STADYN Wave Equation Program 5: Revision of Soil Properties (Modulus of Elasticity)

This is the first post since the presentation of Warrington and Newman (2018) and it introduces what is probably the most significant revision since the original study: the values of the modulus of elasticity for various values of the dimensionless parameters $\xi$ and $\eta$ .

Quick Overview of the “xi-eta” Concept

The basics of the concept are shown below.

The mathematical implementation is shown here:

The results for this (in this case, for the modulus of elasticity) can be seen here:

The original $\xi-\eta$ value matrix was never intended to be “set in stone.” We have already made one revision, by using Jaky’s Equation to estimate the at-rest lateral earth pressure coefficient, and thus Poisson’s Ratio $\nu$ .

The results for this are outlined in Warrington and Newman (2018).

Introduction of Hardin and Black Method for Computing Elastic Modulus

Probably the least satisfactory estimate from the original $\xi-\eta$ scheme was the modulus of elasticity. It was improving this correlation that has led to most of the changes implemented. The research relative to the TAMWAVE project was used extensively in this revision.

First, the Hardin and Black method for estimated the shear modulus of a soil is described here. It allows estimating the small-displacement shear modulus knowing the void ratio, effective stress and the lateral earth pressure coefficient. The last parameter is described above; the void ratio is a function of the specific gravity, the dry unit weight and the saturation state of the soil. If we add the stratigraphic configuration of the soil, we can estimate the effective stress. We can thus estimate the shear modulus–and by extension the elastic modulus–of the soil using some very basic soil properties.

We first turn to the dry unit weight. The TAMWAVE project indicated some revision of how this was computed was in order. It also indicated that, with some generalisation, the dry unit weight could be stated as strictly a function of $\eta$ from all soils. Doing this is a good step to simplifying the scheme and thus improving our chances for success in the optimisation process.

Based on the data, the following correlation was proposed for dry density in $\frac{kg}{m^3}$ :

The dry density essentially varies from $1200 \frac{kg}{m^3}$ at $\eta = -1$ to $2000 \frac{kg}{m^3}$ at $\eta = 1$ .

The specific gravity is unchanged from the original formulation, and is as follows:

Combining the two (we apologise for the mixed graphic formats) yields the following for void ratio:

Because of the nature of the Hardin and Black formula (and of the soils themselves,) the program has an upper limit for the void ratio of 2 and a lower one of 0. This replaces the previous limitations of modulus of elasticity and density.

At this point we need to consider the effect of hyperbolic softening on the shear and elastic moduli. This topic was discussed here and, as was the case with TAMWAVE, we selected a ratio of 0.15. Applying that (and assuming an effective stress of 1 atmosphere) yields the following estimate for elastic moduli in MPa:

In addition to having a stronger basis of fact in the soil properties, by limiting the void ratio the possibility that the correlation goes negative is also avoided.