Structural Engineering & Statics
where the load goes
Concept

Miners rule — where it appears

The assumption that fatigue damage from cycles of different amplitude adds linearly, each cycle contributing the reciprocal of the life it would have alone. Failure is predicted when the sum reaches one, and the rule takes no account of the order the cycles arrived in.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

Three details, and no material anywhere on the plot. Stress range against cycles to failure for three detail categorys — 160, 90, 36 N/mm² at two million cycles. The lines are parallel because they share a slope of three, and the spread between them is a factor of 4.4 in stress and therefore 88 in life. Nothing on this plot depends on the strength of the steel: the same detail in a grade twice as strong lies on the same line. At a stress range of 70 N/mm² the lives are 160: 6.8e+7, 90: 4.3e+6, 36: 2.7e+5 cycles. The knee in each line is the constant-amplitude limit, past which the slope becomes five.

The load that never came near failing anything

A detail survives sixty-eight million cycles at a stress range that another detail in the same steel survives two hundred and seventy thousand of. The two lie a factor of two hundred and fifty apart, and the material is not on the plot anywhere.

materials · Fatigue Two per cent of the traffic and most of the damage. A 120-year traffic spectrum on one detail of category 71, with each band's share of the cycles and its share of the damage. The two bars have almost nothing to do with one another, and the reason is the slope of three: life goes as the inverse cube of the stress range, so a cycle twice as large does eight times the damage and a cycle a third as large does a twenty-seventh of it. A full train is 2% of the crossings and 58% of the damage; and three of the five bands — an ordinary lorry, a van, a car, 90% of the crossings — sit under the cut-off and do none of it at all. The equivalent constant range that would do the same damage in the same number of cycles is 25.5 N/mm², which is the one number a designer is usually handed — and it is a cube-weighted average, so it is nearer the heaviest vehicle than to the average one.

The cycles that do not count

A fatigue spectrum has to be reduced to one number, and the reduction is a cube-weighted average rather than an ordinary one. Two per cent of the traffic does most of the damage, ninety per cent of it does none at all, and the equivalent range that comes out is nearer the heaviest vehicle than the average one.

materials · Fatigue The largest cycle in the record is not between two neighbouring reversals. A 26-second stress record at the mid-span detail of a 20 m road bridge, as seven vehicles cross it and the deck rings at 4 Hz after each: a heavy lorry at 1.0 s, a car at 5.2 s, a van at 8.0 s, an ordinary lorry at 12.0 s, a second lorry meeting it at 12.3 s, a car at 17.0 s, the heaviest vehicle of the record at 20.0 s. With a reversal threshold of 0.5 N/mm² the record has 185 reversals, and rainflow pairs them into 92 cycles. The four largest loops are marked, each by a line joining the two reversals that close it: 74.3 N/mm² between 20.6 s and 21.2 s; 45.1 N/mm² between 1.5 s and 2.2 s; 38.5 N/mm² between 12.6 s and 13.5 s; 16.5 N/mm² between 8.5 s and 8.9 s. The largest is the record's whole range, from its lowest point to its highest. Counted between successive reversals instead, the largest range anywhere in the record is 38.1 N/mm², because the deck's ringing puts reversals on every rise and every fall.

Which reversal closes the loop

A strain-gauge record is a wiggle, not a list of cycles, and its damage depends on how its reversals are paired. Rainflow pairs each one with the reversal that closes its hysteresis loop. Counting the ranges between neighbours breaks every lorry's cycle into pieces, and on one bridge record it gives a detail 424 years where rainflow gives it 34.

materials · Fatigue Half the life is spent growing the first half-millimetre. The crack length against time for a detail starting with a 0.50 mm flaw under 800 cycles a day of the same five-band traffic the S-N calculation used, integrated by Paris's law. It reaches 1 mm after 91.2 years, 2 mm after 131.3, 10 mm after 172.9 and its critical length of 125.7 mm after 195.8. The curve is nearly flat and then nearly vertical, because the rate goes as the cube of ΔK and ΔK goes as the square root of the crack: the crack spends most of its life being too small to find and the rest being too large to ignore. The dashed lines are the crack lengths at which each band of traffic starts doing anything at all.

The loops a crack grows on

A Miner sum reduces a hundred years of traffic to a number and throws away everything below the cut-off — on this bridge, a third of the vehicles doing exactly none of the damage. Integrate the same spectrum as a crack instead and that third grows thirty-seven per cent of the crack, because a cut-off is a statement about a constant-amplitude test and a crack's threshold is a length rather than a stress. The two calculations disagree about the life by a third and about which vehicles matter entirely.

materials · Fatigue The same cycles, and two different lives. Two sequences made of exactly the same cycles — blocks of 6.00M at 40 N/mm² alternating with blocks of 0.22M at 90, on a detail starting with a 0.50 mm flaw. Taking the small cycles first, the crack reaches its critical 87.7 mm after 12.36M cycles; taking the large ones first, after 6.36M. Miner's sum at the moment of failure is 0.99 for the first and 0.68 for the second, so a rule that predicts failure at a sum of one is right to within a per cent about the first and 48 per cent unconservative about the second. The mechanism is on the axes: ΔK rises with the crack, so a large block met late finds a longer crack and does more with it.

The record played backwards

Miner's rule adds damage, and a sum has no order. A crack does, twice over: a large block met late finds a longer crack and does more with it, and an overload leaves a plastic zone that slows everything after it. The same cycles rearranged fail at 12.4 million or at 6.4, and a Miner sum that is right to one per cent about the first is out by half about the second. One cycle in four million can add fifty-four per cent to a life.

materials · Fatigue The two knees the environment takes away. The design S-N curve for a category 71 detail and the two shapes a corrosive environment leaves. In air the slope is three to the constant-amplitude limit at 52.3 N/mm², five to the cut-off at 28.7, and nothing below it. With the cut-off removed the second branch continues. Under free corrosion there is one slope of three all the way down and no knee at all. The faint lines are the five bands of the traffic: three of them sit below the cut-off — an ordinary lorry, a van and a car, 90 per cent of the crossings — which is why they do nothing to a detail in air and something to every other curve on the figure.

The cut-off belongs to the water

An S-N curve's cut-off is the most consequential thing on it: on a category 71 detail under ordinary bridge traffic it deletes ninety per cent of the crossings and leaves two bands doing all the damage. It is a property of steel in air. A crack tip that seawater or de-icing salt can reach has no threshold, no endurance limit and no knee, and the same bridge's life runs from 300 years to 45 depending on which of six defensible calculations is asked.

materials · Fatigue The chimney that locks in most is tuned above the commonest wind. Lock-in cycles a year at the full lock-in amplitude that do the same damage as the whole band (millions), for 1.2 m cylinders of every frequency in hourly mean winds at the top Rayleigh-distributed with scale 5.5 m/s, against the critical speed as a multiple of that scale. Solid, the count through the amplitude envelope; dashed, EN 1991-1-4's closed form 2·T·f₀·ε₀·(Ucr/v0)²·exp(−(Ucr/v0)²) at band widths ε₀ of 0.1 and 0.3. All three peak at a critical speed of √(3/2) = 1.22 times the scale, 6.7 m/s — above the commonest wind (0.71) and the mean (0.89). The envelope is EN's formula with ε₀ = 0.22. The chimney drawn, at 6.0 m/s, collects 4.6 million a year.

The breeze that uses up a chimney

A chimney that locks in to vortex shedding at a breeze is usually told to accept the motion and check its fatigue. Counting the cycles a real wind climate delivers turns that into arithmetic, and the arithmetic has two surprises. The chimney that collects the most cycles is not the one tuned to the commonest wind but to one a fifth stronger than the climate's scale, so stiffening a chimney can shorten its life. And damping does not lengthen the life in proportion: it ends the count, all at once, at the damping that puts the largest stress range under the detail's cut-off.

dynamics · Vortex-shedding

Named alongside it

The objects these essays reach for when they reach for this one.

FatigueCut-off limitSpectrumCrack growthStress rangeDetail categoryEndurance limitInspectionStress intensityCycle countingDamageEquivalent range

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