TRUSS2 — 2-node 3D axial bar
============================

Pure-axial 2-node element.  Carries only axial force — no
bending, no torsion, no shear.  Two end nodes × 3
translational DOFs = 6 DOFs per element.

* **Spec:** ``ELEMENTS.TRUSS2``

Kinematics
----------

Linear 1D Lagrange shape functions on the natural coordinate
:math:`s \in [-1, +1]`:

.. math::

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

The axial-strain measure :math:`\varepsilon_x = \mathrm{d} u_x
/ \mathrm{d} x` reduces to the constant
:math:`(u_{x,2} - u_{x,1}) / L`.

Stiffness
---------

Closed-form local stiffness (no quadrature needed):

.. math::

    \mathbf{K}^{\mathrm{loc}} =
    \frac{E A}{L}
    \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix},

acting on the local axial DOFs.  A direction-cosine 6 × 6
rotation block lifts this into the global 3D frame.

Mass
----

Consistent (Cook Table 16.3-1):

.. math::

    \mathbf{M}^{\mathrm{loc}} =
    \frac{\rho A L}{6}
    \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}.

Lumped: :math:`\rho A L / 2` on each node's axial DOF.

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

* ``REAL[0]`` — :math:`A`, cross-sectional area.

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

* :ref:`sphx_glr_gallery_elements_link180_example_link180.py` — single-bar tip-displacement under axial load.
* :ref:`sphx_glr_gallery_verification_example_verify_axial_rod_nf.py` — fixed-free rod natural frequencies (mixed Dirichlet / Neumann eigenvalue problem).

Implementation: :mod:`femorph_solver.elements.truss2`.

References
----------

* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §2.3 (axial bar), Table 16.3-1
  (consistent mass).
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  §3.4.1 (truss element derivation).