## Task

The following stress state is given:

\[ \newcommand{\myvec}[1]{{\begin{bmatrix}#1\end{bmatrix}}} \]\[ S = \myvec{2 & 1 & 0\\1 & 1 & 0\\0 & 0 & 0} \cdot 100 \, Nmm^{-2} \]

Is it an uniaxial, plane or spatial stress state? Determine the principal stresses and their directions!

## Solution

An uniaxial stress state exists if only σ_{x} or σ_{y} or σ_{z} is given (an exercise for the uniaxial stress state is available here). A plane stress state can have two principal stresses and one shear stress. This is the case here, so it is a plane stress state.

The principal stresses σ_{1} and σ_{2} and the angle φ_{0} can be determined with Mohr's circle of stress, which leads to the following representation:

If the stresses σ_{x}, σ_{y} and τ_{xy} are plotted to scale, the principal stresses and the angle can be measured from Mohr's circle of stresses.

Alternatively, the values can be calculated as follows.

\[ \tag{1} \sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left( \frac{\sigma_x -\sigma_y}{2}\right)^2 + \tau_{xy}^2} \]

\[ \tag{2} \sigma_{1} = 262\,Nmm^{-2} \]

\[ \tag{3} \sigma_{2} = 38\,Nmm^{-2} \]

For the angle φ _{0} applies

\[ \tag{4} tan2\phi_0 = \frac{2 \tau_{xy}}{\sigma_x - \sigma_y} \]

\[ \tag{5} \phi_0 = \frac{1}{2} \cdot arctan \left( \frac{2 \tau_{xy}}{\sigma_x - \sigma_y} \right) \]

\[ \tag{6} \phi_0 = 31.7° \]

φ_{0} is the direction of the principal stress σ_{1}. The direction for σ_{2} is φ_{0} + π / 2. The principal stresses and their directions are thus determined for the plane stress state present here.