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UNRESTRAINED BEAM DESIGN-II

UNRESTRAINED BEAM DESIGN II

1.0 INTRODUCTION

The basic theory of beam bucklin !as e"#laine$ in the #re%ious cha#ter& Doubly

symmetric I- section has been use$ throuhout for the $e%elo#ment of the theory an$

later $iscussion& It !as establishe$ that #ractical beams fail by'

(i) *iel$in+ if they are short

(ii) Elastic bucklin+ if they are lon+ or

(iii) Inelastic lateral bucklin+ if they are of interme$iate lenth&

A conser%ati%e metho$ of $esinin beams !as also e"#laine$ an$ its limitations !ere

outline$&

In this cha#ter a fe! cases of lateral bucklin strenth e%aluation of beams encountere$

in #ractice !oul$ be e"#laine$& ,antile%er beams+ continuous beams+ beams !ithcontinuous an$ $iscrete lateral restraints are consi$ere$& ,ases of monosymmetric beams

an$ non-uniform beams are co%ere$& The bucklin strenth e%aluation of non-symmetric

sections is also $escribe$&

2.0 CANTILEVER BEAMS

A cantile%er beam is com#letely fi"e$ at one en$ an$ free at the other& In the case ofcantile%ers+ the su##ort con$itions in the trans%erse #lane affect the moment #attern& or

$esin #ur#oses+ it is con%enient to use the conce#t of notional effective length, k, !hich

!oul$ inclu$e both loa$in an$ su##ort effects& The notional effecti%e lenth is $efine$as the lenth of the notionally sim#ly su##orte$ (in the lateral #lane) beam of similar

section+ !hich !oul$ ha%e an elastic critical moment un$er uniform moment e.ual to the

elastic critical moment of the actual beam un$er the actual loa$in con$itions&

Recommen$e$ %alues of kfor a number of cases are i%en in Table /& It can be seenfrom the %alues of 0k1 that it is more effecti%e to #re%ent t!ist at the cantile%er e$e rather

than the lateral $eflection&

Generally+ in frame$ structures+ continuous beams are #ro%i$e$ !ith o%erhan at their

en$s& These o%erhans ha%e the characteristics of cantile%er beams& In such cases+ the

ty#e of restraint #ro%i$e$ at the outermost %ertical su##ort is most sinificant& Effecti%e

#re%ention of t!ist at this location is of #articular im#ortance& ailure to achie%e this!oul$ result in lare re$uction of lateral stability as reflecte$ in lare %alues of 0 k+ in

Table /&

2 ,o#yriht reser%e$

Version II 12-1

12


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Beams+ e"ten$in o%er a number of s#ans+ are normally continuous in %ertical+ lateral or

in both #lanes& In the cases+ !here such continuity is not #ro%i$e$ lateral $eflection an$t!istin may occur& Such a situation is ty#ically e"#erience$ in roof #urlins before

sheetin is #ro%i$e$ on to# of them an$ in beams of tem#orary nature& or these cases+ it

is al!ays safe to make no assum#tion about #ossible restraints an$ to $esin them forma"imum effecti%e lenth&

Another case of interest !ith rear$ to lateral bucklin is a beam that is continuous in thelateral #lane i&e& the beam is $i%i$e$ into se%eral sements in the lateral #lane by means

of fully effecti%e braces& The buckle$ sha#e for such continuous beams inclu$e

$eformation of all the sements irres#ecti%e of their loa$in& Effecti%e lenth of the

sements !ill be e.ual to the s#acin of the braces if the s#acin an$ moment #atternsare similar& ?ther!ise+ the effecti%e lenth of each sement !ill ha%e to be $etermine$

se#arately&

To illustrate the beha%iour of continuous beams+ a sinle-s#an beam #ro%i$e$ !ithe.ually loa$e$ cross beams is consi$ere$ (see i& /)&

T!o e.ually s#ace$+ e.ually loa$e$ cross beams $i%i$e the beam into three sementslaterally& In this case+ trueMcrof the beam an$ its bucklin mo$e !oul$ $e#en$ u#on the

s#acin of the cross beams& The critical moment McrBfor any ratio of 1/ b !oul$ lie in

bet!een the critical moment %alues of the in$i%i$ual sements& The critical moments forthe t!o sements are obtaine$ usin the basic e.uation i%en in the earlier cha#ter&

(In the outer sement+ m @ 5&97& Usin / man$ the basic moment+ the critical moment is$etermine$)

(This sement is loa$e$ by uniform moments at its en$s basic case)

Version II 12-3

GJ

E1GJ)IE(

!"1M

#1

#

$1

1cr

+=

GJ

E1GJ)EI(

M #

#

#

$

#

cr#

+=

(/)

(


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Mcr1 an$ Mcr# %alues are #lotte$ aainst 1/ ban$ sho!n in i& < for the #articular case

consi$ere$ !ith e.ual loa$in an$ a constant cross section throuhout&

It is seen that for 1/ b @ 5&=7+ Mcr1 an Mcr# are e.ual an$ the t!o sements are

simultaneously critical& The beam !ill buckle !ith no interaction bet!een the t!o

sements& or any other %alue of 1/ bthere !ill be interaction bet!een the sements

an$ the critical loa$ !oul$ be reater than the in$i%i$ual %alues+ as sho!n in the fiure&or %alues of 1/ b *"+"

The safe loa$ for a laterally continuous beam may be obtaine$ by calculatin allsemental critical loa$s in$i%i$ually an$ choosin the lo!est %alue assumin each

sement as sim#ly su##orte$ at its en$s&

Version II 12-4

'ime

n,ionle,,&riticalMomentM

cr/

Mcr-B

*

#

.

0

*"1 *"# *"+ *".

'imenionle oa 2oition 1/

b

*"!*"+

3ero interaction4oint

Mcr#

Mcr1

5nbrace beam Mcr6 M

cr-B

Fig.2 Interaction beteen !cr1

and !cr2

Mcr&


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It is of interest to kno! the beha%iour of beams+ !hich are continuous in both trans%erse

an$ lateral #lanes& Thouh the beha%iour is similar to the laterally unrestraine$ beams+

their moment #atterns !oul$ be more com#licate$& The beam !oul$ buckle in the lateral#lane an$ $eflect in the %ertical #lane& There is a $istinct $ifference bet!een the #oints of

contrafle"ure in the b-ckle ha4e an$ #oints of contrafle"ure in the eflecte ha4e&

These #oints !ill not normally occur at the same location !ithin a s#an+ as sho!n in i&=& Therefore+ it is !ron to use the $istance bet!een the #oints of contrafle"ure of the

$eflecte$ sha#e as the effecti%e lenth for checkin bucklin strenth&

4.0 EECTIVE LATERAL RESTRAINT

Cro%i$in #ro#er lateral bracin may increase the lateral stability of a beam& 3ateral

bracin may be either $iscrete (e&& cross beams) or continuous (e&& beam

encase$ in concrete floors)& The lateral bucklin ca#acity of the beams !ith$iscrete bracin may be $etermine$ by usin the metho$s $escribe$ in a later

Section& or the continuously restraine$ beams+ assumin lateral $eflection is

com#letely #re%ente$+ $esin can be base$ on in-#lane beha%iour& It is im#ortantto note that in the hoin moment reion of a continuous beam+ if the

com#ression flane (bottom flane) is not #ro#erly restraine$+ a form of lateral

$eflection !ith cross sectional $istortion !oul$ occur&

4.1 Dis!re"e #r$!in%

In or$er to $etermine the beha%iour of $iscrete braces+ consi$er a sim#ly su##orte$ beam#ro%i$e$ !ith a sinle lateral su##ort of stiffness7bat the centroi$+ as sho!n in i& :&

Version II 12-&

M

M

7b

Fig. " #eam it$ single lateral support

1

#

1

#

&ontin-o- beam an loaing

'eflecte ha4e

B-ckle ha4e

Fig. % &ontinuous beam ' deflected s$ape and buckled s$ape

1

1

#

#

81(-l) 8

#(-l)8

#(-l)


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The relationshi# bet!een7ban$Mcris sho!n in i& 9&

It is seen thatMcr%alue increases !ith7b+ until7bis e.ual to a limitin %alue of7b"The

corres#on$in Mcr%alue is e.ual to the %alue of bucklin for the t!o sements of thebeam" Mcr %alue $oes not increase further as the bucklin is no! o%erne$ by the

in$i%i$ualMcr%alues of the t!o sements&

Generally+ e%en a liht bracin has the ability to #ro%i$e substantial increase in stability&There are se%eral !ays of arranin lateral bracin to im#ro%e stability& The limitin

%alue of the lateral bracin stiffness+7b,is influence$ by the follo!in #arameters&

3e%el of attachment of the brace to the beam i&e& to# or bottom flane&

The ty#e of loa$in on the beam+ notably the le%el of a##lication of the trans%erse

loa$

Ty#e of connection+ !hether ca#able of resistin lateral an$ torsional $eformation

Version II 12-'

*1"*

1"!

#"*

#"!

+"*

+"!

# . 0 1* 1# 1. 19on imenional bracing tiffne 7

b+/ .0 E I

$

Mcr8ithbracing

Mcr8itho-tbracing

#G J / E

@#.*

7b

7b

7b

7b

1**

1!

.

Fig. ( Relations$ip beteen )band !

cr


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The #ro#ortion of the beam&

Cro%ision of bracin to tension flanes is not so effecti%e as com#ression flane bracin&Bracin #ro%i$e$ belo! the #oint of a##lication of the trans%erse loa$ !oul$ not be able

to resist t!istin an$ hence full ca#acity of the beam is not achie%e$& or the $esin of

effecti%e lateral bracin systems+ the follo!in t!o re.uirements are essential&

Bracin shoul$ be of sufficient stiffness so that bucklin occurs bet!een the braces

3ateral bracin shoul$ ha%e sufficient strenth to !ithstan$ the force transferre$ by

the beam&

A eneral rule is that lateral bracin can be consi$ere$ as fully effecti%e if the stiffness of

the bracin system is at least


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the nee$e$ restraint by friction an$ bon$& or the case sho!n in i&8( c)+ the metal

$eckin alon !ith the concrete #ro%i$es a$e.uate bracin to the beam& o!e%er+ the

beam is susce#tible to bucklin before the #lacement of concrete $ue to the lo! shearstiffness of the sheetin& Shear stu$s are #ro%i$e$ at the steel-concrete interface to

enhance the shear resistance& The co$al #ro%isions re.uire that for obtainin fully

effecti%e continuous lateral bracin+ it must !ithstan$ not less than / of the ma"imumforce in the com#ression flane&

&.0 BUC*LING O MONOS+MMETRIC BEAMS

or beams symmetrical about the ma>or a"is only e&& une.ual flane$ I- sections+ the

non-coinci$ence of the shear centre an$ the centroi$ com#licates the torsional beha%iour

of the beam& The monosymmetric I-sections are enerally more efficient in resistin loa$s#ro%i$e$ the com#ressi%e flane stresses are taken by the larer flane&

Fhen a monosymmetric beam is bent in its #lane of symmetry an$ t!iste$+ thelonitu$inal ben$in stresses e"ert a tor.ue+ !hich is similar to torsional bucklin of

short concentrically loa$e$ com#ression members& The lonitu$inal stresses e"ert ator.ue+


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The tor.ue $e%elo#e$+


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Non-uniform beams are often use$ in situations+ !here the stron a"is ben$in moment

%aries alon the lenth of the beam& They are foun$ to be more efficient than beams of

uniform sections in such situations& The non-uniformity in beams may be obtaine$ inse%eral !ays& Rectanular sections enerally ha%e ta#er in their $e#ths& I-beams may be

ta#ere$ in their $e#ths or flane !i$ths flane thickness is enerally ke#t constant&

o!e%er+ ste#s in flane !i$th or thickness are also common&

Ta#erin of narro! rectanular beams !ill #ro$uce consi$erable re$uction in minor a"isfle"ural rii$ity+EI$+ an$ torsional rii$ity+ GJ conse.uently+ they ha%e lo! resistance to

lateral torsional bucklin& Re$uction of $e#th in I-beams $oes not affect EI$+ an$ has only

marinal effect on GJ& But !ar#in rii$ity+ E+ is consi$erably re$uce$& Since the

contribution of !ar#in rii$ity to bucklin resistance is marinal+ $e#th re$uction $oesnot influence sinificantly the lateral bucklin resistance of I beams& o!e%er+ re$uction

in flane !i$th causes lare re$uction in GJ+ EI$an$E& Similarly+ re$uction in flane

thickness !ill also #ro$uce lare re$uction in EI$,E, an$ GJin that or$er& or small

$erees of ta#er there is little $ifference bet!een !i$th-ta#ere$ beams an$ thicknessta#ere$ beams& But for hihly ta#ere$ beams+ the critical loa$s of thickness ta#ere$ ones

are hiher& Thus+ the bucklin resistance %aries consi$erably !ith chane in the flane

eometry&

Base$ on the analysis of a number of beams of $ifferent cross sections !ith a %ariety of

loa$in an$ su##ort con$itions+ the elastic critical loa$ for a ta#ere$ beam may be$etermine$ a##ro"imately by a##lyin a re$uction factor r to the elastic critical loa$ for

an e.ui%alent uniform beam #ossessin the #ro#erties of the cross section at the #oint of

ma"imum moment

r 6357

++ (;)

=

233

0

1

1

0

1

0

1

0


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n@ /&9 5&9%m/ %lm/&5 (//)

Fhere%man$%lm are flane areas at the #oints of the smallest an$ larest moment+ an$

m@ /&5

).0 BEAMS O UNS+MMETRICAL SECTIONS

The theory of lateral bucklin of beams $e%elo#e$ so far is a##licable only to $oublysymmetrical cross sections ha%in uniform #ro#erties throuhout its lenth& Many lateral

bucklin #roblems encountere$ in $esin #ractice belon to this cateory& o!e%er+ cases

may arise !here the symmetry #ro#erty of the section may not be a%ailable& Such casesare $escribe$ briefly in this Section&

The basic theory can also be a##lie$ to sections symmetrical about minor a"is only e&&

,hannels an$ -sections& In this section+ the shear centre is situate$ in the a"is of

symmetry althouh not at the same #oint as the centroi$& In the case of channel an$ 3sections+ instability occurs only if the loa$in #ro$uces #ure ma>or a"is ben$in& The

criterion is satisfie$ for the t!o sections if' (/) for the channel section+ the loa$ must actthrouh the shear centre i&6 (a) an$+ (


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