Method for calculating bearing capacity of socketed pile when goaf is located at pile end

Abstract

The invention discloses a method for calculating the bearing capacity of a socketed pile when a goaf is located at the pile end, and relates to the field of socketed pile bearing mechanism analysis under the geological condition of the goaf. The method comprises the following steps: acquiring geometric information of a socketed pile according to geological information of a goaf site; selecting a load transmission model of pile side friction resistance and pile end resistance; according to geological information of a goaf site, load transmission model parameters are determined by combining a mechanical test; establishing a load transfer equation of the socketed pile according to the geometric information of the socketed pile; substituting the pile side frictional resistance load transfer model and parameters thereof into the load transfer equation of the socketed pile to obtain a socketed pile bearing capacity load transfer differential equation when the goaf is located at the pile end, and arranging the equation into an iterative solution form; and by adjusting pile end settlement of the socketed pile, iterative calculation is conducted on the load transfer differential equation till pile top settlement and pile top axial force, meeting the engineering requirements, of the socketed pile when the goaf is located at the pile end are obtained, and the bearing capacity of the socketed pile when the goaf is located at the pile end is represented.

Description

Technical Field

[0001] The invention relates to the field of agriculture, and in particular to a method for calculating the bearing capacity of a rock-embedded pile when a goaf is located at a pile end. Background Art

[0002] With the rapid development of expressways, the number of cases where expressways pass through goafs has gradually increased in recent years. It can be predicted that in the future, passing through goafs will become an inevitable solution for the design of proposed expressways. Goafs exist in the mineral-rich areas of southwest my country, which has an adverse effect on the stability of the foundation. Rock-embedded piles are a common foundation in expressways, with the characteristics of high bearing capacity and small settlement. When rock-embedded piles are directly embedded in the stratum above the goaf, the structural system composed of rock-embedded piles and goafs must have a certain bearing capacity to ensure the safety of the upper structure. The bearing capacity of the rock-embedded pile-goaf structural system is mainly affected by the following factors: (1) pile top load; (2) distance from the goaf to the pile end; (3) nature of the rock in the goaf. When the distance from the goaf to the pile end is small, as the load on the pile top increases, the pile side friction resistance is small, and the pile end rock has generated pile end resistance. Therefore, in this state, the pile end rock is first damaged, and the pile-soil relative displacement or pile-rock relative displacement suddenly increases, causing the pile side friction resistance to play a role. At this time, the pile end resistance accounts for a very small proportion, and the stress condition of the rock-embedded pile is similar to that of a friction pile. When the distance from the goaf to the pile end is large, the pile side friction resistance has been fully exerted, and the pile side friction resistance is constant. Therefore, the failure standard of the rock-embedded pile in this state is the failure of the rock at the pile end. When the distance from the goaf to the pile end is large, the difference between the rock-embedded pile under the goaf condition and the ordinary rock-embedded pile is not obvious. In summary, when the goaf is located at the pile end, the bearing capacity of the rock-embedded pile needs to be considered comprehensively with the distance from the goaf to the pile end.

[0003] In the prior art, when a highway crosses a goaf, the main measure is to avoid it. When it is impossible to avoid it, a large amount of grouting is performed in the goaf to ensure the stability of the goaf. This solution can ensure the bearing capacity of rock-embedded piles under goaf conditions. However, the grouting method mainly uses the uniaxial saturated compressive strength of rock to calculate the bearing capacity of rock-embedded piles. This method cannot reflect the load transfer mechanism of rock-embedded piles, and cannot predict the impact of goaf on the load transfer mechanism of rock-embedded piles in advance, which will cause a waste of resources, and the grouting effect cannot be fully guaranteed.

[0004] Therefore, it is urgent to establish a theoretical method and mathematical model to solve the bearing capacity of rock-embedded piles when the goaf is located at the pile end, so as to quantitatively and accurately evaluate the safety of rock-embedded piles when the goaf is located at the pile end, and provide theoretical guidance for the design of rock-embedded piles in goaf areas. Summary of the invention

[0005] Based on this, it is necessary to provide a method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end to address the above technical problems.

[0006] The present invention adopts the following technical solutions: The present invention provides a method for calculating the bearing capacity of a rock-embedded pile when a goaf is located at the pile end, comprising the following steps: According to the geological information of the goaf, the geometric information of the rock-embedded pile is obtained when the goaf is located at the pile end; According to the geometric information of rock-embedded piles, the load transfer equation of rock-embedded piles is established; Construct the pile side friction load transfer model and pile end resistance load transfer model of rock-embedded piles; and determine the parameters of the pile side friction load transfer model and pile end resistance load transfer model based on the geological information of the goaf and combined with indoor mechanical tests; Substitute the load transfer model and parameters of pile side friction into the load transfer equation of rock-embedded piles, obtain the load transfer differential equation of rock-embedded pile bearing capacity when the goaf is located at the pile end, and organize it into an iterative solution form; The pile end resistance of the rock-embedded pile is calculated using the pile end resistance load transfer model and its parameters. The pile end settlement of the rock-embedded pile is adjusted, and the load transfer differential equation is iteratively calculated through the pile end settlement and pile end resistance until the pile top settlement and pile top axial force of the rock-embedded pile are obtained when the goaf is located at the pile end, which meet the engineering requirements, so as to characterize the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

[0007] Preferably, the geometric information of the rock-socketed pile specifically includes: the circumference of the rock-socketed pile cross section, the elastic modulus of the rock-socketed pile and the cross-sectional area of ​​the rock-socketed pile.

[0008] Preferably, the pile side friction load transfer model is formulated as follows: ; In the formula, τ ( z ) is the pile side friction resistance, m, n are the pile side friction load transfer model parameters; The pile tip resistance load transfer model is formulated as follows: ; In the formula, P b is the pile end resistance, s b is the settlement of the pile top, m b and n b is the load transfer model parameter for pile tip resistance.

[0009] Preferably, the step of obtaining the load transfer differential equation of the rock-embedded pile bearing capacity when the goaf is located at the pile end and arranging it into an iterative solution form specifically includes: Establish the load transfer equation of rock-embedded piles: ; In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, τ ( z ) is the pile side friction resistance, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile; Substituting the pile side friction load transfer model into the load transfer equation of rock-embedded piles, the load transfer differential equation is obtained: ; Arranged as: ; In the formula, s ''( z ) The second-order derivative of the relative displacement between pile and soil or pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m, n are the pile side friction load transfer model parameters; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is solved P b ; The initial boundary conditions of the load transfer differential equation are: ; In the formula, s ( z 1) is the relative displacement between pile and soil or pile and rock, z 1 Vertical coordinate of pile end, E p is the elastic modulus of the rock-embedded pile; Introducing new variables s 0 and s 1: ; Solving for the first derivative of a new variable s '0 and s '1: ; Substitute the rearranged load transfer differential equation and the new variable into the first derivative of the new variable: ; Modify the initial boundary conditions to: ; In the formula, z 1 is the vertical coordinate of the pile end, s b is the pile tip settlement, P b is the pile end resistance, E p is the elastic modulus of the rock-embedded pile; Combining the sorted load transfer differential equation, the introduced new variables and the first-order derivatives of the new variables, the iterative solution form of the load transfer differential equation of the bearing capacity of the rock-embedded pile when the goaf is located at the pile end is obtained, and the formula is: ; In the formula, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m, n Parameters of the pile side friction load transfer model.

[0010] Preferably, the adjusting the pile end settlement of the rock-embedded pile comprises iteratively calculating the load transfer differential equation through the pile end settlement and the pile end resistance, and specifically comprising: Divide the rock-embedded pile into M calculation segments in the vertical direction; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is calculated P b ; Wherein, the pile end is the Mth calculation section of the rock-embedded pile, and s b = s bM , P b = P bM ; set up Δh is the step length, l M is the length of the Mth calculation section of the rock-embedded pile, then the incremental step N = l M / Δh ; The pile end settlement of the Mth calculation section of the rock-embedded pile sbM and pile end resistance P bM Substituting into the equations: ; In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, P b is the pile end resistance; Solve to obtain the pile top settlement of the Mth calculation section of the rock-embedded pile s tM and pile top axial force P tM ; Among them, the pile top settlement of the Mth calculation section of the rock-embedded pile is s tM and pile top axial force P tM That is the M - Pile tip settlement for 1 calculation segment s b(M-1) and pile end axial force P b(M-1) Using the pile tip settlement of the M-1th calculation segment s b(M-1) and pile end axial force P b(M-1) , and substitute it into the equation group to obtain the pile top settlement of the M-1th calculation section of the rock-embedded pile s t(M-1) and pile top axial force P t(M-1) ; Repeat the calculation until M = 1, and obtain the pile top settlement s of the first calculation segment. t1 and the pile top axial force P t1 , at this time s t1 and P t1 It is the pile top settlement and pile top axial force of the entire pile, that is, the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

[0011] At least one of the above technical solutions adopted by the present invention can achieve the following beneficial effects: The present specification provides a method for calculating the bearing capacity of rock-embedded piles when a goaf is located at the pile end. First, the geometric information of the rock-embedded piles is collected based on the on-site geological information of the goaf, and a suitable load transfer model of pile side friction and pile end resistance is selected, and the model parameters are determined in combination with the on-site geological information and indoor mechanical test results; a load transfer equation for the rock-embedded piles is established, and the pile side friction model and its parameters are substituted into the load transfer equation to obtain a load transfer differential equation, which is organized into an iterative solution form, and the pile end settlement is adjusted and iterative calculations are performed until the pile top settlement and pile top axial force of the rock-embedded pile meet the engineering requirements, thereby accurately characterizing the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

[0012] The present invention establishes a load transfer differential equation for solving the bearing capacity of the rock-embedded pile when the goaf is located at the pile end, takes into account the influence of the goaf at the pile end on the bearing capacity of the rock-embedded pile, and calculates the bearing capacity of the rock-embedded pile in combination with an iterative solution method. The result obtained by solving the equation is more accurate and the program is simple to compile. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings: Figure 1 A schematic flow chart of a method for calculating the bearing capacity of a rock-embedded pile when a goaf is located at the pile end provided by the present invention; Figure 2 A schematic diagram of a method for calculating the bearing capacity of a rock-embedded pile when a goaf is located at a pile end provided by the present invention is provided. DETAILED DESCRIPTION

[0014] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with the specific embodiments of the present invention and the corresponding drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0015] The technical solutions provided by various embodiments of the present invention are described in detail below in conjunction with the accompanying drawings.

[0016] Figure 1 A flow chart of a method for calculating the bearing capacity of a rock-embedded pile when a goaf is located at the pile end provided by the present invention comprises the following steps: S101: According to the geological information of the goaf, the geometric information of the rock-embedded pile is obtained, including: Geological information of the goaf site, including: stratum thickness, stratum elastic modulus, Poisson's ratio, cohesion and internal friction angle; Basic parameters of concrete piles, including: concrete elastic model, Poisson's ratio, cross-sectional perimeter, cross-sectional area, and length; Geometric information of rock-socketed piles, including: circumference of rock-socketed pile cross section U p , elastic modulus of rock-embedded pile E p , cross-sectional area of ​​rock-embedded pile A p .

[0017] S102: Select a load transfer model for pile side friction and pile end resistance; and determine the parameters of the load transfer model based on the geological information of the goaf site and combined with indoor mechanical tests, including: Select the pile side friction load transfer model, the mathematical formula is: ; In the formula, τ ( z ) is the pile side friction resistance, m, n are the pile side friction load transfer model parameters; Select the pile end resistance load transfer model, the mathematical formula is: ; In the formula, P b is the pile end resistance, s b is the settlement of the pile top, m b and n b is the load transfer model parameter for pile tip resistance.

[0018] S103: Establishing the load transfer equation of the rock-embedded pile based on the geometric information of the rock-embedded pile; substituting the load transfer model of the pile side friction resistance and its parameters into the load transfer equation of the rock-embedded pile, obtaining the load transfer differential equation of the bearing capacity of the rock-embedded pile when the goaf is located at the pile end and arranging it into an iterative solution form, specifically including: Establish the load transfer equation of rock-embedded piles: ; In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, τ ( z ) is the pile side friction resistance, U p is the cross-sectional perimeter of the rock-embedded pile, Ep is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile; Substituting the pile side friction load transfer model into the load transfer equation of rock-embedded piles, the load transfer differential equation is obtained: ; Arranged as: ; In the formula, is the second-order derivative of the relative displacement between pile and soil or pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m, n are the pile side friction load transfer model parameters; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is solved P b ; The initial boundary conditions of the load transfer differential equation are: ; In the formula, s ( z 1) is the relative displacement between pile and soil or pile and rock, z 1 Vertical coordinate of pile end, E p is the elastic modulus of the rock-embedded pile; Introducing new variables s 0 and s 1: ; Solving for the first derivative of a new variable s ' 0 and s ' 1 : ; Substitute the rearranged load transfer differential equation and the new variable into the first derivative of the new variable: ; Modify the initial boundary conditions to: ; In the formula, z1 is the vertical coordinate of the pile end, s b is the pile tip settlement, P b is the pile end resistance, E p is the elastic modulus of the rock-embedded pile; Combining the sorted load transfer differential equation, the introduced new variables and the first-order derivatives of the new variables, the iterative solution form of the load transfer differential equation of the bearing capacity of the rock-embedded pile when the goaf is located at the pile end is obtained, and the formula is: ; In the formula, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m, n Parameters of the pile side friction load transfer model.

[0019] S104: By adjusting the pile end settlement of the rock-embedded pile, the load transfer differential equation is iteratively calculated until the pile top settlement and pile top axial force of the rock-embedded pile when the goaf is located at the pile end that meet the engineering requirements are obtained to characterize the bearing capacity of the rock-embedded pile when the goaf is located at the pile end, specifically including: Divide the rock-embedded pile into M calculation segments in the vertical direction; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is calculated P b ; Wherein, the pile end is the Mth calculation section of the rock-embedded pile, and s b =s bM , P b = P bM ; set up Δh is the step length, l M For rock-embedded piles M The length of the calculation segment, then the incremental step N = l M / Δh ; The pile end settlement of the Mth calculation section of the rock-embedded pile s bM and pile end resistance P bM Substituting into the equations: ; In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, P b is the pile end resistance; Solve to obtain the pile top settlement of the Mth calculation section of the rock-embedded pile s tM and pile top axial force P tM ; Among them, the pile top settlement of the Mth calculation section of the rock-embedded pile is s tM and pile top axial force P tM That is the M - Pile tip settlement for 1 calculation segment s b(M-1) and pile end axial force P b(M-1) ; Using the pile tip settlement of the M-1th calculation segment s b(M-1) and pile end axial force P b(M-1) , and substitute it into the equation group to obtain the pile top settlement of the M-1th calculation section of the rock-embedded pile s tM-1 and pile top axial force P tM-1 ; Repeat the calculation until M=1, and solve for the pile top settlement s of the first calculation segment. t1 and pile top axial force P t1 , at this time s t1 and P t1 It is the pile top settlement and pile top axial force of the entire pile, that is, the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

[0020] This embodiment takes an indoor model test case where a goaf is located at the pile end as an example. Figure 2 , the model pile length is 1110 mm, the pile diameter is 32 mm, and the mechanical parameters of the soil and rock layers are shown in the following table:

[0021] In order to reasonably determine the parameters of the load transfer model, the results of the indoor model test were analyzed. By fitting the results of the indoor model test, the parameters of the pile side friction resistance within the soil layer are m = 0.00153 and n = 0.15711. When the distance from the goaf to the pile end is different, the parameters of the pile side friction and pile end resistance within the rock formation range are shown in the following table:

[0022] Substitute the relevant parameters into the model for calculation. The calculation results are shown in the table below:

[0023]

[0024] From the calculation results in the table, it can be found that there is a slight difference between the indoor model test results and the calculation results, which shows that the calculation method proposed in the present invention has certain feasibility and can accurately calculate the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

[0025] In summary, through the verification of this embodiment, it is explained that the present invention establishes a load transfer differential equation for solving the bearing capacity of the rock-embedded pile when the goaf is located at the pile end, considers the influence of the goaf at the pile end on the bearing capacity of the rock-embedded pile, and calculates the bearing capacity of the rock-embedded pile by combining the iterative solution method. Compared with the traditional method, the theory is rigorous, the solution result is more accurate, and the program is simple to compile.

[0026] The technical solution of the present invention is not limited to the above-mentioned specific embodiments. All technical variations made according to the technical solution of the present invention fall within the protection scope of the present invention.

Claims

1. A method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end, characterized in that: The method comprises the following steps: According to the geological information of the goaf, the geometric information of the rock-embedded pile is obtained when the goaf is located at the pile end; According to the geometric information of rock-embedded piles, the load transfer equation of rock-embedded piles is established; Construct the pile side friction load transfer model and pile end resistance load transfer model of rock-embedded piles; and determine the parameters of the pile side friction load transfer model and pile end resistance load transfer model based on the geological information of the goaf and combined with indoor mechanical tests; Substitute the load transfer model and parameters of pile side friction into the load transfer equation of rock-embedded piles, obtain the load transfer differential equation of rock-embedded pile bearing capacity when the goaf is located at the pile end, and organize it into an iterative solution form; The pile end resistance of the rock-embedded pile is calculated using the pile end resistance load transfer model and its parameters. The pile end settlement of the rock-embedded pile is adjusted, and the load transfer differential equation is iteratively calculated through the pile end settlement and pile end resistance until the pile top settlement and pile top axial force of the rock-embedded pile are obtained when the goaf is located at the pile end, which meet the engineering requirements, so as to characterize the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

2. A method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end as claimed in claim 1, characterized in that: The geometric information of the rock-socketed pile specifically includes: the circumference of the rock-socketed pile cross section, the elastic modulus of the rock-socketed pile and the cross-sectional area of ​​the rock-socketed pile.

3. The method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end as claimed in claim 1, characterized in that: The pile side friction load transfer model is formulated as follows: ； In the formula, τ ( z ) is the pile side friction resistance, m、n are the pile side friction load transfer model parameters; The pile tip resistance load transfer model is formulated as follows: ； In the formula, P b is the pile end resistance, s b is the settlement of the pile top, m b and n b is the load transfer model parameter for pile tip resistance.

4. The method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end as claimed in claim 1, characterized in that: The method of obtaining the load transfer differential equation of the rock-embedded pile bearing capacity when the goaf is located at the pile end and arranging it into an iterative solution form specifically includes: Establish the load transfer equation of rock-embedded piles: ； In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, τ ( z ) is the pile side friction resistance, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile; Substituting the pile side friction load transfer model into the load transfer equation of rock-embedded piles, the load transfer differential equation is obtained: ； Arranged as: ； In the formula, s ''( z ) is the second-order derivative of the relative displacement between pile and soil or pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m、n are the pile side friction load transfer model parameters; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is solved P b ; The initial boundary conditions of the load transfer differential equation are: ； In the formula, s ( z 1) is the relative displacement between pile and soil or pile and rock, z 1 Vertical coordinate of pile end, E p is the elastic modulus of the rock-embedded pile; Introducing new variables s 0 and s 1: ； Solving for the first derivative of a new variable s '0 and s '1 s ' 1 : ； Substitute the rearranged load transfer differential equation and the new variable into the first derivative of the new variable: ； Modify the initial boundary conditions to: ； In the formula, z 1 is the vertical coordinate of the pile end, s b is the pile tip settlement, P b is the pile end resistance, E p is the elastic modulus of the rock-embedded pile; Combining the sorted load transfer differential equation, the introduced new variables and the first-order derivatives of the new variables, the iterative solution form of the load transfer differential equation of the bearing capacity of the rock-embedded pile when the goaf is located at the pile end is obtained, and the formula is: ； In the formula, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, m、n Parameters of the pile side friction load transfer model.

5. The method for calculating the bearing capacity of rock-embedded piles when the goaf is located at the pile end as claimed in claim 1, characterized in that: The adjustment of the pile end settlement of the rock-embedded pile is performed by iteratively calculating the load transfer differential equation through the pile end settlement and the pile end resistance, and specifically includes: The rock-embedded piles are divided into M calculation segments; Assume the pile tip settlement is s b According to the pile end resistance load transfer model, the pile end resistance is calculated P b ; Wherein, the pile end is a rock-embedded pile M calculation segments, and s b = s bM , P b = P bM ; set up Δh is the step length, l M For rock-embedded piles M The length of the calculation segment, then the incremental step N = l M / Δh ; The rock-embedded pile M Pile tip settlement of each calculation segment s bM and pile end resistance P bM Substituting into the equations: ； In the formula, s ( z ) is the relative displacement between pile and soil or between pile and rock, z is the depth of rock-embedded piles, U p is the cross-sectional perimeter of the rock-embedded pile, E p is the elastic modulus of the rock-embedded pile, A p is the cross-sectional area of ​​the rock-embedded pile, P b is the pile end resistance; Solve for the rock-embedded pile M Pile top settlement of each calculation segment s tM and pile top axial force P tM ; Among them, rock-embedded piles M Pile top settlement of each calculation segment s tM and pile top axial force P tM That is the M - Pile tip settlement for 1 calculation segment s b(M-1) and pile end axial force P b(M-1) ; Use the M - Pile tip settlement for 1 calculation segment s b(M-1) and pile end axial force P b(M-1) , and substitute it into the equations to obtain the rock-embedded pile M -1 calculation segment of pile top settlement s t(M-1) and pile top axial force P t(M-1) ; Repeat the calculation until M = 1, and the pile top settlement s of the first calculation segment is obtained by solving t1 and the pile top axial force P t1 , at this time s t1 and P t1 It is the pile top settlement and pile top axial force of the entire pile, that is, the bearing capacity of the rock-embedded pile when the goaf is located at the pile end.

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