moment inersia

MOMENT OF INERTIA


7.1. Introduction
We have discussed in Art 3-2 that the moment of a force about a point, is the product of the force (P) and perpendicular distance (x) between the point and the line of action of the force (i, e, P, x). This moment is also called first moment of force. If this moment is again multiplied by perpendicular distance (x) between the point and the line of action of the force i, e. (P.x) x = P.x2. Then this quantity is called, moment of the moment of a force or second moment of force or moment of inertia (briefly written as M.I.).
Sometimes, instead of force, the area or mass of a figure or body is taken into consideration, then the second moment is known as second moment of area or second moment of mass. But all such second moments are broadly termed as moment of inertia. In this chapter, we shall discuss the moment of inertia of plane areas only.

7.2. Moment of Inertia of a Plane Area
Consider a plane area, whose moment of inertia is required to be found out. Split up the whole area into a number of small elements.
Moment of Inertia :
Let … = Areas of small elements, and
… = Distance of the elements from the line about which the moment of inertia is required to be found out
Now the moment of inertia :
I =
=

7.3. Units of Moment of Inertia
The units of moment of inertia of a plane area depends upon the units of the area and the length e.g.
1. If area is in m2 and the length is also in m, the moment of inertia is expressed in m4
2. If area is in cm2 and the length is also in cm, then moment of inertia is expressed in cm4
3. If area is in mm2 and the length is also in mm, then moment of inertia is expressed in mm4

7.4. Modulus of Section
The modulus of section (or section modulus) o, a figure is the quantity obtained by dividing the moment of inertia, of the figure, about its e.g. by the distance of the extreme fibre from the centroidal axis. It is generally denoted by Z and the suffixes (XX or YY) indicates the axis, about which the distance is measured.
Consider a rectangular section of width b and depth d s shown in Fig. 7-1.
Let Ixx be the moment of inertia about X-X axis.
Modulus of section about X-X axis :
= (where is the distance of the extreme fibre AE or CD from X-X axis)
Similarly =

7.5. Methods of Finding out Moments of Inertia
The moments of inertia of a body (or an area) may be found, out by any one of the following two methods :
(1) By using Route’s rule, and
(2) By the method of integration

7.6. Moment of Inertia by Routh’s Rule
If a body is symmetrical about three mutually perpendicular axes, then the moment of inertia, according, to Routh’s rule, in a body about any one axis passing through its centre of gravity, is given by :
I = (for a square or rectangular lamina)
= (for a circular elliptical lamina)
= (for a spherical body)


Where :
A = Area of the body
M = Mass of the body, and
S = Sum of the squares of the two semi-axes, other than the axis, about which the moment of inertia is required to be found out.

7.7. Moment of Inertia by the Method of Integration
The moment of inertia of an area or (mass) may also be found out by the method of integration as discussed below :
Consider a plane figure, whose moment of inertia is required to be found out about X-X axis and Y-Y axis as shown in Fig. 7.2. Let us divide the whole area into a number of strips. Consider one of these strips.
Let, A = Area of the strip under consideration
x = Distance of the e.g. of the strip on X-X axis and
y = Distance of the e.g. of the strip on Y-Y axis
We know that the moment of inertia of the strip about Y-Y axis.
= dA.x2
i, e, X-X axis, Y-Y axis or Z-Z axis, and moment of inertia of the whole area may be found out by integrating above equation i, e,
=
Similarly =

7.8. Moment of Inertia of a Rectangular Section
Consider a rectangular section ABCD as shown in Fig. 7-3.
Let b = Width of the section and
d = Depth of the section
Now consider a strip PQ of thickness dx parallel to X-X axis and at a distance x from it as shown in Fig. 7-3.

= b.d.x
M.I. of the strip about Y-Y axis,
= Area x x2 = b.dx) x2
= b.x2.dx
The M.I. of the whole section can be found out by integrating for the whole length of the lamina i, e, from .

=
=
This can also be obtained by Routh’s rule as discussed below :
= ( of rectangular section)
A = b x d
Where area, and sum of the square of semi axis Y-Y and Z-Z,
S =
=

7.9. Moment of Inertia of a Hollow Rectangular Section
Consider a hollow rectangular section, in which ABCD is the main section and EFGH is the cut out section as shown in Fig. 7-4.
Let, b = Breadth AB of the outer rectangle
d = Depth BC of the outer rectangle