Page 395 - Mechanics Analysis Composite Materials

380 Mechanics and analysis of composite materials

follow from the corresponding free body diagrams of the shell element and can be
written as (see, e.g., Vasiliev, 1993)

where z(r)specifiesthe form of the shell meridian, z’ = dz/dr, and

Q = TPO+E (7’ -6) (8.41)
2

Let the shell be made by winding an orthotropic tape a1 angles +4 and -4 with
respect to the shell meridian as in Fig. 8.4. Then, N, and Np can be expressed in
terms of stresses 61, 62 and 212, referred to the principal material coordinates of the
tape with the aid of Eqs. (4.68), i.e.

N, = h(ol cos24 + 62 sin24 - 212sin 2&), (8.42)
NF = h(o1 sin24 + a2 cos24 + q2sin 24) ,

where h is the shell thickness. Stresses q,~,and 212 are linked with the
corresponding strains by Hooke’s law, Eqs. (429, as

while strains E~,EZ,and yI2 can be expressed in terms of the meridional, E,, and
circumferential, ~p,strains of the shell using Eqs. (4.69), i.e.

EI = E, cos24 + ~p sin24, ~2 = E, sin24 + cos24, y12 = (~p- E,)sin 24 .

(8.44)
Because the right-hand side parts of these three equations include only two strains,
E, and ES,there exists a compatibility equation linking E!,~2 and y12. This equation is
(CI - ~2)sin24 +yI2cos2r$= 0 .

Writing this equation in terms of stresses with the aid of Eqs. (8.43) we get

In conjunction with Eqs. (8.42), this equation allows us to determine stresses as