Question

A rectangular block on material is subjected to a tensile stress of 100 N/mm2 on one plane and 

a tensile stress of 50 N/mm2

on a plane at right angles, together with shear stress of 60N/mm2

on the 

faces. Find:

a) The direction of principal planes

b) The magnitude of principal stresses and 

c) Magnitude of greatest shear stress.

Answer 1

Stress in x direction:

$\sigma_x=100\ N/mm^2$

Stress in y direction:

$\sigma_y=50\ N/mm^2$

Shear stress:

$\tau_{xy}=60\ N/mm^2$

Principal stresses:

$\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{(\frac{\sigma_x-\sigma_y}{2})^2+\tau_{xy}^2}$

$\sigma_{1,2}=\frac{100+50}{2}\pm\sqrt{(\frac{100-50}{2})^2+60^2}$

$\sigma_1=140\ N/mm^2$

$\sigma_2=10\ N/mm^2$

Direction of principal plane:

$tan2\theta_p=-\frac{\sigma_x-\sigma_y}{2\tau_{xy}}$

$tan2\theta_p=-\frac{100-50}{2\cdot60}=-5/12$

$\theta_p=-11.31\degree$

The greatest shear stress:

$\tau_{max}=|\sigma_1-\sigma_2|/2$

$\tau_{max}=(140-10)/2=65\ N/mm^2$