# Equivalent Axial Stiffness of Horizontal Stays
## 5 Equivalent Stiffness of Stays Fixed at Their Ends  
If the cable is fixed at its ends (Figure 4), obviously it is \(dx_{B}=da\); therefore, by applying Equations (25), (26) and (29), \(dy\) results  
\[N_{0}\;da+\int\limits_{0}^{L}p\frac{\partial y_{e}}{\partial N_{0}}ds\;dN_{0}+ \int\limits_{0}^{a}p\bigg{(}\frac{\partial y}{\partial N_{0}}dN_{0}+\frac{ \partial y}{\partial a}da\bigg{)}\;\sqrt{1+{y^{\prime}}^{2}}\;dx=\frac{N_{0} \;dN_{0}}{EA_{0}}\int\limits_{0}^{a}\left(1+{y^{\prime}}^{2}\right)^{\frac{3} {2}}\;dx. \tag{38}\]  
Recalling the results of the previous section, and considering that  
\[ \begin{array}{ll}\int\limits_{0}^{a}p\frac{\partial y}{\partial a}\sqrt{1+{ y^{\prime}}^{2}}\;dx\;da&=\frac{p}{2}\int\limits_{0}^{a}\cosh\left(\frac{p\;a}{2 N_{0}}-\frac{p\;x}{N_{0}}\right)\cosh\left(\frac{p\;a}{2N_{0}}-\frac{p\;x}{2N_{0}} \right)\sinh\left(\frac{p\;a}{2N_{0}}\right)dx\;da\\ &=\;N_{0}\;\sinh^{2}\left(\frac{p\;a}{2N_{0}}\right)da,\end{array}  \tag{39}\]  
Equation (24) assumes the form  
\[ \begin{array}{ll}\cosh^{2}\left(\frac{p\;a}{2\;N_{0}}\right)dL&+\Big{[} \frac{1}{4\;p}\sin\left(\frac{p\;a}{N_{0}}\right)-\frac{a}{4\;N_{0}}\,\cos \left(\frac{p\;a}{N_{0}}\right)\Big{]}dN_{0}\\ &=\frac{N_{0}}{6\;EA_{0}\;p}\Big{[}9\sin\left(\frac{p\;a}{2\;N_{0}}\right)+\, \sin\left(\frac{3p\;a}{2\;N_{0}}\right)\\ &-24\,\sinh^{3}\left(\frac{p\;a}{2\;N_{0}}\right)+6\frac{p\;a}{N_{0}}\,\sin \left(\frac{p\;a}{2\;N_{0}}\right)\sin\left(\frac{p\;a}{N_{0}}\right)\Big{]}dN _{0},\end{array}  \tag{40}\]  
from which one derives  
\[ \begin{array}{ll}\frac{1}{E_{ft,\alpha}}=\frac{d\varepsilon_{x}}{d\omega_{0} }&=\frac{1}{\cosh^{2}\left(\frac{\gamma\;a}{2\;\omega_{0}}\right)}\left\{ \frac{\sigma_{0}}{6\;A\;E\gamma}\right[9\,\sin\left(\frac{\gamma\;a}{2\; \omega_{0}}\right)\right.\\ &\left.+\sinh\left(\frac{3\;\gamma\;a}{2\;\omega_{0}}\right)-24\,\sinh^{3} \left(\frac{\gamma\;a}{2\;\omega_{0}}\right)+6\,\frac{\gamma\;a}{\omega_{0}}\, \sin\left(\frac{\gamma\;a}{2\;\omega_{0}}\right)\sin\left(\frac{\gamma\;a}{ \omega_{0}}\right)\right]\\ &-\frac{1}{4\;\gamma\;\gamma}\,\sin\left(\frac{\gamma\;a}{\omega_{0}}\right)+ \frac{1}{4\;\sigma_{0}}\,\cos\left(\frac{\gamma\;a}{\omega_{0}}\right)\end{array}  \tag{41}\]  
A comparison of Equation (41) with Equation (36) demonstrates that the equivalent along the chord stiffness of the cable fixed at its ends is always bigger than the equivalent stiffness of the cable, having the same characteristics, running over a pulley. In fact, as \(\gamma a/\sigma_{0}>0\), the ratio \(\omega\) between \(E_{ft,\alpha}\) and \(E_{pt,\alpha}\) is always bigger than one:  
\[\omega=\frac{E_{ft,\alpha}}{E_{pt,\alpha}}=\cosh\left(\frac{\gamma\;a}{2\; \sigma_{0}}\right)>1. \tag{42}\]  
Evidently, it is  
\[\lim_{\omega_{0}\rightarrow\infty}\omega=1, \tag{43}\]  
therefore, confirming that the cable behavior tends to be independent on the end conditions for \"flat\" and highly stressed cables. The dependence of \(\omega\) on the product \(\gamma a\) and on the horizontal component of the normal stress, \(\sigma_{0}\), is illustrated in Figure 5, where curves are parameterized according the value of the horizontal stress component \(\sigma_{0}\). In the diagram, considering that \(\sigma_{0}>500\) MPa implies \(\omega\approx 1\), values of \(\sigma_{0}\) have been selected in the range 50-500 MPa. More generally, it can be remarked that \(\omega\) differs significantly from one only when the stress level is low or \(\gamma a\) is high. To exemplify, considering a steel cable having a chord length \(\mathrm{a}=1000\) m, for which \(\gamma\mathrm{a}=78.5\) MPa, it results \(\omega<1.1\), provided that the horizontal stress component satisfies the inequality \(\sigma_{0}>88.5\) MPa. It must be emphasized that expressions (36) and (41) are much more relevant than one might assume at first sight.