## 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.

Posted in Soil Mechanics

## More Uses for p-q Diagrams

In our last post on p-q diagrams we discussed their basic concept and application.  In this post we’ll expand on that for two applications: using it to estimate the friction angle and cohesion for multiple triaxial tests, and using it to plot the failure function.

## Processing Triaxial Test Results

The process of determining internal friction angle and cohesion from successive triaxial tests (i.e., those where the confining stress is successively increased) is well known.  In the case of two tests, using the standard $\sigma - \tau$ diagram, the tangent line between the two circles is unique (well, there are two of them, but the slopes and intercepts have opposite signs) as shown below.

If we use the p-q diagram, as we saw earlier, the process is even simpler, as two points have a unique line between them.

But what happens with three tests?  Mathematically there is no guarantee of a unique line, and given the nature of geotechnical testing it is the extraordinary lab which could hit such as result.  It’s also possible that the failure envelope is non-linear, as shown below.

So is there a way to at least get a decent approximation without guesswork or graphics skills?  The answer is “yes” and it involves using p-q diagrams in conjunction with a spreadsheet.  The mathematical concept behind this is here and we have an example to show how it is done.  The problem is taken from Tchebotarioff’s (1951) classic soil mechanics test.  The results of three triaxial tests are as follows, the failure stresses are in tsf:

 Test $\sigma_3$ $\sigma_1$ p q 1 0.2 0.82 0.51 0.31 2 0.4 1.6 1 0.6 3 0.6 2.44 1.52 0.92

We’ve taken the liberty of computing the p and q values for each test.  Now we can plot these in our spreadsheet.

We’ve also taken the liberty to use the spreadsheet’s “trend line” feature to plot a linear “curve fit” for the points.  The slope of the equation $m=tan \delta = sin \phi = 0.6041$, which yields both $\delta = 31.1^\circ$ and $\phi = 37.2^\circ$.  For the intercept $b = c\sqrt {1-\left (\tan(\delta)\right )^{2}} = 0.0001$, which means we can solve for the cohesion, but in this case the quantity is so small it’s probably best to assume that the cohesion is zero.

The $R^2$ value for this problem is very high, so the correlation is good.  We can use this parameter to determine whether we have a good correlation or not.  We can also use least-squares trend line analysis for non-linear failure envelopes, although when we consider the “kink” caused by preconsolidation this may not be as meaningful as one would like.

## Plotting the Failure Function

As mentioned earlier, the Mohr-Coulomb failure function is define in this way:

A little math transforms this into

$f=2\,q-2\,c\sqrt {1-\left (\tan(\delta)\right )^{2}}-2\,p\tan(\delta)$

or

$f=2\,q-2\,c\sqrt {1-\left (\sin(\phi)\right )^{2}}-2\,p\sin(\phi)$

Since $\delta$ and $c$ are known, this suggests that we can plot the failure function three-dimensionally.  Consider the case where $\delta = \frac{\pi}{8}$ and $c = 5$.  The p-q diagram for the failure envelope $f = 0$ is shown below.

If we plot the failure function three-dimensionally, we obtain this result:

The failure envelope of the previous diagram is the contour line which stops at the q-axis at around q = 4.6.  Values below this line are negative and values above it are positive.  Positive values of $f$ indicate failure and an illegal stress state.  The failure function is used extensively in finite element analyses like this one.

Posted in TAMWAVE

## TAMWAVE: Cavity Expansion Theory and Soil Set-Up

One of the things that was attempted in the TAMWAVE project is the use of cavity expansion theory to estimate soil set-up in cohesive soils.  Doing this, however, brought some complications that need some explanation.

Cavity expansion theory is basically the study of what happens when one body expands inside of another.  When this takes places, additional radial stresses (most analyses center around a cylinder or sphere) are generated.  In the case of driven piles, these additional stresses add to the pile’s resistance to load.  It can be argued that cavity expansion is one of the key advantages of driven piles.  In the case of drilled piles such as drilled shafts or auger-cast piles, this does not take place, as the soil is removed either before or during the actual pile installation.  The result of this is that driven piles, for the same length and diameter as a corresponding drilled pile, a driven pile will have a greater resistance to load (ultimate capacity.)

Applying cavity expansion theory to piles has a long history and is detailed in documents such as Randolph, Carter and Wroth (1979) and Yu and Houlsby (1991).  Our particular interest is with clays because, in addition to the changes in the soil from cavity expansion, the pore water pressure increases.  This is the primary (but not the only) reason why the SRD (soil resistance to driving) in cohesive soils is significantly less than the ultimate capacity; this fact inevitably complicates drivability studies.

The increase in pore water pressure is a dynamic phenomenon; it experiences a sudden increase during driving and then gradually dissipates after installation.  How gradually the latter takes place depends on many factors such as the permeability of the soil.  Study of this phenomenon is well represented in the literature; however, for the TAMWAVE project it is not really of interest.  The primary interest here is the value of SRD after the immediate increase of pore water pressures during driving.

Both cavity expansion theory and practice show that excess pore water pressures can easily exceed the effective stresses.  In principle, considering that we have established the beta (effective-stress dependent) method for both cohesionless and cohesive soils, this can mean a complete loss of SRD.  Although dramatic drops in SRD are not unknown, for most piles this is unrealistic.  The reason for this is that, like the dissipation, the build-up of pore water pressures is a dynamic phenomenon, albeit in a much smaller time frame.  Dissipation, hindered as it is by the low permeability of cohesive soils, begins immediately.  Pore water pressures (along with any other stresses induced by cavity expansion) also vary with the distance from the pile.

The result of all this is that prediction of both the increase of pore water pressure and its effect on the SRD of the pile during driving is a complicated phenomenon that is not completely represented by closed-form cavity expansion based solutions.  For a project such as this, what we need is something that will give a reasonable representation of soil set-up for cohesive soils.  To accomplish this, we stick with computing the excess pore water pressure, but with a different methodology.  We assume the following:

1. The basic validity of our effective-stress based beta methods of shaft friction calculation.
2. All of the decrease from static ultimate capacity to SRD takes place due to pore water pressure increase.
3. The excess pore water pressure increases affect the effective stress used to compute the shaft friction.
4. Only those pile segments under the water table are subject to this analysis.

Assuming all that, for a soil set-up factor (from this source, loosely adapted) $S_r$, the excess pore water pressures that affect the effective stress (which in turn determine the shaft friction) are computed by the equation

$\Delta u = \sigma'_{vo}\left( 1 - \frac{1}{S_r} \right)$

This gives identical results to those when the $S_r$ are applied directly.

In practice this phenomenon is still subject to investigation.  Some of the research involves use of numerical methods (such as finite element methods) to simulate cavity expansion effects.  This is doubtless an advance over the closed-form solutions of the past, and a necessity given the complexity of the physics of the problem.  Empirical methods are also still being developed, such as are documented by Wang, Verma and Steward (2009).

Posted in Geotechnical Engineering, Soil Mechanics

## p-q Diagrams and Mohr-Coulomb Failure

Students and practicioners of soil mechanics alike are used to seeing triaxial test results that look like this (from DM 7.01):

Ideally, the Mohr-Coulomb failure line should be straight, but with real soils it doesn’t have to be that way.  With the advent of finite element analysis we also have the failure function to consider, thus (from Warrington (2016)):

All of these involve constructing (or using) a line which is tangent to a circle at failure.  This can be confusing to understand completely.  The biggest problem from a “newbie” standpoint is that the maximum shear defined by the circle of stress (its radius) and the failure shear stress defined by the intersection of the circle with the Mohr-Coulomb failure envelope are not the same.

Is there a better graphical way to represent the interaction of stresses with the Mohr-Coulomb failure criterion?  The answer is “yes” and it involves the use of p-q diagrams.  These have been around for a long time and are used in such things as critical state soil mechanics and stress paths.  A broad explanation of these is found in our new publication, Geotechnical Site Characterization.  The purpose of this article is to present these as a purely mathematical transformation of the classic Mohr-Coulomb diagram.  This is especially important since their explanation is frequently lacking in textbooks.

## The Basics

Consider the failure function, which is valid throughout the Mohr-Coulomb plot.  It can be stated as follows:

$f=\sigma_{{1}}-\sigma_{{3}}-2\,c\cos(\phi)-\left (\sigma_{{1}}+\sigma_ {{3}}\right )\sin(\phi)$

(The main difference between the two formulations is multiplication by 2; the failure function can either be diametral or radial relative to Mohr’s Circle.  With a purely elasto-plastic model, the results are the same.)

Now let us define the following terms:

$p=1/2\,\sigma_{{1}}+1/2\,\sigma_{{3}}$

$q=1/2\,\sigma_{{1}}-1/2\,\sigma_{{3}}$

We should also define the following:

$\sin(\phi)=\tan(\delta)$

The physical significance of the last one is discussed in this post.  In any case we can start with $\phi$ and solve for $\delta$ or vice versa.  Solving for $\phi$ and substituting this and the equations for p and q into the failure functions yields

$f=2\,q-2\,c\sqrt {1-\left (\tan(\delta)\right )^{2}}-2\,p\tan(\delta)$

For the failure line, $f = 0$.  Let us set the p axis as the abscissa (x-axis) and the q axis as the ordinate (y-axis.)  For the failure line, if we substitute for $f$ and solve for q, we have

$q = p\tan(\delta) + c\sqrt {1-\left (\tan(\delta)\right )^{2}}$

This is a classic “slope-intercept” form like $y = mx + b$, where in this case $q = mp + b$, $m = \tan(\delta)$ and $b = c\sqrt {1-\left (\tan(\delta)\right )^{2}}$.  A sample plot of this kind is shown below.

### Some Observations

1. For the case of a purely cohesive soil, where $\phi = \delta = 0$, the failure envelope is horizontal, just like with a conventional Mohr-Coulomb diagram.
2. For the case of a purely cohesionless soil, where $c = 0$, the y-intercept is in both cases through the origin.
3. The two diagrams are thus very similar visually, it’s just that the p-q diagram eliminates the circles and tangents, reducing each case to a single point.

## Examples of Use

### Drained Triaxial Test in Clay

Consider the example of a drained triaxial test in clay with the following two data points:

1. Confining Pressure = 70 kPa; Failure Pressure = 200 kPa.
2. Confining Pressure = 160 kPa; Failure Pressure = 383.5 kPa.

Determine the friction angle and cohesion using the p-q diagram.

We first start by computing p and q for each case as follows:

$p_1 = 200/2+70/2 = 135\,kPa$

$p_2 = 383.5/2 + 160/2 = 271.75\,kPa$

$q_1 = 200/2-70/2 = 65\,kPa$

$q_2 = 383.5/2 - 160/2 = 111.75\,kPa$

The slope is simply

$m = \frac {q_2 - q_1}{p_2 - p_1} = \frac {111.75 - 65}{271.5 - 135} = 0.342 = \tan(\delta)$

from which

$\delta = 18.9^o$

$\phi = sin^{-1}(tan(\delta)) = sin^{-1}(0.342) = 20.03^o$

$b = q - mp = 65 - 0.342 \times 135 = 18.83$ (using values from the first point, just as easy to use the second one.)

$b = c\sqrt {1-\left (\tan(\delta)\right )^{2}} = c \sqrt {1-0.342^{2}} = 0.94 c$

$b = 18.83 = 0.94 c$

$c = 20.03\,kPa$

Use of this method eliminates the need to solve two equations in two unknowns, and the repetition of the quantity $tan(\delta)$ makes the calculations a little simpler.  When $c = 0$, the calculations are even simpler, as $p_1 = q_1 = 0$.

### Stress Paths

As mentioned earlier, p-q diagrams are commonly used with stress paths.  An example of this from DM 7.01 is shown below.

We note that p and q are defined here exactly as we have them above.  (That isn’t always the case; examples of other formulations of the p-q diagram are here.  We should note, however, that for this diagram $\phi" = \delta$)  With this we can track the stress state of a sample from the start (where the deviator stress is zero, at the start of the triaxial test) around to its various points of stress.

As an example, consider the stress path example from Verruijt, A., and van Bars, S. (2007). Soil Mechanics. VSSD, Delft, the Netherlands.  The basic data from Test 1 are below:

 $\sigma_3$ Deviator Stress Pore Water Pressure 40 0 0 40 10 4 40 20 9 40 30 13 40 40 17 40 50 21 40 60 25

Using the p-q diagram and performing some calculations (which are shown in the spreadsheet Stress Paths Verruijt Example)  the stress paths can be plotted as follows:

It’s worth noting that the q axis is unaffected by the drainage condition because the pore water pressures cancel each other out.  Only the p-axis changes.

## Conclusion

The p-q diagram is a method of simplifying the analysis of triaxial and other stress data which are commonly used in soil mechanics.  It can be used in a variety of applications and solve a range of problems.