Construction method of analysis model for seismic stability of unsaturated soil slopes by horizontal slice method

Abstract

The present invention discloses a method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method. Based on the pseudo-dynamic assumption, the present invention proposes an improved horizontal slice method to evaluate the seismic response characteristics of an unsaturated soil slope. This method is used to calculate the external power done by the gravity of the unsaturated soil, the energy dissipation of the capillary cohesion and the anchoring force of the prestressed anchor cable, construct an energy balance equation, and obtain a semi-analytical numerical expression of the safety factor of the soil slope according to the principle of increased unit weight. The minimum safety factor of the slope and the corresponding critical slip surface are searched based on the enumeration method. The analysis model constructed by the present invention combines the advantages of the limit analysis analytical method and the limit analysis finite element method, effectively combines the unsaturated soil strength theory and the limit analysis method, can reasonably explain the action mechanism of the slope suction effect, and has important academic value and practical significance for guiding slope design, construction and reinforcement under complex conditions.

Description

Technical Field

[0001] The present invention relates to a method for analyzing the seismic stability of slopes, and particularly to a method for constructing an analysis model of the horizontal slice method for the seismic stability of unsaturated soil slopes, belonging to the field of engineering disaster prevention and mitigation. Background Technique

[0002] At present, the consideration of the suction effect in the analysis and research of the stability of engineering slopes is not sufficient enough, and the evaluation of the seismic stability of slopes mostly adopts the pseudo-static hypothesis, and the consideration of dynamic characteristics such as the frequency, amplitude, amplification coefficient and damping property of ground motion is not sufficient enough, and it is unable to fully reflect the dynamic characteristics of slopes under seismic action. There are relatively few studies on the limit analysis method, especially on the stability and reinforcement mechanism of unsaturated soil slopes.

[0003] The bottleneck problems existing in the analysis and research of the seismic stability of slopes include: First, the consideration of the suction effect of slopes is not sufficient enough; second, the pseudo-static assumption is mostly adopted, and the consideration of dynamic characteristics such as the frequency, amplitude, amplification coefficient and damping property of ground motion is not sufficient enough, and it is unable to fully reflect the dynamic characteristics of slopes under seismic action.

[0004] The main difficulties in the research on the limit analysis method for the seismic stability of unsaturated soil slopes are: First, it is difficult to construct a stress field permitted by static force and a velocity field permitted by motion in heterogeneous materials; second, due to the spatio-temporal variability of matrix suction and seismic acceleration, it is difficult to construct an energy balance equation for slopes to seek upper and lower bound solutions, and the seismic stability evaluation of conventional slopes is difficult to consider the dynamic characteristics of earthquakes. Summary of the Invention

[0005] Object of the Invention: The present invention aims to provide a method for constructing an analysis model of the horizontal slice method for the seismic stability of unsaturated soil slopes with simple calculation principle and reliable calculation results.

[0006] Technical Solution: The method for constructing an analysis model of the horizontal slice method for the seismic stability of unsaturated soil slopes of the present invention proposes an improved horizontal slice method to evaluate the seismic response law of unsaturated soil slopes, uses this method to calculate the external power done by the gravity and seismic inertia force of unsaturated soil, calculates the power dissipation rate of capillary cohesion and anchoring force, constructs an energy balance equation, and obtains a semi-analytical numerical expression of the safety factor of the soil slope according to the principle of increased unit weight, and searches for the minimum dynamic safety factor of the slope and the corresponding critical slip surface based on the enumeration method.

[0007] It includes the following steps:

[0008] S1 Improved slice method: First, establish a polar coordinate system. Taking the polar angle corresponding to the landslide body as the object, discretize the logarithmic spiral sliding surface, evenly divide it into several parts, and obtain the expressions of the polar angle and polar radius of the discrete points. Secondly, taking the discrete points of the sliding surface as the reference, discretize the sliding soil mass into several horizontally distributed unit soil layers, obtain the areas of the upper and lower surfaces of the unit soil layers, approximate the discrete unit soil layers as trapezoids, and calculate their volumes.

[0009] S2 Calculate the external power done by the gravity of the slope soil mass. Ignoring the characteristic of the local non-linear variation of the soil unit weight within the unit soil layer, considering it as a constant value and representing it by the value at the centroid of the soil layer, the external power done by the gravity of the unit soil layer can be expressed as the product of the gravity of the unit soil layer and the velocity in the direction of gravity at its centroid. The gravity of the unit soil layer can be expressed as the product of its volume and the corresponding unit weight at its centroid.

[0010] S3 Accumulate the external power done by the gravity of all unit soil layers to obtain the external power W done by the gravity of the sliding soil mass. γ′ ;

[0011] S4 Calculate the external power done by the seismic inertial force of the landslide body. The external power done by the seismic inertial force of the landslide body includes the external power done by the horizontal seismic force and the external power done by the vertical seismic force. Use the same method as in step S2 to obtain the external power W s ah done by the horizontal seismic force and the external power W s av ;

[0012] S5 Calculate the total dissipation rate of the damping force of the landslide body. The total dissipation rate of the damping force of the landslide body includes the energy dissipation rate of the damping force during the propagation of the horizontal seismic wave and the energy dissipation rate of the damping force during the propagation of the vertical seismic wave. Use the same method as in step S2 to obtain the energy dissipation rate of the damping force during the propagation of the horizontal seismic wave and the energy dissipation rate of the damping force during the propagation of the vertical seismic wave

[0013] S6 Calculate the energy dissipation rate caused by the apparent cohesion of unsaturated soil, which is expressed as the product of the area of the micro-element velocity discontinuity surface and its corresponding capillary cohesion and tangential velocity component, and integrate it over the entire velocity discontinuity surface to obtain.

[0014] S7 Calculate the energy dissipation rate of the anchoring force of a single prestressed anchor cable, and accumulate to obtain the energy dissipation rate caused by all anchor cables.

[0015] S8 Obtain the expression of the dynamic safety factor FOS of the soil slope according to the principle of increased unit weight, which can be expressed as the ratio of the internal energy dissipation rate of the landslide body to the external force power. The FOS calculation formula is as follows:

[0016]

[0017] S9: Search for the minimum dynamic safety factor of the slope based on the enumeration method and give its corresponding critical slip surface.

[0018] Furthermore, in step S1, the volume S of the unit soil layer i The calculation formula is (taking the longitudinal width of 1 meter for calculation):

[0019]

[0020] In the above formula:

[0021] l m′n′ = r n′ cosθ n′ - r h cosθ h -(r h sinθ h - r n′ sinθ n′ )cotβ

[0022] l mn = r n cosθ n - r h cosθ h -(r h sinθ h - r n sinθ n )cotβ

[0023] r n′ = r0exp(iΔθ)

[0024] θ n′ = θ0 + iΔθ

[0025] r n = r0exp[(i - 1)Δθ]

[0026] θ n = θ0 + (i - 1)×Δθ

[0027] In the above formula:

[0028]

[0029] In the formula: m is the number of discrete points on the slip surface; θ0 and θ h are the initial and end polar angles corresponding to the potential slip surface of the landslide body; β is the slope inclination angle; r0 and r h are the initial and end polar radii corresponding to the potential slip surface of the landslide body; θ n and θ n'is the polar angle corresponding to the discrete points above and below the unit soil layer; r n and r n' are the corresponding polar radii; l mn and l m'n' are the upper and lower surface areas of the unit soil layer.

[0030] Furthermore, in step S2, the gravity of the unit soil layer is calculated by the formula:

[0031]

[0032] where: γ'(z i + z0) is the unit weight of the unit soil layer; z i is the vertical distance from the centroid of the unit soil layer to the toe of the slope, and its calculation formula is:

[0033] z i = r h sinθ h - r n sinθ n - 0.5(r n′ sinθ n′ - r n sinθ n )

[0034] L og and θ g are the polar radius and polar angle from the centroid of the unit soil layer to the rotation axis, and can be respectively expressed as:

[0035]

[0036]

[0037] Furthermore, in step S3, the external power W γ′ done by the gravity of the sliding soil mass is calculated by the formula:

[0038]

[0039] Furthermore, in step S4, the external power W s ah done by the horizontal seismic force is calculated by the formula:

[0040]

[0041] The external power W s av done by the vertical seismic force is calculated by the formula:

[0042]

[0043] where: a h(t, (H - z i )) and a v (t, (H - z i )) are the horizontal and vertical seismic accelerations respectively.

[0044] Furthermore, a h (t, (H - z i )) and a v (t, (H - z i )) are expressed according to the pseudo - dynamic assumption as:

[0045]

[0046]

[0047] Where: V s and V p represent the shear wave velocity and the compression wave velocity respectively; k h and k v are the horizontal and vertical seismic acceleration coefficients respectively; f a is the amplification coefficient; H is the slope height; z is the vertical distance to the slope top; t is the time; g is the acceleration due to gravity, and ω is the angular velocity of the seismic wave (ω = 2π / T, T is the seismic wave period).

[0048] Furthermore, in step S5,

[0049] the energy dissipation rate of the damping force during the propagation of the horizontal seismic wave

[0050]

[0051] the energy dissipation rate of the damping force during the propagation of the vertical seismic wave

[0052]

[0053] Furthermore, in step S6, the energy dissipation rate D c caused by the apparent cohesion of unsaturated soil

[0054]

[0055] Where: c cap (z1) is the capillary cohesion corresponding to the micro - element velocity discontinuity surface; z1 is the vertical distance from the micro - element velocity discontinuity surface to the water table, and can be expressed as:

[0056] z1 = r0exp(θ h - θ0)tanφ′sinθ h-r0exp(θ - θ0)tanφ′sinθ + z0

[0057] Further, in step S7, the energy dissipation rate W of the anchor cable T is calculated by the formula:

[0058]

[0059] where: m T is the number of anchor cables; T i is the anchoring force of the i-th anchor cable; θ i and l i are the polar angle and polar radius of the anchor cable at the slope surface; ξ is the layout angle of the anchor cable; ω is the angular velocity.

[0060] Further, in step S9, an optimization program is developed based on Mathematica software. Taking the geometric parameters of the sliding soil mass as independent variables, the dynamic safety factor of the slope is calculated and compared with the preset value, and the smaller safety factor is stored; the magnitudes of the above independent variables are changed in turn, and each independent variable is changed in turn according to the specified increment size within a single calculation loop. The S1 - S8 calculation steps are repeated to obtain a new safety factor, which is compared with the stored value. This process is repeated until the minimum value is searched, and finally the minimum safety factor and the corresponding geometric parameters of the sliding soil mass are output to obtain the critical sliding surface.

[0061] A logarithmic spiral mechanism is used to simulate the potential sliding surface of the landslide body of the unsaturated soil slope. The sliding surface passes through the toe of the slope and intersects the top of the slope; the soil - water characteristic curve of the landslide body is described by the Fredlund - Xing model, and its volumetric water content can be expressed as a function of the matrix suction. The unit weight of the unsaturated soil can be obtained from the dry unit weight measured from the soil sample.

[0062] The layout angles, heights, applied anchoring forces, and anchor cable spacings of all prestressed anchor cables can be the same or different, and the anchor cables are all anchored in the soil below the landslide body.

[0063] Based on the pseudo - dynamic assumption, the dynamic characteristics of the seismic wave are described by a sine wave, and the horizontal and vertical seismic accelerations can be expressed as sine functions of time and soil layer depth.

[0064] The shear wave velocity and compression wave velocity in the soil can be estimated by empirical formulas based on the shear modulus, density, and Poisson's ratio of the soil. For unsaturated soil, its unit weight is non - linearly distributed along the depth, so the shear wave velocity and compression wave velocity increase non - linearly along the soil layer depth.

[0065] For a potential landslide, its damping is a linear function of the mass and stiffness matrices. The Rayleigh damping coefficient can be determined according to the relationship between the damping ratio and the natural frequency of vibration. In the present invention, it is assumed that the contributions of the mass matrix and the stiffness matrix to Rayleigh damping are the same, and only the influence of the mass matrix is considered to obtain the damping coefficient, and thus the damping force is calculated.

[0066] The energy dissipation rate of the damping force of the unit soil layer can be expressed as the product of the damping force of the unit soil layer and the velocity in the direction of the damping force at its centroid. The damping force of the unit soil layer can be expressed as the product of its mass and the seismic wave velocity at its centroid, and the seismic wave velocity can be obtained by integration based on the horizontal and vertical seismic accelerations.

[0067] The landslide body is discretized into several soil layer units, and the unit weight of the soil layer is regarded as a constant value. The external power done by the unit weight of the soil layer is calculated, and the external power of the entire landslide body is obtained by accumulating all the soil layer units; this assumption is more reasonable for gentle slopes and weak nonlinear problems, with higher calculation accuracy, and the calculation accuracy for steep slopes and strong nonlinear problems is slightly weaker, but it can still meet the actual application.

[0068] Regarding the characteristics that the horizontal and vertical seismic accelerations vary sinusoidally along the soil layer depth, the characteristic of the local nonlinear variation of the seismic acceleration within the unit soil layer is also ignored, regarded as a constant value and represented by the value at the centroid of the soil layer, and the external power done by the seismic inertia force of the soil layer unit is calculated.

[0069] Based on the principle of increased unit weight, an expression for the dynamic safety factor of the slope is obtained. Using an optimization method based on the enumeration method, during the process of searching for the minimum safety factor, the search stops when the increment of the initial and end polar angles corresponding to the landslide body reaches one degree; according to the upper bound theorem of limit analysis, the energy dissipation rate of the sliding soil mass must be greater than or equal to the external power done by the soil gravity. During the process of searching for the global minimum safety factor, when the safety factor is less than 1, the search can automatically stop.

[0070] This method is more suitable for single - order regular slopes with only one slope angle. For multi - order multi - platform slopes with more than two slope angles and working platforms, this method is also applicable, but the method needs to be improved.

[0071] Using this improved horizontal slicing method, the thickness of the soil layer units in the upper part of the slope is relatively large, and the thickness of the soil layer units in the lower part of the slope is relatively small; for steep slopes, the calculation accuracy of this method is relatively high, and for gentle slopes, since the sliding surface is lower than the toe of the slope, the calculation accuracy of this method is slightly poor, but it can still meet the requirements of engineering applications.

[0072] For unsaturated soil slopes under conditions such as slope top overload and pore water pressure action, as well as slopes with complex geometric shapes, the analysis model proposed in the present invention still has a certain degree of applicability, but further modification is required to adapt to new working conditions.

[0073] The improved horizontal slice method analysis model for the seismic stability of unsaturated soil slopes constructed in the present invention based on the pseudo-dynamic assumption combines the advantages of the limit analysis analytical method and the limit analysis finite element method, effectively combines the unsaturated soil strength theory and the limit analysis method, can reasonably explain the action mechanism of the suction effect of slopes, and has important academic value and practical significance for guiding slope design, construction and reinforcement under complex conditions.

[0074] Beneficial effects: Compared with the prior art, the present invention has the following remarkable advantages:

[0075] (1) The analysis model disclosed in the present invention can effectively evaluate the seismic response law of unsaturated soil slopes under seismic action, reveal the influence of the suction effect on the slope stability analysis results, the calculation results are more in line with the actual engineering situation, the analysis results are more reasonable, and the seismic safety and safety reserve of slopes can be judged more accurately;

[0076] (2) The analysis model disclosed in the present invention can effectively consider the influence of environmental factors such as the change of water content in the soil mass and external evaporation and infiltration on the slope stability, effectively deal with the nonlinear problems of geotechnical materials, and obtain analysis results closer to the actual situation, providing theoretical guidance and reference for slope engineering design and construction;

[0077] (3) The analysis model disclosed in the present invention can effectively reveal the strengthening mechanism of the suction effect in unsaturated soil slopes, and has high academic value and theoretical significance;

[0078] (4) Based on the principle of increased unit weight, the present invention uses the enumeration method to optimize the design of slope stability problems, has the advantages of simple calculation principle, high operability, no iterative calculation, high calculation efficiency and high calculation accuracy. A large number of comparative analysis results show that the semi-analytical calculation results and theoretical analysis results obtained by this method have high consistency;

[0079] (5) For slopes that need to take prestressed anchor cable reinforcement measures, because the analysis results obtained by this method are closer to the actual situation, the consumption of anchor cables is less, the slope support cost is more reasonable, and it is more in line with the "dual carbon" goal. Description of the drawings

[0080] Figure 1 It is a simplified calculation diagram of an unsaturated soil slope under seismic action of the present invention;

[0081] Figure 2 It is a simplified calculation diagram of the external power done by the soil weight and seismic inertia force based on the improved horizontal slice method of the present invention;

[0082] Figure 3The modified shear wave velocity and compression wave velocity of the present invention;

[0083] Figure 4 The influence of the suction effect of the present invention on slope stability ((a) is the influence of the friction angle, (b) is the influence of the slope height, WS represents considering suction, and NS represents not considering suction);

[0084] Figure 5 The influence of the seismic dynamic parameters of the present invention on slope stability ((a) is the influence of the horizontal seismic acceleration coefficient, (b) is the influence of the vertical seismic acceleration coefficient). Detailed implementation manners

[0085] Next, the analysis method in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without making creative analysis methods belong to the scope protected by the present invention.

[0086] Improve the seismic stability analysis model of unsaturated soil slopes by the improved horizontal slice method based on the pseudo-dynamic assumption. For the detailed implementation manners, see Figure 1-2 , the height of the slope is H, the slope angle is β, the groundwater level is located at z0 below the ground surface and is horizontally distributed, and the soil mass is divided into a saturated soil layer and an unsaturated soil layer; the initial polar angle and the ending polar angle of the slip surface are θ0 and θ h , θ h has a logarithmic spiral relationship with θ0, that is, the slip surface is a logarithmic spiral. The slope is reinforced by multiple prestressed anchor cables. The layout angles of the anchor cables are all ξ, and the layout height is h i , and the anchoring force is T i , assuming that the anchoring force and the anchor cable spacing are the same and are all anchored in the soil mass below the slip surface. The polar radius and polar angle of the anchor cable at the slope surface are l i and θ i ; the soil strength is described by the generalized Mohr-Coulomb failure criterion, and the strength indexes are the effective internal cohesion c' and the effective internal friction angle φ'. The influence of suction on slope stability can be reflected by regarding it as an apparent cohesion.

[0087] Specifically, Figure 1 For the seismic stability analysis model of unsaturated soil slopes improved by the improved horizontal slice method based on the pseudo-dynamic assumption, an improved horizontal slice method is proposed to calculate the external power done by the gravity and seismic inertia force of unsaturated soil.

[0088] The specific calculation steps are as follows:

[0089] S1: Steps of the improved slice method: First, establish a polar coordinate system. Taking the polar angle corresponding to the sliding soil mass as the object, discretize the logarithmic spiral sliding surface and divide it evenly into several parts ( Figure 2 ), and obtain the expressions of the polar angles and polar radii of the discrete points; Discretize the sliding soil mass into multiple horizontally distributed unit soil layers based on the discrete points of the sliding surface, obtain the areas of the upper and lower surfaces of the unit soil layers, approximate the unit soil layer as a trapezoid, and calculate its volume. The polar coordinates of the discrete points, the upper and lower surface areas and volumes of the unit soil layers can be expressed as:

[0090]

[0091] θ n = θ0 + (i - 1)×Δθ (2)

[0092] r n = r0exp[(i - 1)Δθ] (3)

[0093] θ n′ = θ0 + iΔθ (4)

[0094] r n′ = r0exp(iΔθ) (5)

[0095] l mn = r n cosθ n - r h cosθ h -(r h sinθ h - r n sinθ n )cotβ (6)

[0096] l m′n′ = r n′ cosθ n′ - r h cosθ h -(r h sinθ h - r n′ sinθ n′ )cotβ (7)

[0097]

[0098] In the formula: θ0 and θ h are the initial and end polar angles corresponding to the potential sliding surface of the landslide body, r0 and r h are the initial and end polar radii corresponding to the potential sliding surface of the landslide body, θ n and θ n' are the polar angles corresponding to the upper and lower discrete points of the unit soil layer, r nand r n' is the corresponding polar radius, l mn and l m'n' are the upper and lower surface areas of the unit soil layer, S i is the volume of the unit soil layer.

[0099] S2: Calculate the external power done by the gravity of the slope soil mass. Ignoring the characteristic of the non-linear variation of the soil unit weight within the unit soil layer, considering it as a constant value and representing it by the value at the centroid of the soil layer, the external power done by the gravity of the unit soil layer can be expressed as the product of the gravity of the unit soil layer and the velocity in the direction of gravity at its centroid. The gravity of the unit soil layer can be expressed as the product of its volume and the corresponding unit weight at its centroid, that is

[0100]

[0101] In the formula: γ'(z i +z0) is the unit weight of the unit soil layer; z i is the vertical distance from the centroid of the unit soil layer to the toe of the slope; L og and θ g are the polar radius and polar angle from the centroid of the unit soil layer to the rotation axis, which can be respectively expressed as:

[0102] z i = r h sinθ h - r n sinθ n - 0.5(r n′ sinθ n′ - r n sinθ n ) (10)

[0103]

[0104]

[0105] S3: Accumulate the external power done by the gravity of all unit soil layers to obtain the external power done by the gravity of the sliding soil mass, that is

[0106]

[0107] S4: The external power done by the horizontal and vertical seismic forces can be obtained by the same method. The external power done by the seismic force of the unit soil layer can be expressed as the product of the seismic force of the unit soil layer and the velocity in the direction of the seismic force at its centroid. The seismic inertial force of the unit soil layer can be expressed as the product of its mass and the corresponding seismic acceleration at its centroid. Accumulate all unit soil layers to obtain the external power done by the seismic inertial force of the landslide body, that is

[0108]

[0109]

[0110] where: a h (t, (H - z i )) and a v (t, (H - z i )) are the horizontal and vertical seismic accelerations respectively, and are expressed according to the pseudo-dynamic assumption as:

[0111]

[0112]

[0113] where: V s and V p represent the shear wave velocity and the compression wave velocity respectively, k h and k v are the horizontal and vertical seismic acceleration coefficients respectively, f a is the amplification coefficient, H is the slope height, z is the vertical distance to the slope top, t is the time, g is the acceleration due to gravity, and ω is the angular velocity of the seismic wave (ω = 2π / T, where T is the seismic wave period).

[0114] The shear wave velocity and compression wave velocity in soil can be estimated using empirical formulas based on the shear modulus, density, and Poisson's ratio of the soil. For unsaturated soil, the unit weight of the soil varies non-linearly with depth, so the shear wave velocity and compression wave velocity increase non-linearly with soil layer depth ( Figure 3 ), that is

[0115]

[0116]

[0117] where: G is the shear modulus of the soil, ν is Poisson's ratio, γ d is the dry unit weight of the soil, θ w is the volumetric water content, and γ w is the unit weight of water.

[0118] S5: The energy dissipation rate of the damping force during the propagation of horizontal and vertical seismic waves can be obtained in the same way. The energy dissipation rate of the damping force of the unit soil layer can be expressed as the product of the damping force of the unit soil layer and the velocity in the direction of the damping force at its centroid. The damping force of the unit soil layer can be expressed as the product of its mass and the corresponding seismic wave velocity at its centroid. Summing up the energy dissipation of the damping forces of all unit soil layers, the total dissipation rate of the damping force of the landslide body is obtained;

[0119]

[0120]

[0121] S6: The energy dissipation rate caused by the apparent cohesion of unsaturated soil can be obtained by integrating the product of the area of the micro-element velocity discontinuity surface and its corresponding capillary cohesion and tangential velocity component over the entire velocity discontinuity surface, that is

[0122]

[0123] In the formula: c cap (z1) is the capillary cohesion corresponding to the micro-element velocity discontinuity surface, and z1 is the vertical distance from the micro-element velocity discontinuity surface to the water table, which can be expressed as:

[0124] z1 = r0exp(θ h -θ0)tanφ′sinθ h -r0exp(θ - θ0)tanφ′sinθ + z0 (23)

[0125] S7: Calculate the energy dissipation rate of the anchoring force of a single prestressed anchor cable, and accumulate the energy dissipation rates of all anchor cables, that is

[0126]

[0127] In the formula: m T is the number of anchor cables, T i is the anchoring force of the i-th anchor cable, θ i and l i are the polar angle and polar radius of the anchor cable at the slope surface, ξ is the layout angle of the anchor cable, and ω is the angular velocity.

[0128] S8: According to the principle of bulk density increase, the expression FOS of the dynamic safety factor of the soil slope can be obtained, which can be expressed as the ratio of the energy dissipation rates caused by the apparent cohesion, damping force and anchoring force of the landslide body to the external power done by the gravity and seismic force of the landslide body:

[0129]

[0130] S9: Search for the minimum dynamic safety factor of the slope based on the enumeration method and give its corresponding critical slip surface. Develop an optimization program based on Mathematica software, use the geometric parameters of the sliding soil mass as independent variables, calculate the dynamic safety factor of the slope and compare it with the preset value, and store the smaller safety factor; sequentially change the magnitudes of the above independent variables, and each independent variable changes sequentially according to the specified increment size within a single calculation loop, repeat the calculation steps of S1 - S8, obtain the new safety factor, and compare it with the stored value, and repeat the cycle until the minimum value is searched, and finally output the minimum safety factor and the corresponding geometric parameters of the sliding soil mass to obtain the critical slip surface.

[0131] Specifically, for the optimization enumeration method based on the principle of increasing unit weight, during the search process, the search stops when the increment of the initial and ending angles corresponding to the landslide body is one-hundredth of a degree.

[0132] Table 1

[0133]

[0134] Table 1 shows the comparison between the calculation results of the present invention and those of other methods. Method 1 is cited from the method in Deng D P, Li L, Zhao L H. Stability analysis of slopes reinforced with anchor cables and optimal design of anchor cable parameters. European Journal of Environmental and Civil Engineering, 2021, 25(13): 2425-2440. It can be found that the calculation results of the present invention are highly consistent with those of other calculation methods and theoretical calculation results, indicating the rationality of the calculation method of the present invention. Figure 4 and Figure 5 are partial analysis results, revealing the force effect and seismic response characteristics of unsaturated soil slopes.

[0135] In summary, the present invention creatively proposes an improved horizontal slice method analysis model for the seismic stability of unsaturated soil slopes based on the pseudo-dynamic assumption. Based on the pseudo-dynamic assumption and combined with the strength theory of unsaturated soil and the limit analysis method, an improved horizontal slice method is creatively proposed, which can reasonably explain the action mechanism of the suction effect of slopes, reveal the dynamic response characteristics of unsaturated soil slopes, and has important academic value and practical significance for guiding slope design, construction and reinforcement under complex conditions.

[0136] The above embodiments are merely examples clearly illustrating the present invention, rather than limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all the embodiments here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A method for constructing an analytical model of the seismic stability of an unsaturated soil slope by the horizontal slice method, characterized in that, The steps are as follows: S1 Improve the slice method: First, establish a polar coordinate system. Taking the polar angle corresponding to the landslide body as the object, discretize the logarithmic spiral sliding surface, evenly divide it into several parts, and obtain the expressions of the polar angle and polar radius of the discrete points. Secondly, taking the discrete points of the sliding surface as the reference, discretize the sliding soil mass into several horizontally distributed unit soil layers, obtain the areas of the upper and lower surfaces of the unit soil layers, approximate the discrete unit soil layers as trapezoids, and calculate their volumes. S2 Calculate the external power done by the gravity of the landslide body. Ignoring the characteristics of the local non-linear variation of the soil unit weight within the unit soil layer, regard it as a constant value and represent it by the value at the centroid of the soil layer. The external power done by the gravity of the unit soil layer can be expressed as the product of the gravity of the unit soil layer and the velocity in the direction of gravity at its centroid. The gravity of the unit soil layer can be expressed as the product of its volume and the corresponding unit weight at its centroid. S3 sums up the external power done by the gravity of all unit soil layers to obtain the external power W done by the gravity of the sliding soil mass γ′ ; S4 calculates the external power done by the seismic inertial force of the landslide body. The external power done by the seismic inertial force of the landslide body includes the external power done by the horizontal seismic force and the external power done by the vertical seismic force. The external power W done by the horizontal seismic force is obtained separately using the same method as in step S2 s ah and the external power W done by the vertical seismic force s av ; S5 calculates the total dissipation rate of the damping force of the landslide body. The total dissipation rate of the damping force of the landslide body includes the energy dissipation rate of the damping force during the propagation of the horizontal seismic wave and the energy dissipation rate of the damping force during the propagation of the vertical seismic wave. The energy dissipation rate of the damping force during the propagation of the horizontal seismic wave is obtained by the same method as in step S2 and the energy dissipation rate of the damping force during the propagation of the vertical seismic wave S6 Calculate the energy dissipation rate D caused by the apparent cohesion of unsaturated soil c , which is expressed as the product of the area of the velocity discontinuity surface of the infinitesimal element and its corresponding capillary cohesion and tangential velocity component, and is obtained by integrating over the entire velocity discontinuity surface; The energy dissipation rate D of the anchoring force of a single prestressed anchor cable is calculated by S7 T , and the energy dissipation rates caused by all the anchor cables are accumulated S8 Obtain the expression of the dynamic safety factor FOS of the soil slope according to the principle of increased unit weight. It can be expressed as the ratio of the internal energy dissipation rate of the landslide body to the external power. The FOS calculation formula is as follows: S9 Search for the minimum dynamic safety factor of the slope based on the enumeration method and give its corresponding critical sliding surface.

2. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S1, the volume S of the unit soil layer i The calculation formula is: taking 1 linear meter of longitudinal width for calculation, In the above formula: l m′n′ = r n′ cos θ n′ - r h cos θ h -(r h sin θ h - r n′ sin θ n′ ) cot β l mn = r n cosθ n - r h cosθ h -(r h sinθ h - r n sinθ n ) cotβ r n′ = r0 exp(iΔθ) θ n′ = θ0 + iΔθ r n = r0exp[(i - 1)Δθ] θ n = θ0 + (i - 1) × Δθ In the above formula: Where: m is the number of discrete points on the sliding surface; θ0 and θ h are the initial and end polar angles corresponding to the potential sliding surface of the landslide body; β is the slope angle; r0 and r h are the initial and end polar radii corresponding to the potential sliding surface of the landslide body; θ n and θ n' are the polar angles corresponding to the upper and lower discrete points of the unit soil layer; r n and r n' are the corresponding polar radii; l mn and l m'n' are the upper and lower surface areas of the unit soil layer.

3. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S2, the gravity of the unit soil layer is calculated by the formula: where: γ'(z i + z0) is the unit soil layer weight; z i is the vertical distance from the centroid of the unit soil layer to the slope toe, and its calculation formula is: z i = r h sinθ h - r n sinθ n -0.5(r n′ sinθ n′ - r n sinθ n ) L og and θ g are the polar radius and polar angle from the centroid of the unit soil layer to the rotation axis, which can be respectively expressed as:

4. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S3, the external power W done by the gravity of the sliding soil mass γ′ The calculation formula is as follows:

5. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S4, the external power W done by the horizontal seismic force s ah The calculation formula is as follows: External power W done by vertical seismic force s av The calculation formula is as follows: Where: a h (t, (H - z i )) and a v (t, (H - z i )) are the horizontal and vertical seismic accelerations respectively.

6. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, a h (t,(H - z i )) and a v (t,(H - z i ) are expressed according to the pseudo - dynamic assumption as: Where: V s and V p represent the shear wave velocity and the compression wave velocity respectively; k h and k v are the horizontal and vertical seismic acceleration coefficients respectively; f a is the amplification factor; H is the slope height; z is the vertical distance to the slope top; t is the time; g is the acceleration due to gravity, ω is the angular velocity of the seismic wave, ω = 2π / T, and T is the seismic wave period.

7. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S5, Energy dissipation rate of damping force during horizontal seismic wave propagation The calculation formula is as follows: Energy dissipation rate of damping force during vertical seismic wave propagation The calculation formula is as follows:

8. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S6, the energy dissipation rate D caused by the apparent cohesion of unsaturated soil c is calculated by the following formula: where: c cap (z1) is the capillary cohesive force corresponding to the micro-element velocity discontinuity surface; z1 is the vertical distance from the micro-element velocity discontinuity surface to the water table, which can be expressed as: z1 = r0exp(θ h - θ0)tanφ′sinθ h -r0exp(θ - θ0)tanφ′sinθ + z0。 9. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S7, the energy dissipation rate W of the cable bolt T is calculated by the following formula: Where: m T is the number of cable bolts; T i is the anchoring force of the i-th cable bolt; θ i and l i are the polar angle and polar radius of the cable bolt at the slope surface; ξ is the layout angle of the cable bolt; ω is the angular velocity.

10. The method for constructing an analysis model of the seismic stability of an unsaturated soil slope by the horizontal slice method is characterized in that, In step S9, develop an optimization program based on the Mathematica software. Taking the geometric parameters of the sliding soil mass as independent variables, calculate the dynamic safety factor of the slope and compare it with the preset value, and store the smaller safety factor. Sequentially change the magnitudes of the above independent variables. Each independent variable changes sequentially by the specified increment size within a single calculation cycle. Repeat the calculation steps of S1 - S8 to obtain a new safety factor and compare it with the stored value. Repeat this process until the minimum value is searched. Finally, output the minimum safety factor and the corresponding geometric parameters of the sliding soil mass to obtain the critical sliding surface.

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