Section Modulus & Moment of Inertia Calculator

Section Modulus & Moment of Inertia Calculator

Calculate area moment of inertia (I), section modulus (Z), polar moment of inertia, and radius of gyration (k) for solid rectangles, hollow rectangles, solid circles, round tubes, and symmetric I-beams. Every formula is taken verbatim from Machinery’s Handbook, 29th Edition, pp. 235–245, with MH29’s exact variable letters so cross-checking against the book is one-to-one.

Section modulus (Z = I/c) is a geometric property of a cross-section that measures resistance to bending stress, where I is the area moment of inertia and c is the distance from the neutral axis to the extreme fiber. Bending stress is σ = M/Z, so a larger Z gives lower stress for the same bending moment. Section modulus has units of length cubed (in3 or mm3).

Section Property Formulas

Laid out in the same row format as Machinery’s Handbook 29th Ed., pp. 235–237, with MH29’s exact variable letters. k is MH29’s symbol for radius of gyration (also written r in AISC).

Solid Rectangle

Section Moment of Inertia, I Section Modulus, Z Radius of Gyration, k
A = bd
y = d/2		 bd3 / 12 bd2 / 6 d / √12 = 0.289d
where: b = width parallel to neutral axis; d = depth perpendicular to neutral axis (the dimension in the direction of bending). For bending about the y-axis, swap b and d.

Source: Machinery’s Handbook 29th Ed., p. 235, Square and Rectangular Sections (1st row).

Hollow Rectangle / Rectangular Tube

Section Moment of Inertia, I Section Modulus, Z Radius of Gyration, k
A = bd − hk
y = d/2		 (bd3 − hk3) / 12 (bd3 − hk3) / (6d) 0.289 √[(bd3 − hk3) / (bd − hk)]
where: b = outside width, d = outside depth; h = inside width, k = inside depth. For a uniform wall thickness t: h = b − 2t and k = d − 2t. (MH29 reuses the letter k for both inside depth and the radius-of-gyration column — context disambiguates.)

Source: Machinery’s Handbook 29th Ed., p. 235, Square and Rectangular Sections (last row).

Solid Circle / Round Bar

Section Moment of Inertia, I Section Modulus, Z Radius of Gyration, k
A = πd2 / 4 = 0.7854d2
y = d/2		 πd4 / 64 = 0.049d4 πd3 / 32 = 0.098d3 d / 4
where: d = diameter. Polar moment of inertia J = πd4 / 32 = 2I (perpendicular-axis theorem).

Source: Machinery’s Handbook 29th Ed., p. 237, Circular Sections; polar moment p. 245.

Hollow Circle / Round Tube / Pipe

Section Moment of Inertia, I Section Modulus, Z Radius of Gyration, k
A = π(D2 − d2) / 4 = 0.7854(D2 − d2)
y = D/2		 π(D4 − d4) / 64 = 0.049(D4 − d4) π(D4 − d4) / (32D) = 0.098(D4 − d4) / D √(D2 + d2) / 4
where: D = outside diameter, d = inside diameter. Polar moment of inertia J = π(D4 − d4) / 32 = 2I.

Source: Machinery’s Handbook 29th Ed., p. 237, Circular Sections (hollow row); polar moment p. 245.

I-Beam (Symmetric, Rectangular Flanges)

Section Moment of Inertia, I Section Modulus, Z Radius of Gyration, k
A = bd − h(b − t)
y = d/2		 [bd3 − h3(b − t)] / 12 [bd3 − h3(b − t)] / (6d) √{[bd3 − h3(b − t)] / [12(bd − h(b − t))]}
where: b = flange width; d = total depth; t = web thickness; h = clear height between flanges = d − 2s (with s = flange thickness). x-axis is the strong axis (perpendicular to the web).

Source: Machinery’s Handbook 29th Ed., p. 238, I-Sections (rectangular-flange variant). MH29 expresses these formulas in the form shown above using b, d, t, and h; the input field s (flange thickness) is provided as a convenience — the page substitutes h = d − 2s internally.

How to Use This Calculator

Select a cross-section shape and enter its dimensions. The calculator returns area (A), area moment of inertia about both principal axes (Ix, Iy), section modulus (Zx, Zy), polar moment of inertia (Ip = Ix + Iy), and radius of gyration (kx, ky). Every formula is taken verbatim from Machinery’s Handbook, 29th Edition, pp. 235–245, with MH29’s exact variable letters.

Use Zx when bending occurs about the strong axis (load applied perpendicular to the web of an I-beam, or perpendicular to the long side of a rectangle). Use Zy for bending about the weak axis. To find the maximum bending stress in a beam, plug your section modulus into the beam deflection calculator — it returns both deflection and stress for the most common load cases.

For static structural design, AISC ASD applies an allowable bending stress of 0.6 Fy, which corresponds to a factor of safety of 1.67 against yield. Most CNC and fabrication customer designs target FoS of 2 to 4 against yield depending on load uncertainty and consequence of failure. Compare σ = M/Z against the material’s yield strength Fy divided by your chosen safety factor.

Worked Example

Problem: A machine-mounting bracket is fabricated from a 2 in × 4 in A36 steel bar (4 in dimension vertical, bending about the strong axis). It carries a 25,000 in-lbf bending moment from a side-loaded fixture. Find the moment of inertia, section modulus, and verify whether the bar stays below A36’s allowable stress.

Given: b = 2 in (width parallel to neutral axis), d = 4 in (depth perpendicular to neutral axis), M = 25,000 in-lbf, A36 steel (Fy = 36,000 psi)

A = bd = 2 × 4 = 8 in2

Ix = bd3/12 = 2 × 43 / 12 = 10.667 in4

Zx = bd2/6 = 2 × 42 / 6 = 5.333 in3

kx = d/√12 = 4 / 3.464 = 1.155 in

Maximum bending stress: σ = M/Zx = 25,000 / 5.333 = 4,688 psi.

The factor of safety against yield is 36,000 / 4,688 = 7.7 — well above the typical FoS = 1.67 used in AISC ASD for static structural design. The bracket is safe for this load.

Frequently Asked Questions

What is section modulus?

Section modulus (Z) is a geometric property of a cross-section that measures its resistance to bending stress. It is defined as Z = I/c, where I is the moment of inertia about the neutral axis and c is the perpendicular distance from the neutral axis to the extreme fiber. The bending stress in a beam is σ = M/Z, where M is the bending moment. Section modulus has units of length cubed — for example, in3 or mm3. A larger Z means lower bending stress for the same applied moment.

How do you calculate the area moment of inertia of a rectangle?

For a solid rectangular cross-section, the moment of inertia about the centroidal axis parallel to the base is I = bd3 / 12, where b is the width (parallel to the axis) and d is the depth (perpendicular to the axis — the dimension in the direction of bending). The section modulus is Z = bd2 / 6, and the radius of gyration is k = d / √12 ≈ 0.289d. For example, a 2 in × 4 in bar with the 4 in dimension as the depth gives I = 2 × 43 / 12 = 10.667 in4 and Z = 2 × 42 / 6 = 5.333 in3. These formulas are from Machinery’s Handbook 29th Ed., p. 235.

How do I find the area moment of inertia of an I-beam?

For a symmetric I-beam with flange width b, total depth d, flange thickness s, and web thickness t, the I-beam moment of inertia about the strong (x) axis is Ix = [bd3 − h3(b − t)] / 12, where h = d − 2s is the clear height between the flanges. This is the bounding-rectangle subtraction method — the I-beam is treated as a solid rectangle (b × d) with two open rectangles ((b−t)/2 × h each, one on each side of the web) removed (Machinery’s Handbook 29th Ed., p. 238, I-Sections). The section modulus is Zx = Ix / (d/2). For published wide-flange shapes (W12×26, W8×31, etc.), the AISC Steel Construction Manual lists tabulated values.

What is polar moment of inertia, and how does it differ from the torsional constant?

Polar moment of inertia (Ip or J) is a measure of a cross-section’s resistance to twisting about its longitudinal axis. By the perpendicular axis theorem, Ip = Ix + Iy. For a solid round shaft, Ip = πD4 / 32; for a hollow shaft, Ip = π(D4 − d4) / 32. For circular cross-sections, the polar moment of inertia equals the torsional constant J, and the shear stress under torque T is τ = Tr / J. For non-circular sections (rectangles, I-beams), warping makes the actual torsional constant smaller than Ip — for those shapes, use the torsional constant from a structural manual rather than the polar moment of inertia. Formulas for polar moment of inertia are on Machinery’s Handbook 29th Ed., p. 245 (the introductory definition is on p. 244).

What units does section modulus use?

Section modulus has units of length cubed. In US customary units, this is in3. In SI units, it is mm3 (most common for engineering data) or m3. Moment of inertia has units of length to the fourth power: in4 or mm4. To convert: 1 in3 = 16,387.064 mm3 and 1 in4 = 416,231.4 mm4. Always confirm units when reading a steel manual or material datasheet, since some references list moment of inertia in cm4 rather than mm4.

Related Calculators

References

  • Oberg, E. et al. Machinery’s Handbook, 29th Edition, Industrial Press, 2012, “Moments of Inertia, Section Moduli, and Radii of Gyration,” pp. 234–243. Formulas for solid and hollow rectangles (p. 235), circular sections (p. 237), and I-sections (p. 238).
  • Oberg, E. et al. Machinery’s Handbook, 29th Edition, “Polar Area Moment of Inertia and Section Modulus,” pp. 244–245. Polar moment formulas for solid and hollow shafts.

Data last verified: May 2026

Request a Quote

Pi Fabricators is a CNC and fabrication shop in Salem, Oregon. We deliver precision-machined and fabricated parts and assemblies to your specifications.

Request a Quote

This calculator is provided for reference only and is offered “as is” without warranty of any kind. Pi Fabricators LLC is not liable for any damages or losses arising from the use of this tool. Verify all critical calculations independently with a qualified professional.

Spot an error on this page? Let us know at Contact@pifabricators.com