Calculator
Example data table
| Case | Section | Moment | Dimensions | Typical stress |
|---|---|---|---|---|
| Floor beam | Rectangular | 12 kN·m | b=200 mm, h=300 mm | ≈ 3.0 MPa |
| Handrail post | Circular solid | 0.35 kN·m | d=40 mm | ≈ 35 MPa |
| Pipe member | Circular hollow | 2.5 kN·m | dₒ=90 mm, dᵢ=70 mm | ≈ 90 MPa |
| Steel catalog | Direct S | 18 kN·m | S=120 cm³ | ≈ 150 MPa |
Formula used
For elastic bending about the neutral axis, the flexure formula is: σ = M c / I. Here M is the bending moment, I is the second moment of area, and c is the distance to the extreme fiber.
The section modulus is S = I / c, so stress can also be written as σ = M / S. The calculator derives I, c, and S from your selected inputs.
- Rectangular: I = b h³ / 12, c = h / 2
- Circular solid: I = π d⁴ / 64, c = d / 2
- Circular hollow: I = π (dₒ⁴ − dᵢ⁴) / 64, c = dₒ / 2
How to use this calculator
- Enter the bending moment from your load analysis.
- Select a method that matches available section data.
- For geometry, choose the section type and dimensions.
- For tables, enter I and c, or section modulus directly.
- Submit to view stress in Pa, MPa, and psi.
- Download CSV or PDF for your project records.
Technical article
1) Why bending stress matters
Bending stress governs crack initiation in timber, yielding in steel, and serviceability in reinforced members. It rises with moment demand and with distance from the neutral axis, so outer fibers govern design checks. In beams with holes or notches, local stresses increase and require detailing.
2) Moment demand from loads
The input moment should come from a load-path model covering dead load, live load, wind, seismic, and construction staging. For a simply supported span with a midspan point load, peak moment equals P·L/4; a uniform load gives w·L²/8. Cantilevers peak at the fixed end with P·L or w·L²/2, so sign convention matters.
3) Flexure formula and section modulus
The calculator applies σ = M·c/I. Because S = I/c, the same check becomes σ = M/S, which is convenient when using catalog properties. For example, if M = 18 kN·m and S = 120 cm³, the stress is about 150 MPa. Using S also reduces input error when c is uncertain.
4) Rectangular beams in framing
For rectangles, I = b·h³/12 and c = h/2. Height drives performance strongly because inertia scales with h³; doubling depth reduces stress by roughly four times for the same moment. In floor systems, depth increases may be limited by headroom, so engineers increase width or switch to built-up members, I-joists, or steel sections.
5) Circular members and posts
Solid round members use I = π·d⁴/64, while hollow tubes use I = π(dₒ⁴ − dᵢ⁴)/64. Tubes resist bending efficiently because material sits farther from the neutral axis. In site work, verify wall thickness and corrosion allowance, because diameter losses can reduce inertia and raise stress.
6) Units and realistic ranges
Construction practice mixes kN·m, N·mm, and imperial units. The tool converts to N·m internally and reports stress in Pa, MPa, and psi. Typical bending stresses in structural steel fall between 50 and 250 MPa, while timber bending stresses are often in the single-digit to tens of MPa depending on grade and duration factors.
7) Comparing against limits
Compare computed stress to allowable or factored limits from your governing code and material specification. For concrete, bending stress alone is not sufficient; check reinforcement, cracking, and strain compatibility. For steel, also consider lateral-torsional buckling and section class. For timber, moisture, temperature, and load duration modifiers can govern.
8) Reporting and documentation
Record the moment, section properties, and the axis of bending. Exporting CSV supports audit trails; the PDF is useful for submittals and site reports. Pair stress checks with deflection, shear, bearing, and connection verification for complete compliance. When assumptions change in the field, rerun the calculation and archive the output.
FAQs
1) What is bending stress?
Bending stress is the normal stress produced by a bending moment. It varies linearly across the section, reaching maximum magnitude at the extreme fibers farthest from the neutral axis.
2) Which moment should I enter?
Enter the peak design or service moment at the critical section from your structural analysis, using the same sign convention throughout. If you only need magnitude, enable the absolute stress option.
3) When should I use section modulus?
Use section modulus when your beam properties come from catalogs or manufacturer tables. It lets you compute stress directly with σ = M/S without separately entering I and c.
4) Does this handle non-rectangular shapes?
This version includes rectangular, solid round, and hollow round sections plus direct-property inputs. For I‑beams, channels, or built-up sections, select direct section modulus or enter I and c from published properties.
5) Is the result factored or allowable?
The calculator reports raw stress from the supplied moment and section properties. Apply your code’s load combinations, resistance factors, or allowable stress adjustments separately, based on the material and design method used.
6) Why is my stress negative?
A negative value indicates the chosen moment sign convention produces compression at the reference fiber. The magnitude is still useful for comparison; use absolute stress for quick checks when sign is not required.
7) What other checks should accompany bending stress?
Pair bending stress with shear, deflection, lateral stability, bearing, and connection checks. For concrete and timber, include cracking, duration factors, moisture effects, and detailing requirements from the governing code.