Question

In a two-dimensional stress analysis, the state of stress at a point P is
$\left[\sigma \right]=\left[\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\\ {\tau }_{xy}& {\sigma }_{yy}\end{array}\right]$
The necessary and sufficient condition for existence of the state of pure shear at the point P, is

• A. ${\sigma }_{xx}+{\sigma }_{yy}=0$
• B. ${\tau }_{xy}=0$
• C. ${\sigma }_{xx}{\sigma }_{yy}-{\tau }_{xy}^2=0$
• D. $({\sigma }_{xx}-{\sigma }_{yy}{)}^2+4{\tau }_{xy}^2=0$

Answer 1

The correct option is A ${\sigma }_{xx}+{\sigma }_{yy}=0$

Given,

$\sigma =\left[\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\\ {\tau }_{xy}& {\sigma }_{yy}\end{array}\right]$

For the state of pure shear, the centre of Mohr's circle coincides with origin. Also the principal stresses are equal to shear stress
Hence,

$\frac{{\sigma }_{xx}+{\sigma }_{yy}}{2}\pm \sqrt{{\left(\frac{{\sigma }_{xx}-{\sigma }_{yy}}{2}\right)}^2+{\tau }_{xy}^2}=\sqrt{{\left(\frac{{\sigma }_{xx}-{\sigma }_{yy}}{2}\right)}^2+{\tau }_{xy}^2}$

Let us take '+ve' sign into consideration,

$\Rightarrow \frac{{\sigma }_{xx}+{\sigma }_{yy}}{2}=0$
$\Rightarrow {\sigma }_{xx}+{\sigma }_{yy}=0$