Matched asymptotic expansions for cables with small
bending stiffness: Analytical form-finding of multi-span cable systems
Y. Feng & G.H.M. van der Heijden
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 multi-span 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, what we call, transmission
line problems in which the cable lengths are given and the span shapes are
to be found. The governing algebraic equations are conveniently developed on
a per-span basis. The two types of problem 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.
keywords: multi-span cables, small bending stiffness, matched asymptotic
expansion, boundary layer, catenary
ASCE Journal of Engineering Mechanics 152, 04026047 (2026)