Engineering Optimization: Theory and Practice, Fourth Edition

9.5 Example Illustrating the Calculus Method of Solution 557

Figure 9.11 Four-bar truss.

Memberi pi di=
(stressi)li
E

=

Ppili

xiE

(in.) δi=pidi(in.)

1 − 1. 25 − 1. 25 /x 1 1. 5625 /x 1

2 0.75 0. 9 /x 2 0. 6750 /x 2

3 1.25 1. 25 /x 3 1. 5625 /x 3

4 − 1. 50 − 0. 9 /x 4 1. 3500 /x 4

The vertical deflection of jointAis given by

dA=

∑^4

i= 1

δi=

1. 5625

x 1

+

0. 6750

x 2

+

1. 5625

x 3

+

1. 3500

x 4

(E 2 )

Thus the optimization problem can be stated as

Minimizef (X)=x 1 + 1. 2 x 2 +x 3 + 0. 6 x 4

subject to

1. 5625

x 1

+

0. 6750

x 2

+

1. 5625

x 3

+

1. 3500

x 4

= 0. 5 (E 3 )

x 1 ≥ 0 ,x 2 ≥ 0 ,x 3 ≥ 0 , x 4 ≥ 0

Since the deflection of jointAis the sum of contributions of the various members,
we can consider the 0.5 in. deflection as a resource to be allocated to the various
activitiesxiand the problem can be posed as a multistage decision problem as shown
in Fig. 9.12. Lets 2 be the displacement (resource) available for allocation to the first
member (stage 1),δ 1 the displacement contribution due to the first member, andf 1 ∗(s 2 )
the minimum weight of the first member. Then

f 1 ∗(s 2 ) =min[R 1 =x 1 ]=

1. 5625

s 2

(E 4 )

such that

δ 1 =

1. 5625

x 1

and x 1 ≥ 0