Cantilever beam — tip deflection under uniform distributed load
================================================================

A clamped slender cantilever of length :math:`L` carrying a
uniformly-distributed transverse load :math:`q` (force per unit
length, e.g. self-weight) along its span deflects at the tip by

.. math::

    \delta_{\text{tip}} = \frac{q L^{4}}{8 E I}, \qquad
    I = \frac{b h^{3}}{12}.

Companion to the :doc:`tip-shear <cantilever_beam_tip_deflection>`
and :doc:`tip-moment <cantilever_tip_moment>` cantilever cases.

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
    * - Distributed load ``q``
      - 1 000 N/m (acts in ``-z``)
    * - Expected tip deflection
      - ``q L⁴ / (8 E I) = 1.20 × 10⁻³ m``

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

Integrating the Euler–Bernoulli equation
``EI w'''' = q`` (Timoshenko, *Strength of Materials Part I*,
§5.4) four times with cantilever BCs
(``w(0)=w'(0)=0``, ``EI w'''(L)=0``, ``EI w''(L)=0``) gives
``w(x) = q x² (6L² - 4Lx + x²) / (24 E I)`` and hence
``δ_tip = q L⁴ / (8 E I)``.

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

Ran by :file:`tests/validation/test_cantilever_udl.py` on an
SOLID185 enhanced-strain hex mesh.  The UDL is lumped onto the
top-face nodes with a trapezoidal rule: edge columns carry half-
weight per unit length, interior columns full weight.  The nodal
resultant integrates to ``q * L`` exactly — a regression test
pins this invariant.

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

    * - Discretisation
      - Mesh (``nx × ny × nz``)
      - ``δ_tip`` (m)
      - Error vs Euler–Bernoulli
    * - Coarse
      - 10 × 3 × 3
      - see test
      - < 5 %
    * - Medium
      - 20 × 3 × 3
      - see test
      - ~1 %
    * - Reference
      - 40 × 3 × 3
      - 1.193 × 10⁻³
      - 0.59 %
    * - Refined
      - 80 × 3 × 3
      - see test
      - < 0.2 %

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

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

    * - Source
      - Reported ``δ_tip`` (m)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 1.200 × 10⁻³
      - Timoshenko *SoM Part I* §5.4
    * - Gere & Goodno (2018) §9.3 Table 9-1 case 1
      - 1.200 × 10⁻³
      - UDL cantilever
    * - femorph-solver (reference mesh)
      - 1.193 × 10⁻³
      - :file:`test_cantilever_udl.py`
    * - MAPDL Verification Manual
      - 1.20 × 10⁻³
      - VM-2 *Beam stresses and deflections* (UDL case)
    * - Abaqus Verification Manual
      - 1.20 × 10⁻³
      - AVM 1.5.x cantilever-with-UDL family

Source
------

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

Backing regression test:
:file:`tests/validation/test_cantilever_udl.py` — four-mesh
convergence sweep plus a nodal-load-conservation check.