Catenary configuration and geometric stiffness matrix of inextensible cables: Analytical high-order asymptotic solutions for parametric design

The analytical and geometric study of catenary curves is a classic matter of applied mathematical modeling. The paper systematizes a methodological strategy to achieve fully analytical solutions for the mechanical problem of determining the equilibrium configuration assumed by inextensible inclined cables under gravitational loads. By developing a two-step perturbation scheme, the statically indeterminate equilibrium problem is asymptotically formulated, and asymptotic solutions are obtained in the form of convergent polynomial series of terms with increasing orders of smallness. As a major achievement, the asymptotic analytical solutions are explicit functions of the governing parameters and automatically satisfy the integral compatibility condition, which traditionally requires a numerical solution to assess the hyperstatic unknowns. The mathematical conditions for the existence of admissible solutions and asymptotic consistency of the perturbation scheme are provided, while the characteristic properties of the asymptotic series are recognized or demonstrated. Parametric analyses successfully verify the high approximation accuracy achievable by high-order asymptotic series, truncated to a large but finite number of terms. With the aim of extending the methodology to the largest possible variety of cable structure applications, a fully analytical, although asymptotic, expression of the geometric stiffness matrix of inextensible inclined cables is determined. Parametric analyses confirm that high-order asymptotic expressions can accurately approximate the exact stiffness matrix assessed numerically. This achievement opens up viable opportunities for analytically studying and parametrically designing complex cable structures (including collaborations with other structural elements), within the framework of the direct stiffness method. Finally, the feasibility and effectiveness of the asymptotic direct stiffness method are successfully verified by solving in a fully-analytical way a paradigmatic static problem for the catenary cable-stayed beam.