Finite Element Method for Stress Analysis of an Infinite Plate with an Elliptical Hole Using Functionally Graded Materials

Abstract

This study investigates the stress concentration factor in an infinite steel plate with a thickness of 1 cm, containing an elliptical hole, subjected to biaxial loading at infinity. The elliptical hole has semi-axes a=5.0 cma = 5.0 \, \text{cm}a = 5.0 cm (major axis) and b=2.5 cmb = 2.5 \, \text{cm}b = 2.5 cm (minor axis). The applied stresses at infinity are a tensile stress of σ1=100 kg/cm2\sigma_1 = 100 \, \text{kg/cm}^2= 100 kg/parallel to the major axis and a compressive stress of σ2=−100 kg/cm2\sigma_2 = -100 \, \text{kg/cm}^2= -100 kg/ perpendicular to the major axis. The material properties include Young's modulus E=2.1×106 kg/cm2E = 2.1 \times 10^6 \, \text{kg/cm}^2E = 2,1.  kg/ and Poisson's ratioν=0.3\nu = 0.3  = 0.3. Using analytical solutions from classical elasticity theory, the maximum tangential stress at the edge of the ellipse is calculated as σmax=600 kg/cm2\sigma_{\text{max}} = 600 \, \text{kg/cm}^2= -600 kg/, yielding a stress concentration factor of kσ=σmax/σ=6k_\sigma = \sigma_{\text{max}} / \sigma = 6 =  =6. Additionally, a finite element (FE) analysis based on the Salerno and Sahoni problem for a quarter section of the plate results in kσ=3.1125k_\sigma = 3.1125 = 3.1125 for a configuration with s/r=5s/r = 5s/r = 5, showing a discrepancy of 1.3% compared to the theoretical value of kσ=3.1k_\sigma = 3.1= 3.1 from Peterson's Stress Concentration Factors. The results demonstrate good agreement between the calculated model and theoretical predictions, validating the accuracy of the FE approach for stress concentration analysis in such configurations.