Moments of Inertia
Determination of Moments of Inertia by both analytical and graphical methods.
Contents
Moments Of Inertia
The moment of InertiaInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass.
To calculate the polar moment of inertia
about the centre of the section
:
But
and by the parallel axis theory (For proof of theorem see next section), and since
and
are equal being moments of inertia about a diameter.
For a hollow section with external diameters and internal diameters of
and
.
and
The ratio
max is called the section modulus
, so that the maximum stress
equals
.
The bending moment which can be carried by a given section for a limiting stress is called the Moment of resistance.
A bending moment exists in a structural element when a moment is applied to the element so that the element bends.
The Parallel Axis Theorem

Stated in words:
The moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the centroid plus the area times the square of the distance between the axes.
It should be noted that the moment of inertia through the centroid is the minimum value for any axis in that particular direction.
If it is required to transfer from one axis Calculations Of Moments Of Inertia
a) Rectangular section.
From the diagram,
For a hollow section of outside dimensions
and inside dimensions
)
b) 
Using the dimensions shown, the moment of inertia about
may be obtained by subtracting that for rectangles
wide and
deep from the overall figure for
ties
.
i.e.
Alternatively, for greater accuracy of calculation, the web and flanges may be treated separately using the parallel axis theorem for the flanges.
Hence,
Where
is the distance between the centroid axis of the flange itself and the principle axis of the whole cross section
. The term
is very small and can usually be neglected
The width being the dimension parallel to
and the depth parallel to
.
The Graphical Determination Of Moment Of Inertia.
Assume that it is required to find the Moment of Inertia of an irregular shape about its centroid,

If
is the distance from the centroid then taking Moments about
:
i.e.
Using the Parallel Axis Formula (Equation 1)
Where 
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