Method for calculating the effects of vibrations in an arbitrary undulating terrain

Abstract

A method for calculating the vibration effects of arbitrarily undulating strata belongs to the field of wave propagation analysis in complex sites. Its main principle is based on the wavefield separation principle, decomposing the free wavefield of an irregularly undulating site into a known free wavefield of a regular site (w) and a scattered wavefield to be determined. The difficulty in solving the complex wavefield components is transformed into an internal wave scattering problem. Furthermore, based on the concept of an isolator, a substructure of the stratum boundary is constructed to calculate the equivalent effect of the scattered wavefield in the boundary region of arbitrarily undulating strata. This invention can calculate the wave field of a foundation with arbitrary undulating strata under the incident vibration wave, and accurately describe the non-uniform propagation characteristics of site waves.

Description

Technical Field

[0001] This invention belongs to the field of wave propagation analysis in complex sites, specifically relating to a method for calculating the vibration effect of arbitrarily undulating strata. Background Technology

[0002] Earthquakes or underground engineering blasting can generate intense elastic waves that propagate through the foundation to existing structures, causing severe vibration damage. Actual nuclear power and hydropower projects are generally located in areas far from densely populated areas. The strata in their natural site areas usually exhibit complex and irregular undulating shapes. Vibration waves may undergo a series of refractions, reflections, scattering, and waveform transformations in the foundation, thus forming a complex near-field wave field distribution.

[0003] Site factors and seismic input have become key considerations in nuclear power plant structural design. Relevant standards, such as Chapter 2 of ASCE / SEI 4-16 and Clause 3.3.6 of the Seismic Design Standard for Nuclear Power Plants (GB 50267-2019), explicitly emphasize this. Related studies have confirmed that slope topography has a significant amplification effect on vibrations. Earthquakes are one of the important controlling factors for the development of nuclear and hydropower. Furthermore, the propagation of stress waves caused by blasting impacts within the rock mass can also lead to strong ground vibrations, and even cause damage or collapse of nearby structures.

[0004] The foundation is the medium for wave propagation and a crucial component of the soil-structure dynamic interaction (SSI) system. Global SSI analysis typically uses the foundation's free field as the basis for external wave input. Calculating the complex wave propagation within the site is a fundamental problem. High-precision and applicable vibration wave field analysis methods can quantify these dynamic effects, thereby enabling scientifically reliable vibration control or blast-controlled assessment of structures.

[0005] When considering earthquakes and engineering blasting as wave source problems, seismic design typically uses the given outcrop bedrock response as input data to invert the incident wave. The outward propagation of blast shock waves is primarily in the form of body waves. When the blast source is far from the study area, it resembles a cylindrical body wave; at even greater distances, it approaches a plane body wave. Therefore, accurately calculating the wave field of complex, undulating, layered foundations given the incident vibration wave is crucial for refined vibration assessment of engineering structures.

[0006] The complexity of foundation wavefield analysis stems primarily from two aspects: the irregularity of the surface undulations and the distribution of underground strata. Incident waves undergo reflection, refraction, and scattering under these influences, thereby altering the movement of the structural foundation. The foundation's side elevations exhibit a stepped geometric profile due to topographical differences, extending into an infinite domain. This makes the wave propagation characteristics of the site even more complex and difficult to calculate directly, requiring solutions to the non-uniform wave scattering from the topographic surface and the asymmetry of the foundation's infinite domain.

[0007] Currently, there are few wave analysis methods applicable to irregular undulating layered foundations. The main methods are derived by using boundary conditions of specific regular terrain sites, which has limitations and greatly simplifies the irregular morphology of actual strata, or only studies external wave input models based on approximate wave fields.

[0008] For horizontally layered foundations, free-field analysis can be performed using the one-dimensional harmonic superposition method and the stiffness matrix method. However, incident waves from bedrock, reflected by the strata and undulating surfaces, will generate multi-angle superimposed scattered wave components in the near field. Domestic and international scholars have analytically derived wave scattering solutions for two-dimensional semi-circular, arc-shaped, concave-convex terrains, adjacent multi-arc, V-shaped, or three-dimensional hemispherical terrains based on the wave function expansion method. Alternatively, they have derived the far-field asymptotic expression for wave scattering under cylindrical pores using the matched asymptotic expansion method; and they have used the IBEM method to solve the infinite boundary scattering problem, and by introducing the oblique Green's function, solved the dynamic response of rock slope sites. However, the forms of the above methods and solutions are only applicable to ideal site models with specific regular geometry, and are difficult to apply to actual terrain.

[0009] Another approach is to directly solve the wave propagation problem numerically using external wave excitation. Since the free field of undulating terrain is difficult to obtain, some researchers use approximate wave fields as input for seismic motion, such as: 1) using the free field of a flat terrain with corresponding soil layer height as the input for the side boundaries, and the incident wave field as the input for the bottom boundary; 2) inputting seismic motion as displacement or acceleration on the bedrock surface; 3) inverting ground seismic records to the bottom of the model and inputting them as boundary motion. However, because the non-uniform scattering of incident waves at arbitrary undulating strata boundaries is not fully considered, it is difficult to accurately characterize the propagation and superposition process of near-field waves, and computational errors are unavoidable. Typically, the computational accuracy is only satisfactory when the near-field truncated side boundary is sufficiently far from the target study area.

[0010] As mentioned earlier, for complex heterogeneous layered or interbedded half-space foundations, during vibration input, vibration waves will be reflected or refracted at the interfaces between different foundation media, resulting in corresponding multi-level secondary waves. The corresponding free-field motion must satisfy the boundary condition of zero stress on the foundation surface, as well as the displacement and stress continuity conditions at the medium interface. Compared to homogeneous foundations, the computational complexity increases significantly, making the solution extremely difficult, especially for foundations with arbitrary undulations and layers. Furthermore, wave scattering analysis needs to consider the radiation damping characteristics of asymmetric layered far-field regions. Local artificial boundary models (such as viscoelastic artificial boundaries or perfectly matched layers) or higher-precision global artificial boundary simulations can be used, such as the proportional boundary finite element method based on similar centerlines or the boundary element method; these are collectively referred to as artificial boundaries. The key issue here is how to accurately reflect the equivalent effect of irregular strata boundaries on wave scattering, for which there is a lack of effective methods. Summary of the Invention

[0011] This invention aims to provide a simplified calculation technique for the vibration effects of irregular strata interfaces or non-uniform wave scattering on the ground surface in arbitrarily undulating layered sites under incident vibration waves. Its main principle is based on the wavefield separation principle, decomposing the wavefield to be determined into a known wavefield under regular ground conditions and a scattered wavefield caused by the actual irregular strata distribution. The difficulty in solving for complex wavefield components is transformed into an internal wave scattering problem. Based on the concept of an isolator, a strata boundary substructure is constructed to calculate the equivalent effect of the scattered wavefield in the irregular strata boundary region.

[0012] The technical solution of this invention is:

[0013] A method for calculating the vibration effects of arbitrarily undulating strata, such as Figure 1 As shown, the steps are as follows:

[0014] Step 1. Based on the geometric and material information of any undulating strata, perform finite element modeling of the near field of the irregular undulating site;

[0015] like Figure 2 As shown, the finite domain near the irregular undulating site is discretized using finite element units to obtain a finite domain model. Based on spatial location, nodes are divided into four categories: nodes in the irregular stratigraphic boundary region are denoted as e; internal nodes adjacent to the boundary of the irregular stratigraphic boundary region are denoted as d; other internal nodes are denoted as I; and artificial boundary nodes are denoted as b. The regions corresponding to the nodes are denoted as... , , , ; will the area Boundary between medium and irregular strata The associated part is marked as The rest are marked as ; and The corresponding regions are respectively denoted as and ;

[0016] According to the wave propagation principle, the free wave field of an irregular undulating site is decomposed into a known free wave field of a regular site w and a scattered wave field to be determined, such as... Figure 1 As shown, the corresponding field motion expressions for the two types of fields after partitioning are as follows:

[0017] (1)

[0018] In the formula, the superscript f represents the known free wave field of the regular field w; p represents the scattered wave field generated under the action of additional stress. This represents the free wave field motion of an irregularly undulating field h. Represents the known free wave field motion of a regular field w. Indicates the motion of the scattered wave field;

[0019] Total stress field at any point in the free wave field of an irregular undulating site It is divided into two parts, namely

[0020] (2)

[0021] In the formula, Describe the known free wave field motion of a regular field w. Stress generated on irregular, undulating terrain; Represents the motion of the scattered wave field Stress generated on irregular, undulating terrain;

[0022] Under the influence of external waves, the motion of the free wave field in an irregularly undulating field causes the equivalent nodal forces at each node in the finite field model to increase. Should be The equivalent nodal force, as shown in equation (2), is expressed in the following form:

[0023] (3)

[0024] In the formula, , , They represent stress respectively , , The resulting equivalent nodal forces at each node;

[0025] For known The equivalent nodal forces generated at each node of the finite field are obtained by the following equation:

[0026] (4)

[0027] In the formula, This represents the dynamic stiffness submatrix for the corresponding region, with subscripts indicating the associated region; that is... For the region For the region The dynamic stiffness submatrix reflects the dynamic coupling relationship between the degrees of freedom in the two regions; , ,in, For a known free wave field at a regular field w, at the node The movement of the place, The force generated at node e when the known free wave field motion of a regular field w is mapped to an irregular undulating field;

[0028] In practical calculations, an additional layer of nodes u located in the infinite field is added outside the finite field, solely for computation purposes. The boundary dynamic stiffness is simulated by using viscoelastic artificial boundaries or proportional boundaries calculated by finite element method.

[0029] Step 2. Remove irregular stratigraphic boundaries. and and The unit and node information is used to generate the stratigraphic boundary substructure;

[0030] like Figure 2 As shown, within the range of the outer boundary of a finite domain, an irregular undulating field is considered as a region derived from the corresponding regular field. The replacement was performed, only in this area. The medium distribution of the nodes is inconsistent with that of regular sites; in the finite domain model of irregular undulating sites, in addition to the regions The remaining nodes conform to the free wave field compatibility conditions of a regular field, and their equivalent nodal forces are: Equation (4) can be simplified to the following form:

[0031] (5)

[0032] Combining the concept of isolators, finite element elements and their adjacent elements on irregular stratigraphic boundaries are constructed as a substructure (e.g., Figure 3 (as shown in (a)), including the region , , This is referred to as the stratigraphic boundary substructure; furthermore, the finite element stiffness matrix of the stratigraphic boundary substructure is the coefficient matrix corresponding to equation (5);

[0033] Step 3. Calculate the free wave field motion of the corresponding regular field w under the incident wave. ,Will Mapping the nodes to the stratigraphic boundary substructure yields the motion matrix of equation (5);

[0034] Furthermore, for cases where undulating strata are located at the surface, the aforementioned stratigraphic boundary substructure consists of only one layer of units along the surface boundary, and the equivalent effect of the scattered wave field acts on the site surface, such as... Figure 3 As shown in (b), additional stress will also be generated outside the finite domain of the free field motion calculation of the regular site. However, since the range of the calculation domain is selected to be large, it is assumed that the additional stress outside the calculation domain has little impact on the target study area and can be ignored in terms of engineering analysis accuracy. As the truncated calculation domain is expanded, the calculation results tend to be more accurate.

[0035] Furthermore, the free wave field of a regular site w, which is considered to be horizontally layered, is calculated using a one-dimensional harmonic superposition algorithm or the precise stiffness matrix method; for sites with regular terrain shapes, it is calculated using the wave function expansion method or the boundary element method.

[0036] Step 4. Perform finite element dynamic calculations of the stratigraphic boundary substructure. Calculate the irregular stratigraphic boundary region generated by the scattered wave field using the coefficient matrix, motion matrix, and equation (5). Equivalent nodal force ;

[0037] Step 5. Establish an artificial boundary model to simulate the far-field semi-infinite domain, so that the outward scattered waves generated in the near-field region of the irregular undulating site under the excitation of the endogenous force can be transmitted and absorbed at the outer boundary of the finite domain, and construct a computational model of the overall irregular undulating site.

[0038] Step 6. Take as The reverse force acts on the region. The motion of the scattered wave field in an irregularly undulating terrain is calculated as follows: ;

[0039] Obtained from equation (3) :

[0040] (6)

[0041] As shown in the following formula, The motion of the scattered wave field of the irregular undulating site is obtained by applying it to the nodes of the finite field and using formula (7). :

[0042] (7)

[0043] in, Let be the dynamic stiffness matrix of the far field relative to the near field of the irregularly undulating site, acting on the outer boundary of the finite field near field of the irregularly undulating site. superior;

[0044] Step 7. Calculate the scattered wave field of the irregular undulating site obtained in Step 6. Free wave field motion of the regular field calculated in step 3 By superimposing the data, the total field motion of the irregular undulating site can be obtained. .

[0045] The proposed stratigraphic boundary substructure model makes equation (5) applicable to stratigraphic structures with arbitrary undulations. It only requires pre-calculating the free-field motion of the auxiliary regular site and mapping it to the substructure nodes, making it simple and easy to implement. Finally, the scattered wave field motion obtained from equation (7) is... Substituting into equation (1), and Superposition, or free field motion in irregular sites, can be used as vibration wave input for the dynamic interaction system of structure-complex foundation system within the SSI analysis domain.

[0046] The beneficial effects of this invention are: calculating the wave field of an arbitrary undulating foundation under the incident vibration wave, and accurately describing the non-uniform propagation characteristics of site waves. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the wave field separation model of the present invention, wherein (a) is the free wave field of an irregular undulating site, (b) is the known free wave field of a regular site, and (c) is the scattered wave field.

[0048] Figure 2 This is a schematic diagram of the near-field finite element model of the foundation under irregular strata according to the present invention. Among them, (a) shows the case of undulation of the underlying strata, and (b) shows the case of undulation of the surface strata.

[0049] Figure 3 This is a schematic diagram of the mapping relationship between the regular site free wave field and the substructure nodes of the strata boundary according to the present invention. Among them, (a) represents the case of the underlying strata undulation, and (b) represents the case of the surface strata undulation.

[0050] Figure 4 This is a flowchart of the method for analyzing the subsurface volume wave field of irregularly undulating stratified sites based on the stratigraphic boundary substructure.

[0051] Figure 5 This is a schematic diagram of the calculation model in the embodiment. Among them, (a) represents the case of underlying strata undulation, and (b) represents the case of surface strata undulation.

[0052] Figure 6This is a comparison and verification result of the displacement time history of the observation point under P-wave incident and the reference solution in the embodiment. Among them, (a) is the horizontal displacement under the condition of undulating underlying strata, (b) is the horizontal displacement under the condition of undulating surface strata, (c) is the vertical displacement under the condition of undulating underlying strata, and (d) is the vertical displacement under the condition of undulating surface strata.

[0053] Figure 7 This is the displacement time history distribution along the observation path of the P-wave incident in the embodiment. Among them, (a) is the horizontal displacement under the condition of undulating underlying strata, (b) is the vertical displacement under the condition of undulating underlying strata, (c) is the horizontal displacement under the condition of undulating surface strata, and (d) is the vertical displacement under the condition of undulating surface strata.

[0054] Figure 8 This is a comparison and verification result of the displacement time history of the observation point under S-wave incident and the reference solution in the embodiment. Among them, (a) is the horizontal displacement under the condition of undulating underlying strata, (b) is the horizontal displacement under the condition of undulating surface strata, (c) is the vertical displacement under the condition of undulating underlying strata, and (d) is the vertical displacement under the condition of undulating surface strata.

[0055] Figure 9 This is the displacement time history distribution along the observation path of the S-wave incident in the embodiment. Among them, (a) is the horizontal displacement under the condition of undulating underlying strata, (b) is the vertical displacement under the condition of undulating underlying strata, (c) is the horizontal displacement under the condition of undulating surface strata, and (d) is the vertical displacement under the condition of undulating surface strata. Detailed Implementation

[0056] The specific embodiments of the present invention will now be described in detail with reference to the technical solutions and accompanying drawings.

[0057] Vibrating body waves mainly include two types: longitudinal waves (P-waves) and transverse waves (S-waves). Among them, the propagation of P-waves depends on the volume compression and stretching of the medium, and the vibration direction is consistent with the wave propagation direction; S-waves, on the other hand, propagate through the shear deformation of the medium, and the vibration direction is perpendicular to the wave propagation direction.

[0058] Meanwhile, the surface strata and the underlying strata represent two typical cases of the stratigraphic foundation of the irregular undulating site described in this invention. The following specific embodiments will analyze the underlying undulating site and the surface undulating site under the incidence of P-waves and S-waves, respectively, to verify the effectiveness of the calculation method. The calculation models are as follows: Figure 5 As shown in (a) and (b) in the figure, the parameters of the strata and soil materials are shown in Table 1.

[0059] Table 1 shows the parameters of geological strata and soil materials.

[0060]

[0061] displacement pulse wave As an incident wave vertically upward at an elevation of -145m below the Earth's surface, duration The time step is 0.7 seconds, with a time increment of 0.001 seconds. The expression is as follows:

[0062] (8)

[0063] In the formula, ; ; It is the Heaviside step function.

[0064] (1) P-wave incidence;

[0065] In this embodiment, the reference solution for P-wave incidence is constructed as follows: constrain the horizontal motion of the finite domain truncated boundary nodes, apply vertical incident wave motion to the bottom nodes of the model, and adjust... It is large enough that spurious fluctuations reflected from the constraint boundary do not interfere with the target study area.

[0066] When using the method described in this invention, the calculation process is as follows: 1) Based on the geological geometric material information, establish finite element models of irregular undulating sites under two conditions: underlying strata undulation and surface strata undulation, and appropriately select the cutoff range. ;2) Separate the stratigraphic boundary substructure models for the two cases respectively;3) Calculate the free field of a regular site divided into three layers with each layer flat under the pulse wave incident in the form of P wave using the one-dimensional harmonic superposition method, and map it to the stratigraphic boundary substructure models for the two cases respectively;4) Obtain the forces generated in the stratigraphic boundary region of the irregular undulating site under the two cases respectively using finite element dynamic calculation;5) Establish a near-field-far-field overall calculation model for the underlying undulating site and the surface undulating site;6) Take the forces generated in the stratigraphic boundary region of the irregular undulating site as the reverse force, apply it to the model of the irregular undulating site, and obtain the scattered wave field motion for the two cases respectively;7) Superimpose the scattered wave field motion calculated in step 6 with the free field motion of the regular site calculated in step 3 to obtain the overall vibration wave field of the irregular site.

[0067] The horizontal and vertical displacement results at the observation points calculated by the method described in this invention were compared and verified with the reference solution. They showed a basic agreement. Figure 6 As shown in (a), (b), (c), and (d) in the figure. Furthermore, the observation paths under different conditions were extracted and compared (see...). Figure 5 The displacement time history distribution of each point on the surface, such as Figure 7 As shown, the proposed method can clearly characterize the differences in wave fields under different strata undulations.

[0068] (2) S-wave incident;

[0069] In this embodiment, the reference solution for S-wave incidence is constructed as follows: constrain the vertical motion of the finite domain truncation boundary nodes, apply horizontal incident wave motion to the bottom nodes of the model, and adjust... It is large enough that spurious fluctuations reflected from the constraint boundary do not interfere with the target study area.

[0070] When using the method described in this invention, the calculation process is generally the same as in Example (1), except that step 3 is adjusted. The free field of a regular field divided into three layers with each layer flat is calculated by the one-dimensional harmonic superposition method, and mapped to the stratigraphic boundary substructure model in the two cases respectively.

[0071] The horizontal and vertical displacement results at the observation points calculated by the method described in this invention were compared and verified with the reference solution. The results showed a basic agreement in both cases of undulating underlying strata and undulating surface strata. Figure 8 As shown in (a), (b), (c), and (d) in the figure. Furthermore, the observation paths under different conditions were extracted and compared (see...). Figure 5 The displacement time history distribution of each point on the surface, such as Figure 9 As shown, the proposed method can clearly characterize the differences in wave fields under different strata undulations.

Claims

1. A method for calculating the vibration effect of arbitrarily undulating strata, characterized in that, The steps are as follows: Step 1. Based on the geometric and material information of any undulating strata, perform finite element modeling of the near field of the irregular undulating site; Finite element methods are used to discretize the finite domain in the near field of the irregular undulating site to obtain a finite domain model. Based on spatial location, nodes are divided into four categories: nodes in the irregular stratigraphic boundary region are denoted as e; internal nodes adjacent to the boundary of the irregular stratigraphic boundary region are denoted as d; other internal nodes are denoted as I; and artificial boundary nodes are denoted as b. The regions corresponding to the nodes are respectively denoted as... , , , ; will the area Boundary between medium and irregular strata The associated part is marked as The rest are marked as ; and The corresponding regions are respectively denoted as and ; Based on the wave propagation principle, the free wave field of an irregular undulating site is decomposed into a known free wave field of a regular site w and a scattered wave field to be determined. The corresponding field motion expressions for the two fields after the decomposition are as follows: (1) In the formula, the superscript f represents the known free wave field of the regular field w; p represents the scattered wave field generated under the action of additional stress. This represents the free wave field motion of an irregularly undulating field h. Represents the known free wave field motion of a regular field w. Indicates the motion of the scattered wave field; Total stress field at any point in the free wave field of an irregular undulating site It is divided into two parts, namely (2) In the formula, Describe the known free wave field motion of a regular field w. Stress generated on irregular, undulating terrain; Represents the motion of the scattered wave field Stress generated on irregular, undulating terrain; Under the influence of external waves, the motion of the free wave field in an irregularly undulating field causes the equivalent nodal forces at each node in the finite field model to increase. Should be The equivalent nodal force, as shown in equation (2), is expressed in the following form: (3) In the formula, , , They represent stress respectively , , The resulting equivalent nodal forces at each node; For known The equivalent nodal forces generated at each node of the finite field are obtained by the following equation: (4) In the formula, This represents the dynamic stiffness submatrix for the corresponding region, with subscripts indicating the associated region; that is... For the region For the region The dynamic stiffness submatrix reflects the dynamic coupling relationship between the degrees of freedom in the two regions; , ,in, For a known free wave field at a regular field w, at the node The movement of the place, The force generated at node e when the known free wave field motion of a regular field w is mapped to an irregular undulating field; In practical calculations, an additional layer of nodes u located in the infinite field is added outside the finite field, solely for computation. The boundary dynamic stiffness is simulated by using viscoelastic artificial boundaries or proportional boundaries calculated by finite element method. Step 2. Remove irregular stratigraphic boundaries. and and The unit and node information is used to generate the stratigraphic boundary substructure; Within the outer boundary of a finite domain, an irregular undulating field is considered as a region derived from the corresponding regular field. The replacement was performed, only in this area. The medium distribution of the nodes is inconsistent with that of regular sites; in the finite domain model of irregular undulating sites, in addition to the regions The remaining nodes conform to the free wave field compatibility conditions of a regular field, and their equivalent nodal forces are: Equation (4) can be simplified to the following form: (5) Combining the concept of isolators, finite element elements and their adjacent elements on irregular stratigraphic boundaries are constructed as a substructure, including the region. , , This is referred to as the stratigraphic boundary substructure; furthermore, the finite element stiffness matrix of the stratigraphic boundary substructure is the coefficient matrix corresponding to equation (5); Step 3. Calculate the free wave field motion of the corresponding regular field w under the incident wave. ,Will Mapping the nodes to the stratigraphic boundary substructure yields the motion matrix of equation (5); Step 4. Perform finite element dynamic calculations of the stratigraphic boundary substructure. Calculate the irregular stratigraphic boundary region generated by the scattered wave field using the coefficient matrix, motion matrix, and equation (5). Equivalent nodal force ; Step 5. Establish an artificial boundary model to simulate the far-field semi-infinite domain, so that the outward scattered waves generated in the near-field region of the irregular undulating site under the excitation of the endogenous force can be transmitted and absorbed at the outer boundary of the finite domain, and construct a computational model of the overall irregular undulating site. Step 6. Take as The reverse force acts on the region. The motion of the scattered wave field in an irregularly undulating terrain is calculated as follows: ; Obtained from equation (3) : (6) As shown in the following formula, The motion of the scattered wave field of the irregular undulating site is obtained by applying it to the nodes of the finite field and using formula (7). : (7) in, Let be the dynamic stiffness matrix of the far field relative to the near field of the irregularly undulating site, acting on the outer boundary of the finite field near field of the irregularly undulating site. superior; Step 7. Calculate the scattered wave field of the irregular undulating site obtained in Step 6. Free wave field motion of the regular field calculated in step 3 By superimposing the data, the total field motion of the irregular undulating site can be obtained. .

2. The method for calculating the vibration effect of arbitrarily undulating strata according to claim 1, characterized in that, In step 3, for the case of undulating strata on the surface, the stratum boundary substructure consists of only one layer of units along the surface boundary, and the equivalent effect of the scattered wave field acts on the site surface.

3. The method for calculating the vibration effect of arbitrarily undulating strata according to claim 1, characterized in that, In step 3, the free wave field of a regular site w considered as horizontally layered is calculated using a one-dimensional harmonic superposition algorithm or the precise stiffness matrix method; for sites with regular terrain shapes, it is calculated using the wave function expansion method or the boundary element method.

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