Download - Tabel Properties of Plane Areas
Properties of Plane Areas
Notation: A � areax�, y� � distances to centroid CIx, Iy � moments of inertia with respect to the x and y axes, respectivelyIxy � product of inertia with respect to the x and y axesIP � Ix � Iy � polar moment of inertia with respect to the origin ofthe x and y axesIBB � moment of inertia with respect to axis B-B
1 Rectangle (Origin of axes at centroid)
A � bh x� � �b2
� y� � �h2
�
Ix � �b1h2
3
� Iy � �h1b2
3
� Ixy � 0 IP � �b1h2� (h2 � b2)
2 Rectangle (Origin of axes at corner)
Ix � �b3h3
� Iy � �h3b3
� Ixy � �b2
4h2
� IP � �b3h� (h2 + b2)
IBB � �6(b
b2
3
�
h3
h2)�
3 Triangle (Origin of axes at centroid)
A � �b2h� x� � �
b �
3c
� y� � �h3
�
Ix � �b3h6
3
� Iy � �b3h6� (b2 � bc � c2)
Ixy � �b7h2
2
� (b � 2c) IP � �b3h6� (h2 � b2 � bc � c2)
yc
h
b
C x
x
y
y
x
h
b
O
B
B
y
x
xhy
b
C
D
D1
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4 Triangle (Origin of axes at vertex)
Ix � �b1h2
3
� Iy � �b1h2� (3b2 � 3bc � c2)
Ixy � �b2h4
2
� (3b � 2c) IBB � �b4h3
�
5 Isosceles triangle (Origin of axes at centroid)
A � �b2h� x� � �
b2
� y� � �h3
�
Ix � �b3h6
3
� Iy � �h4b8
3
� Ixy � 0
IP � �1b4h4
� (4h2 � 3b2) IBB � �b1h2
3
�
(Note: For an equilateral triangle, h � �3� b/2.)
6 Right triangle (Origin of axes at centroid)
A � �b2h� x� � �
b3
� y� � �h3
�
Ix � �b3h6
3
� Iy � �h3b6
3
� Ixy � � �b7
2h2
2
�
IP � �b3h6� (h2 � b2) IBB � �
b1h2
3
�
7 Right triangle (Origin of axes at vertex)
Ix � �b1h2
3
� Iy � �h1b2
3
� Ixy � �b2
2h4
2
�
IP � �b1h2� (h2 � b2) IBB � �
b4h3
�
8 Trapezoid (Origin of axes at centroid)
A � �h(a
2� b)� y� � �
h
3
(
(
2
a
a
�
�
b
b
)
)�
Ix ��h3(a2
36�
(a4�
abb�
)b2)
� IBB � �h3(3
1
a
2
� b)�
y
yxh
b
a
B B
C
y
x
h
b
B B
O
y
y x
xh
b
BC
B
B B
C
y
x
b
hy
x
y c
h
bO
B B
x
D2 APPENDIX D Properties of Plane Areas
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9 Circle (Origin of axes at center)
A � pr2 � �p
4d2
� Ix � Iy � �p
4r4
� � �p
6d4
4
�
Ixy � 0 IP � �p
2r4
� � �p
3d2
4
� IBB � �5p
4r 4
� � �5p
64d 4
�
10 Semicircle (Origin of axes at centroid)
A � �p
2r2
� y� � �34p
r�
Ix � �(9p 2
7�
2p
64)r4
� � 0.1098r4 Iy � �p
8r4
� Ixy � 0 IBB � �p
8r 4
�
11 Quarter circle (Origin of axes at center of circle)
A � �p
4r 2
� x� � y� � �34p
r�
Ix � Iy � �p
1r6
4
� Ixy � �r8
4
� IBB � �(9p2
14�
4p
64)r4
� � 0.05488r4
12 Quarter-circular spandrel (Origin of axes at point of tangency)
A � �1 � �p
4��r2 x� � �
3(42�
rp)
� � 0.7766r y� � �(1
3
0
(4
�
�
3
p
p
)
)r� � 0.2234r
Ix � �1 � �51p
6��r4 � 0.01825r4 Iy � IBB � ��
13
� � �1p
6��r4 � 0.1370r4
13 Circular sector (Origin of axes at center of circle)
a � angle in radians (a � p/2)
A � ar2 x� � r sin a y� � �2r
3sia
n a�
Ix � �r4
4
� (a � sin a cos a) Iy � �r4
4
� (a � sin a cos a) Ixy � 0 IP � �a
2r 4
�
14 Circular segment (Origin of axes at center of circle)
a � angle in radians (a � p/2)
A � r2(a � sin a cos a) y� � �23r� ��a � s
siinn3
a
a
cos a��
Ix � �r4
4
� (a � sin a cos a � 2 sin3 a cos a) Ixy � 0
Iy � �1r2
4
�(3a � 3 sin a cos a � 2 sin3 a cos a)
a a
C
O
y
y
x
r
a a
C
O
y
r
x
x
y
x
y
x
r
x
OC
B B
y
y
yx
r
B BC
O
x
y
y xr C
B B
y
xr
d = 2r
C
B B
APPENDIX D Properties of Plane Areas D3
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15 Circle with core removed (Origin of axes at center of circle)
a � angle in radians (a � p/2)
a � arccos �ar
� b � �r2 � a�2� A � 2r2�a � �arb2��
Ix � �r6
4
� �3a � �3ra2b
� � �2a
r 4b3
�� Iy � �r2
4
� �a � �arb2� � �
2ar 4
b3
�� Ixy � 0
16 Ellipse (Origin of axes at centroid)
A � pab Ix � �pa
4b3
� Iy � �p b
4a3
�
Ixy � 0 IP � �p
4ab� (b2 � a2)
Circumference � p [1.5(a � b) � �a�b� ] (a/3 � b � a)
� 4.17b2/a � 4a (0 � b � a/3)
17 Parabolic semisegment (Origin of axes at corner)
y � f (x) � h�1 � �bx2
2��A � �
23bh� x� � �
38b� y� � �
25h�
Ix � �1160b5h3
� Iy � �21h5b3
� Ixy � �b1
2h2
2
�
18 Parabolic spandrel (Origin of axes at vertex)
y � f (x) � �hbx2
2
�
A � �b3h� x� � �
34b� y� � �
31h0�
Ix � �b2h1
3
� Iy � �h5b3
� Ixy � �b1
2h2
2
�
19 Semisegment of nth degree (Origin of axes at corner)
y � f (x) � h�1 � �xbn
n
�� (n � 0)
A � bh��n �
n1
�� x� � �b2((nn
�
�
12))
� y� � �2n
h�
n1
�
Ix � Iy � �3(
hnb�
3n3)
� Ixy ��4(n �
b2
1h)
2
(nn
2
� 2)�
2bh3n3
���(n � 1)(2n � 1)(3n � 1)
y
y
h
bx
x
CO
Vertex
y = f (x)
y
x
b
a a
bC
a
a
y
x
a
2a
br
b
C
D4 APPENDIX D Properties of Plane Areas
y = f (x)
y
x
y
x
C
O b
h
Vertex
C
O
yy = f (x)
y
x
x
b
h
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© 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.
20 Spandrel of nth degree (Origin of axes at point of tangency)
y � f (x) � �hbxn
n
� (n � 0)
A � �n
b�
h1
� x� � �b(
n
n
�
�
2
1)� y� � �
2
h
(
(
2
n
n
�
�
1
1
)
)�
Ix � �3(3
bnh�
3
1)� Iy � �
nh�
b3
3� Ixy � �
4(nb2
�
h2
1)�
21 Sine wave (Origin of axes at centroid)
A � �4p
bh� y� � �
p
8h�
Ix � ��98p� � �
1p
6��bh3 � 0.08659bh3 Iy � ��
p
4� � �
p
323��hb3 � 0.2412hb3
Ixy � 0 IBB � �89bp
h3
�
22 Thin circular ring (Origin of axes at center) Approximate formulas for case when t is small
A � 2prt � pdt Ix � Iy � pr 3t � �p
8d3t�
Ixy � 0 IP � 2pr3t � �pd
4
3t�
23 Thin circular arc (Origin of axes at center of circle)Approximate formulas for case when t is small
b � angle in radians (Note: For a semicircular arc, b � p/2.)
A � 2brt y� � �r si
b
n b�
Ix � r3t(b � sin b cos b) Iy � r3t(b � sin b cos b)
Ixy � 0 IBB � r3t��2b �
2
sin2b� � �
1 � c
b
os2b��
24 Thin rectangle (Origin of axes at centroid)Approximate formulas for case when t is small
A � bt
Ix � �t1b2
3
� sin2 b Iy � �t1b2
3
� cos2 b IBB � �tb3
3
� sin2 b
y
x
t
Cr
d = 2r
APPENDIX D Properties of Plane Areas D5
y
x
x h
bO
C y
y = f (x)
y
yh
b b
xB B
C
y
y
x
B BC
b b
t
r
O
y
x
B B
C
b
t
b
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25 Regular polygon with n sides (Origin of axes at centroid)
C � centroid (at center of polygon)
n � number of sides (n � 3) b � length of a side
b � central angle for a side a � interior angle (or vertex angle)
b � �36
n0°� a � ��n �
n2
��180° a � b � 180°
R1 � radius of circumscribed circle (line CA) R2 � radius of inscribed circle (line CB)
R1 � �b2
� csc �b
2� R2 � �
b2
� cot �b
2� A � �
n4b2
� cot �b
2�
Ic � moment of inertia about any axis through C (the centroid C is a principal point andevery axis through C is a principal axis)
Ic � �1n9b2
4
��cot �b
2���3cot2 �
b
2� � 1� IP � 2Ic
D6 APPENDIX D Properties of Plane Areas
b
b
a
C
BA
R1R2
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© 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.