## STADYN Wave Equation Program 7: Revision of Soil Properties (Results)

In the last post we discussed the change in cohesion in the $\xi-\eta$ interpolation.  Now we apply these to the program to see how they change the results.

In the original study there are three test cases, as noted here.  For this stage the first and the third will be considered.  The results for the second (after adjustments for changes in the $\xi-\eta$ interpolation) are little different from before; we will consider major changes to impact that case in our next round.

That leaves the first and third.  For the first–comparison with a static load test–it was necessary to readjust the values for $\eta$ to reflect changes in the meaning of $\eta$ relative to soil properties, as discussed in the last post.  The changes made resulted in layering with the following characteristics:

 Layer Bottom y-coordinate, m $\xi$ $\eta$ 1 5.18 -1 0 2 7.32 -1 0 3 15.2 0.5 0 4 30.5 0.5 0

The static load test results can be seen below.

Comparison with the original study show little improvement in the failure load correlation; however, the load-deflection relationship before failure is significantly improved.  The system was much softer in the original study and the improvement reflects the better estimate of the elastic modulus.  The blow count is very similar to the original study.

Turning to the inverse case that was also discussed in Warrington and Newman (2018), based on the previous results, it was decided to drop consideration of 1-norm results.  The results are reproduced in outline below for the three cases run (refer to Warrington and Newman (2018) for details.)

 $\eta$ Limiting Difference Sum Static Load, kN Average Shaft $\xi$ Average Shaft $\eta$ Toe $\xi$ Toe $\eta$ +-1 0.0034 673 0.084395 -0.46825 -0.992 0.758 +-2 0.00327 449 -0.455 -1.3375 -0.57 -0.727 +-3 0.00143 269 -0.26775 -2.1775 0.113 1.12

1. The difference sum for the highest $\eta$ variation was the best match we have obtained to date for this case; the velocity match is shown below.  The situation around L/c = 1.5 is still difficult but the rest of the correlation is improved.
2. The static load decreases with broader $\eta$ variation and a lower difference sum.  This is different than Warrington and Newman (2018), where the static load “settled down” to a close range of values.
3. The plot above does not reflect that, on the whole, the variation of $\eta$ along the shaft was less than experienced in the past, especially with the +-3 run.  The stratigraphy of the site suggested relatively uniform soil properties along the shaft, and this is beginning to be seen in the inverse results.
4. The +- 3 run was very heavily “toe weighted” in resistance.

While both of these cases show progress, the time has come to consider the whole issue of pile-soil interface issues, which will be considered in our next updates.

## STADYN Wave Equation Program 6: Revision of Soil Properties (Cohesion)

In our last post we discussed the overhaul of STADYN’s $\xi-\eta$ system relative to the modulus of elasticity, which additionally involved revising the way the program estimated dry unit weight and void ratio.  The last is necessary because the modulus of elasticity is estimated using the Hardin and Black formulation.  In this post we will discuss revision of another parameter, namely soil cohesion.

We based the relationship of $\rho$ to $\eta$ based on work for the TAMWAVE program.  It would doubtless be useful to state the relationship between $\eta$ and the consistency/density of the soil, and this is as follows:

 $\eta$ Cohesive Designation Cohesionless Designation -1 Very Soft Very Loose -0.6 Soft Loose -0.2 Medium Medium 0.2 Stiff Dense 0.6 Very Stiff Dense 1 Hard Very Dense

Doing it this way enabled us to have a linear relationship between $\rho$ and $\eta$.  It is too much to expect for the linear relationship to extend to other variables, and this is certainly the case with cohesion.  Unfortunately, a conventional $\xi-\eta$ interpolation dictates such a relationship.  The original $\xi-\eta$ function for cohesion can be seen below, for values of cohesion in kPa.

Note that the relationship between cohesion and $\eta$ is linear for the purely cohesionless state at $\xi = 1$.  If extended past the bounds of the graph for lower values of $\eta$, the cohesion becomes negative.  STADYN prevents this from happening but this essentially deprives soft soils of any cohesion.

Baseon on the TAMWAVE values, for purely cohesive soils the following approximate relationship can be established for cohesion:

$\frac{c}{p_{atm}} = 0.5e^{1.5\eta},\,\xi=1$

where $c$ is the soil cohesion and $p_{atm}$ is the atmospheric pressure.  The left hand side of the equation is the “normalised” cohesion using the atmospheric pressure.  Doing this for parameters such as effective stress makes for an interesting look at soil properties.  The best known use of this is in the SPT correction for overburden.

For cases where $\xi < 1$, the value can be reduced linearly so that $c = 0$ when $\xi = -1$.  The result of all this can be seen in the graph below.

The curve “flattens out” for lower values of $\eta$, so preventing negative values of cohesion is unnecessary.

In our next post we will look at the results when this is applied to the STADYN program.

## 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.

This slideshow requires JavaScript.

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.