384 Higher Engineering Mathematics

Second moments of areas are usually denoted byIand

have units of mm^4 ,cm^4 , and so on.

Radius of gyration

Several areas,a 1 ,a 2 ,a 3 ,...at distancesy 1 ,y 2 ,y 3 ,...

from a fixed axis, may be replaced by a single area

A,whereA=a 1 +a 2 +a 3 +···at distancekfrom the

axis, such thatAk^2 =

∑

ay^2.

kis called theradius of gyrationof areaAabout the

given axis. SinceAk^2 =

∑

ay^2 =Ithen the radius of

gyration,

k=

√

I

A

The second moment of area is a quantity much used in

the theory of bending of beams, in the torsion of shafts,

and in calculations involving water planes and centres

of pressure.

The procedure to determine the second moment of

areaofregularsectionsabout agivenaxisis(i)tofindthe

second moment of area of a typical element and (ii) to

sum all such second moments of area by integrating

between appropriate limits.

For example, the second moment of area of the rect-

angle shown in Fig. 38.14 about axisPPis found by

initiallyconsidering an elemental strip of widthδx,par-

allel to and distancexfrom axisPP. Area of shaded

strip=bδx.

b

l

x

P

P

x

Figure 38.14

Second moment of area of the shaded strip about

PP=(x^2 )(bδx).

The second moment of area of the whole rectangle about

PPis obtained by summing all such strips betweenx=

0andx=l,i.e.

∑x=l

x= 0 x

(^2) bδx.
It is a fundamental theorem of integration that
limit
δx→ 0
∑x=l
x= 0
x^2 bδx=
∫l
0
x^2 bdx
Thus the second moment of area of the rectangle
aboutPP
=b
∫l
0
x^2 dx=b
[
x^3
3
]l
0

bl^3
3
Since the total area of the rectangle,A=lb,then
Ipp=(lb)
(
l^2
3
)

Al^2
3
Ipp=Ak^2 ppthusk^2 pp=
l^2
3
i.e. the radius of gyration about axesPP,
kpp=
√
l^2
3

l
√
3
Parallel axis theorem
In Fig. 38.15, axisGGpasses through the centroidC
of areaA.AxesDDandGGare in the same plane, are
parallel to each other and distancedapart. The parallel
axis theorem states:
IDD=IGG+Ad^2
Using the parallel axis theorem the second moment of
area of a rectangle about an axis through the centroid
d
G
C
Area A
G
D
D
Figure 38.15