The coincidence reinforced concrete stands on

Every account of reinforced concrete says at some point that steel and concrete have the same coefficient of thermal expansion, and that this is fortunate. Both halves of the sentence are worth checking, because the first is false and the second is a considerable understatement.

Steel expands at about $12\times10^{-6}$ per degree. Concrete expands at about $10\times10^{-6}$, varying with the aggregate between about 7 for limestone and 13 for flint gravel. They are not the same. The mismatch is seventeen per cent of the larger, which for a forty-degree change is eighty microstrain of differential movement between two materials cast onto one another.

What eighty microstrain does #

The temptation is to multiply by a modulus and be alarmed. $E_c \times 80\times10^{-6} = 2.4$ N/mm², which is very nearly the tensile strength of the concrete, and reinforced concrete would then crack every time the weather changed.

That calculation is the fully restrained one, and nothing here is fully restrained. The bar and the concrete are attached to each other and to nothing else, so they strain together to whatever common value their stiffnesses agree on. Compatibility says $\varepsilon_1 = \varepsilon_2$; equilibrium says the two forces cancel; and

$$N = \frac{(\alpha_2 - \alpha_1)\Delta T}{\dfrac{1}{E_1A_1} + \dfrac{1}{E_2A_2}}.$$

For one and a half per cent of steel in concrete, at 40 degrees, that gives 14.9 N/mm² of compression in the steel and 0.223 N/mm² of tension in the concrete — 7.7 per cent of the concrete’s tensile strength, and 1.9 per cent of what full restraint would have delivered.

The reason the number is so small is that the steel is doing nearly all of the accommodating. At $\rho = 0.015$ the concrete’s axial stiffness $E_cA_c$ is $3\times10^9$ and the steel’s $E_sA_s$ is $3.1\times10^8$ — a tenth of it — so the soft member takes most of the differential strain and the stiff one barely notices.

The property of the pair is the strain, not the stress #

The stresses depend on which of the two materials happens to be the reinforcement and how much of it there is. The mismatch strain does not: $(\alpha_2 - \alpha_1)\Delta T$ is a fact about the two materials and the weather, with no geometry in it at all, and it is the right quantity to rank pairs on.

Every other pair on that list is either not bonded, or is bonded and is a known problem. Glass in an aluminium frame is set in a gasket with clearance all round, and the clearance is sized on exactly this number. Aluminium bolted to steel needs slotted holes. Brass on steel is a bimetallic strip, which is a device that exploits the mismatch, and it is the mismatch this table ranks.

What the alternative would look like #

Replace the steel with aluminium — same area, same bond, twice the strength-to-weight — and the mismatch goes from $2\times10^{-6}$ to $13\times10^{-6}$. The concrete’s tension rises from 0.223 to 0.528 N/mm², a factor of 2.36, and it rises linearly with $\Delta T$ so a 90-degree fire-exposure excursion would put it past the tensile strength on the thermal mismatch alone.

Carbon fibre is worse per degree in a different way: its coefficient along the fibres is very nearly zero, so a CFRP bar in concrete has the full 400 µε of mismatch at 40 degrees. Which is one of several reasons carbon reinforcement is used as external strengthening bonded to the surface rather than cast in, where a differential movement can be taken up by the adhesive layer.

The bimetallic case, which is the same physics rewarded #

Put the two materials side by side rather than one inside the other and the pair cannot resolve the mismatch by self-stressing alone: it curves.

$$\kappa = \frac{6(\alpha_2 - \alpha_1)\Delta T\,(1+m)^2}{h\left[3(1+m)^2 + (1+mn)\left(m^2 + \dfrac{1}{mn}\right)\right]}$$

with $m$ the thickness ratio and $n$ the modulus ratio — Timoshenko’s 1925 result, and the whole of a thermostat. A brass-and-steel strip a millimetre thick and 100 mm long lifts its tip 5.1 mm for a 100-degree rise.

That is the same 280 µε of mismatch from the table above, converted into a movement instead of a stress by putting the two materials where they have a lever arm. The geometry decides whether a mismatch becomes a stress or a movement, and the concentric arrangement of a reinforced section is the one that produces the least of both.

Why the number is small twice over #

There are two independent reasons the steel–concrete self-stress is negligible, and separating them is worth doing because only one of them is the coincidence.

The first is the coincidence: the coefficients are nearly equal, so the mismatch strain is 80 microstrain rather than the 400 or 500 every other pair produces. That is a fact about the materials and it is luck.

The second is not luck at all: reinforcement is a small fraction of the section, so the stiff material is the one with almost no area. At $\rho = 0.015$ the steel’s axial stiffness is a tenth of the concrete’s, so nine tenths of the mismatch is taken up by the steel straining and only a tenth by the concrete. Raising the ratio to four per cent — a heavily reinforced column — raises the concrete’s tension from 0.223 to 0.515 N/mm², which is still comfortable and is 2.3 times as much.

The two reasons multiply, and the second is the larger. Even if the mismatch were aluminium’s 13 microstrain per degree, the concrete would carry only 0.53 N/mm² at ordinary reinforcement ratios — uncomfortable rather than fatal. What actually rules aluminium out is not the bulk stress but the pressure at the bar surface, which is a local problem the section calculation cannot see.

Which free body produced the number #

Two, and they have to be drawn together.

Cut the bar out of the concrete. On it: an axial force $N$ from the bond along its surface, and nothing else. Its strain is $\alpha_1\Delta T + N/E_1A_1$.

Cut the concrete away from the bar. On it: $-N$, distributed round the bar’s surface. Its strain is $\alpha_2\Delta T - N/E_2A_2$.

Neither free body determines $N$. Equilibrium is satisfied for any value of it, which is the signature of a self-equilibrating stress — the same signature a residual stress has, and a prestress in a determinate member, and a shrinkage stress. The extra equation is compatibility: the two strains are equal because the two materials are stuck together.

No amount of statics will find this stress, and no equilibrium check will detect it. It is one of the family of internal forces that exist because a structure was made rather than because it was loaded.

What else the match buys #

The self-stress is the obvious consequence and it is not the important one. Three others follow from the same near-equality and each is larger.

Curvature. A section whose two materials expand differently under a uniform temperature change does not merely self-stress — it bows, because the mismatch strain acts at the lever arm between the two materials’ centroids. In a beam reinforced only near its soffit that lever arm is nearly half the depth, so a mismatched pair would produce a temperature-driven curvature and therefore a temperature-driven deflection. At 80 microstrain and a 200 mm lever arm the curvature is $4\times10^{-7}$ per mm, which over an 8 m span is 3.2 mm of movement. At aluminium’s mismatch it would be 21 mm — comparable with the span/250 limit, from the weather.

Cracking under a temperature change alone. Restrained early-age contraction already cracks most walls, and it does so with the reinforcement contracting with the concrete. A reinforcement that contracted differently would add its own crack-inducing strain to that, and the total opening — which is a strain times a length and contains no steel — would rise accordingly.

Fire. In a fire the bar reaches several hundred degrees and the mismatch, whatever it is at 20 degrees, is being multiplied by a $\Delta T$ of six hundred. A pair mismatched by $13\times10^{-6}$ would develop 7,800 microstrain of differential — an order of magnitude past anything the bond could carry — and the reinforcement would debond from the concrete before either material lost much strength.

Where the model stops #

One coefficient for concrete. It is not a material constant: it runs from about 7 for limestone aggregate to 13 for flint, so a concrete made with a siliceous aggregate is better matched to steel than one made with limestone, and the mismatch can have either sign. A designer does not usually know which.

One temperature for both. They are at the same temperature only after they have equalised. During a fire the bar is cooler than the surface concrete and hotter than the core, and during a summer afternoon the reverse; the transient is worse than the steady state and is not this calculation.

The bond is perfect and the section is uncracked. Once the concrete has cracked, the compatibility statement applies only between cracks, and the mismatch is accommodated by slip at the crack faces — which makes the problem much easier and is why nobody worries about it in service.

It is linear in $\Delta T$. Both coefficients change with temperature, concrete’s markedly so above about 200 degrees where the aggregate and the paste move in opposite directions and the mismatch becomes an internal problem of the concrete itself.

The bar is one bar and the section is one section. A real member has bars of several diameters in several layers, links round them at a different orientation, and a cover zone that behaves differently from the core — so the “common strain” is a fiction over a section that has at least three different strain states in it. The fiction is a good one because the mismatch is small, which is the argument of this page arriving at its own limitation.

And the drawing ranks pairs, not designs. A large mismatch is only a problem where the two materials are bonded over a length and cannot slip. Aluminium and steel appear together in thousands of structures with no difficulty at all, because the connection between them is a bolt in a slotted hole — which is a movement budget rather than a compatibility problem, and is the standard answer to every row of the table except the last.

The measurement nobody makes #

There is a practical asymmetry worth ending on. Steel’s coefficient is a property of a controlled alloy and it is known to two figures for every grade in every table. Concrete’s is a property of whatever aggregate the batching plant had that week, it varies by a factor of nearly two between rock types, and it appears in no specification, no test certificate and no delivery ticket.

So the mismatch is the difference of a number that is known and a number that is not, and it is a small difference of two similar quantities — which is the arithmetic most sensitive to an error in either. A limestone concrete at $7\times10^{-6}$ has a mismatch of $5\times10^{-6}$ against steel, two and a half times the nominal; a flint gravel concrete at 13 has a mismatch of $-1\times10^{-6}$, half the nominal and of the opposite sign.

The design value of 10 is the middle of a range that spans the whole answer, and none of it is measured because none of it needs to be. That is what a large margin buys: the freedom not to know an input.

The ladder from here #

Later rungs on this anchor: the aggregate’s own contribution, and why the coefficient of a concrete can be measured but not specified. The transient case in fire, where the bar and the concrete are at different temperatures and the mismatch is a gradient rather than a difference. Restrained shrinkage read as a mismatch of the same kind, with the “coefficient” being a drying strain rather than a thermal one — which is a larger number and the same arithmetic. The differential movement of a composite steel-and-concrete beam, where the two materials are separated by a lever arm and the mismatch produces curvature rather than self-stress. Adhesive joints between dissimilar materials, where the mismatch is taken up in a thin compliant layer and the design question is how thin it can be. And the historical thread: reinforced concrete was patented before anybody had measured either coefficient, and the coincidence it depends on was discovered after the fact rather than designed for.