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This paper investigates the classical buckling problem of rectangular thin plates with two opposite edges free and others rotationally restrained by the finite Fourier integral transform method. The rotationally restrained edges are typically practical boundary conditions in many engineering structures, such as bridge decks. However, these non‐classic edges bring mathematical difficulties in solving boundary value problems of governing partial differential equations of plate. Based on the integral transform theory, the title problem is transformed into solving regular linear algebraic simultaneous equations, which provides a more rational and theoretical solution process than traditional inverse/semi‐inverse approaches. The acquired comprehensive results of critical buckling load factors and mode shapes are in excellent agreement with the ones obtained by finite element method, which verifies the validity and accuracy of the present approach. In addition, classical boundary conditions, such as simply supported and clamped edges, can be investigated by choosing different rotational fixity factors. Furthermore, the features of the employed method that no preselection of trial function and the generality for various boundary conditions enable one to analytically solve static problems for moderately thick plates and thick plates. [ABSTRACT FROM AUTHOR]

Abstract: Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points. [Copyright &y& Elsevier]

A two-dimensional linearized problem on plane buckling modes (BMs) of a rectangular plate with free edges, made of an elastic orthotropic material, underbiaxial tension-compression is considered. With the use of double trigonometric basis functions, displacement functions exactly satisfying all static boundary condition on plate edges are constructed. It is shown that the exact analytic solutions found describe only the pure shear BMs, and if the normal stress in one direction is assumed equal to zero, an analog of the solution given by the kinematic Timoshenko model can be obtained. Upon performing the limit passage to the zero harmonic in the displacement functions of one of the directions, the solution to the problem of biaxial compression can be obtained by equating the Poisson ratio to zero; in the case of uniaxial compression, this solution exactly agrees with that following from the classical Bernoulli-Euler model. [ABSTRACT FROM AUTHOR]

Abstract: The Reissner–Mindlin plate model in presence of free boundary conditions is considered. It is known that in this case the solution can suffer, for small thickness, from some boundary layer effects that reduce the regularity of the solution, in particular of the rotations and shear strains. This causes a loss of convergence order in the numerical approximation. For this reason a new model that modifies the free boundary conditions is considered. The approximation of this new model by means of an element of low degree of the partial selective reduced integration family has been analyzed both from the theoretical and the numerical point of view. Some numerical results have been presented showing the performance of this element when applied to free plates. [Copyright &y& Elsevier]

For the first time, the double half-Sinusoidal series is adopted as integral kernel to construct integral transform pair for new precise vibration analysis of orthotropic rectangular thin plates, supported with two adjacent edges rotationally-restrained and the other edges are free. Impose the defined single/double transforms on both the vibration governing partial differential equation (PDE) and some of the boundaries investigated, which gives the Fourier coefficient for the deflection expressed by some specific unknown constants. Utilizing the inverse formular to satify the remaining boundaries leads to a set of linear algebraic simultaneous equations, which yields the solutions of the natural frequency parameter and the associated mode shapes. Moreover, plates under classical boundary conditions also can be investigated via changing the rotating fixed coefficients introduced. Finally, more than 500 comprehensive analytical solutions were well validated by numerical results presented by finite element method (FEM), which can be treated as new reference results for further studies. Since its straightforward and rational solution procedure, the presented approach is promising to be further extended to solve more complicated boundary value problems (BVPs) of plates. [ABSTRACT FROM AUTHOR]