Element Stiffness in Global axes

The element stiffness is created initially in the local axes of the element. This gives a square symmetric matrix.

$$\mathbf{f}_{e} = \mathbf{K}_{e}\mathbf{u}_{e}$$

This need to be transformed before it is added to the structure stiffness matrix, however some degrees of freedom are ‘released’ and these are retained on the element. This is represented by the matrix equation

$$\begin{Bmatrix} \mathbf{f}_{s} \\ \mathbf{f}_{e} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{K}_{ss} & \mathbf{K}_{se} \\ \mathbf{K}_{es} & \mathbf{K}_{ee} \\ \end{bmatrix}\begin{Bmatrix} \mathbf{u}_{s} \\ \mathbf{u}_{e} \\ \end{Bmatrix}$$

The structure degrees of freedom need to be transformed to global directions, using the direction cosine array $\mathbf{D}$.

$$\begin{Bmatrix} \mathbf{f}_{g} \\ \mathbf{f}_{e} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{D}^{T}\mathbf{K}_{ss}\mathbf{D} & \mathbf{D}^{T}\mathbf{K}_{se} \\ \mathbf{K}_{es}\mathbf{D} & \mathbf{K}_{ee} \\ \end{bmatrix}\begin{Bmatrix} \mathbf{u}_{g} \\ \mathbf{u}_{e} \\ \end{Bmatrix}$$

Offsets in global directions then relate the global structural degrees of freedom to the nodal degrees of freedom through a rigid transformation

$$\begin{Bmatrix} \mathbf{f}_{n} \\ \mathbf{f}_{e} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{T}^{T}\mathbf{D}^{T}\mathbf{K}_{ss}\mathbf{DT} & \mathbf{T}^{T}\mathbf{D}^{T}\mathbf{K}_{se} \\ \mathbf{K}_{es}\mathbf{DT} & \mathbf{K}_{ee} \\ \end{bmatrix}\begin{Bmatrix} \mathbf{u}_{n} \\ \mathbf{u}_{e} \\ \end{Bmatrix}$$