Structural Engineering & Statics
where the load goes
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One support too many — page 7

Essays 145 to 148 of 148 on this thread, in the same order.
A bigger top column attracts more moment, and only the joint stops it. The top storey of the edge column of a braced six-storey frame — 4 m storeys, a 254 UC 89 below the splice and a 203 UC 60 above — carrying a 9 m beam at every floor on partial-strength joints of 261 kN·m, with each of seven columns that might be chosen for it: the moment its floor's edge joint delivers at the design load of 60 kN/m (dark) and at the beams' collapse load (light), against what the column can carry at its top end, its plastic moment reduced for the axial force (line). 203×203 UC 46: 148 and 195 against 167; 203×203 UC 60: 171 and 225 against 230; 203×203 UC 71: 189 and 247 against 283; 203×203 UC 86: 205 and 261 against 347; 254×254 UC 73: 218 and 261 against 352; 254×254 UC 89: 232 and 261 against 435; 254×254 UC 107: 243 and 261 against 527. A stiffer column attracts more of the joint's moment until the joint's 261 kN·m caps it; the lightest column that carries what it attracts at collapse is the 203 × 203 UC 60. Connections

The column on the other side of the joint

A partial-strength joint is sold as a fuse: it caps the moment a beam can put into its support at the joint's own resistance. At an interior column that is true. At the edge of a frame the column is not a support that stays still; it is a spring in series with the joint, so the joint delivers less than its resistance in service and almost exactly what a rigid joint would. The column then carries three times the moment the simple-construction rule gave it, and at the top storey, where it stands alone and is lightest, it is weaker than the joint it was meant to be protected by.

 A secondary holds more than it twists once its connection passes a few tens of kilonewton-metres per radian. The mid-span twist of the 8 m primary beam of 12 kN/m at 75 mm off its shear centre, with a 6 m 356 × 171 × 51 UB framing in at mid-span whose 60 kN reaction acts 60 mm off the shear centre, against the rotational stiffness of the secondary's connection on a logarithmic scale, in series with the secondary's own 3EI/L of 14,453 kN·m/rad. Dashed, the 6.25° with no secondary. On the deck's side the secondary is worse than none below 33 kN·m/rad and better above it: 9.12° at 10, 3.28 at 100, 0.47 at 1,000. On the far side its torque opposes the deck's (0.89° at 10); as a pair, the torques cancel (1.05° at 100). EN 1993-1-8 calls this secondary's connection nominally pinned below 2,409 kN·m/rad (dotted). Deflection

The secondary beam that twists what it holds

A secondary beam framing into a twisting primary at mid-span is the best restraint a primary can have — concentrated exactly where the twist is. It is also a load: its reaction arrives on the primary's web, off the shear centre, as a torque at exactly the same place. Which wins is decided by the connection, and the threshold is low: a fin plate stiffer than about 33 kN·m per radian, a seventieth of what still counts as a pin, makes the secondary hold far more than it twists. The bolts' free play is another matter, because the torque acts from the first millimetre and the restraint only once the play is taken up.

 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.

 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.

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