Theorems on displacement

Displacements of an elastic structure can be found, and checked, with the following theorems.

1. Maxwell's reciprocal displacement theorem

In a linearly elastic structure, the deflection at point A due to a unit load at point B equals the deflection at B due to a unit load at A:

$$\delta_{AB} = \delta_{BA}$$

Illustration. Take a cantilever of length $L$ with A at mid-span and B at the free end.

• Unit load at B: deflection at A $= \dfrac{5L^3}{48EI}$.
• Unit load at A: deflection at B $= \dfrac{(L/2)^2\left(3L - L/2\right)}{6EI} = \dfrac{5L^3}{48EI}$.

The two are equal. The same reciprocity holds for rotations, and for mixed pairs: the rotation at A due to a unit load at B equals the deflection at B due to a unit couple at A.

2. Betti's law

The work done by a first system of loads $P_1$ through the displacements caused by a second system $P_2$ equals the work of $P_2$ through the displacements caused by $P_1$: $\sum P_1\Delta_{12} = \sum P_2\Delta_{21}$. Maxwell's theorem is the special case of two unit loads.

3. Castigliano's second theorem

The displacement at a point in the direction of a load equals the partial derivative of the strain energy with respect to that load: $\Delta = \dfrac{\partial U}{\partial P}$. For a beam, $\Delta = \displaystyle\int \frac{M}{EI}\frac{\partial M}{\partial P}dx$.

4. Unit load (virtual work) theorem

The displacement at a point is $\Delta = \displaystyle\int \frac{M\,m}{EI}\,dx$, where $M$ is the moment due to the real loads and $m$ is the moment due to a unit load at the point and in the direction of the required displacement.

These theorems give the flexibility coefficients and are used to apply the compatibility condition in the force method.