Principal Stresses

Entity names: PS1, PS2, PS3, worstPS

The principal stresses \sigma are named PS1, PS2, PS3. From the three principal stresses \sigma the absolute maximum value will be calculated and named worstPS. For example if a node has the three values 100, 0 and -110 MPa then -110 MPa would be shown. The three principal stresses \sigma_{1} \sigma_{2} \sigma_{3} are derived from the following equation:

\begin{displaymath}\left [\begin{array}{ccc}\sigma_{xx}-\lambda & \sigma_{xy} & \sigma_{xz}\\\sigma_{yx} & \sigma_{yy}-\lambda & \sigma_{yz}\\\sigma_{zx} & \sigma_{zy} & \sigma_{zz}-\lambda\end{array}\right ]\left [\begin{array}{c}nx\\ny\\nz\end{array}\right ]=\left [\begin{array}{c}0\\0\\0\end{array}\right ]\end{displaymath}

They are given by the three roots of the equation (stress tensor is symmetric: \sigma_{xy} = \sigma_{yx} etc.):

\begin{displaymath}\sigma^3 - (\sigma_{xx}+\sigma_{yy}+\sigma_{zz})\sigma^2 + (\sigma_{xx}\sigma_{yy}+\sigma_{yy}\sigma_{zz}+\sigma_{zz}\sigma_{xx}-\sigma_{xy}^2-\sigma_{yz}^2-\end{displaymath}

\begin{displaymath}\sigma_{zx}^2)\sigma - (\sigma_{xx}\sigma_{yy}\sigma_{zz}+2\sigma_{xy}\sigma_{yz}\sigma_{zx}-\sigma_{xx}\sigma_{yz}^2-\sigma_{yy}\sigma_{zx}^2-\sigma_{zz}\sigma_{xy}^2) = 0\end{displaymath}