What is refuted here — page 16

Switching an actuated tuned mass back on is as safe as switching it off, because both states are stable.

What decides it: Letting go leaves the mass on a spring that catches it; taking hold removes the spring that was holding it and releases the mass with whatever velocity it has. In a steady storm at the structure's frequency the stroke after the law takes hold reaches 1.39 to 3.12 times its limit depending on the instant, and waiting for zero stroke gives 2.81.

Tested in Taking hold is harder than letting go, at the figure it turns on · the essays about tuned mass damper.

A drained basement whose pump is running is at its design head.

What decides it: The pump holds the sump at the design head; the slab sees the head in the blanket under it, which stands above the sump by the risen table divided by one plus the drains' conductance over the ground's. New drains thirty times as conductive as the ground leave 0.06 m; silted with a time constant of eight years, the same pump leaves the head at flotation's 0.97 m in the wet season of year 24.

Tested in The drain that fails a season at a time, at the figure it turns on · the essays about uplift.

The toughness of a 25 mm specimen is the right toughness for a surface crack, because its front is shorter than a through crack's.

What decides it: A surface crack's front is its whole curved length, and for a long shallow crack most of it sees nearly the deepest point's stress intensity. A crack 8 mm deep and 80 mm long in a 40 mm plate has a front of 84 mm, worth more than 40 mm of straight front at its largest stress intensity, and the 25 mm toughness puts its failure stress 19 per cent above the weakest link's.

Tested in The front that counts for half its length, at the figure it turns on · the essays about fracture.

A passage through the wall of a hollow shaft should be kept as far from the bore as possible.

What decides it: The torsional stress grows outwards, so a passage near the surface sits in the most stress: centred 21.7 mm out it peaks at 2.84 of a plain shaft's surface stress. Opposite a flat of a hexagonal bore and as near it as it will go, 12.3 mm out, it peaks at 1.13 and raises the bore's corners from 0.54 to 0.63; only opposite a corner does nearness cost, raising the bore to 1.48.

Tested in The bore with corners and the passage beside it, at the figure it turns on · the essays about membrane analogy.

A stiffer core always buys a stiffer storey.

What decides it: Only while the floor can deliver to it. Below a floor stiffness per bay of about 90 times P/h, no core stiffness at all brings the ten-bay storey to a buckling factor of 2, because the floor buckles with the far columns before their push reaches the core.

Tested in The floor the columns lean on, at the figure it turns on · the essays about effective length.

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 essays about arch.

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 essays about plane sections.

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 essays about buckling.

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 essays about bolt group.

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 essays about net section.

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 essays about bearing.

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 essays about weld group.

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 essays about Slip-critical.

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 essays about moment connection.

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 essays about base plate.

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 essays about damping.

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 essays about response spectrum.

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 essays about ductility demand.

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 essays about gust response.

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 essays about crowd synchronisation.

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 essays about tuned mass damper.

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 essays about friction.

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 essays about corner moment.

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 essays about curved beam.

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 essays about punching shear.

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 essays about plan torsion.

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 essays about diaphragm.

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 essays about erection stability.

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 essays about uplift.

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 essays about compatibility torsion.

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 essays about shear truss analogy.

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 essays about arch buckling.

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 essays about Self-weight buckling.

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 essays about outrigger.

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 essays about strain energy.

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 essays about moment redistribution.

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 essays about anchorage zone.

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 essays about sandwich section.

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 essays about sandwich section.

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 essays about prestress limits.

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 essays about branching structure.

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 essays about continuous restraint.

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 essays about mode interaction.

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 essays about counterweight.

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 essays about load arrangement.

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 essays about end rotation.

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 essays about drift components.

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 essays about confinement.

The size effect is statistical: a bigger specimen has more flaws, so it is likelier to contain a bad one.

What decides it: That is Weibull's mechanism and it predicts a straight line on log axes with no size in it anywhere — so it has nothing to say about where a transition happens, and it disagrees with the energetic law by up to 251 per cent across this range. The energetic mechanism has a length scale, and the length scale is the whole usefulness of the answer.

Tested in The bigger one is the weaker one, at the figure it turns on · the essays about size effect.

A closed section is stiff in torsion, so an eccentric load on a box girder is a small matter.

What decides it: It is stiff in torsion and that is the wrong quantity. The eccentric load is equivalent to three cases, and the third carries no torque at all — its edge forces add to nothing about the axis — so no amount of torsional stiffness touches it. On the box drawn here it produces a longitudinal stress of 75 N/mm² against a bending stress of 71, and Bredt's shear flow knows nothing about either.

Tested in The section that will not keep its shape, at the figure it turns on · the essays about box distortion.

Thickening a corrugated web is the way to raise its shear buckling strength.

What decides it: Local buckling improves as the square of the thickness and global buckling as its three quarters, so on the web drawn here — where global governs at 82 N/mm² against local at 424 — thickening the plate is spending money on the mode that is not deciding anything. Deepening the corrugation raises the orthotropic rigidity across the fold and moves the mode that governs; the sweep says by how much.

Tested in The web that carries no bending, at the figure it turns on · the essays about corrugated web.

A framed tube behaves as a hollow box cantilever, so plane sections gives the column forces.

What decides it: A hollow box has a flange; a framed tube has a row of columns joined by spandrel beams, and an axial force reaches the middle of that row only through the frame's in-plane shear. On the tube drawn here the corner column carries 1.52 times the plane-sections stress and the middle column carries 0.40 of it, and the face is doing its work on an effective width of 51 per cent.

Tested in The corner columns take more than their share, at the figure it turns on · the essays about framed tube.

Deflection calculations for concrete are uncertain because the modulus is uncertain.

What decides it: The modulus is one of the better-known properties. What decides the answer just above cracking is the tensile strength, whose scatter is around twenty per cent and which is inferred from a bending test rather than measured — and a beam at 1.2 times its cracking moment moves 80 per cent in deflection for a 20 per cent change in it. Well past cracking the same change is worth 7 per cent.

Tested in Stiffer than its cracked section says, at the figure it turns on · the essays about tension stiffening.

Shear deflection is small, so where it comes from does not matter.

What decides it: It is small on a slender beam and it comes from the opposite half of it. The bending density of a uniformly loaded span is concentrated at mid-span, where the shear is zero; the shear density is concentrated at the supports, where the moment is zero. A member whose two deflection components matter — a sandwich panel, a deep beam, a timber joist — is being stiffened in two different places at once.

Tested in Where a deflection comes from, at the figure it turns on · the essays about deflection distribution.

Shear yield stress is a material property, measured in a torsion test.

What decides it: A torsion test measures a shear stress at which something happened; converting that into a prediction for a web under combined stress needs a criterion, and the criterion is the assumption. Von Mises gives f_y/√3 = 205 N/mm² for this steel and Tresca gives f_y/2 = 178, and no third measurement decides between them — they are two theories fitted to the same uniaxial test.

Tested in The shear strength nobody measured, at the figure it turns on · the essays about yield criterion.

Shear is rarely the governing check in a beam.

What decides it: It is rarely the governing check in a steel beam, and the reason is a ratio of two material strengths. Bending and shear utilisations are equal at a span-to-depth of exactly f_m/f_v — 1.73 for steel and 6.0 for this timber — so a timber beam shallower than about six times its depth is shear-governed, which is an ordinary joist over a corridor.

Tested in The material that has a direction, at the figure it turns on · the essays about anisotropy.

The hoop force in a tank wall is worst at the bottom, where the pressure is.

What decides it: It is zero at the bottom. A wall cast into its base slab cannot move outwards there, and with no hoop strain there is no hoop force — the pressure is carried in vertical bending instead, at 52.6 kNm per metre. The peak sits 2.77 m up a 8 m wall, at 64% of what the membrane triangle predicts, and reinforcement placed by the triangle is in the wrong place.

Tested in The force that is only a radius, at the figure it turns on · the essays about hoop tension.

Making the coupling beams stiffer is always the cheap way to stiffen a coupled wall.

What decides it: It works and it stops working. Going from a 400 mm beam to a 1,000 mm one takes the degree of coupling from 51% to 72% and the drift from 33 mm to 18, but the peak shear in a beam rises from 157 kN to 284 in a member one metre long. The stiffness is bought with a shear demand that arrives faster than the stiffness does, and the limit is the beam rather than the wall.

Tested in Two walls that agreed to be one, at the figure it turns on · the essays about wall coupling.

A numerical section analysis is a fallback for when the closed form runs out.

What decides it: It is the general case and the closed forms are its special cases. The strip loop is told nothing about cracking, yielding, confinement or prestress: it is given a strain profile and a stress-strain law per strip and it sums. Handed an uncracked elastic section it reproduces the transformed-section answer, and handed a cracked one it finds x = 137.0 mm without being told that anything had cracked.

Tested in The section calculation with no formula in it, at the figure it turns on · the essays about fibre model.

A masonry pier is checked by comparing its stress with the strength of the stone.

What decides it: That check is real and it is never the one that governs. The toe stress here is 351 kPa against a masonry strength of several megapascals — a factor of twenty in hand — while the resultant is already outside the middle third. What fails first is a geometric condition on where the resultant lies, and no material property appears in it.

Tested in The weight that makes it safer, at the figure it turns on · the essays about buttress.