Link-Beam Model for Pipeline Buckle Propagation
Authors: Nogueira André Campos 
Published: October 2008

Abstract
A relatively simple model for pipeline quasistatic buckle propagation is presented. It is hypothesized that individually collapsing rings linked longitudinally by beams at the 12 o'clock and 6 o'clock positions can approximately reproduce the three dimensional aspects of the propagating buckle. The model formulates an equilibrium equation of a ring under pressure and concentrated force. This allows closed form equations to describe the force interaction between the beams and the rings. With this approach, the longitudinal bending work contribution to the energy balance equation can be evaluated and several documented aspects of buckle propagation can be explained. Predictions of the propagation pressure are within engineering agreement to experimental values. This novel description of buckle propagation leads to an improved understanding of the physical phenomenon.

Introduction
Multiple modern1 and historical research projects have addressed pipeline collapse. Steward2 was probably the first engineer to submit long metal tubes to uniform external pressure. In 1906, he used a pressure-controlled chamber to collapse 8 cm to 25 cm diameter pipes, up to 6 m long. Undented pipes were mounted in the chamber with clamps at each end such that the cross section remained circular locally. The pressure was then raised until the pipe collapsed. Upon collapse, the manually controlled pressure was decreased as fast as possible. The resulting flattened pipe segments had the opposite inner walls in contact for a few diameters, usually at the middle of the pipe. Therefore, unknowingly, he was the first to experimentally propagate buckles in long pipelines. Stewart's experiments2 determined the collapse pressure, which for the elastic case3 is Pc =2E(tID)3/(l-v2) (see nomenclature for symbol definitions). For E = 200 GPa, v=0.3, and Dlt = 51, Pc = 3.31 Mpa. Had he resumed pressurization, he would have found that at pressures lower than Pc the collapsed section (buckle) would propagate and flatten the entire pipe. The minimum pressure necessary to sustain propagation is known as propagation pressure Pp. This progressive-collapse mode of failure is known as a propagating buckle. Experimental results for steel alloys with yield stress ?o ? 350 Mpa have been fit4 by the equation:(mathematical equation)(available in full paper)
For Dlt = 51, Eq. 1 yields Pp = 720 kPa. For steel pipes with Dlt = 60 to 90 and pressures very close to Pp, velocities of buckle propagation on the order of 150 to 200 m/s have been found5. The pressure necessary to initiate buckle propagation is known as initiation pressure Pi. For dented pipes P1|Pp=1.1–5, whereas Pc|Pp=4-9. Thus, high velocity of propagation, relatively low initiation pressure, and lower propagation pressure are characteristics of buckle propagation. Fig. I shows a half pipe section in which a buckle is propagating. The axes x1, x2, x3 are defined in Fig. 1. The tail of the buckle is the collapsed region where contact takes place. The front is the undeformed region towards which the buckle is propagating. The propagating profile is the transition region that joins the front to the tail. The fiber at the 12 o'clock position (along the plane x2 = 0) is the top generator. This paper initially examines the collapse of the front of the buckle.