Expansion loop with a curved bend — SWBPIPE verification

Part VI · VerificationVI–A4

Part VI

Verification

A. System benchmarks · Expansion loop with a curved bend

A pipe runs between two anchors: three meters up, a quarter-circle elbow, four meters across. A uniform temperature rise of 150 K causes thermal expansion. Both anchors remain fixed. The example calculates anchor forces and moments as the elbow becomes more flexible.

The hand calculation uses the force method, which the 1941 manual applied generally to piping. The benchmark compares this reference with SWBPIPE’s curved bend element and frame solver. The user-entered factor k multiplies the elbow’s in-plane flexibility. No code value is used.

Fig. 3.—Plane L-bend between two anchors

1.Inputs.

Invented inputs that resemble an 8-inch (DN200) Schedule 40 carbon-steel line. No dimensional or material standard supplies these values.

Quantity	Symbol	Value
Outside diameter	D_o	219.1 mm
Wall thickness	t	8.18 mm
Elastic modulus	E	200 GPa
Expansion coefficient	α	12.0 × 10⁻⁶ /K
Temperature rise	ΔT	+150 K
Legs and bend radius	L₁, R, L₂	3.0, 0.5, 4.0 m
Elbow flexibility factor	k	1, 5, 10, 20

2.Method.

Release anchor B. The loop becomes a cantilever from A. Under the uniform temperature rise, the released end B moves by the free thermal strain times its distance from A.

ε_T = αΔT = 1.8 × 10⁻³, Δ_1T = 8.1 mm, Δ_2T = 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 f_ij = ∫_legs m_i m_j ds + k ∫_arc m_i m_j ds(2)

[EI f] {X} = −EI {Δ_T}(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.

Modeling assumptions. Like the classical method, the hand solution counts bending flexibility only and neglects axial and shear flexibility. The recorded solver comparison matches that assumption by making the members axially very stiff (area multiplied by 10⁵). With the true axial stiffness the reactions change by at most 0.55 % at k = 1 and less at higher k. As a separate check, an independent 800-element stiffness model reproduces every reaction to within 5.4 × 10⁻⁶ relative.

3.Results.

Reactions are the forces and moments that the anchors exert on the pipe. Global +X points right, and +Y points up. Positive moments about +Z act counterclockwise in the figure.

Reactions at anchor A, and movement of the tangent point T2
	k = 1	k = 5	k = 10	k = 20
H_A, N	11 636	7 462	6 144	5 282
V_A, N	7 635	4 330	3 302	2 655
M_A, N·m	−21 598	−17 286	−15 893	−14 925
u_x(T₂), mm	−7.200	−7.200	−7.200	−7.200
u_y(T₂), mm	6.692	6.468	6.418	6.419

Path exercised. The benchmark calls the bend element and frame kernel directly, with a dense solve and a thermal load vector formed by the harness. It does not exercise the desktop or product-physics load-generation path. The current product-physics entrypoint supports realized bends on the pressure-free route and under the v3 pressure contract. The v2 pressure contract remains straight-only. Separate product comparisons cover pressure through a realized bend, separately supported closures and pressure, heat and weight between anchors. This unpressurized kernel comparison does not qualify either product route (Status).

The current linear product route can publish actions from corrected member recovery while retaining the original binary64 nodal movements. Its conditional member-relative bound concerns the intended re-formed system. This historical table does not exercise that qualification (Part III A 6).

Recorded agreement. The benchmark source records that the anchor reactions match the hand calculation to within 5.8 × 10⁻⁸ relative, against a tolerance of 5.0 × 10⁻⁷. The recorded relative deviations are 6.0 × 10⁻⁸ for u_x(T₂) and 4.5 × 10⁻⁸ for u_y(T₂).

The recorded force residual is 2.9 × 10⁻⁵ N, rounded here previously to 3 × 10⁻⁵ N. The moment residual is 7.1 × 10⁻⁵ N·m. Both equilibrium tolerances are 1.0 × 10⁻³ in their respective units.

These are recorded comparison figures, not a new execution for this edition. The pinned benchmark source contains the comparison despite stale draft and TBD labels in the hand reference.

For the student

Make the elbow more flexible. Every anchor reaction magnitude falls in this sweep because the bend accommodates more thermal growth. Look at u_x(T₂): the value does not change.

Under the classical bending-only assumption, the top leg changes length only by thermal strain. Anchor B cannot move. Thus, T₂ must move back by exactly ε_TL₂. This independence from k gives a useful solver check. In contrast, u_y(T₂) is not monotonic across the sweep.

For the practitioner

The illustrative factors k are between 5 and 10. The horizontal anchor force H_A is 53 to 64 % of its value at k = 1. A straight chord in place of the elbow is stiffer still. An elbow next to an anchor or nozzle must therefore be modeled with its flexibility factor. The status page explains why the distinction between chord and curved modes matters.

4.Run it yourself.

The following command selects the expansion-loop benchmark at revision 462f66975d. This edition records source evidence; it does not report a new execution of this command.

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

The hand reference for this comparison is validation/hand_calcs/mechanics/expansion_loop_curved_bend_thermal.md. Its inputs and derivation support the table. The comparison is wired in validation/benchmarks/mechanics/src/lib.rs. Neither record establishes production qualification or code compliance.