## 5 Finite Element Analysis of a 6 \(\times\) K31 WS + FC Wire Rope with Broken Wires  
The main damage patterns of a wire rope in working conditions are wire breakage, wear, rust, and kink [32], of which wire breakage is the most common form of wire rope failure [33]. [PERSON] and [PERSON] [34] studied the influence of broken wires on the mechanical characteristics of 6 \(\times\) 7, 7 \(\times\) 7, and 8 \(\times\) 7 wire ropes. However, they did not provide further analysis of wire ropes with more complex structures, which are made of more than 100 wires. More importantly, a wire rope with a complex structure and composed of a large number of wires is at a greater risk of wire breakage. However, there are few studies focusing on this issue, which is vital to ensure its safe operation and maintenance. Wire ropes mainly bear tensile loads, and broken wires can no longer bear any tensile loads. Thus, the broken wires could be omitted when carrying out finite element analysis. Figure 15 shows the finite element model of a 6 \(\times\) K31 WS + FC wire rope with a broken wire in the center of the strand. This simplification is conventional because scholars adopted the same methodology when studying wire breakage problems [34]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Serial Number of** & **Bending Moment** & **Axial Torque \(H_{i}\)** & **Shearing Force \(N_{i}\)** & **Axial Force \(T_{i}\)** & **Uniformly** \\
**Wires** & **G\({}_{i}\) (N\(\cdot\)m)** & **(N\(\cdot\)m)** & **(N)** & **(N)** & **Distributed Radial** \\  & & & & & Load \(X_{i}\) (N/m)** \\ \hline III & \(-2.61\times 10^{-3}\) & \(-1.82\times 10^{-2}\) & \(4.48\times 10^{-2}\) & \(4.55\times 10^{3}\) & \(-5.60\times 10^{3}\) \\ IV & \(-2.22\times 10^{-3}\) & \(-8.15\times 10^{-3}\) & \(3.81\times 10^{-2}\) & \(3.06\times 10^{3}\) & \(-7.05\times 10^{3}\) \\ V & \(-4.56\times 10^{-3}\) & \(-2.24\times 10^{-2}\) & \(7.82\times 10^{-2}\) & \(5.06\times 10^{3}\) & \(-8.78\times 10^{3}\) \\ VI & \(-7.30\times 10^{-3}\) & \(-6.40\times 10^{-2}\) & \(-3.35\times 10^{-2}\) & \(8.61\times 10^{3}\) & \(-2.94\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Mechanical responses of wires in each layer of the strand. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Serial Number of Wires** & **Axial Stress \(T_{ei}\)** & **Normal Stress \(\mathcal{G}_{gi}\)** & **Shear Stress \(H_{ei}\)** & **Equivalent Stress** \\  & **(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline II & 1794.98 & \(\backslash\) & \(-0.013\) & 1794.98 \\ III & 1789.67 & \(-4.56\) & \(-15.91\) & 1785.22 \\ IV & 1776.21 & \(-6.98\) & \(-12.81\) & 1769.30 \\ V & 1784.39 & \(-6.77\) & \(-16.65\) & 1777.72 \\ VI & 1753.45 & \(-4.76\) & \(-20.87\) & 1748.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Stress of wires in each layer of strand. In this part, the mechanical response of a 6 \(\times\) K31 WS + FC wire rope with broken wires is thoroughly investigated by adopting the model proposed in Section 3. A tensile load of 30 kN is applied on one end of the rope, and the other end of the rope is fastened reliably. The other setups of the finite element model are identical as stated in Section 3. Then, 12 scenarios with different broken wire numbers and locations have been carefully studied, and the initial broken wire is determined according to the stress distribution of the intact wire rope.