Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 23

Figure 1.11 Helical spring.

whereGis the shear modulus,Fthe compressive load on the spring,wthe weight of
the spring,ρthe weight density of the spring, andKsthe shear stress correction factor.
Assumethat the material is spring steel withG= 12 × 106 psi andρ= 0. 3 lb/in^3 , and
the shear stress correction factor isKs≈ 1. 0 5.

SOLUTION The design vector is given by

X=







x 1

x 2

x 3







=







d

D

N







and the objective function by

f (X)=weight=

π d^2

4

πDNρ (E 1 )

The constraints can be expressed as

deflection=

F

k

=

8 FD^3 N

d^4 G

≤ 0. 1

thatis,

g 1 (X)=

d^4 G

80 FD^3 N

> 1 (E 2 )

shear stress=Ks

8 FD

π d^3

≤ 01 , 000