Analysis of Stress and Strain Concentrations Around an Elliptical Hole Via Finite Element Method and Response Surface Methodology

The current investigation aims to study the stress and strain concentration factors ( K s and K 3 ) around a central elliptical hole in a rectangular plate under uniaxial and biaxial loads. Finite element and response surface methods (FEM and RSM) were used to predict the values of K s and K 3 in isotropic and orthotropic material rectangular plates. Thus, the ﬁ nite element models were built using ANSYS commercial FEM software to obtain the values of K s and K 3 for some designed cases. Moreover, the response surface methodology was applied to limit the required number of FEM runs based on the number of variables. In addition, RSM was used to obtain the required formulas to predict the values of K s and K 3 . The obtained formulas are accurate and closely compatible with ﬁ nite element analysis results. The obtained formulas may be used to suggest an ef ﬁ cient way to estimate the effect of elliptical holes on plate structures made of isotropic and orthotropic materials.


Introduction
I n numerous industrial applications, fastening and riveting are very common procedures for assembling structural components.The presence of such holes would serve as a stress and strain riser in the area around the hole.These localizations of high stresses and strains are commonly called stress and strain concentrations, and they were measured using stress and strain concentration factors (K s and K e ), respectively (Pilkey et al., 2020;Deghboudj et al., 2017;Vajipeyajula et al., 2023).Mekalke et al. (2012) observed the difference in the results obtained from different meshes on a plate with a circular hole under uniform stress.Darwish et al. (2013) studied K s for countersunk rivet holes in orthotropic laminate plates under a uniaxial stress load using the finite element analysis (FEA).Diany (2013) examined the effects of the position and slope of the hole in the thin plate on the stress concentration factor.Moreover, he presented a comparison between the results of the stress concentration factor values estimated by classical formulas given in a numerical simulation and ulterior studies conducted using commercial software for a thin plate with an eccentric hole.Enab (2014) used finite elements to investigate the stress concentration factors K s due to uniaxial and biaxial loads of an elliptic hole in functionally graded material plates.Goyat et al. (2022) investigated the applicability of inverse distance weighted (IDW) function-based functionally graded materials (FGM) for stress concentration reduction in a finite panel with a circular hole and subjected to biaxial loading.Goyat et al. (2017) used the extended finite element method to present a study of the stress concentration under uniaxial, biaxial, and shear loads in an infinite panel with a rounded rectangular hole reinforced with a functionally graded material layer.Their study proved that the stress concentration can be minimized significantly by controlled variation in the material properties of the layer.Khajehsaeid et al. (2016) used finite element simulations to examine the strain and stress concentrations of elastomers at finite deformations, concentrating on elastomeric rectangular sheets with a central circular hole.Tadepalli et al. (2017) used the finite element method to examine the effect of the diameter-to-height ratio and diameter-to-thickness ratio on K s .Their investigation analyzed the K s results around a circular hole in orthotropic and isotropic rectangular plates.Their study was carried out under uniformly distributed loading conditions.
Mhalla et al. (Makki et al., 2018) used response surface methodology (RSM) to develop mathematical models to predict the stress concentration factor of a carbon fiber-reinforced epoxy.Ahmed et al. (2018) used the finite element (FE) model to study the predamaged stress concentration factor for a composite laminate member under tensile loading with a central circular hole.Their results of FE models were found to have similar empirical results.Gharaibeh et al. (2021) used the FEM and the RSM to analyze the K s of an isotropic plate with a central countersunk hole under uniaxial loading.Mosbah et al. (2020) used a definitive screening design to characterize the effects of the prehole geometry in the countersinking process.Statistical analysis and a parametric study were carried out to investigate the effects of geometrical parameters of the prehole and their interactions in the countersinking process to minimize the K s in countersunk holes in isotropic plates.An optimum countersunk hole and plate design geometry were suggested.
Moreover, Gharaibeh (2020) used FEM and RSM to study the stress concentration factors in two identical countersunk rivet holes in isotropic rectangular plates under uniaxial tension.Their results suggested that thin plates with two well-separated countersunk holes at depths and small angles, as well as a small radius, are designed to reduce the stress concentration caused by uniaxial loading.Patel and Desai (2020) derived a formula for the tangential K s in a large rectangular plate around an elliptical hole subject to variable linear in-plane loading on two opposite edges.Their study was carried out with the aid of Cauchy's integral formula, the conformal mapping method, and FE models using the ABAQUS program.Khechai et al. (2014) studied the stress concentration factors in cross-and-angle-ply laminated composite plates with single circular holes in isotropic plates subjected to uniaxial loading using finite elements.Bhargava and Shivakumar (2008) studied strain concentration under tensile load at a countersunk hole in plates and developed a three-dimensional equation for K e .They used FEM to verify the developed equation for a variety of geometric configurations of the plate and countersunk holes.Jreissat and Gharaibeh (2021) used FEM and RSM to investigate the strain concentration factor under uniaxial tension on a central countersunk hole in an isotropic rectangular plate.Their study suggested an optimal design of plate and countersunk holes to reduce the value of K e .Jiang et al. (2023) used numerical simulations to discuss stress and strain concentrations around a centralized elliptical hole in a monodomain liquid crystal elastomer sheet.They used the FEM to investigate the effects of the shape factor on stress and strain fields using finite element methods.Also, Patel et al. (Patel and Patel, 2023) used FEM to investigate the K s around the polygonal holes in the finite plate, under uniaxial compression load.Badiger and Ramakrishna (Badiger and DS, 2023) used the boundary force method to theoretically investigate the stress in an infinite plate under uniaxial load in circular or elliptical holes.They concluded that the boundary force method can be useful as a first estimate in determining K s in plates with various shapes of discontinuities, irregular shaped holes, and interaction of multiple discontinuities.
Based on the aforementioned review, it is evident that, to the best of the authors' knowledge, no study has used the response surface methodology to examine stress and strain concentrations around an elliptical hole.Using this methodology could lead to more precise evaluations of these critical factors.Therefore, in this research, we present a study of the stress and strain concentration factors (K s and K e ) for isotropic and orthotropic plates with central elliptical holes.Thus, the developed K s and K e as well as the stress and strain distributions under uniaxial and biaxial loading conditions will be calculated using the FEM and the RSM.Accordingly, this study will help suggest an optimal geometrical design of the plate and the elliptical hole to minimize the values of K s and K e .

Parameters and plate geometry
The configuration of the plate geometry contains an elliptical hole under uniaxial or biaxial loading, as shown in Fig. 1.A Cartesian coordinate system is used with the origin located at the center of the elliptical hole.The width of the plate is considered to be 2 W, while its height and thickness are assumed to be 2H and t, respectively.The major and minor axes of the elliptic hole will be considered as 2a and 2 b, respectively (see Fig. 1).K s and K e are studied at different radius-to-width ratios (a/W), thickness-to-width ratios (t/W), and biaxial loading ratios (l) using FEM and RSM.Therefore, the geometrical parameters of the plate and the central elliptical hole are chosen as follows: 2 W ¼ 2H ¼ 100 mm; t ¼ 1 : 10 mm with step 2.25; a ¼ 5 : 25 mm with step 5; b ¼ 5 : 25 mm with step 5; and biaxial loading ratio (l) ¼ e1 : 1 with step 0.5.

Mechanical properties of the plate material
In the current work, the elastic properties of the isotropic plate material are considered to be 210 GPa for the elastic modulus (E ) and 0.3 for Poisson's ratio (n).However, the mechanical properties of the orthotropic plate material of carbon/ epoxy (AS4/3501-6) (Darwish et al., 2013) in the principal directions are 27, and n 23 ¼ 0.54.Note that, mathematically, K s and K e are defined as follows: where s max is the maximum stress, s avg the average stress, 3 max the maximum strain, and 3 avg is the average strain.

Finite element modelling
ANSYS parametric design language (APDL) (ANSYS Engineering analysis system) was used to build the finite element models of the present study.As the plate geometry, boundary conditions, and the load are symmetric, only one symmetrical eighth of the plate has been modeled.The boundary conditions of the one-eighth model were imposed by restricting the x-displacement (u x ) at x ¼ 0 and the ydisplacement (u y ) at y ¼ 0 to calculate the full model symmetry planes.Furthermore, to prevent out-ofplane motion, displacements were restrained in the z-direction (i.e., u z ¼ 0) at the plane of z-axis symmetry.
For mesh properties, Solid186 ANSYS hexahedron element was based on the present model, and care was taken to create a finer mesh near the special region of interest, i.e., near the elliptical hole, and comparatively coarser elsewhere.The adequacy of the finite element mesh was tested using a mesh sensitivity study to ensure the accuracy of the stress and strain results with a minimum solution time.Fig. 2 presents the FE mesh configuration.

Mesh sensitivity analysis
The mesh density plays a significant role in influencing the results of the model; thus, it is imperative to select an appropriate mesh density to ensure result accuracy.In addition, in an effort to expedite calculation times, mesh coarseness increases with the distance from elliptical hole elements.It is important to emphasize that the mesh density in regions experiencing high-stress concentration differs from other areas (Fig. 2).Consequently, a mesh controlling factor for element size has been introduced, given that the maximum stress and strain are concentrated exclusively around the central elliptical hole.Note that increasing the value of mesh control factor results in higher numbers of element and thus more accurate results.Table 1 lists the mesh sensitivity results for a plate with a 15 mm circular hole diameter under uniaxial tension.It can be concluded that there is a negligible difference between models with mesh controlling factors of 1.5 and 2. Therefore, a mesh controlling factor of 1.5 was used for comparing the models of the current study.Furthermore, the developed FE models were fully validated and correlated with certain cases in the literature and experimental data in some previous publications that reproduced the same results (Pilkey et al., 2020;Yang, 2009).

Analysis using response surface methodology
RSM is a set of statistical and mathematical techniques used to develop, improve, and optimize processes.Fig. 3 presents the main steps of the RSM model.The current work used RSM to study and optimize (i.e., minimize) K s and K 3 values as well as the stress and strain distributions around a central elliptical hole in isotropic and orthotropic plates under uniaxial and biaxial loading conditions.Therefore, the MINITAB program was used to perform the optimization process using RSM.As listed in Table 2, four independent variables were studied.These independent variables include ellipse major radius to width ratio (a/w), ellipse minor radius to width ratio (b/w), thickness to radius ratio (t/w), and biaxial loading ratio (l).

Analysis of stress and strain concentration factors
The central composite design (CCD) is usually used as the basis for the RSM, which is considered as a useful tool for generating an empirical model for explaining the dependent variable variation as a function of independent variables.The obtained empirical model is frequently presented as a

Independent variables RSM levels
Low High Ellipse major radius-to-width ratio (a/w) 0.05 0.25 Ellipse minor radiuseto-width ratio (b/w) 0.05 0.25 Thickness-to-width ratio (t/w) 0.01 0.1 Biaxial loading ratio (l) À1 1 polynomial regression equation.Accordingly, equation ( 2) represents a common polynomial model equation that predicts the response variables as a result.This mathematical formula was fitted to the K m response in a second-order response surface: where subscript 'm' represents the stress or strain; b 0 is the constant coefficient; b i , b ii , and b ij are the interaction coefficients of linear, quadratic, and second order terms, respectively; and Z i and Z j are the independent variables.
To determine the statistical significance and regression coefficients of the terms in the empirical model, multiple regression analysis and variance analysis (ANOVA) based on the least-square method were used.ANOVA was used to adapt empirical models with simulation data and predict linear and square response models, resulting in a comprehensive optimal response region The adequacy of the empirical model was determined at a significant level of testing using the F-ratio (the mean square for a factor divided by the mean square error), the goodness of fit, and coefficient of determination (R 2 ) (a statistical measure that quantifies the percentage of variance in the dependent variable that can be explained by the independent variable) analysis.

Isotropic material results
This section of the present study is carried out for isotropic material in the elastic state (i.e., linear problem), and therefore only one equation is sufficient and can be applied to obtain the values of K s and K 3 .Therefore, only the study on stress concentration factors were presented at the next subsections.

Central composite design results
The reduction of K s and K 3 is always favorable for safe mechanical designs.In this research, the data of 31 central composite design runs are summarized in Appendix 1.The maximum K s value (K s ¼ 7.642) is observed in the 14th run for the following geometrical and loading parameters: a/w ¼ 0.05, b/w ¼ 0.15, t/w ¼ 0.055, and l ¼ 0. It is noted that the values of stress and strain concentration factors are coincident with each other as the material is isotropic.For example, the maximum K 3 value of 7.642 (note: K e value coincident with K s value) is observed in the 14th run, where a/w ¼ 0.05, b/w ¼ 0.15, t/w ¼ 0.055, and l ¼ 0. Consequently, these geometric parameters of the plate and the elliptic hole are not recommended.

Variance results and empirical model analysis
Stress concentration factor (K s ): In the present research, the quadratic regression model was the most accurate empirical model for the K s value.In response to surface methodology, all the geometric parameters (a/w, b/w, t/a, l) and their interactions were included in the originally proposed equation form as follows: A model transformation to the natural logarithm form was required to provide the best fitted model correctness.It was also necessary to reduce the model, which was accomplished by removing the inconsequential terms.Therefore, the final form of the K s equation is given by lnðk s Þ¼ 0:901À7:79 a The results of the ANOVA from Eq. ( 4) are listed in Appendix 2. For the fitted K s models, the coefficients of determination were R 2 ¼ 0.9756, indicating accuracy of high models.The residuals of the RSM empirical model (i.e., the error between equation ( 4) and FEA results) should be examined to ensure the validity of the models.Note that a valid response surface methodology model requires these residuals to be nonskewed, unrelated to each other, have constant variance, and be normally distributed.For this reason, Fig. 4 shows the residual plot for the K s response.It is obvious to note that the residuals are nonskewed (histograms, Fig. 4a), are normally distributed (Fig. 4b), have constant variance according to the results of these plots (residuals vs. fits, Fig. 4c), and are unrelated to each other (Fig. 4d).This suggests that the assumptions underlying the ordinary least squares method for deriving equation (4) are completely achieved.As a result, the regular least squares regression yields coefficient estimates that are unbiased and possess minimal variance.

Effects of geometrical parameters
Fig. 5 presents the effect of nondimensional geometric and loading parameters (a/w, b/w, t/w, l) on the values of stress concentration factors (K s ).From this figure, it is obvious to note that the values of K s increase as the ellipse major radius-to-width ratio (a/w) decreases and the ellipse minor radius-towidth ratio (b/w) increases.The plate thickness-towidth ratio (t/w) nearly has a nonsignificant effect on K s values within the proposed range of this ratio,  while increasing loading ratio (l) results in a decrease in the stress concentration factor.
Fig. 6 demonstrates the three-dimensional surface plots of K s as a function of two continuous variables.
The use of these surface plots could facilitate the forecast of the analysis of stress and strain concentration factors at any point within the predetermined parameter range.It is evident to note that the investigated nondimensional geometric parameters affecting elliptical holes played a significant role in predicting the values of stress and strain concentration factors.
Finally, during the analysis phase, it becomes highly convenient to determine plate and elliptical hole configurations that result in the minimization of K s and K 3 values.The optimal design is achieved at a/w ¼ 0.2, b/w ¼ 0.1, t/w ¼ 0.0325, and l ¼ 0.5 with K s ¼ 1.708 and K 3 ¼ 1.708 as evaluated by MINITAB.FEA results for the stress distributions at the abovementioned values as shown in Fig. 7.The K s and K 3 results of this design were examined with the FEA data and were within the limited confidence interval (CI ¼ 95 %).

Central composite design results
In a manner similar to the techniques applied on the isotropic material plates, the data of 31 central composite design runs are summarized in Appendix 3. It is noted that the largest K s and K e values (K s ¼ 15.407 and K 3 ¼ 1.039) are observed in the 28th run for the following geometrical and loading parameters: a/w ¼ 0.1, b/w ¼ 0.2, t/w ¼ 0.033, and l ¼ À0.5.Therefore, these geometric parameters of the plate and the elliptic hole are not recommended.

Variance results and empirical model analysis
The quadratic regression model emerged as the empirical representation with the highest accuracy for the K s and K 3 values.Within the framework of response surface methodology, equation (3) encompassed all geometric and loading parameters (a/w, b/w, t/w, l) along with their interdependencies.Furthermore, a conversion of the model into its natural logarithm form was essential to ensure optimal fitting model accuracy.Besides, the process of model refinement involved the elimination of insignificant terms.Therefore, the final form of the K s equation is given by The outcomes of the ANOVA for Eqs. ( 5) and ( 6) can be found in Appendix 4 and 5, respectively.In relation to the fitted K s and K 3 models described earlier, their coefficients of determination were R 2 ¼ 0.9789 and R 2 ¼ 0.8580, respectively, indicating a significant level of accuracy in the models.To ascertain the credibility of the RSM empirical model, specifically, the disparity between equations ( 5) and ( 6) and the FEA results, the residuals must be scrutinized.Note that a valid response surface methodology model necessitates that these residuals exhibit traits such as nonskewness, independence from each other, uniform variance, and adherence to normal distribution.In light of this, Figs. 8 and 9 present the residual plots for the K s and K 3 responses.Evidently, the residuals exhibit nonskewness (histograms, Fig. 8a and 9a), conform to a normal distribution (Fig. 8b and 9b), display consistent variance according to the outcomes of these plots (residuals vs. fits, Fig. 8c and 9c), and exhibit independence from each other (Fig. 8d and 9d).This collectively suggests that the assumptions underlying the ordinary least squares method in deriving equations ( 5) and ( 6) have been thoroughly met.Consequently, the regular least squares regression method provides coefficient estimates devoid of bias and characterized by minimal variance.

Effects of geometrical parameters
In a similar manner for isotropic material plates, Fig. 10 illustrates how the values of stress and strain concentration factors (K s and K 3 ) for orthotropic material plates are influenced by nondimensional geometric and loading parameters (a/w, b/w, t/w, l).It is evident from this figure that decreasing the major radius-to-width ratio (a/w) of the ellipse and  increasing the minor radius-to-width ratio (b/w) lead to higher values of K s and K 3 .The plate thickness-to-width ratio (t/w) has a negligible impact on K s and K 3 within the specified range.Unlike the situation observed with isotropic material plates, an orthotropic material plate exhibits an inverted curve in the relationship between the biaxial loading ratio and the strain concentration factor.Accordingly, increasing the loading ratio (l) results in a reduction of the stress concentration factor.Furthermore, a lower negative and higher positive loading ratio increases the strain concentration factor.Consequently, under uniaxial loading conditions (l ¼ 0), minimum strain concentration occurs, whereas biaxial loading typically rises K e .
Figs. 11 and 12 exhibit 3D surface plots illustrating the variations in K s and K 3 concerning two continuous variables.These surface plots can greatly aid in predicting stress and strain concentration factors across the entire specified parameter range.Notably, the nondimensional geometric parameters under scrutiny for elliptical holes proved to be influential in forecasting stress and strain concentration factor values.
In the concluding analysis phase, it proves highly advantageous to identify plate and elliptical hole configurations that lead to the minimization of K s and K 3 values.The optimal design is attained when a/w ¼ 0.15, b/w ¼ 0.05, t/w ¼ 0.055, and l ¼ 0, resulting in K s ¼ 2.464 and K 3 ¼ 0.299, as determined using MINITAB.The stress and strain distributions at these specified values are illustrated in Fig. 13 based on FEA results.The K s and K 3 outcomes of this design were cross-verified with the FEA data and were found to fall within the 95 % confidence interval, ensuring their reliability.

Conclusions
The FEM and RSM were used to determine the stress and strain concentration factors (K s and K e ) due to an elliptical hole in an isotropic and orthotropic rectangular plate under uniaxial and biaxial loading conditions.The main results of the current study are outlined below: (1) A comprehensive analysis is conducted to analyze stress and strain concentration factors around an elliptical hole through the utilization of the finite element method.
(2) Three dimensionless geometrical factors, i.e., ellipse major radius-to-width ratio (a/w), ellipse minor radius-to-width ratio (b/w), thicknessetowidth ratio (t/w), and biaxial load ratio (l), were tested for isotropic and orthotropic plates.(3) Using the RSM, second-order equations for K s and K e in terms of normalized parameters were derived and verified with respect to FEA results.(4) These equations can serve the purpose of diminishing K s and K 3 values near an elliptical hole within both isotropic and orthotropic plates.Moreover, they can contribute to achieving an optimal design for the elliptical hole and plate geometries.
(5) The investigated nondimensional geometric parameters affecting elliptical holes played a significant role in predicting the minimum values of stress and strain concentration factors.(6) The generated surface plots offer a valuable tool for predicting stress and strain concentration factors across the entire range of studied parameters at any given point.and M. A. Aboueleaz planned the scheme, initiated the project and suggested the methodology and techniques, and also supervised all the research steps.S. M. Aiad and T. A. Enab developed the finite element (FE) models using the commercial FEM software ANSYS.S. M. Aiad and M. A. Aboueleaz prepared and carried out the statistical analysis using the response surface methodology (RSM).All authors examined and analyzed the results.The manuscript was written through the contribution of all authors.All authors discussed the results, as well as they reviewed and approved the final version of the manuscript.

Conflicts of interest
There are no conflicts of interest.

Fig. 1 .
Fig. 1.Geometry configuration of a plate with a central elliptic hole under biaxial loading conditions.

Fig. 2 .
Fig. 2. Finite element mesh configuration: (a) entire plate and (b) close-up of meshing around the hole.

Fig. 4 .
Fig. 4. Residual plots for the K s response in isotropic material plates: (a) histogram, (b) normal probability, (c) residuals versus fits, and (d) residuals versus order.

Fig. 5 .
Fig. 5. Main effects of nondimensional geometric and loading parameters on K s values.

Fig. 6 .
Fig. 6.Three dimensional response surface plots of the K s versus the different normalized parameters: K s versus (a) a/w and b/w, (b) a/w and l, (c) a/ w and b/w, (d) b/w and t/w, (e) b/w and l, and (f) t/w and l.

Fig. 11 .
Fig. 11.Three dimensional response surface plots of the K s versus the different normalized parameters: K s versus (a) a/w and b/w, (b) a/w and l, (c) a/ w and b/w, (d) b/w and t/w, (e) b/w and l, and (f) t/w and l.

Fig. 12 .
Fig. 12.Three dimensional response surface plots of the K 3 versus the different normalized parameters: K e versus (a) a/w and b/w, (b) a/w and l, (c) a/w and b/w, (d) b/w and t/w, (e) b/w and l, and (f) t/w and l.

Table 1 .
Results of finite element method mesh sensitivity analysis.

Table 2 .
The four independent variables used in response surface methodology design.
Appendix 1. Results of central composite design (CCD) for isotropic materials