Pinched ring — diametrical deflection
========================================

Classical thin-ring benchmark from Castigliano's 1879 derivation.
A circular ring of mean radius :math:`R`, bending stiffness
:math:`EI`, loaded by two equal and opposite point forces
:math:`P` acting along a diameter deflects by

.. math::

    \delta_{\text{diam}} = \frac{P R^{3}}{E I}
        \Bigl(\frac{\pi}{4} - \frac{2}{\pi}\Bigr)
    \approx 0.14868 \, \frac{P R^{3}}{E I}.

The line of action of the loads shortens by :math:`\delta_{\text{diam}}`;
the perpendicular diameter lengthens by the Castigliano
companion value :math:`\delta_{\text{perp}} = P R^{3}(4/\pi - \pi/4) / (E I)`.

Problem
-------

.. list-table::
    :header-rows: 1
    :widths: 30 70

    * - Parameter
      - Value
    * - Mean ring radius ``R``
      - 0.1 m
    * - Cross-section
      - 10 mm × 5 mm (radial = 5 mm)
    * - Young's modulus ``E``
      - 200 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Applied point load ``P``
      - 10 N
    * - Expected ``δ_diam / 2``
      - (one loaded point's inward motion) = 3.57 × 10⁻⁵ m

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

Ran by :file:`tests/validation/test_pinched_ring.py` using the
``BEAM2`` kernel on a quarter-symmetry chain of straight
2-node beam segments discretising the arc ``θ ∈ [0, π/2]``.
BCs at the two cut ends enforce the full-ring symmetries:

* ``θ = 0`` (loaded point ``(R, 0)``): ``UY = UZ = ROTX =
  ROTY = ROTZ = 0``.  ``UX`` is free to register the inward
  deflection; the in-plane rotation ``ROTZ = 0`` enforces the
  x-axis symmetry of the cross-section.
* ``θ = π/2`` (symmetry point ``(0, R)``): ``UX = UZ = ROTX =
  ROTY = ROTZ = 0``.  ``UY`` is free.

Half of the load (``P/2``) is applied at the loaded node
inward along ``-x``; by quarter-symmetry the full-ring
diametrical deflection equals twice the loaded-node ``|u_x|``.

.. list-table::
    :header-rows: 1
    :widths: 30 25 20 25

    * - Refinement
      - Elements (quarter arc)
      - ``|u_x(loaded)|`` (m)
      - Error vs Castigliano
    * - Coarse
      - 20
      - 3.570 × 10⁻⁵
      - −0.02 %
    * - Medium
      - 40
      - 3.573 × 10⁻⁵
      - +0.08 %
    * - Refined
      - 80
      - 3.574 × 10⁻⁵
      - +0.10 %

Converges to within 0.1 % — the small positive drift reflects
the slight extensional stiffness the 2-node BEAM2 introduces
(Castigliano's derivation assumes pure bending with zero
axial strain).  Inside the published 5 % engineering
tolerance.

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

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

    * - Source
      - Reported ``δ_diam / 2`` (m)
      - Problem ID / location
    * - Closed form (Castigliano)
      - 3.57 × 10⁻⁵
      - Timoshenko & Young 1968 §79
    * - Roark's *Formulas for Stress and Strain*
      - 3.57 × 10⁻⁵
      - 8th ed. Table 9.2 Case 13
    * - femorph-solver (finest)
      - 3.574 × 10⁻⁵
      - :file:`test_pinched_ring.py`
    * - MAPDL Verification Manual
      - ≈ 3.57 × 10⁻⁵
      - VM-38 pinched-ring family
    * - Abaqus Verification Manual
      - ≈ 3.57 × 10⁻⁵
      - AVM 1.5.x pinched-ring family
    * - NAFEMS R0011
      - ≈ 3.57 × 10⁻⁵
      - Pinched-ring benchmark (quarter-symmetry)

Source
------

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

Backing regression test:
:file:`tests/validation/test_pinched_ring.py` — asserts
``|u_x|`` matches the closed form to within 0.5 % on every
refinement, not just the finest.