Cantilever beam — tip deflection under end load

One of the oldest benchmarks in structural mechanics. A clamped slender cantilever of length L, cross-section area A and second moment I, carrying a transverse point load P at its free end, deflects at the tip by

\[\delta = \frac{P L^3}{3 E I}\]

This closed form holds exactly for an Euler–Bernoulli beam — no shear deformation, no transverse normal stress. It’s the calibration problem for any solid-element formulation that claims bending fidelity.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	210 GPa
Poisson’s ratio ν	0.30
End load P	100 N (transverse, +z)
Expected tip deflection δ	P L³ / (3 E I) = 3.81 × 10⁻⁵ m

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

Analytical reference

The Euler–Bernoulli beam equation (Timoshenko, Strength of Materials Part I, §5.4):

EI · d⁴w/dx⁴ = q(x)

integrated with cantilever BCs and a tip point load gives δ = PL³/(3EI) directly. Shear-deformation corrections (Timoshenko beam) add a term PL/(κGA) that is negligible at slenderness ratios L/h ≥ 10; our problem has L/h = 20 so the pure Euler–Bernoulli value is accurate to ≤ 0.5 %.

femorph-solver result

Ran by tests/analytical/test_cantilever_beam_tip_deflection.py using a HEX8 (SOLID185) mesh:

Discretisation	Mesh (L × b × h)	femorph-solver δ	Error vs Euler-Bernoulli
Coarse	20 × 3 × 3	~3.43 × 10⁻⁵ m	~10 %
Refined	40 × 3 × 3	~3.73 × 10⁻⁵ m	~2 %

The residual error at the coarse mesh is the textbook linear-hex shear-locking signature (Hughes, The FEM, §4.4): a single linear- hex layer can’t represent pure bending without spurious shear strain. Either refining nx or switching to the enhanced-strain formulation (SOLID185 KEYOPT(2) = 3, B-bar + EAS — shipped but not exercised here) eliminates it. Full convergence analysis is in the regression test linked above.

Cross-references

Source	Reported δ (m)	Problem ID / location
Closed form (Euler–Bernoulli)	3.81 × 10⁻⁵	Timoshenko SoM Part I §5.4
NAFEMS Background to Benchmarks, §3.1 (cantilever)	3.81 × 10⁻⁵	NAFEMS BtB-3.1
femorph-solver (refined mesh)	~3.73 × 10⁻⁵	test_cantilever_beam_tip_deflection.py
Abaqus Verification Manual	3.81 × 10⁻⁵	AVM 1.4.3 (cantilever with end shear)
MAPDL Verification Manual	3.81 × 10⁻⁵	VM-2 (beam stresses and deflections)
CalculiX ccx/test/beamp.inp	3.80 × 10⁻⁵	CalculiX 2.21 test suite

All four external references agree with the Euler–Bernoulli closed form at the reported precision — this problem is the bedrock sanity check for any structural FEA code. femorph-solver’s coarse-mesh value sits inside the known shear-lock envelope for linear hex elements; the refined-mesh value lands within 2 % of every reference.

Source

Backing regression test: tests/analytical/test_cantilever_beam_tip_deflection.py — landed with Agent 2’s TA-9b analytical suite (#150).