# Equivalent Axial Stiffness of Horizontal Stays
## 1 Introduction  
Cable-stayed structures, like cable-stayed building roofs, bridges, extradosed bridges, overhead electrical lines, guyed masts, anchorage of floating devices, cable trusses, etc., are widely used in several engineering fields. From the structural point of view, these structures are extremely appealing; however, since they are often characterized by highly nonlinear behavior, their analysis requires the introduction of refined mechanical models. A very effective simplified mechanical model is commonly considered to simulate the behavior of the stays, where each of them is substituted with a suitable equivalent nonlinear spring or truss element. In the classical approach, as proposed by [PERSON] [1, 2] or [PERSON] [3], the stay is modeled as an equivalent straight bar connecting its ends. The equivalence criterion is that the axial stiffness of the substitutive rod equals the apparent global stiffness of the stay along its chord, i.e., the line connecting its ends. Assuming that the influences of the bending stiffness of the cable on the deformed configuration is negligible with respect to the axial stiffness of the cable, the traditional treatment of the problem is based on the equilibrium equation of a deformed cable, with horizontal chord, subjected to the self-weight. Let \(\overrightarrow{p}\) be the weight of the cable per unit length; since the deformed configuration of the cable belongs to the \(\pi\) plane defined by the two cable ends and by the vector \(\overrightarrow{p}\) lying on it, classical simplified formulae have been deduced, simulating the cable behavior in terms of an equivalent axial stiffness. The so-called equivalent modulus can be expressed in form of an equivalent tangent elastic modulus or an equivalent secant elastic modulus. Let \(a\) be the length of the cable chord; \(A_{0}\) the area of the cross section of the cable; \(E\) the elastic modulus of the material; \(\stackrel{{\rightarrow}}{{N}}\) the tensile force in the cable, whose horizontal and vertical components are \(N_{0}\), and \(N_{y}\), respectively; \(\stackrel{{\rightarrow}}{{N}}=\left(N_{0},N_{y}\right)\), the equivalent elastic modulus of a horizontal stay; \(E_{t,eq}(\sigma_{0})\), is thus given by the so called Dischinger's formula,  
\[E_{t,eq}(\sigma_{0})=\frac{d\sigma_{0}}{d\varepsilon}=\frac{E}{1+\frac{\left( Y\;a\right)^{2}}{12\;\sigma_{0}^{3}}\;E} \tag{1}\]  
which is widely used for structural design purposes. In Equation (1), \(\sigma_{0}\) is the horizontal component of the cable tension, \(d\varepsilon\) is the equivalent strain variation in the chord direction and \(\gamma\) is the ratio between the unit weight \(p\) and the area \(A_{0}\), i.e., the specific weight of the cable material. As known, \(N_{0}\), and \(\sigma_{0}\), are independent on the abscissa of the considered section. Of course, since the problem is governed by geometric nonlinearity only, the material is assumed linear elastic, therefore \(E\) is independent on \(\sigma_{0}\). As summarized in the following, Equation (1) was derived by [PERSON] [1; 2] starting from the expression of the strain variation along the chord direction  
\[d\varepsilon=d\varepsilon_{\varepsilon}+d\varepsilon^{*} \tag{2}\]  
where \(d\varepsilon_{\varepsilon}\) is the variation of the elastic strain, and \(d\varepsilon^{*}\) is the apparent variation of the strain. Following [PERSON], when an inextensible cable, whose length is \(L\), is stretched, the cable configuration changes. If the chord length \(a\) increases, the sag and the difference \((L-a)\) reduce and vice versa, i.e., if the chord length \(a\) decreases, the sag and the difference \((L-a)\) increase; in any case, this causes the apparent strain variation \(d\varepsilon^{*}\). Obviously, it is  
\[d\varepsilon_{\varepsilon}=\frac{d\sigma_{0}}{E},\;d\varepsilon^{*}=\frac{d \sigma_{0}}{E^{*}(\sigma_{0})}, \tag{3}\]  
being \(E^{*}\) the apparent elastic modulus resulting from the variation of the inextensible cable configuration.