Analytic Solution For Dynamic And Static Problems Of Elastic Rectangular Plates

Rectangular thin plates receive wide applications as important structure element in many engineering fields such as civil engineering,aerospace,ocean engineering and mechanical engineering.The dynamic and static analysis of rectangular plates with various boundary conditions has been a subject of broad interest for more than a century.However,very few exact analytical solutions are available due to the mathematical complexity.This study explores analytic bending and vibration solutions of rectangular plates,including thin plates(based on Kirchhoff theory)and moderately thick plates(based on Reissner theory),which utilizes finite Fourier integral transform method and generalized integral transform method.Firstly,a double finite half-sinusoidal integral transform method is introduced to explore the accurate bending analysis of thin plates with two adjacent edges free and the others clamped or simply supported.In the solution process,the Fourier coefficients of deflection with unknown constants are obtained by applying finite integral transformation on the governing equation.The unknown constants can be determined by letting the defelection to satisfy associated boundary conditions,in which one can obtain a fully regular infinite system of simultaneous linear algebraic equations.Then,analytic bending solution is elegantly achieved through the inversion formula.Secondly,the double finite Sine integral transform method is introduced to explore the accurate bending analysis of rectangular thin plates with three different corner supports.Introducing the concept of generalized simply supported edge,the general-form analytical solutions for bending of plates under consideration are obtained by applying the finite integral transform to the governing equation and some of the boundary conditions.The analytical bending solutions of plates under specific boundary conditions are then obtained elegantly by imposing the remaining boundary conditions.Another advantage of the method is that the analytical solutions obtained converge rapidly due to utilization of the sum functions.In addition,analytic free vibration solution of rectangular orthotropic thin plates with various classic boundary conditions is obtained by this method.Rotationally-restrained boundary conditions are much closer to practical engineering,this study employs double finite Sine integral transform method to solve the bendling problems of moderately thick plates with such kind of boundary conditions.Via the integral transformation on the high-order partial differential equations governing the bending of a moderately thick plate,one can obtain the expression of the generalized displacement functions with unknown constants.The unknown constants can be determined by letting the generalized displacement functions to satisfy associated boundary conditions.Then,one can get analytic bending solution by utilizing the inversion formula.Through choosing the value of rotational fixity factor for each edge,various edge flexibilities are investigated,including simply supported and clamped edges as limiting situations.Therefore,analytical bending solutions of moderately thick plates under various classic boundary conditions are also obtained.Finally,according to the boundary conditions of the thin plate the vibrating beam functions are adopted as the integral kernels to construct the integral transform pairs,analytic bending and vibration solutions of orthotropic rectangular thin plates can be obtained by employing theory of integral transform.Comparing with finite Fourier integral transform method,the generalized finite integral transform method also does not require any pre-determined deflection function.By imposing the transform to the governing equation,the title problem can be directly converted to that of solving a system of linear algebraic equations,by which the new analytic and fast convergent solutions are elegantly obtained in a straightforward way.