## 4 Analytical Analysis of a 6 \(\times\) K31 WS + FC Wire Rope
### Analytical Model of a Simple Straight Strand  
First, the mechanical response of a straight strand, which is displayed in Figure 11, is analyzed. In addition, the mechanical response of the wire rope with a more complex double-helix structure can be derived. The original curvatures of the center wire and outer wire are \(\kappa_{2}\) and \(\kappa_{2}^{\prime}\). The torsion per unit length of the outer wire is \(\tau_{2}\), and:  
\[\kappa_{2}=0,\kappa_{2}^{\prime}=\frac{\cos^{2}\alpha_{2}}{r_{2}},\tau_{2}= \frac{\sin\alpha_{2}\cos\alpha_{2}}{r_{2}} \tag{6}\]  
where \(r_{2}\), \(\alpha_{2}\) are the helix radius and helix angle of the outer wire, respectively, which are shown in Figure 11.  
The outer wire is transferred into a new single-helix structure under the load of strain, as shown in Figure 12. \(h\) and \(\overline{h}\) are the initial and final length of the strand. \(\theta_{2}\) and \(\overline{\theta_{2}}\) are the initial and final position angle, respectively, which define the position of the center of the outer wire, and \(\theta_{2}=\frac{h}{r_{2}\tan\alpha_{2}}\), \(\overline{\theta_{2}}=\frac{h}{r_{2}\tan\alpha_{2}}\). \(\xi_{1}\) and \(\xi_{2}\) are the axial strains of the center wire and the outer wire, respectively. The original curvature and the torsion per unit length are transformed into:  
\[\overline{\kappa_{2}}=0,\overline{\kappa_{2}^{\prime}}=\frac{\cos^{2} \overline{\alpha_{2}}}{\overline{r_{2}}},\tau_{2}=\frac{\sin\overline{ \alpha_{2}}\cos\overline{\alpha_{2}}}{\overline{r_{2}}} \tag{7}\]  
where \(\overline{r_{2}}\), \(\overline{\alpha_{2}}\) are the new helix radius and new helix angle of the outer wire, \(\overline{r_{2}}=R_{1}(1-\xi_{1})\) + \(R_{2}(1-\
u\xi_{2})\), \(R_{1}\) and \(R_{2}\) are the radius of the core wire and the outer wire, respectively, \(\
u\) is the Poisson's ratio of the wire, and higher-order variables such as \(\xi_{2}\Delta\alpha\) are ignored. The torsion per unit length \(\tau_{\rm s}\), the axial strain of the center wire \(\xi_{1}\), and the rotational strain of the outer wire \(\beta_{2}\) are:  
\[\left\{ \begin{array}{l}\tau_{\rm s}=\frac{\overline{\kappa_{2}}-\theta_{2} }{\overline{h}}\\ \xi_{1}=\frac{\overline{h}-h}{\overline{h}}=(1+\xi_{2})\frac{\sin\overline{ \alpha_{2}}}{\sin\alpha_{2}}-1\\ \beta_{2}=r_{2}\tau_{\rm s}=\frac{\xi_{2}}{\tan\alpha_{2}}-\Delta\alpha_{2}+ \
u\frac{R_{1}\xi_{1}+R_{2}\xi_{2}}{r_{2}\tan\alpha_{2}}\end{array} \right. \tag{8}\]  
Figure 11: A straight strand with a single-helix structure. The curvature variation \(\Delta\kappa_{2}^{\prime}\) and the variation of torsion per unit length \(\Delta\tau_{2}\) can be linearized and expressed as:  
\[\left\{ \begin{array}{l}R_{2}\Delta\kappa_{2}^{\prime}=\frac{R_{2}}{\tau_{2}} \Big{[}\Big{(}v\frac{R_{1}}{\tau_{2}}\frac{\cos^{2}\alpha_{2}}{\tan\alpha_{2}}- 2\sin\alpha_{2}\cos\alpha_{2}\Big{)}\Delta\alpha_{2}+\big{(}v\cos^{2}\alpha_{ 2}\xi_{2}\big{)}\Big{]}\\ R_{2}\Delta\tau_{2}=\frac{R_{2}}{\tau_{2}}\Big{[}\Big{(}1-2\sin^{2}\alpha_{2}+ v\frac{R_{1}}{\tau_{2}}\cos^{2}\alpha_{2}\Big{)}\Delta\alpha_{2}+\big{(}v\sin \alpha_{2}\cos\alpha_{2}\big{)}\xi_{2}\Big{]}\end{array} \right. \tag{9}\]  
Then, the mechanical responses of the outer wire can be expressed as:  
\[\left\{ \begin{array}{l}H_{2}=\frac{\pi E\kappa_{2}^{3}}{4(1+v)}R_{2}\Delta \tau_{2}\\ G_{2}^{\prime}=\frac{\pi E\kappa_{2}^{3}}{4}R_{2}\Delta\kappa_{2}^{\prime}\\ N_{2}^{\prime}=H_{2}\frac{\cos^{2}\alpha_{2}}{\tau_{2}}-G_{2}^{\prime}\frac{ \sin\alpha_{2}\cos\alpha_{2}}{\tau_{2}}\\ T_{2}=\pi ER_{2}^{2}\xi_{2}\\ X_{2}=N_{2}^{\prime}\frac{\sin\alpha_{2}\cos\alpha_{2}}{\tau_{2}}-T_{2}\frac{ \cos^{2}\alpha_{2}}{\tau_{2}}\end{array} \right. \tag{10}\]  
where \(E\) and \(H_{2}\) are the elastic modulus and axial torque of the outer wire, \(G_{2}^{\prime}\) is the bending moment, \(N_{2}^{\prime}\) is the shearing force, \(T_{2}\) is the axial force, and \(X_{2}\) is the contact force per unit length, as Figure 13 shows. Figure 12: The strand geometries before and after loading.