Structural Engineering & Statics
where the load goes
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Scale changes everything — page 7

Essays 145 to 164 of 164 on this thread, in the same order.
Split into what stays and what visits, the column needs far less. The factor a column may take on n floors' summed characteristic imposed loads, keeping the 50-year load at the 95th percentile, for office imposed load as a sustained part of mean 0.50 kN/m², renewed with each tenancy every five years on average, and intermittent loads of mean 0.2 kN/m² arriving 0.3 times a year on each floor for three days: floors let one by one, in tenancies of 5, and to one tenant; the one-number model with the same tenancies (dotted) and the storey rule (dashed). At 20 floors: 0.27, 0.37 and 0.60, against 0.75 for the one-number model and 0.73 for the rule. One floor's characteristic load is 3.24 kN/m². Equilibrium

The load that stays and the load that visits

An office floor's imposed load is two loads: a sustained one that arrives with a tenant and stays until the next, and intermittent ones — a party, a move, a crowd — that visit one floor for a few days. Simulated over fifty years, a floor's characteristic load is set mostly by its worst visit, and visits on different floors almost never coincide. So twenty floors let one by one need only 0.27 of their summed loads, in tenancies of five 0.37 and to one tenant 0.60, against the storey rule's 0.73. The rule is safe for offices by a wide margin. It stops being safe only where the load stays — an archive or a library let to one occupier — and there its column load is exceeded in a quarter of lives.

 Stressed bay by bay, the beam keeps half its secondary moment. The secondary moment along a 4-span beam of 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, straight between supports: stressed in one operation, stressed one bay at a time, and bay by bay after creep with a coefficient of 2.0. At the first interior support it is 350 kN·m stressed at once, 164 bay by bay and 307 after creep; at the next, 234, 160 and 216. Internal forces

The beam stressed one bay at a time

A long continuous beam is rarely stressed in one pull. It is cast and stressed a bay at a time, each new length of tendon coupled to the last, and at every stage the structure is a different beam. The secondary moments the finished beam carries are the sum of what each stage did to the beam that existed then — about half of what the one-stage analysis gives — and creep spends the following years taking most of the other half back.

 The deck's width over the pier is set by its bars. The local effective width, as a fraction of the overhang, along a deck flange overhanging 3.0 m each side of its girder, continuous over two 20 m spans under a uniform load, drawn where the moment is at least three tenths of its largest: uncracked (dashed), and over the pier, where the deck is in tension, cracked with 1.5% of steel along the girder and 1.5% across it and with half that transverse steel. EN 1994's widths are drawn as the short level lines, Le/8 over the pier and in the spans. Over the pier the uncracked deck works over 0.325 of its overhang, the cracked one with equal bars over 0.328, and with half the transverse steel over 0.281; in the spans, uncracked, the deck works over 0.81. The rule gives 0.417 over the pier whichever way the deck is reinforced. Sections and stress

The deck that cracks over the pier

Over the pier of a continuous composite bridge the concrete deck is in tension and cracks, and what is left of it is a grid of bars and struts. Shear lag does not stop at the cracks: the bars still receive their force through the concrete between them. Whether the cracked deck works over a narrower width than the whole one depends on one ratio, the deck's shear stiffness over its axial stiffness — and once the concrete has cracked, both of those come from the reinforcement, so the answer is set by how the bars are arranged.

 A crack starts at the corner and runs along the grain. The end of a timber beam 400 mm deep and 140 mm wide, notched on its underside so that 200 mm stands over the support, with the notch's corner 50 mm from the line of the reaction, drawn to scale. A crack starts at the corner and runs along the grain towards mid-span. With a shear strength of 3.5 N/mm² the un-notched end would carry 130.7 kN; Eurocode 5's factor kᵥ of 0.40 on the reduced depth leaves 25.8 kN, 20% of it. Sections and stress

The notch that costs more than its depth

Cut a timber beam's underside away at the support, so that half its depth sits on the bearing, and a stress check on what is left says the end has lost half its shear strength. The beam says otherwise. A crack starts at the notch's corner and runs along the grain, and whether it runs is a question of energy, not stress — so the answer has a size in it. A 400 mm glulam beam notched to half its depth keeps a quarter of its shear resistance, an 800 mm one keeps 18 per cent, and the deeper the beam the less a notch leaves. Sloping the notch buys much of it back, and moving the corner away from the support costs it again.

 Double the span and the deck is nearly three times nearer buckling. The factor on its own dead load at which a stayed deck buckles, against the main span, on logarithmic axes, for a deck of 0.50 × 10¹² N·m² under 150 kN/m with stays every 15 m from towers 0.40 of the half-span tall. At 300 m it is 40.7; at 600 m, 14.7; at 1,000 m, 6.9; at 2,000 m, 2.3. Beyond 600 m the line falls with a slope of 1.53; it crosses ten, below which a design has to follow the deck's deflections into its forces, at 805 m. Structural form

The deck the stays squeeze

Every stay of a cable-stayed bridge pulls the deck towards its tower, and the deck carries all those pulls as a compression that is largest at the tower — 95 MN in a 1,000 m span under 150 kN/m. The stays hold it up and hold it straight, as springs under a strut. The compression grows with the square of the span and the springs soften with it, so the factor on its own weight at which the deck would buckle falls as the span to the power one and a half: 41 at 300 m, 6.9 at 1,000 m, 2.3 at 2,000 m. A deck four times stiffer only doubles it, and the long stays' sag, which softens them for traffic, hardly enters.

 A quarter of the visits shared uses up the whole reduction. The factor needed by 20 floors carrying office imposed load — a sustained part of mean 0.50 kN/m² renewed every five years on average, and visits of mean 0.2 kN/m² arriving 0.3 times a year on each floor for three days, against the share of the visits made by the whole building at once: floors let singly (solid) and to one tenant (dashed), and the storey rule's 0.73 (dotted). With no share, 0.26 and 0.60; with a tenth, 0.51 and 0.67; with a quarter, 0.68 and 0.75; with all, 0.96 and 0.97. The rule's factor is reached at a share of 0.34 let singly and 0.21 to one tenant. Equilibrium

The visits that come to every floor at once

A floor's characteristic imposed load is set mostly by its worst visit — a party, a move, a crowd — and visits to different floors almost never coincide, which is why a column carrying twenty office floors needs only about a quarter of their summed loads. Let some of the visits be made by the whole building at once and nothing changes on any one floor; one floor cannot tell. The column can. A tenth of the visits shared takes the twenty-floor factor from 0.26 to 0.51, a quarter to 0.68, and a third uses up the whole of the storey rule's reduction. The rule fails in exactly the buildings whose use synchronises the crowd, and it fails most where the load mostly visits.

 The best compensation is less than the prediction. For every floor of a 40-storey building of 3.6 m storeys, its perimeter columns at 12 N/mm² and its core at 6 under the finished load, cast a floor every seven days, with 25 per cent of the load arriving as finishes 90 days after the frame is complete and creep coefficient 2.2, from its fit-out to thirty years, with the creep anywhere within ±30 per cent of the design value: the largest amount any floor stands high (dashed) and stands low (light), and the worse of the two (solid), against the share of its predicted thirty-year movement each floor is set high by. Uncompensated, a floor ends 31.9 mm low; compensated in full, a floor stands 16.0 mm high at its fit-out; the worst is smallest, 11.0 mm, at 0.79 of the prediction, where level 40 standing high 0 days after its fit-out under creep ×0.70 balances level 33 standing low after thirty years under creep ×1.30. Deflection

The floor that is level on one date

Each floor of a tall concrete building is set high by the amount its columns are predicted to drop against the core, so that it arrives level. Counted in time, the drop does not peak at mid-height and stop at the roof: the worst floor of a forty-storey building is level 31, the roof moves 24 mm, and half the movement is still to come when the floors are fitted out. Set high by all of the prediction, a floor is level on one date thirty years away and 16 mm high at handover. With the creep known to ±30 per cent, setting it high by about four fifths of the prediction makes the worst floor's error 11 mm instead of 16. And none of it changes what the partitions see.

 In a core, the depth has to tune the tank, and only shallow water is slow. The sloshing period of rectangular tanks 6, 10, 15 m long against the depth of their water (logarithmic), with building periods of 4, 5, 6 and 8 s (dotted). Deep, each tank is as fast as it can be: 2.77 s, 3.58 s, 4.38 s. To be tuned to a 5 s building the 10 m tank needs 1.80 m of water; to 4 s, 3.50 m; to 8 s, 0.65 m. A building faster than 3.58 s cannot be reached by the 10 m tank at any depth. Dynamics

The tank that must be shallow to be slow

A deep tank's sloshing period is set by its width alone, so a tank in a tall building's core ought to be tuned by choosing how wide to make it. It cannot be. A deep tank fast enough to fit is too fast for the building: 3.6 s for a core 10 m across, where a 5 s building would need a tank 19.5 m long and an 8 s one 50 m. The depth has to tune it, and only shallow water is slow — 1.73 m for 5 s, 0.63 m for 8. Shallow water is the efficient kind, with four fifths of it sloshing. But at the accelerations the tank exists to reduce, the wave it makes is a quarter of its depth on a 5 s building and seven tenths on an 8 s one, and the linear tuning it was designed by has stopped describing it.

 The corridor's share of a column decides its reduction. The factor on the summed characteristic loads that a column carrying n floors needs, against n, for office imposed load as the earlier essays drew it — a sustained part of mean 0.50 kN/m² and visits of mean 0.2 kN/m² a third of a year apart on each floor — and on the corridor a crowd of mean 1.00 kN/m² at one evacuation a year, on every floor at once, with 0%, 10%, 25%, 50%, 100% of the column's tributary area corridor; dashed, the storey rule. At 20 floors: 0.26, 0.30, 0.49, 0.90, 0.98, against the rule's 0.73. The lines are simulated, 4,000 lives at each point, and carry a few hundredths of sampling noise. Equilibrium

The column under the escape corridor

A column's imposed-load reduction is allowed by the number of floors it carries, on the reasoning that floors are not crowded at the same moment. A column whose tributary area includes escape corridor carries a part of every floor that is crowded at exactly the same moment, every time the building is evacuated. Its reduction follows how much corridor its area holds, not how many floors it carries — a tenth costs little, two-fifths uses up the whole of the storey rule's allowance, and a stair-core column needs almost none. Against the nominal loads a designer writes down, the margin that saves the columns at the corridor's edge has run out by the time the area is mostly corridor.

 Three cores and a specification say more together. The predictive distribution of a core's strength in an existing structure: from a concrete specified to a mean of 38 N/mm² and a coefficient of variation of 15 per cent, that prior carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, before any core is taken (dotted); from three cores of mean 33.0 N/mm² and coefficient of variation 12 per cent alone (dashed); and from both (solid); each with its 5 per cent characteristic value marked. The specification alone gives 27.5 N/mm²; the cores alone, 21.9, because three results cannot pin down a scatter; together, 26.5. Materials

The strength three cores revise

A characteristic strength estimated from three specimens and nothing else is heavily penalised, because three results cannot say how scattered a material is. An existing structure is never nothing else. It was built to a specified grade, from a population of concretes whose scatter is known in general, and three cores drilled from it revise that knowledge rather than replace it. Together they give a characteristic strength that three cores alone would need twenty to match. The same arithmetic catches a concrete much weaker than its specification, because cores that disagree with the prior widen it — unless the prior is given so much weight that it no longer listens.

 More hangers, each swinging through less. Each hanger's force range, plotted at its mid-point along the span, for a tied arch of 100 m span and 17 m rise on 32, 64, 96 crossed hangers of 3,000 mm² bar at 60°, 60 kN/m of dead load and 60 kN/m of live load run on from either end. 32 hangers: the worst 678 kN, the mean 528; 64 hangers: the worst 369 kN, the mean 273; 96 hangers: the worst 241 kN, the mean 184. Structural form

More hangers, and the same steel

A network arch's crossed hangers are the web of a truss, and a live load's shear passes through them, so every hanger swings through nearly all of its force. The obvious cure is more hangers, each crossing more of its neighbours, so that every section's shear is shared among more diagonals. It works hanger by hanger — three times the hangers, a third of the range — and not at all in total: the hangers' fatigue steel, sized for their ranges, is set by the shear the arch carries and barely changes with how many hangers it is divided among. What more hangers buy is elsewhere.

 Counted honestly, cores from one bay hand the estimate to the specification. The share of the structure's concrete that falls below the 5 per cent characteristic value estimated from three cores from one bay of a structure's concrete of mean 33.0 N/mm² and coefficient of variation 12 per cent, updating the specification — a mean of 38 N/mm² and a coefficient of variation of 15 per cent, carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, averaged over 2,000 drillings, against the correlation between two cores from one bay: the cores counted as independent (dashed), counted by their correlation (solid), and three cores drilled from three bays (light); dotted, the 5 per cent an honest estimate gives. At ρ = 0.5, 6.2% and 6.5%, and 4.8%; at 0.9, 7.4% and 7.3%, and 4.8%. Materials

The cores that agree for the wrong reason

Three cores drilled a metre apart in one slab share a pour, a curing and a morning's water, so they agree with each other more closely than they agree with the rest of the building. Counted as three independent specimens, they overstate what is known — and with nothing else to go on, the estimate built on them leaves a quarter of the building's concrete below a value meant to leave a twentieth. Counting them by their correlation repairs that. But once the specification is in the update, the honest count hands the estimate back to the specification, and a concrete weaker than it was meant to be hides behind it. The repair is not in the arithmetic. It is in where the cores are drilled.

 Pumping water in or out keeps the damper, and the lost water barely matters. The peak amplification of a building with 1 per cent damping and a 10 m tank sized for a 5.0 s building, 1.73 m of water weighing 2.0 per cent of the modal mass, its sloshing damped at 7 per cent, against the period the building actually has: the tank left at its design depth (dashed), retuned by its water level with the water's mass following the depth (solid), and retuned with the mass held at the design value (light); dotted, the design period; bare, the building would reach 50. At 4.5 s, 21.7 left alone and 9.6 retuned; at 5.5 s, 22.6 and 10.8; at 6.0 s, 34.0 and 11.7, of which the water let out costs 1.7. Dynamics

The tank that follows the building

A sloshing damper is tuned to a period computed before the building exists, and the building that is built has another one — stiffer at handover, with its partitions and cladding working, softer once its concrete has cracked. A tank whose plan is fixed and whose period is set by its depth can be brought back by opening a valve. Ten per cent off tune, the tank left alone lets the building's peak response more than double; pumped back to the right depth, it holds within a tenth of the design value, and the water it gains or loses barely matters. What does matter is that a slower building needs shallower water, and in shallow water the wave the damper is made of grows towards the depth it is sloshing in.

 One alarm in a few hundred that empties the building sets the column's load. The factor on the summed characteristic loads that a column carrying 20 floors needs, for office load and a corridor crowd at one alarm a year, evacuated three floors at a time, against the share of alarms that escalate to every floor at once (logarithmic), with 25% and 50% of the tributary area corridor; dotted, each with every alarm a full evacuation; dashed, the storey rule's 0.73. 25% corridor: 0.26 with none escalating, 0.27 at one in five hundred, 0.33 at one in a hundred, 0.38 at one in twenty, against 0.49 in full every time; 50% corridor: 0.29 with none escalating, 0.39 at one in five hundred, 0.59 at one in a hundred, 0.75 at one in twenty, against 0.90 in full every time. Equilibrium

The stair that empties three floors at a time

A column beside an escape stair carries the corridor crowd of every evacuation, and when every floor empties at once its floors are loaded together and earn almost no reduction. Tall buildings are usually evacuated in phases instead, three floors at a time, so that the stair is never asked to carry everyone. Under such a stair the column earns its reduction back and more — 0.29 of the summed floor loads, where the storey rule allows 0.73. But a phased strategy is a plan for the usual alarm, and the column's characteristic load belongs to the unusual one. If one alarm in a hundred escalates to a full evacuation, the factor climbs back to 0.59; one in twenty, and it passes the rule.

 The flatter the ribbon, the more the casting temperature matters. For two 100 m spans of a stressed ribbon under 35 kN/m on a pier of 200,000 kN/m, one span's joints cast 20 °C warmer than the other's: the force left on the pier as a share of the force a crowd of 20 kN/m on one span puts there, against the ribbon's sag as a percentage of its span. At a sag of 1 per cent the casting leaves 82 per cent of a crowd (1,902 kN against 2,316); at 2 per cent, 14 per cent; at 3, 3 per cent; at 4, 1 per cent. Structural form

The span cast in the afternoon

A stressed ribbon's joints are cast on whatever day the programme reaches them, and a span cast on a warm afternoon sets around strand that has lengthened in the sun. It cools that night as a concrete band eleven times stiffer than the strand, and the band does not stretch back as the strand would have. On a ribbon with a sag of 2 per cent, a span cast 20 °C warmer than its neighbour ends 72 mm higher than it was set and leaves 713 kN on the pier between them — a seventh of what a crowd on one span puts there. On a flatter ribbon the same afternoon costs more than the crowd. The difference between the two spans is the strand's flexibility against the band's, and only the elastic part of it differs; the part that comes from the sag is the same for both.

 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.

 Between the two counts, and the root's toughness says where. What a 20 mm S355 plate welded from one side to 0.5 of its thickness keeps of a double-sided butt throat, pulled straight across, against the toughness of the metal at its root: assessed as a crack on the strip-yield line (solid), by the toughness alone (dashed), and by the elastic and plastic counts (dotted, 0.25 and 0.41). Below 30 MPa√m the root tears before the elastic count's load; at 46 MPa√m the toughness alone would allow the whole plastic count, but the assessment gives 0.34, and it takes 66 MPa√m to reach 95 per cent of it. At 30, 50 and 100 MPa√m the root keeps 0.25, 0.35 and 0.41. Connections

The land that is already a crack

A partial-penetration weld made from one side leaves the rest of the joint unfused, and pulled across, that unfused land is an edge crack whose tip is the root. Counted as a crack it keeps neither the elastic quarter of a double-sided weld nor the plastic 0.41, but a share between them that the root's toughness chooses — and the toughness of a crack front is the toughness of its length, which for a weld is the whole weld. A long weld in a cold structure can tear below the elastic count, and the same joint drawn twice as large tears sooner, which no count of stresses can see.

 Which way the oval lies decides whether the pressure helps or harms. The limit moment of a steel tube of radius 600 mm and wall 2.5 mm (r/t = 240), whose ring buckles under 4.07 kPa, over its limit with no pressure (796 kN·m), against the angle between the bending plane and the diameter a non-uniform pressure pushes in on, for second harmonics of 0.10, 0.25 and 0.50 of that pressure; lines the closed form, dots the minimised energy. At 0.10, 0.93 aligned and 1.08 across; at 0.25, 0.82 aligned and 1.19 across; at 0.50, 0.65 aligned and 1.40 across; at 45° the oval does not touch the limit at all. Stability

The oval the pressure brings

A uniform pressure changes how stiff a tube's ring is against the oval that bending squashes it into, and nothing else: the tube still gives way at a flattening of two ninths of its radius. A pressure that is larger on one diameter than the other has an oval of its own, and that oval does work on the bending's directly. Lying in the bending plane it has done some of the flattening before the bending starts, and a pressure difference a quarter of the ring's own buckling pressure takes a fifth off the tube's bending capacity; lying across it, it adds a fifth. At forty-five degrees it does nothing at all. And the two ninths turn out to have been a property of a round tube.

 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.

 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.

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