Pipe Expansion Leg (Guided Cantilever)
Computes the minimum free leg length a pipe run needs in order to absorb a thermal expansion without exceeding the allowable stress, using the guided-cantilever method: L = √(3 × E × D × Δ ÷ S_a). The idea is to treat the perpendicular leg as a fixed-end beam with a guided tip that displaces Δ along the direction of expansion; the resulting bending stress falls with the square of the length, so doubling the leg divides the stress by four. The method is conservative and is meant for sizing expansion loops and direction changes ahead of a formal flexibility analysis: the larger the outside diameter, the farther the outer fibre sits from the neutral axis and the more stress the same curvature produces, so the longer the leg needed for the same expansion — stiffness does not enter, and two pipes of equal outside diameter but different wall thickness need the same leg. Enter the modulus of elasticity, the outside diameter, the expansion to be absorbed and the allowable stress.
Result
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Pipe expansion leg by the guided cantilever method
A steam line running at 200 °C grows a few centimetres between two anchors, and that growth has to go somewhere. The piping designer settles it during routing, long before opening the flexibility analysis package: add a perpendicular leg and ask whether it is long enough to flex without blowing past the allowable stress. The guided cantilever method answers in thirty seconds on the desk, and tells you whether the offset fits the pipe rack or whether the job already calls for an expansion joint.
L = √(3 × E × D × Δ ÷ S_a). The leg gets treated as a fixed-end beam with a guided tip: the tip moves Δ without rotating, and the bending stress at the outer fibre falls with the square of the length — double the leg and the stress drops fourfold. With the defaults, E = 200000 MPa, D = 168.3 mm (6 inch), Δ = 50 mm and S_a = 120 MPa, the numerator works out to 5.049 × 10⁹, dividing by 120 leaves 4.2075 × 10⁷ mm², and the square root returns 6486.5 mm, nearly six and a half metres. Doubling the expansion multiplies the leg by √2: 100 mm now calls for 9173.3 mm.
Notice what dropped out of the derivation: wall thickness. The second moment of area cancels, so a Sch 10 and a Sch 80 pipe of the same outside diameter call for exactly the same leg — D enters as the distance from the outer fibre to the neutral axis, not as a measure of stiffness. The method further assumes one leg takes the whole movement with no end rotation, which is conservative, and it ignores stress intensification at bends and tees, weight, pressure, support friction and nozzle load limits. Treat it as pre-sizing, never as a substitute for formal analysis.
Frequently asked questions
How do I work out the Δ that goes into the field?
Which allowable stress belongs in the S_a field?
Does this replace a flexibility analysis in software?
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