Conical elbow calculator (reducing segmented elbow)
Enter the large and small diameter, bend radius, elbow angle and number of pieces. You get the pieces of a reducing segmented elbow, all cut from one truncated cone.
Enter your measurements
Seam
7,429.8mm
Exact geometry
- Cut length
- In the app
- Sheet size
- In the app
Every result is in Trueflat
The free web version shows the main result and the drawing. Trueflat gives you every mark and measurement, and exports a cut sheet. It is free to download.
How to lay it out
Draw the cone pattern
Swing the outer and inner arcs of one truncated cone from a single centre and mark its sector angle between two radius lines.
Divide the arc into generators
Divide the outer arc into equal parts and draw a straight line from each division to the centre. These are the cone's generators.
Mark the miter points
On every generator, measure from the centre the distance to each miter of that piece, from the calculated pattern.
Join the miter curves
Connect the points of each miter with a smooth curve. Each piece is the strip between two neighbouring curves; the seam sits on the throat side.
Cut, roll and turn
Cut the pieces apart and roll them. When you fit them, turn every second piece 180° on its miter before you weld, so the elbow bends and keeps reducing.
Good to know
- FabLayout runs the same code as Trueflat; the result is exact geometry.
- Source: No published source. One truncated cone is cut by planes that bisect the angle between neighbouring piece axes and every second piece is turned 180° on its plane (cut and turn); each piece is developed in closed form on the middle of the sheet. Checked against an independent 3D reference implementation in Trueflat.
Method and formula
A reducing elbow cannot be built from separate cones whose axes follow a bent line: two such cones do not meet on the bisecting plane. Trueflat uses the cut-and-turn method instead. One truncated cone runs from D1 to D2, it is cut by tilted miter planes, and every second piece is turned 180° about its miter plane.
- Miter half-angle β = α ÷ (2N), with N = pieces − 1 and τ = tan β.
- Cone height Lz = 2NRτ. The miter planes cross the axis at z = Rτ(2j − 1).
- Taper k = (r1 − r2) ÷ Lz and ε = kτ. At girth angle θ (θ = 0 on the throat) a lower miter point sits at radius ρ ÷ (1 + ε cos θ) and an upper one at ρ ÷ (1 − ε cos θ), where ρ is the cone radius where that plane crosses the axis.
- Unrolled, each point lies at ρ ÷ sin β_c from the apex, and the sector angle is Ψ = 360° × sin β_c.
The pattern is exact. Because the ellipses on each miter are symmetric about their centres, the turned pieces meet curve to curve and the two end axes sit exactly α apart. The axis steps slightly at each joint; that is part of the method, not an error. With sheet thickness, inside or outside diameters are converted to the sheet middle and the miter planes stay where they are.
Worked example
D1 = 200 mm, D2 = 100 mm, R = 200 mm, α = 90°, 2 pieces: N = 1, β = 45°, τ = 1, so Lz = 400 mm, k = 0.125 and ε = 0.125. On the throat line the first miter sits at z = 200 − 75 ÷ 0.875 = 114.29 mm, radius 85.71 mm. The cone angle β_c = 7.125°, so the sector angle Ψ = 44.65°. The shortest generator is 115.18 mm.
Questions
How is this different from a segmented elbow?
A segmented elbow keeps one diameter all the way round. A conical elbow reduces from D1 to D2 while it turns, so the pieces are slices of a cone instead of a cylinder. Use it where an elbow and a reducer would otherwise sit back to back.
Why turn every second piece?
If you weld the slices as they come off the cone, the part stays straight. Turning a piece 180° on its miter flips the slope of the cut, and the joint now bends by 2β. The miter curve is an ellipse that is symmetric about its centre, so it still meets its neighbour exactly.
What limits the bend radius?
R must be at least D1 and at most 100 × D1. Below D1 the inner generators get too short to cut and weld. Any piece with a generator under 5 mm is rejected.
Similar calculators
More in TrueflatConeTruncated coneCone by triangulationSee what the app does