Matched Asymptotic Expansions for Cables with Small Bending Stiffness: Analytical Form-Finding of Multispan Cable Systems

Long-span cable problems are governed by differential equations with singular perturbations as a result of the small bending stiffness compared to the cable’s self-weight. We use the method of matched asymptotic expansions for such singular problems to construct accurate analytical solutions and use these to formulate a general and geometrically exact theory of analytical form-finding for multispan cable systems. We apply the theory to two classes of problems: suspension bridge problems, in which the span shapes are given and the cable lengths need to be found; and transmission line problems, in which the cable lengths are given and the span shapes need to be found. The governing algebraic equations are conveniently developed on a per-span basis. The two types of problems require different solution approaches: in the former, spans are solved in series, while in the latter they are solved in parallel. The provided analytical expressions could be used for efficient experimentation in early design stages.