## 4 Reinforcement Springback Compensation Method
### Solve the Springback Compensation Formula  
(1) Data collection. The bending angle of steel bars used in construction ranges from \(0^{\circ}\) to \(180^{\circ}\). In this paper, an integrated steel bar straightening and bending hoop machine is used to conduct  
Figure 17: Function graph. bending experiments of steel bars at 9 angles of 20\({}^{\circ}\), 40\({}^{\circ}\), 60\({}^{\circ}\), 80\({}^{\circ}\), 100\({}^{\circ}\), 120\({}^{\circ}\), 140\({}^{\circ}\), 160\({}^{\circ}\), and 180\({}^{\circ}\), respectively, to measure the springback angle. In order to ensure the accuracy of the parameters obtained, the springback angle data were collected 20 times for each angle, and the average value of the bounce angle measured 20 times was taken. The collected data is shown in Table 12.  
(2) Solve the fitting curve function. Matlab mathematical software was used to solve the problem, and the expressions of least square linear fitting function \(y_{1}(x)\), least square fitting function \(y_{2}(x)\), Newton interpolation polynomial \(P_{8}(x)\), and cubic spline interpolation function \(s(x)\) were, respectively,  
\[y_{1}(x)=-0.0174x+7.0069 \tag{9}\]  
\[y_{2}(x)=0.0001x^{2}-0.0369x+7.7214 \tag{10}\]  
\[ \begin{array}{ll}P_{8}(x)=&7.02-0.0195\times(x-20)-7.75\times 10^{-4}\times(x-20 )(x-40)\\ &+2.54\times 10^{-5}\times(x-20)(x-40)(x-60)+3.98\times 10^{-7}\times(x-20)\\ &(x-40)(x-60)(x-80)+\ldots+9.14\times 10^{-15}\times(x-20)(x-40)\ldots(x-180) \end{array}  \tag{11}\]  
\[ \begin{array}{ll}\left\{\begin{array}{ll}-0.003\frac{(x-20)^{3}}{6-20}+(7.02-0)\frac{(40-x)}{20}+(6.63-\frac{-0.003}{6}\cdot 20^{2})\frac{(x-20)}{20},\\ (20<x<40)\end{array}\right.\\ &-0.003\frac{(60-x)^{3}}{6-20}+0.0028\frac{(x-40)^{3}}{6-20}+(6.63-\frac{-0.003 }{6}\cdot 20^{2})\frac{(60-x)}{20}\\ &+(5.62-\frac{0.0028}{6-20}\cdot 20^{2})\frac{(x-40)}{20},(40<x<60)\\ &0.0028\frac{(80-x)^{3}}{6-20}+0.0007\frac{(x-60)^{3}}{6-20}+(5.62-\frac{0.00 28}{6}\cdot 20^{2})\frac{(80-x)}{20}\\ &+(5.21-\frac{0.0007}{20}\cdot 20^{2})\frac{(x-60)}{(60<x<80)}\\ &0.0007\frac{(100-x)^{3}}{6-20}-0.0015\frac{(x-80)^{3}}{6-20}+(5.21-\frac{0.00 07}{20}\cdot 20^{2})\frac{(100-x)}{20}\\ &+(5.09-\frac{0.0015}{6}\cdot 20^{2})\frac{(x-80)}{20},(80<x<100)\\ &-0.0015\frac{(120-x)^{3}}{6-20}+0.0015\frac{(x-100)^{3}}{6-20}+(5.09-\frac{ 0.0015}{6}\cdot 20^{2})\frac{(120-x)}{20}\\ &+(4.73-\frac{0.0015}{6-20}\cdot 20^{2})\frac{(x-100)}{20},(100<x<120)\\ &0.0015\frac{(140-x)^{3}}{6-20}-0.0009\frac{(x-120)^{3}}{6-20}+(4.73-\frac{0.0 015}{6}\cdot 20^{2})\frac{(140-x)}{20}\\ &+(4.62-\frac{0.0009}{6}\cdot 20^{2})\frac{(x-120)}{(20<x<140)}\\ &-0.0009\frac{(160-x)^{3}}{6-20}+0.0001\frac{(x-140)}{6-20}+(4.62-\frac{-0.00 09}{6}\cdot 20^{2})\frac{(160-x)}{20}\\ &+(4.38-\frac{0.0001}{6}\cdot 20^{2})\frac{(x-120)}{(140<x<160)}\\ &0.0001\frac{(180-x)^{3}}{6-20}+(4.38-\frac{0.0001}{6}\cdot 20^{2})\frac{(180-x)}{ 20}+(4.11-0)\frac{(x-160)}{20},\\ &(160<x<180)\end{array}  \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Serial** & \begin{tabular}{c} **Diameter of** \\ **Support Shift** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Diameter of** \\ **Crankshaft** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Turning Radius of** \\ **the Crankshaft** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Bending** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Measurement** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} &
\begin{tabular}{c} **Average** \\ **Springback** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} \\ \hline
1 & 20 & 40 & 80 & 20 & 12.98 & 7.02 \\
2 & 20 & 40 & 80 & 40 & 33.37 & 6.63 \\
3 & 20 & 40 & 80 & 60 & 54.38 & 5.62 \\
4 & 20 & 40 & 80 & 80 & 74.79 & 5.21 \\
5 & 20 & 40 & 80 & 100 & 94.91 & 5.09 \\
6 & 20 & 40 & 80 & 120 & 115.27 & 4.73 \\
7 & 20 & 40 & 80 & 140 & 135.38 & 4.62 \\
8 & 20 & 40 & 80 & 160 & 155.62 & 4.38 \\
9 & 20 & 40 & 80 & 180 & 175.89 & 4.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Steel bar bending angle and springback angle. (3) Draw the fitting curve.