## Lateral Earth Pressure Coefficients for Beta Methods in Sands

In our last post we considered some basic concepts behind beta methods for determining beta coefficients for estimating shaft friction for piles in sands.  The idea is that the unit friction along the surface of the pile can be determined at any point by the relationship

$f_s = \beta \sigma'_{vo}$

where $f_s$ is the unit shaft friction, $\sigma'_{vo}$ is the vertical effective stress, and $\beta$ is the ratio of the two, which can be further broken down as follows:

$\beta = K tan \phi$

where $K$ is the lateral earth pressure coefficient and $\phi$ is the internal friction angle of the soil.  Our last post showed that, when compared with empirically determined values of $\beta$, values of $K$ determined from more conventional retaining wall considerations are not adequate to describe the interaction between the shaft of the pile and the soil.

Needless to say, there has been a good deal of research to refine our understanding of this relationship.  Also, needless to say, there is more than one way to express this relationship.  The formulation we will use here is that of Randolph, Dolwin and Beck (1994) and Randolph (2003), and was recently featured in Han, Salgado, Prezzi and Zaheer (2016).  The basic form of the lateral earth pressure equation is as follows:

$K = K_{min} + (K_{max} - K_{min}) e^{-\mu \frac {L-z}{d}}$

Let’s start on the right end of the equation; the exponential term is a way of representing the fact that the maximum shaft friction (with effective stress taken into account) is just above the pile toe and decays above that point to the surface of the soil.  This was first proposed by Edward Heerema (whose company was instrumental in the development of large steam and hydraulic impact hammers) in the early 1980’s.  (For another paper of his relating to the topic, click here.)

In any case the variables in the exponential term are as follows:

• $\mu =$ rate of exponential decay, typically 0.05
• $L =$ embedded length of pile into the soil
• $z =$ distance from soil surface to a given point along the pile shaft.  At the pile toe, $L = z$ and $L-z = 0$, and the exponential term becomes unity.
• $d =$ “diameter” of the pile, more commonly designated as B in American textbooks.

$K_{min}$ is the minimum lateral earth pressure coefficient.  It, according to Randolph, Dolwin and Beck (1994) “can be linked to the active earth pressure coefficient.” Randolph (2003) states that its value lies in the range 0.2-0.4. We stated in our previous post that

$K_a = \frac {1 - sin \phi} {1 + sin \phi}$

How do these two relate?  Although in the last post we produced extensive parametric studies on these, a simpler representation is to compare the active earth pressure coefficient with Jaky’s at-rest coefficient, which is done below.

The at-rest coefficient from Jaky is in blue and the active coefficient from Rankine is in red.  The range of $0.2 < K_a < 0.4$ approximately translates into $25^\circ < \phi < 45^\circ$, which is a wide range for granular soils but reasonable.

That leaves us $K_{max}$.  Randolph, Dolwin and Beck (1994) state that

$K_{max} = S_t N_q$

$N_q$, of course, is the bearing capacity factor at the toe.  It may seem odd to include a toe bearing capacity factor in a shaft equation, but keep in mind that cavity expansion during pile installation begins (literally) with an advancing toe.  Typically $8 < N_q < 40$ depending upon whether the sand is loose (low end) or dense (high end.)  $S_t$ “is the ratio of the radial effective stress acting in the vicinity of the pile tip at shaft failure to the end-bearing capacity.”  Values for $S_t$ vary somewhat but generally centre around 0.02.  This in turn implies that $0.16 < K_max < 0.8$.  Inspection of the complete equation for $K$ shows that, if $L = z$ and the exponential term is at its maximum, $K_min$ cancels out and the range of $K_{max}$ is a range for $K$.

Comparing this result to the graph above, for larger values of $\phi$ these values of $K$ are greater than those given by Jaky’s Equation, which is what we were looking for to start with.  To compute $\beta$, we obviously will need to multiply this by $tan \phi$ (or $tan \delta$).  For, say, $\delta = 35^\circ$, this leads to $\beta_{max} = 0.8 \times tan 35^\circ = 0.56$.  By way of comparison, using Jaky’s Equation for $K, \beta = (1 - sin 35^\circ) tan 35^\circ = 0.30$.

From this we have “broken out” of Burland’s (1973) limitation on $\beta$, which was useful for him (and will be useful to us) for some soils but creates problems with higher values of $\phi$  Although some empirical methods indicate higher values for $\beta$, if we consider variations in $S_t$ and other factors, this differential can be minimised, and in any case this is not a rigourous excercise but a qualitative one.

One thing we should further note–and this is important as we move forward–is that there is more than one way to compute $K_{max}$.  Randolph (2003) states that, when CPT data is available, it can be computed as follows for open-ended piles:

$K_{max} = 0.01 \frac {q_c}{\sigma'_{vo}}$

where $q_c$ is the cone tip resistance.  Randolph (2003) recommends the coefficient be increased to 0.015 for closed-ended piles.  Making generalisations from this formulation is more difficult than the other, but the possibility of using this in conjunction with field data is attractive indeed.

At this point we have a reasonable method of computing $\beta$ coefficients.  However, we still have the issue of clay soils to deal with, and this will be done in a subsequent post.

### References

• Han, F., Prezzi, M., Salgado, R. and Zaheer, M., (2016), “Axial Resistance of Closed-Ended Steel-Pipe Piles Driven in Multilayered Soil“, Journal of Geotechnical and Geoenvironmental Engineering, DOI: 10.1061/(ASCE)GT.1943-5606.0001589.
• Randolph, M., Dolwin, J., Beck, R. 1994, ‘Design of driven piles in sand’, GEOTECHNIQUE, 44, 3, pp. 427-448.
• Randolph, M. 2003, ‘Science and empiricism in pile foundation design’, GEOTECHNIQUE, 53, 10, pp. 847-875.

## A First Look at Estimating Beta Factors for Determining Pile Shaft Capacity of Driven Piles

In the last posting about STADYN, we put forth considerations for interface elements between the pile shaft and the soil.  Before we formally incorporate these into the model (or whether we will incorporate them or not) some consideration of how the interface actually works.  We will start those considerations by looking at methods by which the static capacity of driven piles is computed, and specifically the so-called “beta” methods which are used for cohesionless and sometimes cohesive soils.

Beta methods assume that the shaft resistance of the pile is a function of the effective stress of the soil along the pile shaft.  They assume that the horizontal stress that results from the vertical stress acts perpendicular to the surface of the pile.  The pile surface thus acts like a block on a surface with some kind of Coulombic friction acting against the downward settlement of the pile.  The beta coefficient is the ratio between the vertical effective stress and the horizontal friction on the pile, or

$f_s = \beta \sigma'_o$

$\beta$ is in turn broken down into two components: the lateral earth pressure coefficient, which is the ratio between the horizontal and vertical stresses,

$K = \frac {\sigma_h}{\sigma'_o}$

and the coefficient of friction, or

$\mu = tan \phi$

We put these together to yield

$\beta = K tan \phi$

At this point let’s make two assumptions.  The first is that the lateral earth pressure coefficient is in fact the at-rest lateral earth pressure coefficient.  (For some discussion of this, you can view this slide presentation.)  The second is that the friction angle between the pile and the soil is in fact the same as the soil’s internal friction angle.  If we use Jaky’s formula for the at-rest condition, these assumptions yield

$\beta = \left (1-\sin(\phi)\right )\tan\phi$

The various components of this equation are plotted below.

The three lines are as follows:

• $K_o = 1 - sin \phi$ is in red.
• $tan \phi$ is in blue.
• $\beta$ is in green.

It’s interesting to note that, as $K_o$ increases, $tan \phi$ decreases, and so $\beta$ is within a surprisingly narrow range of values.  This plot is similar to one shown in Burland (1973), which we will discuss later.

If this were the case in practice, estimating $\beta$ would be a straightforward proposition.  We’ll take two examples to show that this is not the case.

Let’s start with the Dennis and Olson Method for cohesionless soils, which is described here.  To arrive at $\beta$ they do the following:

1. They add a depth factor, which we will not consider.  Depth factors and critical lengths are common in static methods, but they are not well documented in the field.
2. They assume $K_o = 0.8$ if their values for friction angle are used.
3. They vary the friction angle from 15-35 degrees depending upon the type of soil.

Leaving out the depth factor, for this method $\beta$ ranges from 0.21 to 0.56.  This is a considerably wider variation than is indicated above.  Since the depth factor is frequently greater than unity, this range is even larger.

An easier way to see this is to consider the method of Fellenius.  His values for $\beta$ are as follows:

• 0.15-0.35 for clay
• 0.25-0.50 for silt
• 0.30-0.90 for sand
• 0.35-0.80 for gravel

Again the range of values is greater than the figure above would indicate.  Why is this?

Although it’s tempting to use a straight empirical approach, let’s back up and consider the structure of the basic equation about and the assumptions behind it.  There are several ways we can alter these equations in an attempt to match field conditions better by considering these assumptions and seeing what changes might be made.

### The Two Friction Angles Aren’t the Same

The first one is suggested by the notation in Dennis and Olson: the internal friction angle of the soil and that of the soil-pile interface are not the same.  Retaining wall theory (when it considers friction) routinely makes this assumption; in fact, the ratio $\frac {\delta}{\phi}$ routinely appears in calculations.  Let us rewrite the equation for $\beta$ as

$\beta = (1 - sin \phi) tan \delta$

and be defining the ratio

$m = \frac {\delta}{\phi}$

we have

$\beta = (1 - sin \phi) tan (m \phi)$

If we plot this in a three-dimensional way, we get the following result.

$\beta$ is the vertical axis; m is varied from 0.25 to 1.75.  The results show that, for a given $\phi$, if we increase m we will increase $\beta$, and this increase is much more pronounced at higher values of $\phi$.

Although it’s certainly possible to have very high values of $\delta = m \phi$, as a practical matter in most cases m < 1.  Nordlund’s Method, for example indicates that m > 1 only with tapered piles, where a tapered pile face induces some compression in the soil in addition to shear.  In any case is m < 1 this will tend to depress values of $\beta$.  We should also note that using a ratio m does not mean that it will be a constant for any given soil.  This is especially true if $\phi = 0$, where a multiplier is meaningless and we should have recourse to an additive term as well.

### Jaky’s Equation Doesn’t Apply, or At-Rest Earth Pressure Conditions Are Not Present

Another assumption that can be challenged is that Jaky’s Equation doesn’t apply, or we don’t have at-rest earth pressure conditions.  Although Jaky’s Equation has done well, it is certainly not the last word on the subject, especially for overconsolidated soils (which we will discuss below.)  To try to “cover our bases” on this, let’s consider a range of lateral earth pressure coefficients by assuming that Jaky’s Equation is valid for the at-rest condition and that we need to somehow vary between some kind of active state and passive state.  The simplest way to do this is to assume Rankine’s conditions with level backfill, which just happens to be identical to Mohr-Coulomb relationships between confining and driving stresses.  (OK, it’s not all luck here…)  Thus,

$K_a = \frac {1-sin\phi}{1+sin\phi}$

and

$K_p = \frac {1+sin\phi}{1-sin\phi}$

Let us also define an active-passive factor called actpas, where actpas = -1 for the active state, 0 for the at-rest state and 1 for the passive state.  We then plot this equation

$\beta = K(\phi,actpas) tan \phi$

below.  Since we only have K values for three values of actpas, we’ll use a little Lagrangian interpolation in an attempt to achieve a smooth transition between the states.

We note from this the following:

1. The dip in $\beta$ for the high values of $\phi$ and  -1 < actpas < 0 (states tending towards the active) may be more a function of the interpolation than the physics.  OTOH, if we look at NAVFAC DM 7.02, Chapter 3, Figure 1, we see a dip between the at-rest and active states for dense sands, which is what we would expect at higher values of $\phi$.
2. Values of $\beta$ for the active case show little variation.  Given that driven piles are subject to cavity expansion during installation, one would expect some passivity in earth pressures.  Drilled shafts are another story; however, if we look, for example, at O’Neill and Reese, values for $\beta$ can certainly range higher than one sees with the active states above.  Bored piles, however, are beyond the scope of this discussion.
3. For low values of $\phi$, there is little variation between the three states.
4. If we compare these values with, say, those of Fellenius or Dennis and Olson, we cannot say that the fully passive state applies for most reasonable values of $\phi$, undrained or drained.  (Values in Nordlund, however, indicate higher values of K for larger displacements, approaching full passivity for large displacement piles.  Another look at this issue is here.)

### Conclusion

If we compare the results we obtain above with empirical methods for determining $\beta$, we see that none of the variations shown above really allows us to match the theory we’ve presented with the empirical methods we’ve described (and others as well.)  As a general rule, $\delta < \phi$ or $m < 1$, so it’s safe to conclude that our assumption that the $K$ can be determined using Jaky’s Equation only results in values of $\beta$ that are too low.

It’s tempting to simply fall back on an empirical value for $\beta$, but for finite element analysis a more refined approach seems appropriate.  In subsequent posts we’ll look at such an approach, along with the issue of applying $\beta$ methods to cohesive soils as well as cohesionless ones.

### References

In addition to those in the original study, the following reference is mentioned here:

• Burland, J.B. (1973) “Shaft friction of piles in clay – A simple fundamental approach.” Ground Engineering 6(3):30-42, January.

## 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.

## 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.

## Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles

This slideshow requires JavaScript.

This doctoral research project at the University of Tennessee at Chattanooga is now complete, and we are pleased to present the following:

• The Dissertation Itself: Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles

### Abstract

This dissertation discusses the development of an improved method for the static and dynamic analysis of driven piles for both forward and inverse solutions. Wave propagation in piles, which is the result of pile head (or toe) impact and the distributed mass and elasticity of the pile, was analysed in two ways: forward (the hammer is modelled and the pile response and capacity for a certain blow count is estimated) or inverse (the force-time and velocity-or displacement-time history from driving data is used to estimate the pile capacity.) The finite element routine developed was a three dimensional model of the hammer, pile and soil system using the Mohr-Coulomb failure criterion, Newmark’s method for the dynamic solution and a modified Newton method for the static solution. Soil properties were aggregated to simplify data entry and analysis. The three-dimensional model allowed for more accurate modelling of the various parts of the system and phenomena that are not well addressed with current one-dimensional methods, including bending effects in the cap and shaft response of tapered piles. Soil layering was flexible and could either follow the grid generation or be manually input. The forward method could either model the hammer explicitly or use a given force-time history, analysing the pile response. The inverse method used an optimization technique to determine the aggregated soil properties of a given layering scheme. In both cases the static axial capacity of the pile was estimated using the same finite element model as the dynamic method and incrementally loaded. The results were then analysed using accepted load test interpretation criteria. The model was run in test cases against current methods to verify its features, one of which was based on actual field data using current techniques for both data acquisition and analysis, with reasonable correlation of the results. The routine was standalone and did not require additional code to use.

### Dynamic Pile Testing Results Crescent Foundation Demonstration Test Pile – Vulcan SC9 Hammer Kenner, Louisiana

#### Brian Mondello and Sean Killingsworth GRL May 2014

This report presents the results from dynamic pile testing, and related data analysis, performed during the initial drive testing of the subject Test Pile on April 30, 2014, at the above referenced jobsite location in Kenner, Louisiana. The primary test objective was the monitoring of the hammer/driving system performance. Additionally, the testing objectives included the monitoring of dynamic pile driving stresses, pile structural integrity, and pile static bearing capacity. These objectives were met by means of a Pile Driving Analyzer® (PDA), Model PAX, which uses the Case Method for numerical computations. An additional analysis was performed on a selected test record using the CAPWAP® computer program. Discussions on the testing equipment, analytical procedures, theory, application, and limitations are presented in Appendix A. Testing and analysis results are presented in Appendix B.

A video of the SC-9 hammer featured in Mondello and Killingsworth:

### Dynamic Response of Footings and Piles

#### Wing Tai Peter To PhD Dissertation University of Manchester February 1985

Dynamic response analyses can be regarded as stress wave propagation problems. The solution of such by the finite element method entails more consideration than static problems, since sources of inaccuracies such as dispersion, spurious oscillations due to mesh gradation, wave reflection at transmitting boundaries, as well as instability or inaccuracy due to temporal operators and discretisation can arise. The criteria for formulating a finite element model for dynamic response analysis have been investigated. Using the relatively simple von Mises soil model (satisfactory for undrained saturated clay) three categories of problems have been investigated:

1. The dynamic response analyses of surface footings subjected to periodic and impact loading have been performed in order to evaluate the finite element model design criteria. An approximate analysis is also performed in reducing a three-dimensional indirect impact problem to a two-dimensional analysis.
2. Vibratory pile driving is a relatively new but somewhat unreliable technique of pile installation. Penetration is instantaneous if conditions are right, but with the high hire charges and uncertainty in success the technique is unpopular, especially in clays. In the work presented it is shown that vibratory installation is possible in cohesive soils at the fundamental frequency for vertical pile translation, if a high enough dynamic oscillatory force is provided. Penetration mechanisms have also been exploited.
3. On the other hand, impact pile driving is reliable and widely adopted in terrestrial as well as offshore construction. Experience in one dimensional wave equation analysis is discussed, and further numerical evaluation of the parameters involved has been carried out by a more elaborate axisymmetric finite element model. In cohesive soils a closed-ended pile may be driven more easily than an equivalent open ended pile, depending on the level of the internal soil column and the soil properties. In the light of the growing popularity of nondestructive determination of the axial load-carrying capacity of piles by dynamic methods, the possibility of correlating the soil resistance mobilised in dynamic conditions to the ultimate static capacity is queried. The semi-empirical Case method has been assessed in detail.