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 or cable-stayed bridges, and are also adopted in many special structural applications, including cable-roof systems and slender footbridges. Monitoring of the cable axial force, possibly along with other relevant structural parameters, is crucial for early detection of structural damage. Vibration-based identification procedures are widely used for this purpose, and they rely on the measurement of a set of low-order natural frequencies and mode shapes of the cable, along 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 linear dynamics of the cable. Analytical models, if properly formulated, offer accuracy, computational efficiency, and clear insight into parameters 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 with both a classical asymptotic approach to the solution of the algebraic eigenvalue problem and the matched asymptotic expansion method are systematically compared against finite elements simulations and semi-analytical solutions, revealing remarkable accuracy for a broad range of values of Irvine’s parameter. These results offer practical tools for efficient monitoring of cable structures where both bending stiffness and cable extensibility effects are relevant, reducing computational costs in inverse analysis and SHM applications.