What is refuted here — page 4

A material's ranking is a property of the material.

What decides it: Glass is second on the stiff-tie index and seventh on the strong-tie index. Timber is first on three of six and fifth on another. Five of the eight materials here move four places or more between load cases. Nothing about any of them changed; the exponent did.

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

Corrosion damages a structure by removing steel, so the check is a reduced area.

What decides it: The cover splits when the bar has lost 10.7 microns of radius — a section loss of 0.21%, which no strength check would notice and no calculation would report. What removes the capacity is not the missing steel; it is the loss of bond and cover that follows the splitting, and by then the structural quantity that has changed is an anchorage rather than an area.

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

The thirty-degree spread is a derived dispersion angle.

What decides it: It is a fit. Whitmore measured strains on photoelastic models in 1952, found that a thirty-degree spread reproduced them, and published. Moving it to forty degrees changes the assumed area by 34% and the assumed capacity with it. Nothing derives the angle, and finite element work has since agreed with it to about ten per cent, which is why it survives.

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

The foundation flexibility that matters is the settlement under the footing.

What decides it: Sliding contributes k/k_x, a fixed number with no height in it, and it is worth 8% of the lengthening here. Rocking contributes k·h²/k_θ and supplies the other 89%. What decides whether any of this matters is the height of the effective mass, and the mechanism is a rotation nobody drew rather than the sliding everybody pictures.

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

A longer roof upwind of a parapet means proportionally more drift against it.

What decides it: The wedge holds the snow the fetch gave up, so its area grows with the fetch and its depth as the square root. Doubling the roof upwind multiplies the drift by 1.41, four times the roof doubles it, and past a fetch of 28 m the one-metre parapet is full and further roof adds nothing whatever.

Tested in The load that arrives where the wind stops, at the figure it turns on · the snow drift ladder.

The worst snow case is the one carrying the most snow.

What decides it: On a two-span beam the balanced case weighs 17 kN and gives a span moment of 4.6 kNm. Clearing one span leaves 9 kN and gives 6.2 kNm — 1.36 times the moment for half the load. A continuous structure answers the pattern of a load, not its total.

Tested in The load that arrives where the wind stops, at the figure it turns on · the snow drift ladder.

A notional horizontal force is a safety allowance, like a partial factor on the wind.

What decides it: It is an exact substitution. A storey leaning by φ carries its vertical load N along a line offset by φh from the columns, which is a moment Nφh — identical to the moment of a horizontal force φN applied at the storey top. Nothing has been added; a geometry has been replaced by a force that produces the same free body.

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

A longer span needs a stronger material.

What decides it: The limiting span contains σ/ρ, not σ. A timber with an eleventh of mild steel's strength beats it by 8.56 to one on the stiff-beam index, because what a long span asks of a material is capacity per unit of weight and the exponent on the shape is what decides the ranking.

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

A load is a load, and where on the section it is applied is a detailing matter.

What decides it: During buckling the section rotates, so a load attached above the shear centre moves sideways with it and its own weight then adds to the twisting moment. The work it does is proportional to the height of the attachment, and it enters the critical-moment expression as a term with the same standing as the warping stiffness.

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

A member sized for a tension of 400 kN can obviously carry 400 kN of compression.

What decides it: Only if its shape happens to suit. Four profiles of exactly 6000 mm² carry between 58 and 4149 kN as a 4 m pin-ended column — a factor of 71 — in the ratio of the squares of their radii of gyration and of nothing else. A tension design never asks that question, so the answer is whatever the fabricator's stock list happened to give.

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

A buckling analysis and a second-order analysis are two different calculations.

What decides it: They are the same matrix asked two questions. The total stiffness is the elastic stiffness minus the load times a geometric stiffness; a second-order analysis solves that system for a deflection at one load level, and a buckling analysis finds the load level at which it has no solution. One assembly, two questions.

Tested in The stiffness the load takes away, at the figure it turns on · the geometric stiffness ladder.

An assumed buckling shape gives an approximate load that could be too high or too low.

What decides it: It is always too high, and by the square of the error in the shape. A shape assumed is a constraint imposed, and a constraint can only stiffen. On a tapered column with no closed form at all, three plausible shapes give 8.402, 9.000 and 8.338 against a converged 8.2486 — every one of them above, none below.

Tested in The stiffness the load takes away, at the figure it turns on · the geometric stiffness ladder.

A reaction is a known force, so the region above a support is a solved problem.

What decides it: Only its total is known. Its distribution follows the relative stiffness of the beam and what it sits on, and on the beam drawn here 99% of the deflection belongs to the supports rather than to the beam. Statics fixes the sum and settles nothing about the pressure, which is what every check near a support actually needs.

Tested in The support that is not a point, at the figure it turns on · the support width ladder.

The tension bar can stop where the bending moment diagram reaches zero.

What decides it: A cracked concrete beam carries shear as a truss, and the diagonal cut passes through the chord as well as the stirrups, so the chord carries half the shear as tension. On this 8 m beam that is 563 kN at the support where bending says nothing, and the whole force curve is the moment diagram shifted 585 mm toward the support.

Tested in The support that is not a point, at the figure it turns on · the support width ladder.

A simple connection transfers a shear and no moment, so a column carrying simply supported beams carries axial force only.

What decides it: The connection transfers no moment across itself and delivers its shear at an eccentricity from the column axis. A force may be moved anywhere at the price of a couple — 80 kN moved 250 mm is 20 kNm — and that couple is applied to the column whether or not the joint is capable of resisting rotation.

Tested in The moment the beam left behind, at the figure it turns on · the column eccentricity ladder.

An internal column between two equal spans has an eccentric moment from each side, so it gets twice as much.

What decides it: They act in opposite senses and subtract. An internal column between equal, equally loaded spans gets almost nothing; the same column where a 9 m span meets a 6 m one gets the difference of the two reactions times the eccentricity, which is most of what one side would have delivered alone.

Tested in The moment the beam left behind, at the figure it turns on · the column eccentricity ladder.

The stress in a finished beam is the total moment divided by the finished section's modulus.

What decides it: Only if the section was finished before any of the moment arrived. The same 12 m composite beam reaches 146 MPa in the bottom of its steel unpropped and 93 MPa propped — a ratio of 1.57 — because 62% of the unpropped beam's final stress was locked in while the slab was still wet and the bare steel was carrying it alone.

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

Two structures with the same geometry and the same load have the same internal forces.

What decides it: A two-span beam erected as simple spans and made continuous before the rest of the load arrived has a support moment of 225 kNm against 375 for the same beam built continuous, and a span moment of 263 against 188. Both diagrams satisfy equilibrium with the same total load. What separates them is when the joint was made, which appears nowhere on the drawing.

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

The shape of the concrete stress block has to be got right for the answer to be right.

What decides it: A bending calculation asks a stress distribution exactly two questions — how much compression there is and where its resultant acts. The real parabolic-rectangular curve and the design rectangle differ visibly and give the same two numbers: 723 kN at 62.4 mm from the face. A triangle and a full rectangle match neither and are nowhere near.

Tested in Where the steel is, not how much of it, at the figure it turns on · the lever arm ladder.

Cover is a durability requirement with no structural cost.

What decides it: Cover, the link diameter and the bar diameter together decide the effective depth, and capacity is proportional to it. On a 600 mm deep beam, moving from 25 mm of cover to 50 costs about 4.6% of the moment capacity, and a second layer of bars costs about as much again — neither of which appears in any calculation as a structural decision.

Tested in Where the steel is, not how much of it, at the figure it turns on · the lever arm ladder.

Curving a plate to carry a pressure is a refinement worth a factor of two or three.

What decides it: It is a change of exponent. A membrane's required thickness is linear in the pressure and a bending strip's goes as its square root, so the ratio between them is √(3σ/p) — 23.7 for the case here, 11.7 mm of steel against 278. And because it is a square root, the advantage grows as the load falls rather than staying put.

Tested in The same span, four ways, at the figure it turns on · the form selection ladder.

The best material for a long span is the strongest one available.

What decides it: Modulus against density on logarithmic axes turns a performance index into a straight line whose slope is the exponent, and the three lines through mild steel have three different orderings behind them. For a tie the answer is carbon fibre, for a beam it is timber at 8.56 times steel, and for a plate timber wins by more still.

Tested in The same span, four ways, at the figure it turns on · the form selection ladder.

Shear lag in a framed tube is a consequence of the building's plan proportions.

What decides it: It is a consequence of the perimeter frame's racking stiffness. Holding the plan fixed and raising that stiffness by ten — which is what a diagonal across the face buys — takes the corner overstress from 1.71 to 1.21 and the tube's effective second moment from 64 per cent of the gross to 92. The plan did not move.

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

Shrinkage causes a shortening, so it is an axial effect and a deflection calculation can ignore it.

What decides it: It is axial only if the restraint is symmetric. Reinforcement in one face resists the free shrinkage on that side and not on the other, so the section takes a curvature — with no applied moment, no resultant force, and a deflection that adds directly to whatever the load produced.

Tested in The curvature nobody applied, at the figure it turns on · the shrinkage curvature ladder.

A heavily reinforced beam is more affected, because there is more steel restraining the shrinkage.

What decides it: The curvature is the free shrinkage times the steel's first moment about the section's own centroid, divided by the section's second moment — so it is largest where the reinforcement is most one-sided, which is the lightly reinforced case. A heavily and symmetrically reinforced member shrinks nearly straight.

Tested in The curvature nobody applied, at the figure it turns on · the shrinkage curvature ladder.

An approximate method could give an answer that is too high or too low.

What decides it: Not when the approximation is a restriction on the shape. Four guesses at a pin-ended column's buckling mode give 9.870, 9.882, 10.000 and 12.000 against the exact 9.8696 — every one high, none low — because an assumed shape is a constraint and a constraint can only stiffen.

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

A partial factor on a material covers the variability of the material.

What decides it: The variability was spent getting from the mean to the 5% fractile. Dividing that fractile by 1.5 puts the design value at a fractile of 6.4×10⁻⁶ — one specimen in 155,818 — which is far below anything a test programme could ever measure. Whatever the factor is for, it is not the scatter.

Tested in The strength no specimen had, at the figure it turns on · the characteristic strength ladder.

Testing more specimens gives a more precise answer, not a different one.

What decides it: With the scatter estimated from the same specimens, the factor applied to it is a Student t quantile — 7.73 at two specimens against 1.73 at thirty. At three specimens this material's characteristic strength comes out at 18.1 N/mm² against 23.2 for a population known exactly. A small test programme does not report a worse estimate; it reports a worse strength.

Tested in The strength no specimen had, at the figure it turns on · the characteristic strength ladder.

The tensile strength is set to zero because it is small.

What decides it: Concrete's tensile strength is about a tenth of its compressive strength, which is not small enough to be neglected on those grounds alone. It is discarded because it is unreliable — the widest scatter of any property in common use, and size-dependent besides, with three geometrically similar specimens of one mix failing at 3.76, 2.97 and 1.88 N/mm².

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

An anchor bolt's capacity is the tensile capacity of the bolt.

What decides it: Only if the concrete around it is strong enough to hold it, and the failure that decides is a cone of concrete pulled out in tension — a material this collection otherwise assumes has none. The base plate here starts engaging its bolts at 150 kN·m and asks 154 kN of them, and whether that is available is a question about the concrete's tensile strength, its edge distances and the neighbouring anchors.

Tested in The failure that is in the concrete, at the figure it turns on · the anchor breakout ladder.

Doubling the embedment doubles the capacity, because the cone is twice as deep.

What decides it: The projected area of the cone goes as the square of the embedment, so the geometry predicts four times. The measured capacity goes as embedment to the power 1.5, because a larger failure surface is a weaker one — the same size effect that makes a 900 mm member carry 1.20 N/mm² where a 100 mm specimen reads 2.80.

Tested in The failure that is in the concrete, at the figure it turns on · the anchor breakout ladder.

A lap with ten bolts carries ten times what one bolt carries.

What decides it: The two plates strain at different rates along the lap, so the slip between them is largest at the ends and nearly zero in the middle — and a bolt with no slip across it transfers almost nothing. On the lap here the end bolts carry 1.16 of their nominal share and the middle ones 0.89, and at 90 diameters of joint length the end bolt is carrying 2.19 times its share.

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

Materials behave the same however fast they are loaded, so a dynamic check uses static strengths.

What decides it: A testing machine works at about 10⁻⁴ per second and a blast at a hundred, and steel's strength rises by 2.04 across that range. The ultimate strength rises only a third as much, so the ultimate-to-yield ratio closes from 1.21 to 0.95 — the material is stronger and has less warning left in it than it started with.

Tested in The load that is over before it has moved, at the figure it turns on · the blast ladder.

A tank's seismic mass is the mass of its contents.

What decides it: Only the part that moves with the wall behaves that way. On the tank here that is 65% of the liquid; the other 35% sloshes at a period of 4.47 s against the impulsive 0.25, sits far out on the falling branch of the spectrum, and supplies 3% of the base shear. A calculation that uses the whole mass overstates the force by half.

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

A bigger tank gets a bigger wave.

What decides it: Only while the convective period is short enough to sit on the spectrum's constant-velocity branch. Past the corner the spectral acceleration falls as the inverse square of the period while the period rises as the square root of the radius, so the product stops depending on the radius at all and the curve flattens.

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

Shear strength is a material property, so a deeper member is stronger in the same proportion.

What decides it: The stress at failure falls as the member deepens. It runs from 0.79 N/mm² at an effective depth of 150 mm to 0.50 at 3,000 on the same concrete, the same reinforcement ratio and the same everything — a factor of 1.59 given away for nothing but size. The force rises over the same range, from 36 kN to 449, and only one of the two is the number in the check.

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

A shear check has nothing to do with the bending reinforcement.

What decides it: Doubling the flexural ratio from 1.2% to 2.4% raises the shear strength by a quarter, and the reason is dowel action: the bars crossing the crack carry shear by bending across it. Nothing in a truss analogy predicts a dependence on the longitudinal steel at all.

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

The enhancement is free, because it comes from the concrete already there.

What decides it: The spread that produces it is a pair of inclined struts, and a pair of inclined struts has a horizontal component. At a spread of three the bursting tension is a sixth of the applied load — 340 kN on the block drawn, which is 782 mm² of steel. Without it the block splits at the unenhanced strength.

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

A wider block always gives a bigger enhancement.

What decides it: The spread is limited to three times the loaded dimension and to twice the block's depth, whichever is less, because a strut flatter than about one in two is not a strut anybody should believe. Past that the block is wider and the capacity is not.

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

A transverse load has to be applied by something.

What decides it: A plate girder's compression flange, curved by the beam's own deflection, needs F/R per unit length to stay on its curve, and the only thing available to supply it is the web. Nothing has been applied to the girder; the load comes from the deflected shape, which is why a straight beam has none of it and a beam at a plastic hinge has a great deal.

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

A deviation force is a small secondary effect.

What decides it: It is the entire mechanism of a shell, an arch and a pressure vessel. A cylinder under pressure carries pR in its hoops for exactly this reason, and the hoop force has no lever arm at all — the deviation force is not a correction to the load path, it is the load path.

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

The partial factors on the materials cover the uncertainty in a member's strength.

What decides it: They cover the concrete and the steel, both of which are tested and neither of which is the problem. The effective depth is a geometric quantity with a bias and a scatter of its own, it is not tested at all, and on a 225 mm slab a five-millimetre bias plus a nine-millimetre standard deviation take about a ninth off the capacity before any material factor is applied.

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

Errors in bar position are as likely to be up as down.

What decides it: Top steel in a slab is walked on, stood on and pushed down by the pour. The measured distribution is offset downward and the bias is a loss before any scatter is considered — the one term in the calculation with a sign that is known in advance.

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

Rounding up to the next section is a small waste.

What decides it: The steps in plastic modulus across a universal beam series run from twenty-three to fifty-nine per cent, so the average member is about fifteen per cent stronger than it needs to be and the worst is nearly half again — steel that is paid for, carried, welded and painted, and does nothing.

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

Shear stress is the shear force divided by the area of the section.

What decides it: That is the average, and the shear stress is nowhere equal to its average. A rectangle's distribution is a parabola with a peak of exactly 1.5V/A at the neutral axis and zero at both faces, and the 1.5 is a property of the shape with nothing empirical in it.

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

A cable roof is stiff because the cables are strong.

What decides it: Strength is an area times a stress, and the stiffness at small deflection is 8(H_s + H_h)/(s L²) plus an elastic term in the sags — 3.81 and 13.53 kN/m³ on the net drawn. Neither term is a strength. Halving this net's pretension costs eleven per cent of its stiffness and nearly three quarters of the load it takes before a family goes slack, with the same cables throughout.

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

Once a net is designed for its worst load it is safe for anything smaller.

What decides it: The governing event is not the largest load but the one that unloads the hogging family. Past that deflection the net has one family working, no restraint against pattern, and a stiffness that has fallen by most of its value — so the failure is a loss of stiffness at a load below the strength.

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

A sliding bearing releases the structure, so a fixed point can go anywhere convenient.

What decides it: Each sliding bearing delivers μV to the deck, and the fixed support takes whatever does not cancel across it. Fixed at one abutment that residue is the whole of the friction on the viaduct drawn — 752 kN before any wind or braking. Fixed at the friction centroid it is nothing at all.

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

Making a pier stiffer makes it attract more of the thermal load, without limit.

What decides it: Stiffening a pier attracts load and also drags the neutral point toward it, which shortens the lever the temperature acts on. The second effect wins in the limit: the force saturates at the sum of the OTHER piers' stiffnesses times their distances, and contains nothing of the pier itself.

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

An elastic analysis at the design load shows the beam is adequate, so it is.

What decides it: The elastic state is one the frame passes through on its way to the state it will actually be in. Every check made in it is a check on a structure that will not exist by the time the demand arrives, and it passes for exactly that reason.

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

A stronger brace makes the frame safer.

What decides it: The unbalanced force is the tension brace's yield less what the buckled one retains, so a stronger tension brace makes the residue larger. Increasing the brace area without increasing the beam moves the failure into the beam, which is the one member the system cannot afford to lose.

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

A gravity-only column is a member to be checked, not a member that affects others.

What decides it: It contributes axial load to the storey and no lateral stiffness, so it lowers the load at which the whole storey buckles without raising anything. Three of them carrying 70% of the gravity load nearly double the effective length of the column that is holding them up.

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

A column pinned at both ends is designed at K = 1.

What decides it: Only when it is braced against sway. In a sway storey it has no critical load of its own at all, and the effective length that reproduces its share of the storey's buckling load is 2.60 here — two and a half times what a member-by-member design would give it.

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

A web only carries what is applied to it.

What decides it: The compression flange carries 3,800 kN and the beam's curvature bends it to a radius of a few hundred metres, so it needs a radial force of a few newtons per millimetre to stay on that curve. Nothing applied it; the load is a consequence of the deflected shape, and a straight beam has none of it.

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

A thinner web is worse in proportion to its thickness.

What decides it: It is worse twice over. A thinner web has a larger slenderness and a smaller shear area, so it appears on both sides of the inequality — the demand rises as one over t and the limit falls as the square root of t, so the utilisation goes as t to the power of minus one and a half.

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

Bearing capacity is an empirical formula with tabulated coefficients.

What decides it: Two of its three coefficients are exact. N_q is the exponential of π tanφ times tan²(45 + φ/2), which is the stress followed round a ninety-degree fan of radial shear, and N_c is that same result written for a cohesion. Only N_γ has no closed form, and the published expressions for it differ by tens of per cent.

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

A footing on sand has a bearing capacity per unit area, like any material.

What decides it: On clean sand with no surcharge the capacity is ½γBN_γ, which is proportional to the width — so the pressure a footing can carry depends on how big the footing is, and a footing of vanishing width carries nothing at all. Sand resting on sand holds nothing up.

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

If the reactions sum to the applied load the model is right.

What decides it: They sum for any statically admissible set of internal forces, correct or not. A member whose stiffness is out by a factor of ten redistributes the internal forces completely and the global residual is exactly zero, because the wrong answer is in equilibrium with the same loads.

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

A finer mesh gives a more trustworthy answer.

What decides it: Refinement converges toward the answer for the model that was built, and says nothing about whether that model is the structure. The errors this essay is about survive refinement exactly — they are properties of the input, and a converged wrong answer is a wrong answer with smaller error bars.

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

The gap at the top of a wall is a deflection, so a deflection calculation sizes it.

What decides it: The deflection is 3.2 mm of a 33 mm total. Thermal movement, shrinkage, creep and a construction tolerance make up the other thirty, and only one of the five is computed to better than a factor of two.

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