From: "Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes", by N. Silvestre, Dept. of Civil Engineering and Architecture, ICIST-IST, Technical University of Lisbon, Portugal, in Thin-Walled Structures, Vol. 45, No. 2, February 2007, pp. 185-198

http://dx.doi.org/10.1016/j.tws.2007.02.001

ABSTRACT: A formulation of generalised beam theory (GBT) developed to analyse the elastic buckling behaviour of circular hollow section (CHS) members (cylinders and tubes) is presented in this paper. The main concepts involved in the available GBT are adapted to account for the specific aspects related to cross-section geometry. Taking into consideration the kinematic relations used in the theory of thin shells, the variation of the strain energy is evaluated and the terms are physically interpreted, i.e., they are associated with the geometric properties of the CHS. Besides the set of shell-type deformation modes, the formulation also includes axisymmetric and torsion deformation modes. In order to illustrate the application and capabilities of the formulated GBT, the local and global buckling behaviour of CHS members subjected to (i) compression (columns), (ii) bending (beams), (iii) compression and bending (beam-columns) and (iv) torsion (shafts), is analysed. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed.

Shown here is Fig. 5 from the 2007 Silvestre paper:

"Variation of the buckling stress (σb) and buckling mode shape with the column length (L)z'

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