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Three dimensional elasticity solutions for plates

Three dimensional elasticity solutions for plates

A Navier-type method for finding the exact three-dimensional solution for isotropic thick and thin rectangular plates is presented. The extension of this procedure to the case of multilayered plates (composite structures) is straightforward and can be found in the listed references . The method presented in these lectures uses the Mixed Form of Hooke's Law (MFHL) which leads one to write the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. Mixed Form of Hooke's Law is obtained from the Classical Form of Hooke's Law (CFHL). The displacements, stresses and external loads are expressed by using trigonometric functions. It is possible to demonstrate that this particular choice satisfies the simply supported boundary conditions. The elasticity solution must satisfy the equilibrium equations, geometric relations and Hooke's law. After some mathematical derivations it is possible to demonstrate that a set of differential equations in the unknown amplitudes of the displacements and stresses has to be solved. An eigenvalue problem is obtained. Only 2 (over a total of 6) eigenvalues are distinct. Therefore, a basis of eigenvectors is not available and 2 generalized eigenvectors have to be found. The solution is a combination of the eigenvectors and generalized eigenvectors multiplied by functions of the out-of-plane coordinate z. The unknown constants of this combination are determined by imposing that the transverse stresses at the top and bottom surfaces of the plate must match the applied pressures. In a general multilayered structure additional conditions on the interlaminar continuity of the displacements and transverse stresses are required. Once the unknown coefficients are calculated, the elasticity solution is complete. These lectures present an algebraic solution for the particular case of a pressure applied at the top surface. The extension to the case of generic load can be obtained by using Fourier analysis. The lectures also present the FREE software that can be downloaded. The program calculates the amplitudes of the displacements and stresses for a sinusoidal load pressure applied at the top surface of the plate. The amplitude of the pressure distribution, material properties, geometric dimensions of the plate and wave numbers are inputs of the software.

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