Consistent and Inconsistent Higher Order Beam and Plate Theories: Some Surprising Comparisons

Abstract

In 1982 Bickford [1] used Hamilton’s principle to derive a consistent, higher order theory for the elastodynamics of beams based upon the kinematic and stress assumptions previously used by Levinson [2] for beams of rectangular cross-section. In [2] the usual beam equations of motion [3] were used to obtain a higher order beam theory. This latter theory provided a fourth order system of differential equations not too unlike the equations of Timoshenko beam theory with a difference being that in Levinson’s theory the shear stress boundary conditions on the lateral surfaces of the beam were satisfied. Bickford’s variationally consistent theory consists of a sixth order system of differential equations requiring the specification, provided by the variational formulation, of three boundary conditions at each end of the beam. A vectorial formulation of Bickford’s theory is achieved in [1] by defining a posteriori, a “...higher order moment resultant” but, as Bickford himself notes, “it remains to be seen whether or not there can be developed a rational method for constructing the vectorial equations without knowing, a priori, the variational equations.” This latter point is of no consequence since the validity of the variational equations is unquestioned.