What is refuted here — page 16

The error is a coarse-mesh artefact, so it goes away in any model anybody would build.

What decides it: It converges as the square of the mesh size — 33.3 per cent on one element, 8.3 on two, 2.1 on four — so it is still 0.5 per cent at eight and it is being paid for with elements rather than with a load vector. The consistent set is exact on every mesh drawn, including the coarsest.

Tested in Equivalent in work, not in resultant, at the figure it turns on · the essays about force couple.

A weld group is checked by comparing its largest force per unit length with the weld's capacity.

What decides it: The weld has more than one capacity. At 47 degrees to its axis the critical point on the group drawn has 1,612 N/mm available against the 1,460 an along-the-weld check would use and the 1,789 a square pull would give — three numbers for one weld, and the check needs the one that belongs to the direction the force is in.

Tested in The weld that is stronger where it is pulled, at the figure it turns on · the essays about weld strength.

Classifying a joint as pinned, semi-rigid or rigid tells a designer what to do with it.

What decides it: It tells them which analysis is permissible, and it is a set of lines drawn across a continuous curve at 0.5EI/L and 8EI/L. The design question is which point of that curve to buy, and the answer — three quarters of the fixed-end moment — falls inside the semi-rigid band that the classification exists to make people avoid.

Tested in The joint that was chosen, at the figure it turns on · the essays about joint classification.

A moment distribution in which every joint balances is a solved structure.

What decides it: Balancing a joint enforces that the moments meeting there sum to nothing. A frame free to translate has a further equation — that the storey is in horizontal equilibrium — and no joint knows about it. On the portal drawn here the fully balanced no-sway table leaves 37.43 kN unaccounted for, which is a prop holding the frame up.

Tested in Every joint balanced, and the frame still leaning, at the figure it turns on · the essays about moment distribution.

Using 3EI/L for a span with a pinned end is the whole of the modification.

What decides it: The fixed-end moment has to change with it. A span released at one end starts from wL²/8 at the held end and nothing at the release, not wL²/12 at both — 192 and 0 rather than 128 and 128 on the 8 m span here. Taking the stiffness without the moment converges quickly to a different answer.

Tested in Told what the far end is doing, at the figure it turns on · the essays about moment distribution.

Symmetry halves the work because it halves the structure.

What decides it: Halving the structure is not enough on its own: the member crossing the axis needs 2EI/L and a carry-over of −1, because its far end rotates equally and oppositely. On the beam here the halved structure with the modified stiffness converges in one cycle and two numbers, against thirty for the full method.

Tested in Told what the far end is doing, at the figure it turns on · the essays about moment distribution.

An arithmetic slip in an iterative method washes out, because later cycles correct it.

What decides it: It depends on whether the method accumulates or recomputes. A moment distribution adds each cycle to what the last one left, so a wrong carry-over is in every subsequent line. Kani recomputes each contribution from its neighbours' current values, so an entry started 60 kNm wrong is within a kilonewton-metre of the clean run after four sweeps.

Tested in The table that cannot be read halfway, at the figure it turns on · the essays about moment distribution.

The rule exists because of hole clearance, so a fitted bolt in a reamed hole may be added to a weld.

What decides it: Take the clearance to nothing and the pair reaches 532 kN of the 780 — 68 per cent — with the bolt at 35 per cent of its own capacity when the weld lets go. The clearance makes the rule obvious rather than making it true; what makes it true is that a bolt needs millimetres of hole elongation to reach its strength and a weld allows tenths.

Tested in Two fasteners that never arrive together, at the figure it turns on · the essays about weld group.

Moving a bolt closer to the web makes the connection stronger, because the lever arm on the flange shortens.

What decides it: It also shortens the yield-line pattern. Moving the bolt from 45 mm to 25 raises the row from 255 kN to 288 — a gain of 13 per cent rather than the 80 the lever arm alone would suggest, because the effective length fell from 200 mm to 157 at the same time and the pattern changed family on the way.

Tested in How much of the plate is bending, at the figure it turns on · the essays about prying.

The missing-mass correction repairs a truncated analysis.

What decides it: It repairs the base shear, which was already right. On the ten-storey frame with two modes it moves the base shear from 99.7 per cent of the exact answer to 99.8, and the roof force from 93.7 to 94.0. The leftover mass is small and it is not where the error is.

Tested in The modes that were left out, at the figure it turns on · the essays about mode shapes.

The first-mode pattern is the right one, because the first mode dominates the response.

What decides it: It dominates the elastic response. Once the ground storey yields, the tangent first mode is a nearly uniform shape rather than a rising one — 0.122 to 0.127 across eight floors against the elastic 0.062 to 0.165 — because the softened storey deforms and everything above it rides rigidly. The pattern that was right at the start is not the pattern for anything after it.

Tested in The pattern that stopped describing the building, at the figure it turns on · the essays about pushover.

A block that stands on a slope with a margin against sliding has that margin available against any other push as well.

What decides it: The slope has already spent part of the disc. A 100 kN block on a 20 degree slope at μ = 0.5 uses 73 per cent of its grip down the fall line, and a sideways push of 32.2 kN — 69 per cent of μN rather than all of it — starts it sliding.

Tested in Seventy-five per cent each way, at the figure it turns on · the essays about friction.

Enough damping in the support removes stick-slip.

What decides it: Damping takes back the overshoot and cannot touch the drop. At five per cent the swing is 111 kN instead of 120; even at a damping ratio of 0.9 it is still the 60 kN between the two coefficients. Only a surface with a smaller drop gives a smaller swing.

Tested in The pier that moves in jumps, at the figure it turns on · the essays about friction.

The horizontal force in a pier under a sliding bearing follows from the bearing's load and its coefficient of friction.

What decides it: The load and the coefficient give the bound, not the force. A bearing on a pier of 20 kN/mm that ends at the same 4 mm of deck movement under the same 4000 kN leaves the pier carrying 60 kN if the deck moved before the load arrived and 80 kN if it moved after, and both states are inside the bound of 120 kN.

Tested in The order the loads arrived in, at the figure it turns on · the essays about friction.

More supports make a body more stable against overturning.

What decides it: On the same circle, more supports raise the weakest reach and leave the strongest where it was. Three legs 1.5 m out reach 0.75 m toward an edge, four reach 1.06 m and six reach 1.30 m, and all of them reach 1.50 m toward a leg: more legs make the base more uniform, not stronger in its best direction.

Tested in Half as far between the legs, at the figure it turns on · the essays about overturning.

A thicker base plate raises the moment it can resist, so thickness can always buy back the difference.

What decides it: Thickness enlarges the compression zone, and the resistance rises from 31 kN·m at 8 mm until the bolts govern at about 25 mm. Past that it barely moves — 164 kN·m at 60 mm — and it never reaches the 220 kN·m a rigid block gives, because a thicker plate carries more compression without moving it outward.

Tested in The compression that stays under the flange, at the figure it turns on · the essays about base plate.

Once the check that failed has been repaired, the end is fixed.

What decides it: The next check is still there. On the same beam coped 300 mm long, flexure governs at 0.57; the edge plate takes it to 0.31 and hands the governing check to shear at 0.35, which the plate cannot touch, so the repaired end is barely better than one with a doubler.

Tested in The repair that fixes the wrong check, at the figure it turns on · the essays about coped beam.

A rocking wall returns upright if its re-centring ratio, weight and prestress against the bars' yield, is at least one.

What decides it: The ratio of one assumes both bars have yielded back by the time the wall closes, which is true only after a large rotation. After 15 mrad the wall leans up to a ratio of one; after 5 mrad only up to 0.80; after 3 mrad at no ratio tried at all.

Tested in A wall that is allowed to lift, at the figure it turns on · the essays about rocking.

A machine block is checked for sway and for rocking separately, and a machine kept clear of both frequencies is clear of resonance.

What decides it: The 150 tonne block 5 m across and 2 m deep has separate frequencies of 11.2 Hz in sway and 15.0 Hz in rocking. Because its mass sits above its base it has modes at 9.9 and 20.9 Hz instead, one below both and one above both, and at 8.9 Hz — more than a fifth below both checks — it amplifies 2.35 where the checks predict 1.71 and 1.51.

Tested in The frequency below both checks, at the figure it turns on · the essays about vibration isolation.

Tying a long stay to its shorter neighbours lifts its frequency out of the range that excites it.

What decides it: With the line free at its ends, the first frequency of stays 180, 150 and 120 m long rises from 0.73 Hz to 0.82 Hz and stops there however stiff the ties are made — short of the 1.01 Hz of the shortest stay, which no free line can pass. Carried to the deck at both ends, the same line reaches 1.09 Hz.

Tested in The line of ties that stops short, at the figure it turns on · the essays about cable dynamics.

A spring and dashpot in the gap models an impact that loses energy.

What decides it: It loses exactly the energy it is set to, and it ends every contact pulling the two buildings together: 3.5 MN of tension after a peak of 16.8 MN on one collision, and 34 MN of tension in a whole history with a stiffer contact — a force two faces touching cannot carry.

Tested in The force that belongs to the model, at the figure it turns on · the essays about pounding.

A grandstand checked empty is checked on its worst case, because spectators only add damping.

What decides it: Seated spectators of half the stand's modal mass lower its worst response to 40 jumpers from 2.54 to 0.49 m/s². But from 1.64 to 2.32 Hz the occupied stand responds more than the empty one, 1.6 times as much at 2.0 Hz, because they have split it into modes at 3.94 and 7.61 Hz and the lower one sits on the second harmonic of the jump.

Tested in The crowd that is also the structure, at the figure it turns on · the essays about floor vibration.

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 essays about truss.

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 essays about gust response.

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 essays about effective width.