Truss Element

A truss element carries axial force only. In EduBeam there is no separate truss element type: a truss bar is a beam element with both end hinges ticked, which condenses out the bending terms and leaves the axial stiffness below.

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:

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