Piping
Branch on elbow calculator
In short
A branch on an elbow is the intersection of a cylinder and a torus, (√(x² + y²) − R)² + z² = (D/2)². The calculation finds, point by point, where each branch generator meets the elbow, and gives its length from the square end: s(φ) = R + L − t(φ) on the outer side.
Calculation
Example calculated with the tool’s default values.
Inputs
- Elbow (pipe)
- DN150 · 168.3 × 4.5 mm
- Branch
- DN50 · 60.3 × 2.9 mm
- Elbow radius
- Bought long-radius elbow (EN 10253-2 3D or ASME B16.9 LR per pipe)
- Elbow angle
- 90°
- Branch position
- Outer side (in the elbow plane)
- Branch height
- 200 mm
- Layout surface
- Branch outside
- Generators
- 16 (every 22.5°)
Results
- Longest generator
- 121.4 mm
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Shortest generator
- 117.3 mm
- Saddle depth
- 4.1 mm
- Span on the elbow arc
- 11.0°
- Elbow radius
- 229.0 mmEN 10253-2 3D elbow
- Set-on: traced on the branch and elbow outside diameters; bevel and gap per the WPS.
- Case handled: branch at mid-arc, radial axis (outer side) or square to the elbow plane (flank). The hole in the elbow cannot be flattened: scribe it with the branch held in place.
Step-by-step calculation
Elbow surface (torus)
(√(x² + y²) − R)² + z² = (D/2)²
R = 229.0 ; D/2 = 84.2 mm
Contact (outer side)
t(φ) = √((R + √(a² − b² cos² φ))² − b² sin² φ)
φ = 0 : t = 307.6 ; φ = 90° : t = 311.7 mm
Generator
s(φ) = R + L − t(φ)
117.3 to 121.4 mm
Branch generators (from the square end)
| No. | Angle (°) | Length (mm) | Position around (mm) | Ordinate (from the high point) (mm) |
|---|---|---|---|---|
| 1 | 0.0 | 121.4 | 0.0 | 0.0 |
| 2 | 22.5 | 120.8 | 11.8 | 0.6 |
| 3 | 45.0 | 119.3 | 23.7 | 2.1 |
| 4 | 67.5 | 117.9 | 35.5 | 3.5 |
| 5 | 90.0 | 117.3 | 47.4 | 4.1 |
| 6 | 112.5 | 117.9 | 59.2 | 3.5 |
| 7 | 135.0 | 119.3 | 71.0 | 2.1 |
| 8 | 157.5 | 120.8 | 82.9 | 0.6 |
| 9 | 180.0 | 121.4 | 94.7 | 0.0 |
| 10 | 202.5 | 120.8 | 106.6 | 0.6 |
| 11 | 225.0 | 119.3 | 118.4 | 2.1 |
| 12 | 247.5 | 117.9 | 130.2 | 3.5 |
| 13 | 270.0 | 117.3 | 142.1 | 4.1 |
| 14 | 292.5 | 117.9 | 153.9 | 3.5 |
| 15 | 315.0 | 119.3 | 165.8 | 2.1 |
| 16 | 337.5 | 120.8 | 177.6 | 0.6 |
How to use it
- Pick the elbow pipe and the branch pipe: outside diameter and wall thickness.
- Pick the elbow radius (bought elbow by default, EN 10253-2, ASME B16.9, 1.5 D, 1 D or custom) and its angle (10° to 180°).
- Pick the position: outer side (radial axis, in the elbow plane) or flank (square to the elbow plane, with a shift towards the outer side).
- Enter the branch height to the square-cut end, then the layout surface and the number of generators.
- Read the longest generator.
Frequently asked questions
How do I calculate a branch on an elbow?
The elbow surface is a torus: (√(x² + y²) − R)² + z² = (D/2)², with R the elbow radius. On the outer side, branch generator φ meets the elbow at t(φ) = √((R + √(a² − b² cos² φ))² − b² sin² φ), where a is the elbow half OD and b the branch radius on the layout surface; its length from the square end is s(φ) = R + L − t(φ). The calculation is exact point by point, on 144 points.
What does a 60.3 branch on a 168.3 elbow give?
168.3 × 4.5 elbow at 90°, EN 10253-2 type 3D (R = 229 mm, D/2 = 84.2 mm), 60.3 × 2.9 branch on the outer side, 200 mm high: contact is at t = 307.6 mm for φ = 0 and 311.7 mm for φ = 90°. The generators run from 117.3 to 121.4 mm from the square end.
How do I make a pipe branch on elbow template?
The generators are laid out on a template that wraps around the branch, with one ordinate per generator (8, 12, 16, 24 or 32). The hole in the elbow cannot be flattened: scribe it with the branch held in place. Set-on: traced on the branch and elbow outside diameters; bevel and gap per the WPS.
Can the branch go on the elbow flank?
Yes: in the flank position, the branch axis is square to the elbow plane, shifted e towards the outer side (negative towards the inner side). The branch must stay on the flank: |e| + d/2 must not exceed D/2. The height is then measured from the elbow plane.
Limits
Indicative results. Case handled: set-on branch at mid-arc, radial axis (outer side) or square to the elbow plane (flank). The branch cannot be larger than the elbow or run off its arc. The calculation flags a branch almost tangent to the elbow or close to its butt welds.
Sources
- Intersection cylindre / tore : Géométrie analytique : équation du tore, génératrices du piquage (calcul exact point par point, 144 points)
- EN 10253-2:2021+A1:2026 (raccords à souder bout à bout) : Coudes 2D/3D (rayon R), tés égaux (F), réductions (L) ; tableaux Zickwolff, consultés le 08/11/2026 (link)