Numerical analysis of slopes treated by nano-materials

1 Introduction

With the remarkable development in the field of soil mechanics, it has become possible to improve soils using various techniques for soil enhancement and improvement. The conventional improvement techniques such as compaction, drainage methods, vibration methods, grouting and injection, chemical stabilisation, pre-compression and consolidation, soil reinforcement, and geotextiles and geomembranes have been utilised successfully in the field of soil mechanics. These techniques have mainly been used for various objectives to improve soil stiffness (to prevent liquefaction), to increase bearing capacity (shear strength), to reduce excessive settlement, and to minimise and control the volume change (the shrinkage and swelling).

The preceding techniques require, however, experience and equipment availability. It has been necessary, therefore, to adopt a new and inventing technique for soil improvement, i.e. nanotechnology. Nanotechnology, which is an innovating approach, concerns mainly about the processing of matter (the properties of materials) on a molecular scale. Due to its improvement capability, the method has been, therefore, utilised successfully in the field of soil mechanics, [1,2].

The influence of different types of nano-materials (NMs; nano-clay, nano-alumina, and nano-copper) on the volume change was investigated by author of [3]. The study was performed on four types of soil mixed with various amounts of bentonite: 0, 5, 10, and 20%. Three different ratios of 0.3, 0.7, and 0.5% of nano-alumina, nano-copper, and nano-clay, respectively, were added (by dry weight) to the soils, to investigate the swelling and shrinkage behaviours. It was deduced that swell and shrinkage strains and desiccation cracks on surface of compacted samples can be reduced for samples treated by nano-copper.

Changizi and Hadad [1] studied the effect of nano-SiO₂ on the geotechnical properties of cohesive soils. Compaction and unconfined compression (UC) tests were conducted where the maximum dry unit weight and shear strength were increased significantly due to the effect of nano-SiO₂. To put this in context, this was equivalent to an increase of 2.1 and 1.23 folds in angle of shearing resistance (ϕ) and cohesion (c), respectively. The increase in shear strength parameters was attributed to fabric changes where addition of nano-SiO₂ helped to create a viscous gel, which led to the increase in the particles’ bonding.

Yao et al. [2] investigated the effect of nano-MgO (at different percentages) on the strength properties and microstructure characteristics of a soft soil stabilised by cement. A series of unconfined compression tests were conducted to quantify the strength of the improved soil by the nano-MgO. Remarkable increase in shear strength was obtained which was attributed to the improvement of the microstructure of the stabilised soil. It was found that at a certain amount of nano-MgO (15%), changes in soil fabrics were considerable as shown in Figure 1a and b.

Figure 1

Microstructure improvement of the stabilised soil: (a) 0% nano-MgO and (b) 15% nano-MgO [2].

In addition, author of [4] used nano-calcium carbonate and carpet waste fibres to examine the mechanical properties of a clayey soil using UC and unconsolidated undrained (UU) triaxial tests. It was found that cohesion increased by a factor of 1.95 and 1.94 folds for the UU and UC tests when adding 0.4% carpet waste fibre and 1.2% nano-calcium carbonate, respectively. In addition to the preceding scholars, the influence of NM to improve the soil fabric and consequently the shear strength parameters have also been reported in the literature (not presented for conciseness) [5,6,7].

On the basis of the preceding experimental results, the role of NMs on the shear strength enhancement due to changes in the soil fabric was considerable. The aim of this study is, therefore, to examine numerically the influence of soil fabric changes due to the NM enhancement on strength using an upper bound approach. A parametric study using a slope case study at different angles and shear strength parameters is, therefore, carried out.

2 Problem geometry

The analysed problem domain is depicted schematically in Figure 2. A slope of 1 m height (H = 1 m) was modelled in the parametric study using an upper bound approach, discontinuity layout optimisation (DLO). The DLO approach is based on evenly spaced nodes that can determine the collapse load of stability problems. The results of the parametric study were normalised by H; therefore, selecting an arbitrary value of the slope height is not an issue. Seven different slope angles (β): 15, 20, 25, 30, 35, 40, and 45° were used. In the simulation, a satisfactory scale factor (equivalent to the number of nodes) was assigned to 8. Comparison between scale factors of 7 and 8 for a modelled slope of β = 30° at c = 0 kPa and ϕ = 45° yielded a trivial difference of 0.24%. The scale factor of 8, therefore, was used in the following analysis.

Figure 2

Schematic diagram of the analysed problem geometry.

As explained in the preceding section that the influence of the NM on the shear strength enhancement due to soil fabric changes was considerable, the parametric study was, therefore, carried out for a series of ϕ = 0–45° at an interval of 5° and c = 5, 10, 15, 20, and 25 kPa. This was to investigate the influence of NM enhancement on the slope stability. The self-weight of 15 kN/m³ was used in the analysis. Two different types of slopes constructed on purely frictional materials and c–ϕ materials were utilised in the parametric study.

3 Numerical results

3.1 Purely frictional slopes

In cohesionless soils, a slope is safe for any ϕ ≥ β, unless an unstable condition would be obtained for ϕ values less than β, i.e. the steeper the slope, the lower the factor of safety (FOS). In such problems, the stability of the system is independent of the soil self-weight.

Figure 3a and b depicts the normalised results for a frictional slope (c = 0 kPa) for a range of ϕ and β values where the increase in FOS due to any changes in soil fabric (i.e. increase in angle of shearing resistance at a constant cohesion) is considerable. All cases showed a safe condition (FOS > 1) once ϕ ≥ β. As expected, β had an inverse effect on FOS, a decrease of about 27.6% in FOS was obtained at c = 0 kPa and ϕ = 30° when β increased from 30 to 40°. Generally, FOS = 1 represented the critical condition (point of collapse or the slope is at the verge of failure, limit equilibrium). The condition of FOS = tan ϕ/tan β was also plotted for comparison in Figure 3, dotted red curves.

Figure 3

Stability charts for frictional soils (c = 0 kPa) for a range of β from (a) 15–25° and (b) 25–45°.

A 2.1 fold increase in ϕ, for example obtained by [1] as stated previously due to the effect of nano-SiO₂, was equivalent to an increase of more than 113% in FOS at ϕ = 20° and β = 25°. This increase was sufficient to transfer the condition from unstable regime (ϕ = 20° and β = 25° for FOS = 0.9463) to a safe condition with FOS = 2.02 at ϕ = 40°.

3.2 Cohesive-frictional slopes

The design charts for this series are depicted in Figure 4 for a variety of combinations of c and ϕ values at a range of β values. The slope stability of cohesive soils is commonly expressed as a stability number, N, as follows:

(1) N = c FOS × γ × H .

Figure 4

Design charts at different β for c–ϕ soils.

Eq. (1) depends on the soil properties and FOS. Bell [8] proposed a modified stability number N ^* which is independent on FOS as:

(2) N ⁎ = c γ × H × tan ϕ .

The normalised results for this series are presented in this manner: FOS/tan ϕ and c/(H × γ × tan ϕ), also used by the author in [9]. The charts in Figure 4 have an advantage over the available iterative charts in the literature, i.e. FOS not applied to N. The main concern, however, is that for a pure cohesive slope (ϕ = 0), Eq. (2) cannot be used as it leads to infinity. The design charts for this series, therefore, excluded the case of ϕ = 0. Figure 4a also exhibited the exact match of the design charts at two β values against obtained by [9].

A significant rise in FOS was obtained due to any increase in c. For example, FOS was increased by 1.71 fold when cohesion doubled from 5 and 10 kPa at ϕ = 10° and β = 25°, as shown in Figure 4a. Such range of increase in cohesion due to NMs enhancement was reported by the author in [4], c improved by 1.95 fold. This rise was over and above any decrease in H from 25 to 15° (FOS ≈ 1.2) at ϕ = 10° and c = 5 kPa. The enhancement becomes even more increasingly significant for any better improvement in c due to the NM addition.

Figure 5 shows the rate of increase in FOS for the cases where and c increased by 5° and 5 kPa, respectively, due to changes in fabric attributed to the NM enhancement for two specific value of β: 15° and 45°. The letter “i” in the x-axes denotes the difference between, e.g. ϕ _i+1 = 10° and ϕ _i = 5° (∆ϕ = 10–5°). A considerable increase in FOS was obtained for any increase in shear strength parameters, inversely with β. An increase of 45% in FOS due to change in ϕ from 10 to 15° at β = 15° was sufficient to produce a safe slope against collapse (FOS > 1). The increase in c also demonstrated similar results.

Figure 5

Rate of increase in FOS% due to NM enhancement for each 5 interval increase in (a) ϕ and (b) c.

The mechanism generated following the enhancement of NM in shear strength parameters is shown in Figure 6 for two scenarios. The first scenario was considered for an increase of 5° in ϕ (from 0°) at c = 20 kPa and β = 45° and the second scenario was for an increase of 5 kPa in c (from 5 kPa) at ϕ = 0°. The obtained mechanisms in Figure 6 were identified as “midpoint circle,” i.e. failure mechanisms extended beyond the toe. However, comparison between Figure 6a and b for scenario 1 exhibited a wider and deeper failure mechanism for ϕ = 0°. In addition, the case ϕ = 5° produced more slip lines. Despite the similarity in Figure 6c and d, the increase in FOS due to 5 kPa change was considerable, for e.g. in Figure 6d, the soil materials were required to be reduced by a factor of 3.659 in order to bring the system to failure.

Figure 6

Failure mechanisms for two different scenarios at (a and b) different ϕ and constant c and (c and d) different c and constant ϕ.

4 Conclusion

In the last three decades, NM technique has proved its efficiency in improving and enhancing the shear strength characteristics of soils due to changes in soil fabric. This work, therefore, addressed a numerical study on the effect of the soil fabric changes induced by the NM addition for two types of slopes (on purely frictional soils and c–ϕ soils) using an upper bound discretization scheme.

A slope with a range of various angles (β = 15, 20, 25, 30, 35, 40, and 45°) was modelled. The numerical results presented within this study proposed that, on the basis of a set of re-generated design charts, the role of the NM in slope stabilisation was considerable. An increase of ∼6.6 fold in FOS was obtained when c increased from 0 to 25 kPa at β = 15° and ϕ = 30°. In addition, a ∼68.2% increase in FOS was obtained when ϕ increased from 30 to 45° at β = 15° and c = 0 kPa. As expected, the effect of slope angle on the stability was inverse. For example, a decrease of 16.7% in FOS was obtained when β changed from 20 to 30° at ϕ = 30° and c = 10 kPa.

The proposed design charts have advantages over the traditional limit equilibrium methods, e.g. method of slices, which can be used readily without the need of iterative procedures. In addition, the design charts provided a set of extra values of β angle and extended both x and y boundaries when compared with other available charts in the literature. Finally, a considerable influence of the NM on the failure mechanisms, to provide an insight into different failure mechanisms, due to the soil fabric changes, was obtained.