Page 98 - Aircraft Stuctures for Engineering Student

82 Energy methods of structural analysis

The total complementary energy C of the system is given by

C=~L~~dOdM- (4.18)
PA,
in which J, de dM is the complementary energy of an element Sz of the beam. This
element subtends an angle Sf3 at its centre of curvature due to the application of the
bending moment M. From the principle of the stationary value of the total comple-
mentary energy

or

(4.19)

Equation (4.19) is applicable to either a non-linear or linearly elastic beam. To
proceed further, theifore, we require the load-displacement (M - 0) and bending
,
moment-load (M - P) relationships. It is immaterial for the purposes of this illustra-
tive problem whether the system is linear or non-linear, since the mechanics of the
solution are the same in either case. We choose therefore a linear M - 0 relationship
as this is the case in the majority of the problems we consider. Hence from Fig. 4.9
se = KSZ
or

from simple beam theory

where the product modulus of elasticity x second moment of area of the beam cross
section is known as the bending orpexural rigidity of the beam. Also

M = Pz
so that
dM

Substitution for de, M and dM/dP in Eq. (4.19) gives

or

The fictitious load method of the framework example may be employed in the
solution of beam deflection problems where we require deflections at positions on
the beam other than concentrated load points. Suppose that we are to find the tip
deflection AT of the cantilever of the previous example in which the concentrated
load has been replaced by a uniformly distributed load of intensity w per unit