Question

Listed below are varying combinations of stresses acting at a point and referred to axes x and y in an elastic material. Using Mohr's circle of stresses, determine the principal stresses at the point and their directions for each combination. or (N/mm or mm Try (N/mm +5 (i) +54 +30 (ii) +30 +54 +5 36 (iii) -60 (iv) +30 +30 50 om at 11.50 to x axis 29 N/mm (i) ou +55 N/mm Ans. on at 11.5 tox axis (i) on +55 N/mm OIL (iii) on 34.5 N/mm om 61 N/mm ay at 79.5° to x axis. (iv) on --40 N/mm on 60 N/mm2 on at 18.5 to x axis

Answer 1

In the question normal stresses along x-axis,y-axis and shear stress are given

we have to find the principal stresses and its direction (stresse are in N/mm²)

we know that formula for principal stress as

$\sigma_1=\frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_2=\frac{\sigma_x + \sigma_y}{2} - \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$tan 2\theta= (\frac{-(\sigma_x-\sigma_y)/2}{\tau_{xy}})$

(i) $\sigma_x=54 ,\sigma_y=30, \tau_{xy}=5$

on putting value we get

$\sigma_1=\frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_1=\frac{54+ 30}{2} + \sqrt{(\frac{54-30}{2})^2 +(5)^2}$

$\sigma_1=55$

$\sigma_2=\frac{\sigma_x + \sigma_y}{2} - \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_2=\frac{54+ 30}{2} - \sqrt{(\frac{54-30}{2})^2 +(5)^2}$

$\sigma_2=29,$

$tan 2\theta= (\frac{-(\sigma_x-\sigma_y)/2}{\tau_{xy}})$

$\theta=11.48^o$

(ii)

$\sigma_x=30,\sigma_y=54,\tau_{xy}=-5$

$\sigma_1=\frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_1= 55$

$\sigma_2=\frac{\sigma_x + \sigma_y}{2} - \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_2=29$

$\theta= 11.48$ with x-axis

(iii)

$\sigma_x=-60,\sigma_y=-36, \tau_{xy}= +5$

$\sigma_1=\frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_1=-34.5$

$\sigma_2=\frac{\sigma_x + \sigma_y}{2} - \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_2=-61$

$tan 2\theta= (\frac{-(\sigma_x-\sigma_y)/2}{\tau_{xy}})$

$\theta=79.5 ^o$ with x-axis

(iv)

$\sigma_x=30, \sigma_y= -50, \tau_{xy}=30$

$\sigma_1=\frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 +(\tau_{xy})^2}$

$\sigma_1=40,\sigma_2=-60$

$\theta= 18.5^o$ with x-axis