The base that lets go and closes again

A preloaded base is rigid until the bed lets go. Holding-down bolts tightened to 150 kN each clamp a 500 by 400 mm base plate onto its grout, and the clamped joint is 45.6 times the column’s $EI/L$ stiff — half again as stiff as the rule for a rigid base asks — until the moment on the column lifts the plate off the grout at its tension edge, at about 57 kN·m. Past that the base softens towards the unpreloaded joint, about a tenth as stiff.

That essay loaded the base once, from rest. A base under wind is loaded one way and then the other, thousands of times, and each excursion past the decompression moment stretches the bolts on one side beyond their preload while the grout under the other side is pressed harder than the preload pressed it. When the load reverses, it left a question: does the base close again into the joint it was, or does each excursion leave the bolts a little looser, until the storm has turned a preloaded base into an unpreloaded one?

A storm in four cycles #

The base is the same — 500 × 400 × 20 mm, two M24 bolts in each of two rows 60 mm from the plate’s ends, preloaded to 150 kN a bolt, on a bed of grout that pushes back but cannot pull. The difference is that each bolt row is now elastic–plastic: its force follows the clamped joint’s diagram while it is below the bolts’ yield, 226 kN a bolt for the M24’s 353 mm² at 640 N/mm², and once it reaches it the row stretches permanently, with a small hardening.

The first excursion follows the single-loading curve: stiff, then soft once the plate lifts, then, near the top, softer again as the bolts on the tension side reach their yield. Coming back, the base does not retrace that path. The bolts that yielded are longer than they were, so at zero moment they clamp less, and the base begins its next excursion as a looser joint. Under the reversed moment the other row yields in turn.

And then it stops. The second cycle traces almost exactly the first cycle’s return; the third and fourth lie on top of it. The base settles into a new joint and stays there. That is shakedown in its simplest form: the first cycles leave a residual state — here, permanent stretch in the bolts — in which every later cycle is elastic.

The bolt that stretches once #

The bolt’s own history shows where the preload goes.

While the plate is in full contact the bolt hardly feels the moment: the clamped joint shares any change of load between the bolt and the grout in proportion to their stiffnesses, and the grout is much the stiffer. A bolt that was already stretched feels a fifth of every kilonewton applied to its joint until the plates part, and this one feels less. Once the plate lifts on its side, the grout has let go and the bolt carries the change alone, so its force climbs steeply, to 226 kN at about 165 kN·m.

From there the bolt stretches at nearly constant force. When the moment falls away, it unloads along the clamped joint’s steep line — the grout is back in contact — and arrives at no moment with 125 kN, having left 25 kN of its preload as permanent elongation. Under the reversed moment its row is on the compressed side of the plate, where the grout carries the moment and the bolt barely changes. When the next positive excursion arrives it climbs from 125 kN rather than 150, reaches its yield at a little more rotation, and stretches very little further, because it is already nearly as long as the first excursion made it.

The reason the second excursion stretches the bolt so little is the same reason a joint is a set of springs in series and in parallel: the bolt and the grout share the clamp in parallel while the plate is down, and the bolt alone carries the change once it is up. A bolt that has already been stretched to a given length reaches its yield again at almost the same rotation of the plate, because the plate’s rotation is what stretches it, and the rotation needed to carry the same moment is nearly the same as before. Only the small hardening, and the slight loosening of the other row, make the second excursion yield it at all.

It settles; it does not ratchet #

The answer to the question the single loading left is in the preload that survives each half-cycle.

At 160 kN·m nothing is lost at all, through every cycle. At 170 kN·m the first excursion takes 25 kN from a bolt, the second takes another kilonewton because the other row’s yielding has slightly relaxed the clamp, and then nothing more. At 175 kN·m it takes 56 kN and then one more. At 180 kN·m, close to the base’s capacity, it takes 102 kN at once and three more over the next cycles before settling at 45. In every case the loss stops within two cycles, at a level set by the largest excursion. The base does not ratchet; it keeps whatever the worst moment it has seen left it.

That is a design statement. A base whose preload is counted on for its stiffness has to be checked, not for how many cycles it will see, but for the largest moment it will ever see. If that is below the moment that first yields a bolt, the preload is permanent. If it is above, the preload that counts is the residual one, and it can be computed once.

Three times the decompression moment, then a cliff #

The moment that first yields a bolt is the one that decides everything, and it is a long way above the moment that first lifts the plate.

The plate lets go at 57 kN·m. The bolts do not yield until 164 kN·m, 2.9 times that. Everywhere between, the base decompresses and closes again with all of its preload, every cycle: lifting the plate is elastic, and nothing about it uses up the clamp. The question the single loading left assumed that letting go of the bed was the damaging event. It is not; it is the event that makes the base softer while it lasts, and the base is as stiff as ever once the moment falls back below it.

Above 164 kN·m the curve falls off a cliff. Eighteen kilonewton-metres more, a tenth of the moment, takes the kept preload from all of it to 30 per cent, because the base’s capacity — the moment at which both bolts in the tension row are yielding and the plate can only rotate — is at 183 kN·m, only 12 per cent above the moment that first yields a bolt. A preloaded base designed so that its largest moment stays below bolt yield has a factor of nearly three on decompression; one whose largest moment approaches its capacity keeps almost none of what the preload bought.

Past a point, more preload buys less #

The obvious way to enlarge the rigid range is to tighten the bolts harder, since the decompression moment is proportional to the preload. The storm has a view on that.

Up to about 120 kN a bolt, the storm takes nothing: the rigid range after it is the rigid range before it. At 150 kN the storm yields the bolts and the range after it is 46 kN·m, barely more than the 120 kN base kept untouched. At 200 kN the base starts with a rigid range of 75.5 kN·m and leaves the storm with 37.3, less than a base tightened to 120 kN keeps. A bolt tightened nearer to its yield needs a smaller excursion to reach it and is stretched further past it, and the stretch takes away more than the extra tightening put in.

So for a given storm there is a best preload, here about 140 kN a bolt, and it is set by the storm rather than by the bolt. It is a strange specification to write — tighten to a force, but not to more than a force — and it is the same strange specification that the torque that goes into the thread makes hard to meet, since a tightening controlled by torque scatters the preload widely from bolt to bolt.

The window between letting go and yielding #

The two moments that bound the elastic behaviour move in opposite directions as the preload rises.

The decompression moment is the preload’s moment about the plate’s kern, and it rises in proportion to the preload. The yield moment is what the bolts can add on top of their preload before they reach their strength, and it falls as the preload uses up more of that strength. At 80 kN the window between them runs from 30 to 178 kN·m, a factor of six; at 150 kN from 57 to 164, under three; at 200 kN from 76 to 124, barely a factor of 1.6. A heavier preload buys rigidity under small moments by spending the margin under large ones. Where the two lines would meet, at a preload near the bolts’ yield, a preloaded base would lose its preload in the same storm that first lifted it.

What the frame above it notices #

A base’s stiffness matters because a frame stands on it. A base drawn as pinned is not pinned, and a base drawn as fixed is fixed only to the degree its bolts and grout make it: the fixed base the bolts decide is a spring, and the frame’s sway under wind and its buckling load both depend on how stiff a spring it is. Preloading the bolts was the way to make that spring stiff enough to count as rigid.

The storm decides whether it still is. While the moment at the base stays inside the rigid range, the base is 45.6 times the column’s $EI/L$; once the moment passes it, the base softens towards the unpreloaded joint’s 9.8. A frame designed on the preloaded stiffness is right only for the part of its life when the base moments stay below the decompression moment — which, after a storm that yielded the bolts, is smaller than it was. After a storm to 180 kN·m the base that started with a rigid range of 57 kN·m has one of 17, and an ordinary service wind that used to leave it rigid now lifts the plate. The frame drifts more under that wind than it was designed to, and its sway stiffness is that of a frame on much softer bases.

So the rigid range is not one number but a history. It starts at the value the preload gives, stays there through every storm below the bolts’ yield, and steps down once for each storm that exceeds the last. A frame whose stability is credited to its bases’ rigidity is relying on the largest storm those bases have seen having stayed below about three times their decompression moment.

What an inspection can find #

The settled state is invisible. A base that has lost a third of its preload looks exactly like one that has not: the nuts have not turned, the grout has not cracked, and the plate sits on it as before. The stretch is in the bolt shank below the nut, a fraction of a millimetre, and the clamp it has lost is the difference between two forces nobody can see.

The usual check, a wrench on the nut, measures the torque needed to turn it, which depends on the friction in the thread and under the nut far more than on the bolt’s tension; it is the same unreliability the torque that goes into the thread makes of tightening, met again on the way out. An ultrasonic measurement of the bolt’s length, against a record taken when it was tightened, reads the stretch directly; without that record it reads nothing useful. And a load-indicating washer that was compressed at tightening shows, after the storm, whether it is still compressed.

That makes the preload’s survival something to be designed rather than inspected. If the largest moment the base will ever see is below the bolts’ yield moment, the preload is permanent and no inspection is needed. If it is not, the residual preload after the design storm is computable — 124 kN of 150 for a storm of 170 kN·m — and the frame should be designed on the base that storm leaves, not on the base that was tightened.

The window, by hand #

The decompression moment is the moment that brings the grout’s pressure to zero at the plate’s tension edge. With the whole preload $2nP_0$ = 600 kN spread over the plate and the bolts’ share of the moment small, it is close to the plate’s kern moment: $600 \times 500/6 = 50$ kN·m, against the 57 the full calculation gives with the bolts’ small share included.

The yield moment is reached when the bolt row on the tension side carries $2 \times 226 = 452$ kN with the plate lifted. Taking the compression resultant near the plate’s far edge, about 30 mm in, the lever arm from the bolt row to it is $500 - 60 - 30 = 410$ mm; the moment is the bolts’ force times that lever arm less the compression’s share needed to balance their preload, which the full calculation does exactly and which puts it at 164 kN·m. At a preload of 200 kN the bolts begin with 400 kN of their 452, and the same arithmetic leaves very little — which is the window closing.

Elastic grout, and bolts that only stretch #

The calculation rests on choices that limit it.

The grout stays elastic. Under 170 kN·m the grout at the compressed edge carries about 16 N/mm², inside the strength of a good grout; under a larger base or a weaker grout it can crush, and crushing is plastic shortening of the bed, which loses preload on that side exactly as a bolt’s stretch does. The same calculation with a bed that can yield would give a second residual state; the shakedown argument is the same.

The bolts yield in tension only. A real anchor bolt has a reduced section at its thread, and its stretch concentrates there; the elastic–plastic row is an average. The small hardening keeps the stretch finite; a bolt with less would stretch further at the same moment.

Creep is left out. The grout creeps under the preload in the first months and a short bonded anchor loses a fifth of its preload to a tenth of a millimetre of it; that loss adds to the storm’s, and it happens whether or not the storm comes.

The moment reverses between equal limits. A real storm is a sequence of unequal moments, and the result is unchanged in shape: the residual state is set by the largest, and the cycles below it are elastic. Fatigue of the bolts is a separate question: within the window their stress range is a fraction of the joint’s, but every cycle past decompression puts the full change of force through them, and those cycles are the ones that wear a bolt out.

Still open: the bolt that is retightened after the storm #

A base that has settled after a storm has lost a known amount of preload, and the obvious repair is to retighten the bolts. A retightened bolt is a bolt with a permanent stretch already in it, whose yield is no higher than before, so it starts the next storm with its preload restored and less margin above it. Whether retightening after each storm keeps a base rigid indefinitely — or whether each retightening spends some of the bolt’s ductility, so that the bolt that is tightened back to its preload after every storm is the one that eventually breaks — is the question of what the stretch in a yielded bolt costs, and it is answered by the bolt’s elongation at fracture rather than by its yield.