Bending Stress Calculator — Section Modulus & Required Beam Dimension
Calculate maximum elastic bending stress in beams and flexural members from bending moment and section modulus or cross-section geometry. Also determine the required section modulus or minimum solid circular and square section size from an allowable bending stress.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Calculation mode
Bending moment
Allowable stress
Cross-section
Section type
Material properties [Optional]
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Maximum elastic bending stress formula:
σmax = |M| · c / I = |M| / S
S = I / c
The formula gives the maximum normal bending stress at the extreme fiber of the evaluated cross-section.
Required section modulus:
Sreq = |M| / σallow
A selected cross-section should have an elastic section modulus equal to or greater than Sreq about the applicable bending axis.
Required solid circular section diameter:
S = π · d3 / 32
d = √√((32 · Sreq) / π)
Required solid square section side:
S = a3 / 6
a = √√(6 · Sreq)
where:
- σmax — maximum elastic bending stress at the extreme fiber
- M — bending moment at the evaluated cross-section
- I — second moment of area about the bending axis
- c — maximum perpendicular distance from the neutral axis to the extreme fiber
- S — elastic section modulus, equal to I / c
- Sreq — minimum required elastic section modulus
- σallow — allowable bending stress entered by the user
- d — required diameter of a solid circular section
- a — required side length of a solid square section
In SI base units, M is expressed in N·m, I in m4, c in m, S in m3, and stress in Pa. A consistent engineering unit set may also be used; for example, N·mm divided by mm3 gives N/mm2, which is equivalent to MPa.
When to use this calculator
Use this bending stress calculator when the bending moment at the evaluated beam or member cross-section is already known and you need to calculate the maximum elastic normal stress at the extreme fiber.
- Calculate maximum bending stress from a known bending moment and custom elastic section modulus.
- Calculate bending stress from the dimensions of a supported solid or hollow cross-section.
- Determine the minimum required section modulus from bending moment and an entered allowable bending stress.
- Estimate the minimum diameter of a solid circular section required to satisfy the entered bending stress limit.
- Estimate the minimum side length of a solid square section required to satisfy the entered bending stress limit.
- Perform a preliminary yield-based screening check when a positive material yield strength is provided.
- Compare alternative cross-section geometries by their elastic section modulus and resulting bending stress.
The calculator is intended for preliminary analysis of straight beams, shafts, bars, plates treated as beam-like members, and other flexural components operating within the elastic range.
Do not use this calculator to determine bending moment from applied loads, span, or support conditions. It does not calculate reactions, shear-force diagrams, bending-moment diagrams, beam deflection, torsional stress, fatigue life, buckling resistance, plastic moment capacity, or code-specific member resistance.
More detailed analysis is required for curved beams, deep beams, composite sections, anisotropic materials, non-linear material behavior, stress concentrations, local loading, combined loading, or geometries that cannot be represented by the available section models.
How to interpret the result
The bending stress result represents the maximum elastic normal stress magnitude at the extreme fiber of the evaluated cross-section. Under simple bending, one side of the member is in tension and the opposite side is in compression.
Bending stress result
Bending stress increases directly with bending moment and decreases as the elastic section modulus increases:
σmax = |M| / S
For the same bending moment, a cross-section with a larger section modulus produces a lower maximum bending stress. The section properties must be taken about the actual axis of bending.
Required section modulus
Sreq is the minimum elastic section modulus required for the entered bending moment and allowable bending stress. A candidate section should satisfy:
S ≥ Sreq
The allowable stress entered in this mode must already include any safety factors, material reductions, load factors, fatigue reductions, temperature effects, or code requirements applicable to the design.
Required section dimension
The dimension result is the theoretical minimum diameter of a solid circular section or the theoretical minimum side of a solid square section. The selected practical dimension should normally be rounded up and then checked using its actual manufactured dimensions and tolerances.
Yield-based screening status
When a positive yield strength Re is provided in bending stress mode, the calculator evaluates:
n = Re / |σmax|
- Safe — the calculated yield-based ratio is greater than or equal to 1.5.
- Warning — the ratio is greater than or equal to 1.0 but lower than 1.5.
- Unsafe — the calculated bending stress is greater than the entered yield strength and the ratio is lower than 1.0.
- Info — the result was calculated without a yield-based classification, or the selected sizing mode does not use this classification.
- Invalid — one or more required values are missing, non-positive, unsupported, or geometrically invalid.
A Safe status confirms only that the calculator’s simplified yield-based bending criterion is satisfied for the entered values. It does not confirm complete safety, code compliance, acceptable deflection, fatigue resistance, shear capacity, buckling resistance, connection strength, or resistance to combined loading.
Calculation example
Example: bending stress, safety factor, and required section size
A beam cross-section is subjected to a bending moment of 2.5 kN·m. Its elastic section modulus is 50,000 mm3. The material yield strength is 250 MPa.
1. Input data
- M — bending moment: 2.5 kN·m
- S — elastic section modulus: 50,000 mm3
- Re — yield strength: 250 MPa
2. Convert the bending moment
2.5 kN·m = 2,500 N·m = 2,500,000 N·mm
3. Calculate maximum bending stress
σmax = M / S
σmax = 2,500,000 / 50,000
σmax = 50 N/mm2 = 50 MPa
4. Calculate the yield-based safety factor
n = Re / σmax
n = 250 / 50 = 5.00
The calculator classifies this result as Safe because the calculated ratio is greater than 1.5. This classification applies only to the simplified elastic bending yield check.
5. Calculate required section modulus
Assume that the allowable bending stress selected for preliminary sizing is 150 MPa.
Sreq = M / σallow
Sreq = 2,500,000 / 150
Sreq = 16,666.7 mm3
A selected cross-section should therefore provide an elastic section modulus of at least approximately 16,667 mm3 about the applicable bending axis.
6. Equivalent solid section dimensions
For a solid circular section:
d = ∛((32 · 16,666.7) / π) = 55.37 mm
For a solid square section:
a = ∛(6 · 16,666.7) = 46.42 mm
These are theoretical minimum dimensions based only on the entered allowable bending stress. A practical design should use a larger available size and must also be checked for deflection, shear, fatigue, buckling, stress concentrations, combined loading, and applicable design-code requirements.
Assumptions and limitations
The calculator applies the elementary elastic flexure formula to a known bending moment and cross-section. The following assumptions and limitations apply:
- The entered bending moment is the moment magnitude at the specific cross-section being evaluated.
- The calculator expects a positive bending-moment magnitude. A negative physical moment should be entered using its absolute value; the moment sign determines which side of the section is in tension or compression but does not change the stress magnitude.
- The material is assumed to behave linearly elastically within the evaluated stress range.
- The member is assumed to be initially straight, and plane cross-sections are assumed to remain plane during bending.
- The cross-section is assumed to be homogeneous, with section properties evaluated about the correct neutral and bending axis.
- The calculated value is the nominal maximum normal bending stress at the extreme fiber.
- Local stress concentrations caused by holes, notches, shoulders, welds, fillets, keyways, contacts, or abrupt geometry changes are not included.
- Axial stress, transverse shear stress, torsional stress, biaxial bending, residual stress, thermal stress, and other combined-stress effects are not included.
- Plastic redistribution and plastic section modulus are not considered. The calculator uses the elastic section modulus.
- Material yield strength is used only for a simplified yield-based stress ratio and status classification.
- The Safe, Warning, and Unsafe thresholds are calculator screening thresholds and are not substitutes for requirements from an applicable engineering standard.
- The allowable bending stress used for section sizing must be selected by the user and should already include the required safety factors and design reductions.
- Required direct section dimensions are calculated only for solid circular and solid square sections.
- For other cross-sections, calculate Sreq and select a section whose actual elastic section modulus is equal to or greater than that value.
- The calculator does not determine bending moment from loads, span, support conditions, or a bending-moment diagram.
- Beam deflection, shear capacity, fatigue, fracture, local buckling, lateral-torsional buckling, instability, connection behavior, and code-specific resistance checks require separate verification.
- Input units and the selected bending axis must be consistent with the section properties used in the calculation.