What is refuted here — page 24

A crushable filler turns the collision force into a design quantity.

What decides it: Only a filler that never becomes dense. In a 50 mm gap the one that never does crushes at 8 MPa, and it delivers 40 MN whether the concrete behind it is soft or a hundred times stiffer — more than the 35 MN the soft estimate of the bare contact gives.

Tested in The force a filler can promise, at the figure it turns on · the essays about pounding.

The tendon of a post-tensioned rocking wall can be treated as elastic in a large earthquake.

What decides it: At the 147 mrad an elastic tendon let the wall reach under a 1.5 g pulse, the tendon would carry 3,540 kN — two and a half times the 1,370 kN at which the 820 mm² of strand its stiffness implies yields. Given that yield, the wall with bars reaches 90 per cent of its toppling angle and the wall with the tendon alone overturns.

Tested in The tendon that forgets its prestress, at the figure it turns on · the essays about rocking.

Setting a machine block into the ground reduces its vibration.

What decides it: Only through the side soil's radiation. Set half its depth into the ground, a 150 tonne block 3.2 m across peaks at 11 µm per kN if its sides radiate as the side-layer model says, and at 169 µm per kN — at 8.4 Hz rather than the surface block's 5.5 — if they do not.

Tested in The damping that comes through the sides, at the figure it turns on · the essays about vibration isolation.

A block set its whole depth into the ground can be designed with the separate sway and rocking checks.

What decides it: The coupling is weakened, not removed. Fully embedded, the checks give 14.6 and 14.3 Hz and the lower mode is at 13.2 Hz, 8 per cent below both; however stiff the backfill, the centre of stiffness only approaches the centre of mass.

Tested in The damping that comes through the sides, at the figure it turns on · the essays about vibration isolation.

A Williot diagram gives the displacements of a truss's joints directly.

What decides it: Started at the pin with the first bottom-chord member held level, the construction puts the roller 6,227.94 units above the support that holds it and the mid-span joint 1,094.56 units above the pin, on a truss that sags. Every member's extension is honoured and the drawing is still out by a rigid rotation, which is 3.03 times the largest true movement of any joint.

Tested in The drawing that is right except for a rotation, at the figure it turns on · the essays about truss deflection.

Shear deformation changes how far a beam deflects but not its bending moments.

What decides it: True of a statically determinate beam and false of a redundant one. A 300 × 600 steel rectangle fixed at one end and propped at the other carries 0.105 wL² at its wall at a span-to-depth ratio of 2 rather than the 0.125 of bending theory, and 0.070 at a ratio of 1, where the span moment has risen from 0.070 wL² to 0.092 — 31 per cent more than the calculation without shear gives.

Tested in The shear that moves the moments, at the figure it turns on · the essays about shear deflection.

The carry-over factor of a prismatic member is one half.

What decides it: It is (2 − φ)/(4 + φ), with φ = 12EI/(GAs L²). One half is the value for a member rigid in shear. It is zero at φ = 2, which a 300 × 600 steel rectangle reaches at a span-to-depth ratio of 1.25, and −0.5 at φ = 8, where turning one end turns the far end's moment the other way.

Tested in The shear that moves the moments, at the figure it turns on · the essays about shear deflection.

Peak counting is a slightly conservative approximation to rainflow.

What decides it: It is an upper bound, not an approximation, and how far above the answer it sits depends on the record. Pairing the highest peak with the lowest valley whether or not they ever made a loop gives this record a Miner sum of 4.36 against rainflow's 2.97, and a life of 23 years against 34.

Tested in Which reversal closes the loop, at the figure it turns on · the essays about fatigue.

The hottest fire a compartment can produce is the worst one for its structure.

What decides it: For bare steel, yes: its peak rises from 839 °C with an opening factor of 0.02 to 1,074 °C at 0.10. For the same member protected, it runs the other way, from 734 °C to 593 °C, because a hot short fire is out before the board has let its heat through and a cool long one is not.

Tested in The fire that goes out, at the figure it turns on · the essays about fire.

A concrete's creep coefficient is a property of the mix.

What decides it: Part of it is. The part drying does not add is 1.28 after fifty years for this concrete in any member at all. The rest belongs to the member: in air at 50 per cent humidity the same mix, poured on the same day, reaches 2.45 in a 150 mm slab open on both faces and 1.93 in a 1.2 m transfer beam.

Tested in The creep that belongs to the member, at the figure it turns on · the essays about creep.

A partial-penetration butt weld is designed on the fillet-weld rules, so the directional criterion sets its strength.

What decides it: It is designed on those rules, and for a butt throat in transverse tension it is their second condition, σ⊥ ≤ 0.9fu, that governs, in every grade: 324 N/mm² against the criterion's 450 in S235, 441 against 544 in S355, and 486 against 540 in S460.

Tested in The throat that misses the load, at the figure it turns on · the essays about weld strength.

A repair welded under load is only worth the live-load share of its benefit, whatever check is being made.

What decides it: For the checks that read a stress, roughly so: flexure runs from 0.27 with no load present to 0.46 with all of it, and shear and local buckling move the same way. Tear-out does not move at all. A block tearing out of the web is a plastic mechanism, and the doubler takes it from 0.57 to 0.30 at every share.

Tested in The plate that arrives after the load, at the figure it turns on · the essays about coped beam.

The polar distance is a drawing convenience, so it can be picked to suit the paper and forgotten.

What decides it: It can be picked to suit the paper, and it cannot be forgotten, because it is the scale of every moment the drawing will be asked for. The depth under the middle load runs from 15.58 m at a polar distance of 15 kN to 0.97 m at 240, and the product of the two is 233.75 kN·m at every point on that line.

Tested in The pole decides the drawing, not the answer, at the figure it turns on · the essays about graphic statics.

Friction in the pins is a second-order effect, so a construction that ignores it is accurate to within a per cent or so.

What decides it: It depends on what is being asked of the drawing, not on the pins. The same bracket at the same pins gives bands of 1.9, 1.9 and 12.3 per cent on its three links, because the width of a band is set by the geometry and the force in the link is not — so the link carrying least has the largest uncertainty, every time.

Tested in The pin that is not a point, at the figure it turns on · the essays about graphic statics.

An arrangement whose supports nearly meet at a point is a bad idea because the forces are large.

What decides it: The forces are large and they are also unknown. Moving the strut's line from 4.24 m to 0.06 m from the crossing takes its force from 6.1 kN to 459, and takes the widest band from 12.3 per cent of its own force to 1,962 — so the drawing stops being able to state a force at all before the structure stops standing.

Tested in The pin that is not a point, at the figure it turns on · the essays about graphic statics.

A member's creep and shrinkage are described by a coefficient and a strain, so the answer for a slab is those two numbers applied to it.

What decides it: Those two numbers are the mean of a profile, and the profile has a slope in it. A 200 mm slab drying through its top face has a mean shrinkage of 185 microstrain at a year and a curvature of 2.43 per kilometre at the same moment, which lifts the edges of a 6 m panel 10.9 mm above its middle. The mean is in the coefficient and the curvature is not.

Tested in The slab that dries from one face, at the figure it turns on · the essays about creep.

Creep takes a structure made continuous late to the moments it would have had if it had been built monolithically.

What decides it: It takes it part of the way and stops. Stitched at 60 days the redundant reaches 46 per cent of the monolithic value and stays there; stitched at 28 days, the earliest the load allows, it reaches 76. The elastic rotation happened before the joint existed and no amount of creep can recover it.

Tested in The support that had no moment when it was cast, at the figure it turns on · the essays about creep.

Losses are a long-term matter, so a check at transfer and a check at fifty years are enough.

What decides it: Half the loss has arrived by 119 days and nine-tenths by eight years, so the state at a year — 131.5 N/mm² of loss against an eventual 190.8 — is neither of the two checked cases and is when most structures are first loaded in anger.

Tested in The prestress the member takes back, at the figure it turns on · the essays about creep.

A bigger building averages the error out.

What decides it: It shrinks it and never removes it. At two spans the first interior support is at 1.25; at ten it is still 1.13, and the end support has only gone from 0.75 to 0.79. The middle of a long run converges to 1.00 and the ends never do, because there is no span beyond them to hog over the first interior support.

Tested in The column given more than its rectangle, at the figure it turns on · the essays about tributary area.

The difference between the assumptions is an academic one, since the bolts are checked on an interaction that is dominated by shear anyway.

What decides it: On this bracket the shear alone is 0.54 of capacity and the three assumptions take the combined check to 0.80, 0.65 and 0.67. A spread of 0.15 of a utilisation is a bolt grade, and it is decided by a sentence in a calculation rather than by anything on the drawing.

Tested in The bolt group has no neutral axis, at the figure it turns on · the essays about bolt group.

A doubly symmetric plate girder in bending has a web with ψ = −1, so its buckling coefficient is 23.9.

What decides it: Only while the whole web is working. On an 1800 by 12 mm web between 450 by 30 flanges, the reduction removes 278 mm from the compressed half, the neutral axis drops 42.5 mm, and the section settles at ψ = −0.910 and k = 21.63. The value 23.9 belongs to a section the calculation has already stopped describing.

Tested in Classified by a gradient it does not have, at the figure it turns on · the essays about plate buckling.

The group effect is a property of the bolt spacing, so closer rows are always worse.

What decides it: It is a property of the spacing and of which mode governs. At the same 90 mm pitch the group delivers 57 per cent of the sum on a 12 mm plate, 82 per cent on a 20 mm plate and exactly 100 per cent at 30 mm and beyond, because once the bolts govern there is no fold to share.

Tested in The rows have to share one fold, at the figure it turns on · the essays about prying.

Preload is only worth having in a slip-critical joint, where friction is the mechanism.

What decides it: In tension it is worth a factor of 1,141 in fatigue life. The same ±20 kN swing reaches the bolt as 7.4 kN while the plates are clamped and 77.2 kN once they have parted, and an endurance curve of slope three cubes the difference.

Tested in The bolt that was already stretched, at the figure it turns on · the essays about prying.

The square root of the sum of squares is unconservative for closely spaced modes.

What decides it: It is unconservative for responses whose two modal contributions have the same sign and conservative for those whose contributions oppose. On a 24 m floor with its stiffness 0.98 m off its mass it gives 82 per cent of the base shear and 141 per cent of the base torque, from the same two modes.

Tested in The twist the combination rule invents, at the figure it turns on · the essays about mode shapes.

Whether a body overturns is decided by its base, its weight and the push, and the stiffness of its supports does not enter.

What decides it: True if the body does not lean. A 6,000 kN tank with its weight 15 m up on four pads 8 m apart, each settling 75 mm, tips under a push at 18 m of 1,240 kN along an axis, where rigid ground gives 1,333: at the moment of tipping the lean has moved the weight 0.28 m toward the edge it turns about, and the push has lost exactly the fraction s·h/a² = 0.07.

Tested in It tips inside its own hull, at the figure it turns on · the essays about overturning.

The method of sections needs a cut through no more than three members, and a frame without one cannot be solved that way.

What decides it: A K-braced tower face has no three-member cut anywhere, and is solved by sections all the same. The joint at the middle of each horizontal gives the two diagonals equal and opposite forces; the section then has three unknowns, and for panel 3 of an 18 m tower returns 20.83 kN in each diagonal, 26.1 kN of tension and 86.3 kN of compression in the legs, matching a whole-face joint solution.

Tested in The cut that needs a joint first, at the figure it turns on · the essays about method of sections.

An eccentric bolt group is checked for the direction the load is drawn in.

What decides it: Only if that is the only direction the load can take. Six bolts at 75 mm with the load through a point 150 mm across and 150 mm up from their centroid carry 198.5 kN straight down and 135.8 kN at 138 degrees, so a check drawn vertically overstates the bracket by 46 per cent.

Tested in The bracket pushed from the wrong side, at the figure it turns on · the essays about bolt group.

As the load's direction turns, the centre of rotation moves around the group on a closed path.

What decides it: It does not close. Twice a turn the load's line passes through the centroid, a pure translation satisfies equilibrium, and the centre of a translation is at infinity — so the locus is one open arc from infinity to infinity, traversed once every 180 degrees.

Tested in The better method flatters the worse layout, at the figure it turns on · the essays about bolt group.

The column analogy needs the elastic centre, and once the elastic centre is found the redundants are three divisions.

What decides it: Only for a section whose principal axes are the ones the redundants are written about. A portal with columns of 5 m and 3 m has a product of inertia of −18.4 against an Ix of 31.6, so the second and third conditions are coupled: solved as two divisions the left foot reports 17.0 kN·m and solved as a two-by-two it reports 27.8.

Tested in The axes that have to be turned first, at the figure it turns on · the essays about Moment-area.

A fatigue life is a property of the stress spectrum and the detail category.

What decides it: Not once a crack is the mechanism. The same spectrum on the same detail gives an unbounded life for a flaw under 0.26 mm, 341 years at 0.3 mm, 196 at 0.5 and 23 at 10 — and nothing in either the spectrum or the category records the flaw size.

Tested in The loops a crack grows on, at the figure it turns on · the essays about fatigue.

The fatigue thickness correction is a size effect of the usual kind — a larger piece is more likely to contain a large flaw.

What decides it: A statistical argument predicts nothing about the stress field, and the stress field is where most of the effect is. Integrating a crack from the same 0.2 mm flaw through a toe's magnified field gives a strength exponent of 0.13 against the design code's 0.2, with no statistics of any kind in it.

Tested in The rule that points sideways, at the figure it turns on · the essays about fatigue.

A corrosive environment is handled by a knock-down factor on the detail category.

What decides it: A factor scales the curve and leaves its shape alone. What corrosion removes is the shape: the two knees. On the same traffic the air curve gives 300 years, the same curve with its cut-off removed gives 256, and a single slope of three all the way down gives 148 — a change of a factor of two with no knock-down applied anywhere.

Tested in The cut-off belongs to the water, at the figure it turns on · the essays about fatigue.

Putting a pin at the crown makes an arch statically determinate, so its thrust is known exactly.

What decides it: Only if the pins are frictionless. With 300 mm pins at µ = 0.15 the construction has eight answers, and at a rise of a twentieth of the span they spread ±4.4 per cent either side of the frictionless thrust; at µ = 0.4, a pin that has not been greased for decades, ±11 per cent.

Tested in The arch that is only its three hinges, at the figure it turns on · the essays about graphic statics.

A crown hinge always costs an arch bending capacity, because it removes a restraint.

What decides it: Under the loads the rib is shaped for, it costs nothing: nine equal loads on a parabolic rib give thrusts of 100.0 kN with the hinge and 99.2 without, and the rib's bending is small in both. Under a single load at the quarter point it is 187.5 kN·m with the hinge against 166.3 without. The hinge matters in proportion to how far the loads are from the ones the rib was shaped for.

Tested in The polygon that runs out of freedom, at the figure it turns on · the essays about graphic statics.

A careful graphical construction is accurate to about the width of the pencil line.

What decides it: Only where its lines cross at a large angle. Two bands of one pencil width crossing at a right angle meet in a parallelogram 1.4 widths long; at 60° it is 2.0 and at 12° it is 9.6, growing as one over the sine of half the angle. The accuracy belongs to the geometry being drawn, and the same hand is worth ten times less at 12° than at 90°.

Tested in How wrong a drawing is, at the figure it turns on · the essays about graphic statics.

Mode shapes are more sensitive to damage than frequencies, so measuring them solves the problem of finding it.

What decides it: The drift is about ten times as sensitive, but it is the difference of two measured floor movements. With every floor measured to 1 per cent of the roof's movement it reliably finds a 10 per cent loss in one storey of ten; to 0.2 per cent, in nine.

Tested in What a mode shape notices that a frequency does not, at the figure it turns on · the essays about mode shapes.

Making a machine block several times heavier than its machine is the conservative choice.

What decides it: It lowers the natural frequency, which is its purpose, and a lower frequency sits under the cut-off of a deeper layer. The 150 tonne block loses its vertical radiation on layers shallower than 4.5 m; five times as heavy on the same footing, it loses it on layers shallower than 11.7 m.

Tested in The rock that sends the waves back, at the figure it turns on · the essays about vibration isolation.

A soft spring mount that isolates a rotating machine will isolate a hammer at least as well.

What decides it: It lowers the force of each blow, from 1.32 MN on 12.5 Hz soil to 0.42 MN on 4 Hz springs, and lets the block move more, 1.42 mm to 4.42. But at 5 per cent damping a train of blows at 120 a minute builds the springs' movement to 2.14 times one blow's, because 120 blows a minute is every second cycle of a 4 Hz block.

Tested in The blow that has no frequency, at the figure it turns on · the essays about vibration isolation.

A truss sized so every member works at the same stress is a strength design, and its deflection is whatever it happens to be.

What decides it: For the same volume of steel, an eight-panel Pratt truss sized fully stressed deflects at mid-span 1.044 times as much as the stiffest division possible; with equal areas, 1.39 times. A Howe truss gives 1.050 and 1.35, a Warren 1.026 and 1.40.

Tested in The truss that is stiff by accident, at the figure it turns on · the essays about truss deflection.

A frame that has as many unknowns as equations and a non-zero determinant is rigid, and its computed forces describe it.

What decides it: Two bars of 200 MN axial stiffness over an 8 m span, meeting at a joint raised 3 degrees and loaded with 50 kN, pass both tests. Each carries 478 kN, and the joint sags 183 mm under the load on a rise of 210 — 87 per cent of the geometry the forces were computed on.

Tested in Rigid by every test, and still folding, at the figure it turns on · the essays about determinacy.

The quarter-point formula for the moment-gradient factor is conservative, because it was calibrated to err on the safe side.

What decides it: It is conservative on straight-line diagrams, by up to 21 per cent. For a fixed-ended beam under a central point load its quarter-point readings are zero and it gives 1.923, against an exact 1.723: 12 per cent on the unsafe side.

Tested in The formula that reads four numbers, at the figure it turns on · the essays about moment gradient.

A tapered plate should be cut to the beam's rotation on the day it is set on its bearing.

What decides it: Cut to the rotation on the day the beam is set, the plate leaves the bearing anything from 9.4 milliradians to permit, allowance included, for a beam set at three days, to 6.8 for one set after a fortnight; cut to the middle of the range in service it never leaves more than 7.9. A beam set early is matched to a rotation it then leaves, because most of its creep is still to come.

Tested in The angle made in the casting yard, at the figure it turns on · the essays about end rotation.

With plates of unequal thickness the thicker plate is so lightly stressed that its bending can be ignored.

What decides it: The thin plate governs throughout. But the thick plate's peak stress is 1.24 times what a joint that could not rotate would give it at twice the thickness and 1.36 times at three, and it reaches 0.63 of the thin plate's stress against the 0.30 that calculation predicts.

Tested in The thicker plate takes the bend, at the figure it turns on · the essays about single lap.

An anvil on a pad tuned to the foundation acts as a tuned mass damper for the block.

What decides it: Not for a blow, which enters the anvil rather than the block: tuned and undamped, the pad moves a spring-mounted block 7.60 mm per blow against 4.47 on an ordinary pad. Tuned to 0.85 and damped at 0.30, it moves the block about as much as an ordinary pad on one blow (4.61 mm), but under a train of blows its worst steady peak is 6.9 mm against 16.4 — at an anvil movement of 16.9 mm per blow.

Tested in The pad that makes the blow worse, at the figure it turns on · the essays about vibration isolation.

The force a restraint must carry is a fixed share of the force in the member it restrains.

What decides it: With frames of 1,400 N/mm, the frame force is 0.95 per cent of the chord force at a fifth of the chord's critical load, 1.20 per cent at half and 2.35 per cent at eight-tenths. The share is set by how near the chord works to buckling.

Tested in What the frames hold is the bow, at the figure it turns on · the essays about continuous restraint.

In a bolt group carrying shear and tension together, the check is made on the bolt with the largest shear, or equivalently on the bolt with the largest tension, since the same corner bolt is worst at both.

What decides it: True for a load straight down: at 179 kN the top right bolt carries the most shear, 90 kN, and the most tension, 38. At 60° from straight down the most shear is on the top right bolt and the most tension on the top left, and it is the top left that reaches its interaction first, at 227 kN, with the top right at 0.97.

Tested in The bracket with no worst bolt, at the figure it turns on · the essays about bolt group.

Linear elastic fracture mechanics is valid wherever the plastic zone it predicts is small beside the crack.

What decides it: Irwin's estimate of the plastic zone at the transition crack length is 13 mm beside a 20 mm crack, which looks small enough. Dugdale's strip at failure there is 49 mm, 2.4 times the crack, and it is longer than the crack for every crack below 36 mm.

Tested in The crack between the two checks, at the figure it turns on · the essays about fracture.

Closing an open channel into a box moves its shear centre from outside the web to the middle of the section, so a boxed channel can be loaded through its web without twisting.

What decides it: Only if the closing plate is thick. For a 300 by 100 mm channel of 10 mm plate, a closing plate of 0.1 mm leaves the shear centre 31.2 mm outside the web against 33.3 open, three per cent of the way; it is half way at 0.28 of the channel's thickness and reaches the web at 0.22.

Tested in The plate that closes the circuit, at the figure it turns on · the essays about shear centre.

An active damper is always better than a passive tuned mass, because it is not limited by tuning or by the size of its mass.

What decides it: With its gain bounded at 0.30 of critical, velocity feedback reaches a peak amplification of 1.7 with no delay against a 3 per cent passive mass's 8.2. At a fifth of the period it reaches 15.5, worse than a 1 per cent passive mass's 14.2, and at a quarter 44.7 against the bare mode's 50.

Tested in The actuator that arrives late, at the figure it turns on · the essays about tuned mass damper.

Because the work-equivalent load vector is the correct one, it gives the most accurate deflection on any mesh.

What decides it: On a 4 m wall of one serendipity element under its own weight it gives 0.799 of the fine-mesh tip deflection, where an eighth of the weight at each node gives 0.996. What it guarantees is a side: it is below the answer on every mesh and rises toward it, while the equal split crosses the answer at two elements, 1.005.

Tested in The corners the pressure pulls up, at the figure it turns on · the essays about force couple.

An intermediate pier of a multi-span stressed ribbon carries only the small difference between the spans' thrusts, so it can be slender and its stiffness does not matter.

What decides it: The difference is not small under live load: 20 kN/m on one of two 100 m spans asks a rigid pier for 9,440 kN. A pier of 200,000 kN/m carries 4,953 kN of it and lets the loaded span sag 358 mm instead of 195; a very soft pier carries almost nothing and lets it sag 520 mm while the other span rises 393.

Tested in The pier that carries only the difference, at the figure it turns on · the essays about stressed ribbon.

A pipe fixed to the floors of a building is designed for earthquake by giving it the floor's acceleration, from the floor spectrum at its own period.

What decides it: That prices only its inertia. Anchored against rotation at every floor of an eight-storey building drifting two thirds of a per cent, a water-filled DN100 riser carries 131 N/mm² from the drift and 4 N/mm² from its inertia; the drift stress passes the yield of grade B pipe, 241 N/mm², at DN200.

Tested in The pipe held at every floor, at the figure it turns on · the essays about floor spectrum.

Everything spanning between two floors is racked through the drift of the storey it sits in.

What decides it: Only if it is held against rotation at both floors. The same DN100 riser held in line but free to turn at every floor bends only through the change in drift from one storey to the next, and never exceeds 3 N/mm² in the building that puts 131 N/mm² into it anchored.

Tested in The pipe held at every floor, at the figure it turns on · the essays about floor spectrum.

A structure on soft clay is checked against tipping for its finished weight, with the ground's long-term strength, because the ground only gets stronger under load.

What decides it: It gets stronger after the load has arrived, not with it. A 15,000 kN tower on clay of 35 kPa that will consolidate to a factor of 1.93 has 1.33 on the day it is finished if it was built in a year, and 0.98 if it was built in a week.

Tested in The ground that arrives after the weight, at the figure it turns on · the essays about overturning.

A single-sided partial-penetration weld keeps about a quarter of a double-sided weld's capacity at half penetration, so it should not be used for primary welds.

What decides it: A quarter straight across, where the eccentricity loads its root. At half penetration of a 20 mm plate it keeps 0.40 at 45°, 0.86 at 75° and all of it straight along the weld, where the shear has no lever arm about the weld's axis.

Tested in The shear that helps a butt weld, at the figure it turns on · the essays about weld strength.

The strip-yield model's correction to the simple check is at most 23 per cent, whatever the plate.

What decides it: Only at the thickness the toughness was measured at. With each plate's own toughness, the simple check using the 25 mm value overstates a 60 mm flange by up to 36 per cent and a 100 mm plate by 45; a 12 mm web by 14.

Tested in The toughness that belongs to the plate, at the figure it turns on · the essays about fracture.

Resizing a redundant truss until every member is at its allowable stress converges to the fully stressed design of that truss.

What decides it: It converges to a fully stressed design, and which one depends on where it started. From six starting areas for the counters, an eight-panel counter-braced Pratt truss reached six different designs — the first panel's counter from 0.40 to 11.19 — every one needing 1,200 units of steel and deflecting 24.0 at mid-span.

Tested in The truss whose forces follow its sections, at the figure it turns on · the essays about truss deflection.

A pile in layered ground can be designed with an equivalent uniform ground, a modulus averaged over the depth the pile is working in.

What decides it: For the head it can: weighted by the square of the pile's own deflection, the average gives the head deflection within 8 per cent for any thickness of soft or stiff upper layer. For the moment it cannot: with a soft layer two characteristic lengths thick over stiff ground the pile's largest moment is 427 kN·m and the averaged ground gives 308.

Tested in The layer a pile feels, at the figure it turns on · the essays about lateral pile.

A block on ground with a bearing limit tips at a definite push, set by the ground's strength and the block's weight and width.

What decides it: On ground that softens toward its limit without reaching it, a block whose weight is at its base has no peak push at all: at a lean of 0.2 radians the push is 801 kN and still rising toward the 813 kN limit. Its weight 8 m up, it peaks at 676 kN, at a lean of 19 thousandths.

Tested in The ground that never quite gives way, at the figure it turns on · the essays about overturning.

A sandwich panel's resistance to a local load is set by its core's crushing strength.

What decides it: Only when the face is thick enough. Under a strip load 10 mm wide, a steel-faced polyurethane panel with 0.5 mm faces yields its face at 3.4 N per mm of width, before its core crushes at 5.0; with 3 mm faces the core crushes first, at 18.8, and the face yields at 23.7.

Tested in The face that dents and the core that crushes, at the figure it turns on · the essays about sandwich section.