## 1 Introduction  
As the primary load-bearing components of cable-stayed bridges, the durability of the stay cables directly influences the service life of the entire structure [1]. During the operation stage of a cable-stayed bridge, external loads induce vibrations in the stay cables,leading to relative movement among the internal wires and consequently giving rise to fretting wear [2]. This phenomenon results in a reduction in the strength and durability of the cables, markedly impacting the overall lifespan of the cable-stayed bridge. Assessing the degree of wire damage by studying the fretting wear behavior within the stay cable, and timely replacing stay cables with insufficient strength ensure the safety of the cable-stayed bridge during its operation stage. Fretting wear is influenced by various factors such as contact pressure, displacement amplitude, and motion frequency. Among these, contact pressure determines the size of the initial contact area, stress field, and tangential stiffness, etc. [3], making it a crucial parameter in studying the laws of fretting wear. Through experiments involving the wear of two wires, including both vertical point contact [4; 5; 6; 7] and parallel line contact [8; 9], it has been demonstrated that the contact pressure between wires not only affects the developmental patterns of friction coefficients and wear coefficients but also, with an increase in contact pressure, induces alterations in the sliding state of the wire contact interface, consequently further diminishing the service life of the cable [10]. In the actual operation of cable-stayed bridges, the dynamic monitoring of contact pressure proves challenging, often leading to an underestimation of the impact of fretting wear, particularly on cable damage, including corrosion [11; 12] and fatigue [13; 14]. Preliminary findings from relevant experimental studies have elucidated the influence of contact pressure between wires on fretting wear. The theoretical exploration of contact pressure between wires originated from the study of failure modes in wire ropes [15; 16; 17; 18; 19; 20; 21]. [PERSON] et al. [22], by examining the tensile condition and separation trend of wires during stretching and twisting, analyzed the wire contact patterns and derived the contact pressure between the core wire and helical wires. [PERSON] et al. [23], based on [PERSON]'s slender rod equilibrium principles, established a radial contact model for stranded cables, elucidating the impact of contact pressure on axial strain and torsional slip of the wires. When considering multi-layer scenarios, [PERSON] et al. [24], employing Hertz contact theory and contact mechanics, derived a theoretical model for interlayer contact forces. These theoretical models serve as crucial reference tools for computing the contact pressure in wire ropes. Unlike wire ropes, the application of a parallel wire cable system is more prevalent in cable-stayed bridges. Despite numerous theoretical models and experimental measurements [25; 26; 27; 28; 29; 30; 31; 32; 33; 34], the existing methods for calculating the internal contact pressure within parallel wire cable still exhibit certain limitations. This is primarily because most of the previous studies have focused solely on the local pressure distribution at the contact interface of two wires [25; 26; 27; 28; 29; 30; 31], neglecting an analysis of the transmission of contact pressure throughout the entire cable. They overlooked the cumulative effects of contact pressure from the outer layers to the inner layers. In practical parallel wire cables, there are often dozens or even hundreds of wires, with studies thus necessitating consideration of the mutual interactions resulting from their contacts. Furthermore, many studies have overlooked the radial compression effect caused by the cooling of the sheath during the production of the stay cable, as well as the wire extrusion effect due to cable vibration and bending during usage [32; 33; 34].