Structural Engineering & Statics
where the load goes
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Which failure arrives first — page 21

Essays 481 to 495 of 495 on this thread, in the same order.
The more of the load that comes back, the less of it the row can carry for ever. The largest load 20 screws leaning at 45° in a row 56 mm apart carries for ever, as a share of its forward peak, against the share of it that comes back the other way each cycle (0 the same load repeated, 1 fully reversed), for each screw carrying none, half and all of its thread's first loss into the other way. Carrying 0 per cent: 0.96 repeated, 0.96 half reversed, 0.82 fully reversed; carrying 50 per cent: 0.96 repeated, 0.94 half reversed, 0.79 fully reversed; carrying 100 per cent: 0.96 repeated, 0.92 half reversed, 0.76 fully reversed. Connections

The row that is pushed as often as it is pulled

A row of leaning screws in a shear wall is loaded one way by the wind and then the other, and a screw that leans is pulled along its axis one way and pushed the other. Pulled, its axial force clamps the joint and friction helps it; pushed, it prises the joint open and the friction goes. So the row has a weaker side, and under a reversing load it settles exactly as it settles under a repeated one — the reversal does not start the ratchet again — but at a ceiling the pushed side sets, 0.79 of the pulled peak against 0.97. And it settles with a gap in the middle that every reversal crosses carrying nothing.

 Spread over the span, the rotation gathers at the supports every drop shares. The plastic rotation each station of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ has gathered by the last drop, for drops always at mid-span, at 0.3 and 0.7 of the span in turn, and at random. Always at mid-span, the worst station holds 0.193 radians, at 3.0 m, and 3 stations more than a milliradian; at 0.3 and 0.7 in turn, the worst station holds 0.104 radians, at a support, and 4 stations more than a milliradian; at random, the worst station holds 0.128 radians, at a support, and 15 stations more than a milliradian. Dynamics

The weight that lands somewhere else

A beam that yields under a falling weight takes a permanent set from every drop, and if the weight always lands in the same place the set grows by the same amount every time. Let the weight land somewhere different each time — rocks on a canopy, crates on a deck — and the set still grows: the weight's energy fixes how much plastic work every drop does, wherever it lands. What the landing point decides is where the work goes. Half of every drop's hinge rotation goes under the weight and half to the supports, so wandering drops spread the first half along the span and leave the second half where it always was: the worst hinge turns half as fast, and it is at a support.

 A slow release moves the hedge up, and never past half the surge. The best share of the payload for the counterweight to balance, and the mast's design moment over the payload's at that share, against the release time the design can rely on, for the crane lifting 60 kN at 48.3 m, its mast and jib rocking backwards with a period of 2.0 s and a damping ratio of 0.02, with a hoisting factor of 1.20. Let go at once, 0.13 and 1.07; over 1.2 s, 0.36 and 0.84; over 6 s, 0.56 and 0.64. However slow the release, the share cannot pass 0.60, half the hoisting factor, because the payload's own moment is still let go even when its rebound is not. Equilibrium

The release the rope cannot slow

A crane whose counterweight knows the payload balances only a third of it, because a payload can be let go and the crane is then thrown backwards by whatever the counterweight was holding, and half as much again in the rebound. Slow the release and the rebound goes, so the hope was that a device on the rope could slow it. Nothing on the rope can: the rope holds only what the hook holds. How fast a load goes belongs to the payload, and what the crane owns is how fast it answers — its period and its damping. And with no rebound at all, the counterweight still should not balance more than 0.6 of the payload, because the payload's own moment is still let go.

 The first hinge comes latest where the beam's end and its span yield together. The load factor at which the first plastic hinge forms in the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, against the initial stiffness of its full-strength end plates (1.20 times the beam's plastic moment), softening as EN 1993-1-8 has it and kept linear. With softening, 3.39 at 100 EI/L, 3.78 at 25 and 3.91 at 3; kept linear, 3.32, 3.51 and 3.91. Stiff joints hinge first at the inner end and flexible ones in the span; the first hinge comes latest between, 4.52 at 10.0 EI/L with softening and 4.31 at 5.0 kept linear. Rigid joints give 3.25, and the beam collapses at 5.06 for every joint. Internal forces

The joint stronger than its beam

A full-strength joint never reaches its own resistance: the beam beside it hinges first. But an end plate softens from two thirds of its resistance, and a joint only a little stronger than its beam is already on the soft part of its curve before the beam yields. The softening sheds moment from the beam's end into its span, so it delays the first hinge — by 8 per cent for a joint at 1.2 times the beam's strength, 18 for one at exactly the beam's, and not at all from 1.5 on. It never moves the collapse. And too much shedding hands the first hinge to the span, so the first hinge comes latest at the stiffness where the end and the span yield together.

 Over many earthquakes the column line ratchets together or stops together. The compression over the squash load of the interior column (solid) and an exterior column (dashed) against the cycles, up to 400, for the frame of three ground-storey columns under floor loads of 15, 20, 25 kN/m on 6 floors; the dotted line is where the hinge starts to ratchet, 0.15. At 15 kN/m the interior column starts at 0.22 and the average is 0.12; after 400 cycles the interior carries 0.15 and the exterior 0.10. At 20 kN/m the interior column starts at 0.29 and the average is 0.16; after 400 cycles the interior carries 0.16 and the exterior 0.16. At 25 kN/m the interior column starts at 0.36 and the average is 0.19; after 400 cycles the interior carries 0.19 and the exterior 0.19. Where the average lies above the threshold the three end at the average and ratchet together; where it lies below, the interior column sheds load until it stops at the threshold and the exterior columns never start. Materials

The floor that cannot keep up with the column

A column whose hinges ratchet under an earthquake shortens every cycle, and in a frame it does not shorten alone: the floors it carries bend down towards it and lift its load onto the columns beside it. That ought to slow it and speed them, until every column shortens together. It does — over hundreds of cycles. Over the ten or twenty of an earthquake, the floors of a six-storey frame lift an eighth of the load off the interior column, it shortens nine tenths as far as it would alone, and the floors carry the whole difference as bending. The frame equalises its columns eventually, and pays for it in its floors.

 The flange is bent hardest where it is thinnest, not where it curls most. The transverse bending stress at the root of the inner flange, where it meets the web, along a knee curved to 1,000 mm through a quarter turn, its flanges 600 mm apart on a 10 mm web, carrying 500 kN·m all round the curve; the dotted line is a yield stress of 355 N/mm². Uniform, 269 all round. Narrowing, 283 at the rafter end and 207 at the column end, largest where λ is; thinning, 191 at the rafter end and 397 at the column end, largest where λ is too. Tapered both ways, 205 at the rafter end and 325 at the column end: largest at the column end, where λ is 0.94 against 1.14 at the rafter end, because there the flange's own stress has nearly doubled. Sections and stress

The flange bent hardest where it curls least

A curved I-beam's flange curls towards its web and is bent across its own width at the root, by a stress that rises with the curling parameter λ. A portal knee's flange is cut from plate and usually tapers, and a taper changes λ along the curve. So the obvious guess is that the transverse stress peaks where λ is largest. For a flange that only narrows, or only thins, it does. Taper it both ways and the guess fails: λ is smallest at the narrow, thin end, and that is where the flange is bent hardest, because the same force is carried there on half the steel. The force the changing width hands to the web, meanwhile, is a few newtons a square millimetre.

 Where real roots sit on the warping curve. The elastic critical moment of a 3.0 m cantilever of 457 mm universal beam with its tip load at the shear centre, with its root holding the lateral slope and resisting warping through a spring, against the spring as a multiple of EIw/L on a logarithmic scale — from 521 kN·m free to warp to 1,266 built in — with four real roots placed on it. Top storey, column ends: 2.5 times EIw/L, 791 kN·m, 36% of the way; 15 mm plate, column runs on: 5.9 times EIw/L, 945 kN·m, 57% of the way; 40 mm plate, column runs on: 6.8 times EIw/L, 970 kN·m, 60% of the way; 15 mm plate, column boxed: 19.4 times EIw/L, 1,124 kN·m, 81% of the way. Stability

The root that belongs to the column

A cantilever's root must hold its flanges against warping at about fifty times the cantilever's own warping stiffness to count as built in, and doubles its buckling moment if it does. A flush end plate is the usual root, so it ought to be the plate's thickness that decides. It is not. The flanges warp by turning in plan in opposite directions, the plate passes that to the column as two opposite torques, and an open column twists between them more easily than the plate bends: three quarters of the root's flexibility is the column. An ordinary root is worth about six times the cantilever's stiffness, halfway up the curve, and a thicker plate barely moves it. Boxing the column does.

 The dynamic ratio peaks where the modes couple, and falls where the static one climbs. The standards' ratio, the larger edge's displacement over the average of the two edges', for a 30 × 18 m floor swaying with a period of 0.5 s, against Ω, the torsional radius over the floor's radius of gyration, for natural eccentricities of 2.1 and 4.2 m: from a static push at the centre of mass (dashed) and from a response spectrum combined by CQC (solid); the dotted lines are 1.2 and 1.4. At 2.1 m, static 2.05 at Ω = 0.5 falling to 1.08 at 2; dynamic 1.08 at 0.5, highest at 1.42 near Ω = 1.15, and 1.10 at 2. At 4.2 m, static 2.46 at Ω = 0.5 falling to 1.15 at 2; dynamic 1.17 at 0.5, highest at 1.55 near Ω = 1.20, and 1.19 at 2. Structural form

The floor that twists in its own time

Seismic design standards flag a twisting floor by the ratio of its worst edge's movement to the average of its two edges', and a static push saturates that ratio. Give the floor mass and shake it, and it sways and twists in two modes that share the motion. A response-spectrum analysis then reads the same ratio from combined peaks — and gives a different verdict on a third of the plans tried. Floors torsionally stiffer than they sway read worse than a push says, because the coupled modes put twist at the edges. Floors torsionally flexible read better, because combining the modes keeps how far each edge moves and forgets which way, so a twisting floor's two edges both look large.

 An old joint passes the bolts that have lost their preload. The chance that an M20 grade 10.9 bolt holds the inspection torque of 647 N·m without its nut turning, against the tension actually in it as a fraction of the minimum preload, for a joint new and 1 year old, 5 years old and 20 years old. New, a bolt at half the minimum preload passes 0% of the time and one at the minimum 83%; 20 years old, the bolt at half passes 80% of the time and one at a quarter 9%. Connections

The wrench that reads the rust

The obvious check on a preloaded joint that has been in service for years is a wrench on the nut, and it measures the wrong thing. The nut holds against the bolt's tension times a friction bracket, and twenty years of rust raise the bracket until a bolt with half its preload passes four times in five. The washer that could have said more remembers only the day it was crushed.

 The survey knows the creep once most of the floors are set. The doubt about the creep of a 40-storey building cast a floor every seven days, its creep known to ±30 per cent, as the half-width of its 95% band, after a survey of every floor set by then — each read to a standard error of 1.0 mm — against the floor it is made after: from the survey alone (drawn at 100% where it is larger), and combined with the design's ±30%; beside them, the share of the building's floors still to be cast, which are all a survey can correct. The survey alone beats the design's doubt only after floor 25, with 38% of the floors still to cast; the combined doubt halves after floor 31, with 23% still to cast. Deflection

The survey that arrives after the floors

A tall building's lower storeys start shortening months before its upper floors are cast, so a survey every ten floors should tell the creep the building really has and let the floors still to come be set for it. It does tell it, to a millimetre's precision, after the twenty-fifth floor. By then most of the floors whose height it was meant to correct are already set, and the floor that ends furthest from level is the one no survey can help.

 Taking hold of a swinging mass throws it. The largest stroke of a tuned mass of 2 per cent, as a share of its limit, after the free-mass law takes hold of it while it swings on its own spring and damper in a steady storm at the structure's frequency, against the instant in the structure's period at which the law takes hold; dashed, the law waiting for the stroke to cross zero, and for the mass to come to rest relative to the structure. Depending on the instant the stroke reaches 1.39 to 3.12 times the limit; waiting for zero stroke, 2.81; waiting for rest, 2.27. Dynamics

Taking hold is harder than letting go

A tuned mass driven by a law that cancels its spring is safe to let go of: the transient is the same whenever the switch falls, and the passive mass takes over. Switching the law back on is the reverse operation and behaves nothing like it. The mass the spring was holding is thrown up to three times past its stroke limit, by an amount that depends on the instant, and a switch that is allowed to come back in a storm near its threshold comes back hundreds of times.

 The silt hands the head back over decades, a season at a time. The basement is under five storeys, its water table 2.0 m above the design head and swinging ±0.5 m over the year, highest in April; pumped drains under its slab start 30 times as conductive as the ground. The head under the slab above the design head, against the years since the drains were new, for drains silting with time constants of 4, 8 and 16 years; dots, a standpipe read every October on the 8-year drains. With those drains the head reaches flotation in the wet season of year 24.1; the reading first exceeds 0.25 m in year 14.8, 9.4 years before. Equilibrium

The drain that fails a season at a time

A basement held at its design head by drains and a pump can fail suddenly, when the pump stops, or slowly, when fines settle in the drains and the head under the slab creeps up while the pump keeps running. The slow failure gives a standpipe read once a year a decade of warning, in its own time constants — if it is read in the wet season. And long before the silt floats the building on its own, it has made every stop of the pump more likely to.

 Where along its front a surface crack starts to cleave. Where along the front of a surface crack 8 mm deep in a 40 mm plate at −60 °C, 22 degrees below the steel's reference temperature cleavage starts, as a probability per millimetre of front, at the stress that gives a 5 per cent chance of cleavage, against the position on the front, for cracks 80 mm long, 40 mm long and 16 mm long at the surface. The middle third of the front, 60° to 120°, holds 76 per cent, 69 per cent and 28 per cent of the chance. Materials

The front that counts for half its length

A ferritic steel's cleavage toughness depends on how much crack front it has to sample, and for a crack through a plate the front is the plate's thickness. The cracks inspections find are not through cracks; they are thumbnails at weld toes, whose curved front sees a stress intensity that varies round it. Weighted by the weakest link, the front counts for a little over half its length — so counting all of it is safe by 8 per cent, and counting none of it, as the 25 mm toughness does, is unsafe by up to a fifth for a long shallow crack.

 A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54. Sections and stress

The bore with corners and the passage beside it

A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

 A floor that shears lowers the storey's buckling load before the core does. The buckling factor of a storey of eleven lines of leaning columns, ten bays, each column carrying 3,000 kN over 3.6 m and built a three-hundredth out of plumb, held by a floor to a core sized for a buckling factor of 4.0 were the floor rigid, against the floor's shear stiffness per bay as a multiple of a column's sway softening P/h, logarithmic, with the core at one end, in the middle, or split between both ends; dotted, the rigid floor's 4.0. At 100 times the factor is 1.52, 2.80 and 3.02; at 1,000 times, 3.49, 3.84 and 3.88; a concrete slab is several thousand times. Stability

The floor the columns lean on

A storey of pin-ended columns, every one a little out of plumb, leans on its floor, and the floor leans on a core. Count the floor as rigid and the core needs the sum of the columns' demands. A floor that shears in its own plane is a brace in series with the core, and the far columns lean on it before their push arrives. With a floor a hundred times stiffer per bay than a column's own sway softening, the storey's buckling factor falls from 4.0 to 1.5 and the core is asked for twice as much.

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