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

MAPDL-compatible alias: BEAM188.

Kinematics. Two-node straight beam with six DOFs per node (three translations + three rotations); 12 DOFs per element.

Stiffness. Axial + torsion + two independent bending planes, each assembled in closed form in the local frame and rotated to global.

Mass. Consistent mass matrix with translational + rotational (section inertia) contributions.

MAPDL compatibility. Reproduces BEAM188 in the slender (Euler–Bernoulli) limit \(\Phi \to 0\); the shear-dominated Timoshenko range (finite \(\Phi\)) is on the roadmap.

Beam kinematics and the slender limit#

BEAM188 is documented as a Timoshenko beam [Hughes1987] [Bathe2014]: a straight reference axis with six kinematic DOFs at every cross-section (three translations + three rotations), plus the shear correction

\[\Phi = \frac{12\, E\, I}{\kappa\, G\, A_s\, L^2},\]

which collapses to zero when the shear area \(A_s\) is infinite or the length-to-depth ratio is large — the slender (Euler–Bernoulli) limit. The stiffness matrix below is the \(\Phi \to 0\) form; femorph-solver’s current BEAM188 kernel implements exactly this limit.

Local DOF ordering (per node): \((u, v, w, \theta_x, \theta_y, \theta_z)\). Global assembly fills rows / cols 0–5 for node I and 6–11 for node J in MAPDL’s canonical \((U_X, U_Y, U_Z, \text{ROT}_X, \text{ROT}_Y, \text{ROT}_Z)\) order.

Decoupled axial / torsion / bending blocks#

With the local \(\hat{x}\) along I → J, the four elastic contributions to \(K_\text{local}\) are decoupled:

  • Axial (\(u_I, u_J\)):

    \[\begin{split}K_a = \frac{E A}{L} \begin{bmatrix} \,\phantom{-}1 & -1 \\ -1 & \phantom{-}1 \end{bmatrix}.\end{split}\]

  • Torsion (\(\theta_{xI}, \theta_{xJ}\)):

    \[\begin{split}K_t = \frac{G J}{L} \begin{bmatrix} \,\phantom{-}1 & -1 \\ -1 & \phantom{-}1 \end{bmatrix}, \qquad G = \frac{E}{2 (1 + \nu)}.\end{split}\]

  • Bending about local z (\(v_I, \theta_{zI}, v_J, \theta_{zJ}\)):

    \[\begin{split}K_{bz} = \frac{E I_{zz}}{L^3} \begin{bmatrix} \phantom{-}12 & \phantom{-}6L & -12 & \phantom{-}6L \\ \phantom{-}6L & \phantom{-}4L^2 & -6L & \phantom{-}2L^2 \\ -12 & -6L & \phantom{-}12 & -6L \\ \phantom{-}6L & \phantom{-}2L^2 & -6L & \phantom{-}4L^2 \end{bmatrix}.\end{split}\]

  • Bending about local y (\(w_I, \theta_{yI}, w_J, \theta_{yJ}\)):

    \[\begin{split}K_{by} = \frac{E I_{yy}}{L^3} \begin{bmatrix} \phantom{-}12 & -6L & -12 & -6L \\ -6L & \phantom{-}4L^2 & \phantom{-}6L & \phantom{-}2L^2 \\ -12 & \phantom{-}6L & \phantom{-}12 & \phantom{-}6L \\ -6L & \phantom{-}2L^2 & \phantom{-}6L & \phantom{-}4L^2 \end{bmatrix}.\end{split}\]

The sign convention on \(K_{by}\) differs from \(K_{bz}\) because deflection along local \(\hat{z}\) is driven by rotation \(\theta_y\) through the opposite sense of the right-hand rule [Cook2001].

Orientation#

Without an explicit orientation node, femorph-solver picks the local \(\hat{y}\) so that local \(\hat{z}\) stays close to world \(+Z\) whenever the beam is not parallel to the global Z-axis, falling back to world \(+Y\) for vertical beams. For axisymmetric sections (\(I_{yy} = I_{zz}\)) the orientation is immaterial.

Consistent mass#

The 12 × 12 local consistent mass [Cook2001] carries a translational part \(\rho A L\) and a rotational / transverse-inertia part coupling deflection to rotation. The closed-form blocks for each plane are the classical Euler–Bernoulli consistent mass; femorph-solver implements them directly — no quadrature.

Real constants#

Slot

Symbol

Meaning

real[0]

\(A\)

Cross-sectional area.

real[1]

\(I_{zz}\)

Bending inertia about local z (resists y-deflection).

real[2]

\(I_{yy}\)

Bending inertia about local y (resists z-deflection).

real[3]

\(J\)

Saint-Venant torsion constant.

Material#

EX (Young’s modulus), PRXY (Poisson’s ratio; only used to derive \(G = E / (2 (1 + \nu))\) for torsion), DENS (mass density for \(M_e\)).

Validation#

Cantilever first mode. Euler–Bernoulli theory gives the first natural frequency of a clamped–free beam [Timoshenko1937] [Cook2001] as

\[f_1 = \frac{\beta_1^2}{2 \pi L^2}\, \sqrt{\frac{E I}{\rho A}}, \qquad \beta_1 L \approx 1.8751.\]

The BEAM188 cantilever benchmark under tests.elements.beam188 reproduces this to sub-percent on a 20-element mesh.

Axial natural frequency. A free-free bar of length \(L\) has the classical wave modes \(f_n = (n / 2) \sqrt{E / \rho} / L\); the BEAM188 axial block reproduces these exactly (one DOF per node per mode family).

API reference#

class femorph_solver.elements.beam188.BEAM188[source]#

Bases: ElementBase

static ke(coords: ndarray, material: dict[str, float], real: ndarray) → ndarray[source]#

Return the element stiffness matrix in global DOF ordering.

Parameters:

  • coords ((n_nodes, 3) float64)

  • material (mapping with MAPDL property names as keys (EX, PRXY, DENS, ...))

  • real (1-D array of real constants)

static me(coords: ndarray, material: dict[str, float], real: ndarray, lumped: bool = False) → ndarray[source]#

Return the element mass matrix in global DOF ordering.

References#

[Hughes1987]

Hughes, T. J. R. (1987 / Dover 2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Ch. 5 on beams and thin-walled structures. https://store.doverpublications.com/0486411818.html

[Bathe2014]

Bathe, K.-J. (2014). Finite Element Procedures, 2nd ed. Ch. 5 (beams). https://www.klausjurgenbathe.com/fepbook/

[Cook2001] (1,2,3)

Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. (2001). Concepts and Applications of Finite Element Analysis, 4th ed. Wiley. Ch. 4 (beams), Ch. 11 (consistent mass). https://www.wiley.com/en-us/Concepts+and+Applications+of+Finite+Element+Analysis%2C+4th+Edition-p-9780471356059

[Timoshenko1937]

Timoshenko, S. (1937). Vibration Problems in Engineering, 2nd ed. D. Van Nostrand Company (reprinted by Wiley). The derivation of the first-mode Euler–Bernoulli cantilever frequency. https://archive.org/details/vibrationproblem00timo