A Nonlinear Model of Curved Beam for the Analysis of Galloping of Suspended Cables

A nonlinear model of curved, prestressed, no-shear, elastic beam, loaded by wind forces, is formulated. The beam is assumed to be planar in its reference configuration under its own weight and static wind forces. The unit extension, two bending curvatures and the torsional curvature are taken as strains and then expanded up-to second-order terms in the three displacement components and in the angle of twist. The incremental equilibrium equations around the prestressed reference configuration are derived, in which shear forces are condensed via a perturbation procedure. By using a linear elastic constitutive law and accounting for inertial effects, the complete equations of motion are obtained. They are successively strongly simplified by estimating the order of magnitude of all their terms, under the hypotheses of small sagto- span ratio, high slenderness, compact section and by neglecting tangential inertia forces and inertia torsional couples. A system of two integro-differential equations in the two transversal displacements only is drawn. A simplified model of aerodynamic forces is then developed according to the quasi-static theory. The nonlinear, nontrivial equilibrium path of the cable subjected to increasing static wind forces is successively evaluated, and the influence of the angle of twist on the equilibrium is discussed. Then stability is studied by discretizing the equations of motion via a Galerkin approach and analyzing the small oscillations around the nontrivial equilibrium. Analysis of the limit cycle (galloping) arising at the Hopf bifurcation is left for future investigation. Finally, the role of the angle of twist on the dynamic stability of the cable is discussed.