Structural Engineering & Statics
where the load goes
The collection

Every essay — page 25

Essays 577 to 585 of 585, in the same order.
The arm goes in almost the same place whatever shape the wind is. The share of the tip drift removed by one outrigger on a 200 m core of bending stiffness 1.2 × 10¹⁰ kN·m² carrying 6,000 kN of wind in all, with an outrigger of 1.0 × 10⁸ kN·m per radian, against the outrigger's level as a share of the height, for the three profiles of the same total wind. uniform: best at 0.745 of the height, removing 45.6 per cent; power law: best at 0.749 of the height, removing 45.9 per cent; triangle: best at 0.757 of the height, removing 46.4 per cent. The curves lie almost on one another: the triangle, with three quarters of its load in the upper half against the uniform wind's half, moves the best level up by 1.2 per cent of the height. Structural form

The arm that ignores the shape of the wind

The outrigger arithmetic is usually done for a wind that is the same at every height, and real wind is not: it grows with height, and an earthquake's first mode loads a building as a triangle. The natural guess is that a load concentrated towards the top moves the best place for the arm. It barely does. Put the same total wind on a 200 m core as a uniform load, a power-law wind and a triangle, and the best level moves by just over one per cent of the height. What moves is the force in the arm — half as much again under the triangle — and how far the top goes, while the arm's own stiffness shifts its best level thirty times as far as the wind's shape does.

6 figures · Outrigger A counterweight that follows the jib leaves only the payload, halved. The moment in the mast of a crane whose 50 m jib weighs 300 kN and luffs between 15° and 85°, lifting 60 kN at its tip, against the luffing angle, with a counterweight linked to the luff so that it balances the jib and half the payload at every angle: 1,449 kN·m loaded and −1,449 empty at 15°, 131 and −131 at 85°. What is left is the payload's moment less the half the counterweight was set for, plus or minus 1,449 kN·m at most, against 4,744 with a fixed counterweight. Equilibrium

The counterweight that follows the jib

A counterweight that stays put balances one position and leaves the mast a moment at every other. Make it move and it can balance anything that moves slowly enough to be followed. On a luffing crane the thing that moves most is not the payload but the jib itself, whose own moment about the mast changes more than twice as much through its luff as the payload's ever is — and a counterweight linked to the luff cancels that change exactly. What it cannot cancel is the payload, which leaves the ground in a second. The best any counterweight can do with that is wait halfway, and the mast is left with half the payload's moment at the longest radius, whatever else the counterweight does.

6 figures · Counterweight The linear answer has the ground holding the ends down. The contact pressure along a strip 16 m long, EI 5.40 × 10⁵ kN·m², on a bed of 50 × 10³ kN/m² per metre of its length, under a column of 1,000 kN. With springs that pull as readily as they push (dashed) the bed presses 197 kN/m under the column and pulls down on the ends at up to 34 kN/m — tension the ground cannot supply. With the pulling springs released (solid), the strip touches the ground only from 3.97 m to 12.03 m, 8.05 m in all, and presses 213 kN/m under the column, 8 per cent more; beyond that it carries nothing. Internal forces

The ends the ground was holding down

Winkler's springs pull as readily as they push, and on a long footing strip the linear answer quietly uses that: it has the ground holding the strip's ends down. Take the pull away and the correction under the column is modest — eight or nine per cent on the pressure and the moment — but the shape of the answer changes completely. A weightless strip longer than about eight metres keeps exactly eight metres of itself on the ground, whatever length was poured, and the rest rises off as straight cantilevers carrying nothing. The middle-third rule turns out to be a property of rigid footings: an eight-metre strip lifts at a fiftieth of the eccentricity the kern allows.

7 figures · Elastic foundation Below its ceiling, a row of braces is only its total. The critical moment of a beam 24 m long whose unbraced critical moment in uniform bending is 837 kN·m, held by one, two, three or five torsional braces equally spaced along it, against the braces' total rotational stiffness. At 4,336 kN·m/rad in all, the rows of two, three and five give 3,815, 3,726, 3,578 kN·m: how the stiffness is divided hardly matters, while one has already stopped. Each row then stops at its own ceiling, where the braces are stiff enough to be nodes and the beam buckles between them in one more half-wave than it has braces: 2,637 kN·m for one, 5,585 kN·m for two, 9,704 kN·m for three, 21,466 kN·m for five. The dashed line is 5,627 kN·m, the moment the beam is asked to carry; one and two cannot reach it however stiff they are. Stability

The stiffness a row of braces shares out

One torsional brace at midspan has a ceiling, the moment at which the beam gives up twisting it and buckles in two half-waves instead. A row of braces raises the ceiling with every brace added, and the stiffness each must have to reach its ceiling rises with the count — seventy-four times as much for eight braces as for one. That is the right answer to the wrong question. Asked for the moment the beam has to carry rather than for its ceiling, a row of braces needs very nearly the same total stiffness however many it is divided between, so each brace gets softer as the count goes up. The count has one job, which is to put the ceiling above the moment; the stiffness has the other.

6 figures · Beam bracing The preload lifts the young slab and lands on the one below it. The load on one slab of a frame of 250 mm slabs on a 7.5 m grid cast a floor every 7 days at 20 °C, on one level of shores and one of backprops, as a multiple of its own weight, from the day it is struck at 7 days to 28 days, with creep. With its backprops put in snug (dashed) it takes 1.52 w at the cast above it, creeps down to 1.29 by the end of the week, takes 1.48 w when the slab above it is struck and cast on in turn, and creeps up to 1.71. With every backprop jacked to 0.30 w (solid) the young slab starts its week at 1.22 w and ends it at 1.14; a week later the preload of the backprops above lands on it, and it takes 1.78 w, creeping to 1.86. Materials

The backprop jacked tight lands on the floor below

A backprop put in snug shares only the load that arrives after it, and creep then pushes the young slab above onto it anyway. Jack it to a preload instead and the young slab is lifted at once — but the preload survives the week, so the lift lands on the slab below the moment the next level is struck, and about half of it is load creep would have moved within the week regardless. The best preload balances the week-old slab against the fortnight-old one, and with creep it is small: a fourteenth of a slab's weight, about six hundred newtons a prop. Anything over a fifth leaves the frame worse than snug. What a small preload does buy completely is immunity to slack.

6 figures · Maturity The tub girder's shear centre at three stages of its building. A steel trough 1,500 mm deep, 3,000 mm between its web tops and 2,000 mm across its 12 mm bottom flange, with 12 mm webs and a 400 × 20 mm flange on each web, drawn to scale. Open, as it is lifted, its shear centre lies 706 mm below the bottom flange, outside the steel entirely. Closed across the top by a bracing truss equivalent to a plate 0.50 mm thick, which carries shear and no bending stress, it is still 538 mm below. With a 250 mm concrete deck cast and hardened across the web tops it is 1,127 mm above the bottom flange, inside the box. The steel's centroid, 673 mm up, does not move until the deck is cast. Sections and stress

The box that is a trough until its deck is cast

A composite tub girder is a box only once its concrete deck has hardened. Before that it is an open steel trough — two leaning webs and a bottom flange — whose shear centre lies seven hundred millimetres below its bottom flange, in the air, and whose resistance to twisting is the thickness of its plates cubed. A light truss across the top is what holds it together while the deck is poured, and it is a plate for shear and nothing else: an angle of ordinary size is worth half a millimetre of steel. That half-millimetre multiplies the trough's torsion constant two thousandfold, and leaves its shear centre almost exactly where it was. The bracing closes the box for twisting; it does not close it for the shear centre.

6 figures · Shear centre A crack at the cope's corner, and the hole its tip needs. A 457 × 190 beam end coped 150 mm long and 80 mm deep, reacting 350 kN of which 70 per cent comes and goes with traffic, to scale. A fatigue crack has grown 30 mm down the web from the corner where the cope's two cuts meet, the most stressed point of the coped section. At that depth the live load's range of stress intensity at its tip is 15.6 MPa√m, and a hole drilled with the tip inside it stops the crack only if its radius is at least (ΔK/10.5√fy)², 6.2 mm: a 12 mm hole. The crack and the hole together leave 341 mm of the 377 mm coped section below them. Connections

The hole at the tip of the crack

The commonest reason to repair a coped beam end is a fatigue crack at the corner of the cope, and the commonest repair needs no plate at all: a hole drilled so that the crack's tip is inside it, turning a crack into a smooth notch that the traffic can no longer drive. The hole has to be big enough, and how big is set by the crack — its radius grows in step with the crack's length. A 30 mm crack at a stringer's cope needs a 12 mm hole, a 60 mm crack a 24 mm one. The repair stops working at about 100 mm, where the hole outgrows a site drill and the section left beside it runs out of capacity at nearly the same depth. Most of the crack's life is spent far shorter than that, which makes the repair a matter of how early the crack is found.

6 figures · Coped beam A cambered girder is an arch on its slings. A 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, with 100 mm of camber at mid-length, drawn in elevation with the camber exaggerated. The girder's centre of gravity is two-thirds of the camber above the line through its ends, 67 mm; its lifting points, on the arc 2 per cent in from each end, are 8 mm above it. It rolls about the line through the picks (dashed), so the camber has lifted its centre of gravity 59 mm towards the roll axis: the girder hangs as though its hook were that much lower. Deflection

The camber that lowers the hook

A camber is built into a girder so that it ends up level in the finished structure, and nothing about it matters until then — except on the day it is lifted. A cambered girder hanging from its ends is a shallow arch, its centre of gravity two-thirds of the camber above the line through its lifting points, and it rolls about that line as though its hook were that much lower. On a long precast girder already close to its lifting limit, a hundred millimetres of camber takes the factor of safety against roll from 2.3 to 1.4. The camber grows while the girder waits in the yard, so the same girder becomes harder to lift every week it is stored. Pick it a fifth of its length in from each end and the camber drops out of the problem altogether.

6 figures · Camber A filler that knows the speed meets the collision hardest at its start. The contact force against the depth crushed in the hardest collision, 500 t and 300 t closing at 2.85 m/s with 761 kJ, on a 40 mm filler between 500 t and 300 t, crushable over 24 mm: uniform (dashed), with 50 per cent of its strength at that speed depending on the speed, and with all of it. Each is as strong as it must be to stop the collision in 24 mm. Uniform: 30.5 MN at first contact; 50 per cent on speed: 38.4 MN at first contact; all on speed: 63.4 MN at first contact. The crushing speed is highest at the first instant and falls to nothing, so a filler whose strength follows it spends its force at the start; the same area under a falling line needs a higher peak than under a level one, and a purely viscous filler's is twice the uniform's. Dynamics

The filler that knows how fast it is hit

A filler graded through its depth meets the small collisions softly and the hard one with a strong layer that has to be very strong, because every kilojoule the soft layer misses must be absorbed in less depth. A filler whose strength rises with the speed it is crushed at ranks collisions by speed instead, with no layers at all, and treats the middling collisions better than grading does. But it meets every collision hardest at its first instant, when the crushing speed is greatest, and a force that falls as the collision proceeds needs a higher peak to absorb the same energy in the same depth. A purely viscous filler that stops the hardest collision in 24 mm hits it with exactly twice the force of a uniform one.

6 figures · Pounding