CN111310321B - Layered soil single pile horizontal vibration analysis method based on Pasternak foundation model - Google Patents
Layered soil single pile horizontal vibration analysis method based on Pasternak foundation model Download PDFInfo
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Abstract
The invention discloses a layered soil single pile horizontal vibration analysis method based on a Pasternak foundation model, which adopts a layered Pasternak foundation model to simulate the shearing effect of soil around piles, adopts a segmented Timoshenko beam model to simulate piles to consider bending and shearing deformation of the piles, and simultaneously assumes that a pile-soil coupling vibration model meets the condition of small deformation, the pile-soil interface is completely contacted and has no relative sliding, and the pile bottom is fixed-end constraint. On the basis of the assumption, the invention firstly establishes a horizontal dynamic balance equation of the segmented pile body units, secondly establishes the relation between the pile body rotation angle, the bending moment and the shearing force and the horizontal displacement of the pile body, thirdly establishes a transmission matrix of the segmented pile body units according to the horizontal displacement, the rotation angle, the bending moment and the shearing force continuity of the pile, and finally obtains the rigidity of the pile top impedance function and the bending moment and the shearing force on any section of the pile body according to boundary conditions.
Description
Technical Field
The invention relates to the field of civil engineering, in particular to a layered soil single pile horizontal vibration analysis method based on a Pasternak foundation model.
Background
At present, when the problem of pile horizontal vibration dynamic response is solved, the pile periphery 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 the soil body among longitudinal layers, so that the calculation result is not tight in theory. The double-parameter foundation model considers the foundation soil shearing effect on the basis of the Winkler model, and is more practical. In this case, the foundation model of Pasternak is more suitable.
In addition, when the problem of pile body horizontal vibration dynamic response is solved, a classical Bernoulli-Euler theory is adopted for the slender rod pile foundation model, and the theory model only considers pile body bending deformation and ignores the influence of pile foundation shearing deformation. For large-diameter piles, it is important to consider the influence of pile body shear deformation on the dynamic impedance of the pile body, and the pile body is more suitable to adopt a Timoshenko beam model.
How to effectively combine and apply the two to solve the problem of horizontal vibration analysis of the single pile in the layered soil 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 layered soil single pile horizontal vibration analysis method based on a Pasternak foundation model, wherein the Pasternak foundation model is adopted to consider the shearing effect of soil around piles, and the segmented Timoshenko beam model is adopted to simulate piles to consider bending and shearing deformation of the piles.
In order to achieve the above purpose, the technical scheme of the invention is as follows:
a method for analyzing horizontal vibration of a single pile in layered soil based on a Pasternak foundation model comprises
S1: the following assumed conditions are introduced, and a horizontal vibration analysis model of a single pile in the layered soil is established: the depth of the single pile body is consistent with that of the surrounding soil of the pile, and the single pile body is longitudinally divided into n layers; the assumed conditions include: assuming that the single pile body is a homogeneous and uniform-section elastomer, timoshenko Liang Moxing is adopted; assuming that each layer of soil body of the pile surrounding soil adopts a Pasternak foundation model; assuming that all parts of the pile-soil coupling vibration model meet the small deformation condition, the pile-soil interface is completely contacted and has no relative sliding; assuming the pile bottom as a solid end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Pasternak foundation model theory, wherein the dynamic balance equation corresponds to the expression
Meanwhile, according to the assumption in the step S1, a pile-soil coupling vibration model boundary condition is established, and the corresponding expression is:
wherein z is a longitudinal coordinate, a zero point of the longitudinal coordinate is positioned on the free surface, downward is positive, and t is a time coordinate; n is soil around pile and singleThe number of layers of the pile bodies, j is the number of layers of the surrounding soil and the single pile body from top to bottom, and j=1 to n; q (Q) 0 Is the amplitude of the horizontal simple harmonic excitation force of the pile top,
is an imaginary unit;
A p 、G p 、E p 、I p 、m p the cross section area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length of the single pile body are respectively; k' is the shear shape coefficient of the section of the single pile body; b (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;
wherein 
Shear wave velocity of surrounding soil of jth layer of pile, < ->
and />
The elastic modulus, the density, the damping coefficient and the poisson ratio of the surrounding soil of the jth layer of pile are respectively +.>
The dimensionless frequency of the surrounding soil of the jth layer of pile is set; />
The thickness of a shear layer of surrounding soil of the jth layer of pile is the thickness of the shear layer; omega is the excitation circle frequency of the pile top horizontal simple harmonic excitation force;
s3: and (2) solving a dynamic balance equation of the layered pile body unit in the step (S2) to obtain a single pile horizontal vibration analysis parameter in the layered soil, wherein the parameter at least comprises pile top horizontal impedance of the pile top acted by horizontal exciting force and internal force of any section of the pile body.
Further, in the step S3, the process of solving the dynamic balance equation of the layered pile body unit in the step S2 to obtain the horizontal vibration analysis parameters of the single pile in the layered soil includes the following steps
Step S31: the horizontal displacement, the section rotation angle, the pile body shearing force and the pile body bending moment of the mass point of the j-th layer pile body are respectively transformed according to the excitation circle frequency of the pile top horizontal simple harmonic excitation force, and the corresponding transformation formula is as follows:
wherein 
For the horizontal displacement amplitude of the j-th layer pile body, < > for the horizontal displacement amplitude of the j-th layer pile body>
Is the cross-section corner amplitude of the jth layer pile body,
the shear amplitude of the pile body of the j-th layer pile body is +.>
The bending moment amplitude value of the j-th layer pile body;
the boundary condition change formula corresponding to the above transformation formula is:
step S32: simplifying dynamic balance equation of layered pile body unit to obtain four-order normal differential homogeneous equation of horizontal displacement amplitude, horizontal displacement amplitude general solution, corner amplitude general solution, pile body bending moment amplitude general solution and pile body shearing force amplitude general solution,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
the expression corresponding to the horizontal displacement amplitude general solution is as follows:
U j (z)＝A j1 cosh(β j1 z)+A j2 sinh(β j1 z)+A j3 cos(β j2 z)+A j4 sin(β j2 z)；
the expression corresponding to the rotation angle amplitude general solution is as follows:
the expression corresponding to the pile body bending moment amplitude general solution is as follows:
the expression corresponding to the pile body shear amplitude general solution is as follows:
wherein each symbol in the fourth-order ordinary differential homogeneous equation is defined as
W p ＝E p I p J p ＝K′A P G p 
A j1 ，A j2 ，A j3 ，A j4 Coefficients determined based on boundary conditions;
wherein each symbol in each general solution is defined as
Step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using the continuous expressions of horizontal displacement, rotation angle, bending moment and shearing force 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
Solving the expression for obtaining the transfer relation of horizontal displacement, rotation angle, bending moment and shearing force of the top and the bottom of the j-th layer pile body is as follows:
b j ＝R j a j
wherein ,
is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude at the top of the jth layer of pile body>
Is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude of the top of the j+1th layer pile body, h j Is the thickness of the j-th layer pile body; r is R j Is a 4 multiplied by 4 j layer pile body transfer matrix, and has the block form of
The expression of each block array is
In the above formulae
Step S34: the transfer moment of each layer of pile body is combined to obtain the full-length transfer relation between the single pile end state column vector and the bottom state column vector, the full-length transfer relation is
b L ＝Ra 0
Where R is the full length transfer matrix of the mono pile,
is a single pile end state column vector, +.>
Is the column vector of the bottom state of the pile body, and the corresponding blocking form of the transfer matrix R is
Wherein the full-length transfer relationship is in the form of a block
Step S35: giving the following boundary conditions to obtain a top impedance function stiffness matrix and pile top horizontal impedance; the boundary condition is
The top impedance function stiffness matrix is K= -R 12 -1 R 11
The pile top horizontal impedance is
Step S36: the internal force on any section of the pile body is obtained by utilizing the pile top impedance function, and the corresponding calculation formula is that
wherein 
wherein R1 To R m-1 Height h of (3) j (j=1, 2,) m-1) is the thickness of each soil layer, R m Is->
Is the longitudinal coordinate of any section of the pile body, R α In the form of blocks of
According to the technical scheme, the single pile horizontal vibration analysis method in the layered soil based on the Pasternak 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, meanwhile, each part of a pile-soil system is assumed to meet the small deformation condition, the pile-soil interface is completely contacted and has no relative sliding, the pile bottom is fixed end constraint, the pile top impedance function rigidity, the bending moment and the shearing force on any section of the pile body are finally obtained according to the boundary conditions, and the result is suitable for the pile foundation horizontal vibration dynamic response problem under the simple harmonic loading effect and can provide theoretical guidance and reference effect 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 an embodiment;
fig. 2 is a schematic diagram of a model corresponding to the method according to the present invention in the embodiment.
Detailed Description
The following describes the 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, the structures of the present invention are not drawn to a general scale, and the structures in the drawings are partially enlarged, deformed, and simplified, so that the present invention should not be construed as being limited thereto.
In order to solve the problems in the prior art, the method for analyzing the horizontal vibration of the single pile in the layered soil based on the Pastemak foundation model is shown in the figures 1-2, and is characterized by comprising the following steps:
a method for analyzing horizontal vibration of a single pile in layered soil based on a Pasternak foundation model comprises
S1: the following assumed conditions are introduced, and a horizontal vibration analysis model of a single pile in the layered soil is established: the depth of the single pile body is consistent with that of the surrounding soil of the pile, and the single pile body is longitudinally divided into n layers; the assumed conditions include: assuming that the single pile body is a homogeneous and uniform-section elastomer, timoshenko Liang Moxing is adopted; assuming that each layer of soil body of the pile surrounding soil adopts a Pasternak foundation model; assuming that all parts of the pile-soil model system meet the small deformation condition, the pile-soil interface is completely contacted and has no relative sliding; assuming the pile bottom as a solid end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Pasternak foundation model theory, wherein the dynamic balance equation corresponds to the expression
The balance equation gives a section G representing the shear modulus of the pile body compared with the existing Euler Liang Moxing and Winkler foundation models p Is given with a section representing the shear coefficient of soil around the pile
Is the expression form of (a);
to construct the dynamic balance equation of the required layered pile body unit by adding the above matters;
meanwhile, according to the assumption in the step S1, a pile-soil model boundary condition is established, and the corresponding expression is:
wherein z is a longitudinal coordinate, a zero point of the longitudinal coordinate is positioned on the free surface, downward is positive, and t is a time coordinate; n is the number of layers of the pile surrounding soil and the single pile body, j is the number of layers of the pile surrounding soil and the single pile body from top to bottom, and j=1 to n; q (Q) 0 Is the amplitude of the horizontal simple harmonic excitation force of the pile top,
is an imaginary unit;
A p 、G p 、R p 、I p 、m p the cross section area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length of the single pile body are respectively; k' is the shear shape coefficient of the section of the single pile body; b (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;
wherein 
Shear wave velocity of surrounding soil of jth layer of pile, < ->
and />
The elastic modulus, the density, the damping coefficient and the poisson ratio of the surrounding soil of the jth layer of pile are respectively +.>
The dimensionless frequency of the surrounding soil of the j-th layer pile is as follows: />
The thickness of a shear layer of surrounding soil of the jth layer of pile is the thickness of the shear layer; omega is the excitation circle frequency of the pile top horizontal simple harmonic excitation force;
s3: and (2) solving a dynamic balance equation of the layered pile body unit in the step (S2) to obtain a single pile horizontal vibration analysis parameter in the layered soil, wherein the parameter at least comprises pile top horizontal impedance of the pile top acted by horizontal exciting force and internal force of any section of the pile body.
Further, in the step S3, the process of solving the dynamic balance equation of the layered pile body unit in the step S2 to obtain the horizontal vibration analysis parameters of the single pile in the layered soil includes the following steps
Step S31: the horizontal displacement, the section rotation angle, the pile body shearing force and the pile body bending moment of the mass point of the j-th layer pile body are respectively transformed according to the excitation circle frequency of the pile top horizontal simple harmonic excitation force, and the corresponding transformation formula is as follows:
wherein 
For the horizontal displacement amplitude of the j-th layer pile body, < > for the horizontal displacement amplitude of the j-th layer pile body>
Is the cross-section corner amplitude of the jth layer pile body,
the shear amplitude of the pile body of the j-th layer pile body is +.>
The bending moment amplitude value of the j-th layer pile body;
the boundary condition change formula corresponding to the above transformation formula is:
step S32: simplifying dynamic balance equation of layered pile body unit to obtain four-order normal differential homogeneous equation of horizontal displacement amplitude, horizontal displacement amplitude general solution, corner amplitude general solution, pile body bending moment amplitude general solution and pile body shearing force amplitude general solution,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
the expression corresponding to the horizontal displacement amplitude general solution is as follows:
U j (z)＝A j1 cosh(β j1 z)+A j2 sinh(β j1 z)+A j3 cos(β j2 z)+A j4 sin(β j2 z)；
the expression corresponding to the rotation angle amplitude general solution is as follows:
the expression corresponding to the pile body bending moment amplitude general solution is as follows:
the expression corresponding to the pile body shear amplitude general solution is as follows:
wherein each symbol in the fourth-order ordinary differential homogeneous equation is defined as
W p ＝E p I p J p ＝K′A p G p 
A j1 ，A j2 ，A j3 ，A j4 Coefficients determined based on boundary conditions;
wherein each symbol in each general solution is defined as
Step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using the continuous expressions of horizontal displacement, rotation angle, bending moment and shearing force 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
Solving the expression for obtaining the transfer relation of horizontal displacement, rotation angle, bending moment and shearing force of the top and the bottom of the j-th layer pile body is as follows:
b j ＝R j a j
wherein ,
is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude at the top of the jth layer of pile body>
Is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude of the top of the j+1th layer pile body, h j Is the thickness of the j-th layer pile body; r is R j Is a 4 multiplied by 4 j layer pile body transfer matrix, and has the block form of
The expression of each block array is
In the above formulae
Step S34: the transfer moment of each layer of pile body is combined to obtain the full-length transfer relation between the single pile end state column vector and the bottom state column vector, the full-length transfer relation is
b L ＝Ra 0
Where R is the full length transfer matrix of the mono pile,
is a single pile end state column vector, +.>
Is the column vector of the bottom state of the pile body, and the corresponding blocking form of the transfer matrix R is
Wherein the full-length transfer relationship is in the form of a block
At the same time, for the convenience of expression, this partThe subscripts 0 and n being temporarily removed for each parameter, e.g
Namely U p (L)，
Namely U p (0) Etc.;
step S35: giving the following boundary conditions to obtain a top impedance function stiffness matrix and pile top horizontal impedance;
the boundary condition is
The rigidity matrix of the top impedance function is
K＝-R 12 -1 R 11
The pile top horizontal impedance is
Step S36: the internal force on any section of the pile body is obtained by utilizing the pile top impedance function, and the corresponding calculation formula is that
wherein 
wherein R1 To R m-1 Height h of (3) j (j=1, 2,) m-1) is the thickness of each soil layer, R m Is->
Is the longitudinal coordinate of any section of the pile body, R α In the form of blocks of
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (1)
1. A method for analyzing horizontal vibration of a single pile in layered soil based on a Pasternak foundation model is characterized by comprising the following steps:
s1: the following assumed conditions are introduced, and a horizontal vibration analysis model of a single pile in the layered soil is established: the depth of the single pile body is consistent with that of the surrounding soil of the pile, and the single pile body is longitudinally divided into n layers; the assumed conditions include: assuming that the single pile body is a homogeneous and uniform-section elastomer, timoshenko Liang Moxing is adopted; assuming that each layer of soil body of the pile surrounding soil adopts a Pasternak foundation model; assuming that the pile-soil coupling vibration model meets the small deformation condition, the pile-soil interface is completely contacted and has no relative sliding; assuming the pile bottom as a solid end constraint;
s2: establishing a dynamic balance equation of the layered pile body unit according to the Timoshenko beam and the Pasternak foundation model theory, wherein the dynamic balance equation corresponds to the expression
Meanwhile, according to the assumption in the step S1, a pile-soil coupling vibration model boundary condition is established, and the corresponding expression is:
P 1 p (0,t)＝Q 0 e iωt
wherein z is a longitudinal coordinate, a zero point of the longitudinal coordinate is positioned on the free surface, downward is positive, and t is a time coordinate; n is the number of layers of the pile surrounding soil and the single pile body, j is the number of layers of the pile surrounding soil and the single pile body from top to bottom, and j=1 to n; q (Q) 0 Is the amplitude of the horizontal simple harmonic excitation force of the pile top,
is an imaginary unit;
A p 、G p 、E p 、I p 、m p the cross section area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length of the single pile body are respectively; k' is the shear shape coefficient of the section of the single pile body; b (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;
wherein 
Shear wave velocity of surrounding soil of jth layer of pile, < ->
and />
Elastic modulus, density and resistance of surrounding soil of the j-th layer pile respectivelyNylon coefficient and poisson ratio,)>
The dimensionless frequency of the surrounding soil of the jth layer of pile is set; />
The thickness of a shear layer of surrounding soil of the jth layer of pile is the thickness of the shear layer; omega is the excitation circle frequency of the pile top horizontal simple harmonic excitation force;
s3: solving a dynamic balance equation of the layered pile body unit in the step S2 to obtain a single pile horizontal vibration analysis parameter in layered soil, wherein the parameter at least comprises pile top horizontal impedance of a pile top acted by a horizontal exciting force and internal force of any section of the pile body; in the step S3, the process of solving the dynamic balance equation of the layered pile body unit in the step S2 to obtain the single pile horizontal vibration analysis parameter in the layered soil includes the following steps:
step S31: the horizontal displacement, the section rotation angle, the pile body shearing force and the pile body bending moment of the mass point of the j-th layer pile body are respectively transformed according to the excitation circle frequency of the pile top horizontal simple harmonic excitation force, and the corresponding transformation formula is as follows:
wherein 
For the horizontal displacement amplitude of the j-th layer pile body, < > for the horizontal displacement amplitude of the j-th layer pile body>
Is the cross-section corner amplitude of the jth layer pile body,
the shear amplitude of the pile body of the j-th layer pile body is +.>
The bending moment amplitude value of the j-th layer pile body;
the boundary condition change formula corresponding to the above transformation formula is:
P 1 p (0)＝Q 0
step S32: simplifying dynamic balance equation of layered pile body unit to obtain four-order normal differential homogeneous equation of horizontal displacement amplitude, horizontal displacement amplitude general solution, corner amplitude general solution, pile body bending moment amplitude general solution and pile body shearing force amplitude general solution,
the expression corresponding to the fourth-order ordinary differential homogeneous equation of the horizontal displacement amplitude is as follows:
the expression corresponding to the horizontal displacement amplitude general solution is as follows:
U j (z)＝A j1 cosh(β j1 z)+A j2 sinh(β j1 z)+A j3 cos(β j2 z)+A j4 sin(β j2 z)；
the expression corresponding to the rotation angle amplitude general solution is as follows:
the expression corresponding to the pile body bending moment amplitude general solution is as follows:
the expression corresponding to the pile body shear amplitude general solution is as follows:
wherein each symbol in the fourth-order ordinary differential homogeneous equation is defined as
W p ＝E p I p J p ＝K'A p G p 
A j1 ,A j2 ,A j3 ,A j4 Coefficients determined based on boundary conditions;
wherein each symbol in each general solution is defined as
Step S33: establishing a coefficient matrix equation set of the j-th layer pile body by using the continuous expressions of horizontal displacement, rotation angle, bending moment and shearing force 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
Solving the expression for obtaining the transfer relation of horizontal displacement, rotation angle, bending moment and shearing force of the top and the bottom of the j-th layer pile body is as follows:
b j ＝R j a j
wherein ,
is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude at the top of the jth layer of pile body>
Is a state column vector formed by horizontal displacement amplitude, corner amplitude, bending moment amplitude and shearing force amplitude of the top of the j+1th layer pile body, h j Is the thickness of the j-th layer pile body; r is R j Is a 4 multiplied by 4 j layer pile body transfer matrix, and has the block form of
The expression of each block array is
In the above formulae
Step S34: the transfer moment of each layer of pile body is combined to obtain the full-length transfer relation between the single pile end state column vector and the bottom state column vector, the full-length transfer relation is
b L ＝Ra 0
Where R is the full length transfer matrix of the mono pile,
is a single pile end state column vector, +.>
Is the column vector of the bottom state of the pile body, and the corresponding blocking form of the transfer matrix R is
Wherein the full-length transfer relationship is in the form of a block
Step S35: giving the following boundary conditions to obtain a top impedance function stiffness matrix and pile top horizontal impedance; the boundary condition is
The rigidity matrix of the top impedance function is
K＝-R 12 -1 R 11
The pile top horizontal impedance is
Step S36: the internal force on any section of the pile body is obtained by utilizing the pile top impedance function, and the corresponding calculation formula is that
wherein 
wherein R1 To R m-1 Height h of (3) j (j=1, 2,) m-1) is the thickness of each soil layer, R m In (a) and (b)
Is the longitudinal coordinate of any section of the pile body, R α In the form of blocks of
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