Collapsing Criteria And Related Parameters Of Collapsing Soils Biology Essay

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An analytical investigation was conducted in which about 1000 sample points from over 400 borehole locations throughout the Tucson, Arizona basin were used to determine the nature and extent of the variability of selected collapse criteria and collapse-related soil parameters both spatially and with depth. Analysis of seven data sets corresponding to different depth increments below the surface showed high dispersion tendencies as expressed by the value of coefficient of variation (cov). The value of cov was found to increase linearly with depth. All the collapse criteria and collapse-related soil parameters used in the study were found to follow a Gamma distribution except for two collapse-related soil parameters which were found to follow a Weibull distribution. A polynomial regression model was developed for the collapse criterion, Cp, which is defined as the percent volumetric strain occurring in a soil sample when saturated under constant load. The model showed that Cp varies almost linearly with depth. A stepwise regression analysis revealed that Cp is strongly correlated with two collapse-related soil parameters, in-situ dry density, γd and in-situ moisture content, w0. Factor analysis validated this finding by producing two strong factors which described almost 80% of the total variance. These factors were closely related to γd and in-situ degree of saturation, s0 which is directly related to in-situ moisture content w0, in-situ void ratio, e0 and specific gravity of solids, Gs

KEYWORDS: collapsing soil, collapse parameter, factor analysis, regression model

Introduction

A considerable portion of the geotechnical engineer's effort is devoted to the identification of soils and the evaluation of their properties for use in a particular analysis. Because of the variation of soil properties the uncertainty in determining soil parameters can be very high. Even within a limited area, soil properties may vary due to inherent variation or heterogeneity of the soil strata (1). Differences in testing techniques, equipment, and overall human factors also add to the difficulty in evaluating these parameters deterministically. A statistical approach to geotechnical engineering problems provides a rational basis to achieve a more economical solution by avoiding the use of extreme values and by quantifying the uncertainty in a solution (2). In general a soil deposit in a region may be either residual or transported. Also a transported soil may be either alluvial (stream borne) or aeolian (wind borne) or colluvial (gravity transported). When alluvial soils are deposited in an arid or a semi-arid environment, they develop larger voids within their structure due to high evaporation rate during consolidation process. Such soils undergo a large decrease in bulk volume virtually instantaneously upon saturation or load application and are known as collapsing soils. However, even within the context of this definition, it is difficult to identify collapse susceptible soils due to the existence of many different types of clay minerals and many other factors that contribute to the collapse phenomenon. Therefore application of statistical and probabilistic methods in analyzing collapsing soil parameter would provide an optimum solution.

In this study statistical techniques were applied to selected collapse criteria and collapse-related soil parameters for soil in Tucson, Arizona, where the presence of collapse susceptible soils is well documented (3,4). Previous works on this topic was limited only to studies involving either specific areas or specific soil parameters. The purpose of this study was to gather as much information as possible from reliable sources and to use this data with statistical techniques, such as regression and factor analysis, to determine the variation of selected collapse criteria and collapse-related soil parameters in three dimensions. Only the variation of these criteria with depth will be considered here. Other geostatistical aspects of the problem were also studies by the authors and presented elsewhere (5-9).

Collapse Criteria and Related Parameters

When soils are deposited in an arid or semi arid environment, there is insufficient time for them to consolidate under their own weight due to high evaporation rates. They become partially saturated with large voids. Application of typical foundation loads on such soils causes only minor deformation as long as the degree of saturation remains low. As soon as the soil becomes saturated, large deformations take place due to the reduction of volume and collapse of the intergranular structure. If water is readily available, the subsequent volume change and deformation are rapid and the phenomenon is referred to as collapse. In general, collapse-susceptible soils can be identified by a dry density criterion. If the dry density of the soil is sufficiently low to give a void space larger than that required to hold the liquid limit water content, then collapse upon saturation is likely. Otherwise collapse generally occurs only when the soil is loaded. In some cases, collapse susceptible soils are found in residual soils also. In general, these soils are terrace sediments consisting of low density, organic -rich silts, sands, clays of varying percentage and traces of gravels. Their approximate distribution may be in the range of clay 15%, silt 50%, and sand 35%. Figure 1a and Figure 1b show grain size distribution of typical collapsing soil. However, regardless of the formation process and grain size distribution, most collapsible soils are geologically young (10).

Collapsing soils has been recognized throughout the world; particularly in Africa, part of Asia, Europe as well as in the United States. In the United States the severity of the problem has been observed for well over two decades in the Midwestern and Western United States, where soil deposits are generally either aeolian or alluvial.

There are altogether about ten criteria for predicting the collapsing potential of a soil. Some of the criteria are empirical. Others are derived theoretically from consolidation test results. The methods for evaluating collapse susceptibility vary from simple to very complex. Considerable effort has been given to establish criteria for predicting the collapse potential and the critical values for severity of a soil. The more commonly used criteria are described by (11). The two criteria most widely used in the United States are the (12) collapse parameter R and the (13) Double Odeometer (Cp). A third parameter A developed by (10) was also included.

The parameter Cp, R and A are defined as:

(1)

(2)

(3)

Where ∆ec and ∆Hc are changes in void ratio and sample height after saturation under a pressure of 200 kPa, el is the void ratio at liquid limit and H0 is the initial height of sample. Other related parameters are : initial dry unit weight (γd), initial moisture content (w0), initial void ratio (e0), initial porosity (n0), initial degree of saturation (s0) and plastic limit (PL).

Specific cut-off values for collapse susceptibility of each parameter for each collapse criterion (R, Cp, A) and collapse-related parameters (γd, e0, n0, s0, PL) are given below. The critical values for Parameters R, Cp and no were obtained from 4 and given in Table 1. Other critical values were derived from a conventional volumetric-gravimetric relationship among the parameters.

For this study field and laboratory test data were collected from local consulting engineers' offices and from the reports of previous researchers (e.g. 4). In all, data for 992 sample points were collected from 411 different locations within the city of Tucson and its surroundings. The raw data were reduced to obtain parameters in two categories: established criteria, such as Cp , R, A and collapse-related soil parameters, such as, γd,w0,e0,n0,s0 and PL.

The collapse parameter, Cp is obtained from a pseudo-consolidation test (13) and represents the volumetric strain the sample undergoes after it is saturated with water while under a pressure of 200 kPa as shown in Fig.2. Parameters R and A are defined earlier. Some of the parameters listed previously are redundant since they can be calculated from a combination of the others. However, the objective was to investigate the effect of each commonly used collapse-related soil parameters on the collapse phenomenon.

Summary Statistics

In this study a total of 7 to 10 soil parameters were determined for each of the 992 sample points. Sample depths ranged from the surface to about 12.2 m. Table 2 shows the data sets for the range of depths considered and the total number of sample points for each set. Data sets one through six each contain seven parameters: Cp, e0, S0, γd, w0 and n0. Table 3 provides the descriptive statistics for each of the six parameters. Data set-7 contains three additional parameters: R, A, PL. The descriptive statistics of these three parameters are presented in Table 5.

Regression Analysis

For the purpose of obtaining general descriptive parameters, primary attention was given to univariate data sets, i.e. numerical values obtained for a single characteristic of a sample. Linear regression and multiple linear regression analyses were utilized for all data sets in order to investigate possible functional relationships among the variables. Polynomial regression analyses were conducted on each data set to model the variation of the parameters with respect to depth, d.

The functional relationship between the dependent variables (y) and the independent variables (x) was found by the method of least squares. The goodness-of-fit or strength of this relationship was expressed in terms of percent reduction of variance, which is defined as 100 times the sum of the squares of the computed values of y divided by the sum of the squares of the observed values of y.

Table 6 gives the results of the polynomial regression analyses for all data sets in terms of depth expressed in units of feet. The relationships apply for depths varying from 0.5 to 40 feet. Also included in the table is p-value used to test the null hypothesis that the multiple correlation is zero. Table 7 gives the results of a similar analysis for Data set-7 which contains values for all collapse criteria and collapse related soil parameters. The regression equations for Cp, So, d, and w0 given in Table 6 and Table 7 are different because of differences in the number of data points in each of the data sets considered and differences in their dispersion tendencies. The results in Table 6 were derived from 992 data points, whereas those in Table 7 were derived from 219 data points.

A stepwise linear regression analysis was utilized to express the collapse criterion Cp as a function of the collapse related soil parameters. A summary of the results of this analysis is given in Table 8. In general, the analysis shows that the parameters Cp is strongly related to the dry unit weight, γd, and the natural moisture content, w0. It is weakly related to natural void ratio, e0, porosity, n0, degree of saturation, S0, and the depth, d. As has been mentioned previously, the importance of a variable may increase or decrease when it is taken in combination with other variables. Moreover, a linear regression model is not the only model that can be used to explain the data set adequately.

Table 9 contains the results of a similar analysis for R, Cp, and A (Data Set-7). The collapse parameter R is related most strongly to A, PL and d as seen in Step 5. The parameter Cp, as seen in Step 3, is strongly related to S0, A and d. The parameter A, on the other hand, is related to almost all parameters except γd and e0; it is strongly related to R, PL and w0.

Correlation Coefficient Matrix

The correlation coefficient matrices, which show the interrelationships among all the variables, were computed for each of the data sets [Xij] considered in this study. Table 10, which applies to Data set-7,

illustrates these matrices. The correlation coefficient matrix and deviation matrix for a given data set are obtained from the following equations.

(4)

(5)

Where [Z] is the deviation matrix, [R] is the correlation matrix and n is the number of observation in the data set. The correlation coefficient matrix is obtained by left-multiplying the deviation matrix by its transpose, and then multiplying this product n the scalar 1/n.

Since the correlation coefficient is a measure of the linear association among variables, or how well one variable predicts another variable, it is an example of simple linear regression. The value of the correlation coefficient ranges between -1 and +1. If the value of the correlation coefficient is zero, no success in prediction is indicated. Perfect prediction is indicated by either +1 or -1. Based on the information provided by the correlation matrices and the result of several regression analyses for each data set, the following conclusions are made:

No significant correlation was observed between depth and the other parameters for Data Sets 1 through 5 since values of the correlation coefficients were close to zero for these cases. This is to be expected because for each of these data sets the depth increment is small (1.0 or 2.0 feet) as shown in Table 2. The lack of correlation between Cp and depth and the other parameters is also clear from Table 7 where the parameter d does not appear in the regression equation for Data Sets 2 through 5. For Data Set 7, the depth varies from 6.0 to 40.0 feet. The first row of Table 10 gives values of the correlation coefficient sufficiently greater or less than zero to indicate a stronger correlation of d with other parameters than was obtained for any of the other data sets. This conclusion is also supported by the regression equation obtained for Cp from Data Set 7 as presented in Table 7.

The parameter Cp has a moderate negative correlation with γd (-0.40) and w0 (-0.17), as was also seen by the use of regression analysis.

The natural void ratio, e0, exhibits a strong negative correlation with γd(-0.91) and a strong positive correlation with n0 (0.98).

The degree of saturation S0 has a strong positive correlation with w0 (0.83) and a moderate negative correlation with Cp (-0.31) as expected.

As shown in Table 10, the Gibbs collapse parameter R has strong correlation with e0 (0.63), A(0.77), PL (-0.62) and γd(-0.75). Since both R and A are functions of PL, the high correlation is justified and expected. A comparison of the regression equations for R presented in Table 9 with the values of the coefficient of determination (R2) for R presented in Table 10 verifies this observation. The comparison also shows that w0 has only a small effect on explaining variations when it is included in the regression equation. One possible explanation is that when two or more variables are considered together, their effect on a third parameter is markedly different than if either of the two variables is considered alone. Moreover, when there is an extreme value for any one of the parameters, an unusual value in the deviation matrix may be developed. This can result erroneously in a very low or high coefficient in the matrix.

The parameter A was found to be moderately to strongly correlated to R (0.77), Cp (0.33), n0 (0.55), e0 (0.52), S0 (-0.25), PL (-0.52) and γd (-0.62). The parameter yd apparently did not enter into the regression equation. It is often difficult to select the one or two strongest variables for a regression analysis on the basis of the correlation matrices or vice versa, since the importance of a given variable may change when it is entered into the regression analysis in combination with other variables.

Factor Analysis

The various statistical techniques described in the previous section provide for rapid data evaluation, determination of statistical significance, evaluation of parameters, and the fitting of linear models. However, neither the predictor equations nor the correlation coefficients provide a complete characterization of the variability in collapse properties. For this reason a higher order statistical technique called factor analysis was applied to each of the seven data sets. Factor analysis (14) is a powerful multivariate analytical tool which is used to reduce a large number of variables to relatively small numbers of factor. The technique is useful in screening large data sets and helpful in formulating a hypothesis for an observed phenomenon.

Method of Analysis

The mathematical theory for the development of factor analysis is described by (15). Several factor solutions have been development and procedures by which the factor coefficients are computed have been presented. The most general and widely used procedure is the principal-factor solution (15) which is also known as "Method of Axis". In this method, the first axis in an ellipsoid is selected so that the sum of the squares of the distances of points from the axis is minimized. Successive axes, each orthogonal to the preceding ones, are chosen in order to minimize the squares of the distances on the points from the new axis. As a result of this process, the factor loading, which are the correlations between the factors and the variables on each successive axis become smaller and smaller until the number of factors reaches the rank of the correlation matrix satisfactorily.

The factor analysis presented here is based on principal components. Principal component analysis transforms a given set of variables into a new set of variables which are either orthogonal or uncorrelated. There is an essential difference between principal component analysis and factor analysis. In factor analysis, a variable is influenced by certain determinants, some of which are shared by other variables while others are not shared by any other variable. The main purpose of factor analysis is to define a minimum number of hypothetical variables or factors with which the correlation can be reanalyzes.

In factor analysis, the Data matrix [Z], which is n x m matrix containing the values m measurements on n objects, is used to find m new measurements with zero as mean and identity matrix, I, as variance. These measurements, called factors, are linear functions of z scores. With the new measurements, the correlation matrix is recalculated and reanalyzed.

The z scores expressed as a linear function of factors are:

(6)

where i=1,2,3,…m. The correlation coefficient of zi with fm is given by and is known as loading.

With the availability of SPSS and other similar programs, lengthy routine calculations can be shortened considerably. This is particularly important when a large matrix is to be inverted or when the solutions of the characteristic equations used in factor analysis need to be obtained.

The factor model used here is based on the least squares method of the form:

(7)

where aij are factor coefficients (loadings) =as shown in Eqn. (6). In matrix notation Eqn. (7) can be expressed as:

{Z} =[A]{f} (8)

Where [A] is the matrix of common factor coefficients aij and {f} is the vector of factors fj.

Discussion

Factor analysis was applied to all seven data sets for collapse and collapse-related soil parameters (Refer to Table 2). The analysis was also applied to a data set consisting of all the data. The results are presented in the form of rotated factor matrices in Table 11. In grouping the variables with a given factor, factor loadings less than 0.3 were arbitrarily considered to be zero and were reported in order to facilitate reading. An explanation of the significance of the factors for each of the data set follows:

Data Set-1: variable for 125 observations were analyzed. The results in the form of a rotated factor matrix are shown in Table 11(a). Four factors were extracted, however only two are reported. The first two factors accounted for 76% of the total variance. The two factors extracted for the various parameters were :

F1: Cp, e0, n0 ,γd, w0

F2: Cp, S0, γd, w0

Factor 1 is seen to depend on unit weight and other natural properties of the soil deposit. For each factor the variables associated with the factor are quite strong with respect to that factor and negligible for other factors. The inverse relationship between γd (-) and e0 (+) is confirmation of the relationship γd=Gs γw/(1+e0).

Factor 2 can be designed as a factor of the degree of saturation S0. The Cp value found to depend moderately on the degree of saturation in regression analysis. A strong correlation with water content is justified, i.e. Se=wGs.

Data Set-2: For this data set the same variables were considered as for Data Set-1 and 286 observations were analyzed. As shown in Table 11(a) the first two factors account for about 76% of the total variance. The two factors are quite similar to those derived for Data Set-1 except for Cp, which did not come out as strong a factor as was previously obtained for Data Set-1.

Data Set-3: Four factors were extracted and only two are reported in Table 11(a). The Cp value is found to have a strong negative correlation with the degree of saturation S0 within Factor 1. This unusual relationship is probably due to some extreme value in the deviation matrix.

Data Set-4: As shown in Table 11(a), three factors accounted for 91.2% of the total variance, however only two are presented. This is essentially the same as for Data Set-2, except that a weak loading on d is combined with Factor 2.

Data Set-5: As shown in Table 11(a) two factors accounted for 78% of the total variance. The factor loadings and variables are similar to those of Data Set-1.

Data set-6: As shown in Table 11(a), two factors accounted for 77% of the total variance. Factor 1 is the same as was found in all the other sets. Factor 2 appears here as a combination of the other factors.

Data Sets-3, 4, 5, and 6 have the same seven variables and 254, 100, 104 and 123 observations respectively. From the rotated factor matrices of these sets, as given in Table 11(a), it is observed that the factors extracted for each data set are virtually the same. This suggests that all of the collapse-related soil parameters are from the same population.

Data Set-7: A separate analysis was done with the three collapse criteria and seven collapse-related soil parameters, i.e. ten variables altogether for 219 observations. Five factors were found to account for about 96% of the total variance, but only two are shown in Table 11(b). The factors extracted were;

F1 : R, Cp, e0, A, n0, γd

F2 ; d, e0, n0, s0, PL, w0

F3 : d, A

F4 : d, Cp

F5 : d, Cp

Factor 1 is clearly a factor of collapse criteria A and R. The strong negative correlation with γd was expected from their theoretical relationships. Factor 2 is determined by water content, w0, which includes PL and S0. Factor 3 is determined by PL. Factors 4 and 5 are the same as for other data sets.

h) All Data Sets: An additional run was made for all data sets with seven variables for 992 observations. The resulting rotated factor matrix is presented in Table 11(c). Four factors were extracted accounting for 98.7% of total variance, but only two are presented. It is seen that the strongest factor in the factor analysis of all the collapse-related soil parameter data is F1 since it accounts for almost 50% of the variability. It appears most strongly related to e0 and γd, therefore it can be considered as the "unit weight factor". The second strongest factor appears to be F2 since it accounts for an additional 27.5% of the variance. It is most strongly related to S0 can therefore be called the "degree of saturation factor". The third factor accounted for an additional 13% of the variance and was solely related to depth. The fourth factor accounted for an additional 8% variance and was related to Cp.

Table 12 presents a semi-quantitative summary of the results of the factor analysis. It is observed that the cumulative proportion of total variance among the eight sets of data is fairly uniform. There is little difference between the proportions of variance extracted from ten variables vs. the proportion of variance extracted from seven variables.

In order to obtain meaningful associations from a factor analysis in this case, two conditions are required. The first is that the number of variables that characterize the collapse susceptibility should be same for all strata analyzed. The second is that the sample source from different locations and different depths should be identical. In other words, the deposit should be homogenous.

In this analysis the most stable factors appear to be unit weight (F1) and degree of saturation (F2) factors. The introduction of unique factors such as Cp and d into the analysis reflects the independent character of these quantities and does not influence the deduced variables significantly. The parameters R and A may be strong factors as obtained from Data Set-7, but this cannot be investigated fully since they were not included as part of the other data sets due to the unavailability of sufficient data at most depths.

Factor analysis was used in this study as a supplement to multiple regression analysis. In regression analysis, the significance of the regression and correlations can be tested. In factor analysis, there is no significance test for factor loadings. However, factor analysis provides more insight into the understanding of theoretical concepts and procedures than does the multiple regression analysis. In some case, however, rotation in factor analysis may yield meaningless factors (16). The factor analysis performed by producing γd and so as the parameters most closely related to the two strongest factors. In this regard, factor analysis provided to be a sophisticated data reduction technique that confirmed the results of more conventional statistical analysis.

Summary and Conclusions

Values of selected collapse criteria and collapse-related soil parameters obtained from tests performed on about 1000 samples obtained from over 400 borehole locations throughout the Tucson Basin were analyzed statistically to determine the nature and extent of their variability both spatially and with depth. The results of those analyses lead to the following observations and conclusions with respect to variability with depth.

The statistical parameters for all the variables considered in the analysis manifest high dispersion tendencies as evidenced by the high values of the coefficient of variation (cov). The value of the cov was found to increase with depth as indicated in Table 8. The variations of cov with depth were best modeled linearly for Cp, e0, γd, w0 and n0 and non linearly for S0.

In an attempt to fit the theoretical probability distribution function to each parameter; it was found that the distribution for all parameters except γd and n0 were closely approximated by the Gamma distribution function. The distributions for the parameters γd and n0 were found to follow the Weibull distribution function.

All parameters were regressed with depth using a polynomial model in order to take non-linearity within the profile into account. As indicated in Table 5, higher order terms made a very insignificant contribution to the variation and can be ignored.

The result of a stepwise linear regression analysis of collapse parameter Cp presented in Table 8 reveal it to be significantly correlated wit γd and w0. The results of a similar analysis performed on the Gibbs Parameter R and contained in Table 9 show a strong correlation between it and A, L, γd.

Factor analysis enabled the number of variables to be reduced to two independent parameters, γd and S0, that were found to describe approximately 80% of the variation encountered in the data. This suggests that a good estimate of the variation of collapse susceptibility can be obtained from the two collapse-related soil parameters, dry unit weight, γd, and degree of saturation, S0. It also validates the earlier findings revealed by regression analysis.

The data base created with collapse and collapse-related soil parameters can be utilized for information regarding the severity of collapsing problem in a particular location of the city, and for further analysis.

Acknowledgement

The authors wish to acknowledge Western Technologies, Inc., Tucson, Arizona, USA, for allowing the author access to their job files to obtain data for this study at the University of Arizona.

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