Page 268 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 253

where, zi20and 4 80. Introduce a new layer coordinate Zi = (zi +zi-l)/2,which is
the distance between the reference plane of the laminate and the middle plane of the
i-th layer. Then, condition C,, = 0 yields

Now assume that we have a group of identical layers or plies with the same stiffness
coefficientsA,, and thicknesses. For example, the laminate should include a 1.5 mm
thick 0" unidirectional layer which consists of 10 plies (the thickness of an
elementary ply is 0.15 mm). Arranging these plies above (Bi)and below (4) the
reference plane in such a way that

(5.64)
.j= I
we have no coupling for this group of plies. Doing the same with the other layers we
arrive at the laminate without coupling. Naturally, some additional conditions
following from the fact that the laminate is a continuous structure should be
satisfied. But even with these conditions, Eqn. (5.64) can be met with several systems
of the ply coordinates, and symmetric arrangement of plies (2, =3)is only one of
these systems. General analysis of the problem under discussion is presented by
Verchery (1999).
Return to the laminates with pre-assigned stacking sequences of the layers. As
follows from Eqs. (5.63), we can always make one of coupling stiffness coefficients
equal to zero, e.g., taking e = estwhere

(5.65)

we get Cst= 0 (the rest coupling coefficients are not zero).
Another way to simplify the equations for stiffnesses is to take e = 0, Le., to take
the surface of the laminate as the reference plane. In this case, Eqs. (5.28) acquire
the form

In practical analysis, constitutive equations for the laminates with arbitrary
structure are often approximately simplified using the method of reduced or
minimum bending stiffnesses described, e.g., by Ashton (1969), Karmishin (1974),
and Whitney (1987). To introduce this method, consider the corresponding equation
of Eqs. (5.28) for bending stiffnesses, Le.