Hierarchical theories for a linearised stability analysis of thin-walled beams with open and closed cross-section

A linearised buckling analysis of thin-walled beams is addressed in this paper. Beam theories formulated according to a unified approach are presented. The displacement unknown variables on the cross-section of the beam are approximated via Mac Laurin\'s polynomials. The governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the expansion order. Classical beam theories such as Euler-Bernoulli\'s and Timoshenko\'s can be retrieved as particular cases. Slender and deep beams are investigated. Flexural, torsional and mixed buckling modes are considered. Results are assessed toward three-dimensional finite element solutions. The numerical investigations show that classical and lower-order theories are accurate for flexural buckling modes of slender beams only. When deep beams or torsional buckling modes are considered, higher-order theories are required.