J. Cent. South Univ. (2020) 27: 2745-2753

DOI: https://doi.org/10.1007/s11771-020-4495-7

A simple theoretical approach for analysis of slide-toe-toppling failure

Hassan Sarfaraz

School of Mining Engineering, College of Engineering, University of Tehran,PO Box: 14155-6619, Tehran, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: A prevalent kind of failure of rock slopes is toppling instability. In secondary toppling failures, these instabilities are stimulated through some external factors. A type of secondary toppling failure is “slide-toe-toppling failure”. In this instability, the upper and toe parts of the slope have the potential of sliding and toppling failures, respectively. This failure has been investigated by an analytical method and experimental tests. In the present study, at first, the literature review of toppling failure is presented. Then a simple theoretical solution is suggested for evaluating this failure. The recommended method is compared with the approach of AMINI et al through a typical example and three physical models. The results indicate that the proposed method is in good agreement with the results of AMINI et al’s approach and experimental models. Therefore, this suggested methodology can be applied to examining the stability of slide-toe-toppling failure.

Key words: rock slopes; slope stability; slide-toe-toppling; theoretical solution

Cite this article as: Hassan Sarfaraz. A simple theoretical approach for analysis of slide-toe-toppling failure [J]. Journal of Central South University, 2020, 27(9): 2745-2753. DOI: https://doi.org/10.1007/s11771-020-4495-7.

1 Introduction

Toppling failures are classified into primary and secondary kinds (Figure 1) [1]. In the primary toppling failure types, the main reason for instability is the rock column weight. However, secondary toppling failures are motivated by several external factors. According to Goodman Bray’s classification, several types of researches have been studied by analytical methods and physical and numerical models [2-5]. In 1987 and 1992, AYDAN and KAWAMOTO [6, 7] simulated the toppling failure utilizing friction table machine. In 2007, ADHIKARY et al [8] accomplished the centrifugal test for flexural toppling failure, where concrete and glass specimens were used as modelling materials. Many kinds of researches were studied for the numerical modelling of toppling failures [9]. A simple solution was offered for the analyzing flexural toppling failure based on the compatibility principle of cantilever beams [10, 11]. In 2018, ZHENG et al [12, 13] suggested the new technique based on limit equilibrium theory for analyzing flexural toppling failure. Also, for this failure, SARFARAZ [14] presented a new theoretical solution for the calculation of the safety factor by applying Sarma’s method in 2020. In 2012, AMINI et al [15] proposed a technique for the analysis of block-flexural toppling failure by combing the approaches of Goodman & Bray and Aydan & Kawamoto. In 2020, SARFARAZ et al [16] numerically modelled this failure using UDEC software.

Secondary toppling failures are various, and some cases have been presented for these instabilities. Three types of these failures are indicated in Figure 2. Many papers have introduced secondary toppling failures [17-21]. In 2019, SARFARAZ et al [22] modelled slide-head-toppling failure using the finite element method. HAGHGOUEI et al [21] presented an analytical solution for the assessment of toppling- slumping failure. The slide-toe-toppling is one of the conventional kinds of secondary toppling failure that rock columns are having a possibility of toppling at the toe section of slope due to sliding of soil mass in the top section of the slope (as shown in Figure 2(b)). Recently, AMINI et al [17] carried out physical tests for this failure and presented a method based on limit equilibrium. Also, in 2020, SARFARAZ et al [23] modelled this failure with distinct and finite element methods. The results of their study demonstrated that the distinct element method is more acceptable than the finite element method. In this research, a theoretical solution is offered for the slide-toe-toppling failure, and then the results are discussed.

Figure 1 Main toppling failure:

Figure 2 Pictures of secondary toppling failure:

2 Proposed theoretical solution

A schematic picture of the recommended analytical solution for slide-toe-toppling failure is shown in Figure 3. In order to analyze the theoretical solution for this failure, the reaction force between soil mass and rock columns must be specified, and then this force should be applied to the topmost block. In this research, this reaction force is obtained by Coulomb’s theory.

Retaining structures such as retaining wall are encountered in civil engineering, which supports slopes of earth masses. According to Figure 4, the soil element positioned at a depth z is subjected to an effective vertical pressure (σ′₀), and an effective horizontal pressure (σ′_h) that a non-dimensional quantity (K) is defined by σ′_h/σ′₀. There are three modes concerning the retaining wall: 1) at-rest mode; 2) active mode, 3) passive mode.

In slide-toe-toppling failure, soil mass motivates the rock blocks. In this failure, the uppermost rock column performs as a cantilever retaining wall, and soil mass is active mode.

Figure 3 Schematic picture of theoretical solution

Figure 4 Description of at-rest (a), active (b) and passive (c) modes [24, 25]

Therefore the following equations can be used to apply the force from the soil mass to the topmost rock block. According to Coulomb’s theory (as shown in Figures 3 and 4), the failure surface is linear, and the force distribution from the soil mass to the block “n” is triangular. According to Figure 3, the following equations can be written from the law of sines [25]:

          (1)

                  (2)

          (3)

where W is the weight of soil wedge; F is the resultant of shear and normal forces on the surface of failure, BC, which is inclined at an angle of f_s to the normal drawn to the plane BC; P_a is the active force from the soil mass. By combination of Eqs. (2) and (3), the following equation can be written:

                   (4)

To determine the critical value of β for the maximum P_a, dP_a/dβ=0, and the following equation can be obtained:

      (5)

                            (6)

where K_a is the active earth pressure coefficient; f_s is the internal friction angle of soil; f_sb is the angle of interface friction between soil mass and topmost rock column; y_s is the upper surface angle; H_t is the height of topmost rock column.

After calculating the force of P_a in Eq. (6), this force is applied to the rock column with the triangular distribution to determine the slope stability against toppling failure (as shown in Figure 3). According to this figure, rock mass constituting the toe of the slope is prone to blocky, flexural or block-flexural toppling failures. So in general, a rock block in the toppling zone undergoes sliding, toppling, bending or shearing, in which case, the force acting on the right of this block can be obtained, respectively, from the following relationships [15, 17]:

      (7)

 (8)

    (9)

    (10)

where P_i is the normal force of inter-block; Q_i is the shear force of inter-block; f_c is the angle of interface friction in between blocks; f_i is the internal friction angle in intact rock; y_f is the slope face angle; y_b is the normal dip to the dominant discontinuities; y_p is the discontinuities angle; c_i is the cohesive strength of intact rock; c_b is the cohesive strength of rock block base; W_i is the weight force; h_i is the average blocks length; y_i is the application point of “P”; σ_t is the tensile strength of rock blocks; H is the slope height; t is the block thickness; subscript sh refers to shearing.

In block toppling failure, the block either overturns or slides due to cross-joints at the block base, in which case Eqs. (7) and (8) can be used for sliding and toppling, respectively. If P_i-1,s>P_i-1, rock column is capable of sliding failure; but P_i-1,t>P_i-1,s, rock block has a prone of toppling failure. If both forces are not positive, this block is the stable state and does not apply any force to the next block (P_i-1=0). In flexural toppling failure, the block breaks under bending and shearing are due to induced tensile stress and the resultant forces applied to them, respectively. The state of stability can be specified by Eqs. (9) and (10). On the sides of every block, shear stresses can be calculated through Mohr-Coulomb failure law. Finally, the sign of force of column “1” can be used to assess the stability of slide-toe-toppling failure. When P₀>0, the slope is unstable. When P₀<0, the slope is stable, and when P₀=0, the slope is the state of limit equilibrium.

3 Evaluation analysis of typical example with proposed analytical solution

For simplifying the stability analysis of mentioned failure, the recommended method was coded in a program, which gets the slope parameters from the user and executed totally computation related to proposed method and Amini et al’s approach. A typical example was analyzed for assessing the performance of suggested and Amini et al approaches (as shown in Figure 5). The outcomes of these analyses are listed in Tables 1-3. According to Figure 5, for comparing the proposed approach with the AMINI et al’s method, the slide surface initiation in the AMINI et al’s method corresponds to the slide failure plane in the proposed method. In Tables 1-3, the toppling failure forces (blocky, flexural, and block-flexural) were calculated by Eqs. (7)-(10).

Figure 5 Representation diagram of typical example

The values of parameters of the slope set as inputs to the code, which are illustrated in the top of the section in Table 1. Also, in the left and right of this table, the results of AMINI et al and recommended methods are indicated, respectively. In the approach of AMINI et al, the magnitude of reaction force between soil and block “10” is equal to 4.71 MN, and point application of this force is 11.80 m above the block base. Besides in the proposed method, the magnitude and point application of this force are 3.43 MN and 10.167 m, respectively. In both methods, the results show that the slope has a prone to slide-toe-block toppling failure. According to the results shown in this Table, the slope is stable for slide-toe-flexural toppling and slide-toe-block-flexural toppling failures because there are stable zones in the toe section.

4 Analysis of physical tests with suggested method

AMINI et al [17] carried out experimental models for this failure through a tilting table machine (as shown in Figure 6). They modelled three types of failure that are shown in Figure 7. The results of the three tests are listed in Table 4. In this table, B, F and BF represent the blocky, flexural, and block-flexure types, respectively, and the number in the front of B, F, BF is model height in cm.

For validating the offered theoretical solution, its results were compared with the outcomes of three physical models. The results of the suggested analytical approach for analyzing the three physical models are presented in Table 5. Also, the outcomes of Amini et al’s method are listed in this Table.

The errors of the presented method can be obtained by comparing the geometry parameters (Tables 4 and 5), this comparison is illustrated in Table 6. Since the tilting table of the experimental tests had a variable inclination, the inclination angle at the moment of failure was chosen for more assessment. In Table 7, the outcomes of this comparison, and the errors of the analytical methods are presented. Accordingly, these errors may be considered reasonable because the mechanism of the failure is complex. Additionally, the results of this comparison might be indicated graphically, as shown in Figure 8. The graph also indicates that there is a satisfactory agreement between the proposed analytical method and physical results. In other words, the outcomes of this comparison show that the suggested methodology is in good agreement with the outcomes of AMINI et al’s method and physical models. It also demonstrates that both analytical predictions have the similar error ranges. So, the proposed methodology can be used for assessing the analysis of slide-toe-toppling failure.

Table 1 Geotechnical parameters of analyzing typical example

Table 2 Sliding parameters comparison of AMINI et al method and proposed method

Table 3 Toppling parameters of interaction of soil mass and rock column when inclination is 32° and K_a is 0.295

5 Conclusions

In this study, a simple theoretical methodology is suggested for the stability analysis of slide-toe-toppling failure. In the recommended method, the force applied to the topmost rock is first calculated by Coulomb’s theory. Next, this force is applied to the rock column with the triangular distribution to determine the slope stability against toppling failure. In the proposed method, there is no need to know the sliding surface. Furthermore, in the recommended method with simple calculations, the active force is obtained from the soil mass. The suggested approach are compared with AMINI et al’s method using a typical example and three physical tests. Comparison between results of the proposed method, AMINI et al’s theoretical methodology, and experimental models shows that there are appropriate agreements. Also, the outcomes show that the proposed method is conservative. So, both existing and proposed theoretical approaches can be used for evaluating the analysis of slide-toe- toppling failure.

Figure 6 Schematic diagram of tilting table machine [17] (1-Feeder; 2-Model; 3-Camera; 4-Control panel; 5-Motor and gearbox; 6-Sensor)

Figure 7 Slide-toe-toppling models:

Table 4 Geometry of experimental tests at failure moment [17]

Table 5 Geometry of analytical models at failure moment

Table 6 Comparison errors between suggested method and AMINI et al’s approach

Table 7 Comparison between physical outcomes and predictions of suggested and Amini et al approaches

Figure 8 Comparison of physical outcomes with predictions of model suggested and AMINI et al analytical approach

Acknowledgements

The author would like to thank Professor Omer AYDAN from Ryukyu University, and Dr Mehdi AMINI from University of Tehran, for the guidance in this research.

Contributors

Hassan Sarfaraz provided the concept and wrote the draft of the literature review and manuscript. Also, he analyzed the calculated results.

Conflict of interest

Hassan Sarfaraz declares that he has no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

一种分析滑移-底部-顶部失稳的简单理论方法

摘要：失稳坍塌是岩质边坡的一种常见破坏形式。在二次倾倒失稳中，由一些外部因素引起坍塌，其中一种是“滑动-底部-顶部失稳”。在这种失稳情况下，斜坡的顶部和底分别具有滑移和倾倒失稳的可能。通过分析方法和实验对这种失稳进行了研究。在本研究中，首先，对倾倒失稳进行了文献回顾。然后，提出了一种简单的评估这一失稳的理论方法。通过一个典型的例子和三个物理模型，采用AMINI等人的方法与本文提出的方法进行了比较。结果表明，该方法与AMINI等人的方法和实验模型的结果符合得较好。因此，本文的方法可用于评估滑移-底部-顶部失稳。

关键词：岩质边坡；边坡稳定性；滑移-底部-顶部失稳；理论解决方案

Received date: 2020-03-14; Accepted date: 2020-06-12

Corresponding author: Hassan Sarfaraz, PhD Candidate; Tel: +98-9104978395; E-mail: sarfaraz@ut.ac.ir; ORCID: https://orcid.org/ 0000-0002-1100-6921