Engineering Optimization: Theory and Practice, Fourth Edition

758 Practical Aspects of Optimization

3.Solve the second-level optimization problem stated in Eqs. (14.101) and find a

new vectorY∗.

4 .Check for the convergence off∗andY∗(compared to the valueY∗used

earlier).

5.If the process has not converged, go to step 2 and repeat the process until

convergence.

The following example illustrates the procedure.

Example 14.2 Find the minimum-weight design of the two-bar truss shown in Fig. 14.6

with constraints on the depth of the truss (y=h), cross-sectional areas of the members

(z 1 =A 1 ) and (z 2 =A 2 ), and the stresses induced in the bars. Treat the depth of the

truss (y) and the cross-sectional areas of bars 1 and 2 (z 1 andz 2 ) as design variables.

The permissible stress in each bar isσ 0 = 015 Pa, unit weight is 76,500 N/m^3 , h si

constrained as 1 m≤h≤6 m, and the cross-sectional area of each bar is restricted to

lie between 0 and 0.1 m^2.

SOLUTION The stresses induced in the bars can be expressed as

σ 1 =

P

√

y^2 + 63

7 yz 1

, σ 2 =

6 P

√

y^2 + 1

7 yz 2

and hence the optimization problem can be stated as follows:

FindX= {y z 1 z 2 }Twhich minimizes

f(X)= 76 , 500 z 1

√

y^2 + 63 + 76 , 500 z 2

√

y^2 + 1

subjectto

P

√

y^2 + 63

7 σ 0 yz 1

− 1 ≤ 0 ,

6 P

√

y^2 + 1

7 σ 0 yz 2

− 1 ≤ 0

1 ≤y≤ 6 , 0 ≤z 1 ≤ 0. 1 , 0 ≤z 2 ≤ 0. 1

6 m

Bar 1

(area,A 1 = z 1 )

Bar 2

(area,A 2 = z 2 )

P

R

Q

1 m

h=y

P= 1000N

Figure 14.6 Two-bar truss.