BETTI’S THEOREM* (RECIPROCITY THEOREM

Betti’s theorem, discovered by Enrico Betti in 1872 states that for all linear elastic structures subject to two sets of forces Pi and Qi, the work done by the set P though the displacement produced by set Q is equal to the work done by the set Q through displacements produced by set P.

Betti’s theorem example

Consider a beam on which two points 1 and 2 have been defined. First we apply force P at point 1 and measure the vertical displacement of point 2. Then we remove force P and apply force Q at point 2. Betti’s reciprocity theorem states that:

Maxwell–Betti Reciprocal theorem

Consider a simply supported beam of span L as shown in Fig. Let

forces P1 P2 can be loaded by two systems of forces

separately as shown in the figure.

Let u21 be the deflection below the load point P2 when only load P1 is acting. Similarly let u12 be the deflection below load P1, when only load

P2 is acting on the beam .

B & M reciprocal Theorem Continued

Reciprocal theorem

The reciprocal theorem states that the work done by forces acting through displacement of the second system is the same as the work done by the second system of forces acting through the displacements

of the first system. Hence, according to reciprocal theorem ,

Reciprocal theorem for System of forces

Summary

1 .The Castiglione's first theorem has been stated and proved. For statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of the load. This theorem when applied to the statically indeterminate structure results in the theorem of Least work. the theorem of Least Work has been

stated and proved .

2 .The celebrated theorem of Maxwell-Betti’s reciprocal theorem has been sated and proved .