The blank diameter is the first number that decides whether a deep drawn cup is cheap or expensive, because it sets how much sheet metal goes into the press and how much of it ends up on the floor as trimmed rim. It is also the number most often copied from a similar part instead of calculated. This guide covers the classical shop formula, why it overestimates, the surface-area method that gets close enough to quote from, and a worked example you can follow with your own drawing.
How do you calculate blank size for a deep drawn cup?
Blank size for a deep drawn cup is calculated by matching the blank's area to the finished part's surface area, because the material volume does not change during forming. Measure the part on its mid-surface — mean diameter, mean height, mean bottom radius — then add three areas: the flat bottom disc, the quarter torus at the bottom radius and the cylindrical wall. Solve D = sqrt(4A / pi) for the blank diameter and add a trim allowance of roughly 3 to 8 per cent. The classical shop approximation, D = sqrt(d2 + 4 d h), assumes zero wall thinning and a sharp bottom corner, so it overestimates the blank by around 3 per cent on a cup with a generous bottom radius. Where the wall is ironed, or thicker than about 3 per cent of the diameter, use volume constancy instead of area.
Why the blank diameter decides more than the material bill
The blank has to contain exactly enough material to form the finished cup plus a rim that will be trimmed. A blank that is too small starves the drawing operation: the wall thins below specification, the bottom radius cracks, or the part tears at the punch nose before the stroke finishes. A blank that is too large forces the tool to draw surplus material into the die, which raises blank-holder pressure, thickens the rim, increases earing and turns the excess into scrap. Both errors are expensive, but they fail differently — one is a quality problem, the other is a material-cost problem.
The classical formula every shop quotes
For a cylindrical cup without a flange, the traditional blank diameter is D = sqrt(d2 + 4 d h), where d is the cup diameter and h is the cup height. It is quick, it is in every handbook, and it is a first approximation rather than an answer. The formula assumes that the wall keeps the original sheet thickness everywhere and that the bottom corner has no radius, so it ignores both the material that the corner radius saves and the thinning that actually happens at the bottom radius.
It also has to be fed the right diameter. Using the outside diameter of the cup and the outside height, which is what a drawing usually shows, biases the result high by roughly half the sheet thickness on the diameter. Suppliers normally calculate on the mean diameter — outside diameter minus one wall thickness — and treat the result as the mid-surface blank, then add the thickness back if they are cutting a blank for the outside surface.
The surface-area method: close enough to quote from
Forming moves material, it does not create or destroy it, so the area of the blank equals the surface area of the finished part. For a cup with a flat bottom and a radiused corner, that surface has three parts, and adding them takes a minute:
- Bottom disc: area = pi × (d/2 − r)², where d is the mean cup diameter and r the mean bottom corner radius
- Bottom corner: a quarter torus, area = pi² × r × (d/2 − r)
- Cylindrical wall: area = pi × d × (h − r), where h is the mean cup height
- Blank diameter: add the three areas and solve D = sqrt(4 × A / pi)
All dimensions in those three lines are mean-surface dimensions, not drawing dimensions: mean diameter = outside diameter − one wall thickness, mean bottom radius = outside radius − half the wall thickness, mean height = outside height − half the wall thickness. On a thin blank the correction looks small, but on a 2 mm wall in a 40 mm cup it is worth several per cent of blank diameter, which is several per cent of the material bill.
Worked example: 50 mm cup, 40 mm deep, 1.0 mm DC04
Take a cylindrical housing with an outside diameter of 50 mm, an outside height of 40 mm, an outside bottom radius of 5 mm and a 1.0 mm wall in DC04. The mean-surface dimensions are a diameter of 49 mm, a bottom radius of 4.5 mm and a height of 39.5 mm. The three areas come to 1,257 mm² for the bottom disc, 888 mm² for the bottom corner and 5,388 mm² for the wall — 7,533 mm² in total. Solving D = sqrt(4 × A / pi) gives a mean-surface blank of 97.9 mm, so a blank cut from 1.0 mm sheet is about 99 mm across.
- Classical formula on the same mean dimensions: D = sqrt(49² + 4 × 49 × 39.5) = 100.7 mm, about 3 per cent larger than the area method because it ignores the bottom radius
- Classical formula on the drawing dimensions: 102.5 mm — the number a shop would get by reading the drawing literally, and the highest of the three
- With a 4 per cent trim allowance on the area result: about 103 mm, which is a sane blank to buy for a first tool
- Total reduction: roughly 2.0 against the 50 mm finished diameter, which sits right at the usual first-draw limit for DC04
That last line is the useful part of the exercise. A 50 mm diameter cup drawn 40 mm deep is not a comfortable single-draw part; it is a part that a DC04 tool can usually manage in one draw with a generous punch radius and good lubrication, and that 304 stainless will not manage in one draw at all, because 304 work-hardens and its practical first-draw limit is closer to 1.6 to 1.8. Blank size and draw ratio are calculated together, not separately.
Cups with a flange: where the blank grows
If the part keeps a flange instead of being trimmed to a plain rim, the flange is drawn material that stays in the part, so the blank has to cover it. The simplest accurate treatment is to add the flange as an annulus — area = pi × (Df² − d²) / 4, where Df is the outside diameter of the flange — to the cup surface area and solve for the blank again. A flanged housing can need a blank 10 to 25 per cent larger in diameter than the same cup with a trimmed rim, depending on flange width: a 10 mm flange on the 50 mm example above pushes the blank to roughly 110 mm. That is why flange width is one of the first questions a tooling engineer asks.
Trim allowance, earing and why the formula is not the final number
Rolled sheet is anisotropic: it flows more easily in one direction than another, so the rim of a drawn cup comes out wavy rather than level — earing. The standard response is to develop the blank slightly oversize and trim the rim in a later station, which is why a trim allowance of roughly 3 to 8 per cent is added to a calculated blank. The lower end suits thick blanks, deep-drawn quality steel and a well-controlled blank holder; the upper end suits thin blanks, stainless steel and a part where the rim has to be square within a tight tolerance in the first pass.
The calculated blank is therefore a starting point for the tool, not a shipping specification. The blank die is cut to the calculated size, the first trial draws are measured, and the blank or the trim station is adjusted until the rim cleans up in one trim and the wall thickness stays inside tolerance. A supplier who quotes a blank diameter without saying that the number will be developed on trial is quoting a spreadsheet, not a tool.
Thick walls and ironed parts: switch from area to volume
Area-equals-area holds only while the wall keeps its original thickness. As soon as the part is ironed — the classic route for solenoid and sensor housings where the drawn wall is reduced on purpose — the wall is thinner than the blank and the area method overstates the blank. The reliable version is volume constancy: calculate the volume of the finished part on its mean surface, add a trim allowance, and solve for the blank diameter from a blank of known thickness. Balford ironed housings and thick-wall motor cans are developed that way, because the wall reduction is a specified feature rather than an accident.
What blank size costs you
Sheet metal is priced by weight, so blank diameter is a direct multiplier on material cost, and it multiplies twice: once through the material that is consumed, and once through the strip width and nesting efficiency that the blank forces. A cup that needs a 103 mm blank instead of a 99 mm blank is about 8 per cent more material per part for the same finished housing. On a part running at a few million pieces a year, the difference between a calculated blank and a copied blank is a line item worth arguing about, and it is one of the reasons a DFM review should happen before the blank die is cut.
How Balford develops a blank
Balford deep draws up to 250 mm diameter in Zhuji, Zhejiang, on presses from 25 t to 350 t, with tooling designed and built in-house — which matters here because the blank diameter, the blank die, the trim station and the number of drawing stages are one decision, not four. We run DC01 and DC04 mild steel, 304 and 316L stainless steel, 65Mn spring steel, 5052 and 6061 aluminium, and copper and brass. Engineering calculates the blank from your drawing on the mean surface, checks it against the draw-ratio limit for the grade, and returns written DFM feedback before tooling is cut. Where a part needs interstage annealing, that heat treatment is carried out by qualified external partners and declared as outsourced rather than implied as an in-house operation.
Key point
Calculate the blank twice — once with the classical formula and once by surface area on the mean dimensions — and quote the difference. If the two numbers disagree by more than a few per cent, the bottom radius or the wall thickness is doing real work in your part, and that is exactly the detail the tooling review should settle before the blank die is made.
Frequently asked questions
How do you calculate blank size for a deep drawn cup?
Match the blank area to the surface area of the finished part measured on its mean surface: bottom disc, bottom corner radius and cylindrical wall. Add the three areas and solve D = sqrt(4A / pi), then add a trim allowance of about 3 to 8 per cent. The classical D = sqrt(d2 + 4 d h) formula is a quick first approximation that ignores the bottom radius and wall thinning.
Why does the formula give a larger blank than the real part needs?
Because it assumes a sharp bottom corner and a wall that keeps the full sheet thickness. A radiused bottom corner removes material from the calculation, and real drawing thins the wall slightly, so a cup with a generous radius typically needs a blank 2 to 4 per cent smaller than the classical result. The difference is small on thin blanks and significant on thick walls.
Should I add a trim allowance to the calculated blank?
Yes. Rolled sheet is anisotropic, so the drawn rim comes out eared rather than level and has to be trimmed. A trim allowance of about 3 to 8 per cent is normal: the low end for thick blanks and deep-drawing quality steel with a controlled blank holder, the high end for thin blanks, stainless steel or a rim that must be square in the first pass.
Does the blank size change when a part needs more than one draw?
No. The blank stays the same for the whole part, because it holds all the material the finished cup will ever contain. What changes between stages is the preform diameter — the size of the partially drawn cup after each stage. Multi-stage tooling reduces the same blank in steps, and interstage annealing is inserted where the material has work-hardened too far to keep drawing.