OPTIMAL DESIGN OF A CLASS OF WELDED BEAMSTRUCTURES BASED ON DESIGN FOR LATITUDE.

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OPTIMAL DESIGN OF A CLASS OF WELDED BEAM STRUCTURES BASED ON DESIGN FOR LATITUDE

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OPTIMAL DESIGN OP A CLASS

OF WELDED BEAM STRUCTURES

BASED ON DESIGN FOR LATITUDE

BY

Gwang-Hun Gim

A Thesis Submitted to the Faculty of the

DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE WITH A MAJOR IN MECHANICAL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 6 4

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the Univesity Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: /^5^/c

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

May 9, 1984

K.£)M. Ragsdell Professor of Aerospace and Mechanical

Date

ACKNOWLEDGMENTS

The author wishes to thank Dr. K. M. Ragsdell,

Director of the Design Optimization Laboratory, for his

encouragement, guidance, and particularly for the insights

gained from discussions and hours of instruction. I am also

grateful to Dr. Don Claussing of Xerox Corporation for

leading us to the important area of design for latitude,

and for their support of the Design Optimization

Laboratory. 1 thank the other industrial sponsors of the

Laboratory (including Honeywell and IBM) as well for

providing much needed support which makes the research

environment of the Laboratory possible.

I am grateful for the support and encouragement of

Dr. Sangchul Kim, Dr. Yongchul Cho, Professor Sangug An,

Dr. Cnolhee Pak, Dr. Chongbo Kim, Dr. Yongsung Kim, Mr.

Sanghyuk Lee, and my other professors at Inha University,

Korea. Finally may I acknowledge the support and

encouragement of my father and mother, and other family

members and friends.

iii

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS vi

LIST OF TABLES viii

ABSTRACT ix

1. INTRODUCTION 1

2. THE WELDED BEAM 5

2.1 Wela Shear Stress 7 2.2 Bar Bending Stress 10 2.3 Bar Buckling Load 10 2.4 Bar Deflection 11

3. FUNCTIONAL CONSTRAINTS 12

3.1 Constraints in General Forms 12 3.2 Dominant Constraints 14

3.2-1 The first constraint 15 3.2-2 The second constraint 16 3.2-3 The third constraint 17 3.2-4 The fourth constraint 18 3.2-5 The fifth constraint 19 3.2-6 Dominant constraints 20

4. OBJECTIVE FUNCTIONS 22

4.1 Objective Function for Sensitivity of a Design Object 23

4.2 Objective Function for Design Cost . . 27 4.2-1 Welding labor cost 27 4.2-2 Material Cost 28

5. TRADE-OFF STUDY 2 9

5.1 A Trade-off Curve 31 5.1-1 Method of specifications .... 31 5.1-2 Method of decomposition 32

iv

TABLE OF CONTENTS—Continued

v

Page

5.2 Lagrange's Interpolation Formula .... 36 5.2-1 The error of the Lagrangian

formula 36 5.3 Newton's Method 39

5.3-1 Convergence of an iteration process 4-2

5.4 A Compromise Solution 44.

6. GENERALIZED REDUCED GRADIENT METHOD 50

7. NLP FORMULATION 54.

8. RESULTS 56

9. CONCLUSION 76

APPENDIX A: OBJECTIVE FUNCTIONS AWD CONSTRAINTS WITH RESPECT TO VARIATION OF UNCONTROLLABLE PARAMETERS BASED ON DESIGN VARIABLES 77

APPENDIX B: CONTOURS OF F. : SENSITIVITY BASED ON DESIGN VARIABLES 83

LIST OF REFERENCES 87

LIST OF ILLUSTRATIONS

Figure Page

1. Welded Beam Structure 6

2. Weld Shear Stress 8

3. Characteristics of a Trade-off Curve Obtained by a Method of Specifications 33

4. Characteristics of a Trade-off Curve Obtained by a Method of Decomposition 34

5. Newton's Method AO

6. Flowchart of Newton's Method Algorithm . ... 4-1

7. Optimization Problem to find the minimum Distance from Utopia Point to Pareto-Minimal Set A5

8. Strategy for Reduced Gradient Method 51

9. Basic Flowchart of Reduced Gradient Algorithm, OPT, for Fully Constrainted Problem .... 53

10. First Objective Function: Sensitivity with respect to Variation of Uncontrollable Parameters 57

11. Second Objective Function: Design Cost with respect to Variation of Uncontrollable Parameters 58

12. First Dominant Constraint with respect to Variation of Uncontrollable Parameters ... 59

13. Second Dominant Constraint with respect to Variation of Uncontrollable Parameters ... 60

vi

LIST OF ILLUSTRATIONS—Continued

vii

Figure Page

14. Fourth Dominant Constraint with respect to Variation of Uncontrollable Parameters . . . 61

15. Fifth Dominant Constraint with respect to Variation of Uncontrollable Parameters ... 62

16. Contours of F^ and All Dominant Constraints with respect to x^ and 64-

17. Contours of F^: Sensitivity with respect to x^ and x2 65

18. Contours of F.: Sensitivity with respect to and x^ 66

19. Contours of FDesign Cost with respect to x- and x, for the Case satisfying Minimum Sensitivity 67

20. Contours of Fp: Design Cost with respect to x2 and Xo for the Actual Case Minimizing only Design Cost 68

21. Contours of Fp: Design Cost with respect to x^ and xj for the Actual Case Minimizing only Design Cost 69

22. Trade-off Curve of a Welded Beam Obtained by Method of Specifications at v = .05. . 71

23. Trade-off Curve of a Welded Beam Obtained by Method of Decomposition with F = W-| F-j jj + W2?2N an

LIST OF TABLES

Table Page

1. Utopia Point Obtained by a Method of Decomposition, F = W-j F-] n + W2F2N' (v = •®5)« • 74

2. The Chosen Data Point for Lagrange's Interpolation Polynomial, (v = .05) 74

3. Compromise Solution and the Related Design Variables, (v = .05) 74.

viii

ABSTRACT

The traditional optimal design approach has

employed a single objective such as cost, weight, or

reliability. In this thesis, we examine a new approach

called Design for Latitude, where we seek to minimize

sensitivity to changes in uncontrollable parameters. This

new approach is demonstrated by application to the well

known welded beam problem of Ragsdell and Phillips. Next we

examine strategies for trading off cost and sensitivity in

a dual objective formulation. These studies require

identification of dominant constraints using the Wilde and

Papalambros regional monotonicity concepts. The resulting

nonlinear programs are solved using the powerful

generalized reduced gradient code, OPT. The numerical

results suggest the possibility of very useful compromise

between cost and latitude.

ix

CHAPTER 1

INTRODUCTION

Nonlinear programming problems are frequently

encountered in many design activities, especially in the

design of machine elements and systems. The conventional

factor of safety concept in the design of machine elements

and systems has several limitations when material

properties, applied loads, assembly conditions, and other

parameters must be treated simultaneously or have random

characteristics. It appears that mechanical design problems

are best treated using now relatively well developed

nonlinear programming methods.

An optimal design for minimizing cost, volume,

weight, error, or operating time, so far, has been a

traditional optimization problem. The optimal design of

a welded beam for minimizing design cost was previously

examined by Ragsdell and Phillips (1976) for the case of

selected bar and weld materials and by G. Gim (1983) for

three different cases. These cases were (1) optimizing

design variables with chosen materials, (2) optimizing the

variables with six kinds of standard sizes, four kinds of

1

2

veld material, and four kinds of bar material, and (3)

combining (1) and (2). The optimal design performed by

minimizing the factors related to design cost may easily

fail to remain within the feasible region of given

constraints if variations of uncontrollable parameters are

present. Tnat is, a design object optimized by minimizing

design cost factors is usually very sensitive to these

variations.

The functional constraints are the relationships

between design variables for the feasible region of the

design to support applied loads. In case uncontrollable

parameters have variation from negative to positive values,

some of the constraints are more sensitive than others to

these uncontrollable parameters. Therefore, the worst case

for the optimal design should be chosen from all possible

cases whicn are caused by variations of uncontrollable

parameters. In other words, dominant constraints must be

selected for the worst case among all constraints which

include the variations.

Design for latitude means minimizing sensitivity of

a design object in the presence of variation of

uncontrollable parameters. The design for latitude

objective is formulated as the squared sum of the rate of

change of constraint values with respect to changes in

3

uncontrollable parameters. Since cost also is an important

factor as well as sensitivity of a design object, another

objective function for cost is considered. The design cost

for welded beam structure has two main components: (1)

welding labor cost and (2) material cost.

These two objectives conflict in the sense that

minimal sensitivity causes maximal design cost for the

welded beam structure. The best choice for the optimal

design can be made using a trade-off between these

contrasting objectives. To satisfy the multiple-criteria,

we examine two methods to choose design variables

satisfying both minimal sensitivity and design cost: (1)

method of specifications and (2) method of decomposition.

After the trade-off curve is constructed using the

method of decomposition, the compromise solution concept of

Salakvadze (1979) can be used to select the final design.

The compromise solution is obtained by drawing a line from

the Utopia point so that it enters perpendicular to the

Pareto-minimal set. An optimization problem is formulated

to find the closest point on the Pareto-minimal set from

the Utopia point and solved using the "maximum principal"

(Leitmann 1981) after a Lagrangian interpolation polynomial

is formed based on the Pareto-minimal set which is obtained

4

with different weighting factors selected suitably in the

method of decomposition.

This thesis shows how to minimize sensitivity

(maximize stability) of a design object (that is, design

for latitude) if variations of uncontrollable parameters

exist. With dominant constraints selected, two objective

functions for sensitivity of a design object and design

cost are formulated for a NLP problem, which is solved by

using the Generalized Reduced Gradient code, OPT (Gabriele

and Ragsdell 1976), and using a trade-off curve and the

compromise solution concept for the best solution

satisfying both minimal sensitivity and design cost.

Through current research, it is realized not only that

minimizing sensitivity of a design object is efficiently

performed, but also that the sensitivity is one of the most

important factors for an optimal design as well as design

cost.

CHAPTER 2

THE WELDED BEAM

In the practical design activities of machine

elements and systems, the problem of welded beam structure

is frequently encountered. Consider a welded beam, shown in

Figure 1, which is composed of bar and weld materials. The

properties , (T^, P, L, E, G, and Df are design parameters

which are assumed to vary without control, called

"uncontrollable parameters". h, 1, t, and b are design

variables used to satisfy the constraints and reduce the

objectives.

That is,

Design Variables:

x = lx-| ,X2 ,X3 ,x^ ]T (2.1)

x = lh,l,t,b]T

Uncontrollable Parameters;

u = lui ,U2 ,U3 ,u^ ,U5 ,u6 »u7]t (2.2)

where,

u = IT , (T ,P,L,E,G,D 1 T d d f

T d : d e s i g n s h e a r s t r e s s o f w e l d

p p

r B 1 L L

1

L

1

A

L

I H» P" 3

F i g u r e 1 W e l d e d B e a m S t r u c t u r e .

CT>

(Td

E

G

D _

design normal stress for beam material

Young's Modulus

Shearing modulus

design deflection

We assume that uncontrollable parameters may vary over the

range + v.

A welded beam structure has four principle failure

properties: (1) weld shear stress, (2) bar bending stress,

(3) bar buckling, and (4) bar deflection.

2.1 Weld Shear Stress

The weld shear stress (Shigley 1977) has two

components, X ' and x " t where "['is the primary shear stress

_ '/ acting over the weld throat area and t is a secondary

torsional stress, shown in figure 2.

T t / 2

V

H L

1/2-

F i g u r e 2 W e l d S h e a r

9

MR P(L + .51)(.51)

r i i _ x v ** J J

„ U q (U / + .5X2)X2 T (2.4)

y 2 0

MR P(L + .51)(.5t) -r* i ' _ j _ _____________

x 3 3

u3(u^ + .5X2)X3 X. = —— (2.5) x 2 3

where,

Xy' = y directional primary shear stress

= y directional secondary torsional stress

T " = x directional secondary torsional stress x

J = polar moment of inertia of weld group

3 2 i r + 3it

j -r 2»

6 JT

10

Therefore, the weld stress becomes:

1 ° J t x + t y

T - [ z + i y > z

2.2 Bar Bending Stress

The maximum bending stress is

MC PLt/2 (T = — =

( 2 . 6 )

I bt3/12

6u0u, (T--J-4- (2.7)

X3X4

2.3 Bar Buckling Load

in the case of a narrow rectangular cross section

where b/t is a small quantity, the approximate formula for

calculating the critical load can be put in the form

(Timoshenko 1961) :

• cr

4.013 E1C t — ( 1 - —

2L

EI

11

where,

4.013] U5IC

cr U "

(1 -12. 2u

u el ( 2 . 8 )

I = x3X^/12

C = x .x-^.u J'5 3 A o

2.4 Bar Deflection

To calculate the deflection, assume the bar to be a

cantilever of length L.

Thus,

PL? DEL =

DEL =

3EI 3Ebt3/12

4u3u2

U5X4X 3

(2.9)

CHAPTER 3

FUNCTIONAL CONSTRAINTS

The various failure conditions considered; namely,

due to combined stress in the weld, bar bending stress, bar

buckling, and excessive deflection are prevented by

application of inequality constraints which are functions

of the design variables and uncontrollable parameters. The

combined effect of simultaneous variation of the

uncontrollable parameters is difficult to predict. It is

possible that a particular constraint will be less

restrictive (i.e. increase the feasible set) while another

will be more restrictive (decrease the feasible set) when

the uncontrollable parameters do indeed vary. These trends

must be predicted in a meaningful design for latitude

formulation. Accordingly, we use monotonicity analysis to

select the "worst case" constraint set.

3.1 Constraints in General Forms

In the case of no variation of uncontrollable

parameters, 5 inequality constraints are in the form:

g^(x,u) = u1 - T(x-j »*2'x3 ,u3 ,UA) - 0 (3-1)

12

Where,

13

g2(x,u) = u2 - (r(x3,x^,u3,uA) > 0 (3.2)

g^(x) = x^, - x1 > 0 (3.3)

g^(x,u) = Pcr (x^x^u^u^u^) - u3 > 0 (3.4)

g5(x,u) = u? - DEL(x3,x^,u3,u^,u5) > 0 (3.5)

"Cu ,u) = maximum shear stress in weld

0"(x,u) = maximum normal stress in beam

Pcr(x,u) = bar buckling load

DEL(x,u) = bar end deflection

In addition, the design variables have lower and upper

bounds:

(3.6)

hl

VI

X1

VI

hu

*1 < x2 <

< x3 <

bl < x4 < bu

14

3.2 Dominant Constraints

Provided that variations of uncontrollable

parameters exist, dominant constraints have to be chosen

for the worst case among all constraints which include each

variation of uncontrollable parameters. Here we use

"detection of monotonic properties" as given by Papalambros

and Wilde (1980).

The model is put in the normalized form:

(3.7)

j = 1, • • • , J

Rj * 1

If • > 0 ; (3.8)

then is increasing with respect to uj

Tnat is, u^ is dominant; Rj = +

(«i )

If 9 R j

< 0 ; (3.9)

then R. is decreasing with respect to u^ then R. J

That is,

t where ui

15

3.2-1 The first dominant constraint

ll > 1 (3.10)

Normalize ,

then,

R!

T "2 + (T ' + T ")' L x L y L.y < 1 (3.11)

*1

Differentiate with respect to uncontrollable parameters.

Then,

(a) 3*

du.

1 ti'2 + *It + X;1) < 0 (3.12)

(b) 3*1

3»3 2"lJti'2 + It; + ty'>

, , > 0 I I \

(3.13)

where,

Q1 = 1 I I X

x3 (u4 + -5x2)

+ + x^') i jy x2(u^ + .5X2)

x.x 1*2

16

^ U 3 X 3 _ u 3 x 2 12 X-x ' — + ^ ( Xy + Ty ' ) j

3R, 2J y y 2J (c) IZIZ==ZZ=====I===I=^,-• > 0 (3.14)

3u

^ 2ulJ"^x'2 + (ty + "^y')2

Thus,

- + +

R-j = R-i (x,u-| ,U3 ,u^) (3.15)

and dominant constraint of g_^(x,u) becomes:

- + +

91 = 91 (x'u1 'u3'u^} (3.16)

That is, the worst case with respect to constraint one is

when u-| decreases, and u^ and u^ increase simultaneously.

U£, u^, u^, and u-y have no effect on constraint one.

3.2-2 The second constraint

6iuu/ g (x,u) = u ^-r-> 0 (3.17)

*A*3 '

Normalize g2

then,

6u~u, r2 V- < 1

X X u X A 3 2

17

Differentiate R2 with respect to uncontrollable parameters.

Then,

3r2 ^^3^/ (a) 2^ < 0 (3.19)

3U2 X4-X3U2

9r2 = 6uA

3u3 x^x3u2

(b) ^ >0 (3.20)

3R2 6UO (c) ^ > 0 (3.21)

au^ x^x3u2

Thus,

r2 = r2

18

3.2-4 The fourth constraint

4.013xo*^ X3U5 g lx,u) -(UcU^ ) - u3 > 0 (3.25)

6ut 4uA

Normalize g. , 4

then,

o/ 2 24u,Uo R, = * — $ 1 (3.26)

4.013x^x|(4u^ u^u^ - x^u^)

Differentiate R^ with respect to uncontrollable parameters.

Then,

3R, 24u? (a) Sl = = > 0 (3.27)

3u^ 4.013x^x^(4u^ - X3U5)

2 3R; 96U;Uo(2U; UcUi - XqUt)

(b) ^ 43 5 " 2J > 0 ,3.28) 3u^ 4.013x3x^ (4u^ U5U5 - X 3 U 5 )

3 dR/ 24u^u3(2u^ u6/u5 - x3)

(c) - « ^ < 0 (3.29) 4 . 0 1 3 x ^ x ^ ( 4 u ^ u ^ U £ , - X 3 U 5 )

3 3R^ 24u^u3(2u^ U5/U6)

(d) 3 s- < 0 (3.30) 3u6 4.013x3x^ (4u^ u5u£, - *3*5)

19

Thus,

+ + - -r4 = R4(x,u3,u4,u5,u6) (3.31)

and dominant constraint of g^(x,u) becomes:

+ + - -g4 = g4 (x,u3 'uvu5'u6> (3.32)

3.2-5 The fifth Constraint

„ 3 4u3

9r (x ,u) = u7 3 > 0 (3.33)

U5X4-X3

Normalize g_ , 5

then,

„ 3 4u3uA Rr = 3 0 (3.35)

3U3 X^U5U7

3R;- 12u^,u^ (b) J 4 - > 0 (3.36)

au^ *Ax§u5u7

20

3 3Rr 4UOU/

(c) 2_= yl

21

All dominant constraints are finally put in the worst case

form:

(a) g1 (x,u1,u3,u^) = !U1 - f + sy >- (3.41)

where,

Sx "

V- | (V- ] + .5X2)

2J

V 1 U 3 v1u3x2(v1uii + .5x 2 ) S = +

2x-|x2 2J

2 2 X1X2*X2 + 3x3>

(J — """""" 6j2~

6v*u„u

(b) ^2^x ,U2'U3'U^) * V 2 U 2 2 - 0 (3.42)

V3

(c) g^(x) = x^ - x1 > 0 (3.43)

3 , , 4.013x^x";v? x3uc;

(d) g U,u3,u ,u",uj) 4 fTft - ) - v,u3 > 0 * 6V1U, 4v-iU ,

1 A 1 ^ (3.44)

4V^u U"^ (e) g (xfu*#u1",u~,u") = v„u 3 4 > 0 (3.45)

5 3 4 i > / 2 / v u x x 3 " 2 5 4-3

CHAPTER 4

OBJECTIVE FUNCTIONS

For optimal design, design criteria should be

selected first on the basis of which the performance or

design of the system can be evaluated so that the "best"

design or set of operating conditions can be identified.

Cost, so far, has been a traditional objective for numerous

problems. However, the optimal design performed by

minimizing the factors related to cost may easily fail to

remain within the feasible region of given constraints in

case variations of uncontrollable parameters exist. In

other words, a design object optimized for the least cost

is usually very sensitive to these variations. Sensitivity

of a design object as well as design cost, therefore, is an

important factor in practical optimal design activities.

Consider two objective functions for (1) sensitivity of a

design object and (2) design cost.

22

23

4.1 Objective Function for Sensitivity of a Design Object

Assuming a feasible design exists, find the set of

design variables x^ , *2 ' x3 ' anc^ x4 w":i:'-ch gives the

smallest change in constraint function values for arbitrary

unit change in the uncontrollable parameters.

That is,

J I BR, 2 MIN: F1 (X,U) = £ZIZI 1H (4.1)

3=1 i=l dui

subject to gjj.(x,u) > 0 ; k = 1,2,3,. . • ,K

h^(x,u) =0 ; 1 = 1,2,3,• • • ,L

where

R- = normalized dominant constraint functions tj

= uncontrollable parameters

g^ = dominant inequality constraint functions

h-^ = dominant equality constraint functions

Since for the welded beam example considered here the third

constraint is independent of uncontrollable parameters, F^

becomes:

5 7 BR, 2 MIN: F-, (X,U) = nUZ 'L) (4.2)

j = l i = l j*3

24

3R, where —sL are given as follows:

aui

la) 3^

Bu.

2 2 Sx +

V2U1

(4.3)

where,

S = x

V^U^X^tV^U^ + .5X2)

23

S„ = v-jU^ v^u^xgtv-iu^ + -5x2)

2 x 1 x 2 2J

J =

2 „ 2 X-|X2(X2 + 3xo)

6 2

(b) Q1 6 R l

3^^ 2V2U-] + Sy (4.4)

where,

0 . Q ' M ' - 5 V 1 x J

2v v x (v u + .5x ) Syl J-+ —J—2 1-A SL-]

X . X 1 2

25

3R« Q?

lC) T~ = ^ 1 o 2" (4*5) du . 2v0u Js 2 + i 2 1J x y

where, 2 0

= V1U3X3 V1U3X2

x . i y

2 3R0 6v.i UnU,

(d) _2 1_3_L (4 6)

du2 V2U2X4.X3

dR2 6V^ u/ (e) = (4.7)

du3 v2u2xAx3

3R2 6v^U3 (f) = (4>8)

9u4 V2U2XAX3

3r4. 3̂ ( g ) = ( 4 . 9 )

3u3 u3^

where,

A 3 Q3 = 24V1U3U4

Q = 4.013x x3v (4v u u u - x u ) K 3 4 2 14 5 6 35

3r4 4Q3 (2v-[ u5u6 - X3U5) (h) (4.10)

au4 u4Q5

26

where,

= 4

^ 4. 013x^x^2

(i) 1*L= Q3(2v1UA Vu5 - x3> 9 u 5 Q 5

ID 1^1- 3Q6

U^Qy

d R 5 g 6 (m) i =

9 u 5 U ^ Q y

(4.11)

^ a3(2v1u^ "s /"f ,> u; _ " ^ (4.12)

3u6 Q5

R5 q6 (k) 1 = —2_

3u3 U3Q? (4,13)

where,

q6 = 4vtu3uI

&7 = V2X4.X3U5U7

(4.14)

(4.15)

27

4.2 Objective Function for Design Cost

The second objective function is total cost. The

major design cost components for such a weld assembly are:

(1) welding labor cost and (2) material cost.

F2(X,U) = C1 + C2 (4.17)

where,

C1 = welding labor cost

Co = material cost

4.2-1 Welding labor cost

Assume a welding labor rate of $12 per hour

(including operating and maintenance). Moreover, assume

that the machine can consistently lay down 6.54 cubic

inches of wela in an hour (Stewart 1981). This number does

not compare directly to that by Ragsdell and Phillips. The

labor cost C^, thus, becomes

C. = 1.835V (4.18) I W

C1 = 1.835x^ x2 $

where,

. 3 V = volume of weld material (in ) "W

2

28

4.2-2 Material Cost

Material cost contains two components: (1) weld

material and (2) bar material. Thus, material cost is:

C2 - CwVw + CBVB

where,

3 Cw = $ per volume of weld material ($/in )

3 Cg = $ per volume bar stock ($/in )

3 Vg = volume of bar (in )

= *3V\ + V Therefore, in the case of no variation of uncontrollable

parameters the objective function for design cost is:

F2 (x ,u) = C, + C2 (4.20)

= 1.835x^X 2 + C w x^x 2 + C B x 3 x A (u 4 + x 2 )

= (1.835 + Cw)x^x2 + CBx3xA(Uit + x2)

Assumming variations of uncontrollable parameters

exist, the objective function for design cost is put in the

form:

F2(X,U) = (1-835 + Cw)x^x2 + CBx3x^(Vlu^ + x2)

(4.21)

CHAPTER 5

TRADE-OFF STUDY

In optimization work the common practice is to use

an objective function that represents a single design

characteristic, such as cost, volume, weight, error, or

operating time, and so on. However, it frequently occurs in

practical applications that there is more than one design

characteristic which the design variables must satisty.

Very often these combined criteria are conflicting in the

sense that when one is increased for the optimal design,

the other is decreased. The best choice for the optimal

design can be made by using a trade-off (Siddall 1982)

between these contrasting design characteristics (objective

functions).

Trade-off curves have been proposed as a tool for

making the trade-off decision (Bartel and Marks 1974) , and

they do give considerable insight into the decision

problem. Trade-off curves can also be thought of as a plot

of optimum designs corresponding to variations in a design

specification.

29

30

After a trade-off curve is obtained, a compromise

solution (Vincent 1983) can be selected. To find the

compromise solution, four steps are needed:

(1) formulate Lagrange's interpolation polynomial of a

trade-off curve with the data points that are obtained

by suitably choosing and V«2 with the method of

decomposition.

(2) set up an optimization problem to find the minimal

distance from the Utopia point* to the Pareto-minimal

set**, and use the "maximum principle" for the optimal

solution.

*. A point G0gEr is a Utopia point if and only if for each i=l,2.3,• • »,r

G? = inf(G.iy) ly € Y}. i I

**. Definition of a Pareto-minimum (vector cost): A point y*€ Y is a Pareto-minimal point if and only if there does not exist a point y €_Y (y€BOY for local Parato-minimum) such that G(y) < G(y*). The notation < means "partially less than," that is, G^(y) < G^iy') for all i €. 11,• • «,r] and Gj (y) < Gj (y*) for at least one j € L1r * • *»rJ- The constraint set Y defined by Y = {y€Em such that g(y) =0 and h(y) > 0} where € is used to designate "an element of" and Em designates an m-dimensional space of real numbers. For each choice of y £ Y one or more objective functions may be defined by

G(y) = IG-| (y), • • • ,Gr (y) ] .

31

(3) using Newton's Method, find the compromise solution.

(4) suitably adjust and W2 in equation (5.4) to produce

the same objective value as the compromise solution;

then design variables , x2, x^/ and x^ can be

obtained simultaneously for a compromise solution.

5.1 A Trade-off Curve

Consider trade-off analysis for multiple-criteria,

minimizing sensitivity of a design object and design cost.

Two methods can be performed to decide design variables

satisfying minimal sensitivity and design cost: (1) method

of specifications and (2) method of decomposition.

5.1-1 Method of specifications

Trade-off curves give useful information on the

relationship of multiple-criteria and on certain

specifications. Specifications are interaction points with

other parts of the system, and a specification is commonly

an arbitray decision to permit suboptimization. When it is

possible to predict the penalty that is being paid in

trade-offs to achieve these specifications, the

specifications are usually subject to negotiation and are

really target values until the design progresses to the

final stages. By setting initial and final specifications

32

on the trade-off curves, the range of suitable design can

be chosen, as shown in Figure 3. The best solution can,

thereafter, be chosen from the available feasible designs.

5.1-2 Method of decomposition

There is always an interaction with other parts of

the device or system, to a greater or lesser extent. The

effort required to solve the total problem is minimized

when the system is decomposed into subsystems with least

interaction. The decomposition of a system uses the

interaction characteristics and divides the problem into

subproblems which can be handled more easily. We can,

thereafter, optimize a formulation of the whole system that

is composed of subproblems. In other words, the new

objective function composed of multiple-criteria can be

minimized to estabilish the best system design

specification. A trade-off curve can be formulated by

choosing suitable weighting factors, as shown in Figure 4.

33

F i n i a l S p e c i f i c a t i o n S e t t i n g

34

Pareto-minimal Set

/•^Compromise Solution

^N^Utopia Point

S E C O N D O P T I M I Z A T I O N C R I T E R I O N

F i g u r e 4 C h a r a c t e r i s t i c s b y M e t h o d o f

o f a T r a d e - o f f O e c o m p o s i t i o n .

C u r v e O b t a i n e d

35

That is,

MIN: F(x,u) (5.1)

where,

F(x/U) — W_| ^ + ^2^*2N (5.2)

or

1/F(x,u) = W^F^ + W2/F2N (5.3)

and V$2 are weighting factors of F^ and F2,

respectively;

W1 + W2 = 1 (5.4)

F^n and F2N are the normalized functions of F-j and

F„, respectively.

36

5.2 Lagrange's Interpolation Formula

When the independent values are not evenly spaced,

the Lagrangian polynomial (Gerald 1978) is used to fit a

polynomial based on the given data points. The Lagrangian

form is also the most straightforward way to get the

polynomial as an explicit function.

Suppose f(x) and x data pairs are given. The

Lagrangian formula then can be written in the form:

n n x

i=l j=l x± (5.5)

5.2-1 The error of the Lagrangian formula

The error of Pn(x) is zero at the n+1 values of x

that are fitted exactly.

That is,

E(x) = f(x) - Pn(x)

= (x-x1 ) (x-x2)« • • (x-xn+1 )g(x)

The auxiliary function W(t) is:

(5.6)

W(t) = f(t) - Pn(t)

- (t-x1 ) (t-x2 )• . (t-xn+1)g(x) ( 5 . 7 )

37

where W(t) = 0 for t = x-^xgr* * •Xn+i, and at t = x, for a

total of n+2 zeros.

Hence,

W (t) = 0

W ' - i t ) = 0

w(n+1)(t) = 0

Let 3^ be the value of t at which w^n+^) (t) = 0 and

xmin < ̂ < *max •

Then,

w(n+1) (^) = 0 = f(n+1) (^) - 0 - (n+1)Ig(x),

f ( n + 1 ) ( / } g (x) = z~

(n+1)1

The error then is:

£U+1)

E(x) = (x-x. ) (x-x9)... (x-x +1 ) (5.8) 1 ^ n 1 (n+1)!

The error can be bracketed between a maximum and a minimum

value only if we have information on the (n+l)st derivative

of the actual function f(x).

38

Finally, the Lagrangian formula can be used simply

to write out an interpolation polynomial. The main

advantages of the Lagrangian formula are two: (1) to find

any value of a function when the given values of the

independent variable are not equidistant, and (2) to find

the value of the independent variable corresponding to a

given value of the function.

39

5.3 Newton's Method

In practical engineering problems, finding roots of

a system of equations is frequently encountered. One of the

most widely used methods of solving equations is Newton's

method (Ragsdell 1982) because it is rapidly convergent at

least in the near neighborhood of a root, shown in Figure

5. The procedure is known as Newton-Raphson iteration. A

flowchart for Newton's method is given in Figure 6.

In Newton's method it is assumed at once that the

function f(x) is diferentiable. This implies that the graph

of f(x) has a definite slope at each point and hence a

unique tangent line. Starting from an initial estimate

which is not too far from a zero of the equation,

extrapolate along the tangent to its intersection with the

x-axis, and take that as the next approximation. This step

is continued until either the successive x-values are

sufficiently close, or the value of the function is

sufficiently near zero.

Consider the Taylor Series Expansion of f(x) about

a given point "x.

40

f(x)

F i g u r e 5 N e w t o n ' s M e t h o d .

41

(NEWT)

DEFINE: XO.N.EPSI

* •

XOLD = XO

< •

I " LL

• ITER = I

F ( X O L D ) X N E W = X O L D - { }

D F ( X O L D )

I S F ( X N E W ) | < E P S A L P H A = X N E W

X O L D = X N E W

F i g u r e 6 F l o w c h a r t o f N e w t o n ' s M e t h o d A l g o r i t h m .

42

~ _ 0 0 ( x ) ^

f(x) = f (x) + > ( ) (x - x)n (5.9) n=l n!

where "x is a point near x. Truncate all terms of degree

greater than one:

f (x") = f (x) + f' (x) (x - x) (5.10)

and force f ("x) to zero to get the formulation of Newton's

method,

~ - f ( * )

x — x (5.11) f' (x)

In general the iteration prescription is:

f (x^) x. = x — (5.12) J+1 J f'txj)

5.3-1 Convergence of an iteration process

Newton's Method is quadratically converged, in the

sense that the error of each step approaches proportionally

to the square of the error of the previous step. Consider a

nonlinear function of a single variable, f(x) in class C^

which has a single zero in the interval x fc (a,b).

43

Define a class of iteration algorithms of the form:

' 3 + 1 = 9Uj> (5.13)

Thus, from eqns. (5.12) and (5-13)

g(Xj) = X j -"x.i>

f'Uj)

f ' (Xi)f • ( X i ) - f ( X i)f ' ' ( X j ) • I X . ) = 1 i *

l f ' ( X j ) ]

f ( X j )f ' • ( X j )

I f ' ( X j ) ] 2

(5.14)

(5.15)

Successive iteration converge if lg'(x)| < 1.

Therefore,

an interval of convergence for Newton's method is defined

as:

f (x)f " (x)

Lf1(x)] < 1 (5.16)

The method will converge for any initial value x-| in the

interval. The condition is sufficient only if f(x) is

continuous and f'(x) exists.

44

5.4 k Compromise Solution

The compromise solution concept of Salakvadze

(1979) may give useful information on the best decision,or

the recommended solution for a final design choice. The

idea behind this concept is to find the "closest" point on

the Pareto-minimal set to a Utopia point (Vincent and

Grantham 1981) , shown in Figure 4. The closest point, the

compromise solution, on the Pareto-minimal set is obtained

in this case by drawing a line from the Utopia point so

that it enters perpendicular to the Pareto-minimal set.

For finding the compromise solution, set up an

optimization problem to get the minimal distance from the

Utopia point to the Pareto-minimal set on the trade-off

curve, shown in Figure 7.

That is,

Min: ^dt (5.17)

X| = cosU (5.18)

X2 = sinU (5.19)

with target 0= X2 - Pn (Xj ) = 0 (5.20)

where Pn (X-j ) is Lagrangian interpolation polynomial that is

obtained with n+1 numbers of given data points. These data

45

( X ° , X ° )

F i g u r e 7 O p t i m i z a t i o n P r o b l e m t o f i n d t h e m i n i m u m D i s t a n c e f r o m U t o p i a P o i n t t o P a r e t o - M i n i m a l S e t .

46

points are gained by setting different and in eqns.

(5.2) or (5.3).

Integrate (5.18) and (5.19), then

X1 = X° + (cosU)t (5.21)

x2 = X° + (sinU)t (5.22)

tanU = (X2 - X°)/(X1 - X°) (5.23)

Let's use "Maximum Principal" (Leitmann 1981) for an

optimal solution of eqns. (5.17) - (5.20).

H = o + A-jCOsU + 2sinU = 0 (5.24)

)\ o < 0 (5.25)

3H -Ai it) =— = 0

9X-,

J\-| (t) = A-1 = const (5.26)

dH -Aolt) = 0

3x 2

7l2(t) = }\ 2 = const (5.27)

Use a given terminal condition,

Ai(tf) (5.28) ^ X-]

47

\ D ̂ ® A2 (tf} =P (5.29)

9X2

From (5.26), (5.27), (5.28), and (5.29)

JN-, (tf > " (5.30)

^ 2 ( t f } = ̂ 2 ( 5 - 3 1 )

For the minimal solution,

3H * * -_^1sinU +^?cosU = 0 (5.32)

3U '

tanD* = ̂ - = (fl— )/($££-) (5.33)

48

or

* 0 * n (X1 " X1 )

X* = xO (5.37) 2 2 JPnlX^/ax-i

since

*• %

ae _ dix2 - pn(xi)i

ax2 dx2

= 1 (5.38)

96 _ aix2 - Pn (X1) ] _ 3Pn (^ )

3x1 ax-, ax-, (5.39)

To get X^ and ^ , substitute (5.37) into (5.20)

then,

0 = Xg - Pn(X^) = 0

* 0 0 (X-j - X-| ) #

0= lx2 * 1 - pn(xl) - 0 (5-40> 3Pn

49

compromise solution, and P2» suitably adjust and W2

in equation (5.2) or (5.3) to get the same value of the

compromise solution, then simutaneously design variables

x^, x3» and *4 can t>e obtained for a compromise

solution.

CHAPTER 6

GENERALIZED REDUCED GRADIENT METHOD

The reduced gradient method is a technique for

handling constraints in conjunction with any of the methods

*/hich use successive one dimensional minimizations. The

generalized reduced gradient method is one of a class of

algorithms using implicit variable eliminations. This

method was first proposed by Wolfe (1963) who formulated

the method with only linear constraints ana named because

Vrf(x) can be viewed as the gradient of f(x) in the reduced

space of the x variables. McCormick (1969), thereafter,

modified the method by avoiding jamming. The method was

extended to nonlinearly constrained problems by Abablie and

Capentier (1969) who created the first general GRG code.

Gabriele and Ragsdell (1975) also successfully implemented

the generalized reduced gradient method for the solution of

the constrainted nonlinear programming problems, called

OPT.

The main idea (Siddall and Michael 1980) is that

all inequalities are converted to equalities using slack

variables and a search direction is chosen tangent to the

50

51

Contours for U

Equality constraint ^-0

n1

Starting point

F i g u r e 8 S t r a t e g y f o r R e d u c e d G r a d i e n t M e t h o d ,

constraint lines in the direction of reducing the objective

function. In this method two strategies are used, as shown

in Figure 8. A strategy is used to "scamper" back to

feasibility. After returning to feasibility, the same step

in the same direction is repeated until the minimum in the

search direction is bracketed at A and B. Another stategy

is used to pinpoint the minimum between A and B.

The major components of the GRG algorithm are given

in the flowchart of figure 9. Additional details of our

implementation of the generalized reduced gradient method

can be found in the OPT users manual (Gabriele and

Ragsdell, 1976).

53

START

S p e c i f y A n y B o u n d S t a t e V a r i a b l e s a s D e c i s i o n V a r i a b l e

C a l c u l a t e R e d u c e d G r a d i e n t

F o r m P r o j e c t e d R e d u c e d G r a d i e n t

D e t e r m i n e S e a r c h D i r e c t i o n (stoT)

T a k e S t e p A l o n g S e a r c h D i r e c t i o n ( • + —

A d j u s t S t a t e V a r i a b l e s U s i n g N e w t o n ' s M e t h o d

R e d u c e S t e p S i z e

I t e r a t e t o N e a r e s t B o u n d

B o u n d

R e f i n e t o L o c a t e M i n i m u m

N e w t o n M e t h o d

M 1 n i m u m B o u n d e d

Increase Step Size

F i g u r e 9 B a s i c F l o w c h a r t o f R e d u c e d G r a d i e n t A l g o r i t h m , O P T , f o r F u l l y C o n s t r a i n t e d P r o b l e m .

CHAPTER 7

NLP FORMULATION

Assume 1018 HR Steel and E6010 as bar and weld

materials, therefore;

(a) Applied load : P = 500 lb

Length of bar : L = 25 in

(b) Beam material: 1018 HR Steel

3 CB - .06509 $/in

0^ = 6385 psi

(c) Weld material: E6010

3 Cw = .09622 $/in

Xw = 3826 psi

Design Variables:

x = lxi ,X2,X3,X4 ]T = lh,1,t,b]T

Uncontrollable Parameters (mean values)

u = lu1,u2,u^,u^,u5,u6,u7]T

u = I T,r

55

Objective Functions

(7.1)

2 f^xru) = 1.93122x^x 2 + . 06509x^x^ (v-^ u^+ x2) (7.2)

Constraints

(a) Limit of sizes of design variables (4 constraints)

.1 < x-j < 2.0

. 1 < x 2 < 1 0 . 0

.1 < X3 £ 10.0

.1 < x^ £ 2.0

where upper limit of x-| and x^ can be chosen readily as 2.0

inches because x-| and x^ are less than one tenth of X3 in

the case of minimizing design cost.

(b) Limit of design properties (5 dominant constraints):

eqns. (3.41), (3.42), (3.43), (3.44), and (3.45).

CHAPTER 8

RESULTS

The curve of the first objective function

(sensitivity) versus variation of uncontrollable parameters

is shown in Figure 10 . As the variation increases from

-.05 to +.05 (from -5% to +5%), the value of the objective

function also increases, which implies that a design object

is inherently sensitive to uncontrolled variation. Figure

11 shows the second objective function (design cost) versus

the variation. Figure 11 also illustrates that design cost

becomes more expensive as the variation increases. The more

detailed figures are given in Appendix A.

It is shown that the value of constraints, which

include uncontrollable parameters, becomes decreased as the

variation increase from -.05 to +.05, in Figures 12, 13,

14, and 15. These Figures also give the tendency that a

design object becomes more sensitive as the variation

All figures in this thesis requiring contours of objective and constraints were prepared using OFCP (David, 1979).

56

57

.759960

1 0"* 025

F i g u r e 1 0 F i r s t O b j e c t i v e F u n c t i o n : S e n s i t i v i t y w i t h r e s p e c t t o V a r i a t i o n o f U n c o n t r o l l a b l e P a r a m e t e r s .

x = 2 . 0 x = 1 0 . 0 x 1 = 1 0 . 0 x f = 2 . 0 3 4

58

124.439

123.624

122.812 $

/

/J

/ 0.0 .025 .05

Figure 11 Second Objective Function: Design Cost with respect to Variation of Uncontrollable Parameters.

x . = 2 . 0 x _ = 1 0 . 0 = 1 0 . 0 x £ = 2 . 0

59

3700.38'

3599.20

.05 .025 0.0 v

Figure 12 First Dominant Constraint with respect to Variation of Uncontrollable Parameters.

x 1 = 2 . 0 x 2 = 1 0 . 0 X 3 = 1 0 . 0 = 2 . 0

60

6010 .00

5831.39

5652.31

.05 0.0 .025 v

Figure 13 Second Dominant Constraint with respect to Variation of Uncontrollable Parameters.

x = 2 . 0 x = 1 0 . 0 xl = 1 0 . 0 x f = 2 . 0 i 4

61

\ 1 \ \

\

\ \

\

\

i X x1 0 0.0 .025 .05

Figure 14 Fourth Dominant Constraint with respect to Variation of Uncontrollable Parameters.

1 _ 2 . 0 1 0 . 0

x 9 = 1 0 . 0 2.°

62

.05 .025 0.0 v

Figure 15 Fifth Dominant Constraint with respect to Variation of Uncontrollable Parameters.

x = 2 . 0 x ? « 1 0 . 0 x ^ = 1 0 . 0 x ^ = 2 . 0

increases. Finally, Figures 10, 11, 12, 13, 14, and 15

illustrates that an optimal design should be performed at

5% variation of uncontrollable parameters for the worst

case because at the case of 5% variation a design object is

the most sensitive amongst all variations from -5% to +5%.

Figure 16 showns all constraints and contours of

the first objective function. The upper limit of design

variables, = 2.0, X£ = 10.0, x^ = 10-0» and x^ = 2.0

inches, produces the minimal sensitivity of the welded

beam, as shown in Fig. 17 and 18. That is, minimal

sensitivity causes maximal strength and design cost. For

better understanding, more figures are given in Appendix B.

Figures 19 also presents that an optimal design

satisfying only minimal sensitivity is not practical. It is

shown, in Figures 19, 20, and 21, that there exist a big

difference of design cost between the cases for minimizing

only design cost and only sensitivity of a design object,

respectively. Therefore, the need of two objective

criteria, sensitivity and design cost, is readily realized.

64-

D C B A 1 0 . 0

U

0

0.0

1 0 . 0 5.0 x

F i g u r e 1 6 C o n t o u r s o f F - | a n d A l l D o m i n a n t C o n s t r a i n t s w i t h r e s p e c t t o X 3 a n d x ^ .

( v = . 0 5 x - | = . 2 0 X 2 = 5 . 0 )

C O N T O U R 1 0 A = . 0 0 7 B = . 0 0 1 C = . 0 0 2 D = . 0 1

65

D C B A 1 0 . 0

2

.0

.0 1 0 . 0 5.0 0.0 x

F i g u r e 1 7 C o n t o u r s o f F 1 : S e n s i t i v i t y w i t h r e s p e c t t o x 1 a n d x 2 .

( v = . 0 5 x 3 = 1 0 . 0 = 2 . 0 )

C O N T O U R I D A = . 0 0 0 1 3 4 8 2 1 B = . 0 0 0 1 3 7 C = . 0 0 0 1 4 5 D = . 0 0 1

D C B A 1 0 . 0

4

0

0.0

0.0 0 1 0 . 0 x

F i g u r e 1 8 C o n t o u r s o f F 1 : S e n s i t i v i t y w i t h r e s p e c t t o a n d x ^ .

( v = . 0 5 x 2 = 1 0 . 0 x 1 = 2 . 0 )

C O N T O U R I D A = . 0 0 0 1 3 4 8 2 1 B = . 0 0 0 3 5 C = . 0 0 1 0 = . 0 0 5

67

1 0. 0 B C D

5.0

0.0

0.0 5.0 x, 1 0 . 0

F i g u r e 1 9 C o n t o u r s o f F : D e s i g n C o s t w i t h r e s p e c t t o X o a n a X / f o r ' 1 t h e C a s e s a t i s f y i n g M i n i m u m S e n s i t i v i t y

( v = . 0 b

C O N T O U R I D A = 9 0 . 0 C = 1 2 4 . 0

X1 = 2 . 0

B = 1 0 5 . 0 D = 1 4 0 . 0

x 2 " 1 0 . 0 )

68

1 0 . 0

0.0 5.0 1 0 . 0

F i g u r e 2 0 C o n t o u r s o f F 2 : D e s i g n C o s t w i t h r e s p e c t t o X 2 a n d X 3 f o r t h e A c t u a l C a s e M i n i m i z i n g o n l y D e s i g n C o s t

( v = . 0 5

C O N T O U R I D A = 2 . 0 C = 3 . 0

x 1 = . 1 7 . 1 8 )

B = 2 . 5 0 D = 3 . 5 0

69

ABCD 1 0 . 0

1 0 . 0

F i g u r e 2 1 C o n t o u r s o f F 2 : D e s i g n C o s t w i t h r e s p e c t t o X 3 a n d f o r t h e A c t u a l C a s e M i n i m i z i n g o n l y D e s i g n C o s t

( v = . 0 5

C O N T O U R I D A = 2 . 0 C = 7 . 0

*1 . 1 7

B = 3 . 5 0 D = 1 2 . 0

X2 • 6 . 0 )

70

A trade-off curve is formulated based on the

sensitivity and design cost, as shown in Figure 22. After

setting initial and final specification, we can obtain the

recommended solution for the final decision of an optimal

design. For example, let's choose labels 8 and 5 as initial

and final specifications respectively, then the region from

5 to 8 can be used for the suitable design from the

available feasibility. The recommended solution also can be

obtained as follows:

F-| = .0022 + (.004866-.0022)/2

= .003533

= 8.488372 + (13.48532-8.488372)/2

= 10.986846

A/ /V where, F-j and Fg are the values at 50% from initial

to final specifications. From F-] and F2» we can obtain the

recommended solution that is the closest point on the A/

trade-off curve from the point (F"|#F2)* *n this example,

label 6 can be chosen as a recommended solution.

For a recommended solution, another method can be

used as follows: find a an Utopia point based on label 5

and 8, as shown Figure 22. Then get the closest point from

the Utopia point to the trade-off curve, as discussed in

71

F1 F2 1 .045717 3.937362 2 .014748 4.912147 3 .010002 5.878498 4 .006880 7.077603 5 .004866 8.488372 6 .C03581 10.05805 7 .002751 11.73553 8 .002200 13.48532 9 .001826 15.28449 10 .001350 19.07323 11 .000935 26.80917 12 .000846 30.15035 13 .000671 42.90977 14 .000135 124.4390

.0229 "

'2

F 62.22

F i g u r e 2 2 T r a d e - o f f C u r v e o f a W e l d e d B e a m O b t a i n e d b y M e t h o d o f S p e c i f i c a t i o n s a t v = . 0 5

chapter 5. Label 6, in this example, can be chosen as a

recommended solution.

The Pareto-minimal set and Utopia point are

formulated by using a method of decomposition,

F = W-|F-|n + w2f2N' shown in Figure 23. Table 1 shows the

Utopia point. Before performing Lagrangian interpolation,

let's choose 5 points and scale these values, as shown in

Table 2, in order to adjust Lagrangian interpolation

polynomial to the trade-off curve given in Figure 23.

Lagrangian formula can be obtained thereafter. From

eqn.(5.40), a compromise solution and the related design

variables can be obtained, after three more steps are

performed, as discussed in chapter 5.

* A compromise solution, = .08964734 and

X* = .04989566, is obtained and multiplied by

F^ = 124.4390 and F. = .045717. Then the values, 2max 1 max tt tt

F^ = 11.15562 and F^ = .0022811, are almost same as those

of = .5 and W£ = .5.

Finally, a compromise solution and the related

design variables, shown in Table 3, can be obtained as

follows:

73

No

045717 3.937362 .031641

004515 7.779320 .071566

002913 9.990186 .073655

002215 11.30412 .069650

001690 13.05326 .064134

001097 16.71004 .051564

000135 124.4390 .002950 0229

124.U us$

6 2 . 2 2

23 Trade-off Curve of a Welded Beam Obtained bv Method of Decomposition with

F * «1Fm + h2f2N and v " -05

74

T a b l e 1

U t o p i a P o i n t O b t a i n e d b y M e t h o d o f D e c o m p o s i t i o n , F = ^ F ^ N + W 2 F 2 N . ( v = . 0 5 )

F1 F2 F 1 / . 0 4 5 7 1 7 F 2 / 1 2 4 . 4 3 9

. 0 0 0 1 3 5 0 3 . 9 3 7 3 6 2 . 0 0 2 9 5 2 9 . 0 3 1 6 4 1 0

T a b l e 2

T h e C h o s e n D a t a P o i n t f o r L a g r a n g e ' s I n t e r p o l a t i o n P o l y n o m i a l . ( v = . 0 5 )

N o . W i w2 f1N f2N F - | / . 0 4 5 7 1 7 F 2 / 1 2 4 . 4 3 9

1 . 2 5 . 7 5 . 0 0 4 5 1 5 7 . 7 7 9 3 2 0 . 0 9 8 7 5 9 7 . 0 6 2 5 1 5 0

2 . 4 0 . 6 0 . 0 0 2 9 1 3 9 . 9 9 0 1 8 6 . 0 6 3 7 1 8 0 . 0 8 0 2 8 1 7

3 . 5 0 . 5 0 . 0 0 2 2 1 5 1 1 . 3 0 4 1 2 . 0 4 8 4 5 0 2 . 0 9 0 8 4 0 6

4 . 6 0 . 4 0 . 0 0 1 6 9 0 1 3 . 0 5 3 2 6 . 0 3 6 9 6 6 5 . 1 0 4 8 9 6 8

5 . 7 5 . 2 5 . 0 0 1 0 9 7 1 6 . 7 1 0 0 4 . 0 2 3 9 9 5 4 . 1 3 4 2 8 2 9

T a b l e 3

C o m p r o m i s e S o l u t i o n a n d t h e R e l a t e d D e s i g n V a r i a b l e s , ( v = . 0 5 )

U W 1

U 2

F*(«)

. 5 . 5 . 0 0 2 2 1 5 4 1 1 . 3 0 4 1 2

< ( i n ) ( i n ) *3 ( i n ) *

( i n )

. 2 8 6 2 2 3 4 o 7 5 2 8 9 1 0 . 0 0 0 0 . 5 2 2 8 0 7

•*

X1 = .286223 in

* x2 = 4.75289 in

X3 = 10.0000 in

* */, = .522807 in

CHAPTER 9

CONCLUSION

Herein we demonstrate an approach (Design for

Latitude) which minimizes sensitivity (maximizes stability

or latitude) in the presence of changes in uncontrollable

parameters. We conclude that maximum latitude gererally

correspons to maximum cost. The region selected by setting

initial and final specifications on a trade-off curve is

useful for an optimal design satisfying both minimal

sensitivity and design cost. Moreover, a compromise

solution obtained by the concept of Pareto-minimal set and

Utopia point gives an efficient guideline for the best

solution.

Dominant constraints, in the present reseach, are

suitably selected for the worst case. Moreover, two

objective functions for sensitivity of a design object and

design cost are efficiently handled as a NLP problem.

Finally sensitivity is found to be one of the most

important factors for an optimal design as well as design

cost.

76

A P P E N D I X A

O B J E C T I V E F U N C T I O N S A N D C O N S T R A I N T S W I T H R E S P E C T T O V A R I A T I O N O F

U N C O N T R O L L A B L E P A R A M E T E R S B A S E D O N D E S I G N V A R I A B L E S

.0

0

0

1 0 . 0 0.0 5.0 x

X - | = 2 . 0 X 2 = 1 0 . 0

C O N T O U R I D F , | ( X , U ) = . 0 0 0 3 5

1 : v • 0 . 0 2 : v = . 0 5 3 : v = . 1 0

77

1 0 . 0

0

0.0

1 0 . 0 5.0 0.0 X

X - i = . 1 7 = . 1 8

C O N T O U R I D F 2 ( X . U ) = 3 . 2 0

1 : v - 0 . 0 2 : v = . 0 5 3 : v = . 1 0

7.9

1 0 . 0

5.0

0.0 0 . 0 5 . 0 x 3 1 0 . 0

41

•

1 ^3

•

X 1 = . 2 0 X 2 = 5 . 0

C O N T O U R I D Q ^ X . U ) =

1 : v = 0 . 0 2 : v = . 0 5 3 : v =

0 . 0

. 10

80

X 1 = . 2 0 X 2 = 5 . 0

C O N T O U R I D g 2 ( x » u )

1 : v = 0 . 0 2 : v = . 0 5 3 : v

= 0 . 0

= . 10

02 * = A : e SO * = A ' -z 0*0 = A :I

o'o = (n'x)^6 ai ynoiNOO

o * s = z t o z * = L x

0"0 L o*$ 0*0

cafes

9i I £ 2 -

L8

X 1 = . 2 0 X 2 = 5 . 0

C O N T O U R I D g c ( X , U ) 5

1 : v - 0 . 0 2 : v = . 0 5 3 : v

= 0 . 0

= . 1 0

A P P E N D I X B

C O N T O U R S O F F : S E N S I T I V I T Y B A S E D O N D E S I G N V A R I A B L E S

DCB A

•v.

-g,

/

0.0 0 . 0 5 . 0 x 1 1 0 . 0

v = .05 x2 = 10.0 = 2.0

C O N T O U R I D A = . 0 0 0 1 3 4 8 2 1 B = . 0 0 0 4 C = . 0 0 1 D = . 0 0 5

83

84

DCBA

/ g3 /

/

0 . 0 5 . 0 x 1 1 0 . 0

v = . 0 5 x 2 = 1 0 . 0

CONTOUR ID A = .000134821 C = .001

x = 1 0 . 0 3

B = .0003 D = .005

85

1 0 . 0

5.0

0.0

I lv •——

I V . V .

h

/

0.0 5.0 1 0 . 0

v = .05 x-] =2.0

CONTOUR ID A = .000134821 C = .001

H = 2 - °

B = .0005 D = .005

86

1 0 . 0

5.0

0.0

V A

- A r -

A

- A r -V

0.0 5.0 1 0 . 0

v = . 0 5 x - i = 2 . 0 x 3 = 1 0 . 0

CONTOUR ID A = .000134821 C = .001

B = .0003 D = .005

LIST OF REFERENCES

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Bartel, D. L., and Marks, R. W., "The Optimum Design of Mechanical Systems with Competing Design Objectives," ASME Journal of Engineering for Industry, pp. 171-178, (Feb. 1974)

David, J., and K. M. Ragsdell, "OFCP: An Optimization Contour Plotting Package," Design Optimization Laboratory, (1979).

Gabriele, G. A., "Application of the Reduced Gradient Method to Optimal Engineering Design," M.S. Thesis. School of Mechanical Engineering, Purdue University, (Dec. 1975).

Gabriele, G. A., and Ragsdell, K. M., "OPT: A Nonlinear Programming Code in Fortran, Users Manual," School of Mechanical Engineering, Purdue University, (1976) .

Gerald, C. F., Applied Numerical Analysis, (Second Edition) , Addison-wesley, (1978) .

Gim, G., "Optimal Design of a Class of Welded Beam Structures Using the Method of Multipliers," AME-507 Term Paper, Aerospace and Mechanical Engineering, The University of Arizona, (May 1983).

Leitmann, G., The Calculus of Variations and Optimal Control, Plenum Press, New York, (1981). ~

McCormick, G. P., "Anti-Zig-Zagging by Bending," Management Science. Vol.15, pp. 315-320, (1969).

87

88

Papalambros, P. and Wilde, D. J., "Regional Monotonicity in Optimum Design," Journal of Mechanical Design, Transactions of the ASME. Vol. 102, pp. 497-500, (July 1980).

Ragsdell, K. M., Applied Numerical Method, Design Optimization Laboratory Publication, (1982).

Ragsdell, K. M., and Phillips, D. T., "Optimal Design of a Class of Welded Sructures Using Geometric Programming," ASME Journal of Engineering for Industry, Vol. 98, Series B, No. 3, pp. 1021-1025, (August 1976).

Reklaitis, G. V., Ravindran, A., and Ragsdell K. M., Engineering Optimization: Method and Applications, Wiley Interscience, (1981) .

Salakvadze, M. E., Vector-Valued Optimization Problems in Control Theory, Academic Press, New York, (1979).-

Shigley, J. E., Mechanical Engineering Design, Third Edition, McGraw Hill, (1977).

Siddall, J. N., Optimal Engineering Design; Principles and Applications, Marcel Dekker, Inc., (1982).

Siddall J. P., and Michael W. K., "Interaction Curves as a Tool in Optimization and Decision Making," Transactions of the ASME. Vol. 102, pp. 510-516, (July 1980).

Stewart, J. P., The Welder's Handbook. A Prentice Hall Company, (1981).

Timoshenko, S. P., Theory of Elastic Stability, Second Edition, McGraw Hill, (1961).

Vincent, T. L., "Game Theory as a Design Tool," Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 105, pp. 165-170, (June 1983).

Vincent, T. L., and Grantham, W. J., Optimality in Parametric Systems, A Wiley Interscience Publication, (1981).

89

Wolfe, P., "Methods for Linear Constraints," Nonlinear Programming, Abadie, J., ed., North Holland, Amsterdam, (1967).