Page 445 - Aircraft Stuctures for Engineering Student

426 Stress analysis of aircraft components

corresponding to a1 is given by
A1
E1 = - (10.44)
I
and
= El&] (10.45)

where El is the modulus of elasticity of the lamina in the direction of the flament.
Also, using the suffixes f and m to designate filament and matrix parameters, we
have
af = EfEl, 0, = E,E~ (10.46)
Further, if A is the total area of cross-section of the lamina in Fig. 10.66, Af is the
cross-sectional area of the filament and A, the cross-sectional area of the matrix
then, for equilibrium in the direction of the filament
o1A = UfAf + amA,
or, substituting for q, af and a, from Eqs (10.45) and (10.46)
E~E~A EfEfAf + Em€lAm
=
so that
(10.47)

Writing Af/A = vf and A,/A = v,, Eq. (10.47) becomes
El = VfEf + VmEm (10.48)

Equation (10.48) is generally referred to as the law of mixtures.
A similar approach may be used to determine the modulus of elasticity in the
transverse direction (Et). In Fig. 10.67 the total extension in the transverse direction
is produced by q and is given by
Etlt = Em& + Eflf

t t fUit t t

Fig. 10.67 Determination of Et.