Page 113 - Aircraft Stuctures for Engineering Student

4.7 Solution of statically indeterminate systems 97

Hence

Assuming that the fuselage frame is linearly elastic we have, from Eqs (ii) and (iii)

Substituting from Eqs (iv) and (v) into Eqs (vi) gives two simultaneous equations
PR
27r - MA + NAR (vii)
7PR -MA+$NAR (viii)
87r
These equations may be written in matrix form as follows

so that

or
-1/2
{ zf} [ -i/R { -7/8}
=
which gives
PR -3P
MA = -, NA = -
4.rr 47r
The bending moment distribution follows from Eq. (iv) and is
PR
M = -(1 - $cos0 - 0sin0)
27r
The solution of Eqs (ix) involves the inversion of the matrix

[: 3RR/2]
which may be carried out using any of the standard methods detailed in texts on
matrix analysis. In this example Eqs (vi$ and (viii) are clearly most easily solved
directly; however, the matrix approach illustrates the technique and serves as a
useful introduction to the more detailed discussion in Chapter 12.

Example 4.7
A two-cell fuselage has circular frames with a rigidly attached straight member across
the middle. The bending stiffness of the lower half of the frame is 2EI, whilst that of
the upper half and also the straight member is EI.