Tee / Lateral: flat pattern

A branch set onto a run: equal or reducing tees and angled laterals, with the saddle cut and the hole in the run. A round branch set onto a round run, of the same or a smaller diameter, square to it or at an angle: the branch with its saddle cut, and the run with its hole.

Configurations

• Equal tee. The branch has the diameter of the run, is square to it and on its centre.
• Reducing tee. A smaller branch, square to the run, on its centre or offset to one side.
• Lateral. A branch at an angle to the run, of the same or a smaller diameter, on centre or offset.

A worked example

The equal tee configuration with the sizes SheetMorph opens with, in mm. 2 plates: the run with its hole, and the branch, each an unrolled cylinder with 32 divisions: 2 parts, 3 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 branch is set onto the run. Its end is cut to the saddle, the line where it meets the wall of the run, and a hole of the same shape is cut in the run.
• The branch has the diameter of the run, so the saddle reaches half-way round the run and comes to a point on each side.
• Both pipes are cylinders, which are developable: each plate is as wide as the developed circumference. The run is cut along the line opposite the branch, so the hole is in the middle of its plate.
• The saddle is found on the run as it is drawn, with flat strips between its divisions, so the hole and the branch end agree exactly with each other; more divisions bring both closer to the true curve.
• The 3D view shows the run as a whole pipe with the joint drawn on it. The weight is for the plates with the hole cut out.
• Division points lie on the true circles, so each plate width (a sum of chords) is slightly less than pi x D.

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 on the run and 150.5 on the branch (max deviation 8.53e-14).
• Run and branch are cylinders (passed): Every generator of the run is parallel to the run axis and every generator of the branch to the branch axis (max deviation 0).
• Saddle on both pipes (passed): All 32 saddle points lie on the branch cylinder and on the wall of the run (max deviation 8.53e-14).
• Surface connectivity (passed): 128 facets form a run and a branch set onto it: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 384 facet edges compared between the 3D surface and the flat pattern (max deviation 3.61e-12).
• Shared edges and seam closure (passed): 126 internal edges are shared vertex-for-vertex between adjacent facets; 2 seam(s) have mating edges of equal true length (max deviation 2.84e-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 9.08e-9).
• Cut outlines closed and non-overlapping (passed): 2 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 1.87e-11).
• Hole in the run matches the saddle (passed): The hole lies inside the run's plate and its edge is as long as the saddle, 1148.0333 (max deviation 1.29e-14).
• Symmetric about the plane of the axes (passed): Every point has its mirror image across the XZ plane (max deviation 1.46e-13).

Assumptions

• The pattern is developed on the mid-thickness surfaces. Inside or outside diameters are converted to them by one thickness; lengths, the branch angle and the offset are not changed.
• The branch is set onto the run: it ends on the wall of the run and does not pass into it.
• The axis of the branch lies in a plane parallel to the XZ plane; it crosses the run axis unless it is offset.
• The open ends are cut square to their pipes.
• 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 branch cannot be larger than the run; make the larger pipe the run.
• The branch is set onto the run, not passed through it; a cross (two branches) is not covered.
• The saddle is worked out on the mid-thickness surfaces; bevels and the fit through the thickness are not modelled.
• The 3D view and the orthographic views show the run without its hole.
• Plate edges are straight between divisions, not smooth curves.
• No rolling allowance, and no seam, lap, flange, weld or trim allowances.

Conventions

Coordinate system. Right-handed, Z vertical in the views. The run stands on the Z axis: End A is its circle at Z = 0 and End B its circle at Z = run length. The branch leaves toward +X and ends at End C. The Front view shows the run and the branch in the plane of their axes.

Branch position and length. The branch position is measured along the run from End A to the point where the branch axis crosses the run. The branch length is measured along the branch axis from that point (on the run axis, not on its wall) to the centre of the open end. The whole saddle must fall on the run.

Branch angle. The branch angle is measured between the branch and the run toward End B: 90 degrees is a tee, and a lateral can be from 20 to 90 degrees. The side of the branch facing End B is its throat, the short side; the side facing End A is its heel.

Offset. A branch smaller than the run can be moved to one side: the offset is the distance of its axis from the run axis, along Y, positive toward +Y. The branch must stay wholly on the run, so the offset is at most the difference of the two radii.

Saddle and hole. The branch ends on the wall of the run, on a saddle-shaped line. The branch plate is cut to that line at one edge, and a hole of the same line is marked on the plate of the run. An equal branch reaches half-way round the run.

Numbering. The circumference of each pipe is divided into 4N equal parts, N per quarter. On the run, division 0 is toward +X (under the branch), N toward +Y, 2N toward -X. On the branch, division 0 is the heel, N the +Y side, 2N the throat. A and B are the ends of the run, C the open end of the branch and J the saddle. Plate S1 is the run and S2 the branch.

Development method. Parallel-line development. Each pipe is a cylinder: it is cut along its seam and unrolled to a plate as wide as its circumference. The height of the branch plate at each division is the length of the branch wall there, listed as the ordinates.

Seams. The run is cut along the line opposite the branch, so the hole is in the middle of its plate. The branch seam can be on its throat, its heel or a side.

Material thickness. A plate has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your diameters are inside, mid-thickness or outside sizes: inside diameters are increased by one thickness and outside diameters reduced by one thickness. Lengths, the angle and the offset are the same for every surface. The 3D view and the views always show the diameters 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 intersections

• Cross: Two branches set onto opposite sides of a run: equal, reducing or unequal, with both saddle cuts and both holes in the run.
• Manifold: A round run with a row of equal branches at an even pitch: the run unrolled with every hole marked, and the branches with their saddles.
• Cone on Pipe: A cone set onto a round pipe, square as a boot or hopper outlet, or at an angle as a conical lateral, with the hole in the pipe.
• Pipe on Cone: A round branch set square onto a conical shell, such as a nozzle on a hopper, with the hole in the cone.
• Cone on Cone: A cone set square onto a conical shell, such as a reducing branch on a hopper, with the hole in the main cone.
• Rectangular Tee: A rectangular branch set square onto a rectangular duct, straight or with a shoe, with the hole marked on the duct.
• Rectangle on Round: A rectangular branch set square onto a round run, on its centre or offset: the run with its hole, and the branch folded at its corners.