CN116305480A - Single pile horizontal dynamic response analysis method under seismic S shear wave effect - Google Patents

Single pile horizontal dynamic response analysis method under seismic S shear wave effect Download PDF

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CN116305480A
CN116305480A CN202310303826.4A CN202310303826A CN116305480A CN 116305480 A CN116305480 A CN 116305480A CN 202310303826 A CN202310303826 A CN 202310303826A CN 116305480 A CN116305480 A CN 116305480A
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崔春义
辛宇
王本龙
张鹏
刘海龙
王坤鹏
梁志孟
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Dalian Maritime University
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Abstract

The embodiment of the invention discloses a single pile horizontal dynamic response analysis method under the action of seismic S shear waves, which comprises the following steps: s1, creating a soil dynamic balance equation under the influence of foundation soil on seismic S shear waves based on foundation soil characteristic parameters; s2, combining an Euler beam and a Pasternak foundation model to create a dynamic balance equation of the pile body unit; s3, determining the relation between the pile body rotation angle, bending moment, shearing force and the pile body horizontal displacement coefficient and creating a corresponding coefficient equation set; s4, determining a dynamic balance equation corresponding to the pile body displacement and the internal force coefficient of the pile bottom; and S5, determining a dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient, and further completing analysis of a single pile horizontal dynamic response process under the action of the seismic S shear wave. According to the invention, the soil around the pile is simplified into the Pasternak foundation, and the pile foundation horizontal vibration stress analysis of the seismic S shear wave propagation in the layered soil under the action of the axial force of the pile top is considered, so that theoretical guidance and reference effects can be provided for the seismic design of pile foundation engineering.

Description

Single pile horizontal dynamic response analysis method under seismic S shear wave effect
Technical Field
The invention relates to the technical field of pile foundation vibration analysis, in particular to a single pile horizontal dynamic response analysis method under the action of seismic S shear waves.
Background
In recent years, pile foundations are often damaged in past earthquakes. When an earthquake occurs, the pile foundation is subjected to the action of earthquake waves in foundation soil to generate vibration, and meanwhile, the vibration of the pile also changes the movement of the foundation soil around the pile. Pile-soil movement interactions are key factors in studying pile foundation dynamic response under seismic action.
Therefore, more and more students are researching the pile foundation horizontal dynamic response under the action of earthquake waves. The dynamic response study of piles under transverse seismic loading is generally divided into numerical solutions and analytical solutions. Numerical solutions consume significant computer resources and time and do not guarantee mathematical stringency. The analysis method can simply reveal that the soil shear deformation has less influence on the calculation result, but most of the current researches only simplify the pile surrounding soil into a Winkler foundation in the vibration analysis process of the pile foundation, and do not fully consider the influence factors of the shear deformation, in particular the influence factors of the seismic wave action attribute and the pile-soil interaction system.
Disclosure of Invention
Based on the method, in order to solve the defects existing in the prior art, a single pile horizontal dynamic response analysis method under the action of seismic S shear waves is specifically provided.
The method for analyzing the horizontal dynamic response of the single pile under the action of the earthquake S shear wave is characterized by comprising the following steps of:
s1, creating a ground soil dynamic balance equation under the influence of foundation soil on seismic S shear waves based on foundation soil characteristic parameters, wherein the foundation soil characteristic parameters comprise elastic shear modulus, viscous damping coefficient, mass density and horizontal displacement;
s2, based on the land dynamic balance equation, combining an Euler beam and a Pasternak foundation model to create a dynamic balance equation of the pile body unit;
s3, determining the relation between the pile body rotation angle, the bending moment, the shearing force and the pile body horizontal displacement coefficient based on a dynamic balance equation of the pile body unit, and creating a corresponding coefficient equation set;
s4, determining a dynamic balance equation corresponding to the pile bottom and pile body displacement and the internal force coefficient based on the coefficient equation set;
and S5, determining a dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient, and further completing analysis of a single pile horizontal dynamic response process under the action of the seismic S shear wave.
Optionally, in one embodiment, the step of creating a soil dynamic balance equation in S1 based on the foundation soil characteristic parameter under the influence of the foundation soil on the seismic S shear wave includes:
s11, determining characteristic parameters of each soil layer of the foundation soil, wherein the characteristic parameters comprise elastic shear modulus
Figure BDA0004146056460000021
Viscous damping coefficient->
Figure BDA0004146056460000022
And mass Density->
Figure BDA0004146056460000023
Horizontal displacement +.>
Figure BDA0004146056460000024
S12, based on the characteristic parameters, creating a free field soil dynamic balance equation, wherein the corresponding free field soil dynamic balance equation is as follows
Figure BDA0004146056460000025
In the method, in the process of the invention,
Figure BDA0004146056460000026
the shear stress is given, t is time, and z is soil layer depth;
due to
Figure BDA0004146056460000027
Expressed as:
Figure BDA0004146056460000028
viscous damping coefficient of viscoelastic material
Figure BDA0004146056460000029
Omega is the frequency of the excitation circle,
then substituting formula (2) into formula (1) to obtain:
Figure BDA00041460564600000210
s13, based on horizontal displacement
Figure BDA00041460564600000211
The displacement steady state solution in the frequency domain, the deformation of formula (3) is performed, the horizontal displacement +.>
Figure BDA00041460564600000212
The displacement steady state solution in the frequency domain is: />
Figure BDA00041460564600000213
Substituting into (3) to obtain:
Figure BDA00041460564600000214
s14, creating a displacement amplitude of the j-th layer to horizontally displace under the action of the upward and downward shear waves at the depth z
Figure BDA00041460564600000215
Time-independent partial solutions corresponding to the equation
Figure BDA0004146056460000031
Wherein, complex shear modulus
Figure BDA0004146056460000032
C 1 、C 2 The coefficients to be solved;
s15, acquiring an exponential equation of the formula (5), wherein the corresponding equation is as follows:
Figure BDA0004146056460000033
wherein N-1 is greater than or equal to j is greater than or equal to 1, N represents the number of layers of the base stratum, E j 、F j As the amplitude coefficient of the wave,
Figure BDA0004146056460000034
wave number;
s16, creating a conversion relation equation between the top layer, namely the 1 st layer soil and any layer of amplitude coefficient, wherein the corresponding equation creation process is as follows:
due to wave number
Figure BDA0004146056460000035
Complex shear wave velocity +.>
Figure BDA0004146056460000036
Then the formula (6) is
Figure BDA0004146056460000037
Figure BDA0004146056460000038
Substituting the formulas (7) and (8) into the formula (1) to obtain the product:
Figure BDA0004146056460000039
since the shear stress and displacement are continuous at the interface of the j-th layer and the j+1-th layer of soil, namely:
Figure BDA00041460564600000310
h j refers to the thickness of each layer of soil;
substituting the formula (6) and the formula (9) into the formula (10) to obtain:
Figure BDA00041460564600000311
the corresponding recurrence formula is determined as follows:
Figure BDA00041460564600000312
in the method, in the process of the invention,
Figure BDA0004146056460000041
defined as wave number impedance, let the coefficient matrix (2×2) obtained in the above formula be T j The conversion relation equation between the top layer (layer 1 soil) and any layer of amplitude coefficient is obtained as follows:
Figure BDA0004146056460000042
wherein the transfer function
Figure BDA0004146056460000043
Wherein N-1 is greater than or equal to j is greater than or equal to 1, N represents the number of layers of the base stratum, E j 、F j As the amplitude coefficient of the wave,
s17, creating a land dynamic balance equation under the influence of foundation soil considered on seismic S shear waves, wherein the corresponding equation creation process is as follows:
according to the first layer free-ground boundary conditions: τ 1 (0) =0, get E 1 ＝F 1 Let n=n-1, to obtain:
Figure BDA0004146056460000044
and
Figure BDA0004146056460000045
substituting the above formula into formula (6):
Figure BDA0004146056460000046
optionally, in one embodiment, the dynamic balance equation of the pile body unit in S2 is
Figure BDA0004146056460000047
In the method, in the process of the invention,
Figure BDA0004146056460000048
for horizontal displacement of the unit mass point of the j-th pile body>
Figure BDA0004146056460000049
Is free field movement displacement; e (E) p 、I p 、m p The pile body elastic modulus, the section moment of inertia and the mass per unit length are respectively; />
Figure BDA00041460564600000410
And->
Figure BDA00041460564600000411
The rigidity coefficient, the damping coefficient and the foundation shear rigidity of the foundation around the j-th layer of piles are respectively; b (B) 0 =0.9 (1.5d+0.5) is the calculated width of the pile, ε is the reduction coefficient considering the pile and soil release, and 0.6 to 0.9 is taken.
Optionally, in one embodiment, the step of determining the relationship between the pile body rotation angle, the bending moment, the shearing force and the pile body horizontal displacement coefficient in S3 and creating a corresponding coefficient equation set includes;
each parameter is
Figure BDA0004146056460000051
And->
Figure BDA0004146056460000052
The method is determined according to the following formula:
Figure BDA0004146056460000053
Figure BDA0004146056460000054
Figure BDA0004146056460000055
Figure BDA0004146056460000056
in U 0 Maximum displacement for ground movement; pile surrounding soil shearing wave number
Figure BDA0004146056460000057
Shear wave velocity of pile surrounding soil
Figure BDA0004146056460000058
Figure BDA0004146056460000059
And->
Figure BDA00041460564600000510
The modulus of elasticity, density, damping coefficient and poisson ratio of the surrounding soil of the pile are respectively; dimensionless frequency->
Figure BDA00041460564600000511
Figure BDA00041460564600000512
Is the thickness of foundation soil shear layer +.>
Figure BDA00041460564600000513
d refers to the diameter of the pile;
since the pile horizontal displacement is expressed as:
Figure BDA00041460564600000514
in the method, in the process of the invention,
Figure BDA00041460564600000515
for the horizontal displacement amplitude of the pile body mass point,
let W p ＝E p I p ，
Figure BDA00041460564600000516
Substituting the formula (21) into the formula (16) respectively further yields the following equation:
Figure BDA00041460564600000517
in the method, in the process of the invention,
Figure BDA00041460564600000518
E p 、I p 、B 0 respectively the elastic modulus of the pile body, the section moment of inertia of the pile and the calculated width W of the pile p To define the coefficient, N 0 The axial acting force of the pile top is;
obviously, the homogeneous solution corresponding to formula (22) is:
Figure BDA00041460564600000519
in the method, in the process of the invention,
Figure BDA00041460564600000520
coefficient->
Figure BDA00041460564600000521
The value is determined by the set boundary condition;
and the special solution of formula (23) is:
Figure BDA0004146056460000061
the full solution of equation (23) is expressed as:
Figure BDA0004146056460000062
based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement is:
Figure BDA0004146056460000063
Figure BDA0004146056460000064
Figure BDA0004146056460000065
the coefficient equation set obtained by integrating the equation (25) and the equation (26) is as follows:
Figure BDA0004146056460000066
in the method, in the process of the invention,
Figure BDA0004146056460000067
Figure BDA0004146056460000068
when j=1 and z=0, substituting formula (27) to obtain the transfer integral constant of the first pile body is:
Figure BDA0004146056460000071
the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile are continuous at the sections of the j-th section and the j-1-th section, namely:
Figure BDA0004146056460000072
the coefficient matrix equation set obtained by integrating the equation (27), the equation (28) and the equation (29) is as follows:
Figure BDA0004146056460000073
optionally, in one embodiment, the step of determining, in S4, a dynamic balance equation corresponding to the pile body displacement and the internal force coefficient based on the coefficient equation set includes:
synthesizing the formula (30), the formula (25) and the formula (26) to obtain the displacement of the j-th pile bottom and the internal force transfer equation
Figure BDA0004146056460000074
Further according to the continuous condition of pile section boundary, the comprehensive expression equation of pile bottom and pile body displacement and internal force is calculated as follows:
Figure BDA0004146056460000075
considering pile bottom boundary conditions, further generalizing the equation (32) to obtain:
Figure BDA0004146056460000081
in the method, in the process of the invention,
Figure BDA0004146056460000082
Figure BDA0004146056460000083
optionally, in one embodiment, the step of determining the dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient in S5 includes:
definition of the definition
Figure BDA0004146056460000084
Further solving a dynamic balance equation corresponding to pile top displacement and rotation angle by the formula (33):
Figure BDA0004146056460000085
Figure BDA0004146056460000086
the implementation of the embodiment of the invention has the following beneficial effects:
according to the pile foundation horizontal dynamic response analysis method based on the S shear wave action of the earthquake S shear wave of the foundation of the Pasternak, the pile periphery soil body is simplified into the foundation of the Pasternak by adopting an analysis method, pile foundation horizontal vibration stress analysis of the earthquake S shear wave propagation in the layered soil under the action of axial force of the pile top is considered, the constraint action of the pile periphery soil body on the pile body can be better simulated, the pile foundation horizontal vibration dynamic response problem under the action of the earthquake S shear wave can be solved, and theoretical guidance and reference actions can be provided for pile foundation engineering earthquake-proof design.
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In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
Wherein:
FIG. 1 is a flow chart of steps corresponding to the method described in one embodiment;
FIG. 2 is a schematic diagram of vibration analysis corresponding to the method described in one embodiment;
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. It will be understood that the terms first, second, etc. as used herein may be used to describe various elements, but these elements are not limited by these terms. These terms are only used to distinguish one element from another element. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element, without departing from the scope of the present application. Both the first element and the second element are elements, but they are not the same element.
When the applicant performs research on the horizontal dynamic response of the pile foundation under the action of the seismic waves, the foundation soil is found to have the action attribute of amplifying (or shrinking) the seismic waves, and in view of the characteristics, how to reasonably and effectively process the foundation rock input seismic waves to consider the influence of the foundation rock input seismic waves on a pile-soil interaction system; based on the foregoing design concept, in this embodiment, a method for analyzing horizontal dynamic response of a single pile under the action of an earthquake S shear wave is specifically provided, as shown in fig. 1-2, where the method includes:
s1, creating a ground soil dynamic balance equation under the influence of foundation soil on seismic S shear waves based on foundation soil characteristic parameters, wherein the foundation soil characteristic parameters comprise elastic shear modulus, viscous damping coefficient, mass density and horizontal displacement;
s2, based on the land dynamic balance equation, combining an Euler beam and a Pasternak foundation model to create a dynamic balance equation of the pile body unit;
s3, determining the relation between the pile body rotation angle, the bending moment, the shearing force and the pile body horizontal displacement coefficient based on a dynamic balance equation of the pile body unit, and creating a corresponding coefficient equation set;
s4, determining a dynamic balance equation corresponding to the pile bottom and pile body displacement and the internal force coefficient based on the coefficient equation set;
and S5, determining a dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient, and further completing analysis of a single pile horizontal dynamic response process under the action of the seismic S shear wave.
From the above, the foundation soil dynamic balance equation under the influence of foundation soil on earthquake S shear wave is constructed to add the influence factors of foundation rock on earthquake waves into the dynamic balance equation, and the dynamic balance equation corresponding to pile bottom pile body displacement, internal force coefficient and pile top displacement and corner coefficient is established by combining the Euler beam and the Pasternak foundation model, so that the pile-soil interaction analysis process in single pile horizontal dynamic response under the influence of earthquake S shear wave is completed.
In some specific embodiments, in step S1,
elastic shear modulus for setting characteristics of each soil layer
Figure BDA0004146056460000101
Viscous damping coefficient->
Figure BDA0004146056460000102
And mass Density->
Figure BDA0004146056460000103
Horizontal displacement
Figure BDA0004146056460000104
Expressed, the free-site earth dynamic balance equation can be expressed as:
elastic shear modulus for setting characteristics of each soil layer
Figure BDA0004146056460000105
Viscous damping coefficient->
Figure BDA0004146056460000106
And mass Density->
Figure BDA0004146056460000107
Horizontal displacement
Figure BDA0004146056460000108
Expressed, the free-site earth dynamic balance equation can be expressed as:
Figure BDA0004146056460000109
in the method, in the process of the invention,
Figure BDA00041460564600001010
the shear stress is given, t is time, and z is soil layer depth;
due to
Figure BDA00041460564600001011
Expressed as:
Figure BDA00041460564600001012
viscous damping coefficient of viscoelastic material
Figure BDA00041460564600001013
Omega is the frequency of the excitation circle,
then substituting formula (2) into formula (1) to obtain:
Figure BDA00041460564600001014
based on horizontal displacement
Figure BDA00041460564600001015
Performing a deformation of equation (3) by a displacement steady state solution in the frequency domain, the horizontal displacement
Figure BDA00041460564600001016
The displacement steady state solution in the frequency domain is: />
Figure BDA00041460564600001017
Substituting into (3) to obtain:
Figure BDA0004146056460000111
then substituting formula (2) into formula (1) to obtain:
Figure BDA0004146056460000112
creating a displacement amplitude of the j-th layer to horizontally displace under the action of the upward and downward shear waves at the depth z
Figure BDA0004146056460000113
Time-independent partial solutions corresponding to the equation
Figure BDA0004146056460000114
Wherein, complex shear modulus
Figure BDA0004146056460000115
Obtaining an exponential form equation of formula (5), the corresponding equation being:
Figure BDA0004146056460000116
the first term and the second term on the right of the equation can be regarded as the upward and downward dynamic solutions in the media of the j-th layer (N-1 is larger than or equal to j is larger than or equal to 1, N represents the base stratum), E j 、F j Is the amplitude coefficient;
Figure BDA0004146056460000117
wave number;
the conversion relation equation between the top layer, namely the 1 st layer soil and any layer of amplitude coefficient is established, and the corresponding equation establishment process is as follows:
due to wave number
Figure BDA0004146056460000118
Complex shear wave velocity +.>
Figure BDA0004146056460000119
Then the formula (6) is
Figure BDA00041460564600001110
Figure BDA00041460564600001111
Substituting the formulas (7) and (8) into the formula (1) to obtain the product:
Figure BDA00041460564600001112
since the shear stress and displacement are continuous at the interface of the j-th layer and the j+1-th layer of soil, namely:
Figure BDA00041460564600001113
substituting the formula (6) and the formula (9) into the formula (10) to obtain:
Figure BDA0004146056460000121
the corresponding recurrence formula is determined as follows:
Figure BDA0004146056460000122
in the method, in the process of the invention,
Figure BDA0004146056460000123
defined as wave number impedance, let the coefficient matrix (2×2) obtained in the above formula be T j The conversion relation equation between the top layer (layer 1 soil) and any layer of amplitude coefficient is obtained as follows:
Figure BDA0004146056460000124
wherein the transfer function
Figure BDA0004146056460000125
From the above, it can be seen that for a given angular frequency, the transfer function
Figure BDA0004146056460000129
Depending onSoil properties of the soil layer from the first layer to the nth layer are independent of seismic input. Therefore, a land dynamic balance equation under the influence of foundation soil to seismic S shear waves is created, and the corresponding equation creation process is as follows:
according to the first layer free-ground boundary conditions: τ 1 (0) =0, get E 1 ＝F 1 Let n=n-1, to obtain:
Figure BDA0004146056460000126
and
Figure BDA0004146056460000127
substituting the above formula into formula (6):
Figure BDA0004146056460000128
in some specific embodiments, in step S2, the power balance equation of the pile body unit is obtained by integrating the theory related to the Euler beam and the masterak foundation model as follows:
Figure BDA0004146056460000131
in the method, in the process of the invention,
Figure BDA0004146056460000132
for horizontal displacement of the mass point of the j-th section pile body unit of the driving pile I, < ->
Figure BDA0004146056460000133
Is free field movement displacement; e (E) p 、I p 、m p The pile body elastic modulus, the section moment of inertia and the mass per unit length are respectively; />
Figure BDA0004146056460000134
And->
Figure BDA0004146056460000135
The rigidity coefficient, the damping coefficient and the foundation shear rigidity of the foundation around the j-th layer of piles are respectively; b (B) 0 =0.9 (1.5d+0.5) is the calculated width of the pile, ε is the reduction coefficient considering the pile to be separated from the soil, and is generally 0.6 to 0.9.
In some specific embodiments, in step S3:
the method comprises the steps of determining the relation between the pile body rotation angle, bending moment, shearing force and the pile body horizontal displacement coefficient and creating a corresponding coefficient equation set;
each parameter is
Figure BDA0004146056460000136
And->
Figure BDA0004146056460000137
The method is determined according to the following formula:
Figure BDA0004146056460000138
Figure BDA0004146056460000139
Figure BDA00041460564600001310
Figure BDA00041460564600001311
in U 0 Maximum displacement for ground movement; pile surrounding soil shearing wave number
Figure BDA00041460564600001312
Shear wave velocity of pile surrounding soil
Figure BDA00041460564600001313
Figure BDA00041460564600001314
And->
Figure BDA00041460564600001315
The modulus of elasticity, density, damping coefficient and poisson ratio of the surrounding soil of the pile are respectively; dimensionless frequency->
Figure BDA00041460564600001316
Figure BDA00041460564600001317
The value of the foundation soil shear layer thickness is +.>
Figure BDA00041460564600001318
Since the pile horizontal displacement is expressed as:
Figure BDA00041460564600001319
in the method, in the process of the invention,
Figure BDA00041460564600001320
for the horizontal displacement amplitude of the pile body mass point,
let W p ＝E p I p ，
Figure BDA00041460564600001321
Substituting the formula (21) into the formula (16) respectively further yields the following equation:
Figure BDA00041460564600001322
in the method, in the process of the invention,
Figure BDA0004146056460000141
obviously, the homogeneous solution corresponding to formula (22) is:
Figure BDA0004146056460000142
in the method, in the process of the invention,
Figure BDA0004146056460000143
coefficient->
Figure BDA0004146056460000144
The value is determined by the set boundary condition;
and the special solution of formula (23) is:
Figure BDA0004146056460000145
the full solution of equation (23) is expressed as:
Figure BDA0004146056460000146
based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement is:
Figure BDA0004146056460000147
/>
Figure BDA0004146056460000148
Figure BDA0004146056460000149
the coefficient equation set obtained by integrating the equation (25) and the equation (26) is as follows:
Figure BDA00041460564600001410
in the method, in the process of the invention,
Figure BDA00041460564600001411
Figure BDA0004146056460000151
when j=1 and z=0, obtaining the related parameters in the first layer of soil, and substituting the related parameters into formula (27) to obtain the transfer integral constant of the first section of pile body, wherein the transfer integral constant is as follows:
Figure BDA0004146056460000152
the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile are continuous at the sections of the j-th section and the j-1-th section, namely:
Figure BDA0004146056460000153
the coefficient matrix equation set obtained by integrating the equation (27), the equation (28) and the equation (29) is as follows:
Figure BDA0004146056460000154
based on the coefficient equation set, the step of determining a dynamic balance equation corresponding to the pile bottom and pile body displacement and the internal force coefficient comprises the following steps:
synthesizing the formula (30), the formula (25) and the formula (26) to obtain the displacement of the j-th pile bottom and the internal force transfer equation
Figure BDA0004146056460000155
Further according to the continuous condition of pile section boundary, the comprehensive expression equation of pile bottom and pile body displacement and internal force is calculated as follows:
Figure BDA0004146056460000161
considering pile bottom boundary conditions, further generalizing the equation (32) to obtain:
Figure BDA0004146056460000162
in the method, in the process of the invention,
Figure BDA0004146056460000163
Figure BDA0004146056460000164
the step of determining the dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient comprises the following steps: selecting a corresponding matrix
Figure BDA0004146056460000165
Figure BDA0004146056460000166
Is a 4 x 4 matrix, and the coefficients of the rows and columns in the matrix are selected according to the requirement, for example, the subscript 32 is selected, namely, the system of the 3 rd row and the 2 nd column in the matrix is selected;
definition of the definition
Figure BDA0004146056460000167
Figure BDA0004146056460000168
Further solving a dynamic balance equation corresponding to pile top displacement and rotation angle by the formula (33): />
Figure BDA0004146056460000169
Figure BDA00041460564600001610
The implementation of the embodiment of the invention has the following beneficial effects:
the pile surrounding soil is equivalent to a layered Pasternak foundation, and the pile foundation is simplified to an Euler beam model, so that the pile foundation horizontal dynamic response analysis method based on the Pasternak foundation model under the action of seismic S shear waves is applied to the field of pile foundation vibration research for the first time.
The above examples only represent a few embodiments of the present application, which are described in more detail and are not to be construed as limiting the scope of the present application. It should be noted that it would be apparent to those skilled in the art that various modifications and improvements could be made without departing from the spirit of the present application, which would be within the scope of the present application. Accordingly, the scope of protection of the present application is to be determined by the claims appended hereto.

Claims (5)

1. The method for analyzing the horizontal dynamic response of the single pile under the action of the earthquake S shear wave is characterized by comprising the following steps of:
s1, creating a ground soil dynamic balance equation under the influence of foundation soil on seismic S shear waves based on foundation soil characteristic parameters, wherein the foundation soil characteristic parameters comprise elastic shear modulus, viscous damping coefficient, mass density and horizontal displacement;
s2, based on the land dynamic balance equation, combining an Euler beam and a Pasternak foundation model to create a dynamic balance equation of the pile body unit;
s3, determining the relation between the pile body rotation angle, the bending moment, the shearing force and the pile body horizontal displacement coefficient based on a dynamic balance equation of the pile body unit, and creating a corresponding coefficient equation set;
s4, determining a dynamic balance equation corresponding to the pile bottom and pile body displacement and the internal force coefficient based on the coefficient equation set;
and S5, determining a dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient, and further completing analysis of a single pile horizontal dynamic response process under the action of the seismic S shear wave.
2. The method for analyzing horizontal dynamic response of single pile under action of S shear wave of earthquake as set forth in claim 1, wherein,
in the step S1, based on the characteristic parameters of foundation soil, the step of creating a foundation soil dynamic balance equation considering the influence of the foundation soil on the S shear wave of the earthquake comprises the following steps:
s11, determining characteristic parameters of each soil layer of the foundation soil required for creating an equation, wherein the characteristic parameters comprise elastic shear modulus
Figure FDA0004146056450000011
Viscous damping coefficient->
Figure FDA0004146056450000012
And mass Density->
Figure FDA0004146056450000013
Horizontal displacement +.>
Figure FDA0004146056450000014
S12, based on the characteristic parameters, creating a free field soil dynamic balance equation, wherein the corresponding free field soil dynamic balance equation is as follows
Figure FDA0004146056450000015
In the method, in the process of the invention,
Figure FDA0004146056450000016
the shear stress is given, t is time, and z is soil layer depth;
due to
Figure FDA0004146056450000017
Expressed as:
Figure FDA0004146056450000021
viscous damping coefficient of viscoelastic material
Figure FDA0004146056450000022
Omega is the frequency of the excitation circle,
then substituting formula (2) into formula (1) to obtain:
Figure FDA0004146056450000023
s13, based on horizontal displacement
Figure FDA0004146056450000024
Performing a deformation of equation (3) by a displacement steady state solution in the frequency domain, the horizontal displacement
Figure FDA0004146056450000025
The displacement steady state solution in the frequency domain is: />
Figure FDA0004146056450000026
Substituting into (3) to obtain:
Figure FDA0004146056450000027
s14, creating a displacement amplitude of the j-th layer to horizontally displace under the action of the upward and downward shear waves at the depth z
Figure FDA0004146056450000028
Time-independent partial solutions corresponding to the equation
Figure FDA0004146056450000029
Wherein, complex shear modulus
Figure FDA00041460564500000210
C 1 、C 2 The coefficients to be solved;
s15, acquiring an exponential equation of the formula (5), wherein the corresponding equation is as follows:
Figure FDA00041460564500000211
wherein N-1 is greater than or equal to j is greater than or equal to 1, N represents the number of layers of the base stratum, E j 、F j As the amplitude coefficient of the wave,
Figure FDA00041460564500000212
wave number;
s16, creating a conversion relation equation between the top layer, namely the 1 st layer soil and any layer of amplitude coefficient, wherein the corresponding equation creation process is as follows:
due to wave number
Figure FDA00041460564500000213
Complex shear wave velocity +.>
Figure FDA00041460564500000214
Then the formula (6) is
Figure FDA00041460564500000215
Figure FDA00041460564500000216
Substituting the formulas (7) and (8) into the formula (1) to obtain the product:
Figure FDA0004146056450000031
since the shear stress and displacement are continuous at the interface of the j-th layer and the j+1-th layer of soil, namely:
Figure FDA0004146056450000032
substituting the formula (6) and the formula (9) into the formula (10) to obtain:
Figure FDA0004146056450000033
the corresponding recurrence formula is determined as follows:
Figure FDA0004146056450000034
in the method, in the process of the invention,
Figure FDA0004146056450000035
defined as wave number impedance, let the coefficient matrix (2×2) obtained in the above formula be T j The conversion relation equation between the top layer (layer 1 soil) and any layer of amplitude coefficient is obtained as follows:
Figure FDA0004146056450000036
wherein the transfer function
Figure FDA0004146056450000037
Wherein N-1 is greater than or equal to j is greater than or equal to 1, N represents the number of layers of the base stratum, E j 、F j Is the amplitude coefficient;
s17, creating a land dynamic balance equation under the influence of foundation soil considered on seismic S shear waves, wherein the corresponding equation creation process is as follows:
according to the first layer free-ground boundary conditions: τ 1 (0) =0, get E 1 ＝F 1 Let n=n-1, to obtain:
Figure FDA0004146056450000038
and
Figure FDA0004146056450000041
substituting the above formula into formula (6):
Figure FDA0004146056450000042
3. the method for analyzing horizontal dynamic response of single pile under the action of seismic S shear wave according to claim 1, wherein the step of determining the coefficient relation between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement in S3 and creating a corresponding coefficient equation set comprises;
each parameter is
Figure FDA0004146056450000043
And->
Figure FDA0004146056450000044
The method is determined according to the following formula:
Figure FDA0004146056450000045
Figure FDA0004146056450000046
Figure FDA0004146056450000047
Figure FDA0004146056450000048
in U 0 Maximum displacement for ground movement; pile surrounding soil shearing wave number
Figure FDA0004146056450000049
Shear wave velocity of pile surrounding soil
Figure FDA00041460564500000410
And->
Figure FDA00041460564500000411
The modulus of elasticity, density, damping coefficient and poisson ratio of the surrounding soil of the pile are respectively; dimensionless frequency->
Figure FDA00041460564500000412
Is the thickness of foundation soil shear layer +.>
Figure FDA00041460564500000413
d refers to the diameter of the pile;
since the pile horizontal displacement is expressed as:
Figure FDA00041460564500000414
in the method, in the process of the invention,
Figure FDA00041460564500000415
for the horizontal displacement amplitude of the pile body mass point,
order the
Figure FDA00041460564500000416
Substituting the formula (21) into the formula (16) respectively further yields the following equation:
Figure FDA00041460564500000417
in the method, in the process of the invention,
Figure FDA00041460564500000418
E p 、I p 、B 0 respectively the elastic modulus of the pile body, the section moment of inertia of the pile and the calculated width W of the pile p To define the coefficient, N 0 The axial acting force of the pile top is;
obviously, the homogeneous solution corresponding to formula (22) is:
Figure FDA0004146056450000051
in the method, in the process of the invention,
Figure FDA0004146056450000052
coefficient->
Figure FDA0004146056450000053
The value is determined by the set boundary condition;
and the special solution of formula (23) is:
Figure FDA0004146056450000054
the full solution of equation (23) is expressed as:
Figure FDA0004146056450000055
based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement is:
Figure FDA0004146056450000056
Figure FDA0004146056450000057
Figure FDA0004146056450000058
the coefficient equation set obtained by integrating the equation (25) and the equation (26) is as follows:
Figure FDA0004146056450000059
in the method, in the process of the invention,
Figure FDA0004146056450000061
Figure FDA0004146056450000062
when j=1 and z=0, substituting formula (27) to obtain the transfer integral constant of the first pile body is:
Figure FDA0004146056450000063
the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile are continuous at the sections of the j-th section and the j-1-th section, namely:
Figure FDA0004146056450000064
the coefficient matrix equation set obtained by integrating the equation (27), the equation (28) and the equation (29) is as follows:
Figure FDA0004146056450000065
4. the method for analyzing horizontal dynamic response of single pile under the action of seismic S shear wave according to claim 3, wherein the step of determining the dynamic balance equation corresponding to the pile bottom and pile body displacement and the internal force coefficient based on the coefficient equation set in S4 comprises the following steps:
synthesizing the formula (30), the formula (25) and the formula (26) to obtain the displacement of the j-th pile bottom and the internal force transfer equation
Figure FDA0004146056450000071
Further according to the continuous condition of pile section boundary, the comprehensive expression equation of pile bottom and pile body displacement and internal force is calculated as follows:
Figure FDA0004146056450000072
[S n (h n )][S n (0)] -1 [S n-1 (h n-1 )][S n-1 (0)] -1 …[S 2 (h 2 )][S 2 (0)] -1 [H 1 (h 1 )-H 2 (0)]+[S n (h n )][S n (0)] -1 [S n-1 (h n-1 )][S n-1 (0)] -1 …[S 3 (h 3 )][S 3 (0)] -1 [H 2 (h 2 )-H 3 (0)]+…+…+[S n (h n )][S n (0)] -1 [H n-1 (h n-1 )-H n (0)]+[H n (h n )]
considering pile bottom boundary conditions, further generalizing the equation (32) to obtain:
Figure FDA0004146056450000073
in the method, in the process of the invention,
Figure FDA0004146056450000074
Figure FDA0004146056450000075
5. the method for analyzing horizontal dynamic response of a single pile under the action of seismic S shear waves according to claim 4, wherein the step of determining the dynamic balance equation corresponding to the pile top displacement and the rotation angle coefficient in S5 comprises:
definition of the definition
Figure FDA0004146056450000076
Further solving a dynamic balance equation corresponding to pile top displacement and rotation angle by the formula (33):
Figure FDA0004146056450000077
Figure FDA0004146056450000081
CN202310303826.4A 2023-03-24 2023-03-24 Single pile horizontal dynamic response analysis method under seismic S shear wave effect Pending CN116305480A (en)

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