02. an Experimental Verification of the Generalized Beam Theory Applied to Interactive Buckling Problems

8/14/2019 02. an Experimental Verification of the Generalized Beam Theory Applied to Interactive Buckling Problems

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t~ I'E L S E V I E R

T h i n W a l l e d S t r u c t u r e s Vol. 25, No . 1, pp. 61 79, 1996C o p y r i g h t @) 1996 E l s e v i e r S c i e n c e L td

Pr i n te d i n Gr e at B r i ta i n . A l l r i gh ts r e s e r ve d0263-8231/96 5;15.00

0 2 6 3 - 8 2 3 1 9 5 ) 0 0 0 3 1 - 3

A n E x p e r i m e n t a l V e r i f i c a t i o n o f t h e G e n e r a l i z e d B e a mT h e o r y A p p l i e d t o I n t e r a c t i v e B u c k l i n g P r o b l e m s

P . L e a c h J . M . D a v i e sT e l f o r d R e s e a r c h I n s t i t u t e f o r S t r u c tu r e s a n d M a t e r i a ls E n g i n e e r i n g , U n i v e r s i t y of Salford,

S a l f o r d , UKR e c e i v e d 3 November 1994; accepted 21 December 1994

ABSTRACTPrevious papers Refs 1-4) have presented a method of analysis for anyopen unbranched thin walled section considering both rigid body movementand cross section distortion including local buckling). Reference 1 descri-bed how the Generalized Beam Theory GBT) can be used to calculategeneralized section properties fo r all modes, including each of the four rigidbody modes and the distortional modes. The additional section propertiesevolved from GBT were then used in Ref. 2 to consider second order elasticcritical buckling problems.

This paper compares the critical buckling predictions of GBT with theresults obtained in two series of tests carried out on lipped and unlippedchannels subject to a major axis bending moment. These predictions arethen combined with the yield criteria of EC3 to allow a comparison with theanalysis of these tests carried out by Lindner and Aschinger Ref 5). Thepaper concludes that the Generalized Theory is a powerful and effectiveanalysis tool fo r the solution o f interactive buckling problems where bothlocal and overall buckling can occur.

1 I N T R O D U C T I O NT h e G e n e r a l i z e d B e a m T h e o r y ( G B T ) i s a m e t h o d o f a n a l y s i s o r i g i n a l l yd e v e l o p e d b y S c h a r d t 6 a n d s u b s e q u e n tl y b y M i o s g a v fo r o p e n u n b r a n c h e dm e m b e r s w h i c h c o n s id e r s t h e m e m b e r t o h a v e a d e f o r m a b l e cr o s s s e ct io n .T h e t h e o r y i s u s e d t o c a l c u l a t e a n u m b e r o f s e c t i o n p r o p e r t ie s , s o m e r e la -

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6 P. Leach, J. M. Davies

t e d t o r i g i d b o d y d e f o r m a t i o n m o d e s a n d o t h e r s r e l a t e d t o c r o s s s e c t i o nd i s t o r t i o n mo d es . F o r f i r s t o r d e r p r o b l ems , t h e s e C r o s s s ec t i o n p r o p e r t i e sc a n b e u s e d i n th e n o r m a l e q u a t i o n s o f e q u i l i b r iu m g i v e n by t h e s i m p leE n g i n e e r s T h e o r y o f B e n d i n g t o c a l c u l a t e st re s se s a n d d e f l e c ti o n s i n t h esec t ion . 8

E x t e n d i n g t h e t h e o r y t o t h e s e c o n d o r d e r e n a b l e s G B T t o b e u s e d t os o l v e s eco n d o r d e r e l a s t i c c r i t i c a l b u ck l i n g p r o b l ems b y ca l cu l a t i n g ad d i -t io n a l s e c t io n p r o p e r t i e s a n d m o d i f y i n g t h e b a s ic e q u i l i b r iu m e q u a t i o n t oi n c o r p o r a t e t h e s e a d d i t io n a l t e rm s . R e f e r e n c e 1 d e s c ri b e s t h e m e t h o d o fc a l c u l a t in g t h e c o m p l e t e s e c o n d o r d e r c r o s s s e c t i on p r o p e r t i e s a n d d e r iv e st h e s e c o n d o r d e r e q u a t i o n o f e q u i l i b r i u m . R e f e r e n c e 2 u s e s t h e s e p r o p e r -t ie s i n t h e eq u i l i b r i u m e q u a t i o n t o s o l v e a n u m b e r o f c ri ti c a l b u ck l i n gp r o b l e m s .T h i s p a p e r c o m p a r e s t h e r e su l ts o f a s e c o n d o r d e r i n t e ra c t i v e b u c k l i n ga n a l y si s u s in g t h e f i n it e d if f e re n c e a p p r o a c h i n c o n j u n c t i o n w i t h G B T t o as e r ie s o f t e st s c a r r i ed o u t a t t h e U n i v e r s i t y o f S a l f o r d o n a n u m b e r o fl i p p e d a n d u n l i p p e d c h a n n e l s e c t i o n s s u b j e c t t o a m a j o r a x i s b e n d i n gm o m e n t .

U s i n g t h e c r i ti c a l b u c k l i n g m o m e n t s o b t a i n e d i n th i s w a y , t h e l o a dc a r r y i n g c a p a c i t y o f t h e c h a n n e l s w a s c a l c u l a t e d a n d c o m p a r e d w i t h t h ecap ac i t y ca l cu l a t ed b y L i n d n e r a n d A s ch i n g e r , -s w h o an a l y s ed t h e r e s u l t su s i n g f i v e a l t e r n a t i v e d e s i g n p r o ced u r e s .

2 S E C O N D O R D E R E Q U A T I O N O F E Q U I L I B R I U MA cc o r d i n g t o R e f . l , t h e fo l l o w i n g eq u a t i o n c an b e ap p l i ed t o a ll f ir s t an ds e c o n d o r d e r p r o b l e m s i n o r d e r t o c a l c u l a t e t h e s t r e s s d i s t r i b u t i o n i n as e c t io n s u b j e c t to a n a r b i t r a r y l o a d , o r i n o r d e r t o c a l c u l a t e t h e b if u r c a t i o nl o ad o f a s ec t i o n w h en s u b j ec t t o a n y l o ad ( o r s e r ie s o f l o ad s ):

9 [ Jk ckv t ' _ G i k D k v + /k B k v + U k ~c ~( iw i V )j= l i= I( i w J V + 2 i W ' J V ' ) I = k q Q 1 )w h e r e : J kB - - b en d i n g s t i ff n e ss i n mo d e j w i t h r e s p ec t t o l o ad ap p l i ed i nm o d e k , J k c1 - - f ir s t o r d e r l o n g i t u d i n a l st if fn e ss i n m o d e j f o r l o a d a p p l i e di n m o d e k , J k c 2 - - s eco n d o r d e r l o n g i t u d i n a l s t if fn e s s i n mo d e j f o r l o ada p p l ie d i n m o d e k , J k c - - i k c 1 + j k C 2 , ik D ~ - - f i r s t o rder t o r s iona l s t i f f -n e s s i n m o d e j f o r l o a d a p p l i e d i n m o d e k , J k D 2 - - s e c o n d o r d e r t o r s i o n a ls t if f ne s s i n m o d e j f o r l o ad ap p l i ed i n m o d e k , i k D - - 2 ( 1 - v ) / k D l


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GBT appl ied to in terac t ive buck l ing problems 63_ v ( . i k D i + k D 2) , E - - Y o u n g s m o d u l u s = 2 0 00 0 k N / c m 2, G - - s h e a rm o d u l u s = 7 69 2 k N / c m 2, i Jk x ,~ - - s e c o n d o r d e r t e r m s a r i s i n g f r o m l o n g -i t u d i n a l s t r e s s e s , i Jk x ~ - - s e c o n d o r d e r t e r m s f r o m s h e a r s tr e ss e s, YW - -s tr e ss r e s u l t a n t i n m o d e j , J V - - g e n e r a l i s ed d i s p l a c e m e n t f u n c t i o n f o rm o d e j , a n d v - P o i s s o n s ra t i o .

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

I n g e n e r a l b u c k l i n g p r o b l e m s , t h e s h a p e o f t h e b u c k l i n g w a v e i n e a c hm o d e i s u n k n o w n , a s is t h e c o n t r i b u t i o n o f e a c h m o d e t o t h e f a i l u re l o a d .I n o r d e r t o s o l ve s u c h p r o b l e m s , f i n i t e d i f f e re n c e m e t h o d s c a n b e u s e d i nc o n j u n c t i o n w i t h e q n ( 1 ). T o i l l u s t r a t e t h e m e t h o d , i t i s u s e f u l t o f ir s tc o n s i d e r t h e c a s e o f b u c k l i n g i n a s in g l e m o d e ( i n a n a r b i t r a r y b u c k l i n gw a v e s h a p e ) a n d t h e n t o e x t e n d t h e m e t h o d t o th e g e n e r a l c as e o f i n te r -a c ti v e b u c k l in g i n t w o o r m o r e m o d e s .3 1 B u c k l i n g w i t h o u t i n t e r a c ti o nC o n s i d e r e q n (1 ) a p p l i e d t o a b u c k l i n g m o d e k . F o r b i f u r c a t i o n s i t u a -t i o n s , t h e l o a d i n t h e d i r e c t i o n o f t h e d i s p l a c e m e n t i s z e r o s o t h a t t h e r i g h th a n d s i de o f t h e e q u a t i o n v a n i s h e s . F u r t h e r m o r e , f o r b u c k l i n g in a s in g l em o d e , j - - k a n d t h e s u m m a t i o n s o v e r j m o d e s ar e n o t r e q u ir e d . H e n c e :

E k ~ C k V - - G / k D k V + k k B k V = - - 2 ]~ -- ~ i k k K a ( i w k v ) tL i = I

+ ik ~ K~ ( i w k V + 2 i W ' ~ V ) [ .I

2)T h i s c a n b e r e w r i t t e n i n f in i te d i f f e r e n c e f o r m f o r n o d e m a s:

k k S ( m ) = 2 k k F ( m )w h e r e :

k kS (r n) = E k k C k V ( m ) ' ' - G k kD k V ( m ) + k k B k V ( m )a n d :

k k F (r n ) = - ~ [ik kK ~ [ i w k V ( m ) ] - i k k x ~ [ i w k V ( m ) + 2 i w ' k V ( m ) '] t.i = 1

T h e s e c o n d a n d f o u r t h d e r i v a t i v e s o f V c a n b e r e p l a c e d b y t h e i r c e n t r a lf i n i t e d i f f e r e n c e e q u i v a l e n t s :


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64

V ( m ) =

P. Leach J. M. DaviesV ( m - 1 ) - 2 V ( m ) + V ( m + l )

d x 2V ( m ) - V ( m - 2) - 4 V ( m - 1) + 6V (m ) - 4 V ( m + 1) + V ( m + 2)

d x

B o u n d a r y c o n d i t i o n s c a n b e i n s e r te d a s a p p r o p r i a t e t o g iv e a n e q u a t i o nf o r e a c h o f t h e 'p ' n o d e p o i n t s a lo n g t h e l e n g t h o f th e b e a m . T h e b o u n d a r yc o n d i t i o n s r e l a ti n g to s i m p l y s u p p o r t e d o r c l a m p e d e n d c o n d i t i o n s c a n b ed e r iv ed w i th l i tt le e f fo r t f ro m f i rs t p r i n c ip l e s . Us in g t h e s e en d co n d i t i o n s ,a l l t h e p o i n t s a l o n g t h e b e a m c a n b e c o m b i n e d t o g i v e t h e f o l l o w i n ge i g e n v a l u e e q u a t i o n :

k * s - - 2 k * F ) * V = 0w h e r e

kk S =

*ks(1 )**S(2)

a n d

kkF :* * F( I ) *kF 2)

* * S p - l ) **S p)

* * F p - 1 ) **F p)* V = e i g e n v e c t o r s o f m o d e s h a p e s ( ' p ' v e c t o rs e a c h w i t h ' p ' e l e m e n t s) ,2 = m u l t i p l i e r fo r t h e ap p l i ed l o ad ( i. e. ' p ' e i g en v a lu es ).

T h e l o wes t e i g en v a lu e g iv es t h e c r it i ca l b u c k l i n g l o ad fo r a g en e ra ls e c t io n w i t h a r b i t r a r y b o u n d a r y c o n d i t i o n s f o r b u c k l i n g i n a s i n g le m o d e .I t a l s o gi v e t h e n e x t ' p - 1 ' b u c k l i n g lo a d s w i t h t h e ir a s so c i a t e d m o d es h ap es . B ecau s e o f t h e n u m er i ca l i n accu racy o f t h e f i n i t e d i f f e ren cea p p r o x i m a t i o n s f o r t h e d e r i v a ti v e s o f V w h e n t h e m o d e s h a p e i s c o m p l e x ,o n l y t h e f ir st f e w b u c k l i n g lo a d s a n d m o d e s c a n b e u s e d w i th c o n f i d e n c e .T h i s l im i t a t i o n i s i n s ig n i f i can t i n en g in ee r in g p ro b l em s s i n ce t h e h ig h e rm o d e s a r e a l w a y s a s s o c i a te d w i t h c o r r e s p o n d i n g l y h i g h c r i ti c al b u c k l i n gloads .


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G T applied to interactive buckling problem s 6A l t e r n at i v el y , f o r a m o r e c o m p r e h e n s i v e t r e a t m e n t , o f v a ri o u s e n d

c o n d i t i o n s , R e f . 3 c o n s i d e r s a l l p o s s i b l e s u p p o r t c o n d i t i o n s a n d o f f e r s am o r e e f f ic i en t s o l u t i o n m e t h o d f o r t h e a b o v e s y s t e m o f e q u a ti o n s .3 2 Buck ling with interactionW h e n t h e b u c k l i n g f ai lu r e is d u e t o a c o m b i n a t i o n o f a n u m b e r o f di ff er -e n t m o d e s , t h e f a i l u r e l o a d c a n b e c a l c u l a t e d b y c r e a t i n g a n e i g e n v a l u ep r o b l e m w h i c h i n c o r p o r a t e s a ll o f t h e c o m p o n e n t m o d e s . T h e e i g e n v a lu ep r o b l e m t h e n b e c o m e s :

[kkS -- 2kkF ] ~V) = O.k= l

T h e s o l u t i o n o f t h i s e q u a t i o n g i v e s n p ) e i g e n v a l u e s , t h e l o w e s t o f w h i c hi s t h e m u l t i p l i e r f o r t h e i n t e r a c t i v e c r i t i c a l b u c k l i n g l o a d c o n s i d e r i n g a sm a n y o f t h e p o s s i b le i n d i v i d u a l m o d e s a s i s r e q u ir e d .

4 I N T E R A C T I V E B U C K L I N G B E H A V I O U R O F C O L D - F O R M E DC H A N N E L S

4 1 Experim ental investigationA n u m b e r o f te s ts h a v e b e e n c a rr i e d o u t o n c o l d - f o r m e d st ee l c h a n n e ls e c ti o n b e a m s s u b j e c t t o m a j o r a x i s b e n d i n g m o m e n t s in o r d e r t o v a l id a t et h i s m e t h o d o f a n a l y s i s .

L o v e l l9 c a r r i e d o u t a s e ri es o f t es t s o n b o t h l i p p e d a n d u n l i p p e d c h a n -n e ls s u b je c t t o a u n i f o r m m a j o r a x i s b e n d i n g m o m e n t a n d a d d i t i o n a l t es tsw e r e c a r r i e d o u t b y L e a c h , 8 u s i n g a r i g s i m i l a r t o t h a t u s e d b y L o v e l l. I nt h e s e l a t t e r t e s t s , a s e r ie s o f l ig h t g a u g e s t ee l b e a m s w e r e s u b j e c t t o a m a j o ra x i s b e n d i n g m o m e n t a p p l i e d t o o n e e n d o n l y , t h u s p r o d u c i n g a l i n e a r l yv a r y in g d i s t r ib u t i o n o f b e n d i n g m o m e n t a l o n g t h e l e n g t h o f t h e b e a m .L o v e l l te s t e d s ix d i f f e r e n t s ec t i o n s o v e r v a r i o u s l e n g t h s , w h i l e a n o t h e r f o u rd i f f e r e n t s e c t io n s w e r e t e s t e d o v e r v a r i o u s l e n g t h s i n R e f . 8 . A ll o f t h es e c t i o n s h a d s i m i l a r e n d c o n d i t i o n s a s d e t a il e d i n T a b l e 1 b e l o w .

T h u s , t h e e n d s o f t h e m e m b e r s u n d e r t e s t w e r e s im p l y s u p p o r t e d w i t hr e g a r d t o b e n d i n g b u t f u l l y r e s t r a i n e d a g a i n s t t o r s i o n , w a r p i n g a n d c r o s ss e c t i o n d i s t o r t i o n .

I n e a c h t e s t, t h e l o a d d e f l e c t i o n c h a r a c t e r i s t ic s o f t h e b e a m s w e r em e a s u r e d , a s w e r e t h e f a i lu r e l o a d s . I n o r d e r t o d e f i n e t h e b u c k l i n g l o a d , ah o r i z o n t a l d e f l ec t i o n o f s p a n / 1 0 0 0 w a s c o n s i d e r e d t o b e th e m a x i m u m


6/19

6 6 P . L e a c h , J . M . D a v i e sT A B L E

E n d C o n d i t i o n s f o r B o t h T e s t S e r i e s

M o d e E n d c o n d it io nT r a n s l a t i o n a l R o t a t i o n a l

1 - - A x i a l m o v e m e n t F r e e2 - - M a j o r a x is b e n d i n g F i x e d F r e e3 - - M i n o r a x is b e n d i n g F i x e d F r e e4 - - T o r s i o n F i x e d F i x e d5 t o n - - D i s t o r t i o n F i x e d F i x e d

T A B L E 2 aS e c t i o n s T e s t e d b y L o v e l l 9

S e r i e s R e f . D e p t h W i d t h L i p s i z e T h i c k n e s s L e n g t hm m ) r a m ) m m ) m m ) r nr n )

A 9 0 3 6 U n l i p p e d 1 .1 7 8 0 0 - 2 4 0 0B 9 0 5 0 1 5 1 .1 7 9 0 0 - 4 4 0 0C 9 0 2 6 9 1 . 17 5 0 0 2 6 0 0D 1 2 2 3 6 U n l i p p e d 1 .1 7 5 0 0 - 2 5 0 0E 1 2 0 5 0 1 5 1 .1 7 6 0 0 - 4 5 0 0F 1 2 2 3 6 U n l i p p e d 1 -1 7 5 0 0 - 2 5 0 0

T A B L E 2 bS e c t i o n s T e s t e d b y L e a c h 8

S e r i e s R e f . D e p t h W i d t h L i p s i z e T h i c k n e s s L e n g t hr a m ) r a m ) m m ) m m ) r a m )

B 9 0 5 0 1 5 1 - 20 1 5 0 0 - 6 0 0 0E 1 2 0 5 0 1 5 1 . 20 1 5 0 0 - 6 0 0 0H 9 0 4 0 U n l i p p e d 1 . 90 8 0 0 - 3 0 0 0G 1 2 0 5 0 U n l i p p e d 1 .9 0 8 0 0 - 3 0 0 0

allowable deflection before post buckling effects occurred. In mostinstances failure occurred before this limit was reached.4 1 1 T e s t r i gA p h o t o g r a p h s h o w i n g t h e g e n e r a l a r r a n g e m e n t o f t h e te st rig i s s h o w n i nF i g . 1 . I n o r d e r t o a c h i e v e t h e e n d c o n d i t i o n s g i v e n i n T a b l e 1 t h e te s t r igu s e d t w o t u r n t a b l e s u p p o r t f r a m e s a s s h o w n i n F i g . 2 . T r a n s l a t i o n a l f r e e-d o m a l o n g t h e b e a m w a s a c h i ev e d b y m o u n t i n g o n e t u r n ta b l e o n n e e d l e


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G T applied to interactive buckling problems 67

Fig. l. General arrangement of test rig.

r o l le r b e a r in g s . I n o r d e r t o r e s t r a i n t h e s e c t i o n a g a i n s t t w i s t in g a n d w a r p -i n g , t h e e n d s o f t h e b e a m w e r e c l a m p e d w i t h 12 m m t h i c k m i l d s te e l p l a t e so n t o a b o x s e c t i o n f a b r i c a t e d f r o m 1 2 m m s te e l p l a t e s as s h o w n i n F i g . 3 .

C a r e f u l o b s e r v a t i o n o f t h e e n d s o f t h e s p e c i m e n d u r i n g t h e t es ts i n d i-c a t e d th a t t h is a r r a n g e m e n t w a s a g o o d a p p r o x i m a t i o n t o a f ix e d w a r p -i n g c o n d i t i o n w h il s t t h e t u r n t a b l e s a n d r o ll er b e a r in g s a l l o w e d r o t a t i o n a lf r e e d o m w i t h r e s p e c t t o m a j o r a n d m i n o r a x i s b e n d i n g a n d p r e v e n t e d t h ed e v e l o p m e n t o f a n y c a t e n a r y f o rc e s.


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68 P. Leach J M. Davies1 6 m m s h a t t in r o l le r b e a r i n g sg i v i ng r o t a t i ona l f r eedom i nt h e m a j o r a x is

H o r i z o n t a l t u r n t a b hg i v i ng r o t a t i ona lf re e d o m i n t h e m i nax i s d i r ec t i on

Fig 2 End v ie w of t u rn t a b l e suppo r t f r a me

1 2 m m p l a te s c la m p i n gs e c t io n t o b o x f a b r i c a te df r o m 1 2 m m p l a te I

U

Fig 3 C l a m pi ng a r ra nge m e nt p rov i d i ng f ixe d wa rp i ng re s tra i n t

Load was applied to the section by means of a lever arm system anddead weights to each end of the beam. In the Lovell tests both ends o f thebeam were equally loaded to give a constant bending moment along thebeam while in the tests carr ied out by the author one end was left unloa-ded in order to create a linearly varying distribution of bending momentalong the beam.


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G T applied to interactive buckling problems 94 2 T h e o r e t i c a l i n v e s t ig a t i o n

W h e n c a r r y i n g o u t t h e f ir st p a r t o f t h e t h e o r e t i c a l a n a l y s e s a c c o r d i n g t oG B T i n o r d e r t o d e t e r m i n e t h e g e n e r a l is e d s et o f c r o ss s e c t i o n p r o p e rt ie s ,t h e n o d e p o i n t s w i t h in t h e s e c t i o n w e r e t a k e n a s f o ll o w s :

( a) L i p p e d c h a n n e l : six n o d e s , o n e a t e a c h e d g e o f t h e s e c ti o n a n d a t e a c hf o l d . N o i n t e r med i a t e n o d es , i . e .

1

( b ) U n l i p p e d c h a n n e l : s ix n o d e s , o n e a t e a c h e d g e o f t h e s e c ti o n , o n e a te a c h f o l d t o g e t h e r w i t h in t e r m e d i a t e n o d e s a t t h e m i d p o i n t s o f e a c hf lange, i .e .

E a c h o f th e s e a n a l y s e s g a v e ri se t o t h e u su a l f o u r r i g id b o d y m o d e s o fa x i a l e l o n g a t i o n , b e n d i n g a b o u t e a c h p r i n c i p a l a x i s a n d t o r s i o n , t o g e t h e rw i t h t w o h i g h e r o r d e r l o c a l b u c k l i n g / d i s t o r ti o n a l m o d e s . I n t h is w a y , c r o s ss e c t i o n d i s t o r t i o n w a s a c c o u n t e d f o r a n a l s o l o c a l p l a t e b u c k l i n g i n t h eu n l i p p e d f la n g e s w h i c h w e r e e x p e c t e d t o b e s u s c e p t ib l e t o l o c a l b u c k l i n g .C o n v e r g e n c e s t u d ie s in d i c a t e d t h a t 2 4 e le m e n t s a l o n g t h e b e a m w a s t h eo p t i m u m n u m b e r f o r t h e f in i te d if f e r e n c e a n a l y s is a d o p t e d , a l t h o u g h i t h a sb e e n s u b s e q u e n t l y f o u n d t h a t t h e i m p r o v e d s o l u t i o n m e t h o d o f R e f . 8g en e r a l l y co n v e r g es w i t h a s f ew a s s ix e l emen t s .

5 R E S U L T SF i g u r e s 4 - 1 4 s h o w t h e te s t r e su l ts t o g e t h e r w i t h a n u m b e r o f t h e o r e ti c a lcu r v es . Th es e cu r v es a r e :

Yield moment T h e y i e l d m o m e n t o f t h e s e c t i o n c a l c u l a t e d u s in g t h e g r o s ss e c t io n p r o p e r t i e s o f th e b e a m ( i n d e p e n d e n t o f l e n g th ) a n d a y i e ld s tr e sso f 3 1 2 N / m m 2 i n t h e L o v e l l te s ts a n d 2 8 4 N / m m 2 i n t h e L e a c h t es ts .


10/19

7 0

3 0 0

2 5 02 0 0

1 5 0

1 0 0

P. Leach J. M. DaviesM O M E N T ( k N . c m )

5 0

0

G B T M o d e s t - 4GBT All M o d e s

x Test Results

Y i e l d M o m e n t

X X X X

5 0 1 0 0 15 0 2 0 0 2 5 0LENGTH (cm)

F i g . 4 . L o v e l l t e s t s e r i e s A .

M O M E N T ( K N . C M )5 0 0

GBT Modes 1-4- GBT All Modes4 0 0 ~ , Test Re.cu ' s

3 O O

x x x x x Y i e l d M o m e n t

F1 0 0 x x

00 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

L E N G T H (C M )F i g . 5 . L o v e l l t e s t s e r i e s B .

GB T modes 1 4: T h e e l a s ti c b u c k l i n g c u r v e o f t h e r i g i d c r o s s s e c ti o n . T h e s ec u r v e s t h e r e f o r e i n c l u d e m o d e s 2 - 4 . F o r u n i f o r m b e n d i n g , t h e c u r v e i si d e n ti c a l t o t h a t o b t a i n e d b y u s i n g t h e a n al y t ic a l f o r m u l a o f T i m o s h e n k oa n d G e r e ~ f o r a s i m p l y s u p p o r t e d b e a m r e s t ra i n e d a g a i n s t w a r p i n g .GBT all modes: T h e e l a s t i c b u c k l i n g c u r v e o f t h e c r o s s s e c t i o n t a k i n ga c c o u n t o f c ro s s s e c t io n d i s t o r t i o n a s c a l c u la t e d b y G B T .Nethercot F i g s 1 - 4 o n l y ) . T h e e l a s ti c b u c k l i n g c u r v e o f t h e cr o s s


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G T applied to interactive buckl ing problem s 71s e c ti o n a c c o r d i n g t o R e f s 1 1 a n d 12 T h i s m e t h o d d o e s n o t a l l o w fo rc r o s s s e c t i o n d i s t o r t i o n

I t s h o u l d b e n o t e d t h a t t h e t h e o r e t i c a l c r i t i c a l b u c k l i n g c u r v e s a n d t h ey i el d c u r v e a r e e a c h a n u p p e r b o u n d t o t h e r e a l b e h a v i o u r I n th e s e fi g u re sn o a t t e m p t h a s b e e n m a d e t o c o n s i d e r t h e i n t er a c ti v e e ff e c ts o f y ie l dt o g e t h e r w i t h b u c k l i n g T h i s i n t e r a c t i o n i s c o n s i d e r e d a t a la t e r s t a g e int h is p a p e r

3 0 0

2 5 02 0 0150100

500

MOMENT KN.CM)GBT Mode s 1-4

N o t e A l l m o d e s curvea n d modes 1 - 4 curve -- GBT Al l Modescoincide Test Results

X ~ Yiel Mom ent

5 0 1 0 0 1 5 0 2 0 0 2 5 0LENGTH (CM)

F i g . 6 . L o v e l l t e s t s e r i e s C .

3 0 0

3 0 02 5 02 0 01 5 0100

5 00

MOMENT kN.cm)GBT Mo des 1-4

Yield Mom ent GBT Al l Modes Test Results

x

0 50 1 0 0 1 5 0 2 0 0LENGTH (cm)

F i g . 7 . L o v e l l t e s t s e r i e s D .

2 5 0 3 0 0


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7 2

6 0 0

5 0 04 0 03 0 02 0 01 0 0

0

P . L e a c h J . M . D a v i e sMOM ENT (kN.cm)

GBT Modes 1-4GBT All Modes Test Results

y i el d_ M o m e n t . . . . . . . . . . . ~ . . . . . . . . . . . . . . .X X X X ~ ~

5 0 1 1 5 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0LENGTH (cm)F i g . 8 . L o v e l l t e s t s er ie s E .

4 5

3 5 03 0 02 5 0

2 0 01 5 01

5 00

MOMENT (KN.CM)Y_i_el=dMo me n t t G BT Modes 1 -4

GBT A l l Modes Tes t Resul ts

5 0 1 0 0 1 5 0 2 0 0 2 5 0LENGTH (CM)

F i g . 9 . L o v e l l t e s t s e r i e s F .

3 0 0

6 SIMPLIFIED BIFURCATION ANALYSISIt is shown in Refs 2 and 4 that a very much simplified analysis for elasticbuckling is available if the displacement function for each of the activemodes can be assumed to be a half sine wave thus:

k V k a sin ~X.LWith this assumption and when only a single mode is considered a simple


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G B T a p p l i e d to i n t e r a c ti v e b u c k l in g p r o b l e m s 73ex p l ic i t ex p re s s i o n is av a i l ab l e fo r th e b u ck l i n g lo ad an d fo r h i g h e r o rd e rm o d e s t h e b u c k l i n g w a v e l e n g t h . W h e n t w o m o d e s a re i n c lu d e d it isn e c e ss a r y t o s o l v e a q u a d r a t i c e q u a t i o n a n d e a c h a d d i t i o n a l m o d e m e r e l yad d s a fu r t h e r v a r i ab le .A p p a r e n t l y t h e f ix e d b o u n d a r y c o n d i t i o n s f o r t o r s i o n a n d w a r p i n g i np a r t i cu l a r p rec l u d e t h is m e t h o d o f an a ly s i s o f t h e t e st r e s u lt s d i s cu s s edea r l ie r . I t is n ev e r t h e l e s s i n s t ru c t iv e to ap p l y t h e s i m p l i fi ed p ro c ed u re an dt o e x am i n e t h e r e su l ts .As wi th a l l cases o f sec t ions su f fe r ing la te ra l to rs iona l buck l ing as ar e s ul t o f t h e a p p l i c a t io n o f a m a j o r a x is b e n d i n g m o m e n t t h e l at e ra l

MOMENT (KN.CM)4 5 04 0 03 5 03O02 5 02 0 01 5 01 0 0

5 00 2 O O

~\ G B T M o d e s 1 4X ~ GB T A ll M o d e s

\~ ,~ N e t h e r c o t,, Test

Y i e l d M o m e n _ t . . . . . ~

2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0LENGTH (CM)

Fig. 10. Leach test series B .

MOMENT (kN.cm)8 0 07 0 06 0 05 0 04 0 03 0 02 0 01 0 0

02 0 0

GBT Modes 1-4.. - GBT All M o d e s~ \ - N e t h e r c o t

Test

2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0LENGTH (cm)

Fig. 11. Leach test series E .


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7 4 P. Leach J. M. Davies

b e n d i n g m o d e 3 a n d t h e t o r s io n a l m o d e 4 h a v e n o i n d e p e n d e n t s ig n if i-can ce an d a re co u p l ed i n t h e r i g id -b o d y l a t e ra l t o r s i o n a l b u ck l i n g m o d e .T h i s i s ev id en t wh e n t h e iJ kx v a lu es a re c o n s id e red b ecau s e 2 3 3 K 2 4 4 K = 0.F u r t h e r m o r e , h e r e , th e h i g h e r - o r d e r d i s to r t i o n a l m o d e s 5 a n d 6 s im i l ar lyh av e n o i n d e p en d en t si g n if i can ce b ecau s e 2 5 5x = 2 6 6 K = 0 . In d eed , b ecau s eo f t h e c o u p l e d i n t e r a c t io n w h e r e b y t h e b u c k l i n g o f t h e c o m p r e s s i o n fl a n g er e n d e r s t h e s e c t i o n d o u b l y u n s y m m e t r i c a l t h e o n l y m o d e c o m b i n a t i o n swh ich g ive usefu l so lu t ions a re 3 and 4 ( r ig id body) a nd 3 , 4 , 5 and 6 ( r ig idb o d y p lu s b o th c ro s s s ec t i o n d i s t o r ti o n s ) .T h e r e s u lt s o f ap p ly in g t h i s s im p l i f ied an a ly s is t o t h e u n l i p p ed s ec t io n sof L ovel l s tes t se r ies A an d D are g iven in F igs 14 an d 15, respec t ive ly .

MOMENT (KN.CM)1 0 0 0

8 0 0

6 0 0

4 0 0

2 0 0

0 100

G B T M o d e s 1 - 4 \ , x- - G B T A l l M o d e s \- N e t h e r c o t ? . \\X T e s t

Yield Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . .

x

15 2 0 0 2 5 0 3 0 0LENGTH (CM)

F i g . 1 2 . L e a c h t e s t s e r i es G .

MOMENT (KNCM)6O

5 0 04 0 03 0 02 0 0100

0 100

- G B T M o d e s 1 - 4GBT All ModesN e t h e r c o t

1 5 0 2 0 0 2 5 0LENGTH(CM)

F i g . 13 . Le ac h t es t s e ri es H .

3 0 0


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G T applied to interactive buckling problems 75B e c a u s e o f t h e i m p l ie d s im p l y - s u p p o r t e d b o u n d a r y c o n d i t io n s , t h em o m e n t s c a u s i n g r i g i d - b o d y l a t e r a l t o r s i o n a l b u c k l i n g a r e c o n s i d e r a b l yl o w e r t h a n t h e c o r r e s p o n d i n g v a l u e s in F i g s 4 a n d 7 . H o w e v e r , b o t h F i g s1 4 an d 1 5 s h o w a c l ea r l y d e f i n ed l o ca l b u ck l i n g mo d e w i t h a r e l a t iv e l ys h o r t w a v e l e n g t h w h i c h w o u l d n o t b e s i g n i f i c a n t l y i n f l u e n c e d b y t h eb o u n d a r y co n d i t i o n s i n t h e t e s t s . Th u s , s ec t i o n A s u f f e r s l o ca l b u ck l i n g a ta m o m e n t o f 101 k N c m w i t h a h a l f w a v e l e n g t h o f 9 cm . T h e c o r r e s p o n d -i n g v a l u es f o r s ec t i o n D a r e 1 48 k N cm an d 10 cm. T h es e b u ck l i n gm o m e n t s a r e , o f c o u r s e , p r e c i se l y th e v a l u e s i n f l u e n c in g t h e h o r i z o n t a lp o r t i o n s o f t h e G B T a l l m o d e s l in e s i n F i g s 4 an d 6 .

MOMENT (kN.cm)

p

0 50 100 150 200BUCKLING LENG TH (cm)

F i g . 1 4 . L o v e l l t e st s e ri es A : c r i t i c a l b u c k l i n g m o m e n t s .

A l l M odesR i g i d B o d y M odes

2 5 0

MOMENT (kN.cm)1 0 0 0

8 0 0

6 0 0

4 0 0

2 0 0

0

i

7 0 06 0 05 0 04 0 03 0 02 0 0100

0

A l l M odesR i g i d ody M odes

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0BUCKLING LENGTH (cm)

Fig. 15. Lovell test series D : critical buckling moments .


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76 P. Leach J. M. DaviesTh e s i g n i f i can ce o f t h e s e r e s u l t s i s t h a t t h e ex t r eme l y co mp l i ca t ed

b u c k l i n g b e h a v i o u r o f u n l i p p e d c h a n n e l s b e n t a b o u t t h e m a j o r a x is ,w h e r e b y l o c a l b u c k l i n g o f t h e c o m p r e s s i o n f la n g e e f f ec ti v el y m a k e s t h es e c t i o n d o u b l y u n s y m m e t r i c a l , n o w a d m i t s a r e l a t i v e l y s i m p l e s o l u t i o ni n v o l v i n g f o u r u n k n o w n s .

7 I N T E R A C T I O N O F B U C K L I N G W I T H Y I E L D E F F E C T SL i n d n e r a n d A s c h i n g e r 5 h av e s t u d i ed t h e t e s ts c a r r i ed o u t b y L o v e l l an de v a l u a t e d th e t h e o r e ti c a l f a il u re m o m e n t a c c o r d i n g t o t h e G e r m a n D I N 18800 Par t s 1 , 2 and 3 ,13-15 E ur oc od e 3 , t6 AIS117 an d two m odi f i e d p roc edu resp r o p o s ed b y L i n d n e r . 18 A l l o f th e s e p r o ce d u r e s c o n s i d e r ed t h e i n t e r ac t i o n o fl o ca l an d g l o b a l b u ck l i n g w i t h y i e l d e f fec ts . I n each o f t h e d e s i g n p r o ced u r e sco n s i d e r ed d es i g n a t ed A , B , C , D , E ) t h e t h eo r e t i ca l o v e r a ll e l a s ti c c r i ti c a lb u c k l i n g m o m e n t w a s c a l c u la t e d u s i ng a c o m p u t e r p r o g r a m b a s e d o n t h ef in i te s tr i p me t h o d a n d t h en co m b i n e d w i t h y i e l d in g o f t h e s ec t io n u s i n g ani n t e r a c ti o n e q u a t i o n b a s e d o n a P e r r y R o b e r t s o n t y p e o f a p p r o a c h .

A s u m m a r y o f t h e r es u l ts o f t h e s tu d y b y L i n d n e r a n d A s c h i n g e r a r es h o w n i n T a b l e 3 w h e r e M u is th e t h e o r e t ic a l u l ti m a t e m o m e n t o f t h es e c t io n a n d Mexp i s t h e mo men t a t w h i ch t h e s ec t i o n f a i l ed i n t h e t e s t .U s i n g t h e b u c k l i n g m o m e n t s g i v e n i n F i g s 4 - 9 w h i c h a c c o u n t f o r b o t hl o ca l an d g l o b a l b u ck l i n g ) i n th e i n t e r ac t i o n f o r m u l a o f EC 3 P a r t 1 .3 , as i mi l a r c a l i b r a t i o n ex e r c i s e w as ca r r i ed o u t . Th e r e s u l t s o f t h i s c a l i b r a t i o na r e g i v e n i n T a b l e 4 . T h e r e s u l t s c a n b e s u m m a r i z e d a s a m e a n v a l u e o fMu Mexpo f 0-8 56 an d a s t an d a r d d ev i a t i o n o f 0 .1 6 6 . Th es e v a l u es can b es e e n t o c o m p a r e f a v o u r a b l y w i th t h o s e g iv e n i n T a b l e 3 . F u r t h e r m o r e , a n yd ec i s i o n r eg a r d i n g l o ca l p l a t e b u ck l i n g co e f f i c i en t s w h i ch w as r eq u i r ed i nt h e p r o c e d u r e s r e p o r t e d b y L i n d n e r a n d A s c h i n g e r i s n o t r e q u i r e d s i n c el o c a l p l a t e b u c k l i n g is a u t o m a t i c a l l y c o n s i d e r e d b y G B T .

TABLE 3Summary of Results Obtained byLindner and AschingerDesign Mean va lue Standardmethod Mu/ Mex p deviation

A 0 739 0 118B 0 694 0.218C 0.786 0.171D 0.886 0.175E 0 910 0 189


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GBT applied to interact ive buckl ing problemsT A B L E 4

Comparison of Test Results with GBT/EC3 Analysis

77

Lengthm )

GBT/ EC 3 C ompar i sonYield mo men t Buck ling mo men t Interac tive mo men t

k N m ) k N m ) k N m )Test result

k N m ) Mu/Mexp

Series 'A'0.801 1-584 0-870 0.703 1.0291.121 1.583 0-870 0.703 1.0211.281 1.581 0.870 0.702 1-0521.439 1.562 0.870 0.701 0.9941.599 1.604 0.870 0.704 0-9141-761 1.545 0-930 0.736 0.9341.905 1.562 0-940 0.744 0.8562-368 2.586 0-680 0-573 0-690

S e r i e s B0-972 2.351 4.500 1-967 2.1421.319 2.358 3.970 1-915 2.1701.520 2.358 3.720 1.881 2.1611.720 2.368 3.580 1.866 2.1801.922 2.322 3.540 1.834 2.1522.119 2-335 3.440 1.825 2.0822-320 2.412 3.080 1.794 2.0412.904 1.952 2.090 1.345 1.4073-399 1.982 1.550 1.140 1-2793-926 2-007 1.180 0.939 1.1664.372 1.999 0.970 0.802 1.074

Series 'C'0.514 1.463 4.200 1.307 1.5060.810 1.447 3.740 1.275 1.4031.123 1.459 2-430 1.818 1.1851.817 t .436 0.970 0.746 0.7792.217 1.455 0.660 0.551 0.5642.533 1.584 0-510 0.445 0.465

S e r i e s D0.485 2.362 1.280 1.037 1.4950.992 2.367 1.270 1.030 I. 3281.393 2.388 1.260 1.026 1.2481.739 2.394 1.290 1.046 1.2392.045 2.377 1.350 1.083 1.1062.246 2.412 1.110 0.925 1.1232.508 2.410 0-910 0.780 0.86

Section 'E0.6041.090

3.459 7.540 2.968 3.1873.462 5.360 2.747 3.325

0.6830.6880-6680.7050-7710.7880.8700-830

0-9180.8820.8710.8560.8520.8770.8790.9560.8910-8060.746

0.8680.9090.9970-9580-9780.956

0.6930-7760-8220-8440.9790-8240.898

0.9310.826


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78 P. Leach, J. M. DaviesTABLE 4 - - contd

GBT/EC3 ComparisonLength Yield moment Buckling moment Interactive moment Test result Mu/Mexpm) kNm) kNm) kNm) kNm)

Series 'E'(contd)1.408 3.468 4-970 2.687 3.295 0.8161.885 3.454 4-540 2-597 3.175 0.8182.247 3.481 3-870 2-441 2.755 0.8862.502 3.494 3.870 2.446 2.460 0.9942.925 3.366 2.900 2-059 1.922 1.0713.337 3.502 2.250 1-754 1.908 0.9193.924 3.346 1.630 1.346 1.320 1.0204.421 3.341 1.190 1.026 1.112 0.923Series F0.583 2.999 1.300 1.093 1.206 0.9071.003 3.065 1.270 1.075 1.260 0.8531.414 3.065 1.260 1.068 1.029 1.0382.009 3.057 1.350 1.132 0.920 1-2312.507 3.088 0.910 0.800 0.932 0-859

Statistics: Mean; 0.856, Standard Deviation; 0.166.

8 CONCLUSIONSIt is clear from Figs 4-1 6 t hat GB T gives a goo d estimate of the elasticcritical loads of a beam, pa rticularly when local buckling and /or distor-tional effects are dom inant .

When yield of the section is considered, again the met hod compar es wellwith other cur rent method s, as illustrated by the compar ison with Ref. 18.

It should be noted that, in all the above analyses, local buckling effectswere calculated by G BT witho ut rec ourse to effective width fo rmulae orothe r empirical methods. This is particular ly useful in the case of channelsections composed of slender elements, since local buckling leads tounsymmetr ical be nding o f the membe r which is outside the scope of mostanalytical solutions.

R E FE R E N C E S1. Leach, P., The calcula tion of modal cross section propert ies for use in theGeneralized Beam Theory. Thin-Walled Structures 19 (1994) 61-79.2. Davies, J. M. Leach, P., Some applica tions of the General ized BeamTheory. ll th Int. Speciality Conf. on Cold-Formed Steel Structures, Univer-

sity of Missouri Rolla, 1992, pp. 479-501.


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G BT applied to teractive buckling problems 793 . Dav i e s , J . M. Leach , P . , F i r s t - o rde r Gene ra l i s ed Beam Theo ry .J . Cons truc t ional S tee l Research 31 (1994) 187-220.4 . Dav i e s , J . M. Leach , P . , Seco nd-o rd e r Gene ra l i s ed Beam Theo ry .

J . Cons truc t ional S tee l Research 31 (1994) 221-241.5 . L indne r , J . Asch inge r , R . , Loa d -ca r ry ing capac i t y o f co ld - fo rm ed beam ssubjec t to overa l l l a te ra l - tors iona l buckl ing and loca l p la te buckl ing .J . Cons truc t ional S tee l Research 31 (1994) 267 287.6 . Schard t , R . , Verallgemeinerte Technische Bieg etheo rie (Gene ra l i z ed BeamTheory) . Springer , Berl in, 1989.7 . M iosga , G . , Vorw iegend l angsbeansp ruch t e dunn wa nd ige p r i sma t i scheS ta s se und P i a t t en mi t End l i chen E l a s ti s chen V e r fo rmungen . D i s se r t a t ion ,Techn i sche Hochschu l e , Da rms t ad t , 1976 .8 . Leach , P . , The G enera l ized Beam T he ory wi th f in ite d i f fe rence appl ica t ions .Ph D Thes is , U n ive r s i t y o f Sa l fo rd , 1989 .9 . Lovel l , M . H . , La te ra l buck l ing of l igh t gauge s tee l beam s. M Sc Thes is ,Un ivers i ty of Sa l ford , 1983.10. Tim osh enk o, S . P . Ge re , J . M . , Theory o f Elas t ic S tabi l i ty , 2nd edn .M c G r a w - H i l l , N e w Y o r k , 1 96 1.11. Ne the r co t , D . A . Ro ckey , K . C . , A un i fi ed app ro ach t o t he e l a s ti c buck l i ngo f b e a m s . The Structural Engineer, Ju ly 1971.12. Ne the r co t , D . A . Ro ckey , K . C . , La t e r a l buck l i ng o f beam s wi th mixedend cond i t i ons . The Structural Engineer, April 1973.13. D IN 18 800 Teil 1: S tah lbauen ; Bemessung und Kons t ruk t i on (Steel Struc-t u r e s ; de s ign and cons t ruc t i on ) . November 1990 .14. DIN 18 800 Teil 2: Stahlbauten; S tabi l i ta t s fa l le , Knicken yon S taben undS tabwerken (S tee l S t ruc tures ; s tab i l ity , buckl ing of bars and ske le ta l s truc-

t u r e s ) . November 1990 .15. DIN 18 800 Teil 3: Stahlbauten; Stabil i tatsfal le , Plat tenbeulen (Steel Struc-tures ; s tab i l i ty , buckl ing o f p la tes ) . N ov em be r 1990.16 . Eurocode 3 : Des ign o f S t ee l S t ructu re s, Par t 1 . 3 , Co ld Form ed Th in GaugeMe mb ers and Shee t ing . August 1992.17. Pekoz , T . , Deve lopm en t o f a un i f ied app ro ach t o t he de s ign o f co ld - fo rm eds tee l members . Research Repor t C F 87-1 , Ad vi so ry G ro up on t he Speci fi c a-t i on fo r t he de s ign o f Co ld F o rm ed S t ee l S t ruc tu r a l M em ber s , Am er i can I ronand S teel I n s t it u t e , 113 3 15 th S t r ee t , N W W ash ing ton , D C 20005, M arch1987.18. Lindn er , J . Greg ul i , T . , Zu r Trag las t von Biege t ragern , d ie durc h g le ich-ze i t i ge s Auf t r e t en von o r t l i chem Beu lem und B ieged r i l l kn i cken ve r sagen(Loa d ca r ry ing cap ac i t y o f beams sub j ec t t o l oca l p la t e buck l i ng and l a te r a l-tors iona l buckl ing) . Stahlbau, 61 (1992) 9,15.

02. an Experimental Verification of the Generalized Beam Theory Applied to Interactive Buckling Problems

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