There are two type of transformation in GSA: transformation of coordinates from global to local axes and vice versa, and the transformation of results such as displacements, forces, and stresses. The coordinate transformation requires a rotation + translation while the displacement transformation requires only a rotation.
The rotation can be defined by a 3 × 3 direction cosine matrix, so the transformation is
And the reverse transformation is then
The axis transformation can be considered as
And the reverse transformation as