2. BACKGROUND

The SPT is the most common test used in determination and evaluation of the engineering properties of different types of tests in the field. SPT has a number of advantages, including its simplicity and the strength of the mechanical equipment, low cost due to its application in boreholes, and sampling. SPT was used to determine the mechanic and dynamic properties of especially coarse-grained soils, by use of a number of empirical correlations. Although its use in fine-grained soils is still questioned, a number of compressive and undrained shear parameters of fine-grained soils were correlated with this value (Sivrikaya & Toğrol, Reference Sivrikaya and Toğrol2002).

Several correlations were established to set up relationships among the shear strength parameters, consistency, and SPT-N values. Of all these, those tabulated in Table 1 were the most widely known, which was proposed by Terzaghi and Peck (Reference Terzaghi and Peck1967).

Table 1. The dependency of undrained shear strength and SPT-N value on the consistency of fine-grained soils proposed by several researchers

Note: SPT-N, standard penetration test standard penetration resistance value; c _u, undrained shear strength.

Thereafter, many researchers were concentrated on determining the unconfined compressive strength of soils using SPT values (Terzaghi & Peck, Reference Terzaghi and Peck1967; Sanglerat, Reference Sanglerat1972; Hara et al., Reference Hara, Ohta, Niwa, Tanaka and Banno1974; Stroud, Reference Stroud1974; Sowers, Reference Sowers1979; Nixon, Reference Nixon1982; Tomlinson, Reference Tomlinson1986; Ajayi & Balogum, Reference Ajayi and Balogum1988; Décourt, Reference Décourt1990; Kulhawy & Mayne, Reference Kulhawy and Mayne1990; Sivrikaya & Toğrol, Reference Sivrikaya and Toğrol2006; Hettiarachchi & Brown, Reference Hettiarachchi and Brown2009; Mahmoud, Reference Mahmoud2013). The correlations were given in the studies of Sivrikaya and Toğrol (Reference Sivrikaya and Toğrol2009) and Nassaji and Kalantari (Reference Nassaj and Kalantari2011). The idea behind the establishment of these correlations was rational. These expressions were specific to different soil classes and the coefficients were affected from soil plasticity. Established correlations were in the general form of Equation (1):

(1)$$q_{\rm u}=a.N\comma \;$$

where q _u is the unconfined compressive strength of soil, N is the SPT-N value, and a is a coefficient. In the studies referred above, a coefficient ranged between 6.70 and 25. It should be mentioned that, among the relationships depicted above, solely the study of Kulhawy and Mayne (Reference Kulhawy and Mayne1990) included an exponential relationship. These expressions were classified in terms of plasticity and soil class, and some of the relationships were specifically established for certain soils. Of all these, Sivrikaya and Toğrol's (Reference Sivrikaya and Toğrol2006, Reference Sivrikaya and Toğrol2009) relatively recent and impressive studies were specific to Turkish practice. In detail, linear regression and statistical analyses were used to establish simple correlations between the undrained shear strength of soils (S _u) and the SPT number. It was emphasized that the undrained shear strength database was constituted using three types of tests: unconsolidated-undrained triaxial tests, field vane tests, and unconfined compression tests. The relationships were also established for clays of high plasticity, clays of low plasticity, clays, and fine-grained soils. The extensive study on Turkish clays also made a distinction for the coefficient of field N values and N ₆₀ values. Another study to establish empirical correlations for estimation of S _u from different parameters was carried out by Nassaji and Kalantari (Reference Nassaj and Kalantari2011). The study utilized nonlinear regression analyses to establish relationships for estimation of N(SPT), using water content (w _n), liquid limit (w _L), and plasticity index (I _P) parameters. It was emphasized that the standard deviation for the data in their study were lower, in comparison with those obtained in former studies.

Unsupervised clustering algorithms were used to classify several engineering properties of soils using a number of dependent parameters. In this scope, Göktepe et al. (Reference Göktepe, Altun and Sezer2005) used these algorithms to classify the Anatolian soils in terms of their strength and plasticity characteristics. Moreover, another study was conducted to classify the strength development in cement-stabilized clays using their water content and unconfined compressive strength (Göktepe et al., Reference Göktepe, Sezer, Sezer and Ramyar2008). Apart from the applications in civil engineering, particularly geotechnics, these algorithms found applications in soil sampling planning for rubber tree management along with spatial data (Lin et al., Reference Lin, Li, Luo, Lin and Li2013), assessment of pollution using a color index (Amini et al., Reference Amini, Afyuni, Fathianpour, Khademi and Flühler2004), signal processing for positioning systems (Cheng et al., Reference Cheng, Wang, Wu, Wu and Zhang2013), spatial regionalization of hazardous materials (Nourzadeh et al., Reference Nourzadeh, Hashemy, Martin, Bahrami and Moshashaei2013), and even evaluation of sunk cost industries (Arvas & Bozkır, Reference Arvas and Bozkir2013). The algorithms do not necessitate training; in other words, the algorithms perform classification without learning process, and this advantage is benefited in most of studies emphasized above. Along with the advantage of using geographical information systems, these algorithms have the potential in providing classification and characterization of different aspects of systems as well as processes.

Analyzing the short literature survey above, for fine-grained soils, it is clear that the studies were particularly concentrated on establishment of empirical relationships among several index properties and corrected/uncorrected SPT-N values. However, as can be seen in Table 1, a number of investigators concentrated on the classification of these parameters for a better understanding of the relationship between consistency and SPT value, as well as the shear strength of these soils. In this study, unsupervised clustering algorithms were employed to classify the SPT-N parameter in terms of the variables depicted above. This will enable the reader to check out if the former tables that are of use and accepted by the soil scientists worldwide may be useful or not, for a specific region (Table 1). A database constituted by use of test results in northern Izmir was used to answer this question.

3. SPT TESTS, ATTERBERG LIMITS, AND TRIAXIAL TEST DATA

A number of field tests including SPT, cone penetration test, Vane, dilatometer, and pressure meter are used to characterize the underlying soil strata during field investigations. The SPT test is the most frequently used one of these. A thick-walled sample tube is used in the test, which has outside diameter, inside diameter, and length of 50, 35, and 650 mm, respectively. After boring is stopped, at certain depths, this tube is driven into the soil by dynamic effects: a slide hammer with a weight of 63.5 kg falls from a height of 76 cm. After the penetration of a first 15 cm, the number of blows needed for additional two 15 cm penetration is recorded. The sum of blows for the last 30-cm penetration is termed the standard penetration resistance, or the N value. When the soil does not permit 15 cm advance although 50 blows are performed, the penetration after 50 blows is recorded. The blow counts are used in many correlations to determine other properties of soils. The procedure is designated in accordance with ASTM D1586-11 procedure (ASTM, 2011). Although many corrections are performed on test results, in this study, it was decided to use the raw data.

Another input parameter, liquidity index is determined using three tests: water content, liquid limit, and plastic limits tests. In water content test, the specimen is kept in an oven at 105°C for 24 h. The ratio of water weight over weight of dry soil is the water content of the specimen. Liquid limit is the water content of the fine-grained soil in transition from plastic to liquid state. In contrast, plastic limit is the water content that is the point of transition from the semisolid to plastic state. Liquid limit and plastic limit tests can be employed in accordance with the procedure in ASTM D4318-10 (ASTM, 2010). Measuring these parameters, the liquidity index (LI) can be calculated as

(2)$${\rm LI}=\displaystyle{{{\rm \omega} _n - {\rm \omega} _{\rm P} } \over {{\rm \omega} _{\rm L} - {\rm \omega} _{\rm P} }}\comma \;$$

where ω_n, ω_L, and ω_P are the water content, liquid limit, and plastic limit of the soil, respectively (Das, Reference Das2001).

The Mohr–Coulomb strength parameters of the soils are determined by consolidated-undrained triaxial tests, which were carried out in accordance with ASTM D2850 (ASTM, 2007). In unconsolidated-undrained triaxial tests, after application of the confining pressure, deviatoric stresses are suddenly raised to fail the specimen. Drainage is not allowed during the tests. Axial deformation and deviatoric stresses are measured during the shearing phase. As a result, these values are used to determine the Mohr–Coulomb shear strength parameters of the soils, namely, cohesion intercept (c) and internal friction angle (f). It should be emphasized that the relatively low internal friction angles observed in our database (3°–5°) are due to tests conducted on partially saturated fine-grained soils.

The shear strength values are calculated by the Mohr–Coulomb criterion:

(3)$${\rm\tau}=c+{\rm\sigma} \tan {\rm\phi}.$$

Therefore, a database was constituted concerning several parameters related to strength, consistency, and field test results. For clustering, four of these were selected: the percentage of material passing through a number 200 sieve (no. 200), the LI as indicated in Equation (2), the shear strength of soil from laboratory triaxial test results at a certain depth (τ), and the SPT blow counts in the field (SPT-N). In the construction of the database, SPT blow counts were noted for the nearest elevation from which the undisturbed specimen was extracted. In addition, if two SPT tests were close to an undisturbed specimen, attention was paid to recording the SPT test results performed on the same classes of soils with the ones on which the shear strength tests were employed.

4. UNSUPERVISED CLASSIFICATION TECHNIQUES

Clustering algorithms are generally known as mathematical processes that are employed to find out structures and behaviors as well as different groups in a data set. A cluster can therefore be defined as a group of elements of data that are organized similarly. Similarity here is defined by the Euclidean distances of the elements to cluster centers. The methods are advantageous in classification of multidimensional data so that the “distance” term is well defined.

4.1. Hard k-means (HKM) classifier

An HKM classifier segments the data in hand to a predefined number of clusters, and this is accomplished by an iterative procedure. In the calculation procedure, the center points are dynamic. Decision of a data point's cluster is made by the comparison of the Euclidean distances from the point to the center of each cluster, of which their number is fixed a priori. The algorithm moves the cluster centers after every iteration. In HKM, a data point can only belong to a certain cluster (Şen, Reference Şen2004). The first step algorithmically is taking each point in the data set in hand and later associating it to the nearest centroid. Minimization of the squared error objective function (J) in Equation (4) leads to clustering of n data points to c clusters (Ross, Reference Ross1995):

(4)$$\eqalign{J\lpar U\comma \; v\rpar & =\sum\limits_{\,j=1}^n {\sum\limits_{i=1}^c {{\rm\kappa} _{ij} \left\Vert {x_j - v_i } \right\Vert ^2 } }=\sum\limits_{\,j=1}^n {\sum\limits_{i=1}^c {{\rm\kappa} _{ij} \left[{d_{ij}^2 } \right]} }\cr & =\sum\limits_{\,j=1}^n {\sum\limits_{i=1}^c {{\rm\kappa} _{ij} \left[{\sum\limits_{t=1}^m {\left({x_{\,jt} - v_{it} } \right)^2 } } \right]} }\comma \; }$$

where J is the minimization function, U is the partition matrix, v is the matrix of center clusters, m is the number of features in input matrix, d is the similarity measure taking the Euclidean distance calculation as a guide, and κ is a function in the calculation of the partition matrix defined as

(5)$$U={\rm\kappa}_1 {\rm\comma \; }{\rm\kappa}_2 {\rm\comma \; }{\rm . }\,{\rm . }\,{\rm .\,\comma \; }{\rm\kappa}_i$$

for any x _j, so that if κ_i is equal to 1, x _j is the element of ith cluster and the contrary is true if κ_i = 0.

Regarding this knowledge, cluster centers are computed as in Equation (6):

(6)$$v_{it}=\displaystyle{{\sum\nolimits_{k=1}^n {{\rm\kappa} _{ik} \, x_{kt} } } \over {\sum\nolimits_{k=1}^n {x_{ik} } }} \quad\lpar {\rm for}\, t=1\colon m \, {\rm and}\, i=1\colon c\rpar.$$

As a consequence, the difference between obtained dissimilarity matrices is calculated and compared with the error criterion (ε). The iteration is accomplished when the difference is below ε. The elements of the similarity matrix (d) can be updated using the following formulation (Şen, Reference Şen2004):

(7)$$\; \left. \matrix{{\rm if}\; d_{ik} \lpar s\rpar \, ={\rm min}\left[{d_{ik} \lpar s\rpar } \right]^{} {\rm\kappa} _{ij} \lpar s+1\rpar =1 \cr {\rm otherwise\comma \; }\; {\rm\kappa} _{ij} \lpar s+1\rpar =0\; \; \hfill} \right\}\; \forall j \in c\comma \; \;$$

where s is the number of iteration step. Unsuccessful outcomes may be obtained when data is noisy, duplicate, or not convex shaped (Ross, Reference Ross1995).

4.2. Fuzzy c-means (FCM) algorithm

The HKM method is a crisp classification technique, and as can be derived from its formulation, a data point can belong to a certain cluster or not. Instructing the partial belongingness concept by advantage of fuzzification, the FCM method was developed by Bezdek (Reference Bezdek1981). Derived from the Euclidean distance of a data point to a certain cluster center, computation of a membership value enables the determination of a data point's partial belonging to any certain cluster. The primary difference between the crisp and fuzzy classification concepts is sketched in Figure 1. Assuming that there are two cluster centers, from the figure, it is clear that the points A and B belong to the first and second cluster centers, respectively. In the light of this, crisp classification techniques compute the µ_A2 and µ_B1 values as 0. In contrast, µ_A1 and µ_B2 values will be 1. In fuzzy classification, the membership of a point to every cluster is calculated in each calculation step, and the greatest value of membership determines which cluster this point belongs to. Therefore, a comparative evaluation of memberships reveal that µ_A1 > µ_A2 and µ_B2 > µ_B1. As will be mentioned, membership values range between 0 and 1 (Wu & Yang, Reference Wu and Yang2002; Miyamoto et al., Reference Miyamoto, Ichihashi and Honda2008).

Fig. 1. Definition of membership values: x and y are dependent parameters.

An additional parameter instructed to the HKM algorithm is the fuzzification parameter (m′), ranging between 1 and a feature number (n). For every data point, the greatest membership value to any cluster indicates which cluster this data point belongs to. Similar to HKM, the dissimilarity function in Equation (8) is minimized to terminate the algorithm after supplying a satisfactory amount of error e, belonging to the data point of the cluster. The dissimilarity function [Equation (8)] was minimized to create a criterion for the algorithm termination step (Lanhai, Reference Lanhai1998):

(8)$$\min\left[{J_m \left({{\bf U}\comma \; {\bf v}} \right)} \right]=\min\left[{\sum\limits_{k=1}^n {\sum\limits_{i=1}^c {\left({{\rm\mu} _{ik} } \right)^{m^{\prime} } \times \left({d_{ik} } \right)^2 } } } \right]\comma \;$$

where μ_ik is the membership degree of the kth data point in the ith cluster and d _ik is the Euclidean distance between kth data point and ith cluster center, which is depicted in Equation (9).

(9)$$d_{ik}=\Vert x_k-v_i \Vert =\sqrt {\sum\nolimits_{i=1}^{m}\lpar x_{ki}-v_{i}\rpar ^2}.$$

The centroids herein may be calculated by

(10)$${\bi v}_{ij}=\displaystyle{{\sum\nolimits_{k=1}^n {{\rm\mu} _{ik} ^{m^{\prime}}\, \times x_{kj} } } \over {\sum\nolimits_{k=1}^n {{\rm\mu} _{ik} ^{{\bi m}^{\prime}} } }} \quad\lpar {\rm for}\, j=1\colon m \, {\rm and}\, i=1\colon c\rpar \comma \;$$

where v is the matrix composed of cluster centers, x is the data point, and m′ is the fuzzification parameter or weighting coefficient. A series of iterative calculations are used to employ fuzzy partitioning (Bezdek, Reference Bezdek1981):

(11)$$u_{ik} \lpar w+1\rpar = \left[\sum\limits_{\,j=1}^{c} \left({d_{ik}\lpar w\rpar \over d_{ jk}\lpar w\rpar }\right)^{2\over m^{\prime} -1}\right]^{-1}\comma \;$$

where w is the calculation step. In any step, the following criterion is the key to terminate the algorithm:

(12)$$\Vert U^{w+1} - U^{w} \Vert \le {\rm \varepsilon}.$$

Generally speaking, minimization of the objective function, satisfying the criterion in Equation (12), leads to optimized cluster centers.

4.3. Self-organizing maps (SOMs)

SOMs utilize a competitive learning process, which is based on a “winner takes all” rule to categorize features without feedback. Consisting of two layers, input layer is one dimensional, and output neurons are arranged in two dimensions. Briefly, with the aid of the topological neighboring concept, a winning neuron in the output, which is a competitive layer. is determined Haykin (Reference Haykin1996). Many topological shapes could be utilized for neighborhood detection (Şen, Reference Şen2004), but the most common shape, rectangular neighborhood, was used in this study.

In the algorithm, output of each neuron is calculated as follows:

(13)$${\bi O}_j=\sum\limits_{i=1}^{n} {\bi w}_{ij} {\bi X}_{i}\comma \;$$

where O denotes the output vector, w is the weight matrix, and X is the input vector. The weights of the output unit with the highest activation are updated. Euclidean distance is usually leveraged to determine the distance between the winning neuron and the processing neuron in the output layer:

(14)$$d_j =\Vert X_j - w_{ij} \Vert.$$

In Equation (14), d is the lateral distance vector. The best choice for the quantification of the topological neighborhood function is the Gaussian distribution function. Because this function is bell shaped, it can comfortably satisfy the criteria above:

(15)$$h_{ij}=\exp\lsqb\! {-}\lpar d_{ij}^{2}/2 {\rm\sigma}^2\rpar \rsqb \comma \;$$

where h _ij is the topological neighborhood and σ is the neighborhood width parameter. As a consequence, the weights of the neighboring neurons were updated after the determination of the winning neuron via the following equation:

(16)$$w_{ij}\lpar t+1\rpar =w_{ij}\lpar t\rpar +{\rm\eta}\lpar t\rpar h_{ij}\lpar t\rpar {\bf \lfloor} X_{\,j}\lpar t\rpar -w_{ij}\lpar t\rpar \rfloor.$$

In Equation (16), t is the iteration step, and η is the learning rate parameter, which diminishes through the iterative process. The iterative process ends when a stable output lattice is obtained (Kohonen, Reference Kohonen1982, Reference Kohonen1998; Göktepe et al., Reference Göktepe, Sezer, Sezer and Ramyar2008).