Rational approach for higher-order shear deformation beam theories

Abstract

The studies on the higher-order beam theory and its application to functionally graded (FG), laminated composite and sandwich beams have received much research attention over recent years. While significant progress has been made, there is an issue with the selection of the appropriate distribution function for describing section warping due to transverse loading. In the present study, a rational approach is proposed for the determination of correct warping functions for symmetric cross sections. Two new conditions on warping function are suggested: first, the warping function should vanish on the neutral plane; and second, the first derivative of warping function should take unity value on the neutral plane. A rigorous procedure is then developed by using the equilibrium condition and three conditions. An exact higher-order theory is then developed for creating accurate analysis models to consider both material and geometry variations over the beam cross section. A finite element analysis model is presented and a two-node beam element with bubble displacement modes is implemented. Results for sandwich and FG beam problems are shown and numerical results are compared with those of plane stress continuum models to demonstrate the correctness and application of the new theory and finite element model.