Engineering Optimization: Theory and Practice, Fourth Edition

60 Introduction to Optimization

Figure 1.31 Crane hook carrying a load.

whereSis the yield strength,ethe joint efficiency,pthe pressure, andRthe radius.

Formulate the design problem for minimum structural volume usingxi, i= 1 , 2 , 3 , 4 ,as

design variables. Assume the following data:S= 30 ,000 psi ande= 1 .0.

1.30 A crane hook is to be designed to carry a loadFas shown in Fig. 1.31. The hook can

be modeled as a three-quarter circular ring with a rectangular cross section. The stresses

induced at the inner and outer fibers at sectionABshould not exceed the yield strength

of the material. Formulate the problem of minimum volume design of the hook using

ro,ri,b, andhas design variables.Note:The stresses induced at pointsAandBare

given by [1.117]

σA=

Mco

Aero

σB=

Mci

Aeri

whereMis the bending moment due to the load (=F R),Rthe radius of the centroid,

rothe radius of the outer fiber,rithe radius of the inner fiber,cothe distance of the

outer fiber from the neutral axis=Ro−rn,cithe distance of inner fiber from neutral

axis=rn−ri,rnthe radius of neutral axis, given by

rn=

h

In(ro/ri)

Athe cross-sectional area of the hook=bh, andethe distance between the centroidal

and neutral axes=R−rn.

1.31 Consider the four-bar truss shown in Fig. 1.32, in which members 1, 2, and 3 have

the same cross-sectional areax 1 and the same lengthl, while member 4 has an area of