The load that was left on too long

Load a timber beam in a testing machine and it breaks in about five minutes at a stress that goes into the tables. Load a second beam from the same batch to sixty per cent of that stress and leave it, and it will break too — in about fifty years.

Nothing varies. The load does not cycle, so it is not fatigue. The beam need not have deflected appreciably, so it is not a stability failure. It is that a static stress slowly breaks fibres, and the relation between the stress and the time it takes is one of the most striking curves in structural engineering.

Which free body produced the number #

None, and for once that is a statement about the material rather than about the presentation.

The Madison curve is a fit to the Forest Products Laboratory’s ramp-and-hold tests, and it has the form

$$SR(t) = 0.183 + 1.084\,t^{-0.04635}$$

with $t$ in seconds, normalised so that the five-minute standard test gives 1. Every constant in it is a regression coefficient.

What is underneath is a fracture process: micro-cracks in the cell walls of the timber extend under sustained stress, each extension makes the next easier, and the member fails when the crack network percolates. Timber is the material that has a direction, and the direction matters here too — the cracks run along the grain, so a member stressed across it is on a much steeper curve than one stressed along it. That is the same physics as the flaw that sets the strength, running slowly. Concrete does it too, as does glass, as does any material whose strength is governed by defects rather than by a yield criterion.

Steel does not, which is why nobody who works in steel has heard of the effect: a yield stress is a dislocation-motion phenomenon and it is essentially time-independent at ordinary temperatures. The steel that is stronger in a millisecond is the same axis explored in the other direction, where steel does have a rate effect — and the direction of it is opposite to this one.

The asymptote nobody quotes #

Read the curve backwards and it has a floor: 0.183.

A stress below about 18 per cent of the short-term strength has no time to failure at all. The curve never reaches it; the inverse relation returns infinity. So there is a stress below which a member is not on a slow path to failure, it is simply safe, and everything above it is on a clock.

An asymptote of that kind is a familiar shape from elsewhere. A fatigue curve has a cut-off below which cycles do no damage; a creep curve has a stress below which tertiary creep never begins; a stress corrosion problem has a threshold intensity. In each case the mechanism needs a driving force above some value to propagate at all, and below it the defects are stable. That is why the number is worth knowing even though it is quoted nowhere: it is the boundary between a member with a design life and a member without one.

That is a much more useful thing to know than the curve itself, because it changes the question from “how long will this last” to “is this member on the clock or not”. Most timber members in ordinary buildings are stressed under permanent load at 20 to 35 per cent of their short-term strength, which puts them just above the asymptote — where the times to failure are enormous but finite, and where the curve is very steep in the stress ratio.

Why the shortest load does the damage #

Now the part that runs against the name of the effect.

Time to failure is $t_f(s) = ((s - 0.183)/1.084)^{-1/0.04635}$, which is a twenty-first power of the margin above the asymptote. A load a third larger has a time to failure several thousand times shorter.

Integrate a service life as a damage sum, $\alpha = \int dt/t_f(s(t))$ with failure at $\alpha = 1$, and the result is not what “duration of load” leads anybody to expect.

Six hours a year of wind, at a stress ratio of 0.72, consumes more of the member’s life than fifty years of dead load at 0.36. Duration is the cheap variable and stress is the expensive one, and the exponents say so: eleven decades of time buys forty per cent of strength, while a third more stress costs three or four decades of time.

That inverts the intuitive reading completely. The effect is called duration of load and the design system that goes with it awards the longest load the smallest strength — which is exactly right as a design rule, for reasons in the next section, and is the opposite of what the damage integral does with a real history.

The factor is the curve #

Here is a small result that makes the whole system make sense.

The permanent-duration modification factor in timber design is about 0.6. The Madison curve at fifty years is 0.589.

They are the same number, and it is not a coincidence: the factor was chosen so that a member fully utilised under permanent load alone sits at the fifty-year point of the curve. Read that way, a duration factor is not a safety factor covering an uncertainty; it is a measurement, quoted as a factor.

That explains the whole set. The medium-duration factor of 0.8 is the curve at a few weeks. The short-duration factor of 0.9 is the curve at a few days. The instantaneous factor above 1.0 is the curve to the left of the standard test, where a load faster than the test is worth more strength than the test measured.

And it explains the crossover the design system produces. Each combination is checked against a different strength, so the utilisation lines cross: below a certain variable action, the permanent-only case governs even though it is by far the smallest load on the structure. A designer who checks only the largest combination has not checked the member.

What the same effect does to other materials #

Timber is the case the design system handles explicitly, and it is not the only material with the effect. Naming the others is worth doing because the design treatment differs wildly and the physics does not.

Concrete loses about 15 per cent of its short-term compressive strength under sustained load — the well-known factor of 0.85 in front of the cylinder strength in every design stress block is largely this effect, plus a shape allowance. It is applied universally, is almost never explained, and is exactly a duration factor with a different name.

Glass is the extreme case. A pane’s design strength for wind, applied for seconds, is several times its design strength for a permanent load, and glass design is one of the few disciplines where the duration appears explicitly in every calculation as an exponent rather than as a factor.

Masonry and adhesives both have it, the second severely — a bonded connection loaded permanently at a fraction of its short-term capacity will creep and eventually fail, which is why structural adhesives are qualified against sustained-load tests rather than short-term ones.

And steel does not, at ordinary temperatures. Above about 400 °C it does, and then it is called creep rupture and it is the mechanism that decides how long a steel member survives in a fire — the hour that is really a temperature is the same curve, at a temperature where the timescales have come down from decades to minutes.

Why the system and the integral disagree #

Two things have now been said that do not sit together, and it is worth being explicit rather than letting the reader notice.

The damage integral says the short loads do the damage. The design system gives the long loads the smallest strength. Both are defensible, and they are answers to different questions.

The design system is asking: at what stress may this member be held indefinitely. The answer is a point on the curve at the design life, and the longest-duration load is the one that has to be checked against it.

The damage integral is asking: how much life does this particular history consume. The answer depends on the actual stress ratios in the history, and the wind case’s ratio is high because the wind case is what the member was sized for.

The reconciliation is that the two are the same statement viewed from opposite ends. A member sized so that its permanent-only case is at 0.59 has a wind case at something below 1.0, and how far below decides how much life the wind consumes. Sized tightly, the wind case dominates the integral; sized generously, nothing dominates it because nothing is near the curve.

Reading a design check with this in mind #

Put the pieces together and a timber design check has a shape worth recognising, because it is not the shape of a steel one.

A steel member is checked against one strength. Every load combination is compared with the same capacity, the largest combination governs, and finding it is a matter of arithmetic on the actions.

A timber member is checked against as many strengths as there are duration classes, and the governing case is the one with the largest ratio of action to its own capacity rather than the largest action. That is why the crossover exists, why the permanent case can govern a member whose wind case is three times larger, and why a designer who ranks the combinations by size has ranked them by the wrong quantity.

It also changes what a “worst case” means during a scheme. Adding a heavier roof finish raises the permanent action, which is checked against the smallest strength, so it costs more than the same weight of snow would. A member’s sensitivity to a load depends on which duration class the load belongs to, and two loads of the same size are not interchangeable.

Which failure arrives first is usually a question about mechanisms — shear or bending, buckling or squashing. Here it is a question about when: the same member, the same mechanism, and a capacity that depends on how long the question is asked for.

Where the model stops #

The curve is for timber, at a stated moisture content. Wet timber is weaker and creeps more, and the duration effect is stronger in it — which is why the design factors come in a table with service class as well as duration.

The Madison curve is one of several. Later work using damage-accumulation models fitted to the same data gives somewhat different curves, and the differences are large at the fifty-year end where nobody has run a test.

Nothing here is about size. The curve is a stress ratio, and a member’s short-term strength itself depends on its size through the bigger one is the weaker one — so a large member starts lower on the curve as well as travelling along it.

The member was assumed to fail rather than to deflect out of use. Timber creeps as well as weakening, and a beam at 60 per cent of its short-term strength for fifty years will have deflected to two or three times its instantaneous value long before it breaks. The deflection that arrives three years late is the effect that usually decides the member, and the two share a mechanism without sharing a limit state.

And a real history is not three states. The stress ratio varies continuously, the damage integral is over that variation, and the answer depends on the tail of the distribution rather than on its mean. Which makes the whole calculation an exercise of the same kind as the envelope is not a structure: a single worst case is not the input the integral wants, and constructing the input the integral wants means knowing a load history nobody records.

The generalisation #

The habit worth taking away is to ask, of any quoted strength, how long the specimen was under load.

Every strength in every table was measured on a test with a duration, and the duration is a property of the test standard rather than of the material. Five minutes for timber, a couple of minutes for a concrete cube, a fraction of a second for a Charpy impact, several minutes for a steel tensile test. For steel it barely matters. For timber it is worth forty per cent. For concrete under sustained load it is worth about fifteen. For glass it can be a factor of two.

There is a corollary about testing that is worth stating, because it is where the effect does the most damage to intuition. Any accelerated test — a load applied for an hour to represent a lifetime, a proof load, a sample tested to destruction on the day of delivery — is measuring a point on this curve at the test’s own duration and nowhere else. A proof load that a member survives for ten minutes establishes that it survives ten minutes. Extrapolating it to fifty years requires the curve, and if the curve is not known for the material then the extrapolation has no basis at all. The strength no specimen had is about extrapolating across a population; this is about extrapolating across time, and it is the less familiar of the two.

The second habit is about exponents. A relation with a power of twenty in it is not a relation to be evaluated at a best estimate: the answer at the mean of the inputs is nothing like the mean of the answers, so a damage sum computed from average stresses understates the damage badly. The load that never came near failing anything makes the same point about a cube, which is mild by comparison — and the discipline is the same in both. Where the exponent is large, the tail is the answer, and the average is a distraction.