SWBPIPE · The open manual
Part III · VerificationIII–7

Part III

Verification

A. Mechanics  ·  Expansion loop with a curved bend

A pipe runs between two anchors: three metres up, a quarter-circle elbow, four metres across. It is warmed uniformly by 150 K and has nowhere to go. What forces and moments do the anchors feel, and how do they change as the elbow is made more flexible?

The answer is worked here by hand with the force method, the approach the 1941 manual made general, and then computed by the SWBPIPE solver. The elbow’s in-plane flexibility is multiplied by a factor k that you enter; no code value is used.

A plane L-shaped pipe between anchor A at the bottom and anchor B at the right: a 3 metre vertical leg, a quarter-circle elbow of 0.5 metre radius, and a 4 metre horizontal leg, under a uniform temperature rise of 150 kelvin. ANCHOR A ANCHOR B T1 T2 L1 = 3.0 m L2 = 4.0 m R = 0.5 m X Y ΔT = +150 K UNIFORM, NO PRESSURE, NO WEIGHT ELBOW FLEXIBILITY × k
Fig. 3.—Plane L-bend between two anchors

1.Inputs.

All values are invented for this case. They resemble a DN200 line but are taken from no standard.

QuantitySymbolValue
Outside diameterDo219.1 mm
Wall thicknesst8.18 mm
Elastic modulusE200 GPa
Expansion coefficientα12.0 × 10−6 /K
Temperature riseΔT+150 K
Legs and bend radiusL1, R, L23.0, 0.5, 4.0 m
Elbow flexibility factork1, 5, 10, 20

2.Method.

Release anchor B. The loop becomes a cantilever from A, and under a uniform rise it simply grows: B would move by the free thermal strain times its distance from A.

εT = αΔT = 1.8 × 10−3,   Δ1T = 8.1 mm,   Δ2T = 6.3 mm(1)

The anchor must put B back. Its three reactions X1, X2, X3 are the redundants, found from the flexibility of the released loop, with the arc’s share multiplied by k:

EI fij = ∫legs mi mj ds + k ∫arc mi mj ds(2)
[EI f] {X} = −EI {ΔT}(3)

Each coefficient is integrated in closed form and checked against numerical quadrature. The 3 × 3 system is solved for each k, and the reactions at A follow from equilibrium of the whole loop.

3.Results.

Reactions at anchor A, and movement of the tangent point T2
k = 1k = 5k = 10k = 20
HA, N11 6367 4626 1445 282
VA, N7 6354 3303 3022 655
MA, N·m−21 598−17 286−15 893−14 925
ux(T2), mm−7.200−7.200−7.200−7.200
uy(T2), mm6.6926.4686.4186.419

Agreement. The solver matches the hand calculation to within 5.8 × 10−8 relative, against a tolerance of 5.0 × 10−7. Whole-body equilibrium closes to 3 × 10−5 N.

For the student

Make the elbow more flexible and every anchor reaction falls, because the bend takes up more of the growth. But look at ux(T2): it does not change at all. The top leg can only grow by its own thermal strain and B cannot move, so T2 must move back by exactly εTL2. A quantity that cannot depend on k is a very good check on a solver.

4.Run it yourself.

cd projects/chirality-piping
cargo test --manifest-path validation/benchmarks/mechanics/Cargo.toml expansion_loop

The full hand calculation, the fixture and the recorded run are in the repository under validation/hand_calcs/mechanics/.

Note

This case checks that the software solves a stated mechanics problem. It is not a design basis, and it says nothing about compliance with any code.

Contents · The program: swbpipe.com · MIT licence