Betti's theorem, also known as Maxwell-Betti reciprocal work theorem, discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,m and {Qj}, 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 theorem has applications in structural engineering where it is used to define influence lines and derive the boundary element method.
Contents
Betti's theorem is used in the design of compliant mechanisms by topology optimization approach.
Proof
Consider a solid body subjected to a pair of external force systems, referred to as
When the
The work-energy balance associated with the
Now, consider that with the
Conversely, if we consider the
If the work-energy balance for the cases where the external force systems are applied in isolation are respectively subtracted from the cases where the force systems are applied simultaneously, we arrive at the following equations:
If the solid body where the force systems are applied is formed by a linear elastic material and if the force systems are such that only infinitesimal strains are observed in the body, then the body's constitutive equation, which may follow Hooke's law, can be expressed in the following manner:
Replacing this result in the previous set of equations leads us to the following result:
If we subtracting both equations then we obtain the following result:
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