Eigenproblem Versus the Load-Carrying Capacity of Hybrid Thin-Walled Columns with Open Cross-Sections in the Elastic Range.

Zbigniew Kolakowski,Andrzej Teter

doi:10.3390/ma14133468

Abstract

The phenomena that occur during compression of hybrid thin-walled columns with open cross-sections in the elastic range are discussed. Nonlinear buckling problems were solved within Koiter’s approximation theory. A multimodal approach was assumed to investigate an effect of symmetrical and anti-symmetrical buckling modes on the ultimate load-carrying capacity. Detailed simulations were carried out for freely supported columns with a C-section and a top-hat type section of medium lengths. The columns under analysis were made of two layers of isotropic materials characterized by various mechanical properties. The results attained were verified with the finite element method (FEM). The boundary conditions applied in the FEM allowed us to confirm the eigensolutions obtained within Koiter’s theory with very high accuracy. Nonlinear solutions comply within these two approaches for low and medium overloads. To trace the correctness of the solutions, the Riks algorithm, which allows for investigating unsteady paths, was used in the FEM. The results for the ultimate load-carrying capacity obtained within the FEM are higher than those attained with Koiter’s approximation method, but the leap takes place on the identical equilibrium path as the one determined from Koiter’s theory.

Highlights

Experimental investigations [1] on the thin-walled columns made of modern materials with open cross-sections under compression have proved that the multi-mode buckling can be hazardous
Multimodal buckling of hybrid columns with two types of cross-sections, namely: channel (Figure 1a) and top-hat (Figure 1b), made of two isotropic materials of step-variable gradation of material properties, i.e., of material A and material B, in the elastic range and subject to compression, was considered
The results were verified with the finite element method (FEM) in the linear and nonlinear range

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. An unexpected phenomenon of stability loss of C-section beams of medium length subject to bending in the web’s plane was observed, which was interpreted as a sudden influence of the secondary global distortional buckling mode characterized by a significantly higher value of the bifurcation stress than the primary mode [5]. Many bifurcational values, buckling modes, and components of membrane inner forces for various numbers of buckling half-waves along the longitudinal direction (or for the assumed length of the structure) can be determined for the linear eigenproblem, which is possible thanks to, among others, the semi-analytical method (SAM) based on Koiter’s theory [6,7,8,9,11,19]. Piola–Kirchhoff’s stress tensor in Lagrange’s description, and a numerical method of the transition matrix using Godunov’s orthogonalization [8,19,21]

Formulation of the Problem

Channel Column

Top-Hat Column

Conclusions