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Chapter 8. Optimal composite structures 361
with respect to design variables hi, 4; and multipliers A, i.e.
Minimization with respect to I gives, obviously, constraints in Eqs. (8.1) and (8.2),
while the first two of Eqs. (8.4) yield
of)().xcos24; + sin’ +i +A.~? sin 4;cos 4;)= I , (8.5)
hiby)[(A,.- sin 24i + cos 24~= EI A;[(&? - E,) sin 24; + y.yy COS 24;1 .
(8.6)
The solution of Eq. (8.6) is
where c is a constant. Substituting Ivy, and from these equations into Eq. (8.5)
and taking into account Eq. (8.2) we get
This equation has two solutions: oy)= fc.
Consider the first case, Le., cy)= c. Adding up the first two equations of
Eqs. (8.1) and taking into account Eq. (8.3) we have
I
h=-(Nx+iyv) .
C
Obviously, the minimum value of h corresponds to c = 51, where 81 is the ultimate
stress. Thus, the total thickness of the optimal plate is
Taking now o? = 81in Eqs. (8.1) and eliminating 81with the aid of Eq. (8.8) we
arrive at the following two optimality conditions in terms of design variables and
acting forces:
