Cantilever under tip torque — Saint-Venant torsion

First torsion benchmark in the corpus. A prismatic cantilever loaded by a pure tip torque \(T\) about the beam axis twists uniformly along its span. Saint-Venant torsion gives

\[\varphi_{\text{tip}} = \frac{T L}{G J},\]

with \(G = E / (2(1+\nu))\) the shear modulus and \(J\) the torsion constant of the cross-section.

For a rectangular section b × t (b ≥ t):

\[J = \beta(b/t) \, b \, t^{3},\]

where \(\beta\) tabulated from Saint-Venant’s infinite- series solution (Timoshenko & Goodier §109): β = 0.141 at b/t = 1 (square), approaching 1/3 for long thin sections. For a square h × h section, \(J \approx 0.141 \, h^{4}\).

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Poisson’s ratio ν	0.30
Tip torque T	10 N·m
Shear modulus G	E / (2(1+ν)) = 76.92 GPa
Torsion constant J	0.141 × h⁴ = 8.81 × 10⁻⁷ m⁴
Expected tip twist φ	T L / (G J) = 1.475 × 10⁻⁴ rad

femorph-solver result

Ran by tests/validation/test_cantilever_torsion.py using the BEAM188 kernel — femorph-solver’s 3D linear beam with 6 DOFs per node. Clamp all six DOFs at the left end, apply MX = T at the tip (moment about the beam x-axis), read the tip ROTX back.

Refinement	Elements	φ_tip (rad)	Error vs closed form
Coarse	10	1.4752 × 10⁻⁴	< 1 × 10⁻¹²
Medium	20	1.4752 × 10⁻⁴	< 1 × 10⁻¹²
Refined	40	1.4752 × 10⁻⁴	< 1 × 10⁻¹²

BEAM188 integrates the Saint-Venant linear torsion equation exactly: the torsion block of the stiffness matrix produces the closed-form T L / (GJ) twist to machine precision on any mesh. This is the standard signature of a correctly- implemented linear-beam torsion kernel.

Cross-references

Source	Reported φ_tip (rad)	Problem ID / location
Closed form (Saint-Venant)	1.475 × 10⁻⁴	Timoshenko & Goodier 1970 §109
Saint-Venant (1853)	1.475 × 10⁻⁴	Original derivation, β(b/t) series
femorph-solver (any mesh)	1.475 × 10⁻⁴	test_cantilever_torsion.py
MAPDL Verification Manual	1.475 × 10⁻⁴	VM-23 family (rotating-disk / torsion kernel)
Abaqus Verification Manual	1.475 × 10⁻⁴	AVM 1.5.x cantilever-torsion family

Source

Problem class: femorph_solver.validation.problems.CantileverTorsion.

Backing regression test: tests/validation/test_cantilever_torsion.py — asserts φ_tip matches the closed form to within 1 × 10⁻¹⁰ relative error on every mesh in the refinement sweep (not just the finest).