What is refuted here — page 12

The worst moment for a steel beam in a fire is its hottest, so a beam that survives the peak temperature has survived the fire.

What decides it: A beam held at its ends with a tenth of its own axial stiffness reaches 134 N/mm² of compression while heating and 195 N/mm² of tension once the fire is out and the beam is cold — 55 per cent of its yield stress, at 20 °C, hours after the peak. Its connections see the larger force after the fire.

Tested in The tension a fire leaves behind, at the figure it turns on · the essays about fire.

Restraint against thermal expansion only matters while the fire is hot.

What decides it: The restraint matters most afterwards. At 2 per cent of the beam's own stiffness it causes 39 N/mm² of compression and leaves 28 of tension; at 10 per cent, 134 and 195; at 20 per cent, 220 and the full 355 of tension. From 4 per cent upwards the tension left behind is larger than the compression the fire caused.

Tested in The tension a fire leaves behind, at the figure it turns on · the essays about fire.

A group of tension piles holds the sum of what each pile holds, so more piles always mean more resistance to uplift.

What decides it: Only while they are far enough apart. 600 mm piles 15 m long hold 636 kN each; under a 20 × 30 m raft the group can hold no more than the submerged weight of the 15 m block of ground beneath it, 90 MN, which 142 piles at 2.06 m centres already reach. Closer, the group lifts the block and the extra piles add nothing.

Tested in The piles that would lift the ground, at the figure it turns on · the essays about uplift.

Longer tension piles can be placed closer together, because each holds more.

What decides it: The opposite. A pile's pull-out grows as its length squared and the block's weight only as its length, so the spacing below which the block governs grows as the square root of the length: 1.30 m for piles 6 m long, 2.06 m at 15 m, 2.91 m at 30 m. Longer piles must stand further apart to be worth their length.

Tested in The piles that would lift the ground, at the figure it turns on · the essays about uplift.

Specifying more strand than strength requires, at a lower working stress, makes a stressed ribbon's erection easier on its piers.

What decides it: The opposite. A tendon working at a lower stress is stretched less, so less of the hung span's thrust is already held across the empty one: a rigid pier carries 4,109 kN at the stage with tendons at 1,100 N/mm² and 11,045 kN with tendons at 400, and below about 475 N/mm² the stage exceeds the crowd.

Tested in The tendon that pulls before the deck arrives, at the figure it turns on · the essays about stressed ribbon.

Continuity over an interior support reduces a wide-flanged girder's worst flange stress, because it reduces the peak bending moment.

What decides it: For two continuous 20 m spans with a 3 m flange overhang under a uniform load, the peak moment over the support, with the reaction spread over a 1 m bearing, is 0.94 of a single span's mid-span moment; but the flange there works over 0.33 of its overhang against 0.88, and the stress at the web is 2.5 times the single span's peak.

Tested in The reaction that is made of short waves, at the figure it turns on · the essays about effective width.

A fuse in series with a rocking wall's post-tensioning tendon preserves the wall's re-centring by keeping the strand elastic.

What decides it: It keeps the strand elastic and nothing else. With a fuse yielding at 1,000 kN in series with a 1,370 kN strand, the 2 × 8 m wall under a 1.5 g pulse rocks to 110 mrad instead of 89, and its tendon line has no prestress left afterwards — exactly as much as the yielded strand alone kept.

Tested in The fuse that protects the strand and not the wall, at the figure it turns on · the essays about rocking.

The two cover plates of a double-cover splice share the load in proportion to their areas, so any pair whose total area matches the main plate's is as strong as the plate.

What decides it: A 20 mm plate 200 mm wide spliced with a 15 mm and an 8 mm cover: the pair has 15 per cent more area than the plate, but across the gap the 8 mm cover carries 471 kN of 900, not the 313 its area gives it, and its outer face works at 313 N/mm² against 196. It yields at 1,023 kN, 72 per cent of the plate's 1,420.

Tested in The thinner cover carries more, at the figure it turns on · the essays about single lap.

Secondary prestress effects are moments; the shear they produce is negligible.

What decides it: The secondary shear here is 26 kN against an applied shear of 234 — a ninth. But the check is made on what is left after the tendon's relief, and there it is half of everything. Under 40 kN/m the slope-only count says the shear at the end has vanished, when 26 kN remains.

Tested in The relief that belongs to the pressure line, at the figure it turns on · the essays about secondary prestress.

The unit-load integral says how much of a frame's deflection comes from each member, and from bending, axial and shear deformation, so the members contributing least are the ones not worth stiffening.

What decides it: Only for the unit load carried on the frame itself. A fixed portal swaying 10.38 mm gives the left column −3.72 mm on the frame, +10.38 on the left column cantilevered alone, −15.78 on three hinges and −31.43 on a pin and roller; all four totals are 10.38.

Tested in One column is enough to find the sway, at the figure it turns on · the essays about virtual work.

To check a computer's deflection of an indeterminate frame by the unit-load method, the whole frame has to be re-analysed by hand.

What decides it: The unit load may stand on the left column alone. Its sway then needs only that column's real base and top moments, 27 and 149 kN·m, and its shear, 29.3 kN: 10.84 mm of bending less 0.46 of shear is the frame's 10.38.

Tested in One column is enough to find the sway, at the figure it turns on · the essays about virtual work.

An active mass driver is a tuned mass with an actuator added, so it is at least as good as the passive mass and the actuator's force only improves it.

What decides it: The law that keeps it stable cancels the mass's spring, which throws away the tuning. With a 2 per cent mass held to the passive stroke, an actuator allowed a fifth of the applied force gives a peak of 9.34 against the passive mass's 8.72; it needs three tenths to break even.

Tested in The law that throws the tuning away, at the figure it turns on · the essays about tuned mass damper.

A seam that has passed a pressure test is safe under any service load, because pressure is the governing load for a pipe.

What decides it: Pressure is the load that protects the seam. Add an axial pull of one and a half times the hoop stress — a pipe in a bend, a pile in tension — and the 60° seam falls from 78 per cent of the wall's strength to 61.

Tested in The seam that pressure protects, at the figure it turns on · the essays about Heat-affected zone.

A box girder's shear centre lies above its centroid, near the stiffer top flange.

What decides it: Of a rectangle, yes: 39 mm above in a 1.5 m by 3 m box with a heavier top plate. Narrow the bottom flange to 2 m, so that the webs lean at 18 degrees, and it lies 106 mm below the centroid; the two cross at a lean of under 6 degrees.

Tested in The web that leans and lifts the centre, at the figure it turns on · the essays about shear centre.

Engesser's and Haringx's formulas are rival theories, and tests have to decide between them.

What decides it: They are one formula. Engesser's with a shear stiffness of S + P gives the same quadratic as Haringx's with S. What a test decides is which shear stiffness a member has — the one measured with the axial load acting along the deformed axis, or across the rotated sections.

Tested in The bearing Engesser says has buckled, at the figure it turns on · the essays about Built-up column.

Since a collector in a concrete slab carries only a fraction of the uniform-shear force, it can be designed for that fraction.

What decides it: The fraction depends on the slab's shear stiffness. Cracking that halves it raises the collector from 24 to 36 per cent of the rule; a slab cracked to a tenth of its stiffness puts 68 per cent on the collector. The member must be sized for the floor it will be after the first cycles, and the slab around the bay for the floor before them.

Tested in The collector the slab does not need, at the figure it turns on · the essays about diaphragm.

A column base can be made rigid by shortening its holding-down bolts' free length, since the bolts are most of its flexibility.

What decides it: A 500 × 400 × 20 mm base with two M24 bolts a row is 9.8 times the column's EI/L with cast-in bolts, 13.9 with bonded anchors of 136 mm free length, and 19.3 with an implausible 60 mm. The plate and the concrete stay in series with the bolts, and the rigid boundary is 30.

Tested in The base that is rigid until the bed lets go, at the figure it turns on · the essays about base plate.

A cantilever retaining wall moves enough during backfilling to relieve the compaction pressure, so it can be designed for active pressure.

What decides it: A 3 m stem 300 mm thick, compacted lift by lift, ends with a base moment of 83 kN·m a metre: 87 per cent of the 95 a rigid stem carries, against 30 for active pressure. Its base never moves and its top lifts have nothing after them, and both keep their residual.

Tested in The stem that bends away from the roller, at the figure it turns on · the essays about lateral pressure.

A beam's twisting restraint is worth its total stiffness wherever it is applied, so a deck and a single connection of the same total stiffness restrain a beam equally.

What decides it: On an 8 m beam twisting 6.25° unrestrained, a deck of 5 kN·m per metre per radian — 40 kN·m per radian in total — leaves 4.20°; the same 40 concentrated at mid-span leaves 3.13°, and 19.5 at mid-span matches the deck. Concentrated where the twist is largest, a restraint is worth about twice its total spread out.

Tested in The restraint that works where the twist is, at the figure it turns on · the essays about twist serviceability.

A deck moving at random at twice a stay's frequency is as dangerous to the stay as a sine of the same rms.

What decides it: On a 120 m stay with 0.1 per cent damping, a ±10 mm sine at twice the stay's frequency grows it tenfold every 147 s. A deck moving at random with the same 7.1 mm rms, as a mode with 1 per cent damping of its own, lets the stay settle, halving roughly every six minutes; the damping it needs is 0.061 per cent against the sine's 0.348.

Tested in The deck that forgets its own rhythm, at the figure it turns on · the essays about cable dynamics.

The largest deck movement measured is the amplitude to check a stay's parametric stability against.

What decides it: A random deck with a 7.1 mm rms reaches about 30 mm once in an hour. Read as a sine, that asks for 1.04 per cent damping in the stay. The random deck itself needs 0.061 per cent for a typical history and 0.12 per cent for the mean square, because what drives the stay is the share of the deck's motion that holds twice its frequency, not the size of the peak.

Tested in The deck that forgets its own rhythm, at the figure it turns on · the essays about cable dynamics.

A backprop installed snug carries nothing until the next floor is cast, and then only its share of that floor's weight.

What decides it: On a frame of 250 mm slabs cast a floor a week, the backprop under a seven-day slab takes 0.48 of a slab's weight when the next floor is cast and 0.71 by the end of the week, because the struck slab creeps onto it under the weight it took on being struck. The slab below reaches 1.71 times its weight rather than 1.5.

Tested in The backprop that takes load nobody gave it, at the figure it turns on · the essays about maturity.

A small hole well inside a shaft carries no stress concentration, because the torsional stress is low away from the surface.

What decides it: Round its edge the stress is twice what the solid shaft would carry there, the factor any round hole puts into a shear field. A 10 mm hole 15 mm off the axis of a 50 mm shaft raises the shaft's largest stress to 1.70 times its solid surface stress, at the hole's outer edge.

Tested in The island the film floats, at the figure it turns on · the essays about membrane analogy.

A plate in a material with no yield plateau loses buckling strength to its rounded curve in the same proportion as a column of the same material and slenderness.

What decides it: For an austenitic stainless steel with a Ramberg–Osgood exponent of 6, a column buckling by its tangent modulus falls to 0.59 of the sharp envelope where its elastic stress equals its proof stress. A long plate at the same slenderness, by deformation theory, falls only to 0.72, because only one of its three stiffnesses is the tangent modulus.

Tested in The plate the wrong theory gets right, at the figure it turns on · the essays about inelastic buckling.

The more rigorous theory of plasticity gives the better prediction of a plate's inelastic buckling stress.

What decides it: Flow theory, whose stress–strain law is path-dependent as a metal's is, puts a stocky stainless plate at λ̄p = 0.6 at 2.30 times its proof stress, a stress no plate test reaches; deformation theory, whose law is not, gives 0.99. Plate tests sit near deformation theory, and two thirds of the gap between the theories is the twisting stiffness alone.

Tested in The plate the wrong theory gets right, at the figure it turns on · the essays about inelastic buckling.

A warm day during erection unbalances a stressed ribbon's piers by the force the bare tendon loses, EA times α times the warming.

What decides it: Warming both spans of a half-hung 100 m ribbon by 30 °C takes 1,334 kN off the bare tendon and 1,021 kN off the hung span, which at a 2 per cent sag is mostly stretch too, so the pier force moves only from 3,080 to 3,392 kN. It is the bare strand warming more than the shaded one that moves the pier: 30 °C on the bare tendon alone gives 4,126 kN.

Tested in The strand in the sun beside the strand in the shade, at the figure it turns on · the essays about stressed ribbon.

A bare tendon left in the sun during erection can lose its stretch and go slack.

What decides it: At the 2 per cent sag a ribbon is laid at, the bare tendon holds four fifths of its finished thrust as stretch, and it would need to be 381 °C warmer than the day it was cut to lose it. Only a ribbon laid near a 4 per cent sag, where the tendon holds little stretch, could go slack in the sun — at 109 °C at 4 per cent.

Tested in The strand in the sun beside the strand in the shade, at the figure it turns on · the essays about stressed ribbon.

A row of inclined screws, each of which softens after a peak at about two millimetres of slip, loses capacity along its length as a long bolted joint does, so that its effective number falls well below the actual number.

What decides it: Twenty 8 mm screws at 45° along the grain, 56 mm apart, in softwood members with 40 mm of depth per row, reach 0.98 of twenty times one screw's peak: half are past their peak and the rest still climbing when the row peaks, and none is below 0.92 of its own peak. The effective number is 19.6, where n⁰·⁹ gives 14.8.

Tested in Twenty screws that peak together, at the figure it turns on · the essays about dowel yield.

Making some of a frame's joints semi-rigid can only relieve the rest, since the frame as a whole becomes more flexible.

What decides it: Under a sideways push of 20 kN a floor, softening only the interior joints to 3 EI/L raises the roof corner's panel shear by a fifth and the floor exterior's by 7 per cent; softening only the exterior joints raises the roof interior's by 28 per cent and the floor interior's by 11. The sway goes to the joints that are still stiff.

Tested in The panel a soft joint empties into the span, at the figure it turns on · the essays about corner moment.

A pressure inside a tube in bending stiffens it against ovalisation, so the pressurised tube collapses at a smaller flattening than the bare one.

What decides it: It collapses at the same flattening. The pressure adds to the ring's resistance to the oval, so it takes more curvature to reach any given oval, but the peak of the moment is still where d[κ(1 − 3ζ/2)]/dκ = 0 with ζ proportional to κ², which is ζ = 2/9 whatever the pressure. Five pressures, from 0.75 of the ring's buckling pressure outside to three times it inside, all peak at 0.222 of the radius.

Tested in Three kilopascals that halve a tube in bending, at the figure it turns on · the essays about brazier buckling.

For a given total of chord stiffness, how it is split between the top and bottom chords of a Vierendeel girder does not affect the girder's deflection.

What decides it: It does unless the posts are very stiff. With the chords' second moments adding to twice 11,700 cm⁴ and posts as stiff as one chord, equal chords deflect 6.20 mm under 100 kN at mid-span, four fifths in the top chord 6.80 mm, and 95 per cent 8.76 mm. With posts a hundred times stiffer the three are 4.20, 4.22 and 4.26 mm.

Tested in The heavy chord the posts will not let work, at the figure it turns on · the essays about vierendeel.

A partial-strength joint limits the moment a beam puts into its column, so the edge column of a semi-continuous frame carries less moment than the edge column of a continuous one.

What decides it: At the edge it carries about the same. In a braced six-storey frame of 9 m beams at 60 kN/m, partial-strength joints of 261 kN·m deliver 259, 261, 259, 229, 246 and 171 kN·m to the edge column from the first floor to the top, and rigid joints 282, 320, 281, 230, 248 and 156 — within 6 per cent on average. The joints differ at the interior, 261 against 466 kN·m; at the edge the column's own flexibility limits both.

Tested in The column on the other side of the joint, at the figure it turns on · the essays about joint classification.

Autofrettage improves a pressure vessel's fatigue life by reducing the range of stress its bore sees in each pressure cycle.

What decides it: The range is unchanged: a cycle from zero to the working pressure moves the bore's hoop stress by the same Lamé amount with or without the overstrain. What the overstrain changes is where the range sits — in a 2.5 to 1 tube from −900 to 144 N/mm² instead of from 0 to over 1,000 — and a crack at the bore grows far more slowly when it is held shut for most of every cycle.

Tested in The overstrain the release gives back, at the figure it turns on · the essays about unloading.

A composite beam with half the shear connectors that full interaction needs loses half of the moment capacity the slab added.

What decides it: It loses much less. A 457 × 191 × 67 UB under a 130 mm C30/37 slab carries 516 kN·m as steel alone and 1,001 kN·m fully composite; with half the connection its plastic moment is 850 kN·m, 0.85 of full, because the slab's compression block only becomes thinner and the steel's own neutral axis rises to carry the difference. The straight line EN 1994 allows for any connector gives 0.76.

Tested in The slip the plastic moment asks of the studs, at the figure it turns on · the essays about composite action.

The minimum degree of shear connection in EN 1994-1-1 exists so that a composite beam is strong enough.

What decides it: Strength is covered by the stress blocks at any degree of connection. The minimum is about slip: requiring the end studs to slip no more than 6 mm while the beam reaches 95 per cent of its plastic moment gives a least degree of connection of 0.51 at 8 m, 0.69 at 12 m and 0.88 at 20 m, within 0.08 of the Eurocode's line at every span from 8 m up.

Tested in The slip the plastic moment asks of the studs, at the figure it turns on · the essays about composite action.

A prestressed beam whose strands satisfy the four stress limits at its section of greatest moment satisfies them everywhere along its span.

What decides it: Eight 125 kN strands at 220 mm below the centroid of a 300 × 700 mm beam spanning 12 m leave its mid-span top fibre at −0.36 N/mm² at transfer against a limit of −2.5. Bonded straight from the ends, the same strands put the top fibre past its limit from 0.58 to 1.52 m from each end, at up to −3.31 N/mm², because the self-weight moment that relieved the top fibre at mid-span falls to nothing towards the supports while the strands' force does not.

Tested in The strands that are sleeved at the ends, at the figure it turns on · the essays about prestress limits.

A secondary beam on a simple, nominally pinned connection gives the primary beam it frames into no restraint against twist.

What decides it: The connection is pinned for the secondary's bending, which is exactly the rotation the primary's twist imposes on it, but a pin in that classification can be anything below 0.5EI/L — 2,409 kN·m/rad for a 6 m 356 × 171 × 51 UB. At 1,000 kN·m/rad, inside that class, the secondary takes an 8 m primary's mid-span twist from 6.25° to 0.47°, its own torque included.

Tested in The secondary beam that twists what it holds, at the figure it turns on · the essays about twist serviceability.

A control law that keeps most of a tuned mass's spring gets the reliability of the passive mass and most of the performance of the actuated one.

What decides it: On a 2 per cent mass with the actuator held to half the applied force and 1.5 times the passive stroke, the best stable law peaks at 4.65 with the spring cancelled, 5.67 with half of it kept, 7.58 with four fifths, 9.88 with 85 per cent and 12.24 with nine tenths — the last two worse than the passive mass's 8.72 with no law at all.

Tested in Half a spring is neither tuning nor freedom, at the figure it turns on · the essays about tuned mass damper.

If the actuator of a tuned mass whose spring is partly cancelled fails, the structure is left with a mistuned mass.

What decides it: The cancellation is done by the actuator, not by the spring, which is always whole. With the power off the mass is Den Hartog's again, peaking at 8.72 whatever share the law cancelled. It is with the power on that the mass is retuned to √κ of its frequency: with nine tenths of the spring kept, the best law leaves a peak of 12.24 at 1.02 of the structure's frequency.

Tested in Half a spring is neither tuning nor freedom, at the figure it turns on · the essays about tuned mass damper.

Because wind pressure grows with height, an outrigger should be placed differently under a realistic wind profile from where a uniform-load analysis puts it.

What decides it: For the same total wind on a 200 m core, the drift-optimal level is 0.745 of the height under a uniform load, 0.749 under a power-law wind and 0.757 under a triangle. Placed at the uniform optimum, the arm leaves the triangle's tip drift 0.1 mm larger than at its own best, out of 393 mm.

Tested in The arm that ignores the shape of the wind, at the figure it turns on · the essays about outrigger.

The forces in an outrigger under a non-uniform wind can be taken from a uniform-load analysis scaled by the ratio of the base moments.

What decides it: The triangle of the same total force has 33 per cent more base moment than the uniform load, and its outrigger carries 48 per cent more couple — 217 MN·m against 146 — because more of the load is above the arm, where the arm takes it all.

Tested in The arm that ignores the shape of the wind, at the figure it turns on · the essays about outrigger.

A counterweight that moves to follow the load can keep a crane balanced at every position, so the mast need carry only the error in knowing the load.

What decides it: The payload leaves the ground in a second and no counterweight moves that fast, so at pick-up the mast takes the payload's whole moment as a step — 2,898 kN·m for 60 kN at 48 m — whatever the counterweight is doing. The best a counterweight can do is wait at the half-load position, so that the step runs from −1,449 to +1,449 kN·m.

Tested in The counterweight that follows the jib, at the figure it turns on · the essays about counterweight.

A footing stays in full contact with the ground as long as the resultant of its load stays inside the middle third.

What decides it: That is the rigid footing's rule. A flexible 8 m strip on the same bed first lifts under a moment of 27 kN·m on a 1,000 kN column, an eccentricity of 27 mm against a middle third reaching 1,333 mm. Only a strip short in characteristic lengths keeps close to the rule: at 4 m it holds to 609 kN·m, 91 per cent of the kern's 667.

Tested in The ends the ground was holding down, at the figure it turns on · the essays about elastic foundation.

A row of braces designed by its total stiffness loses capacity in proportion when one brace is lost, since the total is what the beam sees.

What decides it: Four braces of 2,710 kN·m/rad carry the girder to 5,627 kN·m. Reading the loss of one as a quarter of the total predicts 4,891 kN·m. The beam drops to 3,704 kN·m without an inner brace and 3,787 without an outer one, a loss of 34 per cent, because the gap is a bay twice as long and the buckle gathers in it.

Tested in The stiffness a row of braces shares out, at the figure it turns on · the essays about beam bracing.

The more a backprop is preloaded, the better protected the young slab above it, so a larger preload is the safer choice.

What decides it: The preload is not lost; it lands on the slab below when the next level is struck. The struck slab's use of strength falls and the lower slab's rises, and they cross at 0.07 w, both at 1.87 against 1.95 snug. At 0.30 w the worst use is 2.04 and at 0.50 w it is 2.21, worse than no preload at all.

Tested in The backprop jacked tight lands on the floor below, at the figure it turns on · the essays about maturity.

A stiff enough bracing truss brings the open trough's shear centre up into the section.

What decides it: It takes an equivalent thickness of 2.9 mm just to bring the shear centre up to the bottom flange. A single diagonal of 100 × 100 × 10 angle across a 3 m trough is worth 0.48 mm in 4 m panels and 0.64 in 2 m panels; crossed in 2 m panels, 1.27. Every practical truss leaves it below the steel.

Tested in The box that is a trough until its deck is cast, at the figure it turns on · the essays about shear centre.

A crack at a coped end is urgent once it is long enough to weaken the section, and harmless until then.

What decides it: The coped tee's static utilisation rises only from 0.52 to 0.58 with a 30 mm crack and its hole, and to 0.71 with a 60 mm one. But a crack spends 85 per cent of its life below 30 mm and 95 per cent below 60, so a crack long enough to worry the static checks has almost no life left in which to be drilled.

Tested in The hole at the tip of the crack, at the figure it turns on · the essays about coped beam.

Camber is a serviceability matter: it is set so that a beam ends up level, and it plays no part in a girder's stability.

What decides it: Hanging from picks near its ends, a cambered girder's centre of gravity sits two-thirds of the camber above the line it rolls about, less the picks' own rise. For a 40 m precast girder picked 2 per cent in, 100 mm of camber lifts it 59 mm towards the roll axis and takes the factor of safety against roll from 2.31 to 1.42.

Tested in The camber that lowers the hook, at the figure it turns on · the essays about camber.

Making a filler's strength depend on speed can only help, because the fast collisions are the ones that need a strong filler.

What decides it: A filler sized to stop the hardest collision is gentler than the uniform filler only below about 1.7 m/s with up to half its strength on speed, and below 1.37 m/s with all of it — 48 per cent of the design speed. Every faster collision meets more force than the uniform filler would have given it.

Tested in The filler that knows how fast it is hit, at the figure it turns on · the essays about pounding.

A small asymmetry in a shallow arch costs it a correspondingly small share of its strength.

What decides it: Above a rise of √22 the loss goes as the two-thirds power of the imperfection, so it is large for small ones: an antisymmetric imperfection of a hundredth of a radius of gyration costs a rise-6 arch 2.9 per cent and a tenth costs it 12 per cent. Below √22 the loss goes as the square: a hundredth costs a rise-4 arch 0.05 per cent.

Tested in The arch that turns aside before it snaps, at the figure it turns on · the essays about Snap-through.

A linear analysis of a network arch is safe once the hangers it finds in compression are checked separately, since a slack hanger simply carries nothing.

What decides it: Taking the slack hangers out changes everything else. With 60 kN/m of dead load and 120 kN/m of live load on half a 100 m span, the linear solution has four hangers in compression and a tie moment of 491 kN·m; solved with hangers that cannot push, eight go slack and the tie carries 998 kN·m, twice the linear answer.

Tested in The deck that keeps the hangers in the truss, at the figure it turns on · the essays about network arch.

A lighter deck is always better for a tied arch, because it reduces every force the arch has to carry.

What decides it: Under the same 120 kN/m of live load on half the span, the tie's worst moment is 2,773 kN·m with 20 kN/m of dead load and fifteen hangers slack, 998 with 60 kN/m and eight slack, and 582 with 120 kN/m and one: the deck's weight is what keeps the hangers in the truss.

Tested in The deck that keeps the hangers in the truss, at the figure it turns on · the essays about network arch.

An arch dam should be curved as tightly as possible, since a smaller radius means a smaller thrust and a thinner arch.

What decides it: Across a fixed gap the thrust pR is least at a semicircle, 180°, but the arch is then longest. The concrete, thickness times length, is least at a central angle of 133.6° and is 14 per cent more at 180° and at 90°; across a 100 m gap 50 m down, 677 m³ per metre of height against 770 for the semicircle and 1,027 for a 60° arch.

Tested in The angle that saves the most concrete, at the figure it turns on · the essays about curved surface pressure.

A preloaded base loses its preload a little with every storm cycle that lifts its plate, until it is no stiffer than an unpreloaded one.

What decides it: Lifting the plate is elastic. A base preloaded to 150 kN a bolt lets go at 57 kN·m and returns its whole preload after every reversal up to 164 kN·m. Past that its bolts yield, and the loss stops within two cycles: at 170 kN·m the weaker row settles at 124 kN after one cycle and stays there.

Tested in The base that lets go and closes again, at the figure it turns on · the essays about base plate.

A section's shape factor is a property of its shape alone, fixed once the section is chosen.

What decides it: Only without axial load. Under a compression n of its squash load a rectangle's first yield falls as 1 − n and its plastic moment as 1 − n², so its shape factor is 1.5(1 + n): 2.10 at n = 0.4 and 2.70 at 0.8. An I-section's moves only from 1.09 to about 1.2.

Tested in The hinge a squeezed section has to work for, at the figure it turns on · the essays about Moment-curvature.

End restraint that is not stiff enough to develop the full arch still develops it, only at a larger deflection.

What decides it: A softer restraint lets the strip sag before it pushes back, and a strip that has sagged has less depth left to arch in: with springs of a twentieth of its axial stiffness the strip peaks at two-thirds of its thickness and at 1.51 times its flexural load, against a quarter of its thickness and 3.89 times held rigidly.

Tested in The edge that cannot hold the arch, at the figure it turns on · the essays about membrane action.

A water tower is in its worst seismic condition when its tank is full.

What decides it: On a flexible tower, more water lengthens the tower's mode past the spectrum's last corner, where a heavier mass draws no more force, and pushes the mass into the long, mostly sloshing mode. On a 2.5 s tower the base shear peaks at 688 kN with the tank 84% full and falls to 665 kN full; the top displacement peaks at the same fill.

Tested in The tank that sloshes the tower, at the figure it turns on · the essays about sloshing.

A bottom-heavy girder loses only a little to local buckling, since its flanges are stocky and only the web is slender.

What decides it: Its compression flange is the thin one, and its neutral axis is low, so the compressed part of its web is deep. The 15/25 girder keeps 78% of its gross modulus and 70% of what the same steel split 22/18 carries; its web loses 538 mm to buckling where the top-heavy girder's loses 322.

Tested in The flange a slender web wants heavier, at the figure it turns on · the essays about Effective cross-section.

A roof is safe against ponding if its joists and its girders are each safe against ponding.

What decides it: The water fills the sum of their sags, so the two members are one system. Two members each with a ponding parameter of 0.53 — each about half-way to its own critical value of 1 — are critical together, and a 12 by 9 m bay at 0.26 and 0.19 has its joists' sag amplified 1.79 times where each member alone says 1.24 and 1.35.

Tested in The roof that ponds on its girders, at the figure it turns on · the essays about serviceability.

A roof whose members meet a span-over-deflection limit is stiff enough not to pond.

What decides it: A member sized to L/n under a load q has a ponding parameter 384γL/(5π⁴nq), proportional to its span. A square bay sized to L/240 under 1.5 kN/m² holds 2.47 times the rain depth at mid-bay at 12 m, 5.25 times at 18 m, and ponds at 24.5 m, every member meeting its deflection limit throughout.

Tested in The roof that ponds on its girders, at the figure it turns on · the essays about serviceability.