# Free Moment of Inertia Calculator

Easily calculate custom section properties including moment of inertia, warping, centroid, and section modulus.

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## How to Use the Free Cross-Section Calculator

The ClearCalcs cross-section calculator allows the user to input the geometry of an arbitrary cross-section using either simple dimensions of common shapes, or fully-custom outline definitions. It then determines the elastic, warping, and/or plastic properties of that section - including areas, centroid coordinates, second moments of area / moments of inertia, section moduli, principal axes, torsion constant, and more!

You can use the cross-section properties from this tool in our free beam calculator.

Signing up for a ClearCalcs account will unlock further advanced features for design and analysis of beams and a variety of other structural elements - and allow the use of these custom cross-sections in those designs. ClearCalcs enables design in steel, concrete and timber, according to Australian, US and EU Standards.

The sheet is divided into two main sections:

1. ‘Key Properties’, where the geometry of the cross-section is defined.
2. ‘Summary’, where the type of analysis is selected and the calculated properties are displayed.

A ‘Comments’ section is also included for the user to leave any specific design notes. Clicking on any of the input/property labels gives a descriptive reference explanation.

### 1. Input Key Properties

First, select the Cross-Section Type from the drop-down menu just below the diagram. When you do so, the input boxes below it will change to those required for the given type of cross-section.

For example, a rectangle has two dimensions to define it: Depth and Breadth. Both dimensions must be greater than zero, but there are no other restrictions.

Some other cross-section types have specific restrictions. For example, an I-section's Depth must be greater must be greater than two times the Flange Thickness plus two times the Inner Radius. Restrictions such as these are merely logical restrictions on the geometry; overlaps or incomplete fillet radii are not physically possible.

The diagram in this section will show the cross-section as it has been input, as well as some of the key properties of that cross-section - including centroid, principal axis orientation, and, if the relevant analysis types have been performed, plastic centroid and shear centre.

### Calculation Summary Outputs and Analysis Types

Four different Analysis Type options are available: “Elastic Only”, “Elastic + Warping”, “Elastic + Plastic”, and “Everything”. The default, and fastest, option is “Elastic Only”, while the other options add warping and/or plastic analyses as well. Note that an elastic analysis is always performed in every option. The various outputs calculated in each of these analysis types are described below:

#### Elastic Analysis

• Angle of Major Principal Axis: The major principal axis (the “1” axis) may be inclined for non-symmetric sections, or it may be at 90 degrees if the section has more lateral than vertical stiffness. This defines its angle, relative to the X-axis. Note that the minor principal axis (the “2” axis) is exactly perpendicular to this. The principal axis orientation is also indicated on the cross-section diagram.
• Area: The cross-sectional area of the section. This value is commonly used in determining the axial strength of a column.
• First Moments of Area: The first moments of area are relevant for certain shear calculations, such as shear flow. Note that the first moments are area are taken about the centroid and the geometric axes.
• Second Moments of Area / Moments of Inertia: The second moments of area, also known in engineering as the moments of inertia, are related to the bending strength and deflection of a beam. Note that all values are taken about the centroid of the cross-section, though values are available for both geometric and principal axes. The Polar Moment of Inertia is identical for both types of axes, as the “Z” axis is always assumed to be the same as the “3” axis. The Product Moment of Inertia is, by definition, zero for principal axes.
• Elastic Section Moduli: The elastic section moduli are equal to the second moments of area / moments of inertia divided by the distance to the farthest fibre in the cross-section perpendicular to the axis of bending. Values are provided for both positive and negative bending, where positive bending is defined as the top-most or left-most portion of the cross-section being in compression. Values are also provided for both geometric and principal axes, and are always about the centroid. N.B. Elastic section modulus is also known as statical section modulus.
• Distance from Centroid to Extreme Fibres: The distance between the centroid of the cross-section and the extreme fibre of the cross-section, perpendicular to the axis of bending. The second moments of area / moments of inertia divided by these distances will equal the elastic section moduli.
• Radii of Gyration: The radii of gyration are the root mean square distances of each fibre in the cross-section relative to the given axis. Values are always about the centroid, and are available about both the geometric and principal axes. The Polar Radius of Gyration is identical for both types of axes, as the “Z” axis is always assumed to be the same as the “3” axis.
• Centroid: The location of the centroid is shown in the cross-section diagram. Mouse over the green circle icon, and a tooltip will display the exact coordinates of the centroid. Note that the origin (0,0) location is indicated by blue crosshairs.

#### Warping Analysis

• St Venant Torsion Constant: The torsion constant is related to how well the cross-section can resist pure torsional forces, and is commonly used in lateral-torsional buckling formulae.
• Warping Constant: As a torsional or eccentric force is applied, the cross-section may not just twist but also warp. This constant is a measure of how easily that warping can happen, and is often used in lateral-torsional buckling formulae.
• Shear Areas: Only some of a cross-section will effectively resist a shear force applied about a given axis (that is, a shear force perpendicular to the given axis). The values shown are based upon a shear flow integration, and as such, they may not exactly match classical calculations based upon areas of a web.
• Monosymmetry Constants: The monosymmetry constants define how close a cross-section is to being symmetric. The constant approaches zero when a cross-section is symmetric about the given axis.
• Shear Centre: The location of the shear centre is shown in the cross-section diagram. Mouse over the star icon, and a tooltip will display the exact coordinates of the shear centre. Note that the origin (0,0) location is indicated by blue crosshairs.

#### Plastic Analysis

• Plastic Section Moduli: The plastic section moduli are shown for both geometric and principal axes. Note that while the plastic section modulus is not dependent upon the direction of bending, the shape factor, which is the ratio of plastic to elastic section moduli, is so dependent.
• Plastic Centroid: The location of the plastic centroid is shown in the cross-section diagram. Mouse over the orange square icon, and a tooltip will display the exact coordinates of the plastic centroid. Note that the origin (0,0) location is indicated by blue crosshairs.