Engineering Optimization: Theory and Practice, Fourth Edition

742 Practical Aspects of Optimization

1

4

2

3

1

3

2

4

y 6

y 5

y 2

y 1

y 4

y 8

y 7

y 3

50 in.

25 in.

1000 lb

100 in.

50 in. 100 in.

Figure 14.2 Crane (planar truss).

Table 14.1

Area of Global node of: Direction cosines of member

Member, cross Length, Corner Corner

e section,Ae le(in.) 1,i 2,j lij=

Xj−Xi

le

mij=

Yj−Yi

le

1 A 1 55.9017 1 3 0.8944 0.4472

2 A 2 55.9017 3 2 0.8944 –0.4472

3 A 3 167.7051 3 4 0.8944 0.4472

4 A 4 141.4214 2 4 0.7071 0.7071

whereAeis the cross-sectional area,Eeis Young’s modulus,leis the length, and

(lij, mij) are the direction cosines of membere.Equation (E 1 ) can be used to compute

thestiffness matrices of the various members using the data of Table 14.1. When the

member stiffness matrices are assembled and the boundary conditions (y 1 =y 2 =y 3 =

y 4 = ) are applied, the overall stiffness matrix becomes 0

[K]=( 30 × 106 )



















( 0. 8 A 1

55. 9017 +

0. 8 A 2

55. 9017 +

0. 8 A 3

167. 7051

) ( 0. 4 A 1

55. 9017 −

0. 4 A 2

55. 9017 +

0. 4 A 3

167. 7051

) (− 0. 8 A 3

167. 7051

) (− 0. 4 A 3

167. 7051

)

( 0. 2 A 1

55. 9017 +

0. 2 A 2

55. 9017 +

0. 2 A 3

167. 7051

) (− 0. 4 A 3

167. 70501

) (− 0. 2 A 3

167. 7051

)

symmetric

( 0. 8 A

3

167. 7051 +

0. 5 A 4

141. 4214

) ( 0. 4 A

3

167. 7051 +

0. 5 A 4

141. 4214

)

( 0. 2 A

3

167. 7051 +

0. 5 A 4

141. 4214

)



















(E 2 )