What is refuted here — page 8

A shallower lacing angle uses less steel per metre and is otherwise neutral.

What decides it: It is penalised three times. Taking the angle from 45° to 30° lowers the critical load from 3,832 kN to 3,456, raises the design shear at 2,000 kN from 32.4 kN to 45.8, and lengthens the diagonal from 1,131 mm to 1,600 — so a larger force is carried by a member buckling over a longer length.

Tested in The lacing decides the force it has to carry, at the figure it turns on · the Built-up column ladder.

A slip coating reduces downdrag in proportion to how much it reduces the friction.

What decides it: It reduces it faster, because the neutral plane moves as well. Taking the interface factor from 0.3 to 0.1 — a third — takes the drag from 1,131 kN to 177, a sixth, because the plane climbs from 18.86 m to 12.93 and sheds the deepest and hardest-pulling metres first.

Tested in The coating that takes the resistance with it, at the figure it turns on · the downdrag ladder.

The smoother the shaft, the better, since the drag is what is being designed against.

What decides it: Past 0.05 on this pile the calculation has no answer at all: the base takes 600 kN and the whole shaft 477, so 1,200 kN at the head has nowhere to go. The coating removes the resistance along the same length it removes the drag, and the pile fails in bearing before the drag has fully gone.

Tested in The coating that takes the resistance with it, at the figure it turns on · the downdrag ladder.

A bigger pile is a safer pile where downdrag is expected.

What decides it: It drags harder. Reducing the diameter from 750 mm to 400 takes the worst force from 2,331 kN to 1,663 — the perimeter that collects the drag has fallen by 47 per cent while the section carrying it has fallen by 72, so the stress goes up, but the force a connection or a splice must carry goes down.

Tested in The coating that takes the resistance with it, at the figure it turns on · the downdrag ladder.

A trapezoidal hook section is a founding and forging convenience.

What decides it: It is worth 39 per cent at the governing fibre. The same trapezoid at the same area, depth and moment gives 223.6 N/mm² at the inner fibre with its wide face inward and 311.6 with the wide face outward, because the shape moves the neutral axis and the section modulus in the same direction.

Tested in The wide side goes inside, at the figure it turns on · the curved beam ladder.

Making a Vierendeel girder deeper stiffens it, as it would a truss.

What decides it: It softens it. The same 21 m girder at 6,000 mm deep deflects 109.09 mm and at 9,000 mm deflects 139.85 — worse by 28 per cent — because the depth lengthens nothing that resists and the chord bending, which is 98 per cent of the movement, depends on the panel length rather than on the depth.

Tested in The frame is a girder stood on end, at the figure it turns on · the vierendeel ladder.

More panels means more members and more joints, so it is the expensive way to stiffen one.

What decides it: It is the only lever that works well. Going from 4 panels to 10 on the same span takes the deflection from 166.19 mm to 65.86 — a factor of 2.5 — and the panel moment from 196.9 kN·m to 78.8, because both go with the panel length and the deflection goes with its cube.

Tested in The frame is a girder stood on end, at the figure it turns on · the vierendeel ladder.

A frame's lateral stiffness comes from its columns' axial stiffness, as a truss's does.

What decides it: Ninety-eight per cent of it is bending. Doubling the chord's second moment of area alone — the axial areas untouched — takes the deflection from 109.09 mm to 55.72, while triangulating the same members takes it to 6.53. The frame is a bending structure wearing a truss's outline.

Tested in The frame is a girder stood on end, at the figure it turns on · the vierendeel ladder.

The tension field is a property of the web panel.

What decides it: It is a property of the panel and its boundary together. The band is anchored by the flanges above and below and by the stiffeners either side, and at the end panel one of those stiffeners has nothing beyond it — which is why an end panel is designed without tension field action or given a stiffener sized as a beam.

Tested in The tension has to pull on something, at the figure it turns on · the tension field ladder.

Shedding is free, since the spandrel is being relieved.

What decides it: It is bought with a rotation. Getting to a quarter of the torsional stiffness means the spandrel has cracked in torsion along its length, and the joint has rotated by whatever that cracking allows — a serviceability state with a crack width and a visible twist, neither of which appears in the strength calculation that licensed the release.

Tested in The torque that should not be shed, at the figure it turns on · the compatibility torsion ladder.

Whether a torque is compatibility or equilibrium is a property of the torque.

What decides it: It is a property of what else is available. The same spandrel carries a 197 kN·m fixed-end moment it can shed and a canopy's 116 kN·m it cannot — and at a 3.6 m canopy that second number is 311, on a member whose whole compatibility torque was 100.

Tested in The torque that should not be shed, at the figure it turns on · the compatibility torsion ladder.

Adding a diaphragm at midspan roughly halves a box girder's distortional stress.

What decides it: It removes a third of it. The bare 60 m box peaks at 73 N/mm²; one diaphragm at midspan takes it to 49, because at 30 m from each end it is further away than the 23.6 m decay length and neither end can feel it. Two take it to 18 and three to 10.

Tested in One diaphragm is nearly none, at the figure it turns on · the box distortion ladder.

A lighter box is a smaller distortion problem, since there is less load in it.

What decides it: It is a much larger one, and its own arithmetic disguises it. Taking the plates from 20 mm to 12 lengthens the decay from 23.6 m to 28.6 — which reads as needing diaphragms further apart — while raising the distortional stress at one decay length's spacing from 20 per cent of the bending stress to 55.

Tested in One diaphragm is nearly none, at the figure it turns on · the box distortion ladder.

Diaphragm spacing is a detailing rule, so a span-over-five sort of number is adequate.

What decides it: The quantity it should be measured against varies by a factor of 1.7 across ordinary boxes. The same span gives a decay length of 16.7 m on a 2.5 m wide box, 23.6 m on a 4 m one, and 28.6 m on the same 4 m box in thinner plate — and the stress at one decay length's spacing runs from 9 per cent to 55.

Tested in One diaphragm is nearly none, at the figure it turns on · the box distortion ladder.

Members that carry no force are redundant and can be taken out.

What decides it: They are the robustness. Removing member 12 from a Pratt truss with its counters leaves a structure at 1.97 times the worst survivor's force, with two previously idle members now working; removing the same member from the same truss without them leaves a mechanism. The counters carried nothing until they were needed.

Tested in A determinate truss has no robustness at all, at the figure it turns on · the robustness ladder.

A deeper truss is a more robust one, since its members are less heavily loaded.

What decides it: Its worst redistribution is larger. The same 8-panel Pratt truss at a depth of 1.6 asks a survivor for 5.51 times what it carried before; at 0.6 it asks for 2.01. A deep truss's members carry small forces, and a small force multiplied is still a large multiple.

Tested in A determinate truss has no robustness at all, at the figure it turns on · the robustness ladder.

Rocking response is a resonance, so a longer pulse at the same acceleration is not worse.

What decides it: It is much worse, and there is no resonance to speak of. Doubling the pulse from 0.8 s to 1.6 at the same 1.0 g overturns both blocks — the small one and the one three times its size — where the shorter pulse left the large one at 46 per cent. The block has no period to be tuned to.

Tested in The only damping is the landing, at the figure it turns on · the rocking ladder.

A masonry arch stands because its shape is right — because it matches the funicular of the loads it carries.

What decides it: Heyman's safe theorem asks for far less: it is enough that *some* line of compression can be drawn inside the stonework, and there are infinitely many candidates. The family is drawn for one arch under one load, with the minimum- and maximum-thrust members both fitting inside the ring — which is why a cracked arch is usually a working one, and why the question 'what is this arch doing?' is both unanswerable and unnecessary.

Tested in The line that must stay inside, at the figure it turns on · the arch ladder.

The bending stress in a beam is My/I.

What decides it: It is, and only because a cross-section plane before bending is still plane afterwards — a kinematic assumption, not a law and not derived from one. Strain is drawn across a cut face at span-to-depth ratios of 8, 4, 2 and 1 against the straight line the theory assumes: they coincide at the first and the real distribution is nothing like a straight line at the last. Everything in the subject that fails does so where that sentence stops holding.

Tested in Plane sections stay plane, and what the assumption costs, at the figure it turns on · the plane sections ladder.

The Euler load is the load a slender column can carry.

What decides it: Euler's load is the load at which a *perfectly straight* column becomes indifferent to being bent, and no column is perfectly straight. Started with an initial bow, the deflection grows continuously from zero load: there is no bifurcation, and the Euler load is an asymptote the column never reaches. The Southwell plot then recovers that unreachable load from readings taken nowhere near it.

Tested in The column that was never straight, at the figure it turns on · the buckling ladder.

The instantaneous centre method is a more accurate version of the elastic vector method, so the difference between them is an error.

What decides it: They are different assumptions about what the plate does, not two attempts at one number. Swept over eccentricity for a 3 × 2 group, the gap is 16% at 40 mm, 13% at 150 mm and 12% at 300 mm — a curve, not the single factor it is usually quoted as — and just off zero eccentricity the instantaneous centre comes out 1.2% below, because it anchors deformation at the furthest bolt while a concentric elastic group has every bolt at exactly its capacity.

Tested in The bolt that carries more than its share, at the figure it turns on · the bolt group ladder.

The s²/4g term is an approximation to the true diagonal net area.

What decides it: It is not an approximation to a length at all. The diagonal path is geometrically longer than the straight one by a factor of sec θ, which at s = 50 and g = 60 is 1.302 — 18.1 mm of extra length where the rule adds back 10.4, and adding back the extra length would be a correction in the wrong direction besides. The rule stands in for the diagonal leg carrying a mixture of tension and shear rather than for its length, and it is a fit to test data with no derivation behind it.

Tested in The tear that goes diagonally, and the correction that has no derivation, at the figure it turns on · the net section ladder.

Bearing on the plate is a check on the material's crushing strength.

What decides it: Only past a corner it usually is not past. Below an end distance of 2.5·3·(d+2) = 165 mm the failure is not crushing at all: the bolt shoves a channel of metal out to the end of the plate, and the capacity is proportional to the end distance rather than to any material property of the region in front of the bolt.

Tested in The hole that goes oval, and the one that tears to the edge, at the figure it turns on · the bearing ladder.

For a weld group under an eccentric load, check the point furthest from the group's centroid.

What decides it: True for a group symmetric about the load's axis and false otherwise. Measured on five shapes: a C, a plain vertical run and a deep C put the peak exactly at the furthest point; an L-shaped group puts it 15.6% higher at a point nearer the centroid; and two horizontal runs put four sampled points at identical radius carrying stresses that differ by a factor of 1.656.

Tested in The corner that is not the worst point, at the figure it turns on · the weld group ladder.

Tightening a bolt harder makes the connection stronger.

What decides it: It makes the joint stiffer and it changes the mechanism; it does not raise the ultimate capacity. Two M20 bolts preloaded to 137 kN each carry 137 kN of slip resistance at μ = 0.5, and 188 kN in bearing once slipped. The ultimate capacity is the second number and preload has not moved it. What preload bought is that the first 137 kN arrive with no movement at all.

Tested in The joint that carries nothing until it slips, at the figure it turns on · the Slip-critical ladder.

Bolt rows share the tension in proportion to their distance from the compression centre.

What decides it: That is the elastic distribution and it is one of two. With three equal rows it gives 142.3 kN·m; letting every row reach its own limit gives 172.8 — a factor of 1.214 for no extra steel. Which is available depends on whether the top row can keep carrying its load while the lower rows catch up, and that is a property of which component governs that row rather than of the arithmetic.

Tested in Making a moment cross a gap, at the figure it turns on · the moment connection ladder.

The middle-third rule is the design limit for a base plate under moment.

What decides it: It marks a lift-off, not a capacity. The regime it opens ends when the peak bearing pressure reaches the concrete's limit, and for this plate that is at e = 200 mm and M = 120 kN·m — well before the bolts are needed at 150. The plate crushes with the bolts still idle, so the governing boundary is neither the kern nor the bolt.

Tested in Where the structure meets the ground, and when the bolts start working, at the figure it turns on · the base plate ladder.

Damping is a material property, so a steel structure and a concrete one can be looked up.

What decides it: Material hysteresis is a small part of it. Most of a real structure's damping comes from friction in bolted joints, from cladding and partitions rubbing against a frame that moves under them, and from energy radiated into the ground — none of which is a property of the material and none of which appears on a drawing. The measured range for nominally identical buildings spans a factor of five.

Tested in The only thing that stops it, at the figure it turns on · the damping ladder.

The spectral acceleration is the acceleration the structure experiences.

What decides it: It is the pseudo-acceleration, ω²·Sd — defined so that mass times it gives exactly the elastic force in the structure at the instant of peak displacement. The peak of the actual total acceleration differs from it, by a fraction of a per cent at 5% damping and more as damping rises. The two are used interchangeably and only one of them gives the base shear exactly.

Tested in The spectrum is not a load, at the figure it turns on · the response spectrum ladder.

A structure that yields in an earthquake has failed.

What decides it: It has been damaged, which is a different statement and is the design intent for every ordinary building. What yielding buys is a large increase in energy dissipation — the hysteresis loop encloses real area at every cycle, and unlike viscous damping it does not care how fast the cycle is. What it costs is measurable: 11.8 mm of permanent offset on the case drawn here, and a structure that will not be exactly where it was.

Tested in The earthquake asks for a displacement, at the figure it turns on · the ductility demand ladder.

A gust that lasts three seconds is what shakes a tall building.

What decides it: A tower at 0.2 Hz needs about forty cycles — three and a half minutes — to build up to a steady resonant response. A three-second gust is over before the first cycle finishes; what it does is deflect the building quasi-statically. The resonant response is driven by the small part of the wind's energy that happens to sit at 0.2 Hz and is sustained for many minutes.

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

The bridge swayed because the crowd happened to walk in step with it — it was resonance with a marching frequency.

What decides it: Resonance needs an excitation that exists whether or not the structure moves. Here the lateral force is proportional to the deck's own velocity: with the deck still there is no net lateral force at all, because the pedestrians are uncorrelated. What synchronises them is the motion, so the excitation is created by the response and the response is not bounded by any magnification factor.

Tested in The bridge that was pushed by its own sway, at the figure it turns on · the crowd synchronisation ladder.

A tuned mass damper absorbs the vibration energy, so a bigger one absorbs more.

What decides it: Almost none of the energy ends up in the absorber's own mass — it ends up in the absorber's dashpot as heat, and the mass is there to have inertia rather than to store anything. The quantity that decides the result is the mass RATIO, and the returns are severely diminishing: 1% gives a factor of 4.3, 3% gives 6.8 and 10% gives 11.5, for more than three times the mass.

Tested in The mass that helps by being late, at the figure it turns on · the tuned mass damper ladder.

A ladder leaning on a wall is solved by taking the wall as frictionless, which gives the forces at its base.

What decides it: That assumption picks one member of a family, and at 60° with μ = 0.4 at the floor and 0.3 at the wall it picks a member that does not exist. Cutting the whole ladder free and taking moments about the base, the frictionless-wall state needs 404.1 N of friction at the floor against a capacity of 0.4 × 1000 = 400 N — a ratio of 1.010. The admissible states run from 344.5 to 390.7 N, and the ladder stands.

Tested in The force that is whatever it needs to be, at the figure it turns on · the friction ladder.

A beam built into something at its ends carries wL²/12 there, so a portal's corner moment is wL²/12.

What decides it: Only if the columns are walls. The corner of a real portal is a spring, and the moment it takes is (wL²/12)·2/(2+N) with N the beam-to-column stiffness ratio. At the frame drawn here N = 1.27 and the corner takes 65.2 kNm — 61% of the 106.7 kNm a built-in beam would have carried. Over three decades of N the closed form and the assembled frame solution agree to 4.1e−6 kNm.

Tested in The moment that goes round the corner, at the figure it turns on · the corner moment ladder.

The neutral axis of a section in pure bending passes through its centroid.

What decides it: True only where the fibres share a length. Zero net force across the face requires ∫σ dA = 0, and with strain going as 1/r that condition gives R_n = A ÷ ∫(dA/r), which is a harmonic mean rather than an arithmetic one. For this hook it puts the axis at 75.04 mm against a centroid at 79.70 — 35.8% of the depth from the inner fibre rather than the 42.4% the centroid sits at.

Tested in The bar that was bent before it was loaded, at the figure it turns on · the curved beam ladder.

Punching is a strength problem, so it is solved by stronger concrete.

What decides it: The resistance contains the cube root of the concrete strength and the first power of a perimeter and a depth. Going from 30 to 50 N/mm² buys 19%; going from 225 to 300 mm of effective depth buys 51%. It is a geometry problem wearing a material's units.

Tested in A check made on a perimeter, not on a section, at the figure it turns on · the punching shear ladder.

A building that twists too much should be stiffened.

What decides it: Stiffness added at the centre of rigidity raises the translational stiffness and adds nothing to the torsional one, so the twist is unchanged while the translation it is added to shrinks. The edge amplification climbs from 1.98 to 2.96 as the added stiffness reaches the whole of what was there. What helps is stiffness far from the centre, which is a question about position and not about quantity.

Tested in The corner that moves most, at the figure it turns on · the plan torsion ladder.

A concrete slab is rigid, so a rigid-diaphragm analysis needs no justification.

What decides it: Rigidity is a comparison, not a property. The criterion is the plate's own in-plane deflection against the deflection of the walls it sits on, and a long narrow plan on stiff cores can fail it with a 200 mm slab. What decides it is the plate's span-to-depth ratio in plan, which is the building's proportions rather than its slab.

Tested in The floor is a beam lying down, at the figure it turns on · the diaphragm ladder.

Erection stability is a contractor's problem, because it concerns temporary works rather than the permanent structure.

What decides it: Every number that decides it is a property of the permanent structure — its effective lengths without its bracing, its joints before they are made rigid, its frames before the diaphragms exist. The temporary works are the answer, and the question belongs to whoever computed the effective lengths.

Tested in The most dangerous day is before it is finished, at the figure it turns on · the erection stability ladder.

A thicker base slab solves an uplift problem, because it resists the pressure.

What decides it: It resists nothing. A base slab under uplift is a slab spanning between whatever holds it down, and thickening it adds bending capacity it may not need and weight it certainly does. The weight is what works: holding this 20 × 30 m substructure down with slab alone needs 2.45 m of concrete, against the 0.9 m it has.

Tested in A basement is a boat, at the figure it turns on · the uplift ladder.

A torque that comes out of a frame analysis is a torque the member has to carry.

What decides it: The spandrel here attracts 100 kNm because its torsional stiffness is 67,500 kNm per radian against the floor beam's 64,800. Reduce it to a quarter — which is what cracking does — and the torque falls to 41 while the floor beam's midspan moment rises from 147 to 177 kNm. The two always add back to the free moment of 197. Nothing about the structure changed except a number in the model.

Tested in The torsion that goes away if you let it, at the figure it turns on · the compatibility torsion ladder.

Shear reinforcement is designed for the shear the concrete cannot carry, so it is a subtraction.

What decides it: In the truss analogy the concrete carries no shear across the crack at all, and the stirrups carry the whole of it — 563 kN here from 1,047 mm² of steel per metre. The number the model produces is not smaller than the additive one because a term was left out; it is larger, because the crack was flattened and the cut crosses 8.25 stirrups instead of 3.3.

Tested in The beam that becomes a truss, at the figure it turns on · the shear truss analogy ladder.

An arch is braced by bracing it against the load it carries.

What decides it: The first mode of the parabolic rib here is antisymmetric — one half rises while the other falls and the crown moves sideways — at a coefficient of 46. The symmetric squash the load looks like arrives at 106, 2.3 times higher. Restraint against sideways movement of the crown is what changes the answer, and restraint against downward movement is not.

Tested in The arch that leans instead of squashing, at the figure it turns on · the arch buckling ladder.

Self-weight can be replaced by an equivalent load at the top.

What decides it: It can, and the equivalent is 3.18 times as much total weight — a column carries its own weight better than it carries somebody else's, because most of it is near the base where the buckle is not. Using the total weight as a tip load is out by a factor of three in the wrong direction.

Tested in Too tall for nothing but itself, at the figure it turns on · the Self-weight buckling ladder.

An outrigger works by making the perimeter columns share the wind shear.

What decides it: It transmits no shear at all. It applies a couple to the core — 146 MNm here, carried as a 4,877 kN tension–compression pair in the columns at 15 m centres — and restrains a rotation. The core still carries the whole of the shear, and the base moment falls from 600 to 454 MNm because the couple opposes it, not because anything was shared.

Tested in The arm that makes the columns work, at the figure it turns on · the outrigger ladder.

Castigliano's theorem and the unit-load method are two different ways of computing a deflection.

What decides it: They are the same sum. ∂U/∂P is Σ N(∂N/∂P)L/EA, and for a linear structure ∂N/∂P is exactly the force a unit load produces — so the two expressions are identical term by term. Computed here by a numerical derivative and by a virtual structure, they agree to 2 × 10⁻¹⁴.

Tested in The deflection that is a derivative, at the figure it turns on · the strain energy ladder.

Redistribution weakens the beam, so the limit on it is a strength requirement.

What decides it: The collapse load of a beam of uniform capacity does not depend on the elastic distribution at all. What redistribution changes is the section that has to be provided: 240 kNm at no redistribution against 168 at thirty per cent, on the same beam under the same load. The cap is a rotation limit — 9.1 milliradians are demanded here — and the rotation is a property of the section's ductility, not of its strength.

Tested in The moment that was moved on purpose, at the figure it turns on · the moment redistribution ladder.

Guyon's coefficient of 0.25 is an empirical constant for concrete.

What decides it: It is a lever arm. Put the strut-and-tie model's tie at half the section depth from the loaded face and T = (P/2)(h − a)/4 ÷ (h/2), which is 0.25P(1 − a/h) identically, for any material whatever. Move the tie to a quarter of a depth and the coefficient becomes 0.5.

Tested in The force that splits what it pushes on, at the figure it turns on · the anchorage zone ladder.

Shear deflection is a small correction that can be added at the end if anyone is worried.

What decides it: It is a small correction for a solid steel beam and it is the answer for a sandwich. The panel here spends 9 per cent of its deflection on shear at a span-to-depth ratio of 39, and 60 per cent at a ratio of 10 — because bending deflection falls as the fourth power of the span and shear deflection as the second, so shortening the panel makes shear dominant rather than negligible.

Tested in Two skins and the space between them, at the figure it turns on · the sandwich section ladder.

The core holds the faces apart, so it is a spacer rather than a structural material.

What decides it: It contributes 1.2 per cent of the section's bending stiffness and all of its shear stiffness, and the shear stiffness is what sets the second term of the deflection and the wrinkling stress of the faces. A panel with a core of half the shear modulus is a different panel; a panel with a core of half the Young's modulus is very nearly the same one.

Tested in Two skins and the space between them, at the figure it turns on · the sandwich section ladder.

A prestressed section is designed by choosing a force and then checking the stresses.

What decides it: The four limits are simultaneous and two of them push in opposite directions, so a force chosen first is a guess that the checks then accept or reject with no indication of which way to move. Plotted against 1/P they are four straight lines whose intersection is every acceptable answer at once: 1,219 to 2,808 kN at an eccentricity of 300 mm on the section here.

Tested in Four inequalities and a wedge, at the figure it turns on · the prestress limits ladder.

A branching column saves material because it shortens the span the roof has to cover.

What decides it: That is what it does for the roof, and it is not what makes the column itself worth building. Measured on the column alone with only strength counted, the tree is never better than a fan of straight struts from the base: the volume comes to P(H + s²/g)/σ, which falls monotonically as the split moves down and is least at the ground.

Tested in The tree that strength does not ask for, at the figure it turns on · the branching structure ladder.

A restraint that is not attached to anything rigid cannot help, because it moves with the member.

What decides it: A U-frame restraint does move, and what matters is its stiffness rather than its immobility. A third of a newton per millimetre per millimetre of length raises this flange's critical load by a factor of 11.4 — and the answer 2√(kEI) contains no reference point at all, only the stiffness of whatever is pushing back.

Tested in Held everywhere, and it forgets its length, at the figure it turns on · the continuous restraint ladder.

Local buckling is a property of the section, so it can be checked once and reused for any length of member.

What decides it: The local critical load is a property of the section. The local reduction is not: it applies to whatever stress the global check has already allowed, so the same section loses 21 per cent at three metres and less at one. A local check made with no length in it is a check on a stub column.

Tested in Two ways of buckling at once, at the figure it turns on · the mode interaction ladder.

A counterweight sized for the mean load balances a moving load well enough.

What decides it: It balances it at exactly one radius. Sized for the mean of a trolley's travel, this jib carries 1,110 kNm one way with the load at maximum radius and 1,290 kNm the other way with nothing on the hook at all — and the second is the larger. An empty crane is a crane leaning backwards.

Tested in Balanced, and four times as heavy, at the figure it turns on · the counterweight ladder.

The bending-moment envelope is the beam's worst bending-moment diagram.

What decides it: It is not a bending-moment diagram at all. Each of the eight arrangements satisfies the free-moment identity to machine precision — mid-span ordinate minus the mean of the end moments is wL²/8 exactly — and the envelope misses it by up to 23 per cent, because its ordinates come from different load cases.

Tested in The envelope is not a structure, at the figure it turns on · the load arrangement ladder.

A connection detailed as a pin does not attract moment, so the beam's end rotation is irrelevant to it.

What decides it: A nominally pinned connection has a real rotational stiffness, and the beam turns through 8.9 milliradians whether the connection likes it or not. The moment that follows is the rotation times that stiffness, and it exists in a joint designed for none.

Tested in The angle nobody limits, at the figure it turns on · the end rotation ladder.

Adding stiffness to a tall building reduces its drift wherever the drift is largest.

What decides it: It depends which stiffness. Adding bending stiffness flattens the top of the profile and does almost nothing at the base; adding shear stiffness does the reverse. On this building bending is 61 per cent of the roof drift and 0 per cent of the bottom storey's, so a measure aimed at the roof figure can leave the governing storey untouched.

Tested in Two motions with one name, at the figure it turns on · the drift components ladder.

Confinement is a way of making concrete stronger.

What decides it: It raises the strength of this column's core by a factor of 1.48 and its ultimate strain by a factor of 8. The area under the stress-strain curve — the toughness — goes up by eleven. The strength gain is the advertised number and the third one is what the hoops are provided for.

Tested in Squeezed sideways into a different material, at the figure it turns on · the confinement ladder.