Structural Engineering & Statics
where the load goes
Concept

Stress intensity — where it appears

The parameter K describing the strength of the stress field at a crack tip, combining the applied stress, the crack length and the geometry. A crack grows unstably when it reaches the material's fracture toughness, so it is the quantity a fracture assessment compares against.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

The crack length at which the strength stops mattering. Failure stress against crack length for a toughness of 100 MPa√m, with two 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 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.

The flaw that sets the strength

A member with a crack twenty millimetres long fails at its yield stress. Make the steel stronger and the crack that does it gets shorter, so the same flaw that was harmless in the weaker grade decides the stronger one.

materials · Fracture 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 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 magnification a crack sees is a ratio, not a depth. The stress magnification at a weld toe against the crack's depth as a fraction of the plate's thickness, on BS 7910's two-branch fit. It is a function of a/t alone, because a weld's own size scales with the plate it is on, so the elevated field is geometrically similar. The dots are the same absolute starting flaw of 0.20 mm in plates of 12, 16, 25, 40, 60, 80, 100 mm: the flaw does not move and its magnification runs from 1.81 to 3.50. A fixed flaw in a thicker plate is a smaller fraction of it, which puts it deeper inside the raised field rather than nearer the edge of it.

The rule that points sideways

Every fatigue code puts the same detail in a thicker plate into a lower category, by a factor of (25/t) to the power 0.2, and explains nothing. It is a strange rule: a detail's strength made to depend on a dimension at right angles to the crack. Integrate a crack through a weld toe's own stress field and the rule falls out — same form, same sign, and an exponent of 0.13 against the design code's 0.2. Remove the toe's magnification and the effect reverses.

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 Between the two lines, a curve below both. The stress at which a plate of grade 355 steel of toughness 100 MPa√m, with an edge-crack factor of 1.12 fails, against the length of the crack in it on a logarithmic scale. Dashed: the fracture-mechanics line, the toughness over Y√(πa), and the yield stress; the usual check takes whichever is lower. Solid: Dugdale's strip-yield model, in which a thin strip ahead of the crack yields and the two mechanisms act together. The two dashed lines cross at the transition length, 20.1 mm, and there the strip-yield stress is 288 N/mm², 0.81 of both: the lower of the two simple answers overstates the plate by 23 per cent. At 20 mm the strip-yield stress is 289 N/mm² against 355 N/mm². Far from the crossing on either side the curve joins the line it approaches.

The crack between the two checks

A cracked plate is checked twice: once for fracture, as if the steel could not yield, and once for yielding, as if the crack could not grow, and whichever answer is lower is taken. Each check is right far from the other. Where they cross — at a crack of twenty millimetres in an ordinary structural steel — the plate fails at four fifths of both, because a strip of yielded steel ahead of the crack is by then two and a half times as long as the crack itself, and neither check has a place for it.

materials · Fracture A crack at the cope's corner, and the hole its tip needs. A 457 × 190 beam end coped 150 mm long and 80 mm deep, reacting 350 kN of which 70 per cent comes and goes with traffic, to scale. A fatigue crack has grown 30 mm down the web from the corner where the cope's two cuts meet, the most stressed point of the coped section. At that depth the live load's range of stress intensity at its tip is 15.6 MPa√m, and a hole drilled with the tip inside it stops the crack only if its radius is at least (ΔK/10.5√fy)², 6.2 mm: a 12 mm hole. The crack and the hole together leave 341 mm of the 377 mm coped section below them.

The hole at the tip of the crack

The commonest reason to repair a coped beam end is a fatigue crack at the corner of the cope, and the commonest repair needs no plate at all: a hole drilled so that the crack's tip is inside it, turning a crack into a smooth notch that the traffic can no longer drive. The hole has to be big enough, and how big is set by the crack — its radius grows in step with the crack's length. A 30 mm crack at a stringer's cope needs a 12 mm hole, a 60 mm crack a 24 mm one. The repair stops working at about 100 mm, where the hole outgrows a site drill and the section left beside it runs out of capacity at nearly the same depth. Most of the crack's life is spent far shorter than that, which makes the repair a matter of how early the crack is found.

connections · Coped beam Between the two counts, and the root's toughness says where. What a 20 mm S355 plate welded from one side to 0.5 of its thickness keeps of a double-sided butt throat, pulled straight across, against the toughness of the metal at its root: assessed as a crack on the strip-yield line (solid), by the toughness alone (dashed), and by the elastic and plastic counts (dotted, 0.25 and 0.41). Below 30 MPa√m the root tears before the elastic count's load; at 46 MPa√m the toughness alone would allow the whole plastic count, but the assessment gives 0.34, and it takes 66 MPa√m to reach 95 per cent of it. At 30, 50 and 100 MPa√m the root keeps 0.25, 0.35 and 0.41.

The land that is already a crack

A partial-penetration weld made from one side leaves the rest of the joint unfused, and pulled across, that unfused land is an edge crack whose tip is the root. Counted as a crack it keeps neither the elastic quarter of a double-sided weld nor the plastic 0.41, but a share between them that the root's toughness chooses — and the toughness of a crack front is the toughness of its length, which for a weld is the whole weld. A long weld in a cold structure can tear below the elastic count, and the same joint drawn twice as large tears sooner, which no count of stresses can see.

connections · Weld strength Where along its front a surface crack starts to cleave. Where along the front of a surface crack 8 mm deep in a 40 mm plate at −60 °C, 22 degrees below the steel's reference temperature cleavage starts, as a probability per millimetre of front, at the stress that gives a 5 per cent chance of cleavage, against the position on the front, for cracks 80 mm long, 40 mm long and 16 mm long at the surface. The middle third of the front, 60° to 120°, holds 76 per cent, 69 per cent and 28 per cent of the chance.

The front that counts for half its length

A ferritic steel's cleavage toughness depends on how much crack front it has to sample, and for a crack through a plate the front is the plate's thickness. The cracks inspections find are not through cracks; they are thumbnails at weld toes, whose curved front sees a stress intensity that varies round it. Weighted by the weakest link, the front counts for a little over half its length — so counting all of it is safe by 8 per cent, and counting none of it, as the 25 mm toughness does, is unsafe by up to a fifth for a long shallow crack.

materials · Fracture

Named alongside it

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

FatigueCrack growthFracture toughnessFractureInspectionSize effectSpectrumBrittle fractureCritical crackDamage toleranceMiners ruleStress concentration

All concepts