Piping
Miter bend calculator
In short
A miter bend of angle Θ is made of m full pieces and two end pieces, so m + 1 welds. Each cut is inclined α = Θ / (2 (m + 1)); a full piece is 2 (R + D/2) tan α long on the back and 2 (R − D/2) tan α on the belly.
Calculation
Example calculated with the tool’s default values.
Inputs
- Pipe
- DN100 · 114.3 × 3.6 mm
- Bend angle
- 90°
- Number of full pieces
- 2
- Centreline radius
- 1.5 × OD
- Straight length at ends
- 0 mm
- Root gap
- 0 mm
- Marks around
- 12
- Template to print
- Full piece
Results
- Cut angle at each joint
- 15.00°
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Number of welds
- 3
- Full piece: outside
- 122.5 mm
- Full piece: centreline
- 91.9 mm
- Full piece: inside
- 61.3 mm
- End piece: outside
- 61.3 mm
- End piece: centreline
- 45.9 mm
- End piece: inside
- 30.6 mm
- Take-out (vertex to face)
- 171.4 mm
- Pipe length on centreline
- 275.6 mm
- Centreline radius used
- 171.4 mm
- Lengths on the outside diameter, no root gap: enter the gap to take it off at each joint.
Step-by-step calculation
Cut angle
α = Θ / (2 (m + 1))
90.0 / (2 × 3) = 15.00°
Full piece, outside
Le = 2 (R + D/2) tan α
2 × 228.6 × tan 15.00° = 122.5 mm
Full piece, inside
Li = 2 (R − D/2) tan α
2 × 114.3 × tan 15.00° = 61.3 mm
End piece, centreline
Lb = R tan α + straight
171.4 × tan 15.00° + 0.0 = 45.9 mm
Take-out
E = R tan(Θ / 2) + straight
171.4 × tan 45.0° + 0.0 = 171.4 mm
Generator no. k
φ = 360° (k − 1) / n ; l = 2 (R + D/2 cos φ) tan α − j
n = 12
Check: centreline arc
A = R × Θ (rad)
171.4 × 1.5708 = 269.3 mm
Pieces in order (first and last = ends)
| Piece | Back (outside) (mm) | Centreline (mm) | Belly (inside) (mm) |
|---|---|---|---|
| 1 | 61.3 | 45.9 | 30.6 |
| 2 | 122.5 | 91.9 | 61.3 |
| 3 | 122.5 | 91.9 | 61.3 |
| 4 | 61.3 | 45.9 | 30.6 |
Length on each generator
| No. | Angle from the back (°) | Position around (mm) | Full piece (mm) | End piece (mm) |
|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 122.5 | 61.3 |
| 2 | 30.0 | 29.9 | 118.4 | 59.2 |
| 3 | 60.0 | 59.8 | 107.2 | 53.6 |
| 4 | 90.0 | 89.8 | 91.9 | 45.9 |
| 5 | 120.0 | 119.7 | 76.6 | 38.3 |
| 6 | 150.0 | 149.6 | 65.4 | 32.7 |
| 7 | 180.0 | 179.5 | 61.3 | 30.6 |
| 8 | 210.0 | 209.5 | 65.4 | 32.7 |
| 9 | 240.0 | 239.4 | 76.6 | 38.3 |
| 10 | 270.0 | 269.3 | 91.9 | 45.9 |
| 11 | 300.0 | 299.2 | 107.2 | 53.6 |
| 12 | 330.0 | 329.2 | 118.4 | 59.2 |
How to use it
- Pick the pipe: outside diameter and wall thickness.
- Enter the bend angle (1° to 180°) and the number of full pieces between the two end pieces (0 to 12).
- Pick the centreline radius: 1, 1.5, 2 or 3 × OD, or custom.
- Add a straight length at the ends and the root gap if needed, then the number of marks around (8, 12, 16, 24 or 32).
- Read the cut angle at each joint.
Frequently asked questions
What is the miter bend calculation formula?
The centreline is the polygon drawn around the arc of radius R. With m full pieces, the cut angle is α = Θ / (2 (m + 1)). On generator no. k, at φ = 360° (k − 1) / n from the back, a full piece is l = 2 (R + D/2 cos φ) tan α − j long. An end piece is R tan α on the centreline, plus any straight length.
What does a 90° mitre bend calculation give?
Example: 114.3 pipe, 90° bend, two full pieces, centreline radius 1.5 × OD (171.4 mm). Cut angle: 90 / (2 × 3) = 15°; 3 welds. Full piece: 2 × 228.6 × tan 15° = 122.5 mm on the back and 2 × 114.3 × tan 15° = 61.3 mm on the belly. End piece: 171.4 × tan 15° = 45.9 mm on the centreline. Take-out: 171.4 × tan 45° = 171.4 mm.
What cut angle for a segmented elbow?
It depends on the number of pieces: α = Θ / (2 (m + 1)). Above 22.5° per cut, the calculation warns that ASME B31.3 § 304.2.3 strongly reduces the allowable pressure of multiple miter bends: add pieces.
Where is the root gap taken off?
Each joint takes the gap j on the line, that is j / 2 from each neighbouring piece. With no gap entered, lengths are on the outside diameter with no root gap. If the pieces get too short on the belly, use fewer pieces, a larger radius or a smaller gap.
How do I check a lobster back bend?
The take-out from vertex to face is E = R tan(Θ / 2) + straight. The centreline arc A = R × Θ (in radians) is a check: 171.4 × 1.5708 = 269.3 mm for the 90° example.
Limits
Indicative results. Lengths are on the outside diameter. The centreline radius must exceed half the diameter; below one diameter, the belly is very short and fit-up and welding are difficult. Allowable pressure of multiple miter bends: ASME B31.3 § 304.2.3, not calculated here.
Sources
- Géométrie du coude recoupé : Polygone circonscrit à l’arc de rayon R ; développé de la section oblique d’un cylindre
- Coudes à coupes multiples : ASME B31.3, § 304.2.3 (angle de coupe θ ≤ 22,5° pour les coudes à coupes multiples)