Structural Engineering & Statics
where the load goes
Concept

Ratcheting — where it appears

The accumulation of a permanent change with each cycle of a load that itself returns to where it started. It is the alternative to shakedown, and it appears wherever a cycle is not reversible — a soil densifying behind a wall, a section straining a little further each time, a joint working loose.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 30% of the plastic moment with a 20°C profile it never yields at all; At 60% of the plastic moment with a 120°C profile it shakes down; At 85% of the plastic moment with a 200°C profile it ratchets, at 1.9% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains.

The structure that settles down, and the one that walks

A load that is safe applied once may not be safe applied ten thousand times. Nothing about that is fatigue — the structure never breaks, it simply arrives somewhere slightly further round every cycle, until it has arrived somewhere unusable.

materials · Shakedown The summer that is worse than the one before it. The earth pressure behind an integral abutment, summer by summer, as a multiple of the at-rest value it started at. A 60 m deck expands by 10.8 mm at each end and pushes the abutment into the backfill. Granular soil under cyclic strain densifies, so the same movement next year needs a higher pressure to achieve, and K climbs from 0.38 toward 0.96 — a factor of 2.49 on the force, reached after about a century. The design load on an integral abutment describes the bridge's whole life rather than a load case, and it is the only load in this collection that gets larger because time has passed rather than because something was added.

The summer that is worse than the last

An expansion joint is a hole in a deck that leaks salt water onto the bearings underneath it. Remove it and the thermal movement does not go away — it goes into the soil behind the abutment, twice a day for a hundred and twenty years, and granular soil under cyclic strain gets denser.

structures · Integral bridge Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 20% of the plastic moment with a 40°C profile it never yields at all; At 50% of the plastic moment with a 150°C profile it shakes down; At 90% of the plastic moment with a 260°C profile it ratchets, at 51.0% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains.

The map with three regions

A structure carrying a constant load and a cycling temperature has three possible fates and only one of them is a collapse. It can stay elastic, it can yield once and then stop, or it can gain a little more deformation every cycle for ever — and the third has no failure load at all.

materials · Shakedown Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither.

The order the loads arrived in

Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

equilibrium · Friction Settling down or walking away, cycle by cycle. The total plastic hinge rotation of the beam after each cycle of loading — span 1, both, span 2, neither — with the midspan load at 0.98, 1.02, 1.05, 1.10 times the shakedown load of 126.3 kN. At 0.98 it stops at 0.59 mrad. At 1.02 it grows 4.57 mrad a cycle. At 1.05 it grows 11.43 mrad a cycle. At 1.10 it grows 22.86 mrad a cycle. Nothing collapses in any single cycle; above the shakedown load the beam walks.

The load it can carry once

A two-span beam whose loads come and go span by span collapses at 150 kN under any one arrangement, and walks at 127. Between the two it can carry every arrangement once and none of them forever: each cycle leaves a few more milliradians of rotation at the support and a midspan fifteen millimetres lower. Melan's theorem finds the limit as the last residual moment line that fits, Koiter's as a mechanism no single load state can drive, and a cycle-by-cycle calculation walks exactly where both say it will.

materials · Shakedown 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.

dynamics · Impact factor The loads a portal can carry for ever. For a fixed-base portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, in the plane of the gravity load's peak V·L/Mp — the load comes and goes — and the wind's H·h/Mp — it reverses: the first-yield boundary (dotted), the shakedown boundary (solid), and the collapse boundary under the same two loads applied once (dashed). With little wind the shakedown boundary is the beam's own mechanism, which the wind does no work in, falling from 8 to 6.0 as the wind grows to H·h/Mp = 2.7; beyond that it is alternating plasticity at a column base, nearly level. The dot is 160 kN of gravity with 60 kN of wind and the ring its shakedown point, a factor of 0.88: the gravity load alone is at the beam's collapse load, and the wind takes 12% off what the pair can carry without doing any work in the beam's mechanism.

The beam the wind walks without pushing

A fixed-base portal whose beam carries exactly its collapse load is safe under that load applied once. Add a wind that comes from either side — a wind that does no work at all in the beam's collapse mechanism — and the beam can no longer carry the load for ever: 12 per cent comes off, and above that the midspan gains fifteen milliradians every cycle while the frame does not sway. Each knee takes its worst moment from a different wind, and the beam assembles a mechanism no single storm drives.

materials · Shakedown Retightened after every storm, the bolts stretch in every storm. The permanent stretch in the worse bolt row of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows, accumulated over ten storms of four cycles to ±170 kN·m, with the nuts retightened after every storm to 200 kN, to the 150 kN they were first given, to the 124 kN the first storm left, or left alone. Left alone, the bolts stretch 0.12 mm in the first storm and not at all after it. Retightened to 150 kN they add 0.12 mm in every storm, 1.25 mm after ten; to 200 kN, 0.43 mm a storm. Retightened only to what the first storm left, they stretch no more than if they had been left alone.

The nut turned after the storm

A preloaded column base that has been through a storm has lost some of its preload, and the obvious repair is to retighten its bolts. Left alone, the bolts stretched once in the first storm and never again; the base settled. Retightened, they are given back the preload the storm could not live with, and the next storm takes it away again by stretching them again. Retightening after every storm turns a base that settles into one that ratchets, a tenth of a millimetre at a time, and the count of storms it has in it is set by the bolt's elongation.

connections · Base plate Pinned bases: weaker, and nothing lost to repetition while gravity leads. For a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, in the plane of the gravity load's peak V·L/Mp and the reversing wind's H·h/Mp: the shakedown boundary (solid) and the collapse boundary under the loads applied once (dashed), on fixed bases and on pinned ones. On pinned bases the two coincide wherever gravity leads, along the beam and combined mechanisms, until the wind reaches about H·h/Mp = 1.6; past that a knee alternates and shakedown falls inside collapse, meeting it again at H·h/Mp = 2 with the wind alone. On fixed bases collapse reaches 8 and 4 and shakedown sits inside it except where the beam's mechanism meets the gravity axis. The dot is 160 kN of gravity at midspan that comes and goes with 60 kN of wind that reverses: its shakedown factor is 0.88 on fixed bases and 0.73 on pinned ones, where it is also the collapse factor.

The base that sends the wind to the knees

A fixed-base portal loses 12 per cent of its load to a wind that reverses, because each knee takes its worst moment from a different storm. Pin the bases and the loss disappears — not because the pinned frame is better, but because its collapse load already contains it. What the bases decide is where the moment goes, and a base that is merely weaker than its column can cost nothing at all, or be made into the part that yields every storm so that nothing else does.

materials · Shakedown Shorter after every cycle, at the same load. The shortening of a column hinge at the end of each cycle, in yield strains, for an I-section of 3,000 mm² and 200 mm depth and the rectangle of the same area, each carrying a constant 30% of its squash load while its curvature cycles between ±3.0 times its first-yield curvature, in elastic–perfectly plastic steel. The I-section shortens by 8.14 yield strains every cycle, 97.1 after 12; the rectangle by 1.60, 19.4. Neither settles: the load has not changed, and the column is shorter after every cycle.

The column that shortens on every cycle

A column hinge bent back and forth by an earthquake carries its share of the building's weight the whole time. In elastic–perfectly plastic steel an I-section at three tenths of its squash load, cycled to three times its first-yield curvature each way, comes back from every cycle shorter by eight yield strains, and goes on doing so: it never settles. The flange that yields in compression shortens, and the other flange, carrying its part of the column's load, never reaches yield in tension to pull it back. A rectangle of the same area shortens a fifth as fast, and the overturning that swings the column's load up and down hardly matters. The weight and the curvature make the ratchet between them.

materials · Moment-curvature The preload a storm can live with, raised by a spring under the nut. The largest preload the bolts of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows can be given and come out of a storm without stretching, against the stiffness of a stack of disc springs under each nut, for storms to ±165, ±170, ±175 kN·m; dashed, the same for the bare base; dotted, the bolt's yield of 226 kN. To ±170 kN·m the bare base keeps at most 128 kN; with a stack of 500 kN/mm, 150; 100 kN/mm, 187; 20 kN/mm, 214; 5 kN/mm, 223.

The spring under the nut

Retightening a column base after every storm makes its bolts ratchet, because a storm can only live with 128 kN of preload and the spanner keeps putting back 150. A stack of disc springs under each nut changes what the storm can live with. Soft enough, the bolts never reach their yield: the compression side keeps its preload, the bed's resultant moves out to the plate's edge, and the moment rides on lever arm rather than on bolt force. A stack as stiff as the bolt itself keeps 150 kN through every storm. Much softer, and the storm the bolts no longer feel lands on the grout.

connections · Base plate Where the knees begin to alternate. The shakedown factor (solid) and collapse factor (thin) of a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, on pinned bases, with 160 kN of gravity that comes and goes and a 1,000 kN roof carried at its knees, against the peak of a wind that reverses, first-order (dashed) and amplified. First-order, the shakedown and collapse factors are one line until the knees alternate, from a wind of 110 kN; amplified, from 90 kN. At 60 kN: shakedown 0.727 first-order and 0.692 amplified, collapse 0.727 and 0.692. At 120 kN: shakedown 0.533 first-order and 0.488 amplified, collapse 0.571 and 0.539. At 200 kN: shakedown 0.348 first-order and 0.326 amplified, collapse 0.400 and 0.366.

The roof that makes the wind worse every storm

A portal frame's shakedown check is made on its elastic moments, and the elastic moments a heavy roof sways are larger than first-order analysis says: each storm's wind is amplified by 1/(1 − N/Ncr), on its way out and on its way back. On pinned bases under a 1,000 kN roof that takes the wind at which the knees start to alternate from 110 kN down to 90, and the shakedown load down by 4.8 per cent where gravity leads and 8.4 where the wind does. Fixed bases are worth more under a heavy roof than without one, but by half again, not twice.

materials · Shakedown 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.

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.

dynamics · Impact factor Over many earthquakes the column line ratchets together or stops together. The compression over the squash load of the interior column (solid) and an exterior column (dashed) against the cycles, up to 400, for the frame of three ground-storey columns under floor loads of 15, 20, 25 kN/m on 6 floors; the dotted line is where the hinge starts to ratchet, 0.15. At 15 kN/m the interior column starts at 0.22 and the average is 0.12; after 400 cycles the interior carries 0.15 and the exterior 0.10. At 20 kN/m the interior column starts at 0.29 and the average is 0.16; after 400 cycles the interior carries 0.16 and the exterior 0.16. At 25 kN/m the interior column starts at 0.36 and the average is 0.19; after 400 cycles the interior carries 0.19 and the exterior 0.19. Where the average lies above the threshold the three end at the average and ratchet together; where it lies below, the interior column sheds load until it stops at the threshold and the exterior columns never start.

The floor that cannot keep up with the column

A column whose hinges ratchet under an earthquake shortens every cycle, and in a frame it does not shorten alone: the floors it carries bend down towards it and lift its load onto the columns beside it. That ought to slow it and speed them, until every column shortens together. It does — over hundreds of cycles. Over the ten or twenty of an earthquake, the floors of a six-storey frame lift an eighth of the load off the interior column, it shortens nine tenths as far as it would alone, and the floors carry the whole difference as bending. The frame equalises its columns eventually, and pays for it in its floors.

materials · Moment-curvature

Named alongside it

The objects these essays reach for when they reach for this one.

ShakedownPlastic hingeResidual stressMechanismSelf-stressBase plateLow-cycle fatiguePortal frameAnchor boltCollapseContinuous beamCyclic loading

All concepts