Beam Column Chen JCSR 1991

7/30/2019 Beam Column Chen JCSR 1991

http://slidepdf.com/reader/full/beam-column-chen-jcsr-1991 1/40

J. Co nstruct. Steel Researc h 18 (1991) 269-308

B e a m - C o l u m n D e s i g n i n S te e l F r a m e w o r k s - -I n s i g h t s o n C u r r e n t M e t h o d s a n d T r e n d s

J . Y . R i cha r d L i e w , D . W . W h i te & W . F . C h e nStructural E nginee ring D epa rtme nt, School of Civil Engineering, Purd ue U niversity, W est

Lafa yette, Indiana 47907, U SA

(Rece ived 20 Septem ber 1990; revised version received and accep ted 13 February 1991)

A B S T R A C T

The f i rs t par t o f the pa per describes the back grou nd behind the develop-

m e n t o f the curren t A IS C -L R F D beam -co l um n in te raction equa ti ons . Th i s

is f o l l ow ed by a r igorous eva l ua t ion o f the L R F D procedures w i th a

spec if ic foc us on the use o f e f fec ti ve l ength fac tors fo r beam -column design .

Va rious me thod s o f co m pu t ing effective length factors are reviewed, an d

characteris tic values obtained f r o m the di f ferent m ethod s are dem ons -

trated. Co m pa rison s are m ad e be tween ul t imate s trength curves repre-sen ted by the LRFD beam-column equat ions and s t rength curves com-

pu ted by sec ond -orde r inelastic analysis . Last ly , the pro cedu re in the

current Canadian S tandard C SA-S1 6.1-M 89, which does no t involve the

use o f e f fec ti ve l engt h f act ors , is pos ed i n L R F D f o rm a t an d com pared t o

the current L R F D metho d , which requires the use o f K fac tors . The goal is

to i llustrate the quali ties an d l imi tat ions o f both types o f design approaches.

N O T A T I O N

A F G e n e r a l t e r m f o r m o m e n t a m p l i f i ca t i o n f ac t o r s

B1 P - 6 m o m e n t a m p l i f i c a t i o n f a c t o r in L R F D ~,

Cm

B 2 P - ~ m o m e n t a m p l i f i c a t io n f a c t o r ,

1/ "oh no qn o,L X

1 - po \ /,H L269

J. Co nstruct. Steel Research 0143-974X/91/$03.50t~) 1991 Elsevier Science Pub lishers L td ,Eng land. Prin ted in G reat Bri tain



270 J. Y. Richard Liew, D. W. White, W. F. Chen

C~EE,

G, GA, GB

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g uM~x, Muy

P c l "

Be

eek

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e .e~r

( equ iva len t to - U2 of Sec t ion 8 .6 .1 , CSA -S 16 .12) o r

1( XPu ) i n e q n ( H 1 - 6 ) o f L R F D ( n o t a l lo w e d

1 - EP,'--"~ in CSA -S16 .12)

E q u i v a l e n t u n i f o r m m o m e n t f a c t o r fo r b e a m - c o l u m n s

M o d u lu s o f e l a s t ic i ty , a s su med eq u a l t o 2 0 0 0 0 0 N /m m 2

C o l u m n t a n g e n t m o d u lu s

R a t io o f t h e b en d in g s t if fn es s o f t h e co lu m n s v e rsu s t h a t o f t h e

b eams a t a b eam- co lu mn jo in t ( su b sc r ip t s ap p ly t o t h e

r e sp ec t iv e en d s o f t h e co lu m n )

H o r i zo n ta l f o r ce a t a f r am e f lo o r l ev e lM o m e n t o f in e r t ia o f t h e b e a m a n d c o l u m n c r o ss -s e ct io n s

E f f ec t i v e len g th f ac to rA c t u a l l en g t h o f a m e m b e r

A c t u a l l e n g th s o f b e a m s a n d c o l u m n s

M a x i m u m m o m e n t i n t h e m e m b e r c a u s e d b y l a t e ra l tr a n s la -

t ion o f the s to ry , ca lcu la ted by a f i r s t -o rder e last ic ana lys i s

M a x i m u m m o m e n t i n t h e m e m b e r a s s u m in g n o s t o ry tr a n s la -

t ion , ca lcu la ted b y a f i r s t -o rder e las tic ana lys isPlas t ic c ross - sec t ion bend ing c apac i ty

R eq u i r ed f l ex u r a l s t r en g th w i th in th e l en g th o f a me m b er

M a x i m u m s e c o n d - o r d e r el a st ic m o m e n t w i th i n th e l e n g t h o f a

m em b er ab o u t t h e s t r o n g an d w eak ax es o f th e c r o s s - sec t io n ,

r espec t ive ly

T h e e l a st ic b u ck l in g l o ad f r o m a b u ck l in g an a ly s isT h e E u le r b u ck l in g l o ad , ,rr2 EFL2

Elas t ic b uck l ing lo ad , 7 r2EI/(KL) 2, based on a va lue o f Ko b t a i n e d f r o m t h e a l i g n m e n t c h a r tE la s t i c b u ck l in g l o ad o f co lu mn i , , x 2 E I i / ( K i L i ) 2 , w i th K i

ca l cu l a t ed f r o m L eMessu r i e r ' s f o r mu la o r t h e mo d i f i ed K

f a c t o r f o r m u l a

N o m i n a l a x i a l c o m p r e s s i v e s t r e n g t h b a s e d o n t h e L R F Dc o l u m n s t re n g t h e q u a t i o n s ,Py(0.658~J) for A c - 1.5 or

R e q u i r e d a x ia l s t re n g t h f o r t h e m e m b e r u n d e r c o n s i d e r a t io n

A x ia l lo ad a t f u l l -y i e ld co n d i t i o n , i. e . , t h e sq u ash lo ad

R a d i u s o f g y r a t i o n



Beam -column design in stee l rame wo rks 271

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L o a d r a t io b e t w e e n a d j a c e n t fr a m e d c o l u m n s

Fi r s t -o rder in t e r - s to ry def l ec t ion

D e f l e c t io n a s so c i a te d w i t h m e m b e r c u r v a t u re , m e a s u r e d f r o m

t h e m e m b e r c h o r dL e n g t h r a ti o b e t w e e n a d j a c e n t f ra m e d c o l um n s

No r m a l i zed co l u m n s l en d e r n es s r a t io ( KL / ,r r ) X / ~y /E

F r a m e b u c k l in g lo a d f a c t o r

S t o r y b u ck l i n g l o ad f ac t o r

L R F D r es i st an ce f ac t o r f o r fl ex u r e , 0 . 9

L R F D r es i st an ce f ac t o r f o r co m p r es s i o n , 0 .8 5

M ax i m u m co m p r es s iv e r e s i d u a l s t re s s , a s su m ed eq u a l t o 0 .3 t ry

for a l l cases

Th e m a t e r i a l y i e l d s t re s s , a s su m ed eq u a l t o 2 50 N/ m m 2 f o r a l l

cases

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

Th e l i m i t - s t a t e s d es i g n o f b eam - co l u m n s i n b u i l d i n g f r am es h as b een a

su b j ec t o f r e sea r ch i n t e r e s t f o r m o r e t h an 2 0 y ea r s . A l m o s t a l l o f t h e

c u r r e n t d e s i g n m e t h o d s a r e b a s e d o n i n t e r a c t i o n e q u a t i o n s w h i c h ,t h r o u g h a co m b i n a t i o n o f an a l y ti ca l an d em p i r ica l m ean s , f it t h e u l t i m a t e

s t r en g t h o f an i n d i v i d u a l m em b er co n s i d e r i n g t h e e f f ec t s o f g eo m et r i c

i m p er f ec t i o n s an d r e s i d u a l s t r e s se s . B eam - co l u m n i n t e r ac t i o n eq u a t i o n s

can e s t i m a t e q u i t e accu r a t e l y t h e u l t i m a t e l o ad - ca r r y i n g cap ac i t y o f a

s i m p l y su p p o r t ed b eam - co l u m n wi t h eq u a l ex t e r n a l l o ad s ap p l ied a t b o t h

en d s . To g en e r a l i ze t h e eq u a t i o n s su ch t h a t t h ey m ay b e ap p l i ed f o r

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

m em b er s i n g en e r a l f r am ewo r k s , ce r t a i n f ac t o r s t h a t ap p r o x i m a t e t h eac t u a l b eh av i o r o f t h ese m em b er s m u s t b e i n t r o d u ced . Th e e f f ec t i v e

l e n g t h is t h e k e y f a c t o r th a t h a s b e e n e m p l o y e d i n m a n y d e s ig n m e t h o d s t o

es t i m a t e t h e i n f l u en ce o f t h e o v e r a l l f r am ewo r k o n t h e s t r en g t h o f

c o m p o n e n t b e a m - c o l u m n m e m b e r s . D u e t o t h e a p p ro x i m a t e a n d e m p ir ic -

a l n a t u r e o f b eam - co l u m n i n t e r ac t i o n eq u a t i o n s , a n d s i n ce t h e i n t e r ac t i o n

b e t ween t h e m em b er s i n an ac t u a l f r am e a t t h e s t r en g t h l i m i t s t a t e i s

c o m p l e x , b e a m - c o l u m n e q u a t i o n s c a n n o t i n g e n e r a l p r o v id e a c c u r at e

p r ed i c t i o n s o f t h e m em b er f a i lu r e lo ad s i n an o v e r a l l s t r u c tu r a l sy s t em .T h i s p a p e r p r e s e n t s s e v e ra l in s ig h ts o n c u r r e n t N o r t h A m e r i c a n

m et h o d s an d t r en d s f o r b eam - co l u m n d es i g n i n s t ee l f r am ew o r k s . S ec t i o n

2 o f t h e p ap e r p r o v i d es an o v e r v i ew o f t h e d es i g n g u i d e l i n es an d

p r o c e d u r e s i n v o lv e d in th e d e v e l o p m e n t o f th e A m e r i c a n I n s t i tu t e o f S te e l

C o n s t r u c ti o n L o a d a n d R e s is ta n c e F a c t o r D e s ig n ( A I S C - L R F D ) i nt er a c-




t i on equa t ions . 1Th e s ign i f icance o f the e f fec tive leng th fac to r i n the L R F D

beam-co lumn in t e rac t ion equa t ions i s d i scussed in Sec t ion 3 . In th i s

se c ti o n , t h e a c c u r a cy o f t h e L R F D i n t e r a c ti o n e q u a t i o n s f o r c h e c k i n g t h e

s t reng th o f beam -co lum ns a s pa r t o f a s truc tu ra l sys t em i s exam ined us ing a

s imple s t ruc tu ra l subassembly .

Se v e r a l m e t h o d s f o r e s t im a t i n g c o l u m n e f fe c ti v e le n g t h s in u n b r a c e d

f rames a re p resen ted in Sec t ion 4 . Cha rac te r i s t i c r e su l t s o f t hese e f fec t ive

l eng th fac to r ca l cu la t ions a re show n fo r simple por t a l f ram es wi th uneq ua l

co lumn s t i f fnesses , fo r l eaned co lumn f rames wi th unequa l ax ia l l oads in

the co lumns , and fo r a gene ra l mul t i - s to ry f rame . The qua l i t i e s and

l i m i ta t io n s o f t h e d i f f e r e n t m e t h o d s a r e e x p l a i n e d u s i n g th e se e x a m p l e s .

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

r e c o m m e n d e d p r o c e d u r e s f o r b e a m - c o lu m n d e s ig n i n t h e A I S C - L R F D 1a n d C a n a d i a n S t a n d a r d s Sp e c if ic a ti o ns . 2 T h e f o r m e r r e q u i r e s t h e u se o f

the e f fec t ive l eng th concep t whereas the l a t t e r e l imina te s i t s use fo r

unbraced f raming sys t ems . A gene ra l t r end o f t hese spec i f i ca t ions i s

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

cho ice fo r ob ta in ing des ign fo rces . Bo th spec i f i ca t ions requ i re the

ca lcu la t ion o f second-o rde r e l a s ti c fo rces in f ram e m em bers . Th e s t ruc tu -

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

e q u a t i o n s .f u n d a m e n t a l l y n o t c o m p a t i b le . O f c o u r se , t h e o n l y wa y o f p r o p o r t i o n i n g

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

r a t i o n a l i s t h r o u g h t h e u se o f s e c o n d - o r d e r i n e la s t ic an a l y si s . H o w e v e r ,

s e c o n d - o r d e r i n e la s t ic a n a ly s i s m e t h o d s a r e o n l y b e g i n n i n g to b e c o m e

p r a c t ic a l f o r a c tu a l d e s i g n as a n a ly s i s m e t h o d s a r e i m p r o v e d a n d g e n e r a l

c o m p u t i n g p o w e r i s i n c r e ase d . T h e l a s t p a r t o f t h e p a p e r d is c u sse s p o s s i b le

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

m e m b e r f o r t h e d e t e r m i n a t i o n o f o v e r a l l sy s t e m p e r f o r m a n c e . T h eprob lem s a ssoc ia t ed w i th the use o f e l a s ti c -pe r fec tly p l a s t ic ( i . e . , p l a s t i c

h inge ) a na lys i s i n gen e ra l f r ame d es ign a re addres sed , and poss ib l e

d i rec t ions fo r use o f ine l a s t i c ana lys i s a s a t oo l fo r m ore ra t iona l des ign o f

s t ee l beam-co lumns a re iden t i f i ed .

2 T H E A I S C - L R F D B E A M - C O L U M N I N T E R A C T I O N

E Q U A T I O N S

Fo r t h e d e s i g n o f st e e l b e a m - c o l u m n s , t h e A I S C - L R FD sp e c if ic a ti o n 1

p r o v i d e s t h e f o l lo wi n g in t e r a c t i o n e q u a t i o n s :

Pu 8 ( Mux MuY ) = 1 " 0 for Pu >-0"2 (1)~bc----P-ff+ ~bb Mnx ~- ¢ b M .y ~b¢- - - ~



Beam--column design in steel rameworks 273

Pu t M__________~_~ Muy = 1.0 for Pu < 0.2 (2)2 5 c P n ¢t~bMnx ¢~bbMny ~c p n

w h e r e P u is h e r e q u i r e d a x i a l t r e n g th ; ~ is h e n o m i n a l ax i a l o m p r e s s i v e

s t r e ng t h ; M ~ , a n d M u y a r e t h e r e q u i r e d f l e x u r a l s t r e n g t h w i t h i n t h e

u n b r a c e d l e n g t h o f t h e m e m b e r ; M . x a n d M n y a r e t h e n o m i n a l f l e x u r a l

s t r en g t h s; a n d ~'c a n d ~ a r e t h e r e s i st a n c e f a c t o r s f o r c o m p r e s s i o n a n d

f l e x u r e .

T h e L R F D s pe ci fi ca ti on u g g e s t s t h e f o l l o w i n g p r o c e d u r e t o e s t i m a t e

t h e s e c o n d - o r d e r el as ti c o m e n t M u i n l i eu f a d i r e c t e c o n d - o r d e r e la s ti c

ana lys i s :

Mu = B1Mnt + B2Mlt (3)

In th i s equa t ion , M~t and M~t a re the m axim um f i rs t -o rde r el a s ti c m om ent s

i n t h e m e m b e r b a se d o n ' n o t r a n s l a t i o n ' a n d ' l a te r a l t r a n s l a t io n ' a n a l y se s

of the f rame re spec t ive ly . B1 and B2 a re moment ampl i f i ca t ion fac to r s

w h i c h a c c o u n t f o r P - 8 a n d P - A e ff ec ts .

2 . 1 D e v e l o p m e n t o f t h e L R F D i n te r a c ti o n e q u a t i o n s - - g r o u n d r u le s

I n t h e d e v e l o p m e n t o f t h e L R F D i n t e r a c t i o n e q u a t i o n s , t h e f o l l o w i n g

g u i d e l i n e s w e r e e s t a b li sh e d : 3

( 1) T h e e q u a t i o n s sh o u l d b e a p p l i ca b l e t o a w i d e r a n g e o f p r o b l e m s

su c h a s s tr o n g - a n d we a k - a x i s b e n d i n g , swa y a n d n o n - swa y f ra m e s ,

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

s l e n d e r n e s s r a t io s a n d d e g r e e s o f e n d r e s t r a in t , a n d l e a n e d c o l u m n

sy s t e m s . T h e y sh o u l d a c c o u n t f o r i n e l a s t i c b e h a v i o r a n d s e c o n d -

orde r e f fec t s .

( 2) T h e e q u a t i o n s sh o u l d c l e a r l y d i s ti n g u i sh b e t w e e n t h e l o a d e f f e c tsand the re s i s t ances so tha t seco nd-o rde r e l a s ti c ana lys i s can be eas i ly

a c c o m m o d a t e d .

( 3 ) T h e e q u a t i o n s sh o u l d b e b a se d o n t h e l o a d e f f e c t s o b t a i n e d f r o m

secon d-orde r e las tic ana lysis s ince , a t the pr esen t t ime, secon d-order

ine la s t ic ana lys i s i s no t r ead i ly access ib l e fo r des ign o f fi ce use .

( 4) T h e e q u a t io n s s h o u l d n o t be m o r e t h a n 5 % u n c o n s e r v a ti v e w h e n

c o m p a r e d t o s e c o n d - o r d e r p l a st ic - z o n e so l u ti o n s .

( 5 ) T h e e q u a t i o n s sh o u l d n o t n e e d t o c o n s i d e r s t r e n g t h a n d s t a b i l i t y

sep a ra t e ly , s ince in gen e ra l , a l l co lum ns o f f in it e l eng th fa il by som e

c o m b i n a t i o n o f i n e l a s ti c b e n d i n g a n d s t a b i li t y e f fe c ts .

( 6 ) T h e s a m e u l t i m a t e s t r e n g t h sh o u l d b e p r e d i c t e d f o r s i m i l a r m e m -

b e r s , su c h a s t h e o n e s sh o w n i n F ig . 1 . T h e b e h a v i o r i n e a c h o f t h e se

prob lem s i s phys ica l ly iden t i ca l in bo th the e l a s ti c an d ine l a s ti c

r a n g e s .



2 7 4 J. Y. Richard Liew, D. W. White, W. F. Chen

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F i g . 1 . B e a m - c o l u m n p r o b l e m s w h i c h h a v e id e n t i c a l s o l u ti o n s .

( 7) A d j u s t m e n t o f e f f e c ti v e l e n g t h b a s e d o n c o l u m n in e l a st ic i ty s h o u l d

b e a l l o w e d .

2.2 Development of the LRFD interaction eq u at io~ procedure

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

g e n e r a t e d a n d r e p o r t e d b y K a n c h a n a la i . 4 T h e L R F D i n te r a c t io n e q u a -

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

o b t a i n e d f r o m t h e s e i n el a st ic a n al y se s . T h e p r o c e d u r e u s e d i n d e v e l o p i n g

t h e L R F D i n t e r a c t i o n e q u a t i o n s is o u t l i n e d b e l o w : 3'5

( 1) U l t i m a t e s t r e n g t h c u rv e s d e t e r m i n e d i n K a n c h a n a l a i ' s w o r k w e r e

p r e s e n t e d i n a n o n d i m e n s i o n a l i z e d f o rm . T h e n o r m a l i z e d a xia l l o a dP/Py w a s p l o t t e d a g a i n st t h e n o r m a l i z e d f i rs t - o rd e r m o m e n t M1/Mp.T y p i c a l r e s u l t s f o r a b e a m - c o l u m n b e n t a b o u t t h e w e a k a x i s a n d

h a v i n g L/r = 1 00 a r e s h o w n i n F i g . 2 . A l l t h e m e m b e r s w e r e

a s s u m e d t o b e p e r f e c t l y s t r a i g h t w i t h a m a x i m u m c o m p r e s s i v e

r e s i d u a l s t r e s s o f tr ~ = 0 .3 0 ry a n d a y i e l d s t r e s s o f O-y = 2 5 0 N / m m 2 .

I t s h o u l d b e n o t e d t h a t t h e b e h a v i o r o f t h e t w o m e m b e r s s h o w n i n

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

( 2 ) T h e n o r m a l i z e d first-order m o m e n t M1/Mp, o b t a i n e d f r o m t h ep l a s t i c - z o n e a n a l y s i s , w a s c o n v e r t e d t o a second-order elasticm o m e n t M2/Mp u s in g t h e s e c o n d - o r d e r e la s ti c m o m e n t a m p lif ic a -

t i o n f a c t o r . F o r t h e b e a m - c o l u m n s s h o w n i n t h e f i g u r e , t h e

s e c o n d - o r d e r e l a s ti c m o m e n t a m p l if ic a t io n f a c t o r is t a n a/a, w h e r e

a = 0-5 ~" V'ffTP~.



Beam -column design in steel ram eworks 275

P

0.8

0,7

0 .6

0 .5

0.4

0 .3

0 .2

0.1

- t P t p

~ ~ Kanch ana la i , 1977 - ~ ~ ~ ~

- x \1 I I •

0.25 0.5 0.75 1.0

M 1 IMp o r M 2 IMp

Fig. 2. Procedure for obtaining the ne t MI/M p and M2[Mp cu rv e s .

( 3) T h e c o l u m n s t r e n g t h o f a n a x ia l ly l o a d e d m e m b e r w i t h L / r = 100 is

g i v e n b y t h e L R F D c o l u m n - s t r e n g t h e q u a t i o n a s 0 -5 91 Py . Ho w e v -

e r , t h e c o l u m n s t r e n g t h ( f o r z e r o b e n d i n g m o m e n t ) c o m p u t e d b y

K a n c h a n a l a i ' s p l a s t ic - z o n e a n a l y s i s is a p p r o x i m a t e l y e q u a l t o0 .7 1 Py. T h i s d is c r e p a n c y is d u e t o t h e f a c t t h a t t h e L R F D c o l u m n

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

i n i ti a l ly c r o o k e d , h a v i n g a m a x i m u m m i d - l e n g t h d e f le c t io n o f

a p p r o x i m a t e l y L / 1 5 0 0 . However , t he ine l a s t i c ana lys i s i s fo r a

p e r f e c t ly s tr a i g h t m e m b e r . T o i n c lu d e t h e ' i m p e r f e c t i o n e f f e ct ' in

t h e b e a m - c o l u m n i n t e r a c t i o n e q u a t i o n s , t h e ' u se d - u p ' f i r s t - o r d e r

m o m e n t c a p a c i ty d u e t o c o l u m n i n i ti a l i m p e r f e c t io n s wa s a s su m e d

t o v a r y l i n e a r l y w i t h t h e a x i al lo a d , a s sh o wn i n F i g . 2 . T h e n e t u se f u lf i r s t - o r d e r m o m e n t c a p a c i t y wa s t h e n o b t a i n e d b y su b t r a c t i n g t h e

' u se d - u p ' m o m e n t f r o m t h e f i r s t - o r d e r m o m e n t M1 . T h i s v a l u e ,

w h e n m u l t ip l i e d b y t h e e l a s ti c m o m e n t a m p l i f ic a t io n f a c t o r y i e ld s

t h e n e t u se f u l s e c o n d - o r d e r m o m e n t c a p a c i t y ( sh o wn a s t h e n e t

ME/M p in Fig . 2) .



276 J . Y . Richa rd L iew , D . W . W h i t e , W . F . Chen

0 .8

P

Py

0.7

0 .6

0 .5

0 .4

0 .3

0 .2

0.1

{ P ~ P

4 - - H

N et M 2 = N e t M I x tan..__..~ 2 O R

Q

Pn 0 .5 9 1 o t = ~ - ~ . ~ H H

t p tP y

" ~ M p ~ L R F D ( E q n s 1.2)

" M r, ~ ' - ~ . . . ~ ~ M 2

L R F D M.I ~ " - ~ -" ~ " ~

• . / " - . ' ~ . - ~ ' ~

N e t @ C u r v e ~" - .~ . ~ * .~a p " ; " ~ ll~ t, ~ ' ~ "

0 .25 0 .5 0 .75 1 .0M

M p

Fig. 3 . C u r v e - fi tt in g o f t h e A I S C -L R a R D b e a m - c o l u m n e q u a t i o n s .

( 4 ) T h e n e t r e su l ts ar e p l o t t e d i n F i g . 3. T h e L R F D e q u a t i o n s a r e

o b t a i n e d b y c u r v e fi tt in g t o th i s n e t s e c o n d - o r d e r m o m e n t . S i m i la r

c u r v e s h a v e b e e n g e n e r a t e d f o r a w i d e v a r ie t y o f c a s e s in c l u d i n gs tr o n g - a n d w e a k - a x i s b e n d i n g , s w a y a n d n o n - s w a y , a n d l e a n e d

c o l u m n s ys te m s . T h e L R F D i n te r a c ti o n e q u a t i o n s ar e n o m o r e t h a n

5 % u n c o n s e r v a ti v e w h e n c o m p a r e d t o a ll o f th e c o m p u t e d r e su lt s

f o r in - p l a n e s t r o n g -a x i s a n d w e a k - a x i s b e n d i n g .

T h e w e a k a x i s L /r = 1 0 0 c a s e w a s t h e m o s t c r i t i c a l o f a l l t h e

c o m p u t e d r e su lt s s f o r th e b e a m - c o l u m n s s h o w n i n F i gs 2 a n d 3 . T h e

L R F D e q u a t i o n s s h o w a n e x c e l l e n t f it t o t h is c u r v e . I n g e n e r a l , t h e

e q u a t i o n s a l s o g i v e a s u p e r b f i t f o r a ll s t r o n g - a x i s c a s e s o f t h e t y p es h o w n i n F i g . 3 fo r L /r r o m 0 t o 1 0 0 , b u t t h e y a r e v e r y c o n s e r v a t i v e

f o r t h e w e a k - a x i s c a s e w h e n L /r r a n g e s fr o m 0 t o 4 0. T h e y a r e

m o d e r a t e l y c o n s e r v a t i v e f o r b o t h a x e s w h e n L /r s g r e a t e r t h a n 1 2 0 .

( 5 ) T h e f i n al L R F D e x p r e s s i o n s a r e o b t a i n e d b y i n c o r p o r a t i n g t h e

r e s i s t a n c e f a c t o r s , (/)b a n d d )c .



B eam-colu mn design n teel ram ewor ks 277

3 S I G N I F I C A N C E O F T H E E F F E C T I V E L E N G T H F A C T O R I N

T H E L R F D B E A M - C O L U M N E Q U A T I O N S

M a n y f o r m s o f b e a m - c o l u m n i n t er a c ti o n e q u a t i o n s h a v e b e e n p r o p o s e d i np r e v i o u s w o r k b a s e d o n t h e u s e o f a n e f f e c t iv e l e n g t h f a c to r o f 1 . 0. F i g u r e 4

c o m p a r e s t h e r e su lt s o b t a i n e d u s in g t h e S S R C P - A m e t h o d 6 w i t h th e

p l a s t ic - z o n e s o l u t i o n 4 f o r a s i m p l e p o r ta l f r a m e . T h e e x a m p l e f r a m e h a s a

r i g id g i r d e r a n d a p i n n e d b a s e . T h e t w o i n t e r a c t i o n e q u a t i o n s o f t h e P - A

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

e x a m p l e . F o r p u r p o s e s o f c o n s i s te n c y w i t h t h e o th e r p r o c e d u r e s d i s c u s se d

in th e p a p e r , t h e L R F D c o l u m n s tr e n g th f o r m u l a s ar e u s e d h e r e t o

c o m p u t e Pn - I n t h e P - A a p p r o a c h , t h e c o u p l e d m e m b e r a n d s y s t e m

s ta b il it y e f fe c t s o n b e a m - c o l u m n s tr e n g t h a re c o n s i d e r e d o n l y i n t h e

P

Py

1 . 0 !

~ P 0.6M*' - - + 1.0

0 .8 P nK ~ O 1 1 - ~ - ) M p~ , g - l o

P M " H Lc- - + = 1 .0 , M*

0.6 ~, Py Mp 1- PL2c / 13E [ c )

Unco nservat i~ % ax is }e'~ j StrongW eak a x i s ~ r~ an ~.an aD a, 1977)

0 ,4 ~ G 0

H lb GB = ooA

0 .0 0.2 0.4 0.6 0.8 1.0

H Lc / 2Mp

Fig. 4. Com parison of the SSR C P - A equations6 with ultima te strength curves based oninitially perfect ge om etry.L




e q u a t i o n w h i c h c o n t a i n s P n a n d P e ( t h e o t h e r e q u a t i o n o n l y c o n t a in s

c r o s s - se c t io n s t r e n g t h te r m s ) . T h e P - A m e t h o d 6 s u g g e s t s t h a t K = 1 .0

m a y b e u s ed to e v a lu a t e P n w h e n M * is b a s e d o n a P - A t y p e o f

s e c o n d - o r d e r e l a st ic a n a ly s is . T h i s is a n a t t e m p t t o s e p a r a t e m e m b e r

s ta b i li ty f r o m f r a m e s t a b i li ty b y c o n s i d e r i n g f r a m e s ta b i li ty o n l y t h r o u g h

t h e c a l c u la t io n o f M * . H o w e v e r , th i s a p p r o a c h d o e s n o t c o r r e la t e w e l l w i t h

' e x a c t ' i n e l a s ti c s t r e n g t h c u r v e s s u c h a s K a n c h a n a l a i ' s . I n a c t u a l it y , t h e r e is

m o r e c o u p l i n g b e t w e e n s y s te m a n d m e m b e r s ta b il it y e f fe c ts t h a n t h e P - A

i n t e ra c t i o n e q u a t i o n s im p l y . F i g u re 4 s h o w s th a t t h e P - A m e t h o d i s i n

g e n e r a l u n c o n s e r v a t i v e c o m p a r e d t o t h e ' ex a c t ' i n e la s ti c s t re n g t h . F u r t h -

e r m o r e , t h e m e t h o d p r o d u c e s d i f f e r e n t r e su l ts f o r t h e d i f f e re n t p r o b l e m s

s h o w n in F ig . 1 .

I f K i s t a k e n a s u n i t y , t h e e r r o r i n u s i n g t h e p r e s e n t A I S C - L R F Di n t e r a c t i o n e q u a t i o n s is a ls o u n a c c e p t a b l y l a r g e ( s e e F ig s 5 a n d 6 ). F i g u r e 5

is a c o m p a r i s o n o f th e s t r e n g t h c u r v e s d e t e r m i n e d b y K a n c h a n a l a i ( w h i c h

a r e b a s e d o n a f r a m e w i t h n o i n it ia l o u t - o f - p lu m b n e s s o r c o l u m n s w i th n o

i ni ti al o u t - o f- s tr a ig h t n e s s) v e rs u s th e A I S C - L R F D b e a m - c o l u m n s tr e n g t h

c u r v e s . F i g u r e 6 c o m p a r e s t h e c u r v e s a f t e r K a n c h a n a l a i ' s d a t a i s a d j u s t e d

b y t h e p r o c e d u r e s d i s c u s s e d i n S e c t i o n 2 . 2 t o a c c o u n t f o r i m p e r f e c t i o n s .

T h e e r r o r s a s s o c i a t e d w i th u s in g K = 1 a r e p a r ti c u l a rl y l a r g e f o r m e m b e r s

o f i n t e r m e d i a t e t o l o n g l e n g t h s s u b j e c t e d t o r e l a ti v e l y h i g h a x ia l l o a d s . O nt h e o t h e r h a n d , f o r t h e f r a m e s u b a s s e m b l a g e c o n s i d e r e d h e r e , t h e

e q u a t i o n s g i v e a g o o d f it t o t h e m o r e e x a c t s o l u ti o n w h e n a K o f 2 . 0 is u s e d .

A l t h o u g h a c u r r e n t t r e n d i s t o a v o i d th e u s e o f e f f e c ti v e l e n g t h f a c to r s i n

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

f o r m u l a t e a n i n t e r a c t i o n e q u a t i o n t h a t i s a c c u r a t e f o r a l l r a n g e s o f a x i a l

l o a d a n d m o m e n t w i t h o u t c o n s i d e r i n g t h e e f f e c ti v e l e n g t h d ir e c tl y . T h i s

a s p e c t i s d i s c u s s e d in m o r e d e t a i l in S e c t i o n 6 .

H o w e v e r , i t s h o u l d b e n o t e d t h a t t h e u s e o f K = 1 .0 is o n l y sl ig h t lyu n c o n s e r v a t i v e a s l o n g a s t h e a x i a l l o a d i n t h e b e a m - c o l u m n i s n o t

e x c e ss iv e . T h e C a n a d i a n S t a n d a r d 2 p r o p o s e s a b e a m - c o l u m n e q u a t i o n

w h i c h u s e s K = 1 . 0 , b u t it i n d i r e c t l y l i m i t s t h e a x i a l c a p a c i t y b y i m p o s i n g a

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

g r a v it y lo a d e d f r a m e s b e s u b j e c t e d t o a m i n i m u m l a t er a l lo a d a s s o c ia t e d

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

C a n a d i a n S t a n d a r d i n t e r a c ti o n e q u a t i o n s f o r b e a m - c o l u m n d e s i g n a r e

d i s c u s s e d in S e c t i o n 6 .

F o r u n b r a c e d f r a m e s , t h e L R F D s p e ci fi ca ti o n r e q u i r e s th a t P n b e

c a l c u l a t e d b a s e d o n K - > 1 . I f a f ir s t - o r d e r e l a s ti c a n a l y s is is e m p l o y e d , B 1

s h o u l d b e d e t e r m i n e d b a s e d o n t h e e la s t ic b u c k li n g l o a d o f t h e m e m b e r ,

P e , w i t h K - 1 ( i . e ., th e c o l u m n is a s s u m e d t o b e p r e v e n t e d f r o m

s i d e - sw a y ) . F o r b r a c e d f r a m e s , i t is a d v i s a b l e t o u se K = I f o r e v a l u a t i o n



B eam-co l u m n desi gn n st eel r am ewo r ks 2 7 9

1 .0

0 .8

P

P y

0 .6

0 .4

0 .2

0 .00 .0

Strong a x i s.../ i ; : _ / W e a k a x 's } ( Ka n ch a na la i' 1 9 77 )

~:~F... / G A - 0~,. Ga-¢°"..~,a(X."-, L/r - 40

" . . " ,. LRF D (K = 1 .0)

L c I e

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

H L c / 2 M p

F i g. $ . C o m p a r i s o n o f t h e A I S C - L R F D b e a m - c o l u m n e q u a ti o n s w i t h e x a c t u l ti m a t e

s t r e n g t h c u r v e s f o r i n i t ia l l y p e r f e c t g e o m e t r y . 4

1 .0

0 .8

P

0 .6

0 .4

0 .2

0 .00 .0

• .. GA " 0GB - ,aD

LR FD (K = 1 . 0 ) L / r = 40

• . . . . . ' " . . . . . . t ,~ °a~gaxa~;s Ka nch an ala,ad j us t ed ,

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

HL~/2M p

F i g. 6 . C o m p a r i s o n o f t h e A I S C - L R F D b e a m - c o l u m n e q u a t io n s w i th t h e u l t im a t e s tr e n g th

c u r v e s4 a d j u s t ed t o a c c o u n t f o r f r a m e i m p e r f e c t io n s .



280 j. Y. Richard Liew, D. W. White, W. F. Chen

of P,. If the designer relies on the beams to restrain the columns (i.e., if

K-< 1 is used in the determination of P.) the beams should be designed to

carry the second-order moments induced by the columns.7

For an unbraced frame, the calculation of B2 by LRFD equation H1-6

involves the calculation of Pek for all the columns in a story which offer

restraint to side-sway buckling. K is generally assumed to be greater than

one for these cases--al though for some of the procedures discussed in the

next section, the computed K factor may be less than one. As an

alternative to the use of the B1 and B2 factors, the second-order elastic

moments may be obtained directly from a second-order elastic analysis.

However, an accurate calculation of member P - 8 moments is generally

not provided by most second-order analysis programs. 8

The P. term in the LRFD interaction equations is governed by the

maximum column slenderness ratio KL/r for strong- or weak-axis

bending. The term Pek, used in the calculation of the B1 and B2 factors, is

always governed by the slenderness ratio in the plane of bending. In

general, the design thus involves the calculation of the effective length for

both axes of bending. Therefore, a correct interpretation of K factors in

the various terms of the interaction equation is essential.

In the following sections, various methods of computing the K factors in

unbraced frames are examined. Their use with the LRFD equations forchecking beam-column strength is discussed using several simple structu-

ral subassemblies.

4 COMPUTATION OF THE EFFECTIVE LENGTH FACTOR

A variety of methods have been suggested for calculation of K factors.

These methods may be grouped according to those associated with: (1) thebuckling of an idealized subassemblage, (2) the buckling of a story, and (3)

the buckling of the overall structural system. Some of the methods are

suitable for hand calculations. Others are obtained typically from com-

puter solutions. All of the methods involve certain assumptions and

simplifications. Some of the most popular methods are briefly reviewed in

the following sections.

4 . 1 A l i g n m e n t c h a r t m e t h o d (b a s e d o n t h e bucklingo f a n i d e a li z e d

s u b a s s e m b l a g e )

The most widely used method for calculating K factors in rectangular

building frameworks is the alignment chart method.9 For an unbraced

frame, this method is based on the buckling of the subassemblage shown in



Beam --column design in steel~rameworks 281

( a )

J l3 /

GA K

oo ' 20.0

100.0 10.050.030.0 5.0

20.0 4.0

10.09 . 0 -

8 . 0 -7 .0 -

6.0

5 . 0 -

4 .0

3 .0

2 .0 -

1.0.

3.0

2.0

1.5

1.0

GB

~oo

-100.0

50.0

30.0

20.0

10.09.0

8 .07.0

6.0

5.0

4.0

3.0

2.0

1.0

( b )

F i g . 7 . ( a ) Subassemblagem o d e l o f a n u n b r a c e d f r a m e ; ( b ) a l ig n m e n t c h a r t K f a c t o r f o r

u n b r a c e d f r a m e s .



282 J. Y. Richard Liew, D . W . W hite , W . F. Chen

F ig . 7 a. T h e b a s ic m e t h o d i n v o l v e s t h e f o l l o w i n g m a i n a s s u m p t i o n s : ( 1 ) A l l

m e m b e r s a r e p r i s m a t i c a n d a ll j o i n t s a r e r ig i d . ( 2 ) T h e c o l u m n s ti ff n e ss

p a ra m e te r L ~ ] s i d en ti c al f o r a l l c o lu m n s . T h a t is , a t i n c ip ie n t

b u c k l in g , n o n e o f t h e o t h e r p o r t i o n s o f t h e f r a m e c o n t r i b u t e s a n y l at e ra l

r e s tr a in t a t t h e b o u n d a r i e s o f t h e s u b a s s e m b l a g e , a n d a ll m e m b e r s a r e b e n t

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

m e m b e r . ( 3 ) A x i a l f o r c e s i n t h e g i r d e r s a r e n e g l i g i b l e . ( 4 ) T h e r a t i o

b e t w e e n t h e b e a m a n d c o l u m n b e n d i n g s ti ff ne ss is u n a f f e c te d b y d i s tr i-

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

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

m e m b e r st if fn e s se s ). ( 5) T h e r o t a ti o n s a t o p p o s i te e n d s o f th e b e a m s a r e

e q u a l in m a g n i t u d e , p r o d u c i n g r e v e r s e c u r v a t u r e b e n d i n g .

A s a r e s u l t o f t h e s e a s s u m p t io n s , t h e b u c k l i n g s o lu t i o n fo r t h e u n b r a c e ds u b a s s e m b l a g e s h o w n in F ig . 7 c a n b e o b t a i n e d f r o m t h e t r a n s c e n d e n t a l

e q u a t i o n ,

GA GB (~/K ) 2 - 36 (1r/K) - 0 (4)

6 ( G A + G s ) t an (Tr/K)

w h e r e G A a n d G s a r e t h e c o l u m n to b e a m s ti ff n es s r a ti o s a t th e t w o

c o l u m n e n d s . T h e s e t e rm s c a n b e w r i tt e n a s

Z ( E l c

G A =

GB ~ °' °E / ( 6 )

T h e s o l u t io n o f e q n ( 4) c a n b e e x p r e s s e d i n t h e c o n v e n i e n t f o r m o f t h e

a l i g n m e n t c h a r t s h o w n i n F ig . 7 b. S i n c e th e a b o v e e q u a t i o n s a s s u m e t h a t

a l l j o i n t s a r e r i g i d , t h e g i r d e r s t i f f n e s s ( I / L ) b n e e d s t o b e m o d i f i e d a s

f o l lo w s i f t h e g i r d e r s h a v e d i f f e r e n t b o u n d a r y c o n d i t i o n s a t t h e i r f a r e n d s .6

F o r u n b r a c e d f r a m e s ,

( i) i f t h e f a r e n d o f t h e g i r d e r is fi x ed , d i v i d e ( I / L ) b by 1 .5 ,( ii ) i f t h e f a r e n d o f t h e g i r d e r is h i n g e d , d i v i d e ( l / L ) b b y 2 . 0 .

I f a c o l u m n i s l o a d e d i n t h e i n e la s ti c r a n g e w h i le t h e a d j a c e n t b e a m s

r e m a i n e l as t i c , th e b e a m s b e c o m e m o r e e f f e c ti v e i n r e s tr a i n in g t h e c o l u m n

a g a in s t b u c k l i n g ; t h e r e f o r e , t h e e f fe c ti v e l e n g t h o f t h e c o l u m n i s r e d u c e d .



Beam-columndesign n steel ram eworks 283

A s a re s u l t , t h e b u c k l i n g l o a d o f t h e c o l u m n i s h i g h e r t h a n t h a t c a l c u l a t e d

b a s e d o n a n e l a s t i c K - f a c t o r . T h e A I S C - L R F D M a n u a l 1 s u g g e s t s th a t i n

d e t e r m i n i n g t h e e f fe c t i v e l e n g th f o r th e P n t e rm i n t h e b e a m - c o l u m n

i n t e r a c t i o n e q u a t i o n s , t h e c o l u m n t o b e a m s t if f n es s r a t io s i n e q n s ( 5 ) a n d( 6 ) m a y b e c a l c u l a t e d a s

G i n e l a s t i c =

E ( EtE E IcL c

/(7 )

t o a c c o u n t f o r th e i n e l a s t i c it y i n t h e c o l u m n s . T h e t e r m E t / E i n th i s

e q u a t i o n i s r e f e r r e d t o a s t h e i n e l a s t i c s ti f fn e s s r e d u c t i o n f a c t o r , a n d i t i s i n

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

in t h e L R F D m a n u a l a s

Et Pn inelast ic 0" 65 8A- - ~ - = < 1 - 0 ( 8 )

E P n e l a st ic 0 " 8 7 7 /A -

w h e r e A i s d e f i n e d i n t e r m s o f P/Py a s

~ l n (eu/Py)A = In 0.658 (9)

T h i s e q u a t i o n is o b t a i n e d b y s o lv i n g t h e L R F D i n e l a s t ic c o l u m n s t re n g t h

e x p r e s s i o n f o r t h e s le n d e r n e s s p a r a m e t e r it c o r r e s p o n d i n g t o P n = P u. T h e

v a l u e o f i t i s l e s s t h a n 1 . 5 f o r c o l u m n s w h i c h f a i l i n t h e i n e l a s t i c r a n g e .

U s i n g e q n ( 9 ), a c o r r e s p o n d i n g it c a n b e o b t a i n e d f o r a g i v e n P J P y . T h ei n e l a s ti c s ti ff n e s s r e d u c t i o n f a c t o r c a n t h e n b e c a l c u l a t e d b y s u b s t i t u t i n g

e q n ( 9 ) i n t o e q n ( 8 ) . T h e r e s u l t i n g v a l u e s o f E t / E a r e s h o w n i n F i g . 8 . I ts h o u l d b e n o t e d f r o m F i g . 8 t h a t t h e i n e l a s t ic st if f n es s r e d u c t i o n f a c t o r i s

l es s t h a n o n e o n l y w h e n Pu/Py >-0 . 3 9 , a n d i t s u s e i s l i m i t e d o n l y t o t h e a x i a l

r e s i s t a n c e t e r m i n t h e b e a m - c o l u m n i n t e r a c t i o n e q u a t i o n . E l a s t i c e f f e c ti v e

l e n g t h s m u s t b e u s e d w i t h t h e B 1 a n d B 2 t e r m s s i n c e s e c o n d - o r d e r elasticm o m e n t s a r e t o b e c o m p u t e d u s i n g t h e s e te r m s .

4 . 2 L e M e s s u r i e r ' s m e t h o d ( b a s e d o n t h e b u c k l i n g o f a s t o r y )

T h e a l i g n m e n t c h a r t m e t h o d is b a s e d o n t h e a s s u m p t i o n t h a t a ll c o l u m n s i n

a s to r y b u c k l e s i m u l t a n e o u s l y , w i t h o u t a n y p a r t ic u l a r c o l u m n s p r o v i d i n g

l a te r a l r e s t ra i n t t o o t h e r c o l u m n s . L e M e s s u r i e r p r o p o s e d a m e t h o d f o r

u n b r a c e d f r a m e s i n w h i c h t h e l a te r a l r e s tr a i n i n g e ff e c t s p r o v i d e d b y t h e

c o l u m n s i n a s t o r y h a v i n g s m a l l e r st if f n es s p a r a m e t e r s ( L ~ / P u / E I ) c a n b e



2 8 4

1.0

0 .9

0 . 8 i

0 .7

0 .6

Et 0.5E

0 ,4

0 ,3

0 .2

0.1

J . Y . R ichard L iew , D . W . White, W. F . Chen

, ,

il j 0 . 3 9 | a i ~ I , ~,0 .1 0 .2 0 .3 0 .4 0 ,5 0 .6 0 .7 0 .8 0 .9 1 .0

P / P y

F i g . 8 . C o l u m n i n e l a s t i c s t i f f n e ss r e d u c t i o n f a c t o r .

a c c o u n t e d f o r . 10 A m o d i f i e d f o r m o f h is f o r m u l a f o r t h e e f f e c t i v e l e n g t h

f a c t o r c a n b e w r i t t e n a s

~ e t , [ ~ eu + yC , eu ]K i 2 = P u i L i 2 L X P L

( 1 0 )

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

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

A F =E P u ) ( 1 1 )

1 - y pL _ Y~ ( CL P u)

w h e r e

e u i

] i

L iPL

= s u m o f a ll v e r t ic a l fo r c e s a c ti n g o n t h e s t o r y a t t h e f a c t o r e d l o a d l e v e l

= a x i al f o r c e o n c o l u m n i

= m o m e n t o f i n e r t i a o f c o l u m n i

= h e i g h t o f c o l u m n i= t h e f o r c e th a t p r o d u c e s a u n i t r o t a t i o n a l d i s p l a c e m e n t A / L o f t h e

m e m b e r . P L m a y b e w r i t te n a s



Beam --column design in steel ram ewo rks 285

CL = stiffness correction factor for a column, w hich accounts for P -effects and can be w ritten as

K2

= column e nd re strain t coefficient defined as

6 (GA + GB) + 36

2(GA + G B ) + G A G B + 3

N o t e t h a t t h e m e m b e r K f a c t o r in th e e x p r e s s i o n f o r t h e C I. t e r m i s

o b t a i n e d f r o m t h e a l i g n m e n t c h a r t p r o c e d u r e . Fo r a g e n e r a l c a se , t h e

p a r a m e t e r s PL a n d C L m a y b e o b t a i n e d a l so b y d i re c t a n a ly s i s.

I f t h e L R FD B 2 f a c t o r ( e q n H1 - 6 o f t h e L R FD sp e c i f i c a t i o n ) i s u se d

i n s t e a d o f e q n ( 1 1 ), t h e n t h e a m p l if ic a t io n f a c t o r m a y b e w r i tt e n a s

A F = B 2 =

Y-Pu ) (12)1 - E Pei

w here E Pet i s t he su m of the e l a s t i c buc k l ing loads o f a ll t he co lum ns in thes t o ry . U s i n g L e M e ssu r i e r ' s a p p r o a c h , t h e l o a d i n a n y i n d iv i d u a l c o l u m n a t

t h e e l as t ic b u c k l i n g o f th e s t o r y m a y b e w r i tt e n a g

E/iP e i = ~ ( I 3 )

Sub s t i tu t ing Ki f rom eqn (10) in to eqn (13) a nd su bs t i t u t ing the re su l t so

o b t a i n e d i n t o e q n ( 1 2 ), t h e m o m e n t a m p l i f ic a t io n f a c t o r in e q n ( 12 ) c a n b e

r e wr i t t e n a s

1A F =

I _ ( E Pu + ~" CL Pu )Z'~L (14)

T h i s e q u a t i o n i s n o t e x a c t ly e q u a l t o e q n ( 11 ). I n f a c t , it c a n b e sh o wn f r o m

L e Me ssu r i e r ' s d e r i v a t i o n t h a t t h i s i s t h e a m p l i f i c a t i o n f a c t o r f o r t h e

f i rs t -o r d e r l a t e r a l d isp l a c e m e n t s . H o w e v e r , t h e C L f a c to r , w h i c h r a n g e sf r o m 0 t o 0 . 2 1 6 , i s u su a l ly sm a ll f o r c o lu m n s i n u n b r a c e d f r a m e s a n d m a y

b e i g n o r e d i n m a n y p r a c t i c a l d es i g ns . Ne g l e c t in g t h e C L t e r m i s e q u i v a l e n t

t o d i s r e g a r d i n g m e m b e r c u r v a t u r e P - 8 e f fe c ts o n t h e a m p l if i ca t io n o f t h e

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

m o m e n t a m p l i f i c a t i o n f a c t o r s g i v e n b y e q n s ( 1 1 ) a n d ( 1 4 ) b e c o m e




equivalent as CL approaches zero. It is important to note that, if CL is

assumed equal to zero, then these equations are identical to the expression

for BE given by eqn (H1-5) of LRFD.

Equations (10) and (11) account accurately for the fact that all the

columns in a story participate in any side-sway buckling mode, and that the

stronger columns with a smaller L ~ ratio brace the weaker columns

with a larger L V ~ u / E I until side-sway buckling occurs. Also, they account

directly for cases in which there are gravity or leaner columns in the story.

An iterative procedure which enables the calculation of inelastic K factors

is also outlined by LeMessurier. 10 Interested readers should refer to this

reference for details. The basic concept of inelastic stiffness reduction

employed by LeMessurier is the same as that of the LRFD procedure

described in Section 4.1 with one exception. LeMessurier's inelasticstiffness reduction is based on the value of Et/E at incipient inelastic

buckling whereas the LRFD procedure is based on Et/E at the factored

load level Pu. If P~ is smaller than the axial load in the column associated

with inelastic buckling, then LeMessurier's procedure results in a smaller

value of Et/E and thus smaller inelastic K factors. This is usually the case

for beam-columns subjected to significant bending action. An iterative

alignment chart procedure also exists for determining inelastic K factors.

This procedure was originally proposed by Yura, 1~ and it is also based onEt/E at incipient inelastic buckling.

It can be argued that the inelastic buckling strength Pn should be

calculated based on Et/E at incipient inelastic buckling. However, this

calculation is necessarily iterative. Also, questions arise with regard to

how a general frame which is subjected to design loads resulting in bending

action should be loaded to calculate this inelastic buckling load. The

simpler and direct LRFD method (based on eqns (8) and (9)) under-

estimates Et/E, and thus it overestimates K when there is significantbending action. This results in a conservative value for P,. In recent work,

Goto et al.12 have provided a refined method of analysis for the elastic

buckling of rigid frames. The method considers primary bending and

member bowing effects. Of course, calculation of the actual member

forces in a frame requires a second-order inelastic analysis of some type.

4 . 3 M o d i f i e d K f a c to r m e t h o d ( b a s e d o n t h e b u c k l i n g o f a st o r y )

A modified K factor may be calculated by assuming that the sum of the

axial loads which causes sway buckling of a story is equal to the sum of the

individual buckling loads for each of the columns which provide story

side-sway resistance. The individual buckling loads are determined using



Be am -colu m n design in steel~rameworks 287

t h e e f f e c ti v e l e n g t h s f r o m t h e a l i g n m e n t c h a r t . T h e r e f o r e , a t th e b u c k l i n g

o f a s t o r y , we h a v e

AsE P . = E Pek (15)

w here As i s t he s to ry buck l ing load fac to r . Th e c or re sp ond ing ax ia l l oad in

a n y c o l u m n i t h a t p a r t ic i p a t e s i n t h e r e s is t a n c e t o s t o r y b u c k l in g is g iv e n b y

CEIiA s P ui = ( K i L i ) 2 ( 1 6 )

El im ina t ing As f rom eqn s (15) an d (16) g ives

n 2 E l i E Pu (17)K 2 = ~ u i X Pek

T h i s e q u a t i o n i s a s i m p l i f i e d f o r m o f L e Me ssu r i e r ' s f o r m u l a ( e q n ( 1 0 ) )

s ince i t neg lec t s t he con t r ibu t io n o f t he CL t e rm. A l so , i t s use does no t

requ i re L X/~ , /E- I t o be iden t i ca l fo r a l l t he co lumns . Hence , i t i s an

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

e f fec t ive l eng th used in ca l cu la t ion o f t he co lumn s t reng th Pn . E f fec t ive

l eng th fac to r s such as those o b ta in ed f ro m e qns (10) o r (17) a re e ssen t i a l

f o r p r o p e r c a l c u la t io n o f Pn f o r m e m b e r s o f t h e l a t e r al - re s i s ti n g sy s t e m i nl e a n e d c o l u m n f r a m e s . I n e l a s ti c K f a ct o r s c a n b e c o m p u t e d f r o m e q n ( 1 7)

i f t he X Pek t e rm i s ca l cu la t ed base d on the ine l a s t ic G fac to r exp ressed b y

eqn s (7 ) to (9 ) and I i i s r ep laced by E t I i / E , i n wh i c h E t / E m a y b e c a l c u l a t e d

f r o m e q n s ( 8 ) a n d ( 9) f o r e a c h c o l u m n .

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

the e l a s t i c K fac to r f rom eq n (17) in to eqn (13) , and th en sub s t i t u t ing P¢i

f rom eqn (13) in to eqn (12). Th i s r e su l t s i n the fo l lowing s imple equ a t ion :

m F ~1 1

[ Pu i = Y+P u1 ( 1 8 )

T h a t i s , t h e A F v a l u e o b t a i n e d f r o m t h e a b o v e p r o c e d u r e i s i d e n t ic a l t o t h e

B 2 v a l u e o b t a i n e d f r o m e q n ( H1 - 5 ) o f th e L R F D Sp e c if ic a ti o n w i t h t h e u se

o f a l i g n m e n t c h a r t K f a c to r s . T h e r e f o r e , t h e u se o f m o d i f ie d K f a c t o rs i n

e q n ( H1 - 5 ) o f t h e L R F D Sp e c i fi c at io n i s n o t n e c e s sa ry . I n m a k i n g t h i s

a s se s sm e n t , i t i s i m p o r t a n t t o r e c o g n i z e t h a t , i f t h e a l i g n m e n t c h a r t

p ro ced ure i s used fo r ca l cu la ting Be , s to ry sum m at ion o f t he Pu /Pek t e rm s i s

r e q u i r e d a n d t h e Pe k v a l u e f o r a l e a n e d c o l u m n m u s t b e t a k e n a s z e r o .

H o w e v e r , i f t h e m o d i f ie d K f a c t o r e q u a t i o n s a r e u se d i n s t e a d o f t h e



288 J. Y. Richa rdLiew , D . IV. Wh ite, W . F. Ch en

a l i g n m e n t c h a r t K v a l u es , n o s t o r y su m m a t i o n i s n e c e s sa ry ( a n d t h e e e i

t e r m f o r a l e a n e d c o l u m n is th e f in i te v a lu e o b t a i n e d b y su b s t i tu t i o n o f e q n

(17) in to eqn (13) fo r t he l eane d co lum n me m ber ) . Th i s is because the

s to ry e f fec t s hav e be en inc luded in the m odi f ied K fac to r , wh ich is de r ived

b a se d o n t h e c o n c e p t o f s t o r y b u c k li n g .

Th e A us t ra l ia n AS 4100 Spec i f ica t ion 13 spec i f ies an am pl i f ica t ion of

f i rs t -o rde r sway m om ent s tha t i s s imi l a r t o eq n (18) :

A F =

0.9

1 - ( E P uEPcr ) (19)

T h e Z P c r t e rm in th i s equ a t ion i s de f ined ge ne r i ca l ly a s the s to ry e l a s t i c

buck l ing load . Eq ua t ion (19) i s app l i cab le wh en E Pu /'Z P ~ i s be tw een 0 .1

an d 0 .25. F or Y. Pu/E Pc~ < 0 .1 , th e P - A effec t i s sm al l an d hen ce can be

n e g l e c te d . Fo r Z P u / ~ Pe r > 0 "2 , a s e c o n d - o r d e r i n e la s t ic a n a l y s is is

r e c o m m e n d e d . I t s h o u l d b e n o t e d t h a t in th e A u s t r a li a n p r o c ed u r e , b o t h

t h e c o lu m n a n d b e a m e n d m o m e n t s a r e a m p li fi ed w h e r e a s in th e L R F D

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

m o m e n t is e x p l ic i tl y r e q u i r e d . T h e a u t h o r s b e l i e v e t h a t t h is a p p r o a c h t o

t h e a m p l i fi ca ti o n o f th e b e a m m o m e n t s s h o u l d b e e x t e n d e d t o t h e L R F DSpec i f ica t ion p roced ure a s we l l.

4 . 4 B u c k l i n g a n a l y s i s o f th e o v e r a ll s tr u c t u r a l s y s t e m

The e f fec t ive l eng th fac to r s may a l so be ca l cu la t ed f rom a buck l ing

ana lys i s o f t he en t i r e s t ruc tu ra l sys t em . Th i s is accom pl i shed by equ a t ing

the ax ia l fo rce in a m em ber a t the inc ip i en t buck l ing o f t he f ram e , Per, to

the buck l ing load a ssoc ia t ed wi th an e f fec t ive l eng th K L . T h e r e su l t i n ge la s ti c e f fec t ive l eng th fac to r i s de t e rm ined a s

~ f P e ~ f ~.2 E I / L 2K = -ff~ = (20)

A n ine la s t ic K fac to r can be c om pu ted f rom e qn (20) i f Per i s ca l cu la t ed

based on ine l a s t ic buck l ing ana lys is and I i s r ep laced by E t l / E fo r t he

p a r t ic u l a r m e m b e r u n d e r c o n s i d er a ti o n .T h e u se o f t h e b u c k l i n g l o a d a n a l y s is f o r a f r a m e w o r k l e a d s to a n o t h e r

p o s s i b le f o r m o f t h e m o m e n t a m p l i fi c a ti o n f a c to r ,

1Av = 1/A----~ (21)



B e a m - . c o l u m n d e s i g n i n t ee l r a m e w o r k s 2 8 9

where Afis the ratio of the elastic buckling load of the entire frame to the

factored load of the frame.In rectangular unbraced frames, the story moment amplification factor,

based on eqn (18), often gives a better prediction of the second-orderelastic moments than the frame moment amplification factor shown in eqn

(21). This is because the amplification effects are not the same for each and

every story throughout the entire framing system. For instance, stories

that are more stiff or more lightly loaded will have smaller amplification

effects than stories that are more flexible and heavily loaded. However,

the frame moment amplifier is applicable for non-rectangular and

irregular frames, and for structures with a mixture of elements providing

restraint against side-sway. For general frames, a buckling analysis of the

entire frame is necessary for accurate determination of the effective length

factors to be used in the axial resistance term of the beam-column design

formulas. The use of the system buckling analysis may lead to unexpected-

ly large K values for members that are lightly loaded (i.e. for members

having small design axial forces). In fact, for a column with zero axial load

Pu, the K factor calculated from eqn (20) is infinity. This results in an

indeterminate form of the axial strength ratio, Pu/Pn = 0/0. However, it

can be shown by algebraic manipulation that the first term of the

beam-column interaction equation (eqn 2) has a finite value of

P~ 1

P---~- = 0-877-----~ (22)

for this case. Equation (22) is derived by writing P~ in eqn (20) both as

Per = Pn/0"877 and Pcr= AfPu for this case. By equating these two

expressions, solving for Pu, and substituting this value into Pu/Pn, the

result shown by eqn (22) is obtained.

If the LeMessurier or the modified K factor procedures are employed

(eqns 10 or 17), a similar situation occurs in the limit as the column axial

force approaches zero. In general, the first term of the beam-column

interact ion equation approaches the value of

Pu 2 2Pu i L i K i

Pn - 0-877 ~r2EIi (23)

for these procedures a s P u i approaches zero. By substituting either eqn 10or eqn 17 into this expression, an expression which is independent of Pui isobtained.

The rationale for the first term of the interaction equations being taken as

the values given by eqn (22) or (23), even if a beam-column has nearly zero

axial force, is that the member may contribute to the buckling resistance of



290 J . Y . Rich ard Lie w , D . IV. W hi te , W . F. Chen

structural system by providing restraint to other members. 14 Yet, in many

cases it can be said that the members which have small axial forces and

large effective lengths from a system buckling analysis are actually not

participating in the buckling of the system at all, and therefore, the systembuckling load will be changed little if the size of these members is changed.

In this case, one might rationalize that such a member 's effective length

should be based on the buckling analysis of an isolated subassemblage of

the frame that includes the lightly loaded member (i.e., it should be based

on a procedure such as the alignment chart method). The member 's K

factor would be smaller if determined in this way. This would be the case

for the beam-columns in the top story of the multi-story frame example

discussed in the next section (see Fig. 11). However , in some situations, a

member may have a large K factor from the system buckling analysis

because it is restraining other more critically loaded members at incipient

buckling of the overall system. If this restraint is counted upon in the

design of the more critical members, then the larger K factor from the

system buckling analysis must be utilized for the more lightly loaded

members.

4 .5 Cha rac ter is t ic re su l t s o f K fac tor ca l cu la t ions

Figures 9 and 10 show the column K factors, computed for two simple

portal frames using the methods described in the previous sections. These

frames have fixed and p inned bases respectively. The results are presented

for G values associated with a range of rigid to very flexible beam

restraints, and also for several values of a, which is defined as the ratio of

the axial loads on the columns. For these examples, eqns (10) and (17)

predict the same variations of the K factor for different values of a. These

equations correspond to LeMessurier's method and the modified K factormethod respectively. Although the curves shown in Figs 9 and 10 have

been developed for the right-hand column in these frames (column CD),

the same curves can also be used for the left-hand column (column AB)

simply by replacing a by 1/%Fa.

It should be observed that the K factors predicted by eqns (10) and (17)

are affected by the load distributions in the columns. The column carrying

the smaller axial load has a larger K factor than the column carrying the

higher axial load. This is due to the fact that the column with the low axial

force effectively braces' the column that has the high axial force until story

buckling occurs. Also, it should be noted that the column with the larger

axial load may have a K factor less than 1.0. When both columns in the

frame carry the same load (i.e., for a = 1-0), the K factors obtained from

eqns (10) and (17) coincide with the K factors obtained from the alignment



Be am -co lum n des ign in s tee tf rameworks 291

2 .5

2 .2

1.9

KCO

1.6

1.3

1.0

0 .7

ecp l P 4.0

T . e z b t D '~ =

I I

0 1 2 3 4 5

G = ( I c / L e ) / ( l b / L b )

Fig . 9 . Charac te r i s t i c re su l t s o f K fac tors for por ta l f rames wi th f ixed base condi t ions .

E q ua t i ons ( 10 ), ( 17 ) a nd ( 20) g i ve i de n ti c a l r e su l t s a nd s how a va r i a t ion o f t he K f a c t o r f o r

d i f f e r e n t a va l ue s . T he K va l ue s ob t a i ne d f r om t he a l i gnm e n t c ha r t p r oc e du r e do no t

acco unt fo r va r i a t ions in c~.

chart method. However, the alignment chart method (eqn 4) is not at all

applicable when a is not equal to 1.0. This is due to the fact that the

alignment chart is based on the explicit assumption that all the columns inthe structural system have the same stiffness parameter (L ~ / ) at

incipient buckling.

Although the assessment of the stability of an unbraced frame must

consider the total buckling strength of each individual column in the story,

there is one limitation; that is, a column cannot carry a load higher than its

own capacity in a non-sway mode. If this is the case, the K factor calculated

based on the story buckling formulae should not be less than that from the

side-sway prevented case.In practical building design, the columns within a frame seldom have the

same stiffness parameter. Figure l l a shows the member sizes, dimensions,

and working loads for a bent of a 10 story building frame. It is assumed that

the spacing between bents is 20 ft (6096 ram). The elastic K factors for the

columns of this frame are reported based on a system buckling analysis in




V5 .5 ~ ~ = 4 , 0

| ~ P p

jo

4 . 5

4 .0

KC D

3 .5

3 . 0

2 . 5

2 .0

1.5 I L I I I

0 1 2 3 4 5

G = ( I c / L c ) / ( I b / L b )

Fig. 10. Characteristic results of K factors for portal frames with hinged base conditions.

Equations (10), (17) and (20) give identical results and show a variation of the K factor fol

different a values. The K values obtained from the alignment chart procedure do not

account for variations in a.

Fig. 11b, and based on the alignment chart method in Fig. 11c. It is

observed that the discrepancy in the K factors is quite high for some of thecolumns.

The difference in K factors is particularly large for the columns in thebottom and the top stories. This is because the fundamental assumptions

of the alignment chart K factor have been violated in these members.

Typically, the members with smaller axial forces from the system buckling

analysis have larger K values. It is interesting to note that changes in thesize of the members which have large K values often have little effect onthe overall System buckling load.

Figure 12 shows an unbraced frame with columns of unequal lengths. A

buckling analysis of this frame has been performed for column lengthratios, 7 = LAB/Lco, ranging from 1.0 to 4.0. Equal axial loads are

applied at the top of each column. For this frame, eqns (10) and (17)predict essentially the same elastic K factors for all the cases considered.

Prior to story buckling, the shorter column braces the longer column, and

thus the shorter column has a larger effective length factor. The use of the



B eam--colum n design n teel r am ewor ks 2 9 3

R o o f t o a d

Live (L r ) 1437 KN /m 2

D e a d (O) 2.395 KN /m =

Floor load

Live (L) 1.916 KN /m 2

Dead (E)) 2.634 KN /m 2

E xte r ior w a l l (story)

= 423KN or1 exter ior

columns

Fac tor ed load

= 1 2D + 16L +0.SL r

Ia ; •

N14x2;

~ do

i ~12x2~~ do1

W16X20

d o

W14x3Q

do

I ! 3 @ 6 0 9 6 m m

2.98

1.83

2.55

2.16

2.43

2.20

2.40

2,24

2.29

2.16

2.05 2.06 2.93 t. 15 1.16 1.37

1.42 1.42 1.83 1.34 1,34 1.67

2.27 2.27 2.55 1.59 1.59 2.05

1.96 1~96 2.16 1.75 1.75 2.35

2.00 2.00 2.43 1.33 1.83 2.59

1.63 1.83 2,20 1.34 1.84 2,59

2,23 2.23 2.40 190 1.90 2,57

2.09 2.09 2.24 1.97 1.99 2.71

1.95 1.95 2.29 1.98 ! 1.98 2,81

( a ) ( b ) ( c )

F i g . 1 1 . ( a ) D i m e n s i o n s , l o a d s , a n d m e m b e r s i z e s f o r a 1 0 - st o r e y 3 - b a y f r a m e ; ( b ) K fa c t o r s

f r o m s y s t e m b u c k l i n g a n a l y s is ; ( c ) K f a c t o rs f r o m t h e a l i g n m e n t c h a r t m e t h o d .

K

B I b D

I G = ( I/Lc /( lb /L )3 .0 i c i c K ~ = K C D / ¥

C o lu m n ~ fi~ 13 m ~ - s ~ y m o d e

A L b G = 5 . O

• olumn CD

~ l ~ - - ~ f ~ J ' ~ ~ o 5.o":'!:°"'°"5 ..~o.o--2.0 } C o l u m n A B

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

C o lu m n A B f a i b i n n o n - s w a y m o d e

O.0 I I I I1 . 0 1 , 6 2 . 0 2 , 1 i a . 0

~ = L e q t h A B / L m s t h C D

F i g . 1 2 . C h a r a c t e r i st ic K f a c t o r r e su l t s f o r p o rt a t f r a m e s w i t h u n e q u a l c o l u m n l e n g t h s .



294

8

J. Y. Richard Liew, D. W. White, W. F. Chen

K C D

o r = 4 . 0a p p L e M e s s u r i e r { E q n . l O ) , ~

~ B [ b ~ D . . . . M o d i f i e d K ( E q n . 1 7) , . ~ ' ~ ; ' ~

. S 20

.,, " " ~ ~ 1.0

, o . ¢ " ~ ~ . s ~ " , ~

o , , . ., " ~ 0 . 5

" ~ " ~ d 0. 0

0 1 2 3 4 5

G = ( l c / L c ) / ( I b / 2 L b )

Fig. 13. Ch aracteristic K facto r results for leaned -colum nframes.

a l i g n m e n t c h a r t i s s tr ic tl y a p p l ic a b l e o n l y w h e n t h e s ti ff n es s p a r a m e t e r s o f

b o t h c o l u m n s a r e t h e s a m e . I t c a n b e o b s e r v e d f r o m F i g . 1 2 t h a t t h e d i r e c t

u s e o f t h e a l i g n m e n t c h a r t w i t h o u t a n y m o d i f i c at io n w il l u n d e r e s t i m a t e t h e

K f a c t o r f o r c o l u m n C D a n d o v e r e s t im a t e th e K f a c t o r f o r c o l u m n A B .H e n c e , c o l u m n C D ' s c o n t r i b u t i o n t o t h e s t r e n g t h o f t h e s y s t e m w i l l b e

o v e r - p r e d i c t e d w h e r e a s c o l u m n A B ' s c o n t r i b u t i o n w il l b e u n d e r p r e d i c t e d .

A s p e c ia l c o n c e r n o f th e w r i te r s w i th r e g a r d t o t h e p r o p e r c a l c u l a t io n o f

K f a c t o r s is t h e c a s e o f l e a n e d c o l u m n f r a m e s , i n w h i c h a l a r g e n u m b e r o f

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

l a t e r a l - f o r c e r e s is t in g s y s te m . T h e s e t y p e s o f f r a m e s a r e q u i t e c o m m o n f o r

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

b u il d in g s . T h e e f fe c t i v e l e n gt h s o f l e a n e d c o l u m n s s h o u l d b e c o m p u t e d

b a s e d o n K = 1. I n o t h e r w o r d s , t h e b e a m - c o l u m n s o f t h e la t e r al - re s i st in g

s y s te m a r e c o u n t e d u p o n t o b r a c e t h e l e a n e d c o l u m n s a t e a c h f lo o r l ev e l.

D u e t o t h e d e s t a b i l i z i n g a c t i o n o f a x i a l f o r c e s a c t i n g o n t h e l e a n e d

c o l u m n s , t h e e f f e c ti v e l e n g t h s o f th e b e a m - c o l u m n s i n t h e l a te r a l- r e s is t in g

s y s t e m a r e i n c r e a s e d . T h i s e f f e c t is i l l u s t ra t e d b y t h e p o r t a l f r a m e s h o w n i n

F i g . 1 3 . T h e p a r a m e t e r a i n t h e f i g u r e is t h e r a t i o o f t h e a x i a l l o a d a p p l i e d



Beam -column design in steel ramew orks 295

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

r e si st in g c o l u m n o n t h e r ig h t s id e . T h i s f r a m e i s e q u i v a l e n t t o a m o r e

g e n e r a l f r a m e w h i c h h a s a l e a n e d c o l u m n s , e a c h w i th a lo a d e q u a l t o P .

T h e s e c o l u m n s a r e ' b r a c e d ' b y t h e s o l e l a te r a l- l o a d re s is t i n g c o l u m n ,

c o lu m n C D .

T h e K f a c to r s o b t a i n e d f r o m L e M e s s u r i e r 's m e t h o d a s w e l l a s f r o m t h e

m o d i f i e d K f a c to r m e t h o d a r e s h o w n i n t h e f ig u re . F o r t h is p r o b l e m ,

L e M e s s u r i e r 's m e t h o d ( e q n ( 1 0) ) g iv e s t h e m o r e a c c u r a te e l a st ic K f a c to r s,

w h e r e a s t h e m o d i f i e d K f a c to r m e t h o d ( e q n ( 1 7) ) alw a y s gi v es a

c o n s e r v a t i v e p r e d i c t i o n o f K ( i .e . , i t g i v e s a l a r g e r K ) . T h e a l i g n m e n t c h a r t

m e t h o d is o n ly v a li d fo r t h e c a s e w h e r e t h e a x ia l l o a d i n th e l e a n e d c o l u m n

is e q u a l t o z e r o (i . e . , a = 0 - 0) . A n y a t t e m p t t o u s e t h e a l i g n m e n t c h a r t t o

c a l c u l a te t h e K f a c t o r s fo r a o t h e r t h a n 0 -0 w i ll l e a d t o u n c o n s e r v a t i v er e s u lt s. T h e e x i s t e n c e o f la r g e a x ia l f o r c e s i n t h e l e a n e d c o l u m n ( o r o f a

l a rg e n u m b e r o f l e a n e d c o l u m n s ) r e su l ts i n l ar g e d i ff e r e n c e s i n t h e

c a l c u la t e d K f a c to r a n d h e n c e a l a rg e d i f f e r e n c e b e t w e e n t h e r e su l ts

o b t a i n e d b y t h e a l i g n m e n t c h a r t a n d t h e m o r e e x a ct m e t h o d s . T h is e f fe c t

w o u l d a l so b e e x h i b i t e d fo r a n y f r a m e i n w h i c h a la r g e n u m b e r o f le a n e r

c o l u m n s a r e r e s t r a i n e d a g a i n s t s i d e -s w a y b y o n l y a f e w c o l u m n s .

5 T H E U S E O F E F F E C T I V E L E N G T H F A C T O R S W I T H T H E

L R F D B E A M - C O L U M N E Q U A T I O N S

F i g u r e 1 4 s h o w s a s y m m e t r ic p i n n e d - b a s e p o r t a l f r a m e w i th e q u a l v e r t ic a l

l o ad s a t t h e t o p o f th e c o l u m n s . T h e L R F D i n te r a c ti o n e q u a t i o n s a r e

e v a l u a t e d f o r th i s p r o b l e m u s i n g b o t h t h e e l a st ic a n d t h e i n e l a s t ic K f a c t o r s

f r o m t h e a l i g n m e n t c h a r t m e t h o d . T h e i n el a st ic K f ac to r s a re e v a l u a t e d

b a s e d o n t h e L R F D M a n u a l p r o c e d u r e ( e q n s ( 8) a n d (9 )) . T h e r e s is ta n c ef a c to r s q ~ a n d ~bc i n th e L R F D e q u a t i o n s a r e t a k e n a s u n i t y t o o b t a i n t h e

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

f r a m e s w e r e g e n e r a t e d b y K a n c h a n a l a i , 4 a s s u m i n g try = 2 5 0 N / r a m 2,

E = 2 0 0 0 0 0 N / m m 2 a n d trr¢ = 0 -3 try . A l l t h e m e m b e r s a r e a s s u m e d t o b e

p e r f e c tl y s tr a ig h t a n d p l u m b i n K a n c h a n a l a l ' s s tu d y .

K a n c h a n a l a i ' s r e s u l t s a r e s h o w n i n t h e f i g u r e f o r b o t h s t r o n g - a n d

w e a k - a x i s b e n d i n g a n d f o r c o l u m n s h a v i n g sl e n d e r n e s s ra t io s o f L / r = 2 0

a n d 4 0 . T h e c u r v e s s h o w n a r e n o t a d j u s t e d t o a c c o u n t f o r i m p e r f e c t i o n

e f f ec t s .

T h e L R F D c u r v e s, a s s h o w n in F ig . 1 4 , a r e g e n e r a l ly c o n s e r v a t i v e

c o m p a r e d t o t h e i n e l a s t ic s t r e n g t h c u r v e s . T h i s is p a r t l y d u e t o t h e f a c t t h a t

t h e L R F D b e a m - c o l u m n e q u a t i o n s i m p l ic i tl y i n c l u d e t h e e f f e c t o f in i ti a l

g e o m e t r i c i m p e r f e c t i o n s . H o w e v e r , t h e d i f f e r e n c e b e t w e e n t h e i n e l as t i c

s t r e n g t h c u r v e s a n d t h e L R F D s t r e n g t h c u rv e s is q u i t e la r g e fo r a s h o r t



2 9 6 J. Y. Richard Liew, D. W. White, W. F. Chen

P

P y

1.0

0 .8

0 .6

0 .4

0 .2

l b

Lb w,#

\

~ m

G A = ( l c / L e ) / ( [ b / L b ) = 3 . 0

G B = 0

\\

J

L / r = 4 0 / " ~ .

S t r o n g a x i s

} ( K a n c h a n a l a i ,1 9 7 7 )W e a k a x i s

L R F D e l a s t i c K ( E q n s . 1 - 2 , 4 - 6 )

L R F D i n e la s t i c K ( E q n s . 1 - 2 , 4 , 7 - 9 )

L / r = 2 0

\

\

\

\ \

\

t I I I

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

H L c / 2 M p

Fig. 14. Effect of using elastic and inelastic K factors on th e AISC -LRF D beam-colum nequations.

c o l u m n b e n t a b o u t t h e w e a k a xis , p a r t ic u l a r l y f o r c o l u m n s s u b j e c t e d t o

l ow a xia l l o ad . T h i s is p r im a r i ly b e c a u s e t h e L R F D b e a m - c o l u m n

i n t e r a c t i o n c u r v e s d o n o t d i s ti n g u is h b e t w e e n s t ro n g - a n d w e a k - a x i s c a s es .

T h a t i s , th e r e i s o n l y o n e n o r m a l i z e d L R F D b e a m - c o l u m n s t re n g t h cu r v e

f o r a g i v e n L/r. H o w e v e r , t h e a c t u a l s t r e n g t h s f o r st ro n g - a n d w e a k - a x is

b e n d i n g a r e d i ff e r en t • N e v e r t h e l e s s , t h e u s e o f o n l y o n e i n t e r a c t i o n

e q u a t i o n f o r s t r o n g - a n d w e a k - a x is b e n d i n g i s c o n s i s t e n t w i t h t h e u s e o f

o n l y o n e c o l u m n s t r e n g t h c u r v e f o r a xia l st r en g t h •

C l o s e e x a m i n a t i o n o f F i g . 1 4 a l so r e v e a l s t h a t t h e u s e o f t h e i n e l a s t i c K

f a c t o r ( c a lc u l a t e d a c c o r d i n g t o t h e p r o c e d u r e in S e c t i o n 4 . 1) i n t h e L R F D

e q u a t i o n s is o n l y b e n e f i c i al fo r s h o r t c o l u m n s s u b j e c t e d t o h i g h a x ia l l o a d s •

T h e r e is e s se n ti a l ly n o d i f f e r en c e b e t w e e n t h e L R F D c u r v es b a s e d o n

e l a s t ic a n d i n e l a s t i c K f a c t o r s f o r t h e L/r = 4 0 c a s e . T h i s is e x p e c t e d s i n c e



Beam--column e s i g n n teel rameworks 297

P

P y

t.O

0. 8

0 .6

0.4

0.2

H ~ ° t P l b : o o ~ P

c l

~ .~ Lb ~1

Q = 3 . 0

G A = 0

G B = ~ o

L / r = 2 0

S t r o n g a x i s ~W e a k a x i s J ( K a n c h a n a la i , 1 9 7 7 )

~ . ~ L R F D e l a s t ic K ( E q n s . 1 - 2 , 1 0 - 1 1 )

. . . . L R F D e l a s ti c K ( E q n s. 1 - 2 , 1 7 - 1 8 )

\\

\

\f . X . . \ . = o o

\

, \

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 .0

H L c I M p

Fig. 15. Comparisonof the AISC-LRF D beam-column quat ionswith ult imatestrengthcurves4 for leaned -colum n ram e w ith GA ---- 0 and L /r -- 20.

t h e i n e l a st ic K f a c to r is u s e d o n l y i n t h e a x ia l r es is t a n c e te r m o f t h e L R F D

e q u a t i o n s . H e n c e , i t o n l y a f f ec t s c o l u m n s s u b j e ct e d to p r e d o m i n a n t l y

a x i a l l o a d s . N e v e r t h e l e s s , i n fr a m e s w h i c h h a v e b o t h h e a v i l y lo a d e d a n d

l ig h t l y l o a d e d b e a m - c o l u m n s , th e i n e l a s ti c K f a c t o rs o b t a i n e d b y L e M e s -

s u r i e r ' s m e t h o d , t h e m o d i f i e d K f a c t o r m e t h o d , o r b y s y s t e m b u c k l i n g

a n a l y s i s c a n b e s i g n i fi c a n t ly d i f f e r e n t th a n t h e c o r r e s p o n d i n g e l a s t ic K

v a l u e s . T h i s i s p a r t ic u l a r l y t h e c a s e a t i n c i p i e n t i n e l a s t i c b u c k l i n g . I f

i n e l a st ic st if f n e ss r e d u c t i o n is e m p l o y e d w i th t h e s e m e t h o d s , t h e e f f e c t i v e

l e n g t h s f o r c o l u m n s w h i c h e x h i b i t s i g n if ic a n t y i e ld i n g w i l l t e n d t o b e

s m a l le r , a n d t h o s e f o r c o l u m n s w h i c h r e m a i n p r e d o m i n a n t l y e l a s ti c w i l l

t e n d t o b e l ar g e r. T h u s , t h e u s e o f i n e l a s ti c s t if fn e s s r e d u c t io n w i t h t h e s e

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

r e s t r a in t h e i n e l a s t i c c o l u m n s a g a i n s t s i d e - s w a y i n s ta b i l it y .

A n i m p l i c it a s su m p t i o n o f u s i n g t h e i n e l a s ti c K f a c to r f o r c o l u m n d e s i g n



298 J . Y . Richard L iew, D . W . W h i te , W . F . Chen

T.0 ~-

0 . 8

P

Py

0 .6

0 .4

0 .2

H

L . . L b

P

A

lc

B

G A = 0

G B = o o

L / r = 4 0

~ m

m B !

S t r o n g ax is )( K a n c h a n a l a i , 1 9 7 7 )

W e a k a x i s )

LRFD e l a s t i c K ( E q n s . 1 - 2 , 1 0 - 1 1 )

L R F D e l a s t i c K ( E q n s , 1 - 2 , 1 7 - 1 8 )

= 0.0

a = 2 .0

Fig . 16 .

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

H L c / M p

Comparison of the AISC-LRFD beam-column equat ions wi th the ul t imates t r e n g t h c u r v e s 4 for leaned-co lum n fram e with G A = 0 and L / r = 4 0.

i s th a t t h e c o l u m n is lo a d e d i n t h e i n e l a s ti c r a n g e w h i l e t h e s u r r o u n d i n g

m e m b e r s r e m a i n e la s t i c . T h i s a s s u m p t i o n i s a d e q u a t e f o r e la s t i c a n a ly s i s /

d e s i g n s i n c e t h e c a p a c i t y o f th e s t r u c tu r a l s y s t e m is i m p l ic i t ly t a k e n a s t h e

l o a d a t w h i c h t h e m o s t c r it ic a l m e m b e r r e a c h e s i ts l i m i t s ta t e . H o w e v e r , i f

p l a s t ic h i n g e f o r m a t i o n i n th e b e a m s a n d s u b s e q u e n t i n e l a s t i c r e d is tr ib u -

t i o n o f fo r c e s is r e l ie d u p o n , a s in p l a s ti c d e s i g n , t h e i n e l a s t i c K f a c t o r

c o n c e p t i s n o t s tr i c t l y a p p l i c a b l e . I n f a c t , t h e u s e o f t h e e l a s t i c K f a c t o r s t oa c c o u n t f o r s t a b i l i t y a s p e c t s i n i n e l a s t i c a n a l y s i s / d e s i g n i s a l s o n o t f u l l y

r a t io n a l . T h e r e f o r e , i f s o m e f o r m o f in e l a s t i c a n a ly s i s / d e s ig n i s e m p l o y e d ,

t h e p r o p e r a c c o u n t i n g o f s ta b i li ty e f f e c t s s h o u l d b e c a r e f u l ly c o n s i d e r e d .

O b v i o u s l y , d i r ec t c o n s i d e r a t i o n o f s t a b i li ty e f f e c t s i n s o m e f o r m o f

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



B eam--co lum n design n teel ram ewor ks 299

P

1.0

0 . 8

0 .6

0 .4

0 .2

H ~ o tp J .b

l c

L.. Lb

(~ = 3.0

Ic

G A = ( l c / L c ) / ( | b / 2 L b) = 4 .0

G B = O ~

L / r = 2 0

S t r o n g a x i s ). ( K a n c h a n a l a i , 1 9 7 7 )

~ - - W e a k a x is J

- - . - - L R F D e l a s t i c K ( E q n s . 1 - 2 , 1 0 - 1 1 )

L R F D e l a s t i c K ( E q n s . 1 - 2 , 1 7 - 1 8 )

~ ot = 0 .0

\

\

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 .0

H L c ] M p

F i g . 1 7 . Comparison o f t h e A I S C - L R F D b e a m - c o lu m n equations w i t h t h e u l t i m a t e

s t r e n g th c u r v e s 4 f o r l e a n e d - c o lu m n f r a m e wi th GA ---- 4 a nd L/r = 20.

T h e L R F D e q u a t i o n s a r e f u r th e r i n v e s t ig a t e d i n F i g s 15 t o 1 8 b yc o m p a r i n g th e p r e d i c t e d st r en g t h o f a b e a m - c o l u m n i n a l e a n e d c o l u m n

f r a m e w i t h t h e r e s u lt s d e t e r m i n e d f r o m K a n c h a n a l a i 's p l a s t ic - z o n e

a n a l y se s . 4 T h e L R F D e q u a t i o n s a r e e v a l u a t e d b a s e d o n t h e e f f e c t iv e

l e n g t h a n d m o m e n t a m p l i fi c a t io n f a c to r s d i s c u s se d i n S e c t i o n s 4 . 2 a n d 4 . 3 .

T h e c o m p a r i s o n s a r e s h o w n f o r s e v e r a l v a l u e s o f L / r , G a n d o r . T h e

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

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

s i m p l e fr a m e s a n d o c c u r m a i n l y i n fr a m e s w i t h h i g h a v a l u e s ( w h i c h a r ee q u i v a l e n t t o f r a m e s w i t h a l a r g e n u m b e r o f l e a n e r c o l u m n s ) . T h e

c o m p a r i s o n s a ls o i n d ic a t e th a t t h e L R F D e q u a t i o n s a r e g en e r a l ly c o n -

s e r v a t iv e f o r a ll c o m b i n a t i o n s o f a a n d G . I f K a n c h a n a l a i ' s c u r v e s a r e

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

c u s s e d i n S e c t io n 2 . 2 , t h e L R F D e q u a t i o n s p r o v i d e a n e x c e l l e n t f i t f o r t h e



300 J. Y. Richard L iew, D. W. White, W. F. Chen

P

t.O

0.8

0.6

H $a P Ib

G A = ( | c / L c ) / { I b / 2 L b ) = 4 .0

G B = oo

L / f = 4 0

. . .. S t r o n g a x i s

(Kanchana{ai ,1977)m W e a k a x is

~ - ~ L R F D e l a s t i c K ( E q n s . 1 - 2 , 1 0 - 1 1 )

0 . 4 . . . . L R F D e l a s t i c K ( E q n s . 1 - 2 , 1 7 - 1 8 )

I l i I I0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

H L c / M p

18, Com parison of the AIS C-L RF D beam-colum n equations with the ultimates t r e n g th c u r v e s4 f o r l e a n e d - c o l u m n f r a m e with GA = 4 and L/r = 40.

s t r o n g - a x i s c a s e s s h o w n . T h e y a r e s t i l l s o m e w h a t c o n s e r v a t i v e f o r t h ec o r r e s p o n d i n g w e a k - a x i s c a s e s.

T h e u s e o f L e M e s s u r ie r ' s K fa c to r e q u a t i o n ( ¢ q n ( 1 0 )) a n d t h e m o m e n t

a m p l if ic a t io n f a c to r s ( e q n ( 1 1 ) ) i n c o n j u n c t i o n w i t h t h e L R F D e q u a t i o n s

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

t h e s e o n e - s t o r y e x a m p l e s ) . T h e m o d i f i e d K f a c t o r ( e q n ( 1 7 ) ) a n d t h e

c o r r e s p o n d i n g m o m e n t a m p l i f i c a t i o n f a c t o r ( ¢ q n ( 1 8 ) ) g i v e a l m o s t i d e n t i -

c a l r e s u l t s t o L e M e s s u r i e r ' s p r o c e d u r e s a n d t h e y a r e s i m p l e r t o u s e .

6 D O W E R E A L L Y N E E D K F A C T O R S ?

I t i s l ik e l y t h a t s o m e r e s e m b l a n c e o f a K f a c t o r i s n e c e s s a r y i n a n y general

b e a m - c o l u m n d e s i g n e x p r e s s i o n s b a s e d o n a m p l i fi e d f ir s t- o rd e r o r d ir e c tl y




determined second-order elastic forces. That is, the K factor is probably

necessary to obtain an accurate fit to results from refined analyses of

simple structural systems for a l l possible combinations of axial force and

end moment. The effective length concept is one method of estimating theinteraction of individual member strength with the overall system

strength. The general validity of the AISC-LRFD approach for columns in

a regular building frame has been demonstrated in the preceding section

by comparisons with the 'exact' strength of columns in simple frames.

However, procedures for determination of the proper effective length

factor for building systems such as exterior tubes with interior leaner

columns, systems with significant interaction between moment-resisting

frames and braced or shear-wall cores; eccentrically braced frames; and

flexibly connected frames can be difficult and the results can be ofquestionable accuracy. Although the engineer can usually make con-

servative assumptions to estimate a K factor for member design, the

process of calculating the K factors for each individual member in a

complex framework can be cumbersome and time consuming. Moreover,

the K factor concept is simply an engineering approximation. When used

to account for the stability of a complex structural system, the simple

concepts upon which K factors are based are not fully rational.

6 . 1 S e c o n d - o r d e r e l a s ti c a n a l y s i s a n d b e a m - c o l u m n d es ig r, c o m p a r i s o n

o f t h e L R F D a n d C a n a d i a n S ta n d a r d s

The current LRFD specification permits the engineer to compute the load

effects directly from second-order elastic analysis. Second-order elastic

analysis is the preferred method in the Canadian Standard. 2 However,

most second-order elastic analysis methods provide only the second-order

moments at the element ends. The maximum moments within themembers (including P - 8 effects) usually require additional calcula-

tions. 8'15 The LRFD specification is silent on how analysis calculations

should be made. It is implicitly assumed that this calculation can be

performed as part of a direct second-order elastic analysis. In Ref. 15, the

authors propose the use of an analytical expression for calculation of the

P - 8 moment amplification effects with a direct second-order elastic

analysis. Interested readers are referred to this reference for details.

The use of direct second-order elastic analysis eliminates the need for K

factors in the determination of Mu. However, the use of a K factor is still

required in the axial resistance term of the LRFD equations for accurate

prediction of the beam-column capacity (see Figs 5 and 6).

The interact ion equations given in the Canadian Standard z effectively

use K = 1-0 for calculation of the axial resistance. The differences in the



302 J . Y . R icha rd L iew , D. W . Whi t e , W. F . Chen

H [bA

1.0

0 . 8 L b

G A = ( I c / L c ) / ( I b / 2 L b ) = 4 .0

G B =0 O

L / r = 2 0

S t r o n g a x i s }

W e a k a x i s ( K a n c h a n a l a i , 1 9 7 7 )

L R F D ( L e M e s s u r i e r K - f a c t o r s )

L RFD (K =1 .0 )

P B a > f . 4 ~ ~ = 0 . 0 ~

~ X ~Py 0 .6 f x ~ ~ - . _ _

B 21 .4 ~ , " ~

x ~" ~ L e M e s s u r i e r K - f a c t o r s0.4 3 .0 %

B 2 > 1.4 ~ x = ~ . = . ~ ~ % % . x~ Q = 3 .0 K = 6 .3 0 2

P = 0 . 14 4 Py - x ~

B 2 < 1 . 4 ~ ~ ~ " ~

012 a =.4 0.6 018 1.0

H L c / M p

F i g . 1 9 . C o m p a r is o n o f th e A I S C - L R F D b c ar a-c o lu m n e q ua tio n s w i th a n d w i th o u t t h e us e

o f K f a c t o r s .

a p p r o a c h e s t a k e n b y t h e t w o s p e c i f i c a t i o n s c a n b e s t b e u n d e r s t o o d b y

w r i t i n g b o t h i n t h e s a m e f o r m a t . P o s i n g t h e C a n a d i a n S t a n d a r d p r o c e -d u r e s i n L R F D f o r m a t , t h e m a x i m u m s e c o n d - o r d e r e l a s t i c m o m e n t i s

c a l c u l a te d e s s e n t i a ll y a s B 1 M f in l i e u o f a m o r e d e t a i le d a n a l y si s, w h e r e M f

is t h e m a x i m u m s e c o n d -o r d e r e la s ti c e n d m o m e n t , a n d B1 is a P - 6

m o m e n t a m p l i fi c a t io n f a c to r i n w h i c h K i s t a k e n a s 1 .0 . T h e u s e o f K = 1 .0

i n t h e B 1 te r m i s a n a t t e m p t t o t r e a t t h e b e a m - c o l u m n s a s i s o la t e d m e m b e r s

w i t h t h e c o m p u t e d s e c o n d - o r d e r f o r c e s b e i n g a p p l ie d a t b o t h e n d s . 15 T h e

C a n a d i a n S t a n d a r d a ls o r e q u i re s t h a t t h e e l a st ic P - A a m p l i fi c a ti o n ,

c a l cu l a te d b a s e d o n a n e q u a t i o n e q u i v a l e n t to e q n ( H 1 - 5 ) o f t h e L R F D ,d o e s n o t e x c e e d 1 .4 ( i . e . , B 2 < 1 . 4 ) . I f B 2 i s g r e a t e r t h a n 1 . 4 , s e c o n d - o r d e r

p l a s t ic - z o n e a n a l y s is is g e n e r a l l y r e c o m m e n d e d .

I t c a n b e d e d u c e d f r o m e q n ( 1 2 ) t h a t t h e c o n s tr a i n t o f B 2 < 1 - 4 i n d i r e c t l y

l im i ts t h e m a g n i t u d e o f th e a x i a l l o a d w h i c h m a y a c t o n t h e c o l u m n s . T h i s

r e s tr i c ts t h e u s e o f t h e in t e r a c t i o n e q u a t i o n s t o a r a n g e in w h i c h t h e u s e o f



Bea m-co lumn design in s tee l f ramew orks 3 0 3

P I P

1 . 0 L G A - 0

t G e = ~I .. R - 4 0

0 . 8 " : :p P/Py 0.768

" \ S t r o n g a x i s

0 . 6 " " . ..

0.4 __ ~ 2 . 0 1P/Py 0.356

° I 1 ' . . . .

0 ' 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

H L c / 2 M p

F ig . 20 . C o m p a r i s o n o f t h e A I S C - L R F D b e a m - c o l u m n e q u a t io n s w i th a n d w i t h o u t th e

inc lus ion of a hor izo nta l load 0 .005 (2P) .

K = 1 -0 i n t h e P n t e r m i s r e a s o n a b l e . T h e c o n c e p t is i l l u s t ra t e d w i t h t h e

L R F D b e a m - c o l u m n e q u a t io n s f o r a l e a n e d c o l u m n f r a m e i n F ig . 1 9, a n d

f o r a s im p l e u n b r a c e d p o r t a l f r a m e w i t h r ig i d c o n n e c t i o n s i n F i g . 20 . I n

t h e s e fi g u re s , t h e L R F D i n t e r a c t i o n e q u a t i o n s a r e p l o t t e d u s i n g K = 1 a s

w e l l a s w i t h K f ac t or s d e t e r m i n e d b a s e d o n L e M e s s u r i e r ' s m e t h o d . T h e

u n a d j u s t e d u l t im a t e s t r en g t h c u r v e s g i v en b y K a n c h a n a l a i 4 a r e a l s o

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

K = 1 is a d e q u a t e , c o m p a r e d t o t h e ' e x a c t ' s t re n g t h f o r a f r a m e w i t hi n i ti a l ly p e r f e c t g e o m e t r y , a s l o n g a s t h e B 2 f a c t o r i s l e ss t h a n a b o u t 1 .4 .

H o w e v e r , if c o m p a r e d t o th e K a n c h a n a l a i d a t a a f t er a d j u s t m e n t f o r

i m p e r f e c t i o n e f fe c t s, th e c u r v e s b a s e d o n K = 1 w i ll a l w a y s b e s o m e w h a t

u n c o n s e r v a t i v e .

F o r g r a v i t y l o a d c a s e s , t h e C a n a d i a n S t a n d a r d a l s o r e q u i r e s t h a t t h e

s e c o n d - o r d e r e n d m o m e n t s i n th e m e m b e r s b e t a k e n a s n o t le ss t h a n t h e

m o m e n t s p r o d u c e d b y l a te r a l lo a d s o f 0.0 0 5 t i m e s t h e f a c t o r e d g r a v it y lo a d

a p p l i e d a t e a c h s t o r y . F o r t h e f r a m e s h o w n i n F i g. 2 0 , i f t h is p r o c e d u r e isa p p l i e d w i t h t h e L R F D b e a m - c o l u m n e q u a t i o n s a n d a W 8 x 3 1 s e c t io n is

c o n s i d e r e d , a n a x ia l l o a d c a p a c i t y o f PIPy = 0 .7 6 8 is o b t a i n e d . P a r t o f t h e

b e a m - c o l u m n c a p a c i t y i s u s e d u p b y t h e m o m e n t a s s o c i a t e d w i t h t h e

a d d i t i o n a l l a t e r a l l o a d o f 0 .0 0 5Y ~ P . W h e n K = 1 .0 w i t h a g e n e r a l

i n t e r a c t i o n e q u a t i o n , t h e a d d i t i o n a l l o a d o f 0-0 05 t i m e s t h e g r a v i t y l o a d s




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

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

m o m e n t t e r m M , m a y b e a d j u s t e d t o b e t te r f it t h e ' e x a ct ' d a t a in l i eu o f

a d j u s t m e n t o f P , . T h e e f fe c t o f u s i n g a n a d d i t i o n a l l a te r a l lo a d o f 0 -0 05 E P

a n d K = 1 .0 i n t h e L R F D i n t e r a c t i o n e q u a t i o n is i l lu s t r a t e d in F i g . 2 0 .

H o w e v e r , t h is t y p e o f a p p r o a c h is s ti ll n o t l i k e ly t o r e s u l t i n a n i n t e r a c t i o n

e q u a t i o n t h a t i s a s a c c u r a t e f o r a ll ra n g e s o f ax ia l f o r c e v e rs u s m o m e n t a s

e q u a t i o n s w h i c h i n c l u d e a n e f f e ct iv e l e n g t h f a c to r .

T h e f u n d a m e n t a l d i ff e re n c e b e t w e e n t h e A I S C - L R F D a n d C a n a d i a n

S p e c i f i c a ti o n s is t h a t t h e f o r m e r r e q u i r e s t h e u s e o f a K f a c t o r i n t h e a x i a l

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

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

l a tt e r l im i ts th e m a x i m u m v a lu e o f th e s e c o n d - o r d e r m o m e n t a m p l if ic a t i o n

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

c o l u m n s s u ch t h a t t h e b e a m - c o l u m n e q u a t i o n s a r e v a l id w i t h o u t t h e u s e o f

a n e f f e c t i v e l e n g t h f a c t o r ( i. e . , K = 1 ). N e v e r t h e l e s s , n e i t h e r o f t h e a b o v e

t w o m e t h o d s a c c o u n t s f o r s y s te m s ta b il it y e ff e c ts i n a c o m p l e t e l y r a t i o n a l

w a y .

7 P R A C T I C A L S E C O N D - O R D E R I N E L A S T I C A N A L Y S I S

I n r e c e n t y ea r s , m a n y e f fo r ts h a v e b e e n d e v o t e d t o t h e d e v e l o p m e n t a n d

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

p r a c t i c e i n t h e U n i t e d S t a t e s i s t o p e r f o r m e l a s t i c a n a l y s i s ( f i r s t - o r d e r

e l a s t i c a n a l y s i s w i t h m o m e n t a m p l i f i c a t i o n f a c t o r s B 1 a n d B E , o r d i r e c t

s e c o n d - o r d e r e l a s t i c a n a l y s i s ) t o o b t a i n t h e r e q u i r e d s t r e n g t h , M , , f o r

m e m b e r d e s ig n . M a t e r ia l n o n l i n e a r i ty a n d m e m b e r i n it i al i m p e r f e c t io n s

a r e a c c o u n t e d f o r im p l i c it ly in t h e d e s i g n i n t e r a c t io n e q u a t i o n s . I t is w e l lk n o w n t h a t a s t h e f r a m e d e f o r m s i n t o t h e i n e la s ti c r a n g e , t h e d i s t r ib u t i o n

o f m o m e n t s a n d a x ia l fo r ce s in t h e m e m b e r s w i ll d i ff e r f r o m t h o s e o b t a i n e d

b y a s s u m i n g t h e f r a m e r e m a i n s f ul ly el as ti c. C o n v e r s e l y , i f a g i v e n s et o f

f o r c e s a r e c o n s i d e r e d o n a n i s o l a t e d m e m b e r , c o m p a t i b i li t y is n o t s a ti sf ie d

b e t w e e n t h e a c t u al in e la s ti c m e m b e r , a s a s su m e d i n th e d e s i g n e q u a t io n s ,

a n d t h e e l a st ic m e m b e r , a s a s s u m e d i n t h e f r a m e a n a l y si s. T h u s , i t c a n b e

s t a t e d t h a t t h e o n l y w a y t o a c c o u n t f o r s y s t e m s t a b i li ty i n a f u l ly r a t i o n a l

w a y is t h r o u g h a s e c o n d - o r d e r i n e la s ti c f r a m e a n a ly s is .A n ' e x a c t ' a n a l y s i s w h i c h i n c l u d e s t h e s p r e a d o f p l a s t i c i t y , r e s i d u a l

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

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

b e a m - c o l u m n i n t e r a c t i o n e q u a t i o n s . T h i s t y p e o f a n a ly s is is a l l o w e d in t h e

n e w A u s t r a l i a n l i m i t s t a t e s s p e c i f ic a t i o n f o r s t e e l d e s i g n 13 a s w e l l a s i n



Beam-column design in steel frameworks 305

Eu r o co d e No . 3 .1 6 H o w ev e r , ' ex ac t ' an a l y s i s i s u su a l ly to o co m p u t a -

t i o n a l ly i n ten s i v e f o r g en e r a l d es i g n u se .

A p o s s i b le a l t e r n a t i v e ap p r o ach i s t o em p l o y a s eco n d - o r d e r p l a s ti c -

h i n g e an a l y s is . Th i s m e t h o d m a k es u se o f co n c en t r a t ed p l a s ti c h i n g es t oap p r o x i m a t e t h e i n e l a st ic b eh a v i o r o f s t ru c t u r a l m em b er s i n a f r am e . Th i s

t y p e o f an a l y s i s u su a l l y i n v o lv es t h e u se o f o n e b eam - co l u m n e l em en t f o r

e a c h f r a m e m e m b e r a n d h a s b e e n s h o w n t o b e e f f ic ie n t f o r a n a ly z i n g

m o d e r a t e l y l a r g e t h r ee - d i m en s i o n a l u n b r ac ed f r am es . 17 Ho w ev e r , a

s i m p l e p la s t ic - h in g e an a l y s i s wh i ch u ses o n e b eam - co l u m n e l em en t f o r

e a c h fr a m e m e m b e r i s o n l y a n a p p r o x im a t e m e t h o d . W h e n u s e d t o a n a ly z e

a s in g l e b eam - co l u m n , t h is m e t h o d d o es n o t n eces sa r i ly p r ed i c t t h e co r r ec t

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

Th i s b e i n g so , a g en e r a l i n e l a s ti c an a l y s is o f a s t r u c t u ra l sy s t em b ased o n a

s i m p l e p la s t ic h in g e m o d e l c an n o t n ece s sa r il y r ep r esen t t h e co r r ec t

i n e l a s ti c p e r f o r m an ce . So m e f o r m o f a d es i g n ch eck i s s ti ll n eces sa r y t o

acco u n t f o r t h e v a r i o u s p o s s i b le m em b er s t ab il it y e f fec ts . A l t e r n a t i v e l y ,

so m e d es i g n g u i d e l i n es m i g h t b e p r o v i d ed t o d e fi n e an ap p r o p r i a t e t y p e o f

p las t i c h ing e a na lys i s and the rang e o f app l i cab i l i ty o f th i s ana lys i s wh ich

d o es n o t r eq u i r e a b eam - co l u m n i n t e r ac t i o n ch eck . Th i s co u l d b e

acco m p l i sh ed b y co m p ar i n g p l a s ti c -h i n g e so lu t io n s w i th r e f in ed s eco n d -

o r d e r p l a s ti c -z o n e b e n c h m a r k s o f sy s te m a n d m e m b e r r e sp o n s e .I n l i g h t o f th e n ee d t o p r o v i d e m o r e co m p r eh en s i v e g u i d e l in es f o r t h e

u se o f d i r ec t s eco n d - o r d e r i n e l a s ti c an a ly s i s i n t h e co n t ex t o f d es i g n

p r o v i s i o n s , t wo g r o u p s h av e r ecen t l y b een e s t ab l i sh ed : t h e Am er i can

I n s t i t u t e o f S t e e l C o n s t r u c t i o n ( A I S C ) T e c h n i c a l C o m m i t t e e 1 1 7 -

I n e l a s t i c An a l y s i s an d Des i g n , an d t h e S t r u c t u r a l S t ab i l i t y R esea r ch

C o u n c i l ( S S R C ) T a s k G r o u p 2 9 S e c o n d - o r d e r I n el a s ti c A n a l y s i s f o r

F r am e Des i g n . So m e o f t h e m a i n o b j ec t i v es f o r t h ese g r o u p s a r e :

1. To p r o v i d e a u n i f o r m d e f i n it io n o f t e r m s wh i ch p e r t a i n t o n o n l i n ea r

b eh av i o r an d an a l y s i s .

2 . To ca r e f u l ly ca t eg o r i ze an d q u an t i f y b eh av i o r a l e ff ec ts an d p h y s i ca l

p a r a m e t e r s wh i ch a f f ec t i n e l a s ti c f r am e s t r en g t h a n d s t ab i l it y , e . g . ,

t h e m ag n i t u d e an d d i s t ri b u t io n o f r e s id u a l s tr e s se s, m e m b er o u t -o f -

s t r a ig h t n es s , s t o r y o u t - o f -p l u m b n e ss , e t c.

3 . To i d en t i f y b en ch m ar k p r o b l em s f o r u se i n v e r if ica t io n an d co m -

p a r i so n o f s eco n d - o r d e r i n e la s t ic an a l y s i s p r o ced u r es .

4 . To i d en t i f y s i tu a t i o n s in wh i ch ad v an c ed an a l y s i s is ap p r o p r i a t e an ds h o u l d b e p e r f o rm e d .

5 . To a s ses s th e accu r acy an d f eas i b i li ty o f v a r io u s ap p r o a ch es f o r

ine l as t i c ana lys i s .

6 . To i d en t i f y i n n o v a t i v e an a l y s i s an d d e s i g n ap p r o ach e s wh i ch i n-

co rpo ra t e exp l i c it con s idera t ion o f i ne l as t ic e f fec t s i n the ana lys i s .




An important goal of further research should be to identify circumst-

ances and situations in which simplifying assumptions with regard to

structural behavior can be applied without incurring noticeable errors. In

recent work by King e t a l . , 18 an inelastic analysis approach based on a

modified plastic-hinge concept has been developed for rigid and semi-rigid

frames. This method assumes that the tangent stiffness of a beam-column

element reduces gradually from the elastic stiffness prior to initial yielding

at the element ends, to the stiffness associated with a real hinge, when the

plastic moment (modified to account for axial force effects) is reached at a

hinge location. The proposed analysis can predict the behavior of isolated

beam-column members, as well as the behavior of many types of frames,

more accurately than conventional plastic hinge methods. However, if

member P - 6 effects are large, more than one element per member is stillrequired to obtain accurate solutions. Research on a number of inelastic

analysis and design procedures is currently being undertaken by theauthors. 19

8 CONCLUSIONS

The first part of the paper provides a state-of-the-art review on thedevelopment of the LRFD beam-column interaction equations. The goal

of this part is to highlight the particular difficulties associated with

development of general equations for design of beam-columns in frames.

The LRFD equations in general provide a good fit to the more exact

strength curves for members in simple portal frames and leaned column

frames. This is true only when the column effective lengths are evaluated

properly. To this end, various methods of computing the effective length

factors and the corresponding second-order elastic moment amplificationfactors have been discussed. LeMessurier's procedure, which is based on

the story buckling concept, and the procedure based on a system buckling

analysis are possibly the most general for calculating effective length

factors. The modified K factor procedure gives essentially identical resultsto LeMessurier's procedure, and it is simpler to use. However, the

alignment chart procedure, which is based on the buckling of an idealized

subassemblage, is limited in its applicability and validity. The use of the

alignment chart effective length factors for leaned column frames or for

simple portal frames with unequal column stiffness parameters can result

in an unconservative estimate of member and/or system capacity.

Inelastic reduction of effective length factors by the procedure adopted

in the AISC-LRFD Manual appears to have little consequence on the

beam-column capacity in many types of frameworks, particularly where




t h e b ea m - co l u m n s a r e n o t su b j ec t ed t o h e av y ax ia l l o ad s. A l so , t h e u se o f

t h e i n e l a st ic K f ac to r s m ay b e q u es t i o n ab l e f o r ca ses i n wh i ch t h e b e am s

m a y b e i n e l a s ti c p r i o r t o th e f ac t o r ed l o ad s b e i n g r each ed . Fo r ex am p l e ,

t h e u se o f i n e l a s t i c K f ac t o r s wo u l d b e a m a t t e r o f co n ce r n i n a p l a s t i c

an a l y s is / d es ig n p r o ced u r e wh i ch t ak e s ad v an t ag e o f in e l a s ti c r ed i st r ib u -

t ion o f fo rces .

T h e C a n a d i a n a p p r o a c h t o b e a m - c o l u m n d e s ig n (w h i c h a v o id s t h e u s e

of K fac to r s ) i s sa fe , bu t t he des ign o f t he m em bers i s impl i c i t ly r es t r i c t ed

t o ce r t a i n r a t i o s o f b en d i n g m o m en t v e r su s ax i a l f o r ce . A l t h o u g h t h e

L R F D i n t e r a c t i o n e q u a t i o n s a r e g e n e r a l f o r t h e f u l l r a n g e o f b e n d i n g

m o m en t v e r su s ax i a l f o r ce , t h ey i n v o l v e t h e u se o f e f fec t iv e l en g t h f ac to r s .

Th e i n t e r p r e t a t i o n o f t h e e f f ec t i v e l en g t h fac t o r s f o r b eam - co l u m n s i n a

c o m p l e x f ra m e w o r k i s a c u m b e r s o m e t a sk . M o r e o v e r , t h e c u r r e n t d e s ig np r o c e d u r e s c o m m o n l y u s e d i n e n g i n e e r in g p ra c ti ce l e a d t o a f u n d a m e n t a l

i n co m p a t i b i li t y b e t w een e l a s ti c f r am e an a l y s i s an d ap p r o x i m a t e i n e l a st ic

m em b e r d es i g n eq u a t i o n s . Th e o n l y f u ll y r a t i o n a l wa y to acco u n t f o r

o v e r a l l f r am e s t ab i li ty is th r o u g h a s eco n d - o r d e r i n e l a s ti c an a ly s i s.

O n e m a j o r ad v a n t ag e o f s eco n d - o r d e r i n e l a s ti c an a l y s is i s t h a t t h e

r ed i s t r i b u t io n o f fo r ces fr o m o v e r l o ad ed m em b er s i s h an d l ed d i r ec t ly . Fo r

a la r g e s tr u c t u r a l sy s t em , o t h e r m e m b er s i n t h e f r am e w o r k m ay e f fec t iv e l y

' b r a c e ' a n o v e r l o a d e d m e m b e r s u c h t h a t t h e m e m b e r a n d t h e o v e r a l ls t r u c t u r e can s t il l r em a i n s t ab l e . Th e r e f o r e , i n g en e r a l t h e b eam - co l u m n

i n t e r ac t i o n eq u a t i o n s ap p ea r t o b e co n se r v a t i v e i n p r o v i d i n g f o r t h e

s t ab i li t y o f l a rg e a n d s t r u c tu r a l sy s tem s . T h e r e f o r e , i t is ex p ec t ed t h a t

f u t u r e r e f i n em en t s i n f r am e an a l y s i s an d d es i g n w il l l i k e ly f o cu s m o r e o n

t h e o v e r a l l sy s t em r e sp o n se an d l e ss o n i n d i v i d u a l m em b er r e sp o n se . Th e

A I S C - L R F D a n d C a n a d i a n S t a n d a r d s p ec if ic a ti on s e x p li ci tl y a ll ow t h e

u se o f d i r ec t s eco n d - o r d e r e l a s ti c an a ly s i s f o r d es ig n . Th e A u s t r a l i an L i m i t

S t a t e s S t ee l Des i g n Sp ec i f ica t io n 13 ( AS4 1 0 0) an d t h e Eu r o co d e No .316

p r o v i d e m o r e ex p l ic i t g u i d an ce f o r u se o f s eco n d - o r d e r i n e la s t ic an a l y s is i n

d e s i g n . A l t h o u g h m u c h p r o g r e s s h a s b e e n m a d e i n r e c e n t y e a r s o n

seco n d - o r d e r i n e l a s ti c an a l y s i s f o r f r am e d es i g n , m u ch m o r e w o r k r em a i n s

t o b e d o n e .

A C K N O W L E D G E M E N T S

T h e s u p p o r t o f th i s r e s e a r c h b y th e N a t i o n a l S ci en c e F o u n d a t i o n u n d e r

g r an t n u m b e r M SM-8 91 44 61 i s g r a t e f u l ly ack n o w l ed g ed . Th e f i rs t au t h o ris o n a n o v e r s e a s g r a d u a t e s c h o l a rs h i p fr o m t h e N a t i o n a l U n i v e r s i t y o !

S i n g a p o r e . T h e f in a n c ia l s u p p o r t p r o v i d e d b y t h e N a t i o n a l U n i v e r s i t y o !

S i n g ap o r e is a lso g r a t e f u l ly ack n o w l ed g ed .



308 J. Y. Richa rd Liew, D. W. W hite, W. F. Ch en

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