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.

Kinematics#

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

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

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

Closed-form local stiffness (no quadrature needed):

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

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

Consistent (Cook Table 16.3-1):

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

Lumped: \(\rho A L / 2\) on each node’s axial DOF.

TRUSS2 — axial truss under end load — single-bar tip-displacement under axial load.
Fixed-free axial rod — natural frequencies — fixed-free rod natural frequencies (mixed Dirichlet / Neumann eigenvalue problem).

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).