Every essay — page 30
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.
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.
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.
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.
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.
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.
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.
Deflection
The pier's share is cut into the members
A truss continuous over a middle support has one reaction that statics cannot find, and the lengths its members are cut to decide it. Cambered to land level under its dead load on three bearings, a two-span truss puts 572 kN on its pier; cambered span by span, as two simple trusses, the same members on the same bearings put 480 kN there and carry the dead load with no moment over the pier at all. The list chooses the moment diagram. And a millimetre of shop tolerance on every member moves the pier's share as much as a bearing set fourteen millimetres out of level, so a reaction the design depends on has to be set by a jack rather than by a drawing.
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.
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.
Internal forces
The slip that peaks where the moment turns
A simply supported composite beam slips most at its supports, and its studs are crowded there. Make the same beam continuous over a middle support and the slip peaks twice: at the end support, and again, just as hard and the other way, at the point of contraflexure — not over the middle support, where the shear is largest but the slip falls to nothing. Studs graded toward the supports, as a simple span's would be, miss the second peak and buy a tenth; graded to the slip they buy more than a third. And the hogging region cannot give its studs away to the span, because its edge is where the second peak is.
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.
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.
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.
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.
Connections
The row that is pushed as often as it is pulled
A row of leaning screws in a shear wall is loaded one way by the wind and then the other, and a screw that leans is pulled along its axis one way and pushed the other. Pulled, its axial force clamps the joint and friction helps it; pushed, it prises the joint open and the friction goes. So the row has a weaker side, and under a reversing load it settles exactly as it settles under a repeated one — the reversal does not start the ratchet again — but at a ceiling the pushed side sets, 0.79 of the pulled peak against 0.97. And it settles with a gap in the middle that every reversal crosses carrying nothing.
Deflection
The check that cannot see the beams
The unit load that finds a frame's sway may be carried on any structure that can carry it, and in a frame of four bays and ten storeys it can be carried down one line of columns alone. Then the roof sway is that line's real moments against the unit load's, a few dozen products by hand — exact for the frame as drawn, for its beams cracked to a third of their stiffness, for its joints made springs and for its bases made springs, the last only once the bases' rotation is given its own term. It is exact because the unit load never enters a beam. Which makes it a check on the analysis's arithmetic that cannot say anything about the analysis's beams.
Dynamics
The weight that lands somewhere else
A beam that yields under a falling weight takes a permanent set from every drop, and if the weight always lands in the same place the set grows by the same amount every time. Let the weight land somewhere different each time — rocks on a canopy, crates on a deck — and the set still grows: the weight's energy fixes how much plastic work every drop does, wherever it lands. What the landing point decides is where the work goes. Half of every drop's hinge rotation goes under the weight and half to the supports, so wandering drops spread the first half along the span and leave the second half where it always was: the worst hinge turns half as fast, and it is at a support.
Equilibrium
The release the rope cannot slow
A crane whose counterweight knows the payload balances only a third of it, because a payload can be let go and the crane is then thrown backwards by whatever the counterweight was holding, and half as much again in the rebound. Slow the release and the rebound goes, so the hope was that a device on the rope could slow it. Nothing on the rope can: the rope holds only what the hook holds. How fast a load goes belongs to the payload, and what the crane owns is how fast it answers — its period and its damping. And with no rebound at all, the counterweight still should not balance more than 0.6 of the payload, because the payload's own moment is still let go.
Internal forces
The joint stronger than its beam
A full-strength joint never reaches its own resistance: the beam beside it hinges first. But an end plate softens from two thirds of its resistance, and a joint only a little stronger than its beam is already on the soft part of its curve before the beam yields. The softening sheds moment from the beam's end into its span, so it delays the first hinge — by 8 per cent for a joint at 1.2 times the beam's strength, 18 for one at exactly the beam's, and not at all from 1.5 on. It never moves the collapse. And too much shedding hands the first hinge to the span, so the first hinge comes latest at the stiffness where the end and the span yield together.
Materials
The floor that cannot keep up with the column
A column whose hinges ratchet under an earthquake shortens every cycle, and in a frame it does not shorten alone: the floors it carries bend down towards it and lift its load onto the columns beside it. That ought to slow it and speed them, until every column shortens together. It does — over hundreds of cycles. Over the ten or twenty of an earthquake, the floors of a six-storey frame lift an eighth of the load off the interior column, it shortens nine tenths as far as it would alone, and the floors carry the whole difference as bending. The frame equalises its columns eventually, and pays for it in its floors.
Sections and stress
The flange bent hardest where it curls least
A curved I-beam's flange curls towards its web and is bent across its own width at the root, by a stress that rises with the curling parameter λ. A portal knee's flange is cut from plate and usually tapers, and a taper changes λ along the curve. So the obvious guess is that the transverse stress peaks where λ is largest. For a flange that only narrows, or only thins, it does. Taper it both ways and the guess fails: λ is smallest at the narrow, thin end, and that is where the flange is bent hardest, because the same force is carried there on half the steel. The force the changing width hands to the web, meanwhile, is a few newtons a square millimetre.
Stability
The root that belongs to the column
A cantilever's root must hold its flanges against warping at about fifty times the cantilever's own warping stiffness to count as built in, and doubles its buckling moment if it does. A flush end plate is the usual root, so it ought to be the plate's thickness that decides. It is not. The flanges warp by turning in plan in opposite directions, the plate passes that to the column as two opposite torques, and an open column twists between them more easily than the plate bends: three quarters of the root's flexibility is the column. An ordinary root is worth about six times the cantilever's stiffness, halfway up the curve, and a thicker plate barely moves it. Boxing the column does.
Structural form
The floor that twists in its own time
Seismic design standards flag a twisting floor by the ratio of its worst edge's movement to the average of its two edges', and a static push saturates that ratio. Give the floor mass and shake it, and it sways and twists in two modes that share the motion. A response-spectrum analysis then reads the same ratio from combined peaks — and gives a different verdict on a third of the plans tried. Floors torsionally stiffer than they sway read worse than a push says, because the coupled modes put twist at the edges. Floors torsionally flexible read better, because combining the modes keeps how far each edge moves and forgets which way, so a twisting floor's two edges both look large.