A steel cubic block of side 200 mm is subjected to hydrostatic pressure of 250 N/mm² . The elastic modulus is 2 × 10⁵ N/mm² and Poisson ratio is 0.3 for steel. The side of the block is reduced by _________ mm (round off to two decimal places).

The correct answer is 0.1.

Step-by-Step Explanation:

1. Identify the given parameters:
We are given a steel cubic block with the following material properties and loading conditions:
• Side of the cube, a = 200 mm
• Hydrostatic pressure, P = 250 N/mm²
• Modulus of Elasticity, E = 2 × 10⁵ N/mm²
• Poisson's ratio, μ = 0.3

2. Determine the stress states:
Since the block is subjected to hydrostatic pressure, the compressive stress is equal in all three perpendicular directions (x, y, and z directions):

${\sigma }_x={\sigma }_y={\sigma }_z=-P=-250{\text{N/mm}}^2$

The negative sign indicates that the stresses are compressive, which leads to a reduction in the dimensions of the block.

3. Calculate the linear strain:
Using generalized Hooke's Law, the linear strain (ε) along any of the directions (for example, the x-direction) is given by:

$\epsilon =\frac{{\sigma }_x}{E}-\mu \frac{{\sigma }_y}{E}-\mu \frac{{\sigma }_z}{E}$

Substituting ${\sigma }_x={\sigma }_y={\sigma }_z=-P$ into the expression:

$\epsilon =\frac{-P}{E}-\mu \left(\frac{-P}{E}\right)-\mu \left(\frac{-P}{E}\right)$

Simplifying the strain equation yields:

$\epsilon =-\frac{P}{E}\left(1-2\mu \right)$

4. Calculate the change in dimension of the side:
The linear strain is also defined as the change in length of the side (δα) divided by the original side length (a):

$\epsilon =\frac{\delta a}{a}$

Therefore, the change in the side of the block is:

$\delta a=a\times \epsilon =-\frac{P\times a}{E}\left(1-2\mu \right)$

5. Substitute the values:
Substitute the given values into the equation:

$\delta a=-\frac{250\times 200\times \left\{1-\left(2\times 0.3\right)\right\}}{2\times {10}^5}$

Calculate the numerator:
$1-(2\times 0.3)=1-0.6=0.4$
$250\times 200\times 0.4=50000\times 0.4=20000$

Divide by the denominator:

$\delta a=-\frac{20000}{200000}=-0.10\text{mm}$

The negative sign denotes a reduction in length. Thus, the magnitude of the reduction in the side of the block is 0.10 mm.