Patching Asymptotics Solution of a Cable with a Small Bending Stiffness

Abstract

The analysis of a cable with a small bending stiffness is a problem encountered in many engineering applications such as the fatigue assessment of stay cables, the modeling of pipeline laying operation, or the determination of bending stresses in drillpipe assemblies. Because this phenomenon is modeled by a singularly perturbed equation, standard numerical techniques fail to solve these problems efficiently. As an alternative, provided the complexity of the analytical developments does not preclude their application, these problems may be tackled with appealing analytical procedures such as matching asymptotics or multiple scales. Otherwise, advanced numerical simulations combining patching asymptotics within a numerical framework are the only possible approach for problems where the governing equations are too complex. Patching asymptotics also feature a number of merits such as the possibility of using a boundary layer with a finite extent. Aiming at a better understanding of this latter technique, the purpose of this paper was to determine the solution of a cable with a small bending stiffness. Interesting details about patchability conditions and about how to restore higher derivative continuity are included. The accuracy of the patching asymptotics approach is also compared with that of matched asymptotics.