Asymptotic solutions for the modal properties of shallow cables with small bending stiffness and compliant supports

Metallic cables are key components of large and strategic infrastructures, such as suspension and cable-stayed bridges, and are also adopted in many special structural applications, including cable-roof systems and slender footbridges. Monitoring the cable axial force, possibly together with other relevant structural parameters, is crucial for the early detection of structural damage. Vibration-based identification procedures are widely used for this purpose. They rely on the measurement of a set of low-order natural frequencies and mode shapes of the cable, together with structural models linking these properties to the cable parameters. The accuracy of the identified parameters is inherently affected by the ability of the underlying structural model to correctly reproduce the essential features of the cable's linear dynamics.

Analytical models, when properly formulated, offer high accuracy, computational efficiency, and valuable insight into parameter sensitivity. This paper addresses a notable gap in the literature by developing asymptotic closed-form solutions for the modal properties of shallow cables that simultaneously account for bending stiffness, sag/extensibility effects, and rotational flexibility at the supports. The results obtained using both a classical asymptotic approach to the algebraic eigenvalue problem and the method of matched asymptotic expansions are systematically compared against finite element simulations and semi-analytical solutions, revealing remarkable accuracy over a broad range of values of Irvine's parameter.

These results provide practical tools for the efficient monitoring of cable structures in which both bending stiffness and cable extensibility play a significant role, while reducing the computational cost of inverse analyses and structural health monitoring (SHM) applications.