Three-dimensional stratified foundation vibration input method based on infinite element artificial boundary

Abstract

The invention relates to the technical field of geotechnical engineering anti-seismic analysis, in particular to a three-dimensional stratified foundation vibration input method based on an infinite element artificial boundary, and the method comprises the following steps: solving the seismic equivalent nodal force expression of a three-dimensional stratified foundation considering the infinite element artificial boundary in a free field; an infinite element and finite element mixed three-dimensional layered foundation model is established, the serial number, the space coordinate and the equivalent node area of each boundary node at the boundary of the finite field are obtained, and the crustal stress field node force matched with the initial stress field is calculated; and calculating the seismic equivalent node force of the grid nodes at the junction of the infinite and finite grids, and modifying the layered foundation model calculation files in batches to realize the initial static force and seismic force input of each boundary node. The seismic oscillation equivalent node force obtained through the method is high in precision, the propagation characteristic of seismic waves in the layered foundation is met, batch application of the heterogeneous foundation seismic node force is achieved, and the method has simplicity and operability.

Description

Technical Field

[0001] The invention relates to the technical field of geotechnical engineering seismic analysis, and in particular to a three-dimensional layered ground vibration input method based on infinite element artificial boundaries. Background Art

[0002] The finite element method has been widely used in the field of seismic analysis of geotechnical engineering. The foundation has semi-infinite characteristics, but in most cases, earthquake analysis only cares about near-field wave problems. Therefore, the finite calculation area of ​​interest needs to be cut out from the infinite domain, and artificial boundary conditions are set to make the propagation characteristics of mechanical waves at the cut boundary consistent with the actual infinite domain situation. As a way to simulate infinite regions, infinite elements and conventional finite elements have similar unit shape functions, so there is an inherent "coordination" between infinite elements and traditional finite elements, which is more advantageous than other dynamic artificial boundary methods such as viscoelastic boundaries and boundary elements.

[0003] Infinite elements can greatly reduce the computational cost of the model, but currently the infinite element dynamic artificial boundary is only suitable for local point source vibration problems within the domain. For external source incident problems such as earthquakes, infinite elements cannot directly handle them. Based on this, many scholars at home and abroad have conducted relevant research on external source incident earthquakes. It should be noted that natural soil layers have stratified characteristics, and the reflection and transmission of waves at the interfaces of each soil layer must be considered, which is very different from the homogeneous foundation model. At present, most of the earthquake input and earthquake dynamic analysis of stratified foundations use the earthquake equivalent node force obtained by the homogeneous foundation model, which only considers the influence of different materials on the wave velocity and ignores the influence of the medium interface on the seismic wave. Although some scholars have also proposed a seismic motion input method that takes into account the stratified characteristics of the foundation, it is too simplified and there is a big difference from the actual situation, which needs further research. Summary of the invention

[0004] In view of the defects in the prior art, the present invention provides a three-dimensional layered ground vibration input method based on infinite element artificial boundaries.

[0005] In order to achieve the above-mentioned purpose, the present invention provides a three-dimensional layered foundation ground vibration input method based on infinite element artificial boundary, the method comprising the following steps: solving the seismic equivalent node force expression of the three-dimensional layered foundation considering the infinite element artificial boundary 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 accuracy, conforms to the propagation characteristics of seismic waves in the layered foundation, and realizes the batch application of seismic node forces of heterogeneous foundations, with simplicity and operability.

[0006] Optionally, the method of solving the expression of equivalent nodal force of earthquake in three-dimensional layered foundation considering infinite element dynamic artificial boundary under free field comprises the following steps: According to the one-dimensional wave equation applicable to plane waves, the unit force balance condition and the dynamic artificial boundary complete absorption boundary condition, the damping coefficient of the normal damper and the damping coefficient of the tangential damper in each layered soil layer are determined; Considering the case of vertical incident seismic waves at the bottom of the foundation, the reflection coefficient and transmission coefficient of the wave are determined according to the amplitude relationship between the incident wave, the reflected wave and the transmitted wave; The seismic equivalent nodal force of the three-dimensional layered foundation is calculated according to the damping coefficient of the normal damper, the damping coefficient of the tangential damper, the reflection coefficient, the transmission coefficient and the velocity time history of the input seismic wave.

[0007] Optionally, the damping coefficient of the normal damper and the damping coefficient of the tangential damper respectively satisfy the following relationship:

[0008]

[0009] in, is the damping coefficient of the normal damper of the i-th soil layer, is the damping coefficient of the tangential damper of the i-th soil layer, is the soil density of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the S wave velocity in the i-th soil layer.

[0010] Optionally, the reflection coefficient satisfies the following relationship:

[0011] in, is the reflection coefficient of the wave, is the amplitude of the incident wave, is the amplitude of the transmitted wave, is the soil density of the soil layer farthest from the ground between two adjacent soil layers, is the wave velocity in the soil layer farthest from the ground between two adjacent soil layers, is the soil density of the soil layer closer to the ground between two adjacent soil layers, It is the wave velocity in the soil layer closer to the ground between two adjacent soil layers.

[0012] Optionally, the transmission coefficient satisfies the following relationship:

[0013] in, is the transmission coefficient of the wave, is the reflection coefficient, is the amplitude of the reflected wave, is the amplitude of the transmitted wave, is the soil density of the soil layer farthest from the ground between two adjacent soil layers, is the wave velocity in the soil layer farthest from the ground between two adjacent soil layers, is the soil density of the soil layer closer to the ground between two adjacent soil layers, It is the wave velocity in the soil layer closer to the ground between two adjacent soil layers.

[0014] Optionally, under the action of P waves, the seismic equivalent nodal force at the bottom boundary of the three-dimensional layered foundation satisfies the following relationship:

[0015] in, is the earthquake equivalent nodal force at the bottom boundary of the three-dimensional layered foundation under the action of P waves, The subscript z indicates the direction of the earthquake equivalent nodal force, The superscript -z indicates that the surface normal is in the plane and the surface normal points to the negative direction of the z-axis. is the damping coefficient of the normal damper of the first soil layer, is the soil vibration velocity at time t, t is time, Control area of ​​the node.

[0016] Optionally, under the action of P waves, the seismic equivalent nodal forces at the front and rear side boundaries of the three-dimensional layered foundation satisfy the following relationship:

[0017]

[0018] in, and are the earthquake equivalent nodal forces at the rear boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the face normal lies and the face normal points to Negative axis direction, and are the earthquake equivalent nodal forces at the front boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the face normal lies and the face normal points to Axis positive direction, , , and Subscript and z both represent the direction of the earthquake equivalent nodal force, is the first Lamé constant of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the transmission coefficient of the wave at interface j, For delay The soil vibration velocity after is, t is the time, is the reflection coefficient of the wave at interface i, For delay The soil vibration velocity after is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at interface n, is the transmission coefficient of the wave in the kth soil layer, is an algebraic expression, , , is a parameter related to i, For delay The soil vibration velocity after is the reflection coefficient of the wave at interface i-1, For delay The soil vibration velocity after , , and There are 4 time delays. Control area of ​​the node.

[0019] Optionally, under the action of P waves, the seismic equivalent nodal forces at the left and right boundaries of the three-dimensional layered foundation satisfy the following relationship:

[0020]

[0021] in, and are the earthquake equivalent nodal forces at the left boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the surface normal is located and the surface normal points to the negative direction of the y-axis. and are the earthquake equivalent nodal forces at the front boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the surface normal is located and the surface normal points to the positive direction of the y-axis. , , and The subscripts y and z both indicate the direction of the earthquake equivalent nodal force. is the first Lamé constant of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the transmission coefficient of the wave at interface j, For delay The soil vibration velocity after is, t is the time, is the reflection coefficient of the wave at interface i, For delay The soil vibration velocity after is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at interface n, is the transmission coefficient of the wave in the kth soil layer, is an algebraic expression, , , is a parameter related to i, For delay The soil vibration velocity after is the reflection coefficient of the wave at interface i-1, For delay The soil vibration velocity after , , and There are 4 time delays. Control area of ​​the node.

[0022] Optionally, the step of establishing a three-dimensional layered foundation model that is a mixture of infinite elements and finite elements, obtaining the numbers, spatial coordinates, and equivalent node areas of each boundary node at the boundary of the finite domain, and calculating the node forces of the geostress field that matches the initial stress field comprises the following steps: Determine the finite domain range of the layered foundation model, perform three-dimensional modeling of the foundation soil by coupling finite elements and infinite elements, and obtain the three-dimensional layered foundation model; Constraining the three-dimensional degrees of freedom of the finite domain boundary, applying unit pressure to the finite domain boundary, and obtaining the number, spatial coordinates and equivalent node area of ​​each boundary node at the finite domain boundary; Determine the soil layer number where the boundary node is located according to the Z coordinate of the spatial coordinate and the soil layer height, and determine the plane where the boundary node is located according to the spatial coordinate; An initial geostress field is applied to the three-dimensional layered foundation model, and the geostress field node force of each boundary node is calculated according to the initial geostress distribution.

[0023] Optionally, the calculating of the seismic equivalent nodal force of the grid nodes at the junction of the infinite grid and the finite grid, batch modification of the layered foundation model calculation file, and realizing the initial static force and seismic force input of each boundary node include the following steps: Determine the input seismic wave at the bottom of the ground, obtain the corresponding velocity time history curve, and calculate the time delay of each of the boundary nodes; Calculating the reflection coefficient and the transmission coefficient, and determining the vibration velocity of each of the boundary nodes at any time, and then calculating the earthquake equivalent nodal force of each of the boundary nodes at any time; Based on the fopen function and fprintf function embedded in Matlab, in accordance with the inp file format and convention of the ABAQUS calculation model, the node sets are created, the amplitude functions are established, and the concentrated node forces are applied in batches to complete the application of the seismic equivalent node forces and the ground stress field node forces of each boundary node.

[0024] The present invention has at least the following beneficial effects: 1. Based on the infinite element dynamic artificial boundary, the present invention proposes a ground vibration equivalent node force input method that can accurately consider the influence of layered heterogeneous characteristics. This method also considers the influence of different materials and medium interfaces on the propagation of seismic waves. Compared with the seismic node force obtained by the conventional homogeneous model and the layered foundation model that only considers one transmission and reflection, the present invention has higher accuracy and is more in line with the propagation characteristics of seismic waves in layered foundations.

[0025] 2. The present invention realizes batch application of earthquake node forces on heterogeneous foundations, is simple and operable, and is very suitable for promotion and use in earthquake dynamic response analysis of interaction between layered heterogeneous soil and structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present application and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without paying creative work.

[0027] Figure 1 It is a flow chart of a method for inputting three-dimensional layered ground vibration based on infinite element artificial boundary according to an embodiment of the present invention; Figure 2 A schematic diagram of a calculation model for earthquake equivalent nodal forces of a three-dimensional layered foundation according to an embodiment of the present invention; Figure 3 A schematic diagram of an example of a three-dimensional layered foundation model according to an embodiment of the present invention; Figure 4 A velocity time history curve of an input seismic wave at the bottom of a three-dimensional layered foundation model according to an embodiment of the present invention; Figure 5 is the power spectrum density of the input seismic wave of the embodiment of the present invention; Figure 6 This is a schematic diagram of verifying the effectiveness of the method according to an embodiment of the present invention; Figure 7 The soil displacement time history diagrams of the embodiments of the present invention are respectively when the method of the present invention, the remote boundary model, the layered foundation model considering only the material layered characteristics and the layered foundation model considering only one interlayer reflection and transmission. DETAILED DESCRIPTION

[0028] The specific embodiments of the present invention will be described in detail below. It should be noted that the embodiments described herein are only for illustration and are not intended to limit the present invention. In the following description, a large number of specific details are set forth in order to provide a thorough understanding of the present invention. However, it is obvious to those of ordinary skill in the art that these specific details do not need to be adopted to implement the present invention. In other examples, in order to avoid confusing the present invention, known circuits, software or methods are not specifically described.

[0029] Throughout the specification, references to "one embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in conjunction with the embodiment or example is included in at least one embodiment of the present invention. Therefore, the phrases "in one embodiment," "in an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily all refer to the same embodiment or example. In addition, particular features, structures, or characteristics may be combined in one or more embodiments or examples in any suitable combination and / or subcombination. In addition, it should be understood by those of ordinary skill in the art that the figures provided herein are for illustrative purposes and that the figures are not necessarily drawn to scale.

[0030] It should be noted in advance that, in an optional embodiment, except for independent explanations, the same symbols or letters appearing in all formulas have the same meanings and values.

[0031] In an alternative embodiment, see Figure 1 The present invention provides a three-dimensional layered ground vibration input method based on infinite element artificial boundaries, the method comprising the following steps: S1. Solve the expression of seismic equivalent nodal force of three-dimensional layered foundation considering infinite element artificial boundaries under free field.

[0032] Wherein, step S1 specifically includes the following steps: S11. According to the one-dimensional wave equation applicable to plane waves, the unit force balance condition and the dynamic artificial boundary complete absorption boundary condition, the damping coefficient of the normal damper and the damping coefficient of the tangential damper in each stratified soil layer are determined.

[0033] Specifically, in this embodiment, in a semi-infinite foundation, the seismic wave can be approximately regarded as a plane wave, and its wave conforms to the one-dimensional wave equation. The displacement component of the plane wave during propagation is only a function of the z coordinate and time t. For the P wave action, the wave equation of the soil microelement of the i-th soil layer can be expressed as the following differential equation:

[0034] in, is the soil displacement of the i-th soil layer in the z-axis direction, i=1, 2, 3, ..., n, n is the total number of soil layers, is the P wave velocity in the i-th soil layer, , is the soil density of the i-th soil layer, and are the shear modulus and the first Lamé constant of the i-th 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 z value represents the distance from a certain point on the soil layer to the coordinate origin. In the subsequent description, the i-th soil layer may also be referred to as soil layer i. In addition, in the subsequent description, if the symbols z, x, and y in the formula are not separately explained, z, x, and y all represent the coordinate values ​​on the corresponding coordinate axes.

[0035] The general solution of the above differential equation is , is the upward wave propagating in the positive direction of the z-axis in the i-th soil layer, is the downward wave propagating in the negative direction of the z-axis in the i-th soil layer. Combined with the theory of elastic mechanics, the soil stress under the P wave is:

[0036]

[0037] in, , and Both represent soil normal stress, , and The first symbol from left to right before the subscript comma indicates the direction of soil stress, and the second symbol indicates the normal direction of the plane where the soil stress lies.

[0038] In dynamic analysis, the infinite element dynamic artificial boundary is essentially a viscous boundary. The additional damping stress of the soil unit at the interface between the infinite element and the finite element under the action of P wave is It can be expressed as ,in is the damping coefficient of the normal damper at the artificial boundary, and u is the soil displacement.

[0039] According to the unit force balance condition and the fully absorbing boundary condition, we can get , is the damping coefficient of the normal damper of the i-th soil layer. Similarly, the damping coefficient of the tangential damper of each soil layer under the action of S waves can be obtained as satisfy , is the damping coefficient of the tangential damper of the i-th soil layer, is the S wave velocity in the i-th soil layer, .

[0040] S12. Considering the situation of vertical incident seismic waves at the bottom of the foundation, the reflection coefficient and transmission coefficient of the wave are determined according to the amplitude relationship among the incident wave, reflected wave and transmitted wave.

[0041] Specifically, in this embodiment, considering the case where the seismic wave is incident vertically at the bottom of the foundation, according to the Fresnel equation, the mathematical relationship between the amplitudes of the incident wave and the transmitted wave and the mathematical relationship between the amplitudes of the reflected wave and the transmitted wave are respectively shown as follows:

[0042]

[0043] in, is the reflection coefficient of the wave, is the wave transmission coefficient, is the amplitude of the incident wave, is the amplitude of the reflected wave, is the amplitude of the transmitted wave, is the soil density of the soil layer farthest from the ground between two adjacent soil layers, is the wave velocity in the soil layer farthest from the ground between two adjacent soil layers, is the soil density of the soil layer closer to the ground between two adjacent soil layers, It is the wave velocity in the soil layer closer to the ground between two adjacent soil layers.

[0044] S13. Calculate the seismic equivalent nodal 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 velocity time history of the input seismic wave.

[0045] Specifically, in this embodiment, see Figure 2 , taking the P wave incident vertically at the bottom of the foundation as an example, if the wave incident vertically at the bottom of the foundation , the vibration speed can be expressed as For the i-th soil layer, the displacement of any soil microelement in it is determined by the transmitted wave and reflected waves Considering the delay of the wave and the two incidences and reflections at the interface, and They satisfy the following relations respectively:

[0046]

[0047] in, is the transmission coefficient of the wave at interface j, For delay After the incident wave, For delay After the incident wave, For delay After the incident wave, For delay After the incident wave, is an algebraic expression, , is the transmission coefficient of the wave in the kth soil layer and , is the reflection coefficient of the wave at interface n, is the reflection coefficient of the wave at interface i, is the reflection coefficient of the wave at interface i-1, , , and There are 4 time delays. , , and The following relations are satisfied in sequence:

[0048]

[0049]

[0050]

[0051] in, is the thickness of the jth soil layer, is the seismic wave velocity in the jth soil layer, is the velocity of the seismic wave in the i-th soil layer, is the distance between the soil and the lower adjacent interface, is the thickness of the i-th soil layer, Figure 2 middle Specifically, it is the distance between soil A and interface i-1.

[0052] In calculation In the relationship of The influence of the transmission effect of all interfaces below the current soil layer on the incident wave is considered, which belongs to the forward propagation of the wave; The reverse re-incident wave after the wave propagates to the surface (nth soil layer) and reflects is considered, which belongs to the negative propagation of the wave. In the relationship of It indicates the reflected wave at the top interface of the i-th layer of soil, which belongs to the negative propagation of the wave. Represents the reflected wave of the reverse re-incident wave from the surface at the bottom interface of the i-th soil layer, which belongs to the forward propagation of the wave. Wherein, the interface refers to the interface between adjacent soil layers, the top interface of the i-th soil layer can be recorded as interface i, the bottom interface of the i-th soil layer can be recorded as interface i-1, the top interface of the i-1-th soil layer can also be recorded as interface i-1, and so on.

[0053] Therefore, the vibration velocity of the layered foundation soil It can be expressed as:

[0054] in, is an algebraic expression, , For delay The soil vibration velocity after For delay The soil vibration velocity after For delay The soil vibration velocity after For delay The soil vibration velocity after is a parameter related to i. It satisfies the following relationship:

[0055] Through the above analysis, it can be deduced that under the action of P waves, the seismic equivalent nodal force at the bottom boundary of the three-dimensional layered foundation, that is, the seismic equivalent nodal force in the z-axis direction, satisfies the following relationship:

[0056] in, is the seismic equivalent nodal force at the bottom boundary of the three-dimensional layered foundation under the action of P waves, The subscript z indicates the direction of the earthquake equivalent nodal force. The superscript -z indicates that the surface normal is in the plane and the surface normal points to the negative direction of the z-axis. is the damping coefficient of the normal damper of the first soil layer, is the soil vibration velocity at time t, Control area of ​​the node.

[0057] Under the action of P waves, the seismic equivalent nodal force at the front and rear side boundaries of the three-dimensional layered foundation, that is, the seismic equivalent nodal force in the x-axis direction, satisfies the following relationship:

[0058]

[0059] in, and are the equivalent nodal forces at the rear boundary of the three-dimensional layered foundation. and Superscript Indicates the plane where the face normal lies and the face normal points to Negative axis direction, and are the equivalent nodal forces at the front boundary of the three-dimensional layered foundation. and Superscript Indicates the plane where the face normal lies and the face normal points to Axis positive direction, , , and Subscript and z both represent the direction of the earthquake equivalent nodal force, For delay The soil vibration velocity after For delay The soil vibration velocity after is an algebraic expression, , is a parameter related to i, For delay The soil vibration velocity after For delay The soil vibration velocity after .

[0060] Under the action of P waves, the seismic equivalent nodal forces at the left and right boundaries of the three-dimensional layered foundation, that is, the seismic equivalent nodal forces in the y-axis direction, satisfy the following relationship:

[0061]

[0062] in, and are the equivalent nodal forces at the left boundary of the three-dimensional layered foundation. and Superscript Indicates the plane where the surface normal is located and the surface normal points to the negative direction of the y-axis. and are the equivalent nodal forces at the front boundary of the three-dimensional layered foundation. and Superscript Indicates the plane where the surface normal is located and the surface normal points to the positive direction of the y-axis. , , and The subscripts y and z both indicate the direction of the earthquake equivalent nodal force.

[0063] In addition, it should be noted that Figure 2 middle, is the transmitted wave at the top layer of the i-th soil layer (interface i), is the reflected wave at the top layer of the i-th soil layer (interface i), is the transmitted wave at the top layer of the i-1th soil layer (interface i-1), is the reflected wave at the top layer of the i-1th soil layer (interface i-1).

[0064] Furthermore, for the S-wave incident vertically at the bottom of the foundation, the derivation process and expression of the earthquake equivalent nodal force are similar to those for the P-wave, which will not be elaborated here.

[0065] S2. Establish a three-dimensional layered foundation model that is a mixture of infinite elements and finite elements, obtain the number, spatial coordinates and equivalent node area of ​​each boundary node at the boundary of the finite domain, and calculate the node force of the ground stress field that matches the initial stress field.

[0066] Wherein, step S2 specifically includes the following steps: S21. Determine the finite domain range of the layered foundation model, perform three-dimensional modeling of the foundation soil by coupling finite elements and infinite elements, and obtain the three-dimensional layered foundation model.

[0067] Specifically, in this embodiment, if Figure 3 As shown in (a), this embodiment is based on the ABAQUS numerical software three-dimensional layered foundation model. The three-dimensional layered foundation model is a cuboid, the z axis is vertically upward, the x and y axes are horizontal axes, and the model size is 105m×55m×55m, of which the finite domain size is 100m×50m×50m. A three-dimensional cube is established by the Extrusion command, and the three-dimensional cube is divided into two regions, the near field and the far field, by the Partition Cell command. Then, the foundation soil is modeled three-dimensionally by coupling finite elements and infinite elements to obtain a three-dimensional layered foundation model. Among them, the near-field finite domain unit type is C3D8R, and the unit type of the far-field infinite domain is changed to CIN3D8 by modifying the keyword *Element.

[0068] The number of soil layers is 5, and the soil layers are numbered 1, 2, 3, 4, and 5 from bottom to top. The thickness of each soil layer is 20 m. The material parameters of each soil layer from bottom to top are as follows: Poisson's ratio v s =[0.3, 0.3, 0.3, 0.3, 0.3]; density ρ = [2.0, 2.0, 2.0, 2.0, 2.0] t / m 3 ; Elastic modulus E = [500000, 300000, 200000, 150000, 60000] kPa; Heavy weight γ = [6.0, 6.0, 6.0, 6.0, 6.0] kN / m 3 The wave velocity information of each soil layer is shown in Table 1.

[0069] Table 1 Wave velocity information of each soil layer Soil layer number Thickness / m E / kPa <![CDATA[c s (m / s)]]> <![CDATA[c p (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 Table 1 is the S-wave velocity, is the P-wave velocity.

[0070] Furthermore, the mechanical parameters of natural soil are depth-dependent. To reduce the computational cost, a non-uniform grid is used along the height direction when constructing a three-dimensional layered foundation model. The following relationship is satisfied:

[0071] in, is the highest frequency component of the earthquake motion, It can be determined according to the power spectrum density of the selected earthquake motion, c is the velocity of the earthquake wave in the soil layer, and c can be calculated according to the calculation provided in step S11 Or calculate The relationship is calculated.

[0072] S22, constraining the three-dimensional degrees of freedom of the finite domain boundary, applying unit pressure to the finite domain boundary, and obtaining the number, spatial coordinates and equivalent node area of ​​each boundary node at the finite domain boundary.

[0073] Specifically, in this embodiment, for Figure 3 For the three-dimensional layered foundation model shown in (a), in the component module of the ABAQUS pre-processing interface, the five faces (including four side faces and one bottom face) on the finite domain boundary are uniformly established as a face set "Surf-1", and the corresponding node set "Set-1" is established; in the analysis step module of ABAQUS, a Static general static analysis step is established for the layered foundation model with finite element-infinite element mesh coupling; in the load module of ABAQUS, a pressure command is used to apply unit pressure to the face set "Surf-1", and the three-dimensional degrees of freedom of the node set "Set-1" are constrained; in the field output requirement of ABAQUS, only the RF variable is output for the node set "Set-1", and finally the task is submitted. After the calculation is completed, the last analysis step is selected in the Report Field Output window of the post-processing module in ABAQUS, and the node reaction force RF of the node set "Set-1" at the finite domain boundary is output, and it is written into the "abaqus.csv" file in csv format, and finally the boundary node number, spatial coordinates and equivalent node area are obtained.

[0074] S23. Determine the soil layer number where the boundary node is located according to the Z coordinate of the spatial coordinate and the soil layer height, and determine the plane where the boundary node is located according to the spatial coordinate.

[0075] Specifically, in this embodiment, the "abaqus.csv" file is opened, the first row is deleted and only the 5th to 8th columns and the 12th to 14th columns are retained, and the data of these columns are stored in the "Nodes_info.txt" file. In the matlab software, the load('Nodes_info.txt') command is used to import the data in the "Nodes_info.txt" file into matlab, and then the soil layer number of each boundary node is determined by taking the z-axis coordinate as the control index and combining the soil layer height.

[0076] Furthermore, through MATLAB programming, the plane to which the node belongs is determined according to the spatial coordinates of each boundary node. The specific idea is as follows: 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 belongs to the front / rear 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 belongs to the left / right boundary (left / right face) of the three-dimensional layered foundation model; 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 belongs to the bottom boundary (bottom face) of the three-dimensional layered foundation model.

[0077] S24, applying an initial geostress field to the three-dimensional layered foundation model, and calculating the geostress field node force of each boundary node according to the initial geostress distribution.

[0078] Specifically, in this embodiment, body force is selected in the load module of ABAQUS, and a body force field is applied to the finite domain, and the value of the body force field is -6. A predefined field is defined in ABAQUS, and an initial geostress field adapted to the body force field is applied to the entire three-dimensional layered foundation model, where the lateral earth pressure coefficient Take 0.65.

[0079] Furthermore, according to the gravity, the depth of the boundary node and the equivalent node area, the geostress field node reaction force of each boundary node matching the initial geostress field is calculated, and then the geostress field node force with the same magnitude and opposite direction as the geostress field node reaction force is obtained. Among them, the depth of the boundary node is the z coordinate of the boundary node minus the z coordinate of the top surface of the three-dimensional layered foundation model.

[0080] S3. Calculate the seismic equivalent nodal force of the grid nodes at the junction of the infinite and finite grids, batch modify the calculation files of the layered foundation model, and realize the initial static force and seismic force input of each boundary node.

[0081] Wherein, step S3 specifically includes the following steps: S31, determining the input seismic wave at the bottom of the earth, obtaining the corresponding velocity time history curve, and calculating the time delay of each of the boundary nodes.

[0082] Specifically, in this embodiment, in the three-dimensional layered foundation model, a consistent incident pulse wave is used as the input seismic wave at the bottom of the foundation. When the data sampling frequency is 100 Hz, the velocity time history curve at the bottom of the three-dimensional layered foundation model is as follows: Figure 4 The pwelch power spectrum function built into the MATLAB software is used to calculate the power spectrum density of the input seismic wave. The result is shown in Figure 5 As shown, it can be seen that in step S21 The value of is 8Hz, so for Figure 3 For the three-dimensional layered foundation model shown in (a), the grid size of the bottom three layers can be set to 2 m, and the grid size of the top two layers can be set to 1 m.

[0083] Further, the time delay of each boundary node is calculated according to the relationship shown in step S13 , , and Then, the Hermite interpolation function built into the MATLAB software is used, combined with the selected ground motion sampling frequency, to obtain the velocity history of each boundary node after considering each time delay, and finally four velocity history are obtained, corresponding to the acquisition , , and Note that in these four speed time series, for each time delay less than , , and At the moment, the soil vibration velocity is 0.

[0084] S32, calculating the reflection coefficient and the transmission coefficient, and determining the vibration velocity of each of the boundary nodes at any time, and then calculating the seismic equivalent nodal force of each of the boundary nodes at any time.

[0085] Specifically, in this embodiment, the relationship shown in step S12 is used to calculate the reflection coefficient and transmission coefficient of different waveforms in each soil layer, and the calculation results are shown in Table 2.

[0086] Table 2 Basic information of each soil layer interface Interface / soil layer number z coordinate of soil top / m z coordinate of soil bottom / m <![CDATA[α p ]]> <![CDATA[β p ]]> <![CDATA[α s ]]> <![CDATA[β s ]]> 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 Table 2, is the reflection coefficient of P wave, is the reflection coefficient of S wave, is the transmission coefficient of P wave, is the transmission coefficient of S wave.

[0087] Furthermore, after calculating the reflection coefficient and the transmission coefficient, the soil vibration velocity is used as the vibration velocity of the boundary node, and the seismic equivalent node force of each boundary node at any time can be calculated according to the calculation formula of the seismic equivalent node force under the action of P waves shown in step S13. Correspondingly, the calculation formula of the seismic equivalent node force under the action of S waves can also be calculated to calculate the seismic equivalent node force of each boundary node at any time. Figure 3 In the three-dimensional layered foundation model shown in (a), the earthquake equivalent node force at a typical boundary node E is as follows: Figure 3 As shown in (b) in .

[0088] S33. Based on the built-in fopen function and fprintf function of matlab, in accordance with the inp file format and convention of the ABAQUS calculation model, batch creation of node sets, establishment of amplitude functions and application of concentrated node forces are performed to complete the application of seismic equivalent node forces and ground stress field node forces of each of the boundary nodes.

[0089] Specifically, in this embodiment, based on the built-in fopen function and fprintf function of matlab, the *Nset keyword is used to batch establish node sets for each boundary node and write them into the "Set_Node.txt" file; the *Amplitude keyword is used to batch establish amplitude functions for the earthquake equivalent node force of each boundary node and write them into the "CLoad-AMP.txt" file; the *Cload keyword is used to apply a node reaction force matching the geostress field to each boundary node and write it into the "GeoF.txt" file, where the node reaction force matching the geostress field is: , .in, and is the lateral reaction force at the boundary node, and The subscript of indicates the load direction, and the load direction always points to the inside of the three-dimensional layered foundation model; is the distance from the boundary node to the ground surface; is the vertical reaction force at the boundary node, The subscript indicates the load direction. Always positive. Use the *Cload keyword to apply the earthquake equivalent concentrated node force F to each boundary node and write it into the "Cload_S.txt" file. The earthquake equivalent concentrated node force F includes , , , , , , , and For further explanation, here are some relevant Matlab codes: "fid=fopen('Output for inp files\CLoad-AMP.txt','w'); fclose(fid); fid=fopen('Output for inp files\CLoad-AMP.txt','a+'); fprintf(fid,['**Please locate:','\n','*End Assembly','\n']); %---------------------------------Cload-------------------------------- fid1=fopen('Output for inp files\Cload_S.txt','w'); fclose(fid1); fid1=fopen('Output for inp files\Cload_S.txt','a+'); fprintf(fid1,['**Put it at the seismic analysis step','\n','**Please locate:','\n','**LOADS','\n','**','\n']); fid2=fopen('Output for inp files\GeoF.txt','w'); fclose(fid2); fid2=fopen('Output for inp files\GeoF.txt','a+'); fprintf(fid2,['**Starting from the geostress step, each subsequent analysis step must write the geostress node force','\n','**Please locate:','\n','** LOADS','\n','**','\n']);” Use matlab to generate the corresponding inp calculation file according to the calculation formula of seismic equivalent nodal force, place all the contents of the "Set_Node.txt" file at the *End Instance keyword; place all the contents of the "CLoad-AMP.txt" file at the *End Assembly keyword; place all the contents of the "GeoF.txt" file at the **Loads keyword copied to each analysis step; place all the contents of the "Cload_S.txt" file at the **Loads keyword of the seismic loading step; at this point, the input of the seismic equivalent nodal force of the three-dimensional layered foundation based on the infinite element dynamic artificial boundary is completed.

[0090] Furthermore, in order to verify the effectiveness of this method, this embodiment degenerates the formula for calculating the equivalent nodal force of earthquakes under the action of P waves into a homogeneous case, and obtains a three-dimensional homogeneous foundation model. 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 history of the top point B and the bottom point C of the three-dimensional homogeneous foundation model is shown in Figure 6 It can be seen that the results obtained based on this method are consistent with the analytical solution, that is, the three-dimensional layered foundation seismic equivalent nodal force of the infinite element dynamic artificial boundary proposed in this embodiment can effectively reflect the wave characteristics of the homogeneous semi-infinite foundation.

[0091] Furthermore, for Figure 3 The soil displacement time history of point D when the three-dimensional layered foundation model shown in (a) of FIG. 1 is respectively used with the method of the present invention, the remote boundary model, the foundation model considering only the layered characteristics of the material, and the layered foundation model considering only the first reflection and transmission between layers is as follows: Figure 7 It can be seen that the error between the displacement time history of the top of the soil layer 3 obtained by the method of the present invention and the remote boundary result is less than 5%, and the soil displacement time history under the remote boundary result can be considered as a true solution. Therefore, compared with the earthquake equivalent nodal force calculation method that only considers the stratification characteristics of the material and the earthquake equivalent nodal force calculation method that only considers one reflection and transmission, the method of the present invention has higher accuracy.

[0092] It should be noted that, in some cases, the actions described in the specification can be performed in a different order and still achieve the desired results. In this embodiment, the order of steps given is only to make the embodiment appear clearer and easier to explain, rather than to limit it.

[0093] In summary, the present invention proposes a method for inputting equivalent node forces for ground vibration based on infinite element dynamic artificial boundaries, which can accurately consider the influence of layered heterogeneous characteristics. This method simultaneously considers the influence of different materials and medium interfaces on the propagation of seismic waves. Compared with the seismic node forces obtained by conventional homogeneous models and layered foundation models that only consider one-time transmission and reflection, this method has higher accuracy and is more in line with the propagation characteristics of seismic waves in layered foundations. In addition, the present invention also realizes the batch application of seismic node forces on heterogeneous foundations, which is simple and operable, and is very suitable for promotion and use in the seismic dynamic response analysis of layered heterogeneous soil-structure interaction.

[0094] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. These modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and specification of the present invention.

Claims

1. A three-dimensional layered ground seismic input method based on infinite element artificial boundary, characterized in that: The steps include: Solve the seismic equivalent nodal force expression of three-dimensional layered foundation considering infinite element artificial boundaries under free field; A three-dimensional layered foundation model that is a mixture of infinite elements and finite elements is established to obtain the number, spatial coordinates, and equivalent node area of ​​each boundary node at the boundary of the finite domain, and to calculate the node force of the ground stress field that matches the initial stress field; The seismic equivalent nodal force of the grid nodes at the junction of the infinite and finite grids is calculated, and the calculation files of the layered foundation model are modified in batches to realize the initial static force and seismic force input of each boundary node.

2. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 1 is characterized in that: The method for solving the expression of the equivalent nodal force of the three-dimensional layered foundation earthquake considering the infinite element dynamic artificial boundary under the free field comprises the following steps: According to the one-dimensional wave equation applicable to plane waves, the unit force balance condition and the dynamic artificial boundary complete absorption boundary condition, the damping coefficient of the normal damper and the damping coefficient of the tangential damper in each layered soil layer are determined; Considering the case of vertical incident seismic waves at the bottom of the foundation, the reflection coefficient and transmission coefficient of the wave are determined according to the amplitude relationship between the incident wave, the reflected wave and the transmitted wave; The seismic equivalent nodal force of the three-dimensional layered foundation is calculated according to the damping coefficient of the normal damper, the damping coefficient of the tangential damper, the reflection coefficient, the transmission coefficient and the velocity time history of the input seismic wave.

3. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 2 is characterized in that: The damping coefficient of the normal damper and the damping coefficient of the tangential damper respectively satisfy the following relationship: ， ， in, is the damping coefficient of the normal damper of the i-th soil layer, is the damping coefficient of the tangential damper of the i-th soil layer, is the soil density of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the S wave velocity in the i-th soil layer.

4. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 2 is characterized in that: The reflection coefficient satisfies the following relationship: ， in, is the reflection coefficient of the wave, is the amplitude of the incident wave, is the amplitude of the transmitted wave, is the soil density of the soil layer farthest from the ground between two adjacent soil layers, is the wave velocity in the soil layer farthest from the ground between two adjacent soil layers, is the soil density of the soil layer closer to the ground between two adjacent soil layers, It is the wave velocity in the soil layer closer to the ground between two adjacent soil layers.

5. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 4 is characterized in that: The transmission coefficient satisfies the following relationship: ， in, is the transmission coefficient of the wave, is the reflection coefficient, is the amplitude of the reflected wave, is the amplitude of the transmitted wave, is the soil density of the soil layer farthest from the ground between two adjacent soil layers, is the wave velocity in the soil layer farthest from the ground between two adjacent soil layers, is the soil density of the soil layer closer to the ground between two adjacent soil layers, It is the wave velocity in the soil layer closer to the ground between two adjacent soil layers.

6. A three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 5, characterized in that: Under the action of P waves, the seismic equivalent nodal force at the bottom boundary of the three-dimensional layered foundation satisfies the following relationship: ， in, is the earthquake equivalent nodal force at the bottom boundary of the three-dimensional layered foundation under the action of P waves, The subscript z indicates the direction of the earthquake equivalent nodal force, The superscript -z indicates that the surface normal is in the plane and the surface normal points to the negative direction of the z-axis. is the damping coefficient of the normal damper of the first soil layer, is the soil vibration velocity, t is the time, Control area of ​​the node.

7. A three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 6, characterized in that: Under the action of P waves, the seismic equivalent nodal forces at the front and rear side boundaries of the three-dimensional layered foundation satisfy the following relationship: ， ， in, and are the earthquake equivalent nodal forces at the rear boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the face normal lies and the face normal points to Negative axis direction, and are the earthquake equivalent nodal forces at the front boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the face normal lies and the face normal points to Axis positive direction, , , and Subscript and z both represent the direction of the earthquake equivalent nodal force, is the first Lamé constant of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the transmission coefficient of the wave at interface j, For delay The soil vibration velocity after is, t is the time, is the reflection coefficient of the wave at interface i, For delay The soil vibration velocity after is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at interface n, is the transmission coefficient of the wave in the kth soil layer, is an algebraic expression, , , is a parameter related to i, For delay The soil vibration velocity after is the reflection coefficient of the wave at interface i-1, For delay The soil vibration velocity after , , and There are 4 time delays. Control area of ​​the node.

8. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 7 is characterized in that: Under the action of P waves, the seismic equivalent nodal forces at the left and right boundaries of the three-dimensional layered foundation satisfy the following relationship: ， ， in, and are the earthquake equivalent nodal forces at the left boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the surface normal is located and the surface normal points to the negative direction of the y-axis. and are the earthquake equivalent nodal forces at the front boundary of the three-dimensional layered foundation, and Superscript Indicates the plane where the surface normal is located and the surface normal points to the positive direction of the y-axis. , , and The subscripts y and z both indicate the direction of the earthquake equivalent nodal force. is the first Lamé constant of the i-th soil layer, is the P wave velocity in the i-th soil layer, is the transmission coefficient of the wave at interface j, For delay The soil vibration velocity after is, t is the time, is the reflection coefficient of the wave at interface i, For delay The soil vibration velocity after is an algebraic expression, , is an algebraic expression, , is the reflection coefficient of the wave at interface n, is the transmission coefficient of the wave in the kth soil layer, is an algebraic expression, , , is a parameter related to i, For delay The soil vibration velocity after is the reflection coefficient of the wave at interface i-1, For delay The soil vibration velocity after , , and There are 4 time delays. Control area of ​​the node.

9. The three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 2 is characterized in that: The method of 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 node force of the ground stress field matching the initial stress field includes the following steps: Determine the finite domain range of the layered foundation model, perform three-dimensional modeling of the foundation soil by coupling finite elements and infinite elements, and obtain the three-dimensional layered foundation model; Constraining the three-dimensional degrees of freedom of the finite domain boundary, applying unit pressure to the finite domain boundary, and obtaining the number, spatial coordinates and equivalent node area of ​​each boundary node at the finite domain boundary; Determine the soil layer number where the boundary node is located according to the Z coordinate of the spatial coordinate and the soil layer height, and determine the plane where the boundary node is located according to the spatial coordinate; An initial geostress field is applied to the three-dimensional layered foundation model, and the geostress field node force of each boundary node is calculated according to the initial geostress distribution.

10. A three-dimensional layered ground seismic input method based on infinite element artificial boundary according to claim 9, characterized in that: The calculation of the seismic equivalent node force of the grid nodes at the junction of the infinite and finite grids, batch modification of the layered foundation model calculation file, and implementation of the initial static force and seismic force input of each boundary node include the following steps: Determine the input seismic wave at the bottom of the ground, obtain the corresponding velocity time history curve, and calculate the time delay of each of the boundary nodes; Calculating the reflection coefficient and the transmission coefficient, and determining the vibration velocity of each of the boundary nodes at any time, and then calculating the earthquake equivalent nodal force of each of the boundary nodes at any time; Based on the fopen function and fprintf function embedded in Matlab, in accordance with the inp file format and convention of the ABAQUS calculation model, the node sets are created, the amplitude functions are established, and the concentrated node forces are applied in batches to complete the application of the seismic equivalent node forces and the ground stress field node forces of each boundary node.

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