# Equivalent Axial Stiffness of Horizontal Stays
## 6 Some Relevant Examples  
To highlight the refinements achieved by using the proposed approach, the new formula (41) has been applied to evaluate the apparent stiffness \(E_{ft,eq}\) in the relevant case of steel cables, considering chord lengths varying in the range 10-3000 m. According to usual design assumptions, the following values have been adopted for the specific weight and the elastic modulus of the steel cable: \(\gamma=78.5\) kN/m\({}^{3}\); \(E=1.8\times 10^{5}\) MPa, thus obtaining the results summarized in Figures 6 and 7.  
In Figure 6, the \(E_{ft,eq}=E_{ft,eq}(\sigma_{0}\;)\) curves are parameterized in terms of the chord length, \(a\); conversely, in Figure 7, \(E_{ft,eq}=E_{ft,eq}(a\;)\) curves are parameterized in terms of the horizontal component of the tension, \(\sigma_{0}\).  
Inspection of the diagrams confirms that, when the stress \(\sigma_{0}\) is low and the chord length \(a\) is large, the equivalent modulus \(E_{ft,eq}\) can be significantly smaller than \(E\). Obviously, the effect is further intensified when the cable is running around a fixed pulley.  
To facilitate the interpretation of the results, given the chord length \(a\), it is useful to associate the value of the stress component \(\sigma_{0}\) with the corresponding cable sag, \(d\), or, better, with the corresponding sag to chord ratio \(k\), given by  
\[k=\frac{d}{a}=\frac{\sigma_{0}}{a\,\gamma}\bigg{(}\mathrm{cosh}\bigg{(}\frac{ \gamma\;a}{2\,\sigma_{0}}\bigg{)}-1\bigg{)} \tag{44}\]  
as illustrated in the bi-logarithmic graph of Figure 8. In Figure 8, curves have been derived considering that, in practice, upper limits exist for the cable sag \(d\), for example, due to the limited height of the  
Figure 5: \(\omega-\gamma a\) curves parameterized in terms of horizontal stress component \(\sigma_{0}\).  
pylons or to the limited elevation of the cable attachments over the roof, over the deck, or over the valley floor. For that reason, in the considered examples, it has been assumed \(d\leq 500\) m, implying that \(k-\sigma_{0}\) curves are limited on the right and that \(\sigma_{0}\) is bounded from below.  
Figure 6: \(E_{ft,eq}-\sigma_{0}\) curves parameterized in terms of chord length, \(a\).  
Figure 7: \(E_{ft,eq}-a\) curves parameterized in terms of horizontal stress component, \(\sigma_{0}\).  
For instance, in the present case, where \(\gamma=78.5\)\(\mathrm{kN}/\mathrm{m}^{3}\), and \(d\leq 500\) m, the lower limit of \(\sigma_{0}\), which is about 50 MPa when \(a=1500\) m, increases to about 85 MPa when \(a=2000\) m, to finally attain values around 130 MPa and 180 MPa for \(a=2500\) m and \(a=3000\) m, respectively. Clearly, when cables are made with materials other than steel, lower limits of \(\sigma_{0}\) vary, depending on the specific weight, \(\gamma\), of the cable material itself.  
To check the capability of the classical  
The diagrams demonstrate that the equivalent stiffness derived via the classical Dischinger's formula is regularly lower than the one derived with the formulation proposed here and that the error increases as soon as the stress in the cable decreases, or, evenly, the sag to chord ratio, \(k\), increases.