Page 269 - Mechanics Analysis Composite Materials

254 Mechanics and analysis of composite marerials

(5.66)

and find coordinate e delivering the minimum value of D,,,. Using the minimum
conditions

d d2
-Dmn = 0, -Dnln > 0 ,
de de2
we get

(5.67)

This result coincides with Eqn. (5.65) and yields C,,,,,= 0. Thus, calculating D,,
and C,, we use for each mn = 1I, 12, 22, 14, 24, 44 the corresponding value e,,l
specified by Eqn. (5.67). Substitution yields

(5.68)

and constitutive equations, Eqs. (5.5) become uncoupled. Naturally, this approach
is only approximate because the reference plane coordinate should be the same for
all stiffnesses, but it is not in the method under discussion. As follows from the
foregoing derivation, coefficients P,,,specified by Eqs. (5.68) do not exceed the
actual values of bending stiffnesses, Le., DL,<D,,,. So the method of reduced
bending stiffnesses leads to underestimation of the laminate bending stiffness. In
conclusion, it should be noted that this method is not formally grounded and can
yield both good and poor approximation of the laminate behavior.

5.10. Stresses in laminates

Constitutive equations derived in the previous sections of this chapter link forces
and moments acting on the laminate with the corresponding generalized strains. For
composite structures, forces and moments should satisfy equilibrium equations,
while strains are expressed in terms of displacements. As a result, a complete set of
equations is formed allowing us to find forces, moments, strains, and displacements
corresponding to a given system of loads acting on this structure. Because the
problem of structural mechanics is beyond the scope of this book and is discussed
elsewhere (Vasiliev, 1993), we assume that this problem has been already solved, Le.,
that we know either generalized strains E, y, and K entering Eqs. (5.5) or forces and
moments Nand M. If this is the case, we can use Eqs. (5.5) to find E, y, and K. Now,