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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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13. Buckling of Columns
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. Buckling of Columns
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$$ort when it is on the verge ofbuckling is called the critical loadPcr.
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13. Buckling of Columns
13.1 CRITICAL LOA
#$ring develo$s restoring forceF = k whilea$$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$$lied through the centroid of the x:section.
;e also assume that the material behaves in alinear:elastic manner and the column buckles orbends in a single $lane.
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13. Buckling of Columns
13.! IEAL COLU"N #ITH PIN SUPPORTS
!n order to determine the critical load and buckledsha$e of column we a$$ly 03n 12:1
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13.! IEAL COLU"N #ITH PIN SUPPORTS
#umming momentsM4 P
,
03n 1%:1becomes
?eneral solution is
#ince 4 < atx4
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13.! IEAL COLU"N #ITH PIN SUPPORTS
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$ressionG
( )
2
2
N/mm5.320
mm5890
N/kN1000kN6.1887
=
==A
Pcrcr
kN5.1472
mm5890N/mm250
22
=
=
P
P
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13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
+rom free:body diagramM4P57. Differential e3n for the deflection curve is
#olving by using boundary conditionsand integration we get
( )7-132
2
EI
P
EI
P
dx
d =+
( )8-13cos1
= x
EI
P
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13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
(hus smallest critical load occurs when n4 1 sothat
-y com$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$$orted column.
( )9-134 2
2
L
EIPcr
=
13 B ckling of Col mns
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13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
0ffective length !f a column is not su$$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.
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13. Buckling of Columns
13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
0ffective length
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13. Buckling of Columns
13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
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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13. Buckling of Columns
13.3 COLU"NS HAVIN' VARIOUS T(PES O) SUPPORTS
0ffective length &ere 5KL/r7 is the columns effective:slenderness
ratio.
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13. Buckling of Columns
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$$ort s$ that the
column does not buckle normaterial exceed the yield stress.
(akeEst4 2
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*13.+ THE SECANT )OR"ULA
(he secant formula aximum stress in column occur when
maximum moment occurs at thecolumns mid$oint.
Fsing 03ns 1%:1% and 1%:1
aximum stress is com$ressive and
( )
( )18-13
2
s"c
max
=
+=
L
EI
PPe
eP
+=+=
2s"c# maxmax
L
EI
P
I
Pec
A
P
I
Mc
A
P
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13. Buckling of Columns
*13.+ THE SECANT )OR"ULA
(he secant formula #ince radius of gyration r2= I/A
max4 maximum elastic stress in column at innerconcave side of mid$oint 5com$ressive7.
P4 vertical load a$$lied to the column. P9Pcr
unless e4
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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. Buckling of Columns
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$$ort a load acting at itsedge or on an angle bracketattached to its side.
(he bending momentM = Pe
caused by eccentric loadingmust be accounted for whencolumn is designed.
Fse of available column formulae
#tress distribution acting over x:sectional area ofcolumn shown is determined from both axial force Pand bending momentM = Pe.
13. Buckling of Columns
*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
Fse of available column formulae
aximum com$ressive stress is
A ty$ical stress $rofile is also shown here. !f we assume entire x:section is subected to uniform
stress max then we can com$are it with allow which
is determined from formulae given in cha$ter 1%.. !f maxallow then column can carry the s$ecifiedload.
( )30-13maxI
Mc
A
P+=
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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!nteraction formula
!t is sometimes desirable to see how the bendingand axial loads interact when designing aneccentrically loaded column.
!f allowable stress for axial load is 5
a7allow thenre3uired area for the column needed to su$$ort theloadPis
( )all#$aa
PA
=
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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!nteraction formula
#imilarly if allowable bending stress is 57allow thensinceI = Ar2 re3uired area of column needed toresist eccentric moment is determined from flexureformula
( ) 2r
Mc
Aall#$b
b
=
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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!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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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!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$$lied momentMM is found from 4Mc/I whereIis the moment of inertia of x:sectional areacom$uted about the bending or neutral axis.
( ) ( )
( ) ( )( )31-131
1
2
2
+
+
r
McP
r
all#$ball#$a
all#$b
b
all#$a
a
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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!nteraction formula
5a7allow4 allowable axial stress as defined by formulaegiven in cha$ter 1%. or by design code s$ecs. Fsethe largest slenderness ratio for the columnregardless of which axis it ex$eriences bending.
57allow4 allowable bending stress as defined by codes$ecifications.
( ) ( )
( ) ( )( )31-131
1
2
2
+
+
r
McP
r
all#$ball#$a
all#$b
b
all#$a
a
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*13 / ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
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*13./ ESI'N O) COLU"NS )OR ECCENTRIC LOAIN'
!nteraction formula
03n 1%:%1 is sometimes referred to as theinteraction formula.
(his a$$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$ecifies
the use of 03n 1%:%1 only when the axial:stress ratioa65a7allow 0.15.
13. Buckling of Columns
E$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$ection 03n 1%:2 must be used 52//.1 8 ,,7.
( )
( ) ( ) ( )[ ] ( ) ( )[ ]1.277
mm80mm40/mm408012/1
mm160023
mm
==
r
KL
( ) ( )MPa92.4
1.277
MPa378125MPa37812522ao
===rKL
13. Buckling of Columns
E$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
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E$A"PLE 13 13
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