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′
⎧
I
±
1
if
r
≤
par
i
i
I
′
=
(3)
⎨
i
′
I
if
r
>
par
⎩
i
i
Constraint handling:
The new harmony vector obtained using the above-mentioned
rules is checked whether it violates design constraints. If this vector is severely infeasi-
ble it is discarded and another harmony vector is sought. However if it is slightly infea-
sible, it is included in the harmony matrix. In this way the slightly infeasible harmony
vector is used as a base in the pitch adjustment operation to provide a new harmony
vector that may be feasible. This is achieved by using large error values initially for the
acceptability of the new design vectors. The error value is then gradually reduced dur-
ing the design cycles until it reaches to its final value. This value is then kept the same
until the end of iterations. This adaptive error strategy is found quite effective in han-
dling the design constraints in large design problems.
Step 4. Update of Harmony Matrix:
After generating the new harmony vector, its
objective function value is calculated. If this value is better (lower) than that of the
worst harmony vector in the harmony memory, it is then included in the matrix while
the worst one is discarded out of the matrix. The updated harmony memory matrix is
then sorted in ascending order of the objective function value.
Step 5. Termination:
The steps 3 and 4 are repeated until a pre-assigned maximum
number of cycles
cy
N
is reached. This number is selected large enough such that
within this number no further improvement is observed in the objective function.
3 Optimum Design of Steel Frames
Optimum design of steel frames necessitates determination of steel profiles from a
standard steel sections table available in the practice for the beams and column of a
frame such that its response under the applied load is within the limitations specified
by steel design codes and its cost or weight is the minimum.
3.1 Discrete Optimum Design Problem
For a steel frame consisting of
nm
members that are collected in
ng
design groups
(variables), the optimum design problem according to BS 5950 [22] code yields the
following discrete programming problem, if the design groups are selected from steel
sections in a given profile list.
Find a vector of integer values
I
(Eq. 1) representing the sequence numbers of
steel sections assigned to
ng
member groups
[
]
T
I
=
I
,
2
I
,...,
I
(4)
1
ng
to minimize the weight (
W
) of the frame.
ng
t
Minimize
∑∑
=
W
=
m
l
(5)
r
s
r
1
s
=
1
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