CN108170898B - Lower limit method for reliability analysis of jointed rock slope - Google Patents

Lower limit method for reliability analysis of jointed rock slope Download PDF

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CN108170898B
CN108170898B CN201711276041.3A CN201711276041A CN108170898B CN 108170898 B CN108170898 B CN 108170898B CN 201711276041 A CN201711276041 A CN 201711276041A CN 108170898 B CN108170898 B CN 108170898B
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rock slope
jointed rock
reliability
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slope
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CN108170898A (en
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张小艳
张立翔
李泽
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Kunming University of Science and Technology
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Abstract

The invention discloses a lower limit method for reliability analysis of a jointed rock slope, which comprises the following steps: step 1, drawing up basic information of the jointed rock slope according to the actual situation of the jointed rock slope; step 2, generating a Monte Carlo random number of the shearing resistance parameter of the structural surface; step 3, establishing a lower limit method random planning model for solving the reliability of the jointed rock slope; step 4, solving numerical values of a lower limit method random programming model for calculating the reliability of the jointed rock slope to obtain a limit state function of the jointed rock slope; and 5, calculating the reliability of the jointed rock slope according to the result of the step 4. The method adopts the Monte Carlo method to solve the random programming problem, and has the characteristics of high precision, high calculation speed and the like; the method solves the linear programming problem by adopting a simplex method, and the calculation program is simple to program.

Description

Lower limit method for reliability analysis of jointed rock slope
Technical Field
The invention relates to a lower limit method for reliability analysis of jointed rock slopes, and belongs to the technical field of reliability analysis of rock slope engineering.
Background
Slope stability analysis of jointed rock slopes is always a research hotspot of the geotechnical engineering community, and the safety factor or the limit load of the slopes is generally solved to predict the safety degree of the slopes. Slope stability analysis methods can be generally classified into deterministic analysis methods in which all parameters are calculated using a certain value, and non-deterministic analysis methods in which some calculation parameters are calculated using random variables. As is known, many calculation parameters of the rock slope are uncertain, that is, any property of the actual engineering slope has no unique certainty, such as rock mass volume weight, distribution of joints, shear strength of rock mass materials, groundwater level and the like, and the parameters have great uncertainty and variability; the definite values used by our conventional calculations are generally difficult to represent the true values of the parameters. Therefore, the development of the stochastic numerical computation method and the improvement of the computer performance have been paid more and more attention to the reliability analysis of the side slope by engineers.
In the research results of the reliability analysis method of the current jointed rock slope, the method mainly comprises two categories: one is reliability analysis based on the limit balance method, and the other is reliability analysis based on the finite element method. The extreme balance method has the characteristics of simple concept, high calculation efficiency and convenient engineering application, and particularly, some extreme balance methods have calculation expressions with explicit safety factors, and the deficiency is that the position of a sliding surface must be assumed in advance. The finite element method has stronger universality than a limit balance method, can be suitable for complicated slope stability analysis, can obtain stress strain distribution of the slope, and can obtain a more reasonable result by adopting the slope reliability analysis of the finite element method; but the defects are that the constitutive equation of the material is imperfect, and the calculation is large and the calculation efficiency is low.
When the reliability of the side slope is analyzed, the method for analyzing the stability of the side slope comprises the following steps: the random simulation methods matched with the limit balance method and the finite element method include Monte Carlo simulation, importance sampling simulation and the like, and the random simulation methods can be suitable for complex function functions and can solve complex problems of high latitude, so the methods are widely applied to reliability analysis of slopes.
In the geotechnical engineering ultimate bearing capacity analysis, an ultimate analysis lower limit method is an efficient solving tool, a static allowable stress field of a geotechnical body can be established according to a lower limit theorem, and a mathematical programming method is adopted to solve the ultimate load. The lower limit analysis method of the rock-soil body mainly includes an analytical method, a lower limit method based on a numerical method, and the like. Particularly, in recent thirty years, the combination of the lower limit method, the finite element discrete idea and the block unit discrete idea makes it possible to construct the static allowable stress field of the large-scale geotechnical structure; and the lower limit solution obtained by the lower limit method is smaller than the true solution (is more safe), so the lower limit solution is also most useful for engineering. The advantages of the lower limit method are: the lower limit method is theoretically tighter than the rigid body limit balance method; meanwhile, the lower limit method of constitutive relation without considering materials is more efficient than the finite element method in numerical calculation; the lower limit method can obtain the safety coefficient and the stress field at the same time. The research results of the deterministic analysis of the rock slope by using the lower limit method are very many, but the research results of the reliability analysis of the rock slope by using the lower limit method are very few.
In view of the above, the invention provides a new lower limit method for reliability analysis of jointed rock slopes based on the research work of the national science fund project (51564026).
Disclosure of Invention
The invention provides a lower limit method for reliability analysis of a jointed rock slope, and provides a new method for reliability calculation of the jointed rock slope.
The technical scheme of the invention is as follows: a lower limit method for reliability analysis of jointed rock slopes comprises the following specific steps:
step 1, drawing up basic information of the jointed rock slope according to the actual situation of the jointed rock slope;
step 2, generating a Monte Carlo random number of the shearing resistance parameter of the structural surface;
step 3, establishing a lower limit method random planning model for solving the reliability of the jointed rock slope;
step 4, solving numerical values of a lower limit method random programming model for calculating the reliability of the jointed rock slope to obtain a limit state function of the jointed rock slope;
and 5, calculating the reliability of the jointed rock slope according to the result of the step 4.
The basic information of the jointed rock slope comprises: the geometrical parameters of the jointed rock slope, the physical and mechanical parameters of the complete rock mass, the geological parameters of the structural plane and the external load information of the slope.
The step 2 specifically comprises the following steps: determining the cohesion c of the structural surfaceskMean value of (a)cStandard deviation σcAnd coefficient of variation deltacAngle of friction
Figure BDA0001496565720000024
Mean value of
Figure BDA0001496565720000025
Standard deviation of
Figure BDA0001496565720000026
And coefficient of variation
Figure BDA0001496565720000027
While setting the cohesion ckAngle of friction
Figure BDA0001496565720000021
Is a normal distribution; generation of cohesion and friction angles for structural plane k using Monte Carlo random number Generation function
Figure BDA0001496565720000022
Random number of
Figure BDA0001496565720000023
The step 3 specifically comprises the following steps: (1) adopting a block unit to discretely joint the rock slope, and discretizing the joint rock slope into a geometric system of the block unit and a structural surface; (2) establishing an objective function: setting a random variable of the volume weight overload coefficient as a target function; (3) establishing a lower limit method constraint condition of the block unit; (4) establishing a limit state function for calculating the reliability of the jointed rock slope; (5) obtaining a lower limit method random planning model for solving the reliability of the jointed rock slope according to constraint conditions of the target function and the lower limit method and by combining the Monte Carlo random number of the shearing parameters of the structural surface and the extreme state function calculated by the reliability of the jointed rock slope; the lower limit method constraint conditions of the block units comprise balance conditions of the block units, yield conditions of structural surfaces and static force boundary conditions of the block units.
The step 4 specifically comprises the following steps: random number of cohesive force
Figure BDA0001496565720000031
Random number of friction angles
Figure BDA0001496565720000032
Substituting the lower limit method random planning model from 1 to N successively; for each one determined
Figure BDA0001496565720000033
The random programming model is changed into a linear programming problem of which the coefficient matrix is a fixed value, and a simplex method is adopted to solve a corresponding objective function lambda (t); substituting the solved N groups of lambda (t), (t is 1, …, N) into the extreme state function to obtain the extreme state function Z is lambda (t), (t is 1, …, N) of the jointed rock slope.
The reliability of the jointed rock slope comprises the mean value, the standard deviation and the reliability index of the unit weight overload coefficient of the jointed rock slope and the failure probability of the jointed rock slope.
The lower limit method stochastic programming model is as follows:
Figure BDA0001496565720000034
where t is 1, …, N is the number of random numbers generated;
a limiting state function Z of the reliability calculation of the jointed rock slope is lambda (t), and an objective function lambda (t) represents a random variable of a volume weight overload coefficient;
Figure BDA0001496565720000035
denotes the equilibrium condition of the bulk unit, miIs the number of structural planes, alpha, in the block unit ikIs n of structural plane kkThe included angle between the axis and the x axis is positive anticlockwise, NkIs structural plane k along outer normal nkNormal force in direction and pulling positive pressure negative, VkIs that the structural plane k is along the tangential direction skThe block rotates anticlockwise to be positive, lambda (t) is the random quantity of the t-th overload coefficient, gamma is the unit weight of the block unit material, AiIs the area of the bulk unit i, fxiIs the equivalent external force in the x direction at the centroid of the ith block unit, fyiIs an equivalent external force in the y direction at the centroid of the ith block element,nbthe number of block units in the jointed rock slope is saved;
Figure BDA0001496565720000041
the yield condition of the structural surface is shown,
Figure BDA0001496565720000042
is the t-th random number of structural surface k cohesion,
Figure BDA0001496565720000043
is the t-th random number, l, of the k-friction angle of the structural planekIs the length of the structural plane k, nsThe number of structural surfaces in the whole jointed rock slope;
Figure BDA0001496565720000044
representing the static boundary conditions of the block unit,
Figure BDA0001496565720000045
is the internal force vector of the structural surface with known boundary conditions, b represents the structural surface with known boundary external force equal to 0 in the jointed rock slope, ncThe number of structural surfaces with external force equal to 0 on the boundary in the whole rock slope is shown;
Figure BDA0001496565720000046
a generating function representing a monte carlo random number of cohesion and friction angles;
Figure BDA0001496565720000047
respectively, are the average values of structural surface cohesion and friction angle,
Figure BDA0001496565720000048
the standard deviation of the structural surface cohesion and the friction angle are respectively.
The invention has the beneficial effects that:
1. the invention provides a new method for analyzing the reliability of the ultimate bearing capacity of the jointed rock slope, and the reliability of the jointed rock slope (the mean value, the standard deviation and the reliability index of the volume weight overload coefficient and the failure probability of the jointed rock slope) can be solved by combining the lower limit method theory, the block unit dispersion idea, the random planning method and the Monte Carlo method for the first time.
2. The method can obtain the normal distribution condition of the volume weight overload coefficient corresponding to the Monte Carlo random number (normal distribution) of the structural surface cohesive force and the friction angle, and can calculate the failure probability of the side slope according to the volume weight overload coefficient less than 1;
3. the method adopts the Monte Carlo method to solve the random programming problem, and has the characteristics of high precision, high calculation speed and the like;
4. the method solves the linear programming problem by adopting a simplex method, and the calculation program is simple to program.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic view of the stress of a jointed rock slope block unit;
FIG. 3 is a schematic view of a structural plane between adjacent block units of a jointed rock slope;
FIG. 4 is a schematic view of the geometry of the jointed rock slopes of example 1;
FIG. 5 is a random number of AC cohesive force of the structural surface of example 1
Figure BDA0001496565720000058
A schematic diagram of normal distribution;
FIG. 6 is a random number of AC rubbing angles of the structural surfaces of example 1
Figure BDA0001496565720000057
A schematic diagram of normal distribution;
FIG. 7 is a discrete schematic view of the jointing rock slope block unit of example 1;
FIG. 8 is a schematic diagram of the normal distribution of the volume-weight overload coefficient of the jointed rock slope in example 1;
FIG. 9 is a schematic view of the geometry of the jointed rock slopes of example 2;
FIG. 10 is a graph showing the random number of cohesive force of the structural surface in example 2
Figure BDA0001496565720000051
A schematic diagram of normal distribution;
FIG. 11 is a random number of the rubbing angles of the structural surfaces of example 2
Figure BDA0001496565720000052
A schematic diagram of normal distribution;
FIG. 12 is a discrete schematic view of the jointing rock slope block unit of example 2;
FIG. 13 is a schematic diagram of the volume weight overload coefficient normal distribution of the jointed rock slopes of example 2.
Detailed Description
Example 1: as shown in fig. 1 to 13, a lower limit method for reliability analysis of jointed rock slopes includes the following steps:
drawing up basic information of jointed rock slope
Example 1 is a rock slope comprising a set of weak structural surfaces, the structure of which is shown in fig. 4, and the main calculation parameters are as follows: the geometrical parameters of the jointed rock slope are as follows: the height is 55.94m, the top width is 36.6m, and the angle of the face is 45 degrees; the volume weight of the complete rock mass is set to be a fixed value and 25kN/m is taken3(ii) a The inclination angle of the structural surface (joint surface) AC is 30 degrees; the side slope is only under the action of the dead load of the rock mass.
(II) generating Monte Carlo random numbers of shear parameters of structural surfaces
Average value of the cohesion of the structural surface ACc100kPa, coefficient of variation δcRespectively taking 0.10, 0.15, 0.20 and 0.25, and taking the average value of the friction angle
Figure BDA0001496565720000059
Coefficient of variation
Figure BDA00014965657200000510
The probability distribution types of the cohesion and the friction angle are normal distribution respectively taken as 0.10, 0.15, 0.20 and 0.25. Generation of AC shear parameters of structural plane by using formula (1)The random number of (2). In this example, 3000N are taken as N, that is, 3000 random numbers of cohesion and friction angle are generated, respectively
Figure BDA0001496565720000053
At the coefficient of variation deltacRandom number of structural surface AC cohesive force when 0.2
Figure BDA0001496565720000054
The normal distribution of (A) is shown in FIG. 5, and the coefficient of variation at the friction angle
Figure BDA0001496565720000055
In the case of (2) random number of AC friction angles of the structural surface
Figure BDA0001496565720000056
The normal distribution of (A) is shown in FIG. 6.
(III) establishing a lower limit method random programming model for solving the reliability of the jointed rock slope
The slope of example 1 was discretized into 2 block units as shown in fig. 7. And (3) establishing a lower-limit method stochastic programming model for solving the reliability of the jointed rock slope in the embodiment 1 according to the formula (9).
(IV) numerical solution of lower-limit method stochastic programming model for reliability calculation of jointed rock slopes
3000 sets of volume weight overload systems lambda (t) (t is 1, …,3000) are calculated by adopting a numerical calculation process of the reliability lower limit of the jointed rock slopes as shown in fig. 1. Mean value of cohesion in structural planec100kPa, coefficient of variation δc0.20, average of angle of friction
Figure BDA0001496565720000061
Coefficient of variation
Figure BDA0001496565720000062
Fig. 8 shows the normal distribution of the calculated slope specific gravity overload coefficients λ (t), (t ═ 1, …,3000) in example 1.
And (V) calculating the reliability of the jointed rock slope.
The reliability of the jointed rock slope of the embodiment 1 obtained by solving the equations (10), (11), (12) and (14) is shown in table 1.
Table 1 example 1 statistical table of reliability of jointed rock slopes
Figure BDA0001496565720000063
In the case where all the parameters are constant values, solving the volume weight overload coefficient of the slope is a statically determined problem, the limit value of the corresponding constant value λ can be determined analytically, and the cohesion c at the structural surface AC isk100000kPa, friction angle
Figure BDA0001496565720000064
In the case of (2), the obtained analysis of the volume-weight overload coefficient is 1.259, the errors of the mean value and the analysis solution of various different variation coefficients calculated by the method of the invention are small (within 1.1%), and the errors of the mean value and the analysis solution increase with the increase of the variation coefficient.
The reliability of the side slope is that the beta mark is reduced along with the increase of the variation coefficient, which shows that the higher the discrete degree of the shear strength parameter is, the lower the reliability of the side slope is.
Probability of failure P of slopefThe coefficient of variation increases, which shows that the higher the dispersion degree of the shear strength parameter, the higher the failure probability of the slope.
Example 2: a lower limit method for reliability analysis of jointed rock slopes comprises the following specific steps:
drawing up basic information of jointed rock slope
Example 2 is a rock slope containing two sets of joints, the geometrical information of which is shown in fig. 9, and the main calculation parameters are as follows: the geometrical parameters of the jointed rock slope are as follows: the height is 30m, the top width is 45m, and the angle of the blank surface is 80 degrees; the volume weight of the complete rock mass is set to be a fixed value and 27kN/m is taken3The distance between the two groups of structural surfaces (joint surfaces) is 6m, and the inclination angles of the two groups of structural surfaces are respectively 25 degrees and 80 degrees; the side slope is only under the action of the dead load of the rock mass.
(II) generating Monte Carlo random numbers of shear parameters of structural surfaces
Example 2: average value of all structural surface cohesionc50kPa, coefficient of variation δcRespectively taking 0.10, 0.15, 0.20 and 0.25, and taking the average value of the friction angles of all structural surfaces
Figure BDA0001496565720000071
Coefficient of variation
Figure BDA0001496565720000072
Respectively taking 0.10, 0.15, 0.20 and 0.25. Random numbers of the shear parameters of all structural surfaces of the slope of example 2 were generated using equation (1). In this example, N is 3000, that is, 3000 random numbers each generating joint cohesion and friction angle
Figure BDA0001496565720000073
At the coefficient of variation deltacRandom number of all structural surface cohesive force in the case of 0.15
Figure BDA0001496565720000074
Is shown in FIG. 10, and the normal distribution of the coefficient of variation
Figure BDA0001496565720000075
Random number of friction angles of all structural surfaces in the case of (2)
Figure BDA0001496565720000076
The normal distribution of (A) is shown in FIG. 11.
(III) solving lower limit method stochastic programming model of reliability of jointed rock slope
The slope of example 2 was discretized into 37 block units as shown in fig. 12. And (3) establishing a lower-limit method random planning model of the reliability of the jointed rock slope in the embodiment 2 according to the formula (9).
(IV) numerical solution of lower-limit method stochastic programming model for reliability calculation of jointed rock slopes
3000 sets of volume weight overload systems λ (t) (1, …,3000) in example 2 were calculated by using the numerical calculation flow of the lower limit of reliability of jointed rock slopes as shown in fig. 1.
Mean value of cohesion in structural planec50kPa, coefficient of variation δc0.15, mean value of rubbing angle
Figure BDA0001496565720000077
Coefficient of variation
Figure BDA0001496565720000078
Fig. 13 shows the normal distribution of the calculated slope bulk density overload coefficients λ (t), (t ═ 1, …,3000) in example 2.
And (V) calculating the reliability of the jointed rock slope.
The reliability of the jointed rock slope of the embodiment 2 obtained by solving the equations (10), (11), (12) and (14) is shown in Table 2.
Table 2 example 2 reliability statistical table for jointed rock slopes
Figure BDA0001496565720000081
The mean value and the standard deviation of the volume weight overload coefficients of the jointed rock slope are increased along with the increase of the variation coefficient, and the reliability index beta is reduced along with the increase of the variation coefficient.
Failure probability P of jointed rock slopefThe coefficient of variation decreased and increased, and the probability of failure was 17.867% when the coefficient of variation assumed 0.1, and increased to 35.7% when the coefficient of variation assumed 0.25.
The working principle of the invention is as follows:
the method comprises the steps of taking a jointed rock slope as a research object, combining a Monte Carlo method, a lower limit method theory, a block unit discrete thought and a random planning method, and solving the reliability of the jointed rock slope; firstly, generating random numbers of structural surface shearing parameters by adopting a Monte Carlo method; then, dispersing the jointed rock slopes into a geometric body of a block unit and a structural surface; constructing a static allowable stress field of the jointed rock slope according to a lower limit method theory, wherein the static allowable stress field simultaneously meets the block unit balance condition, the structural surface yield condition and the static boundary condition, and the volume weight overload coefficient is taken as a target function; a lower limit method stochastic programming model for solving the reliability of the jointed rock slope is established, and a solving method of the stochastic programming model is provided; the reliability of the jointed rock slope (including the mean value, standard deviation and reliability index of the volume weight overload coefficient and the failure probability of the slope) can be obtained through calculation.
The technical scheme of the invention is shown in figure 1.
The technical scheme of the lower limit method for solving the reliability of the jointed rock slope is sequentially carried out according to the following steps:
drawing up basic information of jointed rock slope
According to the actual situation of the jointed rock slope, main calculation parameters for reliability analysis are drawn up, and the calculation parameters comprise: the geometrical parameters of the jointed rock slope, the physical and mechanical parameters (volume weight and the like) of the complete rock mass, the geological parameters of the structural plane, the external load information of the slope and the like.
Second, generating Monte Carlo random numbers of shear parameters of structural surfaces
The bearing capacity of the jointed rock slope is mainly controlled by the strength of the structural surface, and two shear strength parameters (cohesion c) of the structural surfacekAngle of friction
Figure BDA0001496565720000091
Has larger uncertainty and variability, so the invention sets the cohesion c of the structural surfacekAngle of friction
Figure BDA0001496565720000092
Are mutually independent random quantities and conform to normal distribution; and simultaneously setting other parameters such as the volume weight of the rock mass and the like as fixed values. The invention adopts the Monte Carlo method to generate the random quantity of the cohesive force and the friction angle
Figure BDA0001496565720000093
The specific process of the random number is as follows:
1.determining the cohesion c of the structural surfaceskMean value of (a)cStandard deviation σcAnd coefficient of variation deltacAngle of friction
Figure BDA00014965657200000914
Mean value of
Figure BDA0001496565720000094
Standard deviation of
Figure BDA0001496565720000095
And coefficient of variation
Figure BDA0001496565720000096
While setting the cohesion ckAngle of friction
Figure BDA0001496565720000097
Is a normal distribution;
2. generation of cohesion and friction angles for structural plane k using Monte Carlo random number Generation function
Figure BDA0001496565720000098
The random number, cohesion and friction angle monte carlo random number generation functions of (a) are as follows:
Figure BDA0001496565720000099
in the formula, t is (1, …, N), N is the number of generated random numbers and is generally 2000-5000;
Figure BDA00014965657200000910
the t-th random number (the superscript r represents that the variable is a random number variable) of the structural surface cohesion;
Figure BDA00014965657200000911
the t-th random number (the superscript r represents that the variable is a random number variable) which is the friction angle of the structural surface;
Figure BDA00014965657200000912
respectively, are the average values of structural surface cohesion and friction angle,
Figure BDA00014965657200000913
the standard deviation of the structural surface cohesion and the friction angle are respectively.
Third, establishing a lower limit method random planning model for solving the reliability of the jointed rock slope
The lower limit of plasticity limit analysis method can be described as: the ultimate load is the largest among the loads corresponding to all static allowable stress fields. The stress field that simultaneously satisfies the equilibrium condition, the yield condition, and the static force boundary condition is referred to as the static force allowable stress field. In a jointed rock slope structure body, a plurality of static fields meeting the conditions can be constructed; one seeks its corresponding load, where the maximum load is necessarily the closest to the true limit load. The lower bound solution is necessarily smaller than the true solution.
In order to construct the static allowable stress field of the jointed rock slope, the invention adopts block units to discretely joint the rock slope, sets the overload coefficient of the external load of the slope as an objective function, constructs the static allowable stress field of which the block units simultaneously meet a balance equation, a yield condition and a boundary condition, and establishes a lower limit method random programming model for solving the ultimate load of the jointed rock slope. The method comprises the following specific steps:
1. discrete jointing rock slope by adopting block units
The invention adopts the block units to discretely joint the rock slope, the stress of the block units is shown in figure 2, and the stress of the structural surface k of the adjacent block units is shown in figure 3. Wherein (x, y) is the global coordinate system; (n)k,sk) Is a local coordinate system of a structural plane k between adjacent block units i, j, n as shown in FIG. 3kIs the outer normal direction of the structural plane k, skIs tangential to the structural surface; i-shaped center C of block unitiActing on equivalent load force vector
Figure BDA0001496565720000101
fxiIs the equivalent external force in the x direction at the centroid of the ith block unit, fyiIs the equivalent external force in the y direction at the centroid of the ith block element. (ii) a Structural plane k centroid P between adjacent block units i and jkActing on the joint with an internal force vector
Figure BDA0001496565720000102
The other side of the structural surface acts with the reaction force vector thereof
Figure BDA0001496565720000103
NkIs along the structural plane k along the external normal nkNormal force in direction (positive or negative sign: pulling positive or negative), VkIs along the structure plane k along the tangential direction skDirectional shearing force (the sign of which is defined as that the block is rotated counterclockwise to be positive), N'kIs NkReaction force of (V)kIs' is VkThe reaction force of (a).
After the discrete joint rock slope of the block units is adopted, the joint slope becomes a geometric system of the block units and a structural plane, and in order to simplify calculation and facilitate the construction of constraint conditions of a static allowable stress field of a lower limit method, the invention makes the following assumptions: (1) assuming that the block unit only translates and does not rotate; (2) assuming that the block units are rigid bodies, the block units cannot deform or be damaged at all, and the damage of the side slope only occurs on the structural surface between the adjacent block units; (3) assuming that the structural plane material is an ideal rigid plastic material, and (4) assuming that the structural plane in the rock body is only subjected to shear failure.
2. Establishing an objective function
The invention adopts a volume weight overload mode to ensure that the jointed rock slope reaches a limit state, and the actual volume weight of the rock mass of the slope is set as gamma, namely, the slope is in a damaged limit state when the volume weight gamma is gradually increased to gamma'. The invention defines the volume weight overload coefficient as follows:
Figure BDA0001496565720000104
in the above formula: lambda is the volume weight overload coefficient of the jointed rock slope; gamma' is the ultimate volume weight of the block unit material when the jointed rock slope is overloaded; gamma is the current actual volume weight of the block unit material of the jointed rock slope.
According to the lower limit method theory, when the jointed rock slope reaches a limit state, the maximum value gamma' of the volume weight gamma is required; when the random variation of the cohesion force and the random variation of the friction angle of the structural surface are respectively dispersed into N random numbers according to the formula (1), the volume weight overload coefficient lambda is also correspondingly changed into
Figure BDA0001496565720000105
And
Figure BDA0001496565720000106
the relevant random variables are as follows:
Figure BDA0001496565720000107
in the above formula: t is (1, …, N), and λ (t) is a random variable of the joint rock slope unit weight overload coefficient.
According to the theory of the lower limit method, the random variable lambda (t) of the volume-weight overload coefficient is set as the following objective function:
Maximize:λ(t)(4)
3. establishing lower bound constraints for a block unit
Building a static allowable stress field requires that the block units simultaneously satisfy a balance equation, a yield condition and a static boundary condition.
(1) Equilibrium condition of block unit
For any block unit i shown in fig. 2 to be balanced under the action of self weight, external load and adjacent rock, the force balance equation is as follows:
Figure BDA0001496565720000111
in the above formula: n isbThe number of block units in the jointed rock slope is saved; m isiIs the number of structural faces in the block unit i; λ (t) is the t-th overload coefficient random quantity, AiIs the area of the block unit i, and γ is the block unitVolume weight of material, alphakIs n of structural plane kkThe angle between the axis and the x-axis (positive or negative, provided that it is positive counterclockwise), NkIs along the structural plane k along the external normal nkNormal force in direction (positive or negative sign: pulling positive or negative), VkIs along the structure plane k along the tangential direction skShear in the direction (the sign of which defines that the block is rotated anticlockwise to positive), fxiIs the equivalent external force in the x direction at the centroid of the ith block unit, fyiIs the equivalent external force in the y direction at the centroid of the ith block element.
(2) Yield condition of structural surface
For any one of the structural surfaces k shown in FIG. 3, N sets of random numbers are used due to the cohesive force and friction angle of the structural surface
Figure BDA0001496565720000112
The Mohr-Coulomb yield condition for shear slip failure of structural plane k can therefore be written in the form of random numbers of cohesion and friction angle:
Figure BDA0001496565720000113
in the above formula: n issThe number of structural surfaces in the whole jointed rock slope; lkIs the length of structural plane k;
Figure BDA0001496565720000114
a tth random number (the superscript r represents a variable as a random number variable) which is the cohesive force of the structural surface k;
Figure BDA0001496565720000115
the t-th random number (the superscript r represents that the variable is a random number variable) of the friction angle of the structural plane k; n is a radical ofkIs the normal force (sign convention: pulling positive and negative) V of the structural plane kkIs the shear of structural plane k; n is the number of cohesion random numbers and friction angle random numbers.
(3) Static boundary conditions of block units
According to the lower bound law theory, the static allowable stress field must satisfy known static boundary conditions. Considering a structural surface b with a known boundary external force equal to 0 in the jointed rock slope, the boundary condition expression is as follows:
Figure BDA0001496565720000121
in the above formula: n iscThe number of structural surfaces with external force equal to 0 on the boundary in the whole rock slope is shown;
Figure BDA0001496565720000122
is the internal force vector of the structural plane for which the boundary conditions are known.
4. Establishing a limit state function for calculating the reliability of the jointed rock slope
In order to solve the mathematical distribution rule of the volume weight overload coefficient of the jointed rock slope, the invention sets the extreme state function of the reliability calculation of the jointed rock slope according to the formula (3) as follows:
Figure BDA0001496565720000123
in the above formula: z is the extreme state function of the jointed rock slopes.
5. Lower limit method random planning model for establishing joint rock slope reliability calculation
According to the objective function formula (4) of the static allowable stress field and the constraint condition formulas (5), (6) and (7) of the lower limit method, and by combining the condensation force random number and the friction angle random number of the structural surface to generate the function formula (1) and the extreme state function formula (8) for calculating the reliability of the jointed rock slope, the lower limit method random programming model for solving the reliability of the jointed rock slope can be as follows:
Figure BDA0001496565720000124
in the above formula: (t ═ 1, …, N).
Fourthly, numerical solution of lower limit method random programming model for calculating reliability of jointed rock slope
The solution objective of equation (9) is: random number of cohesive force from structural surface
Figure BDA0001496565720000131
And random number of friction angle
Figure BDA0001496565720000132
The mathematical characteristics of the volume-weight overload coefficient random variable lambda (t) are solved. The coefficient of the matrix of the lower limit method model of the formula (9) includes random numbers of the cohesive force and the friction angle of the structural surface
Figure BDA0001496565720000133
Which is a typical random programming problem and thus attempts to go directly through
Figure BDA0001496565720000134
It is very difficult to accurately solve the mathematical characteristics of λ (t), and only approximate numerical calculation methods can be used.
The invention adopts the following numerical method to solve the lower limit method random programming model of the reliability of the jointed rock slope:
(1) cycling t from t to 1 to t to N, and sequentially calculating the random number of the cohesive force
Figure BDA0001496565720000135
Random number of friction angles
Figure BDA0001496565720000136
Substitution formula (9);
(2) for each determination
Figure BDA0001496565720000137
The equation (9) becomes a linear programming problem with a coefficient matrix being a fixed value, and a simplex method is adopted to solve the corresponding objective function lambda (t);
(3) substituting the solved N groups λ (t), (t ═ 1, …, N) into the extreme state function, and calculating the extreme state function Z ═ λ (t), (t ═ 1, …, N) of the jointed rock slope.
The numerical solution flow of the present invention is shown in fig. 1.
And fifthly, calculating the reliability of the jointed rock slope.
Calculating the basic statistics of the extreme state function Z according to the calculated N groups of lambda (t) to obtain the reliability of the jointed rock slope, wherein the reliability comprises the following steps: the mean value, the standard deviation and the reliability index of the volume weight overload coefficient of the jointed rock slope and the failure probability of the jointed rock slope.
The mean value calculation formula of the volume weight overload coefficient of the jointed rock slope is as follows:
Figure BDA0001496565720000138
in the above formula: mu.sλIs the mean value of the volume weight overload coefficient of the jointed rock slope.
The standard deviation calculation formula of the volume weight overload coefficient of the jointed rock slope is as follows:
Figure BDA0001496565720000139
in the above formula: sigmaλIs the standard deviation of the volume weight overload coefficient of the jointed rock slope.
The joint rock slope reliability index calculation formula is as follows:
Figure BDA0001496565720000141
in the above formula: mu.sλIs the mean value, sigma, of the volume weight overload coefficient of the jointed rock slopeλIs the standard deviation of the volume weight overload coefficient of the jointed rock slope, and beta is the reliability index of the jointed rock slope.
When the volume weight overload coefficient of the side slope is more than or equal to 1, the side slope is safe, and when the volume weight overload coefficient of the side slope is less than 1, the side slope is unstable, so that the failure function of the jointed rock side slope is as follows:
Figure BDA0001496565720000142
in the above formula: i (t) is a function of failure of the jointed rock slopes.
The probability of failure P of the slope can be calculated as follows:
Figure BDA0001496565720000143
in the above formula: pfIs the failure probability of the jointed rock slope.
The invention is characterized in that: the method combines a Monte Carlo method, a plastic limit analysis lower limit method, a block unit discrete thought and random planning, provides a lower limit method for calculating the reliability of the jointed rock slope, can calculate and obtain the mean value, the standard deviation and the reliability index of the volume weight overload coefficient of the slope, and can obtain the failure probability of the jointed rock slope. Compared with the traditional limit balance reliability analysis method, the method has higher calculation precision; compared with a finite element reliability analysis method, the method omits the constitutive relation of the materials, so that the calculation speed and efficiency are higher.
While the present invention has been described in detail with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

Claims (1)

1. The utility model provides a joint rock matter side slope reliability analysis lower limit method which characterized in that: the lower limit method comprises the following specific steps:
step 1, drawing up basic information of the jointed rock slope according to the actual situation of the jointed rock slope;
step 2, generating a Monte Carlo random number of the shearing resistance parameter of the structural surface;
step 3, establishing a lower limit method random planning model for solving the reliability of the jointed rock slope;
step 4, solving numerical values of a lower limit method random programming model for calculating the reliability of the jointed rock slope to obtain a limit state function of the jointed rock slope;
step 5, calculating the reliability of the jointed rock slope according to the result of the step 4;
the lower limit method stochastic programming model is as follows:
Figure FDA0003238649630000011
where t is 1, …, N is the number of random numbers generated;
a limiting state function Z of the reliability calculation of the jointed rock slope is lambda (t), and an objective function lambda (t) represents a random variable of a volume weight overload coefficient;
Figure FDA0003238649630000012
denotes the equilibrium condition of the bulk unit, miIs the number of structural planes in the block unit i, akIs n of structural plane kkThe included angle between the axis and the x axis is positive anticlockwise, NkIs structural plane k along outer normal nkNormal force in direction and pulling positive pressure negative, VkIs that the structural plane k is along the tangential direction skThe block rotates anticlockwise to be positive, lambda (t) is the random quantity of the t-th overload coefficient, gamma is the unit weight of the block unit material, AiIs the area of the bulk unit i, fxiIs the equivalent external force in the x direction at the centroid of the ith block unit, fyiIs the equivalent external force in the y direction at the centroid of the ith block unit, nbThe number of block units in the jointed rock slope is saved;
Figure FDA0003238649630000021
the yield condition of the structural surface is shown,
Figure FDA0003238649630000022
is the t-th random number of structural surface k cohesion,
Figure FDA0003238649630000023
is the t-th random number, l, of the k-friction angle of the structural planekIs the length of the structural plane k, nsThe number of structural surfaces in the whole jointed rock slope;
Figure FDA0003238649630000024
representing the static boundary conditions of the block unit,
Figure FDA0003238649630000025
is the internal force vector of the structural surface with known boundary conditions, b represents the structural surface with known boundary external force equal to 0 in the jointed rock slope, ncThe number of structural surfaces with external force equal to 0 on the boundary in the whole rock slope is shown;
Figure FDA0003238649630000026
a generating function representing a monte carlo random number of cohesion and friction angles; mu.sc,
Figure FDA0003238649630000027
Mean values, σ, of structural surface cohesion and friction angle, respectivelyc,
Figure FDA0003238649630000028
Respectively are the standard deviation of the structural surface cohesion and the friction angle;
the step 2 specifically comprises the following steps: determining the cohesion c of the structural surfaceskMean value of (a)cStandard deviation σcAnd coefficient of variation deltacAngle of friction
Figure FDA0003238649630000029
Mean value of
Figure FDA00032386496300000210
Standard deviation of
Figure FDA00032386496300000211
And coefficient of variation
Figure FDA00032386496300000212
While setting the cohesion ckAngle of friction
Figure FDA00032386496300000213
Is a normal distribution; generation of cohesion and friction angles for structural plane k using Monte Carlo random number Generation function
Figure FDA00032386496300000214
Random number of
Figure FDA00032386496300000215
The basic information of the jointed rock slope comprises: the method comprises the following steps of (1) saving geometrical parameters of a rock slope, physical and mechanical parameters of a complete rock body, structural surface geological parameters and slope external load information;
the step 3 specifically comprises the following steps: (1) adopting a block unit to discretely joint the rock slope, and discretizing the joint rock slope into a geometric system of the block unit and a structural surface; (2) establishing an objective function: setting a random variable of the volume weight overload coefficient as a target function; (3) establishing a lower limit method constraint condition of the block unit; (4) establishing a limit state function for calculating the reliability of the jointed rock slope; (5) obtaining a lower limit method random planning model for solving the reliability of the jointed rock slope according to constraint conditions of the target function and the lower limit method and by combining the Monte Carlo random number of the shearing parameters of the structural surface and the extreme state function calculated by the reliability of the jointed rock slope; the lower limit method constraint conditions of the block units comprise balance conditions of the block units, yield conditions of structural surfaces and static force boundary conditions of the block units;
the step 4 specifically comprises the following steps: random number of cohesive force
Figure FDA00032386496300000216
Random number of friction angles
Figure FDA00032386496300000217
Substituting the lower limit method random planning model from 1 to N successively; for each one determined
Figure FDA00032386496300000218
The random programming model is changed into a linear programming problem of which the coefficient matrix is a fixed value, and a simplex method is adopted to solve a corresponding objective function lambda (t); substituting the solved N groups of lambda (t) into the extreme state function to obtain the extreme state function Z of the jointed rock slope as lambda (t);
the reliability of the jointed rock slope comprises the mean value, the standard deviation and the reliability index of the unit weight overload coefficient of the jointed rock slope and the failure probability of the jointed rock slope.
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