Example

Question:

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, and (c) strain on the rod. Young's modulus of structural steel is \(2.0 \times 10^{11}\;\mathrm{N\,m}^{-2}\).

Solution:

We assume the rod is held by a clamp at one end and the force \(F\) is applied at the other end, parallel to the length of the rod.

The stress on the rod is given by: \[ \text{Stress} = \frac{F}{A} = \frac{F}{\pi r^2} \] \[ = \frac{100 \times 10^3\,\mathrm{N}}{3.14 \times (10^{-2}\,\mathrm{m})^2} = 3.18 \times 10^8\,\mathrm{N\,m}^{-2} \]
The elongation, \[ \Delta L = \left(\frac{F}{A}\right) \frac{L}{Y} \] \[ = \frac{3.18 \times 10^8\,\mathrm{N\,m}^{-2} \times 1\,\mathrm{m}}{2 \times 10^{11}\,\mathrm{N\,m}^{-2}} = 1.59 \times 10^{-3}\,\mathrm{m} = 1.59\,\mathrm{mm} \]
The strain is given by \[ \text{Strain} = \frac{\Delta L}{L} = \frac{1.59 \times 10^{-3}\,\mathrm{m}}{1\,\mathrm{m}} = 1.59 \times 10^{-3} = 0.16\% \]