## 5 Finite Element Analysis of a 6 \(\times\) K31 WS + FC Wire Rope with Broken Wires
The 12 scenarios are (a) intact wire rope; (b) the wire that bears the maximum stress in scenario (a) is broken; (c) the core wire of the strand is broken; (d) one wire in the outer layer near the rope core is broken; (e) two wires in the outer layer near the rope core are broken; (f) two wires in the outer layer away from the rope core are broken; (g) three wires in the outer layer near the rope core are broken; (h) three wires in the outer layer away from the rope core are broken; (i) five wires in the outer layer near the rope core are broken; (j) five wires in the outer layer away from the rope core are broken; (k) nine wires in the outer layer near the rope core are broken; (l) nine wires in the outer layer away from the rope core are broken. These 12 scenarios are clearly demonstrated in Figure 16.  
Figure 16: The 12 scenarios of wire breakage. Figure 15: Finite element model of a 6 \(\times\) K31 WS + FC wire rope with a broken wire. After careful observation of Figure 17, while the wire rope is intact as depicted in Figure 16a, the maximum stress \(\sigma^{\rm a}\)max = 658.02 MPa, and it is located at the wire marked by a red circle. According to the simulation results, the maximum stress cross-section appears at both ends of the wire rope in all 12 cases. This is consistent with the facts that wire breakage always occurs at both ends of the wire rope during actual operation. In Figure 16b,c, only one wire is broken, and the broken wire shares the same radius \(R\) = 0.85 mm, which denotes that the decrease in the cross-section area of the wire rope is 0.45%. The corresponding maximum stress are \(\sigma^{\rm b}\)max = 663.10 MPa and \(\sigma^{\rm c}\)max = 678.33 MPa, respectively. The maximum stress shown in Figure 17 is increased by 0.77% and 3.09% when compared with \(\sigma^{\rm a}\)max. Apparently, the location of the broken wire has a dramatic impact on the maximum stress. Figure 21: Stress distribution at the end of a wire rope with a tensile force of 30 kN in cases (a,b). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Serial**} & \multirow{2}{*}{**Number of**} & \multirow{2}{*}{**Cross-Section**} & \multirow{2}{*}{**Maximum**} & \multirow{2}{*}{**Maximum**} & \multirow{2}{*}{**Minimum**} & **Stress** & **Maximum Stress** \\  & & & & & & **in Middle** & **in Middle** \\
**Number** & & & & & & **Cross-Section** & **Cross-Section** \\  & & & & & **(MPa)** & **(Increase)** \\ \hline a & 0 & 0 & 658.02 & / & 311.73 & / \\ b & 1 & 0.45\% & 663.10 & 0.77\% & 305.52 & \(-\)1.99\% \\ c & 1 & 0.45\% & 678.33 & 3.09\% & 317.49 & 1.85\% \\ d & 1 & 0.78\% & 665.61 & 1.15\% & 311.26 & \(-\)0.15\% \\ e & 2 & 1.57\% & 673.79 & 2.40\% & 311.40 & \(-\)0.11\% \\ f & 2 & 1.57\% & 686.33 & 4.30\% & 336.29 & 7.88\% \\ g & 3 & 2.35\% & 680.40 & 3.40\% & 308.11 & \(-\)1.16\% \\ h & 3 & 2.35\% & 722.20 & 9.75\% & 374.09 & 20.00\% \\ i & 5 & 3.92\% & 695.98 & 5.77\% & 324.33 & 4.04\% \\ j & 5 & 3.92\% & 760.30 & 15.54\% & 400.13 & 28.36\% \\ k & 9 & 7.06\% & 771.56 & 17.26\% & 399.52 & 28.16\% \\ l & 9 & 7.06\% & 796.71 & 21.08\% & 420.77 & 34.98\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The cross-section area and corresponding maximum stress in different cases. Figure 20: Stress distribution at the end of a wire rope with a tensile force of 30 kN in cases (a,b). In Figure 16d, one wire \(R\) = 1.35 mm is broken and the maximum stress \(\sigma^{\rm d}_{\rm max}\) = 665.61 MPa, which is larger than \(\sigma^{\rm a}_{\rm max}\) by 1.15%. The increases in maximum stress due to the broken wire in cases (c) and (d) are 3.09% and 1.15%, while the decreases in the cross-section area are 0.45% and 0.78%. This is more evidence that the location of wire breakage is influential to the maximum stress of the wire rope. In addition, it seems that the breakage of a strand core wire could have a more dramatic influence on the maximum stress of the rope than if it was another wire in the strand.