# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 5 Modelling and Calculation of Residual Stress  
The total residual stress of the sheet after levelling consists of two parts: One is the rolling residual stress caused by multi-roll levelling, which is distributed over the entire thickness direction of the sheet and the other is the remaining part of the initial residual stress, which is only distributed in the elastic region of the first valid bending. The calculation model of the rolling residual stress is shown in Figure 6. According to the evolution rule of the residual stress of the sheet in multi-roll levelling, the distribution of the final rolling residual stress after the \(m\)-th valid bending is the region surrounded by \(OO_{1}\cdots O_{m}A_{c}^{(N+m)^{\prime}}\) and \(OA_{c}^{N+m}\). For the convenience of calculation, the abscissa and the ordinate are divided by \(\varepsilon_{e}\) and \(H\)/2, respectively, with the coordinate interchanging. The residual stress distribution after coordinate interchange is shown in Figure 6, where \(OO_{1}\cdots O_{m}O_{m+1}\) (which should be \(OO_{1}\cdots O_{m}A_{c}^{(N+m)^{\prime}}\) according to the naming rules in Figure 4, and \(O_{m+1}\) is an alias of \(A_{c}^{(N+m)^{\prime}}\) for modelling uniformity) represents the initial position and \(OP_{m+1}\) (which should be \(OA_{c}^{N+m}\) and \(P_{m+1}\) is an alias of \(A_{c}^{N+m}\)) represents the actual stress distribution after the last bending and springback. The sheet after levelling is nearly flat, so the strain at point \(P_{m+1}\) is close to zero. If the sheet is completely flat, \(P_{m+1}\) is on the abscissa axis. Since the plate thickness has been normalised, the absolute mean value \(\overline{|\sigma_{r}|}\) of the residual stress is the product of the shaded area in Figure 6 and the stress yield limit \(\sigma_{s}\), that is:  
\[\overline{|\sigma_{r}|}=\frac{1}{H}\int_{-H/2}^{H/2}\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
The residual curvature after levelling is calculated by the bending moment curvature model, so \(OP_{m+1}\) is determined, but the coordinates of \(O_{i}\) need to be further calculated.  
According to Figure 4, the line equation of the strain line corresponding to the curvature of the \(i\)-th valid bending in Figure 6 is:  
\[y=k_{N+i}x \tag{5}\]  
where,  
\[k_{N+i}=\frac{A_{N+1}}{A_{t}} \tag{6}\]  
The line \(O_{i}A_{c}^{(N+i)^{\prime}}\) (i.e., \(O_{i}O_{i+1}\)) is obtained by translating the line \(OA_{N+i}\) in the \(y\) direction with distance 1, and the equation is:  
\[y=k_{N+i}x+(-1)^{N+i+1} \tag{7}\]  
When \(i\geq 2\), \(O_{i}\) is the intersection of line \(O_{i}A_{c}^{(N+i)^{\prime}}\) and line \(O_{i+1}A_{c}^{(N+i+1)^{\prime}}\), and Equation (8) is obtained according to Equation (7):  
\[\left\{ \begin{array}{l}y=k_{N+i}x+(-1)^{N+i+1}\\ y=k_{N+i+1}x+(-1)^{N+i}\end{array} \right. \tag{8}\]  
The solution to Equation (8) is the coordinate \((x_{i},y_{i})\) of \(O_{i}\), which is solved as follows:  
\[\left\{ \begin{array}{l}x_{i}=(-1)^{N+i+1}\frac{2}{k_{N+i+1}-k_{N+i}}\\ y_{i}=(-1)^{N+i+1}\frac{k_{N+i+1}+k_{N+i}}{k_{N+i+1}-k_{N+i}}\end{array} \right. \tag{9}\]  
When \(i=1\), \(O_{1}\) is on the line \(OA_{c}^{0}\) and at the abscissa position of \(O_{1}\), the difference between line \(OA_{c}^{0}\) and line \(OA_{N+1}\) is exactly 1. Then,  
\[\left\{ \begin{array}{l}y_{1}=k_{0}x_{1}\\ y_{1}+(-1)^{N+1}=k_{1}x_{1}\end{array} \right. \tag{10}\]  
The equation for line \(OP_{m+1}\) is \(y=k_{n}x\) and \(Q_{i}\) is the foot of \(O_{i}\) to line \(OP_{m+1}\). The coordinate \(\left(x_{i}^{q},y_{i}^{q}\right)\) of point \(Q_{i}\) is:  
\[\left\{ \begin{array}{l}x_{i}^{q}=\frac{k_{n}y_{i}+x_{i}}{k_{n}^{2}+1}\\ y_{i}^{q}=\frac{k_{n}^{2}y_{i}+k_{n}x_{i}}{k_{n}^{2}+1}\end{array} \right. \tag{11}\]  
Figure 6: Distribution of the residual stress after multi-roll levelling.