Soil Mechanics – vulcanhammer.net

Getting to the Bottom of Terzaghi and Peck’s Lateral Earth Pressures for Braced Cuts

Posted on 12 May 202413 May 2024 by Don Warrington

One of those “things” in geotechnical engineering that looks like “settled science” but may not be is the whole business of lateral earth pressures for braced cuts. (An example of one is shown at the right.) Textbooks of all kinds (including Soils in Construction) show pressure profiles and solution techniques that are “definitive.” Or are they? This article is more about asking questions than delivering another round of “definitive” answers, but hopefully it will at least spark some thought and perhaps make practitioners more careful in their application of these methods.

The Basic Problem

Based on experience, first in Berlin and later in Chicago, Terzaghi (and later also Ralph Peck) developed a set of pressure distributions as shown at the left. These are at variance with those usually associated with retaining walls in general and sheet pile walls in particular. The theory behind these (certainly for the clays) had its genesis in Rankine theory adapted for cohesive soils, but the distribution is rather different. These distributions have been reproduced in many textbooks and reference books, including Soils in Construction and Sheet Pile Design by Pile Buck.

It’s worth noting that there are other pressure distributions that have been formulated other than the ones shown above, as outlined by Boone and Westland (2005).

Along with the distributions came the method of using them: the “hinged method” of analysing the sheet pile wall. Strictly speaking the sheet pile wall is a continuous beam with multiple supports. Since there are usually more than two struts and supports, to use a continuous beam requires a statically indeterminate beam. Applying hinges (as shown at the right) can make the beam statically determinate. Although methods for solving statically indeterminate beams existed in Terzaghi’s day (remember it was geotechnical methods which then and now lag the rest of civil engineering in advancement,) converting the problem to a statically determinate one was convenient for computational purposes.

But then comes the kicker: as generally presented, if the distributions above are used, they must be used with the hinged method, even when analysing a braced cut wall using a braced cut method with a continuous beam is nearly trivial now. Why is this?

How It Came About

Like so many things in civil engineering, the investigation of pressures on braced cuts came about as a result of tragedy. As noted by Rogers (2013):

On the evening of December 1, 1938 Terzaghi delivered a terse lecture titled “The danger of excavating subways in soft clays beneath large cities.” The lecture focused on his recent experiences with construction of the Berlin Subway, which was hampered by a high water table in running sands. These conditions had contributed to the sudden failure of a shored excavation which killed 20 workers in August 1935. He made a convincing case for proper geotechnical oversight during construction if similar tragedies were to be avoided in Chicago.

The lecture with its graphic images of the dead bodies beneath the collapsed bulkhead along the Hermann Goring Strasse succeeded in scaring his audience to death, and promptly found the State Street Property Owners’ Association and City of Chicago bidding for Terzaghi’s services. The City wanted him to advise them on how best to monitor progress of excavations and ground settlement, differentiating what structural or architectural damage was caused by subway construction.

Both Terzaghi and Ralph Peck ended up doing the monitoring. The soils in Chicago were predominantly soft clays, so the earth pressures were different. Much of the theory and application behind this is documented in Peck (1943). (Interesting side note: one of the lines of the subway ran past the site of Vulcan’s old Milwaukee Avenue plant where the first Warrington-Vulcan steam hammers were designed and built.)

One might ask, “How did they come up with the earth pressure distributions?” They did so–and this is the key to the problem–by measuring the reactions on the braces. They did this in the face of the fact that, as is usual with braced cuts, the braces were put in successively with excavations, and that much of the movement of the wall–and thus the mobilisation of the earth pressures–was in place before the braces were installed. (It is easy to forget the importance of that mobilisation, but both Terzaghi and Peck were well aware of it and its effects.) To translate the loads on the braces into a pressure distribution, they adopted Terzaghi’s procedure from the Berlin subway as follows (Peck (1943)):

The vertical members of the sheeting are assumed to be hinged at each strut except the uppermost one, and a hinge is assumed to exist at the bottom of the cut. The abscissas of the pressure diagram “A” represent the intensity of horizontal pressure required to produce the measured strut loads. A study of such diagrams for all of the measured profiles disclosed that the maximum abscissa never exceeded the value K_A Y_a H. Every measured set of strut loads resulted in a different pressure diagram “A,” all of which were found to lie within the boundaries of the trapezoid indicated by the dotted lines. Thus, if strut loads are computed on the basis of this trapezoid, they will most probably be on the safe side.

This, therefore, is the origin of the requirement to use a hinged wall where there were no actual hinges. At the time it was a reasonable solution. As noted earlier, solutions for continuous beams existed but back-figuring the pressures using them would have been a formidable “inverse problem” given the computing power of the day. Doing this, however, raises as many questions as it answers, such as the following:

If a continuous beam had been used, would the pressure distribution have been different?
What is the relationship between the pressure distribution computed by Terzaghi’s method and what is actually experienced by the wall? Put another way, did Terzaghi’s simplification of the structural situation compromise his distribution? (No doubt some conservatism in the pressure distributions offset that problem.)
If the pressure distributions are right for engineering purposes, is it still necessary to use a hinged solution? Especially with beam software, a continuous beam is much simpler to analyse and structurally more representative of the actual sheeting and bracing.

Peck himself was well aware of the limitations of the method; he made the following admission:

It is apparent, therefore, that it is useless to attempt to compute the real distribution of lateral pressure over the sheeting. Of far greater practical importance is the statistical investigation of the variation in strut loads actually measured, in order to determine the maximum loads that may be expected under ordinary construction procedures.

This too raises another question: in developing distributions primarily to determine brace/strut loads, do we compromise the accuracy of determining the maximum moment in the sheeting itself?

Moving Forward

It’s difficult to really know how to answer many of the questions this problem raises. Some suggestions are as follows:

It is hoped that there is enough conservatism in these earth pressure distributions to accommodate either method. That’s likely, as inspection of some of Peck (1943) curves will attest. That likeliness is buttressed by the fact that these methods came out of a deadly accident.
More comparisons of hinged and continuous beam models are needed. There is one in Sheet Pile Design by Pile Buck and another in the post on this site A Simple Example of Braced Cut Analysis. These are simply not enough to establish a trend one way or another, although the results are interesting and hold promise.
A “hand” solution based on parametric studies using FEA or another numerical method would move things forward considerably. Obviously these are limited by the accuracy of the soil modelling but they can be applied to a wider variety of cases.
Field tests should include measurement of actual lateral earth pressures on the sheeting at various points. The use of strut loads, although easier to measure with the technology of the 1930’s and 1940’s, is still indirect. Another interesting approach is to use an inverse method and a continuous beam with existing data, although this is not as satisfactory as direct measurement of earth pressures.

References

Boone, S.J. & Westland, J. (2005) “Design of excavation support using apparent earth pressure diagrams: consistent design or consistent problem?” Fifth International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground, International Conference on Soil Mechanics and Geotechnical Engineering, 809 – 816.
Peck, R.B. (1943) “Earth-Pressure Measurements In Open Cuts, Chicago (Ill.) Subway.” Transactions of the American Society of Civil Engineers, Vol 108, pp 1008-1036.
Rogers, D.A. (2013) “Ralph Peck’s Circuitous Path to Professor of Foundation Engineering (1930-48)” Presented at the Seventh International Conference on Case Histories in Geotechnical Engineering.

Academic Issues, Geotechnical Engineering, Soil Mechanics

Getting to the Legacy of B.K. Hough and his Settlement Method

Posted on 14 March 2024 by Don Warrington

Last year I posted The Sorry State of Compression Coefficients where I a) gave a brief summary of earlier posts on consolidation settlement and b)showed that there was more than one way to express them. An example of the “alternative” (for geotechs in some countries it’s the accepted way) compression coefficient is “Hough’s Method” which is featured in both the Soils and Foundations Reference Manual and the Shallow Foundations manual. Hough’s Method, however, is for cohesionless soils. Why, you ask, can methods usually associated with cohesive soils be applied to cohesionless ones? Because consolidation settlement methods using the logarithmic difference of pressure reflect the fact that the elastic (or shear) modulus of a soil increases as the void ratio/porosity of the material decreases, which I discuss in From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”.

In this post I will attempt to do two things:

The method as presented in the above references has been described as too conservative, i.e., the settlements predicted are too large. I will attempt to explain this and perhaps offer a solution based on Hough’s own works.
Discuss the whole business of bearing capacity vs. settlement failure in shallow foundations, which was perhaps the greatest legacy of Hough’s work and remains an important issue in geotechnical engineering today.

Bearing Capacity vs. Settlement

Terzaghi’s solution (or more accurately his adaptation of Prandtl’s punching shear theory) of the bearing capacity problems was one in a number of solutions that became “reference standard” in geotechnical engineering.

We were regaled with photos of Canadian grain elevators on their side to show that bearing capacity failure was the first thing we should look for in shallow foundation design. Terzaghi’s formula was so highly regarded that for many years it was fashionable for introductory geotechnical courses to require students to learn both Terzaghi’s method and the subsequent improvements/extensions of that method by researchers such as Meyerhof, Vesic, Brinch Hansen, etc..

Up until that time shallow foundations were generally designed using what we call “presumptive bearing capacities” based on soil types and foundation configurations. These were enshrined in the building codes of the day. They were generally purely empirical in nature, as was most of geotech in the era before Terzaghi and his contemporaries. They had one advantage however: because they were derived from actual performance, be it ever so crude, they included the effects of soil settlement under load.

Like any other engineering material, only on a larger basis (because their elastic/shear moduli were several orders of magnitude lower than more conventional materials) soil is deformable under load. That deformation not only allows the foundation to deflect under load, it also affects the failure surfaces as they develop. The latter reality became apparent and so we have the modification of the bearing capacity for punching and local shear. It should be noted that Terzaghi and Peck were well aware of the problem of settlement, and included provision for it in their classic 1948 textbook Soil Mechanics in Engineering Practice.

One possible solution was to use elastic theory for the initial settlement. The implementation of that is discussed in Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads. It is even possible to develop a lower bound solution for the bearing capacity problem, as was discussed in Lower and Upper Bound Solutions for Bearing Capacity. The problem with this is twofold. The first is that the lower bound solution assumes that the footing is a purely flexible foundation, which is not really true with this type of foundation. The second is that, if we went to the other extreme and assumed a purely rigid foundation, by elastic theory the stresses at the corners is infinite for any load. (This is conventionally attributed to foundations in cohesive soil, but it can be shown to be true by elastic theory.)

Hough’s Settlement Method

Hough presented his settlement method in his 1959 ASCE paper. He starts by presenting a graph similar to the following, which illustrates the transition from small, elastic displacement to large inelastic ones:

The solution is in the form (similar to that presented in Verruijt) of

$S=\frac{H_{o}}{C'} \log_{10} \frac{\sigma'_f}{\sigma'_o}$

The coefficient $C'$ is determined using a chart which is below.

The chart itself is basically the same as the one in Hough (1959,) but now things get complicated.

In Shallow Foundations we are informed of the following:

Cheney and Chassie (2000) recommend that the SPT blowcounts be corrected for overburden pressure before correlating the N-values to the bearing capacity index,
C′. An overburden correction by Bazaraa (1967) was recommended by Cheney and Chassie (2000). Since that time, many researchers have studied the effect of overburden stress on the SPT N-value, largely in support of liquefaction hazard assessment procedures. Recent consensus by the 1996 and 1998 National Center for Earthquake Engineering Research (NCEER) (Youd et al., 2001) concluded that the correction proposed by Liao & Whitman (1986) (shown in Figure 5-18) could be used for routine engineering applications. Therefore, the correction by Liao & Whitman is included here as part of the Hough procedure, in particular because it is easy to calculate and can be used without charts in simple computation spreadsheets.

The basic problem with this is that Hough himself never mentions any kind of correction for the variations of the SPT tests, overburden or otherwise, and that in modifying Hough’s Method we run the risk of making some assumptions that may not be applicable.

The idea of using the SPT tests, even with shallow foundations in cohesionless soils such as the ones Hough is concerned about, is admirable on the face because undisturbed samples of these soils are hard to obtain in normal soil testing. The problem with SPT tests–which were coming into acceptance when Hough presented his original method–is that their variations in both configuration and those induced by the overburden are significant. It wasn’t until the 1980’s that this was “formally” sorted out with the correction system that we have today.

With our current method we have two stages of correction. The first stage is for variations in the configuration of the split spoon, the effect of the rod and borehole, and most importantly the efficiency of the hammer itself. The second is for the overburden. In both his original paper and the presentation of the method in the Second Edition of his textbook Basic Soils Engineering (1970,) Hough does not delve into either of these.

The whole point of the N₆₀ correction (the first series) was to harmonise the results to a single mechanical standard, and one that was generally attained “back in the day” so that empirical methods such as Hough’s could be used today. We could assume that Hough’s value are at least N₆₀ values, although we’re not guaranteed of that due to the lack of supporting data.

The business of overburden correction brings up Bazaraa (1967.) At this point credit needs to be given where credit is due: Bazaraa had the thankless task of sorting out the settlement methods which were current in his day, and that task included dealing with the variations of the SPT method. Since his overburden correction method was originally used with Hough’s Method and then changed, let us compare the two. We start be defining

$p_{norm} = \frac{\sigma'_o}{p_{atm}}$

The two correction factors $C_N$ for overburden are shown below.

Except for the region where Liao and Whitman is “flat topped” the two are reasonably close, and so substituting Liao and Whitman’s correction is legitimate, assuming it should be used at all.

It should be noted, however, that Bazaraa’s objective is not to correct Hough’s method, which he did not do: it was to provide a new method of estimating settlements in cohesionless soils, and to advance Terzaghi and Peck’s 1948 method. So we are still “up in the air” about how to apply all of this to Hough’s Method.

If the overburden stress is less than the standard atmospheric pressure (approx. 2 ksf) then the N value used will be increased, the C’ values will likewise increase, and the estimated settlement will decrease, which may be more accurate but moves away from conservatism. If the opposite is true then the result will be more conservative. For shallow foundations such as Hough’s Method the lower overburden stresses will be more of a factor (depending upon the depth D.) At this point it is not clear to me (at least) that including an overburden correction is really significant given the other unsolved problems that exist with this method.

One further complicating factor is that, in the aforementioned Second Edition of Basic Soils Engineering, Hough presents a new chart (again with no explanation of correction of any kind) which is redrawn as follows:

Some trend line work results in a correlation in this form:

$C = A \exp^{BN}$

where the coefficients A and B are below.

Soil Type	A	B
Organic Silt, Little Clay	58.66	0.0225
Inorganic Sandy Silt	37.02	0.0221
Very Well Graded Fine to Coarse Sand	32.85	0.0216
Well Graded Clean Fine to Coarse Sand	28.22	0.0216
Well Graded Silty Sand and Gravel	22.86	0.0203
Uniform Clean Inorganic Silt	18.28	0.021
Very Uniform Clean Medium Sand (Similar to Std. Ottawa)	7.22	0.0229

It should be noted that, where the curves are directly comparable with Hough (1959), they tend to be lower, i.e., the values of C are less. This would increase settlement and thus conservatism. At this point perhaps we should consider Hough’s Methods in the plural rather than the singular.

Beyond Hough’s Method

To start this part consider this, from NAVAC DM 7.1:

We have the conventional C_c based on either liquid limit, initial void ratio or water content. Hough is represented here as well; however, these correlations are for cohesive soils. Hough himself had a broader application for these.

In the Second Edition of Basic Soils Engineering, he presents a function to compute C_c as follows:

$C_c = a(e_0 - b)$

He also establishes a relationship between correlations based on liquid limit and those on void ratio, but getting to that is for another time.

In any case he presents values for this equation which are tabulated below, and include both cohesionless and cohesive soils.

	a	b*
Uniform cohesionless Material (C_u < 2)
Clean Gravel	0.05	0.5
Coarse Sand	0.06
Medium Sand	0.07
Fine Sand	0.08
Inorganic Silt	0.1
Well-graded, cohesionless soil
Silty sand and gravel	0.09	0.2
Clean, coarse to fine sand	0.12	0.35
Coarse to fine silty sand	0.15	0.25
Sandy silt (inorganic)	0.18	0.25
Inorganic, cohesive soil
Silt, some clay; silty clay; clay	0.29	0.27
Organic, fine-grained soil
Organic silt, little clay	0.35	0.5

*The value of the constant b should be taken as e_min whenever the latter is known or can conveniently be determined. Otherwise, use tabulated values as a rough estimate.

From here we can use the well-tried “C_c” formulae (which would include preconsolidated soils) to estimate settlement. This opens up a new vista for using conventional consolidation settlement theory to estimate the settlements of cohesionless soils.

To return to our original objectives, settlement and bearing capacity are, in the process of failure, two sides of the same thing. It is also worth noting that most foundations fail in settlement, although the failure isn’t as “spectacular” as bearing capacity.

The only way to treat them together is to use a method that can combines both phenomena, and finite element methods are capable of doing that. However our profession has been uncomfortable with “black box” methods such as these, although they really apply the same laws we use to formulate closed form solutions. In that respect they have value, and to apply methods such as Hough’s to cohesionless soils with better calibration of the constants would be a good step forward.

References

Bazaraa, A.R.S. (1967). “Use of the Standard Penetration Test for Estimating Settlements of Shallow Foundations on Sand,” Ph.D. Thesis presented to University of Illinois, Urbana.
Hough, B.K. (1959). “Compressibilty as the Basis for Soil Bearing Value,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 85, Part 2.
Hough, B.K. (1970). Basic Soils Engineering. Second Edition. New York: Ronald Press Company.
Kimmerling, R.E. (2002). Shallow Foundations: Geotechnical Engineering Circular No. 6. FHWA-SA-02-054. Washington, DC: Federal Highway Administration.
Terzaghi, K. and Peck, R. (1948). Soil Mechanics in Engineering, 1st Ed., John Wiley & Sons.

Academic Issues, Geotechnical Engineering, Soil Mechanics

Determining the Degree of Consolidation

Posted on 3 November 202211 November 2022 by Don Warrington

This is the last (hopefully) post in a series on consolidation settlement. We need to start by a brief summary of what has gone before. Note: the material for this derivation and those that preceded it have come from Tsytovich with some assistance from Verruijt.

Review

In the post From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”, we discussed the issue of how much soils (especially cohesive ones) settle through the rearrangement of particles. We were able to start with the theory of elasticity and, considering the effects of lateral confinement, define the coefficient of volume compression $m_v$ by

$m_v = \frac{\beta}{E}$ (1)

where E is the modulus of elasticity and $\beta$ is a factor based on Poisson’s Ratio and includes the effects of confinement, be that in an odeometer or in a semi-infinite soil mass. We also showed that, for a homogeneous layer,

$\delta_p = m_v H_o \sigma_x$ (2)

where $\delta_p$ is the settlement of the layer, $H_o$ is the thickness of the layer and $\sigma_x$ is the uniaxial stress on the layer. The problem is that $m_v$ is not constant, and the settlement more accurately obeys the law

$\delta_p = \frac{C_c H_o}{1+e_o} \log{\frac{\Delta p + \sigma_o}{\sigma_o}}$ (3)

where $C_c$ is the compression index, $e_o$ is the initial void ratio of the layer, $\Delta p$ is the change in pressure induced from the surface, and $\sigma_o$ is the average effective stress in the layer.

Turning to the post Deriving and Solving the Equations of Consolidation, we first determined that the change in porosity $\Delta n$ could, for small deflections, be equated to the change in strain $\epsilon$ . From this we could say that

$\Delta n = m_v \Delta \sigma_x$ (4)

The change in porosity, for a saturated soil whose voids are filled with an incompressible fluid (hopefully water) induces water flow,

${\frac {\partial }{\partial x}}q(x,t)=-{\frac {\partial }{\partial t}} {\it n}(x,t)$ (5)

where $q(x,t)$ is the flow of water out of the pores and $n(x,t)$ is the porosity as a function of position and time. The flow of water is regulated by the overall permeability of the soil, and all of this can be combined to yield

${\frac {k{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)}{{\it \gamma_w }}}=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (6)

where $k$ is the permeability of the soil and $\gamma_w$ is the unit weight of water. Defining

$c_v = \frac{k}{m_v \gamma_w}$ (7)

and making some assumptions about the physics, we can determine the equation for consolidation as

$c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}u(x,t)$ (8)

where $latex u(x,t) is the pore water pressure. If we invoke the effective stress equation and solve this for the boundary and initial conditions described, we have a solution

$\sigma_{x}(x,t)=p\left(1-\frac{4}{\pi}\left(\sin(1/2\,{\frac{\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/3\,\sin(3/2\,{\frac{\pi\,x}{h}}){e^{-9/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/5\,\sin(5/2\,{\frac{\pi\,x}{h}}){e^{-{\frac{25}{4}}\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}\cdots\right)\right)$ (9)

The Degree of Consolidation

One thing that our theory presentation demonstrated was the interrelationship between pore pressure, stress and deflection. We know what the ultimate deflection will be based on Equation (3) above (or more complicated equations when preconsolidation is taken into consideration.) But how does the settlement progress in time?

We start by defining the degree of consolidation thus:

$U = \frac{\delta(t)}{\delta_p}$ (10)

where $\delta(t)$ is the settlement at any time before complete settlement. For the specific case (governing equations, initial equations and boundary conditions) at hand, the degree of consolidation–the ratio of settlement at a given point in time to total settlement–can be determined as follows:

$U_{o}=\intop_{0}^{h}\frac{\sigma_{x}(x,t)}{ph}dx$ (11)

In this case the result is divided by the uniform pressure p and the height h. Let us further define the dimensionless time constant

$T_{v}=\frac{c_{v}t}{h^{2}}$ (12)

That being the case, if we integration Equation (9) with Equation (11), we obtain

$U_{o}=1-\sum_{n=1}^{\infty}4\,{\frac {{e^{-1/4\,{\it Tv}\,{n}^{2}{\pi }^{2}}}\left (\cos(n\pi )\cos(1/2\,n\pi )-\cos(n\pi )-\cos(1/2\,n\pi )+1\right )}{{n}^{2}{\pi }^{2}}}$ (13)

otherwise put

$U_{o}=1-8\,{\frac {{e^{-1/4\,{\it Tv}\,{\pi }^{2}}}}{{\pi }^{2}}}-{\frac {8}{9}}\,{\frac {{e^{-9/4\,{\it Tv}\,{\pi }^{2}}}}{{\pi }^{2}}}-{\frac {8}{25}}\,{e^{-{\frac {25}{4}}\,{\it Tv}\,{\pi }^{2}}}{\pi }^{-2}\cdots$ (14)

As was the case with Equation (9), only the odd values of n are considered; the even ones result in zero terms.

It is regrettable that, in defining $T_v$ , the value $\frac{\pi^2}{4}$ was not included, as using Equation (14) would be much simpler. For certain cases, it is possible to use the first two or three terms. In any case the usual method for determining $T_v$ –and by extension the degree of consolidation–is generally done either using a graph or a table, as is shown in the graph at the start of the post (repeated below:)

Degree of Consolidation for Instantaneous Uniform Loading and One-Dimensional Flow. From NAVFAC DM 7.1: Soil Mechanics

The notation is a little different. We use the variable $U_o$ to emphasise that we are dealing with the “standard” case. The above graph also gives approximating equations; it is easy to see that, for $T_v > 0.2$ , the equation given is simply the first two terms of Equation (14). The distinction between the drainage length h ( $H_{dr}$ in the graph above) and the layer thickness H is clear.

Conclusion

We have covered the basic, classic case of consolidation settlement in this post and its predecessors From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” and Deriving and Solving the Equations of Consolidation. We trust that this presentation has been enlightening and informative.

Academic Issues, Geotechnical Engineering, Soil Mechanics

Deriving and Solving the Equations of Consolidation

Posted on 1 November 20224 November 2022 by Don Warrington

In an earlier post we discussed consolidation settlement. For situations where soil a) decreases its volume due to rearrangement of the particles and b) does so over a relatively long period of time due to difficulties in expelling pore water pressures, we need to know how long it takes to reach maximum settlement, in addition to know what that settlement is and what deflections we might achieve along the way.

This is generally a two part process: a) determining the dissipation of pore water pressure and b) determining the amount of settlement at a given time. This post will focus on the first part of the process; we will deal with the second in a later post. This is a well-worn path in geotechnical engineering but hopefully this derivation is a little simpler than others.

Developing the Differential Equation

Relating Porosity to Strain

Let us begin by considering the system above (from Tsytovich (1976)), a uniformly loaded soil layer which is saturated (always) and clay (usually.) The first thing we need to note is that, while the diagram uses z as the variable of length in the vertical direction, from this post we will use the variable x (sorry for the confusion.) The second thing is that we assume the water and the solids to be incompressible. The third thing is that all the changes that take place do so because of changes in the voids where the water is resident. We first note the definition of porosity as

$n={\frac {V_{{v}}}{V_{{t}}}}$ (1)

where

n = porosity
V_v = volume of voids
V_t = total volume

That being the case, the relationship between the porosity of the soil (due to changes in the volume of the voids) and the flow rate of the water can be expressed as

${\frac {\partial }{\partial x}}q(x,t)=-{\frac {\partial }{\partial t}} {\it n}(x,t)$ (2)

We can envision a differential volume having a height x and an area A. Since the problem is one-dimensional, the areas cancel out and the ratio of the void height to the total height is

$n={\frac {x_{{v}}}{x_{{t}}}}$ (3)

where

x_v = height of the voids
x_t = height of the solids

We want to determine the change in porosity from some state 0 to some state 1, just as we did with void ratio in this post. That change can be expressed as follows:

${\frac {x_{{{\it v0}}}}{x_{{{\it t0}}}}}-{\frac {x_{{{\it v1}}}}{x_{{{\it t0}}}-x_{{{\it v0}}}+x_{{{\it v1}}}}}$ (4)

We can assume that the change in void volume x_v0 – x_v1 << x_t0, in which case Equation (4) can be simplified to

$\Delta{{{\it n}}}={\frac {x_{{{\it v0}}}-x_{{{\it v1}}}}{x_{{{\it t0}}}}} = \frac{\Delta x}{x_t}$ (5)

Now we can say, for the small increments we are dealing with here, that the change in strain is equal to the change in porosity,

$\Delta n = \Delta \epsilon$ (6)

From this and our previous post, we can thus use the change in strain to come to the following:

$\Delta n = m_v \Delta \sigma_x$ (7)

Stating this differentially,

${\frac {\partial }{\partial t}}{\it n}(x,t)=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (8)

In this way we emphasise that both porosity and uniaxial stress are functions of depth and time. We now combine Equations (2) and (8) to yield

${\frac {\partial }{\partial x}}q(x,t)=-m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (9)

Including Permeability

Darcy’s Law (or more properly d’Arcy’s Law) states that

$q(x,t) = -k{\frac {\partial }{\partial x}}H(x,t)$ (10)

where k is the coefficient of permeability and H is the hydraulic head. We then combine Equations (9) and (10) to obtain

$k{\frac {\partial ^{2}}{\partial {x}^{2}}}H(x,t)=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (11)

Noting that

$H(x,t) = \frac{u(x,t)}{\gamma_w}$ (12)

where $\gamma_w$ is the unit weight of water and $u(x,t)$ is the excess pore water pressure generated by the decrease in porosity, we substitute this with the result of

${\frac {k{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)}{{\it \gamma_w }}}=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (13)

Putting It All Together

We now define the parameter

$c_v = \frac{k}{m_v \gamma_w}$ (14)

At this point we need to stop and make an observation: this whole process wouldn’t be worth too much if $c_v$ wasn’t constant (or reasonably so.) The reason it is is that the coefficient of permeability $k$ and the coefficient of volume compressibility $m_v$ both decrease as the void ratio/porosity decrease, and do so at roughly the same rate. That being the case, we substitute Equation (14) into Equation (13) to have at last

$c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (15)

Tidying up the Physics, Governing Equation, Boundary and Initial Conditions, and the Solution

It should be evident that getting to Equation (15) was a major triumph for geotechnical theory. But it’s also evident that there’s one glaring problem: the dependent variable on the left hand side is not the same as the one on the right. It is here that we need to clarify some assumptions behind our equations.

A basic assumption in consolidation theory is that, when the load at the surface is applied, all of this additional load is initially borne by the pore water pressure. Because of the aforementioned permeability, the water will want to “head for the exits,” i.e., the permeable boundaries of the layer being compressed. When the particles have rearranged themselves and the excess pore water has been squeezed out, the settlement should stop (until secondary compression kicks in.) During this process the load on the pore water is being progressively transferred to the soil particles until consolidation has stopped (which, in theory, it never does, as we will see) and the load is completely handed off to the particles.

That being the case, the consolidation equation (15) should be rewritten as

$c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}u(x,t)$ (16)

At this point we should invoke the effective stress equation

$\sigma_x(x,t) = p - u(x,t)$ (17)

where p is the applied pressure, and do the following to determine the time and distance history of the vertical stress $\sigma_x(x,t)$ :

Determine the solution of Equation (16), which will be our governing equation.
Determine the initial and boundary conditions for the problem.
Solve the problem for $\sigma_x(x,t)$ using Equation (17).

There are several ways to solve the governing equation for this problem. Verruijt employed Laplace Transforms to accomplish this; these are very useful, as was demonstrated by Warrington (1997) for the wave equation. For this analysis, we will use separation of variables, a method which is not as fundamental as one might like (this is a demonstration of a method that is) but which is easier to follow than most of the others.

The separation of variables begins by assuming the solution is as follows:

$u(x,t) = X(x)T(t)$ (18)

This means that the solution is a product of two functions, one of time and the other of distance, and that those functions are separate one from another. Substituting this into Equation (16) and rearranging a bit yields

${\frac {{\frac {d^{2}}{d{x}^{2}}}X(x)}{X(x)}}={\frac {{\frac {d}{dt}}T(t)}{{\it c_v}\,T(t)}}$ (19)

Since the left and right hand sides are equal to each other, they are equal to a third variable, which we will designate as $\beta^2$ . In reality they are equivalent to the eigenvalue $\lambda = \beta^2$ , but the utility of the squared term will become evident. In any case,

${\frac {{\frac {d^{2}}{d{x}^{2}}}X(x)}{X(x)}}=-{\beta}^{2}$ (20a)

${\frac {{\frac {d}{dt}}T(t)}{{\it c_v}\,T(t)}}=-{\beta}^{2}$ (20b)

The solutions to these equations are, respectively,

$X(x)={\it C_1}\,\cos(\beta\,x)+{\it C_2}\,\sin(\beta\,x)$ (21a)

$T(t)={e^{-{\it c_v}\,{\beta}^{2}t}}{\it C_3}$ (21b)

We need to pause and consider the boundary conditions. As the problem is shown at the top of the article, the boundary conditions are as follows:

$X(0) = 0$ (22a)

$X'(h) = 0$ (22b)

Equation (22a) represents a Dirichelet boundary condition and Equation (22b) represents a Neumann boundary condition. While the equation for $X(x)$ can certainly be solved for this boundary condition, a simpler way would be to do the following:

Mirror the problem shown above at x = h so that you have two permeable boundaries.
You now have a layer of 2h thickness with Dirichelet boundary conditions on both sides. At the centre of this new layer, there is no flow; the water above it flows upward, and the water below flows downward.
Problems in the field can have either one or two permeable boundaries. The distance h is NOT the thickness of the layer but the longest distance the trapped pore water must travel to escape. Confusing h with the thickness of the layer is a common mistake and should be avoided at all costs.

That said, the second boundary condition is now

$X(2h) = 0$ (22c)

Returning to Equation (21a), because of Equation (22a) $C_1 = 0$ as the cosine is by definition unity at this point. If we then set x = 2h and X(2h) = 0, for real, non-zero values of $C_2$ the boundary condition can be satisfied if and only if

$\beta = 1/2\,{\frac {n\pi }{h}}$ (23)

This is the square root of the eigenvalues. Any integer value of n > 0 is valid for this, and we will have recourse to them all to produce a complete orthogonal set (Fourier Series) to solve the problem.

This change will affect both Equations (21a) and (21b). Combining constants into the coefficient $B_n$ , substituting the results into Equation (18) and making it an infinite sum for the Fourier series yields

$u(x,t) = {\it B_{n}}\,\sin(1/2\,{\frac{n\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{n}^{2}{\pi}^{2}t}{{h}^{2}}}}}$ (24)

At this point we need to consider our initial conditions $f(x)$ . Although many different distributions of initial pore pressure are possible, the simplest one–and the one most commonly used–is a uniform pressure p, which is the same as the surface pressure compressing the layer. For a complete orthogonal set, the coefficients $B_n$ can be computed by (Kreyszig (1988)) as

${\it B_n}=\int _{0}^{2\,h}f(x)\sin(1/2\,{\frac {n\pi \,x}{h}}){dx}{h}^{-1}$ (25)

Substituting $f(x) = p$ and performing the integration,

$B_n = 2\,{\frac{p\left(1-\cos(n\pi)\right)}{n\pi}}$ (26)

Substituting this into Equation (24) and taking the complete sum, we have at last

$u(x,t)=\sum_{n=1}^{\infty}2\,p\left(1-\cos(n\pi)\right)\sin(1/2\,{\frac{n\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{n}^{2}{\pi}^{2}t}{{h}^{2}}}}}{n}^{-1}{\pi}^{-1}$ (27)

More simply we can say that

$u(x,t)=\frac{4p}{\pi}\left(\sin(1/2\,{\frac{\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/3\,\sin(3/2\,{\frac{\pi\,x}{h}}){e^{-9/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/5\,\sin(5/2\,{\frac{\pi\,x}{h}}){e^{-{\frac{25}{4}}\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}\cdots\right)$ (28)

The vertical pressure on the soil skeleton is determined by combining Equations (17) and (28) to yield

It is interesting to note that, although Equation (27) includes all positive non-zero values of n, only the odd ones end up in Equations (28) and (29). For even values of n, $B_n = 0$ .

Conclusion

At this point we have derived the equations of pressure dissapation for consolidation settlement. In a future post we will deal with the second part of the problem, namely how much of the total anticipated settlement has taken place at any given time from initial loading.