BEAM2 — 3D 2-node Euler-Bernoulli beam
======================================

The slender-beam (no-shear) limit of the standard 2-node 3D
beam.  Two end nodes × 6 DOFs each = 12 DOFs per element.
Three independent kinematic blocks combine into a 12 × 12 local
stiffness:

* axial (linear shape function),
* torsional (linear),
* bending in two planes (Hermite cubic each).

Axial / torsion / bending decouple in the local frame; a
direction-cosine block-diagonal :math:`\boldsymbol{\Lambda}`
rotates :math:`\mathbf{K}_{\mathrm{loc}}` into the global
frame.

* **Spec:** ``ELEMENTS.BEAM2``

Kinematics
----------

Slender-beam limit of Timoshenko-beam theory: shear parameter
:math:`\Phi = 12\, EI / (\kappa\, A\, G\, L^{2}) \to 0`.  Pure
flexural response — no shear deformation, no rotary inertia.
The Timoshenko form is on the roadmap; the slender form here
is exact for prismatic Euler-Bernoulli beams.

Real constants
--------------

``REAL[0..3]`` supply the cross-section properties:

* ``REAL[0]`` — :math:`A`, cross-sectional area
* ``REAL[1]`` — :math:`I_z`, second moment about local *z*
  (resists local *y*-deflection)
* ``REAL[2]`` — :math:`I_y`, second moment about local *y*
  (resists local *z*-deflection)
* ``REAL[3]`` — :math:`K`, Saint-Venant torsion constant

For circular cross-section, :math:`K = J = I_y + I_z`; for
rectangular cross-section :math:`b \times t`,
:math:`K = \beta(b/t) \cdot b\, t^{3}` (Roark Table 20).
Substituting the polar moment :math:`J` for a rectangular
:math:`K` overstates the torsional stiffness by ~12 % on a
square shaft — see
:ref:`sphx_glr_gallery_verification_example_verify_cantilever_torsion.py`.

Shape functions
---------------

Linear axial / torsion on :math:`s \in [-1, +1]`:

.. math::

    N_1^{L}(s) = \tfrac{1 - s}{2},
    \qquad N_2^{L}(s) = \tfrac{1 + s}{2}.

Hermite cubic for transverse displacement and slope, mapped to
the physical length :math:`L` via :math:`\xi = (s + 1)/2 \in
[0, 1]`:

.. math::

    H_1(\xi) = 2\xi^{3} - 3\xi^{2} + 1,
    \qquad H_2(\xi) = L\, (\xi^{3} - 2\xi^{2} + \xi),

.. math::

    H_3(\xi) = -2\xi^{3} + 3\xi^{2},
    \qquad H_4(\xi) = L\, (\xi^{3} - \xi^{2}).

:math:`H_1` / :math:`H_3` interpolate the two nodal
displacements; :math:`H_2` / :math:`H_4` interpolate the two
nodal slopes.  See
:ref:`sphx_glr_gallery_elements_beam188_example_beam2_reference_geometry.py`
for plots and a verified Kronecker-delta-at-the-endpoints check.

Integration
-----------

**Analytic, closed-form** — no Gauss quadrature.  The Hermite
cubic basis lets the bilinear form
:math:`\int_{0}^{L} E I\, \kappa^{\!\top}\, \kappa\, \mathrm{d}x`
collapse to the canonical 4 × 4 bending block

.. math::

    \mathbf{K}_{\mathrm{bend}} =
    \frac{E I}{L^{3}}
    \begin{bmatrix}
        12 & 6L & -12 & 6L \\
        6L & 4L^{2} & -6L & 2L^{2} \\
        -12 & -6L & 12 & -6L \\
        6L & 2L^{2} & -6L & 4L^{2}
    \end{bmatrix}.

with no quadrature error to track.

Mass
----

* **Consistent** — Hermite-cubic 4 × 4 bending block
  :math:`(\rho A L / 420) \cdot (\ldots)` (Cook Table 16.3-1)
  plus axial / torsion :math:`(\rho A L / 6) \cdot (2, 1; 1, 2)`
  blocks.
* **Lumped** — :math:`\rho A L / 2` on each translational DOF;
  zero on rotational DOFs.  Diagonal, useful for explicit
  dynamics but less accurate for modal work than the
  consistent form.

Verification cross-references
-----------------------------

* :ref:`sphx_glr_gallery_elements_beam188_example_beam188.py` — tip-deflection + first-mode round-trip on a 10-element cantilever.
* :ref:`sphx_glr_gallery_verification_example_verify_cantilever_tip_moment.py` — pure end moment, parabolic deflection profile to machine precision.
* :ref:`sphx_glr_gallery_verification_example_verify_propped_cantilever_udl.py` — statically-indeterminate beam, consistent UDL nodal loading.
* :ref:`sphx_glr_gallery_verification_example_verify_ss_beam_central_load.py` — simply-supported central point load.
* :ref:`sphx_glr_gallery_verification_example_verify_cc_beam_central_load.py` — clamped-clamped central point load.
* :ref:`sphx_glr_gallery_verification_example_verify_cantilever_torsion.py` — Saint-Venant torsion on a rectangular section.

Implementation: :mod:`femorph_solver.elements.beam2`.

References
----------

* Timoshenko, S. P. (1974) *Vibration Problems in Engineering*,
  4th ed., Wiley, §5.3 (Euler-Bernoulli slender beam).
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §2.4 (axial), §2.5 (Hermite
  cubics for bending), §2.6 (torsion), Table 16.3-1
  (consistent mass blocks).
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite
  Element Method: Its Basis and Fundamentals*, 7th ed.,
  §2.5.1 eqs. (2.26)–(2.27) (Hermite cubic shape functions).
* Przemieniecki, J. S. (1968) *Theory of Matrix Structural
  Analysis*, McGraw-Hill, §5 (12-DOF stiffness derivation).
* Roark, R. J. and Young, W. C. (1989) *Roark's Formulas for
  Stress and Strain*, 6th ed., Table 20 (Saint-Venant torsion
  constants for rectangular sections).