# Equivalent Axial Stiffness of Horizontal Stays
## 3 The Virtual Work Theorem for the Cable  
The nonlinear behavior of the cable can be studied by means of the theorem of virtual works, following the advanced method already adopted by the author in [16]. Starting from the deformed configuration of the cable, the virtual work theorem is applied considering a virtual horizontal relative displacement \(dx\) between the cable extremities, \(A\), and \(B\). Obviously, the resulting equation can be also derived assuming that the end \(A\) is fixed and the relative displacement is concentrated in the end \(B\), or vice versa. When the cable end \(B\) is shift by \(dx_{B}\), in \(B^{\prime}\), two significant cases can occur, depending on the actual constraint at the cable end \(B\): in fact, the virtual displacement can affect the length of the cable, or the length of the chord. In the former case, which also reproduces the tensioning phase of a real cable, the cable itself is assumed to run over a fixed pulley in \(B\) (Figure 3): in this case, the length of the chord is not affected by the displacement \(dx_{B}\), which entirely results in a variation \(dL\) of the cable length \(L\). On the contrary, in the latter case, which represents the most frequent operational condition of a structural cable, the cable extremities are assumed fixed in \(A\) and in \(B\): consequently, the displacement \(dx_{B}\) is entirely converted in a variation \(da\) of the cable chord (Figure 4). Figure 4: Cable with ends fixed in \(A\) and in \(B\). Figure 3: Cable running over a fixed pulley in \(B\), i.e., \(a=\text{const}\). Recalling that the cable ordinate \(y(x)\) is expressed by Equation (21), in both situations, the expression of the virtual work equation is  
\[\bigg{(}\overset{\rightarrow}{N_{B}}+d\overset{\rightarrow}{N_{B}}\bigg{)}d \overset{\rightarrow}{x}_{B}+\int\limits_{0}^{L}\overset{\rightarrow}{p} \cdot d\overset{\rightarrow}{y}\ ds\ =\int\limits_{V}\alpha d\varepsilon\ dV, \tag{23}\]  
where \(\sigma\) is the normal stress, \(d\varepsilon\) the variation of the longitudinal strain, \(V\) the volume of the cable, and \(d\overset{\rightarrow}{y}\) the variation of \(y(x)\) caused by the modification of the shape of the cable itself. Obviously, when the deformations of the cable are disregarded, i.e., the cable is assumed to be inextensible, the left-hand side of Equation (23) is zero. Since \(d\overset{\rightarrow}{N_{B}}\cdot d\overset{\rightarrow}{x}_{B}\) is negligible with respect to \(\overset{\rightarrow}{N_{B}}\cdot d\overset{\rightarrow}{x}_{B}\), and \(\overset{\rightarrow}{p}\) is parallel to \(\overset{\rightarrow}{y}\), Equation (23) can be reduced to  
\[\overset{\rightarrow}{N_{B}}\cdot d\overset{\rightarrow}{x}_{B}+\int \limits_{0}^{L}p\ dy\ ds\ =\int\limits_{V}\alpha d\varepsilon\ dV. \tag{24}\]  
Let \(x\) be the abscissa of an arbitrary point of the cable, we can remark that the variation \(dy\) of the ordinate of that point is the result of the relative displacement component along the cable chord, \(da\), as well as of the variation \(dN_{0}\) of the horizontal component of the axial force. The total variation, \(dy\), can be interpreted as the sum of two contributions: one, \(dy_{incx}\), associated with the modification of the shape of the cable, assumed to be inextensible; the other, \(dy_{e}\), due to the variation of the elastic strain produced by \(dN_{0}\):  
\[dy=dy_{incx}+dy_{e}. \tag{25}\]  
The variation \(dy_{incx}\) associated with the configuration change can be obtained by differentiating Equation (25) and considering that, for a given abscissa \(x\), \(y(x)\) depends only on \(N_{0}\) and on \(a\). Consequently, \(dy_{incx}\) results:  
\[dy_{incx}=\frac{\partial y}{\partial a}da+\frac{\partial y}{\partial N_{0}}dN_{ 0}. \tag{26}\]  
Obviously, in case the chord length is fixed, i.e., \(a\) is constant, Equation (26) reduces to:  
\[dy_{incx}=\frac{dy}{dN_{0}}dN_{0}. \tag{27}\]  
On the other hand, the variation \(dy_{e}\) due to the elastic strain variation is a function of the vertical component of the axial force only.