Mechanics of Thin-Walled Rods of Open Profile

Abstract

We consider rods, whose cross section is a thin open-ended strip. The effect of warping, when the points of the rod move axially at torsion, needs to be included in the analysis. Beginning with the discussion of traditional approaches, based on certain hypotheses and approximations, we proceed with the asymptotic study of the corresponding model of a shell. The procedure of asymptotic splitting justifies known equations with an additional force factor (bi-moment) and warping, related to the rate of twist. Moreover, it allows for accurate recovery of the stressed state in a cross section first in terms of the shell theory, and then in the three-dimensional body. Further, we extend the results to the geometrically nonlinear range with the direct approach to a material line with additional degrees of freedom of particles due to the warping. The incremental analysis results in linear equations in the vicinity of a pre-stressed state. Solutions of particular problems show the importance of cubic terms in the strain energy of the rod, which is deduced from the equations of the nonlinear shell theory. Considered examples include linear problems, several types of buckling behavior as well as the analysis of higher order effects in comparison to numerical solutions with the shell model. The presented form of the direct approach was originally suggested by Vetyukov and Eliseev (Sci. Tech. Bull. St. Petersbg. State Polytechn. Univ., 1:49–53, 2007). This chapter quotes extensively from Vetyukov (Acta Mech., 200(3–4):167–176, 2008; J. Elast., 98(2):141–158, 2010) with kind permission from Springer Science and Business Media.