Design and Analysis of Nonlinear Numerical Algorithm for Seismic Response of Structures Based on HVSR Algorithm

: Earthquakes are one of the main factors that cause structural disasters in current buildings. Under the 13 action of earthquakes, adjacent building structures are generally in different vibration phases, and the structures 14 are likely to collide with each other, and collisions may cause serious damage to the structure. In order to prevent 15 damage to the designed buildings under certain earthquakes, this paper mainly focuses on the design and 16 analysis of numerical algorithms for seismic nonlinear structural dynamic responses based on the HVSR 17 algorithm. In the two - dimensional finite element analysis in this paper, the dam rockfill body adopts four - node 18 isoparametric elements, and the equivalent linear model related to dynamic shear strain is used to reflect the 19 dynamic nonlinear characteristics of the dam material. When the dynamic response is used as the constraint 20 condition, the dynamic response can be used to judge whether the design variables meet the requirements; when 21 the dynamic response is used as the optimization target, it is the optimization result of each step. Considering the 22 importance of the instantaneous amplitude of acceleration response in the correction of nonlinear structure model. 23 The acceleration response time series is divided into 15 time periods uniformly, and then the position 24 corresponding to the peak point of the instantaneous amplitude in each time period is selected as the selected 25 data point position. Apply the same seismic load to the bottom of the established nonlinear model, extract the 26 dynamic response data of the top layer of the structure, and then extract the instantaneous amplitude of the main 27 component of the structural dynamic response through the time - varying filter and Hilbert transform based on 28 discrete analytical mode decomposition and the corresponding the instantaneous frequency. Under the action of 29 these four ground motions, the collision force in the range of 0~50kN accounts for more than 87% of the total 30 number of collisions. The results show that the HVSR algorithm can obtain the instantaneous characteristic 31 parameters of the dynamic response of the nonlinear structure and realize the correction of the model.


Introduction
Since the development of structural seismic research for many years, there have been mature theories and calculation methods.Seismic design is basically based on the design concept of bearing capacity, that is, strengthening the structure by strengthening the foundation, main load-bearing members, etc., to ensure that under any seismic conditions the structure remains intact.The energy at the seismic source is transmitted to the surface through the earth's medium in the form of waves, which causes the ground to move (earthquake), which causes the building to vibrate violently, and the structure is damaged or even collapsed due to excessive stress or deformation.With the development of the national economy and the rapid development of my country's economy, the area, number, and scale of structures have continued to increase, and the shapes of structures have become more complex, and the structure has also continued to grow larger and higher, which will inevitably lead to a stronger structural system.The micro-vibration affects the surrounding free wave field with a greater range and intensity.
Comprehensive analysis of PGA, velocity peaks, displacement peaks and other characteristics and laws of different parts of the slope, as well as characteristics such as duration, frequency spectrum, etc., to study the response characteristics of the same altitude and different depths, the influence of ground motion parameters and the direction of propagation, in order to study the effect of slopes on earthquakes The failure mechanism under the following conditions provides reasonable and evidence-based data support, and further provides references for slope support design, tunnel seismic design and related design specifications, which can enrich the theory of slope deformation and failure mechanism under earthquake action, and can also be used for rock and soil.Kinetics provides new technology, which has significant theoretical and practical significance, and has great economic benefits.
In the seismic design of the structure, it is possible to allow the structure to appear certain damage under the action of a strong earthquake (major earthquake) to ensure that the overall building does not collapse.Zhang at al. (2019) believes that in seismic exploration, effectively attenuating seismic noise and preserving seismic signals are key issues in seismic signal processing.The seismic data collected in complex environments has a low signal-to-noise ratio and non-stationary random noise, which makes it difficult to recover accurately.In order to enhance the seismic signal and filter out non-stationary random noise, he used structural adaptive diffusion coefficient to propose a structural adaptive nonlinear complex diffusion method (ANCD) based on regularized complex shock diffusion (CSD).He uses the imaginary value generated by the complex diffusion process as a directional structure index, and guides the diffusion coefficient through the synergy of the spatial variation structure factor related to the eigenvalues and eigenvectors of the structure tensor.In this way, the structural adaptive diffusion term of ANCD allows the diffusion process along the seismic structure, and the resulting imaginary part enables the shock term to enhance seismic signals with steep dips and rapidly changing slopes.He used synthetic seismic data and field seismic data to verify the method.Although ANCD has achieved a good compromise between suppressing complex random noise and preserving seismic structures, his experiments are not detailed.Xiang at al. (2018) studied the effectiveness of the linear response spectrum analysis method of the nonlinear cable net system.He used energy-based parameters to describe the stiffness of the structure, and based on this, he established an equivalent single-degree-of-freedom system with different irregularity cable net structures.He used the pseudo-energy equivalent method to transform the nonlinear equivalent single-degree-of-freedom system into a linear system whose seismic response is a reasonable approximation of the precise nonlinear result.In numerical calculations, he used 80 seismic records to stimulate 4 cable-net models.
Considering 19 earthquakes of different intensities, he conducted 6,080 nonlinear response history analyses and compared them with the results of linear response spectrum analysis.Although his research has small errors, it lacks innovation.Shamsabadi & Kapuskar (2010) studied the mechanical performance of highway skew bridges under the action of high-speed impulse earthquakes.Bridges with longitudinally inclined abutments exhibit strong coupling under lateral loads.The interaction between the foundation of the skew abutment and the backfill has a great influence on the dynamic response of the bridge.Under the action of seismic load, the bridge structure may remain in the linear range, and the local nonlinear behavior at the abutment-embankment interface will cause significant nonlinear bridge structure response.He used nonlinear abutment springs to establish a skew bridge model considering the soil-abutment-structure interaction.He built an integral three-dimensional non-linear finite element model of a monolithic abutment in Rio de Janeiro, California, with an inclination and seismic instrumentation of the painter street overpass.The model adopts the method of nonlinear foundation-soil interaction.According to the soil characteristics obtained by the geotechnical test, the constitutive hardened soil material is used to conduct continuous finite element analysis of the soil backfill of the bridge abutment considering the inclination of the back wall.He compared the bridge response from seismic records with the structural response data.Although the computer model better reflects the overall seismic response of the skew bridge, his research is not accurate enough.Azizi at al. (2019) believes that in most cases, the computational control of high-rise buildings is great, and the control response of the structure is not significantly reduced.In order to overcome these shortcomings, it is necessary to optimize the parameters of the fuzzy system.He studied the optimization problem of a fuzzy controller applied to seismic excitation of nonlinear steel structures.
He proposed an improved whale optimization algorithm and used it as an optimization technique for fuzzy controller parameter tuning.He compared the performance of the improved algorithm with the standard whale optimization algorithm and 8 different meta-heuristic algorithms.Although the improved method can provide competitive results, it lacks specific experimental data.This paper proposes a new site classification standard based on the HVSR spectral shape, and illustrates that the new standard effectively solves the defect that the HVSR method cannot be applied when the HVSR curve has multiple peaks or no peaks, and improves the HVSR method.The classification accuracy rate.The proposed ground motion intensity index can directly reflect the influence of ground motion on the nonlinear response of the structure under rare earthquakes.
Combining the actual seismic damage difference data and the site borehole difference data, the soil equivalent linear model is established, and the synthesized bedrock ground motion time history is input to discuss the site factors that cause the serious seismic damage difference.Finally, combining the strong motion observation data, the difference in earthquake damage, and the site response, it provides reasonable suggestions for the reconstruction of the disaster area.

2.1HVSR Algorithm
Under the action of HVSR, the vibration energy of the main structure can also be transferred from low frequency to high frequency, so that part of the energy of the structural vibration is dissipated by the high-order damping of the structure.Therefore, HVSR can increase the dissipation rate of structural vibration energy under the action of self-damping, thereby effectively reducing the time of structural vibration.In addition, because HVSR contains high-strength viscous damping force, part of the vibration energy is dissipated by HVSR itself.HVSR's strong nonlinear stiffness and high-strength viscous damping can ensure that the vibration energy transferred from the structure will not return to the structure again, so the HVSR energy transfer is unidirectional.In the case of high frequency approximation, the propagation time of the surface wave along the path between the source A and the receiver B can be discretely expressed as (Choinière, Paultre & Léger, 2019):

 
(1) Among them, p S is the group slowness of the AB l  segment along the surface wave propagation path (the reciprocal of the group velocity), and P is the total number of segments of the ray path (d 'Avila & Lopez-Caballero, 2018).
Assuming that the noise field is diffuse, then HVSR can be written as: (3) Among them, E1 and E2 represent the energy density in the horizontal direction, and E3 represents the energy density in the vertical direction (Yi-Gang, et al., 2018).

Seismic Nonlinear Structure
The classification of seismic waves is shown in Figure 1.According to its changes in space, it can be divided into two categories, one is body waves, and the other is surface waves.Body waves are divided into longitudinal waves and transverse waves.Longitudinal wave is also called "density wave", also called P wave.In actual engineering, under the influence of factors such as different earthquake intensity, different foundation types, and different buried depths, the form of separation between the structural foundation and the foundation is different when encountering an earthquake.Some structural foundations only lift off, and some structural foundations only slip, but most of the structures are two non-linear phenomena co-occurring, such as the column foundations in ancient Chinese wooden structures.Slip is a nonlinear phenomenon in the case where the superstructure and the foundation are separable.At present, the research in the analysis of the foundation lift-off seismic response and the rocking isolation is not deep enough, but it is undeniable that the slip does cause the seismic response of the structure (Bai, et al., 2019).In an isotropic, uniform, undamped elastic body, the movement of a particle must satisfy the stress-strain relationship of the medium, the continuity condition and the second law of Newton's motion.From the theory of small deformation elastic mechanics, the basic equation of motion can be derived as (Shen, et al., 2019): Among them, u, v, w are the displacements in the three directions of the rectangular coordinate system x, y, and z; θ is the mass point strain, and its value is is the Laplacian operator; t is the time (Zhao, et al., 2020).
In theory, the wave can be described by the wave equation, which characterizes the change law of the strain of the medium element at any position in space with time t, and the time and space law of disturbance propagation, such as in an ideal uniform isotropic solid elastic medium.The wave equation of elastic wave is (Sachdeva, Chakraborty & Ray-Chaudhuri, 2018): In the formula, F is the force vector; u is the displacement vector of the particle under the action of the force F; θ is the volume change coefficient; ρ is the density of the medium; λ and μ are constants that measure the ability of the medium to resist deformation, called the elastic modulus ; Μ is the shear modulus (Dadkhah, Kamgar & Heidarzadeh, 2021).
When synthesizing near-site vibrations, the large seismogenic fault is divided into small sub-faults, and each sub-fault can be regarded as a point source.Considering the time delay of fault rupture and dislocation rise time, superimpose the earthquake of each sub-source at the target field point.By moving, you can get the ground motion of the entire seismogenic fault at the target field point.The basic manifestation of the strong motion time history is as follows (Konstandakopoulou, et al., 2020): For general weakly nonlinear systems, is a slowly varying function of time, which reflects the changes in the frequency of the structure itself under the excitation of external loads.Therefore, by selecting the appropriate cut-off frequency and using a low-pass filter based on analytical mode decomposition to filter out the fast-changing component in of the structural system can be obtained (Zheng, et al. 2019).

Dynamic Response Value
For stable long-term periodic loads, such as machine vibration, in addition to the initial vibration, if the saturated soil layer of the foundation has better drainage boundary conditions, any additional pore water caused by machine vibration will have enough time to dissipate.At this time, the effect of additional pore water pressure caused by vibration can no longer be considered.Conversely, for sudden impact loads, such as explosions and pilings, due to the short action time, the saturated soils affected at this time can generally be considered as undrained conditions (Shan & Lai, 2019).For periodic loads with a short acting time, such as earthquake loads, general saturated soils (including sand, gravel, fine sand and other soils that are not too permeable) can still be considered as undrained conditions.If necessary, the process of dissipation and diffusion of vibration pore water pressure caused by earthquake and its influence can also be considered according to its boundary conditions.
Assuming that the spatial position, deformation, motion speed, and stress tensor of the medium particle are represented by the vectors i  , the strain rate tensor and rotation tensor can be expressed as (Chen, et al. 2019): ( ) In the case of strain, the mesh is deformed after the node position is updated, and the following corrections are taken: In each step, the initial conditions of the step, such as displacement and speed, and external excitation history, are used to calculate the response of the system.Therefore, each step is analyzed independently, and there will be no superimposed effects.The step-by-step method can easily analyze the nonlinear dynamic response of the system, which is to convert the so-called nonlinear analysis into a dynamic linear analysis [22].For the seismic record CHY036-W acceleration peak level does not exceed 3.0g, the maximum out-of-plane displacement response of the full bridge always occurs at the node where the wind brace of the vault and the arch rib meet.Once the peak acceleration level reaches 3.3g (the incremental amplitude analyzed by IDA is 0.3g), the maximum out-of-plane displacement response of the full bridge will instead occur at the node where the stroke brace and the arch rib are connected, and in the subsequent IDA loading process, The vault is no longer the location where the maximum out-of-plane displacement reaction of the full bridge occurs, but the structure can continue to bear the load without collapsing, which means that the local stiffness of the structure has changed significantly and will cause local dynamic instability [23].

Experimental Equipment
The main test equipment in this paper is a small shaking table, the table size is 0.8m×0.6m, the load is 200kg, the one-way input, the input load method is loading according to the displacement curve, and the pre-input maximum acceleration amplitude is 0.5g.The experimental model is a linear elastic model.In order to prevent the columns in the structure from yielding (to ensure the linear elasticity of the structure), all steel materials are made of high-strength steel.Then the experimental model is placed on the shaking table, and the direction of the ground motion is along the weak axis of the structure [24].

Selection of Seismic Waves
In this paper, 60 seismic waves are selected from the US Seismic Wave Database and the Pacific Earthquake Engineering Center (PEER) Strong Earthquake Database, divided into 2 groups of 30 seismic waves.Among them, the first set of seismic waves are records from different stations in the same earthquake to compare the analysis results under the same focal mechanism and magnitude; the second set of waves uses similar sites as control indicators, that is, based on the site category where the building is located, considering the epicenter distance and intensity at the same time, select the station records with the same or similar site category as input [25].

Earthquake Response
In the two-dimensional finite element analysis in this paper, the dam rockfill body adopts four-node isoparametric elements, and the equivalent linear model related to dynamic shear strain is used to reflect the dynamic nonlinear characteristics of the dam material.The concrete slab of the dam body adopts beam elements and is processed according to the linear elastic model.A Goodman thicknessless contact surface element is set between the concrete panel and the rockfill to simulate the interaction between materials with different properties [26].

Calculation of Dynamic Response
In the process of structural dynamic response optimization, it is necessary to calculate the dynamic response of the structure under given conditions and the sensitivity of the dynamic response to design variables.When the dynamic response is used as the constraint condition, the dynamic response can be used to judge whether the design variables meet the requirements; when the dynamic response is used as the optimization target, it is the optimization result of each step.The calculation of dynamic response sensitivity can provide the direction of structural design variable modification for the next step, which is used to obtain new design variables.Then judge the obtained solution, if it is not the optimal solution, proceed to the next step of optimization, if it is, then complete the optimization; this reciprocating cycle until the optimal solution is obtained or the constraints are reached.

Numerical Model
Assuming that the nonlinear model is a measured structure, then the nonlinear model of the measured

Linear Model Correction
In this paper, the linear parameters of the structure are modified by using the response surface method, and the main modified parameter is the elastic modulus of each unit of the structure.Through correction, the corrected linear model and the actual structure have the same or similar frequencies.
The basic principle of the finite element model correction based on the response surface method is: select appropriate structural parameters and structural responses, calculate the response of the sample points in the design space of the parameters, and on this basis, fit between the structural response and the structural parameters the response surface model is the response surface model; then, taking the measured response as the target value, the response surface model should replace the finite element model for optimization iteration.

Earthquake Nonlinear Response Analysis
Table 1 shows the correction parameters of the nonlinear model under seismic load.Under the influence of 5% Gaussian white noise, the nonlinear structural model correction method proposed in this paper can still accurately correct the nonlinear model; under the two working conditions, the three defined error indicators are all less than 5%.When a nonlinear structure is excited by seismic loads or other wide bandwidth loads, the instantaneous frequency of the main component of the structural dynamic response extracted by the Hilbert transform reflects the frequency change of the nonlinear structure during the vibration process.Using the extracted instantaneous frequency and instantaneous amplitude to construct the objective function, the nonlinear structure model can be corrected accurately.However, when the external excitation of the structure is a harmonic load or other narrow bandwidth load, the instantaneous frequency of the main component of the structural dynamic extracted by the Hilbert transform is often close to the external excitation frequency, and the nonlinear characteristics are mainly hidden.Included in the instantaneous amplitude of the vibration response.The maximum relative displacement between layers is shown in Figure 2. It can be seen from the figure that under the action of frequent earthquakes, the maximum interlayer relative displacement calculated according to the time history analysis method is slightly smaller than the maximum interlayer relative displacement calculated by the bottom shear method, but it does not exceed the elastic interlayer relative displacement.Displacement limit; under rare earthquakes, the relative displacement between layers obtained by the elastoplastic time history analysis and the relative displacement between the elastoplastic layers obtained by the elastic displacement of rare earthquakes as stipulated in the code did not exceed the relative displacement between layers of rare earthquakes.
The limit value is more consistent in judging the weak layer, all of which are 4-6 layers.However, in the elasto-plastic displacement of the bottom layer, the relative displacement between the elasto-plastic layers, which is amplified by the rare earthquake elastic displacement specified in the code, greatly exceeds the relative displacement between the layers obtained by the elasto-plastic time history analysis.This is because the code does not consider elasto-plasticity.It has the characteristics of uneven distribution between layers.It is caused by multiplying elastic displacement by a unified amplification factor.It cannot accurately reflect the maximum relative displacement of the structure under the action of rare earthquakes.However, this treatment The method can make the design have a certain degree of safety, which is applicable in the design.Table 2 shows the peak acceleration of the measuring points of the working conditions CE1, CE2 and CE3.It can be seen that the difference between the numerical simulation results and the shaking table model test results increases with the increase of the input ground motion peak acceleration.The simulation effect of the working condition CE1 is the best, and the simulation effect of the working condition CE2 is slightly worse.In the case of CE3, the difference between the numerical simulation value and the experimental record value is relatively large.This is because during the test of loading condition CE3, the station structure has floated up, and the contact surface between the soil and the structure has separated; in the modeling, the contact surface is used to simulate the contact surface for the dynamic contact between the soil and the station.There is a difference between the mechanical transfer characteristics between the two.The collision force produced by the four earthquakes is divided into 5 ranges, as shown in Table 3.
As can be seen from the table, in the range of 0-50kN, the number of collisions caused by ELC180 ground motion is 210, accounting for 91.7% of the total number of times, and the number of collisions caused by TAF021 ground motion is 70 times, accounting for 89.7% of the total number of times.,The number of collisions caused by KAK000 ground motion is 63 times, accounting for 87.4% of the total times, and the number of collisions caused by CAP000 ground motions is 76 times, accounting for 87.4% of the total times.Under the action of these four ground motions, the collision force in the range of 0~50kN accounts for more than 87% of the total number of collisions.The number of collisions occurring in each range decreases as the range of collision force continues to increase, showing a monotonous decreasing law.The number of collisions greater than 50kN is less than 13% of the total number, and as the collision force increases, the occurrence The number of collisions does not fluctuate much, and is basically maintained within a small range.The relationship between collision force and time is shown in Figure 3.It can be seen from the figure that when the input ground motions are different, the time of the first collision of the structure is different.When three ground motions of ELC180, TAF021, and KAK000 are input to the structure, the time of the maximum impact force pulse appears later than the time of the peak acceleration of the ground motion, and the degree of lag is different.For the CAP000 ground motion, the peak impact force appears earlier than the peak acceleration.

Non-Linear Model Analysis Results
The comparison result of the collision response to the peak displacement is shown in Figure 4.The four ground motions all caused the structure to collide, and the collision suppressed the positive peak displacement response of node 1512.Compared with the non-collision, the peak displacement was reduced by 27.273%, 33.675%, 27.727%, and 37.248%, respectively.The reduction ratio varies with different ground motion records.The impact on the negative peak displacement response of node 1512 is relatively small.Under the action of the TAF021, KAK000, and CAP000 earthquakes, the negative peak displacements of the 1512 node are all enlarged compared with the non-collision time, which is enlarged by 11.321%, 18.030%, and 17.2%, respectively.The peak displacement is suppressed and reduced by 18.856%.displacement is significantly lower than the above-mentioned working conditions.Through the cross-sectional distribution of the displacement peak, we can also see that the displacement peaks at the corresponding positions on both sides of the tunnel axis are asymmetry.When the fault inclination is 0° and 90°, the peak response of the displacement of the corresponding position of the structure is not much different, and there is a good symmetry trend; while at 30° and 60°, the peak on the right is obviously larger than the peak on the left.Therefore, in the process of tunnel anti-vibration design, attention should be paid to the "eccentric load" effect due to the existence of the broken zone.The HVSR experience curve of different site categories is shown in Figure 6.Compared with the spectral ratio curve matching method, except for SCII venues, the classification success rate of other types of venues is slightly improved.This method has the same shortcomings as the spectral ratio curve matching method, so that the success rate has not been greatly improved.This method is essentially a comparison of the shape similarity between the target HVSR curve and the standard HVSR curve of the four types of venues, and does not consider the magnitude of the spectral ratio of the two.The stress-strain relationship curve of the tensile experiment drops sharply after reaching the peak value because the experimental data is measured in a local area.If only the stress-strain relationship is considered in a local area, the absolute value of the negative tangent modulus after reaching the stress peak will decrease.Since the stress curve assumed in this model is constructed based on the difference between two exponential functions, it cannot achieve such a large negative stiffness in the experimental results under tensile load conditions.The results of the trend analysis of the longitudinal response range of the fault following the track are shown in Table 4. From the table, it can be known from the range of various factors that the sensitivity of the three factors to the longitudinal influence range of the fault tunnel is in the order of fault width> fault dip angle> surrounding rock level, that is to say, the width of the fault is within these three factors.The medium has the greatest impact on the longitudinal response range of the fault tunnel.As the width of the fault increases, the longitudinal response range of the fault tunnel also increases.As the width of the fault increases, the scope of its influence is increased.With the gradual increase of the dip angle of the fault, the longitudinal response range of the fault tunnel gradually decreases.This is because with the gradual increase of the fault angle, the part of the front half of the fault is reduced, which leads to the reduction of its influence range.With the gradual increase in the grade of the surrounding rock and the weakening of the physical and mechanical parameters of the surrounding rock, the longitudinal response range of the fault tunnel has increased, but its influence is smaller than the width and dip of the fault.

Conclusions
In recent years, regional site classification methods based on geological and geomorphological data, and site classification methods based on strong motion record response spectrum shape (RSS) and spectral ratio curve (HVSR) for specific site categories have been widely used internationally.In the shaking table test structure of the seven-story shear wall under the excitation of the seismic load, and the shaking table test structure of the transmission structure under the excitation of the harmonic load, the model of the structure is realized by optimizing the parameters of the nonlinear model the modified, modified nonlinear model can further predict the dynamic response of the structure under unknown excitation.Since the maximum interlayer displacement angle can more comprehensively show the overall nonlinear response of the structure, after comparing and analyzing the time consistency and linear correlation, it is considered that the basic frequency of the structure is mainly used, and the high frequency of the structure is given appropriate weight to construct the characteristic peak value.Second, it can more intuitively characterize the impact of ground motion on the nonlinear response of frame structure.As a ground motion intensity index, it is a better choice than the characteristic peak value.
initial reference model or the model obtained after each iteration, ik v is the bilinear interpolation coefficient along the ray path related to the i-th propagation time, group velocity of the k-th grid point on the two-dimensional plane and its perturbation

a
is the ij-th sub-source (point source) ground motion, l N is the number of sub-faults divided along the fault trend, w N is the number of sub-faults divided along the fault trend, and ij t  is the time delay of earthquake propagation (Cancellara & De Angelis, 2019).When analyzing discrete signals, it is assumed that the sampling frequency of discrete signals conforms to the Nyquist-Shannon sampling principle.Therefore, at any point in time, the maximum frequency component of the signal is less than 1/2 of the sampling frequency.The discrete Hill transform of modal response can be written as(Degli, et al., 2019):

]
structure is established, and the nonlinear model of the measured structure is corrected by the structural nonlinear model correction method proposed in this paper.Taking into account the slow-varying characteristics of the instantaneous frequency and instantaneous amplitude of the main component of the nonlinear structural vibration response.Therefore, the representative instantaneous amplitude and corresponding instantaneous frequency value in each principal component can be selected to construct the optimized objective function.Considering the importance of the instantaneous amplitude of acceleration response in the correction of nonlinear structure model.The acceleration response time series is divided into 15 time periods uniformly, and then the position corresponding to the peak point of the instantaneous amplitude in each time period is selected as the selected data point position.Apply the same seismic load to the bottom of the established nonlinear model, extract the dynamic response data of the top layer of the structure, and then extract the instantaneous amplitude of the main component of the structural dynamic response through the time-varying filter and Hilbert transform based on discrete analytical mode decomposition and the corresponding the instantaneous frequency.

Figure 2 .
Figure 2. Maximum relative displacement between layers

Figure 4 .
Figure 4. Comparison results of collision response to peak displacementThe absolute horizontal displacement peak distribution curve of each measuring point is shown in Figure5.When the angle of the fracture zone changes from 0° to 90°, the displacement response of the same part of the tunnel is different.In general, under the condition of 0° (horizontal), the peak displacement of the arch waist and dome of the tunnel is the largest among all working conditions; at 30°, the fault fracture zone passes through the left foot and right shoulder of the tunnel, so the peak displacement is different.The situation is the largest, and the other positions are also at higher magnitudes; at 60°, the fault-affected area tends to invert, at this time the displacement of the inverted arch has a peak, and the remaining displacements are all less than 30°; at 90°, the The peak

Figure 5 .
Figure 5. Absolute horizontal displacement peak distribution curve of each measuring point

Figure 6 .
Figure 6.HVSR experience curve for different venue categories

Funding:
Not applicable Consent to Publish: All authors read and approved the final manuscript Authors Contributions: All authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by [Shuai Wang], [Chao Wang] and [Zongbao Zhang].The first draft of the manuscript was written by [Shuai Wang] and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.Youth Nursery Project of the Liaoning Provincial Education Department, (Research on the construction of wave velocity field of which the ray path and the travel time is independent and the method of seismic source location in it.(LJ2020QNL010)

Table 1 .
Correction parameters of nonlinear model under seismic load

Table 2 .
Peak acceleration of measuring points under working conditions CE1, CE2 and CE3

Table 3 .
Collision force distribution

Table 4 .
Trend analysis results of the longitudinal response range of the fault following the track