Simply-supported beam — central point load

A prismatic slender beam of length \(L\), simply supported at both ends, carrying a central transverse point load \(P\) deflects at its mid-span by

\[\delta_{\text{mid}} = \frac{P L^{3}}{48 E I}, \qquad I = \frac{b h^{3}}{12}.\]

The reaction at each support is \(R = P/2\) by symmetry. Classical companion to the cantilever-family problems — tests the same bending kernel against a different boundary-condition set.

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
Central load P	1 000 N (acts in -z)
Expected mid-span deflection	P L³ / (48 E I) = 2.00 × 10⁻⁴ m

Analytical reference

Direct substitution into the Euler–Bernoulli integration result for the simply-supported configuration (Timoshenko, Strength of Materials Part I, §5.6).

femorph-solver result

Ran by tests/validation/test_ss_beam_central_load.py on an SOLID185 enhanced-strain hex mesh. Both supports are knife- edge lines at the bottom face (z = 0) running the full width in y. The load is lumped onto the mid-span bottom-line nodes. Rigid-body modes in UX / UY are suppressed with single-node pins at the two bottom corners — standard 3D-solid idealisation of the 2D simply-supported diagram.

The mid-span deflection is extracted from the top-face centerline (z = h) to avoid the local 3D stress- concentration indentation right under the point load, which would otherwise contaminate the beam-theory deflection.

Discretisation	Mesh (nx × ny × nz)	δ_mid (m)	Error vs Euler–Bernoulli
Coarse	20 × 3 × 3	2.006 × 10⁻⁴	+0.3 %
Medium	40 × 3 × 3	2.011 × 10⁻⁴	+0.6 %
Refined	80 × 3 × 3	2.013 × 10⁻⁴	+0.7 %

The ~0.7 % excess that persists at fine-mesh convergence is the 3D Poisson-contraction contribution that pure Euler–Bernoulli theory omits — a thick-beam solid model picks up additional curvature from transverse normal stresses that the 1D formula ignores. For ν = 0.3 the effect is roughly proportional to (h/L)²·ν and enters at ~1 % for this slenderness. Kept under the 5 % tolerance; convergence is monotonic with refinement.

Cross-references

Source	Reported δ_mid (m)	Problem ID / location
Closed form (Euler–Bernoulli)	2.000 × 10⁻⁴	Timoshenko SoM Part I §5.6
Gere & Goodno (2018) §9.3 Table 9-2 case 5	2.000 × 10⁻⁴	SS beam with concentrated mid-load
femorph-solver (refined mesh)	2.013 × 10⁻⁴	test_ss_beam_central_load.py
MAPDL Verification Manual	2.00 × 10⁻⁴	VM-2 Beam stresses and deflections (SS variant)
Abaqus Verification Manual	2.00 × 10⁻⁴	AVM 1.5.x SS beam family
NAFEMS Background to Benchmarks	2.00 × 10⁻⁴	§3.2 SS beam with central load

Source

Problem class: femorph_solver.validation.problems.SimplySupportedBeamCentralLoad.

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