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D
Properties of Plane Areas
Notation: A area
苶x, 苶y distances to centroid C
Ix, Iy moments of inertia with respect to the x and y axes,
respectively
Ixy product of inertia with respect to the x and y axes
IP Ix Iy polar moment of inertia with respect to the origin of
the x and y axes
IBB moment of inertia with respect to axis B-B
y
1
Rectangle (Origin of axes at centroid)
A bh
x
h
C
x
y
bh3
Ix 12
b
苶x 2
h
y苶 2
hb3
Iy 12
Ixy 0
bh
IP (h2 b2)
12
b
y
2
Rectangle (Origin of axes at corner)
B
bh3
Ix 3
h
O
B
x
y
bh
IP (h2 + b2)
3
Triangle (Origin of axes at centroid)
c
bh
A 2
x
h
C y
b
b2h2
Ixy 4
b3h3
IBB 6(b2 h2)
b
3
hb3
Iy 3
x
bh3
Ix 36
bc
x苶 3
h
y苶 3
bh
Iy (b2 bc c2)
36
bh2
Ixy (b 2c)
72
bh
IP (h2 b2 bc c2)
36
D1
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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APPENDIX D Properties of Plane Areas
y
4
Triangle (Origin of axes at vertex)
c
B
B
h
O
x
b
y
5
bh3
Ix 12
bh
Iy (3b2 3bc c2)
12
bh2
Ixy (3b 2c)
24
bh3
IBB 4
Isosceles triangle (Origin of axes at centroid)
bh
A 2
x
h
C
y
B
x
B
b
bh3
Ix 36
b
x苶 2
h
y苶 3
hb3
Iy 48
Ixy 0
bh3
IBB 12
bh
IP (4h2 3b2)
144
(Note: For an equilateral triangle, h 兹3苶 b/2.)
y
6
Right triangle (Origin of axes at centroid)
bh
A 2
x
h
C
y
B
x
B
bh3
Ix 36
b
b
苶x 3
hb3
Iy 36
y
Right triangle (Origin of axes at vertex)
B
B
bh3
Ix 12
h
O
hb3
Iy 12
bh
IP (h2 b2)
12
x
b2h2
Ixy 24
bh3
IBB 4
b
y
8
Trapezoid (Origin of axes at centroid)
a
h
b2h2
Ixy 72
bh3
IBB 12
bh
IP (h2 b2)
36
7
h
苶y 3
C
B
h(a b)
A 2
y
x
h(2a b)
苶y 3(a b)
h3(a2 4ab b2)
I B x
36(a b)
h3(3a b)
IBB 12
b
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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APPENDIX D Properties of Plane Areas
y
9
Circle
d = 2r
r
x
B
Ixy 0
B
10
r
pr 4
pd 4
Ix Iy 4
64
pr4
pd4
IP 2
32
5p r 4
5p d 4
IBB 4
64
y
Semicircle (Origin of axes at centroid)
C
pr2
A 2
y
B
x
B
y
11
(Origin of axes at center)
pd2
A pr 2 4
C
D3
4r
y苶 3p
(9p 2 64)r 4
Ix ⬇ 0.1098r4
72p
p r4
Iy 8
Ixy 0
pr4
IBB 8
Quarter circle (Origin of axes at center of circle)
pr2
A 4
x
B
B
C
y
x
O
4r
x苶 苶y 3p
pr4
Ix Iy 16
(9p 2 64)r 4
IBB ⬇ 0.05488r4
144p
r4
Ixy 8
r
y
12
Quarter-circular spandrel (Origin of axes at point of tangency)
B
B
r
冢
冢
x
O
y
13
a angle in radians
A ar 2
C
a a
r
y
x
O
y
a
冢
冣
1
p
Iy IBB r 4 ⬇ 0.1370r4
3
16
(a p/2)
2r sin a
苶y 3a
苶x r sin a
r4
Ix (a sin a cos a)
4
r4
Iy (a sin a cos a)
4
y
Circular segment (Origin of axes at center of circle)
C
a angle in radians
a
A r 2(a sin a cos a)
r
O
冣
Circular sector (Origin of axes at center of circle)
x
x
14
(10 3p)r
⬇ 0.2234r
苶y 3(4 p)
2r
⬇ 0.7766r
苶x 3(4 p)
5p
Ix 1 r 4 ⬇ 0.01825r 4
16
y
C
冣
p
A 1 r 2
4
x
x
Ixy 0
ar 4
IP 2
(a p/2)
冢
冣
2r
sin3 a
苶y 3 a sin a cos a
r4
Ix (a sin a cos a 2 sin3 a cos a)
4
Ixy 0
r4
Iy (3a 3 sin a cos a 2 sin3 a cos a)
12
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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APPENDIX D Properties of Plane Areas
15
y
a
Circle with core removed (Origin of axes at center of circle)
a angle in radians
r
C
a
a arccos r
b
a
x
a
b
(a p/2)
冢
ab
A 2r 2 a r2
b 兹苶
r 2 a苶2
冢
冣
冣
冢
r4
3ab
2ab3
Ix 3a 2 6
r
r4
冣
r4
ab
2ab3
Iy a 2 2
r
r4
Ixy 0
2a
y
16
Ellipse (Origin of axes at centroid)
A pab
b
C
x
b
a
p ba3
Iy 4
pab
IP (b2 a2)
4
Ixy 0
a
pa b 3
I x 4
Circumference ⬇ p [1.5(a b) 兹a苶b苶 ]
⬇ 4.17b /a 4a
2
17 y
冢
2 bh
A 3
y
O
x
b
x
Vertex
h
C
O
y
bh
3b
x苶 4
bh3
Ix 21
y
b2h2
Ixy 12
3h
苶y 1
0
hb3
Iy 5
冢
xn
y f (x) h 1 bn
x
C
y
x
b
b2h2
Ixy 12
Semisegment of nth degree (Origin of axes at corner)
y = f (x)
O
2hb3
Iy 15
hx2
y f (x) b2
x A 3
b
h
2h
y苶 5
Parabolic spandrel (Origin of axes at vertex)
y = f (x)
19
3b
x苶 8
16bh3
Ix 105
y
18
冣
x2
y f (x) h 1 b2
x
C
(0 b a/3)
Parabolic semisegment (Origin of axes at corner)
Vertex
y = f (x)
h
(a/3 b a)
冢
n
A bh n1
冣
冣
(n 0)
b(n 1)
苶x 2(n 2)
2bh3n3
(n 1)(2n 1)(3n 1)
Ix hn
y苶 2n 1
hb3n
Iy 3(n 3)
b2h2n2
Ixy 4(n 1)(n 2)
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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APPENDIX D Properties of Plane Areas
y
20
Spandrel of nth degree (Origin of axes at point of tangency)
hx n
y f (x) bn
y = f (x)
x
h
C
y
O
bh3
Ix 3(3n 1)
y
21
h
C
b
x
B
b
4bh
A p
hb3
Iy n3
冢
冣
冣
8bh3
IBB 9p
p d 3t
Ix Iy pr 3t 8
A 2prt pdt
x
C
冢
4
32
Iy hb3 ⬇ 0.2412hb3
p
p3
Thin circular ring (Origin of axes at center)
Approximate formulas for case when t is small
d = 2r
r
Ixy 0
t
y
t
pd 3t
IP 2pr 3t 4
Thin circular arc (Origin of axes at center of circle)
Approximate formulas for case when t is small
B
B
C
b y
b
r
x
O
b angle in radians
A 2brt
(Note: For a semicircular arc, b p/2.)
r sin b
苶y b
Ix r 3t(b sin b cos b)
Ixy 0
y
24
Iy r 3t(b sin b cos b)
2b sin2b
1 cos2b
IBB r 3t 2
b
冢
冣
Thin rectangle (Origin of axes at centroid)
Approximate formulas for case when t is small
A bt
b
b
C
x
t
B
b2h2
Ixy 4(n 1)
ph
y苶 8
8
p
Ix bh3 ⬇ 0.08659bh3
9p
16
Ixy 0
y
h(n 1)
y苶 2(2n 1)
Sine wave (Origin of axes at centroid)
y
B
22
(n 0)
b(n 1)
x苶 n2
bh
A n1
x
b
23
D5
tb3
Ix sin2 b
12
tb3
Iy cos2 b
12
tb3
IBB sin2 b
3
B
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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APPENDIX D Properties of Plane Areas
25
A
b
R1 R2
C
a
Regular polygon with n sides (Origin of axes at centroid)
B
b
C centroid (at center of polygon)
n number of sides (n 3)
b length of a side
b central angle for a side
360°
b n
a interior angle (or vertex angle)
n2
a 180°
n
冢
冣
a b 180°
R1 radius of circumscribed circle (line CA)
b
b
R1 csc 2
2
b
b
R2 cot 2
2
R2 radius of inscribed circle (line CB)
2
b
nb
A cot 4
2
Ic moment of inertia about any axis through C (the centroid C is a principal point and
every axis through C is a principal axis)
冢
冣冢
冣
nb4
b
b
Ic cot 3cot2 1
192
2
2
IP 2Ic
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.