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J. C onstruct. Steel Resea rch
11 (1988) 41-55
B u c k l i n g o f L a t e ra l ly a n d T o r s io n a l ly B r a c e d B e a m s
Tong G eng-Shu Chen Shao-Fan
Xi'an Institute of M etallurgy and Construction Engineering, Xi'an, Shaanxi,
People's Repub lic of China
(Receiv ed 27 Octob er 1987; accepted 13 January 1988)
A B S T R A C T
T h i s p a p e r r e p o r ts o n a s t u d y o f t h e b u c k l i n g b e h a v i o u r o f s i m p l y s u p p o r t e d
b e a m s u n d e r u n i f o r m m o m e n t . T h e b e a m s a r e b r a c ed la t er a ll y a n d t o r s io n -
a l ly a t m i d s p a n a n d c a n b e d o u b l y - s y m m e t r i c a l o r m o n o s y m m e t r i c a l . A
c losed - form so lu t ion i s ob ta ined the r e la t ions be tween buc k l ing mom en ts
an d bra c ing s t i f fnesses are g iven a nd form ula e fo r the cr i t ica l s ti~]hesses
r e q u i r e d f o r f u l l b r a c in g a r e p r e se n t ed .
1 I N T R O D U C T I O N
T h e s t a b il i ty o f b e a m s c a n b e i m p r o v e d b y l a te r al a n d / o r t o r s io n a l r e s tr a in t s .
T h e s e r e s t r a in t s c a n b e e i t h e r s e c o n d a r y m e m b e r s i n s t r u c t u re s o r b r a ci n g s
i n t e n t i o n a l l y s e t u p t o r e i n f o r c e th e m a i n b e a m s .
M a n y r e s e a r c h e r s h a v e i n v e s t i g at e d t h e b u c k li n g o f b e a m s c o n s t r a i n e d a t
m i d s p a n . ~ T a y l o r a n d O j a l v o 2 s t u d i e d t o rs i o n al ly b r a c e d b e a m s u s in g
n u m e r i c a l i n t e g r a t i o n f o r t h r e e d i f f e re n t l o a d in g c a s e s; N e t h e r c o t 3 a n a l y s e d
l a t e ra l l y o r t o r s i o n a ll y r e s t r a in e d b e a m s b y F E M f o r th r e e l o a d in g c a se s a n d
t h r e e l o a d l e v e ls ; M u t t o n a n d T r a h a i r 4 d e a l t in a n a p p r o x i m a t e m a n n e r w i th
l a te r a l l y a n d t o rs i o na l ly b r a c e d b e a m s f o r b o t h u n i f o rm m o m e n t a n d c o n -
c e n t r a t e d l o a d a t m i d s p a n a p p l ie d a t t h e s h e a r c e n t r e l e v el ; M e d l a n d 5
i n v e s t i g a t e d t h e b u c k l i n g o f i n t e r b r a c e d b e a m s y s te m s .
I t s e e m s t h a t t h e r e s t i l l r e m a i n s m u c h t o b e s t u d i e d f o r b r a c e d b e a m s ,
e s p e c i a l l y f o r t h o s e b r a c e d b y c o m b i n e d l a t e r a l a n d t o r s i o n a l r e s t r a i n t s .
M o r e o v e r , a ll o f t h e a b o v e - m e n t i o n e d r e s e ar c h w a s f o cu s e d o n d o u b l y -
s y m m e t r i c b e a m s ; t o t h e w r it e rs ' k n o w l e d g e t h e r e a r e n o s o l u ti o n s av a i la b l e
41
J. Construct. Steel Research
0143-974X/88/ 03.50 O 1988 Elsevier Science Publishe rs Ltd ,
England. Printe d in Great Britain
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4 2 Tong Geng-Shu, Chen Shao-Fan
f o r m o n o s y m m e t r i c b e a m s . I n t he p r e s e n t p a p e r , c l o s e d - f o rm s o l u t io n s a re
p r e s e n t e d f o r m o n o s y m m e t r i c b e a m s w h i c h a r e b r a c e d l a te r al ly a n d / o r
t o r s i o n a l ly a t m i d s p a n a n d a r e s u b j e c t e d t o u n i fo r m m o m e n t .
2 C R I T I C A L E Q U A T I O N
A s s h o w n i n F i g. 1 , t h e m o n o s y m m e t r i c , s i m p l y s u p p o r t e d b e a m is
s u b j e c t e d t o e q u a l a n d o p p o s i te e n d m o m e n t s M , t h e b e a m is b r a c e d b o t h
l a t e ra l l y a n d t o r s io n a l l y a t m i d s p a n . W h e n M r e a c h e s a c e r t a in v a l u e , t h e
b e a m w i ll b u c k l e b y d e f l e c t i n g la t e r a ll y u a n d t w i s ti n g 0 a n d a r e a c t i v e f o r c e
J ; /
-q-
L
- I
( : , )
F i g 1 B u c k l i n g o f t h e b r a c e d b e a m
" ~ | e
S
0
4
-I
Y ( b )
F a n d a t o r s io n a l m o m e n t M z w i ll b e i n d u c e d in th e b r a c in g s . A s s u m i n g t h a t
t h e t r a n s l a t i o n a l b r a c e i s a c t i n g a t a d i s t a n c e e a b o v e t h e s h e a r c e n t r e
S ( 0 , Y 0 ), a n d t h a t a ll t h e q u a n t i t i e s s h o w n i n F i g. 1 a r e p o s i t i v e , t h e f o l l o w i n g
d i f f e r e n t ia l e q u a t i o n s c a n b e e s t a b l i s h e d f o r t h e b u c k l e d b e a m : 6
O < _ z< -l: E l y u + M O - ½ F z
= 0 ( l a )
E I,~ O ' - ( G J + [ 3x M )O ' + M u ' - ½ F e
-½Mz = 0 ( l b )
l< _ z< --2 1: E l y u + M O - ' . , _ F ( 2 1 - z ) = 0
E I,~ O . . . ( G J + [ 3x M )O ' + M u ' + ½ F e + M z = 0
w h e r e
I r = s eco n d m o m en t o f a r ea ab o u t y - ax is
I~ = wa rp ing cons t an t o f sec t i on
J = S t V ena n t to r s iona l cons t an t
[3,, = - ~ y ( x 2 + y 2 ) d A - 2 y 0
lx = s e c o n d m o m e n t o f a r e a a b o u t x- ax is
E , G = Y o u n g s mo d u l u s an d s h ea r mo d u l u s , re s p ec t iv e l y .
( l c )
~d)
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B u c k l i n g o f l a te r a ll y a n d t o rs io n a ll y b r a c ed b e a m s 43
D e n o t i n g t h e s o l u t i o n s u l , 0 1 f o r 0 _< z - l , u 2 , 0 2 f o r l -< z -< 2 1, f r o m e q n s
( l a ) , ( l b ) a n d ( l c ) , ( l d ) , t h e f o l l o w i n g is o b t a i n e d :
l k 4 F
o ~ v - f l 2 0 { ' - k 4 0 1 - - - - ~ - ~ Z ( 2 a )
012 - f 1 2 0 ~ ' - k 4 0 2 = - ~ k ' F ( 2 1 - z )
( 2 b )
w h e r e / 3 2 =
( G J + / 3 x M ) / E I . , k 4 = M 2 / ( E I y E I . ) .
L e t
r , = [~/ ( f14 + 4 k 4 ) + f l z ] u 2 / ~ / 2
r2 = [ V / / ~ 4 -{ - 4 k 4 ) -
f l Z ] l l 2 / V / 2
w e f i n d
1 F
01 = C 1 c h r l z + C 2 S h r l Z + C a s i n r z z + (?4 c o s r 2 z + ~ ~ z ( 3 a )
02 = D l C h r l z + D z s h r l z + D 3 s i n r 2 z + D 4 c o s r 2 z + ~ ( 2 / - z ) ( 3 b )
B e c a u s e 0 1 = 0 , 0 [ = 0 w h e n z = 0 , s o
C 1 = C 4 = 0
S u b s t i t u t in g e q n s ( 3 a ) a n d ( 3 b ) i n to e q n s ( l b ) a n d ( l d )
i n t e g r a t i n g o n c e , a n d b y u s i n g u l l ~ = 0 = 0 , u z ]~ = 2t = 0 , w e o b t a i n
u l = ~ e + f l~ + ---~ - z + - ~ z + f l x - - ~ -
O t h e r c o n d i t i o n s u s e d a r e :
z = 2 l : 02 = 0 , 0~ = 0 (6a )
z = l : ( 6 b )
( 4 )
r e s p e c t i v e l y ,
D4
~ ) ( D 3 s i n r 2 z +
c o s r 2 z ) ] ( 5 b )
0 1 = 0 2 , 0 ~ = 0 2 , U 1 ~ U 2 , U ~ = U ~
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44 Ton g Geng-Shu,
Ch e n Sh a o F a n
F r o m e q n s 3 ) , 5 ) a n d 6 ) , a ll t h e c o n s t a n t s C 2, C 3 , D 1 , D 2 , D 3 , D 4 c a n b e
d e t e r m i n e d , b u t i n t h e f o l l o w i n g o n l y C 2 a n d C 3 a r e u s e d :
M e M
r 2 _ _ _
F E I ~ 1 M z E I ~ 1 7a)
C2 = 2 M 2 r, r ]+ rZ 2 ) c h q ~ + 2 M r l ~ + ~ ) c h q l
M e M
~ - ~ - _ _
F E I ~ 1 M z E I ~ 1
C 3 = 2 M r 2 r Z+ r~) cosr21 2-M r 2 ~ + ~ ) c o s r2 / 7 b )
A s s u m i n g t h e s t i f f n e s s e s o f t h e l a t e r a l a n d t o r s i o n a l b r a c i n g s t o b e K a n d
K z
r e s p e c t i v e l y , t h e b u c k l i n g d e f o r m a t i o n s a t m i d s p a n t o b e
U z
a n d
Oz
a n d
d i s p l a c e m e n t a t la t e r a l b r a c i n g p o i n t B t o b e d , w e h a v e :
F = K d = K u z + e O z)
8a )
Mz = Kz Oz Sb)
K I
0~ = C2 sh rl I + C3 s n r21 + ~ uz + eOz)
9a)
, , , ( c . , ) K z,
u z = ~ -- ~ e + /3x + - - ~ - u z + e O z) + - ~ O z
c . , r ) c 2 s h r - , . - ( l + c 3 s i n r 2 , ]
9b)
S u b s t i t u t i o n o f e q n s 7 a ) t o 8 b ) i n t o e q n s 9 a ) a n d 9 b ) r e s u lt s i n t w o
h o m o g e n e o u s l i n e ar e q u a t io n s in
u z
a n d
Oz.
B e c a u s e
U z
a n d 0z a r e b u c k l i n g
d i s p l a c e m e n t s , f o r t h e s o l u t i o n t o e x i s t , t h e d e t e r m i n a n t o f t h e s e t w o
e q u a t i o n s s h o u l d b e z e r o . T h u s , w e o b t a i n , a ft er r e a r r a n g em e n t :
T W = Q V
10)
w h e r e
T = 1 - 9 -~ a l + [ 3 x + - - ~ - a 2 / e
i l a )
Kz
W = 1 - ~ - - ~ a 3 / / 3 x + - ~ -
U b )
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Bu ck l ing o f la te ra lly and to rsiona l ly b raced beams
45
K l G J
K z l l l d )
V = e + - ~ - ~ a l
th q l ~ tgr2 l
a 1 1 ~ + ~ q l ~ + ~ r21 1 2 a )
r 4 th rl I r] tgre l
a2 = 1 + / 3 2 ~ + ~ ) r~ l 3 e ~ + ~ ) r2 l 1 2 b )
, ~ e t h r l l
t g r 2 l )
a 3 - d + ~ . r l I re l 1 2 c)
3 R E S U L T S
3 1 C r i t i c a l s ti f fn e s s e s o f f u l l b r a c i n g s
F i r s t w e d e t e r m i n e t h e t h r e s h o l d v a l u e s o f s t if fn e s s es o f fu l l b r a c i n g s w h i c h
a r e j u s t s u f f i c i e n t t o m a k e t h e b e a m b u c k l e in t w o h a l f - w a v e s . I n t h is c a s e ,
w e h a v e
q l = 2 X / 4 + ~ 2 ¢ ) re l = r r 13a )
2 = ( G J + ~ .M c r 2 ) L e / ( n e E I , )Q
1 3 b )
4 r r eE I y [ _ ~ J ( 1 G J L 2
1 ~ ]
1 3 c)
M e r e - L 2 / 3 x /~ xx 4 7re- - - - - - Z y - ~ - y
B y u s i n g
t g r e l
= 0 a n d t h
r l l
= 1 .0 , t h e c r i ti c a l e q u a t i o n e q n 1 0 ) ) i s
s i m p l i f i e d in t o :
T c W e - Q c V c = 0 1 4 )
w h e r e
4Mere ale + + M ~ e h a 2 e 1 5 a )
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46 Tong Geng-Shu, Chen Shao-Fan
W e = 1 - - T K z e , a3~
4
K c ,L [ a l C + h a 3 C l ~ + G J
Q c
4Mcr2
M---c-~h~]
, ° , )
Vc = e + ~ K zc ra lc ~ x + - ~ 2
K z cr = K ~ c r L / ( G J +
B x M c r z )
at~ = 1 - 8 / [ w ( 8 + ~ ) x / ( 4 + a ~ ) ]
a2~
= 1 + 3 2 / [ ~ ( 8 + a ~ ) ~ / ( 4 + ~ ) ]
a3~ = 2O tZc/[T r(8 + O ZZc)X/(4+ oe2~)]
( 1 5 b )
0 5 c )
( 1 5 d )
16)
OVa)
( 1 7 b )
( 1 7 c )
G J 2~ / I~ /2 ) oe2¢
/ 3 ~ + _ 1 8 )
h M = 2h 1 1 + 1 2 4 x / 4 + oe 2¢ )
i n w h i c h 11 a n d 12 a r e s e c o n d m o m e n t s o f a re a a b o u t t h e y - a x is o f t h e
c o m p r e s s i o n a n d t e n s i o n f la n g e s , r es p e c t i v e l y .
3 1 1 Torsiona l bracing
Th e c r i ti c a l s t i f f n e s s i n th i s c a s e i s d e n o t e d b y K z , 0 , f r o m e q n ( 1 4 )
K ~ c~ 0 27 r (8 + oeZc) V / (4+ eez¢)
--
2 1 9 )
~ c
I n F a b l e 1 r e s u l t s fr o m e q n ( 1 9 ) ar e c o m p a r e d w i t h t h o s e o b t a i n e d b y
N e t h e r c o t 3 b y F E M f o r d o u b l y - s y m m e t r i c b e a m s ( /3 ~ = 0 ) ; it i s s e e n t h a t t h e
r e s u l t s o f t h e l a t te r a r e o n t h e u n s a f e s i d e i f a c > 1 .4 a s c o m p a r e d w i t h e q n
( 1 9 ) . A n o t h e r p o i n t w h i c h s h o u l d b e n o t e d i s t h a t t h e d i f f e r e n c e i n r e su l t s
TABLE
Cr i t ica l S t i f fn esse s Kzc~0
c~c 0-5 0.6 2 1.23 1.52 1.82 2.13 3-03 5.0
R ef . 3 297 93 67 53 46 35 26
Eqn
(19 ) 427 . 45 289 . 33 92 -30 70 . 71 58 . 10 50 . 78 42 . 70 44 . 67
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B uc kl in g o f la terally and tors ionally braced beam s
47
b e t w e e n m o n o s y m m e t r i c a n d d o u b l y - sy m m e t r i c s e c t io n b e a m s c a n b e
e l i m i n a t e d b y i n t r o d u c in g f l x M , 2 i n t o a c a s d e f i n e d in e q n 1 3 b ) .
3 . 1 . 2 T r a n s l a t i o n a l r e s tr a i n t
T h e c r i t i c a l s t i ff n e s s is d e n o t e d b y K ~0 b e c a u s e
M¢ 2
rr2Ely
2 -, 2 ~ ( 1 1 h )
20)
w e o b t a i n
, 2 , ,2 N /( 1 1 2 )
2 1 )
i n w h i c h
f ~o = K~oL3 / E[y 22)
x = [ 2 h a , ¢ + (_ _
G J 2 I
+ ~ r 2 h ) a 2 c + h ) a 3 c / ( ~ - ~ + M ~ , z h ] ] G J~
23)
I n T a b l e 2 r e s u l t s f o r e / h = 0 f r o m e q n 2 1) a r e c o m p a r e d w i t h t h o s e
o b t a i n e d b y N e t h e r c o t 3 f o r d o u b l y - s y m m e t r i c b e a m s fix = 0 ) , it c a n b e s e e n
t h a t t h e t w o s e ts o f r e s u lt s a r e i n e x c e l l e n t a g r e e m e n t . N o t e s h o u l d b e t a k e n
o f t h e i n d e p e n d e n c e b e t w e e n K c,~ a n d 11/12, s in ce , i f e / h = 0 , Kc~ ca n b e
e x p r e s s e d a s
16zr3 8 + az¢) 4 + az¢)3/2
K c ~ o = 3 2 +
r c a ~
8 + 2 2
o , , ) X / 4 + ~ c )
24)
I f t h e t r a n s l a t i o n a l b r a c i n g is a c t in g o n t h e c o m p r e s s i v e f l a n ge , t h e n
e / h = 1 2/ 1 1 +
I2 ). I n T a b l e 3 , r e s u l t s f o r
11/12
= 1-0 to oc an d ac = 1 .0 to 8 .0
a r e l i s te d f o r t h i s c a s e . T h e r e s u l t s f o r l d I : = ~ a r e c a l c u l a t e d f r o m e q n 2 4 ),
a n d , c o r r e s p o n d t o T - s e c t i o n b e a m s . I t is s e e n t h a t b o t h oLc a n d 11/12 h a v e
s i g n i f i c a n t e f f e c t s o n ko~0.
I f t h e t r a n s l a t i o n a l b r a c i n g i s p r o v i d e d b y a la t e r a l l y s t i ff e r b r a c i n g b e a m
TABLE 2
Critical Stiffnesses Kcr0 for
e/h = O)
ac 0.5 0.6164 1-2328 1.526 6 1.828 5 2.135 3 3-07 4-00 4.91
Ref. 3 744 436-8 364-8 312.0 278.4 220-8 187.2
Eqn 24) 790-60 719.92 441.30 367.20 315.57 279.46 221.19 196.23 183.66
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4 8
TongG e n g - S h u , C h e n S h a o -F a n
T A B L E 3
C r i t i c a l S t i f f n e s s e s K c ~ f o r e /h = 12 / 11 + 12 ))
I t~12
O c
1 2 3 4 5 6 7 8
1 7 9 . 3 4 8 2 . 2 2 8 7 . 2 5 9 2 . 8 0 9 8 - 0 5 1 0 2 - 7 6 1 0 6 . 9 0 1 1 I .5 3
2 1 0 7 .4 7 1 0 5 .2 3 1 0 6 .8 6 1 1 0 .1 2 1 1 3 .7 0 1 1 7 .0 9 1 2 0 .1 6 1 2 2 .8 8
4 1 4 1 . 7 4 1 3 0 . 4 7 1 2 6 . 6 2 1 2 6 . 6 l 1 2 8 . 0 0 1 2 9 .8 1 1 3 1 . 6 4 1 3 3 . 3 7
8 1 8 1 - 6 2 1 5 6 .4 5 1 4 5 .3 8 1 4 1 .4 5 1 4 0 .4 1 1 4 0 .5 4 1 4 l . 1 4 1 4 1 .9 2
x 5 2 4 .2 4 2 9 3 .7 9 2 2 3 .9 6 1 9 6 - 2 3 1 8 2 - 7 6 1 7 5 .2 8 1 7 t -7 2 1 6 7 .7 4
w h i c h i s p a r a l l e l t o a n d h a s t h e s a m e l e n g t h a s t h e r e s t r a i n e d b e a m , t h e
r e q u i r e d b e n d i n g s t i ff n e ss
E I b
o f t h e b r a ci ng b e a m s h o u l d b e
E l b = K c r o E l y / 4 8
b e c a u s e
l y =
11 + Iz = 11 1
I 2 /I t ,
w e o b t a in
E lb = k . b E l l ) 25a)
K c r b = 1 + 1 2 / I L ) K c ~ o / 4 8
2 5 b )
g crb v a l u e s a r e g i v e n i n T a b l e 4 . T o n g 7 h a s s h o w n t h a t , i n o r d e r t o r e d u c e t h e
e f f e c t iv e l e n g t h o f a s im p l y s u p p o r t e d c o l u m n b y h a l f, t h e r e q u ir e d b e n d i n g
s t i f fn e s s o f t h e b r a c i n g b e a m s h o u l d b e 3 . 2 9 t im e s th a t o f t h e c o l u m n i f t h e
b r a c i n g b e a m h a s t h e s a m e l e n g t h a s t h e c o l u m n ) . I n c u r r e n t d e s i g n p r a c t ic e ,
t h e c o m p r e s s i v e r e g i o n o f a b e a m is o f t e n c o n s i d e r e d a s a c o l u m n a n d t h e
b r a c i n g s y s t e m i s d e s i g n e d a s i f i t b r a c e s a c o l u m n c o m p r e s s e d b y a n a x i a l
T A B L E 4
Cr iticalB endingStiff nessesK~bfor e/h = 1 2 / 1 1 + 1 2 ) )
I t~12
O~c
1 2 3 4 5 6 7 8
1 3 . 3 0 6 3 . 4 2 6 3 . 6 3 5 3 . 8 6 7 4 . 0 8 6 4 . 2 8 2 4 - 4 5 4 4 . 6 0 5
2 3 . 3 5 9 3 . 2 9 0 3 - 3 3 9 3 . 4 4 1 3 . 5 5 3 3 . 6 5 9 3 . 7 5 5 3 . 8 4 0
4 3 . 6 9 1 3 . 3 9 8 3 . 2 9 7 3 - 2 9 7 3 . 3 3 3 3 . 3 8 0 3 . 4 2 8 3 - 4 7 3
8 4 . 2 5 7 3 . 6 6 7 3 - 4 0 7 3 . 3 1 5 3 . 2 9 1 3 - 2 9 4 3 . 3 0 8 3 . 3 2 6
zc 1 0 . 9 2 2 6 . 1 2 1 4 . 6 6 6 4 . 0 8 8 3 . 8 0 8 3 - 6 5 2 3 . 5 5 7 3 . 4 9 5
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Bu ck l ing o f la t era ll y and to rs iona l l y b raced beam s 49
f o r c e N =
M , 2 / h .
B u t , a s c a n b e s e e n i n T a b l e 4 , 3- 29 is o n l y a l o w e r b o u n d
o f t h e r e q u i r e d v a l u e s , s o t h e c u r r e n t c o n c e p t i s n o t n e c e s s a r i l y o n t h e s a f e
s i d e .
3 . 1 . 3 T r a n s l a t io n a l a n d to r s io n a l b r a c in g
I n t h i s c a s e , e q n ( 1 4 ) c a n b e f o r m u l a t e d a s f o ll o w s :
g c r g z c r
D - ¢ ~ g z c r
F 4 -
= 1 . 0 ( 2 6 )
K=0 K=~0 K=0 K =,0
w h e r e
D
= 4 b l / b 2 a 3 c
(27)
2 r r N / 4 + O e 2c ) - 2 ) / [ 6 4 - , 7 - 3 4 + o t2 ~ ) / 2 ]
bl = ac
e 11 12 ale a2co~c
b2 = ~ 2X/(1112) 27r2~/(4 + azc ) 16~ 2(4+ a2c) + 41112 rr2a¢2
I f e / h = 0 , D i s s im p l i f i ed i n to :
¢r ~ /( 4 + c~2¢)(rrx/(4 + a 2) - 2) (8 + a2c)2
D = 217rc~:¢ 8 + a2 ¢)x /(4 + a~ ) + 32 ] (28 )
T h e i n t e r a c t i v e r e l a t i o n g i v e n b y e q n s ( 2 6 ) a n d ( 2 8) is s h o w n i n F i g . 2 a ; t h e
r e s u lt s w h e n
e / h = I 2 / l l + / 2
a r e s h o w n i n F i g. 2b . W h e n
11/12
is i n c r e a s e d ,
t h e c u r v e s w i ll b e m o r e c o n c a v e t o w a r d t h e o r i g i n . I t is d i ff i c u lt t o f i n d a
1 . 0 [ K c r l K c r o
0.5 ~
a c = 4
1
Kzc r
1-0
a )
1 . 0
0 . 5
c r l K c r o
~ R z c r
O . S 1 . 0
a )
Fig. 2. Interactive eactions or criticalstiffnesses.
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5 0 T o n g G e n g - S h u , C h e n S h a o - F a n
s u i t a b l e e x p r e s s i o n t o a p p r o x i m a t e D i n e q n ( 27 ) , b u t i f I~/ I2 = 1 D c a n b e
a p p r o x i m a t e d v e r y a c c u r a te l y by :
D : 2 . 1 4 + a ~ / 8 ( 29 )
I t is a l w a y s o n t h e s a f e s i de t o u s e e q n ( 29 ) w h e n 11/12 > 1 .0 .
3 . 2 I n c o m p l e t e r e s tr a i n ts
I f t h e b r a c i n g is n o t s t if f e n o u g h t o m a k e t h e b e a m b u c k l e w i t h a h a lf - w a v e
l e n g t h e q u a l t o = L / 2 , t h e n i t is c a l le d i n c o m p l e t e , a n d t h e c r i t ic a l m o m e n t
M w i l l b e w i t h i n t h e r a n g e : M c~ l ~ M < M , 2 w i t h
r r E l y
[ l
I ( I G J L
I , , , ~ ]
M o . . - L 2 g f l x f l
7 r E l y l y ]
30)
I n t h is c a s e , w e a r e i n t e r e s t e d i n t h e r e l a t io n s h i p b e t w e e n t h e c r it ic a l
m o m e n t a n d t h e s t if fn e s s o f th e b r a c in g .
D e n o t i n g a 2 = B2L2/rrz , 0~] = GJL2/ rr2EI ,~) , w e h a v e
r , t=T- ,
(31a)
= O~ + + 0~2c) _ a2 (3 1b )
r21 2----X-~ t 1 6 ~ - z z ( 4
2 = 0 1 21 q _ 8 a] 8 4 1 1 I 2
ac 4 1 1 I 2 +
]
1 1 - t- / 2 ) 2 4 3 2 a )
4 I a i 2 a l
a2 = a ¢ [ 1 - h "- ~ M c r 2 h } ]
(32b )
a n d e q n ( 1 0) c a n b e r e f o r m u l a t e d a s fo l lo w s :
2 - - - -
a cK zc~ o K ,,o [ M cr 2 \ 2 _ ~ , ~
64 zr2 (4+ a~ ) ~ ' - -M -- ] ' " ' a2a3) -~ --~- -K,=0 Kc , +
2 - - _ . g z
+ ° t c a 3 K z c r ° ~ = 1 0
4 a 2 Kz¢~0
3 ' K c , o K ~
(33 )
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Buck l ing o f la tera ll y and tors iona l ly braced beam s 51
_
v -
t [ w -
b
b
L- -I
i - - A t
h t 1 5 b
]
Fig. 3. Sections of the beam s.
w h e r e
4 z 2 X / l l I : ) M ) z [ a 2 a 2 2%/(1112)
Y = ~ 1 1 ÷ I 2 ~ / [ 4 ( 4 + a 2 ~ ) 1 1 + 1 2
e M 2a l
h Mc~2 %/(4+a2~)
I n t h e f o l l o w i n g o n l y s e c t i o n I a n d s e c t i o n I I s h o w n i n F i g . 3 a r e a n a l y s e d .
F o r s e c t i o n I ,
[3x /h
= 0 ,
l ~ / I z
= 1 , a n d f o r s e c t i o n I I ,
[3x /h
= 0 . 7 1 5 6 3 ,
1 1 / 1 2 = 8 .
3 . 2 . 1 T o r s i o n a l r e s tr a i n t
T h e c r i ti c a l m o m e n t is d e n o t e d b y M zc, i n t h is c a se a n d e q n ( 33 ) is s i m p l i fi e d
i n t o :
Kz 4c~2
K z c ~ 2
a 3 0 t c
K z ,o
(34)
R e s u l t s f o r s e c t i o n I a n d s e c t i o n I I a r e s h o w n i n F i g s 4 a a n d 4 b , r e s p e c t i v e l y .
F o r s e c t i o n I , N e t h e r c o t 3 s u g g e s t e d t h e f o l lo w i n g fo r m u l a t o a p p r o x i m a t e
t h e r e l a t i o n b e t w e e n t h e b u c k l i n g m o m e n t a n d t h e st if fn e ss o f t h e r e s tr a in t :
Mcr------ - Mcr------ Jr- 1 - - M c r 2 ~ 3 5 )
w h i c h is a l so s h o w n in F ig . 4 a in d o t t e d - a n d - d a s h e d l in e s. A s c a n b e s e e n ,
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52 Tong Geng-Shu Chen Shao-Fan
1-C
0,5
M z c r 1 0
Met2
/ ~ Section I 0.5
~ : o . ~ , E qn 3 4 )
. . . . .
Eqn (36)
. . . . . Eqn (35)
_ M z c r
M c r ~ f
. ~ / ~ 0 . 1 : 4 ' 0
~l~ 0.1 0 '5 Se ction R
Eqn (34)
. . . . . Eqn (37)
R R
zcrO Rzcro
I I
0 5 1'0 0 5 1.0
(a) (b)
Fig. 4. Re lationship between criticalmo men ts and bracing stiffnesses K = 0).
e q n 3 5 ) is a l i tt l e o p t i m i s t i c w h e n
K~/Kz~
is s m a l l ; a m o r e a p p r o p r i a t e
f o r m u l a m i g h t b e
M:cr Mcrl
Mcr2 Mcr2
- - + 1 M . 1
[ 1 + 0 . 4 f i - z - 0 . 4 K _ _ ~ z ) z ] 3 6 )
w h i c h i s a l s o g i v e n i n F i g . 4 a i n d o t t e d l in e s . F o r s e c t i o n I I , t h e c u r v e s c a n b e
a p p r o x i m a t e d b y :
z c rc r 1 ( 1 _ _ c r a ) ( ) O 4 3 5 M c r 2
[ 1 + 0 . 4 K__.~__z 0 . 4 ~ z / 2 ]
K~ 0 \K~ ,0 l
37)
3 2 2 Translational restraint
E q u a t i o n 3 3 ) i s s i m p l i fi e d i n t o
g/Kc~o = y/Kc~o
38)
w h e n
e/h
= 0 a n d r e s u l ts f o r s e c t io n I a r e s h o w n i n F ig . 5 a . T h e f o l l o w i n g
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B u c k l i n g o f l a t er a ll y a n d t o r si o n a ll y b r a c e d b e a m s 53
1 .0
0 . 5
•
S e c t i o n I 0 . 5
_ o o
. M
• ~ .0
R R
K cr O K c r o
J l I
0 5 1 0 0 5 1 ' 0
a ) b )
F i g . 5. R e l a t i o n s h i p b e t w e e n c r it ic a l m o m e n t s a n d b r a c in g s t if f n e s se s K z = 0 ,
e / h = 0 .
1. ]
0 .5
. t w 0
,~c
0-5
e I
~_
~ 'M c r 2
S e c t i o n ] I
e . I i~
h 1 1 * I 2
R _ g
K c r o K c r o
I I I
0 . 5 1 .0 0 . 5 1 . 0
a ) b )
F i g . 6 . R e l a t i o n s h i p b e t w e e n c r i t ic a l m o m e n t s a n d b r a c in g s t if f n e s se s K z = 0 , e / h = 1 2 /
11 + 12)).
f o r m u l a s u g g e s t e d b y N e t h e r c o t 3 is i n e x c e l le n t a g r e e m e n t w i t h t h e a c c u r a t e
r e s u l t s :
M Mcrl 1 + (39)
Mcr2 M ce 48 + 0 02K
R e s u l t s f o r s e c t i o n I I a r e s h o w n i n F i g. 5 b .
R e s u l t s w h e n e / h = I 2 / I 1 + 1 2) a r e s h o w n i n F i g . 6 a a n d F i g . 6 b ; a l i n e a r
r e l a t i o n f o r t h i s c a s e i s s u f f ic i e n t ly a c c u r a t e , s o
M M=~ (
M crl ) g
- - - ~- 1 - 4 0 )
Mot2 2 EE
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54 To ng Geng-Shu. Chen Shao-Fan
1 0
0 5
• M
I . C
~ - - ~ \ / ~ = 0-2
• ]
~ . o.o
c r
I I
0 5 1 0
a )
_~_M
Mcr2
Sect ion g
e
= 0 . 0
Kz/RzcrO
0 1
Rcr
0.5 1 .0
b )
Fig. 7. Relationship between c ritical mo men ts and bracing stiffnesses e /h = 0 ) .
1'0
0 5
- M
M c r 2
• I 2
h = I 1 . I 2
Kz/KzcrO 0'2
_ M
°5 2
-h = f 1 I 2
~ z l ~ z c r O = 0 I
r l I
0.5 1.0 0-5
a ) b )
1 ' 0
_
~ , c r
i
1 0
Fig. 8. Relationship between critical mo me nts and bracing stiffnesses
e/h = 12/ 11 + 12)).
3 . 2 . 3 T r a n s l a t i o n a l a n d to r s io n a l b r a c in g
E q u a t i o n 3 3 ) s h o u l d b e u s e d i n t hi s g e n e r a l c as e . F o r a g i v en s e c ti o n , I~/12
a n d [ 3 x / h a r e k n o w n , i f a~ is s p e c if ie d , a c c a n b e c o m p u t e d b y e q n 3 2 a ) , a n d
a b y e q n 3 2 b ) ; o n th e o t h e r h a n d , D i s a l so d e t e r m i n e d , s o t h e in t e r a c ti v e
r e l a t i o n s h i p o f Kc, /Kc~o a n d K ~ / K ~ , 0 is k n o w n . L e t K ~ / K ~ o = K ~ , / K ~ c ~ o ,
t a k e a c e r t a i n v a l u e , t h e n K ~ r / K ~ is f o u n d f r o m e q n 2 6 ). L e t K / K , t a k e a
v a l u e b e t w e e n 0 . 0 a n d 1 .0 , t h e n M / M ~ r 2 c a n b e f o u n d f ro m e q n 3 3 ) b y
i t e r a t i o n . I f K / K , = O , M / M , z = M , J M ~ a , i f K / K , = 1 , M / M , 2 = 1 .0 .
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uck l ing o f la terally and tors ional ly braced beam s 55
R e s u l t s f o r e / h = 0 a n d e /h = 12/ 11 + 12) a r e s h o w n i n F i g . 7 a n d F i g . 8
r e s p e c t i v e ly ; t h e y c a n b e a p p r o x i m a t e d b y:
Mcr2 Met2 Mcr ] Kcr
(41)
M z , is g i v e n i n e q n s ( 3 6 ) a n d ( 3 7) .
4 C O N C L U S I O N S
T h i s p a p e r s t ud i e s t he b u c k li n g b e h a v i o u r o f s i m p ly s u p p o r t e d b e a m s u n d e r
u n i f o r m m o m e n t s . T h e b e a m s a r e b r a c e d l a t e r a l l y a n d t o r s i o n a l l y a t
m i d s p a n , a n d c a n b e d o u b l y - sy m m e t ri c al o r m o n o s y m m e t r ic a l . C l o s e d-
f o r m s o l u t i o n s a r e o b t a i n e d , t h e r e l at io n s b e t w e e n t h e b u c k l in g m o m e n t s
a n d t h e b r a c i n g s t if f n e s se s a r e g i v e n a n d f o r m u l a e f o r t h e c r it ic a l s t if f n e s se s
o f t h e f u l l b r a c i n g a r e p r e s e n t e d .
R E F E R E N C E S
1. Tra ha i r , N . S . Ne therco t , D . A . , Brac ing requ i re m ents in th in -wal l ed
s t ruc tu res .
Dev elopmen ts in Thin-wa lled Structures--2
(Rh o d es , J . W a l k e r ,
A . C . ( ed s ) ), E l s ev i e r A p p l i ed Sci en ce Pu b l is h e rs L t d , L o n d o n , 1 9 8 4 , p p .
93-130 .
2 . T ay lo r , A . C. Oja lvo , M . , Tors iona l res t ra in t o f l a te ra l buck l ing .
J. Struct.
D i v . , A S C E ,
92(ST2) (A pril 1966) 115-29.
3 . N ethe rco t , D . A . , Bu ck l ing o f l at e ra l ly o r to rs iona lly res t ra ined beam s . J.
En g n g Mech . D iv . , AS C E,
99(EM 4) (A ugu st 1973) 773-91.
4 . M ut to n , B. R. Trah a i r , N . S . , S tif fness requ i rem ents fo r l a te ra l b rac ing . J .
Struc t . Div . , ASCE, 99(ST10) (O ctob er 1973) 2167-82.
5 . M ed l an d , I . C . , B u ck l i n g o f in t e rb raced b eam s y st ems .
Engineering Structures,
2 (A pr il 1980 ) 90--6.
6 . L u , L . W . , Sh en , S . Z . , Sh en , Z . Y . H u , X . R . , The Theory of Stability for
Steel Structural Members.
Ch ina Bui ld ing Indus t r i a l Pub l i sh ing Ho use , Bei j ing ,
1983 ( in Chine se) .
7 . T o n g G e n g -Sh u Ch en Sh ao -Fan , D es i g n ap p ro ach fo r mu l t ip l e l at e ra l
b r ac in g s o f a co l u m n . (T o a p p ea r i n
Proceedings o f the 1988 An nua l Technical
Sessions and Meeting o f the Structural Stability Research Council.)