772 Practical Aspects of Optimization
(e)Bounded objective function method
(f)Lexicographic methodProblems
14.1 Consider the minimum-volume design of the four-bar truss shown in Fig. 14.2 subject
to a constraint on the vertical displacement of node 4. LetX 1 = { 1 , 1 , 0. 5 , 0. 5 }Tand
X 2 = { 0. 5 , 0. 5 , 1 , 1 }Tbe two design vectors, withxidenoting the area of cross section
of bari(i= 1 , 2 , 3 , 4 ). By expressing the optimum design vectors asX=c 1 X 1 +c 2 X 2 ,
determine the values ofc 1 andc 2 through graphical optimization when the maximum
permissible vertical deflection of node 4 is restricted to a magnitude of 0.1 in.
14.2 Consider the configuration (shape) optimization of the 10-bar truss shown in Fig. 14.10.
The (X, Y )coordinates of the nodes are to be varied while maintaining(a)symmetry
of the structure about theXaxis, and(b)alignment of nodes 1, 2, and 3 (4, 5, and 6).
Identify the independent and dependent design variables and derive the relevant design
variable linking relationships.
14.3 For the four-bar truss considered in Example 14.1 (shown in Fig. 14.2), a base design
vector is given byX 0 = {A 1 , A 2 , A 3 , A 4 }T= { 2. 0 , 1. 0 , 2. 0 , 1. 0 }Tin^2. IfXis given by
X= { 0. 4 , 0. 4 ,− 0. 4 ,− 0. 4 }Tin^2 , determine
(a)The exact displacement vectorY 0 = {y 5 , y 6 , y 7 , y 8 }TatX 0
(b)The exact displacement vector(Y 0 +Y)at(X 0 +X)
(c)The displacement vector(Y 0 +Y)whereYis given by Eq. (14.20) with five
terms14.4 Consider the 11-member truss shown in Fig. 5.1 with loadsQ= −1000 lb, R=1000 lb,
andS=2000 lb. IfAi=xi denotes the area of cross section of memberi, and
u 1 , u 2 ,... , u 10 indicate the displacement components of the nodes, the equilib-
rium equations can be expressed as shown in Eqs. (E 1 ) to (E 10 ) of Example 5.1.
Assuming thatE= 30 × 106 psi,l=50 in.,xi=1 in^2 (i= 1 , 2 ,... , 11 ), xi= 0. 1
in^2 (i= 1 , 2 ,... , 5 ), andxi= − 0 .1 in^2 (i= 6 , 7 ,... , 11 ), determine44 56XY75633(^22)
1 1
7
8 10
9
Figure 14.10 Design variable linking of a 10-bar truss.