NAFEMS LE5 — clamped square plate under uniform pressure

Companion to the simply-supported plate — same square Kirchhoff plate but with all four edges fully clamped (zero deflection AND zero rotation along each edge), loaded by uniform transverse pressure q.

The maximum centre deflection has no elementary closed form, but the published double-Fourier coefficient (Timoshenko & Woinowsky-Krieger §32 Table 12; NAFEMS R0015 §2.5 LE5) gives

\[w_{\text{max}} \approx 0.00126 \, \frac{q\, a^{4}}{D}, \qquad D = \frac{E\, h^{3}}{12 (1 - \nu^{2})}.\]

The 0.00126 factor is ~3.2 stiffer than the simply-supported case (0.00406), exactly as expected from the boundary-rotation constraint.

Problem

Parameter	Value
Edge length a	1.0 m
Plate thickness h	0.02 m (a/h = 50)
Young’s modulus E	200 GPa
Poisson’s ratio ν	0.30
Pressure q	100 kPa (uniform)
Expected w_max	0.00126 q a⁴ / D ≈ 8.60 × 10⁻⁴ m

femorph-solver result

Ran by tests/validation/test_clamped_plate_static.py on an SOLID185 enhanced-strain hex slab with all four edge faces fully clamped (every DOF pinned). Lumped uniform-pressure distribution on the top face — same convention as the simply-supported plate problem.

Refinement	Mesh (nx × ny × nz)	w_max (m)	Error vs Timoshenko
Coarse	10 × 10 × 2	6.523 × 10⁻⁴	−24.1 %
Medium	20 × 20 × 2	7.729 × 10⁻⁴	−10.1 %
Refined	30 × 30 × 2	8.050 × 10⁻⁴	−6.4 %

Converging from below — typical for a solid mesh that suffers shear-locking on Kirchhoff plates at modest through-thickness refinement (only 2 elements through h). Tighter convergence needs either thicker through-thickness sampling or a dedicated SHELL181 kernel. The 15 % NAFEMS engineering tolerance absorbs the discretisation gap on the reference 30×30×2 mesh.

Cross-references

Source	Reported w_max (m)	Problem ID / location
Closed form (Timoshenko)	8.60 × 10⁻⁴	Theory of Plates §32 Table 12
NAFEMS R0015 §2.5	8.60 × 10⁻⁴	LE5 Clamped square plate
femorph-solver (refined)	8.05 × 10⁻⁴	test_clamped_plate_static.py
MAPDL Verification Manual	≈ 8.60 × 10⁻⁴	VM-12 family (clamped plate)
Abaqus Verification Manual	≈ 8.60 × 10⁻⁴	AVM 1.4.1 clamped-plate-pressure family

Source

Problem class: femorph_solver.validation.problems.ClampedPlateStatic.

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