Section Modulus Calculator
Compute the section modulus (Z) for rectangular, solid circular, and hollow circular cross-sections.
📐 What is Section Modulus?
Section modulus (Z) is a geometric property of a cross-section that relates the second moment of area (I) to the distance from the centroidal axis to the extreme, outermost fiber of the section (c), using Z = I / c. It is the value that feeds directly into the bending stress formula, sigma = M / Z, where M is the applied bending moment and sigma is the resulting stress at the extreme fiber.
Structural engineers use section modulus to size beams and columns against bending failure, checking that the maximum stress under design loads stays below the material's allowable or yield stress. Mechanical engineers use it for shaft and bracket design under bending loads. Steel section tables (I-beams, channels, angles) list Z directly for every standard rolled shape precisely so this stress check can be done quickly without recomputing I and c by hand.
A common point of confusion is mixing up section modulus with second moment of area. They are closely related but answer different questions: second moment of area (I) measures resistance to bending deflection, while section modulus (Z = I/c) measures resistance to bending stress. Both matter, and both are usually checked for the same beam, deflection for serviceability and stress for strength.
This calculator computes Z for three common cross-sections, a solid rectangle, a solid circle, and a hollow circular tube, in both mm^3 and cm^3, using the same dimension inputs as the Second Moment of Area Calculator so results can be cross-checked between the two.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Rectangular Timber Beam Section
A 200 mm × 400 mm rectangular timber section, the same as used in the Second Moment of Area Calculator
Example 2 — Solid Round Steel Bar
A 150 mm diameter solid steel bar, the same as used in the Second Moment of Area Calculator
Example 3 — Hollow Steel Tube (Pipe Section)
A steel tube with 200 mm outer diameter and 180 mm inner diameter, the same as used in the Second Moment of Area Calculator
❓ Frequently Asked Questions
🔗 Related Calculators
What is section modulus?
Section modulus (Z) is a geometric property of a cross-section that relates the second moment of area (I) to the distance from the centroidal axis to the extreme fiber (c), using Z = I/c. It is used directly in the bending stress formula sigma = M/Z, where M is the applied bending moment.
What is the formula for the section modulus of a rectangle?
For a rectangle of width b and height h, Z = bh^2/6. This comes from I = bh^3/12 divided by c = h/2.
What is the formula for a solid circular section?
For a solid circle of diameter d, Z = pi*d^3/32. This comes from I = pi*d^4/64 divided by c = d/2.
What is the formula for a hollow circular tube?
For a tube with outer diameter D and inner diameter d, Z = pi*(D^4 - d^4)/(32D). This comes from I = pi*(D^4 - d^4)/64 divided by c = D/2.
What is the difference between section modulus and second moment of area?
The second moment of area (I) measures the raw geometric stiffness of a cross-section. Section modulus (Z = I/c) divides that by the distance to the extreme fiber, converting it into a value used directly for bending stress (sigma = M/Z) rather than deflection.
What units does this calculator use?
All dimensions are entered in millimeters (mm), and the result is displayed in both mm^3 and cm^3 (divide by 1,000), whichever is more convenient for your reference tables, material datasheets, or design code.
Why must the outer diameter be greater than the inner diameter for a tube?
The hollow circle (tube) formula subtracts the inner circle's I from the outer circle's I before dividing by c. If the outer diameter were not larger than the inner diameter, there would be no wall material left, so the calculator validates D greater than d before computing.
How is section modulus used to find bending stress?
Once you know the section modulus Z and the applied bending moment M, maximum bending stress is simply sigma = M / Z. Compare this stress against the material's allowable or yield stress to check whether the section is adequate.
Does this calculator use the same dimensions as the Second Moment of Area Calculator?
Yes, it uses the identical rectangle, solid circle, and hollow circle (tube) modes with the same dimension inputs, so you can compute I and Z for the same section and cross-check both properties consistently.
Why is a hollow tube more efficient than a solid bar for bending?
For the same cross-sectional area, moving material away from the centroidal axis (as in a hollow tube) increases both I and Z far more than the area itself increases, giving a much higher bending capacity per unit of material, which is why structural tubes and I-beams outperform solid bars of equal weight.