Page 402 - Mechanics Analysis Composite Materials

Chapter 8. Optimal composite structures 387

i.e., under axial tension, a hyperbolic shell is obtained with the meridian determined
as

r' - 2 tan2 4R = R' .

This meridian corresponds to line 1 in Fig. 8.6. For 4R = 0, the hyperbolic shell
degenerates into a cylinder (line 2). Curve 3 corresponds to T = pr0/2, i.e., to a shell
for which the polar opening of radius r0 is closed. For the special angle
4R = 4o = 54"44', the shell degenerates into a circular cylindrical shell (line 2)
discussed in section 8.1. For T = 0, Le., in case of an open polar hole, the meridian
has the form corresponding to curve 4. The change in the direction of axial forces T
yields a toroidal shell (line 5). Performing integration of Eq. (8.65) and introducing
dimensionless parameters

we finally arrive at

-7
-
z=- - '- F(k, 0) + d=E(k, 0) , (8.67)
d-
where

J
F(k,B) = E(k, 0) = d1 - k2 sin' 0 de
0 0
are the first-kind and the second-kind elliptic integrals and

kl 2 =

As an application of the foregoing equations, consider the optimal structure of the
end closure of the pressure vessel shown in Fig, 4.14. The cylindrical part of the
vessel consists of zk4R angle-ply layer with thickness la^, that can be found from
Eq. (8.64) in which we should take T = pr0/2, and a circumferential (4 = 90") layer
whose thickness is specified by Eq. (8.18), i.e.
h90 = h&cos2 4R - 1) .

The polar opening of the dome (see Fig. 4.14) is closed. So T = pr0/2, fj = 0, and the
dome meridian corresponds to curve 3 in Fig. 8.6. As has already been noted, upon