Concerning the Transverse and Longitudinal Vibrations of a Fluid Conveying Beam and the Pipe Walking Phenomenon

In oil and gas installations, whether on-shore or off-shore, pipes are the primary vessel for the conveyance of either crude or products from one location to another. Under use, the pipes are subjected to both internal and external temperature fluctuations while repeated operational start-up and shut-down procedures triggers vibrations of these pipes, propagates internal waves and results in finite and irreversible longitudinal extension of the pipe over time. This longitudinal extension which is sometimes accompanied by pipe buckling is known as ratcheting and has also been described by some as pipe walking. In view of the complicated and intractable nature of the problem, most attempts to study the behavior of these pipes have been limited to the analysis of some reduced problem based on heuristic arguments and idealizations. Within this context, the transverse vibration and stability of such pipes have been studied while the problem of undamped clamped-pinned pipe conveying fluid has also been tackled numerically. Keiper and Metrikine [2004] however pointed out that such numerical schemes sometimes lead to disputed or controversial results. More importantly, the coupling between the transverse vibration and longitudinal motion has been largely ignored or neglected altogether by most writers. The objective of this paper is to formally derive the governing equations of Euler-Bernoulli beam capturing various effects including temperature variations (within and without), Coriolis acceleration, transverse acceleration, pre-stress, pressurization, rotatory inertia, and cross-sectional area change. In particular, it is shown that the latter effect is what causes pipe walking phenomenon. Most of the other effects were either earlier accounted for by Semler, et al [1994] or recently captured by Reddy and Wang [2004]. Nonetheless earlier contributions neglected the effect of the cross-sectional area change completely, thereby omitting the pipe walking phenomenon. Simple examples are considered to demonstrate the importance of these terms.