Engineering Optimization: Theory and Practice, Fourth Edition

Problems 61

Figure 1.32 Four-bar truss.

cross sectionx 2 and length

√

3 l. The truss is made of a lightweight material for which

Young’s modulus and the weight density are given by 30× 106 psi and 0.03333 lb/in^3 ,

respectively. The truss is subject to the loadsP 1 = 10 ,000 lb andP 2 = 20 ,000 lb. The

weight of the truss per unit value oflcan be expressed as

f= 3 x 1 ( 1 )( 0. 03333 )+x 2

√

3 ( 0. 03333 )= 0. 1 x 1 + 0. 05773 x 2

The vertical deflection of jointAcan be expressed as

δA=

0. 6

x 1

+

0. 3464

x 2

and the stresses in members 1 and 4 can be written as

σ 1 =

5 ( 10 , 000 )

x 1

=

50 , 000

x 1

, σ 4 =

− 2

√

3 ( 10 , 000 )

x 2

= −

34 , 640

x 2

The weight of the truss is to be minimized with constraints on the vertical deflection of

the jointAand the stresses in members 1 and 4. The maximum permissible deflection

of jointAis 0.1 in. and the permissible stresses in members areσmax= 8333 .3333 psi

(tension) andσmin= − 4948 .5714 psi (compression). The optimization problem can be

stated as a separable programming problem as follows:

Minimizef (x 1 , x 2 )= 0. 1 x 1 + 0. 05773 x 2

subject to

0. 6

x 1

+

0. 3464

x 2

− 0. 1 ≤ 0 , 6 −x 1 ≤ 0 , 7 −x 2 ≤ 0

Determine the solution of the problem using a graphical procedure.

1.32 A simply supported beam, with a uniform rectangular cross section, is subjected to both
distributed and concentrated loads as shown in Fig. 1.33. It is desired to find the cross
section of the beam to minimize the weight of the beam while ensuring that the maximum
stress induced in the beam does not exceed the permissible stress(σ 0 )of the material
and the maximum deflection of the beam does not exceed a specified limit(δ 0 ).
The data of the problem areP= 105 N, p 0 = 106 N/m,L=1 m,E=207 GPa, weight
density(ρw)= 76 .5 kN/m^3 , σ 0 =220 MPa, andδ 0 = 0 .02 m.