## 3 Internal Contact Pressure Model Validation
### Validation for Twist Model  
The contact pressure caused by twisting in parallel wire cables are smaller compared to that of spiral wires because of the smaller wire twist angle. However, they remain a significant factor that cannot be neglected when calculating contact pressure between the wires. The \(1\times 7\) wire model shows that plastic deformation begins to develop when the axial strain in the stay cable reaches 0.006, and the cable enters the plastic zone when the strain reaches 0.01. This study primarily focuses on the development of contact between parallel wires in the elastic zone. Therefore, the contact stress situations within the strain of 0.01 are compared with theoretical analysis. In the \(1\times 7\) wire model, the contact includes the contact between the central wire and the helical wire (WC contact) and the contact between helical wires (WW contact). In [PERSON]'s study of the tensile and bending behavior of seven-wire strands [26], a refined model of seven-wire strands was established, and the contact stress patterns between the strands were analyzed. To compare theoretical methods with the results from the finite element model, nodes on the central and helical wires were selected. The relationship between contact stress and wire tensile strain was plotted, and the results were compared with the theoretical approach, the finite element model in this study, and the results from the refined seven-wire model. As Figure 13 shows, the relationship between the contact stress of wires and tensile strain essentially follows the theoretical model during the elastic phase. Figure 13a illustrates the comparison between FEA (finite element analysis) and the PIC in this study, while Figure 13b illustrates the comparison between FEA in [PERSON]'s paper and PIC in this study. At lower tensile strains, the calculated value of the contact stress is greater than that of the finite element simulation because the torsion force is greater and the contact area is smaller. With increasing strain, the contact relationship between the wires gradually stabilizes, and the stress values obtained through PIC closely align with those of FEA, especially in the elastic phase of the wires, where the conformity is notable. During the elastic phase, the contact stress of WW contact is greater than that of WC contact. This is because at the initial tight contact of the strands, and as the tension is applied, surrounding wires come into contact due to the gripping effect, and consequently, the contact stress in the circumferential direction increases rapidly as tension develops. As Figure 14 shows, the distribution of contact pressure caused by the twisting of wires is illustrated, with the total cable force computed as 2296.3 kN. Figure 14a shows the hierarchical basis and numbering rules for the contact points of the wires, while Figure 14b shows the distribution of contact pressure between different layers, which generally exhibits an increasing trend toward the inner layers. Within the same layer, contact points along the diagonal yield the highest contact pressure. This is because contact points along the diagonal lie on the line connecting the upper layer wires and the central wire, facilitating the complete transmission of contact pressure  
Figure 14: Contact pressure distribution caused by twisting: (**a**) number of contact points of different layers; (**b**) the contact pressure caused by twisting; (**c**) distribution of contact pressure at contact point \(\langle 1\rangle\); (**d**) distribution of contact pressure at contact point \(\langle 2\rangle\). Figure 13: The contact pressure of wire due to twisting under different strains: (**a**) FEA and PIC comparison; (**b**) Yu-FEA and PIC comparison. compared to other contact points in the same layer that require reduction. Therefore, these diagonal contact points exhibit a relatively greater pressure.