Engineering Optimization: Theory and Practice, Fourth Edition

776 Practical Aspects of Optimization

Bar 1

(area = A 1 ) Bar 2(area = A

2 )

P = 1000 N

9 m 1 m

Figure 14.14 Two-bar truss.

induced in the bars (σ 1 andσ 2 ). Treaty, A 1 , andA 2 as design variables withσi≤ 105 Pa

(i=1, 2), 1 m≤y≤4 m, and 0≤Ai≤ 0 .2 m^2 (i= 1 ,2). Use multilevel optimization

approach for the solution.

14.17 Find the sensitivities ofx∗ 1 , x 2 ∗, andf∗with respect to Young’s modulus of the tubular

column considered in Example 1.1.

14.18 Consider the two-bar truss shown in Fig. 1.15. The problem of design of the truss for

minimum weight subject to stress constraints can be stated as follows:

Findx 1 , A 1 ,andA 2 which minimize

f= 28. 30 A 1

√

1 +x^2 + 14. 15 A 2

√

1 +x^2

subject to

g 1 =

0. 1768 ( 1 +x)

√

1 +x^2

A 1 x

− 1 ≤ 0

g 2 =

∣∣

∣∣

∣

0. 1768 (x− 1 )

√

1 +x^2

A 2 x

∣∣

∣∣

∣

− 1 ≤ 0

0. 1 ≤x≤ 2. 5 , 1. 0 ≤Ai≤ 2. 5 (i= 1 , 2 )

where the members are assumed to be made up of different materials. Solve this opti-

mization problem using the multilevel approach.

14.19 Consider the design of the two-bar truss shown in Fig. 14.15 with the location of nodes

1 and 2(x) and the area of cross section of bars (A) as design variables. If the weight

and the displacement of node 3 are to be minimized with constraints on the stresses

induced in the bars along with bounds on the design variables, the problem can be stated

as follows [14.34]:

FindX= {x 1 x 2 }Twhich minimizes

f 1 (X)= 2 ρhx 2

√

1 +x^21

f 2 =

P h( 1 +x 12 )^1.^5

√

1 +x 14

2

√

2 Ex^21 x 2