CN112307545B - 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 PDFInfo
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Abstract
The invention provides a large-diameter single pile horizontal vibration analysis method considering axial force action, simplifies a large-diameter single pile into Timoshenko Liang Moxing, and provides a mechanical system model considering pile body shear deformation and pile periphery soil body shear effect simultaneously. In addition, the pile top simultaneously considers the influence of axial force under the action of horizontal load, and the comprehensive consideration of the actions of two loads can be suitable for the problem of horizontal vibration of a large-diameter single pile under the action of complex multidirectional load, can better simulate the pile-soil coupling interaction of a foundation pile under the action of complex load in actual engineering, and can lay a foundation for pile foundation vibration theory; the invention considers the complex load effect of the pile top, the influence of the shearing deformation of the pile body and the shearing deformation of soil mass around the pile and the influence factor of the stressed deformation related to the pile foundation in the actual engineering, and provides theoretical guidance and reference effect for the foundation pile deformation and internal force change rule of the pile foundation under the action of dynamic load based on the model provided by the invention.
Description
Technical Field
The invention relates to the field of civil engineering, in particular to a large-diameter single pile horizontal vibration analysis method considering the action of axial force.
Background
Along with the rapid development of bridge engineering and high-rise buildings in China, pile foundations are increasingly widely applied, and stress analysis of foundation piles under complex multidirectional loads is increasingly important. At present, when the problem of pile horizontal vibration dynamic response is solved, the pile periphery 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 the soil body among longitudinal layers, so that the calculation result is not tight in theory. The dual-parameter foundation model considers the foundation soil shearing effect on the basis of the Winkler model, is more practical, and is more suitable to adopt the Pasternak foundation model. For the slender rod pile foundation model, the classical Bernoulli-Euler theory is adopted, and the theory model only considers the bending deformation of the pile body and ignores the influence of the shearing deformation. For large-diameter piles, the influence of shear deformation of the pile body under the action of axial and horizontal forces on the dynamic response of the pile body is considered to be particularly important, and the pile body is more suitable to adopt a Timoshenko beam model.
Disclosure of Invention
The invention aims to overcome the defect that a Winkler foundation model in the prior art ignores the shearing effect of soil and cannot reflect the continuity of soil bodies among longitudinal layers; the Bernoulli-Euler theoretical model only considers the bending deformation of the pile body, but ignores the defect of influence of shearing deformation.
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, and a large-diameter single pile horizontal vibration analysis model under the action of axial and horizontal forces of the Pasternak layered foundation 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 circular homogeneous constant-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: and 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:
in the formula ,
respectively horizontal displacement and section rotation angle of the mass point of the pile body of the jth layer section; z is the direction of pile foundation along depth, p is the upper mark of representative pile; a is that p 、G p 、E p 、I p 、m p Respectively the pile body sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length; t is time; n (N) 0 Is the axial force acting on the pile top; k' is the shear shape factor; the thickness, rigidity coefficient, damping coefficient and foundation shear coefficient of the j-th layer of soil are respectively h j 、/>
and />
B 0 Calculated width for stake =0.9 (1.5d+0.5) for +.>
and />
The method is determined according to the following formula:
in the formula ,
the shear wave velocity of the j-th layer of soil; />
and />
The elastic modulus, the density, the damping coefficient and the poisson ratio of the j-th layer of soil are respectively; />
Is dimensionless frequency, and omega is excitation circle frequency; />
The thickness of the shear layer of the jth foundation soil is +.>
d is the diameter of the pile;
pile tops are subjected to simple harmonic vibration in steady-state vibration, and pile body horizontal displacement and rotation angle are simplified into:
in the formula ,
for the horizontal displacement amplitude of the j-th layer pile body, < > for the j-th layer pile body>
E is the amplitude of the section angle of the section of the jth pile body iωt Representing a complex frequency domain;
substituting formula (5) into formula (1) to obtain the following equation:
in the formula ,
M p ＝E p I p ，J p ＝K'A p G p ，S p ＝ρ p I p ω 2 ，
the characteristic root corresponding to the formula (6) is
Obtaining displacement general solution of the horizontal displacement amplitude of the j-th layer pile body: />
wherein ,
unknown coefficient A j1 、B j1 、C j1 、D j1 The value of (2) is determined by the boundary conditions of the pile top and the pile bottom;
s3: and (2) solving the dynamic balance equation in the step (S2) to obtain horizontal vibration analysis parameters of the single pile in the layered soil, wherein the parameters at least comprise horizontal displacement, bending moment, shearing force and section rotation angle of the pile body.
Further, step S31: according to displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer section and a dynamic balance equation of the pile body unit, obtaining a section corner of the pile body, wherein the expression is as follows:
when the pile body does not undergo shear deformation, the formula (8) is degraded into
Substituting formula (7) into formula (8) to obtain a cross-section rotation angle general solution:
step S32: determining the interrelationship among the bending moment, shearing force, horizontal displacement and section rotation angle of the pile body;
the relation expression of the bending moment of the pile body and the section rotation angle is as follows:
the relation expression among the bending moment, the shearing force, the horizontal displacement and the section rotation angle of the pile body is as follows:
order the
The predetermined coefficients in the formulae (9), (10) and (11) are expressed as: />
in the formula ：
A j2 、A j3 、A j4 、B j2 、B j3 、B j4 、C j2 、C j3 、C j4 、D j2 、D j3 、D j4 obtaining unknown coefficients by boundary conditions of pile tops and pile bottoms;
A j2 、A j3 、A j4 、B j2 、B j3 、B j4 、C j2 、C j3 、C j4 、D j2 、D j3 、D j4 obtaining unknown coefficients by boundary conditions of pile tops and pile bottoms;
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;
at the sections of the j-th section and the j+1-th section, the continuity expressions of the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile body are as follows:
the coefficient matrix equation set obtained by integrating the equation (12) and the equation (13) is as follows:
in the formula
{T j }＝[A j1 B j1 C j1 D j1 ] T ；
Obtained from formula (14):
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j } (15)
then the corresponding coefficient matrix { T of the mth pile body is calculated by the recurrence relation m Expressed as:
step S34: setting boundary conditions of pile top and pile bottom, and according to the corresponding coefficient matrix { T } of the mth pile body m Obtaining the horizontal displacement, bending moment and shearing force of the pile body;
the boundary conditions of the pile top and the pile bottom are set as follows:
substituting the coefficient expression (12) into the expression (17) for simplification to obtain:
combined equations (16) - (18) to determine { T ] 1 According to the corresponding coefficient matrix { T } of the mth section pile body m And obtaining the 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:
in the formula ,umax (z)、m max (z)、p max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
According to the technical scheme, the large-diameter single pile is simplified into Timoshenko Liang Moxing based on the Pasternak foundation model, and a mechanical system model which simultaneously considers pile body shear deformation and pile periphery soil body shear effect is provided. In addition, the pile top simultaneously considers the influence of axial force under the action of horizontal load, and the comprehensive consideration of the actions of two loads can be suitable for the problem of horizontal vibration of a large-diameter single pile under the action of complex multidirectional load, can better simulate the pile-soil coupling interaction of a foundation pile under the action of complex load in actual engineering, and can lay a foundation for pile foundation vibration theory; the invention considers the complex load effect of the pile top, the influence of the shearing deformation of the pile body and the shearing deformation of soil mass around the pile and the influence factor of the stressed deformation related to the pile foundation in the actual engineering, and provides theoretical guidance and reference effect for the foundation pile deformation and internal force change rule of the pile foundation under the action of dynamic load based on the model provided by the invention.
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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, a large-diameter single pile horizontal vibration analysis method considering the action of axial force 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, and a large-diameter single pile horizontal vibration analysis model under the action of axial and horizontal forces of the Pasternak layered foundation 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 circular homogeneous constant-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: the dynamic balance equation of the layered pile body unit is established according to the Timoshenko beam and the Pasternak foundation model theory, and is different from the prior art in that: the dynamic balance equation considers the axial force acting on the pile top and the shearing effect of soil around the pile; the expression corresponding to the dynamic balance equation is:
in the formula ,
respectively horizontal displacement and section rotation angle of the mass point of the pile body of the jth layer section; z is the direction of pile foundation along depth, p is the upper mark of representative pile; a is that p 、G p 、E p 、I p 、m p Respectively the pile body sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length; t is time; n (N) 0 Is the axial force acting on the pile top; k' is the shear shape factor; the thickness, rigidity coefficient, damping coefficient and foundation shear coefficient of the j-th layer of soil are respectively h j 、/>
and />
B 0 Calculated width for stake =0.9 (1.5d+0.5) for +.>
and />
The method is determined according to the following formula:
in the formula ,
the shear wave velocity of the j-th layer of soil; />
and />
The elastic modulus, the density, the damping coefficient and the poisson ratio of the j-th layer of soil are respectively; />
Is dimensionless frequency, and omega is excitation circle frequency; />
The thickness of the shear layer of the jth foundation soil is +.>
d is the diameter of the pile;
in steady-state vibration, pile body horizontal displacement and section rotation angle in a time domain range are converted into pile body horizontal displacement and section rotation angle in a complex frequency domain range, and the expression is as follows:
in the formula ,
for the horizontal displacement amplitude of the j-th layer pile body, < > for the j-th layer pile body>
E is the amplitude of the section angle of the section of the jth pile body iωt Representing a complex frequency domain;
substituting formula (5) into formula (1) yields the following equation:
in the formula ,
M p ＝E p I p ，J p ＝K'A p G p ，S p ＝ρ p I p ω 2 ，
the characteristic root corresponding to the formula (6) is
The displacement general solution of the horizontal displacement amplitude of the j-th layer pile body can be obtained: />
wherein ,
unknown coefficient A j1 、B j1 、C j1 、D j1 The value of (2) is determined by the boundary conditions of the pile top and the pile bottom;
s3: and (2) solving a dynamic balance equation in the step (S2) to obtain horizontal vibration analysis parameters of the single pile in the layered soil, wherein the parameters at least comprise horizontal displacement, bending moment, shearing force and section rotation angle 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 mono-pile in the layered soil includes the following steps:
step S31: according to displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer section and a dynamic balance equation of the pile body unit, obtaining a section corner of the pile body, wherein the expression is as follows:
when the pile body does not undergo shear deformation, the formula (8) can be degenerated into
Substituting formula (7) into formula (8) can obtain a cross-section rotation angle which is generally solved as:
step S32: determining the interrelationship among the bending moment, shearing force, horizontal displacement and section rotation angle of the pile body;
the relation expression of the bending moment of the pile body and the section rotation angle is as follows:
the relation expression among the bending moment, the shearing force, the horizontal displacement and the section rotation angle of the pile body is as follows:
order the
The coefficients to be determined in the formulae (9), (10) and (11) can be expressed as: />
in the formula ：
A j2 、A j3 、A j4 、B j2 、B j3 、B j4 、C j2 、C j3 、C j4 、D j2 、D j3 、D j4 the unknown coefficient can be obtained by the boundary conditions of the pile top and the pile bottom;
A j2 、A j3 、A j4 、B j2 、B j3 、B j4 、C j2 、C j3 、C j4 、D j2 、D j3 、D j4 the unknown coefficient can be obtained by 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 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;
at the sections of the j-th section and the j+1-th section, the continuity expressions of the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile body are as follows:
the coefficient matrix equation set obtained by integrating the equation (12) and the equation (13) is as follows:
in the formula
{T j }＝[A j1 B j1 C j1 D j1 ] T ；
From formula (14):
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j } (15)
then the corresponding coefficient matrix { T of the mth pile body can be obtained by the recurrence relation m The } can be expressed as:
step S34: setting boundary conditions of pile top and pile bottom, and according to the corresponding coefficient matrix { T } of the mth pile body m Obtaining the horizontal displacement, bending moment and shearing force of the pile body;
the boundary conditions of the pile top and the pile bottom are set as follows:
substituting the coefficient expression (12) into the expression (17) can be simplified to obtain:
the { T } can be obtained by combining (16) - (18) 1 According to the corresponding coefficient matrix { T } of the mth section pile body m And obtaining the 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:
in the formula ,umax (z)、m max (z)、p max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
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 (3)
1. The large-diameter single pile horizontal vibration analysis method considering the axial force action is characterized by comprising the following steps of:
s1: the following assumed conditions are introduced, and a large-diameter single pile horizontal vibration analysis model under the action of axial and horizontal forces of the Pasternak layered foundation 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 circular homogeneous constant-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: and 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:
in the formula ,
respectively horizontal displacement and section rotation angle of the mass point of the pile body of the jth layer section; z is the direction of pile foundation along depth, p is the upper mark of representative pile; a is that p 、G p 、E p 、I p 、m p Respectively the pile body sectional area, the shear modulus, the elastic modulus, the section moment of inertia and the mass per unit length; t is time; n (N) 0 Is the axial force acting on the pile top; k' is the shear shape factor; the thickness, rigidity coefficient, damping coefficient and foundation shear coefficient of the j-th layer of soil are respectively h j 、/>
and />
B 0 Calculated width for stake =0.9 (1.5d+0.5) for +.>
and />
The method is determined according to the following formula:
in the formula ,
the shear wave velocity of the j-th layer of soil; />
and />
The elastic modulus, the density, the damping coefficient and the poisson ratio of the j-th layer of soil are respectively; />
Is dimensionless frequency, and omega is excitation circle frequency;
the thickness of the shear layer of the jth foundation soil is +.>
d is the diameter of the pile;
pile tops are subjected to simple harmonic vibration in steady-state vibration, and pile body horizontal displacement and rotation angle are simplified into:
in the formula ,
for the horizontal displacement amplitude of the j-th layer pile body, < > for the j-th layer pile body>
The section corner amplitude value of the jth pile body is the section corner amplitude value of the jth pile body;
substituting formula (5) into formula (1) to obtain the following equation:
in the formula ,
M p ＝E p I p ，J p ＝K'A p G p ，S p ＝ρ p I p ω 2 ，
the characteristic root corresponding to the formula (6) is
Obtaining displacement general solution of the horizontal displacement amplitude of the j-th layer pile body:
wherein ,
unknown coefficient A j1 、B j1 、C j1 、D j1 The value of (2) is determined by the boundary conditions of the pile top and the pile bottom;
s3: and (2) solving the dynamic balance equation in the step (S2) to obtain horizontal vibration analysis parameters of the single pile in the layered soil, wherein the parameters at least comprise horizontal displacement, bending moment, shearing force and section rotation angle of the pile body.
2. The method according to claim 1, wherein in the step S3, the process of solving the dynamic balance equation in the step S2 to obtain the horizontal vibration analysis parameters of the mono-pile in the layered soil includes the steps of:
step S31: according to displacement general solution of the horizontal displacement amplitude of the pile body of the jth layer section and a dynamic balance equation of the pile body unit, obtaining a section corner of the pile body, wherein the expression is as follows:
when the pile body does not undergo shear deformation, the formula (8) is degraded into
Substituting formula (7) into formula (8) to obtain a cross-section rotation angle general solution:
step S32: determining the interrelationship among the bending moment, shearing force, horizontal displacement and section rotation angle of the pile body;
the relation expression of the bending moment of the pile body and the section rotation angle is as follows:
the relation expression among the bending moment, the shearing force, the horizontal displacement and the section rotation angle of the pile body is as follows:
order the
The predetermined coefficients in the formulae (9), (10) and (11) are expressed as:
in the formula ：
A j2 、A j3 、A j4 、B j2 、B j3 、B j4 、C j2 、C j3 、C j4 、D j2 、D j3 、D j4 obtaining unknown coefficients by boundary conditions of pile tops and pile bottoms;
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;
at the sections of the j-th section and the j+1-th section, the continuity expressions of the horizontal displacement, the rotation angle, the bending moment and the shearing force of the pile body are as follows:
the coefficient matrix equation set obtained by integrating the equation (12) and the equation (13) is as follows:
in the formula
{T j }＝[A j1 B j1 C j1 D j1 ] T ；
Obtained from formula (14):
{T j+1 }＝[F j+1 (z j )] -1 [F j (z j )]{T j } (15)
the coefficient matrix { T { corresponding to the mth pile body is calculated by a recurrence relation m Expressed as:
step S34: setting boundary conditions of pile top and pile bottom, and according to the corresponding coefficient matrix { T } of the mth pile body m Obtaining the horizontal displacement, bending moment and shearing force of the pile body;
the boundary conditions of the pile top and the pile bottom are set as follows:
substituting the coefficient expression (12) into the expression (17) for simplification to obtain:
the { T } is obtained by the combined type (16), (17), (18 a), (18 b) 1 According to the corresponding coefficient matrix { T } of the mth section pile body m And obtaining the horizontal displacement, bending moment and shearing force of each section of the pile body.
3. The method according to claim 2, further comprising, after the step S34: introducing dimensionless parameters;
the dimensionless parameter expression is:
in the formula ,umax (z)、m max (z)、p max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
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| CN115630458A (en) * | 2022-10-25 | 2023-01-20 | 北京铁科特种工程技术有限公司 | Method for casting continuous beam in situ based on elastic foundation beam theory and application thereof |
| CN116244978B (en) * | 2022-11-10 | 2024-04-09 | 中山大学 | Pipeline landslide influence calculation method based on Timoshenko Liang Moxing |
| CN116522724B (en) * | 2023-04-27 | 2024-01-26 | 华东交通大学 | Slope bridge pile dynamic response calculation method based on terrain effect |
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| CN111310264A (en) * | 2020-02-07 | 2020-06-19 | 大连海事大学 | Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model |
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| CN111310264A (en) * | 2020-02-07 | 2020-06-19 | 大连海事大学 | Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model |
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