Structural Engineering & Statics
where the load goes
Concept

Erection — where it appears

The assembly of a structure, during which it is a sequence of different structures and none of them is the one that was analysed. Members can carry their largest force, and stability can be at its worst, on a day that appears on no drawing.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

At thirty degrees each leg carries the whole load. A 100 kN lift on two legs at 60 degrees to the horizontal. Each leg is a two-force member, so its force is the vertical share it carries divided by the sine of its angle: T = W/(2 sin β) = 57.7 kN, which is 0.58 times the whole load in each. The curve on the right is that division, and it is the reason the rule about sling angles exists rather than a convention: at sixty degrees a leg carries 0.58 W, at forty-five 0.71, at thirty exactly 1.00, and at fifteen 1.93. And the horizontal components do not disappear — they run through the thing being lifted, which here carries 57.7 kN of compression between the two pick points. That is what a spreader beam is for: it takes the compression as a designed strut so the legs above it can stand up.

The angle that doubles the force

A crane hook takes the weight and nothing else. What holds the load is a set of legs running down to it at an angle, and each leg is a two-force member — so at thirty degrees from the horizontal each of two legs carries the entire load, and the difference goes into the thing being lifted as compression.

equilibrium · Rigging Five millimetres short, and a hundred kilonewtons in every member. An X-braced bay 6 m by 4 m in which one diagonal was fabricated 5 mm short, with the force that leaves in every member. Nothing is applied to this frame. The forces are the self-stress state the frame's one redundancy supports, scaled so that the diagonal is pulled back to the length it should have been: tension in both diagonals at 100 kN, compression in the four members round the outside, and the whole set in equilibrium with nothing. That is 25% of the force the diagonal was sized to carry, and it is there for the life of the structure. Take one diagonal out and the frame becomes determinate: the short member then simply puts the joint somewhere else, and the structure is in the wrong place instead of under stress. Redundancy is bought, and this is the price.

Built to the wrong length

A redundant structure's members do not have independent lengths. Choose all but one and geometry decides the last, so a member made a different length has to be pulled or pushed into place — and the force required stays in the structure for as long as the structure does. Nothing has been applied to it, there is no load case and no factor, and the members are carrying real force.

connections · Fit-up A preloaded joint, before and after it slips. Four preloaded bolts at 172 kN each, on one friction face at μ = 0.5. The joint carries 344 kN by friction with the bolts in tension and not in shear at all; past that it slips into bearing and carries 362 kN with the bolts now in shear. Two different mechanisms, one joint.

The hole made bigger so the steel would fit

A preloaded joint carries load by friction, and the friction is reduced by the shape of the hole the bolt passes through — not by how much steel the hole removes, but by a coefficient in a table. An oversize hole costs fifteen per cent of the resistance; a long slot costs thirty-seven. Both are provided because the steel would not otherwise line up.

connections · Slip-critical One span hung, the other still bare tendon. The level along two 100 m spans at a sag of 2.00 per cent, 35 kN/m finished and 3.0 kN/m of bare tendon, the tendons at 1100 N/mm² when finished, with the deck hung on the left span only (solid) and as finished (dashed), on a pier of 200,000 kN/m. The hung span sags 2.04 m and pulls 21,442 kN; the bare tendon beside it sags 0.20 m and already pulls 18,363 kN, because it was cut 455 mm shorter than the span so as to carry its share when finished. The pier is asked for the difference, 3,080 kN, and leans 15 mm towards the hung span — against 4,953 kN under 20 kN/m of crowd on one finished span.

The tendon that pulls before the deck arrives

A stressed ribbon is hung one span at a time, and the obvious fear is the stage at which one span carries its deck and the next does not: the pier between them asked for a whole span's thrust. It is not asked for that, because tendons cut to the finished length are already stretched across the empty span and pulling. For two 100 m spans at a fiftieth, the rigid pier's stage force is 4,109 kN against a crowd's 9,440 — unless the tendons were sized generously, when the stage overtakes the crowd.

structures · Stressed ribbon The pier reads the difference between the two spans' temperatures. Contours of the pier force between two 100 m spans at a sag of 2.00 per cent with the deck hung on one, the tendons at 1,100 N/mm² when finished, on a pier of 200,000 kN/m, every half meganewton, over the warming of the hung span's tendons (across) and of the bare tendon's (up) since they were cut. With neither warmer the pier carries 3,080 kN; warming the bare tendon 30 °C alone gives 4,126 kN, the hung span alone 2,342 kN, and both together 3,392 kN. The contours are straight and parallel: the pier gains 35 kN for each degree the bare tendon warms and loses 25 for each degree the hung span does, so a warming both share moves it only 30 per cent as far as the same warming on the bare tendon alone, and what the pier feels is mostly how much warmer one span's strand is than the other's.

The strand in the sun beside the strand in the shade

Halfway through hanging a stressed ribbon, one span is a sagging cable full of segments and the next is bare tendon stretched nearly straight, and the pier between them carries the difference of their pulls. A warm day takes tension off both — and at the flat sag a ribbon is laid at, the hung span is so nearly all stretch that it loses three quarters of what the bare one does. What moves the pier is not the weather but the shade: strand in the sun beside strand under concrete, 35 kN on the pier for every degree between them. No sun slackens the bare tendon. The hour does change something else, which is the level the joints are cast at.

structures · Stressed ribbon The flatter the ribbon, the more the casting temperature matters. For two 100 m spans of a stressed ribbon under 35 kN/m on a pier of 200,000 kN/m, one span's joints cast 20 °C warmer than the other's: the force left on the pier as a share of the force a crowd of 20 kN/m on one span puts there, against the ribbon's sag as a percentage of its span. At a sag of 1 per cent the casting leaves 82 per cent of a crowd (1,902 kN against 2,316); at 2 per cent, 14 per cent; at 3, 3 per cent; at 4, 1 per cent.

The span cast in the afternoon

A stressed ribbon's joints are cast on whatever day the programme reaches them, and a span cast on a warm afternoon sets around strand that has lengthened in the sun. It cools that night as a concrete band eleven times stiffer than the strand, and the band does not stretch back as the strand would have. On a ribbon with a sag of 2 per cent, a span cast 20 °C warmer than its neighbour ends 72 mm higher than it was set and leaves 713 kN on the pier between them — a seventh of what a crowd on one span puts there. On a flatter ribbon the same afternoon costs more than the crowd. The difference between the two spans is the strand's flexibility against the band's, and only the elastic part of it differs; the part that comes from the sag is the same for both.

structures · Stressed ribbon

Named alongside it

The objects these essays reach for when they reach for this one.

PrestressCable stiffnessStressed ribbonThermal movementToleranceConstruction sequenceImposed deformationIndeterminacySag ratioStiffnessBearingBolt hole

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