Engineering Optimization: Theory and Practice, Fourth Edition

Problems 47

Figure 1.15 Two-bar truss.

1.2 The two-bar truss shown in Fig. 1.15 is symmetric about theyaxis. The nondimensional

area of cross section of the membersA/Aref, and the nondimensional position of joints

1 and 2,x/ h, are treated as the design variablesx 1 andx 2 , respectively, whereAref

is the reference value of the area(A)andhis the height of the truss. The coordinates

of joint 3 are held constant. The weight of the truss(f 1 )and the total displacement of

joint 3 under the given load(f 2 )are to be minimized without exceeding the permissible

stress,σ 0. The weight of the truss and the displacement of joint 3 can be expressed as

f 1 (X)= 2 ρhx 2

√

1 +x 12 Aref

f 2 (X)=

P h( 1 +x^21 )^1.^5

√

1 +x^41

2

√

2 Ex 12 x 2 Aref

whereρis the weight density,Pthe applied load, andEthe Young’s modulus. The

stresses induced in members 1 and 2 (σ 1 andσ 2 ) are given by

σ 1 (X)=

P ( 1 +x 1 )

√

( 1 +x 12 )

2

√

2 x 1 x 2 Aref

σ 2 (X)=

P (x 1 − 1 )

√

( 1 +x 12 )

2

√

2 x 1 x 2 Aref

In addition, upper and lower bounds are placed on design variablesx 1 andx 2 as

ximin≤xi≤ximax; i= 1 , 2

Find the solution of the problem using a graphical method with (a)f 1 as the objective, (b)f 2
as the objective, and (c) (f 1 +f 2 ) as the objective for the following data:E= 30 × 106 psi,