Page 383 - Mechanics Analysis Composite Materials

368 Mechanics and analysis of composite materials

k
hi(N,sin’ 4i -N,,cos24i)= 0
i= 1
k
hi[(N.+Ny)sin 4; COS 4; -Nx!,]= O . (8.10)
i= 1

Thus, 2k design variables, Le., k values of hi and k values of (pi, should satisfy three
equations, Eqs. (8.8)-(8.10). All possible optimal laminates have the same total
thickness in Eq. (8.8). As follows from Eq. (8.2), condition cy)= 81is valid, in the
general case, if E.~= = E and yX., = 0. Applying Eqs. (4.69) to determine the strains
in the principal material coordinates of the layers we arrive at the following result
el = 82 = E and yl2 = 0. This means that the optimal laminate is the structure of
uniform stress and strain in which the fibers of each layer coincide with the
directions of principal strains. An important feature of the optimal laminate follows
from the last equation of Eqs. (4.150) which yields 4:. = 4i.Thus, the optimal angles
do not change under loading.
Introducing new variables

and taking into account that

k
Xli;=l (8.11)
i= I
we can transform Eqs. (8.8)-(8.10) that specify the structural parameters of the
optimal laminate to the following final forms:

(8.12)

k k
hicos2 qji = A, Chisin2qj; = Any (8.13)
i= I i= I

hisin 4icos +i= in,y . (8.14)
i= I
For uniaxial tension in the x-direction, we have n,,= nxy =0,,? = 1. Then,
Eqs. (8.13) yield & = 0 (i = 1,2,3, ...,k) and Eq. (8.12) gives the obvious result
h =Nr/O.
To describe tension in two orthogonal directions x and y, we should put = 0.
As follows from Eq. (8.14), the laminate structure in this case should be symmetric,