Structural Engineering & Statics
where the load goes
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Which failure arrives first — page 19

Essays 433 to 456 of 495 on this thread, in the same order.
Caught from both sides without a mode shape. The elastic critical load factor of a one-bay frame of six storeys, 3.5 m high and 6 m wide, columns of 40,000 kN·m² and beams of 60,000, 400 kN on every floor, 14.96 (level line), and the bounds the traces of its flexibility give with no assumed shape, against how many traces are taken: from below, 5.12, 10.82, 13.11, 14.07 for the first four; from above, 22.90, 19.23, 17.39, 16.39. Dotted, the storey estimate of EN 1993-1-1, 12.86, and Rayleigh's quotient on the first-order sway, 15.88. 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.

 The range belongs to the truss. The worst hanger force range against the tie's largest moment in a tied arch of 100 m span and 17 m rise on 32 crossed hangers of 3,000 mm² bar, 60 kN/m of dead load and 60 kN/m of live load run on from either end, as the hangers' stiffness runs from four times a plain bar's to a fiftieth of it (dots), and for the same arch on vertical hangers (the lone dot). Softening the crossed hangers walks the bridge from 678 kN of range and 316 kN·m of tie moment to 557 and 1,154; vertical hangers sit at 194 and 3,505. The line bends nowhere near the vertical arch's point: a soft network is still a truss. 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.

 Shorter after every cycle, at the same load. The shortening of a column hinge at the end of each cycle, in yield strains, for an I-section of 3,000 mm² and 200 mm depth and the rectangle of the same area, each carrying a constant 30% of its squash load while its curvature cycles between ±3.0 times its first-yield curvature, in elastic–perfectly plastic steel. The I-section shortens by 8.14 yield strains every cycle, 97.1 after 12; the rectangle by 1.60, 19.4. Neither settles: the load has not changed, and the column is shorter after every cycle. 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.

 The knee moves out as the braces multiply. The critical load of a pin-ended column, over the Euler load of the segments between its braces, against the braces' stiffness in units of that load over the segment's length, for one, two and four braces. Each curve rises to the segment's Euler load and stays there; the knee where it arrives is at 2.00, 3.00, 3.62. With no braces at all the column buckles at 1/(n + 1)² of the segment's load: 0.25, 0.11, 0.04. 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.

 Whole spans miss the shear inside a span. The shear along three spans of 8.0, 8.0, 8.0 m, under 20 kN/m of dead load everywhere and 15 kN/m imposed: the largest and least when the imposed load stands wherever it raises or lowers the shear at each section (solid), and when it may only cover whole spans (dashed). At the supports the two agree. At the middle of the second span the exact envelope reaches 23.8 kN either way and the whole spans 10.0 kN; at a tenth of the way along it, 122.4 against 122.0. 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.

 A crack starts at the corner and runs along the grain. The end of a timber beam 400 mm deep and 140 mm wide, notched on its underside so that 200 mm stands over the support, with the notch's corner 50 mm from the line of the reaction, drawn to scale. A crack starts at the corner and runs along the grain towards mid-span. With a shear strength of 3.5 N/mm² the un-notched end would carry 130.7 kN; Eurocode 5's factor kᵥ of 0.40 on the reduced depth leaves 25.8 kN, 20% of it. 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.

 Double the span and the deck is nearly three times nearer buckling. The factor on its own dead load at which a stayed deck buckles, against the main span, on logarithmic axes, for a deck of 0.50 × 10¹² N·m² under 150 kN/m with stays every 15 m from towers 0.40 of the half-span tall. At 300 m it is 40.7; at 600 m, 14.7; at 1,000 m, 6.9; at 2,000 m, 2.3. Beyond 600 m the line falls with a slope of 1.53; it crosses ten, below which a design has to follow the deck's deflections into its forces, at 805 m. 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.

 A quarter of the visits shared uses up the whole reduction. The factor needed by 20 floors carrying office imposed load — a sustained part of mean 0.50 kN/m² renewed every five years on average, and visits of mean 0.2 kN/m² arriving 0.3 times a year on each floor for three days, against the share of the visits made by the whole building at once: floors let singly (solid) and to one tenant (dashed), and the storey rule's 0.73 (dotted). With no share, 0.26 and 0.60; with a tenth, 0.51 and 0.67; with a quarter, 0.68 and 0.75; with all, 0.96 and 0.97. The rule's factor is reached at a share of 0.34 let singly and 0.21 to one tenant. 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.

 Allowed to yield, the damaged truss is hardly lighter. The steel in an eight-panel Pratt truss with a counter-diagonal in each interior panel, loaded by 10 at every bottom joint, sized to survive the loss of any one interior diagonal, against the share of the full load it must carry with the diagonal gone: checked elastically (solid) and by the lower-bound theorem, every member allowed to yield (dashed); the intact design's 1,200 as a level line. At a third, 1,221 elastic and 1,206.7 plastic; at two thirds, 1,304 elastic and 1,296.7 plastic; at the full load, 1,555 elastic and 1,545.0 plastic; at four thirds, 2,074 elastic and 2,060.0 plastic. The plastic design saves 7.4 units at two thirds of the load out of the 104 that robustness costs. 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.

 The preload a storm can live with, raised by a spring under the nut. The largest preload the bolts of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows can be given and come out of a storm without stretching, against the stiffness of a stack of disc springs under each nut, for storms to ±165, ±170, ±175 kN·m; dashed, the same for the bare base; dotted, the bolt's yield of 226 kN. To ±170 kN·m the bare base keeps at most 128 kN; with a stack of 500 kN/mm, 150; 100 kN/mm, 187; 20 kN/mm, 214; 5 kN/mm, 223. 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.

 The best compensation is less than the prediction. For every floor of a 40-storey building of 3.6 m storeys, its perimeter columns at 12 N/mm² and its core at 6 under the finished load, cast a floor every seven days, with 25 per cent of the load arriving as finishes 90 days after the frame is complete and creep coefficient 2.2, from its fit-out to thirty years, with the creep anywhere within ±30 per cent of the design value: the largest amount any floor stands high (dashed) and stands low (light), and the worse of the two (solid), against the share of its predicted thirty-year movement each floor is set high by. Uncompensated, a floor ends 31.9 mm low; compensated in full, a floor stands 16.0 mm high at its fit-out; the worst is smallest, 11.0 mm, at 0.79 of the prediction, where level 40 standing high 0 days after its fit-out under creep ×0.70 balances level 33 standing low after thirty years under creep ×1.30. 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.

 The pair holds its clamp longer than the stub it is checked by. The tension a bolt at which the clamp lets go, as a share of the 110 kN preload, against the end plate's lever arm from web to bolt, for an end plate 18 mm thick on a column flange 14 mm thick at 30 mm: the pair (solid), the end plate's stub (dashed) and the column flange's (dotted). With the end plate's bolt at 30 mm the pair and the stubs nearly agree. At 45 mm the pair lets go at 88.2 kN and the end plate's stub already at 81.1, the pair holding 9 per cent more; at 60 mm, 82.0 against 71.2, 15 per cent more. 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.

 Past yield, the notch trades stress for strain. The root of a notch of elastic factor 3.0 in a mild steel whose cyclic curve reaches 0.2 per cent strain at 252 N/mm², under a nominal stress of 200 N/mm²: the cyclic stress–strain curve (solid), the elastic line the root would follow if it stayed elastic (thin), and Neuber's hyperbola σε = (KₜS)²/E (dashed), which the curve crosses at the root's state. Elastic, the root would carry 600 N/mm² at 0.30 per cent. By Neuber it carries 293 at 0.61 per cent; by Glinka's energy rule 272 at 0.44; by the linear strain rule 246 at 0.30. 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.

 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.

 Braces that pull one way never brace the girder to its moment. The critical moment of the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side, against each brace's stiffness (logarithmic), when every brace resists twist both ways, when the three alternate in the direction they resist, when two face one way and one the other, and when all three face the same way; dotted, the 5,627 kN·m the girder must reach. Two-way braces of 2,846 kN·m/rad reach it. At that stiffness the alternating row reaches 3,087, the two-and-one row 1,734 and the row facing one way 947, the girder's unbraced moment; with rigid braces, 4,063, 1,891 and 947. 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.

 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.

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