Cantilever beam — tip deflection under uniform distributed load

A clamped slender cantilever of length \(L\) carrying a uniformly-distributed transverse load \(q\) (force per unit length, e.g. self-weight) along its span deflects at the tip by

\[\delta_{\text{tip}} = \frac{q L^{4}}{8 E I}, \qquad I = \frac{b h^{3}}{12}.\]

Companion to the tip-shear and tip-moment cantilever cases.

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
Distributed load q	1 000 N/m (acts in -z)
Expected tip deflection	q L⁴ / (8 E I) = 1.20 × 10⁻³ m

Analytical reference

Integrating the Euler–Bernoulli equation EI w'''' = q (Timoshenko, Strength of Materials Part I, §5.4) four times with cantilever BCs (w(0)=w'(0)=0, EI w'''(L)=0, EI w''(L)=0) gives w(x) = q x² (6L² - 4Lx + x²) / (24 E I) and hence δ_tip = q L⁴ / (8 E I).

femorph-solver result

Ran by tests/validation/test_cantilever_udl.py on an SOLID185 enhanced-strain hex mesh. The UDL is lumped onto the top-face nodes with a trapezoidal rule: edge columns carry half- weight per unit length, interior columns full weight. The nodal resultant integrates to q * L exactly — a regression test pins this invariant.

Discretisation	Mesh (nx × ny × nz)	δ_tip (m)	Error vs Euler–Bernoulli
Coarse	10 × 3 × 3	see test	< 5 %
Medium	20 × 3 × 3	see test	~1 %
Reference	40 × 3 × 3	1.193 × 10⁻³	0.59 %
Refined	80 × 3 × 3	see test	< 0.2 %

Cross-references

Source	Reported δ_tip (m)	Problem ID / location
Closed form (Euler–Bernoulli)	1.200 × 10⁻³	Timoshenko SoM Part I §5.4
Gere & Goodno (2018) §9.3 Table 9-1 case 1	1.200 × 10⁻³	UDL cantilever
femorph-solver (reference mesh)	1.193 × 10⁻³	test_cantilever_udl.py
MAPDL Verification Manual	1.20 × 10⁻³	VM-2 Beam stresses and deflections (UDL case)
Abaqus Verification Manual	1.20 × 10⁻³	AVM 1.5.x cantilever-with-UDL family

Source

Problem class: femorph_solver.validation.problems.CantileverUDL.

Backing regression test: tests/validation/test_cantilever_udl.py — four-mesh convergence sweep plus a nodal-load-conservation check.