738 Practical Aspects of Optimization
14.2.2 Design Variable Linking Technique
When the number of elements or members in a structure is large, it is possible to
reduce the number of design variables by using a technique known asdesign variable
linking[14.25]. To see this procedure, consider the 12-member truss structure shown
in Fig. 14.1. If the area of cross section of each member is varied independently, we
will have 12 design variables. On the other hand, if symmetry of members about the
vertical (Y) axis is required, the areas of cross section of members 4, 5, 6, 8, and 10
can be assumed to be the same as those of members 1, 2, 3, 7, and 9, respectively.
This reduces the number of independent design variables from 12 to 7. In addition, if
the cross-sectional area of member 12 is required to be three times that of member 11,
we will have six independent design variables only:X=
x 1
x 2
x 3
x 4
x 5
x 6
≡
A 1
A 2
A 3
A 7
A 9
A 11
(14.2)
Once the vectorXis known, the dependent variables can be determined asA 4 =
A 1 , A 5 =A 2 , A 6 =A 3 , A 8 =A 7 , A 10 =A 9 , andA 12 = 3 A 11. This procedure of treat-
ing certain variables as dependent variables is known asdesign variable linking. By
defining the vector of all variables asZT= {z 1 z 2... z 12 }T≡ {A 1 A 2... A 12 }T53432267(^910)
6
12
1 4
1
7 8
11
5
0
Y
Y 6
Y 7
Y 4
X
Figure 14.1 Concept of design variable linking.