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

.. math::

    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
-------

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

    * - 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 :file:`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.

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

    * - 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
----------------

.. list-table::
    :header-rows: 1
    :widths: 35 30 35

    * - 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
      - :file:`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:
:class:`femorph_solver.validation.problems.ClampedPlateStatic`.

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