CN110598262A - Calculation method of vertical impedance of vertical pile - Google Patents
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Abstract
The invention discloses a method for calculating vertical impedance of a vertical pile, which comprises the following steps: vertically dispersing a straight pile body and foundation soil according to the horizontal layering condition of a soil body, and radially dispersing the foundation soil according to the soil disturbance condition around the pile; solving a wave equation of each soil ring layer and calculating the dynamic stiffness on the pile-soil contact surface by a radial transfer matrix method; and calculating the vertical impedance of the straight pile by a vertical transfer matrix method after solving the equation. The advantages are that: the method is used for calculating and considering the vertical impedance of the straight pile under the condition that the saturated soil around the pile is radially heterogeneous due to the construction disturbance effect in the layered foundation, the radial heterogeneity of the layered saturated soil under the construction disturbance effect is considered through a two-way transfer matrix method, meanwhile, the stress-strain relation of the foundation soil is described through a fractional derivative operator, so that the calculation precision is improved, and the transverse inertia effect of the pile body vibration is considered through the Rayleigh rod vibration theory to describe the vertical vibration of the pile body. The method enables the vertical impedance of the vertical pile obtained through calculation to be more in line with engineering practice.
Description
Technical Field
The invention relates to a method for calculating vertical impedance of a vertical pile, in particular to a method for calculating vertical impedance of a vertical pile in disturbed layered saturated soil described by a fractional derivative.
Background
The vertical pile has the advantages of high bearing capacity and small settlement, so that the vertical pile is widely used as a supporting foundation of structures such as bridge engineering, offshore engineering, high-rise buildings and the like. Under the action of dynamic loads such as people, machinery, sea waves and the like, dynamic interaction exists between soil and the vertical piles, and further the dynamic characteristics of the upper structure are changed. It is therefore necessary to consider the effect of soil-to-vertical pile dynamic interaction on the dynamic characteristics of the superstructure when the superstructure is a power sensitive structure that is high-rise and top mass concentrated. In the problem of reflecting the dynamic interaction, determining the vertical impedance describing the relationship between the vibration displacement of the top of the vertical pile and the external excitation load is an important link.
The vertical impedance of the vertical pile is a complex function depending on the frequency of an external load, wherein the real part of the vertical impedance represents a stiffness coefficient, and the imaginary part of the vertical impedance represents a damping coefficient, and the vertical impedance is related to a plurality of factors such as the material characteristic, the geometric shape, the embedding condition, the soil distribution characteristic and the like of the pile foundation. In the current building foundation basic regulations (GB5007-2011) in China, dynamic analysis of a pile foundation is mainly popularized from a theory of bearing static load of the foundation and is corrected by adopting an empirical coefficient, and the theoretical analysis is not strict enough. Therefore, researchers and scientists have conducted research and achieved certain results. At present, vertical impedance of a straight pile mainly comprises two theoretical calculation methods, one is an analytic or semi-analytic method represented by an elastic foundation beam model, and the other is a numerical method represented by a finite element model. The former has clear physical concept and small calculation amount and can meet the requirement of engineering precision, so the method is widely applied to the problem of pile-soil interaction. The elastic foundation beam model generally compares a pile body with an Euler beam, and a spring and a damper are used for simulating the dynamic reaction force of a soil body to the pile body. However, the traditional euler beam ignores the transverse inertia effect of pile body vibration, and in addition, the coefficients of the spring and the damper in the model are generally taken according to a p-y curve or an empirical formula determined by experiments, so that the stress-strain relation of the soil body cannot be accurately described, the saturation heterogeneity of the soil body, the damping effect and the frequency dependence in the pile-soil interaction cannot be reflected, and the radial heterogeneity caused by the construction disturbance of the layered saturated soil cannot be further considered.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a vertical-pile vertical impedance calculation method, which can solve the problems that the stress-strain relation of a soil body cannot be accurately described, the saturated heterogeneity of layered soil and the radial heterogeneity caused by construction disturbance cannot be considered in the conventional vertical impedance calculation method, and therefore, the calculation precision of the vertical impedance of the vertical pile is improved.
In order to solve the technical problem, the invention provides a method for calculating vertical impedance of a vertical pile, which comprises the following steps:
step S1, vertically dispersing the vertical pile body and the foundation soil according to the horizontal layering condition of the soil body, and radially dispersing the foundation soil according to the soil disturbance condition around the pile;
step S2, establishing a wave equation of a discrete saturated soil ring layer for describing the stress-strain relation of saturated soil by using a fractional derivative, solving the wave equation of each soil ring layer and calculating the dynamic stiffness on the pile-soil contact surface by using a radial transfer matrix method;
and step S3, establishing an axial vibration equation of the pile body by using the dynamic stiffness on the pile-soil dynamic interaction contact surface, and calculating the vertical impedance of the straight pile by a vertical transfer matrix method after solving the equation.
Further, the step S1 is specifically:
in the disturbed layered saturated viscoelasticity half-space, a root with radius r is embedded0L in length, and the top is subjected to vertical simple harmonic load PeiωtActing vertical pile foundation, where P is excitation amplitude, omega is excitation frequency, and imaginary numbert is time, e is a natural constant;
dividing the soil body and the pile body into M layers from top to bottom according to the layering condition of foundation soil, and setting the layer height of the soil body in the ith layer and the length of the pile body section to be hi，i＝1,…,M；
Radially discretizing the soil around the pile in the ith layer into a width diAnd further radially dispersing the inner ring disturbance domain into NiSub-layers, numbered 1, 2, …, N from inside to outsideiThe number of the unperturbed domain is Ni+1, outer radius r of j-th turn in i-th layerijExpression is rij＝r0+jdi/NiWhen the inner ring disturbance domain in each layer of soil is radially dispersed into sub-ring layer division number NiWhen the calculated result of the pile top impedance is large enough to converge, the saturated soil in the sub-ring layer is considered to be homogeneous.
Further, the step S2 includes the following steps:
step S21, describing the stress-strain relation of saturated soil by adopting fractional derivatives, and establishing a discrete wave equation of each saturated soil circle layer;
and step S22, solving a wave equation of each soil ring layer, and calculating the dynamic stiffness on the pile-soil contact surface by a radial transfer matrix method.
Further, the step S21 is specifically:
a constitutive model of the foundation soil stress-strain relationship is described by adopting the following fractional derivatives:
in the formula, λS、GSIs the Lame constant, τ, of the soil bodyε、τσIs the parameters of a soil constitutive model, sigma and epsilon are stress and strain tensors of the soil, I is a unit matrix with 1 on a main diagonal and 0 at other places, Dα＝dαThe/dt is an alpha-order fractional derivative operator, wherein alpha is more than or equal to 0 and less than or equal to 1, when alpha is 0, the stress-strain curve of the foundation soil meets the linear elastic relationship, when alpha is 1, the stress-strain curve of the foundation soil meets the traditional Kelvin viscoelasticity relationship, and when the order of the soil fractional derivative model meets 0<α<1, the relation of a stress-strain curve obtained in a soil triaxial test can be better described;
taking a saturated soil unit in the jth circle in the ith layer as a research object, and substituting the fractional derivative viscoelastic constitutive model into a saturated porous medium vertical vibration control equation to obtain:
in the formula, ρSAnd ρFRespectively soil mass density and fluid density; w represents the vertical displacement of the soil body; the superscripts S and F respectively represent a soil framework and a fluid, r is a radial coordinate, the subscripts i and j represent physical parameters in the jth circle of the ith horizontal layer, and solid-liquid coupling is performedCoefficient xi ═ nF)2γF/kDWherein γ isFIs the gravity of the fluid, kDIs the Darcy permeability coefficient, n is the volume fraction, and the volume fraction of the soil body and the fluid satisfies nS+nF＝1；
When the system does simple harmonic vibration, the displacement of the liquid phase and the solid phase meets the following separation variable form:
the wave equation of the discrete saturated soil ring layer described by the fractional derivative obtained by replacing the formula (3) with the formula (2) is as follows:
further, the step S22 is specifically:
firstly, deducing solution of perturbation domain wave equation to internal perturbation domainThe vertical displacement of the soil body in the ith circle can be obtained by the formula (4) as follows:
wherein,shear wave velocityK0() And I0() First and second modified Bessel functions of zero order, respectively;andthe undetermined coefficient is determined by the boundary condition of saturated soil in the jth circle of the ith layer;
according to the differential relation between the soil deformation and the internal force, the soil vertical displacement W in the jth circle of the ith layerijS (r) and its lateral vertical shearing forceExpressed as the following matrix equation:
in the formula,K1() And I1() First-order and second-order modified Bessel functions respectively;
according to the continuous condition of the interface of the adjacent soil ring layer in the disturbance domainT represents transposition, and the relation between boundary displacement and shearing force at two sides of a disturbance domain is established by a radial transfer matrix method:
in the formula, the soil transfer matrix [ TR ]i]＝[TRi1][TRi2]…[TRNi],[TRij]＝[Trij(ri(j-1))][Trij(rij)]-1。
Secondly, deducing the solution of the wave equation of the non-disturbance domain to the non-disturbance domainAccording to the boundary condition that the soil stress and displacement at infinity are zero, solving the vertical displacement of the soil in the non-disturbance domain in the ith layer by the formula (4) as follows:
similar to the disturbance domain, the vertical displacement and the shear force of the soil ring layer in the non-disturbance domain can be obtained by the formula (8):
finally, the vertical dynamic stiffness of the pile-soil contact surface in the ith layer is solved, and on the basis of solving the displacement and the shearing force of a disturbed domain and a non-disturbed domain, the continuous condition of the interface of the two domains is obtainedThe combined type (7) and the formula (9) can obtain:
wherein (TR)i)ijFor soil mass transfer matrix [ TRi]The ith row and the jth column of (g),
the vertical dynamic stiffness of the pile-soil contact surface in the ith layer can be obtained by the formula (10):
further, the step S3 is specifically:
establishing a vertical downward local coordinate z' at the upper section of the ith section of the pile body, and according to the Rayleigh rod vibration theory, axially vibrating and displacing the pile section by wi(z', t) satisfies the following vibration control equation:
wherein E, A, rho, c and v are respectively the elastic modulus, the sectional area, the density, the viscous damping and the Poisson ratio of the pile body, and KSiAnd the vertical dynamic stiffness of the pile-soil contact surface in the ith layer is shown.
When the pile top is simply tunedWhen the load is applied, the displacement of the pile body meets wi(z’,t)＝Wi(z’)eiωtForm, therefore the solution of equation (12) is:
Wi(z')＝Cicosh(δi z')+Disinh(δi z') (13)
in the formula,Ciand DiThe undetermined coefficient is determined by the boundary condition of the pile body;
according to the formula and the differential relation between the force and deformation when the pile body vibrates axially, the relation between the displacement of the upper surface and the lower surface of the i-th layer of pile section and the axial force can be established as follows:
wherein,hithe height of the i-th layer of soil body;
according to the continuity condition of the adjacent pile body interface (W)i(hi),Pi(hi)}T＝{Wi+1(0),Pi+1(0)}TAnd combining the substitution relation of the local coordinates and the global coordinates, establishing the relation between the displacement and the axial force between the pile top and the pile bottom by a vertical transfer matrix method:
wherein, the pile body transmits the matrix [ TZ]＝[TZM][TZM-1]…[TZ1]，[TZi]＝[Tzi(hi)][Tzi(0)]-1。
The boundary condition of the pile bottom is simulated by adopting a group of springs and dampers, and the coefficient k of the pile bottom spring is measuredbAnd damping coefficient cbThe vertical impedance of the vertical pile obtained by substituting formula (15) is as follows:
in the formula (TZ)ijTransferring matrix [ TZ ] for pile body]Element of ith row and jth column in middle, pile bottom spring coefficientDamping coefficientWherein G isb，ρbAnd vbRespectively representing the shear modulus, the density and the Poisson ratio of a pile bottom soil layer and the vertical impedance of a straight pileIs a plurality ofRepresenting stiffness coefficient, imaginary partRepresenting the damping coefficient.
The invention achieves the following beneficial effects:
1) the invention provides a bidirectional transfer matrix method considering the influence of construction disturbance on layered saturated soil, and further considers the radial heterogeneity presented by the soil body around the pile under the disturbance effect and the permeability of fluid in the saturated soil compared with the homogeneous single-phase foundation soil, thereby being more in line with the actual engineering situation;
2) according to the method, the stress-strain relation of the viscoelastic foundation soil is described by adopting the fractional derivative operator, and compared with the traditional integral-order Kelvin viscoelastic constitutive model, the fractional derivative constitutive model can be matched with a foundation soil stress-strain curve obtained in a test in a wider frequency range by using a small number of parameters, so that the calculation accuracy of the vertical impedance of the vertical pile is improved;
3) the vertical vibration of the pile body is described by adopting a Rayleigh rod vibration theory, and the transverse inertia effect in the pile body vibration process can be further considered compared with the traditional Euler beam theory.
Drawings
FIG. 1 is a flow chart of a method for calculating vertical impedance of a vertical pile in disturbed layered saturated soil described by a fractional derivative according to the invention;
FIG. 2 is a schematic diagram of the dynamic interaction of disturbed layered viscoelastic saturated soil and a vertical pile depicted by fractional derivatives;
FIG. 3 is a schematic diagram of radial discrete division of a disturbance domain in the i-th layer of saturated soil;
fig. 4(a) and 4(b) are the real and imaginary parts, respectively, of the vertical impedance of the straight-stake base at different fractional derivative orders.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
As shown in fig. 1, a method for calculating vertical impedance of a vertical pile includes the following steps:
step S1: the method comprises the steps of vertically dispersing a vertical pile body and foundation soil according to the horizontal layering condition of a soil body, and radially dispersing the foundation soil according to the soil disturbance condition around the pile.
Step S2: establishing a wave equation of a discrete saturated soil ring layer for describing a saturated soil stress-strain relation by using a fractional derivative, solving the wave equation of each soil ring layer, and calculating the dynamic stiffness on the pile soil contact surface by using a radial transfer matrix method;
step S3: and establishing an axial vibration equation of the pile body by utilizing the dynamic stiffness on the dynamic interaction contact surface of the pile soil, solving the equation, and calculating the vertical impedance of the straight pile by a vertical transfer matrix method.
In the step S1, the method for vertically dispersing the vertical pile body and the foundation soil according to the horizontal layering condition of the soil body and radially dispersing the foundation soil according to the soil disturbance condition around the pile includes:
consider that in a perturbed lamellar saturated viscoelastic half-space, as shown in FIG. 2, a layer of radius r is embedded0L in length, and the top is subjected to vertical simple harmonic load PeiωtActing vertical pile foundation, where P is excitation amplitude, omega is excitation frequency, and imaginary numberDividing the soil body and the pile body into M layers from top to bottom according to the layering condition of foundation soil, and setting the layer height of the soil body in the ith layer and the length of the pile body section to be hiI is 1, …, M. In order to deal with the radial non-uniformity of the soil around the pile after being disturbed, the soil around the pile in the i-th horizontal layer is radially dispersed into a width d as shown in FIG. 3iAnd further radially dispersing the inner ring disturbance domain into NiSub-layers, numbered 1, 2, …, N from inside to outsideiThe number of the unperturbed domain is Ni+1. Thus, the outer radius r of the jth turn in the ith layerijExpression is rij＝r0+jdi/Ni. Dividing number N of sub-ring layers formed by radially dispersing inner ring disturbance domains in each layer of soiliWhen the calculated result of the pile top impedance is large enough to converge, the saturated soil in the sub-ring layer can be regarded as homogeneous.
The method for establishing the wave equation of the discrete saturated soil circle layer for describing the stress-strain relation of the saturated soil by using the fractional derivative in the step 2, solving the wave equation of each soil circle layer and calculating the dynamic stiffness on the pile-soil contact surface by using a radial transfer matrix method comprises the following steps:
step S21, describing the stress-strain relation of saturated soil by adopting fractional derivatives, and establishing a discrete wave equation of each saturated soil circle layer; and
and step S22, solving a wave equation of each soil ring layer, and calculating the dynamic stiffness on the pile-soil contact surface by a radial transfer matrix method.
In the step S21, the method for describing the stress-strain relationship of the saturated soil by using the fractional derivative and establishing the discrete wave equation of each saturated soil circle layer includes:
because the fractional order derivative operator can well reflect the stress-strain relation of the viscoelastic material, the constitutive model for describing the stress-strain relation of the foundation soil is described by adopting the following fractional derivatives:
in the formula, λS、GSThe Lame constant of the soil body; tau isε、τσThe soil constitutive model parameters are obtained; sigma and epsilon are stress and strain tensors of the soil body; i is a unit matrix with 1 on the main diagonal and 0 at other places; dα＝dαAnd/dt is an alpha order fractional derivative operator, wherein alpha is more than or equal to 0 and less than or equal to 1. When alpha is 0, the stress-strain curve of the foundation soil is shown to satisfy the linear elasticity relationship; when alpha is 1, the stress-strain curve of the foundation soil is shown to satisfy the traditional Kelvin viscoelasticity relationship. However, experimental research shows that compared with an integer order differential operator, when the order of the soil fractional derivative model satisfies 0<α<The relation of the stress-strain curve obtained in the soil triaxial test can be better described in the time 1.
Taking a saturated soil unit in the jth circle in the ith horizontal layer as a research object, and substituting the fractional derivative viscoelastic constitutive model into a saturated porous medium vertical vibration control equation to obtain:
in the formula, ρSAnd ρFRespectively soil mass density and fluid density; w represents the vertical displacement of the soil body; superscripts S and F respectively represent a soil framework and a fluid, and subscripts i and j represent physical parameters in the jth circle of the ith horizontal layer; n is volume fraction, and the volume fraction of the soil body and the fluid satisfies nS+nF1. Coefficient of coupling between solid and liquid xi ═ nF)2γF/kDWherein γ isFIs the gravity of the fluid, kDIs the darcy permeability coefficient.
When the system does simple harmonic vibration, the displacement of the liquid phase and the solid phase meets the following separation variable form:
the wave equation of the discrete saturated soil ring layer described by the fractional derivative obtained by replacing the formula (3) with the formula (2) is as follows:
in the step S22, the method for solving the wave equation of each saturated soil ring layer and calculating the dynamic stiffness of the pile-soil contact surface by using the radial transfer matrix method includes the following steps:
first, a solution to the perturbation domain wave equation is derived. For internal disturbance domainThe vertical displacement of the soil body in the ith circle can be obtained by the formula (4) as follows:
wherein,shear wave velocityK0() And I0() First and second modified Bessel functions of zero order, respectively;andthe undetermined coefficient is determined by the boundary condition of saturated soil in the jth circle of the ith layer.
According to the differential relation between the soil deformation and the internal force, the soil body in the jth circle of the ith layer is vertically displacedAnd lateral vertical shear thereofCan be expressed as the following matrix equation:
in the formula,K1() And I1() The modified Bessel function is first-order and second-order.
According to the continuous condition of the interface of the adjacent soil ring layer in the disturbance domainThe relationship between the displacement and the shearing force of the boundary at two sides of the disturbance domain can be established by a radial transfer matrix method:
in the formula, the soil transfer matrix [ TR ]i]＝[TRi1][TRi2]…[TRNi],[TRij]＝[Trij(ri(j-1))][Trij(rij)]-1。
Second, a solution to the unperturbed domain wave equation is derived. For non-disturbance domainAccording to the boundary condition that the soil stress and displacement at infinity are zero, the vertical displacement of the soil in the non-disturbance domain in the ith layer can be solved by the formula (4) as follows:
similar to the disturbance domain, the vertical displacement and the shear force of the soil ring layer in the non-disturbance domain can be obtained by the formula (8):
and finally, solving the vertical dynamic stiffness on the pile-soil contact surface in the ith layer. Based on the calculation of the displacement and the shearing force of the disturbed domain and the undisturbed domain, the continuous condition of the interface of the two domains is determinedThe combined type (7) and the formula (9) can obtain:
wherein (TR)i)ijFor soil mass transfer matrix [ TRi]Row i and column j.
The vertical dynamic stiffness of the pile-soil contact surface in the ith layer can be obtained by the formula (10):
in the step S3, the method for calculating the vertical impedance of the straight pile by the vertical transfer matrix method after solving the equation by establishing the axial vibration equation of the pile body by using the dynamic stiffness on the pile-soil dynamic interaction contact surface includes the following steps:
establishing a vertical downward local coordinate z' at the upper section of the ith section of the pile body, and according to the Rayleigh rod vibration theory, axially vibrating and displacing the pile section by wi(z', t) satisfies the following vibration control equation:
wherein E, A, rho, c and v are respectively the elastic modulus, the sectional area, the density, the viscous damping and the Poisson ratio of the pile body, and KSiThe vertical dynamic stiffness on the pile-soil contact surface in the ith layer calculated in the formula (11) is obtained.
When the pile top is under the action of simple harmonic load, the displacement of the pile body meets wi(z’,t)＝Wi(z’)eiωtForm, therefore the solution of equation (12) is:
Wi(z')＝Cicosh(δi z')+Disinh(δi z') (13)
in the formula,Ciand DiThe undetermined coefficient is determined by the boundary condition of the pile body;
according to the formula and the differential relation between the force and deformation when the pile body vibrates axially, the relation between the displacement of the upper surface and the lower surface of the i-th layer of pile section and the axial force can be established as follows:
wherein,
according to the continuity condition of the adjacent pile body interface (W)i(hi),Pi(hi)}T＝{Wi+1(0),Pi+1(0)}TAnd the relationship between the displacement and the axial force between the pile top and the pile bottom can be established by a vertical transfer matrix method by combining the substitution relationship between the local coordinate and the global coordinate:
wherein, the pile body transmits the matrix [ TZ]＝[TZM][TZM-1]…[TZ1]，[TZi]＝[Tzi(hi)][Tzi(0)]-1。
The boundary condition of the pile bottom can be simulated by adopting a group of springs and dampers, and the coefficient k of the pile bottom spring is measuredbAnd damping coefficient cbThe vertical impedance of the vertical pile obtained by substituting formula (15) is as follows:
wherein (A), (B), (CTZ)ijTransferring matrix [ TZ ] for pile body]Row i and column j. Spring coefficient of pile bottomDamping coefficientWherein G isb，ρbAnd vbRespectively representing the shear modulus, density and poisson ratio of the pile substrate. Vertical impedance of a vertical pileIs a plurality ofRepresenting stiffness coefficient, imaginary partRepresenting the damping coefficient.
Example (b):
in this embodiment, the pile foundation is buried in a saturated viscoelastic half-space whose surface layer is covered with hard soil of 2m thickness. The parameters of the pile foundation are as follows: pile length L is 10m, pile body radius r00.5m, 0.2 Poisson ratio mu of pile body, 1 × 10 damping c of pile body material5Ns/m2The elastic modulus E of pile body is 3.24X 1010Pa. Unless otherwise specified, the parameters of the saturated soil are as follows: density p of soil bodyS＝1800kg/m3Poisson ratio mu of soil bodyS0.4, volume fraction n of fluid in saturated soilL0.4, permeability coefficient kD＝1×10-3m/s, the parameters of the constitutive model of the fractional derivatives for describing the soil stress-strain relation are all taken as alpha-0.5, tauε1s and τσ3 s. Due to the influence of construction disturbance, the soil around the pile has a disturbance domain with the width same as the radius of the pile body, and the shear wave velocity of the soil body in the disturbance domain is linearly changed. The shear wave velocity of the disturbed inner soil body of the surface hard soil is 200m/s, and the shear wave velocity of the outer soil body and the undisturbed soil body is 300 m/s; lower half-space disturbance domain inner side soil shear waveThe speed is 300m/s, and the shear wave speed of the undisturbed region and the soil at the bottom of the pile is 150 m/s.
The method comprises the following steps:
considering that the viscoelastic half-space surface embedded in the pile foundation is covered with a layer of hard soil, the pile soil system is divided into 2 layers from top to bottom according to the layering condition of foundation soil, the height of the 1 st layer is 2m, and the height of the second layer is 8 m. In order to process the pile-surrounding soil which presents radial non-homogeneity after being disturbed, the pile-surrounding soil in each horizontal layer is annularly dispersed into an inner ring disturbed domain with the width of 0.5m and an outer ring non-disturbed domain, and further, the inner ring disturbed domains in the upper layer and the lower layer of soil are radially dispersed into 30 sub-ring layers, wherein the mark of j is 1, 2, … and 30 from inside to outside, and the mark of j is 31. Thus, the outer radius r of the jth turn in the ith layerijExpression is rij＝0.5+0.5j/30。
Step two:
and establishing a wave equation of the discrete saturated soil ring layers for describing the stress-strain relation of the saturated soil by using fractional derivatives, solving the wave equation of each soil ring layer, and calculating the dynamic stiffness on the pile-soil contact surface by using a transfer matrix method. With vertical dynamic stiffness K on the soil contact surface of the 1 st layer of pilesS1For example, according to equation (11) in step two, one can obtain:
in the formula, K0() And K1() A first class of modified Bessel functions of zero order and first order, respectively; parameter eta1，31，β1，31，V1，31The method can be obtained by calculation by using a formula (5) and parameter description thereof in the invention content; r is1，30I.e. the radius of the 30 th circle in the soil of the 1 st circle, and r can be obtained according to the step I1，301.0; layer 1 foundation soil transfer matrix [ TR ]1]Element (TR) of ith row and jth column1)ijCan be obtained by calculation by using the formula (7) and the parameter description thereof in the summary of the invention.
By the same method, the vertical dynamic stiffness K on the soil contact surface of the pile at the 2 nd layer can be obtainedS2. Vertical dynamic stiffness K on pile-soil contact surfaceS1And KS2Plural, real parts respectively with Re (K)S1) And Re (K)S2) Expressed by Im (K) for imaginary partsS1) And Im (K)S2) And (4) showing. According to the parameters of the foundation soil and the vertical piles in the embodiment, the vertical dynamic stiffness K on the soil contact surface of the piles at the 1 st layer and the 2 nd layer under different external load excitation frequencies can be calculatedS1And KS2As shown in table 1.
TABLE 1 vertical dynamic stiffness at pile-soil contact surface at different excitation frequencies
Step three:
and (3) after the vertical dynamic stiffness on the contact surface of the foundation soil of the 1 st layer and the 2 nd layer and the vertical pile is obtained by calculation in the second step, respectively establishing vertical vibration control equations of the two pile bodies according to a formula (12) in the invention content. According to the derivation of formula (13) -formula (17) in the summary of the invention, the vertical impedance of the vertical pile can be finally obtained as follows:
in the formula, a pile body transfer matrix [ TZ]Element (TZ) of the ith row and the jth column in the middleijThe method can be obtained by calculation by using a formula (15) and parameter description thereof in the invention content; spring coefficient k of pile bottombAnd damping coefficient cbCan be obtained by calculation using the parameter description of formula (16) in the summary of the invention.
Vertical impedance of a vertical pileIs a plurality ofRepresenting stiffness coefficient, imaginary partRepresenting the damping coefficient. Fig. 4 shows the vertical impedance of the straight pile foundation of the fractional derivative constitutive model of the foundation soil of the present example under different orders (α is 0, 0.5, 0.75, 1). It can be seen that: the vertical impedance is fluctuated along with the excitation frequency, and reflects the obvious resonance phenomenon of the pile-soil system under the excitation of external load. The influence of the order alpha of the saturated soil fractional derivative constitutive model on the pile top vertical impedance is mainly concentrated at the resonance peak value. When α is 0.5 and 0.75, the fluctuation form of the pile top vertical impedance is consistent with the results of the elastic foundation (α is 0) and the kelvin viscoelastic foundation (α is 1), but the resonance peak value is between the results of the elastic foundation and the traditional kelvin viscoelastic foundation, which is more consistent with the vertical pile impedance in the actual engineering.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.
Claims (6)
1. A method for calculating vertical impedance of a vertical pile comprises the following steps:
step S1, vertically dispersing the vertical pile body and the foundation soil according to the horizontal layering condition of the soil body, and radially dispersing the foundation soil according to the soil disturbance condition around the pile;
step S2, establishing a wave equation of a discrete saturated soil ring layer for describing the stress-strain relation of saturated soil by using a fractional derivative, solving the wave equation of each soil ring layer and calculating the dynamic stiffness on the pile-soil contact surface by using a radial transfer matrix method;
and step S3, establishing an axial vibration equation of the pile body by using the dynamic stiffness on the pile-soil dynamic interaction contact surface, and calculating the vertical impedance of the straight pile by a vertical transfer matrix method after solving the equation.
2. The method for calculating vertical impedance of a vertical pile according to claim 1, wherein the step S1 specifically comprises:
in the disturbed layered saturated viscoelasticity half-space, a root with radius r is embedded0L in length, and the top is subjected to vertical simple harmonic load PeiωtActing vertical pile foundation, where P is excitation amplitude, omega is excitation frequency, and imaginary numbert is time, e is a natural constant;
dividing the soil body and the pile body into M layers from top to bottom according to the layering condition of foundation soil, and setting the layer height of the soil body in the ith layer and the length of the pile body section to be hi，i＝1,…,M；
Radially discretizing the soil around the pile in the ith layer into a width diAnd further radially dispersing the inner ring disturbance domain into NiSub-layers, numbered 1, 2, …, N from inside to outsideiThe number of the unperturbed domain is Ni+1, outer radius r of j-th turn in i-th layerijExpression is rij＝r0+jdi/NiWhen the inner ring disturbance domain in each layer of soil is radially dispersed into sub-ring layer division number NiWhen the calculated result of the pile top impedance is large enough to converge, the saturated soil in the sub-ring layer is considered to be homogeneous.
3. The method for calculating vertical impedance of a vertical pile according to claim 1, wherein the step S2 includes the steps of:
step S21, describing the stress-strain relation of saturated soil by adopting fractional derivatives, and establishing a discrete wave equation of each saturated soil circle layer;
and step S22, solving a wave equation of each soil ring layer, and calculating the dynamic stiffness on the pile-soil contact surface by a radial transfer matrix method.
4. The method for calculating vertical impedance of a vertical pile according to claim 3, wherein the step S21 is specifically as follows:
a constitutive model of the foundation soil stress-strain relationship is described by adopting the following fractional derivatives:
in the formula, λS、GSIs the Lame constant, τ, of the soil bodyε、τσIs the parameters of a soil constitutive model, sigma and epsilon are stress and strain tensors of the soil, I is a unit matrix with 1 on a main diagonal and 0 at other places, Dα＝dαThe/dt is an alpha-order fractional derivative operator, wherein alpha is more than or equal to 0 and less than or equal to 1, when alpha is 0, the stress-strain curve of the foundation soil meets the linear elastic relationship, when alpha is 1, the stress-strain curve of the foundation soil meets the traditional Kelvin viscoelasticity relationship, and when the order of the soil fractional derivative model meets 0<α<1, the relation of a stress-strain curve obtained in a soil triaxial test can be better described;
taking a saturated soil unit in the jth circle in the ith layer as a research object, and substituting the fractional derivative viscoelastic constitutive model into a saturated porous medium vertical vibration control equation to obtain:
in the formula, ρSAnd ρFRespectively soil mass density and fluid density; w represents the vertical displacement of the soil body; the superscripts S and F respectively represent a soil framework and a fluid, r is a radial coordinate, subscripts i and j represent physical parameters in the jth circle of the ith horizontal layer, and a solid-liquid coupling coefficient xi is (n)F)2γF/kDWherein γ isFIs the gravity of the fluid, kDIs the Darcy permeability coefficient, n is the volume fraction, and the volume fraction of the soil body and the fluid satisfies nS+nF＝1；
When the system does simple harmonic vibration, the displacement of the liquid phase and the solid phase meets the following separation variable form:
the wave equation of the discrete saturated soil ring layer described by the fractional derivative obtained by replacing the formula (3) with the formula (2) is as follows:
5. the method for calculating vertical impedance of a vertical pile according to claim 4, wherein the step S22 is specifically as follows:
firstly, deducing solution of perturbation domain wave equation to internal perturbation domainThe vertical displacement of the soil body in the ith circle can be obtained by the formula (4) as follows:
wherein,shear wave velocityK0() And I0() First and second modified Bessel functions of zero order, respectively;andthe undetermined coefficient is determined by the boundary condition of saturated soil in the jth circle of the ith layer;
according to the differential relation between the soil deformation and the internal force, the soil body in the jth circle of the ith layer is vertically displacedAnd lateral vertical shear thereofExpressed as the following matrix equation:
in the formula,K1() And I1() First-order and second-order modified Bessel functions respectively;
according to the continuous condition of the interface of the adjacent soil ring layer in the disturbance domainT represents transposition, and the relation between boundary displacement and shearing force at two sides of a disturbance domain is established by a radial transfer matrix method:
in the formula, the soil body transfer matrix
Secondly, deducing the solution of the wave equation of the non-disturbance domain to the non-disturbance domainAccording to the boundary condition that the soil stress and displacement at infinity are zero, solving the vertical displacement of the soil in the non-disturbance domain in the ith layer by the formula (4) as follows:
similar to the disturbance domain, the vertical displacement and the shear force of the soil ring layer in the non-disturbance domain can be obtained by the formula (8):
finally, the vertical dynamic stiffness of the pile-soil contact surface in the ith layer is solved, and on the basis of solving the displacement and the shearing force of a disturbed domain and a non-disturbed domain, the continuous condition of the interface of the two domains is obtainedThe combined type (7) and the formula (9) can obtain:
wherein (TR)i)ijFor soil mass transfer matrix [ TRi]The ith row and the jth column of (g),
the vertical dynamic stiffness of the pile-soil contact surface in the ith layer can be obtained by the formula (10):
6. the method for calculating vertical impedance of a vertical pile according to claim 1, wherein the step S3 specifically comprises:
establishing a vertical downward local coordinate z' at the upper section of the ith section of the pile body, and according to the Rayleigh rod vibration theory, axially vibrating and displacing the pile section by wi(z', t) satisfies the following vibration control equation:
wherein E, A, rho, c and v are respectively the elastic modulus, the sectional area, the density, the viscous damping and the Poisson ratio of the pile body, and KSiAnd the vertical dynamic stiffness of the pile-soil contact surface in the ith layer is shown.
When the pile top is under simple harmonic load, the pileBody displacement satisfies wi(z’,t)＝Wi(z’)eiωtForm, therefore the solution of equation (12) is:
Wi(z')＝Cicosh(δiz')+Disinh(δiz') (13)
in the formula,Ciand DiThe undetermined coefficient is determined by the boundary condition of the pile body;
according to the formula and the differential relation between the force and deformation when the pile body vibrates axially, the relation between the displacement of the upper surface and the lower surface of the i-th layer of pile section and the axial force can be established as follows:
wherein,hithe height of the i-th layer of soil body;
according to the continuity condition of the adjacent pile body interface (W)i(hi),Pi(hi)}T＝{Wi+1(0),Pi+1(0)}TAnd combining the substitution relation of the local coordinates and the global coordinates, establishing the relation between the displacement and the axial force between the pile top and the pile bottom by a vertical transfer matrix method:
wherein, the pile body transmits the matrix [ TZ]＝[TZM][TZM-1]…[TZ1]，[TZi]＝[Tzi(hi)][Tzi(0)]-1。
The boundary condition of the pile bottom is simulated by adopting a group of springs and dampers, and the coefficient k of the pile bottom spring is measuredbAnd damping coefficient cbThe vertical impedance of the vertical pile obtained by substituting formula (15) is as follows:
in the formula (TZ)ijTransferring matrix [ TZ ] for pile body]Element of ith row and jth column in middle, pile bottom spring coefficientDamping coefficientWherein G isb，ρbAnd vbRespectively representing the shear modulus, the density and the Poisson ratio of a pile bottom soil layer and the vertical impedance of a straight pileIs a plurality ofRepresenting stiffness coefficient, imaginary partRepresenting the damping coefficient.
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