Truss Element

A truss element is a structural component designed to bear only axial forces, commonly used in bridges and buildings.

Schematic of 2D truss element

Degrees of Freedom

The 2D Truss Element features two DOFs at each of the nodes:

• Translation (Dx): Displacement along the X-axis.
• Translation (Dz): Displacement along the Z-axis.

Local Stiffness Matrix

The local stiffness matrix of a truss element is given by:

$\mathbf{K_l} = \begin{pmatrix} \frac{EA}{L} & 0 & -\frac{EA}{L} & 0 \\[2ex] 0 & 0 & 0 & 0 \\[1ex] -\frac{EA}{L} & 0 & \frac{EA}{L} & 0 \\[2ex] 0 & 0 & 0 & 0 \end{pmatrix}$

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

Transformation Matrix

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

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

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, $\mathbf{K_l}$:

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

The multiplication results into:

$\mathbf{K_g}={ {EA}\over{l}}\left[\begin{array}{cccc} c^2&cs&-c^2&-cs\\ cs&s^2&-cs& -s^2\\ -c^2&-cs&c^2&cs\\ -cs&-s^2&cs&s^2 \end{array}\right];\;\;\begin{array}{c}c=\cos(\alpha)\\s=\sin(\alpha)\end{array}$