buk7-coupled instabilities in mono-symmetrical steel compression members

8/11/2019 Buk7-Coupled Instabilities in Mono-symmetrical Steel Compression Members

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J . Cons truc t . S tee l Research

21 (1992) 71-9 5

oupled Instabilities in M onosym m etrical Steel

om pression M em bers

aV. G ionc u aN. B~ilut bD. D ubin~ aA. M old ov an aC. Pacos te

Building Research Institute, INCERC, Timi~oara Branch, Romania

bTechnical Un iversi ty of Timi~oara, D epartm ent of Steel Structures, Timi~oara, Ro ma nia

A B S T R A C T

Th e paper analyses the in teraction betw een f lexu ral a nd f lex ura l - torsional

buckling in the case of mon osym metrical s teel compression m embers nam ely C ~

L and T sections. Th eor etica l analysis has led to the conclusion that this

in teraction can be c lassi fied as w eak or medium. L abor atory tests were performed

on welded T sections. An a pprox ima te method based on an in terpolation betw een

column curves in order to al low for the detr im ental e f fect o f instabili ty coupling is

sug ges ted fo r practical design.

N O T A T I O N

o q N F

a l

o~

N T

3 ~ t N F ~ / 3

A = C r o ss s e c ti o n a r e a o f t h e c o l u m n

B x E I x

B y E I y

C G I t

Co, = E I~

D r = r y B y

e x ~ ey z = S h e a r s t r a i n s

ez z = A x i a l s t r a i n

E = Y o u n g ' s m o d u l u s o f e l a s t ic i ty

f y = Y i e l d s t r e s s

71

J. Construct. S tee l Resea rch 0143-974X/92/ 05.00 1992 Elsevier Science Publishers Ltd ,

England. Pr inted in Malta


2/25

72

G

2

I c

Ix, I r

I o

J

K

N

N c r

NF

N v x

Nv~

N u

N T

N x

N y

ry

(x

K

V. Gion cu et al.

= E / 2 (1 + v), Shear modulus of elasticity

= ~ [ Y 2 + x - x c ) 2 ] d A

ic

= Moments of inertia

= Torsional warping c onstant of the cross-section

= St Venant's torsion constant of the cross-section

= 1 - \ ~ /

= Length of the member

= Axial load

= Bifurcation critical load

= Critical load for flexural buckling

= Critical load for flexural-torsional buckling

= f y A Full-yield axial load (squash load)

= Ultimate load of the column

C

. 2

I c

rC2Bx

12

= 7 ~ 2 B y

= [y + x-x0qxdA

r y

1

X e

= Factor of ECCS column curves

(cq = 0-206; eb=0 339; ctc=0 489)

= Buckling coefficient

4

1 - N N

f Xc


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Cou pled instabilities in com pression mem bers

73

~2

m w

~ m a x

: J N p l / g m l n

~ rnin = J N p l / N m a x

1 r2

= S h o r t e n i n g o f t h e b a r

= R e d u c e d s l e n d er n e ss o f th e c o l u m n

P o i s s o n ' s r a t i o

O t h e r n o t a t i o n u s e d i n t h i s p a p e r i s s p e c i f ie d i n t h e t e x t o r i n th e f i g u r e s . )

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

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

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

f o r m o f i n s t a b il i ty , w h i c h m a y b e t e r m e d ' i n t e r a c t i v e b u c k l i n g ' o r c o u p l e d

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

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

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

case Of s imple ins tab i l i t ies .

W h i l e t h e t h e o r e t i c a l r e s e a r c h c a r r i e d o u t b y E u l e r i n t h e e i g h t e e n t h

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

a p p r o a c h t o t h e c o m p l e x p h e n o m e n o n o f c o m p o u n d b u c k l in g w o u l d n o t

h a v e b e e n p o s s i b le w i t h o u t t h e p i o n e e r w o r k d o n e b y K o i t e r , 1 '2 '3 C h i l v e r , 4

T h o m p s o n a n d H u n t , s B u d i a n s k y 6 a n d o t h e r s i n t h e s e c o n d h a l f o f th is

c e n t u r y .

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

f l e x u r a l - to r s i o n a l b u c k l i n g o f m o n o s y m m e t r i c a l st ee l m e m b e r s . G r i m a l d i

a n d P i g n a t a r o 7 h a v e s t u d ie d t h e c a s e o f s i m u l t a n e o u s b i fu r c a ti o n s b y

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

s h o w n t h a t t h es e m e m b e r s b e c o m e h i g h ly s e ns it iv e t o g e o m e t r ic a l im p e r -

f ec t io n s . Ho wev e r , t h i s sen s i t i v i ty a l so ex ten d s to th e ca se o f n ea r -

s i m u l t a n e o u s b i f u rc a t i o n ; it r e su l ts i n a n i n t e r a c t i o n c u r v e w h i c h is s i m i l a r

t o t h e o n e s e n v i s a g e d b y T h o m p s o n . 8

T h e n u m e r i c a l a p p l i c a t i o n s g i v e n h e r e i n r e fe r t o 1 , ~ , ( , L a n d T

c o m p r e s s i o n m e m b e r s . P a r ti c u l a r ly , a r e s ea r c h p r o g r a m m e , b o t h t h e o re t i-

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

m e m b e r s a n d a p r a c t ic a l m e t h o d f o r t h e d e s ig n o f t h es e m e m b e r s a r e

p r e s e n t e d . W e l d e d T s e c t i o n s a r e c u r r e n t l y u s e d i n R o m a n i a a s t r u s s

c h o r d s .


4/25

74 ~ G i o n c u e t a l

2 T H E C O U P L E D I N S T A B I L I T Y P H E N O M E N O N

M o n o s y m m e t r i c a l m e m b e r s c o u l d b e si ze d s o t h a t t h e cr it ic a l lo a d s

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

f l e x u r a l - t o r s i o n a l b u c k l i n g ) a r e e q u a l . L o c a l b u c k l i n g i s n o t t a k e n i n t o

a c c o u n t . B a s e d o n t h e s e c o n s i d e r a t i o n s , M a t e e s c u

e t a l 9

h a v e o b t a i n e d a

s e t o f o p t i m a l T s e c t io n s .

I n o r d e r t o a n a l y s e t h e c o u p l e d i n s t a b il it y p h e n o m e n o n , t h e c a se o f

e l a s t i c i n s t ab i l i t y w i l l b e co n s i d e red f i r s t .

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

F i g . l a )) t h e c r it i c a l l o a d c o r r e s p o n d s t o t h e i n t e r s e c t i o n b e t w e e n t h e

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

i o u r , w h i ch i s s t ab l e :

N m i n =

min NF, NFT)

1)

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

c o n t i n u o u s i n c r e a s e i n th e m e m b e r l a te r a l d e f l e c t io n s ; t h is i n c r e a s e is

m a r k e d i n t h e v i c in i ty o f t h e b i f u r c a t i o n p o i n t .

I n t h e c a s e o f c o u p l e d b u c k l i n g F i g. l b )) , t h e s e c o n d a r y p o s t - c r i t ic a l

c u r v e is u n s t a b le . F o r a n a c t u a l m e m b e r , it is f o u n d t h a t :

a) t h e d e f o r m a t i o n s a r e in c r e a s e d in c o m p a r i s o n w i th t h o s e c o r re -

s p o n d i n g t o u n c o u p l e d i ns ta b il it y; a n d

b ) a n u l t i m a t e l o a d N u i n th e e la s ti c ra n g e c o u l d b e d e fi n e d a s th e l o a d

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

o f e q u i l i b r i u m is r e a c h e d .

N

Nmin I

/ ~ - ' - actuat

uncupt.e.d e

tnsfabitffy

N auncoup~ed j ~

/ / / instabitity

u

a) b)

Fig. 1. (a) Uncoupled buckling; (bl coupled buckling.


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Cou pled instabilities in compression members 75

I n t he c a s e o f p l a s ti c de f o r m a t i ons , t w o d i f f e r en t k i nd s o f be ha v i ou r s c a n

b e o b s e r v e d , d e p e n d i n g o n t h e m e m b e r s l e n de r n es s . T w o c u r v e s w e r e

d r a w n i n F i g . 2, the f i rs t one c o r r e s po nd i ng t o t he f i rs t y i e ld a nd t he

s e c on d one t o t he f o r m a t i on o f a p l a st ic h i nge . T h e c u r ve o f e l as t ic

be h a v i o u r i s i n te r s e c t e d w i t h t he s e t w o c u r ve s i n d i ff e re n t w a ys , de pe n d i ng

o n t h e N m i n N p l ra t io .

F o r v e r y s l e n d e r m e m b e r s , w h e r e N m i n ~ N p l (Fig. 2(a)) , the curve

de v i a t e s f r om t ha t c o r r e s pond i ng t o e l a s t i c de f o r m a t i ons a f t e r t he f i r s t

y i el d is p r o d u c e d a n d t h e u l ti m a t e l o a d

u

d o e s n o t d if fe r t o o m u c h f r o m

t ha t c o r r e s pond i ng t o t he e l a s t i c be ha v i ou r .

W h e n N m i n

i s c lose

t o N p h

t he l imi t va lue u d i f f e r s ve r y m uc h f r om

tha t cor re sponding to e l a s t i c de forma t ions (F ig . 2 (b) ) .

T h i s c a s e i s m or e f r e que n t i n p r a c t i c e a nd t he e qu i l i b r i um pa t h i s

pre sen t ed i n F ig . 3 .

G e n e r a l ly , t he d i ff e re nc e be t w e e n t he b i f u r c a ti on l oa d a nd t he l i m i t l oa d

r e p r e s e n t s t he e f fe ct o f e r o s i on a n d de p e nd s on t w o f a ct o rs :

(1 ) t h e m a g n i t u d e o f t h e u n s t a b le c o m p o n e n t o f t h e b i fu r c a ti o n l o a d

( w h i c h c a n a l w a y s b e d e c o m p o s e d i n to a s ta b le a n d a n u n s t a b l e

c om pone n t , t he l a s t one be i ng e r ode d by i m pe r f e c t i ons a s i t w a s

p r o v e d b y G i o n c u a n d I v a n 1 a n d G i o n c u l ) ;

(2 ) t he va l ue s o f t he ge om e t r i c a l a nd m e c ha n i c a l i m pe r f e c t i ons ( w h i c h

a r e r e p l a c e d i n m a n y c a se s by e qu i va l e n t ge om e t r i c a l i m pe r f ec t ions ) .

I I I / I I I r s t y~eLd

sudden increa,ing of deformahon

u

l

b

F i g 2. (a) Elas tic-plas tic buckling of a very slender column, Nmia ,~ Npl; (b) elastic-p lastic

buc kling o f a usual column, Nmln close to N p:.


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76 V ,

G i o n c u e t a l

F i g . 3 . C o u p l e d e l a s t i c - p l a s t i c b u c k l i n g c u r v e .

I t c an b e seen f ro m F ig . 4 b ) t h a t t h e g eo m e t r i ca l e f f ec t i s t h e e ro s io n o f

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

a m o u n t o f e r o s i o n , t h r e e c l as s e s o f e r o s i o n w e r e su g g e s t ed , d e p e n d i n g o n

t h e e r o s i o n p a r a m e t e r f t .

T h e i m p o r t a n c e o f t h is c l a s s i fi c a ti o n r e s u lt s f r o m t h e d i f fe r e n t d e g r e e s o f

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

t h e f ir st c l a ss o f e r o s i o n a v e r y p r e c i s e d e t e r m i n a t i o n o f th e i m p e r f e c t io n s

i s l e s s im p o r t an t t h an fo r th e th i rd c l a ss .

I n t h e c a s e o f s t r a ig h t c o m p r e s s i o n m e m b e r s w i th c o m p a c t c ro s s-

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

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

bifurcation oint

N

\ \ /posfcr~tlcar ~ -~

N h

~ ~ / ~ pOS e tas fica l r tlrV e ~ r

. . . . . .

Nor I

l r o s l o

n ~ ~ u n st te

I 1 4 . ~

gL~motCicat- ~ IL da ss 3

a ) par a~, ter

geon le Hc a[

b parameter

F i g . 4 . a ) E r o s i o n o f c r i t ic a l b i f u r c a t i o n l o a d ; b ) c l a s s e s o f e r o s i o n .


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oupled instabilities in compression members 77

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

c o m p o n e n t p a r t s o f a s t r u c tu r e o t h e r u n s t a b l e c o m p o n e n t s r e s u lt fr o m

e la s t i c su p p o r t s p a r t i a l r e s t ra in t s ).

C o n s e q u e n t l y , c o u p l e d i n s t a b i li ti e s r e p r e s e n t a n i n c r e a s i n g f a c to r f o r i m -

p e r f e c t io n s e n s i t iv i ty a n d a d e c r e a s i n g f a c t o r o f s t r u c t u r a l b e a r i n g c a p a c i t y .

I n t h e c a s e o f c o u p l e d i n s ta b i li ti e s , t h e a m o u n t o f e r o s i o n d e p e n d s o n

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

t h e i r p o i n t o f in t e r s e c t io n .

G i o n c u 11 s u g g e s ts t h e f o l lo w i n g c l a s si fi c a ti o n , d e p e n d i n g o n t h e e r o s i o n

fac tor ~, Fig . 4 b)):

- - w eak in t e r ac t io n , f o r ~ ,~


8/25

7 8 V . G i o n c u e t a l .

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

e~z = w + (u ) 2 + (v )2 + u lu ' + v'iv'; (3a,b,c)

exz = 7xz;

e y z = ~ y z

T h e t e r m s o f g re a t e r o r d e r t h a n 2 w e r e n e g l e ct e d w h e n c o m p u t i n g t h e

a x i a l s tr a in s , w h i l e o n l y t h e l i n e a r t e r m s w e r e c o n s i d e r e d i n t h e e x p r e s s io n s

o f s h e a r d e f o r m a t i o n s

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

t h e e x p r e s s i o n o f a x i a l s t r a i n . B y s u b s t i t u t i n g t h e s p e c i f i c d e f o r m a t i o n s

f r o m e q n ( 3 ) i n t o e q n ( 2 ) a n d a r r a n g i n g t h e t e r m s b y t h e i r e x p o n e n t s , o n e

o b ta in s :

[ I = I - I 2 + I - I 3 + [ I 4 (4)

T h e l a st t e r m c o n t a i n s t h e fo u r t h o r d e r e x p o n e n t s a n d is n e g l e c t e d i n

t h e a s y m p t o t i c s t a b i li t y t h e o r y o f K o i t e r . 1 T h e t e r m s i n t h e p o t e n t i a l

e q u a t i o n (4 ) m a y b e e x p r e s s e d a s:

i - i 2 i j . . . . 2 . . . . 2 . 2 ,2

= t o y u + o x v + C~ ,q 0 + C t p ) d z

N I n - 2 , 2 , ,( u ' 2 + v ' 2 + i c t p - 2 x c v q g ) d z

o

- N [ u ~ u ' + v i ( v ' - x c q ) ' ) + q ) i ( - x c v ' + i ~ q ) ' ) ] d z

l In L , , ,2 , , , , , , ,

- I 3 = - ~ ( D y u q) + 2 B x u v q ~ - 2 B y u v ) d z

_ i 4 f o

~ By f f '2u 2 d2

(5a)

(5b)

(5c)

F o r a s i m p l y s u p p o r t e d m e m b e r , th e g e o m e t r ic a l i m p e r f ec t io n s a n d t h e

i n s t a b i l i t y d e f o r m a t i o n s m a y b e w r i t t e n a s :

m T t z

{ U i / i q i } T = 2 { U i m l V i m l ~ i m } T

sin ~

??

m T ~ z

{uv~0}v= ~ {U,,,lVmlckm} s in

l

m

(6a,b)


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Coupled instabilities in compression members 79

w he r e

V i m

Vim , Ji m,

U r n V m

a nd q~,. a r e no n - d i m e ns i ona l pa r a m e t e r s .

Su bst i tu t ing eq ns (6) into eq ns (5) resul t s (m = 1):

21-1 n ,, 2

7 z l = N r ~ U + ~ [fy Ny Ucp2 + 2 ( N r - N x) U Vdp]

2 2 N 2 V 2 .2 ~b2_ 2:~cV~b)

N y U 2 + N ~ V a + i~ N T C ~ ) - ~ U + + to

- NE UU i + VVi + ic~bi- xc ViO + V[ i)]

(7)

F o r a n i de a l c o l u m n w i t hou t ge om e t r i c a l i m pe r f e c ti ons , t he c r it ic a l loa d

r e s u lt s f r om t he c on d i t i on 6F I = 0, w h e nc e t he s t ab i li ty e qua t i ons

Ny = N F

2 2 -2

Nx -

NFT) NT - - N t - r t c - NF T X c = 0

(8a,b)

I t c a n b e s e en f r o m e q n s ( 8) t h a t f o r a n i d e a l c o l u m n o f m o n o s y m m e t r i c a l

c ross - sec t i on , i ns t ab i l i t y can occur e i t he r by f l exura l buck l ing wi th in t he

p l a n e o f s y m m e t r y o f t h e m e m b e r u n d e r t h e b i f u rc a ti o n l o a d N v , o r b y

f l e xu r a l - t o r s i ona l buc k l i ng unde r t he b i f u r c a t i on l oa d N vr r e s u l t i ng f r om

the i n t e rac t i on equa t ion (F ig . 6 ) :

NFT NFT N2T

K - - 1 (9)

N x N T N ~ N T

F r om t he c u r ve s p l o t t e d i n F i g . 6, i t c a n be s e e n tha t t he c oup l i ng o f t he

t w o f o r m s o f i n s t a b il i ty doe s no t t a ke p l a c e i f K = 1 ( w he n xc = 0 , i.e .

do ub l y s ym m e t r i c c r o ss - se c ti on ). F o r a n i de a l c o l um n , t he l ow e r o f t he t w o

c r i ti c a l b i fu rca t i on l oads , N v or Nva -, wi l l be t h e co l l apse l oad . T hus , fo r an

idea l co lumn, F ig . 7 (a ) i nd i ca t e s t he two c r i t i c a l l oads , whi l e F ig . 7 (b)

s how s t he r a nge s o f the s e t w o i n s ta b i l it y m od e s . F i gu r e s 7 ( c ) a nd 7 (d )

Nr

K = I 0

0 8

=

Fig. 6. Interac tion curve of eqn (6).


10/25

8

K ioncu et

a l

1 1 6

J t o r s p o n a l - f t e x u r a [

I / / / b u c k l i n g

8 - - - j 0 . B

I

f l e x u r a [ 4 a c f u a t b a r

z ~ b u c 0 , ~

1 h

0 . 2 0 ~ , 1 6 0 . 8

a )

t t

h 2

h i

f [ e x u r a { ~ t o r s i o n a l

b u c k i n g ~ f e x u r a t

b

oz o .6 o .~ o . 7 ~

b }

N / ~ 2 t h 3

Q I [

~ b

1 6 1 6 1 . 8

C )

~ . . 0

0 . 2 t

I t S

h 2

t o r s i o n a t

ft xural ~ [ g

b u t k t i n g

i

d )

F i g . 7 . B u c k l i n g o f i d e a l m o n o s y m m e t r i c a l s e c t i o n : ( a ) [ - s e c t i o n b u c k l i n g c u r v e s , ( b) [

s e c t i o n b u c k l i n g r a n g e s , ( c ) T s e c t i o n b u c k l i n g c u r v e s , ( d ) T s e c t i o n b u c k l i n g r a n g e s .

p r e s e n t t h e s a m e r e s u l t s f o r a T c o l u m n . F i g u r e 7 ( a ) a l s o i n d i c a t e s t h e

c o l l a p s e l o a d o f th e a c tu a l c o l u m n c o n s i d e r i n g a g e o m e t r i ca l i m p e r f e ct i o n

e q u a l to 1 / 2 00 . T h e c u r v e i s s e e n t o b e c o n t i n u o u s , t h e p o i n t o f i n t e r s e c t io n

o f t h e t w o f o r m s o f i n s t a b i l it y h a v i n g d i s a p p e a r e d . T h e d e c r e a s e i n th e

c o l l a p s e l o a d a s a re s u lt o f t h e in f l u e n c e o f i m p e r f e c t i o n s is m a x i m u m n e a r

t h e p o i n t o f s im u l t a n e o u s b i fu r c a ti o n . T h i s i s w h y B l e ic h s s i m u l t a n e o u s

b u c k l i n g d e si g n p r i n c i p le p r o v i d e s o p t i m u m s o l u t i o n s o n l y f o r i d ea l

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

T h e d e t r i m e n t a l i n f l u e n c e o f g e o m e t r ic a l i m p e r f e c t io n s c a n b e o b -

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

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

i n g f r o m t h is r e m a r k , G r i m a l d i a n d P i g n a t a r o s t h e o r e t i c a l r e s e a r c h

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

c a t i o n .


11/25

Cou pled ins tabi l it ies in com press ion mem bers 8 1

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

i z in g f l e x u r a l - t o r s i o n a l b u c k l i n g i s g i v e n b y

N F T (10)

V = N x _ N F T X C ~

A s s u m i n g t h i s r e la t i o n s h i p t o b e a l so v a l i d f o r l a rg e d e f o r m a t i o n s a n d f o r

the im pe r f e c t ion s V~ a n d ~b~ the po te n t i a l (7 ) c a n be w r i t t e n :

2 H n 2 U 4 + c q U ~ b 2 + 1 - ~ -~ + 2 \ N F N -v ~2

~ 2 1 N F 32

N N

- - N - - ~ U i U - ~FF 0~2~i1 (I I)

T h e l o a d - d e f o r m a t i o n c u r v e re s u l ts fr o m t h e fo l l o w i n g e q u a t i o n s :

~H t~H

= 0 ; ,,---;- = 0 (12 a,b)

---~

q~c

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

6 3 2 I-I 0 2 I I ( 0 2 I - I / 2

~ U 2 ~ ( ~ 2 ~ k ~ - - - ~ / = 0 ( 1 3 )

F i n a l l y , a s i m p l e i n t e r a c t i o n r e l a t i o n s h i p r e s u l t s ( b y n e g l e c t i n g t h e t e r m s

c o n t a i n i n g U 4 ) i n t h e p o t e n t i a l e n e r g y e x p r e s si o n :

N F J \ ~

= a ' - N ~ v U ' + a z ~

1-NvJ r . J

(14)

T h i s r e l a t i o n s h i p i s o n l y v a l i d f o r th e i n t e r a c t iv e b u c k l i n g o f a n a c t u a l

m e m b e r . I f U i = 0 a n d ~bl = 0, t h e n t h e t w o i n s t a b i l i t y f o r m s b e c o m e i n d e -

p e n d e n t .

C o m p u t a t i o n o f t h e c o e ff i ci e n ts a~ a n d a 2 f o r v a r i o u s t y p e s o f c r o s s -

s e c t io n s s h o w s t h a t t h e f i rs t i s m u c h g r e a t e r t h a n t h e s e co n d . C o n s e q u e n t -

ly , t he e ff ec t o f f l e xu r a l im p e r f e c t io ns i s m uc h g r e a t e r t h a n the e ff ec t o f

t o r s i o n a l i m p e r f ec t io n s , w h i c h i s a ls o s h o w n i n t h e f o l lo w i n g a p p l i c a t i o n s .

F i g u r e s 8 - 1 1 p r e s e n t n u m e r i c a l r e s u lt s w h i c h c a n b e o b t a i n e d b y a p p l y i n g

e q n ( 1 4 ) f o r [ , L , P. a n d T s e c t io n s . S i n c e t h e d e c re a s e i n b e a r i n g


12/25

82 F Gioncu et a l

o6 ~ ur ~-~.lo--

~ N

0 2 0 / * 0 .6 0 .8 1 .0 ~ - -

N T F

F i g 8 i n t e r a c t i v e b u c k l i n g c u r v e f o r [ s e c t i o n

Q8

0 4

0.2

N / / [ / 2

1 03

~o o.. _ ~ oo

f t e x u r l

~

b u c k t i n g \ \ \

\ . 0 . 6

a c t u a l b a r II b u c k t i n o

i 0 2

i

t

a l

/ N T F

1 0

Ui=~i=

_21~3

N

L i . z L ~

0 .2 O # 0 . 6 0 . 8 1 .0 N

b )

F i g 9 I n t e r a c t i v e b u c k l i n g c u r v e f o r L s e c t i o n

h i / ~ E th ;~

A , - x ' ~ ~NTF

~

0 u c k ,,o ~ / . 4 - - ~ I J ~ - -

\\I

8 I [ D u c klin g " I ~ ' ~ I I - ~ I I

0.2 0 ~ 0.6 O. 1.0

a ) b )

F i l l I 0 I n t e r a c t i v e b u c k l in g c u r v e f o r ~ a n d ~ s e c t io n s

c a p a c i t y i s n o t v e r y l a r ge t h i s t y p e o f c o u p l i n g c a n b e c l a s s if i ed a s

we k

i n t e r a c t i o n

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

p l a st ic d e f o r m a t i o n s . T h i s i s w h y a p p r o x i m a t e s o l u t i o n s a r e u s e d in s u c h

c a s e s .


13/25

oup led n stabi l i t i es n compr ession members 8 3

O t ~

0.3

0.2

0.1

. = ~ N F

h ~ / f ~ f

- - l .O

f l e x u r a l

b t ~ k t l n 9

_ 0 9 0

b u c k l i n g

b 0.85

T

l t 1 . 6 1 . 6 1

a

F i g . 1 1 . I n t e r a c t i v e b u c k l i n g c u r v e f o r T s e c t i o n .

1 . 3 8

Z _ = I .5 / b.

1.5 2. 2 5 3.0 3mS

b)

M o l d o v a n a n d G i o n c u 12 h a v e g i v e n a s i m p l e s o l u t i o n f o r th e c a se o f T

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

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

s idered:

1 ) T h e i n s t a b i l i ty o f a m e m b e r , w h i c h h a s i m p e r f e c t i o n s o n l y i n it s p l a n e

o f s y m m e t r y , U i = / / 5 0 0 , i s d u e t o f le x u r a l b u c k l i n g . F i g u r e 1 2 a ) i ll u s tr a t e s

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

a n d t h e c u r v e c o r r e s p o n d i n g t o t h e f o r m a t i o n o f a p la s t ic h i n g e a t th e

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

N

NF

0 8

0 6

O 4

O?

l I

I=~O00mm

p I . a s f i

s l o

~o.lo-~ u

a

N F ~ - f o r s F o l - f l e u r a (

0 .8 \ b u d ~ l l n g i ~

0.6 /~ \~ors tona l f lexUrat IE 3

L / / b v c k l ~ n g o 4 P l e d ~ lllll

. I f w i f h

f ~xuraf

1 ~ ~ X ~ O l ~ C h i n g u a l b u c k l in g

5 . % 0 - ~ I 0 ~ i 0 - . = -

U

b

F i g . 1 2 . a ) I n s t a b i l it y p r o d u c e d b y fl e x u r a l b u c k l i n g ; b ) i n s t a b i li t y p r o d u c e d b y s i m u l t a -

n e o u s b u c k l i n g .


14/25

84

V G i o n c u e t a l

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

a n A y r t o n - P e r r y r e l a t i o n s h i p i n t h e f o r m o f

15)

2 ) I n s t a b i l i t y o c c u r s b y n e a r - s i m u l t a n e o u s b u c k l i n g , b e i n g i n f l u e n c e d

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

F i g . 1 2 b ) i n t e r s e c t s b o t h t h e f i rs t y ie l d c u r v e a n d t h e p l a s ti c h i n g e c u r v e .

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

c a se . B y g e n e r a l i z i n g e q n 1 5), o n e o b t a i n s :

g p l ) k N F ) k NF--~) = 0~ ? gmi-'--- N p I

16)

E q u a t i o n 1 6) is p l o t t e d i n F ig . 1 3 f o r ~ = 0 . 3 3 9 , i n c o m p a r i s o n w i th t h e

E C C S c o l u m n c u r v e s b a n d c . I t c a n b e s e e n t h a t t h e d e c r e a s e i n t h e

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

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

N

NpL

o a N F = N T F

0.2 0.4 06 0.8

Fig. 13. In teraction curve of eqn 16).

N

1 0 N F


15/25

oupled instabilities in com pression memb ers

85

E X P E R I M E N T A L R E SE A R C H

S i nc e t he o r e t i c a l i nve s t i ga t i ons ha ve l e d t o t he c onc l u s i on t ha t c oup l e d

ins t ab i l i ty r e su l t s i n an i nc reas ed sens i ti v i ty to impe r fec t i ons , i t was j u dg ed

ne c e s s a r y t o c a r r y ou t l a bo r a t o r y te st s.

T h e e x p e r i m e n t a l p r o g r a m m e p e r f o r m e d b y t h e a u t h o r s t 3 1 . i n v o l v e d

t he t e s t i ng o f 48 w e l de d T s e c t ion m e m be r s , 36 o f w h i c h w e r e s ub j e c te d t o

buc k l i ng , w h i l e t he o t h e r 12 w e r e u s e d f o r de t e r m i n i ng the m e c ha n i c a l

cha rac t e r i s t i c s and t he re s idua l s t r e s se s (Tabl e 1 ) . The member c ross -

s e c t i ons w e r e c hos e n s o a s t o f a c i l i t a t e t he e xpe r i m e n t a l s t udy o f t he

va r i ous t ype s o f buc k l i ng . T h e t e s t s pe c i m e ns w e r e m a de o f O L 52 g r a de

s te e l s he et , w i t h f y = 36 N / m m 2, u s i ng c u r r e n t i ndus t r i a l t e c hno l ogy .

T h i s e xp e r i m e n t a l p r o g r a m m e i nc l ud e d t he f o ll ow i ng t es ts :

(1) s t ub c o lu m n t e s ts (F igs 14(a ) an d (b)) ;

( 2 ) e xpe r i m e n t a l de t e r m i na t i on o f r e s i dua l s t r e s s e s by m e a ns o f t he

s t r i p -cu t t i ng me thod (F ig . 15) ;

(3 ) de t e r m i na t i on o f t e n s il e y i e ld s t r e ng t h a n d u l t i m a t e s t r e ng t h by t es ts

pe r f o r m e d on s tr ip s ;

(4 ) o p t i ca l d e t e r m i n a t i o n o f g e o m e t r i c a l i m p e r f e ct io n s o f th e m e m b e r

c hos e n t o un de r g o bu c k l i ng t e st s (F i g. 16);

(5) bu ck l ing te s t s ca r r i ed ou t by u s ing a 200 t ons h ydrau l i c j a ck (F ig . 17).

T h e s up po r t de v i c e s u s e d f o r the buc k l i ng t e st s w e r e m e a n t s o a s t o m od e l

a s c l o s e ly a s pos s i b le t he t he o r e t i c a l s upp o r t c ond i t ions , na m e l y :

- - p r e v e n t i o n o f d i s p l a c em e n t s i n a n y d i r e c t io n p e rp e n d i c u l a r to t h e

l o n g i t u d i n a l ax is o f t h e m e m b e r ;

- - f re e f le xu r a l r o t a t i o n i n a ny d i r e c ti on ;

- - p r e v e n t i o n o f t o r s i o n a l r o t a ti o n .

3 h

i

J i x

T A B L E 1

C h a r a c t e r i s t i c s o f th e T e s t S p e c i m e n s

m m m m m m ~ ~ m m

75 BO 4 4. A

T1 2'700 110 160 10 0 .5 4 4 0,566 3 '7 B

107 73 2 . A

T2 2700 80 160 10

0.351 0.626 2 7 B

5 8 3 4 A

T3 2700 140 160 10

0.679 0 .502 3 7 B

80 06 3 6 A

T6 3600 160 210 10

0.506 0.533 4. 7 B

106 77 z 4. A

T5 3600 110 210 10

0.355 0.591 3 7 B

69 90 5 4. A

T6 3600 16G 210 10

0.590 0.48'/ 2 7 B


16/25

86 V G i o n c u e t a l

Fig 14 Stub column test


17/25


18/25

88


a

o-. cl~ o r-~ . . . .


19/25

oupled instabilities in compression mem bers

89

Fig 17 Buckling test

a p p l i c a t i o n o f t h e c o m p r e s s i o n l o a d . T h i s s u p p o r t d e v i ce F i g s 1 8 a ) a n d

b )), i m a g i n e d b y D u b i m t a n d P a c o s t e a n d d e s i g n e d b y D u b in ~ l, 14 g e n e r a l -

i ze s t h a t d e s c r ib e d b y K 1 6 p p e l a n d U n g e r 15 i n t h e s e n s e t h a t a s y s t e m o f

s u p p o r t s a n d g u i d e s in b o t h x a n d y d i r e c t io n s e n a b l e s t h e e n d s e c ti o n s t o

r o t a t e f r ee l y a b o u t t h e s e a x e s t h e p r i n c i p a l a x e s o f i n e r ti a ).

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

e x p e r i m e n t a l r e s u l t s . T h i s i s w h y F i g s 1 9 a ) a n d b ) a n d F i g . 2 0 p r e s e n t i n a

s y n t h e t i c a l m a n n e r o n l y t h e r e s u l t s c o n c e r n i n g t h e b u c k l i n g t e s t s .

T h e f o l lo w i n g c o n c l u s i o n s h a v e b e e n a r r iv e d a t u p o n a n a l y s i n g al l t h e

r e s u lt s r a t h e r t h a n j u s t t h o s e m e n t i o n e d h e r ei n ) o f t h e e x p e r i m e n t a l

p r o g r a m m e :

1 ) A l t h o u g h t h e y d i s p l a y g r e a t s c a tt e r w i t h i n t h e l im i t s c o m m o n

t o s u c h t es ts ), th e v a l u e s o f t h e b u c k l i n g l o a d s f o r t h e m e m b e r s

T 1 a n d T 4 r e v e al b o t h t h e i n t e r a c t i o n o f t h e t w o f o r m s o f i n s t a b il -

i t y d i s c u ss e d h e r e i n a n d a n i n c r e a s e d e r o s i o n o f t h e b i f u r c a t io n

l o a d .


20/25

90


Fig 18 Support device for the buckling tests


21/25

Cou p led in stab i l i t i es in compr ession m embers 9

0 . 2

0 . 1 5

0 . I

Q O _

Ncr~E ._ . . -~ . f te xurat torsionat buckt in

N /' i ' l '~th~ f l e x u r a l t o r s i o n a [ b u c k li n g

c r / , ~

~ r - - .

J th ickness o f w e [ d a : G m m

th i ckn ess o f we l d a-~Trnm

aver ag e va l u e

f f

f c

( f n i c k n e s s o t w e l d a = 6 m m

tn icl~ness of wetcl az?mm

a v e r a g e v a l u e

~ . o ~ . z

1.~

~ . 6 ~ . e z . o ~ o

~ z

1 . 4 ~ . 6 1 . e

a b

2 )

3 )

Fig. 19. Experimental results compared w ith buck ling curves.

1 .0 ~ ~ N y

0 8

0.0

Oh

0.2

0 .2 0 ~

a :Gmm

a :7mm

~ a v e r a g e

o f

a v e r a e o f ==

M a q u o i _Rondal

e q u a l J o n

~ ~c lu rve b (*~-:

0.8 1.0 N~.NF

Fig. 20. Exp erimental results com pared w ith interaction curves.

G i v e n t h e s a m e c r o s s -s e c t io n a n d t h e s a m e v a l u e s o f t h e g e o m e t r ic a l

i m p e r f e c t io n s , t h e m e m b e r s w i t h t h ic k e r w e l d s t y p e B ) g e n e r a l l y

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

m e m b e r s w i t h t h i n n e r w e l d s t y p e A ).

W h e n d i s c u s si n g o n l y t h e b e h a v i o u r o f t y pe A m e m b e r s t he n o r m a l

c a se ), it b e c o m e s a p p a r e n t t h a t i n a l m o s t a l l c a s e s w h e r e o n e o f t h e

i n s t a b il it y f o r m s f le x u r a l b u c k l i n g f o r T 2 A a n d T 5 A , f le x u r a l -

t o r s i o n a l b u c k l i n g f o r T 3 A a n d T 6 A ) p l a y s a d e c i s i v e r o l e , t h e

r e s u l t s a r e a b o v e c u r v e b o r i m m e d i a t e l y b e l o w i t . F i g u r e s 1 9 a n d

2 0 s h o w t h a t in t h e c a se o f T 1 a n d T 4 m e m b e r s , c u r v e c is

b

2 .0


22/25

92 V G i o n c u e t al

4 )

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

c u r v e y i e l d e d b y e q n ( 3 ) c o v e r s f a v o u r a b l y b o t h t y p e s , A a n d B .

I n t h is c o n t e x t , it se e m s r e a s o n a b l e t h a t w h e n o n l y o n e o f th e t w o

f o r m s o f i n s t a b i l i ty p r e v a i l s ( r e g a r d le s s o f w h e t h e r i t is f l e xu r a l

b u c k l i n g i n t h e s y m m e t r y p l a n e o r f l e x u r a l - t o r s i o n a l b u c k l i n g ) , t h e

b u c k l i n g c oe f fi c ie n t K s h o u l d b e t a k e n f r o m c u r v e b , a n d w h e n t h e

t w o f o r m s o f i n s t a b il i ty a r e c o u p l e d , t h i s c o e ff ic ie n t s h o u l d b e t a k e n

f r o m c u r v e c .

5 S U G G E S T I O N O F A F O R M U L A F O R P R A C T I C A L

D E S I G N P U R P O S E S O F T S E C T I O N C O L U M N S

B ~ l u t et a l ~ 7 s u g g e s t s a g e n e r a l i z a t i o n o f M a q u o i a n d R o n d a l ' s 1C' f o r m u l a

s o t h a t w h e n t h e c r i t i c a l l o a d s Nmin a n d N m a x ( c o rr e s p o n d in g t o t h e t w o

b u c k l i n g m o d e s o f t h e i d e a l e l as ti c m e m b e r ) a r e e q u a l , th e b e a r i n g c a p a c i t y

N u f it s c u r v e c ; i n t h e o t h e r e x t r e m e c a s e, w h e n Nmax ~ o, N u is c a l c u l a t e d

o n t h e b a s i s o f c u r v e b .

I n t h i s w a y t h e f o l lo w i n g i n t e r a c t i o n f o r m u l a is o b t a i n e d :

( 1 _ N u ' ~ ( 1

N u X f [ N p ,

- 0 2 ) N u

N p l /~ , N m ~ n ~ --~ 1 ~ k ~ fN r n i~ --n N p ,

17)

w h e r e

) X ,. 2

N m in = 0 . 33 9 + 0 . 1 5 ( - m , , ~

18)

I t w il l b e a s s u m e d t h a t t h e c o u p l i n g o f t h e t w o f o r m s o f in s t a b i li ty c a n b e

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

3 % b e l o w v a l u e o f X c o r r e s p o n d i n g t o ~-max i n c u r v e b .

~ x - -0 3 3 9

f l = X/ O-q5- (19)

t h e c o n d i t i o n t h a t m u s t b e s a t is f ie d in o r d e r f o r th e c o e f f ic i e n t K t o b e

t a k e n f r o m c u r v e b d e p e n d i n g o n ~-m a, is :

Xmin~ ]~-max 20 )

w h e re 2-max an d 2 -mi. r e f e r t o t h e t w o b u ck l i n g m o d es .


23/25

Co u p l ed i n s t a b il i ti e s i n co mp ress io n m emb ers 9 3

0 8

0.6

03

0 2

I

I l ' 0 . t ~ ~ ) ~ m a x ~ < O . 5 1 3= 1 . 7 3 3 - 1 . 8 7 ~ m a

[ 0 . 5 < ~ r n a x ~ < 1 . 0 f l ~ l . 6 1 ~ . - 2 . 1 7 6 ~ m a x *

I

0 . ~ . 0 . 5 1 . 0 I . 5 2 . 0 2 5

1 . 0 8 8 - ~ a ~

~ 0 < - ~ m a x ~ < 3 . 0 r J - - - O . 3 7 3 O . l S 3 ~ a y

F i g 2 1 A p p r o x i m a t i o n o f t h e f l c o e f fi c ie n t s

3 9 ; ~ m a x

W h e n 2-m~x


24/25

94 v. G ioncu et al.

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Rappor t p r61 imina i re , CITCM, p . 125 .

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Th e Buckl in 9 o f S tructures in T he ory and Practice ed . J . M , Th o mp s o n

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N ew Y ork, 18-19 Apr il , 1989 , p . 357 70.

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ax i a l s t ee l compress ion , i n s t ee l s t ruc tu res . Recen t r esearch advances and

the i r app l i ca t i ons t o des ign , Par t I , ed . Nik o la Hajd in . P roc . o f t he In t e rna-

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c o n c e rn i n g t h e w e l d e d T -s e c ti o n c o m p r e s s io n m e m b e r s. I C C P D C - R e s e a r c h

Rep. 1474/1988, Timi~oara, Dec. 1988 ( in Rom anian ) .

14. G ion cu , V. , B~lul~, N. , M old ov an , A. , Pa co ste C. Du bin~ , D. , Th eore t i -

ca l and exper im en ta l r esearch on the i n t e rac ti on be tw een fl exura l and

f l ex u r a l - to r s i o n a l b u ck l i n g o f w e l d ed T - s ec t io n co m p r es s i o n mem b er s . I n t.

Col l. o n Stabi l i ty of Steel St ructures , Bu dap est , 25 -27 Apr il , 199 0, vol. I ,

p . 77-86.

15. K l6ppe l , K. Un ger , B. , Ein Be i t rag zur Erm i t t lung der Tragf / ih igkei t

d i i nnwand iger o f t ener Pro f i l e aus ha l t gefo rmten S t ah l be i Ber i i cks i ch t igung

des Zusammenwi rkes von Knicken , Dr i l l en und Elemen tbeu len e insch l i ess l i ch

der Ber i i cks i ch t igung des i i berk r i t i schen Beu lverha l t ens Ver6 f fen t l i chungen


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buk7-coupled instabilities in mono-symmetrical steel compression members

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