Eccentric reducer layout by triangulation
An eccentric reducer joins two pipes or ducts of different sizes with their centres offset, usually so that one side is flat (flat on top in pump suction lines so no air pocket forms, flat on the bottom on pipe racks and drains). Its flat pattern is not a cone pattern with arcs: it is developed by triangulation, from a table of true lengths. This guide works one through from start to finish.
Why the cone formulas do not work
A concentric reducer is part of a right cone, so every line from the small end to the large end has the same length and the pattern is an annular sector (see the cone flat pattern formula). Offset the small end and those lines change length all the way round: on the flat side the line is vertical and as short as the reducer is long; on the opposite side it is the longest. The surface is an oblique cone, and its development has no single radius. Triangulation handles it exactly, up to how finely you divide the circles.
The flat-side (or "flat back") offset is half the difference in diameters: o = (D − d) ÷ 2. For a 500 mm to 300 mm reducer that is 100 mm. A smaller offset gives a reducer that slopes on both sides but less on one; a larger one makes the small end overhang.
Triangulation, step by step
- Divide both circles into the same number of equal parts, numbered from the seam. 12 is a practical minimum by hand; the calculator's table offers 12 to 32. Station 0 is on the flat (short) side here.
- Find each element's plan length. In the plan view, measure from each station on the large circle to the same-numbered station on the small circle. Also measure each diagonal: from station k on the small circle to station k + 1 on the large one.
- Convert to true lengths. Every line rises by the reducer's length H, so true length = √(H² + plan length²). This is the step a table saves you.
- Lay the triangles down in order. Draw element 0 at its true length. From its small-end point swing the diagonal, and from its large-end point swing one large-end division (the chord between stations); where the arcs cross is station 1 on the large end. Then swing element 1 from there and one small-end chord from the previous small-end point. Repeat round to the seam.
- Join the points with smooth curves (a thin batten), and add seam and end allowances.
Worked example: 500 to 300 mm flat-back reducer, 400 mm long
Two lines can be checked by hand. On the flat side the large-end and small-end stations sit directly above each other (plan length 0), so element 0 is exactly H = 400 mm. On the opposite side the large end's edge is R = 250 mm from the large end's centre, while the small end's edge is only r − o = 150 − 100 = 50 mm past it (its centre moved 100 mm towards the flat side). The plan length is 250 − 50 = 200 mm, so the longest element is √(400² + 200²) = 447.2 mm. The calculator gives the rest:
| Station | Angle | Element (small end to large end) | Diagonal to next station |
|---|---|---|---|
| 0 | 0.0° | 400.0 | 420.4 |
| 1 | 30.0° | 403.3 | 437.1 |
| 2 | 60.0° | 412.3 | 452.8 |
| 3 | 90.0° | 424.3 | 463.7 |
| 4 | 120.0° | 435.9 | 467.3 |
| 5 | 150.0° | 444.2 | 462.7 |
| 6 | 180.0° | 447.2 | 450.9 |
| 7 | 210.0° | 444.2 | 434.8 |
| 8 | 240.0° | 435.9 | 418.4 |
| 9 | 270.0° | 424.3 | 406.3 |
| 10 | 300.0° | 412.3 | 402.2 |
| 11 | 330.0° | 403.3 | 407.5 |
| Chord between stations, large end | 129.4 |
| Chord between stations, small end | 77.6 |
| Shortest element | 400.0 |
| Longest element | 447.2 |
The chords are the straight distances between neighbouring stations, a little shorter than the arc π × D ÷ 12. Neighbouring elements of an oblique cone meet at its apex, so each pair lies in one plane and the points you step out are exactly on the developed edges; only the edge between two points is unknown. Draw a smooth curve through the points rather than straight lines, and the edge length comes out right. The calculator divides the circles finely enough that its template's edges match the circumferences to a few thousandths of a millimetre.
- cut line
- roll guide lines
Change the sizes, the offset or the seam side and the table and pattern update. You can print the template full size on A4 or Letter paper or download a DXF for plasma or laser cutting.
Open this eccentric reducer in the calculatorWhich side should the seam go on?
On the flat (short) side the seam is the shortest possible weld and stays on the straight line, which is easy to fit up and keeps the seam out of the sloping face. On the long side the pattern is more compact on the sheet for some proportions. The calculator offers both and shows how the outline changes.
Checks before cutting
- The large-end edge of the template should measure π × D = 1570.8 mm along the curve, and the small end π × d = 942.5 mm.
- The seam edges must be the same length as each other (here, both are element 0, 400.0 mm).
- For plate, develop on the mean diameter (mid-thickness), as for any rolled shape. Rolling an eccentric reducer also needs the roll guide lines kept parallel to the rolls; they are drawn on the template.
Related shapes
Square or rectangle to round transitions use the same triangulation idea, with flat triangles at the corners: see how to lay out a square to round. If the offset is 0, use the ordinary cone formulas; the calculator switches between the two methods automatically.