Page 120 - Aircraft Stuctures for Engineering Student

104 Energy methods of structural analysis

at the points of application of the complete system of loads are then

(4.28)

or, in matrix form

which may be written in shorthand matrix notation as
-PI = [AI-w

Suppose now that an elastic body is subjected to a gradually applied force P1 at a
point 1 and then, while P1 remains in position, a force P2 is gradually applied at
another point 2. The total strain energy U of the body is given by

The third term on the right-hand side of Eq. (4.29) results from the additional work
done by P1 as it is displaced through a further distance a12P2 by the action of P2. If we
now remove the loads and apply P2 followed by P1 we have

By the principle of superposition the strain energy stored is independent of the order
in which the loads are applied. Hence
u, = u2
and it follows that

a12 = a21 (4.31)
Thus in its simplest form the reciprocal theorem states that:
The deflection at a point 1 in a given direction due to a unit load at apoint 2 in a second
direction is equal to the deflection at the point 2 in the second direction due to a unit
load at the point 1 in the first direction.
In a similar manner, we derive the relationship between moments and rotations,
thus:

The rotation at a point 1 due to a unit moment at a point 2 is equal to the rotation at
the point 2 produced by a unit moment at the point 1.