Mechanical Engineering Principles

FIRST AND SECOND MOMENT OF AREAS 91

Table 7.1 Summary of standard results of the second moments of areas of regular sections

Shape Position of axis Second moment

of area,I

Radius of

gyration,k

Rectangle

lengthd

breadthb

(1) Coinciding withb

(2) Coinciding with d

(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

dh^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

IGG=dh

3

12 whered=^40 .0mmandh=^15 .0mm

Hence IGG=

( 40. 0 )( 15. 0 )^3

12

=11250 mm^4

From the parallel axis theorem,

IPP=IGG+AH^2 ,

where A= 40. 0 × 15. 0 =600 mm^2

and H= 25. 0 + 7. 5 = 32 .5mm,

the perpendicular distance betweenGGandPP.

Hence IPP= 11250 +( 600 )( 32. 5 )^2

=645000 mm^4

IPP=AkPP^2 , from which,

kPP=

√

IPP

area

=

√(

645000

600

)

= 32 .79 mm