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

Key applications

General member deflection under load

Moment of inertia in structural engineering


The basics


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:


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.