A prismatic bar of circular cross-section [Area of cross-section = 700 ², length = 3 m] is loaded by a tensile force of 70 kN (as shown). What will be the change in the volume of the bar? [Poisson's ratio \(=\frac{1}{3}\), Modulus of elasticity = 70 GPa]

Concept:

We use the principles of axial and lateral strain in a prismatic bar under tensile loading to determine the change in volume.

Given:

• Cross-sectional area, \( A = 700 \, \text{mm}^2 \)
• Length, \( L = 3 \, \text{m} = 3000 \, \text{mm} \)
• Tensile force, \( P = 70 \, \text{kN} = 70,000 \, \text{N} \)
• Modulus of elasticity, \( E = 70 \, \text{GPa} = 70,000 \, \text{N/mm}^2 \)
• Poisson's ratio, \( \nu = \frac{1}{3} \)

Step 1: Calculate axial stress

\( \sigma = \frac{P}{A} = \frac{70,000}{700} = 100 \, \text{N/mm}^2 \)

Step 2: Calculate axial strain

\( \epsilon_{\text{long}} = \frac{\sigma}{E} = \frac{100}{70,000} = \frac{1}{700} \)

Step 3: Calculate lateral strain

\( \epsilon_{\text{lat}} = -\nu \cdot \epsilon_{\text{long}} = -\frac{1}{3} \cdot \frac{1}{700} = -\frac{1}{2100} \)

Step 4: Calculate volumetric strain

\( \epsilon_v = \epsilon_{\text{long}} + 2 \epsilon_{\text{lat}} = \frac{1}{700} + 2 \left(-\frac{1}{2100}\right) = \frac{1}{2100} \)

Step 5: Calculate change in volume

Original volume, \( V = A \cdot L = 700 \times 3000 = 2,100,000 \, \text{mm}^3 \)

\( \Delta V = \epsilon_v \cdot V = \frac{1}{2100} \times 2,100,000 = 1000 \, \text{mm}^3 \)