chapitre 07
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ChapterVII
IN-PLANE BENDING
1
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Beam axis
Crosssection( totheaxis)AreaA
DEFINITION OF A BEAM
CROSSSECTIONDIMENSIONS
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INTERNAL FORCES IN CROSS-SECTION
G
EXTERNALFORCESANDREACTIONS CUT stresses(equilibrium )INTERNALFORCES
3
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AxisCrosssection( totheaxis)
DEXTORSUMSYSTEMOFAXES
INTERNAL FORCES IN CROSS-SECTION
4
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IN-PLANE BENDING5
ShorteningCompressivestresses
ElongationTensile stresses
=0somewhereinbetween(neutral axis)
BEAMDEFORMATIONUNDERCONSTANTMOMENT:CIRCLE
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BERNOUILLIS PRINCIPLE6
Crosssectionsremain flatandperpendicular tothebeam axis
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IN-PLANE BENDING7
Sliceofunitlength
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IN-PLANE BENDING8
andso
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IN-PLANE BENDING9
Lineardistributionof and
y
Linear distributionofdeformations
Hookes law
Linear distributionofstresses
y
E yE E
ENA
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IN-PLANE BENDING10
Determinationoftheneutralaxisand Longitudinalequilibrium(alongthebeamaxis)
Neutralaxiscorrespondstothecentreofgravity
0 0A A
EdA ydA
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IN-PLANE BENDING11
Determinationoftheneutralaxisand Equilibriuminrotation
as
2
A A
EydA y dA M 2
A
y dA I1 M
EI
Flexural rigidity
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IN-PLANE BENDING12
Stressdistribution
1 M yand EEI
: MyNavierI
yENA
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IN-PLANE BENDING13
Navier applicabletosymmetricalandnonsymmetricalcrosssectionsaslongastheplaneofbendingcorrespondstooneoftheprincipalaxesofthecrosssection
y
z
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IN-PLANE BENDING14
Navier notdirectlyapplicabletocrosssectionssubjectedtobendingnotappliedaboutthemainaxes
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Navier stillassumedtobevalidforbeamssubjectedtononconstantbendingmomentsalongthebeamlength
IN-PLANE BENDING15
Mmax = pL/8T1 = PL/2
T2 = -PL/2
p
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LimitationsoftheNavier lineardistribution
IN-PLANE BENDING16
b
h
Rectangularcrosssection
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Navier stillassumedtobevalidforbeamssubjectedtononconstantbendingmomentsalongthebeamlength
IN-PLANE BENDING17
Mmax = Pa(L-a)/LT1 = P(L-a)/L
T2 = -Pa/L
P
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LimitationsoftheNavier lineardistribution
IN-PLANE BENDING18
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LimitationsoftheNavier lineardistribution
Inpractice: Localeffect(StVenants principle)
Localyieldingundertheconcentratedload
Transversestiffeners Bearingplateplate
IN-PLANE BENDING19
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Navier stillassumedtobevalidfortaperedbeams(smoothcrosssectionvariation)
IN-PLANE BENDING20
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LimitationsoftheNavier lineardistribution
IN-PLANE BENDING21
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LimitationsoftheNavier lineardistribution
IN-PLANE BENDING22
!Stressconcentration
cfr.tension
- h
- h
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h1
IN-PLANE BENDING Beamcrosssectionverification
11,max 1
1
22,max 2
2
Mh M RI WMh M RI W
1 1 2 2min( ; )M WR W R
h2
G
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IN-PLANE BENDING Adaptcrosssectionshape
Similar valuesofAandh: Icrosssection: W=0,32Ah rect.crosssection: W=0,167Ah
Isectiontwice moreresistant
max
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IN-PLANE BENDING Adaptcrosssectionshape
Similar valuesofAandh:
W=0,32Ah
W=0,5Ah
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IN-PLANE BENDING Adaptcrosssectionshape
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Beamsmadeoftwodifferentmaterials
IN-PLANE BENDING29
E0b modulus
Ea modulus Ea modulus
dy
Equal forcesintheslicedy E0bbdy=Eab1dy
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Beamsmadeoftwodifferentmaterials
IN-PLANE BENDING30
E0b modulus
Ea modulus Ea modulus
0
1b
a
Eb bE
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Beamsmadeoftwodifferentmaterials
IN-PLANE BENDING31
b
a a=Eaa
int a,intEaintb,intE0b/Ea int
bE0b/Ea b,equ
EQUIVALENTSECTION ACTUALSECTION
b,equEab
aEaa
intEaint
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Beamsmadeoftwodifferentmaterials
IN-PLANE BENDING32
0
1 ( tan. ) ( )b ba a
E Eb b ins loading or b after creepE E
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