Structural Engineering & Statics
where the load goes
Dynamics

The weight that lands somewhere else

A beam that yields under a falling weight takes a permanent set from every drop, and if the weight always lands in the same place the set grows by the same amount every time. Let the weight land somewhere different each time — rocks on a canopy, crates on a deck — and the set still grows: the weight's energy fixes how much plastic work every drop does, wherever it lands. What the landing point decides is where the work goes. Half of every drop's hinge rotation goes under the weight and half to the supports, so wandering drops spread the first half along the span and leave the second half where it always was: the worst hinge turns half as fast, and it is at a support.

Assumes The weight that was dropped, After the first yield, which is not the end and The structure that settles down, and the one that walks.

16 min read 7 figures The load that will not hold stillWhich failure arrives firstThe load must go somewhere

The same weight, dropped again found what a falling weight does to a structure allowed to yield. On the first drop, yielding saves strength: the energy goes into bending the steel rather than into force, and a structure allowed a ductility of two and a half needs half the strength of an elastic one. On every later drop the saving is spent again. A perfectly plastic structure adds the same permanent set every time and never stops; a hardening one stops only when it has stiffened itself back to nearly the elastic force it was made weaker to avoid, after moving more than half a metre.

Every drop in that essay landed in the same place and pushed the same way, so its structure had one degree of freedom and one hinge. A weight that lands at different points on a beam — rocks on a canopy under a cliff, crates dropped on a loading deck — makes a different hinge each time, and each hinge’s set bends the beam in a different place. The essay ended by asking 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.

A beam that can hinge anywhere

The beam is 6 m long, fixed at both ends, with a bending stiffness of 42,000 kN m242{,}000\ \mathrm{kN\,m^2} and a plastic moment of 200 kN·m. A weight that brings 4.9 kJ to each drop — half a tonne falling a metre — is the striker. Struck at mid-span, the beam collapses plastically at 8Mp/L=2678M_p/L = 267 kN and stores Pc2/2k=0.95P_c^2/2k = 0.95 kJ elastically on the way, with k=192EI/L3k = 192EI/L^3, so most of each drop’s energy has to go into plastic work.

The beam is divided into 48 stations, and a hinge can form at any of them. Each drop pushes the beam at its landing point until the work done equals the weight’s energy — equal energy, which the earlier essay showed to be exact for a single impact rather than an approximation — and then lets it go. The stations that yielded keep their plastic rotations, and the rotations lock self-equilibrated moments into the fixed-ended beam, which the next drop starts from: the residual moments after the first yield found in a beam loaded once, carried from drop to drop.

Four sequences are compared: every drop at mid-span; every drop at three tenths of the span; drops alternately at three and seven tenths; and drops at random points in the middle three fifths.

A trough instead of a vee

The same weight, dropped in different places, leaves a trough instead of a vee. The permanent deflected shape of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ, after the last drop, for drops always at mid-span, always at 0.3 of the span, at 0.3 and 0.7 in turn and at random. Always at mid-span, 290 mm at its deepest; always at 0.3 of the span, 250 mm at its deepest; at 0.3 and 0.7 in turn, 180 mm at its deepest; at random, 206 mm at its deepest.
Fig. 1 The permanent deflected shape of the 6 m beam fixed at both ends, plastic moment 200 kN·m, after 20 drops of a 4.9 kJ weight, for drops always at mid-span, always at 0.3 of the span, at 0.3 and 0.7 in turn and at random. Always at mid-span, 290 mm at its deepest; always at 0.3, 250 mm; at 0.3 and 0.7 in turn, 180 mm; at random, 206 mm.

After twenty drops the beam struck always at mid-span is a vee 290 mm deep: two straight halves joined at a hinge under the weight. Struck always at three tenths of the span it is a lopsided vee, 250 mm deep, shallower because the beam is stronger there and the same plastic work moves it less. Struck alternately at three and seven tenths it is a trough with a flat bottom 180 mm down, and struck at random a rounded trough 206 mm down.

The trough is the first sign of what wandering does. Each drop at three tenths kinks the beam at three tenths, and the next, at seven tenths, kinks it there; between the two kinks the beam stays straight and is carried down by both. The set is spread over the span instead of piled under one point.

Wandering drops still ratchet

Wandering drops still ratchet; they ratchet more slowly. The largest permanent sag of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ, against the number of drops, for the four sequences. Always at mid-span, 290 mm after the last drop, still growing by 14.2 mm a drop; always at 0.3 of the span, 250 mm after the last drop, still growing by 12.3 mm a drop; at 0.3 and 0.7 in turn, 180 mm after the last drop, still growing by 8.9 mm a drop; at random, 206 mm after the last drop, still growing by 10.6 mm a drop.
Fig. 2 The largest permanent sag of the beam against the number of drops, for the four sequences. Always at mid-span, 290 mm after 20 drops, still growing by 14.2 mm a drop; always at 0.3 of the span, 250 mm and 12.3; at 0.3 and 0.7 in turn, 180 mm and 8.9; at random, 206 mm and 10.6.

None of the four stops. Every sequence’s largest sag is still growing after twenty drops, by 14.2 mm a drop for drops at mid-span and 8.9 mm for the alternating ones. The wandering drops ratchet more slowly in the deepest point of the beam, because they are deepening a trough rather than a single point, but they ratchet.

The weight decides how much the beam yields; the place decides where. The plastic rotation each drop adds to a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ, summed over every station, against the drop number, for the four sequences. After the first two drops every sequence adds between 0.019 and 0.021 radians a drop, whatever its points: the weight's energy, less what the beam stores elastically and gives back, goes into plastic work at the plastic moment, and the plastic moment is the same at every station.
Fig. 3 The plastic rotation each drop adds to the beam, summed over every station, against the drop number, for the four sequences. After the first two drops every sequence adds between 0.019 and 0.021 radians a drop, whatever its points: the weight’s energy, less what the beam stores elastically and gives back, goes into plastic work at the plastic moment, and the plastic moment is the same at every station.

The reason is that the energy decides. A drop brings 4.9 kJ; the beam stores some of it elastically and gives that back as the weight rebounds; the rest is plastic work, done at hinges whose moment is the plastic moment. With the plastic moment the same at every section, the plastic work divided by it is the total hinge rotation, and it is between 0.019 and 0.021 radians a drop for every sequence. The weight decides how much the beam yields; where it lands decides only where. Sharing the yielding among more hinges cannot make the beam yield less in total, so a perfectly plastic beam cannot shake down under wandering drops any more than under repeated ones.

Hardening changes that only slowly, as it did for the single hinge. With hinges whose moment rises, for each radian they turn, by a twentieth of the beam’s EI/LEI/L, twenty drops at mid-span leave 252 mm and alternating ones 170 mm, both still growing. A hinge has to harden until the beam can store the whole 4.9 kJ elastically, which for this beam means a moment more than twice its plastic one — far past any steel’s hardening before it fractures. A canopy struck by falling rocks is a structure with a finite number of drops in it, and that number is set by the ductility of its worst hinge.

Half of every drop goes to the supports

So the question that matters is which hinge is worst.

Spread over the span, the rotation gathers at the supports every drop shares. The plastic rotation each station of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ has gathered by the last drop, for drops always at mid-span, at 0.3 and 0.7 of the span in turn, and at random. Always at mid-span, the worst station holds 0.193 radians, at 3.0 m, and 3 stations more than a milliradian; at 0.3 and 0.7 in turn, the worst station holds 0.104 radians, at a support, and 4 stations more than a milliradian; at random, the worst station holds 0.128 radians, at a support, and 15 stations more than a milliradian.
Fig. 4 The plastic rotation each station of the beam has gathered after 20 drops, for drops always at mid-span, at 0.3 and 0.7 in turn, and at random. Always at mid-span, the worst station holds 0.193 radians, at mid-span, and 3 stations more than a milliradian; at 0.3 and 0.7 in turn, the worst holds 0.104 radians, at a support, and 4 stations more than a milliradian; at random, the worst holds 0.128 radians, at a support, and 15 stations more than a milliradian.

Under drops at mid-span the hinge under the weight gathers 0.193 radians and the two supports 0.097 each: the mechanism of a fixed-ended beam struck at its middle turns the middle hinge twice as far as each end. Under alternating drops each landing point gathers 0.102 radians and each support 0.104 — the supports are now the worst sections in the beam. Under random drops the rotation in the span is spread over fifteen sections, none of them past 0.041 radians, and the left support holds 0.128.

There is a rule underneath it. A fixed-ended beam struck at a point collapses as three hinges, and the hinge under the load turns by exactly the sum of the two supports’ rotations — the angle of the vee is the two halves’ slopes added. Half of every drop’s hinge rotation goes under the weight and half to the supports, wherever the weight lands. Wandering drops move the first half around; they cannot move the second half anywhere, because there are only two supports. Over many drops the supports gather half the total rotation between them, split between the two ends by how close each drop lands, while the span’s half is spread over as many sections as the drops visit.

For a canopy or a deck designed to be struck repeatedly, that turns the design question around. A beam struck always in one place fails under the weight, and its most important detail is the section there. A beam struck all over fails at its supports, and its most important detail is its end connection, which has to turn through half the beam’s whole plastic rotation over its life — the rotation capacity moment redistribution asks of a support once, asked again with every drop — 0.10 radians after twenty drops here, the rotation the cable that pays the factor back found ordinary connections cannot reach.

The worst hinge turns half as fast

The worst hinge turns half as fast when the drops wander. The plastic rotation gathered at the worst station of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ, against the number of drops, for the four sequences. Always at mid-span, 0.193 radians after the last drop; always at 0.3 of the span, 0.201 radians after the last drop; at 0.3 and 0.7 in turn, 0.104 radians after the last drop; at random, 0.128 radians after the last drop.
Fig. 5 The plastic rotation gathered at the worst station of the beam against the number of drops, for the four sequences. Always at mid-span, 0.193 radians after 20 drops; always at 0.3 of the span, 0.201; at 0.3 and 0.7 in turn, 0.104; at random, 0.128.

The worst section’s rotation grows steadily in every sequence, and it grows half as fast when the drops alternate as when they land in one place: 0.104 radians after twenty drops against 0.193 at mid-span and 0.201 at three tenths. A section that can turn through 0.2 radians before it tears survives twenty drops in one place and nearly forty alternating; random drops, which favour neither support, put the worst section between the two, at 0.128.

The saving is real and it is bounded. However the drops are spread, the supports take half the total rotation between them, so the worst support can never gather less than a quarter of the total — the case in which every drop lands at mid-span splits it evenly between the two ends. Spreading the drops brings the worst section down from the landing point’s half of the total towards the supports’ quarter, and no further.

The moment each drop leaves behind

Each drop leaves a moment the next one starts from. The moment locked into a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) by the first drop of the alternating sequence, at 0.3 of the span (solid), and by the first two, the second at 0.7 (dashed), sagging positive; the weight is 4.9 kJ. After the first drop the beam holds 83 kN·m at its left support and −83 at its right; the second drop met that and peaked at 316 kN where the first had peaked at 323, and left −57 and 77.
Fig. 6 The moment locked into the beam by the first drop of the alternating sequence, at 0.3 of the span (solid), and by the first two, the second at 0.7 (dashed), sagging positive. After the first drop the beam holds 83 kN·m at its left support and −83 at its right; the second drop met that and peaked at 316 kN where the first had peaked at 323, and left −57 and 77.

The plastic rotations of a drop are not compatible with a straight fixed-ended beam, and the beam holds them by locking in moments. After the first drop, at three tenths, the left support holds 83 kN·m of sagging moment where the drop had bent it in hogging, and the right support 83 kN·m of hogging. The second drop, at seven tenths, meets a beam already bent against it at its own near support, yields slightly sooner — at 316 kN rather than the 323 its point would need from new — and leaves a different pattern. That is the coupling the earlier essay’s single hinge could not have: each drop’s starting state depends on where the last ones landed.

The coupling changes when each drop yields and by little else. It moves a few per cent of each drop’s energy between elastic and plastic work, and the total plastic work per drop stays within the narrow band the work figure shows. Residual moments in a beam under energy loads redistribute the damage; they do not create or destroy it.

A weight too light to ratchet one place

The 4.9 kJ weight is five times what the beam can store elastically, and nothing about where it landed could have stopped it ratcheting. The question the opening asked is sharper for a lighter weight: one that a beam struck always in the same place would shake down under.

How much energy a fixed-ended beam can take elastically depends on where it is struck. A perfectly plastic beam can hold no more force at a point than its collapse load there, Pc=2MpL/abP_c = 2M_pL/ab for a point aa from one end and bb from the other, and its elastic stiffness at that point is 3EIL3/a3b33EIL^3/a^3b^3, so the most it can store is 23Mp2 ab/EIL\tfrac{2}{3}M_p^2\,ab/EIL: 0.95 kJ at mid-span, 0.80 kJ at three tenths, 0.61 kJ at a fifth. A weight below that at its landing point can be absorbed elastically once the beam has locked in the right residual moment, and drops that always land at that point find the moment by themselves: the first few yield, each leaves a little more of the protecting moment behind, and then the beam stops.

A weight one place shakes down under keeps ratcheting a beam when it wanders. The largest permanent sag of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 200 times by a weight of 0.70 kJ, less than the 0.80 kJ it can store elastically at 0.3 of its span, against the number of drops. Always at 0.3 of the span, 2.1 mm, gained over the first ten drops and none after; at 0.3 and 0.7 in turn, 29.7 mm and still growing by 0.12 mm a drop; at random, 49.4 mm and still growing by 0.21 mm a drop.
Fig. 7 The largest permanent sag of the same beam struck 200 times by a weight of 0.70 kJ, less than the 0.80 kJ it can store elastically at 0.3 of its span. Always at 0.3, 2.1 mm, gained over the first ten drops and none after; at 0.3 and 0.7 in turn, 29.7 mm and still growing by 0.12 mm a drop; at random, 49.4 mm and still growing by 0.21 mm a drop.

Struck by 0.7 kJ always at three tenths of the span, the beam gains 2.1 mm over ten drops and nothing after: it has shaken down, as the structure that settles down did under a fixed load and a cycling temperature. Struck by the same weight alternately at three and seven tenths — two points where it could store 0.80 kJ, more than the weight at either — it does not stop. After two hundred drops its largest sag is 29.7 mm and growing by 0.12 mm a drop. Struck at random it is at 49.4 mm and growing faster, although part of that comes from drops landing near a fifth of the span, where the beam stores less than the weight even when every drop lands there.

The reason is the residual moment. A drop at three tenths yields the left support in hogging, and what it locks in to protect a repeat at three tenths is a sagging moment at the left support and a hogging one at the right, which is the pattern the residual figure shows. A drop at seven tenths hogs the right support, so it meets a beam whose right support is already part of the way to yield in the direction it pushes. It yields there sooner, and in yielding replaces the pattern with its mirror image, which leaves the left support ready to yield under the next drop at three tenths. Each drop undoes the protection the other one built.

Melan’s theorem, which drew the map with three regions, says a structure shakes down under a set of loads only if one residual moment keeps every load in the set elastic at once. The pair at three and seven tenths asks for two moments that are mirror images of each other, and at 0.7 kJ no single moment serves both. At 0.6 kJ one does, and the alternating drops settle at 5.0 mm. This pair’s own shakedown limit lies between those two weights, below the 0.80 kJ that either point allows by itself. A set that contains both points contains each one, so a moment that protects the set protects each point alone: wandering can lower the weight a beam shakes down under, and it can never raise it. It is the shape the beam the wind walks without pushing found in a portal frame — two loads the structure could carry for ever one at a time, and which together walk it.

The rotation per drop by hand

At mid-span the beam collapses at Pc=8×200/6=267P_c = 8 \times 200/6 = 267 kN, and its elastic stiffness there is k=192×42,000/63=37,300k = 192 \times 42{,}000/6^3 = 37{,}300 kN/m. The weight brings 4.9 kJ; the beam gives back 2672/(2×37,300)=0.95267^2/(2 \times 37{,}300) = 0.95 kJ as it rebounds, so each drop does 3.95 kJ of plastic work. At 200 kN·m that is 3.95/200=0.01973.95/200 = 0.0197 radians of hinge rotation, half under the weight and a quarter at each support: 0.0099 at mid-span and 0.0049 at each end, a set of 0.0049×3=14.80.0049 \times 3 = 14.8 mm a drop. Twenty drops: 0.197 radians at mid-span, against the 0.193 the stations gathered.

At three tenths the same arithmetic splits the rotation unevenly. The collapse load is 2×200×6/(1.8×4.2)=3172 \times 200 \times 6/(1.8 \times 4.2) = 317 kN, and the stiffness there 3×42,000×63/(1.83×4.23)=63,0003 \times 42{,}000 \times 6^3/(1.8^3 \times 4.2^3) = 63{,}000 kN/m, so the beam gives back 0.80 kJ and each drop does 4.10 kJ of plastic work, 0.0205 radians. In the mechanism the support nearer the load turns by δ/a\delta/a and the far one by δ/b\delta/b, so the near support takes b/2L=0.35b/2L = 0.35 of every drop’s rotation and the far one a/2L=0.15a/2L = 0.15: after twenty drops 0.144 radians at the left support and 0.061 at the right, against the 0.144 and 0.057 the stations gathered. The near support is the one the weight leans on.

What the energy balance assumes

The weight’s mass is large beside the beam’s. Equal energy treats the beam as massless while the weight decelerates on it. A beam whose own mass is comparable with the weight’s takes some of the energy into its own motion, rings, and yields in more than one mode, and the hinges a drop makes can then be other than the three-hinge mechanism.

The beam stays shallow. At sets of 200 to 300 mm on a 6 m span the beam’s ends, held against moving apart, would carry it partly as a cable, which would raise its resistance and slow every ratchet here. That membrane action, which the force nobody put in the model found holding up slabs their own analysis condemned, is left out, so the sets are larger than a beam held at its ends would show and the ratchets do not slow as they would.

Every section has the same plastic moment. A real beam’s ends are often its connections, weaker than its span and less ductile, which makes the supports’ half of the rotation more dangerous still.

The drops do not overlap. Each drop is over before the next; a second weight landing while the beam is still moving would meet it in a state no static energy balance describes.

Still open: the canopy whose supports are its weak point

Every support here is as strong as the beam. A canopy is usually a beam on two connections that are weaker than it and much less ductile — bolted end plates or fin plates — and the supports’ half of every drop’s rotation then lands on the parts of the structure least able to take it. If the connections yield first, the mechanism changes: a beam with weak ends is struck into two hinges at the ends and one under the load, at a lower collapse load and with more of each drop’s work at the ends. Whether a canopy designed for rocks should then be given partial-strength connections that act as fuses, replaced after a season of drops, or full-strength connections that move the work back into the span, is what the weak supports ask of the weight that was dropped once it starts landing everywhere.

Where this goes next

The force climbs back to the elastic one. The peak force of each of 40 drops of a 500 kg weight dropped 1.0 m onto a structure whose stiffness gives it 2.0 mm of static deflection under the weight, onto a structure yielding at 80 kN — half the 160 kN an elastic structure takes (dotted). Perfectly plastic, every drop peaks at 80 kN and the saving holds, at the price of the ratchet. Hardening at 2 per cent, the peak climbs drop by drop: 82 kN at the first, 91 at the fifth, 131 at the 40th; hardening at 5 per cent, the peak climbs drop by drop: 86 kN at the first, 104 at the fifth, 152 at the 40th. A hardening structure ends up carrying almost the elastic force it was made weaker to avoid. The same weight, dropped again A structure allowed to yield under a falling weight needs only a fraction of the strength an elastic one needs — half of it for a ductility of two and a half — because the energy goes into bending the steel rather than into force. That is the whole saving, and it is spent on the first drop. Drop the same weight again and the structure yields again: a perfectly plastic one adds the same permanent set every time and never stops, and a hardening one stops only when it has stiffened itself back up to nearly the elastic force it was made weaker to avoid, after moving more than half a metre. What a drop height is worth, as a factor on the answer for a weight placed slowly. The peak displacement as a multiple of the static deflection, against the height a weight is dropped from divided by the deflection that weight causes when it is placed. The curve is 1 + √(1 + 2h/δ), which is conservation of energy and nothing else: the weight does work over the height it falls PLUS the distance the structure then gives, and the structure stores work only over the second. At a ratio of 40 the factor is 10.00, and at zero it is exactly 2 — the marked point, where a dropped weight becomes a placed one. The weight that was dropped A half-tonne load lowered onto a beam produces 5 kN. The same load dropped one metre onto the same beam produces 160 kN — and onto a beam ten times softer, 54 kN. The stiff structure is the one that suffers, which is the opposite of nearly every other rule about structures. The collapse mechanism of a propped cantilever. A collapse mechanism, with the hinge position found by searching rather than quoted. Every position gives an upper bound on the collapse load; the lowest is 7.29, at a hinge 58.6 per cent along, which is a coefficient of 11.657 times Mp over the square of the span. After the first yield, which is not the end A steel beam whose extreme fibre has reached yield has not failed. It has started forming a hinge, and collapse waits until there are enough hinges to make a mechanism.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

DuctilityEnergy balanceImpact factorLow-cycle fatigueMechanismPlastic hingeRatchetingResidual stressSelf-stressShakedown