Solid Mechanics

Formulae: Stress Transformation

The necessary formulae for the plane stress case are as follows(ref. Solid Mechanics, Popov).Stresses on the $\theta$-plane.\begin{displaymath} \sigma_{x'}=\frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2} cos(2\theta) + \tau_{xy}sin(2\theta) \end{displaymath}\begin{displaymath} \tau_{x'y'}=-\frac{\sigma_x-\sigma_y}{2} sin(2\theta) + \tau_{xy}cos(2\theta) \end{displaymath}Principal planes\begin{displaymath} tan(2\theta_{1})=\frac{2\tau_{xy}}{\sigma_x-\sigma_y} \end{displaymath}This gives principal planes at $\theta_{11}$ and $\theta_{12}(=\theta_{11}+\pi/2)$. Principal stresses $(\sigma_{1}$ and $\sigma_2)$ corresponding to these planes are found from the formula for $(\sigma_{x'})$ above.Maximum shear stress $(\sigma_1$ ~$ \sigma_2)/2$ and its planes

Planes at 

and,

 

are the planes on which maximum shear stress occurs. Sense of maximum shear stresses corresponding to these planes are found from the formula for $(\tau_{x'y'})$ above.


Principal stresses and maximum shear stress can also be found directly as,