Rectangular Duct Bend: flat pattern

Elbows of rectangular duct, rolled to a radius or in mitered segments: two cheeks, a throat and a heel. Elbow of rectangular duct, with a rolled radius or in mitered segments: two flat cheeks, a throat and a heel, each developed as its own plate.

Configurations

• Radius bend. The throat and the heel are rolled to an arc. The arc is drawn as a chain of short chords.
• Segmented bend. The bend is made of straight pieces joined at miter joints, like a round miter bend; the throat and the heel are bent, not rolled.

A worked example

The radius bend configuration with the sizes SheetMorph opens with, in mm. 4 plates: two cheeks, throat and heel, the arc drawn as 16 chords: 4 parts, 4 seams.

How it is developed

Flat cheeks in true shape; throat and heel unfolded as straight strips. This is a triangulated approximation: the curved surface is divided into small flat triangles, each laid out at its true size. The finer the division, the closer the pattern is to the surface; SheetMorph states the deviation it found.

• The bend is made of four plates: two flat cheeks, parallel to the plane of the bend, and the throat and heel that join them on the inside and the outside of the bend.
• A cheek is flat, so its plate is its true shape. The two cheeks are mirror images; both are drawn outside face up.
• The throat and the heel are rolled to the bend radii. Their plates are straight strips as wide as the depth of the duct. Each arc is drawn as 16 chords, so a strip is slightly shorter than the true arc; the true arc lengths are listed, and more divisions bring the two together.
• The four corners, where a cheek meets the throat or the heel, are joints between plates: no allowance for a seam, lap or flange is added.
• Plates are drawn stacked with gaps; they can be nested closer when cutting.

Checked before it can be exported

Every pattern SheetMorph draws is checked against the 3D part it comes from. For this example:

• Both ends are square-cut rectangles (passed): Both ends are 401 x 301 rectangles square to the duct (max deviation 4.46e-14), 90 deg apart.
• Cheeks flat and parallel (passed): Every corner lies in the plane of its cheek, 150.5 either side of the plane of the bend (max deviation 0).
• Throat and heel on their radii (passed): Every station of the arc has its throat corners at radius 199.5 and its heel corners at 600.5 from the bend axis (max deviation 1.14e-13).
• Surface connectivity (passed): 128 facets form one closed band: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 384 facet edges compared between the 3D surface and the flat pattern (max deviation 1.51e-12).
• Shared edges and seam closure (passed): 124 internal edges are shared vertex-for-vertex between adjacent facets; 0 seam(s) have mating edges of equal true length (max deviation 0).
• Developed facets consistently oriented (passed): All developed facets are counter-clockwise (outside face up); none is mirrored or degenerate.
• Developed area equals surface area (passed): Facet area in 3D and in the flat pattern agree (difference 1.16e-10).
• Cut outlines closed and non-overlapping (passed): 4 closed outline(s); no outline crosses itself or another outline.
• Throat and heel plates are straight strips (passed): The throat and heel plates are rectangles 301 wide, with every line across them square to their long edges (max deviation 1.42e-12).
• Cheeks are mirror images (passed): Every corner has its mirror image across the plane of the bend (max deviation 0), and the two cheek plates have equal area (difference 1.66e-9).

Assumptions

• The pattern is developed on the mid-thickness surface. Inside or outside sizes are converted to it by one thickness on both the width and the depth; the bend radius is to the centreline and is not changed.
• The cheeks are flat and parallel to the plane of the bend. The duct keeps the same width and depth all the way round.
• A radius bend is drawn with its throat and heel arcs as chains of equal chords. A segmented bend has straight pieces between miter planes that pass through the bend axis.
• The open ends are cut square to the duct.
• No allowance for seams, laps, flanges, welds or trimming is added to any outline, and no turning vanes are drawn.
• Plates are drawn outside face up.

Limits of this module

• The duct keeps the same width and depth through the bend; reducing bends are not covered.
• A square throat (throat radius zero) and a square heel are not covered.
• A radius bend is drawn with chords, not true arcs, so plate edges are straight between divisions.
• No allowances for corner seams (Pittsburgh, snap-lock), laps, flanges or welds, and no turning vanes or splitters.
• The fit of the corner joints through the sheet thickness is not modelled.
• Plates are stacked with gaps for clarity, not nested for minimum material.
• Sheet nesting, grain direction and stock-size checks are not performed.

Conventions

Coordinate system. Right-handed, Z vertical in the views. End A is the rectangle at Z = 0, centred on the origin, with the duct leaving along +Z. The bend lies in the XZ plane and turns toward +X about the bend centre, which is at X = R level with the start of the bend. The Front view therefore shows the bend in its own plane, with a cheek facing you.

Width and depth. The width W is the size of the duct in the plane of the bend (along X at End A); the depth D is its size across that plane (along Y). Bending on the width is called an easy way bend when W is the smaller side and a hard way bend when it is the larger; swap the two values to turn the bend the other way.

Throat, heel and cheeks. The throat is the side on the inside of the bend, toward the bend centre (+X at End A). The heel is the side on the outside. The two cheeks are the flat sides parallel to the plane of the bend: cheek 1 on the -Y side and cheek 2 on the +Y side.

Bend angle and radius. The bend angle is the total angle the duct turns through (up to 180 deg). The radius can be given to the duct centreline or to the throat; the centreline radius is the throat radius plus half the width, and the heel radius is the centreline radius plus half the width. The throat radius must be greater than zero.

Radius bend. The throat and heel follow arcs about the bend centre. Each arc is divided into equal parts and drawn as chords, so the outline of a cheek is a polygon and the throat and heel plates are slightly shorter than the true arcs. The true arc lengths are listed with the calculation.

Segmented bend. A bend of M segments has two end pieces and M - 2 full pieces, joined at M - 1 miter joints, as in the round miter bend. The miter angle is the bend angle divided by 2 (M - 1), and each joint turns the duct by twice that. Each piece has straight walls; the joints lie in planes through the bend axis.

Straight lengths. A straight length can be added before the bend at End A and after it at End B. End A stays at Z = 0; the bend starts that much higher. On a radius bend the line where rolling starts is marked on the throat and heel plates.

Numbering. The corners of every station are numbered as seen looking along the duct from End A: 0 heel / -Y, 1 throat / -Y, 2 throat / +Y, 3 heel / +Y. Vertex Rj_c is corner c of station j, counted from End A. The plates are P1 cheek 1, P2 throat, P3 cheek 2 and P4 heel.

Development method. The cheeks are flat, so each plate is the true shape of the cheek. The throat and the heel are strips whose lines across are all parallel, so each unfolds to a straight rectangle as wide as the depth of the duct and as long as the throat or the heel line.

Material thickness. A plate has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your width and depth are inside, mid-thickness or outside sizes: inside sizes are increased by one thickness and outside sizes reduced by one thickness to reach the mid-thickness surface. The centreline radius is the same for every surface. The 3D view and the views always show the sizes as entered.

Change the sizes to your own in SheetMorph, check the drawing, add seam and flange allowances, and export DXF, PDF, SVG, CSV or G-code. Check every pattern against your own standards before cutting.

Other duct bends

• Reducing Elbow: A rectangular bend whose width, or width and depth, change steadily from one end to the other.
• Square-Throat Elbow: A rectangular elbow with a square inside corner, joined on one mitre, with or without turning vanes.
• Offset: A rectangular duct moved sideways between two parallel ends: an ogee of two curves, or a slanted piece on two mitres.
• Round to Rectangle: A square-to-round whose two ends are at an angle: it changes shape and direction in one piece.