Cladding Reducer: flat pattern

The jacket over an insulated reducer, concentric or eccentric: a cone between the outsides of the insulation on the two pipes. The jacket over an insulated reducer, concentric or eccentric: a cone between the outsides of the insulation on the larger and the smaller pipe.

Configurations

• Concentric. The two pipes share one centreline.
• Eccentric (flat on one side). The jacket is flat along one side, on -Y, as over an eccentric reducer laid flat side down.

A worked example

The concentric configuration with the sizes SheetMorph opens with, in mm. Triangulated approximation: the wall is 64 flat facets between 32 rulings: 4 parts, 1 seam.

How it is developed

Triangulation (true-length unfolding). 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.

• Both circles are divided into 32 equal parts and matching divisions are joined by rulings. Each strip between two rulings is split along its shorter diagonal into two triangles, which are unfolded from their true lengths.
• This is a right cone, which develops exactly into a sector of an annulus. The triangulated pattern approximates that sector with straight chords; the closed-form radii and sector angle are listed for comparison.
• Division points lie on the true circles, so each developed round edge (a chain of chords) is slightly shorter than its true circumference. Increase the divisions, or use the matched-circumference model, if the joints to round ducts must close exactly.
• The wall is rolled, not folded: rulings are shown as forming lines, and the four quadrant lines mark the quarters.
• One-piece pattern with 1 seam along a quadrant line.

Checked before it can be exported

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

• End A points on one circle, equally spaced (passed): 32 division points lie in the End A plane at radius 134.5 with equal chords (max deviation 1.28e-13).
• End B points on one circle, equally spaced (passed): 32 division points lie in the End B plane at radius 107.5 with equal chords (max deviation 1.03e-13).
• 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 2.56e-13).
• 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 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 1.75e-10).
• Cut outlines closed and non-overlapping (passed): 1 closed outline(s); no outline crosses itself or another outline.
• Expected symmetry (passed): left/right and top/bottom mirror symmetry holds: every mirrored circle point coincides (max deviation 1.91e-13).

Assumptions

• The pattern is developed on the mid-thickness surface. Inside or outside diameters are converted to it by one thickness; mid-thickness diameters are used unchanged.
• Each circle is represented by equally spaced division points joined by straight chords.
• Matching divisions of the two circles are joined; each strip between rulings is split along its shorter diagonal.
• No rolling allowance, and no seam, lap, flange, weld or trimming allowance, is added to any outline.
• End-plane twist about the Z axis is not supported.
• The pattern is drawn outside face up.

Limits of this module

• The insulation is taken as an even thickness all round; a jacket over insulation of uneven thickness, or over trace heating, is not modelled.
• No swage, bead or crimp is formed on the edges: the laps are flat overlaps added by Fabrication.
• The round edges of the pattern are chains of chords, not smooth arcs.
• No rolling allowance or thickness-dependent rolling correction.
• Inside/outside conversion is one thickness on each diameter; the extra difference on a steeply inclined wall is not modelled.
• No seam, lap, flange, weld or trim allowances; no straight collars on either end.
• No end-plane twist about the Z axis.
• Sheet nesting, grain direction and stock-size checks are not performed.

Conventions

Pipe and insulation. Give the outside diameter of the pipe and the thickness of the insulation. The jacket fits over the insulation, so its diameter is the pipe and twice the insulation, taken as an inside size: leave the sizes as inside sizes, and the plate is worked out on its mid-thickness.

Reducer. The jacket stands on the Z axis with the larger pipe at End A, Z = 0, and the smaller at End B. Eccentric, its two ends are flush on -Y, so the jacket is straight along that side.

Laps. The plates are cut net. The laps, where one sheet runs over the next, come from Fabrication: a cladding module starts with a lap seam, the same overlap on the long seam of each sheet and on one side of each joint between sheets. Change it there to your specification.

Coordinate system. Right-handed. X is horizontal (+X right), Y is vertical (+Y top), Z is the reference axis. End A is centred on the origin; End B is centred on (dx, dy, L). Left, right, top and bottom are as seen looking from End B back toward End A.

Length and offsets. L is the distance between the two end centres measured along Z, not the slant height. dx and dy are the End B centre minus the End A centre: positive dx moves End B right, positive dy moves it up.

End-plane inclination. Each end can be inclined about two axes through its own centre, so the centre does not move. "About X" turns the end about the horizontal axis; positive moves its top toward +Z. "About Y" turns it about the vertical axis; positive moves its right side toward -Z. The rotation about X is applied first. Limit: +/-75 deg. An inclined end stays a true circle.

Flush sides. Making a side flush (left, right, top or bottom) places both circles tangent to one plane on that side, so that side of the cone is straight. The centre offset on that axis is then calculated: without inclination, left gives dx = (DB - DA) / 2 and bottom gives dy = (DB - DA) / 2 (right and top the opposite sign).

Numbering. Each circle is divided into 4N equal parts, N per quarter. Divisions A0, A1, ... and B0, B1, ... run counter-clockwise (seen from End B) from the left quadrant point, so the quadrant points are 0 left, N bottom, 2N right, 3N top. Division k of End A is joined to division k of End B by ruling Gk.

Quarters. The four quarters between quadrant lines are labelled Q0_BL (left to bottom), Q1_BR, Q2_TR and Q3_TL.

Development method. Each strip between two rulings is split along its shorter diagonal into two triangles. All triangles are unfolded in sequence from their true 3D edge lengths, each sharing an edge with the previous one. The round edges of the pattern are chains of straight chords. For a right cone the exact sector (slant height, developed radii, sector angle) is reported alongside for comparison.

Circle model. Inscribed: division points are on the true circles, and each developed round edge is slightly shorter than pi x D (0.16% with 8 divisions per quarter). Matched circumference: points are on marginally larger circles so each developed round edge equals pi x D exactly.

Seam and layout. The seam runs along a quadrant line (left, bottom, right or top). One piece has one seam; two halves have seams on two opposite quadrant lines.

Material thickness. A flat pattern 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 sizes are increased by one thickness and outside sizes reduced by one thickness to reach the mid-thickness surface, on which the pattern is developed. 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 cladding

• Elbow: The jacket over an insulated pipe bend, in mitred gores made to the pipe’s bend radius and the outside of the insulation, with laps between them.
• Tee: The jacket over an insulated tee, equal or reducing: the run with the hole for the branch, and the branch cut to the saddle.
• Flange Box and End Cap: A removable box in two halves over a pair of insulated flanges, or the cap that closes the insulation where it ends on a pipe.