Every essay — page 29
Sections and stress
The beam the haulier holds
A pretensioned beam leaves the yard on a trailer, carried on a turntable and a bogie the haulier places, and bounced on them at road speed. The bounce works both ways: a wheel rising adds to the beam's weight over its overhangs, and a wheel falling away takes weight off — and with it the sag that was relieving the beam's top fibre between its supports. On the yard's own bearers a bounce of 30 per cent takes as much from that fibre as a stack placed 100 mm out of line. A rear overhang of 2.5 m takes five times as much. With the yard's schedule, a trailer may hold this beam only within half a metre and three quarters of a metre of its ends.
Stability
The braces that pull only one way
A brace made of a single tie, or a cross-frame whose diagonals can only pull, resists a girder twisting one way and not the other. A buckling girder has no preference between the two directions, so it takes the one its braces leave free. Three such braces all facing the same way leave a 24 m girder exactly as weak as if they were not there. Facing alternately, they reach 3,087 kN·m at the stiffness that takes two-way braces to 5,627, and no stiffness takes them past 4,063. Opposing the outer pair does not help; one one-way brace anywhere in the row is enough to cap it. Every brace has to resist both ways, and made of ties that costs twice the steel.
Structural form
The arm that is reset by hand
An outrigger connected to a concrete building at topping out collects the core's and the columns' differential creep for fifty years, 3.48 MN in each arm against a design wind of 5.80. A damper lets the creep through and stops the wind, but it is a machine. The older answer is a joint closed with shims and opened every few years to take the creep out. Three visits, at one, three and ten years, hold the arm below 1.40 MN, and spaced like the creep they do it with roughly equal thicknesses of shim each time. Evenly spaced, the same three visits hold 2.48. And a visit that is missed costs exactly the creep it was there to remove.
Dynamics
The forging that cushions the blow
A forge hammer's foundation is usually designed for a blow treated as a fixed impulse: the tup's momentum, multiplied by a factor for its rebound. The blow is not fixed. Between the tup and the anvil lies a hot forging, a plastic body that flows under the tup, stretches the blow to 48 milliseconds and absorbs nearly all of its 49 kJ, so that the anvil's pad feels 0.46 MN. As the part cools it flows less, the blow shortens, the tup rebounds, and the pad force climbs to 5.86 MN when the dies meet with nothing between them. The foundation is designed by the blow that does no work.
Deflection
The joint poured twice over the pier
A continuity joint over a pier can be cast in two stages — the diaphragm and the beams' bottom connection first, the slab weeks later — in the hope that the early pour catches the beams' creep and the late one leaves the slab's shrinkage little to do. It does neither. From the slab's day on the composite section holds exactly what a single pour on that day would have held, and the first stage adds only its own small account, of either sign. It trims a year-old slab's hog by a fifth, and it asks a joint with half the cracking moment to carry the beams through the months between.
Connections
The splice that bends as one section
A plate spliced with two unequal covers shares its load by lever arm about the line the load is held on, and the thinner cover carries more. A beam's flange splice looks like the same joint, and the rule that worked for the plate puts the thin inner covers at the edge of yield. They are working at half that. A beam's halves are not held on any line; they turn, and every splice plate across the gap turns with them, so the plates bend as one section and share the beam's moment by their distances from its neutral axis. The outer cover works hardest, the inner covers less, and the web plates less than the hand design hands them.
Equilibrium
The column under the escape corridor
A column's imposed-load reduction is allowed by the number of floors it carries, on the reasoning that floors are not crowded at the same moment. A column whose tributary area includes escape corridor carries a part of every floor that is crowded at exactly the same moment, every time the building is evacuated. Its reduction follows how much corridor its area holds, not how many floors it carries — a tenth costs little, two-fifths uses up the whole of the storey rule's allowance, and a stair-core column needs almost none. Against the nominal loads a designer writes down, the margin that saves the columns at the corridor's edge has run out by the time the area is mostly corridor.
Internal forces
The footing a prevailing wind leans
A footing rocked by storms that reverse evenly settles into a symmetric bowl and stops. A footing under a frame in a prevailing wind is pushed harder one way than the other, and its ground keeps the difference. It leans — and then it stops leaning, within three storms, at a tilt set by how unequal the storms are and how large the largest has been. Fifty years of storms lean it further only in the years when a storm beats every one before it, and between those years nothing that happens to the footing moves it at all.
Materials
The strength three cores revise
A characteristic strength estimated from three specimens and nothing else is heavily penalised, because three results cannot say how scattered a material is. An existing structure is never nothing else. It was built to a specified grade, from a population of concretes whose scatter is known in general, and three cores drilled from it revise that knowledge rather than replace it. Together they give a characteristic strength that three cores alone would need twenty to match. The same arithmetic catches a concrete much weaker than its specification, because cores that disagree with the prior widen it — unless the prior is given so much weight that it no longer listens.
Sections and stress
The deck whose frame is its crossbeams
A steel box girder resists distortion as a beam on an elastic foundation, the foundation being its walls' transverse bending. When the top flange is an orthotropic deck, that foundation is almost absent between the crossbeams and concentrated at them, so the box sits on a row of springs, not a bed. Smeared into a bed, the springs give the right answer to within a few per cent as long as they are closer together than the bed's own decay length — and the rule holds for every crossbeam stiffness on one curve. Past that spacing the frames act as separate diaphragms and the warping stress between them doubles.
Stability
The prop at the crown of the arch
A shallow arch that turns sideways before it snaps can be braced at its quarter points, which holds the sideways mode and carries almost nothing. A single prop at the crown is cheaper and does something else. The crown does not move in the sideways mode, so the prop cannot stop it; it raises the load at which the arch turns aside only by taking part of the load itself. But behind a prop about three times as stiff as the rib's own first mode, the sideways branch stops falling, and the arch no longer snaps at all — at every rise, by the same prop.
Structural form
More hangers, and the same steel
A network arch's crossed hangers are the web of a truss, and a live load's shear passes through them, so every hanger swings through nearly all of its force. The obvious cure is more hangers, each crossing more of its neighbours, so that every section's shear is shared among more diagonals. It works hanger by hanger — three times the hangers, a third of the range — and not at all in total: the hangers' fatigue steel, sized for their ranges, is set by the shear the arch carries and barely changes with how many hangers it is divided among. What more hangers buy is elsewhere.
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.
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.
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.
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.
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.
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.
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.
Structural form
The span cast in the afternoon
A stressed ribbon's joints are cast on whatever day the programme reaches them, and a span cast on a warm afternoon sets around strand that has lengthened in the sun. It cools that night as a concrete band eleven times stiffer than the strand, and the band does not stretch back as the strand would have. On a ribbon with a sag of 2 per cent, a span cast 20 °C warmer than its neighbour ends 72 mm higher than it was set and leaves 713 kN on the pier between them — a seventh of what a crowd on one span puts there. On a flatter ribbon the same afternoon costs more than the crowd. The difference between the two spans is the strand's flexibility against the band's, and only the elastic part of it differs; the part that comes from the sag is the same for both.
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.
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.
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.
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.