Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 Contents lists available at ScienceDirect Journal of Rock Mechanics and Geotechnical Engineering journal homepage: www.rockgeotech.org Full Length Article Assessment of clay stiffness and strength parameters using index properties Sayed M. Ahmed Structural Engineering Department, Ain Shams University, Cairo, 11517, Egypt a r t i c l e i n f o a b s t r a c t Article history: Received 5 September 2017 Received in revised form 5 October 2017 Accepted 8 October 2017 Available online 20 February 2018 A new approach is developed to determine the shear wave velocity in saturated soft to firm clays using measurements of the liquid limit, plastic limit, and natural water content with depth. The shear wave velocity is assessed using the site-specific variation of the natural water content with the effective mean stress. Subsequently, an iterative process is envisaged to obtain the clay stiffness and strength parameters. The at-rest earth pressure coefficient, as well as bearing capacity factor and rigidity index related to the cone penetration test, is also acquired from the analyses. Comparisons are presented between the measured clay parameters and the results of corresponding analyses in five different case studies. It is demonstrated that the presented approach can provide acceptable estimates of saturated clay stiffness and strength parameters. One of the main privileges of the presented methodology is the site-specific procedure developed based on the relationships between clay strength and stiffness parameters, rather than adopting direct correlations. Despite of the utilized iterative processes, the presented approach can be easily implemented using a simple spreadsheet, benefiting both geotechnical researchers and practitioners. Ó 2018 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). Keywords: Soft to firm clays Atterberg limits Shear wave velocity Small-strain shear modulus Constrained modulus Undrained shear strength Effective friction angle Cone penetration test 1. Introduction Determination of the parameters of compressibility and strength of geomaterials is a difficult yet essential task in geotechnical analyses. Reliable estimates of these parameters allow geotechnical engineers to design structures, roads and utilities to be safe, serviceable and economic. Commonly, geotechnical engineers, in the initial phases of the geotechnical investigation, focus on the index properties such as the liquid limit, LL, plastic limit, PL, natural water content, wn, and gradation tests. These tests are generally cheap, as they do not require expensive capacious equipment or special experience. They are mainly used to serve in the classification of soils. The variation of the natural void ratio, e0, due to changes in the in situ effective vertical stress, s0v0 , may be assessed using measurements of the natural water content, wn, with depth since wn and e0 are interrelated (i.e. e0 ¼ GSwn/100 in saturated soils, where GS is the soil specific gravity). The inferred wn-s0v0 data can be considered as the result of a natural full-scale sustained oedometer test that cannot be fully replicated in the laboratory even with the E-mail address: sayed_mohamed@eng.asu.edu.eg. Peer review under responsibility of Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. use of the most sophisticated equipment and sampling procedures due to the inevitable sample disturbances. Subsequently, distinct clay units (i.e. clay units having different geotechnical characteristics) may be delineated by considering the disclosed wn-s0v0 relationship as well as values of the LL and PL pertaining to a specific study area. This simple approach was utilized to characterize the gravitational compression of different natural clays (Skempton, 1970). It is also employed to differentiate between the subunits of London clays (Hight et al., 2007). Geotechnical engineers commonly utilize clay index properties to estimate the geotechnical parameters. For example, the use of plasticity index, PI, to estimate the effective friction angle, f0 , and the use of the liquidity index, LI, to determine the undrained shear strength, su, are the normal geotechnical practices. Nevertheless, such correlations have a substantial scatter (Kulhawy and Mayne, 1990; Ameratunga et al., 2015). Fig. 1 shows the obviously scattered PI-f0 data, pertaining to different clays (Tanaka, 2002), versus a recent PI-f0 correlation proposed by Sorensen and Okkels (2013). This figure also illustrates that some clays may have a high effective friction angle, f0 , despite of high plasticity index, PI, which contradicts with the commonly related correlations. Moreover, some clays naturally exist with a liquidity index, LI, greater than 100% (i.e. the natural water content, wn, is greater than the liquid limit, LL); yet they have non-trivial undrained shear https://doi.org/10.1016/j.jrmge.2017.10.006 1674-7755 Ó 2018 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 580 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 (i) If DS > 1.123, then we have .    s0p pa ¼ 1:62 s0v0 pa 0:89 ðLLÞ0:12 wn 0:14 (2) (ii) If DS  1.123, then we have . s0p pa ¼ 7:94 s0v0 pa   0:71 ðLLÞ0:53 wn 0:714 (3) where DS is the sample discrimination function given as follows:    DS ¼ 5:152 log10 s0v0 pa 0:061 LL 0:093 PL (4) þ 0:0622 GS wn Subsequently, the overconsolidation ratio, OCR, is estimated as follows: Fig. 1. Relationship between the effective friction angle f0 and plasticity index PI. strength, su, that increases with depth. This contradicts with the common correlations that relate su to LI as they presume such clays to have negligible undrained strength. Examples of these clays include Ariake clay in Japan and Champlain clay in Canada (Rochelle et al., 1974; Tanaka et al., 2001). The above-mentioned discrepancies shed doubts on the reliability of the correlations relating the clay parameters to the index properties. One of the anticipated reasons for these discrepancies is that such correlations are developed based on the data pertaining to particular sites. Hence, inaccuracies may occur when such correlations are applied to other sites with varied clay mineralogy, gradation and/or state of stresses. Additionally, many influencing factors, such as aging, cementation, and anisotropy, cannot simply be incorporated into the correlations that relate clay parameters to the index properties. In this context, site-specific correlations are generally expected to perform better than generic correlations. In this study, the parameters controlling the site-specific relationships between the shear wave velocity, VS, the void ratio, e0, and the effective mean stress, p0 , are obtained using measurements of the clay index properties with depth. Subsequently, the strength and stiffness parameters are determined using relationships interrelating clay stiffness and strength. The presented approach comprises an iterative site-specific methodology rather than a direct correlation. It can be implemented using a simple spreadsheet. Thus, it can be utilized in both geotechnical research and routine engineering practices. 2. Previous related correlations The following correlations are utilized in this study as a part of the presented procedures: (1) The compression index, CC, is related to the plasticity index, PI, as follows (Wroth and Wood, 1978): . OCR ¼ s0p s0v0 (3) Based on the estimated OCR, the at-rest earth pressure coefficient, K0, can be determined as follows (Mayne and Kulhawy, 1982):   K0 ¼ s0h0 s0v0 ¼ 1  0 sin f0 OCRsin f Accordingly, the mean effective stress, mined as follows: (6) p0 , can also be deter- p0 ¼ s0v0 ð1 þ 2 K0 Þ=3 (7) Eqs. (1)e(7) can simply be substituted with alternative sitespecific correlations or factual data without affecting the structure of the presented methodology. 3. Site-specific formulation of the shear wave velocity Recently, the use of shear wave velocity (either independently or with other parameters) has become an increasing trend in geotechnical engineering. The shear wave velocity has been related to many important geotechnical parameters such as the bulk unit weight, effective friction angle, undrained shear strength, and atrest earth pressure coefficient (Mayne, 2014; Hussien and Karray, 2016; L’Heureux and Long, 2016; Moon and Ku, 2016a). Moreover, the indispensable need to consider the small-strain stiffness in geotechnical applications is related to the shear wave velocity, which has been frequently demonstrated (Burland, 1989; Atkinson, 2000; Elhakim, 2005; Benz, 2007; Clayton, 2011). Many studies have shown that the shear wave velocity can be expressed directly as a function of the effective mean stress, p0 , the soil void ratio, e0, and the OCR (Hardin and Richart, 1963; Hardin, 1978). More recently, a different approach has been presented. The shear wave velocity is expressed as a function of certain sitespecific parameters with either the effective mean stress, p0 , or the void ratio, e0 (Santamarina et al., 2001; Ku et al., 2016) as follows: VS;p0 ¼ aðp0 =1 kPaÞ CC ¼ GS PI=200 (5) b (8) (1) (2) The effective pre-consolidation pressure, s0p , is related to the liquid limit, LL, plastic limit, PL, natural water content, wn, and effective vertical stress, s0v0 , as follows (Kootahi and Mayne, 2016): VS;e0 ¼ aeb0 (9) where VS;p0 and VS;e0 are the estimates of shear wave velocity using the mean effective stress, p0 , or the void ratio, e0, respectively. The parameters a, b, a, and b are the site-specific parameters. S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 It is noted that Eqs. (8) and (9) do not provide VS as functions of the effective mean stress p0 or void ratio e0 only, as these formulae depend also on site-specific parameters, i.e. a, b, a, and b. Based on the measurements made in many sites with different geomaterials (including clays and sands) at different states of stresses and OCRs, it is found that the site-specific parameters a and a are related to b and b, respectively, which can be written as follows (Ku et al., 2016; Moon and Ku, 2016b): b ¼ 1:02 0:18 ln a (10) b ¼ 3:253 0:796 ln a 581 (11) Generally, the site-specific parameters (i.e. a, b, a, and b) are determined from in situ measurements of the shear wave velocity VS. In this study, the shear wave site-specific parameters, i.e. a, b, a, and b, are determined by utilizing the variation of the natural water content, wn (or the void ratio e0) with the effective mean stress, p0 . Hence, shear wave velocity VS can be estimated consequently. Fig. 2. Iterative procedure used in this study. Fig. 3. Continuation of the analysis after convergence. 582 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 4. Methodology VS;e0 ¼ 60 expð 4.1. Determination of the shear wave velocity VS b (13) Since VS ¼ VS;p0 ¼ VS;e0 , we have According to Eqs. (8)e(11), VS;p0 and VS;e0 may be expressed as follows: VS;p0 ¼ 290 expð 5:556 bÞ ðp0 =1 kPaÞ 1:256bÞeb0 (12) 1:576    5:556 bv þ bv ln s0v0 1 kPa ¼ 1:256 b þ b ln e0 (14) As lnGS z 1 for the common value range of GS, lne0 in saturated soils may be expressed as follows: Fig. 4. Locations of the five case studies. Fig. 5. Index properties and inferred OCR of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 ln e0 ¼ ln GS þ ln wn ln 100zln wn 3:605 (15) Accordingly, Eq. (14) may be rearranged so that a linear relationship between ln wn and ln p0 is written as follows: ln wn ¼ iw mw lnðp0 =1 kPaÞ (16) 583 where iw and mw are the site-specific parameters representing the intercept and the negative slope of the ln wn ln p0 linear relationship, respectively. The intercept iw and the slope mw in Eq. (16) can be determined using simple regression analysis with common spreadsheet program, such as EXCEL, by assuming a power fitting function with respect to the natural water content, wn, and the mean effective stress, p0 , as follows: wn ¼ Iw ðp0 =1 kPaÞ mw (17) The parameter iw is related to Iw as follows: iw ¼ ln Iw (18) It may be more convenient in some cases to consider the equivalent e0-p0 relationship as e0 ¼ Ie ðp0 =1 kPaÞ ie ¼ ln Ie Fig. 6. Regression analysis of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). Fig. 7. Undrained shear strength su of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). me (19) (20) where ie, me, and Ie are the site-specific parameters representing the intercept of the ln e0 ln p0 linear relationship, the negative exponent of the e0-p0 power relationship, and the constant of the e0-p0 power relationship, respectively. The relationships between iw, ie and mw, me are Fig. 8. At-rest earth pressure coefficient K0 of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). 584 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 iw ¼ ie þ 3:605 (21) mw ¼ me (22) Following the determination of the parameters iw and mw, the shear wave velocity parameters b and b are obtained: b ¼ b¼ 1:576=ð4:861 þ 5:556 mw (23) iw Þ (24) mw b Theoretically, Vs estimated using the formulations of V or VS;e0 should be identical. Nevertheless, as commonly acknowledged, there might be differences between independent expressions for the same geotechnical parameter, especially when they depend on different inputs. Hence, in case of VS;p0 sVS;e0 , it is logical to use the average of them as an estimate of VS. This average is assumed herein as the geometric mean value. Hence, the shear wave velocity VS may be estimated using VS;e0 and VS;p0 as follows: S,p0 VS ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VS;p0 VS;e0 (25) Accordingly, VS may be expressed as VS ¼ 132 expð 2:778 b 0 0:628 bÞðp =1 kPaÞ The geometric mean value, expressed in Eqs. (25) and (26), is preferred over other averaging methods as it yields a similar expression to the direct correlations that relate the shear wave velocity VS to both of the mean effective stress, p0 , and the void ratio, e0. Nevertheless, it is noted that the above equations utilize the site-specific parameters b and b in lieu of constants in the direct correlations. The mean effective stress, p0 , in the above equations is determined in accordance with Eqs. (2)e(7). An initial value for the effective friction angle, f0 , is needed at the start of the analyses. This initial value may be assumed as 30 . Subsequently, an iterative procedure is used to determine the true value of f0 as detailed in the following sections. 4.2. Determination of clay stiffness parameters In this section, the small-strain shear modulus, G0, and the operative moduli, G50, and Eoed, are deduced from the deduced shear wave velocity. Firstly, the small-strain shear modulus, G0, is determined from the shear wave velocity and the soil unit weight as follows (Mayne, 2014): G0 ¼ VS2 g=g b=2 ðe0 Þ b=2 (26) Fig. 9. The shear wave velocity VS of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). (27) where g and g are the clay unit weight and the gravitational acceleration, respectively. Fig. 10. The effective friction angle f0 of Bothkennar clay case study. Sources of data: Allman and Atkinson (1992), Hight et al. (1992, 2003), Nash et al. (1992), McGinty (2006), and Mayne (2016). S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 The secant shear modulus at 50% of the ultimate shear stress, G50, is related to the small-shear modulus, G0, as follows (Krage et al., 2014): G50 ¼ 0:26 G0 (28) The tangential constrained/oedometric modulus, Eoed, is related to G0 as follows (Mayne et al., 2001): Eoed ¼ G0 =10 to G0 =20 (29) 585 The operative shear modulus G in Eq. (30) may be assumed equal to G50 (Kulhawy and Mayne, 1990; Mayne, 2001). Thus, Eq. (30) may be modified as follows:    1 þ e0 1 þ ln OCR G0 su;CIUC ¼ 82:123 M CC OCR0:8 (31) Additionally, based on the critical state mechanics, the CIUC shear strength ratio su,CIUC/s0 v0 may be expressed using the parameters M and OCR as follows (Wroth and Wood, 1978; Kulhawy and Mayne, 1990): 4.3. Determination of the effective friction angle In this section, the effective friction angle, f0 , is determined using the formulation of the rigidity index, Ir, which relates clay strength to its stiffness. The parameter, Ir, is defined as the ratio of the operative shear modulus G to the undrained shear strength su (i.e. Ir ¼ G/su). For the case of the consolidated isotropic undrained triaxial compression (CIUC) test in soft to firm clays, the rigidity index Ir,CIUC may be estimated using the critical state mechanics as follows (Kulhawy and Mayne, 1990; Mayne, 2001):  Ir;CIUC ¼ G su;CIUC ¼ 21:352 M   1 þ e0 1 þ ln OCR CC OCR0:8 (30) where M is a critical state strength parameter. It is related to the sinfcs Þ. critical state friction angle fcs as M ¼ 6 sin fcs =ð3 However, in soft to firm clays, the difference between the peak strength friction angle, f0 , and the critical state friction angle, fcs, may be ignored. Hence, the parameter M may be defined for these studied clays as M ¼ 6 sinf0 /(3esinf0 ) (Kulhawy and Mayne, 1990).  su;CIUC s0v0 ¼ 0:287 MðOCRÞ0:8 (32) By combining Eqs. (31) and (32), the parameter M may be estimated as follows: M ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðG0 CC Þ 23:57 s0vo ð1 þ e0 Þð1 þ ln OCRÞ (33) Subsequently, the effective friction angle, f0 , is estimated using the value of M obtained from Eq. (33) as follows: f0 ¼ sin 1 ½3M=ð6 þ Mފ (34) The outcome of Eq. (34) is considered a better estimate of the effective friction angle f0 than its initial assumption (i.e. 30 ). The effective friction angle, f0 , becomes closer to its real value with more iterations; the value of f0 concluded from an iteration is considered to be the new initial value in the subsequent iteration. The iterations stop when the effective friction angle f0 converges to its anticipated correct value. Fig. 11. Index properties and OCR of Ariake clay case study. Source of data: Tanaka et al. (2001). 586 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 employed in the geotechnical analysis that involves different stress paths such as the analysis of foundations and slopes. The DSS undrained strength su,DSS can be obtained from the estimated effective friction angle, f0 , overconsolidation ratio, OCR, and the effective vertical stress, s0 v0, as follows (Wroth and Wood, 1978; Kulhawy and Mayne, 1990): su;DSS ¼ 1 sin f0 OCR0:8 s0v0 2 (35) The undrained shear strength for other stress paths can also be determined for the same stress and effective strength parameters (Kulhawy and Mayne, 1990). 4.5. Determination of the cone factors Ir and Nkt Fig. 12. Regression analysis of Ariake clay case study. Source of data: Tanaka et al. (2001). 4.4. Determination of the undrained shear strength Based on the estimated values of G50 and su,DSS, the cone rigidity factor, Ir, cone, may be determined as   Ir;cone ¼ G50 su;DSS ¼ 0:26 G0 su;DSS (36) Additionally, The cone undrained strength factor Nkt may be estimated using the results of the above analyses as follows (Yu et al., 2000): The undrained shear strength varies with the stress path. Typically, the undrained shear strength of the triaxial compression (TC) test is the largest and that of the triaxial extension (TE) test is the lowest. The undrained strength of the direct simple shear (DSS) test is nearly the average of these limits, and hence it is often Nkt ¼ 2 ln Ir;cone þ 1:515 Fig. 13. Undrained shear strength su of Ariake clay case study. Source of data: Tanaka et al. (2001). Fig. 14. Small-strain shear modulus G0 of Ariake clay case study. Source of data: Tanaka et al. (2001). 0:915 ð1 K0 Þ s0v0 su;DSS (37) S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 587 4.6. Iterative procedure As mentioned previously, the analysis starts with an assumption of the effective friction angle f0 ¼ 30 . Subsequently, an updated value is obtained from Eq. (34). A new iteration starts with the updated value of f0 until two subsequent iterations yield close values of the effective friction angle, f0 . Fig. 2 shows the flow chart of the proposed methodology before and during the iteration process. Once convergence is reached, the clay stiffness and strength parameters are calculated as demonstrated in Fig. 3. 5. Validation cases Five well-reported case studies are analyzed using the proposed approach. The case studies are from different places around the world (i.e. United Kingdom, Japan, Thailand, South Korea and Canada), as shown in Fig. 4. The consistency of the clays in these case studies varies from very soft to firm. In two of the cases, the natural water content, wn, exceeds the liquid limit, LL (i.e. Ariake and Champlain clays). As illustrated before, a regression analysis is required to determine the power function that relates the natural water content, wn, and the mean effective stress, p0 , in Eqs. (17) and (18). This function is represented by a straight line in the log10 wn log10 p0 space. Data points used in the regression analyses are to be selected in accordance with the following tentative criteria: (1) The natural water content, wn, should decrease with the increase of the mean stress, p0 . Data points having different trends are to be ignored in the regression analyses. (2) Outliers (i.e. points that may substantially reduce the coefficient of determination R2 when considered) are removed from the data utilized for the regression analyses. Fig. 15. Cone factor Nkt of Ariake clay case study. Source of data: Tanaka et al. (2001). Fig. 16. Index properties and OCR of Bangkok clay case study. Source of data: Bergado et al. (2002). 588 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 The measured undrained shear strength of clays in the case studies was mainly determined using vane shear test (VST). In some case studies, clay parameters other than the undrained shear strength (e.g. VS, G0, f0 , K0 and Nkt) were also reported; these parameters are also compared with the results of the analyses for validation of the proposed approach. also noted that the soil granulometry also changes abruptly from silty clay (about 50% silt þ 50% clay) above the depth of 18 m to sandy clay (about 50% sand þ50% clay) below that depth (Tanaka et al., 2001). The regression analysis between the natural water content, wn, and the effective mean stress, p0 , is shown in Fig. 12; the results are Iw ¼ 937.44, iw ¼ 6.843, mw ¼ 0.582 and R2 ¼ 0.89. Hence, the site- 5.1. Bothkennar clay This site is located in Bothkennar, Scotland, United Kingdom. It has been intensively used for the geotechnical researches related to soft clays. An exhaustive geotechnical site investigation program was carried out that comprised undisturbed sampling with different methods (i.e. block samples, Delft continuous samples and piston samples), installation of piezometers, conducting of piezocones (CPTU), seismic piezocone (SCPTU), VST, dilatometer (DMT), and self-boring pressuremeter (SBPM) (Allman and Atkinson, 1992; Hight et al., 1992, 2003; Nash et al., 1992; McGinty, 2006; Mayne, 2016). Fig. 5 shows the liquid limit, LL, plastic limit, PL, natural moisture content, wn, and inferred overconsolidation ratio, OCR. A regression analysis between the natural water content, wn, and effective mean stress, p0 , was conducted as shown in Fig. 6. The following results are obtained: Iw ¼ 572.8, iw ¼ 6.35, and mw ¼ 0.551. The coefficient of determination, R2, is 0.89. The site-specific shear wave parameters b and b are 0.553 and 1.003, respectively. Fig. 7 shows the predicted DSS undrained strength versus the corrected VST undrained strength. In general, there is a good agreement between the predicted undrained strength and the corrected undrained VST strength except for the few meters near the ground surface. It is noted that different in situ investigations (i.e. CPTU, DMT and VST) conducted as a part of the different site investigation campaigns have shown similar contradictions near the ground surface in this particular site (Mayne, 2016). Hence, it is believed that the revealed differences near the ground surface are mainly due to the substantial natural variability of the surficial soil layers at this site. Fig. 8 shows the predicted at-rest earth pressure coefficient, K0, as well as the values inferred from the self-boring pressuremeter SBPM measurements. Fig. 9 shows the predicted and the measured values of the shear wave velocity VS. Generally, there is a good agreement between the measurements and the predicted values of K0 and VS except for few points near the ground surface for the reasons explained above. The commonly reported effective friction angle f0 for Bothkennar clay is 34 . Fig. 10 shows that the predicted values for the effective friction angle, f0 , generally ranges between 32 and 38 , except for positions near the ground surface where higher values are anticipated. It is realized that the measured and predicted ranges of the effective friction angles are mostly in a good agreement. Fig. 17. Regression analysis of Bangkok clay case study. Source of data: Bergado et al. (2002). 5.2. Ariake clays This experimental site is located in Kyushu Island, Japan. It has been used for studying Ariake soft clays by the Japanese research institutes. The natural water content is generally greater than the liquid limit in this site (generally, 110% LI  170%). This special aspect gives Ariake site a special importance as the other Japanese soft clay sites have lower LI values (Tanaka et al., 2001). Fig. 11 shows the liquid limit, LL, the plastic limit, PL, the natural moisture content, wn, and the overconsolidation ratio, OCR, at the Ariake site. The predicted values of OCR are generally in good agreement with the values known for this site, except for the depth of 17 m or deeper (predicted OCR z 1.5; actual OCR z 1.7e2). It is Fig. 18. Undrained shear strength su of Bangkok clay case study. Source of data: Bergado et al. (2002). S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 specific shear wave parameters are b ¼ 0.732 and b ¼ 1.257. Fig. 13 shows the predicted undrained strength as well as the measured undrained strength. Fig. 14 shows the predicted small strain modulus and values inferred from the shear wave velocity. Fig. 15 shows the predicted cone factor, Nkt, as well as the value inferred from the measurement of the cone and the vane shear (Tanaka et al., 2001). In general, the predicted values of su, G0 and Nkt are in good agreement with the measured/reported values except for the depth of 17 m or deeper. The reason for the deviation between the predicted and measured su and G0 values below this depth is the above-discussed abrupt changes in the clay’s OCR values as well as in its granulometry, implying different characteristics below the depths of 17e18 m. 589 5.4. Busan clay This site is located in Busan New Port, southeast coastal region of South Korea. Busan clay extends from a depth of approximately 7 m to the end of the exploration depth at 40 m. The natural water content wn is almost equal to or slightly lower than the liquid limit, LL, for the entire depth. Fig. 19 shows the liquid limit, LL, plastic limit, PL, natural moisture content, wn, and overconsolidation ratio, OCR, at the site (Choo et al., 2016). Regression analyses between the natural water content (wn) and the effective mean stress p0 , as shown in Fig. 20, were carried out. 5.3. Bangkok clay This clay is located approximately 30 km east of Bangkok City. The soil profile at the site consisted of a 2-m thick weathered crust overlying soft to stiff Bangkok clay. A VST was carried out for the top 15 m of the site. Fig. 16 shows the liquid limit, LL, the plastic limit, PL, the natural moisture content, wn, and the overconsolidation ratio, OCR, at the site (Bergado et al., 2002). The results of the regression analysis between the natural water content wn and the effective mean stress p0 , as shown in Fig. 17, are Iw ¼ 2341.8, iw ¼ 7.759, mw ¼ 0.897 and R2 ¼ 0.95. Hence, the sitespecific shear wave parameters are b ¼ 0.677 and b ¼ 0.755. The predicted DSS undrained strength is plotted versus the corrected vane shear strength, as shown in Fig. 18. Generally, a close agreement between the actual and predicted strengths is observed. Fig. 20. Regression analysis of Busan clay case study. Source of data: Choo et al. (2016). Fig. 19. Index properties and OCR of Busan clay case study. Source of data: Choo et al. (2016). 590 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 The results are Iw ¼ 24787, iw ¼ 10.118, mw ¼ 1.235 and R2 ¼ 0.78. Hence, the site-specific shear wave parameters are b ¼ 1.214 and b ¼ 0.983. The predicted DSS undrained strength is plotted versus the corrected vane shear strength, as shown in Fig. 21. A close agreement between the measured and predicted undrained strengths is generally observed. 5.5. Champlain clay Fig. 21. Undrained shear strength su of Busan clay case study. Source of data: Choo et al. (2016). This experimental site is located in Quebec, Canada. The prevailing formation, often termed as Champlain clay, is mainly soft sensitive cemented silty clay that has low-to-medium plasticity. One of the main characteristics of this clay is having natural water content, wn, that is substantially higher than its liquid limit, LL (generally, 140%  LI  270%). Fig. 22 shows the liquid limit, LL, plastic limit, PL, natural moisture content, wn, and the inferred overconsolidation ratio, OCR, at this site. In addition to the undrained strengths inferred from the VST, the undrained shear strength that corresponded to the failure of a full-scale test embankment was also back-calculated. The depth of failure was about 7e9 m from the ground surface. The undrained strength ratio su/s0 p for the clay mass within the failure zone was estimated to be 0.22 (Rochelle et al., 1974). Regression analyses between the natural water content (wn) and the effective mean stress p0 were carried out, as shown in Fig. 23. The results are Iw ¼ 318.47, iw ¼ 5.763, mw ¼ 0.499 and R2 ¼ 0.56. The site-specific shear wave parameters are b ¼ 0.421 and b ¼ 0.843. The predicted DSS undrained strength versus the corrected vane shear strength is plotted in Fig. 24. The predicted strength is generally in a good agreement with the back-calculated undrained strength and corrected VST undrained strength from the ground surface till reaching the depth of the failure surface. At the failure surface (i.e. at the depths of 7e9 m from the ground surface), the analysis predicts the undrained strengths of much lower value Fig. 22. Index properties and OCR of Champlain clay case study. Source of data: Rochelle et al. (1974). S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 than the VST and the back-calculated undrained strengths. This may be due to reaching the post-peak low undrained strength at the failure surface. Below the failure zone (i.e. deeper than 7e9 m from the ground surface), the predicted strength approaches again the VST undrained strength, which is higher than the back-calculated 591 strength. These results are acceptable since the back-calculated strength is limited to the depth of the failure zone (i.e. at depths shallower than 7e9 m from the ground surface). The anticipated relatively high undrained strength below the depth of the failure zone explains the reason that the failed soil mass did not extend below that depth. 6. Limitations and advantage of the proposed approach Based on the results obtained in the five case studies, the suggested methodology may be utilized to determine the stiffness and strength parameters of clay layers, which is however subjected to the following constraints: Fig. 23. Regression analysis of Champlain clay case study. Source of data: Rochelle et al. (1974). (1) The analyzed clay layers should have nearly unified geological setting, clay mineralogy and granulometry. (2) Water content and Atterberg limits should be accurately determined. Low-quality geotechnical site investigations may produce unreliable index parameters. (3) Water content and Atterberg limits should be frequently determined (e.g. at a spacing of 1 m along the clay depth) to ensure that the data are sufficient to conduct a reliable regression analysis. (4) The trend of decreasing water content with increasing effective stress for a number of the points should be included in the results. In the absence of this trend, the presented approach cannot be utilized. (5) The vertical effective stress profile should be determinable. This may limit the use of the presented approach in under-consolidated clays, as the effective stress profiles are varied with depth and time, and thus cannot be easily determined. Conversely, there are advantages of the presented approach as follows: (1) It depends on simple tests that are often carried out for soil classification. (2) It is site-specific, as it depends on a regression analysis relating the water content wn to the mean effective stress p0 for the site under consideration. 7. Conclusions Fig. 24. Undrained strength su of Champlain clay case study. Source of data: Rochelle et al. (1974). A new approach is presented to relate the shear wave velocity, VS, and small-strain shear modulus, G0, to the index properties of saturated soft to firm clays using the site-specific variation of the natural water content, wn, with the effective mean stress, p0 . The clay stiffness and strength parameters as well as the CPT modulus of rigidity Ir and bearing factor Nkt are assessed based on the presented approach. The suggested approach is used as an iterative methodology rather than direct correlations. It utilizes the relationships between clay strength and stiffness inferred from the critical state mechanics. The analysis starts with an assumption of the effective friction angle f0 that is enhanced with the progress of the iterations until convergence is achieved. The proposed iterative scheme, which is described in the flowcharts shown in Figs. 2 and 3, can be performed using a simple EXCEL spreadsheet. Five case studies in the United Kingdom, Japan, Thailand, South Korea, and Canada, respectively, were analyzed using the presented approach. It is shown that the suggested approach provides acceptable estimates of the strength and the stiffness of clays using their index properties. 592 S.M. Ahmed / Journal of Rock Mechanics and Geotechnical Engineering 10 (2018) 579e593 Conflicts of interest su,CIUC The author wishes to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. su,DSS TC TE VS VS,e0 Acknowledgments VS;p0 The author would like to express his sincere gratitude to Prof. F.M. El-Nahhas for his kind advices during the preparations of this study. The author would also like to acknowledge the efforts of the anonymous reviewers of the Journal of Rock Mechanics and Geotechnical Engineering (JRMGE). Their constructive comments helped the author to enhance the contents and the presentation of this paper. Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.jrmge.2017.10.006. VST wn a b f0 fcs g gw s0h0 s0p sv0 s0v0 Undrained shear strength obtained from the CIUC triaxial test Undrained shear strength obtained from the DSS test Triaxial compression Triaxial extension Shear wave velocity Shear wave velocity estimated using a site-specific Correlation with e0 Shear wave velocity estimated using a site-specific correlation with p0 Field shear vane test Natural water content (expressed as a percentage) Parameter in the VS-p0 site-specific relationship Stress exponent in the VS- p0 site-specific relationship Effective friction angle Critical state friction angle Saturated soil unit weight, g ¼ (GS þ e0)/(1 þ e0)gw Water unit weight, gw ¼ 9.81 kN/m3 Effective horizontal stress, s0h0 ¼ K0 s0v0 Effective preconsolidation stress Total vertical stress Effective vertical stress List of symbols a b CC CIUC DS DSS e0 Eoed G g G0 G50 GS Ie ie Ir Ir,CIUC Ir,cone Ir,DSS Iw iw K0 LI LL me mw M Nkt OCR p0 pa PI PL qt R2 SBPM su Parameter related to the site-specific VS-e0 relationship Void ratio exponent in the site-specific VS-e0 relationship Compression index Consolidated isotropic undrained triaxial compression Sample discrimination function to determine s0p Direct simple shear Natural void ratio, e0 ¼ GSwn/100 for saturated soils Constrained/oedometric modulus Operative secant shear modulus, G ¼ G50 Gravitational acceleration, g ¼ 9.81 m/s2 Small-strain shear modulus (i.e. at shear strains less than 10 5) Shear secant modulus at a shear stress equal to 50% of the ultimate shear stress Specific gravity Parameter in p0 -e0 relationship Parameter in p0 -e0 relationship, ie ¼ ln Ie Rigidity index, Ir ¼ G/su Rigidity index for the CIUC case Rigidity index for the cone penetration test, Ir,cone ¼ Ir,DSS Rigidity index for the DSS case Parameter in p0 -wn relationship Parameter in p0 -wn relationship, iw ¼ ln Iw at-rest earth pressure coefficient liquidity index, LI ¼ 100(wn e PL)/PI (expressed as a percentage) Liquid limit (expressed as a percentage) Stress exponent in e0ep0 relationship Stress exponent in wn ep0 relationship Strength parameter in the Cam Clay Model Bearing capacity factor related to the cone penetration test, Nkt ¼ (qt - sv0)/su,DSS Overconsolidation ratio Effective mean stress, p0 ¼ (1þ2K0)s0 v0/3 Atmospheric pressure, pa z 100 kPa Plasticity index (expressed as a parentage), PI ¼ LL e PL Plastic limit (expressed as a parentage) Total cone resistance Coefficient of determination Self-boring pressuremeter Undrained shear strength References Allman MA, Atkinson JH. 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Proceedings of the International Symposium on Coastal Geotechnical Engineering in Practice, vol. 2. Yokohama: A.A. Balkema; 2002. p. 3e25. Tanaka H, Locat J, Shibuya S, Soon TT, Shiwakoti DR. Characterization of Singapore, Bangkok, and Ariake clays. Canadian Geotechnical Journal 2001;38(2):378e 400. Wroth CP, Wood DM. The correlation of index properties with some basic engineering properties of soils. Canadian Geotechnical Journal 1978;15(2): 137e45. Yu HS, Herrmann LR, Boulanger RW. Analysis of steady cone penetration in clay. Journal of Geotechnical and Geoenvironmental Engineering 2000;126(7):594. https://doi.org/10.1061/(ASCE)1090-0241(2000)126:7(594). Dr. Sayed M. Ahmed is an Associate Professor of Geotechnical Engineering at Ain Shams University, Cairo, Egypt. He obtained his PhD from Ain Shams University, Egypt, in 2001. He has a broad experience in both academia and consultancy related to geotechnical engineering. His main interests are soft-ground tunneling, deep excavations, in situ testing and analysis of failures.