Page 387 - Mechanics Analysis Composite Materials

372 Mechanics and analysis of composite materials

Particularly, Eqs. (8.13) and (8.14) do not describe the case of pure shear for which
only shear stress resultant, N,,, is not zero. This is quite natural because strength
condition cry) = 51 under which Eqs. (8.12)-(8.14) were derived is not valid for
shear inducing tension and compression in angle-ply layers.
To study in-plane shear of the laminate, we should use both solutions of Eq. (8.7)
and assume that for some layers, e.g., with i = 1,2,3,...,n- 1, of)= 81 while
for the other layers (i = n,n + I,n -t2,. ..,k),or) = -81. Then, Eqs. (8.1) can be
reduced to the following forms:

N, +N" = -h-) (8.20)

(8.21)

(8.22)

where

n- I k
hf = h- = Ch;
i= I i=n
are the total thicknesses of the plies with tensile and compressive stresses in the
fibers, respectively.
For the case of pure shear (N, =N, = 0), Eqs. (8.20) and (8.21) yield h+ = h-
and #j = f45". Then, assuming that +i= +45" for the layers with hi = hi+, while
q5i = -45" for the layers with hi = hi we get from Eq. (8.22)

The optimal laminate, as follows from the foregoing derivation, corresponds to
4~45'angle-ply structure shown in Fig. 8.2b.

8.2. Composite laminates of uniform strength

Consider again the panel in Fig. 8.1 and assume that unidirectional plies or fabric
layers, that form the panel are orthotropic, i.e., in contrast to the previous section,
we do not neglect now stresses 02 and ~12in comparison with 01 (see Fig. 3.29).
Then, constitutive equations for the panel in plane stress state are specified by the
first three equations in Eqs. (5.35), i.e.