What is strain energy; real work method for beam

Strain energy is the energy stored in a deformed elastic body, equal to the work done by the external loads when they are applied gradually: $U = \tfrac12 P\Delta$. It is recovered on unloading within the elastic limit.

Real work method for beam deflection

The method uses the principle of conservation of energy: the external work done by the load equals the internal strain energy of the beam.

1. Find the bending moment $M$ along the beam due to the single load $P$.
2. Strain energy: $U = \displaystyle\int \frac{M^2}{2EI}\,dx$ (shear energy is usually neglected).
3. External work: $W = \tfrac12 P\,\Delta$, where $\Delta$ is the deflection at the load point.
4. Equate $W = U$ and solve for $\Delta$.

Example. Cantilever of length $L$ with a load $P$ at the free end.

$$M = Px,\qquad U = \int_0^L \frac{P^2x^2}{2EI}dx = \frac{P^2L^3}{6EI}$$ $$\tfrac12 P\Delta = \frac{P^2L^3}{6EI} \;\Rightarrow\; \Delta = \frac{PL^3}{3EI}$$

The method gives the deflection only under a single load, at its point of application, in the direction of the load.