Variational Formulation of Mindlin Plate Equation, And Solution for Deflections of Clamped Mindlin Plates
Journal Title: International Journal for Research in Applied Science and Engineering Technology (IJRASET) - Year 2017, Vol 5, Issue 1
Abstract
In this work, the governing equations for the flexure of linearly elastic, isotropic, homogeneous thick circular plates under static loading are formulated using the methods of the calculus of variations. The total potential energy functional for the thick circular plate, which is the sum of the strain energy functional and the load potential functional was obtained using the equations of material constitutive law, which accounts for the shear deformation of the circular plate and the axi-symmetrical nature of loading and plate. Euler-Lagrange differential equations of equilibrium of the plate were obtained using EulerLagrange conditions. The total potential energy functional was extremized in accordance with the principle of minimization of the total potential energy functional to obtain the governing equations, and the boundary conditions. The governing partial differential equations were then integrated to obtain the deflection of the thick circular plate. The deflection function w(r) obtained was found to be a sum of expressions for deflection due to pure bending and deflection due to shear deformation only. It was observed that when the shear stress coefficient k 4 3, which corresponds to the parabolic distribution of the shear stress over the plate thickness, the solution for deflection becomes identical with the deflection obtained using the rigourous methods of theory of elasticity, and presented by Love. For the case of thick circular plates, with clamped edges, subject to uniformly distributed load, the maximum deflection was found to occur at the plate center. The effect of the plate thickness h on the maximum deflection was investigated by computing the maximum deflection for varying ratios of 0 h r ranging from 0.001 to 1.00 for a Poisson’s ratio, of 0.30. It was observed that for 0 h r less than 0.05, corresponding to thin circular plates, shear deformation has insignificant contribution to the deflection, while when 0 h r is greater than 0.05, shear deformation has a significant effect on the maximum deflection, in agreement with literature.
Authors and Affiliations
IKE C. C. , Nwoji C. U. , Ofondu I. O.
Keywords
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