Forcing a blank into a die cavity reduces the blank’s diameter and generates circumferential compressive stress as the blank deforms over the die’s lead-in radius. The greater the reduction in diameter, the higher the stress. The higher the stress, the greater the flow resistance. If flow resistance exceeds the blank’s tensile strength, the blank will stretch or tear near the nose of the punch.
To avoid this situation, the process must be designed to reduce the blank’s diameter by no more than the material can tolerate. This limiting draw ratio (LDR) varies based on material, thickness, and the number of times the part has already been drawn. Draw ratios typically are included with the raw material’s specifications in a draw reduction chart. If the reduction during a deep draw exceeds this limit, the part will need to be drawn in stages.
To illustrate this concept, we will step through the basic calculations to determine how many draws would be needed to produce a flangeless cylindrical cup that is 6 inches tall with a diameter of 4 inches. A cup with a retained flange or a more complex shape would require more complex calculations, but the underlying concepts remain the same.
First, calculate the blank size. Because sheet thickness remains essentially the same during deep drawing, the surface area of the finished part plus any residual flange material will be equal to the surface area of the blank. In our example of a cylindrical cup with no flange, we can calculate the required blank size with the following formula:
Where Rb = Radius of the blank, Rc = Radius of the cup, and H = height of the cup
For a 6″ by 4″ cup, this calculation returns a blank diameter of 10.58″.
Next, calculate the draw reduction needed to achieve our final part’s diameter. To do so, divide the part diameter by the blank diameter and subtract that number from 1:
Where Dc = Diameter of the cup and Db = Diameter of the blank.
The reduction for our example cup is about 62%. For ease of calculation, we will assume our blank’s LDR is 2.0 for the first draw, 1.5 for second, and 1.25 for the third. These translate to a 50% reduction for the first draw, 30% for second, and 20% for third. Because the cup requires more than 50% total reduction, the process will require more than one stage.
In the first stage, the cup diameter can be reduced 50%, resulting in an intermediate part diameter of 5.29″. In the second stage, the cup diameter can be reduced up to 30%. Drawing to that limit would result in a part diameter of 3.70″. This diameter is smaller than the specified 4″, so the cup can be safely be drawn in 2 stages.
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