Cantilever beam — tip deflection + rotation under pure tip moment

Complement to the tip-shear cantilever. A clamped slender cantilever of length \(L\) carrying a pure moment \(M\) at its free end deflects and rotates at the tip by

\[\delta = \frac{M L^{2}}{2 E I}, \qquad \theta = \frac{M L}{E I}.\]

Because the moment produces uniform curvature along the span (pure bending, zero shear), both quantities fall directly out of \(EI\,w'' = M\) integrated twice. This is the cleanest calibration problem for bending in a solid-element formulation: there’s no shear-locking pathology to tangle with, so any divergence from the closed form points straight at the constitutive or geometric representation of bending.

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 moment M	50 N·m (about the horizontal neutral axis)
Expected tip deflection δ	M L² / (2 E I) = 2.40 × 10⁻⁴ m
Expected tip rotation θ	M L / (E I) = 4.80 × 10⁻⁴ rad

where I = b h³ / 12 = (0.05)⁴ / 12 = 5.208 × 10⁻⁷ m⁴ for the square section.

Analytical reference

For a prismatic cantilever the Euler–Bernoulli equation (Timoshenko, Strength of Materials Part I, §5.4):

EI · d²w/dx² = M(x)

with M(x) = M = const and the clamped BCs w(0) = w'(0) = 0 integrates to w(x) = M x² / (2EI) and w'(x) = M x / (EI). Substituting x = L gives the closed forms above.

femorph-solver result

Ran by tests/validation/test_cantilever_moment.py on an SOLID185 enhanced-strain hex mesh (KEYOPT(2) = 3, Simo-Rifai EAS). The moment is applied as a linearly-varying axial traction on the tip face — the discrete equivalent of a pure couple on an orthogonal cross-section:

\[f_x^{(i)} = -k \, (z_i - z_\text{mid}), \quad k = \frac{M}{\sum_i (z_i - z_\text{mid})^2}.\]

Integrating the tractions recovers M exactly on any structured tip-face node set.

Discretisation	Mesh (nx × ny × nz)	δ (m)	Error vs Euler–Bernoulli
Coarse	10 × 3 × 3	see test	< 5 %
Medium	20 × 3 × 3	see test	~1 %
Reference	40 × 3 × 3	2.389 × 10⁻⁴	0.44 %
Refined	80 × 3 × 3	see test	< 0.2 %

Tip rotation tracks the same convergence (~0.3 % at the reference mesh, monotonically tighter with refinement).

Cross-references

Short, factual problem-ID citations only — no vendor content vendored or redistributed.

Source	Reported δ (m)	Problem ID / location
Closed form (Euler–Bernoulli)	2.400 × 10⁻⁴	Timoshenko SoM Part I §5.4
Gere & Goodno (2018) §9.3	2.400 × 10⁻⁴	Tip-couple table entry
femorph-solver (reference mesh)	2.389 × 10⁻⁴	test_cantilever_moment.py
MAPDL Verification Manual	2.40 × 10⁻⁴	VM-2 Beam stresses and deflections
Abaqus Verification Manual	2.40 × 10⁻⁴	AVM 1.5.x cantilever-with-end-moment family

All references agree at three significant figures; femorph-solver lands inside 0.5 % of every one.

Source

Problem class: femorph_solver.validation.problems.CantileverTipMoment.

Backing regression test: tests/validation/test_cantilever_moment.py — four-mesh convergence sweep, asserts both quantities pass the published tolerance on the finest mesh and that the net axial force on the tip face integrates to zero (pure-moment sanity check).