FLEXURAL ANALYSIS OF THIN RECTANGULAR PLATES USING GALERKIN METHOD

MMADUAKONAM OBIORA OKOYE (REG.NUMBER:NAU/2009226002P)

Abstract


The flexure of thin rectangular isotropic plates subjected to uniformly distributed loads using Galerkin variational method has been studied for five different boundary conditions namely, CCCS, SSSS, CCCC, CCSS and CSCS. The deflected surface was approximated using a grid work of beams. The deflection functions of the deformed surfaces were derived in terms of characteristic coordinate polynomials with unknown coefficients which satisfy the prescribed boundary conditions of the plate. Different approximations of the derived deflection functions corresponding to the first, second, truncated third and third approximations were developed for each case. These deflection functions were substituted into the fourth-order governing differential equation of plate and the Galerkin method reduced the solution of the differential equations to the evaluation of definite integrals of simple functions. The unknown coefficientswere obtained by solving the resulting set of linear functions. These obtained coefficients were then put back into the different approximations of the deflection functions to calculate the deflections and their corresponding span moments. Results were obtained for the five different sets of boundary conditions considered and the aspect ratio (p = b a ) was varied from 1.0 to 2.0for each approximation. The accuracy and pattern of convergence of the present formulations were assessed by comparing them with the results of the classical solutions. The variations of the deflections and span moments with respect to aspect ratios for the different approximations were presented in graphical forms and discussed. For the clamped rectangular plate, it was discoveredthat the accuracy and convergence to the classical solution improved as the approximation increased from the first through third. For instance, the average percentage difference between the present study and the results in literature gave 1.98, 1.96 and 8.46 for deflection, short-span moment and long-span moment respectively for the clamped plate at the third approximation.This level of convergence makes the present study invaluable for the design engineer. Moreover, the present study provides computer algorithm in MATLAB (M-Files) for the different approximations which can be of help to the design engineer for the calculation of the mechanical properties of plates at arbitrary points on the plate surface. In conclusion, the accuracy and convergence of results as the number of terms of deflection function increased were found to bedependent on the boundary condition of the plate for the present formulation.

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References


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