design of end platdfgfge connection.pdf

8/14/2019 design of end platdfgfge connection.pdf

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E L S E V I E R

J Construct Steel Res Vo l . 4 3 , N o s . 1 - 3 , p p . 11 9 - 1 4 0 , 1 9 9 7 1 9 9 7E l s e v i e r S c i e n c e L td A l l r ig h t sr e s e r v e d

P r i n t e d i n G r e a t B r i t a i nP I I : S 0 1 4 3 - 9 7 4 X ( 9 7 ) 0 0 0 1 6 - 3 0 1 4 3 - 9 7 4 X / 9 7 1 7 . 0 0 + 0 . 0 0

Des ign o f o l ted Endp la te Conn ec t ions

P. C . O l s e n

Colbe rg Consult , Fa sanvae nget 124, 2980 Kokkedal , DK -De nm ark

Rece ived 25 January 1996; revised version received 3 January 1997;ccepted23 January 1997)

A B S T R A C T

Design formu lae for bol ted f lush and extended endplates are presented basedon the theory of plasticity. The flush endplate is designed such that pryingforc es are not present. The bolts of the extended endplate are designed suchthat yielding a t the boltline and at the flan ge ca n occur thus reducing theendplate thickness b ut increasing the required bolt forc e capacity. An explicitexpress ion fo r the pryin g ratio is presented. The effect of using seve ral boltsalongside the web is accounted for. The design formu lae fo r flush andextend ed endplates are c hanne lled into on e general set o f design formulae .To mea sure the adequacy o f this simplified design method comp rehensivenum erical analyses of the bolt/endplate connection have been performed. Thenum erical metho d is prese nted briefly. The agreeme nt between the twome thods is excellent with only min or difference s in the ultimate load bearingcapacities. 19 97 Else vier Science Ltd.

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

G r e a t a t t e n t i o n h a s b e e n d r a w n t o t h e d e s i g n o f b o l t e d e n d p l a t e c o n n e c t i o n s .P r im a r i l y t he f o r ce s i n t he bo l t s we re ob j ec t s o f g rea t conce rn wh e rea s on lyl it tl e a t te n t io n w a s g i v e n t o th e d e s i g n o f t h e e n d pl a te . A l t h o u g h v e r y c o m p l i -ca t ed m om en t d i s tr i b u t i on s ar e p r e sen t in t he e nd p l a t e t he endp l a t e is mo s t l yd e s i g n e d b a s e d o n s i m p l e T- s t u b m o d e l s w i t h y i e l d i n g a t t h e r o o t o f t h e T[1] a n d pos s ib ly a t t he b o l t l i n e [ 2 3 ] . I n t he ann ex t o EC 3 [4 ] t he de s ign o fb o l t e d e n d p l a t e s i s b a s e d e n t i r e l y o n t h e m o d e l o f e q u i v a l e n t T- s t u b s . H o w -eve r t he m e t hod s p r e s en t ed i n Eu roco de N o . 3 [ 4 ] a r e a pp l i c ab l e on ly f o rc o n n e c t i o n s w i t h o n e l i n e o f b o lt s o n e a c h s i d e o f t h e w e b o r a l te r n a ti v el yo n e l i ne o f b o l t s o n each s i d e o f t h e t en s ion f l an ge . I n add i t i on t he r egu l a t i ons

119


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120 P C O l s e n

a re d i rec tly app l icab le on ly fo r sm al l ax ia l fo rces . For such s imp le cases theT-s tub mo de l seems to be adequa te a l though the m om ent d i st r ibu t ion at thej u n c t i o n b e t w e e n w e b a n d f l a n g e i s v e r y c o m p l e x .

I t is v e ry im por tan t to cons ide r the c onn ec t ion bo l t s /endp la te in i t s en t ir e ty.Th e d im ens ion o f the endp la te in f luences the bo l t fo rces i .e . a ve ry th ickendp la te r e su lt s in sm al l e r bo l t fo rces wh ereas p ry ing fo rces a re in t roducedfor a th inner endp la te r e su l t ing in l a rge r bo l t fo rces . The ex i s tence o f p ry ingforces i s a ma t t e r o f to wha t ex ten t the endp la te makes con tac t wi th theres is t ing med ia i .e . an opp osi te endp la te for a be am spl ice or a f lange for ab e a m - c o l u m n c o n n e c t i o n .

Fro m th is in t roduc to ry d i scuss ion i t i s obv ious tha t m ore genera l me thods

of ana lys is and des ign o f bo l t ed endp la tes in pa r ti cu la r me thods ph ys ica l lym o r e c o m p r e h e n s i v e a n d s u it ab l e f o r c o m p u t e r i m p l e m e n t a ti o n a r e d e s ir a b le .I n G e b b e k e n e t a l [5] resul t s of numer ica l computa t ions us ing the f in i tee lem ent me tho d and the theory o f e l as t ic i ty a re p resen ted . H ow ever the p rob-lem of con tac t be tween su r faces i s ve ry complex in the theory o f e l a s t i c i tyand requ i res g rea t computa t iona l e ffo r t .

In th i s paper a genera l approach o f ana lys i s and des ign o f bo l t ed endp la teconn ec t ions based on the s t ruc tu ra l l aws o f p las ti c ity i s p resen ted . Bo th end-pla te and bol ts must thus possess suff ic ient capaci ty of p las t ic i ty. The upper

bou nd theory i s app l ied i .e . the y ie ld l ine theory i s app l i ed fo r the endp la te .Con t ra ry to cl a s s ica l y ie ld l ine theory w here a s ing le pa ram ete r de te rminesthe en ti r e m echa n i sm the endp la te i s subd iv ided in to tr i angu la r e l ements thed i sp lacements o f each ve r t ex o f the mesh represen t ing a pa r t i a l mechan i sm.T h e p r o p e r m e c h a n i s m i s d e t e r m i n e d b y m e a n s o f s o l v i n g a l in e a r ly c o n -s t r a ined op t imiza t ion p rob lem. The con tac t p rob lem i s s imply so lved byadding addi t ional cons t ra in ts regarding the d isplacements a t the ver t ices ; th isdoes no t inc rease the complex i ty. The op t imiza t ion p rob lem i s so lved bym eans o f a ve ry e ff ic i en t m e thod d eve lop ed by the au thor. Th i s new ly

deve loped method has a r e sponse t ime so min imal a s to enab le in te rac t ivedes ign.

O n e m a y u t il iz e th is m e t h o d o l o g y i n o n e o f tw o w a y s e i th e r in te r m s o fa sm al l va r i e ty o f poss ib le y ie ld l ine pa t te rns se lec ted by eng in ee r ing judg e-m e n t w i t h fe w p a r a m e t er s d e t e r m i n i n g t h e m e c h a n i s m o r i n t e rm s o f a p p ly i n ga v e r y la rg e n u m b e r o f e l em e n t s i n w h i c h c a s e th e n e c e s s it y o f e n g in e e r i n gj u d g e m e n t b e c o m e s s u p e r f lu o u s a n d w h e r e b y a v e r y a c cu r a te e s ti m a t io n o fthe bear ing c apaci ty i s ensu red. In the f ir s t case the respon se t ime of themethodo logy i s comparab le wi th tha t o f app ly ing a s imple ana ly t i ca l so lu t ion

expressed in te rms o f a fo rmu la w hereas the r e sponse t ime in the second caseespec ia l ly in a PC -env i ron m ent unfor tuna te ly i s unaccep tab le . The dura t ion o fthe ana lyses o f the examples p resen ted in th i s paper was typ ica l ly 3 -10 minw h i c h m u s t b e c o m p a r e d t o a n a c c e p t a b l e r e s p o n s e t i m e o f m a x i m u m 3 0 s .


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De sign of bolted endplate connect ions 121

T h e c o n c l u s i o n d r a w n w a s t o d e v e l o p a n a l y t i c a l d e s i g n a i d s b y m e a n s o fs im ple y i e ld l ine pa t t e rns and to t e s t t he r e su l t s aga ins t t he e l abora t e an d accur-a t e n u m e r i c a l m e t h o d o f u s i n g a v e r y l a rg e n u m b e r o f e l e m e n t s . T h i s m u s tb e s e e n i n c o n t e x t w i t h t h e r e q u i r e m e n t i n E C 3 o f h a v i n g t o d e m o n s t r a t e th ea d e q u a c y o f th e m o d e l b y m e a n s o f p h y s i c al te st s w h e n t h e y i e ld l in e m e t h o di s u s e d . T h e a d e q u a c y o f t h e a n a l y t i c a l s o l u t i o n s p r e s e n t e d i n t h i s p a p e r h a sb e e n p r o v e d b y m e a n s o f a n a c c u ra t e n u m e r i c a l m e t h o d .

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

I t i s w e l l k n o w n [ 6 ] t h a t t h e u l t i m a t e l o a d f a c t o r o f t h e u p p e r b o u n d t h e o r ym a y b e d e t e r m i n e d b y m e a n s o f s o l v in g l in e a r o p ti m i z a ti o n p ro b l e m s . F o r m u -la t ion o f t he y i e ld l i ne theo ry a s a p rob lem o f l i nea r op t imiza t ion i sa c c o m p l i s h e d b y s u b d i v i d i n g t h e p l at e u n d e r c o n s i d e r a t io n in t o a n u m b e r o ft r i angu la r e l emen t s a s shown in F ig . 1 , and app ly ing the p r inc ip l e o f v i r tua lw o r k a n d t h e u p p e r b o u n d t h e o r e m . S u b d i v i s i o n o f t h e d o m a i n i n t o t r i a n g l e si s advan tageous in the sense tha t a l l pa r t i a l mechan i sms au tomat i ca l ly a reg e o m e t r i c a l ly a d m i s s i b le .

Di sp lacemen t s and r e l a t ive ro t a t ions a re a s s igned to a l l ve r t i ces and a l l

t r ia n g u l a r e d g e s , r e s p e c ti v e l y, in t h e s u b d i v i d e d d o m a i n . A l l d i s p l a c e m e n t sa re co l l ec t ed in a vec to r u and a l l ro t a t ions in a vec to r 0 . Compa t ib i l i t yrequ i re s the fo l lowing equa t ion to be sa t i s f i ed :

0 = B u 1 )

w h e r e B i s a r e c ta n g u l a r m a t r i x w i t h th e n u m b e r o f ro w s c o r r e s p o n d i n g t ot h e n u m b e r o f e d g e s a n d t h e n u m b e r o f c o l u m n s c o r r e s p o n d in g t o t he n u m b e ro f v e r ti c e s. W r i ti n g t h e r o t a t io n s a s t h e d i f f e r e n c e b e t w e e n t w o p o s i t iv e n u m -

ber s , eqn 1 ) can be rew r i t t en acco rd ing ly as :

0 + - - 0 - = B u 0 + - - - 0 0 - - - > 0 . 2 )

Fig. 1. Subdivision of part o f an endplate around the boltholes into triangular elements.


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122 P C Olsen

T h e i n t e m a l w o r k is d e t e r m i n e d as t h e p r o d u c t o f t h e y ie l d m o m e n t s a n dt h e ro t a ti o n s . L e t t in g m + a n d i n - r e p r e s e n t t h e y i e ld m o m e n t s , t h e in t e r n a lw o r k is d e t e r m i n e d f r o m e q n 3 ):

W i n t e r n = m 0 + m - 0 - . 3 )

T h e e x t e m a l w o r k i s p ri n c ip a l ly d e t e r m i n e d a s t h e p r o d u c t o f t h e e x t e m a lf o r c e s a n d t h e d i s p l a c e m e n t s . T h i s c a n b e e x p r e s s e d a s a l i n e a r e q u a t i o n , a ss h o w n i n e q n 4 ):

We x t er n C u ( 4 )

w h e r e t h e c o m p o n e n t s o f C r e p r e s e n t t h e e x t e rn a l w o r k f o r u n i t d is p l ac e m e n t s .B y m e a n s o f th e u p p e r b o u n d t h e o r e m , t h e u lt im a t e l o a d f a c to r is d e t e r m i n e das t he quo t i en t :

in te rn- - - 5 )

We x t e r n

F o r e a c h m e c h a n i s m a n u p p e r v a l u e o f t h e u lt im a t e l o a d f a c t o r is d e t e r m i n e da n d t h e p r o b l e m is h e r e a f t e r to d e t e r m i n e t h e p a r t ic u l a r m e c h a n i s m y i e l d i n gt h e s m a l l e s t l o a d f a c t o r. B y r e q u i r i n g t h e e x t e r n a l w o r k t o b e c o n s t a n t , f o re x a m p l e

C u = 1 6 )

i t i s e a s i l y s e e n t h a t t h i s : m e c h a n i s m a n d t h e r e b y t h e u l t i m a t e l o a d f a c t o r a r ed e t e r m i n e d b y s o lv i n g t h e f o l l o w i n g o p t i m i z a ti o n p r o b l e m :

rain[m+ 0 + + m - 0- ] , 0 + - 0 - - B .u = 0 C-u = 1 0 + --- 0 0- --- 0 . 7)

T h e i n c o r p o r a t i o n o f t h e b o lt s re q u i r e s o n l y a s m a l l e x t e n s i o n to e q n 7 ).I n a d d i t i o n t o t h e i n t e r n a l w o r k f r o m t h e e n d p l a t e , t h e i n t e r n a l w o r k d o n e b yt h e b o lt s, w h i c h is s i m p l y t h e p r o d u c t o f b o l t f o r c e a n d d i s p l a c e m e n t , m u s ta l s o b e c o n s i d e r e d . T h u s e q n 3 ) i s a m e n d e d a s f o l lo w s :

Winte rn =m + 0 + + m - 0 - + P b o l t u ( 8 )

w he re Pboxt i s a vec to r ex p re s s in g th e y i e ld c ond i t i on o f t he i nd iv idua l bo l t s.F i n a ll y, th e p r o b l e m o f c o n t a c t b e t w e e n t h e e n d p l a t e a n d t h e r e si st in g m e d i ai s s o l v e d b y r e q u i r i n g t h e d i s p l a c e m e n t s t o b e n o n - n e g a t i v e . A s c a n b e s e e n ,


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esign of bolted endplate connections 123

t h i s d o e s n o t i n c r e a s e t h e c o m p l e x i t y o f t h e p r o b l e m . T h u s , t h e b e a r i n gc a p a c i t y o f t h e a s s e m b l a g e o f e n d p l a t e / b o l t s w i t h a u t o m a t i c c o n s i d e r a t i o n o fp o t e n t ia l p r y i n g f o r c es i s d e t e r m i n e d b y s o l u t io n o f t h e f o l l o w i n g o p t im i z -a t i o n p r o b l e m :

m in[m +0 + + m - 0 - +Pbo~tU],0+ - 0 - - B . u = 0 C . u = l 0 > - 0 0 -> 0. 9)

R e q u i r i n g t h e d i s p l a c e m e n t s to b e n o n - n e g a t i v e is a l s o a r e q u i r e m e n t to t h eres i s t ing med ia in t e rms o f su ff i c i en t s t r eng th . A beam sp l i ce i s r eckoned toh a v e s u f fi c ie n t s t r e n g t h w h e n t h e t h ic k n e s s o f t h e e n d p l a t e s a t o p p o s i t e b e a me n d s i s e q u a l . F o r a b e a m - c o l u m n c o n n e c t i o n t h e m e t h o d i s a l s o a p p l i c a b l ei f o n e m o d e l s b o t h e n d p l a t e a n d c o l u m n f l an g e w i t h t h e a p p r o p ri a te s t re n g t h sa n d r e q u i r e s t h e d i f f e r e n c e i n d i s p l a c e m e n t s b e t w e e n e n d p l a t e a n d f l a n g e t obe no n-neg a t ive . W i th th i s la t t e r t echn ique p rac t ica l ly any type o f bo l t ed con -nec t ion can be ana lysed .

E n d p l a t e c o n n e c t i o n s f o r s y m m e t r i c a l I s e c t i o n s w i l l n o w b e d i s c u s s e d . I nth i s case the t ens i l e s t r e s ses in the f l ange and in pa r t o f t he we b a re t rans fe r r edv ia the endp la t e to the bo l t s , whereas the compress ion s t r e s ses a re t r ans fe r r edb y m e a n s o f c o n t a c t b e t w e e n t h e e n d p l a t e s. H o w e v e r, t h e f la n g e s a n d t h e w e b

a l so suppor t t he endp la t e , i n tha t t he sec t ion p rac t i ca l ly de fo rms r ig id ly. Th i se f f e c t is a c c o m m o d a t e d b y i n t r o d u c i n g in f in i te l y s t r o n g b e a m s . B e c a u s e o fs y m m e t r y, o n l y h a l f o f t h e e n d p l a t e i s c o n s i d e re d , a n d i n o r d e r to r e d u c e t h ec o m p u t a t i o n a l r e s o u r c e s o n l y t h e p a r t o f t h e e n d p l a t e a r o u n d t h e b o l t h o l e s i nthe t ens ion zone i s desc re t i zed . T h i s i s j u s t i f i ed by the f ac t t ha t t he I s ec t ionro ta t e s r ig id ly a round the compress ion f l ange . Th i s r equ i re s , however, t ha tf l anges a re pos i t i oned oppos i t e t o one ano the r. One thus a r r ives a t t he mode lshown in F ig . 2 .

In F ig . 3 the y i e ld l ine pa t t e rns fo r a t h in and a th i ck endp la t e a re shown .

The bo l t s a re in bo th cases des igned a s suming an in f in i t e ly s t rong endp la t e .As can be seen , com ple te ly d i f f e ren t y i e ld l ine pa t t e rns a re dec i s ive in thetwo cases .

3 D E S I G N O F B O LT S A S S U M I N G A N IN F I N IT E LY S T R O N GE N D P L AT E

In the case o f an in f in i t e ly s t rong endp la t e , t he u l t ima te load f ac to r i s ea sy tod e t e r m i n e , a s i t c o r r e s p o n d s t o a m e c h a n i s m , w h e r e th e a s s e m b l a g e o f b e a m sand endp la t e ro t a t e s r ig id ly abou t t he compress ion f l ange a s shown in F ig . 3 .In th i s case the in t e rna l work i s so l e ly due to the y i e ld ing o f t he bo l t s :


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1 2 4 P C O l s e n

F i g . 2 . C o m p u t a t io n a l m o d e l o f b o l t e d e n d p la t e.

a)t = l O m m

k = 0 . 7 2

b )t = 2 0 r a m

k : 1 . 1 3

.Fig . 3 . Yie ld l ine pa t terns for A ) a th in endpla te and B) a th ick endp la te


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D esign o f bolted endp late connections 125

N

Wint~m = O Z P B , i h i (10)

w he re PB.i i s the y ie ld s t rength o f the bol ts , 0 i s the angle o f ro ta tion andii s the d i s t ance be tween the bo l t s and the compress ion f l ange . The ex te rna lw o r k i s d e t e r m i n e d f r o m e q n ( 11 ):

W e x t e r n = Oh(o Tbntfa +1/6(20 T + o'c)twh) 11)

w he re O'T is the stress at the f lan ge in ten sion , Oc is the stress at the f langein com press ion , t~ and bn a re the th ickness and the w id th , r e spec t ive ly, o f thef lange in tens ion, tw is the th ickness of the web, and f ina l ly h i s the he ight ofthe sec t ion [eqn (11) impl ies a l inear s t ress d is t r ibut ion through the sec t ion] .The bo l t s can no w b e des ign ed by mean s o f eqns (5 ), (10) and (11) in thatthe load fac tor must be la rger than one , or, equivalent ly :

N

~ P B . i h i ~ h(o Tbtatn +1/6(2OFT + o'c)twh). (12)

For exam ple , fo r g iven b o l t d imens ions , bo l t s t r eng th and bo l t spac ing , then u m b e r o f r e q u i r e d b o l t s m a y b e d e t e r m i n e d .

4 D E S I G N O F F L U S H E N D P L AT E

The bo l t des ign method o f eqn (12) i s comple te ly t r ad i t iona l . Choos ing anendp la te th ickness l a rge enough to p reven t any y ie ld ing in the endp la te , thebo l t d imens ions thus de te rmined a re a l so the f ina l d imens ions . A ques t ion o fg rea t impo r tance i s , thus , w ha t the low er l imi t o f th ickness o f the en dp la te isin o rde r to p reven t y ie ld ing in the endp la te .

Befo re address ing th i s ques t ion , ho we ver, ano the r l imi t ing va lue o f the end-pla te th ickness , assuming fu l l s t rength u t i l iza t ion of the bol ts , i s de terminedf rom the punch ing shea r r e s i s t ance . Accord ing to EC3/4 / , t he punch ing shea rres i s t ance i s de te rmined on the bas i s o f a c r i t i ca l d iamete r co r respond ing tothe mean dimension dm of the cross f la ts and the cross points of the bol t heador the nut . Requir ing fu l l s t rength u t i l iza t ion of the bol ts , th is leads to the

fo l lowing min imal endp la te th icknesstp:

tp --> (1 3)4dmfp.y


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126 P. C . O l sen

w h e r e d B i s t h e d i a m e t e r o f t h e n e t s t r e s s a r e a o f th e b o l t ,fB yi s t h e y i e l ds t r e n g t h o f t h e b o l t s a n d fp ,y i s th e y i e l d s t r e n g t h o f t h e e n d p l a t e . U s i n g t h es p e c i f i c a t i o n s f o r t h e M - b o l t s ,d m / d B ~ 1 . 9 a n d d N / d B ~ 1 . 1 5 , w h e r e d N i st h e n o m i n a l d i a m e t e r o f th e b o lt . T h i s l e a d s t o t h e f o l l o w i n g r e q u i r e m e n t s :

> / 0 . 5 4 d N . . . . . . . . . 8 . 8 b o l t s )

tr, - - L0 .76dN . . . . . . . . . 10 .9 bo l t s ) 14)

assumingfp y =2 3 5 M P a a n d u t i l iz i n g t h e y i e l d s t r e n g t h o f t h e b o lt o n l y.A s i m p l e a n a l y s i s i n c l u d i n g o n e b o l t o n l y a n d a s s u m i n g t h e s im p l e y ie l d

m e c h a n i s m , w h i c h i s s h o w n i n F i g . 4 , i s n o w p e r f o r m e d . T h e i n t e r n a l w o r k

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

h c h ) h c ) 1Wintern= m 2 + - 1 + 2 (c .-}- x ) ~ ~ p fp y O 15)

X C e

w h e r e t h e s y m b o l s u s e d a r e s e e n i n F i g . 4 . T h e i n t e r n a l w o r k i s m i n i m a l f o r :

x = ,,/m-7. 1 6 )

N -x

Fig. 4. Simp le yield mech anism for an e ndplate with one bolt only.


9/22

esign ofbolted endpl te connection s 1 2 7

T h e e x t er n a l w o r k i s d e te r m i n e d b y m e a n s o f e q n ( 11 ) a n d t hu s th e m i n i m a lth i cknes s o f t he endp la t e i s de t e rmined f rom:

2 h o rTbntn (1/6)(20 T O'c)twh)tp --> . (1 7 )

m 2 ~ - + - - c 1 + 2 e ( c + ) ,y

In F ig . 5 the endpla te th ick ness i s p lo t ted for the IPE-ser ies o f sec t ions ,a s suming the s ec t i ons ca r ry on ly a bend ing momen t caus ing y i e ld ing in bo th

f langes . Also , the des ign y ie ld s t rengths of sec t ion and endpla te a re equa l .W h e n p l a c i n g s e v e r a l b o l t s a l o n g s i d e t h e w e b , t h e y i e l d m e c h a n i s m s h o w nin F ig . 6 i s d ec i s ive . A s a r esu l t o f t he ro t a ti on abou t t he com pres s ion f lange ,the a r ea deno ted ABCD in F ig . 6 can no t de fo rm r ig id ly and mus t t he re fo rebe d iv ided a s shown . The in t e rna l work i s de t e rmined to be :

h c + h )W i n te m = m 2 + 1 + 2

x c

h - c - de

c d x) ~tp 2 f p y O

(18)

where t he symbo l s u sed a r e s een in F ig . 6 . S imi l a r ly t o t he y i e ld mechan i smof F ig . 4 , t he i n te rna l work i s m in im a l fo r t he x -va lue de t e rmined in eqn (16 )a n d th e e x te r n a l w o r k i s d e t er m i n e d b y m e a n s o f e q n ( 11 ) . T h e m i n i m a l e n d -p l a t e t h i cknes s i s t hus de t e rmined f rom:

3 5

t

3 0 . . . . . - . . . . . . . . . . . . . . F - - - -/ ~ - . . . .

~ 2 5 I. . . . . . . . . I - . . . . r - - - - r . . . . . . . .

~Z~ I t I I

x ~ I I I

._ u 2 0 i. . . . . . . . . . . . ~ - . . . . I - . . . . I -( -4 - ,

- - ~ / I P , y : y d- o 1 o . . . . . - / - - - - m = O . ' S b , F ' t

c ' - / iI I I I C : U , J m

e = O S m5 . . . . . . ; - . . . . i -

i ii ii i

o I I I I I0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

I P E

F i g 5 E n d p l a t e t h ic k n e s s fo r t h e I P E s e c ti o n a p p l y i n g e q n1 7 ) .


10/22


11/22

D e si g n o f b o l t ed e n d p l a t e o n n e c ti o n s 129

3 5

3 0 -

o3~ 2 5 -

. o 2 0 -c -

. ~

~ 1 5 -4 .

c~o _

- o 1 0 -c--

I M

5 -

oo

. . . . . . . . . . . . . . . 7 . . . . ? . . . . 7 . . . .

d = O m m I J. ? . ; . ; . ; .Ld = 5 0 m m I I . . . ,~ f ~ ~ _ -

I I II I I

. . . . P . . . . r . . . . I - . . . . . . l - . . . . I

I 1 I I I I

I I ~ I I I. . . . . ~ . . . . ~ ~ - - . . . . ~ . . . . . . . . . .

I

I I I

o f f = ~ y o l. . . . t_ . . . . ~ - ~ ~ C = - y d

I I I - -

~ ~ P y - - F y d_ _ ] _ _ _ I

. . . . P . . . . ~- m = 0 , S b f t

~ ~ C = 0 5 r ni I

, e = 0 . 5 m. . . . . p . . . . ~ p

i II I I

I I I I I1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

I P E

Fig. 7. The effect of placing several bolts alongside the web.

respec t ive ly. I t i s seen tha t the resu l t o f p lac ing severa l bo l t s a longs ide the

web may be pa r t l y a s t r eng then ing , pa r t l y a weaken ing o f t he endp la t e ,depend ing on the d i s t ance be tween the bo l t s and the compres s ion f l ange . Theendpla te i s s t rengthened i f th i s d i s tance i s l a rge .

In what fo l lows , i t i s demons t ra ted tha t p ry ing forces a re no t p resen t a t theu l t ima te l im i t s t a t e when des ign ing the endp la t e t h i cknes s acco rd ing to eqns1 7 ) , 1 9 ) a n d 2 0 ) . T h e y i e ld m e c h a n i s m o f F i g . 8 i s c o n s id e r e d , i n w h i c h

the con tac t su r f ace o f po t en t ia l p ry ing fo rces i s l i f ted a s sho wn , co r r e spond ingto a ro t a t i on abou t t he compres s ion f l ange . The in t e rna l work i s now:

W i n t e r n - ~- W e q n l 8 ) . I

U ~ P B i h i - m 2h c

,c + a + x ) t g f . , ,h c d h \ h c d

- - + - - 1 ) + 2 2 1 )X C e

and i t i s s een tha t x i s once ag a in de t e rmined by eqn 16 ) . T he endp la t eth i cknes s i s deduced f rom the r equ i r emen t t ha t t he de r iva t ive o f t he i n t e rna lwork wi th respec t to the l i f t ing u mus t be non-nega t ive , and i t i s seen tha t ,w hen app ly ing the bo l t f o r ce s o f eqn 12 ) , t he t h i cknes s i s onc e aga in det e rm-ined by eqns 17 ) and 19 ) , t hus conc lud ing tha t p ry ing fo rces a re no t p r e sen t .


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130 P C O l s e n

UI)= u

UA = u 1-~-dc

Fig 8 Lif t ing of the surface of potential prying forces

5 T E S T IN G T H E D E S I G N P R O C E D U R E F O R F L U S H E N D P L AT E S

The com bine d p rocedu re o f des ign ing the bo l ts , a s sum ing an in f in it e ly s trongendp la te and des ign ing the ac tua l endp la te th ickness accord ing to eqns 17) ,1 9 ) a n d 2 0 ) , is n o w t e s te d b y m e a n s o f th e n u m e r i c a l m e t h o d . A n I P E 3 0 0

is se lec ted and the u l t ima te load fac to r i s f i rs tly de te rm ined fo r the p a ramete r sused in the cha r t o f F igs 5 and 7 fo r one and th ree bo l t s , r e spec t ive ly. Sec -

ond ly, fo r ex t raord ina ry bo l t pos i t ions , an end p la te wi th a wid th o f 200 m mwi th the f ir st bo l t pos i t ioned 100 m m be lo w the t ens ion f l ange, in l ine w i ththe t ip o f the f langes i s exam ined . A yie ld s t rength fyd = 183.5 M Pa is appl ied .Th e resu l ts a re sh ow n in F ig . 9 , in w hich the r equ i red to ta l bo l t fo rce capac i tyPB, the en dp la te th ickness tp and th e u l t ima te load fac tor A ca lcu la ted aretabu lated . T he y ie ld l ine pa t te rns show n in th i s f igure ind ica te tha t the des ignresul ts in an u l t ima te load fac tor corres po nd ing to an inf in i te ly s t rong endp la te .Th i s i s , however, no t t rue fo r the endp la te P3 , the r eason be ing tha t the end-p la te is w eak ene d by the miss ing supp or t o f the t ens ion f lange a t the ex tendedpar t o f the endp la te , r e su l ting in a low er u l t ima te load fac to r.

In Fig . 10 the effec t of increas ing the to ta l bol t force capaci ty i s shown. I ti s seen tha t the fo rces in the endp la te to some ex ten t may be red i s t r ibu teddu e to pryin g ac t ion , resul t ing in la rge r u l t imate loa d fac tors . I t i s a lso not ice-


13/22

esign of bol ted endplate connections 131

P1 P2 P3 P4

410 KN 507 KN 539 KN 710 KN

21 51 mm 21.39 Mm 26 60 mm 30 93 mm

1.00 1.00 0 , 9 4 1 00

Fig. 9. Design values ultimate load factors and yielding patterns for varying flush endplate.

15o . . . . . . . . . ? . . . . ?- . . . ? . . . .

&- -~, P2 I i1 . 4 0 5 ~ k ' ~ P 3 L . . . . . . . . . . . . . .

F,- '='=',, '- P 4 I I I I II I t I

. . . . L . . . . . . . . L . . . . L . . . . I

I I I I I

I I I I

I I I I I II I I I I

. . . . . . L . . . . L . . . . L . . . . L . . . . II I I I I

I I I I I II I I I I I

I I I I I I

I I I I I I

II I I I I I

I I I I I

1 . 0 0 L . . . . L . . . . L . . . . L . . . . L . . . . II I I I I II I I I I I

I I I I I I

0 . 9 01 . 0 0 1 . 5 0 2 . 0 0 2 . 5 0 3 . 0 0 3 . 5 0 4 . 0 0

o lt f o r c e f a c t o r

1 . 3 00

q . -

00

~ r - 1 . 2 0

0

0_ . j 1 . 1 0

Fig. 10. The b earing capacity as a function of increase in bolt force capacity.

a b l e t h a t t h e i n c r e a s e i n th e u l t i m a t e l o a d f a c t o r is n o n - p r o p o r t i o n a l w i t h t h ei n c r e a s e i n b o l t f o r c e c a p a c i t y i n d i c a t i n g t ha t t h e c o n n e c t i o n b o l t s / e n d p l a t ei n i ts e n t i r et y b e c o m e s d e c i s i v e f o r t h e b e a r in g c a p a c i t y. T h i s i s a l s o c l e a r lys e e n i n F i g . 11 w h e r e t h e y i e l d l i n e p a t t e r n s f o r a n i n c r e a s e i n b o l t f o r c ec a p a c i t y o f 1 . 2 5 a n d 4 . 0 r e s p e c t i v e ly a re s h o w n .

6 D E S I G N O F E X T E N D E D E N D P L A T E

E x t e n d e d e n d p l a t e s a re c o n s i d e r e d n e x t . T h i s t y p e o f c o n n e c t i o n c o u l d a l s ob e d e s i g n e d s u c h th a t n o p r y i n g f o r c e s o c c u r. In t h is c a s e h o w e v e r o n l yy i e l d i n g a t t h e fl a n g e w o u l d b e p o s s i b l e r e s u l t i n g in r a th e r t h i c k e n d p l a t e s.


14/22

132 P C Olsen

PI P P3 P4

1 25P B

4 00P B

Fig 11 Yie ld l ine pa t tems fo r va ry ing endpla tes and bo l t fo rce capaci ti es

x

Fig 12 Yie ld me chan ism for an ex tended endpla te


15/22

Design of bolted endplate connections 133

U s i n g a s l i g h t l y m o r e c o m p l i c a t e d d e s i g n p r o c e d u r e , t h e c o n n e c t i o n m a y b ed e s i g n e d w i t h y i e l d l in e s a t b o t h t h e b o l t l in e o f t h e e x t e n d e d p a r t o f t h e e n d -p l a te a n d a t t h e f la n g e . To t h is e n d , t h e y i e l d m e c h a n i s m o f F i g . 1 2 is c o n -s i d e re d . T h e i n t e rn a l w o r k is d e t e r m i n e d f r o m e q n ( 2 2) , w h e r e a s t h e e x t e r n a lwork i s s t i l l g iven by eqn (11 ) . A l so , i t i s e a s i l y s een tha t t he min imiz ingva lue o f x is s ti ll g iven by eq n (16 ) . Thus , f ro m eqns (11 ) and (22 ) , t hee n d p l a t e t h i c k n e s s a n d t h e b o l t f o r c e s a r e d e r i v e d a s f o l l o w s i n e q n s ( 2 3 - 2 5 ) :

tp

Wintern = 22)

m 2 + 2 + 2 - 1 + 2 c + d + x ) 0x c a e

+

- v ~ / h i - m 2 - + 2 - 1Intern X C

b e d+ 2 c + d + x )

e

+

+ t~,fp,y.x t2fpy m l -- a

(23)

h crTbntn +(1/6)(20 T + crc)twh)

h - c - d h h ) h - c - dm 2 + 2 - + 2 - 1 + 2~m e c a e

, h - c - d PB ,ihi >-- ~ t~fpy m

Intern ~ e

+ 2 c + d + )e

h+ 2 - - 1

C

(C + d 1- ~ ' ~ ) ~ p , y

(24)

PB ,e >-- ~ t~fey

a2 -

m la a

1 -l

(25)


16/22

134 P. C. Olsen

B y m e a n s o f e q n 2 3 ) t h e e n d p l a te t h ic k n e s s is f ir st d e t e r m i n e d , w h e r e a f t e rthe in t e rna l and the ex te rna l bo lt s ar e des igne d by app ly ing eqn 24) and eqn25) , r e spec t ive ly. On ly one row o f ex te rna l bo l t s i s cons ide red . The des ign

p r o c e d u r e i s i t e ra t iv e i n t h a t t h e d i s ta n c e d d e p e n d s o n t h e n u m b e r o f i n t er n a lbo l t rows .

I t i s s een tha t t he r equ i red bo l t fo rce capac i t i e s o f t he in t e rna l and thee x t e r n a l b o l t s a r e p r o b a b l y n o t t h e s a m e . H o w e v e r, a p p l y i n g i d e n t i c a l b o l t sa n d d e s i g n i n g t h e l e n g t h o f t h e e x t e n d e d p a r t o f t h e e n d p l a t e a c c o r d i n g t oeqn 26) :

mk - 2

26)l - mk -

w h e r e :

c - .m 2 + 2 - 1 + 2 c + d + )

k = 27)i

I n t e r n

y ie lds iden t i ca l r equ i red bo l t fo rce capac i t i e s fo r bo th in t e rna l and ex te rna lbol ts .

F ina l ly, t he p ry ing r a t io i s de t e rmined a s the r a t io be tween the to t a l bo l tfo rce capac i ty o f eqns 24 ) and 25) and the bo l t fo rce capac i ty o f eqn 12)fo r an in f in i t e ly s t rong endp la t e w here n o p ry ing fo rces a re p resen t . The p ry ingr a ti o is t h u s d e t e r m i n e d f r o m e q n 2 8 ):

=

m 2 - a / l ) l~ , h i k + a l - ~ l ) JA l l

m 2 + 2 - + 2 - 1 + 2 c + d + )~ /me c a e

28)

a n d i t is s e e n t h a t th e p r y i n g r a t io d e p e n d s o n l y o n t h e g e o m e t r y o f t h e c o n n e c -

t ion.In F ig . 13 the r equ i red endp la t e th i ckness i s p lo t t ed fo r t he IPE-se r i e s o f

s e c ti o n s , a s s u m i n g t h e s e c t io n s t o c a rr y o n l y a b e n d i n g m o m e n t c a u s in g y i e ld -i n g i n b o t h f l a n g e s . A l s o t h e d e s i g n y i e l d s t r e n g t h s o f s e c t i o n a n d e n d p l a t e


17/22

esign o f bolted endplate connections 135

3 5

3 0

2 5 -0E, , (

. 0 _ 2 0 -c -

q ) 1 5 -

- 0 ~ 1 0 -c

5

0

0

. . . . . . . . . . F . . . . F . . . . F . . . . - ? - - - ~l. . . .I I I . , ~ - - - - _ _ iI

o h - = { v d- r . . . . r . . . . r - - - i . . . . .

c = - % a , , , . Z ' , /1 'I I I

C P, , = , , c 1 / J J ~J ] ] r . . . . r r . . . . .

I t I I:o 5~ , / . , x . J ~ y

, ~ I . . . . . . .a = o , - ~ - r - / ' , ~ _ h ~ , ' - . . . . :

I I I IL L I

I I I I

I I I I

I I I I

. . . . I I I I

i I ~ F l u s h e n d p l a t e. . . . ~ ; . . . . . ; . . . . ; ~ o = O . 2 _ 5 ~

i i s . . ' .. ~ , ~ . ~ a = O . 5 0 mi i t . L L A - - a = l .0 0 mI i I

I I I i I1 O 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

I P E

F i g . 1 3 . E n d p l a t e t h i c k n e s s f o r t h e I P E s e c t i o n s u s i n g e x t e n d e d e n d p l a te s .

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

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

I n F i g . 1 4 t h e r e q u i r e d i n t e r n a l b o l t f o r c e c a p a c i t i e s a r e p l o t t e d t o g e t h e rw i t h t h e r e q u i r e d e x t e r n a l b o l t f o r c e c a p a c i t y f o r v a r i o u s l e n g t h s o f t h e

e x t e n d e d p a r t o f t h e e n d p la t e . A y i e l d s t r e n g t h f o r s e c t i o n a n d e n d p l a t e O f f yd= 1 8 3 .5 M P a i s a s s u m e d .

4 o 0 . . . . . . . . . . . ? . . . . ;- . . . . ? . . . . ; - - - - - / ~I I I I I

= e - / :- i y e l i I , l . / IO ' P = - - ~ - - I i I , P ~ I

z 3 - ' r Y ^ - Ya r . . . . r . . . . r . . . . ~ . . . . ,~ / P I = U , ~ I O D I I I I / I I4 t .

: o S r ~ , , P FC ' I I I / I / i

- d = 0 ' ' ~ / ' ~ '@ t i / J I /o 2oo-- ~ =o 5~ ~ .. . . ~- -~ - ~- -- ~F - - - ~

~ , , ~ = I 8 4 M P ~ , ~ / , J t ~ K ,'~ ' y '*.,I f I / I /

I I I I I

I I I I I I0 i i i i

r ~ 1 0 0 . . . . . . r . . . . . . . .

I n t e r n a lE x t e r n a l , a ~ 1 = 0 . 6 7 ;

i j = J ~ ( d l ~ i ~ , - , ~ 3 ~ . ~ F . . x t e m a l , a ' 1 1 = O , S O i

0 - i I i I I I0 , 0 0 2 0 0 3 0 0 , 0 0 s ~ o 6 0 0

I P E

F i g . 1 4 . B o l t f o r c e c a p a c i t i e s f o r t h e I P E s e c t i o n s u s i n g e x t e n d e d e n d p l a t e s .


18/22

1 3 6 P C O l s e n

PLate P1 P2 P3 P4

PB n 296 KN 296 KN 372 KN 372 KN

PB Ex 149 KN 198 KN 147 KN 196 KN

t p 1 6 . 4 2 ~ 1 6 4 2 m ~ 1 6 . 3 3 M m 1 6 . 3 3

X 1 02 1 . 0 2 1 0 3 1 0 3

F i g . 1 5 . De s ign va lues u l t ima te load f ac to r s and y i e ld l ine pa tt erns fo r va ry ing ex tendede n dp l a t e s .

7 T E S T IN G T H E D E S I G N P R O C E D U R E F O R E X T E N D E DE N D P L AT E S

T h e d e s i g n p r o c e d u r e f o r e x t e n d e d e n d p l a t e s i s n o w t e s t e d b y m e a n s o f t h en u m e r i c a l m e t h o d . A n I P E 3 0 0 i s s e le c t e d a n d th e u l ti m a t e l o a d fa c t o r i s

de t e r mine d fo r t he pa r ame te r s u sed i n F ig . 1 4 . A l so , f o r t h e s a me pa r ame te r sbu t u s ing t h r ee in t e rna l b o l t s i n s t e ad w i t h d = 1 00 m m , t he u lt i m a t e l oa d f a c t o ri s de t e rmined . The r e s u l t s a r e shown in F i g . 15 , i n wh i ch t he r equ i r ed t o t a lbo l t fo r ce cap ac i ty PB,],t and PB,Ext fo r the in te rna l a nd ex te rn a l b o l t s , r espec t -i v e ly, t he endp l a t e t h i ckn es s tp and t he u l t ima t e l oad f ac to r t c a l cu l a t ed a r et abu l a t ed . I t i s s e en t ha t t he de s ign p rocedu re i s s a t i s f ac to r y and l e ads t o t h eex p ec t ed be hav iou r o f t he c o n nec t i on a t t he u l t ima t e l im i t s t a t e .

I n F ig . 1 6 the u l t im a t e l oad f ac to r fo r con n ec t i on P3 i s p lo t t ed a s a f unc t i on

B o l t f o r c e f o c t o r

1 . 5 0 . . . . . . . . . . . . . . . . ~ . . . . / . . . . 7 . . . ii In te rna l i i i i

E n t o m a l i i i n

1 40 L L L k k I

i i J i iI ~ g i i ii i a i i iL i i i :

O1 . 3 0 ~... . IlL. . . . I ' IL. . . I lL. . . Ill I11 . . . IIII i i i i

01 .20 . . . . . . L- - -L . . . . L . . . . t . . . . . L . . . . iI i i i I i

~ o :i i i i a1 1 o . . . . . ~ . . . . L . . . L . . . . . . . . ~ . . . .

1 .oo . . . . . . L . . . . L . . . . L . . . . L . . . . U . . . .I I t U II I = I II I = I I I

I I I O

0 9 0

1 0 0 4.00

Yie t dl ln e p ~ c t e r n s ~ o r v ~ r y l n g I n te r na l b o t

+ t rEe c a p a c l l e s

1 25 PB nt 1.50 PB In 2.0 PB nt 4.0 PB n

r rF i g . 1 6 . T h e b e a r i n g capaci ty as a funct ion of increase in external and in ternal bol t forcecapaci t ies and corresponding y ie ld l ine pa t terns .


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of inc reas ing in t e rna l and ex te rna l bo l t fo rce capac i t i e s , r e spec t ive ly. In th i sf igu re the y i e ld l ine pa t t e rns fo r va ry ing in t e rna l bo l t fo rce capac i t i e s a re a l sos h o w n . I t i s s e e n t h a t t h e b e a r i n g c a p a c i t y d o e s n o t i m p r o v e b y i n c r e a s i n gt h e e x t e r n a l b o l t f o r c e c a p a c i t y a b o v e t h a t g i v e n b y e q n ( 2 5 ) . H o w e v e r, a swi th f lu sh endp la t e s , t he bea r ing capac i ty inc reases by inc reas ing the in t e rna lbo l t fo rce capac i ty, i nd ica t ing tha t t he fo rces in the endp la t e a re , t o someextent , red is t r ibuted .

8 C O N N E C T I O N S W I TH S E V E R A L B O LT S PE R R O W

The wide - f l anged H-se r i e s o f s ec t ions o ffe r s t he poss ib i l i t y o f p l ac ing twob o l t s p e r r o w o n e a c h s i d e o f t h e w e b a n d t h i s ty p e o f c o n n e c t i o n is c o n s i d e r e dn e x t . I n F ig . 1 7 t h e b e a r i n g c a p a c i t y o f a c o n n e c t i o n fo r a n H E B 3 0 0 w i t h a nex tended endp la t e des igned accord ing to eqns (23 ) , (24 ) and (25 ) i s p lo t t edas a func t ion o f i nc reas ing bo l t fo rce capac i ty. I t i s s een tha t t he des ign p ro -cedure s l igh t ly ove res t ima tes the bea r ing capac i ty and tha t t he r e se rves inbea r ing capac i ty in t e rms o f i nc reas ing bo l t fo rce capac i ty a re l e s s than fo rthe conn ec t io n wi th on ly on e bo l t pe r row, a s seen in F igs 10 and 16.

I t i s p robab ly no t poss ib l e by s imple y i e ld l ine pa t t e rns to de r ive a s imple

fo rm ula tha t re f lec t s a be tt e r app rox im a t ion fo r th i s t ype o f conne c t ion . The re -fo re , a s a s imple way o f inc reas ing the bea r ing capac i ty, t he au thor sugges t sapp l i ca t ion o f t he des ign fo rmulae o f eqns (23 ) , (24 ) and (25 ) , i nc reas ing thee n d p l a t e th i c k n e s s o f e q n ( 2 3 ) b y 5 . T h i s h a s b e e n t e st e d o n s e v e r a l c o n n e c -t ions and l eads to sa fe des ign . Th e e ff ec t on the bea r ing capac i ty o f i nc reas ingthe endp la t e th i ckness i s s een in F ig . 17 .

I t m u s t b e e m p h a s i z e d t h a t n o r e d u c t i o n i n e n d p l a t e s t r e n g th d u e t o t h e b o l tho le s has been in t roduced . Th i s i s j u s t i f i ed by the ex ce l l en t ag reem en t fo r the

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

J n e n ~ K I b y ~ x q i i ii i i

. . . . ~ - . . . . [ - . . . . . . . . . . . . . . . . .

t i I i i. . . . L . . . . L . . . . . . . L . . . . L . . . . I

, . . ~ .~ . , , , , , , , , ,i i t t t

1 . 2 0 . . . . . . . L . . . . L . . . . L . . . . k . . . . .t ~ i t

L : ~ t iO IO 1 . 1 0 . . . . L . . . . U . . . . . . . . L . . . . I

i t i t

i J ~ t t

1 . 0 0 . - - ' - - - - ' - - - - ' - - - - ' - - - - . - - - -i i = =i i = ~ iI i i = J

J i ii

o . , o . , .~ o ~ .~ o , .~ o ~ .~ o ~ . ~ o , .~ oo l t f o r c e f a c t o r

Y l e t o i l i n e p Q t e r n s F o r v o r y l n g In e rn ~ L o t

Force c a p a c i i e s u s i n g tp of eq,(23),

1 0 0 PB, n 1,50 PB,]n't 2,00 PB, n 4,00 PB,In

F i g 17. Endplate w ith four bo lts per row . The bearing capacity as a function of increase ininternal bolt force capac ity and correspon ding yieldline patterns.


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21/22


A pre requ i s i te fo r app ly ing these des ign fo rm ulae is tha t the oppos ing end-p la te has the same th ickness a s the one cons ide red and tha t the compress ionf lange i s r ig id ly suppor ted in i t s ent i re length .

U n s t i f f e n e d b e a m / c o l u m n c o n n e c t i o n s h a v e n o t b e e n c o n s i d e r e d i n d e p t hin th i s paper. I f the cond i t ions ab ove a re sa ti sf ied fo r a beam /co lum n conn ec-t ion , i .e . the column f lange th ickness i s not less than the endpla te th icknessand a co lumn f l ange s t i f f ene r i s p rov ided in f ron t o f the compress ion f l angeof the beam, the des ign fo rm ulae a re app l icab le . Supp lemen ta ry inves t iga t ionsm u s t n a tu r a ll y b e d o n e f o r th e c o l u m n w e b . H o w e v e r, a p p l y in g t h e n u m e r i c alapproach desc r ibed in sec t ion 2 , supp lemented wi th a s imi la r approach fo rt h e i n - p la n e f o r c e s, u n s t i ff e n e d b e a m / c o l u m n c o n n e c t io n s c o u l d b e t h o r o u g h l y

inves t iga ted based o n the idea l -p last ic i ty theory. The au thor has un success fu l lyapp l i ed fo r funds to ca r ry ou t th i s work .End p la te th ickness and bo l t fo rce capac i ty a re on ly two o f seve ra l imp or tan t

fac to r s invo lved in a p roper endp la te des ign . Two o the r f ac to r s wi l l be d i s -c u s s e d h e r e , r e q u i r em e n t s f o r t h e w e l d i n g a n d r e q u i re m e n t s r e g a r d i n g t h r o u g h -th ickness effec ts .

Th e d es ign o f endp la te th ickness an d bo l t fo rce capac i ty is based o n a s t re s sd i s t ribu t ion in the sec t ion cor resp ond ing to the N av ie r-B erno u i l l i d is t ribu tion .The endp la te i s cons ide red to be s imply suppor ted a t the f l anges . Thus , the

welds mus t r e s i s t on ly the d i rec t s t r e s ses a r i s ing f rom the ex te rna l fo rces .U s ing the l inea r-e las ti c s tr e ss d i s tr ibu t ion in the des ign o f the w e lds , fu ll com -pa t ib i l i ty wi th the endp la te des ign i s ach ieved .

In order to t ransfer the tens i le s t resses of the sec t ion to the bol ts , la rgeperpend icu la r t ens i l e s t r e s ses a r i se in por t ions o f the endp la te nex t to thewelds . Th ere fo re the endp la te m us t be chec ked fo r su ff i c ien t s t reng th in d i rec -t ions pe rpend icu la r to the endp la te .

1 0 R O TAT I O N A L S T I F F N E S S

The ro ta t ional s t i ffness of the connect ion a t the u l t imate l imi t s ta te i s eas i lyde te rm ined . A dhe r ing to the des ign fo rm ulae o f eqns 30) , 31 ) , 32 ) and 33) ,y ie ld ing o f the bo l t s and y ie ld ing o f the endp la te a re equa l ly l ike ly to occurfo r f lush endp la tes . The s t ra in ene rgy eva lua ted f rom a de fo rm ed endp la te ands t i ff bol ts i s equal to tha t o f a s t ra ight endpla te an d f lexib le bol ts. F or ex tend edendp la tes the s t r a in ene rgy can be eva lua ted pa r t ly f rom the de fo rmat ions o f

the in te rna l bo l t s and pa r t ly f rom the de fo rmat ion o f the ex tended pa r t o f theendp la te . Thus , de te rm in ing the s t r ain ene rg y o f the ex tended pa r t o f the end-p la te f rom s imple beam bend ing p roper t i e s , the ro ta t iona l s t i f fness S a t theul t imate l imi t s ta te can be evaluated as fo l lows:


22/22

140 P. C. Olsen

a i= e + - +a tp

ntern

34)

in which the s t r a ins o f the bo l t s a re based on a l eng th co r respond ing to theendp la te th ickness A i a r e a of bo lt s) . In o rde r to eva lua te eqn 34) in t e rmsof the c l a s s if i ca tion ru les in E C3 [4] , in w hich the s ti ffness o f the con nec t ionis related to the st i ffnessEI / l of the conne c ted beam , eqn 34) is rewr i t t en as:

l ~ ~ ) m ) ~S = ~ p --PP a E a i h 2

n te rn

35)

w here I and l a re the m om ent o f ine r t ia and the l eng th o f the beam, r e spec t -ive ly. Examin ing th i s express ion , fo r example fo r the IPE-se r i e s o f sec t ions ,ind ica tes tha t the conn ec t ions accord ing to EC3 can be c ons ide red as rig id fo rnormal ly app l i ed beam leng ths a s we l l a s fo r b raced as fo r unbraced f rames .

A C K N O W L E D G E M E N T S

This s tudy was ca r r i ed ou t in o rde r to be implemented in a PC s tee l des ign

p r o g r a m m e t o e x p e d i t e t h e d a i l y r o u t i n e w o r k a n d o p t i m i z e t h e d e s i g n o fb o l t ed e n d p l a te s . T h i s p r o g r a m m e h a s b e e n i m p l e m e n t e d b y s e v e r a l D a n i s hs tee l manufac tu r ing compan ies , one o f which i s A-L S th l A/S o f Sk je rn , oneof the l a rges t s t ee l manufac tu re r s in Denmark , p roduc ing s t ee l f r ames andload bear ing s tee l s t ruc tures for the bui ld ing t rade , who suppor ted th is s tudyf inanc ia l ly. The au thor g ra te fu lly ack now ledges th is suppor t.

R E F E R E N C E S

1. Schineis, M ., Vereinfachte Ber echn un g geschraubter R ahmenecken.D er Bauing-enieur, 196 9, 44, 12.

2. Grundy, P., Thomas, I. R. and Bennetts, I. D., Beam-to-column m om ent con nec-tions. Journal of the Structural Division,1980 , 106 , 313-330.

3. Thom sen, K. and Agerskov, H.,Versuche zur Er m itdun g des Tragverhaltens vonKopfplattensti~flen in biegebeanspruchten gew alzten IP E u nd HEB -Profil-Tri~gern.D er Stahlbau, 197 3, H. 8.

4. Eu roc od e No. 3, Design o f Steel Structures, Part 1-1 : General Ru les and Rulesfor Buildings, April 1992.

5. Gebbeken, N., Binder, B. an d Rothert, H.,Zur numerischen Analyse yon

Kopfplatten-Verbindungen.D er Stahlbau, 199 2, H. 9.6. Anderheggen, E. and K n6ppel, H ., Finite elem ent limit analysis using linear pro-gramming. International J ournal o f Solids a nd Structures,1972, 8, 1413-1431.

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