SPRING — 2-node longitudinal spring

A linear longitudinal spring of stiffness \(k\) 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:

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

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 \(\mathbf{M}^{\mathrm{loc}}\) is zero.

Real constants

• REAL[0] — \(k\), axial stiffness in N / m.

Verification cross-references

• SPRING — two springs in series — series of two springs vs analytical equivalent stiffness \(k_{\mathrm{eq}} = k_1 k_2 / (k_1 + k_2)\).

• POINT_MASS — single-DOF spring-mass oscillator — single-DOF spring-mass system, \(f = (1/2\pi)\sqrt{k/m}\).

Implementation: 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).