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BEAM2 · two-node 3D Timoshenko beam

Physical formulation · assumptions, essential equations, full derivation, algorithm, code and verification references.

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Physical formulationROUTE-DEPENDENT · section and slenderness matter

Two nodes with UX, UY, UZ, RX, RY, RZ at each end. The local frame and reference vector are part of the model.

01 Model and scope

Timoshenko kinematics keep transverse shear deformation by treating section rotations as independent unknowns. Axial, torsional, shear and two-plane bending energies build the local stiffness; a frame transformation maps it to global degrees of freedom.

Route assumptions and calculation checks

  1. The section law includes axial, torsional, transverse-shear, and two-plane bending stiffness.
  2. Bending uses Timoshenko shear parameters φ=12EI/(κGA L²).
  3. Consistent mass combines Hermite bending mass with rotary inertia; it is not a generic diagonal mass.

Useful foundations and prerequisites

Interpolation, mapping and assemblyStrain, stress and rotation measures

02 Essential equations and symbols

ϵ=[u′,v′−θz,w′+θy,θx′,θy′,θz′]T\boldsymbol\epsilon=[u',v'-\theta_z,w'+\theta_y,\theta_x',\theta_y',\theta_z']^TN=EAϵx,Qy=κysGAγy,Qz=κzsGAγzN=EA\epsilon_x,\quad Q_y=\kappa_y^sGA\gamma_y,\quad Q_z=\kappa_z^sGA\gamma_zT=GJκx,My=EIyκy,Mz=EIzκzT=GJ\kappa_x,\quad M_y=EI_y\kappa_y,\quad M_z=EI_z\kappa_zϕy=12EIzκysGAL2,ϕz=12EIyκzsGAL2,Kg=TTKlT\phi_y=\frac{12EI_z}{\kappa_y^sGA L^2},\quad \phi_z=\frac{12EI_y}{\kappa_z^sGA L^2},\quad \mathbf K_g=\mathbf T^T\mathbf K_l\mathbf T

Definitions and conventions

εx\varepsilon{}x
Axial strain from the axial displacement gradient.
γy,γz\gamma{}y, \gamma{}z
Transverse shear strains, including section rotations.
κx,κy,κz\kappa{}x, \kappa{}y, \kappa{}z
Torsional and bending curvatures.
TT
Local-to-global degree-of-freedom transformation.
12xyu, v, wθx, θy, θzThe chord defines local x; a reference defines y12 DOFs: 6 at each beam nodeAxial + torsion + two-plane Timoshenko bendingShear parameters depend on section and length
BEAM2 local frame and section responseThe local frame controls the signs and directions of shear, bending, torsion, section properties, and local loads. It is transformed back to the global nodal DOFs by a congruence transformation.
5 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Local frame and 12 beam DOFs

BEAM2 joins two distinct nodes. Each end has three translations and three rotations, giving 12 local DOFs. The local x axis follows the chord; a reference vector is projected onto its normal plane to define the remaining axes. A nearly parallel reference vector is rejected because it cannot define a stable frame.

e1=x2−x1L,L=∥x2−x1∥\mathbf e_1=\frac{\mathbf x_2-\mathbf x_1}{L},\qquad L=\lVert\mathbf x_2-\mathbf x_1\rVerte~2=r−(r⋅e1)e1,e3=e1×e2\widetilde{\mathbf e}_2=\mathbf r-(\mathbf r\cdot\mathbf e_1)\mathbf e_1,\qquad \mathbf e_3=\mathbf e_1\times\mathbf e_2RTR=I,det⁡R=1,qe∈R12\mathbf R^T\mathbf R=\mathbf I,\qquad \det\mathbf R=1,\qquad \mathbf q_e\in\mathbb R^{12}

Because section axes affect Iy, Iz, shear directions, and loads, the reference vector is model data rather than a cosmetic orientation.

2 · Timoshenko strains keep transverse shear

Timoshenko kinematics allow the cross-section rotations to differ from the centreline slope. This retains transverse shear deformation and avoids Euler–Bernoulli's thin/slender assumption. The signs couple v with θz and w with θy in the local frame.

εx=u′,γy=v′−θz,γz=w′+θy\varepsilon_x=u',\qquad \gamma_y=v'-\theta_z,\qquad \gamma_z=w'+\theta_yκx=θx′,κy=θy′,κz=θz′\kappa_x=\theta_x',\qquad \kappa_y=\theta_y',\qquad \kappa_z=\theta_z'

3 · Section resultants and energy

The section material converts generalized strain into axial force, two shear forces, torsion, and two bending moments. The shear correction factors κyˢ and κzˢ depend on cross-section shape and are required engineering inputs. A wrong area, inertia, or local axis changes stiffness directly.

N=EAεx,Qy=κysGAγy,Qz=κzsGAγzN=EA\varepsilon_x,\quad Q_y=\kappa_y^sGA\gamma_y,\quad Q_z=\kappa_z^sGA\gamma_zT=GJκx,My=EIyκy,Mz=EIzκzT=GJ\kappa_x,\qquad M_y=EI_y\kappa_y,\qquad M_z=EI_z\kappa_zδWint=∫0L(δεxN+δγyQy+δγzQz+δκxT+δκyMy+δκzMz) dx\delta W_{int}=\int_0^L(\delta\varepsilon_xN+\delta\gamma_yQ_y+\delta\gamma_zQ_z+\delta\kappa_xT+\delta\kappa_yM_y+\delta\kappa_zM_z)\,dx

4 · Axial/torsional blocks, flexure, and transformation

Axial and torsional stiffness use the exact two-node difference operator. Transverse flexure uses a constant-property Timoshenko solution with dimensionless shear parameters φ; when φ tends to zero the bending block approaches Euler–Bernoulli. The local 12×12 matrix is transformed by congruence so elastic energy is invariant under rotation of the beam in space.

KN=EAL[1−1−11],KT=GJL[1−1−11]\mathbf K_N=\frac{EA}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix},\qquad \mathbf K_T=\frac{GJ}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}ϕy=12EIzκysGAL2,ϕz=12EIyκzsGAL2\phi_y=\frac{12EI_z}{\kappa_y^sGA L^2},\qquad \phi_z=\frac{12EI_y}{\kappa_z^sGA L^2}Keg=TTKelT,(dg)TKegdg=(dl)TKeldl\mathbf K_e^g=\mathbf T^T\mathbf K_e^l\mathbf T,\qquad (\mathbf d^g)^T\mathbf K_e^g\mathbf d^g=(\mathbf d^l)^T\mathbf K_e^l\mathbf d^l

5 · Consistent mass, benchmarks, and limits

The consistent mass combines two-node axial and torsional blocks, a cubic-Hermite transverse-displacement block, and the physical rotary inertia of the section. The cantilever fixture checks axial response, torsion, two-plane bending, shear correction, frame invariance, and modal mass. Beam-theory comparisons are meaningful only when span, section constants, boundary conditions, and slenderness are stated. Strongly varying sections, warping, local buckling, and nonlinear large rotations are outside this linear element derivation.

Max=ρAL6[2112],Mtor=ρJL6[2112]\mathbf M_{ax}=\frac{\rho A L}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix},\qquad \mathbf M_{tor}=\frac{\rho J L}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix}Mb=ρAL420[15622L54−13L22L4L213L−3L25413L156−22L−13L−3L2−22L4L2]\mathbf M_b=\frac{\rho A L}{420}\begin{bmatrix}156&22L&54&-13L\\22L&4L^2&13L&-3L^2\\54&13L&156&-22L\\-13L&-3L^2&-22L&4L^2\end{bmatrix}Mrot,y=ρIyL6[2112],Mrot,z=ρIzL6[2112]\mathbf M_{rot,y}=\frac{\rho I_yL}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix},\qquad \mathbf M_{rot,z}=\frac{\rho I_zL}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix}

The implementation requires positive density for dynamic analysis. A diagonal lumped mass does not reproduce these coupling terms or the mass used by the named modal evidence.

04 Algorithm and implementation

Source excerpt · v0.2.11

Axial, torsional, and two-plane Timoshenko bending blocks are assembled in the local frame, then mapped by a stiffness-preserving congruence transformation.

Functions shown: local_stiffness stiffness

Code
def local_stiffness(self, coords: np.ndarray) -> np.ndarray:
    length = self.length(coords)
    section = self.material
    matrix = np.zeros((12, 12), dtype=float)
    self._pair(matrix, (0, 6), section.E * section.A / length)
    self._pair(matrix, (3, 9), section.G * section.J / length)
    phi_y = 12.0 * section.E * section.Iz / (section.kappa_y * section.G * section.A * length**2)
    phi_z = 12.0 * section.E * section.Iy / (section.kappa_z * section.G * section.A * length**2)
    self._bending_block(matrix, (1, 5, 7, 11), section.E * section.Iz, length, phi_y, sign=1.0)
    self._bending_block(matrix, (2, 4, 8, 10), section.E * section.Iy, length, phi_z, sign=-1.0)
    return 0.5 * (matrix + matrix.T)

def stiffness(self, coords: np.ndarray) -> np.ndarray:
    transform = self.transformation(coords)
    return transform.T @ self.local_stiffness(coords) @ transform
Read the complete versioned implementation →
Implementation

05 Verification and limitations

ROUTE-DEPENDENT · section and slenderness matter

The cantilever checks axial response, torsion, flexure, shear correction, frame invariance, and modal mass.

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

Consolidated qualification registryv0.2.11 verification index
Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

FormulationInterpolation and element matricesTimoshenko shear-beam referenceImplementationRun inputConsolidated qualification registryv0.2.11 verification index

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