CN112307545A - Large-diameter single pile horizontal vibration analysis method considering axial force action - Google Patents

Large-diameter single pile horizontal vibration analysis method considering axial force action Download PDF

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CN112307545A
CN112307545A CN202011193166.1A CN202011193166A CN112307545A CN 112307545 A CN112307545 A CN 112307545A CN 202011193166 A CN202011193166 A CN 202011193166A CN 112307545 A CN112307545 A CN 112307545A
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pile body
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崔春义
辛宇
梁志孟
王本龙
孟坤
刘海龙
裴华富
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Dalian Maritime University
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Abstract

The invention discloses a large-diameter single-pile horizontal vibration analysis method considering axial force action, which simplifies a large-diameter single pile into a Timoshenko beam model and provides a mechanical system model considering pile body shear deformation and pile surrounding soil body shear effect simultaneously. In addition, the influence of the axial force is simultaneously considered under the action of the horizontal load of the pile top, the action of the two loads is comprehensively considered, the problem of horizontal vibration of the large-diameter single pile under the action of complex multi-directional load can be solved, the pile-soil coupling interaction of the foundation pile under the action of complex load in the actual engineering can be better simulated, and a foundation can be laid for the pile foundation vibration theory; the invention considers the influence of the shearing deformation of the pile body and the shearing deformation of the soil body around the pile while considering the complex load action of the pile top, and also considers the influence factors of the stress deformation related to the pile foundation in the actual engineering.

Description

Large-diameter single pile horizontal vibration analysis method considering axial force action
Technical Field
The invention relates to the field of civil engineering, in particular to a large-diameter single-pile horizontal vibration analysis method considering axial force action.
Background
With the rapid development of bridge engineering and high-rise buildings in China, pile foundations are increasingly widely applied, and the stress analysis of foundation piles under complex multidirectional loads is increasingly emphasized. At present, when the problem of horizontal vibration dynamic response of a pile body is solved, a pile soil body is 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, and the Passternak foundation model is more suitable. The method is characterized in that a classic Bernoulli-Euler theory is adopted for a slender rod pile foundation model, and the theoretical model only considers the bending deformation of a pile body and ignores the influence of the shearing deformation. For the large-diameter pile, the influence of the shear deformation of the pile body under the action of axial and horizontal forces on the dynamic response of the pile body is particularly important, and the Timoshenko beam model adopted by the pile body is more suitable.
Disclosure of Invention
The invention aims to overcome the defect that the Winkler foundation model in the prior art ignores the shearing effect of soil and cannot reflect the continuity of soil among longitudinal layers; the Bernoulli-Euler theoretical model only considers the bending deformation of the pile body and neglects the defect of the influence of the shearing deformation, and the invention provides the large-diameter single-pile horizontal vibration analysis method considering the axial force action.
The technical scheme of the invention is as follows:
a large-diameter single pile horizontal vibration analysis method considering axial force action comprises the following steps:
s1: the following assumed conditions are introduced to establish a large-diameter single-pile horizontal vibration analysis model under axial and horizontal forces of the Passternak layered foundation: 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: assuming that the single-pile body is a round 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; the pile-soil coupling vibration model is assumed to meet the small deformation condition, and the pile-soil interface is in complete contact and has 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 Passternak foundation model theory, wherein the corresponding expression of the dynamic balance equation is as follows:
Figure BDA0002753284740000021
in the formula ,
Figure BDA0002753284740000022
respectively the horizontal displacement and the section corner of the pile body point of the jth layer; z is the direction of the pile foundation along the depth, and p is the upper mark representing the pile; a. thep、Gp、Ep、Ip、mpRespectively representing the sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the unit length mass of the pile body; t is time; n is a radical of0Is an axial force acting on the pile top; k' is a shear shape coefficient; the thickness, the rigidity coefficient, the damping coefficient and the foundation shear coefficient of the jth layer of soil are respectively hj、
Figure BDA0002753284740000023
And
Figure BDA0002753284740000024
B0calculated width of the stake for 0.9(1.5d +0.5)
Figure BDA0002753284740000025
And
Figure BDA0002753284740000026
determined by the following formula:
Figure BDA0002753284740000027
Figure BDA0002753284740000028
Figure BDA0002753284740000029
in the formula ,
Figure BDA00027532847400000210
the shear wave velocity of the jth layer of soil;
Figure BDA00027532847400000211
and
Figure BDA00027532847400000212
respectively the elastic modulus, density, damping coefficient and Poisson ratio of the jth layer of soil;
Figure BDA00027532847400000213
dimensionless frequency, ω is excitation circle frequency;
Figure BDA00027532847400000214
the shear layer thickness of the foundation soil of the jth layer is obtained by taking the value as
Figure BDA00027532847400000215
d is the pile diameter;
in steady-state vibration, the pile top is subjected to simple harmonic vibration, and the horizontal displacement and the corner of the pile body are simplified as follows:
Figure BDA00027532847400000216
in the formula ,
Figure BDA00027532847400000217
is the horizontal displacement amplitude of the pile body of the jth layer,
Figure BDA00027532847400000218
is the angle amplitude of the j section of the pile bodyiωtRepresenting a complex frequency domain;
the following equations are obtained by substituting the formula (5) for the formula (1), respectively:
Figure BDA0002753284740000031
in the formula ,
Mp＝EpIp，Jp＝K'ApGp，Sp＝ρpIpω2，
Figure BDA0002753284740000032
Figure BDA0002753284740000033
Figure BDA0002753284740000034
Figure BDA0002753284740000035
the characteristic root corresponding to the formula (6) is
Figure BDA0002753284740000036
Obtaining a displacement general solution of the j-th layer pile body horizontal displacement amplitude:
Figure BDA0002753284740000037
wherein ,
Figure BDA0002753284740000038
is unknownCoefficient Aj1、Bj1、Cj1、Dj1The value of (a) is determined by the boundary conditions of the pile top and the pile bottom;
s3: and solving the dynamic balance equation in the step S2 to obtain the horizontal vibration analysis parameters of the single pile in the laminar soil, wherein the parameters at least comprise the horizontal displacement, the bending moment, the shearing force and the section corner of the pile body.
Further, step S31: according to the displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer and the dynamic balance equation of the pile body unit, the section corner of the pile body is obtained, and the expression is as follows:
Figure BDA0002753284740000039
when the pile body does not generate shearing deformation, the formula (8) is degraded into
Figure BDA00027532847400000310
Substituting formula (7) for formula (8) to obtain a general solution of the cross-section corner:
Figure BDA00027532847400000311
step S32: determining the correlation among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body;
wherein, the relational expression of the bending moment of pile body and cross-section corner is:
Figure BDA00027532847400000312
the relational expression among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body is as follows:
Figure BDA0002753284740000041
order to
Figure BDA0002753284740000042
Figure BDA0002753284740000043
Figure BDA0002753284740000044
The undetermined coefficients in equations (9), (10) and (11) are expressed as:
Figure BDA0002753284740000045
in the formula ：
Figure BDA0002753284740000046
Figure BDA0002753284740000047
Figure BDA0002753284740000048
Figure BDA0002753284740000049
Figure BDA00027532847400000410
Figure BDA00027532847400000411
Aj2、Aj3、Aj4、Bj2、Bj3、Bj4、Cj2、Cj3、Cj4、Dj2、Dj3、Dj4for unknown coefficient, from pile top and pile bottomObtaining the boundary condition of the target;
Aj2、Aj3、Aj4、Bj2、Bj3、Bj4、Cj2、Cj3、Cj4、Dj2、Dj3、Dj4the unknown coefficient is obtained according to the boundary conditions of the pile top and the pile bottom;
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;
at the j section and j +1 section pile body section, the continuity expressions of the horizontal displacement, the corner, the bending moment and the shearing force of the pile body are as follows:
Figure BDA0002753284740000051
then combining equation (12) and equation (13) yields the following system of coefficient matrix equations:
Figure BDA0002753284740000052
in the formula
{Tj}＝[Aj1 Bj1 Cj1 Dj1]T；
Figure BDA0002753284740000053
Obtained from formula (14):
{Tj+1}＝[Fj+1(zj)]-1[Fj(zj)]{Tj} (15)
the coefficient matrix { T) corresponding to the m-th section of pile body is obtained through recursion relationmExpressed as:
Figure BDA0002753284740000054
step S34: setting boundary conditions of the pile top and the pile bottom, and according to the m-th section of pile body corresponding coefficient matrix { T }mSolving horizontal displacement, bending moment and shearing force of the pile body;
setting boundary conditions of the pile top and the pile bottom as follows:
Figure BDA0002753284740000055
the coefficient expression (12) is simplified by substituting formula (17) into:
Figure BDA0002753284740000056
Figure BDA0002753284740000061
determination of { T } in the combination of (16) to (18)1According to the m-th section of pile body corresponding coefficient matrix { T }mAnd obtaining horizontal displacement, bending moment and shearing force of each section of the pile body.
Further, after the step S34, the method further includes: introducing dimensionless parameters;
the dimensionless parameter expression is:
Figure BDA0002753284740000062
in the formula ,umax(z)、mmax(z)、pmaxAnd (z) is the horizontal vibration displacement, the bending moment and the maximum shear force of the pile foundation respectively.
According to the technical scheme, the large-diameter single pile is simplified into the Timoshenko beam model based on the Passternak foundation model, and the mechanical system model considering the shear deformation of the pile body and the shear effect of the soil body around the pile is provided. In addition, the influence of the axial force is simultaneously considered under the action of the horizontal load of the pile top, the action of the two loads is comprehensively considered, the problem of horizontal vibration of the large-diameter single pile under the action of complex multi-directional load can be solved, the pile-soil coupling interaction of the foundation pile under the action of complex load in the actual engineering can be better simulated, and a foundation can be laid for the pile foundation vibration theory; the invention considers the influence of the shearing deformation of the pile body and the shearing deformation of the soil body around the pile while considering the complex load action of the pile top, and also considers the influence factors of the stress deformation related to the pile foundation in the actual engineering.
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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 model diagram corresponding to the method of the present invention in the embodiment.
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, the structure shown in the drawings is not drawn to a general scale and is partially enlarged, deformed and simplified, so that the present invention should not be construed as limited thereto.
In order to solve the problems in the prior art, the method for analyzing the horizontal vibration of the large-diameter single pile considering the axial force action, as shown in fig. 1-2, is characterized by comprising the following steps:
a large-diameter single pile horizontal vibration analysis method considering axial force action comprises the following steps:
s1: the following assumed conditions are introduced to establish a large-diameter single-pile horizontal vibration analysis model under axial and horizontal forces of the Passternak layered foundation: 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: assuming that the single-pile body is a round 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; the pile-soil coupling vibration model is assumed to meet the small deformation condition, and the pile-soil interface is in complete contact and has no relative sliding; assuming that the pile bottom is fixed end constraint;
s2: the dynamic balance equation of the layered pile body unit is established according to the Timoshenko beam and Passternak foundation model theory, which is different from the prior art that: the dynamic balance equation considers the axial force acting on the pile top and the shearing effect of the soil body around the pile; the expression corresponding to the dynamic balance equation is as follows:
Figure BDA0002753284740000071
in the formula ,
Figure BDA0002753284740000072
respectively the horizontal displacement and the section corner of the pile body point of the jth layer; z is the direction of the pile foundation along the depth, and p is the upper mark representing the pile; a. thep、Gp、Ep、Ip、mpRespectively representing the sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the unit length mass of the pile body; t is time; n is a radical of0Is an axial force acting on the pile top; k' is a shear shape coefficient; the thickness, the rigidity coefficient, the damping coefficient and the foundation shear coefficient of the jth layer of soil are respectively hj、
Figure BDA0002753284740000073
And
Figure BDA0002753284740000074
B0calculated width of the stake for 0.9(1.5d +0.5)
Figure BDA0002753284740000075
And
Figure BDA0002753284740000076
determined by the following formula:
Figure BDA0002753284740000077
Figure BDA0002753284740000078
Figure BDA0002753284740000079
in the formula ,
Figure BDA0002753284740000081
the shear wave velocity of the jth layer of soil;
Figure BDA0002753284740000082
and
Figure BDA0002753284740000083
respectively the elastic modulus, density, damping coefficient and Poisson ratio of the jth layer of soil;
Figure BDA0002753284740000084
dimensionless frequency, ω is excitation circle frequency;
Figure BDA0002753284740000085
the shear layer thickness of the foundation soil of the jth layer is obtained by taking the value as
Figure BDA0002753284740000086
d is the pile diameter;
in steady state vibration, the horizontal displacement and the section corner of the pile body in a time domain range are converted into the horizontal displacement and the section corner of the pile body in a complex frequency domain range, and the expression is as follows:
Figure BDA0002753284740000087
in the formula ,
Figure BDA0002753284740000088
is the horizontal displacement amplitude of the pile body of the jth layer,
Figure BDA0002753284740000089
is the angle amplitude of the j section of the pile bodyiωtRepresenting a complex frequency domain;
substituting equation (5) for equation (1) respectively yields the following equations:
Figure BDA00027532847400000810
in the formula ,
Mp＝EpIp，Jp＝K'ApGp，Sp＝ρpIpω2，
Figure BDA00027532847400000811
Figure BDA00027532847400000812
Figure BDA00027532847400000813
the characteristic root corresponding to the formula (6) is
Figure BDA00027532847400000814
Then the displacement general solution of the horizontal displacement amplitude of the j-th layer pile body can be obtained:
Figure BDA00027532847400000815
wherein ,
Figure BDA00027532847400000816
unknown coefficient Aj1、Bj1、Cj1、Dj1The value of (a) is determined by the boundary conditions of the pile top and the pile bottom;
s3: and solving the dynamic balance equation in the step S2 to obtain the horizontal vibration analysis parameters of the single pile in the layered soil, wherein the parameters at least comprise the horizontal displacement, the bending moment, the shearing force and the section corner of the pile body.
In this embodiment, in step S3, the process of solving the dynamic balance equation in step S2 to obtain the horizontal vibration analysis parameters of the single pile in the layered soil includes the following steps:
step S31: according to the displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer and the dynamic balance equation of the pile body unit, the section corner of the pile body is obtained, and the expression is as follows:
Figure BDA0002753284740000091
when the pile body does not generate shearing deformation, the formula (8) can be degraded into
Figure BDA0002753284740000092
The general solution of the cross-sectional angle obtained by substituting formula (7) for formula (8) is:
Figure BDA0002753284740000093
step S32: determining the correlation among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body;
wherein, the relational expression of the bending moment of pile body and cross-section corner is:
Figure BDA0002753284740000094
the relational expression among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body is as follows:
Figure BDA0002753284740000095
order to
Figure BDA0002753284740000096
Figure BDA0002753284740000097
Then formula(9) The undetermined coefficients in (10), (11) and (10) can be expressed as:
Figure BDA0002753284740000098
in the formula ：
Figure BDA0002753284740000101
Figure BDA0002753284740000102
Figure BDA0002753284740000103
Figure BDA0002753284740000104
Figure BDA0002753284740000105
Figure BDA0002753284740000106
Aj2、Aj3、Aj4、Bj2、Bj3、Bj4、Cj2、Cj3、Cj4、Dj2、Dj3、Dj4the unknown coefficient can be obtained according to the boundary conditions of the pile top and the pile bottom;
Aj2、Aj3、Aj4、Bj2、Bj3、Bj4、Cj2、Cj3、Cj4、Dj2、Dj3、Dj4the unknown coefficient can be obtained according to the boundary conditions of the pile top and the pile bottom;
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;
at the j section and j +1 section pile body section, the continuity expressions of the horizontal displacement, the corner, the bending moment and the shearing force of the pile body are as follows:
Figure BDA0002753284740000107
then the system of coefficient matrix equations obtained by combining equation (12) and equation (13) is as follows:
Figure BDA0002753284740000108
in the formula
{Tj}＝[Aj1 Bj1 Cj1 Dj1]T；
Figure BDA0002753284740000109
From formula (14):
{Tj+1}＝[Fj+1(zj)]-1[Fj(zj)]{Tj} (15)
the coefficient matrix { T } corresponding to the mth section of pile body can be obtained through the recursion relationmCan be expressed as:
Figure BDA0002753284740000111
step S34: setting boundary conditions of the pile top and the pile bottom, and according to the m-th section of pile body corresponding coefficient matrix { T }mSolving horizontal displacement, bending moment and shearing force of the pile body;
setting boundary conditions of the pile top and the pile bottom as follows:
Figure BDA0002753284740000112
the coefficient expression (12) is simplified by substituting the formula (17):
Figure BDA0002753284740000113
Figure BDA0002753284740000114
the simultaneous type (16) - (18) can then determine { T }1According to the m-th section of pile body corresponding coefficient matrix { T }mAnd obtaining horizontal displacement, bending moment and shearing force of each section of the pile body.
In this embodiment, after step S34, the method further includes: introducing dimensionless parameters to facilitate subsequent analysis; the dimensionless parameter expression is:
Figure BDA0002753284740000115
in the formula ,umax(z)、mmax(z)、pmaxAnd (z) is the horizontal vibration displacement, the bending moment and the maximum shear force of the pile foundation respectively.
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 (3)

1. A large-diameter single pile horizontal vibration analysis method considering axial force action is characterized by comprising the following steps:
s1: the following assumed conditions are introduced to establish a large-diameter single-pile horizontal vibration analysis model under axial and horizontal forces of the Passternak layered foundation: 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: assuming that the single-pile body is a round 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; the pile-soil coupling vibration model is assumed to meet the small deformation condition, and the pile-soil interface is in complete contact and has 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 Passternak foundation model theory, wherein the corresponding expression of the dynamic balance equation is as follows:
Figure FDA0002753284730000011
in the formula ,
Figure FDA0002753284730000012
respectively the horizontal displacement and the section corner of the pile body point of the jth layer; z is the direction of the pile foundation along the depth, and p is the upper mark representing the pile; a. thep、Gp、Ep、Ip、mpRespectively representing the sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the unit length mass of the pile body; t is time; n is a radical of0Is an axial force acting on the pile top; k' is a shear shape coefficient; the thickness, the rigidity coefficient, the damping coefficient and the foundation shear coefficient of the jth layer of soil are respectively hj、
Figure FDA0002753284730000013
And
Figure FDA0002753284730000014
B0calculated width of the stake for 0.9(1.5d +0.5)
Figure FDA0002753284730000015
And
Figure FDA0002753284730000016
determined by the following formula:
Figure FDA0002753284730000017
Figure FDA0002753284730000018
Figure FDA0002753284730000019
in the formula ,
Figure FDA00027532847300000110
the shear wave velocity of the jth layer of soil;
Figure FDA00027532847300000111
and
Figure FDA00027532847300000112
respectively the elastic modulus, density, damping coefficient and Poisson ratio of the jth layer of soil;
Figure FDA00027532847300000113
dimensionless frequency, ω is excitation circle frequency;
Figure FDA0002753284730000021
the shear layer thickness of the foundation soil of the jth layer is obtained by taking the value as
Figure FDA0002753284730000022
d is the pile diameter;
in steady-state vibration, the pile top is subjected to simple harmonic vibration, and the horizontal displacement and the corner of the pile body are simplified as follows:
Figure FDA0002753284730000023
in the formula ,
Figure FDA0002753284730000024
is the horizontal displacement amplitude of the pile body of the jth layer,
Figure FDA0002753284730000025
and the amplitude of the section corner of the j section of the pile body is shown.
The following equations are obtained by substituting the formula (5) for the formula (1), respectively:
Figure FDA0002753284730000026
in the formula ,
Mp＝EpIp，Jp＝K'ApGp，Sp＝ρpIpω2，
Figure FDA0002753284730000027
Figure FDA0002753284730000028
the characteristic root corresponding to the formula (6) is
Figure FDA0002753284730000029
Obtaining a displacement general solution of the j-th layer pile body horizontal displacement amplitude:
Figure FDA00027532847300000210
wherein ,
Figure FDA00027532847300000211
unknown coefficient Aj1、Bj1、Cj1、Dj1The value of (A) is determined by the boundary conditions of the pile top and the pile bottomDetermining;
s3: and solving the dynamic balance equation in the step S2 to obtain the horizontal vibration analysis parameters of the single pile in the laminar soil, wherein the parameters at least comprise the horizontal displacement, the bending moment, the shearing force and the section corner of the pile body.
2. The analysis method as claimed in claim 1, wherein in the step S3, the process of solving the dynamic balance equation in the step S2 to obtain the analysis parameters of horizontal vibration of the single pile in the stratified soil comprises the following steps:
step S31: according to the displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer and the dynamic balance equation of the pile body unit, the section corner of the pile body is obtained, and the expression is as follows:
Figure FDA00027532847300000212
when the pile body does not generate shearing deformation, the formula (8) is degraded into
Figure FDA0002753284730000031
Substituting formula (7) for formula (8) to obtain a general solution of the cross-section corner:
Figure FDA0002753284730000032
step S32: determining the correlation among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body;
wherein, the relational expression of the bending moment of pile body and cross-section corner is:
Figure FDA0002753284730000033
the relational expression among the bending moment, the shearing force, the horizontal displacement and the section corner of the pile body is as follows:
Figure FDA0002753284730000034
order to
Figure FDA0002753284730000035
Figure FDA0002753284730000036
The undetermined coefficients in equations (9), (10) and (11) are expressed as:
Figure FDA0002753284730000037
in the formula ：
Figure FDA0002753284730000038
Figure FDA0002753284730000039
Figure FDA0002753284730000041
Figure FDA0002753284730000042
Figure FDA0002753284730000043
Figure FDA0002753284730000044
Aj2、Aj3、Aj4、Bj2、Bj3、Bj4、Cj2、Cj3、Cj4、Dj2、Dj3、Dj4the unknown coefficient is obtained according to the boundary conditions of the pile top and the pile bottom;
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;
at the j section and j +1 section pile body section, the continuity expressions of the horizontal displacement, the corner, the bending moment and the shearing force of the pile body are as follows:
Figure FDA0002753284730000045
then combining equation (12) and equation (13) yields the following system of coefficient matrix equations:
Figure FDA0002753284730000048
in the formula
{Tj}＝[Aj1 Bj1 Cj1 Dj1]T；
Figure FDA0002753284730000046
Obtained from formula (14):
{Tj+1}＝[Fj+1(zj)]-1[Fj(zj)]{Tj} (15)
the coefficient matrix { T) corresponding to the m-th section of pile body is obtained through recursion relationmExpressed as:
Figure FDA0002753284730000047
step (ii) ofS34: setting boundary conditions of the pile top and the pile bottom, and according to the m-th section of pile body corresponding coefficient matrix { T }mSolving horizontal displacement, bending moment and shearing force of the pile body;
setting boundary conditions of the pile top and the pile bottom as follows:
Figure FDA0002753284730000051
the coefficient expression (12) is simplified by substituting formula (17) into:
Figure FDA0002753284730000052
Figure FDA0002753284730000053
the simultaneous type (16), (17), (18a), (18b) determines { T }1According to the m-th section of pile body corresponding coefficient matrix { T }mAnd obtaining horizontal displacement, bending moment and shearing force of each section of the pile body.
3. The analysis method according to claim 2, wherein after the step S34, the method further comprises: introducing dimensionless parameters;
the dimensionless parameter expression is:
Figure FDA0002753284730000054
in the formula ,umax(z)、mmax(z)、pmaxAnd (z) is the horizontal vibration displacement, the bending moment and the maximum shear force of the pile foundation respectively.
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