CN119808497A - A method for inputting three-dimensional layered ground seismic data based on infinite element artificial boundaries - Google Patents
A method for inputting three-dimensional layered ground seismic data based on infinite element artificial boundaries Download PDFInfo
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Abstract
本发明涉及岩土工程抗震分析技术领域，尤其是涉及一种基于无限元人工边界的三维成层地基地震动输入方法，所述方法包括如下步骤：求解自由场下考虑无限元人工边界的三维层状地基的地震等效节点力表达；建立无限元与有限元混合的三维层状地基模型，获取有限域边界处各个边界节点的编号、空间坐标以及等效节点面积，并计算与初始应力场匹配的地应力场节点力；计算无限与有限网格交界处网格节点的所述地震等效节点力，对成层地基模型计算文件进行批量修改，实现各个所述边界节点的初始静力与地震力输入。本发明得到的地震动等效节点力精度高，符合地震波在成层地基中的传播特点，并实现了非均质地基地震节点力的批量施加，具有简便性与可操作性。
The present invention relates to the technical field of seismic analysis of geotechnical engineering, and in particular to a three-dimensional layered foundation vibration input method based on infinite element artificial boundaries, the method comprising the following steps: solving the seismic equivalent node force expression of the three-dimensional layered foundation considering the infinite element artificial boundaries under the free field; establishing a three-dimensional layered foundation model mixed with infinite elements and finite elements, obtaining the number, spatial coordinates and equivalent node area of each boundary node at the boundary of the finite domain, and calculating the ground stress field node force matching the initial stress field; calculating the seismic equivalent node force of the grid node at the junction of the infinite and finite grids, batch modifying the calculation file of the layered foundation model, and realizing the initial static force and seismic force input of each of the boundary nodes. The seismic equivalent node force obtained by the present invention has high precision, conforms to the propagation characteristics of seismic waves in the layered foundation, and realizes the batch application of the seismic node force of the heterogeneous foundation, and has simplicity and operability.
Description
Technical Field
The invention relates to the technical field of geotechnical engineering earthquake-resistant analysis, in particular to a three-dimensional layered foundation earthquake-resistant input method based on infinite artificial boundaries.
Background
The finite element method is widely applied to the field of geotechnical engineering earthquake-resistant analysis. The foundation has semi-infinite characteristics, but in most cases, the seismic analysis only concerns the near-field fluctuation problem, so that a limited calculation region of interest needs to be intercepted from an infinite domain, and artificial boundary conditions are set, so that the propagation characteristics of mechanical waves at the intercepted boundary are consistent with the actual infinite domain situation. As a way of simulating an infinite area, the infinite element has a similar unit shape function with a conventional finite element, so that the infinite element and the conventional finite element have 'coordination' with each other, and the infinite element has advantages compared with other dynamic artificial boundary methods such as a viscoelastic boundary, a boundary element and the like.
The infinite element can greatly reduce the calculation cost of the model, but the existing infinite element power artificial boundary is only suitable for the problem of local point source vibration in the field, and the infinite element cannot directly process the problem of external source incidence such as earthquake. Based on the method, a plurality of students at home and abroad develop related researches on the incidence of the external source of the earthquake. It should be noted that the natural soil layer has a layering property, and reflection and transmission of waves at the interfaces of the soil layers need to be considered, which is very different from the homogeneous foundation model. At present, aiming at the earthquake equivalent node force obtained by layering foundation earthquake input and earthquake dynamic analysis by mostly applying a homogeneous foundation model, only the influence of different materials on wave velocity is considered, and the influence of medium boundaries on earthquake waves is ignored. Although some scholars also put forward a seismic input method considering the foundation layering characteristics, the method is too simplified and has a great difference from the actual situation, and needs to be further studied.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a three-dimensional layered foundation earthquake motion input method based on an infinite element artificial boundary.
The invention provides a three-dimensional layered foundation earthquake motion input method based on an infinite element artificial boundary, which comprises the following steps of solving seismic equivalent node force expression of a three-dimensional layered foundation taking the infinite element artificial boundary into consideration in a free field, establishing a three-dimensional layered foundation model with mixed infinite elements and finite elements, obtaining serial numbers, space coordinates and equivalent node areas of all boundary nodes at the finite field boundary, calculating the earth stress field node force matched with an initial stress field, calculating the seismic equivalent node force of grid nodes at the boundary of infinite and finite grids, and modifying batch calculation files of the layered foundation model to realize initial static force and earthquake force input of all boundary nodes. The seismic equivalent node force obtained by the method has high precision, accords with the propagation characteristics of seismic waves in the layered foundation, realizes batch application of seismic node force of the heterogeneous foundation, and has simplicity and operability.
Optionally, the solving the three-dimensional layered foundation seismic equivalent node force expression considering the infinite dynamic artificial boundary in the free field comprises the following steps:
Determining a normal damper damping coefficient and a tangential damper damping coefficient in each stratified soil layer according to a one-dimensional wave equation suitable for plane waves, a cell stress balance condition and a dynamic artificial boundary complete absorption boundary condition;
considering the condition that seismic waves are vertically incident at the bottom of the foundation, and determining the reflection coefficient and the transmission coefficient of the waves according to the amplitude relation of the incident waves, the reflected waves and the transmitted waves;
And calculating the seismic equivalent node force of the three-dimensional layered foundation according to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the speed time course of the input seismic waves.
Optionally, the normal damper damping coefficient and the tangential damper damping coefficient satisfy the following relationships, respectively:
Wherein, The normal damper coefficient of the ith soil layer,The tangential damper coefficient of the ith soil layer,The soil layer density of the ith soil layer,For the P wave velocity in the ith soil layer,Is the S wave velocity in the ith soil layer.
Optionally, the reflection coefficient satisfies the following relationship:
Wherein, As said reflection coefficient of the wave,For the amplitude of the incident wave,For the amplitude of the transmitted wave,Is the soil layer density of one layer of soil layer far away from the ground in two adjacent soil layers,For the wave velocity in one soil layer far away from the ground in two adjacent soil layers,The soil layer density of one soil layer close to the ground in two adjacent soil layers,Is the wave velocity in one soil layer close to the ground in two adjacent soil layers.
Optionally, the transmission coefficient satisfies the following relationship:
Wherein, As said transmission coefficient of the wave,As a function of the said reflection coefficient,For the amplitude of the reflected wave,For the amplitude of the transmitted wave,Is the soil layer density of one layer of soil layer far away from the ground in two adjacent soil layers,For the wave velocity in one soil layer far away from the ground in two adjacent soil layers,The soil layer density of one soil layer close to the ground in two adjacent soil layers,Is the wave velocity in one soil layer close to the ground in two adjacent soil layers.
Optionally, under the action of the P-wave, the seismic equivalent node force at the boundary of the three-dimensional layered foundation bottom satisfies the following relationship:
Wherein, For the seismic equivalent node forces at the bottom boundary of the three-dimensional layered foundation under the action of P-waves,The subscript z of (1) indicates the direction of the seismic equivalent node force,The superscript-z of (a) denotes the plane in which the surface normal lies and the surface normal points in the negative z-axis direction,Is the damping coefficient of the normal damper of the 1 st layer soil layer,The vibration speed of the soil body at the moment t is the time t,The area is controlled for the node.
Optionally, under the action of the P wave, the forces of the equivalent nodes of the earthquake at the front and rear side boundaries of the three-dimensional layered foundation satisfy the following relations:
Wherein, AndAre all the seismic equivalent node forces at the rear side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atIn the negative direction of the axis,AndAre all the seismic equivalent node forces at the front side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atThe positive direction of the axis is set,、、AndSubscript of (2)And z both represent the direction of the seismic equivalent node force,A first lame constant for the ith soil layer,For the P wave velocity in the ith soil layer,For the transmission coefficient of the wave at interface j,Is time delayThe vibration speed of the soil body after the process is finished, t is time,For the reflection coefficient of the wave at the interface i,Is time delayThe vibration speed of the soil body after the process is finished,Is a first-generation number type number,,Is a first-generation number type number,,For the reflection coefficient of the wave at the interface n,Is the transmission coefficient of the wave in the k-th soil layer,Is a first-generation number type number,,,Is a parameter associated with i and is,Is time delayThe vibration speed of the soil body after the process is finished,For the reflection coefficient of the wave at the interface i-1,Is time delayThe vibration speed of the soil body after the process is finished,、、AndFor a time delay of 4 times,The area is controlled for the node.
Optionally, under the action of the P wave, the forces of the equivalent nodes of the earthquake at the boundary of the left side and the right side of the three-dimensional layered foundation satisfy the following relationship:
Wherein, AndAre all the seismic equivalent node forces at the left boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the negative y-axis direction,AndAre all the seismic equivalent node forces at the front side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the positive y-axis direction,、、AndThe subscripts y and z of (c) each represent the direction of the seismic equivalent node force,A first lame constant for the ith soil layer,For the P wave velocity in the ith soil layer,For the transmission coefficient of the wave at interface j,Is time delayThe vibration speed of the soil body after the process is finished, t is time,For the reflection coefficient of the wave at the interface i,Is time delayThe vibration speed of the soil body after the process is finished,Is a first-generation number type number,,Is a first-generation number type number,,For the reflection coefficient of the wave at the interface n,Is the transmission coefficient of the wave in the k-th soil layer,Is a first-generation number type number,,,Is a parameter associated with i and is,Is time delayThe vibration speed of the soil body after the process is finished,For the reflection coefficient of the wave at the interface i-1,Is time delayThe vibration speed of the soil body after the process is finished,、、AndFor a time delay of 4 times,The area is controlled for the node.
Optionally, the step of establishing a three-dimensional layered foundation model with mixed infinite elements and finite elements, obtaining the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the boundary of the finite field, and calculating the ground stress field node force matched with the initial stress field comprises the following steps:
Determining a finite field range of a layered foundation model, and performing three-dimensional modeling of finite element and infinite element coupling on foundation soil to obtain the three-dimensional layered foundation model;
restricting three-dimensional freedom of a finite field boundary, applying unit pressure to the finite field boundary, and acquiring the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the finite field boundary;
Determining a soil layer number of the boundary node according to the Z coordinate of the space coordinate and the soil layer height, and determining a plane of the boundary node according to the space coordinate;
And applying an initial ground stress field to the three-dimensional layered foundation model, and calculating the ground stress field node force of each boundary node according to initial ground stress distribution.
Optionally, calculating the seismic equivalent node force of the grid node at the junction of the infinite grid and the finite grid, modifying the calculation file of the layered foundation model in batches, and realizing the initial static force and seismic force input of each boundary node comprises the following steps:
determining an input seismic wave at the bottom of the foundation, acquiring a corresponding speed time-course curve, and calculating the time delay of each boundary node;
Calculating the reflection coefficient and the transmission coefficient, determining the vibration speed of each boundary node at any moment, and further calculating the seismic equivalent node force of each boundary node at any moment;
Based on the fopen function and fprintf function embedded in Matlab, according to the ABAQUS calculation model inp file format and convention, the node set is created in batches, the amplitude function is built, the node force is applied in a concentrated mode, and the seismic equivalent node force and the ground stress field node force application of each boundary node are completed.
The invention has at least the following beneficial effects:
1. the invention provides a foundation earthquake motion equivalent node force input method which can accurately consider the influence of layering non-homogeneous characteristics based on an artificial boundary of infinite element power, the method simultaneously considers the influence of different materials and medium boundaries on earthquake wave propagation, compared with the seismic node force obtained by a conventional homogeneous model and a layered foundation model which only considers primary transmission and reflection, the method has higher precision and better accords with the propagation characteristics of seismic waves in the layered foundation.
2. The method realizes batch application of the seismic node force of the heterogeneous foundation, has simplicity and operability, and is very suitable for popularization and application in seismic dynamic response analysis of interaction of layered heterogeneous soil and structures.
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In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings that are needed in the embodiments will be briefly described below, it being understood that the following drawings only illustrate some embodiments of the present application and therefore should not be considered as limiting the scope, and other related drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic flow chart of a three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to an embodiment of the invention;
FIG. 2 is a schematic diagram of a seismic equivalent node force calculation model of a three-dimensional layered foundation according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of an example three-dimensional layered foundation model according to an embodiment of the present invention;
FIG. 4 is a graph showing the velocity profile of a bottom-input seismic wave of a three-dimensional layered foundation model according to an embodiment of the present invention;
FIG. 5 is a power spectral density of an input seismic wave of an embodiment of the invention;
FIG. 6 is a schematic diagram of a verification of the validity of the present method according to an embodiment of the present invention;
FIG. 7 is a time chart of soil displacement when the method of the present invention, the remote boundary model, the material layering property only and the primary interlayer reflection transmission layering foundation model only are considered, respectively, according to an embodiment of the present invention.
Detailed Description
Specific embodiments of the invention will be described in detail below, it being noted that the embodiments described herein are for illustration only and are not intended to limit the invention. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, to one skilled in the art that these specific details need not be employed to practice the present invention. In other instances, well-known circuits, software, or methods have not been described in detail in order not to obscure the invention.
Reference throughout this specification to "one embodiment," "an embodiment," "one example," or "an example" means that a particular feature, structure, or characteristic described in connection with the embodiment or example is included in at least one embodiment of the present invention. Thus, the appearances of the phrases "in one embodiment," "in an embodiment," "one example," or "an example" in various places throughout this specification are not necessarily all referring to the same embodiment or example. Furthermore, the particular features, structures, or characteristics may be combined in any suitable combination and/or sub-combination in one or more embodiments or examples. Moreover, those of ordinary skill in the art will appreciate that the illustrations provided herein are for illustrative purposes and that the illustrations are not necessarily drawn to scale.
It should be noted in advance that in an alternative embodiment, the same symbols or alphabet meaning and number are the same as those present in all formulas, except where separate descriptions are made.
In an alternative embodiment, referring to fig. 1, the present invention provides a three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries, the method comprising the steps of:
S1, solving the seismic equivalent node force expression of the three-dimensional layered foundation taking infinite element artificial boundaries into consideration in a free field.
The step S1 specifically includes the following steps:
S11, determining the damping coefficient of the normal damper and the damping coefficient of the tangential damper in each stratified soil layer according to a one-dimensional wave equation suitable for plane waves, a cell stress balance condition and a dynamic artificial boundary complete absorption boundary condition.
Specifically, in the present embodiment, in the semi-infinite foundation, the seismic wave can be approximated as a plane wave whose wave motion conforms to a one-dimensional wave equation, and the displacement component of the plane wave during propagation is only a function of the z-coordinate and the time t. For the P-wave effect, the wave equation of the soil micro-element of the i-th soil layer can be expressed as the following differential equation:
Wherein, For soil displacement of the i-th soil layer in the z-axis direction, i=1, 2, 3.For the P wave velocity in the ith soil layer,,The soil layer density of the ith soil layer,AndThe shear modulus and the first Ramez constant of the ith soil layer, respectively. The positive direction of the z-axis is the upward direction perpendicular to the horizontal plane, and the absolute value of the value of z represents the distance from a point on the soil layer to the origin of coordinates. In the following description, the ith soil layer may also be referred to as soil layer i. In addition, in the following description, if the symbols z, x, and y in the formula are not separately described, z, x, and y each represent coordinate values on the corresponding coordinate axes.
The general solution of the differential equation is as follows,Is an upward wave which propagates in the positive direction of the z-axis for the ith soil layer,Is a descending wave which propagates in the negative direction of the z-axis of the ith soil layer. By combining with the elastic mechanics theory, the soil stress under P wave is as follows:
Wherein, 、AndAll the positive stresses of the soil body are represented,、AndThe first symbol from left to right before the subscript comma indicates the direction of the soil stress, and the second symbol indicates the normal direction of the plane where the soil stress is located.
In dynamic analysis, an infinite element dynamic artificial boundary is essentially a viscous boundary. Additional damping stress applied to soil body unit at junction of infinite element and finite element under action of P waveCan be expressed asWhereinThe damping coefficient of the normal damper at the artificial boundary is represented by u, which is the displacement of the soil body.
Based on the stress balance condition and the complete absorption boundary condition of the unit, the method can obtain,The damping coefficient of the normal damper is the damping coefficient of the i-th soil layer. Similarly, the tangential damper damping coefficient of each soil layer under the action of S wave can be obtainedSatisfy the following requirements,Is the tangential damper damping coefficient of the ith soil layer,Is the S wave velocity in the ith soil layer,。
S12, considering the condition that the earthquake waves are vertically incident on the bottom of the foundation, and determining the reflection coefficient and the transmission coefficient of the waves according to the amplitude relation of the incident waves, the reflected waves and the transmitted waves.
Specifically, in this embodiment, considering the case of vertically incident seismic waves at the bottom of the foundation, according to the fresnel equation, the mathematical relationship between the magnitudes of the incident wave and the transmitted wave and the mathematical relationship between the magnitudes of the reflected wave and the transmitted wave are respectively shown as follows:
Wherein, Is the reflection coefficient of the wave,Is the transmission coefficient of the wave and,For the amplitude of the incident wave,For the amplitude of the reflected wave,For the amplitude of the transmitted wave,Is the soil layer density of one layer of soil layer far away from the ground in two adjacent soil layers,For the wave velocity in one soil layer far away from the ground in two adjacent soil layers,The soil layer density of one soil layer close to the ground in two adjacent soil layers,Is the wave velocity in one soil layer close to the ground in two adjacent soil layers.
S13, calculating the seismic equivalent node force of the three-dimensional layered foundation according to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the speed time course of the input seismic waves.
Specifically, in the present embodiment, please refer to fig. 2, taking the vertical incident P-wave at the bottom of the substrate as an example, if the vertical incident P-wave at the bottom of the substrateThe vibration speed of which can be expressedFor the ith soil layer, any soil micro-element displacement in the ith soil layer is realized by transmission wavesAnd reflected waveComposition is prepared. Considering the retardation of the wave and the two incidences and reflections at the interface,AndThe following relationships are satisfied:
Wherein, For the transmission coefficient of the wave at interface j,Is time delayThe incident wave after that is reflected by the optical fiber,Is time delayThe incident wave after that is reflected by the optical fiber,Is time delayThe incident wave after that is reflected by the optical fiber,Is time delayThe incident wave after that is reflected by the optical fiber,Is a first-generation number type number,,Is the transmission coefficient of the wave in the k-th soil layer,For the reflection coefficient of the wave at the interface n,For the reflection coefficient of the wave at the interface i,For the reflection coefficient of the wave at the interface i-1,、、AndFor 4 time delays.、、AndThe following relationships are satisfied in turn:
Wherein, Is the thickness of the j-th soil layer,Is the wave velocity of the earthquake waves in the j-th soil layer,Is the wave velocity of the earthquake waves in the ith soil layer,Is the distance between the soil body and the adjacent interface of the lower part,For the thickness of the ith soil layer, FIG. 2In particular to the distance between the soil body A and the interface i-1.
In the calculationIn the relation of (2),Considering the influence of the transmission effect of all interfaces below the current soil layer on the incident wave, belonging to the forward propagation of the wave; The reverse re-incident wave after the wave propagates to the earth surface (the nth soil layer) and is reflected is considered, and the wave belongs to the negative propagation of the wave. In the calculation In the relation of (2),Representing the reflected wave at the top interface of the ith earth, belonging to the negative propagation of the wave,Representing the reflected wave of the reverse re-incident wave from the earth's surface at the bottom interface of the ith earth, belonging to the forward propagation of the wave. The interface refers to an interface between adjacent soil layers, the top interface of the i-th soil layer can be marked as an interface i, the bottom interface of the i-th soil layer can be marked as an interface i-1, the top interface of the i-1-th soil layer can be marked as an interface i-1, and the like.
Thus, the vibration speed of the soil body of the layered foundationCan be expressed as:
Wherein, Is a first-generation number type number,,Is time delayThe vibration speed of the soil body after the process is finished,Is time delayThe vibration speed of the soil body after the process is finished,Is time delayThe vibration speed of the soil body after the process is finished,Is time delayThe vibration speed of the soil body after the process is finished,Is a parameter related to i. The following relationship is satisfied:
Through the analysis, under the action of P waves, the earthquake equivalent node force at the boundary of the three-dimensional layered foundation bottom, namely the earthquake equivalent node force in the z-axis direction, can be deduced, and the following relation is satisfied:
Wherein, Is the equivalent node force of the earthquake at the boundary of the bottom of the three-dimensional layered foundation under the action of P waves,The subscript z of (1) indicates the direction of the seismic equivalent node force,The superscript-z of (a) denotes the plane in which the surface normal lies and the surface normal points in the negative z-axis direction,Is the damping coefficient of a normal damper of the 1 st layer soil layer,The vibration speed of the soil body at the moment t,The area is controlled for the node.
Under the action of P waves, the earthquake equivalent node force at the front and rear side boundaries of the three-dimensional layered foundation, namely the earthquake equivalent node force in the x-axis direction, meets the following relation:
Wherein, AndAre all the equivalent node forces of the earthquake at the rear side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atIn the negative direction of the axis,AndAre all seismic equivalent node forces at the front boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atThe positive direction of the axis is set,、、AndSubscript of (2)And z both represent the direction of the seismic equivalent node force,Is time delayThe vibration speed of the soil body after the process is finished,Is time delayThe vibration speed of the soil body after the process is finished,Is a first-generation number type number,,Is a parameter associated with i and is,Is time delayThe vibration speed of the soil body after the process is finished,Is time delayThe vibration speed of the soil body after the process.
Under the action of P waves, the earthquake equivalent node force at the boundary of the left side and the right side of the three-dimensional layered foundation, namely the earthquake equivalent node force in the y-axis direction, meets the following relation:
Wherein, AndAre all the equivalent node forces of the earthquake at the left boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the negative y-axis direction,AndAre all seismic equivalent node forces at the front boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the positive y-axis direction,、、AndThe subscripts y and z of (c) each denote the direction of the seismic equivalent node force.
Note that, in fig. 2,Is a transmitted wave at the top layer (interface i) of the ith earth,Is a reflected wave at the top layer (interface i) of the ith earth,Is a transmitted wave at the top layer (interface i-1) of the i-1 th layer of soil,Is a reflected wave at the top layer (interface i-1) of the i-1 th layer of earth.
Further, for the S wave vertically incident at the bottom of the foundation, the derivation process and expression of the seismic equivalent node force are similar to those of the P wave, and will not be described here.
S2, establishing a three-dimensional layered foundation model with mixed infinite elements and finite elements, obtaining the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the boundary of the finite field, and calculating the ground stress field node force matched with the initial stress field.
The step S2 specifically includes the following steps:
s21, determining a finite field range of the layered foundation model, and performing three-dimensional modeling on foundation soil by coupling finite elements and infinite elements to obtain the three-dimensional layered foundation model.
Specifically, in this embodiment, as shown in (a) of fig. 3, this embodiment is based on ABAQUS numerical software three-dimensional layered foundation model, which is a cuboid, with the z-axis being vertically upward, the x-and y-axes being horizontal axes, the model size is 105m×55m, and the finite field size is 100m×50m. And establishing a three-dimensional cube through an Extrusion command, dividing the three-dimensional cube into a near field region and a far field region through a Partition Cell command, and further performing three-dimensional modeling of finite element and infinite element coupling on foundation soil to obtain a three-dimensional layered foundation model. The unit type of the near-field finite field is C3D8R, and the unit type of the far-field finite field is changed into CIN3D8 by modifying keywords.
The number of soil layers is 5, the numbers of soil layers from bottom to top are sequentially 1, 2, 3, 4 and 5, and the thickness of each soil layer is 20 m. The material parameters of each layer of soil body from bottom to top are poisson ratio v s = [0.3,0.3,0.3,0.3,0.3], density ρ= [2.0,2.0 ] t/m 3, elastic modulus E= [500000,300000,200000,150000,60000] kPa, and gravity γ= [6.0,6.0 ] kN/m 3. Wave velocity information of each soil layer is shown in table 1.
Table 1 wave velocity information for each soil layer
| Soil layer numbering | Thickness/m | E/ kPa | cs(m/s) | cp(m/s) |
| 1 | 20 | 500000 | 310 | 580 |
| 2 | 20 | 300000 | 240 | 449 |
| 3 | 20 | 200000 | 196 | 367 |
| 4 | 20 | 150000 | 170 | 318 |
| 5 | 20 | 60000 | 107 | 201 |
In Table 1The wave velocity of the S-wave is set to be,Is the P wave velocity.
Furthermore, the mechanical parameters of the natural soil body have depth dependence, and in order to reduce calculation cost, a non-uniform grid form is adopted along the height direction when the three-dimensional layered foundation model is constructed. Size of each soil layer gridThe following relation determination is satisfied:
Wherein, Is the highest frequency component of the ground vibration,Can be determined according to the power spectral density of the selected vibration, c is the wave velocity of the seismic waves in the earth, c can be calculated according to the calculation provided in step S11Or calculateIs calculated from the relation of (2).
S22, restraining three-dimensional freedom degree of the finite field boundary, applying unit pressure to the finite field boundary, and obtaining the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the finite field boundary.
Specifically, in this embodiment, for the three-dimensional layered foundation model shown in fig. 3 (a), a surface Set "Surf-1" is built for 5 surfaces (including 4 side surfaces and 1 bottom surface) of the finite field boundary in a unified manner in the component module of the ABAQUS preprocessing interface, and a corresponding node Set "Set-1" is built, in the analysis step module of ABAQUS, a Static general Static analysis step is built for the built layered foundation model of finite element-infinite element mesh coupling, in the load module of ABAQUS, a unit pressure is applied to the surface Set "Surf-1" by using a pressure command, and the degree of freedom of the node Set "Set-1" is constrained, and at the field output requirement of ABAQUS, only an RF variable is selected to be output for the node Set-1", and finally a task is submitted. After the calculation is completed, the last analysis step is selected in Report Field Output windows of a post-processing module in ABAQUS, the node counter force RF of the node Set 'Set-1' at the boundary of the finite field is output, and the node counter force RF is written into an 'abaqus.csv' file in a csv format, so that the serial number, the space coordinate and the equivalent node area of the boundary node are finally obtained.
S23, determining a soil layer number of the boundary node according to the Z coordinate of the space coordinate and the soil layer height, and determining a plane of the boundary node according to the space coordinate.
Specifically, in this embodiment, an "abaqus.csv" file is opened, the first row is deleted, only the 5 th to 8 th columns and the 12 th to 14 th columns are reserved, and the reserved columns are stored in a "Nodes _info.txt" file. In matlab software, a load ('Nodes _info.txt') command is used to import data in a 'Nodes _info.txt' file into matlab, and then the z-axis coordinate is used as a control index, and the soil layer number of each boundary node is determined by combining the soil layer height.
Further, through matlab programming, the plane attribution of the node is determined according to the space coordinates of each boundary node, and the specific idea is that when the x-coordinate of the boundary node is equal to the minimum/maximum x-coordinate of the node in the three-dimensional layered foundation model, the boundary node is attributed to the front/rear side boundary (front/rear) of the three-dimensional layered foundation model, when the y-coordinate of the boundary node is equal to the minimum/maximum y-coordinate of the node in the three-dimensional layered foundation model, the boundary node is attributed to the left/right side boundary (left/right side) of the three-dimensional layered foundation model, and when the z-coordinate of the boundary node is equal to the minimum z-coordinate of the node in the three-dimensional layered foundation model, the boundary node is attributed to the bottom boundary (bottom surface) of the three-dimensional layered foundation model.
And S24, applying an initial ground stress field to the three-dimensional layered foundation model, and calculating the ground stress field node force of each boundary node according to initial ground stress distribution.
Specifically, in this embodiment, physical force is selected in the load module of ABAQUS, and a physical force field is applied to the finite field, and the value of the physical force field is-6. Defining a predefined field in ABAQUS, applying an initial ground stress field adapted to the body force field to the whole three-dimensional layered foundation model, wherein the lateral soil pressure coefficientTake 0.65.
Further, according to the depth of the boundary node and the equivalent node area, the ground stress field node counter force matched with the initial ground stress field of each boundary node is calculated, and then the ground stress field node force with the same magnitude and opposite direction with the ground stress field node counter force is obtained. The depth of the boundary node is the z coordinate of the boundary node minus the top surface z coordinate of the three-dimensional layered foundation model.
S3, calculating the seismic equivalent node force of the grid nodes at the junction of the infinite grid and the finite grid, and modifying the calculation file of the layered foundation model in batches to realize the initial static force and seismic force input of each boundary node.
The step S3 specifically includes the following steps:
s31, determining an input seismic wave at the bottom of the foundation, acquiring a corresponding speed time-course curve, and calculating the time delay of each boundary node.
Specifically, in the present embodiment, in the three-dimensional layered foundation model, the foundation bottom uses a uniform incident pulse wave as an input seismic wave, and the velocity time-course curve of the three-dimensional layered foundation model bottom is shown in fig. 4 when the data sampling frequency is 100 Hz. The power spectrum density of the input seismic wave is calculated by selecting pwelch power spectrum functions built in matlab software, the result is shown in figure 5, and the result is known in step S21For the three-dimensional layered foundation model shown in fig. 3 (a), the bottom three-layer mesh size is set to 2 m and the top 2-layer mesh size is set to 1 m.
Further, calculating the time delay of each boundary node according to the relation shown in step S13、、And. Then, using a Hermite interpolation function embedded in matlab software, combining the sampling frequency of the selected earthquake, obtaining the speed time course of each boundary node after considering each time delay, finally obtaining 4 speed time courses, and respectively obtaining correspondingly、、And. Note that in these 4 speed ranges, for each less than the time delay、、AndThe vibration speed of the soil body is 0.
S32, calculating the reflection coefficient and the transmission coefficient, determining the vibration speed of each boundary node at any moment, and further calculating the seismic equivalent node force of each boundary node at any moment.
Specifically, in this embodiment, the reflection coefficient and the transmission coefficient of different waveforms in each soil layer are calculated using the relational expression shown in step S12, and the calculation results are shown in table 2.
TABLE 2 basic information of soil layer interfaces
| Interface/soil layer numbering | Z coordinate/m of soil layer top | Z coordinate/m of soil layer bottom | αp | βp | αs | βs |
| 1 | 25 | 5 | 0.13 | 1.13 | 0.13 | 1.13 |
| 2 | 45 | 25 | 0.10 | 1.10 | 0.10 | 1.10 |
| 3 | 65 | 45 | 0.07 | 1.07 | 0.07 | 1.07 |
| 4 | 85 | 65 | 0.22 | 1.23 | 0.23 | 1.23 |
| 5 | 105 | 85 | 1 | 1 | 1 | 1 |
In the table 2 of the description of the present invention,Is the reflection coefficient of the P-wave,Is the reflection coefficient of the S-wave,Is the transmission coefficient of the P-wave,Is the transmission coefficient of the S wave.
Further, after the reflection coefficient and the transmission coefficient are calculated, the vibration speed of the soil body is used as the vibration speed of the boundary nodes, and then the seismic equivalent node force of each boundary node at any moment can be calculated according to the calculation formula of the seismic equivalent node force under the action of the P wave shown in the step S13. Correspondingly, the earthquake equivalent node force of each boundary node at any moment can be calculated by a calculation formula for calculating the earthquake equivalent node force under the action of the S wave. For the three-dimensional layered foundation model shown in fig. 3 (a), the seismic equivalent node forces at a typical boundary node E are shown in fig. 3 (b).
S33, based on a fopen function and fprintf functions embedded in matlab, according to an ABAQUS calculation model inp file format and convention, creating node sets, establishing amplitude functions and applying concentrated node forces in batches, and finishing seismic equivalent node forces and ground stress field node forces of all boundary nodes.
Specifically, in this embodiment, based on the fopen function and fprintf function embedded in matlab, a node Set is built in batches for each boundary node by using a Set keyword and written into a "set_node.txt" file, an Amplitude function is built in batches for the seismic equivalent node force of each boundary node by using a Set keyword and written into a "CLoad-amp.txt" file, and a node counter force matched with a ground stress field is applied for each boundary node by using a Set keyword and written into a "geof.txt" file, wherein the node counter force matched with the ground stress field is:, . Wherein, AndAs a side reaction force to the boundary node,AndThe subscript of (2) represents the load direction, and the load direction is constantly directed to the inside of the three-dimensional layered foundation model; distance from boundary node to earth surface; Is the vertical counter-force of the boundary node, The subscript of (c) indicates the direction of the load,Is always positive. Applying a seismic equivalent concentrated node force F to each boundary node using the Cload key, and writing into a "Cload_S.txt" file, the seismic equivalent concentrated node force F comprising、、、、、、、And. For further explanation, the partial correlation matlab code is given here:
“fid=fopen('Output for inp files\CLoad-AMP.txt','w');
fclose(fid);
fid=fopen('Output for inp files\CLoad-AMP.txt','a+');
fprintf (fid, [ '/n','/End Assembly','/n' ]) positioning;
%---------------------------Cload-----------------------------
fid1=fopen('Output for inp files\Cload_S.txt','w');
fclose(fid1);
fid1=fopen('Output for inp files\Cload_S.txt','a+');
fprintf (fid 1, [ ' is placed at the seismic analysis step ',' \n ',' ' is located: ', ' \n ',' ' is load ',' \n ',' ' is n ' ];
fid2=fopen('Output for inp files\GeoF.txt','w');
fclose(fid2);
fid2=fopen('Output for inp files\GeoF.txt','a+');
fprintf (fid 2, [ ') starting from the ground stress step, each subsequent analysis step is to write the ground stress node force ', ' n ', ' please locate: ' n ', ' load ', ' n ' ], respectively:
Generating a corresponding inp calculation file according to a calculation formula of the earthquake equivalent node force by using matlab, placing all contents in a set_node.txt file at END INSTANCE keywords, placing all contents in a CLoad-AMP.txt file at an End Assembly keyword, placing all contents in a GeoF.txt file at the keywords copied to each analysis step, placing all contents in a Cload_S.txt file at the keywords of an earthquake loading step, and completing the input of the earthquake equivalent node force of the three-dimensional layered foundation based on an infinite dynamic artificial boundary.
Further, in order to verify the effectiveness of the method, the embodiment degenerates the formula for calculating the equivalent node force of the earthquake under the action of the P wave into a homogeneous situation, and the three-dimensional homogeneous foundation model is obtained. Except that the mechanical parameters are consistent with the soil layer 1 of the three-dimensional layered foundation model, the other parameters of the three-dimensional homogeneous foundation model are consistent with the three-dimensional layered foundation model. The displacement time course of the top B point and the bottom C point of the three-dimensional homogeneous foundation model is shown in fig. 6. It can be seen that the result obtained based on the method is consistent with the analytic solution, namely the three-dimensional layered foundation earthquake equivalent node force of the infinite dynamic artificial boundary provided by the embodiment can effectively reflect the fluctuation characteristic of the homogeneous semi-infinite foundation.
Further, for the three-dimensional layered foundation model shown in fig. 3 (a), the soil displacement time course of the D-point when the method of the present invention, the remote boundary model, the foundation model considering only the layering characteristics of materials, and the layering foundation model considering only interlayer primary reflection transmission are used, respectively, is shown in fig. 7. It can be seen that the error between the top displacement time interval of the soil layer 3 obtained by the method and the remote boundary result is less than 5%, and the soil body displacement time interval under the remote boundary result can be considered as a true solution, so that compared with an earthquake equivalent node force calculation method which only considers the layering property of materials and an earthquake equivalent node force calculation method which only considers primary reflection and transmission, the method has higher accuracy.
It should be noted that, in some cases, the actions described in the specification may be performed in a different order and still achieve desirable results, and in this embodiment, the order of steps is merely provided to make the embodiment more clear, and it is convenient to describe the embodiment without limiting it.
In summary, the invention provides a foundation earthquake motion equivalent node force input method capable of accurately considering the influence of layering non-homogeneous characteristics based on an infinite element power artificial boundary. The method considers the influence of different material and medium boundaries on the propagation of the seismic waves, and compared with the seismic node force obtained by a conventional homogeneous model and a layered foundation model only considering primary transmission and reflection, the method has higher precision and accords with the propagation characteristics of the seismic waves in the layered foundation. In addition, the invention also realizes batch application of the seismic node force of the heterogeneous foundation, has simplicity and operability, and is very suitable for popularization and application in seismic power response analysis of layering heterogeneous soil-structure interaction.
It should be noted that the above embodiments are only used to illustrate the technical solution of the present invention, but not to limit the technical solution of the present invention, and although the detailed description of the present invention is given with reference to the above embodiments, it should be understood by those skilled in the art that the technical solution described in the above embodiments may be modified or some or all technical features may be equivalently replaced, and these modifications or substitutions do not make the essence of the corresponding technical solution deviate from the scope of the technical solution of the embodiments of the present invention, and all the modifications or substitutions are included in the scope of the claims and the specification of the present invention.
Claims (10)
1. The three-dimensional layered foundation earthquake motion input method based on the infinite element artificial boundary is characterized by comprising the following steps of:
solving the seismic equivalent node force expression of the three-dimensional layered foundation taking infinite element artificial boundaries into consideration in a free field;
Establishing a three-dimensional layered foundation model with mixed infinite elements and finite elements, obtaining the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the boundary of a finite field, and calculating the ground stress field node force matched with an initial stress field;
And calculating the seismic equivalent node force of the grid nodes at the junction of infinite and finite grids, and modifying the calculation files of the layered foundation model in batches to realize the initial static force and seismic force input of each boundary node.
2. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 1, wherein the solving of the three-dimensional layered foundation earthquake equivalent node force expression of infinite element dynamic artificial boundaries under the free field comprises the following steps:
Determining a normal damper damping coefficient and a tangential damper damping coefficient in each stratified soil layer according to a one-dimensional wave equation suitable for plane waves, a cell stress balance condition and a dynamic artificial boundary complete absorption boundary condition;
considering the condition that seismic waves are vertically incident at the bottom of the foundation, and determining the reflection coefficient and the transmission coefficient of the waves according to the amplitude relation of the incident waves, the reflected waves and the transmitted waves;
And calculating the seismic equivalent node force of the three-dimensional layered foundation according to the normal damper damping coefficient, the tangential damper damping coefficient, the reflection coefficient, the transmission coefficient and the speed time course of the input seismic waves.
3. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 2, wherein the normal damper damping coefficient and the tangential damper damping coefficient respectively satisfy the following relations:
,
,
Wherein, The normal damper coefficient of the ith soil layer,The tangential damper coefficient of the ith soil layer,The soil layer density of the ith soil layer,For the P wave velocity in the ith soil layer,Is the S wave velocity in the ith soil layer.
4. A three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 2, wherein the reflection coefficient satisfies the following relationship:
,
Wherein, As said reflection coefficient of the wave,For the amplitude of the incident wave,For the amplitude of the transmitted wave,Is the soil layer density of one layer of soil layer far away from the ground in two adjacent soil layers,For the wave velocity in one soil layer far away from the ground in two adjacent soil layers,The soil layer density of one soil layer close to the ground in two adjacent soil layers,Is the wave velocity in one soil layer close to the ground in two adjacent soil layers.
5. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 4, wherein the transmission coefficient satisfies the following relationship:
,
Wherein, As said transmission coefficient of the wave,As a function of the said reflection coefficient,For the amplitude of the reflected wave,For the amplitude of the transmitted wave,Is the soil layer density of one layer of soil layer far away from the ground in two adjacent soil layers,For the wave velocity in one soil layer far away from the ground in two adjacent soil layers,The soil layer density of one soil layer close to the ground in two adjacent soil layers,Is the wave velocity in one soil layer close to the ground in two adjacent soil layers.
6. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 5, wherein under the action of P waves, the earthquake equivalent node force at the boundary of the bottom of the three-dimensional layered foundation satisfies the following relationship:
,
Wherein, For the seismic equivalent node forces at the bottom boundary of the three-dimensional layered foundation under the action of P-waves,The subscript z of (1) indicates the direction of the seismic equivalent node force,The superscript-z of (a) denotes the plane in which the surface normal lies and the surface normal points in the negative z-axis direction,Is the damping coefficient of the normal damper of the 1 st layer soil layer,Is the vibration speed of the soil body, t is the time,The area is controlled for the node.
7. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 6, wherein under the action of P waves, the earthquake equivalent node forces at the front and rear side boundaries of the three-dimensional layered foundation satisfy the following relationship:
,
,
Wherein, AndAre all the seismic equivalent node forces at the rear side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atIn the negative direction of the axis,AndAre all the seismic equivalent node forces at the front side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing atThe positive direction of the axis is set,、、AndSubscript of (2)And z both represent the direction of the seismic equivalent node force,A first lame constant for the ith soil layer,For the P wave velocity in the ith soil layer,For the transmission coefficient of the wave at interface j,Is time delayThe vibration speed of the soil body after the process is finished, t is time,For the reflection coefficient of the wave at the interface i,Is time delayThe vibration speed of the soil body after the process is finished,Is a first-generation number type number,,Is a first-generation number type number,,For the reflection coefficient of the wave at the interface n,Is the transmission coefficient of the wave in the k-th soil layer,Is a first-generation number type number,,,Is a parameter associated with i and is,Is time delayThe vibration speed of the soil body after the process is finished,For the reflection coefficient of the wave at the interface i-1,Is time delayThe vibration speed of the soil body after the process is finished,、、AndFor a time delay of 4 times,The area is controlled for the node.
8. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 7, wherein under the action of P waves, the earthquake equivalent node forces at the left and right side boundaries of the three-dimensional layered foundation satisfy the following relationship:
,
,
Wherein, AndAre all the seismic equivalent node forces at the left boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the negative y-axis direction,AndAre all the seismic equivalent node forces at the front side boundary of the three-dimensional layered foundation,AndSuperscript of (2)Indicating the plane in which the surface normal lies and pointing in the positive y-axis direction,、、AndThe subscripts y and z of (c) each represent the direction of the seismic equivalent node force,A first lame constant for the ith soil layer,For the P wave velocity in the ith soil layer,For the transmission coefficient of the wave at interface j,Is time delayThe vibration speed of the soil body after the process is finished, t is time,For the reflection coefficient of the wave at the interface i,Is time delayThe vibration speed of the soil body after the process is finished,Is a first-generation number type number,,Is a first-generation number type number,,For the reflection coefficient of the wave at the interface n,Is the transmission coefficient of the wave in the k-th soil layer,Is a first-generation number type number,,,Is a parameter associated with i and is,Is time delayThe vibration speed of the soil body after the process is finished,For the reflection coefficient of the wave at the interface i-1,Is time delayThe vibration speed of the soil body after the process is finished,、、AndFor a time delay of 4 times,The area is controlled for the node.
9. The three-dimensional layered foundation earthquake motion input method based on infinite element artificial boundaries according to claim 2, wherein the steps of establishing a three-dimensional layered foundation model of infinite element and finite element mixture, obtaining the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the finite field boundaries, and calculating the ground stress field node force matched with the initial stress field comprise the following steps:
Determining a finite field range of a layered foundation model, and performing three-dimensional modeling of finite element and infinite element coupling on foundation soil to obtain the three-dimensional layered foundation model;
restricting three-dimensional freedom of a finite field boundary, applying unit pressure to the finite field boundary, and acquiring the serial numbers, the space coordinates and the equivalent node areas of all boundary nodes at the finite field boundary;
Determining a soil layer number of the boundary node according to the Z coordinate of the space coordinate and the soil layer height, and determining a plane of the boundary node according to the space coordinate;
And applying an initial ground stress field to the three-dimensional layered foundation model, and calculating the ground stress field node force of each boundary node according to initial ground stress distribution.
10. The three-dimensional layered foundation earthquake motion input method based on infinite artificial boundaries according to claim 9, wherein the calculating of the earthquake equivalent node force of grid nodes at the junctions of infinite and finite grids, the batch modification of the layered foundation model calculation file, and the realization of the initial static force and earthquake force input of each boundary node comprise the following steps:
determining an input seismic wave at the bottom of the foundation, acquiring a corresponding speed time-course curve, and calculating the time delay of each boundary node;
Calculating the reflection coefficient and the transmission coefficient, determining the vibration speed of each boundary node at any moment, and further calculating the seismic equivalent node force of each boundary node at any moment;
Based on the fopen function and fprintf function embedded in Matlab, according to the ABAQUS calculation model inp file format and convention, the node set is created in batches, the amplitude function is built, the node force is applied in a concentrated mode, and the seismic equivalent node force and the ground stress field node force application of each boundary node are completed.
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