## 2 Materials and Methods
### Approximation of the Deflection of the Simple Strand under Multi-Axial Loadings  
A proper fitting function is in the first place proposed to approximate the deflection of the simple strand. It is essential to find out a suitable function that can match the deflection curves. A reasonable idea is that this function is analogical to the deflection formula relative to the linear bending stiffness. The deflection curve expression corresponding to the linear bending stiffness is, therefore, derived in terms of the free bending of the smooth simple strand under multi-axial loadings. Figure 7 illustrates the schematic of free bending. For simplification, only in-plane deformation is considered here. The bending moment is proportional to the curvature for strands with linear bending stiffness, the expression is, therefore, given:  
\[M=EI\cdot\kappa \tag{12}\]  
where \(M\) denotes the bending moment on the cross-section, \(EI\) denotes the linear bending stiffness, and \(\kappa\) denotes the in-plane curvature of the strand. Figure 7 shows the strand bends from a straight line to an arcuate one under the axial tensile force \(T\) and the lateral force \(P\) whose position is localized on the mid-span. The left end of the strand is constrained with all degrees of freedom, called the fixed-end, and the other end is only released x-translation degree of freedom, called the loading-end. Figure 6: A complete quasi-static FE simulation model for simple strand with length of \(L_{m}\) and loaded with \(T\) and \(P\). Figure 7: Schematic of free bending. Apparently, the deflection of the whole strand is axially symmetric about the vertical line passing through the lateral loading points. The following work, therefore, focuses on the left part of the whole strand. Additionally, the deflection curve of the left part as marked by 1 and 2 is centrosymmetric. A cartesian coordinate system xoy is established with the origin coinciding with the mid-point of the left span and the x-axis parallel with the direction of T. Another cartesian coordinate system XOY is fixed at the left end of the strand and its axis is parallel with xoy. The span of the deformed strand is \(L_{m}\), and the horizontal distance between the lateral loading point and the origin o is \(l\), whose value is a quarter of the total span. In the system xoy, the differential equation that defines the deflection for the beam-like strand under the free bending is given by:  
\[EI\cdot y^{\prime\prime}=Ty-\frac{p}{2}x \tag{13}\]  
where \(y^{\prime\prime}\) represents the curvature \(\kappa\) because these two are very close in value due to the small deformation assumption. The general analytical solution of Equation (13) is  
\[y=c_{1}\cosh(\lambda x)+c_{2}\sinh(\lambda x)+\frac{\zeta}{2}x \tag{14}\]  
where \(\lambda=\sqrt{\frac{\zeta}{EI}}\), \(\zeta\) denotes loading factor and \(\zeta=\frac{p}{T}\). The two unknown constants \(c_{1}\) and \(c_{2}\) are determined by the boundary conditions, which are \(y(0)=0\) and \(y^{\prime}(l)=0\),  
\[\left\{ \begin{array}{c}c_{1}=0\\ c_{2}=-\frac{\zeta}{2\lambda}\cosh(\lambda l)\end{array} \right. \tag{15}\]  
Therefore, the expression of the deflection is:  
\[y=-\frac{\zeta}{2\lambda}\frac{\sin\hslash(\lambda x)}{\cosh(\lambda l)}+ \frac{\zeta}{2}x \tag{16}\]  
If the original reference frame is replaced with another coordinate system XOY, there exists the following transformation rule:  
\[\left\{ \begin{array}{c}x=X-l\\ y=Y+y(-l)\end{array} \right. \tag{17}\]  
Substituting Equation (17) into Equation (16), the expression of the deflection in the coordinate system XOY is:  
\[Y=-\frac{\zeta(1-\tan\hslash(\lambda_{l}))}{4\lambda}e^{\lambda X}+\frac{ \zeta(1+\tan\hslash(\lambda_{l}))}{4\lambda}e^{-\lambda X}+\frac{\zeta}{2}X- \frac{\zeta\tanh\hslash(\lambda_{l})}{2\lambda} \tag{18}\]  
where \(\lambda_{l}\) represents the flexibility factor associated with bending stiffness and \(\lambda_{l}=\sqrt{\frac{\zeta l^{2}}{EI}}\).