SPRING — 2-node longitudinal spring
===================================

A linear longitudinal spring of stiffness :math:`k` connecting
two nodes.  Two end nodes × 3 translational DOFs = 6 DOFs per
element.  No mass contribution.

* **Spec:** ``ELEMENTS.SPRING``

Stiffness
---------

The local 2 × 2 axial stiffness:

.. math::

    \mathbf{K}^{\mathrm{loc}} =
    k\, \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix},

rotated into the global 6 × 6 by a direction-cosine block.

The spring carries **only** the axial component along the
node-to-node direction; transverse motion is unconstrained
(zero stiffness, zero coupling).  When a single SPRING element
is the only member at a free node, the solver's zero-pivot
guard pins the free transverse DOFs automatically.

Mass
----

None.  SPRING is a stiffness-only kernel — every entry in
:math:`\mathbf{M}^{\mathrm{loc}}` is zero.

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

* ``REAL[0]`` — :math:`k`, axial stiffness in N / m.

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

* :ref:`sphx_glr_gallery_elements_combin14_example_combin14.py` — series of two springs vs analytical equivalent stiffness :math:`k_{\mathrm{eq}} = k_1 k_2 / (k_1 + k_2)`.
* :ref:`sphx_glr_gallery_elements_mass21_example_mass21.py` — single-DOF spring-mass system, :math:`f = (1/2\pi)\sqrt{k/m}`.

Implementation: :mod:`femorph_solver.elements.spring`.

References
----------

* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §2.2 (spring element).
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  §3.4.1 (extension to springs).