Betti

11

1. BETTI’S THEOREM* (RECIPROCITY THEOREM
Betti’s theorem, discovered by Enrico Betti in 1872 states that for all
, the work iand Q ilinear elastic structures subject to two sets of forces P
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:

2. Maxwell–Betti Reciprocal theorem
L as shown in Fig. Let Consider a simply supported beam of span
2P 1P forces can be loaded by two systems of forces
separately as shown in the figure.

when only load P1 is 2be the deflection below the load point P 21Let u
, when only load 1be the deflection below load P 12acting. Similarly let u
is acting on the beam. 2P
B & M reciprocal Theorem Continued

3. 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

5. 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.