# STADYN Wave Equation Program 2: Effective Stress, At-Rest Lateral Earth Pressures and Poisson’s Ratio

With the output improved, we can turn to the first topic of interest. Before we do that, we need to discuss our test cases.

## Test Cases

The original study featured several test cases. For this and subsequent installments, we will concentrate on three of those:

1. FINNO2, which features the actual static load test output from Finno (1989).
2. SEASIA, which is a GRLWEAP comparison from a notional offshore pile case in Southeast Asia.
3. MANDK3, which features the inverse solution of an instrumented pile in the New Orleans area. The original GRL report for this is Mondello and Killingsworth (2014). Several soil profiles were analyzed using both standard and annealed polytope methods of analysis. For this study the four-layer annealed polytope case will be featured, as its results a) seemed to be the most reasonable and b) matched the standard results very closely.

Details of the original results are shown in the original study.

## Effective Stresses, Vertical and Horizontal

The concept of effective stress is a foundational one in geotechnical engineering, and is discussed in textbooks such as Fellenius (2015) and Verruijt and van Bars (2007). As is the case with “classical” methods of analysis, it is necessary to compute these for successful geotechnical finite element analysis. However, there are two important considerations that come up with finite element analysis that can usually be ignored with simpler methods.

The first is that it is necessary to apply gravity forces at the start of the run to the elements to simulate the impact of effective stresses on the soil finite elements. This is one of those important steps in analysis that most manuals and discussions of commercial codes mention in passing but do not detail how they are done. It is one of those phenomena that has “gone dark” in the literature. The original study presented an outline on the procedure for computing the effective stresses and applying them to the elements.

The second is that the computation of effective stresses concentrate on the vertical stresses and generally ignore the horizontal ones until retaining walls come into play. Nevertheless, for any three-dimensional continuum such as the semi-infinite soil mass we assume in geotechnical analysis, horizontal stresses are guaranteed to exist, if nothing else via the theory of elasticity. If we use the theory of elasticity, which is valid in an elastic-purely plastic model such as is used in STADYN until the yield point is exceeded, the relationship between the horizontal and vertical stresses is given by the equation (Verruijt and van Bars (2007))

$\frac{\sigma_{x}}{\sigma_{z}}=\frac{\nu}{1-\nu}\$ (1)

where $\nu$ is Poisson’s Ratio, $\sigma_z$ is the vertical stress, and $\sigma_x$ is the horizontal stress.  We normally define the left hand side thus:

$K = \frac{\sigma_{x}}{\sigma_{z}}$ (2)

For our case, the lateral earth pressure coefficient $K$ is the at-rest lateral earth pressure coefficient, generally expressed as $K_o$. This is reasonable for this case because, since we have a semi-infinite soil mass, the soil literally has nowhere to go, thus all of the horizontal strains are zero. This is a key assumption for Equation 1. We can thus combine Equations 1 and 2 to yield

$K_o = \frac{\nu}{1-\nu}\$ (3)

In theory, we could compute the lateral earth pressure coefficient using Equation 3 and “reasonable” values of Poisson’s Ratio.

Turning back to STADYN itself, soil properties in most cases (and especially for inverse problems) are defined using the “ξ – η” system, which in turn uses typical values of various soil properties to reduce computing same for a given typical soil state to two dimensionless variables. Using this system, Poisson’s Ratio is a function of ξ and η, and is thus varied as these dimensionless parameters are varied. The variation of ν with ξ and η is shown in the original study.  As a practical matter, even if Poisson’s Ratio is measured for each project and soil profile (an unlikely situation at best,) the problematic nature of soil elasticity makes accuracy of the parameter equally problematic. Another approach is to begin by considering the following empirical relationship

$K_o = 1 - sin(\phi)$ (4)

This is Jaky’s Equation. It has been shown to be reasonable for normally consolidated soils, although there are other relations in use for both normally and overconsolidated soils. Values of the at-rest lateral earth pressure coefficients are limited to $0\leq K_{o}\leq1$, the upper limit achieved for a purely cohesive soil where $\phi = 0$. As the original study noted, Equation 4 is a common expression to compute horizontal stresses from vertical effective stresses in finite element codes, and is used to compute the horizontal effective stresses in STADYN.

Unfortunately this leaves an inconsistency between the way horizontal stresse sare computed between the effective stress computation and subsequent computations. To remedy this problem, we can combine Equations 3 and 4 and solve for Poisson’s Ratio to yield

$latex \nu=\frac{sin\phi-1}{sin\phi-2}\$ (5)

Poisson’s Ratio is varied here as $0\leq\nu\leq0.5$, where once again the upper limit is for purely cohesive soils. This indicates that these soils act as a fluid, which is nearly true for very soft clays. The main problem with this result is that, when ν = 0.5, the consitutive matrix experiences singularities. The simplest way to deal with this problem is to limit Poisson’s Ratio to a value below this one. In STADYN this value is 0.45.

## Comparison With Previously Generated Values and Forward Test Cases

Having defined a new way of generating values of Poisson’s Ratio, we can compare these values both with the original values and with the two forward test cases. We will leave the inverse test cases for a later post.First, the original “ξ – η” relationship to generate Poisson’s Ratio values is shown in Figure 1.

Figure 1: Poisson’s Ratio “ξ – η” Relationship, Original Configuration

We can see that Poisson’s Ratio is independent of η in this configuration,and
$0.25\leq\nu\leq0.45$ for $-1\leq\xi\leq1$. Computing Poisson’s Ratio based on Equation 5 yields the result shown in Figure 2.

Figure 2 Poisson’s Ratio “ξ – η” Relationship, Jaky’s Equation

There are several differences to note, as follows:

1. The maximum value is ν = 0.5 for the revised relationship. To prevent singularities in the constitutive matrix, in actual application Poisson’s Ratio is limited as described earlier, a similar concept to the “corner cutting” for Mohr-Coulomb failure.
2. For purely cohesive soils, ν is invariant in both cases. As ξ is reduced and internal friction is increased, ν varies with η. In other words, Poisson’s Ratio tends to decrease in cohesionless soils as the relative density of the soil increases.
3. The range of possible values for Poisson’s Ratio in both cases is very much the same; it is simply distributed differently in the continuum.

As far as the forward test cases (the first two) are concerned, SEASIA is the same in both cases because the soil is assumed to be purely cohesive, thus Poisson’s Ratio is the same in both cases.

For FINNO2, the simplest way to compare the two is to compare the hammer blow counts and the Davisson static load test result. That comparison is as follows:

• Original Poisson’s Ratio Computation: Blow count 17.7 blows/300 mm, Davisson failure load 976 kN.
• Revised Poisson’s Ratio Computation: Blow count 17.8 blows/300 mm, Davisson failure load 980 kN.

The differences for this case are not that substantial. The differences which emerge in the inverse case will be discussed in a subsequent post.

One other change that was made in the program was the stopping point for the static load test. The program is capable of interpreting the static load test for several criteria; however, how long the static load test is conducted (in the computer or in the field) depends upon the criteria being used to interpret it. The program now stops the test depending upon when the selected criterion is reached; Davisson’s criterion is the default. It is also interesting to note that, since the Jacobian is fixed, Davisson’s criterion, which generally stops before the others, is probably more suitable for STADYN’s current algorithm.

References are given in the original study.

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