The Static WKB Solution to Catenary Problems with Large Sag and Bending Stiffness

Abstract

Large sag with a bending stiffness catenary is a subject that draws attention in the realm of fatigue analysis, estimation of suspension cable sag for bridge cable hoisting, and ocean engineering of the employment of mooring systems. However, the bending stiffness is the cause of boundary layers at the anchorage of cables, thereby finding a solution of the differential equation can be extremely difficult. Previous studies have tackled this problem with the perturbation method; yet, due to the complexity of the matching process and solution finding, the method might not be an ideal solution for engineering applications. Moreover, the finite difference method and the finite element method in numerical analysis can often be ineffective because of inappropriate parameter configuration and the drastic variation of functions in the boundary layers. Therefore, this study proposed a novel bending moment expression of a large sag catenary. The expression was derived from the sag identified using bending moment equations, and a solution was found by applying the WKB method (Wentzel-Kramers-Brillouin method) to overcome the complex problem of boundary layers. Consequently, a simple solution of various mechanical properties in a cable with bending stiffness and large sag could be obtained.