Effect of Soil Anisotropy on Ground Motion Characteristics

: Soil transverse isotropy results in different stiffness characteristics in horizontal and vertical directions. However, the effect is usually neglected in seismic motion analysis. In this study, an equivalent linear anisotropic soil model was established based on the ﬁnite element method, and we investigated the impact of anisotropic parameters on ground motion at the site under various seismic wave inputs. It was found that the anisotropic parameters have a more signiﬁcant effect on seismic waves, with the dominant frequency being closer to the fundamental frequency of the site. As an example, the soil dynamic parameters in Shanghai Yangshan Port were calibrated by a series of bending elements, resonance columns, and cyclic triaxial tests. The inﬂuences of anisotropy on the peak ground acceleration (PGA) and response spectrum were studied for Yangshan Port. Additionally, the standard design response spectra considering the soil anisotropy were provided. A comparison reveals that the existing isotropic design response spectrum may lead to dangerous seismic design for the structures at Yangshan port.


Introduction
To improve the seismic design of structures, it is necessary to conduct a site-specific seismic motion analysis to understand the characteristics of ground motion at the site.Earthquakes have resulted in significant economic losses and casualties.Earthquakes have the potential to inflict significant destruction upon various types of structures, encompassing inland buildings [1], hydraulic structures [2], and subterranean constructions [3].One of the important causes of the structural damage caused by earthquakes is soil amplification.This is illustrated in damage assessments of buildings in the countries of Turkey [4][5][6], Greece [7], Iran [8], Mexico [9], Korea [10], Pakistan [11], and Nepal [12].Numerous scholars have extensively researched the seismic resilience of engineering structures [13][14][15][16].The analysis of structures for seismic resistance utilizes a range of methodologies, encompassing response spectrum analysis, the base shear method (quasi-static method), and time history analysis.The response spectrum method is extensively utilized as the predominant seismic analysis approach in engineering practice.The response spectrum is obtained by calibrating ground motion characteristics at the specific site.Consequently, numerous scholars have researched the seismic design of structures based on the seismic motion characteristics at the site [16][17][18].
Soil dynamic constitutive models can be categorized into total stress models and effective stress models, considering the aspects of stress transmission and inter-particle contact [19].The effective stress model offers superior capability in addressing the issue of seismic liquefaction [20][21][22].Regarding the issue of seismic ground motion response, researchers commonly employ the total stress model for analysis.The total stress models include the elastic-plastic model [23][24][25][26][27][28][29][30], nonlinear model [22,31,32], and equivalent linear model [33].Traditional elastoplastic models such as the Mohr-Coulomb model [34] and the Drucker-Prager model [35] can be integrated with the boundary interface theory [25][26][27], kinematic hardening theory [23,24], and nested yield surface theory [29,30] to account for the effects of cyclic loading.Although the elastoplastic model theory effectively describes the hysteresis characteristics and nonlinearity of soils, its computational complexity and significant workload make it inconvenient for engineering applications.Researchers employ nonlinear models [31,32] based on the Masing criteria to depict the nonlinearity and hysteresis characteristics of soils.While these nonlinear models simplify the workload associated with elastoplastic models, they still entail a certain level of complexity.The linear model offers the advantages of a low workload and low complexity, allowing for the consideration of soil nonlinearity and hysteresis characteristics through the application of an equivalent concept.Hence, the most widely utilized approach at present is the equivalent linear model, which has evolved from linear models.Schnabel [33] initially pioneered the development of the frequency domain equivalent linear model within the SHAKE program.Idriss et al. [36] modified the expression of damping in the SHAKE program and developed the SHAKE91 program.With the development of commercial software, the SHAKE and SHAKE91 programs have been integrated into software such as EERA 1998 and DEEPSOIL v7.0.33.The SAHKE program demonstrates limited proficiency in effectively addressing anisotropic challenges.
Soil anisotropy has aroused great interest in recent years.Zhang et al. [37,38] and Teng [39] investigated the impact of soil anisotropy on excavation-induced effects in excavations.Wei et al. [40], Soe et al. [41], and Zhang et al. [42] investigated the impact of soil anisotropy on tunnel design and construction.Peric et al. [43] and Ai et al. [44,45] investigated the impact of anisotropy on the design of pile foundations.It has been found that there is anisotropy in the soil under small strain conditions.Bentil [46] has delved into the anisotropy of the small-strain shear modulus by conducting bending element experiments.The consideration of soil anisotropy in the seismic response analysis of soil layers has been scarce among scholars, primarily due to the complexities involved in studying anisotropy, seismic loads, and their associated intricacies.Considering the lack of relevant research and to maintain the coherence and rigor of scientific inquiry, this study investigates the impact of anisotropy on the seismic response of the site using a finite element method with an anisotropic time-domain equivalent linear model.

Linear Viscoelastic Model
The viscoelastic Kelvin model (a spring connected in parallel with a sticky pot) is used to reflect the hysteresis of the soil under cyclic loading.The stress-strain relationship is described by Equation (1): where G is the shear modulus; τ is the shear stress; γ is the shear strain; and η G is the shear viscosity coefficient, as given in Equation (2): where D is the damping ratio and ω is the circular frequency.

Modulus and Damping Models
The key to the effective linearization method is to determine the relationship between the shear modulus ratio and the damping ratio with the shear strain.Hyperbolic models are widely used to describe nonlinear soil behavior under cyclic loading, such as the Pyke model and the Stokoe model [47].In this study, the improved Stokoe model is selected to fit the relationship between the shear modulus ratio and shear strain, as given in Equation (3): (3) where G norm is the normalized shear modulus, r r is the reference strain, and α is the fitting parameter; the definition of r r is different from the Harden-Drnevich model (r r = τ max /G max ).Zhang et al.'s [48] formula is adopted to describe the relationship between D and G norm as follows: where K 1 , K 2 , K 3 are the model fitting parameters.

Effects of Anisotropy
The notation for an anisotropic material used herein is the y-axis (the vertical direction) represents the direction of the anisotropy, and the x, z-plane is the plane of isotropy.The stress-strain increment equation for an anisotropic material can be written as follows (5) [49].
where E v and E h are Young's moduli in the vertical and horizontal directions, respectively; V hh and V vh are Poisson's ratios for horizontal strains from a horizontal and vertical strain, respectively; V hv is Poisson's ratio for vertical strains from a horizontal strain; G vh and G hv are the shear moduli in the vertical plane; and G hh is the shear modulus in the horizontal plane.
The anisotropy ratios AR G and AR E for the shear modulus and Young's modulus are, respectively, defined as: ) Under the undrained condition, these Poisson's ratios [50] need to satisfy the additional relationships (v vh = 0.5, v hh + v hv = 1).Therefore, one can further obtain the following equations [50]:

Simulation of Time-Domain Equivalent Linear Model for Anisotropic Soil Layers
This study takes the actual recorded seismic waves as input conditions, degenerates the model to isotropy, and compares it with EERA for verification.This study focuses on exploring the ground motion response of anisotropic sites with different sedimentary characteristics (different anisotropic parameters).

Input Ground Motion and Finite Element Model
In Eurocode 8 [51], sites are classified according to the average shear wave velocity in the upper 30 m thick soil profile.Therefore, the soil layer thickness is selected as 30 m. Also, three seismic waves are chosen: the Ei-Centro Wave, Shanghai Wave, and Kobe Wave.The amplitudes of these seismic waves were adjusted to have a peak acceleration of 0.1 g. Figure 1 illustrates the three seismic waves.As shown in Figure 2, a two-dimensional plane strain soil finite element model is established based on ABAQUS 6.14.The type of the finite element is CPE4, and the estimated size of the mesh is 1 m × 1 m.The mesh size meets the requirement of less than 1/10 wavelength.The boundary adopts the infinite element boundary, and seismic waves are input to the base.Therefore, the wave vibrates in the x direction and propagates in the y direction.

Simulation of Time-Domain Equivalent Linear Model for Anisotropic Soil Layers
This study takes the actual recorded seismic waves as input conditions, degenerates the model to isotropy, and compares it with EERA for verification.This study focuses on exploring the ground motion response of anisotropic sites with different sedimentary characteristics (different anisotropic parameters).

Input Ground Motion and Finite Element Model
In Eurocode 8 [51], sites are classified according to the average shear wave velocity in the upper 30 m thick soil profile.Therefore, the soil layer thickness is selected as 30 m. Also, three seismic waves are chosen: the Ei-Centro Wave, Shanghai Wave, and Kobe Wave.The amplitudes of these seismic waves were adjusted to have a peak acceleration of 0.1 g. Figure 1 illustrates the three seismic waves.As shown in Figure 2, a twodimensional plane strain soil finite element model is established based on ABAQUS 6.14.The type of the finite element is CPE4, and the estimated size of the mesh is 1 m × 1 m.The mesh size meets the requirement of less than 1/10 wavelength.The boundary adopts the infinite element boundary, and seismic waves are input to the base.Therefore, the wave vibrates in the x direction and propagates in the y direction.

Isotropic Soil Layer
This subsection is used to verify the validity of the established finite element method with an anisotropic time-domain equivalent linear model in the isotropic case studies.The shear modulus model chosen here is the Seed-Idriss mode [52], and the damping curve model used is the Idriss mode [36].Table 1 presents the model parameters.The properties of the soil layers are shown in Table 2.

Isotropic Soil Layer
This subsection is used to verify the validity of the established finite element method with an anisotropic time-domain equivalent linear model in the isotropic case studies.The shear modulus model chosen here is the Seed-Idriss mode [52], and the damping curve model used is the Idriss mode [36].Table 1 presents the model parameters.The properties of the soil layers are shown in Table 2. Figure 3 illustrates the comparison of the peak ground acceleration (PGA) with depth, considering three seismic records, namely the Ei-Centro Wave, Shanghai Wave, and Kobe Wave, along with four soil conditions.They exhibit good agreements with the results obtained from the classical seismic analysis code EERA.
ing curve model used is the Idriss mode [36].Table 1 presents the model parameters.The properties of the soil layers are shown in Table 2.

The Influence of Anisotropic Parameters
It should be noted that this study assumes a consistent relationship between  ℎ −  and  ℎℎ − .Diffserent anisotropic parameter (  = 1.00, 1.30, 1.60;   = 1.05, 1.30, 1.55, 1.80, 2.05) were selected for a 30-meter-thick layer of soft soil with a shear wave velocity of 150 m/s.Detailed parameters are listed in Table 3.

The Influence of Anisotropic Parameters
It should be noted that this study assumes a consistent relationship between G vh − γ and G hh − γ.Diffserent anisotropic parameter (AR E = 1.00, 1.30, 1.60; AR G = 1.05, 1.30, 1.55, 1.80, 2.05) were selected for a 30-meter-thick layer of soft soil with a shear wave velocity of 150 m/s.Detailed parameters are listed in Table 3. PGA aniso /PGA iso is the ratio between the peak ground acceleration obtained at an anisotropic site and that obtained at an isotropic site.Figure 4 shows that, with AR E increasing, PGA aniso /PGA iso gradually decreases.Conversely, as AR G increases, PGA has an obvious increase.Especially when inputting the Kobe wave, the PGA of anisotropic conditions exceeds that of isotropic conditions up to 14% with AR G increasing.Consequently, it is considered that soil anisotropy is essential in analyzing site conditions for seismic response.To assess the impact of anisotropy on the seismic response, the dominant frequency of the seismic waves and the fundamental frequency of the site are further discussed as listed in Table 4, where the fundamental frequency of the site is calculated using Equation (12) [53]: where the symbol "  " represents the fundamental frequency of the site; "  " refers to the site's characteristic period; and the variable "  " denotes the thickness of individual To assess the impact of anisotropy on the seismic response, the dominant frequency of the seismic waves and the fundamental frequency of the site are further discussed as listed in Table 4, where the fundamental frequency of the site is calculated using Equation ( 12) [53]: where the symbol " f g " represents the fundamental frequency of the site; "T g " refers to the site's characteristic period; and the variable "H i " denotes the thickness of individual soil layers, whereas "V Si " represents the shear wave velocity specific to each respective soil layer.As shown in Figure 4, the Ei-Centro and Kobe waves have significant influences on the PGA aniso /PGA iso , while the Shanghai wave has a relatively slight effect.From Table 4, compared with the dominant frequency of the Shanghai wave, those of the Kobe wave and Ei-Centro waves are closer to the fundamental frequency of the site.The waves with dominant frequencies closer to the fundamental frequencies of the sites may result in having a more significant effect in terms of anisotropy.Therefore, detailed investigations on the effects of the soil anisotropy on the seismic response of the Shanghai Yangshan Port site are conducted as an example.

Input Seismic Wave
The seismic characteristics of Yangshan Port are examined in this study through the analysis of artificial seismic wave data from Shanghai.According to the site classification method specified in the "Code for Seismic Design of Buildings" (GB 50011-2010) [54], the Yangshan Port area belongs to the fourth category of site.Based on the linear elastic soil layer, the ground acceleration time history is inverted to bedrock to obtain the bedrock acceleration time history [55].

Calculation Model and Parameters of Soil Layer
Most of the topsoil in Yangshan port is soft clay [56], and there is sand soil in the lower layer.Through the geological investigation, the site of Yangshan Port consists of four typical soil layers: clay layer, silty clay layer, muddy silty clay, and sand soil layer.According to Hou [57], the AR E values for this area range from 1.6 to 2.40.Li [58] reported AR G values ranging from 1.08 to 1.39 for Shanghai soil, while Ng [59] stated that the values were between 1.07 and 1.38.Detailed borehole data for the soil layers are provided in Table 5.The design confining pressures are 200 kPa, 250 kPa, and 300 kPa for sand.Figure 5 shows the details of the cyclic triaxial test and the resonant column test.The experimental results pertaining to silty clay are specifically analyzed in this study.Figure 6 illustrates the S-wave output signals of the bending element in the silty clay specimen subjected to a confining pressure of 150 kPa across different input frequencies.Furthermore, the G 0 value derived from the bending element tests is approximately 1.1 to 1.2 times greater than the results obtained from the resonant column tests, which aligns with the outcomes reported by Yang et al. [60] and Gu et al. [61,62].The reason behind this is that the results of the bending element tests specifically relate to the localized stiffness of the shear wave propagation path, while the resonant column offer insights into the overall stiffness characteristics of the specimens [60][61][62].This confirms the accuracy of the resonant column test.Simultaneously, Yang et al. [60], Gu et al. [61,62], and Youn et al. [63] propose to determine G 0 based on the resonant column test.The cyclic triaxial experiments were performed employing strain control.To mitigate the influence of cyclic loading on subsequent tests, the number of cycles for each loading stage was limited to six. Figure 7 depicts the outcomes of the cyclic triaxial tests conducted on the silty clay.The modulus degrades as the strain increases, while it amplifies with the elevated confining pressure.Through fitting analysis, we obtained the   −  and  −  curves for clay in the Yangshan Port site.Figure 8 depicts the curve of best fit for the silty clay specimen.For the other soil layers, the model parameters calibrated from the resonance column and the cyclic triaxial tests are also provided in Table 6.The cyclic triaxial experiments were performed employing strain control.To mitigate the influence of cyclic loading on subsequent tests, the number of cycles for each loading stage was limited to six. Figure 7 depicts the outcomes of the cyclic triaxial tests conducted on the silty clay.The modulus degrades as the strain increases, while it amplifies with the elevated confining pressure.Through fitting analysis, we obtained the   −  and  −  curves for clay in the Yangshan Port site.Figure 8 depicts the curve of best fit for the silty clay specimen.For the other soil layers, the model param- The cyclic triaxial experiments were performed employing strain control.To mitigate the influence of cyclic loading on subsequent tests, the number of cycles for each loading stage was limited to six. Figure 7 depicts the outcomes of the cyclic triaxial tests conducted on the silty clay.The modulus degrades as the strain increases, while it amplifies with the elevated confining pressure.Through fitting analysis, we obtained the G norm − γ and D − γ curves for clay in the Yangshan Port site.Figure 8 depicts the curve of best fit for the silty clay specimen.For the other soil layers, the model parameters calibrated from the resonance column and the cyclic triaxial tests are also provided in Table 6.

Results
Taking into account the previous discussion regarding the suppressive effect of AR E and the enhancing effect of AR G on PGA performance, we initially selected a value of 1.6 for AR E and a value of 1.4 for AR G to simulate the unfavorable conditions.In this case, study, the Shanghai wave is adjusted to have peak ground acceleration of 0.1 g.Figures 9 and 10 show the changes in peak acceleration along the depth and the ground peak acceleration spectrum, respectively.It is evident that when the proposed calculation model reduces to the isotropic scenario, it aligns well with the results obtained from EERA.However, when considering soil anisotropy, the PGA increases by 19.70%; the peak value of the ground response spectrum undergoes a significant increase of 28.82% at 1 Hz.
Buildings 2023, 13, x FOR PEER REVIEW for   and a value of 1.4 for   to simulate the unfavorable conditions.In t study, the Shanghai wave is adjusted to have peak ground acceleration of 0.1 g.F and 10 show the changes in peak acceleration along the depth and the ground celeration spectrum, respectively.It is evident that when the proposed calculatio reduces to the isotropic scenario, it aligns well with the results obtained from However, when considering soil anisotropy, the PGA increases by 19.70%; the p ue of the ground response spectrum undergoes a significant increase of 28.82% a In order to discuss the influence of anisotropy on the ground acceleration frequency, the ground motion response of the Ei-Centro waves and Kobe waves input at Yangshan port is calculated.The result can be seen in Figure 10.It can be found that anisotropy has little influence on the frequency position of the ground Fourier acceleration spectrum peak.
To further explore the design response spectrum of Yangshan Port, the ground motion responses of the site under different seismic intensities (0.1 g, 0.15 g, 0.2 g, 0.25 g, 0.3 g) are studied.Based on the 5% damped ground acceleration response spectrum of different intensities, the least squares method by Andreotti [64] et al. was used to calibrate the design response spectrum.The shape function of the design response spectrum by Chinese code "GB 51247-2018" is as follows [65]: where T 0 is the period associated with the initial inflection point, established at 0.1 s.T g signifies the characteristic period.Since this study focuses on a certain soil layer in Yangshan Port, the characteristic period (T g ) can be calculated according to Equation (12), which is 0.92 s.T m is the cutoff period, χ represents the attenuation index, and β max denotes the plateau value.In order to discuss the influence of anisotropy on the ground acceleration frequency, the ground motion response of the Ei-Centro waves and Kobe waves input at Yangshan port is calculated.The result can be seen in Figure 10.It can be found that anisotropy has little influence on the frequency position of the ground Fourier acceleration spectrum peak.As well as that given in Equation ( 13) (termed as "Shape Function One"), another two typical commonly used shape functions shown in Figure 11 are also collected for calibrating the design response spectrum for Yangshan Port.Shape Function Two was provided by Deoda and Adhikary (2020) [66], where T D is defined as the value defining the beginning of the constant displacement response range of the spectrum.Shape Function Three is employed by both NZS 1170.5 [67] and Eurocode 8 [51].
The gray lines in Figure 11 are the ground response spectrums at different intensities with consideration of soil anisotropy.The upper and lower bounds of the gray area are the 16th to 84th percentiles, respectively, showing the region with a probability range of 16% to 84% for the occurrence of seismic events.The red line represents the average of the response spectrum curve.Utilizing Shape Function One, the 5% damped design response spectrum for Yangshan Port is calibrated and represented by the blue line.The design response spectrums based on Shape Function Two and Three are depicted as the magenta and orange lines, respectively.
For Shape Function One suggested by GB 51247-2018, the plateau value (β) is 2.97, and the attenuation index (χ) is 0.5.The Pearson correlation coefficient (r value) is 0.63.Additionally, utilizing Shape Function Two results in a plateau value (β) of 3.00 and an attenuation index (χ) of 0.5.This yields a Pearson correlation coefficient (r value) of 0.57.Similarly, when employing Shape Function Three, the plateau value (β) is 3.00, while the attenuation index (χ) is 0.5, with a significantly higher Pearson correlation coefficient (r value) of 0.71.Three shape functions provide a similar plateau value (β), but Shape Function Three exhibits a better overall fitting performance.
shan Port, the characteristic period (  ) can be calculated according to Equation (12), which is 0.92 s.   is the cutoff period,  represents the attenuation index, and   denotes the plateau value.
As well as that given in Equation ( 13) (termed as "Shape Function One"), another two typical commonly used shape functions shown in Figure 11 are also collected for calibrating the design response spectrum for Yangshan Port.Shape Function Two was provided by Deoda and Adhikary (2020) [66], where   is defined as the value defining the beginning of the constant displacement response range of the spectrum.Shape Function Three is employed by both NZS 1170.5 [67] and Eurocode 8 [51].To further investigate the influence of anisotropy on ground response spectra, a series of cross-parameter studies were conducted focusing on anisotropy-related parameters (elastic modulus ratio AR E = 1.60, 2.00, 2.40; shear modulus ratio AR G = 1.10, 1.25, 1.40), resulting in a total of nine combinations.Following the method illustrated in Figure 11, Figure 12 presents nine average response spectrum curves.Utilizing the mean values of these spectrums, the root mean square error (RMSE) across the entire period range for the nine response spectrum curves was calculated and is depicted by the gray line in Figure 12.It is observed that the maximum RMSE among the nine response spectrum curves reaches 34.5% at lower frequencies.The gray lines in Figure 11 are the ground response spectrums at different intensities with consideration of soil anisotropy.The upper and lower bounds of the gray area are the 16th to 84th percentiles, respectively, showing the region with a probability range of 16% to 84% for the occurrence of seismic events.The red line represents the average of the response spectrum curve.Utilizing Shape Function One, the 5% damped design response spectrum for Yangshan Port is calibrated and represented by the blue line.The design response spectrums based on Shape Function Two and Three are depicted as the magenta and orange lines, respectively.
For Shape Function One suggested by GB 51247-2018, the plateau value (β) is 2.97, and the attenuation index (χ) is 0.5.The Pearson correlation coefficient (r value) is 0.63.Additionally, utilizing Shape Function Two results in a plateau value (β) of 3.00 and an attenuation index (χ) of 0.5.This yields a Pearson correlation coefficient (r value) of 0.57.Similarly, when employing Shape Function Three, the plateau value (β) is 3.00, while the attenuation index (χ) is 0.5, with a significantly higher Pearson correlation coefficient (r value) of 0.71.Three shape functions provide a similar plateau value (β), but Shape Function Three exhibits a better overall fitting performance.
To further investigate the influence of anisotropy on ground response spectra, a series of cross-parameter studies were conducted focusing on anisotropy-related parameters (elastic modulus ratio   = 1.60, 2.00, 2.40 ; shear modulus ratio   = 1.10, 1.25,1.40),resulting in a total of nine combinations.Following the method illustrated in Figure 11, Figure 12 presents nine average response spectrum curves.Utilizing the mean values of these spectrums, the root mean square error (RMSE) across the entire period range for the nine response spectrum curves was calculated and is depicted by the gray line in Figure 12.It is observed that the maximum RMSE among the nine response spectrum curves reaches 34.5% at lower frequencies.Finally, the proposed 5% damped standard design response spectrum for Yangshan Port has been developed from the previously described nine combinations of anisotropic parameters and five seismic intensity levels.The design response spectrum for the anisotropic site is refined using Shape Function Three, which is depicted as an orange line in Figure 13.This calibration involves a plateau value (β) of 3.00 and an attenuation index Finally, the proposed 5% damped standard design response spectrum for Yangshan Port has been developed from the previously described nine combinations of anisotropic parameters and five seismic intensity levels.The design response spectrum for the anisotropic site is refined using Shape Function Three, which is depicted as an orange line in Figure 13.This calibration involves a plateau value (β) of 3.00 and an attenuation index (χ) of 0.50.When compared to the isotropic spectrum, the anisotropic spectrum demonstrates a notably elevated design response spectrum.Factoring in anisotropy has led to an increase of 18% in the plateau value (β) for the standard design response spectrum.Consequently, utilizing the isotropic design response spectrum for the seismic design of structures at Shanghai Yangshan Port may result in dangerous results.
Buildings 2023, 13, x FOR PEER REVIEW 14 of 18 (χ) of 0.50.When compared to the isotropic spectrum, the anisotropic spectrum demonstrates a notably elevated design response spectrum.Factoring in anisotropy has led to an increase of 18% in the plateau value (β) for the standard design response spectrum.Consequently, utilizing the isotropic design response spectrum for the seismic design of structures at Shanghai Yangshan Port may result in dangerous results.(χ) of 0.50.When compared to the isotropic spectrum, the anisotropic spectrum demonstrates a notably elevated design response spectrum.Factoring in anisotropy has led to an increase of 18% in the plateau value (β) for the standard design response spectrum.Consequently, utilizing the isotropic design response spectrum for the seismic design of structures at Shanghai Yangshan Port may result in dangerous results.For instance, let us consider a single-story frame with a total seismic weight of 980 kN and a total column lateral stiffness k of 6 MN/m.The structure has a damping ratio of 5%.
The basic acceleration (PGA) by design is 0.1 g.For the conventional response spectrum, the design base shear amounts to 250 kN.However, when the response spectrum is applied with consideration of anisotropy, the design base shear amounts to 294 kN, marking an increase of 18%.

Conclusions
This study establishes an equivalent linear ground motion model for anisotropic sites to analyze the ground motion response characteristics for layered soils with various anisotropy parameters.Further investigations were conducted to study the ground motion response features of anisotropic sites when subjected to different seismic wave inputs.It was found that the anisotropy ratios for the shear modulus (AR G ) have a promoting effect on the peak ground acceleration (PGA), while the anisotropic ratio of undrained Young's modulus (AR E ) has an inhibitory effect on peak ground acceleration (PGA) in the site earthquake response problem.The impact of anisotropy on the ground motion of the site becomes more significant when seismic waves have dominant frequencies closer to the fundamental frequencies of the sites.In the ground motion response problem, anisotropy cannot be ignored when the dominant frequencies of the seismic waves are closer to the fundamental frequencies of the sites.
This study further takes the Yangshan Port site as an example to calibrate the seismic motion parameters (G − γ and D − γ) of the Yangshan Port soil layer based on cyclic triaxial tests and resonant column tests.The results show that the combination of the resonant column test and cyclic triaxial test can better describe the dynamic characteristics of soil from small strains (10 −6 ~10 −3 ) to large strains (10 −3 ~10 −1 ).Based on the calibrated dynamic parameters, the ground motion characteristics of the anisotropic site of Yangshan Port were studied.It was found that the consideration of site anisotropy leads to significant increases in both the peak ground acceleration (PGA) and the peak ground Fourier acceleration spectrum for the Yangshan Port site.Anisotropy has little influence on the frequency position of the ground Fourier acceleration spectrum peak.The frequency region where the amplitude of the ground Fourier acceleration spectrum increases significantly is close to the site fundamental frequency (1 Hz).This demonstrates that site anisotropy can potentially lead to a severe underestimation of the acceleration response in regions proximate to the site's fundamental frequency.
A series of cross-parameter studies regarding anisotropic parameters were also conducted to investigate the influence of anisotropy on ground response spectra.When taking site anisotropy into account, the calibrated design response spectrum plateau value (β) is 3.00.Conversely, when anisotropy is not considered, the calibrated design response spectrum plateau value (β) stands at 2.55.The design response spectrum calibrated for an anisotropic site surpasses that for an isotropic site.Thus, when performing the seismic design of buildings, utilizing the design response spectrum derived from isotropic site ground motion response calibration might present higher risks.

Figure 1 .
Figure 1.Seismic wave time history curve.Figure 1. Seismic wave time history curve.

Figure 1 .
Figure 1.Seismic wave time history curve.Figure 1. Seismic wave time history curve.Buildings 2023, 13, x FOR PEER REVIEW 5 of 18

Figure 6 .
Figure 6.BE S-wave signals in silty clay specimen.

Figure 14
Figure 14  illustrates the application of the response spectrum for a single-degree-offreedom frame model.When utilizing the response spectrum, two key aspects must be contemplated.Firstly, the fundamental frequency ( = 2√  ⁄ ) is derived from the concentration of mass and lateral stiffness.Secondly, the design base shear is computed through  =  × .

Figure 14 .
Figure 14.Application of response spectrum for single-degree-of-freedom systems.For instance, let us consider a single-story frame with a total seismic weight of 980 kN and a total column lateral stiffness k of 6 MN/m.The structure has a damping ratio of 5%.The basic acceleration (PGA) by design is 0.1 g.For the conventional response

Figure 14
Figure 14  illustrates the application of the response spectrum for a single-degree-offreedom frame model.When utilizing the response spectrum, two key aspects must be contemplated.Firstly, the fundamental frequency ( f = 2π √ m/k) is derived from the concentration of mass and lateral stiffness.Secondly, the design base shear is computed through F = mβ × PGA.

Figure 14
Figure 14  illustrates the application of the response spectrum for a single-degree-offreedom frame model.When utilizing the response spectrum, two key aspects must be contemplated.Firstly, the fundamental frequency ( = 2√  ⁄ ) is derived from the concentration of mass and lateral stiffness.Secondly, the design base shear is computed through  =  × .

Figure 14 .
Figure 14.Application of response spectrum for single-degree-of-freedom systems.For instance, let us consider a single-story frame with a total seismic weight of 980 kN and a total column lateral stiffness k of 6 MN/m.The structure has a damping ratio of 5%.The basic acceleration (PGA) by design is 0.1 g.For the conventional response

Figure 14 .
Figure 14.Application of response spectrum for single-degree-of-freedom systems.

Table 2 .
Isotropic soil example.Figure3illustrates the comparison of the peak ground acceleration (PGA) with depth, considering three seismic records, namely the Ei-Centro Wave, Shanghai Wave, and Kobe Wave, along with four soil conditions.They exhibit good agreements with the results obtained from the classical seismic analysis code EERA.

Table 4 .
Site fundamental frequency and seismic wave dominant frequency.

Table 5 .
Soil layer information of Yangshan port.Resonance column and cyclic triaxial tests were conducted to obtain G norm − γ and D − γ curves for clay, silty clay layer, muddy silty clay, and sand.The design confining pressures are 150 kPa, 200 kPa, and 250 kPa for clay, silty clay layer, and muddy silty clay.

Table 6 .
The soil parameters of the Yangshan Port site.

Table 6 .
The soil parameters of the Yangshan Port site.