Rectangular Offset: flat pattern

A rectangular duct moved sideways between two parallel ends: an ogee of two curves, or a slanted piece on two mitres. A rectangular duct moved sideways between two parallel ends: an ogee of two reverse curves, or a slanted piece between two mitres.

Configurations

• Ogee (two curves). Two equal curves, one after the other and turning opposite ways; the radius follows from the offset and the length.
• Mitred (slanted piece). A straight length at each end and a slanted piece between them, joined on two mitres.

A worked example

The ogee (two curves) configuration with the sizes SheetMorph opens with, in mm. 4 plates: two flat cheeks and two wrappers rolled to an S, 12 pieces along: 4 parts, 4 seams.

How it is developed

Parallel-line development: flat cheeks, rolled wrappers. This is an exact development: the surface unfolds without stretching, so the pattern closes exactly.

• The cheeks lie in the planes of the two flat sides and are cut to their true shape.
• The wrappers are strips as wide as the depth, rolled one way and then the other; their length is the developed length of the inside and outside curves. The lines across them are forming lines, not folds.
• Each side is a separate plate. The four corner edges of a duct are joints between plates, drawn as plain edges without allowance.

Checked before it can be exported

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

• Every station is a rectangle (passed): 13 stations: opposite sides and the two diagonals of each compared (max deviation 0).
• Plates flat, rolled or triangulated (passed): 2 flat, 2 folded or rolled from flat pieces (both develop exactly), 0 twisted and split along a diagonal.
• Surface connectivity (passed): 96 facets form one closed duct: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 288 facet edges compared between the 3D surface and the flat pattern (max deviation 2.84e-13).
• Shared edges and seam closure (passed): 92 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 3.49e-10).
• Cut outlines closed and non-overlapping (passed): 4 closed outline(s); no outline crosses itself or another outline.

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside sizes are converted to it by one thickness across each opening.
• Folds are sharp lines; no bend allowance is applied.
• No seam, lap or weld allowance is added.

Limits of this module

• The offset is in one plane only; a compound offset (sideways and up) is not modelled.
• The two curves of the ogee are equal; the curved sides are straight between stations.
• No seam, lap, lock-seam, flange or weld allowance is added unless the module says so; corner joints are drawn as plain edges.
• Bend allowance and springback are not modelled; folds are sharp lines on the mid-thickness surface.
• No stiffening, tie rods, access doors or nesting onto stock sheets.

Conventions

Coordinate system. End A is at Z = 0, centred on the Z axis, with its width along X and its depth along Y. A duct that turns or is offset does so in the XZ plane, toward +X.

Corners and sides. Corners are numbered 0 (-X, -Y), 1 (+X, -Y), 2 (+X, +Y), 3 (-X, +Y), seen from +Z at End A. Rj_c is corner c of station j. Each of the four sides is its own plate; the plates are joined along the four corner edges.

Plates. Plates are drawn outside face up and stacked in the order of their numbers. The blank sizes listed are the bounding rectangle of each plate.

Material thickness. Say whether the sizes are inside, mid-thickness or outside sizes. The pattern is developed on the mid-thickness surface, one thickness larger or smaller across each opening; the views show the sizes as entered.

Offset and length. The offset is how far End B is moved along +X; the length is the distance between the ends along Z. Both ends stay square to Z. For the ogee, each curve turns 2 atan(offset / length) on a centreline radius of length / (2 sin of that angle).

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

• Elbow: Elbows of rectangular duct, rolled to a radius or in mitered segments: two cheeks, a throat and a heel.
• 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.
• Round to Rectangle: A square-to-round whose two ends are at an angle: it changes shape and direction in one piece.