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θ is the relative angle of rotation between the ends of the supported tube.
FROM:
Mohamed Elchalakani, Xiao-Ling Zhao and Raphael Grzebieta (Dept. of Civil Engineering, Monash University, Victoria 3800, Australia),
“A closed-form solution for elastic buckling of thin-walled unstiffened circular cylinders under pure flexure”, (publisher and date not given in the pdf file. Most recent citation is dated 2000), https://www.researchgate.net/publication/239328062
ABSTRACT: To date, despite the significant development in the field of structural mechanics, there still remains a paradox in the solutions available for a classical shell buckling problem. The difference in strength between a cylindrical shell under uniform axial compression and that under pure bending is not quite well investigated. This lack of research is reflected in the wide variations in the elastic bending strength given in current international design standards. This paper presents a theoretical treatment of the stability of unstiffened thin cylinders under pure flexure. The results of the theory are in good agreement with experimental findings. The theoretical results confirm the classical 40% increase in bending strength over axial strength. The current design rules for thin-cylinders are compared, and a newly derived design method is proposed.
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