An optimized artificial neural network modeling for the stress concentration factor in orthotropic plates with central countersunk holes

Mohammad A. Gharaibeh

doi:10.1515/jmbm-2025-0093

Article Open Access

An optimized artificial neural network modeling for the stress concentration factor in orthotropic plates with central countersunk holes

Mohammad A. Gharaibeh

Published/Copyright: January 8, 2026

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From the journal Journal of the Mechanical Behavior of Materials Volume 35 Issue 1

Abstract

Stress concentration factors (SCFs) play a critical role in the structural integrity of engineering structures. Rivets, the commonly used joining components, usually leave geometric discontinuities footprints in the form of countersunk holes. The present study investigates the SCF in orthotropic carbon/epoxy (AS4/3501-6) plates with centrally located countersunk holes under uniaxial tensile loading. A comprehensive 3D finite element analysis (FEA) is employed to evaluate the effects of geometric parameters – such as hole radius-to-plate width ratio (r/w), plate thickness-to-hole radius ratio (t/r), countersinking depth-to-thickness ratio (C _s/t), and countersink angle (θ _c) – as well as material orthotropy, represented by the ply angle (θ _p) and the corresponding infinite plate orthotropic stress concentration factor (K _∞). An artificial neural network (ANN) optimization process is executed to model the relationship between these parameters and the SCF, achieving high accuracy with a root mean square error (RMSE) of 0.0090. The optimal ANN architecture utilizes a single hidden layer with nine neurons, the Levenburg-Marquardt learning algorithm, and a hyperbolic tangent sigmoid activation function. The model is validated against FEA results, demonstrating excellent agreement with errors below 6 % for 94 % of the cases studied. Furthermore, an optimization analysis identifies the geometric and material configuration that minimizes the SCF, yielding a value of (K _t = 1.95) for a plate with (r/w = 0.1), (t/r = 5), (C _s/t = 0.1), (θ _c = 80°), and (θ _p = 50°, K _∞ = 1.90). The findings of this work provide valuable insights for designing orthotropic plates with reduced stress concentrations, enhancing their performance and structural integrity.

Keywords: stress concentration factor; countersunk holes; orthotropic materials; artificial neural networks; optimization; finite element analysis

1 Introduction

Rivets are commonly employed in the structural components joining process. Riveting footprints usually leave geometric discontinuities in the form of countersunk holes throughout the jointed structure. The presence of such discontinuities would act as a stress riser in the area near the countersunk hole. This localization of high stress is commonly known as stress concentration, and it is commonly investigated by the means stress concentration factor (SCF). Mathematically, the value of the stress concentration factor (K _t) is computed as the ratio between the maximum stress (σ _max) to the nominal stress (σ _nom).

Extensive research has been conducted on stress concentration factors in two-dimensional (2D) plates with circular holes under various loading conditions [1]. Shivakumar and Newman [2] conducted a three-dimensional (3D) finite element analysis (FEA) on plates with straight-shank circular holes under tensile loading, revealing that the peak SCF occurs at the mid-thickness of isotropic plates and decreases toward the surfaces. Wu and Mu [3] performed FE simulations on isotropic and orthotropic plates with circular holes under uniaxial and biaxial loads, analyzing SCF in plate structures and pressure vessels. Kotousov and Wang [4] derived analytical solutions for 3D stress distributions near stress concentrators in isotropic plates of varying thicknesses. Later, Kotousov et al., [5] demonstrated that classical plane solutions, which neglect plate thickness, can introduce errors in stress assessments near notches. Li et al., [6] used FE analysis to study elastic stress fields near notches in plates of different thicknesses under tension. Berto et al., [7] proposed an analytical model for notch-root stress fields in plates of arbitrary thickness, examining the influence of thickness and notch geometry on stress distribution, out-of-plane constraints, and strain energy density. She and Guo [8] have applied FEA to assess SCF variations along elliptic holes in isotropic plates under tensile stress.

For SCF in countersunk holes, Wharely [9] was able to visualize localized stresses experimentally using birefringent coatings on aluminum plates. Shivakumar et al., [10] and Bhargava and Shivakumar [11], 12] developed finite element simulations and empirical equations for the evaluations of stress and strain concentration factors in countersunk holes under tensile loads. Later, Darwish et al., [13], Gharaibeh [14], Gharaibeh et al., [15], 16] formulated refined empirical formulas to compute stress and strain concentration factors in countersunk holes in uniaxially loaded isotropic plates. Such equations showed higher accuracy than the equations reported in [10], [11], [12]. In [10], [11], [12], [13], [14], [15], [16], it was reported that stress and strain concentration factors are function of the countersunk hole geometry as well as the plate thickness and width.

For an isotropic plate with two identical countersunk holes, the numerical studies and statistical analysis results of Gharaibeh [17], 18] have shown that the stress as well as strain concentration factors can be highly affected by the separation distance between the two holes, in addition to the hole geometric configuration. Also, Jabri and Gharaibeh have conducted thorough FEA study to investigate the von Mises SCF in countersunk holes due to biaxial loading [19]. Their results showed that von Mises SCF is largely influenced by the ratio between the applied stresses in the in-plane directions, in addition to the hole dimensions.

For stress concentration analysis around geometric discontinuities in orthotopic plates, the problem becomes more complicated due to the directional dependence of the orthotropic mechanical properties of the elastic plate. Lekhnitskii [20] formulated an equation to estimate SCF in thin orthotropic plates with circular holes, which uses in the in-plane material properties. This equation was later by Pipes et al., [21] and Toubal et al., [22] in their experimental studies of SC in orthotropic plate with a circular hole. Whitney and Nuismer [23] introduced two fracture criteria – the point stress criterion and average stress criterion – to predict tensile failure in composite laminates with holes. Potti et al., [24] enhanced the accuracy of these criteria, while Jen et al. [25] adapted the point stress criterion to predict notch strength in composites at high temperatures, achieving strong experimental agreement.

Hong and Crews [26] conducted a 2D finite element analysis on orthotropic laminates with circular holes, studying width and length-to-radius effects on SCF, which was not considered in [20]. Konish and Whitney [27] derived three SCF equations for orthotropic plates with holes, and Tan [28], 29] modified the PSC for elliptical holes. Jain and Mittal [30] used FE analysis to examine the influence of hole-to-width ratios on SCF in isotropic and orthotropic plates under transverse loads. Hayajneh et al., [31] developed modeling strategy based on homogenization of material properties can be adopted in the analyses of orthotropic plates and is reasonably accepted in the strain-based analysis of quasi-isotropic plates. In 2013, Darwish et al., [32] used extensive finite element analysis findings and nonlinear regressions to obtain high-accuracy equation for computing the stress concentration factor in countersunk hole rivetted in orthotropic plates. This equation formulates the SCF in terms of the countersunk hole geometric parameters, plate width and plate orthotropy.

In addition, machine learning techniques have been commonly used for the modeling and analysis of the SCF in various geometric discontinuities and stress risers. For example, Wang et al., [33] employed an Extreme Learning Machine (ELM) to predict fatigue stress concentration factors in U-notched specimens, demonstrating its effectiveness over traditional regression methods. Iqbal et al., [34] investigated stress concentration factors in Carbon fiber-reinforced polymers (CFRP)-reinforced KT-joints under multi-planar bending through combined experimental and numerical methods. In their work, ANN was employed to construct empirical models to predict SCF across diverse load scenarios. The proposed ANN models were validated with experimental data and demonstrated high accuracy with a maximum error of less than 15 % at the location of the SCF. Rasul et al., [35] developed an artificial neural network-based empirical model to predict stress concentration factors in tubular T-joints under complex bending loads, demonstrating superior accuracy over conventional methods.

It can be concluded that the consideration of machine learning methods, like artificial neural networks, highlights the potential of the use of such modern techniques in the analysis of stress concentrations and for structural optimizations in various engineering structures with high accuracy as well as in a computationally effective manner [36], [37], [38]. Also, machine learning methods are used in fatigue studies [39], 40], crack detection [41] and other structural engineering applications [42], [43], [44]. Therefore, the current paper aims to employ finite element analysis and artificial neural networks to study, model, and optimize the stress concentration factor in orthotropic plates with a centrally located countersunk holes under uniaxial tensile loadings. In fact, the countersunk holes in orthotropic plates configurations are less studied than typical circular holes and other notch-shaped discontinuities.

This structure of the current paper first describes the countersunk hole problem in uniaxially-loaded orthotropic plates, defining the geometric and material parameters. The finite element modeling approach is then detailed subsequently, followed by the ANN-based optimization procedure. The optimal ANN architecture is presented, analyzed, and thoroughly validated. Finally, the newly developed ANN model is employed to acquire an optimum design configuration of the plate material and countersunk hole geometry that minimizes the stress concentration factor in the orthotropic plate with a countersunk hole structure.

2 Countersunk hole in orthotropic plates configuration

Figure 1 illustrates the geometric layout of a rectangular plate with a centrally located countersunk hole. The geometric parameters of this configuration are the plate length (2h), the plate width (2w) and the plate thickness (t). The plate thickness consists of the straight shank thickness (b) and the sinking depth (C _s), therefore (t = b + C _s). Also, the straight shank radius is (r) and the countersink angle is (θ _c). In practice, the countersink angle is commonly available in the range of 80°–120° [10].

Figure 1:

The countersunk hole in a rectangular plate structure: (a) x−y plane, (b) y−z plane, (c) x−z plane, and (d) 3-dimensional design.

For the current problem, an orthotropic material configuration of carbon/epoxy (AS4/3501-6) consisting of eight plies with a stacking system of ± θ p 2 s , shown in Figure 2, is considered. The nine orthotropic mechanical properties, i.e., modulus of elasticity (E), modulus of rigidity (G), and Poisson’s ratio (ν), of a single unidirectional laminate of (AS4/3501-6) in the principal directions are E ₁ = 149 GPa, E ₂ = 10.3 GPa, E ₃ = 10.3 GPa, G ₁₂ = 6.9 GPa, G ₂₃ = 3.7 GPa, G ₁₃ = 6.9 GPa, ν ₁₂ = 0.27, ν ₂₃ = 0.54,and ν ₁₃ = 0.27, as reported by [45].

$Figure 2: The 8-ply stacking configuration ± θ p 2 s ${\left[{\pm}{\theta }_{p}\right]}_{2s}$ of an orthotropic plate.$

Figure 2:

The 8-ply stacking configuration ± θ p 2 s of an orthotropic plate.

Utilizing finite element analysis, extensive simulations will be conducted to generate the value K _t in uniaxially loaded orthotropic plates with a central countersunk hole at different countersink angles (θ _c), plate thickness to hole radius ratio (t/r), countersinking depth to plate thickness ratio (C _s/t) and hole radius to plate width ratio (r/w) as well as different ply angles (θ _p). To exclude the influence of the plate length (2h) on K _t, the plate length to the hole radius ratio is kept constant at (h/r = 15) throughout the analysis.

Lekhnitskii’s equation [20], which was originally developed for computing the SCF in an infinite thin orthotropic plate with a circular hole (K _∞), was included in the present analysis as a measure of the plate orthotropy effect, i.e., the ply angle (θ _p) effect. The (K _∞) is written as:

(1) K ∞ = 1 + 2 E x E y − ν xy + E x 2 G xy

where E _x and E _y are the plate’s material elastic moduli in x and y directions, respectively; ν _xy and G _xy are the Poisson’s ratio and shear modulus in the x-y plane, respectively, of the material of the plate. In fact, a previous study have shown that the out-of-plane mechanical properties (E _z, G _xz, G _yz, ν _xz and ν _yz) do not have a significantly contribute to the SCF value in orthotropic plates with countersunk holes [32]. Therefore, the use of the in-plane properties (E _x, E _y, ν _xy and G _xy), designated in K _∞, is appropriate for computing and investigating the stress concentration value in countersunk hole placed in orthotropic plates (K _t).

For an 8-ply plate configuration, the mechanical properties values are indeed function of the ply angle (θ _p), and thus the K _∞ value. Table 1 lists the K _∞ values at different ply angles for an 8-ply carbon/epoxy (AS4/3501-6) orthotropic plate system.

Table 1:

The corresponding K _∞values in an 8-ply carbon/epoxy AS4/3501-6 plate system ± θ p 2 s at different ply angles.

θ _p [degrees]	0	10	20	30	40	45	50	60	70	80	90
K _∞ [−]	6.32	5.34	3.94	2.88	2.18	2.00	1.90	1.92	1.99	2.21	2.41

To sum up, FEA simulations are executed to generate the stress concentration factor values (K _t) in orthotropic plates with a central countersunk hole subjected to tensile loads for various geometric parameters (r/w, t/r, C _s/t, θ _c) as well as different ply angles (θ _p). Again, the influence of the change in the orthotropic material properties for different ply angle configurations is included using the K _∞ metric.

3 Research methods

3.1 Finite element modeling

Comprehensive 3D finite element analysis is performed in the current work to determine the stress concentration factor at the countersunk hole under various geometric and material configurations. ANSYS Mechanical Pro Version 2021 is utilized to create the finite element model, apply loading and boundary conditions and to execute the analysis. Due to the symmetric nature of the current problem, only one quarter model is considered and investigated.

ANSYS SOLID185 element type is used to generate FE mesh with hexahedron 3D elements only, resulting in a mapped mesh consisting of 144,000 elements and 152,971 nodes. The SOLID185 element type is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. SOLID185 is a homogeneous structural solid that uses the simplified enhanced strain formulation to prevent shear locking phenomenon and is widely used for 3D modeling of solid structures [46]. To ensure accurate results for the stress concentration factor around the countersunk hole without increasing mesh density, the mesh was engineered using the mesh gradation technique to have a very fine mesh near the hole and a coarser mesh design elsewhere in the FE model. In this technique, the element size is controlled by having smaller elements (finer mesh) near the region of interest, e.g., near the hole, and coarser mesh away from that region. This model is presented in Figure 3.

Figure 3:

Finite element model mesh design, showing the high mesh density around the hole.

Symmetric boundary conditions were applied on the quarter FE model by restraining the displacements in the x and y directions (u _x = u _y = 0) at the x = 0 and y = 0 planes, respectively. Additionally, the out-of-plane displacement (u _z) was restricted in all directions at a single node located at x = h, y = w, and z = 0. The tensile loading is applied as a unity remote stress (σ _o = 1) on the x = h. The boundary and loading conditions are illustrated in Figure 4.

Figure 4:

Boundary and loading conditions imposed on the symmetric quarter FE model.

In the finite element model, the mechanical properties of the eight plies of the orthotropic angle-ply laminate were homogenized. This form of homogenization affects only the normal shear coupling terms, leaving the other components unchanged. As a result, the plate can be treated as one single thick ply with homogenized orthotropic properties that depend only on the ply angle [21], 23], 28], 31], 32], 47], 48]. Finally, it is important to mention that the current FE model was previously adopted in an earlier publication [32], where the mesh quality and properties were confirmed. Also, the model SCF results were thoroughly validated with literature data, the reader is strongly recommended to refer to this publication for further information.

3.2 Modeling Using ANN

Artificial neural networks (ANNs) are modern computational models and widely employed in engineering applications for problem-solving and high accuracy predictive modeling. A key advantage of ANNs is their ability to operate without prior knowledge of the underlying relationships between input and output variables in a given process. Moreover, they excel at handling complex non-linear regression tasks with high precision. Architecturally, an ANN comprises numerous processing units, or neurons, organized into multiple hidden layers. A typical network includes an input layer, one or more hidden layers, and an output layer. Before deployment, ANNs undergo a training phase where they analyze extensive datasets containing input-output pairs. This process enables the network to identify patterns and adjust the synaptic weights between neurons. These weights are then processed through activation functions, ultimately allowing the output layer to generate accurate predictions.

In the present work, ANN technique is implemented in MATLAB to obtain the underlying relationships between the geometric (r/w, t/r, C _s/t, θ _c) as well as material (K _∞) factors of the orthotropic plate with a central countersunk hole problem, i.e., the input variables, and the response, e.g., the stress concentration factor values (K _t) value due to uniaxial tension. To ensure the effectiveness and robustness of the ANN modeling procedure, optimal ANN prediction model that accurately estimates stress concentration factor values is first established. The ANN optimization procedure included:

A broad spectrum of hidden neurons – ranging from 1 to 25 – was considered to account for varying ANN model complexities. This approach ensures that both simpler (lower neuron count) and more complex (higher neuron count) network architectures are considered, ultimately improving the precision of the predicted SCF values.
Five backpropagation (BP) learning algorithms including, (1) Levenburg-Marquardt (LM), (2) Bayesian Regularizations (BR), (3) Scaled Conjugate Gradient (SCG), (4) One Step Secant (OSS), and (5) Resilient Propagation (RP), were evaluated to identify the learning algorithm with best accuracy.
Eight activation functions were also explored to ensure ANN model robustness. Such activation functions are (1) Log-sigmoid (logsig), (2) hyperbolic tangent sigmoid (tagsig), (3) linear transfer function (purelin), (4) triangular basis (tribas), (5) radial basis (radbas), (6) Elliot symmetric sigmoid (elliotsig), (7) hard-limit (hardlim), and (8) symmetric hard-limit (hardlims). Such transfer functions are readily available built-in functions in MATLAB.

During the optimization process, the FEA-generated dataset was randomly divided into training, validation and testing datasets using the 70/15/15 rule. However, 32 datapoints, to ensure covering the lower and upper limits of each input (r/w, t/r, C _s/t, θ _c), were forced into the training dataset [40]. To examine many training, validation, and testing data sets, and to potentially reach an optimal configuration, the division process has been repeated 10 different times [49], 50]. Additionally, to maintain dimensional standardization in the network parameters and prevent issues such as underfitting or overfitting, the input and output data are first normalized to a [0, 1] scale [50], [51], [52]. This is achieved by dividing each input parameter and the output parameters by the corresponding maximum value observed for that input/output parameter [51], 52].

This overall optimization process has ended up with the evaluation of 10,000 neural networks (1,000 networks for each simulation trial). The performance metric, i.e., the root means squared error (RMSE), of each network configuration has been computed and stored. The RSME between the ANN predictions (SCF_ANN) and FEA values (SCF_FEA) of the stress concentration values for the entire data points (N) is computed as [53]:

(2) RMSE = 1 N ∑ k = 1 N SCF FEA − SCF ANN 2

Consequently, the optimal ANN architecture with the least overall RMSE value is then selected in this paper to formulate the relationship between the geometric and material parameters of the countersunk hole and the orthotropic plate configuration and the K _t value.

4 Results and discussions

4.1 Optimal ANN architecture

As previously noted, this study employs ANN to obtain the relationship between the geometric and material parameters of the countersunk hole and the orthotropic plate problem and the stress concentration factor due to uniaxial loading. To achieve this, 10,000 different ANN architectures were evaluated based on their accuracy, measured using the root mean square error, to obtain an optimal network architecture. From this network optimization process, it was found that network number 322 from the tenth simulation trial has the least RMSE value of 0.0090. This network has nine hidden neurons in the single hidden layer, uses Levenburg-Marquardt (LM) learning algorithm, and employs hyperbolic tangent sigmoid (tagsig) activation function. Figure 5 presents the least RMSE value (0.0090) in each simulation trial, and Figure 6 shows the RMSE value of the 1,000 ANN of the 10th simulation trial, and a zoomed-up view to show the optimal network system of 322.2

Figure 5:

The least RMSE value obtained in each simulation trial.

Figure 6:

RMSE value for the 1,000 ANNs generated in the 10th simulation trial (left), and a zoomed-up view to show that network #322 has the least RMSE of the value 0.0090 (right).

For completion, Figure 7(a) presents the evolution of the RMSE value versus the activation function ID for the network of nine hidden neurons and LM learning algorithm setting. It is clear that the activation function ID = 2, which corresponds to hyperbolic tangent sigmoid (tagsig), results in the least RMSE value (0.0090). Also, Figure 7(b) depicts the evolution of the RMSE value versus the BP learning algorithm for the network of nine hidden neurons and tagsig activation function configuration. Apparently, the algorithm ID = 1, which corresponds to Levenburg-Marquardt (LM) learning algorithm, results in the least RMSE value (0.0090). Furthermore, Figure 7(c) illustrates the value of the RMSE versus the number of hidden neurons for the network with LM algorithm and tagsig activation function network configuration. Evidently, the network with nine neurons lead to the least RMSE value (0.0090). As this network produces the least RMSE value, i.e., optimal network architecture, it will be used for further analysis in the current paper. Again, this network has one hidden layer with nine hidden neurons, uses Levenburg-Marquardt learning algorithm, and employs hyperbolic tangent sigmoid activation function.

Figure 7:

The evolution of the RMSE value versus (a) activation function, (b) learning algorithm, and (c) number of neurons in the 10th simulation trial.

For computational efficiency purposes, Figure 8 plots the time spent, in minutes, in each learning algorithm over all the 10 simulation runs with all number of neurons and activation functions scenarios. Clearly, the Bayesian regularization algorithm takes the longest time (≈61 min), and the one step secant algorithm takes the shortest time (≈7 min). While the optimal algorithm, the Levenburg-Marquardt, takes about 7 min and a half to execute. These further states that while the current optimization procedure obtained the ANN system with best accuracy, the simulation time is kept reasonable.

Figure 8:

Computational time for each learning algorithm executed over all ANN configurations and simulation trials. Total simulation time = 92.77 min.

4.2 Error Analysis

As mentioned previously, the network number 322 of the 10th simulation trial was found as the optimal ANN configuration. This network contains one hidden layer with nine hidden neurons and uses Levenburg-Marquardt learning algorithm as well as hyperbolic tangent sigmoid activation function. The coefficient of determination of this network, i.e., the goodness of fit parameter, was calculated as (R² = 98.9), indicating high ANN fit quality.

Figure 9 presents the ANN-predicted stress concentration factor values versus the FEA values for the training, validation, and testing data sets. The figure reveals that all data points are well clustered around the normal line, demonstrating that the ANN model is well-trained. This close alignment indicates that the presently developed optimal ANN model has successfully learned and identified the underlying patterns and relationships within the inputs and outputs.

Figure 9:

The ANN-predicted versus the FEA-computed (actual) stress concentration values for the training, validation, and testing data sets.

Figure 10(a) illustrates a comparison between the ANN-predicted and the actual stress concentration factor values for all data points used throughout the development of the ANN configuration. Apparently, the fitted and FEA values of the SCF are in good agreement, which further indicates the high-quality optimal ANN configuration. Figure 10(b) shows the absolute errors (|predicted-actual|), the percentage error (|predicted-actual|/actual × 100 %), and the error frequency plot. For the absolute errors, all error values are less than 0.5 except for one data point where the absolute error was 0.51. Considering the percentage error, the majority of the percentage errors are less than 10 %, and there are only a few cases where the error is little higher. The error frequency plot indicates that 92 % of the percentage errors are less than 6 % and only 8 % are higher than that. As a result, the presently optimized ANN configuration can faithfully capture the relationship between the input values of (r/w, t/r, C _s/t, θ _c, K _∞), and the stress concentration factor (K _t) in countersunk holes that are centrally placed in orthotropic plates under uniaxial tension.

Figure 10:

Error analysis: (a) Comparison between ANN-predicted and FEA target SCF values, and (b) analysis for the original dataset.

To further ensure the a ccuracy of the optimal ANN configuration of the present work, the FE model was used to generate 64 more data points considering plate-hole configurations that were not used during the ANN development process. Such 64 cases were plugged into the optimal ANN system to compute the values of the SCF. Figure 11(a) shows the comparison between the ANN-predicted and the FEA-computed stress concentration factor values for the new 64 data points. Again, the ANN-based and FEA values of the SCF are in excellent agreement. Figure 11 Error! Reference source not found. (b) shows the absolute errors, the percentage error, and the error distribution plot. For the absolute errors, all error values of this new dataset are equal to or less than 0.2. The percentage error plot suggests that all errors are <10 %. Also, the error distribution plot implies that 94 % of the percentage errors are equal to or less than 6 %. Therefore, the presently optimized ANN approach can accurately capture the relationship between the geometric and the material factors of the present problem and the stress concentration factor values. Hence, it can be trustfully used for further analysis, i.e., stress concentration factor optimization, as will be discussed in the subsequent subsection.

Figure 11:

New dataset of 64 cases: (a) comparison between ANN-predicted and FEA target SCF values, and (b) error analysis.

4.3 Stress concentration factor optimization

As the final stage of this study, an optimization analysis that focuses on identifying the best combination of input design parameters that minimize the predicted SCF is conducted using MATLAB. Table 2 presents the optimal input variables obtained using the presently tuned ANN prediction model. For completeness, the corresponding FEA-based SCF values for this optimal configuration are also reported in Table 2. Apparently, this plate-hole configuration results with a very low stress concentration factor value (K _t = 1.9491). Also, this minimized value matches very well with the FEA computations, ending up with a very low percentage error of 1.75 %. It is important to emphasize that the ply angle value that corresponds to K _∞ = 1.90, according to Table 1, is equal to θ _p = 50°. Therefore, the current study recommends the use of this hole-plate geometric configuration in an 8-ply carbon/epoxy AS4/3501-6 plate system with a ply angle θ _p = 50°, which ends up with a K _∞ = 1.90, as it is feasible from manufacturing point view and produces very low stress concentration.

Table 2:

Stress concentration factor optimization results.

Optimal input parameters					ANN predicted K _t	FEA K _t	Percentage error
r/w	t/r	C _s/t	θ _c	K _∞	ANN predicted K _t	FEA K _t	Percentage error
0.10	5	0.10	80	1.90	1.9491	1.9839	1.75 %

Figure 12 shows the FEA stress contour plots of this optimal system. Apparently, the maximum stress value occurs at the countersunk hole edge, which is consistent with literature value. Also, the maximum stress value is σ _max = 1.98 Pa, and the nominal (or average) stress value is σ _nom = 1.00 Pa, thus the stress concentration factor value can be conveniently computed as K _t = 1.98. It is worth noting here that this plate configuration contains a thick plate, small hole with short sinking depth and small sinking angle and made of a ply angle with the least K _∞ value.

$Figure 12: The stress contour plots of the optimum orthotropic plate with a central countersunk hole configuration K t , FEA = σ max σ nom = 1.98 1 = 1.98 $\left({K}_{t,\text{FEA}}=\frac{{\sigma }_{\mathrm{max}}}{{\sigma }_{\text{nom}}}=\frac{1.98}{1}=1.98\right)$ .$

Figure 12:

The stress contour plots of the optimum orthotropic plate with a central countersunk hole configuration K t , FEA = σ max σ nom = 1.98 1 = 1.98 .

In conclusion, this paper has presented a methodology based on finite element simulations and artificial neural networks to accurately model the stress concentration factor induced in orthotropic rectangular plates with a centrally riveted counter sunk holes due to uniaxial tensile loading. This methodology has obtained the optimal artificial neural network considering the number of hidden neurons, backpropagation learning algorithm, and the activation function. Finally, this optimal network system was able to obtain the optimal problem parameters that minimize the stress concentration factor.

The methodology presented in this paper can be generalized and used for the analysis of stress concentration factors induced in other forms of stress risers and other orthotropic layups and materials. More importantly, the presently trained ANN model can be reliable for the analysis of the SCF in centrally placed countersunk holes in rectangular plates manufactured from other orthotropic materials or other ply layups as long as the values of the infinite plate orthotropic stress concentration factor (K _∞) are comparable to the values investigated in the current research.

From the sustainability perspectives, the findings of this study could contribute to sustainability in engineering design by enabling more efficient use of advanced composite materials in structural components. By accurately predicting and minimizing stress concentration factors in orthotropic plates with countersunk holes, the proposed ANN-based modeling framework supports the development of lighter, longer-lasting structures. This reduces the need for material overdesign, minimizes waste, and contributes to savings in transportation and infrastructure systems. Additionally, the methodology may assist in extending the lifecycle of components made from carbon/epoxy composites, aligning with broader goals in sustainable manufacturing and engineering.

The presently developed FEA-ANN modeling procedure is accurate, and easy to implement. However, it has some limitations, and further improvements are essentially recommended. First, this study assumed linear elasticity in the FEA simulations used to train the ANN. In reality, however, stress risers can create very high stress localization zones, potentially introducing nonlinear behavior, i.e., stresses beyond the yield point. Consequently, the model’s accuracy may diminish at higher stress concentration values. Second, the current formulation does not account for stress/strain triaxiality which might be significant especially for thicker plate configurations. Finally, experimental validation is strongly recommended to enhance confidence in the current formulations and findings.

5 Conclusions

This paper has presented a robust methodology for analyzing and optimizing stress concentration factors in orthotropic plates with countersunk holes. Finite element simulations were employed to generate stress concentration factor data as a function of plate and the hole geometric parameters, such as hole radius-to-plate width ratio, plate thickness-to-hole radius ratio, countersinking depth-to-thickness ratio, and countersink angle, as well as plate’s orthotropy, represented by the ply angle and the corresponding infinite-plate concentration factor (K _∞). An optimization process was then conducted to create an artificial neural network to effectively characterize the relationship between the input parameters and the stress concentration factor. The ANN model is profoundly validated with FEA results, demonstrating excellent agreement between both solutions. Lastly, the optimized ANN model was employed to identify the geometric and material configuration that minimizes the stress concentration factor in orthotropic plates with a central countersunk hole, offering actionable insights for engineers to design safe and efficient structures with such discontinuities. Nevertheless, the present study assumed linear elastic behavior and did not account for stress triaxiality or nonlinear effects. Experimental validations are strongly recommended to further verify the model’s accuracy.

Corresponding author: Mohammad A. Gharaibeh, Department of Mechanical Engineering, Faculty of Engineering, The Hashemite University, Zarqa, 13133, Jordan, E-mail: mohammada_fa@hu.edu.jo

Funding information: Author states no funding involved.
Author contributions: Author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Author states no conflict of interest.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2025-10-21

Accepted: 2025-11-10

Published Online: 2026-01-08

This work is licensed under the Creative Commons Attribution 4.0 International License.