Fig. shows an element in a stressed body. Determine normal, tangential and resultant stresses on a plane Q-Q inclined as shown. Either use analytical method or graphical method

Analytical Method

Data:

$P_1 = 140 \ Mpa$ (tensile)

$P_2 = 80 \ Mpa$ (tensile)

$\theta = 30°$ with the maximum principal plane.

A] Normal stress $\sigma_n / Pn$ :

$\sigma_n = \frac{P_1 + P_2}{2} + [\frac{P_1 – P_2}{2}] \ cos \ 2 \ \theta$

$= \frac{140 + 80}{2} + (\frac{140 – 80}{2}) \ cos \ 2 \times 30°$

= 110 + 30 cos 60

$\sigma_n = 125 \ Mpa$

B] Tangential stress $\sigma_t/Pt$

$\sigma_t = [\frac{P_1 – P_2}{2}] \ sin \ 2 \theta = (\frac{140 – 80}{2}) \ sin \ 2 \times 30$

$\sigma_t = 25.98 \ Mpa$

C] Resultant stress $\sigma_R / Re$

$\sigma_R = \sqrt{\sigma n^2 + \sigma t^2} = \sqrt{ 125^2 + 25.98^2}$

$\sigma_R = 127.67 \ Mpa$

And its direction or inclination is,

$\phi = tan^{-1} (\frac{\sigma t}{\sigma_n})$

$\phi = tan^{-1} (\frac{25.98}{125}) = 13.04°$