BEAM2 — 3D 2-node Euler-Bernoulli beam#

The slender-beam (no-shear) limit of the standard 2-node 3D beam. Two end nodes × 6 DOFs each = 12 DOFs per element. Three independent kinematic blocks combine into a 12 × 12 local stiffness:

  • axial (linear shape function),

  • torsional (linear),

  • bending in two planes (Hermite cubic each).

Axial / torsion / bending decouple in the local frame; a direction-cosine block-diagonal \(\boldsymbol{\Lambda}\) rotates \(\mathbf{K}_{\mathrm{loc}}\) into the global frame.

  • Spec: ELEMENTS.BEAM2

Kinematics#

Slender-beam limit of Timoshenko-beam theory: shear parameter \(\Phi = 12\, EI / (\kappa\, A\, G\, L^{2}) \to 0\). Pure flexural response — no shear deformation, no rotary inertia. The Timoshenko form is on the roadmap; the slender form here is exact for prismatic Euler-Bernoulli beams.

Real constants#

REAL[0..3] supply the cross-section properties:

  • REAL[0] — \(A\), cross-sectional area

  • REAL[1] — \(I_z\), second moment about local z (resists local y-deflection)

  • REAL[2] — \(I_y\), second moment about local y (resists local z-deflection)

  • REAL[3] — \(K\), Saint-Venant torsion constant

For circular cross-section, \(K = J = I_y + I_z\); for rectangular cross-section \(b \times t\), \(K = \beta(b/t) \cdot b\, t^{3}\) (Roark Table 20). Substituting the polar moment \(J\) for a rectangular \(K\) overstates the torsional stiffness by ~12 % on a square shaft — see Cantilever Saint-Venant torsion — rectangular cross-section.

Shape functions#

Linear axial / torsion on \(s \in [-1, +1]\):

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

Hermite cubic for transverse displacement and slope, mapped to the physical length \(L\) via \(\xi = (s + 1)/2 \in [0, 1]\):

\[H_1(\xi) = 2\xi^{3} - 3\xi^{2} + 1, \qquad H_2(\xi) = L\, (\xi^{3} - 2\xi^{2} + \xi),\]
\[H_3(\xi) = -2\xi^{3} + 3\xi^{2}, \qquad H_4(\xi) = L\, (\xi^{3} - \xi^{2}).\]

\(H_1\) / \(H_3\) interpolate the two nodal displacements; \(H_2\) / \(H_4\) interpolate the two nodal slopes. See BEAM2 reference geometry — Hermite cubic shape functions for plots and a verified Kronecker-delta-at-the-endpoints check.

Integration#

Analytic, closed-form — no Gauss quadrature. The Hermite cubic basis lets the bilinear form \(\int_{0}^{L} E I\, \kappa^{\!\top}\, \kappa\, \mathrm{d}x\) collapse to the canonical 4 × 4 bending block

\[\begin{split}\mathbf{K}_{\mathrm{bend}} = \frac{E I}{L^{3}} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^{2} & -6L & 2L^{2} \\ -12 & -6L & 12 & -6L \\ 6L & 2L^{2} & -6L & 4L^{2} \end{bmatrix}.\end{split}\]

with no quadrature error to track.

Mass#

  • Consistent — Hermite-cubic 4 × 4 bending block \((\rho A L / 420) \cdot (\ldots)\) (Cook Table 16.3-1) plus axial / torsion \((\rho A L / 6) \cdot (2, 1; 1, 2)\) blocks.

  • Lumped — \(\rho A L / 2\) on each translational DOF; zero on rotational DOFs. Diagonal, useful for explicit dynamics but less accurate for modal work than the consistent form.

Verification cross-references#

Implementation: femorph_solver.elements.beam2.

References#

  • Timoshenko, S. P. (1974) Vibration Problems in Engineering, 4th ed., Wiley, §5.3 (Euler-Bernoulli slender beam).

  • Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, §2.4 (axial), §2.5 (Hermite cubics for bending), §2.6 (torsion), Table 16.3-1 (consistent mass blocks).

  • Zienkiewicz, O. C. and Taylor, R. L. (2013) The Finite Element Method: Its Basis and Fundamentals, 7th ed., §2.5.1 eqs. (2.26)–(2.27) (Hermite cubic shape functions).

  • Przemieniecki, J. S. (1968) Theory of Matrix Structural Analysis, McGraw-Hill, §5 (12-DOF stiffness derivation).

  • Roark, R. J. and Young, W. C. (1989) Roark’s Formulas for Stress and Strain, 6th ed., Table 20 (Saint-Venant torsion constants for rectangular sections).