tang

7/17/2019 Tang

http://slidepdf.com/reader/full/tang563db8a0550346aa9a956b85 1/15

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

7/17/2019 Tang


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)

7/17/2019 Tang


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 ~

7/17/2019 Tang


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 )

7/17/2019 Tang


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 )

7/17/2019 Tang


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

7/17/2019 Tang


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

7/17/2019 Tang


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

7/17/2019 Tang


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.

7/17/2019 Tang


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 )

7/17/2019 Tang


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 ,

7/17/2019 Tang


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

7/17/2019 Tang


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

7/17/2019 Tang


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 .

7/17/2019 Tang


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.)

tang

Documents