Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 27

load(x 1 +x 2 +x 3 ) hat can be supported by the system. Assume that the weights oft
the beams 1, 2, and 3 arew 1 , w 2 , andw 3 , respectively, and the weights of the ropes
are negligible.

SOLUTION Assuming that the weights of the beams act through their respective
middle points, the equations of equilibrium for vertical forces and moments for each
of the three beams can be written as

For beam 3:

TE+TF=x 3 +w 3

x 3 ( ) 3 l +w 3 ( ) 2 l −TF( ) 4 l = 0

For beam 2:

TC+TD−TE=x 2 +w 2

x 2 (l)+w 2 (l)+TE(l)−TD( ) 2 l = 0

For beam 1:

TA+TB−TC−TD−TF=x 1 +w 1

x 1 ( ) 3 l +w 1 (^92 l)−TB( ) 9 l +TC( ) 2 l +TD( ) 4 l +TF( ) 7 l = 0

whereTidenotes the tension in ropei.The solution of these equations gives

TF=^34 x 3 +^12 w 3

TE=^14 x 3 +^12 w 3

TD=^12 x 2 +^18 x 3 +^12 w 2 +^14 w 3

TC=^12 x 2 +^18 x 3 +^12 w 2 +^14 w 3

TB=^13 x 1 +^13 x 2 +^23 x 3 +^12 w 1 +^13 w 2 +^59 w 3

TA=^23 x 1 +^23 x 2 +^13 x 3 +^12 w 1 +^23 w 2 +^49 w 3

The optimization problem can be formulated by choosing the design vector as

X=







x 1

x 2

x 3







Since the objective is to maximize the total load

f (X)= −(x 1 +x 2 +x 3 ) (E 1 )

The constraints on the forces in the ropes can be stated as

TA≤W 1 (E 2 )

TB≤W 1 (E 3 )

TC≤W 2 (E 4 )