Section Modulus Calculator - Elastic Modulus S = I / c
Compute the section modulus of a rectangle, circle, or I-beam. Enter the cross-section dimensions to get S = I / c with the second moment of area I and extreme-fiber distance c for bending checks.
Section Modulus Calculator
Results
What This Calculator Does
Section modulus is the elastic modulus S of a beam cross section, and this tool returns it from the geometry you enter. The modulus is the property that links a bending moment to the stress it creates, and it is defined as S = I / c, where I is the second moment of area and c is the distance from the neutral axis to the farthest fiber. It supports rectangles, circles, and wide-flange I-beams so you can compare shapes without hand-deriving integrals.
- • Beam sizing checks: Quickly compare candidate cross sections before running a full bending stress analysis.
- • Homework and exam problems: Verify textbook values for S of standard shapes such as a rectangle or circle.
- • Structural literacy: See why deeper sections and I-beams resist bending better than shallow solid bars.
- • Preliminary design: Screen whether a given profile meets a target modulus before refining in design software.
You enter a shape and its key dimensions, and the calculator returns the modulus in mm^3 and cm^3, along with the second moment of area I (in mm^4) and the extreme-fiber distance c (in mm). Those three numbers are enough to apply the bending stress formula sigma = M / S.
The modulus is a property of geometry only; it does not depend on the material. Steel and aluminum with the same cross section have identical values, even though their allowable stress differs. Keep that split in mind when you move from a number to a real design decision.
Once you know the modulus, you can feed it into a beam bending stress calculator to size the beam against its bending moment.
How the Calculation Works
The calculator applies the elastic modulus definition S = I / c for whichever shape you select. It first builds the second moment of area I for that shape, then divides by c, the distance to the extreme fiber.
- S: Elastic modulus, the cross-section's resistance to bending (length cubed).
- I: Second moment of area about the bending axis (length to the fourth).
- c: Distance from the neutral axis to the most stressed outer fiber.
- b, h: Rectangle width and height (h is the dimension to the extreme fiber for the strong axis).
- d: Circle diameter; c = d / 2 and I = pi d^4 / 64.
- bf, tf, tw, hTotal: I-beam flange width, flange thickness, web thickness, and total depth.
For a rectangle the closed form is S = b h^2 / 6 because I = b h^3 / 12 and c = h / 2. For a circle, S = pi d^3 / 32. For a wide-flange I-beam the calculator subtracts the web void from the outer rectangle: I = (bf * hTotal^3) / 12 - ((bf - tw) * (hTotal - 2 tf)^3) / 12, with c = hTotal / 2.
All inputs are read in millimetres, and the modulus is shown in mm^3 and cm^3 (divide by 1,000). The tool also reports I in mm^4 and c in mm so you can trace every step of S = I / c.
Rectangle 100 x 200 mm
Width b = 100 mm, height h = 200 mm.
I = 100 * 200^3 / 12 = 66,666,666.67 mm^4; c = 200 / 2 = 100 mm; S = I / c = 666,666.67 mm^3.
Elastic modulus S = 666,666.67 mm^3 (666.67 cm^3).
A beam twice as deep yields four times the modulus, which is why depth is the strongest lever in bending design.
Circle diameter 150 mm
Diameter d = 150 mm.
I = pi * 150^4 / 64 = 24,850,488.76 mm^4; c = 75 mm; S = I / c = 331,339.85 mm^3.
Elastic modulus S = 331,339.85 mm^3 (331.34 cm^3).
The round bar has about half the modulus of the rectangle above for a similar footprint, because material sits closer to the neutral axis.
According to Wikipedia: Section modulus, Defines S = I / c and the standard shape formulas used here.
According to Wikipedia: Second moment of area, Defines the second moment of area I used in S = I / c.
Material stiffness from an elastic constants tool sets the allowable stress you compare against sigma = M / S.
Key Concepts Explained
Four ideas decide how the modulus behaves, and the calculator makes each one visible.
Elastic vs plastic modulus
The elastic modulus S = I / c assumes the section stays within the linear stress range. The plastic modulus Z uses the full cross section at yield and is larger than S; this calculator reports the elastic value used in standard sigma = M / S checks.
Modulus vs moment of inertia
I describes how area is spread about an axis; S = I / c adds the distance to the extreme fiber. You use I inside the calculator and S in the final bending stress equation.
Neutral axis and extreme fiber
c is the farthest point from the neutral axis. For symmetric sections it is half the depth, which is why a deeper rectangle has a much larger S than a wider one of equal area.
Strong and weak axes
An I-beam bends far more easily about its weak axis. This calculator reports the strong-axis modulus where c = hTotal / 2; bending about the weak axis uses c = bf / 2 and gives a smaller S.
The area moment of inertia used here differs from the rotational quantity in a mass moment of inertia calculator, which measures resistance to angular acceleration instead of bending.
How to Use This Calculator
Follow these steps to get a modulus you can trust, then carry it into a bending check.
- 1 Pick the shape: Select rectangle, circle, or wide-flange I-beam from the dropdown.
- 2 Enter dimensions: Provide the active inputs for your shape: b and h for a rectangle, d for a circle, or bf, tf, tw, and hTotal for an I-beam, all in mm.
- 3 Read the results: Note S in mm^3 and cm^3, plus I in mm^4 and c in mm.
- 4 Apply the moment: Multiply by your design moment using sigma = M / S to estimate the bending stress.
- 5 Compare to allowable stress: Check the result against the material's allowable bending stress from the relevant design code.
- 6 Compare shapes: Re-run with a different section to see which geometry gives the modulus you need.
Say a simply supported beam sees a mid-span moment of 5,000,000 N*mm. With the 100 x 200 mm rectangle above (S = 666,666.67 mm^3), the bending stress is sigma = 5,000,000 / 666,666.67 = 7.5 MPa, well within typical timber or aluminum allowable limits.
To complete a bending check, first get the moment M from a shear force and bending moment calculator for your span and load.
Benefits of Using This Calculator
This tool pays off whenever you need to weigh geometry against bending performance.
- • Fast shape comparison: Swap rectangle, circle, and I-beam inputs to see which profile meets a target modulus without redrawing integrals.
- • Fewer arithmetic slips: The closed forms for b h^2 / 6 and pi d^3 / 32 are easy to mis-key; the calculator removes that risk.
- • Transparent steps: It reports I, c, and S together, so you can show exactly how S = I / c was evaluated.
- • Better design intuition: Repeated runs reveal that depth dominates: doubling height quadruples the modulus of a rectangle.
- • Lecture and exam support: Students can confirm hand calculations against an independent value in seconds.
Pair the section modulus with a stress calculator to convert the moment M into the actual bending stress sigma = M / S.
Factors That Affect Your Results
These inputs and assumptions move the result, and a few limits keep the output meaningful.
Depth to the extreme fiber (c)
Doubling the overall depth roughly doubles c but raises I by about eight times, so S grows fast with depth. Depth is the single biggest lever on S.
Material placement
An I-beam puts area far from the neutral axis, so for similar weight it beats a solid rectangle. The calculator captures this through the subtracted web void.
Axis of bending
The strong-axis modulus uses c = hTotal / 2, while bending about the weak axis uses c = bf / 2 and yields a smaller S. Make sure you pick the axis that matches your load.
Units
All dimensions are millimetres, and S is reported in mm^3 and cm^3. Mixing in centimetres without conversion is the most common source of wrong answers.
- • This is an educational geometry check; it does not apply code safety factors, lateral-torsional buckling, or shear deflection. Confirm real designs against AISC 360 or the applicable standard.
- • The elastic modulus assumes linear material behavior and a symmetric, homogeneous section. Composite or locally buckled sections need a different treatment.
According to AISC 360-22 Specification for Structural Steel Buildings, Authoritative steel bending-stress design limits referenced for context.
For torsion problems the analogous property is the polar moment, which you can compute with a polar moment of inertia calculator.
Frequently Asked Questions
Q: What is the elastic modulus in simple terms?
A: The modulus is a single number that captures how well a beam cross section resists bending. It equals the second moment of area I divided by the distance c from the neutral axis to the farthest fiber (S = I / c). A larger value means the same bending moment produces less stress, so the beam is less likely to yield.
Q: What is the formula for the elastic modulus?
A: The elastic modulus is S = I / c. I is the second moment of area (sometimes called the area moment of inertia) about the bending axis, and c is the distance from the neutral axis to the most stressed edge. For a rectangle this simplifies to S = b h^2 / 6, and for a circle to S = pi d^3 / 32.
Q: How do you calculate the modulus of a rectangular beam?
A: For a rectangle of width b and height h bent about its strong axis, first compute I = b h^3 / 12 and c = h / 2, then S = I / c = b h^2 / 6. For example, a 100 mm by 200 mm rectangle gives S = 100 * 200^2 / 6 = 666,666.67 mm^3.
Q: What is the modulus of a circular cross section?
A: For a solid circle of diameter d, the second moment of area is I = pi d^4 / 64 and c = d / 2, so S = I / c = pi d^3 / 32. A 150 mm diameter round bar therefore has S = pi * 150^3 / 32 = 331,339.85 mm^3.
Q: Is the modulus the same as moment of inertia?
A: No. The second moment of area I measures how the cross-section area is distributed about a bending axis, while S = I / c folds in the distance to the extreme fiber. S is the value you actually use in the bending stress equation sigma = M / S.
Q: Why does a larger modulus reduce bending stress?
A: Bending stress follows sigma = M / S, where M is the applied bending moment. Because S sits in the denominator, increasing it directly lowers the stress for the same moment. That is why deeper beams and I-sections (which move material far from the neutral axis) carry load more efficiently.