CN108170898A - A kind of jointed rock slope reliability analysis Lower Bound Limit - Google Patents
A kind of jointed rock slope reliability analysis Lower Bound Limit Download PDFInfo
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Abstract
The invention discloses a kind of jointed rock slope reliability analysis Lower Bound Limit, the method for the present invention is：Step 1, the essential information that jointed rock slope is drafted according to the actual conditions of jointed rock slope；Step 2, generating structure face parameters of shear resistant Monte Carlo random number；Step 3 establishes the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope reliability；The numerical solution of step 4, the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating, obtains the limit state function of jointed rock slope；Step 5, the reliability that jointed rock slope is calculated according to the result of step 4.Dynamic Programming Problems are solved using monte carlo method in the method for the present invention, have the characteristics that precision is high, calculating speed is fast；The method of the present invention solves linear programming problem using simplex method, and factorization is simple.
Description
Technical field
The present invention relates to a kind of jointed rock slope reliability analysis Lower Bound Limits, belong to the analysis of rock side slope engineering reliability
Technical field.
Background technology
The slope stability analysis of jointed rock slope is always the research hotspot of geotechnical engineering circle, generally by solving side slope
Safety coefficient or ultimate load predict the degree of safety of side slope.Method for Slope Stability Analysis generally can be divided into deterministic parsing
Method and uncertainty analysis method, the former is to take all parameters to determine that value calculates, and the latter is by certain calculating
Parameter carries out calculating analysis as stochastic variable.It is generally known that many calculating parameters of rock side slope are all probabilistic, i.e.,
Any property of Practical Project side slope does not all have unique certainty, as rock mass unit weight, the distribution at joint, rock mass materials it is anti-
Intensity, groundwater level etc. are cut, these parameters all have very big uncertainty and variability；Used by our conventionally calculations
Determine that value is general it is difficult to the actual value of representation parameter.Therefore, the development of random number computational methods and the promotion of computer performance,
Side slope carries out reliability analysis and is increasingly paid attention to by engineers.
In the achievement in research of the reliability analyzing method of current jointed rock slope, it is largely divided into two classes：One kind is base
In the reliability analysis of limit equilibrium method, another kind of is the reliability analysis based on Finite Element.Limit equilibrium method has general
The characteristics of simple, computational efficiency is high, engineer application is convenient is read, particularly some limit equilibrium methods have the calculating of explicit safety coefficient
Expression formula, deficiency are must to assume the position of sliding surface in advance.Finite Element is more stronger than limit equilibrium method versatility, can be applicable in
In complicated Analysis of Slope Stability, and the Stress distribution of side slope can be obtained, it is reliable using the side slope of FInite Element
Degree analysis can obtain relatively reasonable result；But its deficiency is that material constitutive equation is not perfect, it is low to calculate larger computational efficiency.
In the reliability analysis for carrying out side slope, with Method for Slope Stability Analysis：Limit equilibrium method, Finite Element are matched
The Method of Stochastic of set has Monte Carlo simulation, importance sampling simulation etc., these Method of Stochastic can be adapted for multiple
Miscellaneous power function can solve the challenge of high latitude, therefore these methods are widely used in the reliability point of side slope
In analysis.
In geotechnical engineering analysis of Ultimate, lower bound limit analysis is a kind of efficient solution tool, under
The static(al) permissible stress field of Rock And Soil can be established and solve ultimate load using mathematic programming methods by limiting reason.The lower limit of Rock And Soil
Analytic approach mainly includes analytic method, Lower Bound Limit based on numerical method etc..Particularly in the late three decades, Lower Bound Limit and finite element from
Dissipate the combination of thought, the discrete thought of Rigid Body Element so that it becomes in the static(al) permissible stress field for constructing extensive geotechnical structure
It may；And the lower limit solution that Lower Bound Limit solves is less than true solution (relatively safety), therefore lower limit solution is also most useful to engineering.Under
The advantages of limit method is：Lower Bound Limit is tighter than balance method of rigid-body limit in theory；Simultaneously as do not consider that this structure of material closes
It is that Lower Bound Limit is more more efficient than FInite Element in numerical computations；Lower Bound Limit can obtain safety coefficient and stress field simultaneously.It uses
The achievement in research that Lower Bound Limit analyzes being determined property of rock side slope is very more, but using Lower Bound Limit to rock side slope can
It is also considerably less by the achievement in research of degree analysis.
In consideration of it, the research work the present invention is based on project of national nature science fund project (51564026) proposes one kind newly
Jointed rock slope reliability analysis Lower Bound Limit.
Invention content
The present invention provides a kind of jointed rock slope reliability analysis Lower Bound Limits, are jointed rock slope reliability calculating
A kind of new method is provided.
The technical scheme is that：A kind of jointed rock slope reliability analysis Lower Bound Limit, the method specific steps
It is as follows：
Step 1, the essential information that jointed rock slope is drafted according to the actual conditions of jointed rock slope；
Step 2, generating structure face parameters of shear resistant Monte Carlo random number；
Step 3 establishes the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope reliability；
The numerical solution of step 4, the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating, obtains joint rock
The limit state function of matter side slope；
Step 5, the reliability that jointed rock slope is calculated according to the result of step 4.
The essential information of the jointed rock slope includes：The geometric parameter of jointed rock slope, the physics of rockmass
Mechanics parameter, structural plane geologic parameter, side slope external load information.
The step 2 is specially：The cohesion c of structural plane is determined respectivelykMean μc, standard deviation sigmacAnd variation factor δc,
Angle of frictionMean valueStandard deviationAnd the coefficient of variationSet cohesion c simultaneouslyk, angle of frictionProbability distribution
It is normal distribution；Use the cohesiveness and angle of friction of Monte Carlo random number generation function generating structure face kIt is random
Number
The step 3 is specially：(1) using the discrete jointed rock slope of Rigid Body Element, jointed rock slope is separated into
The geometrical system of Rigid Body Element+structural plane；(2) object function is established：The stochastic variable of unit weight over-loading coefficient is set as target letter
Number；(3) the Lower Bound Limit constraints of Rigid Body Element is established；(4) the limiting condition letter of jointed rock slope reliability calculating is established
Number；(5) according to the constraints of object function, Lower Bound Limit, and the Monte Carlo random number of the parameters of shear resistant in integrated structure face with
And the limit state function of jointed rock slope reliability calculating, the Lower Bound Limit for obtaining solving jointed rock slope reliability are random
Plan model；Wherein the equilibrium condition of the Lower Bound Limit constraints of Rigid Body Element including Rigid Body Element, the yield condition of structural plane,
The static(al) boundary condition of Rigid Body Element.
The step 4 is specially：By cohesiveness random numberAngle of friction random numberLower limit is gradually substituted into from 1 to N
Method Stochastic Programming Model；It is determining to eachStochastic Programming Model becomes the line that a coefficient matrix is definite value
Property planning problem, corresponding object function λ (t) is solved using simplex method；Obtained N group λ (t) will be solved, (t=1 ..., N)
Limit state function is substituted into, obtains the limit state function Z=λ (t) of jointed rock slope, (t=1 ..., N).
The mean value of the reliability of the jointed rock slope including jointed rock slope unit weight over-loading coefficient, standard deviation, can
By degree index and the failure probability of jointed rock slope.
The Lower Bound Limit Stochastic Programming Model is：
In formula, t=1 ..., N, N are the quantity of the random number of generation；
The limit state function Z=λ (t) of jointed rock slope reliability calculating, object function λ (t) represent unit weight overload
The stochastic variable of coefficient；
Represent Rigid Body Element
Equilibrium condition, miIt is the quantity of structural plane in Rigid Body Element i, αkIt is the n of structural plane kkAxis and the angle of x-axis and counterclockwise for just,
NkIt is structural plane k along exterior normal nkThe normal force and drawing positive pressure in direction are born, VkIt is structural plane k tangentially skThe shearing in direction and make
Block generation is rotated counterclockwise as just, and λ (t) is t-th of over-loading coefficient random quantity, and γ is the unit weight of Rigid Body Element material, AiIt is
The area of Rigid Body Element i, fxiIt is equivalent external force at the centre of form of i-th of Rigid Body Element in the x-direction, fyiIt is i-th of Rigid Body Element
The centre of form at equivalent external force in the y-direction, nbIt is the quantity of Rigid Body Element in jointed rock slope；
Represent the yield condition of structural plane,It is t-th of random number of structural plane k cohesiveness,It is t-th of random number of structural plane k angle of frictions, lkIt is structural plane k
Length, nsIt is the quantity of structural plane in entire jointed rock slope；
Represent the static(al) boundary condition of Rigid Body Element,It is the structural plane of known boundaries condition
Interior force vector, b represent jointed rock slope in known boundary external force be equal to 0 structural plane, ncIt is in entire rock side slope
The quantity of structural plane of the external force equal to 0 on boundary；
Represent the generating function of cohesiveness and angle of friction Monte Carlo random number；It is structural plane cohesiveness and the mean value of angle of friction respectively,It is structural plane cohesiveness, the standard deviation of angle of friction respectively.
The beneficial effects of the invention are as follows：
1st, the present invention provides a kind of new method for the Reliability Analysis of Ultimate Bearing Capacity of jointed rock slope, for the first time by lower limit
The discrete thought of law theory, Rigid Body Element, stochastic programming method and Monte Carlo method combine analysis jointed rock slope can
By degree, reliability (mean value, standard deviation, RELIABILITY INDEX and the joint rock matter of unit weight over-loading coefficient of rock side slope can be solved
The failure probability of side slope).
2nd, the method for the present invention can obtain corresponding with structural plane cohesiveness, the Monte Carlo random number (normal distribution) of angle of friction
Unit weight over-loading coefficient normal distribution situation, can the failure probability of side slope be calculated by unit weight over-loading coefficient ＜ 1；
3rd, Dynamic Programming Problems are solved using monte carlo method in the method for the present invention, it is fast with precision height, calculating speed
The features such as；
4th, the method for the present invention solves linear programming problem using simplex method, and factorization is simple.
Description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 is jointed rock slope Rigid Body Element stress diagram；
Structural plane schematic diagrames of the Fig. 3 between jointed rock slope adjacent block unit；
Fig. 4 is the geometry schematic diagram of 1 jointed rock slope of embodiment；
Fig. 5 is the random number of 1 structural plane AC cohesiveness of embodimentNormal distribution schematic diagram；
Fig. 6 is the random number of 1 structural plane AC angle of frictions of embodimentNormal distribution schematic diagram；
Fig. 7 is the 1 discrete schematic diagram of jointed rock slope Rigid Body Element of embodiment；
Fig. 8 is 1 jointed rock slope unit weight over-loading coefficient normal distribution schematic diagram of embodiment；
Fig. 9 is the geometry schematic diagram of 2 jointed rock slope of embodiment；
Figure 10 is the random number of 2 structural plane cohesiveness of embodimentNormal distribution schematic diagram；
Figure 11 is the random number of 2 structural plane angle of friction of embodimentNormal distribution schematic diagram；
Figure 12 is the 2 discrete schematic diagram of jointed rock slope Rigid Body Element of embodiment；
Figure 13 is 2 jointed rock slope unit weight over-loading coefficient normal distribution schematic diagram of embodiment.
Specific embodiment
Embodiment 1：As shown in figures 1-13, a kind of jointed rock slope reliability analysis Lower Bound Limit, specific method step is such as
Under：
(1), the essential information of jointed rock slope is drafted
Embodiment 1 is the rock side slope containing one group of weak structural face, and structure is as shown in figure 4, its main calculating parameter
It is as follows：The geometric parameter of jointed rock slope：Height 55.94m, top width 36.6m, 45 degree of free face angle；The appearance of rockmass
It is reset to definite value and takes 25kN/m3；The inclination angle of structural plane (joint plane) AC is 30 degree；Side slope is only acted on by rock mass gravity load.
(2), the Monte Carlo random number of the parameters of shear resistant in generating structure face
The mean value of the cohesiveness of structural plane AC takes μc=100kPa, variation factor δcTake 0.10 respectively, 0.15,0.20,
0.25, the mean value of angle of friction takesThe coefficient of variationTake 0.10,0.15,0.20,0.25 respectively, cohesiveness and angle of friction
Probability distribution be normal distribution.The random number of structural plane AC parameters of shear resistant is generated using formula (1).In the present embodiment
N=3000 is taken, i.e., generates 3000 random numbers of cohesiveness and angle of friction respectivelyBecoming
Different coefficient δcThe random number of structural plane AC cohesiveness in the case of=0.2Normal distribution as shown in figure 5, angle of friction become
Different coefficientIn the case of structural plane AC angle of frictions random numberNormal distribution it is as shown in Figure 6.
(3), the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope reliability is established
Use Rigid Body Element discrete 1 side slope of embodiment as shown in fig. 7, covariance is 2 Rigid Body Element.According to formula (9)
Constitution and implementation example 1 solves the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability.
(4), the numerical solution of the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating
Using 3000 groups of unit weight overload of jointed rock slope Lower Confidence Limit method numerical computations workflow management as shown in Figure 1
It is λ (t), (t=1 ..., 3000).In the mean μ of structural plane cohesivenessc=100kPa, variation factor δc=0.20, angle of friction
Mean value takesThe coefficient of variationUnder conditions of, the embodiment 1 side slope unit weight over-loading coefficient λ (t), (t that are calculated
=1 ..., 3000) normal distribution situation it is as shown in Figure 8.
(5), the reliability of jointed rock slope is calculated.
The reliability that 1 jointed rock slope of embodiment is obtained by formula (10), (11), (12), (14) solution is as shown in table 1.
The reliability statistical form of 1 embodiment of table, 1 jointed rock slope
In the case where all parameters are all definite value, the unit weight over-loading coefficient for solving this side slope is a statically problem, right
The limiting value of definite value λ answered can be obtained by analytic method, in the cohesion c of structural plane ACk=100000kPa, angle of frictionIn the case of, the obtained parsing of unit weight over-loading coefficient is 1.259, the various differences being calculated by the method for the present invention
The equal very little of the error of mean value and analytic solutions (within 1.1%) under the coefficient of variation, and the error of the two is with the increasing of the coefficient of variation
Add and increase.
The reliability of side slope refers to β marks and reduces with the increase of the coefficient of variation, illustrates that shear strength parameter dispersion degree is got over
It is high then the reliability of side slope is lower.
The failure probability P of side slopefIncrease with the increase of the coefficient of variation, illustrate that shear strength parameter dispersion degree is higher
Then the failure probability of side slope is higher.
Embodiment 2：A kind of jointed rock slope reliability analysis Lower Bound Limit, the method are as follows：
(1), the essential information of jointed rock slope is drafted
Embodiment 2 is the rock side slope for being grouped joint containing two, and geological information is as shown in figure 9, it mainly calculates ginseng
Number is as follows：The geometric parameter of jointed rock slope：Height 30m, top width 45m, 80 degree of free face angle；The unit weight of rockmass is set
27kN/m is taken for definite value3, the spacing of two groups of structural planes (joint plane) is 6m, the inclination angle of two groups of structural planes is 25 degree and 80 respectively
Degree；Side slope is only acted on by rock mass gravity load.
(2), the Monte Carlo random number of the parameters of shear resistant in generating structure face
Embodiment 2：The mean value of the cohesiveness of all structural planes takes μc=50kPa, variation factor δcTake 0.10 respectively, 0.15,
0.20th, 0.25, the mean value of the angle of friction of all structural planes takesThe coefficient of variationTake 0.10 respectively, 0.15,0.20,
0.25.The random number of the parameters of shear resistant of all structural planes of 2 side slope of embodiment is generated using formula (1).N=is taken in the present embodiment
3000, i.e., joint cohesiveness and 3000 random numbers of angle of friction are generated respectively
In variation factor δcThe random number of all structural plane cohesiveness in the case of=0.15Normal distribution such as Figure 10
It is shown, in the coefficient of variationIn the case of all structural plane angle of frictions random numberNormal distribution such as Figure 11 institutes
Show.
(3), the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability is solved
2 side slope of embodiment is discrete as shown in figure 12 using Rigid Body Element, and covariance is 37 Rigid Body Elements.According to formula
(9) the Lower Bound Limit Stochastic Programming Model of 2 jointed rock slope reliability of constitution and implementation example.
(4), the numerical solution of the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating
Using the 3000 of jointed rock slope Lower Confidence Limit method numerical computations workflow management embodiment 2 as shown in Figure 1
Group unit weight overload is λ (t), (t=1 ..., 3000).
In the mean μ of structural plane cohesivenessc=50kPa, variation factor δc=0.15, the mean value of angle of frictionBecome
Different coefficientUnder conditions of, the 2 side slope unit weight over-loading coefficient λ (t) of embodiment that is calculated, (t=1's ..., 3000)
Normal distribution situation is as shown in figure 13.
(5), the reliability of jointed rock slope is calculated.
The reliability that 2 jointed rock slope of embodiment is obtained by formula (10), (11), (12), (14) solution is as shown in table 2.
The reliability statistical form of 2 embodiment of table, 2 jointed rock slope
The mean value and standard deviation of the unit weight over-loading coefficient of jointed rock slope all increase with the increase of the coefficient of variation, reliably
Degree index β reduces with the increase of the coefficient of variation.
The failure probability P of jointed rock slopefSubtract increase with the increase of the coefficient of variation, when the coefficient of variation takes 0.1
Failure probability is 17.867%, and failure probability increases to 35.7% when the coefficient of variation takes 0.25.
The present invention operation principle be：
Using jointed rock slope as research object, by monte carlo method, lower limit law theory, the discrete thought of Rigid Body Element and
Stochastic programming method combines, and solves the reliability of jointed rock slope；First, using monte carlo method generating structure face
The random number of parameters of shear resistant；Then, jointed rock slope is separated into the solid of Rigid Body Element+structural plane；According to Lower Bound Limit
The Theory Construction jointed rock slope meets the quiet of Rigid Body Element equilibrium condition, structural plane yield condition and static(al) boundary condition simultaneously
Power permissible stress field, and using unit weight over-loading coefficient as object function；Establish the Lower Bound Limit for solving jointed rock slope reliability
Stochastic Programming Model, while propose the method for solving of Stochastic Programming Model；By calculate can obtain jointed rock slope can
By spending (failure probability of mean value, standard deviation, RELIABILITY INDEX and side slope including unit weight over-loading coefficient).
The Technology Roadmap of the present invention is as shown in Figure 1.
The technical solution of the Lower Bound Limit of the solution jointed rock slope reliability of the present invention carries out according to the following steps successively：
First, the essential information of jointed rock slope is drafted
According to the actual conditions of jointed rock slope, the main calculating parameter for carrying out reliability analysis is drafted, including：Joint
The geometric parameter of rock side slope, the physical and mechanical parameter (unit weight etc.) of rockmass, structural plane geologic parameter, side slope external load letter
Breath etc..
2nd, the Monte Carlo random number of the parameters of shear resistant in generating structure face
The bearing capacity of jointed rock slope is controlled primarily by the intensity of structural plane, while two shearing strengths of structural plane
Parameter (cohesion ck, angle of frictionWith larger uncertainty and variability, therefore the present invention sets the cohesiveness of structural plane
ck, angle of frictionFor mutually independent random quantity, and meet normal distribution；It is fixed to set the other parameters such as the unit weight of rock mass simultaneously
Value.The present invention is using monte carlo method generation cohesiveness and angle of friction random quantityRandom number, detailed process is as follows：
1. the cohesion c of structural plane is determined respectivelykMean μc, standard deviation sigmacAnd variation factor δc, angle of frictionMean valueStandard deviationAnd the coefficient of variationSet cohesion c simultaneouslyk, angle of frictionProbability distribution be normal distribution；
2. use the cohesiveness and angle of friction of Monte Carlo random number generation function generating structure face kIt is random
Number, cohesiveness and angle of friction Monte Carlo random number generation function are as follows：
In above formula:T=(1 ..., N), N are the quantity of the random number of generation, generally take 2000~5000；It is structure
T-th of random number of face cohesiveness (subscript r represents that variable is random number variable)；Be t-th of structural plane angle of friction with
Machine number (subscript r represents that variable is random number variable)；It is structural plane cohesiveness and the mean value of angle of friction respectively,Point
It is not structural plane cohesiveness, the standard deviation of angle of friction.
3rd, the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope reliability is established
Plastic limit analysis Lower Bound Limit can be stated as：In load corresponding with all static(al) permissible stress fields, limit lotus
It carries maximum.The stress field for meeting equilibrium condition, yield condition and static(al) boundary condition simultaneously is known as static(al) permissible stress field.
In one jointed rock slope structure, countless static(al) fields for meeting above-mentioned condition can be constructed；Seek its correspondence one by one
Load, maximum load is necessarily closest to true ultimate load.Lower limit solution is certainly less than true solution.
In order to build the static(al) permissible stress field of jointed rock slope, the present invention uses the discrete joint Yan Zhi sides of Rigid Body Element
The over-loading coefficient of side slope external load is set as object function by slope, and structure Rigid Body Element meets equilibrium equation, yield condition simultaneously
With the static(al) permissible stress field of boundary condition, the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope ultimate load is established.
It is as follows：
1. use the discrete jointed rock slope of Rigid Body Element
The present invention is using the discrete jointed rock slope of Rigid Body Element, and Rigid Body Element stress is as shown in Fig. 2, adjacent block unit
Structural plane k stress it is as shown in Figure 3.Wherein (x, y) is global coordinate；(nk,sk) it is structure between adjacent block unit i, j
The local coordinate system of face k, n as shown in Figure 3kFor the exterior normal direction of structural plane k, skFor the tangential of structural plane；Rigid Body Element i shapes
Heart CiUpper effect has equivalent load force vectorfxiIt is in the x-direction equivalent at the centre of form of i-th of Rigid Body Element
External force, fyiIt is equivalent external force at the centre of form of i-th of Rigid Body Element in the y-direction.；Structural plane k between adjacent block unit i, j
Centre of form PkPlace's effect has interior force vectorThe effect of structural plane opposite side has its reaction force vector
NkIt is along structural plane k along exterior normal nk(sign provides the normal force in direction：Positive pressure is drawn to bear), VkIt is along structural plane k tangentially sk
(sign provides the shearing in direction：Block generation is made to rotate counterclockwise as just), N'kIt is NkReaction force, Vk' it is VkIt is anti-
Active force.
After the discrete jointed rock slope of Rigid Body Element, Jointed element becomes the geometry system of Rigid Body Element+structural plane
System, in order to simplify the constraints calculated and convenient for building the static(al) permissible stress field of Lower Bound Limit, the present invention makes the following assumptions：
(1) assume that Rigid Body Element is only translatable, will not rotate；(2) assume that Rigid Body Element is rigid body, Rigid Body Element will not occur
Any deformation and failure, the destruction of side slope are only occurred on the structural plane between adjacent block unit；(3) assume structure plane materiel
Expect for rigid-perfectly plastic material, (4) assume that failure by shear only occurs for the structural plane in rock mass.
2. establish object function
The present invention makes jointed rock slope reach capacity state by the way of overloading using unit weight, if the practical appearance of side slope sillar
Weight is γ, i.e., side slope is in the limiting condition destroyed when unit weight γ is incrementally increased to γ '.The present invention defines unit weight over-loading coefficient
It is as follows：
In above formula：λ is the unit weight over-loading coefficient of jointed rock slope；Rigid Body Element when γ ' is jointed rock slope overload
The limit unit weight of material；γ is the currently practical unit weight of the Rigid Body Element material of jointed rock slope.
According to lower limit law theory, jointed rock slope is when reaching capacity state, the maximum value γ ' of demand unit weight γ；Root
According to the cohesiveness stochastic variable and angle of friction stochastic variable of formula (1) structural plane respectively by discrete for N number of random number, then unit weight overload
Coefficient lambda also it is corresponding become withWithRelevant stochastic variable is as follows：
In above formula：T=(1 ..., N), λ (t) are the stochastic variables of jointed rock slope unit weight over-loading coefficient.
According to lower limit law theory, it is as follows that the stochastic variable λ (t) of unit weight over-loading coefficient is set as object function by the present invention：
Maximize:λ(t)(4)
3. establish the Lower Bound Limit constraints of Rigid Body Element
Structure static(al) permissible stress field needs Rigid Body Element to meet equilibrium equation, yield condition and static(al) perimeter strip simultaneously
Part.
(1) equilibrium condition of Rigid Body Element
Conducted oneself with dignity for any one Rigid Body Element i shown in Fig. 2, the effect of external load, adjacent sillar and keep balancing,
The equilibrium equation of its power is as follows：
In above formula：nbIt is the quantity of Rigid Body Element in jointed rock slope；miIt is the quantity of structural plane in Rigid Body Element i；λ
(t) it is t-th of over-loading coefficient random quantity, AiIt is the area of Rigid Body Element i, γ is the unit weight of Rigid Body Element material, αkIt is structural plane
The n of kk(sign provides for axis and the angle of x-axis：Counterclockwise for just), NkIt is along structural plane k along exterior normal nkThe normal force in direction
(sign provides：Positive pressure is drawn to bear), VkIt is along structural plane k tangentially sk(sign provides the shearing in direction：Generate block inverse
Hour hands rotation is just), fxiIt is equivalent external force at the centre of form of i-th of Rigid Body Element in the x-direction, fyiIt is i-th of Rigid Body Element
Equivalent external force at the centre of form in the y-direction.
(2) yield condition of structural plane
For any one structural plane k as shown in Figure 3, since the cohesiveness and angle of friction of structural plane are N groups are random
NumberTherefore the Mohr-Coulomb yield conditions that the shearing slip of structural plane k destroys can use cohesiveness and angle of friction
Random number be written as form：
In above formula：nsIt is the quantity of structural plane in entire jointed rock slope；lkIt is the length of structural plane k；It is structure
T-th of random number of the cohesiveness of face k (subscript r represents that variable is random number variable)；It is the t of structural plane k angle of frictions
A random number (subscript r represents that variable is random number variable)；NkIt is the normal force (sign convention of structural plane k：Positive pressure is drawn to bear), Vk
It is the shearing of structural plane k；N is the quantity of cohesiveness random number and angle of friction random number.
(3) the static(al) boundary condition of Rigid Body Element
According to lower limit law theory, static(al) permissible stress field must satisfy known static(al) boundary condition.Consider joint rock matter
Known boundary external force is equal to 0 structural plane b in side slope, and boundary condition expression formula is：
In above formula：ncIt is the quantity of structural plane of the external force equal to 0 on boundary in entire rock side slope；It is known boundaries item
The interior force vector of the structural plane of part.
4. establish the limit state function of jointed rock slope reliability calculating
In order to solve the mathematical distribution rule of jointed rock slope unit weight over-loading coefficient, according to formula (3), the present invention sets joint
The limit state function of rock side slope reliability calculating is：
In above formula：Z is the limit state function of jointed rock slope.
5. establish the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating
According to the constraint equation (5) of the target function type (4) of static(al) permissible stress field and Lower Bound Limit, (6), (7), and
The cohesiveness random number and angle of friction random number generation function formula (1) and jointed rock slope reliability calculating in integrated structure face
Limit state function formula (8), can arrive solve jointed rock slope reliability Lower Bound Limit Stochastic Programming Model it is as follows：
In above formula：(t=1 ..., N).
4th, the numerical solution of the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating
Formula (9) solves target：By the cohesiveness random number of structural planeWith angle of friction random numberMathematics it is special
Solicit the mathematical feature of solution unit weight over-loading coefficient stochastic variable λ (t).It is included in the coefficient of the matrix of the Lower Bound Limit model of formula (9)
The cohesiveness of structural plane and the random number of angle of frictionIt is a typical Dynamic Programming Problems, therefore is attempted
Directly pass throughMathematical feature accurately solve λ (t) mathematical feature be it is extremely difficult, can only use approximation
Numerical computation method.
The present invention uses the Lower Bound Limit Stochastic Programming Model of following Numerical Methods Solve jointed rock slope reliability：
(1) t is recycled from t=1 to t=N, gradually by cohesiveness random numberAngle of friction random numberSubstitution formula
(9)；
(2) it is determining for eachFormula (9) becomes the linear programming that a coefficient matrix is definite value and asks
Topic solves corresponding object function λ (t) using simplex method at this time；
(3) obtained N group λ (t) will be solved, (t=1 ..., N) substitutes into limit state function, joint rock matter can be calculated
The limit state function Z=λ (t) of side slope, (t=1 ..., N).
The numerical solution flow of the present invention is as shown in Figure 1.
5th, the reliability of jointed rock slope is calculated.
The basic statistics amount that limit state function Z is calculated according to the N group λ (t) being calculated can obtain jointed rock slope
Reliability, including：Mean value, standard deviation, RELIABILITY INDEX and the jointed rock slope of jointed rock slope unit weight over-loading coefficient
Failure probability.
The mean value computation formula of jointed rock slope unit weight over-loading coefficient is as follows：
In above formula：μλIt is the mean value of jointed rock slope unit weight over-loading coefficient.
The standard deviation calculation formula of jointed rock slope unit weight over-loading coefficient is as follows：
In above formula：σλIt is the standard deviation of jointed rock slope unit weight over-loading coefficient.
Jointed rock slope RELIABILITY INDEX calculation formula is as follows：
In above formula：μλIt is mean value, the σ of jointed rock slope unit weight over-loading coefficientλIt is jointed rock slope unit weight over-loading coefficient
Standard deviation, β be jointed rock slope RELIABILITY INDEX.
When the unit weight over-loading coefficient >=1 of side slope side slope be safe, as the unit weight over-loading coefficient ＜ 1 of side slope when side slope send out
Raw unstability, therefore the invalidation functions function of jointed rock slope is as follows：
In above formula：I (t) is the invalidation functions function of jointed rock slope.
Then the failure probability P of side slope can be calculated as follows：
In above formula：PfIt is the failure probability of jointed rock slope.
It is characteristic of the invention that：The present invention is by monte carlo method, plastic limit analysis Lower Bound Limit, the discrete think of of Rigid Body Element
Think and stochastic programming combines, it is proposed that a kind of Lower Bound Limit of jointed rock slope reliability calculating can be calculated
Mean value, standard deviation, the RELIABILITY INDEX of side slope unit weight over-loading coefficient, while the failure probability of jointed rock slope can be obtained.
Relatively traditional limiting equilibrium reliability analysis method its with higher computational accuracy；Compared to Finite Element Reliability Analysis method, by
The constitutive relation of material is had ignored in the method for the present invention, therefore calculating speed and more efficient.
The specific embodiment of the present invention is explained in detail above in conjunction with attached drawing, but the present invention is not limited to above-mentioned
Embodiment, within the knowledge of a person skilled in the art, can also be before present inventive concept not be departed from
It puts and makes a variety of changes.
Claims (7)
1. a kind of jointed rock slope reliability analysis Lower Bound Limit, it is characterised in that：The method is as follows：
Step 1, the essential information that jointed rock slope is drafted according to the actual conditions of jointed rock slope；
Step 2, generating structure face parameters of shear resistant Monte Carlo random number；
Step 3 establishes the Lower Bound Limit Stochastic Programming Model for solving jointed rock slope reliability；
The numerical solution of step 4, the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability calculating obtains joint Yan Zhi sides
The limit state function on slope；
Step 5, the reliability that jointed rock slope is calculated according to the result of step 4.
2. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The joint rock matter
The essential information of side slope includes：The geometric parameter of jointed rock slope, the physical and mechanical parameter of rockmass, structural plane geology ginseng
Number, side slope external load information.
3. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The step 2 has
Body is：The cohesion c of structural plane is determined respectivelykMean μc, standard deviation sigmacAnd variation factor δc, angle of frictionMean value
Standard deviationAnd the coefficient of variationSet cohesion c simultaneouslyk, angle of frictionProbability distribution be normal distribution；It is special using covering
The cohesiveness and angle of friction of Carlow random number generation function generating structure face kRandom number
4. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The step 3 has
Body is：(1) using the discrete jointed rock slope of Rigid Body Element, jointed rock slope is separated into the several of Rigid Body Element+structural plane
What system；(2) object function is established：The stochastic variable of unit weight over-loading coefficient is set as object function；(3) Rigid Body Element is established
Lower Bound Limit constraints；(4) limit state function of jointed rock slope reliability calculating is established；(5) according to object function, under
The constraints of limit method, and the Monte Carlo random number of the parameters of shear resistant in integrated structure face and jointed rock slope reliability meter
The limit state function of calculation obtains solving the Lower Bound Limit Stochastic Programming Model of jointed rock slope reliability；Wherein Rigid Body Element
Lower Bound Limit constraints include equilibrium condition, the yield condition of structural plane, the static(al) perimeter strip of Rigid Body Element of Rigid Body Element
Part.
5. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The step 4 has
Body is：By cohesiveness random numberAngle of friction random numberLower Bound Limit Stochastic Programming Model is gradually substituted into from 1 to N；It is right
Each is determiningStochastic Programming Model becomes the linear programming problem that a coefficient matrix is definite value, using list
Pure shape method solves corresponding object function λ (t)；Obtained N group λ (t) will be solved, (t=1 ..., N) substitutes into limit state function,
The limit state function Z=λ (t) of jointed rock slope are obtained, (t=1 ..., N).
6. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The joint rock matter
The reliability of side slope includes mean value, standard deviation, RELIABILITY INDEX and the joint Yan Zhi sides of jointed rock slope unit weight over-loading coefficient
The failure probability on slope.
7. jointed rock slope reliability analysis Lower Bound Limit according to claim 1, it is characterised in that：The Lower Bound Limit with
Machine plan model is：
In formula, t=1 ..., N, N are the quantity of the random number of generation；
The limit state function Z=λ (t) of jointed rock slope reliability calculating, object function λ (t) represent unit weight over-loading coefficient
Stochastic variable；
Represent the balance strip of Rigid Body Element
Part, miIt is the quantity of structural plane in Rigid Body Element i, αkIt is the n of structural plane kkAxis and the angle of x-axis and counterclockwise for just, NkIt is knot
Structure face k is along exterior normal nkThe normal force and drawing positive pressure in direction are born, VkIt is structural plane k tangentially skThe shearing in direction and produce block
Raw to rotate counterclockwise as just, λ (t) is t-th of over-loading coefficient random quantity, and γ is the unit weight of Rigid Body Element material, AiIt is bulk single
The area of first i, fxiIt is equivalent external force at the centre of form of i-th of Rigid Body Element in the x-direction, fyiIt is the centre of form of i-th of Rigid Body Element
The equivalent external force of place in the y-direction, nbIt is the quantity of Rigid Body Element in jointed rock slope；
Represent the yield condition of structural plane,It is
T-th of random number of structural plane k cohesiveness,It is t-th of random number of structural plane k angle of frictions, lkIt is the length of structural plane k
Degree, nsIt is the quantity of structural plane in entire jointed rock slope；
Represent the static(al) boundary condition of Rigid Body Element,It is the internal force of the structural plane of known boundaries condition
Vector, b represent structural plane of the known boundary external force equal to 0, n in jointed rock slopecIt is in entire rock side slope on boundary
The quantity of structural plane of the external force equal to 0；
Represent the generating function of cohesiveness and angle of friction Monte Carlo random number；μc,
It is structural plane cohesiveness and the mean value of angle of friction respectively, σc,It is structural plane cohesiveness, the standard deviation of angle of friction respectively.
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