Twin-Bend Breeches, Rectangle to Round: flat pattern

A rectangular trunk that changes to round and divides into two round segmented bends that turn opposite ways: a bullhead at 90°, or any angle you give. A trunk that changes from a rectangle at its open end to round, then divides into two round segmented bends that turn opposite ways; with both at 90 degrees, a bullhead. One plate per piece.

Configurations

• Bullhead (90°). Each branch turns 90 degrees, so the two ends face opposite ways, square to the trunk.
• Angled branches. Each branch turns the angle you give.

A worked example

The bullhead (90°) configuration with the sizes SheetMorph opens with, in mm. 5 plates: the trunk, which changes from rectangular to round, and 4 branch pieces unrolled, each with 32 divisions: 5 parts, 10 seams.

How it is developed

Triangulation of the trunk; parallel-line development of the branch pieces. 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 trunk divides into two bends, each turning 90 deg in 2 equal steps of 45 deg.
• The first step of both bends is taken at the junction: there the trunk and the first piece of each branch meet on three flat cuts, as in a Y-piece, with the cut between the two branches coming to a point at the crotch.
• The other 1 step of each branch is an ordinary miter joint, cut at 22.5 deg from square.
• The trunk changes from a rectangle 251 x 151 at End A to the circle at the junction, a transition whose plate is unfolded from the true lengths of the narrow triangles between End A and the cuts of the junction. Every branch piece is a cylinder, unrolled. Branch C is the mirror image of branch B.
• The seams of the branch pieces are on the outside of the Y (the throat of each bend); the trunk seam is on its +X side. Plates are drawn stacked with gaps, the trunk 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:

• End A is a square-cut rectangle and the branch ends square-cut circles (passed): The 32 points of End A lie on the rectangle 251 x 151 at Z = 0, with a point at every corner; the 64 points of the branch ends lie on circles of radius 100.5 square to their pipes (max deviation 4.26e-14).
• Every branch piece is a cylinder (passed): In all 4 branch pieces every generator is parallel to the piece's axis and at the pipe radius from it; the trunk changes from rectangular to round and is not a cylinder (max deviation 1.22e-13).
• Junction and miter joints planar and bisecting (passed): The three joints of the junction and the 2 miter joint(s) each lie in the plane that bisects the two pipes they join; the junction's joints meet at two points beside it (max deviation 3.55e-14).
• Surface connectivity (passed): 320 facets form one trunk dividing into two bends: every edge inside a part joins exactly two facets, every open edge one (V - E + F = -1).
• Developed edges equal true 3D lengths (passed): 960 facet edges compared between the 3D surface and the flat pattern (max deviation 9.45e-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 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 1.86e-9).
• 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 9.09e-13).
• Symmetric, with the two branches mirror images (passed): Every point has its mirror image across the XZ plane, and branch C is the mirror image of branch B across the plane between them (max deviation 1.37e-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 trunk and both branches have one diameter at the junction, and the two bends are alike, turning opposite ways in one plane.
• The first turn of each bend is at the junction, where three flat cuts meet; the others are miter joints that bisect adjacent pieces.
• 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.

Limits of this module

• The trunk is round at the junction, with the diameter of the branches, and the two bends are alike.
• The trunk changes shape, so it is not a cylinder: it is divided into narrow triangles and unfolded from their true lengths.
• The junction is a Y of three flat cuts at the first turn. Two complete bends that start side by side on the trunk and are joined along the plane between them are not covered.
• More turns need a larger bend radius, because the cut between the branches must end within the first piece.
• 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, the open end of the trunk, is the rectangle at Z = 0, centred on the origin. The trunk runs up the Z axis to the junction point. Branch B (End B) turns toward +X and branch C (End C) toward -X; all axes lie in the XZ plane, so the Front view shows the whole piece.

Bends. Each branch turns its bend angle, from 20 to 90 degrees (90 for a bullhead), in a number of equal turns. The bend radius R is to the pipe centreline and must exceed half the diameter. The pieces of a branch are 2 R tan(half a turn) long on their axes, and the last is half of that plus any straight length.

Junction. The first turn of both bends is taken at the junction point itself. There the trunk and the first piece of each branch meet on three flat cuts, exactly as the three pipes of a Y-piece do, and the cut between the two branches comes to a point at the crotch. Each turn must be at least 10 degrees.

Radius and number of turns. The cut between the branches reaches r / tan(turn) along the first piece of each branch and must end before that piece's next joint. Smaller turns make that cut longer and the pieces shorter, so more turns need a larger bend radius; the calculation tells you when the two do not fit.

Numbering. The circumference of every piece is divided into 4N equal parts, counted about the piece from its +X side toward +Y. A is the open end of the trunk; P1, P2, JTB, JTC and JBC are the points of the junction, as in the Y-piece; Bi_k and Ci_k are division k of ring i of each branch, the last ring being its open end. Plate S1 is the trunk, then the pieces of branch B from the junction outward, then those of branch C.

Rectangle to round. End A is a rectangle, its width along X, in the plane of the bends, and its depth along Y. The trunk changes from it to the circle of the junction: each division of the circle is matched to one of the rectangle, the quadrant points to the middles of its sides and a division to each corner, so the trunk is a transition between End A and the three cuts of the junction. The branches are round from the junction to their ends.

Development method. The trunk changes shape, so it is divided into narrow triangles between End A and its cuts and unfolded from their true lengths. Each branch piece is a cylinder, cut along its seam and unrolled.

Material thickness. A plate has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your sizes are inside, mid-thickness or outside sizes. The bend radius is to the centreline, the same for every surface. The 3D view and the views 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 branches

• V-Piece: Breeches pieces: one round inlet dividing into two round branches, equal or unequal, level or angled.
• Y-Piece: A trunk dividing into two branches on flat miter cuts, symmetric or as a lateral, the branches of the trunk diameter or tapering to smaller ends.
• Y-Piece to Rectangle: A round trunk dividing into two branches that change from round at the junction to rectangular at their ends.
• Twin Bend: A trunk dividing into two segmented bends that turn opposite ways: a bullhead at 90°, or any angle you give.
• Twin Bend to Rectangle: A round trunk dividing into two segmented bends that turn opposite ways and change to rectangular duct as they turn: a bullhead at 90°, or any angle you give.
• Three- or Four-Way: A round trunk that divides into three or four equal branches spaced evenly round it: a trifurcation or a four-way piece.
• Rectangular Y: A rectangular trunk dividing into two rectangular legs side by side: a trouser piece, with equal or unequal legs.
• Rectangle to Two Rounds: A rectangular inlet that divides into two round outlets side by side, as two square-to-rounds on one dividing edge.