Structural Engineering & Statics
where the load goes
Materials

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.

Assumes The load that never came near failing anything, The flaw that sets the strength and The detail decides and the steel does not.

16 min read 8 figures Which failure arrives firstScale changes everythingThe load must go somewhere

A fatigue check is a promise about a hundred years of loading made from a category, a stress range and a slope of three. It says a detail will survive. It does not say what happens if it does not, and for a structure whose failure would be serious the second question is the one that gets designed for.

Damage tolerance inverts the promise: assume the crack forms, and arrange that it is found before it matters. That turns the design problem into two calculations neither of which is a fatigue calculation.

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.
Fig. 1 Failure stress against crack length at a toughness of 100 MPa√m, with three steel grades. The falling curve is fracture — K_c over Y√(πa) — and it does not know what the yield stress is. At 275 N/mm² the two cross at 33.5 mm, at 355 at 20.1 and at 460 at 12.0. The stronger grade’s transition is the shorter one. At a working stress of 120 N/mm² the critical crack is 176 mm.

The two ends of the interval

The inspection interval is the time between two crack lengths, and neither of them is a fatigue quantity.

The lower end is the smallest crack that inspection can reliably find. For a visual inspection of a painted weld that is tens of millimetres; for magnetic particle inspection a few; for ultrasonics on a good surface a millimetre or two. It is a property of the method, the access and the surface, and it is the one number in the whole calculation that belongs to somebody other than the designer.

The upper end is the crack at which the member fails — which is the flaw that sets the strength, a fracture calculation with the toughness in it and no fatigue anywhere.

The time between them comes from a crack-growth law, and the interval that gets specified is that time divided by two or more, so that a crack is passed over by at least two inspections before it becomes critical.

Every one of those three quantities is uncertain in a different way, and the interval inherits all three.

Which free body produced the number

The free body is the cracked section, and what makes fracture mechanics different is what is done with it.

An ordinary strength calculation takes the stress on the section and compares it with a strength. That fails here for a specific reason: the stress at a crack tip is infinite for any load at all, so there is no stress to compare with anything.

What is finite is the coefficient of that infinity. The elastic stress field near a sharp crack goes as σ∼K/2πr\sigma \sim K/\sqrt{2\pi r}, and KK — the stress intensity factor — is finite, is proportional to the applied stress, and depends on the crack’s length and geometry:

K=YσπaK = Y\sigma\sqrt{\pi a}

The comparison is then between KK and a measured material property KcK_c, and the free body’s role is to supply σ\sigma and aa. A quantity with units of stress times root length is being compared with another quantity in the same units, which is unlike every other check in this collection and is why fracture mechanics feels foreign on first meeting.

The critical crack follows by rearrangement: ac=(Kc/Yσ)2/πa_c = (K_c/Y\sigma)^2/\pi. It goes as the square of the toughness and inversely as the square of the stress, and both of those exponents matter more than any of the constants.

Why a stronger steel is worse

The hero figure carries a result that is worth stating on its own, because it inverts a very strong intuition.

Raising the grade raises the horizontal line and does nothing to the falling curve, so the crossing moves left. A 460 N/mm² steel reaches its own yield at a 12.0 mm crack where a 275 grade reaches its at 33.5.

That is not a statement that high-strength steel is bad; it is a statement that the two properties are independent. Toughness is not a function of yield stress, and often falls as strength rises within a family, so a designer who upgrades a grade to save weight has raised the working stress, shortened the critical crack twice over, and changed a member from one governed by yielding to one governed by fracture.

The detail decides and the steel does not is the fatigue version of the same statement. Here the steel does matter — but through a property nobody asked for on the order.

The crack length at which the strength stops mattering. Failure stress against crack length for a toughness of 70 MPa√m, with one steel grade 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 355 N/mm² the two cross at a crack 9.7 mm long. Below that the section yields and the crack is irrelevant; above it the crack decides and the 355 is irrelevant. At a working stress of 180 N/mm² the critical crack is 37.9 mm.
Fig. 2 The same 355 N/mm² steel at 70 MPa√m rather than 100. The transition falls from 20.1 mm to 9.7 and the critical crack at a working stress of 180 N/mm² from 78.2 mm to 37.9. A 30 per cent drop in toughness has halved the flaw the member can carry, because the relation is a square.
The crack length at which the strength stops mattering. Failure stress against crack length for a toughness of 100 MPa√m, with one steel grade 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 355 N/mm² the two cross at a crack 20.1 mm long. Below that the section yields and the crack is irrelevant; above it the crack decides and the 355 is irrelevant. At a working stress of 180 N/mm² the critical crack is 78.2 mm.
Fig. 3 The higher toughness at the same working stress, for comparison: 78.2 mm of tolerable flaw against the 37.9 above. Everything about the two members is identical except a property measured on a Charpy specimen and correlated — rather than converted — into the number used here.

There is a second reading of the crossing that decides how a member should be checked at all.

To the left of the crossing the member is yield-governed. A crack shorter than the transition length does not reduce the failure stress below the yield stress, so the member fails by yielding across its remaining section and an ordinary net-section calculation is right. Fracture mechanics is unnecessary.

To the right the member is fracture-governed. The crack decides, the yield stress is irrelevant, and a net-section calculation is unconservative by whatever the ratio of the two curves is.

The transition length is therefore the boundary between two entirely different checks, and knowing which side a member is on is the first question rather than a refinement. For the 355 grade at 100 MPa√m it is 20.1 mm — which is a crack a routine inspection would find, and is also a crack that could sit undetected inside a thick weld for years.

That boundary moves with everything: 33.5 mm at a lower grade, 9.7 mm at a lower toughness, and less again on a cold day. A member can cross it without anything happening to it, which is the single most unsettling property of the subject and is why the design temperature matters as much as the design load.

The toughness is a curve, and its axis is the weather

The most uncomfortable part of the calculation is that KcK_c is not a number.

The same steel, brittle in January. Fracture toughness against temperature for a 40 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = -32 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 220 N/mm², which goes as the square of it. At +20 °C this plate carries a 241 mm flaw and at -10 °C it carries 97 mm — a factor of 2.5 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 29% of the flaw a 10 mm plate does. Loading it in a millisecond shifts the whole curve another forty degrees.
Fig. 4 Fracture toughness against temperature for a 40 mm ferritic plate, from Wallin’s master curve, with the tolerable crack length on its own scale beside it. At +20 °C the plate carries a 241 mm flaw and at −10 °C it carries 97 — a factor of 2.5 for a forty-degree change in the weather, on steel that met its specification on both days.

Ferritic steel has a transition: tough above it, brittle below, with the change happening over a few tens of degrees. The design case for a damage-tolerant structure is therefore the coldest night, and the critical crack on that night can be a third of the one on a warm afternoon.

Two other variables move the same curve and both go the wrong way.

The same steel, brittle in January. Fracture toughness against temperature for a 100 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = -15 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 220 N/mm², which goes as the square of it. At +20 °C this plate carries a 142 mm flaw and at -10 °C it carries 60 mm — a factor of 2.4 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 29% of the flaw a 10 mm plate does. Loading it in a millisecond shifts the whole curve another forty degrees.
Fig. 5 The same steel in a 100 mm plate rather than 40. The reference temperature has moved from −32 °C to −15, and the plate tolerates 29 per cent of the flaw a 10 mm plate of the same steel does. Constraint at the crack tip suppresses the yielding that would have blunted it, so a thicker plate is a more brittle one.
The same steel, brittle in January. Fracture toughness against temperature for a 40 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = 8 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 220 N/mm², which goes as the square of it. At +20 °C this plate carries a 71 mm flaw and at -10 °C it carries 33 mm — a factor of 2.1 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 37% of the flaw a 10 mm plate does. And this is the dynamic curve, shifted 40 °C by the loading rate alone.
Fig. 6 And the same plate loaded in a millisecond rather than slowly. The whole curve has shifted forty degrees to the right: the tolerable flaw at +20 °C has fallen from 241 mm to 71, and at −10 °C from 97 to 33. An impact is a different design case in a way no stress calculation shows.

Thickness, temperature and loading rate all shift the same curve in the same direction, and a structure that is thick, cold and impact-loaded — a bridge in winter under a vehicle collision, a crane in a cold store, an offshore jacket — is at the wrong end of all three at once.

What sets the growth rate

The time between the two crack lengths comes from Paris’s law:

dadN=C(ΔK)m\frac{da}{dN} = C(\Delta K)^m

with mm close to three for steel. Since ΔK=YΔσπa\Delta K = Y\Delta\sigma\sqrt{\pi a}, the growth rate goes as a3/2a^{3/2}, which has one dominant consequence.

Almost all of the life is spent while the crack is small. Integrating from a millimetre to 100 mm, the time to reach 10 mm is the great majority of the total, because the rate at 10 mm is already thirty times the rate at 1. A crack that has become visible has used most of its life.

That is why the inspection interval has to be short relative to the total life, and why the sensitivity of the whole scheme is to the smallest detectable crack rather than to the critical one. Halving the detectable size buys much more interval than doubling the critical one.

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.
Fig. 7 The fatigue side for comparison: three detail categories with a common slope of three, and no material anywhere on the plot. At a stress range of 70 N/mm² the lives are 6.8 × 10⁷, 4.3 × 10⁶ and 2.7 × 10⁵ cycles — a factor of 4.4 in stress and 88 in life.

The two calculations share the exponent of three and share nothing else. The S–N line is an empirical summary of initiation plus growth on a test specimen; the Paris integration is growth alone on the member in hand, and a damage-tolerant design uses the second because the first cannot say where a crack has got to.

An interval, worked

Putting numbers on the three quantities makes the scheme concrete, and shows which of them the answer is sensitive to.

Take a welded detail on a bridge girder at a stress range of 70 N/mm² and a mean stress of 120, in 40 mm plate at a design temperature of −10 °C.

The critical crack at 120 N/mm² and 100 MPa√m is 176 mm from the hero figure, and at −10 °C the toughness has fallen so that the tolerable flaw is under 100 mm. Take 90 mm.

The detectable crack by magnetic particle inspection on a dressed weld toe is about 3 mm with good access and perhaps 10 with poor.

The growth time from 3 mm to 90 mm, integrating da/dN=CΔK3da/dN = C\Delta K^3 with typical constants at Δσ=70\Delta\sigma = 70 N/mm², is of the order of a few million cycles — perhaps fifteen years of the traffic in the spectrum figure.

Divide by two for the requirement that a crack be seen at least twice before it becomes critical, and the interval is about seven years. That is a design output, arrived at from a fracture calculation, an inspection method and a traffic count, and it appears on a maintenance schedule rather than on a drawing.

Now change one input at a time. Doubling the detectable crack from 3 mm to 6 mm removes roughly a third of the interval, because so much of the life is spent below 10 mm. Halving the critical crack from 90 mm to 45 removes about a tenth, because almost nothing happens up there. The scheme is sensitive to the inspection and nearly indifferent to the fracture toughness, which is the opposite of where the design effort usually goes.

What the spectrum does to it

The load is not one range repeated, and the way a spectrum is reduced to one number matters more here than in the fatigue check.

Two per cent of the traffic and most of the damage. A 100-year traffic spectrum on one detail of category 90, 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. The a full train band is 2% of the crossings and 66% of the damage; the smallest band is 35% of the crossings and, being under the cut-off, does none 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.
Fig. 8 A hundred-year traffic spectrum on one detail, 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: the full-train band is 2 per cent of the crossings and 66 per cent of the damage, and the smallest band is 35 per cent of the crossings and does none at all. The equivalent constant range is 25.5 N/mm², a cube-weighted average nearer the heaviest vehicle than the average one.

For crack growth the same weighting applies, so the interval is set by the heaviest few per cent of the traffic — which is both good and bad news. Good, because a small number of vehicles is easier to count than all of them. Bad, because a change in the heavy end of the spectrum that barely moves the total traffic can halve the inspection interval, and the heavy end is the part that changes when a route’s use changes.

The member that is not allowed to be inspected

There is a category of structure where the whole scheme is unavailable, and naming it is the useful boundary.

A fracture-critical member is one whose failure would collapse the structure and which has no alternative load path. Bridge codes name them explicitly — a two-girder deck’s girders, a tie in a tied arch, a hanger in a suspended span — and require them to be fabricated to a higher standard, made of a tougher steel, and inspected more often.

What makes the category necessary is that damage tolerance needs three things and such a member may have none of them: access to the detail, a method that can find a small crack there, and time between detection and failure. An internal weld in a closed box has no access; a detail in a joint has no clean surface; and a member with a stress range near the critical crack’s stress has no time.

For those the design reverts to the first promise — it will not crack — and buys it by lowering the stress range until the fatigue life is several times the design life. That is expensive and it is the honest response, and it is why a fracture-critical member is heavier than its strength requires by an amount that has nothing to do with strength.

The alternative is to provide the alternative load path instead, which removes the member from the category entirely. A three-girder deck, a network of hangers, a redundant tie: each turns a fracture-critical member into an ordinary one, and the cost is usually less than the fabrication premium.

What to carry away

Damage tolerance replaces “it will not crack” with “it will be found first”, and the second is an inspection interval rather than a stress check.

Both ends of the interval come from fracture, not fatigue. The detectable crack is a property of the inspection method and the critical crack of the toughness and the stress.

The critical crack goes as the square of the toughness, and the toughness is a curve whose axes are temperature, thickness and loading rate — all three pointing the same way.

And a stronger steel shortens the critical crack. Yield strength and toughness are independent properties, and raising one while raising the working stress moves a member from a yielding failure to a fracture one.

Where the model stops

The crack is a through-thickness edge crack in an infinite plate. Y is 1.12 for that case; real cracks are semi-elliptical surface flaws at weld toes, and the geometry factor for those comes from a handbook rather than from a formula.

Linear elastic fracture mechanics is assumed. At service temperatures structural steel yields substantially at a crack tip, and the correct treatment is elastic-plastic — CTOD or the J-integral — which gives a larger tolerable flaw and needs a different test.

Charpy is correlated with toughness, not convertible to it. The master curve’s reference temperature is estimated from Charpy data through a fitted relation with real scatter, and the whole of this arithmetic sits on top of that estimate.

Nothing here counts residual stress. A weld toe carries residual tension at yield, so the mean stress at the crack is far higher than the applied one — which does not change the fatigue calculation and does change the fracture one.

The inspection is assumed to be reliable. It is not: detection is a probability rather than a threshold, and a “detectable” crack size is a size at which detection is likely rather than certain. A proper treatment carries a probability-of-detection curve and computes a risk rather than an interval.

And the growth law’s constants are for one material in one environment. Sea water, corrosion or a stress ratio near one each change C by a factor that no interval derived here contains.

What it changes about a drawing

The scheme reaches back into the design in three specific ways, and each is a decision made years before the first inspection.

Access. A detail that cannot be reached cannot be inspected, so a damage-tolerant design has to leave room for a person and an instrument at every fatigue-critical location. That is a geometry decision competing with every other, and it is usually lost.

Redundancy. A structure that survives losing a member has time between a crack and a collapse, and one without it has none. The interval calculation assumes the member is still standing while the crack grows, which is true of a girder and untrue of a tie.

And the specification. The steel’s toughness has to be ordered, since it is not implied by the grade. A subgrade specifying impact energy at a stated temperature is what puts a number under the master curve, and it is a line on a material schedule that costs money and is easy to omit.

All three are cheap at design and impossible afterwards, which is the standing argument for deciding at the outset whether a structure is going to be inspected or is going to be designed never to need it.

Three neighbours decide whether an inspection interval means anything at all. Which cycles in a history actually count sets how much damage the structure is accumulating; the flaw that sets the strength sets how large a crack has to be before it matters; and the same steel, brittle in January decides whether the structure will give a crack to find or a fracture with no warning. And the detail decides the category, not the steel, so the interval is set by a weld’s geometry rather than by anything in the specification.

The ladder from here

Later rungs on this anchor: Paris’s law integrated properly from an initial flaw to a critical one, to produce a life rather than an interval. The geometry factor Y for real configurations — edge cracks, surface cracks, cracks at holes — and how a handbook of them is compiled. Elastic-plastic fracture mechanics with CTOD and the J-integral, which is what applies to structural steel at service temperatures. Rainflow counting, which is the algorithm that turns a measured history into the spectrum this page assumed. Improvement techniques and what each is worth. And leak-before-break, which is damage tolerance designed into a geometry rather than into a schedule.

Damage tolerance came from aviation rather than from civil engineering, and it arrived after the Comet: a fuselage designed for infinite life, cracking from a corner of a window, at a detail whose category nobody had. What the aviation industry concluded — that a structure will crack and the useful question is whether it will be found — took another twenty years to reach bridges, and reached them through the fracture-critical member, which is the same idea with the inspection schedule written into the design.

Where this goes next

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. 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. The same steel, brittle in January. Fracture toughness against temperature for a 25 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = -60 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 200 N/mm², which goes as the square of it. At +20 °C this plate carries a 764 mm flaw and at -20 °C it carries 199 mm — a factor of 3.8 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 27% of the flaw a 10 mm plate does. Loading it in a millisecond shifts the whole curve another forty degrees. The same steel, brittle in January Every other material property in this collection is a number. Toughness is a curve, and the axis it runs along is temperature. Between twenty degrees and minus twenty a structural steel does not get gradually weaker — it changes the mechanism by which it fails, and the flaw it will tolerate falls by a factor of four. 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.

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ConstraintCrack growthDamage toleranceDetail categoryFatigueFractureFracture toughnessFree bodyInspectionStress intensityStress rangeTransition temperature