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 :math:`\Phi \to 0`; the shear-dominated Timoshenko range (finite :math:`\Phi`) is on the roadmap. .. contents:: On this page :local: :depth: 2 ---- 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 .. math:: \Phi = \frac{12\, E\, I}{\kappa\, G\, A_s\, L^2}, which collapses to zero when the shear area :math:`A_s` is infinite or the length-to-depth ratio is large — the slender (Euler–Bernoulli) limit. The stiffness matrix below is the :math:`\Phi \to 0` form; femorph-solver's current BEAM188 kernel implements exactly this limit. Local DOF ordering (per node): :math:`(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 :math:`(U_X, U_Y, U_Z, \text{ROT}_X, \text{ROT}_Y, \text{ROT}_Z)` order. Decoupled axial / torsion / bending blocks ------------------------------------------ With the local :math:`\hat{x}` along I → J, the four elastic contributions to :math:`K_\text{local}` are decoupled: * **Axial** (:math:`u_I, u_J`): .. math:: K_a = \frac{E A}{L} \begin{bmatrix} \,\phantom{-}1 & -1 \\ -1 & \phantom{-}1 \end{bmatrix}. * **Torsion** (:math:`\theta_{xI}, \theta_{xJ}`): .. math:: K_t = \frac{G J}{L} \begin{bmatrix} \,\phantom{-}1 & -1 \\ -1 & \phantom{-}1 \end{bmatrix}, \qquad G = \frac{E}{2 (1 + \nu)}. * **Bending about local z** (:math:`v_I, \theta_{zI}, v_J, \theta_{zJ}`): .. math:: 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}. * **Bending about local y** (:math:`w_I, \theta_{yI}, w_J, \theta_{yJ}`): .. math:: 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}. The sign convention on :math:`K_{by}` differs from :math:`K_{bz}` because deflection along local :math:`\hat{z}` is driven by rotation :math:`\theta_y` through the opposite sense of the right-hand rule [Cook2001]_. Orientation ----------- Without an explicit orientation node, femorph-solver picks the local :math:`\hat{y}` so that local :math:`\hat{z}` stays close to world :math:`+Z` whenever the beam is not parallel to the global Z-axis, falling back to world :math:`+Y` for vertical beams. For axisymmetric sections (:math:`I_{yy} = I_{zz}`) the orientation is immaterial. Consistent mass --------------- The 12 × 12 local consistent mass [Cook2001]_ carries a translational part :math:`\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 -------------- .. list-table:: :widths: 25 15 60 :header-rows: 1 * - Slot - Symbol - Meaning * - ``real[0]`` - :math:`A` - Cross-sectional area. * - ``real[1]`` - :math:`I_{zz}` - Bending inertia about local z (resists y-deflection). * - ``real[2]`` - :math:`I_{yy}` - Bending inertia about local y (resists z-deflection). * - ``real[3]`` - :math:`J` - Saint-Venant torsion constant. Material -------- ``EX`` (Young's modulus), ``PRXY`` (Poisson's ratio; only used to derive :math:`G = E / (2 (1 + \nu))` for torsion), ``DENS`` (mass density for :math:`M_e`). Validation ---------- **Cantilever first mode.** Euler–Bernoulli theory gives the first natural frequency of a clamped–free beam [Timoshenko1937]_ [Cook2001]_ as .. math:: 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 :mod:`tests.elements.beam188` reproduces this to sub-percent on a 20-element mesh. **Axial natural frequency.** A free-free bar of length :math:`L` has the classical wave modes :math:`f_n = (n / 2) \sqrt{E / \rho} / L`; the BEAM188 axial block reproduces these exactly (one DOF per node per mode family). API reference ------------- .. autoclass:: femorph_solver.elements.beam188.BEAM188 :members: 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] 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