moments of areas appendix a moments of areas a.1 first moment of an area; centroid first moments of...
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.1 First Moment of An Area; Centroid
First Moments of the Area A Aboutthe x- and y-Axis are Defined As
AxQ ydAAyQ xdA
x y
The centroid of the area A is defined as the point C of coordinates
and which satisfy the relations
/yx Q A/xy Q A
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS The first moment of an area about its symmetric axis is zero, so, the centroid of the area must be on the symmetric axis. When an area possesses a center of symmetry O, the first moment of the area about any axis through O is zero. In other words, O is the centroid of the area.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
21 1
2 2xQ Ay bh h bh
21 1
2 2yQ Ax bh b b h
If an area has two symmetric axes, the inter-section of the two axes must be the centroid of the area.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.1For the triangular area of Fig. (a), determine (a) the first moment Qx of the area with respect to the x-axis, (b) the
coordinate y of the centroid of the area.
Fig. (a) Fig. (b)
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
dA=udy, h y
u bh
2
0 6Ax
h h y bhydA
hQ y bdy
2 / 6
/ 2 3xQ bh h
yA bh
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASComplement ProblemDetermine the first moment of a semi-circular area about the x-axis, (b) the coordinate of the centroid of the semi-circle.
x AQ ydA 2 22dA R y dy
2 2
0
2 2 3/ 23
0
2
( ) 2
3/ 2 3
R
x
R
Q y R y dy
R yR
3
2
2 4
3( / 2) 3xQ R R
yA R
y
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.1 Determination of The First Moment And Centroid of A Composite Area
1 2 3x A A A A
Q ydA ydA ydA ydA 1 1 2 2 3 3xQ A y A y A y
x i ii
Q A y y i ii
Q A xi i
i
ii
A xX
A
i ii
ii
A yY
A
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.2Locate the Centroid C of the area A shown in Fig. (a).
Fig. (a) Fig. (b)
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
Solution:
1 1 2 2
1 2
y A y AY
A A
A1=80×20=1600 mm2,
A2=60×40=2400 mm2
70 1600 30 240046mm
1600 2400Y
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.3 Referring to the area A of Sample Problem A.2, we consider the horizontal x axis which is through its centroid C. (Such an axis is called a centroidal axis.) Denoting by A the portion of A located above that axis (Fig. a), determine the first moment of A with respect to the x axes.
Fig. (a) Fig. (b) Fig. (c)
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
Solution:
1 1 3 3
3
24 80 20
7 14 40
42320mm
xQ y A y A
4 4
3
23 46 40
42320mm
xQ y A y A
3
0
42320mm
x x x
x x
Q Q Q Y A
Q Q
In fact 0
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1 1 2 2
1 2
i iA x A x A xx
A A A
Complement Problem
1 1 2 2
1 2
i iA y A y A yy
A A A
80
10
10
c(19.7;39.7)
x
y
C1
C2
Determine the centroid of the L-shape area.2
1 1 1700 , 45 , 5A mm x mm y mm 2
2 2 21200 , 5 , 60A mm x mm y mm
120 45 700 5 1200
700 1200
19.7( )mm
)(7.391200700
1200607005mm
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Alternative method: Negative area method
40 9600 45 ( 7700)19.7( )
9600 7700mm
z
y
21 1 19600 , 40 , 60A mm x mm y mm
22 2 27700 , 45 , 65A mm x mm y mm
2C60 9600 65 ( 7700)
39.7( )9600 7700
mm
1C0C
1 1 2 2
1 2
i ic
A x A x A xx
A A A
1 1 2 2
1 2
i iA y A y A yy
A A A
80
120
10
10
x
y
x
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.3 Second Moment, or Moment of Inertia, of An Area; Radius of Gyration
Moment of Inertia of A With Respect To the And x Axis And y Axis are Defined, Respectively, As
2x A
I y dA2
y AI x dA
Define the Polar Moment of Inertia of the Area A With Respect To Point O As the Integral :
2 2 2O x yA A A
J dA y dA x dA I I
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASRadii of Gyration of An Area A with respect to the x and y axis:
2x xI r A
2y yI r A
xx
Ir
A y
y
Ir
A
2O OJ r A O
O
Jr
A
2 2 2O x yr r r
Radii of Gyration With Respect To the Origin O
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.4For the rectangular area of Fig.(a). determine (a) the moment of inertia Ix of the area with respect to the centroidal x axis. (b) the corresponding radius of gyration rx.
Fig. (a) Fig. (b)
2
/ 2 2
/ 2
3
12
x A
h
h
I y dA
y bdy
bh
12x
hr
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.5
For the circular area of Fig. (a), determine (a) the polar moment of inertia JO, (b) rectangular moments of inertia Ix and Iy.
Fig. (a) Fig. (b)
4 42 2
2 32O A
r dJ d
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.4 Parallel-Axis TheoremA.4 Parallel-Axis Theorem
2x A
I y dA 2 2( )x A AI y dA y d dA
2 2( ) 2x A A AI y dA d y dA d dA
2x xI I Ad
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.5 Determination of The Moment of Inertia of a Composite Area.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.6Determine the moment of inertia xI of the area shown with
respect to the centroidal x axis (Fig. a).
Fig. (a) Fig. (b)
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
1 2( ) ( )x x A x AI I I
1
21 1
32 4
( )
80 2080 20 24 974933mm
12
x A xI I A d
2
22 2
32 4
( )
40 6040 60 16 1334400mm
12
x A xI I A d
41 2( ) ( ) 974933 1334400 2309333mmx x A x AI I I
Solution:Solution:
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.6 Product of Inertia for An Area.product of inertia for an element of area located at point (x, y)
is defined as
xy AI xydAxydI xydA
0xyI
If either x or y axis is a symmetric axis, Ixy=0.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
Parallel-Axis theorem
( )( )xy x yI x d y d dA x y x yxydA d ydA d xdA d d dA
xy xy x yI I Ad d
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.7Determine the product of inertia Ixy of the triangle shown in Fig. (a).
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.7
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
xdA hdx
b
23
2
1
2 2y
x x hdI xydA x h h dx x dx
b b b
23
20 2
b
xy xyA A
hI xydA dI x dx
b
2 2
8xy
b hI
Solution:Solution:
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.8 Compute the product of inertia of the beam’s cross-sectional area, shown in Fig. (a), about the x and y centroidal axes.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
Solution:Solution:
80 (300)(100)( 250)(200) 15 10xy xy x yI I Ad d
0 0 0xy xy x yI I Ad d
80 (300)(100)(250)( 200) 15 10xy xy x yI I Ad d
8 8 8( 15 10 ) 0 ( 15 10 ) 30 10xyI
Rectangle B
Rectangle D
The product of inertia for the entire cross section is
mm4
Rectangle A
mm4
mm4
mm4
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.7 Moments of Inertia for An Area About Inclined Axes
cos sinu x y
cos sinv y x
2 2
2 2
( cos sin )
( cos sin )
( cos sin )( cos sin )
u
v
uv
dI v dA y x dA
dI u dA x y dA
dI uvdA x y y x dA
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS2 2cos sin 2 sin cosu x y xyI I I I
2 2sin cos 2 sin cosv x y xyI I I I 2 2sin cos sin cos 2 cos sinuv x y xyI I I I
cos 2 sin 22 2
x y x yu xy
I I I II I
cos 2 sin 22 2
x y x yv xy
I I I II I
sin 2 cos 22
x yuv xy
I II I
0 u v x yJ I I I I
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASPrincipal Moments of Inertia
2 sin 2 2 cos 2 02
x yuxy
I IdII
d
2tan 2 xy
x y
I
I I
1
1
sin 2
cos 22
xyp
x yp
I
RI I
R
2
2
sin 2
cos 22
xyp
x yp
I
RI I
R
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS
maxmin
2
2
2 2x y
xyx y I II I
I I Ra
2
2
2x y
xy
I IR I
2x yI I
a
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.9 Determine the principal moments of inertia for the beam’s cross-sectional area shown in Fig. (a) with respect to an axis passing through the centroid.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASA.8 Mohr’s Circle for Moments of Inertia
2 2
2 2
2 2x y x y
u uv xy
I I I II I I
2 2 2u uvI a I R
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREASSample Problem A.10Using Mohr’s circle, determine the principal moments of inertia for the beam’s cross-sectional area, shown in Fig. (a), with respect to an axis passing through the centroid.
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APPENDIX A
MOMENTS OF AREASMOMENTS OF AREAS