Studying swelling soils by multidimensional statistical analysis methods as the basis for engineering structures foundations in Sudan

Ключові слова: набрякаючі ґрунти; фізичні властивості; ступінь набрякання; несуча здатність; зонування території; фундаменти; кластерний, факторний і кореляційно-регресійний аналізи; переважальні фактори і кореляційні залежності.

Ключевые слова: набухающие грунты; физические свойства; степень набухания; несущая способность; зонирование территории; фундаменты; кластерный, факторный и корреляционно-регрессионный анализы; преобладающие факторы и корреляционные зависимости.

Various analytical methods and mathematical modeling are now used to determine similar physical and mechanical properties of swollen soils on new construction sites and areas where the destruction of buildings and structures has already taken place.

Physical-mechanical properties of swollen soils are the end product of the influence of not one, but a whole set of natural and man-made processes (conditions of formation, flooding, anthropogenic load, etc.). These processes affect the interrelated change in soil properties and the nature of the relationships between them. But these links are not stored in "pure" data sets. Paired correlations between the observed values of variables actually make it impossible to determine the similarity of physical and mechanical properties and which processes are decisive in changes in the engineering and geological properties of swollen soils.

To determine similar properties of swollen soils in different parts of the study area, we used cluster analysis, and to identify processes that are decisive in changes in the engineering and geological properties of these soils - factor analysis. These research methods were developed by (Davis, 1990 and Jereskog et al., 1980). In hydrogeological and engineering-geological research they were widely used by (Reshetov, 2004; Seifeldin, 2007, 2008; Taranov, 2008, 2014; Chomko D.F. et al., 2002, 2012, 2004, 2014 and Chomko F.V. et al., 2014).

The essence of the agglomerative (unifying) cluster procedure is a step-by-step calculation of the Euclidean distance between all pairs of soil samples in different areas and combination of the pair at each step for which a minimum of this distance has been achieved. In our case, it is combining soils physical and mechanical characteristics. Engineering and geological surveys in Khartoum and El Jazeera were used to implement this technique. The sample consisted of 57 test soil samples and included the following predictors: y, w, wL, wp, Ip, ф, c, ssw psw, wsw (thus, each soil sample was interpreted as a point in ten-dimensional space).

Based on the results of cluster analysis of the general data matrix, a diagram of Euclidean distances of the consolidation was constructed step by step (Fig. 4), as well as a dendrogram (Fig. 5) - a one-dimensional graph containing information about the location and numbers of objects and wells, interrelationships between different soil samples. Fig. 4 shows that all samples were combined in 56 steps with a minimum consolidation distance (Euclidean distance) in the first step of 0.000 and a maximum of 292,628 in the last step.

Fig. 4. Diagram of stepwise consolidating distances

Fig. 5. Dendrogram combining 57 objects (10 parameters)

Considering the tree-like hierarchical structure in Fig. 5, it follows that it can be divided into seven clusters (see symbols), split into smaller subclusters. The first four clusters (counting from right to left) are consolidated at short distances, from 0.000 to 99.812, which allows us to speak about the homogeneous properties of these soils. The soils of clusters 5-7 differ from the soils of the first four clusters and from each other: the Euclidean distances at which the consolidation took place, more than 125, indicate a noticeable difference in their physical and mechanical characteristics.

A taxonomy was performed to clarify the results - a hierarchical clustering of two equal matrices (obtained by dividing the total), characterizing the soils of Khartoum region and peripheral areas, which clearly shows a significant difference in their Euclidean distances.

Fig. 6. Factor distribution scheme

In Fig. 6 all points of soil sampling are plotted on the map of Mesopotamia. The cluster of objects in Khartoum and the adjacent area is noteworthy. This is explained, apparently by the factor of capitalism, by their sparseness in the rest of Mesopotamia. However, there is a visible trend, suggesting that the study area can be "divided" into two zones: Northern El Jazeera with Greater Khartoum where the soils are more homogeneous (the first four clusters), and Southern Mesopotamia where the soils are very different (clusters five - seven). Obviously, the boundary (dotted line) between these zones is quite conditional due to the lack of data because of a relatively small volume of construction in the agricultural part of the region.

At the next stage of data processing the overall matrix was subjected to factor analysis.

This method is one of the most effective means of identifying patterns hidden in data sets. As a rule, in engineering geology, there is no possibility to directly observe and measure factor processes. They can be judged only by the final results of the processes manifestation that are reflected in the values of different physical and mechanical characteristics of soils.

The factor analysis is a very effective means of compressing information by moving from the original data to new variables - factors. There is a minimum number of new factors that are linear combinations of the source data, and these new variables contain the same amount of information.

According to the results of factor analysis (R-modifica- tion) of the initial matrix, a number of factors which account for 100% of the total effect on the indicators of physical and technical properties of swollen soils have been identified. Only three factors make a significant contribution, their weight after the rotation of the matrix being more than 10%. They reflect the contribution of each of the factors to the total variance of the sample.

In addition, factor analysis revealed characteristic elements of physical and technical characteristics of swollen soils in each of the factors (Table 3).

Table 3. Factor loads after rotation (R-modification)

The first factor (weight 22.65%) contains the following soil properties: natural moisture (bond strength 0.738074), plasticity number (bond strength 0.728427) and lower limit (bond strength 0.400961). Other elements do not make a significant contribution to this factor due to their bonding. Characteristic elements of the first factor are natural humidity, the amount of plasticity and the lower limit.

The second factor (weight 15.09%) includes swelling pressure (bond strength -0.748681), adhesion (0.717359) and upper limit (-0.512390). The swelling pressure and the upper limit are included in this factor with a negative bond strength. Other elements do not significantly contribute to this factor due to the bonding strength. Characteristic elements of the second factor are the pressure of swelling and adhesion.

The third factor (weight 10.50%) includes the angle of internal friction (bond strength 0.966581) and the lower limit (0.891289), which are the characteristic elements of this factor.

The characteristic elements of each factor are used to compile the regression equation, determining the characteristics of soil strength ф and c in different zones of Mesopotamia.

To analyze the obtained factors distribution on the territory of Mesopotamia, the load of each of them on all wells was determined, using Q-modification of factor analysis.

This allowed to draw a map-scheme of factor loads. Distribution of these three factors values in the territory of Mesopotamia is shown in Fig. 6.

Analyzing the joint distribution of all three factors in Mesopotamia, several areas where two or three factors act simultaneously, can be identified. Thus, the first and second factors intersect in the area of sites № 18 and № 10 in Khartoum. The same is true between № 5 and № 22 in El Jazeera. In the area of sites № 3, № 20 and № 19 in Khartoum, the second and third factors intersect.

The intersection of the same factors is observed in the area between sites № 3 and № 16 in El Jaseera. The first and third factors intersect in the area of site № 9 in Khartoum. The intersection of the same factors is observed in the area between sites № 3 and № 5. This indicates the similarity of engineering and geological processes in these areas.

At the final stage the regression equation was obtained to determine the characteristics of soil strength in the first zone (correlation coefficients are equal to 0.95 and 0.545, respectively) by the method of stepwise regression analysis, using characteristic elements.

ф = 13.86 * y - 0.014 * Ir - 237.48; c = 17.23 * y + 0.133 * Ir - 300.45.

Using these expressions, the values of the internal friction angle ф and the specific adhesion c due to the degree of soil swelling were calculated. The obtained results are given in Table 4 and can be used in the design of buildings and structures in Greater Khartoum and Northern El Jaseera.

Table 4 Normative values of expansive soil characteristics in Greater Khartoum and Northern El Jazeera