The collection

Every essay — page 22

Essays 505 to 513 of 513, in the same order.

Three members after a diagonal goes. The forces in three members of the counter-braced truss when the diagonal 9–2, carrying 242 kN, is removed instantaneously, with 2 per cent damping, over one and a half of the damaged truss's first periods (0.57 s); dashed, the static force each settles to. The counter-diagonal 1–10 goes from −112 kN to −354 kN and peaks at −491 kN, 1.57 times its change. The bottom chord 3–4 ends exactly where it started, 750 kN, and on the way peaks at 1,115 kN. The bottom chord 2–3 settles lower, at 556 kN, after a first swing the other way, to 905 kN.

The factor of two belongs to one mode

A member that fails suddenly hands its force to the structure around it all at once, and the convention is to double the static answer: a load applied suddenly to a spring overshoots to twice its static deflection. A truss is not one spring. Take a diagonal out of a counter-braced truss in an instant and some members swing to three times their change of force, one swings the wrong way first, and a bottom chord whose force does not change at all passes through half as much again as it carries — because every mode overshoots by two, at its own time, and a member is a sum of modes.

5 figures · Robustness

A soft layer over stiff ground, and a moment neither has. The bending moment down a 20 m pile of bending stiffness 120 MN·m² with a free head, pushed sideways by 150 kN (its characteristic length in ground at 5.00 MN/m³ being 1.89 m), in three grounds: all of it at 5.00 MN/m³; all of it at 0.50 MN/m³; and soft ground 2.0 characteristic lengths (3.8 m) thick over the other (dotted: the interface). The largest moments are 219 kN·m, 346 kN·m and 427 kN·m; the head deflections 20.4 mm, 81.4 mm and 61.6 mm. The layered ground gives a larger moment than either ground on its own: the pile bends as a pile in the soft layer would, and then meets ground that holds it.

Internal forces

The layer a pile feels

A pile pushed sideways in uniform ground has one characteristic length, the fifth root of its stiffness over the ground's, and every answer is a multiple of it. Put a soft layer over stiff ground and that length stops existing. The head feels the top of the ground in proportion to the square of its own deflection, so an average weighted that way gets the head deflection within eight per cent. It gets the moment a quarter too low: with soft ground two characteristic lengths thick over stiff, the pile bends harder than it would in either ground alone, and the peak sits where the stiff ground takes hold.

6 figures · Lateral pile

Three grounds, three tipping pushes. The push against the lean it produces, in thousandths of a radian, for the 6 m square, 3,000 kN block on ground that bears 300 kPa at most, half of it mobilised at 7.5 mm of settlement, pushed 8 m up, its weight 5.0 m up, on the three grounds matched at their limit and at half of it; dashed, the rigid-plastic limit, 813 kN. Linear, capped: 776 kN at a lean of 13.2 thousandths; hyperbolic: 701 kN at a lean of 26.2 thousandths; S-shaped: 770 kN at a lean of 15.2 thousandths. The S-shaped ground behaves almost exactly like the capped one; the hyperbola, which never quite reaches its limit, tips 10 per cent sooner at twice the lean.

Equilibrium

The ground that never quite gives way

On a bed of springs capped at the ground's bearing limit, a block pushed sideways tips at a definite push, below the limit the rigid-plastic parabola gives. Real ground is curved: it softens all the way to its limit and never quite reaches it. On such ground the block has no tipping push of its own. Its resistance creeps toward the limit, and the peak it does have is made by its own weight leaning with it — 14 per cent below the limit with the weight 5 m up, nearly at it with the weight at the base, where the lean simply runs away.

5 figures · Overturning $What the actuator pushes against decides its limit. The largest feedback gain, as the damping it would add to the bare mode, at which the loop stays stable, against the delay as a fraction of the period, on a logarithmic scale, for a mode of 1.00 per cent damping carrying a tuned mass of 2.00 per cent of its modal mass at Den Hartog's tuning. Pushing against the ground, fed by the structure's velocity: 2.41 at a delay of 0.05, 0.25 at 0.2. Pushing against the tuned mass, fed by the same velocity: 0.0156 at almost no delay, rising to 0.091 at a quarter of a period. Pushing against the tuned mass, fed by the velocity across it: unlimited with no delay, 0.048 at 0.05, 0.0073 at 0.2.$ Dynamics

The actuator that pushes against the mass

An actuator that damps a structure by feeding back its velocity has to push against something. Against the ground, it is stable up to large gains until a delay of a quarter period removes its damping. Against a tuned mass — the arrangement meant to keep a passive damper underneath — the same feedback goes unstable at a gain of a per cent and a half with no delay at all, makes the structure worse than the passive mass at every gain below that, and drives the mass three or four times as far. What limits it is not when it acts but what it pushes against.

5 figures · Tuned mass damper

Stability

The column with no plateau

A steel column's curve is the lesser of two lines, its yield stress and Euler's, because mild steel keeps its whole stiffness up to yield and then has a plateau to stop on. Aluminium and stainless steel have no plateau: their stiffness starts falling well below the proof stress and never stops. Their column curves are not the steel curve moved; they are a different shape — four tenths below the corner for a stainless steel, above the proof stress for a stocky column — and the shape is set by two numbers, only one of which describes the knee.

6 figures · Inelastic buckling

How far a local load spreads, and what is nearest its limit. Along a steel-faced polyurethane cladding panel, 0.5 mm faces of 320 N/mm² steel on an 80 mm core crushing at 0.12 N/mm², from the middle of a strip load of 3.0 N per mm of width spread over 10 mm: the face's deflection over its value under the load, the core's compressive stress over its crushing strength, and the face's bending stress over its yield stress. The load spreads over a length set by the fourth root of the face's bending stiffness over the core's, 1/β = 20.5 mm; the deflection is 1.44 mm under the load and changes sign beyond 48 mm. The core is at 0.60 of its crushing strength and the face at 0.89 of its yield stress: this panel's face yields at 3.36 N/mm and its core crushes at 5.00.

Sections and stress

The face that dents and the core that crushes

A sandwich panel carries bending as a couple between its faces, and that is a calculation about the whole panel. Put a local load on one face — a foot, a fixing, a dropped tool — and the face becomes a thin beam on a soft bed, spreading the load over a length the fourth root of their stiffnesses sets. Then either the face yields and dents, or the core crushes beneath it, and which comes first is decided by one thickness: below it the face gives, above it the core does, and a steel-faced roof panel is on the wrong side of it for anyone who walks on it.

5 figures · Sandwich section

A deck holds the middle of the span. The twist along an 8 m open section on forks, free to warp, carrying 12 kN/m at 75 mm from its shear centre — 900 Nmm of torque per mm of span. With nothing fastened to it the beam twists 6.25° at mid-span; a deck resisting the top flange's rotation at 5.0 kNm per metre per radian holds it to 4.20°, and one four times as stiff to 2.10°. The deck works where the beam is weakest, in the middle; near the supports the beam's own torsional stiffness does most of the work whatever is fastened to it.

Deflection

A deck is a spring, not a wall

An open-section beam loaded off its shear centre twists, and the usual reassurance is that the deck fastened to its top flange will stop it. The deck resists the flange's rotation with a stiffness per metre of span, and that stiffness has to be compared with the beam's own. On an 8 m beam a screwed deck of ordinary stiffness removes a third of the twist; on a 16 m beam the same deck removes three quarters, because the beam's torsional stiffness falls with the square of its span and the deck's does not.

6 figures · Twist serviceability

Ductile until it was welded. A 2.0 m 6082-T6 member pulled in tension, as supplied and with one weld across it whose heat-affected zone is 60 mm long: the load per unit of parent section against the member's overall strain, to the maximum load, where the weaker part begins to neck. The plain member reaches 310 N/mm² at 7.0% strain, 140 mm of stretch. The welded one reaches 185 N/mm² at 0.62%, 12.3 mm — 8.8% of the plain member's, and the parent never reaches its proof stress of 260 N/mm². (uniform elongations assumed: 7.0% for the parent, 12% for the zone)

Materials

The soft zone that takes all the stretch

The metal beside a weld in heat-treated aluminium is weaker than the rest of the member and more ductile, and it is tempting to read the second fact as a consolation. It is not. A member welded across stretches only where it is weak, and the weak zone is sixty millimetres long; a two-metre 6082-T6 tie that would extend 140 mm before necking extends 12 mm once welded, and the parent metal never yields at all. What decides it is one ratio, the zone's ultimate strength over the parent's, and the ductility returns only when that ratio reaches one.

6 figures · Heat-affected zone

The modes that tilt, and the ones that do not. Johansen's mechanisms for a 12 mm bolt with washers in single shear through 40 and 40 mm of timber of density 350 kg/m³, each with the rope term the fastener's axial resistance of 6.64 kN adds to it. Modes a and b, where the fastener only translates, gain nothing; the modes in which it tilts gain a quarter of its axial resistance, capped at a quarter of their own Johansen value. The joint's capacity rises from 5.02 kN (mode c) to 6.28 kN (mode c), 25 per cent.

Connections

The pull that Johansen left out

Johansen's mechanisms treat a timber fastener as a beam in a bed of crushing wood, bent and pushed sideways and nothing else. A real fastener that tilts across the joint is also pulled along its own axis, and if a washer or a thread resists the pull, it clamps the two members together and adds to what the joint can carry. The addition is a quarter of the axial resistance, it goes only to the modes in which the fastener tilts, and it is capped by fastener type — nothing for a dowel, a quarter for a bolt, all of it for a screw — which is how a screw keeps gaining strength past the thickness at which Johansen's capacity stops.

6 figures · Dowel yield