Engineering Optimization: Theory and Practice, Fourth Edition

468 Nonlinear Programming III: Constrained Optimization Techniques

whereσiis the stress induced in memberi,σ(u)the maximum permissible stress in

tension,σ(l)the maximum permissible stress in compression,x(l)i the lower bound

onxi, andx(u)i the upper bound onxi. The stresses are given by

σ 1 (X)=P

x 2 +

√

2 x 1

√

2 x 12 + 2 x 1 x 2

σ 2 (X)=P

1

x 1 +

√

2 x 2

σ 3 ( X)=−P

x 2

√

2 x 12 + 2 x 1 x 2

Data:σ(u)= 0, 2 σ(l)= − 1 5,xi(l)= 0. 1 (i= 1 , 2 ),xi(u) = 5. 0 (i= 1 , 2 ),P=20,

andE=1.

Optimum design:

X∗ 1 =

{

0. 78706

0. 40735

}

, f 1 ∗=

2. 6335 , stress constraint of

member 1 is active atX∗ 1

X∗ 2 =

{

5. 0

5. 0

}

, f 2 ∗= 1. 6569

7.22.2 Design of a Twenty-Five-Bar Space Truss

The 25-bar space truss shown in Fig. 7.22 is required to support the two load condi-

tions given in Table 7.7 and is to be designed with constraints on member stresses as

well as Euler buckling [7.38]. A minimum allowable area is specified for each mem-

ber. The allowable stresses for all members are specified asσmaxin both tension and

compression. The Young’s modulus and the material density are taken asE= 107 psi

andρ= 0 .1 lb/in^3. The members are assumed to be tubular with a nominal diame-

ter/thickness ratio of 100, so that the buckling stress in memberibecomes

pi= −

100. 01 πEAi

8 l^2 i

, i= 1 , 2 ,... , 25

whereAiandlidenote the cross-sectional area and length, respectively, of memberi.

The member areas are linked as follows:

A 1 , A 2 =A 3 =A 4 =A 5 , A 6 =A 7 =A 8 =A 9 ,

A 10 =A 11 , A 12 =A 13 , A 14 =A 15 =A 16 =A 17 ,

A 18 =A 19 =A 20 =A 21 , A 22 =A 23 =A 24 =A 25

Thus there are eight independent area design variables in theproblem. Three problems

are solved using different objective functions:

f 1 (X)=

∑^25

i= 1

ρAili= eightw

f 2 ( X)=(δ 12 x+δ 12 y+δ^21 z)^1 /^2 + (δ^22 x+δ 22 y+δ^22 z)^1 /^2