How to lay out a lobster-back elbow by hand

What you are making

The worked example: a four-piece 90° elbow in 168.3 pipe (6-inch), centreline radius 1.5 × diameter = 252.45, no straight tangents. For a template wrapped on pipe, develop on the outside diameter; for gores rolled from sheet, on the mean diameter.

1. The angle bookkeeping

Count each end piece as one half-segment and each middle piece as two (Neubecker, pp. 44–46; Coey's elbow rule, pieces × 2 − 2): four pieces make 6. The bend divided by that number is how far every cut is tilted from square: 90° ÷ 6 = 15.00°. Each joint turns twice that, 30.00°, and the three joints make the 90°.

2. Throat and heel

The throat is the inside of the bend, the heel the outside. A middle piece measures 2 × (centreline radius ∓ pipe radius) × tan(cut angle) along them:

The end pieces are half a middle piece plus any tangent. Daugherty finds the same lengths by drawing the throat and heel as quarter circles about the bend's centre and dividing the heel arc (Problem 16).

3. The stretch-out and ordinates

Draw the pipe's centre line as a straight stretch-out π × 168.3 = 528.73 long and divide it into twelve spaces of 44.06, point 0 at the throat. At an angle φ round the pipe from the throat, the cut stands (centreline radius − pipe radius × cos φ) × tan(cut angle) off the centre line of the piece: at the throat (252.45 − 84.15) × tan 15° = 45.10, at the heel (252.45 + 84.15) × tan 15° = 90.19.

The books take the stretch-out spaces off a drawn circle with dividers. Those are chords, and a chord is shorter than its arc (Kidder, p. 66): twelve of them make 522.71, about 6.02 short round the pipe. Work out the circumference and divide it instead.

4. Draw the patterns

For a middle piece, set every ordinate off both sides of the centre line and join each side with a smooth curve. An end piece is one side only, measured from its square end (plus the tangent, if any). The books draw one curve, cut it in metal, and scribe the rest from it, reversing it for the other side (Daugherty, Problem 16). Neubecker likewise develops only the first mitre line (pp. 47–49).

5. Cutting from one pipe

Mark the pieces one after another along one length of pipe, turning the template half a turn each time so a heel follows a throat: the cuts nest with no waste between them. Without tangents the whole elbow takes 405.86 of pipe, before saw kerf. Turning the pieces also puts the seams of rolled gores on alternate sides (Coey, Ch. 19).

Common mistakes

• Dividing the bend by the number of joints for the cut angle — that is the turn per joint, twice the cut angle.
• Making the end pieces as long as the middle ones; they are halves.
• Stepping chords off a drawn circle for the stretch-out.
• Developing a pipe template on the inside diameter.

How this matches the tool

The lobster-back elbow tool computes the same cut angle, throat, heel and offsets exactly, gives a cut list for one pipe, and prints each template at 1:1 or exports DXF. Every number on this page was produced by its own code when the page was built.

Sources

1. B. Coey, Pattern Development: Sheet Metal Level 1 (BCcampus, 2021, CC BY 4.0), Ch. 19 “Round Elbow”, pp. 61–65: the elbow rule, pieces × 2 − 2 gores
2. W. Neubecker, Sheet-Metal Work (1920), pp. 44–49: end pieces count one, each middle piece two; patterns from the first miter line
3. J. S. Daugherty, Sheet-Metal Pattern Drafting and Shop Problems (1922), Problem 16 “Four-Piece 90° Elbow”, pp. 38–40
4. G. W. Kittredge, The New Metal Worker Pattern Book (1901), Problems 41–42, pp. 132–134: four- and five-piece elbows
5. F. S. Kidder, Triangulation Applied to Sheet Metal Pattern Cutting (1917), p. 66: stepped chords very likely come to something less than the circumference; divide the computed circumference instead