J O U R N A L O F M A T E R I A L S S C I E N C E L E T T E R S
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Optimum fiber orientation in filament wound structures
S. GROVEAdvanced Composites Manufacturing Centre, Department of
Mechanical and Marine Engineering, Universityof Plymouth, UKE-mail:
[email protected]
A winding angle of approximately 55 is usually quotedas the
optimum for filament wound pipes and pressurevessels. The
conventional derivation is based on nettinganalysis [1], and
strictly applies only to cylinders underinternal pressure, in which
the ratio of hoop stress/axialstress (h/a) is 2. Netting analysis
neglects the con-tribution of the matrix and only considers the
stress inthe fiber direction (f). For a winding angle (Fig.
1),resolution of direct stresses gives
h = f sin2 a = f cos2
Thush
a= tan2
If h/a= 2, then = tan12= 54.7.
Figure 1 The basic geometry for netting analysis. represents the
wind-ing angle; h, a and f are stresses in hoop, axial and fiber
directions,respectively.
Figure 2 Geometry of unit length wound fiber on unzipped
cylinderof length L , radius r .
An alternative derivation of this angle is obtained
byconsidering the volume of a helically wound cylinder,radius r ,
length L , which comprises inextensible fibersof unit length. The
geometry of such a cylinder is easilyconsidered by unzipping it
along its length (Fig. 2).
Writing the dimensions in terms of winding angle:
r = sin2pi
L = cos
The volume of the cylinder is
V = pir2L = sin2 cos
4pi
The volume tends to zero at= 0 (r 0) and= 90(L 0). The volume is
a maximum at dV/d= 0.Differentiating and solving again gives tan2 =
2, or= 54.7 as before.
The optimum angle is now identified as that whichmaximizes the
volume of the cylinder, in other words,an increase in internal
pressure has the least chance ofincreasing the cylinder volume, and
an equal tendencyto increase length or radius.
Powell [2] has discussed the dimensional changesof arteries,
which result from their helical windings ofcollagen fibers. More
recently, Vogel [3] has identi-fied this maximum volume with
helically wound struc-tures in nature, referring to them as
hydrostats. Thesquid, for example, achieves propulsion by
contractingcircumferential muscles in the mantle. Lengthwise
ex-pansion (which would occur in an isotropic material)tends to
cause helically wound fibers to decrease theirwinding angle. This
necessarily reduces the volume ofthe vessel and water is
expelled.
References1. G . E C K O L D , Design and Manufacture of
Composite Structures
(Woodhead, 1994).2. P . P O W E L L , Engineering with
Fibre-Polymer Laminates
(Chapman & Hall, 1994).3. S . V O G E L , Cats Paws and
Catapults (Norton, 1998).
Received 4 Marchand accepted 23 March 1999
02618028 C 1999 Kluwer Academic Publishers 1203
