CN111950160A - Tunnel fracture slippage type rock burst determination method - Google Patents
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Abstract
The invention discloses a tunnel cracking sliding type rock burst determination method, which comprises the following steps: establishing a tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism; according to the tunnel split sliding type rockburst destruction mechanical model and the constitutive relation of the weak rock layer, obtaining a potential function and a balance curved surface calculation formula of a system formed by the split sliding type rockburst sliding rock body and the weak rock layer; analyzing the system stability of the fracture slippage type rock burst problem by adopting a folding mutation model; and judging the tunnel fracture slippage type rock burst based on the folding mutation theory principle. The tunnel cracking slip type rock burst determination method based on the folding mutation theory has important significance for effectively preventing and controlling rock burst.
Description
Technical Field
The invention belongs to the technical field of underground engineering, and relates to a tunnel cracking sliding type rock burst determination method.
Background
With the social and economic development and the consumption of shallow resources, the underground engineering construction and the resource development gradually move to the deep part, and show the development trend of long length, large quantity, large section and deep burying. The buried depth of the deeply buried underground engineering is increased, so that the deeply buried underground engineering often has a higher ground stress level, for example, the maximum buried depth of a diversion tunnel of a brocade screen secondary hydropower station is 2525m, and the maximum ground stress reaches 72 MPa; the mining depth of Shandong exquisite gold mine reaches kilometer level, the ground stress can reach 56.6MPa, and the ground stress linearly increases along with the increase of the buried depth; the maximum buried depth of the Xikang Qinling railway tunnel is about 1600m, and the maximum ground stress is about 86.2 MPa. In addition, some projects have small burial depth, but are also in environments with higher ground stress levels due to the geological structure, such as a second beach hydropower station, a fishery and creek hydropower station and the like.
Underground works in environments with high ground stress levels often encounter dynamic disasters, typically rock bursts, during excavation. Rock burst is the phenomenon that unstable damage occurs due to the rapid release of surrounding rock energy caused by the redistribution of stress caused by tunnel excavation. Rock burst has burst property and strong destructive property, can cause over-excavation, primary support failure and construction period delay, and can cause earthquake to destroy the whole tunnel or mine pit, thereby causing serious economic loss and threatening the safety of field constructors and equipment. Therefore, the method has important application value and practical significance for effectively preventing and controlling the rock burst.
At present, in the research on rockburst by domestic and foreign scholars, the occurrence mechanism of rockburst is researched mainly through theoretical analysis, numerical simulation or indoor test, the rockburst intensity level is predicted based on engineering practice and test, analytical algorithm or field monitoring data, and the feasibility, optimization space and the like of rockburst prevention and treatment measures are researched, so that abundant research results are obtained, and different rockburst prediction methods are also provided. However, due to the complexity of the rock burst itself and the continuous deepening of the knowledge, the existing prejudgment method has a further improved perfect space, especially for the prejudgment of different types of rock bursts.
Because the occurrence of the rock burst is sudden, the number of influencing factors is large, the relationship among the factors is complex, the rock burst is a sudden change phenomenon of a system balance state, and the rock burst is feasible by analyzing and researching by adopting a sudden change theory to determine that the damaged critical load has certain pertinence. However, the selection of models in mutation theory is conditional, and currently, relatively few studies are conducted on the applicability of the selection of models in mutation theory.
Disclosure of Invention
In view of this, the embodiment of the invention provides a tunnel fracture slippage type rock burst determination method.
The technical scheme adopted by the embodiment of the invention is as follows:
the embodiment of the invention provides a tunnel cracking slip type rock burst determination method, which comprises the following steps:
establishing a tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism;
according to the tunnel split sliding type rockburst destruction mechanical model and the constitutive relation of the weak rock layer, obtaining a potential function and a balance curved surface calculation formula of a system formed by the split sliding type rockburst sliding rock body and the weak rock layer;
analyzing the system stability of the fracture slippage type rock burst problem by adopting a folding mutation model;
and judging the tunnel fracture slippage type rock burst based on the folding mutation theory principle.
Further, according to a tunnel cracking sliding type rock burst generation mechanism, a tunnel cracking sliding type rock burst mechanical model is established, and the tunnel cracking sliding type rock burst mechanical model specifically comprises the following steps:
abstracting a corresponding tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism, wherein a rock block separated from surrounding rocks slides along a weak layer, the length of the sliding surface weak layer is B, the height of the sliding surface weak layer is B, and B & lt B is simplified into a plane strain problem for analysis; according to the unit width study, the length of the sliding rock is B, the height of the sliding rock is H, the inclination angle of the sliding surface is alpha, the mass mg of the rock is rho BHg, the normal pressure acting on the weak layer is N, N is not less than mg cos alpha, the downward sliding force is F, and F is not less than mg sin alpha.
Further, according to the tunnel fracture slippage type rockburst destruction mechanical model and the weak layer constitutive relation, a potential function and a balance curved surface calculation formula of a system formed by the fracture slippage type rockburst rock and the weak layer are obtained, and the method specifically comprises the following steps:
according to the tunnel splitting sliding type rock burst destruction mechanical model, for the weak layer on the sliding surface, the normal strain and the shear strain of the vertical sliding surface are respectively as follows:
in the formula: v is positive displacement, u is tangential displacement, b is the height of the weak layer, is positive strain, and gamma is shear strain;
the weak layer constitutive relation can be as follows:
σ＝E (3)
τ＝Gγexp[-(γ/γ0)m] (4)
in the formula: σ is the normal stress of the soft layer, τ is the shear stress of the soft layer, E is the elastic modulus, G is the shear modulus, γ0The average strain measurement is adopted, m is a coefficient related to the brittleness of the rock mass, and the larger m is, the higher the brittleness of the rock mass is, and the higher the softening degree after the peak is;
if the sliding rock mass can be regarded as a rigid body, the potential function expression of the mechanical system is as follows:
in the formula: II is a system potential function;
using the tangential displacement u of the soft layer as a state variable, the method comprises
The system equilibrium surface M equation can be obtained as follows:
in the formula: u. of0＝bγ0。
Further, based on folding sudden change theory principle, judge tunnel fracture slippage type rock burst, specifically include:
order to
And then f '(u) is 0 and f' (u) is 0, and the tangential displacement u where the second derivative of the potential function is zero is obtained1：
The equilibrium surface M equation (6) is set to u ═ u1Expanding the obtained product by Taylor series and cutting the obtained product to a quadratic term to obtain:
order state variable
Formula (8) is a folding mutant form:
-x2+a＝0
in the formula:
when the F value is large, so that a is less than 0, the system cannot keep balance at the moment;
when the F value is smaller and a is larger than 0, the system has a potential energy maximum value point and a potential energy minimum value point, and the state variable is just started to slide when the rock block begins to slide
At the moment, the potential energy of the system is a minimum value, and the balance state is stable;
when the F value is large and small, the maximum value point and the minimum value point of the system potential energy are coincided, and at the moment
The system is in a critical state, and thus the critical load can be obtained as follows:
according to the technical scheme, the invention has the following beneficial effects:
aiming at tunnel cracking slippage type rockburst, the invention provides a relatively accurate judgment method based on a folding mutation theory, and the judgment method is compared with a mechanical system stability theory for verification, so that the critical load obtained by the mechanical system stability theory is consistent with the critical load obtained by a folding mutation model, and the reliability on a theoretical level is proved when the folding mutation model is used for analysis; and finally, verifying through an actual engineering scheme column. The tunnel cracking slip type rock burst determination method based on the folding mutation theory has important significance for effectively preventing and controlling rock burst.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention and not to limit the invention. In the drawings:
fig. 1 is a flowchart of a tunnel split slip type rock burst determination method according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a tunnel fracture slip type rock burst failure mode in an embodiment of the present invention;
FIG. 3 is a tunnel fracture sliding type rock burst mechanical model in the embodiment of the invention;
FIG. 4 is a diagram of a folded mutation model equilibrium surface, singular point sets, and bifurcation sets according to an embodiment of the present invention;
FIG. 5 is a diagram illustrating analysis of a problem applicable to a cusp mutation model by using a folding mutation model according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the technical solutions of the present application will be described in detail and completely with reference to the following specific embodiments of the present application and the accompanying drawings. It should be apparent that the described embodiments are only some of the embodiments of the present application, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
Fig. 1 is a flowchart of a tunnel split slip type rock burst determination method according to an embodiment of the present invention; the tunnel cracking slip type rock burst determination method provided by the embodiment comprises the following steps:
step S101, establishing a tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism;
specifically, tunnel cracking sliding type rock burst mainly occurs in brittle laminar or blocky rock mass with microcrack development, and a rock burst area is generally positioned at the side wall of the tunnel; after the tunnel is excavated, surrounding rock stress release is carried out along a plurality of groups of microstructure surfaces; under the action of tangential concentrated stress, the wall rock is internally cracked parallel to the face of the hollow surface and is further communicated with the existing discontinuous surface. The mutual connection of the discontinuous surfaces cuts out plate-shaped or block-shaped rock mass which is separated from the surrounding rock system from the surrounding rock; under the high stress state, the internal and upper cracks of the cut rock mass are communicated; under the combined action of gravity and internal acting force, the rock body slides along the structural plane. When the soft layer on the sliding surface is subjected to strain softening, the instability of a system formed by the rock body and the soft layer is generated, so that the rock mass is broken out, the rock burst is generated in a lamellar spalling or wedge burst mode, and the broken surface formed after the breakage is mainly in a step shape and a wedge shape. The tunnel fracture slip type rock burst failure mode is schematically shown in figure 2.
Abstracting a corresponding tunnel split sliding type rock burst mechanical model according to a tunnel split sliding type rock burst generation mechanism, wherein a rock mass separated from surrounding rocks slides along a weak layer, the length of the sliding surface weak layer is B, the height of the sliding surface weak layer is B, and B & lt & gt B is simplified into a plane strain problem for analysis as shown in figure 3; according to the unit width study, the length of the sliding rock is B, the height of the sliding rock is H, the inclination angle of the sliding surface is alpha, the mass mg of the rock is rho BHg, the normal pressure acting on the weak layer is N, N is not less than mg cos alpha, the downward sliding force is F, and F is not less than mg sin alpha.
Step S102, obtaining potential energy and a balance curved surface calculation formula of a system formed by the tension-cracking sliding type rock burst sliding rock mass and the weak layer according to the tunnel tension-cracking sliding type rock burst destruction mechanical model and the constitutive relation of the weak layer of the rock mass;
specifically, according to the tunnel split slip type rock burst destruction mechanics model, to the weak layer on the glide plane, perpendicular glide plane normal strain and shear strain do respectively:
in the formula: v is positive displacement, u is tangential displacement, b is the height of the weak layer, is positive strain, and gamma is shear strain;
the weak layer constitutive relation can be as follows:
σ＝E (3)
τ＝Gγexp[-(γ/γ0)m] (4)
in the formula: σ is the normal stress of the soft layer, τ is the shear stress of the soft layer, E is the elastic modulus, G is the shear modulus, γ0The average strain measurement is adopted, m is a coefficient related to the brittleness of the rock mass, and the larger m is, the higher the brittleness of the rock mass is, and the higher the softening degree after the peak is;
if the sliding rock mass can be regarded as a rigid body, the potential function expression of the mechanical system is as follows:
in the formula: II is a system potential function;
using the tangential displacement u of the soft layer as a state variable, the method comprises
The system equilibrium surface M equation can be obtained as follows:
in the formula: u. of0＝bγ0。
Order to
And then f '(u) is 0 and f' (u) is 0, and the tangential displacement u where the second derivative of the potential function is zero is obtained1And the tangential displacement u at which the third order derivative of the potential function is zero2：
S103, analyzing the system stability of the fracture slippage type rock burst problem by adopting a folding mutation model;
specifically, according to the mutation theory, under the condition that the dimension of the control parameter does not exceed four dimensions, seven elementary mutation models are provided, namely folding, cusp, dovetail, butterfly, elliptic umbilical point, hyperbolic umbilical point and parabolic umbilical point.
When a mutation model is used to explain the mutation phenomenon, the selected model needs to correspond to the main characters of the prototype of the research object. Among the seven elementary mutation models, the elliptic, hyperbolic and parabolic umbilicus point mutation models are suitable for the system description with two-dimensional state variable dimensions. In the model with one-dimensional state variable dimension, the dovetail and butterfly mutation model control parameter dimensions are three-dimensional and four-dimensional respectively, and the dovetail and butterfly mutation model control parameter dimensions are provided with a plurality of stable and unstable balance positions, so that the method is suitable for describing a system with a plurality of mutually independent control parameters. In contrast, the folding mutation model has a stable equilibrium state in one branch and an unstable state in the other branch, and the cusp mutation model has a stable equilibrium state in the upper and lower leaves and an unstable middle leaf. In the theoretical research of the rock burst and other rock body dynamic instability problems, a research object can be generally simplified into a plane problem, the dimension of a state variable is mostly one-dimensional, and a large amount of observation and experimental research shows that the system only has two states before and after one-time sudden dynamic instability, namely unstable balance of a destabilization precursor and new stable balance after the destabilization. Therefore, the folding mutation model and the sharp point mutation model are better suitable for a tunnel cracking sliding type rock burst destruction mode on the whole.
As shown in FIG. 4, the theory of abrupt change is to use the second derivative of the system potential function pi with respect to the system state variable x
To determine the stability of the system state: when the potential energy of the system is at an extreme value, i.e. the system is in a balanced state, if
The system state is stable; if it is
The system state is unstable; if it is
The system is in a critical state, i.e. a state discontinuity. When the sharp point mutation model is adopted to analyze the system state mutation, the standard form corresponding to the equilibrium surface non-model obtained after the derivation of the system potential function needs to be equal to u when x is equal to u2Taylor expansion is performed. The singular point set equation and the formula of the cusp mutation model can be obtained, and the state variable is
And
where a jump occurs. In the range, the coincidence degree of the cubic function formula obtained by Taylor expansion and the original equilibrium curved surface M equation is reduced, and the obtained result has certain difference with the actual situation. In the process of solving the folding mutation model, although Taylor expansion is also carried out on the balanced curved surface to meet the model standard form, the expansion point is consistent with the system stability mutation point, and u is equal to u1(the potential energy second order derivative is zero), the problem that the coincidence degree of the expansion and the original equilibrium surface equation is low does not exist. Therefore, compared with a cusp mutation model, the folding mutation model is more suitable for analyzing the problem that the standard form of the equilibrium curved surface is inconsistent with that of the mutation model.
Indeed, the folding mutation model is the simplest mutation model, including all more complex mutation models. The problem that can be solved with the cusp mutation model can be analyzed with the folding mutation model, and the argument is demonstrated below.
The original balance surface of a certain system or the standard form of a cusp mutation model after Taylor expansion, namely
f(y)＝y3+py+q＝0 (9)
In the formula: y is a state variable, and p and q are control variables.
According to the singularity set equation, when the state variable and the control variable satisfy:
the system will undergo an equilibrium state stability mutation. When p is more than or equal to 0, the system equilibrium state is stable according to the formula (10); when p is less than 0, the equilibrium state of the system is mutated, and the state variable at the mutation position is changed
Will y1,2The value is substituted into the formula (10) above, and the control variable can meet the requirement when mutation occurs:
equation (11) is the cusp mutation model bifurcation set equation, which is the mutation critical condition.
Taylor expansion is carried out on the formula (9) at the position where f' (y) is 0, and the expansion point is
Each corresponding to a mutation point. To be provided with
For example, the equilibrium surface is expressed as:
the formula (12) is simplified into a folding mutation standard form, and when mutation is directly obtained according to an odd point set equation, the following steps are carried out:
expansion point y ═ y1The same is the state mutation point, and the developed equilibrium surface equation has the same consistency with the original equation (9). Will be provided with
Substitution gives the same result as in equation (11) calculated by the cusp mutation model.
The above demonstration process can be represented by figure 5.
In conclusion, a folding mutation model is adopted for stability analysis of the fracture slippage type rockburst problem, so that a more accurate solution is obtained.
And S104, judging the tunnel fracture slippage type rock burst based on the folding mutation theory principle.
Specifically, let
Then f' (u) is 0 and f "(u) is 0, so as to obtain a potential function of twoZero order derivative tangential displacement u1And the tangential displacement u at which the third order derivative of the potential function is zero2：
The equilibrium surface M equation (6) is set to u ═ u1Expanding the obtained product by Taylor series and cutting the obtained product to a quadratic term to obtain:
order state variable
Formula (8) is a folding mutant form:
-x2+a＝0 (15)
in the formula:
when the F value is large, so that a is less than 0, the system cannot keep balance at the moment;
when the F value is smaller and a is larger than 0, the system obtained by the formula (10) has a potential energy maximum value point and a potential energy minimum value point, and the state variable is just started to slide when the rock block begins to slide
At the moment, the potential energy of the system is a minimum value, and the balance state is stable;
when the F value is large and small, the maximum value point and the minimum value point of the system potential energy are coincided, and at the moment
The system is in a critical state, and thus the critical load can be obtained as follows:
theoretical verification:
and introducing a mechanical system stability theory for verifying the correctness of the folding mutation model analysis result. According to the theory of mechanical system stability, if the system potential energy Π (u) is sufficiently smooth, the change in potential energy amount Π (u) around equilibrium can be expressed as:
when the system is in a balanced state, the judgment condition of whether the system is stable is as follows: when the first non-zero term Π in equation (17)(k)(u) when k is an odd number, the equilibrium state of the system is an unstable state; if k is even number, if Π(k)(u) > 0, the equilibrium state is steady state, if Π(k)(u) < 0, the equilibrium state is an unstable state.
For tunnel fracture sliding type rock burst, the critical load can be determined by the method. When the lateral displacement u of the soft layer is less than u1When pi' (u) ═ 0 and pi "(u) > 0 exist, the equilibrium state of the system is a stable state, namely the rock mass makes quasi-static displacement on the weak layer according to the judgment conditions. When the lateral displacement u of the soft layer is more than u1When pi' (u) is 0 and pi "(u) < 0, the equilibrium state of the system is unstable. Extreme point u-u for stress-strain curve1When pi '(u) is 0, pi' ″ (u) < 0, the system is in a critical equilibrium state. Changing u to u1Substituting the equation into the equilibrium curved surface to obtain the critical load obtained according to the stability theory of the mechanical system:
comparing the formula (16) with the formula (18) shows that the analysis using the fold mutation model has reliability on a theoretical level.
Example verification:
the brocade screen secondary hydropower station is positioned in a Liangshan continent of Sichuan province, and is a hydropower station with the highest water head of the elegant Hudgeon and the largest installed scale. 7 parallel tunnels are built in the hydropower station, namely 1-4# diversion tunnels, construction drainage tunnels and A and B auxiliary tunnels. The total length of the 4 diversion tunnels is about 17km, the excavation diameter is 12.40-13.00m, the general burial depth is 1000-2000m, and the maximum burial depth is 2525 m. The maximum value of the field ground stress is 46MPa, and the maximum value of the regression analysis ground stress can reach 72MPa, and belongs to an extremely high ground stress area. Rock burst happens for many times in the excavation process of the long exploratory hole at the exploration stage, and the rock burst frequency and intensity are higher in the excavation process of the two auxiliary holes. A rock burst is actually recorded at a position 3699.5m of a long exploratory pile number of a silk screen secondary hydropower station PD1, and through field investigation, the mechanical mode of the rock burst at the position can be regarded as a tunnel cracking sliding type rock burst.
Determining the geological condition parameters of the area according to indoor tests and field conditions as follows: the lithology is white marble local mixed with glutenite, hard rock and layered structure, and the grade of the surrounding rock is II grade; the elastic modulus E of the weak layer rock mass is 30GPa, the Poisson ratio v is 0.24, the shear modulus G is E/2(1+ v) 12.096GPa, and gamma is00.004, and the bulk density of the rock is 26.8kN/m3(ii) a The inclination angle alpha of the structural plane is 30 degrees, the sliding surface length B is 1m, and the rock height h is 1 m. And (4) calculating and taking constitutive parameter m as 1.
The critical load F calculated by adopting a folding mutation model is 17.8kN/m, the downward sliding force of the rock mass slipping downwards is 26.8kN/m, the downward sliding force is greater than the critical load required by rock burst, namely tunnel fracture sliding type rock burst can occur at the position, and the theoretical analysis result is consistent with the actual situation on site.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (4)
1. A tunnel cracking slippage type rock burst determination method is characterized by comprising the following steps:
establishing a tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism;
according to the tunnel split sliding type rockburst destruction mechanical model and the constitutive relation of the weak rock layer, obtaining a potential function and a balance curved surface calculation formula of a system formed by the split sliding type rockburst sliding rock body and the weak rock layer;
analyzing the system stability of the fracture slippage type rock burst problem by adopting a folding mutation model;
and judging the tunnel fracture slippage type rock burst based on the folding mutation theory principle.
2. The method for judging the tunnel cracking slip type rockburst according to claim 1, wherein a tunnel cracking slip type rockburst mechanical model is established according to a tunnel cracking slip type rockburst generation mechanism, and specifically comprises the following steps:
abstracting a corresponding tunnel fracture sliding type rock burst mechanical model according to a tunnel fracture sliding type rock burst generation mechanism, wherein a rock block separated from surrounding rocks slides along a weak layer, the length of the sliding surface weak layer is B, the height of the sliding surface weak layer is B, and B & lt B is simplified into a plane strain problem for analysis; according to the unit width study, the length of the sliding rock is B, the height is H, the inclination angle of the sliding surface is alpha, the mass mg of the rock is rho BHg, the normal pressure acting on the weak layer is N, N is not less than mgcos alpha, the downward sliding force is F, and F is not less than mgsin alpha.
3. The method for judging the tunnel cracking slip type rockburst according to claim 1, wherein a potential function and a balance surface calculation formula of a system formed by a cracking slip type rockburst rock body and a weak layer are obtained according to a tunnel cracking slip type rockburst destructive mechanical model and a weak layer constitutive relation, and the method specifically comprises the following steps:
according to the tunnel splitting sliding type rock burst destruction mechanical model, for the weak layer on the sliding surface, the normal strain and the shear strain of the vertical sliding surface are respectively as follows:
in the formula: v is positive displacement, u is tangential displacement, b is the height of the weak layer, is positive strain, and gamma is shear strain;
the weak layer constitutive relation can be as follows:
σ＝E (3)
τ＝Gγexp[-(γ/γ0)m] (4)
in the formula: σ is the normal stress of the soft layer, τ is the shear stress of the soft layer, E is the elastic modulus, G is the shear modulus, γ0The average strain measurement is adopted, m is a coefficient related to the brittleness of the rock mass, and the larger m is, the higher the brittleness of the rock mass is, and the higher the softening degree after the peak is;
if the sliding rock mass can be regarded as a rigid body, the potential function expression of the mechanical system is as follows:
in the formula: II is a system potential function;
using the tangential displacement u of the soft layer as a state variable, the method comprises
The system equilibrium surface M equation can be obtained as follows:
in the formula: u. of0＝bγ0。
4. The method for judging the tunnel cracking slip type rockburst according to claim 3, wherein the judgment of the tunnel cracking slip type rockburst is carried out based on a folding mutation theory principle, and specifically comprises the following steps:
order to
And then f '(u) is 0 and f' (u) is 0, and the tangential displacement u where the second derivative of the potential function is zero is obtained1：
The equilibrium surface M equation (6) is set to u ═ u1Expanding the obtained product by Taylor series and cutting the obtained product to a quadratic term to obtain:
order state variable
Formula (8) is a folding mutant form:
-x2+a＝0
in the formula:
when the F value is large, so that a is less than 0, the system cannot keep balance at the moment;
when the F value is smaller and a is larger than 0, the system has a potential energy maximum value point and a potential energy minimum value point, and the state variable is just started to slide when the rock block begins to slide
At the moment, the potential energy of the system is a minimum value, and the balance state is stable;
when the F value is large and small, the maximum value point and the minimum value point of the system potential energy are coincided, and at the moment
The system is in a critical state, and thus the critical load can be obtained as follows:
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN113536531A (en) * | 2021-05-31 | 2021-10-22 | 上海工程技术大学 | Train operation risk prediction method based on mutation theory |
| CN114528617A (en) * | 2022-01-13 | 2022-05-24 | 中铁二院工程集团有限责任公司 | Method for determining railway space line position of high-geostress hard rock burst area |
| CN114778800B (en) * | 2022-04-28 | 2023-08-08 | 中交第一公路勘察设计研究院有限公司 | Multi-factor rock burst prediction method based on analysis method |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN113536531A (en) * | 2021-05-31 | 2021-10-22 | 上海工程技术大学 | Train operation risk prediction method based on mutation theory |
| CN114528617A (en) * | 2022-01-13 | 2022-05-24 | 中铁二院工程集团有限责任公司 | Method for determining railway space line position of high-geostress hard rock burst area |
| CN114528617B (en) * | 2022-01-13 | 2023-04-07 | 中铁二院工程集团有限责任公司 | Method for determining railway space line position of high-geostress hard rock burst area |
| CN114778800B (en) * | 2022-04-28 | 2023-08-08 | 中交第一公路勘察设计研究院有限公司 | Multi-factor rock burst prediction method based on analysis method |
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