Back in 1936, Lazarus White (of Spencer, White and Prentis fame) wrote the following about the dynamic formulae:

I read some papers last night where some of these pile driving formulas were derived, and the result was that my sleep was very much disturbed.

While there are many reasons to lose sleep while writing a Ph.D. dissertation, dynamic formulae aren’t among them. However, in the course of researching, coding, debugging, and writing what ended up as Warrington (2016), it became clear that there are things that induce sleep loss in geotechnical engineering in general and pile dynamics in particular.

As the title indicates, this is intended to be a “lessons learned” type of presentation. Given the manifold uncertainties that accompany any geotechnical engineering topic, the dual temptations to either completely discard any past experience or to dogmatically defend the current “state of the art” should be avoided. Given the environment we work in, coming to “definitive” solutions is not an easy task. The industry’s/profession’s innate conservatism, especially in codifying methods and materials, is well founded. On the other hand, there comes a point where serious issues “out there” need to be discussed and new (or at least different) solutions need to be considered.

The current state of the technology of pile dynamics, which in practical terms begins with Smith (1960) and Rausche (1970), has advanced not only the design and installation of driven piles but, though high-speed dynamic testing applied to bored piles and pile integrity testing, deep foundations in general. Since both (and the improvements that have come since then) are numerical methods with computer implementation, to say that computer power has advanced in the intervening years states the obvious. Numerical methods that existed even then, such as those documented in Carnahan, Luther and Wilkes (1969) can be brought to bear on the problem in new ways. Plasticity in finite element analysis is well established, documented in works such as Owen and Hinton (1980). Optimization techniques such as those described in Gill, Murray and Wright (1981) can be used for signal matching techniques, which are crucial in determining the soil resistance to driving (SRD) from dynamic pile data.

So without further introduction, on to the lessons learned.

### We Have Gone As Far with the Smith Model as We Can Go

Given the state of soil dynamics when Smith (1960) was published, it is remarkable that his model has survived and continued (with modifications) as the “standard” model for soil response in the one-dimensional wave equation for piles. It has been tested (although really meaningful testing is difficult) and has been used in many applications, both in forward (predictive) and inverse (interpretive) application of the wave equation to piles. It is relatively simple to implement and the quantities of quake and damping have been extensively correlated.

However, any numerical model of a physical phenomenon has weaknesses, and the Smith model is no exception. Some of these are as follows:

There is no really accepted correlation between the soil properties as determined using standard testing procedures and the Smith parameters. This is especially critical with the soil damping, which has been shown to be one of the most critical properties to properly quantify in a wave equation analysis (Meseck (1985)).

The Smith model in the forward mode virtually requires the use of static methods to determine the SRD, which have numerous problems of their own. (Their proliferation is reminiscent of that of the dynamic formulae.) An additional complication is the diffuse state of static load test interpretation. While most dynamic methods in the U.S. presume the use of Davisson’s Method, this method is not universal in application.

The nature of Smith damping has never been completely enunciated. When velocity-based damping is used in the modelling of a physical system, generally speaking it refers to some kind of viscous friction, such as in a “dashpot.” Most of the “damping” in a dynamic pile system is the radiation of energy into the distributed mass and elasticity (and plasticity) of the soil surrounding the pile.

There is no good way to integrate many phenomena experienced in pile driving into the Smith model. Perhaps the most important of these is pile set-up, which is currently the “hot” research topic. In most cases the goal of these studies is to determine the ratio of the SRD (used in the wave equation analysis) to the ultimate static capacity of the pile. What is needed is a method to predict pile set-up, principally through the computation of elevated pore water pressures.

The Smith model’s neat division of soil resistance into shaft and toe does not reflect the reality of many driven piles, especially tapered piles, piles with plugging and piles with non-planar toes.

There are many “add-ons” that we use in the application of the Smith model (and other purely 1D wave equation analyses. But many of these are just that: add-ons to a model that in itself does not take into account many of the phenomena experienced in pile driving. Like Vergil in the Divine Comedy, the Smith model has been a faithful guide, but now we must move on if we want to really advance the science of pile dynamics.

### Mohr-Coulomb Isn’t Perfect, But For The Moment It’s The Best We Have

If you realize that I’m advocating for a “3D” model of the hammer-pile-soil system, you’re right. (If we stick with round, axisymmetric piles, we can use a 2D model.) But that raises as many questions as it answers. Some of these relate to the hammer, including the gas pressures in diesel hammers, the modelling of intextensible places such as the hammer-cushion, hammer-cap or cap-pile interfaces, and cushion plasticity. But the key question, as always, relates to the soil. What soil model will we use? How will we take into consideration the inherently non-linear nature of the soil? How will we model the inevitable plasticity? What soil properties will we use?

At this point, in spite of its limitations, the Mohr-Coulomb model, familiar to just about everyone in the geotechnical industry, is probably the best overall soil model to use in pile dynamics. It certainly has its limitations; it is, in some ways, an attempt to render an elasto-plastic simplification of a hyperbolic (Duncan and Chang (1970)) soil response. In that respect it justifies the characterization of Massarsch (1983) as “crude.” But it remains overall the best model to use (Abbo et.al. (2011); McCarron (2013)) for the following reasons:

It is suitable for a wide variety of soils, and piles are driven into diverse stratigraphies, both from one site to the next and along the length of the pile.

The current standard testing methodology is “designed” around Mohr-Coulomb. If we are serious about addressing the lack of correlation between properties used in pile dynamics and soil properties, we need to be able to test for these properties. Until we revise our standard testing methodologies, this limitation will stand.

It is well represented in finite element codes that are used for geotechnical analysis. Warrington (2016) represented the first application of a standalone finite element code for pile dynamics in over twenty years. There are advantages and disadvantages in both approaches, but implementation of Mohr-Coulomb in finite element code is well established.

For these and other reasons, Mohr-Coulomb remains the best model for failure and plasticity that we have available. Like everything else, this is not written in stone; but until other things advance in the industry, this is the best we have.

### We Need To Understand Why Reducing Pile Head Data Is So Difficult

The ability to estimate static pile response from high-velocity dynamic data is the cornerstone of pile dynamics. Although it is “intuitively obvious” (a favourite phrase of academics,) quantifying it has been something of a challenge. Today methods in use for doing this estimation are basically signal-matching techniques. The concept is to have the model return the same force-time and velocity-time histories as come from the field. In processing the data, we compare the two “point by point;” if the sum of the differences (or the square of the differences) goes below some kind of tolerance criterion, we say the signals match and the model replicates the physical system.

Although the results have been reasonable problems remain, requiring a great deal of human intervention. So why is this problem so difficult? The answer, in terms of optimization techniques, is the presence of multiple local minima of the aforementioned “sums.” This is a “fancy” term that needs some explanation.

Optimization techniques attempt to find a minimum (or maximum) value for a function of several variables. Consider the example of the famous Rosenbrock Function, given (in a typical form) by the equation

R(x,y) = (1-x)^{2} + 100(y-x^{2})^{2}

A plot of this is shown below.

This function is often used to test optimization techniques, by starting at a point which away from the known minimum and checking to see how well the technique converges on the actual minimum, which in this case is . Although the function is difficult, it is reasonably “regular.”

Now consider a topographical map of a hilly region, or better the soil profile for a Karst topography. Now there are many local minima; it is easy for an optimization technique looking for the “bottom” of the valley without realizing that there is yet another valley over the hill that is actually lower! Using an automated technique, especially one based on Newton’s Method, can easily get trapped in such a minimum, thinking it has found the solution to the problem when in fact it has not.

Getting to the real minimum has been a central frustration in the development of signal matching techniques for pile dynamics, because the inverse problem in pile dynamics, like the ground the piles are driven into, has many local minima. This problem has impeded the automation of the search process more than any other. The increase in computer power available to engineers is an opportunity to “get to the bottom” of the problem, although in doing so a consistently successful methodology is necessary.

### We Need to Instrument Piles in More Places than the Pile Head

The earliest instrumented pile study was that of Glanville et.al. (1938), and the piles were instrumented at the head, middle and toe. Jorg Osterberg showed us how to test the piles with a load cell at the toe, and Fellenius (2014) has advocated instrumenting piles at locations other than the head for static testing. And yet today, eighty years after it was first done, we generally still instrument the pile head only for both static and dynamic testing. The need to do so for dynamic testing is, if anything, more urgent than with static testing, and with the expendable instrumentation we have available it should not be that big of a problem. Doing so would give us a more complete picture of what’s really going on during driving and make the search for an SRD much simpler.

### The Advance of Numerical Methods Should Occasion Some Changes in Geotechnical Education

At this point we turn away somewhat from the problem at hand and look at geotechnical education in general. If we take a serious look at our basic geotechnical courses, much of the material we have is little different than that which was published in Terzaghi and Peck (1948). While we cannot denigrate those who went before us just because we have found out something new, nevertheless the environment we operate in is vastly different from that which those two giants strode in a half century and more ago. The biggest change is the introduction and proliferation of numerical methods.

Many geotechnical engineers are wary of the advance of these methods. Some will tell you that the results obtained from hand calculations based on experience are no worse (and many cases better) than those from numerical methods. Often this is correct. Part of the problem is that the numerical methods are based on the same theory as the closed form solutions. The advantage of numerical methods is not necessarily in underlying theory but in the ability to better simulate the physical system through geometric replication. But another part of the problem is that most numerical methods, presented in packages, are “black boxes” to many practitioners, who frequently do not have a complete grasp of their inner workings. The temptation is thus great to receive the results uncritically, without support from more conventional methods or experience. It is thus incumbent upon geotechnical educators, starting at the undergraduate level, to give students some understanding of how these methods work, and also in their strengths and weaknesses, so that they can form an intelligent opinion of the results.

The biggest difference between the way geotechnical professionals have employed theory in the past and the way numerical methods handle it is that, in the first case, elasticity and plasticity are handled separately, while in the second the two are simulated in the same model, which alternates from one to the other depending upon the system modelled and the loads placed upon it. For example, in settlement problems we frequently consider elastic settlement separate from consolidation settlement (primary and secondary,) which in turn is separate from bearing capacity. In physical reality all of these are taking place at the same time, and the model that can accurately simulate this is of value.

It is for this reason that this educator has shifted towards a text such as Verruijt and van Bars (2007). Such an approach emphasizes more of the “continuum mechanics” of soil mechanics than is customarily done. This is arguably a more “theoretical” approach to soil mechanics. It is not a complete solution to the problem, but it is a start.

### Conclusion

These are some observations and proposals derived from a long (three year) study on the subject of pile dynamics. Geotechnical engineering tends to be a conservative field of study and practice, but we must not let that impede real progress towards foundations that are more economical and reliable than before. Ultimately numerical methods in general and finite element analysis in particular will advance in usage; if we implement them intelligently and educate our new engineers in their inner workings, the profession will advance.

If we can achieve that goal, we can all sleep better at night.

### References

- Abbo, A.J., Lyamin, A.V., Sloan, S.W. and Hambleton, J.P. (2011). “A C2 continuous approximation to the Mohr-Coulomb yield surface.” International Journal of Solids and Structures, 48(21), 3001-3010.
- Carnahan, B., Luther, H.A. and Wilkes, J.O. (1969).
*Applied Numerical Methods*, John Wiley & Sons, Inc., New York, NY - Duncan, J.M., and Chang, C.Y. (1970). “Nonlinear Analysis of Stress and Strain in Soils.” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, 96(5), 1629-1653.
- Fellenius, B.H. (2014)
*Basics of Foundation Design*. http://www.fellenius.net (March 13, 2015) - Gill, P.E., Murray, W. and Wright, M.H. (1981). Practical Optimization, Academic Press, Inc., London, England.
- Glanville, W.H., Grime, G., Fox, E.N, and Davies, W.W (1938). An Investigation of the Stresses in Reinforced Concrete Piles During Driving, Department of Scientific and Industrial Research, British Building Research Board, London, England.
- Massarsch, K. R. (1983). “Vibration Problems in Soft Soils.” Proceedings of the Symposium on Recent Developments in Laboratory and Field Tests and Analysis of Geotechnical Problems, Asian Institute of Technology, A. A. Balkema, Rotterdam, 539-549.
- McCarron, W.O. (2013). “Numerical Modeling Applications in Offshore Petroleum Developments.” Geo-Strata, 17(3), 34-39.
- Meseck, H. (1985). “Application of a Wave Equation Programme to Establish the Bearing Capacity of Driven Piles.” Proceedings of the International Symposium on Penetrability and Drivability of Piles, Volume 2, Japanese Society of Soil Mechanics and Foundation Engineering, 84-90.
- Owen, D.R.J., and Hinton, E. (1980).
*Finite Elements in Plasticity: Theory and Practice.*Pineridge Press, Swansea, Wales. - Rausche, F. (1970).
*Soil Response from Dynamic Analysis and Measurements on Piles.*Ph.D. Dissertation, Case Western Reserve University, Cleveland, OH. - Smith, E.A.L. (1960). “Pile-Driving Analysis by the Wave Equation.” Journal of the Soil Mechanics and Foundations Division, 127(1) 35-61.
- Terzaghi, K., and Peck, R.B. (1948)
*Soil Mechanics in Engineering Practice*, McGraw-Hill Company, New York, NY. - Verruijt, A., and van Bars, S. (2007).
*Soil Mechanics*. VSSD, Delft, the Netherlands. - Warrington, D.C. (2016).
*Improved Methods For Forward And Inverse Solution Of The Wave Equation For Piles*. Ph.D. Dissertation, University of Tennessee at Chattanooga, Chattanooga, TN.