Page 397 - Mechanics Analysis Composite Materials

382 Mechanics and analysis of composite materials

1
61 +a2=-(N,+Np) ,
h
we arrive at the following relationships:

which coincide with Eqs. (8.31).
Substituting N, from the first equation of Eqs. (8.40) into Eq. (8.49) we get

Q[l + (~7~1
’I2
clh = - . (8.50)
rzz’[n+ (1 - n)cos2 44
Assume that the optimal shell is the structure of uniform stress. Differentiating
Eq. (8.50) with respect to r and taking into account that according to the foregoing
assumption 01 = constunt, we arrive at the following equation in which 2’ is
eliminated with the aid of Eq. (8.48):

d
-{rh[n+(l -n)Cos*(b]}-h[l-(1-n)Cos*(b] =o . (8.51)
dr
This equation specifies either the thickness or the orientation angle of the optimal
shell.
Consider two particular cases. First, assume that a fabric tape of variable width
w(r) is laid up on the surface of the mandrel along the meridians of the shell of
revolution to be fabricated. Then, 4 = 0 and Eq. (8.51) acquires the form
d
-(rh)-nh=O.
dr
The solution for this equation is

h = hR@n-‘ , (8.52)

where hR = h(r =R) is the shell thickness at the equator r =R (see Fig. 8.4).
Assuming that there is no polar opening in the shell (ro = 0)or that it is closed
(T=pr0/2) we have from Eq. (8.41) Q=$/2. Substituting this result in
Eqs. (8.48) and (8.50) we obtain

(8.53)

(8.54)