CN112287444B - Method and system for analyzing horizontal dynamic interaction of adjacent pile foundations in layered Pasternak foundation - Google Patents
Method and system for analyzing horizontal dynamic interaction of adjacent pile foundations in layered Pasternak foundation Download PDFInfo
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Abstract
The embodiment of the application discloses a method and a system for analyzing horizontal dynamic interaction of adjacent pile foundations in a layered Pasternak foundation, wherein the method comprises the following steps: s1, setting assumption conditions corresponding to the simplified mechanical model of horizontal power interaction of adjacent pile foundations; s2, creating a dynamic balance equation of the pile body unit of the active pile; s3, creating a relation equation of horizontal vibration displacement and rotation angle of the pile body of the driving pile and obtaining a corresponding displacement general solution; s4, calculating the pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile; s5, calculating dynamic displacement of the passive pile II caused by the active pile I and calculating adjacent pile interaction factors. The application can better simulate the constraint effect of the soil body around the pile on the pile body; and meanwhile, the horizontal power interaction of the adjacent pile foundations under the action of axial force is considered, so that the method can be suitable for the horizontal vibration problem of the adjacent pile foundations under the action of complex multidirectional loads, is more suitable for the research of pile group interaction systems in actual engineering, and can provide guidance and reference effects for pile foundation power theory.
Description
Technical Field
The application relates to the technical field of pile soil horizontal vibration, in particular to a method and a system for analyzing horizontal dynamic interaction of adjacent pile foundations in a layered Pasternak foundation by considering axial force action.
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. However, 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 strict in theory. And the foundation soil shearing effect is considered on the basis of the Winkler model by using the Pasternak foundation model, so that the pile soil coupling system can more reasonably simulate the actual engineering working condition. At present, a technology for carrying out pile soil horizontal vibration by utilizing a Pasternak foundation model appears, but the technology only considers pile body bending and shearing deformation of a single pile, and can be used for solving the pile body horizontal vibration dynamic response problem under simple harmonic excitation force, but in actual engineering, pile foundations often appear in a pile group mode. The pile group effect can appear under the action of external dynamic load, namely, one pile is influenced by dynamic load to generate vibration, the vibration can enable surrounding soil bodies to vibrate, vibration waves are transmitted in soil media to further influence the vibration of other surrounding piles, and therefore the interaction between adjacent piles is considered when the pile group power problem is solved. And with the rise of large-scale complex buildings, the stress analysis of foundation piles under the simultaneous action of complex multidirectional loads becomes extremely important.
Disclosure of Invention
Based on the above, in order to solve the defects existing in the prior art, the application provides an adjacent pile foundation horizontal power interaction analysis method considering axial force action in a layered Pasternak foundation.
The horizontal dynamic interaction analysis method for the adjacent pile foundations in the layered Pasternak foundation is characterized by comprising the following steps of:
s1, giving assumption conditions corresponding to the horizontal power interaction simplified mechanical model of the adjacent pile foundations, wherein the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be divided into n layers along the longitudinal direction of the pile body, each layer of soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the small deformation condition, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom;
s2, creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considered shaft and the transverse force;
s3, creating a relation equation of horizontal vibration displacement and rotation angle of the pile body of the driving pile and obtaining a corresponding displacement general solution;
s4, calculating the pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
s5, calculating dynamic displacement of the passive pile II caused by the active pile I and calculating adjacent pile interaction factors.
Optionally, in one embodiment, the creating of the model in S2 includes:
and establishing a dynamic balance equation of a jth pile body unit of a single-pile horizontal vibration model of the driving pile, wherein the dynamic balance equation is as follows:
in the formula (1):the horizontal displacement of the mass point of the pile body of the j-th section active pile I is shown; e (E) p 、I p 、m p Respectively the pile body elastic modulus, the section moment of inertia and the mass per unit length, N 0 Is the axial force acting on the pile top; />The shear rigidity of foundation soil around the j-th layer of pile is obtained; b (B) 0 For calculating width of pile, B 0 =0.9 (1.5d+0.5); j is the number of soil layers, j=1, 2, once again, m, n;
wherein , and />Then it is determined as follows:
in the formula ：the shear wave velocity of the soil around the pile; /> and />The elastic modulus, density, damping coefficient and poisson ratio of the pile surrounding soil are respectively; />Is a dimensionless frequency; />The thickness of the shear layer of the jth foundation soil is equal to the value +.>
Optionally, in one embodiment, the equation of the relationship between the horizontal vibration displacement and the rotation angle of the driving pile body in S3 is expressed as:
in the formula ,the horizontal displacement amplitude of the mass point of the j-th section of the pile body of the active pile I is set;
the process of obtaining a general solution of the equation includes:
substituting the formula (5) into the formula (1) respectively, the following equation is obtained:
in the formula ,Wp ＝E p I p ，J p ＝K'A p G p ，
Based on the 4 characteristic roots corresponding to the formula (6)Its equation displacement solution is:
in the formula (7), the amino acid sequence of the compound,coefficient A 1_1j 、B 1_1j 、C 1_1j 、D 1_1j The value of (2) is determined by the following boundary conditions.
Optionally, in one embodiment, in the step S4, based on Euler beam theory, a correlation between a pile body rotation angle, a bending moment, a shearing force and a pile body horizontal displacement of the driving pile is:
order the f 7j ＝2λ j χ j ；
Wherein the expressions corresponding to the respective coefficients in the formulas (9), (10) and (11):
coefficient A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j Determined by boundary conditions;
and finally, calculating the pile body horizontal vibration displacement, the pile body bending moment and the shearing force of the driving pile based on the pile body corner, the bending moment, the shearing force and the pile body horizontal displacement interrelation expression of the driving pile.
Optionally, in one embodiment, in S4, the solving process of each coefficient includes the following steps:
by utilizing the continuous condition of the soil layer junction, the horizontal displacement, the corner, the bending moment and the shearing force of the pile are required to be continuous at the sections of the jth section and the jth+1th section of the pile body, namely:
then the coefficients A are obtained by combining the formula (12) and the formula (13) 1_1j 、B 1_1j 、C 1_1j 、D 1_1j 、A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j The matrix equation set of (2) is as follows:
[F I_j (z j )]{T I_j }＝[F I_j+1 (z j )]{T I_j+1 } (14)
in the formula ：{TI_j }＝[A 1_1j B 1_1j C 1_1j D 1_1j ] T ；
Obtained by the formula (14):
{T I_j+1 }＝[F I_j+1 (z j )] -1 [F I_j (z j )]{T I_j } (15)
then the corresponding coefficient matrix { T of the m-th pile body of the active pile I I_m Expressed as:
further consider the pile top and bottom boundary conditions, which are
And substituting the expressions of displacement, rotation angle, bending moment and shearing force into the expression (17) to simplify the expression:
in the formula ：[TI_1 ]＝[ A1_11 B 1_11 C 1_11 D 1_11 ] T 。
Substituting equation (16) into equation (18 b) yields the value of { T ] I_1 Two equations of { T } are then combined (18 a) to give { T } I_1 Four equations for { T } to find I_1 -a }; obtaining a coefficient matrix { T } corresponding to the pile body of any m sections according to a recurrence formula (16) I_m Then each section of the pile body can be obtained; and according to the pile body horizontal displacement expression, the pile body bending moment and shear force distribution is obtained by utilizing the relation between the pile body bending moment and shear force and the pile body horizontal displacement.
For facilitating the subsequent analysis, the following dimensionless parameters were introduced as follows:
in the formula ,umax (z)、m max (z)、q max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
Optionally, in one embodiment, in S5, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I includes:
s51, setting the geometric dimensions and the material properties of each pile in the active pile I and the passive pile II to be the same;
s52, obtaining the site vibration displacement caused by the active pile I based on the attenuation function of the soil horizontal displacement, wherein the attenuation function of the soil horizontal displacement is as follows:
in the formula (20):θ is the included angle between the connecting line of the two piles and the vibration direction x, and S is the distance between the two piles;
the corresponding field vibration displacement caused by the j-th layer soil by the driving pile I is as follows:
s53, considering the dynamic interaction between the pile and the soil body, the dynamic balance equation of the passive pile II is as follows:
because the horizontal displacement and the rotation angle of each unit of the j-th section of the pile body of the driven pile II are expressed as
Further simplified formula (22):
in the formula ：the solution of equation (23) consists of a general solution and a special solution, and the general solution of the corresponding homogeneous equation is as follows:
in the formula ：λj 、χ j Expression of (2)A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j For undetermined coefficients
The special solution of formula (23) is set as:
in the formula ：γ1j ＝λ j +χ j i；γ 2j ＝λ j -χ j i
Substituting the formula (25) into the formula (23) to obtain:
then the solution of equation (23) is:
s54, determining coefficient A 21 、B 21 、C 21 、D 21 And calculates the displacement of the passive pile II, which specifically comprises:
based on Euler beam theory, the interrelationship between the pile body rotation angle, bending moment, shearing force and the horizontal displacement of the pile body of the passive pile II is as follows:
in the formula ：
in the formula ：
in the formula ：
coefficient expression A in formulas (24) to (30) 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The representation is:
considering the continuous condition of soil layer juncture, in passive stake II pile body jth section and jth+1th section cross section department, horizontal displacement, corner, moment of flexure and shear force of stake need be continuous, namely:
then the coefficients A are obtained by combining the formula (31) and the formula (32) 2_1j 、B 2_1j 、C 2_1j 、D 2_1j 、A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The matrix equation set for (a) is as follows:
[F II_j (z j )]{T II_j }+[R II_j (z j )]＝[F II_j+1 (z j )]{T II_j+1 }+[R II_j+1 (z j )] (33)
in the formula ：the expression of (2) is shown in formula (14)>{T II_j }＝[A 2_1j B 2_1j C 2_1j D 2_1j ] T ，
Obtained by the formula (33):
in the formula ：
using (34) to use the recurrence relation to make the corresponding coefficient matrix { T of the m-th pile body of the passive pile II II_m Expressed as:
considering the condition of the boundary condition pile top constraint corner and pile bottom fixed end, the boundary condition is:
let F 5j ＝F 1j +F 3j ；F 6j ＝F 2j +F 4j Respectively expressing the displacement, the rotation angle, the bending moment and the shearing force of the passive pileSubstituting into (36) to obtain:
in the formula ：[TII_1 ]＝[A 2_11 B 2_11 C 2_11 D 2_11 ] T ；
Obtaining an unknown variable coefficient expression A by using the formulas (35) and (36) 2_1 、B 2_1 、C 2_1 、D 2_1 To solve the four equations of each layerAnd solving a displacement distribution expression of the passive pile, and further solving an expression of the corner and the internal force of the passive pile by using a relational expression among coefficients.
Optionally, in one embodiment, in S5, a calculation formula of the adjacent pile interaction factor is:
in addition, in order to solve the defects existing in the traditional technology, a horizontal dynamic interaction analysis system of adjacent pile foundations in the layered Pasternak foundation is also provided.
A horizontal dynamic interaction analysis system for adjacent pile foundations in a layered masternak foundation, comprising:
the first data acquisition unit is used for giving the assumption conditions corresponding to the adjacent pile foundation horizontal dynamic interaction simplified mechanical model, and the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be divided into n layers along the longitudinal direction of the pile body, each layer of soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the small deformation condition, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom; the first model creation unit is used for creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considering shaft and the transverse force;
the second model creation unit is used for creating a relation equation of horizontal vibration displacement and rotation angle of the driving pile body and obtaining a corresponding displacement general solution;
the first data acquisition unit is used for calculating pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
and the second data acquisition unit is used for calculating the dynamic displacement of the passive pile II caused by the active pile I and calculating the adjacent pile interaction factor.
Optionally, in one embodiment, the creating process of the model in the first model creating unit includes:
and establishing a dynamic balance equation of a jth pile body unit of a single-pile horizontal vibration model of the driving pile, wherein the dynamic balance equation is as follows:
in the formula (1):the horizontal displacement of the mass point of the pile body of the j-th section active pile I is shown; e (E) p 、I p 、m p Respectively the pile body elastic modulus, the section moment of inertia and the mass per unit length, N 0 Is the axial force acting on the pile top; />The shear rigidity of foundation soil around the j-th layer of pile is obtained; b (B) 0 =0.9 (1.5d+0.5) is the calculated width of the stake; j is the number of soil layers, j=1, 2, once again, m, n;
wherein , and />Then it is determined as follows:
in the formula ：the shear wave velocity of the soil around the pile; /> and />The elastic modulus, density, damping coefficient and poisson ratio of the pile surrounding soil are respectively; />Is a dimensionless frequency; />The thickness of the shear layer of the jth foundation soil is equal to the value +.>
Optionally, in one embodiment, the equation of the relationship between the horizontal vibration displacement and the rotation angle of the driving pile body in the second model creation unit is expressed as:
in the formula ,the horizontal displacement amplitude of the mass point of the j-th section of the pile body of the active pile I is set;
the process of obtaining a general solution of the equation includes:
substituting the formula (5) into the formula (1) respectively, the following equation is obtained:
in the formula ,Wp ＝E p I p ，J p ＝K'A p G p ，
Based on the 4 characteristic roots corresponding to the formula (6)Its equation displacement solution is:
in the formula (7), the amino acid sequence of the compound,coefficient A 1_1j 、B 1_1j 、C 1_1j 、D 1_1j Will be determined by the boundary conditions.
Optionally, in one embodiment, based on Euler beam theory, the correlation between the pile body rotation angle, bending moment, shearing force and horizontal displacement of the pile body of the driving pile is:
order the f 7j ＝2λ j χ j ；
Wherein the expressions corresponding to the respective coefficients in the formulas (9), (10) and (11):
coefficient A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j Determined by boundary conditions;
and finally, calculating the pile body horizontal vibration displacement, the pile body bending moment and the shearing force of the driving pile based on the pile body corner, the bending moment, the shearing force and the pile body horizontal displacement interrelation expression of the driving pile.
Optionally, in one embodiment, in the first data acquisition unit, the solving process of each coefficient includes the following steps:
by utilizing the continuous condition of the soil layer junction, the horizontal displacement, the corner, the bending moment and the shearing force of the pile are required to be continuous at the sections of the jth section and the jth+1th section of the pile body, namely:
then the coefficients A are obtained by combining the formula (12) and the formula (13) 1_1j 、B 1_1j 、C 1_1j 、D 1_1j 、A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j The matrix equation set of (2) is as follows:
[F I_j (z j )]{T I_j }＝[F I_j+1 (z j )]{T I_j+1 } (14)
in the formula ：{TI_j }＝[A 1_1j B 1_1j C 1_1j D 1_1j ] T ；
Obtained by the formula (14):
{T I_j+1 }＝[F I_j+1 (z j )] -1 [F I_j (z j )]{T I_j } (15)
then the corresponding coefficient matrix { T of the m-th pile body of the active pile I I_m Expressed as:
further consider the boundary conditions of pile top and pile bottom
And substituting the expressions of displacement, rotation angle, bending moment and shearing force into the expression (17) to simplify the expression:
in the formula ：[TI_1 ]＝[A 1_11 B 1_11 C 1_11 D 1_11 ] T 。
Substituting equation (16) into equation (18 b) yields the value of { T ] I_1 Two equations of { T } are then combined (18 a) to give { T } I_1 Four equations for { T } to find I_1 -a }; obtaining a coefficient matrix { T } corresponding to the pile body of any m sections according to a recurrence formula (16) I_m Then each section of the pile body can be obtained; and according to the pile body horizontal displacement expression, the pile body bending moment and shear force distribution is obtained by utilizing the relation between the pile body bending moment and shear force and the pile body horizontal displacement.
For facilitating the subsequent analysis, the following dimensionless parameters were introduced as follows:
in the formula ,umax (z)、m max (z)、q max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
Optionally, in one embodiment, in the first data acquisition unit, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I includes:
in said S5, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I comprises:
s51, setting the geometric dimensions and the material properties of each pile in the active pile I and the passive pile II to be the same;
s52, obtaining the site vibration displacement caused by the active pile I based on the attenuation function of the soil horizontal displacement, wherein the attenuation function of the soil horizontal displacement is as follows:
in the formula (20):θ is the included angle between the connecting line of the two piles and the vibration direction x, and S is the distance between the two piles;
the corresponding field vibration displacement caused by the j-th layer soil by the driving pile I is as follows:
s53, considering the dynamic interaction between the pile and the soil body, the dynamic balance equation of the passive pile II is as follows:
because the horizontal displacement and the rotation angle of each unit of the j-th section of the pile body of the driven pile II are expressed as
Further simplified formula (22):
in the formula ：the solution of equation (23) consists of a general solution and a special solution, and the general solution of the corresponding homogeneous equation is as follows:
in the formula ：λj 、χ j Expression of (2)A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j For undetermined coefficients
The special solution of formula (23) is set as:
in the formula ：γ1j ＝λ j +χ j i ；γ 2j ＝λ j -χ j i
Substituting the formula (25) into the formula (23) to obtain:
then the solution of equation (23) is:
s54, determining coefficient A 21 、B 21 、C 21 、D 21 And calculates the displacement of the passive pile II, which specifically comprises:
based on Euler beam theory, the interrelationship between the pile body rotation angle, bending moment, shearing force and the horizontal displacement of the pile body of the passive pile II is as follows:
in the formula ：
in the formula ：
in the formula ：
coefficient expression A in formulas (24) to (30) 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The representation is:
considering the continuous condition of soil layer juncture, in passive stake II pile body jth section and jth+1th section cross section department, horizontal displacement, corner, moment of flexure and shear force of stake need be continuous, namely:
then the coefficients A are obtained by combining the formula (31) and the formula (32) 2_1j 、B 2_1j 、C 2_1j 、D 2_1j 、A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The matrix equation set for (a) is as follows:
[F II_j (z j )]{T II_j }+[R II_j (z j )]＝[F II_j+1 (z j )]{T II_j+1 }+[R II_j+1 (z j )] (33)
in the formula ：the expression of (2) is shown in formula (14)>{T II_j }＝[A 2_1j B 2_1j C 2_1j D 2_1j ] T ，
Obtained by the formula (33):
in the formula ：
using (34) to use the recurrence relation to make the corresponding coefficient matrix { T of the m-th pile body of the passive pile II II_m Expressed as:
considering the condition of the boundary condition pile top constraint corner and pile bottom fixed end, the boundary condition is:
let F 5j ＝F 1j +F 3j ；F 6j ＝F 2j +F 4j And (3) substituting the expressions of the displacement, the rotation angle, the bending moment and the shearing force of the passive pile into the expression (36) respectively to obtain:
in the formula ：[TII_1 ]＝[A 2_11 B 2_11 C 2_11 D 2_11 ] T ；
Obtaining an unknown variable coefficient expression A by using the formulas (35) and (36) 2_1 、B 2_1 、C 2_1 、D 2_1 To solve the four equations of each layerAnd solving a displacement distribution expression of the passive pile, and further solving an expression of the corner and the internal force of the passive pile by using a relational expression among coefficients. />
Optionally, in one embodiment, the formula for calculating the adjacent pile interaction factor is:
in addition, to solve the deficiencies of the conventional technology, a computer readable storage medium is also provided, which includes computer instructions that, when executed on a computer, cause the computer to perform the method.
The implementation of the embodiment of the application has the following beneficial effects:
the Pasternak foundation model which can be adopted by the application considers the shearing effect of soil around the pile, and the soil around the pile considers the longitudinal layering characteristic of the soil formed by natural deposition, so that the application can simulate the geological condition of actual engineering more truly; in addition, the influence of axial force is considered under the action of horizontal load, and the model comprehensively considers the actions of two loads, so that the model can be suitable for the problem of horizontal vibration of adjacent pile foundations under the action of complex multidirectional load, and can provide guidance and reference effects for the power theoretical design of the pile foundations under the complex multidirectional load working condition.
Drawings
In order to more clearly illustrate the embodiments of the application 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 application, 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 an implementation technique in one embodiment;
FIG. 2 is a simplified mechanical model of the interaction of adjacent pile foundations of a layered foundation in one embodiment;
FIG. 3 is a simplified computational model diagram of a single pile in one embodiment;
fig. 4 is a schematic diagram of a double pile plan position in one embodiment.
In the figure, W is a rigid substrate.
Detailed Description
The present application 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 application 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 application.
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 application belongs. The terminology used herein in the description of the application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. 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.
In view of the fact that the prior art does not consider complex multidirectional loading, and for the traditional pile soil horizontal vibration system, the shearing effect of the soil body is ignored due to the commonly adopted Winkler foundation model, the continuity of the soil body between longitudinal layers cannot be reflected, and therefore the calculation result is not strict in theory. Therefore, in this embodiment, a method for analyzing horizontal dynamic interaction of adjacent pile foundations based on a layered panernak foundation is specifically provided, and the method is mainly aimed at solving the problem of horizontal vibration of adjacent elongated piles (i.e. double piles) under the action of axial force in layered soil, and solves the problem of horizontal vibration of adjacent pile foundations under the action of complex multidirectional load by considering adjacent pile interaction factors, constructing a simplified mechanical model of the horizontal dynamic interaction of adjacent pile foundations based on a layered panernak foundation model under the action of vertical shaft and transverse force, and other techniques, wherein the adjacent pile interaction factors refer to the ratio of additional dynamic displacement of a driven pile II caused by external load excitation of a driving pile I to the displacement of the driving pile I under the action of self load.
Specific: as shown in fig. 1-4, a method for analyzing horizontal dynamic interaction of adjacent pile foundations in a layered panernak foundation is characterized by comprising the following steps:
s1, giving assumption conditions corresponding to the horizontal power interaction simplified mechanical model of the adjacent pile foundations, wherein the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be divided into n layers along the longitudinal direction of the pile body, each layer of soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the small deformation condition, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom;
s2, creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considered shaft and the transverse force;
s3, creating a relation equation of horizontal vibration displacement and rotation angle of the pile body of the driving pile and obtaining a corresponding displacement general solution;
s4, calculating the pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
s5, calculating dynamic displacement of the passive pile II caused by the active pile I and calculating adjacent pile interaction factors. In summary, the method for researching the horizontal dynamic interaction of the adjacent pile foundations under the action of axial force is considered based on the Pasternak foundation model, the Pasternak foundation model considers the shearing effect of soil around piles, and the soil around the piles considers the longitudinal layering characteristic of the soil due to natural deposition, so that the method can simulate the geological conditions of actual engineering more truly; in addition, the influence of axial force is considered under the action of horizontal load, and the model comprehensively considers the actions of two loads, so that the model can be suitable for the problem of horizontal vibration of adjacent pile foundations under the action of complex multidirectional load, and can provide guidance and reference effects for the power theoretical design of the pile foundations under the complex multidirectional load working condition. The scheme creates a Pasternak foundation model, soil longitudinal layering, axial force effect consideration and double pile interaction, and the slender pile foundation is simplified into an Euler Liang Moxing-mainly aiming at solving modes of horizontal vibration problems of adjacent slender piles (namely double piles) under the axial force effect consideration in layered soil, so that the problem of horizontal vibration of the adjacent pile foundation under the complex multidirectional load effect is applicable.
In some specific embodiments, a simplified mechanical model diagram of horizontal dynamic interaction between adjacent piles, i.e. active pile foundation I and passive pile foundation II, based on a Pasternak foundation model under the action of axial and transverse forces is shown in FIG. 1In the drawings, and />Respectively representing the rigidity coefficient, damping coefficient and foundation shear coefficient of the pile surrounding soil, L and d respectively representing the pile length and pile diameter, and simultaneously, the pile top of the driving pile I is subjected to horizontal harmonic excitation force Q 0 e iωt And bending moment M 0 e iωt Is Q 0 、M 0 The vibration force amplitude and the bending moment amplitude are respectively, omega is the vibration circle frequency, and the vibration circle frequency is +>t is time.
In some specific embodiments, the S2 axis, the elongated mono-pile simplified calculation model based on the pasernak foundation model under the action of the lateral force is shown in fig. 2, and the creating process of the model in S2 includes:
and (3) integrating the related theory of the Euler beam and the Pasternak foundation model, and establishing a dynamic balance equation of a jth pile body unit of a single-pile horizontal vibration model of the active pile, wherein the dynamic balance equation is as follows:
in the formula (1):the horizontal displacement of the mass point of the pile body of the j-th section active pile I is shown; e (E) p 、I p 、m p Respectively the pile body elastic modulus, the section moment of inertia and the mass per unit length, N 0 Is the axial force acting on the pile top; />The shear rigidity of foundation soil around the j-th layer of pile is obtained; b (B) 0 =0.9 (1.5d+0.5) is the calculated width of the stake; j is the number of soil layers, j=1, 2,...,m,...,n；
wherein , and />Then it is determined as follows:
in the formula ：the shear wave velocity of the soil around the pile; /> and />The elastic modulus, density, damping coefficient and poisson ratio of the pile surrounding soil are respectively; />Is a dimensionless frequency; />The thickness of the shear layer of the jth foundation soil is equal to the value +.>
In some specific embodiments, the equation of the relation between the horizontal vibration displacement and the rotation angle of the driving pile body in S3 is expressed as follows:
in the formula ,the horizontal displacement amplitude of the mass point of the j-th section of the pile body of the active pile I is set;
the process of obtaining a general solution of the equation includes:
substituting the formula (5) into the formula (1) respectively, the following equation is obtained:
in the formula ,Wp ＝E p I p ，J p ＝K'A p G p ，
Based on the 4 characteristic roots corresponding to the formula (6)Its equation displacement solution is:
in the formula (7), the amino acid sequence of the compound,coefficient A 1_1j 、B 1_1j 、C 1_1j 、D 1_1j Will be determined by the boundary conditions.
Meanwhile, in the process of establishing and deducing a control equation, based on the horizontal dynamic interaction characteristic of adjacent pile foundations in the layered Pasternak foundation, the shear deformation of soil mass around the pile and the axial force action of the pile top are considered, so that the following contents are introduced:
in one embodiment, in the step S4, based on Euler beam theory, the correlation between the pile body rotation angle, the bending moment, the shearing force and the pile body horizontal displacement of the driving pile is:
/>
order the f 7j ＝2λ j χ j ；
Wherein the expressions corresponding to the respective coefficients in the formulas (9), (10) and (11):
coefficient A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j Determined by boundary conditions;
and finally, calculating the pile body horizontal vibration displacement, the pile body bending moment and the shearing force of the driving pile based on the pile body corner, the bending moment, the shearing force and the pile body horizontal displacement interrelation expression of the driving pile.
In one more specific embodiment, in said S4, the coefficient a 11 、B 11 、C 11 、D 11 The solving process of (1) comprises the following steps:
by utilizing the continuous condition of the soil layer junction, the horizontal displacement, the corner, the bending moment and the shearing force of the pile are required to be continuous at the sections of the jth section and the jth+1th section of the pile body, namely:
the coefficients A obtained by the combination of the formula (12) and the formula (13) 1_1j 、B 1_1j 、C 1_1j 、D 1_1j 、A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j The matrix equation set of (2) is as follows:
[F I_j (z j )]{T I_j }＝[F I_j+1 (z j )]{T I_j+1 } (14)
in the formula ：{TI_j }＝[A 1_1j B 1_1j C 1_1j D 1_1j ] T ；
From formula (14):
{T I_j+1 }＝[F I_j+1 (z j )] -1 [F I_j (z j )]{T I_j } (15)
the corresponding coefficient matrix { T of the m-th pile body of the active pile I can be obtained by a recurrence relation I_m The } can be expressed as:
further consider the boundary conditions of pile top and pile bottom
L is the variable form of pile length L;
and substituting the expressions of displacement, rotation angle, bending moment and shearing force into the expression (17) to obtain the composite material by simplification:
in the formula ：[TI_1 ]＝[A 1_11 B 1_11 C 1_11 D 1_11 ] T 。
Substitution of equation (16) into equation (18 b) yields the correlation { T } I_1 Two equations of { T } are found by the re-association (18 a) I_1 Four equations of { T } can be found I_1 -a }; according to the recurrence formula (16), the corresponding coefficient matrix { T } of any m sections of pile bodies can be finally obtained I_m Then each section of the pile body can be obtained; according to the pile body horizontal displacement expression, the pile body bending moment and shear force distribution can be obtained by utilizing the relation between the pile body bending moment and shear force and the pile body horizontal displacement.
For facilitating the subsequent analysis, the following dimensionless parameters were introduced as follows:
in the formula ,umax (z)、m max (z)、q max And (z) is the maximum value of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively.
In one specific embodiment, in S5, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I includes:
s51, setting the geometric dimensions and the material properties of each pile in the adjacent piles, namely each pile in the driving pile I and each pile in the driven pile II to be the same; the analysis is carried out to obtain a horizontal dynamic response analysis solution of the active pile I under the action of simple harmonic load, and dynamic displacement of the passive pile II caused by the active pile I is further analyzed, wherein the plane position schematic diagram of the two piles is shown in fig. 3, the included angle between the connecting line of the two piles and the vibration direction x is theta, and the distance S between the two piles is equal.
S52, obtaining the site vibration displacement caused by the active pile I based on the attenuation function of the soil horizontal displacement, wherein the attenuation function of the soil horizontal displacement is as follows:
in the formula ：
the field vibration displacement of the j-th layer soil caused by the active pile I is as follows:
the horizontal displacement and the rotation angle of each unit of the j-th section of the driving pile II can be expressed as Considering the dynamic interaction between the pile and the soil body, the dynamic balance equation of the j-th section pile body unit of the passive pile II is simplified into:
further simplification of formula (22) can be obtained:
in the formula ：
the solution of equation (23) consists of a general solution and a special solution, and the general solution of the corresponding homogeneous equation is as follows:
in the formula ：λj 、χ j The expression of A is the same as the expression of A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j Is a coefficient to be determined.
A special solution of formula (23) can be set as:
in the formula ：γ1j ＝λ j +χ j i；γ 2j ＝λ j -χ j i
Substituting the formula (25) into the formula (23) can obtain:
then the solution of equation (23) is:
based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement is:
in the formula ：
in the formula ：
in the formula ：
coefficient expression A in formulas (24) to (30) 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j Reference to (12) is made to A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j To express:
to obtain the unknown coefficient A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j Considering the continuous condition of soil layer juncture, in passive stake II pile body jth section and jth+1th section cross section department, horizontal displacement, corner, moment of flexure and shear force of stake need be continuous, namely:
then the coefficients A are obtained by combining the formula (31) and the formula (32) 2_1j 、B 2_1j 、C 2_1j 、D 2_1j 、A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The matrix equation set for (a) is as follows:
[F II_j (z j )]{T II_j }+[R II_j (z j )]＝[F II_j+1 (z j )]{T II_j+1 }+[R II_j+1 (z j )] (33)
in the formula ：the expression of (2) is shown in formula (14)>
{T II_j }＝[A 2_1j B 2_1j C 2_1j D 2_1j ] T ，
From formula (33):
in the formula ：
the corresponding coefficient matrix { T of the mth section pile body of the passive pile II can be obtained by using the recurrence relation (34) II_m The } can be expressed as:
further consider boundary condition pile top constraint corner, the condition of pile bottom stiff end, namely:
let F 5j ＝F 1j +F 3j ；F 6j ＝F 2j +F 4j And (3) substituting the expressions of the displacement, the rotation angle, the bending moment and the shearing force of the passive pile into the expression (36) respectively to obtain:
in the formula ：[TII_1 ]＝[A 2_11 B 2_11 C 2_11 D 2_11 ] T ；
The unknown variable coefficient expression A can be obtained by using the formulas (35), (36) 2_1 、B 2_1 、C 2_1 、D 2_1 To solve each layer of four equationsThe displacement distribution expression of the passive pile can be obtained, and the expression of the corner and the internal force of the passive pile can be further obtained by utilizing the relation between the coefficients.
According to the definition of the adjacent pile interaction factor expression, the following can be obtained:
based on the same inventive concept, the application also provides an adjacent pile foundation horizontal power interaction analysis system considering the axial force action in the layered Pasternak foundation.
A layered masternak foundation horizontal dynamic interaction analysis system that considers adjacent pile foundations, comprising:
the first data acquisition unit is used for giving the assumption conditions corresponding to the adjacent pile foundation horizontal dynamic interaction simplified mechanical model, and the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be a homogeneous soil layer, the soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the condition of small deformation, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom;
the first model creation unit is used for creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considering shaft and the transverse force;
the second model creation unit is used for creating a relation equation of horizontal vibration displacement and rotation angle of the driving pile body and obtaining a corresponding displacement general solution;
the first data acquisition unit is used for calculating pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
and the second data acquisition unit is used for calculating the dynamic displacement of the passive pile II caused by the active pile I and calculating the adjacent pile interaction factor.
Because the specific implementation of the system is consistent with the design principle and scheme of the method, the detailed description is omitted here.
Based on the same inventive concept, the application also proposes a computer-readable storage medium comprising computer instructions, which when run on a computer, cause the computer to perform the method.
The foregoing examples illustrate only a few embodiments of the application and are described in detail herein without thereby limiting the scope of the application. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the application, which are all within the scope of the application. Accordingly, the scope of protection of the present application is to be determined by the appended claims.
Claims (2)
1. The horizontal dynamic interaction analysis method for the adjacent pile foundations in the layered Pasternak foundation is characterized by comprising the following steps of:
s1, giving assumption conditions corresponding to the horizontal power interaction simplified mechanical model of the adjacent pile foundations, wherein the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be divided into n layers along the longitudinal direction of the pile body, each layer of soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the small deformation condition, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom;
s2, creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considered shaft and the transverse force;
s3, creating a relation equation of horizontal vibration displacement and rotation angle of the pile body of the driving pile and obtaining a corresponding displacement general solution;
s4, calculating the pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
s5, calculating dynamic displacement of the passive pile II caused by the active pile I and calculating adjacent pile interaction factors; the creating process of the model in S2 includes:
and establishing a dynamic balance equation of a jth pile body unit of a single-pile horizontal vibration model of the driving pile, wherein the dynamic balance equation is as follows:
in the formula (1):the horizontal displacement of the mass point of the pile body of the j-th section active pile I is shown; e (E) p 、I p 、m p Respectively the pile body elastic modulus, the section moment of inertia and the mass per unit length, N 0 Is the axial force acting on the pile top; />The shear rigidity of foundation soil around the j-th layer of pile is obtained; b (B) 0 For calculating width of pile, B 0 =0.9 (1.5d+0.5); j is the number of soil layers, j=1, 2, once again, m, n;
wherein , and />Then it is determined as follows:
in the formula ：the shear wave velocity of the soil around the pile; /> and />The elastic modulus, density, damping coefficient and poisson ratio of the pile surrounding soil are respectively; a, a 0 Is of dimensionless frequency-> The thickness of the shear layer of the jth foundation soil is equal to the value +.>And the relation equation of the horizontal vibration displacement and the rotation angle of the driving pile body in the step S3 is expressed as follows:
in the formula ,the horizontal displacement amplitude of the mass point of the j-th section of the pile body of the active pile I is set;
the process of obtaining a general solution of the equation includes:
substituting the formula (5) into the formula (1) respectively, the following equation is obtained:
in the formula ,Wp ＝E p I p ，J p ＝K'A p G p ，
Based on the 4 characteristic roots corresponding to the formula (6)Its equation displacement solution is:
in the formula (7), the amino acid sequence of the compound,coefficient A 1_1j 、B 1_1j 、C 1_1j 、D 1_1j The value of (2) is determined by boundary conditions; based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement of the driving pile is as follows:
order the f 7j ＝2λ j χ j ；
Wherein the expressions corresponding to the respective coefficients in the formulas (9), (10) and (11):
coefficient A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j Determined by boundary conditions;
finally, calculating pile body horizontal vibration displacement, pile body bending moment and shearing force of the driving pile based on the pile body corner, bending moment, shearing force and pile body horizontal displacement interrelation expression of the driving pile;
in S4, the solving process of each coefficient includes the following steps:
by utilizing the continuous condition of the soil layer junction, the horizontal displacement, the corner, the bending moment and the shearing force of the pile are required to be continuous at the sections of the jth section and the jth+1th section of the pile body, namely:
then the coefficients A are obtained by combining the formula (12) and the formula (13) 1_1j 、B 1_1j 、C 1_1j 、D 1_1j 、A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j The matrix equation set of (2) is as follows:
[F I_j (z j )]{T I_j }＝[F I_j+1 (z j )]{T I_j+1 } (14)
in the formula ：{TI_j }＝[A 1_1j B 1_1j C 1_1j D 1_1j ] T ；
Obtained by the formula (14):
{T I_j+1 }＝[F I_j+1 (z j )] -1 [F I_j (z j )]{T I_j } (15)
then the corresponding coefficient matrix { T of the m-th pile body of the active pile I I_m Expressed as:
further consider the boundary conditions of pile top and pile bottom
And substituting the expressions of displacement, rotation angle, bending moment and shearing force into the expression (17) to simplify the expression:
in the formula ：[TI_1 ]＝[A 1_11 B 1_11 C 1_11 D 1_11 ] T ；
Substituting equation (16) into equation (18 b) yields the value of { T ] I_1 Two equations of { T } are then combined (18 a) to give { T } I_1 Four equations for { T } to find I_1 -a }; obtaining a coefficient matrix { T } corresponding to the pile body of any m sections according to a recurrence formula (16) I_m Obtaining the horizontal displacement of each section of the pile body; according to the pile body horizontal displacement expression, the pile body bending moment and shearing force and the relation between the pile body bending moment and the shearing force and the pile body horizontal displacement are utilized to calculate the pile body bending moment and the shearing force distribution;
for facilitating the subsequent analysis, the following dimensionless parameters were introduced as follows:
in the formula ,umax (z)、m max (z)、q max (z) the maximum values of horizontal vibration displacement, bending moment and shearing force of the pile foundation respectively; in said S5, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I comprises:
s51, setting the geometric dimensions and the material properties of each pile in the active pile I and the passive pile II to be the same;
s52, obtaining the site vibration displacement caused by the active pile I based on the attenuation function of the soil horizontal displacement, wherein the attenuation function of the soil horizontal displacement is as follows:
in the formula (20):θ is the included angle between the connecting line of the two piles and the vibration direction x, and S is the distance between the two piles;
the corresponding field vibration displacement caused by the j-th layer soil by the driving pile I is as follows:
s53, considering the dynamic interaction between the pile and the soil body, the dynamic balance equation of the passive pile II is as follows:
because the horizontal displacement and the rotation angle of each unit of the j-th section of the pile body of the driven pile II are expressed as
Further simplified formula (22):
in the formula ：the solution of equation (23) consists of a general solution and a special solution, and the general solution of the corresponding homogeneous equation is as follows:
in the formula ：λj 、χ j Expression of (2)A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j For undetermined coefficients
The special solution of formula (23) is set as:
in the formula ：γ1j ＝λ j +χ j i；γ 2j ＝λ j -χ j i
Substituting the formula (25) into the formula (23) to obtain:
then the solution of equation (23) is:
s54, determining coefficient A 21 、B 21 、C 21 、D 21 And calculates the displacement of the passive pile II, which specifically comprises:
based on Euler beam theory, the interrelationship between the pile body rotation angle, bending moment, shearing force and the horizontal displacement of the pile body of the passive pile II is as follows:
in the formula ：
in the formula ：
in the formula ：
the expression of the coefficients in the formulas (24) to (30)A is a kind of 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The representation is:
considering the continuous condition of soil layer juncture, in passive stake II pile body jth section and jth+1th section cross section department, horizontal displacement, corner, moment of flexure and shear force of stake need be continuous, namely:
then the coefficients A are obtained by combining the formula (31) and the formula (32) 2_1j 、B 2_1j 、C 2_1j 、D 2_1j 、A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The matrix equation set for (a) is as follows:
[F II_j (z j )]{T II_j }+[R II_j (z j )]＝[F II_j+1 (z j )]{T II_j+1 }+[R II_j+1 (z j )](33) Wherein:the expression of (2) is shown in formula (14)>{T II_j }＝[A 2_1j B 2_1j C 2_1j D 2_1j ] T ，
Obtained by the formula (33):
in the formula ：
using (34) to use the recurrence relation to make the corresponding coefficient matrix { T of the m-th pile body of the passive pile II II_m Expressed as:
considering the condition of the boundary condition pile top constraint corner and pile bottom fixed end, the boundary condition is:
let F 5j ＝F 1j +F 3j ；F 6j ＝F 2j +F 4j And (3) substituting the expressions of the displacement, the rotation angle, the bending moment and the shearing force of the passive pile into the expression (36) respectively to obtain:
in the formula ：[TII_1 ]＝[A 2_11 B 2_11 C 2_11 D 2_11 ] T ；
Obtaining an unknown variable coefficient expression A by using the formulas (35) and (36) 2_1 、B 2_1 、C 2_1 、D 2_1 To solve the four equations of each layerSolving a displacement distribution expression of the passive pile, and further solving an expression of the corner and the internal force of the passive pile by using a relational expression among coefficients; the calculation formula of the adjacent pile interaction factor is as follows:
2. a horizontal dynamic interaction analysis system for adjacent pile foundations in a layered masternak foundation, comprising:
the first data acquisition unit is used for giving the assumption conditions corresponding to the adjacent pile foundation horizontal dynamic interaction simplified mechanical model, and the assumption conditions at least comprise: the pile body is simplified into a circular uniform-section and homogeneous Euler beam, the soil body around the pile is set to be divided into n layers along the longitudinal direction of the pile body, each layer of soil body is simplified into a Pasternak foundation model, each part of the model is set to meet the small deformation condition, and the pile-soil interface is completely contacted and has no relative sliding; meanwhile, setting that only horizontal displacement occurs at the pile top and solid end constraint is adopted at the pile bottom; the first model creation unit is used for creating a dynamic balance equation of the active pile body unit based on the Euler beam and the Pasternak foundation model under the action of the considering shaft and the transverse force;
the second model creation unit is used for creating a relation equation of horizontal vibration displacement and rotation angle of the driving pile body and obtaining a corresponding displacement general solution;
the first data acquisition unit is used for calculating pile body horizontal vibration displacement, pile body bending moment and shearing force of the active pile;
the second data acquisition unit is used for calculating the dynamic displacement of the passive pile II caused by the active pile I and calculating the adjacent pile interaction factor; the creation process of the model in the first model creation unit includes:
and establishing a dynamic balance equation of a jth pile body unit of a single-pile horizontal vibration model of the driving pile, wherein the dynamic balance equation is as follows:
in the formula (1):the horizontal displacement of the mass point of the pile body of the j-th section active pile I is shown; e (E) p 、I p 、m p Respectively the pile body elastic modulus, the section moment of inertia and the mass per unit length, N 0 Is the axial force acting on the pile top; />The shear rigidity of foundation soil around the j-th layer of pile is obtained; b (B) 0 For calculating width of pile, B 0 =0.9 (1.5d+0.5); j is the number of soil layers, j=1, 2, once again, m, n;
wherein , and />Then it is determined as follows:
in the formula ：the shear wave velocity of the soil around the pile; /> and />The elastic modulus, density, damping coefficient and poisson ratio of the pile surrounding soil are respectively; a, a 0 Is of dimensionless frequency-> The thickness of the shear layer of the jth foundation soil is equal to the value +.>
The relation equation of the horizontal vibration displacement and the rotation angle of the driving pile body in the second model creation unit is expressed as follows:
in the formula ,the horizontal displacement amplitude of the mass point of the j-th section of the pile body of the active pile I is set;
the process of obtaining a general solution of the equation includes:
substituting the formula (5) into the formula (1) respectively, the following equation is obtained:
in the formula ,Wp ＝E p I p ，J p ＝K'A p G p ，
Based on the 4 characteristic roots corresponding to the formula (6)Its equation displacement solution is:
in the formula (7), the amino acid sequence of the compound,coefficient A 1_1j 、B 1_1j 、C 1_1j 、D 1_1j Will be determined by the boundary conditions; based on Euler beam theory, the interrelationship between pile body rotation angle, bending moment, shearing force and pile body horizontal displacement of the driving pile is as follows:
order the f 7j ＝2λ j χ j ；
Wherein the expressions corresponding to the respective coefficients in the formulas (9), (10) and (11):
coefficient A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j Determined by boundary conditions;
finally, calculating pile body horizontal vibration displacement, pile body bending moment and shearing force of the driving pile based on the pile body corner, bending moment, shearing force and pile body horizontal displacement interrelation expression of the driving pile; in the first data acquisition unit, the solving process of each coefficient includes the steps of:
by utilizing the continuous condition of the soil layer junction, the horizontal displacement, the corner, the bending moment and the shearing force of the pile are required to be continuous at the sections of the jth section and the jth+1th section of the pile body, namely:
then the coefficients A are obtained by combining the formula (12) and the formula (13) 1_1j 、B 1_1j 、C 1_1j 、D 1_1j 、A 1_2j 、A 1_3j 、A 1_4j 、B 1_2j 、B 1_3j 、B 1_4j 、C 1_2j 、C 1_3j 、C 1_4j 、D 1_2j 、D 1_3j 、D 1_4j The matrix equation set of (2) is as follows:
[F I_j (z j )]{T I_j }＝[F I_j+1 (z j )]{T I_j+1 } (14)
in the formula ：{TI_j }＝[A 1_1j B 1_1j C 1_1j D 1_1j ] T ；
Obtained by the formula (14):
{T I_j+1 }＝[F I_j+1 (z j )] -1 [F I_j (z j )]{T I_j } (15)
then driving pile I mth section pile bodyCorresponding coefficient matrix { T I_m Expressed as:
further consider the boundary conditions of pile top and pile bottom
And substituting the expressions of displacement, rotation angle, bending moment and shearing force into the expression (17) to simplify the expression:
in the formula ：[TI_1 ]＝[A 1_11 B 1_11 C 1_11 D 1_11 ] T ；
Substituting equation (16) into equation (18 b) yields the value of { T ] I_1 Two equations of { T } are then combined (18 a) to give { T } I_1 Four equations for { T } to find I_1 -a }; obtaining a coefficient matrix { T } corresponding to the pile body of any m sections according to a recurrence formula (16) I_m Then each section of the pile body can be obtained; according to the pile body horizontal displacement expression, the pile body bending moment and shearing force and the relation between the pile body bending moment and the shearing force and the pile body horizontal displacement are utilized to calculate the pile body bending moment and the shearing force distribution;
for facilitating the subsequent analysis, the following dimensionless parameters were introduced as follows:
in the formula ,umax (z)、m max (z)、q max (z) horizontal vibration displacement and bending moment of pile foundation respectivelyAnd a shear maximum;
in the first data acquisition unit, the process of calculating the dynamic displacement of the passive pile II caused by the active pile I comprises:
s51, assuming that the geometric dimensions and the material properties of each pile in the active pile I and the passive pile II are the same;
s52, obtaining the site vibration displacement caused by the active pile I based on the attenuation function of the soil horizontal displacement, wherein the attenuation function of the soil horizontal displacement is as follows:
in the formula (20):θ is the included angle between the connecting line of the two piles and the vibration direction x, and S is the distance between the two piles;
the corresponding field vibration displacement caused by the j-th layer soil by the driving pile I is as follows:
s53, considering the dynamic interaction between the pile and the soil body, the dynamic balance equation of the passive pile II is as follows:
because the horizontal displacement and the rotation angle of each unit of the j-th section of the pile body of the driven pile II are expressed as
Further simplified formula (22):
in the formula ：the solution of equation (23) consists of a general solution and a special solution, and the general solution of the corresponding homogeneous equation is as follows:
in the formula ：λj 、χ j Expression of (2)A 2_1j 、B 2_1j 、C 2_1j 、D 2_1j For undetermined coefficients
The special solution of formula (23) is set as:
/>
in the formula ：γ1j ＝λ j +χ j i；γ 2j ＝λ j -χ j i
Substituting the formula (25) into the formula (23) to obtain:
then the solution of equation (23) is:
s54, determining coefficient A 21 、B 21 、C 21 、D 21 And calculates the displacement of the passive pile II, which specifically comprises:
based on Euler beam theory, the interrelationship between the pile body rotation angle, bending moment, shearing force and the horizontal displacement of the pile body of the passive pile II is as follows:
in the formula ：
in the formula ：
in the formula ：
coefficient expression A in formulas (24) to (30) 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The representation is:
/>
considering the continuous condition of soil layer juncture, in passive stake II pile body jth section and jth+1th section cross section department, horizontal displacement, corner, moment of flexure and shear force of stake need be continuous, namely:
then the coefficients A are obtained by combining the formula (31) and the formula (32) 2_1j 、B 2_1j 、C 2_1j 、D 2_1j 、A 2_2j 、B 2_2j 、C 2_2j 、D 2_2j 、A 2_3j 、B 2_3j 、C 2_3j 、D 2_3j 、A 2_4j 、B 2_4j 、C 2_4j 、D 2_4j The matrix equation set for (a) is as follows:
[F II_j (z j )]{T II_j }+[R II_j (z j )]＝[F II_j+1 (z j )]{T II_j+1 }+[R II_j+1 (z j )](33) Wherein:the expression of (2) is shown in formula (14)>{T II_j }＝[A 2_1j B 2_1j C 2_1j D 2_1j ] T ，
Obtained by the formula (33):
in the formula ：
using (34) to use the recurrence relation to make the corresponding coefficient matrix { T of the m-th pile body of the passive pile II II_m Expressed as:
considering the condition of the boundary condition pile top constraint corner and pile bottom fixed end, the boundary condition is:
let F 5j ＝F 1j +F 3j ；F 6j ＝F 2j +F 4j And (3) substituting the expressions of the displacement, the rotation angle, the bending moment and the shearing force of the passive pile into the expression (36) respectively to obtain:
in the formula ：[TII_1 ]＝[A 2_11 B 2_11 C 2_11 D 2_11 ] T ；
Obtaining an unknown variable coefficient expression A by using the formulas (35) and (36) 2_1 、B 2_1 、C 2_1 、D 2_1 To solve the four equations of each layerSolving a displacement distribution expression of the passive pile, and further solving an expression of the corner and the internal force of the passive pile by using a relational expression among coefficients; adjacent pile interactionThe calculation formula of the factor is as follows:
/>
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| CN111539050A (en) * | 2020-02-14 | 2020-08-14 | 大连海事大学 | Method for analyzing horizontal vibration of solid pile in bidirectional heterogeneous saturated soil |
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