## 4 Analytical Analysis of a 6 \(\times\) K31 WS + FC Wire Rope
### Analytical Analysis of a 6 \(\times\) K31 WS + FC Wire Rope  
It is difficult to obtain the mechanical response directly according to the classical theory due to the complex structure of a 6 \(\times\) K31 WS + FC wire rope, so appropriate assumptions must be made. First, the 6 \(\times\) K31 WS + FC wire rope can be treated as a straight strand by regarding each strand of the wire rope as a steel wire with a helix structure, as shown in Figure 14a,b. Then, the mechanical response of each strand of the wire rope can be obtained according to Section 4.1. Finally, the mechanical responses of wires in the strand can be obtained by the same method specified in Section 4.1 with an input load calculated in the first step. The helix radii of wires in each layer of strand are \(rr_{3}\), \(rr_{4}\), \(rr_{5}\), and \(rr_{6}\), respectively, which are:  
\[\left\{ \begin{array}{l}rr_{3}=R_{2}+R_{3}\\ rr_{4}=R_{2}+2R_{3}+R_{4}\\ rr_{5}=\sqrt{R_{2}^{2}+R_{3}^{2}}+\sqrt{R_{3}^{2}+R_{5}^{2}}\\ rr_{6}=\frac{\sqrt{(R_{4}+R_{5})^{2}-R_{5}^{2}}+R_{4}+2R_{3}+R_{2}}{\cos\gamma} \end{array} \right. \tag{13}\]  
where \(\gamma\) is the angle between two lines, which is the line between the center of the rope core and the strand core wire, and the line between the center of the strand core wire and the outer wire, and it is illustrated in Figure 14c, \(\gamma\) = 15.14\({}^{\circ}\). If there is no initial torsion in the wire rope (\(\beta\) = 0), and the 6 \(\times\) K31 WS + FC wire rope is loaded with an axial strain \(\varepsilon\) = 0.01, which is located at the center of the end surface of the wire rope, the whole rope is regarded as a 1 \(\times\) 7 simple straight strand and the radius  
Figure 14: Assumptions made in the analytical model (**a**) original structure of a 6 \(\times\) K31 WS + FC wire rope, (**b**) simplified structure of a 6 \(\times\) K31 WS + FC wire rope, (**c**) illustration of angle \(\gamma\). of the center wire (rope core) \(R_{1}\) = 7.4 mm, the radius of the outer wire (strand of rope) \(R_{\rm c}\) = 6.27 mm, and the helix radius of the outer wire \(r_{\rm c}\)\({}^{*}\) = 13.67 mm. The equivalent modulus \(E_{\rm c}\) is used to represent the elastic modulus of each strand [31], \(E_{\rm c}\) can be calculated as:  
\[E_{\rm c}=E\Big{(}R_{2}^{2}+m_{3}R_{3}^{2}+m_{4}R_{4}^{2}+m_{5}R_{5}^{2}+m_{6} R_{6}^{2}\Big{)}/R_{\rm c}^{2} \tag{14}\]  
where \(m_{i}\) is the number of each type of wire in one strand, \(R_{i}\) is the radius of each type of wire in one strand, \(E\) is the elasticity modulus, and \(E_{\rm c}\) = 1.67 \(\times\) 10\({}^{11}\) Pa.  
Then, the following parameters could be calculated as described in Section 4.1. The axial strain of the strand \(\xi_{\rm c}\)\({}^{*}\) = 8.55 \(\times\) 10\({}^{-3}\), the helix angle variation \(\Delta\alpha_{\rm c}\)\({}^{*}\) = 4.06 \(\times\) 10\({}^{-3}\), and the helix radius of the loaded strand \(r_{\rm c}^{*}\) = 13.63 mm. The curvature variation of the strand \(\Delta\kappa_{\rm c}\)\({}^{*}\) = \(-1.65\)\(\times\) 10\({}^{-4}\) mm\({}^{-1}\), the torsion per unit length \(\Delta\tau_{\rm c}\)\({}^{*}\) = \(-1.65\)\(\times\) 10\({}^{-3}\) mm\({}^{-1}\), and the rotational strain of the strand \(\beta_{\rm c}\)\({}^{*}\) = \(-1.76\)\(\times\) 10\({}^{-17}\) N\(\cdot\)m. Then, the mechanical responses of the simplified 6 \(\times\) K31 WS + FC wire rope is figured out, as listed in Table 5.  
Note that the axial stress and bending moment of the rope core are regarded as 0, since the load of the fiber rope core is relatively small compared with the strand in a 6 \(\times\) K31 WS + FC wire rope. Based on the mechanical responses of the simplified 6 \(\times\) K31 WS + FC wire rope, the mechanical responses of each wire in each strand could be figured out, and the axial stress \(T_{\sigma 2}\) = 1.42 \(\times\) 10\({}^{3}\) MPa, the normal stress \(G^{\prime}_{\sigma 2}\) = 178.08 MPa, and the shear stress \(H_{\sigma 2}\) = 68.28 MPa.