Mechanical Engineering Principles

282 MECHANICAL ENGINEERING PRINCIPLES

Summary of standard results of the second moments of areas of regular sections

Shape Position of axis Second
moment
of area,I

Radius

of gyra-

tion,k

Rectangle
lengthd
breadthb

(1) Coinciding withb

(2) Coinciding withd

(3) Through centroid,

parallel tob

(4) Through centroid,

parallel tod

bd^3

3

db^3

3

bd^3

12

db^3

12

d √ 3 b √ 3 d √

12

b

√

12

Triangle
Perpendicular
heighth
baseb

(1) Coinciding withb

(2) Through centroid, parallel

to base

(3) Through vertex,

parallel to base

bh^3

12

bh^3

36

bh^3

4

h

√

6

h

√

18

h

√

2

Circle
radiusr
diameterd

(1) Through centre,

perpendicular to

plane (i.e. polar axis)

(2) Coinciding with diameter

(3) About a tangent

πr^4

2

or

πd^4

32

πr^4

4

or

πd^4

64

5 πr^4

4

or

5 πd^4

64

r √ 2 r 2 √ 5 2

r

Semicircle
radiusr

Coinciding with diameter

πr^4

8

r

2