What is refuted here — page 14

A deep coupling beam is much less stiff than 12EI/l³ says, so a coupled-wall analysis that ignores its shear deformation badly overstates the coupling and the beams' shears.

What decides it: The shear term cuts a 2 m deep beam over a 2.4 m opening to a third of its bending stiffness, but the pair's degree of coupling falls only from 0.798 to 0.777 and the largest beam shear from 351 to 328 kN. A beam that stiff is already past the point at which its stiffness decides the coupling.

Tested in The coupling beam asked for shear, at the figure it turns on · the essays about shear deflection.

Computing each flange of an end-plate joint as a tee stub on a rigid base is an approximation that always overstates the pair's stiffness substantially, since neither flange has a rigid base.

What decides it: For two identical flanges the rigid base is exact: bolted back to back they meet on their plane of symmetry, which does not move. The error is the two flanges' mismatch: 3% for an 18 mm end plate on a 14 mm column flange with lever arms of 45 and 30 mm, and at most 6% for any of the thicknesses drawn at those lever arms.

Tested in The flanges that lean on each other, at the figure it turns on · the essays about prying.

Creep restores a beam built in stages to the moments it would have had if it had been built and stressed all at once.

What decides it: Creep moves the moments towards the one-stage values and stops short of them. With a creep coefficient of 2 and an ageing coefficient of 0.8 it recovers 2/2.6, or 77 per cent, of the difference: the first span's secondary shear rises from 10.3 kN to 19.2, not to 21.9, and a coefficient of 3 still leaves it at 20.5.

Tested in The beam stressed one bay at a time, at the figure it turns on · the essays about secondary prestress.

A base plate weaker than the column it carries is a weak link: under repeated wind it lowers the load the frame can carry.

What decides it: Down to 0.59 of the column's plastic moment the frame's shakedown factor stays at 0.88, because the limit belongs to the beam's ratchet and the base never reaches its range. The collapse factor starts to fall sooner, below 0.76. Only below 0.59 does the base take over, and then it fails by yielding back and forth rather than by letting the beam walk.

Tested in The base that sends the wind to the knees, at the figure it turns on · the essays about shakedown.

Over the pier the concrete has cracked, so the deck's tension is carried by its reinforcement alone, and bars have no shear lag: every bar across the full width of the deck counts.

What decides it: The bars receive their force from the steel girder through the concrete between the cracks, which works in shear exactly as an uncracked slab does, only softer. A 3 m overhang over the pier of two 20 m spans, cracked with 1.5% of steel each way, works over 0.328 of its width — the uncracked deck's 0.325 to within a per cent, not the whole of it.

Tested in The deck that cracks over the pier, at the figure it turns on · the essays about effective width.

A storey whose sway factor α is above ten is stiff enough that second-order effects in it can be neglected.

What decides it: Under a hat truss at the roof of a 20-storey core-and-frame building whose critical factor is 5, the three top storeys read α of 8.7, 11.1 and 15.5, and their storey estimates of the second-order increase in drift are 13, 10 and 7 per cent. The true increase in each is about 24 per cent, because the building buckles in one mode and the top storeys lean with it whatever their own stiffness.

Tested in The hat that hides the lean, at the figure it turns on · the essays about sway stability.

Adding a hat truss or an outrigger stiffens a building, so a storey check that was safe without one stays safe with it.

What decides it: Without a hat the roof storey's estimate is 33 per cent against a true 26 — safe. With a hat of ten times the core's EI/H at the roof it is 6.9 per cent against 24, a quarter of the truth. The building-wide least factor stays safe, 4.14 against 5, but it has moved to storey 9.

Tested in The hat that hides the lean, at the figure it turns on · the essays about sway stability.

The posts beside a Vierendeel girder's supports carry the largest moments, so they are the joints that govern its fatigue life.

What decides it: As a 100 kN load crosses a six-panel girder, the first interior post's corner peaks at 58.8 kNm and the middle post's at 29.6 — half. But the first post's moment swings from −2.3 to 58.8 and the middle post's from −29.6 to 29.6, so their ranges are 61.1 and 59.3. Every interior post is within three per cent of the same range, and fatigue reads the range.

Tested in The corner that counts its crossings, at the figure it turns on · the essays about vierendeel.

A haunch at a Vierendeel joint reduces the stress range there in proportion to its depth, so a deeper haunch always lengthens the joint's fatigue life.

What decides it: A haunch twice the member's depth takes the corner's stress range from 78 to 43 MPa, but the worst range moves to the haunch's toe, 250 mm along the post, and rises there with the haunch's depth — from 55 MPa at 1.1 times to 61 at twice and 63 at three times — because stiffer corners draw more moment into the joints. Past about 1.5 times the depth, more depth makes the joint worse.

Tested in The corner that counts its crossings, at the figure it turns on · the essays about vierendeel.

Counted plastically, a single-sided partial-penetration weld recovers most of what its eccentricity costs.

What decides it: At half penetration of a 20 mm S355 plate, pulled straight across, the single side keeps 0.25 of a double-sided throat elastically and 0.41 plastically. Yielding wins back a fifth of the loss and leaves three fifths of it, and the share stays at 0.41 for every direction up to 22.7°.

Tested in The root that has to stretch for it, at the figure it turns on · the essays about weld strength.

A single-sided partial-penetration weld can never carry as much as a fillet weld with the same throat.

What decides it: Counted plastically, a single-sided throat through 0.7 of a 20 mm S355 plate carries 357 N/mm² of throat with the load 40° off square, against a fillet's 352, and carries more than the fillet for every direction from 38° to 88° — provided its root can strain several times its yield strain without tearing.

Tested in The root that has to stretch for it, at the figure it turns on · the essays about weld strength.

A truss with a counter-diagonal in every panel is robust as designed, because every panel has a second diagonal to take over if one fails.

What decides it: Sized fully stressed for its full load, with every counter given at least one unit of area, the eight-panel counter-braced truss loses a Pratt diagonal beside the support and leaves a vertical at 5.47 times its allowable stress under two thirds of the load. The second diagonal is there; the members that turn the shear into it are not sized for the turn.

Tested in The truss sized to lose a diagonal, at the figure it turns on · the essays about truss deflection.

Designing a truss to survive the loss of any one member costs about the same whichever member is lost.

What decides it: At two thirds of the load, surviving any counter's loss needs 1,213 units of steel, any vertical's 1,236 and any diagonal's 1,304, against 1,200 intact; surviving any top chord's loss needs 2,458. And four members — the end diagonals and the end panels' bottom chords — cannot be survived at any weight.

Tested in The truss sized to lose a diagonal, at the figure it turns on · the essays about truss deflection.

A stroke amplifier lets a nonlinear damper sized for the earthquake keep working in the wind, because it makes the damper move further when the building barely moves.

What decides it: The same α = 0.3 damper behind a lever of ratio 3 acts on the storey as one 4.17 times larger: it is best at 7.5 times the design amplitude and gives 2.3% at the design amplitude instead of 22.7%. Re-sized to be at its best at the design amplitude, it gives the storey exactly the unamplified curve, 1.4% at a hundredth of the design amplitude.

Tested in The lever that moves the damper and not the band, at the figure it turns on · the essays about damping.

A toggle brace's amplification is a fixed ratio set by its geometry.

What decides it: A toggle in a 3.5 m by 6 m bay with its knee 400 mm off the diagonal has a ratio of 2.71 at rest, 2.04 with the storey drifted 30 mm one way and 4.42 drifted 30 mm the other, and its links lie straight — and lock — at 53 mm.

Tested in The lever that moves the damper and not the band, at the figure it turns on · the essays about damping.

A frame that is a mechanism cannot carry a load, and a frame raised a little off its critical form can.

What decides it: Two bars of 200 MN axial stiffness across 8 m, raised 3 degrees and unstressed, snap through under 11.1 kN. The same bars flat — a mechanism by every count — and tensioned to 200 kN carry 11.1 kN at 89 mm of movement and 50 kN at 210 mm, and their stiffness keeps rising.

Tested in The mechanism that stands up anyway, at the figure it turns on · the essays about determinacy.

A tensioned mechanism has a stiffness set by its tension, as reliable as any other stiffness the frame has.

What decides it: The tension is a lack of fit: 200 kN in a 4 m steel bar of 200 MN axial stiffness is a stretch of 4 mm. Warm the bars 42 degrees above their supports and half the stiffness is gone; 83 degrees and all of it. A raised frame's stiffness does not depend on a stretch of millimetres at all.

Tested in The mechanism that stands up anyway, at the figure it turns on · the essays about determinacy.

A joint that has reached its moment resistance limits the moments in the joints around it, because it has stopped taking load.

What decides it: It stops taking load and the frame sends the load elsewhere. In the two-bay frame under a pattern and a push, the loaded beam's inner joint reaches its 120 kN·m at λ = 2.69; the unloaded beam's joint across the same column carries 13 kN·m at λ = 1 and 81 at λ = 3, six times as much for three times the load.

Tested in The joint that softens under its own load, at the figure it turns on · the essays about corner moment.

An elastic analysis with every joint at one reduced stiffness, half its initial value as EN 1993-1-8 allows for an end plate, gives a frame's joint moments and panel shears to a usable accuracy.

What decides it: At λ = 3 it asks the loaded inner joint for 178 kN·m, 1.49 times its resistance, gives the unloaded inner joint 43 kN·m where it carries 81, and gives the floor's exterior panel 142 kN where it carries 185.

Tested in The joint that softens under its own load, at the figure it turns on · the essays about corner moment.

A jacket of wire much stronger than the liner lets a wound tube carry more than twice the liner's first-yield pressure.

What decides it: The wound tube answers its working pressure as one elastic wall, so its bore swings by the whole wall's Lamé range and may fall at rest no further than the liner's own yield. Wire of any strength gives exactly twice the first-yield pressure — 756 N/mm² on a 900 N/mm² liner 2.5 times its bore — where it can be wound at all.

Tested in The wire that cannot push, at the figure it turns on · the essays about unloading.

Winding a liner with wire of its own steel is the many-layer limit of a shrink-fitted compound tube, so it reaches the ceiling on the same walls.

What decides it: Shrunk-on layers of one steel reach the ceiling on walls 2.22 times their bore and up. Wire of the same steel needs 3.01, because on thinner walls the innermost turns would have to rest in compression, and a wire wound under tension cannot push.

Tested in The wire that cannot push, at the figure it turns on · the essays about unloading.

A box girder's diaphragm has to be a stiff plate, welded all round, to hold the section square.

What decides it: It has to be stiff against the girder's own resistance to a point racking load, 15 × 10⁹ N·mm/rad for a 4.0 by 2.5 m box. A cross-frame of two 1,000 mm² diagonals is 26 times that and keeps the warping stress within 10 per cent of a rigid diaphragm's; a 10 mm plate is 524 times it.

Tested in The diaphragm that is a few angles, at the figure it turns on · the essays about box distortion.

A softer cross-frame attracts less force, so a flexible diaphragm can be made lighter in proportion.

What decides it: The most loaded frame's diagonals carry 166 kN at 26 times the girder's point stiffness, 161 kN at a tenth of that stiffness, and a rigid diaphragm's would be 182. The force follows the load, not the stiffness, until the frame is softer than the girder itself.

Tested in The diaphragm that is a few angles, at the figure it turns on · the essays about box distortion.

Rayleigh's quotient on the frame's first-order sway under its notional loads is close to the critical load factor, because the sway shape is close to the buckling mode.

What decides it: Over a bottom storey at three eighths of the others' stiffness the critical mode puts 0.65 of the roof's sway at the first floor and the first-order sway 0.31; Rayleigh's quotient gives 11.70 against 9.91, 18 per cent high. With rigid beams, 24 per cent high.

Tested in The trace that adds up every storey, at the figure it turns on · the essays about stability energy.

Making a network arch's hangers more flexible moves the shear back into the tie and brings the hangers' force ranges down to what their details can survive.

What decides it: A spring that leaves each hanger a tenth of its bar's stiffness takes the worst range from 678 to 635 kN, 6 per cent, and the tie's largest moment from 316 to 424 kN·m. A fiftieth of the stiffness takes the range to 557 and the tie to 1,154. On vertical hangers the range is 194 kN and the tie's moment 3,505.

Tested in The hanger that cannot be made soft enough, at the figure it turns on · the essays about network arch.

A column hinge cycled at a constant axial load settles into a stable loop once its fibres have found their range, so its length is only changed by the first cycle.

What decides it: An I-section of 3,000 mm² at 30% of its squash load, cycled to ±3 first-yield curvatures in elastic–perfectly plastic steel, shortens by 8.14 yield strains in every cycle: 97 after twelve. The rectangle of the same area shortens by 1.60 every cycle. Neither settles.

Tested in The column that shortens on every cycle, at the figure it turns on · the essays about Moment-curvature.

A column's hinge shortens in an earthquake because the overturning swings its axial load up and down with the sway.

What decides it: At a constant 30% of its squash load the I-section shortens 8.14 yield strains a cycle. Swinging the load by ±30% with the curvature takes that down to 6.36; the rectangle's 1.60 hardly moves. The constant part of the load and the reversed curvature make the ratchet.

Tested in The column that shortens on every cycle, at the figure it turns on · the essays about Moment-curvature.

A column divided among more braces asks less of each brace, since the restraint is shared between them.

What decides it: The ideal stiffness of each nodal brace, in units of the load over the segment length, is 2 for one brace, 3 for two, 3.62 for four and 3.91 for ten, approaching 4. Each brace has to stop a zigzag in which both its neighbours push it the same way, and the more braces there are, the nearer the zigzag comes to its worst case.

Tested in The braces that have to hold each other, at the figure it turns on · the essays about effective length.

A row of braces each well above the ideal stiffness can lose one, because the others have stiffness to spare.

What decides it: Four nodal braces at twice the ideal stiffness carry the column to the Euler load of its segments. One of them can drop to 0.68 of the ideal without loss; with it gone the column buckles at 0.52 of that load, over a bay twice as long, whatever the stiffness of the other three.

Tested in The braces that have to hold each other, at the figure it turns on · the essays about effective length.

Loading whole spans in the standard patterns finds the worst shear in a continuous beam, as it finds the worst moment.

What decides it: Three spans of 8 m with 20 kN/m of dead load and 15 kN/m imposed: at the middle of the inner span the imposed load placed from the section to the far support, with the first span loaded too, gives 23.8 kN of shear. The best pattern of whole spans gives 10.0 kN. The moment at the same section is the same either way, 104 kN·m.

Tested in The shear whole spans cannot find, at the figure it turns on · the essays about influence line.

The straight line from the support shear down to qL/8 at mid-span, which is safe for a simple span, is safe for a continuous one.

What decides it: On the simple span the line lies above the exact envelope everywhere and touches it at mid-span. On the inner span of three equal spans the exact imposed shear at mid-span is 0.198 qL, 23.8 kN from 15 kN/m on 8 m — 158% of qL/8 — and at three quarters of the span it is 13.0 kN against the line's 7.5.

Tested in The shear whole spans cannot find, at the figure it turns on · the essays about influence line.

A beam notched at its support to half its depth keeps half its shear strength, since the shear stress on the remaining depth is simply doubled.

What decides it: A glulam beam 400 mm deep and 140 mm wide, with a shear strength of 3.5 N/mm², carries 130.7 kN un-notched. Notched to half its depth with the corner over the reaction, Eurocode 5's kᵥ of 0.50 on the 200 mm left gives 32.7 kN — a quarter, not a half. At 800 mm deep the notched end keeps 18%.

Tested in The notch that costs more than its depth, at the figure it turns on · the essays about shear flow.

A notched end's capacity is a stress check on the reduced section, so two beams of the same proportions are equally safe.

What decides it: The crack that fails a notched end runs when the energy it releases reaches the timber's fracture energy, and that makes the cracking stress fall as one over the square root of the depth. At half the depth left, the cracking stress equals the shear strength at 100 mm deep and is half of it at 400 mm and a quarter at 1,600 mm, whatever the proportions.

Tested in The notch that costs more than its depth, at the figure it turns on · the essays about shear flow.

A cable-stayed deck hangs from its stays, so it is a beam and cannot buckle; only the towers carry compression.

What decides it: The stays' horizontal pulls gather in the deck as a compression of 95 MN at the tower of a 1,000 m span under 150 kN/m. On the stays' springs a deck of 0.5 × 10¹² N·m² buckles at 6.9 times its dead load at that span and at 2.3 times at 2,000 m: below ten from about 800 m, where a design must follow the deck's deflections into its forces.

Tested in The deck the stays squeeze, at the figure it turns on · the essays about stiffening girder.

The sloshing mass of a tank grows with the water in it, so a tall standpipe has a large tuned mass on board.

What decides it: In a tank 10 m across the sloshing mass reaches 170 t at 5 m deep and stops, at 0.455ρπR³ = 179 t, whatever is added: full to 28 m it is 178 t of 2,199. Its period stops changing too, at 3.31 s, set by the radius alone.

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

A screen across a standpipe's water makes the water a damper, as it does in an elevated tank.

What decides it: The standpipe's sloshing runs at 0.12 of its shaft's frequency, where a tuned damper needs 0.75. A screen giving 10% of damping takes the rms drift from 0.82 to 0.76 of the frozen-water value; in a tank on a 2.5 s tower the same screen takes it from 1.01 to 0.40.

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

A floor's characteristic imposed load, measured or surveyed floor by floor, is all a column's reduction needs to know.

What decides it: Sharing a quarter of the visits between all floors leaves one floor's fifty-year characteristic at 3.22 kN/m² against 3.20 — the same within sampling — and moves twenty floors' average at the 95th percentile from 0.85 to 2.18 kN/m². The column's factor depends on how the floors' visits are timed against each other, which no single floor's record contains.

Tested in The visits that come to every floor at once, at the figure it turns on · the essays about live load reduction.

An accidental design may let the damaged structure yield, so checking a member's loss by the lower-bound theorem rather than elastically makes an alternative load path much cheaper.

What decides it: Sized to survive any one interior diagonal's loss at two thirds of the full load, the eight-panel counter-braced truss needs 1,304.1 units of steel checked elastically and 1,296.7 checked by the lower bound — 7.4 of the 104 units robustness costs. Held elastic in the intact state, the plastic design needs 1,297.8. The diagonals are identical in every design, because a cut through a panel that has lost a diagonal fixes the force in the one that remains.

Tested in The section that yielding cannot reach, at the figure it turns on · the essays about truss deflection.

Once the damaged truss may yield, a lost chord's force can be shared among every diagonal in the truss, so the alternative path round a chord stops doubling the steel.

What decides it: Surviving the loss of any top chord at two thirds of the load needs 2,458 units checked elastically and 2,363 checked plastically, 4 per cent less and still about twice the intact truss's 1,200. A section through the panel whose chord is gone cuts the other chord and two diagonals, and its three equations hand the panel's moment to those two diagonals whatever the rest of the truss does.

Tested in The section that yielding cannot reach, at the figure it turns on · the essays about truss deflection.

The only way to stop a preloaded base ratcheting under repeated storms is to leave its bolts at the reduced preload the first storm leaves.

What decides it: A stack of disc springs of 500 kN/mm per bolt under each nut raises the largest preload a storm to ±170 kN·m leaves untouched from 128 kN to 150, and one of 20 kN/mm to 214. Retightened to 150 kN after every storm, the stacked base adds no permanent stretch in ten storms where the bare base adds 1.25 mm.

Tested in The spring under the nut, at the figure it turns on · the essays about base plate.

Setting each floor high by the whole of its predicted shortening leaves a tall building level.

What decides it: It leaves the building level on one date: thirty years on, at the creep designed for. At the fit-out the forty-storey building's floors stand up to 16.0 mm high under the least creep the ±30 per cent band allows, and its worst floor ends 5.4 mm low under the most. Set high by 0.79 of the prediction instead, no floor is ever more than 11.0 mm out.

Tested in The floor that is level on one date, at the figure it turns on · the essays about differential shortening.

The roof of a tall building does not move after it is set, because nothing is built above it.

What decides it: Counted in time, the roof's column drops 24.1 mm against the core in the thirty years after the roof is cast: 8.5 mm of creep still to come from the load already on the columns, and 15.7 mm from the finishes that arrive at the fit-out and their creep.

Tested in The floor that is level on one date, at the figure it turns on · the essays about differential shortening.

An end plate with its bolt far from its web decompresses earlier than its component model says, because its prying lifts the column flange's tip and opens the clamp from its edge.

What decides it: It decompresses later. With the end plate's bolt 45 mm from its web against the column flange's 30, the pair's clamp lets go at 88.2 kN and the end plate's stub, which the component model is governed by, at 81.1; at 60 mm, 82.0 against 71.2. Leaning on a column flange that gives way, the end plate pries 0.38 of the tension instead of its stub's 0.50.

Tested in The clamp the pair holds longer, at the figure it turns on · the essays about prying.

A tuned liquid damper can be designed with linear sloshing theory, as a tuned mass damper is designed with a linear spring.

What decides it: At a peak acceleration of 20 thousandths of g, inside the range the damper exists to control, the linear theory's wave in a 10 m tank tuned to a 5 s building is 0.26 of its depth, and in one tuned to an 8 s building 0.71. A wave that size has stopped being the small disturbance the linear frequency and the linear mass were derived for.

Tested in The tank that must be shallow to be slow, at the figure it turns on · the essays about sloshing.

The largest pressure in a silo's hopper is at its outlet, where the depth of solid above is greatest.

What decides it: In a 20° hopper under a bin 8 m across holding 30 m of grain, the emptying pressure on the wall is 95.7 kPa at the hopper's top edge, 36.7 half way down, and the vertical stress half a metre above the apex is 2.6 kPa. The hopper carries its load at the transition and almost nothing at the outlet.

Tested in The pressure that switches at the hopper, at the figure it turns on · the essays about silo pressure.

A silo empties more slowly as its contents fall, as a tank of water does when its head drops.

What decides it: The stress half a metre above the apex of the emptying hopper is 2.57 kPa with 5 m of grain in the bin and 2.58 kPa with 60 m, where a liquid's would go from 144 to 639 kPa. The flow through the outlet is set by the stress near it, and that stress is set by the hopper's angle and the solid, not by the silo above.

Tested in The pressure that switches at the hopper, at the figure it turns on · the essays about silo pressure.

Once a continuous prestressed beam is complete it carries the same secondary moments whatever order it was built in; the order affects only the stresses during construction.

What decides it: On the day of the last stressing the first interior support of the five-span beam keeps 161 kN·m built from both ends and 215 built from the middle out. After creep at a coefficient of 2 the two are 302 and 314: closer, still different, and both short of the 344 a beam stressed at once carries.

Tested in The beam built towards its own middle, at the figure it turns on · the essays about secondary prestress.

A notch with a stress concentration factor of three triples the stress at its root at any load.

What decides it: Only while the root is elastic, up to a nominal stress of about 84 N/mm² in this mild steel. At a nominal 200 N/mm² the root carries 293, a concentration of 1.47, while its strain is 6.14 times the nominal strain. The factor of three survives only as the geometric mean of the two.

Tested in The notch that trades stress for strain, at the figure it turns on · the essays about stress concentration.

A notch triples a mean stress as it triples any other, so a fatigue check at a notch must count three times the nominal mean at the root.

What decides it: Cycled by ±100 N/mm² about a nominal mean of 150, the root of a notch of factor 3 settles after its first cycle at a mean of 97 N/mm², not 450: the first excursion yields it, and the yielding takes most of the mean away. The root's life falls from 72,500 cycles to 60,300 rather than to the 32,700 an elastic mean would give.

Tested in The notch that trades stress for strain, at the figure it turns on · the essays about stress concentration.

A pretensioned beam that passes its checks on its yard bearers will pass them on a trailer, since a trailer carries it on two supports just as the yard does.

What decides it: On its yard bearers 1 m from each end the beam's top fibre is at −2.98 N/mm² against a limit of −3.0. A road's bounce of 30 per cent on the same bearers takes it to −3.26; a trailer carrying it 1.5 m from its front end and 2.5 m from its rear takes it to −4.68 standing still and −4.88 bouncing. The trailer's supports are where the haulier put them, not where the schedule assumed.

Tested in The beam the haulier holds, at the figure it turns on · the essays about prestress limits.

A road's bounce is a factor on the beam's weight, so it matters only where that weight hogs the beam, over its overhangs.

What decides it: The bounce also lightens the beam, and a lighter beam sags less between its supports, which is where its self-weight was relieving the top fibre. On the yard's bearers a 30 per cent bounce puts the worst top fibre at −3.26 N/mm², 9.8 m from the front end, between the supports — not over them.

Tested in The beam the haulier holds, at the figure it turns on · the essays about prestress limits.

A row of tension-only braces will hold a girder as long as their total stiffness is enough, since a girder can only buckle one way at a time and the braces facing that way will catch it.

What decides it: The girder buckles the way its braces do not resist. Three one-way braces all facing the same way leave the 24 m girder at its unbraced 947 kN·m whatever their stiffness. Alternating, at the 2,846 kN·m/rad each that takes two-way braces to the target of 5,627, they reach 3,087, and rigid they reach 4,063.

Tested in The braces that pull only one way, at the figure it turns on · the essays about beam bracing.

If the outer braces of a one-way row are oriented in opposition, they hold the two-wave shape, and the row then needs nothing more.

What decides it: The two-wave shape twists the outer brace points in opposite directions, and opposed one-way braces at those points are then both slack, whichever way the girder buckles. With the middle brace two-way and the outer pair opposed and rigid, the girder reaches 3,532 kN·m against a target of 5,627. One one-way brace anywhere in the row caps it: the middle one alone gives 4,063.

Tested in The braces that pull only one way, at the figure it turns on · the essays about beam bracing.

Re-shimming an outrigger's joint every few years keeps its creep force as low as a viscous damper in the joint would.

What decides it: A damper with a ten-day time constant lets in about 2 per cent of the creep force. Adjusted at one, three and ten years after it is connected, the arm of a 200 m concrete building still holds up to 1.40 MN of creep force between visits — 40 per cent of the 3.48 MN it reaches left alone. Shims take out the creep that has happened; they cannot stop the creep that is happening.

Tested in The arm that is reset by hand, at the figure it turns on · the essays about outrigger.

The adjustments of an outrigger's joint are best spread evenly through the building's life.

What decides it: Creep slows as the logarithm of time, so a visit is worth most early. Three adjustments at one, three and ten years hold the arm below 1.40 MN; at five, ten and fifteen years, 2.48; at ten, twenty and thirty, 2.93. The geometric spacing also asks for nearly the same thickness of shim at each visit.

Tested in The arm that is reset by hand, at the figure it turns on · the essays about outrigger.

A forge hammer's blow can be taken as a fixed impulse on the anvil, the tup's momentum times a factor for its rebound, whatever is being forged.

What decides it: The forging decides the impulse and, more, its duration. A 2 t tup at 7 m/s hands the anvil 14.2 kN·s over 48 ms through a hot forging flowing at 0.3 MN, and 22.5 kN·s over 2.2 ms when the dies meet. The anvil's pad carries 0.46 MN in the first case and 5.86 MN in the second; the same impulse delivered at once would put 3.70 MN through it for a forging flowing at 0.1 MN that actually puts 0.15.

Tested in The forging that cushions the blow, at the figure it turns on · the essays about vibration isolation.

The heaviest blow a forge hammer's foundation receives is the one that does the most forging.

What decides it: The blow that forges most, into a hot part, delivers almost none of the tup's 49 kJ to the foundation — the forging absorbs 99 per cent of it. The blow that forges nothing, the dies meeting, sends 31 kJ into the anvil and the block and moves the block 1.42 mm against 0.39. The foundation is designed by the blow that does no work.

Tested in The forging that cushions the blow, at the figure it turns on · the essays about vibration isolation.

Casting a continuity joint's diaphragm early and its slab late lets the early pour hold the beams' creep and the late pour escape most of the slab's shrinkage, so the joint ends with little moment either way.

What decides it: From the slab's day on, the composite section holds what a single pour on that day would hold, and the first stage adds only what it gathered on the bare beams. With the first stage at 28 days, a slab at 42 days ends at 139 kN·m against 160 poured once, a slab at 90 days at 41 against 40, and a slab at a year at −165 against −214. The joint balances at 119 days instead of 113.

Tested in The joint poured twice over the pier, at the figure it turns on · the essays about end rotation.

The first stage of a two-stage joint only has to locate the beams until the slab is cast; it carries nothing of consequence.

What decides it: It holds the beams' creep on the bare beam's section, with a cracking moment of 71 kN·m against the finished joint's 140. With the first stage at 28 days and the slab later than about four months, the beams' creep sags it past that before the slab arrives — to 112 kN·m with the slab at a year — and with the slab at six weeks the wet slab's weight, carried continuous, hogs it to 61.

Tested in The joint poured twice over the pier, at the figure it turns on · the essays about end rotation.

A beam's flange splice with an outer cover and thinner inner covers shares the flange force between them by their lever arms about the flange's centre line, as a plate splice with unequal covers does, so the thin inner covers carry the larger share.

What decides it: In a 610 mm beam at 700 kN·m, with a 15 mm outer cover and two 8 mm inner covers, the lever rule gives the inner covers 350 N/mm² and the outer 123. Across the gap the plates bend together with the beam's halves, and as one section the outer cover works at 206 and the inner covers at 186 — the order reversed, and the inner covers at half the lever rule's stress.

Tested in The splice that bends as one section, at the figure it turns on · the essays about single lap.