Question

The state of 2D stress at a point is given by a matrix The maximum shear stress is MPa.
$\left[\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\\ {\tau }_{yx}& {\sigma }_{yy}\end{array}\right]=\left[\begin{array}{cc}120& 80\\ 80& 40\end{array}\right]\text{MPa}$

• A. 89.44

Answer 1

The correct option is A 89.44

${\sigma }_{1/2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$
$=\frac{120+40}{2}\pm \sqrt{{\left(\frac{120-40}{2}\right)}^2+{80}^2}$
$=80\pm 89.44\text{MPa}$
$\Rightarrow {\sigma }_1=169.44\text{MPa}$
and ${\sigma }_2=-9.44\text{MPa}$
$\therefore {\tau }_{max}=\frac{{\sigma }_1-{\sigma }_2}{2}=\frac{169.44-(-9.44)}{2}=89.44\text{MPa}$