Section Modulus Calculator

Elastic section modulus: A geometric property that tells how well a cross-section resists elastic bending. Larger values mean lower bending stress for the same moment.

Second moment of area: Also called area moment of inertia, written as I. It measures how far the area is spread from the chosen axis.

Extreme-fiber distance c: The distance from the neutral axis to the farthest outside surface of the section. Section modulus uses S = I/c.

Bending axis: The centroidal x-axis or y-axis the section bends about. A shape can be strong about one axis and weak about the other.

Neutral axis: The line through the cross-section where bending stress is zero in pure bending.

Maximum elastic bending stress: The stress found from the entered moment and the calculated section modulus using sigma = M/S. Its sign follows the sign of the moment input.

Formula used: The shape-specific equation the calculator applied for your selected geometry and axis.

Mistake: Entering inches in Width, flange width, or outer width (selected unit) while Input length unit is set to mm.
Fix: Set Input length unit before typing dimensions, or re-enter all geometry in one consistent unit.

Mistake: Mixing up the bending direction and picking the wrong Bending axis .
Fix: Use the axis that matches the bending you want to study, then check whether Elastic section modulus gets larger or smaller than expected for that axis.

Mistake: Typing an Inner diameter (selected unit) that is equal to or larger than Diameter or outer diameter (selected unit) .
Fix: Make the inner diameter strictly smaller than the outer diameter so the tube wall has positive thickness.

Mistake: For a hollow rectangle, entering Inner width (selected unit) or Inner height (selected unit) that is not smaller than the outer size.
Fix: Keep both inside dimensions strictly below Width, flange width, or outer width (selected unit) and Height, depth, or outer height (selected unit) .

Mistake: For an I-section, using a Flange thickness (selected unit) so large that two flanges are as deep as or deeper than the whole section.
Fix: Make sure 2 x Flange thickness (selected unit) is strictly less than Height, depth, or outer height (selected unit) , and keep Web thickness (selected unit) smaller than the flange width.

Mistake: Entering a bending moment in one unit but leaving the wrong Moment unit selected.
Fix: Match Moment unit to the number you typed in Bending moment (force-length) , then read Maximum elastic bending stress in the chosen Stress output unit .

• This tool calculates elastic section modulus only. It does not calculate plastic section modulus, shape factors, or code-based capacity.
• All formulas assume standard ideal shapes with centered voids and, for the I-section, equal flanges and symmetry about both centroidal axes.
• The optional stress result uses Maximum elastic bending stress = moment / section modulus. It is a geometry-based check, not a full design check.
• Results depend completely on the selected Bending axis. Changing axis can greatly change the answer for rectangles, hollow rectangles, and I-sections.
• For circles and tubes, x-axis and y-axis are equivalent through the centroid, so axis choice does not change the result.
• Impossible geometry is rejected, including nonpositive dimensions, inside sizes that are not smaller than outside sizes, and I-section dimensions that do not leave a valid web region.
• Output units follow the chosen length unit: section modulus scales as length cubed, and second moment of area scales as length to the fourth power.
• If the applied moment is not about the same axis used for the section properties, the stress result will not represent the intended bending case.

Core idea

The calculator uses the elastic section modulus definition:

S = I / c

Here, I is the second moment of area about the chosen centroidal axis, and c is the distance from that axis to the farthest outer fiber [3]. The tool also returns both supporting values so you can see where the final answer comes from.

Shape formulas used

Rectangle about x-axis: Ix = b*h^3/12, c = h/2, Sx = b*h^2/6

Rectangle about y-axis: Iy = h*b^3/12, c = b/2, Sy = h*b^2/6

Solid circle: I = pi*D^4/64, c = D/2, S = pi*D^3/32

Hollow circle: I = pi*(Do^4 - Di^4)/64, c = Do/2, S = pi*(Do^4 - Di^4)/(32*Do)

I-section about x-axis: Ix = (b*h^3 - (b - tw)*(h - 2*tf)^3)/12, c = h/2, Sx = Ix/c

I-section about y-axis: Iy = (2*tf*b^3 + (h - 2*tf)*tw^3)/12, c = b/2, Sy = Iy/c

Hollow rectangle about x-axis: Ix = (Bo*Ho^3 - Bi*Hi^3)/12, c = Ho/2, Sx = Ix/c

Hollow rectangle about y-axis: Iy = (Ho*Bo^3 - Hi*Bi^3)/12, c = Bo/2, Sy = Iy/c

These are standard elastic section-property relations for common shapes [1].

Optional stress check

If you enter a bending moment, the calculator also computes the maximum elastic bending stress for the selected axis:

sigma = M / S

A larger section modulus gives a smaller stress for the same moment. If you enter a negative moment, the stress output stays signed so the bending sense is preserved.

Worked mini-example

Example:

rectangle, x-axis bending, width = 100 mm, height = 200 mm.

Ix = 100*200^3/12 = 66,666,666.67 mm^4

c = 200/2 = 100 mm

Sx = Ix/c = 66,666,666.67/100 = 666,666.67 mm^3

If the bending moment is 2 kN m, first convert it to 2,000,000 N mm, then divide by the section modulus.

sigma = 2,000,000 / 666,666.67 = 3 N/mm^2 = 3 MPa

Units and conversion logic

The calculator converts geometry to a consistent internal length basis, performs the section-property calculation, then reports results in the unit family you selected. Because of dimensional scaling, section modulus is reported in length cubed and second moment of area in length to the fourth power.

For the optional stress check, moment units are converted to a consistent force-length basis before dividing by section modulus, then the result is converted to the selected stress unit.

Input checks used

Results are blocked if any required visible dimension is zero, negative, missing, or geometrically impossible. Hollow sections require inside dimensions to be strictly smaller than outside dimensions. For I-sections, the web thickness must be smaller than flange width, and two flange thicknesses must be strictly less than the total height.

Assumptions behind the results

The formulas assume ideal, symmetric textbook shapes and centroidal bending axes. Real sections with fillets, tapered walls, off-center holes, rounded corners, or unequal flanges can have different values. The stress output is an elastic estimate only and does not check yielding, buckling, shear, deflection, or code limits.