It is easy to analyze stresses if they are aligned with the given coordinate system, but what do we do it they are not? Well, we can transform the stresses into a different coordinate system that has been adjusted using rotation matrices.
To transform stresses into a different coordinate plane, we can follow the equation
[ 1 ]
$$ [\sigma '] = [A] [\sigma] [A]^T $$
where $ [\sigma '] $ is the stress matrix in the transformed coordinate system, $ [\sigma] $ is the stress matrix in the original coordinate system, and $ [A] $ is rotation matrix.
Now, there are 3 different types of rotation matrices, $ R_x $, $ R_y $, and $ R_z $, each a rotation about an axis. For example, $ R_x $ would be the rotation matrix about the x axis or while in the y-z plane. Let's take a look at how these matrices are derived.
Fig. 1 - Image of 2D coordinate system transformation.
In the system transformation in Figure 1, we can use the cosine and sin relationships between the transformed and original coordinate systems. From these relations, we get that
[ 2 ]
$$ [A] = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} $$
With this matrix, it can be used to transform stress matrices and it has two properties.
We can then say for a 2D transform in the x-y plane, the transformation would look something like this:
[ 3 ]
$$ \begin{bmatrix} \sigma_{xx}' & \tau_{xy}' \\ \tau_{xy}' & \sigma_{yy}' \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} \sigma_{xx} & \tau_{xy} \\ \tau_{xy} & \sigma_{yy} \end{bmatrix} \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$
Multiplying the matrices out, we get a set of equations for $ \sigma_{xx}' $, $ \sigma_{yy}' $, and $ \tau_{xy}' $.
[ 4 ]
$$ \sigma_{x}' = (\frac{\sigma_x + \sigma_y}{2}) + (\frac{\sigma_x - \sigma_y}{2}) \cos 2\theta + \tau_{xy} \sin 2\theta $$ $$ \sigma_{y}' = (\frac{\sigma_x + \sigma_y}{2}) - (\frac{\sigma_x - \sigma_y}{2}) \cos 2\theta - \tau_{xy} \sin 2\theta $$ $$ \tau_{xy}' = -(\frac{\sigma_x - \sigma_y}{2}) \sin 2\theta + \tau_{xy} \cos 2\theta $$
These transformations can be made in 2D and 3D, so don't think they only apply to 2D! This was just an example because 2D transformations will be more common in this guide.