Structural Engineering & Statics
where the load goes
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The load must go somewhere — page 18

Essays 409 to 432 of 450 on this thread, in the same order.
The pressure on the wall jumps where the bin meets the hopper. The normal pressure on the wall of a bin 8 m across holding 30 m of a solid weighing 9 kN/m³, wall friction 0.45 and K = 0.45, over a conical hopper of half-angle 20° with wall friction 0.40, the solid's internal friction 35°, down the bin and then down the hopper: in the bin, K times Janssen's vertical stress (solid) and that times the 1.35 discharge factor (dotted); in the hopper, filling (light) and emptying (dark). At the foot of the bin the wall carries 38.1 kPa, 51.4 with the discharge factor. On the hopper's top edge it carries 75.8 kPa filling and 95.7 emptying — 1.99 and 2.51 times the bin wall's. Half way down the hopper, 46.3 and 36.7 kPa. 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.

 The order of building decides which supports keep the secondary moment. The secondary moment at each interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built and stressed from one end, from both ends towards a closure bay, from the middle outwards, and all at once. From one end: 161, 175, 160, 219 kN·m; from both ends: 161, 172, 172, 161 kN·m; from the middle out: 215, 120, 120, 215 kN·m; all at once: 344, 258, 258, 344 kN·m. 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.

 On the road the beam's top fibre has nothing left. The top fibre along the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, the worst of the two bounces at each section: on its yard bearers standing still and bouncing ±30 per cent, and on a trailer carrying it 1.5 m from its front end and 2.5 m from its rear standing still and bouncing; dotted, the yard's limit of −3.0 N/mm². Yard bearers, still, −2.98 at 2.0 m; yard bearers, ±30 per cent, −3.26 at 9.8 m; trailer, still, −4.68 at 9.5 m; trailer, ±30 per cent, −4.88 at 9.5 m. 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.

 Each new set of shims gives back the creep since the last. The creep force in one arm of a 200 m concrete building built a storey every week and fitted out over six months, with an outrigger at 149 m of 1.0 × 10⁸ kN·m a radian connected on day 385, over fifty years on a logarithmic axis: connected and left (dashed), adjusted once at 1 years, and adjusted at 1, 3 and 10 years, the arm's force falling to nothing at each adjustment; the design wind's 5.80 MN dotted. Left alone the arm ends at 3.48 MN. Adjusted once it ends at 2.58; adjusted at 1, 3 and 10 years it never holds more than 1.40 and ends at 0.91. 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.

 The forging is a pad nobody designed. For a 2.0 t tup at 7.0 m/s, 49 kJ, on the 150 t foundation with its 30 t anvil on a pad: the largest force through the anvil's pad (solid) against the forging's flow force (logarithmic), with the dies meeting at the right; dashed, the pad force if the same impulse arrived all at once, as a fixed-impulse calculation takes it. Through a forging flowing at 0.1 MN the pad carries 0.15 MN, against 3.70 for the same impulse delivered at once; at 1.5 MN, 2.32 against 3.93; with the dies meeting, 5.86 against 5.93. 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.

 A joint poured twice ends near where the slab's day puts it. For two 12 m pretensioned beams, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, made continuous over their pier: the moment in the joint, sagging positive, over thirty years, with its first stage — the diaphragm and the beams' bottom connection — cast at 28 days and the slab at 42, 90 and 365 (solid), against the joint cast once, with each slab (dashed). With the slab at 42 days it ends at 139 kN·m against 160 poured once; with the slab at 90 days it ends at 41 kN·m against 40 poured once; with the slab at 365 days it ends at −165 kN·m against −214 poured once. The first stage starts each history early and the slab's weight kicks it into hog on the day it is cast; after that each solid line runs beside its dashed one. 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.

 Across the gap the splice plates bend as one section. The top half of the splice of a 612 mm deep I-beam with 229 by 19.6 mm flanges, spliced at 700 kN·m by a 15 mm outer cover on each flange, two 8 mm inner covers beside the web under it and two 8 mm web plates 480 mm deep, to scale, cut at the gap where only the plates cross (the beam's own flange and web dashed beyond), and the stress through it. Bent as one section, the outer cover works at 206 N/mm² at its centre and the inner covers at 186, in proportion to their distances from the neutral axis; the hand design's area rule gives both 193; the plate splice's lever rule gives the outer 123 and the inner 350. 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.

 The corridor's share of a column decides its reduction. The factor on the summed characteristic loads that a column carrying n floors needs, against n, for office imposed load as the earlier essays drew it — a sustained part of mean 0.50 kN/m² and visits of mean 0.2 kN/m² a third of a year apart on each floor — and on the corridor a crowd of mean 1.00 kN/m² at one evacuation a year, on every floor at once, with 0%, 10%, 25%, 50%, 100% of the column's tributary area corridor; dashed, the storey rule. At 20 floors: 0.26, 0.30, 0.49, 0.90, 0.98, against the rule's 0.73. The lines are simulated, 4,000 lives at each point, and carry a few hundredths of sampling noise. 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.

 A prevailing wind leans the footing, and the lean stops. The 8 m strip of EI 540 MN·m² on a bed of 50 MN/m² under a 1,000 kN column on ground that unloads and reloads 4 times as stiffly as it first loads, rocked by storms that push to 1,000 kN·m one way and swing back all, three quarters, half, a quarter or none of that the other: the column's tilt at rest after each of eight storms, in milliradians, positive the way the storms push. After eight: 0.011, 0.064, 0.117, 0.171, 0.218. Each settles within three storms; the lean is set by how unequal the storms are, not by how many there have been. 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.

 Three cores and a specification say more together. The predictive distribution of a core's strength in an existing structure: from a concrete specified to a mean of 38 N/mm² and a coefficient of variation of 15 per cent, that prior carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, before any core is taken (dotted); from three cores of mean 33.0 N/mm² and coefficient of variation 12 per cent alone (dashed); and from both (solid); each with its 5 per cent characteristic value marked. The specification alone gives 27.5 N/mm²; the cores alone, 21.9, because three results cannot pin down a scatter; together, 26.5. 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.

 Ring frames close together are a continuous foundation. The corner-angle change along a 60.0 m box 4.0 m wide and 2.5 m deep, its top flange an orthotropic deck of 14 mm plate, its webs 14 mm and its bottom flange 20 mm, under 45 kN/m at 2.0 m from its centre line, in milliradians, with ring frames whose crossbeam has a second moment of 1.7 × 10⁹ mm⁴ every 4.0 m and every 12.0 m (solid), and each row smeared into a continuous foundation of the same stiffness per metre (dashed). At 4.0 m the two are indistinguishable, 1.12 and 1.12 at the worst point; at 12.0 m the box distorts between its frames, to 3.45 against 3.33. Without any frames the same deck distorts 145. 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.

 A stiff enough crown brace stops the arch snapping. The load parameter against the crown's deflection for a shallow pinned arch of rise 6.0 radii of gyration under a sinusoidal load, built with an antisymmetric imperfection of 0.10 radii of gyration, with a crown brace of κc = 0, 1, 2, 4; the dots mark the first peak of the load, where the arch snaps. κc = 0: peaks at 273; κc = 1: peaks at 299; κc = 2: peaks at 327; κc = 4: no peak, the load rising throughout. 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.

 More hangers, each swinging through less. Each hanger's force range, plotted at its mid-point along the span, for a tied arch of 100 m span and 17 m rise on 32, 64, 96 crossed hangers of 3,000 mm² bar at 60°, 60 kN/m of dead load and 60 kN/m of live load run on from either end. 32 hangers: the worst 678 kN, the mean 528; 64 hangers: the worst 369 kN, the mean 273; 96 hangers: the worst 241 kN, the mean 184. 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.

 Counted honestly, cores from one bay hand the estimate to the specification. The share of the structure's concrete that falls below the 5 per cent characteristic value estimated from three cores from one bay of a structure's concrete of mean 33.0 N/mm² and coefficient of variation 12 per cent, updating the specification — a mean of 38 N/mm² and a coefficient of variation of 15 per cent, carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, averaged over 2,000 drillings, against the correlation between two cores from one bay: the cores counted as independent (dashed), counted by their correlation (solid), and three cores drilled from three bays (light); dotted, the 5 per cent an honest estimate gives. At ρ = 0.5, 6.2% and 6.5%, and 4.8%; at 0.9, 7.4% and 7.3%, and 4.8%. 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.

 Closing late beats jacking again on the same date. For the 60 m concrete rib rising 6 m under 60 kN/m, struck at 28 days, on ground that spreads 8 mm at once and as much again as it consolidates, 90 per cent in 10 years: the springing moment after a century, against a date in the arch's first years — left three-hinged and closed on that date (solid), or fixed at striking and jacked back to no moment on that date (dashed); dotted, fixed and jacked at striking only, −173. At three months, −124 against −140; at a year, −84 against −117; at ten years, −15 against −76. The difference is the memory of the years before: the jacked arch's early restraint goes on relaxing after the second jacking, and the open arch had none. 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.

 Pumping water in or out keeps the damper, and the lost water barely matters. The peak amplification of a building with 1 per cent damping and a 10 m tank sized for a 5.0 s building, 1.73 m of water weighing 2.0 per cent of the modal mass, its sloshing damped at 7 per cent, against the period the building actually has: the tank left at its design depth (dashed), retuned by its water level with the water's mass following the depth (solid), and retuned with the mass held at the design value (light); dotted, the design period; bare, the building would reach 50. At 4.5 s, 21.7 left alone and 9.6 retuned; at 5.5 s, 22.6 and 10.8; at 6.0 s, 34.0 and 11.7, of which the water let out costs 1.7. 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.

 One alarm in a few hundred that empties the building sets the column's load. The factor on the summed characteristic loads that a column carrying 20 floors needs, for office load and a corridor crowd at one alarm a year, evacuated three floors at a time, against the share of alarms that escalate to every floor at once (logarithmic), with 25% and 50% of the tributary area corridor; dotted, each with every alarm a full evacuation; dashed, the storey rule's 0.73. 25% corridor: 0.26 with none escalating, 0.27 at one in five hundred, 0.33 at one in a hundred, 0.38 at one in twenty, against 0.49 in full every time; 50% corridor: 0.29 with none escalating, 0.39 at one in five hundred, 0.59 at one in a hundred, 0.75 at one in twenty, against 0.90 in full every time. 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.

 A jack set for the first day is wrong by the last. The secondary moment over the first interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built from both ends towards a closure bay, against the creep coefficient: not jacked; jacked at closure by the whole of what the order took, 183 kN·m there; jacked by 0.60 of it, so as to reach the at-once moment at a creep coefficient of 2.0; and by 0.89, so that the shortfall on the day and the overshoot at the end are equal; dotted, stressed at once, 344. On the day and at 2.0: not jacked, 161 and 302; jacked by the whole, 344 and 372; for the end, 271 and 344; balanced, 324 and 365. 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.

 A concrete rib creeps into snapping long before its prop matters. For the shallow pinned arch of rise 6.0 radii of gyration under a sinusoidal load, propped at its crown by a concrete column of stiffness κc = 3.5, the creep coefficient at which a load that stays on loses equilibrium, against that load as a share of the day's critical load: a steel rib on the creeping column (solid), which loses only the column's share, so that by a creep coefficient of 3 no load below 85 per cent snaps; and a concrete rib whose column creeps with it (dashed), whose lasting load acts as if (1 + φ) times larger, so that it snaps at φ = pcr/p − 1; dotted, a creep coefficient of 2. At half the critical load the concrete arch snaps at a creep coefficient of 1.00, at 70 per cent at 0.43; the steel arch on its column does not snap at either. 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.

 Whether the cross-frames' spacing matters depends on the crossbeams between them. For the 60.0 m orthotropic box 4.0 m wide and 2.5 m deep under 45 kN/m at 2.0 m from its centre line: the largest distortional warping stress against the spacing of cross-frames of two 1,900 mm² diagonals, with nothing between them (dashed), with open tee crossbeams every 4.0 m (light) and with ring frames every 4.0 m (solid); dotted, the ring frames alone, 4.1 N/mm². At 8 m, 3.8, 3.8 and 3.1 N/mm²; at 20 m, 21.8, 21.7 and 7.1; at 30 m, 60.8, 59.2 and 7.3. 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.

 The flatter the ribbon, the more the casting temperature matters. For two 100 m spans of a stressed ribbon under 35 kN/m on a pier of 200,000 kN/m, one span's joints cast 20 °C warmer than the other's: the force left on the pier as a share of the force a crowd of 20 kN/m on one span puts there, against the ribbon's sag as a percentage of its span. At a sag of 1 per cent the casting leaves 82 per cent of a crowd (1,902 kN against 2,316); at 2 per cent, 14 per cent; at 3, 3 per cent; at 4, 1 per cent. 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.

 The worst direction is the long axis of an ellipse. For a bracket of six M20 grade 8.8 bolts in three rows of two at 75 mm, loaded 150 mm to the side of the group and 150 mm in front of the column face: the demand on the governing bolt, top right, per kilonewton of load, drawn as a radius in the direction the load acts — straight down at the bottom — under the elliptical interaction (solid), which in each quadrant of directions is an arc of one ellipse, and under the straight line (dashed), which is not. The ellipse's longest radius, at 16.0°, is its eigenvector: 0.0058 of the bolt's interaction used for every kilonewton, a capacity of 172.5 kN, found without trying a single direction. 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.

 The road is the harshest stretch of the beam's journey. The worst top fibre, at any corner of any section, of the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, in N/mm² with tension negative: on the yard's bearers; on the haulier's trailer, 1.5 m and 2.5 m from its ends with a bounce of ±30 per cent; lifted by loops 1.0 m in from each end on slings at 60° with a snatch of ±20 per cent; slewed at 0.10 g; and set down on bearings sloping 1 per cent across it; dashed, the stored concrete's limit of −3.0. Yard bearers, −2.98; trailer, −4.88; lifted, −2.97; slewed, −3.13; set down, −1.79. 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.

 Two neutral axes in one cut. Stress against height through a girder with a 300 by 30.0 mm top flange, a 229 by 19.6 mm bottom flange and a 573 by 11.9 mm web, spliced at 700 kN·m. The girder's own neutral axis is 374 mm above its underside, and its flanges work at 133 and 201 N/mm². Its splice, each flange's covers sized to that flange's force at 250 N/mm², has its own centroid at 326 mm — 48 mm lower — and across the gap the plates bend about it: the top outer cover at 244 N/mm², the bottom outer at 269, the web plates from 179 at their top to 212 at their foot. Shaded: the covers' heights (darker) and the web plates'. 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.

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