How to calculate geometric properties of a plane section

We will show three methods to calculate geometric properties of a plane section.

1. Use formulas of mechanics
2. Use a CAD system on a computer
3. Use an app, like SecProp for Iphone.

1. Use formulas of mechanics

1.1. Calculate the centroid

The centroid of area for a symmetrical object such as a square or circle is located at the centre of the object, however this is not necessarily the case for other complex objects.

You can use this formulas:

In the equation yi and xi are the coordinates of the centre of mass of the smaller objects (i.e. the region must be divided in smaller objects).

Ai is the area of the smaller object

AT is the total area of the cross section

The products yi x Ai and xi x Ai are the first moments of area relative to the x axis and y axis.

1.2 Calculating the Moment of Inertia

The moment of inertia depends on the dimensions and distribution of mass within an object; it also depends on the objects orientation, therefore the moment of inertia about the x axis is different than the moment of inertia about the y axis.

The moment of inertia about the x axis can be calculated using the equation below.

I x ‘,1 is the moment of inertia of the smaller object about the x axis, located at yi

The variable d is the distance between yi and Y

 

2. Use a CAD system

If you have a CAD software you can (depending from the software) use some commands.

For example in Autocad ® with the AMINERTIA command, you can calculate the moment of inertia of closed contours.

The CAD software is expensive, but if you have it, you can easily use it.

 

3. Use an app

The iOS app SecProp, avaliable on the AppStore is a cheap alternative to calculate geometric properties of a section, avoiding to perform calculations or use a CAD system.