Theory of flexure of orthotropic multi-layer circular sandwich plates
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The theory of flexure of multi-layer orthotropic circular sandwich plates is obtained by extremizing the augmented complementary energy. The complementary energy is the sum of the strain energies of the individual layers minus the work done at the boundary of the plate. The constraint conditions which are imposed on the stresses and stress resultants and those dictated by the equilibrium equations of a differential element are used in conjunction with arbitrary Lagrange multipliers to augment the complementary energy. The sandwich structure contains an arbitrary number n of isotropic and homogeneous membrane type faces that have different thicknesses and elastic properties. The number m of weak cores used in the sandwich construction are orthotropic and possess different moduli of rigidity and thicknesses. The augmented complementary energy is extremized by using the principles of the calculus of variations to obtain a set of Euler-Lagrange equations. A set of N equations in N unknowns becomes available for solution v/here N = 3n + 2m + 16. This set of equations is reduced to two coupled partial differential equations in which the transverse deflection and the shear force per unit length in the rz-plane are the unknowns. The simultaneous solutions of these two equations with the use of the proper boundary conditions is necessary to obtain the stresses, the stress resultants and the displacements at any section of the plate. The surface of deflection for polar symmetric bending is found to be represented by a fourth order non-homogeneous ordinary differential equation with non-constant coefficients. This type of bending is found to be independent of the tangential shear stiffness. When a common Poisson's ratio is assigned to all faces, the system of M equations is reduced to two partial differential equations. One of these equations is uncoupled with the shear force as an unknown while the other contains the transverse deflection and the shear force as unknowns. These two equations are used to obtain the surface of deflection which is represented by a ninth order non-homogeneous partial differential equation with non-constant coefficients. Several practical cases of plates with polar symmetric bending are solved to demonstrate the application of the pertinent equations and the proper boundary conditions. The bending of two hydrostatically loaded cases was obtained to demonstrate the application of the equations obtained for a non-polar symmetric case. Curves are included to show the effect of shear and the degree of orthotropy on the deflection for several cases. The effect of the tangential shear rigidity upon the hydrostatically loaded cases is very snail. The orthotropic plate can be treated as an isotropic plate if the tangential shear stiffness is taken to be equal to the radial shear stiffness. The error involved in this approximation is small. The error would be large, however, if one were to take the radial shear stiffness equal to the tangential shear stiffness.
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Copyright held by author.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.