Real work and virtual work, limitations

Real work method

The real work method equates the external work done by the actual loads (applied gradually) to the strain energy stored in the structure:

$$W_e = U \;\Rightarrow\; \tfrac12 P\,\Delta = \int \frac{M^2}{2EI}\,dx$$

The work is "real" because the loads move through their own actual displacements. Example: a cantilever with end load $P$: $\tfrac12P\Delta = \dfrac{P^2L^3}{6EI}$, so $\Delta = \dfrac{PL^3}{3EI}$.

 P |    /|
   |  /  |  Real work = area = 1/2 P*D
   |/____|
   +-----+--- Displacement
        D

Virtual work method

In the virtual work method, an imaginary (virtual) force system, usually a unit load at the point where the displacement is wanted, is applied. The external virtual work of the virtual forces moving through the real displacements equals the internal virtual work of the virtual internal forces on the real deformations:

$$1\cdot\Delta = \int \frac{M\,m}{EI}\,dx \qquad\text{(beams, frames)}$$ $$1\cdot\Delta = \sum \frac{F\,k\,L}{AE} \qquad\text{(trusses)}$$

Here the virtual load is constant while the real displacement occurs, so the virtual work is $1\times\Delta$ (no factor of $\tfrac12$).

 Real loads          Virtual load
 P1  P2              1 (at the point)
 v   v               v
 ______________      ______________
 -> M, F             -> m, k

Limitations of the real work method

1. Only one load can act on the structure; with several loads the energy equation has several unknown displacements.
2. It gives the displacement only at the point of application of the load, and only in the direction of the load.
3. It cannot give the displacement of a point where no load acts, or a rotation, unless a fictitious load is added.
4. It cannot account for temperature change, lack of fit or support settlement.
5. It is applicable only to linearly elastic structures.