Amirkabir University of Technology Amirkabir Journal of Mechanical Engineering 2008-6032 54 8 2022 10 23 Large-Amplitude Frequency Analysis of Bi-directional Functionally Graded with Non-Uniform Porous Beams using a Higher Order Shear Deformation Theory Large-Amplitude Frequency Analysis of Bi-directional Functionally Graded with Non-Uniform Porous Beams using a Higher Order Shear Deformation Theory 1761 1788 4857 10.22060/mej.2022.20916.7336 FA Mohammadamin Forghani Department of Mechanical Engineering,Shiraz Branch, Islamic Azad University, Shiraz, Iran Yousef Bazargan Lari Mechanical Engineering Department, Shiraz Branch, Islamic Azad University, Shiraz, Iran Parham Zahedinejhad Department of Mechanical Engineering, University of North Texas, North Texas, USA Kazemzadeh-parsi Mohammad-Javad Associate professor, Department of Mechanical Engineering, Shiraz Branch Islamic Azad University Journal Article 2021 12 23 <span style="letter-spacing: .05pt;">This Paper deals with the large amplitude frequency behavior of porous bi-directional functionally graded beams subjected to various boundary conditions which are simply supported, clamped-simply supported, clamped-clamped, and clamped-free utilizing Reddy third-order shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam change according to power and exponential law in both directions. The equations of motion and associated boundary conditions are derived by means of Hamilton’s principle. A generalized differential quadrature method in conjunction with a direct numerical iteration method is selected to solve the system of equations. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Timoshenko beam theory. The results of extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratio, porosity coefficient, Tapered ratio, shear and elastic foundation parameters, and boundary conditions on the Large amplitude vibration frequencies of the bi-directional functionally graded beams. The results reveal that non-linear frequencies increase with the rise of elastic foundation and tapered coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non-dimensional frequencies. The results of this study, while carefully examining the frequencies of variable cross-sectional functionally graded beams, are effective in the optimal design of bi-directional beams and are very effective in predicting and detecting failure modes of these beams.</span> <span style="letter-spacing: .05pt;">This Paper deals with the large amplitude frequency behavior of porous bi-directional functionally graded beams subjected to various boundary conditions which are simply supported, clamped-simply supported, clamped-clamped, and clamped-free utilizing Reddy third-order shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam change according to power and exponential law in both directions. The equations of motion and associated boundary conditions are derived by means of Hamilton’s principle. A generalized differential quadrature method in conjunction with a direct numerical iteration method is selected to solve the system of equations. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Timoshenko beam theory. The results of extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratio, porosity coefficient, Tapered ratio, shear and elastic foundation parameters, and boundary conditions on the Large amplitude vibration frequencies of the bi-directional functionally graded beams. The results reveal that non-linear frequencies increase with the rise of elastic foundation and tapered coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non-dimensional frequencies. The results of this study, while carefully examining the frequencies of variable cross-sectional functionally graded beams, are effective in the optimal design of bi-directional beams and are very effective in predicting and detecting failure modes of these beams.</span> https://mej.aut.ac.ir/article_4857_62459f4e225e2f4f196c9d42f4ad7111.pdf