Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 5, No. 1, 2023 

 

191 

Study	on	Calculation	Method	of	Stability	of	Damaged	
Sliding	Dangerous	Rock	on	Reservoir	Bank	
-- Taking Jianchuandong Dangerous Rock in Wushan County as An Example 

Hang Zhu* 

Hohai College, Chongqing Jiaotong University, Chongqing, 400074, China 

* Corresponding author: Hang Zhu (Email: 1250688026@qq.com) 

 

Abstract: The rapid destruction of dangerous rock produces a large amount of energy, causing heavy casualties and economic 
losses every year. There are various failure modes and stability calculation methods of dangerous rock. The common failure 
modes of dangerous rock are tipping mode, sliding mode and falling mode. For large tower pillar dangerous rock with degraded 
foundation, it is unreasonable to use conventional stability calculation methods to calculate the stability of dangerous rock. This 
paper takes Jianchuandong dangerous rock in Wushan County as the research background, considering the expansion of rock 
cavity of damaged dangerous rock base and the deterioration of rock mass in the base of large tower column reservoir bank, and 
establishes the stability calculation method of dangerous rock suitable for the damage deterioration of the base. It is innovative 
on the basis of the traditional stability calculation method of dangerous rock, which has important practical significance and 
theoretical basis for the calculation of the stability of unconventional dangerous rock. 

Keywords: Damaged dangerous rock, Deterioration of the base, Stability, Cavern extension. 
 

1. Introduction  
The formation and failure of dangerous rock is a long 

process. From the macroscopic concept of time, because the 
weathering of rocks is certain, so is the movement of 
geological structures, all the unstable dangerous rock mass 
will be unstable and fail no matter whether it is treated or not. 
The dangerous rock mass in the stable state at the present 
stage will gradually become unstable until it becomes 
unstable and fails, and the unstable failure of dangerous rock 
is inevitable. However, since the instability and failure of 
dangerous rock in the macroscopic time concept are often 
measured in thousands or even millions of years, the 
formation and failure process of dangerous rock is not 
inevitable from the microscopic time perspective. The 
prevention and control of dangerous rock is of great 
significance to the protection of human production and life 
and property safety. Many scholars have studied dangerous 
rocks, Zhang Zhihua et al [1] According to the standard 
method, the stability of dangerous rock under different 
working conditions is calculated. The calculation shows that 
the dangerous rock is in a basically stable to stable state, and 
the stability coefficient is reduced to 1.081 when the shear 
strength of the base rock mass is used to calculate the stability 
after the dry and wet cycle, and the dangerous rock mass may 
suffer the collapse of the base fracturing slip. Ran Tao et al [2] 
Combined with field investigation, the development form, 
volume and failure mode of the dangerous rock mass in the 
four-layer rock scarp belt of Wanzhou District are 
investigated in detail, and the stable state of the rock mass is 
evaluated through qualitative and quantitative analysis. Hu 
Mingjun et al [3] In this paper, the carbonate bank slope in the 
fluctuation zone of the Three Gorges reservoir is studied, and 
the data of rock mass deterioration and damage under periodic 
water level changes are obtained by means of experiments. 
Zhou Yuntao et al [4] When calculating the stability of 

dangerous rock, the influence of receding rock cavity of 
dangerous rock base on the stability of near horizontal hard 
soft rock interbed is considered. Zhou Peng [5] In this paper, 
the stability of 3D geological body bounded by structural 
plane is studied from the perspective of 3D, and a calculation 
method of 3D stability of sliding dangerous rock considering 
the boundary of structural plane is proposed. Deng Yue et al 
[6] Through the dynamic time-history curve of seismic wave 
attenuation under dynamic action, the function of equivalent 
blasting dynamic load with time is obtained. Combined with 
fracture mechanics, several conventional stability calculation 
methods of dangerous rock under dynamic action are 
established. Li Xiangqing et al [7] In order to study the 
stability of high slope in alpine region, the temperature field 
was established considering frost heave force, gravity and 
crack water pressure, and the stability calculation method was 
established based on fracture mechanics. 

Current research on the stability of dangerous rock, the 
evaluation method is numerous, for the special dangerous 
rock mass, the traditional calculation method of the stability 
of dangerous rock is not fully applicable to the pillar of 
dangerous rock mass base damage type dangerous rock 
stability calculation, such as ZengZi Yan dangerous rock 
crushing destruction, longmen for whole village and 
dangerous rock coffin ridge may occur such as dangerous 
rock crushing destruction, Jianchuandong dangerous rock is a 
typical tower pillar rock cut by structural fissure. The lower 
bedrock is deteriorated by dead weight load and long-term 
circulation soaking of reservoir water. Therefore, it is of great 
significance and practical effect to establish a new calculation 
method for the stability of damaged damaged rock in 
reservoir bank. 

2. Study Area Profile 

2.1. Background of study area 
The study area is located in the joint belt of several tectonic 



 

192 

systems, such as the arcuate structure of Daba Mountain, 
Sichuan-Hubei Hunan-Guizhou uplift fold belt and eastern 
Sichuan fold belt. The lowest point is the side of Peishi 
Yangtze River with an altitude of 175m, and the highest point 
is Taiping Mountain on the southern slope of Daba Mountain 
with an altitude of 2680m, which is a typical karst landform. 
The maximum mountain area in Wushan Mountain is more 
than 97%, and the mountain area is divided into middle 
mountains, low mountains and hills accounting for 60%, 36% 
and 4%, respectively. Overall, the terrain is high and low in 
the middle. The middle mountain area is interspersed with the 
strip of low mountain area with hills scattered between them. 
Jianchuandong dangerous rock body is located on the west 
side of Shennv Peak and on the left bank of the Yangtze River 
in the Wuxia section. It is only 12km away from the county 
town and 112.3km downstream is the Three Gorges Dam. The 
whole base of the dangerous rock mass is immersed in the 
water of the Yangtze River. Once the dangerous rock mass 
becomes unstable and breaks down and falls into the 
waterway of the Yangtze River to form a surge wave, it will 
pose a fatal threat to the life and property safety of 
downstream residents and the safety of the entire waterway 
transportation, wharves and scenic spots. 

 

 
Figure 1. Overall picture of dangerous rock pierced by 

Arrows (picture from the Internet) 

2.2. Geological environment of the study area 
Jianchuandong dangerous rock mass is located in the 

turning section of the axial part and southeast wing of Shennv 
anticline. There is neither fault nor active fault in the axial 
rock mass. The occurrence of the axial rock mass is NE255°-
265°∠5°-7°, and the dip of the southeast wing rock strata is 
NE150°-160° ∠ 10°-24°. Three fissures developed in 
Jianchuandong dangerous rock mass are in the southeast, 
northwest and northeast, and the fissures have been all 
through, showing irregular long strips. Three main structural 
cracks occur in dangerous rock mass: the boundary cracks 
(NE300°-315°∠78°-85°) on the southeast side. The upper 
part of the cracks is not filled, but locally filled with rubble 
soil and residual gravel soil. From top to bottom, the cracks 
gradually converge and point out at the elevation of 153m. 
North west boundary crack (NE325°-335°∠85°-88°); The 
fissure in the northeast side (NE255°-265°∠ 70°-80°) is 
basically filled by gravel, and the fissure has a large opening 
degree, with the maximum reaching 3.15m. 

In terms of spatial form, the engineering geological 
conditions of Jianchuandong dangerous rock show irregular 
hexahedron in geometric form. In addition to the dangerous 

rock being divided by three fissures, Jianchuandong 
dangerous rock also has three adits that were used as 
defensive rock in Wuxia during the Anti-Japanese War, with 
elevations of 157m. The section size and size of the three adits 
are the same. The fractured fissure formed by the dead weight 
of the dangerous rock mass in the upper part of the foundation 
rock mass and the existing adit destroy the integrity of the 
dangerous rock mass and reduce the stability of the dangerous 
rock mass. When the river water level rises and inundate the 
foundation, the river water's erosion, erosion and dissolution 
on the foundation will further soften the argillous strip 
limestone of the foundation, thus further damaging the 
stability of the dangerous rock mass. 

 

 
Figure 2. Geological profile of dangerous rock in Arrow 

hole 

3. Stability Calculation Method of 
Damaged Sliding Dangerous Rock 
on Reservoir Bank 

The development of dangerous rock is influenced by 
internal and external factors. Different lithologic 
combinations, topographic conditions, stratigraphic lithologic 
conditions and hydrological conditions will affect the failure 
mode and development characteristics of dangerous rock. The 
rock of dangerous rock mass base is damaged and 
deteriorated under the action of river water and weathering, 
and the rock strata peel off to form rock cavity. The gravity of 
dangerous rock mass acts on the undamaged base rock and 
the damaged part of base rock, forming the damaged 
dangerous rock on the reservoir bank. The dangerous rock 
mass is subject to seismic force (P), dead weight of dangerous 
rock mass (W), support force of undamaged rock at the base 
(N1), support force of damaged rock at the base (N2), shear 
force of rock mass (T) and support force of main control 
structural plane (N). The damaged sliding mechanical model 
of dangerous rock at the reservoir bank is as follows: 

 

 
Figure 3. Physical model of dangerous rock 



 

193 

 
Figure 4. Mechanical model of dangerous rock 

 
Due to the large opening degree of the main control 

structural plane at the back of the rock pierced by arrows, the 
influence of the water pressure on the fracture of the rear 
reinforced structural plane can be ignored. Chen Hongkai et 
al [8] According to the rule, the pore water pressure of 
dangerous rock is mainly affected by the height of fissure 
water under the natural state of hydrostatic pressure and the 
state of rainstorm. The hydrostatic pressure formula can be 
expressed as follows: 

Natural state: 
 

 
1 1 2( sin )
2 3

Q e
w

          (1) 

 
State of rainstorm: 
 

 21 2
( sin )

2 3wQ e        (2) 

 
Where: (Q) is the hydrostatic pressure of the crack in 

dangerous rock. For the dangerous rock with the opening 
degree of the main control structural plane of 0.2-0.5cm, (e) 
is the length of the crack through dangerous rock, (  ) is the 
dip Angle of the main control structural plane of dangerous 
rock, (   ) is the damage coefficient of the crack water 

pressure, and ( w ) is the bulk density of water. Due to the 

large opening degree of the crack in dangerous rock, the 
maximum is 3.15m. Therefore, the hydrostatic pressure of 
crack in dangerous rock is not considered. 

For the wharf type dangerous rock, different from the water 
pressure calculated by the general dangerous rock mass in the 
posterior structural plane fracture, the wharf type dangerous 
rock has hydrostatic pressure in the sliding direction of the 
dangerous rock, that is, the hydrostatic pressure at the sliding 
open surface can be expressed as: 

 

 2
1

1
h

2 wQ                 (3) 

 

Where: ( 1h ) represents the height of the dangerous rock 

mass immersed in water in the direction of the free face, and 
other symbols have the same meaning as the above formula.  

The dead weight of dangerous rock mass is the product of 
the volume of dangerous rock mass and the bulk weight of 
dangerous rock. Generally, if the unit thickness is 1m, the 
dead weight of dangerous rock can be expressed as: 

 
 W V                  (4) 

Where: (W) represents the gravity of dangerous rock mass, 
( ) represents the bulk density of dangerous rock mass, and 
(V) represents the volume of dangerous rock mass. 

The seismic force of dangerous rock mass can be divided 
into horizontal seismic force and vertical seismic force. In 
general, the horizontal seismic force is mainly considered in 
the calculation of the stability of dangerous rock mass. It is 
the product of the dead weight of dangerous rock mass and 
the seismic force coefficient, which can be expressed as: 

 
 P W                (5) 

 
Where: (P) represents the horizontal seismic force of 

dangerous rock, (W) represents the gravity of dangerous rock, 
and (  ) represents the horizontal seismic force coefficient. 

After rock damage, the internal microstructure of the rock 
is damaged and deteriorated, the physical and mechanical 
properties of the rock are damaged, and the strength is 
damaged. By setting the damage stress function, the damage 
stress of the rock can be expressed as the damage coefficient 
of the rock and the rock: 

 

 ' ck               (6) 

 
Where: ( ' ) represents the damage stress of rock damaged 

by river water, ( c  ) represents the standard value of rock 

compressive strength, and (k) represents the damage 
coefficient of rock that can be obtained through test fitting. 

The regression velocity of rock cavity is introduced into the 
function ( ( )d t ), which can be expressed as: 

 
 ( )d t wt               (7) 

 
Where: ( ( )d t  ) represents the correlation between the 

receding velocity of rock cavity and time and the average 
receding velocity of rock cavity; (w) represents the average 
receding velocity of rock cavity after rock damage; (t) 
represents time. 

Failure criterion of rock foundation conforms to Mohr-

Coulomb strength criterion, and shear force of foundation ( tT ) 

is expressed by Mohr-Coulomb strength criterion as: 
 

 1 2 'tT d d                (8) 

 

 1 2 0d d d d                (9) 

 

Where: ( tT ) represents the shear force of dangerous rock 

mass base, ( 1d  ) represents the depth of rock cavity in 

undamaged part of the base, ( 2d ) represents the depth of rock 

cavity in damaged part of the base, ( 0d ) represents the depth 

of rock base, (d ) represents the depth of rock cavity in rock 
base, ( ) represents the shear strength of undamaged rock 
and ( ' ) represents the shear strength of damaged rock. 

Mohr-Coulomb strength criterion can be used to express 
the shear strength of undamaged rock and damaged rock as 
follows: 

 



 

194 

 tan c                (10) 
 

 tant t tk c              (11) 

 
The meaning of symbols in the formula is the same as 

above 
By substituting equations (11) and (10) into equations (8), 

the shear force of rock base ( tT ), can be expressed as: 

 

 
0 2

2

( )( tan

) ( tan )
t

t t

T d d d

c d k c

 
 

  
  

       (12) 

 

Where: ( tT ) represents the interface shear stress between 

rock base and dangerous rock mass, ( c  ) represents the 
interface cohesion between rock base and dangerous rock 
mass, ( ) represents the internal friction Angle between base 
and dangerous rock mass, (   ) represents the interface 

normal stress between base and dangerous rock mass, ( t ) 

represents the internal friction Angle between damaged rock 

base and dangerous rock mass, ( tc ) represents the interface 

cohesion between damaged rock base and dangerous rock 
mass, The other symbols have the same meaning. 

Similarly, the supporting force between the base of 
dangerous rock pierced by arrows and the main control 
structural plane can be expressed as: 

 

 '
1 2 costN d d l            (13) 

 

Where: ( tN ) represents the supporting force between the 

hazardous rock mass base and the main control structural 
plane; ( l  ) represents the distance between the interface 
between the hazardous rock mass and the main control 
structural plane; (  ) represents the dip Angle of the main 
control structural plane of the hazardous rock mass; other 
formulas have the same meaning as before. 

By substituting equation (6) into equation (13), the 
supporting force of damaged rock base can be expressed as: 

 

 1 2 costN d d k l             (14) 

 
The meaning of the symbol is the same as before. 
Along the main structural plane, the main structural plane 

of dangerous rock can be decomposed into normal stress and 
tangential stress according to the stress of dangerous rock in 
unit length. It can be expressed as: 

Tangential component: 
 

 
( )sin

( )cos
t

t

T W N

P T Q




  

 
          (15) 

 
Normal component: 
 

 
( ) cos

( )sin
t

t

N W N

P T Q




  
 

       (16) 

 

Assuming that the normal stress and tangential stress on the 
main control structural plane of dangerous rock are evenly 
distributed, the average normal stress and average shear stress 
on the main control structural plane are: 

 
 /T l   (                   17) 

 
 /N l                  (18) 

 
 
When equations (16), (18) and (10) are combined, the shear 

force borne by the main control structural plane can be 
expressed as: 

 

 

( ) cos

( )sin
tan

t

t

W N

P T Q
c

l


 

 

 
 

     (19) 

 
The shear strength of the main control structural plane of 

the dangerous rock mass is: 
 

 tanf k c                (20) 

 
The stability coefficient of sliding dangerous rock with 

damage on reservoir bank is expressed by shear force and 
shear strength as: 

 

 f
SF




                   (21) 

 
Equation (21), Equation (20) and equation (19) are 

combined to obtain the stability coefficient of damaged 
sliding dangerous rock on the reservoir bank: 

 

 

tan
( ) cos

( )sin
tan

S
t

t

k c
F

W N

P T Q
c

l

 

 




 
 


      (22) 

 
Equation (22) is the calculation method of stability 

coefficient of damaged sliding dangerous rock on reservoir 
bank. 

Where: (FS) represents the stability coefficient of damaged 
sliding dangerous rock on the reservoir bank, ( ) represents 

the shear stress of the rock master structural plane, ( f  ) 

represents the shear strength of the rock master structural 
plane, (β) represents the dip Angle of the rock master 
structural plane, ( l ) represents the distance from the interface 
between the base and the dangerous rock mass to the bottom 
of the master structural plane, and the other symbols have the 
same meanings as before. 

4. Conclusion 
Jianchuandong dangerous rock is a typical tower pillar rock 

cut by tectonic fissure, and the dangerous rock mass has a 
huge volume. Once it falls into the waterway of the Yangtze 
River, it will inevitably cause surging waves and threaten the 
safety of ships in the waterway of the Yangtze River and cities 



 

195 

in the lower reaches of the Yangtze River. This paper takes 
arrow-pierced dangerous rock as the research background and 
draws the following conclusions: 

(1)The physical and mechanical model of Jianchuandong 
dangerous rock is established based on previous studies and 
Jianchuandong dangerous rock geological profile. The water 
pressure distribution of Jianchuandong dangerous rock is 
different from that of traditional dangerous rock, and its 
influence on the stability of dangerous rock is also different. 
The water pressure of reservoir bank dangerous rock exists in 
the direction of dangerous rock instability and failure surface, 
and the water pressure of dangerous rock fracture may exist. 
So we don't consider the air surface water pressure. 

(2)The traditional calculation method of the stability of 
dangerous rock is unreasonable because of the different stress 
conditions and the different damage conditions of the base. 
By considering the depth of rock cavity and the influence of 
damaged rock mass on the stability of damaged rock mass, a 
method for calculating the stability of damaged damaged rock 
mass is established to calculate the stability of damaged rock 
mass. Calculation methods and verification of other failure 
modes of damaged dangerous rock on reservoir bank are 
presented in a separate paper. 

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