1 Introduction

Liquefaction is an undesirable phenomenon that occurs due to substantial or total loss of shear strength and stiffness of soil during a cyclic or monotonic loading. The loss of shear strength and stiffness of soil may cause because of reduction in effective stress of soil deposit under cyclic loading condition. The liquefaction is dependent on various parameters that vary from one place to another like relative density, soil type, water table depth, and topography, etc. It has generally been observed that loose saturated cohesionless soil deposits are highly susceptible to liquefaction; however, the presence of fines in the sand has always shown a muddling behaviour under seismic loading condition. The soil can be classified as “sand-like” and “clay-like” soil based on its cyclic behaviour [1, 2]. The ‘sand-like’ soils are gravels, sands, and low plastic silts, whereas ‘clay-like’ soils are clays and plastic silts. The ‘sand-like’ soils can experience liquefaction, and its behaviour has become the area of research for the last few decades. The clay-like soils can also experience cyclic failure and significant ground deformation during an earthquake [3, 4].

Case studies from previous earthquakes (e.g., Haicheng in 1975; Tangshan in 1976) indicate that soil having clay content less than 20% had liquefied. Moreover, the field investigation after the Tokachioki earthquake by [5] showed that the presence of a higher proportion of plastic fines increased the liquefaction resistance [6] of sand layers. Therefore, the main focus of this study was to quantify the effect of non-plastic silts intruded in the sand in terms of cyclic behaviour under cyclic loading condition. In the last couple of decades, the impact of non-plastic fines in the sand has been studied by various researchers [7,8,9,10,11,12]. A summary of significant literature to quantify the effect of fines content (FC) on liquefaction resistance (LR) using cyclic triaxial tests are tabulated in Table 1 [9, 11, 13,14,15,16,17,18,19,20,21,22,23, 35, 47].

Table 1 Summary of significant literature showing the requirement of the present study
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There are many factors such as sample preparation technique, confining pressure, loading frequency, global void ratio, intergranular void ratio and type of test (stress- or strain-controlled) that control the outcomes. It can be observed from the literature that no conclusive remark can be drawn from the previous studies on cyclic/monotonic load response of sand-silt mixture. Therefore, due to divergence outcomes of previous studies, the cyclic response of sand impinged by the silt of different percentages (0%, 5%, 10%, 20%, 30% and 100%) by weight under cyclic loading condition has been studied in this research. The main objective was to inspect the cyclic behaviour of locally available sands, i.e. Sone and Ganga sand at different silt content. It has been done through strain-controlled undrained cyclic triaxial tests. A part of the study on Ganga sand mixed with non-plastic silt has already been presented by [15]. A comparison between the cyclic response of Sone and Ganga sand intruded by silt based on mean grain diameter ratio (D50 sand/D50 silt) has been presented at ± 0.5 mm axial displacement amplitude.

Further investigation has been carried out to find out the effect of the applied cyclic strain level. The results show that at higher strain levels (0.66–1.31%) the rate of generation of excess pore water pressure (EPWP) for all three soil specimens are more or less same. Liquefaction strength of a soil depends on the amount and rate of EPWP development and can be estimated using a strain-based mathematical model proposed by [24]. This model calculates the EPWP ratio as a function of the number of loading cycles. The EPWP model parameters (F is one of them) in this model depend on grain size, saturation condition, and void ratio, as documented in the literature [25, 26]. Another novel aspect studied here is the effect of non-plastic silt content on EPWP model parameter F. A graphical relation has been presented for EPWP model parameter F with the percentage of the silt present in the sand matrix.

2 Material Used

The soil used in this study is locally available sand collected from the alluvial deposit. The geology of the region is greatly influenced by the fine sands deposited by Sone and Ganga river [27]. Particle size distribution and specific gravity confirming to IS: 2720 (part-4) [28] and IS: 2720 (part-3) [29] respectively have been carried out. As shown in Fig. 1, the Sone and Ganga river sand get classified into poorly graded, and the silt particle that is being used is having the smallest size of around 10 µm. The other important characteristics related to particle size distribution have been tabulated in Table 2. The specific gravity for all the soil samples was found to be lying in between 2.67 and 2.72. The maximum and minimum dry density of soil is not unique as it is very much dependent on the method used for its determination [30]. The maximum dry densities for all the soil samples were measured using heavy compaction test confirming to IS: 2720 (part-8) [31]. Dry density obtained using heavy compaction yielded little bit higher values than that obtained using the vibratory table. For the minimum dry density of soil samples, ASTM D 4254-16 [32] method C has been used in this study. Using the relationship between dry density and void ratio, maximum and minimum void ratio of all soil (Sone and Ganga sand intruded with various percentages of silt) samples were obtained and is shown in Fig. 2. Consistency limit of silt confirming to IS: 2720 (part-5) [33] was obtained, and the test results suggest that the liquid limit value for the silt that is being used is less than 2%. Furthermore, it was observed that molding the sample into a thread of 3.2 mm was not possible because of very low plasticity. Hence, it has been ascertained that the silt used in the study is non-plastic. Silt used in this study is collected from the top layer of IIT Patna soil. Since the particle size of Ganga sand was smaller than Sone sand hence the presence of space for accommodating silt particles would also be lesser than the Sone sand matrix. The X-ray diffraction (XRD) patterns of various soils obtained from the XRD analysis are shown in Fig. 3. From the XRD analysis, it has been observed that minerals such as quartz, calcite, smectite and plagioclase are present in Ganga sand. Apart from these above mentioned minerals, actinolite is also present in non-plastic silt and Sone sand.

Fig. 1
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Particle size distribution curves

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Table 2 Properties of the Sone and Ganga sand used in the study
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Fig. 2
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Maximum (emax) and minimum (emin) void ratio for Sone and Ganga sand at different percentage of silt

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Fig. 3
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X-ray diffraction pattern and microscopic view of various soils obtained from the XRD and microscopic analysis (magnification factor = 20) a Sone sand, b Ganga sand and c Silt

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3 Experimental Scheme

The cyclic response of Sone and Ganga sand intruded with the different percentages by weight of non-plastic silt has been studied here using the strain-controlled cyclic triaxial test. A series of 18 strain-controlled cyclic triaxial tests were performed at the soil dynamics lab of IIT Patna using the hydraulic controlled cyclic triaxial testing equipment (shown in Fig. 4). These tests were conducted with various combinations of silt percentage mixed with natural Sone or Ganga river sand (details of these experiments are shown in Table 3). All tests were performed in undrained conditions on isotropically consolidated specimens, as per the guidelines provided in ASTM D-5311 [34].

Fig.4
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Cyclic triaxial apparatus used in this study

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Table 3 Details of cyclic triaxial experiments performed in this study (effective confining pressure = 150 kPa)
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3.1 Specimen Preparation

There are several methods for soil specimen preparation, e.g., dry air pluviation, dry and moist tamping and wet tamping etc. According to [35], a higher degree of uniformity can be observed in soil specimen prepared using dry deposition technique, while moist tamping method yields considerable non-uniformity. The steady-state line was nearly the same for the silty sand prepared using dry pluviation and moist tamping method [36, 37]. Researchers [35, 38] suggested that the specimen prepared using a dry deposition technique replicates the natural deposition of sand. Because of these recommendations and findings, the dry air pluviation method has been adopted in this study as it maintains uniformity in soil specimen. The soil samples were prepared in a split mould of 38 mm diameter and 76 mm height at low relative density (Dr). The soil samples were prepared with an almost similar degree of compactness (Loose, Dr was between 10 and 25%) for each test. The effect of silt intrusion in sandy deposits has been studied in the literature by various approaches (e.g., constant gross void ratio, constant sand skeleton or intergranular void ratio, constant relative density, and limiting fine contents). The seismic behaviour of silty sand has been studied here at a constant relative density of the soil sample. As more and more silts are added to the sand, it passes from the sand dominated matrix to silt dominated matrix. The limiting fines content (LFC) is a threshold value over which further addition of silt in the sand matrix led to loosing of intergranular contacts between sand particles. The schematic demonstration of particle arrangement in the sand-silt matrix is shown in Fig. 5. The LFC is very much dependent on particle shape and size and is a unique value for every soil [20]. Using the formula given by [20], the LFC was calculated for both Sone and Ganga river sand and found to be 31 and 29, respectively.

Fig. 5
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Schematic demonstration of particle arrangement in sand-silt matrix

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3.2 Sample Saturation and Consolidation

After depositing the soil inside the split mold, the sample cap having a screwed piston rod was placed over it. A suction of 10 kPa was applied to the sample before removing the split mold. Then the cell chamber was attached, which was filled with water to apply the desired (150 kPa) confining pressure. The pressure difference between confining and back pressure value was kept constant (i.e. 20 kPa) during saturation. The degree of saturation was estimated using the Skempton’s B parameter after every increment of confining pressure of 20 kPa. Once the desired degree of saturation (greater than 99%) was achieved, the soil specimens were isotropically consolidated to an effecting confining pressure of 150 kPa. At the time of consolidation, the volume change was measured from the volume of water, leaving the system through the back pressure line and getting collected into the volume change indicator device. The soil samples were allowed for consolidation until the volume change has become almost constant. The pre- and post-consolidation relative density of Sone and Ganga sand at different silt percentage are shown in Fig. 6.

Fig. 6
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Pre- and post-consolidation relative density of Sone river sand specimens (at axial strain amplitude of ± 0.5 mm) at different percentages of silt

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3.3 Cyclic Loading

Earthquake loading is highly irregular and is applied to a soil deposit in all three directions. Most of the laboratory cyclic triaxial testing generally apply unidirectional uniform cyclic loadings under stress-controlled or strain-controlled condition. Previous studies claimed that the results obtained from the stress-controlled test consist of unacceptable scattered test data and is very sensitive to sample disturbances, unlike in strain-controlled testing [39]. Cyclic shear strain (γ) is a fundamental parameter governing the cyclic response of sand [40,41,42]. According to Muley et al. [13] and Vucetic and Dobry [39], excess pore water pressure generation is controlled mainly by the level of induced shear strains. Hence, the present study has been focused on analyzing the behaviour of sand-silt mixture using strain-controlled cyclic triaxial tests. After attaining constant volume change, all the soil specimens were loaded at different cyclic axial strain displacement amplitudes [i.e., ± 0.1 mm (ε = 0.13%), ± 0.5 mm (ε = 0.66%) and ± 1 mm (ε = 1.31%)] with a frequency of 0.5 Hz. The cyclic shear strains corresponding to axial strain levels of 0.13%, 0.66% and 1.31re 0.2%, 0.99% and 1.97% respectively.

4 Test Results and Discussion

4.1 Effect of Soil Type on Liquefaction Strength

The comparison of liquefaction potential of Sone and Ganga sand has been carried out based on the mean grain size diameter ratio (D50 sand/D50 silt) at the axial strain of 0.66% (± 0.5 mm amplitude). Theoretically, it has been considered that a soil reaches initial liquefaction when the EPWP ratio (ru) becomes 1, but in practice, it is considered that the soil liquefies before the EPWP value reaches unity [43]. The liquefaction initiation is considered here when EPWP ratio reaches equal to 0.9. Figure 7 presents the liquefaction behaviour of Ganga and Sone sand mixed with various percentages of non-plastic silt. Figure 7a, b present the development of maximum EPWP ratio (ru max) [44] in each cycle of loading. The maximum EPWP signifies the maximum EPWP developed at a particular cycle, as shown in the inset figures. It is quite noticeable that the soil sample with 100% silt has the highest rate of generation of pore water pressure, which is then followed by 30% silt soil sample in both the cases. It is also interesting to note that a lower rate of EPWP development is observed for specimens prepared using 10% silt in both Sone and Ganga river sand. The specimens with 5% and 20% silt content shown more or less similar behaviour in both cases. However, in terms of the overall scenario, it can be ascertained that the rate of development of EPWP was higher in Ganga sand specimens as compared to Sone sand specimens. To demonstrate the behaviour of soil specimen when it liquefies deviatoric stress and axial strain curves has been shown in Fig. 7c, d for Ganga and Sone sand mixed with 20% silt. From the results, it can be inferred that soils have been liquefied with in few cycles.

Fig. 7
figure 7

Maximum EPWP ratio (ru max) developed at each cycles of loading for axial strain of 0.66% in a Sone sand and b Ganga sand; deviatoric stress variation with strain for each cycles of loading in c Sone sand with 20% silt and d Ganga sand with 20% silt

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Alternatively, the same can be ascertained from Fig. 8. Figure 8 delineates the number of cycles at which the soil specimens liquefied. The very first observation is that all the specimens are getting liquefied within five loading cycles only, and this is because the samples were prepared at low relative density. It can be easily depicted that the number of cycles required for liquefaction occurrence in Ganga sand specimen started decreasing after increasing the silt content beyond 10% silt in the sample. However, the liquefaction resistance remained the same for samples with 5% silt to 20% silt content in Sone sand and beyond that, the liquefaction potential of specimen increased. The similar behaviour has also been observed by [13] on Saloni river sand impinged with different percentages of silt at an axial strain of 0.75%

Fig. 8
figure 8

No. of cycles of loading requisite for liquefaction due to change in silt proportion at axial strain amplitude of ± 0.5 mm

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A couple of previous studies in the literature documented that the soil having a lower D50 sand/D50 silt ratio would possess higher liquefaction potential at a lower percentage of silt content [22, 23]. As presented in Table 2, the D50 sand/D50 silt ratio was higher for Sone sand. It is evident from Fig. 8 that for every silt-sand mixture, the samples of Ganga sand-silt mixture liquefied earlier than that of Sone sand-silt mixture. As the D50 sand/D50 silt ratio of Ganga sand was smaller than Sone sand, due to which the silt particles found it difficult to fit themselves in the intergranular voids of Ganga sand grains. This peculiar characteristic of Ganga sand results in pushing of its grains apart from each other, which in turn reduces the liquefaction resistance at even lower fines content. The results obtained in this study gives very good affirmation to the findings of Monkul and Yamamuro [23] because at the same relative density, same confining pressure and same fine content, the liquefaction susceptibility is higher for the soil having less D50 sand/D50 silt ratio.

4.2 Effect of Silt Percentage on EPWP Model Parameter F

As discussed in the previous subsection, the liquefaction strength of studied natural soil depends on the strain amplitude and the silt percentage present in that soil. Dobry et al. [40] proposed a mathematical model for EPWP generation in the soil which was based on the observation from a series of strain-controlled cyclic triaxial tests. The proposed equation is as follow:

$$\frac{1}{{r_{{\text{u}}} }} = \frac{1}{p} + \frac{1}{{pFfN\left( {\gamma_{{\text{c}}} - \gamma_{{{\text{tvc}}}} } \right)^{s} }},$$
(1)

where p, F and s are the model parameters, f = 1 or 2 depending on whether the loading is one or two-directional, γc = cyclic shear strain amplitude and γtvc = threshold cyclic shear strain amplitude. Based on the literature, it has been observed that the reciprocals of maximum EPWP ratio and the number of cycles followed a linear relationship between them [25]. The Eq. (1) can be represented using the following expression:

$$\frac{1}{{r_{{\text{u}}} }} = x\frac{1}{N} + y.$$
(2)

In the modified equation, ru is the maximum EPWP ratio in each loading cycle, and N is the number of loading cycles. The intercept y is almost equal to 1 as the maximum EPWP ratio development in all the cyclic triaxial tests reaches close to unity at the end of the test. So, the parameter p, which is reciprocal of y is also close to 1. However, the slope x is different for different strain amplitudes and effective confining stresses. In the present study, the loading was one-dimensional (1-D), so f was considered as 1, the parameter s also considered as a unity, and the threshold shear strain value was considered as 0.015%.

4.2.1 Variation of the Slope of 1/r u and 1/N Plot at Different Silt Content

Figure 9 presents the result of strain-controlled cyclic triaxial tests performed on silt intruded Sone and Ganga sand in terms of 1/N and 1/ru for 0.66% axial strain. At a particular shear strain with a y-intercept of 1/p an inverse slope of \(g\left( {\gamma_{{\text{c}}} } \right)\). exists in the figure (as shown in Fig. 9), where \(g\left( {\gamma_{{\text{c}}} } \right) = F\left( {\gamma_{{\text{c}}} - \gamma_{{{\text{tvc}}}} } \right)^{s}\). The slope has become lowest until the silt content reaches 10% of the total weight of sand in the mixture and increases after that for further increases in the silt content. It is because the silt particles are filling inside the voids of sand skeleton initially. Due to the lower voids, less amount of EPWP has been develop. After a certain percentage (10% in this study) of silt intruded in the sand matrix, sand particles lose contact and start floating in the silt matrix. The variation of the slope with the increase in silt content for various axial train levels has been presented in Fig. 10. It has been observed that this change in the slope due to change in the silt content is less at lower shear strain and more at higher shear strain (shown in Fig. 10). At higher strain, EPWP developed very quickly, which causes a rapid increase in the slope (also observed in [25]).

Fig. 9
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Relationship between 1/ru and 1/N for different percentage of silt intruded (ε = 0.66%) in a Sone sand, b Ganga sand

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Fig. 10
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Variation of the slope of 1/ru and 1/N plot with the variation in the silt content at different cyclic shear strain for Sone sand

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4.2.2 Variation of EPWP Model Parameter F Depending on the Silt Content for Various Sands

The EPWP model parameter F also has been followed a similar trend, as shown in Fig. 11. Tests have been performed at intermediate silt contents also at a shear strain level of 0.99% to identify the variation of EPWP model parameter F. It has been observed from results that for 0.99% shear strain, it decreases up to 10% silt content and increases after that. Further, the variation of EPWP model parameter F has been compared for Ganga sand also (as shown in Fig. 12) for an applied shear strain of 0.99%. It can be observed from the figure that for both Ganga and Sone sand, the EPWP model parameter F decreases initially with the increase in the silt content up to 10% and then increases. The possible reason behind it could be the silt dominated soil skeleton formation at higher silt contents in the soil mixture as discussed in Fig. 5 and also by [45].

Fig. 11
figure 11

Variation of the EPWP model parameter F with the variation in silt content at different shear strain for Sone sand

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Fig. 12
figure 12

Variation of the EPWP model parameter F with the variation in the silt content for different sands at large strain (ε = 0.66%)

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4.3 Effect of Strain Rate

To evaluate the effect of strain rate on cyclic behaviour of sand-silt mixture, the analysis on Sone sand and non-plastic silt mixed samples has been carried out at two other axial displacement amplitudes also, i.e., at ± 0.1 mm and ± 1 mm. The liquefaction resistance of Sone sand has started decreasing after 20% silt content (as shown in Fig. 8) and which is considered here as threshold silt content. Hence to rationalize, the tests were carried out on three specimens (Sone sand with 0%, 20% and 100% silt contents). Table 4 presents the number of loading cycles at which ru max = 0.9 developed at axial displacement amplitudes of 0.1 mm, 0.5 mm and 1 mm. It is evident from the table that soil specimen with 100% non-plastic silt is highly susceptible to liquefaction. However, it has also been observed that at higher axial strain levels (0.66% and 1.31%) the number of cycles required for ru max = 0.9 is almost the same for all three specimens.

Table 4 No. of cycles required for liquefaction at various axial strain levels for Sone sand
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Jiaer et al. [43] have discussed that shear strain (\(\gamma\)) serves as a very good criterion for seismic performance assessment purposes and the relation between shear and axial strain in undrained triaxial tests is “\(\gamma = 1.5\times\epsilon\)”. Thus in an attempt to showcase the results in terms of shear strain, Fig. 13 portrays the generation of maximum EPWP ratio in each loading cycle for the soil samples tested at three different shear strains (0.2%, 0.99% and 1.97%). It is quite obvious that a faster rate of EPWP generation has been observed for specimens tested at higher strain amplitudes, and the same can be observed from Fig. 13b, c. Although at higher strains, the ru max has become 0.9 within few loading cycles only. However, a contrasting rate of EPWP generation behaviour can be observed for the samples tested at a comparatively lower strain level (shown in Fig. 13a). It is because the soil samples have been prepared at very low relative densities (Dr = 10–25%) due to which they are unable to withstand for longer at higher shear strains. Vis-a-vis the effect of silt can be observed at lower shear strains, i.e., increasing silt content, the liquefaction resistance found to be decreasing.

Fig. 13
figure 13

Maximum EPWP ratio (ru max) versus number of loading cycles a ε = 0.13%, b ε = 0.66%, c ε = 1. 31%

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Hazirbaba and Rathje [46] documented that the rate of generation of EPWP for the sand specimen with 0% and 20% silt content at \(\gamma\) = 0.3% were quite identical which is also having very good agreement to the behavior observed in the present study as shown in Fig. 13a. While little discrepancy observed in the behaviour of sand specimen with 20% silt at \(\gamma\) = 0.2% with the study conducted by Hazirbaba and Rathje [46] can be justified by the presence of more percentage of finer silt particles in the latter case. A higher percentage of fines led to higher pore water pressure development per cycle. As can be observed from the particle size distribution curve shown in Fig. 1 that the particle size of silt used in the present study is ranging from 75 µm to about 20 µm while the silt used in the study conducted by Hazirbaba and Rathje [46] has a minimum particle size of 2 µm.

It is evident that despite varying strain levels, the development of pore water pressure with loading cycles was observed more in the soil specimen having 100% non-plastic silt. Thus it can be easily claimed now that the non-plastic fines are highly susceptible to liquefaction. On comparing clean sand and sand specimen with 20% non-plastic silt, the development of pore water pressure was observed more or less similar at a shear strain of 0.9% and 1.75%. However, a small discrepancy has been observed at a shear strain of 1.75% where the rate of pore water pressure generation is a little bit more than that of clean sand. This anomaly in seismic behaviour of sand can be occurred because, at higher strains, sudden large magnitudes of loads are applied. Sudden load application is more like an application of impact loads, and thus higher pore pressure generation takes place in the soil specimen with less percentage of voids.

5 Conclusion

A series of 18 strain-controlled cyclic triaxial tests were performed using the servo-hydraulic controlled cyclic triaxial testing equipment. The major distinction of this present study is modifying the EPWP model parameter F for non-plastic silty sands. The EPWP model parameter F has been expressed here as a percentage of silt present in two different sand matrixes. The cyclic behaviour of sand with varying percentage of non-plastic silt has been assessed and explained in terms of mean grain diameter ratio (D50 sand/D50 silt). Following conclusion can be drawn from this study:

  • Higher rate of excess pore water pressure generation has always been observed for the soil specimen prepared with 100% non-plastic silt at all strain levels. The rate of excess pore water pressure generation was initially found to decrease till 20% and 10% non-plastic silt content in Sone and Ganga sand, respectively. This behaviour was mainly observed because till this time the silt particle could accommodate themselves inside the sand matrix without effecting their intergranular contact.

  • Among Sone and Ganga sand, the liquefaction susceptibility of Ganga sand has been found higher than Sone sand at the same relative density, confining pressure, strain level and fines content. Hence, it can be stated that with the increase in D50 sand/D50 silt ratio, the liquefaction susceptibility of sand decreases.

  • While inspecting the behaviour possessed by Sone river sand at various axial strain levels (i.e., 0.13%, 0.66% and 1.31%), it was found that the effect of non-plastic silt intrusion in the sand could be observed at the lower strain level. The liquefaction resistance of sand decreases with the increase in non-plastic silt proportion at axial strain level of 0.13%. However, at higher axial strain amplitudes (0.66% and 1.31%) the number of cycles required for liquefaction was observed almost the same for the soil specimen with 0%, 20% and 100% non-plastic fines.

  • Based on the Dobry’s work, EPWP model parameter F has been modified here for various percentages of silt for two different sands (Sone and Ganga river sand). The EPWP model parameter F has been reduced initially (up to 10% silt) as a function of non-plastic silts percentage and increased after that.

It can be inferred from this study that the model parameter F can be modified depending on the percentage of silt particles present in the sand matrix. Further study is required to propose EPWP model parameter F for a wider range of silt percentages.