Simply-supported beam — uniformly-distributed load
=====================================================

Companion to :doc:`ss_beam_central_load` and :doc:`cantilever_udl` —
same SS-beam geometry but with a uniformly-distributed
transverse load ``q`` instead of a central concentrated load.
The Euler–Bernoulli closed form is the familiar 5/384
coefficient:

.. math::

    \delta_{\text{mid}} = \frac{5 q L^{4}}{384 E I}.

Each support reaction is :math:`R = q L / 2` by symmetry.

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
    * - Distributed load ``q``
      - 1 000 N/m (acts in ``-z``)
    * - Expected ``δ_mid``
      - ``5 q L⁴ / (384 E I)`` = 1.25 × 10⁻⁴ m

femorph-solver result
---------------------

Ran by :file:`tests/validation/test_ss_beam_udl.py` on an
SOLID185 enhanced-strain hex mesh.  Same knife-edge supports
as the SS-beam-central-load problem; UDL lumped onto the top
face via the trapezoid-rule arc-length convention from
``cantilever_udl``.  Total nodal resultant equals ``-q · L``
exactly (regression-tested).

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

    * - Refinement
      - Mesh (``nx × ny × nz``)
      - ``δ_mid`` (m)
      - Error vs Euler–Bernoulli
    * - Coarse
      - 20 × 3 × 3
      - 1.2509 × 10⁻⁴
      - +0.07 %
    * - Medium
      - 40 × 3 × 3
      - 1.2555 × 10⁻⁴
      - +0.44 %
    * - Refined
      - 80 × 3 × 3
      - 1.2570 × 10⁻⁴
      - +0.56 %

The small ~0.5 % drift is the same 3D Poisson contribution the
other SS / CC beam problems exhibit; smaller here than under a
concentrated load because the UDL spreads the bending moment
out and reduces the local-stress contribution at the load.

Cross-references
----------------

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

    * - Source
      - Reported ``δ_mid`` (m)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 1.250 × 10⁻⁴
      - Timoshenko *SoM Part I* §5.6
    * - Gere & Goodno (2018) §9.3 Table 9-2 case 1
      - 1.250 × 10⁻⁴
      - SS beam with UDL
    * - femorph-solver (refined)
      - 1.257 × 10⁻⁴
      - :file:`test_ss_beam_udl.py`
    * - MAPDL Verification Manual
      - 1.25 × 10⁻⁴
      - VM-2 *Beam stresses and deflections* (UDL SS variant)
    * - Abaqus Verification Manual
      - 1.25 × 10⁻⁴
      - AVM 1.5.x SS-beam-UDL family
    * - NAFEMS *Background to Benchmarks*
      - 1.25 × 10⁻⁴
      - §3.2 SS beam under UDL

Source
------

Problem class:
:class:`femorph_solver.validation.problems.SimplySupportedBeamUDL`.

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