Propped cantilever — central point load
==========================================

A propped cantilever is a beam clamped at one end (``x = 0``)
and simply supported at the other (``x = L``).  The extra
support makes the problem statically indeterminate — the four
reactions (``R_A``, ``M_A``, ``R_B``, plus the compatibility
constraint on displacement at ``B``) cannot be solved from
statics alone; one integration step of the Euler–Bernoulli
equation provides the missing compatibility equation.

Under a central point load :math:`P` at :math:`x = L/2`:

.. math::

    \delta_{\text{mid}} &= \frac{7 P L^{3}}{768 E I}, \\
    R_A &= \frac{11 P}{16} \text{ (fixed end)}, \\
    R_B &= \frac{5 P}{16}  \text{ (simple end)}, \\
    M_A &= -\frac{3 P L}{16} \text{ (fixed-end moment)}.

The 11/16 + 5/16 partition is the classic fingerprint of the
propped cantilever — the fixed end carries more of the load
than the simple end by a factor of 2.2.

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
    * - Central load ``P``
      - 1 000 N
    * - Expected ``δ_mid``
      - ``7 P L³ / (768 E I)`` = 8.75 × 10⁻⁵ m

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

Ran by :file:`tests/validation/test_propped_cantilever.py` on
an ``SOLID185`` enhanced-strain hex mesh.  BCs:

* ``x = 0``: full all-DOF clamp on the left face.
* ``x = L``: knife-edge roller at the bottom line (``UZ = 0``
  along the full width), with ``UY = 0`` pinned at one corner
  to kill the lateral rigid-body mode.  ``UX`` left free so
  axial strain isn't locked.

The central load is lumped across the mid-span bottom-line
nodes; mid-span deflection is extracted from the top-face
centerline (same as the static SS + CC beam problems, to
avoid the local-load 3D stress-concentration indentation).

.. 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
      - 8.713 × 10⁻⁵
      - −0.43 %
    * - Medium
      - 40 × 3 × 3
      - 8.809 × 10⁻⁵
      - +0.68 %
    * - Refined
      - 80 × 3 × 3
      - 8.843 × 10⁻⁵
      - +1.07 %

The ~1 % positive drift at fine-mesh convergence is the same
3D Poisson contribution the SS and CC beam problems exhibit —
a solid-mesh beam picks up curvature from transverse normal
stresses that pure Euler–Bernoulli ignores.

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

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

    * - Source
      - Reported ``δ_mid`` (m)
      - Problem ID / location
    * - Closed form (Gere & Goodno)
      - 8.75 × 10⁻⁵
      - §10.3 Table 10-1 Case 6
    * - Timoshenko (1955)
      - 8.75 × 10⁻⁵
      - *SoM Part I* §5.8
    * - femorph-solver (refined)
      - 8.843 × 10⁻⁵
      - :file:`test_propped_cantilever.py`
    * - MAPDL Verification Manual
      - 8.75 × 10⁻⁵
      - VM-41 family (propped cantilever)
    * - Abaqus Verification Manual
      - 8.75 × 10⁻⁵
      - AVM 1.5.x propped-cantilever family

Source
------

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

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