What is refuted here — page 15

Every column carrying the same number of floors of the same occupancy is entitled to the same imposed-load reduction.

What decides it: On a double-loaded office plan carrying twenty floors, the perimeter column needs a factor of 0.26 on its summed characteristic loads, the column at the corridor's edge, whose area is 29 per cent corridor, 0.56, and the stair-core column 0.98. The storey rule gives all three 0.73, which is three times what the first needs and three quarters of what the last does.

Tested in The column under the escape corridor, at the figure it turns on · the essays about live load reduction.

Evacuations are so rare that they can be left out of a column's imposed load.

What decides it: They are rare on any one day and certain over a building's life, and each one is on every floor at once. With a quarter of the column's area corridor, one evacuation a decade gives a twenty-floor factor of 0.41 and one a month 0.54, against 0.26 with no corridor; the crowd's density matters more than its frequency, 0.34 for a crowd of 0.5 kN/m² and 0.77 for 2.

Tested in The column under the escape corridor, at the figure it turns on · the essays about live load reduction.

A footing rocked year after year by a prevailing wind tilts a little further with every storm, so its lean grows steadily through the building's life.

What decides it: On an 8 m strip under a 1,000 kN column, storms pushing to 1,000 kN·m and swinging back half as far lean it 0.099 milliradians in the first storm, 0.114 in the second and 0.117 in the third, where it stays. Through fifty years of storms of random size the lean moves by thousandths of a milliradian in ordinary years and arrives, 94 per cent of it, in the three years whose storm is the largest yet.

Tested in The footing a prevailing wind leans, at the figure it turns on · the essays about elastic foundation.

A footing that has been rocked into a lean is softer in rocking towards that side, because its ground there has been pressed and has yielded.

What decides it: It is stiffer that way. The ground under the pushed end has been pressed before and reloads along its stiff line, so the push's peak rotation falls from 1.12 milliradians in the first storm to 0.97 in the sixth, 13 per cent; the swing back, onto ground hardly pressed, falls only 6 per cent.

Tested in The footing a prevailing wind leans, at the figure it turns on · the essays about elastic foundation.

Using the specification as a prior makes a weak concrete look as good as it was meant to be.

What decides it: Only if the prior is made heavy. With the specification weighted as three specimens, three cores from a concrete 15 per cent weaker than expected give 23.3 N/mm² against the population's 22.9, because the conflict between cores and prior widens the estimated scatter. Weighted as thirty, it gives 27.4 — 4.5 N/mm² on the unsafe side.

Tested in The strength three cores revise, at the figure it turns on · the essays about characteristic strength.

A box girder with an orthotropic deck has no continuous section frame, so its distortion cannot be found as a beam on an elastic foundation.

What decides it: Its ring frames at the crossbeams are a row of springs, and smeared into a continuous foundation they give the distortion and the warping stress to within 6 per cent at a 4 m spacing, and within 10 per cent at any spacing below about half the smeared decay length, for crossbeams from a third to three times as stiff as drawn.

Tested in The deck whose frame is its crossbeams, at the figure it turns on · the essays about box distortion.

Making the crossbeams stiffer brings an orthotropic box's ring frames towards rigid diaphragms and its warping stress down with them.

What decides it: Ten times the crossbeam's second moment takes the peak warping stress only from 4.3 to 4.2 N/mm², because the ring is limited by its web and bottom stiffeners and because the peak sits where the end diaphragm meets the frames' distortion. Rigid diaphragms at the same 4 m spacing would give 0.9.

Tested in The deck whose frame is its crossbeams, at the figure it turns on · the essays about box distortion.

A prop at the crown of a shallow arch stops it buckling sideways, because it holds the arch's midpoint still.

What decides it: The sideways mode is a full sine wave whose node is at the crown, so the crown does not move in it and a prop there cannot resist it. On an arch rising six radii of gyration the sideways branch leaves at the same crown deflection, 1.53 radii of gyration, with any crown prop; the load at which it leaves rises from 311 to 335 at κc = 1 only because the prop carries 6 per cent of the load.

Tested in The prop at the crown of the arch, at the figure it turns on · the essays about Snap-through.

A crown prop only delays an arch's sideways snap; once the arch turns aside it fails as an unbraced arch would.

What decides it: The prop changes the sideways branch's slope from −48 unbraced to −17 at κc = 2 and +11 at κc = 4. Past about κc = 3.1 an arch built 0.05 radii of gyration out of shape has no peak load at all: it sways with the load still rising, and the same prop does this at every rise from 5 to 12.

Tested in The prop at the crown of the arch, at the figure it turns on · the essays about Snap-through.

Using more hangers in a network arch, each crossing more of its neighbours, reduces the steel its hangers need for fatigue, because every section's shear is shared among more of them.

What decides it: Each hanger's range does fall, almost as one over the number — 678 kN on 32 hangers, 369 on 64, 241 on 96 — but the number of hangers times the range, which is the total bar area sized for that range, stays at 21.7, 23.6 and 23.2 MN of range. The hangers share the same shear; dividing it more finely does not make less of it.

Tested in More hangers, and the same steel, at the figure it turns on · the essays about network arch.

The steeper a network arch's hangers, the more directly they carry the deck's weight, and the smaller their fatigue ranges.

What decides it: On 64 hangers the worst range is 352 kN at 45° from horizontal, 369 at 60° and 511 at 70°: steeper hangers cross fewer neighbours and each carries more of the shear. The flatter arrangement costs little; the steeper one costs a third more.

Tested in More hangers, and the same steel, at the figure it turns on · the essays about network arch.

Cores drilled close together are a conservative sample, because concrete from one place is at least as uniform as concrete from many, so three cores from one bay can only understate the structure's scatter in the safe direction.

What decides it: Understating the scatter is the unsafe direction. Three cores from one bay of a concrete whose cores correlate at 0.9 agree to a tenth of the structure's variance, and counted as independent with nothing else known they give a characteristic value with 26.5 per cent of the structure's concrete below it, against the 5 per cent the value promises. Counted by their correlation they give 5.0 per cent.

Tested in The cores that agree for the wrong reason, at the figure it turns on · the essays about characteristic strength.

Once the correlation between cores is allowed for in a Bayesian update, the specification's protection against a weak concrete is restored, since the update is then honest about what the cores know.

What decides it: Honest about the cores means lighter on them, and the specification takes their place. For a concrete 15 per cent weaker than expected, three cores from one bay counted as independent leave 13.6 per cent of the concrete below the estimate at a correlation of 0.9; counted by their correlation, 24.2 per cent. The same three cores from three bays leave 8.7 per cent at every correlation.

Tested in The cores that agree for the wrong reason, at the figure it turns on · the essays about characteristic strength.

Jacking a fixed arch back to no moment a year after it is closed puts it where an arch closed at a year would be, since in both the movements of the first year have been taken out.

What decides it: The jacked arch remembers its first year. Its early restraint and the first jacking's correction go on relaxing after the second jacking, so at a century its springing moment is −117 kN·m. The same rib left three-hinged and closed at a year, with no jack at all, ends at −84.

Tested in The arch that waits with its hinges open, at the figure it turns on · the essays about rib shortening.

A sloshing damper retuned by draining it for a slower building loses so much of its water that it ends no better than the detuned tank it replaced.

What decides it: Draining the 10 m tank from 1.73 m to 1.16 m of water for a building of 6 s cuts its water from 2.0 to 1.3 per cent of the modal mass, but shallow water sloshes a larger share of itself, and the retuned tank holds the building's peak amplification to 11.7, against 34.0 left at its design depth. Of the 11.7, the water let out accounts for 1.7.

Tested in The tank that follows the building, at the figure it turns on · the essays about sloshing.

A column under a stair used for phased evacuation can be designed for the phase, three floors of crowd at once, since the strategy exists so that the stair is never asked to carry more.

What decides it: The strategy describes the usual alarm, and the column's characteristic load describes the worst life in twenty. With fifty alarms in a life and 2 per cent of them escalating to every floor, 64 per cent of lives contain a full evacuation, and the column carrying twenty floors with half its tributary area corridor needs 0.67 of the summed floor loads, against 0.29 for the phases alone.

Tested in The stair that empties three floors at a time, at the figure it turns on · the essays about live load reduction.

Jacking a staged beam's closure by exactly the secondary moments its construction order cost gives it the at-once moments, and creep then has nothing left to change.

What decides it: Creep still has two things to do, and they do not cancel. It goes on moving the staged moments towards the at-once ones, by φ/(1 + χφ) of the deficit, and it relaxes the jacked moment to 1/(1 + χφ) of itself; the two would balance only if χ were 1. At φ = 2 and χ = 0.8 the first interior support starts at the at-once 344 kN·m and ends at 372.

Tested in The jack that creep carries past the mark, at the figure it turns on · the essays about secondary prestress.

Lowering the two interior supports beside the abutments, where the construction order cost the most, recovers most of the deficit.

What decides it: A pair of jacks makes one shape of moment, and it is not the deficit's. Lowering supports 1 and 4 to put back 0.89 of the deficit puts 93 kN·m there where 163 is wanted, and hogs supports 2 and 3 by 51 where they wanted 76 more sag — leaving them further from the at-once moments than if nothing had been jacked.

Tested in The jack that creep carries past the mark, at the figure it turns on · the essays about secondary prestress.

A concrete prop that holds an arch out of its snapping range on the day it is cast will let it snap once it has crept, since its stiffness then falls below the value the arch needs.

What decides it: Only for the load that stays on. A quick load meets the column at its instantaneous stiffness, κc = 3.5, and the arch's sideways branch still rises at a slope of +4.0, as on the first day. For the lasting load the crept column is a spring of 1.17 at φ = 2, but it lowers that load's critical value only from 390 to 338, and no lasting load below 85 per cent of the day's critical load snaps by φ = 3.

Tested in The column that creeps under the arch, at the figure it turns on · the essays about Snap-through.

If the arch rib is concrete too, its own creep relieves the thrust it carries and offsets its column's creep.

What decides it: A concrete rib and column creeping together keep their ratio of stiffnesses, but the whole arch then holds its lasting load as if it were 1 + φ times larger. Under half its critical load it snaps when its creep coefficient reaches 1.0; under 70 per cent, at 0.43 — long before the column's creep could matter.

Tested in The column that creeps under the arch, at the figure it turns on · the essays about Snap-through.

Adding stiff cross-frames to a box girder already held by ring frames at every crossbeam can only reduce its distortional stresses, since it adds stiffness against distortion and takes none away.

What decides it: It reduces the distortion and raises the peak warping stress. With ring frames every 4 m the box's largest distortional warping stress is 4.1 N/mm², near its ends; with cross-frames every 20 m among them it is 7.1, at each cross-frame, where a stiff frame holds a box that the ring frames let distort gently.

Tested in The stiff frame among soft ones, at the figure it turns on · the essays about box distortion.

A span of a stressed ribbon cast on a warm afternoon will settle back to its designed level as it cools, since strand and concrete shorten together and by the same coefficient.

What decides it: They shorten together, but the band that cools is eleven times stiffer than the strand that warmed. Warming took 570 kN of thrust off the bare strand of a 100 m span at a 2 per cent sag; cooling puts 1,380 kN back on the band. Cast 20 °C warm, the span ends 72 mm higher than it was set, having sat 53 mm low on the afternoon it was cast.

Tested in The span cast in the afternoon, at the figure it turns on · the essays about stressed ribbon.

The least capacity of a bracket loaded from an unknown direction, in its plane and pulled off its column, can only be found by trying directions until the lowest is reached.

What decides it: Under the elliptical interaction it is a closed form. In each quadrant of directions a bolt's demand squared is uᵀMu with M a 2 × 2 matrix of the group's geometry, so the worst direction is M's eigenvector: for six M20 bolts loaded 150 mm to the side and 150 mm in front of the column it gives 172.5 kN at 16.0°, which a sweep at a quarter of a degree reproduces to the last figure.

Tested in The worst direction is an eigenvector, at the figure it turns on · the essays about bolt group.

The effect of slack in a brace's connections can be removed by making the braces stiffer.

What decides it: Stiffer braces take the girder towards the twist and the force that the slack itself sets. At thirty times the stiffness the girder needs, tight braces carry 18.2 kN·m at the middle brace and slack ones 31.4, and the twist at midspan is 5.2 against 8.4 mrad.

Tested in The braces with slack in their bolts, at the figure it turns on · the essays about beam bracing.

Lifting a pretensioned beam by its loops is the harshest moment of its delivery, because it hangs from two points with its whole weight and the crane's snatch on it.

What decides it: Hung from loops at the yard's bearer positions it is on its bearers again: its worst top fibre is −2.97 N/mm² with a ±20 per cent snatch, against −2.98 on the bearers and −4.88 on the haulier's trailer. Only a slew of the crane, at 0.1 g sideways, takes it past its limit of −3.0, to −3.13.

Tested in The beam hung from its loops, at the figure it turns on · the essays about prestress limits.

Inclined slings compress a beam and so should be avoided with a spreader beam; vertical slings are always gentler on a pretensioned beam.

What decides it: Inclined slings pull the loops inwards and, the loops being in the top face, compress the top between them. With loops 1 m in the worst top fibre is −2.97 N/mm² at 60° and −3.16 with a spreader's vertical slings. With loops 2 m in the worst section is over the overhang, which the squeeze never reaches, and the angle makes no difference at all.

Tested in The beam hung from its loops, at the figure it turns on · the essays about prestress limits.

Sizing each post of a Vierendeel girder to its own peak moment is a safe economy, because every post then works exactly as hard as the most heavily loaded post already did.

What decides it: Sized to its peak, every corner of a six-panel girder peaks at 75.4 MPa, as the worst post of the uniform girder did. But the middle post, cut to 0.39 of the chords' section, sweeps a range of 151 MPa as the load crosses — twice the uniform girder's worst of 78 — and at category 71 survives 209 thousand crossings against 1.49 million.

Tested in The post sized by what it must survive, at the figure it turns on · the essays about vierendeel.

Designing a Vierendeel girder's posts for fatigue rather than for strength means making them all alike, and costs steel.

What decides it: Sized to their ranges the posts are not alike: the end posts, bent one way only, drop to 0.59 of the chords' section while the interior posts stay between 0.96 and 1.03. The posts hold 6.14 chord sections' worth of steel against the uniform girder's 7, and every corner reaches the same 1.49 million crossings.

Tested in The post sized by what it must survive, at the figure it turns on · the essays about vierendeel.

A splice in a girder with unequal flanges carries an axial force across its gap unless its plates are proportioned so that the splice's neutral axis falls on the girder's.

What decides it: A cut through the gap carries whatever the girder carries, and a girder in bending carries no net force, so the plates carry none whatever their centroid. With the splice's axis 48 mm below the girder's, a girder held rigidly at both ends 12 m apart develops 0.66 kN across a 10 mm gap — against a squash load of 7,209 kN.

Tested in The splice that finds its own axis, at the figure it turns on · the essays about single lap.

Sizing each flange's covers to that flange's force, and the web plates to the web's moment, gives every splice plate the force it will actually carry.

What decides it: With a 300 × 30 mm top flange and a 229 × 19.6 mm bottom one, the flanges carry 1,126 and 877 kN at 700 kN·m, and the web carries the 249 kN difference as a net compression that this design assigns to no plate. Across the gap the web plates take 127 kN of it and their foot works at 212 N/mm² against the 163 they were designed for; the bottom outer cover works at 269 against 250.

Tested in The splice that finds its own axis, at the figure it turns on · the essays about single lap.

A roof whose primary drains are partly blocked behaves part-way between the roof with working drains and the roof with only its overflows, in proportion to how much of the drains is blocked.

What decides it: On a 12 by 9 m bay at L/240 in a 150 mm/h storm, the drain at the middle of the bay passes the whole storm until it is down to 15 per cent of its clear flow. Half open, the members carry 0.45 of their capacity against 0.37 with the drain clear and 0.99 with it blocked — a sixth of the way from clear to blocked, not half.

Tested in The drain that is mostly blocked, at the figure it turns on · the essays about serviceability.

Against rainwater a stiffer roof is always the safer one, since its sag makes less room for water.

What decides it: Only with the drain nearly or wholly blocked. The sag also puts water over the drain at the bottom of the bay, and a drain's flow rises as that depth to the power 1.5: at 12 m a roof at L/180 needs its drain only 6 per cent open to keep the scupper dry, a rigid roof 46 per cent. Half open, the L/180 roof's members carry 0.44 of their capacity and the rigid roof's 0.48.

Tested in The drain that is mostly blocked, at the figure it turns on · the essays about serviceability.

A gap filler that has been crushed dense before an earthquake's hardest contact arrives has failed, and meets that contact far harder than a filler that kept its depth.

What decides it: Through a six-cycle record peaking at 0.35 g, a filler with half its strength on speed is dense when a 2.88 m/s contact arrives and meets it at 38.7 MN, on the concrete. A graded filler with 12 mm still crushable meets a 2.76 m/s contact at 56.0 MN, because its remaining depth is its strong layer, which crushes at 56 MN whatever is left of it.

Tested in The filler that is used up before the worst, at the figure it turns on · the essays about pounding.

The order in which an earthquake's contacts arrive decides which kind of crushable filler does best.

What decides it: Run the same six cycles last first and each filler's worst contact moves by under a meganewton — 37.2 to 36.5 MN uniform, 56.0 to 56.0 graded, 38.7 to 38.3 rate-sensitive — and the ranking is unchanged. What decides a filler's worst contact is the force its remaining depth crushes at, not the depth itself.

Tested in The filler that is used up before the worst, at the figure it turns on · the essays about pounding.

A basement stem whose ground-floor slab has shrunk away from it is pushed back into its fill every winter by the slab's seasonal expansion, so its propped design is restored for part of every year.

What decides it: A ±3 K seasonal swing moves the end of a 30 m slab ±0.45 mm, riding on a shrinkage that reaches 3.8 mm in six years against the 2.08 mm at which the slab lets go of the stem. The slab last touches the stem 1.3 years after the fill is finished, a 20 m slab after 3.4, and neither ever again.

Tested in The seasons that stop reaching the wall, at the figure it turns on · the essays about lateral pressure.

A slab short enough never to shrink away from its stem keeps the stem propped as it was designed.

What decides it: A 10 m slab touches the stem every year for thirty years, but its 1.5 mm of shrinkage lets the stem's top lean away and the compaction residual relax: the prop's force falls from 22.6 to 8.2 kN a metre and the base moment rises from 26.6 kN·m to between 54 and 60 — most of the way to the 69.6 of a slab gone slack.

Tested in The seasons that stop reaching the wall, at the figure it turns on · the essays about lateral pressure.

A footing lifted clear of the ground in every reverse storm comes down somewhere new each time, and walks.

What decides it: An 8 m strip weighing 320 kN under a column whose load swings by 150 per cent hangs clear of the ground for the last part of every swing back, past 264 kN·m. Its tilt at rest is 0.113 milliradians after the first storm and 0.095 after the eighth, settled within three: while it hangs the ground's memory does not change, and it comes back down into the bowl its presses made.

Tested in The footing that hangs from its column, at the figure it turns on · the essays about elastic foundation.

The more a storm swings a column's load, the more the column's footing leans.

What decides it: Under storms of 300 kN·m each way, the settled tilt is 0.103 milliradians with the load swinging by 50 per cent, 0.091 at 100 and 0.102 at 200 — within a fifth across the whole range. What the swing does is press the footing deeper: its settlement at rest goes from 5.39 mm with no swing to 7.87 at 100 per cent and 10.66 at 200.

Tested in The footing that hangs from its column, at the figure it turns on · the essays about elastic foundation.

A shakedown check made on first-order elastic moments is on the safe side for a portal frame, because repeated loading is checked against the more demanding of two limits already.

What decides it: On pinned bases with a 1,000 kN roof at its knees, the first-order check gives a shakedown factor of 0.727 under 160 kN of gravity and 60 kN of wind, and with each storm's wind amplified for its own gravity the factor is 0.692. With twice the wind, 0.533 against 0.488. At αcr of 3 the first-order check overstates by 7.8 and 13.8 per cent.

Tested in The roof that makes the wind worse every storm, at the figure it turns on · the essays about shakedown.

The elastic count of a single-sided partial-penetration weld, with its root factor of four at half penetration, is the conservative bound, so a weld that passes it cannot fail at its root.

What decides it: Assessed as an edge crack on the strip-yield line, the root of a half-penetration weld in a 20 mm S355 plate tears before the elastic count's load whenever its toughness is below 30 MPa·m½, and with the master curve's 5 per cent toughness at −20 °C for a crack front 300 mm long, the same proportions in a plate thicker than 56 mm keep less than the elastic count's 0.25.

Tested in The land that is already a crack, at the figure it turns on · the essays about weld strength.

Cambering each span of a continuous truss for its own dead-load sag, as if it were simply supported, gives the right camber and the right forces once the truss is continuous.

What decides it: Cut that way the truss stands 35 mm above its middle bearing before any load, and once its dead load closes the gap it carries that load as two simple spans: 480 kN on the pier instead of the 572 the continuous analysis assumed, no hogging over the pier where the analysis has 715 kN·m, and 1,395 kN·m of sagging in each span against 915.

Tested in The pier's share is cut into the members, at the figure it turns on · the essays about camber.

The more heavily damped a deck is, the more irregular the traffic on it may be before its rhythm reaches the stays.

What decides it: The scatter of headways that halves a regular stream's requirement is 28 to 50 milliseconds on a deck damped at 0.5 per cent and 37 to 43 milliseconds on one damped at 2 per cent, at headways from one to eight deck periods: about a tenth of the deck's 497 ms period, whatever the deck's damping.

Tested in The convoy that keeps the deck's time, at the figure it turns on · the essays about cable dynamics.

Under a risen water table the slab of a basement is the part that fails first, since it carries the water less only its own weight while the building's whole weight resists flotation.

What decides it: It is the first to run out of crack width under a heavy building and never under a light one. With the water risen 3 m, the slab's 0.2 mm limit is reached 9.5 hours after the pump stops whatever the building, but the flotation check is used up at once under three storeys, after 1.0 hour under four and 4.7 under five; the slab's bending reaches only 0.61 of its strength at the full rise.

Tested in The building held down by its pump, at the figure it turns on · the essays about uplift.

In a continuous composite beam the shear connection is busiest over the interior supports, where the shear is largest, so that is where the studs should be concentrated.

What decides it: Counted with the studs' own flexibility, the slab force cannot follow the moment's kink over the support, and the slip there falls to nothing over about a metre. Along a 12 m span of a continuous beam with evenly spaced studs the slip is 0.68 mm at the end support and 0.62 mm at the point of contraflexure, 9.5 m along, and these two are the peaks.

Tested in The slip that peaks where the moment turns, at the figure it turns on · the essays about composite action.

The studs in the hogging region of a continuous composite beam carry little, since the cracked slab is only its bars, and can be moved into the sagging span where the slip is.

What decides it: The hogging region starts at the point of contraflexure, where the second slip peak is. Thinning the studs beyond it to a quarter of the density elsewhere, the span's total kept, raises the largest slip from 0.68 to 1.69 mm, and takes the bars' force over the support from 344 to 118 kN.

Tested in The slip that peaks where the moment turns, at the figure it turns on · the essays about composite action.

A bracing member whose stress range is a few tens of newtons per square millimetre is far below its fatigue strength.

What decides it: At 42 N/mm² a lorry and two million such lorries a year, the flow EN 1991-2 assigns to the slow lane of a motorway carrying many lorries, a diagonal's connection of fatigue category 50 lasts two years, category 71 eight and category 90 twenty-five. Where the deck is whole the range is below every category's cut-off.

Tested in The bracing nobody took out, at the figure it turns on · the essays about shear centre.

A non-uniform external pressure on a tube in bending always weakens it, as a uniform external pressure does.

What decides it: It depends on which diameter it pushes in on. For the 1.2 m tube with a 2.5 mm wall, a second harmonic of 1.02 kPa takes its bending limit from 796 to 651 kN·m when it pushes in on the diameter in the bending plane, raises it to 949 when it pushes in on the diameter across it, and leaves it at 796 at 45°.

Tested in The oval the pressure brings, at the figure it turns on · the essays about brazier buckling.

A floor whose beam-to-column connections cannot follow the catenary has no robustness beyond the beams' plastic collapse load.

What decides it: With 252 mm² of mesh a metre in a 16 m square slab, a sudden loss can be held at 1.50 times the two beams' collapse load with connections good for only 0.03 radians, against 0.94 for the beams alone: the connections fracture on the way down and the slab catches the floor alone, 1.16 m down at 1.2 times the load.

Tested in The slab that catches what the joints let go, at the figure it turns on · the essays about robustness.

Each reversal of the load on a row of fasteners resets its slack, so a row under a reversing load ratchets where the same row under a repeated load would settle.

What decides it: A row of twenty screws leaning at 45°, each remembering its history in each direction, settles under a fully reversed load of 0.70 of its forward peak within four cycles, as it settles under a repeated one. What the reversal changes is the ceiling: fully reversed loads are carried for ever up to 0.79 of the forward peak, against 0.965 for loads repeated one way, because the pushed screws are the weaker.

Tested in The row that is pushed as often as it is pulled, at the figure it turns on · the essays about dowel yield.

Checking a frame's computed sway by the unit-load method on one column line confirms the stiffness the analysis assumed for its beams.

What decides it: The unit load carried down one column line puts no moment in any beam, so the check reads only that line's real moments. In a four-bay, ten-storey concrete frame it reproduces the sway exactly with the beams at their gross stiffness, 15.7 mm, at 0.35 of it, 33.3 mm, and at 0.2, 52.0 mm: it confirms each analysis's arithmetic and none of its assumptions.

Tested in The check that cannot see the beams, at the figure it turns on · the essays about virtual work.

A storey's drift comes from the bending of that storey's columns.

What decides it: Measured from its tangent at the floor below, the windward column of the second storey of the four-bay frame bends by −0.18 mm; the storey drifts 2.68 mm because the floor below it turns, carrying 2.85 mm up the storey. Only the ground storey, standing on fixed bases, drifts by its own columns alone.

Tested in The check that cannot see the beams, at the figure it turns on · the essays about virtual work.

A beam struck by a weight at different places each time shakes down sooner than one struck in one place, because the yielding is shared among several hinges.

What decides it: It shakes down later or not at all. A 6 m fixed-ended beam struck by 0.7 kJ, less than the 0.80 kJ it can store elastically at three tenths of its span, settles at 2.1 mm after ten drops landing always there. Struck by the same weight at three and seven tenths in turn it is at 29.7 mm after two hundred drops and still gaining 0.12 mm a drop, because the moment one drop locks in to protect its own point is the moment that makes the other drop yield.

Tested in The weight that lands somewhere else, at the figure it turns on · the essays about impact factor.

A crane whose payload could only ever be released slowly could let its counterweight balance the whole payload.

What decides it: A slow release removes the rebound, not the release. A counterweight holding a share of the payload throws the mast backwards by that share once the payload has gone, however gently it went, so the best share is half the hoisting factor at most: 0.60 for a hoisting factor of 1.2. Over six seconds on a crane whose period is 2 s the best share is 0.56, and balancing all of the payload would leave the mast 1.07 times the payload's moment against 0.64.

Tested in The release the rope cannot slow, at the figure it turns on · the essays about counterweight.

A more flexible joint always delays a beam's first plastic hinge, because it sheds moment from the beam's end.

What decides it: The moment it sheds goes into the span, and below about 10 EI/L the span hinges first. With full-strength end plates at 1.2 times the beam's strength, the first hinge comes at λ = 3.39 for 100 EI/L, 3.78 for 25, 4.52 for 10 — where end and span yield together — and back down to 3.91 for 3, with the hinge in the span.

Tested in The joint stronger than its beam, at the figure it turns on · the essays about corner moment.

A column that settles relieves the floor beams it supports, because it takes moment off them.

What decides it: It takes hogging moment off them and gives them sagging. Over the interior column of the six-storey frame, the floor beam hogs 200 kN·m before the earthquake, −80 after 20 cycles, and sags 266 kN·m once the columns have equalised, while its largest span moment rises from 113 to 355 against a plastic moment of 405.

Tested in The floor that cannot keep up with the column, at the figure it turns on · the essays about Moment-curvature.

As a curved flange's effective width changes along a tapered knee, the web and its welds must take up a significant force from the flange.

What decides it: The flange's force changes by a sixth over a metre and a half of curve, and the shear it puts into a 10 mm web beside the weld is at most 12 N/mm² against a shear yield of 205.

Tested in The flange bent hardest where it curls least, at the figure it turns on · the essays about curved beam.

A cantilever bolted to a column through an end plate can be taken as built in at its root.

What decides it: A flush end plate 15 mm thick on M20 bolts to a 254 mm universal column that runs on above and below holds the root against warping at 5.9 times the cantilever's own warping stiffness, against about fifty for a built-in root. A 3 m cantilever of 457 mm beam buckles at 945 kN·m on it, between 521 free to warp and 1,266 built in; where the column ends at the beam, at 791.

Tested in The root that belongs to the column, at the figure it turns on · the essays about moment gradient.

A response-spectrum analysis confirms a static push's verdict on torsional irregularity, only more accurately.

What decides it: It gives a different verdict on 78 of 256 plans of a 30 × 18 m floor with eccentricities up to a fifth of its length. The static push alone flags 41, mostly torsionally flexible floors; the response spectrum alone flags 37, mostly floors whose torsional radius is a little larger than their radius of gyration. A floor 1.5 m eccentric at Ω = 1.14 reads 1.17 statically and 1.37 dynamically.

Tested in The floor that twists in its own time, at the figure it turns on · the essays about plan torsion.

A nut that does not turn at the inspection torque shows that its bolt still carries its preload.

What decides it: The torque that turns a nut is the preload times the friction bracket the thread has now, plus whatever rust and paint hold the nut by themselves. With the thread's static friction drifted from 0.16 to 0.30 and 147 N·m of rust, an M20 grade 10.9 bolt at half its minimum preload holds the 647 N·m inspection torque 80% of the time, and one at 66 kN — 38% of the minimum — half the time.

Tested in The wrench that reads the rust, at the figure it turns on · the essays about Slip-critical.

A compensation fitted to measurements is safer than one fitted to a prediction.

What decides it: A survey during construction sees creep under loads a few months old. With that creep as designed and the creep long after 1.3 times it, the surveyed compensation leaves its worst floor 14.6 mm out and the design's fixed share 12.3 — the survey has measured the part of the creep that matters least, and the fixed share was hedging against the part it could not see.

Tested in The survey that arrives after the floors, at the figure it turns on · the essays about differential shortening.