Structural Engineering & Statics
where the load goes
Theme

The load that will not hold still — page 7

Essays 145 to 150 of 150 on this thread, in the same order.
Spread over the span, the rotation gathers at the supports every drop shares. The plastic rotation each station of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ has gathered by the last drop, for drops always at mid-span, at 0.3 and 0.7 of the span in turn, and at random. Always at mid-span, the worst station holds 0.193 radians, at 3.0 m, and 3 stations more than a milliradian; at 0.3 and 0.7 in turn, the worst station holds 0.104 radians, at a support, and 4 stations more than a milliradian; at random, the worst station holds 0.128 radians, at a support, and 15 stations more than a milliradian. Dynamics

The weight that lands somewhere else

A beam that yields under a falling weight takes a permanent set from every drop, and if the weight always lands in the same place the set grows by the same amount every time. Let the weight land somewhere different each time — rocks on a canopy, crates on a deck — and the set still grows: the weight's energy fixes how much plastic work every drop does, wherever it lands. What the landing point decides is where the work goes. Half of every drop's hinge rotation goes under the weight and half to the supports, so wandering drops spread the first half along the span and leave the second half where it always was: the worst hinge turns half as fast, and it is at a support.

 Over many earthquakes the column line ratchets together or stops together. The compression over the squash load of the interior column (solid) and an exterior column (dashed) against the cycles, up to 400, for the frame of three ground-storey columns under floor loads of 15, 20, 25 kN/m on 6 floors; the dotted line is where the hinge starts to ratchet, 0.15. At 15 kN/m the interior column starts at 0.22 and the average is 0.12; after 400 cycles the interior carries 0.15 and the exterior 0.10. At 20 kN/m the interior column starts at 0.29 and the average is 0.16; after 400 cycles the interior carries 0.16 and the exterior 0.16. At 25 kN/m the interior column starts at 0.36 and the average is 0.19; after 400 cycles the interior carries 0.19 and the exterior 0.19. Where the average lies above the threshold the three end at the average and ratchet together; where it lies below, the interior column sheds load until it stops at the threshold and the exterior columns never start. Materials

The floor that cannot keep up with the column

A column whose hinges ratchet under an earthquake shortens every cycle, and in a frame it does not shorten alone: the floors it carries bend down towards it and lift its load onto the columns beside it. That ought to slow it and speed them, until every column shortens together. It does — over hundreds of cycles. Over the ten or twenty of an earthquake, the floors of a six-storey frame lift an eighth of the load off the interior column, it shortens nine tenths as far as it would alone, and the floors carry the whole difference as bending. The frame equalises its columns eventually, and pays for it in its floors.

 The dynamic ratio peaks where the modes couple, and falls where the static one climbs. The standards' ratio, the larger edge's displacement over the average of the two edges', for a 30 × 18 m floor swaying with a period of 0.5 s, against Ω, the torsional radius over the floor's radius of gyration, for natural eccentricities of 2.1 and 4.2 m: from a static push at the centre of mass (dashed) and from a response spectrum combined by CQC (solid); the dotted lines are 1.2 and 1.4. At 2.1 m, static 2.05 at Ω = 0.5 falling to 1.08 at 2; dynamic 1.08 at 0.5, highest at 1.42 near Ω = 1.15, and 1.10 at 2. At 4.2 m, static 2.46 at Ω = 0.5 falling to 1.15 at 2; dynamic 1.17 at 0.5, highest at 1.55 near Ω = 1.20, and 1.19 at 2. Structural form

The floor that twists in its own time

Seismic design standards flag a twisting floor by the ratio of its worst edge's movement to the average of its two edges', and a static push saturates that ratio. Give the floor mass and shake it, and it sways and twists in two modes that share the motion. A response-spectrum analysis then reads the same ratio from combined peaks — and gives a different verdict on a third of the plans tried. Floors torsionally stiffer than they sway read worse than a push says, because the coupled modes put twist at the edges. Floors torsionally flexible read better, because combining the modes keeps how far each edge moves and forgets which way, so a twisting floor's two edges both look large.

 The survey knows the creep once most of the floors are set. The doubt about the creep of a 40-storey building cast a floor every seven days, its creep known to ±30 per cent, as the half-width of its 95% band, after a survey of every floor set by then — each read to a standard error of 1.0 mm — against the floor it is made after: from the survey alone (drawn at 100% where it is larger), and combined with the design's ±30%; beside them, the share of the building's floors still to be cast, which are all a survey can correct. The survey alone beats the design's doubt only after floor 25, with 38% of the floors still to cast; the combined doubt halves after floor 31, with 23% still to cast. Deflection

The survey that arrives after the floors

A tall building's lower storeys start shortening months before its upper floors are cast, so a survey every ten floors should tell the creep the building really has and let the floors still to come be set for it. It does tell it, to a millimetre's precision, after the twenty-fifth floor. By then most of the floors whose height it was meant to correct are already set, and the floor that ends furthest from level is the one no survey can help.

 Taking hold of a swinging mass throws it. The largest stroke of a tuned mass of 2 per cent, as a share of its limit, after the free-mass law takes hold of it while it swings on its own spring and damper in a steady storm at the structure's frequency, against the instant in the structure's period at which the law takes hold; dashed, the law waiting for the stroke to cross zero, and for the mass to come to rest relative to the structure. Depending on the instant the stroke reaches 1.39 to 3.12 times the limit; waiting for zero stroke, 2.81; waiting for rest, 2.27. Dynamics

Taking hold is harder than letting go

A tuned mass driven by a law that cancels its spring is safe to let go of: the transient is the same whenever the switch falls, and the passive mass takes over. Switching the law back on is the reverse operation and behaves nothing like it. The mass the spring was holding is thrown up to three times past its stroke limit, by an amount that depends on the instant, and a switch that is allowed to come back in a storm near its threshold comes back hundreds of times.

 The silt hands the head back over decades, a season at a time. The basement is under five storeys, its water table 2.0 m above the design head and swinging ±0.5 m over the year, highest in April; pumped drains under its slab start 30 times as conductive as the ground. The head under the slab above the design head, against the years since the drains were new, for drains silting with time constants of 4, 8 and 16 years; dots, a standpipe read every October on the 8-year drains. With those drains the head reaches flotation in the wet season of year 24.1; the reading first exceeds 0.25 m in year 14.8, 9.4 years before. Equilibrium

The drain that fails a season at a time

A basement held at its design head by drains and a pump can fail suddenly, when the pump stops, or slowly, when fines settle in the drains and the head under the slab creeps up while the pump keeps running. The slow failure gives a standpipe read once a year a decade of warning, in its own time constants — if it is read in the wet season. And long before the silt floats the building on its own, it has made every stop of the pump more likely to.

All themes