Properties of Plane Areas

E
Properties of Plane Areas

Notation: A area x, 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

1

y x h C b y x

Rectangle (Origin of axes at centroid) A Ix bh bh3 12 x Iy b 2 hb3 12 y h 2 Ixy 0 IP bh 2 (h 12 b2)

2

y

B

Rectangle (Origin of axes at corner) Ix bh3 3 Iy hb3 3 Ixy b2h2 4 IP bh 2 (h + b2) 3

h O B b y c x h C y b x x

IBB

b3h3 6(b2 h2)

3

Triangle (Origin of axes at centroid) A Ix Ixy bh 2 bh3 36 bh2 (b 72 x Iy 2c) b 3 bh 2 (b 36 IP c y bc h 3 c2) b2 bc c2)

bh 2 (h 36

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APPENDIX E Properties of Plane Areas

4

y B

c B h

Triangle (Origin of axes at vertex) Ix Ixy bh3 12 bh2 (3b 24 Iy bh (3b2 12 2c) IBB 3bc bh3 4 c2)

O

b y

x

5 x h C B b

Isosceles triangle (Origin of axes at centroid) A y x B

bh 2 bh3 36

x Iy

b 2 hb3 48 3b2)

y

h 3 Ixy IBB 0 bh3 12 3 b/2.)

Ix IP

bh (4h2 144

(Note: For an equilateral triangle, h 6 x C B b y x B y

Right triangle (Origin of axes at centroid) A Ix IP bh 2 bh3 36 bh 2 (h 36 x Iy b2) b 3 hb3 36 IBB y h 3 Ixy bh3 12 b2h2 72

h

7
B h

y B

Right triangle (Origin of axes at vertex) Ix IP bh3 12 bh 2 (h 12 Iy b2) hb3 12 IBB Ixy bh3 4 b2h2 24

O

x b y a

8

Trapezoid (Origin of axes at centroid) A h(a 2 b) y h(2a 3(a b) b) IBB h3(3a b) 12

h B

C b

y

x B

Ix

h3(a2 4ab b2) 36(a 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.

APPENDIX E Properties of Plane Areas

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9

y

d = 2r r

Circle