The base that is rigid until the bed lets go

The fixed base the bolts decide treated a column base as what the component method says it is: three springs in series, the concrete under the compressed flange, the plate bending at the bolt row, and the holding-down bolts stretching over their free length. For a 500 by 400 mm plate 20 mm thick under a 260 mm column, held by two M24 bolts in each of two rows, the base came out at 9.1 times the column’s $EI/L$ — less than a third of the 30 that EN 1993-1-8 asks of a base before it may be called rigid in a sway frame — and the bolts were most of its flexibility. A thicker plate could not close the gap, more bolts and a thicker plate together reached about 22, and a longer plate made it worse.

That essay ended on the remedies it had not tried: stiffening the bolts themselves. They are long because they have to be anchored deep, and their eight diameters of free length are what makes them soft. A bonded anchor grips the concrete almost at once and stretches over a fraction of that length. And any bolt can be preloaded against the plate, so that it carries moment as a stiff clamped joint does rather than as a long spring. Whether either can reach thirty without a base much larger than the column is the question.

The base on its bed #

The component method’s springs are a model of a joint that is already open: one side of the plate pressed down on the grout, the other lifted and held only by the bolts. A preloaded joint is closed, so it needs a model in which the plate can be pressed down everywhere and let go anywhere. Take the plate as rigid, sitting on an elastic bed — the grout and the concrete under it — that pushes back in proportion to how far it is compressed and cannot pull. Give the bed the stiffness EN 1993-1-8 itself gives the concrete under a base, $E_c\sqrt{A}/1.275$ for an area $A$, spread over the whole plate. Make each bolt row a spring: the bolts’ axial stiffness over their free length in series with the plate’s bending at the row. Apply a moment, and find the plate’s sinking and rotation that keep it in equilibrium, letting the bed lose contact wherever it would otherwise be in tension.

With no preload this model gives the base 58,600 kN·m per radian, 9.8 times the column’s $EI/L$, against the component method’s 54,200 and 9.1. The two methods share the bolts and the plate and differ in how they treat the concrete, and they agree within a tenth — which is the check that the bed is the same base.

Shorter bolts help, and stop helping #

The cast-in bolt’s free length is eight diameters plus the grout, the plate, a washer and half a nut — 256 mm for an M24. A bonded anchor, resin-set into a drilled hole, starts gripping almost at the grout’s underside, and its free length is perhaps three diameters plus the same plate and grout, 136 mm. That takes the base from 9.8 to 13.9 times $EI/L$, a gain of two fifths.

It is not a road to thirty. Halving the bolts’ flexibility does not halve the base’s, because the plate bending at the bolt row and the concrete under the compressed flange are still in series with them — and as the bolts get stiffer those two become a larger share of what is left. A free length of 60 mm, shorter than any real anchor could have, reaches 19.3. The curve flattens because springs in series are governed by the softest, and once the bolts stop being the softest, shortening them buys nothing.

Preload, and what it actually changes #

Tighten each bolt to 150 kN against the plate. Before any moment arrives, the four bolts pull the plate down onto the bed with 600 kN, and the bed pushes back with 600 kN spread under the whole plate. The joint is now closed, the way a preloaded bolted joint is closed: the plate and the bed and the bolts are clamped together, and a moment has to unload the bed on one side before it can do anything else.

In that state the base is the bed and the bolts acting in parallel, not in series. The bed resists rotation over the whole plate, not only under one flange; the bolts on both sides add their stiffness to it rather than adding their flexibility. The base becomes 273,000 kN·m per radian, 45.6 times $EI/L$ — half as much again as the rigid boundary asks. Preload does not make the bolts stiffer. It makes the concrete under the plate part of the joint.

The bolt barely notices the moment while this lasts. At 40 kN·m the most heavily loaded bolt carries 156 kN, six more than its preload, because almost all of the moment is carried by unloading the bed on the tension side. That is the same arithmetic that protects a preloaded bolt in fatigue, and it is the reason the base is so stiff: the moment goes into a short, stiff path — the bed — and not into the long, soft one.

Until the bed lets go #

The arrangement lasts exactly as long as the bed under the tension side is still pressed down. The moment that first relieves it completely, at the plate’s tension edge, is the decompression moment — here 57 kN·m. Past it the plate begins to lift at that edge, the bearing length shrinks, the bolts begin to carry the moment as stretched springs, and the base heads back toward the unpreloaded joint it would have been. By 100 kN·m only two thirds of the plate bears, and the bolt carries 173 kN; by 150 kN·m it carries 212, more than an M24 of grade 8.8 is designed to carry in tension, which is the joint’s other limit arriving.

So “rigid” becomes a statement about a range of moments. Every preload gives the same initial stiffness, 45.6 times $EI/L$, because before decompression the stiffness is the bed’s and the bolts’ and has nothing to do with the preload’s size. What the preload sets is how long it lasts: at 50 kN a bolt the base’s secant stiffness falls below thirty at 43 kN·m; at 100 kN, at 80; at 150, at 120. The rigid range is set by the preload and by nothing else.

For a sway frame that is a useful property and a dangerous one. The frame’s drift under wind is computed at moments within that range, and there the base is rigid. Its stability under the full factored load, and its response in an earthquake, involve moments well beyond it, and there the base is the unpreloaded spring — which is the one the frame’s buckling length and its second-order design should have been checked with.

What the frame gains #

The number thirty is a classification boundary, and what a frame actually feels is its drift and its buckling load. For a portal 4 m tall and 8 m wide on these bases, with the column’s $EI/L$ the 5,985 kN·m per radian used throughout, the unpreloaded base gives the frame 79 per cent of the sway stiffness it would have on perfectly fixed bases and 84 per cent of the buckling load. Bonded anchors raise those to 84 and 88. A base exactly at the rigid boundary gives 92 and 94, and the preloaded base, 94 and 96.

So the preload buys about fifteen per cent of drift stiffness over the cast-in base, and about the same in buckling load — enough to matter for a frame whose drift governs, and the reason the rigid boundary exists: past thirty, the frame cannot tell its base from a fixed one to within the few per cent that analysis is good for. It is the same kind of gain a pinned base that is not quite pinned gives at the other end of the scale, and it comes from the same place: the ground the structure meets is part of the frame, and its stiffness is as much a design variable as the column’s.

It is also the same trick a slip-critical bolted joint plays in shear. There, preload clamps two plates together so that friction carries the load with no movement until it slips; here, preload clamps the plate to its bed so that the bed carries the moment with almost no rotation until it lets go. In both the preload is not a strength but a range: the joint is stiff up to a definite load and ordinary beyond it.

The column’s weight does the same thing #

There is a preload every base already has. A column carries its share of the building’s weight down onto the base plate, and that compression presses the plate onto the bed exactly as the bolts’ preload does. The bed cannot tell the difference: 600 kN of column load decompresses at the same 57 kN·m as 600 kN of bolt preload, and below that moment the base is just as rigid.

That is already in the design rules, in a form easy to read past. EN 1993-1-8 gives a base’s rotational stiffness separately for a base whose eccentricity, moment over axial load, is small — both sides of the plate in compression, the stiffness that of two concrete springs and no bolts at all — and for one whose eccentricity is large, the bolts in tension and the component method’s series spring. The fixed base the bolts decide checked the second case at zero axial load, which is the base of a lightly loaded column and the worst base there is. A heavily loaded column in a building with modest moments is sitting in the first case, and its base is rigid without anyone having stiffened anything.

So the case for preloading holding-down bolts is the column whose axial load is small and whose moment is not: a portal frame’s column, a single-storey hall, a sign or a mast, a column at the top of a building where the weight above it is little. There the preload buys a range of rigid behaviour the column’s own weight cannot supply.

The bed creeps, and the short bolt pays #

A steel-to-steel preloaded joint keeps its preload because steel does not creep at room temperature. A base plate is clamped onto grout and concrete, and both creep under sustained compression and shrink as they dry. Every tenth of a millimetre the bed under the plate gives up is a tenth of a millimetre the bolts are allowed to shorten, and a bolt that shortens loses tension in proportion to its stiffness.

So the two ways of stiffening the bolts fight. A cast-in bolt, 256 mm free, loses a seventh of a 150 kN preload when the bed shortens by 0.1 mm; a bonded anchor, 136 mm free and stiffer, loses a fifth. At 0.2 mm the cast-in bolt keeps 73 per cent and the bonded anchor 59. The stiffer the bolt, the faster the bed takes its preload away — exactly as a prestressing tendon loses force to the creep of the member it compresses, and for the same reason: a stiff tendon and a soft one shortened by the same amount do not lose the same force.

Since the decompression moment is proportional to the preload, the rigid range shrinks with it. A base preloaded on the day it was grouted, to be rigid up to 120 kN·m, is rigid up to perhaps 90 a year later if its bolts are short and to 100 if they are long. Re-tightening after the grout has cured and crept is the practical answer, and it is the reason holding-down bolts that are relied on for preload are specified with a re-tightening inspection and long free lengths deliberately — sleeves left unbonded at the top — rather than short ones.

One base, by hand #

The preloaded base’s initial stiffness is two parallel terms. The bed’s modulus is $k = E_c/(1.275\sqrt{A}) = 33{,}000/(1.275 \times 447) = 57.9$ N/mm³, and the plate’s second moment of area is $BL^3/12 = 400 \times 500^3/12 = 4.17 \times 10^9$ mm⁴, so the bed gives $57.9 \times 4.17 \times 10^9 = 2.41 \times 10^{11}$ N·mm per radian. Each bolt row, two M24s of 353 mm² stretching over 256 mm ($2 \times 210{,}000 \times 353/256 = 580$ kN/mm) in series with the plate’s bending at the row (about 1,900 kN/mm), is a spring of 444 kN/mm, 190 mm from the plate’s centre: the two rows give $2 \times 444{,}000 \times 190^2 = 3.2 \times 10^{10}$. Together, $2.73 \times 10^{11}$ N·mm per radian — the 273,000 kN·m of the figures — and the bed is 88 per cent of it.

The decompression moment is the moment whose bed share unloads the tension edge: the preload’s 600 kN over the plate’s area, 3.0 N/mm², times the plate’s section modulus $BL^2/6 = 1.67 \times 10^7$ mm³, is 50 kN·m carried by the bed, and the bed carries 88 per cent of any moment, so the base decompresses at $50/0.88 = 57$ kN·m. Without the bolts in parallel it would be 50 — the moment at which a column carrying 600 kN through the plate’s middle third would lift its edge, which is all the decompression moment is.

A rigid plate, a linear bed and one preload #

The plate is rigid on its bed. A real 20 mm plate bends, and under preload it bends most around the bolts; the bed’s rotational stiffness is therefore an upper bound, and the true preloaded stiffness somewhat lower. The comparison with the component method, within a tenth, says the bound is not far off for this plate.

The bed is linear and is the concrete’s. Grout is often softer than the concrete under it, and a thick grout layer lowers the bed’s stiffness; the bed here uses the concrete’s modulus throughout.

And the preload is what the bolt was tightened to. The torque that goes into the thread decides how much of a tightening becomes tension, and an holding-down bolt tightened by torque on a dusty site has a scatter of preload that the curves here do not carry.

Cracks, grout pads and the second storm #

They cannot show the concrete cracking. Bolts pulled near the edge of a pedestal crack it, and a base whose pedestal has cracked has a bed that is no longer the elastic half-space assumed.

They cannot show a shimmed base. Many bases are set on steel shims or levelling nuts and grouted afterwards, and a base held on levelling nuts has its preload shared between the nuts and the grout in a way that depends on the order of tightening.

And they cannot show the cycle. A base that decompresses in one storm and closes again has bolts that have been stretched beyond their preload and may have yielded; the second storm meets a base with less preload than the first, and a stiffness that has been spent rather than kept.

Stiffen the joint, not the bolt #

Shortening the bolts helps, and then stops. Bonded anchors take the base from 9.8 to 13.9 times $EI/L$, and nothing shorter reaches thirty, because the plate and the concrete are still in series.

Preloading makes the bed part of the joint. The base is then 45.6 times $EI/L$ — rigid — up to the moment that lets the bed go, 57 kN·m at 150 kN a bolt, and stays above thirty in secant to about twice that.

The column’s own load does the same. 600 kN of compression is worth 600 kN of preload, which is why bases under heavy columns are rigid and bases under light ones are not.

And creep spends the preload fastest in the stiffest bolt — a fifth of it from a bonded anchor for every tenth of a millimetre the grout gives up.

Still open: the base that decompresses and closes again #

Every curve here is a single loading from rest. A base under wind or earthquake is loaded one way, then the other, many times, and each excursion past the decompression moment stretches the bolts on one side further than their preload while the other side’s bed is compressed harder than the preload put it. When the load reverses, the stretched bolts do not return to their preload, the bed under them has been worked, and the next excursion begins from a different joint. Whether a preloaded base settles into a smaller rigid range after a few cycles, or whether it ratchets until it is the unpreloaded base it was before the bolts were tightened, is the question the preload has to answer before it can be counted on in a storm.