J. Cent. South Univ. (2019) 26: 1856-1862

DOI: https://doi.org/10.1007/s11771-019-4139-y

A new slope optimization design based on limit curve method

FANG Hong-wei(方宏伟)^1,2, CHEN Yohchia(陈幼佳)^3,4, DENG Xiao-wei(邓晓蔚)⁵

1. School of Geomatics and Prospecting Engineering, Jilin Jianzhu University, Changchun 130118, China;

2. School of Civil Engineering and Architecture, Northeast Electric Power University, Jilin 132012, China;

3. School of Civil Engineering, Chongqing University, Chongqing 400045, China;

4. Department of Civil Engineering, The Pennsylvania State University, Middletown PA 17055, USA;

5. Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong 999077, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: A new method is proposed for slope optimization design based on the limit curve method, where the slope is in the limit equilibrium state when the limit slope curve determined by the slip-line field theory and the slope intersect at the toe of the slope. Compared with the strength reduction (SR) method, finite element limit analysis method, and the SR method based on Davis algorithm, the new method is suitable for determining the slope stability and limit slope angle (LSA). The optimal slope shape is determined based on a series of slope heights and LSA values, which increases the LSA by 2.45°–11.14° and reduces an invalid overburden amount of rocks by 9.15%, compared with the space mechanics theory. The proposed method gives the objective quantification index of instability criterion, and results in a significant engineering economy.

Key words: slope optimization design; limit state; limit curve method; limit slope angle

Cite this article as: FANG Hong-wei, CHEN Yohchia, DENG Xiao-wei. A new slope optimization design based on limit curve method [J]. Journal of Central South University, 2019, 26(3): 1856-1862. DOI: https://doi.org/10.1007/ s11771-019-4139-y.

1 Introduction

Slope optimization design is important for open pits and water dumps, whose main objective is to determine the slope angle under a limit state. This issue has been studied in many countries [1, 2]. Currently, there are three optimization methods available. The first one is the graph theory. KHALOKAKAIE et al [3] developed a computer program for optimal design of open pits based on Lerchs-Grossmann algorithm. RASSAM et al [4] studied three-dimensional (3D) effects on the slope stability of high waste rock dumps and found that the concave surface was stable at an angle at least 2° steeper than an identical slope following a straight line. The second optimization method is the comprehensive evaluation method. LAI et al [5] adopted a 3D physical model to optimize an open pit excavation by using the acoustic emission sensors with crack optical acquirement and ground penetrating radar and found that the slope angle on the downside was bigger than that on the upside. BYE et al [6] designed an open pit by adopting the mining rock mass rating system (MRMR) and performed numerical simulations using the software FLAC [7] and found that the slope angle deceased with an increasing slope height. SINGH et al [8] assumed a slope angle to perform the limit equilibrium analyses and numerical simulations using (FLAC) to determine the optimal slope angle. CAI et al [9] adopted geographic information system (GIS) based on a 3D limit equilibrium analysis to optimize the slope geometry. MALEKI et al [10] employed an empirical method using the software Phase2D [11] to optimize the slope angle and check the slope stability. LU et al [12] developed a comprehensive method based on the engineering analogy, limit equilibrium, and emulations. The third optimization method is some new techniques as follows. WU et al [13] proposed a new safety cleaning bank mode which reduced an invalid overburden amount of rocks by 3%–5%. ZHU et al [14, 15] proposed the space mechanics theory (SMT) for slope optimization design, it is shown that the deeper the furrow pit is excavated, the greater the stable slope angle is. Based on the literature survey, the key for the slope optimization design is the determination of the slope under a limit state [16].

The slip-line field theory [17, 18] can determine limit slope load, limit slope height [19] and limit slope curve (LSC) [20]. VO et al [21] developed the stability charts for curvilinear slopes based on the slip-line field theory. FANG et al [22] found a rule based on the relative position between the LSC and the slope, which is a limit curve method (LCM) capable of checking the slope stability. A new method is proposed based on the LCM to calculate the limit slope angle (LSA) and slope optimization design in this work.

2 Description of new method

2.1 Boundary conditions and calculation formula of LSC

This section describes the LSC which is calculated from the slip-line field theory. Only the boundary conditions and calculation formulas pertaining to the LSC are presented in this work as the derivation details have been reported by SOKOLOVSKI [17] and CHEN [18].

According to the slip-line field theory, the slip line can be divided into two families, α and β. Their intersection points involve four main parameters, x, y, θ and σ, where x and y are the coordinates, q is the intersection angle between the maximum principal stress (s₁) and x axis and s is the characteristic stress. The intersection of slip lines, M (x, y, θ, σ), can be determined by Eqs. (1)–(4), where M_α (x_α, y_α, θ_α, σ_α) is a point on α family slip line and M_β (x_β, y_β, θ_β, σ_β) is a point on β family slip line, m (=π/4–φ/2) is the mean intersection angle between a and b slip lines, and φ is the internal friction angle of soil.

  (1)

               (2)

     (3)

               (4)

Note that Eqs. (2) and (4) yield the same result. In determining the LSC, the origin of the Cartesian coordinate system is set at the top of slope with positive x₀ axis going rightward and positive y₀ axis going downward. While in determining the intersection (i.e., coordinate x₁) between the LSC and x axis, the origin of the coordinate system is shifted to the slope toe with positive x axis going rightward and positive y axis going upward, as shown in Figure 1.

Figure 1 Determination of LSC and x₁

The slip-line field theory defines the slope with three zones:

1) Active zone (i.e., O₁AB in Figure 1). For the known boundary points on O₁A line, M_a and M_b, y=0, and x=Δx·i (i=0-N₁) with Δx being the calculation step size and N₁ being the number of steps. Larger N₁ value consumes more computation time. N₁=999 is taken in this study based on the computer capability. The intersection angle q_I between s₁ and x₀ axis is π/2, where with P as the load at the top of slope.

2) Transition zone (i.e., O₁BC in Figure 1). For the known boundary point O₁, where with k=0–N₂, Δ_θ=θ_III–θ_I, and N₂ is the number of transition zone points, q_III is the intersection angle between s₁ and x axis in passive zone at point O₁.

3) Passive zone (i.e., O₁CD in Figure 1). The point on the LSC (i.e., O₁D line in Figure 1), M_ij(x_ij, y_ij, θ_ij, σ_ij), can be determined by Eqs. (5)–(8), where M′_β(x′_β, y′_β, θ′_β, σ′_β) is the known point on b slip line and M_b(x_b, y_b, θ_b, σ_b) is the known point on the LSC. The first M_b point is the origin at the top of slope (i.e., point O₁), which has x_b=0, y_b=0, and σ_b=σ_III= with c=cohesion.

      (5)

            (6)

                      (7)

                            (8)

Note that Eq. (6) yields the same result. To satisfy the minimum load at the top of slope When there is no external load on the slope, the minimum load is imposed at the top of slope to advance the calculation. As such, the value of N₂ has no effect on the calculation results and it is hence set equal to zero in this study.

2.2 Proposed method based on LCM

The new method based on LCM [22] is defined as: stable condition when x₁>0 (x₁ being the abscissa of the intersection between the LSC and the bottom of slope); limit state condition when x₁=0; unstable condition when x₁<0 (Figure 2). A computer program utilizing MATLAB [23] was developed by the authors to facilitate the computations using Eqs. (1)–(4) and (5)–(8).

Figure 2 New method based on LCM

2.3 Proof of new method

The new method is combined with the SRM [24] to determine the factor of safety (FOS) against the slope stability. The flowchart of the proposed computation method is shown in Figure 3.

Figure 3 Flowchart for calculation of safety factor by proposed method

Material set 2 of the case described by TSCHUCHNIGG et al [25] was adopted in this study to calculate the safety factors (SFs) along with unit weight γ=19 kN/m³, cohesion c=20 kPa, friction angel φ=25°, slope height H=10 m, and slope angle α=15°–60°.

The computed SFs by TSCHUCHNIGG et al [25] and the proposed method are listed in Table 1. In the table, SRFEA1 and FELA1 are respectively the strength reduction (SR) method and finite element (FE) limit analysis method under the “associated flow rule”, SRFEA2 and FELA2 are respectively the SR method and FE limit analysis method under the “non-associated flow rule”, and SRFEA (Davis) represents the SR method based on Davis algorithm [25]. It shows that the SFs by all methods decrease with increasing α value. Relative errors (E) for the SFs and P_min values are summarized in Table 2, which are calculated by

                (9)

Table 1 Comparison of SFs

The proposed method yields 5.99%–18.14% lower SFs than the SR method and –1.32%– 22.27% lower SFs than the FE limit analysis method (Table 2). The computed SFs between the proposed method and the SRFEA (Davis) method are similar. Note that higher P_min value implies lower slope stability and smaller slope safety factor. Relative errors increase with increasing P_min values. The main advantages of the proposed method are that there is no need to assume or search for a critical slip surface to determine if the slope is in a limit state, and it gives the objective quantification index of instability criterion.

Table 2 Relative errors of SFs and P_min

3 Slope optimization design

3.1 Calculation of limit slope angle (LSA)

The following new method is proposed to calculate the limit slope angle (LSA, A_cs): Increasing the slope angle a with constant c and j values, when the limit slope curve (LSC) and the slope intersect at the toe of the slope (i.e., x₁ = 0), the corresponding a is the LSA, as shown Figure 4. The computational flowchart for the LSAs is shown in Figure 5.

Figure 4 Calculation of LSAs (a₁23):

Figure 5 Flowchart for calculation of LSAs by proposed method

The case of TSCHUCHNIGG et al [25] described in Section 2.3 was used. The analysis results show that with an increasing α, the relative position relation between the LSC and the slope line changes from separation to intersection, as shown in Figure 6. At x₁=0, the calculated LSA value by the proposed method is 56.7° versus the approximate 60° by the SRFEA(Davis) method, as shown in Table 1, with an error of 5.5%.

3.2 Comparison with space mechanics theory

The slope optimization can be achieved through the determination of LSA as: for a specific H value, increase a to the LSA until x₁=0 (i.e., a slope limit state) is reached; based on the series of H and LSA values, determine the optimal slope shape using the MATLAB function “plot”. ZHU et al [14, 15] proposed the space mechanics theory (SMT) for slope optimization design according to the limit equilibrium principle, in which the relation between H and r (the set horizontal distance for the SMT) is shown in Eq. (4). The LSA is then calculated by Eq. (5), where r₀ is set at 100 m.

        (9)

Figure 6 Calculated LSAs by proposed method:

                        (10)

where r₁ is the horizontal distance for the proposed method determined by Eq. (6) in which A_LS1 is the limit slope angle calculated by the proposed method.

                          (11)

The comparison of LSA values between the SMT and the proposed method is shown in Table 3, in which γ=27 kN/m³, c=200 kPa and φ=45° are assumed [15]. It shows that the proposed method increases LSA values by 2.45°–11.14°. The comparison of slope optimization designs is shown in Figure 7 with the first coordinates of r₀=100 m and H=0 m at the foot of slope, point A is the foot of slope, points B and C are the top of slope determined respectively by the proposed method and the SMT, and points E and D are the bottom of slope determined respectively by the proposed method and the SMT. The results show that the optimal slope determined by the proposed method is a convex curve that is relatively flat in the upper portion and rather steep in the lower portion (i.e., an up flat-down steep curve). The reduced invalid overburden amount of rocks is 9.15%=S_ABC/S_ACD= (S_ABE+S_BCDE–S_ACD)/S_ACD, where S_ABE+S_BCDE is the amount of rocks calculated by the proposed method and S_ACD is the amount of rocks calculated by the SMT.

Table 3 Comparison of LSAs

Figure 7 Comparison of slope optimization design

4 Conclusions

A new method for slope optimization designs is proposed, which is based on the limit curve method. The slope is in the limit equilibrium state when the limit slope curve calculated from the slip-line field theory and the slope intersects at the toe of the slope. The safety factors for stability computed by the proposed method are found to generally match with those calculated by the SRFEA method. The analysis results show that the relative position relation between the LSC and the slope line changes from separation to intersection with an increasing slope angle. The relative error between the calculated LSA value by the proposed method and that of the SRFEA (Davis) method is merely 5.5%. The proposed method need not require knowing a critical slip surface in advance when determining if the slope is in a limit state and gives the objective quantification index of instability criterion.

The calculated limit slope angles (LSAs) by the proposed method is 56.7° which is close to 60° computed by the SRFEA (Davis) method. Based on the computed various slope heights H and LSA values, an optimal slope shape can then be determined through the use of MATLAB function “plot”. The optimal slope shape by the proposed method is convex, indicating that the computed LSA values increase with decreasing H values. Compared with the space mechanics theory, the proposed method increases the LSA by 2.45°–11.14°, resulting in 9.15% reduction of invalid overburden rocks, which is economically beneficial.

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(Edited by FANG Jing-hua)

中文导读

一种新的基于极限曲线法的边坡优化设计方法

摘要：基于极限曲线法，提出一种新的边坡优化设计方法，当由滑移线场理论计算的极限状态下的边坡坡面曲线与边坡坡面相交于坡脚时，判断边坡处于极限状态。与传统的强度折减法、有限元极限分析法、基于Davis算法的强度折减法相对比，提出的方法适用于计算极限坡角。基于提出方法计算的坡高和坡角得到的边坡优化坡面，与空间理论相对比，坡角提高了2.45°~11.14°，节省剥离废矿岩体体积达9.15%。提出的方法给出了边坡失稳的客观量化指标，能够产生有意义的经济效益。

关键词：边坡优化设计；极限状态；极限曲线法；极限坡角

Foundation item: Project(JJKH20180450KJ) supported by Education Department of Jilin Province, China; Project(20166008) supported by the Science and Technology Bureau of Jilin Province, China

Received date: 2018-01-10; Accepted date: 2018-10-15

Corresponding author: CHEN Yohchia, PhD, Professor; Tel: +86-17179486146; E-mail: yxc2@psu.edu