## 2 Materials and Methods
### Approximation of Tension for Individual Wire of the Simple Strand under Multi-Axial Loadings  
This section has focused on the evaluation of the tension of each wire based on the simulation results. As for a strand loaded with combinations of axial and lateral loads, the tension of each wire can be obtained by the linear superposition of two parts. One part is contributed by axial tension \(T\), and the other is caused by internal friction. \[F_{wi}=T_{wi}+f_{wi} \tag{23}\]  
in which \(T_{wi}\) is the tensile force acting on the individual wire prior to bending and \(f_{wi}\) is the additional tension force induced by friction. \(T_{wi}\) can be calculated if the strand is loaded by a tension \(T\). \[T_{wi}=\frac{A_{w}\cos^{2}\alpha}{mA_{w}\cos^{3}\alpha+A_{c}}T \tag{24}\]  
Figure 8a shows a helical wire \(w\) wound clockwise around a core wire with a rotation angular. The angular position \(\theta_{wi}\) is defined as the angle of the line connecting the centers of the core and the wire with respect to the x-axis of the global Cartesian coordinates. Figure 8: (**a**) The internal tensile force \(F_{wi}\) and its corresponding angular position \(\theta_{wi}\); (**b**) a force cycle (red) fits with the force group. For the node \(i\) of wire \(w\) in the present FE model, \(\theta_{wi}\) can be written as:  
\[\theta_{wi}=\theta_{i}+\theta_{w0} \tag{25}\]  
where \(\theta_{i}=\frac{(i-1)\pi\overline{1}}{36}\) and \(\theta_{w0}\) denotes the initial angular position of wire \(w\). Assume the relative position of the nodes on each cross-section remains unchanged during the bending, a force group \(\left(F_{wi},\theta_{wi}\right)\mid w=1,\ 2,\ \dots,\ 6\) is defined to describe the distributions of tension on the \(i\)th cross-section. \(F_{wi}\) is axial tension for node \(i\) of wire \(w\). The force group on the \(i\)th cross-section is drawn in Figure 8b. A force cycle is subsequently defined as \(T_{wi},\ r_{i},\ \varphi_{i}\) to fit the force group on each cross-section of the strand. \[\left(T_{wi},r_{i},\varphi_{i}\right)=\arg\min\sum_{w=1}^{6}T_{wi}{}^{2}-\left( F_{wi}\cos\theta_{wi}-r_{i}\cos\varphi_{i}\right)^{2}-\left(F_{wi}\sin \theta_{wi}-r_{i}\sin\varphi_{i}\right)^{2},\varphi_{i}\in\left[-\pi,\pi\right] \tag{26}\]  
\(r_{i}\) is the distance between the center and the coordinate origin, and \(\varphi_{i}\) is the angular position of the force center with reference to the x-axis which is limited in \(\left[-\pi,\pi\right]\). Once a sequence of parallel force circles along the strand axis is determined, the magnitude of tension force for each node with angular position \(\theta_{wi}\) is computed as follows,  
\[\hat{F}_{wi}=\sqrt{T_{wi}^{2}+r_{i}^{2}+2T_{wi}r_{i}\cos(\varphi_{i}-\varphi_{ i})} \tag{27}\]  
where \(\varphi_{i}=\theta_{wi}+\arcsin(\frac{r_{i}}{T_{wi}}\sin(\theta_{wi}-\varphi_{ i}))\). The center of each force cycle is denoted as \(r_{ix},r_{iy}\). Here, \(r_{ix}=r_{i}\cos\varphi_{i},r_{iy}=r_{i}\sin\varphi_{i}\). These centers form an analogous Archimedean spiral curve in 3-D space, which is induced by uneven tension distribution of outer layer wires around the core. This curve is found to be further fitted by the following equation set by a series of trial and error. \[\left\{ \begin{array}{l}r_{ix}=C_{1}e^{-C_{3}\theta_{i}}\cos\theta_{i}+C_{2} e^{-C_{3}\theta_{i}}\sin\theta_{i}+C_{4}\\ r_{iy}=-C_{2}e^{-C_{3}\theta_{i}}\cos\theta_{i}+C_{1}e^{-C_{3}\theta_{i}}\sin \theta_{i}\end{array} \right.. \tag{28}\]  
These unknown coefficients \(C_{i}\)\(\left(i=1,\ 2,\ 3,\ 4\right)\) are solved by fitting the \(r_{i}\) curve from force cycles of simulation results. They depend on friction coefficient \(\mu\), axial loading factor T and lateral loading factor \(\zeta\). There are in total 480 sets of sample data consisting of six levels in friction coefficient and eight levels in axial tension as well as 10 levels in lateral loading.