Posted in Soil Mechanics

## Terzaghi “Low Walls” Curve Correlations

Practitioners who design gravity retaining walls are familiar with the existence of Terzaghi’s “low walls” curves to estimate the equivalent fluid pressure on the wall, horizontal and vertical.  The basic chart comes in several versions but the one (for straight backfill) from NAVFAC DM 7.02 is above.  The explanation for it (important when one is using it) is below.

In the “slide rule” days, reading charts like this was routine.  A better way now, with spreadsheets abounding, is to have a formula available.  Some least squares curve fitting correlations are shown below.  The variable $\beta$ is the slope angle as shown in the figure.  As is the case with the chart, the formulas return $K_v$ and $K_h$ in units of $\frac {psf}{ft}$ of wall length.

• $K_v$ (top chart)
• Soil 1 $K_v = 0.0127 \beta^{2.21}$
• Soil 2 $K_v = 0.043 \beta^{1.92}$
• Soil 3 $K_v = 0.109 \beta^{1.71}$
• $K_h$ (bottom chart)
• Soil 1 $K_h = 30 + \frac{exp^{0.139 \beta}}{4}$
• Soil 2 $K_h = 37 + \frac{exp^{0.138 \beta}}{4}$
• Soil 3 $K_h = 48 + \frac{exp^{0.148 \beta}}{5}$
Posted in Soil Mechanics

## Soils in Construction (Sixth Edition) Now Available

It’s here a last: Soils in Construction, the Sixth Edition, now available from Waveland Press.

Many of you (and especially those who are familiar with the companion site vulcanhammer.info) are aware that I’ve spent much of my career in geotechnical engineering and deep foundations dealing with contractors.  As such I am both sympathetic with their situation and also aware that they need good information to make decisions that can “make or break” a job or their company.  It was for that reason and more that I was very happy to be invited to co-author the Fifth Edition fifteen years ago and to help prepare this revision as well.

Soils in Construction is designed to teach the basics of soil mechanics and foundation design to construction management students and to be a reference for those “in the field.”  It takes a practical approach to the subject, and it also deals with “temporary works” such as dewatering and cantilever sheet pile walls that many design engineers are unfamiliar with.

There will be more resources for this book available, both from Waveland Press and on this site as well; I’ll keep you posted.  Waveland has been great to work with and I appreciate the effort they have put into the book.  But the one person I want to say special thanks to is Lee Schroeder, Professor Emeritus (and former Interim Athletic Director) at Oregon State University.  Eminent in his own right outside of the book (he’s the Schroeder of the Schroeder-Maitland method for cellular cofferdams,) his vision for the original work, his practical and experienced implementation of same, and his graciousness and support in allowing me to be a part of this project are deeply appreciated.

You can order the book (and for you academics, request an evaluation copy) here.

Posted in Soil Mechanics

## Van der Merwe’s Method, Adapted for SI Units

Van der Merwe’s method, which was first introduced in the 1960’s, is a simple method for estimating the vertical movement of expansive soils.  The method at its most basic is described in publications such as Foundations in Expansive Soils, and basically looks like this:

The problem with this presentation is that it is entirely in “Imperial” units, which were the standard in the South Africa of van der Merwe’s day.  We need to convert this to make it usable in SI units as well.  First we present the potential expansion chart, from Soil Mechanics:

Then we reconstruct van der Merwe’s equation to make it applicable for SI units as well as US (or Imperial) units:

Note that the swell potential is now in the denominator.  That’s a convenience to make the PE values of a reasonable order of magnitude.  But now it’s dimensionless. Also note that the depth reduction factor has a different exponent for SI units than it does for US units.

Posted in Soil Mechanics

## Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results

In his recent, excellent article on the settlement (and subsidence) of the San Jacinto Monument east of Houston, Briaud (2018) takes an opportunity to vent a “pet peeve” of his relative to the way consolidation tests are reduced and consolidation properties reported:

### A Chance to Share a Pet Peeve

The consolidation e versus log p’ curve is a stress-strain curve.  Typically, stress-strain curves are plotted as stress on the vertical axis and strain on the horizontal axis.  Both axes are on normal scales, not log scales.  It’s my view that consolidation curves should be plotted in a similar fashion: effective vertical stresses on the vertical axis in arithmetic scale, and normal strain on the horizontal axis in arithmetic scale.  When doing so, the steel ring confining the test specimen influences the the measurements and skews the stiffness data.  Indeed the stress-strain curve, which usually has a downward curvature, has an upward curvature in such a plot. (p. 54)

Is this correct?  And is he the only one who thinks this way?  The two questions are neither the same nor linked.  Although this problem will certainly not be solved in one blog post, it deserves some investigation.

## Statement of the Problem

Let’s start with a text we use often here: Verruijt, A., and van Bars, S. (2007). Soil Mechanics. VSSD, Delft, the Netherlands. Early in the presentation on the subject, he presents the following plot:

As Jean-Louis would have us do, the strain (or negative strain, since we’re dealing with compression) is on the abscissa, and the dimensionless stress is on the ordinate.  The difference between the two is that the stress is plotted logarithmically.  But it’s a step.  We’ll come back to that later.

Verruijt defines the relationship between the strain and stress ratio as follows:

$\epsilon = -\frac{1}{C}\ln\frac{\sigma}{\sigma_0}$

This relationship goes back to Terzaghi’s original tests and formulation of settlement and consolidation theory almost a century ago.

From a “conventional” standpoint there are two things wrong with this formulation.  The first is that it is based on strain, not void ratio.  The second is that it uses the natural logarithm rather than the common one.  The last problem can be fixed by rewriting it as follows:

$\epsilon = -\frac{1}{C_{10}}\log\frac{\sigma}{\sigma_0}$

This formulation is essentially the same as is used in Hough’s Method for cohesionless soils, once the strains are converted to displacements by considering the thickness of the layer.  So it is not as strange as it looks.

The first problem can be “fixed” by noting the following:

$\epsilon = \frac{e-e_0}{1+e_0}$

We can substitute this into the equation before it and, with judicious changes of the constants and other subsitutions, come up with the familiar, non-preconsolidated formula for consolidation settlement, or

$\Delta H = \frac{C_c H_0}{1+e_0}\log\frac{\sigma}{\sigma_0}$

When we reverse the axes, we then get the “classic” plot as follows:

But is there a problem with using strain?  Verruijt explains the two conventions as follows:

In many countries, such as the Scandinavian countries and the USA, the results of a confined compression test are often described in a slightly different form, using the void ratio e to express the deformation, rather than the strain ε…It is of course unfortunate that different coefficients are being used to describe the same phenomenon. This can only be explained by the historical developments in different parts of the world. It is especially inconvenient that in both formulas the constant is denoted by the character C, but in one form it appears in the numerator, and in the other one in the denominator. A large value for $C_{10}$ corresponds to a small value for $C_c$. It can be expected that the compression index $C_c$ will prevail in the future, as this has been standardized by ISO, the International Organization.

As is often the case, the simplest way to help sort out this issue is with an example.  Briaud (2018) actually has one, but we will use another.

## Example of Settlement Plotting

An example we have used frequently in our teaching of Soil Mechanics is this one, from the Bearing Capacity and Settlement publication.  It is a little more complex than the theory shown above because it involves a preconsolidated soil.  The plot (with the simplifications for determination of $C_c$ and $C_r$ is shown below.

With this information in hand, we process the data as follows:

1. We convert the void ratio data to strains using the formula above.
2. We convert the stresses to dimensionless stresses by dividing them by the initial stress.
3. We “split” the data up into compression and decompression portions to allow us to develop separate trend lines for both.

First, the strain-dimensionless stress plot, using natural scales for both.

The result is similar to that in Briaud (2018).  The compression portion best fits a second-order polynomial fit.  (Not that we have thrown out the zero point to allow more fit options.)  The decompression portion fits an exponential trend line best.

Below is the same plot with the stress scale now being logarithmic.

This is basically the original graph with the axes reversed.  There is no effect using strain; we will discuss the advantages of doing so below.

Now let us look at the data from another angle: the tangent “modulus of elasticity,” defined of course by

$E = \frac{\Delta\sigma}{\Delta\epsilon}$

We consider natural scales for both modulus and strain.  To obtain the slope, we used a “central difference” technique except at the ends.

It’s interesting to note that, except for the “kink” caused by preconsolidation, in compression the tangent modulus of elasticity increases somewhat linearly with strain, as it does with the decompression.

## Discussion of the Results

There’s a great deal to consider here, and we’ll try to break it down as best as possible.

### Use of Strain vs. Void Ratio

The graphs above show that there is no penalty in using strain instead of void ratio to plot the results.  The advantage to doing so is both conceptual and pedagogical.

In the compression and settlement of soils, we traditionally conceive of it as a three-stage process: elastic settlement, primary consolidation settlement, and secondary consolidation settlement.  Consolidation settlement is nothing more than the rearrangement of particles under load; the time it takes to do so is based in part on the permeability of the soil and its ability to expel pore water trapped in shrinking voids.  Elastic settlement is due to the elastic modulus of the material, the strain induced in the material and the geometry of the system.  This distinction, however, obscures the fact that we are dealing with one soil system and one settlement.  Using strain for all types of settlement would both help unify the problem conceptually and ease the transition to numerical methods such as finite element analysis, where strain is used to estimate deflection.  In the past we were able to use a disparate approach without difficulty, but that option is not as viable now as before.

### The Natural Scale, Consolidation Settlement Stiffness, and the Ring

Both here and in Briaud (2018) the natural stress-strain curve experiences an upward curvature, which is obviously different from what we normally experience in theory of elasticity/plasticity.  This comes into better focus if we consider the variation of the tangent modulus of elasticity, which (except for the aforementioned preconsolidation effect) linearly increases with stress.  There are two possible explanations for this.

The first is to observe that, as soils compress in consolidation settlement, their particles come closer together, and thus more resistant to further packing.

The second, as suggested by Briaud (2018), is that the presence of the confining ring in the consolidation test augments the resistance of the particles to further compression.  The issue of confinement is an interesting one because in other tests (unconfined compression tests, triaxial tests) confinement is either very flexible or non-existent.  It should be observed that consolidation theory, as originally presented, is one-dimensional consolidation theory.  For true one-dimensional consolidation, we assume a semi-infinite case where the infinite boundary “confines” the physical phenomena.  The use of a confining ring assumes that the ring can replicate this type of confinement in the laboratory.  Conditions in the field, with finite loads and variations in the surrounding soils, may not reflect this.  While it would be difficult to replicate variations in confinement in the laboratory, these variations should be kept in mind by anyone using laboratory-generated consolidation data.

### The “Modulus of Elasticity” for Consolidation Settlement

This may strike many geotechnical engineers (especially those in areas where void ratio is used to estimate consolidation settlement) as an odd concept, but if we consider the material strain vs. its deflection, it is a natural one.  Varying moduli of elasticity are nothing new in geotechnical engineering; they have been discussed on this site in detail.  The situation here is somewhat different for a wide variety of reasons, not the least of which is that here we are dealing with a tangent modulus while previously we looked at a secant one.  Also, differing physical phenomena are at work; theory of elasticity implicitly assumes that particle rearrangement is at a minimum, while consolidation settlement (both primary and secondary) is all about particle rearrangement.

A more unified approach to settlement would probably reveal a process where the change in stress vs. the change in strain varies at differing points in the process along a stress path with multiple irreversibilities.  Such an approach would require some significant conceptual changes in the way we look at settlement, but would hopefully result in more accurate results.

## Conclusion

Consolidation settlement is a topic that has occupied geotechnical engineering for most of its modern history.  While the theory is considered well established, changes in computational methodology will eventually force changes in the way the theory is applied.  A good start of this process is to use strain (rather than void ratio) as the measure of the relative deflection of structures, and the example from Briaud (2018), along with the demonstration relative to natural scales, is an excellent start.

References

Briaud, J.-L. (2018) “The San Jacinto Monument.”  Geostrata, July/August.  Issue 4, Vol. 22, pp. 50-55.

Posted in Soil Mechanics

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

For an update and application of this, you should visit this post.

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.