Page 400 - Mechanics Analysis Composite Materials

Chapter 8. Optimal composite StrucIures 385

The solution of this equation that satisfies the boundary condition +(r= R) = 4Ris
presented as follows:

As has been already noted in the previous section, the simplest and rather adequate
model of unidirectional fibrous composites for design problems is the monotropic
model ignoring the stiffness of the matrix. For this model, we should take n = 0 in
the foregoing equations. Particularly, Eq. (8.58) yields in this case
r sin 4(r) = R sin 4R . (8.59)

This is the equation of a geodesic line on the surface of revolution. Thus, in the
optimal filament wound shell the fibers are directed along the geodesic lines. This
substantially simplifies the winding process because the tape placed on the surface
under tension automatically acquires the form of the geodesic line if thcrc is no
friction between the tape and the surface.
As follows from Eq. (8.59), for 4 = 90”, the tape touches the shell parallel of
radius
r0 = R sin 4R (8.60)

and the polar opening of this radius is formed in the shell (see Fig. 8.4).
Transforming Eq. (8.48) with the aid of Eqs. (8.59) and (8.60) and taking n = 0
we arrive at the following equation that specifies the meridian of the optimal
filament wound shell:

(8.61)

where
, , 2T
q- = r; - - r0 . (8.62)
P
Integrating Eq. (8.61) with due regard for the condition l/z’(R) = 0 which,
as earlier, requires that for r = R the tangent to the meridian be parallel to z-axis,
we get

(8.63)