Area Moment of Inertia: Definition, Formula, Unit, Calculation

In this article, you will learn a complete overview of the area moment of inertia or second moment of area such as its definition, formula of different sections, units, calculation, and many more. 

The area moment of inertia is a geometrical property of an area that indicates how its points are distributed about an axis.

It signifies the resistance of an area against the applied moment ( bending moment or twisting moment ) about an axis.

We have also discussed the polar moment of inertia, first moment of area, in our previous article here we will learn only about the area moment of inertia.

So without wasting time let's get started.


Definition of Area Moment of Inertia

The area moment of inertia is a geometrical property of an area that measures how its points are distributed with regard to an arbitrary axis, providing measures of how efficiently the cross-sectional shape can resist bending caused by loading.

The area moment of inertia is also called the second moment of inertia, second moment of area, or second area moment.

It is denoted by I.


Formula of Second Moment of Area

The first thing to note is that the area moment of inertia is not a unique property of a cross-section.

It measures the resistance to bending or twisting about a particular axis, and hence its value changes depending on where we place this reference axis.

We can estimate the area moment of inertia of a cross-section by dividing it into smaller elements shown in the figure.

second moment of inertia

Each element contributes to the total area of ​​inertia, equal to its area dA multiplied by d².

Here d² is the distance from the reference axis.

We can sum up the values ​​for all of the small elements to obtain the area moment of inertia for the entire cross-section.

Mathematically,

I = Σd².dA

We can more precisely define the area moment of inertia using integration like this:

I = ∫d².dA

The area moment of inertia around yy is calculated using distance to the arbitrary axis yy, given the coordinate along xx. As shown in the above figure.

Iᵧᵧ = ∫x².dA

Where,

Iᵧᵧ = Area Moment of Inertia related to the y axis 

x = the perpendicular distance from axis y to the element dA

dA = An elemental area 

and,

The area moment of inertia around xx is calculated using distance to the arbitrary axis xx, given the coordinate along with yy.

Iₓₓ = ∫y².dA

Where,

Iₓₓ = Area Moment of Inertia related to the x-axis

y = The perpendicular distance from axis x to the element dA

dA = An elemental area 


Area Moment of Inertia of about x' and y' Axis

Sometimes it is necessary to calculate the area moment of inertia of a body with respect to the x' and y' axes.

We can calculate the moment of area for the x' and y' axes using two theorems.

  • Parallel Axis Theorem 
  • Perpendicular Axis Theorem

Parallel Axis Theorem

The parallel axis theorem states that The moment of inertia of a plane section about any axis parallel to the centroidal axis is equal to the moment of inertia of the section about the centroidal axis plus the product of the area of the section and the square of the distance between the two axes.

Mathematically, It can be written as,

Iₓ, = Iₓ + Ad²

Where,

A = area of the body

Iₓ = Area Moment of inertia about point x

d = Perpendicular distance between the two lines x and x'.

Iₓ, = Area Moment of inertia about point x'

A similar method can be applied to calculate a moment of area about y' axis.


Perpendicular Axis Theorem

The theorem states that the moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two perpendicular axes lying in the plane of the body such that all three axes are mutually perpendicular and have a common point.

This theorem is also called the polar axis theorem.

Mathematically,

Iz = Iₓ + Iᵧ

Where,

Iz = Moment of Inertia about the Z axis

Iₓ = Moment of Inertia about the X axis

Iᵧ = Moment of Inertia about the Y axis


Unit of Second Moment of Inertia

In SI Unit,

As we know, the second moment of area

I = Σd².dA

So,

I = m² × m²

I = m⁴  or mm⁴

Similarly in the CGS unit, it will be,

I = cm⁴ 

So, The second moment of inertia has the unit of length to the fourth power.


Area Moment of Inertia for Different Shapes

Now step by step we will know the area moment of inertia of different sections.

Show in figure rectangle, hollow rectangle, and square section that the second moment of area is calculated one by one.

second moment of area

Rectangular Section

As we know,

The second moment of inertia in relation to an axis:

I = ∫d².dA

So,

Area Moment of Inertia related to the y axis

Iₓₓ = ∫y².dA

Iₓₓ = ∫y².bdy = [(b.y³)/3]     ( from -h/2 to h/2)

Iₓₓ =  [(b.h³)/12]

Similarly,

Area Moment of Inertia related to the x-axis

Iᵧᵧ = ∫x².dA = [(h.x³)/3]     ( from -b/2 to b/2 )

Iᵧᵧ =  [(h.b³)/12]

Similarly,


Hollow Rectangular Section

Iₓₓ =  [(B.H³)/12] - [(b.h³)/12]

Iᵧᵧ =  [(H.B³)/12] - [(h.b³)/12]


Square Section

Iₓₓ = Iᵧᵧ = a⁴

Now, Shown in the figure isosceles triangle and a right-angled triangle the second moment of Inertia is given below.

second moment of area

Isosceles triangle

Iₓₓ =  [(b.h³)/36]

Iᵧᵧ =  [(h.b³)/48]


Right-Angled Triangle

Iₓₓ =  [(b.h³)/36]

Iᵧᵧ =  [(h.b³)/36]

Now, show in the figure circle, semi-circle, and quarter circle that the second moment of area is calculated one by one.

area moment of inertia

Circle

Let a small element dA.

So,

dA = r.dr.dθ

y = rsinθ

Iₓₓ = ∫y².dA

= 4 ∫ ∫r²sin²θ.r.dr.dθ   ( from 0 to r and 0 to π/2)

= 4 ∫r³dr ∫sin²θ.dθ

After calculating,

Iₓₓ = (πr⁴)/4 = (π/64)d⁴

So,

Iₓₓ = Iᵧᵧ = (π/64)d⁴

Similarly,


Semi Circle

Iₓₓ = Iᵧᵧ = (π/128)d⁴


Quarter Circle

Iₓₓ = Iᵧᵧ = (π/256)d⁴


Calculation of Second Moment of Area

Question

Determine the area moment of inertia about the centroidal x and y axes for the rectangular area of 600 mm height and 400 mm width.

Solution

Given Data

h = 600 mm

b = 400 mm

As we know,

The second moment of area for the rectangular section,

For x-axis,

Iₓₓ =  [(b.h³)/12]

So,

Iₓₓ =  [(400 × 600³)/12]

Iₓₓ =  7200 × 10⁶ mm⁴

For y-axis,

Iᵧᵧ =  [(h.b³)/12]

Iᵧᵧ =  [(600 × 400³)/12]

Iᵧᵧ =  3200 × 10⁶ mm⁴


So here you have to know all aspects related to the second moment of area or the second moment of inertia. 

If you have any doubt then you are free to ask me by mail or on the contact us page.


Thank You.

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