1 Introduction

Many of the geotechnical project sites in the lower Shivalik region of the Indian Himalayas often encounter coarse-grained materials, which include colluviums, fluvial, rockfills, and tailing. These coarse granular materials, termed as soil–rock mixture (S–RM), comprise a mixture of soils (clay, silt, and sand) and rock particles (gravel, cobbles, and boulders). As the proportion of rock particles in the S–RM increases, the shear strength behaviour becomes more complex and poses a great challenge to geotechnical engineers. The presence of rock fragments offers relatively higher frictional resistance during shear and enhances the shear strength. In the process of shearing, the rock particles may interlock, translate, and/or rotate with each other. Thus, the basic friction angle of parent rock particles, particles rearrangement, interlocking (i.e. dilatancy effect), normal stress level, and particle breakage play major role in governing the shear strength response of S–RM.

The determination of shear strength of S–RM in the field is difficult, time-consuming, and costly as testing warrants special large size equipment. As an alternative, different modelling techniques such as parallel gradation [23], scalping technique [32], generation of quadratic size distribution curve [10], and replacement technique [9] are preferred. These modelling techniques aim to scale down the maximum particle size to smaller sizes which can be tested in laboratory. However, the results obtained from modelled materials could potentially be inaccurate because of the inevitable size-dependent dilation and different mechanisms of particle crushing [14].

In addition to the above modelling techniques, a number of research studies have been conducted to investigate shear strength of S–RM response [1, 6,7,8, 11, 15,16,17,18, 20,21,22, 24, 25, 27, 28, 30, 31, 33]. The studies have indicated that as the rock (gravel) proportion increases in the mixture, the frictional resistance of S–RM increases linearly. Some of the researchers [15, 21, 24] have suggested empirical correlations to estimate the shear strength of soil–rock mixtures. In 1981, Barton and Kjaernsli (BK) [5] proposed a model for estimating nonlinear strength of rockfill materials. The present study envisages that there are many similarities between the shear strength response of S–RM and rockfill materials and hence BK model has great potential for modelling nonlinear shear strength behaviour of S–RM. An extensive laboratory study was planned and executed. Direct shear tests with shear box dimensions of 300 × 300 × 200 mm3 were performed on S–RM. Applicability of empirical correlation proposed by Irfan and Tang [15], Lindquist and Goodman [21], and Medley [24] was evaluated. Also, Barton and Kjaernsli [5] shear strength model for rockfills was evaluated for its applicability to S–RM. Modifications are suggested in the BK model to extend the model for predicting the shear strength of soil–rock mixtures.

2 Shear strength of S–RM

Performing laboratory tests on S–RM, especially with substantial gravel/cobble fraction, is difficult and sometimes not feasible also. As an alternative, two different approaches are available to estimate the shear strength behaviour of S–RM. Firstly, the shear strength of S–RM can be estimated using the empirical correlations proposed by the different authors, e.g. Irfan and Tang [15], Lindquist and Goodman [21], and Medley [24]. The friction angle of S–RM is expressed as

$${\phi }_{\mathrm{S}-\mathrm{RM}}={\phi }_{\mathrm{sf}}+\Delta \phi$$
(1)

where \({\phi }_{\mathrm{sf}}\) is friction angle of sand fines matrix and \(\Delta\phi\) is increase in friction angle due to addition of gravel content.

Secondly, there are shear strength models available for describing the shear strength of rockfill materials. One such model is Barton and Kjaernsli [5] model. Barton and Kjaernsli (BK) model was derived from Barton [3] model for rough rock joints. Barton and Kjaernsli [5] model is expressed as:

$${\tau }_{\mathrm{f}}={\sigma }_{\mathrm{n}}\mathrm{tan}\left[R {\mathrm{log}}_{10}\left(\frac{S}{{\sigma }_{\mathrm{n}}}\right)+{\phi }_{r}\right]$$
(2)

where \(\left[R {\mathrm{log}}_{10}\left(\frac{S}{{\sigma }_{\mathrm{n}}}\right)+{\phi }_{\mathrm{r}}\right]\) is the peak friction angle (ϕp) of rockfills, \({\tau }_{\mathrm{f}}\) is peak shear stress, σn is normal stress at the time of failure, \({\phi }_{\mathrm{r}}\) is the residual friction angle, R is equivalent roughness of rockfill, and S is the equivalent strength of rockfill particles.

Empirical charts as suggested by Barton and Kjaernsli [3] were used to obtain R and S values. R is a function of porosity of the rockfills, particle origin, roundedness, and smoothness. The parameter S is a function of mean particle size and uniaxial compressive strength (UCS) of rock materials.

3 Experimental programme

Triaxial, direct shear, and plane strain tests are commonly employed to study the shear strength behaviour of soil–rock mixtures containing large size rock particles. Direct shear test is simple and easy, and it is often used by many researchers to understand the shear strength behaviour of S–RM. The test has some limitations, e.g. plane of shear failure, is predetermined, and nonuniformity of stress may occur in the sample at large displacement. In spite of this, it is used widely to characterize geo-materials with wide range of particles. Based on the researcher’s experience, it was stated that a well-designed direct shear apparatus can successfully minimize any undesirable effect and provide reliable results [26].

Direct shear tests with two different shear box sizes have been performed on four different soil–rock mixtures. Conventional (60 × 60 × 25 mm3) direct shear tests were performed on sand fines matrix. Large size (300 × 300 × 200 mm3) direct shear tests were performed on S–RM samples. S–RM samples were prepared by adding different proportion of gravel content (i.e. rock particles) to sand fines matrix. Rock particles of angular phyllitic, dolomitic limestone, quartzitic limestone, and river bed material with sub-rounded particles were used in the present study. To estimate the characteristics of rock particles, specific gravity and UCS tests on rock particles were performed.

3.1 Materials and S–RM sample preparation

The brief description about each material type such as origin, shape and rock type and physical properties of samples is presented in this section.

Material type 1 comprised of debris material which was collected from a landslide site near Koteshwar dam in the state of Uttarakhand, India. The material comprises of phyllitic rock particles and sand fines matrix. The rock particles are flaky type and angular in shape. These rock particles are little elongated and more prone to particle breakage.

Material type 2 is river bed material (RBM) of fluvial deposit also collected from the landslide site located near Koteshwar dam, Uttarakhand, India. The material comprises of sub-rounded to rounded rock particles and sand fines matrix.

Material type 3 is colluvium collected from a landslide site located at national highway NH-92, near the town Mussoorie in the state of Uttarakhand, India. The material comprises of dolomitic limestone rock particles and sand fines matrix. The dolomitic rock particles are angular in shape.

Material type 4 was collected from a quarry site from western part of India. The material comprises of quartzitic limestone and disintegrated particles passing through 4.75-mm sieve. The shape of particle is categorized as angular with sharp edges.

The S–RM samples for direct shear tests were prepared with a mixture of gravels (G) and sand fines (SF) by adding different proportion of gravel content to the base soil, i.e. sand fines matrix. Figure 1 shows rock particles in size ranging from 4.75 to 40 mm used to prepare S–RM samples. The maximum rock particle size used in the sample was set to be one-sixth the shear box height. The S–RM samples are named as GXX_SFYY. ‘G’ indicates initial of gravels, ‘XX’ indicates the percentage of gravel content used in the S–RM sample, ‘SF’ indicates sand fines, and ‘YY’ indicates the percentage of sand fines quantity (e.g. G10_SF90). Figure 2 shows different particle size distribution for S–RM samples. The mean effective particle size (d50) was obtained from each gradation. The bulk density of 19.26 kN/m3 was maintained for all the tests while preparing the samples. All the samples have the relative densities > 80%. The unit weight of rock particles was determined and was used in calculating the porosity as per the IS: 2386 (Part-III) [13]. To estimate the uniaxial compressive strength (UCS) of individual rock particles, 20 pieces of irregular rock lumps were randomly selected from S–RM and point load strength index tests were performed. The basic friction angle for rock was obtained from the literature as suggested by Barton [3]. The physical characteristics of the different S–RM samples are summarized in Table 1.

Fig. 1
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Rock particles, in size range of 4.75–40 mm, of four different S–RM

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Fig. 2
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Particle size distribution curves for different gravel proportions mixed with sand fines matrix

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Table 1 Physical characteristics of the S–RM materials
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3.2 Conventional direct shear tests

Conventional direct shear tests were performed on sand fines matrix of soil–rock mixtures. The particles passing through 4.75 mm were used for preparing these samples with 5% moisture content. Test specimen was prepared by remoulding sample in the shear box to a density of 19.26 kN/m3. The specimen was prepared in three layers after compacting each layer to maintain uniform density. The dilation, horizontal displacement, and shear load were measured by the test. The rate of shearing is fixed to 1.25 mm/min. Sample was sheared along the predetermined shear plane. The shear load was recorded at regular interval of horizontal shear displacements till failure of the sample. Figure 3 shows typical results of direct shear tests presented in the form of shear stress vs. horizontal shear displacement plots and failure envelope (i.e. shear strength vs. normal stress). The summary of friction angle of sand fines matrix obtained from conventional direct shear tests is summarized in Table 2.

Fig. 3
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Typical results of conventional direct shear tests conducted on sand fines matrix

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Table 2 Summary of friction angle obtained from conventional direct shear tests
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3.3 Large size direct shear test setup and testing procedure

Large size direct shear test apparatus consists of a square shear box of 300 × 300 × 200 mm3. The shear box is an assembly of two halves, top and bottom, and they are screwed together to ensure proper alignment and also to arrest the movement while preparing samples. A plain grid plate over the base plate is placed inside the lower shear box. Another plain grid plate is placed on top of the sample, and then a top base plate which is having a lading cap is placed in the shear box. The lever arm arrangement to transfer the normal load on the sample is aligned properly. The dilation and horizontal shear displacements are measured by LVDTs. A data acquisition system is used to record the data, i.e. shear load and dilation corresponding to horizontal shear displacement (Fig. 4).

Fig. 4
figure 4

Large size direct shear test apparatus setup

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The S–RM samples were prepared inside the direct shear box carefully to maintain the density. The whole S–RM sample was placed in three layers of equal thickness and weight, and mechanical compaction after each layer was done using steel rammer of 24 N weight to obtain the desired density. Pre-decided normal stress was applied on the top plate of the shear box, and the sample was allowed to shear at 1.25 mm/min until the failure of the sample. The shear load and dilation readings were recorded corresponding to horizontal displacements at regular intervals for each test.

3.3.1 Direct shear test results and discussion

A total of 120 large size shear tests on four different types of S–RM samples comprising varying individual rock particles from very angular to sub-rounded were performed. For each S–RM sample, a minimum of five direct shear tests was carried out at normal stress levels of 5, 25, 50, 100, and 150 kPa, respectively. The physical characteristics of the S–RM samples are summarized in Table 1 along with proportion of gravel content and sand fines matrix proportion.

For each test, detail record of shear load and dilation readings was obtained at regular intervals of horizontal shear displacements. The tests data were used to calculate the shear and normal stresses. Area correction was applied while computing shear stress (τ) and normal stress (σn). The typical results of direct shear tests conducted on S–RM sample with 50% gravel fraction are presented in Fig. 5. The test results are presented in the form of stress ratio (τ/σn) and dilation vs horizontal shear displacement plots. The maximum stress that sample can withstand is the peak shear stress (\({\tau }_{\mathrm{f}}\)), called as failure shear stress. At low normal stress levels, the shear stress versus displacement plots show clearly defined peak at failure, while for high stress levels, no distinct peak was generally observed. For the tests where no clearly define peak was observed, failure has been considered corresponding to 13% horizontal shear displacement. From the results it is observed that the shear strength of soil–rock mixtures increases gradually with increase in normal stress and percentage of gravel content. From the dilation vs. horizontal shear displacement plots (Fig. 5), peak dilation was found to coincide with the peak value of shear stress.

Fig. 5
figure 5

Large size direct shear testing results of the soil–rock mixture with 50% gravel content

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3.3.2 Effect of gravel content on the shear strength

Literature suggests that gravel content is one of the most influencing parameter governing the shear strength of deposits like S–RM. The effect of gravel content on the shear strength behaviour of S–RM for the present study can be examined from Fig. 6. The results show that the percentage of gravel content has a significant effect on the shear strength behaviour of S–RM. For the normal stress levels of above 100 kPa, a considerable increase in the shear strength was observed with increase in gravel content for all the material types. For the normal stress level of 5–50 kPa, the increase in the shear strength with increase in gravel content is very small.

Fig. 6
figure 6

The relationship between the shear strength and percentage of gravel content

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Under the stress levels of 100–150 kPa, the role of rock particles on the shear strength behaviour is substantial. While shearing, the sample of S–RM undergoes compression especially at high stress levels and then individual rock particles govern the shear strength behaviour due to the contact forces among rock particles. This behaviour is obvious when the percentage of gravel content is above 50% due to large number of inter-particle contacts. Hence, the shear strength behaviour of S–RM becomes more sensitive to the characteristics of individual rock particles. In the material type 1, there is a gradual increase in shear strength with increasing gravel content. In the material type 2, when the normal stress level is 100 kPa the shear strength gradually increases upto 50% of gravel content and then becomes almost constant with the gravel content above 50%. At 150 kPa, the shear strength is almost constant for the gravel content upto 35%, increases gradually for the gravel content 35% to 50%, and becomes constant beyond the 50% of gravel content. In material type 3, the shear strength behaviour increases linearly for the gravel content upto 50% and then becomes almost constant for the gravel content above 50%, while for stress level of 150 kPa, the rate of increase in shear strength is more prominent while shearing the sample. In the material type 4, the shear strength increases upto gravel content of 50% and then becomes constant. The fluctuations in the shear strength were due to the breakage of sharp edges while shearing as the rock particles are very angular.

4 Modelling of shear strength of S–RM

Shear strength behaviour of S–RM can be modelled through Mohr–Coulomb shear strength parameters (i.e. cohesion and friction angle) or peak friction angle (i.e. secant friction angle). For a true representation of shear strength behaviour, shear strength test would be required on representative samples of S–RM from the field. However, due to presence of large size particles and their wide variations, it is very difficult to test the field samples. The empirical methods are much easier to assess the shear strength parameters. Therefore, the present study focussed to evaluate the empirical correlations including BK model with an objective to predict the shear strength of S–RM.

4.1 Applicability of existing empirical correlations

According to previous studies [15, 21, 24], the shear strength of S–RM increases linearly (Eq. 1) with rock content. The increase in shear strength is attributed to a linear increase in friction angle with increasing gravel content, while cohesion changes very little.

Irfan and Tang [15] reported a linear relationship of Δϕ with gravel content in the range of 25–75%. For gravel content beyond 75%, Δϕ was reported to be almost constant. Lindquist and Goodman [21] also observed almost similar behaviour as observed by Irfan and Tang [15]. A linear increase in Δϕ with gravel content was also reported by Medley [24]; however, the range of gravel content used was limited to 15% to 40%. In the present study also a linear relationship between the percentage of gravel content and increase in friction angle has been investigated. Figure 7 shows the comparative evaluation of increase in friction angle (∆ϕ) with percentage of gravel content for the four different type materials. The variation of ∆ϕ for different material types with increasing gravel content has been elaborated in the following section.

Fig. 7
figure 7

Variation of increase in friction angle with different gravel fraction

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4.1.1 Material type 1

The Δϕ for material type 1 has been found to be linearly increasing with gravel content (Fig. 7). The friction angle of sample with 10% gravel content was found to be almost equal to the friction angle of sand fines matrix (\({\phi }_{\mathrm{sf}}\)). As gravel content increases beyond 10%, the \(\Delta \phi\) was found to be increasing by an amount of 1.1° for addition of every 10% gravel content. The equation deduced to estimate the \(\phi_{{{\text{RSM}}}}\) is expressed as follows

$$\phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{sf}}}} + \Delta \phi \quad ({\text{for}}\,{ }10{\text{\% }} < {\text{gravel \% }} \le 100{\text{\% }})$$
(3)

where \(\Delta \phi = 1.1^{ \circ }\) for the addition of every 10% coarse fraction.

4.1.2 Material type 2

For the material type 2, the variation of \(\Delta \phi\) with increasing gravel content is shown in Fig. 7. The \(\Delta \phi\) increases linearly with the gravel content up to 70% by an amount of 1.7° for the addition of gravel content every 10%. For samples with more than 70% gravel content, \(\Delta \phi\) becomes almost constant or reduces by very little amount. Based on these observations, the equation deduced to predict \(\phi_{{{\text{S}} - {\text{RM}}}}\) of S–RM samples is given below

$$\phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{sf}}}} + \Delta \phi \quad \left( {{\text{for}}\,{\text{gravel \% }} \le { }65{\text{\% }}} \right)$$
(4)

where \(\Delta\phi =1.7^{ \circ }\) for the addition of every 10% of gravel content up to 65%.

4.1.3 Material type 3

The results reveal that when the gravel content was less than 20%, there was a negligible increase of 2°–3° in the \(\Delta \phi\) with an increase in gravel content (Fig. 7). When the gravel content varied from 20 to 50%, the \(\Delta \phi\) increases linearly by an amount of 4.3˚ for the addition of every 10% gravel content. When the gravel content was above 50%, \(\Delta\phi\) did not change significantly. The following equations are obtained.

$$\begin{aligned} & \phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{sf}}}} + 2\quad \left( {{\text{for}}\,{\text{gravel \% }} \le 20{\text{\% }}} \right) \\ & \phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{sf}}}} + \Delta \phi \quad ({\text{for}}\,{ }20{\text{\% }} < {\text{gravel \% }} < 50{\text{\% }}) \\ & \phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{S}} - {\text{RM}}@50}} { } + 1\quad ({\text{for}}\,{\text{gravel \% }} > 50{\text{\% }}) \\ \end{aligned}$$
(5)

where \(\Delta \phi =\) 4.3° for the addition of every 10% coarse fraction; \({\phi }_{\mathrm{S}-\mathrm{RM}@50}\) is the frictional angle of soil–rock mixture sample with gravel content of 50%.

4.1.4 Material type 4

The increase in friction angle increases linearly with increasing gravel content up to 60% by an amount of 3.4° for the addition of gravel content every 10% (Fig. 7). In the samples with gravel content of beyond 60%, the \(\Delta \phi\) is almost negligible. The following equations obtained to predict the friction angle of S–RM are expressed as:

$$\begin{aligned} & \phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{sf}}}} + \Delta \phi \quad \left( {{\text{for}}\,{\text{gravel \% }} \le { }60{\text{\% }}} \right) \\ & \phi_{{{\text{S}} - {\text{RM}}}} = \phi_{{{\text{RSM}}@60}} - 1\quad \left( {{\text{for}}\,{\text{gravel \% }} \ge { }60{\text{\% }}} \right) \\ \end{aligned}$$
(6)

where \(\Delta \phi =\) 3.4° for the addition of every 10% coarse fraction. Above the 60% of gravel content, the \(\Delta \phi\) increases negligibly by an amount of approximately 1°.

Based on these observations it can be inferred that the approach suggested by the researchers [15, 21, 24] is inappropriate as there was no common agreement between increase in friction angle and percentage of gravel content. It is worth to mention here that the present approach is based on only the percentage of gravel content and it does not consider effect of individual particle characteristics like shape and size of particles, porosity, and strength of individual rock particles. These parameters significantly influence the shear strength; hence, there is need for a better model to predict the shear strength of S–RM. Therefore, an attempt was made in the below section to evaluate the applicability of BK model which considers not only the gravel content but also the characteristics of rock particles to model the shear strength of mixture.

4.2 Applicability of the BK model

Barton’s [3] model is the most widely used criterion for predicting the nonlinear shear behaviour of rough rock joints. It was extended to predict the shear strength of rockfill materials by Barton and Kjaernsli [5]. The input parameters used to estimate the shear strength of soil–rock mixtures are discussed in this section.

The BK model uses three parameters, i.e. residual friction angle (ϕr), roughness coefficient (R), and strength coefficient (S). The residual friction angle (ϕr) is expected to be obtained based on very high displacement; however, there are evidences that it can be considered equal to the basic friction angle (ϕb) of the rock [2, 12]. Therefore, in the present study, residual friction angle (ϕr) has been replaced by the basic friction angle (ϕb). The basic friction angle was obtained based on its rock type as suggested by Barton [4]. R and S values were estimated using empirical charts from Barton and Kjaernsli [5] based on porosity, and origin and shape of the particles. By substituting ϕb, R and S in Eq. (2) for all the sample, the peak friction angles (ϕp_cal) was calculated. The laboratory peak friction angles for the samples tested in the laboratory were calculated using Eq. (7). The laboratory peak friction angles are denoted as ϕp_exp.

$${\phi }_{\mathrm{p}\_\mathrm{exp}}={\mathrm{tan}}^{-1}\left(\frac{{\tau }_{\mathrm{f}}}{{\sigma }_{\mathrm{n}}}\right)$$
(7)

A comparison between ϕp_cal and ϕp_exp has been made as shown in Fig. 8. The difference between the calculated and experimental values was used to quantify the per cent error. The per cent error has been obtained as:

Fig. 8
figure 8

Comparison of calculated and experimental peak friction angles

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$$\mathrm{Percent \,error }\left(Pe\right)= \left(\frac{{\upphi }_{{\mathrm{p}}_{\mathrm{cal}}}-{\upphi }_{{\mathrm{p}}_{\mathrm{exp}}}}{{\upphi }_{{\mathrm{p}}_{\mathrm{exp}}}}\times 100\right)\%$$
(8)

Lines are shown on these plots termed as error lines (Fig. 8). The points falling close to the line designated as 50% indicate the corresponding per cent error in prediction. 10%, 20%, and 50% error lines are shown on the plot. The comparison between calculated and experimental peak friction angles is shown in Fig. 8. In the case of material type 2, calculated values are found to be on much lower side with per cent error in the range of + 10 to + 50%, while for the other three material types, the per cent error in calculated values was found to be − 25 to + 25%. Thus, the BK model developed to predict the shear strength of rockfills needs modification in order to extend its applicability to soil–rock mixture.

The BK model predicts the shear strength of rockfills by introducing basic friction angle ϕb, roughness coefficient (R), and strength coefficient (S) as a function of normal stress. The ϕb is independent of normal stress and depends on its rock type. S is estimated based on UCS of individual rock particle. It is a well-known fact that small-sized rock particles exhibit higher strength than larger-sized particles, and this scale effect has already been considered in the BK model. The S value is assigned based on mean effective particle size. The parameter R is the only coefficient that is expected to vary in case of soil–rock mixture. As the S–RM contains a wide range of rock particles with different sizes and shapes and sand fines content, the R may significantly vary with the proportion of gravel content and mean effective particle size for different normal stress levels. Therefore, the study was further extended to investigate the variation of R with the proportion of gravel content, mean effective particle size, and normal stress level.

4.3 Backward calculation of R

The roughness coefficient representing a soil–rock mixture sample can be back-calculated by using the BK model if the shear strength, strength coefficient (S), and basic friction angle (φb) of soil–rock mixture samples are known. The large size direct shear test results performed on different soil–rock mixture samples were used to back-calculate R. The model parameters ϕb and S are obtained based on rock type and the UCS of individual rock particles. The R values are back-calculated and these are denoted as Rexp. The shear strength as per BK model is expressed as:

$${\tau }_{\mathrm{f}}={\sigma }_{\mathrm{n}}\mathrm{tan}\left({\phi }_{\mathrm{b}}+R \,{\mathrm{log}}_{10}\left(\frac{S}{{\sigma }_{\mathrm{n}}}\right)\right)$$
(9)

The back-calculated R is obtained as:

$${R}_{\mathrm{exp}}=\frac{{\mathrm{tan}}^{-1}\left(\frac{{\tau }_{\mathrm{f}}}{{\sigma }_{\mathrm{n}}}\right)-{\phi }_{\mathrm{b}}}{{\mathrm{log}}_{10}\frac{S}{{\sigma }_{\mathrm{n}}}}$$
(10)

where \({\tau }_{\mathrm{f}}\) is laboratory peak shear stress, σn is normal stress at the time of failure, \({\phi }_{\mathrm{b}}\) is the basic friction angle, S is the equivalent strength of rock particles.

To study the variation of R with Rexp, R (i.e. Rcal) and Rexp were plotted together against the normal stress for the different proportions of gravel content (Figs. 9, 10, 11, 12). Despite considering the characteristics of the individual particles of soil–rock mixture, i.e. origin and shape and porosity while depicting R from Barton's standard charts, significant variation is observed between Rcal and Rexp. Therefore, it was inferred that R obtained from the empirical chart developed for the rockfills, does not truly represent the soil–rock mixtures roughness coefficient. Rockfills are rock particles larger than gravel size and may contain only limited to negligible amounts of sand fines mixture. The soil–rock mixture is a mixture of rock particles and sand fines containing a wide range of particle sizes with a variable content of sand fines. For a given rockfill material with a particular porosity and origin, the roughness coefficient will be a constant irrespective of the proportion and size of rock fragments and stress levels, while for soil–rock mixture samples, the Rexp decreases rapidly when normal stress level increases upto 50 kPa and becomes almost constant beyond normal stress level of 50 kPa (Figs. 9, 10, 11, 12). From the same plots, it is also observed that increased gravel content (G %) in the soil–rock mixture samples leads to an increase in Rexp. Mean effective particle size (d50) is usually used to represent the mean particle size of a whole sample, and it is directly proportional to the percentage of gravel content. Therefore, the present study was further focussed on establishing a correlation of R with normal stress, mean effective particle size, and percentage of gravel content. The procedure adopted to modify the R is summarized in the section below.

Fig. 9
figure 9

Comparison of Rcal and Rexp with normal stress and gravel content for material type 1

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Fig. 10
figure 10

Comparison of Rcal and Rexp with normal stress and gravel content for material type 2

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Fig. 11
figure 11

Comparison of Rcal and Rexp with normal stress and gravel content for material type 3

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Fig. 12
figure 12

Comparison of Rcal and Rexp with normal stress and gravel content for material type 4

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4.4 Modifications of coefficient R

The ratio, \(\frac{R}{{R}_{\mathrm{exp}}}\), was calculated for the tests data. The results of these calculations are summarized corresponding to the normal stress level, mean effective particle size (d50), and percentage of gravel content (G%) and are presented in Tables 3, 4, 5, and 6. R-ratio equal to 1 indicates that R is matching 100% with Rexp. Lower values of R-ratio (i.e. \(\frac{R}{{R}_{\mathrm{exp}}}\) <1) indicate underestimation, while higher values of R-ratio (i.e. \(\frac{R}{{R}_{\mathrm{exp}}}\) >1) indicate overestimation. It is observed that R-ratio values increase with an increase in normal stress level and decrease with mean effective particle size (d50) and percentage of gravel content (G%). In the case of material type 3, mean values of \(\frac{R}{{R}_{\mathrm{exp}}}\) are found to be high in the range of 0.73–2.3, while for material type 2 (RBM), they are low in the range of 0.13–0.49. For other material types 1 and 4, \(\frac{R}{{R}_{\mathrm{exp}}}\) values are found to be in the range of 0.5–2.14 and 0.54–2.03, respectively.

Table 3 \(\frac{R}{{R}_{\mathrm{exp}}}\) values obtained for material type 1
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Table 4 \(\frac{R}{{R}_{exp}}\) values obtained for material type 2
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Table 5 \(\frac{R}{{R}_{exp}}\) values obtained for material type 3
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Table 6 \(\frac{R}{{R}_{exp}}\) values obtained for material type 4
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4.5 Correlations of \(\frac{{\varvec{R}}}{{{\varvec{R}}}_{\mathbf{e}\mathbf{x}\mathbf{p}}}\) with σ n, d 50, and G (%)

In the previous sections, it has already been seen that the roughness coefficient of S–RM is influenced by normal stress level, mean effective particle size, and percentage of gravel content. In an attempt to establish correlations, \(\frac{R}{{R}_{\mathrm{exp}}}\) was plotted versus normal stress (σn), mean effective particle size (d50), and proportion of gravel content (G %) (Figs. 13, 14, 15). The correlations obtained are presented in Table 7.

Fig. 13
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Plots showing the correlation between R-ratio and normal stress

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Fig. 14
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Plots showing the correlation between R-ratio and d50

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Fig. 15
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Plots showing the correlation between R-ratio and gravel %

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Table 7 The correlations obtained to adopt correction towards σn, d50, and G%
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Using correlations established, the charts have been generated to suggest the correction factor which is to be applied to roughness coefficient, R. Figure 16 shows the charts for correction factor corresponding to normal stress (σn), percentage of gravel content (G%), and mean effective particle size (d50).

Fig. 16
figure 16

Proposed corrections for R to compute Rsug for soil–rock mixtures

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Using the charts, the correction factor has to be calculated for a sample corresponding to a required normal stress level, gravel percentage, and mean effective particle size, respectively. The maximum value out of these three has to be used to get modified R as expressed below:

$$R_{\bmod } = {\text{CF }} \times \, R$$
(23)

To check whether the Rmod predicts peak friction angle of soil–rock mixtures appropriately or not, peak friction angles have been re-calculated using equation as given below

$$\phi_{{{\text{p}}\_{\text{cal}}}} = \left[ {\phi_{{\text{b}}} + R_{\bmod } \log_{10} \left( {\frac{S}{{\sigma_{{\text{n}}} }}} \right)} \right]$$
(24)

These predicted peak friction angles, i.e. \({\phi }_{\mathrm{p}\_\mathrm{cal}},\) were then again compared with experimental data, i.e. \({\phi }_{\mathrm{p}\_\mathrm{exp}}\). Figure 17 shows the comparison plot of \({\phi }_{\mathrm{p}\_\mathrm{cal}}\) and \({\phi }_{\mathrm{p}\_\mathrm{exp}}\) to check the per cent error with predictions. The results indicate that calculated values are in good agreement. The predicted values after applying correction factor are found to have the per cent error less than 10%.

Fig. 17
figure 17

Comparison of peak friction angles between calculated and experimental

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Further, to quantify an overall average error in the predicted values for each material type, the average percentage of error (avpe) has been estimated. The avpe is expressed as:

$${\text{avpe}} = \sqrt {\frac{1}{npt}\mathop \sum \limits_{i = 1}^{npt} {\text{P}}_{{\text{e}}}^{2} }$$
(25)

where Pe is per cent in error in calculated values as compared to experimental, and ‘npt’ is the number of data points. Figure 18 shows the comparison of avpe for four different rock type materials. The overall avpe in calculated values has been found to vary between 6 and 9%. It is concluded that the Barton and Kjaernsli [5] model can be conveniently used to predict the shear strength of soil–rock mixture by applying some minor correction to the roughness coefficient. The proposed charts would facilitate to estimate that correction factor that is to be applied on R.

Fig. 18
figure 18

Overall average percentage of error with predictions

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4.6 Validation of proposed charts

The charts proposed in the present study have been validated using independent results of large size direct shear tests performed on phyllitic and RBM soil–rock mixtures. Physical properties of the samples are presented in Table 8.

Table 8 Physical properties of soil–rock mixtures
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Step-by-step procedure adopted to apply the correction factor using charts proposed in this study is as follows

  1. i.

    Firstly, the roughness coefficient (R) and strength coefficient (S) of the soil–rock mixtures were assessed using charts proposed by Barton and Kjaernsli [5] model, respectively.

R

5.7

6.1

1.7

1.2

S (kPa)

61,020.00

64,700.00

66,810.00

47,083.80

  1. ii.

    The correction factor (CF) from proposed charts corresponding to normal stress level (σn), proportion of gravel content (G %), and mean effective particle size (d50) was obtained. A maximum value out of these three is used to get modified R.

  2. iii.

    The Rmod (i.e. Rmod = CF × R) has been calculated and obtained values are presented below

R mod

Samples

G%

d50 (mm)

σn (kPa)

25

50

100

150

Phyllitic S–RM

47.2

4.0

8.1

8.1

8.1

8.1

Phyllitic S–RM

52

2.0

7

6.5

6.5

6.5

RBM S–RM

10.5

0.4

8

6.9

6

5.6

RBM S–RM

51.5

5.2

5.6

5.6

5.6

5.6

  1. iv.

    The peak friction angles for the samples corresponding to each normal stress level were calculated using Rmod in the equation of Barton and Kjaernsli [5] model as expressed below

    $$\phi_{{{\text{p}}\_{\text{cal}}}} = \left[ {\phi_{{\text{b}}} + R_{\bmod } \log_{10} \left( {\frac{S}{{\sigma_{E} }}} \right)} \right]$$
    (26)

The experimental and calculated values of peak friction angle are plotted in Fig. 19. It is seen that the predicted peak friction angles are found to be within the ± 10% error lines. Therefore, it can be concluded that the proposed charts may be used to apply correction factor to R to extend the applicability of Barton and Kjaernsli [5] model to soil–rock mixtures.

Fig. 19
figure 19

Validation of proposed charts by comparing calculated peak friction angles of independent tests data with experimental

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Further to check the efficacy of the proposed charts, it was planned to evaluate the proposed charts using shear strength values of soil rock mixture available in the literature [19, 29]. Lee et al. [19] performed direct shear tests on soil rock mixtures of semi-schist rock type procured from a quarry site. A maximum particle size in the samples was 50 and 75 mm. The rock particles are categorized as angular in shape with sharp edges. Authors reported a minimal particle breakage with increasing normal stress level. Wei et al. [29] also studied the shear strength behaviour of weathered basaltic rock soil mixture. These materials were collected from a slope. A maximum particle size of 40 mm was maintained in the samples. A moderate level of particle breakage was reported. For both the soil rock mixtures the input parameters that are required to compute the peak friction angle through proposed approach were obtained. The porosity of the samples was obtained from the unit weight of and specific gravity of the individual rock particles. Shear strength of the samples with different gravel fractions was obtained, and peak (i.e. secant) friction angle for each sample was computed for an applied normal stress level. Physical and shear strength properties of the samples are presented in Table 9. Following the steps i to iv as explained in Sect. 4.6, the peak friction angle for each sample was calculated before and after applying correction factor (Table 9). Figure 20 shows the comparison of peak friction angles (\({\phi }_{\mathrm{p}\_\mathrm{cal}}\)) of soil rock mixture with peak friction angles (\({\phi }_{\mathrm{p}\_\mathrm{exp}}\)) of experimental results. Despite having considerable particle breakage and also increased maximum particle size of 70 mm in the samples, the predicted values are found to be lie within the close range of ± 10% error lines. Therefore, the proposed approach can be used confidently in the true field conditions to predict the shear strength of soil rock mixtures congaing maximum particle > 40 mm. The proposed approach can also be extended to soil rock mixture exhibiting a minimal particle breakage. However, a further investigation on shear strength behaviour of soil rock mixture considering a maximum particle size of > 100 mm and particle breakage is required.

Table 9 Validation of proposed charts using tests data extracted from literature [19, 29]
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Fig. 20
figure 20

Peak friction angles calculated (\({\phi }_{\mathrm{p}\_\mathrm{cal}}\)) vs peak friction angles experimental (\({\phi }_{\mathrm{p}\_\mathrm{cal}}\)) plots before (a) and after (b) applying the correction factor

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5 Concluding remarks

Soil–rock mixture (S–RM) is a composite material consisting of sand fines matrix and gravel content. Estimating the shear strength of S–RM is quite complex and expensive because of wide variation in the size of rock particles and gravel content. The aim of the present study is to suggest an approach to predict the shear strength of S–RM by extending the applicability Barton and Kjaernsli (BK) model. To achieve this, extensive laboratory investigations on four different types of S–RM were carried out. The laboratory investigations included the characterization of individual particles, conventional direct shear tests on sand fines matrix, and a series of large size direct shear tests on sand fines matrix with various gravel content. Large size direct shear tests results were used to investigate the effect of variation in gravel content on the shear strength behaviour. The results were also used to evaluate the empirical correlations and BK nonlinear shear strength model. Findings were used to establish the correlations. Conclusions drawn from the present study are summarized as follows:

  1. 1.

    In general, addition of gravel content to the sand fines matrix brings a substantial increase in shear strength. However, for very low normal stress levels (5–50 kPa) this increase in shear strength with increasing gravel content was found to be less significant.

  2. 2.

    In response to the shear forces, S–RM samples show dilative and contraction behaviour subjected to different normal stress level. At low normal stress i.e. (i.e. ≤ 50 kPa), the behaviour was observed to be dilative for all the sample types irrespective of material types (i.e. pure soil to pure gravels). The dilative behaviour is attributed to relatively free movement of individual particles including sliding, rotation, and overriding. The energy required for dilation is small and hence dilation occurs. When the stress level increases beyond 100 kPa, during initial phase of shearing, there is more interlocking of particles which opposes free movements of the particles and hence contraction occurs. However, with further increase in shear stress the shear force becomes adequate enough to overcome resistance offered by these particles and again sliding/ration/overriding occurs resulting in dilative behaviour.

  3. 3.

    The results of large size direct shear tests were used to evaluate the existing empirical correlations by Irfan and Tang [15], Lindquist and Goodman [21], and Medley [24]. These correlations consider only the gravel content in defining increase in friction angle. The analysis indicated that in addition to gravel content there are other parameters which play significant role in deciding friction angle.

  4. 4.

    BK nonlinear shear strength model, which considers almost all important governing parameters, was also evaluated to predict the shear strength of SRM. When compared with experimental values, the results are found to be on much lower side with an error of about 50% in predictions. Experimental results were analysed to investigate the parameter responsible for variation in the shear strength of soil–rock mixture. The roughness coefficient R varies on adding gravel content to the sand fines matrix as the addition of gravel content increases the mean effective particle size. In addition, the normal stress level also influences the R significantly. Therefore, R is considered to be a lead parameter playing a key role in assessing the shear strength of soil–rock mixture.

  5. 5.

    Based on analysis, simple correlations to quantify the change in R (i.e. Rmod) for normal stress level, mean effective particle size, and percentage of gravel content have been suggested in this study. The shear strength of S–RM has been re-calculated using Rmod in the BK model, and results were compared with experimental one. The per cent error with calculated values has been found to be less than 10% for all material types. Also, the overall average error values were found to be less than 8% indicating the reliability of the proposed modifications.

  6. 6.

    Validation of the proposed charts through independent test results of large size direct shear indicates that the proposed modifications to roughness coefficient R yield satisfactory results. Therefore, the proposed charts would facilitate in extension of the applicability of BK model to predict the shear strength of S–RM by conducting some simple index tests in the field and laboratory.