Geometrical moment of inertia - 2 (Second moment of area)

 

I will explain it following 'Geometrical moment of inertia-1'.

 

The 'I' mentioned in the 'Geometrical moment of inertia - 1'

is also referred to as the 'Right angle moment of inertia'.

On the other hand, there is something called the 'Polar moment of inertia'.

 

The 'Right angle moment of inertia' is based on the x, y-axis,

while the 'Polar moment of inertia' is based on the z-axis.

 

For example, as shown in Fig. 1 below.

 

Fig.1

 

If the 'Right angle moment of inertia (I)' obtained with

the cross section of the object as the x, y-axis is related

to the bending stiffness involved in bending the object about the x, y axis

 

The 'Polar moment of inertia' is related to the torsional stiffness

involved in twisting an object about the z-axis passing through the cross section of the object.

 

Let's check the relation between the 'Right angle moment of inertia'

and the 'Polar moment of inertia'.

 

 

From Fig. 2 below

 

Fig.2

 

'Right angle moment of inertia' for a micro-surface is

 

Eq.1 Right angle moment of inertia for micro-surface

 

The sum of the two moments of inertia

 

Eq.2

 

Because of

 

Eq.3

 

Equation.2 becomes Equation.4.

 

Eq.4

 

As a result,

 

Eq.5 Polar moment of inertia

 

The 'Polar moment of inertia' is called 'J'.

The 'Polar moment of inertia' is equal to the sum of the

two 'Right angle moments of inertia'.

 

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As shown in Fig. 3, if you know the 'Geometrical moment of inertia'

based on the centroid of the object and the reference axis and the center axis do not coincide,

you can use the 'Parallel-axis theorem'.

 

Fig.3

 

Using the 'Parallel axis theorem' in Fig. 3 is as follows.

 

Eq.6

 

The same is true for the y-axis.

 

Eq.7

 

The reference axis and the Centroid must be parallel.