CN111914446B - Supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis - Google Patents
Supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis Download PDFInfo
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Abstract
The invention discloses a supercritical angle oblique incidence earthquake motion input method in finite element numerical analysis. The method is mainly directed to a layered foundation and comprises the following steps: solving the free field response; the earthquake motion input is realized in a mode that equivalent node force is applied to a viscoelastic artificial boundary node to enable displacement and stress on the artificial boundary to be the same as an original free field. Wherein solving for the free-field response further comprises: adopting a layered foundation and a dynamic stiffness matrix with accurate lower half space to establish a free field integral dynamic stiffness matrix; determining outcrop motion of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop motion; solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response; and further solving a free field time domain response solution by combining a Fourier time frequency transformation and an equivalent linearization method. The method solves the problem of inputting the finite element numerical model by the seismic oscillation of the supercritical angle incidence, and has high calculation precision.
Description
Technical Field
The invention belongs to the field of seismic engineering, and particularly relates to a supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis.
Background
In the earthquake simulation analysis by using finite element software, earthquake motion input is divided into vibration input and fluctuation input according to different earthquake action application modes, wherein the former is only suitable for the vertical incidence condition of shear waves, and the latter can consider the oblique incidence condition of plane waves. The wave input is usually realized by solving the free field seismic response, that is, the equivalent node force corresponding to the seismic input is obtained by solving the free field response, and the equivalent node force is applied to the artificial boundary, so that the stress and displacement at the artificial boundary are consistent with the corresponding position of the free wave field. The solution of the free field comprises an analytic method and a numerical method, wherein the analytic method can only solve the problem of elastic uniform half-space, and various numerical methods are needed for the condition of the layered foundation.
The free field numerical calculation methods can be classified into two types, one is a time domain method and the other is a frequency domain method. The time domain calculation method is more widely applied from the aspect of realizing the oblique incidence seismic oscillation input of the layered foundation in the finite element method. But the time domain method cannot simulate SV wave supercritical angle incidence because the surface will not reflect P waves and the particles near the surface will vibrate in the form of Rayleigh surface waves. For seismic input of a frequency domain method, the application of SV supercritical angle seismic input in finite element numerical analysis is not seen yet at present. Therefore, how to input seismic oscillation of the supercritical angle incidence of the SV wave into a finite element numerical model is an urgent problem to be solved.
Disclosure of Invention
The invention aims to solve the problems and provides a supercritical angle oblique incidence earthquake motion input method in finite element numerical analysis.
The method may include: step 1, solving free field response; and 2, applying equivalent node force on the viscoelastic artificial boundary nodes to enable the displacement and stress on the artificial boundary to be the same as those of the original free field to achieve seismic motion input. Said solving for a free-field response further comprises: adopting a layered foundation and a dynamic stiffness matrix with accurate lower half space to establish a free field integral dynamic stiffness matrix; determining outcrop motion of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop motion; solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response; and further solving a free field time domain response solution by combining a Fourier time frequency transformation and an equivalent linearization method.
Step 1.1, adopting a layered foundation and a dynamic stiffness matrix with accurate lower half space to establish a free field integral dynamic stiffness matrix.
In the frequency domain, the dynamic stiffness matrix of the layered foundationK]LCan be expressed as:
whereinP、RRespectively representing horizontal and vertical loads acting on a soil layer interface, subscripts 1 and 2 respectively representing that the loads act on the upper surface and the lower surface of a soil layer, and i is a unit complex number; rigidity matrix [ 2 ]K]LIs a symmetric matrix, the coefficients of whichk ij See the literatureWolf John P. Dynamic soil-structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985；u、wRespectively represent the horizontal displacement and the vertical displacement at the soil layer interface, and subscripts 1 and 2 respectively represent the displacement of the upper surface and the lower surface of the soil layer.
The stiffness matrix of the half-space is [ [ alpha ] ]K]RCan be expressed as:
whereinP 0、R 0Respectively representing the horizontal and vertical loads acting on the surface of a half space, the stiffness matrixK]RIs a symmetric matrix, the coefficients of whichr ij See the literatureWolf John P. Dynamic soil-structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985；u 0 、w 0Respectively, the horizontal displacement and the vertical displacement of the surface of the half space, namely the outcrop movement of the bedrock.
The stiffness matrix of soil layerK]LAnd its lower half space rigidity matrixK]RThe free field integral rigidity matrix can be obtainedK]。
And 1.2, determining outcrop motion of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop motion.
Said exposed end displacement of bed rocku 0, w 0}TFor uniform half-space, a given shift spectrum isU 0(ω) Is seismic-moved byU 0(ω) Amplitude of incoming wave as incident SV or P waveA sv0OrA P0The half-space earth surface motion under the action of the amplitude of the incoming wave is determined by the following formula:
whereinA sv0AndA P0the incoming wave amplitude of the incident SV wave and the incident P wave;
is a quantity related to the incident angle of the seismic wave, as detailed in the literatureWolf John P. Dynamic soil-structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985。
The total external load vector of the free field isQ i }TSince the external load acting on the soil layer is 0, thenQ 1ToQ n2-2Are all 0, andQ n2-1,Q n2}={P 0, R 0}Twherein, in the step (A),nthe number of layers of lying soil on the bedrock.
Said Chinese dictionaryP 0, R 0}TThe outcrop displacement of the bedrock obtained according to the formula (3) is used for the external load applied to the bedrock surface in the layered foundationu 0, w 0}TAnd the result is obtained by substituting the formula (2).
And step 1.3, solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response.
The free field response can be obtained by a Gaussian elimination method:
{U}T=[K]-1{Q}T (4)
whereinU}TThe displacement response of each soil layer interface of the free field is obtained; {Q}TIs the external load vector.
And the coordinates in the soil layer are (x, z) The displacement of (a) can be determined by:
u(x, z)=u(z)exp(－ikx) (5a)
w(x, z)=w(z)exp(－ikx) (5b)
u(z)=l x [A pexp(iksz)+B pexp(－iksz)]－m x t[A svexp(iktz)－B svexp(－iktz)] (5c)
w(z)=－l x s[A pexp(iksz)－B pexp(－iksz)]－m x [A svexp(iktz)+B svexp(－iktz)] (5d)
wherein the content of the first and second substances,A PandB Pis the incoming wave and the wave removing amplitude of P waves in the soil layer,A SVandB SVmoving at a known soil layer interface for incoming wave and wave-removing amplitude of SV wave in soil layerU}TIn time, the displacement of the upper and lower interfaces of the soil layer can be substituted into the formula to reversely calculate the coefficientA p、B pAndA SV、B SV；l x andm x is a parameter related to the angle of incidence of seismic wavesThe number of the first and second groups is,l x = sinθ P , m x = sinθ SV，θ P、θ SVthe incident angles of the P wave and the SV wave respectively;kin terms of the wave number, the number of waves,k=ω/c；cphase velocity in the horizontal direction according to Snell's theoremcAre all equal in each soil layer and meet
. Parameter(s)s、t、 
Respectively as follows:
wherein the content of the first and second substances,c Pandc Srespectively the compression wave velocity and the shear wave velocity of the soil,ζis the damping ratio.
The coordinates in the soil layer are (x, z) The stress at (a) can be determined by:
σ x (x,z)=(1+2iζ)[λ(u ,x +w ,z )+2Gu ,x ] (7a)
σ z (x,z)=(1+2iζ)[λ(u ,x +w ,z )+2Gu ,z ] (7b)
τ xz (x,z)=(1+2iζ) G(u ,z +w ,x ) (7c)
whereinG、λIs the Lame constant.
The displacements and stresses obtained by the formulae (5 a), (5 b), (5 c), (5 d), (7 a), (7 b) and (7 c) are all frequency domain solutions.
And step 1.4, further solving a free field time domain response solution by combining a Fourier time frequency transformation and equivalent linearization method.
The fourier time-frequency transform transforms the frequency domain solution to the time domain solution by inverse fourier transform.
The equivalent linearization is to use a set of equivalent shear modulus and damping ratio to replace the shear modulus and damping ratio at different strain amplitudes, so as to convert the nonlinear problem into a linear problem to solve, and the program flow chart is shown in fig. 2. As the stress state of the free field is actually a two-dimensional plane strain state, the maximum shear strain of any point in the soil layerγ maxFrom the strain state of the spotε x 、ε z 、ε xz Determining:
the equivalent shear strainγ effTaken as maximum shear strainγ max0.65 times the peak time course:
γ eff =0.65max∣γ max(t)∣ (9)
wherein m isax∣γ max(t) is the peak value of the maximum shear strain time course.
For a supercritical angle oblique incidence earthquake motion input method in finite element numerical analysis, step 2 is implemented by applying equivalent node force on artificial boundary nodes to enable displacement and stress on the artificial boundary to be the same as those of an original free field, and the specific implementation steps are as follows:
the viscoelastic artificial boundary can be realized by applying a grounding spring and a damper element which are connected in parallel on the artificial boundary of the computational model, and the related parameters of the spring and the damper element of the viscoelastic artificial boundary are taken as follows:
wherein the content of the first and second substances,K、Cthe stiffness and damping coefficients of the spring, damper element, respectively, the subscripts bn, bt respectively denote the normal and tangential directions applied to the boundary by the element,ρas the density of the medium, it is,ras distance, parameter, of scattering source to artificial boundaryA、BObtained by numerical experiments, and suggestedA=0.8，B= 1.1. In addition, in the application process of the viscoelastic artificial boundary, the area of the influence region of the boundary node also needs to be considered in the parameters of the rigidity coefficient and the damping coefficient.
The seismic motion input can be realized in a mode that equivalent node force is applied to an artificial boundary node to enable the displacement and the stress on the artificial boundary to be the same as those of an original free field. The equivalent node forces applied to the left, right and bottom boundaries of the finite element model are respectively:
left boundary:
(11a)
right border:
(11b)
bottom boundary:
(11c)
wherein the content of the first and second substances,
the horizontal speed and the vertical speed are obtained by derivation of displacement to time;F ix 、F iz for each boundary node, the equivalent node forces in the x and z directionsi=l、r、b)；K、CThe stiffness and damping coefficients of the spring, damper element, respectively, the subscripts bn, bt representing the normal and tangential directions applied to the boundary by the element, respectively; sigma ij Is the nodal stress;A l 、A r 、A b the influence areas of the left, right and bottom boundary nodes of the finite element model are shown.
The invention solves the problem of inputting finite element numerical model by seismic oscillation of supercritical angle incidence, and has high calculation precision.
Drawings
FIG. 1 is a schematic diagram of a free-field model, in whichG i、ρ i、ζ i、ν iRespectively the shear modulus, density, damping ratio and Poisson coefficient of the i-th layer soil,G R、ρ R、ζ R、ν Rrespectively shear modulus, density, damping ratio and Poisson system of the bedrock,θis wavefront and levelxThe angle of the axes.
FIG. 2 is a flow chart of an equivalent linearization method.
FIG. 3 is a soil layer shear modulus ratioG/G max) Damping ratio ofζ) And shear strain ofγ) The relationship is a graph.
FIG. 4 shows the response ratio of the surface acceleration obtained by the method of the present invention (stiffness matrix method), EERA method and Abaqus methodCompare (A) withθ=0°）。
FIG. 5 is a graph showing a comparison of the surface acceleration response obtained by the method of the present invention (stiffness matrix method) and the Abaqus method (equation:θ=38.6°）。
Detailed Description
The following examples are intended to more clearly illustrate the technical solutions of the present invention, and it should be understood by those skilled in the art that any simple changes or equivalent substitutions within the technical scope of the present invention are included in the protection scope of the present invention.
The following describes a specific embodiment of the present invention based on a nonlinear layered foundation model and verifies the calculation accuracy of the present invention, taking SV wave supercritical angle incidence as an example, with reference to the accompanying drawings. The present example uses the inventive method, EERA method and Abaqus finite element method, respectively, to calculate and compare their surface acceleration responses. The elasticity parameters of soil layers and bedrocks are shown in table 1, and the corresponding soil layer dynamic modulus curve and the corresponding damping ratio curve are shown in fig. 3. When SV waves are vertically incident, the method and the EERA method adopt El Centro waves with acceleration peak amplitude modulation of 0.1g as input seismic motion, the Abaqus finite element method adopts the seismic motion input calculated by the method, and the earth surface acceleration response calculated by the method, the EERA method and the Abaqus finite element method is compared. When the SV wave is incident at an angle greater than the critical angle: (θ=38.6 °), comparing the horizontal and vertical acceleration responses of the earth's surface obtained using the method of the invention and using the Abaqus method.
It should be noted that the method of the present invention adopts rayleigh damping in the Abaqus method, which takes into account the damping ratio of the soil layer, which is different from the complex damping theory adopted by the method of the present invention, and the Abaqus method adopts rayleigh dampingC=αM+βK，MIn order to be a quality matrix,Kis a finite element stiffness matrix, parametersα、βTo damping ratioζThe relationship of (a) is as follows:
wherein the content of the first and second substances,ω 1is the first order natural frequency of the field,ω 2is the dominant frequency of input seismic motion.
TABLE 1 soil layer parameters
| Soil layer numbering | Soil series | Thickness of soil layer/m | Density ofρ/(kg/m3) | Poisson ratioν | Shear wave velocityc s/(m/s) | Velocity of compressional wavec p/(m/s) | Critical angleθ cr/(°) |
| 1 | |
10 | 1925 | 0.333 | 175 | 350 | 30.0 |
| 2 | A | 10 | 1950 | 0.333 | 200 | 400 | 30.0 |
| 3 | |
10 | 1975 | 0.333 | 225 | 450 | 30.0 |
| 4 | |
10 | 2000 | 0.333 | 250 | 500 | 30.0 |
| 5 | |
10 | 2025 | 0.333 | 300 | 600 | 30.0 |
| 6 | |
10 | 2050 | 0.333 | 350 | 700 | 30.0 |
| 7 | |
10 | 2075 | 0.333 | 400 | 800 | 30.0 |
| 8 | |
10 | 2100 | 0.333 | 450 | 900 | 30.0 |
| Bed rock | — | — | 2150 | 0.333 | 500 | 1000 | 30.0 |
The specific implementation steps of the supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis comprise: step 1, solving free field response; and 2, implementing seismic motion input in a mode that equivalent node force is applied to the viscoelastic artificial boundary nodes to enable the displacement and stress on the artificial boundary to be the same as the original free field.
For a supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis, firstly, the solution of free field response is carried out, and fig. 1 shows a free field model schematic diagram. The step of solving for the free-field response further comprises: adopting a layered foundation and a dynamic stiffness matrix with accurate lower half space to establish a free field integral dynamic stiffness matrix; determining outcrop motion of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop motion; solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response; and further solving a free field time domain response solution by combining a Fourier time-frequency transformation method and an equivalent linearization method. The specific implementation steps for solving the free field response are as follows:
step 1.1, establishing a free field integral dynamic stiffness matrix by adopting a layered foundation and a dynamic stiffness matrix with accurate lower half space.
Dynamic stiffness matrix of soil layer in frequency domainK]LCan be expressed as:
whereinP、RRespectively representing horizontal and vertical loads acting on a soil layer interface, subscripts 1 and 2 respectively representing that the loads act on the upper surface and the lower surface of a soil layer, and i is a unit complex number; rigidity matrix [ 2 ]K]LIs a symmetric matrix, the coefficients of whichk ij See the literatureWolf John P. Dynamic soil-structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985；u、wRespectively represent the horizontal displacement and the vertical displacement at the soil layer interface, and subscripts 1 and 2 respectively represent the displacement of the upper surface and the lower surface of the soil layer.
The rigidity matrix of the half space is [ 2 ]K]RCan be expressed as:
whereinP 0、R 0Respectively representing effects on halfThe horizontal and vertical loads of the space surface are set as the rigidity matrixK]RSymmetric matrix of coefficients thereofr ij See the literatureWolf John P. Dynamic soil-structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985；u 0 、w 0Respectively, the horizontal displacement and the vertical displacement of the surface of the half space, namely the outcrop movement of the bedrock.
The stiffness matrix of soil layerK]LAnd its lower half space rigidity matrixK]RThe free field integral rigidity matrix can be obtainedK]。
And 1.2, determining outcrop motion of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop motion.
To be provided withA sv0AndA P0as incoming wave amplitudes of the incident SV wave and the incident P wave, for a uniform half space, the half space surface motion under the action of the incoming wave amplitudes is determined by the following formula:
wherein
Is a quantity related to the incident angle of seismic waves, as detailed in the literatureWolf John P. Dynamic soil- structure interaction[M]. Englewood Cliffs: Prentice-Hall, 1985。
For a given shift spectrum isU 0(ω) Is seismic-moved byU 0(ω) Amplitude of incoming wave as incident SV or P waveA sv0OrA P0According to the formula (3), the exposed head displacement of the bedrock can be obtainedu 0, w 0}TSubstituting the above into equation (2) to obtain an external load curve applied to the bed rock surface in the layered foundationP 0, R 0}T。
Since the external load acting on the soil layer is 0, if the distance is equal toQ i }TMarking the total external load vector of the free field, thenQ 1To is thatQ n2-2Are all 0, andQ n2-1,Q n2}={P 0, R 0}Twherein, in the step (A),nthe number of layers of lying soil on the bedrock.
And 1.3, solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response.
The stiffness matrix of soil layerK]LAnd its lower half space rigidity matrixK]RObtaining the free field integral rigidity matrixK]Combined with external load vectorQ}TThe displacement response of each soil layer interface of the free field can be obtained by Gauss elimination methodU}T：
{U}T=[K]-1{Q}T (5)
And the coordinates in the soil layer are (x, z) The displacement of (a) can be determined by:
u(x, z)=u(z)exp(－ikx) (6a)
w(x, z)=w(z)exp(－ikx) (6b)
u(z)=l x [A pexp(iksz)+B pexp(－iksz)]－m x t[A svexp(iktz)－B svexp(－iktz)] (6c)
w(z)=－l x s[A pexp(iksz)－B pexp(－iksz)]－m x [A svexp(iktz)+B svexp(－iktz)] (6d)
wherein, the first and the second end of the pipe are connected with each other,A PandB Pthe amplitude values of the incoming wave and the removed wave of the P wave in the soil layer,A SVandB SVthe coming wave and wave-removing amplitude of SV wave in the soil layer when the soil layer boundary is knownFace displacementU}TIn time, the displacement of the upper and lower interfaces of the soil layer can be substituted into the formula to reversely calculate the coefficientA p、B pAndA SV、B SV；l x andm x is a parameter related to the angle of incidence of the seismic waves,l x = sinθ P , m x = sinθ SV，θ P、θ SVthe incident angles of the P wave and the SV wave respectively;kis a wave number of the wave number,k=ω/c；cphase velocity in the horizontal direction according to Snell's theoremcEqual in each soil layer and meet the requirement
. Parameter(s)s、t、 
Respectively as follows:
wherein the content of the first and second substances,c Pandc Srespectively the compression wave velocity and the shear wave velocity of the soil,ζis the damping ratio.
The coordinates in the soil layer are (x, z) The stress at (a) can be determined by:
σ x (x,z)=(1+2iζ)[λ(u ,x +w ,z )+2Gu ,x ] (8a)
σ z (x,z)=(1+2iζ)[λ(u ,x +w ,z )+2Gu ,z ] (8b)
τ xz (x,z)=(1+2iζ) G(u ,z +w ,x ) (8c)
whereinG、λIs the Lame constant.
The displacements and stresses obtained by the formulae (6 a), (6 b), (6 c), (6 d), (8 a), (8 b) and (8 c) are all frequency domain solutions.
And step 1.4, further solving a free field time domain response solution by combining a Fourier time frequency transformation and equivalent linearization method.
The fourier time-frequency transform transforms a frequency domain solution to a time domain solution by an inverse fourier transform.
The equivalent linearization method is characterized in that a group of equivalent shear modulus and damping ratio is adopted to replace the shear modulus and the damping ratio under different strain amplitudes, so that the nonlinear problem is converted into a linear problem to be solved. The equivalent linearization method comprises the following steps: assuming an initial shear modulus and a damping ratio, and calculating the maximum shear strain of each soil layer by integrating a stiffness matrix; calculating equivalent shear strain, and determining shear modulus and damping ratio according to the equivalent shear strain; calculating the maximum shear strain of each soil layer by integrating a stiffness matrix according to the newly-solved shear modulus and the damping ratio; and circulating two to three steps until the maximum shearing strain of the last two soil layers is less than an error tolerance value. The flow chart of the equivalent linearization procedure is shown in FIG. 2.
Due to the fact that the stress state of the free field is realActually, the strain state of a two-dimensional plane and the maximum shearing strain of any point in the soil layerγ maxFrom the strain state of the spotε x 、ε z 、ε xz Determining:
equivalent shear strainγ effComprises the following steps:
γ eff =0.65max∣γ max(t)∣ (10)
wherein max |γ max(t) is the peak value of the maximum shear strain time course.
For a supercritical angle oblique incidence seismic oscillation input method in finite element numerical analysis, the seismic oscillation input is realized by applying equivalent node force on a viscoelastic artificial boundary node to ensure that the displacement and the stress on the artificial boundary are the same as the original free field, and the specific implementation steps are as follows:
a viscoelastic artificial boundary is applied in the numerical computational model. The viscoelastic artificial boundary can be realized by applying a grounding spring and a damper element in parallel on the artificial boundary of the computational model, and the related parameters of the spring and the damper element of the viscoelastic artificial boundary are taken as follows:
wherein the content of the first and second substances,K、Cthe stiffness and damping coefficients of the spring, damper element, respectively, the subscripts bn, bt denote the normal and tangential directions applied to the boundary by the element,ρas the density of the medium, it is,ris a powderDistance from source to artificial boundary, parametersA、BObtained by numerical experiments, and suggestedA=0.8，B= 1.1. In addition, in the application process of the viscoelastic artificial boundary, the area of the influence region of the boundary node also needs to be considered in the parameters of the rigidity coefficient and the damping coefficient.
And applying equivalent node force on the viscoelastic artificial boundary node to enable the displacement and stress on the artificial boundary to realize seismic motion input in the same way as the original free field. The equivalent node forces applied to the left, right and bottom boundaries of the finite element model are respectively:
left boundary:
(12a)
right border:
(12b)
bottom boundary:
(12c)
wherein the content of the first and second substances,
the horizontal speed and the vertical speed are obtained by derivation of displacement to time;F ix 、F iz for each boundary node, the equivalent node forces in the x and z directionsi=l、r、b)；K、CThe stiffness and damping coefficients of the spring, damper element, respectively, the subscripts bn, bt representing the normal and tangential directions applied to the boundary by the element, respectively;σ ij is the nodal stress;A l 、A r 、A b the influence areas of the left boundary node, the right boundary node and the bottom boundary node of the finite element model are shown.
FIG. 4 shows SV wave at normal incidence: (θ=0 °) comparative graphs of the ground acceleration responses obtained by the stiffness matrix method, the EERA method, and the Abaqus method, and it can be seen that threeThe results of both methods are more consistent (wherein, the method of the present invention has higher consistency with the EERA method because both methods are frequency domain methods).
FIG. 5 shows when SV wave is incident at greater than critical angle: (θ=38.6 °) from a comparison graph of the horizontal and vertical surface acceleration responses obtained by the method of the present invention and the Abaqus method, it can be seen that the results of both methods are well matched.
Claims (1)
1. A supercritical angle oblique incidence earthquake motion input method in finite element numerical analysis is characterized by comprising the following steps:
step 1, solving free field response, wherein SV wave is vertical incidence or supercritical angle incidence;
step 2, implementing earthquake motion input in a mode that equivalent node force is applied to a viscoelastic artificial boundary node in finite element software to enable displacement and stress on the artificial boundary to be the same as an original free field;
solving for a free-field response further comprises the steps of:
step 1.1, establishing a free field integral rigidity matrix by adopting a layered foundation and a rigidity matrix with accurate lower half space;
wherein, in frequency domain, the rigidity matrix [ K ] of the layered foundation]LExpressed as:
p, R respectively represents the horizontal and vertical loads acting on the soil layer interface, subscripts 1 and 2 respectively represent the loads acting on the upper surface and the lower surface of the soil layer, and i is a unit complex number; stiffness matrix [ K ]]LU and w respectively represent horizontal displacement and vertical displacement at a soil layer interface, and subscripts 1 and 2 respectively represent displacement of the upper surface and the lower surface of the soil layer;
stiffness matrix of half-space [ K ]]RExpressed as:
wherein P is0、R0Respectively representing horizontal and vertical loads acting on the surface of the half space; stiffness matrix [ K ]]RIs a symmetric matrix, u0、w0Respectively representing horizontal displacement and vertical displacement of the surface of the half space, namely exposed movement of bedrock;
integrated stiffness matrix [ K ]]LAnd a stiffness matrix [ K ]]RObtaining a free field integral rigidity matrix [ K ]]；
Step 1.2, determining outcrop movement of the bedrock according to the incident waves of the bedrock and calculating an external load vector applied to a free field according to the outcrop movement;
step 1.3, solving a free field motion equation under the action of the load vector by adopting a Gaussian elimination method to obtain free field response;
and step 1.4, further solving a free field time domain response solution by combining a Fourier time frequency transformation and equivalent linearization method.
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| CN112698406B (en) * | 2020-12-22 | 2022-02-15 | 大连理工大学 | Method for simulating seawater velocity potential under seismic excitation of ocean field |
| CN113486543B (en) * | 2021-05-11 | 2023-10-13 | 中国长江三峡集团有限公司 | Dam-base system seismic response and damage analysis method based on extended finite element method |
| CN113358466B (en) * | 2021-05-28 | 2022-10-04 | 燕山大学 | Method and system for determining transfer coefficient of dynamic stress of layered foundation soil layer interface |
| CN114279827B (en) * | 2021-11-16 | 2023-08-29 | 中国科学院武汉岩土力学研究所 | Method for obtaining tunnel damping layer material |
| CN116306177B (en) * | 2023-05-18 | 2023-08-04 | 石家庄铁道大学 | Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108549104A (en) * | 2018-04-10 | 2018-09-18 | 江南大学 | Layered place inclined seismic wave fluction analysis method |
| CN110427680A (en) * | 2019-07-26 | 2019-11-08 | 中国水利水电科学研究院 | The method for obtaining inclined seismic wave effect slope earthquake motive force enlarge-effect |
| CN110794459A (en) * | 2019-11-14 | 2020-02-14 | 福州大学 | Fitting method for seabed near fault seismic oscillation |
| WO2020035822A1 (en) * | 2018-08-17 | 2020-02-20 | Allen John Damian | Control structure with rotary force limiter and energy dissipater |
-
2020
- 2020-07-10 CN CN202010664078.9A patent/CN111914446B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108549104A (en) * | 2018-04-10 | 2018-09-18 | 江南大学 | Layered place inclined seismic wave fluction analysis method |
| WO2020035822A1 (en) * | 2018-08-17 | 2020-02-20 | Allen John Damian | Control structure with rotary force limiter and energy dissipater |
| CN110427680A (en) * | 2019-07-26 | 2019-11-08 | 中国水利水电科学研究院 | The method for obtaining inclined seismic wave effect slope earthquake motive force enlarge-effect |
| CN110794459A (en) * | 2019-11-14 | 2020-02-14 | 福州大学 | Fitting method for seabed near fault seismic oscillation |
Non-Patent Citations (10)
| Title |
|---|
| 土层地震反应等效线性化方法综述;齐文浩等;《世界地震工程》;20071231;第23卷(第4期);全文 * |
| 地下工程抗震分析中地震动输入方法研究;黄胜等;《岩石力学与工程学报》;20100630;第29卷(第6期);全文 * |
| 地震波斜入射时水平层状场地的非线性地震反应;尤红兵等;《岩土工程学报》;20090228;第31卷(第2期);正文第1，3节，图6-图8 * |
| 基于黏弹性边界的地下隧道非线性地震响应分析模型;梁建文等;《地震工程学报》;20140930;第36卷(第3期);正文第1节 * |
| 基于黏弹性边界的拱坝地震反应分析方法;杜修力等;《水利学报》;20060930;第37卷(第9期);全文 * |
| 弹性和柔性地基上动力人工边界及地震动输入方法;董凯等;《水电能源科学》;20200531;第38卷(第5期);全文 * |
| 水平层状饱和场地地震响应分析的;张季等;《工程力学》;20161031;第33卷(第10期);正文第1节 * |
| 结构一地基动力相互作用问题;刘晶波，吕彦东;《土木工程学报》;19980630;第31卷(第3期);正文第三部分 * |
| 近场波动模拟的人工应力边界条件;杜修力等;《力学学报》;20060131;第38卷(第1期);正文第2.2节 * |
| 黏弹性人工边界地震波动输入方法综述;王晓东;《工程科技》;20180228;第3卷(第1期);全文 * |
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