Structural Engineering & Statics
where the load goes
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Scale changes everything — page 5

Essays 97 to 120 of 164 on this thread, in the same order.
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. Materials

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.

 The fire is one curve, and the steel in it is several. Gas temperature and steel temperature against time in a standard fire. The gas curve is the same for every member in the compartment; the three bare steel curves are section factors of 76, 160 and 323 per metre, and they reach 558 °C at 17, 11, 8 minutes — a spread of a factor of two from geometry alone. The fourth curve is the middle section with 15 mm of board on it, which reaches the same temperature at 46 minutes. The kink near 735 °C on the bare curves is not a numerical artefact: steel's specific heat spikes there as its crystal structure changes, and the member spends several minutes absorbing heat at almost constant temperature. Materials

The temperature is a shape

Two members of the same steel in the same compartment, under the same fire and the same load ratio, fail eight minutes apart. Nothing about the material differs and nothing about the fire does. What differs is a perimeter divided by an area, and it is the only number in the whole calculation that a designer chooses.

 The same bridge by the two theories it might have been designed by. Deck moment along a 500 m suspended span with half of it loaded, computed twice. Elastic theory treats the deck as a beam and applies the cable's extra tension as an upward load: 80.1 MN·m. Deflection theory keeps the cable's total tension acting on the deck's own deflected shape — a geometric stiffness — and returns 56.3, which is 30 per cent less. The extra cable tension is very nearly the same in both (6.20 against 6.11 MN), so nothing about the cable is in the difference: it is entirely the H·v″ term the older theory drops. Structural form

The tension that was left out

A suspension bridge's deck sits on a cable pulling hard along it, and a member with a large tension in it is stiffened by that tension. Leaving the term out of the deck's own equilibrium is what elastic theory does, and on a long span it asks for fourteen times the girder.

 What the second, third and fourth arms are worth. Top drift removed against the number of outriggers, each arrangement at its own optimum levels, on a 40-storey core 200 m tall. One arm at 59 per cent of the height removes 82.3 per cent of the drift. A second, with both moved to 35 and 71, takes it to 91.6 — a gain of 9.3 points, which is half of what was left. The third is worth 3.0 and the fourth 1.5, and each one costs a storey of the building's most valuable height. Structural form

What the second arm is worth

One outrigger at its best height removes five sixths of a tall core's drift, which sounds like the end of the argument. A second removes half of what is left, a third half of that, and each of them costs a storey of the most valuable floor area in the building — so the question is not where to put an outrigger but how many the arithmetic still justifies.

 The deflection a building can take, and which way it is bending. The limiting deflection ratio — the sag or hog of a building divided by its length — at a critical tensile strain of 0.075 per cent, against how long the building is compared with its height. Two curves, and the gap between them is the whole finding: hogging is worse than sagging by exactly 2.0 times, because the neutral axis of a hogging building sits near the bottom and the tension face is the full height of the wall. The minimum is at L/H = 1.6, where the two mechanisms cross: a squat building cracks diagonally in shear and a long low one cracks in bending at the extreme fibre. Deflection

The crack is a strain, not a slope

Angular distortion is what every table of settlement limits is written in, and it is a proxy. What cracks a building is a tensile strain in it, and reading the building as a deep beam says which of two mechanisms supplies it, why hogging is exactly twice as damaging as sagging, and why a horizontal stretch of a few millimetres takes half the capacity away before the settlement starts.

 Half of the creep is the member drying out. The creep coefficient of a concrete of mean strength 38 N/mm² loaded at 28 days, in a member of notional size 150 mm, against time since loading: exposed to air at 50 per cent relative humidity, and the same member sealed so that it cannot dry. After a year the exposed member's coefficient is 1.93 and the sealed member's 0.81; after 50 years they are 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member's final creep, is what drying adds. The sealed member's coefficient has no size in it; the drying part is where the member comes in. Materials

The creep that belongs to the member

A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether.

 The humidity through a slab's depth, and the strain it asks for. A slab 200 mm thick drying through its top face only into air at 50 per cent, with its underside sealed by a steel deck. Left, the pore humidity against depth at 28 days — 50 per cent at the surface and 100 at the base; 90 days — 50 per cent at the surface and 100 at the base; 1 year — 50 per cent at the surface and 99 at the base; 5 years — 50 per cent at the surface and 74 at the base; 20 years — 50 per cent at the surface and 51 at the base. Right, the free strain each of those profiles asks for, taken as shrinkage at the local humidity. At 28 days the top wants to be 461 microstrain shorter than it was and the bottom 0; at 20 years the two are 461 and 456, and the gradient that produced the curl has gone. Not one of the profiles is straight, and a section that stays plane cannot deliver any of them. Materials

The slab that dries from one face

A creep coefficient is one number for a member, and a member drying through one face does not have one. Give every depth its own humidity and the section a strain profile it cannot deliver, and two things follow that no single coefficient contains — a six-metre slab lifts eleven millimetres at its edges with nothing on it, and its top surface is past cracking before it has been loaded.

 Where a tendon's force goes, over fifty years. The loss of stress in a tendon stressed to 1300 N/mm² and released at 7 days into a member of notional size 300 mm at 70 per cent humidity, with the three causes stacked. At 28 days the total is 61.9 N/mm²; at a year 131.5; at fifty years 190.8, which is 14.7 per cent of what the tendon started with. Creep supplies 107.7 of that, drying and autogenous shrinkage 50.6, and the steel's own relaxation 32.5. Half the loss has happened by 119 d and nine-tenths by 8 y. Materials

The prestress the member takes back

A tendon is stretched, locked off against the concrete, and then has to hold that extension while the concrete underneath it shortens by itself. Fifteen per cent of the force goes, most of it to creep, and how much goes is decided by the shape of the member and the air it stands in rather than by anything about the steel.

 The coefficient is an envelope, and its scallops are whole half-waves. The plate buckling coefficient against aspect ratio α = a/b. Each faint branch is one half-wave count m: k = (m/α + α/m)², a curve whose own minimum is exactly 4 at α = m. The plate takes whichever branch is lowest, so the answer is the bold envelope — four touching 4 at α = 1, 2, 3 and 4, with cusps between them at α = √(m(m+1)) = 1.41, 2.45, 3.46, 4.47, where the plate is indifferent between m and m+1 half-waves. The first cusp reaches k = 4.50 and every later one is lower — 4.17, 4.08, 4.05. Past α = 1 the envelope never exceeds 4.49, which is why the length of a plate drops out of a formula that is otherwise entirely geometry. Stability

Four was never a fact about plates

The coefficient every plate calculation starts from is quoted as 4, derived nowhere and remembered by everyone. It is the minimum of a quantity that has nothing to do with plates in it — and what the plate supplies is not the four but the restriction that produces the scallops around it.

 The coefficient a web gets depends on where its neutral axis is. The plate buckling coefficient for an internal element against ψ = σ₂/σ₁, the ratio of the stresses at the two edges of the panel. Uniform compression is ψ = 1 and k = 4; a gradient running from compression to zero is ψ = 0 and k = 7.81; pure bending is ψ = −1 and k = 23.92, six times the value a column's flange gets. The 1800 × 12 mm web drawn here starts at ψ = −1.000 and k = 23.92 and ends at ψ = −0.910 and k = 21.63, because losing width from the compressed half drops the neutral axis and deepens the compression zone. The curve is steepest exactly where a bending web sits, so a small movement of the neutral axis costs 2.29 of coefficient. Stability

Classified by a gradient it does not have

A web in bending is the one plate whose buckling coefficient cannot be looked up. It depends on the stress gradient, the gradient depends on where the neutral axis is, and the neutral axis depends on how much of the web the coefficient has just taken away — so the answer is a fixed point, and the calculation everyone does is its first term.

 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. Materials

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.

 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. Materials

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.

 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. Materials

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.

 Two pencil lines, and where they cross. Two lines drawn with a pencil 0.2 mm wide are two bands, drawn here much wider than a pencil so the shape can be seen, and they cross not at a point but in a parallelogram. At 60° apart the parallelogram's long diagonal is 2.0 pencil widths — 0.40 mm; at 12° apart the parallelogram's long diagonal is 9.6 pencil widths — 1.91 mm. The crossing's uncertainty along the bisector is the width divided by twice the sine of half the angle, so it grows without limit as the lines turn parallel, and a construction that finds a point by crossing two lines inherits it. Equilibrium

How wrong a drawing is

A pencil line is a band, and two bands cross in a parallelogram that grows as they turn parallel. The accuracy of a graphical construction is therefore a property of the angles it makes, not of the hand that made it — and the worst case is the shallow arch, the structure the method was most used on. Measured properly, the drawing's error there is the size of the builder's, and the check draughtsmen relied on cannot see it.

 The vertical response on layers of different depth. The vertical amplitude of a 150 tonne block 5.0 m across on soil with a shear wave speed of 178 m/s, under a harmonic force of fixed size, as a multiple of its static deflection on a half-space, against frequency. On a half-space the mode is at 12.5 Hz and the peak is 1.10. On 10.0 m the mode is at 14.3 Hz against a cut-off of 7.4 and the peak is 1.03. On 6.0 m the mode is at 15.5 Hz against a cut-off of 12.3 and the peak is 1.74. On 4.5 m the mode is at 16.3 Hz against a cut-off of 16.4 and the peak is 5.84. The layer stiffens the footing and lifts the frequency a little; what changes the picture is the cut-off, which rises as the layer thins and, once it passes the mode, takes the radiation damping away and leaves only the soil's own 5 per cent. Radiation below the cut-off is taken as nothing and its rise above it as linear, which is the usual idealisation of the exact layered solution. Dynamics

The rock that sends the waves back

A machine block on the ground is damped by the waves it launches, and a half-space lets them all escape. Put rock four and a half metres down and none of them can: below the soil layer's own lowest frequency there is no wave that travels, and a vertical mode damped at 47 per cent of critical keeps the soil's own 5. Its resonance grows fivefold, and the heavy block the textbook rule recommends is the one that loses its damping on the deepest ground.

 Thirty years later, two concretes against one. Stress down the composite section after thirty years — a 160 mm slab cast 6 weeks after a 700 mm pretensioned beam — computed twice: with the slab as a second, younger concrete that creeps and shrinks by its own laws, and with it given the beam's concrete and age. With two concretes the slab ends at −0.15 N/mm² at its top and −0.30 at its bottom, the beam at −4.03 at its top and −7.43 at its soffit. With one, the slab is at −0.29 and −1.06, the beam at −2.22 and −8.37. The slab's own shrinkage has taken its compression away and handed it to the top of the beam, and the soffit — the fibre the prestress was designed to keep in compression — has lost 0.94 N/mm² of it. Compression is negative. Materials

The slab that shrinks onto a finished beam

A precast beam with an in-situ slab cast on it is one member made of two concretes, and they do not age together. The slab's shrinkage is nearly the same whenever it is poured; what changes is how much shrinking the beam has left to share it with. Cast the slab at six weeks and it takes a ninth of the soffit's precompression away over thirty years. Cast it at a year and it takes a quarter, and goes into tension itself.

 The bow the frames have to hold. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN critical at 1548 kN, drawn with its sideways movement exaggerated; its critical load is 1.66 times the chord force. Dashed, the worst of the bows tried: four half-waves, each 12.0 mm high, a five-hundredth of its own length. Solid, the chord once the compression has grown the bow; each arrow is the force the frame there must supply, the largest 13.6 kN — 1.47 per cent of the chord force. Stability

What the frames hold is the bow

A row of U-frames is sized for a force, and the rules give that force as a percentage of the compression chord's force that nobody derives. Derived, it is not a property of the frames at all. It is the chord's initial bow, grown by how near the chord works to buckling — and it is least, not greatest, near the stiffness at which the chord stops using the frames. Stiff frames tend to a limit that is π²/500 of the chord force: the familiar two per cent, arriving from a bow of one five-hundredth.

 The same bridge, anchored to the ground and to itself. The deck's bending moment under the live load, for a 500 m suspension span with a 50 m sag, a deck of EI 1.03 × 10¹² N·m², 100 kN/m of dead load and 20 kN/m of live load on half the span. Anchored to the ground, the deck carries at most 56.3 MN·m, because the cable's whole tension works on its deflected shape. Anchored to the deck's own ends, 83.0 MN·m — 1.47 times as much, and a little more than Rankine's elastic theory, 80.1, which leaves the tension out altogether. The deck's compression acts on the same deflected shape as the cable's tension, with the opposite sign, and cancels it. Structural form

The bridge that pays for its own anchorage

A suspension bridge's deck is light because the cable's tension works on its deflected shape and stiffens it. Tie the cable to the ends of the deck instead of to the ground and the deck must carry the same pull as a compression — which acts on the same deflected shape with the opposite sign and cancels the stiffening exactly. A self-anchored bridge is designed by the theory the long suspension bridge was invented to escape, and the price grows as the square of the span.

 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. Materials

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.

 Weakest the day it is finished. The factor of safety against tipping of a 15,000 kN tower on a 10 m square base over soft clay (undrained strength 35 kPa, gaining 0.22 of the effective stress the tower puts into it, consolidating with a 90 per cent time of 7.1 years), through twelve years, for construction times of three months, one year and four years; its wind push grows as it rises. Built in three months, it bottoms out at 1.14 on the day it is finished; built in one year, it bottoms out at 1.33 on the day it is finished; built in four years, it bottoms out at 1.60 on the day it is finished. After that every curve climbs toward 1.93, the factor once the clay has consolidated. Had the finished weight arrived in an instant the factor would be 0.92; on ground that could not fail it would be 3.00. Equilibrium

The ground that arrives after the weight

A block's resistance to tipping on ground that can yield is a parabola in its weight, with its peak at half of what the ground can bear. On soft clay the ground's capacity is not fixed: it grows as the clay drains under the weight, and it grows years after the weight has arrived. A tower on such ground is at its weakest on the day it is finished — 1.33 against the 1.93 it will have once the clay has drained — and a silo filled in a week walks over the top of the parabola and down the far side, which is the other way a structure can be built.

 One steel, one crack, three plates. The strip-yield failure stress of plates with an edge crack, at −20 °C in a grade 355 steel whose master-curve reference temperature is −38 °C, against the crack length on a logarithmic scale, for plates 12, 25 and 60 mm thick, each with the toughness of its own crack front at the 5 per cent level: 88, 77 and 66 MPa√m. At a 20 mm crack the 12 mm plate fails at 267 N/mm², the 25 mm plate at 242 and the 60 mm plate at 214. Dashed, the simple check with the 25 mm toughness, the lower of fracture and yield: 274 N/mm² at the same crack, for every thickness. Materials

The toughness that belongs to the plate

A steel's cleavage toughness is measured on a specimen 25 mm thick, and a crack through a plate starts at the weakest spot its front passes through — so a crack through a 60 mm flange samples more weak spots than the test did, and is less tough, and one through a 12 mm web samples fewer. Put each plate's own toughness into the strip-yield assessment and the simple check, which overstates a cracked plate by 23 per cent at the specimen's thickness, overstates a 60 mm flange by 36 per cent and a 100 mm plate by 45, while a thin web pays much of the strip-yield debt back.

 The film over a keyed shaft, and the corner it climbs. Contours of Prandtl's stress function — the height of a soap film blown up over a hole of the section's shape — for a 50 mm shaft with a keyway 14 mm wide and 5.5 mm deep, its bottom corners rounded to 0.40 mm; the shear stress is the slope of the film, so it is highest where the contours crowd. Left, the whole shaft: the contours follow the circle and bend round the keyway, and the keyway costs 11 per cent of the plain shaft's torsional stiffness. Right, a window 3.0 mm across around the keyway's corner, solved on its own fine grid with its edges taken from the whole film: the contours crowd into the fillet, where the stress is 2.81 times the plain shaft's surface stress under the same torque. Sections and stress

The corner a soap film cannot finish

Cut a keyway into a round shaft and its torsional stiffness falls by only 11 per cent. The stress does not fall; it moves. It leaves the surface beside the keyway's mouth almost unloaded and crowds into the keyway's bottom corners, where a soap film stretched over the section would have to stand vertical. Computed on finer and finer grids, a sharp corner's stress never settles — it grows by the cube root of two every time the grid is halved — and a fillet of a few tenths of a millimetre is what turns an infinite answer into three times the plain shaft's.

 A soft layer over stiff ground, and a moment neither has. The bending moment down a 20 m pile of bending stiffness 120 MN·m² with a free head, pushed sideways by 150 kN (its characteristic length in ground at 5.00 MN/m³ being 1.89 m), in three grounds: all of it at 5.00 MN/m³; all of it at 0.50 MN/m³; and soft ground 2.0 characteristic lengths (3.8 m) thick over the other (dotted: the interface). The largest moments are 219 kN·m, 346 kN·m and 427 kN·m; the head deflections 20.4 mm, 81.4 mm and 61.6 mm. The layered ground gives a larger moment than either ground on its own: the pile bends as a pile in the soft layer would, and then meets ground that holds it. Internal forces

The layer a pile feels

A pile pushed sideways in uniform ground has one characteristic length, the fifth root of its stiffness over the ground's, and every answer is a multiple of it. Put a soft layer over stiff ground and that length stops existing. The head feels the top of the ground in proportion to the square of its own deflection, so an average weighted that way gets the head deflection within eight per cent. It gets the moment a quarter too low: with soft ground two characteristic lengths thick over stiff, the pile bends harder than it would in either ground alone, and the peak sits where the stiff ground takes hold.

 A column curve with no plateau. The tangent-modulus buckling stress of perfect pin-ended columns of an aluminium alloy of 0.2 per cent proof stress 250 N/mm² and modulus 70,000, as a share of the proof stress, against the slenderness λ̄ — the square root of the proof stress over the Euler stress — for Ramberg–Osgood exponents of 5, 10, 20 and 40; dashed, the sharp envelope a material with a plateau at the proof stress would give, the lesser of the proof stress and Euler's. n = 5: 1.54 at λ̄ = 0.2, 0.66 at 1, 0.41 at 1.5; n = 10: 1.16 at λ̄ = 0.2, 0.74 at 1, 0.45 at 1.5; n = 20: 1.04 at λ̄ = 0.2, 0.82 at 1, 0.45 at 1.5; n = 40: 1.00 at λ̄ = 0.2, 0.88 at 1, 0.45 at 1.5. Stocky columns buckle above the proof stress, because the curve goes on rising past it; slender ones follow Euler; in between every curve sags below the envelope. Stability

The column with no plateau

A steel column's curve is the lesser of two lines, its yield stress and Euler's, because mild steel keeps its whole stiffness up to yield and then has a plateau to stop on. Aluminium and stainless steel have no plateau: their stiffness starts falling well below the proof stress and never stops. Their column curves are not the steel curve moved; they are a different shape — four tenths below the corner for a stainless steel, above the proof stress for a stocky column — and the shape is set by two numbers, only one of which describes the knee.

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