Generally, helically wound cables are made up of a straight core that is surrounded by multiple layers of helical wires. These type of systems are mainly used in overhead power transmission lines, as hoist ropes or as cables in cable stayed and suspended bridges. Analysis of this cable system is crucial as it enables one to obtain effective stiffness of the cable under conditions of tension, bending and torsion. The semi-continuous model type where each layer of the helical wire is treated as a transversely isotropic continuum has been widely employed for such analysis. Unfortunately, this technique has an underlying drawback in that it becomes quite cumbersome to get the effective elastic moduli when the discrete wires are treated as a continuum and incorporating intrinsic interface conditions into the model. Alternative continuum theories have been developed, specifically for the fiber-reinforced material, but are yet to be incorporated in the analysis for simultaneous tension, torsion and bending of helically wound cables.
Dr. Loïc Le Marrec, Mr. Dansong Zhang, Professor Martin Ostoja-Starzewski from the Department of Mechanical Science and Engineering- University of Illinois at Urbana-Champaign developed a new model for the simultaneous tension, torsion and bending derivation of helically wound cables in a Timoshenko beam formalism. They purposed to employ a rod formalism instead of solving 3D equations for elasticity so as to obtain the solutions more easily and explicitly. Their work is currently published in the research journal, Acta Mechanica.
To begin with, the researchers assumed that a helically wound cable could be treated effectively as a 3D solid rod continuum that obeys Spencer’s constitutive law. Next, they employed the Timoshenko beam assumption where the cross section of the helical material was assumed to remain rigid during deformation. They then proceeded to obtain the cross-sectional forces and moments by integrating the stress components over the cross section. The researchers obtained the rod constitutive relations that relate the cross-sectional forces and moments to the rod deformations. Eventually, the applicability of the model to helically wound cables was verified.
From the numerical testing undertaken, the authors noted that by inclusion of CF (a representation of the coupling between tension and torsion, as well as between shearing and bending) and CT (a representation of the difference between shear and torsional rigidities) in the model was essential for the correct description of the vibration of helically wound cables. The researchers also observed that when all parameters were included, the error between the measured and estimated Eigen frequencies was minimal. Moreover, it was numerically verified that with a proper set of αi’s chosen, the lay-angle dependency of the parameters E, G, CF and CT derived for a circular solid rod with uniform lay angle could be applied to the 1 + 6 cables except a shift imposed on shear anisotropy (CT).
The Loïc Le Marrec and colleagues study has presented a novel technique to describe the tension, torsion and bending of helical-fiber-reinforced rods. In this work, a full set of non-dimensional equations for the free vibration of the rod, in the form of tension, torsion and bending, has been derived. Altogether, the study has demonstrated that once the parameters for the rod model are obtained, vibrations of cables of arbitrary lengths and boundary conditions can be solved directly from the rod vibration equations and analytical solutions may be acquired that explicitly describe the behavior of helically wound cables, bypassing the need of solving equations of 3D elasticity.
Loïc Le Marrec, Dansong Zhang, Martin Ostoja-Starzewski. Three-dimensional vibrations of a helically wound cable modeled as a Timoshenko rod. Acta Mechanica, volume 229, pages 677–695 (2018)Go To Acta Mechanica