Deformation analysis method of inclined short anchor pile foundation under elastic theory and deformation compatibility condition
By employing elastic theory and analytical methods under deformation compatibility conditions, the gap in deformation analysis of inclined anchor short pile foundations was filled, enabling the calculation of their displacement and internal forces, revealing influencing factors, completing the design of the ultimate tensile bearing capacity, and providing a basis for construction.
Patent Information
- Application Number
- CN202410485792.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-04-22
- Publication Date
- 2026-02-27
- Estimated Expiration
- 2044-04-22
AI Technical Summary
The lack of existing methods for analyzing the deformation of inclined anchored short pile foundations leads to high construction difficulty, high cost, and limited application.
Using elastic theory and analysis methods under deformation compatibility conditions, the calculation of elastic deformation is simplified by establishing displacement equilibrium equations. The foundation is decomposed into three parts for calculation. Combined with boundary conditions and the pull-out force of the anchor structure, the ultimate bearing capacity and deformation of the inclined anchor short pile are calculated.
The displacement and internal force calculations of inclined anchor short pile foundations were realized, the ultimate tensile bearing capacity design was completed, the main influencing factors of the foundation were revealed, and the basis for stability analysis was provided.
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Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of data processing, in particular to a short inclined anchor pile foundation deformation analysis method under elastic theory and deformation coordination conditions. BACKGROUND
[0002] At present, the power transmission lines in mountainous areas in China mainly adopt the types of hole-digging pile foundation, rock embedded foundation and rock anchor pile foundation. The hole-digging pile foundation and the rock embedded foundation have low requirements on geological conditions and are widely used in engineering. However, due to the large excavation amount of the two types of foundations, the construction is difficult and the cost is high. The rock anchor pile foundation has less material consumption, small excavation amount and low construction difficulty, but it has high requirements on engineering geological conditions, and the application proportion is low at present. In view of the above situation, a new type of composite foundation for power transmission tower, i.e. short inclined anchor pile foundation, is proposed. The short inclined anchor pile foundation is composed of short pile foundation and anchor structure. The pile foundation has good vertical bearing performance, the inclined anchor structure not only has the effect of enhancing the uplift resistance of the foundation and weakening the influence of horizontal load on the uplift performance, but also has the characteristics of simple production process and construction technology. The short inclined anchor pile foundation needs to adopt the type of multiple anchors. The symmetrically arranged anchors can more reasonably resist the horizontal load in each direction, so three anchors with a vertical projection direction included angle of 120° are arranged. The short pile foundation and the anchor of the short inclined anchor pile foundation need to jointly bear the external load, so the reinforcement of the anchor structure is inserted into the short pile foundation to make the two become a whole to jointly resist the external load. The structure of the short inclined anchor pile foundation of the mountainous area power transmission line is shown in Figure 1
[0003] In engineering, it is a relatively reliable method to predict the deformation and bearing capacity of the uplift pile foundation through field test, but it cannot be widely carried out due to construction conditions, economic cost and other factors. Domestic and foreign scholars have carried out a large number of researches on the deformation characteristics and ultimate bearing capacity of the uplift pile through the shear displacement method, elastic theory method, load transfer method and specification calculation method, but there is no research on the short inclined anchor pile foundation at present. SUMMARY
[0004] The purpose of the present application is to provide a short inclined anchor pile foundation deformation analysis method under elastic theory and deformation coordination conditions, aiming at solving the problem of no deformation analysis of the short inclined anchor pile foundation.
[0005] In order to achieve the above purpose, the present application provides a short inclined anchor pile foundation deformation analysis method under elastic theory and deformation coordination conditions, comprising the following steps:
[0006] According to the displacement deformation coordination condition of the short pile and the anchor, the displacement component of the short pile at the pile-anchor node in the displacement direction of the anchor is equal to the displacement amount of the anchor, and the displacement balance equation is established according to the displacement deformation coordination principle;
[0007] Simplify the elastic deformation calculation method of the inclined anchor short pile foundation based on the displacement balance equation according to the anti-pulling single pile elastic deformation analytical analysis method, and obtain a simplified equation;
[0008] Based on the simplified equation, the boundary conditions of the inclined anchor short pile foundation are considered, the whole foundation is divided into three parts, the first part is the pile body from the pile top to the pile-anchor node; the second part is a micro-element section at the pile-anchor node; the third part is the node to the pile bottom; the internal force and displacement of the upper half of the pile body and the internal force and displacement of the lower half of the pile body are calculated to obtain the solving result;
[0009] Based on the solving result, the anti-pulling ultimate bearing capacity design calculation of the inclined anchor short pile is completed, the calculation result is obtained, and the foundation deformation analysis is carried out to obtain the analysis result.
[0010] The displacement balance equation comprises:
[0011] Δ=fcosθ (1)
[0012] In the formula, Δ is the displacement of the anchor rod at the pile-anchor node, f is the displacement of the pile at the pile-anchor node, and θ is the included angle between the anchor rod and the pile body.
[0013] The displacement balance equation comprises:
[0014] The pile side rock mass is simplified as a plurality of independent springs, and the anti-pulling force of the anti-pulling pile is mainly provided by the pile side friction; the pile side shear stress is represented by the following formula:
[0015] τ=k s w (2)
[0016] In the formula, w is the displacement of the single pile, k s is the spring stiffness of the soil, which can be estimated by an empirical formula or theoretical derivation;
[0017] It is assumed that the pile body and the rock mass are both elastic materials, and do not have relative slip under the action of the load; the elastic modulus of the pile body E p , and the area of the pile body A P ; considering the balance of the vertical force of a pile unit separated from the pile body, the following formula can be obtained:
[0018] dF=Uτdz=Uk s wdz (3)
[0019] In the formula, U is the circumference of the pile, z is the length of the pile body from the ground surface, and k s is the ground reaction coefficient;
[0020] The ground reaction coefficient calculation method mainly includes two kinds:
[0021] Determine k using the theoretical methods of mechanics of materials s .
[0022]
[0023] Or it can be obtained based on the shear displacement method;
[0024]
[0025] In the formula: G s ν is the shear modulus of soil. s Let r be the Poisson's ratio of the soil / rock mass, r0 be the radius of the pile, and r m To influence the radius;
[0026] The relationship between the axial force and displacement of a pile can be expressed by the following formula:
[0027]
[0028] Differentiating equation (5) yields:
[0029]
[0030] Combined equations (2) and (6):
[0031]
[0032] It can be simplified to:
[0033]
[0034] in:
[0035]
[0036] The solution to the differential equation (8) is:
[0037] w = c1e -λz +c2e λz (10)
[0038] Substituting equation (10) into equation (6):
[0039] F=λE P A p (c1e -λz -c2e λz (11).
[0040] Wherein, the simplified equation is based on the boundary conditions of the inclined anchor short pile foundation, the whole foundation is divided into three parts, the first part is the pile body from the pile top to the pile-anchor node, the second part is a micro-element section at the pile-anchor node, and the third part is the pile body from the node to the pile bottom; the internal force and displacement of the upper half of the pile body and the internal force and displacement of the lower half of the pile body are calculated to obtain the solving result, including:
[0041] The boundary conditions of the inclined anchor short pile foundation are considered, the whole foundation is divided into three parts, the first part is the pile body from the pile top to the pile-anchor node, the length of which is z1, and the pile bottom axial force is F1; the second part is a micro-element section at the pile-anchor node, the length of which is extremely small and is assumed to be 0 in calculation, and the pile top axial force is F1 and the pile bottom axial force is F2; the third part is the pile body from the node to the pile bottom, and the pile top axial force is F2;
[0042] According to the static equilibrium condition, it is known that:
[0043] F1=F2+T(12)
[0044] F ten is the load applied to the pile top, T is the load applied to the pile body by the anchor rod during loading, z1 is the length of the anchor rod node from the pile top, and L is the length of the pile body;
[0045] The differential equations of the upper and lower pile bodies are solved respectively, and the differential equation of the upper half of the pile body is assumed to be:
[0046]
[0047] In the formula: c1, c2 are the coefficients of the differential equation of the upper half of the pile body;
[0048] The boundary conditions are:
[0049]
[0050] The solution is:
[0051]
[0052] The differential equation of the lower half is:
[0053]
[0054] In the formula: c3, c4 are the coefficients of the differential equation of the upper half of the pile body;
[0055] The boundary conditions are:
[0056]
[0057] The solution is:
[0058]
[0059] Since the displacement of the bottom of the upper half of the pile is the same as the displacement of the top of the lower half of the pile, that is,
[0060]
[0061] Substitute formula (15) and formula (16) into formula (19) and formula (12) to obtain:
[0062]
[0063] According to the deformation coordination condition at the pile-anchor joint, we have:
[0064]
[0065] In the formula: f is the displacement of the joint, that is, the displacement of the top of the lower half of the pile, k bolt The stiffness coefficient of the anchor bar is expressed as:
[0066]
[0067] In the formula: N is the number of anchor bars per anchor, E g is the elastic model of the anchor bar, and l is the length of the anchor bar, A s is the cross-sectional area of the anchor bar, so:
[0068]
[0069] By combining formula (19), formula (21) and formula (23), we can obtain:
[0070]
[0071] The uplift load is F ten The displacement of the top of the uplift pile is:
[0072] w t = c1 + c2 (25)
[0073] In the calculation process, the uplift resistance provided by the anchor bar structure can be solved by formula (24), the internal force and displacement of the upper half of the pile can be solved by combining formula (20), formula (25) and formula (24), and the internal force and displacement of the lower half of the pile can be solved by combining formula (18), formula (20) and formula (24).
[0074] The design calculation of the uplift ultimate bearing capacity of the inclined anchor short pile is completed based on the solving result, the calculation result is obtained, and the foundation deformation analysis is performed to obtain the analysis result, including:
[0075] The load effect acting on the foundation is not greater than the resistance of the foundation, and the uplift stability of the inclined anchor short pile foundation is designed by the following formula:
[0076] γ rf T≤R u (26)
[0077] Formula (26) wherein: R u is the uplift capacity of the inclined anchor short pile foundation; T is the uplift load, γ rf is an additional partial coefficient, which can be obtained by relevant provisions of the regulation;
[0078] The uplift capacity of the inclined anchor short pile foundation is mainly composed of three parts, i.e., the side friction provided by the short pile part, the vertical component of the uplift capacity of the anchor structure and the self weight of the pile body, so:
[0079] R u = η1R1(s min )+ η2R2(s min )+ G (27)
[0080] s min = min {s1, s2} (28)
[0081] Formula (28) wherein: s min is the limit displacement of the inclined anchor short pile foundation when failure occurs, R1(s min ), R2(s min ) are the load values corresponding to the vertical displacement of the short pile and the anchor rod reaching s m i n , η1 is the short pile bearing capacity development coefficient, η2 is the anchor rod bearing capacity development coefficient, G is the self weight of the short pile foundation, and s1 and s2 are the displacement amounts under the limit state of the short pile and the anchor rod respectively.
[0082] The method for analyzing the deformation of the inclined anchor short pile foundation under the elastic theory and deformation coordination condition, by the displacement deformation coordination condition of the short pile and the anchor rod, the displacement component of the short pile at the pile-anchor joint in the displacement direction of the anchor rod is equal to the displacement amount of the anchor rod, the displacement balance equation is established according to the displacement deformation coordination principle; based on the displacement balance equation, the elastic deformation calculation method of the inclined anchor short pile foundation is simplified according to the elastic deformation analysis method of the uplift single pile, and the simplified equation is obtained; based on the simplified equation, the boundary conditions of the inclined anchor short pile foundation are considered, the whole foundation is divided into three parts, the first part is the pile body from the top of the pile to the pile-anchor joint; the second part is a microelement section at the pile-anchor joint; the third part is the pile body from the joint to the bottom of the pile; the internal force and displacement of the upper half of the pile body and the internal force and displacement of the lower half of the pile body are calculated, and the solving result is obtained; based on the solving result, the design calculation of the uplift limit capacity of the inclined anchor short pile is completed, the calculation result is obtained, the foundation deformation analysis is carried out, the analysis result is obtained, and the problem that there is no foundation deformation analysis of the inclined anchor short pile is solved. BRIEF DESCRIPTION OF DRAWINGS
[0083] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, the drawings in the following description only constitute some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained from these drawings without creative effort.
[0084] Figure 1 is a schematic diagram of a inclined anchor short pile foundation.
[0085] Figure 2 is a schematic diagram of pile-anchor node deformation coordination condition.
[0086] Figure 3 is a schematic diagram of pile stress.
[0087] Figure 4 is a schematic diagram of pile stress.
[0088] Figure 5 is a flow chart of a deformation analysis method of a inclined anchor short pile foundation under elastic theory and deformation coordination condition provided by the present application. DETAILED DESCRIPTION
[0089] The embodiments of the present application will be described in detail below, and examples of the embodiments are shown in the drawings, wherein the same or similar notations represent the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the drawings are exemplary and are intended to explain the present application, and cannot be understood as a limitation of the present application.
[0090] Please refer to Figures 1 to 5 , the present application provides a deformation analysis method of a inclined anchor short pile foundation under elastic theory and deformation coordination condition, comprising the following steps:
[0091] S1, according to the displacement deformation coordination condition of the short pile and the anchor rod, the displacement component of the short pile at the pile-anchor node in the displacement direction of the anchor rod is equal to the displacement amount of the anchor rod, and the displacement balance equation is established according to the displacement deformation coordination principle;
[0092] Specifically, the displacement balance equation comprises:
[0093] Δ=fcosθ (1)
[0094] In the formula, Δ is the displacement amount of the anchor rod at the pile-anchor node, f is the displacement amount of the pile at the pile-anchor node, and θ is the included angle between the anchor rod and the pile body.
[0095] S2, based on the displacement balance equation, the elastic deformation calculation method of the inclined anchor short pile foundation is simplified according to the elastic deformation analytical analysis method of the uplift single pile, and a simplified equation is obtained;
[0096] Specifically, the pile side rock mass is simplified as many independent springs, and the uplift resistance of the uplift pile is mainly provided by the pile side friction; the pile side shear stress is expressed by the following formula:
[0097] τ=k s w (2)
[0098] In the formula: w is the displacement of a single pile, k s is the spring stiffness of the soil, which can be estimated by an empirical formula or theoretical derivation;
[0099] It is assumed that the pile body and the rock mass are both elastic materials, and there is no relative slip between them under the action of load; the pile body elastic modulus E p , the pile body area is A P ; considering the balance of the vertical force of a single pile unit separated from the pile body, we can get:
[0100] dF=Uτdz=Uk s wdz (3)
[0101] In the formula: U is the circumference of the pile, z is the length of the pile body from the ground surface, k s is the ground reaction coefficient;
[0102] There are mainly two methods to calculate the ground reaction coefficient:
[0103] The material mechanics theory method is used to determine k s .
[0104]
[0105] or obtained according to the shear displacement method;
[0106]
[0107] In the formula: G s is the shear modulus of the soil, v s is the Poisson's ratio of the rock-soil mass, r0 is the radius of the pile, r m is the influence radius;
[0108] The relationship between the axial force and the displacement of the pile can be expressed by the following formula:
[0109]
[0110] Differentiate formula (5) to get:
[0111]
[0112] Solve formula (2) and formula (6) together:
[0113]
[0114] It can be simplified as:
[0115]
[0116] wherein:
[0117]
[0118] The solution of the differential equation of formula (8) is:
[0119] w = c1e -λz +c2e λz (10)
[0120] Substitute formula (10) into formula (6):
[0121] F = λE P A p (c1e -λz -c2e λz ) (11)。
[0122] S3, based on the simplified equation, considering the boundary conditions of the inclined anchor short pile foundation, the whole foundation is divided into three parts, the first part is the pile body from the pile top to the pile-anchor node, the second part is a micro-element section at the pile-anchor node, and the third part is the pile body from the node to the pile bottom; the internal force and displacement of the upper half of the pile body and the internal force and displacement of the lower half of the pile body are calculated to obtain the solution result;
[0123] Specifically, considering the boundary conditions of the inclined anchor short pile foundation, the whole foundation is divided into three parts, the first part is the pile body from the pile top to the pile-anchor node, the length of which is z1, and the pile bottom axial force is F1; the second part is a micro-element section at the pile-anchor node, the length of which is extremely small and is assumed to be 0 in calculation, and the pile top axial force is F1, and the pile bottom axial force is F2; the third part is the pile body from the node to the pile bottom, and the pile top axial force is F2;
[0124] According to the static equilibrium condition, it is known that:
[0125] F1 = F2 + T (12)
[0126] F ten is the load applied to the pile top, T is the load applied to the pile body by the anchor rod during loading, z1 is the length of the anchor rod node from the pile top, and L is the length of the pile body;
[0127] The differential equations of the upper and lower pile bodies are solved respectively, and the differential equation of the upper half of the pile body is assumed to be:
[0128]
[0129] In the formula: c1, c2 are the coefficients of the differential equation of the upper half of the pile body;
[0130] The boundary conditions are:
[0131]
[0132] Solving the equation, we get:
[0133]
[0134] The lower half of the differential equation is:
[0135]
[0136] In the formula: c3, c4 are the coefficients of the upper half of the pile body differential equation;
[0137] The boundary conditions are:
[0138]
[0139] Solving the equation, we get:
[0140]
[0141] Because the displacement of the bottom of the upper half of the pile body is the same as the displacement of the top of the lower half of the pile body, that is:
[0142]
[0143] Substituting equation (15) and equation (16) into equation (19) and equation (12) can be obtained:
[0144]
[0145] According to the deformation compatibility condition at the pile-anchor joint:
[0146]
[0147] In the formula: f is the displacement of the pile body at the joint, that is, the displacement of the top of the lower half of the pile body, k bolt The stiffness coefficient of the anchor bar is expressed as:
[0148]
[0149] In the formula: N is the number of anchor bars per anchor, E g is the elastic model of the anchor bar, l is the length of the anchor bar, A s is the cross-sectional area of the anchor bar, so:
[0150]
[0151] Solving equation (19), equation (21), and equation (23) together, we get:
[0152]
[0153] The uplift load is Ften The displacement of the top of the pull-out pile is:
[0154] w t =c1+c2 (25)
[0155] During the calculation process, the pull-out force provided by the anchor structure can be solved by equation (24), and the differential equation of the upper half of the pile body can be solved by combining equations (20), (25), and (24) to calculate the internal force and displacement of the upper half of the pile body. The internal force and displacement of the lower half of the pile body can be solved by combining equations (18), (20), and (24).
[0156] S4 completes the design calculation of the ultimate uplift bearing capacity of the inclined anchor short pile based on the solution results, obtains the calculation results, and performs foundation deformation analysis to obtain the analysis results.
[0157] Specifically, the load acting on the foundation should not exceed the foundation's resistance. The pull-out stability of the inclined anchored short pile foundation is designed using the following formula:
[0158] γ rf T≤R u (26)
[0159] In the formula: R u γ represents the pull-out bearing capacity of the inclined anchored short pile foundation; T represents the pull-out load; γ represents the pull-out load. rf Additional component coefficients can be obtained from relevant clauses of the regulations;
[0160] The uplift bearing capacity of inclined anchored short pile foundations mainly consists of three parts: the side friction provided by the short pile, the vertical component of the uplift bearing capacity of the anchor structure, and the self-weight of the pile. Therefore:
[0161] R u =η1R1(s min )+η2R2(s min )+G (27)
[0162] s min =min{s1,s2} (28)
[0163] In the formula: s min R1(s) represents the ultimate displacement when the inclined anchored short pile foundation fails. min R2(s) min The vertical displacement of the short piles and anchors reaches s. m i n The corresponding load values are: η1 is the bearing capacity utilization coefficient of the short pile, η2 is the bearing capacity utilization coefficient of the anchor rod, G is the self-weight of the short pile foundation, and s1 and s2 are the displacements of the short pile and anchor rod under the ultimate state, respectively.
[0164] The present application provides a method for analyzing the deformation of a short inclined anchor pile foundation under the elastic theory and deformation compatibility conditions, which comprises the following steps: according to the displacement deformation compatibility conditions of the short pile and the anchor rod, the displacement component of the short pile at the pile-anchor node in the displacement direction of the anchor rod is equal to the displacement amount of the anchor rod, and a displacement balance equation is established according to the displacement deformation compatibility principle; based on the displacement balance equation, the elastic deformation calculation method of the short inclined anchor pile foundation is simplified according to the elastic deformation analytical analysis method of the uplift single pile, and a simplified equation is obtained; based on the simplified equation, the boundary conditions of the short inclined anchor pile foundation are considered, and the whole foundation is divided into three parts: the first part is the pile body from the top of the pile to the pile-anchor node; the second part is a micro-element section at the pile-anchor node; and the third part is the pile body from the node to the bottom of the pile; the internal force and displacement of the upper half of the pile body and the internal force and displacement of the lower half of the pile body are calculated to obtain the solving result; based on the solving result, the design calculation of the uplift ultimate bearing capacity of the short inclined anchor pile is completed to obtain the calculation result, and the foundation deformation analysis is carried out to obtain the analysis result, thereby solving the problem of no foundation deformation analysis of the short inclined anchor pile.
[0165] The beneficial effects are:
[0166] (1) The displacement and internal force calculation of the new short inclined anchor pile foundation under the elastic theory and deformation compatibility conditions can be realized.
[0167] (2) The design calculation of the uplift ultimate bearing capacity of the short inclined anchor pile is completed.
[0168] (3) It is found that the displacement and internal force of the short inclined anchor pile foundation are mainly affected by the mechanical properties of the rock-soil mass on the side of the pile, the size and material of the short pile, and the size and material of the anchor rod structure. The theoretical value of the uplift bearing capacity is mainly composed of the uplift bearing capacity of the short pile foundation, the uplift bearing capacity of the anchor rod structure and the self weight of the pile body.
[0169] The above only discloses a preferred embodiment of the method for analyzing the deformation of a short inclined anchor pile foundation under the elastic theory and deformation compatibility conditions, and of course cannot limit the scope of the present application, and those skilled in the art can understand that all or part of the above-mentioned embodiments can be implemented, and equivalent changes made according to the claims of the present application still belong to the scope covered by the present application.
Claims
1. A method for analyzing the deformation of inclined anchored short pile foundations under elastic theory and deformation compatibility conditions, characterized in that, Includes the following steps: Based on the displacement and deformation coordination condition between the short pile and the anchor, the displacement component of the short pile in the direction of the anchor displacement at the pile-anchor node is equal to the displacement of the anchor. The displacement balance equation is established based on the displacement and deformation coordination principle. Based on the displacement equilibrium equation, the elastic deformation calculation method of the inclined anchor short pile foundation is simplified according to the analytical analysis method of elastic deformation of the single pile with tensile strength, and a simplified equation is obtained. Based on the simplified equations, considering the boundary conditions of the inclined anchored short pile foundation, the foundation is divided into three parts: the first part is the pile body from the pile top to the pile-anchor node; the second part is a micro-element segment at the pile-anchor node; and the third part is the pile body from the node to the pile bottom. The internal forces and displacements of the upper half of the pile body and the internal forces and displacements of the lower half of the pile body are calculated to obtain the solution results. Based on the solution results, the design calculation of the ultimate uplift bearing capacity of the inclined anchor short pile was completed, and the calculation results were obtained. Then, the foundation deformation analysis was performed, and the analysis results were obtained. Based on the simplified equation, considering the boundary conditions of the inclined anchor short pile foundation, the foundation is divided into three parts: the first part is the pile body from the pile top to the pile-anchor node; the second part is a micro-element segment at the pile-anchor node; and the third part is the pile body from the node to the pile bottom. Calculate the internal forces and displacements of the upper and lower sections of the pile, and obtain the solution results, including: Considering the boundary conditions of the inclined anchored short pile foundation, the foundation is divided into three parts. The first part is the pile body from the pile top to the pile-anchor node, and its length is denoted as . The axial force at the pile bottom is The second part involves a tiny segment at the pile-anchor joint, whose length is extremely small and set to 0 in the calculation. The axial force at the pile top is assumed to be... Axial force at the bottom of the pile The third part is the pile body from the node to the pile bottom, and the axial force at the pile top is... ; According to the static equilibrium condition, we know that: (12) The load applied to the top of the pile, During the loading process, the anchor bolts apply loads to the pile body. This is the length of the anchor node from the top of the pile. The length of the pile; Solve the differential equations for the upper and lower ends of the pile body separately. Let the differential equation for the upper half of the pile body be: (13) In the formula: , The coefficients are the differential equations for the upper half of the pile body. This refers to the displacement of a single pile. The boundary conditions are: (14) Solving for: (15) The second half of the differential equation is: (16) In the formula: , These are the coefficients of the differential equation for the lower half of the pile body; The boundary conditions are: (17) In the formula: Z2: length from the anchor node to the bottom of the pile; Solving for: (18) Since the displacement of the bottom of the upper half of the pile body is the same as the displacement of the top of the lower half of the pile body, that is: (19) Substituting equations (15) and (16) into equations (19) and (12) and solving them together, we get: (20) From the deformation compatibility conditions at the pile-anchor joint, we can know that: (21) In the formula: This refers to the pile displacement at the node, specifically the top displacement of the lower half of the pile. The stiffness coefficient of the anchor bar is expressed as: (22) In the formula: The number of steel bars in each anchor rod. This is the elastic model for the anchor bar. The length of the anchor bar. Let be the cross-sectional area of the anchor bar, therefore: (23) By combining equations (19), (21), and (23), we can obtain: (24) Uplift load is The displacement of the top of the pull-out pile is: (25) During the calculation process, the pull-out force provided by the anchor structure can be solved by equation (24), and the differential equation of the upper half of the pile body can be solved by combining equations (20), (25), and (24) to calculate the internal force and displacement of the upper half of the pile body. The internal force and displacement of the lower half of the pile body can be solved by combining equations (18), (20), and (24).
2. The method for analyzing the deformation of inclined anchored short pile foundations under elastic theory and deformation compatibility conditions as described in claim 1, characterized in that, The displacement equilibrium equations include: (1) In the formula: This represents the displacement of the anchor rod at the pile-anchor joint. This represents the displacement of the pile at the pile-anchor joint. The angle between the anchor rod and the pile body.
3. The method for analyzing the deformation of inclined anchored short pile foundations under elastic theory and deformation compatibility conditions as described in claim 2, characterized in that, The method for calculating the elastic deformation of inclined anchor short pile foundations is simplified based on the displacement equilibrium equation and the analytical analysis method for elastic deformation of single piles resisting uplift, resulting in a simplified equation, including: The rock mass along the pile is simplified as many independent springs. The pull-out resistance of the pull-out pile is mainly provided by the skin friction along the pile side; the shear stress along the pile side is expressed by the following formula: (2) In the formula: This refers to the displacement of a single pile. The spring stiffness of soil can be estimated by empirical formulas or theoretical derivation; Assuming both the pile and the rock mass are elastic materials, and that there is no relative slippage between them under load; the elastic modulus of the pile is... The area of the pile body is By considering a single pile element detached from the pile body and taking the equilibrium of the vertical forces within that element, we can obtain: (3) In the formula: Let z be the perimeter of the pile, and z be the distance from the pile to the ground surface. This is the soil reaction coefficient; There are two main methods for calculating the foundation reaction coefficient: Determine using the theoretical methods of mechanics of materials Or it can be obtained based on the shear displacement method; (4) In the formula: The shear modulus of soil. Poisson's ratio of the rock and soil mass Let be the radius of the pile. To influence the radius; The relationship between the axial force and displacement of a pile can be expressed by the following formula: (5) Differentiating equation (5) yields: (6) Combined equations (2) and (6): (7) It can be simplified to: (8) in: (9) The solution to the differential equation (8) is: (10) Substituting equation (10) into equation (6): (11)。 4. The method for analyzing the deformation of inclined anchored short pile foundations under elastic theory and deformation compatibility conditions as described in claim 1, characterized in that, The design calculation of the ultimate uplift bearing capacity of the inclined anchor short pile is completed based on the solution results, and the calculation results are obtained. Then, the foundation deformation analysis is performed to obtain the analysis results, including: The load acting on the foundation is not greater than the foundation's resistance. The pull-out stability of the inclined anchored short pile foundation is designed using the following formula: (26) In the formula: For the pull-out bearing capacity of inclined anchor short pile foundation; F ten To lift the load, Additional component coefficients can be obtained from relevant clauses of the regulations; The uplift bearing capacity of inclined anchored short pile foundations mainly consists of three parts: the side friction provided by the short pile, the vertical component of the uplift bearing capacity of the anchor structure, and the self-weight of the pile. Therefore: (27) (28) In the formula: This represents the ultimate displacement when the inclined anchored short pile foundation fails. , For the vertical displacement of short piles and anchors to reach The corresponding load value at that time This is the bearing capacity utilization factor for short piles. Here, G is the anchor bolt bearing capacity utilization factor, and G is the self-weight of the short pile foundation. , These represent the displacements under the ultimate conditions of short piles and anchor bolts, respectively.