This document summarizes key concepts related to beam deflection and bending moments:
1. Curvature and bending moment are directly related - as the bending moment increases, so does the curvature of the beam. The elastic curve depicts this relationship graphically.
2. Deflection and rotation angle can be calculated based on the radius of curvature. The rotation angle is equal to the arc length divided by the radius. Deflection is calculated using integrals that relate the slope, rotation angle, and bending moment.
3. The differential equation relating deflection (y), slope (dy/dx), bending moment (M), and flexural rigidity (EI) is: d2y/dx2 = -M
1. T.Chhay
PaBdabrbs;Fñwm
Deflection of beam
1> kMeNag nigm:Um:g;Bt; Curvature and bending moment
enAeBlFñwmsamBaØmYyrgnUvkMlaMgbBaÄrxageRkA kugRtaMgTaj)anekIteLIgenAEpñkxagelIénGkS½NWt
nig kugRtaMgsgát;enAEpñkxageRkam. sésEdlrgkarTaj lUtEvgCagmun ÉsésEdlrgkarsgát; rYjxøICag
mun. EdlkarenHeFVIeGayFñwm mankMeNag b¤dab. CaTUeTA bøg;NWtRtUv)aneKeGayeQμaHfa ExSeGLasÞic
(elastic curve). enaHkaMkMeNagRtUv)anKNnatamrUbxageRkam³
eKman HD' CabMErbMrYlragBIRbEvgedIm C' H O Center of
curvature
enaHeK)an bMErbMrYlrageFob ε = C ' H = JF
HD' HD'
tamc,ab;h‘Ukm:UDuleGLasÞic E = εs dθ
R= Radius of curvature
HD' ( R = OJ = OF )
⇒ sb = Eε = E ( )
JF
Elastic curve
kñúgkrNIkMeNagmantMéltUc/ eRbobeFobRtIekaN G
D' FH nigRtIekaN JOF eK)an
A' B'
c HD'
= J F
R JF c
dθ
c
⇒ sb = E
R C' D' Segment of
H
mü:ageTot tamsmIkarkugRtaMgm:Um:g;Bt; dl
loaded beam
dx
M .c
sb =
I
c c
⇒E =M
R I
1 M
⇒ =
R EI
R=
EI
M
b¤
2> PaBdab nigmMurgVil Deflection and rotation angle
θ
eday k = R = ddl
1 Y
dθ M
⇒ =
dl EI dθ
eday tan θ = dy
dx
R dθ=θ1−θ2
dl
d d2y
⇒ tan θ = 2
dx dx
dθ d 2 y
⇒ (1 + tan 2 θ ) = θ1 θ2
dx dx 2 X
PaBdabrbs;Fñwm 99
2. T.Chhay
d2y dy dθ
⇒ 2
= [1 + ( ) 2 ]
dx dx dx
2
d y
dθ 2
⇒ = dx
dx 1 + ( dy ) 2
dx
mü:ageTot dx
dl dl
=
1
= 2
1
2
=
1
dx + dy 12 [1 + ( dy ) 2 ] 12
( )
dx dx 2 dx
2
d y
dθ dθ dx
dUcenH k=
dl
= ( )( ) =
dx dl
dx 2
dy 3
[1 + ( ) 2 ] 2
dx
kñúgedaytMél dy 2
( ) →0
dx
dθ d 2 y
⇒k = =
dl dx 2
d2y M
⇒ 2 =
dx EI
b¤ y' ' =
M
EI
sMrab;krNI TisedArbs;GkS½ Y eLIgelI
EtebITisedA Y cuHeRkamenaH
2
⇒
d y
dx
=−2
M
EI
b¤ y' ' = − EI
M
Edl y CaPaBdab
eday dy = tan θ = θ edaysar θ CamMurgVilmantMéltUc
dx
d 2 y dθ M
⇒ = =
dx 2 dx EI
b¤ y' = EI
M
PaBdabrbs;Fñwm 100