Miter Bend: flat pattern

Segmented round bends, at a constant diameter or reducing and expanding, one plate per gore. Segmented (lobster-back) bend of round duct, of constant diameter or reducing/expanding: gores between miter cuts, each developed as its own plate.

Configurations

• Centreline radius. The bend radius is given to the duct centreline.
• Throat radius. The bend radius is given to the inside (throat) of the duct; the centreline radius is the throat radius plus half the diameter.

A worked example

The centreline radius configuration with the sizes SheetMorph opens with, in mm. 5 plates, each an unrolled cylinder with 32 divisions: 5 parts, 9 seams.

How it is developed

Parallel-line development (unrolled cylinders). 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 5 gores: two end gores and 3 full gores. Each gore is a short length of round duct cut at 11.25 deg from square on its mitered side(s).
• A cylinder is developable, so each plate is exact apart from the division of the circumference: its width is the developed circumference and its edges follow the miter cuts.
• Division points lie on the true circle, so the plate width (a sum of chords) is slightly less than pi x D. Increase the divisions, or use the matched-circumference model, if the rolled gore must close exactly on the true diameter.
• 4 miter joints and 5 longitudinal seams are to be joined.
• 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 circles (passed): All 64 end points lie in their end plane at radius 150.5 (max deviation 8.53e-14).
• Every gore is a cylinder of the duct diameter (passed): In all 5 gores every generator is parallel to the gore axis and at the duct radius from it (max deviation 1.14e-13).
• Miter joints planar and bisecting (passed): 4 joint(s): every joint is a planar ellipse of the duct radius at that joint (max deviation 8.53e-14) and makes 11.25 deg with the square cut of both gores it joins; the ends are 90 deg apart.
• Surface connectivity (passed): 320 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): 960 facet edges compared between the 3D surface and the flat pattern (max deviation 4.33e-13).
• Shared edges and seam closure (passed): 315 internal edges are shared vertex-for-vertex between adjacent facets; 5 seam(s) have mating edges of equal true length (max deviation 8.53e-14).
• 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-9).
• Cut outlines closed and non-overlapping (passed): 5 closed outline(s); no outline crosses itself or another outline.
• Plates are true unrolled cylinders (passed): Every generator and seam of every plate is perpendicular to the plate's baseline, as on an unrolled cylinder (max deviation 4.55e-13).
• Symmetric about the plane of the bend (passed): Every point has its mirror image across the plane of the bend (max deviation 2.14e-13).

Assumptions

• The pattern is developed on the mid-thickness surface. An inside or outside diameter is converted to it by one thickness; the bend radius is to the centreline and is not changed.
• Miter planes pass through the bend axis and bisect adjacent gores. With a constant diameter each gore is a true cylinder; with different end diameters the duct radius varies linearly along the centreline and each gore is the tapered surface between its two elliptical joints.
• End gores are half of a full gore, plus any straight extension.
• The circumference is divided into equal parts; plate edges are straight between divisions.
• No rolling allowance, and no seam, lap, flange, weld or trimming allowance, is added to any outline.
• Plates are drawn outside face up, End A side down.

Limits of this module

• Round duct only; rectangular miter bends are not covered.
• In a reducing or expanding bend the taper is linear along the centreline, also through any straight length; a constant-diameter collar on a tapered bend is not modelled.
• Plate edges are straight between divisions, not smooth curves.
• No rolling allowance, and no seam, lap, flange, weld or trim allowances.
• The fit of the miter joints through the sheet thickness (bevels) 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 circle 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.

Bend angle and radius. The bend angle is the total angle the duct turns through (up to 180 deg). The centreline radius R is measured to the duct axis; the throat radius is R minus half the diameter and the back radius R plus half the diameter. R must exceed half the diameter.

Reducing and expanding bends. Give a different diameter at End B for a bend that reduces (smaller) or expands (larger). The duct radius then varies linearly along the centreline of the gores from End A to End B. Each joint is the ellipse of the local radius in the same miter plane, so neighbouring gores share their joint curve exactly. A gore is then close to a cone frustum rather than a cylinder: it is not exactly developable and is triangulated. The throat radius is measured at End A.

Segments and miter angle. A bend of M segments has two end gores and M - 2 full gores, joined at M - 1 miter joints. Each end gore is half a full gore. The miter angle is the bend angle divided by 2 (M - 1): it is the angle of each cut from a square cut, and each joint turns the duct by twice that.

Straight lengths. A straight length can be added to each end gore, for a collar or a joint to straight duct. End A stays at Z = 0; the bend starts that much higher.

Numbering. The circumference is divided into 4N equal parts, N per quarter. Division 0 is at the back (outside of the bend), N and 3N at the sides, 2N at the throat (inside of the bend). Vertex Rj_k is division k on ring j, where ring 0 is End A, rings 1 to M - 1 are the miter joints and ring M is End B. Gores are S1 (at End A) to SM (at End B).

Development method. Each gore is a cylinder, which is developable. It is cut along its seam and unrolled: the plate is as wide as the circumference, and its height at each division is the length of the generator there. For a full gore that height is 2 (R - (D/2) cos phi) tan(miter angle), phi being measured from the throat. The listed ordinates are those heights from back to throat; the other half of the plate mirrors them.

Seams. Each gore has one longitudinal seam, on the back, a side or the throat. With staggered seams, alternate gores have theirs on the opposite line so that seams do not meet at a joint.

Circle model. Inscribed: division points are on the true circle, and the plate width is slightly less than pi x D. Matched circumference: points are on a marginally larger circle so the plate width equals pi x D exactly.

Material thickness. A plate has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your diameter is an inside, mid-thickness or outside size: an inside diameter is increased by one thickness and an outside diameter 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 diameter 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 pipe bends

• S-Offset: An S-bend of round pipe: two opposed miter bends with a straight between, from the offset or the straight you give.
• Flat Oval: A segmented bend in flat-oval duct, bent the easy way or the hard way: every gore unrolled, with the mitre at each joint.
• Oval to Rectangle: A segmented bend that changes from flat-oval duct to rectangular duct as it turns, bent the easy way or the hard way: every gore developed, with the mitre at each joint.
• Oval to Round: A segmented bend that changes from flat-oval duct to round duct as it turns, bent the easy way or the hard way: every gore developed, with the mitre at each joint.