Cross: flat pattern

Two branches set onto opposite sides of a run: equal, reducing or unequal, with both saddle cuts and both holes in the run. Two round branches set onto opposite sides of a round run, equal to it or smaller: each branch with its saddle cut, and the run with its two holes.

Configurations

• Equal cross. Both branches have the diameter of the run and are square to it.
• Reducing cross. Both branches have one smaller diameter and are square to the run.
• Unequal branches. Each branch has its own diameter, angle and length.

A worked example

The equal cross configuration with the sizes SheetMorph opens with, in mm. 3 plates: the run with its two holes, and the two branches, each an unrolled cylinder with 32 divisions: 3 parts, 5 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.

• Two branches are set onto opposite sides of the run. Each ends on a saddle, the line where it meets the wall of the run, and a hole of the same shape is cut in the run for each.
• Both branches have the diameter of the run, so each saddle reaches half-way round it and the two meet at a point on each side of the run.
• The run is cut along its +Y line, between the two branches, so each hole lies whole in one half of its plate.
• The branches are alike, so their plates are mirror images of each other.
• The saddles are found on the run as it is drawn, with flat strips between its divisions, so each hole and its branch end agree exactly. The 3D view shows the run as a whole pipe; the weight is for the plates with the holes cut out.

Checked before it can be exported

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

• All four ends are square-cut circles (passed): All 128 end points lie in their end plane, square to their pipe, at the radius of that pipe (max deviation 8.53e-14).
• Run and branches are cylinders (passed): Every generator is parallel to the axis of its pipe (max deviation 0).
• Saddles on the run and on their branches (passed): All 64 saddle points lie on their branch cylinder and on the wall of the run, branch C on its +X side and branch D on its -X side (max deviation 8.53e-14).
• Surface connectivity (passed): 192 facets form a run and two branches 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): 576 facet edges compared between the 3D surface and the flat pattern (max deviation 3.72e-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 6.98e-9).
• Cut outlines closed and non-overlapping (passed): 3 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 1.83e-11).
• Holes in the run match the saddles (passed): Both holes lie within the run's plate and each has an edge as long as its saddle (max deviation 1.64e-14).
• Symmetric about the plane of the axes (passed): Every point has its mirror image across the XZ plane (max deviation 1.57e-13).
• Branches are mirror images (passed): Branch D is the mirror image of branch C across the run (max deviation 1.58e-13).

Assumptions

• The pattern is developed on the mid-thickness surfaces. Inside or outside diameters are converted to them by one thickness; lengths and angles are not changed.
• Both branches are set onto the run; they end on its wall and do not pass into it.
• The axes of the run and both branches lie in one plane, and both branch axes cross the run axis at the same point.
• 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

• Neither branch can be larger than the run, and both lie in one plane with the run, on its centre.
• Both branch axes cross the run at the same point; staggered branches are two tees.
• The branches are set onto the run, not passed through it.
• The 3D view and the orthographic views show the run without its holes.
• 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 at Z = 0 and End B at Z = run length. Branch C leaves toward +X (End C) and branch D toward -X (End D). The Front view shows all three pipes in the plane of their axes.

Branch position and length. Both branch axes cross the run axis at the same point, the branch position from End A. A branch length is measured along its axis from that point (on the run axis, not on its wall) to the centre of its open end. Each saddle must fall wholly on the run.

Branch angle. A branch angle is measured between the branch and the run toward End B: 90 degrees is square to the run, and an unequal branch can be from 20 to 90 degrees.

Saddles and holes. Each branch ends on the wall of the run, on a saddle-shaped line, and a hole of the same line is marked on the plate of the run. Two branches of the run's own diameter each reach half-way round it, so their holes meet at a point on each side of the run.

Numbering. The circumference of each pipe is divided into 4N equal parts. On the run, division 0 is toward +X, N toward +Y, 2N toward -X. On branch C division 0 faces End A; on branch D division 0 faces End B; division N is the +Y side of both. A and B are the ends of the run, C and D the open ends of the branches, JC and JD their saddles. Plate S1 is the run, S2 branch C and S3 branch D.

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 a branch plate at each division is the length of the branch wall there, listed as the ordinates.

Seams. The run is cut along its +Y line, between the two branches, so each hole lies whole in one half of its plate. The branch seams can be on the throat (the side facing End B), the heel or the +Y 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. Lengths and angles are the same for every surface. The 3D view and the views 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

• Tee or Lateral: A branch set onto a run: equal or reducing tees and angled laterals, with the saddle cut and the hole 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.