Structural Engineering & Statics
where the load goes
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The load must go somewhere — page 19

Essays 433 to 450 of 450 on this thread, in the same order.
How much of the drain has to work. The load in the more heavily loaded member, with the roof's own 0.5 kN/m², over its 2.0 kN/m² capacity, for a 12 by 9 m bay in a 150 mm/h storm, with a drain at its middle that passes the storm at 30 mm when clear and a 400 mm scupper at its edge whose sill is 50 mm above the supports, against the share of a clear drain's flow the drain still passes. Members at L/180, L/240, L/360 and rigid: with the drain fully blocked, 1.25, 0.99, 0.84, 0.66; a tenth open, 0.80, 0.77, 0.71, 0.62; half open, 0.44, 0.45, 0.46, 0.48; clear, 0.37, 0.37, 0.38, 0.40. A softer roof is the worse with the drain blocked and very slightly the better once about a fifth of it works. Deflection

The drain that is mostly blocked

A roof's ponding check is made with its primary drains blocked, and a drain is seldom blocked outright: leaves and gravel choke it to a fraction of its flow. A fraction turns out to be nearly enough. On a 12 by 9 m bay sized to L/240, a drain passing a seventh of its clear flow still takes the whole of a 150 mm/h storm, because the water standing over it at the bottom of the sag is the bay's own deflection plus the level. A softer roof's drain works harder still. Only near complete blockage does the scupper take over, and only there does stiffness help.

 Where each filler's depth goes. The crushable depth left in each filler between buildings of 500 t and 300 t, periods 0.8 and 1.2 s, 50 mm apart, through a record of six one-second sine cycles peaking at 0.35 g, for 40 mm fillers crushable over 24 mm, each just strong enough to stop a 2.85 m/s collision from new. Uniform: 5 mm when its hardest contact arrives at 2.76 m/s, 0 mm at the end. Graded: 12 mm when its hardest contact arrives at 2.76 m/s, 6 mm at the end. 50% on speed: 0 mm when its hardest contact arrives at 2.88 m/s, 0 mm at the end. The graded filler's soft layer goes in the first middling contacts and its strong layer is barely reached; the other two lose depth to every middling contact and have little or none left when the largest arrives. Dynamics

The filler that is used up before the worst

A crushable filler between two buildings is sized for the hardest collision it might meet, meeting it new. An earthquake does not oblige: the hardest contact arrives after a dozen middling ones, and each has taken some of the depth. Through a record of six cycles, a filler half of whose strength follows its crushing speed is dense before the worst contact arrives, and a uniform one has five millimetres left. A graded filler still has twelve. Yet the graded filler meets the worst contact at 56 MN and the spent rate-sensitive one at 39. Keeping depth is not the same as being gentle, and the order of the contacts barely matters.

 The seasons ride on the shrinkage. How far the slab's end has moved away from the fill since the fill was finished, over six years, for slabs 10, 20, 30 m between walls at 60 per cent humidity, swinging 3.0 K either way each year, propping a 3 m stem fixed at its base, its top held by the slab cast before its fill went up in 250 mm lifts rolled at 60 kN/m. The slab lets go of the stem at 2.08 mm (dashed). 10 m: a seasonal swing of ±0.15 mm on shrinkage reaching 1.28 mm by year six; it still touches the stem in the sixth year. 20 m: a seasonal swing of ±0.30 mm on shrinkage reaching 2.55 mm by year six; it last touches the stem 3.4 years on. 30 m: a seasonal swing of ±0.45 mm on shrinkage reaching 3.83 mm by year six; it last touches the stem 1.3 years on. Equilibrium

The seasons that stop reaching the wall

A ground-floor slab that has shrunk away from the basement wall it propped moves with the seasons as well, a few tenths of a millimetre at each end every year. It looks like a ratchet waiting to happen: each winter the slab presses the stem back into its fill, and each push might lock a little more pressure in. On a 30 m slab it never gets the chance — the seasons ride on a shrinkage ten times larger, and the slab last touches the wall sixteen months after the fill is finished. On a 10 m slab it touches every year for good, and the stem still loses most of its prop.

 Pressed and pulled, the footing leans once and stops. The column's tilt at rest after each of eight storms reversing fully to 300 kN·m each way, on the 8 m strip of EI 540 MN·m², weighing 320 kN, on a bed of 50 MN/m² under a column carrying 1,000 kN, on ground that unloads and reloads 4 times as stiffly as it first loads, for loads swinging by 0, 50, 100, 150, 200 per cent. 0 per cent: −0.007 milliradians after the first storm and 0.003 after the eighth; 50 per cent: 0.112 milliradians after the first storm and 0.103 after the eighth; 100 per cent: 0.109 milliradians after the first storm and 0.091 after the eighth; 150 per cent: 0.113 milliradians after the first storm and 0.095 after the eighth; 200 per cent: 0.114 milliradians after the first storm and 0.102 after the eighth. The lean is towards the end the heavier half of each storm presses; it stops within three storms whether or not the strip ever hangs clear. Internal forces

The footing that hangs from its column

A footing under a bracing bay's column is pressed hard when the wind pushes one way and pulled when it swings back, and if the pull is larger than the footing's weight it hangs from the column, clear of the ground, for part of every storm. It looks like a footing free to come down somewhere new each time, and walk. It does not walk. The ground under it keeps the memory of every press, and a footing that is not touching the ground cannot change that memory; it comes back down into its own bowl, leans once towards its harder press, and stops within three storms.

 The middle reaction is a dimension, read off in millimetres. The middle reaction under dead load of a Pratt truss continuous over two 24.0 m spans, 3.0 m deep, carrying 60 kN of dead load at each interior bottom joint, against the height of its middle bearing above the level drawn, cut to the continuous list (solid) and as two simple spans (dashed). At the drawn level the continuous list lands with 572 kN, the analysis's value, and the simple-span list with 480 kN, the two spans' own share. Every millimetre the bearing stands high adds 2.6 kN: 10 mm high, 598 kN; 10 mm low, 545; the simple-span list's 92 kN shortfall is the bearing 35 mm low. Deflection

The pier's share is cut into the members

A truss continuous over a middle support has one reaction that statics cannot find, and the lengths its members are cut to decide it. Cambered to land level under its dead load on three bearings, a two-span truss puts 572 kN on its pier; cambered span by span, as two simple trusses, the same members on the same bearings put 480 kN there and carry the dead load with no moment over the pier at all. The list chooses the moment diagram. And a millimetre of shop tolerance on every member moves the pier's share as much as a bearing set fourteen millimetres out of level, so a reaction the design depends on has to be set by a jack rather than by a drawing.

 Under a light building the first thing the pump was holding up is the building. The hours after the pump stops at which each check on the basement is used up — the slab's 0.2 mm cracks and the building's flotation — against the number of storeys over it, with the water table risen 3.0 m and ground recovering in 12 hours; drawn at 48 hours where a check is never used up. The cracks' deadline is 9.5 h whatever the building. Flotation's is at once under 2 storeys, 1.0 h under 4 storeys, 4.7 h under 5 storeys, 10.1 h under 6 storeys, never under 8 storeys; with 5 storeys or fewer it comes first. Equilibrium

The building held down by its pump

A basement whose water table has risen above its design level is often given back its design head by a pump under the slab, and from then on it is safe while the pump runs. When the pump stops the water comes back on the ground's own clock, and the checks the pump was protecting are lost in an order the building's weight decides. Under eight storeys only the slab's cracks were ever at risk; under five, the building's flotation check goes first, four and three-quarter hours after the pump stops, and the slab's strength is never in question at all. A structure that was safe by its weight has become one that is safe by a machine, and the machine's reliability is now one of its loads.

 The hogging region cannot give its studs away. For a 457 × 191 × 67 steel beam under a 130 mm slab, continuous over two 12.0 m spans under 30 kN/m, 40 studs a span: the largest slip (solid, left scale) and the force in the slab's bars over the middle support (dashed, its value in the legend), against the density of studs beyond the point of contraflexure as a share of the density elsewhere, the span's total held at 40. At the same density everywhere the largest slip is 0.68 mm and the bars carry 344 kN; at half the density beyond contraflexure, 1.04 mm and 251 kN; at a quarter, 1.69 mm and 118 kN. The support's hogging moment falls from 448 to 398 kN·m as the bars stop working. Internal forces

The slip that peaks where the moment turns

A simply supported composite beam slips most at its supports, and its studs are crowded there. Make the same beam continuous over a middle support and the slip peaks twice: at the end support, and again, just as hard and the other way, at the point of contraflexure — not over the middle support, where the shear is largest but the slip falls to nothing. Studs graded toward the supports, as a simple span's would be, miss the second peak and buy a tenth; graded to the slip they buy more than a third. And the hogging region cannot give its studs away to the span, because its edge is where the second peak is.

 The more it carries, the less it pulls when the fire is out. For a 9 m 457 × 191 × 67 steel beam held at its ends at 10 per cent of its own axial stiffness through the fire: the tension pulling on its ends once it has cooled (solid) and the largest tension while it is still hot (dashed), against the load it carries as a share of its cold plastic moment; dotted, the straight beam's 1,669 kN. At 0.10, 1,017 kN cold and 41 hot; at 0.39, 613 and 293; at 0.54, 477 and 397. From 0.56 the beam can find no sag that carries its load near the fire's peak and runs away. Materials

The beam that pays the fire back in sag

A steel beam held at its ends and kept straight through a fire yields short on the way up and pulls on its connections with its plastic shortening on the way down. A beam carrying a floor does not stay straight. Squeezed by its own expansion, it forms a hinge at about 500 °C and bows out of the way, hangs as a catenary while still hot, and keeps a metre of sag when it cools — length it no longer has to be pulled for. It ends the fire pulling a third as hard as the straight beam, nearly whatever holds it, and the hard pull comes earlier instead: at 800 °C, on connections that are as hot as it is.

 The diagonals over the support are the girder's shortest-lived members. The years a diagonal's connection of the top bracing of a 1.5 m deep composite tub girder under a 250 mm deck, continuous over two 40 m spans, 480 kN lorries passing 2.5 m off its centre line lasts at two million such lorries a year, along a span, for fatigue categories 50, 71 and 90, logarithmic and drawn at 1,000 years where the range is below the category's cut-off and the life is without limit; dotted, 100 years. Where the deck is whole every category lasts without limit. At the worst diagonal, 34.5 m along, category 50 lasts 2 years, category 71 lasts 8 years, category 90 lasts 25 years. Sections and stress

The bracing nobody took out

The light truss that closes a steel tub girder while its deck is poured is usually left in once the deck has hardened, because taking it out from under a finished slab is more trouble than it is worth. Over most of a span that is harmless: the deck lifts the neutral axis to the top flanges and takes ninety-nine per cent of the shear round the top, and a passing lorry puts five newtons per square millimetre in a diagonal. Over an interior support, where the deck is cracked to its bars, the same diagonals sit six hundred millimetres from the axis and are handed a share of the torque, and each heavy lorry puts forty in them — the shortest-lived connections in the girder, on members nobody designed for anything after the day the concrete went off.

 A floor's robustness belongs to its joints until they break, and to its slab after. For a floor whose interior column is lost, two 8 m-span beams crossing over it (each collapsing at 400 kN) and a slab 16 m square over the four bays round it: the largest floor load a sudden loss can be held at, as a multiple of the two beams' collapse load, against how far the beams' connections can turn before they fracture, for no mesh and for 142, 252 and 393 mm² a metre. No mesh: 0.94 at 0.03 radians and 1.19 at 0.07; 142 mm²: 0.94 at 0.03 radians and 1.29 at 0.07; 252 mm²: 1.50 at 0.03 radians and 1.81 at 0.07; 393 mm²: 2.26 at 0.03 radians and 2.56 at 0.07. Structural form

The slab that catches what the joints let go

When an interior column is lost, the beams that crossed over it sag into cables and the slab over the four bays round it sags with them, into a membrane hanging from the floor that did not lose anything. Whether the floor's robustness belongs to its steelwork or to its concrete turns on one number with no span in it: the slab's mesh only starts to carry its full tension when the beams' ends have turned through about 0.09 radians, more than any ordinary joint can follow. So the membrane arrives after the joints have gone. And then it catches the floor — a metre down, alone — and with an ordinary mesh it holds half as much again as the beams could.

 The column line's sum agrees with every analysis, because it cannot see the beams. The roof sway of a 4-bay, 10-storey concrete frame, 8 m bays and 3.5 m storeys, 600 mm columns and 400 × 700 mm beams, 40 kN/m on every beam and 40 kN of wind at every floor, against the beams' stiffness as a share of their gross value, from the whole frame (line) and from the windward column line's integral of the moments that analysis gives it (dots). At the gross stiffness, 15.7 mm; at 0.5, 25.4; at 0.35, the usual allowance for cracked beams, 33.3; at 0.2, 52.0. The windward column's base moment moves from 126 to 235 kN·m, and the line's integral follows the sway exactly at every one. Deflection

The check that cannot see the beams

The unit load that finds a frame's sway may be carried on any structure that can carry it, and in a frame of four bays and ten storeys it can be carried down one line of columns alone. Then the roof sway is that line's real moments against the unit load's, a few dozen products by hand — exact for the frame as drawn, for its beams cracked to a third of their stiffness, for its joints made springs and for its bases made springs, the last only once the bases' rotation is given its own term. It is exact because the unit load never enters a beam. Which makes it a check on the analysis's arithmetic that cannot say anything about the analysis's beams.

 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.

 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.

 Where the shelter is decides whether the leading row works harder. The share of the cap's shear the leading row and the last row of three rows carry, as a multiple of the average pile's, against the thickness of soft ground over stiffer, in characteristic lengths, with the row factors applied at every depth (solid) or in the soft layer only (dashed). With the factors at every depth the leading row carries 1.24 and the last 0.83 at 2.0 lengths; in the soft layer only, 1.07 and 0.96. In uniform ground the usual reading gives 1.35 and 0.75. Internal forces

The shelter that stops at the interface

Piles in a group shelter one another, and practice counts the shelter as a factor on each row's soil, calibrated on groups in uniform ground. Put the same group in soft ground over stiff and the calibration has to be read one of two ways — the shelter at every depth, or only where the ground is soft — and the two readings disagree about which row works hardest. The first gives the leading pile 17 per cent more moment at the cap; the second gives the last pile 24 per cent more in the ground.

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