Every essay — page 28
Connections
The root that has to stretch for it
A partial-penetration weld made from one side carries its pull off the plates' line, and counted elastically it keeps a quarter of a double-sided weld at half penetration. Let the throat yield and it keeps 0.41 — but the same 0.41 at every direction up to the one where the double-sided weld is strongest, so that weld's peak advantage is never won back. What the yielding does instead is move the single side's own strongest direction out to 40° or beyond, where a deep single-sided throat outcarries a fillet. Every per cent of it is paid for with strain at the root, which is the tip of an unfused land.
Deflection
The truss sized to lose a diagonal
A Pratt truss with a counter-diagonal in every interior panel, sized fully stressed, starves its middle counters to nothing and saves a little steel over the plain Pratt. Ask it to survive the loss of any one diagonal at the accidental two thirds of its load and the family of equal designs collapses to one: every counter comes back, sized by its partner's loss, and the truss needs 1,304 units of steel against 1,200 — 9 per cent for an alternative path the Pratt cannot have at any price. A lost chord can be survived too, by the panel's diagonals turned into a chord, for twice the steel. Four members in the end triangles cannot be survived at all.
Dynamics
The lever that moves the damper and not the band
A damper whose force goes as a low power of its speed is sized for the earthquake and locks in the wind. A toggle or scissor linkage multiplies its stroke, and it seemed it might move the band of amplitudes the damper serves down towards the wind. It cannot. Behind a lever of ratio f the same damper acts on the storey as one f to the power 1 + α larger, so its band moves up, not down; re-sized to the design amplitude, it gives the storey exactly the curve it had without the lever, and what has changed is the damper — a quarter of the size, a third of the force, three times the stroke. What would widen the band is a ratio that grows with the motion, and a toggle has one — until its links lie straight and it locks.
Equilibrium
The mechanism that stands up anyway
Two bars in a straight line between two pins are a mechanism by the count, by the rank and by every measure of how nearly a frame folds. Tension them and they carry load: 200 kN of tension gives the joint the stiffness of the same bars raised 1.8 degrees, and where the bars raised 3 degrees snap through at 11 kN, the tensioned flat pair carries 50 kN and goes on stiffening. The stiffness is a lack of fit of four millimetres, which 83 degrees of warmth takes away; compressed, the same pair buckles to rest crooked on one side or the other; and no tension can reach a mechanism that swings rather than shivers.
Internal forces
The joint that softens under its own load
A bolted end plate is stiff until its moment passes two thirds of its resistance and then softens until, at its resistance, it is a third as stiff. Put such joints at every beam end of a frame under a pattern load and a push, and the most loaded joint softens first and hands its share across the column to the joint on the other side, which carries six times as much for three times the load. The redistribution grows as the frame approaches its strength. Every joint ends at its own stiffness, so the single secant stiffness an elastic analysis is allowed asks the loaded joint for half as much again as it can carry, and gives the exterior panel three quarters of what it gets.
Materials
The wire that cannot push
A tube's liner wound with wire under tension is a compound tube of a hundred layers, each shrunk on by its own winding. It reaches the same ceiling, twice the first-yield pressure, and no further, because the bore's range belongs to the whole wall and its floor to the liner's steel. What the wire's strength decides is where the ceiling can be reached at all. A wire cannot be compressed, so a jacket of the liner's own steel needs a wall three times the bore, where shrunk-on layers of the same steel need 2.22; wire half as strong again needs 1.69; wire twice as strong reaches it on any wall. Every turn is wound harder than it will rest, by what all the turns after it take away.
Sections and stress
The diaphragm that is a few angles
A box girder's diaphragm holds its section square against the distortion an eccentric load drives. It does not have to be rigid; it has to be stiff against the girder's own resistance to a point racking load, which is small — twice the section's frame stiffness times its decay length. A cross-frame of two 1,000 mm² diagonals is twenty-six times that and holds the warping stress at the rigid value, while a 10 mm plate is five hundred times it. The cross-frame lets the section rack three times as far as a plate, which leaves the walls bending at 5 N/mm² instead of 1.5, against 275 with no diaphragm at all. And its diagonals carry nearly a rigid diaphragm's force whatever their stiffness, so it is sized for strength.
Stability
The trace that adds up every storey
A column's buckling load can be bounded from below by the traces of its flexibility, with no shape guessed. A sway frame has a flexibility too — one number per pair of floors — and its traces bound the frame's critical load factor from below and from above, closing in as more are taken: on a six-storey frame whose factor is 14.96, the first four give 14.07 to 16.39. But the first trace is Dunkerley's sum over the storeys, counting each as though its sway mode were the frame's, and it bounds a well-balanced frame worst of all: one over the number of storeys when every storey is alike, which is exactly when Eurocode 3's storey-by-storey estimate is exact.
Structural form
The hanger that cannot be made soft enough
A network arch's crossed hangers swing through nearly all their force as traffic passes, because they are the web of a truss. Make them softer — a spring at each anchorage — and the tie ought to take back some of the shear and the hangers' ranges ought to fall. They hardly do. A tenth of the stiffness takes 6 per cent off the worst range and adds a third to the tie's bending; a fiftieth takes 18 per cent off and multiplies the tie's bending by 3.7. The range belongs to the truss and leaves only when the truss does, which vertical hangers show at 194 kN against 678 and eleven times the bending. So the hanger is sized for its range, eight times the area its strength needs, and the stiffness that comes with it costs the range nothing.
Materials
The column that shortens on every cycle
A column hinge bent back and forth by an earthquake carries its share of the building's weight the whole time. In elastic–perfectly plastic steel an I-section at three tenths of its squash load, cycled to three times its first-yield curvature each way, comes back from every cycle shorter by eight yield strains, and goes on doing so: it never settles. The flange that yields in compression shortens, and the other flange, carrying its part of the column's load, never reaches yield in tension to pull it back. A rectangle of the same area shortens a fifth as fast, and the overturning that swings the column's load up and down hardly matters. The weight and the curvature make the ratchet between them.
Stability
The braces that have to hold each other
One brace at mid-height must have a stiffness of twice the column's load over the length it halves. Spread the same column over four braces and each one must be stiffer, not softer: 3.62 times the load over its segment, rising towards four with every brace added, because the shape the braces have to stop is a zigzag in which every brace point is pushed by both its neighbours. Braces that tie the column to itself instead of to the ground need one, however many there are. A row with surplus stiffness can carry one brace at two thirds of the ideal, but not one that is missing, and the force each brace has to resist grows with the count.
Internal forces
The shear whole spans cannot find
A continuous beam is designed by loading its spans in patterns, whole spans at a time, and for the bending moment that is very nearly exact: the moment's influence line keeps one sign over each span, so a span is either worth loading or not. The shear's influence line changes sign at the section itself, inside the span, and no pattern of whole spans can load one side of a section and not the other. At the middle of the inner span of three equal spans the imposed shear is 0.198 qL; whole spans find 0.083 qL, and the straight line to qL/8 that is safe on a simple span finds 0.125.
Sections and stress
The notch that costs more than its depth
Cut a timber beam's underside away at the support, so that half its depth sits on the bearing, and a stress check on what is left says the end has lost half its shear strength. The beam says otherwise. A crack starts at the notch's corner and runs along the grain, and whether it runs is a question of energy, not stress — so the answer has a size in it. A 400 mm glulam beam notched to half its depth keeps a quarter of its shear resistance, an 800 mm one keeps 18 per cent, and the deeper the beam the less a notch leaves. Sloping the notch buys much of it back, and moving the corner away from the support costs it again.
Structural form
The deck the stays squeeze
Every stay of a cable-stayed bridge pulls the deck towards its tower, and the deck carries all those pulls as a compression that is largest at the tower — 95 MN in a 1,000 m span under 150 kN/m. The stays hold it up and hold it straight, as springs under a strut. The compression grows with the square of the span and the springs soften with it, so the factor on its own weight at which the deck would buckle falls as the span to the power one and a half: 41 at 300 m, 6.9 at 1,000 m, 2.3 at 2,000 m. A deck four times stiffer only doubles it, and the long stays' sag, which softens them for traffic, hardly enters.
Dynamics
The tank that is its own tower
A standpipe holds its water in its own shaft, so the water fills a height several times its width. Only a lens of it near the surface sloshes — 0.455ρπR³, 178 t of the 2,199 t in a steel standpipe 10 m across and 30 m tall — and that lens stops growing once the water is deeper than a radius, so its period is set by the width alone: 3.3 s, against the shaft's 0.38. The lens is 8% of the water but a quarter of the shaft's moving mass, because it sits where the shaft moves most. It is far too slow to be tuned as a damper, and a screen can do little with it; what it does instead is stay still while the shaft sways, which takes nearly a fifth off the drift a frozen-water calculation predicts.
Equilibrium
The visits that come to every floor at once
A floor's characteristic imposed load is set mostly by its worst visit — a party, a move, a crowd — and visits to different floors almost never coincide, which is why a column carrying twenty office floors needs only about a quarter of their summed loads. Let some of the visits be made by the whole building at once and nothing changes on any one floor; one floor cannot tell. The column can. A tenth of the visits shared takes the twenty-floor factor from 0.26 to 0.51, a quarter to 0.68, and a third uses up the whole of the storey rule's reduction. The rule fails in exactly the buildings whose use synchronises the crowd, and it fails most where the load mostly visits.
Deflection
The section that yielding cannot reach
A truss that loses a member in an accident is allowed to yield while it finds another way to stand, so a counter-braced truss sized by the lower-bound theorem ought to need far less steel than one sized elastically. It needs 1,296.7 units against 1,304 — seven of the 104 that robustness costs. One diagonal gone, the damaged panel is determinate at its own section, and no redistribution can change a force a cut decides; yielding can reach only the verticals. What does move the price is a counter that buckles: good for half its tension in compression, it adds 120 units, more than robustness did.
Connections
The spring under the nut
Retightening a column base after every storm makes its bolts ratchet, because a storm can only live with 128 kN of preload and the spanner keeps putting back 150. A stack of disc springs under each nut changes what the storm can live with. Soft enough, the bolts never reach their yield: the compression side keeps its preload, the bed's resultant moves out to the plate's edge, and the moment rides on lever arm rather than on bolt force. A stack as stiff as the bolt itself keeps 150 kN through every storm. Much softer, and the storm the bolts no longer feel lands on the grout.
Deflection
The floor that is level on one date
Each floor of a tall concrete building is set high by the amount its columns are predicted to drop against the core, so that it arrives level. Counted in time, the drop does not peak at mid-height and stop at the roof: the worst floor of a forty-storey building is level 31, the roof moves 24 mm, and half the movement is still to come when the floors are fitted out. Set high by all of the prediction, a floor is level on one date thirty years away and 16 mm high at handover. With the creep known to ±30 per cent, setting it high by about four fifths of the prediction makes the worst floor's error 11 mm instead of 16. And none of it changes what the partitions see.
Connections
The clamp the pair holds longer
A preloaded bolt clamps an end plate to a column flange, and the joint stays closed until the tension relieves the clamp. The joint diagram says that happens at the preload over one minus the bolt's share — 122 kN for a 110 kN preload — but prying adds to the tension at the bolt line and the clamp lets go at 88. A component model checks each flange as a stub on a rigid base and lets the joint go when its weaker stub does, at 81 kN. The real pair, its tips leaning on a column flange that gives way, pries less than the end plate alone would, and holds its clamp nine per cent longer — fifteen when its lever arms are two to one.
Dynamics
The tank that must be shallow to be slow
A deep tank's sloshing period is set by its width alone, so a tank in a tall building's core ought to be tuned by choosing how wide to make it. It cannot be. A deep tank fast enough to fit is too fast for the building: 3.6 s for a core 10 m across, where a 5 s building would need a tank 19.5 m long and an 8 s one 50 m. The depth has to tune it, and only shallow water is slow — 1.73 m for 5 s, 0.63 m for 8. Shallow water is the efficient kind, with four fifths of it sloshing. But at the accelerations the tank exists to reduce, the wave it makes is a quarter of its depth on a 5 s building and seven tenths on an 8 s one, and the linear tuning it was designed by has stopped describing it.
Equilibrium
The pressure that switches at the hopper
Below a silo's bin a conical hopper funnels the grain to its outlet, and the wall pressure there does not keep growing with depth as a liquid's would. It peaks at the hopper's top edge. Filling, the edge carries twice the bin wall's pressure just above it; emptying, the grain converges on the outlet, its largest stress turns from vertical towards horizontal, and the edge carries two and a half times. Half a metre above the apex the stress is 2.6 kPa whether the bin holds 5 m of grain or 60 — which is why a silo empties at the same rate full or nearly empty.
Internal forces
The beam built towards its own middle
A continuous prestressed beam built one bay at a time from one end keeps about half the secondary moment a beam stressed at once would carry, and most of it at the end where it was finished. Built from both ends towards a closure bay, it is symmetric, and the closure's supports keep only a little more than the rest: 172 kN·m against 161. Built from the middle outwards, the first interior supports — the ones the load hogs hardest — keep 215, 63 per cent of the at-once value, for any length of beam. The order of construction is a design variable. Creep then takes back most of the choice.
Materials
The notch that trades stress for strain
A notch with an elastic factor of three triples the stress at its root only while the root stays elastic, which in a mild steel is up to a nominal stress of about 84 N/mm². Past that the root's stress and strain part company: at a nominal 200 the root carries 293 N/mm², not 600, and strains 0.61 per cent, twice what the elastic factor implies. Neuber's rule and Glinka's bracket the truth a factor apart, and in the low-cycle range that factor is a factor of two in life. The trade has two gifts in it: one overload leaves the root in compression, and a mean stress mostly yields itself away.