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Materials processing and joining

Ignoring changes in thickness and the radius at the bottom corner, a cylindrical cup with a bottom, 50 mm in diameter and 40 mm high, is made from a circular blank in a single deep-drawing operation. Assuming the surface area does not change, which value is closest to the diameter of the blank?

  1. AAbout 81 mm
  2. BAbout 102 mm
  3. CAbout 130 mm
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Answer: B. About 102 mm

The surface area of the cup is the sum of the bottom area πd²/4 and the side area πdh. Setting this equal to the blank area πD²/4 gives D² = d² + 4dh = 50² + 4 × 50 × 40 = 2500 + 8000 = 10500, so D ≈ 102 mm. About 130 mm is d + 2h, obtained by simply folding the side wall outward flat, which ignores the circumferential contraction of the outer part of the blank as it is drawn in; about 81 mm, √(d² + 2dh), halves the side area by taking it as πrh (r being the radius) instead of 2πrh. The ratio of blank diameter to cup diameter (the drawing ratio) is about 2.05, and since the limiting drawing ratio of steel sheet is roughly 2 to 2.2 (depending on the r-value and other factors), a single draw is close to the limit, and in practice the work is sometimes split with redrawing.

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