Generalized Lamé problem for a thin elastic tube under non-axisymmetric internal pressure

Source abstract

This study extends the classical Lamé problem to a thin elastic tube subjected to non-axisymmetric static internal loading and develops an asymptotically consistent dimensionally reduced formulation. An exact Airy-stress-function solution is first obtained and used to guide a systematic reduction of the plane-strain equations in the limit of small thickness-to-radius ratio. The resulting asymptotic model provides explicit expressions for displacements, stress resultants and bending stress couples and shows excellent agreement with the full elasticity solution in the thin-walled regime. One of the main observations is that the circumferential stress adopts a parabolic, rather than a uniform, distribution through the thickness. This feature, previously recognized in dynamic analyses of thin cylindrical shells, is here shown to originate purely from geometry and mid-surface near-inextensibility, independent of inertia. Consequently, the circumferential stress resultant is governed by higher-order terms and cannot be captured by classical membrane or semi-membrane shell theories. The consideration is further extended to include a Winkler-type outer boundary condition. While foundation stiffness alters displacement amplitudes and stress magnitudes, it does not modify the fundamental parabolic structure of the circumferential stress component. The resulting formulation provides a clear and efficient basis for analysing thin cylindrical components subjected to non-axisymmetric loading.