The spring under the nut

The nut turned after the storm followed a preloaded column base through ten storms, each four cycles of a moment reversing between ±170 kN·m. Left alone, its holding-down bolts stretched 0.12 mm in the first storm, kept 124 of their 150 kN of preload, and never stretched again: the base shook down. Retightened to 150 kN after every storm, they stretched 0.12 mm in every storm, because the storm could live with at most 128 kN of preload and the spanner kept putting back 150. The base ratcheted, and the count of storms its bolts had in them was set by how far a grade 8.8 bolt can be stretched before it breaks.

The ratchet is a property of what the storm can live with, and that essay ended on a way to change it. A stack of conical disc springs under each nut is a soft spring in series with a stiff one. A bolt that stretches by some amount slackens the stack by the same amount, and a soft spring loses little force for a given movement. The question was whether a base with springs under its nuts can be preloaded above the storm’s limit without yielding, and what the softer row costs.

One more spring in the row #

The base is the one the earlier essays built: a 500 × 400 × 20 mm plate on a grout bed that pushes back but cannot pull, with two M24 grade 8.8 holding-down bolts in each of two rows 60 mm in from its ends, each bolt preloaded to 150 kN. Each bolt is a spring of 290 kN/mm over its 256 mm free length; the plate bending at the row adds a spring of 948 kN/mm per bolt in series; together a row of two is 444 kN/mm. The bolts yield at 226 kN.

A stack of disc springs under each nut is a third spring in series with those two. Its stiffness is chosen by how many discs are stacked and how — discs nested in parallel to carry more force, opposed in series to give more travel — and here it is a single number per bolt, from 2 kN/mm, much softer than anything else in the row, to 1,000 kN/mm, much stiffer than the bolt. The stack is assumed never to close flat below the bolt’s yield, so that up to then the row is one linear spring.

The stack keeps the storm off the bolt. With no stack, the storm’s peak moment takes the bolts on the tension side to their yield, and they stretch. A stiff stack of 500 kN/mm per bolt still lets the storm reach yield, though only just. Softer, the peak falls steadily: 206 kN at 50 kN/mm, 197 kN at 20, 187 kN at 5. Every one of those is a storm that stretches nothing.

Where the moment goes instead #

That is not obvious. A bolt’s force at the peak of a storm looks like a matter of statics — the moment has to be carried by the bolts pulling on one side and the bed pushing on the other — and a softer bolt is no less obliged to pull.

The free body at the peak shows where the difference comes from. The base has two bolt rows, not one, and a bed that can move its resultant.

Bare, at 170 kN·m, the tension-side row carries 465 kN, the compression-side row 232 kN — it has lost 68 of its 300 kN of preload as the plate pressed down around it — and the bed pushes back with 697 kN at 180 mm from the plate’s centre, over the 209 mm of plate still in contact. The three forces balance the moment: $465 \times 0.19 + 697 \times 0.180 - 232 \times 0.19 = 170$ kN·m.

With a stack of 20 kN/mm under each nut, the tension-side row carries 394 kN, the compression-side row 294 kN — it has kept nearly all of its preload — and the bed pushes back with 688 kN at 220 mm from the centre, over only 91 mm of contact. The same moment is carried, but differently: $394 \times 0.19 + 688 \times 0.220 - 294 \times 0.19 = 170$ kN·m.

The stack trades bolt force for lever arm. A stiff row resists every millimetre the plate moves, so the plate cannot turn far; its bed stays in contact over a long length with its resultant well inboard, and the tension bolts make up the moment with force. A soft row lets the plate turn. As it turns, the contact shrinks towards the edge, the bed’s resultant moves out, and each kilonewton it carries is worth more moment. At the same time the compression-side bolts, which a stiff row would have unloaded as the plate pressed down on their side, hardly notice the movement and keep their preload. Both effects take moment off the tension bolts.

A joint made of springs in series found that a connection’s stiffness is set by its softest component. Here a deliberately soft component changes not only the stiffness but where the force goes, and that is what the rest of the argument turns on.

The preload the bare bolts could keep #

The first storm on a bare base found the largest preload the bolts could keep through it: 128 kN for ±170 kN·m. Tightened harder, they yielded and came out with less.

The stack lifts that limit. At 500 kN/mm per bolt, stiffer than the bolt, it is 150 kN, which is exactly the preload the bolts were given; at 100 kN/mm, 187; at 20 kN/mm, 214; and soft enough, the limit runs up against the bolt’s yield itself. For a storm to ±175 kN·m — where the bare base can keep only 101 kN — the same stacks give 130, 176 and 211. For ±165 the bare base’s 146 becomes 164, 194 and 216.

The limit was the storm telling the base how much preload it could keep. With a spring under each nut the storm says something different, because the moment it brings is carried by the base in a different way, and what it says is generous: a stack of moderate stiffness gives the bolts back the full preload the designer wanted, and a soft one gives them nearly everything their yield allows.

Nothing for the spanner to put back #

The ratchet is gone, and not because retightening has become gentler. The stacked base’s bolts never reach their yield in a storm to ±170 kN·m, so a storm takes nothing from them, and a spanner has nothing to put back. A retightened stacked base and a stacked base left alone are the same base. The inspector’s instruction that the earlier essay found had to be rewritten — retighten to what the storm left, not to the specification — loses its point, because the storm leaves the specification.

That is the shakedown the structure that settles down described, arranged in advance. The bare base found its shaken-down state by yielding in its first storm and loosening itself until the storm no longer reached yield. The stacked base starts in a state the storm cannot push past yield at all. It has no first-storm loss to recover from and no stretch to accumulate.

What the stack gives back #

A preload is there for the rigid range: the moment up to which the plate stays in contact with its whole bed, and the base is several times stiffer than it is afterwards. The base that is rigid until the bed lets go found that range proportional to the preload, and the earlier storm essay priced each kilonewton·metre of it in stretch.

The stack lowers the rigid range, and for a definite reason. Before the bed lets go, the plate resists a moment in two ways at once: the bed pushes back harder on one side and less on the other, and the bolt rows, stiff in a bare base, resist the plate’s turning as a pair of springs. A soft row contributes almost nothing to the second. With a stack of 20 kN/mm the plate holds its moment almost entirely by shifting the bed’s pressure, and it lets go when the shift has brought the clamping force’s resultant to the edge of the bed’s middle third — the middle third of a section that cannot take tension. The clamping force is 600 kN and the plate 500 mm long, so that limit is $600 \times 0.5/6 = 50.0$ kN·m. A soft stack approaches it and cannot go below it.

So the stacked base gives back between a twentieth and a ninth of its new rigid range, depending on the stack. What matters is what it keeps. A bare base retightened to 150 kN after every storm has 56.6 kN·m of rigid range and ratchets; a bare base left alone has 46.0 and does not. A base with a stack of 20 kN/mm has 50.5 kN·m and does not ratchet — more rigid range than the shaken-down bare base, for ever, and no stretch.

Where the price is paid #

The moment the bolts no longer carry is carried by the plate turning further, and by the bed carrying more of it near the edge. Both are costs, and they belong to the column and the grout rather than to the bolts.

The loop shows both halves of the trade. Up to the rigid range the two bases are almost the same. Beyond it the stacked base is far softer, because its bolt rows, the only things resisting a plate that has lifted, are springs of 37 kN/mm rather than 444. It goes much further round, and comes back to where it started, with no permanent rotation and no lost preload. The bare base goes round a tighter loop, yields at the tips, and comes out of the storm a little looser than it went in.

At 20 kN/mm the base turns 5.2 times as far as the bare one at the storm’s peak. For a column whose frame was designed on the fixed base the bolts decide, that is the base becoming a much softer spring exactly when the storm is largest, and the column’s top sways further in proportion. For a column designed as nominally pinned, as the pinned base that is not pinned found most are, it costs nothing that was counted.

The second cost is the one that limits how soft a stack may be. The bed’s resultant moved out to carry the moment the bolts gave up, and to move out it had to concentrate. At 20 kN/mm the plate bears on the last 91 mm of its length at a peak of 37.8 N/mm², nearly twice the bearing strength allowed for the grout and concrete under it; the bare base, with its bed’s resultant further in, peaks at 16.7 over 209 mm. The storm the bolts no longer feel lands on the grout. At 5 kN/mm the edge pressure would be three times the strength, and the elastic bed of the model would long since have crushed.

Choosing the stack #

The figures leave a band, and it is not at the soft end.

A stack of about 300 kN/mm per bolt — the same stiffness as the bolt itself — does the job and costs little. The storm then takes the bolts to 222 kN, below their yield of 226, so a 150 kN preload survives a storm to ±170 kN·m and every storm after with no stretch at all. The rigid range is 53.8 kN·m, more than the shaken-down bare base’s 46.0 and within a twentieth of the new one’s. The base turns 1.95 mrad at the peak, 1.4 times the bare base, and the grout’s edge pressure is 19.9 N/mm², inside its strength. The stack is closed 0.50 mm at the preload and 0.74 mm at the storm’s peak, which is the travel a stack must provide, and it must not close flat before the bolt reaches its yield, which means a flat load above 226 kN.

Softer stacks buy margin against a larger storm at the price of rotation and grout pressure; stiffer ones run out of margin. A stack of 100 kN/mm keeps 187 kN through ±170 kN·m — enough to survive a storm to ±175 with a 150 kN preload — but turns the base 2.90 mrad and puts 24.2 N/mm² on the grout. Which is right is a question about the storms the base will see and the frame above it, not about the bolts. The stack’s job is to be about as soft as the bolt, not much softer: halving the row’s stiffness is what moves the compression-side preload and the bed’s resultant enough to keep the tension bolts below yield, and anything beyond that is paid for at the grout.

The series spring, by hand #

A bolt of 290 kN/mm, its share of the plate of 948 kN/mm and a stack of 300 kN/mm in series have a stiffness per bolt of

$$\frac{1}{1/290 + 1/948 + 1/300} = 128\ \text{kN/mm},$$

so a row of two is 255 kN/mm against the bare row’s 444: a little more than half as stiff. At 20 kN/mm the stack dominates, and the row is 37 kN/mm, a twelfth of the bare row.

The preload a bolt loses for each millimetre it stretches is the row’s stiffness per bolt. Bare, 222 kN; with the 300 kN/mm stack, 128 kN; with the 20 kN/mm stack, 18 kN. The bare base lost 26 kN in its first storm, which at 222 kN/mm is the 0.12 mm it stretched. A stacked base that did stretch by the same amount would lose 15 kN at 300 kN/mm and 2 kN at 20 — but at both stiffnesses it does not stretch at all, because the plate’s turning has already carried the storm round its bolts.

A linear stack, a rigid plate and one storm #

The stack is linear and never closes. Real disc springs stiffen as they approach flat, carry some friction between nested discs, and relax a little under sustained load, so a preload set through a stack drifts down over the first months. A stack that closes flat during a storm becomes part of the bolt again and gives the bare base’s ratchet back.

The plate is rigid and the bed is elastic. A 20 mm plate bearing on its last 91 mm is not rigid; it bends over the edge, spreading the pressure further than the model allows and lowering the peak. The grout at the edge, carried past its bearing strength, crushes locally rather than pushing back harder, and a crushed edge keeps a set that the next storm starts from.

The springs are in the open. A holding-down bolt’s nut sits in a pocket, often filled with grout or left to collect water. A disc spring corroding in a wet pocket, or locked solid by grout poured over it, is not a spring. The stack has to be protected for as long as the base is relied on to keep its preload.

And the storm is reversing and always the same. Disc springs cycled through a large travel have a fatigue life of their own, which a few storms a year will not approach — the same distinction the load that never came near failing anything draws between a ratchet counted in events and a crack counted in millions of cycles; a base under a machine that rocks it many times a minute is a different question.

The column, the pocket and the inspector #

They cannot show the column above, which sways further in every storm a stacked base turns further, and which is where the decision to stack a base is really made. They cannot show the nut and the stack as a detail — the washer under the stack, the space the stack takes in the pocket, the thread length it uses up. And they cannot show the inspector, who can no longer read anything useful from a nut that has not moved: on a stacked base the evidence that the preload is still there is the stack’s own height, which is the one thing worth recording.

Still open: the grout that crushes at the edge #

Every bed here is elastic: it pushes back in proportion to how far it is pressed and gives back everything when the plate lifts. A stacked base asks its bed for nearly twice its bearing strength at the edge in a storm, and grout pressed that hard crushes locally, losing height it does not recover. A crushed edge lets the plate turn further in the next storm, which crowds the pressure still further out, onto grout that has already been crushed once. Whether a stacked base’s grout shakes down after a few storms, as the bare base’s bolts did after one, or ratchets in its turn — so that the stack has moved the storm’s ratchet from the steel to the grout rather than removed it — is the question a bed that keeps a set puts to a base whose bolts no longer do.