## 3 Results and Discussion
### Effects of Length, Friction and Tension on the Deflection for Simple Strands
The other two coefficients hardly vary with \(\mu\). Figure 10: Comparison of simulation deflection curves and fitted curves for 6p-length strands with different friction coefficients (**a**) \(\mu=0\) and (**b**) \(\mu=0.2\); (**c**) the corresponding fitting coefficients of the simulation deflection curves. Figure 11: Three-dimensional plot of fitting coefficient of deflection curve for strands with different lengths: (**a**) \(\chi_{1}-\overline{L}_{m}-\zeta\) and (**b**) \(\chi_{2}-\overline{L}_{m}-\zeta\); (**c**) the slope and intercept of \(\chi_{i}-\zeta\) line for each strand with friction coefficients 0.2 and 5 kN in tension. Figure 12: Three-dimensional plot of fitting coefficient of deflection curve for strands with different friction coefficients: (**a**) \(\chi_{1}-\mu-\zeta\) and (**b**) \(\chi_{2}-\mu-\zeta\); (**c**) the slope and intercept of \(\chi_{i}-\zeta\) line for 12-pitch-long strand with 5 kN in tension. Finally, the effect of axial tension on the coefficient array is analyzed. The 3D plot of \(\chi_{1}-\zeta-\lambda_{p}\) exhibits a concave surface, but that of \(\chi_{2}-\zeta-\lambda_{p}\) appears a flat plane (as shown in Figure 13a,b). Figure 13c gives a plot of the variation of the coefficient array with nominal axial tension \(\lambda_{p}\). The slope of the fitting coefficient \(\chi_{1}\) with respect to \(\zeta\), namely \(k_{\chi_{1}}\), has a distinct downward tendency with the increasing of \(\lambda_{p}\). In fact, it is possible that there exists a couple of effects of tension and friction on the slope and intercepts due to the friction inside the strand being affected by these two factors. There is, therefore, a need to observe regularities for strands with different friction coefficients at different levels of tension force except for those discussed above; the results are presented in Figure 14a-f.  
For strands with different friction coefficients, the variation trend of the first two slopes with nominal tensile force remains almost unchanged, while the other two coefficients change greatly, especially for the last one. The last two coefficients keep constant and \(b_{\chi_{1}}\) approaches to 0 in cases where \(\lambda_{p}\geq 4\) and \(\mu\leq 2\), in particular. Moreover, it can be inferred from Figure 12c that \(b_{\chi_{2}}\) is proportional to \(\mu\). Consequently, the generalized fitting equation given in Equation (19) can be further simplified in the following form:  
\[\overline{Y}=\frac{\zeta}{2}(\frac{e^{-\lambda_{p}\overline{X}}-1}{\lambda_{ p}}+\overline{X})-k_{\mu}l\overline{X} \tag{33}\]  
where \(k_{\mu}\) is the proportionality factor of \(b_{\chi_{2}}\) with respect to \(\mu\). Figure 13: Three-dimensional plot of fitting coefficient of deflection curve for strands with different axial tension: (a) \(\chi_{1}-\zeta-\lambda_{p}\) and (b) \(\chi_{2}-\zeta-\lambda_{p}\); (c) the slope and intercept of \(\chi_{i}-\zeta\) line for 12-pitch-long strand with friction coefficient 0.2. For the present simple strand, the proportionality factor \(k_{\mu}=0.01\) and the friction coefficient \(\mu=0.115\).