Clamped-clamped beam — central point load
===========================================

A prismatic slender beam of length :math:`L`, clamped rigidly 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}}{192 E I}, \qquad
    I = \frac{b h^{3}}{12}.

One-quarter the simply-supported deflection: the fixed-end
moments :math:`\pm P L / 8` resist rotation at the supports and
stiffen the beam considerably.

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 / (192 E I) = 5.00 × 10⁻⁵ m``

Analytical reference
--------------------

Integrating Euler–Bernoulli piecewise over ``[0, L/2]`` and
``[L/2, L]`` with clamped end BCs ``w(0)=w'(0)=w(L)=w'(L)=0``
and the mid-span shear jump from :math:`P` gives the closed
form above (Timoshenko, *SoM Part I*, §5.7).

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

Ran by :file:`tests/validation/test_cc_beam_central_load.py` on
an SOLID185 enhanced-strain hex mesh.  Both end faces carry an
all-DOF Dirichlet clamp — no knife-edge idealisation needed since
the fixed-fixed configuration removes every support-side
freedom.  The central load is lumped across the mid-span bottom-
line nodes; mid-span deflection is extracted from the top-face
centerline to sidestep the local 3D stress-concentration.

.. 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
      - 4.967 × 10⁻⁵
      - −0.7 %
    * - Medium
      - 40 × 3 × 3
      - 5.050 × 10⁻⁵
      - +1.0 %
    * - Refined
      - 80 × 3 × 3
      - 5.079 × 10⁻⁵
      - +1.6 %

The persistent ~2 % excess at fine-mesh convergence is the same
3D Poisson contribution noted on the simply-supported beam — a
solid-mesh beam picks up curvature from transverse normal
stresses that pure Euler–Bernoulli ignores.  Well inside the
5 % tolerance.

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

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

    * - Source
      - Reported ``δ_mid`` (m)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 5.000 × 10⁻⁵
      - Timoshenko *SoM Part I* §5.7
    * - Gere & Goodno (2018) §10.3 Table 10-1
      - 5.000 × 10⁻⁵
      - Fixed-fixed beam, central load
    * - femorph-solver (refined mesh)
      - 5.079 × 10⁻⁵
      - :file:`test_cc_beam_central_load.py`
    * - MAPDL Verification Manual
      - 5.00 × 10⁻⁵
      - VM-2 *Beam stresses and deflections* (clamped variant)
    * - Abaqus Verification Manual
      - 5.00 × 10⁻⁵
      - AVM 1.5.x fixed-fixed beam family

Source
------

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

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