Torsion Equations for Lateral Buckling (No. R964)

Abstract

A torsion differential equation previously used for analysing the elastic lateral buckling of simply supported doubly symmetric beams with distributed loads acting away from the centroidal axis omits an expected term and includes an unexpected term. A different equation is derived by two different methods, either by using the calculus of variations with the second variation of the total potential, or by considering the equilibrium of the deflected and twisted beam. Four different methods are used to find solutions for the elastic buckling of beams with uniformly distributed loads. Two of these solve the differential equations numerically, either by using a computer program based on the method of finite integrals, or by making hand calculations with a single term approximation of the buckled shape. These methods produce different solutions for the two torsion differential equations. The two other methods used are based on the energy equation for lateral buckling. The first of these uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program based on cubic deformation fields. Both of these produce solutions which agree closely with the finite integral and approximate solutions for the different differential equation derived in this paper, but are markedly different from the solutions for the previously used equation. It is concluded that the previously used torsion differential equation is in error.