Page 279 - Mechanics Analysis Composite Materials

264 Mechanics and analysis of composite materials

(5.100)

For the unidirectional ply, we take transverse shear moduli G13 = GI* = 4 GPa and
G23 = 3GPa. Using Eqs. (4.72) we get
Ai:) = GI3 cos' q5 + GDsin' 4 = 3.7 GPa and A$ = 3 GPa .

Now, calculation in Eqs. (5.100) yields Dll = 16.9GPamm3 and Sjs =
4.05 GPa mm.
Equilibrium equations, Eqs. (5.93), in conjunction with constitutive equations,
Eqs. (5.96W5.99) compose a set of seven ordinary differential equations including
the same number of unknown functions -N,, Nv, M,, V,, IC, w,and 0,. Thus, the set is
complete and can be reduced to one governing equation for deflection w.
To do this, we integrate the first equilibrium equation in Eqs. (5.93) which shows
that Nx = constant. Because at the cylinder ends N, = -N, this result is valid for the
whole cylinder. Using Eqs. (5.94) and (5.96) we obtain

(5.101)

Substitution into Eq. (5.97) yields

(5.102)

where B =Bll& -BllB21. We can express 0, from Eq. (5.99) and, after differen-
tiation, change for N, with the aid of the last equilibrium equation in Eqs. (5.93).
Substituting N, from Eq. (5.102) we arrive at

(5.103)

where C = 1 - (C21/(S&)). Using Eqs. (5.98) and (5.103) we can express the
bending moment in terms of deflection, Le.,

(5.104)

The governing equation follows now from the second equilibrium equation in
Eqs. (5.93) if we differentiate it, substitute M: from Eq. (5.104), express V, in terms
of 0: and w" using Eq. (5.99) and substitute 0.: from Eq. (5.103). The final equation
is as follows: