The Problem of Size: Gazetas and Stokoe (1991) Revisited

In a recent exchange Dr. Mark Svinkin, who has contributed several well-read articles to this site, pointed out that he had commented on a paper by Gazetas and Stokoe (1991.) The paper, Dr. Svinkin’s comments and their response can be found here:

Although this research was done a long time ago, it’s worth revisiting because of the issue that Dr. Svinkin brings up: the issue of size, that it’s not a straightforward business to extrapolate the results of model tests in controlled environments to full-scale foundations in actual stratigraphies.

In my fluid mechanics laboratory course, I discuss the issue of dynamic similarity, how one can take an airfoil or other flying object on a small scale and, using things such as the Reynolds Number, extrapolate those results to full-scale aircraft. This has proven very useful in the development of aircraft, especially before (and even long after) the development of simulation using computational fluid dynamics.

With geotechnical engineering, it has not been quite as simple. Attempts to use things such as centrifuge testing have not been as successful as, say, wind tunnel testing has been for aeronautics. Part of the problem is, as I like to say, that geotechinical engineering is not non-linear in the same sense as fluids are. Another problem is that the earth is not as homogeneous as the atmosphere, even when altitude and weather effects are considered (and these influence each other in the course of events.) But underneath all of this there are some fundamental issues that have complicated the issue of foundation size, and Dr. Svinkin points this out. My intent is to amplify on that and remind people that these issues are still relevant.

Dr. Svinkin points out the following figure from Tsytovich (1976.) I’ve referenced this text in several recent posts. Tsytovich looks at many problems in soil mechanics differently from our usual view in this country, and his perspective is frequently insightful. (An excellent example is here.) In this diagram he shows the effect of basic foundation size on the settlement of the foundation, and Tsytovich’s own explanation of this follows:

Relationship between settlement of natural soils and dimensions of loading area (from Tsytovich (1976).) The variable F is the area of the foundation, thus the square root of F is the basic dimension of the foundation and, in the case of square foundations, the exact dimension b or B (see below.)

Thus, Fig. 90 shows a generalized curve of the average results of numerous experiments on studying the settlements of earth bases (at an average degree of compaction) for the same pressure on soil but with different areas of loading. Three different regions may be distinguished on the curve: I —the region of small loading areas (approximately up to 0.25 m²) where soils at average pressures are predominantly in the shear phase, with the settlement being reduced with an increase of area (just opposite to what is predicted by the theory of elasticity for the phase of linear deformations); II — the region of areas from 0.25-0.50 m² to 25-50 m² (for homogeneous soils of medium density, and to higher values for weak soils), where settlements are strictly proportional to $\sqrt{F}$ and at average pressures on soil correspond to the compaction phase, i.e., are very close to the theoretical ones; and III — the region of areas larger than 25-50 m², where settlements are smaller than the theoretical ones, which may be explained by an increase of the soil modulus of elasticity (or a decrease of deform ability) with an increase of depth. For very loose and very dense soils these limits will naturally be somewhat different.

The data given can be used for establishing the limits of applicability of the theoretical solutions obtained for homogeneous massifs to real soils, which is of especial importance in developing rational methods of calculation of foundation settlements.
From Tsytovich (1976)

Although much of the discussion centred on Tsytovich and Barkan (1962,) there is evidence elsewhere to underscore this problem, which Tsytovich sets forth in a very succinct manner.

To begin with, let us consider the basic equation for elastic settlement, which was discussed in this post, and is as follows:

$s = \frac{\omega p b (1-\nu^2)}{E}$ (1)

where

$s =$ settlement of the foundation at the point of interest
$\omega = I =$ influence factor, given in the table below
$p =$ uniform pressure on the foundation
$b = B =$ smaller dimension of rectangle or dimension of square side
$\nu =$ Poisson’s Ratio of the soil
$E =$ Modulus of elasticity of the soil

The values of $\omega$ are shown below.

Similar values can be found in both the Soils and Foundations Manual and NAVFAC DM 7.

It is clear that, once one is past the basic soil properties and the pressure applied on the foundation, the settlement is proportional to the basic dimension of the foundation, which is exactly what is taking place in Region II. This is also why the bulk modulus of the soil is not a basic soil property, as I discuss in this lecture. When we consider plate load tests, we must correct them for the difference between the size of the test plate and the size of the foundation, as this slide presentation shows.

Since we are dealing with foundation dynamics, one item that seems to have fallen out of the whole discussion is that of Lysmer (1965). Lysmer’s Analogue, which reduces the response of a soil under the foundation to a simple spring-damper-mass system, defines the spring constant as follows:

$K = \frac{4Gr}{1-\nu}$ (2)

where $K$ is the spring constant of the soil and $r$ is the foundation’s radius. If we break it down further, as is done in Warrington (1997,) and develop a unit area spring constant under the foundation, we have

$k = \frac{4Gr}{F(1-\nu)}$ (3)

where $k$ is the equivalent unit area spring constant under the soil. Equation (3) in particular shows that, for a given unit load on a foundation, the static portion of the reaction is inversely proportional to the basic size of the foundation. (The unit damping constant is actually independent of the area for round foundations.)

These results show that, while the effect of size may differ from one model to the next, it cannot be overlooked in any attempt to extrapolate physical model tests of any kind to actual use. This effect is further complicated by variations in shear modulus due to either strain softening, layered stratigraphy, effective stresses or other factors. The effect of the stratigraphy is further magnified by the fact that larger foundations have larger “bulbs of influence” into the soil and thus layers that smaller foundations would not interact with become significant with larger ones.

“Sand box” tests have other challenges. While they attempt to simulate a semi-infinite space, reflections from the walls of the box are inevitable, especially with periodic loads such as were present in this test. These challenges were documented in the original study. (An interesting study using another one of these boxes is that of Perry (1963).)

The failure of geotechnical engineering to adequately resolve the size issue, both in terms of design and in terms of using laboratory data to simulate full-scale performance, remains a frustration in geotechinical engineering. Hopefully other types of models will help move things forward, along with advances in our understanding of soil behaviour and our ability to replicate it both experimentally and numerically.