Pipe Offset (S-Bend): flat pattern

An S-bend of round pipe: two opposed miter bends with a straight between, from the offset or the straight you give. Two opposed miter bends of round pipe with a straight between them, so the pipe carries on parallel, moved sideways: each piece developed as its own plate.

Configurations

• The offset. You give the sideways offset between the two ends; the straight between the bends is worked out.
• The straight length. You give the straight between the two bends; the offset is worked out.

A worked example

The the offset 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 offset is two miter bends of 45 deg, the second turning back, with a straight between them: 5 pieces joined at 4 miter joints, each cut at 11.25 deg from square.
• The end of the first bend, the straight and the start of the second bend lie on one line, so they are made as one piece, the middle piece.
• Each piece is a cylinder, which is developable: its plate is as wide as 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.
• Every piece has its seam on the +Y side. Plates are drawn stacked with gaps, End A's piece at the bottom.

Checked before it can be exported

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

• Ends square, parallel and offset (passed): Both ends are circles of radius 100.5 square to +Z, and the centreline traced through the bends ends 400 across and 648.52814 up from End A (max deviation 1.14e-13).
• Every piece is a cylinder of the pipe diameter (passed): In all 5 pieces every generator is parallel to the piece's axis and at the pipe radius from it (max deviation 8.53e-14).
• Miter joints planar and bisecting (passed): 4 joints: each is planar and makes 11.25 deg with the square cut of both pieces it joins (max deviation 6.69e-14).
• 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 2.31e-12).
• 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 2.27e-13).
• 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 0).
• Cut outlines closed and non-overlapping (passed): 5 closed outline(s); no outline crosses itself or another outline.
• Pipe plates are true unrolled cylinders (passed): Every generator and seam of every pipe plate is perpendicular to the plate's baseline, as on an unrolled cylinder (max deviation 4.09e-12).
• Symmetric about the plane of the offset (passed): Every point has its mirror image across the XZ plane (max deviation 1.30e-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.
• The two bends have the same radius, angle and number of pieces, and turn opposite ways in one plane, so the two ends are parallel.
• Miter planes bisect adjacent pieces; each piece is a true cylinder.
• 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

• Both bends are alike and lie in one plane; a rolling offset (two planes) is not covered.
• The pipe keeps one diameter; reducing offsets are not covered.
• The fit of the miter joints through the sheet thickness (bevels) is not modelled.
• Plate edges are straight between divisions, not smooth curves.
• No rolling allowance, and no seam, lap, flange, weld or trim allowances.
• Plates are stacked with gaps for clarity, not nested for minimum material.

Conventions

Coordinate system. Right-handed, Z vertical in the views. End A is the circle at Z = 0, centred on the origin, with the pipe leaving along +Z. The first bend turns toward +X and the second turns back, so End B faces +Z again, centred at X = offset and Z = rise. The Front view shows the S of the offset.

Offset, straight and rise. The offset E is the distance between the axes of the two ends. The straight S is the length of the inclined run between the two bends. They are tied by E = 2 R (1 - cos angle) + S sin(angle): give either and the other is worked out. The rise is the distance between the ends along their axes.

Bends. Both bends have the same centreline radius R and the same angle, from 5 to 90 degrees, and the same number of pieces. With 2 pieces per bend each bend is a single miter joint; more pieces make it smoother. R must exceed half the diameter.

Pieces. With M pieces per bend there are 2 M - 1 pieces in all: the end piece at End A, M - 2 gores, the middle piece (the end of the first bend, the straight and the start of the second, in one), M - 2 gores and the end piece at End B. The miter angle is the bend angle divided by 2 (M - 1).

Straight lengths. A straight length can be added before the first bend at End A and after the second at End B. They add to the rise.

Numbering. The circumference is divided into 4N equal parts, N per quarter: division 0 is toward +X, N toward +Y, 2N toward -X. Vertex Rj_k is division k on ring j, where ring 0 is End A, the last ring End B and the others the miter joints. Pieces are S1 at End A upward.

Development method. Parallel-line development. Each piece is a cylinder: it is cut along its seam and unrolled to a plate as wide as the circumference, whose height at each division is the length of the wall there. The plates of the second bend are those of the first, turned half-way round.

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. The bend radius is to the centreline, the same for every surface. The 3D view and the views 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

• Round: Segmented round bends, at a constant diameter or reducing and expanding, one plate per gore.
• 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.