Mechanical Engineering Principles

FIRST AND SECOND MOMENT OF AREAS 89

theradius of gyrationof areaAabout the given
axis. SinceAk^2 =

∑
ay^2 =I then the radius of
gyration,

k=

√

I

A

.

The second moment of area is a quantity much used
in the theory of bending of beams (see Chapter 8),
in the torsion of shafts (see Chapter 10), and in
calculations involving water planes and centres of
pressure (see Chapter 21).
Theprocedure to determine the second moment
of area of regular sectionsabout 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 Figure 7.7 about axis PP is
found by initially considering an elemental strip of
widthδx, parallel to and distancexfrom axisPP.
Area of shaded strip=bδx. Second moment of
area of the shaded strip aboutPP=(x^2 )(bδx).
The second moment of area of the whole rectangle
aboutPP is obtained by summing all such strips
betweenx=0andx=d,

i.e.

x∑=d

x= 0

x^2 bδx

d

Figure 7.7

It is a fundamental theorem of integration that

limit

δx→x

x∑=d

x= 0

x^2 bδx=

∫d

0

x^2 bdx

Thus the second moment of area of the rectangle

aboutPP

=b

∫d

0

x^2 dx=b

[

x^3

3

]d

0

=

bd^3

3

Since the total area of the rectangle,A=db,then

Ipp=(db)

(

d^2

3

)

=

Ad^2

3

Ipp=Akpp^2 thus kpp^2 =

d^2

3

i.e. the radius of gyration about axisPP,

kpp=

√

d^2

3

=

d

√

3

H

Area

Figure 7.8

Parallel axis theorem

In Figure 7.8, axisGGpasses through the centroid

Cof areaA.AxesDDandGGare in the same

plane, are parallel to each other and distanceH

apart. The parallel axis theorem states:

IDD=IGG+AH^2

Using the parallel axis theorem the second moment

of area of a rectangle about an axis through the

centroid may be determined. In the rectangle shown

in Figure 7.9,

IPP=

bd^3

3

(from above)