Posted in Geotechnical Engineering, Soil Mechanics

Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” (or at least clarifying the problem)

Three years ago I posted Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results. Since that time he has been elected President of the American Society of Civil Engineers and I am in the process of retiring from full-time teaching, so our trajectories are a little different. (He’ll catch up, don’t worry.)

Nevertheless his Presidency would go unfinished if some explanation of the pet peeve wasn’t given. To remind my readers it is as follows:

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.

This post won’t be very rigourous or mathematically detailed, but more of a qualitative statement of the problem. Perhaps a proper solution will solve this dilemma; I think it certainly needs it.

To start, let’s pick up where we left off, with the E vs. \epsilon plot below:

It was noted at the time that the apparent elastic modulus increased more or less (that’s about as good as it gets with most geotechnical phenomena) linearly with strain.

From this, it can be noted that the shear modulus can be estimated for a soil (excluding strain-softening effects) as follows:


where the notation is shown in the source. Let’s make some assumptions:

  • Poisson’s ratio remains constant, thus the relationship between strain and elastic modulus is constant.
  • S is constant for a given soil type.
  • Effective stress \sigma_o for a given sample (that’s just about a given for triaxial tests in any event)
  • Other constants, such as p_{atm} and \bar{n} also remain constant.

That leaves the variable F(e) to change. The shear (and by extension the elastic) modulus of a material is a function of the void ratio. From the same source,


We can convert this to strain by noting the following relationship, which is written so that compressive strain is positive:

\epsilon={\frac {{\it e_0}-{\it e_1}}{1+{\it e_0}}}

Substituting that into the equation before it yields

F(e) = -{\frac {1}{\left (1+{\it e0}\right )^{3}\left (-1+\epsilon\right )^{3}}}

Let’s consider the case of e_0 = 1 . Substituting that into the previous equation and plotting it yields the following result:

It’s not perfect, but it’s close to a linear relationship, at least in the strains under consideration. And, of course, it shows an increasing shear modulus with increasing strain (or decreasing void ratio.)

Verruijt makes two important observations that should be noted. The first is his commentary on the image below, in the caption.

When performing the test, it is observed, as expected, that the increase of vertical stress caused by a loading from say 10 kPa to 20 kPa leads to a larger deformation than a loading from 20 kPa to 30 kPa. The sample becomes gradually stiffer, when the load increases. Often it is observed that an increase from 20 kPa to 40 kPa leads to the same incremental deformation as an increase from 10 kPa to 20 kPa. And increasing the load from 40 kPa to 80 kPa gives the same additional deformation. Each doubling of the load has about the same effect. This suggests to plot the data on a semi-logarithmic scale. In this figure log(σ/σ0 ) has been plotted against ε, where σ0 denotes the initial stress. The test results appear to form a straight line, approximately, on this scale. The logarithmic relation between vertical stress and strain has been found first by Terzaghi, around 1930.

The second is his comment on the use of strain vs. void ratio:

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.

The need to treat compression due to settlement completely differently than that of elastic (or elasto-plastic) settlement is one of the anomalies of geotechnical engineering. The observation that the elastic modulus decreases with void ratio (or increases with strain) is a start in putting the two together and presenting a more or less unified theory of soil deformation. Coupled with agreement on using strain in consolidation tests, this would bring us a long way to solving the dilemma of Jean-Louis Briaud’s–and some of the rest of our–pet peeves.

Posted in Soil Mechanics

Soil Mechanics Textbook by Tsytovich Available

Up until now the only comprehensive soil mechanics textbook we offered for download was Verruijt’s. We now add to that N. Tsytovich’s Soil Mechanics. Download is at the link or at the book cover below; information on the book is as follows:

This is a textbook in the course of Soil Mechanics for higher-school students of civil engineering and hydrotechnical engineering, and also for students of other specialties associated with construc­tion of engineering structures, such as road constructors, ameliorators, geologists, soil scientists.

The Author has made an attempt to write a concise course on the basis of a wide synthesis of natural sciences and to present the theore­tical data in the most simple and comprehensive form, without depreciating, however, the general scientific aspect of the problem; his other aim was to present a number of engineering solutions of problems in the theory of soil mechanics (calculations of strength, stability and deformability), which might be widely used in engi­neering.

Some problems in the book are discussed from new standpoints which take into account the principal properties of soils: contact shear resistance, structure-phase deformability (including creep of skeleton), compressibility of gas-containing porous water, and the effect of natural compaction of soils.

The book shows some new methods used for determination of characteristics of soils and gives some new solutions of the theory of consolidation and creep of soils, which can be used for predictions of settlement rates of foundations of structures and their time varia­tions; a separate chapter discusses rheological processes in soils and their significance.

Topics include the following:

    • Geological Conditions of Soil Formation
    • Components of Soils
    • Structural Bonds and Structure of Soils
    • Physical Properties and Classification Indices of Soils
    • Compressibility of Soils. The Law of C o m p a c t i o n
    • Water Perviousness of Soils. The Law of Laminar Filtration
    • Ultimate Contact Shear Resistance of Soils. Strength Conditions
    • Structural-Phase Deformability of Soils
    • Features of the Physical Properties of Structurally Unstable
      Subsidence Soils
    • Stress Distribution in the Case of a Three-Dimensional Problem
    • Stress Distribution in the Case of a Planar Problem
    • Pressure Distribution over the Base of the Foundation of
      Structures (Contact Problem)
    • Stressed State Phases of Soils with an Increase in Load
    • Equations of Ultimate Equilibrium for Loose and Cohesive
    • Critical Loads on Soil
    • Stability of Soils in Landslides
    • Some Problems of the Theory of Soil Pressure on Retaining Walls
    • Soil Pressure on Underground Pipelines
    • Kinds and Causes of Deformations
    • Elastic Deformations of Soils and Methods for Their Determi­nation
    • One-Dimensional Problem of the Theory of Soil Consolidation
    • Planar and Three-Dimensional Problems in the Theory of Fil­tration Consolidation of Soils
    • Prediction of Foundation Settlements by the Layerwise Summa­tion Method
    • Prediction of Foundation Settlements by Equivalent Soil Layer Method
    • Stress Relaxation and Long-Term Strength of Cohesive Soils
    • Creep Deformations in Soils and Methods for Their Description
    • Account of Soil Creep in Predictions of Foundation Settlements
    • Dynamic Effects on S o i l s
    • Wave Processes in Soils under Dynamic Loads
    • Changes in the Properties of Soils Subject to Dynamic Effects
    • The Principal Prerequisites for Taking the Dynamic Proper­
      ties of Soils into Account in Vibrational Calculations of Founda­tions
Posted in Geotechnical Engineering, Soil Mechanics

Mexico City’s surprising crisis: the city is sinking

The city with a metropolitan population of over 20 million is sinking at a rate of almost 50 centimeters (20 inches) per year — and this isn’t stopping anytime soon.

At first glance, you’d be inclined to attribute this to the strong earthquakes that sometimes strike Mexico City. But while earthquakes can cause their own damage, they’re not the main culprit here. Instead, it’s something much more inconspicuous: subsidence.

You can read it all here. Put into geotechnical terms, the bed of old Lake Texcoco has some very high void ratio soils, and as a large city puts pressure on them the void ratio decreases as the voids between the soil grains shrink. Thus the entire city has severe settlement, total and differential.

A diagram, from the Swedish geotechnical engineer and academic Bengt Broms, showing how we consider the volume and mass/weight relationships in soil. The particulate matter of the soil means that the soil mass has three components: solid (particles,) water (in the voids) and gas/air (also in the voids.) That simplification is shown above, along with the definition of void ratio.
A diagram, again from Bengt Broms, illustrating the problem in Mexico City and whenever what we call consolidation settlement takes place. The soil particles have been combined into one mass (hatched area.) As pressure is applied, the particles come closer to each other and the volume of the voids decreases, thus we have settlement.
A photo from Mexico City showing the effects of subsidence many years ago. The top of the pole was originally the ground surface before structures were built on it and subsidence started. The photo and an explanation can be found in the textbook Soils in Construction. Needless to say, it’s only gotten worse in the intervening years. Photo courtesy of J.R. Bell.

My own lecture on the subject of settlement and consolidation is here.

Posted in Academic Issues, Soil Mechanics

The “Line of Optimums” Approach for Compaction

There are some things in geotechnical engineering that don’t get really good (if any) coverage in many textbooks, which means that those who go on into that part of civil engineering are blindsided by their appearance. One of these is the “line of optimums” approach for compaction evaluation. The only formal textbook I know of that covers it is Soils in Construction, for which I must credit my co-author, Lee Schroeder. It also appears in the Soils and Foundations Reference Manual.

The line of optimums approach seeks to answer a key question in compaction: how much compactive energy is necessary to effect a given compaction? We have the Standard Proctor and the Modified Proctor test, but when we’re trying to determine a specific compactive effort for a particular soil and project, we need more flexibility.

I discuss this in my class video for Soil Mechanics: Compaction and Soil Improvement, but let’s consider an example, in this case from Rebrik (1966).

Compaction Chart with Multiple Compactive Energies and Line of Optimums, from Rebrik (1966)

Lines 1, 2, 3 and 4 represent compaction curves for a soil, but with a different number of blows (25, 50, 100 and 150, as shown in the chart.) The energy variation is explained in my video. In any case with the increase in compactive energy is a closer packing of the soil particles. The peak dry density/unit weight also increases with compactive energy, although the water content decreases (which makes sense as the void ratio decreases with greater compaction.) Line 6 through the peaks in the compaction curve is referred to as the “line of optimums.” Once we establish this line we can make a determination of the compactive effort we will need based on the result we are looking for, taking into consideration the degree of relative compaction we are prepared to allow for.

Line 5 is the zero air voids curve.

The line of optimums method is a good one for compaction evaluation, and we hope that this little presentation helps you to understand it.


  • Rebrik, B.M (1966) Vibrotekhnika v burenii (Vibro-technology for Drilling.) Moscow, Russia: Nedra.