CN106874649A - A kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION - Google Patents

Abstract

The invention provides a kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION, it is characterized in, the INSTABILITY CRITERION is by continuous reduction intensive parameter, i.e. cohesive strength c and angle of friction tangent valueWhen the side slope surface curve under the limiting condition being calculated by slip line field theory and Slope upper thread intersect at toe, judge that side slope is in state of limit equilibrium, reduction coefficient now is safety coefficient, intensive parameter after reduction and side slope geometry and boundary condition are updated in finite difference method, critical slip surface is determined by the maximum plastic strain insertion band being calculated.The INSTABILITY CRITERION provides a new analysis method to evaluate slope stability.With scientific and reasonable, engineering practical value is high, the advantages of effect is good.

Description

A kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION

Technical field

The invention belongs to Slope Stability Evaluation field, and in particular to a kind of homogeneous slope stability_intensity reduction method unstability Criterion.

Background technology

Landslide disaster directly influences the various aspects of national economy, social sustainable development and people life, can cause Serious life and property loss.Slope Stability Evaluation is to judge whether whether landslide disaster occurs, need reinforcement and what is taken The Main Basiss of prophylactico-therapeutic measures are planted, therefore the research of Method for Slope Stability Analysis is a great theory value and important practice The research topic of meaning.Strength Reduction Method developed in recent years reaches side slope by constantly reducing Rock And Soil intensive parameter To state of limit equilibrium, so that sliding surface position and side slope safety factor of strength reserve are directly obtained, because the method need not vacation If critical slip surface, therefore as the focus of research.The key issue of Strength Reduction Method is how to judge side slope in certain reduction The determination of instability status, i.e. INSTABILITY CRITERION is on the verge of under coefficient, this is that Strength Reduction Method is applied to Analysis of Slope Stability Famous problem.Current INSTABILITY CRITERION mainly has following three kinds of methods：1. calculate and do not restrain, 2. key point displacement mutation, 3. mould Property area's insertion, the deficiency of above INSTABILITY CRITERION is the absence of objective quantification index, and analysis result is easily affected by human factors, some feelings Inevitably cause mistake under condition.Therefore the INSTABILITY CRITERION on slope stability Strength Reduction Method, at present can only be young Thin observation avoids erroneous judgement, or uses trial-and-error method value.INSTABILITY CRITERION not only determines the result of calculation of safety coefficient, and to true Determine critical slip surface also to have an impact, because plastic zone connection is the necessary not a sufficient condition of slope failure, therefore it is determined that critical , it is necessary to clear and definite side slope is in limiting condition during slip-crack surface, the plastic strain insertion band only under limiting condition could be used as critical Slip-crack surface, and limiting condition depends on the evaluation result of INSTABILITY CRITERION.Side Slope Safety Coefficient is calculated in application Strength Reduction Method With determine critical slip surface when, due to however wreck and subtract intensive parameter, the process is a continuous iterative process, thus relative to Time spent by other methods is more long.

The content of the invention

In view of the shortcomings of the prior art, scientific and reasonable it is an object of the invention to provide one kind, engineering practical value is high, effect Really good homogeneous slope stability_intensity reduction method INSTABILITY CRITERION, can be by the limit slope that is calculated by slip line field theory Whether surface curve meets at basis for estimation of the toe as slope stability with Slope upper thread.

To achieve the above object, the technical solution adopted by the present invention is：A kind of homogeneous slope stability_intensity reduction method is lost Steady criterion, it is characterized in that, it includes herein below：

1) safety coefficient is calculated：

(1) reduction is carried out to intensive parameter by formula (1) first,

C is cohesive strength in formula,It is angle of friction, F_iIt is reduction coefficient, i is natural number；

(2) slip-line field is calculated, if slope top is the origin of coordinates, slopes side is x-axis positive axis, is downwards y-axis positive axis,

M_α(x_α, y_α, θ_α, σ_α) it is point on α races skid wire, M_β(x_β, y_β, θ_β, σ_β) it is point on β races skid wire, adopt With the intensive parameter c after reduction₁WithPoint M (x, y, θ, σ) on slip-line field is solved by formula (2-1)~(2-4), in formulaIt is two race's slip interfacial angle average values, wherein：X transverse and longitudinal coordinate values, y is ordinate value, and θ is maximum principal stress σ₁With The x-axis angle of cut, σ is characterized stress：

(3) the domatic curve of calculating limit：M_b(x_b, y_b, θ_b, σ_b) it is the domatic curve known point of the limit, M '_β(x′_β, y '_β, θ '_β, σ′_β) it is known point on β races skid wire, using the intensive parameter c after reduction₁WithCan be solved by formula (3-1)~(3-4) Point M on the domatic curve of the limit_ij(x_ij, y_ij, θ_ij, σ_ij)：

(4) conditions setting：Active region OAB boundary conditions：Active region α, β races known calculations points M_αAnd M_βHorizontal seat Mark x=Δ xi, Δ x are material calculation, and the smaller finite difference formulations of the numerical value are more accurate, and i is natural number, i=0~N₁, N₁For Step-length number, takes N₁=999, y=0, active region border maximum principal stress and the x-axis angle of cutActive region boundary characteristic stressP is carried for slope top loading, and skid wire intersection point calculation formula is (2-1)~(2-4)；Passive area OCD boundary conditions：Passively Skid wire intersection point calculation formula in area is still (2-1)~(2-4), and the domatic curve of the limit, i.e. OD lines using formula (3-1)~ (3-4), M_bFirst known point is exactly slope top origin numerical value, coordinate value (x_b, y_b)=(0,0),Transition region OBC boundary conditions：Transition region Skid wire intersection point calculation is still formula (2-1)~(2-4), and the feature stresses of transition region boundary point areWhereinK=0~N₂, Δ θ=θ_III-θ_I, N₂It is transition region point subdivision Number, it is to meet Δ θ >=0, then necessaryTherefore slope top loading carries minimum valueWhen side slope example During without external load, side slope slope top applies P_min, nowI.e. Δ θ=0, takes N₂=0；

(5) the domatic curve of calculating limit and bottom of slope ground baseline intersection point abscissa x₁, now transformed coordinate system, if toe is to sit Mark origin, is upwards y-axis positive axis, is x-axis positive axis to side slope body side bottom of slope ground baseline, works as x₁>When 0, judge at side slope In stable state, now increase F_i, and repeat step (1)~(4), work as x₁When=0, i.e. the domatic curve of the limit and Slope upper thread When intersecting at toe, judge that side slope is in limiting condition, now F_iIt is safety coefficient, works as x₁<When 0, judge that side slope is in destruction shape State；

(6) when side slope is without external load, it is necessary to be further applied load minimum value in Po DingThis is cunning The boundary condition of the domatic curve of line field theory calculating limit is moved, is also the boundary condition of INSTABILITY CRITERION, it is determined that critical slip surface When, to keep boundary condition constant, also to apply P on slope top_min；

2) critical slip surface is determined by finite difference calculus：

1. side slope computation model is set up according to side slope geometrical condition, constraints is side slope model both sides horizontal restraint, mould Type bottom is staff cultivation, and evenly load P is applied on slope top_min；

2. using safety coefficient as F_iIntensive parameter c after reduction₁WithComputational plasticity should in being updated to finite difference calculus Become, wherein maximum plastic strain insertion band is critical slip surface.

Compared with prior art, what is had has a kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION of the invention Beneficial effect is：

(1) INSTABILITY CRITERION objective standard quantification is realized, it is to avoid human factor influence；

(2) INSTABILITY CRITERION is calculated into safety coefficient is separately carried out with determination critical slip surface, and plastic strain insertion is side slope The necessary not a sufficient condition of unstability, but if first obtaining the limiting condition of side slope by INSTABILITY CRITERION, then is moulded by finite difference formulations Property strain insertion band, then now plastic strain insertion also meets adequate condition, therefore plastic strain insertion now is as critical Slip-crack surface；

(3) due to directly determining critical slip surface by the plastic strain insertion band under slope limit state, therefore need not Iterative calculation, operation time can be significantly reduced relative to existing Strength Reduction Method；

(4) its is scientific and reasonable, and engineering practical value is high, and effect is good.

Brief description of the drawings

Fig. 1 is：The domatic curve synoptic diagram of slip line field theory calculating limit；

Fig. 2 is：Examination paper a, material calculation Δ x=0.003, works as F_iWhen=0.85, x₁=4.014 INSTABILITY CRITERION safety system Number calculates figure；

Fig. 3 is：Examination paper a, material calculation Δ x=0.003, works as F_iWhen=1.015, x₁=0.0126 INSTABILITY CRITERION safety Coefficient calculates figure；

Fig. 4 is：Examination paper a, material calculation Δ x=0.003, works as F_iWhen=1.15, x₁=-3.2966 INSTABILITY CRITERION safety Coefficient calculates figure；

Fig. 5 is：Dilative angleWhen, examination paper a critical slip surfaces determine figure；

Fig. 6 is：Examination paper a critical slip surface Key for References；

Fig. 7 is：Examination paper b, material calculation Δ x=0.015, works as F_iWhen=1, x₁=8.1869 INSTABILITY CRITERION safety coefficient Calculate figure；

Fig. 8 is：Examination paper b, material calculation Δ x=0.015, works as F_iWhen=1.31, x₁=0.0527 INSTABILITY CRITERION safety system Number calculates figure；

Fig. 9 is：Examination paper b, material calculation Δ x=0.015, works as F_iWhen=1.5, x₁=-5.686 INSTABILITY CRITERION safety system Number calculates figure；

Figure 10 is：Dilative angleWhen, examination paper b critical slip surfaces determine figure；

Figure 11 is：Examination paper b critical slip surface Key for References；

Figure 12 is：INSTABILITY CRITERION calculates safety coefficient and determines critical slip surface flow chart；

Figure 13 is：x₁>When 0, side slope is in stable state schematic diagram；

Figure 14 is：x₁When=0, side slope is in limiting condition schematic diagram；

Figure 15 is：x₁<When 0, side slope is in collapse state schematic diagram.

Specific embodiment

Specific embodiment of the invention is described in further detail below in conjunction with the accompanying drawings.

A kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION of the invention, including herein below：

1) safety coefficient is calculated：

(1) reduction is carried out to intensive parameter by formula (1) first,

C is cohesive strength in formula,It is angle of friction, F_iIt is reduction coefficient, i is natural number；

(2) slip-line field is calculated, if slope top is the origin of coordinates, slopes side is x-axis positive axis, is downwards y-axis positive axis,

M_α(x_α, y_α, θ_α, σ_α) it is point on α races skid wire, M_β(x_β, y_β, θ_β, σ_β) it is point on β races skid wire, adopt With the intensive parameter c after reduction₁WithPoint M (x, y, θ, σ) on slip-line field is solved by formula (2-1)~(2-4), in formulaIt is two race's slip interfacial angle average values, wherein：X transverse and longitudinal coordinate values, y is ordinate value, and θ is maximum principal stress σ₁With x Crossed axis angle, σ is characterized stress：

(3) the domatic curve of calculating limit：M_b(x_b, y_b, θ_b, σ_b) it is the domatic curve known point of the limit, M '_β(x′_β, y '_β, θ '_β, σ′_β) it is known point on β races skid wire, using the intensive parameter c after reduction₁WithCan be solved by formula (3-1)~(3-4) Point M on the domatic curve of the limit_ij(x_ij, y_ij, θ_ij, σ_ij)：

(4) conditions setting：Active region OAB boundary conditions：Active region α, β races known calculations points M_αAnd M_βHorizontal seat Mark x=Δ xi, Δ x are material calculation, and the smaller finite difference formulations of the numerical value are more accurate, and i is natural number, i=0~N₁, N₁For Step-length number, takes N₁=999, y=0, active region border maximum principal stress and the x-axis angle of cutActive region boundary characteristic stressP is carried for slope top loading, and skid wire intersection point calculation formula is (2-1)~(2-4)；Passive area OCD boundary conditions：Passively Skid wire intersection point calculation formula in area is still (2-1)~(2-4), and the domatic curve of the limit, i.e. OD lines using formula (3-1)~ (3-4), M_bFirst known point is exactly slope top origin numerical value, coordinate value (x_b, y_b)=(0,0),Transition region OBC boundary conditions：Transition region Skid wire intersection point calculation is still formula (2-1)~(2-4), and the feature stresses of transition region boundary point areWhereinK is natural number, k=0~N₂, Δ θ=θ_III-θ_I, N₂It is mistake Area's point subdivision number is crossed, it is to meet Δ θ >=0, then necessaryTherefore slope top loading carries minimum value When side slope example is without external load, side slope slope top applies P_min, nowI.e. Δ θ=0, takes N₂=0；

(5) the domatic curve of calculating limit and bottom of slope ground baseline intersection point abscissa x₁, now transformed coordinate system, if toe is to sit Mark origin, is upwards y-axis positive axis, is x-axis positive axis to side slope body side bottom of slope ground baseline, works as x₁>When 0, judge at side slope In stable state, now increase F_i, and repeat step (1)~(4), work as x₁When=0, i.e. the domatic curve of the limit and Slope upper thread When intersecting at toe, judge that side slope is in limiting condition, now F_iIt is safety coefficient, works as x₁<When 0, judge that side slope is in destruction shape State；

(6) when side slope is without external load, it is necessary to be further applied load minimum value in Po DingThis is cunning The boundary condition of the domatic curve of line field theory calculating limit is moved, is also the boundary condition of INSTABILITY CRITERION, it is determined that critical slip surface When, to keep boundary condition constant, also to apply P on slope top_min；

2) critical slip surface is determined by finite difference calculus：

1. side slope computation model is set up according to side slope geometrical condition, constraints is side slope model both sides horizontal restraint, mould Type bottom is staff cultivation, and evenly load P is applied on slope top_min；

2. using safety coefficient as F_iIntensive parameter c after reduction₁WithComputational plasticity should in being updated to finite difference calculus Become, wherein maximum plastic strain insertion band is critical slip surface.

Table 1 is that the safety coefficient of Australia ACM (ACADS) twice side slope examination paper is answered with the reference of critical slip surface Case (old ancestral's illuminate Soil Slope Stability Analysis:Principle Method program [M] Beijing:Chinese Water Conservancy water power publishing house, 2003: 371.) contrast verification INSTABILITY CRITERION of the present invention, is can be used to calculate safety coefficient and determine the correctness of critical slip surface.

The embodiment of the present invention twice homogeneous slope examination paper calculating parameter of table 1

1. first example is examination paper a, and calculating parameter is shown in Table 1, by the calculation flow chart of Figure 12, first by INSTABILITY CRITERION meter Safety coefficient is calculated, material calculation Δ x=0.003 is taken to this example, work as F_iWhen=0.85, x₁=4.014, as shown in Figure 2, now x₁>0, increase reduction coefficient F_i, work as F_iWhen=1.015, x₁=0.0126, as shown in Figure 3, now x₁=0, such as continue to increase reduction Coefficient F_i, work as F_iWhen=1.15, x₁=-3.2966, as shown in Figure 4.Comparison diagram 13- Figure 15 may determine that Side Slope Safety Coefficient is F_i=1.015, the model answer of examination paper a Side Slope Safety Coefficients as shown in Table 1 is F=0.99~1, it is seen that unstability of the invention is sentenced According to correct safety coefficient answer can be given.

Critical slip surface is determined by Figure 12 flow charts, strength degradation parameter is calculated first：

Slope top loading is calculated to carry

Side slope model is set up by the examination paper a side slope geometric parameters of table 1, its mesoslope H=10m high, slope angle α=26.6 °, both sides water Flat constraint, the P that is further applied load is pushed up on bottom of slope staff cultivation, slope_amin, there are some researches show associated flow ruleFlowed with dereferenced Rule (ψ=0) has the inapplicable situation for determining critical slip surface, therefore the present invention takes dilative anglePass through Finite-difference program calculating maximum plastic strain insertion band, as critical slip surface, as shown in Figure 5；Examination paper a critical slip surfaces are joined Examine answer as shown in Figure 6；Analysis by contrast understands, the shape of the critical slip surface that INSTABILITY CRITERION of the invention is given and in side slope In position it is basically identical with existing Key for Reference, it is seen that INSTABILITY CRITERION of the invention can provide correct critical slip surface.

2. second example is examination paper b, and calculating parameter is shown in Table 1, by the calculation flow chart of Figure 12, first by INSTABILITY CRITERION meter Safety coefficient is calculated, Δ x=0.015 is taken to this example, work as F_iWhen=1, x₁=8.1869, as shown in Figure 7, now x₁>0, increase folding Subtract coefficient F_i, work as F_iWhen=1.31, x₁=0.0527, as shown in Figure 8, now x₁=0, such as continue to increase reduction coefficient F_i, work as F_i When=1.5, x₁=-5.686, as shown in Figure 9.Comparison diagram 13- Figure 15 may determine that Side Slope Safety Coefficient is F_i=1.31, by table 1 The Key for Reference for understanding examination paper b Side Slope Safety Coefficients is F=1.5~1.7, the safety coefficient phase that INSTABILITY CRITERION of the invention is given It is less than normal for model answer.

Reason is：The intensive parameter reduction value of examination paper b is：

And slope top loading is carried

Contrast is carried with the slope top loading of examination paper a understand P_bmin>P_amin, slope top loading carry it is bigger, slope stability is lower, therefore examines The safety coefficient for inscribing b is less than normal than model answer, but is in practice relatively safe.

Critical slip surface is determined by Figure 12 flow charts, side slope model is set up by the examination paper b side slope geometric parameters of table 1, wherein Push up the P that is further applied load in slope H=10m high, slope angle α=26.6 °, both sides horizontal restraint, bottom of slope staff cultivation, slope_bmin, dilative angleMaximum plastic strain insertion band is calculated by finite-difference program, as critical slip surface is shown in Figure 10 institutes Show；Examination paper b critical slip surfaces Key for Reference is as shown in Figure 11.Comparative analysis understands that it is critical that INSTABILITY CRITERION of the invention is provided The shape of slip-crack surface is basically identical with existing Key for Reference, but position in side slope is partially deeper than Key for Reference, for example preceding institute of reason State, because being applied with external load P in Po Ding_bmin, the critical slip surface that INSTABILITY CRITERION of the invention is given is reliable.

By two side slope examination paper, INSTABILITY CRITERION of the invention can provide reliable safety coefficient with critical slip Face, due to relative to former boundary condition, as the external load P for applying_minWhen less than normal, safety coefficient and critical slip surface are answered with reference Case is consistent, as the external load P for applying_minWhen bigger than normal, safety coefficient is less than normal relative to existing conclusion, and critical slip surface is partially deep, but It is in practice relatively safe.

The calculating process of twice examination paper shows that INSTABILITY CRITERION of the invention gives the visitor for judging homogeneous slope limiting condition Sight standard, that is, work as x₁When=0, corresponding reduction parameter F_iIt is safety coefficient, it is to avoid the influence of human factor；Directly calculate side Slope is in the maximum plastic strain under limiting condition, and now plastic strain insertion meets the sufficient and necessary condition of slope instability, The plastic strain insertion band as critical slip surface now pushed up from toe to slope, and need not iterate to calculate, relative to strong Degree reduction method substantially saves time cost.

Finally it should be noted that the above embodiments are merely illustrative of the technical solutions of the present invention rather than its limitations, to the greatest extent Pipe has been described in detail with reference to above-described embodiment to the present invention, it should be understood by a person of ordinary skill in the art that still Specific embodiment of the invention can be modified or equivalent, and without departing from any of spirit and scope of the invention Modification or equivalent, it all should cover in the middle of scope of the presently claimed invention.

Claims (1)

1. a kind of homogeneous slope stability_intensity reduction method INSTABILITY CRITERION, it is characterized in that, it includes herein below：

1) safety coefficient is calculated：

(1) reduction is carried out to intensive parameter by formula (1) first,

C is cohesive strength in formula,It is angle of friction, F_iIt is reduction coefficient, i is natural number；

(2) slip-line field is calculated, if slope top is the origin of coordinates, slopes side is x-axis positive axis, is downwards y-axis positive axis,

M_α(x_α, y_α, θ_α, σ_α) it is point on α races skid wire, M_β(x_β, y_β, θ_β, σ_β) it is point on β races skid wire, using folding Intensive parameter c after subtracting₁WithPoint M (x, y, θ, σ) on slip-line field is solved by formula (2-1)~(2-4), in formulaIt is two race's slip interfacial angle average values, wherein：X transverse and longitudinal coordinate values, y is ordinate value, and θ is maximum principal stress σ₁With The x-axis angle of cut, σ is characterized stress：

$x=\frac{x_{\mathrm{\&alpha;}}\&CenterDot;tan({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}-\mathrm{\&mu;})-x_{\mathrm{\&beta;}}\&CenterDot;tan({\mathrm{\&theta;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})-(y_{\mathrm{\&alpha;}}-y_{\mathrm{\&beta;}})}{tan({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}-\mathrm{\&mu;})-tan({\mathrm{\&theta;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})}---(2-1)$

$\left\{\begin{array}{c}y=(x-x_{\mathrm{\&alpha;}})\&CenterDot;\tan ({\mathrm{\&theta;}}_{\mathrm{\&alpha;}}-\mathrm{\&mu;})+y_{\mathrm{\&alpha;}}\\ y=(x-x_{\mathrm{\&beta;}})\&CenterDot;\tan ({\mathrm{\&theta;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})+y_{\mathrm{\&beta;}}\end{array}\right.---(2-2)$

(3) the side slope surface curve under calculating limit state, the domatic curve of the abbreviation limit of the present invention：M_b(x_b, y_b, θ_b, σ_b) it is pole Limit domatic curve known point, M '_β(x′_β, y '_β, θ '_β, σ '_β) it is known point on β races skid wire, joined using the intensity after reduction Number c₁WithPoint M on the domatic curve of the limit can be solved by formula (3-1)~(3-4)_ij(x_ij, y_ij, θ_ij, σ_ij)：

$x_{ij}=\frac{x_b\&CenterDot;{\mathrm{tan\&theta;}}_b-{x^{\&prime;}}_{\mathrm{\&beta;}}\&CenterDot;\tan ({{\mathrm{\&theta;}}^{\&prime;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})+({y^{\&prime;}}_{\mathrm{\&beta;}}+y_b)}{{\mathrm{tan\&theta;}}_b-\tan ({{\mathrm{\&theta;}}^{\&prime;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})}---(3-1)$

$\left\{\begin{array}{c}{y}_{ij}=(x_{ij}-x_b)\&CenterDot;{\mathrm{tan\&theta;}}_b+y_b\\ y_{ij}=(x_{ij}-{x^{\&prime;}}_{\mathrm{\&beta;}})\&CenterDot;\tan ({{\mathrm{\&theta;}}^{\&prime;}}_{\mathrm{\&beta;}}+\mathrm{\&mu;})+{y^{\&prime;}}_{\mathrm{\&beta;}}\end{array}\right.---(3-2)$

(4) conditions setting：Active region OAB boundary conditions：Active region α, β races known calculations points M_αAnd M_βAbscissa x= Δ xi, Δ x are material calculation, and the smaller finite difference formulations of the numerical value are more accurate, and i is natural number, i=0~N₁, N₁It is step-length Number, takes N₁=999, y=0, active region border maximum principal stress and the x-axis angle of cutActive region boundary characteristic stressP is carried for slope top loading, and skid wire intersection point calculation formula is (2-1)~(2-4)；Passive area OCD boundary conditions：Quilt Dynamic area's skid wire intersection point calculation formula is still (2-1)~(2-4), and the domatic curve of the limit, i.e. OD lines using formula (3-1)~ (3-4), M_bFirst known point is exactly slope top origin numerical value, coordinate value (x_b, y_b)=(0,0),Transition region OBC boundary conditions：Transition region Skid wire intersection point calculation is still formula (2-1)~(2-4), and the feature stresses of transition region boundary point areWhereinK=0~N₂, Δ θ=θ_III-θ_I, N₂It is transition region point subdivision Number, it is to meet Δ θ >=0, then necessaryTherefore slope top loading carries minimum valueWhen side slope example During without external load, side slope slope top applies P_min, nowI.e. Δ θ=0, takes N₂=0；

(5) the domatic curve of calculating limit and bottom of slope ground baseline intersection point abscissa x₁, now transformed coordinate system, if toe is former coordinate Point, is upwards y-axis positive axis, is x-axis positive axis to side slope body side bottom of slope ground baseline, works as x₁>When 0, judge that side slope is in steady Determine state, now increase F_i, and repeat step (1)~(4), work as x₁When=0, i.e. the domatic curve of the limit intersects with Slope upper thread When toe, judge that side slope is in limiting condition, now F_iIt is safety coefficient, works as x₁<When 0, judge that side slope is in collapse state；

(6) when side slope is without external load, it is necessary to be further applied load minimum value in Po DingThis is skid wire The boundary condition of the domatic curve of field theory calculating limit, is also the boundary condition of INSTABILITY CRITERION, it is determined that during critical slip surface, being Keep boundary condition constant, also to apply P on slope top_min；

2) critical slip surface is determined by finite difference calculus：

1. side slope computation model is set up according to side slope geometrical condition, constraints is side slope model both sides horizontal restraint, model bottom Portion is staff cultivation, and evenly load P is applied on slope top_min；

2. using safety coefficient as F_iIntensive parameter c after reduction₁WithIt is updated in finite difference calculus and calculates plastic strain, its Middle maximum plastic strain insertion band is critical slip surface.