Engineering Optimization: Theory and Practice, Fourth Edition

660 Stochastic Programming

whereNc is the number of active turns,Qthe number of inactive turns, andρthe

weight density. Noting that the deflection of the spring (δ)is given by

δ=

8 P C^3 Nc

Gd

(E 2 )

whereP is the load,C=D/d, andG is the shear modulus. By substituting the

expression ofNcgiven by Eq. (E 2 ) into Eq. (E 1 ), the objective function can be expressed

as

f (X)=

π^2 ρGδ

32 P

d^6

D^2

+

π^2 ρQ

4

d^2 D (E 3 )

The yield constraint can be expressed, in deterministic form, as

τ=

8 KPC

π d^2

≤τmax (E 4 )

whereτmaxis the maximum permissible value of shear stress andKthe shear stress

concentration factor given by (for 2≤C≤12):

K=

2

C^0 5.^2

(E 5 )

UsingEq. (E 5 ), the constraint of Eq. (E 4 ) can be rewritten as

16 P

π τmax

D^0 5.^7

d^2   5.^7

< 1 (E 6 )

By considering the design variables to be normally distributed with(d, σd)=d( 1 , 0. 05 )

and(D, σD)=D( 1 , 0. 05 ),k 1 = and 1 k 2 = in Eq. (11.100) and using 0 pj= 0. 9 5,

the problem [Eqs. (11.100) and (11.101)] can be stated as follows:

MinimizeF (Y)=

0. 041 π^2 ρδG

P

d

6

D

2 +^0.^278 π

(^2) ρQd^2 D (E
7 )
subjectto
12. 24 P
π τmax

D

0 5. 7

d

2 5. 7 ≤^1 (E^8 )

The data are assumed asP= 510 N,ρ= 78 ,000 N/m^3 ,δ= 0 .02 m,τmax= 0. 306 ×

109 P a and, Q=2. The degree of difficulty of the problem can be seen to be zero and

the normality and orthogonality conditions yield

δ 1 +δ 2 = 1

6 δ 1 + 2 δ 2 − 2. 75 δ 3 = 0 (E 9 )

− 2 δ 1 +δ 2 + 0. 75 δ 3 = 0

The solution of Eqs. (E 9 ) givesδ 1 = 0. 8 1,δ 2 = 0 .19,andδ 3 = 1. 9 , which corresponds

tod= 0 .0053 m,D= 0 .0358 m, andfmin= 2. 2 66 N.