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:

.. math::

    \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
-------

.. list-table::
    :header-rows: 1
    :widths: 30 70

    * - 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
      - :math:`\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 :math:`\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 ``HEX8`` 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
:file:`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
----------------

.. list-table::
    :header-rows: 1
    :widths: 40 25 35

    * - 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 :class:`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:
:file:`tests/analytical/test_simply_supported_plate.py`
— landed with Agent 2's TA-9b analytical suite (#150).