Engineering Optimization: Theory and Practice, Fourth Edition

250 Nonlinear Programming I: One-Dimensional Minimization Methods

It is important to note that an explicit closed-form solution cannot be obtained for

the displacements as the number of equations becomes large. However, given any

vectorX, the system of Eqs. (E 1 ) to (E 10 ) canbe solved numerically to find the nodal

displacementu 1 , u 2 ,... , u 10.

The optimization problem can be stated as follows:

Minimizef (X)=

∑^11

i= 1

ρxili (E 11 )

subject to the constraints

gj( X)=|uj( X)|−δ≤ 0 , j= 1 , 2 ,... , 10 (E 12 )

xi≥ 0 , i= 1 , 2 ,... , 11 (E 13 )

The objective function of this problem is a straightforward function of the design vari-

ables as given in Eq. (E 11 ) The constraints, although written by the abstract expressions.

gj( , cannot easily be written as explicit functions of the components ofX) X. How-

ever, given any vectorXwe can calculategj( numerically. Many engineering designX)

problems possess this characteristic (i.e., the objective and/or the constraints cannot be

written explicitly in terms of the design variables). In such cases we need to use the

numerical methods of optimization for solution.

Example 5.2 The shear stress induced along thez-axis when two spheres are in contact

with each other is given by

τzx

pmax

=

1

2









3

2

{

1 +

(z

a

) 2 }−(^1 +ν)







1 −

z

a

tan−^1







1

z

a



















 (E^1 )

whereais the radius of the contact area andpmaxis the maximum pressure developed

at the center of the contact area (Fig. 5.2):

a=











3 F

8

1 −ν 12

E 1

+

1 −ν 22

E 2

1

d 1

+

1

d 2











1 / 3

(E 2 )

pmax=

3 F

2 π a^2

(E 3 )

whereF is the contact force,E 1 andE 2 are Young’s moduli of the two spheres,ν 1

andν 2 are Poisson’s ratios of the two spheres, andd 1 andd 2 the diameters of the

two spheres. In many practical applications, such as ball bearings, when the contact

load(F )is large, a crack originates at the point of maximum shear stress and prop-

agates to the surface, leading to a fatigue failure. To locate the origin of a crack, it

is necessary to find the point at which the shear stress attains its maximum value.

Formulate the problem of finding the location of maximum shear stress forν=ν 1 =

ν 2 = 0. 3.