Pinched ring — diametrical deflection

Classical thin-ring benchmark from Castigliano’s 1879 derivation. A circular ring of mean radius \(R\), bending stiffness \(EI\), loaded by two equal and opposite point forces \(P\) acting along a diameter deflects by

\[\delta_{\text{diam}} = \frac{P R^{3}}{E I} \Bigl(\frac{\pi}{4} - \frac{2}{\pi}\Bigr) \approx 0.14868 \, \frac{P R^{3}}{E I}.\]

The line of action of the loads shortens by \(\delta_{\text{diam}}\); the perpendicular diameter lengthens by the Castigliano companion value \(\delta_{\text{perp}} = P R^{3}(4/\pi - \pi/4) / (E I)\).

Problem

Parameter	Value
Mean ring radius R	0.1 m
Cross-section	10 mm × 5 mm (radial = 5 mm)
Young’s modulus E	200 GPa
Poisson’s ratio ν	0.30
Applied point load P	10 N
Expected δ_diam / 2	(one loaded point’s inward motion) = 3.57 × 10⁻⁵ m

femorph-solver result

Ran by tests/validation/test_pinched_ring.py using the BEAM188 kernel on a quarter-symmetry chain of straight 2-node beam segments discretising the arc θ ∈ [0, π/2]. BCs at the two cut ends enforce the full-ring symmetries:

• θ = 0 (loaded point (R, 0)): UY = UZ = ROTX = ROTY = ROTZ = 0. UX is free to register the inward deflection; the in-plane rotation ROTZ = 0 enforces the x-axis symmetry of the cross-section.

• θ = π/2 (symmetry point (0, R)): UX = UZ = ROTX = ROTY = ROTZ = 0. UY is free.

Half of the load (P/2) is applied at the loaded node inward along -x; by quarter-symmetry the full-ring diametrical deflection equals twice the loaded-node |u_x|.

Refinement	Elements (quarter arc)	|u_x(loaded)| (m)	Error vs Castigliano
Coarse	20	3.570 × 10⁻⁵	−0.02 %
Medium	40	3.573 × 10⁻⁵	+0.08 %
Refined	80	3.574 × 10⁻⁵	+0.10 %

Converges to within 0.1 % — the small positive drift reflects the slight extensional stiffness the 2-node BEAM188 introduces (Castigliano’s derivation assumes pure bending with zero axial strain). Inside the published 5 % engineering tolerance.

Cross-references

Source	Reported δ_diam / 2 (m)	Problem ID / location
Closed form (Castigliano)	3.57 × 10⁻⁵	Timoshenko & Young 1968 §79
Roark’s Formulas for Stress and Strain	3.57 × 10⁻⁵	8th ed. Table 9.2 Case 13
femorph-solver (finest)	3.574 × 10⁻⁵	test_pinched_ring.py
MAPDL Verification Manual	≈ 3.57 × 10⁻⁵	VM-38 pinched-ring family
Abaqus Verification Manual	≈ 3.57 × 10⁻⁵	AVM 1.5.x pinched-ring family
NAFEMS R0011	≈ 3.57 × 10⁻⁵	Pinched-ring benchmark (quarter-symmetry)

Source

Problem class: femorph_solver.validation.problems.PinchedRing.

Backing regression test: tests/validation/test_pinched_ring.py — asserts |u_x| matches the closed form to within 0.5 % on every refinement, not just the finest.