The failure that is in the concrete

A holding-down bolt is designed like any other steel component: an area, a stress, a division. The number that comes out of that division is almost never the capacity, because the bolt is cast into concrete and the concrete goes first.

That is an unusual situation in this collection, and it inverts the usual reading of a connection that is not a point: the region that matters is not inside the joint but in the material underneath it. Everywhere else a connection’s capacity is decided by the connection — a bolt in shear, a weld throat, a plate in bearing. Here it is decided by the material the connection was attached to, in a mode that material is not usually asked to work in at all.

Which free body produced the number #

Take a cone of concrete with the anchor’s head at its apex and a free surface at its base, and pull the anchor out of it. The forces on that free body are the anchor’s tension, the self weight, and a tensile stress distributed over the conical surface where the cone parts from the rest of the slab.

The whole of anchorage design is that free body, and the two questions it raises are how big the surface is and what stress it can carry.

The surface’s size follows from an angle. Measured breakout cones stand at about 35° to the concrete face, so a cone from an embedment $h_{ef}$ reaches $1.5h_{ef}$ out in every direction and its projected area on the surface is about $9h_{ef}^2$. Multiply by a tensile stress and the capacity goes as $h_{ef}^2$.

It does not. The measured capacity goes as $h_{ef}^{1.5}$, and the missing half-power is the whole of the interesting content.

Where the exponent went #

A breakout cone is a fracture surface, and fracture surfaces get weaker as they get bigger. The energy released by a crack running through the concrete grows with a volume and the energy consumed grows with an area, so the nominal stress at failure falls as the inverse square root of the size — which takes an exponent of 2 down to 1.5 exactly.

That is the bigger one being the weaker one appearing as the exponent of a design formula rather than as a curiosity about test specimens. It is one of very few places in structural design where a size effect is written directly into a capacity equation instead of being buried in a factor, and the reason is that the effect here is large: over the range of embedments used in practice it costs a factor of two or more.

The same power appears in the concrete cone’s shear counterpart and in punching, and it always has the same origin. Wherever a capacity comes from a tensile fracture surface, a geometric exponent is reduced by a half.

The perimeter, and its counterpart #

The related check most designers already know is punching, and it is worth setting alongside because the two behave differently for a reason.

A check made on a perimeter uses a line times a depth; a breakout check uses an area. That difference makes punching resistance nearly linear in the column size and breakout capacity quadratic in the embedment, and it is the reason the two look like different subjects when they are the same failure surface counted two ways.

The comparison is also a reminder of how much of either perimeter is corner. On the slab above, the rounded corners are most of the control perimeter, and on a breakout cone the projected area is dominated by the parts furthest from the anchor — which is exactly the part an edge or a neighbour takes away.

Edges and neighbours #

The cone is a volume, and volumes interfere with things.

An anchor near a free edge has its cone truncated: the concrete that would have carried part of the tension is not there, and the capacity falls in proportion to the projected area lost. Below an edge distance of $1.5h_{ef}$ the reduction is roughly linear, and an anchor at the very edge of a slab has lost about half its cone before anything else is considered.

A group of anchors is the same problem inwards. Two anchors closer together than $3h_{ef}$ have overlapping cones, and the group’s failure surface is one merged cone rather than two — so the group capacity is the projected area of the merged shape, not twice the area of one. Packing four anchors into a small base plate buys four times the steel and very little more concrete.

That is a genuinely counter-intuitive result and it produces the characteristic detail of anchorage design: base plates that are much larger than the column they carry, with the bolts pushed out to the corners, in an arrangement that looks wasteful and is not.

The Whitmore construction is the plate version of the cone, and it is worth putting beside it because the epistemics are identical. A spread angle is assumed, an effective width follows, and the whole check rests on a geometric convention that has been checked against measurement and is not derivable. Both are conventions with error bars, dressed as geometry.

The bolt’s own problems come first #

Before the concrete has a chance, the load has to reach it, and two things happen on the way.

A flexible base plate bends under the bolt tension, its edge bears back against the concrete, and the bolt carries the applied tension plus the prying force. On the tee drawn that is a factor of 1.34, and on a thinner flange it is more. So the tension arriving at the anchor is larger than the tension in the column, and the concrete cone check has to be done on the larger number.

Shear, which fails a different way #

An anchor in shear does not pull a cone out of the surface. It pushes concrete off the edge in front of it, a half-cone on its side, and that failure depends on the edge distance rather than on the embedment.

That gives a bolt group two entirely separate geometries to satisfy — deep enough for tension, far enough from an edge for shear — and the two are not traded against each other. It also means the interaction of tension and shear on an anchor is an interaction of two different concrete failures rather than two components of one steel one.

That last clause is the connection between this essay and every other anchorage question on a project. Shear across a crack works only because the bars crossing it are anchored, and a bar anchored into a cone that has broken out is not anchored at all. The failures cascade.

Working the arithmetic once #

It is worth putting numbers on a base plate, because the ordering of the failures is the whole design and it is not obvious from any of the formulae.

Take four M24 grade 8.8 bolts, one at each corner of a 500 mm square plate, cast 300 mm into a slab. The steel capacity of one bolt in tension is about 250 kN, so the group in tension is 1,000 kN if nothing else intervenes.

The breakout cone from a 300 mm embedment reaches 450 mm in every direction. Four bolts at 400 mm spacing therefore have cones that overlap heavily: the merged projected area is roughly a 1,300 mm square rather than four separate 900 mm circles, which is about 40% of the area four independent cones would give. Multiply by a nominal tensile capacity and the group’s concrete capacity lands in the region of 500 kN.

So the concrete governs by a factor of two, and every kilonewton of extra bolt steel bought for that connection is wasted. Two moves recover it: push the bolts apart, which separates the cones and costs only plate; or embed deeper, which buys capacity as $h^{1.5}$ and costs only a longer bolt. Deepening from 300 mm to 450 raises the concrete capacity by 1.84, and that is usually the cheapest structural intervention available on a whole project.

Neither move appears in a calculation that treats the anchorage as a bolt check, because neither the spacing nor the embedment is in that calculation at all.

Why the whole subject is about ductility #

Everything above is a brittle failure. A concrete cone comes out with no warning, no redistribution and no reserve — the property that appears in none of the equations, absent, and a group of anchors that fails this way fails all at once because the cones are connected.

So the design rule that governs anchorage everywhere is not a capacity at all. It is a hierarchy: make the steel govern. Choose an embedment deep enough, and an edge distance and spacing generous enough, that the concrete’s capacity exceeds the bolt’s yield capacity — and then the connection fails by the bolt stretching, which is visible, gradual and redistributing.

That is the same capacity-design argument that puts a plastic hinge where somebody wants it and keeps the connection elastic around it. Here the thing being protected is the base material rather than a member, and the arrangement is the part that is meant to be weak with the weak part chosen to be the bolt, and the reason is the same: a failure mode is being chosen, because the alternative chooses itself.

Where the model stops #

Cracked concrete is a different material. All the numbers above assume the cone forms in sound concrete. An anchor in a region that is already cracked — which is most of a reinforced concrete member under service load — has a capacity roughly 30% lower, and whether a given anchor is in a cracked region is a question about the member’s bending, not about the anchor.

Reinforcement changes everything. Bars crossing the cone surface hold it together and can be designed to carry the whole tension, at which point the breakout check disappears and is replaced by an anchorage-into-reinforcement problem — which is the force that splits what it pushes on and a strut-and-tie model.

And the plate itself may not be stiff enough to share. Everything above assumed the four bolts carry equal shares of the tension, which requires a plate stiff enough to distribute it. A flexible plate loads its nearest bolt first — the same non-uniformity a long bolted joint has along its length, one dimension over.

Time is not in any of it. A bolt cast into concrete is protected by the concrete around it, and the cover over it fails by splitting rather than by any of the mechanisms above.

The three checks that have to be made, and their order #

A complete anchorage design is a list, and the value of the list is that its items belong to different materials and different mechanisms rather than being refinements of one another.

Steel failure of the anchor. An area times a stress, with the prying force added. Ductile, well characterised, and the one everybody computes.

Concrete cone breakout. A projected area times a tensile stress, reduced for edges, spacing and cracking, with the exponent that carries the size effect. Brittle.

Pull-out and splitting. The head or the bond fails locally without a full cone forming, which is a bearing failure under the head for a cast-in anchor and a bond failure for a bonded one. It governs at shallow embedments, where the cone is small enough that the local mechanism is weaker.

The order matters because the three have different sensitivities to the same decision. Deepening the anchor helps the second and the third and does nothing for the first. Moving it from an edge helps the second and nothing else. Making it thicker helps only the first — and if the first was not governing, it helps nothing at all.

That is why anchorage design is one of the few places in this subject where the answer to “it does not work” is almost never “make it bigger”.

What the picture cannot show #

The cone is drawn as a cone and it is not one. Real breakout surfaces are irregular, follow the aggregate, and are truncated by whatever reinforcement, ducts and services happen to lie in the way. The 35° angle is a fitted average of a scattered population.

Nor does any figure show the installation. A cast-in anchor’s capacity depends on it being where the drawing says, at the depth the drawing says, in concrete that was properly compacted around its head — and the commonest failure of an anchorage is not any mechanism in this essay but a bolt that was 40 mm shallower than intended.

The generalisation #

The habit worth carrying is to ask which material a connection’s capacity belongs to.

A bolted splice’s capacity belongs to the steel. A welded joint’s belongs to the weld metal and the parent plate. An anchorage’s belongs to the concrete, and the concrete is being asked for tension — which is the one thing the rest of the project has agreed it does not have.

That inversion is the reason anchorage is so consistently underestimated. Every instinct built up designing steelwork is about the connected parts, and here the connected part is a passenger: the anchor is a way of addressing a volume of concrete, and its capacity is a property of that volume, its edges, its neighbours, its cracks and its size. Nothing about the bolt appears in the answer at all until the design has been arranged so that it does.