Structural Engineering & Statics
where the load goes
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Which failure arrives first — page 20

Essays 457 to 480 of 495 on this thread, in the same order.
Counted honestly, cores from one bay hand the estimate to the specification. The share of the structure's concrete that falls below the 5 per cent characteristic value estimated from three cores from one bay of a structure's concrete of mean 33.0 N/mm² and coefficient of variation 12 per cent, updating the specification — a mean of 38 N/mm² and a coefficient of variation of 15 per cent, carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, averaged over 2,000 drillings, against the correlation between two cores from one bay: the cores counted as independent (dashed), counted by their correlation (solid), and three cores drilled from three bays (light); dotted, the 5 per cent an honest estimate gives. At ρ = 0.5, 6.2% and 6.5%, and 4.8%; at 0.9, 7.4% and 7.3%, and 4.8%. Materials

The cores that agree for the wrong reason

Three cores drilled a metre apart in one slab share a pour, a curing and a morning's water, so they agree with each other more closely than they agree with the rest of the building. Counted as three independent specimens, they overstate what is known — and with nothing else to go on, the estimate built on them leaves a quarter of the building's concrete below a value meant to leave a twentieth. Counting them by their correlation repairs that. But once the specification is in the update, the honest count hands the estimate back to the specification, and a concrete weaker than it was meant to be hides behind it. The repair is not in the arithmetic. It is in where the cores are drilled.

 Closing late beats jacking again on the same date. For the 60 m concrete rib rising 6 m under 60 kN/m, struck at 28 days, on ground that spreads 8 mm at once and as much again as it consolidates, 90 per cent in 10 years: the springing moment after a century, against a date in the arch's first years — left three-hinged and closed on that date (solid), or fixed at striking and jacked back to no moment on that date (dashed); dotted, fixed and jacked at striking only, −173. At three months, −124 against −140; at a year, −84 against −117; at ten years, −15 against −76. The difference is the memory of the years before: the jacked arch's early restraint goes on relaxing after the second jacking, and the open arch had none. Deflection

The arch that waits with its hinges open

A concrete arch fixed on the day it is struck has to restrain every millimetre its rib creeps and shrinks and its ground settles for the rest of its life, and jacking it again a year later cannot make it forget the year before. Struck as a three-hinged arch and left that way, it takes the fastest part of every movement for nothing, because a determinate arch has no moment to spend on a change of length. Closed at a year it ends the century with half the springing moment of the arch fixed at striking, and a third less than the arch jacked again on the same date. The price is a crown that has dropped 78 mm while it waited, and a rib that is no longer the shape of its own thrust.

 Pumping water in or out keeps the damper, and the lost water barely matters. The peak amplification of a building with 1 per cent damping and a 10 m tank sized for a 5.0 s building, 1.73 m of water weighing 2.0 per cent of the modal mass, its sloshing damped at 7 per cent, against the period the building actually has: the tank left at its design depth (dashed), retuned by its water level with the water's mass following the depth (solid), and retuned with the mass held at the design value (light); dotted, the design period; bare, the building would reach 50. At 4.5 s, 21.7 left alone and 9.6 retuned; at 5.5 s, 22.6 and 10.8; at 6.0 s, 34.0 and 11.7, of which the water let out costs 1.7. Dynamics

The tank that follows the building

A sloshing damper is tuned to a period computed before the building exists, and the building that is built has another one — stiffer at handover, with its partitions and cladding working, softer once its concrete has cracked. A tank whose plan is fixed and whose period is set by its depth can be brought back by opening a valve. Ten per cent off tune, the tank left alone lets the building's peak response more than double; pumped back to the right depth, it holds within a tenth of the design value, and the water it gains or loses barely matters. What does matter is that a slower building needs shallower water, and in shallow water the wave the damper is made of grows towards the depth it is sloshing in.

 One alarm in a few hundred that empties the building sets the column's load. The factor on the summed characteristic loads that a column carrying 20 floors needs, for office load and a corridor crowd at one alarm a year, evacuated three floors at a time, against the share of alarms that escalate to every floor at once (logarithmic), with 25% and 50% of the tributary area corridor; dotted, each with every alarm a full evacuation; dashed, the storey rule's 0.73. 25% corridor: 0.26 with none escalating, 0.27 at one in five hundred, 0.33 at one in a hundred, 0.38 at one in twenty, against 0.49 in full every time; 50% corridor: 0.29 with none escalating, 0.39 at one in five hundred, 0.59 at one in a hundred, 0.75 at one in twenty, against 0.90 in full every time. Equilibrium

The stair that empties three floors at a time

A column beside an escape stair carries the corridor crowd of every evacuation, and when every floor empties at once its floors are loaded together and earn almost no reduction. Tall buildings are usually evacuated in phases instead, three floors at a time, so that the stair is never asked to carry everyone. Under such a stair the column earns its reduction back and more — 0.29 of the summed floor loads, where the storey rule allows 0.73. But a phased strategy is a plan for the usual alarm, and the column's characteristic load belongs to the unusual one. If one alarm in a hundred escalates to a full evacuation, the factor climbs back to 0.59; one in twenty, and it passes the rule.

 A jack set for the first day is wrong by the last. The secondary moment over the first interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built from both ends towards a closure bay, against the creep coefficient: not jacked; jacked at closure by the whole of what the order took, 183 kN·m there; jacked by 0.60 of it, so as to reach the at-once moment at a creep coefficient of 2.0; and by 0.89, so that the shortfall on the day and the overshoot at the end are equal; dotted, stressed at once, 344. On the day and at 2.0: not jacked, 161 and 302; jacked by the whole, 344 and 372; for the end, 271 and 344; balanced, 324 and 365. Internal forces

The jack that creep carries past the mark

A continuous beam built from both ends towards a closure bay keeps less secondary moment at its supports than the same beam stressed all at once, and creep gives back three-quarters of the difference over the years. Bridge builders do not always wait: before the closure is cast, the supports can be lowered on jacks so that the beam starts with the moments it would have had. Jacked by the whole difference, it does start there — and creep, which goes on returning the staged deficit while it relaxes the jacked correction, carries it past them, by a seventh of what was jacked. Jacked by 0.89 of the difference, it starts a ninth short and ends a ninth over. And it takes a jack at every interior support: with fewer, the moment goes to the wrong places.

 A concrete rib creeps into snapping long before its prop matters. For the shallow pinned arch of rise 6.0 radii of gyration under a sinusoidal load, propped at its crown by a concrete column of stiffness κc = 3.5, the creep coefficient at which a load that stays on loses equilibrium, against that load as a share of the day's critical load: a steel rib on the creeping column (solid), which loses only the column's share, so that by a creep coefficient of 3 no load below 85 per cent snaps; and a concrete rib whose column creeps with it (dashed), whose lasting load acts as if (1 + φ) times larger, so that it snaps at φ = pcr/p − 1; dotted, a creep coefficient of 2. At half the critical load the concrete arch snaps at a creep coefficient of 1.00, at 70 per cent at 0.43; the steel arch on its column does not snap at either. Stability

The column that creeps under the arch

A shallow arch propped at its crown by a stiff enough column stops snapping — the sideways branch it would have fallen down turns upwards. If the column is concrete, it creeps under the force it holds, and after a decade it holds that force as a spring a third as stiff, well on the side where the arch snaps. But a load that arrives in seconds does not meet the crept column; it meets the column's instantaneous stiffness, from a rest length that creep has merely moved down, and the arch's branch still rises. Creep reaches only the load that stays on, and there it can take back only the column's own share — 13 per cent of it. A concrete rib is another matter: it creeps itself, and a lasting load at half its critical load snaps it before its creep coefficient reaches one.

 Whether the cross-frames' spacing matters depends on the crossbeams between them. For the 60.0 m orthotropic box 4.0 m wide and 2.5 m deep under 45 kN/m at 2.0 m from its centre line: the largest distortional warping stress against the spacing of cross-frames of two 1,900 mm² diagonals, with nothing between them (dashed), with open tee crossbeams every 4.0 m (light) and with ring frames every 4.0 m (solid); dotted, the ring frames alone, 4.1 N/mm². At 8 m, 3.8, 3.8 and 3.1 N/mm²; at 20 m, 21.8, 21.7 and 7.1; at 30 m, 60.8, 59.2 and 7.3. Sections and stress

The stiff frame among soft ones

A steel box girder with an orthotropic deck is held in shape by a frame at every crossbeam, and its designers usually add a stiffer frame — a pair of diagonals across the box — at every few crossbeams as well. Whether the soft frames between can be left to do the work depends on one length: how far a distortion reaches along a box held by the soft frames alone. Ring frames welded into the webs and bottom flange hold it to six and a half metres, and then cross-frames twenty metres apart barely change the distortion and add a peak of warping stress at each one, where a stiff frame grips a box that was distorting gently. Crossbeams that are only tees under the deck reach twenty-one metres, and then the box between the cross-frames distorts as if the crossbeams were not there.

 The worst direction is the long axis of an ellipse. For a bracket of six M20 grade 8.8 bolts in three rows of two at 75 mm, loaded 150 mm to the side of the group and 150 mm in front of the column face: the demand on the governing bolt, top right, per kilonewton of load, drawn as a radius in the direction the load acts — straight down at the bottom — under the elliptical interaction (solid), which in each quadrant of directions is an arc of one ellipse, and under the straight line (dashed), which is not. The ellipse's longest radius, at 16.0°, is its eigenvector: 0.0058 of the bolt's interaction used for every kilonewton, a capacity of 172.5 kN, found without trying a single direction. Connections

The worst direction is an eigenvector

A bracket whose load can come from any direction in the plane of its bolts has a capacity that is a closed curve, and once the load also pulls the bracket off its column the curve becomes a surface over direction and distance. Finding its lowest point looks like a search. Under the elliptical interaction of shear and tension it is not: inside each quarter-turn of directions every bolt's demand squared is a quadratic form in the load's direction, so the worst direction is that form's eigenvector, and the bracket's least capacity is a 2 × 2 eigenvalue per bolt. Under the straight line EN 1993-1-8 prints it is a search, because a straight line has no curvature to make an ellipse of — and the hand check that sets the worst shear anywhere against the worst tension anywhere is a tenth too low in the plane and a fifth too low once the bracket is pulled.

 A milliradian of slack costs the brace more than a milliradian of crookedness. For the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side by braces of 5,691 kN·m/rad — 2.0 times the 2,846 that just reaches 5,627 kN·m — erected with 5.0 mrad of initial twist: the moment in the middle brace when the girder reaches 5,627 kN·m, against each brace's slack (solid), and against the same amount added instead to the girder's initial twist, the braces tight (dashed). With tight braces and no extra twist, 34.1 kN·m; each milliradian of slack adds 8.0, each milliradian of initial twist 6.8. At 10.0 mrad, 114.0 against 102.4. Stability

The braces with slack in their bolts

A cross-frame bolted into clearance holes resists nothing until the girder it braces has twisted far enough to take up the clearance, and then resists fully. It is a two-way brace that starts late. The girder, erected a little crooked, twists freely into its braces while the load is still small, and from then on the braces hold a girder that is more crooked than the one they were fitted to. The slack acts as an imperfection, and a costly one: each milliradian of it adds more to the middle brace's force than a milliradian of extra crookedness would, and no stiffness takes it back out. On a 24 m plate girder braced at twice the stiffness it needs, three milliradians of slack — a pair of ordinary clearances over its depth — raise the middle brace's force by 70 per cent.

 The road is the harshest stretch of the beam's journey. The worst top fibre, at any corner of any section, of the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, in N/mm² with tension negative: on the yard's bearers; on the haulier's trailer, 1.5 m and 2.5 m from its ends with a bounce of ±30 per cent; lifted by loops 1.0 m in from each end on slings at 60° with a snatch of ±20 per cent; slewed at 0.10 g; and set down on bearings sloping 1 per cent across it; dashed, the stored concrete's limit of −3.0. Yard bearers, −2.98; trailer, −4.88; lifted, −2.97; slewed, −3.13; set down, −1.79. Sections and stress

The beam hung from its loops

A pretensioned beam is cast in a bed, stored on bearers, hauled on a trailer and then lifted by a crane onto its bearings — and at every stage its top fibre, which the strands put in tension, has to stay inside the limit its debonding schedule was drawn for. The trailer, with a bouncing wheel and supports where the haulier put them, was the harshest stage the yard essays found. The last few metres, from the trailer to the bearings, are gentler: hung from loops at the yard's bearer positions the beam is on its bearers again, set down it is on the support its middle was designed for, and only a slew of the crane pushes it a little past its limit. Where the loops are cast decides how much margin the lift has, and the best place for them is not where the yard put its bearers but where the bed held the beam: at its ends.

 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. Structural form

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.

 Two neutral axes in one cut. Stress against height through a girder with a 300 by 30.0 mm top flange, a 229 by 19.6 mm bottom flange and a 573 by 11.9 mm web, spliced at 700 kN·m. The girder's own neutral axis is 374 mm above its underside, and its flanges work at 133 and 201 N/mm². Its splice, each flange's covers sized to that flange's force at 250 N/mm², has its own centroid at 326 mm — 48 mm lower — and across the gap the plates bend about it: the top outer cover at 244 N/mm², the bottom outer at 269, the web plates from 179 at their top to 212 at their foot. Shaded: the covers' heights (darker) and the web plates'. Connections

The splice that finds its own axis

Splice a girder whose top flange is heavier than its bottom one and the splice plates, sized flange by flange, have a centroid 48 mm below the girder's. It looks as if the splice must then carry an axial force the girder does not, or be proportioned to keep the two axes together. It carries none: a cut through the gap carries what the girder carries, and the girder carries no force. Held at both ends, the mismatch makes less than a kilonewton. What the offset does instead is move the web's own force into plates that were not sized for it.

 How much of the drain has to work. The load in the more heavily loaded member, with the roof's own 0.5 kN/m², over its 2.0 kN/m² capacity, for a 12 by 9 m bay in a 150 mm/h storm, with a drain at its middle that passes the storm at 30 mm when clear and a 400 mm scupper at its edge whose sill is 50 mm above the supports, against the share of a clear drain's flow the drain still passes. Members at L/180, L/240, L/360 and rigid: with the drain fully blocked, 1.25, 0.99, 0.84, 0.66; a tenth open, 0.80, 0.77, 0.71, 0.62; half open, 0.44, 0.45, 0.46, 0.48; clear, 0.37, 0.37, 0.38, 0.40. A softer roof is the worse with the drain blocked and very slightly the better once about a fifth of it works. Deflection

The drain that is mostly blocked

A roof's ponding check is made with its primary drains blocked, and a drain is seldom blocked outright: leaves and gravel choke it to a fraction of its flow. A fraction turns out to be nearly enough. On a 12 by 9 m bay sized to L/240, a drain passing a seventh of its clear flow still takes the whole of a 150 mm/h storm, because the water standing over it at the bottom of the sag is the bay's own deflection plus the level. A softer roof's drain works harder still. Only near complete blockage does the scupper take over, and only there does stiffness help.

 Where each filler's depth goes. The crushable depth left in each filler between buildings of 500 t and 300 t, periods 0.8 and 1.2 s, 50 mm apart, through a record of six one-second sine cycles peaking at 0.35 g, for 40 mm fillers crushable over 24 mm, each just strong enough to stop a 2.85 m/s collision from new. Uniform: 5 mm when its hardest contact arrives at 2.76 m/s, 0 mm at the end. Graded: 12 mm when its hardest contact arrives at 2.76 m/s, 6 mm at the end. 50% on speed: 0 mm when its hardest contact arrives at 2.88 m/s, 0 mm at the end. The graded filler's soft layer goes in the first middling contacts and its strong layer is barely reached; the other two lose depth to every middling contact and have little or none left when the largest arrives. Dynamics

The filler that is used up before the worst

A crushable filler between two buildings is sized for the hardest collision it might meet, meeting it new. An earthquake does not oblige: the hardest contact arrives after a dozen middling ones, and each has taken some of the depth. Through a record of six cycles, a filler half of whose strength follows its crushing speed is dense before the worst contact arrives, and a uniform one has five millimetres left. A graded filler still has twelve. Yet the graded filler meets the worst contact at 56 MN and the spent rate-sensitive one at 39. Keeping depth is not the same as being gentle, and the order of the contacts barely matters.

 The seasons ride on the shrinkage. How far the slab's end has moved away from the fill since the fill was finished, over six years, for slabs 10, 20, 30 m between walls at 60 per cent humidity, swinging 3.0 K either way each year, propping a 3 m stem fixed at its base, its top held by the slab cast before its fill went up in 250 mm lifts rolled at 60 kN/m. The slab lets go of the stem at 2.08 mm (dashed). 10 m: a seasonal swing of ±0.15 mm on shrinkage reaching 1.28 mm by year six; it still touches the stem in the sixth year. 20 m: a seasonal swing of ±0.30 mm on shrinkage reaching 2.55 mm by year six; it last touches the stem 3.4 years on. 30 m: a seasonal swing of ±0.45 mm on shrinkage reaching 3.83 mm by year six; it last touches the stem 1.3 years on. Equilibrium

The seasons that stop reaching the wall

A ground-floor slab that has shrunk away from the basement wall it propped moves with the seasons as well, a few tenths of a millimetre at each end every year. It looks like a ratchet waiting to happen: each winter the slab presses the stem back into its fill, and each push might lock a little more pressure in. On a 30 m slab it never gets the chance — the seasons ride on a shrinkage ten times larger, and the slab last touches the wall sixteen months after the fill is finished. On a 10 m slab it touches every year for good, and the stem still loses most of its prop.

 Pressed and pulled, the footing leans once and stops. The column's tilt at rest after each of eight storms reversing fully to 300 kN·m each way, on the 8 m strip of EI 540 MN·m², weighing 320 kN, on a bed of 50 MN/m² under a column carrying 1,000 kN, on ground that unloads and reloads 4 times as stiffly as it first loads, for loads swinging by 0, 50, 100, 150, 200 per cent. 0 per cent: −0.007 milliradians after the first storm and 0.003 after the eighth; 50 per cent: 0.112 milliradians after the first storm and 0.103 after the eighth; 100 per cent: 0.109 milliradians after the first storm and 0.091 after the eighth; 150 per cent: 0.113 milliradians after the first storm and 0.095 after the eighth; 200 per cent: 0.114 milliradians after the first storm and 0.102 after the eighth. The lean is towards the end the heavier half of each storm presses; it stops within three storms whether or not the strip ever hangs clear. Internal forces

The footing that hangs from its column

A footing under a bracing bay's column is pressed hard when the wind pushes one way and pulled when it swings back, and if the pull is larger than the footing's weight it hangs from the column, clear of the ground, for part of every storm. It looks like a footing free to come down somewhere new each time, and walk. It does not walk. The ground under it keeps the memory of every press, and a footing that is not touching the ground cannot change that memory; it comes back down into its own bowl, leans once towards its harder press, and stops within three storms.

 Where the knees begin to alternate. The shakedown factor (solid) and collapse factor (thin) of a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, on pinned bases, with 160 kN of gravity that comes and goes and a 1,000 kN roof carried at its knees, against the peak of a wind that reverses, first-order (dashed) and amplified. First-order, the shakedown and collapse factors are one line until the knees alternate, from a wind of 110 kN; amplified, from 90 kN. At 60 kN: shakedown 0.727 first-order and 0.692 amplified, collapse 0.727 and 0.692. At 120 kN: shakedown 0.533 first-order and 0.488 amplified, collapse 0.571 and 0.539. At 200 kN: shakedown 0.348 first-order and 0.326 amplified, collapse 0.400 and 0.366. Materials

The roof that makes the wind worse every storm

A portal frame's shakedown check is made on its elastic moments, and the elastic moments a heavy roof sways are larger than first-order analysis says: each storm's wind is amplified by 1/(1 − N/Ncr), on its way out and on its way back. On pinned bases under a 1,000 kN roof that takes the wind at which the knees start to alternate from 110 kN down to 90, and the shakedown load down by 4.8 per cent where gravity leads and 8.4 where the wind does. Fixed bases are worth more under a heavy roof than without one, but by half again, not twice.

 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. Connections

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.

 A scatter of a few per cent in the headways and the rhythm is gone. The damping a 120 m stay at 3,500 kN, 1.01 Hz, over a deck mode at 2.01 Hz damped at 1.0 per cent needs to hold a typical history still, under identical vehicles, each giving the deck the impulse with which a random stream of them every 2.0 s on average gives it an rms of 7.1 mm, arriving every 1.987 s on average — 4 deck periods — against the scatter of their headways (coefficient of variation, logarithmic), from integrating the deck and the stay together over four 1,500 s histories at each point. Exactly regular, 0.99 per cent; scattered by 1 per cent, 0.86 per cent; by 2 per cent, 0.58 per cent; by 5 per cent, 0.13 per cent; at random, 0.05 per cent. The requirement halves at a scatter of 2.2 per cent. Dynamics

The convoy that keeps the deck's time

A deck shaken by traffic is shaken by vehicles, and vehicles arrive with a rhythm of their own. The same lorries that, arriving at random, move a stay's deck by 7 mm rms and ask the stay for six hundredths of a per cent of damping, move it by 20 mm and ask for a whole per cent if they arrive every four deck periods exactly. But the rhythm is fragile in a precise way: scatter the headways by two per cent and half the danger has gone, and the timing that has to be kept is a tenth of the deck's period whatever the headway and however damped the deck. Ordinary traffic is counted by a stay, not timed; a convoy driven by a machine might not be.

 Under a light building the first thing the pump was holding up is the building. The hours after the pump stops at which each check on the basement is used up — the slab's 0.2 mm cracks and the building's flotation — against the number of storeys over it, with the water table risen 3.0 m and ground recovering in 12 hours; drawn at 48 hours where a check is never used up. The cracks' deadline is 9.5 h whatever the building. Flotation's is at once under 2 storeys, 1.0 h under 4 storeys, 4.7 h under 5 storeys, 10.1 h under 6 storeys, never under 8 storeys; with 5 storeys or fewer it comes first. Equilibrium

The building held down by its pump

A basement whose water table has risen above its design level is often given back its design head by a pump under the slab, and from then on it is safe while the pump runs. When the pump stops the water comes back on the ground's own clock, and the checks the pump was protecting are lost in an order the building's weight decides. Under eight storeys only the slab's cracks were ever at risk; under five, the building's flotation check goes first, four and three-quarter hours after the pump stops, and the slab's strength is never in question at all. A structure that was safe by its weight has become one that is safe by a machine, and the machine's reliability is now one of its loads.

 The more it carries, the less it pulls when the fire is out. For a 9 m 457 × 191 × 67 steel beam held at its ends at 10 per cent of its own axial stiffness through the fire: the tension pulling on its ends once it has cooled (solid) and the largest tension while it is still hot (dashed), against the load it carries as a share of its cold plastic moment; dotted, the straight beam's 1,669 kN. At 0.10, 1,017 kN cold and 41 hot; at 0.39, 613 and 293; at 0.54, 477 and 397. From 0.56 the beam can find no sag that carries its load near the fire's peak and runs away. Materials

The beam that pays the fire back in sag

A steel beam held at its ends and kept straight through a fire yields short on the way up and pulls on its connections with its plastic shortening on the way down. A beam carrying a floor does not stay straight. Squeezed by its own expansion, it forms a hinge at about 500 °C and bows out of the way, hangs as a catenary while still hot, and keeps a metre of sag when it cools — length it no longer has to be pulled for. It ends the fire pulling a third as hard as the straight beam, nearly whatever holds it, and the hard pull comes earlier instead: at 800 °C, on connections that are as hot as it is.

 The diagonals over the support are the girder's shortest-lived members. The years a diagonal's connection of the top bracing of a 1.5 m deep composite tub girder under a 250 mm deck, continuous over two 40 m spans, 480 kN lorries passing 2.5 m off its centre line lasts at two million such lorries a year, along a span, for fatigue categories 50, 71 and 90, logarithmic and drawn at 1,000 years where the range is below the category's cut-off and the life is without limit; dotted, 100 years. Where the deck is whole every category lasts without limit. At the worst diagonal, 34.5 m along, category 50 lasts 2 years, category 71 lasts 8 years, category 90 lasts 25 years. Sections and stress

The bracing nobody took out

The light truss that closes a steel tub girder while its deck is poured is usually left in once the deck has hardened, because taking it out from under a finished slab is more trouble than it is worth. Over most of a span that is harmless: the deck lifts the neutral axis to the top flanges and takes ninety-nine per cent of the shear round the top, and a passing lorry puts five newtons per square millimetre in a diagonal. Over an interior support, where the deck is cracked to its bars, the same diagonals sit six hundred millimetres from the axis and are handed a share of the torque, and each heavy lorry puts forty in them — the shortest-lived connections in the girder, on members nobody designed for anything after the day the concrete went off.

 Which way the oval lies decides whether the pressure helps or harms. The limit moment of a steel tube of radius 600 mm and wall 2.5 mm (r/t = 240), whose ring buckles under 4.07 kPa, over its limit with no pressure (796 kN·m), against the angle between the bending plane and the diameter a non-uniform pressure pushes in on, for second harmonics of 0.10, 0.25 and 0.50 of that pressure; lines the closed form, dots the minimised energy. At 0.10, 0.93 aligned and 1.08 across; at 0.25, 0.82 aligned and 1.19 across; at 0.50, 0.65 aligned and 1.40 across; at 45° the oval does not touch the limit at all. Stability

The oval the pressure brings

A uniform pressure changes how stiff a tube's ring is against the oval that bending squashes it into, and nothing else: the tube still gives way at a flattening of two ninths of its radius. A pressure that is larger on one diameter than the other has an oval of its own, and that oval does work on the bending's directly. Lying in the bending plane it has done some of the flattening before the bending starts, and a pressure difference a quarter of the ring's own buckling pressure takes a fifth off the tube's bending capacity; lying across it, it adds a fifth. At forty-five degrees it does nothing at all. And the two ninths turn out to have been a property of a round tube.

 A floor's robustness belongs to its joints until they break, and to its slab after. For a floor whose interior column is lost, two 8 m-span beams crossing over it (each collapsing at 400 kN) and a slab 16 m square over the four bays round it: the largest floor load a sudden loss can be held at, as a multiple of the two beams' collapse load, against how far the beams' connections can turn before they fracture, for no mesh and for 142, 252 and 393 mm² a metre. No mesh: 0.94 at 0.03 radians and 1.19 at 0.07; 142 mm²: 0.94 at 0.03 radians and 1.29 at 0.07; 252 mm²: 1.50 at 0.03 radians and 1.81 at 0.07; 393 mm²: 2.26 at 0.03 radians and 2.56 at 0.07. Structural form

The slab that catches what the joints let go

When an interior column is lost, the beams that crossed over it sag into cables and the slab over the four bays round it sags with them, into a membrane hanging from the floor that did not lose anything. Whether the floor's robustness belongs to its steelwork or to its concrete turns on one number with no span in it: the slab's mesh only starts to carry its full tension when the beams' ends have turned through about 0.09 radians, more than any ordinary joint can follow. So the membrane arrives after the joints have gone. And then it catches the floor — a metre down, alone — and with an ordinary mesh it holds half as much again as the beams could.

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