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Structural Steel Rod Problem
A structural steel rod has a radius of 10 mm and a length of 1.0 m. A 100 kN force stretches it along its length. Calculate:
- (a) Stress
- (b) Elongation
- (c) Strain
Young's modulus of structural steel is \(2.0 \times 10^{11} \, \text{N/m}^2\).
Solution
Given data:
Radius of the rod, \( r = 10 \, \text{mm} = 0.01 \, \text{m} \)
Length of the rod, \( L = 1.0 \, \text{m} \)
Applied force, \( F = 100 \, \text{kN} = 100 \times 10^3 \, \text{N} \)
Young’s modulus, \( Y = 2.0 \times 10^{11} \, \text{N/m}^2 \)
(a) Stress
Stress is given by:
\[
\text{Stress} = \frac{F}{A}
\]
Where \( A \) is the cross-sectional area of the rod:
\[
A = \pi r^2 = \pi (0.01)^2 = 3.14 \times 10^{-4} \, \text{m}^2
\]
Substituting the values:
\[
\text{Stress} = \frac{100 \times 10^3}{3.14 \times 10^{-4}} = 3.18 \times 10^8 \, \text{N/m}^2
\]
(b) Elongation
Elongation is given by:
\[
\Delta L = \frac{F L}{A Y}
\]
Substituting the values:
\[
\Delta L = \frac{100 \times 10^3 \times 1.0}{3.14 \times 10^{-4} \times 2.0 \times 10^{11}}
\]
\[
\Delta L = \frac{100 \times 10^3}{6.28 \times 10^7} = 1.59 \times 10^{-3} \, \text{m} = 1.59 \, \text{mm}
\]
(c) Strain
Strain is given by:
\[
\text{Strain} = \frac{\Delta L}{L}
\]
Substituting the values:
\[
\text{Strain} = \frac{1.59 \times 10^{-3}}{1.0} = 1.59 \times 10^{-3}
\]
Final Results:
1. Stress: \( 3.18 \times 10^8 \, \text{N/m}^2 \)
2. Elongation: \( 1.59 \, \text{mm} \)
3. Strain: \( 1.59 \times 10^{-3} \) (dimensionless)
