Simple analytical equations were developed in this study using limit equilibrium analysis. Moment equilibrium was considered for the calculation of FOS, defined here as the ratio of the shear strength capacity and the shear strength required for equilibrium. The complexity of the calculation increases when reinforcement is considered in the limit equilibrium formulation. The additional force induced by the reinforcement has been included in the equilibrium equations (Wright and Duncan, 1991). In this study, bamboo piles are inserted along a line at a distance of 1 m to reinforce the slope and analyze its stability. Various failure modes were considered in the formulation based on the assumption of the failure and circular slip surface. The following three failure modes are considered in this study:

• Mode I: Slope failure considering the shearing of the bamboo along the failure surface.

Analytical equations were formulated to calculate the FOS of the slope, considering the effect of the bamboo piles for these three failure modes. A single row of bamboo piles is considered as reinforcement that provides stability to the slope. These FOS formulations were developed using two different methods (modified ordinary and Bishop method of slices). The equations developed based on the modified ordinary method of slices are reported in previous research (Kumar and Chakrabortty, 2020). The equations presented in this paper are developed based on the modified Bishop method.

In this mode, shear failure of soil mass and bamboo piles is assumed. When failure occurs due to a driving moment caused by the weight of the slice (W_m) and the weight of the bamboo (W_b), an additional resisting moment due to the shear strength of the bamboo ( F r b comes into effect along with a resisting moment due to the soil strength (F_r). The resisting moment can be calculated from the mth slice (shown in Figure 2A) of soil mass as F r = N r ⁡ tan ⁡ φ ˈ F s + c ˈ ∆ L m F s . (1)

The normal reaction (Eq. 2) can be calculated from the equilibrium of the force polygon shown in Figure 2B. The FOS equation (Eq. 3) is obtained from the equilibrium of all wedges, considering the activating moment and the resisting moment. In the following equations, α m denotes the base angle of each slice, while α b denotes the base angle of the slice in which the bamboo is inserted. The first term in the numerator is the same as that in the Bishop method of slices. The second term in the numerator, which incorporates the effect of the shear resistance of bamboo piles, is included in the equation. The second term in the denominator is included because of the weight of the bamboo piles. N r = w m + ∆ T − c ˈ ∆ L m ⁡ sin ⁡ α m F s cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s , (2) F s = ∑ m = 1 P w m + ∆ T tan ⁡ φ ′ + c ˈ ∆ L m sin α m m α m + n F r b F s ∆ L b ∑ m = 1 P w m sin α m + n W b sin α c , (3) Where, m α m = cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s (4)

In this mode of slope failure, bamboo piles were assumed to remain embedded in the base soil after failure, and the upper part of the soil mass was to be moving downward. During the failure, the friction between the soil mass and the upper part of the bamboo pile comes into effect, which is referred to here as the frictional resistance of the bamboo (τ_b). The free-body diagram of the bamboo pile, the mth, and the kth soil slices are shown in Figure 3. The mth slice is the general slice not in contact with the bamboo, and the kth slice is the slice adjacent to the bamboo. The expression for FOS is formulated considering the equilibrium of the soil mass using the modified Bishop method of slices, as shown in Eqs 5–9. The expression for the resisting moment can be calculated from the force polygon of the mth slice of soil mass as F r = N r ⁡ tan ⁡ φ ˈ F s + c ˈ ∆ L m F s , (5) N r = w m + ∆ T − c ˈ ∆ L m F s   sin ⁡ α m cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s . (6)

The normal reaction of the kth slice (Eq. 7) can be calculated from the equilibrium of the force polygon, as shown in Figure 3C. Here, in Eq. 7, h is the height of the bamboo above the slip surface. The equation for FOS (Eq. 8) was obtained from the equilibrium of all slices, considering the activating moment and the resisting moment. In this mode, the weight of the bamboo does not contribute as a destabilizing force and, therefore, does not contribute to the denominator. The second term in the numerator incorporates the effect of the bamboo piles. This ultimately increases the FOS of the slope with bamboo piles as reinforcement. N r ′ = w k + ∆ T + n π r b h τ b − c ′ ∆ L k ⁡ sin ⁡ α k F s cos ⁡ α k + tan ⁡ φ ′ ⁡ sin ⁡ α k F s , (7) F s = ∑ m = 1 P w m + ∆ T tan ⁡ φ ′ + c ′ ∆ L m ⁡ cos ⁡ α m m α m + ∑ m = k k + 1 n π τ b r b h tan ⁡ φ ˈ m α m ∑ m = 1 P w m ⁡ sin ⁡ α m , (8) Where, m α m = cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s . (9)

In this failure mode, it is assumed that the pull-out force is exerted by the soil mass that is above the critical slip surface to pull the bamboo piles out of the base soil. In this failure mode, the friction between the soil mass and the lower part of the bamboo pile comes into effect, namely, the pull-out resistance of the bamboo (τ_b). The free body diagram for the mth soil slices and the bamboo is shown in Figure 4. Based on the equilibrium of forces, the expression for FOS is formulated as shown in Eqs. 10–12.

From the equilibrium of force polygon shown in Figure 4: F r = N r ⁡ tan ⁡ φ ˈ F s + c ˈ ∆ L m F s , (10) N r = w m + ∆ T − c ˈ ∆ L m ⁡ sin ⁡ α m F s cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s . (11)

Therefore, the proposed equation is F s = ∑ m = 1 P w m + ∆ T tan ⁡ φ ′ + c ˈ ∆ L n ⁡ cos ⁡ α m cos ⁡ α m + tan ⁡ φ ˈ ⁡ sin ⁡ α m F s + n π τ b r b H x 1 + x 2 r ∑ m = 1 P w m sin ⁡ α m + n w b ⁡ sin ⁡ α c . (12)

The limit equilibrium based on the modified Bishop method was chosen in SLOPE/W to analyze the aforementioned slope. The analysis was performed for two different models: unreinforced and bamboo-reinforced slope models created using the software (SLOPE/W). The Mohr–Coulomb constitutive model was used to analyze the soil slope. The critical slip surface (CSS) was searched by the grid and radius method. All the probable slip surfaces are shown in Figure 6A. The FOS for the critical slip surface was determined from the analysis to be 0.881, as shown in Figure 6B.

The material parameters were taken from the literature (Bergado et al., 1987) and were also determined experimentally in the laboratory (Kumar, 2018). The shear strength and average tensile strength of bamboo obtained from the literature were 18 MPa and 128 MPa, respectively, and the pull-out resistance of bamboo, 47 kPa, was also taken from the literature. The FOS was calculated for the slip circle which was found to be critical in the case of the unreinforced soil slope. The FOS of the critical circle found for the unreinforced slope was calculated to be 1.268, as shown in Figure 7A.

The analyses were also performed using experimentally determined bamboo material parameters as input parameters for the reinforcement. The pull-out test was conducted to find the behavior of the soil–bamboo interface. In this test, the soil was reinforced with two types of bamboo specimens, namely, bamboo with nodes and bamboo without nodes. The experimental investigation was conducted by varying the normal stress to 30 kPa, 40 kPa, and 50 kPa for a bamboo specimen with nodes and 50 kPa, 60 kPa, and 70 kPa for the specimen without nodes. The average pull-out resistance of a bamboo specimen with nodes is higher (375.4 kPa) than that without nodes (105 kPa). The experimental investigation in the laboratory estimated the tensile strength and shear strength of bamboo to be 218.75 MPa and 33.5 MPa, respectively (Kumar, 2018). A pull-out resistance of 105 KPa found in the experiment using bamboo without nodes was considered in this study (Kumar, 2018). The FOS was calculated for the same slip circle, which was found to be critical in the case of an unreinforced soil slope. The FOS increased in this case and was found to be 2.04, as shown in Figure 7B. From the comparison, it can be concluded that the reinforcement improved the slope stability against failure because the FOS increased from 0.881 to 2.04 after including the bamboo reinforcement. The model was also analyzed to estimate the critical slip circle after the improvement with a single row of piles. It was found that the CSS has a FOS of 1.01, as shown in Figure 8. Therefore, reinforcement location optimization can be considered as a scope of future study in any actual slope design.

In this subsection, the proposed analytical equations were used to calculate the FOS of the critical slip surface. The FOS was calculated for a slope with an angle of 35° for both the reinforced and unreinforced conditions. The results were compared with those obtained from SLOPE/W. The FOS results are presented in Table 1. In the first part, the values of the variables (shear strength of bamboo and pull-out resistance of bamboo reinforcement in sand) in the equations were taken from the literature (Bergado et al., 1987). In the second part, the values of these variables were taken from the experimental investigation conducted in the laboratory, as discussed in Section 3.1. The FOSs were estimated considering all three failure modes. The lowest of these three will be the critical value using this analytical method.

In the first part, when the values of the variables were taken from the literature, using the proposed analytical equation for the three modes, the FOS was evaluated as 3.95, 0.967, and 1.05, respectively. It can be observed from the SLOPE/W results that when reinforcement was provided, the FOS increased from 0.881 to 1.268. When the experimental results were used in the equation, the FOS for mode I, mode II, and mode III were found to be 5.168, 1.07, and 1.243, respectively, as presented in Table 1. It can be observed from Table 1 that the proposed equation gives a higher value of FOS than that obtained from SLOPE/W analyses. The frictional pullout forces were not considered in the SLOPE/W analysis; therefore, it gives a higher FOS. It is also observed that when parameter values taken from laboratory data are used, the FOS is higher than the FOS determined using reinforcement parameters taken from the literature (Bergado et al., 1987). The different bamboo species may be responsible for these differences.