CN112711868A - Pseudo-static method for calculating power safety coefficient of homogeneous slope under earthquake action - Google Patents

Pseudo-static method for calculating power safety coefficient of homogeneous slope under earthquake action Download PDF

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CN112711868A
CN112711868A CN202110151671.8A CN202110151671A CN112711868A CN 112711868 A CN112711868 A CN 112711868A CN 202110151671 A CN202110151671 A CN 202110151671A CN 112711868 A CN112711868 A CN 112711868A
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方宏伟
张雨慧
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Shanghai Haojingshui Information Technology Center (limited partnership)
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Abstract

The invention provides a quasi-static method for calculating the power safety coefficient of a homogeneous slope under the action of an earthquake, which is characterized in that intensity parameters (cohesive force and internal friction angle) are reduced to obtain reduced intensity parameters; according to the earthquake action stress balance equation set and the Morkolem criterion stress expression, a characteristic line method is applied to derive two groups of characteristic line differential equation sets of a glide line field, and the differential equation sets are solved by adopting a finite difference method according to the dynamic boundary conditions of active, transition and passive regions under the earthquake action to obtain a glide line field under the earthquake action and a slope curve (power limit slope curve for short) of a side slope under a power limit state; different power limit slope curves can be calculated according to different reduction strength parameters, the dynamic stability of the side slope under the earthquake action is judged according to the positive and negative of the abscissa of the intersection point of the power limit slope curve and the slope bottom, when the abscissa of the intersection point is zero, the side slope is judged to be in a power limit state, and the reduction coefficient is the power safety coefficient.

Description

Pseudo-static method for calculating power safety coefficient of homogeneous slope under earthquake action
Technical Field
The invention belongs to the field of slope stability evaluation, and particularly relates to a quasi-static method for calculating the power safety coefficient of a homogeneous slope under the action of an earthquake.
Background
China is a country with frequent earthquakes, particularly in mountainous areas and hilly lands, landslides caused by the action of earthquakes often have the characteristics of wide distribution, large quantity, large harm and the like, so that the problem of slope stability under the action of earthquake loads is always a research difficulty and a hotspot in the geotechnical and earthquake engineering circles. The pseudo-static method is simple in calculation and high in practicability, and is brought into corresponding specifications to be applied to slope dynamic stability analysis. The essence of the quasi-static method is that the earthquake inertia force is regarded as the static load and is applied to the slope body, and then the dynamic safety coefficient of the slope under the earthquake action is calculated by adopting a limit balance bar method or a strength reduction finite element method.
When the extreme balance bar quasi-static method is adopted, two problems mainly exist: one is that when the pseudo-static force strip division method divides the soil strips of the slope rock-soil mass, the pseudo-static force method generates errors even unreasonable results are obtained due to different strip division directions; the other is that a critical sliding surface form needs to be assumed or searched, and the power safety coefficient calculation formula considering the earthquake inertia force has a large difference to the results obtained by different critical sliding surface forms, so that the engineering practice application is not facilitated. When the numerical calculation is carried out by adopting the intensity reduction finite element method, although the assumption or search of a critical sliding surface is not needed, how to determine that the slope is in a power limit state is a difficult problem, namely the determination of the instability criterion is a difficult problem, and the slope instability criterion mainly comprises the calculation unconvergence criterion, the displacement mutation criterion and the plastic region through criterion at present. The complexity of a seismic inertia force slope numerical calculation model, the existence of nonlinear problems and other factors can cause calculation non-convergence, the selection of the displacement mutation characteristic point position has no unified standard, sometimes the inflection point of a characteristic point displacement curve is not obvious, subjective factors can exist when the mutation point is judged manually, and the penetration of a plastic zone is a necessary and insufficient condition for slope damage.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide the homogeneous slope dynamic safety coefficient quasi-static method under the earthquake action, which is scientific and reasonable, high in engineering practical value and good in effect.
In order to achieve the purpose, the invention adopts the technical scheme that:
1. a quasi-static method for calculating the power safety coefficient of a homogeneous slope under the action of an earthquake is characterized by comprising the following steps:
1) and (3) reducing the strength parameter:
Figure BSA0000232852750000011
in the formula c0In order to be the initial cohesive force,
Figure BSA0000232852750000012
is an initial angle of friction, clIn order to reduce the cohesive force after the folding,
Figure BSA0000232852750000013
for the post-reduction friction angle, F is the reduction coefficient, Δ F is the increase in the reduction coefficient, and l is a natural number 1, 2, 3.
2) Formula for calculating slip line field by pseudo-static method
Under the action of earthquake, the stress balance equation set is as follows:
Figure BSA0000232852750000021
Figure BSA0000232852750000022
in the formula sigmaxAnd σyDenotes positive stress in x and y directions, respectively, τxyAnd τyxDenotes shear stress in x and y directions, respectively, fx＝γ·kH，fy＝γ·(1-kV) γ represents volume weight, kHAnd kVRepresenting horizontal and vertical seismic coefficients, k, respectivelyV＝ξ·kHAnd xi is a scaling factor.
To give an expression for the normal stress as well as the shear stress in the molar coulomb criterion, the formula for the characteristic stress σ is introduced:
Figure BSA0000232852750000023
the normal and shear stress expressions at this time are:
Figure BSA0000232852750000024
Figure BSA0000232852750000025
in which theta is the maximum principal stress sigma1Intersecting the x-axis at an angle.
And (3) simultaneously substituting the expressions (5) and (6) into the expressions (2) and (3), and obtaining characteristic line differential equations of two families (alpha and beta families) of the glide slope field theory under the action of the earthquake according to a characteristic line method:
Figure BSA0000232852750000026
Figure BSA0000232852750000027
in the formula
Figure BSA0000232852750000028
The mean value of the intersection angles of the two families of slip lines is obtained.
The differential method is adopted to approximately solve the characteristic line equations (7) and (8),
Figure BSA0000232852750000029
Figure BSA00002328527500000210
in the formula Mα(xα，yα，θα，σα) Is a point on the alpha group, Mβ(xβ，yβ，θβ，σβ) Is a point in the beta family, (x, y) are coordinate values,
Figure BSA0000232852750000031
and
Figure BSA0000232852750000032
calculating a point M (x, y, theta, sigma) to be found on the slip line simultaneously by the following equations (9) and (10):
Figure BSA0000232852750000033
Figure BSA0000232852750000034
Figure BSA0000232852750000035
Figure BSA0000232852750000036
the slope curve under the slope dynamic limit state (called dynamic limit slope curve for short) differential equation obtained by the theoretical calculation of the slip line field under the action of earthquake is as follows:
Figure BSA0000232852750000037
the coordinate point M of the curve of the power limit slope can be solved by adopting a finite difference method in conjunction with the equation (8) of the slip line of the beta groupij(xij，yij，θij，σij)：
Figure BSA0000232852750000038
Figure BSA0000232852750000039
Figure BSA00002328527500000310
Figure BSA00002328527500000311
In the formula Mb(xb，yb，θb，σb) And M'β(xβ′，y′β，θ′β，σ′β) The known points of the power limit slope curve and the beta slip line are shown.
3) Pseudo-static method slip line field boundary condition
(1) Active region O1AB boundary conditions
Known calculation point M of alpha and beta families of active regionαAnd Mβ(x, y) is the crest O1A coordinate value, where x is Δ x · i, Δ x is a calculation step, i is a natural number, and i is 0 to N1，N1Step length is the number of steps, the vertical coordinate is the slope height, and the intersection angle of the maximum principal stress of the boundary of the active region and the x axis is as follows:
Figure BSA00002328527500000312
in the formula
Figure BSA0000232852750000041
For the stress deflection angle caused by earthquake dynamic force, according to the formula, the stress deflection angle and the friction angle must satisfy the relational expression
Figure BSA0000232852750000042
Characteristic stress of active zone boundary:
Figure BSA0000232852750000043
formula middle slope top power load
Figure BSA0000232852750000044
Dynamic positive stress sigma0＝P0·(1-kV) Dynamic shear stress tau0＝P0·kH，P0For static load of top of slope，
Figure BSA0000232852750000045
The intersection calculation formulas of the slip lines are (11) - (14);
(2) transition zone O1BC boundary condition
Known boundary point O of transition zone1And (x, y) is a slope shoulder coordinate value, and the characteristic stress is as follows:
Figure BSA0000232852750000046
in the formula
Figure BSA0000232852750000047
k is a natural number, k is 0 to N2，Δθ＝θIII-θI，N2The intersection calculation formula of the transition region sliding lines is (11) - (14) for the subdivision number of the transition region points;
(3) passive region O1CD boundary condition
MbThe first known point is the shoulder O1The (x, y) coordinate value of (c), but the characteristic stress value is
Figure BSA0000232852750000048
By substituting the formula (21)
Figure BSA0000232852750000049
To satisfy Δ θ ≧ 0, θ is necessaryIII≥θIThus minimum value of dynamic load on the top of the hill
Figure BSA00002328527500000410
In this case, Δ θ is 0, the passive region slip line intersection calculation formulas are (11) to (14), and the dynamic limit slope curve OD is calculated using formulas (15) to (18).
4) Dynamic safety coefficient calculated by instability criterion of pseudo-static method
The intersection point of the curve of the power limit slope and the slope bottom is (x)10), based on the abscissa value x1Analysis of slope stability by pseudo-static method under positive and negative judgment seismic actionQualitative criterion of instability: when x is1When the slope is more than 0, the slope is judged to be in a stable state, the reduction coefficient F is increased, namely the increase value delta F is a positive value, and can be 0.01; when x is1When the speed is equal to 0, judging that the slope is in a power limit state, and at the moment, setting a power safety coefficient FS to be F; when x is1If the value is less than 0, the slope is judged to be in a destruction state, the reduction coefficient F is reduced, namely the increase value delta F is a negative value, and can be-0.01.
Compared with the prior art, the quasi-static method for calculating the power safety coefficient of the homogeneous slope under the action of the earthquake has the beneficial effects that:
(1) calculating to obtain a slope surface shape curve (called as a dynamic limit slope surface curve for short) in a dynamic limit state by deducing a slip line field theory under the action of an earthquake, judging the dynamic stability of the slope by the positive and negative of the abscissa of the intersection point of the dynamic limit slope surface curve and the slope bottom, and at the moment, dividing the slope rock-soil body into strips, so that the strip dividing direction of the rock-soil body is not required to be considered;
(2) the dynamic instability criterion of the stability of the homogeneous slope is given, and when the instability criterion is adopted to calculate the dynamic safety coefficient, the slope critical slip fracture surface does not need to be assumed and searched compared with the existing extreme balance bar quasi-static method;
(3) compared with the existing dynamic instability criterion of the slope intensity reduction method, the instability criterion does not need to consider the influence of calculating unconvergence, does not need to select slope characteristic points and judge the displacement reduction curve mutation points of the characteristic points, realizes the objective standard quantification of the dynamic instability criterion, and avoids the influence of artificial subjective factors;
(4) the method is scientific and reasonable, the engineering practical value is high, and the effect is good.
Drawings
FIG. 1 is a schematic diagram of: calculating a curve diagram of a dynamic limit slope surface by a glide slope field theory under the action of an earthquake;
FIG. 2 is a diagram of: the invention discloses a schematic diagram of dynamic instability criterion of a pseudo-static method;
FIG. 3 is a diagram of: the invention relates to a technical flow chart for calculating a power safety coefficient by using a pseudo-static method power instability criterion;
FIG. 4 is a diagram of: when k isH0.1, 0.5 (i.e. k)V0.05) and a reduction factor F11.3, homogeneous slope seismic glide slope line field theory (Δ x 0.5, N)1＝50、N25) calculating the power limit slope curve chart, and x can be obtained1＝5.4846；
FIG. 5 is a diagram of: when k isH0.1, 0.5 (i.e. k)V0.05) and a reduction factor F21.52, homogeneous slope seismic event glide slope field theory (Δ x 0.5, N)1＝50、N25) calculating the power limit slope curve chart, and x can be obtained1＝0；
FIG. 6 is a diagram of: when k isH0.1, 0.5 (i.e. k)V0.05) and a reduction factor F3When the value is 1.8, the sliding line field theory under the action of the homogeneous slope earthquake (delta x is 0.5, N)1＝50、N25) calculating the power limit slope curve chart, and x can be obtained1＝-9.4591；
FIG. 7 is a diagram of: and (4) calculating a result by a limit balance bar simulation static method (simplified bishop method).
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
The invention relates to a quasi-static method for calculating the power safety coefficient of a homogeneous slope under the action of an earthquake, which comprises the following steps:
1. a quasi-static method for calculating the power safety coefficient of a homogeneous slope under the action of an earthquake is characterized by comprising the following steps:
1) and (3) reducing the strength parameter:
Figure BSA0000232852750000051
in the formula c0In order to be the initial cohesive force,
Figure BSA0000232852750000052
is an initial angle of friction, clIn order to reduce the cohesive force after the folding,
Figure BSA0000232852750000053
for the post-reduction friction angle, F is the reduction coefficient, Δ F is the increase in the reduction coefficient, and l is a natural number 1, 2, 3.
2) Formula for calculating slip line field by pseudo-static method
The schematic diagram of the slope curve of the theoretical calculation dynamic limit of the glide slope field under the action of the earthquake is shown in figure 1.
Under the action of earthquake, the stress balance equation set is as follows:
Figure BSA0000232852750000061
Figure BSA0000232852750000062
in the formula sigmaxAnd σyDenotes positive stress in x and y directions, respectively, τxyAnd τyxDenotes shear stress in x and y directions, respectively, fx＝γ·kH，fy＝γ·(1-kV) γ represents volume weight, kHAnd kVRepresenting horizontal and vertical seismic coefficients, k, respectivelyV＝ξ·kHAnd xi is a scaling factor.
To give an expression for the normal stress as well as the shear stress in the molar coulomb criterion, the formula for the characteristic stress σ is introduced:
Figure BSA0000232852750000063
the normal and shear stress expressions at this time are:
Figure BSA0000232852750000064
Figure BSA0000232852750000065
in which theta is the maximum principal stress sigma1Intersecting the x-axis at an angle.
And (3) simultaneously substituting the expressions (5) and (6) into the expressions (2) and (3), and obtaining characteristic line differential equations of two families (alpha and beta families) of the glide slope field theory under the action of the earthquake according to a characteristic line method:
Figure BSA0000232852750000066
Figure BSA0000232852750000067
in the formula
Figure BSA0000232852750000068
The mean value of the intersection angles of the two families of slip lines is obtained.
The differential method is adopted to approximately solve the characteristic line equations (7) and (8),
Figure BSA0000232852750000069
Figure BSA00002328527500000610
in the formula Mα(xα，yα，θα，σα) Is a point on the alpha group, Mβ(xβ，yβ，θβ，σβ) Is a point in the beta family, (x, y) are coordinate values,
Figure BSA0000232852750000071
and
Figure BSA0000232852750000072
calculating a point M (x, y, theta, sigma) to be found on the slip line simultaneously by the following equations (9) and (10):
Figure BSA0000232852750000073
Figure BSA0000232852750000074
Figure BSA0000232852750000075
Figure BSA0000232852750000076
the slope curve (power is called as limit slope curve for short) differential equation under the slope power limit state calculated by the slip line field theory under the action of earthquake is as follows:
Figure BSA0000232852750000077
coordinate point M of power limit slope curve can be solved by combining with beta family slip line equation formula (8)ij(xij，yij，θij，σij)：
Figure BSA0000232852750000078
Figure BSA0000232852750000079
Figure BSA00002328527500000710
Figure BSA00002328527500000711
In the formula Mb(xb，yb，θb，σb) And M'β(x′β，y′β，θ′β，σ′β) The known points of the power limit slope curve and the beta slip line are shown.
3) Pseudo-static method slip line field boundary condition
(1) Active region O1AB boundary conditions
Known calculation point M of alpha and beta families of active regionαAnd Mβ(x, y) is the crest O1A coordinate value, where x is Δ x · i, Δ x is a calculation step, i is a natural number, and i is 0 to N1，N1Step length is the number of steps, the vertical coordinate is the slope height, and the intersection angle of the maximum principal stress of the boundary of the active region and the x axis is as follows:
Figure BSA00002328527500000712
in the formula
Figure BSA0000232852750000081
For the stress deflection angle caused by earthquake dynamic force, according to the formula, the stress deflection angle and the friction angle must satisfy the relational expression
Figure BSA0000232852750000082
Characteristic stress of active zone boundary:
Figure BSA0000232852750000083
formula middle slope top power load
Figure BSA0000232852750000084
Dynamic positive stress sigma0＝P0·(1-kV) Dynamic shear stress tau0＝P0·kH，P0Is the static load of the top of the slope,
Figure BSA0000232852750000085
the intersection calculation formulas of the slip lines are (11) - (14);
(2) transition zone O1BC boundary condition
Known boundary point O of transition zone1And (x, y) is a slope shoulder coordinate value, and the characteristic stress is as follows:
Figure BSA0000232852750000086
in the formula
Figure BSA0000232852750000087
k is a natural number, k is 0 to N2，Δθ＝θIII-θI，N2The intersection calculation formula of the transition region sliding lines is (11) - (14) for the subdivision number of the transition region points;
(3) passive region O1CD boundary condition
MbThe first known point is the shoulder O1The (x, y) coordinate value of (c), but the characteristic stress value is
Figure BSA0000232852750000088
By substituting the formula (21)
Figure BSA0000232852750000089
To satisfy Δ θ ≧ 0, θ is necessaryIII≥θIThus minimum value of dynamic load on the top of the hill
Figure BSA00002328527500000810
In this case, Δ θ is 0, the passive region slip line intersection calculation formulas are (11) to (14), and the dynamic limit slope curve OD is calculated using formulas (15) to (18).
4) Dynamic safety coefficient calculated by instability criterion of pseudo-static method
The intersection point of the curve of the power limit slope and the slope bottom is (x)10), based on the abscissa value x1The instability criterion for analyzing the slope stability by the pseudo-static method under the action of positive and negative judgment earthquake comprises the following steps: when x is1When the slope is more than 0, the slope is judged to be in a stable state, the reduction coefficient F is increased, namely the increase value delta F is a positive value, and can be 0.01; when x is1When the power safety factor FS is equal to 0, the slope is judged to be in the power limit stateF; when x is1If the value is less than 0, the slope is judged to be in a destruction state, the reduction coefficient F is reduced, namely the increase value delta F is a negative value, and can be-0.01.
The technical process of the invention is shown in figure 3, and table 1 shows the geometric and physical parameter values of a homogeneous side slope with 9 degrees of seismic intensity, according to the technical specification GB50330-2013 of building side slope engineering, the seismic intensity is 9 degrees, and the horizontal seismic coefficient k isHThe calculation example can be used for comparing and verifying the correctness of the dynamic safety coefficient calculated by the pseudo-static method instability criterion.
TABLE 1 homogeneous slope examination question calculation parameters (seismic intensity 9 degree) in the embodiment of the invention
Figure BSA0000232852750000091
*Technical Specification of building slope engineering GB50330-2013
According to the calculation flow chart 3, the horizontal earthquake dynamic coefficient kH0.1, 0.5 (i.e. vertical seismic dynamics coefficient k)V0.05), the theoretical boundary condition of the sliding line field under the action of the homogeneous slope earthquake is that deltax is 0.5, and N is1＝50、N2When the reduction factor F is 51When the power limit slope curve chart is calculated as 1.3, x can be obtained15.4846 (see fig. 4); reduction factor F2When 1.52, x is obtained10 (see fig. 5); reduction factor F3When 1.8, x is obtained1-9.4591 (see fig. 6); according to the pseudo-static force instability criterion (shown in figure 2), the obtained dynamic safety coefficient is FS (1.52), and the obtained dynamic safety coefficient is 1.636 (shown in figure 7) together with the calculation result of the extreme balance bar quasi-static force method (simplified bishop quasi-static force method), and the error percentage is 6.7%.
According to examination questions, the instability criterion of the pseudo-static method can provide reliable power safety coefficients, and the calculation process shows that the instability criterion of the pseudo-static method provides an objective standard for judging the power limit state of a homogeneous slope under the action of an earthquake, namely the abscissa x of the intersection point between a power limit slope curve and a slope bottom calculated by a sliding line field theory1When equal to 0, the corresponding reduction parameter F is a power safety systemThe number FS is counted, the situation that characteristic points need to be selected and characteristic displacement curve mutation points need to be judged artificially and subjectively in the traditional dynamic instability criterion of the slope intensity reduction method is avoided, and compared with the existing extreme balance bar quasi-static method, the instability criterion does not need to assume and search a critical slip crack surface under the dynamic condition.
Finally, it should be noted that the above-mentioned embodiments are only used for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the above-mentioned embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions can be made on the specific embodiments of the present invention without departing from the spirit and scope of the present invention, and all the modifications or equivalent substitutions should be covered in the claims of the present invention.

Claims (1)

1.一种计算地震作用下均质边坡动力安全系数的拟静力法，其特征是，它包括以下内容：1. A pseudo-static method for calculating the dynamic safety factor of a homogeneous slope under earthquake action, it is characterized in that it comprises the following content: 1)对强度参数进行折减：1) Reduce the strength parameter:
Figure FSA0000232852740000011
Figure FSA0000232852740000011
 式中c0为初始粘聚力，
Figure FSA0000232852740000012
为初始摩擦角，cl为折减后粘聚力，
Figure FSA0000232852740000013
为折减后摩擦角，F为折减系数，ΔF为折减系数增加值，l＝1，2，3......N为自然数。where c 0 is the initial cohesion,
Figure FSA0000232852740000012
is the initial friction angle, c l is the reduced cohesion,
Figure FSA0000232852740000013
is the friction angle after reduction, F is the reduction coefficient, ΔF is the increase value of the reduction coefficient, l=1, 2, 3...N is a natural number. 2)拟静力法滑移线场计算公式2) Calculation formula of quasi-static slip line field 地震作用力下，应力平衡方程组为：Under the seismic force, the stress balance equations are:
Figure FSA0000232852740000014
Figure FSA0000232852740000014
Figure FSA0000232852740000015
Figure FSA0000232852740000015
 式中σx和σy分别表示x和y方向正应力，τxy和τyx分别表示x和y方向剪应力，fx＝γ·kH，fy＝γ·(1-kV)，γ表示容重，kH和kV分别表示水平和竖直地震系数，kV＝ξ·kH，ξ为比例系数。where σ x and σ y represent the normal stress in the x and y directions, respectively, τ xy and τ yx represent the shear stress in the x and y directions, respectively, f x =γ·k H , f y =γ·(1-k V ), γ represents the bulk density, k H and kV represent the horizontal and vertical seismic coefficients, respectively, kV =ξ·k H , and ξ is the proportional coefficient. 为了给出摩尔库伦准则中正应力以及剪应力的表达式，引入特征应力σ的公式：In order to give the expressions of normal stress and shear stress in the Mohr-Coulomb criterion, the formula of characteristic stress σ is introduced:
Figure FSA0000232852740000016
Figure FSA0000232852740000016
 此时正应力和剪应力表达式为：The normal stress and shear stress are expressed as:
Figure FSA0000232852740000017
Figure FSA0000232852740000017
Figure FSA0000232852740000018
Figure FSA0000232852740000018
 式中θ为最大主应力σ1与x轴交角。where θ is the intersection angle between the maximum principal stress σ 1 and the x-axis. 将(5)和(6)式同时代入(2)和(3)式中，并根据特征线法，可得地震作用下滑移线场理论两族(α和β族)特征线微分方程：Substitute equations (5) and (6) into equations (2) and (3) at the same time, and according to the characteristic line method, the two families (α and β families) characteristic line differential equations of the slip line field theory under earthquake action can be obtained:
Figure FSA0000232852740000019
Figure FSA0000232852740000019
Figure FSA00002328527400000110
Figure FSA00002328527400000110
 式中
Figure FSA0000232852740000021
为两族滑移线交角平均值。in the formula
Figure FSA0000232852740000021
is the average value of the angle of intersection of the two slip lines. 采用差分法近似求解特征线方程(7)和(8)，The difference method is used to approximately solve the characteristic line equations (7) and (8),
Figure FSA0000232852740000022
Figure FSA0000232852740000022
Figure FSA0000232852740000023
Figure FSA0000232852740000023
 式中Mα(xα，yα，θα，σα)为α族上的点，Mβ(xβ，yβ，θβ，σβ)为β族上的点，(x，y)为坐标值，
Figure FSA0000232852740000024
和
Figure FSA0000232852740000025
where M α (x α , y α , θ α , σ α ) is the point on the α family, M β (x β , y β , θ β , σ β ) is the point on the β family, (x, y ) is the coordinate value,
Figure FSA0000232852740000024
and
Figure FSA0000232852740000025
 由公式(9)和(10)联立计算滑移线上的待求点M(x，y，θ，σ)，公式为：The to-be-determined point M(x, y, θ, σ) on the slip line is simultaneously calculated by formulas (9) and (10), and the formula is:
Figure FSA0000232852740000026
Figure FSA0000232852740000026
Figure FSA0000232852740000027
Figure FSA0000232852740000027
Figure FSA0000232852740000028
Figure FSA0000232852740000028
Figure FSA0000232852740000029
Figure FSA0000232852740000029
 由地震作用下滑移线场理论计算得到的边坡动力极限状态下的坡面曲线(简称动力极限坡面曲线)微分方程为：
Figure FSA00002328527400000210
与β族滑移线方程公式(8)联立采用有限差分法可求解动力极限坡面曲线坐标点Mij(xij，yij，θij，σij)：The differential equation of the slope curve under the dynamic limit state of the slope (referred to as the dynamic limit slope curve) obtained by the theoretical calculation of the sliding line field under the action of the earthquake is:
Figure FSA00002328527400000210
Simultaneously with the β family slip line equation (8), the finite difference method can be used to solve the dynamic limit slope curve coordinate point M ij (x ij , y ij , θ ij , σ ij ):
Figure FSA00002328527400000211
Figure FSA00002328527400000211
Figure FSA00002328527400000212
Figure FSA00002328527400000212
Figure FSA00002328527400000213
Figure FSA00002328527400000213
Figure FSA0000232852740000031
Figure FSA0000232852740000031
 式中Mb(xb，yb，θb，σb)和M′β(xβ′，y′β，θ′β，σ′β)为动力极限坡面曲线及第β族滑移线已知点。where M b (x b , y b , θ b , σ b ) and M′ β (x β ′, y′ β , θ′ β , σ′ β ) are the dynamic limit slope curve and the β-family slip Line known points. 3)拟静力法滑移线场边界条件3) Quasi-static method slip line field boundary conditions (1)主动区O1AB边界条件(1) O 1 AB boundary conditions in the active region 主动区第α、β族已知计算点Mα和Mβ的(x，y)为坡顶O1A坐标值，其中横坐标x＝Δx·i，Δx为计算步长，i为自然数，i＝0～N1，N1为步长数，纵坐标为坡高，主动区边界最大主应力与x轴交角：The (x, y) of the known calculation points M α and M β of the α and β groups in the active area are the coordinates of the slope top O 1 A, where the abscissa x=Δx·i, Δx is the calculation step, i is a natural number, i=0～N 1 , N 1 is the number of steps, the ordinate is the slope height, the maximum principal stress of the active zone boundary and the intersection angle of the x-axis:
Figure FSA0000232852740000032
Figure FSA0000232852740000032
 式中
Figure FSA0000232852740000033
为地震动力引起的应力偏角，根据该公式可知，应力偏角和摩擦角必须满足关系式
Figure FSA0000232852740000034
in the formula
Figure FSA0000232852740000033
is the stress declination angle caused by seismic dynamics. According to this formula, the stress declination angle and the friction angle must satisfy the relational expression
Figure FSA0000232852740000034
 主动区边界特征应力：Active zone boundary characteristic stress:
Figure FSA0000232852740000035
Figure FSA0000232852740000035
 式中坡顶动力荷载
Figure FSA0000232852740000036
动力正应力σ0＝P0·(1-kV)，动力剪应力τ0＝P0·kH，P0为坡顶静荷载，
Figure FSA0000232852740000037
滑移线交点计算公式为(11)～(14)；dynamic load at the top of the slope
Figure FSA0000232852740000036
Dynamic normal stress σ 0 =P 0 ·(1-k V ), dynamic shear stress τ 0 =P 0 ·k H , P 0 is the static load at the top of the slope,
Figure FSA0000232852740000037
The calculation formula of the slip line intersection point is (11)～(14); (2)过渡区O1BC边界条件(2) O 1 BC boundary conditions in the transition zone 过渡区已知边界点O1的(x，y)为坡肩坐标值，特征应力为：The (x, y) of the known boundary point O 1 in the transition area is the shoulder coordinate value, and the characteristic stress is:
Figure FSA0000232852740000038
Figure FSA0000232852740000038
 式中
Figure FSA0000232852740000039
k为自然数，k＝0～N2，Δθ＝θIII-θI，N2为过渡区点剖分数，过渡区滑移线交点计算公式为(11)～(14)；in the formula
Figure FSA0000232852740000039
k is a natural number, k=0～N 2 , Δθ=θ III -θ I , N 2 is the fraction of points in the transition zone, and the calculation formulas for the intersection of the slip lines in the transition zone are (11)～(14); (3)被动区O1CD边界条件(3) Boundary conditions of O 1 CD in passive region Mb第一个已知点为坡肩O1的(x，y)坐标值，但特征应力值为
Figure FSA00002328527400000310
代入公式(21)可得
Figure FSA00002328527400000311
为满足Δθ≥0，则必须θIII≥θI，因此坡顶动力荷载最小值
Figure FSA00002328527400000312
此时Δθ＝0，被动区滑移线交点计算公式为(11)～(14)，而动力极限坡面曲线OD采用公式(15)～(18)计算。The first known point of M b is the (x, y) coordinate value of the slope shoulder O 1 , but the characteristic stress value is
Figure FSA00002328527400000310
Substitute into formula (21) to get
Figure FSA00002328527400000311
In order to satisfy Δθ≥0, θ III ≥ θ I must be satisfied, so the minimum value of dynamic load at the top of slope
Figure FSA00002328527400000312
At this time Δθ=0, the calculation formula of the slip line intersection point in the passive area is (11)~(14), and the dynamic limit slope curve OD is calculated with the formula (15)~(18). 4)拟静力法失稳判据计算动力安全系数4) Calculate the dynamic safety factor according to the quasi-static method instability criterion 动力极限坡面曲线与坡底交点为(x1，0)，根据横坐标值x1的正负判断地震作用下拟静力法分析边坡稳定性的失稳判据：当x1＞0时，判断边坡处于稳定状态，折减系数F变大，即增加值ΔF为正值，可取0.01；当x1＝0时，判断边坡处于动力极限状态，此时动力安全系数FS＝F；当x1＜0时，判断边坡处于破坏状态，折减系数F变小，即增加值ΔF为负值，可取-0.01。The intersection point between the dynamic limit slope curve and the bottom of the slope is (x 1 , 0). According to the positive and negative of the abscissa value x 1 , the instability criterion of the quasi-static method to analyze the slope stability under earthquake action is judged: when x 1 > 0 When x 1 = 0, it is judged that the slope is in a stable state, and the reduction coefficient F becomes larger, that is, the increase value ΔF is a positive value, which can be 0.01; when x 1 = 0, it is judged that the slope is in a dynamic limit state, and the dynamic safety factor FS = F ; When x 1 <0, it is judged that the slope is in a damaged state, and the reduction coefficient F becomes smaller, that is, the increase value ΔF is a negative value, which can be taken as -0.01.
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