Engineering Optimization: Theory and Practice, Fourth Edition

470 Nonlinear Programming III: Constrained Optimization Techniques

Constraints:

|σij( X)|≤σmax, i= 1 , 2 ,... , 25 , j= 1 , 2

σij(X)≤pi( X), i= 1 , 2 ,... , 25 , j= 1 , 2

xi(l)≤xi≤xi(u), i= 1 , 2 ,... , 8

whereσijis the stress induced in memberiunder load conditionj,xi(l)the lower

bound onxi, andx(u)i the upper bound onxi.

Data:σmax= 04 ,000 psi,xi(l)= 0. 1 in^2 ,x(u)i = 5. 0 in^2 fori= 1 , 2 ,... ,25.

Optimum solution:See Table 7.8.

7.22.3 Welded Beam Design

The welded beam shown in Fig. 7.23 is designed for minimum cost subject to con-

straints on shear stress in weld (τ), bending stress in the beam (σ), buckling load on

the bar (Pc) end deflection of the beam (, δ), and side constraints [7.39].

Design vector:











x 1

x 2

x 3

x 4











=











h

l

t

b











Table 7.8 Optimization Results of the 25-Bar Truss [7.38]

Optimization problem

Minimization Minimization Maximization

Quantity of weight of deflection of frequency

Design vector,X 0. 1 a 3.7931 0. 1 a

0.80228 5. 0 a 0.79769

0.74789 5. 0 a 0.74605

0. 1 a 3.3183 0.72817

0.12452 5. 0 a 0.84836

0.57117 5. 0 a 1.9944

0.97851 5. 0 a 1.9176

0.80247 5. 0 a 4.1119

Weight (lb) 233.07265 1619.3258 600.87891

Deflection (in.) 1.924989 0.30834 1.35503

Fundamental frequency (Hz) 73.25348 70.2082 108.6224

Number of active behavior

constraints

9 b 0 4 c

aActive side constraint.

bBuckling stress in members, 2, 5, 7, 8, 19, and 20 in load condition 1 and in members 13, 16, and 24 in

load condition 2.

cBuckling stress in members 2, 5, 7, and 8 in load condition 1.