POST-BUCKLING BEHAVIOUR OF A COMPRESSED SLENDER BEAM CONSTRAINED TO A CYLINDRICAL TUBE

The paper deals with the study of the behaviour of a slender beam introduced in a cylindrical tube and subjected to an axial compressive force and to a torque. The beam is very long compared to its transversal dimensions and therefore, it will buckle to very small axial force according to the Euler formula. The post-buckling behaviour is examined, without and with friction. The study has important applications in the buckling of drill strings inside a cylindrical hole in the case of deep drilling, rotary drilling of wells in bedrocks at depths of several kilometres, buckling of a homogeneous cable in a horizontal circular rigid duct subjected an axial compressive load (for instance posing of optical fibre), in petroleum industry, for coiled tubing, especially in the case of drilling in horizontal or inclined wellbores, etc, [1-4]. The slender beam has a constant cross-section that can have any form, although 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. A special isoparametric 3D beam finite element is elaborated. The paper presents only the static case, but extending the presented approach to dynamic analysis is quite natural and it will be presented in another work. The method is very accurate and very rapidly convergent due to the fact that the exact equations, that is written for the deformed configuration, are solved. Small strains and small displacements are assumed as the diameter of the cylindrical tube is much smaller than the length of the beam. Material is considered linear. Large twist angles could occur as an exception, but the specific twist angle is small. The iterative Newton-Raphson method was used to solve the nonlinear differential equations. To characterize the deformed shape of the beam the Euler-Rodrigues quaternion is used: (