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For a rectangular tube with a uniform wall thickness, the centroidal moments of inertia are:
Iₓ = [BH³ − (B − 2t)(H − 2t)³] ÷ 12
Iᵧ = [HB³ − (H − 2t)(B − 2t)³] ÷ 12
Here, H is the outside height, B is the outside width, and t is the wall thickness. Iₓ describes the section’s resistance to bending about the horizontal x-axis, while Iᵧ applies to bending about the vertical y-axis. The result is expressed in units to the fourth power, such as mm⁴, cm⁴, or in⁴.
These equations are useful for preliminary calculations and comparing rectangular tube sizes. For final structural design, use the section properties published for the applicable product standard and have the member verified by a qualified engineer.
What Is the Moment of Inertia of a Rectangular Tube?
The moment of inertia of a rectangular tube describes how the cross-sectional area is distributed around a specified axis. In structural engineering, it is more precisely called the area moment of inertia or second moment of area.
A larger value generally means the section offers greater resistance to curvature and deflection about that axis. However, moment of inertia is a geometric property only. It does not by itself account for steel grade, yield strength, member length, connections, local buckling, or the applied load.
This term should not be confused with mass moment of inertia, which describes resistance to rotational acceleration and includes mass. Rectangular steel tube calculations normally use the area moment of inertia, with units such as mm⁴ or in⁴.
Rectangular hollow structural sections are commonly used as beams, columns, frames, and supports because their closed shape provides useful bending and torsional characteristics.The correct section still needs to be selected according to the governing design standard and project conditions.
Rectangular Tube Dimensions and X-Y Axes
Before using a rectangular tube moment of inertia formula, define the dimensions and axes consistently.
| Symbol | Definition |
| H | Outside height of the rectangular tube |
| B | Outside width of the rectangular tube |
| t | Uniform wall thickness |
| h | Inside height, calculated as H − 2t |
| b | Inside width, calculated as B − 2t |
| Iₓ | Moment of inertia about the horizontal centroidal x-axis |
| Iᵧ | Moment of inertia about the vertical centroidal y-axis |
Figure 1. Rectangular tube cross-section showing outside height H, outside width B, inside height h, inside width b, wall thickness t, and the centroidal x- and y-axes.
For a symmetrical rectangular tube with uniform thickness, both centroidal axes pass through the geometric centre of the section. If the walls have different thicknesses or the hollow opening is offset, the centroid must be located first and the basic equations above may no longer apply directly.
If you need to confirm the designation and available dimensions before calculating, see the rectangular steel tube dimensions and weight chart.
Rectangular Tube Moment of Inertia Formulas
Moment of Inertia About the X-Axis
For bending about the horizontal centroidal x-axis:
Iₓ = (BH³ − bh³) ÷ 12
Because b = B − 2t and h = H − 2t, the equation can also be written as:
Iₓ = [BH³ − (B − 2t)(H − 2t)³] ÷ 12
The first term represents the moment of inertia of the outer solid rectangle. The second term represents the inner rectangular opening. Subtracting the opening leaves the property of the hollow tube.
Moment of Inertia About the Y-Axis
For bending about the vertical centroidal y-axis:
Iᵧ = (HB³ − hb³) ÷ 12
Using the outside dimensions and wall thickness:
Iᵧ = [HB³ − (H − 2t)(B − 2t)³] ÷ 12
The equations look similar, but the cubed dimension changes. This is why a rectangular tube normally has different moments of inertia about its two centroidal axes.
How to Calculate the Moment of Inertia of a Rectangular Tube
Consider an idealized rectangular tube with the following nominal dimensions:
- Outside height H = 100 mm
- Outside width B = 50 mm
- Wall thickness t = 4 mm
This example assumes a uniform wall thickness and sharp mathematical corners. It is intended to demonstrate the calculation rather than reproduce a standard section-table value.
Step 1: Calculate the Inside Dimensions
The inside height is:
h = H − 2t = 100 − 2 × 4 = 92 mm
The inside width is:
b = B − 2t = 50 − 2 × 4 = 42 mm
Step 2: Calculate Ix
Substitute the outside and inside dimensions into the Iₓ equation:
Iₓ = [50 × 100³ − 42 × 92³] ÷ 12
Iₓ = 1,441,258.67 mm⁴
Iₓ ≈ 1.441 × 10⁶ mm⁴
Step 3: Calculate Iy
Repeat the calculation for the y-axis:
Iᵧ = [100 × 50³ − 92 × 42³] ÷ 12
Iᵧ = 473,658.67 mm⁴
Iᵧ ≈ 4.737 × 10⁵ mm⁴
Step 4: Compare the Results
For this orientation, Iₓ is approximately 3.04 times Iᵧ. The section is therefore substantially stiffer in bending about the x-axis than about the y-axis, assuming the same material, member length, support conditions, and load arrangement.
Rotating the tube by 90 degrees swaps the roles of its height and width. That changes which axis provides the greater bending stiffness, even though the tube’s material and weight remain the same.
Rectangular Tube Moment of Inertia Units
Moment of inertia uses units to the fourth power because its calculation combines area with the square of its distance from an axis.
| Unit | Typical use |
| mm⁴ | Metric engineering drawings and calculations |
| cm⁴ | Some European and international section tables |
| in⁴ | North American HSS tables and calculations |
The main conversion relationships are:
1 cm⁴ = 10,000 mm⁴
1 in⁴ = 416,231.43 mm⁴ = 41.6231 cm⁴
For the 100 × 50 × 4 mm example:
| Property | mm⁴ | cm⁴ | in⁴ |
| Iₓ | 1,441,258.67 | 144.13 | 3.46 |
| Iᵧ | 473,658.67 | 47.37 | 1.14 |
Do not convert these values using a simple linear length conversion. Because the units are raised to the fourth power, the conversion factor must also be raised to the fourth power.
Why Height Has a Greater Effect on Moment of Inertia
In the Iₓ equation, the outside and inside heights are cubed. This makes the vertical distribution of material especially important:
Iₓ ∝ H³
Increasing the overall height moves more material farther from the x-axis. As a result, increasing section depth can improve bending stiffness more efficiently than adding the same amount of material close to the neutral axis.
This is why a rectangular tube used as a beam is often installed with its longer side vertical when the main load acts downward. However, a greater height is not automatically the best choice. Available space, lateral stability, local buckling, connections, weight, cost, and loads in the secondary direction must also be considered.
Ix vs Iy and Strong Axis vs Weak Axis
Iₓ and Iᵧ describe the same cross-section about different axes. For a rectangular tube, the larger value is commonly associated with the strong axis and the smaller value with the weak axis.
| Property | Strong axis | Weak axis |
| Moment of inertia | Higher | Lower |
| Bending stiffness | Higher | Lower |
| Deflection under comparable conditions | Lower | Higher |
| Typical beam orientation | Longer side commonly vertical | Longer side commonly horizontal |
For the 100 × 50 × 4 mm example, the x-axis is the strong axis in the illustrated orientation. If the section is rotated, the orientation changes and the strong-axis value follows the deeper dimension.
Axis notation is not identical in every design table. Some standards and software use x-x and y-y differently, while others use major and minor axes. Always check the accompanying cross-section diagram instead of relying on the letter alone.
Moment of Inertia vs Section Modulus
Moment of inertia and section modulus are related, but they answer different engineering questions.
| Property | Moment of inertia | Section modulus |
| Common symbol | I | S, W, or Z, depending on the convention |
| Units | mm⁴ or in⁴ | mm³ or in³ |
| Main use | Bending stiffness and deflection | Elastic bending-stress comparison |
| Depends on | Distribution of the entire cross-sectional area | Moment of inertia and distance to the extreme fibre |
For a symmetrical rectangular tube:
Sₓ = Iₓ ÷ (H ÷ 2)
Sᵧ = Iᵧ ÷ (B ÷ 2)
Moment of inertia is used in flexural rigidity, commonly represented by EI, where E is Young’s modulus. Section modulus appears in the elastic bending relationship σ = M ÷ S. A tube can therefore have a useful moment of inertia but still require separate checks for bending strength, local buckling, shear, stability, and serviceability.
Why Calculated Values May Differ From Standard Section Tables
The simple hollow-rectangle formula is useful for learning and preliminary comparison, but its result may not exactly match a published HSS or RHS table.
Rounded Corners
Manufactured rectangular tubes have rounded internal and external corners. The basic formula models an outer sharp-cornered rectangle minus an inner sharp-cornered rectangle. Standard section-property calculations account for corner geometry according to the applicable methodology, so the area, moment of inertia, section modulus, and radius of gyration can differ from the idealized result.
Nominal and Design Wall Thickness
The thickness shown in a product designation is not always the thickness used to calculate design properties. For example, guidance for ASTM A500 HSS explains that calculations involving wall thickness have traditionally used a design thickness equal to 0.93 times the nominal thickness because of the permitted thickness tolerance.
The applicable factor depends on the specification and design rules. It should not automatically be applied to products manufactured under a different standard.
Manufacturing Tolerances
Actual outside dimensions, wall thickness, corner radii, straightness, and mass can vary within the tolerances permitted by the product specification. These variations explain why a measurement from one physical tube may not reproduce a catalogue value exactly.
Standard-Specific Calculation Methods
Published tables may use different rules for effective dimensions, design thickness, and corner geometry. ASTM A500, for example, covers cold-formed welded and seamless carbon steel structural tubing in round, square, rectangular, and special shapes for structural applications.
For final design, use the section-property table associated with the specified product and design code. Do not substitute an idealized calculation where certified or code-based values are required.
BAOLAI supplies rectangular structural tubing under standards including ASTM A500, EN 10210, and EN 10219. Confirm the required standard, grade, dimensions, and documentation before comparing section properties.
How to Use Moment of Inertia When Selecting a Rectangular Tube
Start by identifying the main load direction and the axis about which bending will occur. Then compare the corresponding moment of inertia for the available sizes and orientations. A section with a greater depth in the main bending plane will often provide better stiffness for its weight, but this is only one part of the selection process.
The final comparison should also consider section modulus, wall slenderness, member length, total weight, connection details, corrosion allowance, fabrication requirements, and the governing design standard. If loads act in both directions, check both Iₓ and Iᵧ rather than selecting the tube from the strong-axis value alone.
For procurement, specify the complete outside dimensions, nominal wall thickness, steel grade, manufacturing standard, length, surface condition, and required inspection documents. This prevents a theoretical section-property comparison from being disconnected from the product actually delivered.
BAOLAI Rectangular Steel Tubes
BAOLAI supplies square and rectangular steel tubes for structural and fabrication projects. Available options cover multiple sizes, wall thicknesses, steel grades, surface finishes, and international standards. Share your project specification with BAOLAI to confirm production availability, tolerances, testing, and documentation before ordering.
FAQs
Does Steel Grade Change the Moment of Inertia?
No. The area moment of inertia is determined by cross-sectional geometry, so changing the steel grade without changing the dimensions does not change Iₓ or Iᵧ. Steel grade affects properties such as yield strength, while Young’s modulus contributes to bending stiffness through EI.
Can Two Rectangular Tubes Have the Same Weight but Different Moments of Inertia?
Yes. Two sections can contain similar amounts of steel but distribute that material differently. A deeper section may place more area farther from the neutral axis and therefore provide a higher moment of inertia about the main bending axis.
Is a Higher Moment of Inertia Always Better?
Not necessarily. A higher value generally reduces curvature and deflection about the relevant axis, but it may also increase depth, weight, cost, or connection difficulty. Strength, local buckling, stability, fabrication, and project constraints must be checked separately.
Does Galvanizing Change the Moment of Inertia?
The structural section properties in standard tables are normally based on the steel section rather than the zinc coating. Galvanizing adds a thin coating and a small amount of mass, but it is generally not treated as structural material when calculating the tube’s moment of inertia.
Should I Use a Calculated Value or a Standard Section Table?
Use the formula for preliminary checks, education, and non-standard geometric comparisons. For final structural design or procurement, use the section properties published for the specified standard and product, then confirm the design with the responsible engineer.
Conclusion
The moment of inertia of a rectangular tube depends on its height, width, wall thickness, and orientation. Because the dimension perpendicular to the bending axis is cubed, rotating the same tube can produce a major difference between Iₓ and Iᵧ.
The sharp-corner formula provides a useful preliminary result, but real section tables may account for rounded corners, design wall thickness, and manufacturing tolerances. Always use the applicable standard data when the calculation affects structural design or purchasing specifications.
