TECHNICAL NOTE

 

Effect of the minimum void ratio on the vertical intercept of the steady state line of non-plastic soils

 

 

L A Torres-Cruz; S Geyer; P R Mackechnie

Correspondence

 

 

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ABSTRACT

The steady state line (SSL) plays a key role in understanding and modelling the mechanical response of soils. Accordingly, understanding how the SSL correlates to soil index properties is of primary importance. A previous study reported that the vertical location of the SSL (Γ₁) in void ratio (e) versus mean effective stress (ρ') space is correlated to the minimum void ratio (e_min). However, the correlation only included soils with narrow particle size distributions (PSD) and low fines content (FC). In the current study, published data corresponding to 30 non-plastic soils were re-processed to further explore the applicability of the r₁-e_mincorrelation. The results indicate that the r₁-e_mincorrelation is linear (R² = 0.85) and valid regardless of the coefficient of uniformity (C_u), FC, and particle shape. The r₁-e_mindataset presented herein was also compared to a previously published dataset, and good agreement was observed. It is proposed that the r₁-e_mjncorrelation can be very useful to understand how the Γ₁ of different non-plastic soils compare to one another, and to minimise the extent of triaxial testing required when characterising a soil deposit from an SSL standpoint. Limitations of the r₁-e_min correlation are also discussed.

Keywords: steady state line, critical state line, minimum void ratio, non‑plastic soils

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INTRODUCTION

Soils reach constant values of void ratio (e), mean effective stress and deviator stress when sheared to large strains (Castro 1969). The locus of steady state (e, p', q) coordinates attained when shearing from different initial states, defines the steady state line (SSL). Because the SSL represents the stress and void ratio conditions towards which a soil evolves when sheared, it plays a key role in defining mechanical response. Two projections are typically used to define the SSL: q-p' (stress plane) and e-p' (compression plane). The q-p' projection, which reflects steady state frictional properties, is strongly dependent on particle shape (Cho et al 2006) and largely independent of particle size distribution (PSD) (Carrera et al 2011; Rahman et al 2014). By contrast, the e-p' projection, representative of stiffness, is affected by particle shape (Cho et al 2006), PSD (Thevanayagam et al 2002; Rahman & Lo 2008; Muir-Wood & Maeda 2008; Li et al 2013), and void ratio limits (Cho et al 2006; Cubrinovski & Ishihara 2000; Hemer et al 2016). Given the greater number of factors that affect the e-p' projection, it has received attention from a significant number of researchers and is the focus of this note. The e-p' projection is commonly modelled with Equation 1:

Where λ _ω is the slope of the SSL in semi-logarithmic space and Γ is the void ratio at p' = 1 kPa (Figure 1). The current note will explore the correlation between the minimum void ratio (e_mir), which is associated with a defined maximum density state, and Γ _ν

Several studies have investigated how the SSL is affected by soil index properties. Thevanayagam et al (2002) tested gap-graded mixtures composed of sand and non-plastic fines and concluded that the vertical location of the SSLs could be explained by the fines content (FC). They noted that as FC increases from zero, the SSL shifts downwards in e-p' space (Γγ decreases), and that beyond a certain FC value it shifts upwards (Γγ increases). The FC value at which the SSL shift reverses direction was termed by Thevanayagam et al (2002) as the threshold FC (TFC). The calculation of the parameters proposed by Thevanayagam et al (2002) to explain the effect of FC required knowledge of the SSLs of the different sand-fines mixtures. Consequently, the framework lacked predictive power. To overcome this, Rahman & Lo (2008) developed semi-empirical equations to calculate, as a function of FC and other PSD descriptors, the parameters in the framework proposed by Thevanayagam et al (2002). This allowed the prediction of the SSL of sands with FC < TFC, provided that the SSL of another sand-fines mixture with FC < TFC was known. More recent works show that the effect of non-plastic fines on the SSL continues to be investigated (e.g. Mohammadi & Qadimi 2015; Rahman et al 2014; Yang et al 2015).

Despite the success of Rahman & Lo (2008) in predicting several SSLs, their framework has limitations that hinder its wider applicability. For example, the framework is limited to soils with FC smaller than the TFC, which tends to be close to 40%. Additionally, the framework cannot explain the differences between the SSLs of soils with no fines but different PSDs or grain shape.

Cho et al (2006) explored the correlation between Γ1 and e_minof 49 natural and crushec sands with mostly little to no fines (only six sands had FC > 12%) and a maximum coefficient of uniformity (CJ of 6.2 (only three sands had C_u> 4). They found that the correlation was linear, independent of particle shape, and of modest strength (R²= 0.54). However, the validity of thecorrelation for high FC values or widely graded soils remains untested. The objective of this note is to present evidence that expands the applicability of the correlation to a wide range of non-plastic soils, regardless of PSD descriptors such as FC and C_M. The wide applicability of this correlation is considered a step towards overcoming the limitations encountered when using FC and TFC to explain the location of the SSL.

 

METHODOLOGY

Datasets from three references were processed to explore the validity of the correlation: Thevanayagam et al (2002), Yang et al (2006) and Li (2013). The dataset from Thevanayagam et al (2002) includes seven soils composed of foundry sand mixed with non-plastic crushed silica fines. The resulting mixtures have FCs varying from 0% to 100% and C_uvarying from 1.7 to 47. The dataset from Yang et al (2006) includes nine soils composed of Hokksund sand mixed with non-plastic Chengbei silt. The resulting mixtures have FCs varying from 0% to 94% and C_uvarying from 2 to 14. The Thevanayagam et al (2002) and Yang et al (2006) datasets were used herein to assess the validity of the Γι-emm correlation over a wide range of FC and C_uvalues. The dataset from Li (2013) includes 14 soils of which six were made of glass balls (spherical particles) and eight of Hostun sand (angular particles). Two soils, one each of glass balls and Hostun sand, had FC =10%, whereas the remaining 12 had FC = 0%. C_uvaried from 1.1 to 20. This data-set was used herein because: (i) some of the soils made up of glass balls had considerably low e_minvalues which allowed a significant extension of the lower bound of the domain of the F]_-e_mincorrelation; (ii) given that the particle shape of the glass balls is distinctly different from that of Hostun sand, this dataset enables a straightforward assessment of whether particle shape affects the r_l-e_miin correlation; and (iii) this dataset also allows assessment of the validity of the r_l-e_miin correlation at different C_uvalues. Figure 2 presents the PSDs of the 30 soils considered.

Values of Γ ι were calculated by fitting Equation 1 to the (p', e) points that defined the SSL of each soil (Table 1). Some of the SSLs reported by Thevanayagam et al (2002) cannot be modelled with Equation 1, due to the curvature of the SSL in e-logKp' which some soils exhibit at high stress levels (e.g. Been et al 1991; Li & Wang 1998). Consequently, some (p', e) points with high p' values were excluded from the fitting process. Similarly, given the experimental difficulties and uncertainties involved in performing triaxial tests at very low values of effective stress, two (p', e) points with p' = 1 kPa were also disregarded when calculating Γ₁(see footnote c in Table 1).

As annotated in Figures 3 to 5, different methods were used to determine the e_min values of each dataset. ASTM D1557 refers to the modified Proctor compaction test, and ASTM D4253 refers to the method of soil densification using a vibratory table. Although the current authors acknowledge that e_minvalues from different methods are not strictly comparable, it is hypothesised that, regardless of the method, the resulting e_minprovides a reasonable indicator of packing efficiency.

 

RESULTS AND DISCUSSION

Figures 3 to 5 suggest strong (R²> 0.90) linear r]-e_mincorrelations. The data point labels further suggest that the correlations are valid regardless of FC or C_u. The independence of the correlation from FC observed in Figures 3 and 4 is at odds with previous works (e.g. Thevanayagam et al 2002; Rahman & Lo 2008; Rahman et al 2014) which have suggested that Γ₁is fundamentally correlated to FC. Furthermore, Figure 3 explicitly shows that essentially the same Г₁-e_mincorrelation is followed regardless of whether FC is smaller or greater than TFC. Additionally, the angular Hostun sand and the glass balls follow the same Γ₁~e_min correlation despite their significantly different particle shapes (Figure 5). This result agrees with Cho et al (2006) who reported the independence from particle shape of the Γ₁-e_mi_ncorrelation for narrowly graded sands with low values of FC.

When all 30 soils are collectively plotted (Figure 6), a single linear correlation emerges (R²= 0.85). The slight decrease in R²(compare to Figures 3 to 5) is likely a consequence of combining SSLs calculated from triaxial tests conducted in different laboratories and following slightly different protocols, and e_minvalues obtained through different procedures. For example, Prochaska & Drnevich (2005) showed that the maximum dry unit weight, which is associated to e_min, can show variances of the order of ± 3% when estimated from different compaction techniques. A unique linear correlation (R²= 0.77) continues to be apparent when the data corresponding to Cho et al (2006) is included (Figure 7). This indicates that the Γ₁-e_miincorrelation observed by Cho et al (2006) in narrowly graded sands with low FC values is approximately the same for soils with significant amounts of non-plastic fines and large C_uvalues, such as those represented in Figures 3 to 5. The data analysed in this study has also been useful to expand the lower bound of the domain of the Γ₁-e_miin correlation reported by Cho et al (2006) (Figure 7).

The authors suggest that the validity of the Γ₁~e_mincorrelation over such a wide range of non-plastic soil types is explained by the similarity in which both Γ₁and e_minare affected by a soil's fundamental properties. For example, they are both directly correlated to particle angularity (Li 2013; Biarez & Hicher 1994; Cho et al 2006), inversely correlated to C_u(Li 2013; Biarez & Hicher 1994; Poulos et al 1985), and respond in a similarmanner to changes in FC (Lade et al 1998; Rahman & Lo 2008). It is also important to acknowledge that the use of e_minas a predictor of Γ ι may be limited by the differences between the remoulding and particle crushing mechanisms that a soil undergoes when sheared to steady state and when compacted to emin^.

The authors are not recommending the use of the correlations in Figures 3 to 7 to replace triaxial testing to determine the SSL, as doing so can result in significant errors. For example, in Figure 7 a deviation from the best fit line of ± 0.1 is observed in Γι, implying a potential error of 0.2. This value is significantly higher than the error of ± 0.01 in void ratio, which was suggested by Jefferies & Been (2006) as a reasonable target error when calculating the SSL experimentally. Notwithstanding these potential errors, the correlation is very useful to qualitatively understand how the Γι values of different non-plastic soil types will compare to one another. Additionally, when studying a heterogeneous soil deposit that includes a variety of soil types, the correlation can help reduce the number of soil types whose SSLs have to be experimentally determined to fully characterise the deposit from an SSL standpoint (e.g. Hemer et al 2016).

 

CONCLUSIONS

The Гr_l-e_miincorrelation of 30 non-plastic soils has been investigated. The results indi-cate that the correlation is linear (R²= 0.85) and valid regardless of FC, C_u, and particle shape. Comparison of the results presented herein (Figure 6) with the Γ₁ versus e_min dataset obtained by Cho et al (2006), indicates that the Г₁-e_mincorrelation originally observed by Cho et al (2006) for narrowly graded sands with small amounts of fines may be applicable to all non-plastic soils (Figure 7). The similarity in the way in which both Γ₁and e_minrespond to changes in the fundamental properties of a soil is believed to be the reason why the Г₁--e_mincorrelation is valid over a wide variety of soil types.

Given that the effect of non-plastic fines on the SSL continues to be intensely researched, it is an important finding of this work that the Г_l-e_miincorrelation is not affected by FC (Figures 3 and 4). This is not entirely surprising though, as it seems unlikely that a fundamental soil property (Γ₁) can be explained by a property such as FC which is arbitrarily defined as the percentage of particles smaller than 75 μηι (or 63 μηι, depending on the standard). Accordingly, it must be expected that the predictive power of FC and associated concepts like the TFC will have important limitations. The results presented herein suggest that a better understanding of the SSL can be achieved by correlating Γ₁to e_minrather than to FC. The authors are currently analysing an extended database and conducting triaxial experiments to continue exploring the strengths and limitations of the Г_l-e_miincorrelation.

 

NOTATION

e = void ratio

p' = mean effective stress (σ'1 + σ'2 + (/3)/3

q = deviator stress (σ'1 - σ'3)

SSL = steady state line

Γ₁ = steady state void ratio that corre sponds to a mean effective stress of 1 kPa

λ_ω = slope of the SSL in e-log₁₀p') space

FC = fines content (< 75 μm)

PSD = particle size distribution

e_min= minimum void ratio

TFC = threshold fines content

C_u= coefficient of uniformity (D₆₀/D₁₀)

 

ACKNOWLEDGEMENTS

The authors gratefully acknowledge Dr Nico Vermeulen (Jones & Wagener) and Dr Irvin Luker (University of the Witwatersrand) for their valuable insights and discussions.

 

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Correspondence:
L A Torres-Cruz
Department of Civil and Environmental Engineering
University of the Witwatersrand
Private Bag 3 Wits
2050
South Africa
T: +27 11 717 7150
E: LuisAlberto.TorresCruz@wits.ac.za

S Geyer
Prime Resources Environmental Consultants
POBox 2316
Parklands
2121
South Africa
T: +27 11 447 4888
E: stephan@resources.co.za

P R Mackechnie
Department of Civil and Environmental Engineering
University of the Witwatersrand
Private Bag 3 Wits 2050
South Africa
T: +27 11 326 0030
E: mackechniep@gmail.com

 

 

 

DR LUIS ALBERTO TORRES-CRUZ obtained his BSc in Civil Engineering from the Universidad del Valle in Cali, Colombia, and completed his PhD in Geotechnical Engineering at the University of the Witwatersrand (Wits). His doctoral research focused on the use of the cone penetration test to assess the liquefaction potential of tailings dams. He is currently a lecturer in Geotechnical Engineering at Wits.

 

 

STEPHAN GEYER obtained his BSc in Civil Engineering from the University of the Witwatersrand. He is currently employed at Prime Resources Environmental Consultants, where he is involved in the geotechnical and design aspects of water and mining residue management, including tailings storage facilities.

 

 

PETER MACKECHNIE obtained his BSc degree in Civil Engineering from the University of the Witwatersrand.