What is refuted here — page 9

A body that is pushed until it slides moves in the direction it was pushed.

What decides it: It moves along the whole tangential demand, which is the push added to whatever the contact was already resisting. A block on a slope using three-quarters of its grip, pushed straight across the slope, leaves at 41.4 degrees from the fall line when it goes — not at 90.

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

When a sliding bearing lets go, the force it passes to the pier falls to the kinetic friction.

What decides it: It falls past it. The pier springs back under the kinetic friction, overshoots the level that friction would hold it at by as much as it started above it, and grips again below. A bearing carrying 3000 kN at 0.05 static and 0.03 kinetic breaks away at 150 kN and is left at 30 kN, not 90 — a swing of 120 kN against a drop of 60.

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

A stiffer pier holds a sticking bearing more firmly, so it suffers less from stick-slip.

What decides it: The swing does not contain the stiffness. The same bearing on a pier of 80 kN/mm swings by the same 120 kN as on one of 20 kN/mm; what the stiffness changes is the jump, 1.5 mm against 6 mm, and how often it comes, thirteen times in twelve hours against three.

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

A bearing that has stopped sliding under a repeated movement is back in the state it started from.

What decides it: It stops sliding by locking a force into the pier. Cycled 5 mm back and forth under a bound of 60 kN, the bearing slides once on the first movement and never again, and every time the deck returns to its starting position the pier is carrying 40 kN the other way — a force no load put there.

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

A body checked against overturning in the two principal directions of its drawing has been checked in every direction.

What decides it: A body tips about an edge of its base, and a tripod's edges are not aligned with any pair of axes. The same 16 kN at 3 m on a 60 kN stand with legs 1.5 m from its centre tips it pushed toward the gap between two legs, at a factor of 0.94, and leaves it standing at 1.88 pushed toward a leg.

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

A bolt tightened to its specified torque carries its specified preload.

What decides it: The torque fixes the product of the preload and a friction bracket, not the preload. A torque of 711 N·m set to give 188 kN in an M20 grade 10.9 bolt at a coefficient of 0.14 gives 150 kN at 0.18 and reaches the bolt's 220 kN yield force at 0.10; over 30 per cent of that range the bolt is below the 171 kN minimum a slip check assumes.

Tested in The torque that goes into the thread, at the figure it turns on · the essays about Slip-critical.

A tension-control bolt measures the tension it installs.

What decides it: Its spline shears off at a torque, so it carries exactly the torque method's dependence on friction, over the narrower range a factory lubricant delivers. Between coefficients of 0.12 and 0.16 the same bolt receives 167 to 216 kN.

Tested in The torque that goes into the thread, at the figure it turns on · the essays about Slip-critical.

Under a moment, a base plate's compression block sits at the plate's edge, which gives the holding-down bolts their longest lever arm.

What decides it: Only a rigid plate can put it there. A 20 mm plate delivers bearing no more than 43 mm past the compressed flange, so under 300 kN and 120 kN·m the compression acts 124 mm from the column's centre rather than 223, the lever arm is 314 mm rather than 413, and the bolts carry 264 kN rather than 128.

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

A doubler plate on the web is a shear repair.

What decides it: A coped tee is mostly web, with its surviving flange at the far end from the critical fibre, so thickening the web raises the section modulus as well. An 8 mm doubler on a 9 mm web takes the same end's flexure from 0.46 to 0.27, its shear from 0.41 to 0.22 and its tear-out from 0.57 to 0.30.

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

A wall that fails the re-centring check is left standing at a large lean after the earthquake.

What decides it: A slow, one-directional unloading leaves the weakly re-centred wall 9.2 mrad off upright. Rocking both ways under a 1.5 g pulse cycles its bars back, and the same wall comes to rest 0.5 mrad off upright — near it, and unable to stand exactly on it.

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

Mounting a machine low on its block, near the base, avoids exciting the rocking.

What decides it: The lower mode turns about a point 3.69 m below the base, so a force anywhere on the block drives it. With the machine on the base the displacement still peaks at 9.3 Hz, 2.51 µm per kN against 5.94 on top. Only the upper mode's centre, 1.40 m up, lies on the block.

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

Cross-ties give a group of stays a sparse spectrum that can be designed around.

What decides it: Each tie or anchor is one spring and can move at most one frequency past any other. Below 3.5 Hz the three stays have 11 frequencies between them; tied together they have 10, and with the line anchored 10.

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

Which contact law is chosen changes how far two pounding buildings move.

What decides it: Under all four laws, each at two contact constants a hundred times apart, the first building's largest displacement stays between 226 and 244 mm and the second's between 400 and 437 mm. Over the same eight analyses the largest contact force runs from 27 to 225 MN.

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

Forty people jumping together deliver forty times one jumper's harmonics.

What decides it: At 50 ms of timing scatter about a 2.25 Hz beat, each jumper keeps 79 per cent of the first harmonic, 40 per cent of the second and 16 per cent of the fourth, which is the floor that out-of-step jumpers adding as noise leave for a crowd of 40.

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

A stay cable can only be driven by a force across it at one of its own frequencies.

What decides it: A deck moving ±10 mm vertically at the anchorage of a 120 m stay puts no force across it and swings its tension by 1.39 per cent. At 2.01 Hz, twice the stay's own 1.01 Hz, the stay's first mode grows tenfold every 147 s from any small start, with only its own 0.1 per cent damping to stop it.

Tested in The stay shaken along its own length, at the figure it turns on · the essays about cable dynamics.

Once a parametric vibration starts, it grows until something breaks.

What decides it: Only in the linear equation. A stay swinging 362 mm has raised its own frequency as far as 23 kN of extra tension would, which is the whole half-width of the band the growth needs, and the amplitude levels off there.

Tested in The stay shaken along its own length, at the figure it turns on · the essays about cable dynamics.

Linking two adjacent buildings protects both of them.

What decides it: It brings them toward one displacement. The softer building's falls from 419 to 280 mm and the stiffer building's rises from 220 to 269 mm, 22 per cent more; with the softer building's period two and a half times the stiffer's, the stiffer building moves 38 per cent more than it would alone.

Tested in The damper that ends up as a joint, at the figure it turns on · the essays about pounding.

A link that stops two buildings colliding has to be as strong as the collision it replaces.

What decides it: The collisions it replaces peak somewhere between 27 and 225 MN depending on the contact law assumed. The link that keeps the gap carries at most 1.6 MN, and 2.9 MN at the largest period ratio drawn.

Tested in The damper that ends up as a joint, at the figure it turns on · the essays about pounding.

Any gap can be given a filler soft enough to keep a collision force low.

What decides it: The crush force times the crushable depth has to cover the energy of the approach. At 2.85 m/s a filler filling the whole of a 50 mm gap needs at least 25 MN, and bringing the largest force down to 10 MN takes 200 mm of filler in a 210 mm gap.

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

More prestress in a rocking wall's tendon gives it more re-centring margin after an earthquake.

What decides it: Only until the tendon yields. Past its yield it keeps its yield force less 20 kN for every milliradian: prestressed to 300, 600 or 900 kN, the same tendon keeps 170 kN after 60 mrad, and with bars of 400 kN all three walls stop re-centring past the same 48.5 mrad.

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

A rocking wall that re-centres after one earthquake will re-centre after the next.

What decides it: A wall with a re-centring ratio of 1.25 comes out of a 1.5 g pulse with its tendon slack, resting 2.5 mrad off upright. The same pulse again takes it to 71 per cent of its toppling angle and leaves it 94 mrad off upright; with a tendon that never yields it rests upright both times.

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

Soil against the sides of a block, acting below its centre of mass, reinforces the coupling between sway and rocking.

What decides it: It lifts the height at which the block's horizontal stiffness acts toward the centre of mass, and the coupling is set by the distance between the two. Fully embedded in backfill as stiff as the ground, that distance falls from 1.60 m to 0.60 m.

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

Mohr's correction diagram is an unavoidable step of the Williot–Mohr method.

What decides it: Started from the central vertical, which a symmetric truss under a symmetric load cannot turn, the construction needs no rotation at all. Every joint is out by the same translation — the mid-span joint's own movement, 2,019.41 down and 307.50 across — and Mohr's truss shrinks to a single point.

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

The whole deflected shape of a truss needs one unit-load analysis for every joint.

What decides it: One construction gives both movements of all sixteen joints. The seven bottom-chord joints free to move, checked by seven separate unit-load sums with seven sets of virtual member forces, agree with it to the last figure: 778.49, 1,409.56, 1,855.70 and 2,019.41 at mid-span.

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

A fixed-ended beam that is soft in shear carries smaller end moments than one that is not.

What decides it: Under a uniform load its end moments are wL²/12 at every slenderness, because symmetry makes them equal and the shear strain then integrates to nothing between the ends; only its deflection grows, by 1.78, 4.11 and 13.44 times at span-to-depth ratios of 4, 2 and 1. Under a point load at a quarter of the span the end nearer the load does fall, from 0.141 PL to 0.105 at a ratio of 1, and the far end rises from 0.047 PL to 0.082.

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

Deepening a fixed-ended beam at its supports reduces the bending moment there.

What decides it: It raises it. Haunched to twice its depth over a fifth of the span at each end, a uniformly loaded member carries 0.099 wL² at its ends instead of 0.083, and its mid-span moment falls from 0.042 to 0.026 wL². A deeper end is a stiffer end, and a fixed-ended member's moment goes to where it is stiff. The haunch still helps, because the section's capacity grows faster than the moment it attracts.

Tested in The fixed-end moment is a column stress, at the figure it turns on · the essays about Moment-area.

The elastic centre of a member haunched at one end lies toward its deep, stiff end.

What decides it: It lies away from it. The analogous column is as wide as 1/EI, so it is narrowest where the member is deepest. Haunched at its left end to 2.5 times its depth over a quarter of its span, the member's elastic centre is 0.587 of the span from the left, and at three times the depth 0.595.

Tested in The fixed-end moment is a column stress, at the figure it turns on · the essays about Moment-area.

A member's stiffness and carry-over factor are the same whichever end is turned.

What decides it: Only for a symmetric member. Haunched at its left end to twice its depth, a member is 7.394 EI/L stiff at that end and 4.603 at the other, and carries 0.463 of a moment over from left to right but 0.743 from right to left. What is equal is the product — 3.42 both ways — which is Maxwell's reciprocal theorem.

Tested in The fixed-end moment is a column stress, at the figure it turns on · the essays about Moment-area.

The ranges between successive peaks and troughs of a stress record are its stress cycles.

What decides it: On the bridge record drawn, the largest range between two successive reversals is 38.1 N/mm² and the heaviest vehicle's loop is 74.3, because the deck's ringing puts reversals on every rise and fall. Counted that way the record's cube-weighted damage is 23 per cent of rainflow's, and a category 71 detail at 800 vehicles a day lasts 424 years instead of 34.

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

Cutting a long record into short files and counting each one changes nothing, because every cycle lies inside some file.

What decides it: A loop lies between its two reversals, and a cut between them destroys it. Counted file by file the record keeps 90 per cent of its damage in files of 3.3 seconds, 62 per cent in files of 1.6, and 7 per cent in files of 0.41. Carrying the count's stack from each file into the next recovers exactly 100 per cent.

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

A 60-minute fire rating means the member survives a fire that burns for an hour.

What decides it: In a compartment whose fire tracks the furnace curve for sixty minutes and then cools, 15 mm of board lets the member reach 714 °C, 88 minutes in, well after the flames began to die. Holding it below its critical 558 °C through the whole fire takes 31.4 mm of board. The 60-minute rating needs 20.9.

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

Better-insulated walls make a compartment safer in a fire.

What decides it: Walls that cannot soak up heat leave it in the gas. With the same fuel and the same opening, linings of thermal inertia 400 give a fire of 1,265 °C against 944 °C for 1160 and 767 °C for 2200, and the protected member peaks at 857 °C against 714 °C and 629 °C.

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

A slab creeps according to its thickness, whatever it is cast on.

What decides it: It creeps according to the surface it dries through. A 200 mm slab open on both faces has a notional size of 200 mm and a fifty-year coefficient of 2.35. Cast on a steel deck, which seals its underside, its notional size doubles to 400 mm and its coefficient falls to 2.12 — the same as an 800 mm square column's.

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

Bigger members creep less because they are better cured or stronger.

What decides it: The concrete is the same; what differs is how fast moisture leaves it. In saturated air, where nothing dries, members of notional size 50 mm and 1,000 mm have the same coefficient, 1.28. At 40 per cent humidity the thin one reaches 3.31 and the thick one 2.01.

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

A stiffer joint lets a beam carry more load.

What decides it: The collapse load is 8(Mp + Mj)/L², and the joint's stiffness is not in it. A 9 m beam with a plastic moment of 522 kN·m on joints of resistance 261 kN·m collapses at 77.3 kN/m on joints of 2 EI/L and on joints of 25 EI/L alike. Stiffness moves the load at which the joints yield, and how far they must turn afterwards.

Tested in The joint that has to keep turning, at the figure it turns on · the essays about joint classification.

A weaker joint needs less ductility, because it carries less moment.

What decides it: It needs more. On joints at the rigid boundary of 8 EI/L, a joint of a quarter of the beam's plastic moment must rotate 19.8 milliradians plastically before the beam collapses, half 14.3, three quarters 8.7 and full strength 3.2. A rigid full-strength joint needs 12.7.

Tested in The joint that has to keep turning, at the figure it turns on · the essays about joint classification.

Rotation capacity matters only for joints nearly as strong as the beam.

What decides it: The joint as strong as the beam at the balanced stiffness needs none. The one that needs most is weak and stiff: at a quarter of the beam's strength and 25 EI/L it must rotate 21.4 milliradians, and a joint that can supply 15 fractures at 51.3 kN/m, 80 per cent of the collapse load.

Tested in The joint that has to keep turning, at the figure it turns on · the essays about joint classification.

A single-sided partial-penetration weld is as strong as a double-sided one with the same throat.

What decides it: The single-sided throat misses the plates' line of action. Joining 20 mm S355 plates at half their thickness, the double-sided weld carries 4,410 kN per metre and the single-sided one 1,103 elastically and 1,827 fully plastic — a quarter and two-fifths.

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

The unfused root of a partial-penetration weld matters only for fatigue.

What decides it: Under a single static pull, the eccentricity of a single-sided weld at half penetration puts its root at four times the mean throat stress and its face into compression. The unfused land is a notch, and the eccentricity loads it in tension before any cycle has been applied.

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

A doubler plate reduces the stress in a coped web by the same amount whenever it is welded on.

What decides it: Welded to a 457 mm beam coped 200 mm long while 60 per cent of its 300 kN reaction is already on it, an 8 mm doubler leaves the free edge of the web at 137 N/mm². Welded before any load, it would leave 97. As coped, the edge is at 164, so the repair has removed 40 per cent of the stress it would have removed from an unloaded beam.

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

Whether the beam is propped while the plate is welded makes no difference once the repair has been designed.

What decides it: Welded under 60 per cent of the reaction the repaired end governs at 0.39, in flexure. Propped, so the plate is in place before the reaction returns, it governs at 0.30. And at 60 per cent the web beside the new weld is carrying 99 N/mm², which the heated steel stops carrying once it passes 680 °C.

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

A body held by four forces cannot be solved by drawing, because four forces do not have to meet at a point.

What decides it: They pair off. For a beam on three links under a 10 kN inclined load, the load and the vertical link cross at P, (0, 5.20) m, and the other two links cross at Q, (6.00, 0) m. Two triangles on the direction PQ give the links 4.33, 6.12 and 0.67 kN, which is what the three equations of equilibrium give.

Tested in The line that pairs four forces, at the figure it turns on · the essays about graphic statics.

Different graphical constructions of the same problem give slightly different answers, so graphic statics is an approximation.

What decides it: The same beam has three Culmann lines, one for each way of pairing the load, and they cross the sheet in three different places. All three give 4.33, 6.12 and 0.67 kN. The construction is exact; a drawing's error is in its pencil, and a second pairing is how the pencil is checked.

Tested in The line that pairs four forces, at the figure it turns on · the essays about graphic statics.

A body with exactly as many supports as it has equations of equilibrium always has finite reactions.

What decides it: The count is satisfied whatever the geometry. Move the strut's line to within 0.04 m of the point where the other two lines cross and the same 10 kN load needs link forces of 511, 735 and 515 kN; through that point, no forces at all will balance it.

Tested in The line that pairs four forces, at the figure it turns on · the essays about graphic statics.

A graphical construction has arbitrary choices in it, so two drafters solving the same problem get two different answers.

What decides it: The pole is genuinely arbitrary and the answer is not. Polar distances of 45, 75 and 130 kN draw polygons 5.61, 3.37 and 1.94 m deep, sharing no vertex and no slope, and the first and last strings of all three cross on the vertical at 5.24 m — which is Σ P x / Σ P for the same loads, computed with no pole in it.

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

Finding reactions graphically needs the loads resolved and moments taken, the same as finding them by arithmetic.

What decides it: It needs one more line. The chord between the funicular's two ends is drawn, a ray through the pole is drawn parallel to it, and where that ray cuts the load line divides the 155 kN of load into 73.75 and 81.25 kN. No force was resolved and no moment was taken.

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

A pinned connection transmits force and no moment, which is what makes it a pin.

What decides it: It transmits the moment its friction can develop. A 300 mm pin at µ = 0.15 carries its reaction 22 mm off centre, so under 5 MN it delivers 111 kN·m to the member it is pinned to, and at µ = 0.4 — a pin left unlubricated for thirty years — 279 kN·m.

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

An unloaded, unrestrained slab carries no stress.

What decides it: It carries a self-balancing one. At 28 days the same slab has 5.28 N/mm² of tension at its drying face, 2.12 of compression inside it and 1.67 of tension at its base, and those integrate to zero force and zero moment. Nothing holds the slab and it is stressed anyway.

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

A moisture gradient is a temperature gradient with a different cause, so it produces the same stresses.

What decides it: It produces the same curvature and roughly half the stress, because it takes years and a temperature gradient takes hours. Computed at the short-term modulus the drying face carries 9.10 N/mm² at 28 days; computed with each depth at its own effective modulus, 5.42 — a relief of 40 per cent, rising to 68 at twenty years. The curvature moves by two per cent.

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

A structure's moments are decided by its finished geometry and its loads, so two identical finished structures under identical loads have identical moments.

What decides it: Two 15 m spans stitched at 60 days carry 155.2 kN·m over the middle support after twenty years; the same pair stitched at 240 days carry 82.3. The drawing of the finished structure is the same drawing and the load never changed — what differs is the day a crane was available.

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

The ageing coefficient χ is a property of the concrete, about 0.8, so the one-line shortcut is as good as the integral it approximates.

What decides it: It is a property of the problem. On one creep function the χ that reproduces the integral is 0.79 to 0.90 for a redundant that grows, and 1.33 for a strain held constant — a factor of 1.5, on the same concrete, between two questions asked of it.

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

Prestress loss is a property of the tendon and its anchorage, so it is the same for one stressing system wherever it is used.

What decides it: Two thirds of it belongs to the concrete. The same tendon at the same stress loses 214.6 N/mm² in a member of notional size 100 mm and 178.6 in one of 900 — and 229.4 at 40 per cent humidity against 126.7 at 100. Only the steel's own relaxation, about 32 N/mm², is unmoved by either.

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

The three long-term losses are separate mechanisms, so the total is their sum.

What decides it: Adding them gives 230.1 N/mm² and letting them interfere gives 190.8 — the sum is 17.1 per cent high. As the tendon loses force the concrete under it unloads, so it creeps less, so the tendon loses less; each loss removes part of what drives the other two.

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

A tributary division that closes exactly hands each column the load that arrives on it.

What decides it: It hands each column the load that stands over it, which is a different thing as soon as the floor is continuous. Two 8 m spans at 5 kN/m give 15.0, 50.0 and 15.0 kN at their three supports against tributary shares of 20, 40 and 20 — ratios of 0.75, 1.25 and 0.75. The three still add to 80 kN.

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

The discrepancy is a correction of a few per cent, so it can be absorbed in the load factors.

What decides it: It is a factor, and on a grid the two directions multiply. On a two-by-two plate the corner column collects 0.56 of its tributary load and the middle one 1.56 — a factor of 2.8 between two columns on one floor, both of which the tributary rule gives 1.00.

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

A bolt group under a moment behaves like a section: the neutral axis passes through its centroid, and the bolt forces follow from M y over sum y squared.

What decides it: That is a section in which every fibre can pull and push. Bolts cannot push, and what pushes is the plate they are bolted through. Solving for where the plate bears puts the axis 40.2 mm above the compression edge rather than 150, and the top row at 24.6 kN rather than 50.0.

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

The centroid assumption is unsafe, because a real bracket pivots lower and that shortens the lever arm.

What decides it: It lengthens it. An axis lower down puts the compression further from the tension bolts, so the same moment needs less force: 50.0 kN at the centroid, 24.6 where the bearing puts it, 21.4 at the extreme of the plate's own edge. The centroid model is conservative on the worst bolt by a factor of 2.33.

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

A frame's redundants have to be solved as simultaneous equations, because each of them appears in every compatibility condition.

What decides it: They do if the origin is chosen carelessly. Measured from the analogous column's own centroid the cross terms vanish, and the two redundants of a symmetric portal come out as one division each — a direct stress of 33.8 kN·m and a bending stress whose gradient is the 6.4 kN thrust. The same frame by stiffness gives the same numbers to five figures.

Tested in The elastic centre is not on the frame, at the figure it turns on · the essays about Moment-area.

The elastic centre of a frame lies nearer whichever member is stiffer, because that member contributes more to the section.

What decides it: The section's width is 1/EI, so it is thin where the frame is stiff. Taking the beam from a fifth of the columns' stiffness to five times it moves the elastic centre from 0.63 m below the beam to 2.24 — away from the beam, the whole way.

Tested in The elastic centre is not on the frame, at the figure it turns on · the essays about Moment-area.

Where the columns of a portal reverse their bending depends on the ratio of beam to column stiffness, like everything else about a redundant frame.

What decides it: It does not move at all. The base moment is exactly half the knee moment, of opposite sign, at every stiffness ratio, height and span tried — so the point of contraflexure sits at a third of the height from the base, always. It is the carry-over factor of a prismatic member, arriving without being asked for.

Tested in The elastic centre is not on the frame, at the figure it turns on · the essays about Moment-area.