Truss, beam and frame element differences

Comparison (2D)

The local frame element matrix is the bar matrix for $u$ combined with the beam matrix for $v, \theta$:

$$[k'] = \begin{bmatrix} \frac{AE}{L} & 0 & 0 & -\frac{AE}{L} & 0 & 0 \\ 0 & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & -\frac{12EI}{L^3} & \frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{4EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{2EI}{L} \\ -\frac{AE}{L} & 0 & 0 & \frac{AE}{L} & 0 & 0 \\ 0 & -\frac{12EI}{L^3} & -\frac{6EI}{L^2} & 0 & \frac{12EI}{L^3} & -\frac{6EI}{L^2} \\ 0 & \frac{6EI}{L^2} & \frac{2EI}{L} & 0 & -\frac{6EI}{L^2} & \frac{4EI}{L} \end{bmatrix}$$

Need for transformation

Members of a truss or frame lie at different angles, but the global equations must be written in one common (global) $x$-$y$ system. The element matrix derived along the member axis (local) is converted by

$$[k] = [T]^T[k'][T]$$

For a truss element, with $c = \cos\theta$ and $s = \sin\theta$:

$$[T] = \begin{bmatrix} c & s & 0 & 0 \\ 0 & 0 & c & s \end{bmatrix}$$

For a frame element, $[T]$ is a $6\times6$ matrix with the $\theta$ rotation unchanged. Beam elements all lie along one axis, so no transformation is needed.