Cantilever under tip torque — Saint-Venant torsion
=====================================================

First torsion benchmark in the corpus.  A prismatic cantilever
loaded by a pure tip torque :math:`T` about the beam axis
twists uniformly along its span.  Saint-Venant torsion gives

.. math::

    \varphi_{\text{tip}} = \frac{T L}{G J},

with :math:`G = E / (2(1+\nu))` the shear modulus and
:math:`J` the torsion constant of the cross-section.

For a rectangular section ``b × t`` (``b ≥ t``):

.. math::

    J = \beta(b/t) \, b \, t^{3},

where :math:`\beta` tabulated from Saint-Venant's infinite-
series solution (Timoshenko & Goodier §109): ``β = 0.141`` at
``b/t = 1`` (square), approaching ``1/3`` for long thin
sections.  For a square ``h × h`` section,
:math:`J \approx 0.141 \, h^{4}`.

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
    * - Tip torque ``T``
      - 10 N·m
    * - Shear modulus ``G``
      - ``E / (2(1+ν))`` = 76.92 GPa
    * - Torsion constant ``J``
      - ``0.141 × h⁴`` = 8.81 × 10⁻⁷ m⁴
    * - Expected tip twist ``φ``
      - ``T L / (G J)`` = 1.475 × 10⁻⁴ rad

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

Ran by :file:`tests/validation/test_cantilever_torsion.py`
using the ``BEAM188`` kernel — femorph-solver's 3D linear
beam with 6 DOFs per node.  Clamp all six DOFs at the left
end, apply ``MX = T`` at the tip (moment about the beam
x-axis), read the tip ``ROTX`` back.

.. list-table::
    :header-rows: 1
    :widths: 30 30 20 20

    * - Refinement
      - Elements
      - ``φ_tip`` (rad)
      - Error vs closed form
    * - Coarse
      - 10
      - 1.4752 × 10⁻⁴
      - < 1 × 10⁻¹²
    * - Medium
      - 20
      - 1.4752 × 10⁻⁴
      - < 1 × 10⁻¹²
    * - Refined
      - 40
      - 1.4752 × 10⁻⁴
      - < 1 × 10⁻¹²

``BEAM188`` integrates the Saint-Venant linear torsion equation
**exactly**: the torsion block of the stiffness matrix produces
the closed-form ``T L / (GJ)`` twist to machine precision on
any mesh.  This is the standard signature of a correctly-
implemented linear-beam torsion kernel.

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

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

    * - Source
      - Reported ``φ_tip`` (rad)
      - Problem ID / location
    * - Closed form (Saint-Venant)
      - 1.475 × 10⁻⁴
      - Timoshenko & Goodier 1970 §109
    * - Saint-Venant (1853)
      - 1.475 × 10⁻⁴
      - Original derivation, ``β(b/t)`` series
    * - femorph-solver (any mesh)
      - 1.475 × 10⁻⁴
      - :file:`test_cantilever_torsion.py`
    * - MAPDL Verification Manual
      - 1.475 × 10⁻⁴
      - VM-23 family (rotating-disk / torsion kernel)
    * - Abaqus Verification Manual
      - 1.475 × 10⁻⁴
      - AVM 1.5.x cantilever-torsion family

Source
------

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

Backing regression test:
:file:`tests/validation/test_cantilever_torsion.py` — asserts
``φ_tip`` matches the closed form to within ``1 × 10⁻¹⁰``
relative error on every mesh in the refinement sweep (not just
the finest).