Simply-supported beam — central point load
===========================================

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

.. math::

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

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

Problem
-------

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

    * - 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 :file:`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.

.. list-table::
    :header-rows: 1
    :widths: 25 20 25 25

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

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

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

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