Structural Engineering & Statics
where the load goes
The collection

Every essay — page 21

Essays 481 to 489 of 489, in the same order.
The strength design, and the least steel that is stiffer. A Pratt truss of eight panels at a depth of 1, carrying 10 kN at each top joint, each member drawn as wide as its section from a catalogue whose sections step by 25 per cent in area, the smallest 10 per cent of the largest member's need. Above, every member at the smallest section that carries its force. Below, the least steel that makes mid-span 1.50 times as stiff with every member still strong enough — the discrete optimum — with the 24 members it made larger drawn in the second colour: six of six top chord members, eight of eight bottom chord members, six of eight diagonals, four of seven verticals. The optimum uses 44 per cent more steel than the strength design, and it puts it where a member's real and virtual forces are both large; members either force leaves small are not touched. Deflection

The calculus answer, rounded, is the worst one

A real truss is built from a catalogue, and every member is rounded up to the next section. That rounding costs its stiffness almost nothing: the few per cent that separate the strength design from the stiffest survive it. What does cost is the next step. When a deflection limit governs, the obvious move — take the continuous optimum and round it up — needs more steel than any other way of stiffening the truss, and the exact discrete answer is within one per cent of a bound no catalogue can beat.

6 figures · Truss deflection The rotation a bearing sees is made before it arrives. The end rotation of a 12 m pretensioned beam, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, from the prestress's release at three days to thirty years, with sagging positive. Release turns each end 4.8 milliradians upward; creep takes it to 7.9 by day 28, when the beam is set on its bearings. The slab brings it back by 0.9, the surfacing by 0.1, and thirty years of creep take it to 8.1 upward. The band after erection is the imposed load (0.9 down) and a night with the top cooler (0.6 down) above the line, and a sunny day with the top warmer (1.0 up) below it. Every value in the band is below zero: under every load it will carry, the beam's ends still point up, and a level bearing is turned the same way for its whole life. Deflection

The angle made in the casting yard

A bridge bearing is designed for the rotation of the beam it carries, and the rotation is listed as a sum of load, temperature, creep and a tolerance. Followed through the life of a pretensioned beam, the largest term is none of those. It is the upward turn the prestress gives the beam's ends in the casting yard, before any bearing exists, and under every load the beam will ever carry its ends still point up.

5 figures · End rotation The thin plate straightens onto the load line. The centre lines of a 10 mm plate lapped on a 20 mm plate, 100 mm wide, over 60 mm, carrying 60 kN with the grips 1200 mm from the overlap, over 480 mm either side of the overlap, with every vertical distance drawn 14 times the lengthwise scale; dotted, where the centres would be if the joint could not rotate. The load pulls along the straight line between the grips, each of which holds its plate on its own centre, and the moment where a plate enters the overlap is the load times the gap between that line and the plate's centre. The thin arm, flexible under tension, has swung toward the line: its gap is 0.49 of the rigid one, e/2. The overlap has turned with it, and the thick arm, too stiff to follow, is left 1.34 of e/2 from the line. Peak stresses: 193 N/mm² in the thin plate and 121 in the thick one, against 330 and 98 if the joint could not turn. Connections

The thicker plate takes the bend

A lap joint's eccentricity puts a moment into both plates, and the rule is that each takes half of the load times the offset. That is true only while the joint cannot turn. Let it turn, and a thin plate lapped on a thick one straightens under its own tension and hands its share across: the thick plate ends up bending a third more than the rule says, the thin one half as much, and in the limit the thick plate would take the whole of it.

5 figures · Single lap The same blow, three seats under the anvil. The movement of the block — the part of a 150 t hammer foundation, 30 t of it the anvil, that lies below the pad — after one blow of 18 kN·s on the anvil, over three periods of the whole foundation on soil (12.5 Hz, damping 0.47). Rigid seat: largest movement 0.86 mm at 16 ms; pad at 4.0 times: largest movement 1.14 mm at 13 ms; pad tuned to the foundation: largest movement 1.18 mm at 35 ms. On the pad at 4.0 times the block rides the anvil's ringing: a ripple at the anvil's own frequency on top of the foundation's swing, whose first crest lands near the swing's peak. On the tuned pad the block receives the blow as one slow push, peaks later and higher, and then goes on ringing long after the rigid seat has settled, because the mode in which anvil and block swing against each other is damped by the pad and hardly at all by the ground. Dynamics

The pad that makes the blow worse

A forge hammer's anvil sits on a pad on its foundation block, and the pad looks like isolation: a spring between the blow and everything below it. For the block it is the opposite. Every pad an anvil can sit on makes the block move more than a rigid seat would, by half again when the pad is tuned near the foundation, and what the pad buys instead is a smaller force under the anvil. The hope that a tuned pad could act as a tuned mass works only against a train of blows, and only at a softness the anvil cannot live with.

5 figures · Vibration isolation The forces the girder can hold with nothing on it. The one self-stress state of a K-braced girder of eight 3 m panels, 3 m deep: member forces in equilibrium at every joint with no load and no reaction. It lives entirely in the two middle panels — twelve members: the four diagonals meeting at the middle joint of the central vertical in tension, and the four central chord members and the half-verticals at the outer side of each middle panel in compression, at 0.89 and 0.45 of the diagonal force. It is symmetric about mid-span, so a symmetric load can call on it as freely as any other: symmetry does not supply the missing equation. Every other member of the girder is idle in it, drawn faint. Equilibrium

The redundancy only the sun can find

A K-braced girder that is symmetric about mid-span has one member more than statics can resolve, and it is at the centre, where two K's meet at one joint. Symmetry does not remove it, because the forces it allows are symmetric themselves. Loads barely touch it — a section calculation that ignores it is exact for a load at almost any joint. What finds it is a millimetre of misfit or a sunlit top chord, which put forces into the middle of the girder that no load calculation contains.

5 figures · Method of sections The date the joint is cast decides the sign. The moment at the pier after thirty years, for two 12 m pretensioned beams, 300 mm wide and 700 mm deep with a 160 mm slab, made continuous over the pier by a joint cast with the slab, against the beams' age when the joint and slab are cast; sagging positive, with its three parts dashed. Cast at 7 days the joint ends at +320 kN·m; at 28, +214; at 90, +59; at a year, −164. It exceeds the joint's cracking moment of 159 kN·m for any joint cast before about 45 days, and it changes sign at about 130 days. From 7 days to a year the prestress's share falls from +555 to +204, because an old beam has made most of its upward creep before it is joined; the differential shrinkage's grows from −57 to −285, because an old beam has finished its own shrinking and the slab's is then all difference; the dead load's eases from −178 to −83. The first two move the joint the same way as the beams age. Materials

The pier that bends the wrong way

Two precast beams are made continuous over a pier by a joint cast with the deck, and the joint is designed for the hogging moment a continuous beam has there. Thirty years later it is sagging, by more than the moment that cracks its underside, because the beams were still cambering upward when they were joined. Whether that happens is decided by two dates — when the beams were cast and when the joint was — and the deck's shrinkage, which pulls the other way, is not enough to stop it.

5 figures · Creep The bow the frames have to hold. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN critical at 1548 kN, drawn with its sideways movement exaggerated; its critical load is 1.66 times the chord force. Dashed, the worst of the bows tried: four half-waves, each 12.0 mm high, a five-hundredth of its own length. Solid, the chord once the compression has grown the bow; each arrow is the force the frame there must supply, the largest 13.6 kN — 1.47 per cent of the chord force. Stability

What the frames hold is the bow

A row of U-frames is sized for a force, and the rules give that force as a percentage of the compression chord's force that nobody derives. Derived, it is not a property of the frames at all. It is the chord's initial bow, grown by how near the chord works to buckling — and it is least, not greatest, near the stiffness at which the chord stops using the frames. Stiff frames tend to a limit that is π²/500 of the chord force: the familiar two per cent, arriving from a bow of one five-hundredth.

5 figures · Continuous restraint A curved I-beam's flanges curl. The cross-section of an I-beam bent to a radius of 3.0 m, its flanges 300 mm wide and 15 mm thick, with the flanges' radial movement exaggerated 47 times; dashed, where the flanges would be if they did not bend across their width. It is bent the way that opens the curve, so the outer flange is in tension and the inner in compression. A tensioned flange round a curve is pulled toward the centre of curvature and a compressed one pushed away from it, so here both are pressed toward the web, and each outstand bends like a cantilever from the web, its tip moving 0.80 mm. Bent the other way, both would curl away from it by the same amount. The bars are the stress along the beam across each flange: the full value at the web, falling toward the tips to 0.72 of it, so that the flange works as if it were 89 per cent as wide. Sections and stress

The flange that curls away from its stress

Bend an I-beam into a curve and each flange, carrying its stress round the bend, is pressed sideways by that stress. The outstands bend like cantilevers from the web, and in bending they move to a different radius and shed the very stress that pushed them. At a tight radius only a strip beside the web works. But the flange's loss of width arrives slowly, and a stress nobody computes for a straight beam arrives at once: the flange bending across its own width, harder than it is stressed along the beam.

5 figures · Curved beam The same bridge, anchored to the ground and to itself. The deck's bending moment under the live load, for a 500 m suspension span with a 50 m sag, a deck of EI 1.03 × 10¹² N·m², 100 kN/m of dead load and 20 kN/m of live load on half the span. Anchored to the ground, the deck carries at most 56.3 MN·m, because the cable's whole tension works on its deflected shape. Anchored to the deck's own ends, 83.0 MN·m — 1.47 times as much, and a little more than Rankine's elastic theory, 80.1, which leaves the tension out altogether. The deck's compression acts on the same deflected shape as the cable's tension, with the opposite sign, and cancels it. Structural form

The bridge that pays for its own anchorage

A suspension bridge's deck is light because the cable's tension works on its deflected shape and stiffens it. Tie the cable to the ends of the deck instead of to the ground and the deck must carry the same pull as a compression — which acts on the same deflected shape with the opposite sign and cancels the stiffening exactly. A self-anchored bridge is designed by the theory the long suspension bridge was invented to escape, and the price grows as the square of the span.

5 figures · Stiffening girder