Higher order effects in thick rectangular elastic beams

Abstract

Governing equations for the rectangular strip are investigated to obtain solutions which show the relationship between classical beam theory and higher order theories. The exact equations of plane elasticity are reduced to a coupled set of ordinary differential equations by expressing all dependent variables as series solutions containing Legendre Polynomials in the thickness coordinate. Legendre Polynomials are particularly advantageous for this analysis because their completeness, convergence and orthogonality properties are well formulated, and because the usual stress resultants of classical beam theory appear naturally as coefficients of the polynomials P0 and P1. The coupled ordinary differential equations are obtained in a form such that proper truncation of the series to obtain approximate theories is immediately apparent. The coupling effects are investigated and a possible method of obtaining an approximate solution to the fully coupled equations without truncation of the series solutions is suggested. A sample problem is worked out in detail to illustrate the application of a new approximate theory.