Passive pile stability analysis method based on the solution of degradable power series

Abstract

The application provides a passive pile stability analysis method based on a degradable power series solution, which comprises a passive pile segmentation theory based on a pile-soil interaction relationship, a passive pile differential control equation establishment, a passive pile power series solution method and a matrix transfer method. First, the differential control equation of each pile segment is established according to the passive pile segmentation theory, and the differential control equation considering the pile side friction, the p~Δ effect, the passive side soil pressure distribution, the pile body self weight and other factors is proposed according to the power series solution method, the corresponding degradable power series solution can be established by deleting the corresponding content of a factor when the action of the factor is not considered, and finally, the passive pile displacement and internal force are calculated according to the matrix transfer method. The method realizes more simple and accurate calculation of the passive pile displacement and internal force.

Description

Technical Field

[0001] This invention belongs to the field of passive pile stability analysis, specifically involving a passive pile stability analysis method based on degenerate power series solutions. Background Technology

[0002] In recent years, sparse piles (L>6d) have been widely used in soft soil foundations due to their excellent settlement control effect. In sparse pile roadbeds, the horizontal displacement of soft soil affects the piles, forming "passive piles." However, accidents involving passive pile slippage caused by excessive load, poor soil properties, or non-standard construction are increasing, making the analysis and research of passive piles urgent.

[0003] The core of solving passive pile problems lies in establishing physical and mechanical differential governing equations based on the pile's displacement mechanics characteristics and determining the corresponding solution method. While the differential governing equations are mostly similar in form, there are various solution methods, commonly including power series methods, finite difference methods, and finite layer methods. The basic principle of the finite difference and finite layer methods is to discretize the pile into n micro-elements, establish micro-element equations on each element, couple each micro-element, and thus obtain the internal force and displacement distribution of the entire pile. The passive pile segmentation theory divides the pile from top to bottom into free segments, passive segments, and active segments. Each segment is discretized into n smaller segments, and the soil resistance and axial force in the micro-elements are treated as constants to establish a finite difference solution, calculating the displacement internal forces of the passive pile under soil lateral displacement. However, the finite difference method is computationally cumbersome and labor-intensive. The power series solution, on the other hand, assumes that the solution to the differential equation is in power series form, substitutes it into the differential governing equation, and obtains the power series coefficients with respect to the initial boundary conditions u0, ... The power series solution of the equation can be obtained by back-substituting the formulas for M0 and V0, which is a simple calculation method. However, the calculation assumes that the soil resistance coefficient is constant and does not consider the influence of the pile's self-weight and friction, resulting in a large error in the calculation results.

[0004] However, establishing and solving the differential governing equations for passive piles requires considering factors such as pile side friction, p-Δ effect, passive lateral earth pressure distribution, and pile self-weight. This invention, based on the piecewise theory of passive piles, establishes differential governing equations and proposes a power series solution considering each influencing factor. When the effect of a certain factor is no longer needed, the corresponding content is deleted, thus establishing the corresponding degenerate power series solution. The degenerate power series solution and matrix transfer method are used to calculate the internal forces and displacements of the pile. Summary of the Invention

[0005] Purpose of the invention: To address the problems existing in the stability analysis of passive piles, this invention proposes a passive pile stability analysis method based on degenerate power series solutions.

[0006] Technical Solution: To achieve the objectives of this invention, the technical solution adopted is: a passive pile stability analysis method based on degenerate power series solutions, specifically including the following steps:

[0007] Step 1: Based on the different relationships between the passive pile and the surrounding soil, the passive pile is divided into free section, passive section and active section;

[0008] Step 2: Establish the differential control equations for the displacement and internal forces of the passive pile based on the basic principles of mechanics of materials;

[0009] Step 3: Consider the forces acting on the passive soil segment, the soil resistance, and the frictional resistance, and establish the differential governing equations; derive the degenerate power series solution using the power series solution method.

[0010] Step 4: The matrix transfer method is used to transfer the solution method of each segment to obtain the displacement and internal force transfer matrix of the pile top and pile end. Combined with the boundary conditions of the pile top and pile end, the displacement and internal force of the pile top and pile end are calculated, and then the stability of the passive pile is analyzed.

[0011] Furthermore, in step 1, based on the different relationships between the passive pile and the surrounding soil, the passive pile is divided into a free segment, a passive segment, and an active segment. The pile is then divided into n micro-units along its shaft, and each segment is assumed to be rigid.

[0012] Furthermore, the method for step 2 is as follows:

[0013] Step 2.1, when considering the pile's self-weight and skin friction, assume that the pile's skin friction increases linearly with depth, and consider the pile's self-weight and skin friction as a unified factor:

[0014] N z =N0+fz

[0015] Where: N z N0 represents the axial force at depth z of the pile, f represents the combined effect of pile side friction and self-weight, and z represents the depth of the pile.

[0016] Step 2.2: Establish the moment equilibrium equation for the micro-element with the midpoint of the bottom of the micro-element at point z of the pile as the origin. Considering the passive load, the moment equilibrium equation is obtained as follows:

[0017]

[0018] In the formula: M represents the bending moment of the micro-element, u represents the displacement of the micro-element, V represents the horizontal force of the micro-element, p(z) represents the soil resistance, and q(z) represents the soil pressure;

[0019] Considering the horizontal static equilibrium of the micro-element and the case of passive loads, the static equilibrium equations are as follows:

[0020] -V+[p(z)-q(z)]dz+V+dV=0

[0021] Ignoring the error caused by the quadratic term, and neglecting the quadratic term in the moment equilibrium equation, p(z) is expressed as kb1u. Combined with the static equilibrium equation, the simplified result is:

[0022]

[0023] In the formula: k represents the foundation resistance coefficient, and b1 is the calculated width of the pile body.

[0024] Step 2.3 further introduces the formula for the relationship between bending moment and displacement from mechanics of materials to calculate the relationship between bending moment and displacement of the pile micro-element:

[0025]

[0026] In the formula: EI is the bending stiffness of the material;

[0027] Substituting the formulas from mechanics of materials into the moment equilibrium equations yields the differential governing equations as follows:

[0028]

[0029] Step 2.4: Based on the above analysis, establish the differential control equations for the free segment, active segment, and passive segment of the pile.

[0030]

[0031] In the formula: V a Q(z) represents the horizontal force at the top of the free segment of the pile, Q(z) represents the surcharge earth pressure on the free segment, K represents the passive segment foundation resistance coefficient, and k represents the horizontal force at the top of the free segment. e k represents the soil resistance coefficient of the active elastic section of the pile body. d This represents the soil resistance coefficient of the active yielding section of the pile.

[0032] When the self-weight and side friction of the pile are not considered, f = 0 is taken, which degenerates to obtain the degenerate micro-unit control equations of the free segment, active segment and passive segment of the pile when the self-weight and side friction of the pile are not considered.

[0033] Furthermore, the method for step 3 is as follows:

[0034] Step 3.1: Based on the above differential governing equations, considering the forces acting on the passive soil segment, the soil resistance, and the skin friction, establish the differential governing equations:

[0035]

[0036] In the formula, K represents the passive section foundation resistance coefficient, which is often determined using the "K0" method or the "m" method in the analysis of passive pile foundations. This method uses the "m" method. Considering the influence of pile top displacement on K, a two-parameter model is introduced:

[0037] K = m(z0 + z) n

[0038] In the formula: z0 represents the equivalent depth at the surface; m is the foundation ratio coefficient; n is the depth index. To facilitate the establishment of the degenerate solution, n = 1, that is, K = m(z0 + z) = C0 + mz, where C0 = mz0;

[0039] Step 3.2, the distribution of the external load q(z) is represented by the superposition of q1, q2, and q3, which are rectangular, triangular, and parabolic distributions, respectively. The formula is q(z) = q1 + q2 + q3 = Tz. 2 +Lz+R represents the quadratic term coefficient, linear term coefficient, and constant term coefficient, respectively.

[0040] The differential equation is:

[0041]

[0042] Step 3.3, let Simplifying the equation, we get:

[0043]

[0044] Step 3.4: Characterize the bottom displacement of the micro-unit as a power series, i.e. Where a i Let be the coefficient of the i-th term. And for u... z By finding the first, second, and fourth derivatives, we can obtain:

[0045]

[0046] Step 3.5 introduces the displacement and internal force calculation equations from mechanics of materials to calculate the rotation angle, bending moment, and displacement of the pile micro-element.

[0047]

[0048] Step 3.6, consider the boundary condition z = 0 at the top of the micro-element:

[0049]

[0050]

[0051] We can obtain:

[0052] a0 = u0;

[0053] Among them, the top displacement, rotation angle, bending moment and horizontal force of the micro-unit are represented by u0, ... M0 and V0 represent the values, and the displacement, rotation, bending moment, and horizontal force at the bottom of the micro-unit are represented by u, respectively. z , M z V z express.

[0054] Further calculations yield the expressions for a4, a5, and a6:

[0055]

[0056] By recursion, we can deduce that when i ≥ 7:

[0057]

[0058] Step 3.7, place a i Write it as containing u0, The expressions for M0 and V0, and using H 0,i Replace the coefficients before u0 with H 1,i Replacement The first coefficient, denoted by H 2,i Replace the coefficients before M0 with H 3,i Replace the coefficients before V0 with H 4,i Substitute the constant term. The result simplifies to:

[0059]

[0060] a i a i-5 a i-4 a i-3 a i-2 Substituting the values ​​into the equations, we can obtain H. j,i Derivation formula for (j=0,1,2,3,4):

[0061]

[0062] H can be seen j,i expression and a i The expressions are consistent and can be based on a i (i<6), H can be calculated j,i The result of (j = 0, 1, 2, 3, 4, i < 6). Simultaneously, substitute into the formula... We can obtain:

[0063]

[0064] Simplifying, we get:

[0065]

[0066] Step 3.8: Substitute the formulas from the mechanics of materials in step 3.5 to obtain the calculated results. M z V z The calculation formula is as follows:

[0067]

[0068] In the formula:

[0069]

[0070]

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] Among them, A i B i C i D i E i (i = 1 to 4) are the solution coefficients of a certain segment of a micro-unit.

[0081] Step 3.9, as shown in the above analysis, H is calculated. j,i The distribution pattern can be observed, thus allowing the use of power series methods for solution. To establish a degenerate power series solution, the expression in step 3.4 is represented in a degenerate form to analyze the meaning of each term, thereby establishing a degenerate solution.

[0082]

[0083] In the formula, The term represents the bending moment term at the top of the micro-element. Characterizing the axial force term at the top of the micro-unit, Characterizing the effect of side friction on micro-units, β 4 u represents the soil resistance calculated using the "m" method, α 5 The zu characterizes the calculation of the soil resistance along the pile using the "K0" method, tz 2 +lz+r represents the force exerted by the passive soil section on the pile.

[0084] Furthermore, the method for step 4 is as follows:

[0085] When using the matrix transfer method to calculate using the above method, it can be expressed as:

[0086]

[0087] In the formula S i For the coefficient transfer matrix, Represents the displacement and internal force state vector at the top of the pile, where u c , M c V c It indicates the displacement, rotation angle, bending moment, and horizontal force at the top of the pile. The vector representing the displacement and internal force state at the end of the pile, where u f , M f V f It represents the displacement, rotation angle, bending moment, and horizontal force at the pile end.

[0088]

[0089] Where A ij B ij C ij D ij (i is the i-th micro-unit i = 1 to n, j = 1 to 4), and is the transfer matrix coefficient.

[0090] The beneficial effects of the present invention: Compared with existing calculation methods, the technical solution of the present invention has the following beneficial technical effects:

[0091] This invention relates to the piecewise theory of passive piles, the establishment of the differential governing equations for passive piles, the power series solution method for passive piles, and the matrix transfer method. It proposes differential governing equations considering multiple factors such as pile side skin friction, p-Δ effect, passive lateral earth pressure distribution, and pile self-weight, and establishes the corresponding degenerate power series solution. The matrix transfer method is then used for calculation. Through the above analysis, the displacement, rotation angle, bending moment, and horizontal force at the pile top and bottom are calculated. Furthermore, based on the actual engineering requirements for pile stability, a passive pile stability analysis is performed. Attached Figure Description

[0092] Figure 1 It is a passive pile segmented model;

[0093] Figure 2 These are the force diagrams of the passive pile micro-unit; (a) is the force diagram of the active segment, (b) is the force diagram of the passive segment, and (c) is the force diagram of the free segment.

[0094] Figure 3 This is a schematic diagram of the external load distribution of passive piles. Detailed Implementation

[0095] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0096] The passive pile stability analysis method based on degenerate power series solutions described in this invention specifically includes the following steps:

[0097] Step 1, see Figure 1 Based on the different relationships between passive piles and the surrounding soil, passive piles are divided into free sections, passive sections, and active sections.

[0098] Step 2: When considering the pile's self-weight and skin friction, assume that the pile's skin friction increases linearly with depth, and consider the pile's self-weight and skin friction as a unified factor:

[0099] N z =N0+fz

[0100] Step 2.1, see Figure 2 By establishing the moment equilibrium equation of the micro-element with the midpoint of the bottom of the micro-element as the origin, and considering the passive load, the moment equilibrium equation is obtained as follows:

[0101]

[0102] Considering the horizontal static equilibrium of the micro-element and the case of passive loads, the static equilibrium equations are as follows:

[0103] -V+[p(z)-q(z)]dz+V+dV=0

[0104] Neglecting the error caused by the quadratic term, omitting the quadratic term in the moment equilibrium equation, and combining it with the static equilibrium equation, we can simplify to obtain:

[0105]

[0106] Step 2.2 further introduces the material mechanics formulas.

[0107]

[0108] Substituting the formulas from mechanics of materials into the moment equilibrium equations yields the following differential governing equations:

[0109]

[0110] Step 2.3: Based on the above analysis, the differential control equations for the free segment, active segment, and passive segment of the pile can be established segment by segment.

[0111]

[0112] Step 3: Based on the above differential governing equations, considering the forces acting on the passive soil segment, the soil resistance, and the skin friction, establish the differential governing equations:

[0113]

[0114] Step 3.1, using the "m" method to determine K, considering the influence of pile top displacement on K, a two-parameter model is introduced:

[0115] K = m(z0 + z) n

[0116] To facilitate the establishment of degenerate solutions, we take n = 1, that is, K = m(z0 + z) = C0 + mz;

[0117] Step 3.2, see Figure 3 The distribution of the external load q(z) can be represented as the superposition of q1, q2, and q3, which are rectangular, triangular, and parabolic distributions, respectively, i.e., q(z) = Tz. 2 +Lz+R. The differential equation is:

[0118]

[0119] Step 3.3, let Simplifying the equation, we get:

[0120]

[0121] Step 3.4, let And for u z By finding the first, second, and fourth derivatives, we can obtain:

[0122]

[0123] Step 3.5 introduces the material mechanics equations for displacement and internal forces.

[0124]

[0125] Step 3.6, consider the boundary conditions at the top of the micro-element (z = 0):

[0126]

[0127]

[0128] We can obtain:

[0129] a0 = u0;

[0130] Step 3.7, in the expression of step 3.4, take the k... 3 By combining like terms in the expression, we can find a. i The expression (i>4).

[0131]

[0132] Step 3.8, similar to the calculation, can further yield the expressions for a5 and a6:

[0133]

[0134] By recursion, we can deduce that when i ≥ 7:

[0135]

[0136] Step 3.9, place a i Write it as containing u0, Expressions for M0 and V0:

[0137]

[0138] a i a i-5 a i-4 a i-3 a i-2 Substituting the values ​​into the equations, we can obtain H. j,i Derivation formula for (j=0,1,2,3,4):

[0139]

[0140] H can be seen j,i expression and a i The expressions are consistent and can be based on a i (i<6), calculate H j,i The result of (j = 0, 1, 2, 3, 4, i < 6). Simultaneously, substitute into the formula... We can obtain:

[0141]

[0142] Simplifying, we get:

[0143]

[0144] Step 3.10: Introduce the formulas from mechanics of materials to obtain the calculated... M z V zThe calculation formula is as follows:

[0145]

[0146] Step 3.11, as shown in the above analysis, H is calculated. j,i The distribution pattern can be observed, thus allowing the use of power series methods for solution. To establish a degenerate power series solution, the expression in step 3.3 is represented in a degenerate form to analyze the meaning of each term, thereby establishing a degenerate solution.

[0147]

[0148] In the formula, The term represents the bending moment term at the top of the micro-element. Characterizing the axial force term at the top of the micro-unit, Characterizing the effect of side friction on micro-units, β 4 u represents the soil resistance calculated using the "m" method, α 5 The zu characterizes the calculation of the soil resistance along the pile using the "K0" method, tz 2 +lz+r represents the force exerted by the passive soil section on the pile.

[0149] Step 4, when using the matrix transfer method to calculate using the above method, can be expressed as:

[0150]

[0151] In the formula S i For the coefficient transfer matrix, Represents the displacement and internal force state vector at the top of the pile, where u c , M c V c It indicates the displacement, rotation angle, bending moment, and horizontal force at the top of the pile. The vector representing the displacement and internal force state at the end of the pile, where u f , M f V f It represents the displacement, rotation angle, bending moment, and horizontal force at the pile end.

[0152]

[0153] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A passive pile stability analysis method based on degenerate power series solutions, characterized in that, The method includes the following steps: Step 1: Based on the different relationships between the passive pile and the surrounding soil, the passive pile is divided into free section, passive section and active section; Step 2: Establish the differential control equations for the displacement and internal forces of the passive pile based on the basic principles of mechanics of materials; Step 3: Consider the forces acting on the passive soil section, the soil resistance, and the frictional resistance, and establish the differential governing equations; derive the degenerate power series solution using the power series solution method. Step 4: The matrix transfer method is used to transfer the solution method of each segment to obtain the displacement and internal force transfer matrix of the pile top and pile end. Combined with the boundary conditions of the pile top and pile end, the displacement and internal force of the pile top and pile end are calculated, and then the stability of the passive pile is analyzed. The specific method for step 3 is as follows: Step 3.1: Based on the above differential governing equations, considering the forces acting on the passive soil segment, the soil resistance, and the skin friction, establish the differential governing equations: ； In the formula, K represents the passive section foundation resistance coefficient. Considering the influence of pile top displacement on K, a two-parameter model is introduced: ； In the formula: z0 represents the equivalent depth at the surface; m is the foundation ratio coefficient; n is the depth exponent, which is taken as n=1 for ease of establishing the degraded solution. ,in ; Step 3.2, the distribution of the external load q(z) is represented by the superposition of q1, q2, and q3, which are rectangular, triangular, and parabolic distributions, respectively. It is represented as follows, where T, L, and R are the coefficients of the quadratic term, the linear term, and the constant term, respectively; The differential equation is: ； Step 3.3, let ; ; ; , , , Simplifying the equation, we get: ； Step 3.4: Characterize the bottom displacement of the micro-unit as a power series, i.e. ,in, Let be the coefficient of the i-th term, and for By finding the first, second, and fourth derivatives, we can obtain: ； Step 3.5 introduces the displacement and internal force calculation equations from mechanics of materials to calculate the rotation angle, bending moment, and displacement of the pile micro-element; ； Step 3.6, consider the boundary condition z=0 at the top of the micro-element: ； ; ; ； have to: ; ; ; ； In this context, the top displacement, rotation angle, bending moment, and horizontal force of the micro-unit are represented by u0, φ0, M0, and V0, respectively, and the bottom displacement, rotation angle, bending moment, and horizontal force of the micro-unit are represented by u0, φ0, M0, and V0, respectively. z φ z M z V z express; Calculation yields , , The expression: ； Therefore, we can deduce the following when i ≥ 7: ； Step 3.7, place a i Write it as an expression with u0, φ0, M0, V0, and use... Replace the coefficient before u0, using Replace the coefficient before φ0, and use Replace the coefficients before M0, using Replace the coefficients before V0 with By substituting the constant term, we can simplify to: ； a i a i-5 a i-4 a i-3 a i-2 Substituting each value into the equation, we get H. j,i The derivation formula: ； j=0,1,2,3,4, it can be seen that H j,i expression and a i The expressions are consistent; when i < 6, according to a i Calculate H j,i The result, simultaneously substituted into the formula ,have to: ； Simplifying, we get: ； Step 3.8: Substitute the formula from the mechanics of materials in step 3.5 to obtain the calculated φ. z M z V z The calculation formula is as follows: ； In the formula: ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； ； Among them, A i B i C i D i E i Solve for the coefficients of a certain segment of a micro-element, i=1~4; Step 3.9, as shown in the above analysis, H is calculated. j , i The distribution pattern can be used to solve the problem using the power series method. In order to establish a degenerate power series solution, the expression in step 3.4 is represented in a degenerate form to analyze the meaning of each term and thus establish a degenerate solution. ； In the formula, The term represents the bending moment term at the top of the micro-element. Characterizing the axial force term at the top of the micro-unit, Characterizing the influence of side friction resistance of micro-units, The "m" method is used to characterize the calculation of soil resistance along the pile. The "K0" method was used to characterize the calculation of the soil resistance along the pile. It represents the force exerted by the passive soil section on the pile.

2. The passive pile stability analysis method based on degenerate power series solutions according to claim 1, characterized in that, In step 1, based on the different relationships between the passive pile and the surrounding soil, the passive pile is divided into a free segment, a passive segment, and an active segment, and the pile is further divided into n micro-units along the pile body, each segment being rigid.

3. The passive pile stability analysis method based on degenerate power series solutions according to claim 1, characterized in that, The specific method for step 2 is as follows: Step 2.1, when considering the pile's self-weight and skin friction, assume that the pile's skin friction increases linearly with depth, and consider the pile's self-weight and skin friction as a unified factor: ； Where: N z N0 represents the axial force at depth z of the pile, f represents the combined effect of pile side friction and self-weight, and z represents the depth of the pile. Step 2.2: Establish the moment equilibrium equation for the micro-element with the midpoint of the bottom of the micro-element at point z of the pile as the origin. Considering the passive load, the moment equilibrium equation is obtained as follows: ； In the formula: M represents the bending moment of the micro-element, u represents the displacement of the micro-element, V represents the horizontal force of the micro-element, p(z) represents the soil resistance, and q(z) represents the soil pressure. Considering the horizontal static equilibrium of the micro-element and the case of passive loads, the static equilibrium equations are as follows: ； Neglecting the error caused by the quadratic term, and omitting the quadratic term in the moment equilibrium equation, p(z) can be expressed as: And by combining this with the static equilibrium equations and simplifying, we get: ； In the formula: k represents the foundation resistance coefficient, and b1 is the calculated width of the pile body; Step 2.3 further introduces the formula for the relationship between bending moment and displacement from mechanics of materials to calculate the relationship between bending moment and displacement of the pile micro-element: ； In the formula: EI is the bending stiffness of the material; Substituting the formulas from mechanics of materials into the moment equilibrium equations yields the differential governing equations as follows: ； Step 2.4: Based on the above analysis, establish the differential control equations for the free segment, active segment, and passive segment of the pile. ； In the formula: V a Q(z) represents the horizontal force at the top of the free segment of the pile, Q(z) represents the surcharge earth pressure on the free segment, K represents the passive segment foundation resistance coefficient, and k represents the horizontal force at the top of the free segment. e k represents the soil resistance coefficient of the active elastic section of the pile body. d This indicates the soil resistance coefficient of the active yielding section of the pile; When the self-weight and side friction of the pile are not considered, f=0 is taken, which degenerates to obtain the degenerate micro-unit control equations of the free segment, active segment and passive segment of the pile when the self-weight and side friction of the pile are not considered.

4. The passive pile stability analysis method based on degenerate power series solutions according to claim 1, characterized in that, The specific method for step 4 is as follows: When using the matrix transfer method to calculate using the above method, it can be expressed as follows: ； In the formula, S i For the coefficient transfer matrix, The vector representing the displacement and internal force state at the top of the pile, where u c φ c M c V c This indicates the displacement, rotation angle, bending moment, and horizontal force at the top of the pile. The vector representing the displacement and internal force state at the end of the pile, where u f φ f M f V f This indicates the displacement, rotation angle, bending moment, and horizontal force at the pile end; ； Among them, A ij B ij C ij D ij The coefficients of the transfer matrix are i, where i is the i-th micro-unit (i=1~n, j=1~4).

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