Page 110 - Aircraft Stuctures for Engineering Student

94 Energy methods of structural analysis

(a) (b)
Fig. 4.18 Doubly symmetric ring.

P/2. The total complementary energy of the system is (again ignoring shear strains)

C=j ring rdOdM-Z(;A) 0

taking the bending moment as positive when it increases the curvature of the ring. In
the above expression for C, A is the displacement of the top, A, of the ring relative to
the bottom, B. Assigning a stationary value to C we have

or assuming linear elasticity and considering, from symmetry, half the ring

Thus since
P dM
M=MA--RsinO, -- -1
2 dMA
and we have

or

from which
PR
MA=-
lr