The same weight, dropped again

The weight that was dropped dropped half a tonne one metre onto a beam and found 160 kN at the beam where lowering it gently gave 5, the force set by the energy the beam had to store and its stiffness. It ended on the two things its elastic calculation could not do. It “cannot handle a structure that yields, which is where the energy absorbed per millimetre changes by an order of magnitude”, and “it cannot handle repeated impacts, where the question is no longer whether the structure survives one but how many it survives.”

The two belong together, because yielding is exactly what makes repetition a question. An elastic structure hit twice by the same weight does the same thing twice. A structure that yielded on the first drop starts the second from a different place.

One drop, two structures #

The structure is the earlier essay’s: stiff enough that the half-tonne weight resting on it deflects it 2 mm. Dropped from a metre, the weight falls a metre and then keeps falling while the structure gives, and the structure stops it when the work it has absorbed equals the weight’s whole fall. An elastic structure stores that work as a triangle under its force–displacement line, and stops the weight at 160 kN and 65 mm.

A structure that yields at 80 kN — half the elastic force — absorbs the same energy as a long, low trapezium instead of a tall triangle. It reaches 86 kN, a little above its yield force because it hardens, at 81 mm, and when the weight is lifted off it springs back along its elastic stiffness and keeps 46 mm of permanent set. It has used a ductility of 2.48 — its greatest displacement is two and a half times the displacement at which it first yielded — and in exchange it has halved the force on everything it is attached to.

That is the case for ductile impact design, and for most accidental impacts it is the right case. A crash barrier, a rock-fall canopy or a lifting beam’s catch structure is meant to be hit once, absorb the energy by deforming, and be inspected or replaced.

Equal energy is the energy balance #

The trade between strength and ductility for a single drop is a single curve. A ductility of 2 needs 58 per cent of the elastic force; 4 needs 38; 8 needs 27. The curve is the equal-energy rule that the earthquake asks for a displacement found for short-period structures shaken by the ground, and here it is not an empirical rule at all: the area under the elastic triangle and the area under the yielding trapezium are both the weight’s fall, and setting them equal is the calculation. The only difference between the two curves on the figure is the weight’s fall through the structure’s own movement, which the equal-energy form leaves out and which is a few per cent here because the drop is fifteen times the movement.

For an impact, equal energy is exact rather than approximate — which is why it is the right way round to teach it. An earthquake is an energy input spread over many cycles of a structure’s motion, and equal energy is a rule that happens to fit the short-period end of the spectrum. A dropped weight is an energy input delivered once, and equal energy is conservation of energy.

Drop it again #

Lift the weight off and drop it again from the same height. The structure is no longer where it was: it carries 46 mm of permanent set, and it has hardened, so its yield force is now the 86 kN it reached on the first drop rather than the 80 it started with. The second drop reloads it elastically from its new position up to 86 kN, and the weight still has energy left, so it yields further.

The six loops walk to the right and climb. Each drop starts where the last one left the structure, reloads it elastically to the force it has already reached, and then yields a little further along its hardening line, because an elastic reload up to 86 kN stores only 1.5 kJ and the weight brings 5. Each loop adds less set than the one before — 46, 40, 37, 33, 30, 27 mm — because each reaches a higher force before it yields, and a higher elastic range absorbs more of the weight’s energy without yielding.

A ratchet, or a climb to the elastic force #

Two behaviours appear, and neither is the one the single-drop design assumed. A perfectly plastic structure ratchets without limit: every drop reloads it to the same 80 kN and pushes it the same 50 mm further, because nothing about it changes between drops except its position. Forty drops move it two metres. A hardening structure’s increments shrink and the set converges — to 620 mm for 5 per cent hardening and 1,600 mm for 2 per cent — which is shakedown: after enough drops the structure’s elastic range has grown large enough to absorb the whole of each drop’s energy without yielding.

The force tells the other half of the story, and it is the half that decides the design. A hardening structure shakes down only when its elastic range can absorb a drop, and the elastic range that absorbs a drop is exactly the elastic structure’s: a triangle up to 160 kN. So the force at shakedown is the elastic force. The 5 per cent hardening structure reaches 104 kN by its fifth drop and 152 by its fortieth, and it is heading for 160; the 2 per cent one gets there more slowly and further from home. The structure that was made weaker so as to carry half the force ends, after enough drops, carrying almost all of it — and has moved more than half a metre to get there.

Which structures meet the same weight twice #

Recurrence is commoner than the single-impact picture suggests, and it comes in three frequencies. A few times in a life: a rock-fall canopy over a road below a cutting takes a boulder in most winters, and the same boulder-sized energy at the same place in each; a crash barrier at a bend is struck at the same post by more than one vehicle before it is replaced. Many times a day: a hopper lining that takes the drop of every load of ore, a deck beside a quay where grabs set down, a loading bay’s dock plate. And continuously: a drop-forge hammer’s foundation, a pile driver’s cap, a test rig. The single-impact ductile design is defensible only for the first group, and even there only if the structure is inspected after each event, since the set a canopy acquires in one winter is the starting point for the next.

The second and third groups are almost never designed as yielding structures, and the reason is the figure above. A structure yielding under an everyday impact reaches its elastic force within a few dozen repetitions anyway, having deformed by half a metre on the way, so the saving in strength is illusory and the deformation is real. Those structures are made either strong enough to stay elastic or soft enough that their elastic force is small — and their critical check is not strength at all but fatigue, since every drop is a full cycle of stress.

How fast the ratchet stops #

Hardening decides how many drops the structure takes to settle and how far it moves on the way, and the two trade against each other. The set at shakedown is inversely proportional to the hardening ratio, and so, roughly, is the number of drops it takes to get most of the way there. With 2 per cent hardening the structure is within a tenth of its final set only after 101 drops, by which time it has moved 1.4 m; with 5 per cent, after 40 drops and 560 mm; with 10 per cent, after 20 drops and 265 mm; with 20 per cent, after 10 drops and 120 mm. A material or a detail that hardens strongly — a cold-formed section, a stainless steel, a crushable tube whose resistance rises as it folds — settles in a handful of drops. Mild steel in a plastic hinge, whose moment barely rises after the hinge forms, is the 2 per cent case, and it is close to the perfectly plastic ratchet for as many drops as anybody is likely to count.

Soft and elastic, instead of strong and ductile #

The comparison that matters for a recurring impact is not between the ductile structure and the strong elastic one. It is between the ductile structure and a soft elastic one. The elastic force of a drop goes roughly as the square root of the stiffness, so a structure half as stiff takes 115 kN at 94 mm, and a quarter as stiff 83 kN at 135 mm — the force of the ductile design, with no permanent set at all and no limit on the number of drops.

That is the logic of every device built to be hit repeatedly: a rubber fender on a quay, a spring buffer at the end of a crane rail, the elastomeric pad under a drop-forge hammer. Each buys its low force with elastic movement rather than plastic movement, and elastic movement comes back. The ductile structure buys the same low force with movement that stays, which is sensible only when the impact is not expected to recur. For a recurring impact, flexibility is the honest substitute for ductility, and stiffness is not strength in the most literal way: the stiffer structure is the one that attracts the larger force.

A fuse that is meant to be used up #

Between the two lies the replaceable element: a part designed to yield under every impact, with a stroke long enough for a known number of them, and replaced when its stroke is spent. The fall-arrest lanyard of the earlier essay is one, a tearing strip that yields at a chosen force and keeps yielding; a crushable tube behind a buffer is another. The arithmetic for such a fuse is the ratchet itself, read as a budget. A perfectly plastic fuse at 80 kN spends 50 mm of stroke on every drop of the half-tonne weight from a metre, so 300 mm of stroke buys six drops, and the sixth leaves nothing for the seventh.

The same idea at the scale of a whole frame is the part that is meant to be weak: a link in an eccentrically braced frame yields so that nothing else has to, and is inspected and replaced after the earthquake that used it. The difference for impacts is only that the energy per event is known in advance, so the fuse’s life can be counted in events rather than guessed.

Why an energy cannot be shaken down to #

The structure that settles down, and the one that walks found the same two outcomes — shakedown and ratcheting — for a structure under a steady load with a cycling one added, and its shakedown happens at the applied loads: a structure shaken down under a given force carries that force elastically ever after. The drop is different in one respect that changes the answer. A falling weight does not apply a force. It brings an energy, and the force is whatever the structure needs to absorb it. A structure that hardens raises its own force, and a structure that yields lowers its own force, but neither changes the energy it has to absorb on every drop — so the only elastic state it can settle into is the one that absorbs that energy elastically, and that is the elastic structure of the earlier essay.

The consequence is a rule worth stating plainly. Ductility reduces the force of an impact only as often as the structure is allowed to be deformed. For a one-off accidental impact, ductility buys strength honestly. For an impact that recurs — a crane grab striking the same coaming, a pile of rocks shedding onto the same canopy, a test rig’s drop weight — the design has two choices: the elastic force, or a replaceable element that is replaced before it has ratcheted too far.

The ratchet by hand #

The perfectly plastic increment follows from one energy balance. With the weight $W = mg = 4.9$ kN, the yield force $R = 80$ kN, the yield displacement $u_y = R/k = 32.6$ mm and the drop $h = 1$ m, the greatest displacement on each drop satisfies

$$\tfrac12 R u_y + R(d - u_y) = W(h + d)$$

so $d = (Wh + \tfrac12 R u_y)/(R - W) = (4.9 + 1.3)/75.1 = 83$ mm, and the set it leaves is $d - u_y = 50$ mm — the same on every drop, because $R$ never changes. The same set is what the residual state after unloading carries between drops. With hardening, the set at shakedown is the plastic stretch needed to raise the force from $R$ to the elastic force $F_{el}$ along a line of slope $bk$: $(F_{el} - R)(1 - b)/(bk) = 80 \times 0.95/(0.05 \times 2{,}452) = 0.62$ m for 5 per cent hardening, and 1.6 m — more than two and a half times as much — for 2 per cent.

What the model assumes #

The structure is a single spring. A beam that yields under a drop forms a plastic hinge, and a hinge’s moment–rotation is not a single bilinear line: the section that yields from the outside in found a rounded transition, and a rounded transition behaves like early hardening, which speeds the climb back to the elastic force.

The hardening is unlimited and the material does not fail. Structural steel reaches an ultimate strength a third to a half above its yield before it necks, and a structure that must climb from half the elastic force to all of it needs a force rise of 100 per cent. Most real structures would therefore fracture, or buckle locally, before shaking down: the ratchet does not stop, the member does. The cycles are few and very large — the low-cycle end of the load that never came near failing anything — and a crack at a hinge after a few dozen drops is the expected outcome, not a surprise.

The weight is lifted off after each drop, and the structure is the same structure. A weight that rests on the structure between drops loads it statically and changes the starting point; a structure whose supports also yield, or whose geometry changes as it sets — a beam that sags into a catenary — gains membrane stiffness and behaves differently at large deflection.

What the pictures cannot show #

That the permanent set is also a change of shape. A canopy that has ratcheted half a metre drains differently, sheds rocks differently, and loads its supports differently from the one that was designed. A crash barrier that has been struck and straightened has been cold-worked twice. Nor can the figures show the inspection that ought to happen between drops. The ratchet is visible: every drop leaves a set that can be measured, and a structure whose set is growing linearly is telling its owner that it has no hardening left.

Still open: the drop that lands somewhere else #

Every drop here lands in the same place and pushes in the same direction. A weight that lands at different points on a beam each time — rocks on a canopy, crates on a deck — yields a different hinge each time, and each hinge’s set bends the beam in a different place. Whether the beam then shakes down faster, because the hardening is shared among several hinges, or ratchets into a mechanism, because each hinge’s set adds a rotation the others must follow, is the question of what a structure after its first yield does under loads that move, and it is the one that sizes a canopy for a slope rather than a lifting beam for a crane.