Initial shear modulus of sandy soils and equivalent granular void ratio

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1 eomechanics and eoengineering: An International Journal Vol. 7, No. 3, September 22, Initial shear modulus of sandy soils and equivalent granular void ratio M.M. Rahman a *, M. Cubrinovski b and S.R. Lo c a School of Natural and Built Environments, University of South Australia, Mawson Lakes, Adelaide, SA 595, Australia; b Department of Civil & Natural Resources Engineering, University of Canterbury, New Zealand; c School of Engineering and Information Technology, UNSW@ADFA, Canberra, Australia eomechanics and eoengineering 22.7: downloaded from Received 2 June 2; final version received 9 August 2) The small strain/initial shear modulus, O, for clean sands has been extensively studied over the last few decades, and it is generally accepted that O is a function of the effective mean stress, p and void ratio, e. However, natural sandy soils often contain some fines particle size 75 μm) and systematic studies on the effect of fines on O are relatively limited. Two different approaches for capturing the effects of fines on O have been reported. However, both approaches assumed that the O values for a range of fines content specific to the sand and fines type are known in advance, and thus they cannot be used as predictive tools. This paper presents an alternative approach for evaluation of O based on the equivalent granular void ratio, e. Unlike earlier attempts, the proposed method does not rely on back-analysis and e can be calculated from e and other conventional grading properties of sandy soils. The proposed approach approximately leads to a single relationship between O and e, p ) irrespective of fines content. Thus, it provides a framework for estimating O at different fines content from data on clean sand. Keywords: initial shear modulus; sand; fines; void ratio. Introduction It is generally accepted that stress-strain behaviour of soil is near-elastic at shear strains smaller than 4 %. The shear modulus at such a small strain is usually called small strain or initial shear modulus, and denoted as O. A realistic estimate of O is required to analyze many soil mechanics problems such as dynamic behaviour of soil deposits, constitutive modelling of soils, soil-structure interaction, etc. The factors affecting O of clean sands have been extensively studied over the last several decades Hardin and Richart 963, Drenevich and Richart 97, Miura et al. 23). It is generally accepted that O is a function of void ratio, e, and mean effective stresses, p as expressed by Equation ) below Hardin and Richart 963, Iwasaki and Tatsuoka 977, Hardin 978, Tatsuoka and Shibuya 99). O C g f e) p / ) ng p pa a [] where is a reference stress taken as kpa, n g is a parameter accounting for the effects of p on O, f e) is an empirical function of void ratio, e, and C g is a material constant for a given f e). Two f e) functions have been commonly used. The first one as proposed by Hardin and Richart 963) and denoted as f H is: f e) f H e) e r e) 2 + e where e r is a soil constant that depends on the angularity of soil particles. Many researchers suggested e r 2.97 for angular sands and 2.7 for rounded sands Iwasaki and Tatsuoka 977, Chien and Oh 998, Chien and Oh 22). An alternative form of f e) as presented by Jamiolkwski et al. 99) and denoted as f J is: [2] f e) f J e) e a g [3] where a g is a regression constant. Natural sands often contain fines size 75 μm), however systematic studies on the effects of fines on O have been relatively limited. These studies have shown that O for sandy soil decreases with an increase in the fines content. Some researchers have quantified this effect by multiplying O of clean sand by a reduction factor Iwasaki and Tatsuoka 977, Chien and Oh 998). Others attempted to model the reduction in O value by using an equivalent void ratio in lieu of e to capture the effect of fines on O Thevanayagam 999). As discussed in the background section of this paper, both approaches relied on back calculations in order to evaluate the effect of fines on O. Thus, these methods are not directly applicable *Corresponding author. mizanur.rahman@unisa.edu.au ISSN print/issn online 22 Taylor & Francis

2 22 M.M. Rahman et al. eomechanics and eoengineering 22.7: downloaded from for estimating O for sand at a given fines content without first knowing the O values at different fines content for that particular sand and fines mixtures. This paper presents an approach based on capturing the effects of fines on O through the equivalent granular void ratio, e. Rahman, Lo and co-workers developed a method to obtain the equivalent granular void ratio, e, from void ratio, e, and other simple grading parameters without the need of any back calculation Rahman et al. 28, 29). The use of e and the associated methodology of converting e into e can successfully capture the effect of fines on the steady state of deformation, cyclic resistance and undrained instability Rahman and Lo 28, Rahman et al. 28, Rahman 29). The objective of this paper is to examine whether the equivalent granular void ratio, e, obtained by the same methodology can be used to evaluate the effect of fines on O, and to formulate a methodology for predicting the O of sand with fines from O values of clean sand. 2. Background The two approaches mentioned in the introduction will be first briefly described in this section. 2. Approach-I In order to quantify the effects of fines on O, Iwasaki and Tatsuoka 977) introduced a reduction parameter B R in Equation ), thus leading to: O B R C g f e) p / ) ng p pa a [4] The relationships between B R and fines content f c ) as reported by Iwasaki and Tatsuoka 977) and as extracted from Chien and Oh 998, 22) and Salgado et al. 2) are presented in Figure, which clearly shows that the B R - f c relationship developed for a specific sand and fines mixture may not be applicable to another sand and fines mixture. This means, a reliable B R - f c correlation can only be obtained if the O values for a range of f c, specific to the given sand and fines type under consideration, were known in advance. 2.2 Approach-II The second approach is based on the concept of the equivalent granular void ratio, e, defined as Thevanayagam et al. 22): e e + b) f c b) f c [5] where f c fines content in decimal and the parameter b represents the fraction of fines that are active in the force structure. Reduction parameter, B R Figure. studies. Iruma sand Iwasaki and Tatsuoka, 977) Yunlin sand Chien and Oh, 998) Ottawa sand Salgado et al. 2) Fines content, f c %) The variation of B R with fines contents reproduced from previous Approach-II assumes that the effect of fines on O can be captured by replacing e with e in Equation ). Thus, for sand with fines this equation takes the following form: O C g f e ) p / ) ng p pa a [6] where C g, n g and f.) are the same as those for clean sand. The physical meaning of Equation 5) implies the fines are not fully active in the force structure. Therefore it requires b and that the sand-fines mixture has a fines-insand matrix Thevanayagam et al. 22). The latter implies Equation 5) is only physically meaningful if f c is less than a threshold value denoted as, f thre, defining the condition for fines-in-sand matrix. For a clean sand, i.e. f c and e e, Equation 6) reduces to Equation ). For f c >, e > e and this leads to a reduction of O with fines when evaluating O by Equation 6). The challenge is how to determine b so that e can be converted to e. Thevanayagam and Liang 2) analysed the effect of fines on OS-F55, Foundary sand using this approach. They measured a secant modulus at a small axial strain ε.5) by conducting monotonic triaxial tests on samples with different fines contents at p kpa. This secant modulus was considered as a close approximation to O or small strain modulus. For the purpose of this study, that is evaluation of effect of fines on O, we consider the above assumption is appropriate, as also done by Thevanayagam and Liang 2). Their work showed that, by selecting b.25, O was correlated to e irrespective of f c as reproduced in Figure 2. In essence, the b-value was obtained from backcalculation and by assuming that, at a given p, there is a single

3 eomechanics and eoengineering: An International Journal 22 eomechanics and eoengineering 22.7: downloaded from Equivalent granular void ratio, e* b.25) f c. f c.7 f c.5 f c.25 OS-F55, Foundary sand Shear modulus, MPa) at ε.5 Figure 2. The relation between e with b.25) and at ε.5 for OS-F55, Foundary sand with fines, after Thevanayagam and Liang 2). relationship between O and e. Thus, the O values at different fines contents have to be known first in order to facilitate the back-calculation. 3. Determination of equivalent granular void ratio e The approach of capturing the influence of fines on O by using e in lieu of e can become a prediction method if one can determine b without the need of a back-calculation. This is illustrated in this section. Previous studies on e unambiguously suggested that b is an adequate approximation at low fines content, but a non-zero value of b should be considered at higher fines contents. This implies b is a function of f c, and this further highlighted the limitations of back calculating a single b-value for a range of f c using back-calculations. By reanalysing the experimental data of Mceary 96) on binary packing studies, Rahman and Lo 27, 28a) concluded that b is a function of the fines content f c ) and diameter ratio D/d) i.e. b Ff c,d/d), where D is the size of a reference sand particle and d is the size of a reference fines particle. Furthermore the functional relationship has to possess a number of mathematical attributes Rahman et al. 28). Since sand and fines are not single-size materials, the function was generalized to b Ff c, χ), where χ D /d 5 and the subscripts denote fractile passing % for the sand-size fractile and 5% for the fines fractile). To simulate the required mathematical attributes to the relationship, Rahman et al. 28, 29) proposed a semi-empirical equation for b expressed as b [ exp.3 f )] c/f thre ) r f ) r c [7] k f thre where, k r.25 ), r /χ and f thre is the aforementioned threshold fines content. An evaluation of published data on f thre suggested that f thre.3 might be assumed as a first approximation. However, f thre may be determined more reliably using Equation 8) below developed by Rahman and Lo 28b) based on calibration with eight published datasets for χ in the range of 2 to 42). f thre.4 + e + ) [8] α βχ χ where α.5 and β.3. Details of the development of the predictive equation for b and f thre can be found in Rahman and Lo 28) and Rahman et al. 28). The above quoted works showed that using the above approach e can successfully capture the effect of fines on steady state behaviour, cyclic resistance and undrained instability of sandy soils Rahman and Lo, 2a; Rahman and Lo, 2b). The question is whether this can be extended to capturing the effects of fines on O. 4. Evaluation of O based on e : scrutiny of experimental data Equation 6) implies a single relationship between O /C g f e ) and p, where C g is a constant irrespective of f c. In evaluating the validity of Equation 6) with a database, one has to first calculate e by: calculating b using Equations 7) and 8), noting that b depends on f c, and substituting the corresponding b-value into Equation 5) to convert e into e. One can then plot the data points in a O /f e )-p space. If the proposed prediction methodology is valid, all data points should be located along a well defined curve irrespective of f c. Furthermore, this should be the case for both mathematical forms of f e ), i.e. for f f H and f f J. Four published experimental datasets were extracted from the literature in order to evaluate the validity of the proposed methodology as above. For all four datasets, f c < f thre, and thus Equations 7) and 8) are applicable. Details of grading characteristic of the considered soils are listed in Table. It should be noted that the data from Chien and Oh 998) were originally reported in terms of relative density, D R. The void ratio, e, was hence calculated using the e max and e min values reported in their paper. The test conditions under which O values of sand fines mixes with different fines content were evaluated are summarized in Table 2. It is apparent that for all soils a wide range of void ratios densities) and mean effective stresses were considered, thus allowing us to scrutinize the accuracy of the proposed relationshind procedure. For each set of test data, the void ratio e was converted into e using a b value calculated using Equations 8) and 7), as

4 222 M.M. Rahman et al. Table. The grading properties of the soils considered in this study. Sand properties Fines properties D /d 5 References Testing Name e max e min D Cu Name d 5 Salgado et al. 2) Bender element Ottawa Sil-co-Sil.22.8 Iwasaki and Tatsuoka 977) Resonant column Iruma Z Iruma X Iwasaki and Tatsuoka 977) Resonant column Iruma W Iruma X.5 3 Chien and Oh 998) Resonant column Yun-Ling Yun-Ling Thevanayagam and Liang 2) Foundary Sil-co-Sil. 7 eomechanics and eoengineering 22.7: downloaded from Table 2. References The testing conditions for the soils considered in this study. Sand Range of void ratio, e Range of confining pressure, p kpa) RMS prediction Errors Fines content %) f H e ) f J e ) Salgado et al. 2) Ottawa ) ) Iwasaki and Tatsuoka 977) Iruma Z Iwasaki and Tatsuoka 977) Iruma W Thevanayagam and Liang 2) Yun-Ling outlined above. The data for a given host-sand were then plotted in terms of O /f H e )-p and O /f J e )-p as shown in Figure 3a and Figure 3b for Ottawa sand with Sil-co-Sil fines Salgado et al. 2). Note that a total of 38 data points were extracted and both forms of f e ) were plotted to ensure that the findings were not affected by the choice of the empirical f e ) function. One can infer, as a first approximation, that the data points in either Figure 3a or Figure 3b can be reasonably approximated by a single trend curve irrespective of f c. The parameters for the best-fit trend curve are also given in the corresponding figures. These parameters can then be substituted into Equation 6) using either f H or f J so that O values can be predicted. The predicted O values can then be compared to the test values to provide an objective measure of the effectiveness of the proposed prediction methodology. This can be achieved statistically by calculating the root-mean-square RMS) prediction error defined as: RMS prediction error N ),predict 2,test x %,test where O,test is the experimental O value and O,predict is the predicted O value. The resultant RMS prediction errors are: 3.93% for using f H and 3.38% for using f J as summarized in Table 2. A careful examination of the calculation revealed that the data points at low confining pressure contributed significantly to the error. Thus the RMS prediction errors were re-calculated by only using data points with p higher than 6kPa. This still retains 65% of the data points. The RMS prediction errors were thus reduced to 23.48% and 23.63% for f H and f J respectively shown in brackets in Table 2). [9] The same procedure as above was followed in the development of the O /f H e )-p and O /f J e )-p relationship for Iruma Z sand Iwasaki and Tatsuoka 977), Iruma W sand Iwasaki and Tatsuoka 977) and Yun Ling sand Chien and Oh 998) as shown in Figure 4a and b, Figure 5a and b and Figure 6a and b respectively. As showing in Table 2, the RMS prediction errors for all cases were well below 2% indicating a good accuracy of the proposed methodology and predictive relationship. Note that the f H and f J function exhibited similar accuracy in the prediction of O. 5. Application of the proposed method This section illustrates how the O values of sand with fines can be predicted from Equation 6) using only O values of clean sand and grading properties as input information. The data of Thevanayagam and Liang 2) will be re-analyzed for this purpose. This database covers a wide range of void ratios from.359 to.86 and f c ranging from % to 25%. All specimens were isotropically consolidated to kpa prior to the measurement of O with triaxial testing. First the data points for % fines content were used to obtain the necessary soil parameters C g and a g of Equation ). Since the angularity of the sand was not reported in the publication, e r of f H could not be determined and therefore f f J was used for fitting the data points. Since all tests were at kpa, the value of n g does not affect the best-fit curve. Hence, n g.5 was chosen which has been used as a default value for many soils Kramer 996). The data points for clean sand were plotted in terms of e O in Figure 7, and were approximated with the following best-fit) relationship:

5 eomechanics and eoengineering: An International Journal 223 eomechanics and eoengineering 22.7: downloaded from MPa) O / 2.7 e*)2 +e*) O / e *).5 MPa) e*) p O.522 e* p + a f c. f c.5 f c. f c.5 f c.2 a) f c. f c.5 f c. f c.5 f c.2.47 Ottawa sand.47.5 O.467 e* ) p p a a b) 2 Ottawa sand Figure 3. Normalized O /f e )-p relation for Ottawa sand with to 2% fines data from Salgado et al. 2); a) for Equation 6) using f H e ); b) for Equation 6) usng f J e ). p o.53 e 3.75 ).5 [] In this way C g.53 and a g 3.75 were adopted for expressing O as a function of p and e in Equation 6). Thus, Equation 6) could be written as: o.53 e ) 3.75 p ).5 [] We use Equation ) to predict the O values for OS-F55, Foundary sand mixed with different fines contents. Since the test data reported by Thevanayagam and Liang 2) even at other fines content were obtained at p kpa, the value of n g being an assigned value will not have any adverse effect on MPa) O / 2.97 e*)2 +e*) O / e *.9 MPa) e*) e* O.).2).4).8).4).4 Iruma Z sand a).4.9 O.665 e* ) p a Iruma Z sand.).2).4).8).4) b) Figure 4. Normalized O /f e )-p relation for Iruma Z sand with to 4% fines data Iwasaki and Tatsuoka 977): a) for Equation 6) using f H e ); b) for Equation 6) using f J e ). the prediction. For calculating e using Equation 5), the values for b and f thre are needed, where b is dependent on fines content. Using Equation 8) with χ 7., α.5 and β.3, the f thre was calculated as: f thre.4 + e + ) [2] Thus yielding the following relationship for b based on Equation 7) [ b exp.3 f )] c/.363).58 f ).58 c [3].9 [ exp.6f c ) ] f c ).58

6 224 M.M. Rahman et al. 2 Iruma Sand W f c.) Iruma Sand Xf c.) e*) p O.73 e* p + a eomechanics and eoengineering 22.7: downloaded from MPa) / 2.97 e*)2 O +e*) O / e *).9 MPa) O 2.97 e*).29 e* p a a) Iruma Sand W f c.) Iruma Sand Xf c.) b) 2 Iruma W sand.46.9 O.65 e* ) p p a a Iruma Z sand... Figure 5. Normalized O /f e )-p relation for Iruma W sand with to.% fines data from Iwasaki and Tatsuoka 977): a) for Equation 6) using f H e ); b) for Equation 6) using f J e ). where r /χ.58, k r.25 ).58 and f thre.363. This is the relationship between b and f c for the particular sand-fines mixture OS 55, Foundary sand and Sil-co-Sil fines) as illustrated in Figure 8. Thus, for any fines content b is first determined, then e is calculated using Equation 5) and eventually O is calculated using Equation ). Thus O - e curves for fines contents of 7%, 5%, and 25% were established using the above algorithm and plotted in Figure 9 for comparison with reported test data. It is pertinent to emphasize that only O data for clean sand were needed as input in predicting these O - e curves at different f c values. The measured O values at different fines content as extracted from Thevanayagam and Liang 2) are also shown in the figure. ood agreement between the prediction and test data was obtained. MPa) / 2.7 e*)2 +e*) O O/ *). MPa) e Yunlin sand f c. f c. f c.6 f c.2 f c.3... a). Yun-Ling sand.62. O.48 e* ) p p a a b) f c. f c. f c.6 f c.2 f c.3. Figure 6. Normalized O /f e )-p relation for Yun-Ling sand with to 3% fines data from Chien and Oh 998): a) for Equation 6) using f H e ); b) for Equation 6) using f J e ). Shear modulus, MPa) o OS-F55, Foundary sand p a ) e pa f c Void ratio, e Figure 7. Normalized O /f e)-p relation for OS-F55, Foundary sand data from Thevanayagam and Liang 2).

7 eomechanics and eoengineering: An International Journal 225 eomechanics and eoengineering 22.7: downloaded from Parameter, b Relation between b and f c for OS-F55, Foundary sand and Sil-co-Sil fines..2.3 Fines content, f c χ5 χ χ7. χ25 χ 5 Figure 8. Relation between b and f c for OS-F55, Foundary sand and Sil-co- Sil fines data from Thevanayagam and Liang 2). Void ratio, e f c. f c.7 f c.5 f c.25 OS-F55, Foundary sand Shear modulus, MPa) Fitted Eqn for % FC Prediction for 7% FC Prediction for 5% FC Prediction for 25% FC Figure 9. Prediction of shear modulus for OS-F55, Foundary sand with different fines content from the normalized O /f e)-p relationship data from Thevanayagam and Liang 2). 6. Conclusions Previous studies on the influence of fines on O of sandy soils can be classified into two approaches. Both approaches rely on back-calculations using data on O for both clean sand and sand-fines mixture with different fines content. This presents practical difficulties because test data on O for a range of fines content for the particular sand and fines under investigation) have to be known in advance. In this paper, a recent methodology for determining the equivalent granular void ratio, e, from void ratio, e, and grading properties of the sand and fines has been adopted. It allows estimating the effects of fines on O by using experimental data on clean sand without the need of back-calculations. Key findings from this evaluation can be summarized as follows: By using e a single relationship between O /f e ) and p could be established irrespective of fines content. Note that the parameter b required for the calculation of e is not a constant but rather a function of f c. Importantly b is not determined by back calculations but rather using grading characteristics of sand-fines mixture D,d 5 and f thre ). Both functional forms of f e ) i.e. f H e ) and f J e )have similar accuracy and hence are applicable within the proposed methodology. O for any fines content below the f thre can be evaluated by a simple procedure using O for clean sand and the above mentioned grading characteristics. It is to be noted that the proposed prediction methodology is restricted to fines contents of less than the threshold value. The effect of particle shapes and mineralogy was not evaluated in this study. Acknowledgement The first author would like to acknowledge that the manuscript of this paper was prepared during his postdoctoral tenure on the project identification and mitigation of geotechnical hazards at University of Canterbury, New Zealand. He would also like to acknowledge that the prediction approach for equivalent granular void ratio framework used in this paper was developed during his PhD at University of New South Wales at ADFA, Australia. References Chien, L.K. and Oh, Y.N., 998. Influence on the shear modulus and damping ratio of hydraulic reclaimed soil in West Taiwan. International Journal of Offshore and Polar Engineering, 8 3), Chien, L.K. and Oh, Y.N., 22. Influence of fines content and initial shear stress on dynamic properties of hydraulic reclaimed soil. Canadian eotechnical Journal, 39 ), Drenevich, V.P. and Richart, F.E.J., 97. Dynamic prestraining of dry sand. Journal of Soil Mechanics and Foundation Division, ASCE, 96 2), Hardin, B. O The nature of stress-strain behaviour of soils. In: Earthquake Engineering and Soil Dynamics Proceedings of the ASCE eotechnical Engineering Division Specialty Conference, ASCE, 9-2 June, Pasadena, California, USA, 3 9. Hardin, B.O. and Richart, F.E.J., 963. Elastic wave velocities in granular soils. Journal of Soil Mechanics and Foundation Division, ASCE, 89 ), Iwasaki, T. and Tatsuoka, F Effects of grain size and grading on dynamic shear moduli of sands. Soils and Foundations, 73), Jamiolkowski, M., Leroueil, S., and Lo Presti, D. C. F., 99. Theme Lecture: Design parameters from theory to practice. In: The International Conference on eotechnical Engineering for Coastal Development Theory and Practice on Soft round eo-coast 9), 3-6 September, Coastal development Institute of technology, Yokohama, Japan, Vol. 2, Kramer, S.L., 996. eotechnical Earthquake Engineering. New Jersey: Prentice Hall. Mceary, R.K., 96. Mechanical packing of spherical particles. Journal of the American Ceramic Society, 44 ), Miura, S., et al., 23. Deformation strength evaluation of crushable volcanic soils by laboratory and in-situ testing. Soils and Foundations, 43 4),

8 226 M.M. Rahman et al. eomechanics and eoengineering 22.7: downloaded from Rahman, M.M., 29. Modelling the influence of fines on liquefaction behaviour. Thesis PhD). University of New South Wales at Australian Defence Force Academy, Australia. handle.unsw.edu.au/959.4/4392 Rahman, M.M. and Lo. S.R., 27. On intergranular void ratio of loose sand with small amount of fines. In: K. Yee et al. 6th South East Asian eotechnical Conference, Eds.8-May,SEAS,Kuala Lumpur, Malaysia Rahman, M.M. and Lo, S.R., 28a. Effect of sand gradation and fines type on the liquefaction behaviour of sand-fines mixtures. In: The 4th Decennial geotechnical earthquake engineering and soil dynamics conference, SP-8, ASCE, 8-22 May, ASCE, Sacramento, California, USA.. doi:.6/ )9. Rahman, M.M. and Lo, S.R., 28b. The prediction of equivalent granular steady state line of loose sand with fines. eomechanics and eoengineering, 3 3), doi:.8/ Rahman, M.M. and Lo, S.R. 2a. Instability behaviour of sandy soils. In: eo-frontiers 2: Advances in eotechnical Engineering, J. Han, and D. E. Alzamora, eds. SP 2), ASCE, Dallas, TX, USA, doi:.6/465397)367. Rahman, M.M. and Lo, S.R. 2b. Equivalent state parameter for loose sand with fines. Acta eotechnica, Springer, Online. doi:.7/s Rahman, M.M., et al., 28. On equivalent granular void ratio and steady state behaviour of loose sand with fines. Canadian eotechnical Journal, 45 ), doi:.39/t8-64. Rahman, M.M., et al., 29. Reply to discussion by Wanatowski, D. and Chu, J. on - On equivalent granular void ratio and steady state behaviour of loose sand with fines. Canadian eotechnical Journal, 46 4), doi:.39/t9-25. Salgado, R., et al., 2. Shear strength and stiffness of silty sand. Journal of eotechnical and eoenvironmental Engineering, 26 5), Tatsuoka, F. and Shibuya, S., 99. Deformation characteristics of soils and rocks from field and laboratory tests. The 9th Asian regional conference on soil mechanics and foundation engineering, Bangkok, Thailand, Vol. 2, 7. Thevanayagam, S., 999. Intergranular contact and shear modulus of non-plastic granular mixes. In: N. Jones and R. hanem, eds. 3th ASCE Conference on Engineering Mechanics. John Hopkins University, Meryland, USA, 6. Thevanayagam, S. and Liang, J., 2. Shear wave velocity relations for silty and gravely soils. In: S.Prakash,ed.4th international conference on soil dynamics & earthquake engineering. March 2, San Diego, CA, USA, 5. Thevanayagam, S., et al., 22. Undrained fragility of clean sands, silty sands, and sandy silts. Journal of eotechnical and eoenvironmental Engineering, 28 ), Appendix. Notations b active fraction of fines in force structure B R reduction factor D sand particle diameter d fines particle diameter D sand particle diameter at % finer d 5 fines particle diameter at 5% finer e void ratio e equivalent granular void ratio e r soil constant that depends on particle angularity f c fines content in decimal f thre threshold fines content in decimal O small strain shear modulus χ particle size ratio, χ D /d 5 r particle size ratio, r /χ) d 5 /D p mean effective stress, p σ +2σ 3 )/3 Reference stress, kpa