This is yet another commentary on a paper which cites my work, in this case Inverse analysis for parameter estimation of sandy soil with axially loaded pile using nonlinear programming, which cites Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles. Since 2026 is the tenth anniversary of the publication of that effort, some remarks on why I did that are in order.
Rationale for My Original Study
The current predominant regime in pile dynamics has been around for over half a century now. With tweaks and improvements in computer power and hardware, it has enabled us (well, most of us) to jettison the problematic dynamic formulae for capacity prediction and verification during installation. The whole system, however, relies on the 1D representation of pile/soil interaction to be accurate and the optimization algorithm to find the solution to the inverse problem. Both of these are subject to the kinds of improvements we see in other fields.
Getting to this point was not an easy or straightforward task, both because of the application itself and the code/regulatory environment in which we operate. Much of the struggle to get the current methodology accepted was an uphill battle against the existing “we’ve always done it this way” mentality which settles in, and no doubt this will be the case with a new generation of pile dynamics methodology. But there are some difficult challenges inherent in the physics of this problem, most of which stem from the nature of soils themselves.
I ran into many of these challenges during my study which led to Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles. One colleague from an institution in a neighbouring state felt that my effort was “too ambitious.” He’s probably right, which is why he’s in administration now. But my objective was for this study–and the subsequent papers to fine tune the method–to be a convesation starter, and this paper’s citation of my work is evidence that this is taking place. I sense that efforts to “move the football down the field” in this discipline are taking place, and am gratified to be a part of that effort.
Comments on the Paper Itself
Strictly speaking this paper only has the inverse method as a commonality with my own study. In this case the researchers are dealing with a drilled shaft and are trying to back analyse static capacity. While this sidesteps the rate-dependent problem between dynamic signals and static response, it brings other factors into play, some of which are definite weaknesses in the paper and others where the jury is still out.
Optimisation Technique
Let’s start with one which falls into the latter category: the optimisation technique they chose, which was the Davidon-Fletcher-Powell method. The purpose of optimisation techniques is to find the minima and maxima of “equations” (often they can be expressed in this way, but in this business frequently they can’t) and thus the best solution to the problem. The classic example of this (and one frequently used to test optimisation techniques) is the Rosenbrock Equation, which is
and is plotted as shown below for a =1 and b = 100.
This has challenged optimisation techniques for a long time. The problem with using something aimed at problems like this is that, in geotechnical engineering, problems look less like this and more like relief maps. The result is having to deal with false minima. For example, if we have a canyon on top of a plateau, a false minimum would be the lowest elevation at the bottom of the canyon rather than the bottom of the cliffs of the plateau, which are generally lower. Multiple false minima are common for problems in this profession, which is one reason why we still use brute force grid optimisation in problems like slope stability. This is why I chose a polytope method for my own study, which is derivative free and “casts a wider net” on the downhill slopes of a problem. It is slow and its results not perfect but I think this is a problem that needs to be addressed if we are to use optimisation techniques for solving geotechinical problems.
A couple of side notes:
- I did use a quasi-Newton routine related to the DFP method in my study “Analysis of Vibratory Pile Drivers using Longitudinal and Rotational Oscillations with a Purely Plastic Soil Model” because I felt the parameters were “regular” enough to justify its use. The routine I used had an option for the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method. And that leads to…
- My last course for my PhD degree was in Optimisation. One day our professor–Dr. Kyle Anderson, one of the most brilliant people I’ve come to know–was going on about these techniques, and as you can see Roger Fletcher’s name comes up in many of them. So I leaned over to one of my classmates and said, “Fletcher sure does play both sides of the street.” Dr. Anderson was irritated at seeing whispering, and made me repeat this to the whole class. When I did he thought for a second and said, “He does play both sides of the street.”
The Capacity Issue
In the paper at hand, the optimisation technique starts with initial values and comes to back-analysed values which are then compared to reference values. The problem here is that the reference values are based on single values of toe capacity and soil parameters, the latter of which are related to static methods of analysis. There are two problems which arise in this approach.
The first is the variability of static methods relative to the actual performance of the deep foundation. This is evidenced by the wide scatter in the results these methods return (it’s not quite as bad with drilled shafts as it is with driven piles, but it’s bad enough.)
The second is that the whole business of the “capacity” of deep foundations–ultimate, allowable or factored–cannot be divorced from the fact that the resistance of piles to load takes place through a distance, or settlement. Capacity doesn’t mean much when it’s decoupled from settlement. For static analysis the Holy Grail needs to be that we can estimate the distribution of pile resistance–both between the shaft and the toe and along the shaft–from static load tests. Using static “capacities” may have made the optimisation method they chose possible to use but it does not really get us to where we need to go. Dynamic testing and methods such as CAPWAP recognise this problem but, as noted earlier, improvements there are possible if not easy to arrive at.
Conclusion
I think this paper is an interesting study as a step towards using optimisation techniques to solve the inverse problem of pile resistance to axial load. But there are many more issues to deal with if we are to come to a workable solution for this problem.
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