CN110147631A - A kind of entity stake Horizontal vibration of piles method in the heterogeneous saturated soil of radial direction - Google Patents
A kind of entity stake Horizontal vibration of piles method in the heterogeneous saturated soil of radial direction Download PDFInfo
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Abstract
The invention discloses entity stake Horizontal vibration of piles methods in a kind of heterogeneous saturated soil of radial direction, the coupling between solid-liquid two-phase is considered by using radial heterogeneous saturation soil model and Biot porous medium theory model, the pile foundation level vibratory response model for establishing and having solved radial heterogeneous saturation soil model under plane strain condition, has obtained the impedance function of entity stake.Plane strain that the present invention uses assume can the more complicated Practical Project situation of simple process, clear concept, theoretical property are strong；Biot porous media model considers the coupling between solid-liquid two-phase simultaneously, is more more nearly actual condition compared with single-phase medium.Radial heterogeneity can consider pile peripheral earth construction disturbance effect, can provide theoretical direction and reference role for the research of increasingly complex saturated soil-stake dynamic interaction problem.
Description
Technical field
The present invention relates to civil engineering fields, more particularly, to entity stake horizontal vibration in a kind of heterogeneous saturated soil of radial direction
Analysis method.
Background technique
Pile Soil coupled vibrations characteristic research is the field of engineering technology such as Anti-seismic Pile Foundation, aseismatic design and dynamic pile detection
Theoretical basis, be also the hot issue of geotechnical engineering and Solid Mechanics all the time.
Research about the horizontal Coupled vibration problem of saturated soil-stake is based on the development of homogeneous saturated soil dielectric model, the mould
Pile peripheral earth is considered as homogeneous or longitudinal layering by type, and during pile foundation construction, due to soil compaction, relaxation and other factors
It influences, in stake week different range, different degrees of change can all occur for the property and parameter of the soil body, i.e., radial heterogeneous effect
It answers.It is more particularly suitable using radial heterogeneous saturation soil model at this time.In addition, Biot porous medium theory model considers solid-liquid
Coupling between two-phase wants complexity more relative to traditional ideal body (single-phase) medium, has more applicability.
Summary of the invention
It is an object of the invention to overcome drawbacks described above of the existing technology, provide in a kind of heterogeneous saturated soil of radial direction
Entity stake Horizontal vibration of piles method is examined by using radial heterogeneous saturation soil model and Biot porous medium theory model
Consider the coupling between solid-liquid two-phase, established and solves radial heterogeneous saturation soil model under plane strain condition
Pile foundation level vibratory response model has obtained the impedance function of entity stake.
To achieve the above object, technical scheme is as follows:
A kind of entity stake Horizontal vibration of piles method in the heterogeneous saturated soil of radial direction, which is characterized in that include the following steps
S1: it is introduced into following it is assumed that establishing under plane strain condition in radial heterogeneous saturated soil entity stake by horizontal drive
Model of vibration:
(1) assume that entity stake is linear elasticity homogeneous cross-section circle Bernoulli-Euler beam model, ignore pile body shearing and become
Shape, stake end use fixed bearing；
(2) pile peripheral earth is divided into interior zone and perimeter, and interior zone divides n ring layer, and each ring layer soil body is equal
Matter, isotropic two-phase are saturated elastic fluid；
(3) stake soil system vibration is small deformation, and Pile Soil interface completely attaches to, no disengagement and sliding phenomenon, and stake soil contacts
Face is waterproof, and each ring layer Soil Interface two sides are displaced continuous, stress equilibrium；
(4) when pile foundation level vibrates, pile peripheral earth is without vertical deformation；
S2: being based on Biot two-phase medium wave theory, establishes the movement side of each ring layer saturation soil body under plane strain condition
Journey and pile body horizontal vibration fundamental equation；
According in step S1 it is assumed that establishing Pile-soil System boundary condition；
S3: using Laplace transform, each ring layer saturation soil body under the plane strain condition established in solution procedure S2
The equation of motion obtains the lateral dynamic response of entity stake, is divided with the horizontal vibration to radial heterogeneous saturated soil pile foundation
Analysis.
Preferably, which is characterized in that in the step S2, the equation of motion of each ring layer saturation soil body under plane strain condition
For
Pile body horizontal vibration fundamental equation is
It is above it is various in, each symbol meaning is as follows:
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, and z-axis is square
To it is vertical downwards, r axis direction is horizontal direction, and zero point is located at the entity stake upper surface center of circle；
It is accorded with for different operators
Number；
urjThe radial displacement of soil skeleton in the soil body, u are saturated for jth ring layerθjSoil skeleton in the soil body is saturated for jth ring layer
Circumferential direction displacement, wrjRadial displacement of the fluid relative to soil skeleton in the soil body, w are saturated for jth ring layerθjThe soil body is saturated for jth ring layer
Circumferential direction displacement of the middle fluid relative to soil skeleton；
ρj=(1-nj)ρsj+njρfjFor jth ring layer saturated soil volume density, wherein ρfjFluid in the soil body is saturated for jth ring layer
Density, ρsjSoil particle density, n in the soil body are saturated for jth ring layerjSoil cracking behavior is saturated for jth ring layer；
mj=ρj/njFor the sticky coefficient of coup of jth ring layer soil skeleton and pore-fluid, bj=ρfjg/kdjFor jth ring layer soil
Body Darcy's law infiltration coefficient, g are acceleration of gravity；
λjSoil body modulus of shearing, the G of the soil body are saturated for jth ring layerjThe Lame constants of the soil body, υ are saturated for jth ring layersjFor
The Poisson's ratio of the jth ring layer saturation soil body；KsjThe bulk compressibility modulus of soil particle in the soil body, K are saturated for jth ring layerfjFor jth circle
The bulk compressibility modulus of fluid, K in the layer saturation soil bodybj=λj+(2/3)×GjThe body of soil skeleton in the soil body is saturated for jth ring layer
Product compression modulus；Kdj=Ksj[1+nj(Ksj/Kfj-1)]；
αj=1-Kbj/Ksj,It is soil particle and fluid in the characterization jth ring layer saturation soil body respectively
The constant of compressibility；
upFor entity stake body horizontal displacement；
EpModulus of shearing, I for entity stakepThe moment of inertia, A for entity stakepFor the cross-sectional area of entity stake, mpFor entity
The linear mass of stake, N1Horizontal force of the soil around pile of axial direction unit length to entity stake body when for horizontal vibration.
Preferably, in the step S2, Pile-soil System boundary condition is entity stake top boundary condition
Entity stake stake bottom boundaries condition
up|Z=H=0
Pile Soil completely attaches to and the waterproof condition in Pile Soil interface
The condition of continuity between ring layer
It is above it is various in, each symbol meaning is
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, and z-axis is square
To it is vertical downwards, r axis direction is horizontal direction, and zero point is located at the entity stake upper surface center of circle；
H is that entity stake stake is long, r1For entity stake radius；
upFor entity stake body horizontal displacement；P (t) is the excitation in entity stake top portion；
urjThe radial displacement of soil skeleton in the soil body, u are saturated for jth ring layerθjSoil skeleton in the soil body is saturated for jth ring layer
Circumferential direction displacement, wrjRadial displacement of the fluid relative to soil skeleton in the soil body, w are saturated for jth ring layerθjThe soil body is saturated for jth ring layer
Circumferential direction displacement of the middle fluid relative to soil skeleton；σrjThe method on the soil body and+1 ring layer saturated soil body interface of jth is saturated for jth ring layer
To direct stress, τθjThe tangential shearing stress on the soil body and+1 ring layer saturated soil body interface of jth is saturated for jth ring layer.
Preferably, in the step S3, the equation of motion and pile body of each ring layer saturation soil body under plane strain condition are solved
Horizontal vibration fundamental equation includes the following steps
S31: potential function is introduced respectively with fluid to the soil skeleton in the jth ring layer saturation soil bodyψsj,ψfj
S32: by potential functionψsj,ψfjThe equation of motion of the jth ring layer saturation soil body under plane strain condition is introduced,
Make Laplace transformation, obtains
ρfjs2Ψsj+mjs2Ψfj+bjsΨfj=0
S33: it solves potential function and obtains
Φsj=[Aj1K1(βj1r)+Bj1I1(βj1r)+Aj2K1(βj2r)+Bj2I1(βj2r)]cosθ
Φfj=[Cj1K1(βj1r)+Dj1I1(βj1r)+Cj2K1(βj2r)+Dj2I1(βj2r)]cosθ
Ψsj=[Aj3K1(βj3r)+Bj3I1(βj3r)]sinθ
Ψfj=[Cj3K1(βj3r)+Dj3I1(βj3r)]sinθ
S34: potential function is taken back, and obtains each displacement and the stress expression formula containing undetermined coefficient
Urj={ Aj1[-βj1K0(βj1r)-K1(βj1r)/r]+Aj2[-βj2K0(βj2r)-K1(βj2r)/r]+Aj3K1(βj3r)/r+
Bj1[βj1I0(βj1r)-I1(βj1r)/r]+Bj2[-βj2I0(βj2r)-I1(βj2r)/r]+Bj3I1(βj3r)/r}cosθ
Uθj={-Aj1K1(βj1r)/r-Aj2K1(βj2r)/r-Aj3[-βj3K0(βj3r)-K1(βj3r)/r]-Bj1I1(βj1r)-
Bj2I1(βj2r)/r-Bj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
Wrj={ Cj1[-βj1K0(βj1r)-K1(βj1r)/r]+Cj2[-βj2K0(βj2r)-K1(βj2r)/r]+Cj3K1(βj3r)/r+
Dj1[βj1I0(βj1r)-I1(βj1r)/r]+Dj2[-βj2I0(βj2r)-I1(βj2r)/r]+Dj3I1(βj3r)/r}cosθ
Wθj={-Cj1K1(βj1r)/r-Cj2K1(βj2r)/r-Cj3[-βj3K0(βj3r)-K1(βj3r)/r]-Dj1I1(βj1r)-
Dj2I1(βj2r)/r-Dj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
τrθj=Aj1Gj{-2[K1(βj1r)]'/r+2K1(βj1r)/r2}sinθ+Aj2Gj{-2[K1(βj2r)]'/r+2K1(βj2r)/r2}sinθ+Aj3Gj{[K1(βj3r)]'/r-2K1(βj3r)/r2-[K1(βj3r)]”}sinθ+Bj1Gj{-2[I1(βj1r)]'/r
+2I1(βj1r)/r2}sinθ+Bj2Gj{-2[I1(βj2r)]'/r+2I1(βj2r)/r2}sinθ+Bj3Gj{[I1(βj3r)]'/r-2I1
(βj3r)/r2-[I1(βj3r)]”}sinθ
S35: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient Aj1, Aj2, Aj3, Bj1, Bj2, Bj3,
Cj1, Cj2, Cj3, Dj1, Dj2, Dj3；
S36: soil around pile is to the horizontal force of pile body
S37: Laplace transform is carried out to pile body horizontal vibration fundamental equation
And it is solved to obtain
Up(z)=χ1cos(ηz)+χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)
Θp(z)=η [- χ1sin(ηz)+χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
Mp(z)=- EpIpη2[-χ1cos(ηz)-χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)]
Qp(z)=- EpIpη3[χ1sin(ηz)-χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
S38: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient
S39: the lateral dynamic response of computational entity stake
KQU=Qp(0)/Up(0)
It is also denoted as the Dimensionless Form of its real and imaginary parts
In above-mentioned expression formula, each symbol meaning is
S=i ω, wherein s is Laplace transform, and i is imaginary unit, and ω is exciting Loading frequency；
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, and z-axis is square
To it is vertical downwards, r axis direction is horizontal direction, and zero point is located at the entity stake upper surface center of circle；
And ψsjThe radial displacement u of soil skeleton in the soil body is saturated for jth ring layerrjWith circumferential displacement components uθjPotential function,
ψfjRadial displacement w of the fluid relative to soil skeleton in the soil body is saturated for jth ring layerrjW is displaced with circumferential directionθjPotential function；
ΦsjIt is potential functionLaplace transform, ΨsjIt is potential function ψsjLaplace transform, ΦfjIt is potential functionLaplace transform, ΨfjIt is potential function ψfjLaplace transform；
ρj=(1-nj)ρsj+njρfjFor jth ring layer saturated soil volume density, wherein ρfjFluid in the soil body is saturated for jth ring layer
Density, ρsjSoil particle density, n in the soil body are saturated for jth ring layerjSoil cracking behavior is saturated for jth ring layer；
mj=ρj/njFor the sticky coefficient of coup of jth ring layer soil skeleton and pore-fluid, bj=ρfjg/kdjFor jth ring layer soil
Body Darcy's law infiltration coefficient, g are acceleration of gravity；
λjSoil body modulus of shearing, the G of the soil body are saturated for jth ring layerjThe Lame constants of the soil body, υ are saturated for jth ring layersjFor
The Poisson's ratio of the jth ring layer saturation soil body；KsjThe bulk compressibility modulus of soil particle in the soil body, K are saturated for jth ring layerfjFor jth circle
The bulk compressibility modulus of fluid, K in the layer saturation soil bodybj=λj+(2/3)×GjThe body of soil skeleton in the soil body is saturated for jth ring layer
Product compression modulus；Kdj=Ksj[1+nj(Ksj/Kfj-1)]；
αj=1-Kbj/Ksj,It is soil particle and fluid in the characterization jth ring layer saturation soil body respectively
The constant of compressibility；
For operator notation；
Indicate that expression formula takes first order derivative to r in bracket；Indicate that expression formula takes second derivative to r in bracket；
It is the simplification symbol in calculating process；
It is also
Simplification symbol in calculating process；
It is also meter
Simplification symbol during calculation；
η4=(mps2+f1)/Ep/IpIt is also the simplification symbol in calculating process；
It is the first rank first kind, the second class modified Bessel function respectively；
Aj1, Aj2, Aj3, Bj1, Bj2, Bj3, Cj1, Cj2, Cj3, Dj1, Dj2, Dj3It is symbol undetermined, and there are following relationships
χ1, χ2, χ3, χ4It is also symbol undetermined；
a5=η [a3cos(ηH)-a1Sin (η H)], a6=-η [a2sin(ηH)+a4Cos (η H)] and
a7=η [sin (η H) cosh (η H)+cos (η H) sinh (η H)] is also the simplification symbol in calculating process.
Preferably, the method for determining jth ring layer soil body modulus of shearing is
The wherein modulus of shearing of G (r) to be jth ring layer soil away from stake soil interface centre distance be soil at r
Wherein f (r) is the function of soil body modulus of shearing variation
Wherein GR is the parameter for describing pile peripheral earth construction disturbance degree, and GR>1 is soil property hardening, and GR<1 is soil softening,
GR=1 is uniform soil quality, and q is positive index, r1It is entity stake radius, b is radial heterogeneous region soil body radius.
It can be seen from the above technical proposal that the present invention, which uses, is based on plane strain model radially heterogeneous saturation soil model
Entity stake horizontal vibration is analyzed, the plane strain hypothesis used being capable of the more complicated Practical Project of simple process
Situation, clear concept, theoretical property are strong；Biot porous media model considers the coupling between solid-liquid two-phase simultaneously, more singly
Phase medium is more more nearly actual condition.Radial heterogeneity can consider pile peripheral earth construction disturbance effect, can be increasingly complex
The research of saturated soil-stake dynamic interaction problem theoretical direction and reference role are provided.
Detailed description of the invention
Fig. 1 is the method for the present invention flow chart；
Fig. 2 is model schematic of the invention.
Specific embodiment
With reference to the accompanying drawing, specific embodiments of the present invention will be described in further detail.
It should be noted that in following specific embodiments, when describing embodiments of the invention in detail, in order to clear
Ground indicates structure of the invention in order to illustrate, spy does not draw to the structure in attached drawing according to general proportion, and has carried out part
Amplification, deformation and simplified processing, therefore, should be avoided in this, as limitation of the invention to understand.
In specific embodiment of the invention below, referring to Fig. 1, Fig. 1 is flow chart of the method for the present invention.As schemed
Show.
A kind of entity stake Horizontal vibration of piles method in the heterogeneous saturated soil of radial direction, which is characterized in that include the following steps
S1: it is introduced into following it is assumed that establishing under plane strain condition in radial heterogeneous saturated soil entity stake by horizontal drive
Model of vibration:
(1) assume that entity stake is linear elasticity homogeneous cross-section circle Bernoulli-Euler beam model, ignore pile body shearing and become
Shape, stake end use fixed bearing；
(2) pile peripheral earth is divided into interior zone and perimeter, and interior zone divides n ring layer, and each ring layer soil body is equal
Matter, isotropic two-phase are saturated elastic fluid；
(3) stake soil system vibration is small deformation, and Pile Soil interface completely attaches to, no disengagement and sliding phenomenon, and stake soil contacts
Face is waterproof, and each ring layer Soil Interface two sides are displaced continuous, stress equilibrium；
(4) when pile foundation level vibrates, pile peripheral earth is without vertical deformation.
The method for determining jth ring layer soil body modulus of shearing is
The wherein modulus of shearing of G (r) to be jth ring layer soil away from stake soil interface centre distance be soil at r
Wherein f (r) is the function of soil body modulus of shearing variation
Wherein GR is the parameter for describing pile peripheral earth construction disturbance degree, and GR>1 is soil property hardening, and GR<1 is soil softening,
GR=1 is uniform soil quality, and q is positive index, r1It is entity stake radius, b is radial heterogeneous region soil body radius.
S2: being based on Biot two-phase medium wave theory, establishes the movement side of each ring layer saturation soil body under plane strain condition
Journey and pile body horizontal vibration fundamental equation；And according in step S1 it is assumed that establishing Pile-soil System boundary condition.
The equation of motion of each ring layer saturation soil body is under plane strain condition
Pile body horizontal vibration fundamental equation is
Pile-soil System boundary condition is entity stake top boundary condition
Entity stake stake bottom boundaries condition
up|Z=H=0
Pile Soil completely attaches to and the waterproof condition in Pile Soil interface
The condition of continuity between ring layer
S3: using Laplace transform, each ring layer saturation soil body under the plane strain condition established in solution procedure S2
The equation of motion obtains the lateral dynamic response of entity stake, is divided with the horizontal vibration to radial heterogeneous saturated soil pile foundation
Analysis.The specific steps are
S31: potential function is introduced respectively with fluid to the soil skeleton in the jth ring layer saturation soil bodyψsj,ψfj
S32: by potential functionψsj,ψfjThe equation of motion for introducing the jth ring layer saturation soil body under plane strain condition, makees
Laplace transformation, obtains
ρfjs2Ψsj+mjs2Ψfj+bjsΨfj=0
S33: it solves potential function and obtains
Φsj=[Aj1K1(βj1r)+Bj1I1(βj1r)+Aj2K1(βj2r)+Bj2I1(βj2r)]cosθ
Φfj=[Cj1K1(βj1r)+Dj1I1(βj1r)+Cj2K1(βj2r)+Dj2I1(βj2r)]cosθ
Ψsj=[Aj3K1(βj3r)+Bj3I1(βj3r)]sinθ
Ψfj=[Cj3K1(βj3r)+Dj3I1(βj3r)]sinθ
S34: potential function is taken back, and obtains each displacement and the stress expression formula containing undetermined coefficient
Urj={ Aj1[-βj1K0(βj1r)-K1(βj1r)/r]+Aj2[-βj2K0(βj2r)-K1(βj2r)/r]+Aj3K1(βj3r)/r+
Bj1[βj1I0(βj1r)-I1(βj1r)/r]+Bj2[-βj2I0(βj2r)-I1(βj2r)/r]+Bj3I1(βj3r)/r}cosθ
Uθj={-Aj1K1(βj1r)/r-Aj2K1(βj2r)/r-Aj3[-βj3K0(βj3r)-K1(βj3r)/r]-Bj1I1(βj1r)-
Bj2I1(βj2r)/r-Bj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
Wrj={ Cj1[-βj1K0(βj1r)-K1(βj1r)/r]+Cj2[-βj2K0(βj2r)-K1(βj2r)/r]+Cj3K1(βj3r)/r+
Dj1[βj1I0(βj1r)-I1(βj1r)/r]+Dj2[-βj2I0(βj2r)-I1(βj2r)/r]+Dj3I1(βj3r)/r}cosθ
Wθj={-Cj1K1(βj1r)/r-Cj2K1(βj2r)/r-Cj3[-βj3K0(βj3r)-K1(βj3r)/r]-Dj1I1(βj1r)-
Dj2I1(βj2r)/r-Dj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
τrθj=Aj1Gj{-2[K1(βj1r)]'/r+2K1(βj1r)/r2}sinθ+Aj2Gj{-2[K1(βj2r)]'/r+2K1(βj2r)/r2}sinθ+Aj3Gj{[K1(βj3r)]'/r-2K1(βj3r)/r2-[K1(βj3r)]”}sinθ+Bj1Gj{-2[I1(βj1r)]'/r
+2I1(βj1r)/r2}sinθ+Bj2Gj{-2[I1(βj2r)]'/r+2I1(βj2r)/r2}sinθ+Bj3Gj{[I1(βj3r)]'/r-2I1
(βj3r)/r2-[I1(βj3r)]”}sinθ
S35: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient Aj1, Aj2, Aj3, Bj1, Bj2, Bj3,
Cj1, Cj2, Cj3, Dj1, Dj2, Dj3；
S36: soil around pile is to the horizontal force of pile body
S37: Laplace transform is carried out to pile body horizontal vibration fundamental equation
And it is solved to obtain
Up(z)=χ1cos(ηz)+χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)
Θp(z)=η [- χ1sin(ηz)+χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
Mp(z)=- EpIpη2[-χ1cos(ηz)-χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)]
Qp(z)=- EpIpη3[χ1sin(ηz)-χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
S38: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient
S39: the lateral dynamic response of computational entity stake
KQU=Qp(0)/Up(0)
It is also denoted as the Dimensionless Form of its real and imaginary parts
It is above it is various in, each symbol meaning is as follows:
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, and z-axis is square
To it is vertical downwards, r axis direction is horizontal direction, and zero point is located at the entity stake upper surface center of circle；
It is accorded with for different operators
Number；
urjThe radial displacement of soil skeleton in the soil body, u are saturated for jth ring layerθjSoil skeleton in the soil body is saturated for jth ring layer
Circumferential direction displacement, wrjRadial displacement of the fluid relative to soil skeleton in the soil body, w are saturated for jth ring layerθjThe soil body is saturated for jth ring layer
Circumferential direction displacement of the middle fluid relative to soil skeleton；σrjThe method on the soil body and+1 ring layer saturated soil body interface of jth is saturated for jth ring layer
To direct stress, τθjThe tangential shearing stress on the soil body and+1 ring layer saturated soil body interface of jth is saturated for jth ring layer；
ρj=(1-nj)ρsj+njρfjFor jth ring layer saturated soil volume density, wherein ρfjFluid in the soil body is saturated for jth ring layer
Density, ρsjSoil particle density, n in the soil body are saturated for jth ring layerjSoil cracking behavior is saturated for jth ring layer；
mj=ρj/njFor the sticky coefficient of coup of jth ring layer soil skeleton and pore-fluid, bj=ρfjg/kdjFor jth ring layer soil
Body Darcy's law infiltration coefficient, g are acceleration of gravity；
λjSoil body modulus of shearing, the G of the soil body are saturated for jth ring layerjThe Lame constants of the soil body, υ are saturated for jth ring layersIt is
The Poisson's ratio of the j ring layer saturation soil body；KsjThe bulk compressibility modulus of soil particle in the soil body, K are saturated for jth ring layerfjFor jth ring layer
It is saturated the bulk compressibility modulus of fluid in the soil body, Kbj=λj+(2/3)×GjThe volume of soil skeleton in the soil body is saturated for jth ring layer
Compression modulus；Kdj=Ksj[1+nj(Ksj/Kfj-1)]；
αj=1-Kbj/Ksj,It is soil particle and fluid in the characterization jth ring layer saturation soil body respectively
The constant of compressibility；
upFor entity stake body horizontal displacement；H is that entity stake stake is long, r1For entity stake radius；P (t) is entity stake top portion
Excitation；
EpModulus of shearing, I for entity stakepThe moment of inertia, A for entity stakepFor the cross-sectional area of entity stake, mpFor entity
The linear mass of stake, N1Horizontal force of the soil around pile of axial direction unit length to entity stake body when for horizontal vibration；
S=i ω, wherein s is Laplace transform, and i is imaginary unit, and ω is exciting Loading frequency；
And ψsjThe radial displacement u of soil skeleton in the soil body is saturated for jth ring layerrjWith circumferential displacement components uθjPotential function,
ψfjRadial displacement w of the fluid relative to soil skeleton in the soil body is saturated for jth ring layerrjW is displaced with circumferential directionθjPotential function；
ΦsjIt is potential functionLaplace transform, ΨsjIt is potential function ψsjLaplace transform, ΦfjIt is potential functionLaplace transform, ΨfjIt is potential function ψfjLaplace transform；
Indicate that expression formula takes first order derivative to r in bracket；Indicate that expression formula takes second derivative to r in bracket；
It is the simplification symbol in calculating process；
It is also the simplification symbol in calculating process；
It is also meter
Simplification symbol during calculation；
η4=(mps2+f1)/Ep/IpIt is also the simplification symbol in calculating process；
It is the first rank first kind, the second class modified Bessel function respectively；
Aj1, Aj2, Aj3, Bj1, Bj2, Bj3, Cj1, Cj2, Cj3, Dj1, Dj2, Dj3It is symbol undetermined, and there are following relationships
χ1, χ2, χ3, χ4It is also symbol undetermined；
a5=η [a3cos(ηH)-a1Sin (η H)], a6=-η [a2sin(ηH)+a4Cos (η H)] and
a7=η [sin (η H) cosh (η H)+cos (η H) sinh (η H)] is also the simplification symbol in calculating process.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto,
Anyone skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its
Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
Claims (5)
1. a kind of entity stake Horizontal vibration of piles method in heterogeneous saturated soil of radial direction, which is characterized in that include the following steps
S1: it is introduced into following it is assumed that establishing vibration of the entity stake by horizontal drive in radial heterogeneous saturated soil under plane strain condition
Movable model:
(1) assume that entity stake is linear elasticity homogeneous cross-section circle Bernoulli-Euler beam model, it is shear-deformable to ignore pile body, stake
End uses fixed bearing；
(2) pile peripheral earth is divided into interior zone and perimeter, and interior zone divides n ring layer, and each ring layer soil body is homogeneous, respectively
Elastic fluid is saturated to the two-phase of the same sex；
(3) stake soil system vibration is small deformation, and Pile Soil interface completely attaches to, no disengagement and sliding phenomenon, and pile-soil interface is not
Permeable, each ring layer Soil Interface two sides are displaced continuous, stress equilibrium；
(4) when pile foundation level vibrates, pile peripheral earth is without vertical deformation；
S2: being based on Biot two-phase medium wave theory, establish under plane strain condition the equation of motion of each ring layer saturation soil body and
Pile body horizontal vibration fundamental equation；
According in step S1 it is assumed that establishing Pile-soil System boundary condition；
S3: using Laplace transform, the movement of each ring layer saturation soil body under the plane strain condition established in solution procedure S2
Equation obtains the lateral dynamic response of entity stake, is analyzed with the horizontal vibration to radial heterogeneous saturated soil pile foundation.
2. analysis method according to claim 1, which is characterized in that in the step S2, respectively enclosed under plane strain condition
Layer saturation the soil body the equation of motion be
Pile body horizontal vibration fundamental equation is
It is above it is various in, each symbol meaning is as follows:
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, z-axis positive direction lead
Straight r axis direction is horizontal direction downwards, and zero point is located at the entity stake upper surface center of circle；
For different operator notations；
urjThe radial displacement of soil skeleton in the soil body, u are saturated for jth ring layerθjThe circumferential position of soil skeleton in the soil body is saturated for jth ring layer
It moves, wrjRadial displacement of the fluid relative to soil skeleton in the soil body, w are saturated for jth ring layerθjFluid in the soil body is saturated for jth ring layer
Circumferential direction displacement relative to soil skeleton；
ρj=(1-nj)ρsj+njρfjFor jth ring layer saturated soil volume density, wherein ρfjFor jth ring layer be saturated the soil body in fluid density,
ρsjSoil particle density, n in the soil body are saturated for jth ring layerjSoil cracking behavior is saturated for jth ring layer；
mj=ρj/njFor the sticky coefficient of coup of jth ring layer soil skeleton and pore-fluid, bj=ρfjg/kdjIt is reached for the jth ring layer soil body
Western law infiltration coefficient, g are acceleration of gravity；
λjSoil body modulus of shearing, the G of the soil body are saturated for jth ring layerjThe Lame constants of the soil body, υ are saturated for jth ring layersjFor jth circle
The Poisson's ratio of the layer saturation soil body；KsjThe bulk compressibility modulus of soil particle in the soil body, K are saturated for jth ring layerfjFor jth ring layer saturation
The bulk compressibility modulus of fluid, K in the soil bodybj=λj+(2/3)×GjThe volume compression of soil skeleton in the soil body is saturated for jth ring layer
Modulus；Kdj=Ksj[1+nj(Ksj/Kfj-1)]；
αj=1-Kbj/Ksj,It is soil particle and fluid compression in the characterization jth ring layer saturation soil body respectively
The constant of property；
upFor entity stake body horizontal displacement；
EpModulus of shearing, I for entity stakepThe moment of inertia, A for entity stakepFor the cross-sectional area of entity stake, mpFor the list of entity stake
Bit length quality, N1Horizontal force of the soil around pile of axial direction unit length to entity stake body when for horizontal vibration.
3. analysis method according to claim 2, which is characterized in that in the step S2, Pile-soil System boundary condition
For entity stake top boundary condition
Entity stake stake bottom boundaries condition
up|Z=H=0
Pile Soil completely attaches to and the waterproof condition in Pile Soil interface
The condition of continuity between ring layer
It is above it is various in, each symbol meaning is
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, z-axis positive direction lead
Straight r axis direction is horizontal direction downwards, and zero point is located at the entity stake upper surface center of circle；
H is that entity stake stake is long, r1For entity stake radius；
upFor entity stake body horizontal displacement；P (t) is the excitation in entity stake top portion；
urjThe radial displacement of soil skeleton in the soil body, u are saturated for jth ring layerθjThe circumferential position of soil skeleton in the soil body is saturated for jth ring layer
It moves, wrjRadial displacement of the fluid relative to soil skeleton in the soil body, w are saturated for jth ring layerθjFluid in the soil body is saturated for jth ring layer
Circumferential direction displacement relative to soil skeleton；σrjThe soil body is saturated for jth ring layer just to answer with the normal direction on+1 ring layer saturated soil body interface of jth
Power, τθjThe tangential shearing stress on the soil body and+1 ring layer saturated soil body interface of jth is saturated for jth ring layer.
4. analysis method according to claim 3, which is characterized in that in the step S3, solve under plane strain condition
The equation of motion and pile body horizontal vibration fundamental equation of each ring layer saturation soil body include the following steps
S31: potential function is introduced respectively with fluid to the soil skeleton in the jth ring layer saturation soil bodyψsj,ψfj
S32: by potential functionψsj,ψfjThe equation of motion for introducing the jth ring layer saturation soil body under plane strain condition, makees
Laplace transformation, obtains
ρfjs2Ψsj+mjs2Ψfj+bjsΨfj=0
S33: it solves potential function and obtains
Φsj=[Aj1K1(βj1r)+Bj1I1(βj1r)+Aj2K1(βj2r)+Bj2I1(βj2r)]cosθ
Φfj=[Cj1K1(βj1r)+Dj1I1(βj1r)+Cj2K1(βj2r)+Dj2I1(βj2r)]cosθ
Ψsj=[Aj3K1(βj3r)+Bj3I1(βj3r)]sinθ
Ψfj=[Cj3K1(βj3r)+Dj3I1(βj3r)]sinθ
S34: potential function is taken back, and obtains each displacement and the stress expression formula containing undetermined coefficient
Urj={ Aj1[-βj1K0(βj1r)-K1(βj1r)/r]+Aj2[-βj2K0(βj2r)-K1(βj2r)/r]+Aj3K1(βj3r)/r+Bj1
[βj1I0(βj1r)-I1(βj1r)/r]+Bj2[-βj2I0(βj2r)-I1(βj2r)/r]+Bj3I1(βj3r)/r}cosθ
Uθj={-Aj1K1(βj1r)/r-Aj2K1(βj2r)/r-Aj3[-βj3K0(βj3r)-K1(βj3r)/r]-Bj1I1(βj1r)-Bj2I1
(βj2r)/r-Bj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
Wrj={ Cj1[-βj1K0(βj1r)-K1(βj1r)/r]+Cj2[-βj2K0(βj2r)-K1(βj2r)/r]+Cj3K1(βj3r)/r+Dj1
[βj1I0(βj1r)-I1(βj1r)/r]+Dj2[-βj2I0(βj2r)-I1(βj2r)/r]+Dj3I1(βj3r)/r}cosθ
Wθj={-Cj1K1(βj1r)/r-Cj2K1(βj2r)/r-Cj3[-βj3K0(βj3r)-K1(βj3r)/r]-Dj1I1(βj1r)-Dj2I1
(βj2r)/r-Dj3[βj3I0(βj3r)-I1(βj3r)/r]}sinθ
τrθj=Aj1Gj{-2[K1(βj1r)]'/r+2K1(βj1r)/r2}sinθ+Aj2Gj{-2[K1(βj2r)]'/r+2K1(βj2r)/r2}
sinθ+Aj3Gj{[K1(βj3r)]'/r-2K1(βj3r)/r2-[K1(βj3r)]”}sinθ+Bj1Gj{-2[I1(βj1r)]'/r+2I1(βj1r)/r2}sinθ+Bj2Gj{-2[I1(βj2r)]'/r+2I1(βj2r)/r2}sinθ+Bj3Gj{[I1(βj3r)]'/r-2I1(βj3r)/
r2-[I1(βj3r)]”}sinθ
S35: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient Aj1, Aj2, Aj3, Bj1, Bj2, Bj3, Cj1,
Cj2, Cj3, Dj1, Dj2, Dj3；
S36: soil around pile is to the horizontal force of pile body
S37: Laplace transform is carried out to pile body horizontal vibration fundamental equation
And it is solved to obtain
Up(z)=χ1cos(ηz)+χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)
Θp(z)=η [- χ1sin(ηz)+χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
Mp(z)=- EpIpη2[-χ1cos(ηz)-χ2sin(ηz)+χ3cosh(ηz)+χ4sinh(ηz)]
Qp(z)=- EpIpη3[χ1sin(ηz)-χ2cos(ηz)+χ3sinh(ηz)+χ4cosh(ηz)]
S38: previous step expression formula is brought into boundary condition, solution obtains undetermined coefficient
S39: the lateral dynamic response of computational entity stake
KQU=Qp(0)/Up(0)
It is also denoted as the Dimensionless Form of its real and imaginary parts
In above-mentioned expression formula, each symbol meaning is
S=i ω, wherein s is Laplace transform, and i is imaginary unit, and ω is exciting Loading frequency；
J=1~n is the number order of ring layer, is 1, n total for ring layer with the adjacent ring layer number of entity stake；
R, θ are the coordinate of cylindrical coordinate, and wherein the zero point of cylindrical coordinate z-axis is located at the entity stake upper surface center of circle, z-axis positive direction lead
Straight r axis direction is horizontal direction downwards, and zero point is located at the entity stake upper surface center of circle；
And ψsjThe radial displacement u of soil skeleton in the soil body is saturated for jth ring layerrjWith circumferential displacement components uθjPotential function,ψfjFor
Jth ring layer is saturated radial displacement w of the fluid relative to soil skeleton in the soil bodyrjW is displaced with circumferential directionθjPotential function；
ΦsjIt is potential functionLaplace transform, ΨsjIt is potential function ψsjLaplace transform, ΦfjIt is potential function's
Laplace transform, ΨfjIt is potential function ψfjLaplace transform；
ρj=(1-nj)ρsj+njρfjFor jth ring layer saturated soil volume density, wherein ρfjFor jth ring layer be saturated the soil body in fluid density,
ρsjSoil particle density, n in the soil body are saturated for jth ring layerjSoil cracking behavior is saturated for jth ring layer；
mj=ρj/njFor the sticky coefficient of coup of jth ring layer soil skeleton and pore-fluid, bj=ρfjg/kdjIt is reached for the jth ring layer soil body
Western law infiltration coefficient, g are acceleration of gravity；
λjSoil body modulus of shearing, the G of the soil body are saturated for jth ring layerjThe Lame constants of the soil body, υ are saturated for jth ring layersjFor jth circle
The Poisson's ratio of the layer saturation soil body；KsjThe bulk compressibility modulus of soil particle in the soil body, K are saturated for jth ring layerfjFor jth ring layer saturation
The bulk compressibility modulus of fluid, K in the soil bodybj=λj+(2/3)×GjThe volume compression of soil skeleton in the soil body is saturated for jth ring layer
Modulus；Kdj=Ksj[1+nj(Ksj/Kfj-1)]；
αj=1-Kbj/Ksj,It is soil particle and fluid compression in the characterization jth ring layer saturation soil body respectively
The constant of property；
For operator notation；
Indicate that expression formula takes first order derivative to r in bracket；Indicate that expression formula takes second derivative to r in bracket；
It is the simplification symbol in calculating process；
It is also meter
Simplification symbol during calculation；
And it calculated
Simplification symbol in journey；
η4=(mps2+f1)/Ep/IpIt is also the simplification symbol in calculating process；
It is the first rank first kind, the second class modified Bessel function respectively；
Aj1, Aj2, Aj3, Bj1, Bj2, Bj3, Cj1, Cj2, Cj3, Dj1, Dj2, Dj3It is symbol undetermined, and there are following relationships
χ1, χ2, χ3, χ4It is also symbol undetermined；
a5=η [a3cos(ηH)-a1Sin (η H)], a6=-η [a2sin(ηH)+a4Cos (η H)] and a7=η [sin (η H) cosh (η H)+
Cos (η H) sinh (η H)] be also calculating process in simplification symbol.
5. analysis method according to claim 1, which is characterized in that in the step S1, determine that the jth ring layer soil body is sheared
The method of modulus is
The wherein modulus of shearing of G (r) to be jth ring layer soil away from stake soil interface centre distance be soil at r
Wherein f (r) is the function of soil body modulus of shearing variation
Wherein GR is the parameter for describing pile peripheral earth construction disturbance degree, and GR>1 is soil property hardening, and GR<1 is soil softening, GR=
1 is uniform soil quality, and q is positive index, r1It is entity stake radius, b is radial heterogeneous region soil body radius.
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| CN112034137B (en) * | 2020-08-14 | 2022-09-23 | 长沙理工大学 | Method for determining contact stress and vertical dynamic flexibility of rigid circular plate on saturated soil layer |
| CN112287445A (en) * | 2020-10-30 | 2021-01-29 | 大连海事大学 | Horizontal dynamic response analysis method and system for adjacent large-diameter piles |
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