Define strain energy; relation for bending

Strain energy

Strain energy $U$ is the energy stored in an elastic body by the work done by external forces in deforming it. It equals the area under the load-displacement curve (up to the displacement):

$$U = \int_0^\Delta P\,d\Delta$$

For linear behaviour, $U = \tfrac12 P\Delta$.

Complementary strain energy

Complementary strain energy $U^*$ is the area between the load-displacement curve and the load axis:

$$U^* = \int_0^P \Delta\,dP$$

 P |        /
   |  U*  /|
   |    /  |      U + U* = P*Delta
   |  /  U |
   +-------+---- Delta

For a linear structure, $U^* = U = \tfrac12 P\Delta$; for a non-linear one they differ. Castigliano's theorems use $U$ (first) and $U^*$ (second).

Strain energy due to bending

An element $dx$ under moment $M$ rotates by $d\theta = \dfrac{M\,dx}{EI}$. The moment grows gradually, so

$$dU = \tfrac12 M\,d\theta = \frac{M^2\,dx}{2EI}$$ $$U_b = \int_0^L \frac{M^2}{2EI}\,dx$$

Using the stress method as a check: $\sigma = \dfrac{My}{I}$, $dU = \displaystyle\int\frac{\sigma^2}{2E}dA\,dx = \frac{M^2}{2EI^2}\Big(\int y^2dA\Big)dx = \frac{M^2dx}{2EI}$.

Example: cantilever with end load $P$: $M = Px$, so $U_b = \displaystyle\int_0^L \frac{P^2x^2}{2EI}dx = \frac{P^2L^3}{6EI}$.