Beam Element

Timoshenko beam theory accommodates shear deformation and rotational bending in thick beams and high-frequency situations.

Schematic of 2D Timoshenko beam

Degrees of Freedom

The 2D Timoshenko Beam, with three degrees of freedom (DOFs) at each node:

• Translation (Dx): Displacement along the X-axis.
• Translation (Dz): Displacement along the Z-axis.
• Rotation (Ry): Rotation about the Y-axis.

The loads are specified in the direction of the DOFs:

• Horizontal Force (Fx): Force applied along the X-axis.
• Vertical Force (Fz): Force applied along the Z-axis.
• Moment (My): Moment applied about the Y-axis.

Local Stiffness Matrix

The beam stiffness matrix in the local coordinates is given by:

$\mathbf{K_l} = \begin{pmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 & \\[2ex] 0 & \frac{12 EI_y}{ L^3 (1+\varphi)} & \frac{-6 EI_y}{L^2 (1+\varphi)} & 0 & \frac{-12 EI_y}{L^3 (1+\varphi)} & \frac{-6 EI_y}{L^2 (1+\varphi)} &\\[3ex] 0 & \frac{-6 EI_y}{L^2 (1+\varphi)} & \frac{(4 + \varphi) EI_y}{L (1+\varphi)} & 0 & \frac{6 EI_y}{L^2 (1+\varphi)} & \frac{(2 - \varphi) EI_y}{L (1+\varphi)} &\\[2ex] -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 &\\[2ex] 0 & \frac{-12 EI_y}{L^3 (1+\varphi)} & \frac{6 EI_y}{L^2 (1+\varphi)} & 0 & \frac{12 EI_y}{ L^3 (1+\varphi)} & \frac{6 EI_y}{L^2 (1+\varphi)} &\\[3ex] 0 & \frac{-6 EI_y}{L^2 (1+\varphi)} & \frac{(2 - \varphi) EI_y}{L (1+\varphi)} & 0 & \frac{6 EI_y}{L^2 (1+\varphi)} & \frac{(4 + \varphi) EI_y}{L (1+\varphi)} \end{pmatrix}$

where:

• $E$ is the Young's modulus of the material
• $A$ is the cross-sectional area of the beam
• $L$ is the length of the beam
• $I_y$ is the second moment of area about the y-axis
• $\varphi$ is the shear correction factor

Transformation Matrix

The element transformation matrix, $\mathbf{T}$, is used to transform the local stiffness matrix to the global coordinate system.

$\mathbf{T} = \begin{pmatrix} \cos(\alpha) & \sin(\alpha) & 0 & 0 & 0 & 0 \\ -\sin(\alpha) & \cos(\alpha) & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \cos(\alpha) & \sin(\alpha) & 0 \\ 0 & 0 & 0 & -\sin(\alpha) & \cos(\alpha) & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}$

Global Stiffness Matrix

The global stiffness matrix, $\mathbf{K_g}$, is obtained by multiplying the element transformation matrix, $\mathbf{T}$, with the local stiffness matrix, $\mathbf{K_l}$:

$\mathbf{K_g} = \mathbf{T}^\mathsf{T} \cdot \mathbf{K_l} \cdot \mathbf{T}$