A Two-Fold Empirical Approach for Estimating the Preconsolidation Stress in Clay Deposits

  • Karim Kootahi
  • Paul W. Mayne
Conference paper


The conventional method for determining preconsolidation stress of clay is via one-dimensional consolidation tests on undisturbed samples but a first-order approximate value can be estimated from simple empirical models that relate preconsolidation stress to easily measured soil properties, such as natural water content, liquid limit, plasticity index, etc. In this paper, a two-fold simple empirical model for predicting preconsolidation stress that bifurcates at an overconsolidation ratio (OCR) of 3 is developed. Two independent data sets which consist of about 2000 samples of fine-grained soils are used as model-building and model-validation data sets. The results of applying the new and existing simple empirical models to validation data set indicate that (1) the proposed model provides quite acceptable estimates for soils with different stress histories and sensitivities, (2) the predictive ability of the new model is quite superior to existing models, and (3) the performance of prior existing models can be classified as unacceptable.


Preconsolidation stress Index properties Empirical models 

1 Introduction

Among important engineering properties of clay, the preconsolidation stress \( ( {\sigma^{{\prime }}_{p} } ) \) is the most important one, as this key parameter affects shear strength and compressibility of fine-grained soils. Specifically, the stress history of the soil is represented by the preconsolidation stress (or “yield stress”), often reported in a dimensionless form as the overconsolidation ratio \( ( {{\text{OCR = }}\sigma^{{\prime }}_{p} /\sigma^{{\prime }}_{vo} } ) \).

Conventionally, \( \sigma^{{\prime }}_{p} \) is determined by performing one-dimensional consolidation tests on undisturbed soil samples and test results are interpreted by the method of Casagrande (1936). A variety of problematic issues may arise during either or both processes (i.e., during sampling and interpreting the results) which makes the determined values of preconsolidation stresses somewhat doubtful. As such, several different graphical or numerical methods have been developed to either better define \( \sigma^{{\prime }}_{p} \) or correct for the effects of sample disturbance. Considering these difficulties, empirical statistics-based models that use various physical and/or index properties as entry information appear to be very beneficial. The virtue of using simple index properties as predictor variables is that (1) these indices are almost insensitive to sample disturbance, (2) they are familiar to all geoengineers, and (3) they can be easily and economically determined. Such empirical models can also be useful in cross-checking and validating laboratory-determined values of \( \sigma^{{\prime }}_{p} \) (Kootahi and Mayne (2017). Simple correlations for \( \sigma^{{\prime }}_{p} \) have been linked to few index properties (e.g., Stas and Kulhawy 1984; Nagaraj and Murthy 1986). Table 1 summarizes the currently available simple empirical models for estimating \( \sigma^{{\prime }}_{p} \).
Table 1.

Available simple empirical models for predicting the preconsolidation stress.




\( \sigma^{{\prime }}_{p} /p_{a} = \, 10^{(1.11 - 1.62LI)} \)

Clays with St < 10

Stas and Kulhawy (1984)

\( \sigma^{{\prime }}_{p} \, ( {\text{kPa}} ) = 10^{{[ {5.97 - 5.32( {wn/LL} ) - 0.25 \, \,{ \log }\,\sigma^{{\prime }}_{vo} } ]}} \)

Overconsolidated uncemented soils

Nagaraj and Murthy (1986)

\( \sigma^{{\prime }}_{p}\, ( {\text{kPa}} ) = 10^{(2.9 - 0.96\,LI)} \)

Onshore and offshore clays

DeGroot et al. (1999)

\( \sigma^{{\prime }}_{p} /p_{a} = 1.070\,LI^{ - 0.295} \)

Sensitive to quick clays

Ching and Phoon (2012)

Notes: wn = natural water content; LL = liquid limit; LI = liquidity index; St = sensitivity; \( \sigma^{{\prime }}_{vo} \) = effective overburden stress; pa = atmospheric pressure (=1 atm ≈ 100 kPa)

Accordingly, this study aimed at developing a robust simple empirical model for estimating the preconsolidation stress of clays. For this purpose, the methodology of this study used a two-phase strategy, consisting of an initial model-building process, followed by a model-validation procedure. That is, a good quality model-building data set is collected first to derive the empirical expression for predicting \( \sigma^{{\prime }}_{p} \). Later, a separate model-validation data set is compiled and it is used for assessment purposes.

2 Model-Building and Model-Validation Data Sets

Two independent data sets of experimental data which consisted of about 2000 samples of fine-grained soils were compiled from different sources. The data set for developing new model (model-building data set) consists of 120 data points from 59 different sites all over the world. An independent model-validation data set which contains 1850 data points from 194 different sites worldwide was compiled for assessing the performance of both newly proposed and existing empirical models. However, after careful screening, data were selected for the model-building and model-validation data sets only if they were obtained from high-quality undisturbed sampling techniques. Due to space limitations, a detailed description of the two data sets is not given here and only summaries of their characteristics are given. Detailed description and discussion of the both model-building and model-validation data sets (with a full list of sources utilized) can be found in Kootahi and Mayne (2016, 2017).

The model-building data set consists of index properties of different clay soils and their corresponding consolidation curves that were taken from different sources and processed consistently. That is, most of the consolidation curves were digitized from the original sources and redrawn again, and then preconsolidation stresses were determined using the method of Casagrande (1936). The observed measurements include natural void ratio en, natural water content wn, liquid limit LL, plastic limit PL, plasticity index PI, liquidity index LI, preconsolidation stress \( \sigma^{{\prime }}_{p} \), and effective overburden stress \( \sigma^{{\prime }}_{vo} \). This database contains clay soils of different physical and mechanical properties. The model-validation data set consists of index properties and preconsolidation values of different fine-grained soils collected from the literature. In order to ensure a robust and effective validation process, the records in the model-validation data set completely differ from those used in developing new and existing models. The model-validation data set covers a wide variety of soil types. The statistics for main geotechnical characteristics of the both model-building and model-validation data sets are summarized in Table 2. It is seen from Table 2 that the parameter ranges for validation data set are wider than those for model-building data set and thus a conclusive validation test is expected. Moreover, model-validation data set covers most conditions encountered in practice.
Table 2.

Statistics for main geotechnical characteristics of the both model-building and model-validation data sets.


Model-building data set

Model-validation data set


Standard deviation




Standard deviation



\( \sigma^{{\prime }}_{p} /p_{a} \)









\( \sigma^{{\prime }}_{vo} /p_{a} \)




































w n


















3 Statistical Analysis and Discussion

3.1 New Model and Its Performance on Model-Building Data Set

The new model for predicting \( \sigma^{{\prime }}_{p} \) was developed by statistically-fitting equations to the model-building data set. More specifically, multiple linear regression analyses have been performed to express the effective preconsolidation \( ( {\sigma^{{\prime }}_{p} } ) \) in terms of the properties influencing \( \sigma^{{\prime }}_{p} \). which were identified through Pearson’s correlation matrix. However, following the finding of Nagaraj and Murthy (1986), which states that estimates of log \( ( {\sigma^{{\prime }}_{p} } ) \) can be obtained based on knowledge of log \( ( {\sigma^{{\prime }}_{vo} } ) \) and ratio of wn/LL, logarithmic transformations of variables (i.e., log \( ( {\sigma^{{\prime }}_{p} } ) \), log \( ( {\sigma^{{\prime }}_{vo} } ) \), log (wn), and so forth) were considered. Moreover, all stresses have been made dimensionless by use of a reference stress equal to atmospheric pressure: pa = 1 atm ≈ 100 kPa.

The correlation matrix, calculated for logarithmic values of variables (Table 3), indicated a very strong positive correlation between log \( ( {\sigma^{{\prime }}_{p} } ) \) and log \( ( {\sigma^{{\prime }}_{vo} } ) \) and a significant correlation between log \( ( {\sigma^{{\prime }}_{p} } ) \) and log (LI). The correlation matrix also indicated mild correlation between log \( ( {\sigma^{{\prime }}_{p} } ) \) and log (wn) or log (LL), but no significant correlation at all between log \( ( {\sigma^{{\prime }}_{p} } ) \) and log (PL) or log (PI).
Table 3.

Correlation matrix for logarithmic values of different soil properties.

Soil property

\( \sigma^{{\prime }}_{vo} /p_{a} \)

w n





\( \sigma^{{\prime }}_{p} /p_{a} \)







\( \sigma^{{\prime }}_{vo} /p_{a} \)






w n














The model-building data set was split into two subsets and regression equations were fitted to each subset of data. An overconsolidation ratio (OCR) of 3 (OCR < 3 and OCR ≥ 3) was chosen as distinguishing parameter between the two subsets because these two ranges of OCR (i.e., OCR < 3 and OCR ≥ 3) represent two different classes of soil behavior (i.e., contractive vs. dilative behavior). Indeed, soils are usually grouped by OCR as follows: normally- to lightly-overconsolidated (NC-LOC) clays and moderately- to heavily-overconsolidated (MOC-HOC) clays. The results of the applied statistical analyses are expressed in the following two-fold model:
$$ { \log }\, ( {\sigma^{\prime}_{p} /p_{a} } ) = 0.21 + 0.89\, { \log }\, ( {\sigma^{\prime}_{vo} /p_{a} } ) + 0.12\, { \log }\, ( {LL} ) - 0.14 \,{ \log } \,( {w_{n} } ) \;\, {{\rm{for}}} \; {\text{OCR}}_{\text{s}} < 3 $$
$$ { \log }\, ( {\sigma^{{\prime }}_{p} /p_{a} } ) = 0.90 + 0.71 \,{ \log }\, ( {\sigma^{{\prime }}_{vo} /p_{a} } ) + 0.53 \,{ \log } \,( {LL} ) - 0.71\, { \log }\, ( {w_{n} } ) \; \, {{\rm{for}}} \; {\text{OCRs }} \ge 3 $$

Based on the results of analysis of variances, both of these regression equations are significant to a confidence level of 99.99%. The signs of regression coefficients in Eqs. 1a and 1b are exactly what are expected. In fact, the larger the preconsolidation stress, the smaller the natural void ratio (or equivalently the smaller the natural water content). The liquid limit is the limiting value for natural water content and thus it should be positively correlated with preconsolidation stress. A positive correlation between \( \sigma^{{\prime }}_{p} \) and \( \sigma^{{\prime }}_{vo} \) is expected for most overconsolidated soils, in which overconsolidation is due to changes in overburden pressure (not due to cementation).

For ease of use, the proposed relationships (Eqs. 1a and 1b) can be retransformed to the original units, and therefore the direct expressions are:
$$ \sigma^{{\prime }}_{p} /p_{a} = 1.62 ( {\sigma^{{\prime }}_{vo} /p_{a} } )^{0.89} ( {LL} )^{0.12} ( {w_{n} } )^{{{-}0.14}} \quad {\text{for}}\quad {\text{OCRs}} < 3 $$
$$ \sigma^{{\prime }}_{p} /p_{a} = 7.94 ( {\sigma^{{\prime }}_{vo} /p_{a} } )^{0.71} ( {LL} )^{0.53} ( {w_{n} } )^{{{-}0.71}} \quad {\text{for}}\quad {\text{OCRs }} \ge 3 $$
Note that these equations are consistent with the model developed by Nagaraj and Murthy (1986). Specifically, the \( \sigma^{{\prime }}_{vo} \) and ratio of wn/LL exist in both models. However, the signs of coefficients in Nagaraj and Murthy’s model are not consistent with what are expected for overconsolidated soils (e.g., the sign of \( \sigma^{{\prime }}_{vo} \) should in fact be positive). Equation (2a) has a coefficient of determination (R2) equal to 0.95 and Eq. (2b) has an R2 of about 0.75. The calculated overall R2 for the complete model (1 ≤ OCRs ≤ 19) is R2 = 0.89. The performance of retransformed model on model-building data set is presented graphically and quantitatively, as shown in Fig. 1. The quantitative measures include coefficient of determination (R2) and standard error of estimate (SEE). It can be seen from Fig. 1 that the proposed model provides a very good fit to the model-building data set. Note that since OCR depends on \( \sigma^{{\prime }}_{p} \), some engineering judgment is needed to decide whether equation (Eq. 2a or 2b) is applicable to a specific depth in a specific deposit. In the next section, an efficient classification scheme for discriminating between NC-LOC clays and MOC-HOC clays is presented.
Fig. 1.

Performance of proposed model on model-building data set.

3.2 Discriminant Function for Differentiating NC-LOC from MOC-HOC Clays

In order to find a mathematical expression for predicting which class (NC-LOC vs. MOC-HOC) a particular sample with known index properties (wn, LL, etc.) and effective overburden stress \( ( {\sigma^{{\prime }}_{vo} } ) \) belongs, a linear discriminant analysis (LDA) was performed. LDA finds a discriminant function (DF) that is linear combination of the variables and best discriminates between predefined groups. The coefficients of the DF are estimated so that the distance between the means of populations is maximized. The populations (or predefined groups), which were intended to make their distance as large as possible, are as follows: NC-LOC clay samples and MOC-HOC clay samples. Discriminant functions with different predictor variables were tried, yielding the following significant function, in which DS is the discriminant score:
$$ DS = 5.152 \,{ \log } \,( {\sigma^{{\prime }}_{vo} /p_{a} } ) - 0.061\, LL - 0.093 \,PL + 6.219 e_{n} $$

This DF is significant to a confidence level of 99.99%. The mean discriminant score (cut-off value) for this discriminant function is 1.123 and NC-LOC clay samples have DS < 1.123 and MOC-HOC clay samples have DS > 1.123. The performance of DF on model-building and model-validation data sets is as follows: In the model-building data set, DF correctly classified 107 of the 120 samples, for 89% correct classification rate. In the model-validation data set, which is of particular interest, DF correctly classified 1668 of the 1850 samples, for 90% correct classification rate.

The fact that the discriminant function presented here works very well over a wide range of inputs suggests that it may provide a good scheme for guessing whether equation (Eq. 2a or 2b) is applicable to a specific depth in a specific deposit. Thus, Eqs. 2a3 is combined into a single model.

3.3 External Model Validation and Comparison with Existing Models

In order to assess the generality of the new model and to assess its relative performance in comparison to existing simple empirical approaches, the newly proposed and existing models were applied to the compiled model-validation data set. In the present study, the following existing empirical models are examined: Stas and Kulhawy (1984), Nagaraj and Murthy (1986), DeGroot et al. (1999), and Ching and Phoon (2012). In the application of the proposed model to model-validation data set, it was assumed that the OCR was not available for each case. Therefore, in order to predict \( \sigma^{{\prime }}_{p} \) for each case, combination of Eqs. 2a3 was used. The performance of the newly proposed model on validation data set is presented graphically and quantitatively, as shown in Fig. 2. Moreover, Fig. 3 shows the performance of the existing empirical models on validation data set. The quantitative measures of model performance, which are used to compare the overall accuracies of models, include coefficient of determination (R2), coefficient of efficiency (E), mean absolute error (MAE), and mean and coefficient of variation of K (μK, COVK), where K is the ratio of predicted \( \sigma^{{\prime }}_{p} \) over the corresponding measured \( \sigma^{{\prime }}_{p} \).
Fig. 2.

Performance of proposed model on model-validation data set.

Fig. 3.

Performance of existing empirical models on model-validation data set.

From Figs. 2 and 3, it is evident that the predicted values of \( \sigma^{{\prime }}_{p} \) using the proposed model are closer to the perfect prediction line compared to the existing models. In the case of the proposed model, the major source of disagreement between predicted and measured preconsolidation stresses is attributable to the misclassified cases for which the wrong equation was used for predicting \( \sigma^{{\prime }}_{p} \) and it is not due to the inherent defects of the proposed expressions (Eqs. 2a and 2b). Fortunately, there are only 182 misclassifications out of the 194 soil deposits in the validation data set (i.e., only one misclassification in each soil deposit) and it should not be a worrisome issue. Moreover, reduction in the number of misclassified cases and consequently improvement in the performance of the proposed model can be obtained by using newer data mining approaches such as artificial neural networks instead of using Eq. 3.

The quantitative measures of model performance in Figs. 2 and 3 demonstrate good performance of the proposed model and its superiority over existing models. It is noted that coefficient of efficiency (E), with values ranging from–∞ to 1, has several advantages over R2 in assessing model performances (See e.g., Kootahi and Mayne 2016). An E-value of 1 indicates perfect agreement between estimated and measured values but values of E < 0 indicate unacceptable model performances. Based on the calculated E values, predictive performances of all existing empirical models can be evaluated as unacceptable, as their E-values range from –0.93 to 0.00. The proposed model also has the smallest COV and COVs for existing models are quite large. Indeed, it may be seen from Fig. 3 that existing models either severely underestimate or severely overestimate the preconsolidation stress.

4 Summary and Conclusions

Using a two-phase strategy, which allows for developing and validating statistical-empirical models, a new approach has been developed for predicting the preconsolidation stress of clays \( ( {\sigma^{{\prime }}_{p} } ) \) from simple index properties (wn, LL, PL) and the effective overburden stress \( ( {\sigma^{{\prime }}_{vo} } ) \). The compiled data were processed consistently to form a good quality model-building data set and then multiple linear regression techniques were applied, which yield a highly significant algorithm that bifurcates at an OCR = 3. Discriminant analysis was also applied to find a discriminant function for discriminating between soils with OCR < 3 and soils with OCR ≥ 3. The performance of the new model, along with existing simple empirical models, was evaluated using an external model-validation data set and four quantitative measures of model performance (namely, R2, E, MAE, and COV) were used to compare the overall accuracies of models. Based on the results of external validation, the performance of the proposed model was an improvement (R2 = 0.88, E  = 0.87, MAE = 68 kPa, COV = 0.51) over any of the existing models (R2 range 0.11–0.29; E range −0.93–0.00, MAE range 119–211 kPa, COV range 1.16–4.00). Furthermore, according to the calculated coefficient of efficiencies, all of the prior existing models have unacceptable performance.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Civil Engineering, Saghez BranchIslamic Azad UniversitySaghezIran
  2. 2.Geosystems Engineering Group, School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA

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