APPLYING THE STRENGTH REDUCTION

METHOD TO STUDY OF STABILITY OF

RESIDUAL MOUNTAINS: A PARTICULAR

APPLICATION

Xin Jin*

College of Art and Design, Shaanxi University of Science and Technology, Xi’an,

Shaanxi, 710021, China

wyjinxin@sust.edu.cn

Yucheng Hua

College of Art and Design, Shaanxi University of Science and Technology, Xi’an,

Shaanxi, 710021, China

Qiao Tang

College of Art and Design, Shaanxi University of Science and Technology, Xi’an,

Shaanxi, 710021, China

Reception: 31/11/2022 Acceptance: 26/11/2022 Publication: 21/01/2023

Suggested citation:

J., Xin, H., Yucheng and T., Qiao. (2023). Applying the strength reduction

method to study of stability of residual mountains: A particular

application. 3C Tecnología. Glosas de innovación aplicada a la pyme, 12(1),

33-52. https://doi.org/10.17993/3ctecno.2023.v12n1e43.33-52

https://doi.org/10.17993/3ctecno.2023.v12n1e43.33-52

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143

Ed.43 | Iss.12 | N.1 January - March 2023

ABSTRACT

Due to huge disaster-caused force, seismic geological disasters primarily induce

residual landslide, collapse and debris flow disasters with far higher hazard extent

than that of earthquake disasters. Wherefore, this paper, with various vibration slopes

caused by the most representative Wenchuan earthquake as research objects,

introduces the strength reduction method to the stability study of residual mountains in

a certain area, puts forward dynamic stability slope evaluation method based on

dynamic and overall strength reduction method to obtain the mountain slope stability

situation in the process of gradual instability, searches out the sliding surface of

progressive expansion making use of dynamic strength reduction method and

calculates dynamic safety index in the process of gradual slope instability pursuant to

the calculation advantages of dynamic strength reduction method, so as to realize the

analysis and regulation of the whole process of slope instability. The results show that

in the stability analysis of a homogeneous slope, the safety index of slope stability

calculated by strength reduction method is 6.7%, 8.8% and 10.5% higher than that

calculated by finite element limit equilibrium method, Bishop method and Janbu

method respectively. In the stability analysis of a multi-layer soil slope, the safety

index of slope stability calculated by strength reduction method is 4.8%, 4.3% and

9.4% higher than that calculated by other three algorithms. While in the stability

analysis of soft interlining slope, the safety indexes of slope stability calculated by the

method proposed in this paper are increased by 26.10%, 29.11% and 26.46%

respectively compared with other three algorithms, indicating that the stability of

landslide residual mountains calculated by strength reduction method proposed in this

paper is the highest.

KEYWORDS

Strength reduction method; Landslide residual mountain; Stability study; Safety index;

Sliding surface

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PAPER INDEX

ABSTRACT

KEYWORDS

1. INTRODUCTION

2. FAILURE CHARACTERISTICS OF MOUNTAIN SLOPES AND IMPROVEMENT

OF STRENGTH REDUCTION METHOD

2.1. Deformation and failure characteristics of residual slope after landslide

2.2. Actual failure process of mountain slope

2.3. Sliding surface search based on dynamic strength reduction

3. DYNAMIC STABILITY ANALYSIS OF MOUNTAIN SLOPE BASED ON

STRENGTH REDUCTION METHOD

4. RESULTS AND ANALYSIS

4.1. Landslide residual mountain conditions in a certain area

4.2. Experiment parameters of mountain slope in a certain area

4.3. Compare with the results of different algorithms

5. DISCUSSION

6. CONCLUSION

7. DATA AVAILABILITY STATEMENT

REFERENCES

CONFLICT OF INTEREST

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1. INTRODUCTION

In mountainous and hilly areas, landslides are more common, and similar to

earthquakes, mudslides and other disasters, generally have relatively large hazard

extent. In the vast land area of China, the geographical conditions are relatively

complex, and the landslide areas are widely distributed, especially in the mountainous

areas of southwest, northwest, and east China [1-2]. When a landslide occurs, partial

or whole pieces of land will appear one after another in a relatively slow speed and a

relatively long cycle, and will intermittently slide. [3-4].

Landslides are extremely harmful, especially that large-scale landslides can

destroy entire villages, cut off rivers, destroy farmland and forests, and even damage

the safety of life and property of people and countries to a large extent, seriously

hindering and destroying the construction process of countries [5]. Landslides in

China have the characteristics of various types, large scale, wide distribution, strong

concealment and strong destructiveness, and most landslides are sudden and

unpredictable, which brings serious harms to the society [6].

In this regard, it is of great significance to implement monitoring and early warning

of landslides. Landslide movement is highly complex and is affected by many factors.

At present, it is difficult to thoroughly understand the internal characteristics of each

landslide, and to accurately predict each landslide [7-9]. However, landslide

monitoring helps to master and analyze the evolution and characteristics of landslide

mass. With the continuous progress of landslide monitoring technology, in order to

obtain more detailed landslide data, a more in-depth understanding of landslides can

be obtained [10-12].

In recent years, many foreign researchers have carried out research on rock

mechanics algorithms. For example, the literature [13] introduced the concept of

damage mechanics in the metal creep fracture research and rock mechanics research

for the first time. The literature [14] used the concept of fracture surface to

theoretically discuss the continuous damage behavior of rock and concrete. Since

then, people have established various damage theories based on different damage

mechanisms and basic theories, and applied them to the study of nonlinear, plastic,

and viscoplastic damage of rock materials. The literature [15] first applied the damage

theory to establish damage mechanics model of rock-concrete continuum. The

literature [16] established corresponding models and theories based on the structural

characteristics of the rock itself. The literature [17] proposed the famous "strain

equivalence hypothesis", which laid the foundation for the study of damage theory.

The literature [18] proposed a new elastic-plastic damage model based on irreversible

thermodynamics and damage mechanics, which comprehensively considered plastic

friction deformation and plastic pore deformation, and adopted damage variables to

describe the development of microscopic defects. The literature[19] defined a damage

variable as the second-order tensor of fracture density, took into account the damage

evolution of this damage variable pursuant to fracture propagation, and thereby

established a corresponding damage model. Based on the homogenization theory,

whereafter, a thermodynamic framework for the meso-mechanical damage model was

built. Zhu Qizhi et al. reckoned that the rock was a heterogeneous material composed

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of an elastic solid matrix and fractures, and proposed a corresponding meso-

mechanical damage model. The literature [20], on the basis of considering the

dynamic propagation of micro-fractures, proposed a corresponding meso-mechanical

model of rock damages, believing that the stress-strain curve, failure strength and

damage development rate of rocks were closely related to the friction index, initial

fracture length and loading rate. The literature [21] came up with a micro-fracture

damage model of brittle rock under uniaxial compression load as well as theories of

micro-fracture fracture mechanics and rock damage mechanics based on the

assumption of random distribution of micro-fractures. To be concrete, the mechanical

properties under the load were analyzed, that is, when the external load reached a

certain level, the micro-fractures began to expand,, the mechanical properties of the

rock changed, and until the macro-fracture of the rock occurred, the fracture growth

rate increased with the external load.

With the in-depth research aiming at foreign researchers, domestic scholars have

also carried out research on the stability of rock and soil slopes. For instance, based

on the principle of object balance, the literature [22] proposed a finite sliding

displacement evaluation method of slope stability according to slope potential

deformation, in which the force exceeding the yield resistance of the sliding body is

used as the calculation standard for the occurrence of the slope and landslide. The

literature [23] used the FLAC valuation method of slope stability according to slope

potential deformation, and carried out the research of seismic slope failure

mechanism combined with the seismic slope numerical simulation software with both

tensile and shear failure functions. Through analysis, it was concluded that the failure

of the seismic slope was composed of the upper tensile failure and the lower shear

failure of the potential rupture zone, and the method for determining the location of the

rupture surface of the earthquake slope was given through various means. The

literature [24] performed a large number of on-site investigations. Precisely, on the

basis of aforementioned above, the special instability phenomena such as vibration

collapse and high-speed ejection of the slope under the strong earthquake load were

studied, and the genesis mechanism of earthquake-triggered collapse and landslide

was classified according to the specific slope structure. Moreover, a calculation

program for the safety index of a landslide numerical simulation software with both

tensile was proposed to study the stability of slopes under the condition of known

sliding surfaces, and a comparative discussion between the traditional seismic slopes

was carried out via FLAC 3D displacement evaluation method. From a mesoscopic

approach, the literature [26] studied the instability mode of slopes via a strength

reduction calculation module in the RFPA software. The literature [27] discussed the

strength reduction method and proposed a more reasonable reduction calculation

method, which promoted actual use numerical simulation software with both tensile.

With the deepening of research, there are more and more improved algorithms for

strength reduction. However, numerous studies have found that in order to study the

stability of residual mountains, it urgently requires to take into account the shear-

tensile composite yield criterion of mountains, as well as single static safety index,

principal stress and cohesion of mountain slopes. For this, numerical simulation

software with both tensile did calculate damaged area of the mountain slope [28-30],

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but did not consider the soil layer of the mountain slope. Not all the soil layers of the

mountains are homogeneous, and some soil layers are multi-layer soil or contain

weak inter-layers, so it is very necessary to study the stability of the residual mountain

according to the conditions of different soil layers.

Pursuant to the fact that the general algorithm does not take into account the pull

fracture and further study landslide residual mountains, in this paper, the shape

change and structural damage characteristics of the residual slopes on the rear wall of

the landslide is analyzed, and the sliding surface searching of progressive expansion

is implemented making use of dynamic strength reduction method. At the same time,

the residual slope of the rear wall of a landslide in a certain area is also a typical

representative and epitome of the many shaking slopes of the "5.12" Wenchuan

earthquake. It can be predicted that in the next 3 to 5 years, it will be a period of high

incidence of collapse, landslides and debris flows in earthquake-stricken areas. The

research ideas, methods and understanding of its post-earthquake stability will also

provide guidance and reference for the evaluation of similar slopes in earthquake-

stricken areas.

2. FAILURE CHARACTERISTICS OF MOUNTAIN

SLOPES AND IMPROVEMENT OF STRENGTH

REDUCTION METHOD

2.1. DEFORMATION AND FAILURE CHARACTERISTICS OF

RESIDUAL SLOPE AFTER LANDSLIDE

According to the interpretation of low-altitude aerial photography near the residual

mountain on the rear wall of a landslide in a certain area by helicopter, and combined

with on-site geological mapping, it shows that after the "5.12" earthquake, there are

mainly three groups of fractures on the surface of the entire back-edge mountains[31].

And the distribution of residual mountains and fractures on the rear wall of the

landslide is primarily: (1) N40°～50°E and N70°E～

EW direction, this group of

fractures is mainly consistent with the boundary of the rear wall of a landslide in a

certain area, with an extension length of about 20～200m and; (2) N10°W ~ near SN

direction ~ N10°E, this group of fractures is the same as the main direction of the

Dashuigou and Xiaoshuigou valleys, with an extension length of about 50 ~ 400m,

and showing that unloading and tension are towards the free surface of the big and

small ditches, but the scale of the side of the small ditch is obviously small; (3) N70°W

direction, this group of fractures are distributed on the ridge line of the highest

watershed and are consistent with the trend of the ridge line, with an extension length

of about about 1000m and belonging to the earthquake-vibration fractures on the

steep slope of the ridge [32].

The above-mentioned fractures are all extensional, with an opening width of 10-50

cm. Since the penetration depth of each fracture cannot be measured, according to

the earthquake in the core area (such as Dujiangyan, Yingxiu, Wolong, Beichuan,

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Qingchuan, etc.), the characteristics of the slope seismic effect (that is, the slope

slump effect caused by the earthquake on the thin ridge with a slope gradient of

obviously more than 50° and without deep tensile fracture) are adopted to predict the

distribution of the residual mountain mass on the rear wall of the landslide in a certain

area. The depth of the pull fracture is generally between 10 and 50m. Except for the

N40°-50° E group fractures near the front edge of the mountain, which has penetrated

into the weakly weathered rock body, the rest of the fractures near the ridge line are

expected to be a deep location within the strongly weathered rock body. [33].

Judging from the distribution of the above-mentioned three groups of fractures with

different extension directions and scales, the shallow surface layer of the entire

residual mountain has basically been disturbed as a whole. Instead, collapse, slide

and pull fractures also occur in the direction of large and small ditches, but the former

has more advantages. Although the superficial integrity of the residual mountain has

been basically completely destroyed, the integrity of the underlying weakly weathered

rock mass has remained basically intact.

2.2. ACTUAL FAILURE PROCESS OF MOUNTAIN SLOPE

The slope instability is a gradual process that evolves from part to the whole, rather

than an instant process. Because of existence of weak surface, the mechanical index

of mountain slope decreases due to rainfall, the load dustribution is unbalanced and

local stress concentration occurs, thereby resulting in some elements are damaged

first [34]. Slowly, the principal stress will continue to extend and adjust its own

concentration after the internal components is damaged, regrouping into a stress

region. At this time, the damage area of the mountain structure increases and

converges, and finally becomes a complete mass through sliding surface. Figure 1

shows the whole process of slope damage.

Figure 1. Schematic diagram of progressive failure of mountain slope

In Figure 1, at points 1 and 2, if the shear force is greater than that of the peak

point, the mechanical index will decrease due to rainfall,. At this time, it can be

Potential sliding

surface

Sliding

belt

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regarded as that the shear stress at point 3 has just peaked, and the shear stress at

point 4 is well below the peak value. Moreover, since the sliding belt will continue to

expand, the stress will be completed at point 4, the highest point will be completed at

point 4, and at the same time, a complete sliding surface will be formed. Obviously,

when the sliding zone is produced, its strength value will change from the highest

point to the lowest point. In this way, the whole process can improve the

characteristics of the formation of the Osian sliding zone.

The damage of slope structure and the process of landslide are not sudden, but

gradual. Also, the slope instability is a gradual unsuccessful process, so it is

necessary to reflect the whole gradual process of slope stability for evaluation [35]. If

the strength value of the slope instability is wanted to decrease, strength reduction

calculation will cause the final plastic zone to be too large.

2.3. SLIDING SURFACE SEARCH BASED ON DYNAMIC

STRENGTH REDUCTION

The whole process deformation characteristics of geotechnical materials are mainly

classified into two types, namely "hardening" and "softening". Regarding the shape

change and mechanism damage characteristics of the sliding belt in the unsafe

process of mountain slope, relevant researchers have verified via theory and

experiments that the sliding belt of the slope has softening characteristics. Thus, the

slope stability analysis must take into account the strain softening characteristics of

geotechnical materials [36].

Based on referred strength reduction criteria and softening characteristics of the

sliding belt, some scholars first proposed a calculation method and then obtained the

parameters for reducing the local slope strength cohesion and internal friction angle,

as shown in Figure 2

Figure 2. Strength reduction calculation process

Figure 2 shows the strength reduction process in detail, in which the minimum

value of principal stress calculated is the same as the value of tensile failure strength.

Concretely, it is first to destroy the reduction element index, and then obtain the

Elastoplastic Mechanics

Calculations

Determining the Slope

Plastic Region

Rreduction factor

Reduced local slope strength

parameters

,cϕ

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parameters for reducing the local slope strength cohesion and internal friction angle,

so as to perform mechanical calculation aiming at the slope plastic region. Assuming

that the deformation strength characteristics of the sliding zone geotechnical materials

conform to the ideal elastic-plastic softening model, by calculating the damaged area

of the mountain slope according to the elastic-plastic criterion, it is found that the area

of slope increases slowly, which stops when the slope state reaches the limit

equilibrium [37].

Tensile failure often occurs in a certain range of landslide crest, so after

determining mechanical strength parameters of the sliding belt shear yielding, it is

very necessary to integrate the Mohr Coulomb tensile failure model into the numerical

study, that is, if the minimum value of principal stress is the same as the value of

tensile failure strength, the rock and soil mass will suffer unsafe state process. Tensile

composite yield criterion can be expressed as:

（1）

A single static safety index can be expressed as:

(2)

In the above formulas, and

are the maximum and minimum principal

stresses, respectively; is the cohesion force; is the internal friction angle; and

is the tensile strength.

The tensile strength of the slope is basically unchanged, so the tensile strength is

not reduced in the strength reduction method [38-39]. Only under the conditions that

earthquake and other basic factors work, the tensile strength of the slope will be and

needs to be reduced. When calculating the damaged area of the slope according to

the elastic-plastic criterion, the mechanical strength parameter of the sliding belt is

narrowed by the reduction factor , that is:

（3）

In the above formula, and are the cohesion force and the internal friction

angle of the local damaged area, respectively

3. DYNAMIC STABILITY ANALYSIS OF MOUNTAIN

SLOPE BASED ON STRENGTH REDUCTION

METHOD

From the above data research and actual operation, it is obvious that whether the

slope is safe and stable as time goes by, the whole process of the slope from local

Fσ σ=−=

1 3

1 sin 2 cos 0

1 sin 1 sin

Fσ σ

+ϕ ϕ

=−+ =

− ϕ − ϕ

tan

arctan

loc

⎧ ⎫

⎪ ⎪

⎨ ⎬

⎪ ⎪

ϕ=

⎪ ⎪

⎩ ⎭

loc

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failure and gradual development to overall instability will change dynamically,

experiencing stable, less stable and unstable evolutionary processes.

The steps for evaluating the dynamic stability of mountain slopes are shown in

Figure 3.

Figure 3. Dynamic evaluation process of mountain slope stability

In the traditional slope stability evaluation method, a single static safety index

used to evaluate the slope stability, but the static safety index

is not conducive to

the stability analysis and regulation of landslide disasters. Therefore, this paper

proposes that the safety index is a dynamic stability evaluation index.

(1) When the plastic zone appears in the local element of the mountain slope, the

mechanical parameters of the element in the plastic zone are reduced to indicate that

the strength of the local element softens and decreases.

(2) The overall strength reduction method and formula (4) are adopted to reduce

the strength parameters and

, of the entire slope. The reduction range includes

not only the softened plastic zone, but also the unyielding slope. For the plastic zone

in step (1), the softened parameter value is reduced. When the reduction reaches a

sudden change in the slope displacement, the safety index

in the current state is

obtained.

(3) According to Figure 3, after the strength parameters and

are reduced, the

elastoplastic mechanical balance calculation is carried out. Since the elements around

the plastic zone of stress concentration will generate a new plastic zone, the plastic

zone elements will increase.

(4) The increase of plastic zone elements indicates the expansion of the slope

sliding surface. At this time, the mechanical parameters of the newly formed sliding

surface are reduced according to steps (1) and (3).

●Edge model ●Initial and Boundary Conditions ●Reduction factor

Safety factor calculation

Dynamic discount calculation

Sliding surface extension

End of dynamic analysis

Reduced Strength

Parameters by K

K↑

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(5) According to step (2), the slope safety index is obtained after sliding surface

expansion.

(6) The dynamic and overall strength reduction method is used to repeat the above

steps to obtain a series of dynamic safety indexes, stop the reduction after the sliding

surface is penetrated, and finish dynamic security and stability evaluation of the slope.

In order to track the evolution law of the slope safety index in the whole process,

this paper, combined with the advantages of dynamic strength reduction method,

selects a representative section to study how to bring the strength reduction

calculation method into the analysis of slope dynamic safety and stability state, in

which the possibility of overall landslide occurrence is small in the gradual failure

process of the slope, but the reduction index obtained by the dynamic strength

reduction method does not represent the safety index of the slope (as in equation

(3)), only representing the degree of reduction. Therefore, the dynamic and overall

strength reduction calculation methods are combined to analyze the slope stability,

and slowly enter the unsafe state process, thereby obtaining the safety index via

calculation, and obtaining the stability state of the slope on the basis of the safety

index.

Concretely, the fractures at an overall back-edge mountain are dislocated to reduce

and a representative section is selected as the entire slope, so as to obtain the safety

index under the limit state, that is:

（4）

In the formula, and are the reduced cohesion force and the internal friction

angle, respectively.

4. RESULTS AND ANALYSIS

4.1. LANDSLIDE RESIDUAL MOUNTAIN CONDITIONS IN A

CERTAIN AREA

In a certain area, there is still a mountain with a vertical length of 460-500m and a

horizontal width (between the big water ditch and the small water ditch) of 460-640 m

at the rear wall of the landslide, which is characterized by a steep front edge and a

gentle back edge. The slope of the front edge (that is, the rear wall of the landslide in

a certain area) is about 45°, and the back edge is the original slope with a gentle

slope of 10° to 15°. It can be seen via qualitative analysis that the residual mountain

mass on the rear wall of a landslide in a certain area will be dominated by local

sporadic collapse and slump in the superficial part, and the possibility of overall

landslide occurrence is small. Therefore, in the quantitative calculation of stability, the

'tan

arctan

⎧ ⎫

⎪ ⎪

⎨ ⎬

⎪ ⎪

ϕ=

⎪ ⎪

⎩ ⎭

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possible sliding surface is formed based on the most unfavorable residual mountain

slope structure and its various structural surfaces, which is regarded as the controlling

boundaries for quantitative calculations of stability. At the same time, the various

residual mountain slopes and superficial residual mountain slopes formed due to the

earthquake will obviously serve as the boundary of the back edge tensile fracture.

Based on comprehensive analysis, it can be inferred that the residual mountain

may have a large-scale failure mode: the back edge through fractures are dislocated

→ the middle layer slides →

the front edge shear failure occurs, so it can also be

attributed to a three-stage mechanism of tension cracking-sliding-shearing. Among

them, the front edge of the slope crest is most likely to collapse.

4.2. EXPERIMENT PARAMETERS OF MOUNTAIN SLOPE IN A

CERTAIN AREA

According to the geological prototype, geometric mode of right bank is built, and to

perform stability calculation, a representative section is selected. Specifically, the

section model of the computational mesh model is set to be 478m long and 470m

high, and the total number of elements and nodes is 44695 and 71396 respectively.

The rock-material constitutive model adopts the Mohr Coulomb rule for the most

suitable elastic-plastic model criterion. The mechanical parameters of the three kinds

of rock dikes in the corresponding stratum distribution on the vertical surfaces on the

left and right sides, and fixed constraints on the bottom boundary are determined

respectively.

Combined with the exposed shape of the section, the slope mass is generalized as

, , , , and

rock-soil mass and concrete after support (C25

strength grade), and the mechanical parameters of the three kinds of rock dikes in the

corresponding stratum distribution are the same as IV-type rock mass. The natural

bulk density, elastic model and Poisson's ratio appearing in this experiment are all

measured by measuring instruments before the experiment. The layer parameter

values are shown in Table 1.

Table 1. Parameters of the calculation model for the slope on the right side of the residual

mountain.

2III

1III

Medium

Natural test

weight /

（kN•m-3）

Elastic Modulus

/Gpa

Poisson's

ratio Cohesion /Mpa Internal friction

angle /（°）

f231 25.8 2.0 0.28 0.9 22.8

Deep

unloading

fracture

26.2 2.0 0.28 2.0 36.0

V2 class 22.1 2.0 0.27 1.8 21.8

V1 class 24.5 4.0 0.27 2.0 26.5

IV class 25.8 6.0 0.26 7.0 38.6

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possible sliding surface is formed based on the most unfavorable residual mountain

slope structure and its various structural surfaces, which is regarded as the controlling

boundaries for quantitative calculations of stability. At the same time, the various

residual mountain slopes and superficial residual mountain slopes formed due to the

earthquake will obviously serve as the boundary of the back edge tensile fracture.

Based on comprehensive analysis, it can be inferred that the residual mountain

may have a large-scale failure mode: the back edge through fractures are dislocated

→ the middle layer slides → the front edge shear failure occurs, so it can also be