Module 17 · Lesson 17.2
Fatigue: the limit state that sizes the girder
A crane girder is one of the few building structures where fatigue governs — and where the cheapest fix is a change of weld rather than a change of section.
Why this matters
The trial girder in the last lesson was at 54% on bending and was nowhere near adequate. The reason is that a crane girder is loaded and unloaded several million times in its life, and steel that is nowhere near its yield strength will still crack if it is cycled often enough. Fatigue is a genuinely different limit state — driven by a different quantity, with a different remedy — and it is the one that sizes the member.
By the end of this lesson you should be able to
- Explain why fatigue depends on stress RANGE, not peak stress
- Apply the cube law and say what doubling the life requires
- Compare improving a detail with increasing a section
- Say why steel grade is irrelevant to fatigue
What you should already know
- Moving loads and the crane girder's moment (previous lesson)
- Weld details and their geometry (Module 12)
- The elastic section modulus (Module 6)
Range, not peak
Fatigue cracks grow because a detail is stretched and released, repeatedly. What drives that growth is the range of stress — the difference between the maximum and the minimum — and not the peak value.
That has a consequence people find hard to accept:
A detail cycling between 100 and 160 N/mm² is in exactly the same fatigue condition as one cycling between 0 and 60.
The first has a peak nearly three times higher and identical fatigue damage. Nothing about the mean stress appears in the calculation.
The S–N curve has a slope of m = 3 in its main region, so the number of cycles a detail survives goes with the cube of the range:
| Change in range | Effect on life |
|---|---|
| Halve it | ×8 |
| Reduce by 21% | ×2 |
| Reduce by 26% | ×2.5 |
Note the second and third rows. To DOUBLE the life the range must fall by 21%, not 26% — because 2^(1/3) = 1.26, so the range divides by 1.26 rather than falling by 26%. Those are different numbers and it is an easy slip to make.
And one thing that does not appear anywhere: the steel grade. Fatigue strength is essentially independent of yield strength, so a stronger steel buys nothing at all here. It is the detail — the shape of the weld, whether it is ground, whether there is a stress concentration — that sets the category.
Try it
Range, cycles, and what actually fixes it
The S–N curve for the detail you choose. The dot is where this detail sits; the vertical line is the life it must survive. Move the maximum and minimum stress independently and watch only their difference matter.
- Maximum stress
- 58 N/mm²
- Minimum stress
- 0 N/mm²
- Stress range
- 58 N/mm²
- Detail category
- 71 N/mm²
- Cycles survivable
- 1.49 M
- Cycles expected
- 5.0 M
- Damage ratio
- 3.35
- Permissible range
- 38.8 N/mm²
- Damage as built
- 3.35
- If the detail improved
- 0.61
- If the range fell 21%
- 1.65
- Better remedy
- detail
A stress range of 58 N/mm² at a category 71 detail survives 1.5 million cycles, against 5.0 million expected — a damage ratio of 3.35. Not adequate. The permissible range for this life is 39 N/mm², so the range must fall by 33% — or the detail must be improved, which is usually easier. With m = 3 the life goes with the CUBE of the range, so a 21% reduction doubles it.
Things worth trying
- Set the maximum to 160 and the minimum to 100. Now set them to 60 and 0. The range is 60 both times and every fatigue number is identical — the peak plays no part at all.
- Start at 58 and 0 with category 71 over 5 million cycles. Damage 3.42: the girder fails, and this is already a 251 kg/m section.
- Now raise the detail category to 125 without touching anything else. Damage falls to 0.63 and it passes. That is a grinding operation.
- Compare the two remedy rows. Improving the detail beats reducing the range on almost any realistic move, which is why the detail is the first thing to look at.
- Halve the stress range and watch the cycles-survivable figure multiply by eight. The S–N slope is 3, so life goes with the cube.
- Now reduce the range by exactly 21% and watch the life double. Reduce it by 26% instead and the life goes up 2.5 times — those are different numbers, and confusing them is an easy slip.
- Take the cycles from 0.5 to 20 million and watch the permissible range fall. A crane that works a shift a day and one that works continuously are different design problems.
- Drop the category to 36 — a poor welded attachment. The permissible range collapses, and no realistic section would fix it. Some details simply cannot be used in a fatigue situation.
Worked example
Sizing the crane girder — by fatigue
Given
- The 12 m gantry girder from the last lesson
- Fatigue loading uses the UNFACTORED crane load with no dynamic factor: 125 kN per wheel
- That gives a fatigue moment of 547 kNm at the same critical position
- 5 million cycles over the design life; γMf = 1.35
- Compare detail category 71 — a plain welded attachment — against 125, a ground full-penetration butt
Find
What section the girder actually needs, and what the detail is worth.
Assumptions
- The S–N curve and the detail categories are code-calibrated and unverified here
- A single stress range is used rather than a full spectrum with a damage-equivalent factor
- The fatigue load is taken as the unfactored static crane load — a simplification of the real damage-equivalent approach
Practice
A detail cycles between 160 and 100 N/mm². What is the stress range, in N/mm²?
Practice
The stress range at a detail is halved. By what factor does the fatigue life increase? Take m = 3.
Practice
By what percentage must a stress range fall to DOUBLE the fatigue life? Take m = 3.
Practice
A category 71 detail with γMf = 1.35 is checked for 5 million cycles. What is the permissible stress range, in N/mm²? Take Δσ = (Δσc/γMf)(2/N)^(1/3) with N in millions.
Check yourself
A crane girder fails fatigue. Which change is likely to be most cost-effective?
Summary
- Fatigue is driven by stress RANGE — 160→100 and 60→0 are identical conditions
- Mean stress does not appear in the calculation at all
- Life goes with the CUBE of the range: halve it for ×8, reduce 21% for ×2
- A 26% reduction gives ×2.5, not ×2 — 2^(1/3) = 1.26 divides, it does not subtract
- Permissible range at category 71 over 5M cycles was only 38.8 N/mm²
- The 123 kg/m trial girder failed fatigue by a factor of 89 at 0.54 on bending
- Fatigue drove it to 384 kg/m, at a bending utilisation of 0.11
- Improving the detail from 71 to 125 allowed 251 kg/m — 133 kg/m saved by a weld change
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint