What is refuted here — page 27

A stiffer base is worth twice as much to a portal under a heavy roof, because it raises the elastic critical load as well as lowering the knee moments.

What decides it: It is worth more, not twice. First-order, fixed bases raise the shakedown factor to 1.21 times the pinned frame's whatever the roof; with the wind amplified it is 1.26 at a 1,000 kN roof and 1.32 at 2,000 — the base's gain grows by half again, because only the wind's share of the envelope is amplified and gravity, which the base does not touch, carries most of it.

Tested in The roof that makes the wind worse every storm, at the figure it turns on · the essays about shakedown.

A partial-penetration weld loaded along its axis puts no stress across its unfused land and so cannot tear there.

What decides it: The shear along the weld does not open the land, but it slides its faces past each other parallel to the root, which is a crack's tearing mode. Counted with the opening by their energy release rates at 51 MPa·m½, a half-penetration root keeps 0.93 of a double-sided throat straight along the weld, against 1.00 if only the opening were counted, and from 78° to straight along it keeps less than the elastic count.

Tested in The land that is already a crack, at the figure it turns on · the essays about weld strength.

A continuous truss cambered so that it is level under its dead load carries that load with the reactions an elastic analysis of the drawn truss gives.

What decides it: Only if every member is cut exactly. With a length tolerance of 1 mm on every member of a two-span truss of 24 m spans and 3 m depth, the pier's reaction scatters with a standard deviation of 36 kN about the analysis's 572, and one truss in forty lands with 78 kN more — the same effect as a middle bearing set 14 mm out of level.

Tested in The pier's share is cut into the members, at the figure it turns on · the essays about camber.

A deck shaken by traffic is a random deck, so its stays need only the small damping a random deck asks of them.

What decides it: Only while the traffic is random in its timing. The same vehicles that, arriving at random every 2 s, give a 2.01 Hz deck mode an rms of 7.1 mm and ask a 120 m stay for 0.06 per cent of damping, give it 20 mm and ask 0.99 per cent if they arrive every 1.987 s exactly, four deck periods; scattered by 2 per cent they still ask 0.58.

Tested in The convoy that keeps the deck's time, at the figure it turns on · the essays about cable dynamics.

A basement whose risen water table is held at its design level by drainage and a pump is as safe as the basement was when it was built.

What decides it: Only while the pump runs. Under five storeys the 20 × 30 m basement's flotation check, at 0.86 with the water at its design level, is used up when the head under the slab has come back 0.97 m, 4.7 hours after the pump stops in ground that recovers in 12 hours; with two outages a year lasting six hours on average, that happens 0.91 times a year.

Tested in The building held down by its pump, at the figure it turns on · the essays about uplift.

An axially restrained steel beam that yields in compression in a fire ends it pulling on its connections with the plastic shortening times the restraint, so the cooling tension is the force to design them for.

What decides it: Only for a beam kept straight. A 9 m 457 × 191 × 67 UB carrying 0.39 of its plastic moment forms a hinge at 501 °C, bows, and keeps 1.13 m of sag; it ends the fire pulling 613 kN against the straight beam's 1,669, and between 497 and 628 kN at every restraint from 2 to 60 per cent of its axial stiffness, where the straight beam's runs from 236 kN to its whole yield in tension, 3,035 kN.

Tested in The beam that pays the fire back in sag, at the figure it turns on · the essays about fire.

A beam's connections are most at risk in the cooling phase of a fire, when the shortened beam pulls on them.

What decides it: For a loaded beam that sags, it depends on how hot the connections get. With connections resisting 600 kN cold and heating 0.6 as much as the beam, the largest demand is 1.05 of their strength, on cooling; heating 0.8 as much, it is 3.79, at 870 °C, from the hot catenary's 228 kN pulling on bolts with a tenth of their strength left.

Tested in The beam that pays the fire back in sag, at the figure it turns on · the essays about fire.

Once the deck of a composite tub girder has hardened, the top lateral bracing left in under it carries nothing, because the deck is many times stiffer in shear and the top flanges lie near the composite neutral axis.

What decides it: True where the deck is whole: a 480 kN lorry puts at most 5.5 N/mm² in a diagonal of a girder continuous over two 40 m spans. Over the middle support, where the deck is cracked to its 1 per cent of bars, the top flanges are 665 mm above the neutral axis and the bracing takes 3.6 per cent of the top shear flow, and the same lorry puts 42 N/mm² in a diagonal — 30 from the flanges' bending and 15 from the torque.

Tested in The bracing nobody took out, at the figure it turns on · the essays about shear centre.

A long thin tube in bending reaches its ovalisation limit at a flattening of two ninths of its radius, whatever the pressure on it.

What decides it: Only for a pressure that is the same all round. A pressure whose second harmonic pushes in on the diameter in the bending plane has flattened the tube before it bends, and the limit is reached at two ninths plus two thirds of that flattening: for a 1.2 m tube with a 2.5 mm wall and a harmonic of a quarter of its ring's buckling pressure, at 0.278 of the radius against 0.222, and at 0.167 when the harmonic pushes in across the bending plane.

Tested in The oval the pressure brings, at the figure it turns on · the essays about brazier buckling.

After a column is lost, the floor slab's tensile membrane action supplements the beams' catenary action, so the two together carry the floor across the gap.

What decides it: Not at the same time. The slab's mesh reaches its yield, and the membrane its full tension, when the beams' ends have turned through twice the root of 8fy/E over π — 0.090 radians for 500 N/mm² mesh, at every span. Connections good for 0.03 to 0.07 radians have fractured before that; at 0.07 radians the slab is carrying 579 kN against the 1,232 it carries once its mesh yields.

Tested in The slab that catches what the joints let go, at the figure it turns on · the essays about robustness.

Crossing inclined screws in pairs, half leaning each way, is a loss, since half the screws are always pushed rather than pulled.

What decides it: It is a loss one way: the crossed row of twenty screws peaks at 245 kN against the leaning row's 263. Under a fully reversed load it is a gain: carried for ever at 220 kN against 208, with half of each thread's first loss carried into the other way.

Tested in The row that is pushed as often as it is pulled, at the figure it turns on · the essays about dowel yield.

The section a falling weight is most likely to destroy is the one under the weight.

What decides it: For a weight that always lands in one place. Half of every drop's hinge rotation goes to the beam's supports whatever its point, so when the weight wanders the supports become the most-rotated sections: after twenty drops alternating between three and seven tenths of a 6 m span, 0.104 radians at a support against 0.102 under either landing point, and with random landings 0.128 at a support against no more than 0.041 anywhere in the span.

Tested in The weight that lands somewhere else, at the figure it turns on · the essays about impact factor.

A release that takes a second or more, like a grab opening, is slow enough to halve the rebound.

What decides it: Only against the crane's own period. The same release over 1.2 s leaves a rebound of 0.19 on a crane that rocks with a period of 1 s, 0.47 with a period of 2 s and 0.81 with a period of 4 s: what counts is how long the release takes against how long the crane takes to answer it.

Tested in The release the rope cannot slow, at the figure it turns on · the essays about counterweight.

A full-strength joint's moment–rotation curve does not matter to the frame's strength, because the hinge forms in the beam rather than in the joint.

What decides it: It does not move the collapse, which comes at λ = 5.06 for every full-strength joint, but it moves the first hinge. End plates of 25 EI/L kept linear put the first hinge at λ = 3.51; softening as EN 1993-1-8 has it, at 3.78 for a joint 1.2 times as strong as its beam and 4.15 for one exactly as strong. From 1.5 times the beam's strength the curve no longer matters, because the joint is still on its straight part when the beam hinges.

Tested in The joint stronger than its beam, at the figure it turns on · the essays about corner moment.

A column that shortens faster than its neighbours hands its load to them and slows down, so the columns of a frame under an earthquake shorten together.

What decides it: They do in the end, but the floors hand the load out over hundreds of cycles and an earthquake has ten or twenty. In a six-storey frame of two 8 m bays the floors lift 180 kN off an interior column carrying 1,500 kN in 20 cycles, and it shortens 41.6 mm against 46.2 alone; the exterior columns, now carrying more, have not begun to shorten.

Tested in The floor that cannot keep up with the column, at the figure it turns on · the essays about Moment-curvature.

In a tapered curved flange, the transverse bending stress at the web peaks where the curling parameter is largest.

What decides it: Only while one dimension tapers. A knee flange narrowing from 280 to 180 mm and thinning from 25 to 15 mm has λ = 1.14 at the rafter end and 0.94 at the column end, but its transverse stress is 205 N/mm² at the rafter end and 325 at the column end: the flange's own stress there has risen from 127 to 246.

Tested in The flange bent hardest where it curls least, at the figure it turns on · the essays about curved beam.

A thicker end plate makes a cantilever's root stiffer against warping.

What decides it: Only once the column behind it has been stiffened. On an open column a plate thickened from 15 to 40 mm raises the root from 5.9 to 6.8 times the cantilever's warping stiffness, because the column twisting between the flange levels is three quarters of the flexibility. With the column boxed between the flanges the same plates give 19.4 and 34.6.

Tested in The root that belongs to the column, at the figure it turns on · the essays about moment gradient.

The more torsionally flexible a floor, the higher its torsional irregularity ratio.

What decides it: Statically, yes: 2.46 at Ω = 0.5 for a floor 4.2 m eccentric. From a response spectrum the same floor reads 1.17, because the combined peaks have no sign: its two edges move 1.41 and 2.02 times a torsion-free floor's, opposite ways in the dominant mode, and their average of 1.71 is twice its centre's 0.83.

Tested in The floor that twists in its own time, at the figure it turns on · the essays about plan torsion.

Row factors calibrated in uniform ground carry over to layered ground by applying them to every layer.

What decides it: Nothing in the calibration says whether the shelter a leading pile gives extends into stiff ground the piles barely move. Applied at every depth, the group's stiffness is 0.74 of its single piles'; applied in the soft layer only, 0.90, and the leading row's share 1.24 against 1.07 of the average.

Tested in The shelter that stops at the interface, at the figure it turns on · the essays about lateral pile.

Truss joints are pins, so truss members carry axial force and nothing else.

What decides it: The same truss solved twice — pin-jointed, and with the joints continuous as welding actually makes them. The axial forces agree to about a per cent, which is why the idealisation is a good one; the bending it omits reaches a quarter of the axial stress in the worst member. It scales with EI/L, so a stocky truss of heavy members suffers more of it than a light one — which is the opposite of the way most reserves of strength work, and is the direction in which the simplification stops erring safe.

Tested in The joint that is not a pin, at the figure it turns on · the essays about truss.

Wind loading is a static problem: work out the pressure, multiply by the area, apply it.

What decides it: True for a stiff structure and badly wrong for a slender one. Split into its parts, a warehouse at 2 Hz takes 15% of its fluctuating response from resonance and a tower at 0.2 Hz takes 90% — and the resonant part is inversely proportional to a damping ratio that no static calculation contains. The static method survives because the gust factor it hides absorbs the difference for ordinary buildings.

Tested in The wind is a spectrum, at the figure it turns on · the essays about gust response.

The effective width of a flange is min(span/8, b) per side, and that expression comes from the elasticity of the problem.

What decides it: The exact elastic ceiling for an infinitely wide flange fed by a sinusoidal shear flow is L/2π = 0.1592L per side, from Airy's solution. The rule caps at L/8 = 0.125L, which is 0.7854 of it — exactly 2π/8. A rule that is a fixed fraction of the exact answer is a fit with a margin in it, not a derivation of it, and it is a straight line where the elasticity is a curve.

Tested in The flange that is not all there, at the figure it turns on · the essays about effective width.

Quoting k = 4 for a long plate is an approximation, so it could be wrong in either direction.

What decides it: It is the floor of the envelope and not an average through it. Every scallop rises above 4 and none falls below it, so the quoted value understates the critical stress by up to 12.5 per cent and never overstates it — and past an aspect ratio of 1.60 the error can never exceed 5 per cent again.

Tested in Four was never a fact about plates, at the figure it turns on · the essays about plate buckling.

One pass of the effective-width calculation is enough, because the correction is small.

What decides it: It is small in the moment and not in the width. One pass reports 278 mm of web removed against the converged 323, understating the loss by 16 per cent, while the moment capacity it reports is 10,079 kNm against 9,999 — an error of 0.8 per cent. The two errors differ by a factor of twenty because the width that goes missing sits near the neutral axis.

Tested in Classified by a gradient it does not have, at the figure it turns on · the essays about plate buckling.