Reliability and Validity Analysis of Correlations on Strength and Consolidation Parameters for Ankara Clay and Proposal for a New Correlation

Abstract

This study aims to assess the accuracy and consistency of existing correlations between undrained shear strength (c_u) and modulus of volumetric compressibility (m_v) with SPT N value, and compression index (C_c) with index parameters for Turkish clays, with a specific focus on Ankara clay using laboratory testing of over 2500 high-quality undisturbed samples from 42 sites. A comparison of correlations for c_u using five different statistical variables indicates that no single method stands out as superior in terms of both accuracy and conservatism and that the quality of the tests in the database has a significant impact on the correlation performance. Even the most effective correlation for c_u appear insufficient for direct application in geotechnical design processes, given its low consistency characterized by coefficient of variations (COV) ranging between 50% and more than 100%. The widely used m_v estimates proposed by Stroud (1974) were found to be notably unconservative when applied to Turkish clays. Consequently, a new trendline, leveraging both SPT N value and Plasticity Index (PI), was introduced to address this discrepancy. Meanwhile, an examination of 283 high-quality undisturbed samples from Ankara Clay revealed existing correlations between initial void ratio (e₀) and compression index (C_c) to be conservative, yet consistently reliable. This underscores their suitability for routine or preliminary design applications. Notably, a newly derived correlation was proposed, which demonstrates equal applicability to the well-established Bowles (1979) equation.

1 Introduction

Data obtained from in-situ tests has great importance in estimating geotechnical parameters due to the difficulty of obtaining sufficient number of undisturbed samples that adequately represent soil properties in the field. Thus, in geotechnical practice, empirical correlations with in-situ test results are frequently used to obtain design parameters. Furthermore, innovative approaches in geotechnical modeling have also been proposed, such as fractional elastoplastic constitutive models for soils, which offer new ways to capture soil behavior [1, 2]. In recent years, the development of open-source algorithms in models is advancing the computational capabilities of the field by increasing the accessibility to advanced numerical methods [3]. Sari Ahmed et al. [4] and Taleb Bahmed et al. [5] stated that existing models can be improved and new correlations can be proposed for the prediction of geotechnical properties of stabilized soil (parameters such as strength, compressibility and plasticity index) in terms of reliability and validity analysis of correlations.

The Standard Penetration Test (SPT), introduced in the late 1930s in the United States, is one of the earliest and most commonly used field tests within the scope of geotechnical exploration programs, owing to its relatively straightforward procedure. In the test, a 63.5 kg hammer falls a distance of 760 mm to drive a split-spoon sampler 450 mm into the bottom of a borehole. The N-value, indicating the number of blows required to advance the sampler the final 300 mm, is recorded. The initial 150 mm of penetration is disregarded to minimize the influence of borehole drilling disturbance on the SPT measurement.

Due to its extensive use, there are large SPT databases from across the world. Therefore, correlations with SPT N value are available for virtually any geotechnical parameter, such as consistency [6], index properties [7], shear strength [8,9,10,11], deformation properties [12], and liquefaction potentials of cohesive soils [13,14,15,16,17,18].

The undrained shear strength (c_u) and consolidation parameters, such as the coefficient of volumetric compressibility (m_v) or the compression index (C_c), play a crucial role in selecting and designing an appropriate foundation system for structures on clays, primarily to meet the requirements of bearing capacity and settlement issues. However, it's important to acknowledge the difficulty in obtaining truly undisturbed soil samples for strength and consolidation testing due to remolding effects. Therefore, utilizing correlations between c_u, m_v, and C_c with SPT-N values could offer advantages in determining design parameters and providing a realistic estimation of soil properties. This approach is particularly beneficial given the challenges associated with in situ sampling, often resulting from sample disturbance and project constraints such as cost, equipment, and time limitations [19].

However, the accuracy and reliability of these correlations, especially those for fine-grained soils, are affected by many parameters. One of the main sources of inconsistency can be attributed to the errors and variabilities of the SPT, which is discussed elsewhere (e.g. [20]). Also, correlations may become less reliable when they are based on results from low-quality soil samples. An additional and significant source of error is the extrapolation of a local correlation to a case where it is not intended to be used, such as to a site with different clay mineralogy. Regrettably, it is a common practice to utilize published correlations, often derived from data collected from diverse locations or, conversely, from a single soil layer, without taking into account the specific characteristics of local soil conditions, for estimating design parameters. Therefore, “localized” or site-specific relationships are more desirable at the design stage [8, 21]. SPT is expected to continue as a fundamental component in site investigation studies, in this context, investigation of the empirical correlations, such as the relationship of the SPT N value with the undrained shear strength [11, 22] and sensitivity analysis of published correlations [23] to examine the importance of the effects of test types or application of different statistical approaches [21, 24] have been performed to come up with the best correlations [25,26,27] in recent studies.

Similarly, in order to propose specific SPT-based correlations for strength and deformation properties of Turkish clays, especially the finer reddish-brown clastics known as "Ankara Clay" in Turkish geotechnical studies [28], it is essential to evaluate the accuracy and reliability of existing published correlations. To achieve the objective, this study utilizes data from a comprehensive database encompassing unconsolidated-undrained triaxial (UU), unconfined compression (UC), and consolidation tests conducted on over 2500 undisturbed samples, alongside their corresponding SPT N values. Quantitative metrics such as root mean square error (RMSE), the ratio of calculated to measured parameters (K), ranking index (RI), ranking distance (RD), and Theil inequality coefficient (TIC) are employed to assess the relative merit of the widely-used correlations between c_u and SPT N value. Next, the relationship between SPT N and m_v is investigated considering soil classification, plasticity index (PI), range of SPT N, and the effective stress (σ_v′) level, and the results are compared with a frequently utilized correlation in geotechnical engineering practice. Furthermore, data from Ankara clay is utilized to assess the effectiveness of correlations linking C_c to various soil index properties including water content (w), plasticity index (PI), and initial void ratio (e₀). Additionally, a new correlation is proposed to estimate C_c based on e₀ using Ankara Clay data only. The authors are not aware of any comprehensive study of this scale on both the undrained strength and the consolidation parameters of Turkish clays or Ankara clay similar to the one conducted herein; thus, the obtained results hold particular significance.

2 Materials and Methodology

The primary objective of this study is to assess the accuracy of established correlations between undrained shear strength (c_u), coefficient of volumetric compressibility (m_v), and compression index (C_c) with SPT N values or soil index properties, focusing exclusively on clays obtained from 42 distinct project sites across Turkiye, as depicted in Fig. 1. To achieve this aim, the study meticulously collected a total of over 2500 high-quality undisturbed samples for shear strength and consolidation testing from these sites. Among these samples, 512 underwent unconsolidated-undrained (UU) triaxial compression testing, 864 underwent unconfined compression (UC) testing, and the remainder was subjected to consolidation testing. During the SPTs conducted at these 42 locations, a single company was tasked with ensuring consistent quality control and standardization. Furthermore, to maintain uniformity, only laboratory test results conducted by the first author were utilized in this study. A summary of the database in terms of location, number of borings, and laboratory tests is presented in Table 1.

Fig. 1

The locations of project sites across Turkiye and in Ankara

Table 1 Summary of database in terms of location and number of borings and laboratory tests

UU triaxial tests were conducted in accordance with the standards [29, 30], and UC tests were performed according to the standards [31, 32] on undisturbed specimens. As per general practice, Mohr circles are plotted to determine the failure envelopes. For UU tests, the undrained shear strength value (c_u) is obtained at the point where the failure envelope intersects the vertical axis. The angle formed by the failure envelope with the horizontal axis represents the total stress friction angle (ϕ). In saturated clay soils, the internal friction angle is theoretically expected to be zero [33]. Since the samples may not be completely saturated, only data obtained from UU tests with an internal friction angle of less than 10° is utilized within the study's scope. Consequently, c_u values were evaluated using two methods (UU-method 1 and UU-method 2). As shown in Fig. 2, in the first method (UU-method 1), c_u1 is directly obtained from the intersection point of the Mohr–Coulomb envelope with the shear stress axis. In the second method (UU-method 2), for each UU test, c_u2 is calculated by averaging the radii of the last two Mohr–Coulomb circles.

Fig. 2

Illustration of Methods 1 and 2 for calculating undrained shear strengths from UU tests

Another major aim of this study is to assess the existing correlations and establish a robust new correlation between the compression index and index parameters specifically tailored to the overconsolidated Ankara Clay. Approximately two-thirds of the Ankara metropolitan area, accommodating roughly three million inhabitants, is situated atop Pliocene-aged Etimesgut Formation, commonly known as "Ankara Clay" [34]. The ongoing migration of people from rural to urban areas has resulted in the construction of numerous engineering structures built on or within this overconsolidated clay. This formation comprises silty clay and gravelly-sandy clay, with locally occurring thin lime layers, lime nodules, and concretions at shallow depths within the clayey layers, often lacking lateral continuity [35]. Erol et al. [36] noted that the thickness of the clayey sequence varies locally but generally exceeds 200 m. Ordemir et al. [28] described Ankara Clay as an inorganic, preconsolidated clay, with natural water content (w_n) and plastic limit (PL) ranging between 20 and 35%, and liquid limit (LL) ranging between 55 and 75%.

Among the 42 project sites examined in this study, 26 (62%) are situated in Ankara. However, it's crucial to acknowledge that within the framework of site investigations carried out in Ankara, besides the Ankara Clay, alluvial-based clays are also encountered in specific regions. To ensure accuracy and obtain a localized correlation, data from these soils are meticulously separated to ensure that all data pertain specifically to Ankara Clay. Consequently, only 15 studies involving Ankara Clay were utilized to derive the local correlation.

The validity of existing correlations is compared using five different statistical variables: RMSE, K, RI, RD, and TIC. As a result of these comparisons, the most and least successful correlations are quantitatively determined for local soil data examined in this study.

Root mean square error (RMSE) measures the model success of a correlation between calculated values and the measured values from laboratory tests [37]. RMSE can be expressed as in Eq. (1):

$${\text{RMSE}} = \sqrt {\sum\nolimits_{1}^{{\text{N}}} {({\text{x}}_{{cal.}} - {\text{x}}_{{{\text{lab}}.}} )^{2} } /N}$$

(1)

where N denotes the number of data; x_cal. is calculated value from empirical correlation and x_lab. is the measured value from the laboratory test. As the x_lab. values approach the x_cal., the RMSE, which is the standard deviation of the errors, decreases and approaches zero. The closer RMSE to zero, the better the correlation at estimating the desired parameter.

The parameter K proposed by Cherubini and Greco [38] can be calculated using Eq. (2) provided below. This parameter represents the ratio of calculated values to measured values:

$$\text{K}={\text{x}}_{\text{cal}.}/{\text{x}}_{lab.}$$

(2)

K value ranges from zero to infinity with the optimum value of one [39]. It has been used by many researchers [39,40,41,42,43] to measure the success of the proposed relationships for bearing capacity, settlement, and compression index. The frequency of the ratios x_cal./x_lab. < 1 and x_cal./x_lab. > 1 are used to compare a method’s conservatism scale [40].

The ranking index (RI) serves as a comprehensive measure of precision and dispersion, utilizing both the mean and standard deviation of the ratio between calculated and measured values. It is defined as shown in Eq. (3):

$$RI=Mean \left|\text{ln} ({x}_{cal.}/{x}_{lab.})\right|+SD \left|\text{ln} ({x}_{cal.}/{x}_{lab.})\right|$$

(3)

It utilizes the mean and the standard deviation (SD) of K (x_cal./x_lab.) with the natural logarithm (ln). A lower RI value indicates a more successful correlation in general.

An alternative index value, as presented in Eq. (4), is the ranking distance (RD), proposed by Cherubini and Orr [44], which assesses the accuracy and quality of a calculation method by considering the distance to the optimal point. This optimal point is characterized by a mean (x_cal./x_lab.) value equal to one and a standard deviation (SD) equal to zero:

$$RD=\sqrt{{\left[1-Mean ({x}_{cal.}/{x}_{lab.})\right]}^{2}+{\left[SD ({x}_{cal.}/{x}_{lab.})\right]}^{2}}$$

(4)

When RD values are low, it indicates that the proposed correlation is both accurate and precise. Unlike RI values, which are on a logarithmic scale and provide a less suitable assessment for accuracy and precision, the RD method, as suggested by Orr and Cherubini [40], offers a more appropriate evaluation. RD considers accuracy more than precision for highly accurate methods (Mean K = 1), and it emphasizes precision more than accuracy for very precise methods (SD = 0), as noted by Cherubini and Orr [44]. Both RI and RD refer to a computational model that assesses accuracy using the mean and distribution through standard deviation. Low RI and RD values signify high accuracy and reliability of the calculation methods. Therefore, the most successful correlation would exhibit the lowest RD and RI values.

Theil inequality coefficient (TIC) is widely used to validate simulation models [45]. This easy-to-use method measures how well the calculated values compare to measured ones (Eq. (5)):

$$\begin{aligned}\text{TIC}&=\left(\sqrt{{\sum }_{1}^{N}{({\text{x}}_{\text{cal}.}-{\text{x}}_{\text{lab}.})}^{2}}\right)/\\ & \qquad \left(\sqrt{{\sum }_{1}^{\text{N}}{({\text{x}}_{\text{cal}.})}^{2}}+\sqrt{{\sum }_{1}^{\text{N}}{({\text{x}}_{\text{lab}.})}^{2}}\right)\end{aligned}$$

(5)

The TIC value ranges from 0 to 1. A value closer to zero signifies a higher degree of conformity between the calculated and measured parameters, indicating a more reliable correlation.

3 Analysis Results

As elaborated earlier, this study utilizes high-quality SPT data obtained from 42 distinct project sites, along with over 2500 laboratory test results conducted on undisturbed samples, to scrutinize the validity of correlations between SPT N and c_u, as well as SPT N and m_v, exclusively from Turkish sites. Additionally, local data is carefully examined to establish a novel correlation between index parameters and C_c tailored specifically for the Ankara clay. This section presents the analysis results.

3.1 Analysis of Undrained Shear Strength (c_u) Correlations

Correlations between SPT N values and c_u values have been established by various researchers over the years, including Hara et al. [46], Kulhawy and Mayne [47], Sowers [48], Stroud [49], and Terzaghi and Peck [50]. The correlations scrutinized in this study are presented in Table 2 in chronological order. Within the scope of the current research, c_u values are derived from 512 unconsolidated-undrained (UU) tests and 864 unconfined compression (UC) tests.

Table 2 Correlations for undrained shear strength (c_u) and compression index (C_c)

Correlations between c_u and SPT N are then examined using data solely from UU tests, and calculated through five previously mentioned statistical parameters (RMSE, K, RI, RD, TIC). It's important to note that only the soil classes specified in the correlations are considered in the analyses. According to the Unified Soil Classification System, the number of data points from the 512 UU tests are as follows: 320 CH, 122 CL, 24 MH, 17 SC, 11 OH, 9 ML, 4 OL, and 1 SM. For instance, statistical parameters for correlations 7a and 7b are evaluated solely for 320 CH soils. The safety intervals and statistical outputs of correlations can be seen clearly in Figs. 3 and 4 for UU-methods 1 and 2, respectively.

Fig. 3

Safety intervals and the statistical outputs of correlations for UU–method 1

Fig. 4

Safety intervals and the statistical outputs of correlations for UU–method 2

According to the results in Fig. 3, it can be observed that the highest estimating performance is obtained from correlations numbered 7d, 7c, 8, 7e, and 5 based on their RMSE values. When the TIC value approaches zero, this indicates a better degree of agreement between the calculated and measured parameters, and the correlation is more reliable. Thus, it can be inferred that correlations 7d, 7e, 7f, 7a, 8, and 7c demonstrate superior results based on their TIC values. Correlations numbered 5, 8, 7e, 7d, 7a have the lowest RI values. Based on ranking distance (RD), the most successful correlations can be ranked as 9a, 9b, 8, 5, and 7a. When both RD and RI values are low, this indicates better accuracy and reliability, as in the case for correlations of 5, 8, and 7a. A visual examination of Fig. 3 also signals that correlations 7e, 7d, 5, 8, and 7c show the highest performance in the estimation of c_u. Considering all statistical evaluations, it can be concluded that the most successful correlations for UU–method 1 are 8, 7e, 5, 7d, and 7c.

According to the results in Fig. 4, the five highest estimating performances are obtained from correlations 7b, 7d, 7f, 1, and 7e based on their RMSE values. TIC values of correlations 7f, 1, 7b, 7d, and 3 tend to be close to zero. For RI, correlations 1, 7b, 7f, 7a, and 7e give the lowest values. Based on RD, the correlations can be ranked as follows: 7d, 5, 8, 7e, and 7c. Correlation 7e is identified as successful based on both RD and RI. Considering all statistical evaluations, it can be concluded that the most successful correlations for UU-method 2 are 1, 7d, 7e, 7f, and 7b.

Correlations between c_u and SPT N are also examined using data measured from unconfined compression (UC) tests, and evaluated using five different statistical metrics as presented in Fig. 5. In the evaluations of the UC experimental data, the lowest five RMSE values are obtained for correlations 5, 7c, 9b, 9a, and 8, respectively, while the success ranking based on TIC values is 5, 7c, 8, 7d, and 7e. Correlations 9b, 7c, 9a, 8, and 5 give the lowest RI values. Correlations numbered 9a, 9b, 7c, 8, and 5 show the most successful performance depending on the values of RD. Correlations 9a, 9b, 7c, 8, and 5 are determined to be successful based on both RD and RI. Considering all statistical evaluations, it can be concluded that the most successful correlations based on UC test results are 7c and 9b, followed by correlations 8, 5, and 9a.

Fig. 5

Safety intervals and the statistical outputs of correlations for UC tests

Based on all the analyses and evaluations conducted regarding the estimation of c_u, it can be concluded that correlations 7c, 7d, 7e [51], 8 [52], and 5 [53] demonstrate superior performance compared to other correlations examined herein for preliminary geotechnical design. It's noteworthy that Sivrikaya and Togrol [51] also utilized data exclusively from Turkish clays; however, there is no intersection between their data and that used in this study. On the other hand, considering UU-method 1, UU-method 2, and UC results, correlations 6b, 6a, and 2a exhibit the least successful performance in estimating c_u.

3.2 Assessment of Stroud [49] Correlation

The validity and accuracy of the widely used correlation proposed by Stroud [49], which stands out from other SPT N-based correlations due to its direct incorporation of PI through factor f₁, are also examined. Stroud [49] conducted 1200 experiments, including 42 field experiments, to establish a correlation between SPT N value and undrained shear strength, which is further related to the PI. The relationship between SPT N, c_u, and PI (Stroud [49]) is expressed as ${c}_{u}\left(kN/{m}^{2}\right)={f}_{1}.N$, where f₁ is a factor inversely proportional to the soil PI, with values typically ranging between about 4.0 to 6.5 kN/m². No information was provided regarding the correction of the SPT value. The data utilized for UU-method 1, UU-method 2, and UC is compared with the results from Stroud [49]. For clarity purposes, the f₁ values obtained from the current study are denoted as f₁, while those from Stroud [49] are referred to as f_{1 Stroud}. The results in terms of f_{1 Stroud}/f₁ are depicted in Fig. 6.

Fig. 6

The distribution of (f_{1 Stroud}/f₁) according to test type

The mean, standard deviation (SD), and coefficient of variation (COV) values of each method's f_{1 Stroud}/f₁ ratio are also computed and utilized as an evaluation criteria. The UU-method 2 displays the lowest standard deviation (SD) and coefficient of variation (COV) values, whereas the UC method exhibits the highest values for f_{1 Stroud}/f₁ ratios. These results suggest that Stroud [49] correlation performs better with higher-quality testing data. The obtained f_{1 Stroud}/f₁ ratios are also evaluated by considering the safety levels suggested by Orr and Cherubini [40], and the results and distribution of the mean (f_{1 Stroud}/f₁) are presented in Table 3 and Fig. 8a. As shown in this table, UU–method 2 has the highest accuracy, with the closest mean to 1. In addition, the result corresponding to 76% of K < 1 shows that the results tend to remain on the safe (conservative) side when Stroud [49] correlation is used.

Table 3 Correlation assessments of (f_{1 Stroud} /f₁) according to Stroud [49]

3.3 Analysis of the Modulus of Volumetric Compressibility (m_v)

The modulus of volumetric compressibility (m_v) is the inverse of the oedometer test deformation modulus M. In this study, 992 consolidation test results are analyzed. Using the unit weights (γ) reported in the laboratory results, m_v values corresponding to the effective stress (σ_v′) at the sampling level were determined for each test. The dataset was also categorized based on soil classification, consistency conditions determined by SPT N values, PI, and σ_v' level. Stroud [49] presents one of the widely adopted SPT N value-based methods for estimating m_v, wherein the relationship between SPT N and m_v is represented by f₂, a variable dependent on PI. A histogram depicting the distribution of the ratio of Stroud [49] suggested f₂ values (f_{2 Stroud}) to the experimentally obtained f₂ values, i.e., f_{2 Stroud} /f₂, is illustrated in Fig. 7.

Fig. 7

The distribution of (f_{2 Stroud}/f₂)

The findings reveal that 79% of the f₂ values estimated by Stroud [49] exceed the measured f₂ values for soil samples from Turkiye. Consequently, employing the f₂ values derived from Stroud [49] correlation in design would typically yield lower m_v values and, consequently, smaller calculated consolidation settlements than realized. No significant improvement in estimation performance was noted when exclusively considering soils classified as CL and CH, as 77% (761) of the f₂ values calculated from Stroud [49] still exceed those measured (Fig. 7).

The distribution of samples based on SPT N values ranging from 4 to greater than 30, along with σ'_v levels and PI ranges, is depicted in Table 4. The safety levels of f_{2 Stroud}/f₂ are also provided in the same table. Figure 7 also presents results for samples classified as CL and CH, which constitute the majority of the dataset.

Table 4 Correlation assessments of data according to SPT N₆₀, σ_v′ and PI for CL, CH [13, 49]

The clays within the SPT N interval ranging from 4 to 15 exhibit the closest mean to 1.00 (1.19) and the lowest standard deviation (SD) value of 0.53, indicating that Stroud [49] correlations are more consistent with the measured results within this range. The correlation demonstrates the highest accuracy with a neutral safety level for clays of 4 < SPT N ≤ 15. As the SPT N increases, the safety level of the correlation transitions from neutral to very unsafe. Notably, approximately 86% of the f₂ values calculated by Stroud [49] surpass the laboratory-based f₂ values for SPT N > 30. Concurrently, the largest SD value (1.40) of f_{2 Stroud}/f₂ ratios is observed for SPT N > 30, suggesting a higher coefficient of variation and indicating that the results obtained with SPT N > 30 deviate significantly from the moduli measured in the laboratory. The percentage of results with K > 1 and the distribution of mean (f_{2 Stroud}/f₂) are depicted in Fig. 8b.

Fig. 8

Accuracy-reliability relationship between a) f_{1 Stroud}/f₁ b) f_{2 Stroud}/f₂

Figure 8b illustrates the percentage of cases with K > 1 as a function of the mean (f_{2 Stroud}/f₂) for various PI levels. For soils with PI < 20, approximately 85% of the f₂ values calculated from Stroud [49] exceed the actual f₂ values measured from oedometer tests. While results for soils with low plasticity suggest a high level of unconservatism, the safety level rises with increasing plasticity. The correlation demonstrates similar success rates considering the SD and mean values for clays with 20 < PI < 30 and PI > 30. Likewise, after PI exceeds 20, the test results tend to align more closely with Stroud [49]. Consequently, the accuracy of Stroud [49] equation improves for soils with PI > 20.

Among various ranges of σ_v', the interval of 400 < σ_v' ≤ 600 kPa exhibits the lowest SD value of 0.47 for f_{2 Stroud}/f₂. For σ_v' < 400 kPa, over 80% of the f₂ values calculated by Stroud [49] surpass those obtained from oedometer test results. As depicted in Fig. 8b, a higher percentage of K > 1 is observed with lower effective stress. Notably, with the increase of σ_v', particularly after approximately 400 kPa, both the accuracy and the safety level of Stroud [49] equation improve.

3.4 Analysis of the Compression index (C_c)

To assess the effectiveness of existing correlations in the literature for local soil conditions and potentially establish a more accurate relationship, the data from project sites in Ankara are utilized in analyzing the compression index (C_c). Table 5 presents the minimum, maximum, mean, and standard deviation values of various geotechnical parameters obtained from laboratory experiments on 283 undisturbed samples of Ankara Clay.

Table 5 Statistical variables of Ankara clay parameters used in the current study

Based on the index properties, 20 published correlations for the compression index are selected to investigate their validity for Ankara Clay, as presented in Table 2 in chronological order. Compression index values calculated from empirical correlations (C_c cal.) versus those measured by consolidation tests (C_c lab.) for the selected correlations are presented in Fig. 9. The performances of correlations for Ankara Clay are compared in Table 6 based on statistical measures K, RI, and RD. Additionally, these comparisons are visually depicted in Fig. 10.

Fig. 9

Calculated vs. measured C_c values for selected correlations and for the suggested correlation for Ankara Clay

Table 6 Statistical analysis compression index estimations based on correlations

Fig. 10

Safety intervals and statistical outputs of correlations of C_c

The analysis of the compression index (C_c) reveals that among the equations depending on a single variable (12 out of 20; correlations numbered 1, 2, 3, 4, 5, 9, 11, 12, 13, 15, 19, 20), the equations that demonstrate better estimation of the C_c for Ankara Clay are those relying on the initial void ratio (e₀). Regarding the safety levels, it is observed that all methods except Eqs. 1- Hough [54] and 11- Bowles [55] tend to overestimate the compression index, generally falling into the category of being very safe. Consequently, employing these equations could lead to overestimated settlements, potentially resulting in erroneous recommendations for soil improvement methods that may incur additional costs. Conversely, correlation 2, proposed by Hough [54] for organic soils, underestimates the compression index and is therefore deemed very unsafe.

According to results summarized in Table 6 and Fig. 10, the first five correlations with higher estimating performance (lower values) are 11, 1, 2, 18, and 15 based on their RMSE values. For the ranking index (RI), lower values indicate more successful estimations; hence, Eqs. 11, 1, 18, 15, and 6, with the lowest values of RI, are determined to be successful. The same can be stated on reliability for Eqs. 11, 1, 2, 18, and 15 considering ranking distance (RD). Correlations 11, 1, 18, 15 with lower values of both RD and RI display higher accuracy and reliability. On the other hand, when Fig. 10 is examined, correlations of 11 and 1 display the highest performance in the estimation of C_c. Considering all statistical evaluations, it can be said that the most successful correlations are 11- Bowles [55] (C_c = 0.156e₀ + 0.0107) and 1-Hough [54] (C_c = 0.29 (e₀-0.27)) in the estimation of the C_c value of Ankara Clay. These are followed by correlations numbered 18, 15, and 2.

3.5 The Proposed Correlation of Compression Index (C_c) for Ankara Clay

It has been observed that applying the general correlations for parameter estimation can be very misleading without considering the suitability of local soil conditions. In this case, considering the C_c and e₀ values used within the scope of the study, in the preliminary designs for Ankara Clay, it is recommended to use the Eq. 11- Bowles [55], and the following correlation (Eq. (6) - numbered 21 in Table 6 and Fig. 10) suggested by authors, based on the compiled data:

$${\text{C}}_{{\text{c}}} \, = \,0.{\text{1455e}}_{0} \, + \,0.0{215}$$

(6)

Equation (6) outperforms other correlations considered, with Bowles [55] showing a comparable level of success, as evidenced by the statistical measures K, RMSE, RI, and RD as illustrated in Figs. 9 and 10.

4 Discussion

A thorough assessment of the correlations between c_u and SPT N, as outlined in Table 3 for undrained shear strength, yielded several noteworthy observations. As previously mentioned, c_u values were derived from 512 UU tests, employing two different methods, and from 864 UC tests. The findings reveal notable disparities, not only in the numerical values of statistical measures but also in the success ranking of correlations depending on the test method utilized.

Firstly, regardless of the test type, no single method emerges as superior in terms of both accuracy and conservatism. Generally, the equations that exhibit higher degree of inaccuracies tend to be either overly conservative, as exemplified by Nassaji and Kalantari [56], or excessively unconservative, as seen in Decourt [57]. As anticipated, the degree of conservatism is somewhat reduced for relatively accurate equations, such as Sivrikaya and Togrol [51].

The most accurate correlations differ based on both the test type and the method of evaluation for UU tests. For UU-method 1, the top three successful correlations are 8, 7e, and 5e, whereas for UU-method 2, they are 1, 7d, and 7e. For UC test data, correlations 2, 7c and 9b emerge as the most effective. The distribution of the test type in the source database for each correlation may be the influencing factor this outcome.

The relative quality of the tests also impacts the correlation performance of the methods. UC tests, which lack the application of any confining pressure, are known to be more susceptible to erroneous results compared to triaxial testing. Conversely, particularly for over consolidated clays potentially not fully saturated, the second method employed in this study is anticipated to yield more dependable undrained shear strengths due to its focus on testing under higher chamber pressures exclusively. This distinction is clearly evident in the results presented here. For instance, when considering the mean of K values from all methods, which serves as a reliable indicator of accuracy, the closest to one is attained using UU test data and method 2 at 1.27. In contrast, the corresponding values for method 1 and UC data are significantly higher: 2.13 and 3.34, respectively. A similar trend is observed in the mean COV of K from all methods, which denotes reliability or consistency. The lowest COV stands at 57.9% for UU test data and method 2, while it escalates to as high as 80.1% and 101.9% when method 1 and UC data are employed, respectively.

Taking into account various statistical metrics assessed in this study alongside the testing methodologies, correlations 7d, 7c, 7e [51], 8 [52], and 5 [53] outperform their counterparts. Notably, the database utilized by Sivrikaya and Togrol [51], although distinct from this study, also focused exclusively on Turkish clays. This underscores the significance of localized data in achieving robust correlation outcomes.

Despite the rigorous testing protocols employed herein exclusively on Turkish clays, even the most effective correlations for c_u appear inadequate for direct application in geotechnical design processes, except perhaps for preliminary assessments given their reliability. For instance, the coefficient of variation (COV) of the K value—indicating the ratio of calculated to measured strengths—ranges between 52 and 62% for correlations utilizing UU test data and method 2. However, these values escalate significantly to fall between 94 and 105% for UC test data alone.

The widely-used correlation by Stroud [49], which directly incorporates the PI, was analyzed separately for c_u due to its unique approach in handling plasticity compared to other correlations. Notably, again, the UU-method 2 demonstrates the lowest standard deviation (SD) and coefficient of variation (COV) values, suggesting that Stroud [49] correlation also performs better with higher-quality testing data. However, this correlation tends to be conservative only when considering undrained shear strengths measured using method 2 exclusively. Moreover, despite its wide utilization in practice, Fig. 7 illustrates that there is minimal or no improvement in prediction accuracy and consistency compared to other correlations, with mean values and COVs ranging between 0.82 and 2.69, and 63% and 146%, for UU and UC test data, respectively. This serves as further evidence supporting the efficacy of local correlations, given that Stroud [49]'s dataset predominantly comprised British clays.

The analysis of the modulus of volumetric compressibility (m_v) reveals that Stroud [49] on average underestimates m_v by nearly twofold, leading to lower consolidation settlement predictions than observed. Even after grouping data based on soil classes, consistency, and effective stress level (as shown in Table 4), this trend remains consistent. However, focusing solely on CL and CH soils with 4 < SPT N < 30 and σ'_v > 200 kPa improves the safety level of the correlation from very unsafe to neutral. The relationship between m_v and PI, along with the distribution histogram of f_{2 Stroud}/f₂ is depicted in Fig. 11, while Table 4 provides a summary of correlation assessments across all intervals for comparative analysis.

Fig. 11

For CH, CL, 4 < SPT N < 30 and σ'_v > 200 kPa: a) Relationship of m_v & PI b) Distribution of (f_{2 Stroud}/f₂)

In Fig. 12, the trend line and half standard error limits (μ ± 1/2σ) are plotted for the f₂ values derived from the data averaged for each site. Notably, the correlation proposed by Stroud [49] tends to align closer to the upper limit. Hence, it is advisable to utilize f₂ values obtained from the mean line depicted in Fig. 12 rather than those suggested by Stroud [49] for preliminary designs involving Turkish clays.

Fig. 12

The suggested relationship between f₂ & PI

Upon analyzing 283 high-quality undisturbed samples of Ankara Clay, it is evident that equations based on the e₀ perform better in estimating the C_c. Upon closer examination of the K values, an average of 2.5 is observed, indicating a tendency for all equations to be conservative predictors, except for Hough [54], Terzaghi and Peck [50], and Bowles [55]. When the database is limited to Ankara clays, notably low coefficients of variation (COVs) are obtained, especially compared to those of c_u and m_v, with a mean of 30%, which can be deemed acceptable for routine designs. In the end, for Ankara Clay, it is advisable to utilize either Eq. 11[55] or the correlation C_c = 0.1455e₀ + 0.0215 suggested by the authors of this study.

5 Conclusion

This study employed 2500 high-quality undisturbed samples for shear strength and consolidation testing, alongside SPT N values from corresponding sample depths at 42 sites. The aim was to assess the accuracy and consistency of correlations found in the literature for local soil conditions in Turkiye, with a specific focus on Ankara Clay. The main findings of this research are outlined below:

After conducting a comprehensive evaluation of the correlations between c_u and SPT N using five different statistical variables, it was observed that no single method stands out as superior in terms of both accuracy and conservatism. The analysis revealed that the quality of the tests has a significant impact on the correlation performance, with the highest accuracy and consistency achieved using UU test data and method 2. However, even the most effective correlations appear insufficient for direct application in geotechnical design processes, given their low consistency as the COV of the ratio of calculated to measured strengths ranges between 52 and 62% for the best-case scenario, and exceeds 100% for UC test data.

Stroud [49] correlation, which directly incorporates the plasticity index (PI) for estimating c_u, also exhibits better performance with higher-quality testing data. However, no improvement in prediction accuracy and consistency was observed compared to other correlations. The relative success of Sivrikaya and Togrol [51], alongside the inefficiency of Stroud [49] in estimating c_u for Turkish clays, underscores the significance of utilizing localized correlations.

The examination of the modulus of volumetric compressibility (m_v) underscores a notable trend: Stroud [49] estimates consistently fall short, typically by nearly half of the actual values. This discrepancy is pivotal as it could lead to considerably lower predictions of consolidation settlement than what would be observed. Therefore, it is prudent to opt for f₂ values derived from the mean line depicted in Fig. 12 over those suggested by Stroud [49] when conducting preliminary designs involving Turkish clays.

The analysis of 283 high-quality undisturbed samples from Ankara Clay highlights the superiority of equations relying on the initial void ratio (e₀) in predicting the Compression Index (C_c). It's notable that all equations tend towards conservatism, with relatively lower coefficients of variation (COVs) compared to those observed for c_u and m_v, averaging at 30%. Consequently, the correlations for C_c hold greater relevance for routine or preliminary design purposes. Despite deriving a new correlation equation (C_c = 0.1455e₀ + 0.0215) exclusively from Ankara Clay data, Eq. 11 [55] emerges as equally applicable.