Engineering Optimization: Theory and Practice, Fourth Edition

Problems 773

(a)Exact displacement solutionU 0 atX 0

(b)Exact displacement solution(U 0 +U)at the perturbed design,(X 0 +X)

(c)Approximate displacement solution,(U 0 +U), at (X 0 +X) using Eq. (14.20)

with four terms forU

14.5 Consider the four-bar truss shown in Fig. 14.2 whose stiffness matrix is given by
Eq. (E 2 ) of Example 14.1. Determine the values of the derivatives ofyiwith respect
to the areaA 1 , ∂yi/∂x 1 (i= 5 , 6 , 7 , 8 )at the reference designX 0 = {A 1 A 2 A 3 A 4 }T=
{ 2. 0 , 2. 0 , 1. 0 , 1. 0 }Tin^2.

14.6 Find the values of∂yi/∂x 2 (i= 5 , 6 , 7 , 8 )in Problem 14.5.

14.7 Find the values of∂yi/∂x 3 (i= 5 , 6 , 7 , 8 )in Problem 14.5.

14.8 Find the values of∂yi/∂x 4 (i= 5 , 6 , 7 , 8 )in Problem 14.5.

14.9 The equilibrium equations of the stepped bar shown in Fig. 14.11 are given by

[K]Y=P (1)

with

[K]=







A 1 E 1

l 1

+

A 2 E 2

l 2

−

A 2 E 2

l 2

−

A 2 E 2

l 2

A 2 E 2

l 2





 (2)

Y=

{

Y 1

Y 2

}

, P=

{

P 1

P 2

}

(3)

IfA 1 =2 in.^2 , A 2 =1 in.^2 , E 1 =E 2 = 30 × 106 psi, 2l 1 =l 2 =50 in.,P 1 =100 lb,

andP 2 =200 lb, determine

(a)Displacements,Y

(b)Values of∂Y/∂A 1 and∂Y/∂A 2 using the method of Section 14.4

(c)Values of∂σ/∂A 1 and∂σ/∂A 2 , whereσ= {σ 1 , σ 2 }Tdenotes the vector of stresses

in the bars andσ 1 =E 1 Y 1 / l 1 andσ 2 =E 2 (Y 2 −Y 1 )/ l 2

Area,A 1

Y 1

l 1 l 2

Y 2 P 2

P 1

Area,A 2

Figure 14.11 Stepped bar.