Structural Engineering & Statics
where the load goes
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Which failure arrives first — page 15

Essays 337 to 344 of 344 on this thread, in the same order.
The cable pays the factor the plateau saved. The force a beam of two 8.0 m spans over the column, 600 mm deep, plastic moment 800 kN·m, its far ends held against rotation and against spreading (plastic collapse load 400 kN) reaches after a sudden loss divided by the load it carries — the dynamic factor — against that load as a share of the plastic collapse load. Below the plateau the beam is a linear spring and the factor is 2. On the bending plateau it falls to 1.18 at 0.90 times the collapse load, because a flat resistance does its work at full strength from the start. Past it the beam becomes a cable, whose resistance is again straight, and the factor climbs back: 1.73 at 1.5 times, 1.88 at 2, 1.97 at 3. Structural form

The cable that pays the factor back

A frame that loses a column carries the floor across the gap first by bending and then, as the beam sags, as a cable. The hope is that a path which stiffens as it deflects needs less than the factor of two a suddenly loaded spring does. On the bending plateau it does — 1.18 at nine tenths of the collapse load. But past one depth of sag the cable is a straight line again, and the factor climbs straight back toward two: 1.73 at one and a half times the collapse load. And the connections are asked for a tenth of a radian on the way.

 What the soft layer costs the strong one. For a 40 mm filler whose outer 50 per cent is the soft layer, the least crush force its inner layer must have for the filler to stop a collision at 2.85 m/s — 761 kJ — without running out, against the soft layer's crush stress. The uniform filler that just does it crushes at 31.7 MN; every softer outer layer needs a stronger inner one, and at 1.5 MPa outside the inner layer needs 56.0 MN. The bare concrete's contact at that speed is 35.0 MN (dashed): only an outer layer of at least 5.7 MPa keeps the inner one below it. Dynamics

The filler that cannot be gentle twice

A crushable filler in the gap between two buildings caps a collision at its crush force, but a filler soft enough to make the everyday contacts gentle runs out on the hardest one. Grade it — soft at the face, strong behind — and the soft layer takes the small contacts while the strong layer stops the large. The energy says otherwise. Every kilojoule the soft layer does not absorb, the strong layer must, in less depth: in a 40 mm filler, an outer half at 1.5 MPa needs an inner half at 56 MN, far above the 35 MN of the bare concrete it was meant to improve on.

 The studs reach out to where the concrete is enough. One column of a 260 mm flat slab (d = 225 mm) carrying 12 kN/m² on 400 × 400 mm internal columns, the moment it hands the column putting the shear 300 mm off centre, on a 10.0 m bay, to scale. The inner check, on the control perimeter 2d out (solid), needs reinforcement: 12 rails of studs, 6 perimeters of them from 0.5d at 0.75d spacing, 1,415 mm² on each perimeter. The outer perimeter (dashed), where the concrete alone carries the shear, stands 5.4d from the face — 9,173 mm long against the control perimeter's 4,427 — and the last studs must be within 1.5d inside it, 3.9d from the face. Internal forces

The studs that send the check outward

Shear studs round a column fix a failing punching check, and they turn one check into three. The studs carry the control perimeter; past them the concrete alone must carry a perimeter long enough to need no help, and that outer perimeter stands further from the column the larger the load, so the studs follow it out. At a 10 m bay they reach four effective depths from the face, and the steel they need grows as the shear to the power 3.5. The third check, at the column face, is the one no stud reaches, and it ends the series at 11.8 m.

 Out in compression, back in tension. The axial stress in a bare S355 beam (section factor 160 per metre) whose ends are held by the structure around it with 10 per cent of its own axial stiffness, against its temperature, through a compartment fire with an opening factor of 0.04 m^½ and 200 MJ/m² of fuel: heating (solid) and cooling (dashed), with the yield stress at each temperature dotted either side. Heating, the beam pushes against its ends and reaches 134 N/mm² of compression at 628 °C, where the yield stress has fallen to meet it; it then yields, shortening plastically as it follows the falling yield stress up to its peak of 940 °C. Cooling, the thermal expansion comes back out and the plastic shortening does not: the beam passes through zero and ends at 20 °C in 195 N/mm² of tension. Materials

The tension a fire leaves behind

A steel beam held at its ends by the structure around it is pushed into compression as a fire heats it, and yields, because its strength is falling while its expansion is not. When the fire goes out the expansion comes back and the yield does not. Held at a tenth of its own stiffness, a beam that peaked at 134 N/mm² of compression on the way up ends the fire at 195 N/mm² of tension — more than it was ever pushed, arriving hours after the fire was out, and on connections that were designed to carry shear.

 Closer piles add pull-out the block cannot pay for. For 600 mm piles 15 m long under a 20 × 30 m raft, in ground whose shaft friction is 0.30 of the effective vertical stress and whose submerged unit weight is 10 kN/m³: the summed pull-out of every pile on a square grid (thin, rising steeply as the spacing closes), the submerged weight of the block of ground the group would lift (dashed, 90 MN whatever the spacing), and the group's capacity, the lesser (solid). The two meet at 2.06 m — 3.4 pile diameters — where s² = π·d·β·L/2. Closer than that the group lifts the block and the extra piles add nothing; the most piles worth having is about 142. Equilibrium

The piles that would lift the ground

A basement too light to stay down can be pinned down with tension piles, each holding by the friction on its shaft. Add enough of them, close enough together, and the ground between them stops being something they grip and becomes something they carry: the group lifts out as a block, and the block's weight is all it can hold. For 600 mm piles 15 m long that happens closer than 2.06 m apart, a spacing with no raft, no water and no load in it — and every pile past that point is paid for twice.

 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. Structural form

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.

 Two models of one buckled web. The shear resistance of a web 1500 mm deep and 8.0 mm thick in steel of 355 N/mm², as a share of its plastic shear 2,460 kN, against the spacing of its intermediate stiffeners as a multiple of the depth, by Basler's tension field, by the rotated stress field with a rigid end post and with a non-rigid one, and by buckling alone; the dots at the right-hand edge are the same web with no intermediate stiffeners. With stiffeners at the depth Basler gives 0.71 and the rotated field 0.50; at three depths 0.38 and 0.42; with none, 0.14 — buckling alone — and 0.41. The two cross at a spacing of about 2.6 depths. Basler's reserve is a band the stiffeners anchor and it narrows to nothing as they move apart; the rotated field's is spread over the whole web and anchored at its ends, and the spacing reaches it only through the buckling stress. Stability

Two reserves for one buckled web

A slender web keeps carrying shear after it buckles, and design uses two models of how: Basler's band of tension anchored on the stiffeners, and the rotated stress field spread over the whole web and anchored at its ends. Given the same 1,500 × 8 mm panel they agree about the buckling and disagree about everything after it — and in particular about what an intermediate stiffener is for. One says stiffeners save 37 per cent of the steel; the other, 1 per cent.

 A fuse caps the force the strand ever sees. The force in the tendon line of a 2.0 × 8.0 m rocking wall of 400 kN, prestressed to 600 kN, with bars of 400 kN, against the opening of the base under the tendon, loaded to 79 mm and back: the strand alone, 20 kN/mm yielding at 1,370 kN, and the strand with a fuse yielding at 1,000 kN with a stiffness of 500 kN/mm in series, together 19.2 kN/mm and capped at 1,000 kN. The strand alone yields at 39 mm and is slack at upright after any opening past 69 mm; the fused line yields at 21 mm and is slack past 52 mm. In the fused line the strand carries the line's force and never more than 1,000 kN, so it stays elastic and all the plastic stretch, 58 mm at this opening, is in the fuse. Dynamics

The fuse that protects the strand and not the wall

A rocking wall's tendon loses its prestress when it yields, so give it a fuse: a short link in series that yields first and is replaced afterwards. The strand then never passes 1,000 kN and stays elastic. But the fuse caps the restoring moment lower, so under the same 1.5 g pulse the wall rocks to 110 mrad instead of 89, and the line loses all its prestress just as the strand did. A new fuse gives the prestress back; the bars' stretch stays, and the next pulse takes the wall to 134 mrad.

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