Question

A cube of side $a$ is subjected to shear stress P as shown in the figure. Determine tangential and normal stress along diagonal AC.

Answer 1

Hint: Stress acting on a body is the force applied per unit area. It causes the shape, area or volume of a body. A shearing force is being applied on the cube; this means that it will change the shape of the cube along its diagonal. The shearing stress is parallel to the area.
Formulas used:
$\tau ={\displaystyle \frac{F}{A}}$

Complete answer:
When a force is applied on a body such that the body gets deformed by slipping between planes, such a force applied per unit area is called the shearing or tangential stress. Its SI unit is $N{m}^{-2}$. It is given by-
$\tau ={\displaystyle \frac{F}{A}}$ - (1)
Here, $\tau$ is the shearing stress
$F$ is the force applied parallel to its side
$A$ is the displaced area
When a tangential stress, $P$ is applied on cube of side a, it changes its shape as sown in the figure below

From the above figure,
$\begin{array}{rl}& h=a\cos {45}^o\\ & \Rightarrow h={\displaystyle \frac{a}{\sqrt{2}}}\end{array}$
The area of the deformed figure will be-
$\begin{array}{rl}& A=b\times h\\ & \Rightarrow A=a\times {\displaystyle \frac{a}{\sqrt{2}}}\\ & \therefore A={\displaystyle \frac{a^2}{\sqrt{2}}}\end{array}$
The area of the figure is ${\displaystyle \frac{a^2}{\sqrt{2}}}$ sq units.
From eq (1), the shearing stress acting on the cube will be-
$\begin{array}{rl}& \tau ={\displaystyle \frac{P}{{\displaystyle \frac{a^2}{\sqrt{2}}}}}\\ & \Rightarrow \tau ={\displaystyle \frac{\sqrt{2}P}{a^2}}\end{array}$
Therefore, the shearing or tangential stress acting on the cube is ${\displaystyle \frac{\sqrt{2}P}{a^2}}$.
As there is no tension or compression acting on the cube, the normal stress acting on it is zero.
Therefore, the tangential stress on the cube is ${\displaystyle \frac{\sqrt{2}P}{a^2}}$ and the normal stress is zero.

Note:
Normal stress occurs when a body is kept under compression or tension due to which its dimensions change. The force applied in a shearing stress is parallel to the cross sectional area. After stress is applied, a restoring force is developed in the body which tends to bring it back to its original shape.