CN106126892A - A kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity - Google Patents

Abstract

The plastic limit analysis upper bound method of a kind of stone masonry retaining wall ultimate bearing capacity, the present invention relates to a kind of plastic Limit Analysis Method solving stone masonry gear soil ultimate bearing capacity, belongs to side slope protection field of engineering technology.The present invention is with stone masonry retaining wall as object of study, theoretical based on plastic limit analysis upper bound method, use the mechanical characteristic of Stiff Block volume elements simulation stone-laying stone, with the rate of displacement of the Block Element centre of form as unknown quantity, and introduce Rankine's earth pressure theory and set up external force overload condition, structure meets rigid block movement unit and screed facial disfigurement compatibility conditions, Plastic Flow constraints, the equal condition of interior external power and the motor-driven license velocity field of velocity boundary conditions, set up the nonlinear mathematics programming model solving stone masonry gear soil ultimate bearing capacity, and use optimized algorithm to solve the minima of ultimate load.The inventive method has definite conception, computational accuracy high, can be applied to the Analysis of Bearing Capacity of stone masonry retaining wall.

Description

A kind of plastic limit analysis upper bound method of stone masonry retaining wall ultimate bearing capacity

Technical field

The present invention relates to the plastic limit analysis upper bound method of a kind of stone masonry gear soil ultimate bearing capacity, belong to side slope protection work Journey technical field.

Background technology

The protection of slope project is one of important content that foundation engineering is built, and stone masonry retaining wall is in slope project one Plant conventional bank protection structures.Stone masonry retaining wall has lot of advantages, such as: advantage of lower cost, strong adaptability, bank protection energy Power is higher, easy construction etc., because these advantage stone masonry retaining walls are favored by a large amount of side slope protection engineering projects.Therefore Being necessary that the mechanics effect of the bearing capacity to stone masonry retaining wall and destruction is studied, this has important practical value And theory significance.

Work stone or the rubble masonry of cement mortar of stone masonry retaining wall general selected shape rule form.Requirement in construction Sleeping blocks of stone layering, the upper and lower fissure of displacement, inside and outside take block, and strictly accomplish " flat, steady, tight, completely " four words.Flat is exactly that each layer is wanted Ask level to rise, contour carry out, do not allow bricklaying surface to cause the discrepancy in elevation excessive because of progress difference, be exactly surely that stone to be built surely, no Easily shake；Being exactly tightly that stone leans on tight with stone, do not have big gap, the mortar filling in space is closely；It it is exactly completely mortar Crack of stone to be filled, prevents dry joint and empty seam.The stone masonry retaining wall built in strict accordance with construction technology code has higher Strength and stiffness, bearing capacity is preferable.For economic, safe purpose, need gear in the stone masonry DESIGN OF RETAINING WALLS stage Wall carries out the design of science, mainly determines its bearing capacity, physical dimension, material parameter etc..Over nearly 40 years, along with computer The development of technology and the further investigation of ground theory, the analysis of stone masonry retaining wall has defined many practical methods, than As: balance method of rigid-body limit, FInite Element, discrete element method etc..Numerous engineers and scholar are from the soil pressure of stone masonry retaining wall The aspects such as power theory, calculation of stability against sliding, deformation calculating and ultimate bearing capacity have carried out system in-depth study, and obtain Plentiful and substantial achievement in research.

Although stone masonry retaining wall is widely used in side slope protection engineering, but stone masonry retaining wall has the power of complexity Learning characteristic, be mainly manifested in: stone masonry retaining wall is typical Discontinuous transmission, the destruction of body of wall typically occurs in mortar aspect On, stone does not destroys.Therefore, current wide variety of analysis method can't describe stone masonry gear soil accurately Wall bearing capacity problem.This respect still has many weak points, is mainly manifested in:

(1) based on limit equilibrium method in the main flow analysis method of present stage stone masonry retaining wall bearing capacity, the limit is put down Although weighing apparatus method has been highly convenient for application under the effort of many scholars, but it is only applicable to known retaining wall post-failure behaviors and (breaks Broken face) situation, i.e. the plane of fracture be calculate before artificial suppose in advance, this has certain difference with practical situation, so pole Limit counterbalanced procedure 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 by force Spending and higher typically will not destroy, the destruction of retaining wall typically occurs in cement mortar aspect, therefore stone masonry retaining wall tool Having the Discontinuous transmission characteristic of height, under limit state, its post-failure behaviors is difficult to determine, it is therefore desirable to one can simulate non-company The method of continuous mechanics characteristic solves its ultimate load.

In consideration of it, present invention research work based on project of national nature science fund project (51564026) proposes a kind of new The method for solving of ultimate bearing capacity of stone masonry retaining wall.

Summary of the invention

It is an object of the invention to provide the computational methods of a kind of stone masonry retaining wall ultimate bearing capacity, it is thus achieved that stone masonry gear soil Limit state when wall destroys, design, calculating for slope earth-retaining wall provide a kind of new method.

The technical scheme of the stone masonry retaining wall ultimate bearing capacity upper bound method of the present invention sequentially includes the following steps: successively and sees Fig. 1 Technology Roadmap

One, draft the calculating parameter of stone masonry retaining wall, according to the practical situation of stone masonry retaining wall, draft it and calculate ginseng Number, specifically includes that the protection geological conditions parameter of side slope, retaining wall build geometric parameter, retaining wall and the material parameter of the soil body The information such as (unit weight, cohesiveness, angle of friction), parameters of loading.

Two, use rigid block movement mesh discretization stone masonry retaining wall, i.e. use rigid block movement mesh discretization method to grout Stone retaining wall is discrete becomes rigid block movement unit+without the mechanical model (as shown in Figure 2) of thickness mortar aspect, with rigid block movement list The speed of unit's 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 aspect group Become, in order to simplify the Upper Bound Solution calculating and seeking accurately its ultimate load, present invention assumes that stone-laying stone is rigidity Block, it will not occur any deformation and failure, and use its mechanical characteristic of rigid block movement unit simulation；Assume screed simultaneously Face is without thickness contact surface, and the destruction of retaining wall only occurs at the mortar aspect between stone, and in deformation process, stone Will not be mutually disengaged out between block.

The variable defined in Rigid Body Element, mortar aspect (contact surface) is as shown in Figure 3.Wherein global coordinate is (X, Y), Local coordinate system in mortar aspect k of adjacent block unit i and Rigid Body Element j is defined as (S_k,n_k), the block i centre of form acts on Velocity vectors beSpeed present on structural plane k is interrupted, and its vector isEach change Amount explanation is as shown in Table 1 below.

Table 1 mortar aspect and the variable of rigid block movement unit

Three, foundation solves the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity

1, object function

The present invention is by unlimited evenly load q on stone masonry retaining wall rolling earth behind retaining wall surface₀As object function, it is assumed that after wall Banketing and extend to unlimited distance, surface of banketing is level, wall back of the body vertical smooth.Solve its ultimate load to be just to solve for retaining wall and exist There is the load of unstable failure critical moment, according to upper bound theorem, need to solve q₀MinimaObjectives function is fixed Justice is as follows:

Minimize:q₀ (1)

Wherein: q₀Unlimited evenly load (as shown in Figure 4) for 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

After the stone masonry discrete geometrical system for rigid block movement unit+without thickness mortar aspect of gear soil, adjacent block Deformation between unit i, j and both mortar aspects k must is fulfilled for deformation compatibility condition.Deformation compatibility condition is as follows:

$\[\mathrm{\δ}{\stackrel{\→}{q}}_k^t\]=\[{\stackrel{\→}{T}}_k^g(\mathrm{\δ}{\stackrel{\→}{u}}_i^t-\mathrm{\δ}{\stackrel{\→}{u}}_j^t)\]---\left(2\right)$

In above formula:For the relative speed between adjacent block.Wherein:For global coordinate and local coordinate system Transition matrix:

${\stackrel{\→}{T}}_k^g=\left[\begin{array}{cc}{\mathrm{cos\α}}_k& -{\mathrm{sin\α}}_k\\ {\mathrm{sin\α}}_k& {\mathrm{cos\α}}_k\end{array}\right]---\left(3\right)$

Then can get:

$\left\{\begin{array}{c}{\mathrm{\δS}}_k\\ {\mathrm{\δn}}_i\end{array}\right\}=\left[\begin{array}{cccc}{\mathrm{cos\α}}_k& -{\mathrm{sin\α}}_k& -{\mathrm{cos\α}}_k& {\mathrm{sin\α}}_k\\ {\mathrm{sin\α}}_k& {\mathrm{cos\α}}_k& -{\mathrm{sin\α}}_k& -{\mathrm{cos\α}}_k\end{array}\right]\left\{\begin{array}{c}{\mathrm{\δu}}_i\\ {\mathrm{\δv}}_i\\ {\mathrm{\δu}}_j\\ {\mathrm{\δv}}_i\end{array}\right\}---\left(4\right)$

In above formula: α_kIt it is the inclination angle (counterclockwise for just) of mortar aspect between two Rigid Body Elements.

Formula (4) availability vector, matrix are abbreviated as:

$\mathrm{\δ}\stackrel{\→}{q}=\stackrel{\→}{D}\mathrm{\δ}\stackrel{\→}{u}---\left(5\right)$

In above formula:

(2) the Plastic Flow constraints of mortar aspect

Present invention assumes that stone block will not destroy, therefore Plastic Flow occurs over just contact surface (the i.e. stone between stone Mortar aspect between block) on, i.e. suppose that speed is discontinuously positioned on the contact edge of two adjacent rigid Rigid Body Elements (such as Fig. 3 Shown in), and assume that screed face 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 limit of speed Local coordinate system (n, s) in can be written as:

In above formula: σ_n,τ_sIt is respectively the normal stress in mortar aspect and tangential stress,For the angle of friction of mortar aspect, C is the cohesiveness of mortar aspect, f₁(σ_n,τ_s)、f₂(σ_n,τ_s) it is the Mohr-Coulomb yield function of mortar aspect.Can by deriving Obtain the velocity discontinuity value determined by yield function:

In above formula: Δ v_n,Δu_sFor the velocity discontinuity value of stone masonry aspect,For plasticity multiplier.

Formula (8) availability vector, matrix are abbreviated as:

$\mathrm{\δ}\stackrel{\→}{q}={\stackrel{\→}{N}}_0\mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}---\left(9\right)$

In above formula:For plasticity multiplier.

According to the associated flow rule of the theory of plasticity, for ideal epistemology model, deformation compatibility condition obtain broad sense The components of strain should be equal to being obtained generalized plasticity strain rate component by associated flow rule and yield condition.By formula (5) and formula (9) combine and just obtain the Plastic Flow constraints of all mortar aspects:

$\stackrel{\→}{D}\mathrm{\δ}\stackrel{\→}{u}-{\stackrel{\→}{N}}_0\mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}=0---\left(10\right)$

Require that plasticity multiplier is non-negative simultaneously:

$\mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}={\left\{\begin{array}{cc}{\stackrel{\·}{k}}_1^k& {\stackrel{\·}{k}}_2^k\end{array}\right\}}^T\≥0---\left(11\right)$

(3) internal strength power condition equal with external work power

Being learnt by the principle of virtual work, virtual power and the dissipated power of object internal energy that external force is done are equal, then have:

${\∫}_{Q^{*}}{\mathrm{\σ}}_{ij}^{*}{\mathrm{\ϵ}}_{ij}^{*}{\mathrm{d\Ω}}^{*}+{\∫}_{{\mathrm{\Γ}}^{*}}{\mathrm{\σ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{\ϵ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{d\Γ}}^{*}={\mathrm{WV}}^{*}+{\mathrm{QV}}^{*}---\left(12\right)$

In above formula:It is continuum internal stress vector,It is the virtual strain within continuum,It it is border upper stress Vector,Being borderline virtual strain, W is block deadweight, and Q is face power load, V^*For virtual velocity；The equation left side is respectively and produces In stone masonry barricade and along the internal dissipation power destroyed along sliding surface, on the right of equation, it is respectively block deadweight W, borderline The equivalent load of face power load Q is in virtual velocity V^*On power.

According to stone-laying stone of the present invention be rigidity it is assumed that the Rigid Body Element of stone will not deform and destroy, internal strength Dissipation only result from the interface (i.e. mortar aspect) between Rigid Body Element, the therefore internal strength power within continuumConvolution (6), (7), in formula (12), Section 1 can be written as:

${\∫}_{{\mathrm{\Γ}}^{*}}{\mathrm{\σ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{\ϵ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{d\Γ}}^{*}=\underset{z\→0}{\lim }{\∫}_0^z{\∫}_0^l({\mathrm{\σ}}_n{\stackrel{\·}{\mathrm{\ϵ}}}_n+{\mathrm{\τ}}_s{\stackrel{\·}{\mathrm{\γ}}}_s)dtds={\∫}_0^{l}c({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)dl=cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)---\left(13\right)$

In above formula: σ_n,τ_sIt is respectively the normal stress in mortar aspect and tangential stress,It is respectively in mortar aspect Normal direction normal strain rate and tangential shear strain rate,It is border upper stress vector,Being borderline virtual strain, l is mortar The length of aspect, c is the cohesiveness of mortar aspect,For plasticity multiplier.

The most only consider that the interior energy that borderline stress and the deadweight of stone block are caused dissipates.For stone masonry retaining wall Borderline load, patent of the present invention mainly considers the unlimited evenly load on the deadweight of the soil body after retaining wall and rolling earth behind retaining wall q₀Effect to the wall back of the body, its action principle is as shown in Figure 4.According to Theory of Rankine Active Earth Pressure, banket case depth Z in distance The active earth pressure strength p of the retaining walls back of the body of place's any point_akExpression formula be:

$p_{ak}={\mathrm{\σ}}_{ak}{k}_a-2c^{\′}\sqrt{k_a}---\left(14\right)$

In above formula,It is coefficient of active earth pressure, σ_ak=γ Z+q₀It is vertically should at degree of depth Z Power,C' is the shearing strength of rolling earth behind retaining wall.

The face power load Q then banketed on surface-boundary can be written as:

$Q=\underset{\mathrm{\Γ}}{\∫}{p}_{ak}=\underset{\mathrm{\Γ}}{\∫}\left(\right(\mathrm{\γ}Z+q_0)k_a-2c^{\′}\sqrt{k_a})---\left(15\right)$

In above formula, Γ is that retaining walls carries on the back integral boundary.

Consider further that the deadweight W of stone block, then (12) formula becomes:

$cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)={\mathrm{WV}}^{*}+\left(\underset{\mathrm{\Γ}}{\∫}\right(\left(\mathrm{\γ}Z+q_0\right)k_a-2c^{\′}\sqrt{k_a}\left)\right)V^{*}---\left(16\right)$

(4) velocity boundary conditions

From upper bound theorem, motor-driven license velocity field must is fulfilled for known velocity boundary conditions in speed edges. Speed in stone masonry retaining wall is that the velocity boundary conditions on the border b of zero is:

$\mathrm{\δ}{\stackrel{\→}{q}}_j^t={\stackrel{\→}{T}}_j^b\left(\mathrm{\δ}{\stackrel{\→}{u}}_i^t\right)=0---\left(17\right)$

In above formulaFor the coordinate conversion matrix of interface j on the b of border:

4, the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is solved

During finding limit load, by unlimited evenly load q on retaining wall rolling earth behind retaining wall surface₀It is set to object function, then solves The upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is:

$\left.\begin{array}{cc}Minimize:& q_0\\ Subjectto:& \stackrel{\→}{D}\mathrm{\δ}\stackrel{\→}{u}-{\stackrel{\→}{N}}_0\mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}=0\\ & \mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}\≥0\\ & cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)={\mathrm{WV}}^{*}+\left(\underset{\mathrm{\Γ}}{\∫}\right(\left(\mathrm{\γ}Z+q_0\right)k_a-2c^{\′}\sqrt{k_a}\left)\right)V^{*}\\ & \mathrm{\δ}{\stackrel{\→}{q}}_j^t=0\end{array}\right\}---\left(18\right)$

Four, the ultimate bearing capacity of stone masonry retaining wall is solved

Mathematical model derived above is a nonlinear mathematics programming model.It is non-that the present invention uses conjugate gradient method to carry out Solving of linear math plan model, obtains result of calculation and includes the ultimate load of retaining wall and the velocity field of correspondence.

The ultimate principle of the present invention is: theoretical based on upper bound method, with stone masonry retaining wall as object of study, by upper bound method, Rigid block movement mesh discretization method, Theory of Rankine Active Earth Pressure and Mathematical Planning means combine, with stone masonry gear soil The unlimited evenly load on wall rolling earth behind retaining wall surface, as object function, uses rigid block movement mesh discretization method by stone masonry gear soil Wall is discrete for block stone rigid block movement unit+without the mechanical model of thickness mortar aspect, builds and 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 Degree field, sets up the nonlinear mathematics programming model solving stone masonry gear soil ultimate bearing capacity, and uses optimized algorithm to solve the limit The minima of load.

The method have the advantages that

1, the analysis of Ultimate that the present invention is stone masonry retaining wall provides a kind of new method, can accurately solve stone masonry The Upper Bound Solution of the ultimate bearing capacity of retaining wall.

2, the present invention use rigid block movement unit+without thickness mortar aspect pattern come discrete stone masonry retaining wall, permissible 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 inventive method definite conception, computational accuracy height, can be applied to the bearing capacity of stone masonry retaining wall side slope Analyze.

Accompanying drawing explanation

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 aspect；

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 speed vector figure that embodiment retaining wall ultimate load Upper Bound Solution is corresponding.

Detailed description of the invention

The invention will be further described with specific embodiment below in conjunction with the accompanying drawings.

Embodiment: the present embodiment uses following steps to solve the ultimate load of a stone masonry retaining wall, 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

Being illustrated in figure 5 a stone masonry retaining wall, it is for grouting work stone retaining wall, and its design parameter is: stone masonry weight Power formula retaining walls height 3.5m, slope, face gradient: 1:0.3, back slope gradient: 1:0.0, the wall slanted floor gradient: 0.0: 1, retaining wall is supporting course based on " argillaceous sandstone ", and retaining wall substrate enters supporting course >=1m；Stone-laying barricade uses cement bonded sand Block stone material built by laying bricks or stones by slurry, and stone material is selected without differentiation, the square work stone of flawless regular shape, and the thickness of work stone is about 300mm, length and width is all not less than 200mm；Masonry allowable compressive stress is more than 2100kPa, and allowable shearing stress is more than 110kPa； Stone masonry barricade is strictly by squeezing slurry method construction, it is ensured that mortar is full.Table 2 is the Material Physics mechanics parameter table of embodiment.

Table 2 embodiment grouts the physical and mechanical parameter table of stone and mortar aspect

(2), use the discrete embodiment of rigid block movement elements method stone masonry retaining wall, it may be assumed that use rigid block movement unit from Dissipate stone, build the motor-driven license velocity field of retaining wall with the speed of the stone centre of form for unknown quantity.The retaining wall of embodiment altogether from Dissipating is 53 rigid block movement unit and 127 mortar aspects, and its discrete schematic diagram is as shown in Figure 6.

(3), foundation solves the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity

Evenly load q of effect on surface of banketing is carried on the back in order to solve retaining walls₀MinimaCan be according to formula (18) The upper bound method nonlinear mathematics programming model of the ultimate bearing capacity of constitution and implementation example.

(4) ultimate bearing capacity of embodiment retaining wall, is solved.

Ultimate load according to the embodiment stone masonry retaining wall that formula (18) is set upUpper bound method nonlinear mathematics optimize Model, uses the nonlinear mathematics programming solver of establishment, calculates the pole under the conditions of mortar aspect takes different parameters of shear resistant Limit load carriesResult of calculation is listed in shown in table 3.This stone masonry retaining wall body of wall is understood at Ultimate Loads by result of calculation Under failure mode as it is shown in fig. 7, Fig. 7 is mortar aspect parameters of shear resistant take c=100kPa,Time speed vector figure, The failure mode being understood embodiment retaining wall body of wall by it is, along mortar aspect AB, failure by shear occurs.At this failure mode known After, the analytic solutions of ultimate load can be calculated by the rigid Limit Equilibrium relation of retaining wallIt is shown in Table 3 equally Shown in.

From result, the inventive method numerical solution is all higher than analytic solutions, shows as Upper Bound Solution character；And by side of the present invention The ultimate load that method obtainsThe ultimate load obtained with analytic solutionsClosely, the inventive method is calculated Numerical solution is less than 1% with the maximum error of analytic solutions, demonstrates correctness and the accuracy of the inventive method.

The ultimate load result of calculation of table 3 embodiment stone masonry retaining wall

Claims (1)

1. the plastic limit analysis upper bound method of a stone masonry retaining wall ultimate bearing capacity, it is characterised in that: with stone masonry gear soil The unlimited evenly load on wall rolling earth behind retaining wall surface, as object function, uses rigid block movement mesh discretization method by stone masonry gear soil Wall is discrete for block stone rigid block movement unit+without the mechanical model of thickness mortar aspect, builds and 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 Degree field, sets up the nonlinear mathematics programming model solving stone masonry gear soil ultimate bearing capacity, and uses optimized algorithm to solve the limit The minima of load；Specifically comprise the following steps that

One, the calculating parameter of stone masonry retaining wall is drafted

According to the practical situation of stone masonry retaining wall, draft it and calculate parameter, including geological conditions parameter, geometric parameter, gear material Material parameter, parameters of loading, material parameter includes unit weight, cohesiveness, angle of friction；

Two, rigid block movement mesh discretization stone masonry retaining wall is used

Rigid block movement mesh discretization method is used to become rigid block movement unit and mortar aspect by discrete for stone masonry retaining wall；Totally Coordinate system is (X, Y), and the local coordinate system in mortar aspect k of adjacent block unit i and Rigid Body Element j is defined as (S_k,n_k), In the block i centre of form, the velocity vectors of effect isδu_iRepresent rate of translation in X direction, δ v_iRepresent along Y side To rate of translation structural plane k on speed interrupt vector beδS_kRepresent along S_kBetween the rate of translation of direction Disconnected, δ n_kRepresent along n_kDirection rate of translation is interrupted；

Three, foundation solves the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity

(1) object function

Unlimited evenly load q with 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 is level, wall back of the body vertical smooth, solves q₀Minima

(2) the Rigid Body Element upper bound method constraint equation of stone-laying stone

A rigid block movement element deformation compatibility conditions

$\[\mathrm{\δ}{\stackrel{\→}{q}}_k^t\]=\[{\stackrel{\→}{T}}_k^g(\mathrm{\δ}{\stackrel{\→}{u}}_i^t-\mathrm{\δ}{\stackrel{\→}{u}}_j^t)\]$

Wherein:For the relative speed between adjacent block,For global coordinate and the transition matrix of local coordinate system；

${\stackrel{\→}{T}}_k^g=\left[\begin{array}{cc}{\mathrm{cos\α}}_k& -{\mathrm{sin\α}}_k\\ {\mathrm{sin\α}}_k& {\mathrm{cos\α}}_k\end{array}\right]$

Can be abbreviated as:

Wherein:α_kIt is the inclining of mortar aspect between two Rigid Body Elements Angle, is just counterclockwise；

The Plastic Flow constraints of b mortar aspect

Assuming that stone block will not destroy, therefore Plastic Flow occurs over just in the contact surface mortar aspect between stone, and false If screed face thickness is zero, on contact edge, discontinuous normal direction and tangential velocity interruption value meet flowing criterion:

σ_n,τ_sIt is respectively the normal stress in mortar aspect and tangential stress,For the angle of friction of mortar aspect, c is mortar aspect Cohesiveness, f₁(σ_n,τ_s)、f₂(σ_n,τ_s) it is the Mohr-Coulomb yield function of mortar aspect；

Can obtain further

Δv_n,Δu_sFor the velocity discontinuity value of stone masonry aspect,For plasticity multiplier；

Can be abbreviated as:

For plasticity multiplier；

According to the associated flow rule of the theory of plasticity, for ideal epistemology model, deformation compatibility condition obtain generalized strain Component should obtain generalized plasticity strain rate component equal to by associated flow rule and yield condition:With Shi Yaoqiu plasticity multiplier is non-negative:

C internal strength power condition equal with external work power

Being learnt by the principle of virtual work, virtual power and the dissipated power of object internal energy that external force is done are equal, then have

${\∫}_{Q^{*}}{\mathrm{\σ}}_{ij}^{*}{\mathrm{\ϵ}}_{ij}^{*}{\mathrm{d\Ω}}^{*}+{\∫}_{{\mathrm{\Γ}}^{*}}{\mathrm{\σ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{\ϵ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{d\Γ}}^{*}={\mathrm{WV}}^{*}+{\mathrm{QV}}^{*}$

It is continuum internal stress vector,It is the virtual strain within continuum,It is border upper stress vector,It it is border On virtual strain, W be block deadweight, Q is face power load, V^*For virtual velocity；The equation left side respectively results from stone masonry barricade Neutralize along the internal dissipation power destroyed along sliding surface, on the right of equation, be respectively block deadweight W, borderline power load Q Equivalent load is in virtual velocity V^*On power；

According to stone-laying stone be rigidity it is assumed that the Rigid Body Element of stone will not deform and destroy, the dissipation of internal strength is only Result from the interface (i.e. mortar aspect) between Rigid Body Element, the therefore internal strength power within continuum

${\∫}_{{\mathrm{\Γ}}^{*}}{\mathrm{\σ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{\ϵ}}_{{\mathrm{\Γ}}^{*}}^{*}{\mathrm{d\Γ}}^{*}=\underset{z\→0}{\lim }{\∫}_0^z{\∫}_0^l({\mathrm{\σ}}_n{\stackrel{\·}{\mathrm{\ϵ}}}_n+{\mathrm{\τ}}_s{\stackrel{\·}{\mathrm{\γ}}}_s)dtds={\∫}_0^{l}c({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)dl=cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)$

σ_n,τ_sIt is respectively the normal stress in mortar aspect and tangential stress,It is respectively the normal direction normal strain in mortar aspect Rate and tangential shear strain rate,It is border upper stress vector,Being borderline virtual strain, l is the length of mortar aspect, c For the cohesiveness of mortar aspect,For plasticity multiplier；

According to Theory of Rankine Active Earth Pressure, the active soil of the retaining walls back of the body of any point at distance bankets case depth Z Intensity of pressure p_akExpression formula be:

$p_{ak}={\mathrm{\σ}}_{ak}{k}_a-2c^{\′}\sqrt{k_a}$

It is coefficient of active earth pressure, σ_ak=γ Z+q₀, it is the vertical stress at degree of depth Z,C' is The shearing strength of rolling earth behind retaining wall；

The face power load Q then banketed on surface-boundary can be written as:

$Q=\underset{\mathrm{\Γ}}{\∫}{p}_{ak}=\underset{\mathrm{\Γ}}{\∫}\left(\right(\mathrm{\γ}Z+q_0)k_a-2c^{\′}\sqrt{k_a})$

Γ is that retaining walls carries on the back integral boundary；

Consider further that the deadweight W of stone block, then can obtain:

$cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)={\mathrm{WV}}^{*}+\left(\underset{\mathrm{\Γ}}{\∫}\right(\left(\mathrm{\γ}Z+q_0\right)k_a-2c^{\′}\sqrt{k_a}\left)\right)V^{*};$

D velocity boundary conditions

Speed in stone masonry retaining wall is that the velocity boundary conditions on the border b of zero is:

$\mathrm{\δ}{\stackrel{\→}{q}}_j^t={\stackrel{\→}{T}}_j^b\left(\mathrm{\δ}{\stackrel{\→}{u}}_i^t\right)=0$

For the coordinate conversion matrix of interface j on the b of border；

(4) the upper bound method nonlinear mathematics programming model of stone masonry retaining wall ultimate bearing capacity is solved

During finding limit load, by unlimited evenly load q on retaining wall rolling earth behind retaining wall surface₀It is set to object function, then solves stone masonry The upper bound method nonlinear mathematics programming model of retaining wall ultimate bearing capacity is:

$\left.\begin{array}{cc}Minimize:& q_0\\ Subjectto:& \stackrel{\→}{D}\mathrm{\δ}\stackrel{\→}{u}-{\stackrel{\→}{N}}_0\mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}=0\\ & \mathrm{\δ}\stackrel{\→}{\mathrm{\λ}}\≥0\\ & cl({\stackrel{\·}{k}}_1+{\stackrel{\·}{k}}_2)={\mathrm{WV}}^{*}+\left(\underset{\mathrm{\Γ}}{\∫}\right(\left(\mathrm{\γ}Z+q_0\right)k_a-2c^{\′}\sqrt{k_a}\left)\right)V^{*}\\ & \mathrm{\δ}{\stackrel{\→}{q}}_j^t=0\end{array}\right\}.$