Beam bending stress calculator
Bending stress from the moment and the shape of the beam's cross-section.
Fill in the fields and the result will appear here automatically.
Calculates the magnitude of maximum normal bending stress for a solid rectangular or circular section. Supply the bending moment from your load analysis. Rectangle depth is measured in the bending plane; the model assumes linear elastic bending in one plane. It does not determine allowable stress or overall load capacity.
How it works
Formula and logic
σmax = M/W. For a rectangle W = b·h²/6; for a solid circle W = πd³/32. Dimensions are in mm and M in N·m is multiplied by 1000, giving N/mm² = MPa. Moment and active dimensions must be positive finite numbers; dimensions of the other shape are ignored.
Example
A 100×200 mm rectangular section under 4.5 kN·m carries a bending stress of 6.75 MPa.
Fields and units
- Bending moment — N·m
- Cross-section — list option
- Section width — mm
- Section height — mm
- Diameter — mm
How to use
- — Enter the bending moment in newton-metres: for a simply supported beam with a central load it is force times span over four.
- — Section sizes are in millimetres.
- — For a rectangle the height is the dimension along the load, that is vertically.
- — Compare the result with the allowable stress of your material — it depends on the grade and is not assumed here.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- MIT, Roylance: elastic stress σ = −My/I and rectangular second moment of area
- Limitation
- Preliminary calculation of the stated elastic model. It does not replace structural checks for applicable requirements, loads and material properties.
FAQ
Why does a board on edge carry so much more?
Because the section height enters the modulus squared. Turning a 50×150 board from flat to on edge triples its resistance to bending.
How does this differ from a tension calculation?
In tension the stress is uniform across the section and equals force over area. In bending it varies linearly from the neutral axis and peaks at the edge, so the shape of the section matters more than its area.
Why is there no allowable stress?
It depends on the steel grade, the timber species and the safety factors of the applicable code. Baking in one number would present a special case as a general rule.
Is this a complete beam check?
No. This is bending stress in the elastic range and in one plane only. Deflection, buckling, shear and torsion are separate calculations.