Y-Piece: flat pattern

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. A round trunk dividing into two round branches, joined on flat miter cuts: branches of the trunk diameter, each developed as its own unrolled plate, or branches that taper from it to smaller ends.

Configurations

• Equal branches. Both branches leave the trunk at the same angle and have the same length.
• Unequal branches. Each branch has its own angle and length. A branch at 0 degrees runs straight on from the trunk, giving a lateral branch.

A worked example

The equal branches configuration with the sizes SheetMorph opens with, in mm. 3 plates, each an unrolled cylinder with 32 divisions: 3 parts, 6 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 Y-piece is made of three pipes of one diameter: the trunk and two branches. Their axes meet at the junction point.
• Pipes of equal diameter whose axes meet join on a flat cut, in the plane that bisects their axes. Each pipe is cut by two such planes, half of its circumference by each, and the three joints all run between the same two points beside the junction.
• The cut between the two branches comes to a point at the crotch; the cuts between the trunk and each branch form the throats.
• A cylinder is developable, so each plate is exact apart from the division of the circumference: its width is the developed circumference, one edge is straight (the open end) and the other follows the two 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 a rolled pipe must close exactly on the true diameter.
• Three miter joints and three longitudinal seams are to be joined: the trunk seam on its +X side, the branch seams on the outside of each branch.

Checked before it can be exported

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

• All three ends are square-cut circles (passed): All 96 end points lie in their end plane, square to their pipe, at radius 150.5 (max deviation 5.68e-14).
• Trunk and branches are cylinders of the pipe diameter (passed): In all three pipes every generator is parallel to the pipe axis and at the pipe radius from it, and the three axes pass through the junction point (max deviation 6.12e-14).
• Joints planar, bisecting, and meeting at two points (passed): The three joints each lie in the plane that bisects their two pipes (max deviation 2.66e-14), each belongs to exactly those two pipes, and all three run between the same two points beside the junction.
• Surface connectivity (passed): 192 facets form one trunk dividing into two branches: every edge inside a part joins exactly two facets, every open edge one (V - E + F = -1).
• Developed edges equal true 3D lengths (passed): 576 facet edges compared between the 3D surface and the flat pattern (max deviation 2.12e-12).
• Shared edges and seam closure (passed): 189 internal edges are shared vertex-for-vertex between adjacent facets; 3 seam(s) have mating edges of equal true length (max deviation 1.14e-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 4.66e-10).
• Cut outlines closed and non-overlapping (passed): 3 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 straight edge, as on an unrolled cylinder (max deviation 4.55e-12).
• Symmetric about the plane of the axes (passed): Every point has its mirror image across the XZ plane (max deviation 2.34e-13).
• Branches are mirror images (passed): Branch C is the mirror image of branch B across the plane between them (max deviation 1.58e-13), and their plates have equal area (difference 1.16e-10).

Assumptions

• The pattern is developed on the mid-thickness surface. An inside or outside diameter is converted to it by one thickness; lengths and angles are measured along the pipe axes and are not changed.
• The trunk and both branches have the same diameter and their axes meet at one point, so every joint is a flat cut in the plane that bisects two axes.
• The open ends are cut square to their pipes.
• 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

• Branches keep the diameter of the trunk, or taper from it at the junction. For branches that start at another diameter, use the V-piece.
• Tapered branches have their seams in fixed places: the trunk on its +X side and each branch on the outside of the Y.
• All three axes meet at one point and lie in one plane.
• The open ends are square cuts; inclined ends are not covered.
• 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.
• Sheet nesting, grain direction and stock-size checks are not performed.

Conventions

Tapered branches. The branches may taper: each starts at the diameter of the trunk at the junction, which is the junction of the Y-piece, and narrows along its own axis to the end diameter given. A tapered branch is part of a cone, developed from its true lengths. The seams are then fixed: the trunk on its +X side and each branch on the outside of the Y.

Coordinate system. Right-handed, Z vertical in the views. End A, the open end of the trunk, is the circle at Z = 0, centred on the origin. The trunk runs up the Z axis to the junction point, where the three axes meet. Branch B (End B) leaves toward +X and branch C (End C) toward -X; all three axes lie in the XZ plane, so the Front view shows the Y.

Lengths. Every length is measured along a pipe axis: the trunk length from End A to the junction point, and each branch length from the junction point to the centre of its open end. The open ends are cut square to their pipes.

Branch angle. The branch angle is measured at the junction point between the trunk axis continued and the branch axis: 0 runs straight on, 90 is square to the trunk. Two equal branches include twice the branch angle. The two angles together must be at least 20 degrees.

Joints. The trunk and both branches have the same diameter, so each pair of pipes joins on a flat cut in the plane that bisects their axes. There are three joints: trunk to branch B, trunk to branch C, and branch B to branch C, whose top is the crotch. All three run between the same two points, one on each side of the junction. Each pipe must be longer than its cuts reach.

Numbering. The circumference of every pipe is divided into 4N equal parts, N per quarter, counted about the pipe from its +X side toward +Y: division 0 toward +X, N toward +Y, 2N toward -X, 3N toward -Y. A, B and C are the three open ends. P1 and P2 are the two points where the joints meet; JTB, JTC and JBC are the points of the three joints, numbered from P1 to P2. Plate S1 is the trunk, S2 branch B and S3 branch C.

Development method. Parallel-line development. Each pipe is a cylinder, which is developable: it is cut along its seam and unrolled. The plate is as wide as the circumference; its height at each division is the length of the pipe wall there, listed as the ordinates. The cut between two pipes reaches r / tan(half the angle between their axes) from the junction point.

Seams. Each pipe has one longitudinal seam. The trunk seam can be on any of its four quadrant lines; on the +Y or -Y side it ends where the three joints meet, which is best avoided. The branch seams are on the outside of the Y or on the crotch side, where the branch is shortest.

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. Lengths and angles are 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 branches

• V-Piece: Breeches pieces: one round inlet dividing into two round branches, equal or unequal, level or angled.
• 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.
• Flat-Oval Tee: A round or flat-oval branch on the flat side of a flat-oval main: the main with its hole, and the branch cut to the main.