Structural Engineering & Statics
where the load goes
The collection

Every essay — page 31

Essays 721 to 728 of 728, in the same order.
An old joint passes the bolts that have lost their preload. The chance that an M20 grade 10.9 bolt holds the inspection torque of 647 N·m without its nut turning, against the tension actually in it as a fraction of the minimum preload, for a joint new and 1 year old, 5 years old and 20 years old. New, a bolt at half the minimum preload passes 0% of the time and one at the minimum 83%; 20 years old, the bolt at half passes 80% of the time and one at a quarter 9%. Connections

The wrench that reads the rust

The obvious check on a preloaded joint that has been in service for years is a wrench on the nut, and it measures the wrong thing. The nut holds against the bolt's tension times a friction bracket, and twenty years of rust raise the bracket until a bolt with half its preload passes four times in five. The washer that could have said more remembers only the day it was crushed.

6 figures · Slip-critical 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.

6 figures · Differential shortening 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.

6 figures · Tuned mass damper 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.

6 figures · Uplift Where the shelter is decides whether the leading row works harder. The share of the cap's shear the leading row and the last row of three rows carry, as a multiple of the average pile's, against the thickness of soft ground over stiffer, in characteristic lengths, with the row factors applied at every depth (solid) or in the soft layer only (dashed). With the factors at every depth the leading row carries 1.24 and the last 0.83 at 2.0 lengths; in the soft layer only, 1.07 and 0.96. In uniform ground the usual reading gives 1.35 and 0.75. Internal forces

The shelter that stops at the interface

Piles in a group shelter one another, and practice counts the shelter as a factor on each row's soil, calibrated on groups in uniform ground. Put the same group in soft ground over stiff and the calibration has to be read one of two ways — the shelter at every depth, or only where the ground is soft — and the two readings disagree about which row works hardest. The first gives the leading pile 17 per cent more moment at the cap; the second gives the last pile 24 per cent more in the ground.

6 figures · Lateral pile Where along its front a surface crack starts to cleave. Where along the front of a surface crack 8 mm deep in a 40 mm plate at −60 °C, 22 degrees below the steel's reference temperature cleavage starts, as a probability per millimetre of front, at the stress that gives a 5 per cent chance of cleavage, against the position on the front, for cracks 80 mm long, 40 mm long and 16 mm long at the surface. The middle third of the front, 60° to 120°, holds 76 per cent, 69 per cent and 28 per cent of the chance. Materials

The front that counts for half its length

A ferritic steel's cleavage toughness depends on how much crack front it has to sample, and for a crack through a plate the front is the plate's thickness. The cracks inspections find are not through cracks; they are thumbnails at weld toes, whose curved front sees a stress intensity that varies round it. Weighted by the weakest link, the front counts for a little over half its length — so counting all of it is safe by 8 per cent, and counting none of it, as the 25 mm toughness does, is unsafe by up to a fifth for a long shallow crack.

6 figures · Fracture A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54. Sections and stress

The bore with corners and the passage beside it

A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

6 figures · Membrane analogy A floor that shears lowers the storey's buckling load before the core does. The buckling factor of a storey of eleven lines of leaning columns, ten bays, each column carrying 3,000 kN over 3.6 m and built a three-hundredth out of plumb, held by a floor to a core sized for a buckling factor of 4.0 were the floor rigid, against the floor's shear stiffness per bay as a multiple of a column's sway softening P/h, logarithmic, with the core at one end, in the middle, or split between both ends; dotted, the rigid floor's 4.0. At 100 times the factor is 1.52, 2.80 and 3.02; at 1,000 times, 3.49, 3.84 and 3.88; a concrete slab is several thousand times. Stability

The floor the columns lean on

A storey of pin-ended columns, every one a little out of plumb, leans on its floor, and the floor leans on a core. Count the floor as rigid and the core needs the sum of the columns' demands. A floor that shears in its own plane is a brace in series with the core, and the far columns lean on it before their push arrives. With a floor a hundred times stiffer per bay than a column's own sway softening, the storey's buckling factor falls from 4.0 to 1.5 and the core is asked for twice as much.

6 figures · Effective length