Structural Engineering & Statics
where the load goes
Concept

Stress concentration — where it appears

The local multiplication of stress at a hole, notch or re-entrant corner, decided by the shape of the discontinuity and not by its size. The factor is a ratio of the feature's depth to its tip radius, so a long sharp notch multiplies the stress and a large round hole barely does — which is why the fix for a crack is to drill its tip.

Named by 16 essays across 4 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 Three times the stress, and it does not matter how big the hole is. The hoop stress around a circular hole in a wide plate pulled at 100 N/mm², from Kirsch's exact solution. At the sides of the hole it is 3.0 times the applied stress — 300 N/mm² — and the factor is the same for a hole of any radius, because the radius cancels. At the top and bottom of the hole it is -1.0 times the applied stress, which is compression in a plate that nothing is pushing. The disturbance dies quickly: the stress is within 5% of the applied value by 3.5 hole radii, which is Saint-Venant's principle with a number on it.

The hole that multiplies the stress by three

The stress at the side of a hole is three times the applied stress whatever the hole's size, and at the top and bottom of the same hole it is minus one times it — compression in a plate that nothing is pushing.

materials · Stress concentration 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 Cut the throat, and the face carries a moment and a tension at once. A crane hook of trapezoid section, 50 to 120 mm radius and 40 to 15 mm wide, carrying 50 kN, drawn beside the section at the cut and the stress across it. The load hangs on a line through the centre of curvature, so cutting the throat and taking everything below the cut as the free body leaves a face carrying a direct tension of 50 kN and a moment of N·R = 3.985 kN·m about the section's own centroid, which sits a full R = 79.70 mm from the load line. The stress is a hyperbola, zero at r = 75.04 mm rather than at the centroid 4.65 mm outside it, reaching 248.7 N/mm² of tension at the inner fibre and 140.6 of compression at the outer. The straight-beam formula, drawn dashed, reports 161.7 N/mm² for the bending part against the true 222.8, and leaves the 26.0 N/mm² of direct tension out altogether — between them, 54% under the real peak, at the fibre where a hook actually breaks. This is why a hook is trapezoidal: both effects are worst inside, so the material goes there.

The bar that was bent before it was loaded

In a curved bar plane sections still stay plane, and the bending formula is wrong anyway. The fibres were different lengths before anything was applied, so an equal rotation of two plane faces produces unequal strain — the stress is a hyperbola, the neutral axis has moved inward, and a crane hook carries half as much again as My/I reports.

sections · Curved beam A truss drawn inside a solid, and solved as one. A deep member 4000 mm between bearings and 2000 mm deep, carrying 1200 kN at mid-span. The model is two struts and one tie, on a lever arm of 1600 mm, and it is solved by the truss solver rather than by a formula: the tie comes back at 750 kN and each strut at 960 kN, at 38.7° to the horizontal. Spread over a strut width of 812 mm the compression is 3.0 N/mm² against a limit of 15.8 for concrete cracked across its own strut, and the tie needs 1724 mm² of steel. A beam calculation on the same member would have asked the tie for 702 kN, which is 7% less than the model does.

When there is no section to design

Beam theory needs a section, and a section needs the strain to be linear across it. Within about a depth of a support, a load, a corner or a hole it is not — and those are the regions structures actually fail in.

internal-forces · Strut-and-tie One point, every plane through it, one circle. A point carrying 140 N/mm² across one face, 0 across the other and 45 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 83.2 centred at 70.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 153.2 and -13.2, on planes 16.4° from the face the 140 acts on; the largest shear on any plane is 83.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 160.2.

The worst stress is not where the worst bending is

Every stress this collection has quoted is a stress on a particular plane, and neither the bending stress nor the shear stress is a property of the point. Turn the plane and both change; one pair of numbers does not, and on a short beam it peaks where neither of them does.

sections · Principal stress A hole in a web is a Vierendeel panel. A 400 × 300 rectangular opening in a 533 deep beam, 15% along a 9 m span carrying 20 per metre — where the moment is 103 kNm and the shear 63 kN. The moment is a couple on the two tees, 301 kN on a lever arm of 343 mm, which is 70.3 N/mm² of uniform stress. The shear has nowhere to go but through the tees, so each carries 32 kN over the opening and bends in double curvature: a Vierendeel moment of 6.3 kNm and 167.6 N/mm² on top. So 70% of the stress at the corner exists because the hole has a LENGTH, and only 30% of the section's second moment has gone.

The hole that costs nothing, and everything

A service opening removes 30% of a beam's second moment and 0.7% of its deflection. What it costs is not that. Across the opening the shear has nowhere to go but through the two tees, and a tee carrying shear over a length bends.

sections · Web opening Three ways to apply the same force, and one depth to forget the difference. Three end loads on a member 400 mm deep, all with the same resultant and the same moment: a point load, the same force spread over a fifth of the depth, and the same force split in two. What is plotted is the difference between each of them and the beam-theory answer — the self-equilibrating remainder — as a fraction of the mean stress. The point load starts at 20 times it and is under a tenth of it by 0.77 depths; all three are under one per cent by about 1.18. That distance is the licence every figure in this collection is drawn under, and the exact strip eigenvalue agrees with it: 2.106 + 1.125i, whose real part puts one per cent at 1.09 depths and whose imaginary part means the remainder changes sign on the way out, which no statement of the principle mentions.

How far a wrong load reaches

Every figure in this collection applies a load as a point, a line or a uniform pressure, and no real load is any of those. The licence is Saint-Venant's, it is usually quoted as a principle, and it is really a statement about a wavelength.

internal-forces · Saint-Venant's principle The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 200 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 659 mm — 3.3 times the bearing, and 70% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to.

The support that is not a point

A reaction is drawn as a single arrow because the equilibrium equations only need its total. Underneath the arrow is a bearing of some width, delivering a pressure over that width, and almost everything a designer would like to know about the region near a support is a consequence of the width the arrow does not have.

internal-forces · Support width 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 edge, and the length over which it is forgotten. A cylinder of radius 4.00 m and wall 12 mm under 0.6 N/mm² of internal pressure, held at its base. Away from the base the wall carries the pressure as pure hoop tension and bends nowhere, which is why a pressure vessel is a cylinder. At the base the hoop force is zero, because the wall cannot grow there, and the difference is made up by a boundary layer of bending that dies out inward. The length it dies out over is 1/β = 170 mm — 0.778√(Rt), a geometric mean of the radius and the thickness — and the moment is under a twentieth of its edge value by 3.07 of them. Nothing in that length is the load. The base moment is p/2β², and the bending stress it produces is 1.82 times the membrane hoop stress the whole design is about, at every pressure, every radius and every thickness: the ratio is √3/√(1 − ν²) and contains none of them. The hoop force overshoots by 4.3% at 3.2 lengths in, which is the wall springing back past where it was going.

The length a structure was never given

A disturbance applied at one place dies out over a distance, and the distance is not something anybody chose. A beam forgets a badly applied load over its own depth. A beam on the ground forgets a point load over the fourth root of its stiffness against the soil's. A shell forgets a held edge over the square root of the radius times the thickness — a geometric mean of two lengths three orders of magnitude apart, which is neither of them and is not near either.

internal-forces · Edge disturbance 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.

materials · Transition temperature The load path has a kink in it, and the kink is a plate thick. Two 10 mm plates lapped over 60 mm and pulled with 60 kN. The two load paths are offset by the thickness of a plate, so the joint carries a moment nobody applied: P × 10.0 mm / 2. Taken at face value that gives a peak stress 4.00 times the mean. The joint rotates under load and the moment falls to 86 per cent of it, leaving 3.57 times — a saving of 11 per cent and not, on a plate this thick, a rescue. The bolt is bent as well as sheared: 382 N/mm² of bending against 191 of shear.

The joint that is crooked by construction

Lap two plates and fasten them and the two load paths are offset by the thickness of a plate. The joint carries a moment nobody applied, the peak stress is four times the mean, and the rotation that is supposed to straighten it out saves eleven per cent — because the rescue works for thin sheet with a long lap and a bolted structural joint is neither.

connections · Single lap The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 120 mm to frame into a girder. What is left is a tee with a section modulus of 3.836e+5 mm³ against the whole section's 1.438e+6 — 27 per cent. The moment at the end of the cope is the reaction on a lever arm of 130 mm: 23.4 kNm, giving 61 N/mm² and a flexural utilisation of 0.17. The web now has a free edge along the cope, so its buckling coefficient collapses from 4 to 0.425 — a factor of 9.4 — and the re-entrant corner has a stress concentration of 5.5 on a 10 mm radius.

The section that is checked is not the one chosen

A beam framing into a girder has its top flange cut away so the two can sit at the same level. What is left is a tee with a quarter of the section modulus, a web with a free edge, and a re-entrant corner — and the beam was selected on a table entry that describes none of it.

connections · Coped beam 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 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

Named alongside it

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

FatigueLoad pathCrack growthDetail categoryPlane sectionsSaint-Venant's principleDisturbed regionDuctilityFractureFree bodyResidual stressStress range

All concepts