## Abstract

Soil classification is one of the most important stages in preliminary studies for design applications in geotechnical engineering. The classification of fine-grained soils is mostly determined using Casagrande's plasticity chart. However, owing to the factors affecting determination of liquid and plastic limits, uncertainties can arise in fine-grained soil classification when using this approach. The uncertainty increases particularly when the points on the chart fall on the lines that separate certain soil classes (A-line and/or 50% liquid limit line) or very close to these lines. In this study, a fuzzy classification routine is proposed to minimize these uncertainties. For this purpose, the spatial distances of the evaluation points on the chart from the lines were used as a controlling unit. The membership degrees that define the fuzzified soil (clay and/or silt), and plasticity (low and/or high plasticity), were evaluated by considering sigmoid functions. As a consequence, the soil types were established by aggregating fuzzified soil and plasticity using the fuzzy operators.

Silts and clays are different fine-grained soils with regard to their engineering behaviour. Therefore, their correct identification is important. Liquid and plastic limits are commonly used for the classification of silts and clays. These index properties of fine-grained soils are usually referred to as the Atterberg limits. In fact, they were first developed for agricultural purposes (Casagrande 1932). Considering the physical definitions of the Atterberg limits, they are basically defined at certain levels of water content of fine-grained soils: the water content value of silts and clays directly controls the mechanical behaviour of the material. For this reason, the correct classification of fine-grained soils considering liquid and plastic limits can be important in geotechnical engineering applications. The plasticity chart proposed by Casagrande (1948) now has wide use in such applications (ASTM 2015*a*) (Fig. 1).

The ordinate axis represents the plasticity index (*I*_{p}) whereas the abscissa axis represents the liquid limit on the plasticity chart. The soil classes are represented with two capital letters. The first capital letter is ‘C’ for clays, ‘M’ for silts and ‘O’ for organic soils. The second capital letter derives from the plasticity property of soils. If the liquid limit is higher than 50%, the letter ‘H’ is used. If the liquid limit is lower than 50%, the letter ‘L’ is used. The plasticity chart can therefore be divided into four zones that are separated by the A-line and a vertical line that defines the liquid limit (LL) value of 50%. Clays (C) are found above the A-line and silts (M) are located beneath this line. Additionally, low-plasticity soils (L) are found on the left of the LL = 50% line and high-plasticity soils (H) are located on the right of this line. The inclined A-line represents an important empirical boundary. The line goes through the points having a liquid limit value of 20% and a plasticity index value of 0 (0, 20), and a liquid limit value of 50% and a plasticity index value of 22 (50, 22). The equation of the A-line is given as follows (ASTM 2015*a*):
(1)The A-line represents the empirical boundary between typical inorganic clays (CL–CH groups), which are generally found above the A-line, and plastic soils containing organic colloids (OL–OH groups), which are located beneath this line. Additionally, the groups that are found below the A-line are typical inorganic silts and clayey silts (ML–MH groups), except for the points having liquid limits less than 30%, for which inorganic silts may be located slightly above the A-line (Casagrande 1948). There is also one more line, which is known as the U-line, on the plasticity chart. The U-line represents the upper limits on the plasticity chart for natural soils. The U-line goes through the points having a liquid limit value of 8% and a plasticity index value of 0 (8, 0), and a liquid limit value of 16% and a plasticity index value of 7 (16, 7). The equation of the U-line is given as follows (ASTM 2015*a*):
(2)The methods that are used in the determination of liquid and plastic limits are different. The Casagrande liquid limit test is the traditional procedure used for the liquid limit determination whereas the rolling-out test is the most common way of determining the plastic limit (ASTM 2015*b*). Additionally, the fall cone is also recommended as the preferred method for liquid limit evaluation, as the test procedure is much less subjective (BS 1377, BSI 1990). There are several factors, mentioned in the recent literature, that can affect the results of the Casagrande liquid limit test in particular (Lee & Freeman 2007; Sivakumar *et al.* 2009; Haigh 2012). The experience of the operator, incorrect frequency of falls of the Casagrande cup and inappropriate soil groove are some of factors affecting the Casagrande liquid limit tests, whereas the pressure applied to soil, the speed of rolling and contamination are some of the factors influencing the rolling-out test for determination of the plastic limit. Factors such as these can give rise to uncertainties in the determination of the liquid and plastic limits of fine-grained soils. Therefore, the soil types obtained from the plasticity chart, considering the liquid limit and plasticity index, can have inherent uncertainties in their application. The points on the plasticity chart sometimes fall on the A-line and/or the 50% liquid limit line or very close to these lines. In such situations, the solution commonly applied is to refer to the soil types using double symbols. However, this is not necessarily the best solution and can double the inherent uncertainties in the classification. In the present study, when considering the inherent uncertainty in the classification of fine-grained soils using the Casagrande plasticity chart, a fuzzy classification routine is introduced. The study was carried out in two stages: first, the method applied in fuzzification is given; then classifications and nomenclatures are established by applying the proposed method using data obtained from the VSB–Technical University of Ostrava (Czech Republic) soil mechanics laboratory database.

## Fuzzification

Fuzzy set theory was first proposed by Zadeh (1965). The most attractive characteristic of fuzzy logic compared with other techniques commonly applied in geosciences is the ability to describe complex and nonlinear multivariable problems in a transparent way (Setnes *et al.* 1998). Perhaps for this reason over the past two decades fuzzy logic has begun to be widely employed in engineering geology (Sakurai & Shimizu 1987; Juang *et al.* 1998; Alvarez Grima & Babuska 1999; Gokceoglu 2002; Kayabasi *et al.* 2003; Nefeslioglu *et al.* 2006; Mishra & Basu 2013). The degree to which an element belongs to a given set is called the membership degree. Membership degrees of elements in a set represent their truth values. In fuzzy set theory, the membership degrees of elements may be any real number between zero and unity. In contrast, in classical set theory, the membership degrees of elements may only be zero or unity. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false (Zadeh 1965). When the Casagrande plasticity chart is examined, it can be seen that the chart represents the classical set theory. If it is assumed that the plasticity chart is a universal set, the regions of CL, CH, ML and MH become the subsets, which are separated by crisp boundaries on the chart. These boundaries can be illustrated by the A-line and the vertical line for a liquid limit value of 50% (Fig. 2a). The term universal set given here according to set theory is defined as a set that contains all sampling points, including itself.

There are four subsets with no overlapping in the illustration given in Figure 2a, the membership degrees of which are zero or unity. If the set CL is considered, the membership degrees of its elements are equal to unity, whereas the membership degrees of elements in the sets CH, ML and MH are equal to zero. Hence, according to classical set theory, the membership degrees of the elements of the sets on the Casangrande chart are assumed to be zero or unity. On the other hand, according to fuzzy set theory, membership degrees of elements in a fuzzy set vary between zero and unity. To fuzzify the plasticity chart, the spatial distances of the evaluation points from the A-line and the line of LL = 50% on the chart were used. For this purpose, the chart was first divided into 6000 grid cells so that each cell has a side dimension of 1%. Each grid cell except for the points above the U-line was considered to be an evaluation point. In other words, the corner points of the cells represent points that have certain liquid limit and plasticity index values. In this study, the plasticity chart was digitized and the evaluation points can be stored in a database. The spatial distance of each point from the lines on the chart was then calculated.

As mentioned above, the A-line represents the boundary of clay (C) and silt (M) soils and the line of LL = 50% represents the boundary of low plasticity (L) and high plasticity (H). According to the fuzzy classification routine suggested in this study, the soil type and the plasticity were both fuzzified. Hence, the procedures performed for the A-line and the line of LL = 50% were evaluated separately. The crisp sets C (clay) and M (silt) were defined by the A-line (Fig. 2b). There are evaluation points having different finite distances from the A-line in both sets. These points constitute the elements of the C and M crisp sets. The fuzzy membership degrees of the evaluation points can be assigned considering the spatial locations of the evaluation points and the perpendicular distances from the A-line on the chart. The fuzzy membership degrees of the elements in the sets C and M for the evaluation points are denoted by *μ*_{C} and *μ*_{M}. Because the perpendicular spatial distance from the line on the chart is taken into consideration in fuzzification, the sum of *μ*_{C} and *μ*_{M} should theoretically be unity:
(3)Additionally, the sets L (low plasticity) and H (high plasticity) were defined by the line of LL = 50% (Fig. 2c). Similarly, the fuzzy membership degrees of the elements in the sets L and H for the evaluation points are denoted by *μ*_{L} and *μ*_{H}. Theoretically, the sum of *μ*_{L} and *μ*_{H} should also be unity:
(4)The uncertainty increases as the evaluation location approaches the lines on the chart. In contrast, beyond a certain distance from the lines on the chart the uncertainty reaches its minimum. When the A-line is considered, it is obvious that the evaluation points that are farther than a certain distance from the line belong to the set C or M depending upon which side they are located. This approach is also valid for the line of LL = 50%; beyond a certain distance the evaluation points belong to the set L or H. The membership degrees of these points are equal to unity. On the other hand, the maximum uncertainty is observed right on the lines, and the membership degrees should be equal to 0.5 in this situation. Therefore, within a specified distance on both sides of the lines where there may be some degree of uncertainty, the membership degrees of the evaluation points change between zero and unity. To evaluate approximate boundaries of the uncertainty zones on the plasticity chart, the uncertainty in water content of the soil material was considered. As mentioned above, the uncertainties owing to the factors affecting the liquid limit and plastic limit tests are the main sources of the uncertainties in fine-grained soil classifications. The liquid limit and plastic limit values are actually water content values, hence it can be considered that the uncertainty owing to water content of soil material is the main source of the uncertainty observed on the plasticity chart. Therefore considering the specific distance from the lines on the plasticity chart allows the uncertainty to be evaluated. Obviously, the exact uncertainty level owing to water content on the plasticity chart cannot be known precisely. However, an experienced engineer can judge the level of uncertainty on the chart in a reasonable interval. In the present study, the level of uncertainty as well as possible range of the uncertainty interval is represented by *k* (Fig. 3a and b).

We considered a value of *k* = 5 during the determination of example classifications and nomenclatures using the proposed method in this study. In other words, the uncertainty level owing to the factors affecting the liquid limit and plastic limit tests is considered to be ±5% on the plasticity chart. According to the method proposed in the present study, an experienced engineer can judge the magnitude of the value *k* and classifications and nomenclatures of fine-grained soils can be established by considering any uncertainty level on the Casagrande plasticity chart. To define the membership functions, which are used to obtain fuzzified soil and plasticity, sigmoid functions were used. A sigmoid function transforms the input having any value between +∞ and −∞ into a reasonable value in the range between zero and unity, and produces a curve with an S-shape (Negnevitsky 2002). Considering the uncertainty to increase when approaching the lines on the plasticity chart and decrease when moving away from these lines, the type of membership functions should be a nonlinear sigmoid function. The independent variables of the membership functions are the spatial distances of the evaluation points from the lines. The sigmoid functions described on the plasticity chart are defined on the interval [−*k*, *k*].

Fuzzy sets that are defined by the A-line and the line of LL = 50% are illustrated in Figure 4a and b, respectively. The boundaries and intersections of the sets C and M, and L and H are given in the figures. The intersection point of the sigmoid functions for the sets C and M coincides with the A-line. The membership values that are higher than *k* and lower than −*k* are equal to unity. In other words, in cases where the spatial distances are higher than |*k*| on the plasticity chart, the evaluation points above the A-line are certainly clay (C) and the points below the A-line are certainly silt (M). On the other hand, the membership degrees of the evaluation points are between zero and unity between the spatial distances having values of −*k* and *k*. Similarly, the intersection point of the sigmoid functions for the sets L and H coincides with the line of LL = 50%. For spatial distances higher than |*k*|, the points on the right side of the line are certainly high-plasticity soil (H) and the points on the left side of the line are certainly low-plasticity soil (L). Hence, in this case, the membership degrees of the evaluation points relating to the sets L and H are zero or unity, and the membership degrees of the points for the sets L and H are between zero and unity between the spatial distances having values of −*k* and *k* as well.

The mathematical notation for the fuzzy sets M and H, and C and L are given in equations (5) and (6), respectively:
(5)
(6)where the variable *x* represents the perpendicular spatial distance of the evaluation points from the lines. The minus sign in front of *k* represents the direction of movement away from the lines on the plasticity chart. Hence, the plus sign defines the movement direction to the right whereas the minus sign expresses the movement direction to the left from the lines on the chart. The fuzzy classification routine suggested in the present study was evaluated by using sample data obtained from the VSB–Technical University of Ostrava (Czech Republic) soil mechanics laboratory database. A total of 13 fine-grained soil samples that are close to the lines on the plasticity chart were investigated (Fig. 5). The perpendicular spatial distances of the points from the lines on the plasticity chart are given in Table 1. Considering an uncertainty level of ±5% (i.e. *k* = 5), the membership degrees of the soil samples in the fuzzy sets C, M, L and H are calculated using sigmoid functions (Table 1). For example, considering the soil sample 29, the distance to the A-line is measured to be unity in terms of water content percentage difference (*w* %). According to equations (6) and (5), the fuzzy membership degrees of this soil sample for the fuzzy sets C and M are calculated to be *μ*_{C} = 0.27 and *μ*_{M} = 0.73, respectively. Additionally, the distance to the line of LL = 50% is measured to be zero (*w* %) for the same soil sample, hence according to equations (6) and (5), the fuzzy membership degrees of the soil sample for the fuzzy sets L and H are calculated to be *μ*_{L} = 0.50 and *μ*_{H} = 0.50.

## Classification and nomenclature

Using the method proposed in this study, classifications and nomenclatures of the fine-grained soil samples were determined by considering the fuzzified soil and fuzzified plasticity. For this purpose, different fuzzy operators were implemented. Fuzzy operators are used to obtain a single number that represents the result of the antecedent evaluation (Negnevitsky 2002). In the present study, the resultant fuzzy values obtained by operating the membership degrees *μ*_{C} and *μ*_{M}, and *μ*_{L} and *μ*_{H} with different fuzzy operators define the resultant fuzzy soil and plasticity classes; *μ*_{Soil} and *μ*_{Plasticity} for the fine-grained soil samples. The main operations on crisp sets are intersection, union and complement. The operations applied on crisp sets are actually a special case of operations applied on fuzzy sets. The results of operations applied on fuzzy sets are equal to the results of operations applied on crisp sets for the extreme values of membership degrees; *μ*(*x*) = 0 or 1. The main difference arises when the value of *μ*(*x*) is between zero and unity. In this situation, different methods for fuzzy set operations can be applied in aggregation of fuzzified membership degrees. For this reason, fuzzy systems researchers have proposed several fuzzy operators (Cox 1999). The most used and basic ones, ‘min’, ‘max’, ‘prod’ and ‘probor’, are evaluated in this study. Mathematical definitions of the fuzzy operators investigated in this study are given as follows. The ‘min’ operator represents ‘and connective’ and intersection (equation (7)), which means intersection of two fuzzy sets defined by the membership function *μ*_{C∩M}(*x*). According to equation (7), *μ*_{Soil} is equal to the minimum value of membership degrees:
(7)The ‘prod’ operator, which is also known as ‘probabilistic and’, is an amplifier of the ‘min’ operator (equation (8)). This operator gives an aggregated value smaller than the minimum:
(8)The ‘max’ operator represents ‘or connective’ and union (equation (9)), which means union of two fuzzy sets defined by the membership function *μ*_{C∪M}(*x*). According to equation (9), *μ*_{Soil} is equal to the maximum value of membership degrees:
(9)The ‘probor’ operator, which is also known as ‘probabilistic or’, is an amplifier of the ‘max’ operator (equation (10)). This operator gives an aggregated value higher than the maximum:
(10)The same equations can also be written for the fuzzy plasticity class. *μ*_{Plasticity} is calculated using the ‘min’ operator as follows:
(11)*μ*_{Plasticity} is calculated using the ‘prod’ operator as follows:
(12)*μ*_{Plasticity} is calculated using the ‘max’ operator as follows:
(13)*μ*_{Plasticity} is calculated using the ‘probor’ operator as follows:
(14)Considering the fuzzy operators given thus far it is evident that different fuzzy soil and plasticity classes can be obtained. For this reason, different fuzzy operators were investigated in this study by comparing the resultant fuzzy soil classes with the crisp soil class. The resultant fine-grained soil classes obtained by using different fuzzy operators are given in Table 2. For example, considering the soil sample 29 again, the membership degrees for the fuzzified soil are given as *μ*_{C} = 0.27 and *μ*_{M} = 0.73. Hence, considering the fuzzy operators ‘min’, ‘max’, ‘prod’ and ‘probor’, the fuzzy soil classes are found to be C_{(0.27)}, M_{(0.73)}, C_{(0.2)} and M_{(0.8)}, respectively. Additionally, the membership degrees for the fuzzified plasticity are given as *μ*_{L} = 0.5 and *μ*_{H} = 0.5 for the soil sample. Therefore, the fuzzy plasticity classes are found to be L_{(0.5)}, H_{(0.5)}, L_{(0.25)} and H_{(0.75)}, respectively, by applying the fuzzy operators. As a result, the soil sample 29 can be classified and named as CL_{(0.27,0.5)}, MH_{(0.73,0.5)}, CL_{(0.2,0.25)} and MH_{(0.8, 0.75)} by considering different fuzzy operators. When comparing the fine-grained soil classes obtained by different fuzzy operators with the crisp soil class for the soil sample 29, it is revealed that the ‘min’ and ‘prod’ fuzzy operators can produce different soil types. On the other hand, the ‘max’ and ‘probor’ fuzzy operators result in similar soil types to the crisp classification. Additionally, in the case of a maximum uncertainty situation, which means that the evaluation point is located on the line(s), the ‘probor’ fuzzy operator is able to clearly reduce inherent uncertainty on the plasticity chart. As a consequence, the ‘probor’ fuzzy operator is suggested for use in the fuzzy classification routine proposed in the present study. The consequent fine-grained soil class obtained through the fuzzy classification routine is found to be MH_{(0.8,0.75)} for the soil sample 29.

## Discussion and conclusions

Considering the inherent uncertainty sometimes encountered in the classification of fine-grained soils when using the Casagrande plasticity chart a fuzzy classification routine is suggested. The method is mainly based on the measurement of the spatial distances of evaluation points from the A-line and line of LL = 50% on the chart. By using the spatial distances and sigmoid functions, fuzzified soil and fuzzified plasticity can be defined. To aggregate the resultant fuzzy soil classes the ‘probor’ fuzzy operator is suggested. By applying the fuzzy classification routine proposed in this study it is possible to define a fine-grained soil class using only one symbol in maximum uncertainty conditions, which means that the evaluation point is located on the line; crisp soil class ML–MH can be defined as fuzzy soil class MH_{(0.8,0.75)}. Additionally, there are uncertainty conditions on the plasticity chart; applying the method proposed in this study, the membership degrees of soil type and plasticity can be evaluated: the fuzzy soil type can be defined as MH_{(0.8,0.75)}, which means that the membership degree of the sample for the set M is equal to 0.8 whereas that for the set H is equal to 0.75. Sample classifications and nomenclatures were determined by using soil data acquired from the VSB–Technical University of Ostrava (Czech Republic) soil mechanics laboratory database. The uncertainty degree, *k*, is assumed to be *k* = 5 in the sample solutions; however, different solutions can also be achieved by considering different values of *k*. Finally, it is suggested that the fuzzy soil classes determined according to the fuzzy classification routine proposed in the present study could be evaluated as input parameters in swelling potential model studies.

- © 2016 The Author(s)