What is refuted here — page 14

A beam's torsional resistance is GJ, so a member with a small torsion constant has little resistance to twist.

What decides it: Only over lengths much greater than the decay length. At the built-in end of the 9 m beam drawn the Saint-Venant mechanism carries exactly nothing and the flanges bending in opposite directions carry the whole 1.2 kN·m; a decay length along, that share has fallen to 37 per cent, and at the far end to 3.2.

Tested in The restraint that beats the gradient, at the figure it turns on · the moment gradient ladder.

Torsion produces shear stress, so a torsion check is a shear check.

What decides it: Warping torsion produces a longitudinal stress and no diagram predicts it. The bimoment reaches 2.610 kN·m² at the held section and raises 53.8 N/mm² at a flange tip against a peak Saint-Venant shear of 27.7 — at the corner where the bending stress is already highest.

Tested in The restraint that beats the gradient, at the figure it turns on · the moment gradient ladder.

The coupling between the modes is a refinement, and taking the lowest of the three uncoupled loads is close enough.

What decides it: It depends entirely on how far the shear centre sits from the centroid. On the channel drawn the coupled root is 4.5 per cent below the lowest uncoupled one; on the tee it is 35.3 per cent below. The coupling takes from the mode that was already lowest and gives to the one that was highest, so it can never help.

Tested in The third root of the cubic, at the figure it turns on · the Flexural-torsional ladder.

Twisting is a short-column problem, so a slender member is safe from it.

What decides it: True, and the crossover is a length rather than a slenderness. The channel switches from twisting to bending at 5,813 mm and the tee at 3,701 — because the torsional load keeps a Saint-Venant term that does not grow as the member is shortened, while both flexural loads grow as the inverse square.

Tested in The third root of the cubic, at the figure it turns on · the Flexural-torsional ladder.

Making a truss deeper reduces its deflection in proportion to the depth.

What decides it: It reduces it as the 1.56 power, and it changes what the deflection is made of at the same time: the chords' share falls from 0.902 at a depth of 0.5 to 0.471 at 2.0. Past some depth the web is supplying most of the movement and stiffening the chords has stopped being the thing to do.

Tested in The member that is not worth stiffening, at the figure it turns on · the truss deflection ladder.

A sandwich's core is a spacer, so its properties barely matter.

What decides it: Its modulus in bending barely matters — it supplies 1.4 per cent of the section's second moment. Its *shear* modulus decides a quarter of the deflection at ordinary proportions, and changing it from 25 to 6 N/mm² takes the shear deflection from 1.1 mm to 4.6 with nothing else altered.

Tested in The stiffness that belongs to the span, at the figure it turns on · the shear deflection ladder.

Two cycles of moment distribution is an engineering answer for any beam.

What decides it: It is for a regular one. On three spans of 7, 9 and 7 m the error after two cycles is 5.92 of 156.9 kNm — under four per cent. On spans of 6, 14 and 6 m it is 36.43 of 328.9 — eleven per cent, and two more cycles are needed to reach the same accuracy.

Tested in Why it converges, and how fast, at the figure it turns on · the moment distribution ladder.

Yielding under a repeated load means the deformation accumulates.

What decides it: Only past the shakedown boundary. Inside it the member yields on the first cycle, leaves a residual stress field behind, and every cycle after that is elastic — the curvature stops growing after two or three cycles and never moves again. That is the whole meaning of shaking down.

Tested in The map with three regions, at the figure it turns on · the shakedown ladder.

A ductile brace dissipates energy whichever way it is loaded.

What decides it: Only if it cannot buckle. An ordinary brace yields at 900 kN in tension and buckles at 482 in compression — 54 per cent — and is at 12 per cent of its first compression capacity by the sixth cycle. A restrained brace yields at the same force both ways and dissipates 2.07 times as much over the same six cycles.

Tested in Yielding one way, and then the other, at the figure it turns on · the ductility ladder.

Raising the mean strength is the direct way to raise the design value.

What decides it: Cutting the scatter is worth more per unit of effort. Going from a coefficient of variation of 0.18 to 0.12 takes the characteristic value from 73 to 82 per cent of the mean — worth the same as raising the mean by 11 per cent, with none of the material's best specimens improved.

Tested in The strength that belongs to the test programme, at the figure it turns on · the characteristic strength ladder.

Creep makes a structure worse, because deflections grow.

What decides it: It makes a force-controlled action worse and a displacement-controlled one better. A restrained shrinkage strain of 320 microstrain produces 10.88 N/mm² of tension elastically — three times the concrete's tensile strength — and creep relaxation leaves 1.72 after 55 years. The same coefficient that trebles a deflection removes six-sevenths of that stress.

Tested in The stress that leaks away, at the figure it turns on · the creep ladder.

The worst point in a weld group is the one furthest from its centroid.

What decides it: True for a symmetric group and false in general. On an L-shaped group carrying 150 kN at 200 mm the peak throat stress is 2.17 kN per mm and the point at maximum radius carries 1.99 — the radius rule is 9.17 per cent low, because the two stress components add at a different point from where either is largest.

Tested in The radius rule, and where it fails, at the figure it turns on · the weld group ladder.

A thicker flange is a stiffer connection and a marginal improvement.

What decides it: It is a change of regime rather than a margin. Below 22.75 mm the flange is a mechanism and the bolt is not what decides; above 31.91 mm there is no prying at all and the bolt carries exactly what is applied. The nine millimetres between them contain the whole of the subject.

Tested in The thickness that decides who fails, at the figure it turns on · the prying ladder.

Keeping a storey within the code's eighty per cent limit keeps the frame regular.

What decides it: It bounds the concentration rather than removing it. At exactly 80 per cent the ground storey still takes 2.40 per cent drift against 1.11 next to it — a concentration of 2.9 against the regular frame's 1.7, from a frame that passes every irregularity check written.

Tested in Weaker in one place, and better on every average, at the figure it turns on · the pushover ladder.

A long thin column is worse than a square one of the same area for punching, because its perimeter resists the twist less well.

What decides it: True for one orientation and false for the other, and the perimeter beats both. At 300 mm of eccentricity a 450 × 450 column runs at 1.009 of its resistance; the same area as 800 × 250 runs at 0.974 and as 250 × 800 at 0.921. The rectangle wins in both orientations because it is 300 mm longer round.

Tested in Turn the column, and the slab passes, at the figure it turns on · the punching shear ladder.

The shear in a built-up column's lacing can be taken as the axial load over a hundred.

What decides it: It is right at low utilisation and increasingly wrong as the load approaches the critical one. At 1,500 kN on this column the rule gives 15 kN against a computed 17.4; at 2,000 kN it gives 20 against 32.4 — a factor of 1.62 — because the second-order amplification is not linear in the load and the rule is.

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

A curved bar is checked by finding the bending stress and correcting it.

What decides it: Not for a hook, where the free body carries a direct tension as well. At 100 kN the bending alone gives 445.5 N/mm² at the inner fibre and the direct tension adds a uniform 51.9 — the peak is 497.5, and the straight-beam calculation that omits both is 54 per cent under it.

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

Closer stiffeners make a girder web more efficient, at the cost of a few more stiffeners.

What decides it: They also multiply what the flange is asked to hold. Taking the stiffener spacing from 4,500 mm to 800 on a 1,500 mm web raises the panel's shear capacity from 716 kN to 2,498 and the pull on the flange from 72.5 N/mm to 602.8 — a factor of 8.3, on a member sized for bending.

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

A compatibility torsion can be ignored, because the structure does not need it.

What decides it: It can be shed and it cannot be ignored. Releasing the spandrel takes its torque from 100 kN·m to 41, and the floor beam's midspan moment from 147 kN·m to 177 — the two always add to the same free moment of 197, so what is shed arrives somewhere that has to be checked for it.

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

Robustness is a property a structure has more or less of.

What decides it: Not below a threshold. All 31 members of an 8-panel Warren truss leave a mechanism when removed, and all 29 of a Howe. There is no redistribution anywhere in either to measure, so the quantity a member-removal study reports is not small — it does not exist, and the answer is the same for every member.

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

A rocking block's survival is decided by how hard the ground shakes.

What decides it: It is decided as much by what happens between shakes. The same block under the same 1.0 g pulse overturns at a restitution of 0.93 and survives at 87 per cent of the toppling angle at 0.75 — the ground motion identical, and the difference entirely in how much energy each landing takes out.

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

A slender block is worse at rocking than a stocky one, because it topples at a smaller angle.

What decides it: It is worse twice over and the second reason is the impact. A 4.4-to-1 block loses 14 per cent of its energy per landing and a 2.4-to-1 block loses 38 — because the classical restitution falls with the toppling angle, so the slender block is both nearer to going over and worse at losing the energy that would stop it.

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

Truss joints are pins, so truss members carry axial force and nothing else.

What decides it: The same truss solved twice — pin-jointed, and with the joints continuous as welding actually makes them. The axial forces agree to about a per cent, which is why the idealisation is a good one; the bending it omits reaches a quarter of the axial stress in the worst member. It scales with EI/L, so a stocky truss of heavy members suffers more of it than a light one — which is the opposite of the way most reserves of strength work, and is the direction in which the simplification stops erring safe.

Tested in The joint that is not a pin, at the figure it turns on · the truss ladder.

Wind loading is a static problem: work out the pressure, multiply by the area, apply it.

What decides it: True for a stiff structure and badly wrong for a slender one. Split into its parts, a warehouse at 2 Hz takes 15% of its fluctuating response from resonance and a tower at 0.2 Hz takes 90% — and the resonant part is inversely proportional to a damping ratio that no static calculation contains. The static method survives because the gust factor it hides absorbs the difference for ordinary buildings.

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

The effective width of a flange is min(span/8, b) per side, and that expression comes from the elasticity of the problem.

What decides it: The exact elastic ceiling for an infinitely wide flange fed by a sinusoidal shear flow is L/2π = 0.1592L per side, from Airy's solution. The rule caps at L/8 = 0.125L, which is 0.7854 of it — exactly 2π/8. A rule that is a fixed fraction of the exact answer is a fit with a margin in it, not a derivation of it, and it is a straight line where the elasticity is a curve.

Tested in The flange that is not all there, at the figure it turns on · the effective width ladder.