Betti's theorem

Betti's theorem, which was discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces {P_i} i=1,...,m and {Q_j}, j=1,2,...,n, the work done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This result is also known as the Maxwell-Betti reciprocal work (or reciprocity) theorem, and has applications in structural engineering where it is used to derive the Boundary element method.

Betti's theorem is used in the design of compliant mechanisms by topology optimization approach.

Example

For a simple example let m=1 and n=1. Consider a horizontal beam on which two points have been defined: point 1 and point 2. First we apply a vertical force P at point 1 and measure the vertical displacement of point 2, denoted $Delta\_\{P2\}$. Next we remove force P and apply a vertical force Q at point 2, which produces the vertical displacement at point 1 of $Delta\_\{Q1\}$. Betti's reciprocity theorem states that:

:$P\; ,Delta\_\{Q1\}=Q\; ,Delta\_\{P2\}$