## 4 Analytical Analysis of a 6 \(\times\) K31 WS + FC Wire Rope
### Analytical Analysis of a 6 \(\times\) K31 WS + FC Wire Rope
According to the previous assumption, the axial strain \(\xi_{\rm c}\)* and helix angle variation \(\Delta\alpha_{\rm c}\) of the simplified center wire are equal to the axial strain \(\xi_{2}\) and the helical angle variation \(\Delta\alpha_{2}\) of the simplified outer wire, that is, \(\xi_{2}\) = \(\xi_{\rm c}\)\({}^{*}\) = 8.55 \(\times\) 10\({}^{-3}\), \(\Delta\alpha_{2}\) = \(\Delta\alpha_{\rm c}\) = 4.06 \(\times\) 10\({}^{-3}\). Table 6 lists the axial strain and helix angle variation of wires in each layer of the strand. Table 7 lists the mechanical responses of wires in each layer of a strand of a 6 \(\times\) K31 WS + FC wire rope, which are obtained by Equation (10). \begin{table}
\begin{tabular}{c c c c} \hline
**Serial Number of Wires** & **Axial Strain \(\xi_{i}\)** & **Helix Angle Variation** & **Curvature Variation** & **Torsion per Unit** \\  & \(\Delta\alpha_{i}\) & \(\Delta\kappa_{i}\) (m\({}^{-1}\)) & Length \(\Delta\tau_{i}\) (m\({}^{-1}\)) \\ \hline III & \(8.52\times 10^{-3}\) & \(5.29\times 10^{-4}\) & \(-2.41\times 10^{-2}\) & \(-0.219\) \\ I & \(8.46\times 10^{-3}\) & \(9.92\times 10^{-4}\) & \(-4.49\times 10^{-2}\) & \(-0.214\) \\ V & \(8.50\times 10^{-3}\) & \(7.47\times 10^{-4}\) & \(-3.39\times 10^{-2}\) & \(-0.217\) \\ VI & \(8.35\times 10^{-3}\) & \(1.47\times 10^{-3}\) & \(-1.81\times 10^{-2}\) & \(-0.207\) \\ \hline \end{tabular}
\end{table} 
Table 6: Parameters of the wire in each layer of the strand. \begin{table}
\begin{tabular}{c c} \hline
**Mechanical Response** & **Result** \\ \hline Bending moment \(G_{\rm c}\)\({}^{*}\) & \(-37.87\) N\(\cdot\)m \\ Axial torque \(H_{\rm c}\)\({}^{*}\) & \(-29.04\) N\(\cdot\)m \\ Shearing force \(N_{\rm c}\)\({}^{*}\) & 637.76 N \\ Axial force \(T_{\rm c}\)\({}^{*}\) & \(1.87\times 10^{5}\) N \\ Uniformly distributed radial load \(X_{\rm c}\)\({}^{*}\) & \(-1.54\times 10^{6}\) N/m \\ Total axial force of strand \(F_{\rm c}\)\({}^{*}\) & \(1.06\times 10^{6}\) N \\ Total torque of strand \(M_{\rm c}\)\({}^{*}\) & \(4.89\times 10^{3}\) N\(\cdot\)m \\ \hline \end{tabular}
\end{table} 
Table 5: Mechanical response of the simplified 6 \(\times\) K31 WS + FC wire rope. The maximum stress of the 6 \(\times\) K31 WS + FC wire rope \(S_{\rm max}\) = 1794.98 MPa, while axial strain \(\varepsilon\) = 0.01. As stated before, the maximum stress obtained from finite element simulation is 1497.65 Mpa. Obviously, the maximum stress obtained by the analytical analysis is 297.33 Mpa bigger, the deviation is 19.85%. The reasons accounting for the deviation could be summarized as: (1) The assumption that the fiber core does not bear any load in the analytical model may lead to a bigger result; (2) The equivalent modulus \(E_{\rm c}\) may overestimate the stiffness of the assumed strand; (3) The strand of wire rope is not straight thus our assumption will certainly increase the stiffness of the strand. Based on the above analysis, it can be safely said that the proposed finite element model of the 6 \(\times\) K31 WS + FC wire rope is accurate, and it can be used in further analysis of the wire rope.