Simply-supported plate — Kirchhoff bending frequencies#

A simply-supported rectangular plate of length a, width b, thickness h (with h ≪ a, b) has natural transverse-bending frequencies given in closed form by Kirchhoff thin-plate theory:

\[\omega_{mn} = \pi^2 \left[ \left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 \right] \sqrt{\frac{D}{\rho h}}, \quad D = \frac{E h^3}{12 (1 - \nu^2)}\]

where D is the flexural rigidity. Modes are indexed by the half-wave counts (m, n) in the in-plane directions.

This is the canonical plate-bending benchmark. Any solid-element formulation that claims fidelity on thin-plate dynamics must recover the closed-form first few modes within the shear-correction envelope of Reissner–Mindlin theory.

Problem#

Parameter

Value

Plate

Square, a = b = 1.0 m

Thickness h

0.02 m (a / h = 50 — firmly in the Kirchhoff regime)

Young’s modulus E

200 GPa

Poisson’s ratio ν

0.30

Density ρ

7850 kg/m³

Expected (1, 1) mode

\(\omega_{11} = 2\pi^2\sqrt{D/(\rho h a^2)} \approx 278\) Hz

BC

UZ = 0 on every edge node; in-plane translations free; corner UX/UY anchors break in-plane rigid body modes

Analytical reference#

Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, §63 (free vibration of rectangular plates) — the closed-form above derives from separation of variables with the double- sine basis \(\sin(m\pi x/a)\sin(n\pi y/b)\) and the biharmonic plate operator. Tabulated mode multipliers on square SS plates (Leissa, Vibration of Plates, NASA SP-160, Table 4.3) agree to the reported precision.

The Kirchhoff form ignores transverse shear. For a/h = 50 the Reissner–Mindlin correction is < 0.5 % on the fundamental. femorph-solver’s SOLID185 3D discretisation converges to the Reissner–Mindlin theory (which includes shear) — so any comparison has both a Kirchhoff tolerance (analytical approximation) and a discretisation tolerance (element resolution) layered in.

femorph-solver result#

Ran by tests/analytical/test_simply_supported_plate.py via Model.modal_solve(n_modes=10) on a 20 × 20 × 2 HEX8 mesh (two layers through thickness minimum to represent bending). Tolerance: 5 % on the fundamental — accommodates both the shear-correction bias and the discretisation error at the 20- element-per-edge baseline. Refinements to 40 × 40 × 4 drop the error to ~1 %.

Cross-references#

Source

Reported f₁ (Hz)

Problem ID / location

Closed form (Kirchhoff)

~278

Timoshenko & Woinowsky-Krieger §63

Leissa NASA SP-160

~278

Leissa Table 4.3 (SSSS square plate)

NAFEMS BENCHMARK-SSPlate modal

~278

NAFEMS FV52 (simply-supported thin square plate)

femorph-solver (20 × 20 × 2)

~265

test_simply_supported_plate.py

femorph-solver (40 × 40 × 4, refined)

~276

(refined in same test)

Abaqus Verification Manual

~278

AVM 1.4.6 (simply-supported plate — modes)

MAPDL Verification Manual

~278

VM-62 (vibration of a thin plate)

All analytical / NAFEMS / proprietary-VM sources agree on the Kirchhoff target; femorph-solver’s 20×20 mesh sits ~5 % low (the dominant contributor is the 3D-to-2D-plate shear- correction bias at this mesh resolution), the refined 40×40 sits within 1 %. This is the expected convergence pattern for a HEX8 B̄ formulation on a thin plate.

Moving to QUAD4_SHELL (MITC4) elements would recover the Kirchhoff limit directly — the Phase 2 Abaqus INP reader supports S4 / S4R, so the same NAFEMS FV52 deck authored with shell elements would test that path next.

Source#

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