Posted in Geotechnical Engineering, Soil Mechanics

## A Simple Example of Braced Cut Analysis

Most retaining walls are designed with active or passive earth pressures derived from Rankine, Coulomb or Log-Spiral theories.  One notable exception to that are braced cuts.  The development of the earth pressure distributions is attributable to Karl Terzaghi and Ralph Peck.  In the process of developing those, the way the wall is modelled was simplified to avoid statically indeterminate structures.  Although this is not the problem that it was in their day, the method is still dependent upon those statically determinate structures.

The example below is a simple example in that the supports are symmetrically placed and there is no sheeting toe penetrating the bottom of the excavation.  It’s primarily intended to illustrate the concepts, both geotechnical and structural, of the design of these structures.

# Overview of the Example

Let us consider a braced cut excavation which is 45′ deep and which has supports at a depth of 5′, 17′, 28′ and 40′.  The soil behind the wall is uniform with c = 1100 psf and γ = 110 pcf.  The water table is at the bottom of the excavation and does not enter into our calculations.  To show how this lays out we’ll use Pile Buck’s SPW 911 sheet pile software.  We’ll assume PZ-27 sheeting is being used, and that there is no surcharge on the wall.

The options for earth pressure distribution behind braced cuts are shown below, from NAVFAC DM 7.2. or Sheet Pile Design by Pile Buck.

We obviously have a clay soil, thus our selection will be either (b) or (c).  Whether the soil is soft to medium or stiff depends upon the stability number $N_o$, which is computed as follows:

$N_o = \frac{\gamma H}{c} = \frac{110 \times 45}{1100} = 4.5$

This is between (b) and (c), we are thus supposed to use the “larger” of the two diagrams.  The earth pressure coefficient for (b) is

$K_a= 1 - m \frac {4c}{\gamma H}$

Assuming m = 1,

$K_a = 1 - m \frac {4c}{\gamma H} = 1 - \frac{4 \times 1100}{110 \times 45} = 0.11$

and thus

$\sigma_h = K_a \gamma H = 0.11 \times 110 \times 45 = 550\,psf$

If we turn to Case (c) and assume that

$\sigma_h = 0.3 \gamma H = 0.3 \times 110 \times 45 = 1485\,psf$

this is obviously “larger” than Case (b), so we will use Case (c), even when using a “medium” case between the two extreme pressure profiles.

We thus have a pressure distribution that can be described as follows:

1. Beginning at the top, it linearly rises from zero to the maximum value of 1485 psf at a point a quarter down the wall, or 45/4 = 11.25′.
2. From that point until a quarter from the bottom of the wall, or 0.75 * 45 = 33.75′, it is a constant pressure of 1485 psf.
3. From that point until the bottom of the wall, it linearly decreases to a value of zero at the bottom of the wall.

# Guidelines for Structural Analysis of Wall

Turning to the structural aspects of the wall, the guidelines for dividing the wall up are as follows:

1. If the wall is cantilevered at either end, then the endmost support and the one next to it form a simply supported beam with a cantilever at one end and a distributed load.
2. Segments in the middle are analysed as simply supported beams with a distributed load.
3. If there’s a support at the top or the bottom of the wall, the beam at that location is analyzed as a simply supported beam.
4. Reactions are computed for each beam.  For supports where two segments meet, you simply add the two reactions from each beam for a total reaction for the support.
5. Maximum moments are computed for each beam; the largest of these maximum moments is the maximum moment of the system and the one used to size the sheeting.

This was Terzaghi and Peck’s attempt to make the calculations simple.  If the distributions are simple, then “handbook” type formulas can be used.  The trout in the milk takes place (as it does here) when the break points in the distribution don’t coincide with the supports, in which case you end up with a more complicated distribution.  There are two ways of dealing with this problem.

The first is to reduce the distributed loads to point load resultants.  This is a favourite tactic among geotechnical engineers and is used extensively with shallow foundations.  For purely hand calculations, it makes sense.  The moments will be higher (which is conservative) but the reactions will be identical, assuming the concentration of the moments went off according to plan.

The second is to employ beam software to analyse each segment.  Although there’s a lot of beam software out there, being the old coots we are, we’ll use CFRAME, a DOS program for two-dimensional structures.  It gets the job done and is fairly easy to use.  (Note: because of some bad interaction between CFRAME and DOSBox, we ran it on a Windows XP installation.  The manual for CFRAME: Computer Program with Interactive Graphics of Plane Frame Structures is here.)

# Implementation in CFRAME

The first thing we need to do is to specify the distributed loads.  CFRAME, like most finite element programs, considers the beam between each support (and the beams from the outermost supports to the cantilever element) as one element.  So there are six elements.  CFRAME asks us to specify the distributed load (constant or linearly varying) for each element, and requires us to specify the constant loads and the varying loads separately.

But here we run into something that trips up students.  Sheet piles are analysed as beams, but they’re “infinite” beams; we analyse them in terms of moment of inertia per length of wall, section modulus per length of wall, load per unit length of wall, etc.   The good news is that, for distributed loads, the pressure at any point is the load per unit length!  Pressure is expressed, in this case, as lb/ft^2 of wall, when in reality it’s lb/ft/ft of wall.  That makes things simpler; as long as we enter the moment of inertia and cross sectional area in terms of “per foot of wall” (which any US unit section should furnish us) then we’re good.  In this case for PZ-27 the moment of inertia is 184.2 in^4/ft of wall and the cross-sectional area is 7.94 in^2/ft of wall, and these are entered directly into CFRAME.

With that technicality out of the way, for are areas of constant earth pressure (the middle) we’re also good; it’s just 1485 psf, and we enter this directly into CFRAME.  With the ramped portions, they increase from the top and bottom of the wall at a rate of 1485/11.25 = 132 psf/ft from the end.  Looking at the topmost element, which we enter into CFRAME as (surprise!) element 1, the pressure at the topmost support is 132 * 5 = 660 psf, which we enter as the maximum pressure for the “triangle load” on the top element.

For element 2, we have two loads.  The first is a continuation of the ramped load from 660 psf at the top end of the beam to 1485 psf at a point 11.25′ from the top of the wall or 11.25′ – 5′ = 6.25′ from the top end of the beam.  The second load is simply a constant load to the bottom end of the beam.

The middle element 3 has a constant distribution across its entire length.  The bottom two elements are mirror images of the top two elements.

# Results from CFRAME

We entered the data into CFRAME via a small text file.  First we present the model itself.

Now we show the results.

The individual element results are shown below.  The tabular results of the program are here.

# Analysing the Results

First let’s look at the reactions at the supports, which come from the element results.  They are as follows:

1. Support 1 (Node 2):  The reaction/shear at that point from element 1 is 1650 lb/ft of wall and from element 2 7009 lb/ft of wall, summing it comes to 8659 lb/ft of wall.
2. Support 2 (Node 3): The reaction/shear at that point from element 2 is 8233 lb/ft and from element 3 8168 lb/ft, summing it comes to 16401 lb/ft.
3. Support 3 (Node 4) is the same as Node 3 by symmetry.
4. Support 4 (Node 5) is the same as Node 2 by symmetry.

Thus the maximum brace load is on Supports 2 and 3, 16401 lb/ft.  We have for convenience ignored the sign conventions and simply added the reactions, since they’re all in the same direction.

The maximum moment is actually in Element 2 (or 4,) and is 273,900 in-lb/ft of wall.  Since the elastic section modulus for PZ-27 is 30.2 in^3/ft of wall, the maximum bending stress is 273,900/30.2 =  9070 psi, which is well within most allowable specifications.  A lighter section can probably be employed, depending upon the allowable deflection and other requirements.

As a quick check, for a uniformly distributed load on a simply supported beam, the maximum moment is given by the equation

$M_{max} = \frac{wl^2}{8}$

Substituting the values for Element 3, we have

$M_{max} = \frac{wl^2}{8} = \frac {1485 \times 11^2}{8} = 22,461\,\frac{ft-lb}{ft}$

Now we compare these with SPW 911, whose output is as follows:

The differences are minor (SPW 911 and the hand calculation report the maximum moment in ft-lb/ft of wall, not in-lb/in of wall.)  Some discussion of eliminating the additional pins in the simply supported spans is given in Sheet Pile Design by Pile Buck.

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.