Rolling is not bending
On a press brake the material is worked hardest at one line and the neutral axis creeps toward the inside of the bend, which is why K-factors of 0.4 and below are normal there. Rolling works the whole wall through a gentle continuous curve, and the neutral surface stays essentially at mid-wall. That is the reason a rolled shell is developed on the mean diameter rather than the outside.
The size of the error is exactly pi times the wall thickness, whatever the diameter. On 16 gauge that is 0.19 inches, which a fitter will pull together and never mention. On quarter inch plate it is 0.785 inches of unwanted overlap at the seam on every shell, regardless of whether the shell is a foot across or ten feet.
| Shell | Blank on the mean diameter | Blank on the outside diameter | Error |
|---|---|---|---|
| 12 in OD, 0.0598 wall | 37.511 in | 37.699 in | 0.188 in |
| 12 in OD, 0.25 wall | 36.914 in | 37.699 in | 0.785 in |
| 48 in OD, 0.25 wall | 150.011 in | 150.796 in | 0.785 in |
Developing a cone
A cone unrolls into a flat sector of an annulus, and three numbers describe it: an outer radius, an inner radius and an included angle. The outer radius is the slant distance from the imaginary apex out to the large end. The inner radius is that same distance to the small end, which is the outer radius minus the slant height of the piece you are actually making. And the included angle is whatever makes the outer arc come out equal to the circumference of the large end.
Written out: the slant height is the square root of the vertical height squared plus half the diameter difference squared. The outer radius is the slant height times the large diameter divided by the difference of the diameters. The inner radius is the outer radius less the slant height. The included angle in degrees is 180 times the difference of the diameters divided by the slant height.
Two checks fall out of it and both are worth doing. The outer arc, which is the angle in radians times the outer radius, must equal pi times the large diameter. The inner arc must equal pi times the small diameter. If either fails, something has been transposed. The calculator prints both alongside what they should be.
Where the material goes
A cone development is a curved shape and it nests badly. The bounding rectangle of a typical transition is a good deal larger than the sector itself, and the middle of the ring, the part inside the small radius, is scrap unless something else fits there. Two cones of the same size nest reasonably well head to tail; a cone and a rectangular blank do not. Worth knowing before quoting a job on square feet of sheet, because the sheet consumed and the metal in the part can differ by half.
When the development is larger than the stock, splitting it into equal segments is the standard move. Each segment is the same annular sector at the same two radii with the angle divided by however many pieces, and every extra segment adds a longitudinal seam. Shallow cones, where the included angle approaches a full circle, are usually split anyway because rolling a nearly flat ring is awkward and because the offcut in the middle is enormous.
What this does not develop
Everything here assumes a right cone: both ends round, both ends parallel, and the axis square to both. Real ductwork and pipework are full of things that are not that. An eccentric reducer, where one side stays flat, is an oblique cone and every element line down its surface is a different length, so the development is triangulated rather than swept. A round to rectangular transition is the same problem with more faces. Those are laid out element by element, which is a genuinely different exercise, and doing it with the formula on this page produces a part that does not close.
Straight rectangular duct take-off, seams and joints are covered by the duct fabrication take-off calculator, and sizing the duct in the first place by the duct size calculator.
Practical layout notes
Scribing a large radius needs a beam compass or a trammel and a centre point that is often well off the sheet. For a shallow cone the outer radius can be several times the length of the blank, so the usual method is to lay the arc out by offsets, computing the rise at intervals along a chord, or to let the cutting machine draw it. Both arcs share a centre, which is the one thing that makes the layout self-checking: if the two arcs are not concentric, the seam will not close.
For cost and weight of the stock itself rather than the blank, the metal weight calculator covers sheet, plate and tube, and the nesting calculator handles rectangular blanks on a sheet.
Questions people ask
What length blank do I need to roll a cylinder?
Pi times the mean diameter, which is the outside diameter less one wall thickness. Rolling curves the whole wall rather than concentrating the work at one line, so the neutral surface sits at mid-wall and the mean diameter is what the blank has to match. Using the outside diameter makes every blank long by pi times the wall thickness, which is about three quarters of an inch on quarter inch plate no matter how big the shell is.
How do I lay out a cone flat pattern?
Work the slant height first: the square root of the vertical height squared plus half the difference between the diameters squared. The outer radius of the development is the slant height times the large diameter divided by the difference between the diameters. The inner radius is that minus the slant height. The included angle in degrees is 180 times the difference between the diameters divided by the slant height. Draw both arcs from one centre, and check that the outer arc length equals pi times the large diameter before you cut anything.
Why does my rolled cylinder come out too big?
Nearly always because the blank was cut to pi times the outside diameter instead of the mean. The other candidates are a K-factor assumption that does not match what the machine does, and the flat ends that a plate roll leaves where the material has not yet been gripped by all three rolls. That last one is a machine characteristic rather than a layout mistake, and it is dealt with by pre-bending the ends, using a starter strip, or trimming after rolling.
Can this lay out an eccentric reducer?
No, and the difference matters. This develops a right cone, meaning both ends are round and parallel and the axis is square to both, which unrolls into a simple annular sector. An eccentric reducer keeps one side of the cone straight, so it is an oblique cone whose surface has a different element length at every station around the circumference. That gets laid out by triangulation, station by station, and a sector development will not close on it.
What K-factor should I use for rolling?
Half, unless your own results say otherwise. Rolling distributes the deformation across the whole wall instead of concentrating it, so the neutral surface stays near mid-thickness and 0.5 is the working assumption almost everywhere. Press brake values of 0.33 to 0.46 belong to a different process and do not transfer. If rolled shells consistently come out a little large in your shop, trimming K down slightly shortens the blank; that is a calibration against your own machine rather than a property of the metal.