Structural Engineering & Statics
where the load goes
The collection

Every essay — page 26

Essays 601 to 624 of 728, in the same order.
The shadow an opening casts on every perimeter. One column of a 260 mm flat slab (d = 225 mm) carrying 12 kN/m² on 400 × 400 mm columns, the shear 300 mm off centre, on a 10.0 m bay, to scale, with a 300 × 300 mm opening 225 mm (1.0d) from the column. The two tangents from the column's centre to the opening's corners (dashed) enclose 38.9 degrees, and every perimeter loses the part between them: 458 mm of the control perimeter 2d out (solid), 1,093 mm of the outer perimeter (dashed), the lost parts drawn faint. With the opening the outer perimeter stands 6.13d from the face instead of 5.36d, and the studs need 7 perimeters instead of 6; one of the 12 rails runs into the hole. Internal forces

A hole takes a tenth of every perimeter

A 300 mm riser cut a slab's depth from a column throws a shadow from the column's centre, and EN 1992-1-1 deletes whatever part of a punching perimeter falls inside it. The outer perimeter, being longest, loses the most length — but it loses the same share as the control perimeter, a tenth of each, because the shadow is an angle. The cost arrives somewhere less obvious: a seventh ring of studs, 24 per cent more steel, and studs needed in a bay a third of a metre shorter than before.

6 figures · Punching shear The loads a portal can carry for ever. For a fixed-base portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, in the plane of the gravity load's peak V·L/Mp — the load comes and goes — and the wind's H·h/Mp — it reverses: the first-yield boundary (dotted), the shakedown boundary (solid), and the collapse boundary under the same two loads applied once (dashed). With little wind the shakedown boundary is the beam's own mechanism, which the wind does no work in, falling from 8 to 6.0 as the wind grows to H·h/Mp = 2.7; beyond that it is alternating plasticity at a column base, nearly level. The dot is 160 kN of gravity with 60 kN of wind and the ring its shakedown point, a factor of 0.88: the gravity load alone is at the beam's collapse load, and the wind takes 12% off what the pair can carry without doing any work in the beam's mechanism. Materials

The beam the wind walks without pushing

A fixed-base portal whose beam carries exactly its collapse load is safe under that load applied once. Add a wind that comes from either side — a wind that does no work at all in the beam's collapse mechanism — and the beam can no longer carry the load for ever: 12 per cent comes off, and above that the midspan gains fifteen milliradians every cycle while the frame does not sway. Each knee takes its worst moment from a different wind, and the beam assembles a mechanism no single storm drives.

6 figures · Shakedown A girder softer to short waves sheds a little of its support moment. For two continuous 20 m spans of a girder whose flanges are 6.0 m wide and 30 mm thick, 1.20 m apart on 24 mm of web — the flanges 97% of its stiffness — under a uniform load, each interior support 1.0 m long: the bending moment along the girder per unit load, sagging up — for a prismatic girder (dashed), which is what a global analysis gives with one effective width per span, and for the girder whose every wave of moment bends it with that wave's own effective flange (solid). The interior reaction falls by 0.88%, and because the support moment is the difference of two much larger moments, that becomes 4.6% of the support moment, 47.0 to 44.8; the sagging peak rises by 2.9%, 28.1 to 28.9. Sections and stress

The reaction moves one per cent and the moment five

Over the support of a wide-flanged continuous girder the flange works over a third of its width, so the girder there should be softer and shed moment into its spans. It does — but it is the reaction that the softness changes, by less than one per cent, and the support moment, a small difference of two large ones, moves by five. The stress spike over the support follows the reaction rather than the moment and eases by two. A prismatic analysis is right about the moments to a few per cent, and the spike stays a problem of the section.

6 figures · Effective width The room left between two planes of braces. The surface of a 168.3 mm chord unrolled flat — along the chord across, round it up — with the footprints where 114.3 mm braces at 45 degrees are welded to it: two from a K joint in one plane (solid), two from a K joint in a plane 90 degrees round the chord (dashed), both nodes at the same section, each with a 12.6 mm gap between its own toes. The narrowest gap between a brace of one plane and a brace of the other is 6.5 mm, against the 12.6 mm two welds need: too little to weld. Each footprint spans 85.6 degrees of the chord's circumference. Connections

Two planes of braces and one circumference

A tower leg or a triangular girder's chord receives K joints from two planes at once, and each can be checked as a planar joint: its gap, its eccentricity, its band. What the planar checks cannot see is that the two planes' braces are welded to the same tube. A 114 mm brace takes up 86 degrees of a 168 mm chord's circumference whatever its gap, so a square tower's two planes leave 6.5 mm between them where the welds need 12.6, and a triangular girder's braces land on each other. And two eccentricities at the edge of their band give the chord √2 or √3 times the moment the band was written for.

7 figures · Noding eccentricity The gaps that open after the arch is closed. For the 60 m fixed concrete rib rising 6 m under 60 kN/m, closed at 28 days with its abutments jacked apart to take up the day's gap: the gap between the rib and its springings that each movement opens after closure, in millimetres, against the years since, on a log scale — the rib's creep under its thrust, its shrinkage, and the ground consolidating under the thrust by 1.0 times its 8 mm immediate spread with 90% done at 10 years. After a hundred years they are 24.8, 16.2 and 8.0 mm, 48.9 in all, against the 13.6 mm the thrust shortened the rib on the day. The rib resists them while it creeps itself, so the gap it acts as if fixed against is 20.9 mm (fine dashes). Deflection

The jacks the arch forgets

Jack a fixed concrete arch's abutments apart before the crown is closed, by exactly the gap its thrust and its footings open on the day, and the springings start with no moment at all. A century later they carry 173 kN·m — half as much again as the 112 the jacks were set to remove. The thrust's own moment never relaxes; the jacks' correction does, as the rib creeps; and shrinkage and the ground's consolidation arrive after the jacks have gone. No single setting cancels movements that run on different clocks.

6 figures · Rib shortening The water splits the tower's resonance, and a baffle flattens it. For a 12 m tank 8 m deep on a tower of period 2.5 s with 2% damping, 150 t of container, full: the base shear per unit of harmonic ground acceleration against the shaking's angular frequency, as a share of the frozen-water tower's peak. Frozen to its container, the water makes one resonance; free, it is a second mass on a spring tuned 0.69 times the tower's frequency, against Den Hartog's optimum of 0.71, and it splits the resonance in two — with half a per cent of damping both peaks are higher than the frozen one, 1.83 times it. A baffle giving 10% brings the peak to 0.21; the best damping for the rms, 27%, to 0.11. Dynamics

The water tower that could damp itself

A full water tower's sloshing water is a second mass on a spring, hung from the container at very nearly the tuning a designed tuned-mass damper would be given, and it weighs two-fifths as much as what it hangs from. Plain water has half a per cent of damping, so it splits the tower's resonance into two sharp peaks and helps nothing. A screen across the tank that gives it a tenth makes the water a damper: the tower's rms base shear falls to four-tenths of what it would be with the water frozen, and the same screen keeps working as the town empties the tank.

6 figures · Sloshing Propped before it is filled, the stem keeps the roller's pressure. For a 3 m stem of stiffness 67,500 kN·m² a metre, fixed at its base, propped at its top by a slab cast first, backfilled in 250 mm lifts with a 60 kN/m roller: the lateral pressure on the stem after the last lift, in kPa, against depth — propped at its top before the fill (solid), the same stem as a free cantilever (dashed), a stem too stiff to move (dotted) and the active pressure. The propped stem moves at most 0.15 mm, in its middle, where the free cantilever's top moved 2.61 mm; it relaxes almost nothing, and its pressure lies on the stiff stem's along the whole height — the roller's residual, 27 kPa from half a metre down, four times the active pressure at mid-height. Equilibrium

The prop that holds what the roller left

A basement wall whose ground-floor slab is cast before the backfill spans between its base and the slab, and as the fill is compacted against it the wall bows by a sixth of a millimetre. A free cantilever stem leaned sixteen times as far and shed an eighth of the roller's pressure; the propped wall sheds one per cent. Its base moment falls to a third, which looks like a gain, but the pressure has not gone: the slab now carries 22.6 kN a metre against the 6 an active-pressure design gives it and the 9 of soil at rest.

6 figures · Lateral pressure Heave puts the pile in tension where settlement put it in compression. For a 24 m pile 600 mm across carrying 200 kN, in clay whose top 18 m moves, by 100 mm at the surface: the axial force down the pile, compression positive, when the ground heaves (solid) and when it settles as far (dashed). Settling, the pile's compression peaks at 910 kN at 17.2 m and its toe carries 324 kN. Heaving, it is in tension below 8.9 m, peaking at 520 kN at 17.3 m, and its toe carries nothing: lifted, it parts from the soil beneath, so everything holding the pile down comes from the shaft below the plane. Internal forces

The pile the ground pulls apart

Clay that swells after its trees are felled rises past the piles in it, and the same compatibility that put a pile into compression in settling ground puts it into tension in heaving ground, at nearly the same depth. Nearly everything else changes. A lifted toe parts from the soil and holds nothing, so the shaft alone anchors the pile; the head load that added to the downdrag now subtracts from the tension; and the tension peaks 17 metres down, far below the short cage a lightly loaded pile is given. A house's piles in swelling clay are tension members nobody reinforced.

6 figures · Downdrag Two ways to put a bore in compression. For a tube 2.5 times its bore in a 900 N/mm² steel: the stress difference σθ − σr through the wall, as a share of the yield strength, at rest (dashed) and at the working pressure (solid) — for the wall shrunk together from two layers so that every layer's bore yields together at work, and for the same wall autofrettaged to its best overstrain (thin lines). The layered wall carries 540 N/mm², 1.43 times the first-yield pressure, and at rest its bore sits at −0.43 of yield; the autofrettaged wall carries 756, 2.00 times, its bore at rest at −1.00. Materials

The ceiling the bore sets

A thick tube can be put into compression at its bore two ways: overstrain it past yield once and let it spring back, or build it from layers each shrunk on to the one inside. The second never yields the steel at all. Two layers carry 1.43 times a plain tube's pressure where an overstrain carries 2; four layers 1.75; and no number of layers, no overstrain and no combination of the two carries more than twice. The ceiling is not in either method. It is the range the bore's stress swings through, which belongs to the assembled wall, and the floor that range may reach, which belongs to the steel.

6 figures · Unloading Out of the bed, the beam loses its own weight's help. For the 12 m pretensioned beam, 300 × 700 mm, with eight strands at 220 mm below its centroid: its self-weight moment along its length in kN·m, sagging positive — in the casting bed, which holds it at its ends (dashed), and on bearers 1.0 m in from each end (solid). Over the cantilevers the stored beam hogs, 2.6 kN·m at the bearers; between them its moment is the bed's less a constant 31.5 kN·m, the bearers' reactions acting a metre closer in. Mid-span drops from 94.5 to 63.0. That constant is the relief the top fibre loses everywhere between the bearers: 1.29 N/mm² for each metre of inset. Sections and stress

The beam that leaves the bed

A pretensioned beam's debonding is designed for the moment its strands are cut, when the casting bed holds it at its ends and its own weight is the largest relief its top fibre will get. Then it is lifted by loops a little in from its ends and stored on bearers a little further in. Setting the supports a metre in takes 31.5 kN·m off the self-weight moment along the whole length between them, not just at the ends, and the schedule that passed transfer with 0.07 N/mm² to spare has room for 51 mm of it. In the yard's conditions the bearers may come in 0.57 m; a metre in needs a third strand sleeved, and past 2.2 m nothing does.

6 figures · Prestress limits The deck moves the frames' feet, and the chord goes with them. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, in plan, sideways movements drawn 6 times the scale of the length: each frame's foot pushed 7.5 mm inward by its cross-girder's rotation under the deck (arrows), the chord's movement under that push alone (solid), and under the worst bow alone, four half-waves at 12.0 mm (dashed). The chord follows the push, and under its compression goes a little further — 9.1 mm at its middle frames against the 7.5 mm the feet moved — because a chord 24 m long is far softer sideways than the frames holding it; the frames that carry most are the end ones, beside the chord's fixed ends. Stability

The frames the deck pushes first

A half-through girder's U-frames are the ends of the cross-girders that carry its deck. Load the deck and every cross-girder sags, its ends rotate, and every frame's foot swings inward by millimetres before the compression chord has asked the frames for anything. It looks like a second imperfection on top of the chord's bow. It is a much weaker one: the frames all move together and the chord simply goes with them, so a 7.5 mm push adds 1.4 kN to frames that the bow loads with 13.6, and stiffer frames make it smaller, not larger. What the push leaves is mostly in the frames' own members, not in what they ask of the chord.

6 figures · Continuous restraint The columns and the core shorten apart for decades. For a 200 m concrete building built a storey every week, its perimeter columns at 12 N/mm² and its core at 6 at the base, with an outrigger at 149 m of 1.0 × 10⁸ kN·m a radian: how much the columns (solid) and the core (dashed) have shortened below the outrigger's level, and the difference between them (heavy), in mm, against the years since the first storey was cast, on a log scale. When the building tops out, in week 55, the difference is 7.0 mm; after fifty years it is 30.5. 77% of the difference arrives after the frame is finished — creep under the finishes and the floors above, and the thick columns' slower drying, most of which comes after the core's thinner walls have done theirs. Structural form

The arm connected late

An outrigger is often left unconnected until a concrete building is finished, so that the columns and the core can shorten by their different amounts without it. Most of their difference has not happened yet. Three-quarters of fifty years' differential shortening arrives after the frame tops out, and an arm connected then ends up carrying 3.5 MN of it — 60 per cent of what the design wind puts into the same arm, permanently, and in the same direction on both sides of the core. The connection date is the largest load the arm has, and a stiffer arm collects it faster than it collects the wind.

6 figures · Outrigger A doubler is put to work through its welds, and most of the work is at its ends. A 457 mm beam coped 200 mm long and 50 mm deep, carrying 300 kN, with a 8 mm doubler 393 mm deep on its web, the plate welded along its two long edges and running 200 mm past the end of the cope. The shaded bands along the two welds are the shear flow each passes into the plate, averaged over 50 mm: the top weld passes up to 1,056 N/mm and the bottom 821, both largest where the plate ends. At the cut through the cope's end the plate gives 92 per cent of the relief a plate joined at every point would give: the free edge is at 94 N/mm², against 152 as coped and 88 fully joined. Connections

How far a doubler has to run

A doubler on a coped web is joined to the beam only along its welds, and the force reaches it through those welds from wherever it is welded. Stopped at the end of the cope, a plate welded along its two long edges gives about two fifths of the relief a fully joined plate would give at the very section it was put there for. Run on past the cope by four tenths of its own depth, it gives nine tenths. Welded all round, it is fully joined at any length.

6 figures · Coped beam A building that crosses the trough at an angle stands on a warped surface. Plan of a 30 × 12 m building 10 m high, its walls 300 mm thick, over a trough 30 mm deep with its inflection 15 m from the tunnel's axis, its long axis 45° from square across the tunnel. The lines are the greenfield trough's settlement contours, 5, 10, 15, 20, 25 mm, parallel to the tunnel's axis (dashed). The corners settle A 27.4, B 18.4, C 27.4, D 18.4 mm, so they do not lie in a plane: A and C together settle 18.1 mm more than B and D. A building spanning square across the trough has no such warp, and one lying along it has none either; between the two, the trough twists the building as well as bending it. Deflection

The building the trough twists

A building that crosses a tunnel's settlement trough at an angle stands on ground that is warped as well as sagged. Its four corners do not settle into a plane, and beyond about twenty degrees of skew the warp is larger than the sag. Whether the building follows that warp or resists it depends on a property its bending never needed — whether its walls and floors close into a tube — and the same walls, closed or open, differ a thousandfold in their stiffness against it.

6 figures · Differential settlement A branch braced along its length holds the riser to a floor. Elevation along the branch: a water-filled DN100 riser guided at every floor of the eight-storey building, with a DN50 branch taken off 500 mm below the first floor's slab and run 2.0 m to a brace on the slab above. The riser leans with the storey's drift between its guides; the branch's brace holds the branch's far end to its slab. Under the building's two-way record the riser alone carries 2.8 N/mm² at the tee's height. With the branch braced so that it cannot slide along its length, it becomes a strut tying the riser to the slab: the riser carries 71 N/mm² at the tee and the branch 10. Let the branch slide at its brace and the riser is back to 4.3, while the branch, now turned by the riser's lean, carries 54. Dynamics

The branch that holds the riser to a floor

A riser guided at every floor leans with each storey's drift and is barely bent by it. Take a branch off it and brace the branch to a slab, and the branch becomes a strut that holds the riser to that slab partway up the storey. The riser, which was carrying 3 N/mm², carries 71 at the tee. Let the branch slide at its brace and the riser is spared, but the branch is then bent by the riser's lean. The lean has to go somewhere, and the brace decides where.

6 figures · Floor spectrum The water that rises after the basement is finished pushes on its slab first. Section through a line of columns of a 20 × 30 m basement dug 9 m, its 0.9 m slab held down at six columns and by 8 storeys above, designed for a water table 3 m below the ground. The water table as designed is 3 m below the ground, a 6 m head under the slab; dashed, the same water risen 1.0 m. The slab bows up between its columns and walls under the uplift less its own weight (exaggerated). As designed, the building holds itself down at 0.65 of its weight's capacity, the slab is at 0.37 of its bending strength and its cracks are 0.140 mm against the 0.200 mm long set for every crack in concrete that must keep water out. With the water 1.0 m higher: 0.76, 0.45, and 0.176 mm against the same 0.200. Equilibrium

The basement that leaks before it floats

A water table that rises after a basement is finished is a load that grows through the building's life, and the building's flotation check is the last thing it troubles. A metre of rise adds a sixth to the uplift on the whole building but over a quarter to the net load on the slab, which carries the water less its own weight. The slab's cracks, which a watertight basement is designed to, are the first thing to run out of room — and the limit for the cracks that pass right through it tightens as the water rises.

6 figures · Uplift Three braces do more where the moment is. The 24 m plate girder, its minor-axis second moment 2.67 × 10⁸ mm⁴ and its warping constant 1.00 × 10¹⁴ mm⁶, under a uniform load, held by three torsional braces asked to bring its peak moment to 5,627 kN·m: above, the moment diagram; below it, the twist of the buckled girder with the braces equally spaced, 6.0 m apart, and clustered 4.0 m either side of midspan, each at the stiffness that just reaches the moment. Equally spaced the row needs 8,537 kN·m/rad in all; clustered, 6,325, 26 per cent less. The clustered braces sit where the girder twists most, and the bays they leave at the ends carry little moment. Stability

Three braces where the moment is

A row of torsional braces is spaced equally because that is how it is drawn, and under a uniform moment equal spacing is very nearly the best there is. Under a moment that peaks at midspan it is not. Moving the outer two of three braces toward the middle cuts the total stiffness a 24 m girder needs by a quarter under a uniform load and a third under a point load, and the bays it leaves at the ends cost nothing, because they carry so little of the moment.

6 figures · Beam bracing The same studs moved toward the supports slip less. Half of a 12.0 m composite beam, a 457 × 191 × 67 UB in S355 under a 130 mm C30/37 slab, with 50 per cent shear connection, 19 studs from the support (left) to mid-span (right): spaced evenly, and graded so that they crowd toward the support. Below, the slip along the half-span with the mid-span moment at 95 per cent of the beam's plastic moment: evenly spaced, the end studs must slip 8.8 mm; graded, 4.4 mm. The dashed line is the 6 mm a ductile stud is trusted to give. The plastic moment is the same for both, because the studs are the same in number and each reaches its capacity. Internal forces

The same studs, moved to where the slip is

A composite beam with half the studs full interaction needs reaches 85 per cent of the full plastic moment, if its end studs can slip as far as the beam asks — nearly 9 mm on a 12 m span, past the 6 mm a stud is trusted to give. Keep the same nineteen studs and crowd them toward the supports, and the end studs need 4.4 mm. Nothing about the beam's strength changes; what changes is how soon the slab's force builds up, and the least connection the slip allows falls from 0.69 to 0.40.

6 figures · Composite action Slackening a backprop gives the young slab back more than the preload. The load on one slab of a frame of 250 mm slabs cast a floor every 7 days at 20 °C, on one level of shores and one of backprops, as a multiple of its own weight, from the day it is struck at 7 days to 21 days, with creep. Snug, it takes 1.52 w at the cast above it. With its backprops jacked to 0.15 w it takes 1.37. Slackened to snug 1.0 day later, the backprops give it back everything they carry — the preload, their share of the slab cast above and what creep has added — and its load jumps from 1.19 to 2.00 w. Let down by the preload alone, it goes from 1.23 to 1.38. A week after it is struck the slab becomes the slab below, and the backprops jacked above it land on it whatever is done to its own. Materials

The preload that cannot be taken back

A backprop jacked tight under a young slab relieves it and loads the slab below by the same amount at the same instant. The obvious repair is to slacken the backprop once the young slab is past its worst, so that the preload never reaches the slab below. It cannot work. Slackened to snug, a backprop lets go of everything it carries, not only its preload, and the young slab's load jumps from 1.19 to 2.00 times its weight. Let down by the preload alone, it gives back a load the slab below has already carried at its worst.

6 figures · Maturity Two islands in one film: level with each other, or not. Contours of Prandtl's stress function for a 50 mm shaft bored with two 8 mm holes, their edges 2.0 mm apart; the shear stress is the film's slope, and each hole is an island, shaded, floating where the film's pull round it balances the pressure on it. Left, the holes side by side at 12.0 mm from the axis: by symmetry both islands float at the same height, 219, the film between them stays nearly flat, and the steel there carries almost nothing. The shaft keeps 95 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.13 times the solid shaft's surface stress. Right, one hole inside the other on one radius: the inner island floats at 237 and the outer at 156, so the film between them is steep and the steel between the holes carries up to 1.96 times; the shaft's largest stress, 2.03 times, is at the outer hole's outer edge, with the steel between the holes close behind. Sections and stress

Two islands in one film

Bore a hole along a shaft and its torsion film acquires an island that floats at a height of its own. Bore two and there are two islands, and the steel between them is a strip of film stretched from one to the other. Side by side at one radius, the islands float level, the strip is flat, and two holes cost twice one hole's stiffness and none of its stress. One inside the other on a radius, the islands float at different heights, the strip is steep, and the steel between the holes becomes the shaft's weak point.

6 figures · Membrane analogy A load between the posts racks the panel it stands in. Elevation of a Vierendeel girder of six 2 m panels, 1.5 m deep, its deck chord as stiff as its other chord and its posts, the load standing in the middle of its second panel; the racked shape exaggerated, the unloaded girder faint. The panel the load stands in carries most of its shear, and the deck chord there bends both as a Vierendeel chord and as a beam between its posts: 38 kN·m against 27 with the load at mid-span. The posts either side carry 56 and 11 kN·m, and the heaviest post 56. Structural form

The load that visits every panel

A Vierendeel girder checked for a load at mid-span is checked for one position of a load that, on a bridge, stands everywhere in turn. Crossing it, a 100 kN load puts 1.7 times the mid-span check's moment into the end panels' chords, loads the middle post that a central load leaves idle, and makes the deck chord a beam between its posts in every panel where its share of the shear is small. Equal chords stay the stiffest arrangement; a vehicle whose axles span a panel cannot stand in the middle of one.

6 figures · Vierendeel Every day taken off the cycle is paid for in the youngest slab. For a frame of 250 mm slabs on a 7.5 m grid at 20 °C, on one level of shores: the largest use of strength of any slab against the floor cycle, with one level of backprops snug or jacked to its best preload, and with two levels snug or jacked to theirs; dotted, a seven-day cycle on snug backprops. On one level of snug backprops the worst use is 1.95 at seven days, 2.14 at five and 2.53 at three. With the best preload, 1.87, 2.01 and 2.30. With two levels, 1.79, 1.95 and 2.31; with two levels and their best preload, 1.67, 1.78 and 1.99. Materials

The day taken off the floor cycle

A frame cast a floor a week loads its slabs with the same multiples of their weight whether the cycle is seven days or four. What a shorter cycle changes is how strong each slab is when its load arrives, and it changes that unequally, costing the youngest slab more than the one below it. That is why a faster cycle makes a jacked backprop more useful rather than less, and why a frame that wants four days needs both a second level of backprops and a preload to be as safe as a week on one.

6 figures · Maturity Braces that cannot move can still be made stiffer where the moment is. The 24 m plate girder, its minor-axis second moment 2.67 × 10⁸ mm⁴ and its warping constant 1.00 × 10¹⁴ mm⁶, under a uniform load, held by braces at midspan and 6.0 m either side and asked to bring its peak moment to 5,627 kN·m: above, the moment diagram; below it, the twist of the buckled girder with the three braces equally stiff and with the middle one given 61 per cent of the row's stiffness, each row at the total that just reaches the moment; the ticks are drawn thicker for a stiffer brace. Equal, the row needs 8,537 kN·m/rad, 2,846 in each brace; graded, 7,067, 4,280 at midspan and 1,394 at each side, 17 per cent less. Stability

The braces that stay where the framing puts them

Three cross-frames on a girder usually go where the floor beams or the bridge's bays put them, and cannot be moved toward the moment. They can still be made unequal. Giving the middle one three-fifths of the row's stiffness saves a sixth of the total under a uniform load and more than a quarter under a point load — most of what moving them would save. Moved and graded together, they save no more than moved alone, because both are ways of making every brace twist by the same amount.

6 figures · Beam bracing When the slab lets go, the stem is a cantilever again. For a 3 m stem of stiffness 67,500 kN·m² a metre, fixed at its base and propped at its top by a slab cast before the fill, backfilled in 250 mm lifts with a 60 kN/m roller: the bending moment down the stem in kN·m a metre, the base's moment positive — propped as the fill left it; after the slab has shrunk 2.08 mm away and gone slack; and with the slab tied to the stem and shrunk 4.5 mm, pulling it; dashed, the stem as a cantilever from the start. Propped, the base carries 26.6. Slack, 69.6, 2.6 times as much and close to the cantilever's 82.9. Tied, 119.9, more than the cantilever ever carried, because the slab that propped the top now pulls it. Equilibrium

The slab that shrinks away from the wall

A basement wall propped by its ground-floor slab before it is backfilled keeps the roller's pressure, and its slab takes it. Then the slab dries. Two millimetres of shrinkage at each end — about a year for a 30 m slab indoors — and a slab that only bears on the wall has let go of it, leaving a stem designed as propped to carry 2.6 times the base moment it was given. Tying the slab into the wall to stop that makes it worse: the slab goes on shrinking and pulls, and the base moment passes the free cantilever's.

6 figures · Lateral pressure