1. T.Chhay
m:Um:g;sþaTicénépÞrab
First moment of area
1> niymn½y (Definition)
m:Um:g;sþaTicénépÞrab A eFobnwgGkS½ X CaGaMgetRkalépÞGnnþtUc dA éncMgayBITIRbCMuTMgn;énépÞ
GnnþtUcenaHeTAGkS½ X . ¬rUbTI1¦
QX = ∫ y.dA
( A)
Edl - m:Um:g;sþTicénépÞrab A eFobnwgGkS½ X
QX
y - cMgayBITIRbCMuTMgn;énépÞGnnþtUc dA eTAGkS½ X
dA -épÞGnnþtUc
dUcKña m:Um:g;sþaTicénépÞrab A eFobnwgGkS½ Y Ca
GaMgetRkalépÞGnnþtUc dA éncMgayBITIRbCMuTMgn;énépÞGnnþ
tUcenaHeTAGkS½ Y . ¬rUbTI1¦
QY = ∫ x.dA
( A)
Edl QY- m:Um:g;sþTicénépÞrab A eFobnwgGkS½ Y
x - éncMgayBITIRbCMuTMgn;énépÞGnnþtUc dA eTAGkS½ Y
2> épÞsmas (Composite area)
épÞsmas CaépÞEdlpSMeLIgedayrUbFrNImaRtsamBaØCaeRcIn. ebIépÞsmas A mYypSMeLIg
edayépÞsamBaØ A / A />>>/ A EdlmanTIRbCMuTMgn;erogKña c / c />>>/ c ehIy y / y />>>/ y CacMgay
1 2 n 1 2 n 1 2 n
erogKñaBITIRbCMuTMgn;TaMgenaHdl;GkS½ X . épÞrab A manTIRbCMuTMgn; c EdlenAcMgay y BIGkS½ X .
m:Um:g;sþaTicénépÞsmas A eFobnwgGkS½ X KW³
n
QX = A. y = ∑ Ai . yi
i =1
dUcKñaenHEdr
n
QY = A.x = ∑ Ai .xi
i =1
Edl A - épÞrab b¤épÞsmas
A - épÞsamBaØTI i
i
y - cMgayBITIRbCMuTMgn;énépÞsmasmkGkS½ X
y - cMgayBITIRbCMuTMgn;énépÞsamBaØTI i mkGkS½ X
i
m:Um:g;sþaTicénépÞrab 42

2. T.Chhay
- cMgayBITIRbCMuTMgn;énépÞsmasmkGkS½ Y
x
x - cMgayBITIRbCMuTMgn;énépÞsamBaØTI i mkGkS½ Y
i
cMNaM³ m:Um:g;sþaTicénépÞrab A eFobnwgGkS½Edlkat;tamTIRbCMuTMgn;rbs;vamantMélesμIsUnü.
]TahrN_³ cUrrkm:Um:gsþaTicénépÞrabxageRkameFobGkS½kUGredaen.
dMeNaHRsay³
m:Um:g;sþaTicénépÞrabeFobGkS½ X
QX = A. y
Edl A = 6 × 4 = 24cm 2
h 4
y= = = 2cm
2 2
⇒ QX = 24 × 2 = 48cm3
m:Um:g;sþaTicénépÞrabeFobGkS½ Y
QY = A.x
Edl A = 6 × 4 = 24cm 2
b 6
x= = = 3cm
2 2
⇒ QY = 24 × 3 = 72cm3
]TahrN_³ cUrrkm:Um:gsþaTicénépÞrabxageRkameFobGkS½kUGredaen.
dMeNaHRsay³
m:Um:g;sþaTicénépÞrabeFobGkS½ X
n
QX = ∑ Ai . yi
i =1
A1 = 66 × 10 = 660cm 2
10
y1 = 20 + 24 + 30 + = 79cm
2
A2 = 30 × 40 = 1200cm 2
30
y2 = 20 + 24 + = 59cm
2
A3 = 24 × 56 = 1344cm 2
24
y3 = 20 + = 32cm
2
⇒ QX = 660 × 79 + 1200 × 59 + 1344 × 32 = 165948cm3
m:Um:g;sþaTicénépÞrabeFobGkS½ Y
m:Um:g;sþaTicénépÞrab 43

3. T.Chhay
n
QY = ∑ Ai .x i
i =1
66
A1 = 66 × 10 = 660cm 2 x1 = 30 + = 63cm
2
40
A2 = 30 × 40 = 1200cm2 x2 = 30 + = 50cm
2
56
A3 = 24 × 56 = 1344cm 2 x3 = 30 + = 58cm
2
⇒ QY = 660 × 63 + 1200 × 50 + 1344 × 58 = 179532cm3
3> TIRbCMuTMgn;énépÞrab nigTIRbCMuTMgn;énm:as (Centroid and center of gravity)
TIRbCMuTMgn;énépÞrab CacMnucEdlrkSalMnwgrbs;épÞrab. TIRbCMuTMgn;énm:as
CacMnucEdlrkSalMnwgrbs;m:as. ebIépÞsmasmYymanRkLaépÞ A nigmanm:assrub M enaH
y cMgayBITIRbCMuTMgn;eTAGkS½ X nig x cMgayBITIRb CMuTMgn;eTAGkS½ Y .
∑ Ai . y i ∑ Ai . y i
∫ y.dA QX
( A)
y= = = =
∑ Ai
cMgayBITIRbCMuTMgn;énépÞeTAGkS½ X nig GkS½ Y A A A
∑ Ai .x i ∑ Ai .x i
∫ x.dA Q
( A)
x= = = Y
=
∑ Ai A A A
∑ M i . yi ∑ M i . yi
y= =
∑ Mi M
cMgayBITIRbCMuTMgn;énm:aseTAGkS½ X nig GkS½ Y
∑ Ai .xi ∑ Ai .xi
x= =
∑ Mi M
cMNaM³ -sMrab;rUbFatuesμIsac; (homogeneous) TIRbCMuTMgn;énm:as nigTIRbCMuTMgn;énépÞsßitenAelIcMnucEtmYy.
sMrab;rUbFatusmas TIRbCMuTMgn;énm:as nigTIRbCMuTMgn;énépÞsßitenAelIcMnucBIepSgKña. ¬rUbTI2¦
-sMrab;rUbFatuesμIsac; cMeBaHépÞrabEdlmanGkS½qøúHTIRbCMuTMgn;rbs;vasßitenAelIGkS½qøúHTaMgenaH.
¬rUbTI3¦ Centroid Center of gravity
0.25m
Y Y
0.38m 0.38m
b
Wood Steel h/2
X h
rUbTI2 Centroid and center of gravity
h/2
b/2 b/2 e b/2 b/2 e
GkS½ X; Y CaGkS½qøúHrbs;va GkS½ Y CaGkS½qøúHrbs;va
rUbTI3
m:Um:g;sþaTicénépÞrab 44

4. T.Chhay
taragTI1³ RkLaépÞ nigTIRbCMuTMgn;
Vertex
h
h h
h/2 2
h/3 h
5
b/2 b/3 3
b
b b 8
b
Rectangle Right Triangle Second- degree palabola
2
A=b.h A=b.h/2 A = b.h
3
Vertex
3 n +1
h h
10 4n + 2 h
h h
Vertex Vertex h.n
b 2 n +1
b/4
b b n+2 b.( n+1)
2( n+ 2) b
Second- degree palabola th
n - degree palabola
th
n - degree palabola
1 b.h n.b.h
A= b.h A= A=
3 n +1 n +1
a b
2h
h 5 h
4R
h/3 3π
1 ( a + L) 1 (b + L) b/2 R R
3 3
L b
Triangle Second- degree palabola Semi-cicle
A=L.h/2 2
A = b.h πR 2
A=
3 2
R
θ
θ
2 R. sin θ R
4R R 3θ
3π
R
Circular sector
Quarter-cicle Circle
A = R 2 .θ
πR 2 A = π .R 2
A=
4 Note: θ is in radians
m:Um:g;sþaTicénépÞrab 45

5. T.Chhay
]TahrN_³ cUrrkTItaMgTIRbCMuTMgn;eFobnwgkUGredaenEdleKeGay Y
dMeNaHRsay³
A1 = 12 × 2 = 24cm 2 x1 = 1cm y1 = 6cm
A2 = 2 × 8 = 16cm 2 x2 = 6cm y2 = 1cm 10cm
TItaMgTIRbCMuTMgn;eFobGkS½kUGredaen C(3,4)
A1.x1 + A2 .x2 24 × 1 + 16 × 6
x= = = 3cm 2cm
A1 + A2 24 + 16 X
O
A . y + A2 . y2 24 × 6 + 16 × 1
y= 1 1 = = 4cm
2cm 8cm
A1 + A2 24 + 16
]TahrN_³ cUrrkTItaMgTIRbCMuTMgn;eFobnwgkUGredaenEdleKeGay Y
dMeNaHRsay³ A2
A1 = 18 × 2 = 36cm 2
x1 = 11cm y1 = −1cm 6cm
A2 = 12 × 6 = 72cm 2 x2 = 6cm y2 = 11cm
A3 A1
A3 = 8 × 6 = 48cm 2
x3 = 3cm y3 = 4cm
8cm
TItaMgTIRbCMuTMgn;eFobGkS½kUGredaen
A1.x1 + A2 .x2 + A3 .x3 36 × 11 + 72 × 6 + 48 × 3 X
x= = O
A1 + A2 + A3 36 + 72 + 48
⇒ x = 6.23cm 10cm
A1. y1 + A2 . y2 + A3 . y3 36 × (− 1) + 72 × 11 + 48 × 4
y= = 6cm 4cm 2cm
A1 + A2 + A3 36 + 72 + 48
⇒ y = 6.07cm
4> karrkTIRbCMuTMgn;énépÞrabedayGaMgetRkal (Solving centroid by integration)
kñúgkarviPaKrkTIRbCMuTMgn; eKRtUvBinitüFatuGnnþtUc dA EdlCabnÞHQr b¤bnÞHedk.
- rktambnÞHQr (Verticle element strip)
Edl dA = y.dx
L
y y
∫ yele .dA
( A)
∫) 2 .dA
(A
∫ 2 y.dx
y= = = 0
∫ dA ∫ dA
L
( ) A ( )
A ∫ y.dx
0
m:Um:g;sþaTicénépÞrab 46

6. T.Chhay
L
y2
∫ 2
dx dx
⇒y= 0
L
x,y
∫ y.dx y=f(x)
0 dA=y.dx
dUcKñaEdr C(x,y/2)
L
H
∫x
( A)
ele .dA ∫ x. y.dx C(x/2,y) x,y
dy
x= = 0
∫ dA
L
dA=x.dy
( A) ∫ y.dx
0
- rktambnÞHedk (Horizontal element strip) L
Edl dA = x.dy
H
x
∫ xele .dA
( A)
∫ 2 .x.dy
x= = 0
∫ dA
H
( ) A ∫ x.dy
0
H 2
x
∫ 2 dy
⇒x= 0
H
∫ x.dy
0
H
∫ yele .dA
( A)
∫ y.x.dy
y= = 0
∫ dA
H
( A) ∫ x.dy
0
]TahrN_³ kMNt;TIRbCMuTMgn;rbs;épÞrab
dUcbgðajkñúgrUb.
y = f (x) =
H
dMeNaHRsay³ L2
x2
FatuGnnþtUcman H
dx
y
dA = y.dx xele = x yele =
2 yele =y/2
TItaMgTIRbCMuTMgn; C
L L L
H 2 x4
∫x
( A)
ele .dA ∫ x. y.dx ∫ x. L2 x .dx 4 3
xele=x
x= = 0
= 0
= 0
= L L
∫ dA
L L 3 L
H 4
∫ y.dA ∫ L2 x.dx
x
( A)
0 0 3 0
m:Um:g;sþaTicénépÞrab 47

7. T.Chhay
L L L
y H2 4 H 2 x5
∫ yele .dA
( A)
∫ 2 y.dx ∫ 2 L4 x .dx .
2 L4 5 3H
y= = 0
= 0
= 0
=
∫ dA
L L 3 L
H 10
∫ y.dx ∫L x 2 .dx H x
( A) 2 .
0 0 L2 3 0
dUcenH 3 3
C ( L; H )
4 10
m:Um:g;sþaTicénépÞrab 48

8. T.Chhay
lMhat;
1> cUrkMNt;rkTIRbCMuTMgn;nUvragFrNImaRtxageRkamedayviFIGaMgetRkal
k.
x>
K>
X>
2> cUrkMNt;nUvTIRbCMuTMgn;épÞsmasxageRkam
k> edayKitBIRCugxagelI nigxageqVg
m:Um:g;sþaTicénépÞrab 49

9. T.Chhay
x> edayeFobnwgGkS½ X nig Y
K> edayeFobnwgGkS½ X nig Y
X> edayeFobnwgGkS½ X nig Y
g> edayeFobnwgGkS½ X nig Y
m:Um:g;sþaTicénépÞrab 50

10. T.Chhay
c> edayeFobnwgGkS½ X nig Y
q> edayeFobnwgGkS½ X nig Y
C> edayeFobnwgGkS½ X nig Y
m:Um:g;sþaTicénépÞrab 51

11. T.Chhay
Q> cUrkMNt;TIRbCMuTMgn;énrgVg;TaMgR)aMtamGkS½Qr edayeFobnwgépÞxagelIrbs;ctuekaN
d> cUrkMNt;TIRbCMuTMgn;rbs;EdkxageRkamEdlmanrUbragdUcbgðajxageRkam
m:Um:g;sþaTicénépÞrab 52