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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.

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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 $Δ P 2$ . Next we remove force P and apply a vertical force Q at point 2, which produces the vertical displacement at point 1 of $Δ Q 1$ . Betti's reciprocity theorem states that:

$P \,\Delta_{Q1}=Q \,\Delta_{P2}$