Using the PiP model for fast calculation of vibration from a railway tunnel in a multi-layered half-space

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1 Using the PiP model fo fast calculation of vibation fom a ailway tunnel in a multi-layeed half-space M.F.M. Hussein a, H.E.M. Hunt b, L. Rikse c, S. Gupta c, G. Degande c, J.P. Talbot d, S. Fancois c, and M. Schevenels c. a School of Civil Engineeing, Univesity of Nottingham, Univesity Pak, Nottingham, NG7 2RD,UK, Tel: , Fa: mohammed.hussein@nottingham.ac.uk b Engineeing Depatment, Cambidge Univesity, Tumpington Steet, Cambidge, CB2 1PZ, UK, Tel: , Fa: hemh@eng.cam.ac.uk c Depatment of Civil Engineeing, K.U. Leuven, Kasteelpak Aenbeg 40, B-3001 Leuven, Belgium, Tel: , Fa: geet.degande@bwk.kuleuven.be Abstact d Atkins Consultants, Bunel House, RTC Business Pak, London Road, Deby, DE1 2WS, UK, Tel: , Fa: ames.talbot@atkinsglobal.com This pape pesents a new method fo calculating vibation fom undegound ailways buied in a multi-layeed half-space. The method assumes that the tunnel s nea-field displacements ae contolled by the dynamics of the tunnel and the laye that contains the tunnel, and not by layes futhe away. Theefoe the displacements at the tunnel-soil inteface can be calculated using a model of a tunnel embedded a full space. The Pipe-in-Pipe (PiP) model is used fo this pupose whee the tunnel wall and its suounding gound ae modelled as two concentic pipes using the elastic continuum theoy. The PiP model is computationally efficient on account of unifomity along and aound the tunnel. The fa-field displacement is calculated by using anothe computationally efficient model that calculates Geen s functions fo a multi-layeed half-space using the diect stiffness method. The model is based on the eact solution of Navie's equations fo a hoizontally layeed half-space in the fequency adial-wavenumbe domain. The esults and computation time of the new method ae compaed with those of an altenative coupled Finite-Element-Bounday-Element (FE-BE) method that accounts fo a tunnel in a multilayeed half-space. It is shown that the esults of the two methods ae in a good ageement fo typical paamete values of a tunnel. The new method is moe computationally efficient (equies significantly less unning-time on a pesonal compute with much less use of memoy). 1. Intoduction Vibation geneated by undegound tains can have a significant envionmental impact on neaby buildings. Vibation is geneated at the wheel-ail inteface and popagates to neaby buildings whee it may cause annoyance to people and the malfunctioning of sensitive equipment. Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 1

2 Modelling this poblem has ecently gained moe inteest on account of inceasing public sensitivity to noise and vibation and the intoduction of new undegound lines in uban aeas. A numbe of models fo calculating vibation fom undegound ailways have been pesented in the liteatue. These models ange fom simple models (e.g. based on a few numbe-of-degees-offeedom o two-dimensional plane-stain models) to thee-dimensional compehensive models. Enginees tend to use simple models because compehensive thee-dimensional models ae consideed impactical; they take long unning-time and equie substantial computational esouces. Howeve, simple models only wok in cetain situations giving only qualitative esults and a good undestanding of the physics of each situation is equied befoe taking a decision based on thei esults. Thee is an inceasing demand fo a computationally efficient model that accounts fo the essential dynamics of a tunnel and its suounding gound. The coupled FE-BE method accounts fo a tunnel embedded in a multi-layeed half-space. The computation efficiency of this method has significantly impoved in the last few yeas by accounting fo peiodicity in the tunnel diection. This has been achieved by incopoating the discete wavenumbe method [1] o by using the Floquet tansfomation [2,3]. Even with this ecent development, the unning time of the model is still long and it equies significant computational esouces. The model is valuable fo eseach puposes but still computationally epensive to be used by enginees as a pediction tool. The PiP model is pesented to account fo a tunnel embedded in a full-space [4-6]. The model consists of two concentic pipes, both of infinite length. The inne pipe epesents the tunnel wall and can be modelled using the thin shell theoy [4] o the elastic continuum theoy [7]. The oute pipe, with its oute adius being set to infinity, epesents an infinite soil with a cylindical cavity and is modelled using the elastic continuum theoy. The PiP model is computationally efficient on account of the unifomity along and aound the tunnel. The model is validated against the coupled FE-BE model fo the case of a tunnel embedded in a full-space [7]. A good ageement is achieved between esults of the two models. A softwae application based on the PiP model has been built with a use-fiendly inteface [8]. The softwae is available on the intenet as a feewae [9] and it accounts fo a tain unning on a floating-slab tack in the tunnel. The softwae calculates the Powe Spectal Density (PSD) of the vetical displacement at any selected point in the soil fo a oughness ecitation of a unit value (i.e. white noise). The softwae also calculates the Insetion Gain (IG) which is the atio between the PSD displacement befoe and afte changing paametes of the tack, tunnel o soil. The latest vesion of the softwae plots the displacement contous aound the tunnel and accounts fo a bedock laye below the tunnel using the mio-image method [8-10]. A model based on the PiP model, Geen s functions fo a full-space and Geen s functions fo a half-space is pesented to calculate vibation fom a tunnel embedded in a half-space [11]. The model assumes that the tunnel nea-field vibation is not influenced by the eistence of a feesuface. This allows calculating the displacements at the tunnel-soil inteface using a model of a tunnel embedded in a full-space. This model is futhe developed in [12] to impove the computations involved. A vaiant to the PiP model is employed to model a continuum and Geen s functions in closed-fom ae deived in the wavenumbe-fequency domain fo a load in the feesuface. These functions ae used to calculate vibation only in the fee-suface by using Betti theoy of ecipocity. This esults in a significant impovement in the unning-time of the model. In this pape, the wok pesented in [12] is etended to account fo a tunnel embedded in a multilayeed half-space. The fa-field displacement is calculated in thee steps. In the fist step, the displacements at the tunnel-soil inteface ae calculated with the PiP model consideing a tunnel embedded in a full-space. In the second step, the PiP model is used to account fo an infinite space which has the same popeties of the soil that contains the tunnel. Tansfe functions ae calculated Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 2

3 fo two concentic cylindical sufaces in the full-space, with the adius of the oute suface equal to the etenal adius of the tunnel wall. The tansfe functions ae used to calculate foces at the intenal suface that would poduce displacements at the oute suface equal to those calculated by the model of the tunnel in a full-space. In the thid step, the fa-field displacement is calculated using the intenal foces calculated in the second step and Geen s functions fo a multi-layeed half-space. The est of this pape falls into thee sections. Section 2 pesents an outline of the model and descibe the thee steps followed to calculate vibation in the fa-field as mentioned in the pevious paagaph. In Section 3, the main assumption used in these calculations is checked and esults of the model pesented in this pape ae compaed to those of an altenative coupled Finite-Element- Bounday-Element (FE-BE) method. Finally Section 4 povides conclusions fo the wok pesented in this pape. 2. Desciption of the model In this pape, the vetical displacement at any point in a multi-layeed half-space due to a load ~ i( ξ +ωt ) applied on the tunnel invet is calculated. The load takes the fom F = Fe as shown in Fig. 1 ( ) and the displacement takes the fom ~ i ξ +ωt u z = u ze and it is measued along any line paallel to the -ais and passing though a point (y,z). These foms of foce and displacement ae the basis of the analysis in the wavenumbe-fequency domain, which can be used to calculate the displacement fo any sot of loading such as a concentated hamonic load, see [13] fo eample fo moe details. z y d ~ i( ξ +ωt) Fe Fig. 1. Layout of the model descibed in this pape. Analysis is pefomed fo a tunnel embedded in a multi-layeed half-space. The figue is showing a single laye on a half-space. Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 3

4 As descibed befoe, to calculate the fa-field displacement fom a tunnel embedded in a multilayeed half-space, thee steps ae followed. These steps ae descibed in the following subsections Calculating displacements at the tunnel-soil inteface The fist step in the calculations is descibed in this section. The PiP model is employed to calculate the displacements at the tunnel-soil inteface using a model of a tunnel embedded in a fullspace. The PiP model is shown in Fig. 2 and consists of two concentic pipes. The inne pipe accounts fo the tunnel wall with inne adius t and oute adius c. The oute pipe accounts fo the suounding soil with an inne adius c and oute adius of infinite etent. In this pape, both the tunnel wall and the suounding soil ae modelled using the theoy of elastic continuum in cylindical coodinates. Note that the tunnel wall can be modelled as a thin cylindical shell as in efeence [4]. The theoy of elastic continuum is adopted as it gives moe accuacy with no significant loss of the computational efficiency. A summay of the elastic continuum equations is given in [4,6,12]. θ t c Tunnel Soil Fig. 2. Modelling a tunnel embedded in a full-space using the PiP model Calculating the intenal souce in a full-space The PiP model is used in this step to model a full-space as shown in Fig. 3. The inne pipe epesents a solid cylinde with adius < and the oute pipe is an infinite domain with a cylindical cavity with intenal adius i i c and etenal adius of infinite etent. The obective of the Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 4

5 second step is to calculate the stesses at = i that poduce the same displacement at = c as calculated by the fist step. Full details about this step can be found in [12]. θ i Solid Cylinde A full-space with a cylindical cavity Fig. 3. Modelling a full-space using the PiP model Calculating the fa-field displacements The pevious sections show the calculations of the stesses at a vitual cylinde with adius in the full-space. These stesses ae integated at M numbe of lines to give the foces as shown in Fig. 4. Note that due to symmety, only foces at the left half ae calculated. These foces ae moved to a multi-layeed half-space model to calculate the fa-field displacements. The Geen s functions fo a multi-layeed half-space ae calculated following the diect stiffness method [14], the dynamic equilibium of the layeed half-space is epessed in the fequency-adial wavenumbe domain ( k, ω) as: ~ Ku~ = p ~ (1) whee K ~ e is the stiffness mati and it is assembled fom element stiffness matices K ~ in a simila e way as in the Finite Element method. The element stiffness matices K ~ elate the displacements and the tactions at the boundaies of a homogeneous laye o a half-space element. The vectos u ~ and p ~ denote the displacements and the etenal tactions at the intefaces between elements. Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 5 i

6 y z F F M M 1 F F 2 θ F 1 Fig. 4: The foces ae calculated at M lines at the left half of the of the full-space model as those on the ight half can be ~ i( ξ +ωt ) ~ ~ ~ ~ T computed by symmety. F = F, F = [,,. denotes the component in the longitudinal diection, i.e. into the page. e F F F ] y z Eq. (1) is used to calculate the Geen's functions ~ G u ( z', k, z, ω i ) of the layeed half-space fo each adial wavenumbe k and each angula fequency ω. These functions epesent the displacements at a depth z in the diection e due to a unit load applied at a depth z' in the diection e i. The Geen's functions uˆ G i ( z',, y, z, ω) in the fequency-spatial domain ae subsequently obtained by means of an invese Hankel tansfomation in the adial diection and an invese Fouie seies epansion in the cicumfeential diection [14]. These functions epesent the displacements at the position (, y, z) in the diection e due to a hamonic point load applied at the position ( 0,0, z' ) in the diection ei. Finally, the Geen's functions uˆ G i ( z',, y, z, ω) ae tansfomed to the fequency-wavenumbe domain ( k,, ω) using the following equation k y u~ G i ( z', k, k y, z, ω) = 2π 0 0 uˆ G i ( z', cosθ, sinθ, z, ω) e ik cosθ + ik sinθ y dθd (2) Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 6

7 The invese Hankel tansfomation yielding the Geen's functions in the fequency-spatial domain and the integal tansfomation in Eq. (2) ae elaboated analytically in ode to obtain a elation between the Geen's functions ~ G u ( z', k, z, ω i ) and ~ G u ( z', k, k, z, ω i y ). In this way, the numeical evaluation of integal tansfomations is avoided, allowing fo an efficient implementation. The Geen s functions ae used along with the foces calculated fom step 2 to calculate the displacements at any point in the gound. This is done in the same way as descibed in [12] taking advantage of symmety aound the tunnel centeline. Table 1 Tunnel Paametes Etenal adius Thickness P-wave velocity S-wave velocity Density Loss facto (m) (m) (m/s) (m/s) (kg/m 3 ) Results and discussions In this section, the nea-field and fa-field displacements of a tunnel embedded in a layeed halfspace ae calculated fo a hamonic load applied at the invet of the tunnel, at (=0,y=0,z=16.75). The displacements ae calculated by the new model and then compaed with those calculated by the coupled FE-BE model. The coss-section of the tunnel and the gound at =0 is shown in Fig. 4. Paametes of the tunnel and the gound ae given in Tables 1 and 2 espectively. The suface laye has popeties of type 1 of soil and the half-space has popeties of type 2 as pesented in Table 2. The loss factos given in the tables ae associated with both Lame s constants. The displacements ae calculated at 4 points: A (=0 m, y=0 m, z=17 m); B (=0 m, y=0 m, z=11 m); C (=10 m, y=10 m, z=6 m); and D (=10 m, y=10 m, z=0 m). y 6 m 8 m z Soil (type 1) Soil (type 2) 3 m i t e ω Fig. 5: A coss-section of the tunnel and the gound at =0. A hamonic load is applied at the tunnel invet at (=0,y=0,z=16.75) and the displacements ae calculated at 4 points A (0,0,17), B (0,0,11), C(10,10,6), and D(10,10,0). Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 7

8 Table 2 Soil Paametes Type P-wave velocity (m/s) S-wave velocity (m/s) Density (kg/m 3 ) Loss facto The fequency ange of the esults pesented hee is 1-80 Hz with a step of 1 Hz. Results of the new model ae calculated using a pesonal compute (PC) with 2 GB RAM and 2.0 GHz pocesso. The FE-BE esults ae calculated by one pocesso of the high pefomance cluste (HPC) at K.U. Leuven. Highe fequencies ae not attempted due to the high computational cost of the FE-BE model as a esult of the fine mesh equied at high fequencies. The cuent fequency ange is the peceptible ange fo gound-bone vibation in buildings and it is sufficient fo the sake of validation of the new model. Fo fequencies above 80 Hz, the new model is believed to poduce moe accuate esults because the stiffness at the tunnel-soil inteface is moe dominated by the stiffness of the tunnel athe than the stiffness of the soil. The stiffness of the tunnel and the tunnelsoil inteaction ae well accounted fo by the PiP model. Figs. 6.a and 6.b show the displacement at the tunnel-soil inteface at the tunnel invet and the tunnel ape espectively. The esults ae calculated by the PiP model and the coupled FE-BE method. A good ageement is obseved which confims that the nea-field vibation is not influenced by layes away fom the tunnel (a) point A (b) point B disp. [db ef m/n] disp. [db ef m/n] fequency [Hz] fequency [Hz] Fig. 6: The nea-field displacements at (a) point A with coodinates (0, 0, 17) and (b) at point B with coodinates (0, 0, 11) as calculated by the PiP model (continuous) and the coupled FE-BE model (dotted). Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 8

9 200 point C 200 point D disp. [db ef m/n] (a) fequency [Hz] disp. [db ef m/n] (b) fequency [Hz] Fig. 7: The fa-field displacements at (a) point C with coodinates (10, 10, 6) and (b) at point D with coodinates (10, 10, 0) as calculated by the new model (continuous) and the coupled FE-BE model (dotted). Figs. 7.a and 7.b show the fa-field displacements at point C at the inteface of the two types of soil and point D at the fee-suface. These esults confim that the new model calculates the fa-field vibation fo a tunnel embedded in a layeed-gound with a good accuacy. The unning time fo the new model to poduce the esults in Fig. 7.a is appoimately 1.5 minutes on the pesonal compute descibed above. The same esults ae calculated using the coupled FE-BE method in appoimately 17 hs on the HPC. This demonstates clealy the computational efficiency of the new method. The eample above confims that the nea-field vibation is contolled by the dynamics of the tunnel and the soil laye suounding the tunnel. The wok is cuently unde development to study the effect of deceasing the depth of the tunnel and/o the distance between the tunnel and the neaest laye that does not contain the tunnel on the nea-field vibation. This will help detemining the limitations of the method. This method will be put fowad fo the new vesion of the PiP softwae [9]. 4. Conclusions A new method fo calculating vibation fom a ailway tunnel embedded in a multi-layeed halfspace has been pesented. The method is based on the PiP model whee the nea-field vibation is calculated using a model of a tunnel embedded in a full-space. The fa-field displacements ae calculated using Geen s functions fo a multi-layeed half-space. The new method is validated against the coupled FE-BE method fo a tunnel embedded in a layeed half-space. The new method shows a good accuacy with a significant eduction in the unning time and the computational equiements compaed to the coupled FE-BE method. Hussein, Hunt, Rikse, Gupta, Degande, Talbot, Fancois and Schevenels 9

10 Refeences [1] X. Sheng, C.J.C. Jones and D.J. Thompson, Pediction of gound vibation fom tains using the wavenumbe finite and bounday element methods. Jounal of Sound and Vibation 293,3-5, (2006) [2] D. Clouteau, M. Anst, T.M. Al-Hussaini, G. Degande, Feefield vibations due to dynamic loading on a tunnel embedded in a statified medium. Jounal of Sound and Vibation 283, 1-2, (2005) [3] G. Degande, D. Clouteau, R. Othman, M. Anst, H. Chebli, R. Klein, P. Chatteee and B. Janssens, A numeical model fo gound-bone vibations fom undegound ailway taffic based on a peiodic finite element bounday element fomulation. Jounal of Sound and Vibation 293, 3-5, (2006) [4] J.A. Foest, H.E.M. Hunt, A thee-dimensional model fo calculation of tain-induced gound vibation. Jounal of Sound and Vibation 294, 4-5, (2006) [5] J.A. Foest, H.E.M. Hunt, Gound vibation geneated by tains in undegound tunnels. Jounal of Sound and Vibation 294, 4-5, (2006) [6] M.F.M. Hussein, H.E.M. Hunt, A numeical model fo calculating vibation fom a ailway tunnel embedded in a full-space. Jounal of Sound and Vibation, in pess. [7] S. Gupta, M.F.M. Hussein, G. Degande, H.E.M. Hunt and D. Clouteau, A compaison of two numeical models fo the pediction of vibations fom undegound ailway taffic. Soil Dynamics and Eathquake Engineeing 27, 7, (2007) [8] M.F.M. Hussein, H.E.M. Hunt, The PiP model, a softwae application fo calculating vibation fom undegound ailways, Poceeding of the fouteenth Intenational Congess on Sound and Vibation (ICSV14), 9-12 July 2007, Cains, Austalia. [9] The PiP model: [10] L. Rikse, H.E.M. Hunt, M.F.M. Hussein, G. Degande and S. Gupta, A model fo calculating vibation fom a ailway tunnel buied in a full-space including igid bedock. Poceeding of the fouteenth Intenational Congess on Sound and Vibation (ICSV14), 9-12 July 2007, Cains, Austalia. [11] M.F.M. Hussein, S. Gupta, H.E.M. Hunt, G. Degande and J.P. Talbot, An efficient model fo calculating vibation fom a ailway tunnel buied in a half-space. Poceeding of the thiteenth Intenational Congess on Sound and Vibation (ICSV13), 2-6 July 2006, Vienna, Austia. [12] M.F.M. Hussein, S. Gupta, H.E.M. Hunt, G. Degande and J.P. Talbot, A computationally efficient model fo calculating vibation fom a ailway tunnel buied in a half-space. Intenational Jounal fo Numeical Methods in Engineeing, submitted fo publication. [13] M.F.M. Hussein, H.E.M. Hunt, Modelling of floating-slab tacks with continuous slabs unde oscillating-moving loads, Jounal of Sound and Vibation 297, 1-2, (2006) [14] E. Kausel and J.M. Roesset. Stiffness matices fo layeed soils. Bulletin of the Seismological Society of Ameica, 71, 6, (1981), Hussein, Hunt, Rikse, Gupta, Deg ande, Talbot, Fancois and Schevenels 10