A design method for the structural surface of small-diameter piles reinforced under post-loading conditions

Abstract

This invention relates to a design method for reinforcing rock mass structures with small-diameter piles under post-load conditions in the field of slope protection. The method first obtains the inclination angle θ of the outward-dipping structural surface, the internal friction angle and cohesion c, the sliding mass weight γ, the sliding mass height H, the stress ratio n between the pile and the rock mass at the structural surface, the vertical load q applied to the upper part of the sliding mass after the construction of a single deeply embedded small-diameter anti-slide pile, the sliding mass width b, and the horizontal extension length l. Then, using these parameters, it calculates the initial axial force of the deeply embedded small-diameter anti-slide pile before sliding, the deflection angle of the deeply embedded small-diameter anti-slide pile when the anti-slide force is at its maximum, the minimum compressive stiffness of the small-diameter anti-slide pile, and the axial force and anchorage length of the deeply embedded small-diameter anti-slide pile caused by compression after minor deflection deformation at the structural surface. This invention saves resources compared to empirical design methods.

Description

Technical Field

[0001] This invention relates to the field of slope protection, and in particular to a design method for small-diameter pile reinforcement of rock mass structure under post-load conditions. Background Technology

[0002] In recent years, the use of deeply embedded small-diameter anti-slide piles in slope engineering has increased significantly. Unlike ordinary anti-slide piles, deeply embedded small-diameter anti-slide piles are characterized by small pile diameter, large slenderness ratio, low bending stiffness, and complex pile-soil interactions with the surrounding soil and rock mass during the anti-slide process. Therefore, the stress mode and failure mechanism of deeply embedded small-diameter anti-slide piles are more complex than those of ordinary anti-slide piles during slope reinforcement. Furthermore, there is a lack of design methods for deeply embedded small-diameter anti-slide piles; currently, designers can only estimate the design based on experience, making precise design impossible and leading to resource waste. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a method for designing the structural surface of a small-diameter pile reinforced rock mass under post-loading conditions.

[0004] This invention provides a method for designing the structural surface of a small-diameter pile reinforced rock mass under post-loading conditions, comprising the following steps:

[0005] Obtain the dip angle θ and internal friction angle of the rock mass's external dip structural surface. Cohesion c, sliding body weight γ, sliding body height H, stress ratio of pile to rock mass at the structural surface n, vertical load q applied to the upper part of the sliding body after the construction of a single deep-buried small-diameter anti-slide pile, sliding body width b, and horizontal extension length l;

[0006] The initial axial force of a deeply buried small-diameter anti-slide pile when it has not slipped is calculated using the following formula:

[0007]

[0008] In the formula, A is the cross-sectional area of ​​the deeply buried small-diameter anti-slide pile;

[0009] The deflection angle β1 of the deeply buried small-diameter anti-sliding pile when the anti-sliding force is maximum is calculated. The value of β when the following formula holds is calculated using Newton's method iteratively, and is set to β1.

[0010]

[0011] The above formula is called the deflection angle calculation formula, where the initial value of β is assigned to 0°, and E is the composite elastic modulus of the deeply buried small-diameter anti-slide pile;

[0012] The minimum compressive stiffness of a deeply buried small-diameter anti-slide pile is calculated using the following formula;

[0013]

[0014] The above formula is called the formula for calculating compressive stiffness, where F st The slope stability safety factor;

[0015] By iteratively solving the aforementioned formulas for calculating compressive stiffness and deflection angle, the minimum value of EA that satisfies the anti-slip requirement is calculated, and the section parameters are determined using the following formula:

[0016] EA = E s A p +E m (AA p )

[0017]

[0018]

[0019] Where D is the diameter of the deeply buried small-diameter anti-slide pile; d is the diameter of the core reinforcement of the deeply buried small-diameter anti-slide pile; E s The elastic modulus of the steel reinforcement used for deep-buried small-diameter anti-slide piles; E m The elastic modulus of the mortar used for deep-buried small-diameter anti-slide piles; A p The cross-sectional area of ​​the reinforcing steel used for deep-buried small-diameter anti-slide piles; F st The slope stability safety factor;

[0020] The following formula is used to calculate the axial force F of the deeply buried small-diameter anti-slide pile body caused by compression of the deeply buried small-diameter anti-slide pile after the small flexural deformation at the structural surface;

[0021]

[0022] The anchorage length of deeply buried small-diameter anti-slide piles can be calculated using the following two formulas;

[0023]

[0024]

[0025] Among them, l a K is the anchorage length of the deeply buried small-diameter anti-slide pile; K is the bond safety factor of the deeply buried small-diameter anti-slide pile; f rbk f represents the standard value of the ultimate bond strength between the rock layer and the deeply buried small-diameter anti-slide pile. b This is the standard value for the bond strength between steel bars and mortar.

[0026] The method further includes:

[0027] Use the following formula to verify the compressive bearing capacity of the section of a small-diameter anti-slide pile;

[0028] F0+F≤A p f y +(AA p )f c

[0029] Among them, f c The compressive strength of the mortar used for deep-buried small-diameter anti-slide piles; f y The compressive strength of the steel bars used for deep-buried small-diameter anti-slide piles.

[0030] Wherein, the stress ratio n between the pile and the rock mass at the structural surface is the value under vertical load conditions, considering only the vertical deformation.

[0031] Among them, the bonding safety factor K of the deeply buried small-diameter anti-slide piles is taken as 2.6, 2.4 and 2.2 respectively according to the safety level of the slope: Level I, Level II and Level III.

[0032] The step of iteratively solving the compressive stiffness calculation formula and the deflection angle calculation formula to calculate the minimum value of EA that satisfies the anti-slip requirement includes:

[0033] First, given an initial value for EA, substitute it into the deflection angle calculation formula to solve for β1, and then substitute β1 into the compressive stiffness calculation formula to solve for EA. Then, substitute it into the deflection angle calculation formula to solve for β1, and then substitute β1 into the compressive stiffness calculation formula to solve for EA. Repeat this process multiple times until the EA obtained from two adjacent solutions are equal, then end the loop and take the EA obtained from the last solution as the minimum value of EA.

[0034] The initial value of EA is within the range of [190852kN, 5089380kN].

[0035] The embodiments of the present invention have the following beneficial effects:

[0036] The design method for reinforcing rock mass structures with small-diameter piles under post-load conditions provided in this embodiment of the invention takes into account the upper load already borne by the pile before the slider slides and the positive effect of the axial force of the deeply buried small-diameter anti-slide pile on anti-slide, and derives the maximum value of the anti-slide force that the deeply buried small-diameter anti-slide pile can provide during the deformation process. Compared with empirical design methods, this method can save resources.

[0037] Of course, implementing any product or method of the present invention does not necessarily require achieving all of the advantages described above at the same time. Attached Figure Description

[0038] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0039] Figure 1 This is a flowchart illustrating a method for designing the structural surface of a small-diameter pile reinforced under post-loading conditions, according to an embodiment of the present invention.

[0040] Figure 2 This is a schematic diagram illustrating the slight deformation of the pile axis caused by the slider sliding down in an embodiment of the present invention. Detailed Implementation

[0041] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. It should be noted that, without conflict, the embodiments and features in the embodiments of the present invention can also be combined with each other.

[0042] It should be noted that similar reference numerals and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. In the description of the invention, the terms "first," "second," "third," "fourth," etc., are used only to distinguish descriptions and should not be construed as merely or implying relative importance.

[0043] In practice, the application of the superstructure load can be categorized into two scenarios: application before and application after the anti-slide pile construction. When the superstructure load is applied before construction, the anti-slide pile has already undergone deformation and coordination, and load transfer only occurs due to the sliding deformation of the slider. However, when the superstructure load is applied after construction, during the application of the vertical load, because the pile's stiffness is much greater than that of the rock mass, the pile will bear most of the vertical load during the synchronous deformation of the pile and the rock mass. Even before the slider has slid, the pile has already borne part of the superstructure load. For this second scenario, the static equilibrium method in existing standards is no longer applicable. To address this issue, this invention provides a corresponding design method for small-diameter pile reinforcement of rock mass structures under loading conditions.

[0044] This invention provides a design method for reinforcing rock mass structures with small-diameter piles under post-loading conditions. The method first derives the anti-slip force calculation formula based on the load borne by the small-diameter pile before slippage, using static equilibrium, the Mohr-Coulomb criterion, and the geometric relationship after slight deflection of the deeply buried small-diameter pile. The anti-slip force is differentiated with respect to the deflection angle of the deeply buried small-diameter pile, and the derivative is set to 0 to obtain the maximum and maximum values ​​of the anti-slip force. The design compressive stiffness and axial force of the deeply buried small-diameter anti-slip pile are calculated by ensuring the anti-slip force is greater than or equal to the product of the slope stability safety factor and the sliding force. Based on the axial force, the cross-sectional compressive bearing capacity of the deeply buried small-diameter anti-slip pile is verified, and the anchorage length of the deeply buried small-diameter anti-slip pile is calculated. This invention considers the positive effect of the upper load already borne by the pile and the axial force of the deeply buried small-diameter anti-slip pile on anti-slip before the slider slides, deriving the maximum anti-slip force that the deeply buried small-diameter anti-slip pile can provide during deformation. Compared with empirical design methods, this method can save resources.

[0045] like Figure 1 As shown in the embodiment of the present invention, a method for designing the structural surface of a small-diameter pile reinforced under post-loading conditions includes the following steps:

[0046] Step S1: Obtain the dip angle θ and internal friction angle of the rock mass's outward-dipping structural surface. Cohesion c, sliding body weight γ, sliding body height H, stress ratio n between pile and rock mass at structural surface, vertical load q applied to the upper part of the sliding body after the construction of a single deep-buried small-diameter anti-slide pile, sliding body width b, and horizontal extension length l.

[0047] Among them, the dipping angle θ and internal friction angle of the rock external dipping structural surface are obtained. Cohesion c, sliding body weight γ, and sliding body height H can be obtained through on-site measurement and experiments. n is the value under vertical load conditions, considering only vertical deformation.

[0048] Step S2: Calculate the initial axial force value of the deeply buried small-diameter anti-slide pile body when it has not slid down;

[0049] Considering only vertical deformation, the sliding body does not slide down. Based on the balance of vertical forces at the sliding surface, the following equation holds:

[0050] qbl=npA+p(bl-A) (1)

[0051] In the formula, q is the vertical load on the upper part of the sliding body, p is the additional vertical load on the lower rock mass at the sliding surface caused by the upper load when only vertical deformation is considered, and A is the cross-sectional area of ​​the deeply buried small-diameter anti-slide pile.

[0052] From equation (1), the load borne by a single deep-buried small-diameter anti-slide pile can be expressed by equation (2):

[0053]

[0054] Step S3: Calculate the deflection angle β1 of the deeply buried small-diameter anti-sliding pile when the anti-sliding force is maximum;

[0055] Because deeply buried small-diameter anti-slide piles have low bending stiffness and minimal influence of bending moment during deformation, and because they play an anti-slide role, their bending moment is neglected for conservative design and calculation convenience. For example... Figure 2 The diagram shows the slight deformation of the anti-slide pile axis caused by the sliding block. When the sliding body is subjected to a load, it will undergo a slight downward displacement along the sliding surface. During the sliding process, the deeply buried small-diameter anti-slide pile will undergo slight deflection. Assume that the angle between the pile body and the vertical direction after deflection is β. According to the force balance in the direction perpendicular to the sliding body, we can obtain equation (3):

[0056] (F0+F)cos(θ-β)+F N =(γH+q)blcosθ (3)

[0057] Where F is the axial force of the deeply buried small-diameter anti-slide pile caused by compression after slight flexural deformation; F0 is the load borne by the deeply buried small-diameter anti-slide pile during the application of the superload; F N This represents the supporting force of the lower rock mass on the sliding body.

[0058] Assuming the distance between the bottom surface of the sliding body and the top surface of the lower rock mass (i.e., the width of the structural surface) is h, during the slight sliding process of the sliding body, the distance between the bottom surface of the sliding body and the top surface of the lower rock mass remains unchanged. Based on the geometric relationship before and after deformation, the strain of the deeply buried small-diameter anti-sliding pile at the structural surface caused by the sliding of the sliding block can be calculated by equation (4).

[0059]

[0060] In the formula, ε is the strain of the deeply buried small-diameter anti-slide pile at the structural surface caused by the sliding of the slider, l is the length of the deeply buried small-diameter anti-slide pile located in the structural surface before deformation, and Δl is the deformation of the deeply buried small-diameter anti-slide pile in the structural surface before and after deformation.

[0061] The axial force F of the deeply buried small-diameter anti-slide pile caused by compression after slight deflection at the structural surface can be expressed by equation (5):

[0062] F=εEA (5)

[0063] Where E is the composite elastic modulus of the deeply buried small-diameter anti-slide pile.

[0064] When the distance between the bottom surface of the sliding body and the top surface of the lower rock mass is large, the anti-sliding force caused by shear force can be ignored. According to the Mohr-Coulomb strength criterion and the force decomposition of the axial force (F0+F) of the deeply buried small-diameter anti-sliding pile, the anti-sliding force R along the sliding surface direction can be expressed by equation (6):

[0065]

[0066] Solving equations (3), (4), (5), and (6) simultaneously, we can obtain the anti-slip force R along the sliding surface direction, which is expressed by equation (7):

[0067]

[0068] To determine the maximum anti-slide force generated by the anti-slide pile under the condition that the angle between the pile body and the vertical direction after different deflections is β after increasing the depth of the small-diameter anti-slide pile, we can obtain equation (8) by differentiating R with respect to β.

[0069]

[0070] make

[0071] Equation (9) can be obtained.

[0072]

[0073] The value of β when equation (9) holds true is calculated using the iterative method. Let it be β1, where the initial value of β is 0°. For ease of description, equation (9) is called the deflection angle calculation formula.

[0074] Substituting the calculated β1 value into equation (7) yields the maximum anti-skid force R. max Then the stability coefficient of the slider is:

[0075]

[0076] Guarantee the stability coefficient f obtained from the calculation s ≥Stability coefficient f determined by the safety level of the slope st The stability coefficient f is used to determine the safety level of a slope. st It can be determined according to the standard "Technical Specification for Building Slope Engineering GB 50330-2013".

[0077] Step S4: Calculate the minimum compressive stiffness and section parameters of the deeply buried small-diameter anti-slide pile.

[0078] From equations (7) and (10), we can obtain the formula for calculating the compressive stiffness EA of deeply buried small-diameter anti-slide piles, equation (11):

[0079]

[0080] For ease of description, equation (11) is referred to as the formula for calculating compressive stiffness. Solving EA by combining equations (9) and (11) requires an iterative method. Specifically, first, an initial value for EA is given, and β1 is solved by substituting it into equation (9). Then, β1 is substituted into equation (11) to solve EA. This process is repeated multiple times until the EA obtained from two adjacent solutions is equal, at which point the loop ends and the minimum value of EA is obtained. The initial value of EA is selected as follows: The existing small-diameter pile diameters are mostly 90mm to 180mm, with 150mm being the most common. Based on the common cross-sectional forms of small-diameter piles, the range of initial values ​​for the compressive stiffness iteration of small-diameter piles is given: The lower limit of the initial value for the compressive stiffness of small-diameter anti-slip piles can be selected as a 90mm plain concrete small-diameter pile, with the compressive elastic modulus of plain concrete being 3×10⁻⁶. 7 Its compressive stiffness EA can be calculated by the following formula:

[0081]

[0082] The upper limit of the initial value of the compressive stiffness of small-diameter piles can be selected as a 180mm solid steel pile, and the compressive elastic modulus of the steel can be taken as 2×10. 8 Its compressive stiffness EA can be calculated by the following formula:

[0083]

[0084] Therefore, when starting the iterative calculation, the initial value of the compressive stiffness EA of the small-diameter pile can be any value in the range [190852kN, 5089380kN].

[0085] The compressive stiffness of deeply buried small-diameter anti-slide piles can be expressed by equations (13), (14), and (15).

[0086] EA = E s A p +E m (AA p (13)

[0087]

[0088]

[0089] Where E is the composite elastic modulus of the deeply embedded small-diameter anti-slide pile; A is the cross-sectional area of ​​the deeply embedded small-diameter anti-slide pile; D is the diameter of the deeply embedded small-diameter anti-slide pile; d is the diameter of the core reinforcement of the deeply embedded small-diameter anti-slide pile; E s The elastic modulus of the steel reinforcement used for deep-buried small-diameter anti-slide piles; E m The elastic modulus of the mortar used for deep-buried small-diameter anti-slide piles; A p The cross-sectional area of ​​the steel bars used for deep-buried small-diameter anti-slide piles.

[0090] The cross-sectional parameters d and D can be determined using the above equations (13), (14), and (15).

[0091] Step S5: Calculate the axial force F of the deeply buried small-diameter anti-slide pile body caused by compression of the deeply buried small-diameter anti-slide pile after the slight flexural deformation at the structural surface.

[0092] From equations (4), (5), (7), and (10), the calculation formula for the axial force F of the deeply buried small-diameter anti-slide pile caused by compression after slight deflection deformation at the structural surface can be obtained, as shown in equation (12):

[0093]

[0094] Use the following formula to verify the compressive bearing capacity of the section of a small-diameter anti-slide pile.

[0095] F0+F≤A p f y +(AA p )f c (16)

[0096] Among them, f c The compressive strength of the mortar used for deep-buried small-diameter anti-slide piles; f y The compressive strength of the steel bars used for deep-buried small-diameter anti-slide piles.

[0097] Step S6: Calculate the anchorage length of the deeply buried small-diameter anti-slide pile.

[0098] The anchorage length of deeply buried small-diameter anti-slide piles can be calculated using the following two formulas.

[0099]

[0100]

[0101] Among them, l a denoted as , where is the anchorage length of the deeply buried small-diameter anti-slide pile; K is the bond safety factor of the deeply buried small-diameter anti-slide pile, which can be taken as 2.6, 2.4, and 2.2 respectively, depending on the slope safety level (Level I, Level II, Level III); f rbk f represents the standard value of the ultimate bond strength between the rock layer and the deeply buried small-diameter anti-slide pile. b The standard value for the bond strength between steel bars and mortar can be determined according to the specification "Technical Specification for Building Slope Engineering GB 50330-2013".

[0102] Taking actual data from a certain project as an example, this paper introduces the method for determining the structural plane of small-diameter piles reinforced under the above-mentioned post-load conditions. The weight of the landslide rock mass is 26.5 kN / m. 3The sliding body is 5m thick, with an additional top load of 624.7kPa. The width of the sliding body is 1m, and its horizontal extension is 1m. The outward tilt angle of the structural surface is 11°, the internal friction angle is 6°, the cohesion of the structural surface is 20kPa, the slope safety level is Grade I, and the stress ratio between the piles and the rock mass at the structural surface is 10. The preliminary design specifies a deep-buried small-diameter anti-slide pile diameter of 0.15m.

[0103] Substituting into equation (2), we get:

[0104]

[0105] Here, it is assumed that the diameter of the core reinforcement of the deeply buried small-diameter anti-slide pile is 0.05mm, and the elastic modulus of the reinforcement is taken as 2 × 10. 8 kPa, the elastic modulus of the mortar is taken as 3×10 kPa. 7 kPa, substituting into equations (14), (15), and (16), we get:

[0106]

[0107]

[0108] EA = 2 × 10 8 ×1.9635×10 -3 +3×10 7 ×(0.017671-1.9635×10 -3 )=864735kN

[0109] Substituting into equation (10), we get:

[0110]

[0111] Solving for:

[0112] β1=β=2.3178°

[0113] Substituting the above equation and the specific parameters into equation (12), we get:

[0114]

[0115] Substituting EA = 279775.88kN into equation (10), we get:

[0116]

[0117] Solving for:

[0118] β1=β=2.2804°

[0119] Substituting into equation (12), we get

[0120]

[0121] Substituting EA = 279711.59kN into equation (10), we get:

[0122]

[0123] Solving for:

[0124] β1=β=2.2804°

[0125] Substituting into equation (12), we get

[0126]

[0127] Since the calculated value of EA is 279711.59kN for two consecutive times, we take β1=β=2.2804° and EA>279711.59kN.

[0128] If the diameter of the deeply buried small-diameter anti-slide pile is taken as 0.15m, then substituting into equations (14), (15), and (16) yields:

[0129]

[0130] Solving for the problem, we get: the diameter of the reinforcing bar d ≥ 0 mm.

[0131] Substituting into equation (13), the axial force of the deeply buried small-diameter anti-slide pile can be obtained as follows:

[0132]

[0133] If the elastic compressive strength of the steel bar is taken as 300 MPa, and the elastic modulus of the mortar is taken as 14.3 MPa, then we can get the result by substituting into equation (14).

[0134]

[0135] Solving for the diameter, we find that the diameter of the reinforcing bar, d, is ≥ 88.84 mm. Therefore, we take the diameter of the reinforcing bar as 90 mm.

[0136] Substituting into equations (14), (15), and (16), we get:

[0137]

[0138]

[0139] Taking the standard value of the ultimate bond strength between the rock and the anchor body as 1000 kPa, according to equation (12), we can obtain:

[0140]

[0141] Taking the standard value of the bond strength between the steel bar and the mortar as 2100 kPa, according to formula (13), we can obtain:

[0142]

[0143] The anchorage length is set at 11.2m.

[0144] This invention provides a design method for reinforcing rock mass structures with small-diameter piles under post-loading conditions. Based on the load borne by the small-diameter pile before slippage, the anti-slip force calculation formula is derived according to static equilibrium, the Mohr-Coulomb criterion, and the geometric relationship after slight deflection of the deeply buried small-diameter pile. The anti-slip force is differentiated with respect to the deflection angle of the deeply buried small-diameter pile, and the derivative is set to 0 to obtain the maximum and maximum values ​​of the anti-slip force. The design compressive stiffness and axial force of the deeply buried small-diameter anti-slip pile are calculated by ensuring that the anti-slip force is greater than or equal to the product of the slope stability safety factor and the sliding force. Based on the axial force, the cross-sectional compressive bearing capacity of the deeply buried small-diameter anti-slip pile is verified, and the anchorage length of the deeply buried small-diameter anti-slip pile is calculated. This invention considers the positive effect of the upper load already borne by the pile and the axial force of the deeply buried small-diameter anti-slip pile on anti-slip before the slider slides, deriving the maximum anti-slip force that the deeply buried small-diameter anti-slip pile can provide during deformation. Compared with empirical design methods, this method can save resources.

[0145] The above description is merely a preferred embodiment of the present invention and an explanation of the technical principles employed, and is not intended to limit the scope of the claimed invention, but merely to illustrate preferred embodiments of the invention. Those skilled in the art should understand that the scope of the invention is not limited to the specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the inventive concept. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

Claims

1. A design method for reinforcing a rock mass structure surface by a small-diameter pile under a post-loading condition, characterized in that, The method comprises the following steps: Obtaining the dip angle θ of the outer-dipping structural plane of the rock mass, the internal friction angle cohesion c, the bulk density γ of the sliding mass, the height H of the sliding mass, the stress ratio n of the pile and the rock mass at the structural plane, the vertical load q applied on the upper part of the sliding mass after the construction of a single deep-buried small-diameter anti-slide pile, the width b of the sliding mass, and the horizontal extension length l; An initial axial force value of the deep-buried small-diameter anti-slide pile is calculated using the following formula when there is no sliding: In the formula, A is the sectional area of the deep-buried small-diameter anti-slide pile; A flexural inclination β1 of the deep-buried small-diameter anti-slide pile is calculated when the anti-slide force is maximum, wherein the value of β is iteratively calculated using the Newton method until the following formula is satisfied, and the value is β1; The formula is referred to as a flexural inclination calculation formula, wherein the initial value of β is 0°, and E is the composite elastic modulus of the deep-buried small-diameter anti-slide pile; The minimum compressive stiffness of the deep-buried small-diameter anti-slide pile is calculated using the following formula: The above formula is called a compressive stiffness calculation formula, where F st is a safety factor for slope stability; The minimum value of EA satisfying the anti-slide requirement is calculated by iteratively solving the compressive stiffness calculation formula and the flexural inclination calculation formula, and the sectional parameters are determined using the following formula: EA = E s A p +E m (A-A p ) Wherein, D is the diameter of the deep-buried small-diameter anti-slide pile; d is the diameter of the core steel of the deep-buried small-diameter anti-slide pile; E s is the elastic modulus of the steel used for the deep-buried small-diameter anti-slide pile; E m is the elastic modulus of the mortar used for the deep-buried small-diameter anti-slide pile; A p is the cross-sectional area of the steel used for the deep-buried small-diameter anti-slide pile; F st is the safety factor of the slope stability; The axial force F of the deep-buried small-diameter anti-slide pile caused by compression after a slight flexural deformation of the structural plane is calculated using the following formula: The anchoring length of the deep-buried small-diameter anti-slide pile is calculated using the following two formulas: Wherein, l a is the anchoring length of the deep-buried small-diameter anti-slide pile; K is the bonding safety factor of the deep-buried small-diameter anti-slide pile, f rbk is the standard value of the ultimate bonding strength between the rock layer and the deep-buried small-diameter anti-slide pile, f b is the standard value of the bonding strength between the steel bar and the mortar.

2. The design method of reinforcing structural plane of rock mass by small diameter pile under post-loading condition according to claim 1, characterized in that, The method further comprises: The compressive bearing capacity of the small-diameter anti-slide pile is checked using the following formula: F0+F≤A p f y +(A-A p )f c wherein f c is the compressive strength of the mortar used for the deep-buried small-diameter anti-slide pile; f y is the compressive strength of the steel used for the deep-buried small-diameter anti-slide pile.

3. The method according to claim 1, wherein, The stress ratio n of the pile and the rock mass at the structural plane is the value under the vertical deformation condition under the vertical load condition.

4. The method according to claim 1, wherein, The bonding safety factor K of the deep-buried small-diameter anti-slide pile is 2.6, 2.4, and 2.2 respectively according to the safety levels of the slope, i.e., first level, second level, and third level.

5. The method according to claim 1, wherein, The step of iteratively solving the minimum value of EA satisfying the anti-slide requirement by using the compressive stiffness calculation formula and the flexural inclination calculation formula comprises: An initial value of EA is first given, β1 is solved by bringing the initial value into the flexural inclination calculation formula, EA is solved by bringing β1 into the compressive stiffness calculation formula, β1 is solved by bringing the solved EA into the flexural inclination calculation formula, EA is solved by bringing β1 into the compressive stiffness calculation formula, and the cycle is repeated until the solved EA is equal to the EA obtained in the previous cycle, the cycle is ended, and the EA obtained in the last cycle is taken as the minimum value of EA.

6. The method according to claim 5, wherein, The initial value of EA is in the range of [190852 kN, 5089380 kN].

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