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A numerical model for rapid determination of plasticity of fine-grained soils

By David A Fall, Engineering Geology Research Group, School of Earth, Environmental and Physical Sciences, University of Portsmouth

Determination of the Atterberg Limits of a cohesive soil is a fundamental geotechnical characterisation test. BS1377:1990: Part 2 (Methods of test for civil engineering purposes) recommends two methods to establish these parameters: the Casagrande cup method, largely superseded by the cone penetrometer method (also recommended), to determine the liquid limit and a process of 'hand rolling' of the sample to determine the plastic limit. From these two values the plasticity index is then calculated.

The plastic limit test is time-consuming, relying on hand-drying the sample, and highly subjective.Such subjectivity can introduce errors into the final calculation of the plasticity index of a soil.

This paper describes the development of a numerical model that allows the plasticity index of a fine-grained soil to be calculated from the results of the liquid limit test alone. There is no need to determine the plastic limit in the laboratory, saving analytical time and reducing the risk of introducing further error into the plasticity index calculation.


Plasticity is an important characteristic of fine-grained soils. It is a measure of the soils' ability to undergo irrecoverable deformation at constant volume without fracture. Plasticity is due to the presence of clay minerals or organic material. Depending on the moisture content, a soil may exist in a liquid, solid or plastic state.

The plasticity index of a soil is a fundamental geotechnical characterisation test. It defines the range of moisture content where the soil exists in a plastic state, ie between the liquid limit - the moisture content at which the soil acts as a fluid - and the plastic limit - the moisture content at which the soil starts to behave as a brittle solid.

It is widely used to predict the engineering behaviour of a soil, as well as being a measure of the materials' compressibility (Tomlinson, 1993) and an input parameter in the determination of the Activity Index (Skempton,1953).

BS 1377 (1990) describes two methods for determining the liquid limit: the Casagrande cup method and the cone penetrometer method, which since 1975 has largely superseded it in the UK. In the latter, a cone of known geometry and weight is dropped into the soil and the depth of penetration recorded.

The moisture content of the soil is systematically increased, by the addition of deionised water, until a minimum of four penetrations between 15mm and 25mm deep are achieved.These results are plotted on a graph of moisture content against depth of penetration and a line of best fit, termed the liquid limit line in this paper, drawn through the data points.The liquid limit is then defined as the moisture content at which 20mm deep penetration occurs (Figure 1).

The plastic limit of the material is determined by rolling the soil into a thread, on a glass plate, using light finger pressure. The soil is said to have reached its plastic limit when it begins to crumble at a thread diameter of 3mm.The plasticity index (I

P) is a simple measure of the difference between the plastic limit (wP) and the liquid limit (wL): IP= wL- w P1The plastic limit test involves hand-drying the sample and can be time-consuming. This can lead to shortcuts in the procedure, which in turn can introduce errors in the overall calculation of the plasticity index.

By definition, the plasticity index of a soil is a measure of the affinity to water of its clay mineral constituents.Although the clay mineral cation and the pore-fluid chemistry will have an effect on this (Barbour, 1990), the gradient of a liquid limit line must reflect the amount and type of clay mineral present in the soil.

Using the variations in this gradient, the author has developed a mathematical model for determining the plasticity index for fine-grained soils from the results of the liquid limit test results, reducing the risk of introduced errors in the plasticity index calculation, as well as shortening the process because the plastic limit test does not need to be carried out.


Liquid and plastic limit test data from 100 clay and silt samples from different soil formations in southern England were examined to establish the relationship between the gradient of the liquid limit line and the plasticity index.

These observations allowed a mathematical function to be developed that would predict the plasticity index from liquid limit data.To ensure minimum variability in this highly subjective test, laboratory testing and the plasticity index calculation was carried out by a single operative, ensuring that any errors were systematic and reproduced throughout the dataset.

Empirical verification of the model was achieved by undertaking a series of comparative analyses. Simultaneous standard test and predictive methods were carried out on 10 of the soil samples. The plasticity index predicted from the new algorithm was compared with that calculated using the previously recommended British Standard method.


Table 1 shows the development of the numerical model and illustrates the shortcomings of each of the equations discussed below. The first relationship identified is that the plasticity index of a soil must be inversely proportional to the gradient of the liquid limit line. In general terms:

1IP a g21IP= k (g ) g is the gradient of the liquid limit line k is the constant of plasticity

In order to take into account the variations in the clay mineralogy of individual soil samples equation 2 was modified to include the liquid limit value:1IP= wL ( g)

x k 3Rearranging equation 3 to find k: IPk = 13awL ( g ) Examining the laboratory test data and dividing the laboratory determined plasticity indices by the right-hand side of equation 3a, an initial value 0.75 was adopted for k as this gave results closely resembling those calculated using the traditional methods (Table 1).

However, for soil in the higher plasticity range, the inverse of the gradient of the wLline becomes greater than 1. In such cases equation 3 predicts plasticity indices which are higher than the associated liquid limit values (Table 1).

To overcome this inconsistency, a method of reducing the gradient function to a value close to unity in the higher IPrange - without any major alteration to the I Pvalues in the lower range - was required.

The obvious way to achieve this was the inclusion of a root function, and by a process of substitution of different root values into equation 3 the cube root was found to give the best results (Table 1).Equation 3 can then be rewritten as:1IP= wL ( g)1x k . 43The revised equation gives good results for soils with I

Pvalues lower than 55% (Table 1).

However, for soils with an I Pvalue above this, the error between the modelled value and the value attained during testing progressively increased, the model consistently predicting values higher than those determined from laboratory testing.Therefore, a means of gradational reduction in the value of k needed to be introduced.

Soils with high liquid limit values tend to have high I Pvalues (equation 1), so the gradational reduction factor includes the w

Lvalue, which is multiplied by a numerical constant (0.001) and the product subtracted from the constant of plasticity k, and equation 4 is rewritten to include this gradational correction factor:1IP= wL ( g)1x (0.67 - 0.001wL) 5 3After the inclusion of a correction factor it became necessary to reduce the value of k from 0.75 to 0.67. This new value was selected by an iterative reduction, from the initial 0.75, until a good correlation was achieved over a wide range of IPvalues.


Table 2 shows the frequency and magnitude of the predictive error obtained by the numerical model. Of the 100 samples tested, all the predicted plasticity indices were within 5% of the value attained from a standard laboratory procedure, with 86% of results within 3%. To statistically analyse the correlation between the model and British Standard method, the data was plotted on a XY scatter plot (Figure 2). This illustrates the close correlation between the two sets of I Pvalues, further evidenced by a correlation coefficient of 0.981. Figure 3 illustrates the distribution of the predictive error about the laboratory calculated value.This normal distribution, with no noticeable skewness, suggests that the value of the function (0.67 - 0.001LL) is correct.

The results of the simultaneous verification tests are shown in Table 3.On this small sample the model worked well with all the predictive results within 2% of the laboratory determinations.This closer agreement is further evidence of the possible errors that can be introduced into the plasticity determination when commercial pressure is applied. The verification tests were carried out by the author under no time pressure, whereas the 100 original samples were taken from ground investigation reports carried out at a commercial soils laboratory.ConclusionsThe numerical model for the determination of the plasticity index (IP) of fine grained soils is:1IP= wL ( g)1x (0.67 - 0.001wL)

3It has been developed based on data from 100 fine-grained soil samples. Verification of the model was achieved by simultaneous prediction and British Standard determination of the I Pof 10 fine-grained soil samples (Table 3). Statistical analysis of the model showed a high correlation between predicted and measured values over the range of I Pvalues modelled.

With further verification, the author believes this test will be a valuable alternative to the current British Standard method, saving analytical time and reducing the possibility of errors being introduced into the plasticity index determination.


Thanks to Len Pimenta of the University of Portsmouth for supplying the plasticity data.


Barbour SL, 1990. The impact of sodium-chloride solutions on the geotechnical properties of clay-soils: a report for the minerals and groundwater programme.Department of Civil Engineering, University of Saskatchewan. BS1377: 1995.

Methods of Test for Soil Engineering Practice.BSI, London.

Skempton AW,1953.The colloidal activities of clays.Proc.3rd Int.Conf. ISSMFE,1, Zurich.

Tomlinson MJ,1993.Foundation design and construction.Longman Scientific and Technical, Harlow.

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