Post-buckling behaviour of a slender beam in a circular tube, under axial load

This paper deals with the study of the behaviour of a slender beam introduced in a cylindrical tube and subjected to an axial compressive force. The beam is very long compared to its transversal dimensions and therefore it will buckle to a very small axial force. The post-buckling behaviour is examined. The study has important applications in the petroleum industry, for coiled tubing in the case of drilling in horizontal or inclined wellbores. The slender beam has a constant cross-section that can have any form, although a circular cross-section is the most used in practice. The problem has a geometrical non-linearity to which the non-linearity caused by the friction has to be added. Rotations could be large and a special isoparametric 3D beam finite element is elaborated: the EulerRodrigues quaternion was preferred to describe the finite cross-section rotations. The paper presents only the static case, but extending the presented approach to dynamic analysis is quite natural. The method is very accurate and it is rapidly convergent due to the fact that the exact equations, written for the deformed configuration, are solved. The iterative Newton-Raphson method was used to solve the nonlinear differential equations.