Calculation method of load and thrust of sunken anti-slide pile under overtopping failure mode

Abstract

The present application relates to a kind of overtop failure mode under-embedded anti-slide pile load and under-embedded section thrust calculation method.Pile body stress analysis of under-embedded pile is mainly assisted by model test and numerical simulation, and lacks in-depth calculation.This method defines the under-embedded anti-slide pile body subjected to landslide thrust as load section, and the distance between the top of under-embedded anti-slide pile and slope surface as under-embedded section;determines the overtop sliding surface, establishes the equation of overtop sliding surface and its normal stress function in polar coordinate system;obtains the tangential stress function of overtop sliding surface, calculates the resultant force of overtop sliding surface pile rear section;calculates the force on each boundary of arc-shaped rigid block, calculates the force on each boundary of adjacent pile rigid block according to the force on each boundary of arc-shaped rigid block, and obtains the load section thrust;integrally calculates the horizontal resultant force of overtop sliding surface pile top section, which is the under-embedded section thrust.This application separately calculates the under-embedded section thrust and load section thrust, which is very close to the result of FLAC3D, and has important practical significance for pile internal force optimization and under-embedded depth determination.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering technology, specifically to a method for calculating the load and thrust of the buried section of a submerged anti-slide pile under the over-the-top failure mode. Background Technology

[0002] Anti-slide piles have been used for over 70 years, with single-row piles being the most common structural form. However, with the increasing scale of landslide (slope) control and the improvement of design theories, the technology is gradually developing towards larger diameter, multi-row piles. Control schemes are becoming more comprehensive, structural forms are becoming more diverse, and the stress distribution of anti-slide piles is becoming more rational.

[0003] The design of slope reinforcement with anti-slide piles mainly includes pile stress analysis and slope stability evaluation after reinforcement. Traditional anti-slide piles typically have their tops located on the slope surface, meaning the pile must bear the entire landslide thrust transmitted from the rear side. This results in the pile's resistance to the landslide in front of it not being effectively utilized, and the pile experiencing excessive stress. By embedding the pile tops into the slope surface at a certain depth while keeping the pile position unchanged (referred to as embedded piles), research has shown that this structure can effectively overcome the shortcomings of full-length piles.

[0004] Determining landslide thrust is crucial for the design of anti-slide piles. Currently, there is limited research on the stress analysis of driven piles, and studies on pile stress, particularly the thrust in the loaded section, mainly rely on model tests and numerical simulations, lacking more in-depth analysis. Therefore, there is an urgent need to establish a method for calculating landslide thrust using driven piles to meet the needs of engineering design and construction. Summary of the Invention

[0005] The purpose of this invention is to provide a method for calculating the load and thrust of the buried section of a buried anti-slide pile under the over-the-top failure mode, so as to provide a reliable calculation basis for the stress analysis of buried anti-slide piles under the potential over-the-top failure mode.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A method for calculating the load and thrust of the buried section of an embedded anti-slide pile under the over-the-top failure mode, the method comprising:

[0008] The pile body of the buried anti-slide pile subjected to landslide thrust is defined as the load-bearing section, and the distance between the top of the buried anti-slide pile and the slope surface is defined as the buried section.

[0009] Determine the slip surface that crosses the top, and establish the equation of the slip surface and its normal stress function in polar coordinates;

[0010] Based on the normal stress function of the slip surface, the tangential stress function of the slip surface is obtained, and the resultant force of the pile in the rear section of the slip surface is calculated.

[0011] By combining the resultant force of the rear section of the top sliding surface pile, the force on each boundary of the arc-shaped rigid block is calculated. Based on the force on each boundary of the arc-shaped rigid block, the force on each boundary of the rigid block of the adjacent pile is calculated, and thus the thrust of the loaded section is obtained.

[0012] The resultant force in the horizontal direction of the top section of the over-slip surface pile is calculated by integration, which is the thrust of the buried section.

[0013] Furthermore, the equations for the slip surface and its normal stress function in polar coordinates are established, including:

[0014] Establish the equations for the apex-crossing slip surface in polar coordinates:

[0015]

[0016] Establish the normal stress function of the slip surface in polar coordinates:

[0017]

[0018] in:

[0019] ρ(θ) is the equation for the apex slip surface;

[0020] σ(θ) is the normal stress function;

[0021] θ is the polar angle;

[0022] Fs is the stability coefficient corresponding to the over-the-top failure mode;

[0023] γ represents the sliding mass degree;

[0024] c represents cohesion;

[0025] φ is the internal friction angle;

[0026] A and B are integration constants.

[0027] Furthermore, based on the normal stress function of the slip surface, the tangential stress function of the slip surface is obtained, including:

[0028]

[0029] in:

[0030] τ(θ) is the shear stress function of the slip surface.

[0031] Simplified shear stress function for the top slip surface:

[0032]

[0033] Furthermore, the resultant force of the latter section of the overpass sliding surface pile is calculated, including:

[0034] Let the length increment of the sliding surface above the top be dl, then:

[0035]

[0036] Let the increments of the tangential and normal forces at any point on the slip surface be dT and dN, respectively. Then:

[0037]

[0038] Let the increments of the x-direction and y-direction forces on the slip surface be dF respectively. x and dF y ,but:

[0039]

[0040] Using the pole as the moment center, calculate the resultant force of the latter part of the slip surface pile that crosses the top:

[0041]

[0042] in:

[0043] F xBD The resultant force is the horizontal force on the upper section of the top sliding surface pile;

[0044] F yBD The vertical resultant force on the rear section of the top sliding surface pile;

[0045] M BD This is the torque about the pole.

[0046] Furthermore, combining the resultant force of the rear section of the sliding pile, the forces on each boundary of the arc-shaped rigid block are calculated, including:

[0047] Taking the pole as the moment center, from the static equilibrium condition, we get:

[0048]

[0049] in:

[0050] c m This represents the cohesive force on the initial sliding surface.

[0051] The initial internal friction angle on the sliding surface;

[0052] T1 is the tangential force on the adjacent side of the arc-shaped rigid block and the initial sliding surface;

[0053] N1 is the normal force on the adjacent side of the arc-shaped rigid block and the initial sliding surface;

[0054] l BH The length of the adjacent side of the curved rigid block and the initial sliding surface;

[0055] If we simplify the rear section of the sliding pile over the top to a straight edge, then the gravity G1 of the curved rigid block is:

[0056]

[0057] in:

[0058] b is the length of the adjacent side between the arc-shaped rigid block and the adjacent pile rigid block;

[0059] η is the horizontal angle between the curved rigid block and the adjacent side of the initial sliding surface;

[0060] ω is the horizontal angle between the adjacent sides of the arc-shaped rigid block and the adjacent rigid block of the pile;

[0061] Establish the equations of the adjacent sides of the curved rigid block and the initial sliding surface, as well as the equations of the perpendicular lines passing through the midpoints of the adjacent sides of the curved rigid block and the initial sliding surface:

[0062]

[0063] in:

[0064] (x B y B ), (x D y D ), (x H y H () represents the coordinates of the three vertices of the curved rigid block;

[0065] (x1, y1) are the coordinates of the midpoints of the adjacent sides of the arc-shaped rigid block and the initial sliding surface;

[0066] (x2, y2) are the coordinates of the midpoints of the adjacent sides of the arc-shaped rigid block and the adjacent pile rigid block;

[0067] Calculate the distances l1 and l2 from the pole to the adjacent side of the curved rigid block and the initial sliding surface, and the perpendicular line passing through the midpoint of the adjacent side of the curved rigid block and the initial sliding surface; and the distances l3 and l4 from the pole to the adjacent side of the curved rigid block and the adjacent pile rigid block, and the perpendicular line passing through the midpoint of the adjacent side of the curved rigid block and the adjacent pile rigid block, to obtain:

[0068]

[0069] in:

[0070] T2 is the tangential force on the adjacent side of the arc-shaped rigid block and the adjacent pile rigid block;

[0071] N2 is the normal force on the adjacent side of the arc-shaped rigid block and the adjacent pile rigid block;

[0072] d1 is the horizontal distance from the center of gravity of the arc-shaped rigid block to the pole;

[0073] Calculate the forces on each boundary of the curved rigid block:

[0074]

[0075] Furthermore, based on the forces on each boundary of the curved rigid block, the forces on each boundary of the adjacent pile rigid block are calculated, thereby obtaining the thrust of the loaded section, including:

[0076] Taking the midpoint of the loaded section of the pile as the moment center, from the static equilibrium condition, we get:

[0077]

[0078] in:

[0079] δ is the horizontal angle between the rigid block of the adjacent pile and the adjacent side of the initial sliding surface;

[0080] ξ is the angle between the adjacent side of the rigid block of the adjacent pile and the vertical direction and the curved rigid block; G2 is the weight of the rigid block of the adjacent pile;

[0081] T3 is the tangential force on the adjacent edge of the rigid block of the adjacent pile and the initial sliding surface;

[0082] N3 is the normal force on the adjacent edge of the rigid block of the adjacent pile and the initial sliding surface;

[0083] T p This refers to the tangential force on the loaded section of the pile.

[0084] N p For the thrust of the pile body under load;

[0085] j is the length of the adjacent side of the rigid block of the adjacent pile and the initial sliding surface;

[0086] The forces on each boundary of the rigid block of the adjacent pile are obtained by solving:

[0087]

[0088] Then calculate the thrust N of the loaded section. p :

[0089]

[0090] Furthermore, the resultant force in the horizontal direction of the top section of the pile that crosses the slip surface is calculated by integration, which is the thrust of the buried section, including:

[0091]

[0092] in:

[0093] F xDC The resultant force in the horizontal direction of the top section of the over-slip surface pile;

[0094] θ C Cut the horizontal angle between the endpoint and the pole of the ramp surface;

[0095] θD Cut the horizontal angle between the starting point and the pole of the top sliding surface.

[0096] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0097] The method of this invention is applicable to both cohesive and non-cohesive soils. For bedrock-overburden type landslides reinforced with driven piles, based on the potential overtopping failure mode, the overtopping slip surface is divided into two segments: the overtopping slip surface pile top segment and the overtopping slip surface pile rear segment, determined by the pile top location. The resultant force in the horizontal direction of the overtopping slip surface pile top segment is the thrust of the driven segment, which can be obtained by integration. The solution method for the forces in each direction of the overtopping slip surface pile rear segment is similar. Based on this, the rigid body limit equilibrium theory is used to analyze the forces on the slip body enclosed by the pile rear segment and the loaded segment, thus obtaining the formula for calculating the thrust of the loaded segment. The obtained thrust values ​​for the driven and loaded segments are very close to the FLAC3D results, which has significant practical implications for optimizing pile internal forces and determining the driving depth, and will contribute to the further promotion and application of this pile type. Attached Figure Description

[0098] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained from these drawings without creative effort.

[0099] Figure 1 This is a schematic diagram of the cross-section of a landslide reinforced with buried piles.

[0100] Figure 2 This is a diagram of the failure mode of a buried pile exceeding its top.

[0101] Figure 3 This is a force analysis diagram of the BDC (Best Directional Control) surface.

[0102] Figure 4 This is a diagram of the BDH analysis model for an arc-shaped rigid block.

[0103] Figure 5 This is a diagram of the HDE analysis model of the rigid block of the adjacent pile.

[0104] Figure 6 This is a cross-sectional view of a landslide along the Baoji-Chengdu Railway, part of a project to reinforce the landslide with embedded piles.

[0105] Figure 7 This is a numerical model diagram of landslide reinforcement using buried piles.

[0106] Figure 8 It is a critical sliding surface diagram obtained from numerical simulation.

[0107] Figure 9This is a horizontal stress cloud diagram of a landslide reinforced by buried piles.

[0108] Figure 10 This is a graph showing the variation of thrust in the loaded section and the submerged section with the submerged ratio η.

[0109] Figure 11 This is a graph showing the variation of the total landslide thrust at the pile location with the settlement ratio η.

[0110] Figure 12 It is a graph showing the relationship between the ratio of thrust in the submerged section and the loaded section and the submerged ratio. Detailed Implementation

[0111] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0112] It should be noted that similar reference numerals and letters indicate similar items; therefore, once an item is defined in one embodiment, it does not need to be further defined and explained in subsequent embodiments. Furthermore, the terms "comprising" and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or device that includes a series of steps or modules is not necessarily limited to those steps or modules explicitly listed, but may include other steps or modules not explicitly listed or inherent to such processes, methods, products, or devices.

[0113] Furthermore, the steps "S1", "S2", etc. involved in the embodiments of the present invention are only for the convenience of describing the technical solution and for understanding the specific steps and contents of the embodiments. They should not be construed as a limitation on the order of steps. Any change in the order of steps should be within the protection scope of the present invention.

[0114] like Figure 2 The following geometric information is involved in the force analysis of the specific implementation method: arc BDC is the top sliding surface, arc BHE is the initial sliding surface, geometric body BDH is the arc-shaped rigid block, geometric body DHE is the adjacent pile rigid block, the arc-shaped rigid block and the adjacent pile rigid block are located between the top sliding surface, the initial sliding surface and the load-bearing section of the pile, BH is the adjacent side of the arc-shaped rigid block and the initial sliding surface, HD is the adjacent side of the arc-shaped rigid block and the adjacent pile rigid block, DE is the adjacent side of the adjacent pile rigid block and the load-bearing section of the pile, i.e. the load-bearing section of the pile, HE is the adjacent side of the adjacent pile rigid block and the initial sliding surface, and O' is the pole of the polar coordinates.

[0115] This invention provides a method for calculating the load and thrust of the buried section of a buried anti-slide pile under the over-the-top failure mode, specifically including:

[0116] S1: As Figure 1 As shown, part of the landslide thrust transmitted from the rear edge of the buried anti-slide pile acts on the pile. The buried anti-slide pile body subjected to the landslide thrust is defined as the load-bearing section. The other part of the landslide thrust passes over the top of the pile and continues to be transmitted forward. The resistance is provided by the stability of the front side of the sliding body. The area between the top of the buried anti-slide pile and the slope surface is defined as the buried section.

[0117] S2: As Figure 2 As shown, the BDC (Bridge Flow Surface) of the slip surface is determined, and the equations of the BDC and its normal stress function in polar coordinates are established, including:

[0118] S201: Establish the BDC equations for the top-crossing sliding surface in polar coordinates:

[0119]

[0120] S202: Establish the normal stress function of the BDC overpass surface in polar coordinates:

[0121]

[0122] in:

[0123] ρ(θ) is the equation for the apex slip surface;

[0124] σ(θ) is the normal stress function;

[0125] θ is the polar angle;

[0126] Fs is the stability coefficient corresponding to the over-the-top failure mode;

[0127] γ represents the sliding mass degree;

[0128] c represents cohesion;

[0129] φ is the internal friction angle;

[0130] A and B are integration constants.

[0131] For ease of description, the top of the pile is used as the boundary. The slip surface below the top of the pile is defined as the rear section of the slip surface, and the slip surface above the top of the pile is defined as the top section of the slip surface. For example... Figure 3 The force analysis model of the rear section BD and the top section DC of the overpass sliding pile is mainly based on the thrust calculation of the loaded section. Figure 4 and Figure 5 Force analysis of the mid-arc rigid block BDH and the adjacent pile rigid block DHE. The following assumptions are made:

[0132] (1) The initial sliding surface BHE between the top sliding surface and the load-bearing section of the pile is in the limit state and satisfies the Coulomb strength criterion.

[0133] (2) There is only one inflection point H on the initial smooth surface BHE. The case without an inflection point is its particular solution. The solution method for multiple inflection points is similar to that for a single inflection point.

[0134] (3) Both the arc-shaped rigid block BDH and the adjacent pile rigid block DHE are rigid bodies. Except for the top sliding surface section BD, the force application points on the other boundaries are located at the midpoint of the corresponding action surface.

[0135] S3: Based on the normal stress function of the slip surface, obtain the tangential stress function of the slip surface, and calculate the resultant force of the pile segment after the slip surface, including:

[0136] S301: Based on the BDC normal stress function of the top-slip surface and the Mohr-Coulomb strength theory, the tangential stress function of the top-slip surface is obtained, including:

[0137]

[0138] in:

[0139] τ(θ) is the shear stress function of the slip surface.

[0140] Simplified shear stress function for the top slip surface:

[0141]

[0142] S302: Calculate the resultant force of the rear section of the top pile of the overpass sliding surface pile, including:

[0143] Let the length increment of the sliding surface above the top be dl, then:

[0144]

[0145] Let the increments of the tangential and normal forces at any point on the slip surface be dT and dN, respectively. Then:

[0146]

[0147] Let the increments of the x-direction and y-direction forces on the slip surface be dF respectively. x and dF y ,but:

[0148]

[0149] Using the pole O' as the moment center, calculate the resultant force on the rear section of the overpass sliding pile:

[0150]

[0151] in:

[0152] F xBD The resultant force in the horizontal direction of the rear section of the top sliding surface pile;

[0153] FyBD The vertical resultant force of the rear section of the top sliding surface pile;

[0154] M BD Let be the torque about point O'.

[0155] S4: Combining the resultant force of the rear section of the overpass sliding surface pile, calculate the forces on each boundary of the arc-shaped rigid block BDH. Based on the forces on each boundary of the arc-shaped rigid block BDH, calculate the forces on each boundary of the adjacent pile rigid block DHE, and thus obtain the thrust of the loaded section, including:

[0156] S401: Calculate the forces on each boundary of the curved rigid block BDH by combining the resultant force of the rear section of the sliding surface pile, including:

[0157] With O' as the moment center, from the static equilibrium condition, we get:

[0158]

[0159] in:

[0160] c m This represents the cohesive force on the initial sliding surface.

[0161] The initial internal friction angle on the sliding surface;

[0162] T1 is the tangential force on BH;

[0163] N1 is the normal force on BH;

[0164] l BH The length of BH;

[0165] ∑F x =0 indicates that the x-forces are balanced, ∑F y =0 indicates that the y-force is balanced, ∑M O' =0 indicates that the torque is balanced at the pole O';

[0166] If we simplify the front section of the top of the sliding surface pile to a straight edge, then the gravity G1 of the arc-shaped rigid block BDH is:

[0167]

[0168] in:

[0169] b is the length of HD;

[0170] η is the angle between BH and the horizontal plane;

[0171] ω is the angle between HD and the horizontal plane;

[0172] Establish the equations of line BH and the perpendicular line passing through the midpoint of BH:

[0173]

[0174] in:

[0175] (x B y B ), (x D y D ), (x H y H () represents the coordinates of the three vertices of the arc-shaped rigid block BDH;

[0176] (x1, y1) are the coordinates of the midpoint of BH;

[0177] (x2, y2) are the coordinates of the midpoint of HD;

[0178] Calculate the distances l1 and l2 from O' to BH and the perpendicular line CH passing through its midpoint:

[0179]

[0180] The equations of line HD and the perpendicular line passing through the midpoint of HD can be obtained. The distances l3 and l4 from O' to line segment HD and the perpendicular line to the midpoint of HD can be calculated.

[0181] get:

[0182]

[0183] in:

[0184] T2 is the tangential force on HD;

[0185] N2 is the normal force on HD;

[0186] d1 is the horizontal distance from the center of the arc-shaped rigid block BDH to O';

[0187] Calculate the forces on each boundary of the curved rigid block BDH:

[0188]

[0189] S402: Based on the forces on each boundary of the curved rigid block BDH, calculate the forces on each boundary of the adjacent pile rigid block DHE, and thus obtain the thrust of the loaded section, including:

[0190] Taking the midpoint of DE as the moment center, from the static equilibrium condition, we get:

[0191]

[0192] in:

[0193] δ is the angle between HE and the horizontal plane;

[0194] ξ is the angle between HD and the vertical direction;

[0195] G2 is the gravity of the rigid block DHE adjacent to the pile;

[0196] T3 is the tangential force on HE;

[0197] N3 is the normal force on HE;

[0198] T p The tangential force on DE (the loaded section of the pile);

[0199] N p For the thrust of the loaded section;

[0200] j is the length of HE;

[0201] The forces on each boundary of the rigid block DHE adjacent to the pile are obtained by solving:

[0202]

[0203] Then calculate the thrust N of the loaded section. p :

[0204]

[0205] S5: Integral calculation of the resultant force in the horizontal direction of the top section of the overpass sliding pile, which is the thrust of the buried section, including:

[0206]

[0207] in:

[0208] F xDC The resultant force in the horizontal direction at the rear section of the top of the sliding surface pile;

[0209] θ C Cut the horizontal angle (the angle with the horizontal direction) between the line connecting the endpoint and the pole of the overpass slope.

[0210] θ D Cut the horizontal angle between the line connecting the starting point and the pole of the top sliding surface.

[0211] The above method provides a rigorous solution for the thrust of the buried section and the loaded section under the over-the-top failure mode. This solution process can be easily completed with the help of MATLAB.

[0212] The following are specific examples of calculations and numerical simulations to verify the method of this invention:

[0213] Figure 6The figure shows the main cross-section of a landslide involving moderately weathered marble overlying gravelly soil along the Baoji-Chengdu Railway in Sichuan Province. The main physical and mechanical parameters of the slope, obtained through on-site geological investigation and laboratory tests, are shown in Table 1. The natural slope stability coefficient, calculated using the transfer coefficient method, is 1.06. With a design safety factor of 1.20, it is proposed to reinforce the slope using driven piles. The pile cross-section is 2m × 3m, and the out-of-plane pile spacing is 5m. The pile locations are shown in [Table 1]. Figure 7 The thickness of the sliding body at this location is 15.1m.

[0214] For ease of expression, we introduce the definition of the burial ratio (denoted by η), i.e.

[0215] Settlement ratio (η) = Pile top embedment depth / Thickness of sliding body at pile point

[0216]

[0217] Anti-slide piles are considered as elastic bodies, so their cohesion and internal friction angle are not considered.

[0218] Using the method of this invention, when the burial depth is 5.5m (using Yan's method, when the design safety factor is 1.20, the burial ratio η = 0.364, that is, the maximum burial depth of the pile top is 5.5m. For details of Yan's method, please refer to YAN YP, XIAOS GA calculation method for the embedded depth of stabilizing piles in reinforced slopes. International Journal for Numerical and Analytical Methods in Geomechanics, 2020, 44(7): 1077-1092. Based on the over-the-top failure mode that may occur in landslides reinforced by embedded piles, combined with the transfer coefficient method and the variational method under the limit equilibrium theory framework, an analytical solution for the burial depth of the pile top of the rear row, which is applicable to both cohesive and non-cohesive soils, is derived). The thrust of the burial section and the thrust of the loaded section are 439.9kN / m and 2949.6kN / m, respectively, and their ratio is 0.149.

[0219] To further verify the rationality of the theoretical calculation method, numerical simulations were performed using FLAC3D. The numerical model is as follows: Figure 7As shown, the model contains 47,436 8-node hexahedral elements. The slope is modeled as an ideal elastoplastic constitutive model obeying the Mohr-Coulomb yield criterion and the associated flow law. The piles are treated as elastic materials and simulated using structural elements. The four boundaries of the model (front, back, left, and right) are constrained by horizontal displacement, while the bottom surface is constrained by both horizontal and vertical displacement. Numerical simulation analysis using the strength reduction method yields a stability coefficient of 1.21 for a settlement depth of 5.5m. The critical sliding surface at this depth is also shown below. Figure 8 As shown, the result is very close to Yan's calculation result.

[0220] Figure 9 The figure shows the horizontal stress distribution contour map of the landslide under the overtopping failure mode obtained from the numerical simulation. It is easy to see that the thrust in the submerged section and the loaded section are 428.9 kN / m and 2788.9 kN / m, respectively, with a ratio of 0.154. It is evident that the numerical results are in excellent agreement with the theoretical calculation results, demonstrating the rationality of the theoretical calculation method.

[0221] Figure 10 The figure shows the variation of the thrust in the loaded section and the thrust in the submerged section with the submersion ratio obtained by the theoretical method and FLAC3D. It can be seen that the FLAC3D results are very close to the theoretical calculation results. As the submersion ratio increases from 0 to 1, the thrust in the submerged section increases nonlinearly from 0 to about 2000 kN / m, while the thrust in the loaded section decreases nonlinearly from 4300 kN / m to 0. Simultaneously, when the submersion ratio is within the range of 0.7 to 0.8, the thrust in the loaded section and the submerged section decrease and increase sharply, respectively.

[0222] By superimposing the thrust of the buried section with that of the loaded section, the total thrust variation curves at the pile location for different burial ratios can be obtained, as shown in the figure. Figure 11 As shown, the total thrust at the pile location obtained by the two methods is very close, and the trends are consistent. When the settlement ratio is 0, the total landslide thrust at the pile location is significantly greater than when the settlement ratio is 1, with the ratio being approximately 2.13. This can be attributed to... Figure 12 The explanation is that the slope stability coefficient is the highest when the entire length of the pile is used, while it is the lowest when there are no piles.

[0223] Furthermore, the ratio of thrust in the buried section to the loaded section obtained by the two methods is plotted as a function of burial depth. Figure 12It can be seen that the ratios obtained by the two methods increase non-linearly with the settlement ratio, and the trends are very consistent. When the settlement ratio is in the range of 0–0.67, the ratio of the thrust in the settled section to the thrust in the loaded section behind the pile increases slowly from 0 to about 0.3–0.5. When the settlement ratio is in the range of 0.67–0.8, this ratio increases sharply to 1.47–2.12. At this point, the effect of the anti-slide pile is very small, indicating that this settlement depth should be avoided in engineering. Furthermore, the figure also shows that when the ratio of the thrust in the settled section to the thrust in the loaded section behind the pile is 1, the settlement ratio is approximately 0.72–0.74. Comprehensive analysis shows that when the settlement depth is reasonable, the thrust in the settled section is less than the thrust in the loaded section.

[0224] 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 calculating the load and thrust of the buried anti-slide pile under the over-the-top failure mode, characterized in that: The pile body of the buried anti-slide pile subjected to landslide thrust is defined as the load-bearing section, and the area between the pile top and the slope surface is defined as the buried section. Determine the slip surface that crosses the top, and establish the equation of the slip surface and its normal stress function in polar coordinates; Based on the normal stress function of the slip surface, the tangential stress function of the slip surface is obtained, and the resultant force of the pile in the rear section of the slip surface is calculated. By combining the resultant force of the rear section of the top sliding surface pile, the force on each boundary of the arc-shaped rigid block is calculated. Based on the force on each boundary of the arc-shaped rigid block, the force on each boundary of the rigid block of the adjacent pile is calculated, and thus the thrust of the loaded section is obtained. The resultant force in the horizontal direction of the top section of the overpass sliding pile is calculated by integration, which is the thrust of the buried section; in: Establish the equations for the slip surface crossing the top in polar coordinates and its normal stress function, including: Establish the equations for the apex-crossing slip surface in polar coordinates: Establish the normal stress function of the slip surface in polar coordinates: in: ρ(θ) is the equation for the apex slip surface; σ(θ) is the normal stress function; θ is the polar angle; Fs is the stability coefficient corresponding to the over-the-top failure mode; γ represents the sliding mass degree; c represents cohesion; φ is the internal friction angle; A and B are integration constants; Based on the normal stress function of the slip surface, the tangential stress function of the slip surface is obtained, including: in: τ(θ) is the tangential stress function of the slip surface. Simplified tangential stress function of the top slip surface: ； Calculate the resultant force of the pile segment after it crosses the top sliding surface, including: Let the length increment of the sliding surface above the top be dl, then: Let the increments of the tangential and normal forces at any point on the slip surface be dT and dN, respectively. Then: Let the increments of the x-direction and y-direction forces on the slip surface be dF respectively. x and dF y ,but: Using the pole as the moment center, calculate the resultant force of the latter part of the slip surface pile that crosses the top: in: F xBD The resultant force is the horizontal force on the upper section of the top sliding surface pile; F yBD The vertical resultant force on the rear section of the top sliding surface pile; M BD The torque about the pole; Based on the resultant force of the rear section of the sliding pile, calculate the forces on each boundary of the curved rigid block, including: Taking the pole as the moment center, from the static equilibrium condition, we get: in: c m This represents the cohesive force on the initial sliding surface. φ m The initial internal friction angle on the sliding surface; T1 is the tangential force on the adjacent side of the arc-shaped rigid block and the initial sliding surface; N1 is the normal force on the adjacent side of the arc-shaped rigid block and the initial sliding surface; l BH The length of the adjacent side of the curved rigid block and the initial sliding surface; If we simplify the rear section of the sliding pile over the top to a straight edge, then the gravity G1 of the curved rigid block is: in: b is the length of the adjacent side between the arc-shaped rigid block and the adjacent pile rigid block; The horizontal angle between the curved rigid block and the adjacent side of the initial sliding surface; The horizontal angle between the adjacent sides of the arc-shaped rigid block and the adjacent rigid block of the pile; Establish the equations of the adjacent sides of the curved rigid block and the initial sliding surface, as well as the equations of the perpendicular lines passing through the midpoints of the adjacent sides of the curved rigid block and the initial sliding surface: in: (x) B y B ), (x D y D ), (x H y H ( ) represents the coordinates of the three vertices of the curved rigid block; (x1, y1) are the coordinates of the midpoints of the adjacent sides of the curved rigid block and the initial sliding surface; Calculate the distances l1 and l2 from the pole to the adjacent side of the curved rigid block and the initial sliding surface, and the perpendicular line passing through the midpoint of the adjacent side of the curved rigid block and the initial sliding surface; and the distances l3 and l4 from the pole to the adjacent side of the curved rigid block and the adjacent pile rigid block, and the perpendicular line passing through the midpoint of the adjacent side of the curved rigid block and the adjacent pile rigid block, to obtain: in: T2 is the tangential force on the adjacent side of the arc-shaped rigid block and the adjacent pile rigid block; N2 is the normal force on the adjacent side of the arc-shaped rigid block and the adjacent pile rigid block; d1 is the horizontal distance from the center of gravity of the arc-shaped rigid block to the pole; Calculate the forces on each boundary of the curved rigid block: ； Based on the forces at each boundary of the curved rigid block, the forces at each boundary of the adjacent pile rigid block are calculated, thereby obtaining the thrust of the loaded section, including: Taking the midpoint of the loaded section of the pile as the moment center, from the static equilibrium condition, we get: in: δ is the horizontal angle between the rigid block of the adjacent pile and the adjacent side of the initial sliding surface; ξ is the angle between the adjacent side of the rigid block and the curved rigid block and the vertical. G2 is the gravity of the rigid block of the adjacent pile; T3 is the tangential force on the adjacent edge of the rigid block of the adjacent pile and the initial sliding surface; N3 is the normal force on the adjacent edge of the rigid block of the adjacent pile and the initial sliding surface; T p This refers to the tangential force on the loaded section of the pile. N p For the thrust of the pile body under load; j is the length of the adjacent side of the rigid block of the adjacent pile and the initial sliding surface; The forces on each boundary of the rigid block of the adjacent pile are obtained by solving: Then calculate the thrust N of the loaded section. p : ； The resultant force in the horizontal direction at the top of the pile that crosses the slip surface is calculated by integration, which is the thrust of the buried section, including: in: F xDC The resultant force in the horizontal direction of the top section of the over-slip surface pile; θ C Cut the horizontal angle between the endpoint and the pole of the ramp surface; θ D Cut out the horizontal angle between the starting point and the pole of the slope surface above the top of the slip surface.

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