Structural Engineering & Statics
where the load goes
The collection

Every essay — page 18

Essays 409 to 416 of 416, in the same order.
Nine-tenths of a tightening torque stretches nothing. Where the torque applied to the nut of an M20 grade 10.9 bolt goes, against the coefficient of friction in its thread and under its nut, taken as equal. The bottom band is the thread's lead — the only part of the work that stretches the bolt — the middle band is friction in the thread and the top band is friction under the nut. At μ = 0.14 the lead takes 11% of the torque, the thread 39% and the nut face 50%. At μ = 0.06 the lead's share is 22% and at 0.24 it is 6%, so a coefficient nobody measured decides how much of a specified torque arrives in the bolt as preload. Connections

The torque that goes into the thread

A preload specified as a torque is a preload specified through two coefficients of friction that nobody measures. Nine-tenths of the torque on a bolt is spent turning against its own thread and the face of its nut, so a change in the grease moves the clamping force by half — and the one method that escapes it does so by yielding the bolt on purpose.

7 figures · Slip-critical The compression is under the flange, not at the edge of the plate. A 500 × 400 mm base plate, 20 mm thick, under a 260 mm column carrying 300 kN and 120 kN·m, with holding-down bolts 60 mm from the tension edge. A plate this thick can deliver bearing only 43 mm past the compressed flange, so the compression sits under that flange, 124 mm from the column's centre, and the lever arm to the bolts is 314 mm: the bolts carry 264 kN and the flange zone 564 kN. The dashed block is the rigid-plate answer, 54 mm deep at the plate's far edge with its resultant 223 mm out, which needs only 128 kN in the bolts. Both satisfy equilibrium; only one is a plate that can carry the pressure where it is drawn. Connections

The compression that stays under the flange

Put a moment on a base plate and a pressure-block calculation pushes the compression out to the plate's edge, where the lever arm to the holding-down bolts is longest. A plate that is not rigid cannot put it there. The compression stays within a few tens of millimetres of the compressed flange, the lever arm shrinks, and the bolts carry twice what the pressure block said.

7 figures · Base plate Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 200 mm, carrying a reaction of 300 kN through three bolts, as utilisations, for the end as coped and with three repairs: an 8 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.46, shear 0.41, tear-out 0.57 and local buckling 0.46. The doubler thickens the web, which is most of what a coped tee is, and lowers all four; the edge plate gives the tee back a flange and lowers only flexure and buckling, leaving shear and tear-out exactly where they were. What governs: as coped, tear-out at 0.57; doubler, tear-out at 0.30; edge stiffener, tear-out at 0.57; both, tear-out at 0.30. Connections

The repair that fixes the wrong check

A coped beam end that fails its checks is usually repaired by welding a plate along the edge the cope left, because the cope removed a flange and the plate puts one back. On ordinary proportions that repair restores the check that was not failing. The check that governs is carried by the web, and only a plate on the web reaches it.

7 figures · Coped beam The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing. Dynamics

A wall that is allowed to lift

A block that rocks inherits everything that decides whether it survives — a restoring moment set by its weight and shape that falls as it leans, and a loss of energy set by its proportions at each landing. Put a tendon through a wall and yielding bars across its base and both become design quantities, and the ratio between them decides whether the wall comes home.

8 figures · Rocking Two separate checks, and the block that is neither. Amplification of the steady displacement at a machine 2.0 m above the base of a 150 tonne block 5.0 m across and 2.0 m deep, against forcing frequency. Checked separately, the block sways at 11.2 Hz with 29 per cent of critical damping and rocks at 15.0 Hz with 13 per cent, and the rocking check's peak is 3.99. Because its mass sits above its base the two are one system, with modes at 9.9 Hz and 20.9 Hz — one below both checks and one above — and the block's own peak is 2.49 at 9.7 Hz. Dynamics

The frequency below both checks

A machine block is checked for sway and for rocking as if they were two oscillators, each with its own spring, its own radiation damping and its own natural frequency. A block whose mass sits above its base is one oscillator with two modes, and neither of them is a sway or a rocking — one sits below both checks, one above both, and the lower one is where the machine resonates.

8 figures · Vibration isolation However stiff the ties, a free line stops short. The first three frequencies of stays 180, 150 and 120 m long joined by cross-ties, against the stiffness of each tie from 1.0 kN/m to 1000 MN/m, with the line free at its ends and, dashed, anchored to the deck at both ends with the same stiffness. With the line free the first frequency rises from 0.73 Hz and levels off at 0.82 Hz, however stiff the ties are made — short of the 1.01 Hz of the 120 m stay, which a free line cannot pass. Anchored, the first frequency reaches 1.09 Hz. At the 6.0 MN/m marked, the free line gives 0.82 Hz and the anchored line 1.09 Hz. Dynamics

The line of ties that stops short

Cross-ties are the one intervention on a stay cable that changes its frequencies rather than damping them, and the usual account says they lift the stays out of the range that excites them. A line of ties that is not anchored to anything cannot lift the first frequency above the shortest stay's own, however stiff the ties are made. What lifts it is carrying the line to the deck.

8 figures · Cable dynamics One collision, four contact laws, one impulse. Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The elastic linear spring peaks at 19.7 MN and lasts 58 ms; the Hertz law peaks at 22.0 MN over 61 ms. Set to a restitution of 0.65, the spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN, a tension two faces in contact cannot carry, while the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 instead. The areas under the curves are the impulses: 600 kN·s for the spring and dashpot, which is what momentum requires at 0.65, 646 for the damped Hertz law, and 727 for both elastic laws. Dynamics

The force that belongs to the model

When two buildings meet, momentum decides what each of them feels, and no contact law can change it. What the contact law decides is the force, and the force it reports is the contact stiffness somebody assumed, raised to a power. Four laws and a hundredfold change of stiffness move the buildings by a few per cent and the force by a factor of eight.

7 figures · Pounding The rates a seated crowd makes worse. The stand's steady rms acceleration under 40 people jumping, against the rate they jump at, for a stand of 30 t modal mass at 6 Hz with 2 per cent damping, empty and with 15 t of spectators sitting on it. Empty, the worst rate is 3.00 Hz, where the second harmonic of the jump lands on the stand's own frequency, and the stand reaches 2.54 m/s², 25.9 per cent of gravity. Occupied, the worst anywhere from 1.5 to 3.5 Hz is 0.49 m/s² at 3.50 Hz. But from 1.64 to 2.32 Hz, shaded, the occupied stand responds more than the empty one — 1.6 times as much at 2.00 Hz. Dynamics

The crowd that is also the structure

A crowd jumping to music loads a grandstand with harmonics several times those of walking, and nothing about their size is measured: they follow from a pulse that has to average one body weight. But the people who jump arrive with the people who sit, and the seated crowd is mass, stiffness and damping bolted to the stand. It lowers the worst case by a factor of five and makes the most common jumping rates worse.

7 figures · Floor vibration