Each one stops the other failing

Two materials in one member usually means one of two arguments. Either they are made to act together so that the section is stiffer than the sum of its parts, or one of them is pretended away by transforming its area.

A concrete-filled steel tube is neither, and the difference is worth stating in one sentence before any arithmetic. Each material’s failure mode requires a movement that the other one prevents.

Concrete in compression fails by splitting: it dilates laterally and cracks apart. A steel tube around it will not let it dilate. A thin steel tube in compression fails by local buckling, which requires the wall to move inward somewhere. The concrete will not let it.

That is a mutual arrangement rather than a summation, and its two halves behave completely differently.

The half that does not switch off #

Take the local buckling first, because it is the simpler and the more useful.

An unfilled cylinder in compression buckles at a stress $\alpha E t / (r\beta\sqrt3)$, where $\alpha$ is the knockdown factor that says how far short of the classical stress a real shell falls. Setting that equal to the yield stress gives the proportion past which the tube cannot reach its own material strength: for S355 that is $d/t = 64$.

Fill it and two things change. The buckling mode can no longer have inward lobes — the concrete is there — so only outward, axisymmetric modes remain, and those are stiffer. And the wall is supported against the imperfections that produce the knockdown in the first place, so the effective $\alpha$ rises.

The result is a limit at $d/t = 115$ instead of 64: nearly double. For the section drawn at $d/t = 80$ the empty tube buckles at 286 N/mm², well below its 355 yield, and the filled one reaches 508 and yields instead.

That benefit does not depend on slenderness, on the load level, or on how much the concrete has been strained. It is present in every filled tube from the moment the concrete has set.

The half that does switch off #

Confinement is the celebrated half and it is the conditional one.

At high strain the concrete dilates, the tube goes into hoop tension, and equilibrium of a half-ring gives the radial pressure

$$\sigma_r = \frac{2 t \sigma_\theta}{d - 2t}.$$

Richart’s result — a confined cylinder’s strength is its unconfined one plus about 4.1 times the confining pressure — turns that into strength.

But the tube pays for it. It is now in a biaxial state, hoop tension and axial compression together, and von Mises constrains them:

$$\sigma_a^2 - \sigma_a \sigma_\theta + \sigma_\theta^2 = f_y^2.$$

Every newton of confinement is bought from the wall’s own axial capacity. Keeping 84 per cent of the axial fixes the hoop stress at 0.266 $f_y$ — that is von Mises, not a choice — which gives a radial pressure of 2.4 N/mm² and a concrete strength gain of 24 per cent.

For the stocky column drawn at a relative slenderness of 0.18, the net is a bonus of 11 per cent over the plain sum. At 0.42, which is an ordinary building column, it is nothing at all.

The section’s own arithmetic #

It is worth putting numbers on the section drawn before going further, because the proportions of a filled tube are not what an intuition trained on reinforced concrete expects.

A 400 mm tube with a 5 mm wall has a steel area of 6,205 mm² and a concrete area of 119,459 — nineteen times as much. At S355 and C40 the two contribute 2,203 kN and 4,778 kN, so the steel provides 32 per cent of the capacity from 5 per cent of the area.

That ratio is the type’s whole economy. The steel is doing three jobs at once — carrying a third of the load, containing the concrete, and being the formwork — and it does them with a wall thin enough that on its own it could not reach its own yield stress.

The reinforcement ratio, read the way a concrete engineer would read it, is 5.2 per cent, which is at the top of what a conventionally reinforced column can accommodate and is here achieved with no bars, no links and no cover.

Why slenderness kills it #

Confinement requires the concrete to want to dilate, and concrete does not dilate appreciably until it is close to its own crushing strain — around 0.003, which is a long way up the stress-strain curve.

A stocky column reaches that strain. A slender one does not: it deflects, the second-order moment grows, and it fails by buckling at a strain the concrete would have called early. The mechanism needs a deformation the failure mode does not permit, which is why the rules switch confinement off above a relative slenderness of 0.5 and why the switch is a cliff rather than a taper.

The same argument disposes of confinement for an eccentrically loaded column. Bending puts part of the section into tension, the dilation is not uniform, the hoop tension cannot develop uniformly either, and the benefit disappears long before the eccentricity is large.

So the honest summary is: confinement is a stub-column phenomenon, and most filled tubes in buildings get none of it. What they get is the local buckling benefit, the composite stiffness, and a permanent shutter.

Which free body produced the number #

The confining pressure has a free body worth drawing, and it is the same one a pipe’s hoop tension comes from.

Cut the tube along a diameter and take half the ring. Crossing the two cut faces are the hoop forces $\sigma_\theta t$ per unit length. Acting on the inside of the half-ring is the radial pressure from the concrete, over a projected width equal to the diameter. Horizontal equilibrium gives $\sigma_r d_c = 2 \sigma_\theta t$, which rearranges to the expression above.

Nothing about that free body is unusual. What is unusual is the second free body, which is the concrete core: it is in triaxial compression — axial from the load, radial and hoop from the tube — and triaxial compression is the one state in which concrete has no failure mode at all in the usual sense. It cannot split because nothing can move outward; it can only crush by a mechanism that requires volume change, and the confining pressure suppresses that too.

A confined cylinder does not have a strength so much as a strength that depends on what is holding it, which is why the relation is written as an increment on the unconfined value rather than as a property.

The runaway, and what it says about optimisation #

Sweeping the hoop stress and taking the best total produces a maximum, and the maximum is worth looking at because it is instructive rather than useful.

It puts the whole wall into hoop tension, leaves the steel carrying no axial load at all, and relies on Richart’s linear relation to produce a confined concrete strength two and a half times the unconfined one — for a claimed bonus of 32 per cent.

That is not a design. Richart’s coefficient was fitted to plain cylinders under modest confining pressures, and extrapolating a linear fit to a pressure comparable with the concrete’s own strength is outside anything it was measured on. The optimisation does not know that, because the formula does not carry its own range.

An unconstrained optimisation of a calibrated formula finds the formula’s extrapolation, not the structure’s optimum, and it is worth keeping this example because the failure is so visible: the answer hands a structural member’s entire steel content to a secondary function.

The calibrated alternative is to fix the split from test data — 84 per cent of the axial retained at zero slenderness — and let von Mises supply the hoop stress that goes with it. Doing so gives a confining pressure at which Richart’s 9.9 N/mm² and the column-calibrated 9.4 agree to five per cent, which is the only cross-check either of them has.

What it is actually for #

The reasons filled tubes get built are mostly not the ones on this page, and it is worth listing them so the mechanics is kept in proportion.

No formwork and no reinforcement cage. The tube is the shutter, the longitudinal reinforcement and the transverse reinforcement at once. On a site with a tight programme that is the whole argument.

A small column. A filled tube of a given capacity is substantially smaller than a reinforced concrete column of the same capacity, because the confinement and the efficient placement of the steel are both working. In a building where floor area is worth more than steel, that is decisive.

Fire. This is the one that surprises people. A filled tube in a fire loses its steel quickly — the tube is unprotected and heats fast — and the concrete core, insulated by its own mass, carries the load. The load path transfers from the outside inward as the fire develops, and a filled tube can achieve substantial fire ratings unprotected where a bare hollow section achieves almost none.

What the two halves are worth, side by side #

Setting the two mechanisms against each other settles which one deserves the attention, and the answer is not the one the literature gives.

Confinement buys 11 per cent of the plain sum, for stub columns only, at zero eccentricity, and its size depends on empirical coefficients used near the edge of their range. It is the half that appears in every paper on the subject.

Local buckling suppression buys the difference between a section that can reach 355 N/mm² and one that can reach 286 — a quarter of the steel’s contribution, which on this section is 8 per cent of the total. It applies at every slenderness, at every eccentricity, and it rests on a much simpler argument: the wall cannot go inward because something is there.

The two are of comparable size and only one of them is conditional. A designer who remembers one thing about filled tubes should remember the second, and the reason the first gets the attention is that it has a formula with a coefficient in it and the second has a change of classification.

The problems that are specific to it #

Three difficulties belong to the type rather than to either material.

Load introduction. A load applied to the steel alone has to get into the concrete, and there is no shear connection between them except friction and whatever adhesion survives shrinkage. The transfer happens over a length, the bond stresses available are small, and heavily loaded connections need shear studs welded to the inside of the tube or a plate bearing directly on the core.

Shrinkage. The concrete shrinks and the steel does not, so the core pulls away from the wall. That reduces the friction available for load introduction and — more importantly — means the confinement described above has to close a gap before it can do anything. It is one reason the measured confinement benefit is smaller than the mechanics predicts.

Inspection. The concrete cannot be seen, cannot be cored without a diver’s problem, and cannot be checked for voids by any routine means. A filled tube is the one structural element whose principal material is permanently invisible.

The last of those is the one that decides how the type is used in practice. The response is procedural rather than structural: self-compacting concrete, placement from the bottom by pumping rather than from the top by dropping, and vent holes at every change of section. A member whose material cannot be inspected has to be one whose placement cannot go wrong, and that constraint reaches back into the detailing of every connection that interrupts the tube.

The square section, which loses most of it #

A rectangular hollow section filled with concrete is a common member and it is a different structure from the round one, in a way that follows entirely from the free body above.

Hoop tension in a circular tube is a membrane force: the wall carries it by stretching, with no bending anywhere, which is why a thin wall can develop a large confining pressure. A flat wall cannot do that. To resist a pressure on its inside face it has to bend, and a 5 mm plate spanning 400 mm in bending can resist almost nothing.

So a square filled tube confines its concrete only near the corners, where the two walls meet and there is something to develop a membrane force against. The confined region is four small triangles rather than the whole core, and the average benefit falls to a fraction of the circular value.

The local buckling half survives, and survives well: a flat wall backed by concrete still cannot buckle inward, and the classification relaxation for filled rectangular sections is comparable with the circular one. The half that depends on membrane action is lost with the curvature and the half that depends on the infill being there is not, which is the same division this page has been making throughout, applied to a change of shape.

That is also why the confinement literature is almost entirely about circular sections while the buildings are mostly square: the shape that is easiest to connect to is the shape that gets the smaller half of the benefit.

The interaction diagram, where the two materials swap over #

Everything above is a column in pure compression. Add a moment and the section’s behaviour changes character, because the two materials contribute to bending in a way they do not contribute to axial load.

The concrete carries axial load well and bending badly — it cracks on the tension side and half of it stops working. The steel tube is the opposite: it is a thin ring at the extremity of the section, which is exactly the arrangement that maximises a second moment, so almost all of the section’s bending resistance is in the wall.

The interaction diagram that results has an unusual shape. At pure compression the concrete dominates; at pure bending the steel does; and in between there is a bulge — a region where the section carries more moment at moderate axial load than at none, because the compression closes the cracks and lets more of the concrete work.

That bulge is real, it is characteristic of composite sections generally, and it is the one feature of a filled tube’s behaviour that a designer meets before any of the mechanics on this page.

Where the model stops #

The confinement relation is a fit. Richart’s coefficient and the calibrated split are both empirical, and they are being used at pressures near the edge of their data.

The concrete is treated as a material with a strength. Under triaxial compression it is better described by a surface than by a number, and the yield criterion for a material with no tensile strength is not the one used here.

The local buckling limits are code-implied. The empty and filled $d/t$ limits used here are back-figured from the classical stress and the knockdown each case implies, which is the right shape and not a measurement.

The two halves are computed with different tools. The confinement uses a plasticity criterion and an empirical strength relation; the local buckling uses an elastic shell result with a knockdown. Neither is wrong and they are not commensurable, so the eight and eleven per cent quoted above are not really the same kind of number.

And the tube is assumed to stay in contact. Shrinkage, high eccentricity and load introduction all open gaps, and every one of them removes some of the arrangement this page is about.

Where the ladder goes #

Later rungs on this anchor: the interaction diagram of a filled tube in combined axial load and bending, where the two materials’ contributions cross over. Load introduction and shear connection inside a tube. Fire behaviour and the inward transfer of the load path. Shrinkage and the gap it opens. Square and rectangular filled sections, where confinement is much weaker because a flat wall cannot develop hoop tension. High-strength concrete in filled tubes, which is brittle unconfined and ductile confined. Double-skin tubes. And the general shape of the argument: two components whose failure modes are incompatible, which is a stronger form of composite action than sharing a strain.