Results of a critical state line testing round robin programme

The authors made a most valuable contribution to the technical literature on the experimental aspects of critical state line (CSL) determination (Reid et al., 2021). The discusser wishes to address two points. The first point is to request further information regarding the methods used to perform the cross-sectional area correction required for the calculation of the deviator stress, q. In the discusser's experience, q is generally calculated by assuming that the specimen deforms as a right circular cylinder (RCC). However, this assumption is often not met, even when using lubricated and enlarged platens. Since the paper provides an indication of the state of practice across geotechnical laboratories around the world, it would be interesting to know the extent to which the participating laboratories adopted the RCC assumption, or the alternative methods they may have used to perform the area correction.

The second point is to elaborate on the uncertainty of Γ and λ_e, a topic that is seldom discussed in the technical literature. The logarithmic fit equation can be expressed as

where Γ_p* is the void ratio at p′ = p*; λ_e is the slope; and p* is an arbitrary stress that ensures dimensional consistency by cancelling the units in the argument of the logarithm. The authors effectively adopted p* = 1 kPa, which is perhaps the most common choice in the literature. The choice of p* is inconsequential to the regressed CSL as it does not affect the underlying values of e and p′. However, p* does affect the uncertainty of Γ_p* because this uncertainty decreases when p* is close to the stress levels of the underlying (e, p′) points of the CSL. This leads to Γ₁₀₀ generally having a lower uncertainty than Γ₁ since, as in the case of the authors’ data, CSLs typically have underlying points closer to p′ = 100 kPa than to p′ = 1 kPa. Beyond statistical considerations, the practical implication is that Γ₁₀₀ correlates more strongly than Γ₁ to soil index properties (Torres-Cruz, 2019). Accordingly, the current discussion uses Γ₁₀₀ ( = Γ₁– λ_e × ln(100)) instead of the more widely adopted Γ₁. It is worth noting that a base-10 logarithm is also commonly used in equation (1) and that this produces a different slope λ₁₀ = 2·303λ_e.

The results of the round robin programme allow an estimate of the uncertainty of Γ₁₀₀ and λ_e by calculating the standard deviation of the multiple non-excluded measurements of these parameters listed in Table 4. However, geotechnical practitioners cannot count on Γ₁₀₀ and λ_e estimates from multiple laboratories and must instead rely on one set of (e, p′) points from a single source. In this case, the standard error of Γ₁₀₀ and λ_e, which depends partly on the scatter of the data and the number of experimental (e, p′) points that define the CSL, provides an estimate of their standard deviation. For instance, a recent analysis computed the standard error of Γ₁₀₀ and λ₁₀ of 91 CSLs with known underlying (e, p′) points (Torres-Cruz & Santamarina, 2020). To enable comparisons, the standard errors of both parameters were normalised by the size of the range of values observed in non-plastic soils. Based on an analysis of 160 CSLs, Γ₁₀₀ appears to vary from 0·2 to 1·2, yielding a range of 1·0; and λ_e from 0 to 0·13, yielding a range of 0·13. The normalised standard error (NSE) of λ_e (NSE-λ_e), and of Γ₁₀₀ (NSE-Γ₁₀₀) was also calculated herein for the 13 round robin CSLs whose (e, p′) data points are available in the online supplementary material provided by the authors. Fig. 17 shows that the data points of the 91 CSLs and of the round robin CSLs exhibit the same trend: NSE-λ_e is always greater than NSE-Γ₁₀₀, with the ratio of NSE-λ_e to NSE-Γ₁₀₀ mostly varying from 3 to 12. That is, estimates of Γ₁₀₀ are generally more reliable than estimates of λ_e. More specifically, NSE-Γ₁₀₀ varies predominantly from 0·2 to 2% with maximum values of ∼5%, while NSE-λ_e varies predominantly from 1 to 10% with maximum values of ∼17%. It should be noted that Fig. 17 is independent of the logarithmic base used in equation (1) – that is, NSE-λ_e =  NSE-λ₁₀.

These results point to the importance of inspecting the underlying (e, p′) points that define a CSL to, at least, qualitatively assess the expected precision of Γ₁₀₀ and λ_e. Ideally, the standard errors of Γ₁₀₀ and λ_e should be formally calculated to enable the assessment of how their uncertainty propagates into other calculations. However, it is important to note that the standard errors reflect only the precision of the CSL parameters, not their accuracy. The latter can only be ensured by adopting good laboratory practices as emphasised by the work of the authors.

The authors thank the discusser for his interest in their work. The authors are pleased that the data produced in their work could be included in the database collected by the discusser to provide further insight into the trends observed for normalised standard errors for CSL parameters.

With respect to the area correction applied in the program, all participants independently elected to use the RCC approach in their work. This was generally justified on the basis of the use of oversized lubricated end platens by many of the participants. However, this represents an interesting area for further discussion and study, as even when using oversized lubricated end platens it is not uncommon for specimens still to exhibit a slight bulging pattern at large strains that is not entirely consistent with the RCC assumption. This may be consistent with the observations of Tatsuoka & Haibara (1985) and Ueng et al. (1988) that lubricated platens with silicone grease still have an interface friction angle in the order of 1–2°. The visual observation of bulging, even when using oversized lubricated end platens, probably warrants further investigation numerically. For example, Sheng et al. (1997) saw significant differences when comparing an idealised ‘perfect’ end platen to a frictional one with a 20° interface friction angle. Further analysis of this type using more realistic interface friction angles for oversized lubricated end platens may provide guidance on the effect such small friction may have on divergence of specimens from ideal conditions and how this may affect parameters inferred from global test measurements.

e

void ratio

NSE-Γ₁₀₀

normalised standard error of Γ₁₀₀

NSE-λ_e

normalised standard error of λ_e

p′

mean effective stress

p*

arbitrary stress value in the same units as p′

q

deviator stress

Γ

void ratio of the CSL at p′ = 1 kPa, as defined by the authors

Γ_p*

void ratio of the CSL at p′ = p*

λ_e

slope of the logarithmic fit critical state line (CSL) equation when using a natural logarithm

λ₁₀

slope of the logarithmic fit CSL equation when using a base-10 logarithm