Twisted transitions and helical elbows

Every fitting so far has had a plane of symmetry: a plane that cuts it into mirror-image halves, so that half a pattern served for the whole. The fittings in this chapter have none. A rectangular duct whose end is rotated about its own axis twists as it goes, and an elbow that climbs as it turns follows a helix, which has no mirror plane at all. Both are made in the shop regularly, both are approached with suspicion, and both yield to the same coordinates and true lengths as everything else.

This chapter develops the two parts of F6. The first is a twisted rectangular transition, made as folded triangles. The second is a helical elbow in round duct, made as mitred cylinders whose cuts twist from one joint to the next. It ends with a helical elbow made as if it were flat.

F6a: the twisted transition

This is an illustration. Where DX-3 meets the collector’s inlet, a 300 × 200 mm rectangular duct must connect to an inlet spigot of the same size that is rotated 30° about the duct’s axis, because the collector was set on its pad at an angle to the building. The two ends are parallel and the same size; one is turned relative to the other. The fitting is 400 mm long.

The axes put the bottom end in the plane z = 0, centred at the origin, with its 300 mm sides along x and its 200 mm sides along y. Its corners are at (±150, ±100, 0). The top end is the same rectangle rotated 30° anticlockwise about z and raised to z = 400. Rotating a point (x, y) by 30° gives (x cos 30° − y sin 30°, x sin 30° + y cos 30°), so the corner (150, 100) moves to (79.9, 161.6), and the others follow.

Faces that twist

Each face of the transition joins one edge of the bottom rectangle to the matching edge of the top. The two edges are the same length and parallel to their own planes, but not to each other: the top edge is turned 30° from the bottom. A surface joining two straight lines that are not parallel and do not meet is a hyperbolic paraboloid, a saddle. It is made entirely of straight lines, but it is warped, and it has no flat pattern.

There are two practical ways to make it. The face can be formed as a smooth twist, rolled or stretched into its saddle shape, which suits thin sheet and small twists. Or it can be folded along a diagonal into two flat triangles, which needs no forming at all and gives an exact pattern for what is made. F6a is folded.

The triangles

Every corner line, from a bottom corner to the top corner it becomes, has the same true length, because every corner moves through the same rotation. The corner (150, 100, 0) goes to (79.9, 161.6, 400): differences of −70.1, 61.6 and 400, and a true length of 410.7 mm.

Each face has two possible diagonals. The shorter one is used, because it makes the flatter fold and keeps the triangles closer to the twisted surface they replace.

Figure 1F6a's faces, folded into triangles

Face	Edge	Corner lines	Diagonals, long and short	Fold along the short diagonal
Wide faces, 300 mm	300	410.7	526.0 and 465.5	about 35° from flat
Narrow faces, 200 mm	200	410.7	483.1 and 416.4	about 31° from flat

Each wide face folds into two triangles with sides 300, 410.7 and 465.5 mm; each narrow face into two with sides 200, 410.7 and 416.4 mm. The angle in each wide triangle between its 300 mm edge and its corner line is 80.2°, and in each narrow triangle 77.6°. A square would put those angles at 90°, and the difference is the twist made visible on the pattern.

The pattern

The eight triangles are laid out in one strip, starting from a corner line as the seam: the first wide face’s two triangles, then the first narrow face’s, then the second wide face’s and the second narrow face’s, ending at the seam’s other copy. The result is a zigzag band about 982 mm long and 599 mm high, with the bottom rectangle’s 1,000 mm perimeter along one edge, broken at each corner, and the top’s along the other. Its area is about 0.40 m², almost exactly that of an untwisted 400 mm length of the same duct, because folding along diagonals neither stretches nor shrinks the sheet. At 0.8 mm it weighs about 2.5 kg.

The diagonals are marked as fold lines and the corner lines as corner bends. Folded, the band closes into the twisted transition, with each face creased along its diagonal. The creases all run the same way round the fitting, from each bottom corner up to the top corner ahead of it in the direction of twist.

How much twist a fold can take

The fold angles, about 35° and 31°, are large. They come from the 30° twist over a 400 mm length, and they would be smaller on a longer fitting or a smaller twist. A 15° twist over the same length would give corner lines of 402.8 mm and short diagonals of 483.3 and 430.5 mm, with much flatter folds. In ductwork, a crease is acceptable on a fitting this size, but a long, shallow twist is smoother to the air and easier to seal. This is an illustration: the shop’s own rule of thumb is a twist of no more than about 1° per 20 mm of length, and this fitting, at 1.5° per 20 mm, exceeds it. The connection was too tight for more length, and the folds were accepted.

F6b: the helical elbow

Inside the workshop, the DX-3 branch from the edge sander climbs round a building column to clear a crane rail, turning 90° in plan while rising 300 mm. This is an illustration. The duct is Ø200, and its centreline follows a helix of 400 mm radius round the column’s axis. The helix’s slope is constant: over a quarter turn of 400 × π ÷ 2 = 628.3 mm in plan, it rises 300 mm, a slope of 25.5°. Its length along the helix is √(628.3² + 300²) = 696.3 mm.

The elbow is made in four pieces of straight Ø200 duct, each along a chord of the helix between points a quarter of the way round. Each chord is 173.2 mm long, and the four total 692.6 mm. The ducts on either side arrive and leave along the helix’s direction at its ends, sloping at 25.5°, so the elbow joins two sloping ducts that are 79.3° apart in space.

Joints that twist

At each joint, two straight pieces meet at an angle, and the joint is a mitre: each piece is cut by the plane that bisects the angle between them. On a flat elbow, every joint bends in the same plane. On a helical elbow, each joint bends in its own plane, and each plane is rotated a little from the last.

The calculation gives the angles from the pieces’ directions.

• Deflection. The angle between neighbouring pieces is 20.26° at the three joints between pieces, and 10.14° at the two end joints, where a piece meets the sloping duct. The mitre cuts are half those, 10.13° and 5.07°.
• Twist. The plane of each bend is rotated from the plane of the previous bend by 9.85° between the inner joints, and by 6.55° between each end joint and its neighbour.

The deflections add to 81.05°, not 79.3°, because they are measured in different planes; like the tilts in Chapter 1, angles in different planes do not simply add.

Figure 2F6b's joints

Joint	Deflection	Mitre cut	Twist from the previous joint
Inlet duct to piece 1	10.14°	5.07°	(first joint)
Piece 1 to piece 2	20.26°	10.13°	6.55°
Piece 2 to piece 3	20.26°	10.13°	9.85°
Piece 3 to piece 4	20.26°	10.13°	9.85°
Piece 4 to outlet duct	10.14°	5.07°	6.55°

Each piece’s pattern

Each piece is a cylinder cut at both ends, and its pattern is a stretch-out of 628.3 mm with a cut curve along each end. On a flat elbow, both curves peak at the same place, the heel. On F6b, the far end’s curve is shifted along the stretch-out by the twist: for pieces 2 and 3, by 9.85 ÷ 360 × 628.3 = 17.2 mm, and for pieces 1 and 4, by 6.55 ÷ 360 × 628.3 = 11.4 mm.

Calculated directly, by intersecting each line along piece 2 with its two joint planes, the piece’s lengths at 30° steps round from the first joint’s heel are 208.6, 205.4, 193.5, 176.2, 158.1, 144.0, 137.7, 140.9, 152.8, 170.1, 188.2 and 202.3 mm. A flat elbow’s piece with the same cuts would have lengths of 208.9, 204.1, 191.1, 173.2, 155.3, 142.3 and 137.5 mm, rising and falling symmetrically. The helical piece’s lengths are lopsided: longer on one side of the heel than the other, by up to 3 mm, which is the twist showing in the pattern.

Figure 3Piece 2's length round its girth, helical against flat

Helical, 30° past the heel205.4 mm

Flat, 30° either side of the heel204.1 mm

Helical, 30° before the heel202.3 mm

Helical, 90° past the heel176.2 mm

Flat, 90° either side173.2 mm

Helical, 90° before the heel170.1 mm

Why a helix, and the alternative

The branch could have climbed round the column another way: a flat elbow to turn it towards the column’s side, a straight sloping length, and a second flat elbow to turn it onward, a combination usually called a rolling offset. Each elbow would be an ordinary segmented elbow, and the straight would be plain duct. That is simpler to make, and many shops would choose it.

The helical elbow earns its extra work in three situations, and the DX-3 branch met the first of them. It is more compact, because it turns and climbs at once, which mattered here under the crane rail. It is smoother to the air than two elbows close together, because it turns steadily instead of twice. And it looks deliberate, which matters on exposed ductwork. The same geometry serves other helical work: spiral chutes for bulk materials and ductwork that winds round a column or a tank. In all of them, the pieces are straight and the twist lives in the cuts.

Assembly

The pieces are rolled and seamed with their seams on a common line marked on each pattern, and assembled in order, each turned so that its cut matches the previous piece’s cut. Matching is unambiguous when the cuts are right: each joint closes all round only in one rotation. A piece that will not close in any rotation has been cut with the wrong twist, or with its twist in the wrong direction, and the fault is in its pattern, not in the assembly. The elbow is checked by its ends, not its joints: the outlet duct’s centre must be 400 mm from the column’s axis, 300 mm above the inlet’s, and a quarter turn round, and its direction must slope at 25.5° in the right sense.

The error: a helix made flat

This is an illustration. The first F6b was made by a worker who calculated the deflections correctly, cut every piece as for a flat elbow with the same mitres, and planned to twist the pieces against each other at the joints during assembly.

It does not work. A piece cut as for a flat elbow has both its cuts tilting the same way, and turning it about its axis turns both cuts together. However the pieces were turned, each joint only closed when the next piece’s bend lay in the same plane as the last, and the assembled elbow came out flat: a four-piece elbow in one plane, turning 81° instead of climbing round the column. Its outlet ended 76.4 mm from where the duct needed it, and pointed 19.8° away from the duct’s direction.

It was found at the trial assembly against the column, and remade with twisted cuts. The lesson is that the twist must be in the patterns. It cannot be added at assembly, because a mitre joint has only one position in which it closes.

Exercise: a gentler twist

For F6a, suppose the collector could be set so that the twist was 15° instead of 30°. Find the corner lines’ true length and the two possible diagonals of a wide face. Which is used?

Worked answer.

• The corner (150, 100) rotated 15° goes to (150 cos 15° − 100 sin 15°, 150 sin 15° + 100 cos 15°) = (119.0, 135.4).
• The corner line’s differences are −31.0, 35.4 and 400, so its true length is √(31.0² + 35.4² + 400²) = 402.8 mm.
• For the wide face from (150, 100, 0) and (−150, 100, 0), the diagonals run to the rotated corners (−170.8, 57.8, 400) and (119.0, 135.4, 400). The diagonal from (150, 100, 0) to (−170.8, 57.8, 400) is 514.5 mm; the one from (−150, 100, 0) to (119.0, 135.4, 400) is 483.3 mm.
• The shorter, 483.3 mm, is used.

The gentler twist brings the corner lines within 3 mm of the untwisted 400 mm and the folds much closer to flat.