Posted in Soil Mechanics

## Relating Hyperbolic and Elastic-Plastic Soil Stress-Strain Models: A More Complete Treatment

In an earlier post, we discussed this topic.  This is meant as a follow-up to that post; in a sense we left the reader “hanging” because the solution, although informative, was incomplete.  This should “tie some loose ends” and make the result, although it’s still theoretical, more useful.  The concept for most of this is the same but the implementation more closely follows the physical reality of stress-strain.

Let us begin by considering a modified version of the original graphic which compares the hyperbolic and elastic-purely plastic stress-strain models.

We need to make a few definitions.

First, let’s begin by defining two strains.  The first strain is the strain at failure (we’re assuming perfectly plastic failure here) if the small-strain elastic or shear modulus could be maintained to failure (i.e., if linear elasticity would hold until failure.)  That strain is

$\epsilon_0=\frac{\sigma_u}{E_1}$

In this case we are making the dashed line a single failure stress $\sigma_u$, the ordinate $\sigma$ and the strain $\epsilon$.  Although elastic modulus E is habitually used, this treatment could apply to shear modulus G as well.

The second is the failure strain at a reduced modulus assuming an elastic-purely plastic deformation characteristic, or

$\epsilon_1=\frac{\sigma_u}{E_2}$

If we use $\epsilon_0$ as a “reference” strain, we can make the problem dimensionless as follows:

$\hat \epsilon=\frac{\epsilon_1}{\epsilon_0}$

In any case the equation for the hyperbolic stress-strain curve for a given strain is

$\sigma=\frac{E_1 \epsilon_0^2}{\epsilon_0+\epsilon}$

Integrating the area above this curve to the failure stress and $\epsilon_1$ yields

$A_1 = \ln\left( \epsilon_0 + \epsilon_1 \right)E_1\epsilon_0^2-\ln(\epsilon0)E_1\epsilon_0^2$

Defining

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

the area above the elastic region of the elasto-plastic deformation line is

$A_2 = \frac {\epsilon_1^2AE_1}{2}$

We need to do the following:

1. Equate the areas.
2. Solve for the modulus ratio A.
3. Substitute the dimensionless strain ratio $\hat \epsilon$.

Doing all of this yields

$A = 2\,{\frac {\ln (1+{\it \hat\epsilon})}{{{\it \hat\epsilon}}^{2}}}$

Plotting this yields the following:

Although the notation is different, this is basically the same result we got before.  It also has the same problem: it “blows up” as the strain ratio approaches zero .  For high-strain problems (which is our own chief field of interest) this is not a problem, but it still needs to be addressed.  The basic problem is that the whole “area ratio” concept itself breaks down as the strains approach zero.  At zero strain the moduli should be the same and the modulus ratio unity, but the area ratio does not represent this.

This can be seen if we look at a more experimentally-based treatment of the problem, which is summarised in this graph, taken from this publication:

Although it’s certainly possible to do the usual empirical correlation on a curve like this, the higher strain portion and our theoretical presentation resemble each other.  The smaller strain region is the problem.  In many ways this resembles the Euler column buckling problem familiar to structural engineers, where two regions are defined with two equations which meet at a point where both their slope and their value are the same.

But what equation to use for the small-strain region?  Whatever equation we use needs to come to unity at zero strain and decrease from there.  A simple function for this purpose is the cosine function, modified as follows:

$A = \cos(\beta \hat\epsilon)$

To find the meeting point, we need to find the point where both the values of A and the derivatives are the same.  Without going into the algebra, for the second equation $\beta = .495$ and the meeting point is $\hat\epsilon = 1.947$ and $A = 0.571$.  This is plotted below.

Although a more rigourous analysis is necessary, the two plots look very similar.  The biggest difference–and this is not insignificant–is that the empirical plot above is semi-logarithmic in nature, while the theoretical one is linear.

From all this, we can conclude the following:

1. The “area ratio” concept, while useful for larger strains, breaks down with smaller strains.
2. The quantities $\epsilon_1$ and $\hat\epsilon$ are very useful in generalising strains in soils, although the former is physically impossible.
3. “Stitching together” the two equations yields a theoretical construct that shows potential to representing reality in soil stress-strain relationships.  The biggest difference, as noted, is the logarithmic vs. linear nature of the plots; this probably indicates an underlying principle that needs to be addressed.
4. The actual values of the ratio of small-strain shear or elastic modulus to elasto-plastic modulus is very application dependent.  Since quantifying both elastic and shear modulus is more important in geotechnical engineering (primarily due to finite element analysis) than in the past, the need to establish values of this ratio for various applications is great.

## 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.

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.