Characteristic model of movement trajectory of rockfall along slope under earthquake and its construction method

Abstract

The present invention discloses a characteristic model of the movement trajectory of rockfall along a slope under earthquake action and a construction method thereof. By comparing and analyzing the movement trajectory of rockfall under earthquake action and non-earthquake action, it can be seen that the movement distance of rockfall under earthquake action is greater than the movement distance of rockfall without earthquake action, and the difference will also increase with the increase of the rolling time of rockfall. Therefore, according to the results of theoretical analysis, it can be known that the kinetic energy of rockfall under earthquake action is greater than that of non-earthquake action. Therefore, when estimating the movement distance of rockfall caused by slope collapse in earthquake zone, earthquake action should be considered, and the corresponding prediction of the disaster range caused by rockfall should also be expanded.

Description

Technical Field

[0001] The invention belongs to the technical field of slope protection, and specifically relates to a characteristic model of the movement trajectory of collapsed rocks along a slope under earthquake action and a construction method thereof. Background Art

[0002] Slope collapse and rockfall often destroy water conservancy and hydropower engineering facilities and damage transportation facilities such as roads and railways. In resource mining projects, the collapse and rockfall of open-pit mine slopes can also cause serious damage and personal injury to mining machinery and operators. The prediction of the movement trajectory of rockfall is a key content to solve all the above problems, which plays a very important role in the reasonable demarcation of safe avoidance distances and the optimization design of protective structures in the later stage. At present, theoretical studies on the movement trajectory of rockfall almost all assume that the rocks move on a stationary slope, and there are few reports on theoretical studies on the movement trajectory of rockfall under earthquake action. Summary of the invention

[0003] In order to solve the above problems, the present invention proposes a movement trajectory model of collapsed rocks along a slope under earthquake action and a construction method thereof.

[0004] S1. Solve the rebound energy of the rockfall impacting the slope in a static state:

[0005] The calculation formula of the slope rebound energy is as follows:

[0006]

[0007] Where E0 is the kinetic energy of rockfall rebound, V′ is the rockfall rebound velocity, and E ul is the rebound modulus, δ max is the maximum compression deformation of the falling rock and the ground when they collide, m is the mass of the falling rock, Y is the yield strength of the rock and soil on the slope, and R is the radius of the spherical falling rock;

[0008] S2. Solving the seismic wave input energy: The seismic wave input energy is:

[0009]

[0010] Where W is the input energy of the seismic wave, m is the mass of the falling rock, and a EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a moment before the ith collision;

[0011] S3. Calculate the rebound velocity of rockfall under the modified seismic wave: The rebound velocity of rockfall under the modified seismic wave is

[0012]

[0013] Where V′ is the rebound velocity of the rockfall, E is the total rebound kinetic energy of the rockfall after correction, and m is the mass of the rockfall. The calculation model of the corrected total rebound kinetic energy of the rockfall is:

[0014]

[0015] Where E is the total rebound kinetic energy of the rockfall after correction, E0 is the kinetic energy of the rockfall rebound, W is the input energy of the seismic wave, m is the mass of the rockfall, and a is EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a moment before the ith collision, and t i is the contact time of the i-th collision between the falling rock and the ground, E ul is the rebound modulus, Y is the yield strength of the slope rock and soil, δ max is the maximum compression deformation of the falling rock and the ground when they collide, and R is the radius of the spherical falling rock;

[0016] S4. Calculate the normal velocity and tangential velocity of the slope;

[0017] S5. Calculate the time the rock falls in the air, its maximum height, and the distance it moves along the slope.

[0018] In some embodiments, the slope normal velocity and the slope tangential velocity are respectively

[0019]

[0020] Where, (V1) n is the normal velocity of the rockfall when it first hits the slope, (V1) t is the tangential velocity of the rockfall when it first hits the slope, g is the acceleration of gravity, h0 is the vertical height of the initial position of the rockfall from the slope, and θ is the slope angle.

[0021] In some embodiments, the rockfall air movement time is

[0022]

[0023] Where, T i is the flying time of the rockfall after the ith collision, (V i ') n is the normal rebound velocity of the ith rebound velocity, g is the acceleration of gravity, and θ is the slope angle.

[0024] In some embodiments, the maximum height of the rockfall is

[0025]

[0026] In the formula, h iis the maximum height of the rockfall after the ith collision, (V i ') n is the normal rebound velocity of the ith rebound velocity, g is the acceleration of gravity, and θ is the slope angle.

[0027] In some embodiments, the flying distance of the falling rock along the slope is

[0028]

[0029] In the formula, l i is the flying distance of the rockfall along the slope after the ith collision, (V i ') t is the tangential rebound velocity of the i-th rebound velocity, T i is the flight time of the rockfall after the ith collision, g is the acceleration of gravity, and θ is the slope angle.

[0030] The movement distance along the slope includes a flying distance and a rolling distance. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is the collision process between falling rocks and the ground based on Hertz contact theory.

[0032] Figure 2 It is the collision model of falling rocks and slope.

[0033] Figure 3 is the force-displacement relationship in the elastic-plastic loading stage.

[0034] Figure 4 This is a cross-section diagram of a dangerous rock mass.

[0035] Figure 5 It is a prediction map of the trajectory of falling rocks. DETAILED DESCRIPTION

[0036] The embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to be used to explain the present invention, but should not be understood as limiting the present invention.

[0037] like Figure 1 As shown in the figure, the bouncing behavior of falling rocks on the slope can be analyzed based on the Hertz contact mechanics theory. The collision between falling rocks and the slope is an elastic-plastic deformation behavior, which can be divided into three stages according to its motion characteristics and mechanical properties: elastic contact stage, elastic-plastic contact stage and unloading stage.

[0038] When there is an earthquake, the seismic waves will exert inertial seismic force on the falling rocks. According to classical Newtonian mechanics, two objects can only have a force when they are in contact with each other. Therefore, only when the falling rocks come into contact with the slope will they be affected by the seismic force. At the time t when the falling rocks collide with the slope, i In the case of a rockfall, the rockfall and the slope can be considered as a whole, vibrating together under the action of seismic waves. After the collision process, the rockfall will generate an initial velocity in the direction of the inertial seismic force, such as Figure 2 As shown in (b) in the figure. Due to the inertial force of the earthquake, the rebound speed of the rockfall will increase compared to the rebound speed of the rockfall without the earthquake, which will change the trajectory of the rockfall. This is also the direct reason why the distance of rockfall under earthquake is often longer. In order to predict the trajectory of rockfall under earthquake, it is necessary to revise the existing rockfall slope motion theory that does not consider the earthquake effect.

[0039] 1. First calculate the rebound velocity of falling rocks when there is no earthquake.

[0040] The collision behavior of falling rocks with slopes can be divided into three stages: elastic, elastoplastic and unloading stages. Now we will conduct a theoretical analysis of the mechanical behavior of each stage.

[0041] (1) Elasticity stage

[0042] For two radii R i (i=1,2), elastic modulus is E i (i=1,2), Poisson's ratio is v i When two spheres (i=1,2) touch each other, Figure 1 As shown in (a), the Hertz contact compressive stress is:

[0043]

[0044] Where p(r) is the contact compressive stress, a is the contact radius of the two spheres, P is the contact normal pressure, and r is the distance from a point on the contact surface to the contact center.

[0045] The relationship between the contact normal pressure P and the contact radius a is:

[0046]

[0047] In the formula, P is the contact pressure between the two spheres, E * is the equivalent elastic modulus, R * is the equivalent radius, δ is the contact deformation of the two spheres, which consists of two parts, namely: δ = δ1 + δ2, δ1 and δ2 are the deformations of contact spheres 1 and 2 respectively.

[0048] The relationship between the contact deformation δ and the contact surface radius a is:

[0049] a 2 =Rδ (4)

[0050] If two objects collide with each other at a certain speed V, then there is a critical speed V y Make at least one of the objects enter the yield stage, and the yield speed satisfies the relationship:

[0051]

[0052] In the formula, m * is the equivalent mass, m1 and m2 are the masses of the two contacting objects. y is the yield speed. If the impact speed is less than the yield speed, that is, at V <V y In the area of ​​​​a, the falling rock has a completely elastic collision on the slope, and its rebound speed and rebound energy are consistent with those before the collision; y is the radius of the plastic region, and its value is:

[0053]

[0054] Here, the yield stress σ is introduced y concept:

[0055]

[0056] Combining the above formulas, we can get:

[0057]

[0058] In particular, if a sphere hits a plane, then R * =R,m * =m, V y can be simplified to:

[0059]

[0060] In this example, σ y The value is taken as 2.5MPa according to relevant literature, and ρ is the mass density of the sphere.

[0061] (2) Elastic-plastic stage

[0062] When the object yields, the normal contact pressure can be expressed as:

[0063]

[0064] Where P is the positive pressure applied to the two spheres, that is, the normal contact pressure, P eis the elastic pressure, indicating that the same contact radius and contact area can be produced under this elastic pressure; a p for Figure 1 The maximum contact plastic radius, σ, shown in (b) y is the yield stress. Simplifying formula (10) yields:

[0065]

[0066] Usage contact yield pressure P y and plastic radius a y The yield stress σ y :

[0067]

[0068] Contact radius a and sphere equivalent radius R * , elastic pressure P e , equivalent elastic modulus E * The relationship can be expressed as:

[0069]

[0070] Combining equations (12) and (13), we can obtain the contact radius a and the maximum contact plastic radius a. p and the initial yield radius a y The relationship is:

[0071]

[0072] Substituting equation (13) and equation (14) into equation (11), we can obtain:

[0073]

[0074] The parameter δ is introduced to describe the degree of mutual approach between the two mass centers, that is, the compression deformation of the two contacting spheres. The force-displacement relationship in the plastic loading stage can be expressed as:

[0075] P=P y +πσ y R * (δ-a y ) (16)

[0076] Figure 3 This is the force-displacement curve in the plastic loading stage. It can be observed that it satisfies a linear relationship.

[0077] (3) Unloading stage

[0078] The velocity generated in the unloading stage of the collision between the rockfall and the slope is the rebound velocity of the rockfall. Since the plastic stage produces permanent and irreversible deformation, the curvature of the sphere in the unloading stage is smaller than the curvature of the sphere in the initial stage, that is: In the unloading stage, the force-displacement curve can be regarded as the maximum compression curvature According to the Hertz contact theory, when unloading at this point, the contact area is reduced from the actual maximum normal pressure P * Produces, reduces curvature Numerically equal to an equivalent elastic pressure P e * The deformation produced, therefore, can be obtained from formula (13):

[0079]

[0080] Where:

[0081]

[0082] In the formula, is the equivalent radius of the plastic zone of the contact surface, P * is the equivalent contact pressure of the two spheres, R * is the equivalent radius of a sphere, P e * is the equivalent elastic pressure, E * is the equivalent elastic modulus, P y is the normal contact yield pressure, σ y is the yield stress, δ * is the equivalent approximation degree of the center of mass of the two spheres, a * is the equivalent radius of the contact surface.

[0083] When the impact speed V i ≤V y When V y is the yield velocity, no plastic deformation occurs, and the energy loss caused by elastic vibration is ignored. At this time, the rebound velocity is equal to the impact velocity, and the recovery coefficient e = 1.0. i >V y When , the rebound velocity is equal to the work done during elastic unloading, so:

[0084]

[0085] Where V' is the rebound velocity of rockfall; and

[0086]

[0087] Combining equations (19) and (20), the formula for calculating the kinetic energy of rockfall rebound is:

[0088]

[0089] Where E0 is the kinetic energy of rockfall rebound when there is no earthquake, m is the mass of rockfall, V' is the rockfall rebound velocity, α p is the radius of the plastic zone of the contact surface.

[0090] In He Siming's research work, the formula is:

[0091]

[0092] In the formula, E ul is the rebound modulus; δ max is the maximum compression deformation of the rockfall and the ground when they collide, m is the mass of the rockfall, V' is the rebound velocity of the rockfall, Y is the yield strength of the slope rock mass, and R is the radius of the spherical rockfall. Its value can be estimated according to formula (25):

[0093]

[0094] Where δ is the compression deformation of the two contacting spheres, P is the contact normal pressure of the two spheres, R is the radius of the spherical rockfall, and E is the elastic modulus of the rockfall.

[0095] 2. Correct the rebound speed of falling rocks under earthquake action.

[0096] The inertial force of the earthquake will increase the initial velocity of the falling rock, and its value can be calculated by the acceleration formula V = at i Calculate, a is the earthquake acceleration, t i is the action time of the seismic force during the ith collision, which refers to the contact time between the rockfall and the slope, and can be solved by the formula proposed in relevant literature:

[0097]

[0098] Where V i is the velocity before the ith collision, δ * is the equivalent compression, δ is the compression deformation, E * is the equivalent elastic modulus, R is the radius of the spherical rockfall, and m is the mass of the rockfall.

[0099] According to the impulse theorem, the impact force P of the i-th rockfall on the slope is i for:

[0100]

[0101] Substituting equation (27) into equation (25) can calculate the contact compression, and substituting it into equation (24) can calculate the rebound kinetic energy E0 when the falling rock collides with the stationary slope.

[0102] Due to the inertial force of the earthquake, the initial horizontal velocity and vertical velocity of the falling rock are:

[0103]

[0104] Where: V h V is the initial horizontal velocity of the east-west earthquake wave. v is the vertical initial velocity of the seismic wave, a EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a moment before the ith collision, and t i is the contact time between the falling rock and the ground during the i-th collision, that is, the action time of the earthquake force.

[0105] The input energy of seismic waves is:

[0106]

[0107] At this time, the total rebound kinetic energy of the falling rock is corrected to:

[0108]

[0109] The corrected rockfall rebound velocity under the action of seismic waves is:

[0110]

[0111] Where E is the total rebound kinetic energy of rockfall under earthquake action, E0 is the kinetic energy of rockfall rebound without earthquake action, W is the input energy of seismic wave, m is the mass of rockfall, and a is EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a certain moment before the ith collision, V' is the rebound velocity of the rockfall, Y is the yield strength of the slope rock and soil, δ max is the maximum compression deformation of the falling rock and the ground when they collide, and R is the radius of the spherical falling rock.

[0112] 3. Prediction of rockfall slope movement under the action of seismic waves.

[0113] The rock falls away from the bedrock and falls freely. The initial collision speed with the slope is:

[0114]

[0115] Where V1 is the first impact velocity of the rockfall, g is the acceleration of gravity, and h0 is the vertical height of the initial position of the rockfall from the slope.

[0116] The velocity components of the collision velocity along the normal and tangential directions of the slope are:

[0117]

[0118] Where, (V1) n is the normal velocity of the rockfall when it first hits the slope, (V1) t is the tangential velocity of the rockfall when it first hits the slope, g is the acceleration of gravity, h0 is the vertical height of the initial position of the rockfall from the slope, and θ is the slope angle.

[0119] The collision restitution coefficient is introduced to describe the collision behavior between falling rocks and slopes. The collision restitution coefficient is defined as:

[0120] e n =V n ' / V n (36)

[0121] e t =V t ' / V t (37)

[0122] In the formula, e n is the normal collision restitution coefficient, e t is the coefficient of restitution for tangential collision, V n is the normal incident velocity, V n ' is the normal rebound velocity, V t is the tangential incident velocity, V t ' is the tangential rebound velocity.

[0123] The formula for the coefficient of restitution for normal rockfall collision in relevant literature is as follows:

[0124]

[0125] In the formula, e n is the normal collision restitution coefficient, V n is the normal incident velocity, V y The rockfall velocity required to cause the slope rock and soil to yield.

[0126] Substituting the result of formula (38) into formula (36) can calculate the normal velocity of rockfall rebound. From the relationship between rebound velocity, normal rebound velocity and tangential rebound velocity, it can be known that:

[0127]

[0128] Where V' is the rebound velocity, V n ' is the normal rebound velocity, V t ' is the tangential rebound velocity.

[0129] The tangential velocity of rockfall rebound can be solved by equation (39), and then the tangential restitution coefficient of rockfall can be solved by equation (37).

[0130] After the i-th rebound, the rockfall moves in a parabola, and the time it spends in the air can be calculated:

[0131]

[0132] Where, T i is the flying time of the rockfall after the ith collision, (V i ') n is the ith normal rebound velocity, g is the gravitational acceleration, and θ is the slope angle.

[0133] The maximum height that the falling rock can reach after the i-th collision is:

[0134]

[0135] The flying distance of the rockfall along the slope after the i-th collision is:

[0136]

[0137] In the formula, l i is the distance that the rockfall flies along the slope after the ith collision, (V i ') t is the ith tangential rebound velocity.

[0138] Since the rockfall is on the slope during the collision and rebound process, there is no displacement in the normal direction. Therefore, the normal velocity of the rockfall at the i+1th impact is equal to the normal velocity of the i-th rebound, and its value is:

[0139] (V i+1 ) n =(V i ') n (43)

[0140] The incident tangential velocity of the i+1th collision is:

[0141]

[0142] The incident angle (angle with the slope) of the i+1th collision is:

[0143]

[0144] 4. Embodiment

[0145] Theoretical analysis of the trajectory of rockfall is carried out on a dangerous rock mass. Figure 4 Assume that the dangerous rock mass falls in free fall, and the calculation parameters are shown in Table 1. The seismic wave acceleration is a = 0.2g.

[0146] Table 1 Calculation parameters for theoretical analysis of rockfall trajectory

[0147]

[0148] Taking the first collision with the slope, rebound, and the second collision with the slope after the instability of the dangerous rock mass as an example, the calculation steps of the theoretical prediction of the trajectory of falling rocks under the action of seismic waves are as follows:

[0149] ① Solve the critical plastic velocity according to formula (9);

[0150] ② According to equations (33)-(35), solve the rockfall impact velocity, normal velocity, and tangential velocity;

[0151] ③ According to formula (26), solve the contact time between the rockfall and the slope surface;

[0152] ④ According to formula (27), solve the impact force of falling rocks on the slope during the first impact;

[0153] ⑤ Solve the compression amount according to formula (25), and then solve the rebound energy of the rockfall impacting the slope surface under static state according to formula (24);

[0154] ⑥ Solve the seismic wave input energy according to equations (28), (29) and (30);

[0155] ⑦ According to equations (31) and (32), calculate the rebound velocity of falling rocks under the action of seismic waves;

[0156] ⑧ According to formula (38), the normal coefficient of restitution of the slope surface is calculated, and then the normal rebound velocity is calculated according to formula (36); then the tangential rebound velocity can be calculated according to formula (39), and finally the tangential coefficient of restitution can be calculated according to formula (37);

[0157] ⑨ According to equations (40), (41), and (42), calculate the flying time of the falling rock in the air, the maximum height of the falling rock, and the flying distance along the slope respectively;

[0158] ⑩ According to equations (43) and (44), the normal velocity and tangential velocity of the next rockfall are calculated, and the total incident velocity is calculated by equation (39);

[0159] Start the second impact process calculation and repeat steps ②-⑩.

[0160] The standard for ending the rockfall collision trajectory analysis is: when V n ' or h i When it is smaller, it means that the falling rock will no longer bounce after the collision, or will only jump very slightly, and the movement form becomes pure rolling.

[0161] The above is the theoretical calculation method of the trajectory of rockfall under the action of seismic waves. If the seismic effect is not considered, steps ⑥ and ⑦ are omitted, and the other steps remain unchanged. The speed solved in step ⑤ is the rebound speed. In order to compare and analyze the influence of seismic waves on the trajectory of rockfall, two working conditions are calculated respectively without considering seismic waves and considering seismic waves, as shown in Tables 2 and 3.

[0162] Table 2 Calculation of rockfall trajectory without considering earthquake action

[0163]

[0164] Table 3 Calculation of rockfall trajectory considering earthquake action

[0165]

[0166] From the calculation results shown in Table 2, it can be seen that after about 4 collisions, the normal velocity and flight height of the falling rock are already very small. After the fourth collision, the movement of the falling rock changes to pure rolling, and the tangential velocity increases. In this section, the calculation results of the velocity change law and restitution coefficient of the falling rock motion trajectory are very close to those of the relevant literature, so it can be shown that the proposed rockfall motion trajectory prediction method is reasonable.

[0167] From the comparison between Table 2 and Table 3, it can be seen that the effect of seismic waves is extremely limited during the rockfall bounce process. This is because the time of contact between the rockfall and the slope is very short, resulting in each bounce only increasing the kinetic energy by about 5J, and almost no change in its rebound speed. Therefore, the effect of seismic waves can be ignored during the rockfall bounce process. However, after the fourth bounce, the rockfall will turn into a pure rolling motion. The contact time between the rockfall and the slope is approximately equal to the rolling time of the rockfall on the slope. At this time, t i It can reach several seconds or even tens of seconds, so the energy input by the seismic wave can reach hundreds or even thousands of joules. In order to further illustrate the impact of seismic waves on the rolling state of falling rocks, it is assumed that the above-mentioned falling rocks continue to move in the form of pure rolling after the fourth bounce, with an initial velocity of 4.3m / s. The velocity changes after 1s, 2s, and 5s of movement are shown in Table 4. The movement trajectories of falling rocks under earthquake action and non-earthquake action are shown in Table 4. Figure 5 shown.

[0168] Table 4 Comparison of rockfall trajectory under earthquake and non-earthquake action (pure rolling state)

[0169]

[0170] By comparing and analyzing the movement trajectories of rockfall under earthquake and non-earthquake action, it can be seen that the movement distance of rockfall under earthquake action is greater than that without earthquake action, and the difference will increase with the increase of rolling time. In summary, the effect of seismic waves in the bouncing stage of rockfall is small and can be ignored. In the rolling state, seismic waves have a greater impact on the movement distance and speed of rockfall. Therefore, according to the results of theoretical analysis, the kinetic energy of rockfall under earthquake action is greater than that of non-earthquake action. Therefore, when estimating the movement distance of rockfall caused by earthquake collapse in earthquake zones, the earthquake effect should be considered, and the corresponding prediction of the disaster range caused by rockfall should also be expanded.

[0171] Although the above embodiments have been shown and described, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations on the present invention. Changes, modifications, substitutions and variations of the above embodiments by those of ordinary skill in the art are all within the scope of protection of the present invention.

Claims

1. A method for constructing a characteristic model of the movement trajectory of rockfall along a slope under earthquake, characterized in that: The following steps are involved: S1. Establish a rebound energy model of rockfall impacting the slope surface in a static state, that is, first calculate the rebound velocity of rockfall without earthquake action; S2. Obtaining a seismic wave input energy model and correcting the rebound velocity of falling rocks under earthquake action; S3. Construct a modified rockfall rebound velocity model under the action of seismic waves to achieve the prediction of rockfall slope motion under the action of seismic waves, and predict the rebound motion of rockfall after it breaks away from the bedrock and collides with the slope; S4. Calculate the normal velocity and the tangential velocity of the slope, that is, the normal velocity and the tangential velocity of the slope along the slope when the rock falls and the slope collide for the first time; S5. Construct a rockfall air movement time model, a maximum movement height model, and a movement distance model along the slope; that is, the rockfall breaks away from the bedrock and makes a free fall movement. The parabolic movement of the rockfall after the i-th rebound includes the air movement time, the maximum movement height of the rockfall, and the flight distance of the rockfall on the slope; The time model of the falling rock in the air is: The maximum height model of rockfall motion is: The flying distance model of the falling rock along the slope is: Where, T i is the flying time of the rockfall after the ith collision, (V i ') t is the tangential rebound velocity of the i-th rebound velocity, g is the acceleration of gravity, θ is the slope angle; m is the mass of the falling rock, a EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a certain moment before the ith collision, E ul is the rebound modulus, Y is the yield strength of the slope rock and soil, δ max is the maximum compression deformation of the falling rock and the ground when they collide, R is the radius of the spherical falling rock; h i is the maximum height of the rockfall after the ith collision; l i is the flying distance of the rockfall along the slope after the i-th collision, t i is the contact time of the i-th collision between the falling rock and the ground.

2. The method for constructing a characteristic model of the movement trajectory of rockfall along a slope under an earthquake according to claim 1, characterized in that: The calculation formula of the rebound energy model of the rockfall impacting the slope surface in the static state in step S1 is as follows: Where E0 is the kinetic energy of rockfall rebound, V′ is the rockfall rebound velocity, and E ul is the rebound modulus, δ max is the maximum compression deformation of the falling rock and the ground when they collide, m is the mass of the falling rock, Y is the yield strength of the slope rock and soil, and R is the radius of the spherical falling rock.

3. The method for constructing a characteristic model of the movement trajectory of rockfall along a slope under an earthquake according to claim 1, characterized in that: The seismic wave input energy model is: Where W is the input energy of the seismic wave, m is the mass of the falling rock, and a EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a moment before the ith collision, and t i is the contact time of the i-th collision between the falling rock and the ground.

4. The method for constructing a characteristic model of the movement trajectory of rockfall along a slope under an earthquake according to claim 1, characterized in that: The modified rockfall rebound velocity model under the action of seismic waves is: Where V′ is the rebound velocity of the rockfall, E is the total rebound kinetic energy of the rockfall after correction, and m is the mass of the rockfall; The calculation model of the modified total rebound kinetic energy of falling rocks is: Where E is the total rebound kinetic energy of the rockfall after correction, E0 is the kinetic energy of the rockfall rebound, W is the input energy of the seismic wave, m is the mass of the rockfall, and a is EW is the horizontal earthquake acceleration, a UD is the vertical seismic wave acceleration, t is a moment before the ith collision, and t i is the contact time of the i-th collision between the falling rock and the ground, E ul is the rebound modulus, Y is the yield strength of the slope rock and soil, δ max is the maximum compression deformation of the falling rock and the ground when they collide, and R is the radius of the spherical falling rock.

5. The method for constructing a characteristic model of the movement trajectory of rockfall along a slope under an earthquake according to claim 4, characterized in that: The relationship between the rockfall rebound velocity and the normal rebound velocity and the tangential rebound velocity is: Where V' is the rebound velocity, V n ' is the normal rebound velocity, V t ' is the tangential rebound velocity.

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