Introduction

Most important engineering properties of sands, such as stress–strain/strength behavior, volume change and pore pressure developments and the variation in permeability, depend on the integrity of the particles or the amount of particle crushing (Lade et al. 1996). Crushing or particle breakage of soil grains occurs due to extremely high stresses that may be generated during pile driving, construction of high earth or rock fill dams, laying foundations of offshore gravity structures, impact of projectiles, drilling at great depths for petroleum extraction etc. (e.g., Bishop 1966; Lo and Roy 1973; Miura and O-hara 1979; Hardin 1985; Zheng et al. 1990; Coop and Lee 1993). Extensive laboratory investigations have been conducted by researchers to assess the compression and crushing behavior of granular material (e.g., Miura et al. 1984; Hagerty et al. 1993; Morland et al. 1993; Ghosal and Self 1995; Mcdowell et al. 1996; Anand and Gu 2000; Nakata et al. 2001; Feda 2002; Chuhan et al. 2003; Gupta et al. 2004; Bartake and Singh 2005, 2007). These studies demonstrate that some of the main factors which influence the crushing of grains are particle-size distribution, grain mineralogy, shape, hardness, density or void ratio, aspect ratio etc. (e.g. Miura et al. 1984; Hardin 1985; Feda 2002; Chuhan et al. 2003).

Bartake and Singh (2007) have investigated the relationship between the crushing strength, particle size characteristics and G of granular material including sands, cenospheres and glass beads. Using these results, a generalized relationship for estimating its crushing strength has been proposed, Eq. 1, in which D x corresponds to x percent finer fraction of the granular material.

$$ {{\sigma_{CR} } \mathord{\left/ {\vphantom {{\sigma_{CR} } G}} \right. \kern-\nulldelimiterspace} G} = \frac{{20 \cdot D_{15} \cdot D_{45} \cdot D_{50} \cdot D_{80} }}{{(D_{10} )^{1.6} \cdot D_{30} \cdot D_{60} \cdot (D_{90} )^{1.2} }} $$
(1)

Erzin and Yılmaz (2008) have investigated the influence of factors that affect the crushing resistance of three different Anatolian sands. They determined mineralogical, particle shape and size characteristics of the sands and conducted triaxial compression tests. It has been observed that particle crushing occurs in all shear tests. However, less particle crushing occurs in samples which contain higher proportions of quartz and lower proportions of calcite. Particle crushing is also observed to increase with increase in density.

A similar study has been undertaken using seven different Aegean sands collected from western Turkey (Fig. 1). The samples were tested for mineralogy, particle shape and size, and specific gravity. Crushing strength tests were performed on specimens with different aspect ratios (H/d = 0.50, 0.75, 1.00, 1.25 and 1.50) by employing the methodology proposed by Bartake and Singh (2007).

Fig. 1
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Location map of the study area and the sampling points

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Aegean sands have a small range of sphericity and roundness. Thus, additional crushing tests were performed on five Anatolian sands, having different particle shapes and compositions, used by Erzin (2004), with a H/d = 1.00. It was found that Anatolian sands have higher crushing strength values than Aegean sands and again, higher proportions of quartz minerals gave high crushing strength values. Based on these results, a multiple regression analysis (MRA) equation was developed for predicting the crushing strength value of sands based on their mineralogical composition, particle shape, aspect ratio, specific gravity, coefficient of uniformity and coefficient of curvature. Coefficient of correlation (R), variance account for (VAF), root mean square error (RMSE) and scaled percent error (SPE) were calculated to check the prediction capacity of the equation and confirmed an acceptable accuracy for use as a rapid, inexpensive substitute for laboratory techniques in the preliminary stages of site investigations.

Investigation

The sands collected from the locations shown in Fig. 1 were air-dried and the fraction passing a 4.76 mm sieve (ASTM No. 4) was separated into representative sub-samples by passing through the riffle box.

In order to establish gradational characteristics of these sands, tests were performed in triplicate on each sample following the guidelines presented by ASTM D 422–63. The average values were used for developing the particle size distribution curve for the sample (Fig. 2). Table 1 presents the soil parameters and the specific gravity (G) which was established following ASTM D 5550.

Fig. 2
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Particle size distribution characteristics of the sand samples

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Table 1 Physical characteristics of the samples of Aegean sands
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X-ray diffraction analyses were carried out in two stages: whole-rock powder and oriented clay fraction. Semi-quantitative weight percentages of samples were calculated by using mineral intensity factors based on Gundogdu (1982). In this method, samples are mounted in the same way, and the characteristic peak intensities (I) of minerals are normalized to that of the reflection of dolomite. In other words, a K factor for each mineral, including clays with peaks between 19 and 20°2q, is determined as K dolomite/I mineral in a 1:1 dolomite–mineral mixture by weight. Percentages of the minerals are calculated from the following equation:

$$ a = \left( { 100K_{\text{a}} I_{\text{a}} } \right)/\left( {K_{\text{a}} I_{\text{a}} + K_{\text{b}} I_{\text{b}} + \cdots + K_{\text{n}} I_{\text{n}} } \right) $$
(2)

where a is the percentage of the mineral; and K a,b,…,n and I a,b,…,n are the K factor and intensity of each mineral identified. This method was preferred over other methods because it is simple to perform.

According to the XRD results, quartz is the most prominent mineral in samples C to G, with feldspar, calcite and clay minerals found in small quantities. In sample A, higher proportions of feldspar and clay minerals were observed. Amorphous silicate minerals (85 %) were dominant in sample B (85 %) with 15 % quartz. The results are summarised in Tables 2, 3.

Table 2 Whole rock powder diffraction analysis results (% fraction by weight) for the sand samples
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Table 3 Results of XRD oriented clay-size fraction diffraction analysis results (% by weight) for the sand samples
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Particle shape analyses were carried out on 120 grains of each sand sample. The roundness (R F ) developed by Dobkins and Folk (1970), given by Eq. (3), and sphericity (S R ) described by Riley (1941), given by Eq. (4) were determined and the average values were considered as representative.

$$ R_{F} = {\text{ d}}_{\text{k}} /{\text{d}}_{\text{i}} $$
(3)
$$ S_{R} = {\text{d}}_{\text{i}} /{\text{d}}_{\text{c}} $$
(4)

where d k is the diameter of curvature of the sharpest corner, d i is the diameter of the inscribed circle, and d c is the diameter of the circumscribing circle (Fig. 3). A very well rounded particle has a maximum R F of 1.0 while, an extremely angular particle has a value close to 0.0. A perfectly spherical particle has an S R value of 1.0; smaller values mean departure from a spherical shape and the theoretical extreme is 0.0. The results are presented in Table 4.

Fig. 3
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Definitions of angularity and sphericity (Dobkins and Folk 1970)

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Table 4 Results of the particle shape analyses on the sand samples
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Apparatus

In order to study the influence of H/d (=0.5, 0.75, 1.0, 1,25 and 1.5) on the measured crushing strength σ CR of the samples, a laboratory test was set up as depicted in Fig. 4. This apparatus was fabricated by Bartake and Singh (2005) to determine crushing characteristics of the sands, cenospheres and glass beads. It consists of a 110 mm high stainless steel mold with an inner diameter of 33 mm and outer diameter of 51 mm, in which granular material can be filled to achieve a sample of the desired aspect ratio (i.e., height to diameter ratio, H/d) and density ρ. Two 32 mm diameter stainless steel pistons (top and bottom) were used for crushing the sample. A collar was provided at the bottom of the mold on which two removable stainless steel clips of thickness 10 mm can be fixed. These clips were removed before commencing the test to achieve loading of the sample from both the ends.

Fig. 4
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The test setup used for determining crushing strength of sands

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Methodology

A 10 kN load cell compression testing machine supplied by Humboldt, USA was employed in this study. The tests were performed at a strain rate of 1.25 mm/min (Karner et al. 2004). The load cell was attached to a digital readout unit to record the load (P) transmitted to the sample at 10 s intervals. Prior to loading of the sample, silicon grease was applied on the two pistons to minimize the side friction. The sample was poured into the mold by adopting the rainfall technique (i.e., using a funnel and maintaining a constant height of fall) to achieve a certain H/d.

Results and discussion

Typical load-deformation (P-δ) characteristic curves were developed for Samples A to G, corresponding to different H/d ratios (=0.5, 0.75, 1.0, 1.25 and 1.5). The results pertaining to sample C (with H/d = 1.0) are presented in Fig. 5. After an initial elastic response (depicted by portion AB of the P-δ characteristics), rearranging and repacking of the particles starts (depicted by portion BC). Point C corresponds to the point on the P-δ characteristics with the maximum curvature. Beyond point C, sands lose their integrity, hence a tangent drawn to the P-δ characteristics at point C intersects its upper linear portion (represented as DE) at F, which would yield σ CR . As depicted in the figure, the crushing load of the sample is 5.0 kN, which corresponds to σ CR  = 6.6 MPa.

Fig. 5
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Typical load-deformation characteristics of the Sample F (H/d = 1.0)

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To check the reproducibility of the P-δ characteristics, two tests were performed on Sample A by maintaining H/d = 1. The results are depicted in Fig. 6 which indicates the two trials yield very similar trends.

Fig. 6
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Reproducibility of the load-deformation characteristics of Sample A (H/d = 1.0)

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The influence of H/d on σ CR was also investigated. As an example, the P-δ characteristics of Samples B and G are given in Figs. 7, 8, respectively. By employing the methodology in Fig. 5, the σ CR of Samples A to G was computed as shown in Fig. 9. It can be seen that initially, σ CR increases non-linearly with H/d and thereafter it attains a nearly constant value. The transition point between these two trends corresponds to H/d = 1. Hence, for determining the σ CR of the Aegean sands, tests should be conducted with H/d > 1.

Fig. 7
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Typical load-deformation characteristics of Sample B

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Fig. 8
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Typical load-deformation characteristics of Sample G

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Fig. 9
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A comparison of crushing strength of different sand samples

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A comparison of the σ CR values of Samples A to G (for H/d = 0.5) indicates that σ CR is highest (7.2 MPa) for Sample D. Similar trends were observed for other H/d ratios. This can be attributed to the fact that Sample D contains the highest proportion of quartz (refer Table 2), and has the highest C u value (=6.55, refer Table 1), supporting the work of Erzin and Yilmaz (2008).

It is of note that for H/d = 0.5, 0.75, 1.0, 1.25 and 1.5, higher σ CR values were obtained for Sample F compared with Sample G which has a similar R F and S R . However, Sample F has a higher C u (Table 1) and higher proportions of quartz and feldspar with lower proportions of clay minerals compared to Sample G (Table 2). In addition, Sample G contains 16 % calcite.

In addition, the σ CR for Samples A to G were computed from Eq. 1 and compared with the experiment results for H/d = 1.0 (Fig. 10). It can be seen that Eq. 1 yields higher σ CR results; the lowest difference (0.15 MPa) being obtained for Sample G and the highest (6.52 MPa) for sample E. This discrepancy can be attributed to the fact that the particle size distribution characteristics on which Eq. 1 is based, is not the only factor which influences the crushing strength of the Aegean sands but particle shape, mineralogical composition and aspect ratio should also be taken into account.

Fig. 10
figure 10

A comparison of the measured and predicted crushing strengths of different sand samples (H/d = 1.0)

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As the Aegean sands used in this study have a small range of sphericity and angularity (Table 4), crushing strength tests (H/d = 1.0) were also performed on five Anatolian sands with different particle shapes and compositions, as used by Erzin (2004); see Table 5. X-ray diffraction analyses were carried out on two of the Anatolian sand samples (samples H and J) while for the other samples the results were were taken from Erzin and Yilmaz (2008). The whole rock powder diffraction analysis results (% fraction by weight) for the Anatolian sand samples is given in Table 6 and the particle shape analyses in Table 7 (from Erzin and Yilmaz 2008).

Table 5 Physical characteristics of the samples of the Anatolian sands (Erzin 2004)
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Table 6 Whole rock powder diffraction analysis results (% fraction by weight) for the Anatolian sand samples
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Table 7 Results of the particle shape analyses on the Anatolian sand samples (Erzin 2004)
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The crushing strength of the Aegean and Anatolian sands (H/d = 1.0) are compared in Fig. 11. Although the σ CR values of Anatolian sands are higher than those for Aegean sands, again the proportion of quartz is significant.

Fig. 11
figure 11

A comparison of crushing strength of sand samples A to L

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A model based on multiple regression analysis (MRA) was developed to predict σ CR based on mineralogical composition, particle shape, aspect ratio, specific gravity, coefficient of uniformity and coefficient of curvature, using SPSS 8.0.0 (Statistical Package for the Social Sciences. The experimentally obtained G, R F , S R , C u , C c , H/d and the proportions of quartz, feldspar, calcite and clay minerals (Q, F, Ca and Clay respectively) were used and the following equation was obtained.

$$ {{\sigma_{CR} } \mathord{\left/ {\vphantom {{\sigma_{CR} } G}} \right. \kern-\nulldelimiterspace} G} = 5. 4 + 1.0 \cdot R_{F} - 6. 4\cdot S_{R} - 0. 6\cdot C_{u} + 1. 9\cdot C_{c} + 0. 9H/d + 0. 8\cdot Q + 0. 8\cdot F + 0. 6\cdot Ca - 0. 2\cdot {\text{Clay}} $$
(5)

The computed σ CR values from Eq. (5) are compared vis-à-vis experimentally obtained σ CR values in Fig. 12. The solid diagonal line in Fig. 12 represents a perfect prediction line. The lower line represents a 100 % over-prediction bound and the upper line represents a 50 % under prediction bound. Approximately 80 % of the predictions fall onto or are very close to the 1:1 line and all of the predictions fall between these two prediction lines. The high coefficient of correlation (R = 0.96) indicates a strong correlation between the computed and measured σ CR values. Variance (VAF), represented by Eq. (6), and the root mean square error (RMSE) represented by Eq. (7), were also computed (Gokceoglu 2002; Erzin 2007; Erzin et al. 2008).

$$ VAF = \left[ {1 - \frac{{{\text{var}} \left( {y - \hat{y}} \right)}}{{\text{var} \left( y \right)}}} \right] \times 100 $$
(6)
$$ RMSE = \sqrt {\frac{1}{N}\sum\nolimits_{i = 1}^{N} {\left( {y_{i} - \hat{y}_{i} } \right)^{2} } } $$
(7)

where var denotes the variance, y is the measured value, ŷ is the predicted value, and N is the number of the sample. If VAF is 100 % and RMSE is 0, the model is treated as excellent. Values of VAF and RMSE were found as 91.9 % and 1.7 MPa.

Fig. 12
figure 12

Comparison of experimentally obtained and computed values of σ CR for Aegean and Anatolian sands

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In addition to the performance indices, a graph between the scaled percent error (SPE), given by Eq. (8), as employed by Kanibir et al. (2006), and cumulative frequency was also drawn in Fig. 13 to show the performance of the models developed.

$$ SPE = \frac{{\left( {\left( {\sigma_{CR} } \right)_{p} - \left( {\sigma_{CR} } \right)_{m} } \right)}}{{\left( {\left[ {\left( {\sigma_{CR} } \right)_{m} } \right]_{\max } - \left[ {\left( {\sigma_{CR} } \right)_{m} } \right]_{\min } } \right)}} $$
(8)

where \( \left( {\sigma_{CR} } \right)_{p} \,\text{and} \) \( \left( {\sigma_{CR} } \right)_{m} \) are the predicted and measured crushing strengths; and \( \left[ {\left( {\sigma_{CR} } \right)_{m} } \right]_{\max }\,\text{and}\, \left[ {\left( {\sigma_{CR} } \right)_{m} } \right]_{\min }^{{}} \) are the maximum and minimum measured crushing strengths, respectively. As seen from Fig. 13, about 90 % of the crushing strengths predicted from the MR model are ±10 % of the SPE.

Fig. 13
figure 13

The relationship between the scaled percent error and cumulative frequency

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Conclusions

In this study, the influence of factors affecting the crushing strength of Aegean sands has been investigated. Samples from seven different locations were analyzed for mineralogical, particle shape and size, specific gravity and crushing strength tests and the influences of aspect ratio, particle composition, particle shape and size on the crushing strength were examined. It has been observed that the crushing strength, σ CR , initially increases non-linearly with H/d and thereafter attains a nearly constant value for each sample. The highest σ CR was obtained for the sample with the highest C u value and containing the highest proportion of quartz. Due to the fact that Aegean sands used in this study have a small range of sphericity and angularity, crushing strength tests were also performed on five Anatolian sands for H/d = 1.0. The σ CR values of Anatolian sands are found to be higher than those for Aegean sands but the significant factor was again the proportion of quartz.

An MRA equation was developed for predicting the σ CR value of sands based on their mineralogical composition, particle shape, aspect ratio, specific gravity, coefficient of uniformity and coefficient of curvature and the applicability of the equation demonstrated using R, VAF and RMSE.

It is concluded that for these sands the proposed equation is sufficiently accurate for use as a rapid, inexpensive substitute for laboratory techniques in the preliminary stages of site investigations.