Planar Filament Winding
Filament winding a surface of revolution by using planar wound fibers (filaments placed along the lines defined by the intersection of the surface with a plane making an angle with the axis of revolution, Fig. 3) was first examined by Hartung (1963). He developed an optimum dome contour for an internally loaded, planar wound, pressure vessel assuming a closed shell. His work is based on netting theory which assumes the resultant load from the meridian and hoop forces is carried only by the fibers in the fiber direction, and the transverse and shear stresses are negligibly small. This relationship is written as

(6) 
Furthermore, the fiber angle, , (see Fig. 3) for any axisymmetric filament wound shape can be determined by taking the dot product of the unit tangent vectors and , giving

(7) 
where is the initial winding angle and y_{e} is the horizontal offset, both constant. Combining Eqs. 6 and 7 will give a second order differential equation that can be solved to give the netting dome contour. Figure 4 compares the netting dome contour with an ellipsoid and spheroid dome shapes that are examined and compared in this paper.
If the dome contour is known, such as an ellipsoid, r^{2}/a^{2} + z^{2}/b^{2} = 1, then the fiber angle can be determined directly. Letting y_{e} = 0 gives the fiber angle, , at any point on the ellipsoid shell as

(8) 
For a spheroid, the winding angle simplifies down to

(9) 
As fibers are wound onto the shell, the thickness of the shell does not remain constant, as mentioned previously, but will increase as approaches due to the accumulation of fibers near the dome opening. At any given cross section, the fiber area remains constant, giving

(10) 
where A_{f} is the area of an arbitrary circumferential cross section and is the total number of fibers at that section. The area A_{f} is also equivalent to . One can now equate the cross section area at the initial radius, r = a, where the thickness is known, t_{a}, to an arbitrary point on the dome, giving

(11) 
Because there must always be two plys, , due to the filament winding process, the initial shell thickness, t_{a}, will be twice the initial ply thickness, t_{p} giving

(12) 
Equation 12 is based on the assumption that the fibers are infinitesimally thin, where as actual filament wound structures are constructed from groups of fibers that form a band. This simplified equation is accurate except at the polar opening edge. Other more complex equations have been developed by Gramoll (1990a) but, for simplicity, Eq. 12 is used in this paper.
© 1997 Kurt Gramoll, The University of Oklahoma, All Rights Reserved.