Evaluation of state parameter interpretation methods using CPT calibration chamber data Open Access

$Bq$

normalised excess pore water pressure$u$

CPT

cone penetration test

CPTu

cone penetration test with pore pressure measurement

$Dr$

relative density

$e$

void ratio

$emax$

maximum void ratio

$emin$

minimum void ratio

$Fr$

normalised friction ratio

$fs$

sleeve friction

$G$

elastic shear modulus

$H$

NorSand’s plastic hardening modulus

$Ic$

soil behaviour type

$Ir$

elastic shear rigidity

$K0$

coefficient of earth pressure at rest

$k, m$

soil-specific constants

$k¯, m¯$

soil-specific constants, similar to $k, m$

$ksph, msph$

soil-specific constants in cavity expansion

$Mtc$

friction ratio at the critical state

$N$

volumetric coupling coefficient

$p0$

in situ total mean stress

$p0′$

in situ effective mean stress

$Qp$

normalised tip resistance based on mean total and effective stress

$Qsph$

normalised tip resistance in cavity expansion

$Qt1$

normalised tip resistance based on $σv0$ and $σv0′$

$Qtn,cs$

equivalent clean sand normalised cone tip resistance

$Qtn$

normalised tip resistance based on $Qt1$ and a stress exponent$n$

$qc$

tip resistance

$qt$

corrected tip resistance to compensate for an unequal end-area effect

$u$

pore water pressure

$u0$

in situ pore pressure

$Vs$

shear wave velocity

$ν$

Poisson’s ratio

$λ10$

the slope of CSL in$e-log⁡p′$ space

$λe$

the slope of CSL in$e-ln⁡p′$ space

$σh0′$

in situ effective horizontal stress

$σv0$

in situ total vertical stress

$σv0′$

in situ effective vertical stress

$ψ$

state parameter

$ψ0$

in situ state parameter

$ψcc$

state parameter measured in calibration chamber tests

$ψe$

estimated state parameter from interpretation methods

$φcv$

constant volume (critical state) friction angle

Γ

intercept of CSL in$e-log⁡p′$ space measured at $p′$ = 1 kPa

Soil liquefaction is a loss of stiffness and shear strength due to the generation of excess pore water pressure caused by monotonic or seismic loading. It is a concern for structures constructed on saturated granular soils (e.g., sand, silt, gravel, and their combinations) and a design problem for large manufactured geotechnical structures such as tailings storage facilities and earth dams. Liquefaction can result in a sudden failure with minimal advanced indicators. The key to assessing liquefaction potential is determining the in situ density (i.e., void ratio, relative density, or the state parameter) explicitly (Been et al., 1986; 1987a) or implicitly (Idriss and Boulanger, 2008).

The relative density $Dr$ has been the de facto density index, but it can be problematic due to practical limitations in determining it in the laboratory (ASTM D4253-16 and ASTM D4254) and its high sensitivity to gradation and variations in the field (Tavenas, 1973). Density alone cannot capture soil response adequately without accounting for confining stress. An alternative index that captures the effects of the density and confining stress is the state parameter $ψ$ proposed by Been and Jefferies (1985). The state parameter is the difference between the void ratio at a given mean effective stress and the critical state void ratio at the same mean effective stress. Soils with the same$ψ$ display similar behaviour and tend to have the same static and cyclic liquefaction potential, among other attributes (Manmatharajan et al., 2023a). The state parameter represents the in situ state of soil: a positive value $ψ$ indicates contractive and strain-softening behaviour in undrained shear, whereas a negative $ψ$ indicates dilative and strain-hardening response. There is a consensus that if the soil is looser than$ψ=$ −0.05, then static liquefaction will likely happen under the right loading circumstances (Jefferies and Been, 2015).

Cohesionless soil samples can be easily disturbed during sampling, and obtaining reasonably undisturbed samples is prohibitively expensive. In situ penetration tests have thus become the default approach for characterising cohesionless soils. The cone penetration test with pore pressure measurement (CPTu) is widely used for determining the in situ state parameter $ψ0$ and assessing liquefaction potential because it provides continuous data measurement and excellent repeatability at a relatively low cost.

There are several methods of interpreting the state parameter from the CPTu, including those proposed by Fear and Robertson (1995), Konrad (1997), Russell and Khalili (2002), and Shuttle and Jefferies (2016). More commonly used methods are those proposed by Plewes et al. (1992), Shuttle and Jefferies (1998), and its generalised version Ghafghazi and Shuttle (2008). These are progressions of the Been et al. (1986) approach, which accounted for the mechanical properties of soils in determining the state parameter. They all require some accompanying triaxial compression test data, even though they also offer in situ approximations. The cavity expansion methods, Shuttle and Jefferies (1998) and Ghafghazi and Shuttle (2008), require some numerical modelling, while Shuttle and Jefferies (1998) also offered a set of simplified equations to circumvent the modelling. Another popular empirical method proposed by Robertson (2010) is based on the engineering judgement of the author. It does not require any accompanying testing or information as it is entirely based on the CPT’s tip resistance and sleeve friction data. These methods require different levels of information on soil properties, analysis procedures, and theoretical bases. Therefore, unsurprisingly, experience shows that these methods produce different outcomes in practice (Schafer et al., 2019).

This paper evaluates the interpretation of the state parameter from CPT tip resistance through the methods proposed by Been et al. (1986), Plewes et al. (1992), Shuttle and Jefferies (1998), Ghafghazi and Shuttle (2008), and Robertson (2010) by examining them against a database of calibration chamber tests on clean sands. All these methods were developed based entirely or partially on the same database. Therefore, performing well against this database is a minimum requirement for their application in practice. The performance of these methods was evaluated by comparing the estimated $ψ0$ from each method with the actual value of$ψcc$ measured in calibration chamber tests. Their differences and limitations including how they normalise overburden stress are discussed. The preferred method is identified, and use of some methods is cautioned against, so engineers can avoid extensive errors in determining in situ state parameter and applying it in liquefaction susceptibility.

The most basic CPTu probe provides three channels of data: tip resistance (⁠$qc$⁠), sleeve friction (⁠$fs$⁠), and pore water pressure (⁠$u$⁠). The pore water pressure is most commonly measured immediately behind the cone in what is referred to as the $u2$ position. The measured tip resistance (⁠$qc$⁠) is corrected to $qt$ to compensate for an unequal end-area effect (ASTM D5778-20).

In comparing various methods of interpreting the state parameter, the effectiveness of overburden normalisation directly contributes to the quality of the interpretation. Following the ideas of Schmertmann (1976), Wroth (1984) and Houlsby (1988) proposed that the tip resistance can be normalised by total and effective vertical stresses (⁠$σv0$ and $σv0′$⁠) using Equation 1

Variations of Equation 1 that include a stress normalisation factor have been proposed by Olsen and Malone (1988), Robertson and Wride (1998), Zhang et al. (2002), Idriss and Boulanger (2004), Moss et al. (2006), Cetin and Isik (2007), Robertson (2009), and Sadrekarimi (2016).

Robertson and Wride (1998) defined $Qtn$ by adding a stress exponent $n$ in a power function, expressed with Equation 2

where $pa$ is the atmospheric pressure in the same unit as $qt$⁠. The stress normalisation exponent $n$ (typically between 0.5 and 1.0) that likely captures the effect of increasing stiffness (among other factors) is a function of soil behaviour type $Ic$ (Been and Jefferies, 1992; Robertson and Wride, 1998) and vertical effective stress $σv0′$⁠, and can be calculated from Equations 3 and 4, according to Robertson (2009).

Houlsby and Hitchman (1988) showed that the effective horizontal stress can normalise tip resistance of a clean sand much better than the vertical stress. $Qtn$ does not account for in situ horizontal effective stress $σh0′$ or the coefficient of earth pressure at rest $K0$⁠. Been et al. (1986) considered the effects of both horizontal and vertical effective stresses on the cone resistance and proposed an improved form of Equation 1 based on mean effective stress with Equation 5.

where $p0$ is the mean total stress and $p0′$ is the mean effective stress.

Wroth (1984) and Houlsby (1988) also proposed two other dimensionless cone parameters, which were widely adopted by others: the normalised friction ratio $Fr$ and the normalised excess pore pressure $Bq$⁠, expressed in Equations 6 and 7, respectively:

where $u0$ is in situ pore pressure. $Bq$ defined by Equation 7 conceptually captures the influence of excess pore water pressure development due to CPT penetration in low permeability soils, which mostly reduces tip resistance. $Bq$ varies from 0 to 1 for fully drained penetration in sands to fully undrained penetration in fines-rich soils. Negative values of $Bq$ may be observed and are often taken as a sign of the dilative behaviour of soil.

Like the state parameter itself, interpretation of the state parameter has classically been a sand problem, and CPT tip resistance during drained penetration has been one of the most significant input parameters. However, fines (particles smaller than #200 sieve) are common in deposits, and from a practical perspective, liquefaction of fines-rich tailings is a significant area of interest for state parameter interpretation. Hence undrained and partially drained penetration need to be considered in the interpretation process. Consistent with its ‘bare minimum’ evaluation, the rest of this work only considers fully drained penetration (⁠$Bq$ = 0). This is the only possibility with the existing calibration chamber database, which lacks anything but drained penetration.

Been et al. (1986) studied the existing calibration chamber database and triaxial test results for fine to medium sands and found the correlation between the stress-normalised CPT tip resistance $Qp$ and state parameter $ψ$ in Equation 8.

where $k$ and $m$ are soil-specific constants that depend on soil compressibility represented by $λ10$⁠, the slope of the critical state line in the $e-log10p′$ space. $λ10$ can be determined by conducting triaxial tests on reconstituted soil specimens.

The problem with this approach was that there are significant variations in particle size distribution in the field, which would translate into significant variations in $λ10.$ Thus, various gradations of soil make the method difficult to use in practice, given the large number of triaxial tests needed to characterise $λ10$ adequately. Plewes et al. (1992) solved this problem by correlating $λ10$ to the normalised friction ratio $Fr$ through Equation 9

Reid (2015) reviewed this correlation, expanded the database, and demonstrated that it holds valid for many natural soils and tailings, albeit with significant scatter.

Been et al. (1988) noted that the soil-specific constant $k$ in Equation 8 is normalised tip resistance $Qp$ at the critical state (⁠$ψ$ = 0). Therefore, it is reasonable to expect that it would be a function of the friction ratio at the critical state $Mtc$ as well, which can be determined from triaxial tests on reconstituted soil specimens. Unlike $λ10$⁠, $Mtc$ is not sensitive to variations of particle size distribution in the field (Manmatharajan et al., 2023b). The relation between $Qp$ and $ψ$ was further improved by incorporating $Mtc$ into the interpretation, using Equations 10, 11, and 12

where soil-specific constants $k¯$ and $m¯$ are similar to $k$ and $m$ in Equation 8. They are functions of the stress ratio at the critical state $Mtc$ and the slope of the critical state line $λ10$⁠. Equations 10 to Equations 12 were initially developed by Been et al. (1988) and improved successively by Plewes et al. (1992) and Been and Jefferies (1992). The abbreviation ‘P’ will be used in the rest of this manuscript for Plewes et al. (1992).

Compared with Equation 8, the addition of the term $1-Bq$ in Equation 10 accounts for the effect of drainage conditions during cone penetration; thus, Equation 10 can be used for various soils, from sand to silt and even clay. The addition of the ‘+1’ term in Equation 10 was suggested by Houlsby (1988) to avoid a zero normalised tip resistance for fully undrained penetration in clays and is supported by the cavity expansion theory from Shuttle and Cunning (2007).

Although the methods of Been et al. (1986) and Plewes et al. (1992) are simple, Sladen (1989) found that the Been et al. (1986) method, and by extension, Plewes et al. (1992), has a bias with stress level. Shuttle and Jefferies (1998) used cavity expansion analyses to explore this problem. They found that the bias is caused by missing the elastic shear rigidity (⁠$Ir=G/p0′$⁠) in the interpretation process, where $G$ is the elastic shear modulus and $p0′$ is the in situ or consolidation mean effective stress.

Shuttle and Jefferies (1998) used the critical state–based soil constitutive model ‘NorSand’ (Jefferies, 1993) to obtain a normalised tip resistance$Qsph$⁠, which tracks the trend of $Qp$ but is about one order of magnitude smaller than $Qp$ obtained from chamber tests and in situ. Therefore, an empirical scaling function became necessary. The scaling function accounts for the different geometry between cavity expansion and the real CPT, as shown in Equation 13.

Shuttle and Jefferies (1998) found that Equation 8 may be used to recover state parameter$ψ$ in calibration chamber tests on Ticino sand, provided that the parameters $k$ and $m$ are functions of soil properties represented by calibration parameters of NorSand.

Ghafghazi and Shuttle (2008) applied the spherical cavity expansion of Shuttle and Jefferies (1998) to a database of calibration chamber tests (similar to the one presented later in this paper) using an updated version of NorSand (Ghafghazi, 2011). They proposed an analytical solution for calculating the state parameter from CPT tip resistance using spherical cavity expansion analyses.

Similar to Shuttle and Jefferies (1998), Ghafghazi and Shuttle (2008) require in situ measurement of elastic shear modulus $G$ through the shear wave velocity $Vs$ as well as triaxial laboratory testing on reconstituted samples to calibrate NorSand. It also correlates $Qsph$ to $QCPT$ through a scaling factor, as shown in Equation 14:

The abbreviation ‘CE’ will be used in the rest of this manuscript for cavity expansion solutions (Ghafghazi and Shuttle, 2008; Shuttle and Jefferies, 1998).

The method outlined by Shuttle and Jefferies (1998) requires detailed cavity expansion analyses, so they also provided a set of simple equations by adding fitting trendlines through a parametric study on their cavity expansion analyses (Jefferies and Been, 2015). The approximate general equations are in closed form and readily computable.

Equations 15 and 16 capture the effects of soil properties and rigidity on $ksph$ and $msph$ through the parameters of NorSand:

where fitted equations $f1-f12$ are simple to use in a spreadsheet, $N$ is a volumetric coupling coefficient, $H$ is NorSand’s plastic hardening modulus, $λ$ is the slope of the critical state line in the $e-log10p′$ space, and $ν$ is Poisson’s ratio.

For each calibration chamber test, having $Qsph$ determined from Equation 14, and $ksph and msph$ from Equations 15 and 16, the state parameter $ψ$ can be inferred from Equation 17:

where $ksph$ and $msph$ are introduced from cavity expansion analysis and equivalent to $k$ and $m$ in previous equations.

The abbreviation ‘EQ’ will be used in the rest of this manuscript for the simplified equations of Shuttle and Jefferies (1998).

Robertson (2010) extended the chart proposed in Robertson (2009) to suggest a relation between$ψ$ and the equivalent clean sand normalised cone tip resistance $Qtn,cs$⁠, as shown in Equation 18. The correlation is valid in uncemented Holocene age soils and applicable for low-risk projects and initial screening for high-risk projects. Although no data were presented, the method is stated to be based on Jefferies and Been (2006), Shuttle and Cunning (2007), and Wride et al. (2000) from CANLEX site data. These references include calibration chamber data, drained and undrained spherical cavity expansion, and frozen ground sampling. This method relies on the CPT measurements of $qc$ and $fs$⁠, but requires no index or advanced laboratory testing or knowledge of geostatic ratio $K0$⁠. The state parameter is obtained from clean sand corrected normalised tip resistance, which is based on the soil behaviour type parameter $Ic$ (Robertson, 2009).

where $Qtn,cs$ is equivalent to clean sand normalised cone tip resistance. For the clean sand database used here, $Qtn,cs$ is equal to $Qtn$⁠.

Table 1 summarises the stress normalisation factor of each method, soil properties that each method accounts for, and critical inputs for data processing in the interpretation of the state parameter by various methods. The abbreviation ‘R’ will be used in the rest of this manuscript for Robertson (2010).

In the past 40 years, calibration chamber tests have played a key role in CPT interpretation, with two alternatives, undisturbed sampling in the field and numerical modelling as an auxiliary role.

Undisturbed sampling is technically tricky and cost-prohibitive, except for a few large projects. Moreover, spatial variability inevitably limits the comparability of CPT-interpreted state parameters to measured values. Numerical modelling has only had partial success due to the difficulties in theoretical and numerical modelling of CPT that are caused by the complexity of soil behaviour and large deformation during cone penetration. Successful numerical models need to be validated against measurements, and to date, this validation has included some forms of empirical ‘scaling’ as described earlier.

Calibration chambers are large circular steel tanks, typically between 0.8 and 1.5 m in diameter and height. Smaller calibration chambers that use miniature cones exist, but these have had limited success due to uncertainties around the influence of cone to particle diameter ratio and difficulties measuring sleeve friction and pore water pressure on miniature cones. Approximately 1.0 to 2.5 tonnes of sand is deposited at a controlled density and consolidated to desired stress levels in a full-size calibration chamber. Then, a CPT is pushed into the sample following similar configurations and settings used in the field. Calibration chamber studies cover a range of densities and confining stress levels. From these tests, it is possible to correlate tip resistance $qc$ and density represented by void ratio $e$⁠, relative density$Dr$⁠, or the state parameter$ψ$⁠. In engineering practice, the in situ state of soil is then obtained by substituting the field CPT measurements at the estimated in situ stress in the correlation determined in the calibration chamber.

Ghafghazi and Shuttle (2008) summarised a database of nine sands for which triaxial compression tests were also available. It includes two natural sands (Da Nang sand and Erksak sand) and five research standard sands (Hokksund sand, Ottawa sand, two variations of Ticino sand, and Toyoura sand) as well as two mine tailings (Syncrude oil tailings and Hilton Mines). Table 2 summarises the index properties of these sands and their critical state parameters. The critical state parameters were obtained from drained or undrained triaxial compression tests on reconstituted samples, with test conditions summarised in Table 3.

Most of these sands are quartz-based materials and have sub-rounded to rounded grain shapes, except for two tailings with heavy metal minerals and the more angular grain shapes. As shown in Figure 1, these are all medium to fine sands with minimal fines passing the #200 sieve. Calibration chamber tests have been conducted in different chambers with different boundary conditions and chamber-to-cone diameter ratios; hence, a correction must be applied to raw results to account for boundary effects. Ghafghazi and Shuttle (2008) used the correction factors recommended by Been et al. (1987b). Table 4 summarises the chambers’ description and test conditions, such as the range of lateral earth pressure ratio $K0$⁠, consolidation vertical effective stress$σv0′$⁠, mean effective stress $p0′$⁠, state parameter $ψ$⁠, and elastic shear rigidity $Ir$⁠. Table 5 summarises the calibrated NorSand parameters for the nine sands from Ghafghazi and Shuttle (2008).

The calculation of $ψ$ from tip resistance $qt$ by Been et al. (1986) and Plewes et al. (1992) is similar, except that Plewes et al. (1992) additionally consider the effect of $Mtc$ on $k$ . Therefore, only the Plewes et al. (1992) results are continually presented in the paper and compared with other methods for simplicity and to avoid redundancy. For Plewes et al. (1992), given that the critical state parameters $λ10$ and $Mtc$ for selected sands in the calibration chamber database are available from triaxial tests, the soil-specific constants, $k¯$ and $m¯$⁠, can be determined from Equations 11 and 12. The studied sands in the database are medium to fine sands, mainly tested dry, and have virtually zero fines contents, resulting in a fully drained penetration with $Bq$ = 0. Therefore, the estimated value of $ψ$ can be calculated from Equation 10.

The calculation of $ψ$ by Shuttle and Jefferies (1998) and Ghafghazi and Shuttle (2008) involves cavity expansion analyses. For the equations provided by Shuttle and Jefferies (1998), the list of soil-specific constants $ksph and msph$ are obtained from substituting the NorSand parameters calibrated from triaxial tests into the approximated fitting equations. Once the $ksph and msph$ are determined, and $Qsph$ is calculated from $Qcc$ by using the scaling factor as shown in Equation 13, the estimated value of $ψ$ can be calculated from Equation 17. A similar calculation process applies to Ghafghazi and Shuttle (2008), but with the scaling factor between $Qsph$ and $Qcc$ from Equation 14.

The calculation of $ψ$ by Robertson (2010) is different from the other methods because $qt$ is only normalised by the vertical effective stress and expressed as $Qtn$⁠, instead of being normalised by the mean effective stress into $Qp$⁠. Since the sleeve friction data (⁠$fs$⁠) were not available in the database of the calibration chamber tests, the soil behaviour type parameter $Ic$ had to be assumed before using Equation 18 to estimate $ψ$⁠. $Ic$ = 1.57 was assumed as a reasonable value for clean sands calculated by assuming $n$ = 0.5 at $σv0′$ = 100 kPa from Equation 3. Then, the stress exponent $n$ and the normalised tip resistance $Qtn$ were updated based on Equations 2 and 3 with$Ic$ = 1.57 and $σv0′$⁠. Finally, the estimated value of $ψ$ was calculated from Equation 18.

In comparing various methods of interpreting the state parameter, the effectiveness of overburden stress normalisation directly contributes to the quality of the interpretation. Plewes et al. (1992), Shuttle and Jefferies (1998), and Ghafghazi and Shuttle (2008) all adopt $Qp$ for stress normalisation, while Robertson (2010) uses $Qtn$⁠. Figure 2(a) shows an example of comparing Q_tn – ψ and Q_p – ψ correlations for Da Nang sand in a semi-logarithmic space. The $K0$ value of each calibration chamber test is shown near its data point. $Qp$ does a better job of normalising data with less scatter (R² = 0.91), compared with$Qtn$ (R² = 0.67). An obvious bias in the Q_tn – ψ is the influence of $K0$⁠, given that $Qtn$ does not account for horizontal stresses.

Figure 2(a) suggests that $Qtn$ values associated with higher $K0$ values (1.6 to 1.8) are roughly double those associated with lower $K0$ values (0.4 to 0.5). Figure 2(b) shows another example of Hilton Mines tailings that have been recognised as a compressible material and were tested at a narrower range of $K0$ values (0.4 to 0.5). In this case, $Qp$ also produces much less scatter (R² = 0.91) than $Qtn$ (R² = 0.63). In Figure 2(b), the $σv′$ value of each calibration chamber test is shown near its data point, and sometimes one number represents multiple data points (in brackets). The $Qtn$ values corresponding to the test at higher $σv′$ value fall higher than those at lower $σv′$ values. For instance, the tests consolidated to ∼270 kPa vertical stress produced $Qtn$ values that were roughly double those of tests performed at 50 to 60 kPa.

Table 6 summarises the coefficients of determination (R²) of the semi-logarithmic trendlines for Q_tn – ψ and Q_p – ψ relations and the range of other soil properties for all studied sands. Clearly, $Qp$ does a better job for stress normalisation than $Qtn$ by comparing the R² values, representing the level of scatter. It appears that $Qtn$ performs more poorly for materials that have been tested at a broader range of $K0$ values, such as Da Nang and Erksak sands, confirming that ignoring horizontal stress’s influence is detrimental to the overburden normalisation. Also, $Qtn$ performs poorly in materials with a higher value of $λ10$ (higher compressibility), such as Hilton Mines and Syncrude oil tailings. For the materials tested at a narrow range of $K0$ values, such as Ottawa and Ticino 9, $Qp$ does a slightly better job for stress normalisation than $Qtn$ does, while $Qtn$ performs slightly better than $Qp$ for Ticino 4 and Toyoura 160 sands. Hokksund calibration chamber data were collected from different sources and had the greatest scatter in Q_tn – ψ and Q_p – ψ relations among the database sands.

Ghafghazi and Shuttle (2008) found that the tip resistance measured in two calibration chamber tests can differ by more than a factor of two for the same soil (i.e., Ticino 4), while both were tested at virtually identical conditions in terms of density and stress conditions. They also found that the best prediction accuracy of the state parameter was obtained for those soils with $G$ directly estimated from shear wave velocity measurements (i.e., Ottawa, Syncrude oil tailings, and Toyoura 160).

The performance of each method is assessed by comparing its interpreted state parameter with that measured in calibration chamber tests, as shown in Figure 3. The measured state parameter $ψcc$ is obtained by subtracting the void ratio of the sample prepared in the calibration chamber at the consolidation stress from the associated critical state void ratio. The one-to-one line of equivalence is plotted, representing a perfect interpretation. Two boundary lines are drawn beside the equivalency line in Figure 3, at ±$0.04$ and ±$0.07$ error margins. There are significant differences in how methods perform in general and on different materials. For example, Robertson (2010) tends to interpret looser than the actual state parameter, while Shuttle and Jefferies (1998) simplified equations tend to interpret a denser than the actual state parameter across the database. Plewes et al. (1992) and the cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998) tend to perform better and occupy the space near the equivalency line for most materials. The performance of methods among different soils is also different. For example, while all methods performed well on Erksak and Ottawa sands in Figures 3(b) and 3(e), Robertson (2010) completely missed Hilton Mines and Ticino 4 in Figures 3(c) and 3(g), and Shuttle and Jefferies (1998) simplified equations missed Da Nang and Toyoura sands in Figures 3(a) and 3(i), respectively.

Table 7 summarises the number of tests with $ψ$ calculated within ±$0.04$ and ±$0.07$ error margins and the confidence level as percentages that various methods can interpret $ψ$ within either error margin. Ghafghazi and Shuttle (2008) identified these two margins as levels at which a significant jump in confidence occurred in the database. From the 279 tests analysed, Plewes et al. (1992) and the cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998) are the best performers, with 180 (64.5%) and 193 (69.2%) predictions producing $Δψ$ errors less than ±0.04 and 238 (85.3%) and 249 (89.2%) predictions producing$Δψ$ errors less than ±0.07, respectively. Robertson (2010) produced poor interpretations with only 95 (34.1%) tests within ±$0.04$ error margins and 158 (56.6%) tests within $Δψ$ less than ±$0.07$⁠, respectively. The simplified equations of Shuttle and Jefferies (1998) produced similarly poor interpretations with 68 (24.4%) tests and 113 (40.5%) tests within$Δψ$ less than ±$0.04$ and ±$0.07$ error margins, respectively.

The interpretation errors of the four methods plus Been et al. (1986) are presented as a histogram in Figure 4, confirming the earlier observations. Figure 4(a) shows the distribution of the errors by giving equal weight to each of the 279 data points. In contrast, Figure 4(b) removes the bias caused by some materials (e.g., Ticino 4 with 68 tests) having many more tests than others (e.g., Syncrude oil tailings with eight tests). Materials are given equal weight in Figure 4(b) by summarising the percentages of the errors in each material. The differences between Figures 4(a) and 4(b) are inconsequential. In Figure 4, an error (⁠$Δψ=ψe-ψcc$⁠) of zero is a perfect interpretation, a positive error interprets a looser state than reality (conservative), and a negative error is unconservative. Been et al. (1986), Plewes et al. (1992), and cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998) produce the best results. The simplified equations of Shuttle and Jefferies (1998) are grossly unconservative, and Robertson (2010) is grossly conservative.

The performance of these methods on individual materials can be better understood by separating bias from scatter, as illustrated in Figure 5. The scatter is represented by the R² value of the trendline fitted through the interpretations. The bias is the difference between the state parameter estimated from the trendline and the assumed state parameter. For the scatter, in Figure 5(a), Plewes et al. (1992) and cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998) produce near-perfect trendlines with a high value of R² = 0.95 and 0.96, respectively. In Figure 5(b), Plewes et al. (1992) produce the best trendline with R² = 0.91, followed by cavity expansion methods with R² = 0.89. Shuttle and Jefferies (1998) simplified equations produce a scatter as low as the last two methods for two sands, but with significant bias. Robertson (2010) produces poor trendlines with the lowest R² = 0.86 and R² = 0.63 for two sands and significant bias.

The methods using $Qp$ for overburden stress normalisation show similar R² values, while Robertson (2010) using $Qtn$ for overburden stress normalisation shows lower R² values with higher scatter. The R² values of the interpretation trend lines produced by each method are summarised in Table 8. Interestingly, the interpretation R² values for the added trendlines fitted through the predictions produced by Plewes et al. (1992) and Robertson (2010) in Table 8 are almost identical to the R² values for Q_p – ψ and Q_tn – ψ relations in Table 6, which implies that the primary source of the difference in scatter in these methods is their stress normalisation approach. In Table 8, the R² values of the cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998) and Shuttle and Jefferies (1998) simplified equations are close to that of Plewes et al. (1992), with some differences caused by the influence of other soil properties (NorSand parameters) on the interpretation. As explained earlier, Shuttle and Jefferies (1998) demonstrated that half of the scatter in $Qp$ values come from the influence of overburden on elastic shear rigidity, while the other half is due to the repeatability of calibration chamber tests themselves. As demonstrated earlier, the primary source of higher scatter in $Qtn$⁠, compared with $Qp$⁠, is due to ignoring the influence of the horizontal stress or $k0$ on CPT tip resistance. The scatter in state parameter interpretation is not similar across methods or materials. A review of Table 8 highlights Hokksund as a material with poor R² across all methods. This observation is not surprising given that Hokksund is the second-largest data set in the database (51 tests), but unlike the most extensive set of data (Ticino 4 with 77 tests), its data come from three different references. Human factors and different details of tests may have caused a higher amount of scatter in Hokksund data than in other sands. It is also one of the materials with the elastic modulus assumed instead of measured to the detriment of the cavity expansion-based interpretation process (Ghafghazi and Shuttle, 2008). While the methods using $Qp$ have high R² values for all other materials (R² = 0.79 to 0.95), Robertson (2010) produces poor R² values for Da Nang and Erksak sands as well as Hilton Mines and Syncrude oil tailings (R² = 0.53 to 0.72). As explained earlier, this high scatter stems from the methods’ stress normalisation approach, as reflected in the near-identical R² values of the Q_tn – ψ correlations and those of the $ψe$ versus $ψcc$ in Tables 6 and 8, respectively.

In addition, it is very informative to understand how the error bias is distributed across different densities to evaluate the scatter error in the interpretation of state parameters in the four methods. As can be seen in Table 9, the error in the state parameter interpretation (⁠$Δψ$⁠) is reported at three densities; a dense state (⁠$ψ$ = −0.3) and a loose state (⁠$ψ$ = 0.1) as well as the generally accepted threshold for contractive behaviour (⁠$ψ$ = −0.05). The average error is near zero for all three densities for Plewes et al. (1992) and cavity expansion methods (Ghafghazi and Shuttle, 2008,; Shuttle and Jefferies, 1998). Shuttle and Jefferies (1998) simplified equations have their worst average performance on dense sands (⁠$Δψ$ = −0.13), while Robertson (2010) produces the most significant error for loose sands (⁠$Δψ$ = 0.14). Focusing on the state parameter separating contractive from dilative behaviour,$ψ$ = −0.05, Robertson (2010) would interpret a significantly looser state parameter (Δψ = 0.06 to 0.17) than reality in all sands. The Shuttle and Jefferies (1998) simplified equations would miss the threshold by (⁠$Δψ$ = −0.05 to −0.09) in three of the nine sands. Plewes et al. (1992) would only miss the threshold by (⁠$Δψ$ =−0.06 and −0.07) for the Da Nang and Hilton Mines tailings in the database while performing well for the rest of the sands. The cavity expansion methods (Ghafghazi and Shuttle, 2008 ; Shuttle and Jefferies, 1998) perform well within ± 0.02 for all materials except for Da Nang (⁠$Δψ$ = −0.07) and Erksak sand (⁠$Δψ$ = 0.04).