Calculation Method for Ultimate Displacement in Design of Landslide Retaining Piles for Slope Reinforcement

Abstract

The present invention discloses a calculation method for the design ultimate displacement of a landslide - resisting pile for slope reinforcement, including the steps: Step 100: Vertically divide the landslide into several slices and number each slice in sequence from the back to the front; Step 200: Establish the corresponding static equilibrium equations for each slice. The static equilibrium equation of the slice where the loaded section of the landslide - resisting pile is located needs to include the internal force of the landslide - resisting pile at the potential slip surface; Step 300: Form the relationship expressions of the bending moment and axial force with the shear force as the variable and the relationship expressions of each internal force with the design safety factor of slope stability as the independent variable; Step 400: Establish the displacement expressions of each slice; Step 500: Express the shear force of the slice bottom along the direction of the potential slip surface; Step 600: Determine the inter - slice shear force and inter - slice normal force between adjacent slices; Step 700: Solve to obtain the vertical design ultimate displacement at the top of the landslide; Step 800: Solve to obtain the horizontal design ultimate displacement and the tangential design ultimate displacement along the potential slip surface of each slice.

Description

Technical Field

[0001] The present invention relates to the technical field of landslides and engineering slopes. Specifically, it relates to a method for calculating the design ultimate displacement of a slope reinforced by anti-slide piles. Background Art

[0002] The stability state of a slope (landslide) reinforced by anti-slide piles is closely related to the slope displacement. When the slope reaches the ultimate state of sliding, certain displacement will inevitably occur. The displacement that the slope can generate when it reaches the ultimate state (including the design ultimate state) (referred to as the ultimate displacement here) is of great significance for fully understanding the potential displacement development of the slope. However, except for numerical simulation methods, most existing slope stability analysis methods do not link stability with displacement, nor can they reasonably analyze the design ultimate displacement of the slope. Although there were a few stability analysis methods related to displacement in the past, the relevant analytical calculation methods were all for slopes without reinforcement treatment and did not involve the problem of slopes reinforced by anti-slide piles, and the design ultimate displacement of slopes reinforced by anti-slide piles could not be solved by analytical calculation methods. Generally, only through numerical simulation methods can this problem be approximately analyzed.

[0003] Although numerical simulation methods can be used to analyze the ultimate displacement of slopes, in fact, by using the shear strength reduction technique, the shear strength of the sliding zone soil of the slope is continuously reduced in the numerical simulation until the slope of the numerical model reaches the defined ultimate state. At this time, the slope displacement is considered as its corresponding sliding ultimate displacement. However, numerical simulation methods have two defects:

[0004] One is that a numerical model needs to be established first, and the rationality of the numerical model depends on elements such as model parameters, mesh accuracy, material constitutive model, and boundary conditions. Not only is the modeling process complex, but there are also subjective human operation interferences, and it is difficult to have "inheritance" (different people need to start from the modeling operation). It can be used as a reference means for studying complex problems, but it is not conducive to the rapid analysis and operation of actual engineering technicians.

[0005] The second is that during the strength reduction process, the deformation parameters of the slope should also be adjusted synchronously and adaptively with the strength parameters. However, there is currently no reasonable synchronous adjustment strategy, so this method has a conceptual defect. Summary of the Invention

[0006] The main purpose of the present invention is to provide a method for calculating the design ultimate displacement of a slope reinforced by anti-slide piles to solve the technical problem in the prior art that the design ultimate displacement of a slope reinforced by anti-slide piles cannot be reasonably analyzed by analytical calculation.

[0007] To achieve the above purpose, the present invention provides a method for calculating the design ultimate displacement of a slope reinforced by anti-slide piles, and the technical solution is as follows:

[0008] The calculation method of the design limit displacement of the anti-slide pile reinforced slope includes the following steps:

[0009] Step 100: For the slope reinforced with anti-slide piles, based on the basic geometric and physical mechanical parameters of the slope, the pile design parameters, and the specified slope stability design safety factor, the location of the potential sliding surface of the slope and the length of the load-bearing section of the anti-slide pile are determined. The sliding body is vertically divided into a number of strips from the trailing edge of the sliding body forward, and each strip is numbered from the back to the front. The bottom length, bottom surface inclination, and deadweight of each strip are determined. The load-bearing section of the anti-slide pile is included in a certain strip.

[0010] Step 200: Under the specified slope stability design safety factor, for each strip, a corresponding static equilibrium equation is established based on the static equilibrium conditions of horizontal force, vertical force, and moment. For the strip where the load-bearing section of the anti-slide pile is located, the static equilibrium equation must include the internal forces of the anti-slide pile at the potential sliding surface, which include shear force, bending moment, and axial force.

[0011] Step 300: Establishing calculation expressions for various internal forces of the anti-slide pile at the potential sliding surface, forming relationship expressions for bending moment and axial force with shear force as a variable, and relationship expressions for various internal forces with slope stability design safety factor as an independent variable;

[0012] Step 400: Based on the sliding displacement compatibility conditions between the blocks, the displacement expressions of the blocks are established sequentially from the trailing edge of the sliding body forward.

[0013] Step 500: In the static equilibrium equation of each bar, the shear force on the bottom surface of the bar along the potential sliding surface is expressed using the shear stress-shear displacement relationship equation;

[0014] Step 600: Determine the relationship between the inter-strip tangential force and the inter-strip normal force between two adjacent blocks based on the Morgenstern-Price inter-strip force assumption.

[0015] Step 700: Under the specified slope stability design safety factor, the static equilibrium equations and constraints of all strips are solved to obtain the vertical design limit displacement of the top of the sliding body of the anti-slide pile reinforced slope;

[0016] Step 800: Substitute the vertical design limit displacement of the top of the sliding body into the displacement expression of each strip. Under the specified slope stability design safety factor, the horizontal design limit displacement and the tangential design limit displacement along the potential sliding surface of each strip of the anti-slide pile reinforced slope can be obtained.

[0017] The prominent advantages of the calculation method for the design ultimate displacement of the landslide - resistant pile - reinforced slope in the present invention are as follows: It fully considers key control factors such as the stability design safety factor of the slope reinforced by the landslide - resistant pile, the internal force action of the landslide - resistant pile, static equilibrium, deformation coordination, and shear constitutive relationship. The calculation principle is clear, the calculation process is simple and easy to operate, and the calculation result is relatively accurate. It avoids the complex modeling analysis steps and time - consuming calculation process of the numerical simulation method. It can not only reasonably reflect the stability and deformation coordination relationship of the slope reinforced by the landslide - resistant pile, but also has a simplified solution process, providing a convenient and effective method for the treatment design and analysis of landslides or engineering slopes, taking into account both technical significance and engineering practical value.

[0018] The following further describes the present invention in conjunction with the accompanying drawings and specific embodiments. The additional aspects and advantages of the present invention will be partially given in the following description, partially become obvious from the following description, or be understood through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The drawings forming a part of the present invention are used to assist in understanding the present invention. The content provided in the drawings and the related description in the present invention can be used to explain the present invention, but do not constitute an improper limitation to the present invention. In the drawings:

[0020] Figure 1 It is a schematic diagram of the vertical segmentation strip of the sliding body in the slope reinforced by the landslide - resistant pile.

[0021] Figure 2 It is a schematic diagram of the force analysis of a strip containing the load - bearing section of the landslide - resistant pile in the sliding body of the slope reinforced by the landslide - resistant pile.

[0022] Figure 3 It is a schematic diagram of the force analysis of an arbitrary strip without the landslide - resistant pile in the sliding body of the slope reinforced by the landslide - resistant pile.

[0023] Figure 4 It is a schematic diagram of a certain landslide - resistant - pile - reinforced slope in the embodiment.

[0024] Figure 5 It is a schematic diagram of the vertical segmentation strip of the sliding body in the embodiment.

[0025] Figure 6 It is a comparison diagram of the tangential design ultimate displacement of each point on the sliding - body surface along the potential slip surface obtained by numerical simulation in the embodiment and the calculation result of the present invention.

[0026] Figure 7 It is a comparison diagram of the horizontal design ultimate displacement of each point on the sliding - body surface obtained by numerical simulation in the embodiment and the calculation result of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0027] The present invention will be described clearly and completely in conjunction with the accompanying drawings. Those of ordinary skill in the art will be able to implement the present invention based on these descriptions. Before describing the present invention in conjunction with the accompanying drawings, it should be particularly noted that:

[0028] In the present invention, the technical solutions and technical features provided in each part including the following description can be combined with each other without conflict.

[0029] In addition, the embodiments of the present invention involved in the following description are usually only part of the embodiments of the present invention, rather than all of the embodiments. Therefore, all other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts should fall within the scope of protection of the present invention.

[0030] Regarding the terms and units in the present invention. The terms "include", "have" and any variations thereof in the specification, claims and relevant parts of the present invention are intended to cover non-exclusive inclusion.

[0031] The specific implementation manner of the design ultimate displacement calculation method for the anti-slide pile reinforced slope of the present invention includes steps 100 - 800, specifically as follows:

[0032] Step 100: For the slope reinforced by anti-slide piles, based on the basic geometric and physical and mechanical parameters of the slope body, the pile body design parameters, and the specified slope stability design safety factor, determine the position of the potential slip surface of the slope body and the length of the loaded section of the anti-slide pile. Vertically divide the sliding body into several blocks in sequence from the rear edge of the sliding body and number each block in sequence from the rear to the front. Determine the bottom length, bottom inclination angle, and the self-weight of each block. Among them, the loaded section of the anti-slide pile is included in a certain block. Specifically as follows:

[0033] Among them, through calculation, determine the potential slip surface of the slope reinforced by anti-slide piles under the specified slope stability design safety factor. The loaded section of the anti-slide pile refers to the part of the anti-slide pile above this potential slip surface. Among them, the calculation can adopt the limit equilibrium method or the numerical simulation method. The former such as: Morgenstern-Price method, transfer coefficient method; the latter such as: finite element strength reduction method.

[0034] Figure 1 It is a schematic diagram of the vertical division of the sliding body in the slope body reinforced by anti-slide piles.

[0035] Such as Figure 1As shown in the figure, 10 is an anti-slide pile, 20 is a landslide mass, 30 is a potential slip surface, 40 is the surface of the landslide mass, and 50 is the first slice of the rear edge of the landslide mass. The total number of slices n for vertically slicing the landslide mass 20 should be large enough. Preferably, n≥15. The slices are numbered sequentially from 1 to n one by one from the rear edge to the front edge of the landslide mass 20.

[0036] Step 200: Under the specified slope stability design safety factor condition, for each slice, according to the static equilibrium conditions of horizontal force, vertical force, and moment, establish the corresponding static equilibrium equations. Among them, for the slice where the loaded section of the anti-slide pile is located, the internal forces of the anti-slide pile at the potential slip surface, including shear force, bending moment, and axial force, need to be included in the static equilibrium equations. Specifically as follows:

[0037] Figure 2 is a schematic diagram of the force analysis of a slice j containing the loaded section of the anti-slide pile in the landslide mass. As Figure 2 shown, for the force analysis of slice j, according to the static equilibrium conditions of horizontal force, vertical force, and moment, three static equilibrium equations can be obtained as:

[0038]

[0039] In the formula, W j is the self-weight of slice j; N j is the normal force acting on the bottom surface of slice j; T j is the tangential force acting on the bottom surface of slice j; H j is the inter-slice tangential force acting on the vertical slicing interface on the front side of slice j; P j is the inter-slice normal force acting on the vertical slicing interface on the front side of slice j; q j is the uniformly distributed load on the top surface of slice j; z j is the distance from the inter-slice normal force P j along the vertical slicing interface direction to the bottom surface of slice j; b j is the width of slice j; N p , M p and Q p are respectively the axial force, bending moment, and shear force of the anti-slide pile (the pile body cross-section at the bottom end of the loaded section) at the potential slip surface; α j is the inclination angle of the bottom surface of slice j.

[0040] Figure 3 is a schematic diagram of the force analysis of any slice i in the landslide mass that does not contain the loaded section of the anti-slide pile. As Figure 3 shown, for the force analysis of slice i, according to the static equilibrium conditions of horizontal force, vertical force, and moment, three static equilibrium equations can be obtained as:

[0041]

[0042] Where \(i\) is the number of the slice excluding the loaded section of the anti-slide pile, \(1\leq i\leq n\), and \(i\neq j\), \(n\) is the total number of slices; \(W\) i is the self-weight of slice \(i\); \(N\) i is the normal force acting on the bottom surface of slice \(i\); \(T\) i is the tangential force acting on the bottom surface of slice \(i\); \(H\) i is the inter-slice tangential force acting on the vertical sub-slice interface in front of slice \(i\); \(P\) i is the inter-slice normal force acting on the vertical sub-slice interface in front of slice \(i\); \(q\) i is the uniformly distributed load on the top surface of slice \(i\); \(z\) i is the distance of the inter-slice normal force \(P\) i from the bottom surface of slice \(i\) along the vertical sub-slice interface direction; \(b\) i is the width of slice \(i\); \(\alpha\) i is the inclination angle of the bottom surface of slice \(i\).

[0043] Step 300: Establish the calculation expressions for the internal forces of the anti-slide pile at the potential slip surface, form the relationship expressions between the bending moment and the axial force with the shear force as the variable, and the relationship expressions between the internal forces with the design safety factor of slope stability as the independent variable. Specifically as follows:

[0044] According to the plastic deformation mode of the soil on the pile side proposed by Japanese scholars Ito et al. (Ito T, Matsui T. Methods to estimate lateral force acting on stability piles[J]. Soils and Foundations, 1975, 15(4): 43–59.), under the specified design safety factor \(F\) s condition, the simplified relationship expression of the design thrust \(p(y)\) (linear distributed force along the height) of the soil on the pile side is:

[0045]

[0046] Where \(D\) is the pile diameter or the side length of the pile cross-section, \(y\) represents the depth from the pile top, \(\gamma\) represents the soil unit weight; \(\eta_1\), \(\eta_2\), \(\eta_3\), are dimensionless coefficients; \(c_0\) and are the calculated cohesion and calculated internal friction angle of the potential slip surface soil considering the design safety factor of slope stability respectively;

[0047] The calculation expressions of \(\eta_1\), \(\eta_2\), \(\eta_3\), are respectively:

[0048]

[0049] The calculation expressions of \(c_0\), are respectively:

[0050]

[0051] In the formula, c and are the actual average cohesion and average internal friction angle of the soil mass on the potential slip surface respectively; F s is the safety factor for stability design.

[0052] Thus, for the loaded section of the anti-slide pile, from the horizontal static equilibrium relationship, the designed shear force Q of the pile body at the potential slip surface can be obtained as: p The expression is:

[0053]

[0054] Then, substituting Equation (3) into Equation (4), the relationship expression of the shear force Q of the anti-slide pile at the potential slip surface with the safety factor for slope stability design as the independent variable can be obtained. p

[0055] From the static equilibrium condition of the loaded section of the anti-slide pile, the axial force N and bending moment M of the anti-slide pile at the potential slip surface can be obtained, and there are the following relationship expressions between N, M and the shear force Q: p p p

[0056]

[0057] M p = κ p h0Q p (6)

[0058] In the formula, γ p , A p and S p represent the unit weight, cross-sectional area and pile spacing of the anti-slide pile respectively; μ represents the pile-soil friction coefficient; κ p is the coefficient of the net lateral thrust acting on the pile side in the loaded section of the anti-slide pile, which is related to the distribution pattern of the net lateral thrust, and its value range is between 1 / 3 and 1 / 2, and generally 0.35 can be taken; h0 represents the pile length in the loaded section of the anti-slide pile.

[0059] Then, substituting Equation (4) into Equations (5) and (6), the relationship expressions of the axial force N and bending moment M of the anti-slide pile at the potential slip surface with the safety factor for slope stability design as the independent variable can be obtained. p p

[0060] Step 400: According to the sliding displacement compatibility condition between blocks, the displacement expressions of each block are established successively from the trailing edge of the sliding mass to the front. Specifically as follows:

[0061] Regarding each block as a rigid body, when the slope body undergoes sliding displacement, due to the displacement coordination between blocks, the sliding displacement ω of the k-th block can be deduced. k The relationship with the vertical displacement of the first block at the rear edge of the sliding body, or the vertical displacement v0 at the top of the sliding body, is as follows:

[0062]

[0063] In the formula, k is the sequential number of each block starting from 1 from the rear edge to the front edge of the sliding body, 1 ≤ k ≤ n; ψ k is the dilation angle of the soil mass at the potential sliding surface at the bottom of block k, and is taken as

[0064] Based on the dilatancy of the soil mass in the slip zone, the tangential (shearing) displacement δ of block k along the potential sliding surface can be further deduced. k The horizontal displacement u k The relationship expression with the vertical displacement v0 of the first block is as follows:

[0065]

[0066] Step 500: In the static equilibrium equations of each block, for the shear force along the potential sliding surface at the bottom of the block, the shear stress - shear displacement relationship equation is used to express it. Specifically as follows:

[0067] According to the results of soil shear tests, the soil shear stress τ k and the shear displacement δ k The relationship equation expression is:

[0068] τ k = G k δ k (1 + δ k ε / s k ) ρ (10)

[0069] In the formula, τ k is the tangential stress of the soil mass at the bottom of block k along the potential sliding surface; G k represents the linear shear modulus or shear stiffness of the soil mass at the bottom of block k along the potential sliding surface; ε and ρ are the strain softening coefficient and strain softening index respectively, which are fitting parameters of the test curve and are both dimensionless coefficients, and can be obtained by fitting according to the shear test data. Among them, the value of ε is generally taken as 2, and the value of ρ generally ranges from -0.5 to -1.0; s k is a derived quantity related to the normal stress at the bottom of block k, and represents the peak strength through the Coulomb strength theorem. The Coulomb strength theorem is that the soil shear strength is equal to the cohesion c k and the normal stress Nk / l k Multiply by the tangent of the internal friction angle The sum of s k The calculation expression is:

[0070]

[0071] Among them, c k , l k are the cohesion, internal friction angle and length of the potential sliding surface at the bottom of strip k, respectively, and l k =b k / cosα k .

[0072] Step 600: Based on the Morgenstern-Price inter-strip force assumption, determine the relationship between the inter-strip tangential force and the inter-strip normal force between two adjacent blocks. The details are as follows:

[0073] According to the Morgenstern-Price interstrand force H k Assumptions:

[0074] H k =λf k P k (12)

[0075] Where λ is a constant to be determined; f k is a function that characterizes the inter-strip force coefficient and is a sinusoidal function related to the position of the strip k. k The expression is:

[0076]

[0077] Among them, L x x is the horizontal projection length of the entire potential sliding surface, which can be obtained from the potential sliding surface position determined in step 100; k is the horizontal distance between the front inter-strip interface of strip k and the starting point of the potential sliding surface of the trailing edge of the sliding body, and its expression is:

[0078]

[0079] Where m is the number of the block from the 1st block to the kth block; l m is the bottom length of the mth block.

[0080] Step 700: Under the specified slope stability design safety factor, the static equilibrium equations and constraints of all strips are combined to obtain the vertical design limit displacement v of the top of the sliding body of the anti-slide pile reinforced slope. 0u The details are as follows:

[0081] For slice block k, the tangential force along its bottom surface is:

[0082] T k = τ k l k (14)

[0083] By combining equations (1), (2), (8), (10), (12), and (14), the control equation set can be obtained as:

[0084]

[0085] In the formula, is a symbolic function for facilitating simplified expression, where x p represents the horizontal distance from the central axis of the anti-slide pile to point O at the slope toe;

[0086] The expression of the intermediate variable χ k is:

[0087]

[0088] The expression of the intermediate variable ξ k is:

[0089]

[0090] The expressions of the intermediate variables A k , B k , R k , U k are respectively:

[0091]

[0092] T k = χ k [(W k + q k )sinα k - N p sinα k - Q p cosα k (18)

[0093]

[0094] In addition, taking the remaining thrust of the last slice block at the front edge of the sliding mass as zero as the constraint condition. After further derivation from equations (15) to (19), the calculation expression of the vertical displacement v0 at the top of the sliding mass can be obtained as:

[0095]

[0096] Wherein, r is the block number between the k-th block and the n-th block.

[0097] Under the specified design safety factor of slope stability, substituting it into Equation (20) for solution, the vertical design ultimate displacement value v of the top of the sliding mass corresponding to the design safety factor of slope stability can be obtained. 0u .

[0098] Step 800: Substitute the vertical design ultimate displacement v of the top of the sliding mass 0u into the displacement expressions of each block, and the horizontal design ultimate displacement u ku and the tangential design ultimate displacement δ ku along the potential slip surface of each block of the slope reinforced by anti-slide piles under the specified design safety factor of slope stability can be obtained. Specifically as follows:

[0099] Substitute the vertical design ultimate displacement value v of the top of the sliding mass 0u into Equations (8) and (9), and the tangential design ultimate displacement and horizontal design ultimate displacement values of each corresponding block along the potential slip surface under the specified design safety factor of slope stability (positive towards the front edge of the sliding mass 20) can be obtained respectively. The calculation expressions are:

[0100]

[0101] Wherein, δ ku , u ku are respectively the tangential design ultimate displacement and horizontal design ultimate displacement of block k along the potential slip surface.

[0102] The beneficial effects of the calculation method of the present invention are illustrated below through examples.

[0103] Figure 4 is a schematic diagram of a soil slope reinforced by an anti-slide pile in an embodiment. As Figure 4 shown, the geometric parameters of the slope include: the slope height is 10m, the horizontal inclination angle of the slope surface is 30°, the top surface of the slope is horizontal, the soil body is a homogeneous material (i.e., the physical and mechanical parameters of each block are the same), and its basic physical and mechanical parameters include: soil unit weight γ, average cohesion c and average internal friction angle are respectively 20kN / m 3 , 16kPa and 12°, the dilation angle ψ k is 1 / 3 of the average internal friction angle , that is, ψ k = 4°; the shear stiffness G, strain softening coefficient ε and strain softening index ρ of the soil body are respectively taken as 10kPa / mm, 2 and -0.7, and the load q on the top surface of the sliding mass = 0kPa. The design safety factor F s of slope stability = 1.25.

[0104] A single-row anti-slide pile with a side length of 1.5 m is set at a distance of 10 m from the toe of the slope, that is: the horizontal distance x from the central axis of the anti-slide pile to the point O at the toe of the slope p = 10 m, the side length D of the pile cross-section = 1.5 m, and the cross-sectional area A of the pile p = 2.25 m 2 , the friction coefficient μ between the pile side and the soil mass = 0.3, the pile spacing S p = 6.8 m, the unit weight of the pile is γ p = 25 kN / m 3 , the coefficient κ of the net thrust on the pile side in the loaded section p = 0.35.

[0105] The following is the calculation process of the embodiment:

[0106] Step 100: Based on the basic geometric, physical and mechanical parameters of the slope body, the pile design parameters, and the specified design safety factor for slope stability, first calculate the position of the potential slip surface of the slope body and the length of the loaded section of the anti-slide pile according to the Morgenstern-Price method. Then divide the sliding mass.

[0107] Figure 5 It is a schematic diagram of the vertical division strips of the sliding mass in the embodiment. As Figure 5 shown, the sliding mass 20 is divided into 20 strips with equal widths, that is, n = 20. The inclination angle of the bottom surface, the length of the bottom surface, and the self-weight of each strip are listed in Table 1. Among them, the loaded section of the anti-slide pile 10 is included in the 11th strip, that is, j = 11, and the length h0 of the loaded section of the anti-slide pile is obtained by measuring from the geometric relationship in the figure as 5.45 m.

[0108] Table 1

[0109] Slice number k Length of the bottom surface (m) Inclination angle of the bottom surface (°) Self-weight of the slice (kN / m) k = i = 1 2.321 <![CDATA[64.330(α1)]]> 21.04 k = i = 2 1.856 57.191 57.76 k = i = 3 1.607 51.273 85.94 k = i = 4 1.449 46.051 101.74 k = i = 5 1.338 41.289 109.44 k = i = 6 1.257 36.856 114.22 k = i = 7 1.194 32.668 116.60 k = i = 8 1.146 28.669 116.94 k = i = 9 1.108 24.819 115.47 k = i = 10 1.078 21.085 112.37 k = j = 11 1.054 17.443 107.77 k = i = 12 1.036 13.873 101.77 k = i = 13 1.022 10.358 94.44 k = i = 14 1.013 6.881 85.83 k = i = 15 1.007 3.430 75.98 k = i = 16 1.006 -0.009 64.91 k = i = 17 1.007 -3.447 52.62 k = i = 18 1.013 -6.899 39.12 k = i = 19 1.022 -10.375 24.37 k = i = 20 1.036 -13.891 8.34

[0110] Step 200: According to Equations (1) and (2), it can be obtained that:

[0111] For the strip where the anti-slide pile is located, that is, taking j = 11, there is:

[0112]

[0113] For the strips without anti-slide piles, that is, taking i≠11, there is:

[0114]

[0115] Step 300: From Equation (3e), it can be obtained that:

[0116] c0 = 16 / F s

[0117]

[0118] Thus, according to Equations (3a)-(3d) and Equations (3), (4), it can be obtained that: the shear force Q of the anti-slide pile at the potential slip surface p and the design safety factor F of slope stability s The relationship between them, and then from Equations (5), (6), the axial force N of the anti-slide pile at the potential slip surface can be obtained p and the bending moment M p and the design safety factor F of slope stability s The relationship between them.

[0119] Step 400: According to Equations (7), (8), (9), it can be obtained that:

[0120]

[0121] Step 500: According to Equations (10), (11), it can be obtained that:

[0122] τ k = 10Δ k (1 + Δ k 2 / s k ) -0.7 (kPa)

[0123]

[0124] Step 600: It can be calculated according to Table 1 that L x = 20.111m, then from Equation (13), the expression of f k is:

[0125]

[0126] Step 700: Substitute the relevant parameters such as F s = 1.25, D = 1.5m, A p = 2.25m 2 , μ = 0.3, S p = 6.8m, γ p = 25kN / m 3 , κ p = 0.35, ψ k = 4° into Equation (20), solve Equation (20), and the vertical design ultimate displacement value v s at the top of the sliding mass corresponding to the design safety factor F of slope stability can be obtained 0u = 20.61mm.

[0127] Step 800: According to Equations (21), (22), the expressions of the tangential design ultimate displacement and the horizontal design ultimate displacement along the potential slip surface of each slice are respectively:

[0128]

[0129] Thus, the design ultimate displacements along the tangential direction of the potential slip surface and the design ultimate displacements in the horizontal direction of each block can be obtained, as shown in Table 2.

[0130] Table 2

[0131] Slice number k <![CDATA[u ku (mm)]]> <![CDATA[δ ku (mm)]]> Slice number k <![CDATA[u ku (mm)]]> <![CDATA[δ ku (mm)]]> k = i = 1 11.74 23.66 k = j = 11 12.97 13.30 k = i = 2 12.06 20.07 k = i = 12 13.02 13.19 k = i = 3 12.25 18.02 k = i = 13 13.08 13.13 k = i = 4 12.40 16.66 k = i = 14 13.14 13.12 k = i = 5 12.52 15.69 k = i = 15 13.19 13.16 k = i = 6 12.61 14.98 k = i = 16 13.25 13.25 k = i = 7 12.70 14.44 k = i = 17 13.30 13.38 k = i = 8 12.77 14.02 k = i = 18 13.36 13.57 k = i = 9 12.84 13.70 k = i = 19 13.42 13.82 k = i = 10 12.91 13.47 k = i = 20 13.49 14.14

[0132] Comparison and verification:

[0133] The above example is numerically simulated and analyzed by using the three-dimensional numerical simulation method of the commercial geotechnical mechanics analysis software FLAC3D. The displacement when the slope reaches the design ultimate state is determined by the strength reduction method. The design ultimate state mentioned here is the ultimate state corresponding to the stability design safety factor.

[0134] Figure 6 Fig. is a comparison diagram of the design ultimate displacements along the tangential direction of the potential slip surface of each point on the sliding mass surface obtained by numerical simulation and the calculation results of the present invention. Figure 7 Fig. is a comparison diagram of the design ultimate displacements in the horizontal direction of each point on the sliding mass surface obtained by numerical simulation and the calculation results of the present invention. As Figure 6 , Figure 7 shown, the distribution curves of the two are relatively close and respectively have basically the same variation characteristics.

[0135] The numerical simulation result of the vertical design ultimate displacement at the top of the slope sliding mass is 24.32 mm, and the calculation result of the present invention is 20.61 mm. The absolute deviation between the two is 3.71 mm. The absolute value of the deviation of the algorithm of the present invention relative to the numerical simulation result is 15.3%. It should be noted that one of the main reasons for the deviation between the two is the difference in the numerical simulation stress state.

[0136] It can be seen that the calculation method of the present invention includes the stability design safety factor for strengthening the slope by anti-slide piles, the internal force action of anti-slide piles, calculates the shear force of the anti-slide pile at the potential slip surface according to the plastic deformation mode of the soil mass on the pile side, and realizes the specific expression of the three internal forces of the anti-slide pile in the form of shear force through the relationship between the axial force, bending moment and shear force. On this basis, three basic equations, namely the static equilibrium equation of the sliding mass, the shear constitutive equation of the soil mass on the potential slip surface, and the displacement compatibility equation between blocks, are established for the sliding mass. Then, combined with the boundary condition that the remaining thrust at the front edge of the sliding mass is zero, the vertical design ultimate displacement value at the top of the sliding mass corresponding to the stability design safety factor of the slope is calculated. Then, the vertical design ultimate displacement at the top of the sliding mass is substituted into the displacement expression of each block to obtain the horizontal design ultimate displacement and the design ultimate displacement along the tangential direction of the potential slip surface of each block of the slope strengthened by anti-slide piles under the specified stability design safety factor of the slope. This calculation method is reasonable and has the following advantages:

[0137] (1) The mechanical concept of the calculation method of the present invention is clear, the principle is concise, the internal force action of the anti-slide pile is comprehensively considered, and all three internal forces are expressed as quantities closely related to the shear force of the pile body. The three internal force actions of the anti-slide pile are reasonably incorporated into the slope stability control equation related to displacement, forming a stability control equation set for the anti-slide pile to reinforce the slope related to displacement.

[0138] (2) The calculation method of the present invention determines the shear force of the pile body at the potential slip surface by calculating the plastic deformation thrust of the soil on the pile side. A slope stability design safety factor is introduced therein, and the design shear force of the pile body corresponding to this design safety factor can be obtained. Thus, the slope stability control equation set for the anti-slide pile to reinforce the slope contains the slope stability design safety factor, and the solution of the design ultimate displacement of the anti-slide pile to reinforce the slope under the condition of the specified design safety factor can be realized.

[0139] (3) The calculation method of the present invention can calculate and determine the design ultimate displacement of the anti-slide pile to reinforce the slope under different stability design safety factor conditions, and realize the one-to-one correspondence between the design ultimate displacement of the anti-slide pile to reinforce the slope and the stability design safety factor.

[0140] (4) The calculation method of the present invention calculates and obtains the design ultimate displacement of the anti-slide pile to reinforce the slope by a simple computer program in a quick manner. The calculation process is simple, the calculation time is less, and the result is relatively accurate. It avoids the complex modeling analysis steps and time-consuming calculation process of the numerical simulation method, and provides a convenient and effective means for the analysis and judgment of the design ultimate displacement of the anti-slide pile to reinforce the slope and related engineering designs.

[0141] The above has described the relevant content of the present invention. Those of ordinary skill in the art will be able to implement the present invention based on these descriptions. Based on the above content of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.

Claims

1. Design ultimate displacement calculation method for anti-slide pile reinforced slope, characterized in that: Including steps: Step 100: For the slope reinforced with anti-slide piles, based on the basic geometric and physical mechanical parameters of the slope, the pile design parameters, and the specified slope stability design safety factor, the location of the potential sliding surface of the slope and the length of the load-bearing section of the anti-slide pile are determined. The sliding body is vertically divided into a number of strips from the trailing edge of the sliding body forward, and each strip is numbered from the back to the front. The bottom length, bottom surface inclination, and deadweight of each strip are determined. The load-bearing section of the anti-slide pile is included in a certain strip. Step 200: Under the specified slope stability design safety factor, for each strip, a corresponding static equilibrium equation is established based on the static equilibrium conditions of horizontal force, vertical force, and moment. For the strip where the load-bearing section of the anti-slide pile is located, the static equilibrium equation must include the internal forces of the anti-slide pile at the potential sliding surface, which include shear force, bending moment, and axial force. Step 300: Establishing calculation expressions for various internal forces of the anti-slide pile at the potential sliding surface, forming relationship expressions for bending moment and axial force with shear force as a variable, and relationship expressions for various internal forces with slope stability design safety factor as an independent variable; Step 400: Based on the sliding displacement compatibility conditions between the blocks, the displacement expressions of the blocks are established sequentially from the trailing edge of the sliding body forward. Step 500: In the static equilibrium equation of each bar, the shear force on the bottom surface of the bar along the potential sliding surface is expressed using the shear stress-shear displacement relationship equation; Step 600: Determine the relationship between the inter-strip tangential force and the inter-strip normal force between two adjacent blocks based on the Morgenstern-Price inter-strip force assumption. Step 700: Under the specified slope stability design safety factor, the static equilibrium equations and constraints of all strips are solved to obtain the vertical design limit displacement of the top of the sliding body of the anti-slide pile reinforced slope; Step 800: Substitute the vertical design limit displacement of the top of the sliding body into the displacement expression of each strip to obtain the horizontal design limit displacement and the tangential design limit displacement along the potential sliding surface of each strip of the anti-slide pile reinforced slope under the specified slope stability design safety factor; In step 300, the relationship between the bending moment and the axial force with the shear force as the variable is expressed as follows: M p = κ p h0Q p Wherein, N p , M p and Q p are respectively the axial force, bending moment and shear force of the anti-slide pile at the potential slip surface; γ p , A p and S p respectively represent the unit weight, cross-sectional area and pile spacing of the anti-slide pile; μ represents the pile-soil friction coefficient; κ p is the coefficient of the net thrust on the pile side in the loaded section of the anti-slide pile, and its value range is between 1 / 3 and 1 / 2; h0 represents the pile length in the loaded section of the anti-slide pile; The calculation expression of shear force is: Where y represents the depth from the pile top; p(y) is the design thrust of the pile on the soil side; The calculation expression of p(y) is: Wherein, D is the pile diameter or the side length of the pile body cross-section; γ represents the unit weight of soil; η1, η2, η3, are dimensionless coefficients; c0 and are respectively the calculated cohesive force and the calculated internal friction angle of the soil mass on the potential slip surface considering the design safety factor of stability; η1, η2, η3, The calculation expressions are respectively as follows: c0, The calculation expressions are respectively: c0 = c / F s where c and are respectively the actual average cohesion and average internal friction angle of the potential slip surface soil mass; F s is the safety factor for stability design.

2. The design ultimate displacement calculation method for the landslide-resistant pile to reinforce the slope according to claim 1, characterized in that: In step 100: the potential sliding surface of the slope is calculated and determined using a limit equilibrium method or a numerical simulation method; the total number of strips formed by dividing the sliding body is ≥ 15, and they are numbered in sequence from the trailing edge to the leading edge of the sliding body.

3. The design ultimate displacement calculation method for anti-slide pile reinforced slopes as described in claim 2, characterized in that: In step 200: The static equilibrium equation of the strip with the anti-slide pile load section is: In the formula, j is the number of the slice containing the loaded section of the anti-slide pile; W j is the self-weight of slice j; N j is the normal force acting on the bottom surface of slice j; T j is the tangential force acting on the bottom surface of slice j; H j is the inter-slice tangential force acting on the vertical sub-slice interface in front of slice j; P j is the inter-slice normal force acting on the vertical sub-slice interface in front of slice j; q j is the uniformly distributed load on the top surface of slice j; z j is the distance from the bottom surface of slice j along the direction of the vertical sub-slice interface for the inter-slice normal force P j ; b j is the width of slice j; α j is the inclination angle of the bottom surface of slice j; The static equilibrium equation of the strip without the anti-slide pile load section is: where \(i\) is the number of the slice excluding the loaded section of the anti-slide pile, \(1\leq i\leq n\), and \(i\neq j\), \(n\) is the total number of slices; \(W\) i is the self-weight of slice \(i\); \(N\) i is the normal force acting on the bottom surface of slice \(i\); \(T\) i is the tangential force acting on the bottom surface of slice \(i\); \(H\) i is the inter-slice tangential force acting on the vertical sub-slice interface in front of slice \(i\); \(P\) i is the inter-slice normal force acting on the vertical sub-slice interface in front of slice \(i\); \(q\) i is the uniformly distributed load on the top surface of slice \(i\); \(z\) i is the distance of the inter-slice normal force \(P\) i from the bottom surface of slice \(i\) along the vertical sub-slice interface direction; \(b\) i is the width of slice \(i\); \(\alpha\) i is the inclination angle of the bottom surface of slice \(i\).

4. The design ultimate displacement calculation method for the landslide-resistant pile to reinforce the slope according to claim 3, characterized in that: In step 700, the calculation expression for the vertical displacement of the top of the sliding body of the anti-slide pile reinforced slope is: In the formula, k is the sequential number of each block from the trailing edge to the leading edge of the sliding mass, starting from 1, where 1 ≤ k ≤ n; r is the block number between the k-th block and the n-th block; the intermediate variables A k , B k , R k , U k , χ k , ξ k are expressed as follows: T k = χ k [(W k + q k )sinα k - N p sinα k - Q p cosα k ​ In the formula, \(v_0\) is the vertical displacement of the top of the landslide mass of the slope reinforced by anti-slide piles; \(\lambda\) is a constant to be determined; \(f\) k is a sine function related to the position of block \(k\) that characterizes the inter-slice force coefficient, \(L\) x is the projected length of the entire potential slip surface in the horizontal direction; \(x\) k is the horizontal distance from the front inter-slice interface of block \(k\) to the starting point of the potential slip surface at the rear edge of the landslide mass, \(m\) is the block number between the 1st block and the \(k\)th block, \(l\) m is the bottom length of the \(m\)th block; \(\psi\) k is the dilation angle of the soil mass on the potential slip surface at the bottom of block \(k\), taken as \(\rho\) is the strain softening index, ranging from -0.5 to -1.0; \(s\) k is a derived quantity related to the normal stress at the bottom of block \(k\).

5. The design ultimate displacement calculation method for the landslide-resistant pile to reinforce the slope according to claim 4, characterized in that: In step 800, the horizontal design limit displacement and the tangential design limit displacement along the potential sliding surface of each block of the anti-slide pile reinforced slope are: where v 0u is the vertical design ultimate displacement value at the top of the sliding mass; δ ku , u ku are respectively the tangential design ultimate displacement and the horizontal design ultimate displacement of slice k along the potential slip surface.

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