# Equivalent Axial Stiffness of Horizontal Stays
## 4 Equivalent Stiffness of Stays Running over a Fixed Pulley on One End  
Cables running around a fixed pulley on one end are considered first (Figure 3). In that case, the chord length \(a\) is not varying, and \(dy_{incx}\) is given by Equation (27). Hypothesizing a linear elastic material, Equation (24) becomes  
\[N_{B}\;dL+\int\limits_{0}^{L}p\frac{dy}{dN_{0}}dN_{0}\;\;ds=\int\limits_{0}^{L} \frac{N}{EA_{0}}\;dN\;ds. \tag{30}\]  
Recalling Equations (19) and (29), Equation (30) converts in  
\[\begin{array}{ll}N_{0}\;\sqrt{1+y^{\prime}(a)^{2}}\;dL&+\int\limits_{0}^{L}p \frac{\partial y_{c}}{\partial N_{0}}ds\;dN_{0}+\int\limits_{0}^{a}p\frac{dy} {dN_{0}}dN_{0}\;\;\sqrt{1+y^{\prime 2}}\;dx\\ &=\frac{N_{0}\;dN_{0}}{EA_{0}}\int\limits_{0}^{a}\Bigl{(}1+y^{\prime 2} \Bigr{)}^{\frac{3}{2}}\;\;dx.\end{array} \tag{31}\]  
Given that  
\[\frac{dy}{dN_{0}}=-\frac{\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}-\frac{p\;x}{N_{0}} \Bigr{)}}{p}+\frac{\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}}{p}-\frac{\bigl{(} x-\frac{a}{2}\bigr{)}\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}}-\frac{p\;x}{N_{0}} \Bigr{)}}{N_{0}}-\frac{a\,\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}}{2\,N_{0} }, \tag{32}\]  
and that  
\[\begin{array}{ll}\int\limits_{0}^{L}p\frac{\partial y_{c}}{\partial N_{0}} ds&=p\,\Bigl{(}\frac{y^{\prime}(0)}{E\,A_{0}}+\frac{N_{0}}{E\,A_{0}} \frac{dy^{\prime}(0)}{dN_{0}}\Bigr{)}\int\limits_{0}^{L}s\;ds\\ &=\frac{p\;l^{2}}{2\,E\,A_{0}}\,\Bigl{(}\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}} \Bigr{)}-\frac{p\;a}{2\,N_{0}}\,\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)} \Bigr{)},\end{array} \tag{33}\]  
Equation (31) reduces to  
\[\begin{array}{ll}N_{0}\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}\;dL&+\frac {p\;L^{2}}{2\,E\,A_{0}}\,\Bigl{[}\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}- \frac{p\;a}{2\,N_{0}}\,\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}\Bigr{]}\\ &-\int\limits_{0}^{L}\bigl{(}\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}-\frac{p\;x}{N_ {0}}\Bigr{)}\Bigr{[}\frac{p\;a}{2\,N_{0}}\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}} \Bigr{)} \tag{34}\]  
which, integrating and simplifying, leads to  
\[\begin{array}{ll}\cosh\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}\;dL&+\Bigl{[} \frac{1}{4\,p}\,\sin\hbar\Bigl{(}\frac{p\;a}{N_{0}}\Bigr{)}-\frac{a}{4\,N_{0}} \,\cos\hbar\Bigl{(}\frac{p\;a}{N_{0}}\Bigr{)}\Bigr{]}dN_{0}\\ &=\frac{N_{0}}{6\,E\,A_{0}}\Bigl{[}9\,\sinh\Bigl{(}\frac{p\;a}{2\,N_{0}} \Bigr{)}+\,\sin\hbar\Bigl{(}\frac{3\,p\;a}{2\,N_{0}}\Bigr{)}\\ &-24\,\sinh^{3}\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}+6\frac{p\;a}{N_{0}}\,\sin \hbar\Bigl{(}\frac{p\;a}{2\,N_{0}}\Bigr{)}\sin\hbar\Bigl{(}\frac{p\;a}{N_{0}} \Bigr{)}\Bigr{]}dN_{0}.\end{array} \tag{35}\]  
Referring the deformation to the \(x\) axis, the apparent variation of the chord length \(dx_{B}\) equals the variation of the cable length \(dL\), \(dx_{B}=dL\), so that a kind of equivalent tangent elastic modulus, \(E_{pt,eq}\), can be derived  
\[\begin{array}{ll}\frac{1}{E_{pt,eq}}=\frac{dx_{x}}{dx_{0}}&=\frac{1}{\cosh \Bigl{(}\frac{y^{\prime}\;a}{x_{0}}\Bigr{)}}\Bigl{\{}\frac{\partial\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;where \(d\varepsilon_{x}\) is the apparent strain variation of the chord, \(\sigma_{0}\) the horizontal component of the tension and \(\gamma\) the specific weight of the cable material:  
\[d\varepsilon_{x}=\frac{dx_{B}}{a},\sigma_{0}=\frac{N_{0}}{A_{0}},\ \ \gamma=\frac{p}{A_{0}} \tag{37}\]