What is refuted here — page 13

Under a half-span live load a tied arch's rib amplifies its bending as a free rib does, and the deck only shares the load.

What decides it: The deck is the tie, in tension equal to the thrust, and with vertical hangers the thrust's lever arm changes only by the difference between the rib's deflection and the deck's. The 60 m rib that carries 1,303 kN·m free, against 576 at first order, carries 524 hung over a deck a tenth as stiff — 0.99 times its first-order moment.

Tested in The rib that cannot lean on its own tie, at the figure it turns on · the essays about arch buckling.

A tied arch's in-plane buckling load is the free rib's, since the rib is the same.

What decides it: To sway, the hung rib has to tilt every hanger, each in tension, and stretch them. Under the same load the free rib buckles at 1.78 times it and the hung rib at 28.6 times; with hangers a hundredth as stiff as a 50 mm rod, at 7.5 times.

Tested in The rib that cannot lean on its own tie, at the figure it turns on · the essays about arch buckling.

The shape of the canyon decides only the rings' radii; the load path is the same arch action in any valley.

What decides it: The same 100 m dam's crown cantilever carries 10% of the reservoir in a V-shaped canyon and 51% in a U-shaped one, 87% of the water at 90 m depth, and its heel bending stress rises from 1.09 to 3.90 N/mm².

Tested in The water the valley floor takes, at the figure it turns on · the essays about curved surface pressure.

The larger the viscous damper, the more damping it adds to the structure.

What decides it: Held by a brace as stiff as the structure, a damper adds the most — 20.7%, for 22.7% in all — at one size, and less either side of it. Ten times larger it adds almost nothing: the structure's damping falls back to 3.2% at a damper seventeen times the best, and 1.6% at one whose force is a hundred times the brace's, because the damper hardly strokes and the brace carries the motion as a spring.

Tested in The damper is only as good as its brace, at the figure it turns on · the essays about damping.

The effective width tw + 2r + 7k·tf describes how a plate's force is shared along its weld to an unstiffened column flange.

What decides it: For a 250 × 20 mm plate on a 300 × 19 mm flange, the elastic share puts 47% of the force through the 65 mm over the web and fits a width of 69 mm, against the rule's 191 mm. The flange root first yields at 409 kN, 30% of the rule's 1,359 kN: the rule's width is reached only after the joint has redistributed.

Tested in A quarter of the weld takes half the pull, at the figure it turns on · the essays about weld group.

A wider plate spreads its force over more of an unstiffened flange.

What decides it: From 100 to 298 mm of plate on the same flange the elastic width grows from 52 to 70 mm, and the rule's width does not change at all. Past 264 mm of plate the rule's 191 mm falls below (fy/fu)·bp and the joint must be stiffened.

Tested in A quarter of the weld takes half the pull, at the figure it turns on · the essays about weld group.

An opening that removes a tenth of the control perimeter adds about a tenth to the shear reinforcement round the column.

What decides it: On a 10 m bay it raises the demand on the control perimeter by 11.5% and the steel on each ring of studs by 6%, but it pushes the outer perimeter from 5.36d to 6.13d, which needs a seventh ring: 10,490 mm² of studs against 8,490, 24% more.

Tested in A hole takes a tenth of every perimeter, at the figure it turns on · the essays about punching shear.

A wind load that does no work in a portal beam's collapse mechanism cannot reduce the load that beam can carry.

What decides it: Applied once, it does not: 160 kN on the beam of a 10 × 5 m portal with 200 kNm plastic moments collapses at a factor of 1.00 with or without 60 kN of wind. Applied repeatedly, with the wind reversing, it does: the shakedown factor falls to 0.88, because each knee takes its worst hogging moment, 216 kNm against gravity's 160, from the wind in a different direction.

Tested in The beam the wind walks without pushing, at the figure it turns on · the essays about shakedown.

Under combined gravity and wind, a portal frame fails by the combined mechanism, so that is the mechanism to check when the loads are repeated.

What decides it: The combined mechanism governs collapse in the middle of the load plane, but in every frame drawn it never governs shakedown: its hinges are all credited by the wind blowing the same way, so reversing the wind earns it nothing. Shakedown is bounded by the beam mechanism, which collects a knee moment from each direction, or by alternating plasticity at a base.

Tested in The beam the wind walks without pushing, at the figure it turns on · the essays about shakedown.

The effective width over a continuous girder's support and the moment there have to be found by iterating between them, because each depends on the other.

What decides it: The girder is linear. Each wave of moment bends it with its own effective flange — 0.95 of the flange for an 80 m wave, 0.05 for a 2 m one — so its stiffness depends on the wavelength of what bends it, not on how large the load is, and one compatibility solve gives the reactions. The width over the support is read off afterwards.

Tested in The reaction moves one per cent and the moment five, at the figure it turns on · the essays about effective width.

Shear lag sheds moment from a continuous girder's support into its spans, and that relieves the stress spike over the support in proportion.

What decides it: For two 20 m spans with 6 m flanges it sheds 4.6% of the support moment and takes 2.3% off the flange stress at the web there; for 12 m flanges, 14.9% and 5.3%. The spike is made of the reaction's short waves, and the reaction changes by under one per cent and three per cent respectively.

Tested in The reaction moves one per cent and the moment five, at the figure it turns on · the essays about effective width.

A multiplanar tubular joint whose K joints are each legal in their own plane can be built.

What decides it: With 114.3 mm braces at 45 degrees on a 168.3 mm chord, each plane's K is legal with a 12.6 mm gap between its own toes. With the two planes 90 degrees apart and both nodes at one section, a brace of one plane comes within 6.5 mm of a brace of the other, against the 12.6 mm two welds need; 60 degrees apart, they overlap. No gap in either plane changes either result.

Tested in Two planes of braces and one circumference, at the figure it turns on · the essays about noding eccentricity.

Keeping each plane's noding eccentricity inside the band in which its moment may be neglected keeps the chord's bending within what that band allows.

What decides it: Each plane's eccentricity bends the chord in its own plane, and the two moments add as vectors at the planes' angle. Equal eccentricities in both planes give the chord 1.41 times one plane's bending stress when the planes are 90 degrees apart and 1.73 times when they are 60 apart.

Tested in Two planes of braces and one circumference, at the figure it turns on · the essays about noding eccentricity.

Jacking a fixed arch's abutments apart at closure, by the gap its thrust and its footings open, removes its shortening moment.

What decides it: For the 60 m concrete rib under 60 kN/m, closed at 28 days, the springing moment starts at nothing and is −173 kN·m a century later, against −112 for the fixed rib's whole shortening. The jacks' correction relaxes as the rib creeps (−81), shrinkage after closure adds −59 and the ground's consolidation −32.

Tested in The jacks the arch forgets, at the figure it turns on · the essays about rib shortening.

A preset chosen for the long-term movements makes a jacked arch right for its life.

What decides it: Every millimetre of preset puts 8.3 kN·m into the springings on the day, most of which the rib then lets go, while the movements after closure arrive whatever the preset. The best single setting, 16.3 mm beyond the day's gap, leaves +135 kN·m at closure and −135 a century later.

Tested in The jacks the arch forgets, at the figure it turns on · the essays about rib shortening.

The water sloshing in an elevated tank damps the tower that carries it.

What decides it: With the half per cent of damping a plain tank's water has, a full 12 m tank on a 2.5 s tower gives an rms base shear under broadband shaking 1.01 times the same tower's with the water frozen to its container, and a peak under harmonic shaking 1.83 times as high. The water splits the resonance in two and damps neither half.

Tested in The water tower that could damp itself, at the figure it turns on · the essays about sloshing.

A tuned liquid damper works only at the depth it was tuned for, so a water tower whose level changes every day cannot use its own water as one.

What decides it: Emptying the tank slows the sloshing, which detunes it, and makes the water a larger share of what the tower carries, which needs less exact tuning. One screen giving 10% of damping keeps the tower's rms base shear at 0.37 to 0.64 of the frozen-water tower's from a tenth full to full, and above a third full within 41% of the best each fill could have.

Tested in The water tower that could damp itself, at the figure it turns on · the essays about sloshing.

A basement wall propped by its ground-floor slab before it is backfilled puts into the slab a share of the soil's active or at-rest pressure.

What decides it: Backfilled in 250 mm lifts with a 60 kN/m roller, the 3 m stem propped first puts 22.6 kN a metre into its prop: 3.8 times the 6.0 active pressure gives and 2.5 times the 9.0 of soil at rest. Every lift's residual near the top is locked against a stem that never moves there.

Tested in The prop that holds what the roller left, at the figure it turns on · the essays about lateral pressure.

A lightly loaded compression pile needs reinforcement only near its head, where the connection and any bending are.

What decides it: In clay heaving 100 mm the 24 m pile is in tension from 8.9 m to its toe, and the tension asks for 1,196 mm² of steel at 17.3 m — all of it below a cage that stops at 6 m. Plain concrete there would carry 1.8 N/mm² of tension across its whole section.

Tested in The pile the ground pulls apart, at the figure it turns on · the essays about downdrag.

An autofrettaged liner with a shrunk-on jacket combines the two methods' gains and can carry a tube past the factor of two that limits each alone.

What decides it: Assembled, the tube answers its working pressure as one elastic wall, so the bore's stress swings by the whole wall's Lamé range whatever made the prestress. The bore may rise to yield at work and fall only to its reverse yield at rest — βσy for a liner that has been yielded — so the combination is capped at (1 + β) times the first-yield pressure, like the liner alone.

Tested in The ceiling the bore sets, at the figure it turns on · the essays about unloading.

A pretensioned beam whose debonding keeps it inside its stress limits at transfer can be lifted and stored without a further check.

What decides it: The 12 m beam's schedule leaves its worst top fibre at −2.43 N/mm² at transfer against a limit of −2.5. Stored on bearers a metre in from each end the same section reaches −3.71, past even the −3.0 the older concrete allows. In transfer's conditions the bearers may come in 51 mm; with 5% of the prestress lost and that −3.0 limit, 0.57 m.

Tested in The beam that leaves the bed, at the figure it turns on · the essays about prestress limits.

Supporting a beam a little in from its ends changes the stresses only near the ends, where the overhangs hog.

What decides it: Between the supports the self-weight moment falls by a constant waL/2 along the whole length — 31.5 kN·m for each metre of inset on the 12 m beam, 1.29 N/mm² off the top fibre's relief at every section between the bearers. Mid-span, which no debonding can relieve, reaches the transfer limit at 1.66 m of inset.

Tested in The beam that leaves the bed, at the figure it turns on · the essays about prestress limits.

The cross-girders' end rotation under the deck load pushes a half-through girder's compression chord sideways and so adds a force to its U-frames comparable to the chord's initial bow.

What decides it: For a 24 m chord at 930 kN held by 1,400 N/mm frames every 4 m, a 7.5 mm push at every frame's foot loads the worst frame with 1.37 kN, against 13.63 kN from the worst bow; the two together give 14.99. The chord follows the push rather than resisting it, and the push would have to reach about 75 mm before it loaded a frame as much as the bow.

Tested in The frames the deck pushes first, at the figure it turns on · the essays about continuous restraint.

Stiffer U-frames take more of the force the deck's push puts on the chord.

What decides it: The push's largest frame force falls as the frames stiffen — 1.93 kN at 700 N/mm, 1.37 at 1,400, 1.07 at 60,000 — because frames that all move together carry the chord with them, and what resists is the chord's own bending between its fixed ends and the first frames.

Tested in The frames the deck pushes first, at the figure it turns on · the essays about continuous restraint.

An outrigger connected after the building has topped out is free of the differential shortening between the core and the columns.

What decides it: For a 200 m concrete building built a storey a week, 77% of the fifty-year differential at the arm's level arrives after topping out, and an arm of 10⁸ kN·m a radian connected then ends with 3.47 MN in it — 60% of the 5.8 MN the design wind puts there. Waiting five years leaves 1.35 MN.

Tested in The arm connected late, at the figure it turns on · the essays about outrigger.

A doubler plate welded over the coped length of a web reinforces the section at the end of the cope as soon as it reaches that section.

What decides it: Welded along its two long edges and stopped at the end of the cope, an 8 mm doubler on a 457 mm beam carrying 300 kN brings the free edge, read 15 mm inside the cut, from 152 N/mm² to 124 — about two fifths of the way to the 88 N/mm² of a plate joined to the web at every point. Its own bending moment falls to nothing at its unwelded end, which is the section the check is made at.

Tested in How far a doubler has to run, at the figure it turns on · the essays about coped beam.

A riser guided at every floor is bent only by the change of drift from storey to storey, wherever its branches are taken off.

What decides it: Guided alone, a water-filled DN100 riser in the ground storey of the eight-storey building carries 2.8 N/mm² at 3.0 m above the floor. Take a DN50 branch off it there and brace the branch 2 m away to the slab above, so that it cannot slide along its length, and the riser carries 71 N/mm² at the tee: the branch holds it to the slab half a metre below the floor it was guided at.

Tested in The branch that holds the riser to a floor, at the figure it turns on · the essays about floor spectrum.

A basement whose flotation check passes with a comfortable margin can tolerate a rise of the water table up to that margin.

What decides it: A 20 × 30 m basement under eight storeys, designed for a 6 m head, uses 0.65 of its weight against flotation and would tolerate more than the 3 m rise to the ground. Its 0.9 m slab, reinforced with a quarter more steel than a 0.2 mm crack limit needed, passes that limit after a rise of about 1.6 m.

Tested in The basement that leaks before it floats, at the figure it turns on · the essays about uplift.

A backprop jacked to relieve a young slab can be slackened back to snug once the slab is past its worst, so that the preload is never passed to the slab below.

What decides it: Slackened to snug a day after being jacked to 0.15 of a slab's weight, the backprops let go of 0.81 w — the preload, their share of the slab cast above and what creep has added — and the young slab's load jumps from 1.19 to 2.00 times its weight. The frame's worst use of strength rises from 1.95 with snug backprops to 2.49.

Tested in The preload that cannot be taken back, at the figure it turns on · the essays about maturity.

Letting a jacked backprop down by exactly its preload, with a load cell, returns the frame to what snug backprops would have given.

What decides it: It hardly changes the frame at all. Jacked to 0.15 w and let down by 0.15 w a day later, the worst use is 1.94; left jacked, 1.93; snug throughout, 1.95. The slab below takes the preload the instant the jack is turned, and its use is at its peak within hours, before any release.

Tested in The preload that cannot be taken back, at the figure it turns on · the essays about maturity.

Two holes bored side by side along a shaft behave in torsion like one hole of their combined width, with a stress concentration in the steel between them.

What decides it: Two 8 mm holes at 12 mm from the axis of a 50 mm shaft, their edges 2 mm apart, keep 95 per cent of its stiffness and peak at 1.13 times its surface stress; the steel between them carries 0.03 of it. One 16 mm hole there keeps 87 per cent and peaks at 1.83, and one hole of the pair's total area keeps 94 and peaks at 1.37.

Tested in Two islands in one film, at the figure it turns on · the essays about membrane analogy.

A Vierendeel girder that carries a point load at mid-span carries the same load anywhere along its span.

What decides it: Six 2 m panels, 1.5 m deep, chords and posts alike: a 100 kN load at mid-span puts 26 kN·m into the end panels' deck chord and nothing into the middle post. Crossing the girder, it puts 44 kN·m into those chords, with the load at the first post, and 30 kN·m into the middle post. The second post reaches 59 against 46.

Tested in The load that visits every panel, at the figure it turns on · the essays about vierendeel.

Shortening the floor cycle from seven days to four costs a frame nothing, because each slab carries the same multiples of its own weight on either cycle.

What decides it: The loads are the same — a struck slab peaks at 1.52 of its weight on a seven-day cycle and 1.52 on a four-day one — but the slab has 78 per cent of its 28-day strength when it is struck at seven days and 66 per cent at four. Its use of strength rises from 1.95 to 2.29, a sixth more, with nothing about the loads having changed.

Tested in The day taken off the floor cycle, at the figure it turns on · the essays about maturity.

A faster cycle makes a jacked backprop less useful, because the slab below that takes the preload is younger as well.

What decides it: The slab below is younger, but the struck slab is younger by more in strength: one cycle old against two, its strength falls from 0.86 of the slab below's to 0.80 at three days. The preload that balances them rises from 0.07 of a slab's weight at seven days to 0.14 at three, and what it saves rises from 0.08 to 0.23.

Tested in The day taken off the floor cycle, at the figure it turns on · the essays about maturity.

The best way to grade a row of braces under a moment that peaks at midspan is to put nearly all of the stiffness into the brace at the peak.

What decides it: Past a share of 68 per cent the outer braces cannot go below 1,238 kN·m/rad each, because the girder then buckles in two opposite waves about a midspan that does not twist and the middle brace does nothing. With 81 per cent in the middle, the row needs 13,026 kN·m/rad in all against 7,067 at the best share, 61 per cent.

Tested in The braces that stay where the framing puts them, at the figure it turns on · the essays about beam bracing.

Grading braces in stiffness and moving them toward the moment are separate savings, so a row that is both moved and graded saves the sum of the two.

What decides it: Under a uniform load, moving the three braces to 3.75 m either side of midspan cuts the total from 8,537 to 6,282 kN·m/rad, and grading them as well leaves it at 6,282. Under a point load, 3,920 moved and 3,915 moved and graded. Both savings come from evening out the twist at the braces, and once it is even there is nothing left to grade.

Tested in The braces that stay where the framing puts them, at the figure it turns on · the essays about beam bracing.

A ground-floor slab cast before the backfill props the basement wall for the life of the building.

What decides it: It props the wall only until it has shrunk 2.08 mm away from it at each end. A 30 m slab, 250 mm thick and drying from its upper face at 60 per cent humidity, gets there after a year; a 60 m slab after five months. From then on the stem is a cantilever, with a base moment of 69.6 kN·m a metre against the 26.6 it had propped.

Tested in The slab that shrinks away from the wall, at the figure it turns on · the essays about lateral pressure.

Tying the slab into the top of the wall keeps the wall propped while the slab shrinks.

What decides it: A tied slab does not stop shrinking; it pulls. Past 2.08 mm the slab is in tension and the stem is pulled away from the fill: at 4.5 mm the slab pulls at 24.8 kN a metre and the base moment is 119.9 kN·m a metre, more than the 82.9 of the same stem as a free cantilever.

Tested in The slab that shrinks away from the wall, at the figure it turns on · the essays about lateral pressure.

The restraint moment at a continuity joint can be estimated by holding the free end rotation that is still to come with the composite section's elastic stiffness.

What decides it: It overestimates by about half at every casting age. For a joint cast at 28 days, 425 kN·m elastic against 211 with the beams' creep relaxing the restraint as it arrives; at a year, −402 against −214.

Tested in The joint cast over the pier, at the figure it turns on · the essays about end rotation.

Retightening a column base's holding-down bolts after a storm returns the base to the state it was built in.

What decides it: It returns the preload and the rigid range — the bed lets go at 57 kN·m again — but not the bolts. Each storm to ±170 kN·m stretches a retightened bolt row permanently by another 0.12 mm, and after ten storms it has 1.25 mm of permanent stretch where bolts left alone after the first storm have 0.12.

Tested in The nut turned after the storm, at the figure it turns on · the essays about base plate.

Tightening a base's holding-down bolts harder leaves more preload in them after a storm.

What decides it: Only up to the preload the storm can live with. To ±170 kN·m, a bolt tightened to 128 kN keeps 128 after the storm; one tightened to 150 keeps 124; one tightened to 200 keeps 112. Above that point every extra kilonewton of preload is taken by the storm, and more besides.

Tested in The nut turned after the storm, at the figure it turns on · the essays about base plate.

A friction joint in an outrigger, set to slip below the force creep would put into the arm, lets the arm ignore the differential shortening while it still carries the wind.

What decides it: Set to slip at 2 MN, the joint is brought to its slip load by creep after 3.7 years and holds it for the rest of the building's life. The arm on the side where the wind adds to the creep force then has no margin at all before the joint slides, against a design wind of 5.8 MN; the slip load that would hold that wind, 9.3 MN, is never reached by creep, so the joint is a rigid arm with a bolted connection in it.

Tested in The arm that slips on purpose, at the figure it turns on · the essays about outrigger.

A damper that flows under the slow creep of the core and columns must also flow under a storm, so it cannot carry the wind.

What decides it: A viscous damper whose time constant is ten days lets the arm collect at most 2 per cent of the force a rigid arm collects from creep, and holds 99 per cent of a storm's sustained wind after three hours. Any time constant from about 2.4 to 45 days does both jobs.

Tested in The arm that slips on purpose, at the figure it turns on · the essays about outrigger.

Setting the upper beams' bearers further in than the bottom beam's, so that the stack sags it, protects the stack.

What decides it: It helps the bottom beam by 0.03 N/mm², because its worst section is then over its own bearer, which a load landing inside cannot reach. It costs the beam above 1.29 N/mm² for every metre its own bearers move in: 500 mm in, its top fibre is at −3.63 against the limit of −3.0.

Tested in The beam at the bottom of the stack, at the figure it turns on · the essays about prestress limits.

A tuned mass whose actuator switches off falls back to being the passive mass, so a switch can never leave the structure moving more than the passive mass would.

What decides it: Under a steady force at 0.9 of the structure's frequency the law holds the structure to 4.63 times its static deflection and the passive mass would hold it to 6.97. In the cycles after the actuator lets go, it reaches 8.78 — more than either, and more than the passive mass's own worst peak of 8.72 — before settling to 6.97.

Tested in The switch that hands the mass back, at the figure it turns on · the essays about tuned mass damper.

Fading the actuator's force out over several periods, rather than cutting it at once, softens the change from the active law to the passive mass.

What decides it: Cut at once, the structure peaks at 8.53 after the switch. Faded over ten periods, 10.23; over a hundred, 13.48, with the stroke at 1.53 times its limit. A fading law passes through every law that cancels only part of the spring, which are the laws that pump the mass's own mode.

Tested in The switch that hands the mass back, at the figure it turns on · the essays about tuned mass damper.

A damper that gives a building 20% of critical damping in its design earthquake will damp the much smaller motions of everyday wind at least as well, since a small motion is an easier one to damp.

What decides it: A damper of velocity exponent 0.3, sized to its brace's best at the earthquake's amplitude, gives 22.7% of critical damping there and 1.9% at a tenth of that amplitude — less than the structure's own 2% — because at small velocities its force is relatively so large that it hardly strokes, and the brace carries the motion as a spring.

Tested in The damper sized for the earthquake, at the figure it turns on · the essays about damping.

A counterweight that measures the payload and balances all of it leaves the mast with nothing but the error in the measurement.

What decides it: Balanced for all of its payload, the crane is thrown backwards by the payload's whole moment when the load is let go, and by half as much again in the rebound: 1.5 times the payload's moment with a rebound factor of 0.5, the largest of every option — the linked crane's half gives 1.0 and a counterweight balancing 0.35 of the payload gives 0.85.

Tested in The counterweight that knows the load, at the figure it turns on · the essays about counterweight.

A flexible roof is always more vulnerable to rainwater than a stiff one, because its sag makes room for more water.

What decides it: Only when the water's surface is fixed. With a drain at the middle of each bay whose inlet goes down with the roof, the surface follows the sag, and a 24 m square bay sized to L/240 puts 0.69 of a rigid roof's load in its members and allows 221 mm of head, where the same bay with a fixed scupper allows 3 mm.

Tested in The roof that drains towards its sag, at the figure it turns on · the essays about serviceability.

A row of screws that has carried a load once will carry it again, so a load below the row's peak can be repeated indefinitely.

What decides it: Twenty leaning screws whose reloading regains 85% of the load at the slip each had reached, loaded again and again from new to 0.97 of the row's first peak, settle on no cycle: the end screw's slip grows from 2.65 mm to 3.73 and the row cannot reach the load at cycle 21. At 0.965 and below it settles, after a few cycles of creeping slip.

Tested in The row that remembers its overload, at the figure it turns on · the essays about dowel yield.

An overload that takes a row's end screws past their peak leaves the row weaker against repeated loads than loads that never reached the peak.

What decides it: After one overload to the end slip at which the row peaked, the row settles under every repeated load drawn, up to 0.99 of its first peak; from new it settles only up to 0.963. The overload damaged the end screws once; loads repeated near the peak push them back onto their curve and damage them every cycle.

Tested in The row that remembers its overload, at the figure it turns on · the essays about dowel yield.

Repeated rocking compacts the soil under a footing's ends into a bowl the footing then hangs over, so that its contact shortens and its bending grows from one storm to the next.

What decides it: On ground that unloads and reloads four times as stiffly as it loads, an 8 m strip rocked to ±1,000 kN·m leaves 0.81 mm of permanent set at each end, and the strip settles 1.1 mm deeper overall and follows it down: at rest it is on the ground over its whole length after every storm, and its largest bending moment at the top of the storm changes by 2%.

Tested in The footing the storms press down, at the figure it turns on · the essays about elastic foundation.

A column hinge whose axial load rises as it rotates follows the moment–curvature curve of its final axial load, so it needs that load's larger curvature before it develops a hinge at all.

What decides it: An I-section whose compression rises from 0.1 to 0.6 of its squash load as its curvature goes to twelve times first yield carries 0.86 of its unloaded plastic moment at twice the first-yield curvature, where the curve at 0.6 carries 0.43 and needs 6.4 times the curvature to develop 98% of it. The hinge forms at the starting load and then follows the plastic moment at whatever load it has reached.

Tested in The hinge squeezed while it turns, at the figure it turns on · the essays about Moment-curvature.

Σbt³/3 underestimates the torsion constant of every rolled open section, because it leaves out the fillets.

What decides it: For a channel 200 mm deep and 90 mm wide with a 7 mm web, 14 mm flanges and 12 mm roots, the film over the rolled outline gives 0.992 of Σbt³/3: the sum is high by 0.8%. Its four free flange tips cost more than its two L-shaped junctions with one fillet each give back. An I-section with the same plates, whose T junctions have two fillets each, is 8.3% above its sum.

Tested in The corner with one fillet, at the figure it turns on · the essays about torsional constant.

A fillet at a junction adds to the torsion constant in proportion to its own area, wherever it sits.

What decides it: With the same 7 mm web, 14 mm flanges and 12 mm roots, each fillet at the channel's L junctions adds 0.65 cm⁴ and each fillet at the I-section's T junctions adds 1.02 cm⁴ — the same area of steel, worth half as much again where two plates meet a third.

Tested in The corner with one fillet, at the figure it turns on · the essays about torsional constant.

The stiffer the brace on an arch's sideways mode, the more load the arch carries.

What decides it: A brace raises the perfect arch's critical load only until the sideways branch no longer comes first. For a rise of eight radii of gyration that is at 1.75 times the rib's antisymmetric stiffness, where the arch returns to its symmetric limit point, 844; any stiffer brace leaves it there.

Tested in The brace that holds a mode, at the figure it turns on · the essays about Snap-through.

A floor stiff in shear sends its lateral load to the nearest brace bay, whatever the bays' own stiffnesses.

What decides it: A 150 mm concrete slab, the stiffest floor drawn, gives a braced bay of 200 kN/mm 16% of its line's force when the bay at the line's other end is ten times as stiff — close to the 9% a rigid floor gives by stiffness and far from the 50% that each bay's nearer half would give. It is the steel deck, the softest floor, that comes nearest to halves, at 37%.

Tested in The floor that feeds two bays, at the figure it turns on · the essays about diaphragm.

Sizing a braced bay for its share under a rigid diaphragm is safe, since a real floor is stiff enough to share the load as a rigid one would.

What decides it: The rigid floor gives the braced bay 9% of its line, 86 kN. The uncracked slab gives it 155 kN, the slab cracked 291 kN and a steel deck 353 kN: the same bay under the same storey force carries up to four times its rigid-floor share, and the cracked slab — the state an earthquake leaves it in — more than three times.

Tested in The floor that feeds two bays, at the figure it turns on · the essays about diaphragm.

A branch that turns through an elbow before its brace is flexible, so it neither loads the riser nor carries much stress itself.

What decides it: The elbow does free the riser — from 71 N/mm² at the tee to 4 with 300 mm of second leg — but the storey's lean still has to be taken somewhere, and it bends the branch's first leg and the elbow: the tee carries 82 N/mm² once intensified and the elbow 84, against 16 at the tee of the straight held branch.

Tested in The elbow that takes the strut away, at the figure it turns on · the essays about floor spectrum.

The storey rule is conservative for every kind of building it is applied to.

What decides it: Where the sustained load dominates — a floor whose settled use carries 2 kN/m² on average, as an archive or library may — and the building has one occupier, the column's fifty-year load exceeds the rule's in 24.9% of lives against the intended 5%: the factor the averaging allows is 1.01, the rule's 0.73.

Tested in The load that stays and the load that visits, at the figure it turns on · the essays about live load reduction.