Betti's theorem
The set of Betti (also Reziprozitätsatz of Betti ) was formulated in 1872 by Enrico Betti. It applies to static supporting structures, which are subjected to a load by forces and thus subject to deformations.
The structural system is assumed to be linearly elastic and is in equilibrium. It is further assumed that the system is loaded with two static forces or Kraft-/Lastgruppen that can be applied independently. The first load produces a displacement of the point at which the second load is applied, and also generates the second load comprises a displacement at the point at which the first load is applied. You also have to know that physical work is defined as force times distance.
The sentence then reads: The physical work done by the force on the path is equal to the work done by the force on the way by:
This is true even if not two forces, but two power groups act as well as for moments and twist angles.
In other words, one can also say that the reciprocal outer works of two systems that are in equilibrium are equal. The set of Betti is therefore also referred to as a set of reciprocity of shift work. It is also called the " set of Maxwell and Betti ".
The set of Betti has in engineering mechanics, especially the structural analysis, significance. He is also a basis of the boundary element method.
Example
We consider a horizontally mounted beam on which the points 1 and 2 are arbitrarily defined, but not just in the bearings ( because that would result in a trivial case ). First, we have a vertical force P acting at point 1 and measure the vertical lowering of the point 2, which we call. Next, we remove the force P again and now put a force Q to point 2 This produces a reduction in point 1:. After Betti now applies: