## 5 Finite Element Analysis of a 6 \(\times\) K31 WS + FC Wire Rope with Broken Wires
However, the wires in the outer layer of a strand are more susceptible to damage, such as striking, scraping, rusting, etc., by interacting with other components or the environment. Thus, the influence of the number and position of the broken wires in the outer layer of a strand on the wire rope's performance is thoroughly studied. Figure 16e,f demonstrates the stress distribution when two wires of a strand are broken. The maximum stresses shown in Figure 18 are \(\sigma^{\rm e}_{\rm max}\) = 673.79 MPa and \(\sigma^{\rm f}_{\rm max}\) = 686.33 MPa, the corresponding increases in the maximum stress are 2.40% and 4.30%, while the decrease in the cross-section area is 1.57%. Figure 16g,h demonstrates the stress distribution when three wires of a strand are broken. The maximum stresses shown in Figure 19 are \(\sigma^{\rm g}_{\rm max}\) = 680.40 MPa and \(\sigma^{\rm h}_{\rm max}\) = 722.20 MPa, the corresponding increases in maximum stress are 3.40% and 9.75%, while the decrease in the cross-section area is 2.35%. Figure 16i,j demonstrates the stress distribution when five wires of a strand are broken. The maximum stresses shown in Figure 20 are \(\sigma^{\rm i}_{\rm max}\) = 695.98 MPa and \(\sigma^{\rm i}_{\rm max}\) = 760.30 MPa, the corresponding increases in the maximum stress are 5.77% and 15.54%, while the decrease in the cross-section area is 3.92%. Figure 16k,l demonstrates the stress distribution when nine wires of a strand are broken. The maximum stresses shown in Figure 21 are \(\sigma^{\rm k}_{\rm max}\) = 771.56 MPa and \(\sigma^{\rm i}_{\rm max}\) = 796.71 MPa, the corresponding increases in maximum stress are 17.26% and 21.08%, while the decrease in the cross-section area is 7.06%. The simulation results are also listed in Table 9.  
In Figures 17-21, the maximum stress seems to always appear adjacent to the strand with a broken wire. In addition, there is a strong correlation between the decrease in the cross-section area of the wire rope (due to wire breakage) and the increase in the maximum stress of the wire rope. If the cross-section area of the wire rope decreases, there will be a remarkable increase in the maximum stress of the wire rope. For example, in case (j), a decrease in the cross-section area of 3.92% leads to an increase in the maximum stress of 15.54%. This indicates that wire breakage has a critical influence on the maximum stress of the wire rope in operation. Moreover, the breakage of the wire away from the core in the outer layer of the strand has a greater effect on the maximum stress of the wire rope than the breakage of the wire near the core. Figure 22 displays the maximum stress and cross-section area of the wire rope along different wire-breaking conditions. Figure 23 shows the stress distribution in the middle cross-section of the wire rope under 12 scenarios and the details are listed in Table 9. In the middle of the wire rope, the maximum stress is located at either the core or the inner wire of the strand, and the maximum stress in the middle of the wire rope is about half of that at the end of the wire rope. As the number of broken wires increases, the maximum stress of the middle cross-section also increases, and it most likely appears adjacent to the strand with the broken wire. It seems that the bearing capacity of the strand with broken wires gradually decreases, and it is shown in Figure 23 that the stress of the remaining wires in the strand with broken wires is less than that of the wires in the same position in the intact strands. Figure 23i,j demonstrates that the bearing capacity of the wire rope will greatly decrease when the number of broken wires in a strand reaches five and the stress distribution of the wire rope is no longer symmetrical, which means that the excellent bearing capacity is damaged.