## Introduction

The second moment of area, also known as the moment of inertia, is a geometrical property that measures a cross-sectional shape’s resistance to bending and deflection. The second moment of area is typically denoted by ‘I’, and its unit of dimension is a unit value of length to the fourth power (L4), i.e. mm4, m4, in4.
You can use the Moment of Inertia Calculators to the right to:
1. Choose the cross-section you want to evaluate
2. Determine cross-sectional properties, including Area, Centroid and Section Modulus
3. Obtain the results for the Moment of Inertia about both axes

## Explanation

### Significance

The moment of inertia is used in member analysis, calculating bending stresses and when calculating flexural and torsional rigidity. It is also important to note that it is not a fixed value that is unique to each cross-section; it changes depending on the location of the bending (or reference) axis.
The second moment of inertia is one of a structural elements' most important material properties because it is a determining factor in an engineer's section selection. Since this value is used to predict the resistance of beams to bending, torsion and deflection, you'll see it in quite a few engineering concepts and formulas, in particular the ones to the right.
Bending stress: the stress that an object encounters when it is subjected to a large load at a particular point, causing it to bend and become fatigued.
﻿
Deflection: is also known as displacement and can occur from externally applied loads or the structure's self-weight. If the deflection is large enough, you can visualise the degree to which an element/structure changes shape or is displaced from its original position.
﻿

### Parallel Axis Theorem

The parallel axis theorem involves relating the moment of inertia of a cross-section about an arbitrary reference axis to its moment of inertia about a parallel centroidal axis. This theorem is useful because knowing the centroidal moment of inertia of a shape means its moment of inertia can be calculated about any parallel axis by adding a correction factor.
This theorem is particularly useful when the moment of inertia of a complex shape with multiple parts needs to be calculated.
The parallel axis theorem is as follows:
﻿
Where:
﻿
With this equation, the moments of inertia of a composite shape, which is made up of multiple elementary shapes, can be found. The moment of inertia of each individual shape about its own centroidal axis is found first; then, the parallel axis terms are added to determine the moment of inertia of the composite area.

### Combined Shapes

For composite shapes made up of smaller shapes or subparts, the moment of inertia for the entire cross-section is equivalent to the sum of the moments of inertia of the individual shapes, 1, 2 and 3, about the global N.A. in the diagram shown here. Combining sections for compound shapes
﻿ Subtracted sections for a hollow section
If there are any holes or voids within the cross sections, their moment of inertia must also be calculated but subtracted from the total.
﻿
﻿
﻿