Page 93 - Aircraft Stuctures for Engineering Student

78 Energy methods of structural analysis

Fig. 4.7 Determination of the deflection of a point on a framework by the method of complementary energy.

where Xi is the extension of the ith member, Fi the force in the ith member and A, the
corresponding displacement of the rth load P,. From the principle of the stationary
value of the total complementary energy

(4.16)

from which

(4.17)

Equation (4.16) is seen to be identical to the principle of virtual forces in which virtual
forces SF and SP act through real displacements X and A. Clearly the partial
derivatives with respect to P2 of the constant loads PI, P2,. . . , P, vanish, leaving
the required deflection A, as the unknown. At this stage, before A, can be evaluated,
the load-displacement characteristics of the members must be known. For linear
elasticity

where Li, Ai and Ei are the length, cross-sectional area and modulus of elasticity of the
ith member. On the other hand, if the load-displacement relationship is of a non-
linear form, say
Fi = b(Xi)c
in which b and c are known, then Eq. (4.17) becomes

The computation of A, is best accomplished in tabular form, but before the proce-
dure is illustrated by an example some aspects of the solution merit discussion.
We note that the support reactions do not appear in Eq. (4.15). This convenient
absence derives from the fact that the displacements AI, A,, . . . , A, are the real
displacements of the frame and fulfil the conditions of geometrical compatibility
and boundary restraint. The complementary energy of the reaction at A and the