Bending of geometrically nonlinear cantilever beam. Results obtained by Cosserat - Timoshenko and Kirchhoff's rod theories

Abstract

The problem of verification of different program suites for structural analysis has recently become an important component of the construction science. One of the most extensively used benchmark problem is a classical geometrically nonlinear problem of deflection of the cantilever beam of linear elastic material, under the action of external vertical concentrated load at the free end. In fact, the solution for Kirchhoff's rod is used as an analytical result. This rod is inextensible and Kirchhoff's rod theory disregards flexibility of the rod in tension and shear. But in modern program suites Cosserat-Timoshenko rod is often used because Cosserat-Timoshenko rod theory is a geometrically exact theory. It considers not only bending strain but also shear and tensile strain. This means that it is necessary to get a model solution for Cosserat - Timoshenko rod, which can be used for verification of different software suites. This paper presents solutions of the geometrically nonlinear problem obtained by Cosserat - Timoshenko and Kirchhoff's rod theory with comparison of those results. The findings can be used as a benchmark problem for verification of software suites.