
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13036-8
10.1016/j.heliyon.2024.e37005
e37005
Research Article
Research on upper bound limit analysis of slope stability in open-pit mine based on least square principle
Wang Zhen hhstu_wz@163.com
⁎
Li Ang
Yang Tianhao
School of Engineering, Huanghe Science and Technology College, Zhengzhou, China
⁎ Corresponding author. hhstu_wz@163.com
29 8 2024
15 9 2024
29 8 2024
10 17 e3700526 6 2024
24 8 2024
26 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
So as to extend the application of upper bound limit analysis for slope stability in open-pit mine, this paper analyzed failure mechanism of rock mass when shear failure occurred based on plastic mechanics principles, and combined with discrete mathematical theory and least square principle to propose a visualization method of logarithmic spiral. At the same time, the visualization method was applied to engineering practice for slope stability in open-pit mine, and cycle algorithm of stability analysis for open-pit mine slope was proposed through energy balance conditions, then stability analysis results were evaluated reasonably through uniting with limit equilibrium method. The results show that visualization method of logarithmic spiral proposed in this paper is highly consistent with original curve, and can ensure correlation coefficient square is greater than 0.9995. What is more, upper bound limit analysis method has a superhigh accuracy in slope engineering. At the same time, the most dangerous sliding surface obtained by this method can satisfy the requirements of velocity separation, which solved velocity problems during the process of slope evolution. Therefore, the method proposed in this paper has a stronger application in the field of open-pit mining.

Keywords

Upper bound limit analysis
Slope stability
Open-pit mine
Least square principle
Strength reduction
==== Body
pmc1 Introduction

For any line passing through the rotation center, its intersection angle is always constant for logarithmic spiral in polar coordinate system. Logarithmic spiral is also called isometric spiral for this reason. Due to this perfect geometric and mathematical characteristics, logarithmic spiral is widely used in many fields such as water conservancy environment, machinery manufacturing, astronomy meteorology, geotechnical engineering and so on [1]. Among them, associated flow law in geotechnical engineering requires that direction angle between relative velocity changed and slip surface tangential is internal friction angle [[2], [3], [4]]. This satisfy the requirements of velocity separation are highly consistent with characteristics of logarithmic spiral, which leads to widely used for logarithmic spiral in geotechnical engineering [5].

For shear failure of rock mass, the morphology of logarithmic spiral is satisfied with requirements of velocity separation. Therefore, taking logarithmic spiral as shear failure mechanism of slope rock mass, an important method named upper bound limit analysis method is formed in slope stability analysis [6]. On the premise of stable materials, Drucker proposed associated flow law with yield criterion, which created a quantum leap in plastic mechanics. On this basis, some scholars studied on the difference and connection between stress field and velocity field, and established limit analysis for solving values of critical load [[7], [8], [9], [10], [11]]. For limit analysis, it contained two method, upper bound limit analysis and lower bound limit analysis [12]. The former solves upper values of real load through creating kinematic allowable velocity field [13]. While the latter solves lower values of real load through creating static allowable stress field [14]. In comparison, construction of static allowable stress field is more complexity than that of kinematic allowable velocity field, which leads to a wider engineering application of upper bound limit analysis [15,16]. Chen introducedlimit analysis into structure analysis of rock mass, and analyzed calculation principle and application process of upper bound limit analysis method in rock mass mechanism in detail [17]. In the following decades, this method had become a significant subject in geotechnical engineering, and a large number of scholars had applied this method to solve many topics such as underground space engineering, slope and foundation engineering, surrounding rock collapse and stability analysis, and had achieved lots of remarkable results [18]. Leca analyzed the influence of section shape on failure mechanism of deep buried tunnel, and researched corresponding stabilities under different failure mechanisms through upper bound limit analysis method, then evaluated accuracy of calculated results combined with centrifugal model experiments [19]. Michalowski divided slope rock mass into several vertical rigid strips, and established velocity field which satisfied condations of deformation coordination, then proposed slope stability analysis method which met conditions of velocity separation and mechanical equilibrium [20]. Lysmer converted slope stability problems of upper bound limit analysis into linear programming problems through combining with modern mathematical theory, and combined with mechanism minimization of Lagrange's increasing term method to solve universal function, which improved accuracy of stability analysis results effectively [21]. Nowadays, upper bound limit analysis is widely used in the field of engineering practice. But this method believes that the failure trajectory of rock mass is not limited by circular arc, but by logarithmic spiral. And this method has provided a perfect answer to stability problem of homogeneous slope, but its application to stability problem in the field of heterogeneous slope is greatly limited because of logarithmic spiral can't be visualization in plane rectangular coordinate system [22]. As a result, upper bound limit analysis method is rarely used in slope stability analysis in open-pit mine. For open-pit mine slope, limit equilibrium method is widely used in slope practice owing to its simple calculation process and strong convergence of calculation results [23]. On the other side, physical modeling, numerical modelling and other analytical methods are also widely used, and the problems of slope design, analysis and calculation of open-pit mine are effectively solved [[24], [25], [26]]. However, with the purpose of solving the problem of slope stability in open-pit mine through upper bound limit analysis, a large number of scholars have to seek solutions to this problem from the angle of combining upper bound limit analysis and finite element technology [[27], [28], [29], [30], [31]]. Through the efforts of some scholars, although the problem has been solved to a certain extent, because of the limitation of finite element technology, the calculation time is very long and the convergence of calculation results is poor [32,33].

Upper bound limit analysis method has the characteristics of strict theoretical basis and rigorous derivation process, and its advantages in slope stability analysis are obvious. With the purpose of extending the application of this method in slope engineering, this paper proposed a visualization method of logarithmic spiral in plane rectangular coordinate system by combining discrete mathematical theory and least square principle. On this basis, established energy balance equation and proposed cycle algorithm for sliding surface searching and stability coefficient calculating, and evaluated accuracy of analysis results by limit equilibrium method. This paper solved engineering application problem of upper boud limit analysis from the perspective of mathematics theory, compared with traditional analysis method, which effectively improved calculation speed and accuracy, and had important theoretical value and engineering practice significance.

2 Methods

In accordance with the theory of upper bound limit analysis, tangential velocity changed of any point on velocity discontinuity plane is always accompanied by separation velocity and the angle between velocity changed direction and sliding plane direction should be internal friction angle when shear failure occurs [34]. The requirement of velocity separation can be fully satisfied when the path of shear failure is logarithmic spiral. In polar coordinate system, logarithmic spiral equation can be expressed as:(1) r=a·eθtanϕ

In formula (1), r (m) is the length between the point of rotation center and any point on logarithmic spiral; a is an undetermined parameter in formula (1), and a>0; θ (rad) is the horizontal angle of inclined line starts with the point of rotation center and ends with any point on logarithmic spiral; φ (rad) is tangential angle between rotation center and logarithmic spiral. In geotechnical engineering field, according to shear failure characteristics of rock mass, the tangential angle of inclined line starts with the point of rotation center and ends with any point on logarithmic spiral is internal friction angle. Obviously, for formula (1) in polar coordinate system, it should be converted into plane rectangular coordinate system in order to convenient for slope energy calculation. With rotation center as coordinates origin, horizontal direction is selected as X-axis and right direction is selected as positive, vertical direction is selected as Y-axis and upward direction is selectrd as positive, then a plane rectangular coordinate system is established. In accordance with relation between different coordinate systems, logarithmic spiral equation can be expressed as:(2) {x=r·cosθ=a·eθtanϕ·cosθy=−r·sinθ=−a·eθtanϕ·sinθ

Formula (2) is an implicit function equation of logarithmic spiral in plane rectangular coordinate system, and it is very difficult to visualize the implicit function. Therefore, in this paper, the implicit function is visualized by combining discrete mathematics theory and least squares principle. According to slope engineering practice, the slope failure occurs in the fourth quadrant and in the form of increasing interval. Meanwhile, formula (2) shows that for any x/a and y/a, they are both implicit functions and only related to θ. In the fourth quadrant, θ ranges from [0, π/2], and for any θ in this interval, there are corresponding definite x/a and y/a. Therefore, θ in this interval is discretized, and increasing interval can be determined according to each discrete point coordinate. According to the distribution of discrete points in increment interval, each discrete points can be fitted, and finally the visualization of logarithmic spiral can be realized.

In the following, φ = 20° is used to illustrate the visualization process. It means tanφ = 0.364 when φ = 20°, and formula (1) can be expressed as r = a·e0.364θ. Converts it into the form of plane rectangular coordinate system, and be expressed as:(3) {x=r·cosθ=a·e0.364θ·cosθy=−r·sinθ=−a·e0.364θ·sinθ

According to formula (3), for any horizontal angle θ in range of [0, π/2], there are definite x/a and y/a corresponding to it. In this paper, several θ values in range of [0, π/2] are selected, x/a is taken as horizontal axis and y/a is taken as vertical axis to describe distribution characteristics of each discrete point, it is shown in Fig. 1.Fig. 1 Distribution morphology of discrete points on logarithmic spiral.

Fig. 1

Fig. 1 shows distribution morphology of discrete points on logarithmic spiral in the fourth quadrant when φ = 20°. In accordance with the distribution morphology of discrete points, there is only a inflection point (1.067, −0.388) in this range. Taken y/a as independent variable and x/a as dependent variable, when y/a∈[−1.771, −0.388], x/a increases with the increasing of y/a, indicating that the increasing interval is [−1.771a, −0.388a]. When y/a = −0.388, x/a achieves the maximum value of 1.067. When y/a∈[−0.388, 0], x/a decreases with the increasing of y/a, indicating that the decreasing interval is [−0.388a, 0]. In order to facilitate slope energy calculation, polynomial fitting is performed for discrete points in increment interval in the fourth quadrant based on least square principle. The least square method is an optimization technique in the field of modern mathematics. It determines the best matching function through minimizing the sum of errors squares. The least square method can be used to obtain unknown data easily, and ensure the sum of errors squares between obtained data and actual data is minimum. The least square method is widely used to fit discrete points data in engineering. Linear function fitting result is shown in Fig. 2(a), quadratic function fitting result is shown in Fig. 2(b), cubic function fitting result is shown in Fig. 2(c), quartic function fitting result is shown in Fig. 2(d).Fig. 2 The least square fitting results.

Fig. 2

It can be seen from Fig. 2 that for polynomial fitting of discrete points in increment interval in the fourth quadrant, corresponding correlation coefficient squares are respectively 0.8862, 0.9952, 0.9993 and 0.9999 when linear function, quadratic function, cubic function and quartic function are used for polynomial fitting. In this paper, R2 ≥ 0.9995 is selected as polynomial fitting requirement, and it is suggested to select quartic function for polynomial fitting. The fitting result is as follows:(4) x/a=−0.391·(y/a)4−1.3273·(y/a)3−2.0653·(y/a)2−1.1994·(y/a)+0.8392

A function expression about x can be obtained through simplifying formula (4), and the result is as follows:(5) x=f(y)=−0.391y4/a3−1.3273y3/a2−2.0653y2/a−1.1994y+0.8392a

Formula (5) can be approximated as visualization form of logarithmic spiral in plane rectangular coordinate system, and it can be applied to calculate energy of heterogeneous slope, and then realize heterogeneous slope stability analysis.

3 Results and discussion

3.1 Engineering geological

In this section, a typical slope consists of stope slope and dump slope in open-pit mine is selected to explain the process of stability analysis. The dump slope consists of two steps, the height and the slope angle of single step are 20m and 30°, and the dump plate width of single step is 35m. The plate width between stope slope and dump slope is 25m. The stope slope consists of sandstone, mudstone and coal, and thickness of each rock layer is 15m, 35m and 10m respectively. The stope slope is layered horizontally, the height and the slope angle of single step are 10m and 60°, and the stope plate width of single plate is 16m. Rock mass strength parameters in the open-pit mine are shown in Table 1.Table 1 Rock mass strength parameters.

Table 1rock mass	bulk density/kN·m−3	cohesion/kPa	internal friction angle/°	height/m	
abandoned material	21	35	16	40	
sandstone	20.5	65	19	15	
mudstone	23	57.5	17.5	35	
coal	18	90	17	10	

3.2 Calculation model establishment

According to slope engineering experience, for a slope consists of multiple steps, landslide position must occur at the toe point of any step, that is the corresponding location of A1, A2, A3 … A8 in Fig. 3 below. When landslide occurs at the toe point of slope A8, the corresponding stability analysis process is the most complex. Therefore, the following takes landslide occurs at A8 as an example to clarify the slope stability analysis process. For landslide occurs at other locations, the calculation process can be obtained by above. When landslide occurs at A8, its failure mechanism is consists of logarithmic spiral locates in coal, mudstone, sandstone and abandoned material. The slope morphology is shown in Fig. 3.Fig. 3 Slope morphology and failure mechanism diagram.

Fig. 3

For slope failure mechanism, according to visualization method of logarithmic spiral described above, and combined with rock mass strength parameters, polynomial fitting of each logarithmic spiral can be performed separately. For logarithmic spiral locates in coal, mudstone, sandstone and abandoned material, increasing intervals in the fourth quadrant are respectively [−1.617a1, -0.314a1], [−1.640a2, -0.331a2], [−1.717a3, -0.364a3] and [−1.570a4, -0.298a4]. Among them, a1, a2, a3 and a4 are undetermined parameters for logarithmic spiral of coal, mudstone, sandstone and abandoned material. According to analysis results of increasing interval, polynomial fitting of discrete points in increasing interval can be carried out, and fitting results are as follows:(6) x=f1(y)=−0.6369y4/a13−1.9772y3/a12−2.6738y2/a1−1.3463y+0.8258a1

(7) x=f2(y)=−0.6031y4/a23−1.9122y3/a22−2.6481y2/a2−1.3720y+0.8168a2

(8) x=f3(y)=−0.4752y4/a33−1.5766y3/a32−2.3366y2/a3−1.2963y+0.8241a3

(9) x=f4(y)=−0.7550y4/a43−2.2876y3/a42−2.9693y2/a4−1.4383y+0.8129a4

(5), (6), (7), (8), (9) are least square fitting results of logarithmic spiral in coal, mudstone, sandstone and abandoned material, and corresponding correlation coefficient squares are respectively 0.9998, 0.9998, 0.9999 and 0.9998. The above fitting results all satisfy to R2 ≥ 0.9995, which is perfect satisfaction for engineering practice. Fitting result of logarithmic spiral in coal is shown in Fig. 4(a), fitting result of logarithmic spiral in sandstone is shown in Fig. 4(b), fitting result of logarithmic spiral in mudstone is shown in Fig. 4(c), fitting result of logarithmic spiral in abandoned material is shown in Fig. 4(d).Fig. 4 Fitting results of logarithmic spiral in slope.

Fig. 4

For slope morphology, altitude of coal floor is taken as y = y0, and coordinate of slope toe point can be obtained as (f1(y0), y0). On this premise, slope morphology equation can be expressed in form of piecewise function according to parameters of steps and plates in the slope. The expression is as follows:(10) x=g(y)={0.577·(y−y0)+f1(y0)y0≤y≤y0+100.577·(y−y0)+f1(y0)+16y0+10<y≤y0+200.577·(y−y0)+f1(y0)+320.577·(y−y0)+f1(y0)+480.577·(y−y0)+f1(y0)+640.577·(y−y0)+f1(y0)+801.732·(y−y0)+f1(y0)+35.7181.732·(y−y0)+f1(y0)+70.718y0+20<y≤y0+30y0+30<y≤y0+40y0+40<y≤y0+50y0+50<y≤y0+60y0+60<y≤y0+80y0+80<y≤y0+100

Formula (10) is form equation of slope morphology, and slope energy can be calculated according to form equation of slope morphology and failure mechanism.

3.3 Slope energy calculation

In accordance with upper bound limit analysis, for any slope failure mechanism, stability coefficient can be calculated according to the condition that external force power and internal energy dissipation power are equal. Therefore, so as to realize slope stability analysis for open-pit mine, external force power and internal energy dissipation power should be calculated respectively based on slope failure mechanism.

In natural state, rock mass of the slope is only affected by gravity [35,36]. So external force power of rock mass in potential failure area can calculate by:(11) E=∫sγ·vdS

In formula (11), E (kW) is external force power of rock mass for the slope; S (m2) is area of slope potential failure; dS (m2) is an arbitrary area element in this area; according to definition of definite integral, dS = dx·dy, γ (kN/m3) and v (m/s) are bulk density and gravity velocity corresponding to the area element. In accordance with the relationship between linear velocity and angular velocity, the component velocity in direction of gravity can be expressed as v = ω·r·cosθ = ω·x. Then formula (11) can be converted into:(12) E=ω·∫γ·xdxdy

Due to bulk density of each rock mass layer is different, rock mass in slope potential failure area should be stratified horizontally along roof and floor, and external force power can be calculated respectively. For coal in slope potential failure area of the open-pit mine, corresponding lower limit is y = y0 and upper limit is y = y0+10 in y direction, and corresponding lower limit is x = g(y) and upper limit is x = f1(y) in x direction. Therefore, the failure area can be expressed as:(13) {g(y)≤x≤f1(y)y0≤y≤y0+10

Taking the failure area expressed by formula (13) into formula (12), and external force power of coal in slope potential failure area can be expressed as:(14) E1=ω·γ1·∫y0y0+10dy∫g(y)f1(y)xdx

In formula (14), γ1 is bulk density of coal, γ1 = 18 kN/m3. This definite integral can be calculated by taking formula (6) and formula (10) into formula (14). Similarly, external force power of mudstone, sandstone and abandoned material in slope potential failure area can be expressed as:(15) E2=ω·γ2·∫y0+10y0+45dy∫g(y)f2(y)xdx

(16) E3=ω·γ3·∫y0+45y0+60dy∫g(y)f3(y)xdx

(17) E4=ω·γ4·∫y0+60y0+100dy∫g(y)f4(y)xdx

In (15), (16), (17), γ2, γ3 and γ4 are bulk density of mudstone, sandstone and abandoned material, corresponding to γ2 = 23 kN/m3, γ3 = 20.5 kN/m3 and γ4 = 21 kN/m3. Meanwhile, it should be emphasized that since slope equation g(y) is a piecewise function, it should be divided into different intervals when calculated definite integral. External force power of mudstone, sandstone and abandoned material in slope potential failure area can be solved by taking (7), (8), (9), (10) into (15), (16), (17). Finally, external force power can be calculated through adding together external force power of each layer of rock mass, and it can be expressed as:(18) E=E1+E2+E3+E4

When upper bound limit analysis method is used to calculate internal energy dissipation power of slope, the following two assumptions are adopted [37]. Firstly, rock mass is regarded as a rigid body, that is to say, internal energy dissipation due to velocity changed is considered but volume changed is not considered. Secondly, slip surface is regarded as velocity discontinuity surface, that is to say, internal energy dissipation only occurs on slip surface but not occurs inside or outside slip surface. It means that internal energy dissipation only occurs on combined logarithmic spiral. At the same time, based on basic principle of plastic mechanics, since friction energy dissipation and hypothetical shear expansion energy dissipation are exactly offset, led to internal energy dissipation is only furnished by cohesion force.

According to the above analysis, internal energy dissipation entirely furnished by cohesion force on combined logarithmic spiral. For potential failure mechanism in coal of slope, take r·dθ/cosφ1 on logarithmic spiral as microelement length and the corresponding cohesion force can be expressed as c1·r·dθ/cosφ1. Then internal energy dissipation power in coal of slope can be expressed as:(19) D1=∫θ0θ1c1·v·cosϕ1·rdθcosϕ1=c1·ω·∫θ0θ1c1·v·cosϕ1·rdθcosϕ1=c1·ω·∫θ0θ1r2dθ=c1·ω2tanϕ1·(r12−r02)

Similarly, internal energy dissipation power in mudstone, sandstone and abandoned material can be expressed as:(20) D2=c2·ω2tanϕ2·(r22−r12)

(21) D3=c3·ω2tanϕ3·(r32−r22)

(22) D4=c4·ω2tanϕ4·(r42−r32)

In (19), (20), (21), (22), c1, c2, c3 and c4 are cohesion of coal, mudstone, sandstone and abandoned material respectively, corresponding to c1 = 90 kPa, c2 = 57.5 kPa, c3 = 65 kPa and c4 = 35 kPa. Meanwhile, φ1, φ2, φ3 and φ4 are internal friction angles of coal, mudstone, sandstone and abandoned material, corresponding to φ1 = 17°, φ2 = 17.5°, φ3 = 19°, φ4 = 16°. And r0, r1, r2, r3 are distance between the point of rotation center and the starting point of logarithmic spiral in coal, mudstone, sandstone and abandoned material, r4 is distance between the point of rotation center and the end point of logarithmic spiral in abandoned material. According to distance calculation method, it can be obtained:(23) {r0=y02+f12(y0)r1=(y0+10)2+f12(y0+10)r2=(y0+45)2+f22(y0+45)r3=(y0+60)2+f32(y0+60)r4=(y0+100)2+f42(y0+100)

Internal energy dissipation power of coal, mudstone, sandstone and abandoned material in slope potential failure area can be solved by taking formula (23) into (19), (20), (21), (22). On the same way, internal energy dissipation power can calculate by:(24) D=D1+D2+D3+D4

3.4 Stability analysis results

The calculation method of external force power and internal energy dissipation power of open-pit mine slope is proposed above. In order to describe limit equilibrium condition of slope, ratio function ξ is established and defined as follows:(25) ξ=(D/E)min

Obviously, when slope is in limit equilibrium status, corresponding ratio function ξ = 1. For slope in non-limit equilibrium state, strength parameters consist of cohesion and internal friction angle of rock mass should be reduced repeatedly until to equilibrium [38]. And reduction formula of rock mass is as follows:(26) {c′=cFtanϕ′=tanϕF

In formula (26), c and c' are cohesion of rock mass before and after strength reduction, φ and φ' are internal friction angle of rock mass before and after strength reduction. According to the change law of strength reduction, reduction coefficient F is obviously negatively correlated with ratio function ξ. Therefore, ratio function ξ can be approximated to 1 by gradually increasing reduction coefficient F, and the open-pit mine slope stabilitycan be finally analyzed.

The formulas for calculating external force power and internal energy dissipation power without strength reduction were analyzed above. In order to solve corresponding ratio function ξ at this time, the constraint conditions should be properly analyzed in combination with slope engineering practice, and mainly including the following aspects.

First of all, rock layers should be located in increasing interval for corresponding logarithmic spiral, and it can be expressed as:(27) {y0≥−1.617a1−1.640a2≤y0+10≤−0.314a1−1.717a3≤y0+45≤−0.331a2−1.570a4≤y0+60≤−0.364a3y0+100≤−0.298a4

Secondly, at interface of rock layers, coordinates of two adjacent logarithmic spiral are consistent, and it can be expressed as:(28) {f1(y0+10)=f2(y0+10)f2(y0+45)=f3(y0+45)f3(y0+60)=f4(y0+60)

Finally, potential failure mechanism should be located inside edge slope, it means that coordinates of logarithmic spiral in each rock layer should be larger than edge slope in x direction, and it can be expressed as:(29) {f1(y)−g(y)>0y∈(y0，y0+10]f2(y)−g(y)>0y∈[y0+10，y0+45]f3(y)−g(y)>0y∈[y0+45，y0+60]f4(y)−g(y)>0y∈[y0+60，y0+100]

According to above constraint conditions, when strength parameters of rock mass are not reduced, corresponding ratio function ξ > 1. Gradually increasing strength reduction coefficient can form a relationship curve between strength reduction coefficient F and ratio function ξ. The corresponding reduction coefficient is same with stability coefficient when ratio function ξ = 1, and the most dangerous sliding surface can be obtained in accordance with corresponding value of each parameter. Similarly, stability analysis corresponding to landslides occurs at A1, A2, A3, A4, A5, A6, A7 and A8 can be solved respectively. Stability analysis results when landslide occurs at A1 and A2 are shown in Fig. 5(a), stability analysis results when landslide occurs at A3 and A4 are shown in Fig. 5(b), stability analysis results when landslide occurs at A5 and A6 are shown in Fig. 5(c), stability analysis results when landslide occurs at A7 and A8 are shown in Fig. 5(d).Fig. 5 Results of slope stability analysis.

Fig. 5

It can be seen from Fig. 5, when landslide occurs at the position of A1, A2, A3, A4, A5, A6, A7 and A8, the corresponding stability coefficient are 1.44, 1.53, 1.72, 1.57, 1.49, 1.36, 1.31 and 1.28. In accordance with the above analysis results, when landslide occurs at A8, corresponding stability coefficient is the minimum, which is stability coefficient of open-pit mine slope.

3.5 Morphological parameters influence

In this part, slope stability coefficients were calculated through changing slope angles and plate widths of single step, so as to reveal the influence law of morphological parameters on open-pit mine slope stability, and furnished a theoretical basis for slope morphological optimization. For slope angles and plate widths of single step, selecte 10 % decrease, 5 % decrease, 5 % increase and 10 % increase, and applies stability analysis method proposed above to calculate the corresponding stability coefficient. Influence of step angles on slope stability is shown in Fig. 6(a), influence of plate widths on slope stability is shown in Fig. 6(b).Fig. 6 Influence of morphological parameters on slope stability.

Fig. 6

It can be seen that slope stability coefficient increased with the decreased of step angles or the increase of plate widths. On the contrary, slope stability coefficient decreased with the increased of step angles or the decrease of plate widths. For step angles, the most significant area of influence is A1 position, and for plate widths, the most significant area of influence is A8 position. As a conclusion, it is necessary to be attentive to dump step stability in the optimization of step angles, and it is necessary to be attentive to integral slope stability in the optimization of plate widths.

3.6 Discussion

So as to verify applicability of this method, limit equilibrium method is selected to evaluate accuracy of stability analysis results. Limit equilibrium method analyzes slope stability from the perspective of force and moment [39], while upper bound limit analysis method analyzes slope stability from the perspective of velocity and energy [40]. Although analysis angles of two methods are different, calculation results must be accordant. Calculation results of slope stability analysis are shown in Fig. 7. Among them, step angels decreased 10 % is shown in Fig. 7(a), step angels decreased 5 % is shown in Fig. 7(b), initial morphology of slope is shown in Fig. 7(c), step angels increased 5 % is shown in Fig. 7(d), step angels increased 10 % is shown in Fig. 7(e), plate widths decreased 10 % is shown in Fig. 7(f), plate widths decreased 5 % is shown in Fig. 7(g), plate widths increased 5 % is shown in Fig. 7(h), plate widths increased 10 % is shown in Fig. 7(i).Fig. 7 Results of slope stability analysis.

Fig. 7

It can be seen from Fig. 7, for slope stability calculation results, upper bound limit analysis method is a little higher than that of limit equilibrium method. This is because upper bound limit analysis method selects allowable velocity field as allowable condition, and the functional value obtained is upper limit of real load, which leads to conservative calculation results of stability coefficient. While, limit equilibrium method usually ignores tangential force on the side interface of strips, and calculation results of stability coefficient are generally considered to be dangerous. However, error rate of stability coefficient calculated by upper bound limit analysis method and limit equilibrium method is less than 5 %, which is fully verified accuracy of two methods [41,42].

At the same time, the stability coefficient results calculated by upper bound limit analysis are upper limit solutions for stability coefficient, and the error is easy to eliminate. What is more, the most dangerous sliding surface obtained by upper bound limit analysis can satisfy the requirements of velocity separation. Therefore, upper bound limit analysis method has a wide application prospect in engineering practice.

4 Conclusions

(1) Failure mechanism of slope rock mass is analyzed when shear failure occurs, increasing interval of logarithmic spiral in plane rectangular coordinate system is solved combined with discrete mathematical theory, and method for visualization of logarithmic spiral in increasing interval is proposed based on least squares principle.

(2) Combined with slope engineering practice in open-pit mine, energy calculation methods and energy balance equation of slope are established, cycle algorithm for slope stability analysis is proposed in accordance with upper bound limit analysis theory and strength reduction principle.

(3) The stability coefficients at the conditions of different slope angles and plate widths of single step were analyzed, the influence of slope morphology on stability coefficient was explained, and the key factors to be considered in slope morphology optimization were analyzed.

(4) The accuracy of stability analysis method proposed in this paper is evaluated reasonably by using limit equilibrium method. The error rate of stability coefficient calculation results of two analysis methods is less than 5 %, and it is fully verified wide applicability of the method in engineering practice.

Data availability statement

The data that has been used is confidential.

CRediT authorship contribution statement

Zhen Wang: Writing – original draft. Ang Li: Methodology, Data curation. Tianhao Yang: Methodology.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was supported by Scientifc and Technological Project in Henan Province (242102320222 ).
==== Refs
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