Library / Manufacturing / KEVOS® original book
Branched Fittings by the Numbers
Oblique cones, breeches, unequal and angled prongs, multi-way manifolds and pipes into cones, developed from coordinates
Develop branched sheet metal fittings with numbers: oblique cones and frustums, breeches with equal, unequal and angled prongs, three-way manifolds, and a pipe joining a cone.
Start readingAbout this book
A duct that must serve two machines, a chute that must feed two bins, a manifold that must collect from three hoods: each needs a fitting that splits one round section into several. The simplest is a tee, a branch pipe cut to sit on the side of the main. The smoothest is a fitting that divides the main gradually into prongs, each prong turning and shrinking towards its own duct, and meeting its neighbour along a crotch. Those fittings, called breeches, forks, wyes or manifolds, are some of the most satisfying work in sheet metal and some of the most feared.
This book develops them with numbers. It starts with the oblique cone, which is the building block of most branched fittings, and its frustum. It then develops two-pronged fittings with parallel ends, prongs at an angle, unequal prongs, prongs at different angles and fittings with three prongs, and ends with a round pipe joining an oblique cone and with making and checking the set. Along the way it proves a useful result: when every prong of a fitting is a cone standing on the main’s circle, every crotch lies in a plane, and the hardest line in the fitting can be found exactly. Each chapter calculates with one running case, includes a plausible mistake and how it was found, and ends with an exercise and its worked answer.
It is for sheet metal workers, ductwork and chute fabricators, apprentices and drafters, and for engineers who check branched fittings. It assumes school algebra and trigonometry and the basics of pattern development. The Library’s Sheet Metal Patterns by the Numbers covers tees and branches on round pipe, and Transitional Elbows by the Numbers covers the coordinate method and true lengths in more detail.
The running case
Every chapter follows one fictional job. This is an illustration. A ductwork fabricator is making the fittings for a fume-extraction system, called FX-4 on its drawings, in a steel fabrication shop with two welding bays. The system takes fume from wall hoods in each bay, combines the branches, and runs a Ø355 mm main to a filter unit. Its branched fittings are:
- G1, a wall hood that tapers from a Ø600 mm skirt to a Ø250 mm duct, flat on the wall side, an oblique cone frustum;
- G2, a breeches from the Ø355 main to two Ø250 branches with ends parallel to the main’s;
- G3, a breeches from the Ø355 main to two Ø250 branches angled 30° outward;
- G4, a breeches with unequal prongs, Ø300 and Ø200, for bays of different sizes;
- G5, a wye with one Ø300 prong straight on and one Ø200 prong at 45°;
- G6, a three-way manifold from a Ø400 duct to three Ø230 branches;
- G7, a Ø150 pipe joining the side of G1’s hood.
The fittings are made in 0.8 mm galvanised steel sheet. The fabricator, the shop, the dimensions and the mistakes are invented. None describes KEVOS® work or a real business.
How to use the book
Read Chapters 1 to 3 first: they set out branched fittings, the oblique cone and its frustum, which every later chapter uses. Chapters 4 to 8 develop the breeches and manifolds in order of difficulty, and each relies on the one before. Chapter 9 deals with a pipe meeting a cone, and Chapter 10 with allowances, assembly, checks and the material list. Values are given to 0.1 mm so that the arithmetic can be followed; patterns are usually marked to the nearest half millimetre. Every number in the book was calculated by running it.
Source limits and professional responsibility
The cone patterns in this book are exact, and the triangulated prongs are as accurate as their divisions allow. Seam, lap and flange allowances are illustrations, to be calibrated with trial pieces in the shop’s own sheet. Fume-extraction systems must be designed for their duty and for the hazards of the fume they carry, and the balance of flow between branches is a matter for the system’s designer. This book teaches pattern geometry. It is not a design for any system, and qualified review governs real work.
Who this book is for
Sheet metal workers, ductwork and chute fabricators, apprentices and drafters who develop patterns for fittings that split one duct into several, and engineers who check them.
What you will learn
- Develop an oblique cone and its frustum by radial lines from coordinates, including one that is flat on one side.
- Find where the prongs of a branched fitting meet, and prove when that crotch is a flat curve and when it is not.
- Develop breeches with equal, unequal and angled prongs, and manifolds with three or more.
- Find the line where a round pipe meets an oblique cone, and cut both the pipe and the hole in the cone's pattern.
- Allow, assemble, check and list a set of branched fittings so that each prong meets its duct.
Inside this book
Chapters
- 01Branched fittings and their partsTees, wyes, breeches and manifolds compared; prongs, crotches and the area rule; the cone as building block; FX-4's seven fittings; and a breeches whose prong ends collided.
- 02The oblique coneFind an oblique cone's apex from its two circles, calculate its elements from coordinates, lay out its pattern by radial lines through a 120.21° fan, and catch a right-cone formula applied to it.
- 03The frustum of an oblique coneCut G1's cone at the duct to make the hood: frustum elements from the apex or straight from coordinates, a two-piece pattern that fits the sheet, and elements read from the elevation 30.7 mm short.
- 04Two prongs with parallel endsDevelop G2, a Ø355 to 2 × Ø250 breeches with parallel ends: each prong an oblique frustum, the crotch an ellipse in the plane of symmetry, prong patterns by radial lines, and prongs scribed in place.
- 05Breeches with prongs at an angleDevelop G3, a Ø355 to 2 × Ø250 breeches with prong ends tilted 30° outward: triangulated prongs, a crotch in the plane of symmetry peaking at 370.8 mm, and a peak taken from the parallel-end formula.
- 06Unequal prongsDevelop G4, a breeches to Ø300 and Ø200 prongs: why two cones on one base meet in a plane, formulas for the crotch's feet and peak, both prongs' patterns, and a crotch on the centre line.
- 07Prongs at different anglesDevelop G5, a wye with a straight Ø300 prong and a Ø200 prong at 45°: a cone meeting a triangulated prong, a crotch within 1.1 mm of a plane, and a division that skipped the foot.
- 08Any number of prongsDevelop G6, a Ø400 manifold to three Ø230 prongs round its axis: three flat crotches meeting at a triple point 367.3 mm up, one pattern for all three prongs, and a row manifold set out as a ring.
- 09A pipe joining an oblique coneFind where G7, a Ø150 pipe, meets G1's oblique cone from its circular sections, cut the pipe's end and the hole in the hood's pattern, and catch a pipe cut square to a back that only looks flat.
- 10Making and checking branched fittingsFinish the FX-4 fittings: allowances, crotch seams and the order of riveting, a check sheet for every fitting, sheet and mass for the job, and a crotch riveted from the peak down.
Apply This in Practice
No tool has been matched to this book yet. Browse the related tools: Manufacturing & Production tools
References and source notes
The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 26 October 2026.
- F. S. Kidder, The Sheet Metal Publication Company, New York. Triangulation Applied to Sheet Metal Pattern Cutting (sixth edition) (1924). Supplied source, first published 1917, read as a subject map of branched fittings, in particular its oblique cones and frustums, two-pronged fittings, breeches, fittings with unequal and radiating prongs, fittings with any number of prongs, and a pipe joining an oblique cone. No text, figures, constructions, tables or exercises are reproduced; the coordinate method, the running case, its numbers, the planar-crotch result and the checks are this book's own.
- Wolfram MathWorld, E. W. Weisstein. Cone (2026). Reference for right and oblique cones and their elements, used in Chapters 2 and 3.
- Wolfram MathWorld, E. W. Weisstein. Conic Section (2026). Reference for plane sections of a cone, used in Chapters 4 to 8 to explain why the crotch of two cones on a common base lies in a plane.
- Wolfram MathWorld, E. W. Weisstein. Developable Surface (2026). Reference for which surfaces unroll without stretching, used to separate the exact cone prongs from the triangulated ones.
- Standards Australia. AS 4254.2 Ductwork for air-handling systems in buildings, Part 2: Rigid duct (edition as called up). Named by number and scope as the Australian standard for rigid ductwork in buildings. The running case is an industrial extraction system, and nothing in this book claims compliance with it.
- Standards Australia. AS 1397 Continuous hot-dip metallic coated steel sheet and strip (edition as called up). Named by number and scope as the product standard for galvanised sheet of the kind used in the running case. No values are quoted from it.