A new first-order shear deformation theory (FSDT) with pure bending deflection and shearing deflection as two independent variables is presented in this paper for free vibrations of rectangular plate. In this two-variable theory, the shearing deflection is regarded as the only fundamental variable by which the total deflection and bending deflection can be expressed explicitly. In contrast with the conventional three-variable first-order shear plate theory, present variationally consistent theory derived by using Hamiltonian variational principle can uniquely define the bending and the shearing deflections, and give two rotations by the differentiations of bending deflection. Due to more restrictive geometrical constraints on rotations and boundary conditions, the obtained natural frequencies are equal to or higher than those by conventional FSDT for the rectangular plate with at least one pair of opposite edges simply supported. This new theory is of considerable significance in theoretical sense for giving a simple two-variable FSDT which is variational consistent and involve rotary inertia and shear deformation. The relation and differences of present theory with conventional FSDT and other relative formulations are discussed in detail.
This paper proposes an improved differential quadrature finite element method (DQFEM) by combining the virtual boundary spring technique with the standard DQFEM, in order to deal with free vibration of Mindlin plates with arbitrary elastic constraints. The incorporation of the virtual boundary spring technique makes it easy to impose general elastic constraints including some classical boundary conditions and avoids the deficiencies of the classical elimination method, which is widely used to process boundary conditions. The improved DQFEM formulation for rectangular and curvilinear quadrilateral elements is established. Convergence characteristics of the present approach are discussed, and the minimum number of nodes to derive convergent results and the appropriate value of the boundary spring stiffness to simulate classical boundary are obtained. Numerical examples are carried out for Mindlin plates with various boundary conditions and thickness ratios, covering both rectangular and irregular geometries. By numerical comparisons, the high accuracy and the remarkable efficiency of the present method are demonstrated. Additionally, adopting the virtual spring boundary can improve the efficiency to some extent compared with the standard DQFEM.
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