CN112287574A - Pile foundation dynamic stability analysis and calculation method under wave load action - Google Patents

Pile foundation dynamic stability analysis and calculation method under wave load action Download PDF

Info

Publication number
CN112287574A
CN112287574A CN202011023577.6A CN202011023577A CN112287574A CN 112287574 A CN112287574 A CN 112287574A CN 202011023577 A CN202011023577 A CN 202011023577A CN 112287574 A CN112287574 A CN 112287574A
Authority
CN
China
Prior art keywords
pile
load
amplitude
equation
frequency
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN202011023577.6A
Other languages
Chinese (zh)
Other versions
CN112287574B (en
Inventor
张震
徐旭
姚文娟
程泽坤
丁刚
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
University of Shanghai for Science and Technology
Original Assignee
University of Shanghai for Science and Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by University of Shanghai for Science and Technology filed Critical University of Shanghai for Science and Technology
Priority to CN202011023577.6A priority Critical patent/CN112287574B/en
Publication of CN112287574A publication Critical patent/CN112287574A/en
Application granted granted Critical
Publication of CN112287574B publication Critical patent/CN112287574B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/13—Architectural design, e.g. computer-aided architectural design [CAAD] related to design of buildings, bridges, landscapes, production plants or roads
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/04—Constraint-based CAD
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Geometry (AREA)
  • Theoretical Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Computer Hardware Design (AREA)
  • Evolutionary Computation (AREA)
  • General Engineering & Computer Science (AREA)
  • Structural Engineering (AREA)
  • Computational Mathematics (AREA)
  • Civil Engineering (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Architecture (AREA)
  • Foundations (AREA)

Abstract

The invention discloses a method for analyzing and calculating dynamic stability of a pile foundation under the action of wave load, which comprises the following steps: and obtaining a non-homogeneous Mathieu equation through arrangement, and solving to obtain the critical frequency and the instability load of the parameter resonance. By comparing the analytic solution of the invention with the calculation result of finite element simulation, the correctness of theoretical derivation is verified. Analysis of the calculation results shows that: pile top load P0The influence of the depth of the pile body into water on the unstable load is larger, but the unstable load is also influenced by the resonance frequency; resistance coefficient of pile-side soil bodyThe influence of M on the parameter resonance frequency is maximum, the instability frequency of the parameter resonance is rapidly increased along with the increase of M, the stability of the pile body can be effectively improved, and the influence of the pile body mass M and the soil body shear rigidity G on the instability frequency of the pile body is relatively small but can not be ignored; the influence of factors such as pile diameter, wavelength and wave height on the amplitude is different, and the influence can be adjusted according to concrete engineering practice to improve pile body stability.

Description

Pile foundation dynamic stability analysis and calculation method under wave load action
Technical Field
The invention belongs to the technical field of geotechnical engineering, and particularly relates to a pile foundation dynamic stability analysis and calculation method under the action of wave load.
Background
At present, research on dynamic stability analysis of the pile foundation bearing wave load in the sea at home and abroad is not reported yet. Most research focuses on the analysis of the scouring action of waves and the interaction between piles and soil, and relatively few researches are conducted on the problem of pile foundation power under the action of bearing complex loads. In actual engineering, regarding pile foundation construction in the ocean, a finite element method is generally adopted for simulation calculation, the calculation amount is large, the calculation is complex, and research on theoretical aspects is still incomplete. In recent years, more and more buildings, various cross-sea bridges, wind power generation single-pile foundations, offshore drilling platforms and the like are developed, for the marine buildings, the pile foundations are taken as the foundations, the importance is self-evident, the environment in the sea is complex, the wave load is one of the main loads acting on the pile body, and the problem of the dynamic stability of the pile foundations under the combined action of the wave load and other loads is urgently researched.
Disclosure of Invention
In order to solve the problems in the prior art, the invention aims to overcome the defects in the prior art, and provides a pile foundation dynamic stability analysis and calculation method under the action of wave load, which considers the wave load as a horizontal dynamic load applied to a pile side, simultaneously considers the influence of wave scouring action, calculates the resistance of a pile side soil body by adopting a two-parameter method, obtains a pile body stability dynamic differential equation on the basis of a Mathieu-Hill equation, probes the pile body dynamic stability problems under four different conditions, obtains corresponding critical frequency and theoretical solutions of instability load, performs a series of sensitivity parameter analysis, probes the influence factors of the pile body critical frequency and the amplitude change of the pile body under four different load actions, and provides specific calculation steps.
In order to achieve the purpose of the invention, the invention adopts the following technical scheme:
a pile foundation dynamic stability analysis and calculation method under the action of wave load adopts a double-parameter foundation model to replace a traditional winkler model, considers the action of the wave load and the softening of the pile side soil body rigidity, and comprises the following steps:
(1) theoretical analytic calculation of pile foundation dynamic stability;
(2) comparing the analytic solution with the finite element simulation calculation result;
(3) calculating a destabilizing load;
(4) and (5) analyzing the parameter sensitivity.
Preferably, the specific operation steps of step (1) are as follows:
(1-1) adopting a two-parameter foundation method to equivalently calculate the resistance of the soil body on the pile side as a spring, bearing vertical static load and vertical simple harmonic load on the pile top, bearing wave load in the horizontal direction, calculating the wave force by adopting a diffraction theory, and simultaneously scouring the pile body:
wave load f on pile sidezThe calculation is as follows:
Figure BDA0002701450270000021
wherein: rho is the density of the seawater and is 1030kg/m 3;
g is gravity acceleration, and is 9.8m/s2；
H is the wave height;
Figure BDA0002701450270000022
L1is the wavelength;
z is water depth, dlThe depth of the pile body into water is not included;
Figure BDA0002701450270000023
J1' is a first order Bessel function of the first kind, Y1' is a first order Bessel function of the second kind;
omega is the frequency of the wave load;
d is the pile diameter;
the foundation reaction force of the pile side soil body is calculated by adopting a two-parameter method, which is shown as the following formula:
Figure BDA0002701450270000024
wherein k is m0b1，G＝Gpb1，m0Is the resistance coefficient of the pile-side soil body, GPThe shear stiffness of the soil body is the pile side soil body resistance coefficient m under the action of cyclic load because the pile foundation vibrates under the dynamic load0Is calculated as follows:
Figure BDA0002701450270000025
in the formula TNThe period for loading the load, N is the cycle number of the load, m0' value reference Specification[18]The recommended empirical formula to calculate:
Figure BDA0002701450270000026
Figure BDA0002701450270000027
c is the internal friction angle and cohesive force of the soil body, vbFor transverse displacement, c is the cohesive force m of the soil mass0Taking the weighted average value of the layer thicknesses of different soil layers; b1For calculation, when the pile diameter d is more than or equal to 1, b10.9(d + 1); when d < 1, b1＝0.9(1.5d+0.5)，
GPShear stiffness of the soil body, GPThe value of (A) is the fitted shearing steelDegree empirical formula:
Figure BDA0002701450270000031
v in the formulasIs the Poisson's ratio of the foundation soil, EsIs the modulus of elasticity, h, of the foundation soilgThe thickness of the shear layer of the foundation soil; taking eleven times of pile diameter as the thickness of the shear layer of the foundation soil;
the boundary condition of the pile foundation is simplified into free upper part and embedded lower end, and the horizontal displacement of the pile body is assumed as follows:
Figure BDA0002701450270000032
(1-2) when only the vertical simple harmonic load of the pile top is considered, and the transverse wave load is not considered, i.e. fzWhen the energy principle is equal to 0, the control equation of the pile body according to the energy principle is shown as the following formula:
Π＝U+V+T+D
wherein U is the internal force potential energy of the pile body, and U is equal to US+Up，UpIs strain energy of pile body, UsThe strain energy of the soil body on the pile side is calculated by adopting a two-parameter method, and the method is as follows:
Figure BDA0002701450270000033
Figure BDA0002701450270000034
v is external force potential energy, comprises three parts, pile top load potential energy Vp，Vq0Is hydrostatic pressure potential energy and bending moment potential energy of pile top
Figure BDA0002701450270000035
Namely:
Figure BDA0002701450270000036
wherein:
Figure BDA0002701450270000037
Figure BDA0002701450270000038
Figure BDA0002701450270000039
in the formula M0Is the initial bending moment of the pile top, q0Is hydrostatic pressure; p0For static loading of pile top, PtThe load amplitude is simple harmonic; h is the partial length of the part of the pile body in the water;
t is the kinetic energy, and only the kinetic energy of the transverse vibration of the pile body is considered here, so the kinetic energy T is:
Figure BDA0002701450270000041
wherein m is the mass of a unit pile length and l is the pile length;
d is damping potential energy:
Figure BDA0002701450270000042
wherein C is a damping parameter;
the complete pile body energy control equation is as follows:
Figure BDA0002701450270000043
the above formula is obtained by using Hamilton principle:
W(t)＝T(t)-U(t)，
Figure BDA0002701450270000044
obtaining a non-homogeneous Mathieu equation after finishing:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+rcosθt
wherein:
Figure BDA0002701450270000045
m is the pile body mass;
Figure BDA0002701450270000046
Figure BDA0002701450270000047
Figure BDA0002701450270000048
Figure BDA0002701450270000049
Figure BDA0002701450270000051
q0＝ρgh
Figure BDA0002701450270000052
Figure BDA0002701450270000053
the Mathieu equation above is solved according to equation type and semi-inverse solution in the form:
f(t)＝a0+asin(θt)+bcos(θt)
substituting the above formula into the previous nonhomogeneous mathieu equation to obtain:
Figure BDA0002701450270000054
solving the non-homogeneous linear equation system to obtain:
Figure BDA0002701450270000055
the amplitude in the first case is obtained from the above equation:
Figure BDA0002701450270000056
as can be seen from the foregoing inference process, the critical frequency of the parametric resonance at this time is θ ═ Ω;
(1-3) when only the horizontal simple harmonic load is considered, namely only the horizontal wave load and the vertical static load are considered, and the vertical simple harmonic load is not considered, the external force potential energy in the energy equation at the moment is changed, the rest potential energy is unchanged, and the external force potential energy is changed into:
Figure BDA0002701450270000057
in the formula, VP1For vertical loads without simple harmonic terms, VbThe remaining two items are the same as the above;
Figure BDA0002701450270000058
Figure BDA0002701450270000061
the rest processes are the same as the above, and in the second case, the complete pile body energy equation is as follows:
Figure BDA0002701450270000062
and the dynamic differential equation at this time becomes the following equation:
f″(t)+ξf′(t)+Ω2f(t)＝r0+rcosθt
in this time scheme
Figure BDA0002701450270000063
The dynamic differential equation becomes a second-order heterogeneous linear ordinary differential equation, and f (t) is given by the following formula:
f(t)＝ah0+ahsin(ωt)+bhcos(ωt)
substituting the above equation into the dynamic differential equation to obtain:
Figure BDA0002701450270000064
the pile body amplitude in the second case is obtained according to the solution of the equation:
Figure BDA0002701450270000065
at this time, the critical frequency of the pile body is still omega, but the amplitude at this time is different from the former case;
(1-4) considering the situation that the transverse and vertical simple harmonic loads exist simultaneously and have the same frequency, the energy equation form of the pile body is the same as that of the pile body, but the external force potential energy is changed as follows:
Figure BDA0002701450270000066
at this time, the complete energy equation of the pile body is as follows:
Figure BDA0002701450270000067
the power differential equation at this time is still as follows after being collated:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1cosθt
but in this equation
Figure BDA0002701450270000071
As before, the form of the solution to the dynamic differential equation at this time is still:
f(t)＝a00+a11 sin(θt)+b11 cos(θt)
substituting the above formula into a dynamic differential equation to obtain the following parameter expression:
Figure BDA0002701450270000072
substituting the parameters into an equation, and obtaining the amplitude calculation formula under the condition after arrangement as follows:
Figure BDA0002701450270000073
as can be seen from the above formula, when θ ═ Ω, the parameter resonance occurs, and at this time, θ ═ Ω is the critical frequency, and the amplitude at this time is also different from the previous one;
(1-5) considering the situation that horizontal and vertical simple harmonic loads exist at the same time but the load frequencies are different, the external force load potential energy in the energy equation of the pile body is changed, the rest potential energy is unchanged, and the external force potential energy is changed as follows:
Figure BDA0002701450270000074
the complete pile body energy equation is:
Figure BDA0002701450270000075
the dynamic differential equation then becomes of the form:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1 cosωt
the solution to the above equation is obtained in the form of an equation and a semi-inverse solution:
f(t)＝a012+a12sin(θt)+b12cos(θt)+c12sin(ωt)+d12cos(ωt)
substituting the coefficient into a differential equation to obtain the coefficient as follows:
Figure BDA0002701450270000081
the amplitude calculation formula in this case is:
Figure BDA0002701450270000082
from the above formula, when θ ═ Ω or ω ═ Ω, the pile foundation will have parameter resonance, which is one more possible case than the previous case, and at this time, the amplitude is still different from the previous cases.
Preferably, in the step (2), a model of the pile foundation in the seawater is established based on a finite element method, wave force is applied to the side of the pile in a manner of applying load to the side surface, and the critical frequency theoretical solution and the finite element solution are compared and verified under the conditions that the pile foundation simultaneously bears vertical and horizontal simple harmonic loads and the loads have the same frequency; in the model, the soil body adopts a mole-coulomb model, the pile body and the soil body both adopt three-dimensional entity units, the unit type is C3D8R, the reduction integral is calculated, and the total unit number is 289014.
Preferably, the unstable load calculation in the step (3): the unstable load under the action of the simple harmonic load is obtained by changing according to an amplitude formula when N istWhen the critical load is reached, the amplitude tends to infinity, and the vertical simple harmonic load and the wave load are converted and sorted according to an amplitude formula calculated under the condition that the vertical simple harmonic load and the wave load exist simultaneously to obtain the lossThe expression for the steady load is:
Figure BDA0002701450270000083
unstable load NtLoad P along with pile top0The depth h, k of the pile body entering water is the resistance coefficient of the soil body on the pile side, and the displacement is drawn along with NtAnd the time t, from which the destabilizing load N can be observedt(iii) occurrence of (a);
with P0Increase of (2), destabilizing load NtcrDecrease linearly, but when P0When the frequency of the simple harmonic load is the same as the resonance frequency of the structural parameters under a certain condition, the unstable load can generate sudden change, and NtcrWill momentarily decrease to almost 0; n is increased along with the increase of the depth h of the pile body into the watertcrWill decrease rapidly and then tend to stabilize; change of pile side soil body horizontal resistance coefficient k to NtcrWhen the frequency of the simple harmonic load is less than the critical frequency, the unstable load increases linearly with the increase of k.
Preferably, the parameter sensitivity analysis in step (4); the stability and the amplitude of the pile body are influenced by a plurality of factors, such as pile diameter, pile length, pile body scouring depth, pile side soil resistance and the like, different factors have different influences on the parameter resonance frequency and the amplitude of the pile foundation, the influence degree of various factors on the critical frequency and the amplitude of the pile body is obtained through analyzing the factors, and in the actual engineering, the amplitude of the pile foundation is controlled by adjusting the proper influence factors according to local conditions so as to improve the stability of the pile body; the analysis of the above influencing factors shows that the following conclusions can be mainly obtained through parameter analysis:
(4-1) the horizontal resistance coefficient k of the soil body on the pile side has great influence on the critical frequency, and the resistance of the soil body on the pile side is increased to increase the critical frequency, so that the resistance is far away from the frequency of external load, and the probability of an unstable area is reduced; secondly, the influence of the pile body mass m is caused, the critical frequency is increased along with the increase of the pile body mass, and the influence of the soil body shear stiffness G on the critical frequency is relatively small but can not be ignored; different measures are taken in an actual engineering structure according to the influence of different parameters, so that parameter resonance and unstable regions are avoided when the external load frequency is the same as the critical frequency;
(4-2) the relationship between the amplitudes under four simple harmonic loading conditions is: when the longitudinal and transverse simple harmonic loads exist at the same time and have the same frequency, the amplitude is the largest, then the situation that only the transverse simple harmonic load exists, and the situation that only the vertical simple harmonic load exists when the amplitude is the smallest; the existence of the vertical simple harmonic load can inhibit the amplitude to a certain extent, and the transverse simple harmonic load plays a main role in the amplitude;
(4-3) the influence of the pile length on the critical frequency is related to the soil body on the pile side, in the initial stage, the soil body constraint action is weaker, the critical frequency is firstly reduced along with the increase of the length of the pile in the soil, and then the critical frequency is increased along with the increase of the pile length after the constraint action of the soil body is gradually enhanced; when the pile diameter is smaller, the amplitude slightly increases with the increase of the pile diameter d, because the wave load increases with the increase of the pile diameter, but the final amplitude decreases with the increase of the pile diameter;
(4-4) the wave height and the wavelength change in the wave load can have certain influence on the amplitude of the pile body, the amplitude of the pile body is linearly increased along with the wave height along with the increase of the parameters, and is nonlinearly increased along with the wavelength, but the relatively increased amplitude is smaller than the change of other factors; in actual engineering, the change of waves cannot be controlled, but the vibration amplitude of the pile foundation is controlled by using the results of various parameter sensitivity analyses obtained in the invention and starting from other factors, so that the pile foundation is kept in a stable state.
Compared with the prior art, the invention has the following obvious and prominent substantive characteristics and remarkable advantages:
1. the dynamic stability of the pile foundation under the action of the wave load is researched by a variational method and a Hamilton principle, the scouring action of waves is considered in the research process, a two-parameter method which is more consistent with the reality is adopted to calculate the foundation counterforce, a dynamic stability differential equation under four conditions (considering that longitudinal-transverse simple harmonic loads exist at the same time and have the same frequency, longitudinal-transverse simple harmonic loads exist at the same time but have different frequencies, only transverse simple harmonic wave loads and only vertical simple harmonic loads) is obtained by formula derivation, the critical load frequency and the unstable load are obtained by solving, and the accuracy of a theoretical model is verified by finite elements; the pile foundation in seawater bears complex load and is influenced by various factors, the wave load is used as a main dynamic load, and when the wave load impacts the pile body, the stability problem is particularly important; the length, the radius, the wave load, the pile top load, various parameters of the soil body on the pile side and the like of the pile body can bring different influences on the stability of the pile body, wherein the parameters of the soil body on the pile side play an important role, the parameter resonance frequency can be effectively increased, the probability of an unstable area is reduced, and the stability of the pile body is improved; the load size and the load form of the pile top can influence the stability of the pile body to a certain extent;
2. for actual engineering, various parameters can be adjusted in a targeted manner according to specific requirements through the influence degrees of different factors, so that the aims of improving the stability of the pile body and being economical are fulfilled;
3. the method has the advantages of high pile foundation dynamic stability analysis precision, low cost and suitability for popularization and application.
Drawings
FIG. 1 is a diagram of a computational model of the present invention.
FIG. 2 is a diagram of a finite element model according to the present invention.
FIG. 3 is a diagram illustrating finite element and theoretical solution verification according to the present invention.
FIG. 4 shows the destabilizing load N according to the present inventiontFollowing pile top static load P0A variation diagram of (2).
FIG. 5 shows N in the present inventiontThe change of the depth h of the pile body in water is shown.
FIG. 6 shows N in the present inventiontcrGraph with increasing k.
FIG. 7 shows the shift as a function of N in the present inventiontAnd a displacement response plot over time t.
FIG. 8 is a graph of the amplitude versus frequency of the present invention.
FIG. 9 is a graph of the displacement versus time frequency in the present invention.
Fig. 10 is a graph showing the effect of vertical frequency on amplitude in the presence of both longitudinal and transverse loads as described in the present invention.
FIG. 11 is a graph showing the effect of transverse frequency on amplitude in the presence of both longitudinal and transverse loads as described in the present invention.
FIG. 12 is a graph of critical resonance frequency as a function of mass m for the parameters described in the present invention.
FIG. 13 is a graph showing the effect of the pile side soil resistance coefficient k on the critical frequency in the present invention.
Fig. 14 is a graph showing the effect of pile side soil shear stiffness G on critical frequency in the present invention.
FIG. 15 is a graph illustrating the effect of the flush depth h on the critical frequency in the present invention.
Fig. 16 is a graph of the effect of pile length on critical frequency in the present invention.
Fig. 17 is a graph showing the effect of pile diameter on amplitude in the present invention, where the longitudinal and lateral loads are at the same frequency.
FIG. 18 is a graph of the effect of wave height on amplitude in the present invention.
FIG. 19 is a graph of the effect of wavelength on amplitude in accordance with the present invention.
FIG. 20 is a graph illustrating the effect of initial bending moment on amplitude in accordance with the present invention.
Fig. 21 is a graph showing the variation of the displacement with the depth h and frequency of the pile body in the water.
Detailed Description
The above-described scheme is further illustrated below with reference to specific embodiments, which are detailed below:
the first embodiment is as follows:
in this embodiment, a method for analyzing and calculating dynamic stability of a pile foundation under the action of a wave load adopts a two-parameter foundation model to replace a traditional winkler model, and considers the action of the wave load and the softening of the stiffness of a soil body on a pile side, and includes the following steps:
(1) theoretical analytic calculation of pile foundation dynamic stability;
(2) comparing the analytic solution with the finite element simulation calculation result;
(3) calculating a destabilizing load;
(4) and (5) analyzing the parameter sensitivity.
The embodiment researches pile body parameters and gives specific calculation steps.
Example two:
this embodiment is substantially the same as the first embodiment, and is characterized in that:
in this embodiment, referring to fig. 1 to 21, a method for analyzing and calculating dynamic stability of a pile foundation under the action of a wave load includes the following specific operation steps of step (1):
(1-1) adopting a two-parameter foundation method to equivalently calculate the resistance of the soil body on the pile side as a spring, bearing vertical static load and vertical simple harmonic load on the pile top, bearing wave load in the horizontal direction, calculating the wave force by adopting a diffraction theory, and simultaneously scouring the pile body:
wave load f on pile sidezThe calculation is as follows:
Figure BDA0002701450270000111
wherein: rho is the density of the seawater and is 1030kg/m 3;
g is gravity acceleration, and is 9.8m/s2；
H is the wave height;
Figure BDA0002701450270000112
L1is the wavelength;
z is water depth, dlThe depth of the pile body into water is not included;
Figure BDA0002701450270000121
J1' is a first order Bessel function of the first kind, Y1' is a first order Bessel function of the second kind; d is the pile diameter;
omega is the frequency of the wave load;
the foundation reaction force of the pile side soil body is calculated by adopting a two-parameter method, which is shown as the following formula:
Figure BDA0002701450270000122
wherein k is m0b1，G＝Gpb1，m0Is the resistance coefficient of the pile-side soil body, GPThe shear stiffness of the soil body is the pile side soil body resistance coefficient m under the action of cyclic load because the pile foundation vibrates under the dynamic load0Is calculated as follows:
Figure BDA0002701450270000123
in the formula TNThe period for loading the load, N is the cycle number of the load, m0' value reference Specification[18]The recommended empirical formula to calculate:
Figure BDA0002701450270000124
Figure BDA0002701450270000125
c is the internal friction angle and cohesive force of the soil body, vbFor transverse displacement, m0Taking the weighted average value of the layer thicknesses of different soil layers; b1For calculating the width, when the pile diameter d is more than or equal to 1, b10.9(d + 1); when d < 1, b1＝0.9(1.5d+0.5)，
GPShear stiffness of the soil body, GPThe value of (2) is obtained by adopting a fitted shear stiffness empirical formula:
Figure BDA0002701450270000126
v in the formulasPoisson as foundation soilRatio, EsIs the modulus of elasticity, h, of the foundation soilgThe thickness of the shear layer of the foundation soil; taking eleven times of pile diameter as the thickness of the shear layer of the foundation soil;
the boundary condition of the pile foundation is simplified into free upper part and embedded lower end, and the horizontal displacement of the pile body is assumed as follows:
Figure BDA0002701450270000127
(1-2) when only the vertical simple harmonic load of the pile top is considered, and the transverse wave load is not considered, i.e. fzWhen the energy principle is equal to 0, the control equation of the pile body according to the energy principle is shown as the following formula:
Π＝U+V+T+D
wherein U is the internal force potential energy of the pile body, and U is equal to US+Up，UpIs strain energy of pile body, UsThe strain energy of the soil body on the pile side is calculated by adopting a two-parameter method, and the method is as follows:
Figure BDA0002701450270000131
Figure BDA0002701450270000132
v is external force potential energy, comprises three parts, pile top load potential energy Vp，Vq0Is hydrostatic pressure potential energy and bending moment potential energy of pile top
Figure BDA0002701450270000133
Namely:
Figure BDA0002701450270000134
wherein:
Figure BDA0002701450270000135
Figure BDA0002701450270000136
Figure BDA0002701450270000137
in the formula M0Is the initial bending moment of the pile top, q0Is hydrostatic pressure; p is a radical of0For static loading of pile top, PtThe load amplitude is simple harmonic; h is the length of the part of the pile body in the water;
t is the kinetic energy, and only the kinetic energy of the transverse vibration of the pile body is considered here, so the kinetic energy T is:
Figure BDA0002701450270000138
wherein m is the mass of a unit pile length and l is the pile length;
d is damping potential energy:
Figure BDA0002701450270000139
wherein C is a damping parameter;
the complete pile body energy control equation is as follows:
Figure BDA0002701450270000141
the above formula is obtained by using Hamilton principle:
W(t)＝T(t)-U(t)，
Figure BDA0002701450270000142
obtaining a non-homogeneous Mathieu equation after finishing:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+rcosθt
wherein:
Figure BDA0002701450270000143
m is the pile body mass;
Figure BDA0002701450270000144
Figure BDA0002701450270000145
Figure BDA0002701450270000146
Figure BDA0002701450270000147
Figure BDA0002701450270000148
q0＝ρgh
Figure BDA0002701450270000149
Figure BDA00027014502700001410
the Mathieu equation above is solved according to equation type and semi-inverse solution in the form:
f(t)＝a0+asin(θt)+bcos(θt)
substituting the above formula into the previous nonhomogeneous mathieu equation to obtain:
Figure BDA00027014502700001411
solving the non-homogeneous linear equation system to obtain:
Figure BDA0002701450270000151
the amplitude in the first case is obtained from the above equation:
Figure BDA0002701450270000152
as can be seen from the foregoing inference process, the critical frequency of the parametric resonance at this time is θ ═ Ω;
(1-3) when only the horizontal simple harmonic load is considered, namely only the horizontal wave load and the vertical static load are considered, and the vertical simple harmonic load is not considered, the external force potential energy in the energy equation at the moment is changed, the rest potential energy is unchanged, and the external force potential energy is changed into:
Figure BDA0002701450270000153
in the formula, VP1For vertical loads without simple harmonic terms, VbThe remaining two items are the same as the above;
Figure BDA0002701450270000154
Figure BDA0002701450270000155
the rest processes are the same as the above, and in the second case, the complete pile body energy equation is as follows:
Figure BDA0002701450270000156
and the dynamic differential equation at this time becomes the following equation:
f″(t)+ξf′(t)+Ω2f(t)＝r0+rcosθt
in this time scheme
Figure BDA0002701450270000157
The dynamic differential equation becomes a second-order heterogeneous linear ordinary differential equation, and f (t) is given by the following formula:
f(t)＝ah0+ahsin(ωt)+bhcos(ωt)
substituting the above equation into the dynamic differential equation to obtain:
Figure BDA0002701450270000161
the pile body amplitude in the second case is obtained according to the solution of the equation:
Figure BDA0002701450270000162
at this time, the critical frequency of the pile body is still omega, but the amplitude at this time is different from the former case;
(1-4) considering the situation that the transverse and vertical simple harmonic loads exist simultaneously and have the same frequency, the energy equation form of the pile body is the same as that of the pile body, but the external force potential energy is changed as follows:
Figure BDA0002701450270000163
at this time, the complete energy equation of the pile body is as follows:
Figure BDA0002701450270000164
the power differential equation at this time is still as follows after being collated:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1 cosθt
but in this equation
Figure BDA0002701450270000165
As before, the form of the solution to the dynamic differential equation at this time is still:
f(t)＝a00+a11 sin(θt)+b11 cos(θt)
substituting the above formula into a dynamic differential equation to obtain the following parameter expression:
Figure BDA0002701450270000171
substituting the parameters into an equation, and obtaining the amplitude calculation formula under the condition after arrangement as follows:
Figure BDA0002701450270000172
as can be seen from the above formula, when θ ═ Ω, the parameter resonance occurs, and at this time, θ ═ Ω is the critical frequency, and the amplitude at this time is also different from the previous one;
(1-5) considering the situation that horizontal and vertical simple harmonic loads exist at the same time but the load frequencies are different, the external force load potential energy in the energy equation of the pile body is changed, the rest potential energy is unchanged, and the external force potential energy is changed as follows:
Figure BDA0002701450270000173
the complete pile body energy equation is:
Figure BDA0002701450270000174
the dynamic differential equation then becomes of the form:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1 cosωt
the solution to the above equation is obtained in the form of an equation and a semi-inverse solution:
f(t)＝a012+a12sin(θt)+b12cos(θt)+c12sin(ωt)+d12cos(ωt)
substituting the coefficient into a differential equation to obtain the coefficient as follows:
Figure BDA0002701450270000181
the amplitude calculation formula in this case is:
Figure BDA0002701450270000182
from the above formula, when θ ═ Ω or ω ═ Ω, the pile foundation will have parameter resonance, which is one more possible case than the previous case, and at this time, the amplitude is still different from the previous cases.
In this embodiment, in the step (2), a model of a pile foundation in seawater is established based on a finite element method, wave force is applied to the pile side in a manner of applying load to the side surface, and the pile foundation is selected to simultaneously bear vertical and horizontal simple harmonic loads, and the critical frequency theoretical solution and the finite element solution are compared and verified under the condition that the loads are the same in frequency; in the model, the soil body adopts a mole-coulomb model, the pile body and the soil body both adopt three-dimensional entity units, the unit type is C3D8R, the reduction integral is calculated, and the total unit number is 289014.
In this embodiment, the unstable load calculation in step (3): the unstable load under the action of the simple harmonic load is obtained by changing according to an amplitude formula when N istWhen the critical load is reached, the amplitude tends to infinity, and the expression of the unstable load obtained by converting and arranging according to the amplitude formula calculated under the condition that the vertical simple harmonic load and the wave load exist simultaneously is as follows:
Figure BDA0002701450270000183
unstable load NtLoad P along with pile top0The depth h, k of the pile body entering water is the resistance coefficient of the soil body on the pile side, and the displacement is drawn along with NtAnd a time t change displacement response graph, and the unstable load N is observed from the grapht(iii) occurrence of (a);
with P0Increase of (2), destabilizing load NtcrThe linear reduction is realized, but when the frequency of the simple harmonic load is increased to a certain value, the frequency of the simple harmonic load is the same as the frequency of the structural parameter resonance, the simple harmonic load is subjected to sudden change, and the frequency is instantly reduced to be almost 0; n is increased along with the increase of the depth h of the pile body into the watertcrWill decrease rapidly and then tend to stabilize; change of pile side soil body horizontal resistance coefficient k to NtcrWhen the frequency of the simple harmonic load is less than the critical frequency, the unstable load increases linearly with the increase of k.
In this embodiment, the parameter sensitivity analysis in step (4); the stability and the amplitude of the pile body are influenced by a plurality of factors, such as pile diameter, pile length, pile body scouring depth, pile side soil resistance and the like, different factors have different influences on the parameter resonance frequency and the amplitude of the pile foundation, the influence degree of various factors on the critical frequency and the amplitude of the pile body is obtained through analyzing the factors, and in the actual engineering, the amplitude of the pile foundation is controlled by adjusting the proper influence factors according to local conditions so as to improve the stability of the pile body; the analysis of the above influencing factors shows that the following conclusions can be mainly obtained through parameter analysis:
(4-1) the horizontal resistance coefficient k of the soil body on the pile side has great influence on the critical frequency, and the resistance of the soil body on the pile side is increased to increase the critical frequency, so that the resistance is far away from the frequency of external load, and the probability of an unstable area is reduced; secondly, the influence of the pile body mass m is caused, the critical frequency is increased along with the increase of the pile body mass, and the influence of the soil body shear stiffness G on the critical frequency is relatively small but can not be ignored; different measures are taken in an actual engineering structure according to the influence of different parameters, so that parameter resonance and unstable regions are avoided when the external load frequency is the same as the critical frequency;
(4-2) the relationship between the amplitudes under four simple harmonic loading conditions is: when the longitudinal and transverse simple harmonic loads exist at the same time and have the same frequency, the amplitude is the largest, then the situation that only the transverse simple harmonic load exists, and the situation that only the vertical simple harmonic load exists when the amplitude is the smallest; the existence of the vertical simple harmonic load can inhibit the amplitude to a certain extent, and the transverse simple harmonic load plays a main role in the amplitude;
(4-3) the influence of the pile length on the critical frequency is related to the soil body on the pile side, in the initial stage, the soil body constraint action is weaker, the critical frequency is firstly reduced along with the increase of the length of the pile in the soil, and then the critical frequency is increased along with the increase of the pile length after the constraint action of the soil body is gradually enhanced; when the pile diameter is smaller, the amplitude slightly increases with the increase of the pile diameter d, because the wave load increases with the increase of the pile diameter, but the final amplitude decreases with the increase of the pile diameter;
(4-4) the wave height and the wavelength change in the wave load can have certain influence on the amplitude of the pile body, the amplitude of the pile body is linearly increased along with the wave height along with the increase of the parameters, and is nonlinearly increased along with the wavelength, but the relatively increased amplitude is smaller than the change of other factors; in actual engineering, the change of waves cannot be controlled, but the vibration amplitude of the pile foundation is controlled by using the results of various parameter sensitivity analyses obtained in the invention and starting from other factors, so that the pile foundation is kept in a stable state.
The method for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load considers the wave load as a horizontal dynamic load applied to the pile side, simultaneously considers the influence of the wave scouring action, calculates the resistance of the soil body on the pile side by adopting a two-parameter method, obtains a pile body stability dynamic differential equation on the basis of a Mathieu-Hill equation, explores the pile body dynamic stability problem under four different conditions, obtains a theoretical solution of corresponding critical frequency and instability load, analyzes a series of parameter sensitivity, explores influence factors of the pile body critical frequency and amplitude change of the pile body under the action of four different loads, and gives specific calculation steps. For actual engineering, according to specific requirements, various parameters are adjusted in a targeted manner according to the influence degrees of different factors, so that the purposes of improving the stability of the pile body and being economical are achieved.
Example three:
this embodiment is substantially the same as the above embodiment, and is characterized in that:
in this embodiment, a method for calculating a theoretical solution of dynamic stability load and parameter resonance frequency of a pile foundation bearing a wave load is provided. In this embodiment, the length l is 35m, h is 15m, the diameter D is 0.8m, and E is 3 × 104Mpa，I＝πd4/64，N0＝1000kN，NtIn order to consider the extreme case, the initial wave parameter is H3 m, the L wavelength is 20m, and the wave load frequency is ω 0.2, and the calculation model graph is shown in fig. 1.
The specific calculation steps are as follows:
(1) firstly, the wave load f is carried outzThe calculation of (2) is carried out by adopting a diffraction theory:
Figure BDA0002701450270000201
rho is the density of the seawater and is 1030kg/m 3;
g is gravity acceleration, and is 9.8m/s2；
H is the wave height;
Figure BDA0002701450270000202
L1is the wavelength;
z is water depth, dlThe depth of the pile body into water is not included;
Figure BDA0002701450270000203
J1' is a first order Bessel function of the first kind, Y1' is a first order Bessel function of the second kind;
omega is the frequency of the wave load and d is the pile diameter.
(2) The resistance of the pile side soil body is calculated by adopting a two-parameter method, and the calculation formula is as follows:
Figure BDA0002701450270000204
k＝m0b1，G＝Gpb1，
m0is the pile-side soil coefficient, GPBecause the pile foundation vibrates under dynamic load and the wave load on the pile side is a cyclic periodic load, the cyclic reciprocating action of the cyclic periodic load can weaken the rigidity of the soil body, and the resistance coefficient m of the soil body on the pile side under the action of cyclic load is considered0Is calculated as follows:
Figure BDA0002701450270000205
in the formula TNThe period for loading the load, N is the cycle number of the load, m0The' value can be calculated by referring to an empirical formula recommended by building codes:
Figure BDA0002701450270000211
Figure BDA0002701450270000212
c is the internal friction angle and cohesive force of the soil body, vbFor transverse displacement, m0The values may be weighted averages of the layer thicknesses of the different soil layers. b1For calculating the width, when the pile diameter d is more than or equal to 1, b10.9(d + 1); when d < 1, b1＝0.9(1.5d+0.5)。
GPShear stiffness of the soil body, GPThe value of (b) can adopt a fitted shear stiffness empirical formula:
Figure BDA0002701450270000213
v in the formulasIs the Poisson's ratio of the foundation soil, EsIs the modulus of elasticity, h, of the foundation soilgThe thickness of the shear layer of the foundation soil, hgAccording to the results of previous researches, 11 times of pile diameter is taken for calculation.
The boundary condition of the pile foundation is simplified into free upper part and embedded lower end, and the horizontal displacement of the pile body under the boundary condition is assumed as follows:
Figure BDA0002701450270000214
(3) the method comprises the steps of establishing a pile body dynamic differential equation, obtaining a Mathieu type dynamic differential equation through a pile body energy equation and a Hamilton principle, obtaining an expression of pile body displacement through calculation, and obtaining a parameter working vibration frequency and a destabilizing load through the expression of displacement.
The energy equation is shown below:
Π＝U+V+T+D
according to the dynamic differential equations under the four different conditions, the pile body displacement under the four conditions can be obtained, and the critical frequency and the instability load under the four conditions can be obtained by substituting the specific numerical value for calculation.
The analysis of the critical frequency of the pile body is shown in fig. 8-16, the critical frequency is an important parameter in the vibration process of the pile body, the period of the external load is generally large, the frequency of the load action is small, for example, the wave load period is generally 5-7 s, so that the critical frequency of the pile body is increased, the frequency of the external load can hardly reach the value of the critical frequency, the probability of parameter resonance is reduced, and the stability of the pile body is improved. The horizontal resistance coefficient k of the pile side soil body has great influence on the critical frequency, the constraint effect of the pile side can be enhanced by increasing the resistance of the pile side soil body, the critical frequency is effectively increased, and therefore the stability of the pile body is improved; the pile body mass m also has a relatively large influence on the critical frequency, the critical frequency is increased along with the increase of the pile body mass, and the influence of the soil body shear stiffness G on the critical frequency is relatively small, but the influence is also considered; the influence of the pile length on the critical frequency is related to the soil body on the pile side, the soil body constraint action is weaker in the initial stage, the critical frequency is firstly reduced along with the increase of the length of the pile in the soil, and then the critical frequency is increased along with the increase of the pile length after the constraint action of the soil body is gradually enhanced; when the pile diameter is smaller, the amplitude may slightly increase with the increase of the pile diameter d, because the wave load increases with the increase of the pile diameter, resulting in a slight increase of the amplitude, but the final amplitude decreases with the increase of the pile diameter; the wave height and wavelength have less, but not negligible, effect on the amplitude.
Analysis of amplitude as shown in fig. 17 to 21, the relationship between amplitudes for four simple harmonic loads is: when the longitudinal and transverse simple harmonic loads exist at the same time and have the same frequency, the amplitude is the largest, then the situation that only the transverse simple harmonic load exists, and the situation that only the vertical simple harmonic load exists when the amplitude is the smallest; the existence of vertical simple harmonic load can have certain inhibitory action to the amplitude, and horizontal simple harmonic load plays the main effect to the amplitude.
The embodiment is used for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load. The method comprises the following steps: based on a variational principle, an energy equation and a Hamilton principle, a dynamic differential equation of a pile body under four different conditions (vertical simple harmonic load, horizontal simple harmonic load, longitudinal and horizontal same-frequency simple harmonic load and longitudinal and horizontal different-frequency simple harmonic load) is deduced by a double-parameter foundation model, a non-homogeneous form Mathieu equation is obtained through arrangement, and the critical frequency and the unstable load of parameter resonance are obtained through solution. By comparing the analytic solution of the invention with the calculation result of finite element simulation, the correctness of theoretical derivation is verified. Analysis of the calculation results shows that: pile top load P0The influence of the depth of the pile body into water on the unstable load is larger, but the unstable load is also influenced by the resonance frequency; the pile side soil resistance coefficient k has the largest influence on the parameter resonance frequency, the instability frequency of the parameter resonance is rapidly increased along with the increase of k, the stability of the pile body can be effectively improved, and the pile body mass M and the soil body shear rigidity G have smaller relative influence on the instability frequency of the pile body but can not be ignored; the influence of factors such as pile diameter, wavelength and wave height on the amplitude is different, and the influence can be adjusted according to concrete engineering practice to improve pile body stability.
The embodiments of the present invention have been described with reference to the accompanying drawings, but the present invention is not limited to the embodiments, and various changes and modifications can be made according to the purpose of the invention, and any changes, modifications, substitutions, combinations or simplifications made according to the spirit and principle of the technical solution of the present invention shall be equivalent substitutions, as long as the purpose of the present invention is met, and the present invention shall fall within the protection scope of the present invention without departing from the technical principle and inventive concept of the present invention.

Claims (5)

1. A pile foundation dynamic stability analysis and calculation method under the action of wave load is characterized in that a double-parameter foundation model is adopted to replace a traditional winkler model, the action of the wave load and the softening of pile side soil rigidity are considered, and the method comprises the following steps:
(1) theoretical analytic calculation of pile foundation dynamic stability;
(2) comparing the analytic solution with the finite element simulation calculation result;
(3) calculating a destabilizing load;
(4) and (5) analyzing the parameter sensitivity.
2. The method for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load according to claim 1, wherein the specific operation steps of the step (1) are as follows:
(1-1) adopting a two-parameter foundation method to equivalently calculate the resistance of the soil body on the pile side as a spring, bearing vertical static load and vertical simple harmonic load on the pile top, bearing wave load in the horizontal direction, calculating the wave force by adopting a diffraction theory, and simultaneously scouring the pile body:
wave load f on pile sidezThe calculation is as follows:
Figure FDA0002701450260000011
wherein: rho is density of the seawater and is 1030kg/m3；
g is gravity acceleration, and is 9.8m/s2；
H is the wave height;
Figure FDA0002701450260000012
L1is the wavelength;
z is water depth, dlThe depth of the pile body into water is not included;
Figure FDA0002701450260000015
J1' is a first order Bessel function of the first kind, Y1' is a first order Bessel function of the second kind;
omega is the frequency of the wave load;
d is the pile diameter;
the foundation reaction force of the pile side soil body is calculated by adopting a two-parameter foundation method, which is shown as the following formula:
Figure FDA0002701450260000014
wherein k is m0b1，G＝Gpb1，m0Is the resistance coefficient of the pile-side soil body, GPThe shear stiffness of the soil body is the vibration of the pile foundation under the wave load, so the resistance coefficient m of the soil body on the pile side is considered under the condition that the wave is taken as the effect of weakening the stiffness of the soil body by the cyclic load0Is calculated as follows:
Figure FDA0002701450260000021
in the formula TNThe period for loading the load, N is the cycle number of the load, m0The value is calculated according to an empirical formula recommended by pile foundation specifications:
Figure FDA0002701450260000022
Figure FDA0002701450260000023
c is the internal friction angle and cohesive force of the soil body, vbFor transverse displacement, m0Taking the weighted average value of the layer thicknesses of different soil layers; b1For calculating the width, when the pile diameter d is more than or equal to 1, b10.9(d + 1); when d is<1 time, b1＝0.9(1.5d+0.5)，
GPShear stiffness of the soil body, GPThe value of (2) is obtained by adopting a fitted shear stiffness empirical formula:
Figure FDA0002701450260000024
v in the formulasIs the Poisson's ratio of the foundation soil; esThe modulus of elasticity of the foundation soil; h isgTaking eleven times of pile diameter as the thickness of the shear layer of the foundation soil.
The pile foundation boundary condition simplifies to be upper portion freedom, and the lower extreme is inlayed and is fixed, then the lateral displacement of pile body assumes:
Figure FDA0002701450260000025
(1-2) when only the vertical simple harmonic load of the pile top is considered, and the transverse wave load is not considered, i.e. fzWhen the energy principle is equal to 0, the control equation of the pile body according to the energy principle is shown as the following formula:
Π＝U+V+T+D
wherein U is the internal force potential energy of the pile body, and U is equal to US+Up，UpIs strain energy of pile body, UsThe strain energy of the soil body on the pile side is calculated by adopting a two-parameter method, and the method is as follows:
Figure FDA0002701450260000026
Figure FDA0002701450260000027
v is external force potential energy, comprises three parts, pile top load potential energy Vp，Vq0Is hydrostatic pressure potential energy and bending moment potential energy of pile top
Figure FDA0002701450260000028
Namely:
Figure FDA0002701450260000031
wherein:
Figure FDA0002701450260000032
Figure FDA0002701450260000033
Figure FDA0002701450260000034
in the formula M0Is the initial bending moment of the pile top, q0Is hydrostatic pressure, p0For static loading of pile top, PtThe load amplitude is simple harmonic; h is the length of the part of the pile body in the water;
t is the kinetic energy, and only the kinetic energy of the transverse vibration of the pile body is considered here, so the kinetic energy T is:
Figure FDA0002701450260000035
wherein m is the mass of a unit pile length and l is the pile length;
d is damping potential energy:
Figure FDA0002701450260000036
wherein C is a damping parameter;
the complete pile body energy control equation is as follows:
Figure FDA0002701450260000037
the above formula is obtained by using Hamilton principle:
W(t)＝T(t)-U(t)，
Figure FDA0002701450260000038
obtaining a non-homogeneous Mathieu equation after finishing:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r cosθt
wherein:
Figure FDA0002701450260000039
m is the pile body mass;
Figure FDA00027014502600000310
Figure FDA0002701450260000041
Figure FDA0002701450260000042
Figure FDA0002701450260000043
Figure FDA0002701450260000044
q0＝ρgh
Figure FDA0002701450260000045
Figure FDA0002701450260000046
the Mathieu equation can be solved according to equation types and a semi-inverse solution in the form:
f(t)＝a0+a sin(θt)+b cos(θt)
substituting the above formula into the previous nonhomogeneous mathieu equation to obtain:
Figure FDA0002701450260000047
solving the non-homogeneous linear equation system to obtain:
Figure FDA0002701450260000048
the amplitude in the first case is obtained from the above equation:
Figure FDA0002701450260000049
as can be seen from the foregoing inference process, the critical frequency of the parametric resonance at this time is θ ═ Ω;
(1-3) when only the horizontal simple harmonic load is considered, namely only the horizontal wave load and the vertical static load are considered, and the vertical simple harmonic load is not considered, the external force potential energy in the energy equation at the moment is changed, the rest potential energy is unchanged, and the external force potential energy is changed into:
Figure FDA0002701450260000051
in the formula, VP1For vertical loads without simple harmonic terms, VbThe remaining two items are the same as the above;
Figure FDA0002701450260000052
Figure FDA0002701450260000053
the rest processes are the same as the above, and in the second case, the complete pile body energy equation is as follows:
Figure FDA0002701450260000054
and the dynamic differential equation at this time becomes the following equation:
f″(t)+ξf′(t)+Ω2f(t)＝r0+r cosθt
in this time scheme
Figure FDA0002701450260000055
The dynamic differential equation becomes a second-order heterogeneous linear ordinary differential equation, and f (t) is given by the following formula:
f(t)＝ah0+ah sin(ωt)+bh cos(ωt)
substituting the above equation into the dynamic differential equation to obtain:
Figure FDA0002701450260000056
the pile body amplitude in the second case is obtained according to the solution of the equation:
Figure FDA0002701450260000057
at this time, the critical frequency of the pile body is still omega, but the amplitude at this time is different from the former case;
(1-4) considering the situation that the transverse and vertical simple harmonic loads exist simultaneously and have the same frequency, the energy equation form of the pile body is the same as that of the pile body, but the external force potential energy is changed as follows:
Figure FDA0002701450260000061
at this time, the complete energy equation of the pile body is as follows:
Figure FDA0002701450260000062
the power differential equation at this time is still as follows after being collated:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1cosθt
but in this equation
Figure FDA0002701450260000063
As before, the form of the solution to the dynamic differential equation at this time is still:
f(t)＝a00+a11sin(θt)+b11cos(θt)
substituting the above formula into a dynamic differential equation to obtain the following parameter expression:
Figure FDA0002701450260000064
substituting the parameters into an equation, and obtaining the amplitude calculation formula under the condition after arrangement as follows:
Figure FDA0002701450260000065
as can be seen from the above formula, when θ ═ Ω, the parameter resonance occurs, and at this time, θ ═ Ω is the critical frequency, and the amplitude at this time is also different from the previous one;
(1-5) considering the situation that horizontal and vertical simple harmonic loads exist at the same time but the load frequencies are different, the external force load potential energy in the energy equation of the pile body is changed, the rest potential energy is unchanged, and the external force potential energy is changed as follows:
Figure FDA0002701450260000066
the complete pile body energy equation is:
Figure FDA0002701450260000071
the dynamic differential equation then becomes of the form:
f″(t)+ξf′(t)+Ω2(1-2μcos(θt))f(t)＝r0+r1cos(ωt)
the solution to the above equation is obtained in the form of an equation and a semi-inverse solution:
f(t)＝a012+a12sin(θt)+b12cos(θt)+c12sin(ωt)+d12cos(ωt)
substituting the coefficient into a differential equation to obtain the coefficient as follows:
Figure FDA0002701450260000072
the amplitude calculation formula in this case is:
Figure FDA0002701450260000073
from the above formula, when θ ═ Ω or ω ═ Ω, the pile foundation will have parameter resonance, which is one more possible case than the previous case, and at this time, the amplitude is still different from the previous cases.
3. The method for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load according to claim 1, wherein in the step (2), a model of the pile foundation in the sea water is established based on a finite element method, the wave force is applied to the pile side in a manner of applying the load to the side surface, and the critical frequency theoretical solution and the finite element solution are compared and verified under the conditions that the pile foundation simultaneously bears the vertical and the horizontal simple harmonic loads and the loads have the same frequency; in the model, the soil body adopts a mole-coulomb model, the pile body and the soil body both adopt three-dimensional entity units, the unit type is C3D8R, the reduction integral is calculated, and the total unit number is 289014.
4. The method for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load according to claim 1, wherein the unstable load calculation in the step (3) comprises the following steps: the unstable load under the action of the simple harmonic load is obtained by changing according to an amplitude formula when N istWhen the critical load is reached, the amplitude tends to infinity, and the expression of the unstable load obtained by converting and arranging according to the amplitude formula calculated under the condition that the vertical simple harmonic load and the wave load exist simultaneously is as follows:
Figure FDA0002701450260000081
unstable load NtLoad P along with pile top0The depth h, k of the pile body entering water is the resistance coefficient of the soil body on the pile side, and the displacement is drawn along with NtAnd the time t, from which the destabilizing load N can be observedt(iii) occurrence of (a);
with P0Increase of (2), destabilizing load NtcrDecrease linearly, but when P0When the frequency of the simple harmonic load is the same as the resonance frequency of the structural parameters under a certain condition, the unstable load can generate sudden change, and NtcrWill momentarily decrease to almost 0; n is increased along with the increase of the depth h of the pile body into the watertcrWill decrease rapidly and then tend to stabilize; change of pile side soil body horizontal resistance coefficient k to NtcrWhen the frequency of the simple harmonic load is less than the critical frequency, the unstable load increases linearly with the increase of k.
5. The method for analyzing and calculating the dynamic stability of the pile foundation under the action of the wave load according to claim 1, wherein the parameter sensitivity in the step (4) is analyzed; the stability and the amplitude of the pile body are influenced by a plurality of factors, such as pile diameter, pile length, pile body scouring depth, pile side soil resistance and the like, different factors have different influences on the parameter resonance frequency and the amplitude of the pile foundation, the influence degree of various factors on the critical frequency and the amplitude of the pile body is obtained through analyzing the factors, and in the actual engineering, the amplitude of the pile foundation is controlled by adjusting the proper influence factors according to local conditions so as to improve the stability of the pile body; the following conclusions are mainly drawn by parametric analysis:
(5-1) the horizontal resistance coefficient k of the soil body on the pile side has great influence on the critical frequency, and the resistance of the soil body on the pile side is increased to increase the critical frequency, so that the resistance is far away from the frequency of external load, and the probability of an unstable area is reduced; secondly, the influence of the pile body mass m is caused, the critical frequency is increased along with the increase of the pile body mass, and the influence of the soil body shear stiffness G on the critical frequency is relatively small but can not be ignored; different measures are taken in an actual engineering structure according to the influence of different parameters, so that parameter resonance and unstable regions are avoided when the external load frequency is the same as the critical frequency;
(5-2) the relationship between the amplitudes under four simple harmonic loading conditions is: when the longitudinal and transverse simple harmonic loads exist at the same time and have the same frequency, the amplitude is the largest, then the situation that only the transverse simple harmonic load exists, and the situation that only the vertical simple harmonic load exists when the amplitude is the smallest; the existence of the vertical simple harmonic load can inhibit the amplitude to a certain extent, and the transverse simple harmonic load plays a main role in the amplitude;
(5-3) the influence of the pile length on the critical frequency is related to the soil body on the pile side, in the initial stage, the soil body constraint action is weaker, the critical frequency is firstly reduced along with the increase of the length of the pile in the soil, and then the critical frequency is increased along with the increase of the pile length after the constraint action of the soil body is gradually enhanced; when the pile diameter is smaller, the amplitude slightly increases with the increase of the pile diameter d, because the wave load increases with the increase of the pile diameter, but the final amplitude decreases with the increase of the pile diameter;
(5-4) the wave height and the wavelength change in the wave load can have certain influence on the amplitude of the pile body, the amplitude of the pile body is linearly increased along with the wave height along with the increase of the parameters, and is nonlinearly increased along with the wavelength, but the relatively increased amplitude is smaller than the change of other factors; in actual engineering, the change of waves cannot be controlled, but the vibration amplitude of the pile foundation is controlled by using the results of various parameter sensitivity analyses obtained in the invention and starting from other factors, so that the pile foundation is kept in a stable state.
CN202011023577.6A 2020-09-25 2020-09-25 Pile foundation dynamic stability analysis and calculation method under wave load action Active CN112287574B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202011023577.6A CN112287574B (en) 2020-09-25 2020-09-25 Pile foundation dynamic stability analysis and calculation method under wave load action

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202011023577.6A CN112287574B (en) 2020-09-25 2020-09-25 Pile foundation dynamic stability analysis and calculation method under wave load action

Publications (2)

Publication Number Publication Date
CN112287574A true CN112287574A (en) 2021-01-29
CN112287574B CN112287574B (en) 2021-10-08

Family

ID=74421336

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202011023577.6A Active CN112287574B (en) 2020-09-25 2020-09-25 Pile foundation dynamic stability analysis and calculation method under wave load action

Country Status (1)

Country Link
CN (1) CN112287574B (en)

Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112948939A (en) * 2021-03-12 2021-06-11 上海大学 Single-pile dynamic stability analysis method and system under ship impact load action
CN113177243A (en) * 2021-03-12 2021-07-27 上海大学 Method and system for analyzing dynamic response and dynamic impedance of pile group
CN113821950A (en) * 2021-09-10 2021-12-21 同济大学 Vibration measurement method for size of deep water pile foundation scour pit
CN113887109A (en) * 2021-10-20 2022-01-04 上海大学 Analytic calculation method for stability analysis of pile foundation under explosive dynamic load
CN114166673A (en) * 2021-12-01 2022-03-11 福建工程学院 Method for predicting rising displacement of seabed arc oscillation shearing damage soil body under wave action
CN117763925A (en) * 2024-02-22 2024-03-26 中铁建设集团华北工程有限公司 Method for analyzing foundation pit support by using prestressed hollow pile

Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102776875A (en) * 2012-08-19 2012-11-14 张望喜 Method for field measurement for foundation bed coefficient of double-parameter foundation by virtue of static load test for rigid plates
CN105424466A (en) * 2015-11-19 2016-03-23 中国石油天然气集团公司 Method for evaluating resistance of sandy soil on embedded submarine pipeline under action of waves
RU2690411C1 (en) * 2018-04-09 2019-06-03 Сергей Николаевич Косарев Method of forming specified parameters of curvature of a modified double-layer graft with pile surface for treating progressing myopia
CN111241621A (en) * 2020-01-13 2020-06-05 西南交通大学 Stability evaluation method for high-strength pile composite foundation under embankment
CN111310264A (en) * 2020-02-07 2020-06-19 大连海事大学 Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model
CN111382517A (en) * 2020-03-16 2020-07-07 上海大学 Pile foundation buckling critical load analytical solution analysis method based on double-parameter foundation model
CN111460547A (en) * 2020-03-16 2020-07-28 上海大学 Deformation analysis method of expanded-base uplift pile group considering reinforcement effect

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102776875A (en) * 2012-08-19 2012-11-14 张望喜 Method for field measurement for foundation bed coefficient of double-parameter foundation by virtue of static load test for rigid plates
CN105424466A (en) * 2015-11-19 2016-03-23 中国石油天然气集团公司 Method for evaluating resistance of sandy soil on embedded submarine pipeline under action of waves
RU2690411C1 (en) * 2018-04-09 2019-06-03 Сергей Николаевич Косарев Method of forming specified parameters of curvature of a modified double-layer graft with pile surface for treating progressing myopia
CN111241621A (en) * 2020-01-13 2020-06-05 西南交通大学 Stability evaluation method for high-strength pile composite foundation under embankment
CN111310264A (en) * 2020-02-07 2020-06-19 大连海事大学 Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model
CN111382517A (en) * 2020-03-16 2020-07-07 上海大学 Pile foundation buckling critical load analytical solution analysis method based on double-parameter foundation model
CN111460547A (en) * 2020-03-16 2020-07-28 上海大学 Deformation analysis method of expanded-base uplift pile group considering reinforcement effect

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
LI YA ET AL: "Dynamic response of pile-cap structure under random sea wave action", 《PROCEDIA ENGINEERING》 *
ZHANG YONGMOU ET AL: "Comparison of dynamic performance of laterally loaded hollow pile with solid pile", 《THE 2ND INTERNATIONAL CONFERENCE ON INFORMATION SCIENCE AND ENGINEERING》 *
丁加明: "桥梁基桩的屈曲及其可靠度分析方法研究", 《中国优秀硕士学位论文全文数据库电子期刊 工程科技II辑》 *
姚文娟: "边载作用下高桩码头桩基承载力性能及加固研究", 《水运工程》 *

Cited By (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112948939A (en) * 2021-03-12 2021-06-11 上海大学 Single-pile dynamic stability analysis method and system under ship impact load action
CN113177243A (en) * 2021-03-12 2021-07-27 上海大学 Method and system for analyzing dynamic response and dynamic impedance of pile group
CN113177243B (en) * 2021-03-12 2022-07-08 上海大学 Method and system for analyzing dynamic response and dynamic impedance of pile group
CN113821950A (en) * 2021-09-10 2021-12-21 同济大学 Vibration measurement method for size of deep water pile foundation scour pit
CN113821950B (en) * 2021-09-10 2024-02-27 同济大学 Vibration measurement method for deep water pile foundation scour pit size
CN113887109A (en) * 2021-10-20 2022-01-04 上海大学 Analytic calculation method for stability analysis of pile foundation under explosive dynamic load
CN114166673A (en) * 2021-12-01 2022-03-11 福建工程学院 Method for predicting rising displacement of seabed arc oscillation shearing damage soil body under wave action
CN117763925A (en) * 2024-02-22 2024-03-26 中铁建设集团华北工程有限公司 Method for analyzing foundation pit support by using prestressed hollow pile
CN117763925B (en) * 2024-02-22 2024-04-30 中铁建设集团华北工程有限公司 Method for analyzing foundation pit support by using prestressed hollow pile

Also Published As

Publication number Publication date
CN112287574B (en) 2021-10-08

Similar Documents

Publication Publication Date Title
CN112287574B (en) Pile foundation dynamic stability analysis and calculation method under wave load action
Zhang et al. Numerical study of an oscillating wave energy converter with nonlinear snap-through power-take-off systems in regular waves
Lian et al. Analysis on flow induced motion of cylinders with different cross sections and the potential capacity of energy transference from the flow
CN113177243B (en) Method and system for analyzing dynamic response and dynamic impedance of pile group
Jalbi et al. Closed form solution for the first natural frequency of offshore wind turbine jackets supported on multiple foundations incorporating soil-structure interaction
Ueda et al. Centrifuge model tests and effective stress analyses of offshore wind turbine systems with a suction bucket foundation subject to seismic load
Jalbi et al. Concept design of jacket foundations for offshore wind turbines in 10 steps
CN112698406A (en) Method for simulating seawater velocity potential under seismic excitation of ocean field
Xu et al. Dynamic stability analysis of pile foundation under wave load
Jia et al. The seismic response of composite bucket foundation for offshore wind turbines under scour conditions
Jalbi et al. A comparison between advanced and simplified methods to predict the natural frequency of offshore wind turbines incorporating soil-structure interaction
Yu et al. A Rayleigh-Ritz solution for high order natural frequencies and eigenmodes of monopile supported offshore wind turbines considering tapered towers and soil pile interactions
CN110889245A (en) Fluid-solid coupling seismic response analysis method for tower column structure coupling system of hydraulic ship lift
CN110598306A (en) Vibration isolation modeling method considering coupling of workshop beam foundation and machine tool foundation
Manjula et al. Response of a slender cylindrical member under breaking wave impact
Rong et al. Analytical solution for natural frequency of monopile
Liu et al. Study of combined pile raft foundations for heavy dynamic equipment
Zhang et al. Bearing characteristics of mono-column composite bucket foundation in sand for offshore wind turbines
Gorini et al. Predicting the dynamic response of friction dissipative foundations using a modified Newmark model
Li et al. Research on the equivalent stiffness of bucket foundations for offshore wind power
CN111962572B (en) Horizontal-swing vibration impedance determination method for partially-embedded pile foundation
Shibata et al. Seismic resistant design method for open-type wharf with pneumatic caisson foundation
Ravichandran et al. Comparison of settlement response of piled-raft foundation subjected to combined loads computed from finite element and analytical models
Duong An investigation into the performance of anchors under dynamic loading
Yuan et al. Effects of Scour on Stiffness of Wide Shallow Bucket Foundation and 1st Natural Frequency of Offshore Wind Turbine

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
CB03 Change of inventor or designer information
CB03 Change of inventor or designer information

Inventor after: Yao Wenjuan

Inventor after: Xu Xu

Inventor after: Zhang Zhen

Inventor after: Cheng Zekun

Inventor after: Ding Gang

Inventor before: Zhang Zhen

Inventor before: Xu Xu

Inventor before: Yao Wenjuan

Inventor before: Cheng Zekun

Inventor before: Ding Gang

GR01 Patent grant
GR01 Patent grant