Analytical solution of ground-borne vibration due to a spatially periodic harmonic moving load in a tunnel embedded in layered soil

Lihui XU ,Meng MA✉

1Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China

2School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

Abstract: In this study,we propose a novel coupled periodic tunnel–soil analytical model for predicting ground-borne vibrations caused by vibration sources in tunnels.The problem of a multilayered soil overlying a semi-infinite half-space was solved using the transfer matrix method.To account for the interactions between the soil layer and tunnel structure,the transformation characteristics between cylindrical waves and plane waves were considered and used to convert the corresponding wave potentials into forms in terms of the Cartesian or cylindrical coordinate system.The induced ground-borne vibration was obtained analytically by applying a spatially periodic harmonic moving load to the tunnel invert.The accuracy and efficiency of the proposed model were verified by comparing the results under a moving constant and harmonic load with those from previous studies.Subsequently,the response characteristics under a spatially periodic harmonic moving load were identified,and the effects of a wide range of factors on the responses were systematically investigated.The numerical results showed that moving and Doppler effects can be caused by a spatially periodic harmonic moving load.The critical frequency and frequency bandwidth of the response are affected by the load type,frequency,velocity,and wavenumber in one periodicity length.Increasing the tunnel depth is an efficient way to reduce ground-borne vibrations.The effect of vibration amplification on the free surface should be considered to avoid excessive vibration levels that disturb residents.

Key words: Coupled tunnel–soil model;Spatially periodic load;Transfer matrix method;Wave transformation;Parametric analysis

1 Introduction

With the rapid construction of subway tunnels in metropolitan cities,the environmental problem of ground-borne vibrations induced by underground trains has attracted widespread attention in recent decades(Lopes et al.,2016;Zhang et al.,2018;Liu et al.,2022;Ma et al.,2022;Xu et al.,2022).Investigations on this topic are helpful for evaluating and easing the impact of train-induced vibration on adjacent resi‐dents,historical buildings,and sensitive instruments(Hussein and Hunt,2009;Ma et al.,2016;Jin et al.,2022;Zou et al.,2022).A train-induced discretely dis‐tributed force through fasteners satisfies the periodicity conditions.Thus,it can be further decomposed into a series of spatially periodic harmonic moving loads(Doyle,1997) and is a periodic problem.To solve this problem,we propose a tunnel–soil coupled analytical model that is periodic in the longitudinal direction.The model and the characteristics of ground-borne vibra‐tions under a spatially periodic harmonic moving load are presented in this study.

Numerical methods are most commonly used to investigate ground-borne vibrations from underground sources.Two main issues need to be addressed in numeri‐cal simulations (Xu et al.,2022): (1) boundary trun‐cation error,and (2) calculation efficiency.In previous studies,different inspiring models have been proposed to overcome these issues.For instance,the boundary truncation error was avoided by the infinite element(IFE) technique (Lin et al.,2016;Yang et al.,2017,2021;Yang and Li,2022),and the calculation efficiency was improved by using 2.5D approaches (Sheng et al.,2005;François et al.,2010;Gao et al.,2011;Zhou et al.,2019;Ma et al.,2021).Periodic numerical models were also proposed (Clouteau et al.,2005;Degrande et al.,2006;Ma et al.,2017;Xu et al.,2022).More exhaus‐tive summaries of models for predicting vibrations from underground sources have been reported in the litera‐ture (Kouroussis et al.,2014;Lombaert et al.,2015;Ma et al.,2020).However,numerical calculation still requires huge computer memory and computational time.

An analytical model is the most efficient way to investigate ground vibrations from underground sources accurately and rapidly as there are no boundary trun‐cations or approximations.Various analytical models have been proposed in previous studies.A 2D analyti‐cal model (Metrikine and Vrouwenvelder,2000) and the pipe-in-pipe (PiP) semi-analytical model (Forrest and Hunt,2006a,2006b;Hussein et al.,2014) are the most well-known.Recently,Yuan et al.(2017) proposed a closed-form semi-analytical model to obtain a bench‐mark solution for ground vibration from an under‐ground point source moving in a tunnel embedded in a homogeneous half-space.In this model,the soil–tunnel interaction was represented by transformations between ascending and descending plane waves and outgoing and regular cylindrical waves,as proposed by Boström et al.(1991).This approach was then extended to cover the case in which a tunnel was embedded in a multi‐layered half-space (He et al.,2018).A coupled train–track model was also considered (Yuan et al.,2021),and the interactions between twin tunnels were simu‐lated (He et al.,2019).Notably,all these analytical models were homogeneous in the longitudinal direc‐tion.No analytical periodic models for coupled tunnel–soil systems have yet been proposed.

In the dynamic problem of a coupled tunnel–soil system,two main issues remain and need to be ad‐dressed when the coupled system is periodic in the lon‐gitudinal direction.These issues include further improve‐ment of the calculation efficiency of numerical periodic models,and accurate solution of the periodicity prob‐lem in analytical models.This work focused on the 2nd issue.To this end,in this study we extended the cou‐pled tunnel–soil analytical model proposed by Yuan et al.(2017) and He et al.(2018) by investigating,with the aid of generalised modal function series,the case in which a tunnel–soil system is periodic in the longi‐tudinal direction.This study is the first attempt to solve the periodicity problem of a coupled tunnel–soil system analytically.

We propose a coupled periodic tunnel–soil ana‐lytical model to characterise dynamic ground-borne vibration under a spatially periodic harmonic moving load.Based on the generalised modal functions and Fourier transform,the general solutions of the displace‐ments and tractions for each soil layer and tunnel struc‐ture were derived in the frequency-generalised wave‐number domain.The horizontally layered soils were then modeled using the transfer matrix method to obtain the relationship between the coefficients for each layer.The interaction between the soil and tunnel was realised by considering the properties of the transformation between plane and cylindrical waves.The dynamic response for each part of this model was calculated by applying a spatially periodic harmonic moving load to the inverted arch of the inner surface of the tunnel.Subsequently,the accuracy and efficiency of the model were demonstrated by comparing the results under a moving constant and harmonic load with those from the literature (Yuan et al.,2017;He et al.,2018).Finally,the results of the ground-borne vibrations under spa‐tially periodic harmonic moving loads are presented,along with those from a systematic investigation of the effects of a wide range of factors on the vibration responses.Some conclusions are drawn based on the analyses and discussion.

2 Formulation of the coupled periodic tunnel–soil analytical model

2.1 Model description

Fig.1 illustrates a tunnel embedded in a multi‐layered half-space in the global coordinate system.The model is periodic and comprisesN+1 parts,includingN−2 standard interior soil layers where both ascending and descending waves exist,one interior layernwith a cavity where ascending,descending,and outgoing(irregular) waves exist,one semi-infinite regionNwhere only descending waves exist,and one hollow cylinder for the tunnel where outgoing and regular waves exist.In each part,the interfaces are bonded with their adjoin‐ing parts,implying that tractions and deflections can be directly transferred to the adjoining parts.An exter‐nal forcepthat is periodic in space with periodicity lengthLand harmonic in time with circular frequencyωlis applied at the bottom of the inner surface of the hollow cylinder and moves toward the positivez-axis at a constant speed ofv.The material of each part is assumed to be isotropic,homogeneous,and viscoelas‐tic.Therefore,integral transformation and superposi‐tion techniques can be applied in this case.Because the applied force is periodic in thez-direction,the entire system is periodic in thez-direction.

Fig.1 Tunnel embedded in a multilayered half-space subjected to a spatially periodic harmonic moving load p in a global coordinate system.The interface st represents the connection between the hollow cylinder and the layer with a cavity

Fig.2 Geometry,local coordinate system,and state variable at the corresponding interface of the soil layer: (a) layer above the tunnel (in);(c) semi-infinite region,layer N;(d) layer n with a cavity;(e) hollow cylinder for tunnel lining

2.2 Formulation of the analytical periodic model

The motion of the isotropic,homogeneous,and viscoelastic continuum is governed by the free elasto‐dynamics equation,expressed in vector form as (Sheng et al.,2002):

whereuis the displacement vector expressed in Carte‐sian coordinatesu=[ux,uy,uz]T,or in cylindrical coordi‐natesu=[ur,uφ,uz]T.ρis the density of the material.The symbol ‘¨’ denotes the 2nd-order derivative with respect to timet.λandμare the Lamé constants.Con‐sidering nondimensional material dampingζ,the Lamé constants can be rewritten asλ=λ(1+iζ) andμ=μ(1+iζ).

To solve this equation in the frequency-wave‐number-generalised modal space,the Fourier trans‐forms with respect to timetand coordinatey,and the generalised model superposition technique (Hussein and Hunt,2009;Ma and Liu,2018) regarding coordi‐natezare used as follows:

Considering the boundary,compatibility,and equi‐librium conditions,the relationships of the unknown coefficients of each layer can be derived by applying the transfer matrix method.The compatibility and equi‐librium conditions are

The relationship between the unknown coeffi‐cientsA1of the layer 1 andAn−1of the layern−1 can be derived as follows:

whereT(1,n−1)is the transfer matrix.

Similarly,the relationship between the coeffi‐cientsANof the layerNandAn+1of the layern+1 can be derived as follows:

whereT(N,n+1)is the transfer matrix.

To couple the standard layer and the layer with a cavity,the outgoing cylindrical waveshould be con‐verted into ascending or descending plane waves,obey‐ing the following relationships proposed by Boström et al.(1991):

To couple the layer with a cavity and hollow cylin‐der,the ascending and descending plane wave poten‐tials should be expanded in terms of regular cylindrical wave potentials,expressed as (Boström et al.,1991):

whereεm=1 whenm=0 andεm=2 whenm≠0.

The displacement and traction vector along the outer interface (r=R+h) of the cavity can then be ob‐tained.According to the compatibility and equilibrium conditions along the interface,the unknown coeffi‐cientsBmfor the tunnel structure andAofor the layer with a cavity are related by:

2.3 Moving load applied at the inner interface of the tunnel structure

The applied external load is periodic in space with a periodicity lengthL,harmonic in time with a circular frequencyωl,and moves at a constant speed ofvin thez-direction.The force applied at the inverted arch of the tunnel structure can be mathematically expressed as (Xu and Ma,2022):

whereδis the Dirac delta function.

The origin of the moving load is located at (R,π,0 m).Performing a Fourier transform and considering the orthogonality of the generalised modal function,the external load vector at the tunnel invertcan be expressed as follows:

According to the stress boundary condition of the inner interface of the tunnel structure,the following formulation can be obtained:

Substituting Eq.(15) into Eq.(18) yields the fol‐lowing equation:

This results in three equations for eachm.After consideringm=0,1,2,…,M,there are 3(M+1) equa‐tions where there are 3(M+1) unknownAoas well.Thereafter,the unknown coefficients for the tunnel structureBmand the first layerA1can be derived based on Eqs.(15) and (13),respectively.The formulations derived above were programmed in MATLAB,whereM=12 was considered to obtain the convergence result.

More detailed derivations can be obtained from Section S1 in the ESM.

3 Model validation

As observed in Eqs.(5) and (16),if the wave‐number in one periodicity length satisfies the condi‐tionnl=0,the coupled periodic tunnel–soil analytical model is reduced to a model that is homogeneous in the longitudinal direction,as shown in many previous studies.To demonstrate the efficiency and accuracy of our proposed model,the ground-borne vibrations from the model in which the tunnel is embedded in a homo‐geneous and multilayered half-space were compared with those from the literature (Yuan et al.,2017;He et al.,2018).Note that the tunnel is entirely embedded within a single soil layer.Validation cases,where the tunnel is embedded in a homogeneous and multilayered half-space,are illustrated in Fig.3.The corresponding parameters involved in the validations are from the literature (Yuan et al.,2017;He et al.,2018).Further details are given in Section S2 in the ESM,where addi‐tional validation results are also presented.

Fig.3 Validation cases of a tunnel embedded in a homogeneous half-space (a) and a multilayered half-space (b)

Fig.5 Comparison of velocity history at (0 m,10 m,0 m)owing to moving constant load (v=30 m/s,f0=0 Hz): (a)vertical direction,vx;(b) longitudinal direction,vz

First,the results from the model in which the tunnel is embedded in a homogeneous half-space(Fig.3a) were compared with those obtained by Yuan et al.(2017),where the analytical method was adopted.In the longitudinal direction,the Fourier transform was applied because the soil was homogeneous in that direc‐tion according to Yuan et al.(2017).

Comparisons of the vibration responses at (0 m,0 m,0 m) and (0 m,10 m,0 m) of the ground surface under a constant load with frequencyf0=0 Hz moving at a speed ofv=30 m/s with those from the analytical solution (Yuan et al.,2017) are illustrated in Figs.4 and 5,respectively.As expected,the results were satis‐factory in both the vertical and longitudinal directions.Impulsive effects were observed to be induced by the moving load.The vertical velocity reached zero,while the longitudinal velocity reached a maximum when the load moved to the position immediately beneath the observation section (t=0 s).The relative errors of the maximum absolute responses at (0 m,0 m,0 m) and(0 m,10 m,0 m) are listed with those from the analyti‐cal solutions (Yuan et al.,2017) in Table 1.The maxi‐mum relative error was about 2.39%,confirming the high accuracy of the method.

Table 1 Relative errors in the maximum absolute response obtained from the current method and the referenced solution

Second,validation was performed by comparing the results of the tunnel embedded in a multilayered half-space from the present method with those from the reference (He et al.,2018),as shown in Fig.3b.In the analytical method by He et al.(2018),the coupled tunnel–soil system was regarded as homogeneous;thus,a Fourier transform was used.

A comparison of the vertical displacement fre‐quency spectrum at (0 m,10 m,0 m) subjected to a moving load withv=50 m/s and frequencies off0=10 and 20 Hz with the analytical solution (He et al.,2018) is plotted in Fig.6.This indicates that the results from the present method are in good agreement with those from the reference.The frequency spreads mainly around the excitation frequency of the harmonic load.The Dop‐pler effect induces a difference between the dominant frequencies of the response and excitation.

Fig.6 Comparison of the vertical displacement frequency spectrum of the ground surface from a tunnel embedded in a multilayered half-space owing to a moving harmonic load with v=50 m/s: (a) f0=10 Hz;(b) f0=20 Hz

The proposed analytical model was run on a com‐puter with an Intel(R) Core(TM) i7-7700K central pro‐cessing unit (CPU) @ 4.20 GHz processor.Its compu‐tational time was about 39 s in the case of one calcula‐tion while that from the corresponding numerical model is several hours.The memory requirement was about 0.5 GB of random access memory (RAM).Therefore,this model requires low computer memory and com‐putational time.These features have the potential to be applied in a quick environmental vibration assessment.

By comparing the results with those from previous studies,the accuracy and efficiency of the present ana‐lytical model,which is periodic in the longitudinal direction,were fully validated.

4 Numerical results and discussion

The ground vibration responses from the tunnel embedded in the multilayered half-space under a spa‐tially periodic harmonic load were systematically inves‐tigated using the proposed coupled tunnel–soil periodic model.The dynamic parameters of the soil and tunnel were selected from the study of Xu and Ma (2020),in which a statistical survey of the geological parameters in Beijing was conducted.The soil parameters are listed in Table 2.Young’s modulus of the tunnel structure wasE=32 GPa,Poisson’s ratioν=0.2,material densityρ=2400 kg/m3,and hysteretic material dampingζ=0.02.The axis of the tunnel was located in the 2nd soil layer at a depth ofd=15 m from the ground surface.The tunnel structure had an inner radius ofR=3 m and a thickness ofh=0.3 m.Unless otherwise specified,these parameters were used in the following calculations.The number of trigonometric terms,M=12,was con‐sidered to provide a convergence result.When perform‐ing the inverse Fourier transform,a frequency range of[−2fcr,2fcr],where the response in the frequency range off<0 Hz is conjugate with the counterpart at a frequency off>0 Hz,was considered to yield a real response in the time domain.The critical frequencyfcrwas defined as the frequency around which the response frequency mainly spreads.A moving load was imposed at the inverted arch of the tunnel.Note that some additional results and discussion are presented in Section S3 in the ESM.

Table 2 Parameters of soil layers used in the investigation

4.1 General results

The general response of the ground surface under a spatially periodic harmonic moving load (nl=1,f0=5 Hz,v=25 m/s) is presented in this section.The load velocity approximately corresponds to the maximum design velocity of the metro in Beijing,China.

The displacement responses at pointsA(0 m,0 m,0 m) andB(0 m,10 m,0 m) on the ground surface in both the time and frequency domains are shown in Fig.7.In the time domain,the ground point experi‐ences an about 2-s dynamic vibration because the propa‐gating waves radiate outwards from the tunnel.Both the vertical and longitudinal displacements are not per‐fectly symmetric with respect tot=0 s because of damp‐ing and Doppler effects.The vertical displacement exhibited a different time history from that of the lon‐gitudinal displacement.At the time instantt=0 s,when the load moved to the point immediately beneath pointA,the vertical vibration amplitude reached a maximum,while the longitudinal vibration reached zero.The verti‐cal vibration was stronger than the longitudinal vibra‐tion because the applied load at the tunnel invert was vertical.Overall,the displacement responses exhibited a similar time history envelope at pointsAandB,and the displacement vibrations at pointBwere weaker than those at pointAowing to damping effects.These phenomena can also be observed in the frequency domain results.Besides,it can be found from the results in the frequency domain that the response frequency spreads within a narrow frequency band around the criti‐cal frequency.The critical frequency band is deter‐mined by two effects (Xu and Ma,2022): the moving effect,fcr=nlv/L+f0,and the Doppler effect,fband=fcr/(1±v/cR) (frequency band),wherecRis the Rayleigh wave velocity.The moving effect increased the critical fre‐quency and the Doppler effect broadened the frequency band.The critical frequencyfcrwas 46.7 Hz.Further‐more,at this critical frequency,the vertical vibration reached a maximum,whereas the longitudinal vibra‐tion reached a minimum.

Fig.7 Vertical displacement (a) and longitudinal displacement (b) in time (ux and uz) and frequency (ũx and ũz) domain at A (0 m,0 m,0 m) and B (0 m,10 m,0 m) of the ground surface

For a clear inspection of the vibration attenuation characteristics of the ground surface,Fig.8 illustrates the absolute instantaneous displacement response on the ground surface at timet=0 s under a spatially periodic harmonic load.The area concerned has dimensions ofy∈[−40 m,40 m] andz∈[−40 m,40 m] in they–zplane,and the load is immediately beneath (0 m,0 m)att=0 s.Overall,the exciting ground vibration is dis‐tributed mainly in the areay∈[−20 m,20 m] andz∈[−20 m,20 m] and decays undulatingly in the far field.Many troughs and peaks were observed in the entire field.A closer inspection of Fig.8a shows that the wavelength behind the load is longer than that in front of the load,owing to the Doppler effect.In addition,the wavelength in thez-direction (the load moving direction) is longer than that in they-direction,and concentric waves have imperfect circular distributions.This is because the waves generated in the tunnel struc‐ture travel much faster than those in the soil and con‐tribute to the ground vibration.At this time instant,the maximum response appeared at a certain distance from(0 m,0 m),and a vibration shadow emerged above the tunnel.The vertical vibration was distributed mainly in the area within a certain lateral distance from (0 m,0 m),whereas the longitudinal vibration was distribut‐ed mainly in the area within a certain longitudinal dis‐tance from (0 m,0 m).Notably,the absolute longitu‐dinal displacement was approximately zero alongz=0 m,as the propagating waves atz=0 m were perpen‐dicular to the axisy=0 m at the time instantt=0 s.

Fig.8 Absolute instantaneous vertical displacement ux (a) and longitudinal displacement uz (b) on the ground surface at the time instant t=0 s

4.2 Parametric analysis

A wide range of factors that affect the vibration response characteristics on the ground were systemati‐cally investigated,including the load type,wavenum‐ber in one periodicity lengthnl,load frequencyf0,load velocityv,tunnel thicknessh,buried depth of tunnel axisd,and number of soil layersns.The significance of our results to engineering applications is highlighted.

First,the effects of load type on the vibration re‐sponses were studied.Fig.9 shows the vertical velocity responses atA(0 m,0 m,0 m) on the ground under a moving constant load (nl=0,f0=0 Hz),moving harmonic load (nl=0,f0=5 Hz),and moving spatially periodic harmonic load (nl=1,f0=5 Hz) in both the time and fre‐quency domains.The responses under a constant load showed quasi-static behaviour,where the response fre‐quency spread within the frequency range off<2 Hz.Among the three types of loads,the moving harmonic load led to the strongest response on the ground sur‐face.A denser time history was observed as the load type changed.The critical frequencies for the three cases were 0.23,5.00,and 46.67 Hz,respectively.The Dop‐pler effect was observed in all three cases,whereas the moving effect was found only in the case of the periodic harmonic load because it is caused by the spatial peri‐odicity of the moving load.Also,the frequency band within which the response frequency mainly spread was wider in the case of the periodic harmonic load.

Fig.9 Vertical velocities at A (0 m,0 m,0 m) on the ground surface under moving constant (a and d),harmonic (b and e),and spatially periodic harmonic (c and f) load in both time (a–c) and frequency (d–f) domains

The effects of the wavenumber in one periodicity lengthnlon the vertical displacement frequency spec‐trum and maximum vertical displacement along they-axis on the ground are plotted in Fig.10.Note that the vertical coordinates are logarithmic to allow better visualisation.In the frequency domain,as the wave‐number in one periodicity lengthnlincreased,the criti‐cal frequency increased consistently and the frequency band spread wider.Generally,a higher wavenumber results in a weaker displacement response on the ground surface.This is because more vibration energy in the higher frequency range is dissipated by soil.Except in the casenl=0,the vertical displacements exhibited undulating behaviour along they-axis,and vibration amplification effects were observed in some areas.This finding agrees with results from field measurements of vibration induced by metro train passaging that showed a ground vibration amplification area at a certain dis‐tance from the tunnel axis.

Fig.10 Effects of the wavenumber in one periodicity length nl on vertical displacement frequency spectrum at A (0 m,0 m,0 m) (a) and the maximum vertical displacement(ux,max) along the y-axis on the ground surface (b)

Fig.11 shows the effects of the load frequencyf0on the displacement frequency spectrum and the maximum vertical displacement along they-axis on the ground surface.As a general trend,an increase in the load fre‐quency results in an increase in the displacement re‐sponse critical frequency and frequency band owing to the moving and Doppler effects,but a decrease in the displacement response amplitude on the free surface.The displacement attenuation on they-axis under the spatially periodic harmonic moving load exhibits undu‐lating behaviour owing to the propagating waves in the free surface.The undulating behaviour becomes less obvious at higher load frequencies.The vibration ampli‐fication area seems to move toward the tunnel axis asf0increases.This may be caused by the shorter wave‐lengths of the ground propagating waves at higher fre‐quencies.Fig.12 is a plot of the amplitude of the dis‐placement responses atA(0 m,0 m,0 m) andB(0 m,10 m,0 m) on the ground surface,owing to the spa‐tially periodic harmonic load at various frequencies.Generally,under a spatially periodic harmonic moving load,the ground points vibrate more violently in the vertical direction than in the longitudinal direction.This is because the loads applied at the inverted arch of the tunnel are in the vertical direction.Typically,an increase inf0results in a decrease in the displacement response in both the vertical and longitudinal direc‐tions.However,the vibration attenuation shows a fluc‐tuating behaviour with increasing load frequency.

Fig.11 Effects of load frequency f0 on vertical displacement frequency spectrum at A (0 m,0 m,0 m) (a) and the maximum vertical displacement along the y-axis on the ground surface (b)

Fig.12 Amplitudes of vertical and longitudinal displacements at A (0 m,0 m,0 m) (a) and B (0 m,10 m,0 m) (b) on the ground surface due to the spatially periodic harmonic load with various frequencies

Fig.13 presents the vertical displacement fre‐quency spectrum atA(0 m,0 m,0 m) and the instan‐taneous vertical displacement at time instantt=0 s along they-axis on the ground surface under various load velocities.Similarly,owing to the moving and Dop‐pler effects,a higher load velocity results in a higher critical frequency and a wider frequency band.Gener‐ally,the amplitudes of the displacement responses decrease with an increase in the load velocity (Figs.13a and 13b).As shown in Fig.13b,the wavelengths of the propagating waves on the ground surface are shortened by increasing load velocities.This is because the criti‐cal frequencies of the propagating waves are raised by the increase in load velocity,thus leading to shorter wavelengths.

Fig.13 Effects of load velocity on vertical displacement frequency spectrum ũx at A (0 m,0 m,0 m) (a) and real part of instantaneous vertical displacement ux,real at the time instant t=0 s along the y-axis on the ground surface (b)

Figs.14 and 15 show the effect of the tunnel thick‐ness and depth on the displacement frequency spectrum atA(0 m,0 m,0 m) and the instantaneous vertical displacement along they-axis on the free surface.Varia‐tions in the tunnel thickness and depth have little impact on the critical frequency and frequency bands.As a general trend,an increase in tunnel depth reduces the level of vibration,because the soil dissipates more of the energy of the propagating waves emanating from the deeper vibration source.Fig.14b shows that a phase lag is caused by the variation in tunnel thickness,but this effect becomes insignificant in the far field.In the near field,the vibration level is amplified by the tunnel thickness.In practical applications,increasing the tunnel depth is an efficient measure to reduce the ground-borne vibration induced by underground vibra‐tion sources,whereas thickening the tunnel lining does little to attenuate the level of ground vibration.

Fig.14 Effects of tunnel thickness on vertical displacement frequency spectrum at A (0 m,0 m,0 m) (a) and real part of instantaneous vertical displacement along the y-axis on the free surface (b)

Fig.15 Effects of tunnel depth on vertical displacement frequency spectrum at A (0 m,0 m,0 m) (a) and real part of instantaneous vertical displacement along the y-axis on the free surface (b)

Finally,the effect of the number of soil layers was investigated.Fig.16 illustrates the effects of the number of soil layers on the ground-borne vibration under a spatially periodic harmonic moving load.In the casens,the material and geometric parameters of the firstnssoil layers were adopted,where the last soil layer extended to infinity.Variation in the number of soil layers had no effect on the critical frequency and fre‐quency band.Generally,the ground-borne vibration level underns=1 was the highest.However,the vibra‐tion levels underns=2 and3 were almost the same,implying that the characteristics of the soil layer under the tunnel had little impact on the ground response.The wavelengths underns=2 and 3 were longer than those underns=1 because the propagating waves travelled faster in soil layer 2 underns=2 and 3,as listed in Table 2.This played an important role in the ground vibration characteristics.Variation in soil properties also led to the amplification of the vibration levels aty=20 m.In engineering applications,increasing the stiffness of the soil layers containing and beneath the tunnel may amplify the ground-borne vibration levels in some areas.

Fig.16 Effects of the number of soil layers on vertical displacement frequency spectrum at A (0 m,0 m,0 m) (a)and real part of instantaneous vertical displacement along the y-axis on the free surface (b)

4.3 Discussion

A coupled periodic tunnel–soil analytical model was proposed to calculate the ground vibration response under a spatially periodic harmonic moving load.Based on this model,the general characteristics of the re‐sponses were investigated.Further,a parametric anal‐ysis was conducted to study the effects of a wide range of factors on responses.The moving effects and Dop‐pler effects were observed.

Due to the generalised modal function applied in the derivation,this model is applicable only to the case under a moving point load.However,this periodic analytical model can be extended to cover the case of a fixed-point load using the generalised wavenum‐ber technique.Then,the frequency response functions(FRFs) can be calculated and compared with those from the impact experiment.

For the assessment of metro train-induced envi‐ronmental vibration,the track slab and moving train load should be further considered in the presented periodic analytical model.Then,the ground borne vibra‐tions from underground trains can be obtained and com‐pared with field measurements.As the analytical model requires little computer memory and computational time,it has the potential to assess the vibration levels of many sensitive sites along the metro line,which can provide vital information for vibration mitigation mea‐sures.This work will be conducted in future studies.

5 Conclusions

In this study,a novel coupled periodic tunnel–soil analytical model was proposed for predicting the ground-borne vibrations generated from a tunnel em‐bedded in a multilayered half-space.General solutions for all parts were derived,which were regarded as periodic in the longitudinal direction.The multilayered soils overlying the semi-infinite half-space were mod‐eled using the transfer matrix method.The interactions between the tunnel and multilayered half-space were modeled by considering the transformation properties between the cylindrical and plane waves.The pro‐posed model was validated by comparing the results under a moving constant and harmonic load with those from the literature.The characteristics of ground-borne vibrations under spatially periodic harmonic moving loads,and the effects of a wide range of factors on vibra‐tion responses,were systematically investigated.The following conclusions were drawn from the results.

(1) The proposed coupled periodic tunnel–soil ana‐lytical model is highly accurate,computationally effi‐cient,and can be used to predict the ground-borne vibrations induced by train operations within a tunnel.

(2) In the time history,the vertical response reaches a maximum,whereas the longitudinal response reaches a minimum at the time instant when the load is imme‐diately beneath the observation point.Both moving and Doppler effects can be excited by a spatially periodic harmonic moving load.

(3) Owing to moving and Doppler effects,the load type,load frequency,load velocity,and wavenumber in one periodicity length alter the critical frequency and frequency bandwidth.Variations in tunnel depth,tunnel thickness,and soil layer number have little impact on these features but give rise to a change in vibration amplitude.Increasing the tunnel depth was found to be an efficient way to reduce the level of ground-borne vibration.

(4) Under a spatially periodic harmonic moving load,a vibration amplification area exists at a certain horizontal distance from the tunnel axis.This should be considered to avoid potential excessive train-induced vibration that disturbs residents.

The current periodic analytical model is applicable only for the case of a moving point load.Future work could extend this model to cover the case of a fixedpoint load,and to consider the track slab and the mov‐ing train load to assess metro train-induced environ‐mental vibration levels.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities of China (No.2021JBM031) and the National Natural Science Foundation of China (No.51978043).

Author contributions

Lihui XU wrote the first draft of the manuscript.Meng MA revised and edited the final version.

Conflict of interest

Lihui XU and Meng MA declare that they have no con‐flict of interest.