Moment of Inertia of Hollow Rectangle

A structural member with a hollow rectangle cross-section is quite common. Understanding the moments of inertia of a hollow rectangle about the x-axis and the y­-axis for such a cross section can help evaluate how the member will respond to bending loads. Most often, a hollow rectangle cross-section will be axisymmetric about both axes, although there may be some special cases in which the cross-section is not axisymmetric about one (or both) axes.

Calculating The Moment Of Inertia For A Hollow Rectangle That Is Axisymmetric

For a rectangular tube or hollow rectangle that is symmetric about both its x-axis and its y-axis, the equations for its moments of inertia are:

where:

• I_x is the moment of inertia about the x-axis, with SI units of mm⁴
• I_y is the moment of inertia about the y-axis, with SI units of mm⁴
• b is the exterior width (base) of the rectangle, with SI units of mm
• h is the exterior height of the rectangle, with SI units of mm
• b_h is the interior width of the rectangle, with SI units of mm
• h_h is the interior height of the rectangle, with SI units of mm

The interior dimensions of the rectangle are calculated as follows:

where t is the thickness of the rectangle, with SI units of mm.

A graphical representation of these dimensions is shown in the following figure:

Because the thickness t remains constant around the entire rectangle, the structure is axisymmetric about both its x-axis and its y-axis.

Example Calculation

Take a hollow rectangular tube with the following dimensions as an example to demonstrate calculating the moments of inertia:

• b = 20 mm
• h = 55 mm
• t = 5 mm

The moment of inertia about the x-axis is calculated using the following steps:

1. Calculate the interior dimensions of the rectangular tube:

1. Calculate the moment of inertia about the x-axis:

The moment of inertia about the y-axis is calculated as follows:

Variation Of Calculation With Different Thicknesses

Although the cross-section may be axisymmetric about both axes, the thickness of the top and bottom may be different from that of the two sides. In that case, when calculating the interior dimensions, it will be necessary to use the top and bottom thickness for determining the interior height and the side thickness for determining the interior width.

Calculating The Moment Of Inertia For A Non-Axisymmetric Hollow Rectangle

If the hollow rectangular tube is designed such that it is not axisymmetric about one (or both) axis, it is necessary to utilize the parallel axis theorem to evaluate the moment of inertia about the axis running through the centroid of the cross-section. An example of a non-axisymmetric hollow rectangle is shown in the following figure:

The cross-section shown is not axisymmetric about its x-axis due to the thickness at the bottom (t₂) being larger than at the top (t₁). Also, this larger thickness at the bottom will change the calculation of the interior height dimension, which will become:

Calculation Of The Y-Axis Moment Of Inertia

The calculation of the moment of inertia about the y-axis uses the same equation with the new value for h_h. For a cross-section with t₁=t=5 mm and t₂=8 mm, the calculation of the moment of inertia is:

Calculation Of The X-Axis Moment Of Inertia

As stated, because the cross-section is no longer axisymmetric about its x-axis, the moment of inertia will need to be calculated about an axis parallel to the x-axis, which runs through the centroid of the cross-section. To do this, the parallel axis theorem is applied, which is defined as:

where:

• I_x’ is the moment of inertia about the axis parallel to the centroid axis, with SI units of mm⁴
• I_x is the moment of inertia about the centroid axis, with SI units of mm⁴
• A is the area of the cross-section, with SI units of mm²
• d is the distance from the axis through the centroid to the x-axis, with SI units of mm

To determine the moment of inertia about the axis of rotation, it is necessary to divide the cross-section into multiple segments, as follows:

To find the location of the centroid, the following equation is used:

where:

• y̅_c is the location of the centroid in the y-direction, with SI units of mm
• y̅_i is the location of the centroid of section i, with SI units of mm
• A_i is the area of section i, with SI units of mm²

Using the location of the centroid for the entire cross-section, the individual distances of each section from that location can be determined and used along with the area of each section to calculate each section’s moment of inertia about the total centroid. Then, these moments of inertia can be added together to determine the total moment of inertia, as follows:

Example Calculation

To demonstrate calculating the moment of inertia about the x-axis of the non-axisymmetric hollow rectangle above, the following steps are carried out:

1. Calculate the area of each section:

1. Calculate the individual centroid of each section:

1. Calculate the overall centroid of the cross-section:

1. Calculate the distances of the individual centroids from the cross-section centroid:

1. Calculate the moment of inertia about the x-axis for each section:

1. Apply the parallel axis theorem for each section:

1. Sum all the moments of inertia for the individual sections to find the total moment of inertia about the centroid:

Overall, the moments of inertia for a hollow rectangle are key values when determining how a structure with such a cross-section will respond to various loads. Examples of the use of hollow rectangles as structural members include bridge trusses and building girders.