What is refuted here — page 11

A curved I-beam can be checked with the whole of its flanges effective, as a straight one is, until its radius is small enough for the curved-bar correction to matter.

What decides it: A flange 300 mm wide and 15 mm thick bent to 2 m works over 78 per cent of its width and to 1 m over 60 per cent; the outstands curl toward or away from the web and shed the stress at their tips.

Tested in The flange that curls away from its stress, at the figure it turns on · the essays about curved beam.

When a curved flange keeps nearly all of its effective width, its curling can be ignored.

What decides it: At λ = 0.8 the flange still works over 93 per cent of its width, and the bending stress across it at the web already equals the stress along the beam; fully curled it is 1.82 times that stress. The same 300 × 15 flange at 5 m keeps 96 per cent of its width and bends across it at 0.82 of the stress along.

Tested in The flange that curls away from its stress, at the figure it turns on · the essays about curved beam.

Anchoring a suspension bridge's cable to its own deck instead of to the ground changes the anchorage and nothing about how the deck carries load.

What decides it: For a 500 m span with a 50 m sag and a deck of 1.03 × 10¹² N·m², the largest live-load moment is 56.3 MN·m anchored to the ground and 83.0 MN·m anchored to the deck — 1.47 times as much, and above Rankine's elastic theory's 80.1, which leaves the cable's tension out altogether.

Tested in The bridge that pays for its own anchorage, at the figure it turns on · the essays about stiffening girder.

At the tip of a cantilever, as on a span, the lateral restraint belongs on the compression flange, which for a hogging cantilever is the bottom one.

What decides it: A 3 m cantilever of 457 mm universal beam, built in: holding the bottom flange at the tip multiplies its critical moment by 1.00 with the load at the shear centre and 1.28 with it on the top flange; holding the top flange, which is in tension, multiplies it by 2.47 and 2.02.

Tested in The flange that swings is in tension, at the figure it turns on · the essays about moment gradient.

A cantilever's effective length is a fixed multiple of its length once the root, the tip and the load's height are known.

What decides it: For the same section, root free to warp and load on the top flange, the effective length runs from 6.13 times the length at 1.5 m to 1.32 times at 9 m. Built in with the load at the shear centre it stays between 0.70 and 0.82, which is why a single factor looks adequate for the one case and not the other.

Tested in The flange that swings is in tension, at the figure it turns on · the essays about moment gradient.

Combining the group's largest shear with its largest tension, as if one bolt carried both, is an exact check whenever the load is concentrated at a single point.

What decides it: It is exact only while one bolt is worst at both. For the six-bolt bracket loaded 150 mm to the side and 150 mm in front of the column, the shortcut gives 201 kN at 55° against a true 221, under-reading the group by a factor of 1.10 exactly where the governing bolt changes.

Tested in The bracket with no worst bolt, at the figure it turns on · the essays about bolt group.

Where the derivative of the strain energy is wrong for a nonlinear member, it errs on the side of overstating the deflection, so a deflection found that way is safe.

What decides it: Only for a member that softens. A 10 m cable pretensioned to 60 kN and loaded across its midpoint with 10 kN sags 212 mm; the derivative of its strain energy gives 107 mm, 0.51 of it, and with no pretension the route returns a third of the true sag at every load.

Tested in The sag the strain energy halves, at the figure it turns on · the essays about strain energy.

A cracked member is safe if it passes both the fracture check, with the toughness against the stress intensity, and the yield check on the same section.

What decides it: A grade 355 plate of toughness 100 MPa√m with a 20 mm edge crack passes both checks up to 355 N/mm² — the fracture line and the yield line cross there — and fails at 288 N/mm² on the strip-yield model, 0.81 of it. At a working stress of 302 N/mm², cracks of 20 and 25 mm pass both checks and fail.

Tested in The crack between the two checks, at the figure it turns on · the essays about fracture.

A thin plate closing an open section adds little, because it is thin; its contribution to the section's torsional stiffness is in proportion to its thickness.

What decides it: A closing plate a thousandth of the channel's thickness multiplies the torsion constant by 1.7, a hundredth by 8.1 and a tenth by 63. The rate of twist under a load through the web falls to 0.12 of the open channel's with the hundredth-thickness plate, while the load's lever is still 31.1 mm of the original 33.3.

Tested in The plate that closes the circuit, at the figure it turns on · the essays about shear centre.

Velocity feedback is unconditionally stable: an actuator that always pushes against the velocity can only remove energy, so the higher the gain the better.

What decides it: Only with no delay in the loop. With a delay of a fifth of the period the largest stable gain is 0.25 of critical damping; with a quarter-period, 0.079; with three tenths, 0.028. A gain of 0.10 set moving a quarter-period late grows 1.3-fold over twelve periods instead of decaying to 0.071.

Tested in The actuator that arrives late, at the figure it turns on · the essays about tuned mass damper.

The work-equivalent nodal loads of a uniform pressure on an element are shares of the load, so each is positive and each node receives roughly the load on the area around it.

What decides it: On an eight-node serendipity element the corners receive −1/12 of the load each and the mid-sides 1/3: the corners are pulled against the pressure. On a six-node triangle the corners receive nothing and the mid-sides a third each.

Tested in The corners the pressure pulls up, at the figure it turns on · the essays about force couple.

A stressed ribbon whose spans are proportioned so that the dead-load thrusts balance puts no horizontal force on its piers.

What decides it: Only at the temperature it was balanced at. Spans of 80 and 120 m with sags of 1.28 and 2.88 m balance under dead load, and a rigid pier between them carries 2,155 kN on a 30° cooling and 1,314 kN on a 30° warming; two equal spans carry nothing.

Tested in The pier that carries only the difference, at the figure it turns on · the essays about stressed ribbon.

Rayleigh's quotient from a sensible smooth shape is close enough for any column that has no closed form, because its error is the square of the error in the shape.

What decides it: Only while the shape is close. A pin-ended column whose outer quarters keep a tenth of the middle's rigidity buckles at 3.225 EI₀/L²; a half sine gives 8.256 by Rayleigh's quotient, 156 per cent high, and the column's own sag shape 8.041. The same two shapes give 3.745 and 3.712 by Timoshenko's quotient.

Tested in The bound from underneath, at the figure it turns on · the essays about stability energy.

There is no practical lower bound on a buckling load, only upper bounds, so a critical load from an energy method can never be shown to be safe.

What decides it: The traces of the column's flexibility bound it from below with no assumed shape: 6, √90 = 9.487, 9.813 and 9.859 EI/L² for a uniform pinned strut whose load is 9.870. For the column with soft outer quarters the fourth trace gives 3.171 against 3.225.

Tested in The bound from underneath, at the figure it turns on · the essays about stability energy.

Pausing construction on soft ground buys safety in proportion to the time spent waiting, wherever the pause falls.

What decides it: A year's pause inserted in a one-year build leaves the finished tower at 1.34 if taken after 5 per cent of the weight and 1.55 after 95 per cent; building steadily over the same two years gives 1.45.

Tested in The ground that arrives after the weight, at the figure it turns on · the essays about overturning.

Adding shear to a weld that is already carrying a transverse pull can only reduce what it can carry.

What decides it: Not for a butt throat, whose normal stress is capped at 0.9fu. In S355 its resultant capacity rises from 441 N/mm² straight across to 478 when the shear is 0.42 of the pull, 22.7° off square, before it falls to 314 in pure shear.

Tested in The shear that helps a butt weld, at the figure it turns on · the essays about weld strength.

A steel's fracture toughness, measured on the standard specimen, is the toughness of every plate of that steel at the same temperature.

What decides it: In the transition range cleavage starts at the weakest spot along the crack front, so the toughness depends on the front's length. At −20 °C in a steel whose reference temperature is −38 °C, the 5 per cent toughness is 88 MPa√m for a crack through a 12 mm plate, 77 at the specimen's 25 mm and 66 at 60 mm.

Tested in The toughness that belongs to the plate, at the figure it turns on · the essays about fracture.

Enlarging members to stiffen a truss can only lower the stresses in it.

What decides it: In a redundant truss the enlarged members attract force. Making the Pratt diagonals of the fully stressed counter-braced truss half as large again takes the mid-span deflection from 24.0 to 22.7 and puts an unaltered vertical at 1.58 times its allowable stress.

Tested in The truss whose forces follow its sections, at the figure it turns on · the essays about truss deflection.

A keyway weakens a shaft in torsion in proportion to the material it removes, so a keyway that costs a tenth of the torsional stiffness raises the stress by about a tenth.

What decides it: In a 50 mm shaft a 14 by 5.5 mm keyway keeps 0.88 of the stiffness, and its bottom corners, rounded to 0.4 mm, carry 2.81 times the plain shaft's surface stress under the same torque. Only the far side of the shaft sees the tenth: 1.11.

Tested in The corner a soap film cannot finish, at the figure it turns on · the essays about membrane analogy.

A finer analysis of a sharp corner gives a more accurate value of its peak stress.

What decides it: At a sharp re-entrant corner the elastic stress is infinite, so there is no value to converge to. The computed peak at the keyway's corner goes 3.38, 4.19, 5.23 and 6.55 times the plain stress as the grid spacing halves from 0.06 to 0.0075 mm; with a 0.5 mm fillet it settles at 2.66.

Tested in The corner a soap film cannot finish, at the figure it turns on · the essays about membrane analogy.

The dynamic effect of a sudden member loss is covered by doubling the static change of force in each surviving member.

What decides it: Doubling is exact for one mode. Removing a diagonal of a counter-braced Pratt truss in an instant, with 2 per cent damping, a bottom chord changes by 171 kN and passes through 508, 2.97 times; another settles 110 kN lower after first swinging 239 kN higher; and a chord whose force does not change swings by 365 kN.

Tested in The factor of two belongs to one mode, at the figure it turns on · the essays about robustness.

A member whose force is the same after the loss as before it is unaffected by the loss.

What decides it: Only statically. The mid-span bottom chord of the same truss carries 750 kN before and after the loss of a diagonal near the support, and passes through 1,115 kN on the way.

Tested in The factor of two belongs to one mode, at the figure it turns on · the essays about robustness.

A pile's largest moment in layered ground lies between its largest moments in the two layers' grounds taken alone.

What decides it: Soft ground 3.8 m thick over stiff ground gives 427 kN·m; the same pile in the stiff ground alone gives 219 kN·m and in the soft ground alone 346.

Tested in The layer a pile feels, at the figure it turns on · the essays about lateral pile.

Two descriptions of the same ground that agree on its strength and on its stiffness at half load give the same tipping push.

What decides it: Only if they soften alike near the limit. Matched at 300 kPa and at 150 kPa for 7.5 mm, a capped linear spring and an S-curve tip the 3,000 kN block at 776 and 770 kN, a hyperbola at 701. With the block at four fifths of the ground's capacity the hyperbola gives 286 kN against the capped spring's 573.

Tested in The ground that never quite gives way, at the figure it turns on · the essays about overturning.

Putting the actuator of an active damper on a tuned mass makes it safer, because the passive mass underneath is stable on its own and the actuator only adds to it.

What decides it: Fed by the structure's velocity, an actuator pushing against a tuned mass of 2 per cent goes unstable at a gain that would add 1.56 per cent of damping to the bare mode, with no delay; the same actuator pushing against the ground is stable to a gain of 2.41 with a delay of a twentieth of a period.

Tested in The actuator that pushes against the mass, at the figure it turns on · the essays about tuned mass damper.

Any velocity feedback with no delay adds damping, so a little of it can only help.

What decides it: Pushing against the tuned mass, a gain of half a per cent raises the structure's peak response from the passive mass's 8.72 to 13.42 and one per cent to 27.01, and the mass's stroke from 45 to 74 and 183 times the static deflection.

Tested in The actuator that pushes against the mass, at the figure it turns on · the essays about tuned mass damper.

A column is designed on the lesser of its material's proof stress and its Euler stress, whatever the material.

What decides it: Only for a material with a plateau. A perfect aluminium column (proof stress 250 N/mm², Ramberg–Osgood exponent 10) buckles at 0.74 of that envelope at λ̄ = 1, and an austenitic stainless steel with exponent 5 at 0.56; a stocky aluminium column with exponent 5 buckles at 1.54 times its proof stress.

Tested in The column with no plateau, at the figure it turns on · the essays about inelastic buckling.

Two materials with the same Ramberg–Osgood exponent have the same column curve once slenderness is normalised by the proof stress.

What decides it: At n = 8 an aluminium alloy of proof stress 250 N/mm² buckles at 0.72 of it at λ̄ = 1 and a stainless steel of 230 at 0.64, because the 0.2 per cent offset is 0.56 of the aluminium's elastic strain at proof and 1.74 of the stainless steel's.

Tested in The column with no plateau, at the figure it turns on · the essays about inelastic buckling.

A longer beam needs a stiffer deck to hold it straight.

What decides it: The same deck does more on a longer beam. At 5 kNm per metre per radian the deck removes 6 per cent of the twist of a 4 m beam and 73 per cent of the twist of a 16 m one, and the restraint that halves the twist falls from 81.7 kNm/m/rad at 4 m to 1.9 at 16 m, because the beam's own torsional stiffness falls at least as the square of its span.

Tested in A deck is a spring, not a wall, at the figure it turns on · the essays about twist serviceability.

A bolt's rope effect draws on the bolt's own tensile strength.

What decides it: It draws on whatever holds the bolt along its axis, and that is the washer bearing on the timber. A 12 mm bolt could carry 47.6 kN in tension; a washer three diameters across, bearing at three times the timber's crushing strength across the grain, holds 6.64 kN. A washer of about 3.6 diameters is enough to reach the bolt's cap, and past it a stronger bolt adds nothing to the rope term.

Tested in The pull that Johansen left out, at the figure it turns on · the essays about dowel yield.

Added damping lengthens a lock-in fatigue life in proportion, so doubling the damping of a chimney that is failing doubles its life.

What decides it: Doubling it from 0.40 to 0.80 per cent halves the stress range and lengthens the life 8.6 times, from 22 to 187 days, because damage goes as the cube of the range or faster. At 1.80 per cent the life is 35 years and at 1.92 per cent it is unlimited, because the largest range has fallen under the detail's cut-off.

Tested in The breeze that uses up a chimney, at the figure it turns on · the essays about Vortex-shedding.

A remedial regrade designed on back-analysed strengths is safe whichever pair of back-analysed strengths is used, because every pair reproduces the failure.

What decides it: Regraded to 25°, the slope has F = 1.51 on the pair at 40° and F = 1.07 on the pair with no friction, though both reproduce the failure exactly. For a factor of 1.3 the face must go to 27.7° on one pair and 16.6° on another.

Tested in A failed slope gives a line, not a soil, at the figure it turns on · the essays about slope stability.

An interior column joint, with a beam framing in on each side, has twice the panel shear of an exterior joint.

What decides it: Under full gravity it has none: the two beams hog into it with 123.4 kN·m each and their flange couples cancel, while the exterior floor joint carries 143 kN. Under sway alone it has 1.67 times the exterior's, 89 kN against 53, because it rotates less than a joint held by one beam.

Tested in The panel that gravity leaves empty, at the figure it turns on · the essays about corner moment.

More levels of props under the slab being cast spread its weight over more floors, so each floor carries less.

What decides it: As shores that carried the wet concrete, more levels make the worst floor carry more: 2.00, 2.25, 2.37 and 2.44 times a slab's own weight for one to four levels. Only when the extra levels are backprops put in snug after striking do more of them mean less: 1.50, 1.33, 1.25.

Tested in Loaded twice over before it is a month old, at the figure it turns on · the essays about maturity.

The slab most heavily loaded during construction is the lowest floor, since it is under the whole stack.

What decides it: With three levels of shores the peaks on floors one to six are 1.33, 1.78, 2.37, 1.83, 1.99 and 2.06 times a slab's weight: the first floor's shores bear on the ground, which takes the wet weight, and the worst floor is the third, the first slab to be the base of a full stack that does not reach the ground.

Tested in Loaded twice over before it is a month old, at the figure it turns on · the essays about maturity.

The largest torsional shear stress in an I-section is T·t/J in its thickest plate, the flange.

What decides it: It is on the fillet between web and flange, at 1.49 times T·t/J (t the flange's thickness) in the 457 × 191 × 67 UB and 1.45 to 1.65 times across seven standard sections — close to T·D/J, with D the diameter of the circle inscribed in the junction, 19.0 mm against a 12.7 mm flange.

Tested in The corner the rolling mill rounds, at the figure it turns on · the essays about torsional constant.

The storeys of a tall building that drift least are the least affected by P-Δ.

What decides it: In the bending tower the ground storey drifts 0.07 of the worst storey and its second-order amplification is 1.199, against the top storey's 1.262: every floor's lean is amplified by close to the single factor 1.25, whatever its own storey's drift.

Tested in The storey that cannot see the building lean, at the figure it turns on · the essays about sway stability.

A plan whose stiffness is moved further off-centre always reads more torsionally irregular.

What decides it: With a torsional radius of 0.35 of the plan's length the ratio rises to 1.76 at a natural eccentricity of 0.30 of the length and then falls, to 1.72 at 0.45, while the worst edge's movement rises from 3.3 to 4.8 times a torsion-free plan's over the same range.

Tested in A measure of twist that divides by the twist, at the figure it turns on · the essays about plan torsion.

Leaving anchor bolts unbonded in sleeves, to give them tolerance for erection, does not affect the base's behaviour once they are grouted and tightened.

What decides it: The stiffness of a bolt in tension is its area over the length it stretches over, and a sleeve adds its length to that. A 300 mm sleeve takes the base from 9.1 to 5.3 EI/L, and the portal's drift from 1.28 to 1.45 times a fixed-based portal's.

Tested in The fixed base the bolts decide, at the figure it turns on · the essays about base plate.

Fabrication tolerances of a millimetre on member lengths are negligible against a camber of fifty millimetres.

What decides it: Each millimetre arrives at mid-span multiplied by the member's unit-load force, up to 2.50 for the middle chords. With a millimetre of random error on every member the mid-span camber scatters by 6.4 mm, 13 per cent of the 48.8 mm camber, and 44 per cent of trusses miss by more than a tenth.

Tested in The camber that lives in the member lengths, at the figure it turns on · the essays about camber.

A structure designed to yield under an impact, and so to carry a fraction of the elastic impact force, carries that fraction whenever the impact recurs.

What decides it: Only if it cannot harden, and then it ratchets without limit: 50 mm of permanent set on every drop of the half-tonne weight. With 5 per cent hardening the peak force climbs from 86 kN on the first drop to 104 on the fifth and 152 on the fortieth, towards the 160 kN an elastic structure would have taken from the start.

Tested in The same weight, dropped again, at the figure it turns on · the essays about impact factor.

A section with more area and a larger moment of inertia can always be prestressed for a load that a smaller section carries.

What decides it: On a 12 m span with 20 kN/m imposed, the 700 mm tee (260 × 10³ mm², a stiffer section) has no Magnel region at all, while the symmetric I (180 × 10³ mm²) needs 1,147 kN at an eccentricity of 270 mm. The tee's bottom modulus is 22.7 × 10⁶ mm³ against the 24.0 its bottom fibre's two limits need.

Tested in The flange the prestress cannot use, at the figure it turns on · the essays about prestress limits.

The point of a weld group that governs its static check is the point to check for fatigue.

What decides it: For the C-shaped group with a 5 mm throat and the load through its centroid, the static peak is on the vertical run, loaded along its length, and a fatigue check there allows a load range of 144 kN at two million cycles. The returns, loaded across their length, allow 64.8 kN — 2.2 times less.

Tested in Strongest across, and first to crack across, at the figure it turns on · the essays about weld group.

A fixed arch's secondary moments from rib shortening, temperature and shrinkage are zero at the springings, where it is designed, so they matter only at the crown.

What decides it: On the 60 m concrete rib at a 6 m rise under 60 kN/m, fixing the springings gives a shortening moment of 112 kN·m hogging at each springing and 58 kN·m sagging at the crown. The lost thrust acts at the elastic centre, 3.96 m above the springings, not along the springing line.

Tested in The springings that make shortening worse, at the figure it turns on · the essays about rib shortening.

Fixing the springings stiffens an arch, so it reduces the arch's secondary moments from its own shortening.

What decides it: Fixed, the rib loses 0.63 per cent of its thrust to shortening; two-hinged, 0.11. The fixed rib's crown moment from shortening is twice the two-hinged rib's and its springing moment four times, because the fixity takes away the springings' freedom to rotate while leaving the rib's axial stiffness where it was.

Tested in The springings that make shortening worse, at the figure it turns on · the essays about rib shortening.

If an edge column fails its punching check, a larger column will fix it.

What decides it: On a 7.2 m end span the general check is 1.23 at 400 mm, 1.41 at 450 and 1.53 at 550, because the stiffer column draws more of the slab's end moment, from 170 kN·m to 279. The column that passes is the smaller one: 300 mm, at 0.81, whose moment puts the resultant almost on the perimeter's centroid.

Tested in The perimeter whose middle is not the column's, at the figure it turns on · the essays about punching shear.

A fixed arch on a spread footing only behaves as fixed if the footing is on rock.

What decides it: The springing moment from shortening is nine tenths of the fixed value once each springing's rotational stiffness reaches 102 times the rib's EI/L. A 4 × 3 m footing on stiff clay (shear modulus 20 MPa) gives the 60 m rib 104 times its EI/L, and keeps 101 kN·m of the fixed rib's 112.

Tested in The abutment that spreads before it turns, at the figure it turns on · the essays about rib shortening.

A flexible foundation relieves an arch's secondary moments, because it lets the springings move.

What decides it: Turning relieves them a little and spreading adds to them a great deal. On dense sand the footing's rotation alone takes the springing moment from 112 to 108 kN·m; its 8.0 mm of spread under the thrust takes it to 237. On stiff clay the figures are 101 turning and 457 with 23.8 mm of spread.

Tested in The abutment that spreads before it turns, at the figure it turns on · the essays about rib shortening.

A weld straight across a member is the worst orientation for a soft heat-affected zone, because the whole of the load passes through the softened metal.

What decides it: Straight across the load, the parent on either side stops the zone contracting along the weld and it carries 214 N/mm², 15 per cent above its own uniaxial 185. At 54.7 degrees nothing along the zone has to stretch whatever the zone does, and the member carries 185: the inclined weld is the weaker.

Tested in The weld that is weakest at fifty-five degrees, at the figure it turns on · the essays about Heat-affected zone.

A soft zone running along a member costs it ductility as well as strength, because the zone yields first and takes the strain.

What decides it: In parallel the zone cannot strain more than the parent beside it. With the zone a quarter of the section, the member keeps 90 per cent of the parent's strength and reaches its greatest load at 7.8 per cent strain against the parent's 7.0, because the zone goes on hardening after the parent alone would have necked.

Tested in The weld that is weakest at fifty-five degrees, at the figure it turns on · the essays about Heat-affected zone.

An inclined-screw joint has the same capacity whichever way the load acts along it.

What decides it: Pulled, the screw's axial force clamps the joint and adds friction; pushed, the clamping comes off. At 45 degrees the joint carries 13.4 kN one way and 11.4 the other, 85 per cent; at 70 degrees, 8.9 and 6.6, 75 per cent — before any buckling of the pushed screw is counted.

Tested in The screw that is pulled instead of bent, at the figure it turns on · the essays about dowel yield.

A building let to one tenant over several floors gets the same live load reduction as one let floor by floor, since the rule counts storeys.

What decides it: The rule counts storeys; the averaging counts independent loads. Twenty floors let in tenancies of five are four independent loads, and the reduction they can pay for is 0.75 — against the rule's 0.73 and the 0.61 that twenty independent floors would allow.

Tested in The floors that fill up together, at the figure it turns on · the essays about live load reduction.

Where a building's lateral system changes partway up, the storey-by-storey stability check is least reliable at the interface, because two kinds of behaviour meet there.

What decides it: With the spine stopping at storey 15 of 20, the first storey above it reads a sway factor of 4.999 against the building's critical factor of 5.0, and its storey estimate of the second-order drift equals the true value. The interface is where the check is exact; the storeys it gets wrong are the ones inside the spine.

Tested in The storey above the core decides, at the figure it turns on · the essays about sway stability.

An interior web behaves like a closing plate: a thin one changes the torsion and leaves the shear centre where it was.

What decides it: It does the reverse. A web a tenth as thick as the walls already moves the shear centre 5 per cent of the way to itself and carries 8 per cent of the shear; a web as thick as the walls moves it 27 per cent and carries 38. Through all of that the torsion constant changes by less than one per cent.

Tested in The web that moves the centre and not the twist, at the figure it turns on · the essays about shear centre.

Once a beam that has lost its column goes into catenary, its resistance stiffens with deflection, so the dynamic effect of a sudden loss is much less than the factor of two of a linear spring.

What decides it: Past one depth of sag the catenary's resistance is proportional to the sag, a straight line through the origin, and a straight line under a sudden load overshoots like any spring. The force the beam reaches over the load it carries is 1.73 at 1.5 times the plastic collapse load, 1.88 at twice and 1.97 at three times. It is least, 1.18, on the bending plateau, before the cable forms.

Tested in The cable that pays the factor back, at the figure it turns on · the essays about robustness.

A filler graded from soft at its face to strong behind can keep small collisions gentle and still stop the hardest one, because each layer is sized for the collisions that reach it.

What decides it: The energy of the hardest collision, 761 kJ for 500 t and 300 t closing at 2.85 m/s, must be absorbed in the same 24 mm of crushable depth whatever the grading. A uniform filler does it at 31.7 MN; with the outer half softened to 1.5 MPa over 5 m², the inner half must crush at 56 MN. Of every two-layer grading that stops it, none is gentler than the uniform one to both a 1.94 m/s and a 2.85 m/s collision.

Tested in The filler that cannot be gentle twice, at the figure it turns on · the essays about pounding.

Softening the face of a gap filler can only reduce the forces between the buildings.

What decides it: It reduces the light contacts — 4.6 to 5.8 MN against 7.5 to 9.3 for a uniform 4 MPa filler in the same pulse — and raises the rest. The hardest contact, at 2.74 m/s, meets 31.0 MN against 23.0, and the contacts after it 26.2 against 20.6.

Tested in The filler that cannot be gentle twice, at the figure it turns on · the essays about pounding.

A flat slab that fails its punching check can always be made to pass with enough shear studs.

What decides it: Only while the column face holds. For a 260 mm slab carrying 12 kN/m² on 400 mm columns with the shear 300 mm off centre, studs are needed from a 7.0 m bay and can carry the control perimeter at any demand up to an 11.8 m bay, where the face, which no reinforcement reaches, runs out.

Tested in The studs that send the check outward, at the figure it turns on · the essays about punching shear.

The amount of shear reinforcement a slab needs grows in proportion to the shear it must carry beyond the concrete's own resistance.

What decides it: Per perimeter, roughly so. But the outer perimeter beyond the studs must be long enough for the concrete alone, so the studs run further out as the load grows: from two perimeters at a 7.2 m bay to six at 10 m and ten at 12 m. The total steel round the column grows as the shear to the power 3.5.

Tested in The studs that send the check outward, at the figure it turns on · the essays about punching shear.

A redundant truss cannot be cambered without locking force into its members, because it cannot take a change of length for free.

What decides it: Cut every member short by its own elongation under the dead load and the eight-panel counter-braced truss is level under that load with no member force changed at all. Those length changes are the negative of a real elongation field, so they are compatible with every one of its six self-stress states.

Tested in The counter fitted after the load, at the figure it turns on · the essays about camber.