CN114547837A - Method for calculating internal force and displacement of single inclined slope pile based on configuration point method - Google Patents

Method for calculating internal force and displacement of single inclined slope pile based on configuration point method Download PDF

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CN114547837A
CN114547837A CN202111536011.8A CN202111536011A CN114547837A CN 114547837 A CN114547837 A CN 114547837A CN 202111536011 A CN202111536011 A CN 202111536011A CN 114547837 A CN114547837 A CN 114547837A
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赵学亮
何润财
赵星云
龚维明
王洪庆
朱文波
陈信升
戴国亮
邓温妮
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Southeast University
China Energy Engineering Group Guangdong Electric Power Design Institute Co Ltd
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Abstract

The invention belongs to the field of design and construction of a pile foundation in geotechnical engineering, and relates to a calculation method for internal force and displacement of a single inclined steep slope pile based on a configuration point method. The single pile axial force, the slope thrust and the soil resistance of the rock-socketed single pile on the inclined steep slope are calculated; obtaining single pile axial force, slope thrust and soil resistance to establish infinitesimal deflection differential equations of a free section, a loaded section and an embedded section of the rock-embedded single pile; forming a 4-order linear variable coefficient non-homogeneous ordinary differential equation of the whole single inclined slope pile; and solving the 4-order linear variable coefficient non-homogeneous ordinary differential equation by adopting a 4-order implicit Runge Kutta algorithm and combining a configuration point algorithm to obtain an inner force and displacement matrix of the single inclined steep slope pile. The method can solve the single-pile deflection differential equation more conveniently and is convenient to use in practical engineering and academic research.

Description

Method for calculating internal force and displacement of single inclined slope pile based on configuration point method
Technical Field
The invention belongs to the field of design and construction of a pile foundation in geotechnical engineering, and relates to a calculation method for internal force and displacement of a single inclined steep slope pile based on a configuration point method.
Background
Due to the influence of hydrogeological conditions, foundation piles sometimes have to be arranged on the steep slope sections in practical engineering. According to relevant research, the bearing characteristics of the inclined steep slope section bridge pile foundation are obviously different from the conventional horizontal underground pile foundation, and the existing design calculation theory needs to be further optimized and corrected so as to be suitable for the design of the inclined steep slope pile foundation.
For the problem, a related research topic group provides a related inclined steep slope section stress analysis model by researching the bearing mechanism, the failure mode, the stress deformation rule and the like of the inclined steep slope section bridge pile foundation under different load characteristics and different structural characteristics and combining with the previous research, establishes a corresponding deflection differential equation for the model, and provides a corresponding numerical solution through a power series or finite difference method and the like. Unfortunately, the related research on the load bearing mechanism of the single pile on the inclined steep slope lacks related engineering cases and numerical simulation analysis support, and the proposed stress analysis model part is assumed to be too simple.
Disclosure of Invention
The invention provides a method for calculating the internal force and displacement of a single inclined slope pile based on a configuration point method.
The invention adopts the following technical scheme:
the invention relates to a method for calculating the internal force and displacement of a single inclined slope pile based on a configuration point method, which comprises the following calculation steps:
dividing the rock-socketed single pile on the inclined steep slope into a free section, a loaded section and an embedded section, and calculating to obtain the single pile axial force, slope thrust and soil resistance of the rock-socketed single pile;
secondly, acquiring single pile axial force, slope thrust and soil resistance in the first step to establish infinitesimal flexural differential equations of a free section, a loaded section and an embedded section of the rock-embedded single pile;
thirdly, simultaneous integration and induction are carried out through deformation coordination conditions between the loaded section bending differential equation and the embedded section bending differential equation established in the second step, and a 4-order linear variable coefficient heterogeneous ordinary differential equation of the whole single inclined slope pile is formed; the deformation coordination conditions are as follows: the condition that ensures integrity and continuity of the deformed object is called deformation coordination. The deflection, the corner, the shearing force and the bending moment of the contact surfaces of the free section, the loaded section and the embedded section are equal;
fourthly, calculating to obtain the pile top load of the rock-socketed single pile transmitted by the upper structure according to the third step, wherein the pile top load comprises a vertical load, a horizontal load and an eccentric bending moment; determining boundary conditions of the rock-socketed single pile;
and fifthly, solving the 4-order linear variable coefficient non-homogeneous ordinary differential equation by adopting a 4-order implicit Runge Kutta algorithm and combining a configuration point algorithm to obtain an internal force and displacement matrix of the single inclined steep slope pile.
According to the calculation method of the internal force and displacement of the single inclined steep slope pile based on the configuration point method, the free section pile top mainly bears the load of the pile top;
the loaded section is used for bearing the load transmitted by the free section and bearing the residual sliding force of a slope body, the resistance of a pile side rock-soil body and the friction resistance of the pile side;
the embedded section mainly bears the load transmitted by the loaded section, the resistance of the pile side rock-soil body and the friction resistance of the pile side.
According to the method for calculating the internal force and displacement of the single inclined slope pile based on the configuration point method, in the first step, the axial force P (z) of the single socketed pile is represented as follows:
P(z)＝P0+fz
in the formula, P0Is the pile top axial force and z is the depth from the pile top. The part f of the free section is gamma z, the part f of the loading section and the embedding section is gamma z-tau Cz, C is the perimeter of the pile, tau is the friction resistance of the side of the pile, and gamma is the weight of the pile body;
in the step one, the gliding thrust generated by the load-bearing section inclined steep slope body; the distribution form q (z) of the slope thrust of the monopile axial force of the socketed monopile is expressed as:
q(z)＝az2+bz+c
in the formula, a, b and c are undetermined coefficients;
respectively calculating the soil resistance of the front soil body of each segmental pile, and calculating by adopting an m method, wherein the soil body resistance and the horizontal displacement of the pile in the existing specification of the m method are calculated by adopting the method, and the soil body resistance p (x, z) is expressed as follows:
p(x,z)＝K(z)b0x
wherein the embedding section K (z) m2z, loaded section K (z) m1z，b0Calculate width, m, for the pile1Is the value of m in the loaded section, m2Is the value of the embedded section m, and x is the horizontal displacement;
the flexural differential equation of the free-section infinitesimal is as follows:
Figure BDA0003412588510000021
the loaded section infinitesimal flexural differential equation is as follows:
Figure BDA0003412588510000031
the flexural differential equation of the embedded section infinitesimal element is as follows:
Figure BDA0003412588510000032
forming a 4-order linear variable coefficient non-homogeneous ordinary differential equation of the whole single inclined slope pile based on the free section infinitesimal bending differential equation, the loaded section infinitesimal bending differential equation and the embedded section infinitesimal bending differential equation; the equation is expressed as:
Figure BDA0003412588510000033
wherein L is the pile length, L is L1+l2+l3，0～l1Is a free segment,/1～l2Is a loaded section, /)2～l3The other parameters are as follows:
Figure BDA0003412588510000034
Figure BDA0003412588510000035
Figure BDA0003412588510000036
Figure BDA0003412588510000037
Figure BDA0003412588510000038
Figure BDA0003412588510000039
according to the method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method, the boundary condition is determined according to the pile top load transmitted by the overlying structure obtained through calculation in the third step; the boundary conditions may be expressed as:
Figure BDA0003412588510000041
the pile top load comprises a pile top axial force P0Pile top shearing force Q0And pile top bending moment M0。
The invention relates to a calculation method of the internal force and displacement of a single inclined steep slope pile based on a configuration point method, which converts a free section infinitesimal flexural differential equation, a loaded section infinitesimal flexural differential equation and an embedded section infinitesimal flexural differential equation into a differential equation set form, wherein the expression form is as follows:
Figure BDA0003412588510000042
wherein the content of the first and second substances,
Figure BDA0003412588510000043
solving the system of differential equations, let x(j) iWhere j is 0,1,2,3, the solving equations for these four equations are listed:
Figure BDA0003412588510000044
introducing boundary conditions and differential equation sets of pile top load into the solving format; when j is 0,1, the problem is the edge value of the flexural differential equation, when j is 2,3, the problem is the initial value of the first order differential equation set, and the solution is formulated into a matrix form by combining boundary conditions, wherein:
Figure BDA0003412588510000051
the formula of the configuration point method is used for simultaneous solving, and the following can be obtained:
Figure BDA0003412588510000052
at this time, the solution of the original equation can be obtained, and can be expressed as:
Figure BDA0003412588510000053
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003412588510000054
b＝[b1 b2 b3 b′4]T，
Figure BDA0003412588510000055
e is an n-order unit matrix, and then the pile internal force displacement matrix can be obtained
Figure BDA0003412588510000056
The invention relates to a calculation method of the internal force and displacement of a single inclined slope pile based on a configuration point method, wherein undetermined coefficients a, b and c in a distribution form q (z) expression of the calculation method can be obtained according to the following table
TABLE 1 slope glide thrust distribution function
Figure BDA0003412588510000057
In table H1The length of the pile is above the sliding surface, E is the resultant force of the gliding thrust, and k is the form factor.
The invention relates to a method for calculating the internal force and displacement of a single inclined slope pile based on a configuration point method,
in the step (4), the value of the value m in the expression of the soil resistance p (x, z) is introduced into a reduction fitting formula about the gradient theta based on the specification:
mθ＝n1n2n3(m-k·tanθ)
wherein n is1Is a soil property parameter correction coefficient, n2Is the pile length correction factor, n3The pile diameter correction coefficient is obtained; and k is a correction coefficient obtained by fitting.
The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method adopts a solving format of Lobatto III A formula with 4-order precision, the essence of the method is a 4-order implicit Runge Kutta formula, and the values of coefficients are as follows:
Figure BDA0003412588510000061
obtained single-pile internal force displacement matrix of inclined steep slope
Figure BDA0003412588510000062
The matrix and formula contained in (1) are as follows:
Figure BDA0003412588510000063
Figure BDA0003412588510000064
b1＝[0 … 0]T
b2＝[0 … 0]T
b3＝[M0/EI … 0]T
Figure BDA0003412588510000065
Figure BDA0003412588510000066
Figure BDA0003412588510000067
u＝[1 0 … 0]1×n+1
Figure BDA0003412588510000068
Figure BDA0003412588510000071
Figure BDA0003412588510000072
O＝[0 0 … 0]1×2+1
Figure BDA0003412588510000073
Figure BDA0003412588510000074
Figure BDA0003412588510000075
Figure BDA0003412588510000076
Figure BDA0003412588510000077
advantageous effects
The method is based on the configuration point method capable of converting continuous function solution into specific point solution, can conveniently solve the single-pile deflection differential equation, and is convenient to use in practical engineering and academic research.
The method is considered from various factors such as the stress segmentation of the single inclined steep slope pile, the slope thrust calculated based on the anti-slide pile theory, the m method based on the slope reduction and the like, so that the solution accuracy of the deflection differential equation of the single inclined steep slope pile is obviously improved.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is a schematic diagram of force analysis of a single inclined slope pile according to the present invention;
fig. 3 (a) is a schematic diagram of the thickness of the stabilized surrounding rock of the single pile foundation of the inclined steep slope;
FIG. 3 (b) is a schematic view of the thickness of the stabilized surrounding rock of the flat single-pile foundation;
FIG. 4 is a schematic view showing a relationship between a slope gradient and a value m of the present invention;
FIG. 5 is a schematic diagram comparing the solution of the present invention for internal force and displacement on a 30 ° slope;
fig. 6 is a schematic diagram comparing the solution of the internal force and displacement under a 45 ° slope according to the present invention.
Detailed Description
In order to make the purpose and technical solution of the embodiments of the present invention clearer, the technical solution of the embodiments of the present invention will be clearly and completely described below with reference to the drawings of the embodiments of the present invention. It is to be understood that the embodiments described are only a few embodiments of the present invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the described embodiments of the invention without any inventive step, are within the scope of protection of the invention.
A method for calculating internal force and displacement of a single inclined slope pile based on a configuration point method comprises the following steps:
(1) according to the different stress conditions of each section of the rock-socketed single pile on the inclined steep slope, the rock-socketed single pile is divided into a free section, a loaded section and an embedded section. The pile top of the free section mainly bears the load of the pile top(including pile tip axial force P0Pile top shearing force Q0And pile top bending moment M0) The load-bearing section is used for bearing the load transmitted by the free section, and also bearing the residual sliding force of the slope body, the pile side rock-soil body resistance and the pile side frictional resistance, and the embedded section is mainly used for bearing the load transmitted by the load-bearing section, the pile side rock-soil body resistance and the pile side frictional resistance; the stress analysis of the single socketed pile on the inclined steep slope is shown in figure 2;
(2) respectively calculating the axial force of each subsection slope single pile according to the stress analysis of the slope single pile in the step (1); the monopile axial force p (z) is expressed as:
P(z)＝P0+fz
in the formula, P0Is the pile top axial force and z is the depth from the pile top. The part f at the free section is γ z, and the part f at the loaded section and the embedded section is γ z- τ Cz. C is the perimeter of the pile, tau is the side frictional resistance of the pile, and gamma is the weight of the pile body;
as shown in fig. 3; the side frictional resistance of the slope abrupt slope pile foundation can be neglected above the stable thickness of the surrounding rock, and the calculation of the side frictional resistance of the slope foundation pile is converted into the solution of the flat condition so as to achieve the effect of conservative design.
(3) Calculating the gliding thrust generated by the inclined steep slope body at the loaded section according to the stress analysis of the inclined steep slope single pile in the step (1); the profile of the ramp thrust q (z) is expressed as:
q(z)＝az2+bz+c
in the formula, a, b and c are undetermined coefficients; the calculation of the slope thrust introduces the theory of the slide-resistant pile, and undetermined coefficients a, b and c in a distribution form q (z) expression of the slide-resistant pile can be obtained according to the following table:
TABLE 1 slope glide thrust distribution function
Figure BDA0003412588510000081
Note: in table H1The length of the pile is above the sliding surface, E is the resultant force of the gliding thrust, and k is the form factor.
(4) Respectively calculating the resistance of the soil body in front of each segmented pile according to the stress analysis of the single inclined steep slope pile in the step (1); the soil resistance p (x, z) is calculated by using an m method and is expressed as:
p(x,z)＝K(z)b0x
wherein the embedding section K (z) m2z, loaded section K (z) m1z，b0Calculate width, m, for the pile1Is the value of m in the loaded section, m2Is the value of the embedded section m, and x is the horizontal displacement;
the value of the m value in the expression of the soil resistance p (x, z) is introduced into a reduction fitting formula about the gradient theta based on the specification:
mθ＝n1n2n3(m-k·tanθ)
wherein n is1Is a soil property parameter correction coefficient, n2Is the pile length correction factor, n3The pile diameter correction coefficient is obtained;
k is a correction coefficient obtained by fitting, and it can be known from the relationship between slope gradient and m value in fig. 4 that m value is too greatly reduced when the slope is greater than 50 degrees under the condition of natural stability of the slope, and m can be conservative designed50°When considered to be 0, k can be calculated according to the following formula
Figure BDA0003412588510000091
In the formula m, m0The value is taken according to the design Specification of highway bridge foundation and foundation (JTG 3363-2019).
(5) Respectively analyzing each section according to the stress section of the single inclined steep slope pile in the step (1), the step (2), the step (3) and the axial force, the gliding thrust and the soil resistance calculated in the step (4) to establish a deflection differential equation; wherein the step (5) comprises the sub-steps of:
(5.1) the flexural differential equation of the free-section infinitesimal is:
Figure BDA0003412588510000092
(5.2) the flexural differential equation of the loaded section infinitesimal is:
Figure BDA0003412588510000093
(5.3) flexural differential equation of the infill infinitesimal is:
Figure BDA0003412588510000094
(6) performing simultaneous integration and induction on the flexural differential equation of each section in the step (5) through the continuity condition and the deformation coordination condition among the sections to form a 4-order linear variable coefficient non-homogeneous ordinary differential equation of the single inclined steep slope pile; the equation is expressed as:
Figure BDA0003412588510000095
wherein L is the pile length, L is L1+l2+l3，0～l1Is a free segment,/1～l2Is a loaded section, /)2～l3As an embedded section, the other parameters are as follows:
Figure BDA0003412588510000101
Figure BDA0003412588510000102
Figure BDA0003412588510000103
Figure BDA0003412588510000104
Figure BDA0003412588510000105
Figure BDA0003412588510000106
the flexural differential equations of the free section, the loaded section and the embedded section of the single inclined steep slope pile are integrated and summarized into a 4-order linear variable coefficient non-homogeneous ordinary differential equation capable of representing the whole single inclined steep slope pile through continuity conditions and deformation coordination conditions, and the complex process and the discontinuity of solution caused by conventional segmented solution are avoided.
(7) According to the pile top load that superstructure obtained by the calculation transmitted, superstructure is: loading of the bridge and the pier on the pile; (including vertical load P0Horizontal load Q0And eccentric bending moment M0) Determining a boundary condition; the boundary conditions may be expressed as:
Figure BDA0003412588510000107
(8) solving the equation by adopting a 4-order implicit Runge Kutta formula and a configuration point formula; wherein the step (8) comprises the sub-steps of:
(8.1) differentiating the deflection of the pile in said step (6) into the form of a system of differential equations:
Figure BDA0003412588510000111
wherein the content of the first and second substances,
Figure BDA0003412588510000112
(8.2) referring to the solving format of Lobatto III A formula 4 order precision for the differential equation system form in the step (8.1), note x(j) iWhere j is 0,1,2,3, the solving equations for these four equations are listed:
Figure BDA0003412588510000113
and applying the Lobatto III A formula to solving the internal force displacement of the pile foundation. A solving format of Lobatto III A formula with 4-order precision is adopted, the essence of the solving format is a 4-order implicit Runge Kutta formula, and the values of coefficients are listed according to the following numbers:
Figure BDA0003412588510000114
(8.3) according to the solving format in the step (8.2), substituting the boundary conditions in the step (7), and carrying out iterative solution on the differential equation set in the step (8.1) by means of a matrix;
specifically, when j is equal to 0,1, the problem is an edge value problem of the flexural differential equation, and when j is equal to 2,3, the problem is an initial value problem of a first order differential equation set, and the solution is formulated into a matrix form by combining boundary conditions, wherein:
Figure BDA0003412588510000115
the formula of the configuration point method is used for simultaneous solving to obtain:
Figure BDA0003412588510000121
at this time, the solution of the original equation can be obtained, and can be expressed as:
Figure BDA0003412588510000122
wherein the content of the first and second substances,
Figure BDA0003412588510000123
b＝[b1 b2 b3 b′4]T，
Figure BDA0003412588510000124
e is an n-order unit array. At the moment, the internal force displacement matrix of the pile can be obtained
Figure BDA0003412588510000125
Obtained single-pile internal force displacement matrix of inclined steep slope
Figure BDA0003412588510000126
The detailed description of the matrix and formula contained in (1) is as follows:
Figure BDA0003412588510000127
Figure BDA0003412588510000128
b1＝[0 … 0]T
b2＝[0 … 0]T
b3＝[M0/EI … 0]T
Figure BDA0003412588510000129
Figure BDA00034125885100001210
Figure BDA00034125885100001211
u＝[1 0 … 0]1×n+1
Figure BDA0003412588510000131
Figure BDA0003412588510000132
Figure BDA0003412588510000133
O＝[0 0 … 0]1×2+1
Figure BDA0003412588510000134
Figure BDA0003412588510000135
Figure BDA0003412588510000136
Figure BDA0003412588510000137
Figure BDA0003412588510000138
in order to verify the reasonability of the obtained results, a field single-pile horizontal static load test of an actual site of the power transmission line in the mountainous area of Sichuan in the documents of Lixiaming and the like is selected, numerical simulation is carried out according to the test results, the measured related parameters are respectively substituted into the theoretical calculation method and the conventional calculation method which does not reduce the m value in the specification, and the numerical simulation of the test results and the theoretical calculation and the standard calculation of the invention are compared and analyzed. A comparison of the inner force versus displacement solution for a mono pile under a steep incline of 30 deg., 45 deg. slope is shown in fig. 5-6 below.
From the data of fig. 5-6, it can be seen that, in comparison of the bending moment distribution diagram and the shear distribution diagram of two slope tests, the maximum bending moment and the shear force obtained by the standard calculation method are both different by 20-40%, and the maximum errors of the maximum bending moment and the shear force of the pile body and the horizontal displacement of the pile top calculated by adopting the method are controlled within 4%, so that the error of the internal force and the displacement of the single pile of the inclined steep slope calculated by the method is smaller, and the method has higher reliability for different slopes. Under the slopes of 30 degrees and 45 degrees, the calculation method for reducing the m value is greatly different from the calculation method for not reducing the m value which is adopted according to the specification, and along with the increase of the slope, the displacement and the maximum bending moment which are obtained by not reducing the m value are greatly smaller than those of numerical simulation; the reduction method introduced in the method still keeps larger relative measured value except for underestimation of pile top displacement, so that a relatively conservative design result can be obtained. Therefore, the achievement of the invention has more reliability in practical engineering application.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (8)

1. A method for calculating the internal force and displacement of a single inclined slope pile based on a configuration point method is characterized by comprising the following steps of: the calculation steps are as follows:
dividing the rock-socketed single pile on the inclined steep slope into a free section, a loaded section and an embedded section, and calculating to obtain the single pile axial force, slope thrust and soil resistance of the rock-socketed single pile;
secondly, acquiring single pile axial force, slope thrust and soil resistance in the first step to establish infinitesimal flexural differential equations of a free section, a loaded section and an embedded section of the rock-embedded single pile;
thirdly, simultaneous integration and induction are carried out through deformation coordination conditions between the loaded section bending differential equation and the embedded section bending differential equation established in the second step, and a 4-order linear variable coefficient heterogeneous ordinary differential equation of the whole single inclined slope pile is formed;
fourthly, calculating to obtain the pile top load of the rock-socketed single pile transmitted by the upper structure according to the third step, wherein the pile top load comprises a vertical load, a horizontal load and an eccentric bending moment; determining boundary conditions of the rock-socketed single pile;
and fifthly, solving the 4-order linear variable coefficient non-homogeneous ordinary differential equation by adopting a 4-order implicit Runge Kutta algorithm and combining a configuration point algorithm to obtain an internal force and displacement matrix of the single inclined steep slope pile.
2. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 1, wherein the method comprises the following steps: the free section pile top mainly bears the load of the pile top;
the loaded section is used for bearing the load transmitted by the free section and bearing the residual sliding force of a slope body, the resistance of a pile side rock-soil body and the friction resistance of the pile side;
the embedded section mainly bears the load transmitted by the loaded section, the resistance of the pile side rock-soil body and the friction resistance of the pile side.
3. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 1, wherein the method comprises the following steps: the single pile axial force P (z) of the rock-socketed single pile in the first step is represented as follows:
P(z)＝P0+fz
in the formula, P0Is the pile top axial force and z is the depth from the pile top. The part f of the free section is gamma z, the part f of the loading section and the embedded section is gamma z-tau Cz, C is the perimeter of the pile, tau is the friction resistance of the side of the pile, and gamma is the weight of the pile body;
in the step one, the gliding thrust generated by the load-bearing section inclined steep slope body; the distribution form q (z) of the slope thrust of the monopile axial force of the socketed monopile is expressed as:
q(z)＝az2+bz+c
in the formula, a, b and c are undetermined coefficients;
respectively calculating the resistance of the soil body in front of each segmental pile by adopting an m method, wherein the resistance p (x, z) of the soil body is expressed as follows:
p(x,z)＝K(z)b0x
wherein the embedding section K (z) m2z, loaded section K (z) m1z，b0Calculate width, m, for the pile1Is the value of m in the loaded section, m2Is the value of the embedded section m, and x is the horizontal displacement;
the flexural differential equation of the free-section infinitesimal is as follows:
Figure FDA0003412588500000021
the loaded section infinitesimal flexural differential equation is as follows:
Figure FDA0003412588500000022
the flexural differential equation of the embedded section infinitesimal element is as follows:
Figure FDA0003412588500000023
wherein EI is the bending rigidity of the pile body;
forming a 4-order linear variable coefficient non-homogeneous ordinary differential equation of the whole single inclined slope pile based on the free section infinitesimal bending differential equation, the loaded section infinitesimal bending differential equation and the embedded section infinitesimal bending differential equation; the equation is expressed as:
Figure FDA0003412588500000024
γzx-(a1z2+b1z+c1)＝0,z∈[0,L]
wherein L is the pile length, L is L1+l2+l3，0～l1Is a free segment,/1～l2Is a loaded section, /)2～l3The other parameters are as follows:
Figure FDA0003412588500000025
Figure FDA0003412588500000026
Figure FDA0003412588500000027
Figure FDA0003412588500000028
Figure FDA0003412588500000031
Figure FDA0003412588500000032
4. the method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 1, wherein the method comprises the following steps: determining boundary conditions of the pile top load transmitted by the overlying structure obtained by calculation in the third step; the boundary conditions may be expressed as:
Figure FDA0003412588500000033
the pile top load comprises a pile top axial force P0Pile top shearing force Q0And pile top bending moment M0。
5. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 3 or 4, wherein the method comprises the following steps: converting a free section infinitesimal flexural differential equation, a loaded section infinitesimal flexural differential equation and an embedded section infinitesimal flexural differential equation into a differential equation set form, wherein the expression form is as follows:
Figure FDA0003412588500000034
wherein the content of the first and second substances,
Figure FDA0003412588500000035
solving the system of differential equations, let x(j) iWhere j is 0,1,2,3, the solving equations for these four equations are listed:
Figure FDA0003412588500000041
introducing boundary conditions and differential equation sets of pile top load into the solving format; when j is 0,1, the problem is the edge value of the flexural differential equation, when j is 2,3, the problem is the initial value of the first order differential equation set, and the solution is formulated into a matrix form by combining boundary conditions, wherein:
Figure FDA0003412588500000042
the formula of the configuration point method is used for simultaneous solving to obtain:
Figure FDA0003412588500000043
at this time, the solution of the original equation can be obtained, and can be expressed as:
Figure FDA0003412588500000044
wherein the content of the first and second substances,
Figure FDA0003412588500000045
b＝[b1 b2 b3 b′4]T，
Figure FDA0003412588500000046
e is an n-order unit matrix, and at the moment, the pile internal force displacement matrix can be obtained
Figure FDA0003412588500000047
6. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 3, wherein the method comprises the following steps: the undetermined coefficients a, b and c in the distribution form q (z) expression can be obtained according to the following table
TABLE 1 slope glide thrust distribution function
Figure FDA0003412588500000048
Figure FDA0003412588500000051
In table H1The length of the pile is above the sliding surface, E is the resultant force of the gliding thrust, and k is the form factor.
7. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 3, wherein the method comprises the following steps:
in the step (4), the value of the value m in the expression of the soil resistance p (x, z) is introduced into a reduction fitting formula about the gradient theta based on the specification:
mθ＝n1n2n3(m-k·tanθ)
wherein n is1Is a soil property parameter correction coefficient, n2Is the pile length correction factor, n3The pile diameter correction coefficient is obtained; and k is a correction coefficient obtained by fitting.
8. The method for calculating the internal force and displacement of the single inclined steep slope pile based on the configuration point method according to claim 5, wherein the method comprises the following steps: a solving format of Lobatto III A formula with 4-order precision is adopted, the essence of the solving format is a 4-order implicit Runge Kutta formula, and the values of coefficients are listed according to the following numbers:
Figure FDA0003412588500000052
obtained single-pile internal force displacement matrix of inclined steep slope
Figure FDA0003412588500000053
The matrix and the formula contained in the method are as follows:
Figure FDA0003412588500000054
Figure FDA0003412588500000055
b1＝[0…0]T
b2＝[0…0]T
b3＝[M0/EI…0]T
Figure FDA0003412588500000061
Figure FDA0003412588500000062
Figure FDA0003412588500000063
u＝[1 0…0]1×n+1
Figure FDA0003412588500000064
Figure FDA0003412588500000065
Figure FDA0003412588500000066
O＝[0 0…0]1×n+1
Figure FDA0003412588500000067
Figure FDA0003412588500000068
Figure FDA0003412588500000069
Figure FDA00034125885000000610
Figure FDA00034125885000000611
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115017592A (en) * 2022-06-27 2022-09-06 河南大学 Calculation method for rigid pile of push-type landslide pre-stressed anchor cable

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115017592A (en) * 2022-06-27 2022-09-06 河南大学 Calculation method for rigid pile of push-type landslide pre-stressed anchor cable

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