## STADYN Wave Equation Program 5: Revision of Soil Properties (Modulus of Elasticity)

This is the first post since the presentation of Warrington and Newman (2018) and it introduces what is probably the most significant revision since the original study: the values of the modulus of elasticity for various values of the dimensionless parameters $\xi$ and $\eta$.

## Quick Overview of the “xi-eta” Concept

The basics of the concept are shown below.

The mathematical implementation is shown here:

The results for this (in this case, for the modulus of elasticity) can be seen here:

The original $\xi-\eta$ value matrix was never intended to be “set in stone.”  We have already made one revision, by using Jaky’s Equation to estimate the at-rest lateral earth pressure coefficient, and thus Poisson’s Ratio $\nu$.

The results for this are outlined in Warrington and Newman (2018).

## Introduction of Hardin and Black Method for Computing Elastic Modulus

Probably the least satisfactory estimate from the original $\xi-\eta$ scheme was the modulus of elasticity.  It was improving this correlation that has led to most of the changes implemented.  The research relative to the TAMWAVE project was used extensively in this revision.

First, the Hardin and Black method for estimated the shear modulus of a soil is described here.  It allows estimating the small-displacement shear modulus knowing the void ratio, effective stress and the lateral earth pressure coefficient.  The last parameter is described above; the void ratio is a function of the specific gravity, the dry unit weight and the saturation state of the soil.  If we add the stratigraphic configuration of the soil, we can estimate the effective stress.  We can thus estimate the shear modulus–and by extension the elastic modulus–of the soil using some very basic soil properties.

We first turn to the dry unit weight.  The TAMWAVE project indicated some revision of how this was computed was in order.  It also indicated that, with some generalisation, the dry unit weight could be stated as strictly a function of $\eta$ from all soils.  Doing this is a good step to simplifying the scheme and thus improving our chances for success in the optimisation process.

Based on the data, the following correlation was proposed for dry density in $\frac{kg}{m^3}$:

The dry density essentially varies from $1200 \frac{kg}{m^3}$ at $\eta = -1$ to $2000 \frac{kg}{m^3}$ at $\eta = 1$.

The specific gravity is unchanged from the original formulation, and is as follows:

Combining the two (we apologise for the mixed graphic formats) yields the following for void ratio:

Because of the nature of the Hardin and Black formula (and of the soils themselves,) the program has an upper limit for the void ratio of 2 and a lower one of 0.  This replaces the previous limitations of modulus of elasticity and density.

At this point we need to consider the effect of hyperbolic softening on the shear and elastic moduli.  This topic was discussed here and, as was the case with TAMWAVE, we selected a ratio of 0.15.  Applying that (and assuming an effective stress of 1 atmosphere) yields the following estimate for elastic moduli in MPa:

In addition to having a stronger basis of fact in the soil properties, by limiting the void ratio the possibility that the correlation goes negative is also avoided.

## Inverse Method for Pile Dynamics Using a Polytope Method: IFCEE 2018

This paper–which is part of the STADYN project–was presented at the IFCEE 2018 conference in Orlando, FL, 7 March 2018. The slide presentation for the paper is below.

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The preprint for this paper can be found at ResearchGate.

## Relating Hyperbolic and Elasto-Plastic Soil Stress-Strain Models

Note: this post has an update to it with a more rigourous and complete treatment here.

It is routine in soil mechanics to attempt to use continuum mechanics/theory of elasticity methods to analyse the stresses and strains/deflections in soil.  We always do this with the caveat that soils are really not linear in their response to stress, be that stress axial, shear or a combination of the two.  In the course of the STADYN project, that fact became apparent when attempting to establish the soil modulus of elasticity.  It is easy to find “typical” values of the modulus of elasticity; applying them to a given situation is another matter altogether.  In this post we will examine this problem from a more theoretical/mathematical side, but one that should vividly illustrate the pitfalls of establishing values of the modulus of elasticity for soils.

Although the non-linear response of soils can be modelled in a number of ways, probably the most accepted method of doing so is to use a hyperbolic model of soil response.  This is illustrated (with an elasto-plastic response superimposed in red) below.

The difficulties of relating the two curves is apparent.  The value E1 is referred to as the “tangent” or “small-strain” modulus of elasticity.  (In this diagram axial modulus is shown; similar curves can be constructed for shear modulus G as well.)  This is commonly used for geophysical methods and in seismic analyses.

As strain/deflection increases, the slope of the curve decreases continuously, and the tangent modulus of elasticity thus varies continuously with deflection.  For larger deflections we frequently resort to a “secant” modulus of elasticity, where we basically draw a line between the origin (usually) and whatever point of strain/deflection we are interested in.

Unfortunately, like its tangent counterpart, the secant modulus varies too.  The question now arises: what stress/strain point do we stop at to determine a secant modulus?  Probably a better question to ask is this: how do we construct an elasto-plastic curve that best fits the hyperbolic one?

One solution mentioned in the original study is that of Nath (1990), who used a hardening model instead of an elastic-purely plastic model.  The difference between the two is illustrated below.

Although this has some merit, the elastic-purely plastic model is well entrenched in the literature.  Moreover the asymptotic nature of the hyperbolic model makes such a correspondence “natural.”

Let us begin by making some changes in variables.  Referring to the first figure,

$y=\frac{\sigma}{\sigma_1-\sigma_3}=\frac{\sigma}{\sigma_0}$

and

$x = \epsilon$

Let us also define a few ratios, thus:

$A_1 = \frac {E_1}{\sigma_0}$

$A_2 = \frac {E_2}{\sigma_0}$

$A = \frac {E_2}{E_1}$

Substituting these into the hyperbolic equation shown above, and doing some algebra, yields

$y=\frac{x A_1}{1+x A_1}$

One way of making the two models “close” to each other is to use a least-squares (2-norm) difference, or at least minimising the 1-norm difference.  To do the latter with equally spaced data points is essentially to minimise the difference (or equate if possible) the integrals of the two, which also equates the strain energy.  This is the approach we will take here.

It is easier to equate the areas between the two curves and the $\sigma_0$ line than to the x-axis.  To do this we need first to rewrite the previous equation as

$y'=1-\frac{x A_1}{1+x A_1}$

Integrating this with respect to $x$ from 0 to some value $x_1$ yields

$A_{hyp} = \frac{ln\left( 1+x_1 A_1 \right)}{A_1}$

Turning to the elastic-plastic model, the area between this “curve” and the maximum stress is simply the triangle area above the elastic region.  Noting that

$E_2 = \frac {\sigma_0} {x_2}$,

employing the dimensionless variables defined above and doing some additional algebra yields the area between the elastic line and the maximum stress, which is

$A_{ep} = \frac {1}{2 A A_1}$

Equation the two areas, we have

$ln\left( 1+x_1 A_1 \right) - \frac {1}{2 A} = 0$

With this equation, we have good news and bad news.

The good news is that we can (or at least think we can) solve explicitly for $A$, the ratio between the elastic modulus needed by elasto-plastic theory and the small-deflections modulus from the hyperbolic model.  The bad news is that we need to know $A_1$, which is the ratio of the small deflections modulus to the limiting stress.  This implies that the limiting stress will be a factor in our ultimate result.  Even worse is that $x_1$ is an input variable, which means that the result will depend upon how far we go with the deflection.

This last point makes sense if we consider the two integrals.  The integral for the elasto-plastic model is bounded; that for the hyperbolic model is not because the stress predicted by the hyperbolic model is asymptotic to the limiting stress, i.e., it never reaches it.  This is a key difference between the two models and illustrates the limitations of both.

Some additional simplification of the equation is possible, however, if we make the substitution

$x_1 = n x_2$

In this case we make the maximum strain/deflection a multiple of the elastic limit strain/deformation of the elasto-plastic model.  Since

$x_2 = \frac {sigma_0}{E_2} = \frac {1}{A_2} = \frac {1}{A A_1}$

we can substitute to yield

$ln\left( 1+\frac{n}{A} \right) - \frac{1}{2A} = 0$

At this point we have a useful expression which is only a function of $n$ and $A$.  The explicit solution to this is difficult; the easier way to do this is numerically.  In this case we skipped making an explicit derivative and use regula falsi to solve for the roots for various cases of $n$.  Although this method is slow, the computational time is really trivial, even for many different values of $n$.  The larger value of $n$, the more deflection we are expecting in the system.

The results of this survey are shown in the graph below.

The lowest values we obtained results for were about $n = \frac{x_1}{x_2} = 0.75$.  When $n = \frac{x_1}{x_2} = 1$, it is the case when the anticipated deflection is approximately equal to the “yield point.”  For this case the ratio between the elasto-plastic modulus and the small-strain hyperbolic modulus is approximately 0.4.  As one would expect, as $n$ increases the elasto-plastic system becomes “softer” and the ratio $A = \frac {E_2}{E_1}$ likewise decreases.  However, as the deflection increases this ratio’s increase is not as great.

To use an illustration, consider pile toe resistance in a typical wave equation analysis.  Consider a pile where the quake ($x_2$) is 0.1″.  Most “traditional” wave equation programs estimate the permanent set per blow to be the maximum movement of the pile toe less the quake.  In the case of 120 BPF–a typical refusal–the set is 0.1″, which when added to the quake yields a total deflection of 0.2″ of a value of $n = 2$.  This implies a value of $A = 0.2139950$.  On the other hand, for 60 BPF, the permanent set is 0.2″, the total movement is 0.3″, and $n = 3$, which implies a value of $A = 0.1713409$.  Cutting the blow count in half again to 30 BPF yields $n = 5$ or $A = 0.1383195$.  Thus, during driving, not only does the plastic deformation increase, the effective stiffness of the toe likewise decreases as well.

Based on all this, we can draw the following conclusions:

1. The ratio between the equivalent elasto-plastic modulus and the small-strain modulus decreases with increasing deflection, as we would expect.
2. As deflections increase, the effect on the the equivalent modulus decreases.
3. Any attempt to estimate the shear or elastic modulus of soils must take into consideration the amount of plastic deformation anticipated during loading.  Use of “typical” values must be tempered by the actual application in question; such values cannot be accepted blindly.
4. The equivalence here is with hyperbolic soil models.  Although the hyperbolic soil model is probably the most accurate model currently in use, it is not universal with all soils.  Some soils exhibit a more definite “yield” point than others; this should be taken into consideration.

## Shaft Friction for Driven Piles in Clay: Alpha or Beta Methods?

In a previous post we discussed beta methods for driven pile shaft friction in sands, which are pretty much accepted, although (as always) the values for $\beta$ can vary from one formulation to the next.  With clays, also as always, things are more complicated.

Since the researches of Tomlinson in the 1950’s, the shaft friction of piles in clays has been thought to be a function of the undrained shear strength of the clay multiplied by an adhesion factor $\alpha$, thus

$f_s = \alpha c_u$

This was seriously challenged by Burland (1973) who noted the following:

Whereas the use of undrained shear strength for calculating the end bearing capacity of a pile appears justified there seems little fundamental justification for relating shaft adhesion to undrained strength for the following reasons:

1. the major shear distortion is confined to a relatively thin zone around the pile shaft (Cooke and Price (1973)).  Drainage either to or from this narrow zone will therefore take place rapidly during loading;

2. the installation of a pile, whether driven or cast-in situ, inevitably must disturb and remould the ground adjacent to the pile shaft;

3. quite apart from the disturbance caused by the pile there is no simple relationship between the undrained strength and drained strength of the ground.

Burland buttressed his case by noting that

$\beta = K tan \phi$

and presenting a graph similar to the following:

where, as seen earlier,

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

Since, for the ranges of drained friction angles for clay (20-25 deg.) the value for $\beta$ was relatively constant, value of $\beta$ were relatively invariant with friction angle, and thus could be estimated with relative accuracy.  His empirical correlation was very successful with soft clays, not as much with stiff ones.

The year after Burland made his proposal, McClelland (1974) noted the following:

It is not surprising that there is a growing dissatisfaction with attempts to solve this problem through correlations of $\alpha$ with $c_u$.  This is accompanied by a growing conviction that pile support in clay is frictional in character–that load transfer is dependent upon the effective lateral pressure acting against the side of the pile after it is driven.

However, $\beta$ methods–which would embody McClelland’s preferred idea–have never been universally accepted for pile shaft friction in clays.  A large part of the problem, as noted by Randolph, Carter and Wroth (1979) is that the lateral pressure itself is dependent upon the undrained shear strength of the soils.  It is thus impossible to completely discount the effect of undrained shear strength on the shaft friction, even with the remoulding Burland and others have noted.

This has led to the “hybrid” approach of considering both undrained shear strength and effective stress.  This is embodied in the American Petroleum Institute (2002) specification.  A more advanced version of this is given in Kolk and van der Velde (1996).  They give the $\alpha$ factor as

$\alpha = 0.9\left( \frac {L-z} {d} \right)^{-0.2} \left( \frac {c_u} {\sigma'_{vo}} \right)^{-0.3} \leq 1$

The notation is the same as in this post except that we add $c_u$, which is the undrained shear strength.

In this case the unit shaft friction is given by the equation

$f_s = 0.9\left( \frac {L-z} {d} \right)^{-0.2} \left( \frac {c_u} {\sigma'_{vo}} \right)^{-0.3} c_u$

The first is that we can transform this into a $\beta$ method of the form

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

with the following multiplication

$f_s = 0.9\left( \frac {L-z} {d} \right)^{-0.2} \left( \frac {c_u} {\sigma'_{vo}} \right)^{0.7} \sigma'_{vo}$

(A similar operation appears in Randolph (2005).)

in which case

$\beta = 0.9\left( \frac {L-z} {d} \right)^{-0.2} \left( \frac {c_u} {\sigma'_{vo}} \right)^{0.7}$

The only thing we would have to do is to find a way to incorporate the limiting condition for $\alpha$, which we will discuss shortly.

The second thing is that the term $\left( \frac {L-z} {d} \right)$ appears in both this formulation and that for sands in this post.  The difference is that, while Kolk and van der Velde (1996) use the term in a power relationship, Randolph (2005) uses it in an exponential way.  The basic concept in both is the same: the term is at a maximum at the pile toe and decays toward the mudline.

The two are compared in the figure below.

Here the quantity $\left( \frac {L-z} {d} \right)$ is at the x-axis and the following is at the y-axis:

• Kolk and van der Velde Method for Clays, $\left( \frac {L-z} {d} \right)^{-0.2}$ in red.
• Randolph Method for Sands, $e^{-\mu \left( \frac {L-z} {d} \right)}$ in blue, where $\mu = 0.05$.
• $e^{-\mu \left( \frac {L-z} {d} \right)}$ in green, where $\mu = 0.02$.

The graph illustrates the problem (from a computational standpoint) with the Kolk and van der Velde method: there is a singularity in their coefficient using the power relationship at the pile toe, while the exponential relationship yields a value of unity at this point.  The last correlation in green is approximately the best fit of the exponential relationship with the power relationship of Kolk and van der Velde, using either 1-norm or 2-norm methods.  It is not very good; it would be interesting, however, to see what kind of value for $\mu$ might result if this had been in Kolk and van der Velde’s original statistical correlation equation.

In view of all this, perhaps the best way to enforce the limit is to do so as follows:

$\left( \frac {L-z} {d} \right)\geq1$

From all this, we can say that it is certainly possible to compute shaft friction for driven piles with a $\beta$ method provided we include the effects of the undrained shear strength.

### References

In addition to the original study and previous posts, the following references are noted:

Kolk, A.J., and van der Velde, A. (1996) “A Reliable Method to Determine Friction
Capacity of Piles Driven into Clays.” Proceedings of the 28th Offshore Technology Conference, Houston, TX, 6-9 May.  OTC 7993.

McClelland, B. (1974) “Design of Deep Penetration Piles for Ocean Structures.”  Journal of the Geotechnical Engineering Division, ASCE, Vol. 111, July.

## 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.