CN114662341A - Rock mass critical sliding surface limit analysis method - Google Patents
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Abstract
The invention discloses a rock mass critical sliding surface limit analysis method, which collects the distribution ranges of rock masses, structural surfaces and weak layers with different properties in a target area through geological survey; measuring physical and mechanical parameters of rock masses, structural surfaces and weak layers with different properties; dividing the cells; taking the stress components of all nodes as independent variables, constructing a unit static balance constraint equation, a node stress known boundary condition constraint equation, a node stress yield criterion constraint equation and an objective function, and forming a mathematical model for rock mass critical sliding surface limit analysis; and solving the optimal solution of the mathematical model and the node where the tight constraint of the mathematical model is located through a computer program, and determining the rock mass critical sliding surface. The method has the advantages that the method for quantitatively determining the critical sliding surface of the rock mass is provided, the defect that the stability of the rock mass can only be qualitatively judged in the prior art is overcome, the theoretical basis is tight, the applicability is strong, the calculation conclusion is more consistent with the engineering practice, and the method has better engineering practical value.
Description
Technical Field
The invention relates to the field of stability analysis of geotechnical engineering, in particular to a limit analysis method for a rock critical sliding surface.
Background
The stability analysis of rock engineering widely exists in projects such as water conservancy and hydropower, highway and rail traffic, industrial and civil buildings, geological mineral development, ecological environment management and the like. Rock mass stability analysis mainly comprises three problems: analyzing the stability of a rock slope, analyzing the stability of surrounding rocks of an underground cavern and analyzing the bearing capacity of a rock foundation; the method mainly relates to two aspects: and (4) judging the stability of the rock mass and calculating the critical sliding surface of the rock mass. The method mainly solves the problem that whether the rock slopes, the underground caverns and the rock foundations can be kept stable under certain external force conditions. If the rock slopes, the underground caverns and the rock foundations cannot be kept stable under certain external force conditions, the rock mass is usually required to be reinforced in order to meet the requirements of engineering construction. In order to reinforce unstable rock masses, it is necessary to further know where the rock mass is unstable, i.e. the location of the critical sliding surface of the rock mass.
In the prior art, in the rock stability analysis, the commonly used limit balance method can only analyze the rock stability, and in the stability judgment, the critical sliding surface is supposed to be known, so that the calculation of the rock critical sliding surface cannot be carried out. Although other rock stability analysis methods such as a slip line method, a finite element method and a limit analysis method can calculate the rock critical sliding surface, the slip line method has poor applicability and is not suitable for actual engineering calculation; the finite element method needs to be used for the constitutive relation which is most difficult to be clarified in the rock mass, and the extreme analysis method needs to introduce the assumption of stress discontinuity or velocity discontinuity, so that the theoretical tightness is insufficient.
The patent of rock mass stability limit analysis method, which is filed by the inventor in 2021, 12 and 29, has the application numbers: 202111610587.4, it has no need of introducing constitutive relation which is most difficult to be clarified in rock mass, and no need of introducing stress discontinuity or velocity discontinuity assumption, and its applicability is strong, but it can only judge whether rock mass is stable, and can not give rock mass critical sliding surface.
Therefore, the method for analyzing the rock mass critical sliding surface is designed without introducing a stress-strain relation, without introducing stress discontinuity or speed discontinuity hypothesis, and has strict theoretical basis and strong applicability, and has important significance for reinforcing rock slopes, underground caverns, rock foundations and the like.
Disclosure of Invention
The invention aims to provide a rock mass critical sliding surface limit analysis method.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention relates to a rock mass critical sliding surface limit analysis method, which comprises the following steps:
s1, collecting distribution ranges of rock masses, structural surfaces and weak layers with different properties in the target area through geological survey;
s2, measuring physical and mechanical parameters of the rock mass, the structural surface and the weak layer with different properties;
s3, dividing units according to the distribution ranges of rock masses, structural surfaces and weak layers with different properties;
s4, constructing a unit static balance constraint equation, a node stress known boundary condition constraint equation, a node stress yield criterion constraint equation and an objective function by taking the stress components of all nodes as independent variables to form a mathematical model for rock mass critical sliding surface limit analysis;
s5, solving the optimal solution of the mathematical model;
and S6, solving the node where the mathematical model is tightly constrained according to the optimal solution, wherein the region formed by the node is the rock mass critical sliding surface. The rock mass critical sliding surface can have different expression forms for different problems, can be expressed as a two-dimensional curve, and can also be expressed as a two-dimensional area.
Further, in step S2, the rock physical and mechanical parameters include natural volume weight, dry volume weight, saturated volume weight, water content, tensile strength, compressive strength, internal friction angle, and viscosity coefficient;
the physical and mechanical parameters of the structural surface comprise an internal friction angle and a viscosity coefficient;
the physical and mechanical parameters of the soft layer comprise natural volume weight, dry volume weight, saturated volume weight, water content, internal friction angle and viscosity coefficient.
Further, in step S3, the cells in the cell division are quadrilateral cells.
Further, the unit static balance constraint equation is as follows:
wherein, i represents the ith unit, and the value range is all units;
the resultant force of the positive stress component representing the peripheral node of the ith unit on the unit in the X direction;
representing the resultant force of the shear stress component of the peripheral node of the ith unit on the unit in the X direction;
representing the physical strength of the ith cell in the X direction;
the resultant force of the positive stress component representing the peripheral node of the ith cell on the cell in the Y direction;
representing the resultant force of the shear stress component of the peripheral node of the ith unit on the unit in the Y direction;
representing the physical strength of the ith cell in the Y direction.
Further, the constraint equation of the known boundary condition of the node stress is as follows:
wherein j represents the jth node, and the value range is the node within the known stress boundary range;
represents the positive stress of the jth node in the X direction;
represents the positive stress of the jth node in the Y direction;
represents the shear stress of the j-th node;
represents the positive stress of the jth node in the boundary direction;
representing the shear stress of the jth node in the boundary direction;
、
、
and is a calculation parameter.
Further, the node stress yield criterion constraint equation is:
wherein k represents the kth node, and the value range is all the nodes;
represents the positive stress of the k node in the X direction;
represents the positive stress of the k node in the Y direction;
represents the shear stress of the kth node;
，
represents an internal friction angle;
representing the viscous force;
and is a calculation parameter.
Further, the objective function is:
wherein k represents the kth node, and N represents the total number of nodes;
represents the positive stress of the k node in the X direction;
represents the positive stress of the k node in the Y direction;
represents the shear stress of the kth node;
representing positive stress in the X direction of the kth node
An effect coefficient to the objective function;
represents the positive stress in the Y direction of the kth node
An effect coefficient on the objective function;
representing shear stress at the k-th node
The coefficient of effect on the objective function.
Further, the optimal solution refers to the independent variable which satisfies the unit static balance constraint equation, the node stress known boundary condition constraint equation and the node stress yield criterion constraint equation at the same time and enables the objective function to obtain a maximum value.
Further, the tight constraint refers to a constraint that the node stress yield criterion constraint equation takes an equal sign for any optimal solution.
Further, in the three-dimensional rock mass stability analysis, the units are divided by adopting hexahedral units, and according to the unit static balance constraint equation, the node stress known boundary condition constraint equation, the node stress yield criterion constraint equation and the establishment thought of the objective function, a three-dimensional unit static balance constraint equation, the node stress known boundary condition constraint equation, the node stress yield criterion constraint equation and the objective function are established to form a mathematical model of rock mass critical sliding plane limit analysis; solving the optimal solution of the mathematical model; and under the optimal solution, tightly constraining the node where the mathematical model is located, wherein the region formed by the node is the rock mass critical sliding surface. In the three-dimensional analysis, the rock mass critical sliding surface can have different expression forms for different problems, can be expressed as a three-dimensional curve, and can also be expressed as a three-dimensional area.
The method has the advantages that the method for quantitatively determining the critical sliding surface of the rock mass is provided, the defect that whether the rock mass is stable can be judged only qualitatively in the prior art is overcome, the theoretical basis is tight, the applicability is strong, the calculation conclusion is more consistent with the engineering practice, and the method has better engineering practical value.
Drawings
FIG. 1 is a schematic diagram of a rock mass critical sliding surface under the action of strip-shaped uniform load in embodiment 1 of the invention.
Fig. 2 is a flow chart of the method of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely below, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, 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 invention.
Example 1: as shown in fig. 1, the present embodiment takes the problem of the bearing capacity of the foundation under the action of the strip-shaped uniform load as an example, and describes the method for calculating the rock mass critical sliding surface according to the present invention, and the flow is shown in fig. 2.
As shown in figure 1, a strip-shaped uniform load 2 acts within 16m of the top of a foundation 1, the viscous force c of the foundation 1 is 200kPa, and the internal friction angle is
Zero, physical strength. The method according to the invention is used below to solve the critical sliding surface of the foundation 1.
S1, collecting the distribution ranges of rock masses, structural surfaces and weak layers with different properties in the target area; for the embodiment, the foundation 1 is a homogeneous rock mass, and rock masses with different properties, structural surfaces and weak layers do not exist in the foundation range;
s2, obtaining physical and mechanical parameters of rock masses, structural surfaces and weak layers with different properties; the viscous force c of the rock mass of the foundation 1 is 200kPa, the internal friction angle phi is zero, and the volume weight of the rock mass is zero;
s3, dividing the unit; the total horizontal length of the foundation 1 is 40m, wherein the left and right strip-shaped uniform loads 2 are 12m respectively, the vertical thickness of the foundation 1 is 8m, and the unit division is carried out by adopting a quadrilateral unit. The horizontal length range is divided into 40 units and 41 nodes, and the horizontal length dx =1m of the units; 8 units and 9 nodes are divided in the vertical thickness range, and the vertical height of the unit is dy =1 m. The model is divided into a total of 320 units, 369 nodes. The nodes are numbered sequentially from top to bottom and from left to right.
And S4, constructing a constraint equation and an objective function by taking the stress components of all the nodes as independent variables.
For this embodiment, the arguments are:
Wherein,
represents the first
Individual nodes, whose value range is all nodes, i.e.
，
；
Respectively representing the positive stress of the ith node in the X direction, the positive stress of the ith node in the Y direction and the shear stress of the ith node; there were 369 × 3=1107 independent variables.
The constraint equations comprise a unit static balance constraint equation, a node stress known boundary condition constraint equation and a node stress yield criterion constraint equation.
The unit static balance constraint equation is constructed as follows: for this embodiment, the unit horizontal length
Vertical height of unit
The physical strength of the unit in the X direction and the Y direction is zero. By adopting the method, the unit static balance constraint equation is obtained as follows:
wherein,
is taken to be all units, i.e.
。
The node stress known boundary condition constraint equation is constructed as follows: in the present embodiment, the known boundary conditions of the node stress include the normal stress and the shear stress of the upper surface of the foundation 1 on the left side of the load 2, the normal stress and the shear stress of the upper surface of the foundation 1 on the bottom of the load 2, and the normal stress and the shear stress of the upper surface of the foundation 1 on the right side of the load 2.
The node of the upper surface of the left-side foundation 1 of the load 2 has the known boundary condition of node stress
Wherein,
the jth node of the table is a node of the upper surface of the foundation 1 on the left side of the load 2; according to the number of the node in this embodiment, the specific value of j is:
。
the node of the upper surface of the bottom foundation 1 of the load 2 has the known boundary condition of node stress
Wherein,
the jth node of the table is a node of the upper surface of the bottom foundation 1 of the load 2 in a value range; according to the number of the node in this embodiment, the specific value of j is:
。
the node of the upper surface of the foundation 1 on the right side of the load 2 has the known boundary condition of the node stress
Wherein,
the jth node of the table is a node of the upper surface of the foundation 1 on the right side of the load 2 in a value range; according to the present embodiment, the specific value of j to the node number is:
。
the above equations (3), (4) and (5) are the constraint equations of the known boundary conditions of the base point stress in this embodiment.
The node stress yield criterion constraint equation is constructed as follows: for this embodiment, the viscous force c of the rock mass of foundation 1 is 200kPa, the internal friction angle phi is zero,
the physical force is zero, and the node stress yield criterion constraint equation is
The value range of k is as follows:
。
the above formula (2), formula (3), formula (4), formula (5) and formula (6) are the constraint equations of the present embodiment, and thus the construction of the constraint equations of the present embodiment is completed.
And taking the stress components of all the nodes as independent variables, and constructing an objective function as follows:
wherein k represents the kth node, the value range is all the nodes, and N represents the total number of the nodes;
represents the positive stress of the k node in the X direction;
represents the positive stress of the k node in the Y direction;
represents the shear stress of the kth node;
representing positive stress in the X direction of the kth node
An effect coefficient on the objective function;
represents the positive stress of the k-th node in the Y direction
An effect coefficient to the objective function;
representing shear stress at the k-th node
The coefficient of effect on the objective function.
In this embodiment, the objective function is:
wherein,
respectively representing the positive stress of the 1 st, 13 th and Nth nodes in the X direction;
respectively representing the positive stress of the 1 st, 13 th and Nth nodes in the Y direction;
the distribution represents the shear stress of the 1 st, 13 th and Nth nodes;
according to the determination method of the effect coefficients, the effect coefficients of the positive stress in the X direction, the positive stress in the Y direction and the shear stress of the 1 st to 12 th nodes to the target function are all 0; the effect coefficients of the positive stress in the X direction, the positive stress in the Y direction and the shear stress of the 14 th to 369 th nodes on the objective function are all 0; the effect coefficients of the positive stress and the shear stress of the 13 th node in the X direction on the objective function are both 0, and the effect coefficient of the positive stress in the Y direction on the objective function is-1.
The constraint equation and the objective function jointly form a mathematical model of the rock mass sliding surface limit analysis of the embodiment;
s5, solving the optimal solution of the mathematical model through a computer program;
and meanwhile, a unit static balance constraint equation, a node stress known boundary condition constraint equation and a node stress yield criterion constraint equation are satisfied, and the objective function obtains the independent variable of the maximum value, which is the optimal solution of the mathematical model. And the independent variables simultaneously satisfying the unit static balance constraint equation, the node stress known boundary condition constraint equation and the node stress yield criterion constraint equation exist, and the stability of the rock mass is explained. And if the independent variables which simultaneously satisfy the unit static balance constraint equation, the node stress known boundary condition constraint equation and the node stress yield criterion constraint equation do not exist, the rock mass is proved to be unstable. The critical sliding surface of the rock mass can be analyzed only in a state where the rock mass is stable, and if the rock mass itself is unstable, the critical sliding surface does not exist.
S6, according to the optimal solution solved in the step S5, the node where the tight constraint of the embodiment is located is continuously solved through a computer program; the region where the node is located is the critical sliding surface of the rock mass. As shown in fig. 1, the gray node 3 in the figure is the node 3 where the tight constraint of the embodiment is located; the area where the node 3 is located is the gray shaded portion in the figure, which is the critical sliding surface in this embodiment. Aiming at different problems, the critical sliding surface can have different expression forms, can be expressed as a two-dimensional curve, and can also be expressed as a two-dimensional area.
In fig. 1, the number 4 is the rock mass theoretical critical sliding surface 4 of the present embodiment calculated according to the plastic mechanics theory. Compared with the distribution of the nodes 3 where the tight constraint is located, the rock mass critical sliding surface obtained by the method is consistent with the rock mass theoretical critical sliding surface, and the method can well meet the engineering practice requirements.
Example 2
In the calculation of the critical sliding surface of the three-dimensional rock mass, the units are divided by adopting hexahedral units, and a mathematical model for calculating the critical sliding surface of the rock mass is established according to the establishment thinking of the static balance constraint equation of the units, the known boundary condition constraint equation of the node stress and the stress yield criterion constraint equation of the nodes; and solving all tight constraints through a computer program, and further solving all tight nodes to obtain the critical sliding surface of the rock mass.
In summary, the rock mass critical sliding surface limit analysis method provided by the application provides a rock mass critical sliding surface numerical calculation method which does not need to introduce a stress-strain relation, does not need to introduce stress discontinuity or speed discontinuity hypothesis, has tight theoretical basis and strong applicability, and has better engineering practical value, and the calculation conclusion is in accordance with engineering practice.
Claims (10)
1. A rock mass critical sliding surface limit analysis method is characterized by comprising the following steps: the method comprises the following steps:
s1, collecting the distribution ranges of rock masses, structural surfaces and weak layers with different properties in the target area through geological survey;
s2, measuring physical and mechanical parameters of the rock mass, the structural surface and the weak layer with different properties;
s3, dividing units according to the distribution ranges of rock masses, structural surfaces and weak layers with different properties;
s4, constructing a unit static balance constraint equation, a node stress known boundary condition constraint equation, a node stress yield criterion constraint equation and an objective function by taking the stress components of all nodes as independent variables to form a mathematical model for rock mass critical sliding surface limit analysis;
s5, solving the optimal solution of the mathematical model through a computer program;
and S6, solving a node where tight constraint of the mathematical model is located through a computer program according to the optimal solution, wherein a region formed by the node is a rock mass critical sliding surface.
2. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: s2, the physical and mechanical parameters of the rock mass comprise natural volume weight, dry volume weight, saturated volume weight, water content, tensile strength, compressive strength, internal friction angle and viscosity coefficient;
the physical and mechanical parameters of the structural surface comprise an internal friction angle and a viscosity coefficient;
the physical and mechanical parameters of the soft layer comprise natural volume weight, dry volume weight, saturated volume weight, water content, internal friction angle and viscosity coefficient.
3. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: in step S3, the cells in the cell division are quadrilateral cells.
4. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the unit static balance constraint equation is as follows:
wherein, i represents the ith unit, and the value range is all units;
the resultant force of the positive stress component representing the peripheral node of the ith unit on the unit in the X direction;
representing the resultant force of the shear stress component of the peripheral node of the ith unit on the unit in the X direction;
representing the physical strength of the ith cell in the X direction;
the resultant force of the positive stress component representing the peripheral node of the ith cell on the cell in the Y direction;
representing the resultant force of the shear stress component of the peripheral node of the ith unit on the unit in the Y direction;
representing the physical strength of the ith cell in the Y direction.
5. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the constraint equation of the known boundary condition of the node stress is as follows:
wherein j represents the jth node, and the value range is the node within the known stress boundary range;
represents the positive stress of the j node in the X direction;
represents the positive stress of the jth node in the Y direction;
represents the shear stress of the j node;
represents the positive stress of the jth node in the boundary direction;
representing the shear stress of the jth node in the boundary direction;
、
、
and is a calculation parameter.
6. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the node stress yield criterion constraint equation is as follows:
wherein k represents the kth node, and the value range is all the nodes;
represents the positive stress of the k node in the X direction;
represents the positive stress of the k node in the Y direction;
represents the shear stress of the kth node;
，
represents an internal friction angle;
representing viscosityForce;
and is a calculation parameter.
7. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the objective function is:
wherein k represents the kth node, and N represents the total number of nodes;
represents the positive stress of the k node in the X direction;
represents the positive stress of the k node in the Y direction;
represents the shear stress of the kth node;
representing positive stress in the X direction of the kth node
An effect coefficient on the objective function;
represents the positive stress of the k-th node in the Y direction
An effect coefficient on the objective function;
representing shear stress at the k-th node
The coefficient of effect on the objective function.
8. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the optimal solution refers to the independent variable which simultaneously satisfies the unit static balance constraint equation, the node stress known boundary condition constraint equation and the node stress yield criterion constraint equation and enables the objective function to obtain the maximum value.
9. The critical sliding surface limit analysis method of rock mass according to claim 1, characterized in that: the tight constraint refers to the constraint that the node stress yield criterion constraint equation takes equal sign for any optimal solution.
10. The method for analyzing the limit of a critical sliding surface of a rock mass according to claims 1 to 8, wherein: in the three-dimensional rock mass stability analysis, the units are divided by adopting hexahedral units, and a three-dimensional unit static balance constraint equation, a node stress known boundary condition constraint equation, a node stress yield criterion constraint equation and a target function are established according to the establishment thinking of the unit static balance constraint equation, the node stress known boundary condition constraint equation, the node stress yield criterion constraint equation and the target function, so that a mathematical model for rock mass critical sliding surface limit analysis is formed; solving the optimal solution of the mathematical model; and under the optimal solution, tightly constraining the node where the mathematical model is located, wherein the region formed by the node is the rock mass critical sliding surface.
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2023124664A1 (en) * | 2021-12-27 | 2023-07-06 | 黄河勘测规划设计研究院有限公司 | Rock mass stability limit analysis method |
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