Method for acquiring stratified site seismic oscillation time history under action of supercritical angle incident SV wave

Abstract

The invention discloses a method for obtaining a stratified site seismic oscillation time history under the action of a supercritical angle incident SV wave, and the method specifically comprises the following steps: calculating a wave field in each stratum in a time domain through a recursion method under the condition that the seismic SV wave is incident to a stratified stratum at a supercritical angle, and obtaining a seismic wave time history incident to the ground. According to the method, under the action of the supercritical angle incident SV wave, the stratified field seismic oscillation time history can be directly obtained in the time domain without time-frequency domain conversion and numerical simulation.

Description

Technical Field

[0001] This invention belongs to the field of earthquake engineering and disaster prevention and mitigation technology in the water conservancy and civil engineering industry, and relates to a method for obtaining the ground motion time history of a layered site under the action of supercritical angle-incidence SV waves. Background Technology

[0002] When assessing the seismic safety of major projects such as dams, nuclear power plants, and subway tunnels, accurately simulating the propagation of seismic waves in layered sites and obtaining the ground motion time history is crucial. Compared to P-waves, SV waves have a more significant impact on the seismic response of structures, especially when the incidence angle is supercritical (the angle of incidence is greater than the critical angle), generating non-uniform waves within the strata and leading to complex wave phenomena. Currently, when traditional time-domain calculation methods such as the finite element method simulate the propagation of seismic waves in layered sites, the ground motion input under the supercritical incidence angle of SV waves becomes a critical challenge, thus necessitating the use of frequency-domain simulation to obtain the ground motion time history. Such methods not only require operation in the frequency domain but also necessitate processing of the computational model, including size determination, mesh generation, boundary setting, and stability analysis, thereby affecting computational accuracy and efficiency. To overcome the shortcomings and deficiencies of these methods, a method for obtaining the ground motion time history of layered sites under the supercritical incidence angle of SV waves needs to be proposed. Summary of the Invention

[0003] The purpose of this invention is to provide a method for obtaining the ground motion time history of a layered site under supercritical angle-incidence SV wave action, which solves the problem that the ground motion time history of a layered site can be directly obtained in the time domain without time-frequency domain conversion and numerical simulation under supercritical angle-incidence SV wave action.

[0004] The technical solution adopted in this invention is a method for obtaining the seismic motion time history of a layered site under the action of supercritical angle incident SV waves, which specifically includes the following process: when the seismic SV waves are incident on the layered strata at a supercritical angle, the wave field in each stratum is calculated in the time domain by a recursive method, and then the seismic wave time history incident on the ground is obtained.

[0005] The invention is further characterized by: Specifically, the steps include the following: Step 1: Determine the number of soil layers at the site based on the geotechnical investigation results. n Density of strata ρ i Longitudinal wave velocity vp i transverse wave velocity vs i and stratum thickness h i ; Step 2, select the incident angle of the SV wave. βn , and sin β n <vs n / vp i ; Step 3: Calculate the P-wave incident angle for each stratum. α i and SV wave incident angle β i ; Step 4, calculate the P-wave incident angle α n sine α n Cosplay of Cosine α n ; Step 5: Calculate the reflection coefficients of P-waves and SV-waves at the ground. R 1,0 pp , R 1,0 ps , R 1,0 sp and R 1,0 ss ; Step 6, Calculate the coefficient matrix A i,i-1 ; Step 7, Calculate the coefficient matrix A i-1,i ; Step 8, calculate the P wave and SV wave from the first... i Layer incident to the first i Reflectance at -1 layer R i,i-1 pp , R i,i-1 ps , R i,i-1 sp , R i,i-1 ss and transmission coefficient T i,i-1 pp , T i,i-1 ps , T i,i-1 sp , T i,i-1 ss ; Step 9, calculate the P wave and SV wave from the first... i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i pp , R i-1,i ps , R i-1,i sp , R i-1,i ss and transmission coefficient T i-1,i pp , T i-1,i ps , T i-1,i sp , T i-1,i ss ; Step 10: Determine the incident wave function and duration. t w ; Step 11, determine the duration of the seismic response. t l ; Step 12, determine the time sampling interval ; Step 13, Calculate the number of source sampling points nt w Number of ground motion sampling points nt And P waves and SV waves in the first i Number of propagation time samples in layer time np i , ns i ; Step 14, for the incident wave function Discretize the data to form information about... k Timing function ; Step 15: Calculate the seismic waves within the site strata using a recursive method; Step 16: Calculate the seismic P-wave incident on the ground. u p ( k ) and seismic SV waves u s ( k ); Step 17: Obtain the horizontal component of the earthquake motion time history based on the calculation results of Step 16. u x (k and vertical component u y ( k ).

[0006] In step 3, the P-wave incident angle of each stratum is calculated using the following formula (1). α i and SV wave incident angle β i : (1) in, i = 1, 2, ..., n -1.

[0007] In step 4, the incident angle of the P-wave is calculated according to equations (2) to (3). α n sine α n Cosplay of Cosine α n ; (2) (3) in, j It is the imaginary unit.

[0008] In step 5, the reflection coefficients of P-waves and SV-waves at the ground are calculated according to equations (4) to (5). R 1,0 pp , R 1,0 ps , R 1,0 sp and R 1,0 ss : (4) (5) Since only reflected waves exist at the ground and no transmitted waves are transmitted, the transmission coefficient is set to... T 0,1 pp = T 0,1 ps = T 0,1 sp = T 0,1 ss =0.

[0009] In step 6, the coefficient matrix is ​​calculated according to equation (6). A i,i-1 , i = 2, ..., n : (6) In step 7, the coefficient matrix is ​​calculated according to equation (7). A i-1,i : (7) Where i = 2, ..., n.

[0010] In step 8, the P-wave and SV-wave are calculated from the first according to equations (8) to (9). i Layer incident to the first i Reflectance at -1 layer R i,i-1 pp , R i,i-1 ps , R i,i-1 sp , R i,i-1 ss and transmission coefficient T i,i-1 pp , T i,i-1 ps , T i,i-1 sp , T i,i-1 ss : (8) (9) In step 9, the P-wave and SV-wave are calculated from the first wave according to equations (10) to (11). i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i pp , R i-1,i ps , R i-1,i sp , R i-1,i ss and transmission coefficient T i-1,i pp , Ti-1,i ps , T i-1,i sp , T i-1,i ss : (10) (11) in, i = 2, ..., n .

[0011] In step 11, the duration of the seismic motion is determined using the following formula (12). t l , t l The following formula should be satisfied: (12) In step 12, the time sampling interval is determined by the following formula (13). , The following formula should be satisfied: (13) in, f max yes The cutoff frequency.

[0012] In step 13, the number of source sampling points is calculated according to equations (14) to (17). nt w Number of ground motion sampling points nt And P waves and SV waves in the first i Number of propagation time samples in layer time np i , ns i : (14) (15) (16) (17) Here, int is the integer arithmetic function. i = 1, ..., n ; In step 14, the incident wave function is determined according to equation (18). Discretize the data to form information about... k Timing function : (18) in, k = 1, ..., nt .

[0013] In step 15, the upward wave incident on the ground is calculated using the recursive method of the following formula (19). Pu 1 and Su 1: (19) in, (20) (twenty one) Equation (19) is a set of wave functions Pu i ( k ), Su i ( k ), Pd i ( k ), Sd i ( k )about k The recursive operation, where, Pu i ( k ), Su i ( k ) is the incident point to the i The upward P-wave and upward SV-wave of each interface. Pd i ( k ), Sd i ( k ) is the incident point to the i The downward P-wave and downward SV-wave of each interface, and satisfying when k When <0, Pu i ( k ) = Su i ( k ) = Pd i ( k ) = Sd i ( k = 0.

[0014] The beneficial effect of this invention is that, when seismic SV waves are incident on layered strata at a supercritical angle, the wavefield within each stratum is calculated in the time domain using a recursive method, thereby obtaining the time history of the seismic waves incident on the ground. Since the reflection and transmission coefficients are complex numbers, the calculated seismic waves are also complex numbers. Therefore, the real-domain seismic waves need to be obtained through Hilber transform, and then the ground motion time history is obtained through synthesis calculation. This method effectively avoids conventional numerical calculations, improves calculation accuracy and efficiency, and can be applied to the seismic response analysis of large-scale engineering projects such as dams, tunnels, and pile foundations where the influence of the incident angle needs to be considered. Attached Figure Description

[0015] Figure 1(a) is a site response model of the method for obtaining the ground motion time history of a layered site under supercritical angle-incident SV wave action according to the present invention. Figure 1(b) shows the seismic waves at the interface of the method for obtaining the seismic motion time history of a layered site under supercritical angle-incidence SV wave action according to the present invention. i Reflection and transmission at the point; Figure 2 This is the Reckon wave time history of the method for obtaining the ground motion time history of a layered site under supercritical angle-incidence SV wave action according to the present invention; Figure 3 This is the Reck wave amplitude spectrum of the method for obtaining the ground motion time history of a layered site under supercritical angle-incident SV wave action according to the present invention; Figure 4 This invention relates to the method for obtaining the seismic motion time history of a layered site under supercritical angle-incident SV wave action, specifically the seismic wave time history arriving at the ground under inclined incident Reck wave action. Figure 5 This invention relates to the method for obtaining the ground motion time history of a stratified site under supercritical angle-incident SV wave action, specifically the ground motion time history under inclined incident Reck wave action. Figure 6 The method for obtaining the ground motion time history of a stratified site under supercritical angle incident SV wave action of the present invention is based on the horizontal component of the ground motion time history under different incident angles of the Reck wave. Figure 7 The method for obtaining the ground motion time history of a stratified site under supercritical angle-incident SV wave action of the present invention is based on the vertical component of the ground motion time history under different incident angles of the Reck wave. Figure 8 The method for obtaining the ground motion time history of a stratified site under supercritical angle incident SV wave action is the ground motion time history when the Reck wave is incident at an angle smaller than the critical angle. Figure 9 The method for obtaining the ground motion time history of a layered site under supercritical angle incident SV wave action of the present invention is based on the ground motion time history when the Reck wave is greater than the critical angle incident. Figure 10The method for obtaining the ground motion time history of a stratified site under supercritical angle incident SV wave of the present invention is the ground motion amplitude spectrum when the Reck wave is incident at a less than critical angle. Figure 11 The method for obtaining the ground motion time history of a stratified site under supercritical angle incident SV wave of the present invention is based on the ground motion amplitude spectrum when the Reck wave is incident at an angle greater than the critical angle. Figures 12(a) and 12(b) show a comparison of the ground motion time histories of the method for obtaining ground motion time histories of stratified sites under supercritical angle incident SV waves according to the present invention, with SV waves incident at the critical angle. Figures 13(a) and 13(b) show the comparison of ground motion time histories when the SV wave is greater than the critical angle of incidence for obtaining ground motion time histories of stratified sites under the action of supercritical angle incident SV wave of the present invention. Figure 14 Elcentro wave is the method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to the present invention; Figures 15(a) to 15(b) show the ground motion time history of El Centro waves when the El Centro waves are incident at a greater than the critical angle, as described in the method for obtaining the ground motion time history of a layered site under the action of supercritical angle incident SV waves of the present invention. Figure 16 This invention relates to the relationship between the peak ground acceleration and the incident angle when El Centro wave is obliquely incident, in the method for obtaining the ground motion time history of a stratified site under supercritical angle-incident SV wave. Detailed Implementation

[0016] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0017] The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action, as described in this invention, includes the following specific steps: Step 1: Determine the number of soil layers at the site based on the geotechnical investigation results. n Density of strata ρ i Longitudinal wave velocity vp i transverse wave velocity vs i and stratum thickness h i ,in i = 1, 2, ..., n ;density ρ i The unit is kg / m³ 3 Longitudinal wave velocity vp i and transverse wave velocity vs i The unit is m / s; Step 2, select the incident angle of the SV wave. β n (Figure 1(a)), and sin β n <vs n / vp i ,in i = 1, 2, ..., n- 1. To prevent slip waves from appearing inside the strata; Step 3, Equation (1) is Snell's theorem in seismology, which can be used to calculate the P-wave incidence angle at each stratigraphic interface. α i and SV wave incident angle β i (Figure 1(b)); (1) in, i = 1, 2, ..., n -1; Step 4: Calculate the P-wave reflection angle according to equations (2) to (3). α n sine α n Cosplay of Cosine α n ; (2) (3) in, j It is the imaginary unit.

[0018] It is worth noting that when sin β n vs n / vp n At that time, sin α n > 1. cosplay α n The value is an imaginary number. In this case, the incident angle of the seismic SV wave is greater than the critical angle, that is, the SV wave is incident at a supercritical angle, and the reflected P wave is a non-uniform sliding wave (Fig. 1(a), Fig. 1(b)). Step 5: Calculate the reflection coefficient of the P-wave at the ground according to equations (4) and (5). R 1,0 pp , R 1,0 ps The reflection coefficient of SV waves at the groundR 1,0 sp and R 1,0 ss : (4) (5) Since only reflected waves exist at the ground and no transmitted waves exist, to facilitate the recursive calculation of the wave field, the transmission coefficient is set to... T 0,1 pp = T 0,1 ps = T 0,1 sp = T 0,1 ss =0; Step 6: Calculate the coefficient matrix according to equation (6). A i,i-1 ( i = 2, ..., n ): (6) This matrix is ​​used to calculate the P-wave and SV-wave of an earthquake from the th... i Layer incident to the first i Reflectance and transmission coefficients at layer -1; Step 7, calculate the coefficient matrix according to equation (7). A i-1,i ( i = 2, ..., n ): (7) This matrix is ​​used to calculate the P-wave and SV-wave of an earthquake from the th... i -1 layer incident to the first i The reflection and transmission coefficients of the layer; Step 8, calculate the P-wave from the first wave according to equation (8). i Layer incident to the first i Reflectance at -1 layer R i,i-1 pp , R i,i-1 ps Transmission coefficient T i,i-1 pp , T i,i-1 ps ; Calculate the SV wave from the first according to equation (9) iLayer incident to the first i Reflectance at -1 layer R i,i-1 sp , R i,i-1 ss Transmission coefficient T i,i-1 sp , T i,i-1 ss : (8) (9) in, i = 2, ..., n .

[0019] Step 9, calculate the P-wave from the first wave according to equation (10). i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i pp , R i-1,i ps Transmission coefficient T i-1,i pp , T i-1,i ps ; Calculate the SV wave from the first wave according to equation (11) i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i sp , R i-1,i ss Transmission coefficient T i-1,i sp , T i-1,i ss : (10) (11) Step 10, Select the incident wave time history f w ( k ), k = 0, 1, 2, ..., nt -1. There are two types of incident wave time histories: one is the measured seismic record, such as the famous Elcentrop wave and Taft wave, whose sample number... nt wSampling interval Determined by measured values; another is the wave function, such as the Rick wave, bell wave, or pulse wave, with its sampling interval... Number of sampling points nt w The sampling interval is determined based on the discretization result of the wave function. It does not exceed half the reciprocal of the cutoff frequency, that is .

[0020] Step 11, determine the number of ground motion time history sampling points. nt Normally, nt ≥ nt w If the incident wave is a measured seismic record, nt = nt w If the incident wave is a wave function, nt > nt w The specific time can be determined based on the time when the earthquake tail shock approaches zero.

[0021] Step 12, calculate the P wave at the 1st epoch according to equations (12) to (13). i Number of propagation time samples in layer time np i SV wave in the first i Number of propagation time samples in layer time ns i : (12) (13) Here, int is the integer arithmetic function. i = 1, ..., n ; Step 13, calculate the seismic waves within the site strata according to the recursive formula (14): (14) in: (15) (16) Equations (14) to (16) are a set of wave functions Pu i ( k ), Su i ( k ), Pd i ( k ), Sd i (k )about k The recursive operation, i =2,..., n -1; k = 0, 1, 2, ..., nt -1. Pu i ( k )and Su i ( k ) is the incident point to the i The upward P-wave and upward SV-wave of each interface. Pd i ( k )and Sd i ( k ) is the incident point to the i The downward P-wave and downward SV-wave of each interface, and satisfying when k When <0, Pu i ( k ) = Su i ( k ) = Pd i ( k ) = Sd i ( k = 0. Note the correspondence between the interface and the formation (Figure 1(a), Figure 1(b)). When i= At time 1, the first interface is the ground; when i When ≥ 2, the first i The first interface is the... i -1st floor and the i Stratigraphic interfaces of layers; Step 14: Calculate the seismic P-wave incident on the ground according to equations (17) and (18). u p ( k ) and seismic SV waves u s ( k ): (17) (18) in, k = 0, 1, 2, ..., nt -1.

[0022] Equations (17) to (18) show that if sin α n≤ 1. The reflection coefficient and transmission coefficient are real numbers. Pu 1( k )and Su 1( k If ) is also a real number, then u p ( k )= Pu 1( k ), u s ( k )= Su 1( k If sin α n > 1. The reflection coefficient and transmission coefficient are complex numbers. In this case... Pu 1( k )and Su 1( k () is a complex function of time, which requires Hilbert transform to obtain its value. u p ( k )and u s ( k ).

[0023] Step 15: Calculate the horizontal component of the earthquake motion time history according to equation (19). u x ( k and vertical component u y ( k ): (19) in, k = 0, 1, 2, ..., nt -1.

[0024] Example 1 The specific steps for calculating the seismic time history of a two-story site are as follows: (1) According to the geotechnical engineering investigation, the number of soil layers at a certain site is... n = 2, density of the strata ρ i Longitudinal wave velocity vp i transverse wave velocity vs i and stratum thickness h i The parameters are shown in Table 1, where i = 1, 2; Table 1. Stratigraphic parameters of the two-layer site

[0025] (2) Take the incident angle of the seismic SV wave β n = β 2 = 35°, and perform the following verification:

[0026]

[0027] The angle of incidence satisfies sin β n <vs n / vp i ; (3) By Snell's theorem, we get

[0028] (4) Calculate the incident angle of the P-wave. α n sine α n Cosplay of Cosine α n ;

[0029]

[0030] (5) Calculate the reflection coefficient of seismic waves at the ground surface;

[0031]

[0032]

[0033] This calculation result and vp 1 = 670 m / s vs Substituting 1 = 300m / s into equations (4) and (5), we obtain two systems of linear equations in two variables concerning the reflection coefficient:

[0034]

[0035] Solving for the given information yields:

[0036]

[0037] (6) iSubstituting 1 into equation (6), we obtain the coefficient matrix. A 2,1 :

[0038] formation parameters ρ 1 = 1890 kg / m 3 , vp 1 = 670 m / s vs 1 = 300 m / s; ρ 1 = 2100 kg / m 3 , vp 2 = 1500 m / s vs 2 = 810 m / s; and about α 1. β 1. α 2. β Substituting the trigonometric function values ​​into the above formula, we obtain the following matrix:

[0039] (7) i Substituting 1 into equation (7), we obtain the coefficient matrix. A 1,2 :

[0040] Using the same method as step (6), we can obtain:

[0041] (8) i Substituting 1 into equations (8) and (9), we obtain two matrix expressions for a system of four linear equations in four variables related to the calculation of the reflection coefficient and the transmission coefficient:

[0042]

[0043] Substituting the values, we get:

[0044]

[0045] Solving the system of equations, we get:

[0046]

[0047]

[0048]

[0049] (9) i Substituting 1 into equations (10) and (11), we obtain two matrix expressions for a system of four linear equations in four variables concerning the calculation of the reflection coefficient and the transmission coefficient:

[0050]

[0051] Substituting the values, we get:

[0052]

[0053] Solving the system of equations, we get:

[0054]

[0055]

[0056]

[0057] (10) The incident wave is selected using the Reich wave function ( Figure 2 According to the incident wave time history amplitude spectrum (see...) Figure 3 ), to obtain the cutoff frequency Therefore, take ,satisfy Discretizing the Reichs-Warshall function yields... nt w = 1000, the corresponding duration is 1 second, the calculation expression is:

[0058] in, k = 0, 1, 2, ..., nt -1.

[0059] (11) Number of ground motion sampling points The corresponding duration is 3 seconds.

[0060] (12) Calculate the P wave and SV wave at the th minute according to (12) and (13). i Number of propagation time samples in layer time np i , ns i ;

[0061] (14) will n Substituting 2 into equation (14), we can obtain the following recurrence relation.

[0062] After eliminating redundant zero-value terms and further simplifying, we get

[0063] Based on the above calculation results, the relevant calculation parameters in the formula can be determined as follows:

[0064]

[0065] Based on this calculation, Pu 1( k ), Su 1( k ), k = 0, 1, 2, ..., nt -1; (13) Because sin α n =sin α 2 = 1.0622 > At 1 o'clock, Pu 1( k ), Su 1( k Since ) is a complex number, the following Hilbert transformation is required:

[0066]

[0067] in, k = 0, 1, 2, ..., nt -1.

[0068] u p ( k )and u s ( k () represents the P-wave and SV-wave incident on the ground. The calculation results are as follows: Figure 4 As shown.

[0069] (17) Calculate the horizontal component of the earthquake motion time history according to equation (19). u x ( k and vertical component u y ( k ).in,

[0070]

[0071]

[0072]

[0073] Therefore, the formula for calculating the earthquake motion time history is:

[0074]

[0075] u x ( k )and u y ( k This refers to the horizontal and vertical components of the earthquake motion time history. u p ( k )and u s ( k Substituting the values, we can obtain the following: Figure 5 The calculation results are shown.

[0076] Example 2 In Example 1, the critical angle of the SV wave is 32.68°. Therefore, the incident angles at and near the critical angle are taken as 25°, 30°, 32.68°, 30°, and 40°, respectively. Following the calculation steps described above, the ground motion time histories under different incident angles are obtained as follows: Figure 6 and Figure 7 As shown. Among them, Figure 6 It is the horizontal component of the earthquake motion time history. Figure 7 It is the vertical component of the earthquake motion time history.

[0077] As shown in the figure, when the SV wave is obliquely incident near the critical angle, the horizontal component of the ground motion is approximately three times that of the vertical component, exhibiting characteristics of horizontal motion. Another characteristic of the ground motion time history caused by the supercritical angle incident SV wave is the coma phenomenon. When the incident angle is equal to 40°, the horizontal component of the ground motion time history ( Figure 6 ) and vertical component ( Figure 7 The tail oscillation of the ) is strong and decays very slowly over time, with its horizontal component being particularly prominent.

[0078] The peak value of the horizontal component of the ground motion is related to the incident angle. When the incident angles of the SV wave are 25°, 30°, 32.68°, 30°, and 40°, the corresponding peak ground motion time histograms are 3.0142 m / s². 2 2.9443m / s 2 3.0283m / s 2 3.1998m / s 22.6528m / s 2 Therefore, the peak ground motion caused by supercritical SV wave incidence may be larger than that caused by incidence at less than the critical angle.

[0079] Example 3 In Example 1, incident angles of 32.6° and 32.7° were taken, where 32.6° is slightly less than the critical angle and 32.7° is slightly greater than the critical angle. The seismic time history obtained according to the above calculation steps is as follows: Figure 8 and Figure 9 As shown, its Fourier transform amplitude spectrum is as follows Figure 10 and Figure 11 As shown. Therefore, Figure 11 A needle-like peak appears in the amplitude at 6.3 Hz, and this frequency component corresponds to... Figure 9 Tail-wave phenomenon in earthquake time history, and Figure 10 and Figure 8 However, the earthquake motion does not have this characteristic. This example shows that even if the incident angles differ by 0.1°, the resulting earthquake motions will be quite different due to the different relationship between the incident angle and the critical angle.

[0080] Example 4 To verify the correctness of this invention, we cite the double-layer site model in the article "Implementation of Layered Ground Motion Input under Oblique Incidence of SV Wave Supercritical Angle in ABAQUS (Finite Element Analysis Software)", by Zhang Ji et al., Engineering Mechanics, 2021, 38(04): 200-210. The stratum parameters are shown in Table 2. According to the stratum parameters, the critical angle of SV wave is 30°.

[0081] Figures 12(a) and 12(b), and Figures 13(a) and 13(b) show the ground motion time histories when the SV wave is incident at 30° and 60°, respectively. Figures 12(a) and 13(a) are the ground motion time histories of a two-story site model from the literature, while Figures 12(b) and 13(b) are the ground motion time histories calculated according to the above steps. As can be seen from the figures, 12(a) and 12(b), and 13(a) and 13(b) are very close; the absolute value of the peak ground acceleration (PGA) between them has an error of less than 0.1%, thus proving the correctness of the calculation method of this invention.

[0082] Table 2 Stratigraphic parameters of the literature site

[0083] Example 5 To verify the practicality of this invention, a more complex 9-layer site model was selected, with the geological parameters shown in Table 3. The incident wave was represented by the east-west component of the EICentro wave acceleration time history. Figure 14The holding time was 30 seconds, the peak value was 0.1 g, and the incident angle was a critical angle of 30°. The earthquake time histories calculated according to the above steps are shown in Figures 15(a) and 15(b). Figure 15(a) shows the horizontal component, and Figure 15(b) shows the vertical component. Overall, the horizontal earthquake motion is more than twice that of the vertical earthquake motion, and the horizontal component |PGA| reaches 0.502 g, still exhibiting a predominantly horizontal motion characteristic.

[0084] Table 39: Stratigraphic parameters of the site

[0085] Example 6 By taking incident angles of 0.1°, 0.2°, ..., 89.9° respectively, and calculating according to the steps of Example 5, the |PGA| for different incident angles can be obtained. Figure 16 As shown in the figure, the relationship between the SV wave incident angle and |PGA| is complex for both the horizontal and vertical components of the earthquake motion. The main characteristic is that when the SV wave is incident at an angle greater than the critical angle, |PGA| fluctuates with increasing incident angle. As shown in the figure, the horizontal component |PGA| fluctuates between 30° and 40° with the incident angle, corresponding to variations between 0.445g and 0.554g; the vertical component |PGA| fluctuates between 30° and 50° with the incident angle, corresponding to variations between 0.228g and 0.335g. This indicates that when the SV wave is incident at an angle greater than the critical angle, the peak value of the resulting earthquake time history is more than twice the peak value of the incident wave, making the earthquake motion more intense than that produced by near-vertical incident waves.

Claims

1. A method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action, characterized in that: Specifically, the process includes the following steps: when seismic SV waves are incident on layered strata at a supercritical angle, the wave field in each stratum is calculated in the time domain using a recursive method, thereby obtaining the time history of the seismic waves incident on the ground.

2. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to claim 1, characterized in that: Specifically, the steps include the following: Step 1: Determine the number of soil layers at the site based on the geotechnical investigation results. n Density of strata ρ i Longitudinal wave velocity vp i transverse wave velocity vs i and stratum thickness h i ; Step 2, select the incident angle of the SV wave. β n , and sin β n <vs n / vp i ; Step 3: Calculate the P-wave incident angle for each stratum. α i and SV wave incident angle β i ; Step 4, calculate the P-wave incident angle α n sine α n Cosplay of Cosine α n ; Step 5: Calculate the reflection coefficients of P-waves and SV-waves at the ground. R 1,0 pp , R 1,0 ps , R 1,0 sp and R 1,0 ss ; Step 6, Calculate the coefficient matrix A i,i-1 ; Step 7, Calculate the coefficient matrix A i-1,i ; Step 8, calculate the P wave and SV wave from the first... i Layer incident to the first i Reflectance at -1 layer R i,i-1 pp , R i,i-1 ps , R i,i-1 sp , R i,i-1 ss and transmission coefficient T i,i-1 pp , T i,i-1 ps , T i,i-1 sp , T i,i-1 ss ; Step 9, calculate the P wave and SV wave from the first... i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i pp , R i-1,i ps , R i-1,i sp , R i-1,i ss and transmission coefficient T i-1,i pp , T i-1,i ps , T i-1,i sp , T i-1,i ss ; Step 10: Determine the incident wave function and duration. t w ; Step 11, determine the duration of the seismic response. t l ; Step 12, determine the time sampling interval ; Step 13, Calculate the number of source sampling points nt w Number of ground motion sampling points nt And P waves and SV waves in the first i Number of propagation time samples in layer time np i , ns i ; Step 14, for the incident wave function Discretize the data to form information about... k Timing function ; Step 15: Calculate the seismic waves within the site strata using a recursive method; Step 16: Calculate the seismic P-wave incident on the ground. u p ( k ) and seismic SV waves u s ( k ); Step 17: Obtain the horizontal component of the earthquake motion time history based on the calculation results of Step 16. u x ( k and vertical component u y ( k ).

3. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to claim 2, characterized in that: In step 3, the P-wave incident angle of each stratum is calculated using the following formula (1). α i and SV wave incident angle β i : （1） in, i = 1, 2, ..., n -1.

4. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to claim 3, characterized in that: In step 4, the P-wave incident angle is calculated according to equations (2) to (3). α n sine α n Cosplay of Cosine α n ; （2） （3） in, j It is the imaginary unit.

5. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to claim 4, characterized in that: In step 5, the reflection coefficients of P-waves and SV-waves at the ground are calculated according to equations (4) to (5). R 1,0 pp , R 1,0 ps , R 1,0 sp and R 1,0 ss : （4） （5） Since only reflected waves exist at the ground and no transmitted waves are transmitted, the transmission coefficient is set to... T 0,1 pp = T 0,1 ps = T 0,1 sp = T 0,1 ss =0.

6. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incident SV wave action according to claim 5, characterized in that: In step 6, the coefficient matrix is ​​calculated according to equation (6). A i,i-1 , i = 2, ..., n : （6） In step 7, the coefficient matrix is ​​calculated according to equation (7). A i-1,i : （7） Where i = 2, ..., n.

7. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incident SV wave action according to claim 6, characterized in that: In step 8, the P-wave and SV-wave are calculated from the first according to equations (8) to (9). i Layer incident to the first i Reflectance at -1 layer R i,i-1 pp , R i,i-1 ps , R i,i-1 sp , R i,i-1 ss and transmission coefficient T i,i-1 pp , T i,i-1 ps , T i,i-1 sp , T i,i-1 ss : （8） （9） In step 9, the P-wave and SV-wave are calculated from the first wave according to equations (10) to (11). i -1 layer incident to the first i Layer-time reflection coefficient R i-1,i pp , R i-1,i ps , R i-1,i sp , R i-1,i ss and transmission coefficient T i-1,i pp , T i-1,i ps , T i-1,i sp , T i-1,i ss : （10） （11） in, i = 2, ..., n .

8. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incident SV wave action according to claim 7, characterized in that: In step 11, the duration of the seismic motion is determined by the following formula (12). t l , t l The following formula should be satisfied: （12） In step 12, the time sampling interval is determined by the following formula (13). , The following formula should be satisfied: （13） in, f max yes The cutoff frequency.

9. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incident SV wave action according to claim 8, characterized in that: In step 13, the number of source sampling points is calculated according to equations (14) to (17). nt w Number of ground motion sampling points nt And P waves and SV waves in the first i Number of propagation time samples in layer time np i , ns i : （14） （15） （16） （17） Here, int is the integer arithmetic function. i = 1, ..., n ; In step 14, the incident wave function is calculated according to equation (18). Discretize the data to form information about... k Timing function : （18） in, k = 1, ..., nt .

10. The method for obtaining the time history of ground motion in a layered site under supercritical angle-incidence SV wave action according to claim 9, characterized in that: In step 15, the upward wave incident on the ground is calculated using the recursive method of the following formula (19). Pu 1 and Su 1: （19） in, （20） （21） Equation (19) is a set of wave functions Pu i ( k ), Su i ( k ), Pd i ( k ), Sd i ( k )about k The recursive operation, where, Pu i ( k ), Su i ( k ) is the incident point to the i The upward P-wave and upward SV-wave of each interface. Pd i ( k ), Sd i ( k ) is the incident point to the i The downward P-wave and downward SV-wave of each interface, and satisfying when k When < 0, Pu i ( k ) = Su i ( k ) = Pd i ( k ) = Sd i ( k = 0.

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