Cantilever beam — tip deflection under end load
=================================================

One of the oldest benchmarks in structural mechanics.  A clamped
slender cantilever of length ``L``, cross-section area ``A`` and
second moment ``I``, carrying a transverse point load ``P`` at its
free end, deflects at the tip by

.. math::

    \delta = \frac{P L^3}{3 E I}

This closed form holds exactly for an Euler–Bernoulli beam — no
shear deformation, no transverse normal stress.  It's *the*
calibration problem for any solid-element formulation that claims
bending fidelity.

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``
      - 210 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - End load ``P``
      - 100 N (transverse, ``+z``)
    * - Expected tip deflection ``δ``
      - ``P L³ / (3 E I) = 3.81 × 10⁻⁵ m``

where ``I = b h³ / 12 = (0.05)⁴ / 12 = 5.208 × 10⁻⁷ m⁴`` for the
square section.

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

The Euler–Bernoulli beam equation (Timoshenko, *Strength of
Materials Part I*, §5.4)::

    EI · d⁴w/dx⁴ = q(x)

integrated with cantilever BCs and a tip point load gives
``δ = PL³/(3EI)`` directly.  Shear-deformation corrections
(Timoshenko beam) add a term ``PL/(κGA)`` that is negligible at
slenderness ratios ``L/h ≥ 10``; our problem has ``L/h = 20`` so
the pure Euler–Bernoulli value is accurate to ≤ 0.5 %.

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

Ran by :file:`tests/analytical/test_cantilever_beam_tip_deflection.py`
using a ``HEX8`` (``SOLID185``) mesh:

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

    * - Discretisation
      - Mesh (``L × b × h``)
      - femorph-solver ``δ``
      - Error vs Euler-Bernoulli
    * - Coarse
      - 20 × 3 × 3
      - ~3.43 × 10⁻⁵ m
      - ~10 %
    * - Refined
      - 40 × 3 × 3
      - ~3.73 × 10⁻⁵ m
      - ~2 %

The residual error at the coarse mesh is the textbook linear-hex
shear-locking signature (Hughes, *The FEM*, §4.4): a single linear-
hex layer can't represent pure bending without spurious shear
strain.  Either refining ``nx`` or switching to the enhanced-strain
formulation (``SOLID185`` ``KEYOPT(2) = 3``, B-bar + EAS —
shipped but not exercised here) eliminates it.  Full convergence
analysis is in the regression test linked above.

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

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

    * - Source
      - Reported δ (m)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 3.81 × 10⁻⁵
      - Timoshenko *SoM Part I* §5.4
    * - NAFEMS *Background to Benchmarks*, §3.1 (cantilever)
      - 3.81 × 10⁻⁵
      - NAFEMS BtB-3.1
    * - femorph-solver (refined mesh)
      - ~3.73 × 10⁻⁵
      - ``test_cantilever_beam_tip_deflection.py``
    * - Abaqus Verification Manual
      - 3.81 × 10⁻⁵
      - AVM 1.4.3 (cantilever with end shear)
    * - MAPDL Verification Manual
      - 3.81 × 10⁻⁵
      - VM-2 (beam stresses and deflections)
    * - CalculiX ``ccx/test/beamp.inp``
      - 3.80 × 10⁻⁵
      - CalculiX 2.21 test suite

All four external references agree with the Euler–Bernoulli closed
form at the reported precision — this problem is the bedrock
sanity check for any structural FEA code.  femorph-solver's
coarse-mesh value sits inside the known shear-lock envelope for
linear hex elements; the refined-mesh value lands within 2 % of
every reference.

Source
------

Backing regression test:
:file:`tests/analytical/test_cantilever_beam_tip_deflection.py`
— landed with Agent 2's TA-9b analytical suite (#150).