Engineering Optimization: Theory and Practice, Fourth Edition

Problems 691

Figure 12.7 Circular annular plate under load.

whereDis the flexural rigidity of the plate,wthe transverse deflection of the plate,ν

the Poisson’s ratio,Mthe radial bending moment per unit of circumferential length, and

Qthe radial shear force per unit of circumferential length. Find the differential equation

and the boundary conditions to be satisfied by minimizingπ 0.

12.6 Consider the two-bar truss shown in Fig. 12.8. For the minimum-weight design of the
truss with a bound on the horizontal displacement of nodeS, we need to solve the
following problem:
FindX= {x 1 x 2 }Twhich minimizes
f (X)=

√

2 l(x 1 +x 2 )=

√

2 60(x 1 +x 2 )

subject to

g(X)=

P l

2 E

(

1

x 1

+

1

x 2

)

−Umax

= 10 −^3

(

1

x 1

+

1

x 2

)

− 10 −^2 ≤ 0

0 .1 in.^2 ≤xi≤ 1 .0 in.^2 , i= 1 , 2

Find the solution of the problem using the optimality criteria method.

12.7 In the three-bar truss considered in Example 12.5 (Fig. 12.6), if the constraint is placed
on the resultant displacement of nodeS, the optimization problem can be stated as

FindX=

{

x 1

x 2

}

which minimizes

f (X)= 80. 0445 x 1 + 28. 3 x 2

subject to

√

U 12 +U 22 =

P 1 l

E

[

1

x 12

+

1

(x 1 +

√

2 x 2 )^2

] 1 / 2

≤Umax

or

g(X)=

[

1

x 12

+

1

(x 1 +

√

2 x 2 )^2

] 1 / 2

≤Umax

where the vertical and horizontal displacements of nodeSare given by

U 1 =

P l

E

1

x 1 +

√

2 x 2

andU 2 =

P l

E

1

x 1

Find the solution of the problem using the optimality criteria method.