copy of chapter11

8/21/2019 Copy of Chapter11

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UNRESTRAINED BEAM DESIGN-I

UNRESTRAINED BEAM DESIGN – I

1.0 INTRODUCTION

Generally, a beam resists transverse loads by bending action In a ty!ical b"ilding #rame,main beams are em!loyed to s!an bet$een ad%acent col"mns& secondary beams $'en

"sed ( transmit t'e #loor loading on to t'e main beams In general, it is necessary to

consider only t'e bending e##ects in s"c' cases, any torsional loading e##ects beingrelatively insigni#icant T'e main #orms o# res!onse to "ni-a)ial bending o# beams are

listed in Table *

Under increasing transverse loads, beams o# category * +Table* $o"ld attain t'eir #"ll

!lastic moment ca!acity T'is ty!e o# be'avio"r 'as been covered in an earlier c'a!ter

T$o im!ortant ass"m!tions 'ave been made t'erein to ac'ieve t'is ideal beam be'avio"r

T'ey are

♦ T'e com!ression #lange o# t'e beam is restrained #rom moving laterally, and

♦ Any #orm o# local b"c.ling is !revented

I# t'e laterally "nrestrained lengt' o# t'e com!ression #lange o# t'e beam is relativelylong as in category / o# Table *, t'en a !'enomenon, .no$n as lateral buckling or lateral

torsional buckling o# t'e beam may ta.e !lace T'e beam $o"ld #ail $ell be#ore it co"ld

attain its #"ll moment ca!acity T'is !'enomenon 'as a close similarity to t'e E"ler b"c.ling o# col"mns, triggering colla!se be#ore attaining its s0"as' load 1#"ll

com!ressive yield load2

3ateral b"c.ling o# beams 'as to be acco"nted #or at all stages o# constr"ction, to

eliminate t'e !ossibility o# !remat"re colla!se o# t'e str"ct"re or com!onent 4or e)am!le, in t'e constr"ction o# steel-concrete com!osite b"ildings, steel beams are

designed to attain t'eir #"ll moment ca!acity based on t'e ass"m!tion t'at t'e #looring$o"ld !rovide t'e necessary lateral restraint to t'e beams 5o$ever, d"ring t'e erection

stage o# t'e str"ct"re, beams may not receive as m"c' lateral s"!!ort #rom t'e #loors as

t'ey get a#ter t'e concrete 'ardens 5ence, at t'is stage, t'ey are !rone to lateral b"c.ling,$'ic' 'as to be conscio"sly !revented

Beams o# category 6 and 7 given in Table * #ail by local b"c.ling, $'ic' s'o"ld be

!revented by ade0"ate design meas"res, in order to ac'ieve t'eir ca!acities T'e met'odo# acco"nting #or t'e e##ects o# local b"c.ling on bending strengt' $as disc"ssed in an

earlier c'a!terIn t'is c'a!ter, t'e conce!t"al be'avio"r o# laterally "nrestrained beams is described in

detail 8ario"s #actors t'at in#l"ence t'e lateral b"c.ling be'avio"r o# a beam are

e)!lained T'e design !roced"re #or laterally "nrestrained beams is also incl"ded

9 :o!yrig't reserved

Version II 11-1

11


2/32


Table 1 Main failure modes of hot-rolled beams

Category Moe Co!!ents

* E)cessive bendingtriggering colla!se T'is is t'e basic #ail"re mode !rovided 1*2 t'e beam is !revented

#rom b"c.ling laterally,1/2 t'e

com!onent elements are at leastcom!act, so t'at t'ey do not b"c.le

locally S"c' ;stoc.y< beams $ill

colla!se by !lastic 'inge #ormation

/ 3ateral torsional

b"c.ling o# long beams $'ic' are

not s"itably braced

in t'e lateraldirection1ie ;"nrestrained< beams2

4ail"re occ"rs by a combination o#

lateral de#lection and t$ist T'e !ro!ortions o# t'e beam, s"!!ort

conditions and t'e $ay t'e load is

a!!lied are all #actors, $'ic' a##ect#ail"re by lateral torsional b"c.ling

6 4ail"re by local b"c.ling o# a

#lange in

com!ression or $eb d"e to s'ear

or $eb "nder

com!ression d"eto concentrated

loads

Unli.ely #or 'ot rolled sections,$'ic' are generally stoc.y

4abricated bo) sections may re0"ire

#lange sti##ening to !revent !remat"re colla!se

=eb sti##ening may be re0"ired #or

!late girders to !revent s'ear b"c.ling

3oad bearing sti##eners aresometimes needed "nder !oint

loads to resist $eb b"c.ling

7 3ocal #ail"re by1*2 s'ear yield o#

$eb 1/2 local

cr"s'ing o# $eb162 b"c.ling o#

t'in #langes

S'ear yield can only occ"r in verys'ort s!ans and s"itable $eb

sti##eners $ill 'ave to be designed

3ocal cr"s'ing is !ossible $'en

concentrated loads act on

"nsti##ened t'in $ebs S"itable

sti##eners can be designed

T'is is a !roblem only $'en very

$ide #langes are em!loyed=elding o# additional #lange !lates

$ill red"ce t'e !late b > t ratio and

t'"s #lange b"c.ling #ail"re can beavoided

Version II 11-"

Box section

Plate girder in shear

Plate girder in shear

W

Buckling of thin flanges

Crushing of web

Shear yield

W


3/32


".0 SIMI#ARIT$ O% CO#UMN BUCING AND #ATERA# BUCING

O% BEAMS

It is $ell .no$n t'at slender members "nder com!ression are !rone to instability ='en

slender str"ct"ral elements are loaded in t'eir strong !lanes, t'ey 'ave a tendency to #ail

by b"c.ling in t'eir $ea.er !lanes Bot' a)ially loaded col"mns and transversely loaded beams e)'ibit closely similar #ail"re c'aracteristics d"e to b"c.ling

:ol"mn b"c.ling 'as been dealt $it' in detail in an earlier c'a!ter In t'is section, lateral

b"c.ling o# beams is described and its close similarity to col"mn b"c.ling is bro"g't o"t

:onsider a sim!ly s"!!orted and laterally "ns"!!orted 1e)ce!t at ends2 beam o# ;s'ort-

s!an< s"b%ected to incremental transverse load at its mid section as s'o$n in 4ig* 1a2

T'e beam $ill de#lect do$n$ards ie in t'e direction o# t'e load +4ig *1b2

T'e direction o# t'e load and t'e direction o# movement o# t'e beam are t'e same T'is is

similar to a s'ort col"mn "nder a)ial com!ression ?n t'e ot'er 'and, a ;long-s!an< beam +4ig/ 1a2, $'en incrementally loaded $ill #irst de#lect do$n$ards, and $'en t'e

load e)ceeds a !artic"lar val"e, it $ill tilt side$ays d"e to instability o# t'e com!ression

#lange and rotate abo"t t'e longit"dinal a)is +4ig /1b2

Version II 11-'

Undeflected position

Deflected position

Fig. 1(a) Short span beam, (b) Vertical deflection of the beam.

(a) (b)

W W

ori!ontal

"o#e"ent

θ

Fig. 2(a) ong span beam, (b) aterall! deflected shape of the beam

(a) $fter buckling

Before

buckling

% e r t i c a l " o # e " e n t

&wisting

(b)

W W

W


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T'e t'ree !ositions o# t'e beam cross-section s'o$n in 4ig /1b2 ill"strate t'e

dis!lacement and rotation t'at ta.e !lace as t'e midsection o# t'e beam "ndergoes lateral

torsional b"c.ling T'e c'aracteristic #eat"re o# lateral b"c.ling is t'at t'e entire crosssection rotates as a rigid disc $it'o"t any cross sectional distortion T'is be'avio"r is

very similar to an a)ially com!ressed long col"mn, $'ic' a#ter initial s'ortening in t'e

a)ial direction, de#lects laterally $'en it b"c.les T'e similarity bet$een col"mn b"c.ling and beam b"c.ling is s'o$n in 4ig 6

In t'e case o# a)ially loaded col"mns, t'e de#lection ta.es !lace side$ays and t'e col"mn

b"c.les in a !"re #le)"ral mode A beam, "nder transverse loads, 'as a !art o# its cross

section in com!ression and t'e ot'er in tension T'e !art "nder com!ression becomes"nstable $'ile t'e tensile stresses else$'ere tend to stabili@e t'e beam and .ee! it

straig't T'"s, beams $'en loaded e)actly in t'e !lane o# t'e $eb, at a !artic"lar load,

$ill #ail s"ddenly by de#lecting side$ays and t'en t$isting abo"t its longit"dinal a)is+4ig6 T'is #orm o# instability is more com!le) 1com!ared to col"mn instability2 since

t'e lateral b"c.ling !roblem is 6-dimensional in nat"re It involves co"!led lateral

de#lection and t$ist ie, $'en t'e beam de#lects laterally, t'e a!!lied moment e)erts ator0"e abo"t t'e de#lected longit"dinal a)is, $'ic' ca"ses t'e beam to t$ist T'e bending

moment at $'ic' a beam #ails by lateral b"c.ling $'en s"b%ected to a "ni#orm end

moment is called its elastic critical "o"ent (' cr ) In t'e case o# lateral b"c.ling o#

beams, t'e elastic b"c.ling load !rovides a close "!!er limit to t'e load carrying ca!acityo# t'e beam It is clear t'at lateral instability is !ossible only i# t'e #ollo$ing t$o

conditions are satis#ied

• T'e section !ossesses di##erent sti##ness in t'e t$o !rinci!al !lanes, and

• T'e a!!lied loading ind"ces bending in t'e sti##er !lane 1abo"t t'e ma%or a)is2

Similar to t'e col"mns, t'e lateral b"c.ling o# "nrestrained beams, is also a #"nction o# its

slenderness

Version II 11-(

B

B

u

P

P

*

+

Section B,B

Colu"n buckling

'

θ

u

'

Section B,B

Bea" buckling

-. x /-.

y

-. x /01

Fig. " Similarit! of column buc#ling and beam buc#ling

B

B


5/32


'.0 IN%#UENCE O% CROSS SECTIONA# S)A*E ON #ATERA#

TORSIONA# BUCING

Str"ct"ral sections are generally made "! o# eit'er o!en or closed sections E)am!les o#

o!en and closed sections are s'o$n in 4ig 7

:ross sections, em!loyed #or col"mns and beams 1I and c'annel2, are "s"ally o!en

sections in $'ic' material is distrib"ted in t'e #langes, ie a$ay #rom t'eir centroids, to

im!rove t'eir resistance to in-!lane bending stresses ?!en sections are also convenientto connect beams to ad%acent members In t'e ideal case, $'ere t'e beams are restrained

laterally, t'eir bending strengt' abo"t t'e ma%or a)is #orms t'e !rinci!al design

consideration T'o"g' t'ey !ossess 'ig' ma%or a)is bending strengt', t'ey are relatively$ea. in t'eir minor a)is bending and t$isting

T'e "se o# o!en sections im!lies t'e acce!tance o# lo$ torsional resistance in'erent in

t'em No do"bt, t'e 'ig' bending sti##ness 1 -. x2 available in t'e vertical !lane $o"ldres"lt in lo$ de#lection "nder vertical loads 5o$ever, i# t'e beam is loaded laterally, t'e

de#lections 1$'ic' are governed by t'e lo$er -. y rat'er t'an t'e 'ig'er -. x2 $ill be very

m"c' 'ig'er 4rom a conce!t"al !oint o# vie$, t'e beam 'as to be regarded as an element'aving an en'anced tendency to #all over on its $ea. a)is

In contrast, closed sections s"c' as t"bes, bo)es and solid s'a#ts 'ave 'ig' torsionalsti##ness, o#ten as 'ig' as * times t'at o# an o!en section T'e 'ollo$ circ"lar t"be is

t'e most e##icient s'a!e #or torsional resistance, b"t is rarely em!loyed as a beam element

on acco"nt o# t'e di##ic"lties enco"ntered in connecting it to t'e ot'er members and lesser

Version II 11-+

Wide 2lange Bea" Channel $ngle

$pen sections

%losed sections

&ubular Box

Fig. & $pen and closed sections

Standard bea" &ee


6/32


e##iciency as a #le)"ral member T'e in#l"ence o# sectional s'a!es on t'e lateral strengt'

o# a beam is #"rt'er ill"strated in a later Section

(.0 #ATERA# TORSIONA# BUCING O% S$MMETRIC SECTIONS

As e)!lained earlier, $'en a beam #ails by lateral torsional b"c.ling, it b"c.les abo"t its$ea. a)is, even t'o"g' it is loaded in t'e strong !lane T'e beam bends abo"t its strong

a)is "! to t'e critical load at $'ic' it b"c.les laterally +4ig 1a2 and 1b2

4or t'e !"r!ose o# t'is disc"ssion, t'e lateral torsional b"c.ling o# an I-section is

considered $it' t'e #ollo$ing ass"m!tions

* T'e beam is initially "ndistorted/ Its be'avio"r is elastic 1no yielding2

6 It is loaded by e0"al and o!!osite end moments in t'e !lane o# t'e $eb

7 T'e loads act in t'e !lane o# t'e $eb only 1t'ere are no e)ternally a!!lied lateral or torsional loads2

T'e beam does not 'ave resid"al stresses

C Its ends are sim!ly s"!!orted vertically and laterally

?bvio"sly, in !ractice, t'e above ideal conditions are seldom met 4or e)am!le, rolled

sections invariably contain resid"al stresses T'e e##ects o# t'e deviations #rom t'e ideal

case are disc"ssed in a later Section

Version II 11-,

Fig. '(a) $riginal beam (b) laterall! buc#led beam

'

Plan

-le#ation

'

Section

(a)

θ

3ateral Deflection

y

(b)

&wisting

x

$

$

Section $, $


7/32


T'e critical bending moment ca!acity attained by a symmetric I beam s"b%ected to e0"al

end moments "ndergoing lateral torsional b"c.ling bet$een !oints o# lateral or torsional

s"!!ort is a #"nction o# t$o torsional c'aracteristics o# t'e s!eci#ic cross-section t'e !"retorsional resistance "nder "ni#orm torsion and t'e $ar!ing torsional resistance

' cr + 1torsional resistance2/

1 $ar!ing resistance 2/

*>/

4

5

4

y

4

y

. - 6 1 0 . -

6

cr '

+=

Γ *1a2

T'is may be re$ritten as

( ) 4

5

45

1 0

- 5 1 0 . - '

4

4

ycr

+=

Γ π π *1b2

$'ere, -. y is t'e minor a)is #le)"ral rigidity

01 is t'e torsional rigidity

- Γ is t'e $ar!ing rigidity

T'e torsion t'at accom!anies lateral b"c.ling is al$ays non-"ni#orm T'e critical bending

moment, ' cr is given by E0n* 1a2

It is evident #rom E0n* 1a2 t'at t'e #le)"ral and torsional sti##ness o# t'e member relate

to t'e lateral and torsional com!onents o# t'e b"c.ling de#ormations T'e magnit"de o#

t'e second s0"are root term in E0n* 1b2 is a meas"re o# t'e contrib"tion o# $ar!ing tot'e resistance o# t'e beam In !ractice, t'is val"e is large #or s'ort dee! girders 4or long

s'allo$ girders $it' lo$ $ar!ing sti##ness, Γ ≈ and E0n *1b2 red"ces to

An I-section com!osed o# very t'in !lates $ill !osses very lo$ torsional rigidity 1since Fde!ends on t'ird !o$er o# t'ic.ness2 and bot' terms "nder t'e root $ill be o# com!arable

magnit"de T'e second term is negligible com!ared to t'e #irst #or t'e ma%ority o# 'ot

rolled sections B"t lig't ga"ge sections derive most o# t'e resistance to torsional

de#ormation #rom t'e $ar!ing action T'e beam lengt' also 'as considerable in#l"ence"!on t'e relative magnit"des o# t'e t$o terms as s'o$n in t'e term π 4 - Γ 7 401 S'orter and dee! beams 1π 4 - Γ 7 401 term $ill be large2 demonstrate more $ar!ing resistance,$'ereas, t'e term $ill be small #or long and s'allo$ beams E0n 1*2 may be re$ritten in

a sim!ler #orm as given belo$

4

5

4

44

5

y B

6 5

6 1 0 . - cr '

162

Version II 11-

4

5

y 1 0 . -

6

cr '

=

1/2


8/32


$'ere B4 8 4 0 1 7 - Γ 61a2

' cr 8 α (- . y 0 1)574 γ 172

$'ere γ 8 π 7 (59π 4 7 B4 )574 71a2

E0n 172 is a !rod"ct o# t'ree terms t'e #irst term, α : varies $it' t'e loading and s"!!ortconditions& t'e second term varies $it' t'e material !ro!erties and t'e s'a!e o# t'e beam&

and t'e t'ird term, γ , varies $it' t'e lengt' o# t'e beam E0n 172 is regarded as t'e basice0"ation #or lateral torsional b"c.ling o# beams T'e in#l"ence o# t'e t'ree terms

mentioned above is disc"ssed in t'e #ollo$ing Section

+.0 %ACTORS A%%ECTING #ATERA# STABI#IT$

T'e elastic critical moment, ' cr : as obtained in t'e !revio"s Section, is a!!licable only to

a beam o# I section $'ic' is sim!ly s"!!orted and s"b%ected to end moments T'is case isconsidered as t'e basic case #or #"t"re disc"ssion In !ractical sit"ations, s"!!ort

conditions, beam cross section, loading etc vary #rom t'e basic case T'e #ollo$ingsections elaborate on t'ese variations and ma.e t'e necessary modi#ications to t'e basic

case #or design !"r!oses

+.1 S//ort onitions

T'e lateral restraint !rovided by t'e sim!ly s"!!orted conditions ass"med in t'e basiccase is t'e lo$est and t'ere#ore ' cr is also t'e lo$est It is !ossible, by ot'er restraint

conditions, to obtain 'ig'er val"es o# ' cr , #or t'e same str"ct"ral section, $'ic' $o"ld

res"lt in better "tili@ation o# t'e section and t'"s saving in $eig't o# material As lateral b"c.ling involves t'ree .inds o# de#ormations, namely lateral bending: twisting andwarping , it is #easible to t'in. o# vario"s ty!es o# end conditions B"t, t'e s"!!orts

s'o"ld eit'er com!letely !revent or o##er no resistance to eac' ty!e o# de#ormation

Sol"tions #or !artial restraint conditions are com!licated T'e e##ect o# vario"s s"!!ortconditions is ta.en into acco"nt by $ay o# a !arameter called effecti#e length: $'ic' is

e)!lained, in t'e ne)t Section

+." Eeti2e 3engt4

T'e conce!t o# e##ective lengt' incor!orates t'e vario"s ty!es o# s"!!ort conditions 4or

t'e beam $it' sim!ly s"!!orted end conditions and no intermediate lateral restraint, t'ee##ective lengt' is e0"al to t'e act"al lengt' bet$een t'e s"!!orts ='en a greater amo"nt

o# lateral and torsional restraints is !rovided at s"!!orts, t'e e##ective lengt' is less t'an

t'e act"al lengt' and alternatively, t'e lengt' becomes more $'en t'ere is less restraintT'e e##ective lengt' #actor $o"ld indirectly acco"nt #or t'e increased lateral and torsional

rigidities !rovided by t'e restraints As an ill"stration, t'e e##ective lengt's a!!ro!riate

#or di##erent end restraints according to BS are given in Table / T'e destabili@ing#actor indicated in Table / is e)!lained in t'e ne)t Section

Version II 11-5


9/32


Table 2 ffecti*e length

Eeti2e #engt46 e6 or 7ea!s 6 7et8een s//orts

Conitions at s//orts #oaing onitions

Normal Destabilising

Beam torsionally "nrestrained

:om!ression #lange laterally "nrestrained

Bot' #langes #ree to rotate on !lan

*/1 9 4D2 *71 9 4D2

Beam torsionally "nrestrained

:om!ression #lange laterally "nrestrained:om!ression #lange only #ree to rotate on

!lan

*1 9 4D2 */1 9 4D2

Beam torsionally restrained

:om!ression #lange laterally restrained:om!ression #lange only #ree to rotate on

!lan

* */

Beam torsionally restrained:om!ression #lange laterally restrained

Bot' #langes !artially #ree to rotate on

!lan1ie !ositive connections to bot' #langes2

H *

Beam torsionally restrained

:om!ression #lange laterally restrainedBot' #langes N?T #ree to rotate on !lan

H

is t'e lengt' o# t'e beam bet$een restraints D is t'e de!t' o# t'e beam

+.' #e2e3 o a//3iation o trans2erse 3oas

T'e lateral stability o# a transversely loaded beam is de!endent on t'e arrangement o# t'eloads as $ell as t'e level o# a!!lication o# t'e loads $it' res!ect to t'e centroid o# t'e

cross section 4ig C s'o$s a centrally loaded beam e)!eriencing eit'er destabilising or

restoring e##ect $'en t'e cross section is t$isted

A load a!!lied above t'e centroid o# t'e cross section ca"ses an additional overt"rning

moment and becomes more critical t'an t'e case $'en t'e load is a!!lied at t'e centroid?n t'e ot'er 'and, i# t'e load is a!!lied belo$ t'e centroid, it !rod"ces a stabilisinge##ect T'"s, a load a!!lied belo$ or above t'e centroid can c'ange t'e b"c.ling load by

± ;


10/32


Jrovision o# intermediate lateral s"!!orts can conveniently increase t'e lateral stability o#

a beam =it' a central s"!!ort, $'ic' is ca!able o# !reventing lateral de#lection and

t$isting, t'e beam s!an is 'alved and eac' s!an be'aves inde!endently As a res"lt, t'erigidity o# t'e beam is considerably increased T'is as!ect is dealt in more detail in a later

c'a!ter

+.( In3ene o ty/e o 3oaing

So #ar, only t'e basic case o# beams loaded $it' e0"al and o!!osite end moments 'as been considered B"t, in reality, loading !atterns $o"ld vary $idely #rom t'e basic caseT'e t$o reasons #or st"dying t'e basic case in detail are 1*2 it is analytically amenable,

and 1/2 t'e loading condition is regarded as t'e most severe :ases o# moment gradient,

$'ere t'e end moments are "ne0"al, are less !rone to instability and t'is bene#icial e##ectis ta.en into acco"nt by t'e "se o# + eui*alent uniform moments. In t'is case, t'e basic

design !roced"re is modi#ied by com!aring t'e elastic critical moment #or t'e act"al case

$it' t'e elastic critical moment #or t'e basic case T'is !rocess is similar to t'e e##ectivelengt' conce!t in str"t !roblems #or ta.ing into acco"nt end #i)ity

+.(.1 #oaing a//3ie at /oints o 3atera3 restraint

='ile considering ot'er loading cases, t'e variation o# t'e bending moment $it'in asegment 1ie t'e lengt' bet$een t$o restraints2 is ass"med to be linear #rom ' "ax at one

end to ' "in at t'e ot'er end as s'o$n in 4ig

Version II 11-10

' "in

Fig. /on uniform distribution of bending moment

' "in

' "ax

' "in

Positi#eβ

' "in

>egati#eβ

' "ax

' "ax

' "ax

w w w

Botto" flangeloading

Shear center

loading

To! #lange loading

%alue of 4 0 1 7 - Γ

5< 5


11/32


T'e val"e o# β is de#ined as

β ' "in 7 ' "ax .1 .1

T'e val"e o# β is !ositive #or o!!osing moments at t'e ends 1single c"rvat"re bending2and negative #or moments o# t'e same .ind 1do"ble c"rvat"re bending2 4or a !artic"lar

case o# β , t'e val"e o# ' at $'ic' elastic instability occ"rs can be e)!ressed as a ratio" involving t'e val"e o# ' cr #or t'e segment ie t'e elastic critical moment #or β 8 *T'e ratio may be e)!ressed as a single c"rve in t'e #orm

" 8


12/32


T'is a!!ro)imation 'el!s in !redicting t'e b"c.ling o# t'e segments o# a beam, $'ic' is

loaded t'ro"g' transverse members !reventing local lateral de#lection and t$ist Eac'

segment is treated as a beam $it' "ne0"al end moments and its elastic critical momentsmay be determined #rom t'e relations'i! given in E0n T'e critical moment o# eac'

segment can be determined and t'e lo$est o# t'em $o"ld give a conservative

a!!ro)imation to t'e act"al critical moment

Beam and loads Act"al bending

moment

' "ax " E0"ivalent

"ni#ormmoment

' *

'

' 76

W 7; 7

W 4 7@ HH

W 7; C

It may be noted 'ere t'at t'e val"es o# " a!!ly only $'en t'e !oint o# ma)im"m

moment occ"rs at one end o# t'e segments o# t'e beams $it' "ni#orm cross section and

e0"al #langes In all ot'er cases "85


13/32


As disc"ssed earlier, t'e s'a!e o# t'e moment diagram in#l"ences t'e lateral stability o# a

beam A beam design "sing "ni#orm moment loading $ill be "nnecessarily conservative

In order to acco"nt #or t'e non-"ni#ormity o# moments, a modi#ication o# t'e momentmay be made based on a com!arison o# t'e elastic critical moment #or t'e basic case

T'is can be done in t$o $ays T'ey are

(i) Use e0"ivalent "ni#orm moment val"e ' " ' "ax (' "ax is t'e larger o# t'e t$o

end moments2 #or c'ec.ing against t'e b"c.ling resistance moment ' b

(ii) ' b val"e is determined "sing an e##ective slenderness ratio λ 3& 8 λ 3& " 1$'ere λ 3& is t'e lateral torsional slenderness ratio and λ 3& is t'e e##ective lateraltorsional slenderness ratio2

T'e idea o# lateral torsional slenderness λ3T is introd"ced 'ere to $rite t'e design ca!acity

' b as

=

/

*

3& p

b f '

'

λ 12

$'ere ' p is t'e #"lly !lastic moment

Version II 11-1'

Fig. 1 Moment capacit! of beams

3ateral J torsional slenderness λ 3&

' o " e n t c a p a c i t y f a c

t o r ' 7 '

p

5;;


14/32


T'e 0"antity λ 3& is de#ined by

cr

p

y

3& '

'

p

- /π λ = 1*2

4or a !artic"lar material 1ie !artic"lar - and p y2 t'e above e0"ation can be considered as

a !rod"ct o# c constant and ( ) 3& cr

p

'

' λ T'e 0"antity 3& λ is called as t'e ne$ de#ined

slenderness ratio

B"c.ling resistance moment, ' b is al$ays less t'an t'e elastic critical moment, ' cr

T'ere#ore, t'e second met'od is more conservative es!ecially #or lo$ val"es o# λ 3& T'et$o met'ods are com!ared in 4ig *, $'ere #or t'e #irst case ' "ax is to be c'ec.edagainst ' b 7 " and #or t'e second case against ' b only Met'od 1i2 is more s"itable #or

cases $'ere loads are a!!lied only at !oints o# e##ective lateral restraint 5ere, t'eyielding is restricted to t'e s"!!orts& conse0"ently, res"lts in a small red"ction in t'elateral b"c.ling strengt' In order to avoid overstressing at one end, an additional c'ec.,

' "ax H ' p s'o"ld also be satis#ied In certain sit"ations, ma)im"m moment occ"rs

$it'in t'e s!an o# t'e beam T'e red"ction in sti##ness d"e to yielding $o"ld res"lt in a

smaller lateral b"c.ling strengt' In t'is case, t'e !rediction according to met'od 1i2 based on t'e !attern o# moments $o"ld not be conservative& 'ere t'e met'od 1ii2 is more

a!!ro!riate In t'e second met'od, a correction #actor n is a!!lied to t'e slenderness ratio

λ 3& and design strengt' is obtained #or nλ 3& It is clear #rom t'e above t'at n 8 " T'eslenderness correction #actor is e)!lained in t'e ne)t section

+.(.' S3enerness orretion ator

4or sit"ations, $'ere t'e ma)im"m moment occ"rs a$ay #rom a braced !oint, eg $'en

t'e beam is "ni#ormly loaded in t'e s!an, a modi#ication to t'e slenderness, λ 3& : may be"sed T'e allo$able critical stress is determined #or an e##ective slenderness, nλ 3& : $'eren is t'e slenderness correction #actor, as ill"strated in 4ig ** #or a #e$ cases o# loading

4or design !"r!oses, one o# t'e above met'ods ( eit'er t'e moment correction #actor

met'od 1" met'od2 or slenderness correction #actor met'od 1n met'od2 may be "sed I# s"itable val"es are c'osen #or " and n: bot' met'ods yield identical res"lts T'e

di##erence arises only in t'e $ay in $'ic' t'e correction is made& in t'e n #actor met'od

t'e slenderness is red"ced to ta.e advantage o# t'e e##ect o# t'e non- "ni#orm moment,$'ereas, in t'e " #actor met'od, t'e moment to be c'ec.ed against lateral moment

ca!acity, ' b : is red"ced #rom ' "ax to ' by t'e #actor " It is al$ays sa#e to "se " 8 n

* basing t'e design on "ni#orm moment case In any sit"ation, eit'er " 8 * or n8 *, ieany one met'od s'o"ld be "sed

Version II 11-1(


15/32


Slenderness correction #actor, n

3oad !attern Act"al bending moment n E0"ivalent "ni#orm

moment

' '

*

' '

C

W

HC

w7" 7

W W

7

7

+.+ Eet o ross-setiona3 s4a/e

T'e s'a!e o# t'e cross-section o# a beam is a very im!ortant !arameter $'ile eval"atingits lateral b"c.ling ca!acity In ot'er $ords, lateral instability can be red"ced or even

avoided by c'oosing a!!ro!riate sections T'e e##ect o# cross-sectional s'a!e on lateralinstability is ill"strated in 4ig */ #or di##erent ty!e o# section $it' same cross sectional

area

T'e #ig"re s'o$s t'at t'e I-section $it' t'e larger in-!lane bending sti##ness does not

'ave matc'ing stability Bo) sections $it' 'ig' torsional sti##ness are most s"itable #or

Version II 11-1+

7;

7;

>;


16/32

'

C r K ' C r


17/32


T'ere are t'ree distinct regions in t'e c"rve as given belo$

5 Beams $it' 'ig' slenderness 1 */>cr '

p ' ) T'e #ail"re o# t'e beam is by elastic

lateral b"c.ling at ' cr

4 Beams o# intermediate slenderness 7 L */<cr '

p ' 2, $'ere #ail"re occ"rs by

inelastic lateral b"c.ling at loads belo$ ' p and above ' cr

F Stoc.y beams 1 A7<cr '

p ' 22, $'ic' attain ' p $it'o"t b"c.ling

Version II 11-1

Strain distribution

Stress distribution

Spread of yield

(-lastic Jperfectly plastic "aterial beha#iour is assu"ed )

Fig 1" Strain 6 Stress 7istribution and !ielding of section

.nelastic buckling (no residual stress) 'H' Cr

Plastic failure ' 8 ' p

5<

' y 7 '

P

' yr

7 ' P

' o " e n t r a t i o ' 7 '

p

'odified Slenderness


18/32


,." Resia3 stresses

It is normally ass"med t'at a str"ct"ral section in t'e "nloaded condition is #ree #romstress and strain In reality, t'is is not tr"e D"ring t'e !rocess o# man"#act"re o# steel

sections, t'ey are s"b%ected to large t'ermal e)!ansions res"lting in yield level strains in

t'e sections As t'e s"bse0"ent cooling is not "ni#orm t'ro"g'o"t t'e section, sel#-e0"ilibrating !atterns o# stresses are #ormed T'ese stresses are .no$n as residual

stresses Similar e##ects can also occ"r at t'e #abrication stage d"ring $elding and #lame

c"tting o# sections A ty!ical resid"al stress distrib"tion in a 'ot rolled steel beam sectionis s'o$n in 4ig*

D"e to t'e !resence o# resid"al stresses, yielding o# t'e section starts at lo$er momentsT'en, $it' t'e increase in moment, yielding s!reads t'ro"g' t'e cross-section T'e in-

elastic range, $'ic' starts at ' yr increases instead o# t'e elastic range T'e !lastic

moment val"e ' p is not in#l"enced by t'e !resence o# resid"al stresses

,.' I!/eretions

T'e initial distortion or lac. o# straig'tness in beams may be in t'e #orm o# a lateral bo$or t$ist In addition, t'e a!!lied loading may be eccentric ind"cing more t$ist to t'e

beam It is clear t'at t'ese initial im!er#ections corres!ond to t'e t$o ty!es o#

de#ormations t'at t'e beam "ndergoes d"ring lateral b"c.ling Ass"ming ' cr


19/32

2irst yield of initially defor"ed bea"s at

'H' cr (no residual stress)

-lastic buckling

'H' cr

.nitial

defor"ationsincreasing

'odified Slenderness

> o n d

i " e n s i o n a l a p p l i e d " o " e n t


20/32


T'ree distinct regions o# be'avio"r may be noticed in t'e #ig"re T'ey are

• Stoc.y, $'ere beams attain ' p, $it' val"es o# 3& L L 7

• Intermediate, t'e region $'ere beams #ail to reac' eit'er ' P or ' cr & 7L 3& L L*/

• Slender, $'ere beams #ail at moment ' cr M 3& L */

As !ointed o"t earlier, lateral stability is not a criterion #or stoc.y beams 4or beams o# t'e second category, $'ic' com!rise o# t'e ma%ority o# available sections, design is based

on inelastic b"c.ling acco"nting #or geometrical im!er#ections and resid"al stresses

.1 Conser2ati2e esign /roere

T'e lateral b"c.ling moment ca!acity o# a section can be e)!ressed as

' b 8 pb S x 1** 2

$'ere : pb is t'e bending strengt' acco"nting #or lateral instability S ) is t'e a!!ro!riate !lastic section mod"l"s

T'e slenderness o# t'e beam λ 3& is de#ined asN

λ 3& 8 3& L y p - 46

1*/2

T'is 'as close similarity to t'e slenderness associated $it' com!ressive b"c.ling o# a

col"mn T'e relation bet$een pb and λ 3& is s'o$n in 4ig*H

In t'e case o# slender beams : pb is related to λ 3& λ 3& can be determined #or a given section by t'e #ollo$ing relations'i!

Version II 11-"0


21/32


λ 3& 8n u # e 7 r y 1*62

$'ere : n is t'e slenderness correction #actor

u is b"c.ling !arameter #rom steel tables 1 #or rolled beams and c'annels and * #or ot'er sections2

# is slenderness #actor and f( 7r y : x): given in Table *7 o# BS !art *& b"t

a!!ro)imated to * #or !reliminary calc"lations

x is t'e torsional inde) $'ic' is !rovided in BS !art *

x 8( )

2

1

1 $h ABCC #or bi-symmetric sections and sections symmetric abo"t

minor a)is, and

x 82

1

1 . $

y**6/

#or sections symmetric abo"t ma%or a)is

$'ere

$ is t'e cross sectional area o# t'e member

. y is t'e second moment o# t'e area abo"t t'e minor a)is is t'e $ar!ing constant

1 is t'e torsion constant

h is t'e distance bet$een t'e s'ear center o# t'e #langes

4or com!act sections, #"ll !lasticity is develo!ed at t'e most 'eavily stressed section

Unli.e !lastic design, moment redistrib"tion is not considered 'ere 4or e)am!le, #or a

!artic"lar grade o# steel and #or 3& L


22/32


A good design can be ac'ieved by determining t'e val"e o# λ 3& and t'ereby pb moreacc"rately ' b can be determined "sing E0n** E##ective lengt's o# t'e beam may be

ado!ted as !er t'e g"idelines given in Table / 4or beams, and segments o# beams bet$een lateral s"!!orts, e0"ivalent "ni#orm moments may be calc"lated to determine

t'eir relative severity o# instability T'e lateral stability is c'ec.ed #or an e0"ivalent

moment ' given by

' 8 " ' "ax 1*72

$'ere " is t'e e0"ivalent "ni#orm moment #actor

I# ' b ' , t'e section c'osen is satis#actory At t'e 'eavily stressed locations, local

strengt' s'o"ld be c'ec.ed against develo!ment o# ' p

' "ax > ' p 1*2

5.0 SUMMAR$

Unrestrained beams t'at are loaded in t'eir sti##er !lanes may "ndergo lateral torsional

b"c.ling T'e !rime #actors t'at in#l"ence t'e b"c.ling strengt' o# beams are t'e "n braced s!an, cross sectional s'a!e, ty!e o# end restraint and t'e distrib"tion o# moment

4or t'e !"r!ose o# design, t'e sim!li#ied a!!roac' as given in BS Jart-* 'as been

!resented T'e e##ects o# vario"s !arameters t'at a##ect b"c.ling strengt' 'ave beenacco"nted #or in t'e design by a!!ro!riate correction #actors T'e be'avio"r o# real

beams 1$'ic' do not com!ly $it' t'e t'eoretical ass"m!tions2 'as also been described

In order to increase t'e lateral strengt' o# a beam, bracing o# s"itable sti##ness and

strengt' 'as to be !rovided

9.0 RE%ERENCES

* Timos'en.o S, KT'eory o# elastic stability McGra$ 5ill Boo. :o, *st Edition *6C

/ :lar.e AB and :overman, KStr"ct"ral steel $or.-3imit state design, :'a!man and5all, 3ondon, *H

6 Martin 35 and J"r.iss FA, KStr"ct"ral design o# steel $or. to BS , Ed$ardArnold, */

7 Tra'air NS, KT'e be'avio"r and design o# steel str"ct"res, :'a!man and 5all

3ondon, *

Oirby JA and Net'ercot DA,Design #or str"ct"ral stability, Granada J"blis'ing,

3ondon, *

Version II 11-""


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=or.ed e)am!leN 5Made by SSI Date*>6>/

:'ec.ed by S$1 Date 7F7 47"

@

? Q4;

@

w '

44

"ax ==

8 5 " H 54E "

ence ' b / ' "ax

.S'B ;< is adeGuate against lateral torsional buckling

Version II 11-"


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