Pipe Cut at an Angle: flat pattern

A length of round pipe with one or both ends cut at an angle: mitered ends, wedges and oblique lengths. A length of round pipe with one or both ends cut at an angle: a mitered end, a wedge or an oblique length, developed as one unrolled plate.

Configurations

• One end cut. End A is square; End B is cut at an angle.
• Both ends cut. Each end is cut at its own angle. Equal angles give an oblique length with parallel ends; opposite angles give a wedge.

A worked example

The one end cut configuration with the sizes SheetMorph opens with, in mm. 1 plate: an unrolled cylinder with 32 divisions: 1 part, 1 seam.

How it is developed

Parallel-line development (unrolled cylinder). 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.

• A cylinder is developable, so the plate is exact apart from the division of the circumference: it is as wide as the developed circumference, and its height at each division is the length of the pipe wall there.
• A cut at an angle is an ellipse on the pipe and a sine-shaped curve on the plate, drawn here as straight lines between divisions.
• Division points lie on the true circle, so the plate width (a sum of chords) is slightly less than pi x D.
• The seam is on the -X side of the pipe.

Checked before it can be exported

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

• Both ends lie in their cut planes (passed): All 64 end points lie on the pipe, at radius 100.5, in planes cut at 0 deg (End A) and 30 deg (End B) from square (max deviation 3.55e-14).
• The pipe is a cylinder (passed): Every generator is parallel to the pipe axis (max deviation 0).
• Surface connectivity (passed): 64 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): 192 facet edges compared between the 3D surface and the flat pattern (max deviation 1.88e-12).
• Shared edges and seam closure (passed): 63 internal edges are shared vertex-for-vertex between adjacent facets; 1 seam(s) have mating edges of equal true length (max deviation 5.68e-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 2.36e-9).
• Cut outlines closed and non-overlapping (passed): 1 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 5.80e-12).
• Symmetric about the plane of the cuts (passed): Every point has its mirror image across the XZ plane (max deviation 8.53e-14).

Assumptions

• The pattern is developed on the mid-thickness surface. An inside or outside diameter is converted to it by one thickness; the length and the cut angles are not changed.
• Both cuts are tilted about the Y axis, so the pipe is symmetric about the XZ plane.
• 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 the outline.
• The plate is drawn outside face up, End A side down.

Limits of this module

• Both cuts are tilted about the same axis; cuts turned relative to each other about the pipe are not covered.
• The bevel of a cut through the sheet thickness is not modelled.
• 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 pipe stands on the Z axis, End A centred on the origin and End B centred at Z = length. Both cuts are tilted about the Y axis, so the Front view shows them as straight lines.

Length. The length is measured on the axis of the pipe, between the centres of the two ends. The wall is longer on one side and shorter on the other by half the diameter times the difference of the tangents of the two cut angles.

Cut angle. A cut angle is measured from a square cut, from -75 to 75 degrees. A positive angle raises the +X side of the cut and lowers the -X side, at either end. So equal angles at the two ends give parallel cuts, and opposite angles a wedge. The two cuts must not meet.

Numbering. The circumference is divided into 4N equal parts, N per quarter: division 0 is toward +X, N toward +Y, 2N toward -X, 3N toward -Y. A<k> and B<k> are division k at End A and End B.

Development method. Parallel-line development. The pipe is cut along its seam and unrolled: the plate is as wide as the circumference and its height at each division is the length of the wall there, L + (D/2) cos(angle) (tan aB - tan aA). The listed ordinates are those heights.

Views. A square end is dimensioned with its diameter. A cut end is an ellipse: the Front view gives its length up the slope, D / cos(angle), and its angle.

Material thickness. The 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 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 pipe ends and nozzles

• Pipe on Plate: A round pipe standing on a flat or sloping plate, such as a roof: the pipe cut to the slope, and the plate with its hole.
• Nozzle on Head: A round nozzle set onto a dished or hemispherical head, on its centre or off it: the nozzle unrolled with its saddle cut.
• Pipe on Elbow: A round branch set onto a segmented bend, on its outside or on its side: the branch unrolled with the saddle that fits the gores.
• Nozzle on Sphere: A round nozzle on a sphere made in zones, radial or upright: the nozzle unrolled with its cut, and the zone it stands on with the hole marked.
• Bellmouth: A flared inlet to a round duct or a fan, with a straight spigot: made in rings, each part of a cone, or in petals pressed to the curve.
• Roof Flashing: The flashing where a flue or vent passes through a pitched roof: the base plate on the slope with its hole, and the sleeve cut to the pitch.