1 Introduction

In most geotechnical structures, soil elements are subjected to complex anisotropic conditions with different directions of principal stress and unequal principal stresses [1]. Therefore, soil’s response to loading will reflect its inherent anisotropic structure and, consequently, depend on the orientation and continuous rotation of the principal stresses and the intermediate principal stress parameter (b) that is observed in the construction of almost all types of geotechnical structures. Soil anisotropy is one of the most important parameters that influences soil behaviour [2,3,4]. Parameter α° is used to indicate the principal stress rotation or inclination angle, and studies carried out so far show a significant effect of inclination angle on the shear strength parameters of the soil [5, 6].

The triaxial apparatus and the hollow cylindrical apparatus (HCA) are used to investigate the anisotropic behaviour and the influence of the inclination angle (α°) in laboratory tests. The triaxial apparatus allows testing only at inclination angles (α°) equal to 0° and 90°, while the hollow cylindrical apparatus allows testing at any angle α°. Many experiments have been performed using the hollow cylindrical apparatus to show the effect of inclination angle on the behaviour of sand [7,8,9,10,11,12,13,14,15]. The effect of inclination angle has been studied on Toyoura sand [16]. When the inclination angle (α°) becomes larger with respect to the deposition direction, the behaviour clearly becomes strain-softening and shows more contractive behaviour. Based on this study, at the principal stress direction of 15°, 20% of the additional pore water pressure ratio was created, and the behaviour is still hardening, but under the principal stress direction of 75°, the pore water pressure increased to 90% and a strain-softening behaviour was observed. Similar results have been reported by some researchers [6, 9, 13, 17,18,19,20,21,22]. This significant dependence of the behaviour of sand on the principal stress direction indicates the inherent anisotropy in the sand, which reflects the environmental and geological conditions during soil deposition. However, the effect of inclination angle on the shear strength of soil is often neglected in laboratory tests related to geotechnical engineering projects because of the difficulty of achieving this phenomenon in laboratory conditions.

In engineering projects, soil consisting of a sand-fine grained mixture is often encountered [23,24,25,26,27]. These mixtures are created all over the world during natural sediments and filled soils in human activities. A good understanding of the behaviour of these soils requires in situ and laboratory studies. In recent years, the effect of fine grains (silt) on sand behaviour has been discussed. [28] stated that the presence of a small percentage of silt (7%) causes the reverse behaviour of Nevada sand. They argued that the major reason for this behaviour is the difference between the particle structure of silty sand and clean sand. According to this hypothesis, sand containing 7% fine silt is more stable than pure sand, and the liquefaction potential is improved. Therefore, even a low fine content may significantly affect the behaviour of sands [28,29,30,31]. On the other hand, a sand-silt mixture with a high silt content (40%) behaves differently than clean sand or a sand-silt mixture with a low silt content [32]. Several laboratory studies have been conducted on sands containing fine grains, and it seems that they are sometimes contradictory. Some studies reported a decrease in the shear strength with the increase in fine content [33,34,35]; other studies have reported an increase in the shear strength with the addition of fine content [31, 36].

Although several researchers have studied the effect of silt content on sand behaviours, they are generally concentrated in the range above 10%. The evaluation of field studies conducted after large earthquakes and case histories of actual soil behaviour in different places worldwide has shown that many soils have a low percentage of fine grains (< 10%) [31]. Therefore, it is necessary to study the effect of low fine content on the behaviour of sand, especially in anisotropic conditions.

However, there are limited studies on the effect of silt on the anisotropic behaviour of sand, and such studies have also focused on high fine content. [18] conducted a comprehensive study using a torsional shear hollow cylindrical apparatus on Firoozkuh silica sand and different percentages of silt (15%, 30% and 70%). The value of α° varied between 15° and 75°. Their results showed that adding different silt content to the host sand increases the strain-softening behaviour and decreases the effect of anisotropy. In another study on Hamadan and Tehran sand [37], similar results were reported. Both studies focussed on high silt content, and the fine percent variation was very high.

On the other hand, in the studies of soils containing fine grains, it is observed that with increasing the amount of silt, the global void ratio (\(e\)) of the specimens is decreased, and as a result, the relative density (\({D}_{r}\)) will be increased, but this trend of increasing density does not lead to an increase in the strength of the specimen. Therefore, in order to create a comparative basis for the specimens, the equivalent intergranular void ratio (\({e}_{g(eq)}\)) parameter and the new concept of the equivalent intergranular relative density (\({D}_{r}^{*}\)) are utilized.

Based on the cases mentioned, this research studied the effect of low silt content (< 10%) on the inherent anisotropy of sand based on torsional shear undrained hollow cylinder tests and provided several interpretations of the mechanisms. Tests were performed on Firoozkuh sand with (0%, 3%, 5%, 7% and 10%) silt contents. In these tests, the inclination angle (α°) values are changed from 15° to 60°.

2 Materials Description, Testing Apparatus and Methods

2.1 Materials Description

Uniformly graded Firoozkuh sand called F161, which has a golden yellow colour and medium angular grains, was selected as the test material (Fig. 1a). Figure 1c illustrates that the gradation curve of the used sand is located inside the limits for the most liquefiable soil, which indicates the high liquefaction susceptibility of the sand. Based on laboratory observations using a microscope, this sand contains crushed and rounded particles (Fig. 2). Firoozkuh silt was utilized as fines in this research. Firoozkuh silt is brown in colour (Fig. 1b) with homogenous grading and can be considered non-plastic because its plasticity index is less than 5%. The physical properties of soils used for this study are summarized in Table 1.

Fig. 1
figure 1

Materials used in this study: a Firoozkuh sand -F161, b Firoozkuh silt, c Grain size distribution curves of tested materials

Full size image
Fig. 2
figure 2

Scanning electron microscopic (SEM) images (Firoozkuh sand and silt)

Full size image
Table 1 Physical properties of tested materials
Full size table

2.2 Testing Apparatus (Torsional Shear Hollow Cylindrical Apparatus)

Soil behaviour basically depends on the applied stress path, which includes the rotation of the principal stress direction. It is not possible to control the principal stress directions in a normal triaxial shearing apparatus. With a torsional shear hollow cylindrical apparatus, the three principal stresses can be individually controlled by varying the applied stresses and torque. Therefore, it is possible to better investigate the inherent anisotropic behaviour of the soil and its effects on the stress–strain and stress path curves. Figure 3 illustrates the hollow cylindrical apparatus of Urmia University used in this study. With this apparatus, torsional strain control is programmed using a direct current motor to attain the post-peak behaviour. A performed torsional force speed of 0.5°/min was applied in the tests.

Fig. 3
figure 3

Hollow cylindrical apparatus: a HCA apparatus of Urmia University with a specimen in the test, b Schematic view of torsional shearing hollow cylindrical (HCA)

Full size image

In order to study the effect of inherent anisotropy, α and b are kept constant (to reach the desired stress paths) during torsional shear, the general equations of hollow cylindrical apparatus are summarised in Table 2. Geometric characteristics and stress conditions are illustrated in Fig. 4.

Table 2 General equations of hollow cylindrical apparatus (HCA) [18, 37]
Full size table
Fig. 4
figure 4

Geometric characteristics and stress conditions applied to HCA

Full size image

2.3 Equivalent Intergranular Relative Density

The relative density of sand (\({D}_{r}=\frac{{e}_{max}-e}{{e}_{max}-{e}_{min}}\)) is the effective factor in the general behaviour of soils [38], which indicates the specifications of the specimen at the end of the consolidation stage. By adding silt to the host sand, \({e}_{max}\) and \({e}_{min}\) are changed and relative density calculation is different from the host sand. Therefore, \({e}_{max}\) and \({e}_{min}\) should be obtained for different silt contents [32, 39, 40]. The obtained values (Firoozkuh sand–silt mixtures) are plotted in Fig. 5. ASTM-D4253 [41] and ASTM-D4254 [42] were used to measure these parameters. The minimum void ratio was decreased up to 30–35% of fines and then increased at higher silt percentages.

Fig. 5
figure 5

Minimum and maximum void ratios for the Firoozkuh sand–silt mixtures

Full size image

In sand-silt mixtures, two new concepts have been presented: the intergranular void ratio and the equivalent intergranular void ratio \({(e}_{g}\mathrm{ and }{e}_{g(eq)})\). These variables can be calculated using Eqs. (1) and (2). It should be noted that the mentioned parameters explain the behaviour of sand-silt mixtures [24, 43,44,45,46,47,48,49,50].

$${e}_{g}=\frac{e+(FC)}{1-(FC)}$$
(1)
$${e}_{g(eq)}=\frac{e+\left(1-\lambda \right)(FC)}{1-\left(1-\lambda \right)(FC)}$$
(2)

\(e\): Global void ratio; FC: Fines content.

Factor \(\lambda\) indicates the fraction of fine grains that contribute to the force structure. The value of \(\lambda\) ranges from 0 to 1. Fines completely act as voids when \(\lambda\) = 0. However, fines behave like the host sand when \(\lambda\) = 1. It should be noted that when using Eq. (2), the fine content must be less than the threshold fine content (FC < FCth). The presence of fine particles in a sand-silt mixture affects its overall structure. When fine grains are added to sand, two phases with a transition point are created. Below this transition point, the soil structure is generally sand with silt in the sand skeleton (fines in coarse). Meanwhile, beyond this point, due to the presence of enough fine grains, the sand grains have loose contact with each other, and hence, the mechanical response of the soil is dominated by fine particles (coarse in fines). This limiting value (transition point) is called the threshold fines content (FCth) or limiting fines content (LFC) [51, 52]. The reported values of FCth in the literature range between 30 and 40% for most mixed soils [53]. The threshold fine content for the Firoouzkuh sand-silt mixture is about 30% (Fig. 5).

Two methods have been proposed to determine the value of \(\lambda\). Some researchers have recommended determining this parameter by back-calculation [44, 45, 53,54,55], while others researchers have suggested using the provided equations [51, 56]. In this research, the back-calculation method was adopted to calculate \(\lambda\) values based on the results of experimental tests in order to evaluate the effect of silt grain performance. By using \(\lambda\), the equivalent intergranular void ratio and equivalent intergranular relative density (\({D}_{r}^{*}=\frac{{e}_{max}-{e}_{g(eq)}}{{e}_{max}-{e}_{min}}\)) were calculated. This parameter can be used as a comparative basis for the specimens. In the back-calculation method, the initial value is assigned to \(\lambda\) and the steady state data is calculated in terms of modified \({e}_{{\varvec{g}}({\varvec{e}}{\varvec{q}})}\) for each fine content. Then the mean squared error (MSE) of these data is calculated. By changing the value of \(\lambda\) from 0 to 1 in increments of 0.02, the change in MSE with the value of \(\lambda\) can be determined. The optimal \(\lambda\) value for mixed soil corresponds to the lowest MSE [53, 55], as shown in Fig. 6.

Fig. 6
figure 6

Example for determining the factor λ using test data for FC = 10%

Full size image

2.4 Sample Preparation and Experimental Procedure

There are different sample preparation methods for granular soils at a laboratory scale. In this study, the specimens were prepared in the laboratory with the dry deposition method. This method prepares more uniform silty sand specimens [18, 37, 57].

For the saturation stage, carbon dioxide (using a pressure of approximately 3 kPa) and de-aired water were passed through the specimen. The process of saturating the specimen was done by increasing the confining pressure in several steps and measuring the pore water pressure. The specimen was considered saturated when the Skempton’s parameter (B) was greater than 0.95, according to ASTM-D 4767–11 [58]. In our tests, the value of B exceeded 0.97 for all specimens. Note that saturation greatly affects the shear strength of soil, and as the saturation degree is decreased, the liquefaction resistance of sand is increased [59, 60]. The consolidation of the specimens was performed isotropic (P’c = 100 and 200 kPa) to avoid the induced anisotropy effect [18]. The shear stage (with a rate of torque speed of 0.5°/min) started after the consolidation stage. The void ratio of the specimens was measured at the end of the test. The dimensions of the specimen were 12 cm high, with inner and outer diameters of 6 cm and 10 cm. Figure 7 illustrates the prepared specimen.

Fig. 7
figure 7

Specimen: a Specimen preparation, b Specimen at the end of the test

Full size image

3 Results

The aim of this study was to investigate the effect of low silt content on the anisotropic properties of sand-silt mixtures. Thirty undrained shear tests (CU) were conducted on a Firoozkuh sand (F161) using confining stress of 100 and 200 kPa and silt content of 0 to 10%. The inclination angle (α°) varied from 15° to 60°. In all the tests, parameter b was 0.5. By increasing the amount of silt (up to 10%), the global void ratio (e) of the specimens was decreased from 0.738 (clean sand) to 0.69 at P’c = 100 kPa. As a result, the relative density (\({D}_{r}\)) was increased from 41.7% to 52.3%. The equivalent intergranular void ratio (\({e}_{g(eq)}\)) (0.738 for clean sand and 0.736 for sand with 10% silt) and the concept of the equivalent intergranular relative density (\({D}_{r}^{*}\)= 41.7% for clean sand and 41.6% for sand with 10% silt) were utilized in order to create a suitable comparative basis for the specimens. Information is given in Table 3. The test results are displayed in the form of stress path and stress–strain curves, and the comparison of the results was performed based on these curves using the classification of the undrained behaviour of sand (Fig. 8) [16].

Table 3 Reported list of conducted tests
Full size table
Fig. 8
figure 8

Classification of the undrained behaviour of sands [16]

Full size image

3.1 Effect of Inclination Angle (α o) on Clean Sand’s Behaviours

Figure 9 illustrates the results of two series of tests on clean sand (in the first series, P’c = 100 kPa; in the second series, P’c = 200 kPa). Base sand demonstrates a limited flow deformation with strain-softening and hardening behaviour throughout undrained shearing towards the ultimate steady state. This type of behaviour is observed in loose sands (according to Fig. 8), which is characterized by the initial maximum shear strength at a small strain. Then, the shear strength decreases to the minimum value at the medium strain, and after reaching the minimum shear strength, it increases to its maximum value, and the pore water pressure decreases to its minimum value at a large strain. The minimum shear strength is called the quasi-steady state point and is defined as the point when the undrained behaviour is changed from contractive to dilative.

Fig. 9
figure 9

Effect of inclination angle (α°) on the behaviour of clean sand (0% silt)

Full size image

In general, the inclination angle has a significant effect on the undrained shear strength of sands. The specimen at α = 15° showed the highest undrained peak shear strength (\({q}_{peak}\)=54 kPa) compared with α = 30° (\({q}_{peak}\)=42 kPa) and α = 60° (\({q}_{peak}\)=27.5 kPa) when subjected to P’c = 100 kPa for the same relative density (\({D}_{r}={D}_{r}^{*}\approx 42\)). Therefore, as the inclination angle (α°) became higher, the behaviour became softer and flow began to appear (Fig. 9), and the minimum strength occurs when α = 60°. Such strain-softening has been related to the inherent anisotropy in the sand fabric during deposition.

The entanglement between sand grains along the long axis parallel to the direction of the shear plane is the worst case. In other words, sand particles have the greatest tendency to slip on each other when the major principal stress is parallel to the direction of the longitudinal axis of the sand particles [22, 61]. This explanation can give a good reason for reducing the resistance of the specimen with an increase in α°. This behaviour has been described by other researchers [13, 16, 18].

One of the basic parameters that has a significant effect on the mechanical behaviour of soils is the initial confining stress (P’c). The peak strength is increased as the consolidation stress is increased from 100 to 200 kPa. Axial strain at peak strength is also increased by increasing P’c. For example, for α = 15° (compressional loading), the peak resistance reaches a value of 53 kPa at P’c = 100 kPa and 0.34% strain, whereas at P’c = 200 kPa and 0.37% strain, the peak resistance is increased to 92 kPa (Fig. 9).

3.2 Effect of Silt Content on the Mechanical Behaviour of Sand-Silt Mixtures

In Fig. 10, the effect of adding different percentages of silt (3%, 5%, 7%, and 10%) with an inclination angle of 15° (compressional loading) can be evaluated on the sand behaviour. Adding silt particles to the host sand up to 5% increases the peak shear strength of the specimen (18.5%, 12% and 7.7% for α = 15°, 30° and 60°, respectively). Then, the strength of the specimen decreases with the increase in silt content up to 10%. It should be noted that with a silt content of 10%, the strength of the specimen was lower than that of the host sand (about 12%). This behaviour was observed for both 100 and 200 kPa of initial confining stress. Similar results have been presented by other researchers, who observed increased strength in sands by increasing the amount of fine grains at low fine content (about 5%) [28, 31, 36]. At a low percentage of fine grains (up to 5%), the particles are placed in the voids between the grains; as a result, the excess pore water pressure is reduced during shear, which leads to an increase in shear strength and a decrease in the contractive behaviour of sand-silt mixtures. With the increase in fine content (from 5 to 10%), the process is reversed, the fine grains have a negative effect on the behaviour of the sand structure, and the behaviour becomes more contractive. It should be noted that, by adding low silt content, the overall structure of the sand remains the same and the specimens can be evaluated based on the general behaviour of the host sand.

Fig. 10
figure 10

Effect of silt content on the mechanical behaviour of sand’s mixture at α = 15°

Full size image

The same results can be achieved at different inclination angles, such as α = 30° and α = 60° (Figs. 11 and 12). These results admit the optimum strength theory at about 5 per cent silt content.

Fig. 11
figure 11

Effect of silt content on the mechanical behaviour of sand’s mixture at α = 30°

Full size image
Fig. 12
figure 12

Effect of silt content on the mechanical behaviour of sand’s mixture at α = 60°

Full size image

3.3 Combined Effect of Silt Content and Inclination Angle (α o) on the Mechanical Behaviour of Sand-Silt Mixture

The anisotropic behaviour of different silt percentages in the host sand was investigated by testing the sand-silt specimens under variable principal stress directions. The stress–strain and stress path curves of Firoozkuh sand with different silt contents at α = 30° and α = 60° (compressional torsional loading and extensional torsional loading, respectively) are shown in Figs. 11 and 12. By adding silt (up to 5%), the behaviour is stronger (same as the direction of 15°) and the shear strength of the specimen is increased under initial confining stresses of 100 and 200 kPa. However, the effect of silt particles on the behaviour of sand (increasing the strength and decreasing the contractive behaviour) is much greater at a lower inclination angle (α = 15°).

Figure 13 depicts the effect of increasing the peak strength of the mixed specimens compared to the peak strength of the host sand. Based on the results, it can be concluded that with the increase of α° in the anisotropic behaviour, the effect of silt on increasing the strength and decreasing the contractive behaviour of the specimens is decreased. For example, the increase in the peak strength of the specimen with 5% silt compared to pure sand at α = 15° is equal to 1.18, but this ratio is decreased to 1.07 at α = 60°, and it can be seen that with the increase of α, the influence of fine grains as an important parameter in sand-fines mixtures is decreased. The ratio of increasing the strength (by adding silt content) is the lowest at α = 60°. At higher percentages of silt (i.e., 10%) for all three inclination angles (15°, 30° and 60°), the behaviour of the specimens is more contractive than that of the host sand, and the strength of the specimen decreased by about 12%.

Fig. 13
figure 13

Comparison of the peak strength increase ratio of sand-silt mixture samples to the host sand at different principal stress directions

Full size image

3.4 Evaluation of Dimensionless State Indices

3.4.1 Effect of Silt Content on Steady-State Friction Angle

To determine the steady-state specifications according to Fig. 8 based on the laboratory results and drawn stress path, the slope of the steady state line (\({M}_{ss}\)) is calculated and then the steady-state friction angle (\({\varphi }_{ss}\)) can be calculated according to Eq. (3).

$${{\text{Sin}}(\varphi }_{ss})=\frac{\left(3{M}_{ss}\right)}{\left(6+{M}_{ss}\right)}$$
(3)
$${q}_{ss}={M}_{ss}. {p{\prime}}_{ss}$$

Based on the results, the steady-state friction angle (\({\varphi }_{ss}\)) increases slightly with the increase of silt as a result of creating a better structure with the combination of silt and sand. Such behaviour has also been reported by some other researchers [62]. The results are shown in Table 4.

Table 4 Steady state characteristics of sand-silt mixture
Full size table

3.4.2 Brittleness Index

Some scholars have used the brittleness index \({I}_{B}\) to describe the degree of strain softening and magnitude of the decrease in undrained shear strength [63], defined according to Eq. (4):

$${I}_{B}=\frac{{q}_{p}-{q}_{m}}{{q}_{p}}$$
(4)

where (\({q}_{p} {\text{and}}\) \({q}_{m}\)) are the peak shear strength prior to the quasi-steady state and minimum shear strength. IB can be considered a good index for the flow potential of contractive soils [64, 65]. The value of \({I}_{B}\) ranges from 0 to 1, and non-flow (non-brittle) behaviour is observed when \({I}_{B}\)= 0. However, complete static liquefaction (brittle behaviour) is associated with \({I}_{B}\)= 1. In a study conducted on Fraser River sand (very loose sand), a significant increase in the \({I}_{B}\) index was reported with an increase in the inclination angle in the samples [9].

As illustrated in Fig. 14, the brittleness index for mixtures is decreased when the confining pressure increases and is increased when the inclination angle increases, such that at α = 60°, the greatest increase in the index is observed. This increase in the brittleness index demonstrates that the liquefaction susceptibility of sandy soil is increased as the inclination angle increases, which is consistent with the results of other researchers [9, 18]. In silty samples, by increasing the percentage of silt up to 7% at α = 15°, the brittleness index is decreased and then increased. However, this turning point occurs with an increase of α (30° and 60°) at 5% of silt. In other words, increasing the fine grains increases the brittleness index and liquefaction sensitivity, which are more noticeable at higher inclination angles. Also, according to Fig. 14, it can be concluded that the inclination angle has a more significant effect on the brittleness index than the silt content.

Fig. 14
figure 14

Effect of principal stress direction (α°) and silt content on the brittleness index

Full size image

3.4.3 Peak Strength Index

This dimensionless parameter (\(q(peak)/{P}_{c}\mathrm{^{\prime}}\)) is defined to compare the strength of samples before softening while large deformation occurs. The values of the peak strength index for the tests conducted on Firoozkuh sand are shown in Fig. 15. With the increase of silt up to 5%, the peak strength index is increased (e.g., 0.54 for host sand and 0.64 for sand with 5% silt at P’c = 100). This index decreases with an increase in silt content up to 10%. It should be noted that with a silt content of 10%, the peak strength index was lower than that of the host sand.

Fig. 15
figure 15

Effect of principal stress direction (α°) and silt content on the peak strength index

Full size image

Regarding the effect of the inclination angle, it can be observed that the peak strength index decreases with the increase of the inclination angle (α°), and it has the smallest value at the inclination of 60°. On the other hand, with the increase of the inclination angle, the changes in this index are decreased, which indicates the reduction of the effect of fine grains at higher inclination angles (Fig. 15).

3.4.4 Anisotropy Ratio

Equation (5) indicates the anisotropy ratio. This parameter evaluates the effects of anisotropy of the specimens on different values of the inclination angle [66]. The values of the anisotropy ratio for the experiments conducted on Firoozkuh sand are shown in Fig. 16. According to the figure, with the increase of initial confining stress, the anisotropy ratio is decreased, and with the increase of silt content, the values of the anisotropy ratio are decreased. For example, the anisotropy ratio at α = 15 for the host sand is equal to 6.85, and the index is decreased to 6 with the increase of fine grains to 10% (P’c = 100).

$$AR=\frac{{q}_{ss}(\alpha )}{{q}_{ss}(\alpha ={60}^{o})}$$
(5)

where (\({q}_{ss}(\alpha^\circ )\)) is the shear strength of steady-state at α° and (\({q}_{ss}(\alpha ={60}^{o})\)) represents the shear strength of steady-state at α = 60°.

Fig. 16
figure 16

Effect of principal stress direction (α°) and silt content on the anisotropy ratio

Full size image

4 Conclusions

This paper presents an experimental investigation of the anisotropic stress–strain behaviour of sands with low silt content (< 10%) in undrained conditions using a torsional shear hollow cylinder apparatus (HCA). The main findings and conclusions of this article can be summarized as follows:

  • The behaviour of the examined sand is a strain softening-hardening behaviour, which becomes softer and more contractive with the increase of the inclination angle (α°), and the pore water pressure is increased during the undrained shear. Under the inclination angle of 60°, the behaviour of the specimen is strongly contractive and the greatest decrease can be observed in the strength of the sand specimen. However, in the laboratory tests related to geotechnical engineering projects, due to the difficulty of achieving this phenomenon in laboratory conditions, the effect of inclination angle on the shear strength of the soil is often ignored. This leads to the fact that in the calculation of the shear strength parameters of soil in geotechnical projects. Often, only one value of the shear strength parameters determined at the angle α = 0 is used. Considering the value of shear strength parameters determined at an angle of inclination angle (α°) for the entire subsoil often leads to overestimations of the true and incorrect values of the shear strength parameters of the subsoil.

  • The evaluation of field studies conducted after large earthquakes and case histories of actual soil behaviour in different places worldwide shows that many soils have a low percentage of fine grains (< 10%). Therefore, it is necessary to study the effect of low fine content on the behaviour of sand.

  • As silt content is increased, the global void ratio of the specimens is decreased, and as a result, the relative density will be increased, but this trend of increasing density does not lead to an increase in the strength of the specimen. Therefore, for a comparative basis of the specimens, equivalent intergranular void ratio (\({e}_{g(eq)}\)) and equivalent intergranular relative density (\({D}_{r}^{*}\)) are used. Based on the results of experimental tests, the back-calculation method was adopted to calculate the λ parameter involved in the relation of equivalent intergranular relative density in order to evaluate the effect of silt grain performance.

  • By adding a low percentage of non-plastic fine grains (up to 5%), the particles are placed in the voids between the grains so the excess pore water pressure is reduced during shear, which leads to an increase in shear strength and a decrease in the contractive behaviour of sand-silt mixtures. With the increase in fine content (from 5 to 10%), the process is reversed, the fine grains have a negative effect on the behaviour of the sand structure, and the behaviour becomes more contractive.

  • By adding low silt content, the overall structure of the sand remains the same, and the specimens can be evaluated based on the general behaviour of the host sand.

  • Under the principal stress direction of 15° (compressive loading) with the addition of 5% silt, the peak strength of the specimen reaches its maximum value (\({q}_{peak}\)=54 kPa) and the reduction of contractive behaviour is observed, but with the addition of more silt content, the process is decreased. At 7% silt, the behaviour is still more resistant compared to the host sand. Therefore, the value of 5% is the turning point of the specimen behaviour, which has been observed under initial confining stresses of 100 and 200 kPa. Nevertheless, with 10% silt content, the strength of the specimen is lower than the host sand.

  • The specimens are evaluated under the principal stress direction of 30° (compressional torsional loading) and 60° (extensional torsional loading) to investigate the simultaneous effect of inclination angle (α°) and silt content. Under these principal stress directions, the behaviour is more resistant than the host sand with the addition of silt up to 5%, such as 15° (18.5%, 12% and 7.7% for α = 15°, 30° and 60°, respectively), but the amount of this increase in the specimen’s strength has an inverse relationship with the increase in the inclination angle. In other words, it can be seen that with the increase of α, the influence of fine grains as an important parameter in sand-fines mixtures is decreased.

  • The brittleness index for mixtures is increased with the increase of the inclination angle such that at α = 60°, the greatest increase in the index is observed. This increase in the brittleness index demonstrates that the liquefaction susceptibility of sandy soil increases with increasing inclination angle. In silty samples, by increasing the percentage of silt up to 7% at α = 15°, the brittleness index is decreased and then increased. However, this turning point occurs with an increase of α° (30° and 60°) at 5% silt. In other words, increasing the fine grains increases the brittleness index and liquefaction sensitivity, which are more noticeable at a higher inclination angle. Also, it can be concluded that the inclination angle has a more significant effect on the brittleness index than the silt content.

  • In the assessment of the anisotropy ratio parameter, it is observed that the increase in initial confining stress causes a decrease in the anisotropy ratio. Moreover, with the increase in silt content, the index values of the anisotropy ratio decreased slightly. In other words, with the increase in silt content, the effect of the inclination angle decreases.