## 3 Results and Discussion
### An Analytical Model from [PERSON]'s Theory and Comparision Analysis
\[M=(6 EI_{w}\cos\alpha+EI_{c})\kappa+\sum_{i=1}^{6}T(\theta_{w})R_{h}\sin\theta _{w}\cos\alpha \tag{47}\]  
where \(M\) is the bending moment on a specified cross-section of the strand, \(EI_{w}\) denotes the bending stiffness of the outside wire, \(EI_{c}\) denotes the bending stiffness of the core wire. The bending moment \(M\) can be assumed to be a function with respect to the curvature \(\kappa\) as given by:  
\[M=f(\kappa) \tag{48}\]  
Assuming that the continuous strand is divided into \(N\)-1 segments, an orthogonal coordinate system \(\alpha xy\) is defined as shown in Figure 17. Note, that the origin of the coordinate is located at the point where the bending moment is equal to zero. The deflection  
Figure 16: An arbitrary cross-section of the single-layered strand and the angular position for each wire cross-section: (**a**) the initial angular position of wire 1; (**b**) the definition of the angular position. for the deformed strand is expressed as \(\mathbf{y}=\{y_{1},y_{2},\cdots,y_{N}\}^{T}\) and then, the bending moment \(M_{i}\) at \(x_{i}\) is given by:  
\[M_{i}=Ty_{i}-0.5 Px_{i} \tag{49}\]  
Only the small deformation is considered here, the corresponding curvature \(\kappa_{i}\) is, therefore, approximated by the central difference method:  
\[\kappa_{i}=\frac{y_{i-1}+y_{i+1}-2y_{i}}{h^{2}} \tag{50}\]  
where \(h=\frac{L_{m}}{N-1}\). Since both two ends of the strand are clamped, the boundary conditions are:  
\[y^{\prime}(x_{1})=y^{\prime}(x_{N})=0 \tag{51}\]  
An N-dimensional set of equations \(\mathbf{F}(\mathbf{y})\) with respect to \(\mathbf{y}\) is obtained,  
\[F_{i}=\left\{ \begin{array}{c}1.5y_{1}-2y_{2}+0.5y_{3},i=1;\\ y_{i-1}+y_{i+1}-2y_{i}-h^{2}f^{-1}(M_{i}),i=2\sim N-1;\\ 1.5y_{N}-2y_{N-1}+0.5y_{N-2},i=N.\end{array} \right. \tag{52}\]  
The true deflection \(\mathbf{y}^{*}\) is satisfied with the following expression:  
\[\mathbf{F}(\mathbf{y}^{*})=\mathbf{0} \tag{53}\]  
[PERSON]'s iteration method is applied to solve the above equations, the criterion is expressed as follows and the iteration termination condition is \(\parallel\mathbf{y}^{(k+1)}-\mathbf{y}^{(k)}\parallel\leq 1\times 10^{-6}\). \[\left\{ \begin{array}{c}\mathbf{y}^{(k+1)}=\mathbf{y}^{(k)}-\mathbf{G}^{-1}\Big{(}\mathbf{y}^ {(k)}\Big{)}\mathbf{F}\Big{(}\mathbf{y}^{(k)}\Big{)}\\ \mathbf{y}^{(0)}=\mathbf{y}_{0}\end{array} \right. \tag{54}\]  
where \(\mathbf{G}\) is the Jacobian Matrix and its component \(G_{ij}=\frac{\partial F_{i}}{\partial y_{j}}\), \(\mathbf{y}_{0}\) is the deflection corresponding to the minimum bending stiffness \(EI_{min}\), and \(k\) denotes the \(k\)-th iteration. Figure 18 compares the predicted values to analytical values associated with minimum bending stiffness and theoretical values obtained from [PERSON]'s model as well as the experimental measurements. The results demonstrate that the proposed model is quite satisfactory to estimate the bending deflections for frictional simple strands subjected to  
Figure 17: Differential schematic for the left half-span of the free bending strand. multi-axial loads. Figure 19 presents the relative error between the results from the present model and the experimental measurements. It can be seen that the maximum error is less than 10%. Figure 18: Comparisons of deflection obtained from the present model, the theoretical values corresponding to the minimum bending stiffness, the theoretical study of [PERSON] and experimental measurements for simple strand subjected to a variable \(T\) ranging from 1 kN to 5 kN and a lateral concentrate force: (**a**) \(P=39.2\) N (4 kg); (**b**) \(P=78.4\) N (8 kg); (**c**) \(P=117.6\) N (12 kg); (**d**) \(P=156.8\) N (16 kg); (**e**) \(P=196\)N (20 kg). Figure 21a compares the predicted values of \(r_{x}\) and \(r_{y}\) to that of simulation results for the case where \(\mu=0.1\) and \(T=5\) kN as well as \(\zeta=0.1\). The results show that the variation regularity of projections of the summation of tension force of outer wires on the global x-axis and y-axis is well predicted.