The fixed base the bolts decide

The compression that stays under the flange found that a base plate carrying a moment cannot put its compression at its edge, as the rigid picture of it does, because a plate that bends delivers its bearing under the column flange. The bolts therefore work on a lever arm of 314 mm rather than 413, and carry about twice the force the rigid calculation said. It ended on the same base as a spring: its rotational stiffness “goes as the square of the lever arm between its compression and tension components, divided by the sum of their flexibilities”, so the shorter lever arm alone would make it about six tenths as stiff as the rigid picture — and it left open whether the flexibilities move that estimate, and whether the frame above notices.

Both have answers, and the second is the one that matters to a designer who drew the base as fixed.

Three springs in series #

The base is the same one: a 260 mm column on a plate 500 by 400 by 20 mm, with a row of two M24 anchor bolts 60 mm from the plate’s edge on the tension side. Turned by a moment, it rotates because three things deform one after another, and EN 1993-1-8’s component method gives each a stiffness in its own right. A joint made of springs in series set out the method for beam-to-column joints; the base has three of the springs.

The first is the concrete and grout under the compressed flange. The flange bears through a T-stub of plate whose effective width is set by the plate’s thickness — 98 by 346 mm here, the effective area of the plate that is only as big as it is thick — and that patch of concrete compresses like a stiff pad. The second is the plate bending under the bolt row, a T-stub in tension whose stiffness goes as its thickness cubed over the cube of the bolt’s distance from the flange, 53 mm. The third is the bolts, which stretch over a free length of eight diameters of their embedment plus the grout, the plate, a washer and half a nut: 256 mm for an M24.

The rotational stiffness is the lever arm squared over the sum of the three flexibilities, times the steel’s modulus:

$$S = \frac{E\,z^2}{1/k_{13} + 1/k_{15} + 1/k_{16}}$$

With the lever arm at 314 mm this base is 54,200 kN·m per radian. The column above, over a 4 m storey, has an $EI/L$ of 5,985 kN·m per radian, so the base is 9.1 times the column’s EI/L. EN 1993-1-8 calls a base in a sway frame rigid only above 30.

The bolts are most of the spring #

The split is the argument. The anchor bolts are three fifths of the base’s flexibility, the plate less than a third, the concrete an eighth. A bolt in tension is a spring whose stiffness is its area over the length it stretches over, and an anchor bolt stretches over a long length for its area: 256 mm for each 353 mm² of steel, a spring about a fifth as stiff as the patch of concrete it works against.

That changes what the obvious remedies buy. Doubling the plate to 40 mm cuts its own flexibility by nearly an order of magnitude, and the base goes from 9.1 to 13.7 EI/L — the bolts are then 77 per cent of what is left. Bolts of 36 mm instead of 24 more than double the bolts’ area but also lengthen their embedment, and the base goes to 11.7. Neither reaches the rigid boundary, and neither is close.

Why the bolts are long #

The anchor bolts are soft because they are long, and they are long for a reason that has nothing to do with stiffness. A bolt cast into concrete has to be anchored deep enough that the cone of concrete it would pull out, if it failed by pulling out rather than by breaking, is larger than the bolt’s own strength — and that cone’s resistance grows with its depth to the power one and a half. So the embedment is set by the bolt’s strength, and a stronger bolt is embedded deeper. The component method’s free length of eight diameters is the standard allowance for a cast-in bolt with a nut or plate at its foot, stretching over the part of its length the concrete does not grip.

The consequence runs against intuition. Anchoring a bolt more securely, by embedding it deeper, makes the base softer, since the bolt stretches over more of its length before the concrete holds it. The M36 bolts drawn above are more than twice the area of the M24s, but their eight diameters of stretching length are half as long again, and the base gains less than a third. Stiffness and anchorage pull the embedment in opposite directions, and on every base drawn as fixed the anchorage has won, because the anchorage is checked and the stiffness is not.

A thicker plate cannot make it rigid #

Swept over the plate’s thickness the stiffness climbs steeply and then flattens against a ceiling. The ceiling is what the base would be with a plate that did not bend at all — the concrete and the bolts alone — and it is 15.3 EI/L, half the rigid boundary. No plate thickness makes this base rigid; only the bolts can, and a base whose bolts are its weakest spring is a base whose stiffness was decided when the bolt diameter and embedment were chosen, usually by the strength check, without anybody looking at stiffness at all.

There is a small step at 27 mm. Below it the plate is thin enough that the bolt row levers against the plate’s far edge — prying — and the method charges the bolts with a lower stiffness and the plate with a higher one. Above it the plate is too stiff to pry and the charges reverse. The same threshold, set by the bolts’ length against the plate’s stiffness, is the one the force the bolt never saw applied found governing a tee’s strength.

The sleeve put in for tolerance #

Anchor bolts are commonly cast into the foundation inside tubes or cones, a couple of hundred millimetres or more, so that each bolt’s top can be moved a few millimetres to meet the holes in a base plate fabricated elsewhere. The sleeve is grouted afterwards — sometimes — and when it is not, or when the grout does not bond to the bolt, the bolt stretches over the sleeve’s length as well as over its embedment. The stiffness of the bolts falls in proportion. A 300 mm sleeve takes the base from 9.1 to 5.3 EI/L, and the bolts are then three quarters of its flexibility.

It is the clearest case in this subject of a detail chosen for one purpose deciding a property it was never considered for. The sleeve is an erection tolerance, specified by the steel erector’s needs and drawn by whoever details the foundation, and it halves the rotational stiffness of the base the frame analysis assumed was fixed.

What the rigid picture was worth #

The question the earlier essay left can now be answered in two factors. Moving the compression from the plate’s edge to the flange’s centre line, with every component’s stiffness unchanged, scales the base by the lever arms squared, 0.58, as the earlier essay estimated. The components do not stay unchanged: a plate stiff enough to put its compression at its edge would also be stiff enough not to bend under the bolts or pry on them, and letting it bend takes another factor of 0.59. The rigid picture of a base plate overstates its rotational stiffness by a factor of nearly three, and the overstatement is half geometry and half the plate.

What the frame notices #

The portal is the corner essays’ frame — 457 mm beam, 254 mm columns, 8 m span, 4 m height — standing on bases of every stiffness from a pin to a wall. Its sway under a load at the knee, and the effective length of its columns when it buckles sideways, both move smoothly between the pinned and the fixed values, and the useful range is the middle decade.

On the base as detailed, the frame drifts 28 per cent more than on fixed bases, and its columns buckle at 83 per cent of the fixed-base load, with an effective length factor of 1.23 rather than 1.13. On sleeved bolts, 45 per cent more drift and 75 per cent of the load. Even a base that just meets the design rule’s rigid boundary costs the frame 9 per cent of drift and 6 per cent of its buckling load: the boundary is where the difference becomes small enough to accept, not where it vanishes. Neither pinned nor rigid made the point for beam-to-column joints; for column bases it is sharper, because the base carries the whole storey’s sway moment and the frame’s stability depends on it directly.

A frame designed on fixed bases and built on these has a sway stiffness a fifth lower than its analysis, which is a serviceability shortfall, and a critical load factor a sixth lower, which feeds straight into the load that makes itself worse: a frame whose analysis said $\alpha_{cr} = 10$ and needed no second-order check has $\alpha_{cr}$ of about 8.3 and does.

What the boundary of thirty is for #

EN 1993-1-8’s rigid boundary is not a claim that a base of thirty times the column’s EI/L behaves as a fixed one. It is the stiffness above which treating the base as fixed changes the frame’s behaviour by an amount the design rule is prepared to ignore — here 9 per cent of drift and 6 per cent of buckling load, which is about what the boundary was chosen to allow. Below it, EN 1993-1-8 does not say the base must be made stiffer. It says the base must be modelled as what it is: a rotational spring of the stiffness the component method gives, entered into the frame analysis in place of the fixed support.

That is the practical answer to this essay, and it costs nothing. A frame analysis program takes a rotational spring at a support as easily as it takes a fixed one, and the component calculation above is a page. The designer who enters 54,200 kN·m per radian gets a frame that drifts as the real one will and a critical load factor that is the real one’s, and can then decide whether the drift and the second-order effects are acceptable. The designer who enters “fixed” has made the frame look a fifth stiffer than it is, in exactly the quantities — drift and stability — that a sway frame is usually designed by.

What it would take to reach thirty #

The component method also says what a genuinely rigid base would need, and it is more than a thicker plate. Doubling the bolts in the row, to four M24s, lifts the base from 9.1 to 12.9 EI/L; four bolts and a 40 mm plate together reach about 22 — a conservative figure, since the plate’s effective length is held at the two-bolt value. Still short of thirty, and now the bolts and the plate are sharing the flexibility, so neither alone can be improved to close the gap.

The obvious way to buy stiffness — a longer plate, putting the bolts further out on a longer lever arm — makes it worse. The lever arm enters squared, but the plate’s bending flexibility grows as the cube of the bolt’s distance from the flange, and an unstiffened plate 700 mm long with its bolts 60 mm from the end is only 12.4 EI/L, the plate now nearly three quarters of the flexibility. That is why genuinely rigid bases have gusset plates: vertical stiffeners welded between the column flange and the plate’s projecting end, which stop the plate bending and turn the long lever arm into stiffness. A rigid base is a stiffened base with a long lever arm and short, stiff bolts — a different detail from the flat plate that is usually drawn and called fixed.

In a braced frame it hardly matters #

All of this is about a sway frame, whose bases carry the storey’s sway moment. In a braced frame the bracing takes the sway, the columns do not rely on their bases to stand up, and a base’s stiffness enters only through the columns’ own buckling between floors. The rigid boundary there depends on the column’s slenderness: seven times EI/L for a column of relative slenderness one, and nothing at all for a stocky one, so the same base, at nine, would count as rigid in a braced frame and as semi-rigid in an unbraced one. A base’s classification is a statement about the base and the frame together — the conclusion of neither pinned nor rigid for joints, and it holds for bases as strongly.

One base, by hand #

The whole calculation fits on a page. The concrete’s T-stub is $c = t\sqrt{f_y/3f_{jd}} = 20\sqrt{275/60} = 43$ mm beyond the flange each way, so its effective area is $(12.5 + 43 + 43) \times (260 + 86) = 98 \times 346$ mm², and $k_{13} = E_c\sqrt{98 \times 346}/(1.275E) = 33{,}000 \times 184/267{,}750 = 22.7$ mm. The plate’s T-stub has an effective length of 200 mm — half the plate’s width governs — and a bolt distance $m = 190 - 130 - 6.8 = 53$ mm, so $k_{15} = 0.85 \times 200 \times 20^3/53^3 = 9.0$ mm. The bolts are $2 \times 353$ mm² over 256 mm: $k_{16} = 1.6 \times 706/256 = 4.4$ mm. The flexibilities are 0.044, 0.111 and 0.227 per mm, summing to 0.382, and

$$S = \frac{210{,}000 \times 314^2}{0.382} = 5.42 \times 10^{10}\ \mathrm{N\,mm/rad} = 54{,}200\ \mathrm{kN\,m/rad}$$

which over the column’s $210{,}000 \times 1.14\times10^8/4{,}000 = 5{,}985$ kN·m per radian is 9.1.

What the method assumes #

The moment acts alone. Axial compression presses the base closed and stiffens it: EN 1993-1-8’s formula multiplies the stiffness by $e/(e + e_k)$ in the partly-open regime and switches to a much stiffer both-sides-in-compression lever arm when the load’s eccentricity is small, which is how a pinned base that is not pinned resists moment it was never designed for. A base carrying a heavy column load with a small moment is stiffer than the figures here; a base under wind uplift is softer.

The stiffness is the initial one. EN 1993-1-8 reduces it as the moment approaches the base’s resistance — by up to a factor of three at full resistance — so a base designed to its strength is softer again in the load case that uses the strength. The frame’s stability is checked in exactly that case.

The foundation is rigid. The concrete block the bolts are cast into rotates on the ground beneath it, and a pad footing’s rotational stiffness on soil is of the same order as the base plate’s or lower; the two are springs in series, and the frame sees the softer.

What the pictures cannot show #

The grout. Every number here assumes the grout under the plate is full, dense and in contact; a base grouted with voids under the compressed flange has a concrete component orders of magnitude softer until the plate has bent down into contact, and no calculation of stiffness survives a void. Nor can they show the bolts’ own tightening: an anchor bolt preloaded against the plate is stiff until its preload is overcome, which the bolt that was already stretched showed for tee connections, and most anchor bolts are only snugged.

Still open: stiffening the bolts rather than the plate #

The bolts decide this base, and there are ways of stiffening them that do not involve larger bolts: shorter free lengths through bonded anchors or embedded plates at shallow depth, more bolts on a longer lever arm, or preloading them against the plate so that they carry moment at their full stiffness until the preload is overcome. Whether any of these can reach the rigid boundary of EN 1993-1-8 without a base plate much larger than the column — and whether a preloaded anchor bolt keeps its preload through the creep of the grout under it and the concrete around it, which a steel-to-steel joint never has to ask — is the question that turns the component method from a check on a base into a way of designing one to be rigid.