Abstract
Cables are typically used in engineering applications as tensile members. Relevant examples are the main cables of suspension bridges, the stays of cable-stayed bridges, the load-bearing and stabilizing cables of tensile structures, the anchor cables of floating mooring structures, the guy-ropes for ship masts, towers, and wind turbines, the copper cables of electrical power lines. Since cables are characterized by non-linear behavior, analysis of cable structures often requires advanced techniques, like non-linear FEM, able to consider geometric non-linearity. Nevertheless, a traditional simplified approach consists in replacing the cable with an equivalent tie rod, characterized by a suitable non-linear constitutive law. Currently used equivalent constitutive laws have been derived by Dischinger, Ernst and Irvine. Since the equivalence is restricted to taut cables, characterized by small sag to chord ratios, these traditional formulae are not appropriate for uniformly loaded sagging cables: the main cables of suspension bridges are a particularly emblematic case. Despite some recent attempts to find more refined solutions, the problem is still open, since closed form solutions of general validity are not available. In the paper, general analytical formulae of the non-linear constitutive law of the equivalent tie rod are proposed, distinguishing two relevant cases, according as the length of the cable can vary or not. The expressions, derived by applying the general form of the theorem of virtual work, can be applied independently on the material, on the sag to chord ratio, on the load intensity and on the stress level, so allowing the replacement of the whole cable with a single equivalent tie rod. The expressions are critically discussed referring to a wide parametric study also in comparison with the existing formulae, stressing the influence of the most relevant parameters.
Highlights
Cables are widely used in engineering fields as typical load bearing tensile members: relevant examples are the main cables of suspension bridges, the stays of cable-stayed bridges, the load-bearing and stabilizing cables of tensile structures, the anchor cables of floating mooring structures, the guy-ropes for masts, ship masts, towers, and wind turbines, the copper cables of electrical power lines, and so on
A traditional and very effective approach to simplify the structural model consists in replacing the cable, whose behavior is governed by geometric nonlinearity, with an equivalent tie rod connecting the ends of the cable, characterized by a suitable non-linear constitutive law
In the common case of cables with fixed ends, (a) when the virtual length Le is calculated by means of the simplified expression, Equation (11), Irvine’s formula always underestimates the equivalent elastic modulus, leading to satisfactory results only for very small values of the sag to chord ratio f /a: in this field, it is more accurate than the Dischinger’s formula, being the ratio Et,eq,Irvs/Et,eq,a practically insensitive to the stress level; (b) when the virtual length Le is calculated by means of the accurate expression, Equation (10), Irvine’s formula can be adopted when the f /a is small
Summary
Cables are widely used in engineering fields as typical load bearing tensile members: relevant examples are the main cables of suspension bridges, the stays of cable-stayed bridges, the load-bearing and stabilizing cables of tensile structures, the anchor cables of floating mooring structures, the guy-ropes for masts, ship masts, towers, and wind turbines, the copper cables of electrical power lines, and so on. (b) if cable ends, A and B, are fixed, the effect of the relative displacement is a variation of the cable chord a: da = dxB (Figure 4) Referring to both previously mentioned cases, the virtual work equation for the cable can be expressed in the form,. The total variation of the ordinate dy of a point of the cable, whose abscissa is x, depends on the variation of the chord length da, and on the variation of the horizontal component of the normal force dN0 From another perspective, the total variation of the ordinate dy can be seen as sum of two contributions, the one associated with the variation of the configuration of the cable, assumed inextensible, dyin, the other associated with the elastic stretching of the cable, dyel.
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