SOME APPLICATIONS OF INTEGRATION 383H
from the centroid of the area to the fixed axis is
squared.
Second moments of areas are usually denoted by
Iand 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 areaAaboutthe 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 area of regular sections about a given axis is (i) to
find the 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
rectangle shown in Fig. 38.14 about axisPPis found
by initially considering an elemental strip of width
δx, parallel to and distancexfrom axisPP. Area of
shaded strip=bδx.
Figure 38.14
Second moment of area of the shaded strip about
PP=(x^2 )(bδx).
The second moment of area of the whole rectan-
gle aboutPPis obtained by summing all such strips
betweenx=0 andx=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 centroid
Cof 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
Figure 38.15