Tabela de CG e Momentos de Inércia

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225
⎯
x
O C
r
Shape Area
⎯
x
⎯
x
⎯
x
⎯
y
⎯
y
Triangular area
Quarter-circular
 area
Semicircular area
Quarter-elliptical
 area
Semielliptical
 area
Semiparabolic
 area
Parabolic area
Parabolic spandrel
General spandrel
Circular sector
C
CC
⎯
y
⎯
y CC
⎯
x
⎯
x
CC
⎯
y
⎯
y
h
3
bh
2
b
2
b
2
4r
4r
3a
4
3a
8
3h
5
3h
10
2ah
3
3h
5
4ah
3
ah
3
4b
4
4r
2
2
4b
4
4a
O
O
O
O
O
O
O
O
r
h 
0
a
a
a
a
a
b
y = kx2
h
h
n + 1
n + 2
a n + 1
4n + 2
h ah
n + 1
C
0
0
0
⎯
x
⎯
y
y = kxn
h
C
3α
2r sin α r2α
3�
3�
3�
3�3
3�
r2�
r2�
ab�
ab�
�
�
�
Fig. 5.8A Centroids of common shapes of areas.
5.4 First Moments of Areas and Lines
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483
Quarter circle
C
Rectangle
Triangle
Circle
Semicircle
Ellipse
b
y y'
x'
x
1
12
⎯Ix' = bh3
1
12
⎯Iy' = b3h
1
8
Ix = Iy = r4
1
4
J
O
 = r4
1
4
⎯Ix =⎯Iy = r 4
1
2
J
O
 = r4
1
36
⎯Ix' = bh3
1
12
Ix = bh3
1
3
Iy = b3h
1
12
J
C
 = bh(b2 + h2)
1
3
Ix = bh3h
b
x'
x
x
r
O
y
h C
h
3
xO
C
y
r
xO
C
y
r
x
b
y
a
1
16
Ix = Iy = r4
1
8
J
O
 = r4
1
4
⎯Ix = ab3
1
4
⎯Iy = a3b
1
4
J
O
 = ab(a2 + b2)
O
�
�
�
�
�
�
�
�
�
Fig. 9.12 Moments of inertia of common geometric shapes.
determination of moments of inertia is thus a prerequisite to the 
analysis and design of structural members.
 It should be noted that the radius of gyration of a composite area 
is not equal to the sum of the radii of gyration of the component areas. 
In order to determine the radius of gyration of a composite area, it is 
first necessary to compute the moment of inertia of the area.
9.7 Moments of Inertia of Composite Areas
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