Steps of virtual (unit) load method

The virtual work (unit load) method finds the displacement of a chosen point by applying a unit virtual load at that point, in the direction of the required displacement, on the same structure.

Steps

1. Analyse the structure under the real loads and find the internal forces: $M$ (bending moment), $N$, $V$ (for trusses, the member forces $F$).
2. Remove the real loads and apply a unit load (or unit couple for rotation) at the point and in the direction of the required displacement.
3. Analyse the structure under this unit load and find the virtual internal forces $m$ (and $n$ or $k$ for a truss).
4. Write the virtual work equation: external virtual work = internal virtual work.
5. Integrate (or sum) over all members to get the displacement.

For beams and frames (bending only):

$$1\cdot\Delta = \int \frac{M\,m}{EI}\,dx$$

For trusses:

$$1\cdot\Delta = \sum \frac{F\,k\,L}{AE}$$

6. A positive result means the displacement is in the direction of the unit load; a negative result means the opposite.

Example

Find the deflection at the free end of a cantilever of length $L$ with a point load $P$ at that end.

• Real moment: $M = -Px$ (x from the free end).
• Apply a unit load at the free end: $m = -1\cdot x$.
• Then

$$\Delta = \int_0^L \frac{(-Px)(-x)}{EI}dx = \frac{PL^3}{3EI}$$

The sign is positive, so the deflection is downward, in the direction of the unit load.

Other effects (temperature, lack of fit, support settlement) are included by using the virtual forces with the corresponding deformations: $\Delta = \sum k\,\alpha\,\Delta T\,L$ for temperature and $\sum k\,e$ for lack of fit.