The plate that is only as big as it is thick

Where the structure meets the ground treats a base plate as a free body with a pressure block under it: the bearing stress is uniform or linear, the kern decides whether the holding-down bolts are working, and the resultant balances the column’s load and moment.

Every one of those statements is about a plate stiff enough to impose a plane displacement on the grout below it. No base plate is.

Which free body produced the number #

The free body is a strip of the plate, one millimetre wide, cantilevering outward from the face of the column flange.

Crossing the cut at the column face is a moment and a shear. Bearing on the underside of the strip is the grout, pushing up at $f_{jd}$. There is nothing on top of the strip at all — the column is beside it, not on it.

So the strip is a cantilever of length $c$ carrying a uniform pressure, its moment at the root is $f_{jd}c^2/2$ per unit width, and the most it can carry is the plate’s own bending capacity, $t^2f_y/6$. Setting the two equal:

$$c = t\sqrt{\frac{f_y}{3f_{jd}}}$$

Past $c$ the strip cannot deliver the pressure. It is not that the grout there is not strong enough; it is that the plate has no way of getting the force from the column out to it.

That single line is the whole of the argument, and three things follow from it that a bearing-area calculation does not contain.

The plate’s overall size stops mattering. Everything more than $c$ from the column is decoration. A 700 mm square plate on this column at this thickness delivers exactly what a 400 mm one does.

The thickness is the design variable. $c$ is proportional to $t$, and the effective area grows roughly as $c^2$ until the strips from the flanges and the web merge, so doubling the thickness roughly doubles or trebles the capacity.

And the grout strength appears twice, with opposite signs. Stronger grout raises the pressure that can be delivered and shortens the length over which the plate can deliver it, because $c$ has $f_{jd}$ in its denominator. The two nearly cancel, and a base plate on very strong grout is not much better than one on ordinary grout.

The three areas #

Reading the numbers as a sequence makes the size of the effect clear, and it is worth setting them out because the middle one is the only one that is real. All three are areas multiplied by the same bearing strength, and a bearing strength is itself a confinement argument rather than a material property.

The column’s own footprint — the steel section’s area, 168 kN at this bearing strength. That is what would be delivered by a column with no plate at all, standing on the grout.

The effective area — 78,176 mm², 1,564 kN. Nine times the column, which is what the plate is worth.

The plate’s outline — 200,000 mm², 4,000 kN. Thirty times the column, which is what the plate is not worth and what an area-based calculation assumes.

The gap between the middle number and the top one is where base plate design actually lives, and it is closed by thickness rather than by plan area.

Which is why the drawing looks the way it does #

A base plate detail has three features that look like conventions and are consequences of the argument above.

Plates are square-ish and not much larger than the column. A plate whose overhang exceeds $c$ is buying nothing, so the sensible plan dimension is the column plus about $2c$ — which for ordinary thicknesses is the column plus 60 to 100 mm.

Thick plates are common on lightly loaded columns. A 25 mm plate under a 254 UC carrying 800 kN looks over-specified against a bearing check and is exactly right against this one.

And stiffeners appear when neither works. A gusset welded between the column flange and the plate gives the plate a second support, turning a long cantilever into a shorter one or into a two-way panel — which raises the effective area without raising the thickness. That is why a heavily loaded base has stiffeners and a lightly loaded one does not, and why the stiffener is on the plate rather than on the column.

What a stiffener is worth #

The third of those detailing consequences is worth computing rather than asserting, because the answer decides whether a stiffener is a sensible intervention or a fabrication cost.

A gusset welded between the column flange and the plate does two things at once. It gives the plate a line of support where there was none, so the cantilever that was reaching out from the flange now spans between the flange and the gusset — a much shorter distance. And it lets the plate act two-way over the panel between them, which raises the capacity again.

The arithmetic is the same $c = t\sqrt{f_y/3f_{jd}}$ applied to a different geometry: the effective area becomes the region within $c$ of anything stiff, and the gussets are stiff. Two gussets per flange turn a 260 mm column into an object with a perimeter half as long again, and the effective area rises in proportion to that perimeter rather than to anything about the plate.

So a stiffener buys perimeter and a thickness buys reach, and which is cheaper is a fabrication question rather than a structural one. On a small base, thickness wins: a 30 mm plate is a purchase and four gussets are eight welds. On a large heavily loaded base, gussets win, because the thickness needed to reach across a 600 mm column is 60 or 70 mm and nobody wants to weld it.

The same argument, in the other direction #

The effective-area idea appears twice in a base connection and the second time it runs the other way.

On the compression side, the plate reaches out from the column and the effective area is what it can reach. On the tension side — a base with holding-down bolts in tension under a moment — the plate is a T-stub in bending, the bolt pulls down on it, and the prying force is decided by the same competition between the plate’s bending capacity and the length it has to span.

Both are the same object: a plate cantilevering from a stiff member, with its own bending strength deciding how far it can act. The compression case gives an area and the tension case gives a force, and the two are the compression and tension halves of the same connection, usually checked by two different sets of expressions that do not look alike.

That is a good example of a general habit worth having. When two checks on one component have the same free body, they usually have the same governing quantity, and here it is the plate thickness in both.

What it means for the numbers a designer quotes #

There is a small piece of practice that follows from all of this and it is worth stating because it inverts the usual order of a base-plate calculation.

The conventional sequence is: take the column load, divide by the bearing strength, get an area, choose a plate to suit, then check its thickness against the overhang. That sequence can produce a plate that is large, thin, and delivering a fraction of what its area says.

The sequence the effective-area argument suggests is: choose a plate about the column plus a hundred millimetres, then solve for the thickness that makes its effective area sufficient, and accept whatever plan size falls out. It gives a smaller, thicker plate, which is nearly always cheaper — one flame cut instead of a larger one, less weld, and a plate that fits inside the column’s own footprint plus an edge distance for the bolts.

Both routes are in use, and the difference between them shows on a drawing: a thin large plate is a design done the first way, and a thick small one is a design done the second. The second is right and the first is not wrong in a way any check will find, which is the recurring hazard of a component whose governing quantity does not appear in its own headline calculation.

There is a check hiding inside the effective-area calculation that is worth making explicit, because it is the one that decides whether the whole model applies. The cantilever has to be able to carry the shear as well as the moment. At the column face the strip is delivering f_jd·c of shear per unit width, and for the thicknesses and reaches on this page that is a small fraction of the plate’s shear capacity — a 20 mm plate at 43 mm of reach carries 860 N/mm against a shear capacity of several thousand. So bending governs comfortably, which is why the reach expression contains only a bending capacity and why nobody checks the shear. On a very thick plate with a very short reach the two would compete, and the arrangement that produces it — a thick plate on a small column — is exactly the one where the effective area has already reached the plate’s edges and the question has stopped mattering.

Where the model stops #

The cantilever is one-way. The corners of the effective region are quarter circles because the plate spans diagonally there, and a real corner is a two-way plate with a higher capacity than the one-way strip assumed — so the effective area is slightly conservative, by a few per cent.

The grout is uniform and the plate is flat. Grout under a base plate is placed through a hole or from one side, it shrinks, and it is not in contact everywhere. A plate bedded on high spots delivers its load through them, and none of this arithmetic applies.

The column is welded all round. A plate connected to the column by fillet welds only on the flanges has no path for the web’s load into the plate at all near the web, and the effective area attributed to the web strip is not delivered.

And there is no moment. Under combined axial load and moment the compression zone is part of the plate, the effective area is that part grown by $c$, and the whole calculation has to be done on the region actually in bearing — which the rung below computes as a pressure block on the assumption this page is denying.

What the pictures cannot show #

Whether the plate is flat enough to bear at all. A base plate is a flame-cut piece of plate that has been welded to a column on one face, so it has weld distortion in it — the plate cups toward the welds, and a 500 mm plate may be out of flat by a millimetre or two.

That is why the specification says bedded on grout rather than in contact, and why the grout is a structural material rather than a filler. The effective area calculated here is delivered only if the grout is there and has cured; a plate bearing on packing shims during erection is delivering its load through four small areas, at a pressure that has nothing to do with $f_{jd}$.

They also cannot show the foundation. The bearing strength $f_{jd}$ is the concrete’s value enhanced by the confinement of the surrounding material, which is the reason a small pad on a large block is worth three times a large one — so the number this whole page divides by is itself a function of how much concrete surrounds the plate.

The assumption the figure rests on #

That the plate yields rather than the grout.

The expression for $c$ comes from equating the plate’s plastic bending capacity to the moment the bearing pressure applies, so it assumes the pressure reaches $f_{jd}$ everywhere within $c$ and that the plate is at yield at the column face. Both are limit-state statements about a component that is, in service, elastic and delivering much less than either.

The elastic distribution is quite different: the pressure peaks under the column and decays outward, without the sharp edge at $c$ that the figures draw. The effective-area model replaces that smooth distribution with a uniform one over a smaller area — the same substitution an effective width makes for a buckled plate or for a wide flange, chosen so the totals agree at the limit state.

Which is worth remembering when reading a base plate’s actual behaviour: the effective area is a design fiction that gives the right capacity, not a description of where the pressure is. In service the corners are carrying something; at failure they are not.

The history, which is a rule of thumb that turned out to be a derivation #

Base plates were designed by rules of thumb for a century before the cantilever argument was written down, and the interesting thing is how close the rules were.

The traditional method takes the plate’s overhang from the column face as a cantilever carrying the uniform bearing pressure, and sizes the thickness so that the cantilever’s moment is within capacity. That is the same free body as the one on this page, solved for $t$ given the overhang instead of for the reach given $t$ — and it gives the same relationship rearranged.

What the old method assumed was that the whole plate was effective and the overhang was whatever the plate happened to have. The modern one assumes the plate is as effective as it can be and computes how much that is. Both are the same cantilever; they differ in which variable is taken as given, and the modern version is more useful because it says what happens when the plate is larger than the cantilever can serve, which the old one has no answer for.

The T-stub language arrived with the component method for joints in the 1990s, and it is the reason a base plate in a modern code is presented as a T-stub in compression rather than as a plate in bending. That reframing is worth more than it looks: it puts the base plate in the same family as an end plate and a flange cleat, so one set of expressions covers all three, and the effective length of a T-stub is the same idea as the effective area here.

What to carry away #

Three things, and the third is the one that transfers to other components.

A base plate delivers bearing over the column’s outline grown by $c = t\sqrt{f_y/3f_{jd}}$, and everything beyond that is carrying nothing. On an ordinary plate that is a third to a half of the area the drawing shows.

The thickness is the design variable and the plan size is not. Making the plate larger changes nothing once the overhang exceeds $c$; making it thicker raises the reach in proportion and the area faster than that.

And a component’s capacity is bounded by how far its own stiffness can reach, not by how much of it there is. That is the same statement as the strip of web that acts with a bearing stiffener, as an effective width, and as the effective length of a T-stub — four names for one idea, which is that a flexible component participates over a length its own bending decides.

The ladder from here #

Later rungs on this anchor: the base plate under combined axial load and moment, where the compression zone is part of the plate and the effective area is computed within it. Stiffened bases, where a gusset divides the cantilever and the effective area is a two-way plate problem. The anchor bolts in tension, with prying, and the concrete cone that decides them. Column bases as rotational springs, which is what the whole connection is to the frame above it and is what a pinned base is not. And the grout itself: its strength, its shrinkage, the hole it is placed through, and the difference between a bedded plate and one on shims.