The bars that take the column over

The columns are shorter than the core worked out how far a concrete tower’s columns shorten past its core and where the worst of it falls, and it ended on a confession: reinforcement had been ignored. The floor that is level on one date repeated the confession with more interest, and noted that a heavily reinforced column shortens noticeably less than its concrete would alone. Both were treating the steel as a correction. Steel in a column is 2 per cent of the area and on the first day, at a modular ratio of about six, it adds something like 10 per cent to the stiffness, and nobody loses sleep over 10 per cent in a calculation whose creep coefficient is uncertain by a third.

The first day is the wrong day to look at. A column’s bars are the one part of it that does not creep and does not shrink. The concrete around them wants to carry on shortening, and as it does so it drives the bars into higher strain and passes them load. Nothing outside the column has changed. The building above has been there for years, and the force on the column is exactly what it always was, but the share of it going through the steel has been climbing the whole time.

The bars and the concrete have one strain and two histories #

For the arithmetic, take a column of concrete area $A_c$ with bars of area $\rho A_c$ and modular ratio $n = E_s/E_c$, under a load that arrives at once and stays. On the first day the concrete carries $\sigma_0 = \sigma_N/(1 + n\rho)$ and the bars $n\sigma_0$. Afterwards the concrete’s stress changes by an amount $\Delta\sigma$ that is unknown. If it changed all at once, the concrete would respond to it as to any new load. It does not change all at once, and that is why the age-adjusted effective modulus exists: a stress that arrives gradually over an interval of creep is taken at the concrete’s modulus reduced by $1 + \chi\phi$, with $\chi$ near 0.8, because it has not been on for the whole interval. The strain that was imposed derived the same adjustment for relaxation and found its limits.

Compatibility says the concrete’s strain and the bars’ strain are one strain, and equilibrium says the bars’ gain equals the concrete’s loss. Trost’s and Bažant’s result for this section comes in one line:

$$\Delta\sigma = -\frac{n\rho\,\bigl(\sigma_0\phi + E_c\varepsilon_{sh}\bigr)}{1 + n\rho\,(1 + \chi\phi)}.$$

The numerator is what the concrete would like to do freely, in stress units. The denominator measures how much the steel can resist it. Take $E_c = 34$ GPa, $E_s = 200$ GPa, a creep coefficient of 2.2 and a shrinkage of 300 microstrain, the first three as the earlier essays had them. At 12 MPa on the gross concrete with 2 per cent steel, the concrete starts at 10.7 MPa and ends at 7.7, and the bars start at 63 MPa and end at 213.

So a bar that carried a seventh of the load on its first day carries more than a third by the end. Most of the change arrives inside the first few years, as creep and drying are fastest then, and the log scale along the bottom makes the slow tail look more active than it is. At 4 per cent steel the bars end with 57 per cent of the load, and the column has quietly become a steel column with concrete around it for restraint. The calculation that sized this column shared the load by areas on the first day, and it is only right at the beginning.

A hand check at 2 per cent shows the size of it. $n\rho = 5.88 \times 0.02 = 0.118$. The numerator is $0.118 \times (10.74 \times 2.2 + 34{,}000 \times 300 \times 10^{-6}) = 0.118 \times 33.8 = 3.98$, and the denominator is $1 + 0.118 \times 2.76 = 1.325$. So $\Delta\sigma = -3.0$ MPa, and the concrete falls from 10.7 to 7.7. The bars gain what the concrete loses, $3.0/0.02 = 150$ MPa, and go from 63 to 213.

The first-day number was asking the wrong question #

The lower line is what an elastic calculation sees, and it is rightly unimpressed: going from no steel to 4 per cent takes the first-day strain from 353 to 286 microstrain, a fifth. The upper line is the one that shortens a building. From 1,429 microstrain for plain concrete, it falls to 1,222 at 1 per cent, 1,067 at 2 per cent, 849 at 4 and 600 at 8. So at 2 per cent the steel has removed a quarter of the total, more than twice the proportion it removes on the first day, and the same 4 per cent that took a fifth off the elastic figure takes two fifths off the final one.

The reason is that the steel’s effect is multiplied by the time-dependent strain it resists, while its first-day effect is measured against the elastic strain only. Against a creeping material, even a small stiff element has plenty of time to make itself felt. In ratio terms, the column’s long-term modulus is $E_c/(1 + \phi)$ or less, around a third of the short-term value, so the modular ratio that matters over its life is closer to 18 than to 6.

The least steel works hardest #

If the concrete sheds into the bars, a column with fewer bars sheds into less steel, and each bar has to take more. That is the opposite of how a designer thinks of a light reinforcement, as bars that are hardly working.

The bars’ stress is simply the column’s strain times $E_s$, so each line is the upper line of the previous figure rescaled, and it rises as the steel ratio falls. At a very low steel ratio, the column’s strain approaches the plain concrete’s, so a thin scatter of bars is forced to whatever strain the concrete would have gone to, regardless of how much of them there is. That strain was 1,429 microstrain at 12 MPa, equal to 286 MPa of steel stress, with no sign of a limit.

Here is where a code rule comes from that is often read as a matter of minimum robustness. The commentary to the American concrete code gives its 1 per cent minimum in a column a reason in exactly these terms: creep and shrinkage transfer load from the concrete to the bars, the stress increase is larger the smaller the ratio, and without a floor the bars could reach yield under sustained service loads. That rule comes from column tests at Illinois and Lehigh around 1930, when the bars were mild steel yielding near 275 MPa. On the figure, that line is crossed by the 12 MPa curve near 0.25 per cent and by the 16 MPa curve near 2 per cent. So the old rule is a statement about the top left corner of this figure, and the fact that it is written as a percentage of the area hides that it is a statement about strain.

A modern 500 MPa bar has moved the yield line well clear of any of these curves, and a modern column is not going to yield its bars from creep alone. The rule survives on other grounds, some of which were always there: bending from load that is not quite central, and how much tension the section can supply once it cracks. But the original reason has not gone away so much as gone out of range, and it comes back in any column with a low steel ratio and a very high sustained stress. Note also what a yielded bar would actually do. It would stop accepting load, and the concrete would have to take back the rest, creeping again under a higher stress than it had. That is a redistribution with no collapse in it, since strength stays roughly where it was, and the same steel in a redistribution of another kind behaves in the same way. It is the serviceability picture that changes, because the shortening carries on.

Then the load comes off #

Now take the load away. The surprising result lies here, in a case most column designers never consider, since columns are not usually unloaded. Yet they sometimes are. A building is partly demolished, a transfer is jacked, a heavy tank is drained, a storey is taken off and rebuilt, or a backprop is struck.

Taking off the load removes it from the section elastically. An unloading of $\sigma_N$ on the gross concrete, on the composite section, relieves the concrete by the same $\sigma_N/(1 + n\rho)$ it first received and the bars by $n$ times that. The concrete had 10.7 MPa given to it on the first day and has 7.7 left, so taking away 10.7 leaves it at minus 3.0. Elastic recovery cannot give back what creep and shrinkage have taken, so the concrete is too short, and the bars, which are not, are holding it stretched.

So the column with the most steel is the one in most danger of this. At 2 per cent, a typical heavily loaded column, the concrete is left at about its own tensile strength. At 4 per cent it is well past that, and the concrete will crack, in tension, with no force on the column at all, and the bars will stay in compression at 113 MPa. At 8 per cent it would be at nearly 6 MPa. This has been seen. The same early column tests recorded cracking across specimens with heavy reinforcement when they were unloaded, and that is a recognised argument for an upper limit on column steel, along with the practical one about fitting the bars in and getting the concrete past them.

This is a case of a self-equilibrating stress state showing up. The bars push out, the concrete pulls in, and the whole section sums to zero. It is what the movement nobody applied calls a stress with no load behind it, and what the coincidence reinforced concrete stands on looks at for temperature. Here the cause of the mismatch is the column’s own history, and what it means is that a reinforced column remembers it. Two columns identical on the day of inspection, one carrying load for thirty years and one not, are in very different states once they are both unloaded.

It would be easy to overstate this. A column with cracks through it at some spacing and no load on it is a column that will close those cracks again the moment the load returns, and its axial strength depends little on a crack that is held shut. Over the months after unloading, part of the creep recovers and the tension relaxes as well, so the instantaneous number here is the worst of it. What is lost is the concrete’s contribution to shear and to bending stiffness until the load comes back. An engineer deciding to strip a building down to its columns and rebuild on them has a reason to look.

What this does to the tower #

The original question was a tower’s columns against its core, so that is where the result should be returned. Take the 40-storey tower of the first essay, with 12 MPa at the base of the columns and 6 MPa at the base of the core, stress falling linearly with height, creep 2.2, shrinkage 300 microstrain in each. That earlier calculation had the columns not shrinking at all, which suited its argument. Keeping the shrinkage on both makes the plain tower’s total difference at the roof 42 mm.

A core is usually much less reinforced than a column, at perhaps 0.5 per cent, with a low stress, and a perimeter column may have 2 or 4 per cent. Thus the steel falls unequally on the two sides of the calculation.

Three fifths of the difference has gone. The column shortens 95 mm rather than 127, while the lightly reinforced core goes from 85 to 78, so the steel has been far more effective on the side with more stress and more steel. That is the size of the error when a compensation schedule is worked from plain-concrete shortening. The survey that arrives after the floors found that the doubt about the creep coefficient is worth a comparable amount, and so here is a known and calculable effect sitting in the same place as a large unknown one. If the steel ratio is left out, part of what a survey will report as a surprise about the creep is really the column’s bars.

The next figure shows the other side. With a lightly reinforced core, somewhere between 3 and 4 per cent of steel in a column, the column and core end up the same length at the roof, and beyond that the column has stopped being the shorter one. At that point, the sign of the correction a contractor must build reverses, and an outrigger between the two would be pushed the opposite way from the one it was designed for. Steel ratios of 4 per cent and over are common low down in tall concrete buildings, and it is low down that the shortening accumulates. Real columns have reinforcement that tapers up the height, whereas the figure holds one ratio all the way up.

The second line shows that the balance has nothing magical about it. Put 2 per cent in the core, as in a heavily loaded wall or a composite core, and the difference comes back to about 12 mm with the column at 4. What matters is the difference between what the two sides have, weighted by their stress, and it can be set deliberately. A designer with a free choice of bars could, in principle, specify the column and the core to shorten by the same amount by using steel ratios alone. Nobody chooses bars this way, since bars are chosen for strength first, and the cost of 2 per cent more steel is high. It would be a strange reason to spend it. As an accounting matter, though, every steel ratio in a tower is already a decision about its differential shortening, whether anyone said so or not.

What the single section leaves out #

One age and one creep coefficient. The load is treated as arriving all at once on concrete of a single age. A real column gets its load floor by floor, and what the first floors do has been carried in an earlier essay. A staged load in this model would be a small change to the $\phi$ on each increment, and the size of the steel effect would not change much, as the first load on a column is the youngest and has the largest creep coefficient anyway.

The steel stays elastic and straight. Compression bars held by ties, at the stresses here, are fine, but a bar at 300 MPa in a lightly tied column is closer to its own buckling than anyone intended, with the cover spalled, and that is not in this.

The ageing coefficient is taken as 0.8. That value suits most loaded ages and moderate creep. The result is not very sensitive to it, because $\chi$ only enters through the denominator’s product with $n\rho\phi$, and at 2 per cent steel a change from 0.7 to 0.9 moves the final concrete stress by a few per cent.

Shrinkage is uniform across the section. A big column dries from outside in, and its surface zone shrinks first, so the bars near the surface pick up their share early and the core of the column follows. That moves when the load transfers and not, much, how much in the end.

The column is centrally loaded. Any eccentricity would shift bending into the mix, with one face creeping more than the other, and the column would bow further with time, the route to the failure the column that fails years later describes.

The figure cannot show the crack. The tension when the load is removed is a mean stress in the concrete, assuming it stays whole. In fact the concrete at a bar is more highly stressed than the mean, and the tension is relieved at the first crack in a pattern set by bond. The number given is thus a test of whether cracking will begin. It gives no width.

Still open: the column that was jacked #

A column is sometimes deliberately loaded and later unloaded, when a tower is jacked onto a new transfer, a damaged column is replaced by propping, or a structure’s load is moved to temporary works. In those cases, the moment of unloading is chosen, and so is the speed. An unloading spread over months lets the concrete relax some of its tension as it arises, and a sudden one does not. Whether a programme can be arranged to keep a heavily reinforced column below cracking by taking the load off slowly enough, how long that takes, and what its partial reloading does to the shortening already built into the floors above, is a calculation that needs creep recovery modelled in its own right, and the single stage of unloading here treats recovery as something that happens only later.