Steel Stability
5
4P E . However, the load has no chance of getting there because at P ¼ P E the deflection of the column snaps from an S-shape into a half sine wave. A similar problem was also encountered in Chapter 1. The unexpected occu- rrence of such a snap-through event during the test of a full-sized column in the laboratory is something not easily forgotten. 2.6.2 Column with Eccentric Load (Figure 2.8b) The differential equation is determined by equating the internal moment EIv 00 and the external moment Pðe þ vÞ, where e is the eccentricity of the axial load (see Figure 2.8b) and v is the deflection at the point of interest: EIv 00 ¼ Pðe þ vÞ v 00 þ k 2 v ¼k 2 e e is the load eccentricity (Figure 2.8b), v is the deflection at location e, P is the axial force, and EI is the product of the elastic modulus and the moment of inertia, respectively. The second line is a rearrangement after introducing k 2 ¼ P EI The deflection is found from the general solution v ¼ A sin kx þ B cos kx e with the boundary conditions vð0Þ¼ vðLÞ¼ 0: v ¼ e cos kz þ 1 cos kL sin kL sin kz 1 (2.29) The maximum deflection occurs at the center at z ¼ L=2 and it is equal to: vðL=2Þ¼ e 1 cos kL 2 cos kL 2 ! (2.30) When P ¼ P E ¼ p 2 EI L 2 ; k ¼ p L ; cos kL 2 ¼ cos p 2 ¼ 0; vðL=2Þ¼ e 0 !1 Thus, the deflection is infinite when P equals the elastic buckling load, as expected. The center deflection of a beam with a moment M at each end is equal to (from linear structural analysis): v 1 ðL=2Þ¼ ML 2 8EI ¼ PeL 2 8EI ¼ p 2 8 ðP=P E Þe 48 ELASTIC BUCKLING OF PLANAR COLUMNS More free ebooks : http://fast-file.blogspot.com If we divide the deflection of the column (equation 2.30) by the linear beam deflection above we obtain a magnification factor, MF, that defines the effect of the reduction of stiffness, and thus the increase of deflection, due to the axial force: MF ¼ 8 p 2 ðP=P E Þ 1 cos p 2 ffiffiffiffi P P E q cos p 2 ffiffiffiffi P P E q 2 6 4 3 7 5 (2.31) 2.6.3 Column with Distributed Load (Figure 2.8c) A distributed load is not specifically an imperfection; however, the trans- verse loading on the column creates deflection due to bending and is thus similar to the imperfection cases described in the previous sections. The dif- ferential equation for this case is equation 2.9 with the foundation modulus a ¼ 0: EIv iv þ Pv 00 ¼ q (2.32) This equation is then rearranged and solved for the deflection: v 0 v þ k 2 v 00 ¼ q EI v ¼ A þ Bz þ C sin kz þ D cos kz þ qz 2 2P With the boundary conditions vð0Þ¼ vðLÞ¼ v 00 ð0Þ¼ v 00 ðLÞ¼ 0, the deflection at any location z and at the center of the member is, respectively: v ¼ q Pk 2 1 cos; kL sin kL sin kz þ cos kz þ ðkzÞ 2 2 k 2 Lz 2 1 " # vðL=2Þ¼ q Pk 2 1 cos kL 2 ðkLÞ 2 8 1 ! (2.33) The first-order deflection is 5qL 4 384EI , and therefore the magnification factor MF for this case is equal to MF ¼ 384 5k 4 L 4 1 cos kL 2 ðkLÞ 2 8 1 ! (2.33) The variations of the magnification factors of equations 2.31 (for the eccentric axial load) and 2.33 (for the column with a distributed load) are 2.6 THE EFFECT OF IMPERFECTIONS 49 More free ebooks : http://fast-file.blogspot.com
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Transcript of Steel Stability
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4PE.However,theload
has
nochance
ofgettingtherebecause
atP
PE
thedeflectionofthecolumnsnapsfrom
anS-shapeinto
ahalfsinewave.A
similar
problem
was
also
encounteredin
Chapter1.Theunexpectedoccu-
rrence
ofsuch
asnap-througheventduringthetestofafull-sized
columnin
thelaboratory
issomethingnoteasily
forgotten.
2.6.2
ColumnwithEccentricLoad(Figure
2.8b)
Thedifferential
equationis
determined
byequatingtheinternal
moment
EIv00andtheexternal
momentPe
v,whereeistheeccentricityofthe
axialload
(see
Figure
2.8b)andvisthedeectionatthepointofinterest:
EIv00
Pe
vv00
k2vk
2e
eistheload
eccentricity(Figure
2.8b),visthedeflectionat
locatione,
Pis
theaxialforce,andEIistheproductoftheelasticmodulusandthemoment
ofinertia,respectively.Thesecondlineisarearrangem
entafterintroducing
k2
P EI
Thedeflectionisfoundfrom
thegeneralsolutionv
Asinkx
Bcoskx
ewiththeboundaryconditionsv0
vL
0:
v
ecoskz1coskL
sinkL
sinkz1
(2.29)
Themaxim
um
deflectionoccurs
atthecenterat
z
L=2anditisequal
to:
vL=2
e1coskL 2
coskL 2
!
(2.30)
When
P
PEp2EI
L2
;kp L
;coskL 2
cosp 2
0;vL
=2
e 0!1
Thus,thedeflectionis
infinitewhen
Pequalstheelasticbucklingload,as
expected.
Thecenterdeectionofabeam
withamomentM
ateach
endisequalto
(from
linearstructuralanalysis):
v 1L
=2
ML2
8EI
PeL
2
8EIp2 8P
=PEe
48
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Ifwedividethedeflectionofthecolumn(equation2.30)bythelinearbeam
deflectionaboveweobtain
amagnificationfactor,
MF,that
defines
the
effectofthereductionofstiffness,andthustheincrease
ofdeflection,dueto
theaxialforce:
MF
8
p2P
=PE
1cosp 2
P P Eqcosp 2
P P Eq2 6 4
3 7 5(2.31)
2.6.3
ColumnwithDistributedLoad(Figure
2.8c)
Adistributedload
isnotspecicallyan
imperfection;however,thetrans-
verse
loadingonthecolumncreatesde
ectiondueto
bendingandis
thus
similar
totheim
perfectioncasesdescribed
intheprevioussections.Thedif-
ferential
equationforthiscase
isequation2.9
withthefoundationmodulus
a
0:
EIv
iv
Pv00
q(2.32)
Thisequationisthen
rearranged
andsolved
forthedeflection:
v0v
k2v00
q EI
v
A
Bz
Csinkz
Dcoskz
qz2 2P
With
the
boundary
conditions
v0
vL
v00 0
v00 L
0,the
deflectionatanylocationzandatthecenterofthemem
ber
is,respectively:
v
q
Pk2
1cos;kL
sinkL
sin
kzcoskzkz
22
k2Lz
21
"#
vL=2
q
Pk2
1
coskL 2
kL
28
1
!(2.33)
Thefirst-order
deflectionis
5qL4
384EI,andtherefore
themagnificationfactorMF
forthiscase
isequal
to
MF
384
5k4L4
1
coskL 2
kL
28
1
!
(2.33)
Thevariationsofthe
magnification
factors
ofequations2.31
(forthe
eccentric
axialload)and2.33(forthecolumnwithadistributedload)are
2.6
THEEFFECTOFIMPERFECTIONS
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plotted
inFigure
2.11againstP/P
E.Alsoplotted
isthemagnificationfactor
forthecase
ofinitialcurvature
(equations2.24):
MF
1
1
P=PE
(2.34)
Itis
evidentfrom
comparing
the
curves
inFigure
2.11
that
they
are
essentially
ontopofeach
other,andthusthesimplerequation2.34canbe
usedforallthreecases.This
fact
has
beenusedformanyyears
indesign
standardsallover
theworld.
Historically,theidea
ofassumingan
initialout-of-straightnessoran
acci-
dentalload
eccentricitywas
usedto
arriveat
form
ulasforthedesignof
columns.
They
arenam
ed,variously,
theRankineform
ula,orthePerry-
Robertsonform
ula.Since
such
form
ulaswereusedextensivelythroughout
thelast150years,itisusefulto
giveabrief
derivationatthispoint.
Thecommonfeature
oftheseform
ulasistheassumptionthat
themaxi-
mum
strength
ofthecolumnis
reached
when
thesum
oftheaxialnorm
alstress
andtheexuralnorm
alstress
equalstheyield
stress
ofthematerial.
Thisisaveryim
practicalassumption,ofcourse,as
wewillshowin
Chapter
3,butsince
theinitialdeectioniscalibratedso
thattheform
ulapredictsthe
actual
strength
obtained
bycolumntests,
ausefulandsimple
methodof
designisachieved.ThederivationoftheRankineform
ulaisgiven
rst:
P APv S
P A1
vA S
sy
Mag
nific
atio
n F
acto
r
02
46
810
P/P
E
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Initi
al c
rook
edne
ss
Ecc
entr
ic a
xial
load
Dis
trib
uted
load
Fig.2.11
Comparisonofmagnicationfactors.
50
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Pistheaxialload,Aisthearea
andSistheelasticsectionmodulusofthe
cross-section.syistheyield
stress
andvistheinitialdeflectionatthecenter
ofthecolumn.Note
that
Av S
Acv I
cv r2
cv L2
L r 2,whereIis
themomentof
inertia,cisthedistance
from
theneutralaxisto
theouterfiber
ofthecross-
section,ristheradiusofgyration,andListhecolumnlength.Theform
ula
isthen
inthefollowingform
:
scr
P A
sy
1 cv L2
L r2
sy
1a L r
2(2.35)
Thecoefficientais
obtained
bycalibrationto
test
data.
Forexam
ple,the
columnform
ula
inthe1923Specificationforsteelbuildingstructuresof
theAmerican
InstituteofSteelConstruction(A
ISC)was
ofthisform
.There
weremanyvariants
inuse,andthereprobably
arestillextantsomecodes
that
use
it.In
the1923AISCSpecification,theform
ula
isas
followsfora
yield
stress
of33ksiandafactorofsafety
of33=18:
scr
FS
18scr
33
18
1
118;000
L r2
15ksi
(2.36)
ThePerry-Robertsonform
ula
has
beenusedin
manycountries,
anditis
stillin
the
currentAustralian
steeldesign
standard.It
isderived
asfollows:
v
v o
1
P=PE
v o
1
scr
sE
v osE
sEscr
syscr
1h
sE
sEscr
Solvingforthecritical
stress,oneobtains
scr
1 2s
ysE1h
1 2sysE1h
2
sysE
s(2.37)
Empirically
intheAustralian
code,h
0:003L r.
2.6
THEEFFECTOFIMPERFECTIONS
51
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2.7
STABILITYOFARIGID
FRAME
Nextweconsider
theeffect
ofelasticend-restraintonthecritical
load
ofa
column.Thestructure
isshownin
Figure
2.12.Thecolumnhas
apin
atits
bottom,anditisrestrained
atthetopbyan
elasticbeam
thathas
axed
end
atitsfarend.
Theboundaryconditionsat
thebottom
ofthecolumnat
z
Lareequal
tovL
v00 L
0.Atthetopthereisnodeection,andtheslopeat
the
topofthecolumnequalstheslopeattheendofthebeam.Thebendingmo-
mentsoppose
each
other.From
structuralanalysisitcanbedetermined
that
attheendofthebeam
MAB
4EI B
LB
uAauAav00
Thesymbolaisaspringconstantthatisa
4EI B
LB
when
thefarendisfixed,
anda
3EI B
LB
when
thefarendispinned.Themomentat
thetopendofthe
AB
C
I B LB
I CL C
P
A
A
A
MAB
MAB
/2
z
Fig.2.12
Restrained
column.
52
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columnequals
MACE
I Cv00 0
From
theequilibrium
conditionM
AB
MAC
0,wethen
get
thefourth
boundarycondition.Thefourboundaryconditionsaresummarized
next:
v0
0
av00
EI Cv00 0
0
vLC
0
v00 L
C
0
Substitutionofthedeflectionequationv
A
Bz
Csinkz
Dcoskz
and
itsderivatives
gives
fourhomogeneoussimultaneousequations.Settingthe
determinantofthecoefficientsequal
tozero
10
01
0a
ak
P
1L
sinkL
CcoskL
C
00
k2sinkL
Ck
2coskL
C
0
leadsto
thefollowingeigenfunction:
tankL
C
akL
C
PLCa
gkL
C
kLC2g
gaLC
EI C
kLC
PL2 C EI Cs(2.39)
Equation2.39is
thebucklingequationforacolumnwithapinned
endat
oneendandan
elasticspringat
theother
end.When
I Bag
0,the
endrestraintvanishes
andwehaveapinned-endcolumn;that
is,sinkL
C
0!
Pcrp2EI C
=L2 C.When
thetopendisfixed,thefollowingholds:
I Bag1
tankL
C
kLC
kLC2
g1
kL
C
Pcr
20:19EI C L2 C
2.7
STABILITYOFARIGID
FRAME
53
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Comparingtheseextrem
esto
thecorrespondingcasesin
Table2.1
itisseen
that
thesameansw
ersareobtained.Thevariationofthecritical
load
with
thespringconstantbetweenthetwoextrem
esisshownin
Figure
2.13.
Theupper
graphshowsthat
thecritical
load
isPEwhen
g
0,andit
approaches
2.045PEas
thevalueofthespringconstantapproaches
innity.
Thelower
graphillustratesthevariationoftheeffectivelength
factorKfrom
1.0
(pinned
end)to
0.7
(xed
end).Thereisan
importanttrendthat
canbe
deducedfrom
thesecurves:Ontheonehand,when
therestraintis
small,
largeincreasesofthebucklingload
resultfrom
smallincreasesofthespring
constanta.Ontheother
hand,when
abecomes
verylarge,
averysm
all
changein
thebucklingload
resultsfrom
verylargechanges
inthespring
=
LC /
EIC
020
4060
8010
0
P / P
E
1.0
1.2
1.4
1.6
1.8
2.0
2.2
=
LC /
EIC
020
4060
8010
0
Effective Length K
0.5
0.6
0.7
0.8
0.9
1.0
Fig.2.13
Effectofend-restraint.
54
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constant.A
little
restraintgoes
alongway,
butcomplete
restraintis
not
worthattaining.Thisprinciple
isageneral
characteristic
ofbucklingsolu-
tionsdiscussed
inmore
detailin
Chapter5.
2.8
END-RESTRAINEDCOLUMNS
Inthis
sectionweconsider
thegeneral
case
ofprism
atic
columnsthat
are
restrained
byelasticspringsat
theirends.Byconsideringrestrained
ends,
wecandevelopafeel
fortheim
pactofendrestraintonthebucklingload
ofthecolumn.This
situationis
similar
toacolumnrestrained
bybeams
ofnitestiffness,
whichis
discussed
indepth
inChapter5.Westartthe
discussionwithacompressionmem
ber
whose
endsdonottranslatewith
respectto
each
other
(often
called
anon-swaycase)andthat
haveelastic
springsateach
end.Thecolumnanditsendboundaryconditionsareshown
inFigure
2.14.
Substitutionofthefourboundaryconditionsinto
equation2.12resultsin
fourhomogeneoussimultaneousequations.Thedeterminantofthecoef-
cientsoftheconstantsA,B,C,Disequalto
10
01
1L
sinkL
coskL
0aT
aTk
EIk
2
0a
Ba
BkcoskL
EIk
2sinkL
aBksinkL
EIk
2coskL
0
v
z
L
P P
T
B
Tv
'(0)
Bv
'(L)
E
Iv''(0)
E
Iv''(L
)
Boun
dary
con
ditio
ns:
v (0) =
0
Tv'(0
) EI
v''(0)
= 0
v(L)
= 0
Bv'(L
) E
Iv''(L
) = 0
Fig.2.14
Thenonsw
ayrestrained
column.
2.8
END-RESTRAINEDCOLUMNS
55
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Rem
embering
that
k
P EIqand
introducing
thenondim
ensional
spring
constantratios
RTaTL
EI
RBaBL
EI
(2.40)
thealgebraic
decompositionofthedeterminantresultsin
thefollowing
eigenfunction: 2RTRBsinkLR
TRBkL
kLR
T
RB
kL3
coskL2R
TRBkL
2 RT
RB
0(2.41)
Solvingequation2.41numerically
forthesm
allest
kLgives
thecritical
bucklingload.Thelimitingcasesofthisequationarethecasesofboth
ends
pinned
(aTaB
0!
RT
RB
0),andofboth
endsfixed
(aTaB
1!
RT
RB1).
After
somealgebraic
andtrigonometricmanipula-
tionsitcanbedem
onstratedtheeigenfunctionforthepinned
endcolumnis
equal
tosinkL
0andforthefixed
endcolumnitequalssinkL 2
0.These
areindeedthesamefunctionsas
areshownin
Table
2.1
forCases
IandIII.
Thus,
equation2.41enclosesalltheinterm
ediate
conditionsbetweenthe
totallypinned
endsandthetotallyfixed
ends.Thecritical
load
thusvaries
from
Pcr
PEto
4PE,andtheeffectivelength
variesfrom
K
1:0
to0.5.
Thebucklingconditionofequation2.41isdirectlyapplicable
forthesit-
uationwheretheelasticrotational
springconstantsaTandaBareknown.
Following,weconsider
thespecializationoftheexpressionforthecase
ofa
planar
rigid
fram
e.Such
anapplicationiswithin
theeveryday
task
ofstruc-
turaldesignengineers.Anexam
ple
isillustratedin
Figure
2.15.Weassume
that
thefarendsofthetopandbottom
beamshavethesameslopeas
the
nearends.Thisisnotthecorrectsituationforthisgiven
problem,butitis
theassumptionthat
governstheeffectivelength
determinationin
theAISC
Specication(A
ISC2005).
Thetopandbottom
springconstantsare:
aT
2EI g
T
LgT
!RT
2I g
T=LgT
I C=LC
(2.42)
aB
2EI g
B
LgB
!RB
2I g
B=LgB
I C=LC
(2.43)
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