Mitred elbows: cheeks, wrappers and the cut angle

A mitred elbow is the simplest rectangular fitting that changes direction, and the one that shows most plainly how rectangular patterns work. Its two cheeks are flat L-shapes cut along a straight mitre line. Its two wrappers are flat panels folded once at the corner. Every edge is straight and every dimension can be written down from the duct’s size and the elbow’s leg lengths. The one number that is easy to get wrong is the cut angle, which is always half the angle the elbow turns through.

This chapter develops the running case’s 90° mitred elbow, used in a corner too tight for a radius elbow, and a 45° two-piece mitred elbow, then deals with the turning vanes that supply duct mitres usually carry. It ends with a 45° elbow whose pieces were cut at 45° and which turned the duct through 90°.

The 90° mitred elbow

This is an illustration. Near the air handler, the duct turns 90° in plan around a column, and there is no room for a radius elbow. The elbow has the duct’s 600 mm dimension in the plane of the bend, so its cheeks are the top and bottom faces, each 600 mm across, and its wrappers are the two side faces, each 400 mm high. Each leg runs 600 mm along the duct’s centreline from the point where the two centrelines cross.

With the centrelines crossing at a point O, the throat corner, on the inside of the bend, lies 300 mm from each centreline, half the 600 mm width, and the heel corner on the outside lies 300 mm beyond each centreline the other way. So each leg is 600 − 300 = 300 mm along the throat and 600 + 300 = 900 mm along the heel.

The cheeks

Each cheek is an L-shape: a 900 × 900 mm square with a 300 × 300 mm square removed from its inside corner. Its outer edges are the heel lines, 900 mm each; its inner edges are the throat lines, 300 mm each; its two end edges are 600 mm, the duct’s width. The mitre line runs diagonally from the throat corner to the heel corner, 600 × √2 = 848.5 mm long, and divides the cheek into its two legs. For a one-piece cheek, the mitre line is not cut; it is the line along which the turning vane rail is fixed.

The cheek’s area is 0.900² − 0.300² = 0.72 m², and its perimeter, the length of seam around it, is 3,600 mm. Both numbers go into the take-off in Chapter 10.

Top and bottom cheeks are the same shape. They are mirror images in the assembled elbow, which matters only if the seam allowances differ between edges: a cheek with a Pittsburgh flange on its outer edges and a pocket on its inner edges must be marked so that both cheeks have the flanges in the same place relative to the elbow, not relative to the blank.

Figure 1Cheek of the 90° mitred elbow

The wrappers

The throat wrapper runs along the inside of the bend: 300 mm along one leg, a fold of 90° at the throat corner, and 300 mm along the other leg, 600 mm long in all and 400 mm high. The heel wrapper runs along the outside: 900 + 900 = 1,800 mm long, folded at the heel corner, and 400 mm high. A 1,800 mm wrapper is longer than the shop’s folder, so it is made in two 900 mm pieces seamed at the heel corner.

Both wrappers are rectangles. All the shape of the elbow is in the cheeks, which is what makes mitred elbows quick to make: one L-pattern used twice, two rectangles, and seams along every edge.

Figure 2Pieces of the 90° mitred elbow, net sizes

Piece	Quantity	Net size, mm	Notes
Cheek	2	L-shape, 900 × 900 less 300 × 300	mitre line 848.5 mm for the vane rail
Throat wrapper	1	600 × 400, folded at mid-length	300 mm each leg
Heel wrapper	2	900 × 400 each	seamed at the heel corner
Turning vane rail	2	848.5 mm	one on each cheek, along the mitre line

Where the heel wrapper’s two pieces meet, at the heel corner, the joint is a corner seam like any other, and it lines up with the end of the vane rails on the cheeks. Some shops prefer to make the heel wrapper in one 1,800 mm piece on a longer folder, to avoid a seam right where the airflow strikes the heel; others accept the seam and seal it well. Either way, the joint position is chosen before the cheeks are cut, because the cheek’s flange allowance at the heel corner depends on whether a seam or a fold meets it there.

Turning vanes

Air turning a sharp 90° corner separates from the throat and piles up at the heel, and a plain mitred elbow has a high pressure loss. Supply duct specifications commonly require mitred elbows to carry turning vanes: a row of curved blades set along the mitre line, which split the flow into many small, well-guided turns. Ductwork by the Numbers deals with what vanes do to the pressure loss; this chapter deals with where they go.

The vanes are held by two rails, one on each cheek, fixed along the mitre line. Each rail is the mitre line’s length, 848.5 mm, and the vanes are slotted into it at the spacing the vane maker specifies for the vane profile used. The fabricator’s job is to fix the rails exactly on the mitre line, square to the cheek, so that the vanes run straight across the duct from cheek to cheek, 400 mm long here. A rail set off the mitre line by even 20 mm puts the vanes at the wrong place in the turn and leaves a gap at one wrapper.

The 45° two-piece mitred elbow

A two-piece elbow turning the duct through 45° has one mitre joint, and the joint plane bisects the angle between the two legs. Each piece is therefore cut at 22.5° from square to its own axis: half the 45° turn.

For a cheek 600 mm across, the cut line on each piece is 600 ÷ cos 22.5° = 649.4 mm long, and the heel edge of each piece is longer than its throat edge by 600 × tan 22.5° = 248.5 mm. If each piece is to be 300 mm long at the throat, it is 548.5 mm long at the heel.

The wrappers are rectangles again: the throat wrapper’s two pieces, each 300 mm long, and the heel wrapper’s, each 548.5 mm long, all 400 mm high. A two-piece 45° elbow could also be made with one-piece wrappers folded at 45°, if the shop’s folder can make the bend at the joint.

Figure 3Cut angle and cut length for a mitred rectangular elbow

cut angle from square = elbow angle ÷ (number of joints × 2)

cut line on the cheek = w ÷ cos(cut angle) heel − throat per piece = w × tan(cut angle)

w

Cheek width, the duct dimension in the plane of the bend

1. 90° elbow, one joint: cut 45°, cut line 848.5 mm, heel − throat 600 mm per piece
2. 45° elbow, one joint: cut 22.5°, cut line 649.4 mm, heel − throat 248.5 mm per piece
3. The cut angle is half the turn per joint, never the turn itself

Three-piece mitred elbows

Between a two-piece mitre and a radius elbow sits the three-piece mitred elbow: two end pieces and one middle piece, with two joints. For a 90° turn, each joint turns the duct 45°, so each cut is 22.5° from square, the same cut as the two-piece 45° elbow. The middle piece is cut at 22.5° at both ends, and its cheek is a trapezoid: if it is 150 mm long at the throat, it is 150 + 2 × 248.5 = 647.1 mm long at the heel, with two cut lines of 649.4 mm.

A three-piece elbow has a softer turn than a two-piece mitre and often needs no vanes, at the cost of one more joint to seam all round. It is also a useful compromise where a radius elbow’s curved wrappers cannot be formed in the shop, because every piece is still flat and every edge straight. Its pieces are developed exactly as above, with the cut rule applied at each joint.

Checking the assembled elbow

Three checks on the assembled elbow, before it leaves the shop, catch most faults.

1. The angle. With one leg’s end flat against a straightedge on the bench, a large square or a 3-4-5 triangle shows whether the other leg is at 90°. A 3-4-5 triangle of 600, 800 and 1,000 mm is easy to mark on the bench with a tape.
2. The leg lengths. Each leg’s centreline length, from the point where the centrelines cross to the end, should be 600 mm. On the assembled elbow, that is easiest to check as the distance between the two ends’ centres, which for two 600 mm legs at 90° is 600 × √2 = 848.5 mm, the same as the cheek’s mitre line.
3. The ends. Each end must be square and true to the 600 × 400 duct, with equal diagonals of 721.1 mm, or it will not join the next straight section squarely.

An elbow that passes all three can still have its vanes in the wrong place, so the vane rails are checked separately: each should lie on the line from throat corner to heel corner, within a few millimetres, on both cheeks.

Seams around the cheeks

Every edge where a cheek meets a wrapper is a corner seam, and in the running case these are Pittsburgh locks like the straight duct’s. The shop puts the pockets on the wrappers and the flanges on the cheeks, because the cheek’s L-shape is harder to feed through the roll former and a flange is simpler to form on its inside corner.

That decision changes the cheek’s blank. The 8 mm flange allowance goes on every edge of the L where a wrapper will meet it: the two 900 mm heel edges and the two 300 mm throat edges. At the throat corner, the inside corner of the L, the two flanges would collide, so the corner is notched with a small relief before forming. The end edges, where the elbow meets the straight duct, get the transverse joint’s allowance instead. Chapter 9 sets out the allowances in full; the point here is that a cheek’s blank is its net L plus different allowances on different edges, and the pattern must show which.

The error: a 45° elbow cut at 45°

This is an illustration. A second two-piece 45° elbow was cut by a hand who reasoned that “a 45° elbow is cut at 45°” and set the guillotine’s gauge to cut each piece’s end at 45° from square.

Each piece turned the duct 45° relative to the joint plane, and the two together turned it through 90°. The cut lines on the cheeks were 848.5 mm long instead of 649.4 mm, and each piece’s heel was 600 mm longer than its throat instead of 248.5 mm. The fault was obvious the moment the two pieces were offered together, because the cheeks met along a line 199 mm longer than the straight duct’s width would have suggested and the legs stood at right angles.

The rule that prevents it is the one in the formula above, the same rule as for round segmented elbows: the cut is half the turn at each joint. The shop’s job sheet for every mitred fitting now gives three numbers, “TURN 45°, ONE JOINT, CUT 22.5° FROM SQUARE”, so that nobody derives the cut at the guillotine.

Figure 4Mitred elbow checks before cutting

• Do: Dimension in the plane of the bend stated600 for this elbow
• Do: Cut angle is half the turn per joint22.5° for a two-piece 45°
• Do: Cut line on the cheek649.4 mm for 22.5°, 848.5 mm for 45°
• Do: Heel minus throat per piece248.5 mm for 22.5°
• Do: Vane rails on the mitre line, if specified848.5 mm on the 90° elbow
• Avoid: Cut angle taken as the elbow's anglethe 45° elbow that turned 90°

Mitring with the other dimension in the plane

The same arithmetic applies with the 400 mm dimension in the plane of the bend, for a mitred elbow that turns the duct from horizontal to vertical. The cheeks are then the side faces, 400 mm across; the mitre line on a 90° elbow is 400 × √2 = 565.7 mm; and the wrappers, the top and bottom faces, are 600 mm wide. Vanes, if fitted, run 600 mm across the duct between the cheeks.

Which way round the elbow is made affects its weight and its vanes as well as its fit. A mitred elbow with the narrower dimension in the plane of the bend has smaller cheeks and longer vanes; one with the wider dimension in the plane has larger cheeks and shorter vanes. Neither is better in general. The drawing decides, and the job sheet records it, as Chapter 1 insisted.

Exercise: a 30° mitred elbow

A two-piece mitred elbow turns the 600 × 400 duct through 30°, with the 400 mm dimension in the plane of the bend. Each piece is to be 250 mm long at the throat. Find the cut angle, the length of the cut line on each cheek, each piece’s heel length, and the wrappers’ net sizes.

Worked answer.

• One joint, so the cut is 30 ÷ 2 = 15° from square.
• Cut line on the cheek: 400 ÷ cos 15° = 414.1 mm.
• Heel minus throat per piece: 400 × tan 15° = 107.2 mm, so each piece is 250 + 107.2 = 357.2 mm at the heel.
• Wrappers: the throat wrapper’s pieces are 250 × 600 mm each and the heel wrapper’s 357.2 × 600 mm each, since the wrappers are the 600 mm faces when the 400 mm dimension is in the plane of the bend.

Check: across the joint, the throat pieces meet on a cut line that, seen from the side, is 414.1 mm long and inclined at 15°; its projection along the duct is 400 × tan 15° = 107.2 mm, the heel-throat difference, as it must be.