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Moment of Inertia of a Rectangle

  At some stage in your education as an engineer, you will inevitably ask yourself the questions:

What does the moment of inertia mean?

What is the moment of inertia of a rectangle?

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 as 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.

Moment of Inertia of a Rectangle Formula

The general formula used when determining how to find moment of inertia of a rectangle is:

[math] I_{xx}=\dfrac{BD^3}{12} , I_{yy}=\dfrac{B^3D}{12} [math]

Where the xx and yy refer to the particular axis, or direction, being considered.

It is a common structural engineering convention that B refers to the width of the rectangle, parallel to a conventionally horizontal x-axis.

Similarly, D refers to the depth of the rectangle, parallel to a conventionally vertical y-axis.

moment of inertia of a rectangle, moment of inertia, rectangle moment of area

It may be confusing at first, but when structural engineers refer to Ixx they are in fact referencing the strength of a section about the x-axis, meaning in a direction parallel to the D dimension, or y-axis. Similarly, Iyy refers to the strength about the y-axis, meaning in a direction parallel to the B dimension, or x-axis.

moment of inertia of a rectangle, moment of inertia, rectangle moment of area

Rectangular Hollow Sections (RHS)

While engineers could hypothetically use solid rectangular sections when designing, this would use a significantly greater amount of raw material, with corresponding increases in weight and cost. It is much more common to userectangular hollow sections (commonly referred to as an RHS). Here we can utilise the same equation defined above for the general rectangular case, however we must subtract the inner hollow area of the rectangle:

[math] I_{xx}=\dfrac{BD^3}{12} – \dfrac{bd^3}{12} [math]

In this instance, lowercase b and d denote the size of the hollow area within the rectangle that we must subtract from the outer dimensions of the shape, being uppercase B and D. The difference between each corresponding dimension refers to the thickness of the material in that dimension – i.e. B – b = total thickness of the material parallel to the x-axis.

moment of inertia of a rectangle, rhs, rectangular hollow section

Besides the clear examples of weight and material use, why are hollow sections often described as more efficient than their solid counterparts?

Consider a beam undergoing a vertical load downwards. We expect the topmost fibres of the material to undergo a compressive force, while the corresponding bottom fibres will undergo a tensile force. The fibres along the neutral axis of the section (parallel to the section’s centroid) however, will experience neither compression nor tension, hence the name neutral axis.

Importantly, the magnitude of these compressive or tensile forces depend on the distance away from this neutral axis – material closer to the neutral axis needs to resist less force.

moment of inertia of a rectangle, moment of inertia, rectangle moment of area

As such, the inner material of a fully solid section is resisting a relatively small amount of force while occupying a relatively large amount of area because the outermost material is working the hardest! Removing this inner part of the section and making it hollow consequently improves the efficiency of the section in regard to its weight, cost, and material use.

Moment of Inertia of a Rectangle – A detailed breakdown

The moment inertia is important for both Bending Moment Force/Stress and Deflection. This is evident considering their formula, wherein in both cases, I (Moment of Inertia) is in the denominator:

moment of inertia of a rectangle, bending moment formula,

Source: Bending Stress Formula

moment of inertia of a rectangle, moment of inertia, rectangle moment of area

Source: Equation of Deflection in a Cantilever Beam

Free Moment of Inertia Calculator

Check out our Free Moment of Inertia Calculator where you can experiment with the above calculations, or have a read of our Moment of Inertia of a Circle to see how circular and rectangular section shapes compare with one another.

For an even more detailed analysis, sign up to get started with our full SkyCiv Section Builder version today!

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