this field. Zhao et al. [32] investigated crack propagation on structures with central holes using a simulation method. The analysis clearly reveals the critical stress regions that occur around the hole. Similarly, in a study by Daricik [33], the interlaminar damage behavior occurring around open holes in composite laminates was analyzed using the FEA method. Bozkurt [34] emphasized the effectiveness of artificial neural networks in predicting the behavior of composite structures. In this context, the ANN model in this study is methodologically similar. Uzun and Bozkurt [35] performed FEA-based stress analysis on pressure vessels designed in accordance with European Union standards. This approach can contribute to the modeling of perforated plates subjected to pressure. Finally, the experimental study by Shen et al. [36], focusing on the stress relaxation behavior of GFRP pipes under circumferential deformation conditions, provides significant contributions to the literature on the modeling of time-dependent mechanical responses. These studies support the methodological approach of the study and provide a scientific basis for the combined use of numerical analysis and artificial intelligence algorithms.

In this study, the stress concentration factor (SCF) of circular plates with circular holes subjected to internal and external pressure was investigated. Based on Peterson’s SCF tables, graphical data were converted to numerical values ​​by means of high-precision software, and dimensional parameters were calculated according to dimensionless ratios. These values ​​obtained were used in the creation of parametric CAD models in the ANSYS environment, and then mesh optimization and finite element analysis (FEA) were performed by applying appropriate boundary conditions. The SEA results were compared with theoretical values ​​and experimental data to verify the accuracy of the analyses. Artificial neural network (ANN) models were developed using Matlab and STATISTICA software to improve the modeling process. Various ANN algorithms were tested and evaluated according to statistical performance measures such as mean error, mean percentage error (MPE%), and R². ANN models provided high accuracy and efficiency by surpassing experimental and SEA approaches in terms of computational speed and cost effectiveness.

3 Materials and methods

This study focuses on the investigation of stress concentration factors (SCF) in circular plates with circular holes. Finite element method (FEM) and artificial neural network modeling (ANNM) were used to calculate SCF. The process consists of three main steps:

1. Collection and validation of data related to the problem,

2. Development of parameterized finite element analysis (FEA) model and performance of its analysis,

3. Determination and validation of the actual artificial neural network (ANN) model.

In the first step, graphical data related to stress concentration factor curves related to center and off-center circular holes on the circular plate were converted into numerical values ​​using high-precision computing tools and special software. The original data were represented as curves classified and archived by Peterson in his book Stress Concentration (Pilkey & Pilkey) [37]. Then, the curves related to the problem, the curves obtained from experimental results that were not previously mathematically formulated, were converted into numerical data sets.

One of the primary difficulties was the lack of numerical values ​​or equations that define the dimensionless ratios of the problem. To address this, graphical data were processed and sample dimensions were determined. Using these dimensionless parameters as references, a geometric model of the main problem was created based on the dimension ratios. The stress concentration factor is a coefficient that represents the sudden increase in stress caused by a disruption in the geometric continuity of a material. The presence of features such as holes, channels, voids, or protrusions on a component results in sharp increases in stress at these geometrically distinct locations. As a result, maximum stress is observed at the points where the geometric continuity is disrupted or changed. From an engineering perspective, this value is expressed as shown in Equation (1). The total stress experienced by the material is denoted as σnom and is calculated depending on the type(s) of stress to which the material is exposed. The value σmax represents the allowable stress of the material under safe operating conditions.

with σ _max: yield strength (Mpa), σ _nom: maximum bending stress (MPa), and K _t : stress concentration constant.

In order to model the tensile/compressive stresses in a circular element with circular holes, the applied load and the material variational shapes need to be defined. The relevant variables, which are essential for the definitions and content analyses, are presented in Table 1.

Table 1:

Circular perforated plate forms studied in the graphs.

Scenerio	Stress booster form	Description
I		Stress concentration factors K t , R·R0 −1 = 0.625 for a circular member with a central circular hole and four or six circular holes without a center. a: Radius of small holes Ri: Radius of the hole in the center: R: Radius of the axis of the small holes R0: Radius of the outer circle
II		Stress concentration factors Kt for a perforated flange with internal pressure, Ri·R0 −1 = 0.8, R·R0 −1 = 0.9 a: Radius of small holes Ri: Radius of the hole in the center: R: Radius of the axis of the small holes R0: Radius of the outer circle

Scenario I: There are 4 curves in the graph extracted with Optical Character Recognition (OCR) (Figure 1). For a circular element consisting of a hole in the center and four or six holes, the Kt value is obtained as the response of the aˑR₀ ⁻¹ ratio. In the case where all holes are the same size, a = R_i, the outer diameter of the element is 1/4 (R_iˑR₀ ⁻¹ = 1/4). Here, the radius a is assumed to be 5 mm hypothetically, and the R, R_i and R₀ values ​​are calculated accordingly. Certain points are selected from the values ​​read from the graph to be equally spaced. The numerical data obtained from the graph are classified in the Excel table, and a table is created. This classification is calculated parametrically for the circular plate dimensions with the help of dimensionless parameters. The dimensional sizes of the circular plate with holes are produced according to the reference dimension ratios (Table 2). All data are processed in the Excel software at each point.

Figure 1:

Mesh structures, a) circular plate with 4 holes, b) circular plate with 4 holes, c) circular plate with 6 holes, and d) circular plate with 6 holes [38].

Scenario II: There are 4 curves in the graph extracted by Optical Character Recognition (OCR) (Figure 1). Each curve represents a different number of holes (8, 16, 32, 48). For the case of a ring flange (R = 0.9 R₀), the maximum K _t values ​​as a function of the hole size and the number of holes are shown in Figure 2. K _t is defined by dividing σ _max by σnom, which is the average tensile stress on the net radial section passing through a hole. In Kraus’s article, K _t factors are given for R_iˑR₀ ⁻¹ and other RˑR₀ ⁻¹ values. Here, the radius a is assumed to be 5 mm hypothetically, and the R, R_i and R₀ values ​​are calculated accordingly. Certain points are selected from the values ​​read from the graph to be equally spaced. The numerical data obtained from the graph are classified in the Excel table, and a table is created. This classification is calculated parametrically for the circular plate dimensions with the help of dimensionless parameters. The perforated circular plate dimensional sizes are produced according to the reference dimension ratios (Table 1). All data were processed in Excel software at each point.

Figure 2:

Mesh structures, a) circular plate with 8 holes, b) circular plate with 16 holes, c) circular plate with 32 holes, and d) circular plate with 48 holes [38].

Within the scope of the problem, a parametric 3D solid model was designed using ANSYS Workbench software. Mesh optimization was required because the graphic data had a wide range of values ​​in different test sample sizes. In the mesh optimization process, the unit element size was taken as a reference. In order to determine the optimum mesh size, the ANSYS [38] user guide was used (Table 1,2).

4 Parametric modeling using FEM

ANSYS software was used for parametric solid modeling of tensile stress in a circular element with circular holes. Dimensional ratios and design parameters were converted to numerical form by taking Peterson’s graphic data as reference. Parametric solid models were modeled parametrically using ANSYS Design Modeler according to design parameters via Finite Element Method. Boundary conditions appropriate to the problem were applied with ANSYS Workbench software. In order to obtain accurate results from the analysis, mesh optimization of the solid model was prepared in ANSYS environment and optimized using quadratic and then tetrahedral elements. Mesh structure is presented in Figures 1 and 2, and quality parameters of the mesh are presented in Table 3. A comprehensive mesh optimization was performed. During mesh creation, unit element size was set to a certain value, and a finer mesh structure was applied in the regions with holes. Necessary mesh size and number of elements were determined by evaluating mesh metric values. Deformation, strain, and stress values were calculated in computer environment under loading conditions Figures 3 and 4 appropriate to experimental parameters. The original solid model was created with a parametric approach using ANSYS software [38], and the parameters extracted from the graph were assigned to the ANSYS model to ensure consistency.

Figure 3:

Boundary conditions, a) circular plate with central hole 4 circular holes, and b) circular plate with central hole 6 circular holes [38].

Figure 4:

Boundary conditions, a) circular plate with 8 holes, b) circular plate with 16 holes, c) circular plate with 32 holes, and d) circular plate with 48 holes [38].

5 Modeling using artificial neural networks (ANNs) for circular plates with circular holes

Equations (2) through (6) demonstrate the tansig, logsig, and purelin functions that make up the ANN model:

with Net O: ANN result, w_i,j: weights of each layers, x_k: input variable of the layers, b_m: bias variable of each layer, and v_m,n: weight of the bias variable of each layer.

6 Results and discussion

ANSYS software was used to model the circular plate with circular holes subjected to pressure applied inside the hole and negative pressure loads applied to the plate outer surface. In this model, stress concentration around the hole is taken into account, and this causes tensile stress in the plate. Analyses were performed according to the parameters given by ANSYS Static Structural. The analysis results of the plate with 4 and 6 holes are given in Figures 5 and 6, respectively. The analysis results of the flange with 8, 16, 32, and 48 holes belonging to the second graph are given in the Figures 7–10, respectively.

Figure 5:

FEA solutions, a) equivalent stress (MPa), b) normal stress (MPa), c) equivalent elastic strain, and d) normal elastic strain [38].

Figure 6:

FEA solutions, a) equivalent stress (MPa), b) normal stress (MPa), c) equivalent elastic strain, and d) normal elastic strain [38].

Figure 7:

Equivalent stress (MPa) for, a) 8 holes, b) 16 holes, c) 32 holes, and d) 48 holes [38].

Figure 8:

Normal stress (MPa) for, a) 8 holes, b) 16 holes, c) 32 holes, and d) 48 holes [38].

Figure 9:

Equivalent elastic strain for, a) 8 holes, b) 16 holes, c) 32 holes, and d) 48 holes [38].

Figure 10:

Normal elastic strain for, a) 8 holes, b) 16 holes, c) 32 holes, and d) 48 holes [38].

Regression analysis was used to compare all computations and analytical findings. Results from regression, ANN, FEA, and experiments were compared with one another using a statistical method. Regression analysis was performed using Equations (7) through (9). Scenario I and Scenario II show that the FEA regression analyasis results, and Scenario III shows the ANN results regression analysis results (Table 4).

6.1 ANN results

Various artificial neural network (ANN) models are tested, and then the best performing ANN model is determined as the true model (Table 5).Varied ANN models have been tested and comted the performance. Tried ANN models and performances of the models have been shown in Table 5.

Table 5:

Performance comparison of different ANN models in Statistica [39].

Index	Net. name	Training perf.	Test perf.	Training error	Test error	Training algorithm	Error function	Hidden activation	Output activation
1	MLP 19–31–7	0.986347	0.956518	0.003271	0.061198	BFGS 107	SOS	Tanh	Exponential
2	MLP 19–77–7	0.985142	0.950169	0.003553	0.082063	BFGS 110	SOS	Exponential	Identity
3	MLP 19–73–7	0.986304	0.949436	0.003329	0.079734	BFGS 127	SOS	Exponential	Identity
4	MLP 19–8–7	0.958901	0.935169	0.008847	0.073695	BFGS 51	SOS	Logistic	Exponential
5	MLP 19–26–7	0.977875	0.938107	0.005130	0.092747	BFGS 72	SOS	Exponential	Identity
6	MLP 19–100–7	0.951708	0.869475	0.009373	0.148395	BFGS 41	SOS	Identity	Identity
7	MLP 19–100–7	0.930536	0.869118	0.013185	0.180511	BFGS 33	SOS	Identity	Tanh
8	MLP 19–100–7	0.968484	0.911466	0.006867	0.133370	BFGS 49	SOS	Identity	Logistic
9	MLP 19–100–7	0.934947	0.920386	0.012572	0.112152	BFGS 25	SOS	Identity	Exponential
10	MLP 19–100–7	0.951549	0.860363	0.009341	0.158257	BFGS 40	SOS	Identity	Sine
11	MLP 19–100–7	0.522202	0.468931	0.207416	0.565544	BFGS 28	SOS	Identity	Softmax
12	MLP 19–100–7	0.951760	0.868455	0.009385	0.148991	BFGS 41	SOS	Tanh	Identity
13	MLP 19–100–7	0.934430	0.863685	0.012614	0.181375	BFGS 35	SOS	Tanh	Tanh
14	MLP 19–100–7	0.987020	0.932616	0.003167	0.117935	BFGS 122	SOS	Tanh	Logistic
15	MLP 19–100–7	0.933005	0.931339	0.012841	0.094673	BFGS 25	SOS	Tanh	Exponential
16	MLP 19–100–7	0.951834	0.864743	0.009359	0.155681	BFGS 41	SOS	Tanh	Sine
17	MLP 19–100–7	0.448002	0.457841	0.207423	0.565657	BFGS 29	SOS	Tanh	Softmax
18	MLP 19–100–7	0.954555	0.872156	0.009046	0.149743	BFGS 47	SOS	Logistic	Identity
19	MLP 19–100–7	0.947752	0.846005	0.010186	0.174364	BFGS 49	SOS	Logistic	Tanh
20	MLP 19–100–7	0.981824	0.883877	0.004262	0.128971	BFGS 77	SOS	Logistic	Logistic
21	MLP 19–100–7	0.926581	0.904411	0.013660	0.118542	BFGS 35	SOS	Logistic	Exponential
22	MLP 19–100–7	0.953165	0.862885	0.009498	0.160327	BFGS 45	SOS	Logistic	Sine
23	MLP 19–100–7	0.440742	0.416763	0.212292	0.568368	BFGS 38	SOS	Logistic	Softmax
24	MLP 19–100–7	0.985302	0.955377	0.003558	0.079638	BFGS 118	SOS	Exponential	Identity
25	MLP 19–100–7	0.978323	0.895809	0.005211	0.129064	BFGS 84	SOS	Exponential	Tanh
26	MLP 19–100–7	0.953976	0.900601	0.009590	0.130024	BFGS 34	SOS	Exponential	Logistic
27	MLP 19–100–7	0.929516	0.873539	0.013272	0.154219	BFGS 35	SOS	Exponential	Exponential
28	MLP 19–100–7	0.970960	0.854944	0.006176	0.144920	BFGS 66	SOS	Exponential	Sine
29	MLP 19–100–7	0.526736	0.455255	0.209552	0.566496	BFGS 110	SOS	Exponential	Softmax
30	MLP 19–100–7	0.951724	0.872736	0.009384	0.147566	BFGS 40	SOS	Sine	Identity
31	MLP 19–100–7	0.928959	0.872677	0.013385	0.179777	BFGS 32	SOS	Sine	Tanh
32	MLP 19–100–7	0.966351	0.898841	0.007695	0.136383	BFGS 45	SOS	Sine	Logistic
33	MLP 19–100–7	0.935736	0.929794	0.012566	0.102306	BFGS 26	SOS	Sine	Exponential
34	MLP 19–100–7	0.950944	0.862659	0.009495	0.158803	BFGS 40	SOS	Sine	Sine
35	MLP 19–100–7	0.492821	0.474123	0.207574	0.565453	BFGS 28	SOS	Sine	Softmax
36	LM 13 19 29 7	0.99999	0.99999	0.00001	0.00002	LM	SOS	Tansig, tansig, logsig	Purelin

The Multi-Layer Perceptron – Feed Forward Back Propagation (MLP-FFBP) model, which was designed as 24 + 13+19 + 29+7 + 7, was shown to be the best model for the problem (Figure 11). The model training performance in this model has been calculated to be R = 0.99999. The figure indicates that the training performance is excellent. The number is really near to 1.

Figure 11:

ANN model.

The ANN model’s training, testing, and validation results are displayed in Figure 12. The graphic shows that the plate’s training value was 1, its testing value was 0.99999, and its validation value was almost 1. Average R² = 1 was determined for the problem based on the overall outcome.

Figure 12:

ANN model comparison between the training, test, and validation performance [39].

Figures 13 and 14 are graphs comparing the stress concentration factor (Kt) values obtained for different hole numbers and geometric ratios using FEA, experimental, and ANN methods. An examination of the graphs reveals a high level of agreement between the results of the three different methods. In particular, the ANN model successfully represents both the FEA and experimental data. The accuracy of the ANN model was evaluated using regression analysis, and an accuracy value of 0.99999 was obtained. The graphs also clearly demonstrate the impact of increasing the number of holes and changing geometric parameters on Kt. This supports the model’s ability to reflect physical reality. ANN performance was evaluated in detail using the statistical metrics provided in Table 6, and the results reinforce the model’s reliability.

Figure 13:

Stress concentration factor (SCF) curves of perforated circular plates with circular holes outer negative presssure obtained by different solution methods (experimental, ansys (FEA), and ANN).

Figure 14:

Stress concentration factor (SCF) curves of perforated circular plates with circular holes outer negative presssure obtained by different solution methods (experimental, ansys (FEA), and ANN).

Table 6:

Results of regression analysis of ANN model [40].

Circular plates ANN model	Model	% MEP	RMS	R2
Scenario I and II	L-M 13 19 29 7	0.289244033	0.315245102	0.999999

According to Figure 15, the relationship between the parameters and K_t shows a nonlinear distribution. Figure 15a shows the effect of a·R_i ⁻¹ and a·R₀ ⁻¹ on K_t. Kt reaches its highest value (>8) when a·R_i ⁻¹ is between 0.6 and 0.8 and a·R₀ ⁻¹ is low (0.1–0.2). In contrast, when a·R_i ⁻¹ is below 0.5, K_t decreases significantly. This finding indicates that the positioning ratio of the holes has a significant effect on the stress concentration. Figure 15b shows the interaction of a·R_i ⁻¹ and a·R⁻¹ ratios with Kt and exhibits a convex behavior. Kt reaches its highest value (>6) when a·R_i ⁻¹ is between 0.6 and 0.8 and a·R⁻¹ increases. However, when a·R_i ⁻¹ and a·R⁻¹ are at low values, the Kt value can decrease to negative values. This finding indicates a significant transition process in the stress distribution. Figure 15c shows the effect of a·R_i ⁻¹ and R_iˑR₀ ⁻¹ parameters on Kt and the graph follows an exponential increasing distribution. With increasing R_iˑR₀ ⁻¹, Kt increases significantly (>140). Especially when a·R_i ⁻¹ is in the range of 0.8–1, the effect of the R_iˑR₀ ⁻¹ parameter becomes more pronounced and causes the stress concentration to reach peak values. This finding indicates that the R_iˑR₀ ⁻¹ parameter plays a decisive role on the stress distribution, especially when aˑR_i ⁻¹ is high.

Figure 15:

Stress concentration factor (SCF) circular plates with circular holes Kt change according to various proportion and parameters, a) a·R_i ⁻¹ and a·R₀ ⁻¹ versus K_t, b) a·R_i ⁻¹ and a·R⁻¹ versus K_t, and c) a·R_i ⁻¹ and R_i· R₀ ⁻¹ versus K_t [39].

Figure 16 shows the influence of the geometrical parameters on the stress concentration factor (Kt) and exhibits a clear nonlinear behavior. Figure 16a shows the influence of the Kt parameters a·R₀ ⁻¹ and RiˑR₀ ⁻¹. The graph shows a convex trend indicating that Kt reaches maximum levels (>8). This is especially observed when a· R_i ⁻¹ is at low values (∼0.1–0.2) and a·R₀ ⁻¹ is in the range of about 0.8–1.0. However, as a·R_i ⁻¹ increases, Kt decreases significantly. This finding reveals that the stress concentration decreases significantly with increasing distance between the holes. Figure 16b shows the effect of the parameters a·R₀ ⁻¹ and Ri·R₀ ⁻¹ on the graph, which exhibits a symmetrical concave distribution. When both parameters are at intermediate values, Kt reaches minimum (<1) levels. However, if a·R₀ ⁻¹ or Ri·R₀ ⁻¹ take extremely small or large values, Kt exceeds 7, resulting in high stress concentrations. The obtained findings reveal that the interaction of the geometric ratios with the distance between the holes has a decisive influence on the stress distribution. They also emphasize the necessity of selecting these parameters in optimum ranges in order to minimize the stress concentration.

Figure 16:

Stress concentration factor (SCF) circular plates with circular holes Kt change according to various proportion and parameters, a) a·R₀ ⁻¹ and a·R_i ⁻¹ versus K_t and b) a·R₀ ⁻¹ and R_i R₀ ⁻¹ versus K_t [39].

Figure 17 shows the influence of the geometrical parameters on the stress concentration factor (Kt) and shows a nonlinear behavior. Figure 17a shows the effect of a·R⁻¹ and R_iˑR₀ ⁻¹ on Kt, and the graph shows a concave distribution. Minimum Kt values (<0) occur when both parameters are balanced. However, when a·R⁻¹ or R_i·R₀ ⁻¹ values increase significantly, Kt increases above 8. This indicates that higher values result in increased stress concentrations. Figure 17b shows the effect of the parameters a·R⁻¹ and a_i·R⁻¹ on Kt. The graph reveals a convex trend indicating that the a_i·R⁻¹ parameter reaches maximum levels (>6) in the range of low values (∼0.05–0.1) and a·R⁻¹ reaches maximum levels (>6) when the value of a·R⁻¹ is about 1.0. However, with increasing value of a_i·R⁻¹, Kt decreases significantly. This indicates that increasing the distance between the holes significantly reduces the stress concentration. The obtained findings reveal that the interaction of the distance between the holes and the geometrical parameters has a decisive effect on the stress distribution. It also emphasizes the necessity of selecting these parameters in optimum ranges in order to minimize the stress concentration.

Figure 17:

Stress concentration factor (SCF) circular plates with circular holes Kt change according to various proportion and parameters, a) a·R⁻¹ and R_i·R₀ ⁻¹ versus K_t and b) a·R⁻¹ and a_i·R⁻¹ versus K_t [39].

Figure 18a shows the effect of the parameters R_i·R₀ ⁻¹ and a·R₀ ⁻¹ on Kt. The graph shows a concave trend. When R_i·R₀ ⁻¹ is in the range 2–4, Kt is at its minimum value. When the value of R_i·R₀ ⁻¹ exceeds the range 2–4, the value of Kt increases and exceeds 7. Figure 18b shows a concave trend similar to Figure 18a. When R_i·R₀ ⁻¹ is in the range of about 2–4, Kt remains at low levels. However, as this value increases, the Kt value increases significantly and exceeds 8 in the red region. Figure 18c shows a different behavior with a steeper gradient than the other graphs. When R_i·R₀ ⁻¹ is at low levels, Kt exhibits a decreasing trend, reaching negative values. However, as R_i·R₀ ⁻¹ increases, the Kt value rises rapidly and exceeds 140, showing a clear exponential increasing trend.

Figure 18:

Stress concentration factor (SCF) circular plates with circular holes Kt change according to various proportion and parameters, a) R_i· R₀ ⁻¹ and a·R₀ ⁻¹ versus K_t, b) R_i R₀ ⁻¹ and a·R⁻¹ versus K_t, and c) R_i·R₀ ⁻¹ and a·R_i ⁻¹ versus K_t [39].

7 Conclusions

In this study, the stress concentration factor of circular plates with circular holes was investigated using various methods such as converting into numerical values from the graphical curves, verification of the experiemntal data, FEM, and ANN. Pilkey’s graphical curves were used in the study. In this study, unlike the studies of Buivol and Hulbert [5], [6], [7], there is also a circular hole in the center of the circular plates. Pilkey used the results of many experimental studies while creating these graphs in his book. This study consists of three stages. The first is to convert the graphical data into numerical values, the second is to create and analyze the finite element model of the problem, and the third is to create the artificial neural network model of the problem. The perforated flange and radially stressed circular element with a central circular hole and a ring of four or six noncentral circular holes were used as a reference for dimensioning. Afterward, other parameters were calculated proportionally and dimensionally, remaining faithful to the original graph. The finite element analysis model created according to the determined parameters was analyzed and verified. Then, an artificial neural network model was developed with a Matlab software. The created artificial neural network model was evaluated with a statistical approach. In line with the obtained parameters, the stress values ​​were calculated with ANSYS software, and then the software developed in the Matlab environment was used to perform accuracy analysis. Matlab-based artificial neural network (ANN) results have a higher reliability than finite element analysis (FEA) results. While FEA results have an accuracy of R = 0.9991 and R = 0.9999, ANN model results have an accuracy of R = 0.99999. The ANN model offers a practical, time-efficient, and effective solution method for the problem in question. The use of this model allows the designer to realize the fastest and most accurate designs.