Question

The principal stresses at a point in a strained material are
${}^{\prime }{p}_1^{\prime }and{}^{\prime }{p}_2^{\prime }$. The resultant stress ${}^{\prime }{p}_r^{\prime }$ on the plane carrying the maximum shear stress would be

• A. $\frac{{(p_1^2+p_2^2)}^{1/2}}{2}$
• B. ${\left[\frac{p_1^2+p_2^2}{2}\right]}^{1/2}$
• C. ${\left[2(p_1^2+p_2^2)\right]}^{1/2}$
• D. $2{[p_1^2+p_2^2]}^{1/2}$

Answer 1

The correct option is B

${\left[\frac{p_1^2+p_2^2}{2}\right]}^{1/2}$

On the maximum shear stress plane
$Normalstress,\sigma =\left(\frac{p_1+p_2}{2}\right)$

$Shearstress,\tau =\left(\frac{p_1-p_2}{2}\right)$

The resultant stress, $p_r=\sqrt{{\sigma }^2+{\tau }^2}$
The other method is

$p_r=\sqrt{p{}_1^2{\cos }^2\theta +p_2^2{\sin }^2\theta }$
For maximum shear stress plane

$\theta =45{}^{\circ }$

$p_r=\sqrt{\frac{p_1^2+p_2^2}{2}}$