The bay that is cast last

The steel decides how many, not how much is about a wall cast against a base that has already hardened. The base holds the bottom of the wall, the restraint decays with height, and the total movement the wall cannot make is divided into cracks whose number the reinforcement sets and whose sum it does not.

That is edge restraint, and it is one of two mechanisms. The other one is what happens to a bay cast between two panels that have already hardened, and almost nothing about it is the same.

A horizontal line where a proportional one was expected is the whole of this page, and the reason for it is a different mechanism rather than a different coefficient.

What the two mechanisms have in common #

Before the differences run away with the page it is worth naming what does not change, because it is the part that makes both of them tractable.

In both cases the movement is imposed rather than applied. Nothing is pushing the wall; something is preventing it from contracting, and the force that appears is whatever prevention costs. That is the property the strain that was imposed is about, and its consequence is the same here as everywhere: stiffness generates the action rather than resisting it, so a stiffer wall is a worse one and a member that can move has no problem at all.

In both cases the concrete cracks when its tensile strain capacity is exceeded, and that capacity is small — 50 microstrain or so at early age, against 380 of free contraction. The margin is a factor of seven, which is why early-age thermal cracking is normal rather than exceptional and why the design question is what the cracks will be like rather than whether there will be any.

And in both cases the steel does nothing until the concrete has cracked. Before the first crack it is carrying a few per cent of the force, in proportion to the modular ratio; afterwards it is carrying all of it at that section. Reinforcement provided against restraint is doing nothing at all until the event it exists for has already happened, which is an unusual relationship between a component and its purpose and is why the quantity is decided by a threshold rather than by a stress.

Which free body produced the number #

The free body is a length of the wall containing one crack, cut at the crack and at some distance either side of it.

Crossing the crack is a force, and it is carried entirely by the steel, because the concrete there has none of its tension left. Crossing the two outer cuts is the same force, carried by the steel and the concrete together.

The question is how far it takes to hand the force back. The steel is bonded to the concrete, so along the length between the crack and the outer cut the bond stress transfers force from one to the other, and the concrete’s share builds up. At some distance from the crack the concrete is back to its full tensile force, and if the member is still being pulled, that is where the next crack forms.

The crack width is the difference between how much the steel has stretched over that length and how much the concrete has. And the force in the steel at the crack is not a free variable: it is whatever the concrete released, $A_c f_{ct}$, because the crack formed at the instant the concrete reached its tensile strength.

So

$$w = s_{r,\text{max}} \cdot \frac{0.5\,\alpha_e\,k_c k\,f_{ct,\text{eff}}}{E_s}\left(1 + \frac{1}{\alpha_e \rho}\right)$$

and there is no imposed strain anywhere in it. The concrete’s tensile strength is in it, the steel ratio is in it, and the movement that caused the crack is not.

Where the movement goes instead #

If the width is fixed, the movement has to go somewhere, and it goes into the number of cracks.

The wall drawn is asked for 304 microstrain over 20 m, which is 6.1 mm of movement it cannot make. Divided into cracks 0.32 mm wide, that is nineteen of them. Ask for twice the movement and there are thirty-eight, the same width.

That is a genuinely different design conversation from the edge-restrained one. Under edge restraint the question is how wide will the cracks be, and the answer improves with reinforcement. Under end restraint the question is how many will there be, and the width is decided by the concrete’s own strength and the steel ratio — so a colder pour, a longer bay, or a hotter mix changes the count and not the appearance of any one crack. The variables the rung below spends its length on have moved from one side of the answer to the other.

A designer who has met only the edge-restrained case will expect the wrong thing on site, and the expectation fails in a specific direction: the cracks are wider than a strain-based calculation predicts, and there are fewer of them.

The threshold, which is not a curve #

The other half of end restraint is a discontinuity, and it is the reason minimum reinforcement exists as a concept.

Everything in that figure assumes the steel can carry the force the concrete let go of. If it cannot, the sequence stops at the first crack.

The steel yields the instant the crack forms. Yielded steel transfers no more force into the concrete however far along it goes, so the concrete never gets back to its tensile strength, so no second crack can form. The member has one crack, and the whole 6.1 mm goes into it.

$$A_{s,\text{min}} = \frac{k_c k f_{ct,\text{eff}} A_{ct}}{f_{yk}}$$

is the condition for that not to happen, and reading it as a detailing rule misses what it is. It is a statement about which of two mechanisms occurs, and the two mechanisms differ by a factor of twenty in the outcome. A wall with 95 per cent of the minimum does not have cracks 5 per cent worse than one with 100.

That is why the steel the concrete asks for is written the way it is: as an inequality between the steel’s yield force and the concrete’s cracking force, with no reference to any load, any crack width, or any exposure class. It is the same inequality here, arrived at from restraint rather than from bending.

Which restraint a member actually has #

Given that the two mechanisms behave so differently, the practical question is which one applies, and the answer is usually both.

A wall on a base is edge-restrained near the base and progressively less so higher up, and the rung below traces that decay. Its ends, if they are built against something, are end-restrained.

An infill bay between two completed panels is end-restrained, near enough uniformly, at a restraint factor around 0.8. Its base, if it is cast on a slab, is edge-restrained as well.

A suspended slab between two cores is end-restrained and nothing else, which makes it the cleanest case and the one where the mechanism was first recognised as distinct. It is also the case where the restraint is genuinely stiff, since a core is very much stiffer than the slab pulling on it — the ratio that decides how much of an imposed movement becomes a force.

The practical rule is that restraint at the ends dominates whenever it exists, because it acts over the whole section rather than decaying, and a member with both is designed for the end case. On a large pour laid out in bays, that puts the infill bays — the ones cast last, between two that have hardened — in a different design case from the ones cast first.

What that means for a pour sequence #

The distinction has an immediate consequence for how a large wall or slab is built, and it is one of the few places where a construction decision can remove a design problem rather than manage it.

A bay cast against nothing is unrestrained and cracks not at all. A bay cast against one hardened neighbour has some end restraint. A bay cast between two has the full case. So a sequence that pours alternate bays and comes back to fill the gaps puts every infill bay in the worst arrangement available.

The alternative — pouring sequentially along the wall, each bay against one hardened neighbour — halves the restraint on every bay but the first, and requires more movement joints or a longer programme.

Which is chosen depends on the same trade the structure that was never complete is about: the finished structure’s stresses are a property of the order it was built in, and the drawings do not record the order. A wall with an unexpected pattern of wide cracks in every third bay is very often a wall whose pour sequence was decided on site — the same class of finding as a structure whose forces depend on the order it was built in.

The numbers on a real basement #

Putting the two mechanisms on a building makes the size of the difference concrete, and the arrangement is one every job meets.

A basement box, 60 m long, walls 400 mm thick and 3.5 m high, cast in 20 m bays against a slab already in place. The corners and the cross-walls are cast first; the long walls are filled in afterwards, between them.

Every long-wall bay is end-restrained by the transverse walls at its ends, and edge-restrained by the slab beneath it. The end restraint governs, so the crack width is 0.32 mm at 16 mm bars at 150 in two layers, and there are nineteen of them in the bay.

The requirement is usually 0.2 mm for a basement designed as water-resistant without a membrane, so the arrangement fails and the steel has to increase. Reading the second figure, taking the bar size down to 12 mm at 100 raises the ratio, shortens the crack spacing and brings the width under the limit — at more steel and more fixing, and with the number of cracks rising as the width falls, since the total movement is unchanged.

That last exchange is the one worth carrying. Under end restraint, halving the crack width doubles the number of cracks — the movement has to go somewhere and the reinforcement decides only how it is divided. A wall detailed to a tight crack-width limit is a wall with a great many fine cracks in it, and a client expecting an uncracked wall has been told the wrong thing by everybody.

Why it took so long to be separated out #

The two mechanisms were treated as one for most of the period in which reinforced concrete has been built, and the reason is worth a paragraph because it is the usual one.

Crack-width rules grew out of bending, where the steel stress is known from the applied moment and the crack width follows from it. Applying the same expressions to restraint needs a steel stress, and the obvious way to get one is from the restrained strain — which gives the edge-restrained answer, and gives it for both cases.

The end-restrained case does not have a steel stress obtained that way. Its steel stress is whatever the cracking force divided by the steel is, and that is a different quantity arrived at by a different argument. Recognising it required someone to notice that a member restrained at its ends cracks differently, which is easier to see on a slab between two cores than on a wall on a base, and to write the two expressions separately.

The distinction is now in the codes and it is recent — CIRIA’s guidance separated them in the 1990s and the European rules followed. Buildings from before that were designed with one expression for two mechanisms, and the ones that were end-restrained were designed with the more optimistic of the two. Which is a specific and checkable thing to look for in an existing basement: full-depth cracks in the bays that were cast last.

Where the model stops #

The restraint factor is taken as a number. In reality it is the stiffness of what is holding the ends against the stiffness of the member, and a bay held between two thin panels that can themselves bend is much less restrained than one between two cores. R = 0.8 is a convention for the fully restrained case, and computing it properly is a stiffness ratio.

The cracking is early-age. Everything here is the heat of hydration leaving over a few days, with the concrete at a fraction of its 28-day strength and a large creep relief. Long-term drying shrinkage arrives over years, adds to a member that has already cracked, and widens the cracks that exist rather than making new ones — so the two events have to be added rather than treated as one.

The steel is elastic and bonded. A bar at a crack whose bond has been damaged by an earlier event transfers force over a longer length, which increases the spacing and the width together.

And the section is in pure tension. A wall restrained at its ends and also carrying bending — a basement wall with earth on it, a slab with load on it — has a neutral axis, and the effective tension area the steel ratio is computed over is a fraction of the section rather than all of it.

What the pictures cannot show #

Whether the crack goes right through. Under edge restraint the cracking is partial — it stops where the restrained strain falls below the concrete’s capacity, which is part way up the wall. Under end restraint the whole section is in tension, so every crack is a full-depth crack, and a full-depth crack in a water-retaining or basement wall is a leak rather than an appearance problem. A wider crack on the same steel is a serviceability question when it is a mark and a durability one when it is a path.

That is arguably the most important practical difference between the two mechanisms and it is not a difference in width. A 0.17 mm partial-depth crack and a 0.32 mm full-depth one are not two points on one scale; one of them is a mark on a wall and the other is a path.

They also cannot show the joint. Most end-restrained pours have a construction joint at each end, with a water bar, a shear key, or continuity steel through it — and the joint’s own behaviour decides whether the restraint is 0.8 or 0.5, whether the movement concentrates at the joint rather than in the bay, and whether the first crack forms at the joint or away from it.

The assumption the figure rests on #

That the member cracks at all.

Both curves assume the restrained strain exceeds the concrete’s strain capacity somewhere, and the whole of end-restraint design assumes the section will crack and asks what the cracks will be like. That is nearly always right for a member of any length, and it is worth knowing what the alternative looks like: a short, lightly restrained, well-cured pour with a low-heat mix can stay uncracked, and every calculation on this page then returns numbers about an event that does not occur.

The reason the assumption is safe to make is asymmetric. Designing an uncracked member for cracking costs a little steel; designing a cracked member as though it were uncracked costs a leak. Where a threshold decides which of two mechanisms applies, the design has to be on the far side of it rather than at it, which is the same reasoning that puts minimum reinforcement above the cracking force rather than at it.

The ladder from here #

Later rungs on this anchor: the restraint factor computed from a stiffness ratio rather than taken from a table, which is what decides whether a bay is at 0.5 or 0.8. The temperature history itself, from cement content, section thickness and formwork type — the one input a designer genuinely controls. Sequential against alternate bay construction, priced. 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 and adding to what is already there. And the water-retaining case in full, where a crack is a leak, the limit is 0.2 mm or less, and autogenous healing is relied on for the rest.