Ventilation Branch: Correcting a Development Template
A 90° branch connection is being drawn for a ventilation duct. A Ø50 mm branch meets the side of a Ø90 mm main, with both cylinder axes in the same vertical plane and intersecting at right angles. The branch is to be developed by the parallel-line method at 1:1 on A2. The candidate divides the branch circumference into 12 equal parts, projects the division points straight down onto the branch elevation, and takes the cut line from the branch's own outline. The resulting saddle is 6 mm too shallow at the crown when checked against the main pipe. A template already cut from this development does not fit. The drawing is to be reissued with a corrected development and a check on the developed length.
1. Explain why the cut line was wrong and describe the correct construction of the intersection curve.
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The branch outline in elevation meets the main cylinder surface only at the two extreme sides. Between those points the real intersection curve lies inside the main pipe, so taking the cut line from the branch's own outline gives a shallow, incorrect saddle.
The correct construction uses the branch end view:
- Divide the branch circle in the end view (looking along the branch axis) into 12 equal parts and number the points.
- Project each point parallel to the branch axis into the elevation, giving 12 generator lines along the branch.
- Project the same points horizontally onto the main cylinder outline in the elevation, using the main pipe's centre line and profile.
- Where each generator meets the corresponding element of the main cylinder surface, mark the intersection point. Joining them gives the true curve, symmetrical about the vertical centre line.
- Use these heights, not the branch outline, to set out the development.
2. Justify the choice of 12 divisions and state the accuracy it gives for this duct.
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Each division subtends 360/12 = 30°, so half a division is 15°. The maximum radial error from approximating the circle by straight steps is r(1 − cos 15°) = 25 × 0.0341 ≈ 0.85 mm. For a 50 mm duct template cut at 1:1 and fitted on site, that is within a millimetre, so 12 divisions is acceptable.
The choice is a judgement, not a rule: use 24 divisions if the branch were larger, if the saddle had to be welded and leak-tight, or if the drawing were scaled down, because a fixed number of divisions on a larger circle gives a greater absolute error. Always divide the circumference, never the diameter, and keep the divisions equally spaced around the full 360°.
3. Calculate the developed length of the branch and explain how the template should be checked, including the error a stepped chord layout would cause.
Read the worked answer
The mean circumference is πD = π × 50 = 157.1 mm. The development base line must be this length, divided into 12 equal spaces of 157.1/12 = 13.09 mm.
If the candidate steps the chords instead of the arcs, each chord is 2 × 25 × sin 15° = 12.94 mm, and 12 chords total 155.3 mm. The template would then be about 1.8 mm short and would not close around the pipe, which matches the reported misfit. The fix is to lay out equal arc spaces of 13.09 mm along the base line and to set heights on the 13 generator lines, with the curve symmetrical about the crown generator.
Check the finished template by measuring its overall length against 157.1 mm, confirming symmetry, and wrapping it around the branch pipe before cutting metal.