# Free Moment of Inertia Calculator

+ Centroid Calculator

## Calculate Moment of Inertia, Centroid, and Section Modulus for a wide variety of shapes

## What SkyCiv Moment of Inertia Calculator Offers

This simple calculator will determine the moment of inertia, centroid, and other important geometric properties for a variety of shapes including rectangles, circles, hollow sections, triangles, I-Beams, T-Beams, angles and channels. We also have some articles below about how to calculate the moment of inertia, as well as more information on centroids and section modulus.

You can solve up to three sections before you're required to sign up for a free account - which also gives you access to more software and results. Our paid account will show the full hand calculations of how the tool got to this result. Refer below the calculator for more information on this topic, as well as links to other useful tools and features SkyCiv can offer you.

## How to use SkyCiv Moment of Inertia Calculator

Simply choose the cross-section shape you want to evaluate from the drop-down list, enter the dimensions of your chosen section and click Calculate.

This free multi-purpose calculator is taken from our full suite Structural Analysis Software. It allows you to:

- Calculate the Moment of Inertia (I) of a beam section (Second Moment of Area)
- Centroid Calculator used to calculate the Centroid (C) in the X and Y axis of a beam section
- Calculate the First moment of area (Statical Moment of Inertia) (Q) of a beam section (First Moment of Area)
- A Section Modulus Calculator to calculate the Section Modulus (Z) of a beam section
- Calculate the Torsion Constant (J) of a beam section

More information on these values is provided below under "Section Properties Explained".

## Section Properties Explained

The moment of inertia calculator will accurately calculate a number of important section properties used in structural engineering, including:

- Area of Section (A) - Section area is a fairly simple calculation, but directly used in axial stress calculations (the more cross section area, the more axial strength)
- Moment of Inertia (Iz, Iy)–also known as second moment of area, is a calculation used to determine the strength of a member and it’s resistance against deflection. The higher this number, the stronger the section. There are two axis here:
- Z-Axis (Iz)–This is about the Z axis and is typically considered the major axis since it is usually the strongest direction of the member
- Y-Axis (Iy)–This is about the Y axis and is considered the minor or weak axis. This is because sections aren’t designed to take as much force about this axis
- Also worth noting that if a shape has the same dimensions in both directions (square, circular etc..) these values will be the same in both directions. See Moment of Ineria of a circle to learn more.
- Centroid (Cz, Cy)–this is the center of mass for the section and usually has a Z and Y component. For symmetrical shapes, this will be geometric center. For non-symmetrical shapes (such as angle, Channel) these will be in different locations. Learn how to calculate the centroid of a beam section. The above calculator also acts as a centroid calculator, calculating the X and Y centroid of any type of shape.
- Statical Moment of Inertia (Qz, Qy)–Also known as First Moment of Area, this measures the distribution of a beam section’s area from an axis. Like the Moment of Inertia, these are in both the Z and Y direction. These are typically used in shear stress calculations, so the larger this value the stronger the section is against shearing. The calculator will provide this value, but click here to learn more about calculating the first moment of area.
- Elastic Section Modulus (Sz, Sy in America. Zz, Zy in Britain or Australia)–Also known as statical section modulus, and are used in bending stress calculations. They are usually calculated to the top and bottom fibres section. For instance, Szt is the section modulus about the Z axis to the top fibre of the section.
- Torsion Constant (J)

Other Parameters–These are calculated by the full SkyCiv Section Builder:

- Product of Inertia (about Z and Y Axis)
- Angle of Rotation of Principle Axis
- Radius of Gyration (about Z and Y Axis)
- Plastic Section Modulus (about Z and Y Axis)
- Plastic Neutral Axis (about Z and Y Axis)
- Shear Area (about Z and Y Axis)
- Distance of Shear Centre to Centroid (in both Z and Y Axis)
- Torsion Constant (Using FEA)
- Torsion Radius
- Warping Constant
- Monosymmetry Constant (about Z and Y Axis)

## About Moment of Inertia, Centroids and other Section Properties

We've also compiled more information to calculating the moment of inertia of sections. This complete guide should help provide a comprehensive knowledge base for all things related to moment of inertia, centroids, section modulus and other important geometric section properties. In the below segments, we include what is moment of inertia, how to calculate the centroid and moment of inertia and common MOI equations.

## What is Moment of Inertia?

The Moment of Inertia (more technically known as the moment of inertia of area, or the second moment of area) is an important geometric property used in structural engineering. It is directly related to the amount of material strength your section has.

Generally speaking, the higher the moment of inertia, the more strength your section has, and consequently the less it will deflect under load. The Moment of Inertia of a rectangle, or any shape for that matter, is technically a measurement of how much torque is required to accelerate the mass about an axis - hence the word inertia in its name.

## How to Calculate Moment of Inertia

#### Step 1. Segment the Beam

When calculating the area moment of inertia, we must calculate the moment of inertia of smaller segments. Try to break them into simple rectangular sections. For instance, consider the following I-beam section,which we have chosen to split into 3 rectangular segments.

#### Step 2. Calculate the Centroid and Neutral Axis

The Neutral Axis (NA) or the horizontal XX axis is located at the centroid or center of mass. We simply need to use the centroid equation for calculating the vertical (y) centroid of a multi-segment shape.

We will take the datum or reference line from the bottom of the beam section. Now let's find Ai and yi for each segment of the I-beam section shown above so that the vertical or y centroid can be found.

Using the individual segment calculations below we can determine the Neutral Axis location of the entire shape, this calculation is shown to the right.

#### Step 3. Calculate Moment of Inertia

To calculate the total moment of inertia of the section we need to use the "Parallel Axis Theorem" as defined to the right.

Since we have split it into three rectangular parts, we must calculate the moment of inertia of each of these sections. It is widely known that the moment of inertia equation of a rectangle about its centroid axis is simply:

Now we have all the information we need to use the "Parallel Axis Theorem" and find the total moment of inertia of the I-beam section. In our moment of inertia example:

### Moment of Inertia Equations

Simple equations can also be used to calculate the Moment of Inertia of common shapes and sections. These are quick moment of inertia equations that provide quick values and are a great way to cross reference or double check your results. Focusing on simple shapes only, the below diagram shows some of these equations:

## What is a Centroid?

A centroid, also known as the 'geometric center' or 'center of figure', is the center of mass of an object that has uniform density. In other terms, it is the mean position of all points on the surface of a figure. A colloquial understanding of the centroid would be to consider the location at which you would need to place a pencil to make it balance on your finger. The location at which the pencil is balanced and does not fall off your finger would be the approximate location of the centroid of the pencil. That is the location where the mass of the pencil is equal on both sides of your finger, and therefore represents the 'center of mass' of the pencil.

## More about Section Modulus

As noted earlier, this free tool also provides you with a calculation of Elastic Section Modulus, however if you're starting out as an engineer you may not understand what the Section Modulus is.

Put simply, the Section Modulus is represented within a flexural stress calculation (such as in the design of beams) As you may know, we typically calculate flexural stress using the equation:

The Elastic Section Modulus is represented in this equation as simply:

After defining this, we can re-arrange our flexural stress formula as follows:

There are two kinds of Section Modulus: Elastic and Plastic. In America, S is typically used to refer to the Elastic Section Modulus while Z is used to refer to the Plastic Section Modulus.

In Britain and Australia, these are typically reversed. Elastic Section Modulus is typically referred to with a letter Z, while the Plastic Section Modulus is referred to with a letter S.

In general, the Elastic Section Modulus is used for section design because it's applicable up to the yield point for most metals. Metals are not typically designed to go beyond the material's yield point.

## Additional Documentation

Please refer to the following documentation pages for more detailed information on Moment of Inertia, centroids, and how to calculate them for various shapes:

More Free Tools Available

SkyCiv also offers other tools such as I beam size tool and free structural design software. The dynamic section drawer will also show you a graphical representation of your beam section. So if you want to calculate the moment of inertia of circle, moment of inertia of a rectangle or any other shapes, feel free to use the below software or our all-inclusive SkyCiv Section Builder.

SkyCiv offers a wide range of Cloud Structural Analysis and Design Software for engineers. As a constantly evolving tech company, we're committed to innovating and challenging existing workflows to save engineers time in their work processes and designs.