The buckling load of cylindrical shells under axial compression depends on the cross-section curvature

TL;DR

This paper investigates how the buckling load of cylindrical shells under axial compression varies with the cross-section curvature, revealing new scaling laws that depend on the shape's geometric properties.

Contribution

It extends previous work by proving that the buckling load depends on the cross-section curvature, establishing new asymptotic scaling laws for different curvature conditions.

Findings

Convex cross sections with positive curvature have buckling load scaling as h.

Cross sections with mostly positive curvature have buckling load between h^{8/5} and h^{3/2}.

The results generalize previous theories to non-circular shells.

Abstract

It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter's formula predicts linear dependence of the buckling load $\lambda(h)$ of the shell thickness $h$ ($h>0$ is a small parameter), one observes the dependence $\lambda(h)\sim h^{3/2}$ in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in [\textit{J. Nonl. Sci.,} Vol. 26, Iss. 1, pp. 83--119, Feb. 2016], that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling $\lambda(h)\sim h^{5/4},$ while shape imperfections are likely to result in the scaling $\lambda(h)\sim h^{3/2}.$ In this work we prove, that in fact the buckling load $\lambda(h)$ of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then $\lambda(h)\sim h,$ and when the the cross section curve has positive curvature except at finitely many points, then $C_1h^{8/5}\leq \lambda(h)\leq C_2h^{3/2}$ for $h$ small thickness $h>0.$