Why is every gore a different pattern?

Because the diameter grows along the bend. On a constant-diameter bend the middle gores are all alike and come on one sheet cut s − 1 times; here each has its own size. That is why the gore count is capped at twelve — twelve gores is already thirteen sheets.

Does the assembly order matter?

Very much. The gores go in the order they come out, smallest to largest. Assembled out of order the bend does not close — and there is no telling from the patterns, because each one closes on its own. The printed sheet carries that warning.

How is the part built geometrically?

By inscribed spheres: one sphere at each joint, on the same tangent polygon the ordinary bend uses, with growing radii. Each gore is the frustum of the cone circumscribed about two neighbouring spheres — and it is that construction which makes the joints plane ellipses, which is what lets the mitre close.

Are the two diameters given the openings?

They are — circles perpendicular to the axis, which is what you measure and what meets the next pipe. Worth saying because it changes the arithmetic: with that definition the taper enters through the tangent of the cone's angle, not the sine, which is what you get if the diameters are taken at the joints.

Tapered Segmented Bend development

What it is

The mitred bend that also reduces: it goes in at one diameter and comes out at another, along the same arc. It is the part nobody wants to mark out by hand, because here the gores are ALL DIFFERENT — s + 1 patterns instead of the two a constant-diameter bend takes.

Where it is used

Flues and extraction ducts that taper as they rise, curved transitions in process piping, dust collection, fan outlets that change size through the bend.

Measurements the calculation needs

D1 Smaller diameter (inlet mouth)
D2 Larger diameter (outlet mouth)
H Bend radius (on the axis)
ANG Angle of the bend
S Number of gores
DV Divisions
THK Plate thickness

Worked example

The values the form comes pre-filled with, run through the tool itself:

Smaller diameter (inlet mouth)	300 mm
Larger diameter (outlet mouth)	500 mm
Bend radius (on the axis)	800 mm
Angle of the bend	90°
Number of gores	4
Divisions	32
Plate thickness	6 mm
The flat pattern fits a plate of	1490 × 579 mm
Flat patterns produced	5

Result of the example

Plate 1	941 × 358 mm
Plate 2	1089 × 516 mm
Plate 3	1237 × 547 mm
Plate 4	1384 × 579 mm
Plate 5	1490 × 490 mm

Tapered Segmented Bend flat pattern: drawing of the template with the worked example dimensions — Cut lines in purple, welds in green and reference lines in orange. Plate 1490 × 579 mm.

Assembly notes

The gores are ALL DIFFERENT and go in the order they come out, smallest to largest. Assembled out of order the bend does not close — and there is no way to tell from the patterns, because each one closes on its own.

Weld the sheets with the longitudinal seams 180° apart, not lined up. Lined up, the piece assembles into a different shape — and the longitudinal seam crosses the circumferential one.

Frequently asked questions

Why is every gore a different pattern?: Because the diameter grows along the bend. On a constant-diameter bend the middle gores are all alike and come on one sheet cut s − 1 times; here each has its own size. That is why the gore count is capped at twelve — twelve gores is already thirteen sheets.
Does the assembly order matter?: Very much. The gores go in the order they come out, smallest to largest. Assembled out of order the bend does not close — and there is no telling from the patterns, because each one closes on its own. The printed sheet carries that warning.
How is the part built geometrically?: By inscribed spheres: one sphere at each joint, on the same tangent polygon the ordinary bend uses, with growing radii. Each gore is the frustum of the cone circumscribed about two neighbouring spheres — and it is that construction which makes the joints plane ellipses, which is what lets the mitre close.
Are the two diameters given the openings?: They are — circles perpendicular to the axis, which is what you measure and what meets the next pipe. Worth saying because it changes the arithmetic: with that definition the taper enters through the tangent of the cone's angle, not the sine, which is what you get if the diameters are taken at the joints.

Tapered Segmented Bend dimension drawing: D1 (Smaller diameter (inlet mouth)), D2 (Larger diameter (outlet mouth)), H (Bend radius (on the axis)), ANG (Angle of the bend), S (Number of gores), DV (Divisions), THK (Plate thickness), marked on the part

Calculate a Tapered Segmented Bend

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