Structural Engineering & Statics
where the load goes
Concept

Stress range — where it appears

The difference between the largest and smallest stress in a cycle, which is what a fatigue life depends on. The mean stress drops out for a welded detail because the residual stress there is already at yield, so the whole applied cycle happens in tension whatever is applied.

Named by 14 essays across 4 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 Four details, and no material anywhere on the plot. Stress range against cycles to failure for four detail categorys — 160, 112, 71, 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 62 N/mm² the lives are 160: unlimited, 112: 2.1e+7, 71: 3.0e+6, 36: 3.9e+5 cycles. The knee in each line is the constant-amplitude limit, past which the slope becomes five.

The detail decides and the steel does not

A fatigue check contains no material strength anywhere. The same detail in a steel twice as strong lies on exactly the same line, because a fatigue life is decided by the geometry of a weld and by the stress range it sees — and two per cent of the traffic does most of the damage, because life goes as the inverse cube of the range.

connections · Detail category The crack length at which the strength stops mattering. Failure stress against crack length for a toughness of 100 MPa√m, with three steel grades drawn. The falling curve is fracture — Kc divided by Y times the root of pi a — and it does not know what the yield stress is. The horizontal lines are the grades. At 275 N/mm² the two cross at a crack 33.5 mm long; At 355 N/mm² the two cross at a crack 20.1 mm long; At 460 N/mm² the two cross at a crack 12.0 mm long. The stronger grade's transition is the shorter one — raising the yield stress does not raise the strength of a cracked member, it only shortens the crack that takes it away. At a working stress of 120 N/mm² the critical crack is 176.0 mm.

Designed to be found in time

A fatigue design that promises a detail will not crack is making a claim about a hundred years of traffic. A damage-tolerant one assumes it will crack, and sets the inspection interval from how long a crack takes to grow from the smallest size anybody can find to the largest the section can survive.

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 What a crack makes of a notch, against how sharp the notch is. The fatigue notch factor against the root radius, at a fixed elastic factor of 3 in a 430 MPa steel. K_t is a property of the shape and does not move along this axis at all — the dashed line — while the factor fatigue actually feels climbs toward it from below. At a 1 mm root the answer is 2.40, which is 30 per cent of the notch relieved; at 0.1 mm it is 1.38, and a notch that concentrates by 3 elastically is barely felt. Nothing has changed about the stress field: what has changed is that the peak is confined to a smaller volume than the material's own process size, so the crack starts against an average rather than against a maximum.

The notch a crack does not feel in full

The elastic concentration factor is a property of shape and knows nothing about size, which is what makes it so useful and so misleading. A fatigue crack starts against an average over a volume the material owns, so two notches with the same factor and different radii have different fatigue strengths — and the stronger the steel, the less of that relief it gets.

materials · Stress concentration 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 The bolt force is the larger of two lines. What an M20 bolt in a 25 mm tee flange actually carries, against the tension applied to the flange. Preloaded to 171 kN it starts there and climbs at Φ = 0.185 — the bolt's own stiffness over the bolt's plus the clamped plates', 857 against 3781 kN/mm — so 18 per cent of every kilonewton applied reaches it and the rest is unloading the contact. At 138 kN the contact runs out and the line joins the one an ordinary bolt has followed from the start, climbing at 2.12. The two lines meet, so the strength is the same either way; what differs is the slope, by a factor of 11.5. The flange's own mechanism is at 172 kN, comfortably past the crossing.

The bolt that was already stretched

Prying is a lever that needs the flange to lift before it can act, and a preloaded bolt does not let it. The bolt carries a fifth of every kilonewton applied until the plates part, and everything after that is the ordinary calculation — which is why preload changes the fatigue answer by a factor of a thousand and the strength answer by nothing at all.

connections · Prying 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 Where each check bites. The c shape group with a 5.0 mm throat drawn to scale, loaded 1 mm from its vertical run. Offset from each weld, the use of each point: solid for the static directional check, dashed for fatigue at 2 × 10⁶ cycles, each scaled to its own worst point, which is marked. The static check's worst point is where the force runs most nearly along a weld; the fatigue check's is where it runs most nearly across one. Here the two checks are governed at different points, and a fatigue check made only at the static peak allows a load range 2.25 times the group's.

Strongest across, and first to crack across

A fillet weld pulled across its length is √1.5 stronger than one pulled along it, and every static check of a weld group uses that. In fatigue the same two directions are rated the other way round: the part of the force across the weld is checked against a detail category of 36 and the part along it against 80. The point of a group that governs the static check need not govern the fatigue one, and checking fatigue there can allow more than twice the load range the weld will take.

connections · Weld group A crossed hanger swings through nearly all of its force. The largest and smallest force in each crossed hanger of a tied arch of 100 m span and 17 m rise on 32 hangers, with 60 kN/m of dead load and a live load of 60 kN/m running on from either end, each bar running from the one to the other. The worst range is 678 kN, from 49 to 727, in hanger 11; the largest force anywhere is 756 kN. 6 of the 32 go slack at some stage, and the worst range is 90% of the largest force anywhere: the live load's shear passes through the crossed hangers as through a truss's web, and swings them from nearly nothing to nearly their most.

The hangers that carry the shear twice

Crossing a tied arch's hangers turns them into the web of a truss, which is why the tie stops bending. It also means a live load's shear passes through them, so as a load runs on from either end each crossed hanger swings from nearly nothing to nearly its most: on a 100 m arch the worst swings through 678 kN, 90% of the largest force in any hanger, where a vertical hanger swings through 194, half of its own. More dead load raises the force and leaves the range alone. The arrangement that is best for strength is the one that hands its hangers to fatigue.

structures · Network arch A haunch moves the worst stress range to its toe. The stress range up the height of the first interior post of a Vierendeel girder of six 2 m panels, 1.5 m deep, every member of the same section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load, over one crossing: the plain post (dashed), and the post with 250 mm haunches 2.00 times as deep at both corners (solid); with the constant-amplitude limits 52.3 MPa for detail category 71 (dotted). Plain, the range is 78.3 MPa at the corner. Haunched, it is 43.3 at the corner and jumps to 61.2 at the toe, 250 mm along the post — the largest anywhere in it, 61.2.

The corner that counts its crossings

As a load crosses a Vierendeel girder, the moment at a post's corner changes sign when the load passes that post. A static check reads the peak, and the peak says the posts beside the supports work twice as hard as the middle one. Fatigue reads the range, and the range says every interior post works the same. A haunch at the corner lowers the corner's range and moves the worst of it to the haunch's toe — so a deeper haunch soon buys nothing, a longer one keeps paying, and a toe two detail categories worse than the corner cancels the haunch outright.

structures · Vierendeel The range belongs to the truss. The worst hanger force range against the tie's largest moment in a tied arch of 100 m span and 17 m rise on 32 crossed hangers of 3,000 mm² bar, 60 kN/m of dead load and 60 kN/m of live load run on from either end, as the hangers' stiffness runs from four times a plain bar's to a fiftieth of it (dots), and for the same arch on vertical hangers (the lone dot). Softening the crossed hangers walks the bridge from 678 kN of range and 316 kN·m of tie moment to 557 and 1,154; vertical hangers sit at 194 and 3,505. The line bends nowhere near the vertical arch's point: a soft network is still a truss.

The hanger that cannot be made soft enough

A network arch's crossed hangers swing through nearly all their force as traffic passes, because they are the web of a truss. Make them softer — a spring at each anchorage — and the tie ought to take back some of the shear and the hangers' ranges ought to fall. They hardly do. A tenth of the stiffness takes 6 per cent off the worst range and adds a third to the tie's bending; a fiftieth takes 18 per cent off and multiplies the tie's bending by 3.7. The range belongs to the truss and leaves only when the truss does, which vertical hangers show at 194 kN against 678 and eleven times the bending. So the hanger is sized for its range, eight times the area its strength needs, and the stiffness that comes with it costs the range nothing.

structures · Network arch More hangers, each swinging through less. Each hanger's force range, plotted at its mid-point along the span, for a tied arch of 100 m span and 17 m rise on 32, 64, 96 crossed hangers of 3,000 mm² bar at 60°, 60 kN/m of dead load and 60 kN/m of live load run on from either end. 32 hangers: the worst 678 kN, the mean 528; 64 hangers: the worst 369 kN, the mean 273; 96 hangers: the worst 241 kN, the mean 184.

More hangers, and the same steel

A network arch's crossed hangers are the web of a truss, and a live load's shear passes through them, so every hanger swings through nearly all of its force. The obvious cure is more hangers, each crossing more of its neighbours, so that every section's shear is shared among more diagonals. It works hanger by hanger — three times the hangers, a third of the range — and not at all in total: the hangers' fatigue steel, sized for their ranges, is set by the shear the arch carries and barely changes with how many hangers it is divided among. What more hangers buy is elsewhere.

structures · Network arch Lightened where it reverses, and the range follows. The stress range each post's corner sweeps over one crossing of a Vierendeel girder of six 2 m panels, 1.5 m deep, its chords all of one section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load: uniform posts, posts sized to their peaks, and posts sized to both their peaks and their ranges. Uniform, every interior corner ranges 76.1 to 78.3 MPa. Sized to peaks, the middle post's range is 150.8 MPa and its neighbours' 101.6, against the uniform girder's worst of 78.3. Sized to both, no corner exceeds 78.3 MPa.

The post sized by what it must survive

A Vierendeel girder's posts are usually all one section, and a crossing load gives them peaks that differ by a factor of two and ranges that barely differ at all. Size each post to its peak and the steel falls by a quarter — and the middle post, lightened to two-fifths of the chords, sweeps twice the stress range the whole girder used to, and lasts a seventh as long. Size each post to its range instead and the girder needs less steel than the uniform one, not more, because the posts that can be lightened are the ones at the ends.

structures · Vierendeel

Named alongside it

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

FatigueDetail categoryCut-off limitMiners ruleStress concentrationCrack growthEndurance limitFree bodyHangerInfluence lineInspectionLoad sharing

All concepts