Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells

TL;DR

This paper rigorously derives the classical buckling load formula for axially compressed cylindrical shells using asymptotic analysis, providing explicit displacement mode expressions and contributing to understanding shell stability.

Contribution

It offers a mathematically rigorous proof of the buckling load formula and explicit mode amplitude expressions for cylindrical shells, advancing shell buckling theory.

Findings

Rigorous proof of classical buckling load formula.

Explicit expressions for displacement modes in buckling.

Analysis of sensitivity to imperfections.

Abstract

The goal of this paper is to apply the recently developed theory of buckling of arbitrary slender bodies to a tractable yet non-trivial example of buckling in axially compressed circular cylindrical shells, regarded as three-dimensional hyperelastic bodies. The theory is based on a mathematically rigorous asymptotic analysis of the second variation of 3D, fully nonlinear elastic energy, as the shell's slenderness parameter goes to zero. Our main results are a rigorous proof of the classical formula for buckling load and the explicit expressions for the relative amplitudes of displacement components in single Fourier harmonics buckling modes, whose wave numbers are described by Koiter's circle. This work is also a part of an effort to quantify the sensitivity of the buckling load of axially compressed cylindrical shells to imperfections of load and shape.