CN111310264B - Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model - Google Patents

Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model Download PDF

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CN111310264B
CN111310264B CN202010084264.5A CN202010084264A CN111310264B CN 111310264 B CN111310264 B CN 111310264B CN 202010084264 A CN202010084264 A CN 202010084264A CN 111310264 B CN111310264 B CN 111310264B
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pile
pile body
layer
soil
amplitude
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CN111310264A (en
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崔春义
辛宇
梁志孟
孟坤
王本龙
刘海龙
张鹏
赵九野
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Dalian Maritime University
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Abstract

The invention discloses a method for analyzing horizontal dynamic response of single piles in layered soil based on a Pastnak foundation model, which adopts the layered Pastnak foundation model to simulate the shearing effect of a soil body around a pile, adopts a segmented Timoshenko beam model to simulate a pile body so as to consider the bending and shearing deformation of the pile body, and simultaneously assumes that the pile-soil model meets a small deformation condition, a pile-soil interface is in complete contact and has no relative sliding, and a pile bottom is in fixed-end constraint. On the basis of the assumption, firstly, a horizontal dynamic balance equation of the segmented pile body unit is established, secondly, the relation between the pile body corner, the bending moment, the shearing force and the pile body horizontal displacement is established, thirdly, a coefficient matrix equation set of the segmented pile body unit is established according to the horizontal displacement, the corner, the bending moment and the shearing force continuity of the pile, and finally, the horizontal displacement of each segment of the pile body and the bending moment and the shearing force on any section of the pile body are obtained according to boundary conditions. The invention can provide theoretical guidance and reference for pile foundation power detection.

Description

Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model
Technical Field
The invention relates to the field of civil engineering, in particular to a method for analyzing horizontal dynamic response of a single pile in layered soil based on a Passternak foundation model.
Background
At present, when the problem of horizontal vibration dynamic response of a pile body is solved, a pile soil body is generally simplified into a Winkler model for convenient calculation. The Winkler foundation model ignores the shearing effect of soil, and cannot reflect the continuity of soil among longitudinal layers, so that the calculation result is not strict in theory. The double-parameter foundation model considers the shearing effect of the foundation soil body on the basis of the Winkler model, and is more practical. In this case, the use of the Passternak foundation model is more appropriate.
In addition, when the problem of horizontal vibration dynamic response of the pile body is solved, a classic Bernoulli-Euler theory is adopted for the slender rod pile foundation model, the theoretical model only considers the bending deformation of the pile body, and the influence of the shearing deformation of the pile foundation is ignored. For large-diameter piles, the influence of pile body shear deformation on the dynamic impedance is particularly important to consider, and the model of the Timoshenko beam (iron-wood sinco beam) adopted by the pile body is more suitable.
How to effectively combine and apply the two to solve the problem of horizontal vibration dynamic response of the pile body is one of the important points of research in the field.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a method for analyzing horizontal dynamic response of a single pile in layered soil based on a Pastnak foundation model.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a method for analyzing horizontal dynamic response of a single pile in layered soil based on a Passternak foundation model is characterized by comprising the following steps:
s1: establishing a single-pile horizontal dynamic response model in the layered soil, wherein the depth of a single-pile body is set to be consistent with that of soil around the pile, and the single-pile body and the soil around the pile are longitudinally divided into n layers; the following assumptions were also introduced: assuming that a single pile body is a homogeneous elastic body with a uniform cross section, and adopting a Timoshenko beam model; assuming that each layer of soil body of the soil around the pile adopts a Passternak foundation model; assuming that the pile-soil models all meet the small deformation condition, the pile-soil interfaces are in complete contact and have no relative sliding; and assuming that the pile bottom is fixed end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Passternak foundation model theory, wherein the corresponding expression of the dynamic balance equation is as follows:
Figure BDA0002380783700000021
meanwhile, according to the assumed conditions in step S1, a pile-soil model boundary condition is established, and the corresponding expression is:
Figure BDA0002380783700000022
Figure BDA0002380783700000023
P 1p ( 0，t)＝Q 0 e iωt
φ 1 p (0，t)＝0
in the formula, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; n is the number of layers of soil around the pile and the single pile body, j is the number of the layers of the soil around the pile and the single pile body from top to bottom, i.e. j is 1-n, Q 0 Is the amplitude of the horizontal simple resonance exciting force of the pile top,
Figure BDA0002380783700000024
is an imaginary unit;
A p 、G p 、E p 、I p 、m p respectively representing the sectional area, the shear modulus, the elastic modulus, the section inertia moment and the unit length mass of the single pile body; k' is a single pileA shear shape factor of a cross section of the body; b 0 0.9(1.5d +0.5) is the calculated width of the single-pile body, d is the diameter of the single-pile body, and the length of the single-pile body is L;
Figure BDA0002380783700000025
and
Figure BDA0002380783700000026
respectively the horizontal displacement and the section corner of a j-th layer of pile body point;
Figure BDA0002380783700000027
and
Figure BDA0002380783700000028
is the horizontal displacement and the corner of the bottom of the pile body, P 1 p (0, t) and φ 1 p (0, t) is the shear and corner of the pile bottom top;
Figure BDA0002380783700000029
respectively is the stiffness coefficient, the damping coefficient and the foundation shear coefficient of the soil around the jth layer of pile, and respectively corresponding calculation formulas are
Figure BDA00023807837000000210
Figure BDA00023807837000000211
Figure BDA00023807837000000212
In the formula (I), the compound is shown in the specification,
Figure BDA00023807837000000213
is the shear wave velocity of the soil around the jth layer of pile, and
Figure BDA00023807837000000214
Figure BDA0002380783700000031
and
Figure BDA0002380783700000032
respectively the elastic modulus, the density, the damping coefficient and the Poisson ratio of the soil around the jth layer of pile,
Figure BDA0002380783700000033
dimensionless frequency of the soil around the jth layer of piles;
Figure BDA0002380783700000034
the thickness of the shear layer is the thickness of the soil around the jth layer of piles; omega is the excitation circle frequency of the horizontal simple resonance excitation force of the pile top;
s3: and (4) solving the dynamic balance equation of the layered pile body unit in the step (S2) to obtain parameters required for single-pile horizontal dynamic response analysis, wherein the parameters are the pile top horizontal impedance of the horizontal excitation force acting on the pile top and the internal force of any section of the pile body.
Optionally, in one embodiment, in the step S3, the solving process of solving the dynamic balance equation of the layered pile unit in the step S2 includes the following steps:
step S31: and (3) respectively converting the horizontal displacement, the section corner, the pile body shearing force and the pile body bending moment of the j-th layer of pile body particles according to the excitation circle frequency of the horizontal simple resonance excitation force of the pile top, wherein the corresponding conversion formula is as follows:
Figure BDA0002380783700000035
Figure BDA0002380783700000036
Figure BDA0002380783700000037
Figure BDA0002380783700000038
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002380783700000039
is the horizontal displacement amplitude of the j-th layer pile body,
Figure BDA00023807837000000310
is the section corner amplitude of the j-th layer of pile body,
Figure BDA00023807837000000311
is the pile body shearing force amplitude of the j layer of pile body,
Figure BDA00023807837000000312
the bending moment amplitude of the j-th layer pile body is obtained;
the boundary condition change formula corresponding to the transformation formula is as follows:
Figure BDA00023807837000000313
step S32: simplifying the dynamic balance equation of the layered pile body unit to obtain a fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude, a general solution of the corner amplitude, a general solution of the pile body bending moment amplitude and a general solution of the pile body shearing amplitude,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
Figure BDA00023807837000000314
while modifying the expression to
Figure BDA00023807837000000315
Wherein the content of the first and second substances,
Figure BDA0002380783700000041
the expression corresponding to the general solution of the horizontal displacement amplitude is as follows:
Figure BDA0002380783700000042
the expression corresponding to the corner amplitude general solution is as follows:
Figure BDA0002380783700000043
the corresponding expression of the pile body bending moment amplitude general solution is as follows:
Figure BDA0002380783700000044
the general solution of the shear amplitude of the pile body is represented by the following expression:
Figure BDA0002380783700000045
wherein each symbol is defined as:
W p ＝E p I p d p ＝K′A p G p
Figure BDA0002380783700000046
Figure BDA0002380783700000047
Figure BDA0002380783700000048
Figure BDA0002380783700000049
in the formula, A j1 ，B j1 ，C j1 ，D j1 ，A j2 ，B j2 ，C j2 ，D j2 ，A j3 ，B j3 ，C j3 ，D j3 ，A j4 ，B j4 ，C j4 ，D j4 Are determined based on boundary conditions, and the following relationships exist among the coefficients
Figure BDA0002380783700000051
Wherein the content of the first and second substances,
Figure BDA0002380783700000052
Figure BDA0002380783700000053
Figure BDA0002380783700000054
Figure BDA0002380783700000055
Figure BDA0002380783700000056
Figure BDA0002380783700000057
Figure BDA0002380783700000058
Figure BDA0002380783700000059
step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using a horizontal displacement, corner, bending moment and shearing force continuity expression between the j-th layer pile body and the j + 1-th layer pile body; the expression of the coefficient matrix equation set of the j-th layer pile body is
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }; wherein, the first and the second end of the pipe are connected with each other,
{T j }＝[A j1 B j1 C j1 D j1 ] T
Figure BDA0002380783700000061
wherein, the expressions of horizontal displacement, corner, bending moment and shearing continuity between the jth layer pile body and the jth +1 layer pile body are respectively
Figure BDA0002380783700000062
Step S34: the coefficient matrix equation sets of each layer of pile body are combined to obtain the mth section of coefficient matrix equation set, and the expression of the mth section of coefficient matrix equation set is
Figure BDA0002380783700000063
Step S35: based on each boundary condition, a top coefficient matrix equation set and a bottom coefficient matrix equation set are created, and corresponding expressions are respectively
Figure BDA0002380783700000064
Figure BDA0002380783700000065
Step S36: using a recursive relationship T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j The undetermined coefficient { T ] in the bottom coefficient matrix equation set obtained in the step S35 is used n Converting into undetermined coefficient (T) 1 And combining the coefficients with a top coefficient matrix equation set to obtain 4 undetermined coefficients (T) 1 -a system of equations, said system of equations being solved in parallel to obtain a undetermined coefficient T 1 And then according to a recurrence formula { T } j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j Solving other coefficients to be determined in turn { T } 2 }～{T n And finally, substituting each coefficient to be determined into a coefficient matrix corresponding to each section of the pile body respectively, so as to obtain the horizontal displacement, the bending moment and the shearing force of each section of the pile body.
According to the technical scheme, the method for analyzing the horizontal vibration of the single pile in the layered soil based on the Paternak foundation model can simultaneously consider the shearing effect of the soil body around the pile and the bending and shearing deformation of the pile body, simultaneously supposes that all parts of a pile-soil system meet small deformation conditions, the pile-soil interface is in complete contact and has no relative sliding, the pile bottom is restrained by a fixed end, and finally a top coefficient matrix is obtained according to boundary conditions, so that the horizontal displacement, the bending moment and the shearing force on any section of the pile body are calculated, the result is suitable for the problem of horizontal vibration dynamic response of the pile foundation under the action of simple harmonic load, and theoretical guidance and reference action can be provided for pile foundation dynamic detection.
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FIG. 1 is a flow chart of core steps corresponding to the method of the present invention in the embodiment;
FIG. 2 is a schematic view of a pile-soil model corresponding to the method of the present invention in the example.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
In the following detailed description of the embodiments of the present invention, in order to clearly illustrate the structure of the present invention and to facilitate explanation, it should be understood that the structure shown in the drawings is not drawn to general scale and is partially enlarged, modified or simplified, so that the present invention is not limited thereto.
The method for analyzing the horizontal dynamic response of the single pile in the layered soil based on the Passternak foundation model is characterized by comprising the following steps of:
s1: the following assumed conditions are introduced to establish a single-pile horizontal dynamic response model in the stratified soil: the depth of the single pile body is consistent with that of soil around the pile, and the single pile body and the soil around the pile are longitudinally divided into n layers; the assumed conditions include at least: assuming that a single pile body is a homogeneous elastic body with a uniform cross section, and adopting a Timoshenko beam model; assuming that each layer of soil body of the soil around the pile adopts a Passternak foundation model (a two-parameter foundation model); supposing that all parts of the pile-soil model meet the small deformation condition, and the pile-soil interfaces are in complete contact and have no relative sliding; assuming that the pile bottom is fixed end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Passternak foundation model theory, wherein the corresponding expression of the dynamic balance equation is as follows:
Figure BDA0002380783700000071
it should be noted that: in the above equations, the shaft cell balance equations described herein are compared to existing Euler beam models and Winkler ground models to add the shaft shear modulus component G p And the shear coefficient part of the soil body around the pile
Figure BDA0002380783700000081
To construct a balance equation;
meanwhile, according to the assumed conditions in step S1, a pile-soil model boundary condition is established, and the corresponding expression is:
Figure BDA0002380783700000082
Figure BDA0002380783700000083
P 1 p (0，t)＝Q 0 e iωt
φ 1 p (0，t)＝0
in the formula, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; n is the number of layers of soil around the pile and the single pile body, j is the number of the layers of the soil around the pile and the single pile body from top to bottom, i.e. j is 1-n, Q 0 Is the amplitude of the horizontal simple resonance exciting force of the pile top,
Figure BDA0002380783700000084
is an imaginary unit;
A p 、G p 、E p 、I p 、m p the sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the unit length mass of the single pile body are respectively; k' is the shear shape coefficient of the section of the single pile body; b is 0 0.9(1.5d +0.5) is the calculated width of the single-pile body, d is the diameter of the single-pile body, and the length of the single-pile body is L;
Figure BDA0002380783700000085
seed of a plant
Figure BDA0002380783700000086
Respectively the horizontal displacement and the section corner of a j-th layer of pile body point;
Figure BDA0002380783700000087
and
Figure BDA0002380783700000088
is the horizontal displacement and the corner of the bottom of the pile body, P 1 p (0, t) and phi 1 p (0, t) is the shear force and the corner of the top of the pile bottom;
Figure BDA0002380783700000089
respectively is the stiffness coefficient, the damping coefficient and the foundation shear coefficient of the soil around the jth layer of pile, and respectively corresponding calculation formulas are
Figure BDA00023807837000000810
Figure BDA00023807837000000811
Figure BDA00023807837000000812
In the formula (I), the compound is shown in the specification,
Figure BDA00023807837000000813
is the shear wave velocity of the soil around the jth layer of pile, and
Figure BDA00023807837000000814
Figure BDA00023807837000000815
and
Figure BDA00023807837000000816
respectively the elastic modulus, the density, the damping coefficient and the Poisson ratio of the soil around the j-th layer of pile,
Figure BDA00023807837000000817
dimensionless frequency of the soil around the jth layer of piles;
Figure BDA00023807837000000818
the thickness of the shear layer is the thickness of the soil around the jth layer of piles; omega is the excitation circle frequency of the horizontal simple resonance excitation force of the pile top;
s3: and (4) solving the dynamic balance equation of the layered pile body unit in the step (S2) to obtain parameters required for single-pile horizontal dynamic response analysis, wherein the parameters are the pile top horizontal impedance of the horizontal excitation force acting on the pile top and the internal force of any section of the pile body.
Optionally, in one embodiment, in the step S3, the process of solving the dynamic balance equation of the layered pile unit in step S2 includes the following steps:
step S31: and (3) respectively converting the horizontal displacement, the section corner, the pile body shearing force and the pile body bending moment of the j-th layer of pile body particles according to the excitation circle frequency of the horizontal simple resonance excitation force of the pile top, wherein the corresponding conversion formula is as follows:
Figure BDA0002380783700000091
Figure BDA0002380783700000092
Figure BDA0002380783700000093
Figure BDA0002380783700000094
wherein the content of the first and second substances,
Figure BDA0002380783700000095
is the horizontal displacement amplitude of the j-th layer pile body,
Figure BDA0002380783700000096
is the section corner amplitude of the j-th layer of pile body,
Figure BDA0002380783700000097
is the pile body shearing force amplitude of the j-th layer of pile body,
Figure BDA0002380783700000098
the bending moment amplitude of the j-th layer pile body is obtained;
the boundary condition change formula corresponding to the transformation formula is as follows:
Figure BDA0002380783700000099
step S32: simplifying the dynamic balance equation of the layered pile body unit to obtain a fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude, a general solution of the corner amplitude, a general solution of the pile body bending moment amplitude and a general solution of the pile body shearing amplitude,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
Figure BDA00023807837000000910
in addition, J in the above expression p When it tends to infinity, then
Figure BDA00023807837000000911
When the expression approaches 0, the expression can be converted into a Winkler-Euler model;
while modifying the expression to
Figure BDA00023807837000000912
Wherein the content of the first and second substances,
Figure BDA00023807837000000913
the expression corresponding to the general solution of the horizontal displacement amplitude is as follows:
Figure BDA0002380783700000101
the expression corresponding to the corner amplitude general solution is as follows:
Figure BDA0002380783700000102
the expression corresponding to the general solution of the pile body bending moment amplitude is as follows:
Figure BDA0002380783700000103
the expression corresponding to the general solution of the shear amplitude of the pile body is as follows:
Figure BDA0002380783700000104
wherein each symbol is defined as:
W p ＝E p I p J p ＝K′A p G p
Figure BDA0002380783700000105
Figure BDA0002380783700000106
Figure BDA0002380783700000107
Figure BDA0002380783700000108
in the formula, A j1 ，B j1 ，C j1 ，D j1 ，A j2 ，B j2 ，C j2 ，D j2 ，A j3 ，B j3 ，C j3 ，D j3 ，A j4 ，B j4 ，C j4 ，D j4 Are determined based on boundary conditions, and the following relationship exists between the coefficients
Figure BDA0002380783700000109
Wherein the content of the first and second substances,
Figure BDA0002380783700000111
Figure BDA0002380783700000112
Figure BDA0002380783700000113
Figure BDA0002380783700000114
Figure BDA0002380783700000115
Figure BDA0002380783700000116
Figure BDA0002380783700000117
Figure BDA0002380783700000118
step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using a horizontal displacement, corner, bending moment and shearing force continuity expression between the j-th layer pile body and the j + 1-th layer pile body; the expression of the coefficient matrix equation set of the j-th layer pile body is
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }; wherein the content of the first and second substances,
{T j }＝[A j1 B j1 C j1 D j1 ] T
Figure BDA0002380783700000119
wherein, the expressions of horizontal displacement, corner, bending moment and shearing continuity between the jth layer pile body and the jth +1 layer pile body are respectively
Figure BDA0002380783700000121
Step S34: the coefficient matrix equation sets of each layer of pile body are combined to obtain the mth section of coefficient matrix equation set, and the expression of the mth section of coefficient matrix equation set is
Figure BDA0002380783700000122
Step S35: based on each boundary condition, a top coefficient matrix equation set and a bottom coefficient matrix equation set are created, and corresponding expressions are respectively
Figure BDA0002380783700000123
Figure BDA0002380783700000124
Step S36: using a recursive relationship T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j Is determined by the bottom line obtained in step S35Undetermined coefficient in number matrix equation set T n Converting into undetermined coefficient (T) 1 And combining the coefficients with a top coefficient matrix equation set to obtain 4 undetermined coefficients (T) 1 -a system of equations, said system of equations being solved in parallel to obtain a undetermined coefficient T 1 According to a recurrence formula { T } j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j Solving other coefficients to be determined in turn { T } 2 }～{T n And finally, substituting each coefficient to be determined into a coefficient matrix corresponding to each section of the pile body respectively, so as to obtain the horizontal displacement, the bending moment and the shearing force of each section of the pile body.
In conclusion, the method for analyzing horizontal vibration of the single pile in the layered soil based on the Passternak foundation model can simultaneously consider the shearing effect of the soil body around the pile and the bending and shearing deformation of the pile body, simultaneously supposes that all parts of the pile-soil system meet the small deformation condition, the pile-soil interface is in complete contact and has no relative sliding, the pile bottom is restrained by the fixed end, and finally the top coefficient matrix is obtained according to the boundary condition, so that the horizontal displacement, the bending moment and the shearing force on any section of the pile body are calculated, the result is suitable for the problem of horizontal vibration dynamic response of the pile foundation under the action of simple harmonic load, and theoretical guidance and reference action can be provided for pile foundation dynamic detection.
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 person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims (2)

1. A method for analyzing single-pile horizontal dynamic response in layered soil based on a Passternak foundation model is characterized by comprising the following steps:
s1: establishing a single-pile horizontal dynamic response model in the layered soil, wherein the depth of a single-pile body is set to be consistent with that of soil around the pile, and the single-pile body and the soil around the pile are longitudinally divided into n layers; the following assumptions were also introduced: assuming that a single pile body is a homogeneous elastic body with a uniform cross section, and adopting a Timoshenko beam model; assuming that each layer of soil body of the soil around the pile adopts a Passternak foundation model; assuming that the pile-soil models all meet the small deformation condition, the pile-soil interfaces are in complete contact and have no relative sliding; and assuming that the pile bottom is fixed end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Passternak foundation model theory, wherein the corresponding expression of the dynamic balance equation is as follows:
Figure FDA0002380783690000011
meanwhile, according to the assumed conditions in step S1, establishing boundary conditions of the pile-soil model, where the corresponding expression is:
Figure FDA0002380783690000012
Figure FDA0002380783690000013
P 1 p (0，t)＝Q 0 e iωt
Figure FDA0002380783690000014
in the formula, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface, the downward direction is positive, and t is a time coordinate; n is the number of layers of soil around the pile and the single pile body, j is the number of the layers of the soil around the pile and the single pile body from top to bottom, i.e. j is 1-n, Q 0 Is the amplitude of the horizontal simple resonance exciting force of the pile top,
Figure FDA0002380783690000015
is an imaginary unit;
A p 、G p 、E p 、I p 、m p respectively representing the sectional area, the shear modulus, the elastic modulus, the section inertia moment and the unit length mass of the single pile body; k' is the shear shape coefficient of the section of the single pile body; b 0 0.9(1.5d +0.5) is the calculated width of the single-pile body, d is the diameter of the single-pile body, and the length of the single-pile body is L;
Figure FDA0002380783690000016
and
Figure FDA0002380783690000017
respectively the horizontal displacement and the section corner of a j-th layer of pile body point;
Figure FDA0002380783690000018
and
Figure FDA0002380783690000019
is the horizontal displacement and the corner of the bottom of the pile body, P 1 p (0, t) and
Figure FDA00023807836900000110
is the shear force and the corner of the top of the pile bottom;
Figure FDA00023807836900000111
respectively is the stiffness coefficient, the damping coefficient and the foundation shear coefficient of the soil around the jth layer of pile, and respectively corresponding calculation formulas are
Figure FDA0002380783690000021
Figure FDA0002380783690000022
Figure FDA0002380783690000023
In the formula (I), the compound is shown in the specification,
Figure FDA0002380783690000024
is the shear wave velocity of the soil around the jth layer of pile, and
Figure FDA0002380783690000025
Figure FDA0002380783690000026
Figure FDA0002380783690000027
and
Figure FDA0002380783690000028
respectively the elastic modulus, the density, the damping coefficient and the Poisson ratio of the soil around the jth layer of pile,
Figure FDA0002380783690000029
the dimensionless frequency of the soil around the jth layer of pile is obtained;
Figure FDA00023807836900000210
the thickness of the shear layer of the soil around the jth layer of pile is shown; omega is the excitation circle frequency of the horizontal simple resonance excitation force of the pile top;
s3: and (4) solving the dynamic balance equation of the layered pile body unit in the step (S2) to obtain parameters required for single-pile horizontal dynamic response analysis, wherein the parameters are the pile top horizontal impedance of the horizontal excitation force acting on the pile top and the internal force of any section of the pile body.
2. The analysis method according to claim 1, wherein in the step S3, the solving process for solving the dynamic balance equation of the layered pile unit in the step S2 includes the following steps:
step S31: and (3) respectively converting the horizontal displacement, the section corner, the pile body shearing force and the pile body bending moment of the j-th layer of pile body particles according to the excitation circle frequency of the horizontal simple resonance excitation force of the pile top, wherein the corresponding conversion formula is as follows:
Figure FDA00023807836900000211
Figure FDA00023807836900000212
Figure FDA00023807836900000213
Figure FDA00023807836900000214
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00023807836900000215
is the horizontal displacement amplitude of the j-th layer pile body,
Figure FDA00023807836900000216
is the section corner amplitude of the j-th layer of pile body,
Figure FDA00023807836900000217
is the pile body shearing force amplitude of the j layer of pile body,
Figure FDA00023807836900000218
the bending moment amplitude of the j-th layer pile body is obtained;
the boundary condition change formula corresponding to the transformation formula is as follows:
Figure FDA00023807836900000219
step S32: simplifying the dynamic balance equation of the layered pile body unit to obtain a fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude, a general solution of the corner amplitude, a general solution of the pile body bending moment amplitude and a general solution of the pile body shearing amplitude,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
Figure FDA0002380783690000031
while modifying the expression to
Figure FDA0002380783690000032
Wherein the content of the first and second substances,
Figure FDA0002380783690000033
the expression corresponding to the general solution of the horizontal displacement amplitude is as follows:
Figure FDA0002380783690000034
the expression corresponding to the corner amplitude general solution is as follows:
Figure FDA0002380783690000035
the expression corresponding to the general solution of the pile body bending moment amplitude is as follows:
Figure FDA0002380783690000036
the general solution of the shear amplitude of the pile body is represented by the following expression:
Figure FDA0002380783690000037
wherein each symbol is defined as:
W p ＝E p I p J p ＝K′A p G p
Figure FDA0002380783690000038
Figure FDA0002380783690000039
Figure FDA00023807836900000310
Figure FDA00023807836900000311
in the formula, A j1 ，B j1 ，C j1 ，D j1 ，A j2 ，B j2 ，C j2 ，D j2 ，A j3 ，B j3 ，C j3 ，D j3 ，A j4 ，B j4 ，C j4 ，D j4 Are determined based on boundary conditions, and the following relationship exists between the coefficients
Figure FDA0002380783690000041
Wherein the content of the first and second substances,
Figure FDA0002380783690000042
step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using a horizontal displacement, corner, bending moment and shearing force continuity expression between the j-th layer pile body and the j + 1-th layer pile body; the expression of the coefficient matrix equation set of the j-th layer pile body is
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }; wherein the content of the first and second substances,
{T j }＝[A j1 B j1 C j1 D j1 ] T
Figure FDA0002380783690000051
wherein, the expressions of horizontal displacement, corner, bending moment and shearing continuity between the jth layer pile body and the jth +1 layer pile body are respectively
Figure FDA0002380783690000052
Step S34: the coefficient matrix equation sets of each layer of pile body are combined to obtain the mth section of coefficient matrix equation set, and the expression of the mth section of coefficient matrix equation set is
Figure FDA0002380783690000053
Step S35: based on each boundary condition, a top coefficient matrix equation set and a bottom coefficient matrix equation set are created, and corresponding expressions are respectively
Figure FDA0002380783690000054
Figure FDA0002380783690000055
Step S36: using a recursive relationship T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j The undetermined coefficient { T ] in the bottom coefficient matrix equation set obtained in the step S35 is used n Converting into undetermined coefficient (T) 1 And combining the coefficients with a top coefficient matrix equation set to obtain 4 undetermined coefficients (T) 1 Solving the equation set of the undetermined coefficient { T } in parallel to obtain the undetermined coefficient { T } 1 And then according to a recurrence formula { T } j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j }{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j Solving other coefficients to be determined in turn { T } 2 }～{T n And finally, substituting each coefficient to be determined into a coefficient matrix corresponding to each section of the pile body respectively, so as to obtain the horizontal displacement, the bending moment and the shearing force of each section of the pile body.
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