The steel decides how many, not how much

A retaining wall is poured against a base slab that went in three weeks earlier. Over the next two days the wall heats itself to about forty degrees above ambient, because cement hydrating is an exothermic reaction and four hundred millimetres of concrete has nowhere to put the heat. Over the following week it cools back down. It would like to get shorter by three or four hundred microstrain while it does so.

The base will not let it. The base is cold, hard, and considerably stiffer in its own plane than the wall is, and it is connected to the wall by a kicker and a mat of starter bars. So the wall cracks — vertically, at intervals of a few metres, right through its thickness, before anything has been backfilled behind it and before any load has been applied to it at all.

This is the commonest structural crack in the world and it is not caused by a load. The rest of this essay is about what that changes.

The free body is the wall, and the thing acting on it is a length #

Cut the wall free of its base and lay it out. It has a length it wants to be — the length it was cast at, less the contraction — and a length it is allowed to be, which is the length of the base. The difference is imposed, and the whole problem is contained in that word.

$$\varepsilon_{\text{free}} = \alpha\,\Delta T + \varepsilon_{cs} = 10\times10^{-6}\times32 + 60\times10^{-6} = 380\ \mu\varepsilon$$

with 84 per cent of it thermal and the rest early drying and autogenous shrinkage. That is what the concrete would do unrestrained. What it is permitted to do is $(1-R)$ of it, and the rest turns into strain the material has to find somewhere.

The restraint factor $R$ is the number the whole calculation hangs on, and it is neither one nor zero. A member held rigidly at both ends carries $E\alpha\Delta T$ with no dimension in it at all — a stress independent of length, area and second moment. That is the case at $R = 1$, and it is not this case. A base slab is stiff but not rigid; it can strain a little in its own plane, and the wall can slide on it slightly, and the joint between them is a construction joint rather than a weld.

For a wall on a base, measured values cluster around 0.5 at the joint, and creep at early age relieves roughly a third of whatever stress does develop, because concrete two days old is very much more creep-prone than concrete two years old. The stress that actually arrives is

$$\sigma = K R \varepsilon_{\text{free}} E_{cm} = 0.65 \times 0.5 \times 380\times10^{-6} \times 31{,}000 = 3.83\ \text{N/mm}^2,$$

against $E\alpha\Delta T = 67$ N/mm² for the fully restrained case with no creep. A factor of seventeen separates the textbook expression from the number, and the factor is made of two things neither of which is in the expression.

It still cracks #

3.83 N/mm² against a tensile strength around 2.4 is not a close call, and the comparison is better made in strain than in stress because the strength is not the quantity that varies most.

$$\varepsilon_{ctu} = \frac{K f_{ctm}}{E_{cm}} = \frac{0.65 \times 2.4}{31{,}000} = 50\ \mu\varepsilon$$

against a restrained strain of $R \varepsilon_{\text{free}} = 190\ \mu\varepsilon$: a factor of 3.78. There is no arrangement of reinforcement that prevents this, and no realistic concrete mix either, because the tensile strength gains more slowly than the heat arrives — the peak temperature is at two days and the strength at two days is about two thirds of its 28-day value.

So the wall is going to crack. The design decision is not whether.

The conservation statement #

Here is the argument that makes this problem different in kind from every other crack calculation on this site.

A crack in a beam opens because the steel across it stretches under a stress the steel has to carry. Give the beam more steel and the stress falls, the strain falls, and the crack narrows: the same steel in smaller bars gives a narrower crack and more of it gives a narrower one still. The width is bought.

In a restrained member the applied quantity is a strain. The wall is 20 m of concrete that has to end up 3.30 mm shorter than it wants to be, and every millimetre of that has to appear somewhere: as elastic strain in the uncracked concrete, or as crack. Sum along the length:

$$\sum w = \left(R\varepsilon_{\text{free}} - \tfrac12 \varepsilon_{ctu}\right) L = 164.8\times10^{-6} \times 20{,}000 = 3.30\ \text{mm}.$$

There is no steel in that expression. No area, no diameter, no spacing, no yield strength. The total opening is a strain times a length, and both are properties of the pour and the ground plan.

So the reinforcement is not resisting anything. It is a divider. It sets the crack spacing, the spacing sets the width, and the number of cracks is whatever the total requires. This is why the check on a restrained member is written as a crack width and never as a stress, and why the answer to the cracks are too wide is more bars and smaller ones rather than more steel.

The spacing is a bond length #

What the steel actually controls is how far along the wall the concrete has to go before the bars have handed enough force back into it to crack it again. That distance is

$$s_{r,\max} = 3.4c + \frac{0.425\,k_1k_2\,\phi}{\rho_{p,\text{eff}}},$$

a cover term and a bar term. For the wall drawn — 40 mm cover, 16 mm bars at 150 on both faces — it is 623 mm, of which the cover contributes 136 and the bar 487.

The bar term dominating is why bar size is the effective variable rather than bar area. Halving the diameter at constant area halves the spacing, halves the width and doubles the count, and costs nothing except more bars to fix.

The steel that is needed for a different reason #

There is a separate and prior check, and it is the one that decides whether the wall gets thirty cracks or one.

For the wall here, $A_{s,\min} = 0.8 f_{ctm} A_{ct} / f_{yk} = 922$ mm²/m against 2,681 provided, a ratio of 2.91. This is the check that makes the conservation statement useful rather than terrifying: the total opening is fixed either way, and the only thing standing between thirty-two cracks of a tenth of a millimetre and one crack of three and a quarter is enough steel to survive the first one.

Why a short wall does not crack #

The restraint factor falls with height, and the rate it falls at is not a material property or a joint property. It is geometry.

$$R(z) = R_0\,k^{z/h}, \qquad k = \frac{L/h - 2}{L/h + 1}$$

At a length-to-height ratio of 2 the base of that exponent is exactly zero: the top of the wall is entirely free, because a wall as short as that can rotate and bow on its base rather than being held by it. At 6.67 the base is 0.609 and 30 per cent of the restraint survives to the top of the wall. The consequences are large and discrete:

A wall is restrained by being long. The length does not appear in the strain — a 6 m wall and a 60 m wall cool by the same amount and want to contract by the same 380 microstrain each — but it appears twice in the consequences: once in how much of the height is held, and once as the multiplier on the total opening.

This is where movement joints come from, and it explains the spacing they are put at. A joint every two wall heights takes every panel to $L/h = 2$ and the top of the exponent to zero. That is not a rule of thumb: it is the point where the ACI distribution goes to zero, and the reason the rule is stated as a multiple of the height rather than as an absolute length.

It also explains an observation that puzzles people on site, which is that the deepest walls crack least. A 6 m basement wall in a building with a 40 m plan is at $L/h = 6.7$ and cracks throughout; the same 40 m of plan in a 1.2 m upstand is at $L/h = 33$ and cracks four times as often. Height is working in the numerator of the restraint and in the denominator of the crack count at once, and both directions favour the deep wall. Nothing about the concrete has changed.

The joint itself is worth being honest about. It is a plane of weakness deliberately introduced so that the movement collects at a place where it can be sealed rather than distributing itself where it cannot, which puts it in the same family as a hinge put into an arch on purpose — a restraint given up in exchange for control of where the consequence lands. And like that hinge it is not free: a joint is a maintenance item, a leakage path and a discontinuity in the reinforcement, and the alternative of accepting many fine cracks and relying on autogenous healing is a legitimate design and increasingly the usual one.

The width the wall actually gets #

The number that goes on the drawing is the crack width, and it is worth seeing what moves it.

For a wall retaining water the limit is often 0.2 mm and for one retaining soil 0.3, and the wall drawn is at 0.10 with a factor of two in hand. Halving the bars to 8 mm at 75 would take it to 0.063 and fifty-three cracks; going to 25 mm at 365 would take it to 0.148 and twenty-two. Both are the same wall moving by the same 3.30 mm.

Where the model stops #

The temperature drop is one number. It is not: the wall has a temperature profile through its thickness and a history in time, the surface cools first and the core later, and there is an internal self-equilibrating stress from that gradient on top of the external restraint. A thermal gradient produces stress with no restraint at all, and the two effects superpose at exactly the age when the strength is lowest.

The restraint factor is a distribution and it is drawn as a curve. Real edge restraint also varies along the length — highest at mid-length, falling toward a free end — so the first crack forms near the middle and the ends may never crack. The ACI expression captures the height variation and says nothing about the length variation.

Creep is one factor. $K = 0.65$ is a whole relaxation calculation compressed to a number, and it is a function of the age at loading, the maturity, the mix and the section thickness. It is also the number the answer is most sensitive to that is least measured.

The crack spacing formula is borrowed. It was derived and calibrated for flexural cracking under sustained load, where a steel stress exists to be plugged into it. Here there is no steel stress until after the crack forms, and using the same expression is a convention rather than a derivation.

The wall is treated as a bar in tension. It is not: it is a plate held along one edge, so the strain it is prevented from making varies over its face in two directions and the cracks that result are tapered rather than parallel-sided — widest at the base and closing toward the top, in the way a member restrained continuously rather than at points distributes its response over a length it chooses for itself. The single width reported here is the width at the joint, which is the right place to report it and is not the width anywhere else.

And the section is uncracked in bending throughout. A retaining wall gets backfilled, and the flexural cracks that then arrive are on the earth face at the base — the same face and the same region where the restraint cracks already are. The two crack patterns superpose, the code check treats them separately, and nothing in either calculation knows that the concrete it is about has already been divided into thirty-two pieces by the other one.

And the pictures cannot show the wall doing it. Every figure here is a state — a restraint profile, a width, a count. The event is a week long, the cracks arrive one at a time in an order the drawings do not attempt, and the second crack forms where the first one has finished handing its force back. A drawing of thirty-two evenly spaced cracks is a drawing of the end state of a sequence, and the sequence is where the spacing actually comes from.

The ladder from here #

Later rungs on this anchor: the temperature history itself, computed rather than assumed, from cement content, section thickness and formwork type — which is the one input a designer genuinely controls. End restraint as against edge restraint, where a wall cast between two completed panels has $R$ near 0.8 uniformly and behaves entirely differently. The sequence question: bay sizes, pour order, and whether the infill bay or the first bay is the one at risk. Restraint from piles rather than from a base, where the wall is held at points and the crack pattern is a fan. Long-term shrinkage arriving years after the thermal event, adding to a member that has already cracked and widening what is there rather than making new ones. And the same argument wherever a strain is imposed on something that cannot move — a composite deck restrained by its studs, a continuous slab held by friction on the ground, and every jointless bridge deck ever built.