Structural Engineering & Statics
where the load goes
Theme

Which failure arrives first — page 18

Essays 409 to 432 of 495 on this thread, in the same order.
A core drilled into a slab the studs were laid out for without it. 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, its studs laid out for the slab without a hole — six rings of 14 mm studs on twelve rails, the last ring 4.25d from the face — and a 150 mm core 225 mm (1.00d) from the face. Its shadow from the column's centre takes 20.3 degrees of every perimeter. The studs' outer perimeter stands where the last ring leaves it, 5.75d out (solid); with the core the slab needs it at 5.74d (dashed), so the outer check is at 1.00 of its capacity, and the control check's studs at 0.79. No rail runs into it. Internal forces

The core drilled after the slab was built

A building in use acquires new services, and a core drilled through a flat slab beside a column is a routine instruction. The studs round the column were laid out for a slab without the hole, and the drill finds whatever is there. Whether it cuts a stud matters less than it looks. What decides is the outer perimeter, which the hole pushes outward while the studs stay where they are, and whether the slab survives is set by a margin nobody chose: the room left when the studs were rounded up to a stock size and the rings to a whole number.

 The moment the law lets go. The response of a structure of one mode with 1 per cent damping and a tuned mass of 2 per cent of its mass, its spring cancelled by an actuator held to 0.50 of the applied force and a stroke of 1.50 times the passive mass's, per unit static deflection, under a steady harmonic force at 0.90 of the structure's frequency, through a health check failing after sixty periods, when the actuator lets go and the mass is left on its own spring. With the law running the structure moves 4.63; left to the passive mass in a steady state it would move 6.97. In the cycles just after the switch it moves 8.78, more than either: the free mass's large stroke is handed to a spring that holds it at a frequency the forcing is not at, and the two beat before the passive state settles. Dynamics

The switch that hands the mass back

An actuated tuned mass that cancels its own spring does far better than the passive mass, and the passive mass is always underneath it, so the obvious safeguard is a switch: when the stroke nears its limit or the controller fails, let go and leave the mass on its spring. The switch does not land on the passive mass. For several periods after it, the structure moves more than it would with the law or with the passive mass alone — up to the passive mass's own worst peak, at frequencies where the passive mass would have been comfortable. Letting go gradually is worse still, and a switch that is slow lets a growing storm through.

 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 shear term cuts the beam's stiffness, not its share. For 20 storeys of two 6.0 m walls 0.35 m thick either side of a 2.4 m opening, coupling beams 0.35 m wide, 60 kN at every floor, against the coupling beams' depth: each beam's stiffness with its shear deformation counted over its stiffness in bending alone, the degree of coupling likewise, and the largest beam's shear likewise. At a depth of 0.6 m (span over depth 4.0) the stiffness is 0.85 of the bending figure, the coupling 0.977 and the shear 0.961; at 2 m (1.2), 0.33, 0.973 and 0.937. Where the shear term is large the beam is stiff enough that its stiffness has stopped deciding anything. Deflection

The coupling beam asked for shear

The coupling beams of a pair of shear walls are the stubbiest members in a building, and the shear term in their deflection is largest exactly there: a beam two metres deep over a 2.4 m opening is a third as stiff as bending alone says. The walls hardly notice. The pair's degree of coupling moves by under 3%, the beams' shears by under 7%, because a beam that stiff has stopped deciding how much of the overturning the walls carry as a couple. What decides it is the shear the beams can give. A 600 mm beam limited to 2 N/mm² starts to yield at 1.9 times the storey force; past the last yield the pair is two cantilevers again, and the beams supply rotation instead.

 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.

 Pinned bases: weaker, and nothing lost to repetition while gravity leads. For a 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 and the reversing wind's H·h/Mp: the shakedown boundary (solid) and the collapse boundary under the loads applied once (dashed), on fixed bases and on pinned ones. On pinned bases the two coincide wherever gravity leads, along the beam and combined mechanisms, until the wind reaches about H·h/Mp = 1.6; past that a knee alternates and shakedown falls inside collapse, meeting it again at H·h/Mp = 2 with the wind alone. On fixed bases collapse reaches 8 and 4 and shakedown sits inside it except where the beam's mechanism meets the gravity axis. The dot is 160 kN of gravity at midspan that comes and goes with 60 kN of wind that reverses: its shakedown factor is 0.88 on fixed bases and 0.73 on pinned ones, where it is also the collapse factor. Materials

The base that sends the wind to the knees

A fixed-base portal loses 12 per cent of its load to a wind that reverses, because each knee takes its worst moment from a different storm. Pin the bases and the loss disappears — not because the pinned frame is better, but because its collapse load already contains it. What the bases decide is where the moment goes, and a base that is merely weaker than its column can cost nothing at all, or be made into the part that yields every storm so that nothing else does.

 A hat makes the top storeys look stiff. Each storey's own sway factor α up the height of a 20-storey building of a core to the roof and a racking frame (αH 1.0), its gravity scaled to an elastic critical factor of 5.0, under an equal lateral load at every floor: with no hat, with a hat truss at the roof holding the core's rotation with 10 times the core's EI/H, and with the same restraint at storey 10; the building's critical factor 5.0 as a vertical line. With no hat the top storeys read the least, 4.06 at the roof. With the hat the roof storey reads 15.47 and the least, 4.14, has moved down to storey 9. With the outrigger at mid-height the least is back at the top, 3.85. Stability

The hat that hides the lean

A hat truss on a core-and-frame building holds the top of the core, nearly doubles the building's critical load, and makes its top storeys look three or four times as stiff as they did. The storey check believes them. Under the hat the roof storey reads a sway factor of 15 and estimates 7 per cent of second-order drift; the building still buckles leaning, and the roof storey's true increase is 24 per cent. The hat turns the storey check from pessimistic to unsafe at exactly the storeys it was built to stiffen.

 A haunch moves the worst stress range to its toe. The stress range up the height of the first interior post of a Vierendeel girder of six 2 m panels, 1.5 m deep, every member of the same section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load, over one crossing: the plain post (dashed), and the post with 250 mm haunches 2.00 times as deep at both corners (solid); with the constant-amplitude limits 52.3 MPa for detail category 71 (dotted). Plain, the range is 78.3 MPa at the corner. Haunched, it is 43.3 at the corner and jumps to 61.2 at the toe, 250 mm along the post — the largest anywhere in it, 61.2. Structural form

The corner that counts its crossings

As a load crosses a Vierendeel girder, the moment at a post's corner changes sign when the load passes that post. A static check reads the peak, and the peak says the posts beside the supports work twice as hard as the middle one. Fatigue reads the range, and the range says every interior post works the same. A haunch at the corner lowers the corner's range and moves the worst of it to the haunch's toe — so a deeper haunch soon buys nothing, a longer one keeps paying, and a toe two detail categories worse than the corner cancels the haunch outright.

 The single side's strongest direction is not the double side's. The resultant a millimetre of throat can carry in S355 against the direction of the load: a double-sided butt throat, a fillet throat, and a 20 mm plate's single-sided throat at penetrations of 0.5 and 0.7, counted plastically. The double-sided butt is strongest at 22.7°, 478 N/mm². The single side at 0.5 rises from 183 straight across to 314 straight along and is strongest along the weld; at 0.7 it rises from 291 to 357 at 40° and falls to 314, and from 38° to 88° it carries more than the fillet. Connections

The root that has to stretch for it

A partial-penetration weld made from one side carries its pull off the plates' line, and counted elastically it keeps a quarter of a double-sided weld at half penetration. Let the throat yield and it keeps 0.41 — but the same 0.41 at every direction up to the one where the double-sided weld is strongest, so that weld's peak advantage is never won back. What the yielding does instead is move the single side's own strongest direction out to 40° or beyond, where a deep single-sided throat outcarries a fillet. Every per cent of it is paid for with strain at the root, which is the tip of an unfused land.

 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 mechanism outcarries the frame that is nearly one. The load on the joint of two bars of 200 MN axial stiffness across 8.0 m against how far it moves down, followed to large displacement: the flat pair with its bars tensioned to 0, 50 and 200 kN with nothing applied, and the same bars raised 3 degrees with no tension, a rise of 210 mm. The raised frame is stiffer at first and snaps through at 11.1 kN, 89 mm down. The flat pair at 200 kN starts at 100 N/mm, carries 11.1 kN at 89 mm and 50 kN at 210 mm, and goes on stiffening; at 50 kN of tension, 50 kN at 242 mm; with none, at 252 mm. Equilibrium

The mechanism that stands up anyway

Two bars in a straight line between two pins are a mechanism by the count, by the rank and by every measure of how nearly a frame folds. Tension them and they carry load: 200 kN of tension gives the joint the stiffness of the same bars raised 1.8 degrees, and where the bars raised 3 degrees snap through at 11 kN, the tensioned flat pair carries 50 kN and goes on stiffening. The stiffness is a lack of fit of four millimetres, which 83 degrees of warmth takes away; compressed, the same pair buckles to rest crooked on one side or the other; and no tension can reach a mechanism that swings rather than shivers.

 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.

 The wire's strength buys the wall. The thinnest wall, as a multiple of the bore, on which a wound tube can carry the ceiling, against the wire's strength over the liner's. Wire as strong as the liner needs 3.01; 1.2 times as strong, 2.30; 1.5 times, 1.69; from about 2.00 times, any wall. Dotted, 2.22: the thinnest wall on which shrunk-on layers of the liner's own steel reach the ceiling, where σy·ln k is twice the first-yield pressure — thinner than the same steel wound, because a layer can carry compression and a wire cannot. Materials

The wire that cannot push

A tube's liner wound with wire under tension is a compound tube of a hundred layers, each shrunk on by its own winding. It reaches the same ceiling, twice the first-yield pressure, and no further, because the bore's range belongs to the whole wall and its floor to the liner's steel. What the wire's strength decides is where the ceiling can be reached at all. A wire cannot be compressed, so a jacket of the liner's own steel needs a wall three times the bore, where shrunk-on layers of the same steel need 2.22; wire half as strong again needs 1.69; wire twice as strong reaches it on any wall. Every turn is wound harder than it will rest, by what all the turns after it take away.

 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.

All themes