Structural Engineering & Statics
where the load goes
Internal forces

Three times as strong under a smaller pad

Press a small plate onto a large block of concrete and it will carry three times the stress a cylinder of the same concrete fails at. The enhancement is a ratio of areas with no material property in it, which should be a warning: what has actually been measured is not the concrete's strength but the strength of a tie holding it together.

Assumes When there is no section to design, The force that splits what it pushes on and How far a wrong load reaches.

13 min read 7 figures The load must go somewhereGeometry beats materialWhich failure arrives first

A concrete cylinder is crushed in a testing machine at 30 N/mm² and the number goes on the drawings. Then a 250 mm square steel plate is set on a 700 mm square block of the same concrete and pressed until something happens, and the plate is at 51 N/mm² when it does.

Nothing has changed about the concrete. The mix is the same, the age is the same, the curing is the same. What has changed is what is around the loaded area, and the difference is worth a factor of three.

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 2.80 for the 250 mm pad on a 700 mm block drawn — 47.6 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 16.1% of the load. It has to be tied. 1099 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. 1 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 two curves are one statement.

The design rule that captures it is

fRd=fcdA2/A1≤3fcdf_{Rd} = f_{cd}\sqrt{A_2/A_1} \le 3f_{cd}

with A1A_1 the loaded area and A2A_2 the area it is allowed to spread into. It has no material property in it beyond the one being enhanced, and a rule of that shape is always a rule about geometry wearing a material’s clothes.

Which free body produced the number

Cut the block on a horizontal plane a little below the pad and take everything above it. The load PP comes in over the small area; the same PP leaves over a larger one, spread across the width of the block. Between the two the compression has moved sideways, and a compression that moves sideways is not travelling in a straight line.

Draw the two struts that carry it — one to each side, from the edges of the pad down and out to the edges of the spread area — and the free body is complete except for one thing. Each strut is inclined, so each has a horizontal component, and the two point in opposite directions. Something must hold them apart.

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 4e-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 2.36 N/mm² 0.44 depths in. The tension integrates to 228 kN and the compression to -228: 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 281 kN, which is 1.23 times the tensile resultant — a design model deliberately above what the field says.
Fig. 2 The same argument drawn as a model: two inclined struts spreading a concentrated force, and the tie that stops them from separating. The tie is the only tension in the picture and the only thing that is not concrete.

The tension in that tie, from the strut-and-tie model of a symmetric block, is

T=14P(1−a1a2)T = \tfrac{1}{4}P\left(1 - \frac{a_1}{a_2}\right)

and at a spread ratio of three it is one sixth of the applied load. On the block drawn — a 200 mm pad and a 600 mm spread — that is 340 kN, and at a design steel stress of 435 N/mm² it is a little under a thousand square millimetres of reinforcement.

That number is the enhancement. Not a metaphor for it: the whole of the extra 20 N/mm² the pad can carry is bought by putting that steel where the strut-and-tie model says the tension is, and a block without that steel splits along the line the tie was supposed to occupy, at a load very close to the unenhanced one.

Why a square root

The enhancement is A2/A1\sqrt{A_2/A_1} rather than A2/A1A_2/A_1, and the reason is that the spread does not go on forever.

If the load genuinely dispersed into the whole of A2A_2, the stress there would be P/A2P/A_2 and the pad could carry fcdA2/A1f_{cd}A_2/A_1 times as much — a linear ratio, unbounded. It does not, because the spread is not free. Getting the compression sideways costs the tie force above, and the tie force rises with the spread. The square root is what is left after those two effects are traded against each other, and it is the reason the rule has the form it does rather than being a simple statement about area.

There is a second reading, which is squeezed sideways into a different material. Concrete under triaxial compression is a stronger and much more ductile material than concrete under uniaxial compression, and the material around a small loaded area supplies exactly that: a lateral restraint proportional to how much of it there is. The two readings are not rivals. The confinement is what raises the local strength; the tie is what makes the confinement possible.

The same concrete, held sideways. Two stress-strain curves for one concrete. The lower is a cylinder test: it peaks at 35 N/mm² near a strain of 0.002 and has nothing left by 0.0035, because it fails by splitting apart sideways. The upper is the same material inside a 12 mm hoop at 75 mm centres, which cannot stop it expanding but can make the expansion stretch steel: the lateral pressure of 3.58 N/mm² — 10% of the strength it is multiplying — takes the peak to 55.1 and the ultimate strain to 0.030. The strength gain is 1.58 times and the strain gain 8.5; the area under the curve, which is the toughness, goes up by 13. It is the third number the confinement is provided for.
Fig. 3 What confinement does to concrete when it is supplied deliberately. The peak rises and the tail changes completely, and the second is usually worth more than the first.

The cap of three, which is not about concrete

Every version of this rule caps the enhancement, usually at 3.0 and sometimes at 3.3. The cap looks like a statement that concrete stops responding to confinement past a point, and it is not.

It is a statement about strut angle. Spread a 200 mm pad into a 1,200 mm width and the struts run one across for every one down over the depth available — a slope of about 45 degrees at the outside, which is already flat for a strut in a discontinuity region. Push further and the model asks the concrete to turn the compression through an angle at which the “strut” is no longer a plausible flow of stress and the transverse tension is spread over a length that has nothing to anchor it against.

The lower bound theorem, which is what licenses a strut-and-tie model at all, is silent about credibility. Two ways of being wrong is the essay about what those theorems guarantee, and the guarantee here is: any stress field in equilibrium with the load that nowhere exceeds the material strength gives a safe estimate. A very flat strut satisfies that on paper. It also asks a concrete block to behave in a way no test has been run to confirm, so the cap is where the profession stopped believing its own drawing.

The same logic bounds A2A_2 geometrically as well: three times the loaded dimension in each direction, and no more than the loaded dimension plus twice the depth available. A block that is wide and thin cannot deliver a spread its depth has no room for.

Where this is met, which is nearly everywhere

The rule reads as a specialised check for a bearing pad, and it is not. The same argument, with the same square root and the same tie, appears in half a dozen places this collection has already been, and in every one of them the load arrives over an area much smaller than the member it is entering — which is the definition of a discontinuity region, and the reason when there is no section to design is the method rather than a sectional calculation.

A post-tensioning anchorage is a small plate delivering a very large force into the end of a member, and it is the case where the tie is drawn explicitly and called bursting reinforcement. The force that splits what it pushes on is that essay, and the geometry is identical — the anchorage is a bearing plate with a very high stress on it and a very tight spiral around it.

The force spreads, and the spreading needs a tie. The end block behind an anchorage of 1800 kN on a 260 mm plate, in a section 900 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 160 mm and needs a transverse tie to turn it. Placing the tie 0.4 depths from the face makes that tie force 400 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 17.3 N/mm² against 5.0 once the force has spread.
Fig. 4 An end block behind an anchorage. The transverse tension has a distribution rather than a single value, and where the bar is put matters as much as how much of it there is.

A column on a pad foundation is a small area on a large one, and the enhancement is why a column can deliver 40 N/mm² into a base cast from 30 N/mm² concrete without a steel plate under it.

A bearing on a pier is the case the rule is named after, and it is the one where the block is often not much larger than the pad — an edge bearing has spread on three sides and none on the fourth, and A2A_2 has to be taken as the area actually available, which is frequently barely more than A1A_1.

A masonry pier under a beam end is the same statics with a material that has no tie at all, which is why a spreader plate under a beam bearing on brickwork is not a detailing nicety. Masonry’s version of the rule is the middle third applied twice over — once to keep the resultant inside the kern so the whole bearing is in contact, and once to keep the pressure under it below what the pier can carry with no tension available anywhere.

And a bolt bearing on a plate is the metal version, where the enhancement is real for the same reason and is limited by something else entirely: the hole that goes oval is a check on deformation rather than on strength, because steel confined against a bolt does not split, it flows.

The failure it prevents, and what it looks like

A bearing failure without a tie is a splitting failure, and the crack is vertical, under the middle of the pad, running down into the block.

That is not what most people expect. The load is a compression and the intuition is that the concrete under it is crushed, so the picture in the mind is a cone of powder. It happens, at very high local stresses under a stiff plate, but the ordinary failure of an unreinforced block is a clean split down the line of maximum transverse tension — which the strut-and-tie model located before anything was built, about a tenth of the block’s width below the pad and running for about the block’s width.

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 2344 kN at a lever arm of 40% of the depth down to 987 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 A family of models for the same block. Every one of them is in equilibrium with the load, every one is a lower bound, and the tie force differs between them — so the model chosen decides how much steel is needed and where it goes.

How far a wrong load reaches fixes the length scale. Saint-Venant’s principle says the disturbance from a concentrated load dies away in about one dimension of the member, and the transverse tension lives entirely inside that distance. Below it the section is in uniform compression and there is nothing to reinforce.

Which of the several possible models to draw is a real choice, and the lower bound theorem does not make it. Any of them is safe; they differ in how much steel they ask for and where they put it, and a model that puts the tie in the wrong place is safe on paper and produces a block that cracks where nothing was provided. The convention — a tie at between a tenth and a whole spread width below the pad — is the model that most nearly follows the elastic stress trajectories, which is the sensible tie-breaker: put the steel where the material was going to want it anyway, so that the crack it controls opens onto reinforcement rather than beside it.

So the bursting steel is a local detail with a length, a position and an amount, and all three come from the same model. The amount is TT; the position is between about one tenth and one whole spread width below the pad; the length is the spread width. Put it anywhere else and it is steel in a region with no tension in it.

The measurement problem underneath all of this

There is something uncomfortable about a “strength” that depends on the size of the plate used to apply the load, and it is worth stating rather than moving past.

A cylinder test gives 30 N/mm² because the specimen is loaded across its whole section by platens that also confine its ends — which is why a cylinder fails in a cone-and-split pattern rather than by crushing uniformly, and why a cube test gives a higher number than a cylinder of the same concrete. The “strength” is already a property of the test arrangement.

A bearing test gives 51 N/mm² because the specimen is loaded across a fraction of its section by a small plate, with the rest of the block confining it. Also a property of the test arrangement.

Neither number is the strength of concrete, because concrete does not have one: it has a failure surface in three stress dimensions, and any single number quoted for it is that surface evaluated at whatever stress state the test happened to produce. The bigger one is the weaker one is the other half of the same discomfort — a strength that depends on the size of the specimen as well as on the shape of the loading. The shear strength nobody measured makes the same point about a different corner of the same surface.

A metal does not care what pressure it is under; nothing else agrees. Strength against hydrostatic pressure, for a metal and for a pressure-dependent material. Von Mises's criterion contains only stress differences, so squeezing a metal equally in three directions does nothing at all to it and its locus is a cylinder along the hydrostatic axis — the flat line. A granular material's is a cone: its strength rises with pressure at a rate fixed by its friction angle, 30°, and this is the same statement as the confinement argument, where a lateral pressure of a twelfth of the concrete's strength raises that strength by half. The two are not variants of one theory; they disagree about whether a quantity appears at all.
Fig. 6 The surface a single strength is one point on. Concrete’s is strongly pressure-dependent, which is exactly why confinement is worth so much and why one number cannot describe it.

Where the model stops

One more consequence of the length scale is worth stating because it decides a detail rather than a number. The bursting steel has to be distributed over that depth rather than concentrated at the point of maximum tension: a single large bar at the peak is correct by the model and wrong in practice, because it controls a crack at one level and lets the concrete above and below it split. Several small bars over the spread width is the same area of steel doing a job the model cannot express, which is holding a region together rather than resisting a force.

The block was assumed symmetric. A pad near an edge or a corner has spread on fewer sides, and the correct A2A_2 is the symmetric area centred on the pad that actually fits — which for a corner bearing can be barely larger than the pad. Getting this wrong is the commonest error in applying the rule, because the formula does not object.

The load was assumed concentric and static. An eccentric bearing loads part of the pad and unloads the rest, so A1A_1 is the effective area rather than the plate area; and a bearing that moves grinds the surface it is on.

Nothing here is about local crushing under a very stiff plate. A thick steel plate does not distribute its load uniformly at all; it is stiffer than the concrete and the pressure peaks under its edges. A thin plate does the opposite.

The pressure runs away outside the middle third. Peak bearing pressure under a 4 × 3 m base carrying 1400 kN, against the eccentricity of the load. Inside the middle third the line is straight and the pressure has doubled by the time it reaches the edge of it: 117 kPa at the centre, 233 kPa at e = B/6. Beyond that the base lifts, the contact length shortens, and the curve turns upward without limit — at e = 1.54 m the peak is 674 kPa on 1.38 m of base.
Fig. 7 What an eccentric bearing actually delivers. Once the resultant leaves the middle third the contact area is smaller than the plate, and the effective bearing area is smaller with it.

And the tie has to be anchored. A bursting bar is in a discontinuity region and it reaches its tension a short distance from where it starts, so it needs a hook, a loop, or a length that the block does not have. The force that arrives along a length is what a straight bar relies on, and a block 700 mm across does not contain an anchorage length for a 16 mm bar at either end of it. Steel calculated correctly and lapped in the wrong place is not steel.

Nor is anything here about what happens after first cracking. The tie is drawn as though it were there from the start, and it does nothing at all until the block has cracked along the line it crosses — so the enhancement is available only to a member that is allowed to crack, which is a serviceability decision made by a strength calculation.

The generalisation

The result to carry away is the habit of asking, whenever a strength is enhanced by geometry, which tension paid for it.

Enhancement by confinement is one of the very few free lunches in structural engineering, and it is not free. It is bought by the material around the loaded area, and that material can only supply the restraint if it is held together — by a tie, by a hoop, by the surrounding structure, or by a self-weight that happens to be there. Where the holding-together is absent, the enhancement is absent with it, and the formula that predicted it is a formula about a different block.

The other half is geometry beats material in its purest form: two blocks of identical concrete, differing only in the size of the plate on top, differ in strength by a factor of three. Nothing was added. The stress state was changed, and the stress state is what a strength is a property of.

Where this goes next

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. A truss drawn inside a solid, and solved as one. A deep member 4000 mm between bearings and 2000 mm deep, carrying 1200 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 750 kN and each strut at 960 kN, at 38.7° to the horizontal. Spread over a strut width of 812 mm the compression is 3.0 N/mm² against a limit of 15.8 for concrete cracked across its own strut, and the tie needs 1724 mm² of steel. A beam calculation on the same member would have asked the tie for 702 kN, which is 7% less than the model does. When there is no section to design Beam theory needs a section, and a section needs the strain to be linear across it. Within about a depth of a support, a load, a corner or a hole it is not — and those are the regions structures actually fail in. The same concrete, held sideways. Two stress-strain curves for one concrete. The lower is a cylinder test: it peaks at 30 N/mm² near a strain of 0.002 and has nothing left by 0.0035, because it fails by splitting apart sideways. The upper is the same material inside a 12 mm hoop at 100 mm centres, which cannot stop it expanding but can make the expansion stretch steel: the lateral pressure of 2.48 N/mm² — 8% of the strength it is multiplying — takes the peak to 44.4 and the ultimate strain to 0.028. The strength gain is 1.48 times and the strain gain 8.0; the area under the curve, which is the toughness, goes up by 11. It is the third number the confinement is provided for. Squeezed sideways into a different material Concrete in a cylinder test fails by splitting apart sideways under a load pushing it down. Put a hoop round it and the splitting has to stretch steel — and a lateral pressure of a twelfth of the strength raises the strength by half and the ultimate strain by eight. Three ways to apply the same force, and one depth to forget the difference. Three end loads on a member 400 mm deep, all with the same resultant and the same moment: a point load, the same force spread over a fifth of the depth, and the same force split in two. What is plotted is the difference between each of them and the beam-theory answer — the self-equilibrating remainder — as a fraction of the mean stress. The point load starts at 20 times it and is under a tenth of it by 0.77 depths; all three are under one per cent by about 1.18. That distance is the licence every figure in this collection is drawn under, and the exact strip eigenvalue agrees with it: 2.106 + 1.125i, whose real part puts one per cent at 1.09 depths and whose imaginary part means the remainder changes sign on the way out, which no statement of the principle mentions. How far a wrong load reaches Every figure in this collection applies a load as a point, a line or a uniform pressure, and no real load is any of those. The licence is Saint-Venant's, it is usually quoted as a principle, and it is really a statement about a wavelength.

Named alongside this one

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

The objects this essay names

Each one links to every other essay that touches it.

Anchorage zoneBearingBearing stressBurstingConfinementDiscontinuity regionFree bodyLoad pathLower bound theoremSaint-Venant's principleSplittingStress trajectoryStrut and tieTieTriaxial stress