CN106126892B - A kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity - Google Patents
A kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity Download PDFInfo
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- CN106126892B CN106126892B CN201610440635.2A CN201610440635A CN106126892B CN 106126892 B CN106126892 B CN 106126892B CN 201610440635 A CN201610440635 A CN 201610440635A CN 106126892 B CN106126892 B CN 106126892B
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- E—FIXED CONSTRUCTIONS
- E02—HYDRAULIC ENGINEERING; FOUNDATIONS; SOIL SHIFTING
- E02D—FOUNDATIONS; EXCAVATIONS; EMBANKMENTS; UNDERGROUND OR UNDERWATER STRUCTURES
- E02D29/00—Independent underground or underwater structures; Retaining walls
- E02D29/02—Retaining or protecting walls
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Abstract
A kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity, the present invention relates to a kind of plastic Limit Analysis Methods solving stone masonry earth-retaining ultimate bearing capacity, belong to slope protection project technical field.The present invention is using stone masonry retaining wall as research object, based on plastic limit analysis upper limit law theory, the mechanical characteristic of stone-laying stone is simulated using Stiff Block volume elements, using the rate of displacement of the Block Element centre of form as unknown quantity, and it introduces Rankine's earth pressure theory and establishes external force overload condition, construction meets rigid block movement unit and mortar layers defonnation compatibility conditions, Plastic Flow constraints, the motor-driven license velocity field of interior external power equal condition and velocity boundary conditions, establish the nonlinear mathematics programming model for solving stone masonry earth-retaining ultimate bearing capacity, and the minimum value of ultimate load is solved using optimization algorithm.The method of the present invention has the characteristics that definite conception, computational accuracy are high, can be applied to the Analysis of Bearing Capacity of stone masonry retaining wall.
Description
Technical field
The present invention relates to a kind of plastic limit analysis upper bound methods of stone masonry earth-retaining ultimate bearing capacity, belong to side slope protection work
Journey technical field.
Background technology
The protection of slope project is one of important content of builds to foundation engineering, and stone masonry retaining wall is one in slope project
The common bank protection structures of kind.Stone masonry retaining wall has lot of advantages, such as：Advantage of lower cost, adaptable, bank protection energy
Higher, easy construction of power etc., because these advantage stone masonry retaining walls are favored by a large amount of slope protection project projects.Therefore
It is necessary to the mechanics effects of bearing capacity and destruction to stone masonry retaining wall to study, this is with important practical value
And theory significance.
The work stone or rubble of the general selected shape rule of stone masonry retaining wall are formed with masonry of cement mortar.It is required in construction
Stone layering crouch blocks, the upper and lower fissure of displacement, it is inside and outside take blocks, and strictly accomplish " flat, steady, tightly, completely " four words.Flat is exactly that each layer is wanted
Horizontal rising, contour progress is asked not to allow bricklaying surface height difference caused by progress difference excessive, being exactly surely stone will build surely, no
Easily shake；Tight is exactly that stone leans on tightly with stone, not big gap, and the mortar filling in gap is close；Completely it is exactly mortar
Crack of stone is filled, prevents dry joint and empty seam.Have in strict accordance with the stone masonry retaining wall that construction technology regulation is built higher
Strength and stiffness, bearing capacity is preferable.For the purpose of economic safety, needed to gear in the stone masonry DESIGN OF RETAINING WALLS stage
Wall carries out the design of science, mainly determines its bearing capacity, geometric dimension, material parameter etc..In the late four decades, with computer
The further investigation of the development of technology and ground theory, the method that the analysis of stone masonry retaining wall has formd many practicalities, than
Such as：Balance method of rigid-body limit, FInite Element, discrete element method etc..The soil pressure of numerous engineers and scholar from stone masonry retaining wall
Power theory, calculation of stability against sliding, deformation calculating and ultimate bearing capacity etc. have carried out system in-depth study, and obtain
Plentiful and substantial achievement in research.
Although stone masonry retaining wall is widely used in slope protection project, stone masonry retaining wall has complicated power
Characteristic is learned, is mainly manifested in：Stone masonry retaining wall is typical Discontinuous transmission, and the destruction of wall typically occurs in mortar level
On, stone does not destroy generally.Therefore, current widely applied analysis method can't accurately describe stone masonry earth-retaining
Wall bearing capacity problem.Still there are many shortcomings for this respect, are mainly manifested in：
(1) at this stage in the mainstream analysis method of stone masonry retaining wall bearing capacity based on limit equilibrium method, the limit is flat
Although weighing apparatus method has been highly convenient for applying under the effort of many scholars, it is (broken that it is only applicable to known retaining wall post-failure behaviors
Broken face) the case where, i.e., the plane of fracture artificially assumes that this has certain difference with actual conditions, so pole in advance before calculating
Limiting balancing method has certain limitation.
(2) stone masonry retaining wall is mainly formed by different block stone and masonry of cement mortar, and wherein block stone is due to strong
Degree is higher will not generally to be destroyed, and the destruction of retaining wall typically occurs in cement mortar level, therefore stone masonry retaining wall has
There is the Discontinuous transmission characteristic of height, its post-failure behaviors is difficult to determine under limiting condition, it is therefore desirable to a kind of to simulate non-company
Continue the method for mechanics characteristic to solve its ultimate load.
In consideration of it, the present invention is based on the research work of project of national nature science fund project (51564026) to propose one kind newly
Stone masonry retaining wall ultimate bearing capacity method for solving.
Invention content
The object of the present invention is to provide a kind of computational methods of stone masonry retaining wall ultimate bearing capacity, obtain stone masonry earth-retaining
Limiting condition when wall destroys provides a kind of new method for design, the calculating of slope earth-retaining wall.
The technical solution of the stone masonry retaining wall ultimate bearing capacity upper bound method of the present invention carries out according to the following steps successively：Referring to
Fig. 1 Technology Roadmaps
One, the calculating parameter for drafting stone masonry retaining wall drafts it and calculates ginseng according to the actual conditions of stone masonry retaining wall
It counts, includes mainly：Protect geological condition parameter, the material parameter of earth-retaining type wall-type geometric parameter, retaining wall and the soil body of side slope
The information such as (unit weight, cohesiveness, angle of friction), parameters of loading.
Two, using rigid block movement mesh discretization stone masonry retaining wall, that is, use rigid block movement mesh discretization method that will grout
Stone retaining wall is discrete to become rigid block movement unit+mechanical model (as shown in Figure 2) without thickness mortar level, with rigid block movement list
The rate of first centre of form is the motor-driven license velocity field that unknown quantity builds retaining wall.Stone masonry retaining wall is by stone and mortar level group
At in order to simplify the Upper Bound Solution for calculating and accurately seeking its ultimate load, present invention assumes that stone-laying stone is rigidity
Any deformation and failure will not occur for block, and using its mechanical characteristic of rigid block movement unit simulation；Assume screed simultaneously
Face is no thickness contact surface, and the destruction of retaining wall only occurs at the mortar level between stone, and in deformation process, stone
It will not be mutually disengaged open between block.
The variable defined on Rigid Body Element, mortar level (contact surface) is as shown in Figure 3.Wherein global coordinate is (X, Y),
Local coordinate system on the mortar level k of adjacent block unit i and Rigid Body Element j is defined as (Sk,nk), it acts in the block i centres of form
Velocity vectors beRate present on structural plane k is interrupted, and vector isIt is each to become
Amount explanation is as shown in Table 1 below.
The variable of table 1 mortar level and rigid block movement unit
Three, the upper bound method nonlinear mathematics programming model for solving stone masonry retaining wall ultimate bearing capacity is established
1, object function
The present invention is by the unlimited evenly load q on stone masonry retaining wall rolling earth behind retaining wall surface0As object function, it is assumed that after wall
It bankets and extends to unlimited distance, surface of banketing is level, and wall carries on the back vertical smooth.It solves its ultimate load and is just to solve for retaining wall and exist
The load that unstable failure critical moment occurs needs to solve q according to upper bound theorem0Minimum valueObjectives function is fixed
Justice is as follows：
Minimize:q0 (1)
Wherein：q0For the unlimited evenly load (as shown in Figure 4) on retaining wall rolling earth behind retaining wall surface.
2, the Rigid Body Element upper bound method constraint equation of stone-laying stone
(1) rigid block movement element deformation compatibility conditions
Discrete for after rigid block movement unit+geometrical system without thickness mortar level, the adjacent block by stone masonry earth-retaining
Deformation between unit i, j and the mortar level k of the two must satisfy deformation compatibility condition.Deformation compatibility condition is as follows：
In above formula：Relative speed between adjacent block.Wherein：For global coordinate and local coordinate system
Transition matrix：
Then it can be obtained：
In above formula：αkThe inclination angle of mortar level between two Rigid Body Elements (counterclockwise for just).
Formula (4) availability vector, matrix are abbreviated as：
In above formula：
(2) the Plastic Flow constraints of mortar level
Present invention assumes that stone block will not destroy, therefore Plastic Flow occurs over just contact surface (the i.e. stone between stone
Mortar level between block) on, that is, assume speed discontinuously (such as Fig. 3 on the contact edge of two adjacent rigid Rigid Body Elements
It is shown), and assume that mortar slice thickness is zero, in order to meet the condition of motor-driven license, on contact edge discontinuous normal direction and
Tangential velocity interruption value has to comply with flowing criterion.Mohr-Coulomb yield function f (σ, τ) is determined by the discontinuous side of speed
It can be written as in local coordinate system (n, s)：
In above formula：σn,τsNormal stress and tangential stress respectively in mortar level,For the angle of friction of mortar level,
C is the cohesiveness of mortar level, f1(σn,τs)、f2(σn,τs) be mortar level Mohr-Coulomb yield function.It can by deriving
Obtain the velocity discontinuity value determined by yield function：
In above formula：Δvn,ΔusFor the velocity discontinuity value of stone masonry level,For plasticity multiplier.
Formula (8) availability vector, matrix are abbreviated as：
In above formula：For plasticity multiplier.
According to the associated flow rule of the theory of plasticity, for ideal epistemology model, broad sense is obtained by deformation compatibility condition
The components of strain should be equal to obtains generalized plasticity strain rate component by associated flow rule and yield condition.By formula (5) and formula
(9) the Plastic Flow constraints for just obtaining all mortar levels is combined：
It is non-negative to require plasticity multiplier simultaneously：
(3) internal strength power condition equal with external work power
It is learnt by the principle of virtual work, the dissipated power of virtual power and object internal energy that external force is done is equal, then has：
In above formula:It is non-individual body internal stress vector,It is the virtual strain inside non-individual body,It is boundary upper stress
Vector,It is borderline virtual strain, W is block dead weight, and Q is face power load, V*For virtual velocity；The equation left side is respectively to generate
The internal dissipation power on destruction sliding surface is neutralized in stone masonry barricade, is respectively block dead weight W on the right of equation, borderline
The equivalent load of face power load Q is in virtual velocity V*On power.
It is rigidity it is assumed that the Rigid Body Element of stone will not be deformed and be destroyed according to stone-laying stone of the present invention, internal strength
Dissipation only result from the interface (i.e. mortar level) between Rigid Body Element, therefore the internal strength power inside non-individual bodyConvolution (6), (7), first item can be written as in formula (12)：
In above formula：σn,τsNormal stress and tangential stress respectively in mortar level,Respectively in mortar level
Normal direction normal strain rate and tangential shear strain rate,It is boundary upper stress vector,It is borderline virtual strain, l is mortar
The length of level, c are the cohesiveness of mortar level,For plasticity multiplier.
Only consider that the interior energy caused by borderline stress and the dead weight of stone block dissipates herein.For stone masonry retaining wall
Borderline load, patent of the present invention mainly consider the dead weight of the soil body and the unlimited evenly load on rolling earth behind retaining wall after retaining wall
q0Effect to the wall back of the body, action principle are as shown in Figure 4.According to Theory of Rankine Active Earth Pressure, banket case depth Z in distance
Locate the active earth pressure strength p of the retaining walls back of the body at any pointakExpression formula be：
In above formula,It is coefficient of active earth pressure, σak=γ Z+q0It is vertically answering at depth Z
Power,C' is the shearing strength of rolling earth behind retaining wall.
The face power load Q then to banket on surface-boundary can be written as：
In above formula, Γ is that retaining walls carry on the back integral boundary.
Consider further that the dead weight W of stone block, then (12) formula becomes：
(4) velocity boundary conditions
From upper bound theorem, motor-driven license velocity field must satisfy known velocity boundary conditions in speed edges.
The velocity boundary conditions on the b of boundary that rate in stone masonry retaining wall is zero are：
In above formulaFor the coordinate conversion matrix of interface j on the b of boundary：
4, the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is solved
When finding limit load, by the unlimited evenly load q on retaining wall rolling earth behind retaining wall surface0It is set as object function, then is solved
The upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is：
Four, the ultimate bearing capacity of stone masonry retaining wall is solved
Mathematical model derived above is a nonlinear mathematics programming model.The present invention is carried out non-using conjugate gradient method
The solution of linear math plan model obtains the ultimate load and corresponding velocity field that result of calculation includes retaining wall.
The present invention basic principle be：Based on upper limit law theory, using stone masonry retaining wall as research object, by upper bound method,
Rigid block movement mesh discretization method, Theory of Rankine Active Earth Pressure and Mathematical Planning means combine, with stone masonry earth-retaining
The unlimited evenly load on wall rolling earth behind retaining wall surface is as object function, using rigid block movement mesh discretization method by stone masonry earth-retaining
Discrete wall is block stone rigid block movement unit+mechanical model without thickness mortar level, and structure meets rigid block movement unit and mortar
The motor-driven license speed of layers defonnation compatibility conditions, Plastic Flow constraints, the equal condition of interior external power and velocity boundary conditions
Field is spent, establishes the nonlinear mathematics programming model for solving stone masonry earth-retaining ultimate bearing capacity, and the limit is solved using optimization algorithm
The minimum value of load.
The invention has the advantages that：
1, the present invention provides a kind of new method for the analysis of Ultimate of stone masonry retaining wall, can accurately solve stone masonry
The Upper Bound Solution of the ultimate bearing capacity of retaining wall.
2, the present invention using rigid block movement unit+pattern without thickness mortar level come discrete stone masonry retaining wall, can be with
The mechanical characteristic of accurate simulation stone masonry retaining wall Discontinuous transmission, can accurately obtain its ultimate load and corresponding destruction machine
Structure.
3, the method for the present invention definite conception, computational accuracy are high, can be applied to the bearing capacity of stone masonry retaining wall side slope
Analysis.
Description of the drawings
Fig. 1 is the Technology Roadmap of the present invention；
Fig. 2 is stone masonry barricade rigid block movement unit and the discrete schematic diagram of mortar level；
Fig. 3 is the velocity mode of stone masonry rigid block movement unit；
Fig. 4 is that stone masonry retaining wall Rankine Active Earth Pressure calculates schematic diagram；
Fig. 5 is the geometry schematic diagram of embodiment retaining wall；
Fig. 6 is embodiment stone masonry retaining wall rigid block movement mesh discretization schematic diagram；
Fig. 7 is the corresponding speed vector figure of embodiment retaining wall ultimate load Upper Bound Solution.
Specific implementation mode
The invention will be further described in the following with reference to the drawings and specific embodiments.
Embodiment：The present embodiment solves the ultimate load of a stone masonry retaining wall using following steps, and is obtaining
The failure mode of barricade analyzes with conventional method result of calculation later.
(1), the calculating parameter of retaining wall is drafted
It is illustrated in figure 5 a stone masonry retaining wall, to grout work stone retaining wall, design parameter is：Stone masonry weight
Power formula retaining walls height 3.5m, face slope gradient:1:0.3, back slope gradient:1:0.0, the wall slanted floor gradient:0.0:
1, retaining wall is basic supporting course with " argillaceous sandstone ", and retaining wall substrate enters supporting course >=1m；Stone-laying barricade uses cement bonded sand
It starches to build blocky stone material, rectangular work stone of the stone material selection without differentiation, the regular shape of flawless by laying bricks or stones, the thickness of work stone is about
300mm, length and width are not less than 200mm；Masonry allowable compressive stress is more than 2100kPa, and allowable shearing stress is more than 110kPa；
Stone masonry barricade strictly by slurry method construction is squeezed, ensure that mortar is full.Table 2 is the Material Physics mechanics parameter table of embodiment.
2 embodiment of table grouts stone and the physical and mechanical parameter table of mortar level
(2), using the stone masonry retaining wall of the discrete embodiment of rigid block movement elements method, i.e.,：Using rigid block movement unit from
Stone is dissipated, the motor-driven license velocity field of retaining wall is built using the rate of the stone centre of form as unknown quantity.The retaining wall of embodiment altogether from
It dissipates for 53 rigid block movement units and 127 mortar levels, discrete schematic diagram is as shown in Figure 6.
(3), the upper bound method nonlinear mathematics programming model for solving stone masonry retaining wall ultimate bearing capacity is established
The evenly load q acted on surface of banketing is carried on the back in order to solve retaining walls0Minimum valueIt can be according to formula (18)
The upper bound method nonlinear mathematics programming model of the ultimate bearing capacity of constitution and implementation example.
(4), the ultimate bearing capacity of embodiment retaining wall is solved.
According to the ultimate load for the embodiment stone masonry retaining wall that formula (18) is establishedUpper bound method nonlinear mathematics optimization
Model is calculated mortar level and is taken pole under the conditions of different parameters of shear resistant using the nonlinear mathematics programming solver of establishment
Limit load carriesResult of calculation is listed in shown in table 3.This stone masonry retaining wall wall is in Ultimate Loads known to result of calculation
Under failure mode as shown in fig. 7, Fig. 7 be mortar level parameters of shear resistant take c=100kPa,When speed vector figure,
Failure mode of embodiment retaining wall wall known to it is that failure by shear occurs along mortar level AB.In this known failure mode
After, the analytic solutions of ultimate load can be calculated by the rigid Limit Equilibrium relationship of retaining wallEqually it is listed in Table 3 below
It is shown.
By result it is found that the method for the present invention numerical solution is all higher than analytic solutions, Upper Bound Solution property is shown as；And press present invention side
The ultimate load that method obtainsThe ultimate load obtained with analytic solutionsClosely, the method for the present invention is calculated
The worst error of numerical solution and analytic solutions is less than 1%, demonstrates the correctness and accuracy of the method for the present invention.
The ultimate load result of calculation of 3 embodiment stone masonry retaining wall of table
Claims (1)
1. a kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity, it is characterised in that：With stone masonry earth-retaining
The unlimited evenly load on wall rolling earth behind retaining wall surface is as object function, using rigid block movement mesh discretization method by stone masonry earth-retaining
Discrete wall is block stone rigid block movement unit+mechanical model without thickness mortar level, and structure meets rigid block movement unit and mortar
The motor-driven license speed of layers defonnation compatibility conditions, Plastic Flow constraints, the equal condition of interior external power and velocity boundary conditions
Field is spent, establishes the nonlinear mathematics programming model for solving stone masonry earth-retaining ultimate bearing capacity, and the limit is solved using optimization algorithm
The minimum value of load；It is as follows：
One, the calculating parameter of stone masonry retaining wall is drafted
According to the actual conditions of stone masonry retaining wall, its calculating parameter, including geological condition parameter, geometric parameter, gear material are drafted
Expect parameter, parameters of loading, material parameter includes unit weight, cohesiveness, angle of friction；
Two, using rigid block movement mesh discretization stone masonry retaining wall
It is using rigid block movement mesh discretization method that stone masonry retaining wall is discrete as rigid block movement unit and mortar level；It is overall
Coordinate system is (X, Y), and the local coordinate system on the mortar level k of adjacent block unit i and Rigid Body Element j is defined as (Sk,nk),
The velocity vectors acted in the block i centres of form areδuiIndicate rate of translation in X direction, δ viIt indicates along the side Y
To rate of translation, the rate interrupt vector on structural plane k isδSkIt indicates along SkBetween the rate of translation of direction
It is disconnected, δ nkIt indicates along nkDirection rate of translation interruption；
Three, the upper bound method nonlinear mathematics programming model for solving stone masonry retaining wall ultimate bearing capacity is established
(1) object function
With the unlimited evenly load q on stone masonry retaining wall rolling earth behind retaining wall surface0As object function, it is assumed that rolling earth behind retaining wall extends to
Unlimited distance, surface of banketing are level, and wall carries on the back vertical smooth, solves q0Minimum value
(2) the Rigid Body Element upper bound method constraint equation of stone-laying stone
A rigid block movement element deformation compatibility conditions
Wherein：Relative speed between adjacent block,For the transition matrix of global coordinate and local coordinate system；
It can be abbreviated as：
Wherein：αkMortar level inclines between two Rigid Body Elements
Angle, counterclockwise for just；
The Plastic Flow constraints of b mortar levels
Assuming that stone block will not destroy, therefore Plastic Flow occurs over just in the contact surface mortar level between stone, and false
If mortar slice thickness is zero, discontinuous normal direction and tangential velocity interruption value meet flowing criterion on contact edge：
σn,τsNormal stress and tangential stress respectively in mortar level,For the angle of friction of mortar level, c is mortar level
Cohesiveness, f1(σn,τs)、f2(σn,τs) be mortar level Mohr-Coulomb yield function；
It can further obtain
Δvn,ΔusFor the velocity discontinuity value of stone masonry level,For plasticity multiplier；
It can be abbreviated as:
For plasticity multiplier；
According to the associated flow rule of the theory of plasticity, for ideal epistemology model, generalized strain is obtained by deformation compatibility condition
Component should be equal to obtains generalized plasticity strain rate component by associated flow rule and yield condition：Together
Shi Yaoqiu plasticity multipliers are non-negative：
C internal strengths power condition equal with external work power
It is learnt by the principle of virtual work, the dissipated power of virtual power and object internal energy that external force is done is equal, then has
It is non-individual body internal stress vector,It is the virtual strain inside non-individual body,It is boundary upper stress vector,It is side
Virtual strain in boundary, W are block dead weights, and Q is face power load, V*For virtual velocity；The equation left side is respectively to result from stone masonry gear
It is respectively block dead weight W, borderline face power load Q that wall, which neutralizes on the right of the internal dissipation power destroyed on sliding surface, equation,
Equivalent load in virtual velocity V*On power；
According to stone-laying stone to be rigid it is assumed that the Rigid Body Element of stone will not be deformed and be destroyed, the dissipation of internal strength is only
It results from the interface between Rigid Body Element, therefore the internal strength power inside non-individual body
σn,τsNormal stress and tangential stress respectively in mortar level,Normal direction respectively in mortar level is just answered
Variability and tangential shear strain rate,It is boundary upper stress vector,It is borderline virtual strain, l is the length of mortar level
Degree, c are the cohesiveness of mortar level,For plasticity multiplier；
According to Theory of Rankine Active Earth Pressure, the active soil for the retaining walls back of the body at any point at case depth Z of banketing in distance
Intensity of pressure pakExpression formula be：
It is coefficient of active earth pressure, σak=γ Z+q0, it is the vertical stress at depth Z,C' is
The shearing strength of rolling earth behind retaining wall；
The face power load Q then to banket on surface-boundary can be written as：
Γ is that retaining walls carry on the back integral boundary；
The dead weight W for considering further that stone block, then can obtain：
D velocity boundary conditions
The velocity boundary conditions on the b of boundary that rate in stone masonry retaining wall is zero are：
For the coordinate conversion matrix of interface j on the b of boundary；
(3) the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is solved
When finding limit load, by the unlimited evenly load q on retaining wall rolling earth behind retaining wall surface0It is set as object function, then solves stone masonry
The upper bound method nonlinear mathematics programming model of retaining wall ultimate bearing capacity is：
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107341285B (en) * | 2017-05-26 | 2020-08-25 | 昆明理工大学 | Lower limit method for analysis of joint rock slope toppling damage limit bearing capacity under action of seismic force |
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