From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”

Note: I have extensively revised this to boost the rigor of the equation derivation and to clarify the relationship between void ratio and shear/elastic modulus.

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.

Theory of Elasticity Considerations

Dr. Briaud notes that consolidation specimens are confined by a ring when they are tested in an odeometer. That’s to simulate the fact that one-dimensional consolidation theory (and the settlement theory that goes along with it) is based on the assumption that a) you have a uniform surcharge and b) the layer experiencing consolidation settlement is “confined” by the semi-infinite mass we assume the soil to be. That assumed, for theory of elasticity purposes we assume uniaxial strain conditions, which I discuss in my post Constitutive Elasticity Equations: Uniaxial Cases.

The uniaxial strain (assuming the x-direction is vertical) is

$\epsilon_{x}={\frac{\sigma_{{x}}\nu}{\left(1-\nu\right)\lambda}}$ (1)

Since, from Constitutive Elasticity Equations: Three-Dimensional Formulation,

$\lambda = \frac{\nu E}{(1+\nu)(1-2\nu)}$ (2)

Substituting for $\lambda$ ,

$\epsilon_x = \frac{\sigma_x(1+\nu)(1-2\nu)}{E(1-\nu)}$ (3)

which can be rewritten (Tsytovich (1976))

$\epsilon_x = \frac{\sigma_x}{E} \beta$ (4)

where

$\beta = 1-\frac{2\nu^2}{1-\nu}$ (5)

The variable $\beta$ is a measure of the effect of lateral confinement of either the odeometer specimen or the compressed layer in the field. At $\nu = 0$ , $\beta = 1$ and there is no confinement effect. At $\nu = 0.5$ , $\beta= 0$ , and the confinement effect is total: the “fluid” (which what it is in reality) is incompressible.

If we define, assuming that $\sigma_x$ is the uniaxial stress applied to the specimen/soil,

$m_v = \frac{\epsilon_x}{\sigma_x}$ (6)

then combining Equations (4) and (6) yields

$E = \frac{\beta}{m_v}$ (7)

Strain vs. Void Ratio

Using theory of elasticity involves stress vs. strain. Unfortunately, as Verruijt observes:

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.

So we must relate strain to void ratio. To do this, we need to consider the compression of the soil from one void ratio to another, as is shown in the diagram below (from Broms):

Manipulating the equation in the diagram, the relationship of strain to void ratio is as follows:

$\epsilon=\frac{\delta_p}{H_o} = {\frac {{\it e_0}-{\it e_1}}{1+{\it e_0}}}$ (8)

From Equation (6),

$m_v = \frac{e_0 - e_1}{\sigma_x(1+e_0)}$ (9)

Combining and rearranging Equations (8) and (9),

$\delta_p = m_v H_o \sigma_x$ (10)

At this point we can make the following observations:

The modulus of elasticity $E$ and the coefficient of volume expansion $m_v$ are linearly, inversely proportional to each other.
The physical difference between the two is that coefficient of volume expansion includes the effect of confinement while the elastic modulus does not. (Consider that most tension and compression tests on materials are conducted in an unconfined way.) This is the reason why we can write Equation (10) the way we can.
The coefficient of volume expansion is also important when we consider time consolidation.
If we assume $\nu$ to be constant and both of these parameters are constant for the range of void ratios under consideration, consolidation settlement can be considered in a linear elastic way.

Facing Reality About Consolidation Settlement

Unfortunately this is not the case. Let’s pick up where we left off in our last post on the subject, with the E vs. $\epsilon$ plot below:

It was noted at the time that the apparent elastic modulus increased more or less linearly with strain. Since void ratio and strain are linearly dependent, we would expect a similar looking result if we did a void ratio plot.

To solve this problem, we first observe that the elastic and shear modulus $G$ are linearly proportional via $\nu$ (as are $E$ and $m_v$ .) From this, it can be noted that the shear modulus can be estimated for a soil (excluding strain-softening effects) as follows:

$\frac{G_{0}}{p_{atm}}=SF\left(e\right)\left(\frac{\sigma_{0}}{p_{atm}}\right)^{\bar{n}}$ (11)

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,

$F\left(e\right)=\left(1+e\right)^{-3}$ (12)

We plot this as follows, including a logarithmic trend line:

Plot of Equation (12) with Logarithmic Trend Line

The logarithmic correlation isn’t perfect; however, as Verruijt notes in his commentary on his 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.

This, of course, leads us to the classic equation we use in one-dimensional consolidation theory for normally consolidated soils,

$\delta_p = \frac{C_c H_o}{1+e_o} \log{\frac{\Delta p + \sigma_o}{\sigma_o}}$ (13)

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 increasing void ratio (or 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.