## 3 Results and Discussion  
### Effects of Length, Friction and Tension on the Deflection for Simple Strands  
Given a loading condition that the axial tension \(T\) = 5 kN and lateral force \(P\) increasing from 0 to 500 N with an increment of 100 N, two simulations were conducted for two 6-pitch-long strands with different friction coefficients of 0 and 0.2, respectively. A comparison of simulation deflection curves to fitted curves over a quarter of the whole length for four cases in which loading factors \(\zeta=0.01,\ 0.04,\ 0.07\) and 0.1, respectively, see Figure 9(a),b. It shows that the fitting curves agree well with the simulation results for both strands. Figure 9(c) presents the fitting coefficients of the deflection curve under the different loading conditions for strands without friction and with friction coefficient \(\mu\) = 0.2, respectively. Obviously, it can be seen that there is a linear relationship between the coefficients and the loading factor \(\zeta\), no matter whether the strand is smooth or not. Therefore, the coefficients of the fitting function for the deflection curve can be further expressed with respect to \(\zeta\). \[\chi_{i}=k_{\chi_{i}}\zeta+b_{\chi_{i}},i=1,2 \tag{32}\]  
where \(k_{\chi_{i}}\) and \(b_{\chi_{i}}\) denote the slope and intercept of the \(\chi_{i}-\zeta\) lines, respectively. Similarly, the coefficients can be solved by linear fitting the \(\chi_{i}-\zeta\) lines. The effects of several factors on the bending deformation of the strand are studied. These factors are strand length, friction coefficient, axial tension and lateral force. Nominal strand length \(\overline{L}_{m}\) ranges from 1 to 36, frictional coefficient changes from 0 to 0.3 with an increment of 0.05, and axial tension increases from 1 kN to 20 kN. The nominal lateral loading factor \(\zeta\) varies from 0.01 to 0.1. Table 3 lists the factors and levels for the parametric analysis on the effect of lateral deformation of spiral strands under free bending. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\(P\) = 4 kg} & \multicolumn{2}{c}{\(P\) = 8 kg} & \multicolumn{2}{c}{\(P\) = 12 kg} & \multicolumn{2}{c}{\(P\) = 16 kg} & \multicolumn{2}{c}{\(P\) = 20 kg} \\ \hline
**T/kg** & \(y_{max}\)/mm & **T/kg** & \(y_{max}\)/mm & **T/kg** & \(y_{max}\)/mm & **T/kg** & \(y_{max}\)/mm & **T/kg** & \(y_{max}\)/mm \\ \hline
102 & 5.74 & 100 & 11.97 & 101 & 18.57 & 99.5 & 24.79 & 101 & 30.57 \\
127 & 4.79 & 123 & 10.21 & 126 & 15.33 & 125 & 20.64 & 126 & 25.70 \\
149 & 4.18 &Figure 11a,b illustrates 3D plots of the fitting coefficients \(\chi_{1}\) and \(\chi_{2}\) of the general deflection curve equation for strands with different lengths at a constant axial tension of 5 kN and varying lateral loadings, respectively. These two coefficients are sensitive to strand length in cases \(\overline{L}_{m}<3\), but the lines gradually tend to stabilize after the strand length exceeds six pitches. It indicates the effects of boundary condition on deflection gradually declines as the length increases. Figure 11c presents the slope and intercept defined in Equation (32) for each strand with a specified length. It is found that both the slope and intercept of the fitting coefficients of the deflection curve almost keep constant as the strand length exceeds 6 pitches. Figure 12a,b displays 3D plots of \(\chi_{1}\) and \(\chi_{2}\) for strands with different friction coefficients ranging from 0 to 0.3. The nominal strand length is specified as 12 pitches, as mentioned before, this strand length has little effect on the \(\chi_{i}\). The axial tension is set to 5 kN. From the figures, it can be seen that these three variables, which are \(\chi_{i}\), \(\zeta\) and \(\mu\), show a linear relationship with each other. The slope and intercept of the fitting coefficients are further obtained from the 3D plot, see Figure 12c. The parameter \(b_{\chi_{2}}\) of the coefficient array, changes most significantly with an increased friction coefficient as a slight variation as \(\mu\) increases.