A linear curved-beam model for the analysis of galloping in suspended cables

Abstract

A linear 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 incremental equilibrium equations around the prestressed state are derived, in which shear forces are condensed. By using a linear elastic constitutive law and accounting for damping and inertial effects, the complete equations of motion are obtained. They are then greatly simplified by estimating the order of magnitude of all their terms, under the hypotheses of small sag-to-span ratio, order-1 aspect ratio of the (compact) section, characteristic section radius much smaller than length (slender cable), small transversal-to-longitudinal and transversal-to-torsional wave velocity ratios. A system of two integrodifferential equations is drawn in the two transversal displacements only. A simplified model of aerodynamic forces is then developed according to a quasisteady formulation. 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. Finally, the role of the angle of twist on the dynamic stability of the cable is discussed for some sample cables.

Statics of Shallow Inclined Elastic Cables under General Vertical Loads: A Perturbation Approach

“…We note that a small coefficient multiplies the cubic power of H 0 in Equation (39), so that the local compatibility equation turns out to manifest a singular nature. As a further comment, up to this this order, the obtained equations coincide with the solution of the simplified model given in Equation (21).…”

“…Besides nonlinear dynamic analysis, exact static equations considering also the Poisson effect are obtained in [15], where a numerical procedure is suggested in the case of a generic load, while the catenary solution is evaluated when the sole weight is applied. Furthermore, inclined cables are the object of attention for researchers analyzing combined dynamic excitation, as a result of, for example, steady wind-inducing galloping and/or base motion [16][17][18][19][20], while a consistent model of a cable-beam to take into account the swing and the twist of iced cables under steady wind is presented in [21][22][23]. Perturbation methods are extensively used in [24,25] for the dynamic control of taut stings and in [26] for nonlinear dynamic analysis of suspended cables.…”

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

“…The papers [44][45][46][47][48][49][50][51][52] are specifically devoted to modeling problems of structural interest; the main original results are summarized in the following.…”

Dynamic instability of inclined cables under combined wind flow and support motion

“…The boundary value problem so far obtained is not reported here, since the equations are too complicated. They, however, assume a much simpler form if an order-of-magnitude analysis is performed (see [24,26]) and only the leading terms are retained in each equation. The analysis is based on the following assumptions: the initial curvatureκ(s), assumed uniform, allows one to introduce a nondimensional small parameter δ :=κ ; calling EA, GJ, EI the axial, torsional and bending stiffnesses respectively, the nondimensional prestress τ :=T /EA is small of order δ 3 , whereT is uniform; the nondimensional characteristic inertia radius R = r/ of the section is also small of order δ 3 ; the nondimensional stiffness parameter χ := GJ/EI is of order 1; the transversal displacements v and w are of the same order, while O(u/v) = δ and O(ϑ δ/w) = 1; the translations vary on a typical scale of length (since they vanish at the ends, if these are hinged) while the twist varies on a much greater length (since it does not vanish at the ends).…”

“…In [24][25][26] a consistent model of cable-beam accounting for the (small) curvature of the cable, as well as for bending and torsional stiffness was formulated. By retaining only the leading terms in each equations, nonlinear reduced equations were obtained, which are identical to those of the perfectly flexible model, plus an additional equation, of static nature, accounting for both bending and torsion.…”