The load that comes from inside

Every action in this collection so far has arrived from outside the structure. A weight. A pressure. An imposed movement. A temperature change. Even the ones nobody applied are responses to something external.

Corrosion is not applied at all, and it is usually filed under durability — a maintenance subject, adjacent to structures rather than part of them. The reason it belongs here is that the mechanism which does the damage is a load, generated inside the concrete by a chemical change, and it is carried by a free body that is entirely ordinary.

Which free body produced the number #

The cover over a bar, taken as a thick-walled cylinder: inner radius the bar, outer radius the concrete surface, loaded by an internal pressure.

The pressure comes from the corrosion product. Rust occupies between two and six times the volume of the steel it consumed — call it three — so a bar losing a depth $x$ of radius has to find room for $(\alpha - 1)x$ of expansion, and the only place to find it is by pushing the cylinder outwards.

Elastic theory for a thick cylinder gives the circumferential tension at the bore, and setting it to the concrete’s tensile strength gives the pressure at which cracking starts:

$$p_{\text{bore}} = \frac{f_t (b^2 - a^2)}{a^2 + b^2} = 2.63\ \text{MPa}$$

for a 20 mm bar at 35 mm cover. That is not the pressure at which the cover comes off, and taking it as such is the commonest error in this calculation.

A cracked cylinder goes on carrying. The cracks run outwards from the bar and the uncracked ring beyond them keeps working, so the cover fails only when the cracking reaches the surface. Taking the tension as uniform over the cover at that point, equilibrium of half the cover gives

$$p_{\text{crack}} = \frac{2 c f_t}{d} = 10.15\ \text{MPa}$$

Four times the bore value, and the difference matters enormously: using the elastic one predicts a cover that spalls after a micron of section loss, which is a hundredth of what any measurement has ever shown.

The section loss, which is nothing #

Work backwards from that pressure. The bore of the cylinder moves out by 8.8 microns to generate it. Add the porous band — a real layer of voids left by bleed water under a bar, ten to fifteen microns thick, that the first rust simply fills without pressing on anything — and the free expansion needed is 21.3 microns.

At $\alpha = 3$ that is a section loss of

$$x = \frac{d_0 + u}{\alpha - 1} = 10.7\ \mu\text{m}$$

which is 0.213% of the bar’s area.

Two parts in a thousand. No strength check anywhere in this collection would notice it, no measurement of a member’s capacity would detect it, and no calculation would report it as anything at all. And the cover over the bar has split.

What that means for what fails #

If the section loss is negligible, what has actually gone wrong?

Bond. The cover is what confines a bar and lets it develop its force. A split cover confines nothing: the bar’s bond strength falls sharply, its anchorage lengthens, and the member’s capacity is now limited by a development length rather than by a bar area. That is a strength loss, and it arrives through a mechanism the corrosion did not directly touch.

And then everything else. A split cover admits water and oxygen directly to the bar, so the corrosion accelerates. The spalled concrete is gone, so the cover is now less than it was, so the acceleration continues. The process is self-reinforcing from the moment the crack reaches the surface, which is why the observed history of a corroding structure is a long quiet period followed by a rapid one.

So the honest structural statement is: corrosion does not reduce a section, it destroys an anchorage — and it does so at a section loss that no inspection based on measuring bar diameters would find in time.

Why the clock is a square #

Splitting is the second half of the life. The first half is initiation: the time for the chloride front, or the carbonation front, to reach the bar at all.

Fick’s second law with a surface concentration held constant gives

$$t_i = \frac{c^2}{4D\left[\text{erf}^{-1}(1 - C_{th}/C_s)\right]^2}$$

and the $c^2$ is the entire durability argument. The rest of the expression is a diffusion coefficient known to about a factor of two and a threshold concentration known to rather less.

Doubling the cover from 35 to 70 mm takes the initiation from 9.1 years to 36.2 — exactly a factor of four, because it is exactly a square. The propagation barely moves, from 0.92 to 1.26 years.

Cover appears twice and the concrete’s own strength appears once, linearly, in the smaller of the two terms. That asymmetry is why every durability clause in every code is written about cover and crack width and hardly at all about grade, and it is a conclusion that comes out of the arithmetic rather than out of practice. Size effects do not rescue the grade either: the tensile strength that matters here is a local one at a bar, on a scale where the specimen value is optimistic.

The bar size, which is not a free variable #

A larger bar is more durable in one sense and much less in another, and the two are usually conflated.

It takes longer to lose a given fraction of itself, since the same corrosion rate removes the same depth from any bar. That is real.

And it splits its cover sooner, because $p_{\text{crack}} = 2cf_t/d$ carries the diameter in its denominator. A 40 mm bar at 35 mm cover cracks at 5.08 MPa against a 20 mm bar’s 10.15, and it does so at a section loss of 0.11% rather than 0.21%.

So the governing quantity is the ratio $c/d$, not the cover and not the bar. Specifying 35 mm of cover means something different for a 12 mm bar and a 40 mm one, and every durability rule that quotes a cover without reference to bar size is quoting half a parameter. The better-written ones quote a minimum cover and a minimum of one bar diameter, which is the same statement.

The product, which is not one thing #

$\alpha$ is the ratio of the corrosion product’s volume to the steel’s, and treating it as 3 is a convenience.

It runs from about 2 for the densest oxides to 6 for the most hydrated, and which one forms depends on the oxygen available. A bar in a wet, oxygen-poor environment produces dense products and expands little; the same bar in a wet-dry cycling zone produces bulky ones and expands a great deal.

That is why the splash zone of a marine structure is the worst place on it, worse than permanent immersion. Fully submerged steel corrodes slowly and the product is dense; steel that is alternately wet and drying has both the chloride supply and the oxygen, and the expansion is at the top of the range.

The consequence is that the same bar at the same cover in the same concrete can have five times the propagation life depending on where it is in the structure, and none of the difference is visible in any material property.

What can be done about it, which is a short list #

The arithmetic above narrows the interventions to a handful, and it is worth reading them off in order of what the exponents say they are worth.

Cover. Twice, as a square on the initiation and linearly on the pressure. It is the largest single lever and it is essentially free at design stage and impossible afterwards.

Bar size for the same steel area. More smaller bars at the same cover raise $c/d$, so a layer of 16 mm bars is more durable than the same area in 25 mm ones. It costs fixing time and nothing else.

Keeping the chlorides out, by a coating, a membrane or a denser concrete. This attacks $D$, which sits in a denominator alongside the square — halving the diffusion coefficient doubles the initiation, which is worth the same as a 41% increase in cover.

Cathodic protection, which attacks the corrosion rate directly and is the only intervention that works on a structure that has already initiated. Everything else on this list is a decision that has to be made before the concrete is poured.

Crack width limits, which are the item most often quoted and are the hardest to place in this arithmetic. A crack that reaches a bar short-circuits the diffusion path entirely, so the initiation calculation above does not apply to a bar under a crack. That is why the codes limit crack widths and specify cover: the two rules are protecting against different halves of the same life, and neither substitutes for the other.

Where this model stops #

The cylinder is a fiction near a surface. A bar with cover on one side and a great deal of concrete on the other does not split a symmetric ring; it drives a crack to the near face and lifts a plane of cover off. The model above is a reasonable average and it is not a picture of the geometry.

Corrosion is not uniform. Chloride attack is pitting — deep local loss at a few points rather than a general reduction — and a pit can remove a fifth of a bar’s area locally while the average loss is under a per cent. That is a genuine strength loss and this model has none of it, which is why chloride-induced corrosion is the more dangerous of the two mechanisms — the ductility that every method here depends on is what a pitted bar has lost first even though carbonation is the more common.

Prestressing steel is a different problem entirely. A high-strength tendon at 70% of its ultimate stress is susceptible to stress corrosion and hydrogen embrittlement, which are brittle fracture mechanisms rather than section-loss ones — and a tendon that loses its force has removed a load rather than a capacity and can act with no visible expansion at all. Everything in this essay is about ordinary reinforcement.

And the rate is the least known number. One microamp per square centimetre removes 11.6 microns a year, which is the conversion used throughout — but the actual current density in a real structure varies by two orders of magnitude with moisture and temperature, and at 5 µA/cm² the propagation here falls from 0.92 years to 0.18.

What the picture cannot show #

The figure plots a service life against a cover, which makes durability look like a design calculation with an answer. It is not. Every input in it is known to a factor of two or worse: the diffusion coefficient, the surface concentration, the threshold, the expansion ratio, the corrosion rate. Multiplying five such numbers together gives a life whose uncertainty is larger than its value.

What the calculation is actually good for is the exponents. Cover squared. Cover over diameter. A section loss of two parts in a thousand. Those survive the uncertainty because they are structural rather than empirical, and they are what tells a designer that the cover is the variable and the grade is not.

Two mechanisms wearing one word #

Everything above has been written as though “corrosion” were one process. It is two, they have different clocks, and they produce different damage.

Carbonation is the slow neutralisation of the concrete’s alkalinity by atmospheric carbon dioxide, advancing as a front. It is general rather than local: once the front passes a bar, the whole bar loses its passivity together and corrodes more or less uniformly. That is the mechanism this essay’s arithmetic describes, it is the commoner of the two, and it is comparatively benign — the damage is visible as cracking and spalling long before any capacity has gone.

Chloride attack does not need a front to arrive everywhere. Chlorides break down the passive film at points, and what follows is pitting: a small anode surrounded by a large cathode, driving deep local loss at a few places while the rest of the bar is untouched. The section loss at a pit can reach a fifth of the bar with an average loss under one per cent, and the expansion is small because there is little product — so the cover may not crack at all.

That second sentence is the important one. The mechanism this essay describes is the one that gives warning. The mechanism that removes capacity is the one that does not, and the two look identical from outside until a bar breaks.

It is the flaw that sets the strength arriving in a member nobody was inspecting for flaws, and it is why the structures that fail from corrosion are marine and de-iced rather than merely old.

The generalisation #

The habit worth carrying is a question about actions: where does this one get its energy?

Almost every load in this collection is external and its magnitude is given. A few are not, and they behave differently: a restrained thermal strain produces a force proportional to the stiffness resisting it, a prestress is a load put in on purpose, and corrosion is a volume change that generates whatever pressure the surrounding material can take before it cracks.

All three share a property that external loads do not: stiffening the structure makes them worse. A stiffer cover attracts a higher pressure from the same expansion, exactly as a stiffer restraint attracts a higher force from the same thermal strain. That is a small class of actions, it is easy to miss because it inverts the usual instinct, and it is worth being able to recognise on sight.