CN106557608B - A kind of plasticity limit analysis upper bound method of the jointed rock mass discrete based on Mix Amount - Google Patents
A kind of plasticity limit analysis upper bound method of the jointed rock mass discrete based on Mix Amount Download PDFInfo
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Abstract
The present invention relates to a kind of plastic limit Upper bound analysis methods of non across jointed rock mass, belong to Rock Bearing Capacity analysis field in rock mechanics.The present invention is based on the upper limit law theories in plastic limit analysis, using the discrete non-through jointed rock slope of mixed cell, use rigid block movement mesh discretization sillar, using the discrete rock bridge of finite elements, and using Rigid Body Element centroid speed and triangular element node speed as unknown quantity, it constructs while meeting block and structural plane deformation compatibility condition, Plastic Flow constraint condition, the equal condition of interior external power, the motor-driven license velocity field of Rigid Body Element and triangular element interface plastic flowing conditions and velocity boundary conditions, establish the linear math plan model for solving non across jointed rock mass ultimate load, and linear math plan model is solved using interior-point algohnhm, obtain the Upper Bound Solution of non across jointed rock mass ultimate load.The present invention has the characteristics that definite conception, computational accuracy are high.
Description
Technical field
The present invention is a kind of plastic limit Upper bound analysis method of non across jointed rock mass, in particular to a kind of based on mixing
The upper-bound theorem of numerical discretization belongs to Rock Bearing Capacity analysis technical field in rock mechanics.
Background technique
Rock mass is the complex geologic body being made of sillar and structural plane (joint, crack, tomography etc.), and wherein joint plane is deposited
There is larger impact in the bearing capacity to rock mass.In rock mass some joints be perforation, some be non-through, non-through joint
Rock mass has following two important feature: first is that rock mass is cut into complicated rock mass by the joint of perforation, rock mass
Destroy the destruction of structural plane between mainly sillar；On the other hand non-through joint part has the mechanics effect of rock bridge, rock
Bridge destroys mainly shearing and tensile failure.The bearing capacity of non across jointed rock mass is controlled primarily by three factors: joint
The mechanics effect of geometry distribution, the intensity at joint and rock bridge.Therefore, the bearing capacity that accurately solve non across jointed rock mass is
A complex problem in rock mechanics.
In the latest 20 years, numerous scholars have carried out fruitful research work to the Carrying Capacity of non across jointed rock mass
Make.For example the rock bridge of the direct shear test of non-through jointed rock mass, non-through joint weakens mechanical model, non-through joint rock
Criterion of strength and deformation rule of body etc..In addition for the bearing capacity aspect of non-through jointed rock mass, many scholars are proposed
Balance method of rigid-body limit, and the numerical analysis method based on continuous media or Discontinuous transmission, as finite element, manifold method method,
Distinct element method, DDA, block element method etc..Nevertheless, the bearing capacity of non across jointed rock mass is studied there is also some shortcomings,
Such as:
(1) as limit equilibrium method need in advance artificially assume a slip-crack surface, then according to hypothesis it is multiple not
Same slip-crack surface carries out multiple tentative calculation and seeks least favorable sliding surface, and rock mass a fairly large number of for joint, calculation amount is larger, and not
It is easy to get least favorable sliding surface；
(2) Finite Element, discrete element method etc. are in the complicated Joint network simulation of simulation, the constitutive relation of jointed rock mass complexity
And there are certain difficulties in solution safety coefficient.
Plastic limit analysis upper bound method is to solve for a kind of high efficiency method of Rock And Soil ultimate bearing capacity, extensive
Analysis of Bearing Capacity applied to the geotechnical structures object such as ground, side slope.But currently both at home and abroad in the research achievement in upper bound method field
The middle research contents overwhelming majority is all using soil-slope, ground or simple rock side slope as research object, and with non-through section
It is considerably less to manage the research achievement that rock mass is research object, main reason is that: Yao Yunyong plasticity limit analysis upper bound method principle is established non-
There are certain difficulties for the motor-driven license velocity field of perforation jointed rock mass.
The present invention is based on the research work of project of national nature science fund project (51564026), by the plastic limit analysis upper limit
Law theory, mixed cell discrete technology and Mathematical Planning means combine, and propose a kind of non-through jointed rock mass bearing capacity
The upper bound method of calculating.
Summary of the invention
The object of the present invention is to provide a kind of plasticity limit analysis upper bound methods of jointed rock mass discrete based on Mix Amount, to obtain
Non across jointed rock mass ultimate bearing capacity is obtained, provides a kind of new method for the design of jointed rock mass, stable calculation.
Basic principle of the invention is: as shown in Figure 1, plastic limit analysis upper limit law theory is based on, with non-through joint
Rock mass research object combines plastic limit analysis Lower Bound Limit, mixed cell discrete method, mathematic programming methods, uses
The discrete non across jointed rock mass of hybrid-element method is unknown with the node speed of Rigid Body Element centroid speed, rock bridge finite elements
Amount, construct while meeting block condition equal with structural plane deformation compatibility condition, Plastic Flow constraint condition, interior external power,
The motor-driven license velocity field of Rigid Body Element and triangular element interface compatibility of deformation item and velocity boundary conditions, and in joint
The ultimate load of rock mass is objective function, thus establishes the linear math rule for solving the ultimate load of non-through jointed rock slope
Model is drawn, and linear math plan model is solved using interior-point algohnhm, obtains non across jointed rock mass ultimate load
Upper Bound Solution.
The technical solution of the non across jointed rock mass upper bound method discrete based on Mix Amount of the invention successively presses following step
It is rapid to carry out:
One, the calculating parameter of jointed rock mass is drafted
According to the actual conditions of non across jointed rock mass, the calculating ginseng that the analysis of plastic limit analysis upper bound method needs is drafted
Number, specifically include that geological condition parameter, the geometric parameter of rock mass, rock mass materials, the parameter of jointed material (bulk density, cohesiveness,
Angle of friction), rock mass parameters of loading information.
Two, using the discrete non-through jointed rock mass of mixed cell method
Rock mass is cut into discrete rock mass by the joint of perforation, and the rock mass in the non-through region in joint has rock bridge
Effect, the present invention can simulate the cutting or drawing crack of rock bridge to can either simulate rock mass stress destruction, using mixing
The discrete non across jointed rock mass of elements method, as shown in Fig. 2, (a) simulates sillar using Rigid Body Element, with Rigid Body Element centroid speed
For unknown quantity；(b) finite element triangulated linear unit simulation rock bridge is used, is unknown with the node speed of rock bridge triangular unit
Amount；(c) interface of Rigid Body Element and triangle finite elements must satisfy deformation compatibility condition.
1, Rigid Body Element is discrete
For sillar made of the joint cutting that is penetrated through in rock mass, the present invention using rigid block movement unit come discrete,
It is made of Rigid Body Element and structural plane, as shown in figure 3, the variable defined on Rigid Body Element, structural plane is as shown in table 1.It is wherein total
Body coordinate system is (x, y), and the local coordinate system on structural plane k Rigid Body Element i adjacent with Rigid Body Element j is defined as (Sk,nk)。
The velocity discontinuity vector acted in structural plane k centroid isThe force vector acted in Rigid Body Element i centroid isThe velocity vector acted in Rigid Body Element i centroid isSee Table 1 for details for each variable declaration
It is shown.
2, finite element triangular element is discrete
For the rock bridge region in rock mass, the present invention is using finite element Linear Triangular shape unit come discrete, discrete signal
Figure is as shown in Figure 4, Figure 5.Triangular element uses three node linear units, i.e., the speed of hypothesis triangular element is inside unit
Linear to be distributed, the velocity component of any point is represented by the linear function of three node speed components in unit；Triangle simultaneously
Shape unit is using not conode unit mode, therefore each node is pertaining only to some discrete cell, i.e. different units node can
Coordinate having the same；In addition permissible velocity is interrupted between adjacent triangle.The speed of three nodes of triangular element i is distinguished
It is expressed asSee Table 1 for details for speed variables explanation.
It is calculated to simplify, the present invention makes the following assumptions: (1) using the mechanical characteristic of rigid block movement unit simulation sillar,
Assuming that sillar is rigid body, any deformation and failure will not occur, destruction only occurs on the structural plane between adjacent block；
(2) only consider the translation effect of block, and in deformation process, will not be mutually disengaged between sillar；(3) assume rock bridge finite element
The material of triangular element is rigid-perfectly plastic material, does not generate any deformation before stress reaches yield stress, once it reaches
To after yield stress, strain will increase without limitation.
The variable acted on 1 Rigid Body Element i of table and structural plane k and finite element triangular element
Three, the upper bound method model for solving non across jointed rock mass ultimate bearing capacity is established
1, objective function
The purpose of the present invention is solving the ultimate bearing capacity of non across jointed rock mass, that is, solve ultimate load.Ultimate load
It is just to solve for the load that time of unstable failure occurs for rock mass, the present invention solves limit lotus by the way of solving over-loading coefficient
It carries.Over-loading coefficient K is defined herein are as follows:
Qc=KQa (1)
Wherein, QcFor rock mass ultimate load, QaFor the external load of the currently practical application of rock mass structure.
2, rock mass Rigid Body Element upper bound method constraint condition
After rock mass using the discrete non across jointed rock mass of Rigid Body Element, it is several to obtain Rigid Body Element+structural plane
He Ti, in order to construct motor-driven license velocity field, all Rigid Body Elements and structural plane must satisfy following three item constraints condition:
Present invention assumes that Plastic Flow occurs over just on the structural plane of adjacent block unit, i.e. hypothesis speed is discontinuously located at
In the common edge of two adjacent block units, as shown in figure 3, and assuming that structural plane with a thickness of zero, needs to meet between adjacent block
The condition of motor-driven license has to comply with associated flow standard along normal direction and tangential velocity interruption value on the structural plane between adjacent block
Then.It is assumed that rigid block movement and structural plane are ideal epistemology model, then according to the associated flow rule of the theory of plasticity, by deforming
Compatibility conditions obtain generalized strain rate component should be equal to by associated flow rule and yield condition obtain generalized plasticity strain
Rate component, the Plastic Flow constraint condition of available all structural planes:
(1) the Plastic Flow constraint condition of Rigid Body Element and structural plane
In above formula: It is non-negative for the plasticity multiplier of adjacent block cellular construction face k, and requires δUi,δViFor the rate of block i centroid, δ Uj,δVjFor the rate of block j centroid, αk
For nkThe angle (being positive counterclockwise) in direction and the direction x, nkFor the quantity of structural plane,For the angle of friction of structural plane.
(2) the internal strength power of Rigid Body Element condition equal with external work power
Due to present invention assumes that Rigid Body Element will not be deformed and be destroyed, the internal strength dissipation work inside Rigid Body Element
Rate is equal to 0, and internal strength dissipation only occurs on the adjacent structural plane of Rigid Body Element, owns according in associated flow rule Rigid Body Element
Internal strength power on structural planeAre as follows:
In above formula: k=(1 ..., nk), nkFor all Rigid Body Element structural plane quantity,Plasticity for structural plane k multiplies
Son, lkFor the length of structural plane k, ckFor the cohesiveness of structural plane k.
In Rigid Body Element region, the Gravitative Loads of Rigid Body Element are considered, then Rigid Body Element is self-possessed W in virtual velocity V*Upper generation
External work powerAre as follows:
In above formula: W is Rigid Body Element self weight, V*For virtual velocity
In Rigid Body Element region, consideration acts on the borderline external force Q of Rigid Body Element, then on the boundary in Rigid Body Element region
Face power load Q in virtual velocity V*The external work power of upper generationAre as follows:
In above formula: Q is the surface force load of Rigid Body Element, V*For virtual velocity.
(3) velocity boundary conditions of Rigid Body Element
By upper bound theorem it is found that motor-driven license velocity field must satisfy known velocity boundary conditions in speed edges,
The boundary condition on the b of boundary that rate in jointed rock mass is zero are as follows:
In above formula:For the coordinate conversion matrix of the Rigid Body Element j on the b of boundary,αjFor
The angle (being positive counterclockwise) in boundary exterior normal direction and the direction x,For Rigid Body Element j on boundary
Speed variables, nBFor the quantity of Rigid Body Element zone velocity boundary interface.
3, rock bridge finite element triangular element upper bound method constraint condition
It is as shown in Figure 3, Figure 4, false using the rock bridge region of finite element Linear Triangular shape mesh discretization non across jointed rock mass
The speed for determining triangular element is linearly distributed inside triangular element, in order to construct motor-driven license velocity field, all three
Corner shaped elements need to meet following constraint condition:
(1) finite element triangular element Plastic Flow constraint condition
Rock mass materials are assumed into rigid-perfectly plastic body first, and standard is surrendered to Mohr-Coulomb with circumscribed regular polygon
Surrender circle then carries out approximation, then the Plastic Flow constraint condition of finite element triangular element i can be indicated with matrix form:
Wherein, For plasticity multiplier, and requireneFor the quantity of triangular element,For the area of triangular element i, bi=yj-yk, ci=-xj+xk, bj=yk-
yi, cj=-xk+xi, bk=yi-yj, ck=-xi+xj, (xi,yi), (xj,yj), (xk,yk) saved to be upper in respectively triangular element
The position coordinates of point i, j, k. Ck=2sin (2 π k/
P), (k=1 ..., p),For the angle of friction of triangular element i material, p is the surrender circle to Mohr-Coulomb yield criterion
Carry out the number of edges of approximate circumscribed regular polygon.
(2) the discontinuous constraint condition of the speed of common edge between triangular element
The present invention allows in the common edge of adjacent triangle unit there are velocity discontinuity, and common edge is as shown in fig. 6, adjacent three
The matrix that speed on corner shaped elements common edge i discontinuously constrains is expressed as:
Wherein:
For four plasticity multipliers of triangular element common edge, and requireI=(1 ..., ng) indicate common edge serial number, ngFor the quantity of triangular element common edge,For triangular element
The inclination angle of common edge,For the cohesiveness of the material of triangular element common edge.
(3) the internal strength power of finite element triangular element condition equal with external work power
For isotropic material, the internal strength power of all triangular elementsIt can be calculated as follows:
Wherein, neFor the quantity of triangular element, i=(1 ..., ne), The respectively cohesiveness and angle of friction of triangular element i,It is three
The area of corner shaped elements i.
For the discontinuous common edge of speed adjacent between all triangular elements, internal strength powerIt can be as the following formula
It calculates
Wherein, For i-th triangle public affairs
The length on side altogether,For the cohesiveness of i-th triangle common edge.
External work power is the external work power that triangular nodes load generates on node speed (displacement increment)For
WhereinFor the velocity vector of boundary load Operational node,
For the column vector of triangular element joint load,
Nb is external load Operational node sum, vectorIn pxi,pyi, (i=1 ..., nb) is external load Operational node i on boundary
It is upper respectively along the external load size in the direction x and y.
(4) velocity boundary conditions of finite elements
In order to meet motor-driven permissive condition, the velocity field of calculating must satisfy known boundary condition.If node i be located at
X-axis angle is on the boundary θ, and known tangential velocity and normal velocity are on the boundaryThe velocity component of node i at this time
It must satisfy following equation:
In above formula: i=(1 ..., nb),For coordinate conversion matrix, nbFor known boundaries speed
Triangular element boundary node quantity,For the inclination angle of boundary i,For the size of the known speed of boundary i,It is known speed node along normal direction and tangential velocity magnitude,For the speed of the boundary node of known speed
Spend vector.
4, the Plastic Flow constraint condition of the interface of Rigid Body Element and finite element triangular element
The interface j of Rigid Body Element j and triangular element m as shown in Figure 7m, Rigid Body Element and each adjacent three
There is velocity discontinuity in corner shaped elements.According to plastic flow rule, the interface of Rigid Body Element and finite element triangular element
Plastic Flow constraint condition are as follows:
In above formula:
For Rigid Body Element j and triangle finite elements m interface jmNon-negative plasticity multiplier, and require
δUj,δVjFor the speed of Rigid Body Element j centroid,For
The speed of triangular element m centroid, For triangle
The speed of first node of unit m,For the speed of second node of triangular element m,For triangle list
The speed of the third node of first m, (Sj,nj) be interface j local coordinate system, αjFor njThe angle in direction and the direction x is (inverse
Hour hands are positive),For the quantity of the block adjacent with finite element triangular element,For the finite element adjacent with Rigid Body Element j
Triangular element quantity.
There is velocity discontinuity in Rigid Body Element and each adjacent triangular element, then there are between speed on this interface
It is disconnected, according to plastic flow rule, the internal strength power that is generated on the interface of all Rigid Body Elements and triangular element are as follows:
Wherein, For triangular element m and Rigid Body Element j interface
Length,For the cohesiveness of triangular element m and Rigid Body Element j interface.
5, Rigid Body Element condition equal with external work power with the internal strength power of finite element triangular element
It is learnt by the principle of virtual work, the dissipation of external force is done in Rigid Body Element, finite elements virtual power and object internal energy
Power is equal, and considers over-loading coefficient formula (1), in the internal strength power condition equal with external work power for being to solve for over-loading coefficient are as follows:
In order to avoid solving nonlinear programming problem, it is assumed thatThen above formula is writeable are as follows:
6, the upper bound method model of non across jointed rock mass ultimate bearing capacity
When finding limit load, the over-loading coefficient K in formula (1) is set as objective function, set constraint conditional (2), formula
(6), formula (7), formula (8), formula (12), formula (13), formula (16), then solve the upper limit normal of non across jointed rock mass ultimate bearing capacity
Property mathematical programming model are as follows:
Four, the ultimate load of non across jointed rock mass is solved
Model derived above is a large-scale linear programming model, and the hair of over half a century is passed through in linear programming
Exhibition, the solution of linear programming model formd the algorithm of a variety of comparative maturities, such as: simplex method, interior-point algohnhm and have
Set based algorithm etc. is imitated, the present invention solves the linear programming model of generation using interior-point algohnhm, and calculated result is joint rock
The ultimate load of body.
The invention has the characteristics that bearing capacity is controlled primarily by joint since non across jointed rock mass has rock bridge effect
Geometry distribution, the intensity at joint and rock bridge mechanics effect.Therefore, existing analysis method cannot be solved accurately non-through
The bearing capacity of rock mass.The present invention combines plastic limit analysis upper bound method, mixed cell discrete technology, Mathematical Planning means
Come, establish the motor-driven license velocity field of non across jointed rock mass, has obtained solving the linear of non across jointed rock mass ultimate load
Mathematics for programming model.Using mixing discrete technology, so that the method for the present invention can simulate the non-continuum mechanics of rock mass
Characteristic, and the continuous media characteristic of rock bridge can be simulated, solve a problem in rock mechanics.
The invention has the following advantages:
1, the present invention provides a kind of new method to solve the ultimate bearing capacity of non across jointed rock mass, can accurately solve to obtain
The Upper Bound Solution of non across jointed rock mass bearing capacity；
2, the present invention uses the discrete non across jointed rock mass of mixed cell, can simulate the Discontinuous transmission of rock mass simultaneously
The Continuum Mechanics characteristic of mechanical characteristic and rock bridge；
3, the method for the present invention definite conception, computational accuracy are high, engineer application is easy, can be applied to non-through joint rock
The Analysis of Bearing Capacity of body.
Detailed description of the invention
Fig. 1 the technology of the present invention route map；
The discrete schematic diagram of the mixing of Fig. 2 non across jointed rock mass Rigid Body Element and finite elements；
The velocity mode of Fig. 3 Rigid Body Element；
The not conode unit mode of Fig. 4 finite element triangular element；
The velocity mode of Fig. 5 finite element triangular element；
The velocity discontinuity schematic diagram of Fig. 6 adjacent triangle unit common edge；
Fig. 7 Rigid Body Element and triangular element interface schematic diagram；
The geometry schematic diagram of 1 non across jointed rock mass of Fig. 8 embodiment；
1 non across jointed rock mass of Fig. 9 embodiment shears sample stress diagram；
The 1 discrete schematic diagram of non across jointed rock mass mixed cell of Figure 10 embodiment.
Specific embodiment
The present invention will be further explained below with reference to the attached drawings and specific examples.
Embodiment 1: using the technical solution in the content of present invention, a non across jointed rock mass examination is solved by formula (17)
The ultimate load of sample, and analyze with analytic solutions.
(1), the calculating parameter of jointed rock mass is drafted
It is illustrated in figure 8 a non across jointed rock mass staight scissors sample, length 30cm, width and height are 20cm, in
There is a non-through joint in portion, and joint penetrates through region at left and right sides of rock mass, and centre is rock bridge, joint connected ratio kj=60%.
As shown in figure 9, the effect of rock mass lower edges has normal force σn, left and right side effect cut load τ, the purpose of the present embodiment is to ask
Solve different direct stress σnUnder the conditions of Ultimate Shear load size.Table 2 is the Material Physics mechanics parameter table of embodiment 1, table 3
For the calculating parameter table of embodiment 1.
2 embodiment of table, 1 non across jointed rock mass physical and mechanical parameter table
3 embodiment of table, 1 non across jointed rock mass staight scissors sample calculating parameter
(2), using the non-through jointed rock mass of the discrete embodiment 1 of mixed cell
Using the discrete rock mass of Rigid Body Element, using Rigid Body Element centroid rate as unknown quantity；Using finite element triangle
Mesh discretization rock bridge constructs the motor-driven license velocity field of embodiment 1 using the rate of triangular element node as unknown quantity.Implement
The rock mass covariance of example 1 is 4 Rigid Body Elements, rock bridge region covariance be 134 finite element triangular elements, adopt
It is as shown in Figure 10 with the discrete schematic diagram of hybrid-element method.
(3), the linear math plan model for solving non-through jointed rock mass upper bound method is established
After the discrete non across jointed rock mass of mixed cell, is established according to formula (17) and solve non across jointed rock mass
The upper bound method linear math plan model of Ultimate Shear load τ.
(4), the ultimate load of 1 non across jointed rock mass of embodiment is solved
According to the upper bound method linear math plan model of the Ultimate Shear load of the embodiment 1 of formula (17) foundation, using volume
The Optimization Solution program of system solves different direct stress σnUnder the conditions of Ultimate Shear load size.Calculated result such as 4 institute of table
Show.
For the present embodiment, under the premise of assuming that test specimen is cut along joint plane direction and only considers that pure shear destroys,
The Ultimate Shear load and normal stress σ of test specimenn, connected ratio kjBetween there are following analytic solutions relationships:
Wherein, kjFor the connected ratio at joint, σnFor normal stress, AjFor the shear surface area of jointed rock mass sample,For section
The angle of friction in reason face,For the angle of friction and cohesiveness of rock mass rock bridge material
The different direct stress σ being calculated according to analytic solutions formula (18)nUnder the conditions of Ultimate Shear load result simultaneously
It is listed in shown in table 4.
4 embodiment 1 of table is the same as direct stress σnUnder the conditions of Ultimate Shear load calculated result
As can be seen from the results, the ultimate load Q being calculated by upper bound method of the present inventioncThe ultimate load Q obtained with analytic solutionsa
Closely, worst error 9.85% demonstrates the accuracy of the method for the present invention.
Specific embodiments of the present invention are explained in detail above in conjunction with attached drawing, but the present invention is not limited to above-mentioned realities
Example is applied, it within the knowledge of a person skilled in the art, can also be without departing from the purpose of the present invention
Various changes can be made.
Claims (1)
1. a kind of plasticity limit analysis upper bound method of the jointed rock mass discrete based on Mix Amount, it is characterised in that: be based on plastic limit
Upper limit law theory is analyzed, with non across jointed rock mass research object, by plastic limit analysis upper bound method, the discrete side of mixed cell
Method, mathematic programming methods combine, using the discrete non across jointed rock mass of hybrid-element method, with Rigid Body Element centroid speed,
The node speed of rock bridge finite elements is unknown quantity, constructs while meeting block and structural plane deformation compatibility condition, plasticity stream
Moving constraint condition, the equal condition of interior external power, Rigid Body Element and triangular element interface compatibility of deformation item and speed edges
The motor-driven license velocity field of condition, and using the ultimate load of jointed rock mass as objective function, it thus establishes and solves non-through joint
The linear math plan model of the ultimate load of rock side slope, and linear math plan model is asked using interior-point algohnhm
Solution obtains the Upper Bound Solution of non across jointed rock mass ultimate load；Specific step is as follows:
(1) calculating parameter of jointed rock mass is drafted
According to the actual conditions of non across jointed rock mass, the calculating parameter that the analysis of plastic limit analysis upper bound method needs, packet are drafted
It includes: geological condition parameter, the geometric parameter of rock mass, rock mass materials, rock mass parameters of loading, jointed material parameter, jointed material ginseng
Number includes bulk density, cohesiveness and angle of friction；
(2) the discrete non-through jointed rock mass of mixed cell method is used
Using rigid block movement unit come the sillar made of the cutting of perforation joint in discrete rock mass, global coordinate is (x, y), block
Local coordinate system on structural plane k body unit i adjacent with Rigid Body Element j is defined as (Sk,nk), it acts in structural plane k centroid
Velocity discontinuity vector isδSkIt indicates along SkDirection rate of translation interruption, δ nkIt indicates along nkDirection translation speed
Rate interruption；The force vector acted in Rigid Body Element i centroid isfXiIndicate power in X direction, fYiIt indicates along Y
The power in direction；The velocity vector acted in Rigid Body Element i centroid isδUiIndicate translation speed in X direction
Degree, δ ViIndicate the translational velocity along Y-direction；
Speed using the rock bridge region in finite element Linear Triangular shape mesh discretization rock mass, on triangular element i-node 1,2,3
It is expressed as
(3) the upper bound method model for solving non across jointed rock mass ultimate bearing capacity is established
(a) objective function
Ultimate load, Q are solved by the way of solving over-loading coefficientc=KQa, wherein QcFor rock mass ultimate load, QaFor rock mass knot
The external load of the currently practical application of structure, K are over-loading coefficients；
(b) rock mass Rigid Body Element upper bound method constraint condition
1. the Plastic Flow constraint condition of Rigid Body Element and structural plane
Wherein, For the plasticity multiplier of adjacent block cellular construction face k, and δUi,δViFor the rate of Rigid Body Element i centroid, δ Uj,δVjFor Rigid Body Element j centroid
Rate, αkFor nkThe angle in direction and the direction x, wherein be counterclockwise positive, nkFor the quantity of structural plane,For structural plane
Angle of friction；
2. the internal strength power of Rigid Body Element condition equal with external work power
Assuming that Rigid Body Element will not be deformed and be destroyed, therefore the internal strength dissipated power inside Rigid Body Element is equal to 0, interior power consumption
It dissipates and only occurs on the adjacent structural plane of Rigid Body Element, according to the internal strength on all structural planes in associated flow rule Rigid Body Element
PowerAre as follows:Wherein, k=(1 ..., nk), nkFor all Rigid Body Element structural plane quantity,For the plasticity multiplier of structural plane k, lkFor the length of structural plane k, ckFor the cohesiveness of structural plane k；
In Rigid Body Element region, the Gravitative Loads of Rigid Body Element are considered, then Rigid Body Element is self-possessed W in virtual velocity V*Outside upper generation
Function powerAre as follows:Wherein, W is Rigid Body Element self weight, V*For virtual velocity；
In Rigid Body Element region, consideration acts on the borderline external force Q of Rigid Body Element, then the borderline face in Rigid Body Element region
Power load Q is in virtual velocity V*The external work power of upper generationAre as follows:Wherein, Q is the surface force lotus of Rigid Body Element
It carries, V*For virtual velocity；
3. the velocity boundary conditions of Rigid Body Element
The boundary condition on the b of boundary that rate in jointed rock mass is zero are as follows:Wherein:For side
The coordinate conversion matrix of Rigid Body Element j on boundary b,αjFor boundary exterior normal direction and the direction x
Angle, be counterclockwise positive,For the speed variables of Rigid Body Element j on boundary, nBFor Rigid Body Element
The quantity of zone velocity boundary interface；
(c) rock bridge finite element triangular element upper bound method constraint condition
1. finite element triangular element Plastic Flow constraint condition
Rock mass materials are assumed into rigid-perfectly plastic body, and the surrender with circumscribed regular polygon to Mohr-Coulomb yield criterion
Circle carries out approximation, then the Plastic Flow constraint condition of finite element triangular element i can be indicated with matrix form:
Wherein, For plasticity multiplier, and requireneFor the quantity of triangular element,For the area of triangular element i, bi=yj-yk, ci=-xj+xk, bj=yk-
yi, cj=-xk+xi, bk=yi-yj, ck=-xi+xj, (xi,yi), (xj,yj), (xk,yk) saved to be upper in respectively triangular element
The position coordinates of point i, j, k； Ck=2sin (2 π k/
P), k=1 ..., p,For the angle of friction of triangular element i material, p be surrender circle to Mohr-Coulomb yield criterion into
The number of edges of the approximate circumscribed regular polygon of row；
2. the discontinuous constraint condition of the speed of common edge between triangular element
The matrix that speed on adjacent triangle unit common edge i discontinuously constrains is expressed as:
Wherein:
For four plasticity multipliers of triangular element common edge, and requireI=
(1,…,ng) indicate common edge serial number, ngFor the quantity of triangular element common edge,For inclining for triangular element common edge
Angle,For the cohesiveness of the material of triangular element common edge；
3. the internal strength power of finite element triangular element condition equal with external work power
For isotropic material, the internal strength power of all triangular elementsAre as follows:Wherein, neIt is three
The quantity of corner shaped elements, i=(1 ..., ne), The respectively cohesiveness and angle of friction of triangular element i,For
The area of triangular element i；
For the discontinuous common edge of speed adjacent between all triangular elements, internal strength powerAre as follows:Wherein, It is i-th three
The length of angular common edge,For the cohesiveness of i-th triangle common edge；
External work power is the external work power that triangular nodes load generates on node speedAre as follows:Wherein,For the velocity vector of boundary load Operational node,For the column vector of triangular element joint load, nb is outer lotus
Carry Operational node sum, vectorIn pxi,pyi, i=1 ..., nb, be on boundary on external load Operational node i respectively along x and
The external load size in the direction y；
4. the velocity boundary conditions of finite elements
If it is on the boundary θ that node i, which is located at x-axis angle, known tangential velocity and normal velocity are on the boundaryAt this time
The velocity component of node i must satisfy following equation:WhereinSquare is converted for coordinate
Battle array, nbFor the triangular element boundary node quantity of known boundaries speed,For the inclination angle of boundary i,For boundary
The size of the known speed of i,It is known speed node along normal direction and tangential velocity magnitude,For known speed
The velocity vector of the boundary node of degree；
(d) the Plastic Flow constraint condition of the interface of Rigid Body Element and finite element triangular element
The interface of Rigid Body Element j and triangular element m is jm, the interface of Rigid Body Element j and finite element triangular element m
Plastic Flow constraint condition are as follows:
Wherein: For Rigid Body Element j and triangle finite elements m interface jmNon-negative plasticity multiplier, and require δUj,δVjFor the speed of Rigid Body Element j centroid,For triangle list
The speed of first m centroid, For triangular element m's
The speed of first node,For the speed of second node of triangular element m,It is the of triangular element m
The speed of three nodes, (Sj,nj) be interface j local coordinate system, αjFor njThe angle in direction and the direction x, counterclockwise
It is positive,For the quantity of the block adjacent with finite element triangular element,For the finite element triangle adjacent with Rigid Body Element j
Element number；
There is velocity discontinuity in Rigid Body Element and each adjacent triangular element, then there are velocity discontinuity, roots on this interface
According to plastic flow rule, the internal strength power that is generated on the interface of all Rigid Body Elements and triangular element are as follows:
Wherein, For triangular element m and bulk single
The length of first j interface,For the cohesiveness of triangular element m and Rigid Body Element j interface；
(e) Rigid Body Element condition equal with external work power with the internal strength power of finite element triangular element
It is learnt by the principle of virtual work, the dissipated power of external force is done in Rigid Body Element, finite elements virtual power and object internal energy
It is equal, the internal strength power of over-loading coefficient condition equal with external work power are as follows:
Assuming thatThen above formula is writeable are as follows:
(f) the upper bound method mathematical programming model of non across jointed rock mass ultimate bearing capacity is established
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| CN114925423B (en) * | 2022-05-12 | 2023-02-07 | 中南大学 | Upper limit solution analysis method, device and medium for double-layer foundation ultimate bearing capacity |
| CN115203900B (en) * | 2022-06-17 | 2023-04-18 | 中国科学院武汉岩土力学研究所 | Fractured rock mass modeling method based on discrete elements |
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