Optimum design methods
Item Type text; Thesis-Reproduction (electronic) Authors Trondsen, Torvald, 1933-
Publisher The University of Arizona.
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OPTIMUM DESIGN METHODS
by
Torvald Trondsen
A Thesis Submitted to th e F ac u lty o f th e
DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS In P a r t i a l F u lf i l l m e n t o f th e Requirements
For th e Degree o f MASTER OF SCIENCE
WITH A MAJOR IN CIVIL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA
1 9 6 9
STATEMENT BY AUTHOR
This t h e s i s has been su b m itted in p a r t i a l f u l f i l l m e n t o f r e quirements f o r an advanced degree a t The U n iv e rsity o f Arizona and i s d e p o s ite d in the U n iv e r s ity L ib rary t o be made a v a i l a b l e to borrowers under r u l e s o f the L ib rary .
B r ie f q u o ta tio n s from t h i s t h e s i s a re allow able w ith o ut s p e c i a l p erm issio n , p rovided t h a t a c c u ra te acknowledgment o f source i s made.
Requests f o r perm issio n f o r extended q u o ta tio n from o r re p ro d u c tio n o f t h i s m anuscript in whole o r in p a r t may be g ra n ted by th e head o f th e major department o r th e Dean o f th e Graduate College when in h i s ju d g ment the proposed use o f th e m a te r ia l i s i n th e i n t e r e s t s o f s c h o l a r s h ip . In a l l o t h e r i n s t a n c e s , however, perm ission must be o b ta in e d from th e a u th o r.
SIGNED:
srmmJU/
APPROVAL BY THESIS DIRECTOR This t h e s i s has been approved on th e d ate shown below:
I. RALPH MK RICHARD
P r o f e s s o r o f C iv il Engineering
ACKNOWLEDGMENT
The a u th o r wishes t o express h i s g r a t i t u d e t o Dr. R. M. Richard f o r h i s guidance and su g g estio n s which g r e a t l y aided i n th e development and p r e s e n t a t i o n o f t h i s t h e s i s . The au th o r i s a ls o in d eb ted t o Dr.
Richard fo r th e use o f th e a n a ly s is program used in t h i s stud y .
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS. ... V
LIST OF TABLES... v i
ABSTRACT. ... v i i
1. INTRODUCTION... 1
2. THE OPTIMIZATION PROBLEM. ... 4
Concept o f Design Space . ... 4
An Elementary Design P ro b lem . ... 8
3. OPTIMUM DESIGN... 14
O p tim ization P ro ced u re... 15
Numerical Example. ... 19
4. FULLY STRESSED DESIGN ... 24
Formulation o f th e F u lly S tr e s s e d Design Method... 25
C ondition o f O p t i m a l i t y . . . ... 26
5. NUMERICAL EXAMPLES. ... 28
Four Bar T r u s s ... 29
Three Bar T r u s s ... 31
Case 1 ... 31
Case 2 . . . . ... 33
Case 3 ... ■... 35
Twenty-Five Member Space T r u s s ... 35
Crossed Braced T r u s s ... 42
Wing Box S e c t i o n . ... 44
6. SUMMARY AND CONCLUSIONS. ... 50
APPENDIX A - FLOWCHART FOR COMPUTER PROGRAM OF THE FULLY STRESSED METHOD. ... 53
APPENDIX B - NOTATION. ... . . . 55
LIST OF REFERENCES... 57
SELECTED BIBLIOGRAPHY. ... 58 iv
LIST OF ILLUSTRATIONS
Figure Page
1. Three Dimensional Design Space w ith Planes o f Constant
Weight... ... ... ... 6
2. C o n s tra in t Surface f o r S in g le Load C o n d itio n ... 6
3. Typical C o n s tra in t Surface f o r M u ltip le Load C o n d itio n 7 4. S ectio n Through Minimum Weight Design P o i n t ... 7
5. C a n tile v e r Box Beam... ... 8
6. Design Space f o r Box Beam Example ... 10
7. U sab le-F e asib le S e c t o r ... 17
8. P a r t i a l Design Space f o r Box Beam ... 21
9. Four Bar Truss ... 30
10. Three Bar T ru s s, Case 1... 32
11. Three Bar T ru ss, Case 2 ... 34
12. Three Bar T r u s s , Case 3 ... 36
13. Twenty-Five Member T r u s s ... 37
14. Twenty-Five Member T ru ss. Areas > 0 . ... 40
15. Twenty-Five Member T r u s s , Areas > 0 . 1 ... 41
16. Cross Braced T r u s s ... 43
17. Wing Box S e c t i o n ... 45
18. Act/Allow S t r e s s e s . Wing Box S e c t i o n ... 47
v
LIST OF TABLES
Table Page
1. Loads on Twenty-Five Member T r u s s ... 35
2. Member S izes f o r Twenty-Five Member T r u s s ... 39
3. Loads on Cross Braced T r u s s ... 42
4. Cross Braced T r u s s , Member A reas... 44
5. Wing Element T h ic k n e ss e s ... 48
6. Thicknesses o f Elements No. 27 and 2-9... 49
v i
ABSTRACT
Methods o f s t r u c t u r a l s y n th e s i s a re p r e s e n te d and compared. A f u l l y s t r e s s e d design in which the members o f a s t r u c t u r e a re p ro p o r
tio n e d such t h a t th e s t r e s s in each member i s equal to th e allow able s t r e s s in a t l e a s t one load c o n d itio n i s n o t n e c e s s a r i l y a minimum-weight d esig n . A method i s o u t l i n e d to determ ine when a f u l l y s t r e s s e d design has minimum w eight.
The o p tim iz a tio n problem i s one o f d eterm ining a s e t o f design v a r i a b l e s so t h a t th e v alue o f the o b j e c t i v e f u n c tio n becomes optimum s u b je c t to c e r t a i n c o n s t r a i n t c o n d itio n s . I f a c e r t a i n s t r u c t u r a l design i s co n side red as a p o in t in an n-dim ensional hyperspace, o p tim iz a tio n can be thought o f as a search through t h i s hyperspace t o f in d th e p o in t
r e p r e s e n t i n g th e b e s t d e s ig n .
A program f o r design s y n th e s i s by th e i t e r a t i v e f u l l y s t r e s s e d method i s p r e s e n te d and th e r e s u l t s o f s e v e r a l num erical examples s t u d i e d .
v i i
CHAPTER 1
INTRODUCTION
S t r u c t u r a l design i s an i t e r a t i v e p ro ced u re. For a s t a t i c a l l y d ete rm in a te s t r u c t u r e in which th e s t r e s s i n one member i s not a f f e c t e d by the s t r e s s e s in any o t h e r member, a l l s t r e s s e s are determ ined
d i r e c t l y by s t a t i c s . The members are th e n p ro p o rtio n e d f o r the f o rc e s which a c t on them and th e design i s completed in one i t e r a t i o n .
For a s t a t i c a l l y in d e te rm in a te s t r u c t u r e , an i n i t i a l design i s made on th e b a s is o f ex p erien ce w ith s i m i l a r s t r u c t u r e s , " r u l e o f thumb", and perhaps some simple p r e lim in a r y c a l c u l a t i o n s . A d e t a i l e d a n a l y s is i s then made and th e s t r u c t u r e m odified as n e c e s s a r y . The m odified s t r u c t u r e i s then re -a n a ly z e d , th e r e s u l t s examined, and th e s t r u c t u r e m odified a g a in , and so on. I f th e s t r u c t u r e has "normal"
s t r u c t u r a l action* a s a t i s f a c t o r y desig n i s g e n e r a lly a r r i v e d a t a f t e r a few design c y c le s .
Normal s t r u c t u r a l a c tio n has been d e fin e d (Cross, 1936) in g en eral as th e a c tio n o f those s t r u c t u r e s in which i t i s p o s s i b l e to determ ine w itho u t re c o u rse to rig o ro u s in d e te rm in a te methods, the
approximate magnitude o f th e fo rc e s in a c t i o n , and in which the magnitude o f th e se fo rc e s i s a f f e c t e d com paratively l i t t l e by the r e l a t i v e s t r e s s i n t e n s i t y i n th e p a r t s o f th e s t r u c t u r e . A ll s t a t i c a l l y d eterm in a te and most o f th e c l a s s i c a l forms o f in d e te r m in a te s t r u c t u r e s a c t norm ally.
I f , however, th e geometry o f a s t r u c t u r e i s such t h a t two o r more p a r t s p a r t i c i p a t e i n c a r ry in g loads to such an e x te n t t h a t i f the s t r e n g t h o f one p a r t i s changed, th e fo rc e s a c t i n g on th e o t h e r p a r t s are l a r g e l y a f f e c t e d , th e s t r u c t u r e w i l l have "hybrid" a c tio n and i t may take a g r e a t many design cycles to a r r i v e a t an e f f i c i e n t d esig n . C e r ta in ly th e r e i s some i n t e r a c t i o n in a l l s t a t i c a l l y in d e te rm in a te s t r u c t u r e s so th e d i f f e r e n c e between normal and h y b rid a c tio n i s one o f degree only and the number o f design i t e r a t i o n s may vary over a wide range.
A nalysis i s th e most d i f f i c u l t and time consuming p a r t o f the design c y c le . However, th e development o f th e h ig h -sp e ed d i g i t a l computer with i t s a b i l i t y to handle la rg e amounts o f num erical d a ta in a very s h o r t time has r e l i e v e d th e design e n g in eer o f many hours o f tiresom e la b o r w ith desk c a l c u l a t o r o r s l i d e r u l e and made i t p o s s ib le to solve many problems w ith a degree o f accuracy t h a t would have been im p o ssib le a few y ears ago. Methods o f f i n i t e elements and the use o f m a trix a lg e b ra a re w e ll s u i t e d to d i g i t a l computers. Using th e m a trix displacem ent method (P rz e m ie n ie c k i, 1968) th e computer can be used f o r th e e n t i r e a n a l y s is procedure from th e i n i t i a l in p u t o f d a t a , through c a l c u l a t i o n o f element s t i f f n e s s p r o p e r t i e s , g e n e ra tio n o f th e o v e r a l l s t r u c t u r a l s t i f f n e s s m a trix , s o l u t i o n o f th e sim ultaneous e q u a t i o n s , to
f i n a l o u tp u t which may in clu d e s t r e s s and fo rc e d i s t r i b u t i o n s , d e f l e c t i o n s , and in flu e n c e c o e f f i c i e n t s .
Since th e h ig h -sp e ed d i g i t a l computer has proven to be such a v a lu a b le t o o l f o r a n a l y s i s , th e ne x t l o g i c a l s te p i s th e com pletely
- 3 automated g e n e ra tio n o f a s t r u c t u r e , t h a t i s s t r u c t u r a l s y n t h e s i s . Work done in t h i s a r e a in clu d es t h a t by Schmit and h i s a s s o c i a t e s (1960, 1963, 1964, 1965), G e l l a t l y and G allag h e r (1964, 1966), Razani (1965), and o t h e r s .
The purpose o f t h i s t h e s i s i s t o p r e s e n t th e r e s u l t s o f a stu d y o f s t r u c t u r a l s y n th e s is methods. An o p tim iz a tio n procedure i s given and an i t e r a t i v e f u l l y s t r e s s e d design method i s p re s e n te d and a p p lie d to s e v e r a l t r u s s problems and to an a ir p la n e wing s e c t i o n .
The symbols used in t h i s t h e s i s are d e fin ed in Appendix B.
CHAPTER 2
TOE OPTIMIZATION PROBLEM
The o b j e c t i v e o f any s t r u c t u r a l d e s ig n e r i s to design a
s t r u c t u r e t h a t w i l l s a f e l y and econom ically sup p o rt any system o f loads t h a t the s t r u c t u r e may be s u b je c te d t o , w hile rem aining w ith in c e r t a i n l i m i t s o f s t r e s s , d e f l e c t i o n , and s t a b i l i t y . S a fe ty i s o f primary im portance, b u t economy i s a ls o a major c o n s id e r a t i o n .
For a given s t r u c t u r e , th e most economic design i s g e n e r a lly th e minimum weight d esig n . This i s q u i t e obvious in aerospace s t r u c t u r e s , where every pound saved on v e h ic le weight means added pay lo ad , b u t i t a p p lie s e q u a lly w ell to most o t h e r types o f s t r u c t u r e , where c o n s tr u c tio n c o s ts can be f ig u r e d on a p e r pound b a s i s .
Concept o f Design Space
A given s t r u c t u r a l design can be co n sid e red as a p o i n t in an n-dim ensional hyperspace where each dimension r e p r e s e n t s a design v a r i a b le . The design o f a given s t r u c t u r e can be thought o f as a se arc h through t h i s hyperspace to f in d the optimum p o in t r e p r e s e n t i n g th e b e s t d esig n .
Design v a r i a b l e s can be member s i z e s , v a r i a b l e dim ensions, v a r i able m a te r ia l p r o p e r t i e s , and d i f f e r e n t s t r u c t u r a l systems o r t y p e s . In concept any o f th e s e may be used as v a r i a b l e s in an automated d esign p ro c e d u re , however the type o f s t r u c t u r e and the m a t e r ia l are g e n e r a lly
5
s p e c i f i e d ahead o f time and the geometry i s o f te n s e t by th e requirem ents o f th e s t r u c t u r e which leaves the s i z e s o f th e member as the most d e s i r ab le param eters to u se.
I f c r o s s - s e c t i o n areas o r member th ic k n e ss e s a re chosen as p a r a meters and are d e s ig n a te d as x^, x^, th e weight o f the s t r u c t u r e i s a l i n e a r f u n c tio n o f the design param eters and can be ex p ressed by the eq u atio n
W » C1 x1 ♦ C2 x2 ♦ ... cn xn
where C^, C2> ... are c o n s ta n ts depending on th e geometry o f the s t r u c t u r e and the d e n s ity o f th e m a t e r i a l . I t i s th e weight W we wish to minimize. Planes o f co n s ta n t-w e ig h t f o r a th re e -d im e n s io n a l design h y per
space are shown in Figure 1.
The v a r i a b l e s x^, x2 , ... xfi are s u b je c t to c e r t a i n c o n s t r a i n t s such t h a t u n s a t i s f a c t o r y b eh av io r in each load c o n d itio n i s p rev en ted . There may be f u n c tio n a l c o n s t r a i n t s such as a s p e c i f i e d t a p e r in the th ic k n e ss o f th e skin o f an a i r p la n e wing. Geometric c o n s t r a i n t s may be based on allow able minimum an d /o r maximum a re as o f th ic k n e s s e s , and f a b r i c a t i o n l i m i t a t i o n s . Maximum valu es can a lso be p lace d on d i s p l a c e ments and t e n s i l e and compressive s t r e s s e s . In a d d i t i o n , c o n s t r a i n t s may be r e q u ire d to co n fin e th e design to th e reg io n where th e methods o f
a n a ly s is employed are v a l i d . Figure 2 shows a c o n s t r a i n t s u rf a c e f o r a s i n g l e load c o n d itio n w hile Figure 3 shows the more ge n eral case o f a c o n s t r a i n t s u rf a c e f o r a m u ltip le load c o n d itio n . Side c o n s t r a i n t s such t h a t x^, x2 , and x^ remain p o s i t i v e are a ls o in d i c a t e d in Figures 2 and 3.
Figure 4 shows a s e c t io n through a minimum w eight desig n p o i n t .
6
Figure 1. Three Dimensional Design Space w ith Planes o f C onstant Weight
Side C o n s tra in t
x3
F igure 2 . C o n s tr a in t S u rfa ce f o r S in g le Load C o n d itio n
7
Side C o n s tra in t
x3
Figure 3. Typical C o n s tra in t Surface f o r M u ltip le Load C ondition
C o n s tra in t S urface
Minimum Weight Design P o int
F igure 4 . S e c t io n Through Minimum W eight D esign P o in t
8 An Elementary Design Problem
To i l l u s t r a t e th e concept o f design sp ac e, the fo llo w in g example i s p r e s e n te d .
Consider a c a n t i l e v e r box beam w ith a uniform cro ss s e c t io n (see Figure 5) s u b je c t to a c o n c e n tra te d load P o f 5,000 pounds a p p lie d a t th e end, 12 inches from the lo n g i t u d i n a l a x i s . Let the len g th L equal
F igure 5. C a n tile v e r Box Beam
48 in c h e s , the modulus o f e l a s t i c i t y , E * 30,000,000 pounds p e r square in c h , the modulus o f e l a s t i c i t y in s h e a r , G * 11,500,000 pounds p e r square in c h , th e d e n s ity P eq uals 0.283 pounds p e r cubic in c h . Let the depth h o f th e box s e c t i o n equal 6 inches and the width b equal 3 in c h e s . Let the w all th ic k n e s s e s be denoted x^ and Xg as in d i c a t e d in Figure 5.
Find Xj and x^ and the w eight p e r inch W o f th e minimum weight design such t h a t :
tv ^
9 1. and Xg a re g r e a t e r than o r equal to 0.125 i n c h e s .
2. The bending s t r e s s ( te n s io n o r compression) i s le s s than o r equal to 20,000 pounds p e r square in ch .
3. The s h e a rin g s t r e s s ay i s le s s than o r equal to 13,000 pounds p e r square in ch .
4. The v e r t i c a l d e f l e c t i o n a t th e p o i n t o f a p p l i c a t i o n o f th e load i s l e s s than o r equal to 0.33 in c h e s .
Consider " S t. Venant t o r s i o n " on ly. The m a te r ia l and the o v e r
a l l dimensions have been p re a s s ig n e d so t h a t leaves only th e th ic k n e ss e s Xj and x^ as v a r i a b l e s to be determ ined.
The reg io n o f a l l p o s s i b l e p o s i t i v e values o f x^ and x^ can be viewed g e o m e tric a lly in Figure 6. The re g io n i s immediately reduced by exclu din g values o f x^ and X2 le s s than 0.125 inches ( l i n e s a - a ) .
The s h e a r s t r e s s e s in a clo se d r e c t a n g u l a r s e c t i o n s u b je c t to t o r s i o n a re p a r a l l e l to the s id e s o f th e s e c t i o n . The s h e a r flow around th e s e c t io n has a uniform v alue and assuming t h a t x^ w i l l be le s s than X2 , th e maximum s h e a r s t r e s s due to t o r s i o n w i l l be
° v t -
mnq
( 2 - 2)where M is the ap p lie d t o r s i o n a l moment, P x e.
There w i l l a ls o be s h e a rin g s t r e s s due to th e v e r t i c a l load P, equal to
°vb “
z f h q
( 2 - 3)inches
10
. 0
0 . 8
20,000 p s i
0 . 6 def.** 0.33 inches
d e f .= 0.25 inches
0 .4
lbs
13,000 p s i 0 . 2
0
0 . 8 0 . 6
0.4 0 . 2
0
Xg, inches
F ig u re 6 . D esign Space f o r Box Beam Example
which w i l l give a t o t a l s h e a r s t r e s s o f
Mt p
av ~
mnq
* 2h (2*4)Solving Eq. (2.4) f o r y i e l d s
x = (2.5)
1 2 a EFT
v
S u b s t i t u t i n g the given num erical values = 60,000 i n - l b s (5,000 l b s . x 12 i n . ) , b ** 3 i n . , P ■ 5,000 l b s . , o^ = 13,000 I b s / s q . i n . , and h = 6 inches in t o Eq. ( 2 . 5 ) , the l i n e along which the s h e a rin g s t r e s s eq u als 13,000 pounds p e r square inch ( l i n e b-b) i s given by the e x p re ssio n
*1 * K W '-x- T Y T = - 16 i n ' <2 - 6 >
The region below l i n e (b-b) i s excluded in o r d e r to avoid o v e r
s t r e s s due to s h e a r.
Bending s t r e s s can be expressed as
°b ' TT - (2-7)
The moment o f i n e r t i a I i s expressed in terms o f x^ and X2 by , 3 . , 2
x. h bx9 h
I = , ( 2 . 8 )
c equals .5h and th e bending moment M equals P times L. S u b s t i t u t i n g th e se values f o r I , M and c in t o Eq. (2.7 ) y i e l d s
— n— j a ».»)
12 S u b s t i t u t i n g the a p p r o p ria te num erical v alues and s o lv in g Eq. (2.9) f o r
in terms o f y i e l d s the l i n e (c - c ) , 3 x ,
x x = 1 - ( 2 . 1 0 )
along which th e bending s t r e s s equals th e maximum allow able value o f 20,000 pounds p e r square in ch . The reg io n below and to the l e f t o f l i n e (c - c ) , i s excluded in o r d e r to avoid o v e r s t r e s s due to bending.
The d e f l e c t i o n i s given by
d ' 3ET + “iT T ' ( 2 a l )
The t o r s i o n a l c o n s ta n t J i s ex pressed in terms o f x^ and x2 by
, 2 , 2
J = 2 *i x2 b h (2.12)
b x2 + h Xj
The d e f l e c t i o n can now be ex p ressed in terms o f x^ and x2 by s u b s t i t u t i n g Eqs. (2 .8 and 2.12) i n t o Eq. ( 2 .1 1 ) .
3 Pe2L(bx + h x )
d = -^ --- 2 y - -v - - (2.13)
E(xx ti > 3bx2 h ) 2Gx1 x2 bz
S u b s t i t u t i n g the given num erical values and s o lv in g Eq. (2.13) f o r x^ in terms o f x2 gives th e q u a d r a t i c eq uatio n
(47.5 x2 - 4) Xj2 + (71.3 X2 - 32.56 x^) Xj - 3 X2 = 0. (2.14)
By assuming v alues f o r x2 and s o lv in g Eq. (2.14) f o r x^ y i e l d s the l i n e (d - d) along which the d e f l e c t i o n equals .33 in c h e s . The a re a below and to the l e f t o f l i n e (d - d) i s excluded because o f e x c e ssiv e d e f l e c t i o n .
13 The weight o f th e r e c t a n g u l a r box beam i s expressed as follow s:
W = 2p(hXj + bx2) . (2.15)
TTie l i n e along which the weight p e r inch W equals two pounds i s given by the eq u atio n
xx
= 0.59 - 0.50 x2 . (2.16)This contour (W = 2 l b s . ) i s p l o t t e d as l i n e (e - e) in Figure 6. A second weight contour (W = 1.5 l b s . ) i s p l o t t e d as l i n e ( f - f ) .
I t i s app are n t from Figure 6 t h a t th e minimum weight design
s a t i s f y i n g th e v ario u s s t a t e d l i m i t a t i o n s l i e s a t p o in t X where x^ = .16 in c h , x2 = .56 in c h , and W = 1.50 pounds p e r in ch . Figure 6 i s a
geom etric r e p r e s e n t a t i o n o f t h i s simple two v a r i a b l e optimum design problem. P l o t t i n g the c o n s t r a i n t s and contours o f c o n s ta n t weight makes i t p o s s i b l e to scan the e n t i r e s e t o f p o s s i b l e design p o in ts in th e x^, x2 design space and immediately seek out the optimum design a t P o in t X.
I t should be noted t h a t P o in t X l i e s a t the v e r t e x o f n = 2 maximum s t r e s s c o n s t r a i n t s u rf a c e s (c^ = 20,000 p s i f o r the fla n g e s and
0 ^ = 13,000 p s i f o r the webs) so P o in t X i s a f u l l y s t r e s s e d as w ell as a minimum weight d esig n . However, i f th e d e f l e c t i o n had been l im ite d to 0.25 inches as i n d i c a t e d by l i n e (g - g ) , Figure 6, th e minimum weight design would be a t P o in t Z, x^ = 0.19 inch and x2 = 0.70 inch and would n o t be a f u l l y s t r e s s e d design.
While the concept o f p l o t t i n g th e c o n s t r a i n t s and contours o f Figure 6 may be u s e f u l , th e method i s n o t p r a c t i c a l f o r design problems having more than th re e design v a r i a b l e s .
CHAPTER 3
OPTIMUM DESIGN
Given enough time and money any design can be op tim ized . The o b je c t o f s t r u c t u r a l s y n th e s is however i s to a r r i v e a t an optimum design as q u ic k ly and e f f i c i e n t l y as p o s s i b l e . Many methods to achieve t h i s end have been form ulated from r e p e a te d a p p l i c a t i o n o f f u l l y s t r e s s e d design to c o n s id e ra tio n o f the s t r a i n energy d i s t r i b u t i o n in the s t r u c t u r e . The i n t e r e s t e d r e a d e r i s r e f e r r e d to th e b ib lio g ra p h y .
The o p tim iz a tio n problem i s one o f fin d in g the v e c t o r D c o n s i s t ing o f the s e t o f design v a r i a b l e s x ^ (i = l , n ) such t h a t the o b j e c t i v e fu n c tio n M(D) i s minimized s u b je c t to a system o f i n e q u a l i t y c o n s t r a i n t s
gj(D) > 0 j = 1 , 2 , . ...m (3 .1 )
where n equals th e number o f design v a r i a b l e s and m eq uals the number o f c o n s t r a i n t s . The c o n s t r a i n t s , gj(D ), are such t h a t u n s a t i s f a c t o r y
b e h a v io r in each f a i l u r e mode i s p reven ted and the range o f values o f th e design v a r i a b l e s are k ep t w ith in acce p ta b le l i m i t s .
In th e n-dim ensional design hyperspace (Figure 1) each equation gj(D) = 0 r e p r e s e n ts a s u rf a c e d iv id in g th e space i n t o two p a r t s , one o f which s a t i s f i e s gj(D) > 0. The f e a s i b l e domain i s t h a t reg io n o f the n -dim ensional space in which a l l the c o n s t r a i n t s (Eq. 3.1) are s a t i s f i e d . The s o lu t i o n i s a p o i n t in the f e a s i b l e domain p ro v id in g a minimum v alue o f the o b j e c t i v e fu n c tio n .
14
15 O p tim izatio n Procedure
A g r a d ie n t p r o j e c t i o n method o f o p tim iz a tio n w i l l be p r e s e n te d . The s o lu t i o n of an o p tim iz a tio n problem r e q u ir e s an i n i t i a l design as a s t a r t i n g p o i n t . Any p o i n t , f e a s i b l e o r n o t , may be used f o r th e i n i t i a l t r i a l d esign . However, i t i s d e s ir a b l e to use a reaso nab le one such as a f u l l y s t r e s s e d design (Chapter 4 ). The f u l l y s t r e s s e d design may t u rn out to be the optimum o r very c lo se to i t , in which case a saving in computer time can be r e a l i z e d .
S t a r t i n g from a given design p o i n t , s te p s a re taken le a d in g to a reduced value o f th e o b j e c t i v e f u n c tio n .
Dq+1 = Dq ♦ a ; q (3.2)
The d i r e c t i o n o f movement ( i q) and s te p le n g th ( a ) in each i t e r a t i o n depends on the l o c a t i o n o r th e c u r r e n t p o in t w ith r e s p e c t to the con
s t r a i n t s u r f a c e s .
From a c u r r e n t design p o in t in th e f e a s i b l e domain which does n o t l i e on any c o n s t r a i n t s u rf a c e a s t e e p e s t d escen t p ath i s used. The d i r e c t i o n v e c t o r in t h i s case i s th e n e g a tiv e o f th e u n i t g r a d ie n t o f the o b j e c t i v e f u n c tio n .
s q = - VM(D) (3.3)
The g r a d ie n t o f a f u n c tio n has the d i r e c t i o n o f the normal to a le v e l su rf a c e o f the fu n c tio n taken in the sense o f in c r e a s in g v a lu e . There
fo re to minimize M(D) the n e g a tiv e o f th e g r a d ie n t i s used.
16 The s te p le n g th from the c u r r e n t design p o in t to a new design p o in t i s s e t by moving in th e d i r e c t i o n i n d i c a t e d u n t i l a c o n s t r a i n t s u rf a c e i s encountered. The i n t e r s e c t i o n i s found by the method o f Binary Chopping, i . e . , a s te p i s ta k e n , the c o n s t r a i n t s a re checked and i f none are v i o l a t e d an o th e r s te p i s tak en , doubling th e i n t e r v a l from the l a s t s t e p . The p ro cess i s re p e a te d u n t i l one o r more c o n s t r a i n t s are v i o l a t e d , th e l a s t s te p len g th i s then h a lv e d , the v i o l a t e d con
s t r a i n t s checked and a p p r o p ria te c o r r e c t i o n s made u n t i l the i n t e r s e c t i o n i s found.
M athem atically th e method o r b in a ry chopping can be ex p ressed :
a = (2^ - l ) a 0 i = 1 , 2 , 3 . . . (3.4)
where a Q i s the len g th o f th e f i r s t t r i a l s te p and i s dependent on the range o f v alu es o f th e design v a r i a b l e s o f a given problem.
For a c u r r e n t design p o in t lo c a te d on a boundary s u r f a c e , one o r more c o n s t r a i n t s are e q u a l i t y s a t i s f i e d (g^(D) = 0 ). These c o n s t r a i n t s are a c t i v e c o n s t r a i n t s a t t h a t p o i n t . The d i r e c t i o n s^ must not v i o l a t e the a c t i v e c o n s t r a i n t s a t th e c u r r e n t design p o in t D^. That i s i t must be f e a s i b l e , thus
s^'V gj (D^) < 0 (3.5)
where j denotes the c r i t i c a l c o n s t r a i n t s . Also, in o r d e r to decrease the value o f the o b j e c t i v e f u n c tio n , the d i r e c t i o n s q must s a t i s f y
s q *VM(Dq ) < 0 (3.6)
17 t h a t i s Sq must be a u sa b le d i r e c t i o n . Figure 7 shows th e u s a b le -
f e a s i b l e s e c t o r f o r a two dimension design space f o r an o b j e c t i v e fu n c tio n M(D) w ith one a c t i v e c o n s t r a i n t , gj(D) = 0.
The f e a s i b l e d i r e c t i o n o f s t e e p e s t descent w i l l be used f o r sq and i s o b tain e d by sweeping out o f the d i r e c t i o n o f s t e e p e s t d e s c e n t, the components along th e g r a d ie n ts o f th e c o n s t r a i n t s t h a t are a c t i v e a t p o in t Dq . The sweeping process i s ord ered in accordance w ith th e degree
Us a b l e - f e a s i b l e s e c t o r
Figure 7. U s ab le-F e asib le S ecto r
to which th e c o n s t r a i n t s l i m i t movement. This makes i t p o s s i b l e to f in d th e descent d i r e c t i o n c l o s e s t to the n e g a tiv e o f th e g r a d ie n t o f the o b j e c tiv e fu n c tio n and avoid sweeping i n e f f e c t i v e a c t i v e c o n s t r a i n t s .
Consider a g en eral s te p o f sweeping beginning with a u n i t d i r e c t i o n v e c to r s? which has i- 1 c o n s t r a i n t s swept o u t, b u t s t i l l may v i o l a t e k remaining a c tiv e c o n s t r a i n t s , w ith u n i t g r a d ie n ts Vgr , ( r = l , k ) .
18 The s c a l a r p r o d u c t, s9 * Vg^, i s c a l c u l a t e d f o r each o f th e remaining g r a d ie n ts o f th e c o n s t r a i n t s . The most n e g a tiv e prod u ct gives the c o n s t r a i n t g r a d ie n t Vg^ which most c o n s tr a in s movements in th e s?
d i r e c t i o n . I f s? • Vg^ i s - 1 , a t l e a s t a lo c a l minimum o r sad d le p o in t has been reached and f u r t h e r re d u c tio n o f th e o b j e c t i v e f u n c tio n i s not p o s s i b l e .
I f s? • Vg^ i s between -1 and zero , th e c o n s t r a i n t g r a d ie n t Vg^
i s swept from th e d i r e c t i o n v e c t o r s^ to give a new d esce n t d i r e c t i o n which does n o t v i o l a t e th e c o n s t r a i n t g^.
s i * i * 5 i - v V 5 i '
( 3 . 7 )
■ unit n . i
In o r d e r t h a t f u r t h e r sweeping o p e ra tio n s w i l l no t r e in tr o d u c e components o f th e descen t d i r e c t i o n in th e d i r e c t i o n d efin ed by Vg^, i t i s swept from the k-1 remaining a c t i v e c o n s t r a i n t g r a d ie n ts a l s o .
Gr = Vgr - Vgw(Vgr • Vgw)
( 3 . 8 ) gr - u n i t Gr
The sweeping p ro cess i s co ntinued u n t i l a l l a c t i v e c o n s t r a i n t g r a d ie n ts have been swept o u t o f i j .
I f 5? • Vg^ i s p o s i t i v e , no a c t i v e c o n s t r a i n t s w i l l be v i o l a t e d and:
I t » 5? . ( 3 . 9 )
19 When th e component o f the d i r e c t i o n v e c to r along the g r a d ie n t o f an a c tiv e c o n s t r a i n t i s swept o u t, the r e s u l t i n g v e c t o r (th e f e a s i b l e d i r e c t i o n o f s t e e p e s t descent) i s ta n g e n t to t h a t c o n s t r a i n t s u r f a c e . I f the c o n s t r a i n t s u rf a c e i s curved, movement in th e d i r e c t i o n o f th e ta n g e n t may lead to a v i o l a t i o n o f the c o n s t r a i n t . In t h i s case o r in g en eral i f one o r more c o n s t r a i n t s a re v i o l a t e d a t an occupied p o i n t , movement back to the f e a s i b l e domain tak es p lac e in th e d i r e c t i o n o f the r e s u l t a n t o f the u n i t g r a d ie n ts o f th e v i o l a t e d c o n s t r a i n t s . The method o f b in a ry chopping i s used to a d j u s t the s te p len g th u n t i l no c o n s t r a i n t i s v io la te d and one o r more o f th e p r e v io u s ly v i o l a t e d c o n s t r a i n t s are a c t i v e . This method o f moving from th e n o n f e a s ib le to th e f e a s i b l e domain a ls o makes i t p o s s i b l e to s t a r t from a n o n f e a s ib le i n i t i a l design p o i n t .
Using t h i s p roced u re, when th e d i r e c t i o n v e c t o r becomes zero , a minimum has been reached and th e p ro cess ends. I f a l l o f the c o n s t r a i n t fu n c tio n s and th e o b j e c t i v e fu n c tio n s are convex, then th e above i s
s u f f i c i e n t to e s t a b l i s h the c o n s tra in e d optimum as a glo b al optimum. For nonconvex problems th e r e s u l t i n g s o lu t i o n may be a l o c a l and n o t a
g lobal minimum. However, working a problem s e v e r a l tim es s t a r t i n g from widely s e p a ra te d i n i t i a l designs should i n d i c a t e which s o lu t i o n i s g lo b a l.
Numerical Example
To i l l u s t r a t e t h i s procedure f o r s o lv in g o p tim iz a tio n problems, co n s id e r again the box beam with an e c c e n t r i c load shown in Figure 5.
The o b je c t fu n c tio n comes from Eq. (2.15) and can be w r i t t e n :
M(D) » W « 1.7 x2 ♦ 3.4
xl
. (3.10)20 Only the two l i m i t i n g c o n s t r a i n t s (Eqs. 2 .6 , 2.10) w i l l be considered h e r e :
gj(D) * Xj - 0.16 > 0
g2 (D) = 1.5 + Xj - 1 > 0 . (3.11)
P a r t o f Figure 6 has been reproduced in Figure 8 to r e p r e s e n t th e problem g e o m e tric a lly . The f e a s i b l e domain i s the a r e a to th e r i g h t and above th e c o n s t r a i n t l i n e s g^ and g ^ .
The p o in t D = ( .5 , .5) i s chosen as th e i n i t i a l d esig n . The value o f th e o b j e c t i v e fu n c tio n a t t h i s p o in t i s 2.55 . E v a lu a tin g g^ and
g2 a t D* i n d i c a t e s p o i n t D* i s in th e f e a s i b l e domain and not on any c o n s t r a i n t s , t h e r e f o r e the u n i t d i r e c t i o n v e c to r i s
s 1 = - VMCD1) = (-0 .4 4 7 , - 0 .8 9 5 ) . (3.12)
Taking th e length o f the f i r s t t r i a l s te p aQ, to be 0.1 and moving in the d i r e c t i o n o f s* (Eq. 3.2) th e p o in t (0.455, 0.410) i s reached. E v a lu a tin g th e c o n s t r a i n t f u n c tio n s a t t h i s p o in t y i e l d s gj = + 0.25 and g2 = + 0.93 which i n d i c a t e s n e i t h e r i s v i o l a t e d . There
fo re th e i n t e r v a l i s doubled, th e t r i a l s te p becomes 0 .3 and the p o in t (0.366, 0.231) i s checked. The values o f g^ and g2 are + .071 and - .22 r e s p e c t i v e l y . The n e g a tiv e value i n d i c a t e s c o n s t r a i n t g2 i s v i o l a t e d , so the le n g th o f th e l a s t i n t e r v a l i s halved (a = 0.2) to check i n t e r
mediate p o i n t s . The i t e r a t i o n co ntinues in t h i s manner u n t i l the p o in t (0.429, 0.358) i s reached where th e v alue o f g2 i s zero , i n d i c a t i n g g2 i s a c t i v e a t t h a t p o i n t .
F ig u re 8
21
6
.5
(0.455, 0.410) 4
D = (0 .4 2 9 , 0.358)
•H .03
1 -4 *3
(0.366, 0.231) . 2
. 1 3 4 .5 . 6 .7
gi (D)
x2 , in c h e s
P a r t i a l D esign Space f o r Box Beam
22 The p o i n t = (0.429, 0.358) i s th e new design and the value of the o b j e c t i v e f u n c tio n a t t h i s p o in t i s 1.95.
The n e g ativ e o f th e u n i t g r a d ie n t i s again (-0 .4 4 7 , -0.895) and f o r the sweeping p ro c e ss i s denoted by 5^. The s c a l a r p ro du ct s^ • Vg^
( th e p r o j e c t i o n o f 5^, along Vg2) i s e v a lu a te d and i s - .8 6 9 . _ 2 The negative s ig n i n d i c a t e s g2 must be swept from s^ in o rd er to p re v e n t v i o l a t i o n o f c o n s t r a i n t g2 . A p p lic a tio n o f Eq. (3.7) gives
52 » (-0 .4 4 7 , -0.895) - ( - 0 .8 6 9 )( 0 .8 3 2 , 0.555)
= (0 .2 7 5 ,-0 .4 1 3 ) (3.14)
and i 2 = u n i t S2 o r (0 .5 5 5 , - 0 .8 3 2 ) .
Taking a s te p o f len g th .238 (determ ined by b in a r y chopping as above) in t h i s d i r e c t i o n , th e p o in t a r r i v e d a t i s
D3 = (0.429, 0.358) + 0 .2 3 8 (0 .5 5 5 , -0.832) = (0 .5 6 , 0 .1 6 ) . (3.15) The value o f th e o b j e c t i v e fu n c tio n a t t h i s p o in t i s 1.50. The v alu es o f g^ and g2 are both zero a t D3 so both c o n s t r a i n t s are a c t i v e . The u n i t g r a d ie n ts a t D3 are Vg^ = (0, 1 ), Vg2 * (0.832, 0 .5 5 5 ), and -VM(D3) « s 3 = (-0 .4 4 7 , - 0 .8 9 5 ) . For n o n lin e a r f u n c tio n s Vg^, Vg2 , and s^ would vary from p o in t to p o i n t in s t e a d o f remaining c o n s ta n t as in t h i s example.
The p r o j e c t i o n o f s 3, along v e c to r Vg^ i s equal to -0.895 while i t s p r o j e c t i o n along Vg2 i s equal to -0 .8 6 9 . Therefore the component o f s 3 along Vg^ must be swept out f i r s t .
53 = (-0 .4 4 7 , -0 .895) - (0, 1 ) [ ( - 0 . 4 4 7 , -0.895) • (0, 1)]
= (-0 .4 4 7 , 0 .0 ) (3.16)
23 i 2 » u n i t § 2 = ( -1 , 0 ) . Next th e component o f Vg^ i s swept from Vg2 .
G2 * (0.832, 0.555) - (0, 1 ) [( 0 .8 3 2 , 0.555) • (0, 1)]
= (0.832, 0) (3.17)
g2 = u n i t G2 » (1, 0 ) .
The component o f s^ along g2 i s found from t h e i r s c a l a r products to be - 1 .0 . T herefore sweeping t h i s component from s^ w i l l y i e l d
§3 = (-1-, 0) - (1. 0 ) [ ( - l , 0) . (1, 0 )] = (0, 0) (3.18)
which i n d i c a t e s a minimum has been reached and th e problem i s solved.
The r e s u l t s are i d e n t i c a l to those found in Chapter 2.
CHAPTER 4
FULLY STRESSED DESIGN
I f constraints are placed only on element stresses for a struc
ture under multiple load conditions, an ite ra tiv e procedure can be used to redesign the structure so that each element reaches lim iting stress under at least one load condition. I f analysis indicates that a given element is overstressed in a c r itic a l load condition, the fu lly stressed design increases the area or thickness of that element s u ffic ie n tly to remove the overstress. I f a member is understressed, the opposite cor
rection is made. The design variables for this ite ra tiv e fu lly stressed design converge to a vertex of the n hypersurfaces representing n con
straints on the stresses.
In s ta tic a lly indeterminate structures, i f the area or stiffness of a certain element is increased, there is a tendency for that element to a ttra ct a larger portion of the load. Therefore several redesign cycles are required to obtain a fu lly stressed design.
The ite ra tiv e fu lly stressed design method usually changes the configuration of the structure considerably, for the f ir s t few cycles and then the changes in configuration become less violent and the difference between two successive designs becomes small.
I f convergence is slow, the repetition of analysis and fu lly stressed redesign may tend to eliminate some members of the structure and result in a simpler structure. However, to prevent this elimination
24 .
25 of lig h tly loaded, but necessary members such as secondary or bracing members, minimum allowable areas or thicknesses can be added as con
s traints. The resulting design w ill have certain members reduced to the allowable minimum size and the rest of the members w ill be fu lly stressed.
For the redesign of an axial force element such as a bar, the reference stress is merely the axial tensile or compressive stress. For compression members a lower stress can be used and the individual member designed to give a satisfactory fc/r ra tio for that stress. For plate elements where multiple stresses are generated (a , o , t ) , the
x y xy
stresses are combined in some manner (such as Von Mises) to give a single reference stress.
Formulation of the Fully Stressed Design Method
Starting with an i n i t i a l vector of design variables D°, the stress in each element is found for each load condition. The structure is then redesigned and the area (or thickness) of the i ^ 1 element becomes
x ? = x ? x ( 4 . 1 )
1 1 a ll
where a ., is the actual stress in element i due to the c r itic a l load ik
condition k and a . . is the allowable stress in element i . a ll
The difference in area of two successive designs is
Axi = Xi +* " xi (4.2)
for the i**1 element. Due to this change in area there w ill be a
redistribution of stresses necessitating another redesign. After several such cycles this procedure should converge to a fu lly stressed design.
26 Condition of Optimality
Unless deflection or other constraints control a design, for s ta tic a lly determinate structures, the fu lly stressed design is the minimum-weight design. For s ta tic a lly indeterminate structures i t can generally be concluded that for rapidly converging ite ra tiv e fu lly stressed designs such as in the case of structures with normal action, the results are most lik e ly optimum. For slowly converging iterations as in the case of structures with hybrid action, the result are less lik e ly to be optimum.
Razani (1965) has formulated the following method for v e r if i
cation of optim ality of a fu lly stressed design.
For a structure of n elements, le t B be an nxn design variation matrix. At the fin a l point of convergence the numerical value of the elements of matrix B are found as follows:
The area of the i ^ member is increased by a small amount Ax^
and a new analysis and fu lly stressed redesign is made. Element b ^ of matrix B is given by
- %°) b j i = ^ A 3 T ^ -
where x^ and x? are the new and old areas of the j ^ element, respective-
J 3
ly. This procedure is carried out for i = 1 ,2 ,3 , n.
After determination of matrix B, optimality coefficients X are found.
X = (I - B1) " 1 pL (4.4)
Where I is an nxn unit matrix, B* is the transpose of matrix B, p is the unit weight of the structural m aterial, and L is a 1 x n matrix of element lengths for bar elements or areas for plate elements. I f a ll of the optimality coefficients are positive, then the fu lly stressed design is also the minimum-weight design.
CHAPTER 5
NUMERICAL EXAMPLES
A program f o r design s y n th e s is by th e i t e r a t i v e f u l l y s t r e s s e d method i s p r e s e n te d and s e v e r a l num erical examples s tu d i e d .
The program (see Flow chart, Appendix A) i s w r i t t e n around an a n a ly s is program which uses th e d i r e c t s t i f f n e s s method and co n tain s fo u r types o f e le m e n ts :
1. P i n - j o i n t e d b a r element
2. T r ia n g u la r p l a t e element ( in - p la n e fo rc e s ) 3. R ecta ng u lar p l a t e element ( in - p la n e fo rc e s ) 4. Spring elem ent.
For th e t r i a n g u l a r and r e c t a n g u l a r p l a t e elem en ts, th e m u l t i p l e s t r e s s e s ox , a , and g n e ra te d are combined to give a s in g l e e f f e c t i v e s t r e s s u sing th e Von Mises formula:
f 2 2 _ 2 . 1 / 2
° e f f " x + ay ~ °x ay + t xy ' (5 .1 )
Various v alu es f o r th e i n i t i a l design v e c to r were t r i e d b ut had l i t t l e e f f e c t on speed o f convergence. For t h i s reason and a ls o to p ro vid e a b e t t e r b a s i s f o r comparison, a l l elements are given an i n i t i a l
a re a o r th ic k n e s s o f u n i t y .
For convergence o f th e f u l l y s t r e s s e d d e s ig n , th e follow ing e x i t c o n d itio n i s used , f o r th e j**1 i t e r a t i o n :
0.99 > A 4 " 1
> 1 .0 1 . i = 1 , 2 , n (5.2)
28
29 The re d e sig n i s c a r r i e d on f o r a maximum o f te n i t e r a t i o n s . I t i s f e l t t h a t by t h i s p o in t any design t h a t i s optimum w i l l converge and meet th e c r i t e r i a o f ( 5 . 2 ) , and any design t h a t i s not optimum w i l l r e q u i r e many more cy cles to converge. Razani (1965) found f o r a th r e e b a r p lan e t r u s s , t h a t under two s e t s o f loads f o r which th e design was optimum, convergence was in 8 and 9 cy cles w hile load c o n d itio n s which cause th e f u l l y s t r e s s e d design to be o th e r than optimum, 99 cycles were r e q u ire d f o r convergence.
Four Bar Truss
The f i r s t s t r u c t u r e s tu d ie d i s th e fo ur member space t r u s s
shown in Figure 9. This t r u s s design has been optim ized w ith r e s p e c t to weight by Venkayya, V. B . ; Khot, N. S . ; and Reddy, V. S. (to be p u b lish e d )
and thu s p ro v id es a comparison o f th e f u l l y s t r e s s e d and minimum weight d e s ig n s . T h e ir fin d in g s give a design weight o f 9.09 pounds w ith
Xj = 0.132, x2 = 0 .1 9 3 , Xg = 0.122 and x^ = 0.097 square in c h e s . The m a t e r ia l i s aluminum and th e maximum allo w ab le s t r e s s in te n s io n o r compression i s 24,000 pounds p e r square in ch . There are no c o n s t r a i n t s on d e f l e c t i o n . The modulus o f e l a s t i c i t y E, equals
10,000,000 pounds p e r square inch and th e weight o f th e aluminum i s taken as 0.1 pounds p e r cubic i n c h . A lower l i m i t o f 0.10 square inches i s p lace d on member s i z e . There are t h r e e loads o f 5,000, 5,000 and 7,500 pounds a p p lie d one a t a time a t j o i n t f iv e as shown in Figure 9.
Convergence t o a f u l l y s t r e s s e d design r e q u ir e s only four c y c le s . This i s th e lowest number f o r any s t a t i c a l l y in d e te rm in a te s t r u c t u r e s tu d i e d . Member number 4 i s minimum allo w able s i z e o f 0.10 square in c h e s .
30 5 ,0 0 0 lb s
7,500 lbs
96
, 2
60 144
5,000 lbs
5.000 lbs
120
F ig u re 9 . Four Bar T russ
31 The areas o f th e o t h e r members are = 0.131 square in c h e s , x^ = 0.192 square in c h e s , and x^ = 0.120 square in c h e s . The weight o f th e f u l l y s t r e s s e d s t r u c t u r e i s 9.086 pounds. These areas and t h i s weight are very clo se to th o se found by Venkayya (above) f o r th e minimum weight optimum d esig n , t h e r e f o r e f o r t h i s case th e minimum weight and f u l l y s t r e s s e d design a re the same as th e r a p id convergence o r th e i t e r a t i v e procedure i n d i c a t e s .
, Three Bar Truss
Three d i f f e r e n t cases o f loading f o r th e t h r e e b a r plane t r u s s shown in F igures 10, 11, and 12 are s t u d i e d . The t r u s s i s s t e e l w ith a maximum allow able s t r e s s o f 36,000 pounds p e r square in ch . There a re no
c o n s t r a i n t s on d e f l e c t i o n s . The modulus o f e l a s t i c i t y i s 30,000,000 pounds p e r square inch and th e weight o f th e s t e e l i s taken as 0.31 pounds p e r cubic in ch . The loads are a p p lie d one a t a time g iv in g as many load c o n d itio n s as t h e r e are loads f o r each case.
Case 1
Case 1 loading and convergence are shown in Figure 10. Load co n d itio n 1 i s = 20,000 pounds a c tin g t o th e r i g h t and load c o n d itio n 2 i s Pg = 30,000 pounds a c tin g downward to th e l e f t a t an angle o f 60°
t o th e h o r i z o n t a l . The weight o f th e s t r u c t u r e i s given f o r each i n t e r - a t i o n and th e r a t i o o f th e a c t u a l s t r e s s t o th e allo w ab le s t r e s s ( a c t / allow) f o r each member i s p l o t t e d as th e o r d in a te and th e number o f i t e r a t i o n s as th e a b s c i s s a .
This s t r u c t u r e i s w ell behaved and converges smoothly to a f u l l y s t r e s s e d desig n a t :
Stress: Act/Allow
32
Weight
45
2 0 , 0 0 0
30,000
(a) Dimensions and Loading
11.9 5.25 5.13 5.07 5.05 5.03 5.03 5.02 5.02 1 . 1 2
1.08
Element No. 2 1.04
1 . 00
- Element No. 1
— Element No. 3 92
. 88
7 8 9
5 6
4 3
1 2
Number o f I t e r a t i o n s (b) Act/Allow S tre s s
F igure 1 0 . Three Bar T r u ss, Case 1
33 Xj = 0.341 sq. i n .
Xg = 0.584 sq. in . Xg = 0.313 sq. i n .
Convergence in only n in e i t e r a t i o n s i n d i c a t e s t h i s i s more than l i k e l y a ls o an optimum d e s ig n .
The weight o f t h i s s t r u c t u r e behaved normally and changed l e s s than 3.0% from the t h i r d i t e r a t i o n to th e n i n t h .
Case 2
For Case 2, th e loads from Case 1 a re in te rc h a n g e d , i . e . , the 30,000 pound load a c t s to th e r i g h t and th e 20,000 pound load a c ts downward to th e l e f t .
Figure 11 shows th e loaded t r u s s , th e weight f o r each i t e r a t i o n , the a c t / a l l o w s t r e s s curves and a ls o th e v a r i a t i o n in a re a o f th e c e n te r member (No. 3) f o r each i t e r a t i o n .
The t r u s s design does no t converge to a f u l l y s t r e s s e d design in th e ten cy cles allowed. A fte r ten i t e r a t i o n s th e a c tu a l s t r e s s in member No. 3 i s only 95.9% o f th e allo w ab le. Also Figure 11 shows t h a t the a re a o f member No. 3 i s being d riv e n to zero as was p r e d i c t e d in Chapter 4.
I f member No. 3 i s removed and th e r e s u l t i n g s t a t i c a l l y d eterm in ate s t r u c t u r e i s an alyzed , the r e q u ire d s i z e s o f members No. 1 and 2 are found to be 0.59 square inches which i s e s s e n t i a l l y what th ey have been from cy cle 2 on. T h erefo re f o r t h i s s e t o f lo ad s, the f u l l y s t r e s s e d i t e r a t i v e design method i n d i c a t e s t h a t a sim p le r s t r u c t u r e could be used.
34
45°
30,000 60°
2 0 ,0 0 0
(a) Dimensions and Loads
Weight 43
1.04
Element No. 1 1 . 0 0
— Element No. 2 .96
.92 . 8 8
Element No. 3
Element No. 3, Area
No. of Iterations (b) Act/Allows Figure 11. Three Bar Truss, Case 2
35 Case 3
Case No. 3 is shown in Figure 12. Case No. 3 is sim ilar to Case No. 1 except there are three more loads for a to ta l of five.
The greater number of load conditions makes no difference in the rate of convergence. Case No. 3 also converges in nine cycles.
Twenty-Five Member Space Truss
The next structure studied is the space truss shown in Figure 13.
There are three load conditions as given in Table 1.
Table 1. Loads on Twenty-Five Member Truss
Condition 1 A = 50,000 lbs
B = 0
Condition 2 A = 40,000 lbs
B = 20,000 lbs
Condition 3 A = 0
B = 20,000 lbs
The position and direction of A and B are shown on the drawing.
The material is steel with E = 30,000,000 pounds per square inch and an allowable stress of 36,000 pounds per square inch. The weight of the steel is taken as 0.31 pounds per cubic inch which includes an
allowance for connections.
36
45°
2 0 ,0 0 0
60°
10 ,00 0
2 0 , 0 0 0 15,000
30,000 (a) Dimensions and Loads
Weight 11.9
1 . 1 2
1.08
Element No. 1 1.04
1. 00
.96
Element No. 2 Element No. 3 .92
+j
in
. 8 89 7 8
5 6 4
2 3
1
Number of Iterations (b) Act/Allow Stresses
Figure 12. Three Bar Truss, Case 3
37
EL. 0
EL. 250
EL. 150"
Figure 13. Twenty-Five Member Truss
38 The program i s run f i r s t with no l i m i t on th e minimum allow able a r e a o f th e members (except t h a t th e areas remain p o s i t i v e ) and then again w ith a minimum a re a o f 0.10 square in c h e s . Table 2 compares the two r e s u l t i n g designs and Figures 14 and 15 show th e a c t / a l l o w s t r e s s curves f o r c e r t a i n r e p r e s e n t i v e members.
With no minimum th e areas o f some o f th e members become very small (see Table 2) and th e s t r u c t u r e i s very p o o rly behaved. A f te r te n cycles th e s t r e s s i n seven teen o f th e members does n o t converge to the allo w ab le w ith in th e l i m i t s o f Eq. ( 5 .2 ) .
With th e 0.10 square inch minimum, convergence i s achieved in only e ig h t c y c le s . Ten o f th e members are d riv e n down t o th e minimum allo w ab le s i z e . I t i s o f i n t e r e s t t o n o te t h a t on th e t h i r d c y c le , members 10 and 12 are d riv e n t o th e minimum s i z e . Upon r e d i s t r i b u t i o n
o f s t r e s s e s , by th e f i f t h cy cle bo th members 10 and 12 become over s t r e s s e d and from t h a t p o i n t on remain above th e minimum a re a and con
verge norm ally t o f u l l s t r e s s by th e e ig h th c y c le .
The members t h a t are d riv e n t o minimum s i z e are th e secondary and b ra c in g members, and although th e y may be l i g h t l y loaded, in a
p r a c t i c a l problem, th ey are s t i l l r e q u ire d . S p e cify in g minimum allow able s i z e s n o t only p re v e n ts th e s e members from b ein g "designed out" b u t a ls o seems t o have a s t a b i l i z i n g e f f e c t on th e convergence o f th e e n t i r e s t r u c t u r e .
For th e design w ith ou t minimum s i z e c o n s t r a i n t s , th e weight o f th e s t r u c t u r e i s reduced each c y c le , through th e te n cy cles ru n , to 355.2 pounds. However f o r th e c o n s tr a in e d d e sig n , a minimum weight o f
39 Table 2. Member Sizes f o r Twenty-Five Member Truss
Area (sq. i n . ) _____
Member Run 1 Run 2
Area > 0 .0 Area > 0 . 1
1 0.2346 0.1992
2 0.5753 0.6005
3 0.5753 0.6005
4 0.4936 0.4851
5 0.4936 0.4851
6 0.2041 0.1733
7 0.2041 0.1733
8 0.5854 0.6003
9 0.5854 0.6003
10 0.1279 0.1203
11 0.0050 0.1000
12 0.1279 0.1203
13 0.0077 0.1000
14 0.7477 0.6961
15 0.0260 0.1000
16 0.0480 0.1000
17 0.0260 0.1000
18 0.7477 0.6961
19 0.0480 0.1000
20 0.0477 0.1000
21 1.0103 0.9718
22 0.0335 0.1000
23 0.0335 0.1000
24 1.0103 0.9718
25 0.0477 0.1000
Stress: Act/Allow 1 .3 0
Elements 1 . 20
Elements 14 S 18
—Elements 1 .1 0
1 . 0 0
Elements 2 6 3 Elements 20 6 25 .90
.80
—Elements 15 6 17 .70
9 10 7 8
5 6 3 4
2
No. o f I t e r a t i o n s
Figure 14. Twenty-Five Member T r u s s . Areas > 0
Stress: Act/Allow
41
1 .3 0
Elements 14 G 18 Elements 10 § 12
1 . 0 0
Elements 2 § 3 Elements 4 $ 5 .90
Elements 20 & 25 Elements 15 $ 17 .80
20 § 25
.70
7 8
5 6 3 4
2
No. o f I t e r a t i o n s
F ig u re 15. T w en ty-F ive Member T r u ss. Areas > 0 .1
42 371.9 pounds i s reached a t the f i f t h cy cle and as more and more members are h e ld a t the minimum, th e weight s t a r t s to in c r e a s e s l i g h t l y to 380.5 pounds a t cy cle e i g h t .
Cross Braced Truss
The t r u s s shown in F igure 16 i s s tu d ie d under th r e e load co n d itio n s as in d i c a t e d i n Table 3.
Table 3. Loads on Cross Braced Truss
Condition 1 A = 20,000 lbs
B = 0 C = 0
Condition 2 A = 0
B = 40,000 lbs C = 0
Condition 3 A = 0
B = 0
C = 15,000 lbs
The p o s i t i o n and d i r e c t i o n o f A, B, and C are shown in F igure 16.
The m a te r ia l i s s t e e l w ith E = 30,000,000 pounds p e r square inch and an allow able s t r e s s in both te n s io n and compression o f 36,000 pounds p e r square in ch . The weight o f th e s t e e l i s 0.31 pounds p e r cubic inch.
No minimum c o n s t r a i n t i s p la c e d on member s i z e . The r a t e o f convergence f o r s e v e ra l t y p i c a l members i s shown in Figure 16 and th e s i z e o f a l l members a f t e r te n i t e r a t i o n s i s shown in Table 4.
17
(a) Layout
67.10 36.75 35.99 35.79 35.72 35.67 35.64 35.61 35.58 55.57 Weight
Element No. 6 1 . 1 2
Element No. 2
1.08 lement No. 12
1.04 1. 0 0
.96 .92
^—Element No. 15
^-Element No. 9 - Elements No. 1 G 3 . 88
.84
9 10
7 8
5 6
2 3 4
1
No. o f I t e r a t i o n s (b) Act/Allow S tre s s
F igu re 16. Cross Braced Truss
44 Table 4. Cross Braced T ru ss, Member Areas
Member Area
sq. i n . Member Area
sq. in .
1 0.203 12 0.892
2 0.306 13 0.264
3 0.261 14 0.570
4 0.807 15 0.686
5 1.673 16 0.182
6 0.383 17 0.296
7 0.865 18 0.594
8 0.527 19 0.474
9 0.342 20 0.371
10 1.349 21 0.370
11 0.268
Although the design does not converge in th e te n cycle run, the r a t e o f convergence as shown by Figure 16 i n d i c a t e s th e f u l l y s t r e s s e d c o n d itio n would probably be reached w ith in an o th e r te n i t e r a t i o n s . This would p la c e th e s t r u c t u r e in th e range where i t i s d i f f u c u l t to p r e d i c t whether th e design would be optimum o r n o t a t f u l l s t r e s s .
Wing Box S e ctio n
A wing s e c t i o n i s now p re s e n te d as an example using t r i a n g u l a r and r e c t a n g u l a r p l a t e elem ents. The layout i s shown in Figure 17.
The m a t e r ia l i s aluminum w ith E = 10,000,000 pounds p e r square inch and an allow able e f f e c t i v e s t r e s s o f 40,000 pounds p e r square in ch . The weight o f th e aluminum i s taken as 0.1 pounds p e r cubic inch and P o is s o n 's r a t i o i s 0.25 . A minimum allow able th ic k n e ss o f 0.01 inches i s used and due to th e g r e a t e r number o f ele m e n ts, only 8 cy cles are allowed.
18"
90 = No. Upper Element 91 = No. Lower Element
117 115 116 90
91
mCN
o
m in
<N CN
CN
114
to
CN CN CN
to CN CN
00 112
to
CN CN
CN 38 63 113
13
109/1 86 84
61 59 in
S 57 58
60 K 56 54
55
S
101
CN
S ection A-A Figure 17. Wing Box S ectio n
