EngineeringFormulaCivil & Structural Engineering
Section Modulus Calculator
Calculate elastic section modulus for common structural cross-sections, including rectangles, tubes, round sections, and symmetric I-sections.
Section modulus measures how effectively a cross-section resists bending. For elastic bending about a given axis, it is defined as the second moment of area divided by the distance from the neutral axis to the most extreme fiber. A larger section modulus means lower bending stress for the same bending moment. This calculator finds the elastic section modulus for several common shapes from their geometry, so you can compare section efficiency before moving on to stress or deflection checks.
b · Overall width of a rectangular section.
h · Overall depth used as the bending dimension for rectangles and I-sections.
Solution
Choose a section shape, enter its geometry, and calculate the elastic section modulus.
S = b h² / 6
Formula Sheet
- SElastic Section Modulus
- ISecond Moment of Area
- cExtreme-Fiber Distance
- bWidth
- hHeight / Overall Depth
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| S | Elastic Section Modulus | Cross-sectional bending property used with elastic stress checks such as σ = M / S. | mm³, cm³, m³, in³, ft³ |
| I | Second Moment of Area | Geometric resistance to bending about the chosen axis, also called the area moment of inertia. | mm⁴, cm⁴, m⁴, in⁴, ft⁴ |
| c | Extreme-Fiber Distance | Distance from the neutral axis to the farthest point of the section in bending. | mm, cm, m, km, in, ft |
| b | Width | Overall width of a rectangular section. | mm, cm, m, km, in, ft |
| h | Height / Depth | Overall depth used as the bending dimension for rectangles and I-sections. | mm, cm, m, km, in, ft |
| t | Wall Thickness | Uniform wall thickness for hollow rectangular or hollow circular sections. | mm, cm, m, km, in, ft |
| D | Outer Diameter | Outside diameter of a solid or hollow circular section. | mm, cm, m, km, in, ft |
| b_f | Flange Width | Overall width of each flange in the symmetric I-section model used here. | mm, cm, m, km, in, ft |
| t_f | Flange Thickness | Thickness of each flange in the symmetric I-section model used here. | mm, cm, m, km, in, ft |
| t_w | Web Thickness | Thickness of the web connecting the two flanges in the symmetric I-section model. | mm, cm, m, km, in, ft |
How to Use This Calculator
- 01Choose the cross-section shape first. The calculator switches to the matching geometry inputs and formula automatically.
- 02Enter the overall dimensions in the units you want to work in. Hollow sections require outer size plus wall thickness; the I-section requires overall depth, flange width, flange thickness, and web thickness.
- 03Select Calculate to get the elastic section modulus S, plus the supporting second moment of area I and extreme-fiber distance c used in S = I / c.
- 04Use the result as a section-property input for later bending checks. If you already know a bending moment M, elastic bending stress follows from σ = M / S, provided the assumptions of simple elastic bending are appropriate.
- 05Match the bending axis to the real beam orientation. A section's strong-axis and weak-axis section modulus can be very different, so rotating the same shape can change stress by a large factor.
How the Formula Works
Elastic section modulus is the geometric bridge between section shape and bending stress. The core definition is S = I / c, where I is the second moment of area about the bending axis and c is the distance from the neutral axis to the outermost fiber. Because c sits in the denominator, deeper sections usually gain bending efficiency quickly: increasing depth tends to raise I much faster than it raises c.
For standard symmetric shapes, S can be written directly from geometry. A rectangle becomes S = b h² / 6, a solid round becomes S = π d³ / 32, and hollow sections are found by subtracting the inner void from the outer shape before dividing by the extreme-fiber distance. This calculator reports the elastic section modulus only, not the plastic section modulus used in plastic-design checks.
Worked Example 01
Solid rectangular section
Known
- Width (b): 200 mm
- Height (h): 300 mm
Formula
S = b h² / 6
Substitution
S = 200 × 300² / 6
Result
S = 3,000,000 mm³
A 200 mm by 300 mm rectangle has a relatively large section modulus because the 300 mm depth enters squared in the direct rectangular formula.
Worked Example 02
Rectangular hollow section
Known
- Outer Width (b): 200 mm
- Outer Height (h): 300 mm
- Wall Thickness (t): 10 mm
Formula
S = (b h³ − b_i h_i³) / (6 h)
Substitution
S = [200×300³ − 180×280³] / (6×300)
Result
S = 804,800 mm³
Subtracting the inner void reduces the section modulus substantially compared with the solid rectangle, but the deeper outside dimensions still help the tube resist bending efficiently.
Worked Example 03
Solid round bar
Known
- Diameter (d): 120 mm
Formula
S = π d³ / 32
Substitution
S = π × 120³ / 32
Result
S ≈ 169,646 mm³
A 120 mm solid round section has a smaller elastic section modulus than a deep rectangle of similar overall size because more of its area sits closer to the neutral axis.
Worked Example 04
Round hollow section
Known
- Outer Diameter (D): 150 mm
- Wall Thickness (t): 10 mm
Formula
S = π (D⁴ − d_i⁴) / (32 D)
Substitution
S = π × (150⁴ − 130⁴) / (32 × 150)
Result
S ≈ 144,409 mm³
The tube keeps more material away from the neutral axis than a solid bar of the same weight would, which is why tubular sections are often efficient in bending and torsion.
Worked Example 05
Symmetric I-section
Known
- Overall Depth (d): 300 mm
- Flange Width (b_f): 150 mm
- Flange Thickness (t_f): 12 mm
- Web Thickness (t_w): 8 mm
Formula
S = [b_f d³ − (b_f − t_w)(d − 2 t_f)³] / (6 d)
Substitution
S = [150×300³ − (150−8)×(300−24)³] / (6×300)
Result
S ≈ 591,395 mm³
The I-section places much of its material in the flanges, away from the neutral axis, which is why I-shapes are so common for beams.
Applications
- 01Comparing how efficiently different cross-sections resist bending
- 02Estimating section properties before running beam stress or deflection checks
- 03Checking hand calculations against published steel, aluminum, or timber section-property tables
Assumptions
- 01The calculator returns elastic section modulus about the strong centroidal bending axis implied by the formulas shown.
- 02The selected sections are symmetric enough that the neutral axis stays at mid-depth for the formulas used here.
- 03Dimensions describe idealized geometry with uniform thickness where applicable.
- 04Results are geometric properties only; no material yielding, local buckling, or code-reduction effects are included.
Where This Model Stops
- 01This is not a full design check. Final member design still requires bending moment, stress limits, stability checks, and applicable code provisions.
- 02Only the listed shapes are supported. Unequal-flange sections, channels, angles, tees, composite sections, and weak-axis properties require different formulas.
- 03Reports elastic section modulus only, not plastic section modulus, radius of gyration, torsional properties, or buckling capacity.
- 04Does not apply AISC/ACI/timber design factors, compactness limits, local buckling checks, lateral-torsional buckling, or load combinations.
- 05For standard rolled steel shapes, manufacturer or AISC table values should normally control over hand-entered approximate dimensions.
References
- [1]NIST Guide to the SI, Appendix B.9
National Institute of Standards and Technology
Reference for consistent SI conversion factors used for length-derived section-property units.
- [2]3.4.1 Simplified Wall Design
Engineering LibreTexts
Shows the general relation S = I / (h/2) and the rectangular-section form derived from it.
- [3]7.2.2 Structural design of shelf angles
Engineering LibreTexts
Provides a rectangular section modulus expression and uses elastic bending stress in the form f_b = M / S.
- [4]AISC Shapes Database v16.0
American Institute of Steel Construction
Practical reference for published section properties when you are checking standard rolled steel shapes instead of entering geometry manually.
Frequently Asked Questions
Is this elastic section modulus or plastic section modulus?
This page returns elastic section modulus S, based on S = I / c. Plastic section modulus Z uses a different concept and is not included here.
Can I use this directly to find bending stress?
Yes, for simple elastic bending checks you can combine a bending moment M with the result using σ = M / S. Just make sure the section, loading, axis, and material assumptions fit that simplified model.
Why does section depth matter so much?
Because the second moment of area increases strongly as material is placed farther from the neutral axis. In many common formulas, depth appears squared or cubed before the division by c is applied, so deeper sections usually gain bending efficiency quickly.
How is this different from the Area Moment of Inertia Calculator?
Area moment of inertia I measures geometric stiffness against bending and is used directly in deflection formulas. Section modulus S = I/c turns that same geometry into the property used for elastic bending stress, σ = M/S.
Can I compare two materials using section modulus alone?
Section modulus is geometry only. It helps compare shape efficiency, but material strength, stiffness, allowable stress, and deflection limits still need separate checks before choosing a member.