Helical Postbuckling Configuration of a Weightless Column Under the Action of m Axial Load

Abstract

Abstract The helical postbuckled configuration of a weightless, circular column confined within a cylinder was used by Lubinski et al. in 1962 to develop frequently applied equations describing the deformation of oilfield tubing. This paper presents an extension of their work in the form of another expression for the relation between applied force and helical pitch in the buckled state. Based on simple laboratory experiments and stability arguments it appears that the previous force-pitch equation applies during load application and the one given in this paper applies during unloading. Introduction The helical postbuckled configuration of a radially confined tubular is a phenomenon of significant import to the petroleum industry. For the case phenomenon of significant import to the petroleum industry. For the case of smaller tubulars such as most tubing, the radial clearance boundary can be large enough to both allow significant axial shortening and induce bending stresses of a magnitude sufficient to yield the tube body. For larger tubulars such as casing, large radial clearances normally are associated only with washed out or overgauge sections of the wellbore. However, even for small radial clearances, and with particular regard to intermediate casing strings, the helical configuration can present a hazard to the integrity, of the well by promoting wear during drilling operations. The majority of recent work on helical buckling can be traced to the classic paper by Lubinski et al. on helical buckling of tubing sealed in packers. Applications of this early work to a variety of downhole packers. Applications of this early work to a variety of downhole configurations have appeared in the literature; however, with the exception of some recent work on the influence of the packer on the helical configuration, the basic relation between applied compressive axial force and helical pitch has remained unaltered since its introduction. The intent of our study is to reconsider the helical postbuckled configuration by applying the principle of virtual work to deviation of the column from its straight, prebuckled configuration. The result of this effort will be two force-pitch relations (one of which is the relation derived by Lubinski et al.) that differ by a factor of two. The remainder of the discussion is then devoted to the significance of the a alternate solutions and the effect of the dual solution on tubular designs. A description of the postbuckled configuration of a tubular can be quite complex. In fact, if one includes from the outset the axial component of stress arising from the distributed weight of the tubular along its length, the resulting expressions become unwieldy. As an alternative, this discussion is restricted to an analysis of the postbuckled geometry of a weightless rod. Adjustment of the results of this analysis to include the effects of both the distributed weight of the tubular and internal and external pressure are covered in detail in Ref. 1. Geometry of the Helix At the instant a compressed, straight rod buckles, the lateral displacements will be sinusoidal in nature. However, if the lateral displacements are constrained to be less than or equal to some predetermined value (i.e., a wall constraint concentric with the undeformed predetermined value (i.e., a wall constraint concentric with the undeformed position of the rod), contact of the buckled rod with the constraint will position of the rod), contact of the buckled rod with the constraint will induce a rearrangement of the rod such that it assumes the form of a helix. Consider Fig. 1, which illustrates the geometry of a representative length of the helically buckled rod and several important variables to be used throughout the analysis. Notice in particular the definition of the pitch being the length between repetitions of the helical configuration. pitch being the length between repetitions of the helical configuration. Since the rod is assumed here to be weightless and infinite in length, the pitch will be constant throughout. pitch will be constant throughout. Now let a cartesian coordinate system be placed at some representative point along the length of the helix. point along the length of the helix. SPEJ p. 467