What is refuted here — page 17

A finer mesh fixes it, since the cells are smaller.

What decides it: It helps as a cube — halving the spacing multiplies a rigid-jointed grid's shear stiffness by eight, not four, because smearing puts a third power of the spacing in. And eight times 0.35% is 2.8%: the racking falls from 1,205 mm to 151, still above the limit, at twice as many members and four times as many joints.

Tested in A shell only if the grid takes shear, at the figure it turns on · the essays about gridshell.

A beam hung from a point above its centre of gravity cannot roll over.

What decides it: True of a rigid body and false of a beam. Tilting puts a lateral component of the self weight on a member that is soft about its weak axis, and the resulting bow moves the centre of gravity out by z̄θ. Both moments grow with the tilt; if z̄ exceeds the roll height there is no equilibrium at any angle, and for this section that happens at 44 m with the hook 0.9 m up.

Tested in Hung from above and still unstable, at the figure it turns on · the essays about lift stability.

Keeping the resultant in the middle third means the wall is comfortable.

What decides it: The middle third is where the joint stops opening, not where the capacity is intact. At e = t/6 the capacity is exactly two thirds of the squash load — a third has been given away before slenderness has been mentioned at all, and the relationship is a straight line so there is no region where the cost is small.

Tested in It does not buckle, it runs out of width, at the figure it turns on · the essays about wall slenderness.

Internal pressure is a detail that can be ignored for the overall stability check.

What decides it: For the horizontal resultant it cancels exactly — 842 kN sealed and 842 kN with the door open, to the newton. For the roof it does not cancel at all: the uplift goes from 108 kN to 648 against a roof weighing 360, so a building that was comfortably held down is 288 kN short. Ignoring it is right for one check and wrong for the other.

Tested in Most of it is suction, at the figure it turns on · the essays about wind pressure.

A larger structure just means a larger flexibility matrix.

What decides it: It means a matrix whose fill depends on the release. Six equal spans released at the moments give a tridiagonal matrix at 52% fill; released at the reactions, 100%, because a unit reaction anywhere deflects every other point on a single long span. The first is solvable in a column of figures and the second is not.

Tested in Choose what to take away, at the figure it turns on · the essays about force method.

The index is a rule of thumb, so the exponents are approximate.

What decides it: The exponents are exact and they are the only exact thing in the calculation. They come from eliminating one free dimension between a mass and a constraint: a beam of square section gives I proportional to A squared, so E^(1/2) exactly. What is approximate is the property data and the assumption that the shape is free at all.

Tested in The ranking belongs to the load case, at the figure it turns on · the essays about material index.

A larger bar is more durable, since it takes longer to corrode through.

What decides it: It takes longer to lose a given fraction of itself and it splits its cover sooner. The cracking pressure goes as cover over diameter, so a 40 mm bar at the same cover cracks at 5.08 MPa against a 20 mm bar's 10.15 — and it does so at a section loss of 0.11% rather than 0.21%. Bar size and cover are one parameter, not two.

Tested in The load that comes from inside, at the figure it turns on · the essays about corrosion.

A gusset plate is checked by dividing the brace force by its cross-sectional area.

What decides it: There is no cross-sectional area. A gusset is a plate with a connection somewhere on it and no boundary that says where the member ends. The area used is invented — 367 mm of assumed width on a brace connected over 90, four times what anything is actually attached to — and the invention is the calculation rather than an input to it.

Tested in The width nobody drew, at the figure it turns on · the essays about gusset.

A fixed-base model is conservative, so soil-structure interaction can be left out.

What decides it: It is conservative for one answer and not for the other. On this 0.6 s building at 200 m/s the period lengthens 1.31 times, the base shear falls 12%, and the displacement rises 51%. The check that is easy to do is the one that was already safe, and the drift — which breaks the cladding, the services and the gap to the neighbour — is the one that got worse.

Tested in The ground is a spring, at the figure it turns on · the essays about Soil-structure.

A frame designed for wind has plenty of horizontal capacity, so the notional force never governs.

What decides it: True for the wind combination and irrelevant to the one that governs. The notional force is proportional to the gravity load and is present in every combination, including the one with no wind — where it is the only horizontal load there is, and where a bracing system whose connections were sized for a case that "has no horizontal load" has nothing to resist it.

Tested in The load that is really a lean, at the figure it turns on · the essays about notional load.

Self weight is a first estimate that gets corrected once the sizes are known.

What decides it: The correction is a fixed-point iteration whose convergence ratio is (L/L*)² — the same number. At an ordinary span it converges in two passes because that ratio is a fifth; at three quarters of the limiting span each pass fixes only 44% of the remaining error, and the "first estimate" is not converging so much as crawling toward an answer that is already unbuildable.

Tested in The weight that has to be known before it can be found, at the figure it turns on · the essays about dead load.

A restraint at midspan doubles a beam's lateral-torsional capacity as long as it is stiff enough.

What decides it: Only if it is on the right flange. On the compression flange the critical moment climbs from 143 kNm to a plateau of 447 and reaches 99% of it at 366 kN/m. At the shear centre the same plateau costs 2252 kN/m — 6.2 times as much. On the tension flange it never arrives at all — at the stiffness that worked on the other flange it has bought a factor of 1.022, and more stiffness buys the same nothing.

Tested in The load that moves with the twist, at the figure it turns on · the essays about load height.

Reversal is a wind problem, so a structure in a sheltered site does not have one.

What decides it: Uplift is one source of reversal and pattern loading is another, which is present whenever anything is imposed rather than permanent. A Pratt truss under full load has 14 members in tension, 13 in compression and 2 carrying nothing; move the load onto half the span and the membership of those three lists changes, with no wind involved.

Tested in The tie that spends an afternoon as a strut, at the figure it turns on · the essays about load reversal.

Creep is a deflection problem, so a strength calculation can ignore it.

What decides it: In a section built in stages, creep moves stress between the parts — the concrete relaxes and the steel picks up what it sheds, so the stress distribution keeps changing after the load has stopped. The deflection multiplier reaches 3.11 after a year on this member, and the redistribution behind it is a change in the very quantities a strength check divides by.

Tested in The section that changed while it was being loaded, at the figure it turns on · the essays about staged section.

A single optimum angle exists for the diagonals.

What decides it: The diagonals carry the storey shear and the overturning moment at once, and the two want different angles — shallow for shear, steep for moment. The ratio between the demands changes up the height, so the optimum angle changes with it, and a uniform angle is a compromise whose cost is largest at the two ends.

Tested in The columns that lean, at the figure it turns on · the essays about diagrid.

How the model is set up is a matter of convenience and does not affect anything.

What decides it: It affects the cost enormously and the answer not at all, and the two halves of that sentence are what a designer needs. Renumbering the nodes of one frame gives bandwidths of 22 and 20 and identical displacements. Choosing which redundant to release turns a full flexibility matrix into a tridiagonal one — which is the difference between a problem that can be solved on paper and one that cannot.

Tested in The answer that depends on how it was divided, at the figure it turns on · the essays about discretisation.

Working inside the middle third means the section is comfortably loaded.

What decides it: A no-tension material's capacity is exactly 1.5f(t − 2e) — a straight line to zero at the face — and at the edge of the kern it is already at 67% of the squash load. A wall loaded to the limit of the middle third has given away a third of its capacity before slenderness has been mentioned at all.

Tested in The strength thrown away on purpose, at the figure it turns on · the essays about no tension.

A splice placed at a point of contraflexure carries no moment, so it can be nominal.

What decides it: The moment is zero for one load case and moves for every other, the shear is at its largest near a contraflexure point, and the stiffness requirement is unaffected by where the splice is. The point of contraflexure on the two-span beam here shifts as soon as the load is patterned, and the splice is then in a region carrying moment.

Tested in The joint that has to be as good as the member, at the figure it turns on · the essays about splice.

The sloshing mass is negligible because it takes so little force.

What decides it: It takes almost no force and decides the freeboard, which is a displacement rather than a force. A wave that reaches the roof turns the convective mass back into an impulsive one against a shell that was never designed for it, and the height of that wave is proportional to a spectral displacement.

Tested in The liquid has a period of its own, at the figure it turns on · the essays about sloshing.

A short beam is a beam with a large shear.

What decides it: Below about two and a half depths the load does not travel along the member as shear at all. It arches directly to the support, the shear span stops being the variable and the strut's crushing strength becomes it, and the capacity rises steeply rather than falling.

Tested in The strength with no mechanism in it, at the figure it turns on · the essays about concrete shear.

Concrete cannot carry a stress above its cylinder strength.

What decides it: A cylinder test loads the whole section, so nothing is holding it together sideways. A small pad on a large block is confined by the material around it, and carries the design strength times the square root of the ratio of areas — up to three times, and the limit is a strut angle rather than a property of the concrete.

Tested in Three times as strong under a smaller pad, at the figure it turns on · the essays about bearing stress.

A prestressing force is an axial force in the member.

What decides it: Only where the tendon is straight. A parabolic tendon of drape d over a span L applies an upward distributed load of 8Pd/L² along the whole span and a pair of downward forces at the anchorages, and that set of loads is exactly equivalent to the prestress — which is why the method is called load balancing and why it needs no section properties at all.

Tested in The load that comes from changing direction, at the figure it turns on · the essays about deviation force.

A dimensional tolerance is a small quantity compared with a structural dimension.

What decides it: It is small compared with the depth of the member and not small compared with the depth of the member that matters. Ten millimetres is 4.4% of a 225 mm slab and 5.3% of its 187 mm effective depth, and capacity is very nearly proportional to the second.

Tested in The dimension nobody can measure, at the figure it turns on · the essays about effective depth.

A more accurate calculation gives a more efficient structure.

What decides it: Only if the extra accuracy crosses a step. The section chosen is a step function of the moment required, so over most of a tread a five per cent refinement changes nothing that can be bought, and at a riser a one per cent refinement changes the section, the weight and the depth.

Tested in The answer is continuous and the catalogue is not, at the figure it turns on · the essays about available sections.

The lightest section for a moment is the one with the smallest plastic modulus above it.

What decides it: Within one series, yes, because modulus per kilogram rises monotonically with depth. Across a real catalogue it does not: a column section of the same capacity weighs nearly twice a beam section's, and a depth limit is what forces the choice.

Tested in The answer is continuous and the catalogue is not, at the figure it turns on · the essays about available sections.

A section has one area, and the checks divide by it.

What decides it: It has a gross area for axial force, a shear area for shear, and — for the same section — two different shear areas depending on whether the question is strength or deflection. The one in the tables is the one for strength, and using it in a deflection calculation gives the wrong answer for a reason nothing in the calculation reveals.

Tested in The section has two areas, at the figure it turns on · the essays about shear area.

The flanges carry the moment and the web carries the shear only approximately, so the two checks interact.

What decides it: The separation is better than approximate. On a rolled I-section the web is about a fifth of the area and carries around ninety per cent of the shear, while contributing under a tenth of the second moment — so the two checks are made on two different pieces of the same member and the interaction is flat until one of the pieces has been used up.

Tested in The section has two areas, at the figure it turns on · the essays about shear area.

A hanging cable finds the right shape for its load, so a net does too.

What decides it: It finds the shape for the load it has. Change the load pattern and a single family changes shape to suit, which is a mechanism rather than a structure. The second family is what refuses the change, and it can only refuse it while it is still in tension.

Tested in Two curvatures of opposite sign, at the figure it turns on · the essays about cable net.

An integral bridge has no movement problem because it has no joints.

What decides it: It has the same movement and no joints to put it in, so every millimetre is taken by bending a pier or pushing on an abutment. The deck still gets 45 mm longer; what changes is that the movement has become a force rather than a gap.

Tested in Where the structure is allowed to move, at the figure it turns on · the essays about articulation.

The vertical components of the two braces cancel, so the beam carries only gravity.

What decides it: They cancel while both braces are elastic and equal. The compression brace buckles at 445 kN and the tension brace yields at 1,065, so the vertical residue at the apex is 659 kN — a midspan point load on a beam sized for a uniformly distributed 25 kN/m.

Tested in The force the brace leaves behind, at the figure it turns on · the essays about chevron brace.

Effective length is a property of a member and its two ends.

What decides it: On a sway frame it is a property of the storey. The braced column here has K = 1.99 alone and 3.63 with three gravity columns leaning on it, and nothing about the column, its ends or its length changed between the two.

Tested in The column that leans on its neighbours, at the figure it turns on · the essays about storey buckling.

The code limit on web slenderness is an empirical rule.

What decides it: Its form is derivable in four lines: set the flange's radial force against the web's plate buckling resistance and the result is h/t ≤ k(E/f)√(Aw/Afc) exactly. Only the constant is empirical, and it corresponds to a stated flange strain rather than to a safety margin.

Tested in The web that is crushed from inside, at the figure it turns on · the essays about flange induced.

Collapse loads in soil are estimates.

What decides it: The undrained case is one of the very few problems in plasticity where the bounds close. A two-zone stress field gives 4, Prandtl's mechanism gives 2 + π = 5.1416, and the best circular slip gives 5.52 — so the answer is squeezed between an unimprovable mechanism and an unimprovable stress field and is known exactly.

Tested in The ground is a mechanism, at the figure it turns on · the essays about bearing capacity.

An equilibrium check is a weak check, so a compatibility check is the real one.

What decides it: Neither is sufficient alone and they fail on different errors. Equilibrium catches everything that changes what the structure is carrying; compatibility catches everything that changes how it is shared. A model that satisfies both can still be a model of a structure nobody intends to build.

Tested in The check that cannot see the error, at the figure it turns on · the essays about equilibrium check.

Adding the extremes is conservative, which is the safe thing to do.

What decides it: It is conservative and it asks for a joint nearly twice the size a root-sum-square gives, on an assumption whose probability is a few per cent. A joint too large is a detail that leaks, rattles and looks wrong, so the conservatism has a failure mode of its own.

Tested in The gap nobody computed, at the figure it turns on · the essays about movement budget.

Movements combine the way loads do.

What decides it: They should and they do not. The combination rules for actions take one leading term at its full value and the rest at coincidence factors, and the same reasoning applied to movements gives a number between the sum and the root-sum-square — which is where the answer is, and which almost nobody computes.

Tested in The gap nobody computed, at the figure it turns on · the essays about movement budget.

Leaving stiffness out of a model is conservative.

What decides it: It is conservative for deflection, where less stiffness means more movement. It is unconservative for vibration, where less stiffness means a lower frequency and therefore a prediction that lands closer to the footfall range than the building ever will — so the check that fails is the one about a floor that would have passed.

Tested in Stiffer than the model said, at the figure it turns on · the essays about measured stiffness.

The extra stiffness can be counted, since it is measurably there.

What decides it: It is there and it cannot be relied on: partitions are removed, cladding is replaced, a nominally pinned connection may be built pinned. Counting it for strength would be indefensible, so the correction that would make the vibration prediction right is exactly the one nobody may make.

Tested in Stiffer than the model said, at the figure it turns on · the essays about measured stiffness.

A duration factor is a safety factor for long-term loading.

What decides it: It is the curve itself, read at the load's own duration. The permanent-duration factor of 0.6 is the Madison curve at fifty years, which is 0.589 — the factor is not covering an uncertainty, it is quoting a measurement.

Tested in The load that was left on too long, at the figure it turns on · the essays about duration of load.

A work-hardened temper is stronger than an annealed one.

What decides it: As delivered, by a factor of two. Welded, the other way round: 5083-H22 keeps 44 per cent and ends at 110 N/mm² while H111 keeps all of its 125. The heat undoes exactly the work, so the alloy that started twice as strong finishes weaker.

Tested in The strength the welder gives back, at the figure it turns on · the essays about Heat-affected zone.

A weld reduces a member's capacity by the softening ratio.

What decides it: Across the member, yes — the whole section is in the softened zone. Along it, only a strip is, so a 200 mm member with two longitudinal welds keeps 85 per cent and a 600 mm one keeps 95. The same weld costs a narrow member nearly everything and a wide one very little.

Tested in The strength the welder gives back, at the figure it turns on · the essays about Heat-affected zone.

The design is governed by the heaviest vehicle.

What decides it: It is governed by a cube-weighted sum. A cycle twice as large does eight times the damage, so two per cent of the crossings at the largest range consume most of the life while a third of them at the smallest, under the cut-off, consume none at all.

Tested in The detail decides and the steel does not, at the figure it turns on · the essays about detail category.

A structure has a natural period, which is what a spectrum is entered with.

What decides it: A rocking block has no stiffness, so it has no period. Its half-cycle time is Housner's arccosh function of the release amplitude, which is 1.6 s at a tenth of the toppling angle and infinite at the toppling angle itself — a block set exactly at balance never returns.

Tested in The block that is safer for being bigger, at the figure it turns on · the essays about rocking.

The point of contraflexure is at mid-height, near enough.

What decides it: It is at 0.61 of the height in the bottom storey of this frame and 0.36 in the top one, and the average over the whole frame is 0.475. The assumption is right on average and wrong in every storey — and the two storeys it is most wrong in are the two a hand check is most likely to be made in.

Tested in The analysis that assumes the answer, at the figure it turns on · the essays about portal method.

Moment redistribution is an allowance, granted by a code, that could have been granted differently.

What decides it: The amount is not granted at all. Take thirty per cent off both support moments and statics says exactly what must reappear at mid-span — the identity fixes it to the last kilonewton metre. What a code grants is permission to *use* a distribution other than the elastic one, and the price is rotation capacity, not moment.

Tested in Two of these move and the third cannot, at the figure it turns on · the essays about static moment.

A thicker wall is less troubled by its own edges, because it is stiffer.

What decides it: The ratio of edge bending stress to membrane stress is root three over root one minus nu squared — 1.82 at nu = 0.3 — with no thickness, no radius and no pressure in it. Doubling the wall changes it by exactly nothing. What it does change is the reach, which grows as the square root of the thickness, so a thick wall is troubled over a longer length by the same relative amount.

Tested in The length a structure was never given, at the figure it turns on · the essays about edge disturbance.

Saint-Venant's principle covers this: a local effect dies out locally.

What decides it: Saint-Venant's decay length is the member's own depth, so a 400 mm beam has forgotten a badly applied load in about 400 mm. A shell's is a geometric mean of two very different lengths: a 4 m tank wall 12 mm thick forgets its base over 170 mm, and a 30 m concrete dome 100 mm thick takes 1.1 m. Same principle, an entirely different number, and the number is what a design needs.

Tested in The length a structure was never given, at the figure it turns on · the essays about edge disturbance.

A stronger concrete makes the reinforcement requirements easier.

What decides it: It makes this one harder. f_ctm goes as f_ck to the two-thirds, so C60 has a tensile strength 44% above C30's and demands 44% more minimum steel — in a member whose bending capacity was never limited by the concrete and has therefore barely moved.

Tested in The steel the concrete asks for, at the figure it turns on · the essays about minimum reinforcement.

The plastic modulus is the elastic one times a shape factor of about 1.15.

What decides it: That is true of an I-section and of nothing else in this collection. A rectangle is exactly 1.5, a channel about 1.20, a tee 1.79 and a composite deck 1.80 — and the reason the range is so wide is that the two moduli are taken about two different axes, which for a tee are 23% of the depth apart.

Tested in The axis that moves when the section yields, at the figure it turns on · the essays about Equal-area axis.

A section has a neutral axis.

What decides it: It has one while it is elastic and a different one when it is fully plastic, and a continuous family of them in between. It also has two elastic section moduli, one to each extreme fibre, which for this tee differ by a factor of 2.78 — while the plastic section has one, because a fully yielded section does not care which fibre reached yield first.

Tested in The axis that moves when the section yields, at the figure it turns on · the essays about Equal-area axis.

Cracks are controlled by putting in more reinforcement.

What decides it: More steel helps by lowering its stress, which is one of two factors. The other is the crack spacing, and that is a bond length proportional to the bar diameter — so the same 1,340 mm² per metre gives 0.17 mm in 8 mm bars and 0.37 mm in 25 mm bars, a factor of 2.15 with the steel area held identical. Rearranging the same steel is free; adding steel is not.

Tested in The same steel, and a wider crack, at the figure it turns on · the essays about crack width.

More cover is better, because it protects the bar.

What decides it: It protects the bar and widens the crack that lets the environment reach it. The cover enters the crack spacing directly — about 3.4 times it — so raising cover from 15 mm to 70 nearly trebles the calculated crack width. Two durability requirements written in the same clause pull in opposite directions.

Tested in The same steel, and a wider crack, at the figure it turns on · the essays about crack width.

Stiffer props are safer props.

What decides it: A prop's load is an imposed-deformation effect: the wall moves toward the hole and the prop refuses. So the force is proportional to the prop's own stiffness, exactly as a lack-of-fit force is. A stiffer prop takes more load and gives the soil less chance to mobilise its own strength — and the wall's bending moment falls with the movement it is allowed.

Tested in Every prop has its own worst day, at the figure it turns on · the essays about propped excavation.

A group of n piles carries n times what one carries.

What decides it: For capacity, at ordinary spacings, very nearly — the block-failure check on this three-by-three group at five diameters gives a ratio of 4.5, so it does not govern at all, and it only begins to bite at two diameters with many piles or in soft clay with long piles. For stiffness it is nowhere near: the group settles 3.9 times what one pile settles at the same load per pile, and a hundred-pile group settles thirteen times.

Tested in Nine piles, and four times the settlement, at the figure it turns on · the essays about pile group.

The group efficiency factor is the important number.

What decides it: It is the number that is always computed and rarely governs. What governs nearly every pile group ever designed is settlement, and settlement is not in the efficiency formula at all — the interaction that softens the group is an elastic effect over the whole depth of the stress bulbs, not a strength effect at the pile shafts.

Tested in Nine piles, and four times the settlement, at the figure it turns on · the essays about pile group.

A stronger lining is a safer lining.

What decides it: A lining stiff enough to refuse the ground's distortion absolutely collects the whole free-ring moment, p₂R²/3. One flexible enough to go with it collects nothing, because there is no curvature change left to resist. The moment rises monotonically with the lining's own bending stiffness over six orders of magnitude of it, and thickening the lining moves it up the curve.

Tested in The lining that is stronger for being weaker, at the figure it turns on · the essays about tunnel ring.

The load on a tunnel lining is the weight of the ground above it.

What decides it: The thrust is the mean stress times the radius, and the mean is (σ_v + σ_h)/2 — so the horizontal stress is half of it, and at K₀ = 0.6 the "weight above" over-states the mean by a quarter. The bending, which is what usually sizes the section, comes from the *difference* between the two stresses and vanishes entirely at K₀ = 1 however deep the tunnel is.

Tested in The lining that is stronger for being weaker, at the figure it turns on · the essays about tunnel ring.

A structure is stable up to the load at which its stiffness matrix becomes singular.

What decides it: For a load with a potential, yes. Ziegler's two-bar column under a tangential end load has a stiffness determinant of exactly k² at every load — it varies over the whole axis by one part in 10¹⁶, which is round-off — and the column nevertheless loses stability at Pℓ/k = 2.0858. The static criterion is not a poor approximation there; it has no root at all.

Tested in The load it cannot buckle under, at the figure it turns on · the essays about follower force.

Energy methods give an upper bound on the critical load, so they are safe to use.

What decides it: They give a bound when a potential exists. Symmetrising this column's stiffness matrix — which is what treating the follower force as a dead weight amounts to — gives divergence at 2.000, which happens to be near 2.0858 and is near it by luck: it is the answer to a different question about a different structure.

Tested in The load it cannot buckle under, at the figure it turns on · the essays about follower force.

Centrifugal force is a real force acting outward on the body.

What decides it: It is the body's own mass times its own acceleration, moved across the equals sign. Nothing applies it and no reaction to it exists. The move is legitimate and it is exactly half done when the inertia term is added to the free body and the acceleration is left on the other side as well — which counts it twice.

Tested in The force that is really an acceleration, at the figure it turns on · the essays about centrifugal load.