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Abstract

A variational derivation of the von Kàrmàn equations for thin plates is presented and extended to the derivation of the nonlinear equations for thin cylindrical shells. The description in cylindrical coordinates is derived from the invariant form of the equations because the nonlinear strain terms retained in the treatment include a term arising from the angular differentiation of a base vector. It is shown that this term results in a significant reduction in the uniform radially directed critical pressure that causes instability of the shell. The equations for the cylindrical shell are reduced to those for the cylindrical ring by taking the limit of the very narrow shell. The resulting equations are used in the treatment of a cylindrical arch with a uniform radially directed pressure applied. The critical pressure that causes instability of the arch is obtained simply by solving an ordinary boundary value problem, without employing any unusual conditions.

Keywords

thin plates thin cylindrical shells cylindrical rings pin-ended cylindrical arch nonlinear interactions

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References

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On the Analytical Description of the Elastic Stability of Rectilinear and Cylindrical Configurations

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