1. Work Done by Gradually Applied Load: When a load is applied gradually from zero to its final value $W$, the average force is $\frac{1}{2}W$. The work done (and thus the strain energy stored) is:
$$U = \frac{1}{2} \times \text{Load} \times \text{Extension} = \frac{1}{2} W \delta L$$
2. Extension Formula: From Hooke's Law, the extension ($\delta L$) of a bar under axial load is given by:
$$\delta L = \frac{WL}{AE}$$
3. Deriving the Strain Energy Equation: Substituting the extension formula into the work equation:
$$U = \frac{1}{2} W \left( \frac{WL}{AE} \right)$$
$$U = \frac{W^2 L}{2AE}$$
This energy can also be expressed in terms of stress ($\sigma = W/A$):
$$U = \frac{\sigma^2}{2E} \times (AL) = \frac{\sigma^2}{2E} \times \text{Volume}$$
This shows that strain energy is proportional to the square of the applied load and the length, while being inversely proportional to the stiffness of the bar.