Engineering Uncertainty

Apparently someone called James Amrhein said that Structural Engineering is:

The art and science of moulding materials we do not fully understand; into shapes we cannot precisely analyse; to resist forces we cannot accurately predict; all in such a way that the society at large is given no reason to suspect the extent of our ignorance.

Modern design concepts break this complex statement into two halves, as it were. The forces we cannot accurately predict are one half, “Load”, everything else is the other half, “Resistance”. When I was a lad, this was taught as “The Design Equation”:

  • Load =< Resistance

It was taught as the fundamental objective of design, and the final statement in any given design outline. In real life there is more than a solitary load so we end up with the simultaneous solution of a range of design equations for Gravity and Wind and whatever else.

Resistance, on the other hand, is generally similar irrespective of load. Your beam doesn’t know whether it’s supporting a house or an elephant, just what forces it is experiencing. This leads to a simplifying process for design, where instead of progressing through a sequence of cases typified by “Gravity Load =< Gravity Resistance” followed by “Wind Load =< Wind Resistance” we tend to consider “Worst of Gravity and Wind =< General Resistance”.

Uncertainty exists on both sides of these equations. Modern design approaches discuss that uncertainty in statistical terms. In the AS/NZS codes and EuroCodes variable loads like wind are discussed in terms of an intersection between “recurrence intervals” and “design life”. Unvarying loads are discussed in terms more like “working loads” – someone’s “reasonable” derivation of load magnitude.

What ends up hidden by this nomenclature is the real comparison between the loads and resistance, what used to be called a “Safety Factor”. That is, if you strip out all the factors and compare “raw” loads with “raw” strengths, you get a proportion of resistance to load, a margin of safety where your strengths are higher than your loads. If your factored loads are the same as your reduced resistance, your design equation looks as though it’s a close-run thing, that your structure is economised to the limit, whereas, there’s actually a great likelihood that you have considerable spare capacity in your system. Typically loads are based on a 1:500 return interval and a 50-year design life, and resistances are based on the lower-5% of probable strengths.

This can throw up some quirky results. If you compare a steel beam with a timber beam designed for the same conditions, you probably have an effective “safety factor” for the steel of something like 1.7 while you might have a margin of more like 2.5 for timber. The reason is that timber is a highly variable material, while steel is quite predictable, so that lower 5% value for strength is a different proportion of the mean material strength. If you’re concerned that your loads may be exceeded, you may be better to opt for a Timber member, knowing that the real strength will be higher than the real strength of a Steel member with the same nominal capacity.

One area where this difference becomes important is the interaction between soil and structures. Soil behaviour can’t be quantified in the same way as other materials because at the point of design there is insufficient information to form the kind of probabilistic curve that we use for timber, concrete and steel. Code timber strengths are based on thousands of samples and includes a component of sorting. Soil strengths are generally based on inference from a small handful of tests that can’t often can’t directly test the component of strength that’s really of interest to the designer, so there’s a huge amount of approximation that goes on to make a guess at something like an average strength. This strength is often converted into something like a design resistance by semi-arbitrarily dividing mean strengths by a “safety factor” of 3 for non-seismic situations and 2 for seismic situations. In effect, what you can end up with is a very precisely designed structure connected to what is essentially a “guestimated” soil mass.

As you can appreciate, in all of these scenarios you end up with a healthy dose of engineering uncertainty. This uncertainty is ameliorated through empirical testing – engineers build real structures and observe the performance of those structures over time. Engineers and engineering practices build into their methods kerbs and boundaries. For example, my old practice in Wellington specialised in high-end residential work, and we knew from experience that for Wellington wind conditions, you need to install steel members around windows and doors once they get over a certain size because capillary action around the standard flashings pumps water from the outside to the inside once movement gets above a barely-measurable threshold. You can’t demonstrate that necessity via calculation, because the uncertainty in all the input variables is larger than the critical outputs from the design.

In geotechnical design this empiricism relates to the general soil strata of a particular region. Geotechnical engineers are naturally more localised than structural engineers. I know from extensive experience how the soil behaves in Wellington under a variety of load conditions and I’m aware of the usual construction techniques used in the region, so I can make fairly good estimates for the purposes of design. In London, I have lost all that local knowledge, which I am slowly picking up. For example, most of the quite deep piles I’ve done in Wellington have been in areas where there is a young and poorly-graded soil strata. Piles become long and rely on friction along the sides of the piles more than end-bearing on the base of the pile. In part that’s because debris is usually left at the bottom of the hole, in part it’s because many of these piles don’t ever hit something definitively solid. In London, however, the ancient clay layer can be cleanly excavated so end-bearing is available, but side-friction can’t be quite as well developed in clays.

This difference in the origin of resistance leads me to a key concept that is not explicitly treated in any of the design equations, and is only explicitly treated at all in seismic design: the consequences of failure. Engineers are used to thinking about the design equation as outlined above in fairly black-or-white terms, but most failures are partial and progressive. The classic example I encountered all the time in Wellington was the inadequate retaining wall. I used to go out all the time to look at retaining walls that had tilted over to an alarming degree, where clearly the soil behind the wall was slowly pushing the wall over. These walls had failed, because they were clearly providing less resistance than was being applied by the soil, but the immediate consequence of that failure was actually not too serious. At some point, that situation is going to suffer a complete collapse, but for now, the soil is actually held in place.

In New Zealand there is the “Earthquake Commission” which provides funds to repair landslips and retaining wall failures (up to a point), and one recurring discussion I had with their agents was to argue that a wall had failed because it was providing inadequate resistance, to which their response was, roughly speaking, to point at the wall and ask me how they could do that if it had failed. I had a couple of very frustrating encounters with cash-strapped clients and failed retaining walls where I had to advise a client that they could spend $10,000 to stabilise a wall, and if they didn’t have the money that the EQC would kick in the first $25,000 to rebuild it (though the cost to them might still exceed the $10K). Frustratingly, only wealthy clients ever went for the prevention.

Once you start to think about the consequences of failure, the design equation becomes a lot more flexible. Early in my career, I would design any given structure to be “code compliant”, but over time I began to see the two other necessary conditions – over and under engineering. From 2006 onward, I consciously “over-engineered” cladding elements because the consequences of failure in those elements are so severe. Once capillary water gets past flashings, it’s typically the end of a building’s life. Conversely, I started to advise against doing some exterior retention work, because the consequences of a soil failure were too low to merit it. The most extreme case I had was a client who wanted to prevent fretting of a tall bank along their driveway. They were losing a couple of cubic metres of soil every couple of years, with an average annual clean-up cost of $1500-$3000. In order to prevent those slips, it would be necessary to build a sprayed concrete wall whose cost was in the order of $50,000. Given my client was in his late 70s, it made no kind of sense to spend the money to build the wall.

Once we put the consequence of failure into play, there are some interesting economies available to the designer. For example, one common way of retaining soil is to install a steel “soil nail” horizontally into the soil mass and then protect the face of the soil bank with a sprayed concrete face connected to those “soil nails”. When, say, an earthquake happens, the soil mass presses against the concrete face which pulls on the nails. If they’re deep enough, they transfer the load back into the soil mass. There are two main ways that a soil nail can fail. The first is that there is a failure at the connection between the concrete and the nail – the nail could fracture or the concrete could crumble. The second is that the nail can be pulled out of the soil. If you rigorously apply the the “design equation” as outlined above, you ensure that nothing fails. If, instead, you think about the two different kinds of failure, you might decide that you don’t want the first failure, because that would lead to a sudden collapse of the wall… but you might be okay with the second kind of failure because while the wall will move, it doesn’t present a hazard to life or other property.

I said that the consequences of failure are only explicitly treated in Seismic Design, and by that I in fact mean that the whole concept of Load versus Resistance is discarded for seismic design. The design equation is based on the idea that you can compute a maximum load and then design a structure to be bigger than that, but in seismic design there is no practical upper limit to the amount of load the ground can apply to the building. In other words, the earthquake is definitely going to cause some kind of failure in your structure. As a designer, you need to decide what the consequence of that failure is going to be. You can refer back to earlier posts on Capacity Design if you’re interested in more information there.

Engineering is a very complex activity. Frankly, in order for any design to get done at all, most engineers end up having to apply Amrhein’s statement not to the general public but to ourselves. If we used truly stochastic/probabilistic models for every aspect of design, everything would take forever and with no measurable improvement 99% of the time. However, more than once in my career I’ve seen (or even been in my youth) an engineer almost chasing their own tail trying to fine-tune a solution beyond the point justified by the improvement in certainty that’s created. At the worst, I’ve seen engineers discard perfectly workable solutions trying to improve their 95% solution into a 100% solution, a somewhat pessimistic decision given the uncertainties involved. On the other hand, we have all heard of or seen significant engineering failures caused by a designer not adequately predicting loads or resistance. I see both kinds of design mis-step originating in the same fundamental lack of awareness in what the uncertainty is, where it comes from, and what the consequences of a failure will be.

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