Question

In a plane stress problem there are normal tensile stresses ${\sigma }_{x}and{\sigma }_y$ accompained by shear stress ${\tau }_{xy}$ at a point along orthogonal Cartesian co-ordinates x and y respectively. If it is observed that the minimum principal stress on certain plane is zero then

• A. ${\tau }_{xy}=\sqrt{{\sigma }_x+{\sigma }_y}$
• B. ${\tau }_{xy}=\sqrt{{\sigma }_x-{\sigma }_y}$
• C. ${\tau }_{xy}=\sqrt{{\sigma }_x{\sigma }_y}$
• D. ${\tau }_{xy}=\sqrt{\frac{{\sigma }_x}{{\sigma }_y}}$

Answer 1

The correct option is C

${\tau }_{xy}=\sqrt{{\sigma }_x{\sigma }_y}$

Principal stresses,

${\sigma }_1/{\sigma }_2=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$
Since minimum principal stress is zero, therefore

${\left(\frac{{\sigma }_x+{\sigma }_y}{2}\right)}^2={\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2$

${\tau }_{xy}=\sqrt{{\sigma }_x{\sigma }_y}$