## 2 Analytical Method
### Plastic Deformation  
Usually, the plastic flow of steels is concave down in both tension and compression loading. Generally, in order to capture the hardening behaviour of these materials, isotropic hardening rules, which can reproduce concave down curves, are applied. In this study, [PERSON] and Swift hardening laws were considered and are expressed in Equations (7) and (8), respectively.  
\[\sigma_{eq}^{t}=\sigma_{t}+B\Big{\{}1-exp\Big{(}-C\varepsilon_{eq}\Big{)}\Big{\}} \tag{7}\]  
\[\sigma_{eq}=K_{g}\Big{(}\varepsilon_{0}+\varepsilon_{eq}\Big{)}^{n} \tag{8}\]  
where \(\sigma_{t}\) is the tensile yield point and \(B\), \(C\), \(K_{s}\) and \(\varepsilon_{0}\) are material parameters determined from the tensile SS curve. According to Equations (4), (6), and (7), and considering \(1/\beta_{t}=\gamma_{t}\) and \(-1/\beta_{c}=\gamma_{c}\), the bending stress during plastic deformation based on Voce hardening can be defined as:  
\[\sigma_{b}=\sigma_{1}=\left\{\begin{array}{ll}\gamma_{t}\Big{(}\sigma_{t}+B \Big{\{}1-exp\Big{(}-C\gamma_{t}\Big{(}\frac{y-d}{R}-\gamma_{t}\frac{\sigma_{t }}{E^{t}}\Big{)}\Big{)}\Big{\}}\Big{)},&\sigma_{1}>0\\ \gamma_{c}\Big{(}\sigma_{t}+B\Big{\{}1-exp\Big{(}-C\gamma_{c}\Big{(}\frac{y-d }{R}-\gamma_{c}\frac{\sigma_{t}}{E^{t}}\Big{)}\Big{)}\Big{\}}\Big{)},&\sigma _{1}<0\end{array}\right. \tag{9}\]  
Similarly, the bending moment based on Swift hardening rule can be expressed as:  
\[\sigma_{b}=\sigma_{1}=\left\{\begin{array}{ll}\gamma_{t}K_{s}\Big{(} \varepsilon_{0}+\gamma_{t}\Big{(}\frac{y-d}{R}-\gamma_{t}\frac{\sigma_{t}}{E^ {t}}\Big{)}\Big{)}^{n},&\sigma_{1}>0\\ \gamma_{c}K_{s}\Big{(}\varepsilon_{0}+\gamma_{c}\Big{(}\frac{y-d}{R}-\gamma_{ c}\frac{\sigma_{t}}{E^{t}}\Big{)}\Big{)}^{n},&\sigma_{1}<0\end{array}\right. \tag{10}\]  
In order to write the equilibrium condition for a sheet under bending (Figure 1), the force acting on the strip with a unit width and thickness of \(dy\) is considered. Therefore, from the equilibrium condition, the tension \(T\) and the moment of the force element \(M\) can be written as:  
\[T=\int_{-t/2}^{t/2}\sigma_{1}dy \tag{11}\]  
\[M=\int_{-t/2}^{t/2}\sigma_{1}ydy \tag{12}\]  
where \(t\) is the thickness of the sheet during bending.  
Substituting Equation (9) into Equations (11) and (12) and considering the elastic and plastic deformations through the thickness the bending moment based on Voce hardening rule is defined as:  
\[\begin{split} M=\int_{y_{c}}^{y_{t}}E^{\prime}(\frac{y-d}{R})ydy &+\int_{y_{t}}^{t_{0}}\gamma_{t}(\sigma_{t}+B\Big{(}1-exp\Big{(}-C \gamma_{t}(\frac{y-d}{R}-\gamma_{t}\frac{\sigma_{t}}{E^{t}}\Big{)}\Big{)} \Big{)})ydy\\ &+\int_{-\frac{y_{t}}{2}}^{y_{c}}\gamma_{c}(\sigma_{t}+B\Big{(}1- exp\Big{(}-C\gamma_{c}(\frac{y-d}{R}-\gamma_{c}\frac{\sigma_{t}}{E^{t}}\Big{)} \Big{)}\Big{)})\end{split} \tag{13}\]  
where \(t_{0}\) is the initial sheet thickness and \(y_{t}\) and \(y_{c}\) are the coordinates from the mid-surface to the interface of elastic and plastic zones in tension and compression regions, respectively, given as:  
\[y_{t}=R\Big{(}\gamma_{t}\frac{\sigma_{t}}{E^{\prime}}\Big{)}+d \tag{14}\]  
\[y_{c}=R\Big{(}\gamma_{c}\frac{\sigma_{t}}{E^{\prime}}\Big{)}+d \tag{15}\]  
\[\begin{split} T&=E^{\prime}(y_{t}-y_{c})\Big{(}\frac{(y_{t} +y_{c}-2d)}{2K}+\varepsilon_{T}\Big{)}\\ &\quad+\frac{RK_{t}}{n+1}\Big{(}\big{(}\varepsilon_{0+\gamma} \gamma t\Big{(}\frac{t_{0}-2d}{2K}+\varepsilon_{T}-\gamma_{t}\frac{\sigma_{t} }{E^{\prime}}\Big{)}\Big{)}^{n+1}\\ &\quad\quad-\Big{(}\varepsilon_{0+\gamma}\gamma t\Big{(}\frac{y_ {t}-d}{K}+\varepsilon_{T}-\gamma_{t}\frac{\sigma_{t}}{E^{\prime}}\Big{)}\Big{)} ^{n+1}\\ &\quad+\frac{RK_{t}}{n+1}\Big{(}\Big{(}\varepsilon_{0}+\gamma c \Big{(}\frac{y_{t}-d}{K}+\varepsilon_{T}-\gamma_{t}\frac{\sigma_{t}}{E^{\prime }}\Big{)}\Big{)}^{n+1}\\ &\quad\quad-\Big{(}\varepsilon_{0}+\gamma c\Big{(}\frac{t_{0}- 2d}{2K}+\varepsilon_{T}-\gamma_{c}\frac{\sigma_{t}}{E^{\prime}}\Big{)}\Big{)} ^{n+1}\Big{)}\end{split} \tag{20}\]