CN117312796A - Integral equation general solution method for vertical bearing characteristics of non-equilong pile composite foundation - Google Patents

Integral equation general solution method for vertical bearing characteristics of non-equilong pile composite foundation Download PDF

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CN117312796A
CN117312796A CN202311326526.4A CN202311326526A CN117312796A CN 117312796 A CN117312796 A CN 117312796A CN 202311326526 A CN202311326526 A CN 202311326526A CN 117312796 A CN117312796 A CN 117312796A
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pile
foundation
vertical
soil
unit
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姜开渝
刘丰盛
何建
熊礼全
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Chongqing Three Gorges University
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention discloses an integral equation general solution method for vertical bearing characteristics of a non-isometric pile composite foundation, which comprises the following steps: decomposing a pile soil system into extension soil and virtual pile units, and dividing the system and a cushion layer into a series of subunits by taking a pile diameter as a side length; obtaining the vertical strain of any depth of the virtuality based on a one-dimensional elastic foundation beam theory; obtaining the vertical strain of any point in the expansion soil based on the Mindlin basic solution and superposition principle; combining the vertical strain coordination conditions of the virtual pile unit and the expansion soil at the corresponding positions to obtain a second class Fredholm integral control equation taking the axial force of the virtual pile as a parameter to be solved; vertical rigidity expressions of cushion units are combined with a generalized Hooke law; and obtaining vertical rigidity expressions of each pile and each soil subunit by using deformation coordination conditions of the rigid foundation and the top of the cushion layer and boundary conditions of the underlying bedrock surface, and solving the vertical bearing characteristics of the non-isometric pile composite foundation under the rigid foundation by introducing an iterative algorithm.

Description

Integral equation general solution method for vertical bearing characteristics of non-equilong pile composite foundation
Technical Field
The invention relates to the fields of constructional engineering, road engineering and bridge engineering, in particular to an integral equation general solution method for vertical bearing characteristics of non-isometric pile composite foundations.
Background
The rock and soil bodies in the natural foundations of the mountain areas in China are generally unevenly distributed, and the underlying bedrock surfaces are generally obliquely arranged. The non-isometric pile composite foundation can obviously reduce the settlement and uneven settlement of the foundation, so that the non-isometric pile composite foundation is widely applied to foundation reinforcement engineering.
In actual engineering, the bearing characteristics and the deflection mechanism of the pile composite foundation are usually researched through on-site load tests and finite element methods. For large pile group foundations or pile body composite foundations, in-situ load tests are difficult to load into a failure stage. Along with the gradual transition from bearing capacity control to displacement control of foundation design in China, more scholars put forward different analysis methods for bearing characteristics of pile composite foundations and pile group foundations under an elastic theoretical frame, and develop a series of researches, such as a shear displacement method, a load transfer method, a simplification method, a variation method and the like, so as to overcome the problem of the deficiency of the existing calculation method, and for this purpose, we put forward an integral equation general solution method for vertical bearing characteristics of non-equal-length pile composite foundations.
Disclosure of Invention
(one) solving the technical problems
Aiming at the defects of the prior art, the invention provides an integral equation general solution method for the vertical bearing characteristics of a non-isometric pile composite foundation, which solves the technical problem.
(II) technical scheme
In order to achieve the above purpose, the invention is realized by the following technical scheme:
the integral equation general solution method for the vertical bearing characteristics of the non-isometric pile composite foundation comprises the following steps:
step one: decomposing a pile soil system into extension soil and virtual pile units, and dividing the system and a cushion layer into a series of subunits by taking a pile diameter as a side length;
step two: obtaining the vertical strain of any depth of the virtuality based on a one-dimensional elastic foundation beam theory;
step three: obtaining the vertical strain of any point in the expansion soil based on the Mindlin basic solution and superposition principle;
step four: combining the vertical strain coordination conditions of the virtual pile unit and the expansion soil at the corresponding positions to obtain a second class Fredholm integral control equation taking the axial force of the virtual pile as a parameter to be solved;
step five: vertical rigidity expressions of cushion units are combined with a generalized Hooke law; and obtaining vertical rigidity expressions of each pile and each soil subunit by using deformation coordination conditions of the rigid foundation and the top of the cushion layer and boundary conditions of the underlying bedrock surface, and solving the vertical bearing characteristics of the non-isometric pile composite foundation under the rigid foundation by introducing an iterative algorithm.
Preferably, the specific steps of the first step are as follows:
decomposing a bedding layer and a pile body composite foundation (comprising the bedding layer) into m units by taking the diameter of a pile as the side length, wherein n units in the composite foundation are occupied by foundation pile units, m-n units are composed of soil bodies among piles, n units in total interact with a foundation and foundation piles for the bedding layer, m-n units interact with the foundation and soil bodies among piles, and h r Base rock burial depth at the r (r=1, 2, …, m) th cell. In particular, for the n units occupied by the foundation pile units, the length thereof is made up of two parts, i.e. the i-th foundation pile length l i And the distance delta between the pile end and the bedrock i And has h i ＝l i +Δ i (i＝1,2,…,n)。
Preferably, the specific steps of the second step are as follows:
taking the ith virtual pile as a research object, and obtaining according to the vertical stress-strain relationship, the static balance condition and the vertical strain-settlement relationship:
wherein the method comprises the steps ofAnd A represents the vertical strain, sedimentation and cross-sectional area of the ith virtual pile at depth z, E * The modulus of elasticity of the virtual pile, which can be expressed as E * ＝E p -E s ，
The settling of the ith virtual pile at depth z can be expressed as:
in the method, in the process of the invention,is the pile top displacement of the ith virtual pile.
Preferably, the specific steps of the third step are as follows:
the extended soil is taken as a research object,and->Respectively representing the vertical stress, vertical strain and sedimentation of the (r=1, 2, …, m) th expansive soil unit at the depth z, which are obtained by the superposition principle:
wherein the method comprises the steps ofAnd->Average vertical stress, vertical strain and settlement generated by uniform load of the j-th virtual pile unit on the section of the (z) th unit with the action resultant force of the section of the (xi) th unit as unit 1 respectively> And->Respectively the average vertical stress, the vertical strain and the sedimentation generated by the uniform load of the soil unit between the kth piles on the section of the (n) (z) of the (r) th unit with the action resultant force of the top as the unit 1,
substituting into the formula and combiningAnd->Finite jump characteristic at z=ζ, using fractional integration method will +.>And->Expressed as:
wherein, based on Hook's law and equilibrium conditions, it is available:
preferably, the specific steps of the fourth step are as follows:
the control equation can be obtained by combining the vertical strain coordination conditions of the expansive soil and the virtual piles at the corresponding positions as follows:
the equation is a second class of Fredholm integral equations for each virtual pile element axis force.
Preferably, the specific steps of the fifth step are as follows:
taking a cushion layer of the pile top of the ith foundation pile as a research object, and obtaining the compression amount by the generalized Hooke lawThe method comprises the following steps:
the vertical stress of the cushion layer is large main stress, and the ratio of the radial strain to the vertical strain is recorded as alpha under the limited triaxial compression working condition i The side pressure coefficient of the unit can be expressed as:
the vertical rigidity of the pile top cushion layer of the i-th foundation pile can be obtained by the combined type sum type:
where A is the cross-sectional area of the cushion unit.
Preferably, the cushion layer at the top of the kth soil body unit is taken as a research object, and the ratio of the radial strain to the vertical strain is recorded as beta k The vertical stiffness thereof can be expressed as:
with respect to alpha i 、β k The value of (2) can be combined with the horizontal radial stress of two adjacent units to be equally solved, and the penetrating effect of the pile top to the cushion layer can be approximately simulated according to the difference of the compression amounts of the cushion layer at the top of the pile unit and the cushion layer at the top of the adjacent soil unit; and the difference between the pile unit end and the settlement of the adjacent soil unit at the pile end can be regarded as the pile end penetration amount.
Preferably, if the top load of each subunit is known, the control equation is a fixed solution equation, and the axial force of each virtual pile can be directly obtained based on the equation(i=1, 2, …, n), and further, the axial force P of each real foundation pile and each inter-pile soil unit can be obtained by an superposition method r (z) (r=1, 2, …, m) and sedimentation distribution, and finally obtaining the vertical bearing characteristic and deflection mechanism of the variable pile length pile body composite foundation, wherein the real axial force of all sub units can be represented by virtual pile axial force->And expand soil axis force->And (3) superposition, namely:
wherein P is r (z) positive with pressure. When z=0 in the formula, the pile top load P of each subunit can be obtained r (0)。
For the rigid foundation working condition, the vertical load at the top of each unit is unknown, so that a control equation cannot be directly solved, and the deformation coordination condition and the static balance condition between the foundations are required to be supplemented.
Preferably, the precipitation of the nth subunit at the depth z is set asThe settlement of the ith virtual pile unit at depth z is +.>When r=i, the strain coordination condition at the corresponding position of the expansive soil and the virtual pile is available:
let the bedrock depth corresponding to the r-th subunit be h r The true subsidence of the r-th subunit, which is obtained from no subsidence at the underlying inclined bedrock, is:
when z=0 in the formula, the settlement amount at the top of each subunit can be obtained.
Preferably, the foundation rigidity is uneven due to the influence of the inclined bedrock surface and the unequal length foundation piles, and the rigid foundation is easy to generate uneven settlement, thus assuming the flatness of the foundationAll subside to w, the slope is theta, the cushion layer compression quantity corresponding to the nth subunit isThen the top of each subunit settles down to W r (0) Compression amount of cushion layer at corresponding position->The sum and the rigid base satisfy the deformation coordination condition, which can be expressed as:
the equilibrium conditions for a rigid foundation are as follows:
where Ω is the position coordinate matrix of each subunit, which can be derived from the coordinate system shown in figure 2,
the axial force of each virtual pile can be solved by combining the formula and the formulaAnd axial force and sedimentation at any depth of each unit, so as to analyze the compression amount of each cushion unit, pile-soil stress ratio and pile-soil side friction resistance at any depth of each unit, etc.,
in particular, for a rigid foundation, the top load P of the r-th unit is determined r (0) And sedimentation W r (0) Then, the vertical rigidity K of the top part of the (r) th cushion layer unit can be obtained r (0) As shown in the following formula,
(III) beneficial effects
1. According to the invention, from the angle of universality of non-equal length pile body composite foundation vertical bearing characteristic solutions on an inclined bedrock surface, the integral control equation is converted into an expanded algebraic equation through pile and soil units by combining with the characteristic of a second class Fredholm integral control equation, so that axial force and settlement half-resolution solutions of the pile and soil units are obtained, and further, target parameters such as bedding compression amount, pile and soil load sharing ratio, foundation settlement and the like are obtained; the calculation method can consider the combined working conditions of the underlying inclined bedrock surface, the non-equilong foundation piles, different cushion layer rigidities and the like, and meanwhile, the expanded algebraic equation is easy to program and calculate, has very good adaptability and universality, and can well solve the pile body composite foundation bearing characteristic problem under various complex foundation working conditions in geotechnical engineering.
2. However, the shear displacement method ignores a vertical stress along depth variation item for simplifying analysis in the condition of establishing micro-unit body balance, so that the shear displacement method cannot be directly applied to analysis of a composite foundation; the traditional elastic theory method cannot reasonably consider the reinforcement effect and the blocking effect of piles, and the pile-pile and pile-soil interaction coefficients of the traditional elastic theory method are often larger. The integral equation method can reasonably consider the reinforcement and shielding effects of foundation piles in composite foundations and pile group foundations, but the related research is mainly applicable to foundation working conditions of horizontal bedrock surfaces or semi-infinite spaces. The bearing characteristics of the non-equal length pile composite foundation on the inclined bedrock face have not been widely studied.
Drawings
The foregoing description is only an overview of the present invention, and is intended to provide a better understanding of the present invention, as it is embodied in the following description, with reference to the preferred embodiments of the present invention and the accompanying drawings.
FIG. 1 is a flowchart of an iterative algorithm analysis;
FIG. 2 is a non-isometric pile composite foundation on an inclined bedrock face;
FIG. 3 shows the bottom load distribution of the mat;
FIG. 4 is an exploded schematic view of a pile soil system;
FIG. 5 is a pile shaft axial force distribution diagram;
FIG. 6 is a pile side friction resistance profile;
FIG. 7 is a 3X 1 non-isometric pile composite foundation and parameters;
fig. 8 is a pile shaft axial force and sedimentation distribution diagram.
Detailed Description
According to the method for solving the integral equation of the vertical bearing characteristics of the non-isometric pile composite foundation, the explained calculation method can consider the combined working conditions of the underlying inclined bedrock face, the non-isometric foundation piles, different cushion stiffness and the like, and meanwhile, the expanded algebraic equation is easy to program and calculate, so that the method has good adaptability and universality, and the problem of bearing characteristics of the pile composite foundation under various complex foundation working conditions in geotechnical engineering can be well solved.
According to the invention, from the perspective of universality of non-equal length pile body composite foundation vertical bearing characteristic solutions on an inclined bedrock surface, the integral control equation is converted into an expanded algebraic equation through pile and soil units by combining with the characteristic of a second class Fredholm integral control equation, so that axial force and settlement half-resolution solutions of the pile and soil units are obtained, and further, target parameters such as bedding compression amount, pile and soil load sharing ratio, foundation settlement and the like are obtained, and the method has good popularization and application values. The method comprises the following specific steps:
(1) And establishing a vertical bearing force analysis model of the non-isometric pile body composite foundation on the inclined bedrock surface, as shown in figure 2. In the figure, a cushion layer and a pile body composite foundation (comprising the cushion layer) are respectively decomposed into m units by taking a pile diameter as a side length, wherein n units in the composite foundation are occupied by foundation pile units, m-n units are composed of soil bodies among piles, n units in total interact with a foundation and foundation piles for the cushion layer, and m-n units interact with the foundation and soil among piles. h is a r Base rock burial depth at the r (r=1, 2, …, m) th cell. In particular, for the n units occupied by the foundation pile units, the length thereof is made up of two parts, i.e. the i-th foundation pile length l i And the distance delta between the pile end and the bedrock i And has h i ＝l i +Δ i (i＝1,2,…,n)。
(2) Taking the ith virtual pile as a research object, and obtaining according to the vertical stress-strain relationship, the static balance condition and the vertical strain-settlement relationship:
wherein the method comprises the steps ofAnd a represents the vertical strain, settling, and cross-sectional area of the ith virtual pile at depth z, respectively. E (E) * The modulus of elasticity of the virtual pile, which can be expressed as E * ＝E p -E s 。
The settling of the ith virtual pile at depth z can be expressed as:
in the method, in the process of the invention,is the pile top displacement of the ith virtual pile.
(3) The extended soil is taken as a research object,and->Respectively representing the vertical stress, vertical strain and sedimentation of the (r=1, 2, …, m) th expansive soil unit at the depth z, which are obtained by the superposition principle:
wherein, based on Hook's law and equilibrium conditions, it is available:
(4) The control equation can be obtained by combining the vertical strain coordination conditions of the expansive soil and the virtual piles at the corresponding positions as follows:
the equation is a second class of Fredholm integral equations for each virtual pile element axis force.
(5) Taking a cushion layer of the pile top of the ith foundation pile as a research object, and recording the ratio of radial strain to vertical strain as alpha i The vertical rigidity of the pile top cushion layer of the ith foundation pile can be obtained by the generalized Hooke law:
where A is the cross-sectional area of the cushion unit.
Similarly, taking a cushion layer at the top of a kth soil body unit as a research object, and recording the ratio of radial strain to vertical strain as beta k The vertical stiffness thereof can be expressed as:
let the settling amount of the nth subunit at the depth z beThe settlement of the ith virtual pile unit at depth z is +.>When r=i, the strain coordination condition at the corresponding position of the expansive soil and the virtual pile is available:
let the bedrock depth corresponding to the r-th subunit be h r The true subsidence of the r-th subunit, which is obtained from no subsidence at the underlying inclined bedrock, is:
when z=0 in the formula, the settlement amount at the top of each subunit can be obtained.
Because of the influence of the inclined bedrock surface and the unequal-length foundation piles, the foundation rigidity is often uneven, and the rigid foundation is easy to generate uneven settlement, the average settlement of the foundation is assumed to be w, the inclination slope is θ, and the cushion compression amount corresponding to the r subunit isThen the top of each subunit settles down to W r (0) Compression amount of cushion layer at corresponding position->The sum and the rigid base satisfy the deformation coordination condition, which can be expressed as:
the equilibrium conditions for a rigid foundation are as follows:
where Ω is a positional coordinate matrix of each subunit, which can be derived from the coordinate system shown in fig. 2.
The axial force of each virtual pile can be solved by combining the formula and the formulaAnd axial force and sedimentation at any depth of each unit, so that the compression amount of each cushion unit, the pile-soil stress ratio at any depth of each unit, the pile-soil side friction resistance and the like can be analyzed.
In particular, for a rigid foundation, the top load P of the r-th unit is determined r (0) And sedimentation W r (0) Then, the vertical rigidity K of the top part of the (r) th cushion layer unit can be obtained r (0) As shown in the following formula.
(6) The specific iterative flow is shown in figure 1. Assuming that the initial value of the top load of each unit is P i (0)＝Q k (0)＝N 0 M, the assumption satisfying the vertical static equilibrium condition; (2) Solving axial force of each virtual pile unit byFurther deriving the axial force and sedimentation P of each unit from the sum of the formulas r (z) and W r (z) calculating the vertical stiffness K of the top of each cushion unit r (0) The method comprises the steps of carrying out a first treatment on the surface of the (3) Combining the sum to obtain the average sedimentation W and the inclination slope theta of the rigid foundation, and obtaining the sedimentation W of the top of each pile-soil unit again by using the formula r (0) _new The method comprises the steps of carrying out a first treatment on the surface of the (4) By substituting into K r (0) And W is r (0) _new The units being recoverableLoad at the top P r (0) The method comprises the steps of carrying out a first treatment on the surface of the (5) If the second norm of the top settlement of each unit obtained in the step 2 and the step 3 is larger than the given value delta (delta=1 mm, for example), namelyThen P obtained in step 4 is calculated r (0) Substituting step 1 again for iteration until +.>Ending the iteration to finally obtain the axial force P of each unit along the depth distribution r (z) and sedimentation W r (z)。
According to the invention, from the angle of universality of non-equal length pile body composite foundation vertical bearing characteristic solutions on an inclined bedrock surface, the integral control equation is converted into an expanded algebraic equation through pile and soil units by combining with the characteristic of a second class Fredholm integral control equation, so that axial force and settlement half-resolution solutions of the pile and soil units are obtained, and further, target parameters such as bedding compression amount, pile and soil load sharing ratio, foundation settlement and the like are obtained; the calculation method can consider the combined working conditions of the underlying inclined bedrock surface, the non-equilong foundation piles, different cushion layer rigidities and the like, and meanwhile, the expanded algebraic equation is easy to program and calculate, has very good adaptability and universality, and can well solve the pile body composite foundation bearing characteristic problem under various complex foundation working conditions in geotechnical engineering.
Example 1
Taking a single vertical loaded pile as a research object, respectively considering different pile-soil stiffness ratios k p (k p 100,5000) and different aspect ratio l/d (l/d=25, 50), discussing axial force, pile side friction and sedimentation distribution of single pile along pile body under vertical load, wherein the pile-soil rigidity ratio k p ＝E p /E s Soil poisson ratio v=0.4.
The analysis method provided by the invention is combined to analyze the single pile bearing characteristics, and the stiffness ratios k of different piles and soil are different p The axial force distribution of the lower pile body and the pile side friction resistance distribution are respectively shown in fig. 5 and 6. Wherein, all parameters are represented by dimensionless coefficients.
Example 2
The rationality of the method provided by the invention is further compared and verified by taking a 3 multiplied by 1 non-equal-length pile body composite foundation as a research object and combining a finite element method. The 3×1 non-equal length pile composite foundation on the inclined bedrock face and key parameters are shown in fig. 7. The pile body axial force and sedimentation distribution obtained by the two analysis methods are shown in figure 8, and the analysis results of the two methods are very good in agreement, so that the accuracy and the computational rationality of the analysis general solution algorithm for the bearing characteristics of the non-isometric pile body composite foundation, which can be used for the influence of bedrock in the presence, are shown.
However, the shear displacement method ignores a vertical stress along depth variation item for simplifying analysis in the condition of establishing micro-unit body balance, so that the shear displacement method cannot be directly applied to analysis of a composite foundation; the traditional elastic theory method cannot reasonably consider the reinforcement effect and the blocking effect of piles, and the pile-pile and pile-soil interaction coefficients of the traditional elastic theory method are often larger. The integral equation method can reasonably consider the reinforcement and shielding effects of foundation piles in composite foundations and pile group foundations, but the related research is mainly applicable to foundation working conditions of horizontal bedrock surfaces or semi-infinite spaces. The bearing characteristics of the non-equal length pile composite foundation on the inclined bedrock face have not been widely studied.
Finally, it should be noted that: it is apparent that the above examples are only illustrative of the present invention and are not limiting of the embodiments. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art. It is not necessary here nor is it exhaustive of all embodiments. And obvious variations or modifications thereof are contemplated as falling within the scope of the present invention.

Claims (10)

1. The integral equation general solution method for the vertical bearing characteristics of the non-isometric pile composite foundation is characterized by comprising the following steps of: the method comprises the following steps:
step one: decomposing a pile soil system into extension soil and virtual pile units, and dividing the system and a cushion layer into a series of subunits by taking a pile diameter as a side length;
step two: obtaining the vertical strain of any depth of the virtuality based on a one-dimensional elastic foundation beam theory;
step three: obtaining the vertical strain of any point in the expansion soil based on the Mindlin basic solution and superposition principle;
step four: combining the vertical strain coordination conditions of the virtual pile unit and the expansion soil at the corresponding positions to obtain a second class Fredholm integral control equation taking the axial force of the virtual pile as a parameter to be solved;
step five: vertical rigidity expressions of cushion units are combined with a generalized Hooke law; and obtaining vertical rigidity expressions of each pile and each soil subunit by using deformation coordination conditions of the rigid foundation and the top of the cushion layer and boundary conditions of the underlying bedrock surface, and solving the vertical bearing characteristics of the non-isometric pile composite foundation under the rigid foundation by introducing an iterative algorithm.
2. The method for solving the integral equation of the vertical bearing characteristics of the non-isometric pile composite foundation according to claim 1, which is characterized in that: the specific steps of the first step are as follows:
decomposing a bedding layer and a pile body composite foundation (comprising the bedding layer) into m units by taking the diameter of a pile as the side length, wherein n units in the composite foundation are occupied by foundation pile units, m-n units are composed of soil bodies among piles, n units in total interact with a foundation and foundation piles for the bedding layer, m-n units interact with the foundation and soil bodies among piles, and h r Base rock burial depth at the r (r=1, 2, …, m) th cell. In particular, for the n units occupied by the foundation pile units, the length thereof is made up of two parts, i.e. the i-th foundation pile length l i And the distance delta between the pile end and the bedrock i And has h i ＝l i +Δ i (i＝1,2,…,n)。
3. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 2, which is characterized in that: the specific steps of the second step are as follows:
taking the ith virtual pile as a research object, and obtaining according to the vertical stress-strain relationship, the static balance condition and the vertical strain-settlement relationship:
wherein the method comprises the steps ofAnd A represents the vertical strain, sedimentation and cross-sectional area of the ith virtual pile at depth z, E * The modulus of elasticity of the virtual pile, which can be expressed as E * ＝E p -E s ，
The settling of the ith virtual pile at depth z can be expressed as:
in the method, in the process of the invention,is the pile top displacement of the ith virtual pile.
4. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 3, which is characterized in that: the specific steps of the third step are as follows:
the extended soil is taken as a research object,and->Respectively representing the vertical stress, vertical strain and sedimentation of the (r=1, 2, …, m) th expansive soil unit at the depth z, which are obtained by the superposition principle:
wherein the method comprises the steps ofAnd->Average vertical stress, vertical strain and settlement generated by uniform load of the j-th virtual pile unit on the section of the (z) th unit with the action resultant force of the section of the (xi) th unit as unit 1 respectively> And->Respectively the average vertical stress, the vertical strain and the sedimentation generated by the uniform load of the soil unit between the kth piles on the section of the (n) (z) of the (r) th unit with the action resultant force of the top as the unit 1,
substituting the formula intoTo combine withAnd->Finite jump characteristic at z=ζ, using fractional integration method will +.>And->Expressed as:
wherein, based on Hook's law and equilibrium conditions, it is available:
5. the method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 4, which is characterized in that: the specific steps of the fourth step are as follows:
the control equation can be obtained by combining the vertical strain coordination conditions of the expansive soil and the virtual piles at the corresponding positions as follows:
the equation is a second class of Fredholm integral equations for each virtual pile element axis force.
6. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 5, which is characterized in that: the specific steps of the fifth step are as follows:
taking a cushion layer of the pile top of the ith foundation pile as a research object, and obtaining the compression quantity s by a generalized Hooke law c i The method comprises the following steps:
the vertical stress of the cushion layer is large main stress, and the ratio of the radial strain to the vertical strain is recorded as alpha under the limited triaxial compression working condition i The side pressure coefficient of the unit can be expressed as:
the vertical rigidity of the pile top cushion layer of the i-th foundation pile can be obtained by the combined type sum type:
where A is the cross-sectional area of the cushion unit.
7. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 6, which is characterized in that: taking a cushion layer at the top of a kth soil body unit as a research object, and recording radial strain and vertical stressThe ratio of the transformation is beta k The vertical stiffness thereof can be expressed as:
with respect to alpha i 、β k The value of (2) can be combined with the horizontal radial stress of two adjacent units to be equally solved, and the penetrating effect of the pile top to the cushion layer can be approximately simulated according to the difference of the compression amounts of the cushion layer at the top of the pile unit and the cushion layer at the top of the adjacent soil unit; and the difference between the pile unit end and the settlement of the adjacent soil unit at the pile end can be regarded as the pile end penetration amount.
8. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 7, which is characterized in that: if the top load of each subunit is known, the control equation is a constant solution equation, and the axial force P of each virtual pile can be directly obtained based on the equation * i (z) (i=1, 2, …, n), and further, the axial force P of each real foundation pile and each inter-pile soil unit can be obtained by an superposition method r (z) (r=1, 2, …, m) and sedimentation distribution, and finally obtaining the vertical bearing characteristics and displacement mechanism of the composite foundation of the variable pile length pile body, wherein the real axial force of all the subunits can be represented by the virtual pile axial force P * i (z) and extending the soil axial forceAnd A, superposition and obtaining, namely:
wherein P is r (z) positive with pressure. When z=0 in the formula, the pile top load P of each subunit can be obtained r (0)。
For the rigid foundation working condition, the vertical load at the top of each unit is unknown, so that a control equation cannot be directly solved, and the deformation coordination condition and the static balance condition between the foundations are required to be supplemented.
9. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 8, which is characterized in that: let the settling amount of the nth subunit at the depth z beThe settlement of the ith virtual pile unit at depth z is +.>When r=i, the strain coordination condition at the corresponding position of the expansive soil and the virtual pile is available:
let the bedrock depth corresponding to the r-th subunit be h r The true subsidence of the r-th subunit, which is obtained from no subsidence at the underlying inclined bedrock, is:
when z=0 in the formula, the settlement amount at the top of each subunit can be obtained.
10. The method for solving the integral equation of the vertical bearing characteristic of the non-isometric pile composite foundation according to claim 9, which is characterized in that: because of the influence of the inclined bedrock surface and the unequal-length foundation piles, the foundation rigidity is often uneven, and the rigid foundation is easy to generate uneven settlement, the average settlement of the foundation is assumed to be w, the inclination slope is θ, and the cushion compression amount corresponding to the r subunit isThen the top of each subunit settles down to W r (0) Compression amount of cushion layer at corresponding position->The sum and the rigid base satisfy the deformation coordination condition, which can be expressed as:
the equilibrium conditions for a rigid foundation are as follows:
wherein omega is the position coordinate matrix of each subunit, and the combination, the formula and the formula can solve the axial force P of each virtual pile * i (z) and axial force and settlement at any depth of each unit, and further analyzing compression amount of each cushion unit, pile-soil stress ratio and pile-soil side friction resistance at any depth of each unit, and obtaining top load P of the r-th unit for a rigid foundation r (0) And sedimentation W r (0) Then, the vertical rigidity K of the top part of the (r) th cushion layer unit can be obtained r (0) As shown in the following formula,
CN202311326526.4A 2023-10-13 2023-10-13 Integral equation general solution method for vertical bearing characteristics of non-equilong pile composite foundation Pending CN117312796A (en)

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