THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)—RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.

Analytical solutions for a uniformly loaded circular plate with clamped edges

A bi-harmonic potential function was constructed in this study. Love solution was employed to obtain analytical solutions of uniformly loaded plates with two different types of clamped edges. The treatment of clamped boundary conditions was the same as that adopted by Timoshenko and Goodier (1970). The analytical solution for the first type of clamped boundary condition is identical with that obtained by Luo et al.(2004), and the solutions for both types were compared with the FEM results and the calculations of thin plate theory.

THE BENDING OF A THICK RECTANGULAR PLATE WITH THREE CLAMPED EDGES AND ONE FREE EDGE

The exact solution of the bending of a thick rectangular plate with three clamped edges and one free edge under a uniform transverse load is obtained by means of the concept of generalized simply-supported boundary[1] in Reissner's theory of thick plates. The effect of the thickness h of a plate on the bending is studied and the applicable range of Kirchhoffs theory for bending of thin plates is considered.

Some Aspects of the Behavior for Orthotropic Thin Plates with Clamped Edges

In most structural applications, composite structures can be idealized as beams, plates or shells. The analysis is reduced from three-dimensional elasticity problem to a one-dimensional, or two-dimensional problem, based on certain simplifying assumptions that can be made because the structure is thin. In this article is presented the mathematical model properly thin orthotropic plates, based on simplifying assumptions Love- Kirchhoff and small deformations. Proposed analytical solutions are considered both for solving equation orthotropic rectangular plates and for modal analysis, in the case of plates with clamped edges. The purposed solutions were analysed considering a FEM solution for comparison and the experimental test results.

THE APPLICATION OF WEINSTEIN-CHIEN′S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED PORTIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.

Analytical Approaches of Composite Rectangular Thin Plates

The purpose of the present study was to explore and subsequently establish a technique for determination of analytical solutions for the differential equation for composite thin plates. The considered reasons for the solutions were to exactly satisfy the boundary conditions and to verify as close as possible the differential equation of the plate. There are studied two solutions for orthotropic plate with clamped edges, and made comparisons with the solutions presented by Reddy [1] and with the exact solution by Timoshenko and Woinowsky. The models are based on the CLPT （classical laminated plate theory）. The Ritz method, in conjunction with the weighted residue method for the coefficients calculation, is used to analytically determine the bending solutions of orthotropic laminated plates subjected to uniform pressure on the bottom laminate. The purposed solutions were critically analysed considering a FEM （finite element method） solution for comparison. Finally, it is presented the experimental device and the experimental test results, as well. Fabrics have been incorporated into two composite plates were required scalps on one direction, thus ensuring different elasticity modules on both directions. Thorough comparison between analytical solutions, numerical results and experimental data is performed and a good agreement is obtained.