On Some New Analytic Solutions For Buckling Of Rectangular Plates

Buckling often occurs when plates are under compression or temperature rise,which can result in the failure of structures,especially when the thickness of plates becomes sufficiently small with respect to the sizes in another two directions.In traditional analytic methods,thedisplacements are often chosen as the variables in order to solve buckling problems of rectangular plates,where higher order partial differential equations have to be dealt with.However,these methods can hardly be used if the boundary conditions become complex.Based on the traditional and first-order shear deformation plate theories,the Hamiltonian system for buckling of rectangular plates is deduced by introducing new variables.Applying the method of separation of variables,thesymplectic eigen expansion and other mathematical methods,one does not need to deal with higher order partial differential equations,which reduces the complexity of the problems.For complex boundary conditions,the symplectic superposition method can be used by dividing the original problems into several sub-problems to simplify the original boundary conditions,e.g.,to simplify the clamped and free edges to simply supported and slidingly clamped ones,respectively.By using the method of superposition,the original problems' critical buckling loads and mode shapes can be then obtained analytically.In detail,applying the symplectic superposition method,the buckling problems of rectangular thin plates with two adjacent edges free(another two edges clamped or simply supported),all edges free,and thermal buckling of orthotropic plates with all edges clamped are solved.The results agree well with those obtained by the finite element method(FEM).Meanwhile,the convergence of the given examples was studied,which verifies the efficiency of the symplectic superposition method.Besides,the Hamiltonian system for buckling of moderately thick plate based on the first-order shear deformation plate theory is deduced,and the critical buckling loads of simply supported plates are obtained.The method in this thesis is rational and analytic,which avoids assuming any trial solutions,thus,the mathematical derivation is rigorous.The present results are expected to serve as the benchmarks for validation of the other analytic and numerical methods.Furthermore,by using the symplectic superposition method,buckling problems of moderately thick plates with complex boundary conditions can also be solved.