Question

A machine element develops principal stresses of magnitudes 2P and P. What is the maximum magnitude of P before the material reaches the yield stress $f_y$ as per Distortion Shear Energy Theory?

• A. $f_y$
• B. $\frac{f_y}{2}$
• C. $\frac{f_y}{2\sqrt{3}}$
• D. $\frac{f_y}{\sqrt{3}}$

Answer 1

The correct option is D $\frac{f_y}{\sqrt{3}}$

As per distortion energy theory:

$\frac{1}{2}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2]\le f_y^2$

$\Rightarrow \frac{1}{2}[{(2P-P)}^2+{(P-0)}^2+{(0-2P)}^2]\le f_y^2$

$\Rightarrow \frac{1}{2}[P^2+P^2+4P^2]\le f_y^2$

$\Rightarrow P\le \frac{f_y}{\sqrt{3}}$