2-D Plane Stress Transformations

Question

My strength of materials book provides the following equation to determine the x-component of the normal stress acting on all possible planes of a two dimensional element.

$$ \sigma_{x^{\prime}} = \frac{1}{2}(\sigma_x + \sigma_y ) + \frac{1}{2}(\sigma_x - \sigma_y )\cos(2\theta) + \tau_{xy}\sin(2\theta)$$

My question is -- in what orientation are the non-primed variables assumed? My instinct is that the initial orientation that the non-primed quantities are given in is irrelevant, only the numerical value of the components matter and the angle from which you measure the new components (primed quantities).

For example, if you are given a 2-D element that is initially inclined at 60 degrees and you have the values of $\sigma_x$, $\sigma_y$, and $\tau_{xy}$ in that orientation and you want to find $\sigma_{x^{\prime}}$ at the 0 degree orientation you would compute the equation above using a value of $\theta = -60$ because you have to measure relative to where you "start".

Is that correct or do I have this completely confused?

Answer 1

The transformation relation for 2D stresses in matrix form is $$ \boldsymbol{\sigma}' = \mathbf{Q}^T\, \boldsymbol{\sigma}\, \mathbf{Q} $$ where $$ \boldsymbol{\sigma} = \begin{bmatrix} \sigma_{xx} & \tau_{xy} \\ \tau_{xy} & \sigma_{yy} \end{bmatrix} ~,~~ \mathbf{Q} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} $$ In writing the stresses in matrix form, we assume that the stress components have been expressed in terms of a coordinate system with base vectors ($\mathbf{e}_x, \mathbf{e}_y$) and that the angle $\theta$ is measured counter-clockwise with respect to the $\mathbf{e}_x$ vector.

Therefore, the angle $\theta$ is always zero when you are looking along the $\mathbf{e}_x$ vector, by definition.

If you work out the algebra, you will find that (assuming I've made no mistakes) $$ \begin{bmatrix} \sigma_{xx}' \\ \sigma_{yy}' \\ \tau_{xy}' \end{bmatrix} = \begin{bmatrix} \cos^2\theta & \sin^2\theta & \sin2\theta \\ \sin^2\theta & \cos^2\theta & -\sin2\theta \\ -\tfrac{1}{2}\sin 2\theta & \tfrac{1}{2}\sin 2\theta & \cos 2\theta \\ \end{bmatrix} \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \tau_{xy} \end{bmatrix} $$

The inverse relation is $$ \mathbf{Q} \,\boldsymbol{\sigma}' \, \mathbf{Q}^T= \boldsymbol{\sigma} $$ In expanded form, $$ \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \tau_{xy} \end{bmatrix} = \begin{bmatrix} \cos^2\theta & -\sin^2\theta & -\sin2\theta \\ \sin^2\theta & \cos^2\theta & \sin2\theta \\ \tfrac{1}{2}\sin 2\theta & -\tfrac{1}{2}\sin 2\theta & \cos 2\theta \\ \end{bmatrix} \begin{bmatrix} \sigma_{xx}' \\ \sigma_{yy}' \\ \tau_{xy}' \end{bmatrix} $$ If the initial stress components are with respect to a set of initial axes that are not aligned with ($\mathbf{e}_x, \mathbf{e}_y$) and you want to compute the components along $\mathbf{e}_x$ etc., you will have to use the inverse relationship.