What the calculator does
In deep drawing a flat round blank is pulled by a punch through a die and becomes a cup. The sheet thickness changes only a little: the wall thins near the bottom radius and thickens towards the rim, and on average the thickness stays close to the original. The usual design rule follows from that: the surface area of the finished part, measured on the mean fibre (mid-thickness), equals the area of the blank. The blank diameter is the circle with that area, D = √(4A/π).
The calculator converts the drawing dimensions to the mean fibre, adds up the surface area of the bottom, the two radii, the wall and the flange, and gives the blank diameter. It then works out how many draws are needed with the draw ratio limits you set, the diameter and height after each draw, and the forces.
From drawing dimensions to the mean fibre
With thickness s, the mean wall diameter is d = outside diameter − s = inside diameter + s. The mean radius of a bend is its inside radius + s/2, so the bottom radius is R₁ = rp + s/2 and the flange radius R₂ = rd + s/2. With H measured outside, the straight part of the wall is
with flange: h = H − s − R₁ − R₂ without flange: h = H − s/2 − R₁ + trim allowance
The flat bottom has diameter d₁ = d − 2R₁, the flange starts at d₃ = d + 2R₂ and, with a trim allowance a on the radius, ends at d₄ = flange diameter + 2a.
Blank diameter formula
Each surface is written as its contribution to D². The radii are quarter tori; their area follows from Pappus’ theorem (arc length × path of its centroid):
| Surface | Contribution to D² |
|---|---|
| Flat bottom | d₁² |
| Bottom radius (centroid at d₁/2 + 2R₁/π) | 2π·R₁·d₁ + 8R₁² |
| Cylindrical wall | 4·d·h |
| Flange radius (centroid at d₃/2 − 2R₂/π) | 2π·R₂·d₃ − 8R₂² |
| Flange ring | d₄² − d₃² |
D = √(d₁² + 2πR₁d₁ + 8R₁² + 4dh + 2πR₂d₃ − 8R₂² + d₄² − d₃²)
Without radii and flange it reduces to the familiar D = √(d² + 4dh). When both radii are equal (R₁ = R₂ = R) the two radius terms add up to 2πR(d₁ + d) + 4πR², the form found in many die-design notes. As a check, the calculator also integrates the surface numerically along the profile and shows the difference, which should be essentially zero.
Draw ratio and number of draws
The draw ratio of an operation is m = dn/dn−1, the new mean diameter divided by the previous one (the blank for the first draw). Its inverse β = D/d is the drawing ratio. If the total ratio d/D is smaller than the first-draw limit m₁, one draw is not enough. The calculator then reduces the diameter step by step with m₁, m₂, m₃ … until it reaches the finished diameter, which gives the number of draws n. Because the last step would usually overshoot, all ratios are raised by the same factor k = (d/D ÷ m₁m₂…mₙ)1/n, so every draw stays within its limit and the last one ends exactly at d.
The height after each draw comes from the same area balance: with the diameter fixed, the wall height is whatever uses up the blank. For a cup with flange the flange is drawn to its full diameter in the first draw and then held, as is usual practice; the later draws only reduce the diameter and lengthen the wall.
Limiting ratios differ between handbooks and depend on the material, the relative thickness 100·s/D, the die and punch radii, lubrication and whether the part is annealed between draws. That is why every limit is editable. As a general first-draw check, Groover gives three limits for a cup drawn in one operation: drawing ratio D/Dp ≤ 2.0 (Dp is the punch diameter), reduction (D − Dp)/D below 0.5, and thickness above 1 % of the blank diameter. The calculator shows them for the finished cup.
Drawing force and blank-holder force
The drawing force uses the empirical formula from Groover, with punch diameter Dp (the inside diameter of the cup), thickness s and tensile strength Rm:
F = π · Dp · s · Rm · (D/Dp − c) c = 0.7
The constant c corrects for friction. Sharma’s textbook examples use 0.65 and 0.66; you can change it. The blank-holder force, also from Groover, uses the yield strength Re and the die radius rd:
Fh = 0.015 · Re · π · [D² − (Dp + 2.2s + 2rd)²]
The press must supply at least F + Fh for the first draw, plus the margin your shop uses. Both formulas are approximations for planning and press selection, not a simulation; the real blank-holder force is set on the press during tryout, just high enough to stop wrinkles. If the blank does not reach beyond the die radius, the holding formula has no area to act on and the calculator shows 0.
Worked example: DC04 cup with flange
A cup of DC04 steel, s = 1 mm, outside diameter 40 mm, outside height 30 mm to the top of the flange, inside bottom radius 4 mm, flange radius 5 mm on the die side, flange diameter 70 mm with a trim allowance of 2 mm on the radius. Rm = 310 N/mm², Re = 180 N/mm². Press Load worked example to see it.
- Mean fibre: d = 40 − 1 = 39 mm; R₁ = 4 + 0.5 = 4.5 mm; R₂ = 5 + 0.5 = 5.5 mm; h = 30 − 1 − 4.5 − 5.5 = 19 mm; d₁ = 39 − 9 = 30 mm; d₃ = 39 + 11 = 50 mm; d₄ = 70 + 2·2 = 74 mm.
- Terms: bottom 30² = 900; bottom radius 2π·4.5·30 + 8·4.5² = 848.23 + 162 = 1010.23; wall 4·39·19 = 2964; flange radius 2π·5.5·50 − 8·5.5² = 1727.88 − 242 = 1485.88; flange ring 74² − 50² = 2976 (all mm²).
- D² = 9336.11 mm², so D = 96.62 mm. Without the trim allowance it would be 93.60 mm. The blank area is 7332.56 mm²; the two flange terms take 48 % of it.
- Total ratio m = 39/96.62 = 0.404, relative thickness 100·1/96.62 = 1.03. With the limits 0.55, 0.75, 0.80: 96.62 → 53.14 → 39.86 → 31.89 mm, so three draws are needed (two would stop at 39.86 mm, just above 39).
- Even distribution: k = (0.4036/(0.55·0.75·0.80))1/3 = 1.069, giving m = 0.588, 0.802, 0.856 and mean diameters 56.83, 45.58 and 39.00 mm (outside 57.83, 46.58, 40.00 mm). With the 74 mm flange held from the first draw, the outside heights are 22.25, 26.43 and 30.00 mm.
- First draw force: F = π·55.83·1·310·(96.62/55.83 − 0.7) = 56.0 kN. Blank-holder force: Fh = 0.015·180·π·(96.62² − (55.83 + 2.2 + 10)²) = 39.9 kN. Press force for the first draw ≈ 96.0 kN before margin.
Groover’s single-draw check confirms the result: for the finished cup D/Dp = 96.62/38 = 2.54, above 2.0, so it cannot be drawn in one operation.
Common mistakes
Using the outside or inside diameter as if it were the mean
The area balance holds on the mean fibre. For a plain cup 40 mm outside diameter and 40 mm high in 2 mm sheet, using the outside dimensions gives a blank about 3.6 mm too large. Select which diameter you entered and the calculator converts it.
Forgetting the trim allowance
The rim of a drawn cup is never even: anisotropy of the sheet makes ears, typically four, and the top must be trimmed. Add an allowance on the height of a plain cup or on the flange radius. In the example it adds 3 mm to the blank diameter.
A die radius that is too small
A sharp die radius raises the drawing force and tears the bottom. Ramahi and Saleh give 4 to 8 times the thickness. A radius that is too large lets the sheet wrinkle where it leaves the blank holder.
Taking one handbook’s draw ratios as universal
Tables of limiting ratios differ by several hundredths between books, and the right value depends on material and tooling. Use the values you trust and check the first draw in tryout.
Ignoring the need for annealing
Each draw work-hardens the material. After two or three draws, steels and brass often need intermediate annealing before the next draw, and stainless steel hardens much faster. The calculator does not decide this for you.
Questions
Why does the calculator use the mean diameter for the draw ratio?
The area balance and the draw sequence are computed on the mean fibre, so the ratios are too. Some books use the punch diameter instead. For thin sheet the difference is small; Groover’s limits on the punch diameter are shown separately in the single-draw check.
Does a wide flange change the first-draw limit?
Yes. When much of the blank stays in the flange, less material is drawn into the die in the first draw, and handbooks allow a smaller first-draw ratio for wide-flange cups. Enter the value from your handbook as m1. The calculator keeps the flange diameter constant after the first draw. When the flange is too narrow for that, it says there is no valid plan: such cups are usually drawn as a plain cup and flanged in the last operation, so plan them without the flange.
Why is the blank-holder force so large compared with the drawing force?
Groover’s formula is written with π, not π/4, in front of the diameters, so it corresponds to a pressure of about 0.06·Re on the ring of the blank outside the die radius. Groover also notes that the holding force is usually about one-third of the drawing force, so the formula tends to give a high value. Lower the holding factor if your experience or your handbook gives a lower blank-holder pressure.
Can I use it for rectangular or conical parts?
No. This calculator covers cylindrical cups with a flat bottom, with or without a flat flange. Other shapes need their own area formulas.
How do I know the calculations are right?
The engine reproduces the printed results of four textbook and journal examples: the blank diameter, thickness ratio, first-draw depth and press capacity of Sharma’s example 2.15, the three draw heights of his example 2.16, the drawing force of example 2.9, and the 14 kN blank-holder force of Ramahi and Saleh. It also reproduces every number of the worked example above. The blank areas and the heights after each draw were checked against an independent Python calculation that integrates the surface numerically and solves each height by root finding, on 16 cups with and without flange, together with the draw plans and forces: the largest relative difference was below one part in a million million.
Is my data uploaded?
No. Everything is calculated in your browser. Your inputs are kept only in this browser’s local storage, and a share link carries them inside the link itself.
Related tools: the tolerance stack-up calculator for punch, die and part dimension chains, and the process capability calculator to judge measured cup heights or diameters from production.
References
- M. P. Groover, Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, Wiley, chapter on sheet metalworking (drawing ratio, reduction and thickness limits, drawing force and holding force).
- P. C. Sharma, A Textbook of Production Engineering, S. Chand, chapter 2 (press tool design), examples 2.2, 2.9, 2.15 and 2.16.
- Ramahi and Saleh, “Design, Fabrication and Experimentation of a Deep Drawing Machine”, Journal of Mechanics Engineering and Automation 4 (2014) 804–812.
- EN 10130 (cold-rolled low-carbon steel flat products for cold forming), EN 10088-2, EN 1652 and EN 485-2 for the ranges behind the typical material values.