Page 115 - Aircraft Stuctures for Engineering Student

4.7 Solution of statically indeterminate systems 99

or assuming linear elasticity

(iii)

In AB

In DB

aiw dM
M=S,x and -- -0, -=x
as, as,
In CB

Thus
dM
aM
-- - -r sin r$ and - -r sin q5
=
as, as,
Substituting these expressions in Eqs (iii) and integrating we have
3.356SA + S, = Mo/r (iv)
SA + 2.178s~ = Mo/r (4
which, with Eq. (ii), enable SA, SD and S, to be found. In matrix form these equations
are written

from which we obtain

(vii)

which give
SA = O.186M0/r, S, = 0.44MO/r, Sc = 0.373Mo/r
Again the square matrix of Eqs (vi) has been inverted to produce Eqs (vii).
The bending moment distribution with directions of bending moment is shown in
Fig. 4.22.
So far in this chapter we have considered the application of the principles of the
stationary values of the total potential and complementary energies of elastic systems
in the analysis of various types of structure. Although the majority of the examples
used to illustrate the methods are of linearly elastic systems it was pointed out that
generally they may be used with equal facility for the solution of non-linear systems.