This document discusses Betti's and Maxwell's laws of reciprocal deflections for linearly elastic structures. Betti's theorem states that the work done by a set of external forces Pm acting through displacements ∆mn produced by another set of forces Pn is equal to the work done by Pn acting through displacements ∆nm produced by Pm. Maxwell's law of reciprocal deflection states that the deflection of point n due to a force P at point m is numerically equal to the deflection of point m due to the same force P applied at point n. The total external work done on a structure by two sets of forces Pm and Pn applied in different sequences must be the same according to the principle

4. BETTI’S AND MAXWELL’S LAWS OF
RECIPROCAL DEFLECTIONS
Suppose a linearly elastic structure shown in
Fig. is in equilibrium under two separate and
independent systems of forces—system of
forces Pm and system of forces Pn.
Consider that the Pm system of forces is
gradually applied first. Let the deflections at
the point and direction of forces Pm be
represented by

5. If the second system of forces Pn is applied on
the structure while Pm forces are present, the
structure will deform once more and will rest
in equilibrium in its final deformed shape as
shown.
Let
represent the deformations at the point and in
the direction of forces Pn.
The total external work done by these forces
would then be

6. where ∆mn represents the deflection of the
point of application of one of the Pm forces
caused by the Pn force system.
The first and last terms in Eq. 1 represent the
work done by Pm and Pn forces respectively as
they are gradually applied.
The middle term, however, represents the
work done by Pm forces riding along the
deflections caused by Pn forces.

7. Suppose now the loading sequence is reversed,
that is, Pn forces are first applied and Pm forces
later. Pn forces in full ride along the deflections
caused by
Pm forces. The final deformed shape of the
structure will be the same as before which is
shown in dotted line in Fig.1. The total external
work done in this case will be

8. According to the principle of superposition, the
total external work done in either of the
sequences of loading should be same
Hence equating Eqs. 1 and 2, we get
This is known as a Betti’s theorem and may be stated as follows.
For a linearly elastic structure, the work done by a set of external forces Pm acting
through displacements ∆mn produced by another set of forces Pn is equal, to the
work done by the second set of external forces Pn acting through displacements ∆nm
produced by forces Pm.

9. Suppose now that both Pm and Pn systems
consist of a single load P having the magnitude
but not necessarily in the same direction as
shown in Fig. 1, then, from Eq. 3,
This is known as Maxwell’s law of reciprocal deflection and states that:
The deflection of point n due to force P at point m is numerically equal to
the deflection of the point m due to force P applied at point n.