1 Introduction

In geotechnical engineering, description of the stress–strain behavior of soils is a key issue. It is known that the strain level affects the shear modulus (G) of geomaterials. At low strain levels (i.e., about 10−6 or less), G is approximately equal to the small-strain shear modulus (G0). Once G0 is obtained, it can be applied to estimate the shear response at different strain levels. The shear wave velocity (\(V_{\text{s}}\)) is always related to G0 based on elastic theory, which can be expressed as \(G_{0} = \rho V_{\text{s}}^{ 2}\), where ρ is the soil density. Thus, it is essential to estimate the \(V_{\text{s}}\) value of soils to determine G0. Moreover, \(V_{\text{s}}\) can also be used for liquefaction analysis, soil stratigraphy, and in situ strength estimation (Schneider et al. 2001; Andrus et al. 2004; Long and Donohue 2007; Cunning et al. 1995; Cha and Cho 2007; Tang et al. 2016; Oh et al. 2017).

\(V_{\text{s}}\) can be measured by laboratory or field tests. Laboratory methods include bender element, resonant column, the piezoelectric ring-actuator technique, etc. (Kim et al. 2013; Yang and Gu 2013; Karray et al. 2015), but the accuracy of such \(V_{\text{s}}\) measurements is strongly related to sample disturbance. Freezing samples is expensive, hindering application of this method. Meanwhile, in situ \(V_{\text{s}}\) measurements are also widely performed by geophysical field tests. In situ \(V_{\text{s}}\) measurements are widely performed by invasive testing (e.g., down-hole, cross-hole, and up-hole tests) as well as noninvasive geophysics methods (e.g., spectral analysis of surface waves, multichannel analysis of surface waves tests, etc.). The seismic piezocone penetration test (SCPTU) is a new kind of down-hole method, and the measured \(V_{\text{s}}\) is independent of the operator (Campanella et al. 1986; Cai et al. 2010; Mayne 2007). In the absence of direct testing or when in situ testing is not economically feasible for some low-risk projects, values of \(V_{\text{s}}\) can be estimated based on empirical correlations.

In previous studies, relationships between \(V_{\text{s}}\) and geotechnical parameters were developed using direct \(V_{\text{s}}\) measurements (Kulkarni et al. 2010; L’Heureux and Long 2017). However, it seems that the established empirical correlations suggested by prior studies were merely based on rather selective and limited laboratory and/or in situ data. Chang and Cho (2010) proposed a method to estimate the geotechnical engineering parameters of clays using \(V_{\text{s}}\) from laboratory tests, but the method is only applicable to reclaimed clays. Due to differences in soil type and soil variability, the established empirical correlations are not constant and will vary from place to place. Therefore, their application might be limited to site-specific conditions but not be suitable for other areas. On the other hand, although direct in situ measurements of \(V_{\text{s}}\) are preferable to indirect estimation, there are some disadvantages (e.g., the requirement for specialized equipment and experience). Thus, relationships between \(V_{\text{s}}\) and other parameters can be used for site investigation when in situ \(V_{\text{s}}\) measurements are not available.

The primary purpose of this study is to develop reliable relationships between geotechnical properties of Jiangsu clays and Vs values from SCPTU soundings. In this study, a database from 19 Jiangsu clay sites (in eight cities) was studied and an effort made to interlink the \(V_{\text{s}}\) values with engineering indices to obtain further understanding of their mutual correlations. Based on this database, some well-defined correlations were established. Relationships among \(V_{\text{s}}\) and the vertical effective stress (\(\sigma_{\text{v}0}^{{\prime }}\)), unit weight (γ), preconsolidation stress (\(\sigma_{\text{p}}^{{\prime }}\)), site-specific parameters (α, β), undrained shear strength (su), and piezocone penetration parameters are presented and compared with existing correlations. su and \(\sigma_{\text{p}}^{{\prime }}\) were both obtained from laboratory tests conducted on thin-wall tube samples and high-quality block samples. The relationships established herein could be used to evaluate basic soil properties using known \(V_{\text{s}}\) values, and vice versa. It is believed that such relationships represent an important development for reliable estimation of Jiangsu clay site characterization.

2 Testing Method and Database

2.1 Site Description

Geotechnical investigations were conducted in Jiangsu Province, China. Eight cities (Nanjing, Lianyungang, Yancheng, Huai’an, Taizhou, Zhenjiang, Nantong, and Suzhou) that contain sensitive clay deposits were selected to perform SCPTU. Figure 1 shows a map of Jiangsu Province with the approximate study site locations. Average values of soil engineering properties are presented in Table 1.

Fig. 1
figure 1

Distribution map of study sites

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Table 1 Summary of soil engineering properties in this study
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2.2 Testing Equipment and Data Processing

SCPTU field tests were conducted using a lightweight truck with a 20-ton-capacity hydraulic system, which is in accordance with international standards (ASTM D5778, 2012). The penetration rate was set as 20 mm/s, and nearly continuous data were produced at intervals of about 50 mm. \(V_{\text{s}}\) could also be measured by pushing the cone at 1.0-m intervals. Hammer impact was used for wave triggering, while two down-hole geophones measured the shear wave. The time delay between two shear wave arrivals at consecutive depths was determined by cross-correlation methods, then \(V_{\text{s}}\) values were determined for the midpoint of the two consecutive depths.

2.3 Sampling and Laboratory Testing

High-quality samples were taken at various depths corresponding to the depths where \(V_{\text{s}}\) data were obtained at each investigated site. The soil samples were collected using a stationary piston sampler with diameter of 76 mm at intervals of 1.0 m from ground level to the depth of penetration. When the fixed-piston sampler was removed from the borehole, the soil sample was sealed using wax for laboratory testing. Many measures were taken to ensure that little or no disturbance occurred during transportation.

The laboratory testing program with respect to the basic geotechnical properties of soils included the water content (w), unit weight (γ), Atterberg limits, void ratio (e), su, \(\sigma_{\text{v}0}^{{\prime }}\), and \(\sigma_{\text{p}}^{{\prime }}\). Note that su was measured by direct simple shear (DSS), anisotropically consolidated undrained triaxial compression (CAUC), and field vane tests. For the CAUC tests, each specimen was isotropically consolidated to an all-round stress (about 1/6–1/4 of \(\sigma_{\text{v}0}^{{\prime }}\)) equal to the suction pressure applied on the specimens. Then, it was isotropically consolidated to the stress corresponding to the in situ stress. The strain rate was 0.096 %/min. \(\sigma_{\text{p}}^{{\prime }}\) was determined using the Casagrande method by conventional oedometer tests. Once specimens were placed in the oedometer cell, vertical stress was applied. The settlements were measured by a gauge with precision of 0.001 mm. Each load was doubled relative to the previous load, and the duration of each loading step was about 24 h at least. When settlement did not proceed, the settlement was recorded. Finally, \(\sigma_{\text{p}}^{{\prime }}\) was obtained from e and \(\sigma_{\text{v}0}^{{\prime }}\) on logarithmic scale. Using the above-mentioned methods, soil parameters were obtained for the 19 study sites over the depth range for which shear wave velocity and high-quality sample data were available.

The values of w of Jiangsu clay ranged between 20 and 80% (Fig. 2a), with a large proportion being in the range of 30–40%. The majority of the plasticity index values varied between 10 and 40%, as shown in Fig. 2b, with the range of 20–30% accounting for a large proportion.

Fig. 2
figure 2

Summary of soil properties from the database of Jiangsu clays: a water content, b plasticity index

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The sample depth of the different Jiangsu clay samples analyzed in this study is shown in Fig. 3a. The in situ \(\sigma_{\text{v}0}^{{\prime }}\) values for these depths can be determined from the total stress and hydrostatic pressure when the water table and unit weight are known. Figure 3b presents a histogram of the in situ \(\sigma_{\text{v}0}^{{\prime }}\) values for the samples, showing values ranging from 17 to 225 kPa. Note from this figure that the greatest proportion of values lie around 100 kPa. The \(\sigma_{\text{p}}^{{\prime }}\) value can be used to assess the overconsolidation ratio (OCR) when combined with the current \(\sigma_{\text{v}0}^{{\prime }}\), as shown in Fig. 3c.

Fig. 3
figure 3

a Sampling depth, b in situ vertical effective stress, and c preconsolidation stress for samples in the database of Jiangsu clays

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2.4 In Situ Shear Wave Velocity

Figure 4 summarizes all the available \(V_{\text{s}}\) data in this study. It can be observed from Fig. 4 that the variation of Vs with increasing depth shows a very similar linear trend overall, differing only at depths close to the ground surface. The reason for these results is the effect of the increase in effective stress, which indicates that \(V_{\text{s}}\) depends on the effective stress, as mentioned above. It can also be noted that the magnitude of \(V_{\text{s}}\) is higher or lower at some sites compared with others. The very soft clay areas have lower \(V_{\text{s}}\) values. These may be affected by overconsolidated soil or high water content and organic clay, respectively.

Fig. 4
figure 4

Summary of all Vs data for the Jiangsu clay database

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The relationship between \(V_{\text{s}}\) and depth can be expressed using an equation of the following form proposed by Teachavorasinskun and Lukkunaprasit (2004):

$$(V_{\text{s}} )_{z} = (V_{\text{s}} )_{\text{g}} + mz,$$
(1)

where \(\left( {V_{\text{s}} } \right)_{z}\) (m/s) is the value at depth z (m), \(\left( {V_{\text{s}} } \right)_{\text{g}}\) (m/s) is the value close to the ground surface, and m (s−1) is the slope of the trend line of Vs against depth. The value m = 9 was adopted in this study, based on regression analysis of the collected Jiangsu clay database.

Figure 5 depicts a comparison of the measured \(V_{\text{s}}\) values versus those predicted using Eq. (1) with m = 9. Note that \(V_{\text{s}}\) is a function of the depth of the soil and can be estimated at any depth based on the known \(V_{\text{s}}\) value at the ground surface.

Fig. 5
figure 5

Comparison of measured Vs with predictions based on Eq. (1)

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3 Correlations between Shear Wave Velocity and Geotechnical Parameters

3.1 State Characteristics

3.1.1 Vertical Effective Stress

It is known that \(V_{\text{s}}\) is affected by \(\sigma_{\text{v}0}^{{\prime }}\), ρ, e, the soil structure and fabric, etc. (Hardin and Black 1968; Fumal 1978; Fumal and Tinsley 1985; Sully and Campanella 1995; Santamarina et al. 2001; Moon and Ku 2016). Figure 6 depicts all the collected data and the relationship between the in situ \(V_{\text{s}}\) and \(\sigma_{\text{v}0}^{{\prime }}\) obtained at each investigated site in the collected database of Jiangsu clays. Note that \(V_{\text{s}}\) increases with increasing \(\sigma_{\text{v}0}^{{\prime }}\). A good linear relationship between \(V_{\text{s}}\) and \(\sigma_{\text{v}0}^{{\prime }}\) can be seen in the fitting of the data. The best fit equation for the data (R2 = 0.68) is

Fig. 6
figure 6

Relationship of Vs versus σ ′v0

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$$V_{\text{s}} = 0.99\sigma^{\prime}_{\text{v}0} + 49.68.$$
(2)

Most of the data fall within 40 % of Eq. (2). The main reason why the other data fall outside this range is that there are some uncertainties in the evaluation of \(\sigma_{\text{v}0}^{{\prime }}\) in the field. The relationship between \(V_{\text{s}}\) and \(\sigma_{\text{v}0}^{{\prime }}\) indicates that the magnitude of \(V_{\text{s}}\) in geomaterials is closely linked to the vertical effective stress.

In addition, this relation between \(V_{\text{s}}\) and \(\sigma_{\text{v}0}^{{\prime }}\) is closely related to the fact that in situ \(V_{\text{s}}\) values are often normalized by \(\sigma_{\text{v}0}^{{\prime }}\) at 1 atmospheric pressure for evaluation of soil liquefaction (Kayen et al. 2013). Thus, the in situ \(\sigma_{\text{v}0}^{{\prime }}\) can be estimated from the measured \(V_{\text{s}}\) using Eq. (2), and vice versa.

3.1.2 Unit Weight

Some empirical correlations between γ and \(V_{\text{s}}\) are listed in Table 2. Because the estimation of γ proposed by Lunne et al. (1997) had some limits in terms of applicable soil types, various empirical relationships for CPT-based γ estimation were developed thereafter (Mayne et al. 2009; Robertson and Cabal 2010; Moon and Ku 2016).

Table 2 Summary of unit weight and shear wave velocity relationships
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Although most of the empirical correlations suggested in previous studies seem reasonable, there is still a need for specific research to overcome the limited test sites and suggest new correlations for Jiangsu clays. Nineteen sites were tested and readings of \(V_{\text{s}}\) were measured at intervals of 1 m. The values of \(V_{\text{s}}\) obtained from the SCPTU readings and the values of γ obtained from laboratory tests are plotted on semilog scale in Fig. 7. It can be observed that γ increases with increase in \(V_{\text{s}}\). Regression analysis revealed that the relationship between \(V_{\text{s}}\) and γ could be appropriately fit (R2 = 0.83) using the following expression:

Fig. 7
figure 7

Relationship between shear wave velocity and unit weight

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$$\gamma\, { = }\, 4.96 { + } 5.97 \cdot \lg \,V_{\text{s}}.$$
(3)

The correlation coefficient is 0.83, indicating relatively high fitting accuracy. In addition, this correlation between \(V_{\text{s}}\) and γ is feasible. As mentioned above, \(V_{\text{s}}\) can be treated as an effective stress parameter and the total overburden stress (\(\sigma_{\text{v}0}\)) can determine \(\sigma_{\text{v}0}^{{\prime }}\). The value of \(\sigma_{\text{v}0}\) can be calculated from the accumulation of γ with depth (\(\sigma_{\text{v}0}\, { = }\,\int \gamma \,{\text{d}}z\)), which indicates that the \(\sigma_{\text{v}0}^{{\prime }}\) value at any depth is a function of the γ of the soil. Thus, \(V_{\text{s}}\) is very closely related to γ and the developed relationship can be used to estimate \(V_{\text{s}}\) from γ, and vice versa.

3.1.3 Preconsolidation Stress

Soil behavior in terms of strength, compressibility, and permeability is distinguished by preconsolidation stress. The first step to determine the behavior of a soil formation is to obtain a nearly continuous yield stress profile. Thus, \(\sigma_{\text{p}}^{{\prime }}\) is an important focus for soil behavior and has been considered to be an important geotechnical parameter. Based on compression measurements, many methods for evaluating \(\sigma_{\text{p}}^{{\prime }}\) have been proposed and the results obtained by plotting methods and curve-fitting procedures. The first and most common method for determining \(\sigma_{\text{p}}^{{\prime }}\) was the graphical method proposed by Casagrande (1936). Thereafter, other researchers attempted to improve this by developing new and more definitive methods, e.g., Schmertmann’s (1953) reconstruction method, Janbu’s (1969) constrained modulus method, Butterfield’s (1979) logarithmic methods, the Becker et al. (1987) work-energy method, and Wang and Frost’s (2004) dissipated strain energy method.

However, the small-strain behavior was not considered in the commonly used methods mentioned above, and those methods that use the effective stress–void ratio relationship only reflect the global settlement. It is well known that the \(V_{\text{s}}\) profile can be applied in both static and dynamic geotechnical analyses, as it provides the small-strain shear modulus. Yoon et al. (2011) developed a reliable method for evaluating \(\sigma_{\text{p}}^{{\prime }}\) based on \(V_{\text{s}}\) while considering the small-strain behavior, but the \(V_{\text{s}}\) values were measured using bender elements in laboratory tests. Evaluation of \(\sigma_{\text{p}}^{{\prime }}\) based on in situ \(V_{\text{s}}\) measurements was studied by L’Heureux and Long (2017), but the proposed correlations are limited to the test sites. Therefore, it is very important and necessary to develop a new relationship between \(V_{\text{s}}\) and \(\sigma_{\text{p}}^{{\prime }}\) including the small-strain behavior of \(V_{\text{s}}\).

\(\sigma_{\text{p}}^{{\prime }}\) can be estimated using empirical relationships or analytical solutions with in situ \(V_{\text{s}}\) measurements. Figure 8 presents the relationship between \(\sigma_{\text{p}}^{{\prime }}\) and \(V_{\text{s}}\) (\(\sigma_{\text{p}}^{{\prime }}\) determined by the traditional Janbu method). Note that \(\sigma_{\text{p}}^{{\prime }}\) increases with an increase in \(V_{\text{s}}\). As expected, the correlation between these parameters is strong, since \(V_{\text{s}}\) is strongly related to the maximum past vertical effective stress experienced by clays. There is generally a satisfactory power function agreement between \(\sigma_{\text{p}}^{{\prime }}\) and \(V_{\text{s}}\). Some of the scatter and variation in Fig. 8 may be caused by highly overconsolidated clay, indicating that the fit may not be good for OCR, consistent with the study by L’Heureux and Long (2017).

Fig. 8
figure 8

Relationship between shear wave velocity and preconsolidation stress

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$$\sigma_{\text{p}}^{{\prime }} = 0.1097 \cdot (V_{\text{s}} )^{1.3575} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} R^{2} = 0.85.$$
(4)

Note that the correlation between \(\sigma_{\text{p}}^{{\prime }}\) and Vs is satisfactory overall and the fit trendline has a reasonable R2 value of 0.85. This is helpful given the sensitivity of settlement calculations to the \(\sigma_{\text{p}}^{{\prime }}\) value. However, the magnitude of OCR will affect the fit trendline, and highly overconsolidated clays behave differently and were excluded from this trendline. This is because Vs would be expected to represent the current state of stress, not at any higher stress stiffness. Therefore, the OCR effect will affect the fit trendline and cause data scatter. This finding is also reflected in Fig. 9.

Fig. 9
figure 9

Values of σ ′p measured and predicted from the new expression (Eq. 14)

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The measured \(\sigma_{\text{p}}^{{\prime }}\) values and those predicted using Eq. (4) are shown in Fig. 9. Here, the \(\sigma_{\text{p}}^{{\prime }}\) values were calculated from Eq. (4) based on in situ \(V_{\text{s}}\) measurements. Notably, the measured and predicted \(\sigma_{\text{p}}^{{\prime }}\) values are in good agreement (Fig. 9). Therefore, the developed relationship could be used as a first-order estimate of stress history when only shear wave velocity data are available rather than any other geotechnical investigations.

3.1.4 Site-Specific Parameters

Previous works have illustrated that the magnitude of \(V_{\text{s}}\) is affected by the effective confining stress (\(\sigma_{\text{m}}^{{\prime }}\)), e, fabric, etc. \(V_{\text{s}}\) can be treated as an important parameter to quantify the compression and stiffness properties of geomaterials. As the simplest correlation among the various relationships between \(V_{\text{s}}\) and the influential factors, the expression relating the stress and void to \(V_{\text{s}}\) can be expressed as (Santamarina et al. 2001)

$$V_{\text{s}} = \alpha (\sigma_{\text{m}}^{{\prime }} /1\,{\text{kPa}})^{\beta },$$
(5)

where the parameter \(\alpha\) (m/s) and the exponent \(\beta\) are material constants. The values of \(\alpha\) and \(\beta\) can be obtained based on experimental data using Eq. (5).

In fact, as shear waves describe the interparticle contact behavior, the parameter \(\alpha\) and exponent \(\beta\) are correlated with the contact behavior between the particles and their packing type, even at small strain (Lee et al. 2015). Thus, it has been suggested that relationships between the coefficient \(\alpha\) and exponent \(\beta\) can be used to characterize soil behavior at small strain. Table 3 summarizes the \(\alpha\) and \(\beta\) relationships reported in previous research. Note that \(\sigma_{\text{m}}^{{\prime }}\), as a factor affecting \(V_{\text{s}}\), can be categorized as (1) the mean normal stress, \(\sigma_{\text{m}}^{{\prime }} = (\sigma_{ 1}^{{\prime }} + \sigma_{ 2}^{{\prime }} + \sigma_{ 3}^{{\prime }} )/3\), where \(\sigma^{\prime}_{1}\), \(\sigma^{\prime}_{2}\), and \(\sigma^{\prime}_{3}\) are the effective stresses in x, y, and z direction; (2) the individual stress, \(\sigma^{\prime}_{\text{m}} = \sigma^{\prime}_{x} \cdot \sigma^{\prime}_{y},\) where \(\sigma^{\prime}_{x}\) and \(\sigma^{\prime}_{y}\) are the principal effective stresses in the direction of propagation and polarization; (3) the individual stress, \(\sigma^{\prime}_{\text{m}} = (\sigma^{\prime}_{x} + \sigma^{\prime}_{y} )/2\); and (4) the effective vertical stress, \(\sigma^{\prime}_{\text{m}} = \sigma^{\prime}_{v0}\). In previous studies, inverse relationships between \(\alpha\) and \(\beta\) were primarily obtained based on selective laboratory testing and some limited in situ testing (Bate et al. 2013; Cha et al. 2014; Lee et al. 2015; Ku et al. 2016). In this study, the compiled Jiangsu clay database was applied to investigate the relationships between \(\alpha\) and \(\beta\) observed in situ. In this work, it is proposed that the relationship between \(\alpha\) and \(\beta\) is based on the vertical effective stress (e.g., the in situ Vs–\(\sigma_{\text{v}0}^{{\prime }}\) model) because measurement of in situ horizontal effective stresses is a very difficult task.

Table 3 Summary of relationships between α and β
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The in situ parameters \(\alpha\) and \(\beta\) can be determined by plotting \(V_{\text{s}}\) versus the model vertical effective stress when geostatic stress conditions are evaluated, as discussed above. After careful examination of the correlations for all sites in the present study, the \(\alpha\) and \(\beta\) values are plotted with trend lines in Fig. 10. Each data point expressed as a value of the coefficient \(\alpha\) and corresponding exponent \(\beta\) in Fig. 10 represents critical reference information on the stress dependence for each test site. The relationship between \(\alpha\) and \(\beta\) can be appropriately fit (R2 = 0.98) using the following expression:

Fig. 10
figure 10

Compilation of in situ α–β computed for each site based on the effective vertical stress

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$$\beta = 1.07 - 0.20\ln (\alpha ).$$
(6)

It can be seen from Fig. 10 that the log-linear relationship shows good performance. The overall range of exponent \(\beta\) values obtained from in situ testing is between 0.3 and 0.85. This relationship between \(\alpha\) and \(\beta\) is in agreement with the study by Ku et al. (2016). Note that four clay sites (e.g., Lianyungang site1, Lianyungang site 2, Lianyungang site 3, and Suzhou site 2) resulted in high values of the exponent \(\beta\) in the range of 0.68–0.94. It can also be seen that the in situ measurement data show a rather wide range of \(\beta\) values (e.g., high \(\beta\)) compared with the correlation suggested by Cha et al. (2014) and Kang et al. (2014). This is not surprising, as both of those works were based on laboratory tests, in which a predetermined stress path will be exerted on an undisturbed sample. The undisturbed sample will undergo void ratio changes along the selected stress path. Overall, the type of test used, the test conditions, the actual aging effect, the inherent variability of soil, etc. will affect the resulting values of α and β. It is therefore difficult to simulate the actual situation in laboratory tests. This is one possible reason for the higher values observed in in situ conditions. Moreover, the relationship between α and β obtained from the shear wave velocity–stress models listed in Table 3 will also affect the values of α and β even for the same type of test. Therefore, it is suggested that separate equations should be used for laboratory-based and in situ measurements for better estimation of β values.

3.2 Strength Characteristic

As mentioned above, the \(V_{\text{s}}\) of soils mainly depends on \(\sigma_{\text{v}0}^{{\prime }}\) and e. The properties of \(V_{\text{s}}\) and \(s_{\text{u}}\) depend on common parameters, and \(V_{\text{s}}\) can also be directly used to estimate \(s_{\text{u}}\). Many studies have been done by researchers to develop relationships between \(V_{\text{s}}\) and \(s_{\text{u}}\) (Blake and Gilbert 1997; Yun et al. 2006; Chang and Cho 2010; Kulkarni et al. 2010; Taboada et al. 2013; Agaiby and Mayne 2015; Oh et al. 2017; L’Heureux and Long 2017; etc.). An overview of some correlations between \(V_{\text{s}}\) and \(s_{\text{u}}\) for clays all over the world is presented in Table 4. Note that most of the expressions have the same format, but different correlation coefficients. The primary reason for this phenomenon is that the value of \(s_{\text{u}}\) depends on the testing method used. Therefore, it is of great significance to know the origin of the data used to reach such conclusions. The same format can be expressed as

Table 4 Examples of available su–Vs correlations for clays
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$$s_{\text{u}} = aV_{\text{s}}^{{b}},$$
(7)

where a and b are correlation parameters that are often significantly related to site-specific conditions. Although the results of previous studies produced relationships between \(s_{\text{u}}\) and Vs with good performance, further studies of different regions are required. Figure 11 presents the relationship between \(s_{\text{u}}\) and Vs for Jiangsu clay, where the \(s_{\text{u}}\) values were obtained from the CAUC test. Note that \(s_{\text{u}}\) increases with increase in \(V_{\text{s}}\), and the power function fit shows better performance. The best fit relationship is given by the following equation:

Fig. 11
figure 11

Relationship between shear wave velocity and undrained shear strength

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$$s_{\text{u}}\, { = }\, 0.162V_{\text{s}}^{ 1. 5 0} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} R^{2} = 0.89.$$
(8)

As expected, the correlation between these two parameters is strong with correlation coefficient R2 of 0.89. Comparing the results presented in Table 4 and Fig. 11, the a and b values are found to match well with values reported in literature, especially those of Agaiby and Mayne (2015) for a correlation based on soils worldwide. This illustrates that Eq. (8) can be employed for evaluation of \(s_{\text{u}}\) of such clays if values of \(V_{\text{s}}\) are known.

The trend lines corresponding to the correlations of Kulkarni et al. (2010) and L’Heureux and Long (2017) are also displayed in Fig. 11. The value of the correlation coefficient (0.89) in this study is higher than those of Kulkarni et al. (2010) or L’Heureux and Long (2017). The trend line in this study is located below the lines obtained in other studies (as shown in Fig. 11). These differences may be caused by: (1) the fact that the correlations developed by both Kulkarni et al. (2010) and L’Heureux and Long (2017) are from laboratory measurements of \(V_{\text{s}}\), while in this study the \(V_{\text{s}}\) measurements were obtained by in situ testing, and (2) the fact that regional differences cause some soils to show different engineering properties.

The relationship between \(s_{\text{u}}\) and \(V_{\text{s}}\) also belongs to one kind of stiffness–strength correlation. Although large- and small-strain phenomena are not causally related and correspond to different particle-level processes, it is important and interesting that these two parameters can be related through the effective stress variable. These results show that \(s_{\text{u}}\) can be approximately estimated from \(V_{\text{s}}\), and vice versa.

3.3 CPTU Characteristic Parameter

The CPTU is a robust, simple, and economical test that gives rapid, continuous soil profiling for site investigation and design (Lunne et al. 1997; Cai et al. 2010). Meanwhile, \(V_{\text{s}}\) is an important geotechnical characteristic for soil properties. Therefore, it is important to develop relationships between \(V_{\text{s}}\) and various CPTU parameters, as these two techniques can be used complementarily. A number of empirical correlations between \(V_{\text{s}}\) and CPTU parameters have been developed by various researchers. The relevant CPTU parameters are the tip resistance (qc), corrected tip resistance (qt), qnet, sleeve friction (fs), pore pressure (u2), pore pressure parameter (Bq), etc.

Table 5 summarizes empirical correlations between \(V_{\text{s}}\) and CPTU parameters from literature. Note that the correlations have a similar form worldwide. Although the correlation equations have the same form, there is also a special need to include different factors into the equation to obtain a good fit for specific sites. Moreover, use of qt, which is slightly better than qc, to improve the fit of the data and using Bq instead of the soil index property (e0 or w) can also improve the performance of the correlation equations. Among the basic measurements obtained from CPTU, the variations in decreasing order are u2, qt, and fs (Powell and Lunne 2005; Long et al. 2008; Long and Donohue 2010) and the relationships developed based on qt or Bq are more reliable than those based on fs.

Table 5 Examples of available qt–Vs correlations for clays
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Even though these relationships are limited to the test sites only, the proposed relationships seem reasonable because (1) \(V_{\text{s}}\) can be related to the penetration resistance in CPTU and the penetration resistance is also affected by e, the density (ρ), and \(\sigma_{\text{v}0}^{{\prime }}\), and (2) \(V_{\text{s}}\) strongly depends upon \(\sigma_{\text{m}}^{{\prime }}\) and e.

It is known that the parameter qt is considered as the primary CPTU parameter for statistical regression analyses. The correlations related to qt or Bq and \(V_{\text{s}}\) for Jiangsu clays were studied by Cai et al. (2014), but correlations related to the parameter qnet have not yet been studied. The parameter qnet is equal to qt − σv0, which is related to the vertical total stress (σv0). Multiple statistical regression analyses were carried out in the current study to develop various forms of power function to relate \(V_{\text{s}}\) and qnet. The \(V_{\text{s}}\) values of the clays showed reasonable agreement when using a power function of qnet expressed as

$$V_{\text{s}} = 11.58q_{\text{net}}^{0.382}.$$
(9)

The R2 value was 0.60, and the number of datasets was 205 (Fig. 12). The prediction given by Eq. (9) can be improved by introducing the vertical effective stress (\(\sigma_{\text{v}0}^{{\prime }}\)), as shown in Fig. 13a, resulting in the expression (R2 = 0.68)

Fig. 12
figure 12

Comparison of measured Vs with that predicted from the net cone resistance (qnet) in this study

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Fig. 13
figure 13

Comparison of measured Vs with that predicted using the expressions a for the net cone resistance (qnet) and effective stress (\(\sigma_{\text{v}0}^{{\prime }}\)) in this study, b of L’Heureux and Long (2017), c of Taboada et al. (2013)

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$$V_{\text{s}} = 9.337(q_{\text{net}} )^{0.2306} \left( {\sigma_{\text{v}0}^{{\prime }} } \right)^{0.2721}.$$
(10)

The correlation coefficients in decreasing order are 0.60 and 0.68, respectively, revealing that a better correlation between the \(V_{\text{s}}\) and CPTU parameters for Jiangsu clay in this study can be obtained using the combination of qnet with \(\sigma_{\text{v}0}^{{\prime }}\). Note that the predicted value of Vs is highly consistent with the measured \(V_{\text{s}}\). Moreover, Jiangsu clays are highly structured and sensitive, so estimation of \(V_{\text{s}}\) directly using fs or combined with other CPTU parameters may be unreasonable.

The measured \(V_{\text{s}}\) values and those predicted using the original expressions of L’Heureux and Long (2017) and Taboada et al. (2013) are presented in Fig. 13b and c, respectively. The values of \(V_{\text{s}}\) predicted using their expressions deviate from the \(V_{\text{s}}\) values measured for Jiangsu clays. Due to their dependence on soil type and some uncertainties (soil heterogeneity and geologic origin, inherently) of the established empirical correlations, they cannot always be applied at different locations. Thus, new relationships developed for specific sites are significant.

It can be observed from Fig. 13a that methods based on CPTU parameters (qnet) and \(\sigma_{\text{v}0}^{{\prime }}\) match the measured data well. The most important point is that measurements of \(V_{\text{s}}\) are independent from the CPTU data. It appears that good predictions of \(V_{\text{s}}\) can be obtained based on CPTU parameters for Jiangsu clays and the developed relationships can be used for cross-checking with each other. Thus, the \(V_{\text{s}}\) value of clays can be estimated using measured CPTU parameters.

4 Discussion

It is well known that a shear wave can only cause shear deformation and that its velocity (Vs) can be treated as an effective stress parameter (Hussien and Karray 2015). Calculation of geotechnical parameters is mostly based on the principle of effective stress, thus fundamental relationships exist between \(V_{\text{s}}\) and geotechnical parameters of soils.

Figure 14 presents a flowchart for estimating geotechnical parameters based on \(V_{\text{s}}\). Note that the in situ \(V_{\text{s}}\) value is generally superior to the \(V_{\text{s}}\) value measured by laboratory tests (with very little or no soil disturbance at lower cost) (Cai et al. 2010). The specific process is as follows:

Fig. 14
figure 14

Flowchart for Vs-based assessment of geotechnical parameters

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  1. 1.

    Equation (1) with m = 9 can be used to roughly estimate Vs at any depth based on a known Vs value at the ground surface. However, the value of Vs on the ground surface is influenced by many factors and the measured value is not always very accurate;

  2. 2.

    The relationships between Vs and \(\sigma_{\text{v}0}^{{\prime }}\), and site-specific parameters (α, β) are given by Eqs. (2) and (6), respectively;

  3. 3.

    The relationships between G0 or Vs and γ are given by Eq. (3);

  4. 4.

    The relationships between Vs and \(\sigma_{\text{p}}^{{\prime }}\) or su are given by Eqs. (4) and (8), respectively;

  5. 5.

    The relationships between Vs and CPTU parameters (qnet or qnet combined with \(\sigma_{\text{v}0}^{{\prime }}\)) are given by Eqs. (9) and (10), respectively.

Although geotechnical parameters vary with depth or location, these reliable correlations have been established based on a large number of data for preliminary estimation or design. Some of the developed relationships may not be satisfied at sites where clays are heavily overconsolidated or with soil and geologic origin heterogeneity. Overall, for investigation of a region of interest at a site, if in situ tests are conducted to obtain the Vs profile, the described correlations could be used to give first-order estimates of soil properties at any depth for engineers. It would be worthwhile to carry out further experimentation to revise these correlations using a larger number of test data.

5 Conclusions

The aims of this study are to provide engineers with guidelines for estimating \(V_{\text{s}}\) of Jiangsu clays when site-specific data are absent or for first-order estimates of soil properties when \(V_{\text{s}}\) data are known and available prior to other geotechnical investigations. To achieve these aims, a database of Jiangsu clays was compiled and statistically analyzed to produce relevant correlations. Some of the reliable and important results are as follows:

  1. 1.

    It was found that the soil unit weight of Jiangsu clays could be approximately estimated from the measured \(V_{\text{s}}\). The new expression developed may be affected by the stress level, which is a stress-independent expression for the soil γ; It was also observed that the values of α and β for the stress-dependent model could be determined site-specifically in Jiangsu clays. The relationship between α and β can be expressed specifically as \(\beta = 1.07 - 0.20\ln (\alpha )\) for Jiangsu clays of this study

  2. 2.

    It was noted that the in situ \(V_{\text{s}}\) correlates satisfactorily with the σ ′p or \(s_{\text{u}}\) values, and that these correlations can be used to evaluate soil parameters such as σ ′p or \(s_{u}\) from the measured \(V_{\text{s}}\), and vice versa. The relationship between \(V_{\text{s}}\) and \(s_{\text{u}}\) shows better performance than the others, and the developed relationship between \(V_{\text{s}}\) and \(s_{\text{u}}\) also belongs to one kind of stiffness–strength correlation.

  3. 3.

    The suggested correlations based on CPTU parameters could be used for preliminary estimation of \(V_{\text{s}}\) for Jiangsu clays. The new relationships for Jiangsu clays expressed by \(V_{\text{s}} = 9.337(q_{\text{net}} )^{0.2306} \left( {\sigma_{\text{v}0}^{{\prime }} } \right)^{0.2721}\) containing the \(\sigma_{\text{v}0}^{{\prime }}\) parameter can be used for preliminary estimation of Vs in geotechnical engineering investigations.