12.4 Optimality Criteria Methods 685
whereJdenotes the number of displacement (equality) constraints,y∗jthe maximum
permissible value of the displacementyj, andaj iis a parameter that depends on the
force induced in memberidue to the applied loads, length of memberi, and Young’s
modulus of memberi. The Lagrangian function corresponding to Eqs. (12.58) and
(12.59) can be expressed as
L(X, λ 1 ,... , λJ)=f 0 +
∑n
i= 1
cixi+
∑J
j= 1
λj
( n
∑
i= 1
aj i
xi
−yj∗
)
(12.60)
and the necessary conditions of optimality are given by
∂L
∂xk
=ck−
∑J
j= 1
λj
aj i
x^2 k
= 0 , k= 1 , 2 ,... ,n (12.61)
Equations (12.61) can be rewritten as
xk=
∑J
j= 1
(
λj
aj i
ck
)
1 / 2
, k= 1 , 2 ,... ,n (12.62)
Note that Eq. (12.62) can be used to iteratively update the variablexkas
xk(j+^1 )=
∑J
j= 1
(
λj
aj i
ck
)
1 / 2
(j )
, k= 1 , 2 ,... ,n (12.63)
where the values of the Lagrange multipliersλjare also not known at the beginning.
Several computational methods can be used to solve Eqs. (12.63) [12.7, 12.8].
12.4.3 Reciprocal Approximations
In some structural optimization problems, it is convenient and useful to consider the
reciprocals of member cross-sectional areas (1/Ai) s the new design variables (a zi) If.
the problem deals with the minimization of weight of a statically determinate structure
subject to displacement or stress constraints, the objective function and its gradient
can be expressed as explicit functions of the variablesziand the constraints can be
expressed as linear functions of the variableszi. If the structure is statically indeter-
minate, the objective function remains a simple function ofzibut the constraints may
not be linear in terms ofzi; however, a first-order Taylor series (linear) approximation
of the constraints denote a very high-quality approximation of these constraints. With
reciprocal variables, the optimization problem with a single displacement constraint
can be stated as follows:
FindZ = {z 1 z 2... zn}Twhich minimizesf(Z) (12.64)
subject to
g(Z)= 0 (12.65)
