On the accuracy of a beam theory (Q1097085)

This paper provides further information about the consistency and accuracy of Levinson's theory [e.g. \textit{M. Levinson}, ibid. 12, 1-9 (1985; Zbl 0568.73064)] by imbedding it in the two-dimensional linear theory of elasticity. To this end, we construct adequate displacement and stress distributions over the beam depth in terms of the one-dimensional displacement and stress variables of Levinson's theory. Using then the hypersphere theorem of \textit{W. Prager} and \textit{J. L. Synge} [Q. Appl. Math. 5, 241-269 (1947; Zbl 0029.23505)], these distributions are shown to represent reasonably good approximations to exact plane elasticity solutions. We prove that the corresponding relative mean square error is, in general, proportional to the square of the beam depth, with its shear- related component being proportional to the depth cubed - a remarkable fact in the context of composite beams that often exhibit increased shear deformability. Our analysis confirms the coherence of Levinson's theory and, in paricular, the correctness of the equations of equilibrium, despite their variational inconsistency disclosed in \textit{W. B. Bickford} [Developments Theor. Appl. Mech. 11, 137 ff. (1982)]. Levinson's displacement hypotheses, however, are found to be too poor to adequately describe the two-dimensional displacement pattern, although they yield the same one-dimensional equations as those derived here. The stress distributions we present are also believed to be a valuable complement to the results of Levinson.