Betti's law and Maxwell's reciprocal theorem

Betti's law

Statement. In a linearly elastic structure acted on by two separate systems of loads, the work done by the forces of the first system through the displacements produced by the second system equals the work done by the forces of the second system through the displacements produced by the first system.

If system 1 has loads $P_i$ giving displacements $\Delta_{i1}$ and system 2 has loads $Q_j$ giving displacements $\Delta_{j2}$ at the same points, then

$$\sum P_i\,\Delta_{i2} = \sum Q_j\,\Delta_{j1}$$

Proof (outline). Apply $P$ first, then $Q$: the total work is $U_P + U_Q + \sum P\,\Delta_{P(Q)}$. Apply $Q$ first, then $P$: the total work is $U_Q + U_P + \sum Q\,\Delta_{Q(P)}$. Strain energy depends only on the final loads, not on the order, so the two cross terms are equal.

Maxwell's reciprocal theorem

Statement. The displacement at point A in the direction of a unit load applied at B equals the displacement at B in the direction of a unit load applied at A:

$$\delta_{AB} = \delta_{BA}, \qquad \text{or in general } f_{ij} = f_{ji}$$

It follows from Betti's law by taking each system as a single unit load. It holds for linear displacements, rotations and mixed pairs (force and couple).

Example. For a simply supported beam of span $L$, the deflection at the centre due to a unit load at $L/4$ equals the deflection at $L/4$ due to a unit load at the centre, $\dfrac{11L^3}{768EI}$.

Uses

1. It makes the flexibility matrix $[F]$ and the stiffness matrix $[K]$ symmetric, which halves the work of finding the coefficients in the force method.
2. It checks the computed flexibility coefficients ($f_{12}$ against $f_{21}$).
3. It gives influence lines for deflection: the deflected shape of the structure under a unit load placed at A is the influence line for the deflection at A. This idea is the basis of the Muller-Breslau principle.
4. It lets the deflection at a point be found from the deflection curve of another loading, which is easier to compute.