Approximate theory for locally loaded plane orthotropic beams

Abstract

Governing equations for the orthotropic, rectangular strip are investigated to obtain solutions illustrating the relationship between classical beam theory and higher order theories. The exact equations of generalized plane elasticity are reduced to coupled sets of ordinary differential equations by using series representations in Legendre polynomials for all dependent variables. The coupled differential equations are obtained in a form such that a proper truncation scheme to extract any higher order theory is easily formulated. Sample problems involving uniformly and locally loaded beams are worked out using typical values for the material constants. Comparison of results with existing solutions shows that problems involving slowly varying loads with material orthotropy can be handled by classical thin beam theory; however, local loadings, coupled with materials orthotropy, preclude the use of any classical beam theory, even for thin beams.