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ergamon

S0952-1976(96)00021-8

Engng App l ic Ar t i f I n te ll Vol. 9, No. 3, pp. 309-319, 1996

Copyright ~ 1996 Elsevier Science Ltd

Printed in Gr eat Britain. All rights reserved

0952-1976/96 $15.00 + 0.00

ontr ibu ted Paper

N e u r a l n e t w o r k A p p l i ca t io n s i n P r e d ic t in g M o m e n t c u r v a t u r e

P a r a m e t e r s f r o m E x p e r i m e n t a l D a t a

MANSOUR NASSER JADID

King Faisal University, K ingdom o f Saudi Arabia

DANIEL R. FAIRBAIRN

University of Edinburgh, Scotland

Received October 1994; in rev ised for m February 1996)

The obje ct ive o f th is s tudy is to demonstrate a concept and a methodology, rather than to bui ld a

fu l l- sca le knowledge-based sys t em m odel by incorpora ting m os t o f the fundam enta l aspec ts o f a

neura l ne twork to so lve the complex non- l inear mapp ing for a beam-co lumn jo in t . Th i s paper

presents the conce pt o f paral lel d is tr ibuted processing base learning in arti ficial neural netw orks , in

assis ting wi th experime ntal evidence to predic t mo me nt-cur vatur e parameters that are usual ly

acco mplishe d solely by experime ntal work . General ly, i t ma y be possible to ident i fy certain

parameters , and a l low the neura l ne twork to deve lop the model , t hus accoun t ing for the observed

beha viour wi thou t relying on a part icular algori thm, but depending ent irely on the manipulat ion of

num erical data. Cop yright 0 1996 Elsevier Science Lt d

K e y w o r d s : N e u r a l n e t w o r k s , b a c k p r o p a g a t i o n , s t ru c t u ra l , r e in f o rc e m e n ,t , b e a m s , m o m e n t -

c u r v a t u r e , l o a d - d e fl e c ti o n , b e a m - c o l u m n j o i n ts .

1 . I N T R O D U C T I O N

C o n v e n t i o n a l m e t h o d s b a s e d o n t h e v o n N e u m a n n

pr og r a m m i ng i n s t r u c ti on ha v e e s t a b l i s he d t he b a s i s f o r

a l l c o m p u t a t i o n . H o w e v e r , t h i s m e t h o d i s c o m p u -

t a t i ona l l y i n t e ns i ve a nd s e r i a l i n i m p l e m e n t a t i on , s o

t h a t i t d o e s n o t p a r a l le l t h e h u m a n w a y o f th i n k in g a n d

a b s t ra c t i o n . T h e s y m b o l i c - b a s e d l e a rn i n g a p p r o a c h

r e d u c e s t h e c o m p u t a t i o n a l p r o c e s s i n th e p r e s e n c e o f a

h u g e a m o u n t o f d a t a a n d h a s a w e a k n e s s in k n o w l e d g e

r e p r e s e n t a t i on . S ym b o l i c l og i c - b a s e i m p l e m e n t a t i on i s

e x t r e m e l y d i ff ic u l t, pa r t i c u l a r ly i n t he a r e a o f i n te ns i ve

m a t h e m a t i c a l c o m p u t a t i o n . N e u r a l - n e t w o r k f i e l d s

p r o v i d e a c o m p u t a t i o n a l p a r a d i g m c h a l le n g e w h i c h h a s

r e s u lt e d f r o m f o u r d e c a d e s o f i n te n s iv e r e s e a r c h a n d

i n v e s t m e n t . T h e f u n c t io n o f t h e h u m a n b r a i n c o n t a in s

m a ny f e a t u r e s t ha t c a n b e s i m u l a t e d i n a m a c h i ne t o

pe r f o r m c e r t a i n t a s k s t ha t a r e d i f f i c u l t t o a c h i e ve b y

c o n v e n t i o n a l m e t h o d s a n d a sy m b o l i c a p p r o a c h . O n e o f

orrespondence shou ld b e s e n t t o : D r D . R . F a i r b a i r n , D e p a r t m e n t

o f C i vi l a n d E n v i r o n m e n t a l E n g i n e e r i n g , T h e U n i v e r s i t y o f

E d i n b u r g h , C r e w B u i l d i n g , T h e K i n g ' s B u i l d i n g s , E d i n b u r g h E H 9

3 J N , S c o t l a n d . E m a i l : e n e i 0 1 @ c a s t l e . e d . a c . u k .

309

t he p r i m a r y i n t e n t i ons o f the d i s t r i b u t e d c om m u n i t y is

t o d e s i g n n e w h a r d w a r e a n d s o f t w a r e o n a c o m p u t e r

t ha t c a n s i m u l a t e hu m a n t h i nk i ng .

2 . G E N E R A L V I E W O F A P P L I C A T I O N S T O C I V I L

E N G I N E E R I N G

Inte res t in l ea rning in the f i e ld of c iv i l engineer ing

da t e s a s f a r b a c k a s 1966 , w h e n S p i l le r s I pu b l i s he d a

pa pe r e n t i t l e d " A r t i f i c i a l I n t e l l i ge nc e a nd S t r u c t u r a l

D e s i g n " . I t w a s a n a t t e m p t t o d e m o n s t r a t e t h e m o s t

e l e m e n t a r y l e a r n i ng c a pa b i l i t y i n s t r u c t u r a l de s i gn . H i s

p r i m a r y i n t e r e s t w a s t o a pp l y t o s t r u c t u r a l de s i gn , a nd

h i s m a i n c ons i de r a t i on w a s t o de m ons t r a t e t he pos s i b i -

l i t y o f u s i ng e x a m pl e s t o ge ne r a t e r u l e s . A s he

s u gge s t e d :

" E x a m p l e s p l a y a n i m p o r t a n t p a r t i n te a c h i n g a n d

l e a r n i ng i n hu m a ns a nd i t i s na t u r a l t o a s k w ha t

pa r t t he y s hou l d p l a y i n t he f o r m a l a da p t i ve

s ys t e m . B e c a u s e s t u de n t s a r e t a u gh t u s i ng e x a m -

p l e s , sh o u l d c o m p u t e r s b e t a u g h t t h r o u g h t h e u s e

o f e x am p l e s , a n d h o w ? A g a i n , t h e a n s w e r i s n o t

n o w a v a i l a b le b u t p e r h a p s e x a m p l e s m a y b e u s e d

a s c o m p l e t e s e t s o f i n d e p e n d e n t v e c t o r s a r e n o w

u s e d t o r e p r e s e n t f u n c t io n s . "



3 10 M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E

T hi s s how s a n e a r l y a t t e m p t b y S p i ll e r s z t o u s e

e x a m p l e s a s a m e a n o f ge ne r a t i ng r u l e s, a nd c on f i rm s

the f ac t tha t wha t i s ava i l ab le today in l ea rning appl i -

c a t i ons i s m e r e l y a n i l lu s t ra t i on o f w ha t w a s s a i d b e f o r e .

L e a r n i ng f r om b r i dge s t r u c t u r a l fa i l u re w a s p r o pos e d

b y S t one , 2 w h e r e b y a h i e r a r c h ic a l k now l e dge - b a s e

c ou l d b e de ve l ope d t o p r e d i c a t e b r i dge f a i l u r e f r om a

s t u dy o f h i s to r i c a l da t a . T he m e t hod o f le a r n i ng

e m p l o y e d w a s b a s e d o n a n a l g o r i t h m d e v e l o p e d b y

N or r i s e t aL T he r oo t s o f t h is l e a rn i ng t e c hn i q u e

e m e r ge d f r om t he m e d i c a l d i a gnos is fi e ld w h e r e t he

r e l a t i on b e t w e e n t he s ym pt om s a nd t he d i s e a s e i s

ob s e r ve d f o r a nu m b e r o f pa t i e n t c a s e h i s to r i e s . T he

l e a r n i ng p r oc e s s i nvo l ve s t w o pha s e s : a

d i s c r i m i n a t i o n

a na l ys i s t ha t i s k now n a s t he s e r i a l a pp r oa c h , a nd t he

c o n n e c t i v i t y a l go r i t hm w h i c h a dop t s a pa r a l l e l

a pp r oa c h . I n t he d i s c r im i na t i on c a s e , a s e a r c h is c a r r i e d

ou t f o r a s i ng l e f e a t u r e , w h i l e t he c onne c t i v i t y a l go r -

i t hm s e a r c h i s f o r a g r ou p o f fe a t u r e s . T h e p r e s e nc e o r

a b s e nc e o f s u c h a f e a t u r e o r f e a t u r e s , i nd i c a t e s t ha t a n

e l e m e n t o r e l e m e n t s b e l ong t o one o r a no t he r c l a s s . I n

c a s e s o f u n c e r ta i n t y e n c o u n t e r , t h e m e t h o d o f s u p p o r t

logic i s used .

T he a pp l i c a t i on o f ne u r a l ne t w or k s t o s tr u c t u r a l

a na l y s i s w a s c a r r i e d o u t b y J a d i d a nd F a i r b a i r n 4 i n t he

f o r m o f a da p t i ve fi n it e e l e m e n t m e s h ge ne r a t i on . T he

s t u d y c o n c e n t r a t e d o n t h e p r o c e s s o f a d a p t i v e r e -

m e s h i ng o f a n i de a l i z e d s q u a r e s ha pe a n d a n i nd i v i du a l

t r ia ng l e b y u s ing t r i a ngu l a r e l e m e n t s . S u pe r v i s e d t r a i n -

i ng w a s u s e d i n t he a pp l i c a t ion o f the b a c k - p r op a ga t i on

l e a r n i ng a l go r i t hm w h i c h de a l s w i th t he p r ob l e m o f r e -

meshing s t ruc tu ra l e l ements in a s t ruc tu ra l ana lys i s .

T h e m a i n o b j e c t i v e o f th e a u t h o r s ' s t u d y w a s t o d e m o n -

s t r a t e t he a b i l i ty o f ne u r a l n e t w or k s t o e m p l oy r e - m e s h

s t r u c t u r a l e l e m e n t s w i t hou t u s i ng nu m e r i c a l ly i n t e ns i ve

c o m p u t a t i o n s . T h e f u n d a m e n t a l r e q u i r e m e n t w a s t h e

s e l e c t ion o f a f e a s i b l e a nd a p p r op r i a t e d om a i n t o ge ne r -

a t e t r a i n i ng a nd t e s t da t a .

A m o r e c o m p r e h e n s i v e r e s e a rc h w o r k u s in g n e u r a l-

n e t w o r k t e c h n i qu e s w a s u n d e r t a k e n b y J a d i d 5 t o s t u d y

p r e v i o u s e x p e r i m e n t a l s t u d i e s o f b e a m - c o l u m n j o i n t

pa r a m e t e r s f r om a d i f f e r e n t a ng l e , a nd e s t a b l i s h a

on ept a n d m e t h o d o l o g y t h a t w o u l d p r o v i d e r a p i d a n d

e c onom i c b e ne f i t s f o r f u t u r e e x pe r i m e n t a l r e s e a r c h .

3 . F O R M U L A T I O N O F T H E B A C K P R O P A G A T I O N

A L G O R I T H M

A r t i fi c ia l ne u r a l n e t w or k s a r e i n s p i r e d b y hu m a n

b i o l og ic a l ne u r a l ne t w or k s , w he r e b y t he y c a p t u r e t he

b r a i ny f u nc t i on m a n i pu l a t i on t o a pp r oa c h a s pe c i f i c

p r ob l e m b y a pp l y ing c e r t a i n r u l e s to a c h i e ve r e a s ona b l e

resu l t s . The s tudy of a r ti fi ci al neura l ne two rks i s

f ou nde d on a s e m i - e m pi r i c a l b a s e t o m ode l t he b e -

ha v i ou r o f t he b i o l og i c a l ne r ve c e ll s t ru c t u r e . T he

p r oc e s s i ng e l e m e n t i n a n a r ti fi c ia l ne t w o r k i s a na l ogou s

t o t he ne r ve c e l l i n t he hu m a n b r a i n . T he b r a i n i s

c om pos e d o f de ns e ne r ve c e l l s w h i c h a r e h i gh l y

i n t e r c onn e c t e d a nd e s t i m a t e d t o t o t a l 100 b i ll i on ( ne u r -

ons ) o f d i f f e r e n t t ype s , w h i c h a r e c ons t a n t l y s e nd i ng

a nd r e c e i v i ng m e s s a ge s . T he s e ne r ve c e l l s a r e f u nda -

m e n t a l e l e m e n t s t o t he c e n t r a l ne r vou s s ys t e m , a nd

de t e r m i ne a ny a c t i on w h i c h i s t a k e n .

B a c k p r o p a g a t i o n n e t w o r k s a r e c o n s i d e r e d t o b e t h e

m os t r e l i a b l e a nd m os t a pp l i c a b l e o f al l. O ne s u r ve y ha s

s how n t ha t a b ou t 80% o f al l a pp l i c a t i ons u s e d b a c k -

p r opa ga t i on , du e t o t he m a t he m a t i c a l de s i gn o f l e a r n -

i ng c om pl e x non l i ne a r r e l a t i ons h i ps . E ve n t he i npu t

da t a i s l e s s p r e c i s e o r no i s y . A b a c k p r opa ga t i on

ne t w or k ha s t he a b i l i t y t o m i n i m i z e t he m e a n - s q u a r e d

e r r o r b y a pp l y i ng a g r a d i e n t - de s c e n t a l go r i t hm t ha t

f o l l ow s t he g r a d i e n t e r r o r c u r ve dow nw a r d a c r os s a ll

t he i npu t pa t te r n s . T h i s i s m a t he m a t i c a l l y c om p u t e d b y

t a k i ng t he pa r t i a l de r i va t i ve s o f the e r r o r w i t h r e s pe c t

t o t he w e i gh t s . A s u pe r v i s e d l e a r n i ng t e c hn i q u e i s

e s s e n t ia l i n the l e a r n i ng p r oc e s s , d u e t o t he p r e s e nc e o f

i n p u t a n d o u t p u t d a t a w h i c h e n s u r e s t h e d e v e l o p m e n t

o f an i n t e r na l m o de l t ha t de s c r i b e s t he ob j e c t i ve

r e q u i r e m e n t s .

T h e b a c k p r o p a g a t i o n a l g o ri th m i n v o lv e s a f o r w a r d

p r opa ga t i on s t a r t , w he n a s e t o f inpu t pa t t e r n s i s

p r e s e n t e d t o t h e n e t w o r k , a n d t h e b a c k w a r d e r r o r

a c t i va ti on b e g i ns a t t he ou t p u t l a ye r w he n e r r o r s p r opa -

ga t e t h r ou gh t he i n t e r m e d i a t e l a ye r s t ow a r d t he i npu t

l ay e r . T h e p r o c e s s o f f o r w a r d a n d b a c k w a r d p r o p a -

ga t i on c on t i nu e s u n t i l t he e r r o r i s r e du c e d t o a n a c c e p -

tab le l eve l , or has run for a spec i f i ed t ime . F igure 1

s how s a s i m p l i fi e d s ing l e p r oc e s s i ng e l e m e n t o f a b a c k -

p r o p a g a t i o n n e t w o r k w i th i t s s u m m a t i o n a n d a c t iv a t io n

f u nc t i ons w i t h i n a t yp i c a l b a c k p r opa ga t i on ne t w or k o f

t h r e e l a ye r s , e a c h o f w h i c h i s c on ne c t e d t o t he p r oc e s s -

i ng e l e m e n t s i n t he ne x t l a ye r .

4 . F U N D A M E N T A L M A T H E M A T I C A L

F O R M U L A T I O N

T h e f u n d a m e n t a l m a t h e m a t i ca l f o r m u l a t i o n o f th e

b a c k p r o p a g a t i o n n e t w o r k r e q u i r es e a c h p r o c e s s i n g e le -

m e n t t o p e r f o r m f o u r m a i n s t e ps :

1 . I npu t c onne c t i ons , w h i c h a r e a na l ogou s t o t he

s y n a p s e s , r e c e i v e i n f o r m a t i o n f r o m o t h e r p r o -

c e s s i ng e l e m e n t s o r s t a r t w i t h k now n i npu t

da t a .

2 . A s u m m a t i on f u nc t i on , w h i c h invo l ve s the a c t i-

va t i on o f e a c h p r oc e s s i ng e l e m e n t w i t h i t s

weight .

3 . A t h r e s ho l d f u nc t i on , w h i c h i s a p r oc e s s o f

c onve r t i ng t he s u m m a t i on i npu t a c t i va t i on da t a

t o a n ou t pu t a c t i va t i on da t a b y u s i ng a s pe c i f i c

f u nc t i on .

4 . O u t pu t p r oc e s s i ng e l e m e n t s , w h i c h r e s e m b l e



MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CUR VATURE 311

• F o r w a r d A c t i v a t i o n , ~

e r

k

J

Y l , d ,

y2

Y3 a

I n p u t L a y e r

O u t p u t L a y e r

~1 B a c k w a r d A c t i v a t i o n - -

Fig. 1. Backpropagation network.

t h e a x o n i n t h e h u m a n b r a i n , a n d r e s u l t f r o m

t h e p r e v i o u s p r o c e s s .

T h e o u t p u t t o p r o c e s s i n g e l e m e n t y j, d e t e r m i n e d b y

t h e w e i g h t e d s u m o f t h e i n p u t s i g n a l s ( a s e t o f i n p u t

s ignals , b l ,

b2 .. . ,

b , , ap p l i ed t h r o ug h a s e t o f a s -

s o c i a t ed w e i g h t s , w ~ , w ~ , w ~ , . . . , w j~ , b e c o m e s t h e

i n p u t t o t h e a c t i v a ti o n fu n c t i o n . I n m a t h e m a t i c a l te r m s :

s j = ~ b i w ~ i

i l

y j = f s j ) .

F o r a s i g m o i d ac t i v a t i o n f un c t i o n ,

b e c o m e s :

(1)

(2)

e q u a t i o n ( 2 ) ,

1

y j

=f (sj ) = 1 + e-S/ ' (3)

a n d f o r a h y p e r b o l i c a c ti v a t io n t a n g e n t f u n c t i o n , e q u a -

t i o n ( 2 ) beco m es :

eSj_ e-Sj

y j

= f(s j) = eS + e_Sj (4)

w h e r e

b i = in p u t s i g n a ls t o t h e b a c k p r o p a g a t i o n n e t w o r k ,

y j = o u t p u t s i gn a ls o f t h e b a c k p r o p a g a t i o n n e t w o r k ,

d ~ = d e s i r e d s ig n a ls o f t h e b a c k p r o p a g a t i o n n e t w o r k ,

w ~ = b a c k p r o p a g a t i o n w e i g h t a d j u s t m e n t b e t w e e n

t h e i n p u t an d o u t p u t s i g n a l s .

4 1 Weight correct ion for the output l a y e r

T h e p r o c e ss i n g e l e m e n t s a t t h e o u t p u t l a y e r p r o d u c e

a s i n g l e r e a l n u m b e r f o r e a c h Y l k , Y 2 k , • • . , Y q * ) , w h i c h

a r e t h e n c o m p a r e d t o t h e d e s i r e d o u t p u t d l k , d 2 k , • • . ,

d qg ) t o o b t a i n t h e e r r o r s i g n a l . T h e e r r o r s i g n a l, E q k , is a

m e a s u r e o f th e n e t w o r k ' s p e r f o r m a n c e f o r o n e p r o c e s s -

i n g e l e m e n t in t h e o u t p u t l a y e r , t h a t c a n b e d e t e r m i n e d

f r o m :

E q k = d q k - - Y q k ) ,

(5)

w h e r e

q = a b a c k p r o p a g a t i o n p r o c e ss i n g e l e m e n t in t h e

o u t p u t l a y e r ,

k = r e f e r t o t h e o u t p u t l a y e r i n t h e b a c k p r o p a g a -

t i o n ,

E q k - - - - - a b a c k p r o p a g a t i o n e r r o r s i g n a l f o r o n e p r o c e s s-

i n g e l e m e n t i n t h e o u t p u t l a y e r ;

d q k

= t h e d e s i r e d b a c k p r o p a g a t i o n o u t p u t ,



312 MAN SOUR NASSER JADID and DAN IEL R. FAIRBA IRN: MOMEN T-CURVATURE

qk - - - - - t h e

a c t ua l b a c k p r o p a g a t i o n o u t p u t f o r o n e p r o -

c e s s ing e l e m e n t .

T he e r r o r va l u e , 6q k, c a n b e ob t a i n e d b y m u l t ip l y i ng

the e r ror s igna l , E q k o b t a i n e d f r o m e q u a t i o n ( 5 ), b y a n

a c t i va t ion f u nc t i on d e r i va t ive :

ofk

6qk= ~sqkEok. (6)

T he e r r o r va l u e , Oq k ob t a i ne d f r om e q u a t i on ( 6 ) , i s

t he n m u l t i p l i e d b y yp j , t he ou t pu t o f one p r oc e s s i ng

e l e m e n t i n t he h i dde n l a ye r , t o p r ov i de t he c onne c t i on

b ( k now n a s t he ge ne r a l i z e d

e i gh t c o r r e c t i on , AWqp k

de l t a r u l e ) . T h i s c o r r e c t e d w e i gh t i s c om pu t e d a s :

A w ~ p.k = r] ~ E q k y p j (V)

w h e r e

Awb e . j

= a d j u s t e d w e i gh t b e t w e e n t he q t h p r oc e s s i ng

e l e m e n t i n t h e o u t p u t l a y e r a n d t h e p t h

p r oc e s s i ng e l e m e n t i n t he h i dde n l a ye r ,

yp~ = b a c k p r opa g a t i on ou t p u t f o r one p r oc e s s i ng

e l e m e n t i n t he h i dde n l a ye r .

4 . 2 . W e i g h t c o r r e c t i o n f o r t h e h i d d e n l a y e r

B e c a u s e o f t he a b s e nc e o f de s i r e d ou t pu t s i n the

h i d d e n l a y e r , t h e p r e v i o u s p r o c e d u r e c a n n o t b e

a do p t e d . T h e r e f o r e , t he e r r o r va l u e , 6p j, f o r t he h i dde n

l a ye r i s ge n e r a t e d w i t hou t t he d e s i r e d ou t pu t s . T h i s is

a c c om pl i s he d b y c a l c u l a t i ng e a c h p r oc e s s i ng e l e m e n t ' s

e r r o r v a l u e i n t h e o u t p u t l a y e r a s o b t a in e d f r o m e q u a -

t ion (6) , 6 q k . T he s e a r e u s e d t o c o r r e c t t he w e i gh t s

g o i n g in t o t h e o u t p u t l a y e r, w h e r e t h e y p r o p a g a t e t o

t he h i dde n l a ye r t o ge ne r a t e 6p j , f o r t he h i dde n l a ye r ,

w h i c h i s c om pu t e d a s :

O f j { ~ - -~W b ~ q k ) .

I n a s i m i l a r w a y , t he h i dde n l a ye r i s c o r r e c t e d b y :

O f j ( + W ~ ] ~ q k)Y o i

w h e r e

A w ~ o . i = t h e

b a c k p r o p a g a t i o n c o r r e c t e d

Y o i

i

8 )

9 )

w e i gh t

b e t w e e n t he p t h p r oc e s s i ng e l e m e n t i n h id -

de n l a ye r a nd t he o t h e l e m e n t i n t he i npu t

l a ye r ,

= t h e b a c k p r o p a g a t i o n o u t p u t f o r o n e p r o c e s s -

i ng e l e m e n t i n t he i npu t l a ye r ,

= r e f e r s to t he i npu t l a ye r i n b a c k p r opa g a t i on ,

= r e f e r s t o t h e h i d d e n l a y e r i n b a c k p r o p a g a -

t ion ,

= a b a c k p r opa ga t i on p r oc e s s i ng e l e m e n t i n t he

inpu t l ayer .

5 . R E Q U I R E M E N T S F O R N E U R A L N E T W O R K

I M P L E M E N T A T I O N S

T he ope r a t i on o f a ne u r a l - ne t w or k t oo l r e q u i r e s t he

s e t ti ng u p o f tr a i n ing a nd t e s t da t a f o r e a c h i nd i v idu a l

t a s k , a nd c o r r e c t pa r a m e t e r s t ha t t he ne t w or k r e q u i r e s

t o p r ov i de a r e a s ona b l e a nd a c c e p t a b l e t r a i ne d

ne t w or k . T he s e c ons i s t o f c o l l e ct i ng da t a , s c a li ng da t a

a nd c hoos i ng t he ne t w or k .

5 . 1 . C o l l e c t i o n o f d a t a

T he da t a i s s e pa r a t e d i n t o t w o s e t s , one f o r t r a i n i ng

a nd t he o t he r f o r t e s t i ng . T he t e s t i ng da t a i s no r m a l l y

t a k e n a s 10% o f t he t r a in i ng d a t a , s u c h t ha t t he 10 th

e l e m e n t o f e a c h t r a i n i ng s e t i s r e s e r ve d f o r t he t e s t i ng

da t a w h i c h w il l p r ov i d e t he b es t p i c t u r e r e p r e s e n t a t i ons

a nd i nc r e a s e t he c on f i de nc e i n t he pe r f o r m a nc e o f t he

t r a i ne d ne t w or k . G e ne r a l l y , t he m or e t r a i n i ng da t a

u s e d , t he b e t t e r t he ne t w or k w i l l pe r f o r m .

5 . 2 . S c a l i n g o f d a t a

T he ne t w o r k a c c e p t s va l u e s on l y f r om 0 t o 1 f o r t he

s i gm oi da l f u nc t i on , a nd - 1 t o 1 f o r t he hype r b o l i c

t a nge n t f u nc t i on . T he p r o c e s s invo l ve s t he c om pu t a t i on

o f l ow a nd h i gh va l u e s o f e a c h t r a i n i ng e x a m pl e da t a

f ield in the selected data f i les .

5 . 3 . C ho i c e o f

n e t w o r k

T h e b a c k p r o p a g a t i o n n e t w o r k u s e s a n o n l i n e a r

r e g r e s s i on t e c hn i q u e t ha t a t t e m p t s t o m i n i m i z e t he

g l ob a l e r r o r , a nd ha s t he a b i l i t y t o p r ov i de c om pa c t

d i s t r ib u t i on r e p r e s e n t a t i ons o f c om pl e x da t a a nd i ts

po t e n t i a l t o m a n i pu l a t e m u l t i p l e - d i m e ns i ona l f u nc -

t ions . Three main aspec t s a re es sent i a l in se lec t ing a

s pe c if ic ne t w or k pa r a d i gm t ha t d i c t a t e s t he c ha r a c t e r i s -

t ic s o f a g ive n ne t w or k . T he s e l e c t ion o f a n a pp r op r i a t e

ne t w o r k i s b a s e d on t he t h r e e f o l low i ng c on f i gu r a t ions :

a r c h i t e c t u r e , t opo l og y a nd ne u r odyna m i c s .

5 .3 .1 . N e tw o rk a rch i tec tu re

B a c k pr opa ga t i on w a s s e l e c t e d f o r t h i s r e s e a r c h

accord ing to the ava i l ab i l i ty of s e r i es pa t t e rn pa i rs ,

w he r e e a c h pa i r c ons i s t s o f a n i npu t pa t t e r n w i t h a

de s i r e d ou t pu t pa t t e r n . T he l e a r n i ng t e c hn i q u e o f f e r e d

b y b a c k p r opa ga t i on i s s u pe r v i s e d l e a r n i ng .

5 . 3 . 2 . N e t w o r k t o p o l o g y

T h e n e t w o r k t o p o l o g y c on s is ts o f th e n u m b e r o f

i npu t a nd ou t p u t l a ye r s , t he nu m b e r o f p r oc e s s ing

e l e m e n t s ( P E s ) t h e y c o n t a in , t h e n u m b e r o f h i d d e n

l a ye r s a nd p r oc e s s i ng e l e m e n t s , t he i r i n t e r c onne c t i v i t y ,

a nd t he p r ope r t i e s o f t he ge om e t r i c a l c on f i gu r a t i ons .

A s a ge ne r a l r u l e, t he a m o u n t o f i npu t da t a t ha t c a n b e

u s e d a s a n u p p e r b o u n d f o r th e n u m b e r o f P E s in t h e

hidden l ayer i s as fo l lows :

R o w ,

UpE Ra ng e * (outpE + ineE) ' (10)

w h e r e



MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CURV ATURE 313

UpE

Row,

OUtpE

inpE

= upper bound for the number of processing

elements in the hidden layer,

= number of rows in training data,

= number of processing elements in the out-

put layer,

= number of processing elements in the input

data,

Range = range between 5 and 10,

PE = number of processing elements.

5.3.3. Netw ork neurodynamics

The generalized delta rule developed by Rumelhart

et al.6

is the most popular learning rule used by back-

propagation networks. Two popular extensions of the

generalized delta rule are implemented in the

NeuralWorks®7 tool, viz the cumulative delta rule

which accumulates weight changes over several exam-

ples, and the normalised cumulative delta rule.

Variants of the backpropagation networks include the

extended delta bar delta (EDBD) , which uses a heuris-

tic and encourages positive learning trends and reduces

oscillation.

5.3.4. Training and testing the network

An iteration process starts after presentation of the

input data with the desired output to the network, and

continues until the network converges to acceptable

levels, or has run for a specified time. The number of

iterations is specified as the number of learnings in the

run menu, or as an acceptable error in the r.m.s.

diagnosis tool. Once the network is trained and con-

verges, the test set is presented to the network sequen-

tially only once, to increase the confidence of the

network performance and account for accuracy.

5.3 .5 . Netw ork per formance

The network's performance is monitored by

root-

mean-square r .m .s .) , weight histogram,

and

confusion

matrix diagnostic instruments provided by the tool to

achieve a better understanding of the network's perfor-

mance. The r .m.s , error is computed from: 8

where

?p~t E (do.t-YouO:

oPE

r . m . s .

nptnoPE

(11)

r.m.s. = root-mean-square,

pt

= summation over all patterns in the training

set,

=summation over all output processing ele-

oPE

ments,

dour = desi red output,

Yout =

network output,

npt ----

number of patterns in the training set,

hoPE

- - - - n u m b e r

of processing elements in the out-

put layer.

The weight histogram provides a normalized histo-

gram of all the variables in the network that change

during the training session that is used to monitor the

network's performance. The x-axis along the confusion

matrix

provides the network output, and the y-axis is

the desired output. The interior quadrants are discre-

tized into bins to show the network outputs. A value of

one means an excellent correlation between the desired

and network outputs. The

epoch size

is one o f the main

factors that control the convergence of the network,

where weight changes in the network are accumulated.

The epoch size can be tuned and adjusted to provide

better learning procedures by monitoring it as it

evolves.

6 R E L A T I O N S H IP B E T W E E N M O M E N T A N D

C U R V A T U R E

Previous experimental work was carried out by

Nirjar9 to investigate the structural behaviour of cast-

in-situ beam-column joints under static loading con-

ditions. The study investigates the relationship between

the behaviour of beam-column joints and geometrical

shape, amount and size of steel reinforcement, fixed

beam and column cross-sectional dimensions and con-

crete strength. Tests were carried out on a total of 34

specimens under the following conditions:

1. Series NN is based on variations of column

loading conditions, from 10 to 60 of the

ultimate column load.

2. Series NM is based on variations of column

longitudinal reinforcement, Pc, tested at 10 of

the ultimate load loading.

3. Series NO is based on variations of column

longitudinal reinforcement, Pc, tested at 50 of

the ultimate column loading.

4. Series NP is based on variations in the area of

tensile reinforcement in the beams, tested at

10 of the ultimate column load.

5. Series NQ had the same reinforcement varia-

tions as the NP series, but tested at 60 of the

ultimate column load.

6. Series NR involved variations in the area of

transverse reinforcement in the column and

joint, tested at 10 of the ultimate load.

7. Series NS had variations in the area of lateral

reinforcement in the beams using different

spacing of stirrups.

8. Series NT covered five dif ferent concrete

grades, viz. 20, 30, 35, 40 and 45 N/mmz.

The research work presented is part of a complete

and comprehensive investigation into the behaviour of

corner beam-column joints under biaxial bending

moments. The work carried out follows previous exper-

imental work done by Nirjar 9 into the investigation of



314 MANSOURNASSERJAD ID and DAN IEL R. FA IRBAIRN:MOMENT-CURVATURE

34 s pe c i m e ns . T he p r i m a r y w or k de s c r i b e d he r e i s on l y

a pp l i e d t o i nve s t i ga t e t he m om e n t - c u r va t u r e r e l a t i on -

s h i p fo r t he f le x u r al m e m b e r . A p a r t o f t he s t ra t e gy

i m p l e m e n t e d t o c a r r y ou t th i s r e se a r c h w or k i s s how n i n

F i g . 2 , u nde r t he p r e pa r a t o r y s t a ge .

A u n i a x i a l l o a d , m o m e n t a n d c u r v a t u r e p r o c e d u r e

I

I a s e

o n

E x p e r i m e n t

R e c e n t R e s e a r c h

N e w C o d e o f P r a c t ic e

N o

, [Num.r ic l l N.u r l l ~ . /

R ~ ~w S_~gy_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .__~___.

I

o

a s e o n R e c e n tR e s e a r c h

NewCodeofVrectice ~

~Numerical NN m l

/ No~o,-bSas. I"

I I M IM o d e l I

s*op 3

Fig. 2. Strategies or the implementationof predictive stages.



M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A IR N : M O M E N T - C U R V A T U R E

p M

p M ~ . . . . . . . . . . . . . . . . . . . . . . . . . . c

e,

o a o, (a) Load De ~,nf l~ 4 , L , # , # , ÷

( b ) M o m e n t - C u r v a t u re

315

F i g . 3 . Id e a l i z e d l o a d - d e f l e c ti o n a n d m o m e n t - c u r v a t u r e r e l a t io n s h i p s .

w a s d e v e l o p e d b y P f a n g

et al. 1°

T h e y d e s c r i b e d a

p r o c e d u r e t o e v a l u a t e t h e r e l a t io n s h i p b e t w e e n u n i a x ia l

l o a d , m o m e n t a n d c u r v a t u r e b a s e d o n u s i n g a t o o l f o r

t he a na l y t i c a l p r oc e du r e . A s i m i l a r p r oc e du r e w a s

a d o p t e d b y K r o e n k e et al . u w h o a l s o c o n s i d e r e d t h e

e f f e c t s o f a s y m m e t r i c a l r e i n f o r c e m e n t p l a c e m e n t a n d

t he i nc l u si on o f s t r a in ha r de n i ng . A n i nc r e m e n t a l - t ype

m e t h o d f o r t h e d e t e r m i n a t i o n o f t h e l o a d - d e f o r m a t i o n

r e l a t i ons h i p w a s de ve l op e d b y E I - M e t w a l l y a nd C he n .12

T h i s m e t h o d h a d b e e n e x p e r i m e n t a l l y v e r if ie d , a n d c a n

b e a p p l i e d t o a no n l i ne a r a na l y s i s o f f a m e s t r u c t u r e s .

A r e l a t io n s h i p b e t w e e n l o a d - d e f o r m a t i o n a n d

m o m e n t - c u r v a t u r e c a n b e d e t e r m i n e d b y a p p ly i n g t h e

p r i nc i p l e o f e q u i l i b r i u m o f t he i n t e r na l f o r c e s a nd

c om pa t i b i l i t y o f t he s t r a in s . T he m e m b e r u s u a l l y u nde r -

g o e s t h r e e s t a g e s o f b e h a v i o u r d u r i n g a p p l i c a t io n o f t h e

l oa d . F i gu r e 3 s how s t yp i c a l

i d e a l i z e d

l oa d - de f l e c t i on

a n d m o m e n t - c u r v a t u r e r e la t io n s h i p s.

T he l oa d i ng s t a ge s a r e c ha r a c t e r i z e d b y :

( a ) A n e l a s t i c s t a ge b e t w e e n 0 a nd a : t h i s s t a ge i s

u s u a l l y r e f e r r e d t o a s t h e " u n e r a c k e d s t a g e " .

T he t e ns i l e r e i n f o r c e m e n t a t t h i s s t a ge i s b a s i -

ca l ly inac t ive .

( b ) T h e y i e ld s ta g e b e t w e e n a a n d b .

( c ) T he u l t i m a t e s t a ge b e t w e e n b a nd c .

6 1 Y i e ld m o m e n t a n d c u r v a t u r e

A s t he l oa d i nc r e a s e s , t he c r a c k s f o r m i ng i n t he

t e n s i o n z o n e p r o p a g a t e u p w a r d t o w a r d t h e n e u t r a l a x is .

T he c onc r e t e i n t he t e ns i on z one i s i na c t i ve , a nd t he

t e ns i l e s t e e l r e s i st s t he e n t i r e t e ns i on . A s i l l u s t r a t e d i n

F i g . 3 ( b ) o f t he m om e n t - c u r va t u r e r e l a t i ons h i p , b is t he

po i n t on t he c u r v e w h i c h de f i ne s y i e l d i ng o f t he t e ns i on

r e i n f o r c e m e n t w i t h f u r t he r i nc r e a s e i n t he de f l e c t i on ,

w h i le t h e a p p l i e d l o a d r e m a i n s n e a r l y c o n s t an t . A s l o n g

a s t he c r o s s - s e c t i on i s u nde r - r e i n f o r c e d , t he t e ns i l e

r e i n f o r c e m e n t r e a c h e s y i e l d b e f o r e t h e c o n c r e t e

c r u s he s . T he y i e l d c u r va t u r e , ~ y , ge ne r a l l y de f i ne d a s

t he c u r va t u r e a t w h i c h t he t e ns i l e r e i n f o r c e m e n t

r e a c he s i t s y i e l d po i n t s t r e s s , i s de t e r m i ne d f r om F i g .

4 ( b ) b y s t r a in c om p a t i b i l it y a s f o l low s :

gy fy

~ Y = d - x d - Es(1 - x ) d (12)

F o r a d o u b l y r e i n f o r c e d b e a m w i t h a t e n si le r e i n fo r c e -

m e n t r a t i o o f :

AS

P = b--d; (1 3)

a nd c om pr e s s i ve r e i n f o r c e m e n t r a t i o o f :

AS'

p ' = - - ( 14 )

b d

t he f o l l ow i ng e x p r e s s i on c a n b e de r i ve d :

x = X /(p + p' )2n2 +

2 n ( p + p ' a ) - n ( a + p ' )

w h e r e

( 15 )

E~

n = mod ula r ra t io , equ a l to ~ -~,

E s - - m o du l u s o f e l a s ti c i ty o f s t e e l , N / m m 2,

d = e f f e c ti ve de p t h o f t he c r o s s - s e c t i on , m m ,

a d

= d i s t a n c e f r o m t h e c o m p r e s s i o n f a c e o f m e m b e r

t o t he c e n t r i od o f c om pr e s s i on s t e e l , r a m ,

A s = a r e a o f t e ns i on s t e e l r e i n f o r c e m e n t , m m 2 ,

A ¢ = a r e a o f c om p r e s s i on s t e e l r e i n f o r c e m e n t , m m 2 ,

/ 9= t e ns i le r e i n f o r c e m e n t r a t i o o f s t e e l ,

p ' = c om pr e s s i ve r e i n f o r c e m e n t r a t i o o f s t e e l .

T he f ina l e x p r e s s ion f o r t he c u r va t u r e a t y i e l d , ~ y , is



316 MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CURVATURE

t h e n e s t a b l i s h e d b y c o m b i n i n g e q u a t i o n s ( 1 2 ) a n d ( 1 5 )

t o o b t a i n :

Cpy = E s [ n ( p + p ) - V ( p + p ) Z n 2 + 2 n ( p + p a ) + 1 ]d '

(16)

T h e s t r a i n i n th e c o m p r e s s i o n s t e e l i s d e t e r m i n e d f r o m

Fig . 4(b) as :

es, = ( xd - ad )~Oy = d (x - a ) ¢ y .

W h e r e t h e c o m p r e s s i o n s t e e l d o e s n o t y i e l d , i . e .

es, < e y ,, th e c o m p r e s s i v e f o r c e d u e t o t h e c o m p r e s s i o n

steel is C~, where:

C~, = e~,E e4s,.

W here the com press ion s tee l y ie ld s , i . e . e s , -> E y , , t h e

c o m p r e s s i v e f o r c e d u e t o t h e c o m p r e s s i o n s t e e l i s C , ,

w h e r e :

¢~ =As L .

A p p l y i n g i n t e r n a l e q u i l i b r i u m :

T = C c + C ¢

so tha t :

C c = T - C s , = A s f y - A s , f , , .

T he in te rna l r es i s t ing m om en t o f the c r i t ica l c ro ss -

s e c t i o n i s t h e r e f o r e o b t a i n e d b y t a k i ng t h e m o m e n t s o f

t h e i n t e r n a l f o r c e s o f t h e c o m p r e s s i v e c o n c r e t e a n d t h e

c o m p r e s s i v e r e i n f o r c e m e n t a b o u t t h e t e n s i l e r e i n f o r c e -

m e n t :

Subs t i tu t ing fo r C e and C s, in to eq ua t i on (22 ) , the

y i e ld m o m e n t , My,is g iven as:

M y = d [ ( Z s f y - a s , f s , ) ( 1 - 3 ) + Z s , f s , ( 1 - a ) ] . ( 2 3 )

T h e s t r e s s i n t h e c o m p r e s s i o n r e i n f o r c e m e n t , f s , , c a n

b e o b t a i n e d f r o m F i g . 4 , a n d f r o m t h e c o m p r e s s i o n

s t r a in ob ta ined ear l ie r , a s :

( x - a ) (24)

(17 ) f~, = Eses, = Esey (1 - x ) '

w h e n t h e s t r e s s i n t h e c o m p r e s s i o n r e i n f o r c e m e n t

r eac hes the y ie ld s t r es s ( f~ , = fy , ) and

C~, =f~,A~,. (25 )

( 1 8) P r e v i o u s c o m p u t a t i o n s p r o v i d e d e x p r e s s i o n s w h i c h

r e l a t e t h e c u r v a t u r e a n d m o m e n t a t t h e y i e l d s t a g e f o r a

d o u b l y r e i n fo r c e d b e a m .

F o r a s in g l y r e i n f o r c e d b e a m , t h e e x p r e s s i o n f o r r a t i o

x can be s im p l i f ied to :

(19) x = X / ( p 2 n z + 2 n p ) - n p . (26)

T h e d i s t a n c e o f t h e n e u t r a l a x i s f r o m t h e c o m p r e s s i v e

f a c e b e c o m e s :

(20)

x d

= (X/(p2n 2 +

2 n p ) - n p ) d .

(27)

T he cu rv a tu re a t y ie ld i s s im p l i f ied to :

L

(21) q~Y E s [ n p - X /((p Zn 2 + 2 n p ) + 1 ]d ' ( 28 )

a n d t h e m o m e n t f o r a s i n g l y r e i n f o r c e d b e a m b e c o m e s :

(22)

6 .2 . P r e par ator y s tage for y i e ld mom e nt and c ur vatur e

F o r th e p r e p a r a t o r y s t a ge , t h e p r o c e d u r e a d o p t e d i n

t h i s s t u d y t o d e t e r m i n e t h e m o m e n t c u r v a t u r e r e l a t i o n -

d

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a) Cr oss sectio n CO) Stra ins

(C) elastic stress dis tr ibution

Fig. 4. Elastic stress distribution condition at yielding of the tensile reinforcement.



M A N S O U R N A S S E R J A D ID a nd D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E

T a b l e 1 . C o m p a r i s o n s o f m o m e n t s a n d c u r v a t u r e s at y i e l d b y N i rj a r 9 a n d b y J a d i d5 for the preparatory s tage

317

S p e c i m e n s

N i r jar 9 N e u r a l n e t w o r k s5

E r r o r = A b s ( 1 N e t w o r k ~ • 1 0 0

\

( N ) ( N )

q~y

* 10 - s My My C y * 10 - s My (N) w . r . t ,

w.r.t .

f'~ p

C o m p u t e d C o m p u t e d E x p e r i m e n t a l

( N ) ( N ) ( C ) ( C ) ( E )

( N / m m 2 ) ( % ) ( C ) ( C ) ( E ) ( m m - ~ ) ( k N m m ) C y M y M y

NN1 30 .0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 9 .17

NN2 30 .0 1 .28 2120 2829 294 0 2062 .99 2615 .82 2 .69 7 .54 11 .03

NN~ 30 .0 1 .28 2120 2829 2850 2062 .99 2615 .82 2 .69 7 .54 8 .22

NN4 30 .0 1 .28 2120 2829 2790 2062 .99 2615 .82 2 .69 7 .54 6 .24

NNs 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22

NN 6 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22

NM7 30 .0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 9 .17

NM s 30 ,0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 8 .22

NM9 30 ,0 1 .28 2120 2829 2820 2062 .99 2615 .82 2 .69 7 .54 7 .24

NM10 30 .0 1 .28 2120 2829 2820 2062 .99 2615 .82 2 .69 7 .54 7 .24

NO n 30 ,0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22

NO l 2 30 ,0 1 .28 2120 2 829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22

NO13 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22

NO t 4 30 .0 1 .28 2120 2829 2910 2062 .99 2615 .82 2 .69 7 .54 10 .11

NPI5 30.0 0.72 1900 1637 1650 1881.73 1632.30 0.96 0.29 1.07

NPI6 30 .0 2 .00 2360 4 310 4 200 24 4 5 .08 5067 .66 3 .60 17 .58 20 .66

NPI7 30 .0 2 .55 254 0 4 312 5260 2563 .34 5870 .79 0 .92 36 .15 11 .61

N P l s

30 .0 2 .99 2670 6281 5760 2595 .78 6105 .05 2 .78 2 .80 5 .99

NQ19 30 .0 0 .72 1900 1637 1680 188 1 .73 1632 .30 0 .96 0 .29 2 .84

NQ~0 30 .0 2 .00 2360 4 310 4 200 24 4 5 .08 5067 .66 3 .60 17 .58 20 .66

NQ21 30 .0 2 .55 254 0 54 12 5160 2563 .34 5870 .79 0 .92 8 .4 8 13 .78

NQ22 30 .0 2 .99 2670 6281 5690 2595 .78 6105 .05 2 .78 2 .80 7 .29

NR 23 30 .0 2 .99 2670 6281 5880 2589 .51 6057 .88 3 .01 3 .55 3 .03

NR 24 30 .0 2 .99 2670 6281 5820 2589 .51 6057 .88 3 .01 3 .55 4 .09

NR 25 30 .0 2 .99 2670 6281 5760 2589 .51 6057 .88 3 .01 3 .55 5 .17

NR 26 30 .0 2 .99 2670 6281 5760 2589 .51 6057 .88 3 .01 3 .55 5 .17

NS27 30 .0 1 .28 2120 282 9 2700 2062 .99 2615 .82 2 .69 7 .54 3 .12

NS28 30 .0 1 .28 2120 2829 2730 2062 .99 2615 .82 2 .69 7 .54 4 .18

NS ~ 30 .0 1 .28 2120 2829 2700 2062 .99 2615 .82 2 .69 7 .54 3 .12

NS~0 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 ,22

NT31 4 0 1 .28 204 2857 2910 1985 .51 264 1 ,02 2 .67 7 .56 9 .24

NT32 35 1 .28 2070 284 6 2850 2027 .80 2623 .67 2 .04 7 .81 7 ,94

NT33 25 1 .28 2170 2812 2790 2107 .50 2607 .58 2 .88 7 .27 6 .54

NT34 20 1 .28 2250 2790 2700 2161 .82 2623 .14 3 .92 5 .98 2 .85

sh ip i s to se l e c t a ne twor k o f (6 ,40 ,30 ,2 ) . Thi s r e pr e -

s e n t s t h e n u m b e r o f i n p u t P E s , t h e h i d d e n P E s i n t h e

f i r s t l aye r , the h idde n P Es in the se c ond h idde n laye r

a n d t h e P E s i n t h e o u t p u t l a y e r , r e s p e c t i v e l y . T h e

r .m.s , wa s se t to 0 .02 , and pr ov ide d sa ti s factor y

r e s u l t s . T h e E D B D l e a r n i n g r u l e a d o p t e d i n c l u d e d a

h e u r i st i c a d j u s t m e n t t o t h e m o m e n t u m t e r m , s e t t o 0 . 4 .

The l e ar ning r a te s we r e se t to 0 .3 for the f i r s t h idde n

laye r , 0 .25 for the se c ond h idde n laye r and 0 .15 for the

output l aye r . Input and de s i r e d pat te r ns we r e pr e -

se nte d to the ne twor k r andomly in the t r a in ing f i l e and

se que nt ia l l y in the t e s t f i l e . The t r a in ing and te s t f i l e s

w e r e g e n e r a t e d w i t h t h e F O R T R A N p r o g r a m b y

s e l e c t i n g a p o s s i b l e c o m b i n a t i o n o f 3 4 s p e c i m e n s .

Th r e e c onc r e te c y l inde r s tr e ngths , 20 , 30 and 45 N/mm :

we r e use d , wi th two d i f f e r e nt bar ar r ange me nts , 2 and

4 , as we l l a s two d i f f e r e nt bar d iame te r s , 6 and 12 mm ,

t o g e n e r a t e t h e d a t a . T h e t o t a l n u m b e r o f c o m b i n a t i o n s

o b t a i n e d w a s t h e r e f o r e 4 0 8 , o f w h i c h o n l y 2 4 3 p a t t e rn s

we r e use d for t r a in ing and 29 pat te r ns for the t e s t ing

f i l e . T h e r e m a i n d e r d i d n o t c o n f o r m t o t h e C o d e o f

P r ac t i c e . The pr e l iminar y t r a in ing and te s t pat te r n

va lue s obta ine d fr om a nume r ic a l ana lys i s for the c ur -

vatur e , ~y , w e r e ve r y smal l , a s obta ine d fr om us ing the

F O R T R A N p r o g r a m . T h e r e f o r e , t o e n h a n c e t h e

n e t w o r k p e r f o r m a n c e , t h e v a l u e s o f t h e c u r v a tu r e , ~ y ,

we r e magn i f ie d by 08 and the s t e e l p e r c e ntage , p , b y 04 .

T h i s p r o v i d e d a b e t t e r n e t w o r k p e r f o r m a n c e . T h e

r e sul t s pr e dic te d by the ne ur a l ne twor ks for the y i e ld

c ur vatur e s and mome nts ar e in c lose agr e e me nt wi th

those obta ine d by N ir jar . 9 Th e se r e sul ts ar e tabu la te d

in Table 1 , toge the r wi th the pe r c e ntage e r r or s .

T h e p r e d i ct e d f o r m u l a e o b t a i n e d b y n e u r a l n e t w o r k s

for the c ur vatur e c an be pr e se nte d in the for m of :

your.p _ ~out.p (478 .41) + 2206 .8 (30 )

78y ~'~7 8, y

x78.ut.~p _- tanh(Z x~s, y . (31 )

T h e m o m e n t a t y i e ld o b t a i n e d c a n b e e x p r e s s e d a s:

Y 7 9 , ° u t y

_- x79.°ut'y . (2 8 2 4 .0 5 6 ) + 3 8 7 7 .1 9 5 (32)

X79,oUty ----

an h (,~r~9. y

(33)

in w hic h out, . . . . t ,p

78,y

and . ,79,y are the scaled curvature and

mome nt a t y i e ld , r e spe c t ive ly , for the pr e par ator y



3 18 M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E

r t m u m t - C u e v a t u r e Ro a t Ion ; C~c o=4 74?9 ; NSS :. _em~__; r51 :1. ; ~ : 1. ; cR : 1.15

j

j L . . . = _ I

id ro t ~ lbt't, l~

2

. . h . . . I

I C o n .

l ~ t a - l x

1

a ) Tr a ine d Ne t wo r k .

r l m e o n ~ C u ~ a t u e e R e l a t t o n ; C v c 1 o = 47 4 79 ; N ~ : . ~ ; C r 1 :1 . ; C R : . 8 ; T e e t = 2 9

IC, nf . Ma t.rlx I

I c o n e M a t r i x 2

b) Tr a ine d a nd Te s t e d Ne t wo r k .

F i g. 5 . C u r v a t u r e a n d m o m e n t a t y i el d f o r t h e p r e p a r a t o r y s t a g e . 7



MA N SOUR N A SSE R JA DID and DA N IE L R . F A IRB A IRN : MOME N T- C URV A TURE 319

s t a g e . T h e n e t w o r k a r c h i t e c t u re , t o p o l o g y a n d n e u r o -

d y n a m i c s a r e s h o w n i n F i g . 5 f o r t h e t r a i n e d a n d t e s t

n e t w o r k s . T h e n e t w o r k c e a s e d l e a r n i n g a t a 4 7 , 4 7 9

c y c l e s, w i t h e x c e l l e n t c o r r e l a t i o n s o f 1 .0 f o r b o t h t h e

t r a i n e d a n d t e s t n e t w o r k s i n t h e

c o n f u s i o n m a t r i c e s .

T h e c las s i f i ca t ion ra te w a s 1 . 0 , w h i c h i n d i c a t e s t h a t t h e

n e t w o r k c o r r e c t l y c la s si fi e d 1 0 0 % o f th e g o o d c a t e -

g o r i e s a n d d i d n o t m i s - c la s s if y a n y o t h e r c a t e g o r i e s .

T h u s , t h e c u r v a t u r e a n d m o m e n t f o r a re i n f o rc e d

b e a m a t th e y i e ld s t a ge c a n b e c o m p u t e d f r o m e q u a -

t i o n s ( 1 6 ) a n d ( 2 3) r e s p e c t i v e l y , a s p r o p o s e d b y N i r j a r , 9

o r a l t e r n a t iv e l y b y th e n e u r a l n e t w o r k a s p r o p o s e d b y

e q u a t i o n s ( 3 0 ) a n d ( 3 2 ) ,

7 . C O N C L U S I O N

N e u r a l n e t w o r k s c a n p r o v i d e a n a l t e r n a t i v e t o c o n -

v e n t i o n a l m e t h o d s b y p r o v i d i n g a n i n s id e r e l a t io n s h i p

in t h e f o r m o f g e n e ra l iz a t io n s b e t w e e n t h e p a r a m e t e r s

i n v o l v e d . T h e r o l e o f n e u r a l n e t w o r k s i n e x p e r i m e n t a l

i n v e s t ig a t i o n s h a s b e e n d i s c u s s e d , a n d t h e a u t h o r s h a v e

d e m o n s t r a t e d t h e i r p o t e n t i a l i t y i n ass i s t ing e x p e r i m e n -

t al w o r k . T h e p r i m a r y o b j e c t iv e w a s d e m o n s t r a t e d b y

s t u d y in g f e w e r t e st s p e c i m e n s a n d m a p s o f n -

d i m e n s i o n a l s p a c e t o t r a c k t h e r e m a i n i n g d a t a , w h i c h

u l t i m a t e l y r e su l t s i n re d u c i n g t h e n u m b e r o f s p e c i m e n s

t e s t e d , w i t h c o n s e q u e n t e c o n o m i c b e n e f i t s f o r th e

e x p e r i m e n t a l w o r k . T h e

c o n c e p t

a n d

m e t h o d o l o g y

c a n

b e a l s o i m p l e m e n t e d t o p e r f o r m a d d i t i o na l p r e l i m i n a r y

e x p e r i m e n t a l w o r k t o p r o v i d e i n f o r m a t i o n o n t h e r e la -

t i o n s h i p s b e t w e e n t h e m a t e r i a l s , l o a d i n g a n d g e o m e t r i -

c a l s h a p e .

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A U T H O R S B I O G R A P H I E S

D r J a d i d w a s a w a r d e d a B S C E ( C iv i l E n g i n e e r in g ) d e g r e e f r o m t h e U n i v e rs i ty o f W a s h i n g to n , S e a t tl e i n 1 9 81 a n d a n M S C E

( St r u c t u r a l E ng i nee r i ng) degr e e f r om Pennsy l van i a S t a t e Uni ve r s i t y in 1987 . In 1994 , he su ccess fu ll y com pl e t ed h i s Ph .D .

t hes i s a t E d i nb u r gh Uni ve r s i t y . H e i s cu r r en t l y an A ssoc i a t e Pr o fe s so r a t Ki ng F a i s a l Un i ve r s i t y , Sau d i A r ab i a . Rec en t j o i n t

pu b l i ca t i ons have b een i n Artificial Intelligence fo r E ngineering Desig n, a n d Expert Sys tems with Applicat ions .

D r F a i r b a i r n is D e p u t y H e a d o f th e D e p a r t m e n t o f C i v il a n d E n v i r o n m e n t a l E n g i n e e r i n g a t t h e U n i v e r s it y o f E d i n b u r g h . P a s t

pu b l i ca t i ons have i nc l u ded pap e r s i n t he A 71 Structural Journal, Magazine of Concrete Research a n d Ma sonry International.

neural-network applications

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