Example 8.1 A structural steel rod has a radius of 10 mm and a length of \( 1.0 \mathrm{~m} . \mathrm{A} \) 100 kN force stretches it along its length. Calculate \( (\mathrm{a}) \) stress, (b) elongation, and (c)

### (a) Stress 1. **Determine the cross‐sectional area.** The rod has a radius of \[ r=10\,\text{mm}=0.01\,\text{m}. \] The area is given by \[ A=\pi r^2=\pi (0.01)^2=1.0\times10^{-4}\pi\,\text{m}^2. \] 2. **Calculate the stress.** Stress is defined as \[ \sigma=\frac{F}{A}, \] where the force is \[ F=100\,\text{kN}=1.0\times10^5\,\text{N}. \] Thus, \[ \sigma=\frac{1.0\times10^5}{1.0\times10^{-4}\pi}=\frac{1.0\times10^5}{\pi\times10^{-4}}=\frac{1.0\times10^9}{\pi}\,\text{Pa}. \] Numerically, \[ \sigma\approx \frac{1.0\times10^9}{3.1416}\approx3.18\times10^8\,\text{Pa}. \] This is equivalent to about 318 MPa. --- ### (b) Elongation 1. **Formula for elongation.** Using Hooke's law for axial deformation, the elongation \( \Delta L \) is given by \[ \Delta L=\frac{F\,L}{A\,E}, \] where - \( F=1.0\times10^5\,\text{N} \), - \( L=1.0\,\text{m} \), - \( A=1.0\times10^{-4}\pi\,\text{m}^2 \), - \( E \) is the Young's modulus. 2. **Assume Young's modulus for structural steel.** Typically, \[ E=200\,\text{GPa}=200\times10^9\,\text{Pa}. \] 3. **Plug in the values:** \[ \Delta L=\frac{1.0\times10^5\times1.0}{1.0\times10^{-4}\pi\times200\times10^9}. \] Simplify the denominator: \[ 1.0\times10^{-4}\times200\times10^9 =200\times10^5\pi. \] Hence, \[ \Delta L=\frac{1.0\times10^5}{200\pi\times10^5}=\frac{1}{200\pi}\,\text{m}. \] Numerically, \[ \Delta L\approx\frac{1}{628.32}\approx1.59\times10^{-3}\,\text{m}=1.59\,\text{mm}. \] --- ### (c) Strain 1. **Strain definition.** Strain is the ratio of elongation to the original length, \[ \epsilon=\frac{\Delta L}{L}. \] 2. **Calculate strain:** \[ \epsilon=\frac{1.59\times10^{-3}}{1.0}=1.59\times10^{-3}. \] --- ### Final Answers - \( \textbf{(a) Stress: }\sigma\approx3.18\times10^8\,\text{Pa} \) (or 318 MPa) - \( \textbf{(b) Elongation: }\Delta L\approx1.59\,\text{mm} \) - \( \textbf{(c) Strain: }\epsilon\approx1.59\times10^{-3} \)