Using the rigid body limit equilibrium method to derive the length of slope instability, the energy conservation theorem is employed to calculate the distance of slope movement.
3.2.1. Instability length
The excavation usually leads to local instability of the slope at the foot of the leading edge and tension cracks created at the trailing edge, which the state can be seen as understable at this time. Besides, the instability length calculated in this paper refers to the distance from the farthest crack at the trailing edge to the top of the excavated slope at the leading edge. A calculation model for instability length is developed based on a typical bedding rock slope (
Figure 2a). By the limit equilibrium theory of rigid bodies:
Where K denotes the stability coefficient, W
0 stands for the weight of the sliding mass per unit width and
represents the inclination angle of the rock stratum.
and
indicates the length of df and af, while c
0 and φ
0 denote the cohesion and internal friction angle respectively. σ represents the equivalent tensile strength perpendicular to the af. W
0 can be obtained from the following equation:
Where S
T represents the cross-sectional area of the sliding mass, and S
cde, S
abf, and S
bcef are the areas of triangular
cde,
abf, and rectangular
bcef, respectively.
are the weight of the sliding body. Besides, assuming
=L, establish the side length relationship 3 to 5 from the geometric relationship:
From equations 3-5, obtain the S
cde, S
abf, and S
bcef:
Substituting equations 6-8 into 2 yields:
substituting equations 9 into 1, and the following can be derived:
Substituting equations 11 through 14 into 10 results in:
Getting the final instability length
:
Where is the slope excavation angle, θ represents the angle between the trailing edge crack and the horizontal plane. Additionally, h refers to the excavation height of the sliding mass.
3.2.2. Runout distance
The calculation model for slope instability to stability, divided into three stages (I~III), is illustrated in
Figure 2b-2d. The water head heights of the trailing edge fractures in stages II and III are represented by h1 and h
2 respectively, while the lengths of the bottom surface of the sliding mass along the sliding direction in the latter two stages are denoted by L
1 and L
2. W
1 and W
2 refer to the unit width weight of the slope in the aforementioned stages, and b0 represents the length of the fracture along the sliding surface prior to slope instability. H stands for the excavation height, x denotes the distance traveled by the sliding mass on the slope, and d is the average thickness of the sliding body. V
0 represents the speed at which the sliding mass moves at distance x, while G denotes the moving distance of the sliding body on the slope in stage III.
During intense rainfall conditions, a considerable amount of rainwater infiltrates the tension cracks at the trailing edge, creating hydrostatic pressure. Furthermore, water infiltration induces uplift pressure along the sliding surface. During landslide motion, four forces should be considered: self-gravity W1, anti-sliding force f1, hydrostatic pressure Pd (x), and uplift pressure Pu (x). Additionally, the sliding mass decreases continually during the movement owing to the formation of a free face resulting from front edge excavation, and result in these forces undergo constant changes during stage II, which is distinct from the scenario where the impact of excavation factors is not taken into account.
Assuming that the volume of water in the cracks at the rear edge of the landslide remains constant, and θ is nearly vertical. The water-filled cross-sectional areas s0 and s1 in the cracks during the I and II stages can be expressed by Equations 17 and 18, respectively:
When s
0=s
1, the relationship between h
0 and h
1 and the motion distance x can be expressed as follows:
Set N=,So .
The relationship between W
0, W
1, and x can be expressed:
The relationships among L
0, L
1, and L
2 are established through the following equations:
In the II stage, Equation 22 depicts the hydrostatic pressure P
d (x) and the uplift pressure P
u (x). The hydrostatic pressure P
d (x) can be divided into two forces, namely a parallel force P
d1 (x) and a perpendicular force P
d2 (x), as illustrated in Equation 23. Additionally, the anti-sliding force f
1 is as shown in Equation 24:
The shear strength parameter of the sliding belt in Equation 24 should be the residual strength parameter.
Assuming that the sliding mass comes to a halt at a distance of G on the slope. The kinetic energy of the landslide when it moves a distance x (0<x<G) is determined by two parts: the gravitational potential energy required to overcome resistance and the hydrostatic pressure that performs work. Therefore, the law of energy conservation can be applied to derive the following equation:
In Equation 26-29, m1 represents the mass of the sliding body at a given moving distance x. WG denotes the work that is accomplished by the gravity of the sliding mass during this movement. Additionally, W
f1 represents the work that is done by the anti-sliding force, while W
pd(x) refers to the work that is accomplished by the hydrostatic pressure at the trailing edge. The details are as follows:
Where g represents the acceleration of gravity, by substituting Equation 20 into 27, we can obtain W
G:
By substituting Equations 20, 23, and 24 into 28 and 29, we obtain the following expressions for W
f1 and W
pd(x):
Substituting Equations 30 to 32 into 25 yields:
Order I=
, and V
0 can be expressed:
When the moving distance reach G, in accordance with the principle of energy conservation:
Substitute x=G into Equation 35 to obtain:
G can be calculated using Equation 36, with relevant parameters derived from real scenarios. The total displacement of a landslide comprises three components. Firstly, the horizontal distance X
0 covered by the slider during its initial movement in the air. Secondly, the horizontal displacement X
1+... X
n of the sliding body as it collides with the ground repeatedly (assuming n collisions) until its normal velocity reaches 0. Lastly, there is the displacement X
n+1 generated by the block sliding on the ground, as depicted in
Figure 3.
Command Y=
,we can obtain:
T
0, X
0, and V
1 can be expressed as:
When a block collides with the ground at a velocity of V
1, the incomplete elastic collision results in a loss of energy and a decrease in the block's velocity. To describe this phenomenon, we introduce the normal restitution coefficient R1n and the tangential restitution coefficient R1t. The relationship between the initial normal velocity V
1n and tangential velocity V
1t of the block prior to the collision, and the resulting normal velocity V
2n and tangential velocity V
2t of the block after the collision can be expressed as follows:
V
1n and V
1t can be expressed:
Where
is the angle between V
1 and the horizontal plane (
Figure 3):
X
1 represents the movement distance of the block between the first collision and the second collision, which can be expressed as following:
Further, X
1 can be described as:
Similarly, X
2 … X
n can be expressed as:
The second part of the movement distance can be calculated as following:
At the n+1 collision, the normal velocity of the block attenuates to 0, and the block slides on the ground at a velocity of V
(n+1) t, and the associated motion equation can obtain:
The third stage movement distance X
n+1 can be expressed as:
The final runout distance X
t of the landslide can be gained as follows:
Substituting Equation 38 and 40 into 47 yields:
Generally speaking, the velocity of landslide will undergoes an initial increase until reach maximum followed by a subsequent decrease. The maximum speed occurs when the anti sliding force equals the sliding force once again, and can be expressed:
Where G
0 represent the movement distance of the sliding mass on the slope when the block speed reaches the maximum, which can be obtained by substituting the relevant parameters into Equation 51, and Vmax of the sliding mass on the slope can be obtained as follows:
Moreover, given the monotonically increasing correlation between X
t and V, the ultimate displacement distance X
t of the landslide can be formulated as follows: