An elasticity analysis for the generally and specially orthotropic beams subjected to concentrated loads

A classical elasticity solution for the stress and displacement fields in loaded orthotropic beams is presented. The corresponding stress equations, derived from a stress function which satisfies the orthotropic analogue of the biharmonic equation, are written as Fourier series. From these expressions, the stress distributions in a series of centrally loaded generally and specially orthotropic beams are calculated. A detailed discussion of these stress distributions at selected beam sections and their symmetry about the loading axis is also presented. The stress fields are found to be unique relative to what is calculable from beam theory; for example, the stress distribution about the loading axis of a generally orthotropic beam is not symmetric. Finally, regions where beam theory coincides with the classical theory are identified.