Simply-supported beam — uniformly-distributed load

Companion to Simply-supported beam — central point load and Cantilever beam — tip deflection under uniform distributed load — same SS-beam geometry but with a uniformly-distributed transverse load q instead of a central concentrated load. The Euler–Bernoulli closed form is the familiar 5/384 coefficient:

\[\delta_{\text{mid}} = \frac{5 q L^{4}}{384 E I}.\]

Each support reaction is \(R = q L / 2\) by symmetry.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Distributed load q	1 000 N/m (acts in -z)
Expected δ_mid	5 q L⁴ / (384 E I) = 1.25 × 10⁻⁴ m

femorph-solver result

Ran by tests/validation/test_ss_beam_udl.py on an SOLID185 enhanced-strain hex mesh. Same knife-edge supports as the SS-beam-central-load problem; UDL lumped onto the top face via the trapezoid-rule arc-length convention from cantilever_udl. Total nodal resultant equals -q · L exactly (regression-tested).

Refinement	Mesh (nx × ny × nz)	δ_mid (m)	Error vs Euler–Bernoulli
Coarse	20 × 3 × 3	1.2509 × 10⁻⁴	+0.07 %
Medium	40 × 3 × 3	1.2555 × 10⁻⁴	+0.44 %
Refined	80 × 3 × 3	1.2570 × 10⁻⁴	+0.56 %

The small ~0.5 % drift is the same 3D Poisson contribution the other SS / CC beam problems exhibit; smaller here than under a concentrated load because the UDL spreads the bending moment out and reduces the local-stress contribution at the load.

Cross-references

Source	Reported δ_mid (m)	Problem ID / location
Closed form (Euler–Bernoulli)	1.250 × 10⁻⁴	Timoshenko SoM Part I §5.6
Gere & Goodno (2018) §9.3 Table 9-2 case 1	1.250 × 10⁻⁴	SS beam with UDL
femorph-solver (refined)	1.257 × 10⁻⁴	test_ss_beam_udl.py
MAPDL Verification Manual	1.25 × 10⁻⁴	VM-2 Beam stresses and deflections (UDL SS variant)
Abaqus Verification Manual	1.25 × 10⁻⁴	AVM 1.5.x SS-beam-UDL family
NAFEMS Background to Benchmarks	1.25 × 10⁻⁴	§3.2 SS beam under UDL

Source

Problem class: femorph_solver.validation.problems.SimplySupportedBeamUDL.

Backing regression test: tests/validation/test_ss_beam_udl.py.