12.4 Optimality Criteria Methods 683
12.4 Optimality Criteria Methods
The optimality criteria methods are based on the derivation of an appropriate criteria for
specialized design conditions and developing an iterative procedure to find the optimum
design. The optimality criteria methods were originally developed by Prager and his
associates for distributed (continuous) systems [12.6] and extended by Venkayya, Khot,
and Berke for discrete systems [12.7–12.10]. The methods were first presented for
linear elastic structures with stress and displacement constraints and later extended to
problems with other types of constraints. We will present the basic approach using only
displacement constraints.
12.4.1 Optimality Criteria with a Single Displacement Constraint
Let the optimization problem be stated as follows:
FindXwhich minimizesf (X)=
∑n
i= 1
cixi (12.46)
subjectto
∑n
i= 1
ai
xi
=ymax (12.47)
whereciare constants,ymaxis the maximum permissible displacement, andaidepends
on the force induced in memberidue to the applied loads, length of memberi, and
Young’s modulus of memberi. The Lagrangian function can be defined as
L(X, λ)=
∑n
i= 1
cixi+λ
( n
∑
i= 1
ai
xi
−ymax
)
(12.48)
At the optimum solution, we have
∂L
∂xk
=ck−λ
ak
xk^2
+λ
∑n
i= 1
1
xi
∂ai
∂xk
= 0 , k= 1 , 2 ,... , n (12.49)
It can be shown that the last term in Eq. (12.49) is zero for statically determinate as
well as indeterminate structures [12.8] so that Eq. (12.49) reduces to
ck−λ
ak
xk^2
= 0 , k= 1 , 2 ,... , n (12.50)
or
λ=
ckxk^2
ak
(12.51)
Equation (12.51) indicates that the quantityckxk^2 /ak is the same for all the design
variables. If all the design variables are to be changed, this relation can be used.
However, in practice, only a subset of design variables are involved in Eq. (12.49).
Thus it is convenient to divide the design variables into two sets: active variables [those
determined by the displacement constraint of Eq. (12.51)] and passive variables (those
