The column that fails years later

Two facts about concrete are taught in different courses and are almost never put together.

The first is that concrete creeps: hold a stress on it and the strain goes on growing for years, reaching two or three times its instantaneous value. The second is that a compression member’s deflection is amplified by $1/(1 - N/N_{cr})$, because the axial load acts on the deflection it has already produced.

Put them together and the amplifier has a denominator that closes with time.

Which free body produced the number #

Cut the column at mid-height and take the lower half. The moment there is the axial force times the total eccentricity, and the total eccentricity is the first-order one plus whatever the column has already bent.

For a pin-ended column with a sinusoidal imperfection or a sinusoidal first-order moment, that closes into the standard amplifier

$$e = \frac{e_0}{1 - N/N_{cr}}$$

which is exact for the sine shape and near enough for everything else. The only quantity in it that time can touch is $N_{cr}$, and $N_{cr}$ is $\pi^2 EI/L^2$.

Creep touches $E$. The effective-modulus method is the simplest honest way to say how: a stress held constant produces a strain that grows by the factor $(1 + \varphi)$, so the member behaves as though its modulus were $E/(1+\varphi)$, and

$$N_{cr,\infty} = \frac{N_{cr,0}}{1 + \varphi_\infty}$$

For an ordinary creep coefficient of 2.5, that is 29 per cent of the day-one value. A column checked at a comfortable third of its buckling load on the day it is propped is above its long-term buckling load.

Two regimes, and a boundary that is a fraction #

The consequence is a clean division.

Below $N_{cr,0}/(1+\varphi_\infty)$ the column settles. Its deflection grows — by a factor of 2.4 on the column drawn, from 25 mm to 60 — and stops, because $\varphi$ approaches a limit and so does the amplifier. That is a serviceability outcome: a column visibly bent, its moment larger than the design one, and stable.

Above it there is nothing to settle to. $\varphi$ passes the value at which $N = N_{cr}(t)$, the amplifier passes through infinity, and the column fails at a moment fixed by a material property rather than by an event.

The date is not found by watching a number run away. It is found by inverting the creep law: the divergence happens when

$$\varphi(t) = \frac{N_{cr,0}}{N} - 1$$

and the creep law is a known function of time, so the answer comes back directly. At 46 per cent of the day-one critical load it is 55 days. At 38 per cent it is 123 days. At 57 per cent it is 34.

Which is why the check is written the way it is #

The design rules for slender concrete columns look arbitrary until this is in view.

They ask for a creep-adjusted stiffness, computed with an effective creep ratio that is not the material’s $\varphi_\infty$ but $\varphi_\infty$ scaled by the fraction of the load that is permanent — because a column carrying half its load as imposed floor loading only creeps under the half that stays. They ask for an imperfection, because the amplifier has no answer for a perfectly straight column and every real column is not. And they ask for a nominal stiffness or a nominal curvature rather than a section analysis, which is a way of admitting that the stiffness in the denominator is not a property anyone can compute exactly.

Every one of those is a response to the same structure: an amplification whose denominator depends on a stiffness that is falling.

The same shape in three materials #

The structure of this argument is not about concrete. It is about a compression member whose stiffness falls while it is loaded, and there are at least three of those in this collection.

Timber has a duration-of-load factor — a strength reduction for permanent loading — and a creep factor that does exactly what $\varphi$ does here, with the added difficulty that both depend on moisture content. A timber column’s long-term buckling load is a smaller fraction of its short-term one than concrete’s.

Steel at temperature is the same equation with the clock running much faster. The hour that is really a temperature is the strength side of it; the stiffness side is that steel’s modulus falls faster than its strength does above about 400 °C, so a steel column in a fire loses buckling capacity before it loses squash capacity, and a slender one fails in minutes at a load a stub column would carry.

And any structure with a soil under it. A footing on clay is a spring whose stiffness falls as consolidation proceeds, and a frame whose sway stiffness depends on that spring is a frame whose $N_{cr}$ falls with time — the same divergence, with the creep in the ground rather than in the member.

The load that is permanent, and the load that is not #

The strongest simplification in the analysis above is that all of the load creeps, and correcting it is where the design rule earns its complexity.

Creep responds to the stress that is held. A column carrying 2,200 kN of which 1,400 is dead load and 800 is a floor loading that comes and goes creeps under something much closer to 1,400 — so its effective creep ratio is the material’s coefficient scaled by the ratio of permanent to total.

That scaling has a consequence worth stating plainly: two columns carrying the same total load can have long-term buckling loads a factor of two apart, according to which of them is holding a warehouse and which is holding a car park. The load case that governs the day-one check is the full one; the load case that governs the long-term check is the permanent part; and the two are different structures asked different questions.

It also means the mode is worst in exactly the buildings where it is least expected. A heavily loaded transfer column under a residential tower is nearly all permanent load, creeping at nearly the full coefficient, for the whole life of the structure.

The failure has no cause and no warning #

The most useful thing to say about this mode is what it looks like from outside.

Nothing happens. The load has not changed, no new load has been applied, no material has been substituted, and the weather is the same as it was. A column bends slowly for months and then bends quickly, and the interval between “visibly out of plumb” and “gone” is short because the amplifier’s growth is not linear — it is a reciprocal approaching a zero.

There is therefore no event to investigate. A collapse of this kind produces an inquiry that finds no cause, because the cause was the absence of a check performed twenty years earlier at a moment when the structure was demonstrably safe.

That is the argument for the check being about a fraction rather than about a stress. A stress-based check on the column drawn passes comfortably at every age; the load is a fifth of the day-one buckling load and the concrete is nowhere near its strength. What fails is a comparison between a load and a stiffness, and stiffness is the quantity structural checks are least often written in.

The number a designer can actually move #

There are four quantities in $N_{cr,\infty} = \pi^2 EI/(1+\varphi)L^2$, and they are not equally available.

$\varphi$ belongs to the concrete and the environment, and the range a designer controls is narrow — a lower water-cement ratio, a later age at loading, a larger member. $E$ moves with the aggregate and hardly at all with the grade. $L$ is the storey height, which is architecture.

What is left is $I$, and it is the one that responds quadratically. Taking the 400 mm column drawn to 500 mm multiplies its second moment by 2.4 and its long-term critical load with it, and costs a hundred millimetres of plan.

That is why the practical response to a slender concrete column is almost never a stronger concrete. A grade increase buys a little modulus — $E$ goes roughly as the cube root of the strength — and buys nothing at all in the quantity that is falling. A creep problem is solved with geometry. It is the same conclusion the section-shape comparison reaches by a completely different route, and it holds here more strongly, because the property being lost is stiffness rather than strength.

The reason no test finds it #

A structural test is a load applied over minutes. Creep buckling is a load applied over years, and that difference is why the mode is known from theory and from collapses rather than from laboratories.

The obvious response — hold a column at a load for a decade and watch — has been done a handful of times and is not a routine. What is done instead is to accelerate: load at a high stress ratio, where the creep coefficient is reached in months rather than decades, and extrapolate. That works for the arithmetic and imports its own assumption, because concrete’s creep is not linear in stress above about forty per cent of its strength, and the accelerated test is run in exactly that region.

So the design rule for slender concrete columns rests on a chain: a creep model fitted to short-term specimens, an effective modulus that assumes a constant stress, and an amplification derived for an elastic member. Each link is defensible and the chain has never been tested end to end at the timescale it is written for. That is worth knowing when reading a code clause that looks as settled as a strength check.

Where the model stops #

The effective-modulus method is the crudest of the creep formulations. It is exact for a stress held constant from a single instant and approximate for anything else; a load applied in stages, or partly removed, needs a superposition method or an age-adjusted modulus, and both give a slower loss of stiffness than the number here.

Cracking is a fixed factor. The 0.6 applied to $EI$ stands in for a section that is partly cracked, and a real column’s cracked stiffness depends on the moment it is carrying — which is the moment this calculation is trying to find. The honest version is iterative and the answer moves in the unhelpful direction: more moment, more cracking, less stiffness, more moment.

The creep coefficient is treated as a property. It depends on the humidity, the member’s surface-to-volume ratio, the age at loading and the mix, and a factor of two either way across those is ordinary. Since it appears as $1/(1+\varphi)$, that factor of two is a factor of nearly two in the long-term critical load.

The creep law is a curve fitted to tests. The expression used here reaches its final value asymptotically with a time constant of a few hundred days, which is the ordinary shape, and the date it returns for a divergence is only as good as that shape near the value of $\varphi$ that matters. Two creep models that agree about the thirty-year answer can disagree by months about when a particular coefficient is reached, and it is the second that this calculation asks for.

And the column is pin-ended and prismatic. A real column is in a frame, and its effective length depends on members whose own stiffness is creeping — so the buckling length is also a function of time, and not necessarily in the safe direction.

What the pictures cannot show #

The time axis is logarithmic, which is the only way to draw a process that is half over in a year and continues for thirty. It makes the divergence look like an event at a date; on a linear axis it would look like a curve that becomes vertical, which is closer to the truth and impossible to plot next to the first month.

Nor can any figure show the thing that decides whether this failure mode ever arrives, which is what else is holding the structure up. A column in a braced frame that fails in creep buckling sheds its load to its neighbours; a column that is the only thing under a transfer beam does not. The mode is a property of a member and the consequence is a property of the structure, and nothing on this page is about the second.

The assumption the figure rests on #

The load is assumed to be constant from the day it is applied.

Almost no load is. A building is loaded in stages as it is built, its imposed load comes and goes, its prestress relaxes, and the concrete’s own creep redistributes force from the members that creep most to the members that creep least. Each of those makes the effective creep coefficient smaller than the material’s, which is why the check in practice uses a ratio of permanent to total load rather than the coefficient itself.

The one case where the assumption is exactly right is the one that matters most. A prop, a temporary column, a falsework leg under wet concrete: loaded at once, at an age when the concrete is young and creeps fastest, at a load fraction chosen for a short-term check, and left in place for as long as the programme requires. That is where creep buckling has actually happened, and it is not a category of structure anybody designs with a long-term stiffness in mind.