Engineering Optimization: Theory and Practice, Fourth Edition

1.4 Statement of an Optimization Problem 11

i

Figure 1.6 Tubular column under compression.

The behavior constraints can be expressed as

stress induced≤yield stress

stress induced≤buckling stress

The induced stress is given by

induced stress=σi=

P

π dt

=

2500

π x 1 x 2

(E 3 )

The buckling stress for a pin-connected column is given by

buckling stress=σb=

Euler buckling load

cross-sectional area

=

π^2 EI

l^2

1

π dt

(E 4 )

where

I=second moment of area of the cross section of the column

=

π

64

(do^4 −di^4 )

=

π

64

(do^2 +di^2 )(do+di)(do−di)=

π

64

[(d+t)^2 + (d−t)^2 ]

×[(d+t)+(d−t)][(d+t)−(d−t)]

=

π

8

dt (d^2 +t^2 )=

π

8

x 1 x 2 (x^21 +x 22 ) (E 5 )