Canonical relationships between bending solutions of classical and shear deformation beam and plate theories

Abstract

In a series of papers and a book the author and his colleagues have developed algebraic relationships between the solutions (e.g., deflections, buckling loads, and frequencies) of a given shear deformation theory of beams or plates and the corresponding classical theory solutions. The bending relationships, for example, can be used to generate the generalized displacements and forces according to the particular shear deformation theory from the known generalized displacements and forces of the same problem according to the classical theory. In the present study relationships between the bending solutions of several shear deformation beam and plate theories and the classical beam and plate theories are presented in a canonical form, i.e., one set of relationships contains several theories and they can be specialized to a specific theory by assigning values to the parameters appearing in the relationships. Numerical examples of bending solutions for beams and rectangular plates with various boundary conditions are presented to show how the relations can be used to determine the deflections, bending moments, and shear forces for various theories. The relationships are validated by comparing the numerical results obtained using the relationships for the first-order plate theory against analytical solutions or those computed using the ABAQUS finite element program.