Deep Drawing Force (Swift)

The punch that draws a cup has to do two things at once: pull the flange in through the die, and carry that pull through the cup wall that has already formed. H. W. Swift's 1952 analysis produced the form still quoted everywhere — $F = \pi d t \cdot UTS \cdot (D/d - 0.7)$ — and it is worth reading as three pieces. $\pi d t$ is the cross-sectional area of the cup wall. $UTS$ is the strength of the material that area is made of. The bracket is a measure of how much flange has to be dragged in, scaled so that the whole thing comes out as a fraction of what the wall could carry before tearing.

The 0.7 is empirical. It is not derived from plasticity theory; it is a fitted allowance for the friction at the blankholder and over the die profile radius, and for the work of bending the material round that radius and straightening it again. Different sources carry it anywhere from 0.6 to 0.8. Treat the answer as an estimate with genuine width to it — good enough to size a press, not good enough to argue about a tenth of a ton.

Three more things stand between this number and what the machine has to deliver. This is the punch force only. The blankholder that stops the flange wrinkling needs its own force, commonly something like a quarter to a third of the drawing force, and the press has to supply both simultaneously — on a single-action press through a die cushion, on a double-action press through a separate slide. The force is not constant through the stroke. It climbs as the flange draws in and the material work-hardens, peaks partway down, then falls away as the flange runs out; this equation is aiming at that peak, which is what determines the press. And the peak moves with the lubricant, the blankholder pressure and the die radii, none of which appear here.

The most important caution is that force is not the failure criterion. A draw does not fail because the press ran out of tonnage. It fails because the flange demands more pull than the cup wall can carry, and the wall tears — almost always at the punch nose radius, where the wall is thinnest and the load is highest. Whether that happens is decided by the drawing ratio and the material's plastic strain ratio, not by the size of the machine. A thousand-ton press will tear a part drawn past its limiting ratio exactly as reliably as a hundred-ton one. Size the press from this equation; decide whether the part is makeable from the limiting drawing ratio.

Read backwards, the equation is more useful than it looks. Because $\pi d t \cdot UTS (D/d - 0.7)$ is just $\pi t \cdot UTS (D - 0.7d)$, the cup diameter comes out of a linear relation. And backing a strength out of a measured punch force, while a poor way to find out what a material is, is a good way to watch a die age: same tooling, same material, same lubricant, and a recovered figure that starts climbing is friction telling you the die needs attention.

Springback applies here as everywhere in forming. The cup relaxes when the punch withdraws, so the diameter you measure is not the punch diameter, and a deep cup can also distort as residual stresses redistribute after trimming.