Squeezed sideways into a different material

A concrete cylinder in a testing machine fails by coming apart sideways. The load is pushing down; the cracks run vertically; the specimen splits into columns of its own and then loses everything at once. A compression test on concrete is a tension failure, and the tension is in the direction nobody applied a load in.

That immediately suggests a remedy. Stop the sideways expansion and the splitting cannot happen — and stopping it does not require anything like the force being applied downward, because the transverse strains are a fifth of the longitudinal ones and the transverse stresses needed to stop them are smaller still.

A closed hoop of 12 mm bar at 100 mm centres around a 450 mm column delivers an effective lateral pressure of 2.5 N/mm², which is a twelfth of the concrete’s strength. What it buys is not a twelfth of anything.

Which free body produced the number #

Cut the hoop and take half the core.

The hoop, at yield, pulls with $A_h f_{yh}$ at each of the two cut ends. That force has to be balanced by the pressure acting on the projected area of the half-core over the length the hoop is responsible for — its spacing $s$. So

$$f_l = \frac{2 A_h f_{yh}}{d_s\,s}$$

For the column here, 3.2 N/mm². That is the pressure the steel can supply, at the level of the hoop, and it is an upper bound on what the concrete gets.

Between hoops the confinement arches. The pressure at a hoop is the full value; midway between two hoops the concrete is held only by whatever arching action spans between them, and near the surface it is held by nothing. Mander’s coefficient for a circular hoop is

$$k_e = \frac{(1 - s'/2d_s)^2}{1 - \rho_{cc}}$$

with $s'$ the clear spacing, and for this column it is 0.79. The effective pressure is 2.5 rather than 3.2.

The rectangular case is worse, because the arching happens in two directions and between longitudinal bars as well as between hoops: the same steel in a square column gives $k_e = 0.61$. A circular hoop confines by hoop tension alone and a rectangular one relies on its corners, which is why spirals outperform ties by a wide margin for the same weight of steel.

Two placements of one picture, and the difference between them is a shape rather than a quantity of steel. The same bar, the same spacing, the same yield stress: only the geometry the tension is delivered through has changed, and a fifth of the pressure has gone. It is worth reading against the toughness, which is the number the hoops are provided for — that stays at eleven, because the strain gain has risen from 8.0 to 8.4 while the strength gain fell. The rectangular hoop loses strength and keeps deformation, which is the trade a designer would have chosen if asked.

The two gains, which are not the same size #

Mander’s expression for the confined strength is

$$f_{cc} = f_{co}\left(-1.254 + 2.254\sqrt{1 + \frac{7.94 f_l'}{f_{co}}} - \frac{2f_l'}{f_{co}}\right)$$

which is a fitted form for a triaxial test surface. At $f_l'/f_{co} = 0.083$ it gives $f_{cc}/f_{co} = 1.48$.

The peak strain follows a much steeper rule, $\varepsilon_{cc} = \varepsilon_{co}[1 + 5(f_{cc}/f_{co} - 1)]$, which multiplies it by 3.4. And the ultimate strain — set by the hoop fracturing, when the energy stored in the steel equals the energy the concrete needs — comes out at 0.028 against an unconfined 0.0035, a factor of eight.

The area under the curve, which is the toughness and is the physically meaningful thing, goes up by a factor of eleven.

So the three numbers a designer might quote are 1.5, 8 and 11, and they are quantities of quite different kinds. The first is a strength and appears in strength equations; the second and third are deformation quantities and appear nowhere in a strength equation at all. Confinement is a ductility measure that has a strength side effect, and treating it the other way round is the mistake this whole essay exists to prevent.

The cover leaves first #

There is a step in the behaviour that the curve above hides, and it is the reason confined design is not simply better.

The concrete outside the hoops is not confined. It has nothing holding it and it fails at the unconfined strain of about 0.0035 — at which point it spalls off, taking the cover with it. So a confined column loses its cover before the core has begun to demonstrate what it can do.

The consequence is a drop in capacity at a strain of 0.0035, when the section’s area suddenly becomes the core’s area rather than the gross one. For a 450 mm column with 40 mm cover the core is 71 per cent of the gross area, so the section has to make up a 29 per cent loss of area with a 48 per cent gain in strength — which it does, but only just, and for a thicker cover it does not.

That is why confined sections are analysed with two materials: cover concrete on an unconfined curve that ends at 0.0035, and core concrete on the confined one. The moment-curvature curve of such a section has a notch in it where the cover goes.

What the hoops are actually resisting #

A detail worth noticing: the hoop is not loaded by anything until the concrete tries to expand, and the concrete does not try to expand much until it is near failure.

At working stress the transverse strain in concrete is about $\nu$ times the longitudinal one, with $\nu \approx 0.2$, so the hoop is picking up a strain of a few tens of microstrain and a stress of a few newtons per square millimetre. The confinement steel is doing nothing at service load.

As the concrete approaches its unconfined strength the internal cracking begins and the apparent Poisson’s ratio rises steeply, past 0.5 — which is to say the material starts increasing in volume as it is compressed. That is when the hoop is loaded, and it reaches yield only when the core is well past its unconfined peak.

So a confining hoop is a component whose entire working life happens in the last few per cent of the structure’s history, and which is inert until then. That is unusual and it has a design consequence: the hoop’s own strength is irrelevant to the service behaviour, and specifying a higher-grade steel for it does raise the confinement — one of the few places on this site where a stronger material genuinely helps.

The comparison is worth making because it is so unlike the usual answer. Raising the grade of a longitudinal bar buys strength in proportion and buys nothing else; raising the grade of a hoop buys pressure in proportion and the pressure buys deformation at the exchange rate the fits above describe. Set beside the rectangular hoop above, the two moves are of much the same size and opposite in sign — the geometry took the pressure from 2.5 N/mm² to 1.93 and the grade takes it to 2.98 — which says that the shape a hoop is bent to is worth about as much as a hundred newtons per square millimetre on the steel it is bent from, and it is also why the argument has a ceiling: the confining pressure is set by the hoop’s yield stress, so the gain stops the moment the steel supplying it is not ductile enough to reach that stress and hold it, and what is left when the load comes off in the hoop is a strain the concrete has already spent.

The reason any of this is designed for #

A column that reaches its strength and then loses it is a column that fails a building. A column that reaches its strength and holds it through a large rotation is a member that can redistribute, that can be part of a mechanism, and that can absorb energy.

Every argument in this collection that depends on a structure being able to reach a chosen distribution — moment redistribution, plastic collapse, a strut-and-tie model, a seismic mechanism — is an argument that depends on rotation capacity, and rotation capacity in a concrete member is bought almost entirely with confinement.

That is why confinement requirements appear where they do. They are heaviest at the ends of columns and beams, where hinges are expected; they are heaviest in seismic design, where a mechanism is the design intent; and they are absent in a member designed to stay elastic, where nothing needs to rotate.

The seismic case is the one where the third number is being bought outright. A member removes energy from a structure by enclosing an area in a load-deflection cycle, and the area a member encloses is what the whole of that design method is spending. A confined section can trace that loop repeatedly; an unconfined one loses its cover on the first excursion and its core on the second, so its second loop encloses very little and its third encloses nothing. That is the same statement as the eleven-fold toughness gain, read once per cycle rather than once — and it is why the earthquake asks for a displacement rather than for a force, since a displacement is what a hoop makes available and a force is what it barely changes.

The number that decides how much is needed #

There is one quantity that governs how much confinement a column needs, and it is not the concrete strength or the seismic zone. It is the axial load ratio — the axial force divided by the section’s squash load.

The reason is visible in the fibre computation. A column at a low axial load has a small compression zone, and the strain at the extreme fibre for a given curvature is small; a column at a high axial load has a compression zone covering most of the section, and the same curvature demands a far larger extreme-fibre strain. So the strain capacity needed to deliver a given rotation grows steeply with axial load, and confinement is what supplies it.

That relationship is why codes tie confinement requirements to the axial load ratio rather than to anything about the earthquake, and why a heavily loaded column in a tall building needs far more confinement than a lightly loaded one in a low-rise frame — and why, past an axial load ratio of about 0.5, no realistic amount of confinement will give a column useful rotation capacity and the design has to reduce the load instead.

It also gives the cleanest statement of why the whole subject sits in the materials field rather than in sections. The section’s geometry is fixed; the axial force is given; what is being adjusted is the material’s own stress-strain curve, and the hoops are the mechanism for adjusting it.

Where the model stops #

The hoop is at yield. The pressure calculation assumes it, and it is true only near ultimate. At intermediate strains the pressure is lower and the confined curve the section is following is not the one drawn.

The model is for monotonic loading. Under cycling the concrete degrades, the hoop strains ratchet outward, and the effective confinement falls with each cycle. A confined section’s tenth cycle is not its first.

And the arching coefficients are geometric idealisations. The parabolic arches that $k_e$ is derived from are a picture, fitted to tests. They are reasonable for the spacings used in practice and become meaningless for spacings far outside them.

The hoop has a second job #

The spacing of a hoop is decided twice, by two mechanisms that have nothing to do with each other, and the second is usually the one that governs.

The first is confinement: closer hoops mean less arching loss and a higher $k_e$, which is the whole of the argument above. The second is that the longitudinal bars are in compression and will buckle, outward, between the hoops that restrain them — and when they do, they push the cover off, the hoop opens, and the confinement ends.

The arithmetic of that is not the ordinary column one. A 25 mm bar restrained at 300 mm centres has a slenderness of $L/r = 300/6.25 = 48$, and elastically it would buckle at $\pi^2 E/48^2 = 857$ N/mm² — well above yield, so elastically it is safe. But the bar is being asked to work at strains far past yield, where its tangent modulus has collapsed from 200,000 to a few thousand. At a strain-hardening modulus of 4,000 the same slenderness gives a critical stress of 17 N/mm², and the bar goes at once.

So the restraint spacing has to be measured in bar diameters rather than in fractions of the section, which is exactly how every confinement rule is written: hoops at no more than six or eight longitudinal bar diameters in a hinge region. That number looks like a detailing convention and is a buckling calculation on a strut whose stiffness has been destroyed by the strain the confinement was provided to permit.

That figure is the two rules agreeing, which is convenient rather than coincidental. A 250 mm spacing is about ten diameters for a 25 mm bar, so it fails the buckling rule; and it delivers a toughness gain of four where 100 mm delivers eleven, so it fails the confinement one. The wide spacings are bad for both reasons at once, and the narrow ones are good for both, which is why a single number in bar diameters can be written into a code without anybody having to say which mechanism it came from.

The two rules therefore constrain the same variable from two directions, and a designer who satisfies the arching requirement and ignores the bar-buckling one has built a column whose confined behaviour ends at the first excursion — with the hoops correctly sized, correctly spaced for confinement, and intact.

Two fits to the same surface #

Mander’s expression is a fit, and it is worth putting beside the older and simpler one to see how much of the answer is physics and how much is calibration.

Richart’s 1928 result, from triaxial tests on cylinders, is a straight line:

$$f_{cc} = f_{co} + k\,f_l', \qquad k \approx 4.1$$

At the 2.5 N/mm² of confinement here that gives $30 + 10.3 = 40.3$ N/mm² against Mander’s 44.4. The two differ by 10 per cent, which is small; what is not small is the coefficient hiding inside them. Mander’s expression at this pressure has a secant slope of 5.8 rather than 4.1, and near zero pressure its slope is nearly 7 — it is concave, so the first newton of confinement is worth more than the tenth.

Both are fits to a failure surface in three-dimensional stress space rather than derivations, and the coefficient $k$ is the slope of that surface: how much more axial stress the material carries for each unit of lateral pressure applied. A slope of four to seven is a very steep exchange rate, and it is the whole reason the mechanism is worth having — 2.5 N/mm² of pressure, an eighth of the concrete’s own tensile capacity, buying 14 N/mm² of compressive strength.

The disagreement between the two fits is a reminder of what kind of quantity is being quoted. Neither is wrong; they were fitted to different test populations at different pressure ranges, and a value quoted to three figures from either is quoting the calibration rather than the concrete.

Three ways of holding it in #

The hoop is one of three ways of applying the same lateral pressure, and comparing them shows what the mechanism actually requires.

Transverse steel. Bars or spirals around the core, loaded in tension when the concrete expands. Cheap, familiar, and limited by arching between hoops and — in a rectangular section — between longitudinal bars.

A steel tube. A concrete-filled hollow section confines continuously, with no arching loss at all, so $k_e = 1$. It is the most efficient form by a wide margin, and its complication is that the tube is also carrying axial load and therefore is not free to be at yield in hoop tension at the same time: the two demands share the steel’s capacity through a yield criterion.

Fibre wrap. A composite jacket applied to an existing column, which confines elastically rather than at yield — so the pressure grows with the expansion rather than saturating, and the confined curve rises continuously instead of levelling off. It is the standard retrofit for a column built before anybody required hoops.

The three produce visibly different confined curves from one mechanism, and the differences are all about when the pressure arrives: at yield and then constant, at yield with arching losses, or growing without limit until the jacket ruptures.

What they share is the property that made the mechanism worth finding. None of them carries any of the axial load in the direction it is applied. All of them work by refusing to let the material do the one thing it needs to do in order to fail.

What the pictures cannot show #

The confined curve is drawn as one curve for one material, and the section is two materials at once — core and cover — with the cover’s contribution ending abruptly partway along. Nothing on the stress-strain figure indicates that the section it belongs to loses a third of its area at a strain of 0.0035.

Nor can any of these figures show the hoop doing its work, which is a three-dimensional stress state in a material with no visible sign of it. The confined core looks exactly like the unconfined one until the moment they behave differently.

A third thing outside the figures is the construction. Hoops at 100 mm centres in a column already congested with longitudinal bars, laps, and a beam’s reinforcement arriving from four directions is a detail that has to be built, and confinement requirements are among the most frequently compromised on site. A hoop whose 135° bend has been made at 90° for ease of fixing opens under the pressure it was provided to resist, and the confinement it was calculated to give is not there — which is a failure with no visible symptom until the column is asked to rotate.

The assumption the figure rests on #

The hoop’s rupture strain is taken as 0.12, and the ultimate strain of the concrete is proportional to it. That is a property of the bar — a mild reinforcing steel with good elongation — and it is the assumption most likely to be wrong in a real structure. Cold-worked or high-strength hoop steel has a lower rupture strain, and a hoop that fractures early ends the confinement at once: the core is unconfined from that instant and the column has no post-peak behaviour at all. The confinement’s benefit is bounded by the ductility of the steel providing it, which is a longer chain of dependence than the tidy expression suggests.

The steel is also being asked to do it repeatedly. A hoop holds its tension while the core expands and contracts around it, cycle after cycle, at strains well past yield, and each reversal softens it a little through the Bauschinger effect — so the confining pressure available on the tenth excursion is not the one this page computed for the first. Confinement is the property that appears in none of the equations made purchasable, and like that property it is spent as it is used.

The ladder from here #

Later rungs on this anchor: the cyclic degradation of confined concrete and why the tenth cycle differs from the first. The spiral against the rectangular tie, and the factor of arching that separates them. Confinement by steel tube rather than by bars, which is what a concrete-filled hollow section is. The interaction between confinement and axial load ratio, which decides how much rotation a column can deliver at a given level in a building. And the same triaxial argument outside concrete — the reason a soil under a footing is stronger than the same soil in a shear box, and the reason a rock under confinement behaves like a ductile material.