# Equivalent Axial Stiffness of Horizontal Stays
## 8 Conclusions  
The nonlinear behavior of horizontal cables under the self-weight is the subject of a number of studies, often aiming to find suitable expressions for the equivalent along the chord stiffness. The topic is very relevant in view of practical applications, as the results allow important simplifications of nonlinear analysis of cable structures. For example, in finite element analysis (FEA), replacing the cables or parts of them with suitable equivalent nonlinear spring or truss elements considerably increase the speed and the efficiency of the analysis. On the other hand, the knowledge of the equivalent stiffness can be a robust practical tool to rapidly check results of nonlinear FEA. Several simplified formulae have been proposed over the years, starting from the classical expression proposed by [PERSON], still frequently used in current design practice, and arriving to much more refined solutions, like that due to [PERSON]. In a previous study, starting from the classical Bernoulli equation of the catenary, and applying the virtual work theorem, accurate and general expressions of the along the chord tangent stiffness of inclined cables subject to the self-weight were derived, as a function the nature of the relative displacement of the cable extremities. Depending on the boundary conditions of the stay, two relevant situations can occur, corresponding to distinct solutions: in fact, it could happen that both ends of the cable are attached, like in normal working conditions, so that the unstrained length of the cable is fixed; or, alternatively, that one end is fixed and the other is running around a fixed pulley, like in the cable tensioning phase, when the unstrained length of the cable varies as a function of the applied tensile force. The study demonstrates that the equivalent stiffness of the cable in the former case is bigger than in the latter. It must be underlined that not only classical simplified formulae, but also more refined approaches proposed over the years, are unable to make such a fine distinction, depending on the restraints at the stay ends. The abovementioned innovative general closed form solutions, suitable to be used in modeling equivalent spring or truss elements, are also valid for horizontal cables, but they are so complex and the calculations so complicated, that their reduction to that particular case was not easy. For that reason, the above-recalled method has been applied to directly deduce the equations of the equivalent stiffness of horizontal stays. The outcomes of the procedure are new expressions of the equivalent tangent modulus of the horizontal cable, depending on the nature of the boundary conditions. As they accurately consider all influencing parameters, these rather compact innovative formulae can be again applied without restrictions, independently on the inclination of the cable, on the sag of the catenary, and on the cable material, thus allowing a generalized application of the equivalent stiffness approach, in particular in the low stress region. Of course, the formulae given here for horizontal cables, which can be also applied for slightly inclined cables, coincide with those derived from the expressions for inclined cables when \(h\) tends to zero. A relevant case study, involving horizontal steel stays with fixed ends, and chord length \(a\) varying in the range \(10-3000\) m, is discussed in detail, also aiming to estimate the approximations inherent with the [PERSON]'s approach. The outcome is that [PERSON]'s formula underestimates the actual stiffness of the cable, especially when the value of the horizontal component of the stress \(\sigma_{0}\) is small. Results have been extended to other significant cases, taking into account various natural or artificial cable materials, so highlighting that the magnitude of nonlinear effects in the cable depends not only on the specific stiffness, but also on the elastic modulus and on the density of the material.