Mechanical Engineering Principles

7

First and second moment of areas

At the end of this chapter you should be

able to:

• define a centroid


• define first moment of area


• calculate centroids using integration


• define second moment of area


• define radius of gyration


• state the parallel axis and perpendicular
  axis theorems


• calculate the second moment of area and
  radius of gyration of regular sections using
  a table of standard results


• calculate the second moment of area forI,
  Tand channel bar beam sections

7.1 Centroids

Alaminais a thin flat sheet having uniform thick-
ness. Thecentre of gravityof a lamina is the point
where it balances perfectly, i.e. the lamina’scentre
of moment of mass. When dealing with an area (i.e.
a lamina of negligible thickness and mass) the term
centre of moment of areaorcentroidis used for
the point where the centre of gravity of a lamina of
that shape would lie.

7.2 The first moment of area

Thefirst moment of areais defined as the product
of the area and the perpendicular distance of its
centroid from a given axis in the plane of the area.
In Figure 7.1, the first moment of areaAabout axis
XXis given by (Ay) cubic units.

X

C

X

y

Area A

Figure 7.1

7.3 Centroid of area between a curve

and thex-axis

(i) Figure 7.2 shows an areaPQRS bounded

by the curve y = f(x),thex-axis and

ordinates x= aandx= b. Let this area

be divided into a large number of strips,

each of widthδx. A typical strip is shown

shaded drawn at point (x,y)onf(x). The

area of the strip is approximately rectangular

and is given byyδx. The centroid,C,has

coordinates

(

x,

y

2

)

.

y

y

x= a x= b x

dx

0

x

S

P

C(x, )

R

Q

y = f(x)

y

− 2

Figure 7.2