Parallel axis theorem

What is the parallel axis theorem?

The parallel axis theorem states that “The moment of inertia about the axis parallel to the centroidal axis is given by the sum of moment of inertia about a centroidal axis and product of area and square of perpendicular distance between two axes.”

centroidal axis

Parallel axis theorem formula:

As per the statement of parallel axis theorem,

`I_{1}=I_{C}+Ah^{2}`

Where,
`I_{C}` = Moment of inertia about the centroidal axis
`I_{1}` = Moment of inertia about an axis parallel to the centroidal axis
h = Perpendicular distance between two axis
A = Area of the plane lamina

When to use the parallel axis theorem?

Here are the conditions for the use of the parallel axis theorem:-

  1. Both axes should be parallel to each other.
  2. One of the axes should pass through the centroid of the body.

How to use parallel axis theorem?

Here are the steps for finding the moment of inertia by parallel axis theorem:-

Step 1] Find a moment of inertia about the centroid of the body by using standard formulae.

Step 2] Find the area of the object (A) and perpendicular distance (h) between the centroidal axis and axis parallel to the centroidal axis.

Step 3] Use the parallel axis theorem equation to find a moment of inertia about the parallel axis.

Parallel axis theorem numerical:

The numerical is the best way to clear the concepts, therefore we have provided numerical based on parallel axis theorem that will help you to understand the theorem.

For the given cross-section, calculate the moment of inertia about the x and y-axis.

Parallel axis theorem

Solution:-

A) Moment of inertia about x-axis:-

Step 1] Calculate the moment of inertia about the centroidal x-axis:

For the square section, the moment of inertia about the centroidal x-axis is given by,

`I_{XC}` = `\frac{bd^{3}}{12}`=`\frac{2\times 2^{3}}{12}`

`I_{XC}` = 1.33 cm⁴

Step 2] Area (A) and perpendicular distance (h):

Area = 2 × 2 = 4 cm²

Distance between centroidal axis and x axis,

h = 1 + `\frac{2}{2}` = 2 cm

Step 3] Moment of inertia about x-axis:

By parallel axis theorem,

`I_{X}=I_{XC}+Ah^{2}`

`I_{X}=1.33+(4\times 2^{2})`

∴ Moment of inertia about the x-axis = `I_{X}` = 17.33 cm⁴

B) Moment of inertia about y-axis:-

Step 1] Calculate the moment of inertia about the centroidal y-axis:

For the square section, the moment of inertia about the centroidal y-axis is given by,

`I_{YC}`=`\frac{bd^{3}}{12}`=`\frac{2\times 2^{3}}{12}`

`I_{YC}` = 1.33 cm⁴

Step 2] Area (A) and perpendicular distance (h):

Area = 2 × 2 = 4 cm⁴

Distance between centroidal axis and y-axis,

h = 1 + `\frac{2}{2}` = 2 cm

Step 3] Moment of inertia about y-axis:

By parallel axis theorem,

`I_{Y}=I_{YC}+Ah^{2}`

`I_{Y}=1.33+(4\times 2^{2})`

∴ Moment of inertia about the y-axis = `I_{Y}` = 17.33 cm⁴

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