# Design of a Combined Redrawing-Ironing Process to Manufacture a CNG Pressure Vessel Liner
## 2 Design of a Combined and Redrawing-Ironing Process
### New Combined Redrawing-Ironing Process
#### 2.2.1 Force Analysis  
A new theoretical formula to obtain the combined redrawing-ironing load was suggested by adding the drawing load (\(F_{d}\)) to the ironing load (\(F_{i}\)), which is the sum of the tensile force acting on the workpiece (\(F_{1}\)) and the load acting on the punch (\(F_{2}\)). Figure 4 shows stresses in a slab element of a workpiece during an ironing process with force calculated from Equation (15).  
\[\begin{array}{c}\sum F_{z}=2\pi r(h+dh)(\sigma_{z}+d\sigma_{z})-2\pi rh \sigma_{z}\\ +[\tau_{1}(2\pi r)d_{z}+\tau_{2}(2\pi r)d_{z}]+\\ [p(2\pi r)tana-p(2\pi r)tan\beta]=0\end{array} \tag{15}\]By applying the plane strain yielding condition (\(Y^{\prime}\)), which does not take into account thickness deformation in the radial direction, Equation (15) can be expressed as Equation (16).  
\[\frac{dh}{h}=\frac{d\sigma_{z}}{B\sigma_{z}-Y^{\prime}(1+B)} \tag{16}\]  
where \(B=\frac{\mu_{1}-\mu_{2}}{tana-tan}\), \(Y^{\prime}=\frac{2}{\sqrt{3}}\).  
By integration of Equation (16) with the boundary condition of no back tension, the ironing stress in the z-direction (\(\sigma_{z}\)) is rewritten as Equation (17). The sum of the tensile force (\(F_{1}\)) is calculated by multiplying the ironing stress by the cross-sectional area of the product obtained after ironing, as shown in Equation (18). \(D_{f}\) is the diameter of the workpiece of the neutral plane after ironing, and \(h_{f}\) is the workpiece thickness after ironing.  
\[\sigma_{z}=Y\frac{1+B}{B}\Bigg{[}1-\left(\frac{h_{f}}{h_{o}}\right)^{B}\Bigg{]} \tag{17}\]  
\[F_{1}=\pi D_{f}h_{f}Y\frac{1+B}{B}\Bigg{[}1-\left(\frac{h_{f}}{h_{o}}\right)^ {B}\Bigg{]} \tag{18}\]  
The punch load (\(F_{2}\)) is calculated by adding the force from the pressure of the die to the force from the shear stress, as shown in Equation (19).  
\[F_{2}=\int_{0}^{1}D_{p}tan\beta\;d_{z}+\int_{0}^{1}\pi D\mu_{2}p\;d_{z} \tag{19}\]  
It can be assumed that \(\mu_{1}\mu_{2}\) and \(\beta=0\) in the actual process [15], so the ironing load is expressed as Equation (20), where \(D_{f}\) and \(R_{f}\) are the diameter on the neutral surface and radius on the neutral surface after ironing, respectively.  
\[\begin{gathered} F_{i}=F_{1}+F_{2}\\ =\pi D_{f}^{\prime}h_{f}Y^{\prime}\frac{1+B}{B}\Bigg{[}1-\left( \frac{h_{o}}{h_{f}}\right)^{B}\Bigg{]}+\pi D_{f}^{\prime}C\int_{h_{f}}^{h_{o} }\;pdh\\ =\pi D_{f}^{\prime}h_{f}Y^{\prime}ln\frac{h_{o}}{h_{f}}+\pi D_{f} ^{\prime}C\int_{h_{f}}^{h_{o}}\;pdh\\ =2\pi R_{f}^{\prime}h_{f}Y^{\prime}\big{(}1+\frac{\mu}{tana}\big{)} ln\frac{h_{o}}{h_{f}}\end{gathered} \tag{20}\]  
where \(C=\frac{tan\beta+\mu_{2}}{tana-tan\beta}=\frac{H}{tana}\).  
By adopting \(R_{f}^{\prime}=R_{f}+h_{f}/2\), where \(R_{f}\) is the inside radius of the product after ironing, the ironing load (\(F_{i}\)) can be rewritten as Equation (20):  
\[F_{i}=F_{1}+F_{2}=\pi\Big{(}2R_{f}+h_{f}\Big{)}h_{f}Y^{\prime}\Big{(}1+\frac{ \mu}{tana}\Big{)}ln\frac{h_{o}}{h_{f}} \tag{21}\]  
Figure 4: Stresses in a slab element during ironing process.  
The existing drawing load in the second stage is based on an empirical equation with constant thickness, as shown in Equation (22a) [15], while the new combined redrawing-ironing process has thickness reduction (\(h_{o}\to h_{f}\)), as shown in Figure 5. The drawing load was modified as shown in Equation (22b). The combined redrawing-ironing load was derived by adding the ironing load (Equation (21) to the drawing load (Equation (22b)), as shown in Equation (23), where \(k\) is the thickness reduction ratio (\(k=h_{f}/h_{o}\)).  
\[F_{d,\ original}=2\pi R_{f}h_{o}(UTS)\left(\frac{R_{o}}{R_{f}}-0.7\right) \tag{22a}\] \[F_{d} =2\pi\left(R_{f}+\frac{h_{f}}{2}\right)h_{o}(UTS)\left(\frac{R_{o}+\frac{h_{ f}}{2}}{R_{f}+\frac{h_{f}}{2}}-0.7\right)\] (22b) \[F_{t}=F_{i}+F_{d} =\pi D_{f}h_{o}\left[(UTS)\left(\frac{D_{o}}{D_{f}}-0.7\right)-kY ^{\prime}\left(1+\frac{\mu}{tan\alpha}\right)\ln k\right] \tag{23}\]