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Some Things I Would Say if Giving the E.A.L. Smith Award Lecture for the Pile Driving Contractors Association

I found it intriguing that the Pile Driving Contractors Association has instituted the E.A.L. Smith Award with lecture following.  It looks like I’ve already made a contribution to the effort: the graphic they used for the LinkedIn announcement probably comes from my piece on E.A.L. Smith and his contribution.

I don’t anticipate actually doing this, but that’s the result of some choices I’ve made along the way.

To begin with, I allowed my technical membership in PDCA to lapse many years ago, although the organisation I work for certainly is a member.  As the musicians say, you can’t sing the blues if you don’t pay your dues, and that’s as true in deep foundations as it is in jazz music.

Beyond that, most people who have been in the driven pile industry for a while know that Vulcan Iron Works passed from the Warrington family in 1996, and that things really didn’t get better for some time thereafter.  My years in the equipment business convinced me that equipment people could not remain uninvolved in the whole business of pile dynamics, which led me to start this site twenty years ago.  Unfortunately that involvement was not accompanied by a sponsoring organisation or budget, so I had to use the emerging internet to do what I felt needed to be done: furnish information on pile dynamics and driven piles, and ultimately geotechnical and marine engineering in general, without paywall or restriction.  That effort has been successful; the award for it is in geotechnical engineers who can get their work done in a better way, students who can learn about this part of the profession, and those in countries which lack the resources to purchase materials.  If I’ve helped them, I’ve succeeded, and that’s award enough.

In any case part of that effort was my piece on E.A.L. Smith and his development of the wave equation program, and I think my piece has been just about the only one on the subject for a long time.  In looking back at the piece and the whole effort behind it, I’d like to make some observations that hopefully with shed some light on Smith’s effort.

The first is that Smith was Raymond’s Chief Mechanical Engineer.  Raymond was an organisation that defined vertical integration: they not only drove the piles, but made or modified the equipment that did the work.  Smith worked during an era when geotechnical engineering was coming into maturity as a science; Terzaghi and Peck’s Soil Mechanics in Engineering Practice was published in 1947.  Nevertheless it took a mechanical engineer to crack the forward problem in pile dynamics.  That’s because civil engineers in general and geotechnical engineers in particular don’t really like or understand things that move, but moving things (like pile driving equipment) are the centrepiece of a mechanical engineer’s work.  That simple fact invited the interdisciplinary approach to the problem that Smith took, but it also has made pile dynamics a “black box” to many of the civil engineers who work with the problem, and that in turn has guided the way the solution of the problem has been implemented.

The second is that the physics of Smith’s wave equation program is a classically mechanical engineer’s solution to a problem.  Spring-dashpot-mass systems are the core building blocks of any vibrating system; Smith basically took this, made the springs elasto-plastic, and strung them together into the system he developed.  It is a tribute to Smith’s ability to see the big picture of the system to relate the parameters of the system to the soil he was driving into, and not permit himself to get lost in the soil mechanics of his geotechnical peers.

The third is that Smith developed his numerical method at a time when numerical simulation of physical systems was itself in its infancy.  He had the feedback of the likes of W.E. Milne and the collaboration of crosstown IBM’s computers and expertise in the development.  It’s also worth noting that civil engineering, although it has pushed forward finite element modeling with people such as T.J.R. Hughes and D. Vaughn Griffiths, is still content with using “classical” methods for many of its designs, especially in the transportation field.  The singularity of Smith’s achievement needs to be seen in the context of both the situation at the time and afterwards.

The fourth is that Smith’s wave equation program was the result of an extended effort that lasted at least a decade and probably more.  That was facilitated by Raymond itself, a large organisation with considerable resources and the ability to test the model with its own work.  It was done at a time when U.S. corporations were more inclined to engage in long-term research projects and to share those results.  The government’s involvement only entered in the wake of Smith’s seminal ASCE paper, and that too was an extended effort.

So Smith’s effort is certainly worthy of celebration and commemoration, and to learn some lessons from it.  Smith’s basic model of the pile has endured to this day, finding application in both forward and inverse solutions of the wave equation for piles.  But is Smith’s model the last word on the subject?  Probably the best answer came from Smith himself.  In his ASCE paper he noted the following, in his discussion of soil mechanics:

When future investigators develop new facts, the mathematical method explained herein can be modified readily to take account of them…

It’s unreasonable to expect that Smith’s model cannot be improved on beyond tweaking the parameters.  And there are fundamental problems: Smith, wise to initially bypass much conventional soil mechanics, developed a model where the relationship between the parameters he used and the properties of the soil he was driving into is not clear.  Solving that problem might, for example, reduce the importance of the sensitivity issue of soil damping on the results of the wave equation.  Efforts have been made to solve the model-soil properties issues but they are neither as widely perfected or implemented as one would like.

That’s a special problem when he consider the inverse implementation of the wave equation for piling.  Use of Smith’s model brings with it uniqueness issues (and there are enough of those with problems involving plasticity like this one) that need to be addressed.

Numerical methods and computer power have both vastly improved since Smith’s day.  So is it possible to see another paradigm shift in the way we perform forward and inverse pile dynamics?  The answer is “yes,” but there are two main obstacles to seeing that dream become a reality.

The first is the nature of our research system.  As noted above, Smith’s achievement was done in a large organisation with considerable resources and the means to make them a reality.  It was also a long-term effort.  Today the piecemeal nature of our research grant system and the organisational disconnect among between universities, contractors and owners incentivises tweaking existing technology and techniques rather than taking bolder, riskier steps with the possible consequence of a dead-end result and a disappointed grant source.

The second is the nature of our standard, code and legal system.  Getting the wave equation accepted in the transportation building community, for example, was an extended process that took longer than developing the program in the first place.  Geotechnical engineering is a traditionally conservative branch of the profession.  Its conservatism is buttressed by our code and standard system (which is also slow-moving) and the punishment meted out by our legal system when things go wrong, even when the mistake was well-intentioned.  Getting a replacement will doubtless be a similar extended process.  And of course we should consider having been “written into the specs.”  Vulcan was certainly the beneficiary of that phenomenon, although the process was driven more by the ubiquity of the product than an effort by the company.    That last point is certainly not the case here; general acceptance would have never taken place had it been so.

However, we need to face the reality that, sooner or later, the ball will move down the field and newer techniques will be developed.  The question in front of us is whether it will be done on these shores, as was the case with Smith, or somewhere else.  As I like to say, it’s our move: we need to make it.

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STADYN Wave Equation Program 4: Eta Limiting, and More on Norm Matching

In our last post we broached the subject of different norm matching methods for the actual and computed velocity-time histories at the pile top. In this post we will go into $\eta$ limiting, while at the same time running both norms to get a better feel for the differences in the results.

Before we begin, one clarification is in order: CAPWAP’s Match Quality and the use of the 1-norm in STADYN are similar in mathematical concept but different in execution. That’s because the Match Quality weights different part of the force-time history (in their case) differently, whereas STADYN goes for a simple minimum sum difference.

One characteristic of the inverse case both in the original study and in the modifications shown in the last post are very large absolute values of $\eta$. These are products of the search routine, but they are not very realistic in terms of characterising the soil around the pile. To illustrate, we bring back up one of the results from the last post, showing the optimisation track using the 2-norm and phi-based Poisson’s Ratio (which will now be the program standard):

Note that the #8 track ($\eta$ for the lowest shaft layer) has a value approaching -30; this is obviously very unrealistic.

In principle, as with $\xi$, the absolute value of $\eta$ should not exceed unity; however, unlike $\xi$ there is no formal reason why this should be the case. But how much should we vary $\eta$? To answer this question, and to continue our investigation of the norm issue, we will examine a matrix of cases as follows:

1. $\eta$ will be run for values of 1, 2, 3 and unlimited (the last has already been done.)
2. Each of these will be run for both the 1-norm and 2-norm matching.

A summary of the results are shown below

 Changed Parameter Difference Static Load, kN Average Shaft $\xi$ Toe $\xi$ Toe $\eta$ Norm 1 2 1 2 1 2 1 2 1 2 |$\eta$| < 1 0.3364 0.003690 811 1490 -0.364 -0.149 -0.62 -0.311 -0.175 0.611 |$\eta$| < 2 0.2381 0.002626 278 223 -0.091 -0.06 -0.588 -0.316 -0.781 -0.0385 |$\eta$| < 3 0.1806 0.001707 172 207 0.324 0.42 -.832 0.823 -1.01 1.45 Unrestricted $\eta$ 0.1344 0.001456 300 218 -0.329 -0.183 -0.491 0.804 8.19 1.52 $\nu = f(\xi,\eta)$ 0.1484 0.001495 278 187 -0.383 -0.53 0.792 0.366 3.116 1.814

To see how this actually looks, consider the runs where |$\eta$| < 3.  We will use the 2-norm results.

The results indicate the following:

1. The average shaft values of $\xi$ tend to be negative.  This is contrary to the cohesive nature of the soils.  The interface issue needs to be revisited.
2. The toe values do not exhibit a consistent pattern.  This is probably due to the fact that they are compensating for changes in values along the shaft.
3. As values of |$\eta$| are allowed to increase, with the 2-norm the result of the simulated static load test become fairly consistent.  This is not the case with the 1-norm.  Although limiting |$\eta$| to unity is too restrictive, it is possible to achieve consistent results without removing all limits on $\eta$.
4. The velocity (actually impedance*velocity) history matching is similar to what we have seen before with the unlimited eta case.
5. The optimisation track starts by exploring the limits of $\eta$, but then “pulls back” to values away from the limits.  This indicates that, while limiting values “within the box,” i.e., the absolute values of $\eta$ < 1, is too restrictive, reasonable results can be obtained with some $\eta$ limiting.

Based on these results, $\eta$ limiting will be incorporated into the program.  The next topic to be considered are changes in the soil properties along the surface of the pile, as was discussed in the last post.

STADYN Wave Equation Program 3: Match Quality vs. Least Squares Analysis

Having broached the subject of Poisson’s Ratio and how it is computed for forward methods, we can turn to how it affects inverse methods.  However, at the same we need to consider an issue that is vital to understanding either this method or methods such as CAPWAP: how the actual pile head signal is matched with the signal the model proposes.  There is more than one method of doing this, and the method currently used by CAPWAP is different than what is widely used in many engineering applications.  Is this difference justified?  First, we need to consider just what we are talking about here, and to do that we need a brief explanation of vector norms.

Vector Norms

A vector is simply a column (or row) of numbers.  We want to compare vectors in a convenient way.  To do this we must aggregate the entries in the vector into a scalar number, and we use what we call norms to accomplish this.  In theory there are an infinite number of ways to do this: according to this reference, there are three types of norms in most common use, they are as follows:

So how do use norms in signal matching? We reduce the force-time (or in our case the velocity-time) history at the pile top after impact into a series of data points, and then for each point of time of each data point we compute the results our proposed model gives us and subtract it from the actual result.  In the equation above each data point is a value $x_j$.  When we have all the differences in hand, we take them and compute a vector of differences, and then in turn take the norm of those differences.  We do this successively by changing parameters until we get a norm value which is the minimum we can reach.  For the STADYN program, we use the $\xi - \eta$ values as parameters and iterate using a polytope method (standard or annealed, for our test case the latter.)

For our purposes the infinity norm can be eliminated up front: in addition to having uniqueness issues (see Santamarina and Fratta (1998), we have enough of those already) it only concerns itself with the single largest difference between the two data sets.  Given the complexities of the signal, this is probably not a good norm to use.

That leaves us with the 1-norm and 2-norm.  To keep things from getting too abstract we should identify these differently, as follows:

1. 1-norm = “Match Quality” for CAPWAP (see Rausche et. al. (2010))
2. 2-norm = Least Squares or Euclidean norm (think about the hypotenuse of a triangle.)  This relates to many methods in statistics and linear algebra, and has a long history in signal matching (Manley (1944).)  This is what was used in the original study.

One thing that should be noted is that the norm we actually use is modified from the above formulae by division of the number of data points.  This is to prevent mishap in the event the time step (and thus the number of data points) changes.  However, for the Mondello and Killingsworth (2014) pile, the wall thickness of the steel section drove the time step, which did not change with soil changes; thus, this division is immaterial as long as it is done every time, which it was.

Application to Test Case

As noted earlier, we will use the four-layer case using the annealed polytope method of matching.  Let us start at the end, so to speak, by showing the static load test data that the program runs with the final configuration:

 Davisson Load, kN Original $\nu$ $\phi$ -based $\nu$ % Change 1-norm 278 300 7.91% 2-norm 187.1 218 16.52% % Change 48.58% 37.61%

The runs were done for both the original Poisson’s Ratio ($\nu$) and that computed using the internal friction angle $\phi$.  The CAPWAP run done on this project recorded a Davisson simulated static load capacity of 146.3 kN.

Changing the way $\nu$ is computed produces larger variations in SRD (soil resistance to driving) for each of the norms than was evident in the last post.  This is because the values of $\nu$ now vary with $\phi$, which overall tends to increase the value of $\nu$ for the same value of $\phi$.  We will discuss this in detail below.

The most dramatic change took place with the norm was changed; the value for SRD is a third to a half higher with the Least Squares solution, depending upon the way $\nu$ is computed.

 xi results Layer Original nu, 1-norm Original nu, 2-norm Phi-based nu, 1-norm Phi-based nu, 2-norm Shaft Layer 1 -0.708 -0.812 -0.686 0.471 Shaft Layer 2 -0.709 -0.751 -0.845 -0.96 Shaft Layer 3 -0.71 -0.984 0.966 -0.439 Shaft Layer 4 -0.586 0.428 -0.75 0.196 Pile Toe -0.69 0.366 -0.491 0.804 Average -0.681 -0.351 -0.361 0.014

The values of $\xi$ (degree of cohesion) tend to decrease for the Match Quality but the opposite for the Least Squares method.  it is interesting to note that the Least Squares $\phi$ based $\nu$ is the only run to venture into predominately cohesive territory ($\xi > 0$, which is interesting in a soil which is generally characterized as cohesive.

 eta results Layer Original nu, 1-norm Original nu, 2-norm Phi-based nu, 1-norm Phi-based nu, 2-norm Shaft Layer 1 -1.71 -0.622 -8.68 -1.08 Shaft Layer 2 -1.62 -1.38 3.29 -0.117 Shaft Layer 3 -0.838 -4.373 -1.86 -5.85 Shaft Layer 4 -1.74 -28.363 -14 -27.5 Pile Toe -1.29 1.814 8.19 1.52 Average -1.440 -6.585 -2.612 -6.605

The values of $\eta$ (consistency or density) are all low, but more so for the Least Squares cases than the Match Quality cases.  Low values of $\eta$ are to be expected in a soil like this, but these tend to be extreme.  Although limiters such as for elastic modulus are included to prevent serious misadventure in the soil properties, the existence of extreme values of $\eta$ is something that needs to be re-examined.  ($-1 < \xi < 1$ by physical necessity.)

 Poisson’s Ratio Result Layer Original nu, 1-norm Original nu, 2-norm Phi-based nu, 1-norm Phi-based nu, 2-norm Shaft Layer 1 0.279 0.269 0.45 0.45 Shaft Layer 2 0.279 0.275 0.158 0.312 Shaft Layer 3 0.279 0.252 0.45 0.45 Shaft Layer 4 0.291 0.393 0.45 0.45 Pile Toe 0.281 0.387 0 0.45 Average 0.282 0.315 0.302 0.422

As was the $\eta$ values, the values of $\nu$ tend to increase with the $\phi$ based values.  The Match Quality $\phi$-based values are highly irregular, which in turn reflect the wide swings in $\eta$ with less cohesive values of $\xi$.

Now let us present the optimization tracks for each of these cases.

The original study discusses the numbering system for the xi and eta parameters.  In short, tracks 1-6 are for the shaft and 7-8 are for the toe.  From these we can say the following:

1. The Match Quality runs tend to converge to a solution more quickly. The x-axis is the number of steps to a solution.
2. The Match Quality run tended to eta values that were more “spread out” while the Least Squares solution tended to have one or two outliers in the group.
3. The runs go on too long.  This is because, in the interest of getting a working solution, the priority of stopping the run at a convergence was not high.  This needs to be addressed.

Now the norms themselves should be examined as follows:

 Final Norm Original Nu Phi-Based Nu % Change 1-norm 0.148395682775873 0.134369614266467 -9.45% 2-norm 0.001494522212204 0.001456397402301 -2.55%

In both cases the difference norms decreased with the $\phi$ -based $nu$, the Match Quality difference was more pronounced.  The difference norm for the Match Quality is higher than the Least Squares solution, which is to be expected.

We finally look at the tracks compared with each other for the four cases.

It’s tempting to say that the Match Quality results “track more closely” but the whole idea of using a norm such as this is to reduce the subjective part of the analysis.  However, this brings us to look at why one norm or the other is used.

The Least Squares analysis is widely used in analyses such as this.  It is the basis for almost all regression analysis.  However, the Match Quality has some advantages.  It is considered more “robust” in that it is less sensitive to outliers in the data.  In this case, the most significant outlier is the region around L/c = 1.5, which was discussed in the original study.  Situations such as this reveal two sources of uncertainty in the model: the integrity of the mounting of the instrumentation, and the accuracy of the pile data (lengths, sizes, acoustic speed of the wood, etc.) The Match Quality certainly can help to overcome deficiencies caused by this and other factors.  Whether this is at the expense of accuracy has yet to be determined.

So we are left with two questions:

1. If we were to improve the quality of the data by addressing the present and other issues, would we be better off if we used Least Squares?  The answer is probably yes.  Getting this in the field on a consistent basis is another matter altogether.
2. Will the two methods yield different results?  With STADYN this is certainly the case; the use of the Match Quality with STADYN however yields results that are double those of CAPWAP.  With CAPWAP we have no way of comparing the two; the Match Quality is all we have.

Conclusions

Based on all of this we conclude the following:

1. The use of a $\phi$ based $\nu$ leads to an improvement in the signal matching, due probably to the reduction in the number of real parameters being considered.  It will probably remain as the default option.
2. Any final conclusions on this topic depend upon limiting the values of $\eta$ “within the box” to prevent serious outliers.  This will be the topic of future study.
3. We also need to address the issue of stopping the runs at a more appropriate point.
4. The results for $\xi$ bring up again the question of the soil properties at the soil-pile interface vs. those in the soil body.  We will discuss this in a later post.

References

Other than those in the original study, the following work was cited:

• Santamarina, J.C., and Fratta, D. (1998) Introduction to Discrete Signals and Inverse Problems in Civil Engineering.  ASCE Press, Reston, VA.
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Celebrating Twenty Years of vulcanhammer.net

It’s official: twenty years ago today, this website had its beginning.  That’s a long time on the internet, and there have been many changes.

Ten years ago I commemorated that anniversary starting with this:

Ten years ago today, I went online, logged onto my new GeoCities site, and uploaded the first page and images of “The Wave Equation Page for Piling,” my first website.  That website—which is still a part of the companion site vulcanhammer.net—was the beginning of a long odyssey which led to the site as it is today.

You can read about the site’s first decade in that post.  The purpose of the site is unchanged, so it’s time to bring you up to date on our progress.

The first big change took place a few months after that post when vulcanhammer.info was split off from vulcanhammer.net.  The basic idea was to give the Vulcan Iron Works material its own site.  Later the driven pile material was moved there also, to feature it separately.  Perhaps that site’s history can be featured later.

The second was the growth of our printed materials at pz27.net.  This site has always been about free stuff and continues to offer everything that way.  But many want printed books for one reason or another, and so many of the publications offered on this and the companion sites are now available at pz27.net.  The most popular of these have been NAVFAC DM 7.01 and 7.02; putting these back into print and make them available to the geotechnical engineering community has been well received and popular.  For a while we also offered CD-ROM compilations of our documents, but these fell out of favour with increasing bandwith; by the time our publisher discontinued offering optical media, they had stopped selling.  Even with this, the revenue from these sales continues to underwrite the hosting and domain expenses of this site.

That brings us into the early years of this decade.  Although updates and additions to the material available on this site have been ongoing, in 2011 I began the pursuit of my PhD and, to be frank, the site’s progress stalled a bit during those years.  But my MS pursuit was part of the genesis of this site, and the spinoff from the latest effort can be seen, from the page on finite element analysis in geotechnical engineering to the ongoing series on the STADYN wave equation program.  But not all slowed down: I continued to teach at the University of Tennessee at Chattanooga, which meant that the course materials section of the site continued to grow with each semester.

And that leads us to the most recent major change in the site: in January of this year the site was moved to the WordPress platform. The reasons for this are discussed here (along with the change in the marine documents) in the inaugural post.  The result has been a site with interactivity, both on the site and with social media (the vulcanhammer.net Facebook group is still active.)  It is also secure (as has been the case with Positive Infinity for a long time) and adaptive to mobile devices, both of which enhance the site’s search engine draw.  And finally there is evidence that the documents on the site download more quickly, which is the whole point of the site.

It’s easy to say that this site has pretty much accomplished what it set out to do: to provide geotechnical information in an affordable format to those which many not have the resources to purchase them, both in our universities (which keep getting more expensive) and in countries around the world.  It is true that now there are many sites that offer information such as this, including obviously the U.S. government sites where most of this information came from to start with (although its presence there comes and goes.)  But we still claim to offer it with the fewest strings attached, and that’s saying something.

So once again we thank you for your visiting this site and your support, and may God richly bless you.

STADYN Wave Equation Program 2: Effective Stress, At-Rest Lateral Earth Pressures and Poisson’s Ratio

With the output improved, we can turn to the first topic of interest. Before we do that, we need to discuss our test cases.

Test Cases

The original study featured several test cases. For this and subsequent installments, we will concentrate on three of those:

1. FINNO2, which features the actual static load test output from Finno (1989).
2. SEASIA, which is a GRLWEAP comparison from a notional offshore pile case in Southeast Asia.
3. MANDK3, which features the inverse solution of an instrumented pile in the New Orleans area. The original GRL report for this is Mondello and Killingsworth (2014). Several soil profiles were analyzed using both standard and annealed polytope methods of analysis. For this study the four-layer annealed polytope case will be featured, as its results a) seemed to be the most reasonable and b) matched the standard results very closely.

Details of the original results are shown in the original study.

Effective Stresses, Vertical and Horizontal

The concept of effective stress is a foundational one in geotechnical engineering, and is discussed in textbooks such as Fellenius (2015) and Verruijt and van Bars (2007). As is the case with “classical” methods of analysis, it is necessary to compute these for successful geotechnical finite element analysis. However, there are two important considerations that come up with finite element analysis that can usually be ignored with simpler methods.

The first is that it is necessary to apply gravity forces at the start of the run to the elements to simulate the impact of effective stresses on the soil finite elements. This is one of those important steps in analysis that most manuals and discussions of commercial codes mention in passing but do not detail how they are done. It is one of those phenomena that has “gone dark” in the literature. The original study presented an outline on the procedure for computing the effective stresses and applying them to the elements.

The second is that the computation of effective stresses concentrate on the vertical stresses and generally ignore the horizontal ones until retaining walls come into play. Nevertheless, for any three-dimensional continuum such as the semi-infinite soil mass we assume in geotechnical analysis, horizontal stresses are guaranteed to exist, if nothing else via the theory of elasticity. If we use the theory of elasticity, which is valid in an elastic-purely plastic model such as is used in STADYN until the yield point is exceeded, the relationship between the horizontal and vertical stresses is given by the equation (Verruijt and van Bars (2007))

$\frac{\sigma_{x}}{\sigma_{z}}=\frac{\nu}{1-\nu}\$ (1)

where $\nu$ is Poisson’s Ratio, $\sigma_z$ is the vertical stress, and $\sigma_x$ is the horizontal stress.  We normally define the left hand side thus:

$K = \frac{\sigma_{x}}{\sigma_{z}}$ (2)

For our case, the lateral earth pressure coefficient $K$ is the at-rest lateral earth pressure coefficient, generally expressed as $K_o$. This is reasonable for this case because, since we have a semi-infinite soil mass, the soil literally has nowhere to go, thus all of the horizontal strains are zero. This is a key assumption for Equation 1. We can thus combine Equations 1 and 2 to yield

$K_o = \frac{\nu}{1-\nu}\$ (3)

In theory, we could compute the lateral earth pressure coefficient using Equation 3 and “reasonable” values of Poisson’s Ratio.

Turning back to STADYN itself, soil properties in most cases (and especially for inverse problems) are defined using the “ξ – η” system, which in turn uses typical values of various soil properties to reduce computing same for a given typical soil state to two dimensionless variables. Using this system, Poisson’s Ratio is a function of ξ and η, and is thus varied as these dimensionless parameters are varied. The variation of ν with ξ and η is shown in the original study.  As a practical matter, even if Poisson’s Ratio is measured for each project and soil profile (an unlikely situation at best,) the problematic nature of soil elasticity makes accuracy of the parameter equally problematic. Another approach is to begin by considering the following empirical relationship

$K_o = 1 - sin(\phi)$ (4)

This is Jaky’s Equation. It has been shown to be reasonable for normally consolidated soils, although there are other relations in use for both normally and overconsolidated soils. Values of the at-rest lateral earth pressure coefficients are limited to $0\leq K_{o}\leq1$, the upper limit achieved for a purely cohesive soil where $\phi = 0$. As the original study noted, Equation 4 is a common expression to compute horizontal stresses from vertical effective stresses in finite element codes, and is used to compute the horizontal effective stresses in STADYN.

Unfortunately this leaves an inconsistency between the way horizontal stresse sare computed between the effective stress computation and subsequent computations. To remedy this problem, we can combine Equations 3 and 4 and solve for Poisson’s Ratio to yield

$latex \nu=\frac{sin\phi-1}{sin\phi-2}\$ (5)

Poisson’s Ratio is varied here as $0\leq\nu\leq0.5$, where once again the upper limit is for purely cohesive soils. This indicates that these soils act as a fluid, which is nearly true for very soft clays. The main problem with this result is that, when ν = 0.5, the consitutive matrix experiences singularities. The simplest way to deal with this problem is to limit Poisson’s Ratio to a value below this one. In STADYN this value is 0.45.

Comparison With Previously Generated Values and Forward Test Cases

Having defined a new way of generating values of Poisson’s Ratio, we can compare these values both with the original values and with the two forward test cases. We will leave the inverse test cases for a later post.First, the original “ξ – η” relationship to generate Poisson’s Ratio values is shown in Figure 1.

Figure 1: Poisson’s Ratio “ξ – η” Relationship, Original Configuration

We can see that Poisson’s Ratio is independent of η in this configuration,and
$0.25\leq\nu\leq0.45$ for $-1\leq\xi\leq1$. Computing Poisson’s Ratio based on Equation 5 yields the result shown in Figure 2.

Figure 2 Poisson’s Ratio “ξ – η” Relationship, Jaky’s Equation

There are several differences to note, as follows:

1. The maximum value is ν = 0.5 for the revised relationship. To prevent singularities in the constitutive matrix, in actual application Poisson’s Ratio is limited as described earlier, a similar concept to the “corner cutting” for Mohr-Coulomb failure.
2. For purely cohesive soils, ν is invariant in both cases. As ξ is reduced and internal friction is increased, ν varies with η. In other words, Poisson’s Ratio tends to decrease in cohesionless soils as the relative density of the soil increases.
3. The range of possible values for Poisson’s Ratio in both cases is very much the same; it is simply distributed differently in the continuum.

As far as the forward test cases (the first two) are concerned, SEASIA is the same in both cases because the soil is assumed to be purely cohesive, thus Poisson’s Ratio is the same in both cases.

For FINNO2, the simplest way to compare the two is to compare the hammer blow counts and the Davisson static load test result. That comparison is as follows:

• Original Poisson’s Ratio Computation: Blow count 17.7 blows/300 mm, Davisson failure load 976 kN.
• Revised Poisson’s Ratio Computation: Blow count 17.8 blows/300 mm, Davisson failure load 980 kN.

The differences for this case are not that substantial. The differences which emerge in the inverse case will be discussed in a subsequent post.

One other change that was made in the program was the stopping point for the static load test. The program is capable of interpreting the static load test for several criteria; however, how long the static load test is conducted (in the computer or in the field) depends upon the criteria being used to interpret it. The program now stops the test depending upon when the selected criterion is reached; Davisson’s criterion is the default. It is also interesting to note that, since the Jacobian is fixed, Davisson’s criterion, which generally stops before the others, is probably more suitable for STADYN’s current algorithm.

References are given in the original study.