StressTransformation

# Cylindrical Components from Cartesian

The transformation matrix is

$\begin{array}{c}\mathbf{Q}=\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]\end{array}$$\mathbf{Q} = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$

Displacement transformation

$\begin{matrix} u_{r} & = u\cos\theta + v\sin\theta \\ u_{\theta} & = - u\sin\theta + v\cos\theta \\ u_{z} & = w \\ \end{matrix}$

Stress transformation

${\sigma }_{rr}={\sigma }_{xx}{\mathrm{cos}}^{2}\theta +{\sigma }_{yy}{\mathrm{sin}}^{2}\theta +2{\sigma }_{xy}\mathrm{sin}\theta \mathrm{cos}\theta$$\sigma_{rr} = \sigma_{xx}\cos^{2}\theta + \sigma_{yy}\sin^{2}\theta + 2\sigma_{xy}\sin\theta\cos\theta$

${\sigma }_{\theta \theta }={\sigma }_{xx}{\mathrm{sin}}^{2}\theta +{\sigma }_{yy}{\mathrm{cos}}^{2}\theta -2{\sigma }_{xy}\mathrm{sin}\theta \mathrm{cos}\theta$$\sigma_{\theta\theta} = \sigma_{xx}\sin^{2}\theta + \sigma_{yy}\cos^{2}\theta - 2\sigma_{xy}\sin\theta\cos\theta$

${\sigma }_{zz}={\sigma }_{zz}$$\sigma_{zz} = \sigma_{zz}$

${\tau }_{r\theta }=-{\sigma }_{xx}\mathrm{sin}\theta \mathrm{cos}\theta +{\sigma }_{yy}\mathrm{sin}\theta \mathrm{cos}\theta +{\sigma }_{xy}\left({\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta \right)$$\tau_{r\theta} = - \sigma_{xx}\sin\theta\cos\theta + \sigma_{yy}\sin\theta\cos\theta + \sigma_{xy}\left( \cos^{2}\theta - \sin^{2}\theta \right)$

${\sigma }_{\theta z}={\sigma }_{yz}\mathrm{cos}\theta -{\sigma }_{zx}\mathrm{sin}\theta$$\sigma_{\theta z} = \sigma_{yz}\cos\theta - \sigma_{zx}\sin\theta$

${\sigma }_{zr}={\sigma }_{yz}\mathrm{sin}\theta +{\sigma }_{zx}\mathrm{cos}\theta$$\sigma_{zr} = \sigma_{yz}\sin\theta + \sigma_{zx}\cos\theta$