Page 366 - Mechanics Analysis Composite Materials

Chapter 7. Environmental, special loading,and manufacfuring effects 35I

(7.66)

we can conclude that for the case of complete overlap (Fig. 7.44(b)) we have R = 1.
Initial position of the tape placed with overlap wo as in Fig. 7.44(a) is shown in this
Figure with a broken line, while the final position of the tapes is shown with solid
lines. Assume that after the winding and curing are over, the resulting structure is a
unidirectionally reinforced ring which is removed from the mandrel and loaded with
internal pressure, so that the ring radius being R before the loading becomes RI.
Decompose the resultant force acting in the ring cross-section into two components,
i.e.

F=F’+F’’ (7.67)

and introduce the apparent stress acting along the fibers of the ring as
F
GI=-, (7.68)
A
where A = 2w6 is the cross-sectionalarea of the ring made from two tapes as shown
in Fig. 7.44. Force F‘corresponds to part BC of the ring (Fig. 7.44(a)) and can be
found as
RI-R
F’ =A‘EI-
R ’
where A’ = (w+w0)6is the cross-sectional area of this part of the ring and El is the
modulus of elasticity of the cured unidirectional composite. To calculate force F”
that corresponds to part CD of the ring (Fig. 7.44(a)), we should take into account
that the fibers start to take the load only when this part of the tape reaches the
position indicated with broken lines, i.e.

R1 - (R + S)
F’’ = A”E,
R I
where A” = (w - wo)6.With due regard to Eqs. (7.66), (7.67), and (7.68) we can
write the result of the foregoing analysis in the following form:

(7.69)

Here, €1 = (R1 -R)/R is the apparent strain in the fiber direction. For complete
overlap in Fig. 7.44(b), /1 = 1, and o1 = Elel. It should be noted that there exists
also the so-called tape-to-tape winding for which A = 0. This case cannot be
described by Eq. (7.69) because of assumptions introduced in derivation, and the
resulting equation for this case is 61= Elel.