In this paper, the buckling equation and natural boundary conditions are derived with the aid of calculus of variations. The natural and geometric boundary conditions are used to determine the proper solution that represents the post-buckling configuration. Effects of friction and boundary conditions on the critical load of helical buckling are investigated. Theoretical results show that the effect of boundary conditions on helical buckling becomes negligible for a long pipe with dimensionless length greater than 5. Velocity analysis shows that lateral friction becomes dominant at the instant of buckling initiation. Thus, friction can increase the critical load of helical buckling significantly. However, once buckling is initiated, axial velocity becomes dominant again and lateral friction becomes negligible for post-buckling behavior and axial-load-transfer analysis. Consequently, it is possible to seek an analytical solution for the buckling equation. Analytical solutions for both sinusoidal and helical post-buckling configurations are derived, and a practical procedure for modeling of axial load transfer is proposed. To verify the proposed model and analytical results, the authors also conducted experimental studies. Experimental results support the proposed solutions.
IntroductionIt is well known that, when a pipe is subjected to an axial compressive load, the pipe will shorten because of axial compression. As the axial load increases, the pipe will change its configuration from straight to a sinusoidal-wave-like shape and eventually a helical shape. These three different configurations are all static equilibrium states of a pipe subjected to axial compressive load. However, for a given axial load, only one configuration is stable. The critical load beyond which a pipe will change its configuration from a straight line to a sinusoidal-wave-like shape is called the critical load for sinusoidal buckling. The critical load beyond which a pipe will change its configuration from a sinusoidal-wave-like shape to a helical shape is called the critical load for helical buckling.We need to address the difference between stability or buckling analysis and post-buckling analysis of a pipe constrained in a wellbore. Buckling analysis is focused on determination of the critical condition (or critical load) under which a pipe will change its configuration from one type into another; for example, the critical load beyond which a pipe will change from a straight line to a sinusoidal shape or from a sinusoidal shape to a helix. On the other hand, post-buckling analysis is focused on analyzing the relationship between parameters that define the post-buckling configuration and axial load; for example, the relationship between amplitude of the sinusoidal wave and axial load for sinusoidal postbuckling analysis or the relationship between pitch of the helix and axial load for helical post-buckling analysis. The principle of virtual work and the energy-conservation law apply for both buckling and post-buckling analysis.