ChapterVII

IN-PLANE BENDING

1

Beam axis

Crosssection( totheaxis)AreaA

DEFINITION OF A BEAM

CROSSSECTIONDIMENSIONS

INTERNAL FORCES IN CROSS-SECTION

G

EXTERNALFORCESANDREACTIONS CUT stresses(equilibrium )INTERNALFORCES

3

AxisCrosssection( totheaxis)

DEXTORSUMSYSTEMOFAXES

INTERNAL FORCES IN CROSS-SECTION

4

IN-PLANE BENDING5

ShorteningCompressivestresses

ElongationTensile stresses

=0somewhereinbetween(neutral axis)

BEAMDEFORMATIONUNDERCONSTANTMOMENT:CIRCLE

BERNOUILLIS PRINCIPLE6

Crosssectionsremain flatandperpendicular tothebeam axis

IN-PLANE BENDING7

Sliceofunitlength

IN-PLANE BENDING8

andso

IN-PLANE BENDING9

Lineardistributionof and

y

Linear distributionofdeformations

Hookes law

Linear distributionofstresses

y

E yE E

ENA

IN-PLANE BENDING10

Determinationoftheneutralaxisand Longitudinalequilibrium(alongthebeamaxis)

Neutralaxiscorrespondstothecentreofgravity

0 0A A

EdA ydA

IN-PLANE BENDING11

Determinationoftheneutralaxisand Equilibriuminrotation

as

2

A A

EydA y dA M 2

A

y dA I1 M

EI

Flexural rigidity

IN-PLANE BENDING12

Stressdistribution

1 M yand EEI

: MyNavierI

yENA

IN-PLANE BENDING13

Navier applicabletosymmetricalandnonsymmetricalcrosssectionsaslongastheplaneofbendingcorrespondstooneoftheprincipalaxesofthecrosssection

y

z

IN-PLANE BENDING14

Navier notdirectlyapplicabletocrosssectionssubjectedtobendingnotappliedaboutthemainaxes

Navier stillassumedtobevalidforbeamssubjectedtononconstantbendingmomentsalongthebeamlength

IN-PLANE BENDING15

Mmax = pL/8T1 = PL/2

T2 = -PL/2

p

LimitationsoftheNavier lineardistribution

IN-PLANE BENDING16

b

h

Rectangularcrosssection

Navier stillassumedtobevalidforbeamssubjectedtononconstantbendingmomentsalongthebeamlength

IN-PLANE BENDING17

Mmax = Pa(L-a)/LT1 = P(L-a)/L

T2 = -Pa/L

P

LimitationsoftheNavier lineardistribution

IN-PLANE BENDING18

LimitationsoftheNavier lineardistribution

Inpractice: Localeffect(StVenants principle)

Localyieldingundertheconcentratedload

Transversestiffeners Bearingplateplate

IN-PLANE BENDING19

Navier stillassumedtobevalidfortaperedbeams(smoothcrosssectionvariation)

IN-PLANE BENDING20

LimitationsoftheNavier lineardistribution

IN-PLANE BENDING21

LimitationsoftheNavier lineardistribution

IN-PLANE BENDING22

!Stressconcentration

cfr.tension

h

h

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

IN-PLANE BENDING Adaptcrosssectionshape

Similar valuesofAandh: Icrosssection: W=0,32Ah rect.crosssection: W=0,167Ah

Isectiontwice moreresistant

max

IN-PLANE BENDING Adaptcrosssectionshape

Similar valuesofAandh:

W=0,32Ah

W=0,5Ah

IN-PLANE BENDING Adaptcrosssectionshape

Beamsmadeoftwodifferentmaterials

IN-PLANE BENDING29

E0b modulus

Ea modulus Ea modulus

dy

Equal forcesintheslicedy E0bbdy=Eab1dy

Beamsmadeoftwodifferentmaterials

IN-PLANE BENDING30

E0b modulus

Ea modulus Ea modulus

0

1b

a

Eb bE

Beamsmadeoftwodifferentmaterials

IN-PLANE BENDING31

b

a a=Eaa

int a,intEaintb,intE0b/Ea int

bE0b/Ea b,equ

EQUIVALENTSECTION ACTUALSECTION

b,equEab

aEaa

intEaint

Beamsmadeoftwodifferentmaterials

IN-PLANE BENDING32

0

1 ( tan. ) ( )b ba a

E Eb b ins loading or b after creepE E