Structural Engineering & Statics
where the load goes
Connections

The failure that is in the concrete

An anchor bolt is a steel component and its capacity is usually decided by something else entirely — a cone of concrete pulled out around it, failing in tension, in a material every other calculation on the project has assumed cannot take tension at all. The exponent in the capacity says so: it is not the square the geometry implies.

Assumes Where the structure meets the ground, and when the bolts start working, A check made on a perimeter, not on a section and The bigger one is the weaker one.

20 min read 8 figures Which failure arrives firstScale changes everythingThe load must go somewhere

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.

A base plate, and when the bolts start working. A 500 × 500 mm plate carrying 600 kN and 180 kN·m, so the resultant sits 300 mm from the centre against a kern of 83.33 mm. The plate is in bolts engaged: bearing over 150.88 mm at a peak of 20 N/mm², with the holding-down bolts carrying 154.42 kN. The plate lifts at 50 kN·m and crushes at 126 kN·m, and the bolts are not needed until 150 kN·m.
Fig. 1 A 500 mm square base plate carrying 600 kN and 180 kN·m, which puts the resultant 300 mm from the centre against a kern of 83.33 mm. The plate lifts at 50 kN·m, crushes at 126 and does not need its bolts until 150 — past which they are asked for 154.42 kN, over a bearing zone 150.88 mm long at a peak of 20 N/mm². Everything in that sentence is a property of the plate and the steel. Whether the 154 kN is available is a question about the concrete.

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 reason it has to be a cone at all, rather than a plane, is that no plane is available. Every calculation in this collection that begins by choosing a section assumes strain varies linearly across it, and within about a depth of a concentrated force it does not. An anchor head is a concentrated force delivered into the middle of a solid: there is no section to take it on, and the surface has to be assumed before anything can be computed.

Where a section exists, and where it does not. The same beam divided into the regions the two theories own. Within about one depth of a support, a concentrated load, a corner or an opening, the strain is not linear across the section and every calculation on this site that begins by choosing one is inapplicable — those are the D-regions, marked here. What is left between them is the B-region, where beam theory is exact enough to have been trusted for two centuries. On a beam this deep the D-regions are most of it, which is the practical reason the strut-and-tie model exists at all: 38% of this span is a region a section cannot describe.
Fig. 2 Where a section exists and where it does not, on the member this collection uses to make the distinction. Within roughly one depth of a support, a concentrated load, a corner or an opening the strain is not linear across the section, and every calculation that begins by choosing one is inapplicable there; on the beam drawn, 38% of the span is region of that kind. An anchor head is that condition in its purest form — a force introduced into a solid with no section to introduce it to.

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 hefh_{ef} reaches 1.5hef1.5h_{ef} out in every direction and its projected area on the surface is about 9hef29h_{ef}^2. Multiply by a tensile stress and the capacity goes as hef2h_{ef}^2.

It does not. The measured capacity goes as hef1.5h_{ef}^{1.5}, and the missing half-power is the whole of the interesting content.

Where the exponent went

A strength that is a property of the specimen. Nominal strength against size for geometrically similar specimens of one material. On the left the specimen is too small for a crack to run and the strength is a plateau — a plastic limit, and the regime laboratory specimens sit in. On the right a crack releases more energy than it consumes as soon as it starts and the strength falls as the inverse square root of size, which is the regime real structures sit in. The turn happens at D₀ = 80 mm. A 100 mm specimen reads 2.80 N/mm² and a 900 mm member of the same material carries 1.20: the test overestimates the structure by a factor of 2.33.
Fig. 3 Nominal strength against size for geometrically similar specimens. A 100 mm specimen reads 2.80 N/mm² and a 900 mm member of the same material carries 1.20 — the test overestimates by a factor of 2.33.

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. A punching perimeter drawn two effective depths out from a 300 mm square column, at d=180d = 180 mm, is 4×300+4πd=3,4624 \times 300 + 4\pi d = 3{,}462 mm long against 1,200 mm round the column itself — so two thirds of it is the rounded corners rather than the faces. A breakout cone is dominated the same way by the parts furthest from the anchor, which is exactly the part an edge or a neighbour takes away.

What that area is asked to carry has no counterpart anywhere else in the design. The cone parts from the slab in tension, across a surface no reinforcement was placed to cross, in a material every other calculation on the project has agreed has no useful tensile strength at all. It is worth seeing that tension computed rather than asserted, and the cleanest place to see it is the block behind a post-tensioning anchorage, where the same kind of concentrated force enters the same material and the transverse stress it produces falls out of equilibrium alone.

Compression under the plate, tension behind it. The transverse stress along the axis of an end block, computed from an assumed spread of the longitudinal stress and the two equilibrium equations — no elasticity anywhere in it, and the far face closes to 1e-11 N/mm², which is what says the assumed flow is an equilibrium field rather than a sketch. It is compressive right under the plate, crosses zero 0.31 depths in, and peaks in tension at 0.67 N/mm² 0.44 depths in. The tension integrates to 101 kN and the compression to -101: nothing pushes the block sideways, so the two are the same force, and the residual on that identity is 0.0%. Guyon's tie for the same block is 125 kN, which is 1.23 times the tensile resultant — a design model deliberately above what the field says.
Fig. 4 A 1,000 kN force delivered through a 500 mm plate into a metre-deep block — the same plate as the base plate above, and roughly the group tension four M24 bolts could supply — with the transverse stress plotted along its axis. It is compression directly under the plate, crosses zero 0.31 depths in, and peaks in tension at 0.67 N/mm² at 0.44 depths. That tension integrates to 101 kN against 101 kN of compression and the far face closes to 1e-11 N/mm², which is what says the assumed flow is an equilibrium field rather than a sketch: nothing pushes the block sideways. A load that is compressive everywhere it is applied has asked concrete for a hundred kilonewtons of tension.

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 1.5hef1.5h_{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 3hef3h_{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.

Every number in the last three paragraphs came out of an assumed geometry. The 35° angle is assumed, the reach of 1.5hef1.5h_{ef} follows from it, and the merged shape of two overlapping cones is a decision about where to draw a line. None of it is derived from anything. That is not a defect peculiar to anchorage — it is the standing condition of every calculation made where no section exists, and in such a region the models available form a family rather than an answer.

Every one of these models is safe, and they disagree by a factor of two. The tie force in a strut-and-tie model of the same region, against the lever arm the model assumes, as a fraction of the depth. It runs from 1500 kN at a lever arm of 40% of the depth down to 632 kN at 95%, and every model in the shaded band is in equilibrium with the same load. The lower-bound theorem says all of them are safe if the structure is built to carry what they ask for, so choosing one is not a calculation — it is a decision about where the reinforcement goes and how much the concrete has to be trusted. The band is where the strut angle stays between 25° and 65°, outside which the model stops resembling anything the concrete will do.
Fig. 5 What a chosen geometry is worth, on the model this collection uses when a section has failed: the tie force in a strut-and-tie model of one region, against the lever arm the model assumes as a fraction of the depth. It runs from 1,500 kN at 40 per cent of the depth down to 632 kN at 95 per cent — a factor of 2.4 — and every model in the shaded band is in equilibrium with the same load, so the lower-bound theorem says every one of them is safe to build to. Choosing one is a decision about where the reinforcement goes rather than a calculation. The assumed cone angle is the same kind of decision, with the same kind of error bar, made by somebody else and written into a formula.

The band ends where the strut angle leaves 25° to 65°, which is the point at which the model stops resembling anything the concrete will do — and that limit, too, is a convention rather than a derivation. Both constructions 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.

The first is prying. 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 a 20 mm flange carrying 180 kN that comes to 241.88 kN, 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.

The second is that a plate flexible enough to pry is also a plate that does not share. It loads its nearest bolt first, so the cone check that assumed four equal quarters is being made on a tension smaller than the one the corner bolt actually has. Both errors run the same way, and both are made before the concrete has been reached at all.

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.

There is a third geometry underneath both of them, and it is what connects this essay to every other anchorage question on a project. Shear across a crack is carried by the roughness of the crack faces rather than by the bars: the bars supply the normal force that presses the two faces together, and a bar that is not anchored on both sides supplies none of it. 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.

The shape of that result is not peculiar to cones. Any capacity that comes from a concentrated force spreading into a larger area behaves the same way: it grows with the area available to spread into rather than with the area loaded, and it stops growing when the spread becomes too steep to believe in.

An enhanced strength that is the strength of a tie. Bearing strength as a multiple of the design cylinder strength, against how far the load is allowed to spread, with the bursting tension the spread creates on the same axis. The enhancement is √(A₂/A₁) and it reaches 2.60 for the 500 mm pad on a 1300 mm block drawn — 44.2 N/mm² against a design strength of 17.0. There is no material property in that statement beyond the one being enhanced, and the reason is on the second curve: a load that spreads does so along inclined struts, a pair of inclined struts has a horizontal component, and that component is 15.4% of the load. It has to be tied. 3908 mm² of steel is what the enhancement actually is, and the cap of three is not a property of concrete — it is the angle past which nobody believes the strut.
Fig. 6 The same arithmetic where a code writes it down: bearing strength as a multiple of the design cylinder strength, against how far the load is allowed to spread. The enhancement is √(A₂/A₁) and reaches 2.60 for the 500 mm pad spreading into 1,300 mm drawn here — the plate and the merged cone of the paragraph above — which is 44.2 N/mm² against a design strength of 17.0. The second curve is what the enhancement costs. The inclined struts that do the spreading have a horizontal component of 15.4% of the load, and carrying it takes 3,908 mm² of steel, so the enhanced strength is the strength of a tie. The cap of three is not a property of concrete either; it is the angle past which nobody believes the strut.

An anchor group is that statement without the code clause. The capacity follows the area the cones can reach into, and packing the anchors closer does not enlarge it.

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 h1.5h^{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.

The anchor that was not cast in

Everything above is a cast-in anchor: a bolt placed before the pour, with a head bearing on concrete that was compacted around it. A great many anchors are installed into concrete that is already hard, and the mechanisms change with the installation.

Expansion anchors are held by friction. A wedge or a sleeve is forced against the side of a drilled hole, and the anchor’s tension is resisted by that contact. The mechanism is an inequality rather than an equation, with the consequence familiar from every friction connection on this site: the capacity depends on a clamping force that can relax, and a load reversal or a vibration that lets the wedge slip does not put it back.

Undercut anchors cut a recess at depth and bear mechanically into it, so they behave essentially like cast-in anchors and the cone check above applies unchanged.

Bonded anchors — a threaded rod set in resin or cementitious grout — transfer load along their whole embedded length by adhesion to the hole’s wall, and they have a failure mode the cast-in anchor does not: the rod pulls out with a cylinder of resin and concrete around it, sometimes with a shallow cone at the surface on top of it.

That mode has a different exponent, and it is the reason bonded anchors are used at all. A bond failure’s surface is πdhef\pi d h_{ef} — a cylinder — so its capacity grows linearly with embedment, where cone breakout grows as hef1.5h_{ef}^{1.5} but from a much larger constant. The two curves cross, and past the crossing the bond governs; below it the cone does. So a deep bonded anchor is a bond problem and a shallow one is a cone problem, and the design has to establish which before any formula is applied.

The hazard is one no mechanical anchor has. Resin creeps. A polymer holding a sustained tension at an elevated temperature relaxes over months and years, and the anchor slowly pulls out at a load it carried indefinitely in a short-term test. That is what took down 26 tonnes of ceiling panels in the Boston I-90 connector tunnel in 2006: epoxy anchors, sustained tension, and a creep rupture that no static capacity check contained.

Three requirements follow from it and all three look like paperwork until the mechanism is understood. Anchors used in sustained tension have to be qualified for sustained load, which is a long-duration test rather than a calculation. They carry temperature limits, because creep rate is strongly temperature-dependent and a resin in a plant room is not the resin in the laboratory. And the hole has to be cleaned, thoroughly, because the bond is to the hole’s wall and drilling dust is a debonding layer — which makes brushing and blowing a drilled hole a structural operation with a capacity attached, performed by somebody holding a brush.

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.

Part of why that hierarchy is necessary rather than merely prudent is that the two materials are not variants of one theory. A metal’s strength does not depend on the pressure it is under and a granular material’s does, so there is no common yield surface on which one can be traded against the other; they disagree about whether a quantity appears at all. A designer whose intuition was built sizing steel has nothing to bring across.

What the hierarchy looks like once it is drawn is a bar taking over a tension the concrete was being asked for.

The force spreads, and the spreading needs a tie. The end block behind an anchorage of 1000 kN on a 500 mm plate, in a section 1000 mm deep. Half the force enters at the quarter point of the plate and leaves at the quarter point of the section, so a strut between the two rises 125 mm and needs a transverse tie to turn it. Placing the tie 0.5 depths from the face makes that tie force 125 kN — and at exactly half a depth this reproduces Guyon's 0.25P(1 − a/h) to the digit, which makes that famous coefficient a lever arm somebody chose rather than a property of concrete. The bearing stress under the plate is 4.0 N/mm² against 2.0 once the force has spread.
Fig. 7 The same 1,000 kN through the same 500 mm plate into the same metre-deep block, drawn as the truss it is. Half the force enters at the quarter point of the plate and leaves at the quarter point of the section, so the strut between them rises 125 mm and cannot turn without something pulling it round; a tie half a depth from the face carries 125 kN doing that. At exactly half a depth this reproduces Guyon’s 0.25P(1 − a/h) to the digit, which makes that famous coefficient a lever arm somebody chose rather than a property of concrete. The bearing stress falls from 4.0 N/mm² under the plate to 2.0 once the force has spread — and the transverse tension, 101 kN of it on the previous figure, is now a bar’s problem rather than the concrete’s.

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.

A truss drawn inside a solid, and solved as one. A deep member 4000 mm between bearings and 2000 mm deep, carrying 800 kN at mid-span. The model is two struts and one tie, on a lever arm of 1600 mm, and it is solved by the truss solver rather than by a formula: the tie comes back at 500 kN and each strut at 640 kN, at 38.7° to the horizontal. Spread over a strut width of 750 mm the compression is 2.1 N/mm² against a limit of 15.8 for concrete cracked across its own strut, and the tie needs 1149 mm² of steel. A beam calculation on the same member would have asked the tie for 468 kN, which is 7% less than the model does.
Fig. 8 The replacement, drawn as what it is: a truss inside a solid, solved by the truss solver rather than by a formula. A member 4,000 mm between bearings and 2,000 mm deep carrying 800 kN returns a tie of 500 kN and struts of 640 kN at 38.7°, which over a strut width of 750 mm is 2.1 N/mm² against a limit of 15.8 for concrete cracked across its own strut, and 1,149 mm² of steel in the tie. A beam calculation on the same member would have asked the tie for 468 kN — 7% less than the model does, which is the price of admitting there is no section.

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 — on a clock set by the square of the cover, and by no strength appearing anywhere in this essay.

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.

Where this goes next

A base plate, and when the bolts start working. A 500 × 400 mm plate carrying 600 kN and 90 kN·m, so the resultant sits 150 mm from the centre against a kern of 83.33 mm. The plate is in partial contact: bearing over 300 mm at a peak of 10 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 50 kN·m and crushes at 120 kN·m, and the bolts are not needed until 150 kN·m. Where the structure meets the ground, and when the bolts start working Push a base plate off its middle third and it lifts off the foundation. The holding-down bolts then carry exactly nothing, and go on carrying nothing until the plate has crushed the concrete underneath it. A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658. A check made on a perimeter, not on a section Every shear check in this collection is made on a plane cut through a member. A slab sitting on a column has no such plane, because the shear leaves in every direction at once — so the check is made on a closed line, and a line grows with the column while the load grows with the square of the bay. A strength that is a property of the specimen. Nominal strength against size for geometrically similar specimens of one material. On the left the specimen is too small for a crack to run and the strength is a plateau — a plastic limit, and the regime laboratory specimens sit in. On the right a crack releases more energy than it consumes as soon as it starts and the strength falls as the inverse square root of size, which is the regime real structures sit in. The turn happens at D₀ = 120 mm. A 100 mm specimen reads 3.10 N/mm² and a 1500 mm member of the same material carries 1.14: the test overestimates the structure by a factor of 2.71. The bigger one is the weaker one Two geometrically similar beams of the same concrete should fail at the same nominal stress, because a strength is supposed to be a material property. They do not. The large one fails at less, and the reason is that a crack releases energy in proportion to a volume and consumes it in proportion to an area. The force spreads, and the spreading needs a tie. The end block behind an anchorage of 1200 kN on a 200 mm plate, in a section 700 mm deep. Half the force enters at the quarter point of the plate and leaves at the quarter point of the section, so a strut between the two rises 125 mm and needs a transverse tie to turn it. Placing the tie 0.5 depths from the face makes that tie force 214 kN — and at exactly half a depth this reproduces Guyon's 0.25P(1 − a/h) to the digit, which makes that famous coefficient a lever arm somebody chose rather than a property of concrete. The bearing stress under the plate is 20.0 N/mm² against 5.7 once the force has spread. The force that splits what it pushes on A prestressing tendon delivers its whole force through a plate a fraction of the section deep. One depth further along the stress is uniform, and the spreading in between requires a transverse tension nobody applied — the force that splits end blocks, and the only number in the design that no equilibrium equation on the member can see.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

AnchorageBase plateBreakoutBrittlenessCapacity designCoverDuctilityEdge distanceLoad pathPryingPunching shearShear frictionSize effectStrut-and-tieTensile strength