In this research we present a new analytical solution for finding the buckling loads of thin isotropic and orthotropic rectangular plates in which all four corners are supported. This new solution is also capable of solving all cases where few or all the edges are supported. The methods that are currently known in the literature for finding the buckling loads of plates are mainly numerical. Although some plates with specific boundary conditions have analytical solutions, a comprehensive analytical method providing analytical solutions that fit all possible combinations of boundary conditions is lacking. The solution method in this study is based on the development of a static solution for a plate. The physical meaning of buckling is the loss of stiffness, and it is found as the value of the in-plane loading intensity at which a zero force on the plate surface will generate infinite displacement. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. Using this new method, exact buckling loads and buckling modes of many new cases of classical boundary conditions are found. Results are given for several stiffness ratios in both directions of the plate, and for uni-directional and bi-directional loading. K E Y W O R D Sbi-axial buckling load, isotropic plates, orthotropic plates, uni-axial buckling load

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In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: Cclamped, S -Simply supported, Ffree, and Gguided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to¯nding the values of the negative elastic foundation that will yield in¯nite de°ection under a point load on the plate. The solution is composed of three parts, the sum of which satis¯es exactly both the¯eld equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the¯rst six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the¯rst time.