## 5 Conclusions  
In this study, the contact pressure between parallel wires in the stay cable is classified into three types based on their sources: contact pressure caused by wire twisting, contact pressure caused by sheath compression, and contact pressure caused by cable bending. Theoretical derivation and finite element analysis of contact pressure calculation at the six contact points around an individual wire are conducted. A comparison with relevant existing research is performed to validate the effectiveness of PIC model. Finally, based on the theoretical derivation results and considering the influence of loads and cycle counts, a simulation analysis of fretting wear between two wires is conducted. The main conclusions are as follows. * For parallel wires, although the helix angle is relatively small, the contact pressure caused by twisting is still significant. Due to the different axial tension and twist angle, the magnitude of the contact pressure from twisting provided to the inner wire is mainly related to the position of the outer wire. The transmission of contact pressure has a cumulative effect, resulting in greater contact pressure for wires closer to the inner side. Within the same layer, the contact pressure is maximized at the points along the diagonal. * The distribution of contact pressure between wires caused by sheath temperature shrinkage can be calculated according to the Boussinesq distribution. This assumption is reasonable when not considering variations in force chains caused by manufacturing errors. The results of calculations under this assumption show that the contact pressure between wires exhibits distinct layering, which is consistent with the results of finite element simulations. * The contact pressure caused by the bending of the wire fluctuates with the variation in wire curvature during the vibration process of the stay cable and is also related to the axial tension of the wire. The cable exhibits distinct layering during the bending process, with relatively minor variations in contact pressure within the same layer. The wire curvature is the most significant factor influencing the contact pressure, showing an exponential relationship between the two. For cables with higher axial tension, the contact pressure is also greater, providing a lateral explanation for the delayed occurrence of slipping in stay cables with higher tensile forces. The location of the initial slipping within the cable cannot be determined to occur at the neutral axis or the outermost layer, owing to the different distributions of tangential force and interlayer frictional resistance within the stay cable. * Based on the existing calculation results, fretting wear between two wires was simulated in finite element analysis. According to [PERSON]'s theory of wear, it is possible to make certain estimations regarding the development of wear depth. Finite element analysis indicates that with an increase in cycle count, wear at the contact interface gradually becomes more uniform. The main reason for the averaging effect is the reduction in the disparity of contact pressure at various points. In addition, the width of the contact area at the interface increases in proportion to the increase in contact pressure. However, such an increase is not unlimited, as excessively high contact loads can change the contact mode from gross slip to partial slip, which complicates the wear analysis. Conceptualization, Z.Z. and T.F. ; methodology, Z.Z. and T.F. ; validation, T.F. ; writing--original draft preparation, T.F. ; writing--review and editing, Z.Z. and T.F. ; supervision, Z.Z. All authors have read and agreed to the published version of the manuscript. The authors acknowledge the support from the National Key R&D Program of China (grant No. 2020 YFA0711700), the National Natural Science Foundation of China (grant Nos. U23A20659,52122801, 11925206, 51978609 and U22A20254) and the Foundation for Distinguished Young Scientists of Zhejiang Province (grant No. LR20E080003).