Learn: Timber bending utilization
A simple timber bending check compares the calculated bending stress with the applicable design bending strength. It is a stress utilization for one axis, not a complete check of every way the member can fail.
What the formula is saying
First calculate σm = M/W using the relevant elastic section modulus. Then divide by fd. A ratio above 1 exceeds the supplied bending-strength limit for this isolated check.
Read the symbols in plain language
- MEd
- Design moment
Design moment. Divide moment by the elastic section modulus about the same bending axis.
N·mmUse N·mm as the base unit shown here. M is N·mm, W is mm³ and fd is N/mm². M/W becomes N/mm² and the final ratio is dimensionless. A moment in kN·m must be multiplied by 1,000,000 before use in this base-unit form.
- W
- Section modulus
Section modulus. Divide moment by the elastic section modulus about the same bending axis.
mm³Cubic millimetres here describe a section modulus; they are a length-cubed unit.
- fm,d
- Design bending strength
Design bending strength. The ratio compares matching stress quantities for this one timber bending check.
N/mm²One N/mm² equals one MPa.
- Utilization
- Result to find
Timber bending utilization. The ratio compares matching stress quantities for this one timber bending check.
ratio / no unit
Sort out the units first
M is N·mm, W is mm³ and fd is N/mm². M/W becomes N/mm² and the final ratio is dimensionless. A moment in kN·m must be multiplied by 1,000,000 before use in this base-unit form.
Assumptions before calculating
Use moment magnitude, the appropriate section modulus about the checked axis and design strength already modified for the intended timber property, service conditions and duration.
This is a first-generation Eurocode teaching relationship or an explicitly simplified coefficient calculation. The numbers supplied here are exercise data, not a recommendation for any country. Check the adopted edition, relevant clause, National Annex, applicability conditions and all other limit states before any real design.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Utilization and explain the result in the stated output unit.
- MEd · Design moment
- 5000000 N·mm
- W · Section modulus
- 300000 mm³
- fm,d · Design bending strength
- 18 N/mm²
Calculate bending stress magnitude
Divide moment by the elastic section modulus about the same bending axis.
(5000000) ÷ (300000) ≈ 16.66666667 N/mm²Compare with design bending strength
The ratio compares matching stress quantities for this one timber bending check.
(16.66666667) ÷ (18) ≈ 0.9259259259
Does this worked answer make sense?
Doubling moment doubles utilization. Doubling W or fd halves it, provided all other applicability assumptions remain unchanged.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- MEd · Design moment
- 4000000 N·mm
- W · Section modulus
- 250000 mm³
- fm,d · Design bending strength
- 16 N/mm²
Calculate bending stress magnitude
Divide moment by the elastic section modulus about the same bending axis.
(4000000) ÷ (250000) = 16 N/mm²Compare with design bending strength
The ratio compares matching stress quantities for this one timber bending check.
(16) ÷ (16) = 1
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- MEd · Design moment
- 6000000 N·mm
- W · Section modulus
- 400000 mm³
- fm,d · Design bending strength
- 18 N/mm²
Find: Learn: Timber bending utilization
A hint, not the answer
First calculate σm = M/W using the relevant elastic section modulus. Then divide by fd. A ratio above 1 exceeds the supplied bending-strength limit for this isolated check.
M is N·mm, W is mm³ and fd is N/mm². M/W becomes N/mm² and the final ratio is dimensionless. A moment in kN·m must be multiplied by 1,000,000 before use in this base-unit form.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Calculate bending stress magnitude
Divide moment by the elastic section modulus about the same bending axis.
(6000000) ÷ (400000) = 15 N/mm²Compare with design bending strength
The ratio compares matching stress quantities for this one timber bending check.
(15) ÷ (18) ≈ 0.8333333333
Avoid the common trap
Do not use second moment I instead of section modulus W. Do not divide by characteristic strength when the check requires design strength, or use a signed negative moment to create negative utilization.
When this method applies — and when it does not
Lateral instability, combined axial force, biaxial bending, notches, shear, bearing, creep and deflection remain separate checks. A ratio below 1 does not certify the timber member or its connections.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
One idea understood. Keep going.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Timber bending utilization. Timber design values and bending relationships: distinguish the strength modifier kmod from the deformation factor kdef, and apply the correct service class and load duration.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.