Variational Formulations of Exponential Shear Deformable Plate Buckling Problems and Analytical Solutions Using Double Fourier Series Method

This study presents exponential shear deformable plate buckling formulation of a thick plate using variational method. The formulation considered transverse shear deformation and used a rigorous procedure to construct the transverse shear stresses in exponential function form such that and vanish at the top and bottom surfaces. The resulting differential equations (DEs) obviates the transverse shear stress correction factors. The DEs are three in number and have three unknown displacements – two rotations and one transverse displacement w (x, y). Double Fourier series method (DFSM) is used in this work to derive analytical solutions to the DEs for the case of simply supported boundary conditions (BCs). The in-plane loading cases considered are: (i) uniaxial loading in the x and y directions, (ii) biaxial compressive loading in both x and y directions, (iii) simultaneously applied compressive load in the y direction for different plate material properties. It was found that the DFSM converted the DEs to a system of three homogeneous algebraic eigenvalue equations in terms of three unknowns. The method of solving homogeneous algebraic equations by requiring the vanishing of the determinant of the coefficient matrix is used to obtain the eigenvalues from which the critical buckling loads are determined for each case. Comparison with previous solutions confirmed the DFSM gave accurate buckling load solutions. This study presents for the first time an exact mathematical framework for constructing exponential shear deformable thick plate buckling problems under in-plane uniaxial and biaxial loading. It also presents closed form solutions using double Fourier series method for the formulated thick plate buckling problems under in-plane uniaxial and biaxial loadings for simply supported boundary conditions.