Bending Stress Calculator
The moment against the section modulus — the outer-fibre stress, the yield comparison and the utilisation, in one pass.
Bending Stress Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Rectangular sections; elastic linear theory
- Single moment — no envelopes
In short: A 45 kN·m moment on a 100 × 200 mm rectangular section bends the outer fibres at 67.500000 MPa — 45,000,000 N·mm over a section modulus of 666,666.667 mm³. Against 250 MPa yield steel that is 27.000000% utilisation: comfortable, and honestly so — the outer millimetre works hardest while the middle does almost nothing, which is the quiet argument for I-beams everywhere.
Formula
σ = Mc/I = M/S ··· S = bh²/6 (rectangle)
Bending stress grows linearly from zero at the neutral axis to its maximum at the outer fibre, so the section's resistance condenses into one number: the modulus S = I divided by the extreme-fibre distance. For a rectangle that is bh²/6 — notice the square on depth. The page prices the outer fibre because that is where yield begins, and prints the utilisation so the margin is a number, not a feeling.
Worked Example
- Take the peak moment from the beam load page.
- Enter the section's width and depth.
- Set the steel's yield strength.
- Read stress, modulus and utilisation — and say the margin out loud.
Defaults: 45 kN·m on 100 × 200 → 67.500000 MPa, 27.000000% of S235. S355 drops utilisation to 19.014085%. Double the depth at the same area — 100 wide, 400 deep — and the stress quarters to 16.875000.
Strengths & Limits Of This Model
Where this engine is strong
- Utilisation printed beside the stress
- The modulus shown, not hidden
Where it stops
- No rolled-shape tables
- No lateral-torsional checks
Practical Use Cases
Member checks
the first strength verdict
Section choice
depth vs grade trade-offs
Teaching
where the section actually works
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Bending Stress Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does depth beat width so decisively?
Because the section modulus squares the depth and keeps width linear — bh²/6. Moving material away from the neutral axis does double duty: more lever arm for the couple that resists the moment. That is the entire economic argument for I-sections, which push every possible millimetre to the outer fibres where the stress actually lives.
The stress is linear across the depth — so what is the average fibre doing?
Very little. At mid-depth the bending stress is zero by construction; the inner half of the section carries a fraction of what its area suggests. Utilisation numbers overstate how hard a solid rectangle works — the outer fibre yields first while the core watches, which is also why compact sections tolerate a little overstress gracefully in the real, strain-hardening world.
Is 27% utilisation too comfortable to be true?
It is honest arithmetic on one load case with no factors applied — design loads carry amplification, combinations and pattern cases that multiply the raw moment, and the code resistance factors divide capacity again. The page's utilisation is the truth of this moment against this yield; the design check is the code's version of the same division with more inputs.
What happens above 100% utilisation?
The outer fibre passes yield: the linear arithmetic stops being a promise and the section starts developing a plastic hinge — deflection runs away before collapse does. The page prints the over-yield verdict plainly instead of a bigger stress number, because past yield the formula describes a strain the linear theory can no longer price.
Which moment do I enter?
The maximum bending moment on the member for the case under study — from the beam load page's peak for these ideal cases, or the moment envelope's maximum in real frames. Entering the midspan moment for a cantilever check under-reports by the support's share, which is how optimistic numbers are born.
Does the grade matter as much as it looks?
Less than the brochure suggests. Higher yield buys strength linearly but no stiffness whatsoever — E stays 200 GPa across grades — so S355 fixes a stress problem and leaves a deflection problem untouched. The page prints utilisation against the grade you set; the deflection page is where the grade's silence becomes audible.
Can I enter a non-rectangular section?
Not directly — this page owns the rectangle and shows its work. For rolled sections take the tabulated S value and divide it into your moment: σ = M/S is one division, and the section modulus page can verify the rectangle you are comparing against. The formula never changes; only the modulus's address does.
Why MPa and N·mm?
Because they cancel perfectly: N·mm over mm³ is N/mm², which is a megapascal. Structural engineering runs on this quiet unit conspiracy — mm sections, kN·m moments shifted six places, stresses in MPa — and every unit error in beam design is a slip in one of those three shifts. The page keeps the labels on every input for exactly that reason.