Structural Engineering & Statics
where the load goes
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The load must go somewhere — page 17

Essays 385 to 408 of 450 on this thread, in the same order.
Sized for one amplitude, a nonlinear damper damps only a band around it. The damping of a structure with 2.0% of its own and a brace as stiff as the structure, against the storey's amplitude over the amplitude the damper was sized for, on a logarithmic scale, for dampers whose force goes as the velocity to the power α, each sized to the brace's best at the design amplitude. All four give 22.7% there. The linear damper gives it at every amplitude. A tenth of the design amplitude, α = 0.5 gives 6.4%, α = 0.3 1.9% and α = 0.15 1.4%; a hundredth of it, 1.9%, 1.4% and 1.4% — below the structure's own, because the locked brace has stiffened it. Dynamics

The damper sized for the earthquake

A fluid damper whose force goes as its velocity to the power 0.3 is chosen because it caps the force a large earthquake can push through its brace. The same exponent makes its effective size depend on how far it strokes. Sized to its brace's best at the earthquake's amplitude, it adds 20.7% of critical damping there, 5.4% at three-tenths of it, and nothing at all at a tenth — where the brace has locked, the building is stiffer, and the motion its occupants feel meets a resonance that is as sharp as an undamped frame's. Giving half the brace to a linear damper keeps 9.3% under the wind for 1.2 points at the earthquake.

 Balancing the payload is a hedge between lifting it and losing it. The mast's design moment over the payload's own moment, against the share of the payload a counterweight driven from the load cell balances, for a crane whose 50 m jib weighs 300 kN, lifting 60 kN at 48.3 m, with a hoisting factor of 1.20 and a rebound factor of 0.50 for a payload let go at once. Lifting off, the mast carries 1.20 less the share balanced; let go, the share balanced plus 0.50, backwards. The larger of the two is least at a share of 0.35, where it is 0.85. The linked counterweight's half gives 1.00; balancing the whole payload gives 1.50, the worst of the three, because a crane balanced for its load is thrown backwards by all of it when the load goes. Equilibrium

The counterweight that knows the load

A counterweight driven from the hoist's load cell can follow the payload as the rope takes it up, so the step at pick-up that defeated every linked counterweight becomes a ramp a motor can keep pace with. Balancing the whole payload is then the worst thing to do with it. A crane balanced for its load is thrown backwards by all of it when the load is let go — 1.5 times the payload's moment with a rebound factor of 0.5, against 1.0 for the linked crane's half. The best share is a hedge between lifting the payload and losing it: 0.35 of it, for 0.85, which buys 18% more at the tip from the same mast.

 Where the outlet sits decides whether the roof ponds at all. The load the water puts in the more heavily loaded member of a square bay, over what the same head of water would put in it on a rigid roof, against the bay's side, with girders and joists each sized to deflect L/240 under 1.5 kN/m². With the level fixed by a scupper it is 2.01 at 12 m and 3.92 at 18 m, and the bay ponds at 24.5 m. With a drain on the girder line it is 1.24 and 1.41; with a drain at the bay's middle, 0.82 and 0.75, falling as the bay grows, because the water's surface goes down with the roof. Deflection

The roof that drains towards its sag

The depth of water a ponding check takes as given is set by the drainage: an overflow's height above the roof, plus the head a storm needs to leave through it. Which head a roof can carry depends less on how stiff it is than on where the overflow sits. With the level fixed by a scupper in the parapet, a 12 by 9 m bay allows 85 mm of water and an 18 m square bay 39. With a drain at the middle of the bay, whose inlet goes down as the roof sags, the same bays allow 182 and 205 mm — more than a rigid roof would — because the water's surface follows the sag down instead of filling it.

 Below a limit the row settles; above it, it ratchets. The end screw's slip at the top of each cycle, against the number of cycles, for 20 leaning screws 56 mm apart with 40 mm of member depth per row loaded again and again from new to four levels, each screw, reloaded, reaching the slip it had been to with 85% of the load it had there. At 0.94 it settles at 2.50 mm after 3 cycles; at 0.96 it settles at 2.83 mm after 7 cycles; at 0.97 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 21; at 0.99 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 8. The row's first peak is 263 kN; every level drawn is below it. Connections

The row that remembers its overload

A row of twenty leaning screws reaches 98% of twenty screws' peak, once. Loaded again, a screw that has been past its peak comes back along a softer path and reaches the slip it had been to with less load than it carried there. So a row loaded again and again neither simply evens out nor simply ratchets down: below 0.965 of its first peak it settles after a few cycles with the middle screws catching up, and above it the end screws are pushed back onto their curve every cycle, lose a little more each time, and the row fails at a load it carried the first time. One large overload costs it less than many smaller ones near the peak.

 Ground already loaded rocks stiffer than springs that are not. The moment on the column against its rotation through a storm that takes the moment to ±1,000 kN·m and back, for an 8 m strip of EI 540 MN·m² on a bed of 50 MN/m² under a 1,000 kN column: on elastic springs that cannot pull, and on ground that unloads and reloads 4 times as stiffly as it first loads, in the first storm and the third. On elastic springs the column turns 1.418 milliradians at the peak, a rocking stiffness of 0.70 GN·m/rad, the same every storm. On the remembering ground it turns 1.116 in the first storm and 0.868 in the third — 0.90 and 1.15 GN·m/rad — because rocking unloads half the strip, and ground that has carried the column unloads stiffly. Internal forces

The footing the storms press down

A footing strip under a column that a storm rocks one way and then the other was expected to dig itself a bowl: its ends loaded half the time, compacted, and left hanging over the hollow. On ground that unloads and reloads four times as stiffly as it first loads, the bowl appears and the strip fills it. The column settles 26% more over three storms and then stops; the footing rocks 22% less than it did in its first storm and is 64% stiffer than on springs that forget, because rocking unloads half the strip and ground that has carried the column unloads stiffly. The storms change the settlement and the stiffness, and leave the moments almost where they were.

 Squeezed while it bends, the hinge rides down its plastic moment. Moment against curvature for an I-section of 3,000 mm² and 200 mm depth in elastic–perfectly plastic steel, the moment over its unloaded plastic moment and the curvature over its unloaded first-yield curvature: at a constant 10% of its squash load, at a constant 60%, and with the compression rising from the first to the second as it bends (dashed: the plastic moment at the compression the rising path has reached). At constant loads it settles at 0.96 and 0.44. The rising path carries 0.86 at twice the first-yield curvature, 0.70 at 6.0 times it and 0.43 at 12.0: it forms its hinge early, at the lower load, and then loses moment as fast as the plastic moment falls with the load. Materials

The hinge squeezed while it turns

The leeward column of a frame overturning in an earthquake gains compression as its base hinge rotates. Followed fibre by fibre, with every fibre's plastic strain remembered, the hinge does neither of the things the constant-load curves suggest. It forms early, at the low load it started with, and then rides down the section's plastic moment as the load rises: an I-section whose compression goes from a tenth to six-tenths of its squash load holds 0.86 of its unloaded plastic moment at twice its first-yield curvature and 0.43 at twelve times it. The steel never softens, and the hinge still has a falling branch, set by statics. The relieved column on the other side gains what this one loses, less the curvature of the interaction curve.

 The film over an angle's corner. Contours of Prandtl's stress function over the corner of an equal angle 100 × 100 × 10 mm with a 12.0 mm root radius and 6.0 mm toe radii: left as rolled, right with sharp corners. Along each leg the contours run parallel to its faces; at the corner they bulge into the metal the fillet adds, round the outside of the L. The rolled angle's J is 6.80 cm⁴ and the sharp one's 6.18 cm⁴. The dot marks the steepest slope: on the fillet in the rolled angle, in the re-entrant corner in the sharp one, where it grows without limit as the grid is refined. Sections and stress

The corner with one fillet

A rolled I-section's web meets its flanges in two T junctions, four fillets, and its torsion constant is 7 to 12 per cent above Σbt³/3 because of them. An angle meets itself in one L, and a channel's web meets its flanges in two. Solved over the real outline, an L junction's fillet adds about two-thirds of what a T junction's does with the same plates. The sum then errs on either side: low by 8% for an I-section with a channel's plates, high by 0.8% for the channel itself, and anywhere from 5% low to 2% high for angles, depending on which length of leg is counted. The fillet's peak stress is still the inscribed circle's, to a few per cent.

 A brace on the sideways mode gives back the snap. The load parameter against the crown's deflection, both in radii of gyration, for a shallow pinned arch whose rise is 8.0 radii of gyration, built 0.1 of a radius of gyration out of shape in its sideways mode, unbraced and with a brace on the sideways mode of 0.88, 1.75, 3.50 times the rib's own antisymmetric bending stiffness. Unbraced it peaks at 410; with 0.88 it peaks at 658; with 1.75 it peaks at 794; with 3.50 it peaks at 840. The perfect arch's limit point, which a brace can only return it to, is 844; unbraced, it turns sideways at 461. Stability

The brace that holds a mode

A shallow arch rising eight radii of gyration turns sideways at a load of 461 when its symmetric snap would not come until 844. A member that only stops the quarter points moving in opposite senses — a brace on the sideways mode, not on any one point — gives the whole difference back once it is 1.75 times as stiff as the rib's own resistance to that mode, and nothing more after that. The stiffness needed grows as the square of the rise. And the brace is not the light member the argument suggests: an arch built a tenth of a radius of gyration out of shape puts an eighth of its whole load through it at the peak.

 A stiff floor shares by stiffness, a soft one by distance. The share of a brace line's force that reaches its soft bay, against the other bay's stiffness over the soft bay's on a logarithmic scale, for a 48 by 24 m floor carrying 40 kN per metre of span, each brace line 960 kN, with a 6 m braced bay of 200 kN/mm at one end of each line and a 6 m bay 10 times as stiff — for three floors (dashed: a rigid floor, which shares by stiffness, and the tributary half). With the other bay ten times as stiff the rigid floor gives the soft bay 9%; the concrete slab 16%, the cracked slab 30% and the steel deck 37%. Structural form

The floor that feeds two bays

A brace line with a braced frame at one end and a core wall ten times as stiff at the other is analysed one of two ways. As a rigid floor, which shares the line's force between the bays by their stiffness and gives the frame 9%. Or as a flexible floor, which gives each bay the half of the line nearest it, 50%. Solved as a membrane, an uncracked concrete slab gives the frame 16%, the same slab cracked 30%, and a steel deck 37%. The floor does not send its shear to the nearer bay; a stiff floor sends it to the stiffer one. The frame's design force depends on which state the floor is in when the storey force arrives, by more than a factor of two.

 A short leg past the elbow ends the strut and moves the stress. Peak stresses in the two-direction earthquake against the length of the leg past the elbow, for a DN50 branch off a water-filled DN100 riser, its tee 500 mm below the first floor's slab, run 2.0 m to an elbow of 76 mm radius and on to a brace on the slab above: the riser at the tee, the branch at the tee with the welding tee's stress intensification of 1.58, and the elbow with its own of 1.73. Straight and held, the riser carries 71 N/mm² (dashed). With a 100 mm leg it carries 19, the tee 82 and the elbow 146; with 300 mm, 4, 82 and 84; with 1 m, 3, 53 and 39. From a leg of about 400 mm the tee carries more than the elbow. Dynamics

The elbow that takes the strut away

A branch braced to a slab is a strut that holds a guided riser to that slab, and the riser, carrying 3 N/mm² alone, carries 71 at the tee. Turn the branch through an elbow before its brace and the strut is gone with 300 mm of second leg: the riser falls to 4. The storey's lean does not vanish. It bends the first leg and the elbow instead. With the stress intensification the piping rules give each fitting, the elbow carries 146 N/mm² with a 100 mm leg and matches the tee's 82 at 300 mm. Past that the tee governs again. An elbow can rescue the riser, but it cannot rescue a brace on the farther slab, where it reaches 347 N/mm².

 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.

 The series sum is exact for matched flanges and errs by their mismatch. The pair's stiffness over the series of its two flanges as tee stubs on rigid bases, each with half the bolt's length, against the end plate's lever arm from web to bolt, for end plates 14, 18, 25 mm thick on a column flange 14 mm thick with its bolt 30 mm from its web, tips 35 mm beyond the bolt. Where the end plate matches the column flange — 14 mm thick at 30 mm — the ratio is exactly one: two identical flanges back to back meet on their plane of symmetry, which is the rigid base. A 14 mm end plate is 0.95 of the series with its bolt at 45 mm and 0.85 at 65; an 18 mm end plate is 0.97 of the series with its bolt at 45 mm and 0.86 at 65; a 25 mm end plate is 1.00 of the series with its bolt at 45 mm and 0.93 at 65. Connections

The flanges that lean on each other

A bolted end-plate joint's stiffness is built from springs in series: the end plate as a tee stub on a rigid base, the column flange as another, the bolt shared between them. The rigid base is not a fiction. It is exactly the plane of symmetry between two identical flanges bolted back to back, so for matched flanges the series sum is exact. A real end plate and column flange are not matched, and each tip bears on a tip that moves differently from its own. Solved as a pair, the error is the mismatch: 3% for an 18 mm end plate on a 14 mm column flange with similar lever arms, 12–15% when the end plate's bolt is twice as far from its web.

 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.

 What it costs depends on the load it must still carry. The steel in an eight-panel Pratt truss with a counter-diagonal in each interior panel, loaded by 10 at every bottom joint, sized to survive the loss of any one interior diagonal, against the share of the full load it must carry with the diagonal gone; the intact fully stressed design's 1,200 and the Pratt truss's 1,220 as level lines. At a third of the full load the robust truss needs 1,221, at a half 1,252, at two thirds 1,304, at the full load 1,555, and at four thirds — two thirds with the loss counted as sudden, doubled — 2,074. From about a half the interior diagonals grow in proportion to the load, each sized by its partner's loss; past the full load the chords grow too, because the damaged truss is then carrying more than the intact one was sized for. Deflection

The truss sized to lose a diagonal

A Pratt truss with a counter-diagonal in every interior panel, sized fully stressed, starves its middle counters to nothing and saves a little steel over the plain Pratt. Ask it to survive the loss of any one diagonal at the accidental two thirds of its load and the family of equal designs collapses to one: every counter comes back, sized by its partner's loss, and the truss needs 1,304 units of steel against 1,200 — 9 per cent for an alternative path the Pratt cannot have at any price. A lost chord can be survived too, by the panel's diagonals turned into a chord, for twice the steel. Four members in the end triangles cannot be survived at all.

 The loaded end stops; the unloaded one takes over. The moments at the four floor beam ends of the frame of two 8 m bays and two 4 m storeys, its beam ends joined to the columns by end plates of 3.0 EI/L initial stiffness and 120 kN·m resistance, under λ times 20 kN/m on the left bay and 20 kN at each floor, against λ: on the softening joints (solid) and on joints that keep their initial stiffness (dashed). At λ = 1 the loaded beam's inner end carries 78 kN·m and the unloaded beam's inner end, across the same column, 13. The loaded end reaches its resistance at λ = 2.69; at λ = 3 the four ends carry 89, 120, 81, 88 kN·m, against 97, 235, 39, 68 on stiff joints. The unloaded inner end has gone from 13 to 81, 6.2 times as much for three times the load. Internal forces

The joint that softens under its own load

A bolted end plate is stiff until its moment passes two thirds of its resistance and then softens until, at its resistance, it is a third as stiff. Put such joints at every beam end of a frame under a pattern load and a push, and the most loaded joint softens first and hands its share across the column to the joint on the other side, which carries six times as much for three times the load. The redistribution grows as the frame approaches its strength. Every joint ends at its own stiffness, so the single secant stiffness an elastic analysis is allowed asks the loaded joint for half as much again as it can carry, and gives the exterior panel three quarters of what it gets.

 Rigid enough long before it is a plate. A box 4.0 m wide and 2.5 m deep on a 60 m span, 45 kN/m at 2.0 m off its axis, with three diaphragms, each diaphragm of racking stiffness K, against K over the girder's own stiffness against a point distortional load, 15.0 × 10⁹ N·mm/rad: the largest warping stress (solid) and the largest distortion, which drives the walls' transverse bending (dashed), each over its value with rigid diaphragms. At ten times the girder's stiffness the warping stress is 1.05 of the rigid value and the distortion 8.37 times it; at a hundred, 0.89 and 1.50; at a thousand, 0.99 and 1.05. Marked: a cross-frame of two 1,000 mm² diagonals (26 times) and a 10 mm plate (524 times). Sections and stress

The diaphragm that is a few angles

A box girder's diaphragm holds its section square against the distortion an eccentric load drives. It does not have to be rigid; it has to be stiff against the girder's own resistance to a point racking load, which is small — twice the section's frame stiffness times its decay length. A cross-frame of two 1,000 mm² diagonals is twenty-six times that and holds the warping stress at the rigid value, while a 10 mm plate is five hundred times it. The cross-frame lets the section rack three times as far as a plate, which leaves the walls bending at 5 N/mm² instead of 1.5, against 275 with no diaphragm at all. And its diagonals carry nearly a rigid diaphragm's force whatever their stiffness, so it is sized for strength.

 The range belongs to the truss. The worst hanger force range against the tie's largest moment in a tied arch of 100 m span and 17 m rise on 32 crossed hangers of 3,000 mm² bar, 60 kN/m of dead load and 60 kN/m of live load run on from either end, as the hangers' stiffness runs from four times a plain bar's to a fiftieth of it (dots), and for the same arch on vertical hangers (the lone dot). Softening the crossed hangers walks the bridge from 678 kN of range and 316 kN·m of tie moment to 557 and 1,154; vertical hangers sit at 194 and 3,505. The line bends nowhere near the vertical arch's point: a soft network is still a truss. Structural form

The hanger that cannot be made soft enough

A network arch's crossed hangers swing through nearly all their force as traffic passes, because they are the web of a truss. Make them softer — a spring at each anchorage — and the tie ought to take back some of the shear and the hangers' ranges ought to fall. They hardly do. A tenth of the stiffness takes 6 per cent off the worst range and adds a third to the tie's bending; a fiftieth takes 18 per cent off and multiplies the tie's bending by 3.7. The range belongs to the truss and leaves only when the truss does, which vertical hangers show at 194 kN against 678 and eleven times the bending. So the hanger is sized for its range, eight times the area its strength needs, and the stiffness that comes with it costs the range nothing.

 Whole spans miss the shear inside a span. The shear along three spans of 8.0, 8.0, 8.0 m, under 20 kN/m of dead load everywhere and 15 kN/m imposed: the largest and least when the imposed load stands wherever it raises or lowers the shear at each section (solid), and when it may only cover whole spans (dashed). At the supports the two agree. At the middle of the second span the exact envelope reaches 23.8 kN either way and the whole spans 10.0 kN; at a tenth of the way along it, 122.4 against 122.0. Internal forces

The shear whole spans cannot find

A continuous beam is designed by loading its spans in patterns, whole spans at a time, and for the bending moment that is very nearly exact: the moment's influence line keeps one sign over each span, so a span is either worth loading or not. The shear's influence line changes sign at the section itself, inside the span, and no pattern of whole spans can load one side of a section and not the other. At the middle of the inner span of three equal spans the imposed shear is 0.198 qL; whole spans find 0.083 qL, and the straight line to qL/8 that is safe on a simple span finds 0.125.

 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 screen damps a tank on a tower and barely touches a standpipe. The rms drift under broadband ground shaking, as a share of the same structure with its water frozen to the wall, against the damping a screen gives the sloshing: a steel standpipe 10 m across and 30 m tall with a 12 mm shell, full (solid), and the water tower of the earlier essay, a 12 m tank 8 m deep on a tower of 2.5 s (dashed). Plain water, at half a per cent: 0.82 and 1.01. A screen giving 10%: 0.76 and 0.40. The standpipe's gain comes almost entirely from leaving the lens out of the shaft's motion, which plain water already does. Dynamics

The tank that is its own tower

A standpipe holds its water in its own shaft, so the water fills a height several times its width. Only a lens of it near the surface sloshes — 0.455ρπR³, 178 t of the 2,199 t in a steel standpipe 10 m across and 30 m tall — and that lens stops growing once the water is deeper than a radius, so its period is set by the width alone: 3.3 s, against the shaft's 0.38. The lens is 8% of the water but a quarter of the shaft's moving mass, because it sits where the shaft moves most. It is far too slow to be tuned as a damper, and a screen can do little with it; what it does instead is stay still while the shaft sways, which takes nearly a fifth off the drift a frozen-water calculation predicts.

 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.

 Allowed to yield, the damaged truss is hardly lighter. The steel in an eight-panel Pratt truss with a counter-diagonal in each interior panel, loaded by 10 at every bottom joint, sized to survive the loss of any one interior diagonal, against the share of the full load it must carry with the diagonal gone: checked elastically (solid) and by the lower-bound theorem, every member allowed to yield (dashed); the intact design's 1,200 as a level line. At a third, 1,221 elastic and 1,206.7 plastic; at two thirds, 1,304 elastic and 1,296.7 plastic; at the full load, 1,555 elastic and 1,545.0 plastic; at four thirds, 2,074 elastic and 2,060.0 plastic. The plastic design saves 7.4 units at two thirds of the load out of the 104 that robustness costs. Deflection

The section that yielding cannot reach

A truss that loses a member in an accident is allowed to yield while it finds another way to stand, so a counter-braced truss sized by the lower-bound theorem ought to need far less steel than one sized elastically. It needs 1,296.7 units against 1,304 — seven of the 104 that robustness costs. One diagonal gone, the damaged panel is determinate at its own section, and no redistribution can change a force a cut decides; yielding can reach only the verticals. What does move the price is a counter that buckles: good for half its tension in compression, it adds 120 units, more than robustness did.

 The preload a storm can live with, raised by a spring under the nut. The largest preload the bolts of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows can be given and come out of a storm without stretching, against the stiffness of a stack of disc springs under each nut, for storms to ±165, ±170, ±175 kN·m; dashed, the same for the bare base; dotted, the bolt's yield of 226 kN. To ±170 kN·m the bare base keeps at most 128 kN; with a stack of 500 kN/mm, 150; 100 kN/mm, 187; 20 kN/mm, 214; 5 kN/mm, 223. Connections

The spring under the nut

Retightening a column base after every storm makes its bolts ratchet, because a storm can only live with 128 kN of preload and the spanner keeps putting back 150. A stack of disc springs under each nut changes what the storm can live with. Soft enough, the bolts never reach their yield: the compression side keeps its preload, the bed's resultant moves out to the plate's edge, and the moment rides on lever arm rather than on bolt force. A stack as stiff as the bolt itself keeps 150 kN through every storm. Much softer, and the storm the bolts no longer feel lands on the grout.

 The pair holds its clamp longer than the stub it is checked by. The tension a bolt at which the clamp lets go, as a share of the 110 kN preload, against the end plate's lever arm from web to bolt, for an end plate 18 mm thick on a column flange 14 mm thick at 30 mm: the pair (solid), the end plate's stub (dashed) and the column flange's (dotted). With the end plate's bolt at 30 mm the pair and the stubs nearly agree. At 45 mm the pair lets go at 88.2 kN and the end plate's stub already at 81.1, the pair holding 9 per cent more; at 60 mm, 82.0 against 71.2, 15 per cent more. Connections

The clamp the pair holds longer

A preloaded bolt clamps an end plate to a column flange, and the joint stays closed until the tension relieves the clamp. The joint diagram says that happens at the preload over one minus the bolt's share — 122 kN for a 110 kN preload — but prying adds to the tension at the bolt line and the clamp lets go at 88. A component model checks each flange as a stub on a rigid base and lets the joint go when its weaker stub does, at 81 kN. The real pair, its tips leaning on a column flange that gives way, pries less than the end plate alone would, and holds its clamp nine per cent longer — fifteen when its lever arms are two to one.

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