Derive virtual work principle

Principle of virtual work (for deformable bodies): If a deformable structure in equilibrium under a real load system is given a virtual displacement field (compatible, small and imaginary), the external virtual work done by the real forces equals the internal virtual work done by the real internal stresses on the virtual strains. In the dual form (virtual forces), virtual loads are applied and move through the real displacements; this is the form used below.

Derivation (unit load form)

Consider a structure under real loads $P_1, P_2,\dots$ producing real internal forces and real deformations. To find the displacement $\Delta$ at a point, apply a virtual unit load at that point in the required direction. Let this virtual load produce internal forces: moment $m$, axial force $n$ and shear $v$.

1. External virtual work. The unit load, kept constant, moves through the real displacement $\Delta$:

$$W_e = 1\times\Delta$$

2. Internal virtual work. In an element of length $dx$, the virtual internal forces move through the real deformations of the element. For a linearly elastic member, the real deformations are

$$d\theta = \frac{M\,dx}{EI},\qquad d\delta_{axial} = \frac{N\,dx}{AE},\qquad d\gamma = \frac{\alpha V\,dx}{GA}$$

so the internal virtual work of the element is

$$dW_i = m\,d\theta + n\,d\delta_{axial} + v\,d\gamma = \frac{mM}{EI}dx + \frac{nN}{AE}dx + \frac{\alpha vV}{GA}dx$$

3. Equate $W_e = W_i$ (the virtual force system is in equilibrium, and the real deformations are compatible):

$$1\cdot\Delta = \int \frac{mM}{EI}dx + \int\frac{nN}{AE}dx + \int\frac{\alpha vV}{GA}dx$$

For beams and frames, usually only the first term is kept. For trusses, $\Delta = \sum \dfrac{kFL}{AE}$.

The general principle ($W_e = W_i$, i.e. $1\cdot\Delta = \int m\,d\theta + \int n\,d\delta + \int v\,d\gamma$) holds for any material behaviour, provided the real deformations are small and compatible and the virtual forces satisfy equilibrium. The final form with $EI$, $AE$ and $GA$, however, uses $d\theta = M\,dx/EI$ and similar relations, so it applies only to linearly elastic members.