Analytical Buckling Solutions Of Rectangular Plates Using Integral Transform Techniques

The existence and importance of plate as a structure element in many practical engineering applications such as civil engineering,mechanical engineering,aerospace and ocean engineering lead to various techniques and methods to solve plate problems.With the development of modern industrial technology,it's of great significance to analyze the buckling behavior of plate.The buckling solution of plate problems have been one of the important research topics in engineering in the last few decades.However,due to the mathematical complexity the analytical solutions of the buckling problems are hard to find.In this dissertation,the classical Kirchhoff plate theory(thin plate theory)and Mindlin plate theory(moderately thick plate theory)are analyzed.Two analytical approaches have been developed namely;the novel two-dimensional generalized integral transform method,and classical finite integral transform method.The complete theoretical system of solving rectangular plates is established.By the proposed analytical methods,the exact axial buckling solutions of the rectangular thin plates,moderately thick rectangular plates and shear buckling solutions of rectangular thin plates,under a class of complex boundary value problems are obtained,which can serve as a benchmark for validation of other analytical and numerical/approximate solutions.Firstly,taking the classical Kirchhoff plate theory the axial buckling solutions of rectangular thin plates with different boundary conditions are obtained by double finite sine integral transform method.The high order partial differential equation representing the buckling of plate is transformed into a system of linear algebraic equations by integral transformation,where the exact solution is achieved by solving the linear equations and corresponding inverse transformation.The method is further extended to the axial buckling analysis of rectangular thin plates with rotationally-restrained edges subjected to biaxial compression loads.Compared with the semi-inverse method(Navier's method,Levy's method,and superposition method)the classical finite integral transform method does not need to predetermine the deformation function arbitrary and proceed directly from the basic governing equation which simplifies the solving process.Therefore,the solutions obtained are reasonable and theoretical.The buckling of moderately thick plate with clamped edges is another complex problem where the process of transforming the basic governing equation by classical finite integral transform method and deriving the expression of generalized displacement image function is rather complicated.The uniaxial and biaxial buckling solutions of moderately thick rectangular plates are achieved by double finite integral transform,with focus on typical non-Levy-type fully clamped plates that are not easy to solve in a rigorous way by the other analytical methods.Solving the governing higher-order partial differential equations with prescribed boundary conditions,is elegantly reduced to processing four sets of simultaneous linear equations.The existence of non-zero solutions of which determines the buckling loads and associated mode shapes.Furthermore,for the first time the two-dimensional generalized integral transform method is proposed to the buckling problems of thin plates.It is well-known that the previous studies mostly adopted one-dimensional generalized integral transform method for some thermodynamics and fluid mechanics problems.This study presents a first endeavor to extend the one-dimensional generalized integral transform to two-dimensional generalized integral transform for new analytic axial buckling solutions of rectangular thin plates,with focus on non-Levy-type plates which is difficult to solve by other analytic methods.The method is further extended to the shear buckling problems of clamped rectangular thin plates.Compared with the axial buckling the shear buckling problem of plate is mathematically described by differential equations having a term odd-order of derivatives with respect to each of the planar spatial coordinates which make the solution more difficult to solve by traditional inverse/semi-inverse methods.In solution procedure,according to the boundary conditions of the plate the vibrating beam functions are adopted as the integral kernels to construct the integral transform pair.Then the integral transformation is applied on the basic governing high order partial differential equation of the plate,utilizing some inherent properties of beam function and transformed the title problem into a system of a linear algebraic equation directly,where the exact analytical solution is obtained in a straightforward way.The main advantage of the employed methods is that the predetermination of trial function is not required.Both the methods provide a unified solution procedure for buckling problems of plate which is much simpler and effective than other traditional methods.The present solutions are confirmed to be highly accurate with fast convergence,which agrees very well with both the finite element analysis results using(ABAQUS)software as well as the available literature.The succinct but effective techniques presented in this study can provide an easy-to-implement theoretical tool to seek more analytical solutions of complex boundary value problems.