Question

Figure below shows a state of plane stress.


If the minimum principal stress is $-7MN/m^2$ then what is the value of ${\sigma }_x$?

• A. 30 $MN/m^2$
• B. 68 $MN/m^2$
• C. 98 $MN/m^2$
• D. 105 $MN/m^2$

Answer 1

The correct option is D 105 $MN/m^2$

Minimum principal stress is given by

${\sigma }_2=\frac{{\sigma }_x+{\sigma }_y}{2}-\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{xy}\right)}^2}$

$\Rightarrow -7=\frac{{\sigma }_x+21}{2}-\sqrt{{\left(\frac{{\sigma }_x-21}{2}\right)}^2+{\left(56\right)}^2}$

$\Rightarrow {\left(\frac{{\sigma }_x+35}{2}\right)}^2={\left(\frac{{\sigma }_x-21}{2}\right)}^2+{\left(56\right)}^2$

$\Rightarrow \frac{{\left({\sigma }_x\right)}^2+{\left(35\right)}^2+70{\sigma }_x}{4}$

$=\frac{{\left({\sigma }_x\right)}^2+{\left(21\right)}^2-42{\sigma }_x+12544}{4}$

$\Rightarrow 112{\sigma }_x=11760$

$\Rightarrow {\sigma }_x=105MN/m^2$