Deep drawing blank size, hydroforming, spinning

(a) Deep drawing calculation

Blank diameter from equal surface area of blank and cup (thickness constant, sharp corner):

$$D = \sqrt{d^2 + 4dh} = \sqrt{50^2 + 4\times50\times70} = \sqrt{16\,500} = 128.5\ \text{mm}$$

Draw ratio:

$$DR = \frac{D}{d} = \frac{128.5}{50} = 2.57$$

This is greater than the limiting draw ratio (LDR) of 2.0, so a single draw is not possible. First draw cup diameter $d_1 = D/2.0 = 64.2$ mm. The second draw (redraw) from 64.2 mm to 50 mm is a reduction of $1 - 50/64.2 = 22\%$, which is within the usual redraw limit of 25-30 percent. So two draws are needed.

Force of the first draw (approximate formula):

$$F = \pi d_1\, t\,(UTS)\left(\frac{D}{d_1} - 0.7\right) = \pi\times64.2\times1\times350\times(2.0-0.7) = 91.8\ \text{kN}$$

Blank-holder force is about one third of the drawing force, about 30.6 kN, to prevent wrinkling.

(b) Short notes

Hydroforming: a sheet or tube is pressed into a die by pressurised fluid (oil or water) instead of a rigid punch. In sheet hydroforming, a rubber diaphragm or fluid pressure forces the blank around the punch. In tube hydroforming, high internal pressure plus axial feeding expands the tube into the die (automobile frames, exhaust parts). It gives uniform thickness, complex shapes, less springback, and good finish, but the cycle time is longer and the equipment is costly.

Spinning: a flat disc or tube is clamped against a rotating mandrel and a roller or tool forces the metal over it. In conventional spinning the thickness stays nearly constant (cones, hemispheres, cookware, reflectors); in shear spinning the wall thins according to the sine law $t_f = t_0\sin\alpha$. Tooling is cheap and suits small batches of axisymmetric shapes.

(c) Formability assessment

• Tensile test: n-value (stretchability) and anisotropy ratio $r$ (deep drawability; larger $r$ is better).
• Limiting dome height / Erichsen cupping test: a punch stretches a clamped sheet until fracture; the depth indicates stretchability.
• Limiting draw ratio test (Swift cup test) for drawability.
• Forming limit diagram (FLD): plot of major strain against minor strain at which necking begins, obtained from grid-marked specimens; strains in the part must lie below the curve.

Answer: blank diameter $= 128.5$ mm, draw ratio $= 2.57$, two draws, first-draw force $= 91.8$ kN.