steel th 6-design of compression members

7/26/2019 Steel Th 6-Design of compression members

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Chapter 03 A

Design of CompressionMembers


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INTRODUCTION

When a load tends to squeeze or

shorten a member, the stresses

produced are said to be compressive in

nature and the member is called a

compression member. P

P

Figure 3.1. A Simple Compression ember


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!"amples are struts #short compression

members $ithout chances o% buc&ling',

eccentricall( loaded columns, top chordso% trusses, bracing members, compression

)anges o% beams and members that are

sub*ected simultaneousl( to bending andcompressive loads.

+he term column is usuall( used %or

straight vertical member $hose length is

considerabl( greater than the cross

sectional dimensions.


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Short vertical members sub*ected to

compressive loads are o%ten called struts

or simpl( compression members.+here are t$o signi-cant dierences

bet$een the behaviour o% tension and

compression members, e"plained asunder/

1. +here are no chances o% buc&ling in

tension members, $hereas the strength o%

a compression member most dominantl(

depends on buc&ling phenomenon.


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+he tensile loads tend to hold a

member straight even i% the member is

not initiall( in one line and is sub*ectedto simultaneous bending moments.

0n contrast, the compressive loads

tend to bend the member out o% the

plane o% the loads due to

imper%ections, simultaneous bendingmoment or even $ithout all these.


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+ests on ma*orit( o% practical columns

sho$ that the( $ill %ail at a"ial stresses

$ell belo$ the elastic limit o% thecolumn material because o% their

tendenc( to buc&le.

For these reasons, the strength o%

compression members is reduced in

relation to the danger o% buc&ling

depending on length o% column, end

conditions and crosssectional

dimensions.


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+he longer a column becomes %or the

same crosssection the greater is its

tendenc( to buc&le and the smaller isthe load it $ill support.

When the length o% a compression

member increases relative to its cross

section, it ma( buc&le at a lo$er load.

A%ter buc&ling the load cannot besustained and the load capacit( nearl(

approaches zero.


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+he condition o% a column at its critical

buc&ling load is that o% an unstable

equilibrium as sho$n in Figure 3..


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+he three possible states o%

equilibrium are sho$n in the same

-gure.

2e%erring to part #a' o% Figure 3., i% the

ball is given movement and released,

it comes bac& to the original position

sho$ing a Stable Eqilibrim.

0% ball is displaced and released in part#b', it retains its ne$ position but do

not return to its original position. +his

condition is called Netral


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+he ball in part #c' is Unstable

because i% the ball is displaced and

released it do not return bac& to itsoriginal position and do not retain its

ne$ position.

0n the -rst case, the restoring %orces

are greater than the %orces tending to

upset the s(stem.ue to an in-nitesimal small

displacement consistent $ith the

boundar( conditions or due to small


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At the same time, due to stress in the

material, restoring %orces are also

developed to bring the column bac& toits original shape.

0% restoring %orce is greater than the

upsetting moment, the s(stem is

stable but i% restoring %orce is lesser

than the upsetting moment, thes(stem is unstable.

2ight at the transition point $hen

restoring %orce is e"actl( equal to the


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+he %orce associated $ith this

condition is the !riti!al or b!"ling

loa#.

2eturning bac& to the behaviour o% a

compression member, relativel( rigid

end conditions o% the member, not

allo$ing the member to rotate %reel(

at these points, reduce the eect o%length up to certain e"tent ma&ing the

load carr(ing capacit( a little

improved.


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4ther %actors, such as the eccentricit(

o% load application, imper%ection o%

column material, initial croo&edness o%columns, erection stresses and

residual stresses %rom manu%acture,

help to buc&le the column at a lesserload.


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. +he presence o% rivet or bolt holes in

tension members reduces the area

available %or resisting loads5 but incompression members the rivets or bolts

are assumed to -ll the holes and the

entire gross area is available %or resistingload.

+he ideal t(pe o% load on a column is aconcentric load and the member

sub*ected to this t(pe o% load is called

!on!entri!all$ loa#e# !olmn.


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+he load is distributed uni%orml( over

the entire crosssection $ith the centre

o% gravit( o% the loads coinciding $iththe centre o% gravit( o% the columns.

ue to load patterns, the live load on

slabs and beams ma( not be

concentricall( trans%erred to interiorcolumns.


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Similarl(, the dead and live loads

trans%erred to the e"terior columns

are, generall(, having largeeccentricities, as the centre o% gravit(

o% the loads $ill usuall( %all $ell on the

inner side o% the column.

0n practice, ma*orit( o% the columns areeccentricall( loaded compression

members.


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Slight initial croo&edness, eccentricit(

o% loads, and application o%

simultaneous transverse loads producesigni-cant bending moments as the

product o% high a"ial loads (P)

multiplied $ith the eccentricit(, e.

+his moment, P x e, %acilitates buc&ling

and reduces the load carr(ing capacit(.!ccentricit(, e, ma( be relativel(

smaller, but the product #P x e) ma( be

signi-cantl( larger.


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+he A0SC Code o% Standard Practice

speci-es an acceptable upper limit on

the outo%plumbness and initialcroo&edness equal to the length o% the

member divided b( 677.

Stb !olmn is de-ned as a short

compression test specimen that is

long enough to allo$ strainmeasurements but short enough to

avoid elastic and plastic buc&ling.


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RESIDUA% STRESSES

2esidual stresses are stresses that

remain in a member a%ter it has been

%ormed into a -nished product.

+hese are al$a(s present in a membereven $ithout the application o% loads.

+he magnitudes o% these stresses are

considerabl( high and, in some cases,

are comparable to the (ield stresses

#re%er to Figure 3.8'.


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+he causes o% presence o% residual

stresses are as under/

1. 9neven cooling $hich occurs a%ter hot

rolling o% structural shapes produces

thermal stresses, $hich are

permanentl( stored in members.

+he thic&er parts cool at the end, and

tr( to shorten in length.


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While doing so the( produce

compressive stresses in the other parts

o% the section and tension in them.

4verall magnitude o% this tension and

compression remain equal %or

equilibrium.

0n 0shape sections, a%ter hot rolling,

the thic& *unction o% )ange to $ebcools more slo$l( than the $eb and

)ange tips.


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Consequentl(, compressive residual

stress e"ists at )ange tips and at mid

depth o% the $eb #the regions that cool%astest', $hile tensile residual stress

e"ists in the )ange and the $eb at the

regions $here the( *oin.


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. Cold bending o% members be(ond their

elastic limit produce residual stresses

and strains $ithin the members.

Similarl(, during %abrication, i% some

member having e"tra length is %orced to

-t bet$een other members, stresses

are produced in the associated

members.

3. Punching o% holes and cutting

operations during %abrication also

produce residual stresses.


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8. Welding also produces the stresses

due to uneven cooling a%ter $elding.

Welded part $ill cool at the end

inviting other parts to contract $ith it.

+his produces compressive stresses inparts a$a( %rom $elds and tensile

stresses in parts closer to $elds.


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SECTIONS USED &OR CO%UMNS

Single angle, double angle, tee,

channel, Wsection, pipe, square

tubing, and rectangular tubing ma( be

used as columns.

ierent combinations o% these

structural shapes ma( also be

emplo(ed %or compression members toget builtup sections as sho$n in

Figure 3.6.


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:uiltup sections are better %or

columns because the slenderness

ratios in various directions can becontrolled to get equal values in all the

directions.

+his ma&es the column economical as

%ar as the material cost is concerned .

;o$ever the *oining and labour cost isgenerall( higher %or builtup sections.


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+he total cost o% these sections ma(

become less %or greater lengths.

+he *oining o% various elements o% a

builtup section is usuall( per%ormed

b( using lacing.

%IMITIN' S%ENDERNESS RATIO

+he slenderness ratio o% compression

members should pre%erabl( not e"ceed

(00.


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%o!al Instabilit$

uring local instabilit(, the individual

parts or plate elements o% cross

section buc&le $ithout overall buc&ling

o% the column.

Width


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=ocal buc&ling should never be allo$ed

to occur be%ore the overall buc&ling o%

the member e"cept in %e$ cases li&e$eb o% a plate girder.

An Un-stifened Element is a

pro*ecting piece $ith one %ree edge

parallel to the direction o% the

compressive %orce.

+he e"ample is hal% )ange A:in -gure

3.>.


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A Stifened Element is supported

along the t$o edges parallel to the

direction o% the %orce.

+he e"ample is $eb AC in the same

-gure.

For unstiened )ange o% -gure, b is

equal to hal% $idth o% )ange (bf/2)and

tis equal to tf.


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For stiened $eb, his the $idth o% $eb

and tw is the thic&ness o% $eb and the

corresponding value o% or b/tratio ish/tw, $hich controls $eb local buc&ling.


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O+erall Instabilit$

0n case o% overall instabilit(, the column

buc&les as a $hole bet$een thesupports or the braces about an a"is

$hose corresponding slenderness ratio

is bigger as sho$n in Figures 3.? to 3.@.


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U # % h


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Unspporte# %ength

0t is the length o% column bet$een t$o

consecutive supports or braces denoted

b( Lu" or Lu( in the " ( directions,

respectivel(.

A dierent value o% unsupported length

ma( e"ist in dierent directions and

must be used to calculate thecorresponding slenderness ratios.


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+o calculate unsupported length o% a

column in a particular direction, onl( the

corresponding supports and braces areto be considered neglecting the bracing

preventing buc&ling in the other

direction.

E, ti % th Of C l


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E,e!ti+e %ength Of Colmn

+he length o% the column corresponding

to onehal% sine $ave o% the buc&led

shape or the length bet$een t$o

consecutive in)ection points or supports

a%ter buc&ling is called the eective

length.


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)UC-%IN' O& STEE% CO%UMNS


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)UC-%IN' O& STEE% CO%UMNS

:uc&ling is the sudden lateral bending

produced b( a"ial loads due to initial

imper%ection, outo%straightness, initial

curvature, or bending produced b(

simultaneous bending moments.

Chances o% buc&ling are directl( related

$ith the slenderness ratio KL/r andhence there are three parameters

aecting buc&ling.

1. !ective length %actor #B', $hich

9 b d l th % l #L ' hi h


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. 9nbraced length o% column #Lu', $hich

ma( be the unbraced length in strong

direction or unbraced length in $ea&direction, $hichever gives more ans$er

%or KLu/r.

3. 2adius o% g(ration #r', $hich ma( be rx

or ry #strong and $ea& direction' %or

unia"iall( or bia"iall( s(mmetricalcrosssections and least radius o%

g(ration #rz' %or uns(mmetrical cross

sections li&e angle sections.

: &li ill t & l b t


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a. :uc&ling $ill ta&e place about a

direction %or $hich the corresponding

slenderness ratio is ma"imum.b. For unbraced compression members

consisting o% angle section, the total

length and rzare used in the calculation

o% KL/rratio.

c. For steel braces, bracing is considered

the most eective i% tension is

produced in them, due to buc&ling.

d :races that provide resistance b(


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d. :races that provide resistance b(

bending are less eective and braces

having compression are almostineective because o% their small "

sections and longer lengths.

e. +he brace is considered eective i% itsother end is connected to a stable

structure, $hich is not undergoing

buc&ling simultaneousl( $ith the

braced member.

% +he braces are usuall( provided


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%. +he braces are usuall( provided

inclined to main members o% steel

structures starting %rom midspans toends o% the ad*acent columns.

g. :ecause bracing is most eective in

tension, it is usuall( provided on bothsides to prevent buc&ling on either

side.

h. :racing can be provided to prevent

buc&ling along $ea& a"is. KL/r should

be calculated b( using Ky

, unbraced


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i :racing can also be provided to


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i. :racing can also be provided to

prevent buc&ling along the strong a"is.

KL/r in this case should be calculatedb( using Kx, the unbraced length along

strong a"is and rx.

*. +he end condition o% a particular

unsupported length o% a column at an

intermediate brace is considered a

hinge. +he reason is that the rotation

becomes %ree at this point and onl( the

lateral movement is prevented.

E&&ECTI.E %EN'T/ &ACTOR -1


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E&&ECTI.E %EN'T/ &ACTOR -1

+his %actor gives the ratio o% length o%

hal% sine $ave o% de)ected shape a%ter

buc&ling to %ullunsupported length o%

column.

0n other $ords, it is the ratio o% eective

length to the unsupported length.

+his depends upon the end conditions o%the column and the %act that $hether

sides$a(is permitted or not.

reater the K value greater is the


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reater the K-value, greater is the

eective length and slenderness ratio

and hence smaller is the buc&ling load.

K-value in case o% no sides$a( is

bet$een 7.6 and 1.7, $hereas, in case

o% appreciable sides$a(, it is al$a(s

greater than or equal to 1.7

u

e

ue

L

LKor

KLL

=

=

Si#es2a$


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Si#es2a$

An( appreciable lateral or side$ard

movement o% top o% a vertical column

relative to its bottom is called

si#es2a$, s2a$or lateral #rift.

0% sides$a( is possible, Bvalue increases

b( a greater degree and column buc&les

at a lesser load.Sides$a( in a %rame ta&es place due to/

=engths o% dierent columns are

Sections o% columns have dierent


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Sections o% columns have dierent

crosssectional properties.

=oads are uns(mmetrical.

=ateral loads are acting.

0

0

0

Figure 3.11. Causes o% Sides$a( in a :uilding Frame

Sides$a( ma( be prevented in a %rame


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Sides$a( ma( be prevented in a %rame

b(/

Providing shear or partition $alls.

Fi"ing the top o% %rame $ith ad*oining

rigid structures.

Provision o% properl( designed li%t $ell or

shear $alls in a building, $hich ma( act

li&e bac&bone o% the structure reducingthe lateral de)ections.

Shear wall ! a !tru"tural wall that re!!t

!hear for"e! re!ult# fro% the a le'

Provision o% lateral bracing $hich ma(


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Provision o% lateral bracing, $hich ma(

be o% %ollo$ing t$o t(pes/

1. iagonal bracing, and

. =ongitudinal bracing

#bra"e' *ra%e ! that fra%e # wh"h

the re!!ta#"e to lateral loa' !

&ro'e' by the be#'#$ re!!ta#"e of

fra%e %e%ber! a#' ther "o##e"to#!

wthout a#y a''to#al bra"#$.

- &a!tor for Colmns ha+ing


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-&a!tor for Colmns ha+ing

4ell De5ne# En# Con#itions

!ective length %actor and the buc&led

shape o% some e"ample cases are given

in Figure 3.1.

esign Aids ma( be used %or other end

conditions.

Pa$e + o# 'e!$# a'!


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Make correction


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0


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1

-&a!tor for &rame or 6artiall$


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-&a!tor for &rame or 6artiall$

Restraine# Colmns

Consider the e"ample o% column A:

sho$n in Figure 3.13. +he ends are not

%ree to rotate and are also not per%ectl(

-"ed.

0nstead these ends are partiall( -"ed

$ith the -"it( determined b( the ratio o%relative )e"ural stiness o% columns

meeting at a *oint to the )e"ural

stiness o% beams meeting at that *oint.

+his ratio is denoted b( or and is


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+his ratio is denoted b( or and is

determined %or each end o% the column

b( using the e"pression given belo$/

( )

( )

=beamsofLEI

columnsofLEIendeachatGor


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Alignment charts, given in esign Aids,


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Alignment charts, given in esign Aids,

are then used to -nd the eective length

%actors.+he method to use these charts is

e"plained in Figure 3.18. #+his Figure

does not give the actual values'.

First step is to select the alignment chart

depending upon the presence or

absence o% the sides$a(.

De"t, points are mar&ed on t$o outer

lines %or values o% or at end A and :


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Page 104 on design aids

+hese points are then *oined b( a


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+hese points are then *oined b( a

straight edge and the Bvalue is read

%rom the central line according to itsgraduations.

-.ale for Trss 7 )ra!e#

&rames Members

+he eective length %actor, B, is

considered equal to 1.7 %or members o%the truss braced %rames columns. 0n

case the value is to be used less than

one %or %rame columns, detailed buc&ling

E%ASTIC )UC-%IN' %OAD &OR


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E%ASTIC )UC-%IN' %OAD &OR

%ON' CO%UMNS

A column $ith pin connections on both

ends is considered %or the basic

derivation, as sho$n in the Figure 3.16.

+he column has a length equal to land

is sub*ected to an a"ial compressive

load, P.

:uc&ling o% the column occurs at a

critical compressive load, P"r.


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+he lateral displacement %or the buc&led


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p

position at a heighty%rom the base is u.

+he bending moment at this point is

= P"rx u (+)

+his bending moment is %unction o% thede)ection unli&e the double integration

method o% structural anal(sis $here it is

independent o% de)ection.

+he equation o% the elastic curve is

given b( the !uler:ernoulli !quation,

)(-2

2

IIud

EI =


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0

constant

0

0

)(

2

2

2

2

2

2

2

2

2

(V)uCdy

ud

(IV)is awhere C=CEI

PLet

(III)uEI

P

dy

udor

uPdy

ud

EIor

IIdy

EI

cr

cr

cr

=+

=+

=+

+he solution o% this dierential equation


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q

is/

u = "o! (0 x y) 1 !# (0 x y)(3I)

$here, A and : are the constants o%

integration.

ou#'ary 0o#'to# No. +:

Aty = , u =

= "o! (4) 1 !# (4) =

u = !# (0 x y) (3II)

ou#'ary 0o#'to# No. 2:


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y

Aty = l, u =

*ro% 56. (3II): = !# (0l)

5ther = or !# (0l) =

(3III)0% = , the equation becomes u = ,

giving unde)ected condition. 4nl( the

second alternate is le%t %or the buc&ledshape.

Pcr


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;ence %rom !q. 0E/

+he smallest value o% P"ris %or # = +, andis given belo$/

(X)...2,1,0,=nwherenor),......(3,2,0,=for0=sin

(IX)0=sin=)sin(

!radians!!!""

lEI

PCl

cr

(XI)=

=

2

22

l

EI!nP

!nlEI

P

cr

cr

(XII)= 2

2EI!

P


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For other columns $ith dierent end

conditions, $e have to replace l b( theeective length, le Kl.

+he same e"pression ma( be converted

in terms o% area o% crosssection and

radius o% g(ration using the e"pressionI=r2.

(XII) 2l

Pcr

( )(XIII)= 2

2

Kl

EI!Pcr

=

22E$r!

P


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!quations E00 and E0G give the !ulerelastic critical buc&ling load %or long

columns. 0t is important to note that the

buc&ling load determined %rom !ulerequation is independent o% the strength

o% steel used.

( )

( )

( ) (XV)=and

(XIV)AF==

=

2

2

2

2

2

rKl

E!#

rKl

E$!

KlP

e

e

cr

+he most important %actor on $hich this


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p

load depends is the Kl/r term called the

slenderness ratio.!uler critical buc&ling load is inversel(

proportional to the square o% the

slenderness ratio.

With the increase in slenderness ratio,

the buc&ling strength o% a column

drasticall( reduces.

0n the above !quations/


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Kl/r = slenderness ratio

P"r = !ulerHs critical elastic buc&ling

load

*e = !ulerHs elastic critical buc&lingstress

=ong compression members %ail b(

elastic buc&ling and short compression

members ma( be loaded until the

material (ield or perhaps even goes into

;o$ever, in the vast ma*orit( o% usual


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situations %ailure occurs b( buc&ling

a%ter a portion o% crosssection has(ielded.

+his is &no$n as inelastic buc&ling.

+his variation in column behavior $ith

change o% slenderness ratio is sho$n in

Figure 3.1>.

#o&'ression iedin*


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Fcr

KLr (!)

F"#

$

%

A

!c

Ineastic %+c,in* (strai*ht ine or a 'ara-oic ine

Is ass+&ed

+er/s %+c,in* (astic %+c,in*)

astic %+c,in*

0. F"a''roi&ate"

hort

#o+&ns

Inter&ediate

#o+&ns

Lon*

#o+&ns(KLr)&a


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IuestionsJJJJJJJJJJ

steel th 6-design of compression members

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