Evaluation method for stability of embankment slope under rainfall and cyclic vehicle-mounted action

Abstract

The invention discloses an embankment slope stability evaluation method under rainfall and cyclic vehicle-mounted action, and belongs to the technical field of slope stability evaluation of road engineering. According to the method, a three-stage limit analysis framework is adopted, and closed expression and rapid evaluation of the safety coefficient of the embankment are achieved through unsaturated seepage field solving, stress field coupling and spiral failure mechanism analysis. In general, through combination of multi-physics field coupling, a nonlinear strength criterion and an energy balance principle, the stability evaluation precision and calculation efficiency of the embankment slope under the complex environment and load effect are remarkably improved, and the method has important theoretical value and engineering application prospects.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of geotechnical engineering slope stability evaluation, and particularly relates to a method for evaluating the stability of an embankment slope under the combined action of rainfall and cyclic vehicle load. BACKGROUND

[0002] In the construction of high-grade highways in mountainous areas, the problem of lack of high-quality fillers often arises, and cohesive coarse-grained soil (CCGS) excavated on site is often used as embankment filling material in engineering. Such material can form apparent cohesion by matrix suction under optimal moisture content after compaction, and has a certain shear strength. However, under the condition of continuous heavy rainfall, water infiltration leads to loss of matrix suction and increase in soil self-weight, and the repeated action of traffic load will cause cumulative shear deformation, which will jointly induce deep spiral sliding failure. Traditional slope stability analysis methods are mostly based on the assumption of saturated soil or the linear Mohr-Coulomb strength criterion, and use empirical coefficients to approximately correct the influence of rainfall infiltration, which fails to fully consider the coupling effect between suction and effective stress, and cannot accurately reflect the nonlinearity of strength and the cumulative effect of cyclic dynamic load under unsaturated conditions.

[0003] In the prior art, some studies have simulated pore pressure evolution by using the transient Richards equation, and then calculated the stability by using the limit equilibrium method or the finite element method, but still belong to the "weak coupling" strategy, and do not explicitly include suction as a strength enhancement term in the failure criterion. In addition, transient seepage calculation is sensitive to grid density and time step, and has low calculation efficiency, which is difficult to meet the rapid evaluation requirements in engineering design. In terms of vehicle load, previous methods often simplify it as static equivalent or single pulse load, ignoring the cumulative effect and dynamic coupling characteristics of cyclic load, especially failing to accurately reflect the stress path change on the deep spiral sliding surface. In terms of failure mechanism, the classical method often assumes a circular or polygonal sliding surface and uses a linear strength criterion, which cannot accurately describe the spiral-diagonal failure mode commonly seen in practice and the coupling relationship between suction decay and particle crushing.

[0004] Therefore, the prior art has the following deficiencies: first, the unsaturated suction and stress state are not fully coupled in the strength model; second, the cumulative amplification effect of cyclic vehicle load and the coupling mechanism of humidity change are not comprehensively considered; third, there is a lack of an efficient analytical model that can describe the nonlinear strength behavior and spiral failure mechanism simultaneously. It is urgent to develop a limit analysis framework that can uniformly handle unsaturated seepage, suction-stress coupling, cyclic load amplification, and nonlinear energy dissipation, in order to realize rapid and accurate stability evaluation of embankment slopes under extreme conditions. SUMMARY

[0005] The application provides a method for evaluating the stability of embankment slopes under the combined action of rainfall and cyclic vehicle loads, which significantly improves the stability evaluation accuracy and calculation efficiency of embankment slopes under complex environmental and load conditions by combining multi-physical field coupling, nonlinear strength criterion and energy balance principle, and has important theoretical value and engineering application prospect, thereby effectively solving at least one technical problem involved in the background technology.

[0006] To achieve the above-mentioned purpose, the technical scheme of the application is: A method for evaluating the stability of embankment slopes under the combined action of rainfall and cyclic vehicle loads, comprising the following steps: In the first stage, based on the steady-state Richards equation in the polar coordinate system, the van Genuchten-Mualem model is used to solve the saturation field, seepage intensity field and matric suction field of the embankment slope; In the second stage, the saturation field is converted into the Bishop-type suction participation coefficient, and the cyclic vehicle load is equivalent to an enhanced static strip load through an amplification coefficient for non-saturated stress field analysis; In the third stage, the Lade-Duncan nonlinear failure criterion with suction offset term is used, combined with the spiral-diagonal rigid body sliding mechanism and the energy balance principle, to derive the critical polar radius, limit slope height and critical line load, and give the closed-form expressions of the height safety factor and load safety factor.

[0007] The application has the following beneficial effects compared with the prior art: 1. Comprehensive coupling of unsaturated seepage and stress field: By introducing the van Genuchten-Mualem model to solve the steady-state Richards equation, the saturation, seepage intensity and matric suction field are obtained explicitly, and then the saturation field is converted into the Bishop-type suction participation coefficient, realizing weak coupling analysis of the unsaturated seepage field and stress field, and significantly improving the accuracy of slope stability evaluation under extreme rainfall conditions.

[0008] 2. Accurate characterization of the cumulative effect of cyclic vehicle loads: A single amplification coefficient is used to equivalent the multi-scale cumulative effect of vehicle cyclic loads to an enhanced static strip load, overcoming the shortcomings of traditional methods that simplify vehicle loads as static or single pulse loads, and more accurately reflecting the fatigue cumulative damage effect of heavy traffic on slopes.

[0009] 3. Introduction of nonlinear failure criterion and spiral sliding mechanism: The Lade-Duncan nonlinear failure criterion with suction offset term is used, combined with the spiral-diagonal line rigid body sliding mechanism and the energy consistency principle, which can more accurately describe the failure behavior of cohesive coarse-grained soil under high deviatoric stress and unsaturated state, avoiding the overestimation of strength in high suction area by traditional Mohr-Coulomb criterion.

[0010] 4. Provides closed-form analytical solutions for high computational efficiency: The entire framework integrates multiple effects such as unsaturated seepage, suction-stress coupling, cyclic load amplification, self-weight increment, and nonlinear helical failure at the analytical level, deriving closed-form expressions for critical radius, ultimate slope height, and critical load. It is suitable for rapid iteration and safety assessment in the design phase without relying on mesh generation and time step limitations in numerical simulation.

[0011] 5. Unified definition of safety factor: It provides explicit formulas for height safety factor (Ns) and load safety factor (Fs), which can evaluate slope stability from two dimensions: slope height and load, respectively, providing a clear and consistent safety evaluation standard for engineering practice.

[0012] 6. Applicable to extreme service conditions: This method is particularly suitable for stability assessment of highway embankments in mountainous areas under the coupled action of "extreme rainfall - cyclic vehicle load", providing a theoretical basis and practical tool for the safety design and real-time monitoring of special soil slopes such as embankments.

[0013] In summary, this invention significantly improves the accuracy and computational efficiency of stability assessment of embankment slopes under complex environments and loads by combining multi-physics coupling, nonlinear strength criteria, and energy balance principles. It has significant theoretical value and promising engineering applications. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein: Figure 1 A diagram of the wedge-shaped geometric model of the embankment slope provided by the present invention; Figure 2 The seepage intensity diagram of the Richards unsaturated steady-state seepage response field for the wedge-shaped embankment provided by the present invention; Figure 3 The saturation diagram of the Richards unsaturated steady-state seepage response field for the wedge-shaped embankment provided by the present invention; Figure 4 The matrix suction diagram of the Richards unsaturated steady-state seepage response field of the wedge-shaped embankment provided by the present invention; Figure 5 Gravity diagram of Richards' unsaturated steady-state seepage response field for wedge-shaped embankments provided by the present invention; Figure 6The diagram shows the additional stress response field caused by vehicle wheel loads within a wedge-shaped embankment without considering rainfall, as provided by this invention. Figure 7 The additional stress response field diagram of vehicle wheel loads within a wedge-shaped embankment considering rainfall, provided by the present invention; Figure 8 Safety factor diagrams under different rainfall intensities provided by this invention; Figure 9 This invention provides critical load diagrams under different rainfall intensities. Figure 10 This invention provides a critical height diagram for different rainfall intensities. Detailed Implementation

[0015] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0016] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0017] Furthermore, in this invention, descriptions involving "first," "second," etc., are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0018] In this invention, unless otherwise explicitly specified and limited, the terms "connection," "fixed," etc., should be interpreted broadly. For example, "fixed" can mean a fixed connection, a detachable connection, or an integral part; it can mean a mechanical connection; it can mean a direct connection or an indirect connection through an intermediate medium; it can mean the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0019] Furthermore, the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are feasible for those skilled in the art. If the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0020] This invention provides a method for assessing the stability of embankment slopes under rainfall and cyclic vehicle loads, comprising the following steps: In the first stage, based on the steady-state Richards equations in polar coordinates, the van Genuchten–Mualem (VG-M) model was used to solve for the saturation field, seepage intensity field, and matrix suction field of the embankment slope. In the second stage, the saturation field is converted into a Bishop-type suction participation factor, and the cyclic vehicle load is equivalent to an enhanced static strip load through an amplification factor, and an unsaturated stress field analysis is performed. In the third stage, the Lade-Duncan strength criterion with suction offset term is adopted. Combined with the helical-diagonal rigid body sliding mechanism and energy balance principle, the critical extreme radius, ultimate slope height and critical line load are derived, and closed-form expressions for height safety factor and load safety factor are given.

[0021] In the first stage, the soil is assumed to be an isotropic homogeneous medium. A polar coordinate system is used for geometric abstraction, and the steady-state Richards equations are solved through variable separation and Newton's iteration method. Specifically, this includes: 1. Assumptions for Nonlinear Richards Steady-State Seepage Field Analysis (1) Isotropic permeability Assuming the microporosity of the soil is isotropic, with its size and connectivity being consistent in all directions, the permeability coefficient varies only with matrix suction and is independent of direction.

[0022] (2) Homogeneous medium Porosity, van Genuchten-Mualem (VG-M) model parameters, and saturated permeability are treated as constants to ensure that the coefficients in the governing equations depend only on matrix suction and are independent of spatial coordinates.

[0023] (3) Steady-state flow Assuming that rainfall infiltration remains constant over time and the pore pressure field does not change over time, the time derivative term can be ignored, simplifying the partial differential equation to a purely elliptic form.

[0024] (4) Two-dimensional polar coordinate domain By utilizing the geometry of the embankment cross-section, the problem is simplified to a two-dimensional radial-angular formula, while the longitudinal direction remains unchanged. Three-dimensional distribution is neglected.

[0025] (5) Axisymmetric long embankment Assume the embankment is long enough in the longitudinal direction to cause the end effect to decay rapidly, and apply a zero gradient boundary condition in the circumferential direction.

[0026] (6) VG-M water retention-permeability relationship A continuous closed-loop function (VG-M) is used to describe the relationship between saturation and permeability relative to suction, thus avoiding the discontinuities introduced by piecewise fitting.

[0027] (7) Direction of constant gravity Assume that gravity is perpendicular to the ground, constant in magnitude, and is projected onto the polar coordinate system through geometric mapping.

[0028] (8) Incompressible soil and water bodies The coupling between consolidation and pore water pressure-volume change is ignored, and the porosity is kept constant in steady-state calculations.

[0029] Second, the steady-state Richards equations In the unsaturated zone, the motion of water is driven by both gravity and capillary attraction. The fundamental governing equations include the law of conservation of mass, Darcy's law, and Richards' equations.

[0030] Soil porosity is defined as The volumetric water content is = ,in Let be the saturation level. The conservation of water mass can be expressed by formula (1).

[0031] (1).

[0032] In the formula, This represents the Darcy flux in the unsaturated zone. Darcy's law defines the water flux. With head gradient The relationship between them is specifically shown in formula (2).

[0033] (2) In the formula, Matrix suction head (m); For gravitational momentum, take a positive value upwards; The unsaturated hydraulic conductivity (m / s) depends on Or effective saturation.

[0034] Substituting formula (2) into formula (1) yields Richards' formula, see formula (3).

[0035] (3) Incorporating steady-state seepage settings, i.e. Then formula (3) can be degenerated into formula (4).

[0036] (4) Using the van Genuchten–Mualem (VG–M) model, we can obtain and The closed-form expression is shown in formulas (5) to (8): (5) (6) (7) (8) In the formula, Residual saturation; , , These are parameters related to soil properties; It represents the saturated hydraulic conductivity.

[0037] Formula (4) can be used in polar coordinates Expanded below, this becomes formula (9).

[0038] (9) In the formula z = r sin θ.

[0039] 3. Boundary Setting and Dimensionlessness 1. Boundary settings An impermeable pavement (θ=0) is represented by the following formula (10).

[0040] (10) Infiltration slope ( ) is represented by the following formula (11).

[0041] (11) Far-field similarity ( ) is represented by the following formula (12).

[0042] (12) 2. Dimensionless Selecting the feature length (e.g., crest radius) and head scale Then, the definition is: (13) The steady-state Richards equation (i.e., equation (9)) can be rewritten as equation (14): (14) Boundary settings can be dimensionless as follows: (15) IV. Separation of Variables and Ordinary Differential Equations Under the constraints of material homogeneity and uniformity in the angular direction, the solution can be assumed to be in the form of equation (16): (16) but: (17) Substituting the assumed solution into the Richards equations in polar coordinates, we obtain two independent ordinary differential equations (ODEs): (1) Radial ODE (18) Combining the infiltration boundary setting at θ=β [i.e., formula (11)], formula (18) can be rewritten as formula (19): (19) In the formula, The specified infiltration flux (m / s) is used. This is the density of water.

[0043] Therefore, the formula for the radial derivative can be obtained, as shown in equation (20): (20) It can be solved, but it requires far-field decay to be satisfied, i.e., g′(∞)=0.

[0044] (2) Angle differential equation (twenty one) In the formula; This is the integration unit.

[0045] The symmetry θ=0 at the impermeable pavement [i.e., formula (10)] means that the integral constant is zero, therefore: (twenty two) So: (twenty three) constant Absorbed It is decomposed in the middle.

[0046] (twenty four) 5. Newton's Iteration Method for Nonlinear Solutions Formula (20) can be derived from the van Genuchten water retention function. ,by Rephrasing in form to explain The non-linear dependence on saturation is as follows: Formula (25) can be obtained using the chain rule: (25) in: (26) From formula (24), we can derive: (27) Since formula (24) is derived from the boundary setting at θ = 0, and by applying formula (20), formula (27) can be rewritten as formula (28): (28) Combining equations (25), (26), and (28), we obtain equation (29): (29) To numerically solve formula (29), we can... Discretized into nodes The node spacing is Then, the derivative can be approximated using backward finite difference. The residual function can be defined by formula (30): (30) According to Newton's method, it can be deduced that: (31) The Jacobian term can be expressed by formula (32): (32) The iteration continues until the maximum residual is reached. | When the maximum value drops below 10⁻⁷, the node saturation value is... Convergence. Then, a smooth saturation field S(r,θ) can be generated through spline interpolation.

[0047] Saturation Once confirmed, It can be determined according to formula (5), and all remaining saturation fields can be obtained accordingly.

[0048] (1) Matrix suction (33) (2) Seepage intensity (34) (35) (36) (3) Saturation-related self-weight stress (37) (38) In the second stage, the analysis of the unsaturated stress field specifically includes: I. Assumptions for Stress Field Analysis Given a saturation field, the soil skeleton is treated as an isotropic linear elastic material under small deformation plane strain conditions. This assumption allows for the superposition of self-weight and external loads using the Airy stress function. The effect of matric suction is incorporated into the mean effective stress via a Bishop scaling factor, with unit weight increasing linearly with saturation. External loads such as vehicular traffic are idealized as slot-type linear loads. The hydrodynamic coupling is simplified to a unidirectional interaction: the humidity field is solved first, and then the resulting saturation distribution is incorporated into the stress field calculation. This weak coupling strategy preserves a clear analytical structure while still explicitly capturing the suction-stress interaction. The assumptions for the stress field analysis are detailed below: (1) Small deformation plane strain Assuming that the displacement changes only slightly with respect to the slope scale and remains constant along the length of the embankment, a two-dimensional Airy potential can be used for analysis.

[0049] (2) Isotropic elasticity Assume that the elastic modulus and Poisson's ratio of the soil skeleton remain constant throughout the region and are unaffected by water content.

[0050] (3) Bishop effective stress The matrix suction under unsaturated conditions is converted into additional average effective stress proportionally, and the proportionality coefficient ξ changes linearly with the degree of saturation.

[0051] (4) Linear self-weight The unit weight is expressed as a linear weighted sum of dry weight and saturated weight, so as to facilitate the direct superposition of the self-weight volume force.

[0052] (5) Simplification of line loads The actual distributed load is approximated as a single uniform line load applied to the road surface, ignoring its variation along the longitudinal and width directions.

[0053] (6) Weak coupling strategy The seepage field is solved independently to obtain the saturation, and then it is superimposed as a known field into the stress calculation, eliminating the need for iteration between consolidation and seepage.

[0054] (7) Ignore transient pore pressure Assuming the loading rate is slow enough or the drainage conditions are good, the pore water pressure remains stable throughout the loading process.

[0055] II. Characterization of Dual-Track Tire Load Vehicle wheel loads can be characterized using a quasi-static method, specifically expressed as an amplification factor reflecting the cumulative effect of cycles. The product of the bilinear static load and the amplification factor for the cyclic vehicle load. Calculated using the following formula: (39) in, , , , , , , , These are empirical parameters, calibrated through a combination of dynamic weighing data, on-site acceleration records, and indoor tests. The initial recommended values ​​are as follows: ≈ 0.30、 ≈ 0.25 ≈ 0.4 ≈0.3、 ≈ 2.0、 ≈ 1.2、 ≈ 1.0、 ≈ 0.5; For the design traffic volume of the first Number of load cycles; For the first The ratio of the average axle load to the standard axle load of 80kN; Dominant frequency and reference value =2π×10rad / s ratio; Vehicle speed and reference speed =80km / h ratio; International flatness index and reference value =2m / km ratio.

[0056] The line load pair representing the vehicle wheels on the road surface (θ = 0) can be given by formula (40): (40) In the formula, This represents the unamplified line load strength. and The radial positions of the two wheel-rail centers are located at... Place; The uniform width of each line load in the radial direction; This is a rectangular function that equals 1 within a specified interval and 0 within other intervals.

[0057] III. Governing Equations and Solutions (1) Governing equation For an unsaturated soil skeleton, neglecting the contributions of inertia and pore pressure gradient in volumetric stress, the momentum balance governing equation can be simplified to formula (41): (41) Introducing scalar potential function (i.e., the Airy stress function), the stress components can be expressed as formula (42): (42) scalar potential It satisfies the two-dimensional biharmonic equation, i.e., formula (43): (43) (2) Separation of variables and determination of characteristic indices We can assume the potential function Power series expansion is allowed, as shown in formula (44): (44) in, Indicates undetermined real roots (only principal positive roots are retained), constant - Determined by the applied load.

[0058] The surface has no traction in either the normal or tangential directions, as shown in the boundary settings in formula (45): (45) Substituting equation (45) into equation (43) and simplifying, we obtain the eigenvalue equation, as shown in equation (46): (46) Formula (46) is a transcendental equation, which can be solved using the Newton-Raphson iteration method in this invention.

[0059] (3) Coefficient matching and stress field solution Substituting formula (44) into formula (43) for the radial stress components In the process, the potential function is obtained. The contributions of the four terms to the radial stress component are shown in formula (47): (47) The basis functions for the θ direction of the wedge are shown in equation (48): (48) At the boundary θ = 0, cos(λ·0) = 1 and sin(λ·0) = 0. Therefore, the sinusoidal terms involving B and D disappear, and cannot accommodate non-zero normal stress. Furthermore, if a shear stress σrθ = 0 is applied to the free surface (θ = β), then B = D = 0.

[0060] The following boundary settings must be satisfied when θ = 0: (49) Combining equations (40), (47), and (49), we obtain equation (50): (50) To determine the closed-form coefficients, a weighted inner product matching method (i.e., the classic Muskhelishvili method) can be used. Since using a single weight function would cause the integral to diverge, a different weighting function is chosen. w A = r λ and w C = r -(λ+2) As orthogonal weighting functions, the former converges only to term A, and the latter converges only to term C, thus achieving decoupling of the two coefficients. Specifically, this can be expressed as formulas (51) to (54): (51) (52) item It can be obtained by fitting equations (51) and (52) (i.e.) I(L) A = I(R) A ), specifically as formula (53): (53) Similarly, items can be derived. C i As shown in formula (54): (54) Additional stress caused by dual-track tire load on embankment wedges and It can be calculated using formula (55): (55) Fourth, Bishop stress superposition In unsaturated soil, matrix suction The tension of pore water causes the soil skeleton to develop compressive strength due to cementation. Suction is expressed through the Bishop coefficient. The stress is applied to each physical stress surface with equal amplitude offset, merging all components into a unified effective stress frame, which facilitates subsequent application of the Lade-Duncan strength criterion and energy-based limit analysis. The specific calculation method is shown in formula (56): (56) in, It is a factor involved in attraction; These are the components of the total stress tensor; For Bishop effective stress components; For Kroneck δ; Saturation; Residual saturation; Bishop's attraction participation factor; The matrix suction is shown in equation (33).

[0061] The corresponding superimposed self-weight stress field, additional stress field, and total effective principal stress are given by equations (57), (58), and (59), respectively: (57) (58) (59) The third phase specifically includes: I. Assumptions of Nonlinear Limit Analysis To evaluate the limit height and critical load settings, the potential sliding surface is idealized as a helical curve controlled by a single geometric parameter, and the sliding body is considered as a rigid block undergoing rigid body motion. Based on the principle of energy equivalence, external work (from self-weight and line load) is balanced with internal dissipation (from shear resistance and volumetric compression along the sliding surface). Shear strength can be defined using the Lade-Duncan criterion considering suction effects. By specifying a fixed ratio between the friction arm and the extreme radius, the external-internal work equations can be analytically solved to find the root. This framework yields unified formulas for the height safety factor Fs and the load safety factor Ns. The assumptions for the nonlinear limit analysis are detailed below.

[0062] (1) Two-dimensional strip hypothesis Assume the failure mechanism extends infinitely along the longitudinal direction of the embankment, and ignore the influence of three-dimensional boundary and end effects on stability.

[0063] (2) Geometric model of helical sliding surface Assuming that the potential failure surface expands exponentially with the polar angle, it can be described by a single slope parameter, which facilitates volume-area integral analysis.

[0064] (3) Rigid slider The slider is considered a rigid body without internal deformation, and the relative displacement only occurs at the boundary interface, thus eliminating the strain energy inside the slider.

[0065] (4) Shear friction resistance arm model The shear resistance moment along the failure surface is expressed as the product of the fixed arm length and the polar radius, which simplifies the calculation of the moment in the sliding area integral.

[0066] (5) Energy balance criterion The limiting state is determined by the equality of external power and internal energy dissipation, avoiding the direct application of shear stress-shear strain distribution.

[0067] (6) Lade-Duncan strength criterion A nonlinear Lade-Duncan failure criterion based on three stress invariants is adopted, and a displacement term caused by suction is introduced into the first stress invariant to account for unsaturation effects.

[0068] (7) Approximate work of volume compression The self-weight internal energy is calculated based on the cubic polar radius and volume integral coefficient, without the need for detailed subdivision of layered energy dissipation.

[0069] (8) Standardize the safety factor The safety factors for height and load are defined as the ratio of the ultimate bearing capacity to the actual working condition, in order to ensure the consistency between numerical and conceptual criteria.

[0070] II. Helical sliding surface The failure surface of the wedge-shaped embankment slope can be assumed to be the spiral sliding surface defined by formula (60), under the coupled action of cyclic vehicle load and rainfall infiltration.

[0071] (60) In the formula, Let O be the distance from a point on the failure surface to the pole O. For the destruction surface and The radius of the intersection point; For spiral growth parameters; The angle of inclination is the helix angle / the angle of friction.

[0072] The arc length of the failure surface can be calculated using formula (61): (61) In the formula, Let be the differential arc length along the fracture surface.

[0073] The velocity of a rigid block can be expressed by formula (62): (62) In the formula, For the radial velocity component of the sliding body in polar coordinates; The tangential velocity component is in polar coordinates. The angular velocity of the rigid block; It represents the infinitesimal normal displacement on the failure surface.

[0074] Third, the energy balance principle (1) External power According to equations (55) to (59), the effective Bishop radial stress and shear stress caused by paired line loads are given by equations (63) and (64), respectively. Then, by combining the distribution of the effective Bishop radial stress and shear stress along the failure surface, the external power and internal power can be obtained.

[0075] (63) (64) In the energy-based limiting analysis framework, external power It includes three additional contributions. (1) Dry weight power Indicates unit dry weight Through normal components The power generated by the action on the spiral surface. (2) Saturation incremental power The increase in water quality after rainfall was quantified. (3) Additional work done by the vehicle line load. From bilinear loads with effective amplitude, these loads are transmitted through a wedge transfer function. The work is transferred to the spiral surface and done. Each integral is multiplied by a common geometric factor. ( (where is the angular velocity) and sum them up to obtain the total external power. Specifically, see formulas (65) to (69).

[0076] (65) (66) (67) (68) In the formula: (69) (2) Internal Skills An exponential cyclic degradation coefficient can be introduced. The vehicle's cyclic shear fatigue and moisture-induced softening are simultaneously mapped into strength degradation, as shown in formula (70).

[0077] (70) in, For calibration parameters; The equivalent number of axle load cycles within the design service life; Steady-state saturation; This represents the residual saturation.

[0078] It should be noted that, The decay index of the laboratory fit can be adjusted. This index is obtained through cyclic triaxial tests or ring shear tests under varying humidity conditions, with the applied load typically being 2 × 10⁻⁶. -4 ~2×10 -3s-1 . The larger the value, the faster the strength decays under wetting and high-cycle loading. In the absence of measured data, a value of 1×10⁻⁶ can be used for drought periods, normal operation, and heavy loads after extreme rainfall. -4 5×10 -4 and 1×10 -3 The conservative value. Therefore, Automatically follows within the slope range Variations (e.g., tending to 1 in arid regions, significantly decreasing in saturated regions), and related to Bishop participation factors. Together, they capture the dual weakening effect of suction loss and cyclic fatigue on anti-shear strength.

[0079] The reduced shear stress can be expressed by formula (71).

[0080] (71) In the limit analysis of helical failure, the shear capacity of the entire sliding wedge can be equivalently expressed as the combined shear capacity distributed along the failure surface and its moment about the rotation center O. Integrating the shear stress generated after cyclic degradation, the combined shear capacity distributed along the arc length of the failure surface is obtained [i.e., formula (61)], as shown in formula (72).

[0081] (71) The mobilized shear stress is based on the radius =rcekθ weighted and integrated to obtain the combined shear resistance torque with respect to O, as shown in formula (73).

[0082] (73) Synthetic shear stress on the helical sliding surface Relative to the origin of the spiral The dimensionless position can be defined by formula (74). Between 0 and 1.

[0083] (74) Multiply formula (75) by The actual lever arm can be obtained, as shown in formula (75).

[0084] (75) lever arm It can be achieved through helical geometry parameters The dynamic updating of shear stress distribution allows the internal dissipation moment t(1+k2)1 / 2 to accurately reflect the resistance of shear stress to the rotation of the sliding wedge.

[0085] Total internal power dissipation It can be determined by geometric coefficients and energy weights. The product is given.

[0086] (76) in, This is achieved by cyclically reducing the mobilization shear stress along the failure surface when the spiral wedge rotates at an angular velocity. Integrate and use The weights are obtained.

[0087] (77) (3) Energy balance equation Under the limiting condition, the total external power of the embankment wedge can be equal to the total internal dissipation power, as shown in equation (78).

[0088] (78) Substituting formulas (65) and (76) into formula (78), we can obtain formula (79).

[0089] (79) So: (80) The internal power dissipation caused by shear along the spiral failure surface integral can be equivalently represented as the resultant force. Rotating lever arm The work done at that point. Since the shear force acts in a balanced manner on the upper and lower surfaces of the rigid slider during rotation, the effective displacement path is... This results in an equivalent internal dissipation power term, as shown in formula (81).

[0090] (81) Subsequently, formula (80) can be further expressed as formula (82).

[0091] (82) In addition, it can be used Convert the terms on the right-hand side to a weighted form to ensure consistency with the integral energy formula, so that they can be defined in formula (83). .

[0092] (83) Therefore, formula (82) can be transformed into formula (84).

[0093] (84) Formula (85) can be obtained by combining formula (68) and formula (84).

[0094] (85) , but: (86) IV. Improved Lade-Duncan strength criterion This invention improves the Lade-Duncan strength criterion to more accurately characterize the nonlinear failure behavior of unsaturated coarse-grained soils under complex stress paths [see formulas (87)~(91)]. This criterion is based on stress invariants and uses triaxial mean stress. with second-order invariants The ratio of the principal stresses is used to capture the nonlinear evolution of the yield surface as a function of the principal stress ratio, especially under high deviatoric stress and complex stress states. Compared with the traditional Mohr-Coulomb criterion, this criterion introduces a suction participation factor and a dilatation control parameter, thereby avoiding numerical instability caused by iterative adjustment of the friction angle.

[0095] (87) in, The first stress invariant is defined as the sum of the three principal stresses; This is the second deviatoric stress invariant; The nonlinear exponent of the yield criterion; It is a constant and remains unchanged upon yielding; It is a factor involved in attraction.

[0096] in: (88) (89) (90) In the formula, The Bishop suction participation factor [see formula (56)].

[0097] Stress substitution variables and It can be introduced under the radial sliding mechanism according to the definition of formula (91).

[0098] (91) Therefore, by = 3pM and Substituting 3q² / 2 into formula (87), we can obtain formula (92).

[0099] (92) Among them, variables , , , Can be regarded as function.

[0100] Within the proposed limit analysis framework, the strength control of the helical failure mechanism can be achieved using the Lade-Duncan criterion. By establishing the energy integral term and the equivalent shear function, piecewise nonlinear control of the failure surface and accurate calculation of the safety factor under unsaturated conditions can be realized.

[0101] 5. Limiting Equilibrium Equations Based on the predetermined rotation radius Based on the adopted nonlinear strength criterion and the principle of shear energy consistency, a limit equilibrium equation was established. This equation calculates the external disturbance force caused by vehicle load, the internal shear dissipation related to strength reduction, and the change in gravitational potential energy. Then, the ultimate load was derived using this method. The closed-form expression was then obtained. Subsequently, the corresponding safety factor expression was obtained for slope analysis and evaluation.

[0102] To characterize the shear dissipation energy along the helical failure path, a nonlinear shear dissipation factor is used. and using a uniform weighting factor Constructing the shear dissipation energy term The integral form of the equation reflects the degree of dissipation when the sliding surface angle changes, as shown in formula (93).

[0103] (93) Based on the principle of equal power, this dissipated energy term is related to the external work input of the rotary sliding mechanism, thus yielding the total external shear power caused by the cyclic vehicle load. The definition is shown in formula (94).

[0104] (94) In the formula, The shear lever ratio; The curvature coefficient of the disrupted path.

[0105] In the unsaturated spiral failure mechanism, internal power dissipation This characterizes the power dissipated by the body weight along the failure surface. It is derived from the dry weight integral. and saturation adjustment integral Multiply by the helical volume factor The composition reflects the spatial energy distribution, as shown in formula (95). Integral and The weighted contributions of dry weight and saturation function S(θ) in the spiral coordinate system are represented by equations (66) to (67), respectively.

[0106] (95) Under extreme conditions, the system's external input power is balanced by its internal dissipation power, with the former provided by the vehicle load and the latter including shear dissipation generated by slippage along the helical failure surface. and the dissipation of gravitational potential energy during deformation Based on this, the system's total energy conservation equation can be expressed as equation (96).

[0107] (96) in, Indicates width is Spacing is The sum of external power input per unit thickness when two line loads are applied at r = r0 ± d / 2.

[0108] Sixth, standardize the safety factor. ultimate load It can be obtained by substituting formulas (94) and (95) into formula (96), as shown in formula (97).

[0109] (97) The total height of the failure surface (i.e., the critical height) ) can be determined by formula (98).

[0110] (98) Load safety factor It can be defined as the ratio of the critical (ultimate) load strength that the slope can withstand to the actual applied load strength, and is used to assess whether the current load state is within the safe range. See formula (99) for details.

[0111] (99) In the formula, This represents the actual applied unit load intensity. >1 indicates that the slope is stable; = 1 indicates the critical limit state; <1 indicates that the slope is unstable or about to fail.

[0112] High safety factor This represents the ratio of the maximum allowable critical slope height to the actual slope height under the limit state, assuming all other parameters (soil strength, slope angle, load) remain constant. See formula (100) for details.

[0113] (100) In the formula, This is the critical slope height; This represents the actual slope height. Specifically, Indicates slope stability; Corresponding to the critical limit state; It indicates instability or impending destruction.

[0114] Within the energy balance framework, load intensity and slope height are interdependent (e.g., vehicle loads may alter the geometry of the shear path). Therefore, despite the load safety factor... and high safety factor They are usually evaluated separately, but they can be implicitly linked through parametric analysis or numerical analysis.

[0115] The following is a detailed description of the embankment slope stability assessment method under rainfall and cyclic vehicle load provided by the present invention, using specific embodiment 1 as an example.

[0116] Example 1 Please see Figure 1 As shown, the embankment is idealized as a wedge shape with a slope angle of 135° (corresponding to a 1:1 slope ratio). The horizontal projection from the top of the slope to the toe is 5m, and the design height is... Also 5m. Soil dry unit weight. = 20kN·m-3, saturated bulk density = 22.5 kN·m⁻³, the unit weight of water = 9.81 kN·m⁻³. Unsaturated seepage behavior follows the Mualem-van Genuchten (VG-M) model: = 0.5 m-1, =1.3 (therefore) ≈0.231), residual saturation =0.08. Porosity When the initial moisture content is fixed at 0.35 and the initial water content w = 0.084, the corresponding initial saturation is... ≈0.25. Saturated permeability coefficient Take 1×10 -6 m·s-1.

[0117] Traffic load is based on a standard 120kN axle, corresponding to tire-road contact pressure. =0.83MPa (830kN·m-2). Dual-wheel contact bandwidth 1.8m, dimensionless width coefficient. =0.22. To account for dual-wheel superposition, repeated traffic loads, speed-road roughness coupling, and moderate overload, the cyclic load amplification factor is evaluated using an empirical formula. Design traffic volume For 8 million standard axles per lane per year; overload factor =1.20, reflecting that the dynamic and static weighing statistics often exceed the standard by 20%; frequency ratio =1.2; speed ratio =1.20 (120 km·h⁻¹, relative reference 100 km·h⁻¹); International Road Surface Roughness Index =2.0 m·km⁻¹, the same as the benchmark value (i.e. =1.0). Magnification factor Approximately 1.50 ( =0.30, =0.25, =0.4, =0.3, =2.0, =1.2, =1.0, =0.5). After width conversion and amplification, the tire pressure can be converted into an equivalent line load of 274 kN·m-1, acting on the center of the embankment top ( =5m).

[0118] Rainfall boundary conditions using peak infiltration flux = 2.43×10 -6 m·s⁻¹ (8.7417 mm·h⁻¹) represents the heavy rain to torrential rain condition.

[0119] Figures 2~5The steady-state unsaturated seepage response following Richards' equations within a wedge-shaped embankment is specifically given. Figure 2 Medium seepage intensity, Figure 3 Medium saturation Figure 4 medium matrix suction and Figure 5 Spatial distribution of midgravity stress. This field reveals a local flux of approximately 1.2 × 10⁻⁶ at the slope crest bend. -5 m·s⁻¹, rapidly decaying along the slope to 5 × 10⁻¹ near the toe of the slope. -6 The velocity is around m·s⁻¹, while the velocity inside the embankment remains at 1 × 10⁻¹. -6 Below m·s⁻¹, it can be considered quasi-static. This near-surface drainage mechanism maintains saturation at around 0.48 in the rear and bottom regions, while decreasing to 0.20–0.25 near the slope surface; isosaturation contour lines are roughly parallel to the slope surface, outlining the main pathways for water migration to the surface. Matric suction is inversely proportional to saturation, peaking at approximately 1400 kPa in the shallow near-slope layer and rapidly decreasing to near 0 kPa in the interior, reflecting the greater negative pore pressure required to balance the low water content region. Infiltration generates quantifiable disturbances in the self-weight stress field. Within 0–2 m of the upper slope surface, the per unit weight increases from a dry value of 20 kN·m⁻¹. -3 The infiltration intensity was increased to a near-saturation value of 22.5 kN·m⁻³, resulting in an increase of approximately 5–8 kPa in the near-surface self-weight stress relative to the dry state estimate. Continued wetting at the slope toe caused a slight arching effect, reducing the self-weight stress at a depth of 10 m by approximately 3 kPa compared to the linear projection. Although the total stress still increased almost linearly with depth, from near 0 kPa at the surface to approximately 110 kPa at the bottom of the slope toe, infiltration introduced approximately ±5% deviations in both the shallow and base layers. In summary, stable rainfall infiltration creates a high-flux, low-suction drainage zone on the slope surface, fostering distinct wet-dry zones within the embankment and subtly but measurably altering the self-weight stress field through localized density variations. These coupling effects provide more realistic initial seepage and stress conditions for subsequent slope stability and dynamic response analyses.

[0120] Figure 6 Demonstrates wheel load in dry conditions Additional stress caused The maximum stress is concentrated directly below the tire contact area and decays rapidly in both the lateral and longitudinal directions, forming a symmetrical "onion-shaped" ellipsoidal pattern. At a depth of about 3 meters below the load center, the stress decreases to a negligible level, indicating that the relatively high stiffness of the dry soil confines the load effect to the near-surface area. Figure 7The corresponding stress field considering rainfall infiltration is illustrated. Wetting at the peak reduces near-surface stiffness, causing the stress sphere to extend downwards and outwards: the high-stress core shifts downwards by approximately 0.5 meters, and wider arcuate regions form on either side of the peak. Although the peak stress decreases slightly, the volume covered by the isostress lines is much larger, indicating that rainfall-induced softening redistributes the load over a wider area. A comparison of the two figures shows that rainfall not only alters the location of the stress peak but also significantly expands the affected area, a factor that must be considered in subsequent stability assessments.

[0121] To study the impact of rainfall intensity on embankment slope stability, seven peak infiltration fluxes were analyzed, namely... =5.3, 6.92, 11.6, 16.2, 24.3, 32.4 and 39.4×10 -7 m·s⁻¹; these fluxes correspond to moderate rain, moderate to heavy rain, heavy rain, heavy rain turning into torrential rain, torrential rain, extremely heavy rain, and extremely heavy rain, respectively. Under the same boundary conditions, the load safety factor, critical wheel load, height safety factor, and critical sliding surface geometry were determined for each scenario, and the results are summarized in... Figures 8~Figure 10 middle.

[0122] With peak infiltration flux From 5.3×10 -7 m·s⁻¹ increased to 39.4 × 10⁻⁶ -7 In m·s⁻¹, a systematic deterioration in load-bearing stability and geometric stability can be observed. Figure 8 The load safety factor is shown. The value dropped sharply from 8.62 to 2.93 (a decrease of approximately 65%), while the corresponding critical wheel load... The shear strength decreased from nearly 6000 kPa to approximately 2000 kPa. This trend indicates that rainfall significantly weakened the near-surface shear strength, causing the slope to reach its ultimate state under lower external loads. In contrast, Figure 9 The response of the displayed geometric parameters is relatively smooth: high safety factor The critical height only decreased from 1.07 to 0.94. The decrease from 3.20m to 2.83m indicates that rainfall has less impact on the geometrical damage caused by gravity than on the bearing capacity. Figure 10 The evolution of the geometry of the failure surface was quantified. With... The increase in intersection radius It increased from 6.78m to 7.52m, while The reduction remains limited. These findings are consistent with... Figure 10 The sliding surface pattern is consistent, meaning that higher rainfall intensity will cause the rotational slip arc to extend slightly towards the toe of the slope, thereby expanding the potential damage area, but the overall single rotation mechanism remains unchanged.

[0123] Therefore, increased rainfall intensity significantly reduces the load safety factor and critical external load, making bearing capacity the most sensitive stability indicator to rainfall, while its impact on geometric stability is relatively small. A moderate increase in the radius of the sliding surface implies a deeper failure depth, but the failure mode does not fundamentally change. Therefore, priority should be given to reducing the amplified vehicle load and reinforcing or draining the shallow crest area to restore bearing capacity under heavy rainfall conditions.

[0124] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0125] Furthermore, it should be noted that the scope of the methods and systems in the embodiments of the present invention is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. In addition, features described with reference to certain examples may be combined in other examples.

[0126] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of the present invention.

Claims

1. A method for assessing the stability of embankment slopes under rainfall and cyclic vehicle loads, characterized in that, Includes the following steps: In the first stage, based on the steady-state Richards equations in polar coordinates, the van Genuchten–Mualem model was used to solve for the saturation field, seepage intensity field, and matrix suction field of the embankment slope. In the second stage, the saturation field is converted into a Bishop-type suction participation factor, and the cyclic vehicle load is equivalent to an enhanced static strip load through an amplification factor, and an unsaturated stress field analysis is performed. In the third stage, the Lade-Duncan strength criterion with suction offset term is adopted. Combined with the helical-diagonal rigid body sliding mechanism and energy balance principle, the critical extreme radius, ultimate slope height and critical line load are derived, and closed-form expressions for height safety factor and load safety factor are given.

2. The method according to claim 1, characterized in that, In the first stage, the soil is assumed to be an isotropic homogeneous medium. A polar coordinate system is used for geometric abstraction, and the steady-state Richards equations are solved by variable separation and Newton's iteration method.

3. The method as described in claim 1, characterized in that, In the second stage, the amplification factor of the cyclic vehicle load. Calculated using the following formula: ； in, , , , , , , , These are empirical parameters, calibrated through a combination of dynamic weighing data, on-site acceleration records, and indoor tests.

4. The method according to claim 1, characterized in that, In the second stage, the suction force is introduced into the stress field using Bishop's effective stress formula: ； in, It is a factor involved in attraction; For matrix suction; These are the components of the total stress tensor; For Bishop effective stress components; For Kronek ; Steady-state saturation; Residual saturation; This refers to the Bishop attraction factor.

5. The method according to claim 1, characterized in that, In the third stage, the helical sliding surface is defined by the following formula: ； in, Let O be the distance from a point on the failure surface to the pole O. For the destruction surface and = The radius of the intersection point; For spiral growth parameters; The angle of inclination is the helix angle / the angle of friction.

6. The method according to claim 1, characterized in that, In the third stage, the Lade–Duncan intensity criterion is expressed by the following equation: ； in, This is the first stress invariant; This is the second deviatoric stress invariant; It is a non-linear exponent; It is a constant; It is a factor involved in attraction.

7. The method according to claim 1, characterized in that, In the third stage, the high safety factor and load safety factor Calculated using the following formulas respectively: ， ； in, This is the critical slope height; This is the actual slope height; This is the critical unit load; This represents the actual applied unit load intensity.

8. The method according to claim 4, characterized in that, In the third stage, an exponential cyclic degradation coefficient is introduced. The formula for quantifying the strength degradation caused by the combined effects of vehicle cyclic load and moisture softening is as follows: ； in, For calibration parameters; The equivalent number of axle load cycles within the design service life.

9. The method according to claim 1 or 7, characterized in that, In the third stage, in the energy balance principle, external power From dry weight power saturation incremental power and vehicle load power It consists of three parts and is calculated using the following integral formula: ； ； ； in, This refers to the dry density of the soil. This is the saturated bulk density; The saturation distribution function; For wedge-shaped transfer functions; For spiral growth parameters; The slope angle; The critical polar radius; The onboard external power was not amplified; This represents the unamplified line load strength. This is the integration unit.

10. The method according to claim 1, characterized in that, The method is applicable to the stability assessment of embankment slopes filled with cemented coarse-grained soil under the coupled effects of extreme rainfall and heavy traffic.

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