ABSTRACT
It is well known that the soil properties variability have a great influence on the assessment of the swelling parameters. The present work aims to analyze the bi-dimensional (2D) variability of the swelling parameters of a soil located in ex-ITE (Tlemcen) through a probabilistic method based on a random description of the soil properties variability. This method allows the identification of the swelling parameters in many areas where no information on the soil properties is available.
Keywords: stochastic; variability; soil; swelling.
INTRODUCTION
A number of research were conducted during the 15 last years to estimate the swelling parameters (Aissa Mamoune S. M 2002), which are described by the amplitude and the pressure of swelling. The stability of the structures founded on expansive soil represents one of the priorities of the engineer. Several models devoted to the indirect estimation of these parameters, based on a statistical analysis (Djedid et al., 2001), were developped.
This indirect estimation allows, in the preliminary phase of the project, to know the swelling parameters in terms of a number of physical parameters. Thus if these latter are available then one could easily assess the potential of swelling without any difficulty. However, the ground is characterized by a great variability and consequently the information on the physical parameters is not always available.
To overcome this restriction, we have developed a probabilistic approach to estimate the swelling parameters which can be not only perceived as complementary to deterministic approach previously decribed but also because this approach could give information on the potential of swelling in area where no information on physical parameters are available. We will focus hence on spatial variability of the physical soil parameters (Bekkouche et al., 2003a). With this goal in mind, one consider that the parameters of soil ground are characterized by random fields defined by a Lognormal distribution function. The probabilistic approach that estimates the swelling parameters, developed in this paper, will be used to assess the effects of the variability of the physical soil parameters on the swelling parameters.
DESCRIPTION OF THE SWELLING PARAMETERS STATISTICAL MODEL
Several authors proposed models of estimation of the swelling parameters (Bekkouche et al 2003b). hence some proposed correlations between these parameters and the physical parameters of the soil (Aissa Mamoune, 2002). These correlations were obtained using statistical analysis of data known also as deterministic approach. For instance, for the model established for the marls of Tlemcen, the pressure (sg in Bars) and amplitude (eg in %) of swelling are in term of the elements lower than 0.02mm "C (in %)", the water content "wn (in %)", and the dry unit weight "gd (in t/m3, 1 t/m3 = 10 kN/m3)," the plasticity index "Ip %", and the depth (m). From these validated relations, it is possible to make indirect estimation of the swelling parameters and this by using only parameters of ground easily and quickly measurable. Thus, the parameters of swelling are estimated by:
![]() | (1) |
![]() | (2) |
Both models (Eq. 1 and 2) were obtained for the case of the marls of Tlemcen (Bekkouche et al. 2003b). The properties of the ground, determined for 4 boreholes located in the site ex-ITE Tlemcen (Fig. 1), was used for the assessment of the swelling parameters at this site using the above models
Figure 1. Location of borehole in the site ex-ITE
GENERAL METHODOLOGY WITH THE SPATIAL VARIABILITY OF SOIL PARAMETERS
In soil mechanics, the probabilistic approach was used in estimating the rate of consolidation of the soil defined as a random media (Gordon et al., 2002). Some authors modeled the problems of the calculation of the safety coefficient by considering that the ground has a variability of its characteristics (Favre et al, 1985); whereas the problem of the consolidation was recently studied (Magnan et al, 1995 and 1996) using the same appraoch. This same approach is used in this work in order to describe the spatial variability of the soil properties.
Decomposition of spatial variability using random variables
A function g(y,z) could be decomposed using the formula:
![]() | (3) |
where is a deterministic function called also derived component and
is the complementary component of variable
.
If we know precisely the function in all points, independently of the actual values in the soil mass (what is not usually the case), then it is easy the study the variability of the soil using only the variable
, whose values at the points of measurement are known, with any form of the derive. Two factors are in fact of great importance when using:
- The current methods that allow the estimation of the relations between the variable and "Y and applicable only for polynomial shape.
- The number of the points of measurements of the variable which is often too low to allow the determination of many coefficients in the function.
The technique used for the determination of the parameters of the function is identical to that used in the linear (or polynomial) regressions between parameters. Table 1 show the deterministic functions but also the standard deviation used in the statistical models (1) and (2) to calculate the pressure and amplitude of swelling.
Table 1. Data Introduced in the Calculation Parameters
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Parameters | Deterministic function (R > 500) |
Standard deviation |
![]() | ||
Dry unit weight gd(t/m3) | ![]() |
0.113 |
Elements lower than 0.02 gmm (%) | ![]() |
8.9 |
Water content wn (%) | ![]() |
4.57 |
The plasticity index Ip (%) | ![]() |
6.44 |
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The random component is obtained by subtraction of the derive from the measured values
![]() | (4) |
Once the effect of the derived extracted and removed, the residue can be regarded as the stationary realization of a random function in a large view. Then the autocorrelation function can be calculated and consequently allows studying the space variability of the parameter under study.
Generation of the vector of the correlated random variables
The theory of the local averages was established by VanMarcke (1983) for one-, two- and three-dimensional random fields. For the needs for our study we will treat the two-dimensional random field. After performing the discretization of the random field of the soil parameter in elements (vertical and horizontal zones), it is replaced by a vector of random variables for which one must generate values to carry out the simulation of Monté-Carlo. This latter consists in carrying out a great number of calculations of the parameter studied using the parameters of the statistical models (amplitude and pressure of swelling) in calculation as random variables.
The algorithm used (Fenton, 1994) allow the generation of a random vector of central Normal distribution, and the covariance matrix [COV]. This matrix was calculated on the basis of methodology which follows: The function of correlation used is a "triangular Model" (VanMarcke.1983):
![]() | (5) |
The corresponding variance function is:
![]() | (6) |
With i = 1 or 2
D1 is the distance of the horizontal zone. D2 is the distance of the vertical zone.
q1 is the scale of fluctuation of the horizontal zone.
q2 is the scale of fluctuation of the vertical zone
The covariance between gA and gA' (Figure 2) is:
![]() | (7) |
![]() | (8) |
For this study one used the nonconditional case (the two scales of fluctuation are independent), consequently the function of variance is written as follow:
![]() | (9) |
Figure 2. Definitions of the interval Dij in a bidimensionnel profile (Vanmarcke, 1983)
This algorithm generates positive as well as negative values. The latter are inappropriate to represent the parameters of the ground whose values are always positive. To avoid this difficulty, the proposed algorithm was adapted so that the generated values are of a Lognormal distribution.
In order to generate S simulations of a vector {V} of N random variable Vi (i=1, 2,.., N), we use the following stages:
- Decomposition of the matrix [COV] by the Cholesky method into two matrixes, a triangular inferior and it transposed matrix?
- Generation of a vector {Y} of N uncorrelated random variable Yi Normaly distributed. This vector contains the random values which were generated by means of the Matlab5.3 software using the command: normrnd(m, Var, N, S)
m is the mean of the value gA(z) found using Eq (4).
var is the variance of the value gA(z) found using Eq (4).
N is the number of elements of the profile.
S is the number of simulations.
The command normrnd generates random values which follow normal law with a mean m and a variance var, for each soil property used in the calculation. The computation of the vector {V} is done by the following formula
![]() | (10) |
A transformation (Eq. 11) is done for each variable Vi to obtain from {V} a vector {g} of N random variable whose distribution is lognormal with an average m and a variance of s2.
![]() | (11) |
where mlns and slng are mean and variance of lng. They are derived from m and s2 through:
![]() ![]() | (12) |
Note that the same procedure is applied for elements lower than 0.02mm and water content wn.
Thus thanks to this methodology, in all point of the profile of soil, the three geotechnical parameters (dry unit weight, elements lower than 0.02mm and water content wn of the soil can be determined and injected into the equations (1) and (2) and hence will allow us to deduce the swellings parameters at any point of the soil mass.
APPLICATION FOR THE SITE OF EX-ITE TLEMCEN
the soil profile used in this survey is Ex-ITE composed primarily of marls. This profile, of 10m of depth and 35m of width, was broken up into 1200 quadratic elements. In addition, for each element the characteristics of the ground were simulated starting from developed methodology and injected into the statistical models (1) and (2) to obtain the swelling.
These values were randomly generated by considering following assumption:
- Laws of distribution modelled starting from the data available,
- Linear derive with respect to depth
- Matrix of variance which corresponds to the triangular autocorrelation function which is strongly influenced by the scales of fluctuation (vertical and horizontal) but also by the number of elements of the studied profile.
In addition, we adopted the approximation method by the average distance between two successive intersections of the values of the property studied with its average. This method allows us to estimate the scale of fluctuation (Jaksa, 1995)
![]() ![]() | (13) |
where qv: vertical scale fluctuation and qh: horizontal scale fluctuation
The swelling parameters are obtained in all points of the soil mass. The results are shown in figures 3 (pressure of swelling) and 4 (amplitude of swelling) with an interpolation of the computed values for 1200 elements of the soil mass. Careful examination of the results shows clearly that the swelling parameters are characterized by a great variability with respect to X and Z. It should be noted that this result was obtained although the variation of the soil properties used to calculate the swelling parameters was considered only in-depth.
It is noticed that the pressure of swelling increases with depth whereas the amplitude of swelling decreases with depth. On the other hand, one notes that the variation of the pressure for the same depth is very important whereas for amplitude, this important variation is not obtained at the surface. This result could be explained by the nature of the variation of the swelling parameters obtained from actual measurements. Indeed, figure 5 shows that amplitude decrease linearly according to the depth. In turn, it is very difficult to identify a relation between the pressure of swelling and the depth, which could be due to poor representative drive. Consequently, the fluctuation of the pressure (figure 3) is important.
Figure 3. Pressure of swelling (in bars)
Figure 4. Amplitude of swelling (in %)
Figure 5. Variation of the amplitude and the pressure of swelling obtained from the actual measurment
The results show that the variability of the soil properties has great effect on the variation of the swelling parameters. Indeed, it is completely clear that the traditional method based on interpolation between points of measurements, i.e. only between points where information is recorded cannot take into account the space variability of the parameters of swelling because of lack of information (Bekkouche et al, 2003a). In fact information on these points can not be easily generalizable to the whole profile. The results obtained by traditional approach, appear to be over- estimated (Bekkouche et al, 2003a). In turn probabilistic approach developed in this work can give valuable results, if the probability laws are well established. Indeed, developed methodology is easily usable if other probability laws than the laws normal or lognormal.
CONCLUSIONS
The estimation of the swelling parameters is an important stage in the design of structures. Several indirect estimating models of the swelling parameters based on the geotechnical parameters easily and quickly measurable were developed. However, the soil is characterized by a great variability and consequently the use of these models remains tributary of the availability of the information on the soil properties which are generally very rare. This work aims to overcome this restriction by the establishment of a methodology that allows the probabilistic generation of the soil properties.
The results obtained show that estimating the swelling parameters, without taking onto account the variability of the physical soil properties, led to an over-estimate of these parameters and consequently can have considerable effects on the stability of the structures.
REFERENCES
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