Analysis method for slope stability under earthquake action by considering performance degradation of anchor cable

Abstract

The invention discloses a slope stability analysis method considering anchor cable performance degradation under the action of an earthquake. The method comprises the following steps: constructing an anchored rock slope multi-slider geometric model based on a limit analysis upper-bound method; based on the safety coefficient, the anchor cable limit state and the failure mode, the slope dynamic response is divided into three stages of anchor cable elastic coordination response, elastic response and anchor cable plastic yield and failure; establishing a multi-degree-of-freedom concentrated mass-spring-damping power model; the performance degradation characteristic of the anchor cable is introduced, and the displacement of each sliding block under the earthquake action is calculated; by analyzing the relative acceleration and displacement between the adjacent sliding blocks, the action mode between the sliding blocks is judged in real time, and integrated solution of sliding body displacement, dynamic evolution of the safety coefficient and anchor cable performance degradation and failure judgment under the earthquake action is achieved. According to the method, the elastic-plastic evolution and performance degradation of the anchor cable and the dynamic coupling effect among the multiple sliding blocks under the near-fault earthquake action can be effectively represented, and the accuracy of anchoring slope stability evaluation under the earthquake action is remarkably improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of slope stability analysis, and particularly relates to a slope stability analysis method under earthquake action considering performance degradation of anchor cables. BACKGROUND

[0002] Near-fault ground motion is prone to induce severe dynamic response and cumulative damage of the anchoring system of rock slopes, and even induce failure of the anchoring system and instability of the slope, causing serious harm.

[0003] Although prestressed anchor cables are widely used as an efficient anti-seismic reinforcement measure, there are still significant limitations in the failure discrimination of the anchoring system under seismic conditions and the slope stability analysis method in engineering practice and theoretical research: the existing engineering monitoring means are insufficient in data reliability due to cost and environmental limitations; the traditional theoretical model often simplifies the anchor cable as a constant force item, ignoring the elastoplastic evolution of the anchor cable under seismic action, the strength change of the anchor cable, the dynamic time-varying attenuation of the stiffness of the anchor cable, and the potential progressive failure mechanism.

[0004] In view of this, the present application provides a slope stability analysis method under earthquake action considering performance degradation of anchor cables, which combines the limit analysis upper bound method to construct an anchor cable performance degradation model considering damage accumulation and a multi-slide block cooperative motion model. By establishing multi-stage dynamic response control equations, the integrated coupling solution of the stiffness damage degradation of the anchor cable, the dynamic tension response under seismic action, and the dynamic evolution of the cumulative plastic slip of the slope and the safety factor is realized, so as to accurately quantify and predict the complete evolution path of the anchoring system from work to fracture failure and then induce overall instability of the slope. SUMMARY

[0005] The present application provides a slope stability analysis method under earthquake action considering performance degradation of anchor cables to solve the problems of simplifying the action of anchor cables, ignoring the strength change of anchor cables, the stiffness attenuation of anchor cables, and the failure of anchor cables in the prior art.

[0006] According to a first aspect, a slope stability analysis method under earthquake action considering performance degradation of anchor cables is provided in an embodiment, and the method comprises: Step S1, constructing an anchoring rock slope geometric model and setting the geometric parameters of the slope, the physical and mechanical parameters of the slope, and the anchor cable support parameters; Step S2, determining the critical sliding surface by using the limit analysis upper bound method, and discretizing the sliding body into a plurality of slide block units with the anchoring slope surface position as the center, wherein each slide block unit corresponds to an anchor cable, combining the virtual work principle to solve the upper bound solution of the safety factor and the initial critical yield acceleration coefficient, and dividing the slope dynamic response stage; Step S3, a multi-sliding block coordinated dynamic calculation model of the anchored rock slope is established, a multi-degree of freedom lumped mass-spring-damper system is constructed based on a plurality of discrete sliding block units, and a dynamic response control equation of the system is established; Step S4, an earthquake motion is applied, a Newmark- β method is used to solve the dynamic response of each sliding block anchor cable, and a Newmark sliding block displacement method is used to solve the displacement of each sliding block; Step S5, based on the instantaneous acceleration and relative displacement relationship of adjacent sliding blocks, the interaction mode between the sliding blocks is determined in real time, the interaction force between the sliding blocks is calculated, and the stability state of the adjacent sliding blocks is iteratively updated; Step S6, a multi-mode anchor cable failure discrimination model is established, and the whole process of dynamic instability of the slope is evaluated by combining the upper bound method of limit analysis.

[0007] Further, the step S2 specifically comprises: Step S21, according to the functional relationship of the upper bound method of limit analysis, the calculation formula of the minimum energy consumption of the critical sliding surface of the earthquake slope is solved as follows:

[0008] In the formula, is the strain rate in the kinematic admissible velocity field, is the effective stress tensor related to is the surface force vector, is the distributed force vector, is the velocity vector, and are the loading boundary and volume respectively; By solving the minimum energy consumption, the most dangerous critical sliding surface can be selected from a plurality of potential sliding surfaces; Step S22, based on the minimum energy consumption of the critical sliding surface, the upper limit solution of the safety factor of each sliding block , the initial critical yield acceleration coefficient and the upper limit solution of the safety factor of the slope are solved, specifically as follows: According to the virtual work principle, the internal energy consumption is equal to the external power:

[0009] In the formula, is the gravity power of the sliding block, is the horizontal seismic force power of the sliding block, is the work done by the pre-stressed anchor cable tension of the sliding block, is i +1 sliding block to i sliding block interaction force, is i ​-1 slider pair i Work done by slider force, Power done for energy consumption on the sliding surface of the slider bottom, Power done for energy consumption on the inclined interface between the sliders; Based on the upper bound theorem of limit analysis, combined with the safety factor of each slider calculated by strength reduction method The implicit function expression is as follows:

[0010] And further expand the above formula:

[0011] In the formula, The safety factor of each slider, The unit weight of the rock mass of the slider, The volume of the slider, The angle between the bottom of the slider and the horizontal line, The absolute velocity of the slider The angle between the sliding surface and the sliding surface, The velocity field that meets the geometric compatibility condition, The horizontal seismic force coefficient, The prestressed anchor tension of the slider, The anchor inclination, The cohesion of the slider, i +1 slider pair i Slider force, i -1 slider pair i Slider force, The length of the bottom of the slider, The equivalent cohesion of the slider interface, The length of the inclined interface between adjacent sliders, The relative velocity The angle between the inclined surface and the inclined surface, The relative velocity between the two sliders; The yield acceleration coefficient of each slider The calculation formula is:

[0012] Upper bound solution of slope safety factor The implicit function expression is as follows:

[0013] Step S23, based on the upper bound solution of the slope safety factor, combined with the anchor ultimate state, the slope dynamic response stage is divided: When ​​When t < t 1, it is the elastic coordination response stage of anchor cable, i.e. stage one, the slider in the current stage has no displacement, the anchor cable is in the elastic coordination vibration stage, the deformation energy of the anchor cable can be recovered, and the anchor cable tension is constant as the prestress ; When t < t 1, it is the elastic coordination response stage of anchor cable, i.e. stage one, the slider in the current stage has no displacement, the anchor cable is in the elastic coordination vibration stage, the deformation energy of the anchor cable can be recovered, and the anchor cable tension is constant as the prestress ; When t < t 1, it is the elastic coordination response stage of anchor cable, i.e. stage one, the slider in the current stage has no displacement, the anchor cable is in the elastic coordination vibration stage, the deformation energy of the anchor cable can be recovered, and the anchor cable tension is constant as the prestress ; When t < t 1, it is the elastic coordination response stage of anchor cable, i.e. stage one, the slider in the current stage has no displacement, the anchor cable is in the elastic coordination vibration stage, the deformation energy of the anchor cable can be recovered, and the anchor cable tension is constant as the prestress ; Wherein, K 0 is the initial prestress of the anchor cable, K 1 is the tensile bearing capacity of the anchor cable, and K 2 is the safety factor of the slope under the initial prestress of the anchor cable. The safety factor of the slope under the tensile bearing capacity of the anchor cable is calculated by bringing K 1 and K 2 into the upper limit solution of the safety factor of the slope .

[0014] Further, the step S3 specifically comprises: Step S31, using n equivalent concentrated mass blocks generated by the sliding body discretization, based on the D'Alembert principle, a multi-degree-of-freedom system motion equation formula containing block interaction is established as follows:

[0015] In the formula, M is a mass matrix, C is a damping matrix, and K is a stiffness matrix. , , , are displacement, velocity and acceleration vectors; e is a unit vector, a is a horizontal seismic acceleration, g is a gravity load vector, T is an anchor force load vector, and F is a block interaction force vector, which is defined as positive downward and negative upward along the sliding surface. Step S32, the component equation formula of the multi-degree-of-freedom system motion equation in the tangential direction along the sliding surface (s direction) and the normal direction (n direction) is established as follows: s direction:

[0016] n direction: ​​​​​​​​​​​​

[0017] wherein, is the mass matrix, and are the damping matrices in the tangential direction of the slip surface, i.e. s direction, and in the normal direction of the slip surface, i.e. n direction, respectively, and are the stiffness matrices in the s direction and n direction, respectively; , , are the displacement, velocity and acceleration vectors in the tangential direction of the slip surface, i.e. s direction, , , are the displacement, velocity and acceleration vectors in the normal direction of the slip surface, i.e. n direction, is the unit vector, is the horizontal seismic acceleration, is the gravity acceleration, is the cable force load vector, for stage one: , is the interaction force vector between blocks, is the angle between the bottom of the block and the horizontal line, is the angle of the prestressed cable.

[0018] Further, the step S4 specifically comprises: a step S41 of inputting seismic motion and solving the dynamic response of each sliding block cable by using the Newmark method, i.e. solving the axial tension of the cable vibration, and the calculation formula is as follows: β In the above formula,

[0019] is the total axial tension of the cable; and are the static elongations of the cable in the s and n directions under the action of prestress, respectively; and are the dynamic displacement responses of the cable in the s and n directions under the action of earthquake, respectively, which are obtained by using the Newmark method to solve the time history integral of the component equations of the motion equation of the multi-degree-of-freedom system in the s and n directions; is the angle between the bottom of the block and the horizontal line; β is the angle of the prestressed cable; a step S42 of solving the calculation formula of the acceleration, velocity and displacement of each sliding block based on the Newmark sliding block displacement method:

[0020]

[0021] ​​

[0022] The formula for calculating cumulative permanent displacement is:

[0023] In the formula, for t The relative acceleration of the slider along the sliding surface at any given moment. g It is the acceleration due to gravity. for t The horizontal seismic acceleration coefficient input at any time, for t The critical yield acceleration coefficient of the slider at time t. The angle between the bottom of the slider and the horizontal line. for t The relative velocity of the slider along the sliding surface at any given moment. For the current calculation time, For time step, For the outer integral variable (time). For the previous moment The initial velocity of the slider at the end, This represents the displacement increment within the current time step. For the current moment t The cumulative permanent displacement, For the previous moment The cumulative permanent displacement.

[0024] Furthermore, in step S41, the formula for calculating the static elongation of the anchor cable in the s and n directions under prestress is as follows:

[0025]

[0026] In the formula, For the initial prestress of the anchor cable force, l The length of the free section of the anchor cable. E The elastic modulus of the anchor cable. A The cross-sectional area of ​​the free section of the anchor cable. The angle between the bottom of the slider and the horizontal line. The inclination angle of the prestressed anchor cable.

[0027] Further, step S5 specifically includes: Step S51: Based on the instantaneous acceleration and relative displacement relationship between adjacent sliders, determine the interaction mode between the sliders in real time. when ,but i slider pair i +1 slider generates a traction effect; when ,but i slider pair i +1 slider creates a blocking effect; in, Let be the relative acceleration of slider i along the sliding surface at time t; Let be the relative acceleration of slider i+1 along the sliding surface at time t; Step S52, through i slider and i +1 slider t Calculate and update the inter-block forces based on the displacement and relative displacement at time t:

[0028]

[0029] In the formula, for t time i +1 slider pair i The force applied by the slider for t -1 hour i +1 slider pair i The force applied by the slider For the current calculation time, For time step, for i slider and i +1 Stiffness coefficient between the contact surfaces of the slider for i slider and i +1 Damping coefficient between the slider contact surfaces for t time i slider and i +1 Relative displacement of slider For time t i slider and i +1 speed difference of the slider for t time i slider pair i +1 The force applied by the slider; Step S53, according to The interaction forces between the sliders are solved at all times, and the stability state of itself and its adjacent sliders is updated iteratively, including various safety factors, yield acceleration coefficients, anchor cable stiffness and tension state; like:

[0030] Then in i +1 slider actioni slider and i +1 slider is stable; like:

[0031] express i The slider cannot be blocked i The downward trend of the +1 slider, then i slider and i +1 sliders all became unstable; In the formula, The power of the slider's gravity. The horizontal seismic force power of the slider. for t The work done by the anchor cable tension at all times for t time i +1 slider pair i The work done by the slider's force for i -1 slider pair i The work done by the slider's force The power used to account for energy consumption on the sliding surface at the bottom of the slider. The energy consumption power of the inclined interface between the sliders; The formulas for updating the anchor cable stiffness and tension state are as follows:

[0032]

[0033] In the formula, for t The cumulative axial force of the anchor cable at any given moment For the initial prestress of the anchor cable, for t Anchor stiffness at any given time for t The cumulative plastic displacement of the slider at each moment, The elastic modulus of the anchor cable. for t The cross-sectional area of ​​the anchor cable at time . The length of the free segment of the anchor cable. for t The dynamic axial force of the anchor cable at any given moment. for t The axial tension of the anchor cable vibration at a given moment; Step S54: The updated safety factor, yield acceleration coefficient, anchor cable stiffness, and tension state are used as the next time step. + The initial conditions are used for recursive calculation.

[0034] Further, the step S6 specifically comprises: Step S61, establishing a multi-mode anchor cable failure discrimination model: For brittle failure:

[0035] In the formula, is the cumulative axial force of the anchor cable, is the initial prestress of the anchor cable, is the tensile stiffness of the anchor cable, is the cumulative plastic displacement of the sliding block, is the elastic modulus of the anchor cable, is the cross-sectional area of the anchor cable, is the free section length of the anchor cable, is the dynamic axial force of the anchor cable, is the axial tensile amount of the anchor cable vibration, the brittle failure limit bearing capacity of the anchor cable; For ductile failure:

[0036] In the formula, is the tensile bearing capacity of the anchor cable, is the maximum elongation of the anchor cable; Step S62, real-time state determination, the cumulative plastic displacement of the sliding block , the axial tensile amount of the anchor cable vibration and the updated anchor cable stiffness , the dynamic axial force of the anchor cable and the cumulative axial force of the anchor cable calculated at each time t are brought into the established multi-mode anchor cable failure discrimination model for discrimination, to determine whether brittle failure or ductile failure of the anchor cable occurs.

[0037] The application provides a slope stability analysis method under seismic action considering anchor cable performance degradation, which has the following beneficial effects: the application avoids the problems of calculation effect and accuracy caused by simplification of the anchor cable into a constant force term, ignoring the influence of anchor cable strength change and stiffness attenuation and the like in the stability analysis of the anchored slope under seismic action by using the limit analysis method and the limit equilibrium method, and an anchor cable performance degradation model considering damage accumulation and a multi-sliding block cooperative motion model are constructed based on the upper limit method of limit analysis. By establishing a multi-stage dynamic response control equation, the integrated coupling solution of anchor cable stiffness damage degradation, dynamic tension response, slope cumulative plastic slip and dynamic evolution of safety factor under seismic action is realized, the accuracy and reliability of the progressive failure mechanism analysis and whole-process dynamic stability evaluation of the anchored slope under complex seismic working conditions are significantly improved, and a scientific theoretical basis is provided for seismic design and engineering maintenance. BRIEF DESCRIPTION OF DRAWINGS

[0038] Figure 1 A flow chart of a slope stability analysis method under earthquake action considering performance degradation of anchor cables is provided for an embodiment of the present application. Figure 2 A slide block division model schematic diagram in a slope stability analysis method under earthquake action considering performance degradation of anchor cables is provided for an embodiment of the present application. Figure 3 A mass block division schematic diagram in a slope stability analysis method under earthquake action considering performance degradation of anchor cables is provided for an embodiment of the present application. DETAILED DESCRIPTION

[0039] The application will be further described in details through specific embodiments combined with the drawings. In different embodiments, similar elements are associated with similar element labels. In the following embodiments, many details are described in order to make the application better understood. However, those skilled in the art can easily recognize that some features can be omitted or replaced by other elements, materials or methods in different cases. In some cases, some operations related to the application are not shown or described in the specification in order to avoid the core part of the application being overwhelmed by too much description, and it is not necessary to describe these related operations in detail for those skilled in the art based on the description in the specification and general technical knowledge in the art.

[0040] In addition, the features, operations or characteristics described in the specification can be combined in any appropriate way to form various embodiments. At the same time, the steps or actions in the method description can also be sequentially adjusted or adjusted in a manner that is obvious to those skilled in the art. Therefore, the order in the specification and drawings is only for clear description of one embodiment, and does not mean the necessary order, unless otherwise stated that a certain order must be followed.

[0041] The slope stability analysis method under earthquake action considering performance degradation of anchor cables provided by the first embodiment of the present application will be described in detail below. Figure 1

[0042] S1, a geometric model of an anchored rock slope is constructed, and the geometric parameters of the slope, the physical and mechanical parameters of the slope, and the anchor support parameters are set.

[0043] S2, the critical sliding surface is determined by using the upper limit method of limit analysis, and the sliding body is divided into multiple slide block units with the anchor cable position as the center, wherein each slide block unit corresponds to one anchor cable (such as Figure 2 ​As shown in the figure, the dashed line is the dividing line of the slider (the red line represents the anchor cable). The upper limit solution of the safety factor and the initial critical yield acceleration coefficient are solved by combining the principle of virtual work, and the dynamic response stages of the slope are divided.

[0044] The specific steps of step S2 are as follows: Step S21, based on the energy relationship of the upper limit method of limit analysis, the calculation formula for the minimum energy consumption of the critical slip surface of the seismic slope is as follows:

[0045] In the formula, It is the strain rate in the kinematically permissible velocity field. Is with The relevant effective stress tensor, It is a surface force vector. It is a vector of distributed forces. It is a velocity vector. and These are the loading boundary and the volume, respectively. Slope stability analysis can be transformed into a problem of searching for the most dangerous slip surface of a slope. The critical slip surface refers to the slip surface that minimizes the energy dissipation (or minimizes the safety factor) of the slope system, and is the most dangerous and most likely to occur. The purpose of solving for the minimum energy dissipation is to filter out the most dangerous "critical slip surface" from countless possible potential slip surfaces. It defines the mathematical optimization objective that needs to be approximated and solved in the subsequent numerical calculation step S22, thereby determining the true safety factor of the slope.

[0046] Step S22: Based on the minimum energy consumption of the critical slip surface, solve for the upper limit of the safety factor of each slider. Initial critical yield acceleration coefficient Solution to the upper limit of slope safety factor The details are as follows: According to the principle of virtual work, internal energy consumption is equal to external power:

[0047] In the formula, The power of the slider's gravity. The horizontal seismic force power of the slider. The work done by the tension of the prestressed anchor cable on the slider. for i +1 slider pair i The work done by the slider's force for i -1 slider pair i The work done by the slider's force The power used to account for energy consumption on the sliding surface at the bottom of the slider. The energy consumption power of the inclined interface between the sliders; Based on the upper bound theorem of limit analysis and the strength reduction method, the safety factors of each slider are calculated. The implicit function expression is as follows:

[0048] And by further expanding the above equation, we get:

[0049] In the formula, For the safety factor of each slider, The unit weight of the rock and soil mass for the slider. Let be the volume of the slider. The angle between the bottom of the slider and the horizontal line. absolute velocity of the slider The angle between the sliding surface and the sliding surface To satisfy the geometric compatibility condition for the velocity field, This is the horizontal seismic force coefficient. For the tension of the prestressed anchor cable of the slider, Anchor cable inclination angle, The cohesive force of the slider. for i +1 slider pair i Sliding force, for i -1 slider pair i Sliding force, The length of the bottom of the slider. The equivalent cohesive force at the slider interface. The length of the inclined interface between adjacent sliders. relative velocity The angle between the inclined plane and the inclined plane It is the relative velocity between the two sliders; Yield acceleration coefficient of each slider The calculation formula is:

[0050] The design horizontal seismic coefficient, which is a known input condition, is substituted into the formula to solve for the safety factor of the current slope. It is the critical yield acceleration coefficient calculated in reverse, representing the upper limit of seismic force that the slope can withstand to maintain stability (i.e., the slope's seismic capacity reserve); the two have different physical meanings, only differing when the slope is just in a state of limit equilibrium (i.e. The values ​​are equal only under the specific condition that =1).

[0051] upper limit solution of slope safety factor The implicit function expression is as follows:

[0052] Step S23: Based on the upper limit solution of the slope safety factor and combined with the ultimate state of the anchor cable, divide the dynamic response stages of the slope: when At this point, it is the elastic coordinated response stage of the anchor cable, i.e., stage one. In this stage, the slider has no displacement, the anchor cable is in the elastic coordinated vibration stage, the anchor cable deformation energy recovers, and the anchor cable tension is constant at the prestress level. ; when At this point, it is the elastic response stage, or stage two. In this stage, the slider undergoes relative sliding, and the slider produces plastic displacement. The increase in anchor cable tension is not actively applied, but rather passively forced by the displacement of the slider, resulting in the anchor cable elongating and the anchor cable tension becoming... ; when At this time, it is the plastic yielding and failure stage, i.e., stage three, which is used for anchor cable failure determination; in, For the initial prestress of the anchor cable The safety factor of the slope below, For anchor cable tensile bearing capacity The slope safety factor is determined by... and Substitute the upper limit solution of the slope safety factor from step S22 into the solution. The implicit function expression is calculated.

[0053] S3. Establish a multi-slider collaborative dynamic calculation model for anchored rock slopes, construct a multi-degree-of-freedom lumped mass-spring-damping system based on discrete multiple slider elements, and establish the dynamic response control equations of the system. The specific steps of step S3 are as follows: Step S31, using the n equivalent lumped mass blocks generated by the sliding body discretization (e.g. Figure 3 As shown), based on d'Alembert's principle, the equations of motion for a multi-degree-of-freedom system including inter-block coupling are established as follows:

[0054] In the formula, For the quality matrix, Here is the damping matrix. Here is the stiffness matrix; , , These are vectors representing displacement, velocity, and acceleration. It is a unit vector. For horizontal seismic acceleration, The gravity load vector. The anchor cable force load vector. This is the force vector between the blocks, where forces acting downwards along the sliding surface are defined as positive and forces acting upwards as negative. Step S32, establish the component equations of the multi-degree-of-freedom system's motion equations along the tangential (s) direction and the normal (n) direction of the sliding surface: s direction:

[0055] n direction:

[0056] In the formula, For the quality matrix, and These are the damping matrices along the tangential direction (s-direction) and the normal direction (n-direction) of the sliding surface, respectively. and These are the stiffness matrices in the s and n directions, respectively; , , Let S be the displacement, velocity, and acceleration vector along the tangential direction (s) of the sliding surface. , , Let n be the vector of displacement, velocity, and acceleration along the normal direction of the sliding surface, i.e., the n-direction. It is a unit vector. For horizontal seismic acceleration, Gravitational acceleration, For stage one, the anchor cable force load vector is: , This represents the force vector between the blocks. The angle between the bottom of the slider and the horizontal line. The inclination angle of the prestressed anchor cable.

[0057] S4. Apply seismic vibration, using Newmark- β The dynamic response of each slider anchor cable is solved by the method of Newmark slider displacement method; The specific steps of step S4 are as follows: Step S41, input the ground motion, using Newmark- β The method for solving the dynamic response of each slider anchor (the lumped mass model binds the sliders corresponding to the anchor's influence range to the anchor) is as follows:

[0058] In the above formula, This represents the total axial tension of the anchor cable. and These represent the static elongation of the anchor cable in the s and n directions under prestressing, respectively. and These represent the dynamic displacement responses of the anchor cables in the s and n directions under seismic loading, respectively. This was achieved using Newmark- β The method is to solve the component equations of the multi-degree-of-freedom system in the s and n directions by time history integration in step S32; The angle between the bottom of the slider and the horizontal line; The inclination angle of the prestressed anchor cable; The formulas for calculating the static elongation of the anchor cable in the s and n directions under prestressing are as follows:

[0059]

[0060] In the formula, For the initial prestress of the anchor cable force, l The length of the free section of the anchor cable. E The elastic modulus of the anchor cable. A The cross-sectional area of ​​the free section of the anchor cable. The angle between the bottom of the slider and the horizontal line. The inclination angle of the prestressed anchor cable.

[0061] Step S42, the calculation formulas for the acceleration, velocity, and displacement of each slider based on the Newmark slider displacement method are as follows:

[0062]

[0063]

[0064] The formula for calculating cumulative permanent displacement is:

[0065] In the formula, for t The relative acceleration of the slider along the sliding surface at any given moment. g It is the acceleration due to gravity. for t The horizontal seismic acceleration coefficient input at any time, for t The critical yield acceleration coefficient of the slider at time t. The angle between the bottom of the slider and the horizontal line. for t The relative velocity of the slider along the sliding surface at any given moment. For the current calculation time, For time step, For the outer integral variable (time). For the previous moment The initial velocity of the slider at the end, This represents the displacement increment within the current time step. For the current moment t The cumulative permanent displacement, For the previous moment The cumulative permanent displacement.

[0066] S5. Based on the instantaneous acceleration and relative displacement relationship between adjacent sliders, the interaction mode between sliders is determined in real time, the interaction force between sliders is calculated, and the stability state of adjacent sliders is iteratively updated. The specific steps of step S5 are as follows: Step S51: Based on the instantaneous acceleration and relative displacement relationship between adjacent sliders, determine the interaction mode between the sliders in real time. when ,but i slider pair i +1 slider generates a traction effect; when ,but i slider pair i +1 slider creates a blocking effect; in, Let be the relative acceleration of slider i along the sliding surface at time t; Let be the relative acceleration of slider i+1 along the sliding surface at time t; Step S52, through i slider and i +1 slider t Calculate and update the inter-block forces based on the displacement and relative displacement at time t:

[0067]

[0068] In the formula, for t time i +1 slider pair i The force applied by the slider for t -1 hour i +1 slider pair i The force applied by the slider For the current calculation time, For time step, for i slider and i +1 Stiffness coefficient between the contact surfaces of the slider for i slider andi +1 Damping coefficient between the slider contact surfaces for t time i slider and i +1 Relative displacement of slider For time t i slider and i +1 speed difference of the slider for t time i slider pair i +1 The force applied by the slider; Step S53, according to The interaction forces between the sliders are solved at all times, and the stability state of itself and its adjacent sliders is updated iteratively, including various safety factors, yield acceleration coefficients, anchor cable stiffness and tension state; like:

[0069] Then in i +1 slider action i slider and i +1 slider is stable; like:

[0070] express i The slider cannot be blocked i The downward trend of the +1 slider, then i slider and i +1 sliders all became unstable; In the formula, The power of the slider's gravity. The horizontal seismic force power of the slider. for t The work done by the anchor cable tension at all times for t time i +1 slider pair i The work done by the slider's force for i -1 slider pair i The work done by the slider's force The power used to account for energy consumption on the sliding surface at the bottom of the slider. The energy consumption power of the inclined interface between the sliders; The formulas for updating the anchor cable stiffness and tension state are as follows:

[0071]

[0072] In the formula, for t The cumulative axial force of the anchor cable at any given moment For the initial prestress of the anchor cable, for t Anchor stiffness at any given time for t The cumulative plastic displacement of the slider at each moment, The elastic modulus of the anchor cable. for t The cross-sectional area of ​​the anchor cable at time . The length of the free segment of the anchor cable. for t The dynamic axial force of the anchor cable at any given moment. for t The axial tension of the anchor cable vibration at a given moment; Step S54: Use the updated safety factor, yield acceleration coefficient, anchor cable stiffness, and tension state from step S53 as the data for the next time step. + The initial conditions are used for recursive calculation.

[0073] S6. Establish a multi-mode anchor cable failure discrimination model and combine it with the upper limit method of limit analysis to evaluate the entire process of slope dynamic instability.

[0074] The specific steps of step S6 are as follows: Step S61, establish a multi-mode anchor cable failure discrimination model: For brittle fracture:

[0075] In the formula, Accumulate axial force for anchor cables, For the initial prestress of the anchor cable, For the tensile stiffness of the anchor cable, Accumulate plastic displacement for the slider. The elastic modulus of the anchor cable. Let be the cross-sectional area of ​​the anchor cable. The length of the free segment of the anchor cable. For the axial force of the anchor cable, This represents the axial tension caused by anchor cable vibration. Ultimate bearing capacity under brittle failure of anchor cables; For ductile failure:

[0076] In the formula, For the tensile bearing capacity of the anchor cable, This is the maximum elongation of the anchor cable; Step S62, real-time state determination, calculate the cumulative plastic displacement of the slider at each time t. Axial tension of anchor cable vibration and the updated anchor cable stiffness Anchor cable dynamic axial force and cumulative axial force of anchor cables The model is used to determine the failure mode of anchor cables, which is established in step S61. The determination is based on whether the failure exceeds the ultimate bearing capacity of the corresponding failure mode. or To determine whether the anchor cable has suffered brittle or ductile failure.

[0077] The above examples illustrate the present invention only to aid in understanding it and are not intended to limit the scope of the invention. Those skilled in the art can make various simple deductions, modifications, or substitutions based on the principles of this invention.

Claims

1. A method for slope stability analysis under seismic loading considering the deterioration of anchor cable performance, characterized in that, The method includes: Step S1: Construct a geometric model of the anchored rock slope and set the slope's geometric parameters, physical and mechanical parameters, and anchor cable support parameters. Step S2: Determine the critical slip surface using the upper limit method of limit analysis, and discretize the sliding body into multiple slider elements with the anchored slope position as the center. Each slider element corresponds to an anchor cable. Solve the upper limit solution of the safety factor and the initial critical yield acceleration coefficient by combining the principle of virtual work, and divide the slope dynamic response stages. Step S3: Establish a multi-slider collaborative dynamic calculation model for anchored rock slopes. Construct a multi-degree-of-freedom lumped mass-spring-damping system based on discrete multiple slider elements, and establish the dynamic response control equations of the system. Step S4, apply ground vibration, using Newmark- β The dynamic response of each slider anchor cable is solved by the method of Newmark slider displacement method; Step S5: Based on the instantaneous acceleration and relative displacement relationship between adjacent sliders, the interaction mode between sliders is determined in real time, the interaction force between sliders is calculated, and the stability state of adjacent sliders is iteratively updated. Step S6: Establish a multi-mode anchor cable failure discrimination model and combine it with the upper limit method of limit analysis to evaluate the entire process of slope dynamic instability.

2. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 1, characterized in that, Step S2 specifically includes: Step S21, based on the energy relationship of the upper limit method of limit analysis, the calculation formula for the minimum energy consumption of the critical slip surface of the seismic slope is as follows: In the formula, It is the strain rate in the kinematically permissible velocity field. Is with The relevant effective stress tensor, It is a surface force vector. It is a vector of distributed forces. It is a velocity vector. and These are the loading boundary and the volume, respectively. By solving for the minimum energy consumption, the most dangerous critical slip surface can be screened from multiple potential slip surfaces; Step S22: Based on the minimum energy consumption of the critical slip surface, solve for the upper limit of the safety factor of each slider. Initial critical yield acceleration coefficient Solution to the upper limit of slope safety factor The details are as follows: According to the principle of virtual work, internal energy consumption is equal to external power: In the formula, The power of the slider's gravity. The horizontal seismic force power of the slider. The work done by the tension of the prestressed anchor cable on the slider. for i +1 slider pair i The work done by the slider's force for i -1 slider pair i The work done by the slider's force The power used to account for energy consumption on the sliding surface at the bottom of the slider. The energy consumption power of the inclined interface between the sliders; Based on the upper bound theorem of limit analysis and the strength reduction method, the safety factors of each slider are calculated. The implicit function expression is as follows: And by further expanding the above equation, we get: In the formula, For the safety factor of each slider, The unit weight of the rock and soil mass for the slider. Let be the volume of the slider. The angle between the bottom of the slider and the horizontal line. absolute velocity of the slider The angle between the sliding surface and the sliding surface To satisfy the geometric compatibility condition for the velocity field, This is the horizontal seismic force coefficient. For the tension of the prestressed anchor cable of the slider, Anchor cable inclination angle, The cohesive force of the slider. for i +1 slider pair i Sliding force, for i -1 slider pair i Sliding force, The length of the bottom of the slider. The equivalent cohesive force at the slider interface. The length of the inclined interface between adjacent sliders. relative velocity The angle between the inclined plane and the inclined plane It is the relative velocity between the two sliders; Yield acceleration coefficient of each slider The calculation formula is: upper limit solution of slope safety factor The implicit function expression is as follows: Step S23: Based on the upper limit solution of the slope safety factor and combined with the ultimate state of the anchor cable, divide the dynamic response stages of the slope: when At this point, it is the elastic coordinated response stage of the anchor cable, i.e., stage one. In this stage, the slider has no displacement, the anchor cable is in the elastic coordinated vibration stage, the anchor cable deformation energy recovers, and the anchor cable tension is constant at the prestress level. ; when At this point, it is the elastic response stage, or stage two. In this stage, the slider undergoes relative sliding, and the slider produces plastic displacement. The increase in anchor cable tension is not actively applied, but rather passively forced by the displacement of the slider, resulting in the anchor cable elongating and the anchor cable tension becoming... ; when At this time, it is the plastic yielding and failure stage, i.e., stage three, which is used for anchor cable failure determination; in, For the initial prestress of the anchor cable The safety factor of the slope below, For anchor cable tensile bearing capacity The slope safety factor is determined by... and Substitute each into the upper limit solution of the slope safety factor. The implicit function expression is calculated.

3. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 2, characterized in that, Step S3 specifically includes: Step S31: Using the n equivalent lumped mass blocks generated by the sliding body discretization, and based on d'Alembert's principle, the equations of motion for the multi-degree-of-freedom system including inter-block coupling are established as follows: In the formula, For the quality matrix, Here is the damping matrix. Here is the stiffness matrix; , , These are vectors representing displacement, velocity, and acceleration. It is a unit vector. For horizontal seismic acceleration, The gravity load vector. The anchor cable force load vector. This is the force vector between the blocks, where forces acting downwards along the sliding surface are defined as positive and forces acting upwards as negative. Step S32, establish the component equations of the multi-degree-of-freedom system's motion equations along the tangential (s) direction and the normal (n) direction of the sliding surface: s direction: n direction: In the formula, For the quality matrix, and These are the damping matrices along the tangential direction (s-direction) and the normal direction (n-direction) of the sliding surface, respectively. and These are the stiffness matrices in the s and n directions, respectively; , , Let S be the displacement, velocity, and acceleration vector along the tangential direction (s) of the sliding surface. , , Let n be the vector of displacement, velocity, and acceleration along the normal direction of the sliding surface, i.e., the n-direction. It is a unit vector. For horizontal seismic acceleration, Gravitational acceleration, For stage one, the anchor cable force load vector is: , This represents the force vector between the blocks. The angle between the bottom of the slider and the horizontal line. The inclination angle of the prestressed anchor cable.

4. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 3, characterized in that, Step S4 specifically includes: Step S41, input the ground motion, using Newmark- β The dynamic response of each slider anchor cable is solved by the following method, which is to solve for the axial tension of the anchor cable vibration: In the above formula, This represents the total axial tension of the anchor cable. and These represent the static elongation of the anchor cable in the s and n directions under prestressing, respectively. and These represent the dynamic displacement responses of the anchor cables in the s and n directions under seismic loading, respectively. This was achieved using Newmark- β The method is to solve the component equations of motion of a multi-degree-of-freedom system in the s and n directions by time history integration. The angle between the bottom of the slider and the horizontal line; The inclination angle of the prestressed anchor cable; Step S42, the calculation formulas for the acceleration, velocity, and displacement of each slider based on the Newmark slider displacement method are as follows: The formula for calculating cumulative permanent displacement is: In the formula, for t The relative acceleration of the slider along the sliding surface at any given moment. g It is the acceleration due to gravity. for t The horizontal seismic acceleration coefficient input at any time, for t The critical yield acceleration coefficient of the slider at time t. The angle between the bottom of the slider and the horizontal line. for t The relative velocity of the slider along the sliding surface at any given moment. For the current calculation time, For time step, For the outer integral variable (time). For the previous moment The initial velocity of the slider at the end, This represents the displacement increment within the current time step. For the current moment t The cumulative permanent displacement, For the previous moment The cumulative permanent displacement.

5. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 4, characterized in that, In step S41, the formula for calculating the static elongation of the anchor cable in the s and n directions under prestressing is as follows: In the formula, For the initial prestress of the anchor cable force, l The length of the free section of the anchor cable. E The elastic modulus of the anchor cable. A The cross-sectional area of ​​the free section of the anchor cable. The angle between the bottom of the slider and the horizontal line. The inclination angle of the prestressed anchor cable.

6. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 4, characterized in that, Step S5 specifically includes: Step S51: Based on the relationship between the instantaneous acceleration and relative displacement of adjacent sliders, determine the interaction mode between the sliders in real time. when ,but i slider pair i +1 slider generates a traction effect; when ,but i slider pair i +1 slider creates a blocking effect; in, Let be the relative acceleration of slider i along the sliding surface at time t; Let be the relative acceleration of slider i+1 along the sliding surface at time t; Step S52, through i slider and i +1 slider t Calculate and update the inter-block forces based on the displacement and relative displacement at time t: In the formula, for t time i +1 slider pair i The force applied by the slider for t -1 hour i +1 slider pair i The force applied by the slider For the current calculation time, For time step, for i slider and i +1 Stiffness coefficient between the contact surfaces of the slider for i slider and i +1 Damping coefficient between the slider contact surfaces for t time i slider and i +1 Relative displacement of slider For time t i slider and i +1 speed difference of the slider for t time i slider pair i +1 The force applied by the slider; Step S53, according to The interaction forces between the sliders are solved at all times, and the stability state of itself and its adjacent sliders is updated iteratively, including various safety factors, yield acceleration coefficients, anchor cable stiffness and tension state; like: Then in i +1 slider action i slider and i +1 slider is stable; like: express i The slider cannot be blocked i The downward trend of the +1 slider, then i slider and i +1 sliders all became unstable; In the formula, The power of the slider's gravity. The horizontal seismic force power of the slider. for t The work done by the anchor cable tension at all times for t time i +1 slider pair i The work done by the slider's force for i -1 slider pair i The work done by the slider's force The power used to account for energy consumption on the sliding surface at the bottom of the slider. The energy consumption power of the inclined interface between the sliders; The formulas for updating the anchor cable stiffness and tension state are as follows: In the formula, for t The cumulative axial force of the anchor cable at any given moment For the initial prestress of the anchor cable, for t Anchor stiffness at any given time for t The cumulative plastic displacement of the slider at each moment, The elastic modulus of the anchor cable. for t The cross-sectional area of ​​the anchor cable at time . The length of the free segment of the anchor cable. for t The dynamic axial force of the anchor cable at any given moment. for t The axial tension of the anchor cable vibration at a given moment; Step S54: The updated safety factor, yield acceleration coefficient, anchor cable stiffness, and tension state are used as the next time step. + The initial conditions are used for recursive calculation.

7. The slope stability analysis method under seismic loading considering anchor cable performance degradation as described in claim 5, characterized in that, Step S6 specifically includes: Step S61, establish a multi-mode anchor cable failure discrimination model: For brittle fracture: In the formula, Accumulate axial force for anchor cables, For the initial prestress of the anchor cable, For the tensile stiffness of the anchor cable, Accumulate plastic displacement for the slider. The elastic modulus of the anchor cable. Let be the cross-sectional area of ​​the anchor cable. The length of the free segment of the anchor cable. For the anchor cable dynamic axial force, This represents the axial tension caused by anchor cable vibration. Ultimate bearing capacity under brittle failure of anchor cables; For ductile failure: In the formula, For the tensile bearing capacity of the anchor cable, This is the maximum elongation of the anchor cable; Step S62, real-time state determination, calculate the cumulative plastic displacement of the slider at each time t. Axial tension of anchor cable vibration and the updated anchor cable stiffness Anchor cable dynamic axial force and cumulative axial force of anchor cables The results are then fed into the established multi-mode anchor cable failure discrimination model to determine whether the anchor cable has experienced brittle or ductile failure.

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