Segmented (Lobster-Back) Elbow Calculator

A segmented elbow replaces a smooth bend with straight pieces joined at mitre cuts — the classic parallel-line development of sheet-metal pattern drafting, which Kittredge calls parallel forms or mitre cutting, with four- and five-piece elbows in Problems 41–42 (G. W. Kittredge, The New Metal Worker Pattern Book, 1901, pp. 132–134; public domain). Everything here follows from one fact: every joint plane passes through the bend axis.

The cut angle

With P pieces there are P − 1 joints, each turning the pipe by the same share of the bend Φ. A joint that turns the pipe by an angle is two cuts each tilted half of it, so every cut is α = Φ / (2 (P − 1)) from square.

Heel, throat and the flat pattern

Take one piece with its centreline square to the bend axis at the centreline radius R_c. Going round the pipe by an angle φ from the heel, a line along the pipe surface sits at R_c + R·cos φ from the bend axis. A plane through the bend axis tilted by α meets it at a distance h(φ) = (R_c + R·cos φ)·tan α from the square section. A middle piece runs from −h to +h, so it is 2(R_c + R)·tan α long at the heel and 2(R_c − R)·tan α at the throat. Unrolling the pipe is exact — arc length around it becomes distance along the paper — so the flat pattern is that curve plotted against R·φ, exactly π·D wide.

Why the pieces nest on one pipe

Each middle piece's centreline is tangent to the true bend at its middle, and each end piece's at its square end, so the pieces form a polygon around the bend that starts and ends at the same tangent points as a smooth elbow — the centre-to-end dimension is R_c·tan(Φ/2), the same as a smooth bend. Cut a straight pipe with planes tilted alternately one way and the other, turn every second piece half a turn, and the elbow assembles. The stock needed is the sum of the centreline lengths, and the cut list gives where each cut crosses two lines half a turn apart.

How this tool is checked

Hand-derived cases (above, and two more in the test suite), and invariants: the joints always add up to the bend, every pattern is π·D wide, and two neighbouring pieces — decoded from the flat pattern itself, then placed in 3D — meet exactly. A second implementation, written separately in Python, builds the elbow in 3D by rotating pieces about the bend axis and intersecting every surface line with every joint plane, without using the formula above; it must reproduce every length, the cut list and every vertex in the exported DXF, read back with ezdxf. It runs on every release.

Exact versus approximate

The mitre curve is solved analytically; each flat pattern is a sampled polyline of that curve, held within the maximum deviation you set.

What is not covered

Pressure design of mitre bends (a matter for your piping code, for example ASME B31.3 paragraph 304.2.3), weld preparation, kerf and shrinkage, and schemes with unequal pieces.