1. T.Chhay
m:Um:g;niclPaBénépÞrab
Area moment of inertia
1> niymn½y Definition Y
m:Um:g;niclPaBénépÞrabeFobnwgGkS½ X épÞrab A
CaGaMgetRkalEdlmanrag x épÞGnnþtUc dA
IX = ∫y
2
dA
( A)
dUcKñaEdr m:Um:g;niclPaBénépÞrabeFobnwgGkS½ Y
CaGaMgetRkalEdlmanrag y
IY = ∫x
2
dA
( )
A
X
O
Edl dA - épÞGnnþtUc rUbTI1 m:Um:g;niclPaBénépÞrab A
- cMgayBITIRbCMuTMgn;énépÞGnnþtUcmkGkS½ Y
x
y - cMgayBITIRbCMuTMgn;énépÞGnnþtUcmkGkS½ X
m:Um:g;niclPaBmanxñatKitCa mm / cm / m . 4 4 4
2> m:Um:g;niclPaBénépÞragctuekaN Moment of inertia of rectangle
- krNIeFobnwgGkS½EdlRtYtsIuKñanwgRCugNamYyrbs;va Y dx
h
IX = ∫y dA = ∫ y 2 .bdy
2
( ) A 0
3
bh dy
⇒ IX =
3 h dA=b.dy
dUcKñaenHEdr eyIgman y
dA=h.dx
b
IY = ∫x dA = ∫ x 2 .hdx X
2
x
( ) A 0
3 b
h.b
⇒ IY =
3
- krNIeFobnwgGkS½Edlkat;tamTIRbCMuTMgn;rbs;va ehIyRsbnwgRCugBIreTot
h
2
Y dx
IX = ∫ y dA = 2∫ y .bdy
2 2
( A) 0
h
dy
b. y 3 2
b.h 3 h/2
⇒ IX = 2 = y dA=b.dy
3 0
12 X
dUcKñaenHEdr h/2 dA=h.dx
x
m:Um:g;niclPaBénépÞrab b/2 b/2 53

2. T.Chhay
b
2
IY = ∫ x dA = 2 ∫ x .hdx
2 2
( A) 0
b
3
h.b 3
2
h.x
⇒ IY = 2 =
3 0
12
3> kaMniclPaB Radius of gyration
kaMniclPaBénépÞrab A eFobnwgGkS½ X CaRbEvgEdlkaerrbs;vaesμInwgpleFobrvagm:Um:g;niclPaB
én GkS½ X elIRkLaépÞénmuxkat;enaH.
IX
rX =
2
A
kaMniclPaBénépÞrab A eFobnwgGkS½ Y
CaRbEvgEdlkaerrbs;vaesμInwgpleFobrvagm:Um:g;niclPaBén GkS½ Y elIRkLaépÞénmuxkat;enaH.
IY
rY2 =
A
xñatrbs;vaKitdUcxñatRbEvg mm / cm / m .
4> RTwsþIbTGkS½Rsb Parallel axis theorem
edIm,IkMNt;m:Um:g;niclPaBrbs;épÞrabénGkS½ Edlminkat;tamGkS½RbCMuTMgn;rbs;va
EtGkS½enaHRtUvEtRsb CamYynwgGkS½TIRbCMuTMgn;
enaHeKGaceRbIRTwsþIbTGkS½Rsb. ¬rUbTI2¦
épÞrab A manTIRbCMuTMgn;Rtg;cMnuc C . dA
CaépÞ
GnnþtUc Edlman X CacMgayBITIRbCMuTMgn;rbs;vaeTAGkS½
Y nigman Y CacMgayBITIRbCMuTMgn;rbs;vaeTAGkS½ X .
x CacMgayBITIRbCMuTMgn;rbs;épÞGnnþtUceTAGkS½ y nig
man y CacMgayBITIRbCMuTMgn;rbs;épÞGnnþtUceTAGkS½
X.
mü:ageTot l CaKMlatBIGkS½ y eTAGkS½ Y ehIy d Ca
KMlatBIGkS½ x eTAGkS½ X .
eyIgman I = ∫ y dA = ∫ (d + Y ) dA
x
2
2
( A) ( A)
I x = ∫ Y dA + 2d ∫ YdA + d 2 ∫ dA
2
( A) ( A) ( A)
eday ∫ YdA = 0 m:Um:g;énépÞrabkat;tamGkS½TIRbCMuTMgn;
( ) A
m:Um:g;niclPaBénépÞrab 54

3. T.Chhay
⇒ I x = I X + A.d 2
dUcKña I = I + A.l
y Y
2
]TahrN_³ kMNt;m:Um:g;niclPaBrbs;épÞrabdUcbgðajkñúgrUb.
dMeNaHRsay³ 4cm
kMNt;m:Um:g;niclPaBeFobGkS½ X C
tamRTwsþIbTGkS½RsbeeyIg)an
x
d
I X = I x + A.d 2
X
4 × 83 1cm
⇒ IX = + 8 × 4 × 3 2 = 458.67cm 4 4cm
12
]TahrN_³ kMNt;m:Um:g;niclPaBrbs;épÞrabdUcbgðajkñúgrUb. 6cm
dMeNaHRsay³
kMNt;m:Um:g;niclPaBeFobGkS½ x 4cm X
tamRTwsþIbTGkS½RsbeeyIg)an
I x = I X + A.d 2
2cm
6 × 43
⇒ Ix = + 6 × 4 × 4 2 = 416cm 4 x
12
5> m:Um:g;niclPaBb:UElr Polar moment of inertia Y
m:Um:g;niclPaBb:UElrénépÞrab A eFobnwgcMnuc O épÞrab A
b¤b:Ul CaGaMgetRkalEdlmanrag épÞGnnþtUc dA
IO = JO = ∫ r dA
2 x
( A)
eday r 2
= x2 + y2
dUcenH I O = JO = ∫x
2
dA + ∫y
2
dA
r y
( )
A ( )
A
⇒ IO = J O = I X + IY X
O
krNIrgVg;Ggát;p©it D kaM R rUbTI3 m:Um:g;niclPaBb:UElrénépÞrab A
y
IO = ∫ r dA
2
( A)
A
R
⇒ I O = ∫ r .2πr.dr 2
dA=2πrdr
0
R
r4
⇒ I O = 2π . r
4 0 x
o
π .R 4 π .D 4
⇒ IO = = dr R
2 32
m:Um:g;niclPaBénépÞrab 55

4. T.Chhay
πD 4
⇒ Ix = Iy =
64
taragTI1³ lkçN³énmuxkat;
Shape Area (A) Moment of inertia (I ) Radius of gyration (r) Polar moment of inertia (J )
Y Yo
b.h3 h
IXo = rX o =
12 12
( )
CG
h Xo h.b3 b bh 2
h A = b.h IYo = rYo = J CG = h + b2
2 12 12 12
X
b.h3 h
rX =
b
2 IX =
b
3 3
Rectangle
2h
b.h3 rX o =
h
IXo =
h CG 3
b.h
A=
Xo
X 36 18
b 2
b.h3 rX =
h
IX =
Triangle 12 6
Yo
R
CG
π .d 2 π .d 4 d π .d 4
rX o = rYo =
Xo
A= I X o = IYo = J CG =
4 64 4 32
d
Circle
m:Um:g;niclPaBénépÞrab 56

5. T.Chhay
Shape Area (A) Moment of inertia (I ) Radius of gyration (r) Polar moment of inertia (J )
Yo
4R
3π
R
CG I X o = 0.1098 R 4 rX o = 0.264 R J CG = 0.5025R 4
πR 2
Xo
O
d
X
A= I Yo = I X =
πR 4 rYo = rX =
R
JO =
πR 4
2 2 4
8
Semi-cicle
d
d1
IXo =
(
π d 4 − d1
4
) d 2 + d1
2
Xo
A=
(
πd 2
− d1
2
) I Yo = I X o
64 rX o =
4
J CG =
(
π d 4 − d1
4
)
d 4 rYo = rX o 32
2
Yo
Hollow Circle
Yo
d
bd 3 − b1d1 bd 3 − b1d1
3
2
IXo = rX o =
CG
12 A J CG = I X o + I Yo
A = b.d − b1.d1
d d1 Xo
12
b 3 d − b1 d1
3 b 3 d − b1 d1
3
I Yo = rYo =
12 12 A
b1
b
Hollow Rectangle
Yo
I X o = I Yo = 0.0549 R 4 rX o = rYo = 0.2644 R
45° Xo
πR 2 J CG = 0.1098 R 4
X
A= πR 2 rX = 0.5 R
4R
4 IX =
3π
R 16
Quarter-cicle
6> plKuNniclPaB Production of inertia
plKuNniclPaBénépÞrab A eFobnwgGkS½ X GkS½ Y CaGaMgetRkalEdlmanrag I XY = ∫ xydA .
( A)
GacesμIsUnü b¤FMCagsUnü b¤tUcCasUnü eTAtamTItaMgéntMruy
I XY
Y
X , Y eFobnwgépÞrab A . plKuNniclPaBesμIsUnü kalNamanGkS½NamYy
¬GkS½ X b¤GkS½ Y ¦ CaGkS½qøúHénépÞrabenaH. X
kñúgkrNIenH I = 0 eRBaHGkS½ Y CaGkS½qøúH.
XY
b/2
b
m:Um:g;niclPaBénépÞrab 57

6. T.Chhay
RTwsþIbTGkS½RsbsMrab;m:Um:g;bUElr épÞrab A
I O = I C + A.d 2
Edl d = OC TIRbCMuTMgn; C
I m:Um:g;niclPaBb:UElrkat;tamTIRbCMuTMgn;
C
d
O
Y
RTwsþIbTGkS½RsbsMrab;plKuNniclPaB y épÞrab A
I xy = I XY + A.d x .d y
dx
X
C
dy
x
O
y Y
]TahrN_³ kMNt;plKuNniclPaBrbs;épÞrabeFobGkS½ x, y dUcbgðajkñúgrUb.
dMeNaHRsay³
kMNt;plKuNniclPaBeFobGkS½ x, y h
CG
X
tamRTwsþIbTGkS½RsbeeyIg)an
x
I xy = I XY + A.d x .d y b
eday I XY
b
/
= 0 dx =
2
dy =
h
2
/ / A = b.h
2 2
b h b .h
⇒ I xy = b.h. . =
2 2 4 Y
n
épÞrab A
7> rgVilGkS½ épÞGnnþtUc dA
m
RbBn§½kUGredaen X ,Y RtUv)anrgVil x
y.c
mMu α )anRbBn§½kUGredaenfμI m, n . n
os α
m
α
eyIg)anm:Um:g;niclPaBénépÞrab A eFob y
os α
x.s
GkS½ m EdlCaGkS½em (Principle axis) x.c α in α
in
y.s
α
X
I m = ∫ n 2 dA O
( A)
m:Um:g;niclPaBénépÞrab 58

7. T.Chhay
eday n = y.cosα − x.sin α
cos 2α = 2 cos 2 α − 1 = 1 − 2 sin 2 a
cos 2α + 1
⇒ cos 2 α =
2
sin 2α = 2 sin α . cos α
I m = ∫ ( y. cos α − x.sin α ) dA
2
( A)
Im = ∫y cos 2 αdA − 2 sin α . cos α ∫ x. ydA + ∫x sin 2αdA
2 2
( )
A ( A) ( )
A
I m = I X cos 2 α − I XY sin 2α + I Y sin 2 α
cos 2α + 1 1 − cos 2α
Im = I X − I XY sin 2α + IY
2 2
Im =
IX
2
IY
cos 2α − I XY sin 2α − cos 2α + O
2
I
2
eday I O = I X + IY
⎛I I ⎞ I
I m = ⎜ X − Y ⎟ cos 2α − I XY sin 2α + O
⎝ 2 2⎠ 2
I m = I X cos 2 α − I XY sin 2α + I Y sin 2 α
edayeGayDIeprg;Esülénm:Um:g;niclPaBeFobGkS½ m tammMu α esμIsUnüeyIg)an
= (IY − I X )sin 2α − 2 I XY cos 2α = 0
dI m
dα
2 I XY
tan 2α =
IY − I X
edayyk θ = α critical CamMuemEdleKGacTTYl)antMélm:Um:g;niclPaBFMbMputenAeBlrgVilGkS½.
1 ⎛ 2 I XY ⎞
⇒θ = tan −1 ⎜
⎜I −I ⎟
⎟
2 ⎝ Y X ⎠
8> rgVg;m: Mohr circle I XY
rgVg;m: CaviFIRkaPickñúgkarKNna
m:Um:g;niclPaBtamGkS½em.
X(IX ;I XY)
eKalbMNgrbs;rgVg;m: KWcg;rk
m:Um:g;niclPaBtamGkS½emGtibrma nigmMu IXY
2θ
I X ; IY
rgVilemRKITic θ . P(IP2;0)
2
C D P(IP1;0)
1
eyIgKUstMruyEdlmanGkS½edkCa -IXY
GkS½m:Um:g;niclPaB I ; I nigGkS½QrCa
X Y
Y(I Y;-IXY )
IY
GkS½plKuNniclPaB I . XY IX
m:Um:g;niclPaBénépÞrab 59

8. T.Chhay
p©itrgVg;m:RtUvenAelIGkS½edk.
edIm,Isg;düaRkamrgVg;m:)an eKRtUvsÁal; I ; I ; I X Y XY
sg;cMnuc X ( I ; I ) nig Y ( I ;− I )
X XY Y XY
+ IY ⎞
rYcP¢ab;cMnucTaMgBIr kat;GkS½edkeKTTYl)ancMnuc C . ykcMnuc C⎛ I
⎜
⎝
X
2
;0 ⎟
⎠
Cap©itrgVg;m: .
( I X − IY ) 2
dUcenHkaMrgVg;m: CX = CY = R = 2
+ I XY
2
rYcsg;rgVg;m:p©it C kaM R .
eyIg)an tan 2θ = CD = I I − I
XD
X
XY
Y
=
2 I XY
I X − IY
2
⎛ 2 I XY ⎞
dUcTisedArbs;GkS½em 1
⇒ θ = tan −1 ⎜
⎜I −I ⎟
⎟ Y n
2 ⎝ X Y ⎠
enAkñúgrgVg;m: ebIeyIgepþImBIcMnuc X mkrkcMnuc P eyIg 1 épÞrab A
RtUvqøgkat;tammMu 2θ . dUcenH eyIgRtUvcl½tknøHmMu 2θ RtUvnwg
mMu θ tamTisRTnicnaLika cab;BIGkS½ X énépÞrab A eyIgTTYl
X
)anTisedArbs;GkS½em m . m:Um:g;niclPaBeFobGkS½em m man O θ
tMélGtibrma I = I . m , max P1
edIm,ITTYl)anGkS½em n eyIgRKan;EtKUsEkgnwgGkS½em m
m enAelIépÞrabCakareRsc. m:Um:g;niclPaBeFobGkS½em n mantMélGtibrma I n , max = I P2 .
]TahrN_³ eKmanbnÞHEdkBIrpSaP¢ab;KñadUcbgðajkñúgrUb. cUrkMNt; Y
- TIRbCMuTMgn;rbs;muxkat; C ( x; y )
10mm
y
1
- m:Um:g;niclPaBeFobGkS½ X ;Y
- m:Um:g;niclPaBeFobGkS½TIRbCMuTMgn; x; y 150mm
- plKuNniclPaBeFobGkS½TIRbCMuTMgn; x; y C(x,y)
x
- mMurgVilemRKITic
- m:Um:g;niclPaBeFobGkS½eménGkS½TIRbCMuTMgn; O 100mm
2 10mm
X
dMeNaHRsay³
- TIRbCMuTMgn;rbs;muxkat; C ( x; y)
140 10
A1 = 140 × 10 = 1400mm 2 y1 = + 10 = 80mm x1 = = 5mm
2 2
10 100
A2 = 100 × 10 = 1000mm 2 y2 = = 5mm x2 = = 50mm
2 2
m:Um:g;niclPaBénépÞrab 60

9. T.Chhay
Edl A = A + A ⇒ A = 2400mm
1 2
2
∑ Ai . yi 1400 × 80 + 1000 × 5
y= ⇒y= = 48.75mm
A 2400
∑ Ai .xi 1400 × 5 + 1000 × 50
x= ⇒x= = 23.75mm
A 2400
C (23.75;48.75)
- m:Um:g;niclPaBeFobGkS½ X ;Y
I X = I X1 + I X 2
A A
10 × 1403
I X1 =
A
+ 10 × 140 × 80 2 = 11.25 × 106 mm 4
12
100 × 103
A
I X2 = = 0.03 × 106 mm 4
3
⇒ I X = 11.28 × 106 mm 4
140 × 103 10 × 1003
IY = + = 3.38 × 106 mm 4
3 3
- m:Um:g;niclPaBeFobGkS½TIRbCMuTMgn; x; y
I x = I xA1 + I xA2
10 × 1403
I xA1 = + 10 × 140 × (80 − 48.75) = 3.654 × 106 mm 4
2
12
100 × 103
I xA2 = + 10 × 100 × (48.75 − 5) 2 = 1.922 × 106 mm 4
12
⇒ I x = 5.576 × 106 mm 4
I y = I yA1 + I yA2
140 × 103
I yA1 = + 140 × 10 × (23.75 − 5) 2 = 0.504 × 106 mm 4
12
10 × 1003
= + 10 × 100 × (50 − 23.75) = 1.522 × 106 mm 4
2
I yA2
12
⇒ I y = 2.026 × 106 mm 4
- plKuNniclPaBeFobGkS½TIRbCMuTMgn; x; y
I xy = I xy1 + I xy2
A A
I xy1 = 10 × 140 × 31.25 × (− 18.75) = −0.820 × 106 mm 4
A
I xy2 = 10 × 100 × (− 43.75) × 26.25 = −1.148 × 106 mm 4
A
I xy = −1.968 × 106 mm 4
- mMurgVilemRKITic
m:Um:g;niclPaBénépÞrab 61

10. T.Chhay
1 ⎛ 2I ⎞
θ = tan −1 ⎜ xy
⎜I −I ⎟
⎟
2 ⎝ y x ⎠
⇒θ =
1 ⎛
tan −1 ⎜
(
2 × − 1.968 × 106 ⎞)
⎜ 2.026 × 106 − 5.576 × 106 ⎟ = 24
o
⎟
2 ⎝ ⎠
- m:Um:g;niclPaBeFobGkS½eménGkS½TIRbCMuTMgn;
tamvIFIrgVilGkS½
I m = I x cos 2 θ − I xy sin 2θ + I y sin 2 θ
( )
⇒ I m = 5.576 × 106 × cos 2 24o − − 1.968 × 106 sin(2 × 24o ) + 2.026 × 106 × sin 2 24o = 6.45 × 106 mm 4
I n = I y cos 2 θ + I xy sin 2θ + I x sin 2 θ
( )
⇒ I n = 2.026 × 106 × cos 2 24o + − 1.968 × 106 sin(2 × 24o ) + 5.576 × 106 × sin 2 24o = 1.15 × 106 mm 4
tamviFIrgVg;m:
I XY
1.968
2θ=48
I X ; IY
1.151 2.026 5.575 6.45
-1.968
m:Um:g;niclPaBénépÞrab 62

11. T.Chhay
lMhat;³
1> cUrKNnam:Um:g;niclPaBénragFrNImaRtxageRkameFob GkS½ X − X nig GkS½ Y − Y kat;tamGkS½
TIRbCMuTMgn;rbs;va
k> x> K>
2> KNnam:Um:g;niclPaBénépÞragRtIekaNxageRkam eFobGkS½ X − X kat;tamTIRbCMuTMgn; nigeFobGkS½
x − x RtYtsIuKñaCamYy)atRtIekaN.
3> KNnam:Um:g;niclPaBénépÞsmaseFobGkS½kUGredaenkat;tamTIRbCMuTMgn;xageRkam
k x>
K>
m:Um:g;niclPaBénépÞrab 63

12. T.Chhay
X>
g>
c>
q>
C>
m:Um:g;niclPaBénépÞrab 64

13. T.Chhay
Q>
4> cUrkMNt;³
- TIRbCMuTMgn;rbs;muxkat; C ( x; y)
- m:Um:g;niclPaBeFobGkS½ X ;Y
- m:Um:g;niclPaBeFobGkS½TIRbCMuTMgn; x; y
- plKuNniclPaBeFobGkS½TIRbCMuTMgn; x; y
- mMurgVilemRKITic
- m:Um:g;niclPaBeFobGkS½eménGkS½TIRbCMuTMgn;
nUvragFrNImaRtxageRkam³
k> x>
K>
m:Um:g;niclPaBénépÞrab 65