Finite displacement theory of straight beams under configuration-dependent uniform loads

Abstract

A general field transfer matrix for nonlinear elastic curved beams has already been given by Fujii and Gong [ J. struct. Engng, ASCE 114, 675–692 (1988)]. For a straight member, this field transfer matrix can be considerably simplified for practical purposes. Firstly, the simplified field transfer matrix is given explicitly. The load terms are also evaluated in explicit form for liquid pressure, wind pressure and gravity-type loading. Secondly, by assuming the element length to be infinitesimally small, the governing differential equations of finite deflection theory are obtained. The nonlinear coefficients in the differential equations are summarized in a 6 × 6 sparse matrix. It is suggested that a higher order field transfer matrix can be derived analytically by utilizing this sparseness. Thirdly by solving the transfer equations for the nodal forces, total stiffness equations are derived for a nonlinear elastic beam element. Finally, an incremental/iterative nonlinear scheme is proposed to solve the total stiffness equations. Numerical examples including looping and snap-back are computed to examine the formulation, and the effects of different uniform loads are also compared.