160 140 Abasis 120 Bbasis 100 80 60

160 140 A-basis 120 B-basis 100 80 60 40 20 0 420 440 460 480 500 520 540 560 580 Failure strength (MPa) 6. 5 6. 0 Knockdown factor 5. 5 5. 0 4. 5 3. 41 4. 0 3. 5 2. 36 3. 0 1. 93 2. 5 1. 78 2. 0 1. 5 0 5 10 15 20 25 30 35 40 45 50 Number of coupons L = 100 in h w F = 2, 000 lb

y 3 x 2 L 3 1 1 L 2 1, 000 N t r b t b h h h t w w (a) (b) (c) (d) (e) Cx r r Cx Cy Cy (a) Initial design r (a) Initial mesh (b) Perturbed design r (b) Perturbed mesh

Allowable stress Acceptable region H (d. H/db)4 (d. H/db)3 (d. H/db)2 (d. H/db)1 b 0 b 1 b 2 b 3 b 4 b Physical engineering problem Structural modeling Design parameterization Performance definition (objective, constraints) Structural model update Structural analysis (FEM, BEM, CFD…) Design sensitivity analysis Optimized? Yes Stop No

w w t 1 t 2 t 1 b h (a) Independent designs t 2 h (b) Dependent designs D H Feasible set f(b) b 1 b 2 b 4 b 3 b 5 b 6 b 18 16 f=500 Volume constraint f=400 14 H f=300 12 Feasible set f=200 10 8 3. 5 Optimum design 4 4. 5 5 5. 5 6 D 6. 5 7 7. 5 8

1 L 1 3 2 2 L F 4 3 4 2 F

b 5 b 3 1 8 y 7 10 x 6 5 3 6 1 2 9 4 4 2 P 1 P 2 b

Fx b 7 b 6 b 8 b 9 b 5 b 4 b 3 b 1 b 2 b 11 b 10 60 k. N 2 Pin joint 9 cm Roller support 3 1 12 cm y x 1 2 3 2 1 2 L F 2 L F 3 p 0 = 120 N/m C = – 50 N-m 1 2 E, I, L C P

6 5 400 lb 9' 2 1 3 12' 4 12' 1, 200 lb A B C RL 0. 25 m y RR F 0. 4 m 1, 500 lb 1 E, I 1 2 3 E, I 2 10 in x 25 in p 0 E, I, L w t 1 t 2 h