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``

.. raw:: html

    <span class="accuracy-badge accuracy-pass">
    Verified vs analytical series-spring + spring-mass natural frequency
    </span>

Cross-vendor mapping
--------------------

==========================  ==============================  ============================================
Solver                       Element name                    Notes
==========================  ==============================  ============================================
femorph-solver               ``ELEMENTS.SPRING``             linear axial spring; stiffness-only
ANSYS Mechanical APDL        ``COMBIN14``                    longitudinal spring-damper (no damper here)
NX / MSC Nastran             ``CELAS1`` / ``CELAS2``         PELAS property carries stiffness; CELAS2 inline
Abaqus                       ``SPRING1`` / ``SPRING2``       SPRING2 = two-node axial (matches our case)
LS-DYNA                      ``ELEMENT_DISCRETE``            DRO=0 (translational discrete element)
==========================  ==============================  ============================================

Restrictions
------------

Use a different element when:

* **Rotational / torsional stiffness** is needed — SPRING here
  carries only translational stiffness; the rotational analogue
  isn't shipped yet (use a stiff BEAM2 segment as a workaround).
* **Damping** is required for transient / harmonic analysis —
  SPRING is stiffness-only; Rayleigh damping at the model level
  is the current path.

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).