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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

{x~y~z~}=[c11c12c13c21c22c23c31c32c33]{xyz}\begin{Bmatrix} \widetilde{x} \\ \widetilde{y} \\ \widetilde{z} \\ \end{Bmatrix} = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \\ \end{bmatrix}\begin{Bmatrix} x \\ y \\ z \\ \end{Bmatrix}


x~=Cx\widetilde{\mathbf{x}} = \mathbf{Cx}

And the reverse transformation is then

x=C1x~=CTx~\mathbf{x} = \mathbf{C}^{- 1}\widetilde{\mathbf{x}} = \mathbf{C}^{T}\widetilde{\mathbf{x}}

The axis transformation can be considered as

x~=C(xo)\widetilde{\mathbf{x}} = \mathbf{C}\left( \mathbf{x} - \mathbf{o} \right)

And the reverse transformation as

x=CTx~+o\mathbf{x} = \mathbf{C}^{T}\widetilde{\mathbf{x}} + \mathbf{o}