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A numerical model for ground-borne vibrations from underground railway traffic based on a periodic FE-BE formulation

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A numerical model is developed to predict vibrations and re-radiated noise in buildings from excita- tion due to metro trains in tunnels. The three-dimensional dynamic tunnel-soil interaction problem is solved with a subdomain formulation, using a
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  A numerical model for ground-borne vibrations from undergroundrailway traffic based on a periodic FE-BE formulation D. Clouteau, R. Othman, M. Arnst, H. Chebli  Ecole Centrale de Paris, LMSSMat, F-92295 Chˆ atenay-Malabry, France G. Degrande, R. Klein, P. Chatterjee, B. Janssens K.U.Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium Abstract A numerical model is developed to predict vibrations and re-radiated noise in buildings from excita-tion due to metro trains in tunnels. The three-dimensional dynamic tunnel-soil interaction problem issolvedwithasubdomainformulation,usingafiniteelementformulationforthetunneland aboundaryelement method for the soil. The periodicity of the geometry is exploited using the Floquet transform,limiting the discretization to a single bounded reference cell. The responses of two different types of tunnel due to a harmonic load on the tunnel invert are compared, both in the frequency-wavenumberand spatial domains. The tunnel of the line RER B of RATP in the Cit´e Universitaire in Paris isa shallow cut-and-cover masonry tunnel embedded in layers of sand. The tunnel of the Bakerlooline of London Underground in Regent’s Park is a deep cut-and-cover tunnel with a cast iron liningembedded in London clay.1  Introduction Within the frame of the EC-Growth project CONVURT [1], a modular numerical prediction tool isdeveloped to predict vibration and re-radiated noise in buildings from excitation due to metro trainsin tunnels for both newly built and existing situations [2].The three-dimensional dynamic tunnel-soil interaction problem is solved with a subdomain for-mulation, using a finite element formulation for the tunnel and a boundary element method for thesoil. The periodicity of the tunnel and the soil is exploited using the Floquet transform, limiting thediscretization to a single bounded reference cell of the tunnel [3, 4].The model will be validated by means of in situ experiments that have been performed at a sitein the Cit´e Universitaire on the line RER B of RATP in Paris and a site in Regent’s Park on theBakerloo line of London Underground. The tunnel in Paris is a shallow cut-and-cover masonry tunnelwith two tracks, embedded in layers of sand, gravel and marl, while the tunnel in London is a deepcut-and-cover tunnel with a cast iron lining and a single track, embedded in homogeneous Londonclay.After a review of the governing system of equations, details on the geometry and construction of both tunnels are presented. The response due to a harmonic load on the tunnel invert is compared,allowing to draw conclusions on the dynamic behaviour of both tunnel-soil systems. It is demon-strated how the vibration isolation efficiency of a floating slab track can efficiently be computed usinga Craig-Bampton substructuring technique [5].1  2  Dynamic tunnel-soil interaction model 2.1 Problem outline The three-dimensional dynamic soil-tunnel interaction problem is assumed to be periodic with period   in the longitudinal direction ✁ ✂   along the tunnel axis and can be restricted to periodic fields of thesecond kind defined on a reference cell ✄ ☎   (figure 1). The boundary ✆ ✄ ☎   of this domain is decomposedinto the free surface ✄ ✝ ✞   and the boundaries ✟ ✠   and ✟ ✡   on which periodic conditions are imposed.The generic cell ✄ ☎   is decomposed into two subdomains: the soil ✄ ☎☛   and the tunnel ✄ ☎ ☞   . The interfacebetween these subdomains is denoted by ✄ ✟ ☞ ☛   . The boundary ✄ ✝ ☛ ✞   is the free surface of the soil, whilea surface force ✌ ✍ ☞   is applied on ✄ ✝ ☞ ✞   (figure 1).Figure 1: Problem outline and notations.The position vector ✎   of any point in the problem domain ☎   is decomposed as ✎✏ ✌ ✎✑✒  ✁ ✂   ,where ✌ ✎   is the position vector in the reference cell ✄ ☎   and ✒   is the cell number. The Floquet trans-formation ✄ ✓✔ ✌ ✎✕✖✗   of a non-periodic function ✓✔ ✎✗✏ ✓✔ ✌ ✎✑✒  ✁ ✂ ✗   defined on a three-dimensionaldomain ☎   , that is periodic in the direction ✁ ✂   with period    , transforms the distance ✒    between the ✒  -th cell and the reference cell ✄ ☎   to the wave number ✖   and is defined as [4]: ✄ ✓✔✌ ✎✕✖✗✏ ✘✙ ✚   ✛✜✢ ✙ ✓✔✌ ✎✑✒  ✁ ✂ ✗✣ ✤✥ ✔ ✑ ✦ ✒  ✖✗  (1)The function ✄ ✓✔ ✌ ✎✕✖✗   is periodic of the first kind with respect to ✖   with a period ✧ ★✩    and periodic of the second kind with respect to ✌ ✎   : ✄ ✓✔ ✄ ✪ ✕ ✕✄ ✫ ✕✖✗✏✣ ✤✥ ✔ ✬ ✦ ✖  ✗ ✄ ✓✔ ✄ ✪ ✕✭✕✄ ✫ ✕✖✗  (2)The function ✓✔ ✌ ✎✑✒  ✁ ✂ ✗   can be reconstructed for any ✎✏ ✌ ✎✑✒  ✁ ✂   using the inverse Floquettransform: ✓✔ ✌ ✎✑✒  ✁ ✂ ✗✏  ✧ ★✮  ✘ ✯✰ ✡ ✢ ✯✰ ✡ ✄ ✓✔ ✌ ✎✕✖✗✣ ✤✥ ✔ ✬ ✦ ✒  ✖✗  d ✖   (3)2  2.2 Navier equations Using the Floquet transformation, all displacement and traction fields    ✔ ✎✕ ✁   ✗   and ✂  ✔ ✎✕ ✁   ✗   defined onthe periodic domain ☎   are transformed to the fields ✌     ✔ ✌ ✎✕✖✕ ✁   ✗   and ✌  ✂  ✔ ✌ ✎✕✖✕ ✁   ✗   defined on the genericcell ✄ ☎   .The Navier equations in the soil domain ✄ ☎☛   and the boundary conditions on ✄ ✝ ☛ ✞   are written asfollows for every frequency ✁✄   IR and wave number ✖  ✄ ☎  ✬ ★✩  ✕✑  ★✩   ✆   :div ✌  ✝   ☛✔ ✌     ☛ ✗✏  ✬  ✞  ☛  ✁✟  ✌     ☛   in ✄ ☎☛   (4) ✌  ✂  ☛✔ ✌     ☛ ✗✏   0  on ✄ ✝ ☛ ✞   (5) ✌     ☛✔ ✌ ✎✗✏✣ ✤✥ ✔ ✬ ✦ ✖  ✗ ✌     ☛✔ ✌ ✎ ✬  ✁ ✂ ✗  on ✟ ✡   (6)together with the radiation conditions on the displacement field.ThefollowingNavierequations and boundary conditionsholdin the tunneldomain ✄ ☎ ☞   with bound-ary ✄ ✝ ☞ ✞   :div ✌  ✝   ☞ ✔✌     ☞ ✗✏  ✬  ✞  ☞  ✁✟  ✌     ☞   in ✄ ☎ ☞   (7) ✌  ✂  ☞ ✔ ✌     ☞ ✗✏  ✌ ✍ ☞  on ✄ ✝ ☞ ✞   (8) ✌     ☞ ✔ ✌ ✎✗✏✣ ✤✥ ✔ ✬ ✦ ✖  ✗ ✌     ☞ ✔ ✌ ✎ ✬  ✁ ✂ ✗  on ✟ ✡   (9)Continuity of displacements and equilibrium of stresses must hold on the tunnel-soil interface ✄ ✟ ☞ ☛   : ✌     ☞ ✏  ✌     ☛  on ✄ ✟ ☞ ☛   (10) ✌  ✂  ☞ ✔✌     ☞ ✗✑  ✌  ✂  ☛✔✌     ☛ ✗✏   0  on ✄ ✟ ☞ ☛   (11) 2.3 Weak variational formulation Multiplying the Navier equations (4) and the boundary equations (5) and (6) with the complex con- jugate of any virtual field ✌  ✠   ☞ ✔✌ ✎✕✖✗   , integrating over the problem domain and the boundaries, andintegrating by parts, the following weak variational form is obtained for the tunnel: ✮   ✡ ☛☞  ✌  ✌  ✔ ✌  ✠   ☞ ✗  ✍ ✌  ✝   ✔ ✌     ☞ ✗  ✎✏   ✬  ✁✟  ✮   ✡ ☛☞ ✞  ☞ ✌  ✠   ☞  ✑ ✌     ☞  ✑✎✏   ✏  ✮   ✒ ✡ ☛☞  ✌  ✠   ☞  ✑ ✌  ✂  ☞ ✔ ✌     ☞ ✗  ✎✓   ✑  ✮   ✡ ✔ ☞ ✕  ✌  ✠   ☞  ✑ ✌ ✍ ☞  ✎✓   (12)where the boundary ✆ ✄ ☎ ☞   can be decomposed into the tunnel-soil interface ✄ ✟ ☞ ☛   and the two boundaries ✟ ✠  and ✟ ✡   at the two edges ✄  ✖  ✏   ✗    ✩ ✧   of the generic cell. As the actual and the virtual displacementfields are periodic of the second kind, the contribution of the integral on the sum of the boundaries ✟ ✠  and ✟ ✡   vanishes [3]. Accounting for the stress equilibrium (11) along the tunnel-soil interface ✄ ✟ ☞ ☛   ,the weak variational equation becomes: ✮   ✡ ☛☞  ✌  ✌  ✔ ✌  ✠   ☞ ✗  ✍ ✌  ✝   ✔ ✌     ☞ ✗  ✎✏   ✬  ✁✟  ✮   ✡ ☛☞ ✞  ☞ ✌  ✠   ☞  ✑ ✌     ☞  ✑✎✏   ✑  ✮   ✡ ✘ ☞✙  ✌  ✠   ☞  ✑ ✌  ✂  ☛✔ ✌     ☛  ✚  ✔ ✌     ☞ ✗✗  ✎✓   ✏  ✮   ✡ ✔ ☞ ✕  ✌  ✠   ☞  ✑ ✌ ✍ ☞  ✎✓   (13)where ✌     ☛  ✚  ✔ ✌     ☞ ✗   denotes the wave field that is scattered by the tunnel into the soil that obeys displace-ment continuity (10) along the tunnel-soil interface ✄ ✟ ☞ ☛   .3  2.4 Coupled periodic finite element - boundary element formulation Asthetunnelisbounded,thedisplacementfield ✌     ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   can bedecomposedonabasisoffunctions ✌  ✁   ✔ ✌ ✎✕✖✗  that are periodic of the second kind: ✌     ☞ ✔✌ ✎✕✖✕ ✁   ✗✏  ✂   ✚    ✁   ✜  ✄  ✌    ✁   ✔✌ ✎✕✖✗  ☎  ✁   ✔ ✖✕ ✁   ✗✏ ✌  ✆   ☞  ✝   ☞  (14)The modes ✌    ✁   ✔ ✌ ✎✕✖✗   are periodic of the second kind and constructed as follows from the periodic (of the first kind) eigenmodes ✌     ✠  ✁   ✔ ✌ ✎✗   of the reference cell: ✌    ✁   ✔ ✌ ✎✕✖✗✏✣ ✤✥ ✔ ✬ ✦ ✖✁ ✂  ✑ ✌ ✎✗ ✌     ✠  ✁   ✔ ✌ ✎✗  (15)The soil displacements ✌     ☛✔✄   x ✕✖✕ ✁   ✗   can be written as the superposition of waves that are radiated bythe tunnel into the soil: ✌     ☛✔ ✌ ✎✕✖✕ ✁   ✗✏ ✌     ☛  ✚  ✔ ✌     ☞ ✗ ✔ ✌ ✎✕✖✕ ✁   ✗ ✏  ✂   ✚    ✁   ✜  ✄  ✌     ☛  ✚  ✔ ✌    ✁   ✗ ✔ ✌ ✎✕✖✕ ✁   ✗  ☎  ✁   ✔ ✖✕ ✁   ✗✏  ✂   ✚    ✁   ✜  ✄  ✌     ✞ ✁   ✔ ✌ ✎✕✖✕ ✁   ✗  ☎  ✁   ✔ ✖✕ ✁   ✗  (16)The numerical solution of the dynamic tunnel-soil interaction problem is obtained using the clas-sical domain decomposition approach based on the finite element method for the structure and theboundary element method for the soil. The displacements in the structure ✌     ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   are interpolatedas: ✌     ☞  ✟✠     ☞ ✏  N ☞ ✌     ☞ ✏   N ☞ ✌  ✆   ☞  ✝   ☞   (17)Employing the same approximation for the virtual displacements ✌  ✠   ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   , the following system of equations is finally obtained: ✡  K ☞ ✔ ✖✗ ✬  ✁  ✟   M ☞ ✔ ✖✗✑   K ☛✔ ✖✕ ✁   ✗  ☛ ✝   ✔ ✖✕ ✁   ✗✏   F ☞ ✔ ✖✕ ✁   ✗   (18)where  K ☞ ✔ ✖✗   and  M ☞ ✔ ✖✗   are the projections of the finite element stiffness and mass matrices on thetunnel modes: K ☞ ✔ ✖✗✏ ✌  ✆   T ☞  K FE ☞ ✌  ✆   ☞ ✏ ✌  ✆   T ☞ ✮   ✡ ☛☞  ✔  LN ☞ ✗   T D ✔   LN ☞ ✗  ✎✏   ✌  ✆   ☞  M ☞ ✔ ✖✗✏ ✌  ✆   T ☞  M FE ☞ ✌  ✆   ☞ ✏ ✌  ✆   T ☞ ✮   ✡ ☛☞  N T ☞  ✞   ☞  N ☞  ✎✏   ✌  ✆   ☞   (19) F ☞ ✔ ✖✕ ✁   ✗   is the generalized force vector applied on the tunnel invert: F ☞ ✔ ✖✕ ✁   ✗✏ ✌  ✆   T ☞ ✮   ✡ ✔ ☞ ✕  N T ☞ ✌ ✍ ☞  ✎✓   (20)and  K ☛✔ ✖✕ ✁   ✗   is the dynamic stiffness matrix of the soil: K ☛✔ ✖✕ ✁   ✗✏ ✮   ✡ ✘ ☞✙  ✌  ✆   T ☞  N T ☞ ✌  ✂  ☛✔ ✌     ☛  ✚  ✔  N ☞ ✌  ✆   ☞ ✗✗  ✎✓    (21)The stresses ✌  ✂  ☛✔ ✌     ☛  ✚  ✔   N ☞ ✌  ✆   ☞ ✗✗   on the tunnel-soil interface are calculated with a periodic boundary ele-ment formulation with Green-Floquet functions defined on the periodic structure with period    alongthe tunnel [2, 3].4  2.5 Craig-Bampton substructuring method  In order to analyze the influence of different track structures in the tunnel on the vibrations generated,it is advantageous to differentiate between the degrees of freedom of the tunnel invert and the track.Therefore, the displacement vector ✌     ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   of the tunnel is discretized alternatively as followsusing a Craig-Bampton substructuring method [5]: ✌     ☞  ✟✠     ☞ ✏   N ☞ ✌     ☞ ✏   ✡   N ☞     N ☞  ☞  ☛  ✁  ✌     ☞    ✌     ☞  ☞  ✂  ✏   ✡   N ☞     N ☞  ☞  ☛  ✄  ✌  ✆   ☞    ✌  ✆   ☛ ☞    0 ✌  ✆   ☞  ☞  ☎ ✁  ✝   ☞    ✝   ☞  ☞  ✂  (22)where the subscripts ✆✝   and ✆  ☞   refer to the track and the tunnel invert, respectively. The modes inequation (22) are periodic of the second kind and constructed from periodic modes of the first kind: ✄  ✌  ✆   ☞    ✌  ✆   ☛ ☞    0 ✌  ✆   ☞  ☞  ☎  ✏   ✄✞  ☞    ☞    00 ✞  ☞  ☞  ☞  ☞  ☎✟  ✌  ✆   ✠ ☞    ✌  ✆   ☛ ✠ ☞    0 ✌  ✆   ✠ ☞  ☞  ✠  (23)where the diagonal matrices ✞  ☞    ☞     and ✞  ☞  ☞  ☞  ☞   are constructed according to equation (15). The modes ✌  ✆   ✠ ☞    are the eigenmodes of the track clamped at the tunnel invert. The modes ✌  ✆   ✠ ☞  ☞   are the modes of the free tunnel without track. The displacements ✌  ✆   ☛ ✠ ☞     are the quasi-static transmission of the tunnelmodes into the track, computed as: ✌  ✆   ☛ ✠ ☞    ✏  ✬  ✡  K FE ☞    ☞  ☛  ✢  ✄  K FE ☞    ☞  ☞  ✌  ✆   ✠ ☞  ☞  (24)where  K FE ☞    ☞    and  K FE ☞    ☞  ☞  are block submatrices of the finite element stiffness matrix  K FE of the tunnel.Introducing the decomposition (22) into equation (13) results: ☞ ✌ ✍ ✄  ✌  ✆   ☞    ✌  ✆   ☛ ☞    0 ✌  ✆   ☞  ☞  ☎  T ✎ ✄  K FE ☞    ☞    K FE ☞    ☞  ☞  K FE ☞  ☞  ☞    K FE ☞  ☞  ☞  ☞  ☎  ✬  ✁  ✟  ✄  M FE ☞    ☞    M FE ☞    ☞  ☞  M FE ☞  ☞  ☞    M FE ☞  ☞  ☞  ☞  ☎✏ ✄  ✌  ✆   ☞    ✌  ✆   ☛ ☞    0 ✌  ✆   ☞  ☞  ☎  ✑   ✄  0 00 K ☛  ☎ ✂ ✁  ✝   ☞    ✝   ☞  ☞  ✂  ✏   ✁  F ☞    0 ✂   (25) K ☛✔ ✖✕ ✁   ✗   is the dynamic stiffness matrix of the soil: K ☛✔ ✖✕ ✁   ✗✏  ✮   ✡ ✘☞✙  ✌  ✆   T ☞  ☞  N T ☞  ☞  ✌  ✂  ☛✔ ✌     ☛  ✚  ✔  N ☞  ☞  ✌  ✆   ☞  ☞  ✗✗  ✎✓    (26) F ☞    ✔ ✖✕ ✁   ✗   is the generalized force vector applied on the track: F ☞    ✔ ✖✕ ✁   ✗✏  ✌  ✆   T ☞    ✮   ✡ ✔ ☞ ✕  N T ☞    ✌ ✍ ☞    ✎✓   (27)The impedanceofthe soilin equation (25)is only influenced by the tunnelmodes and does not changewhen a calculation is made for another track structure in the tunnel. 2.6 Wave propagation in the soil When thedisplacements ✌     ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   and thestresses ✌  ✂  ☞ ✔ ✌ ✎✕✖✕ ✁   ✗   onthetunnel-soilinterfaceare known,the incident wave field ✌      inc ✔ ✌  ✑  ✕✖✕ ✁   ✗  is obtained by application of the dynamic representation theoremin the unbounded soil domain corresponding to the reference cell: ✄  ✒  inc ✓  ✔ ✌  ✑  ✕✖✕ ✁   ✗✏  ✮   ✡ ✘ ☞✙  ✄  ✒   GF ✓✔  ✔ ✌  ✑  ✕✌ ✎✕✖✕ ✁   ✗ ✄  ✆  ☞  ✔  ✔ ✌ ✎✕✖✕ ✁   ✗ ✬ ✄  ✆   GF ✓✔  ✔ ✌  ✑  ✕✌ ✎✕✖✕ ✁   ✗ ✄  ✒  ☞  ✔  ✔ ✌ ✎✕✖✕ ✁   ✗  ✎   ✟  (28)with ✄  ✒   GF ✓✔  ✔ ✌  ✑  ✕✌ ✎✕✖✕ ✁   ✗  and ✄  ✆   GF ✓✔  ✔ ✌  ✑  ✕✌ ✎✕✖✕ ✁   ✗  the Green-Floquet tensors [3, 4]. The incident wave field inthe soil is obtained by evaluating the inverse Floquet transform (3).5
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