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13. Buckling of Columns

CHAPTER OBJECTIVES

Discuss the behavior ofcolumns.

Discuss the buckling ofcolumns.

Determine the axial loadneeded to buckle an idealcolumn.

Analyze the buckling with

bending of a column. Discuss inelastic buckling of a column. Discuss methods used to design concentric and

eccentric columns.

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CHAPTER OUTLINE

1. Critical oad

2. !deal Column with "in #u$$orts

%. Columns &aving 'arious (y$es of #u$$orts

). *(he #ecant +ormula

,. *!nelastic -uckling

. *Design of Columns for Concentric oading

/. *Design of Columns for 0ccentric oading

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13.1 CRITICAL LOA

ong slender members subected to axialcom$ressive force are called columns.

(he lateral deflection that occurs iscalled buckling.

(he maximum axial load a column cansu$$ort when it is on the verge ofbuckling is called the critical loadPcr.

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13.1 CRITICAL LOA

#$ring develo$s restoring forceF = k whilea$$lied load Pdevelo$s two horizontalcom$onentsPx= P tan which tends to $ush the$in further out of e3uilibrium.

#ince is small4 5L627 and tan . (hus restoring s$ring

force becomesF = kL/2 anddisturbing force is2Px4 2P.

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13.1 CRITICAL LOA

+or k

L/

2 8 2P,

+or kL/2 9 2P,

+or kL/2 4 2P,

me3uilibriustable4

kLP

me3uilibriuneutral4

kLPcr=

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13.! IEAL COLU"N #ITH PIN SUPPORTS

An ideal column is $erfectly straight before loadingmade of homogeneous material and u$on whichthe load is a$$lied through the centroid of the x:section.

;e also assume that the material behaves in alinear:elastic manner and the column buckles orbends in a single $lane.

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!n order to determine the critical load and buckledsha$e of column we a$$ly 03n 12:1

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#umming momentsM4 P

,

03n 1%:1becomes

?eneral solution is

#ince 4 < atx4

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Disregarding trivial soln for C14

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E$A"PLE 13.1 %SOLN&

Fse 03n 1%:, to obtain critical load withEst4 2

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E$A"PLE 13.1 %SOLN&

(his force creates an average com$ressive stress inthe column of

#ince cr9 Y 4 2,< "a a$$lication of 0ulers e3n

is a$$ro$riate.

( )

( ) ( )[ ]MPa100N/mm2.100

mm7075

N/kN1000kN2.228

2

222

==

==

A

Pcrcr

13 B kli f C l

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Ix4 ),.,1

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E$A"PLE 13.! %SOLN&

;hen fully loaded average com$ressive stress in

column is

#ince this stress exceeds yield stress 52,< E6mm27the load " is determined from sim$le com$ressionG

( )

2

2

N/mm5.320

mm5890

N/kN1000kN6.1887

=

==A

Pcrcr

kN5.1472

mm5890N/mm250

22

=

=

P

P

13 B kli f C l

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13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS

+rom free:body diagramM4P57. Differential e3n for the deflection curve is

#olving by using boundary conditionsand integration we get

( )7-132

2

EI

P

EI

P

dx

d =+

( )8-13cos1

= x

EI

P

13 B kli f C l

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(hus smallest critical load occurs when n4 1 sothat

-y com$aring with 03n 1%:, a column fixed:

su$$orted at its base will carry only one:fourth thecritical load a$$lied to a $in:su$$orted column.

( )9-134 2

2

L

EIPcr

=

13 B ckling of Col mns

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0ffective length !f a column is not su$$orted by $inned:ends then

0ulers formula can also be used to determine thecritical load.

HLI must then re$resent the distance between thezero:moment $oints.

(his distance is called the columns effective length

Le.

13 Buckling of Columns

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0ffective length


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0ffective length any design codes $rovide column formulae that

use a dimensionless coefficientK known as theeeffective:length factor.

(hus 0ulers formula can be ex$ressed as( )10-13KLLe=

( ) ( )

( )( )12-13

11-13

2

2

2

2

rKL

E

KL

EI

P

cr

cr

=

=


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0ffective length &ere 5KL/r7 is the columns effective:slenderness

ratio.


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E$A"PLE 13.3

A ;1, mlong and is fixed at its ends asshown. !ts load:carrying ca$acityis increased by bracing it aboutthey-yaxis using struts that areassumed to be $in:connectedto its mid:height. Determine theload it can su$$ort s$ that the

column does not buckle normaterial exceed the yield stress.

(akeEst4 2

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*13.+ THE SECANT )OR"ULA

(he secant formula aximum stress in column occur when

maximum moment occurs at thecolumns mid$oint.

Fsing 03ns 1%:1% and 1%:1

aximum stress is com$ressive and

( )

( )18-13

2

s"c

max

=

+=

L

EI

PPe

eP

+=+=

2s"c# maxmax

L

EI

P

I

Pec

A

P

I

Mc

A

P


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*13.+ THE SECANT )OR"ULA

(he secant formula #ince radius of gyration r2= I/A

max4 maximum elastic stress in column at innerconcave side of mid$oint 5com$ressive7.

P4 vertical load a$$lied to the column. P9Pcr

unless e4

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) m 4 >

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g

E$A"PLE 13., %SOLN&

x-x axis yieldingG

#olving forPxby trial and error noting that argumentfor secant is in radians we get

#ince this value is less than 5Pcr7y4 ,1% kE failurewill occur about thex-xaxis.

Also 4 )1@.)1< mm24 ,,.% "a 9 Y 4 2,< "a.

kN4.419N419368 ==xP


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g

*13.- INELASTIC BUCLIN'

A$$lication of 0ulers e3uation re3uires that the

stress in column remain -0; the materials yield$oint when column buckles. #o it only a$$lies tolong slender columns.

!n $ractice most areintermediate columns so wecan study their behavior bymodifying 0ulers e3uation to

a$$ly for inelastic buckling. Consider a stress:strain

diagram as shown.


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*13.- INELASTIC BUCLIN'

"ro$ortional limit is pl and

modulus of elasticityEisslo$e of lineA.

A $lot of 0ulers hy$erbola is

shown having a slendernessratio as small as 5KL/r7plsince at this $t cr4 pl.

;hen column about to buckle

change in strain that occurs iswithin a small range soEfor material can be taken asthe tangent modulusE!.


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,==r

KL


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E$A"PLE 13.11

A board having x:sectional

dimensions of 1,< mm by )< mmis used to su$$ort an axial loadof 2< kE.

!f the board is assumed to be$in:su$$orted at its to$ and basedetermine its greatest allowablelengthLas s$ecified by the E+"A.


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E$A"PLE 13.11 %SOLN&

-y ins$ection board will buckle about the y axis. !n

the E+"A e3ns d4 )< mm.Assuming that 03n 1%:2@ a$$lies we have

( )

( )( ) ( ) ( )

mm1336

mm40/1

N/mm3718

mm40mm150

N1020

/

MPa3718

2

23

2

=

=

=

L

L

dKLA

P


13 11 %SO &

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E$A"PLE 13.11 %SOLN&

&ere

#ince 2 9KL/d,

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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'

A column may be re3uired to

su$$ort a load acting at itsedge or on an angle bracketattached to its side.

(he bending momentM = Pe

caused by eccentric loadingmust be accounted for whencolumn is designed.

Fse of available column formulae

#tress distribution acting over x:sectional area ofcolumn shown is determined from both axial force Pand bending momentM = Pe.


*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'

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Fse of available column formulae

aximum com$ressive stress is

A ty$ical stress $rofile is also shown here. !f we assume entire x:section is subected to uniform

stress max then we can com$are it with allow which

is determined from formulae given in cha$ter 1%.. !f maxallow then column can carry the s$ecifiedload.

( )30-13maxI

Mc

A

P+=



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!nteraction formula

!t is sometimes desirable to see how the bendingand axial loads interact when designing aneccentrically loaded column.

!f allowable stress for axial load is 5

a7allow thenre3uired area for the column needed to su$$ort theloadPis

( )all#$aa

PA

=



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!nteraction formula

#imilarly if allowable bending stress is 57allow thensinceI = Ar2 re3uired area of column needed toresist eccentric moment is determined from flexureformula

( ) 2r

Mc

Aall#$b

b

=



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!nteraction formula

(hus total areaAfor the column needed to resistboth axial force and bending moment re3uires that

( ) ( )

( ) ( )

( ) ( )( )31-131

1

/

2

2

2

+

+

+=+

r

McP

r#r

AArMcP

AA

all#$ball#$a

all#$b

b

all#$a

a

all#$ball#$aba



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!nteraction formula

a4 axial stress caused by forcePand determinedfrom a4P/A whereAis the x:sectional area of thecolumn.

4 bending stress caused by an eccentric load ora$$lied momentMM is found from 4Mc/I whereIis the moment of inertia of x:sectional areacom$uted about the bending or neutral axis.

( ) ( )

( ) ( )( )31-131

1

2

2

+

+

r

McP

r

all#$ball#$a

all#$b

b

all#$a

a



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!nteraction formula

5a7allow4 allowable axial stress as defined by formulaegiven in cha$ter 1%. or by design code s$ecs. Fsethe largest slenderness ratio for the columnregardless of which axis it ex$eriences bending.

57allow4 allowable bending stress as defined by codes$ecifications.

( ) ( )

( ) ( )( )31-131

1

2

2

+

+

r

McP

r

all#$ball#$a

all#$b

b

all#$a

a



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!nteraction formula

03n 1%:%1 is sometimes referred to as theinteraction formula.

(his a$$roach re3uires a trial:and:check $rocedure.

Designer needs to choose an available column andcheck to see if the ine3uality is satisfied. !f not a larger section is $icked and the $rocess

re$eated. American !nstitute of #teel Construction s$ecifies

the use of 03n 1%:%1 only when the axial:stress ratioa65a7allow 0.15.


E$A"PLE 13 1!

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E$A"PLE 13.1!

Column is made of 2

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E$A"PLE 13.1! %SOLN&

K4 2. argest slenderness ratio for column is

-y ins$ection 03n 1%:2 must be used 52//.1 8 ,,7.

( )

( ) ( ) ( )[ ] ( ) ( )[ ]1.277

mm80mm40/mm408012/1

mm160023

mm

==

r

KL

( ) ( )MPa92.4

1.277

MPa378125MPa37812522ao

===rKL


E$A"PLE 13 1! %SOLN&

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E$A"PLE 13.1! %SOLN&

Actual maximum com$ressive stress in the column is

determined from the combination of axial load andbending.

Assuming that this stress is uniform over the x:section instead of ust at the outer boundary

( )

( ) ( ) ( )( ) ( ) ( )

P

PP

I

cPe

A

P

00078125.0

mm80mm4012/1mm40mm20

mm80mm40 3

max

=

+=

+=

kN30.6N6.6297

00078125.092.4#maxao

====P

P


E$A"PLE 13 13

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