CN111651900B - Pseudo-power upper limit method for calculating stability of rock slope of Xigeda stratum - Google Patents

Pseudo-power upper limit method for calculating stability of rock slope of Xigeda stratum Download PDF

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CN111651900B
CN111651900B CN202010610993.XA CN202010610993A CN111651900B CN 111651900 B CN111651900 B CN 111651900B CN 202010610993 A CN202010610993 A CN 202010610993A CN 111651900 B CN111651900 B CN 111651900B
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rock slope
slope
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刘文连
李泽
眭素刚
李江龙
陆得志
许汉华
唐果
宁飞
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Kunming University of Science and Technology
Kunming Prospecting Design Institute of China Nonferrous Metals Industry Co Ltd
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Abstract

The invention discloses a quasi-dynamic upper limit method for calculating the stability of a rock slope of a Xigeda stratum, which takes the rock slope of the Xigeda stratum as a research object, adopts finite element discrete Xigeda stratum, assumes seismic waves as simple harmonic waves, calculates the seismic acceleration of a finite element by using a quasi-dynamic method principle, and establishes a quasi-dynamic upper limit method nonlinear mathematical programming model for the stability of the rock slope of the Xigeda stratum under the seismic action according to the upper limit method principle; and circularly solving a nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum by using an interior point algorithm to obtain a relation curve of the safety coefficient of the rock slope stability under the earthquake action and time. The method has strict theory, calculation efficiency and simple engineering application, and can be applied to the field of calculating the stability of the rock slope of the Xigeda stratum under the action of earthquake.

Description

Pseudo-power upper limit method for calculating stability of rock slope of Xigeda stratum
Technical Field
The invention relates to a pseudo-dynamic upper limit method for calculating the stability of a rock slope of a Xigeda stratum, and belongs to the technical field of slope stability analysis.
Background
The Xigeda group stratum is a river and lake facies sedimentary stratum, is mainly formed between late-update and the third line, and is widely distributed in the branch climbing area of China. The siltstone stratum is formed by half-diagenetic of siltstone and claystone, the compression strength, the tensile strength and the shear strength of the siltstone stratum are low, and the siltstone stratum is easy to soften when meeting water, and belongs to extremely soft rock. The earthquake is frequent in Sichuan Panzhihua and Xichang areas, the earthquake intensity reaches more than 7 degrees, and the earthquake intensity reaches 9 degrees in individual areas. Landslide disasters often occur on a rock slope formed by the Xigeda stratum under the action of an earthquake, and challenges are brought to the safe construction and operation of engineering.
Seismic effects are an important cause of instability of the rock slope of the Xigeda stratum. The stability evolution rule of the rock slope of the Xigeda stratum is a comprehensive subject related to the intersection of multiple disciplines such as geological engineering, seismic engineering, rock-soil dynamics, computational mechanics and the like. It is generally considered that the instability of the side slope under the action of the earthquake is mainly caused by the fact that the sliding force of the side slope is increased due to the inertia force of the earthquake, and the overall safety of the side slope is reduced. A great number of scholars develop research work with rich effects on the aspects of calculation methods of side slope seismic load under the action of earthquake, dynamic response rules of side slopes, instability mechanisms, stability evaluation and the like, and research methods such as a quasi-static force method, a quasi-dynamic force method, a dynamic time-course analysis method and the like are formed for the research of side slope stability under the action of earthquake.
In the technical Specification of construction side slope engineering (GB 50330-2013), the side slope stability calculation under the action of an earthquake is mainly carried out by adopting a quasi-static method. The quasi-static force method simplifies the earthquake action into static earthquake inertia force acting on the mass center of the rock-soil mass, so that the earthquake vibration effect is simplified into static effect, and the stable safety coefficient of the side slope is solved based on the limit balance principle. However, the quasi-statics method simplifies the seismic effect into a statics effect, does not consider the space-time effect of seismic wave propagation, does not consider the amplitude and frequency of the earthquake, the dynamic and damping characteristics of the rock-soil material, and the calculation result has a larger difference from the actual effect of the earthquake.
Disclosure of Invention
The invention provides a pseudo-power upper limit method for calculating the stability of a rock slope of a Xigeda stratum, which is used for obtaining a safety coefficient distribution rule of the stability of the rock slope of the Xigeda stratum through the upper limit method.
The technical scheme of the invention is as follows: a pseudo-power upper limit method for calculating the stability of a rock slope of a Xigeda stratum takes the rock slope of the Xigeda stratum as a research object, adopts finite element discrete Xigeda stratum, assumes seismic waves as simple harmonic waves, calculates the seismic acceleration of finite elements of the rock slope of the Xigeda stratum by using a pseudo-power method principle, establishes an objective function, a plastic flow constraint condition of the finite elements under the seismic action, a plastic flow constraint condition of a public edge of the finite elements, a finite element boundary condition and a functional balance constraint condition according to the upper limit method principle, and further establishes a pseudo-power upper limit method nonlinear mathematical programming model for the stability of the rock slope of the Xigeda stratum under the seismic action; and solving a nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum.
The method comprises the following specific steps:
step one, setting parameters for calculating the stability of the rock slope of the Xigeda stratum;
step two, adopting finite element discrete Xigeda stratum rock slope, and calculating geometric characteristic parameters of the finite element;
step three, calculating the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum according to the principle of a pseudo-power method;
step four, combining the seismic acceleration, and establishing a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the seismic action according to an upper limit method principle;
and fifthly, circularly solving a quasi-dynamic upper limit method nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of the earthquake by using an interior point algorithm, and obtaining a relation curve of the safety coefficient of the rock slope stability under the action of the earthquake and time.
The parameters for planning the calculation of the stability of the rock slope of the Xigeda stratum comprise: (1) determining geometrical parameters of a rock slope of the Xigeda stratum; (2) determining distribution conditions of siltstone stratum and claystone stratum in the Xigeda stratum; (3) determining physical and mechanical parameters of the Xigeda stratum rock mass material; (4) and determining seismic parameters of the rock slope of the Xigeda stratum.
The geometrical parameters of the Xigeda stratum rock slope comprise: the height H of the rock slope, the width of the rock slope and the coordinates of slope geometry control points; the distribution of the formation includes: the thickness of each layer of the siltstone and claystone strata, the inclination angle of the interface of the siltstone and claystone; the physical mechanical parameters include: the volume weights of the siltstone and the claystone, the internal friction angle and the cohesion of the siltstone, and the internal friction angle and the cohesion of the claystone; the seismic parameters include: earthquake vibration period T and earthquake amplification factor f s Shear wave velocity V of rock mass of side slope s Longitudinal wave velocity V of side slope rock mass p Seismic acceleration coefficient k in horizontal direction h Seismic acceleration coefficient k in vertical direction v 。
The rock slope of the limited-unit discrete Xigeda stratum is specifically:
(1) Establishing a coordinate system of a rock slope of the Xigeda stratum, taking a slope toe of the slope as a coordinate origin, taking a horizontal axis as an x axis of the coordinate system, taking the horizontal right as positive, taking a vertical axis as a y axis of the coordinate system, and taking the vertical upward as positive;
(2) Using finite element discrete Xigeda stratum rock slope, any one finite element i has three nodes, and the kth node of the ith finite element has horizontal velocity
Figure BDA0002560919150000031
And vertical speed->
Figure BDA0002560919150000032
Wherein i= (1, …, N) e )，k＝(1,2,3)，N e Is the number of finite elements in the rock slope of the Xigeda stratum; common edges between adjacent finite elements: each common edge has four nodes, and the h node of the g common edge has horizontal speed +.>
Figure BDA0002560919150000033
And vertical speed->
Figure BDA0002560919150000034
Wherein g= (1, …, N) g )，h＝(1,2,3，4)，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum; the volume weight, the internal friction angle and the cohesive force of the finite element of the siltstone area are valued according to the physical and mechanical parameters of the siltstone material, and the volume weight, the internal friction angle and the cohesive force of the finite element of the claystone area are valued according to the physical and mechanical parameters of the claystone material;
(3) Calculating geometric feature parameters of the finite element, comprising: vertical height of centroid of finite element to toe of slope, area of finite element:
the vertical height from the centroid of the finite element to the toe of the slope is calculated as:
Figure BDA0002560919150000035
wherein: h i Is the vertical height from the centroid of the ith finite element to the toe of the slope;
Figure BDA0002560919150000036
is the y-coordinate of node 1 of the ith triangle element,/, for>
Figure BDA0002560919150000037
Is the y-coordinate of node 2 of the ith triangle element,/, for>
Figure BDA0002560919150000038
Is the y coordinate of the 3 rd node of the i-th triangle unit;
the area of the finite element is calculated as:
Figure BDA0002560919150000039
wherein: s is S i Is the area of the i-th finite element;
Figure BDA00025609191500000310
is the x coordinate of node 1 of the ith triangle element,/>
Figure BDA00025609191500000311
Is the x coordinate of node 2 of the ith triangle element,/, for>
Figure BDA00025609191500000312
Is the x coordinate of node 3 of the ith triangle element.
The seismic acceleration of the limited unit of the rock slope of the Xigeda stratum is calculated according to the principle of a pseudo-power method, and specifically comprises the following steps:
(1) Calculating the seismic acceleration of the finite element in the horizontal direction according to the principle of a pseudo-power method:
Figure BDA00025609191500000313
wherein:
Figure BDA00025609191500000314
is the j-th time step Xigeda stratum rock slopeSeismic acceleration of the ith finite element in the horizontal direction; i= (1, …, N e )，N e Is the number of finite elements in the rock slope of the Xigeda stratum; j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; t is the earthquake vibration period; k (k) h Is the seismic acceleration coefficient in the horizontal direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the migda stratum; f (f) s The seismic amplification coefficient of the slope rock mass; pi is the circumference ratio; v (V) s Is the shear wave velocity of the rock mass of the side slope;
(2) Calculating the seismic acceleration of the finite element in the vertical direction according to the principle of a pseudo-power method:
Figure BDA0002560919150000041
wherein:
Figure BDA0002560919150000042
is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; k (k) v Is the seismic acceleration coefficient in the vertical direction; v (V) p Is the longitudinal wave velocity of the rock mass of the side slope.
The quasi-dynamic upper limit method nonlinear mathematical programming model for establishing the stability of the rock slope of the Xigeda stratum under the earthquake action is specifically as follows:
(1) Establishing an objective function:
the safety coefficient of the stability of the rock slope of the Xigeda stratum is taken as an objective function, and the minimum value of the safety coefficient is solved, specifically as follows:
Minimize:K j
wherein: j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; k (K) j Is the safety coefficient of the jth time-step Xigeda stratum rock slope under the earthquake action; minimum represents "min";
(2) Establishing a finite element plastic flow constraint condition:
Figure BDA0002560919150000043
wherein:
Figure BDA0002560919150000044
is the geometrically compatible constraint matrix of the ith finite element,
Figure BDA0002560919150000045
is the plastic flow constraint matrix of the ith finite element; i= (1, …, N e )，N e Is the number of finite elements in the rock slope of the Xigeda stratum; s is S i Is the area of the i-th finite element; />
Figure BDA0002560919150000046
The 6 form function coefficients of the i-th finite element; />
Figure BDA0002560919150000047
m= (1, 2,., 8) are the 1 st to 8 th plastic flow matrix coefficients of the finite element, respectively,/-j>
Figure BDA0002560919150000051
m= (1, 2,., 8) are respectively the 9 th to 16 th plastic flow matrix coefficients of the finite element, C m+16 =2sin (2pi m/8), m= (1, 2,., 8), 17 th to 24 th plastic flow matrix coefficients of finite element, respectively; />
Figure BDA0002560919150000052
Is the internal friction angle: when in a siltstone stratum->
Figure BDA0002560919150000053
Taking the internal friction angle of siltstone and locating at clay rock stratum +.>
Figure BDA0002560919150000054
Taking an internal friction angle of claystone;
Figure BDA0002560919150000055
is the speed vector of the i-th finite element,
Figure BDA0002560919150000056
is the plastic multiplier vector of the i < th > finite element; />
Figure BDA0002560919150000057
Is the horizontal velocity, k= (1, 2, 3), of the kth node of the ith finite element; />
Figure BDA0002560919150000058
Is the vertical velocity of the kth node of the ith finite element, k= (1, 2, 3); />
Figure BDA0002560919150000059
An mth finite element plastic multiplier that is an ith finite element;
(3) Establishing a plastic flow constraint condition of a public edge of the finite element:
Figure BDA00025609191500000510
wherein:
Figure BDA00025609191500000511
is the geometry-consistent constraint matrix of the g-th finite element common edge,
Figure BDA00025609191500000512
is the plastic flow constraint matrix of the g-th finite element common edge; g= (1, …, N g )，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum;
Figure BDA00025609191500000513
θ g is the inclination angle of the common side of the g-th finite element, theta g Take anticlockwise as positive;
Figure BDA00025609191500000514
is the velocity vector of the g-th finite element common edge,
Figure BDA00025609191500000515
is the plastic multiplier vector of the g-th finite element common edge; />
Figure BDA00025609191500000516
Is the horizontal velocity of the h node of the g-th finite element common edge, h= (1, 2,3, 4); />
Figure BDA00025609191500000517
Is the vertical velocity of the h node of the g-th finite element common edge, h= (1, 2,3, 4); />
Figure BDA00025609191500000518
An nth common edge plastic multiplier, n= (1, …, 4), which is the nth finite element common edge;
(4) Establishing finite element boundary conditions:
A b u b ＝0
wherein: a is that b Is the coordinate transformation matrix of the finite element b on the boundary of the rock slope of the Xigeda stratum; u (u) b Is the velocity vector of the finite element b on the boundary in the rock slope of the Xigeda stratum; b= (1, …, N b )，N b Is the number of finite elements with a velocity equal to 0 on the boundary of the rock slope of the Xigeda stratum;
(5) Establishing a functional balance constraint condition of the finite element:
Figure BDA0002560919150000061
wherein: gamma is the volume weight: the volume weight of the gamma-claystone is measured when the gamma-claystone is positioned in the siltstone stratum and the volume weight of the gamma-claystone is measured when the gamma-claystone is positioned in the claystone stratum;
Figure BDA0002560919150000062
the seismic acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the horizontal direction; />
Figure BDA0002560919150000063
Is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; c s Is the cohesive force: c when in a siltstone formation s Taking the cohesive force of siltstone and c when the siltstone is positioned in a claystone stratum s Taking the cohesive force of claystone; l (L) g Is the length of the common edge of the g-th finite element, g= (1, …, N) g )，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum;
(6) Establishing a pseudo-power upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake:
integrating an objective function, a finite element plastic flow constraint condition, a finite element common side plastic flow constraint condition, a finite element boundary condition, a functional balance constraint condition and an earthquake acceleration equation to obtain a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the earthquake action, wherein the quasi-dynamic upper limit method nonlinear mathematical programming model comprises the following steps:
Figure BDA0002560919150000071
wherein: t is the earthquake vibration period; k (k) h Is the seismic acceleration coefficient in the horizontal direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the migda stratum; f (f) s The seismic amplification coefficient of the slope rock mass; pi is the circumference ratio; v (V) s Is the shear wave velocity of the rock mass of the side slope; k (k) v Is the seismic acceleration coefficient in the vertical direction; v (V) p Is the longitudinal wave velocity of the rock mass of the side slope.
The quasi-dynamic upper limit method nonlinear mathematical programming model for solving the stability of the rock slope of the Xigeda stratum under the action of earthquake is specifically as follows: the known parameters are set from j=1 to j=n t CirculationThe quasi-dynamic upper limit method nonlinear mathematical programming model type of the stability of the rock slope of the Xigeda stratum under the action of an earthquake is brought into, and the quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of the earthquake is solved by using an interior point algorithm, so that N is obtained by solving t Safety factor K of stability of rock slope of personal Xigeda stratum j The method comprises the steps of carrying out a first treatment on the surface of the Then, drawing a safety coefficient K of the stability of the rock slope of the Xigeda stratum under the action of earthquake by taking the safety coefficient as a vertical axis and taking time as a horizontal axis j And time jT/N t Is a relationship of (2); wherein j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; t is the seismic vibration period.
The beneficial effects of the invention are as follows: the invention takes a Xigeda stratum rock slope as a research object, adopts a finite element discrete Xigeda stratum rock slope, assumes a seismic wave as a simple harmonic wave, calculates the seismic acceleration of the finite element by using a pseudo-dynamic method principle, and establishes a pseudo-dynamic upper-limit method nonlinear mathematical programming model of the stability of the Xigeda stratum rock slope under the seismic action according to an upper-limit method principle; and circularly solving a nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum by using an interior point algorithm to obtain a relation curve of the safety coefficient of the rock slope stability under the earthquake action and time. The method has strict theory, calculation efficiency and simple engineering application, and can be applied to the field of calculating the stability of the rock slope of the Xigeda stratum under the action of earthquake.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic diagram of a finite element of a Xigeda formation rock slope;
FIG. 3 is a schematic view of a common edge of adjacent finite elements of a Xigeda formation rock slope;
FIG. 4 is a schematic diagram of the geometry of an example Xigeda stratum rock slope (unit: m);
FIG. 5 is a schematic view of finite element discrete and the numbering of finite elements of a Xigeda formation rock slope according to an embodiment;
FIG. 6 is a plot of seismic acceleration versus time for an 86 th finite element of a Xigeda formation rock slope along the horizontal direction according to an example;
FIG. 7 is a plot of seismic acceleration versus time for the 86 th finite element of the Xigeda formation rock slope along the vertical direction according to an example;
FIG. 8 is a plot of seismic acceleration versus time for a 74 th finite element of a Xigeda formation rock slope along the horizontal direction according to an example;
FIG. 9 is a plot of seismic acceleration versus time for a 74 th finite element of a Xigeda formation rock slope along the vertical direction according to an example;
FIG. 10 is a plot of seismic acceleration versus time for a 12 th finite element of a Xigeda formation rock slope along the horizontal direction according to an example;
FIG. 11 is a plot of seismic acceleration versus time along the vertical direction for the 12 th finite element of the Xigeda formation rock slope of an example;
FIG. 12 is a graph of safety coefficient versus time for a rock slope of a Xigeda formation under the action of an earthquake.
Detailed Description
The invention will be further described with reference to the drawings and examples, but the invention is not limited to the scope.
Example 1: 1-12, a pseudo-power upper limit method for calculating the stability of a rock slope of a Xigeda stratum takes the rock slope of the Xigeda stratum as a research object, adopts a finite element discrete Xigeda stratum rock slope, assumes a seismic wave as a simple harmonic wave, calculates the seismic acceleration of a finite element of the rock slope of the Xigeda stratum by using a pseudo-power method principle, establishes an objective function, a plastic flow constraint condition of the finite element under the seismic action, a plastic flow constraint condition of a public edge of the finite element, a finite element boundary condition and a functional balance constraint condition according to the upper limit method principle, and further establishes a pseudo-power upper limit method nonlinear mathematical programming model for the stability of the rock slope of the Xigeda stratum under the seismic action; and solving a nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum. The flow of the invention is shown in figure 1.
Further, the method may be provided with the specific steps of:
step one, setting parameters for calculating the stability of the rock slope of the Xigeda stratum;
step two, adopting finite element discrete Xigeda stratum rock slope, and calculating geometric characteristic parameters of the finite element;
step three, calculating the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum according to the principle of a pseudo-power method;
step four, combining the seismic acceleration, and establishing a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the seismic action according to an upper limit method principle;
and fifthly, circularly solving a quasi-dynamic upper limit method nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of the earthquake by using an interior point algorithm, and obtaining a relation curve of the safety coefficient of the rock slope stability under the action of the earthquake and time.
Further, the invention combines the specific steps to provide the following procedures:
step one, setting parameters for calculating the stability of the rock slope of the Xigeda stratum.
According to the actual condition of the rock slope of the Xigeda stratum, stability calculation parameters are formulated, and specifically:
(1) determining geometrical parameters of the rock slope of the Xigeda stratum, wherein the geometrical shape of the slope is shown in fig. 4, and the method comprises the following steps: the height H=30m of the side slope, the width of the side slope is 50m; the coordinates of the slope geometry control points are: coordinates of O point (0, 0), coordinates of a point (50, 0), coordinates of B point (50, 30), coordinates of C point (17.32,30);
(2) determining the distribution of siltstone formations and claystone formations in the Xigeda formation, comprising: as shown in fig. 4, the slope stratum of the embodiment is divided into 5 layers from top to bottom, wherein the layers are respectively siltstone, claystone, siltstone, claystone and siltstone, the corresponding thicknesses are respectively 4.87m, 10m and 6.13m, and the inclination angle of the interface between the siltstone and the clatstone is 25 degrees;
(3) determination of physical forces of Xigeda formation rock mass materialA study parameter comprising: the volume weight of the siltstone is 2550kN/m 3 The volume weight of claystone is 2450kN/m 3 The method comprises the steps of carrying out a first treatment on the surface of the The internal friction angle of the siltstone is 26 degrees, and the cohesion of the siltstone is 120kPa; the internal friction angle of the claystone is 24 degrees, and the cohesive force of the claystone is 180kPa.
(4) Determining seismic parameters of a rock slope of a Xigeda stratum, comprising: seismic vibration period t=0.2 s; seismic amplification factor f s =1.1; shear wave velocity V of rock mass of side slope s =3275m/s; longitudinal wave velocity V of side slope rock mass p =5925 m/s; seismic acceleration coefficient k in horizontal direction h =0.15, seismic acceleration coefficient k in vertical direction v ＝0.075。
Step two, adopting finite element discrete Xigeda stratum rock slope, and calculating geometric characteristic parameters of the finite element, wherein the method comprises the following steps:
(1) The method comprises the steps of establishing a coordinate system of a rock slope of a Xigeda stratum, taking a slope toe of the slope as a coordinate origin, taking a horizontal axis as an x axis of the coordinate system, taking a horizontal right axis as positive, taking a vertical axis as a y axis of the coordinate system, taking a vertical upward axis as positive, and establishing the coordinate system as shown in xoy in fig. 4.
(2) Finite element discrete embodiment Xigeda stratum rock slope is used, the finite element discrete schematic diagram of the embodiment Xigeda stratum rock slope and the serial numbers of the finite elements are shown in figure 5, and the slope is divided into 232 finite elements 696 public sides. Any one finite element i has three nodes, and the kth node of the ith finite element has a horizontal velocity
Figure BDA0002560919150000101
And vertical speed->
Figure BDA0002560919150000102
Wherein i= (1, …, N) e )，k＝(1,2,3)，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e =232; the common edges between adjacent finite elements are shown in FIG. 3, each common edge has four nodes, and the h node of the g-th common edge has horizontal velocity +.>
Figure BDA0002560919150000103
And vertical speed->
Figure BDA0002560919150000104
Wherein g= (1, …, N) g )，h＝(1,2,3，4)，N g Is the number of the public edges of the finite element in the rock slope of the Xigeda stratum, N g =696. The volume weight, the internal friction angle and the cohesive force of the finite element of the siltstone area are valued according to the physical and mechanical parameters of the siltstone material, and the volume weight, the internal friction angle and the cohesive force of the finite element of the claystone area are valued according to the physical and mechanical parameters of the claystone material;
(3) Calculating geometric feature parameters of the finite element, comprising: the centroid of the finite element is up to the vertical height at the toe of the slope, and the area of the finite element.
The vertical height from the centroid of the finite element to the toe of the slope is calculated as:
Figure BDA0002560919150000105
wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；H i Is the vertical height from the centroid of the ith finite element to the toe of the slope;
Figure BDA0002560919150000106
is the y-coordinate of node 1 of the ith triangle element,/, for>
Figure BDA0002560919150000107
Is the y-coordinate of node 2 of the ith triangle element,/, for>
Figure BDA0002560919150000108
Is the y coordinate of the 3 rd node of the i-th triangle unit;
the area of the finite element is calculated as:
Figure BDA0002560919150000109
wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；S i Is the area of the i-th finite element;
Figure BDA0002560919150000111
is the x coordinate of node 1 of the ith triangle element,/>
Figure BDA0002560919150000112
Is the x coordinate of node 2 of the ith triangle element,/, for>
Figure BDA0002560919150000113
Is the x-coordinate of node 3 of the ith triangle element,/, for the triangle element>
Figure BDA0002560919150000114
Is the y-coordinate of node 1 of the ith triangle element,/, for>
Figure BDA0002560919150000115
Is the y-coordinate of node 2 of the ith triangle element,/, for>
Figure BDA0002560919150000116
Is the y-coordinate of node 3 of the ith triangle element.
And thirdly, calculating the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum according to the principle of a pseudo-power method.
The invention assumes the seismic wave as simple harmonic wave, adopts the principle of pseudo-power method to calculate the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum, and specifically comprises the following steps:
(1) And calculating the seismic acceleration of the finite element in the horizontal direction according to the principle of a pseudo-dynamic method.
Figure BDA0002560919150000117
Wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；j＝(1,…,N t )，N t Is the number of time steps in the earthquake vibration cycle, N t Taking 20; t is the seismic vibration period, t=0.2 s; Δt=t/N t The time length of the time step is 0.01s for deltat;
Figure BDA0002560919150000118
the seismic acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the horizontal direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the siegeda formation, h=30m; f (f) s Is the seismic amplification coefficient of the slope rock mass, f s ＝1.1；k h Is the seismic acceleration coefficient, k, in the horizontal direction h =0.15; pi is the circumference ratio, pi is 3.14; v (V) s Is the shear wave velocity of the rock mass of the side slope, V s ＝3275m/s；
According to the above, the seismic acceleration of all finite elements of the rock slope of the Xigeda stratum in the horizontal direction at all time steps is calculated
Figure BDA0002560919150000119
The seismic accelerations of the 86 th, 74 th and 12 th finite elements in the horizontal direction are shown in fig. 6, 8 and 10, respectively.
(2) And calculating the seismic acceleration of the finite element in the vertical direction according to the principle of a pseudo-dynamic method.
Figure BDA00025609191500001110
Wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；j＝(1,…,N t )，N t Is the number of time steps in the earthquake vibration cycle, N t Taking 20; t is the vibration period of the earthquake, t=0.2 s;
Figure BDA0002560919150000121
is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the siegeda formation, h=30m; f (f) s Is the seismic amplification coefficient of the slope rock mass, f s ＝1.1；k v Is the seismic acceleration coefficient, k, in the vertical direction v =0.075; 9.81 is the gravitational acceleration in m/s 2 The method comprises the steps of carrying out a first treatment on the surface of the Pi is the circumference ratio, pi is 3.14; v (V) p Is the longitudinal wave velocity of the rock mass of the side slope, V p ＝5925m/s。
According to the above, the seismic acceleration of all finite elements of the rock slope of the Xigeda stratum in the vertical direction at all time steps is calculated
Figure BDA0002560919150000122
The seismic accelerations of the 86 th, 74 th and 12 th finite elements in the vertical direction are shown in fig. 7, 9 and 11, respectively.
And step four, establishing a pseudo-dynamic nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of an earthquake according to an upper limit method principle.
The method comprises the following steps:
(1) An objective function is established. According to the upper limit theory, the safety coefficient of the stability of the rock slope of the Xigeda stratum is taken as an objective function, and the minimum value of the safety coefficient is solved. The method comprises the following steps:
Minimize:K j (5)
wherein: j= (1, …, N t )，N t Is the number of time steps in the earthquake vibration cycle, N t Taking 20; k (K) j Is the safety coefficient of the jth time-step Xigeda stratum rock slope under the earthquake action; minimum means "minimum.
(2) Establishing a finite element plastic flow constraint condition:
Figure BDA0002560919150000123
wherein:
Figure BDA0002560919150000124
is the geometrically compatible constraint matrix of the ith finite element,
Figure BDA0002560919150000125
is the plastic flow constraint matrix of the ith finite element; i= (1, …, N e )，N e Is the number of finite elements in the rock slope of the Xigeda stratum; s is S i Is the area of the i-th finite element; />
Figure BDA0002560919150000126
The 6 form function coefficients of the i-th finite element; />
Figure BDA0002560919150000127
m= (1, 2,., 8) are the 1 st to 8 th plastic flow matrix coefficients of the finite element, respectively,/-j>
Figure BDA0002560919150000131
m= (1, 2,., 8) are respectively the 9 th to 16 th plastic flow matrix coefficients of the finite element, C m+16 =2sin (2pi m/8), m= (1, 2,., 8), 17 th to 24 th plastic flow matrix coefficients of finite element, respectively; />
Figure BDA0002560919150000132
Is the internal friction angle of the finite element, when the finite element is located in the siltstone formation +.>
Figure BDA0002560919150000133
Taking the internal friction angle of siltstone and +.>
Figure BDA0002560919150000134
Taking an internal friction angle of claystone;
Figure BDA0002560919150000135
is the speed vector of the i-th finite element,
Figure BDA0002560919150000136
is the plastic multiplier vector of the i < th > finite element; />
Figure BDA0002560919150000137
Is the horizontal velocity, k= (1, 2, 3), of the kth node of the ith finite element; />
Figure BDA0002560919150000138
Is the vertical velocity of the kth node of the ith finite element, k= (1, 2, 3); />
Figure BDA0002560919150000139
An mth finite element plastic multiplier that is an ith finite element;
(3) Establishing a plastic flow constraint condition of a public edge of the finite element:
Figure BDA00025609191500001310
wherein: wherein:
Figure BDA00025609191500001311
is the geometric compatibility constraint matrix of the common edge of the g-th finite element, < >>
Figure BDA00025609191500001312
Is the plastic flow constraint matrix of the g-th finite element common edge; g= (1, …, N g )，N g Is the number of the public edges of the finite element in the rock slope of the Xigeda stratum, N g ＝696；
Figure BDA00025609191500001313
θ g Is the inclination angle of the common side of the g-th finite element, theta g Take anticlockwise as positive;
Figure BDA00025609191500001314
is the velocity vector of the g-th finite element common edge,
Figure BDA00025609191500001315
is the plastic multiplier vector of the g-th finite element common edge; />
Figure BDA00025609191500001316
Is the horizontal velocity of the h node of the g-th finite element common edge, h= (1, 2,3, 4); />
Figure BDA00025609191500001317
Is the vertical velocity of the h node of the g-th finite element common edge, h= (1, 2,3, 4); />
Figure BDA00025609191500001318
An nth common edge plastic multiplier, n= (1, …, 4), which is the nth finite element common edge; />
Figure BDA00025609191500001319
The internal friction angle of the rock mass is 26 degrees for siltstone and 24 degrees for claystone.
(4) Establishing finite element boundary conditions:
A b u b ＝0 (8)
wherein: a is that b Is the coordinate transformation matrix of the finite element b on the boundary in the rock slope of the Xigeda stratum, u b Is the velocity vector of the finite element b on the boundary in the rock slope of the Xigeda stratum; b= (1, …, N b )，N b Is the number of finite elements with the speed equal to 0 on the boundary of the rock slope of the Xigeda stratum, N b ＝54。
(5) Establishing a functional balance constraint condition of the finite element:
Figure BDA0002560919150000141
wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；j＝(1,…,N t )，N t Is the number of time steps in the earthquake vibration cycle, N t Taking 20; n (N) g Is the number of the public edges of the finite element in the rock slope of the Xigeda stratum, N g ＝696；K j Is the safety coefficient of the jth time-step Xigeda stratum rock slope under the earthquake action; s is S i Is the area of the i-th finite element;
Figure BDA0002560919150000142
the seismic acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the horizontal direction; />
Figure BDA0002560919150000143
Is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; />
Figure BDA0002560919150000144
Is the horizontal velocity, k= (1, 2, 3), of the kth node of the ith finite element; />
Figure BDA0002560919150000145
Is the vertical velocity of the kth node of the ith finite element, k= (1, 2, 3); l (L) g Is the length of the common edge of the g-th finite element, g= (1, …, N) g )，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum; />
Figure BDA0002560919150000146
An mth finite element plastic multiplier, m= (1, …, 8), which is an ith finite element; />
Figure BDA0002560919150000147
An nth common edge plastic multiplier, n= (1, …, 4), which is the nth finite element common edge; gamma is the volume weight of the finite element, and gamma takes out siltstone when the finite element is in the siltstone formationGamma-claystone volume weight when the finite element is in the claystone formation; />
Figure BDA0002560919150000148
Is the internal friction angle of the finite element when the finite element is in the siltstone formation
Figure BDA0002560919150000149
Taking the internal friction angle of siltstone and +.>
Figure BDA0002560919150000151
Taking an internal friction angle of claystone; c s Is the cohesive force of the finite element c when the finite element is in the siltstone formation s Taking the cohesive force of the siltstone and c when the finite element is in the claystone stratum s Taking the cohesive force of claystone; the volume weight of the siltstone is 2550kN/m 3 The volume weight of claystone is 2450kN/m 3 The method comprises the steps of carrying out a first treatment on the surface of the The internal friction angle of the siltstone is 26 degrees, and the cohesion of the siltstone is 120kPa; the internal friction angle of the claystone is 24 degrees, and the cohesive force of the claystone is 180kPa.
(6) And establishing a quasi-dynamic upper limit nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake. Integrating an objective function, a finite element plastic flow constraint condition, a finite element common side plastic flow constraint condition, a finite element boundary condition and a functional balance constraint condition to obtain a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake, wherein the quasi-dynamic upper limit method nonlinear mathematical programming model comprises the following steps:
Figure BDA0002560919150000152
wherein: i= (1, …, N e )，N e Is the number of finite units, N, in the rock slope of the Xigeda stratum e ＝232；g＝(1,…,N g )，N g Is the number of the public edges of the finite element in the rock slope of the Xigeda stratum, N g ＝696；j＝(1,…,N t )，N t Is within the period of the earthquake vibrationNumber of steps, N t Taking 20;
and fifthly, using an interior point algorithm to circularly and iteratively solve a quasi-dynamic upper limit method nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of the earthquake, and obtaining the safety coefficient of the slope stability under the action of the earthquake.
The method comprises the following specific steps: the known parameters are set from j=1 to j=n t The quasi-dynamic upper limit method nonlinear mathematical programming model type (10) of the rock slope stability of the Xigeda stratum under the action of an earthquake with the number of 20 circulation belt in, and the quasi-dynamic upper limit method nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of the earthquake is solved by using an 'interior point algorithm', so that N is obtained by solving t Safety factor for stability of rock slope of personal migda stratum, where j= (1, …, N) t )，N t Is the number of time steps in the earthquake vibration cycle, N t =20; the safety coefficient of the rock slope stability of the Xigeda stratum under the action of 20 earthquakes is obtained by solving is shown in the table 1, and the safety coefficient K of the rock slope stability of the Xigeda stratum under the action of the earthquakes is drawn j The time dependence is shown in fig. 12.
TABLE 1 safety coefficient of dynamic stability of rock slope of Xigeda stratum
Figure BDA0002560919150000161
While the present invention has been described in detail with reference to the drawings, the present invention is not limited to the above embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

Claims (7)

1. A pseudo-power upper limit method for calculating the stability of a rock slope of a Xigeda stratum is characterized by comprising the following steps of: taking a Xigeda stratum rock slope as a research object, adopting a finite element discrete Xigeda stratum rock slope, assuming a seismic wave as a simple harmonic wave, calculating the seismic acceleration of a finite element of the Xigeda stratum rock slope by using a pseudo-power method principle, establishing an objective function, a plastic flow constraint condition of the finite element under the seismic action, a plastic flow constraint condition of a public edge of the finite element, a finite element boundary condition and a functional balance constraint condition according to an upper limit method principle, and further establishing a pseudo-power upper limit method nonlinear mathematical programming model of the stability of the Xigeda stratum rock slope under the seismic action; solving a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake;
the quasi-dynamic upper limit method nonlinear mathematical programming model for establishing the stability of the rock slope of the Xigeda stratum under the earthquake action is specifically as follows:
(1) Establishing an objective function:
the safety coefficient of the stability of the rock slope of the Xigeda stratum is taken as an objective function, and the minimum value of the safety coefficient is solved, specifically as follows:
Minimize:K j
wherein: j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; k (K) j Is the safety coefficient of the jth time-step Xigeda stratum rock slope under the earthquake action; minimum represents "min";
(2) Establishing a finite element plastic flow constraint condition:
Figure FDA0004074430350000011
wherein:
Figure FDA0004074430350000012
is the geometrically compatible constraint matrix of the ith finite element,
Figure FDA0004074430350000013
is the plastic flow constraint matrix of the ith finite element; i= (1, …, N e )，N e Is the number of finite elements in the rock slope of the Xigeda stratum; s is S i Is the area of the i-th finite element; />
Figure FDA0004074430350000014
The 6 form function coefficients of the i-th finite element; />
Figure FDA0004074430350000015
m= (1, 2,., 8) are the 1 st to 8 th plastic flow matrix coefficients of the finite element, respectively,/-j>
Figure FDA0004074430350000016
The 9 th to 16 th plastic flow matrix coefficients, C, of the finite element respectively m+16 =2sin (2pi m/8), m= (1, 2,., 8), 17 th to 24 th plastic flow matrix coefficients of finite element, respectively; />
Figure FDA0004074430350000021
Is the internal friction angle: when in a siltstone stratum->
Figure FDA0004074430350000022
Taking the internal friction angle of siltstone and locating at clay rock stratum +.>
Figure FDA0004074430350000023
Taking an internal friction angle of claystone;
Figure FDA0004074430350000024
is the speed vector of the i-th finite element,
Figure FDA0004074430350000025
is the plastic multiplier vector of the i < th > finite element; />
Figure FDA0004074430350000026
Is the horizontal velocity, k= (1, 2, 3), of the kth node of the ith finite element; />
Figure FDA0004074430350000027
Is the firstVertical velocity of the kth node of i finite elements, k= (1, 2, 3); />
Figure FDA0004074430350000028
An mth finite element plastic multiplier that is an ith finite element;
(3) Establishing a plastic flow constraint condition of a public edge of the finite element:
Figure FDA0004074430350000029
wherein:
Figure FDA00040744303500000210
is the geometry-consistent constraint matrix of the g-th finite element common edge,
Figure FDA00040744303500000211
is the plastic flow constraint matrix of the g-th finite element common edge; g= (1, …, N g )，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum;
Figure FDA00040744303500000212
θ g is the inclination angle of the common side of the g-th finite element, theta g Take anticlockwise as positive;
Figure FDA00040744303500000213
is the velocity vector of the g-th finite element common edge,
Figure FDA00040744303500000214
is the plastic multiplier vector of the g-th finite element common edge; />
Figure FDA00040744303500000215
Is the horizontal velocity of the h node of the g-th finite element common edge, h= (1, 2, 3),4)；/>
Figure FDA00040744303500000216
Is the vertical velocity of the h node of the g-th finite element common edge, h= (1, 2,3, 4); />
Figure FDA00040744303500000217
An nth common edge plastic multiplier, n= (1, …, 4), which is the nth finite element common edge;
(4) Establishing finite element boundary conditions:
A b u b ＝0
wherein: a is that b Is the coordinate transformation matrix of the finite element b on the boundary of the rock slope of the Xigeda stratum; u (u) b Is the velocity vector of the finite element b on the boundary in the rock slope of the Xigeda stratum; b= (1, …, N b )，N b Is the number of finite elements with a velocity equal to 0 on the boundary of the rock slope of the Xigeda stratum;
(5) Establishing a functional balance constraint condition of the finite element:
Figure FDA0004074430350000031
wherein: gamma is the volume weight: the volume weight of the gamma-claystone is measured when the gamma-claystone is positioned in the siltstone stratum and the volume weight of the gamma-claystone is measured when the gamma-claystone is positioned in the claystone stratum;
Figure FDA0004074430350000032
the seismic acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the horizontal direction; />
Figure FDA0004074430350000033
Is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; c s Is the cohesive force: c when in a siltstone formation s Taking the cohesive force of siltstone and c when the siltstone is positioned in a claystone stratum s Taking the cohesive force of claystone;l g is the length of the common edge of the g-th finite element, g= (1, …, N) g )，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum;
(6) Establishing a pseudo-power upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake:
integrating an objective function, a finite element plastic flow constraint condition, a finite element common side plastic flow constraint condition, a finite element boundary condition, a functional balance constraint condition and an earthquake acceleration equation to obtain a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the earthquake action, wherein the quasi-dynamic upper limit method nonlinear mathematical programming model comprises the following steps:
Figure FDA0004074430350000041
wherein: t is the earthquake vibration period; k (k) h Is the seismic acceleration coefficient in the horizontal direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the migda stratum; f (f) s The seismic amplification coefficient of the slope rock mass; pi is the circumference ratio; v (V) s Is the shear wave velocity of the rock mass of the side slope; k (k) v Is the seismic acceleration coefficient in the vertical direction; v (V) p Is the longitudinal wave velocity of the rock mass of the side slope.
2. The pseudo-dynamic upper limit method for calculating the stability of the rock slope of the Xigeda stratum according to claim 1, wherein the method comprises the following steps of: the method comprises the following specific steps:
step one, setting parameters for calculating the stability of the rock slope of the Xigeda stratum;
step two, adopting finite element discrete Xigeda stratum rock slope, and calculating geometric characteristic parameters of the finite element;
step three, calculating the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum according to the principle of a pseudo-power method;
step four, combining the seismic acceleration, and establishing a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the seismic action according to an upper limit method principle;
and fifthly, circularly solving a quasi-dynamic upper limit method nonlinear mathematical programming model of the rock slope stability of the Xigeda stratum under the action of the earthquake by using an interior point algorithm, and obtaining a relation curve of the safety coefficient of the rock slope stability under the action of the earthquake and time.
3. The pseudo-dynamic upper limit method for calculating the stability of the rock slope of the Xigeda stratum according to claim 2, wherein the method comprises the following steps of: the parameters for planning the calculation of the stability of the rock slope of the Xigeda stratum comprise: (1) determining geometrical parameters of a rock slope of the Xigeda stratum; (2) determining distribution conditions of siltstone stratum and claystone stratum in the Xigeda stratum; (3) determining physical and mechanical parameters of the Xigeda stratum rock mass material; (4) and determining seismic parameters of the rock slope of the Xigeda stratum.
4. A pseudo-dynamic upper limit method for calculating the stability of a rock slope of a Xigeda stratum according to claim 3, wherein: the geometrical parameters of the Xigeda stratum rock slope comprise: the height H of the rock slope, the width of the rock slope and the coordinates of slope geometry control points; the distribution of the formation includes: the thickness of each layer of the siltstone and claystone strata, the inclination angle of the interface of the siltstone and claystone; the physical mechanical parameters include: the volume weights of the siltstone and the claystone, the internal friction angle and the cohesion of the siltstone, and the internal friction angle and the cohesion of the claystone; the seismic parameters include: earthquake vibration period T and earthquake amplification factor f s Shear wave velocity V of rock mass of side slope s Longitudinal wave velocity V of side slope rock mass p Seismic acceleration coefficient k in horizontal direction h Seismic acceleration coefficient k in vertical direction v 。
5. The pseudo-dynamic upper limit method for calculating the stability of the rock slope of the Xigeda stratum according to claim 1 or 2, wherein the method comprises the following steps of: the rock slope of the limited-unit discrete Xigeda stratum is specifically:
(1) Establishing a coordinate system of a rock slope of the Xigeda stratum, taking a slope toe of the slope as a coordinate origin, taking a horizontal axis as an x axis of the coordinate system, taking the horizontal right as positive, taking a vertical axis as a y axis of the coordinate system, and taking the vertical upward as positive;
(2) Using finite element discrete Xigeda stratum rock slope, any one finite element i has three nodes, and the kth node of the ith finite element has horizontal velocity
Figure FDA0004074430350000051
And vertical speed->
Figure FDA0004074430350000052
Wherein i= (1, …, N) e )，k＝(1,2,3)，N e Is the number of finite elements in the rock slope of the Xigeda stratum; common edges between adjacent finite elements: each common edge has four nodes, and the h node of the g common edge has horizontal speed +.>
Figure FDA0004074430350000053
And vertical speed->
Figure FDA0004074430350000054
Wherein g= (1, …, N) g )，h＝(1,2,3，4)，N g Is the number of common edges of the finite elements in the rock slope of the Xigeda stratum; the volume weight, the internal friction angle and the cohesive force of the finite element of the siltstone area are valued according to the physical and mechanical parameters of the siltstone material, and the volume weight, the internal friction angle and the cohesive force of the finite element of the claystone area are valued according to the physical and mechanical parameters of the claystone material;
(3) Calculating geometric feature parameters of the finite element, comprising: vertical height of centroid of finite element to toe of slope, area of finite element:
the vertical height from the centroid of the finite element to the toe of the slope is calculated as:
Figure FDA0004074430350000055
wherein: h i Is the vertical height from the centroid of the ith finite element to the toe of the slope;
Figure FDA0004074430350000056
is the y-coordinate of node 1 of the ith triangle element,/, for>
Figure FDA0004074430350000061
Is the y-coordinate of node 2 of the ith triangle element,/, for>
Figure FDA0004074430350000062
Is the y coordinate of the 3 rd node of the i-th triangle unit;
the area of the finite element is calculated as:
Figure FDA0004074430350000063
wherein: s is S i Is the area of the i-th finite element;
Figure FDA0004074430350000064
is the x coordinate of node 1 of the ith triangle element,/>
Figure FDA0004074430350000065
Is the x coordinate of node 2 of the ith triangle element,/, for>
Figure FDA0004074430350000066
Is the x coordinate of node 3 of the ith triangle element.
6. The pseudo-dynamic upper limit method for calculating the stability of the rock slope of the Xigeda stratum according to claim 1 or 2, wherein the method comprises the following steps of: the seismic acceleration of the limited unit of the rock slope of the Xigeda stratum is calculated by using the principle of a pseudo-power method, and the method specifically comprises the following steps:
(1) Calculating the seismic acceleration of the finite element in the horizontal direction according to the principle of a pseudo-power method:
Figure FDA0004074430350000067
wherein:
Figure FDA0004074430350000068
the seismic acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the horizontal direction; i= (1, …, N e )，N e Is the number of finite elements in the rock slope of the Xigeda stratum; j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; t is the earthquake vibration period; k (k) h Is the seismic acceleration coefficient in the horizontal direction; h i Is the vertical height from the centroid of the ith finite element to the toe of the slope; h is the height of the rock slope of the migda stratum; f (f) s The seismic amplification coefficient of the slope rock mass; pi is the circumference ratio; v (V) s Is the shear wave velocity of the rock mass of the side slope;
(2) Calculating the seismic acceleration of the finite element in the vertical direction according to the principle of a pseudo-power method:
Figure FDA0004074430350000069
wherein:
Figure FDA00040744303500000610
is the earthquake acceleration of the ith finite element of the jth time-step Xigeda stratum rock slope along the vertical direction; k (k) v Is the seismic acceleration coefficient in the vertical direction; v (V) p Is the longitudinal wave velocity of the rock mass of the side slope.
7. The pseudo-dynamic upper limit method for calculating the stability of the rock slope of the Xigeda stratum according to claim 1 or 2, wherein the method comprises the following steps of: the saidSolving a quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of an earthquake, which comprises the following specific steps: the known parameters are set from j=1 to j=n t The quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of the circulating belt-in earthquake is used for solving the quasi-dynamic upper limit method nonlinear mathematical programming model of the stability of the rock slope of the Xigeda stratum under the action of the earthquake by using an interior point algorithm, and N is obtained by solving t Safety factor K of stability of rock slope of personal Xigeda stratum j The method comprises the steps of carrying out a first treatment on the surface of the Then, drawing a safety coefficient K of the stability of the rock slope of the Xigeda stratum under the action of earthquake by taking the safety coefficient as a vertical axis and taking time as a horizontal axis j And time jT/N t Is a relationship of (2); wherein j= (1, …, N t )，N t Is the number of time steps in the seismic vibration cycle; t is the seismic vibration period.
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