Page 111 - Aircraft Stuctures for Engineering Student

4.7 Solution of statically indeterminate systems 95

Fig. 4.19 Distribution of bending moment in a doubly symmetric ring.

The bending moment distribution is then

and is shown diagrammatically in Fig. 4.19.
Let us now consider a more representative aircraft structural problem. The circular
fuselage frame of Fig. 4.20(a) supports a load P which is reacted by a shear flow q
(i.e. a shear force per unit length: see Chapter 9), distributed around the circumference
of the frame from the fuselage skin. The value and direction of this shear flow are
quoted here but are derived from theory established in Section 9.4. From our previous
remarks on the effect of symmetry we observe that there is no shear force at the
section A on the vertical plane of symmetry. The unknowns are therefore the bending
moment MA and normal force NA. We proceed, as in the previous example, by
writing down the total complementary energy C of the system. Thus, neglecting
shear strains
Io
M
C = jring dB dM - PA

in which A is the deflection of the point of application of P relative to the top of the
frame. Note that MA and NA do not contribute to the complement of the potential
energy of the system since, by symmetry, the rotation and horizontal displacements
at A are zero. From the principle of the stationary value of the total complementary
energy

(ii)

and
(iii)