Strain energy, gradual vs sudden load, shear

Strain energy is the energy stored in a deformed elastic body due to the work done by external loads in deforming it. It is recovered when the load is removed (within the elastic limit). For a direct load, $U = \dfrac{\sigma^2}{2E}\times\text{Volume}$.

Gradually and suddenly applied direct loads

For sudden loading, the work done $P\delta_{max}$ equals the strain energy $\tfrac12 k\delta_{max}^2$, so $\delta_{max} = 2P/k$: the stress and deflection are twice those for a gradually applied load.

Strain energy due to shear force in a beam

For an element of length $dx$ with shear force $V$, the shear stress at a distance $y$ from the neutral axis is $\tau = \dfrac{VA'\bar y}{Ib}$, and the energy per unit volume is $\dfrac{\tau^2}{2G}$. Integrating over the area, for a rectangular section:

$$dU = \int \frac{\tau^2}{2G}\,dA\,dx,\qquad \tau = \frac{V}{2I}\left(\frac{d^2}{4}-y^2\right)$$ $$dU = \frac{V^2 b}{8GI^2}\cdot\frac{d^5}{30}\,dx = \frac{6}{5}\cdot\frac{V^2}{2GA}\,dx$$

For any section:

$$U_s = \int_0^L \frac{\alpha V^2}{2GA}\,dx$$

where $\alpha$ is the shear form factor ($\alpha = 1.2$ for a rectangle, $10/9$ for a solid circle).