Use of fall cones to determine Atterberg limits: a review

The authors have presented an interesting review of the use of fall cones to determine Atterberg limits (O'Kelly et al., 2018). Wood (1983: p. 80) recognised that the fall-cone test was not suited to determine the true plastic limit (PL): ‘It is not clear how the cone penetrometer plastic limit gives an indication of the water content at which soil changes from the brittle to the plastic state’. The discusser wishes to comment on the strength–water content relationship and the plastic strength limit (PSL, an acronym that is preferable to PL to reinforce that this is a strength-based value with no relation to the true plastic limit), which is based on

Wood & Wroth (1978: p. 37) stated that ‘Studies … have shown that it is reasonable to assume that soil can be assigned a unique strength at the liquid limit’ and suggested a strength of 1·7 kPa was satisfactory. Wood (1983: p. 79) stated, ‘it seems that if round numbers are shown for convenience then R [R_MW] = 100 and C_LL [s_u(LL)] = 1·7 kN/m² is not too bad a first shot’. This produces s_u(PSL100) = 170 kPa. The shear strength at the liquid limit (LL) is not unique, see Table 3.

It is doubtful that the fall-cone factor K in Hansbo's expression (equation (1)) is constant throughout the range of water contents between the liquid and plastic limits. K for the 80 g 30° cone would be 0·867 for s_uFC(LL) = 1·7 kPa. Adopting the cone factor of 0·8 from BS EN ISO 17892-6 (BSI, 2017) s_uFC(LL) would be 1·57 kPa. Koumoto & Houlsby (2001) showed that, theoretically, K could vary from about 1 for a rough cone to 2·0 for a smooth cone and, experimentally, between 0·85 and 1·2. Brown & Huxley (1996) and Brown & Downing (2001) found that, although Hansbo's expression with K ≈ 0·8 applies to water contents near the liquid limit, at lower liquidity indices much lower cone factors were appropriate.

Based on Schofield & Wroth (1968) and Wroth & Wood (1978) the strength gain factor R_MW has been assumed as 100 by many researchers, but the authors’ Figs 3 and 4 show this assumption to be false. A wide variation of R_MW has been found by several other authors (Karlsson, 1961; Whyte, 1982; Wood, 1983; Wijeyakulasuriya, 1990; Brown & Downing, 2001; Nagaraj et al, 2012; Vardanega & Haigh, 2014).

There appears to be a mineralogical effect on R_MW. Dumbleton & West (1970) reported different suctions and vane shear strengths at the plastic limit and liquid limit for kaolinite and montmorillonite, also giving different R_MW values, 25 and 110, respectively. Soils with a high proportion of kaolinite would not comply with the Wroth & Wood (1978) relationship (R_MW = 100). Prakash & Sridharan (2006) recognised significant differences in cone penetration behaviour between soils of low plasticity, being dominated by kaolinite, and those of high plasticity, containing montmorillonite. The strengths at the plastic limit have been shown to be much lower for soils that plot below the A-line on Casagrande's plasticity chart compared to above (Barnes, 2013a).

From a large database of fall-cone tests on 101 soils Vardanega & Haigh (2014) reported a semi-logarithmic model between fall-cone undrained shear strength, s_uFC and water content, w

Given the liquid and plastic limits for each soil these can be plotted as s_uFC against I_L, Fig. 11. With 101 soils there are 101 relationships. A range of s_uFC(LL) (1·55–2·35 kPa) is obtained from the S and H values and a wide range of R_MW was obtained (7–152, mean = 33). The PSL and I_L for each soil can be determined from equations (17)–(20) and are plotted on Fig. 11.

The authors’ fall-cone consistency index (equation (12)) equals 1 for each value of PSL₂₅ and equation (14) then gives the s_uFC–I_L relationship for each soil, Fig. 11. As the authors point out, PSL₁₀₀ mostly lies in the brittle region, well below the plastic limit. Even PSL₂₅ can give a wide range of consistencies from brittle to plastic, sticky to non-sticky. Nevertheless, below the plastic limit the s_u–I_L relationship follows a significantly different path (Marinho & Oliveira, 2012; Vinod et al., 2013).

Interpolation for strengths between the PSL and the liquid limit depends on the assumption of linearity of the semi-logarithmic relationship. However, these relationships are distinctly curved, as shown in the original data in Skempton & Northey (1952) (Fig. 4), and confirmed several times since (Wood, 1985; Wasti & Bezirci, 1986; Harison, 1988; Wijeyakulasuriya, 1990; Stone & Phan, 1995; Feng, 2000; Koumoto & Houlsby, 2001; Muntohar & Hashimi, 2005).

Harison (1988) demonstrated a bilinear semi-logarithmic relationship. From analysis of several sources (Dumbleton & West, 1970; Black & Lister, 1979; Wasti & Bezirci, 1986; Northmore et al, 1992; Stone & Phan, 1995; Feng, 2000; Koumoto & Houlsby, 2001), the discusser (in work to be published) has shown the relationship from fall-cone and vane tests to be multi-linear with a typical example given in Fig. 12. With data available only near the liquid limit R_MW would be obtained from line A, giving a significant underestimate of the PSL. Sources such as Sherwood & Ryley (1970) were not included because the cone penetrometer data would only provide line A.

Extrapolation from single lines through A and B or A, B and C would also underestimate the PSL. Vardanega & Haigh (2014) point out that equation (16) only applies to 0·2 < I_L < 1·1 because fewer data are available below I_L = 0·2. Using the discusser's apparatus (Barnes, 2009, 2013a, 2013b) a stiff transition in the toughness–liquidity index relationship in the region 0 < I_L < ∼0·2 was found providing an additional line close to the plastic limit, Fig. 13.

The shear strength at the liquid limit, the fall-cone factor and the strength gain factor are not constants, but vary significantly; the relationship between shear strength and water content is not linear but curved, or with enough data can be interpreted as multi-linear. A large published database shows that the consistency at the plastic strength limit PSL₁₀₀ is mostly brittle and even at PSL₂₅ many soils would also be brittle.

The authors thank the discusser for his interest in the review paper (O'Kelly et al., 2018). The authors agree with many of the statements in the discussion and in this reply give more elaboration and clarification on some of the key points raised. In his concluding statement, the discusser states ‘The shear strength at the liquid limit, the fall-cone factor and the strength gain factor are not constants, but vary significantly’. Each of these elements will now be studied in turn.

It is well documented that the undrained shear strength at the Casagrande liquid limit (⁠$su(LLcup)$⁠) has a relatively wide range of typically 1–3 kPa (Nagaraj et al., 2012; O'Kelly, 2019). O'Kelly (2019) explained that some very high deduced values of $su(LLcup)$ are perhaps due to insufficient data in the original regressions used for their determination, the curve-fitting approach adopted and various measurement inaccuracies arising from the strength apparatus employed. The value of the Casagrande liquid limit (LL_cup) is also partly dependent on the hardness of base material of the percussion-cup apparatus used (Haigh, 2016; O'Kelly, 2019), such that some variation in the value of LL_cup and hence the associated $su(LLcup)$ is therefore expected for apparatuses with different base hardness.

However, for the fall-cone liquid limit (LL_FC), a standardised fall-cone device with given cone mass, apex angle and smooth cone surface produces values of LL_FC that correspond to a set (predefined) value of undrained shear strength (i.e. $suFC(LL)$⁠). For example, the BSI 30°–80 g fall cone, with LL_FC defined as the water content coincident with 20 mm of cone penetration, would mobilise an $suFC(LL)$ value of ∼1·7 kPa, implying a cone factor (K) value of 0·867. It is true that the pertinent value of K either needs to be calculated theoretically (equation (7): cf. Koumoto & Houlsby (2001)) or calibrated by assigning the cone penetration depth value at LL_FC. However, LL is a somewhat arbitrary criterion as the transition from plastic to liquid state is not abrupt (unlike the change from plastic to brittle state). Hence, once a criterion for LL is established (assigned), LL_FC thus implies a fixed $suFC(LL)$ (or a variable K) value if one invokes Hansbo's (1957) equation (equation (1)).

The fall-cone test can be examined based on plasticity analysis (Koumoto & Houlsby, 2001) to demonstrate a relationship between the cone mass and soil undrained shear strength (Hansbo, 1957) if the soil is assumed to be a perfectly plastic material. For soil with this simplified constitutive behaviour, the cone factor (K) should be constant. Undrained shear strength (⁠$su$⁠) values measured using different strength test methods for saturated, remoulded, physically identical specimens of a given soil inevitably show some experimental variation. Therefore, calibration of $suFC(LL)$ against $su$ data measured by other means (e.g. laboratory vane shear tests) may show some variation in the ‘strength’ at LL owing to the differences between the stress paths, strain rates and other parameters utilised by different test methodologies (see Haigh et al., 2013; O'Kelly, 2013a, 2013b, 2014). The discusser's Table 3 shows this variability, although it is possibly exaggerated by the inclusion of data for undrained shear strength at the LL_cup, with the value of LL_cup often not coincident with LL_FC measured for a given soil. The datasets summarised by the discusser in Table 3 also include data for organic fine-grained (sediment) soils, although the Atterberg limit concepts are not appropriate for those soils containing fibrous organic material (O'Kelly, 2015, 2016). The discusser uses equation (16) (from Vardanega & Haigh, 2014) to establish an undrained shear strength range of 1·55–2·35 kPa at LL_FC. This strength variability largely arises from the individual fitting functions used, so some statistical variability is expected here, since a semi-logarithmic function was imposed on the data. This does not imply that the undrained shear strength at LL_FC has a range of values, but is an artefact of the assumption of linearity in the semi-logarithmic relationship which, as the discusser rightly points out, is only approximate. When establishing the regressions to the whole dataset considered, the undrained shear strength at LL_FC was set to 1·7 kPa following the suggestion of Wroth & Wood (1978): it is pleasing that 1·7 kPa is roughly in the middle of the range quoted by the discusser.

The discusser states that ‘Interpolation for strengths between the PSL and the liquid limit depends on the assumption of linearity of the semi-logarithmic relationship. However, these relationships are distinctly curved.’ The authors agree with this statement. In their paper, however, they do not say that the assumption of the semi-logarithmic relationship was theoretically justified. As pointed out in Vardanega & Haigh (2014), there is no theoretical reason to favour the semi-logarithmic fitting over another, with power-law fitting shown to be statistically more acceptable. Hence, the latter approach was adopted in formulating equations (12)–(14) presented in O'Kelly et al. (2018) (see also Kodikara et al. (1986) and Feng (2000, 2001) who showed the use of the power-law fitting for undrained shear strength variation with water content in the plastic range). One may also use logarithmic liquidity index to predict fall-cone undrained shear strength (Koumoto & Houlsby, 2001). Should enough data be available to map the undrained shear strength variability over the water content range of interest, multilinear interpolation may well be a useful tool for the understanding of soil behaviour. However, for applications in which extrapolation of data needs to be undertaken to interpolate soil undrained shear strength for a given water content, sufficient data are usually not available to accurately predict the form of the pertinent multilinear relationship.

If one accepts the multilinear behaviour shown in Fig. 12 as representative, then semi- or bi-logarithmic regression analysis of $su$–w data and extrapolation along the single line to the standard (thread-rolling) PL value for a given soil would deduce a value of undrained shear strength at the plastic limit that was lower (possibly by a substantial amount) than its actual $su$ value at the standard PL. The authors hypothecate the following.

• (a)

  It would therefore follow that deduced $su$ values for the standard PL from semi- or bi-logarithmic regression are (possibly grossly) conservative and consequently this may reduce the possibility of the few soils identified by the discusser being brittle at their PL₂₅ [PSL₂₅] water contents;

• (b)

  The range of strength gain factor (⁠$RMW$⁠) values identified considering the LL_cup and thread-rolling PL is conservative; the actual $RMW$ range would be greater.

The authors wish also to comment on the introduction of the toughness parameter by the discusser.

For water contents within the adhesive plastic region, fine-grained soil exhibits zero toughness, but has a measurable undrained shear strength. The authors deliberately avoided discussion of strength for liquidity indices below 0·2, as soil plasticity behaviour is required for analysis of the fall-cone test using equation (1). Once soils potentially become brittle, the analysis using equation (1) becomes questionable at best. Whereas the fall-cone approach does not measure toughness, the ‘Barnes apparatus’ (Barnes, 2009, 2013a, 2013b) allows for an analysis of soil toughness in the plastic and ductile state, thus providing for a different method of analysis of soil behaviour for water contents below the brittle transition point. The data shown in Fig. 13 do indeed show a rapid increase in soil toughness for reducing water content close to PL. Whether this is better interpreted as a bilinear relationship, as shown in Fig. 13, or as a non-linear function such as a power law is worthy of further study.

H

slope of a linear regression line fitted to ln(s_uFC) versus w data (as defined in the paper by Vardanega & Haigh (2014))

I_L

liquidity index

I_L25

liquidity index computed with PSL₂₅ (also denoted as PL₂₅)

I_L100

liquidity index computed with PSL₁₀₀ (also denoted as PL₁₀₀)

K

cone factor

LL_cup

Casagrande liquid limit

LL_FC

fall-cone liquid limit

PSL₂₅

water content corresponding to s_uFC = 25 × s_uFC(LL) (also denoted as PL₂₅)

PSL₁₀₀

water content corresponding to s_uFC = 100 × s_uFC(LL) (also denoted as PL₁₀₀)

R_MW

strength gain factor

S

intercept of a linear regression line fitted to ln(s_uFC) versus w data (as defined in the paper by Vardanega & Haigh (2014))

s_u

saturated remoulded undrained shear strength

$suFC$

fall-cone shear strength

$suFC(LL)$

fall-cone shear strength at LL

s_uFC(PSL)

fall-cone shear strength at the PSL

s_u(LL)

undrained shear strength at the LL

s_u(LLcup)

undrained shear strength at the LL_cup

s_u(PSL100)

undrained shear strength at the PSL₁₀₀

w

water content