Question

The required diameter of the blank for the deep drawing of a cup of diameter $d$ and height $h$ is given by:

• A. $\sqrt{d^2 - 2dh}$
• B. $\sqrt{d^2 + 2dh}$
• C. $\sqrt{d^2 - 4dh}$
• D. $\sqrt{d^2 + 4dh}$

Hint:

For deep drawing cups, always apply $D = \sqrt{d^2 + 4dh}$ to find the required blank diameter assuming constant thickness and no material loss.

Answer 1

The Correct Option is D

In deep drawing operations, the blank diameter $D$ is calculated based on the principle of volume conservation.
The volume of the initial flat blank and the final drawn cup must be equal (neglecting thickness variation).
For a cylindrical cup of diameter $d$ and height $h$, the surface area of the side becomes part of the material from the blank.
The total area (and hence material volume) needed to form the base and side wall is: \[ \text{Blank area} = \pi \left( \frac{D}{2} \right)^2 = \pi \left( \frac{d}{2} \right)^2 + \pi d h \]
Solving this gives: \[ \frac{D^2}{4} = \frac{d^2}{4} + dh ⇒ D^2 = d^2 + 4dh ⇒ D = \sqrt{d^2 + 4dh} \]