2.5 D BEM modelisation of ground structure interaction

Transcription

1 paper ID: 510/p D BEM modelisation of ground structure interaction Philippe JEAN CSTB, 24 rue Joseph Fourier, Saint Martin d Hères, France, jean@cstb.fr 2.5 D Green functions of continuous media have been implemented in a 2D BEM code. In 2D, sources are infinite coherent lines, whereas real sources such as tramways are closer to finite incoherent lines excitations. Taking into account 2.5D Green functions written for a given wavenumber along the infinite direction corresponds to the arrival of incoming waves with a non perpendicular angle. Coincidence effects can then be considered. Point excitations are obtained by summing the solutions obtained for different wavenumbers. Applications to ground/structure problems are given. 1. INTRODUCTION: 2D AND 2.5 D MODELS The study of ground structure interaction due to surface waves (trains or tramways) can be undertaken using a full BEM approach both for the ground and structure media. In [1], Ph. JEAN presents a 2D FEM/BEM program called MEFISSTO and several applications. The problem is sub-divided into several domains, each finite domain can be modelled either with FEM (Finite Element Method which imply surface meshings, or BEM (Boundary Element Method) which lead to boundary-line meshings in 2D. Doing 2D calculations has several obvious advantages: domain or boundary meshings are easily implemented and computations times are reasonable so that large scale problems can be studied for the frequency range of interest, which, when dealing with railway-like problems, concerns frequencies between 20 and 200 Hz. Two dimensional xy- problems suppose that the problem is invariant along a third perpendicular z direction; which imply infinite geometries of constant shape along z and probably more important, that excitations are constant along z, generating cylindrical waves: a 2D point excitation is in fact a coherent line excitation along z, so that all waves will impinge on the structure at a normal angle. No angular effects can be modelled with a 2D approach. It is clear that a real source a train for instance- will inject power at several positions along the track, along a finite length and certainly not in a coherent fashion. On the other hand, a 2.5D model refers to a 2D geometry with point excitations (and consequently incoherent lines). In [2], such a description is presented for aero-acoustics: computations for a point source is simply obtained by doing a Fourier-like integration over

2 paper ID: 510/p.2 solutions obtained from 2D calculations. In [3], TADEU proposes, for a solid medium, a 2.5 D Green function (in terms of 3x3 displacement and stress tensors) written for a given wave number k z along the infinite z direction. The behaviour at each node is thus described by 3 components of displacement and/or stress but meshings remain 2D. For a given value of k z, the integral expression of the 2.5 D displacement u(k Z ) is written has u( M ). I ( M ) = [ T ( Q, M ). u( Q) G( Q, M ). t( Q)] ds + h( E, M ) (1) S where t represents the stress vector at point Q, and G and T are the 3x3 displacement and stress tensors; h is the free field solution due to an excitation placed at point E. Post treatment is, as for the acoustic problem, simply carried by integrating the solutions for the different values of k z. The displacement U at a point x,y,z is expressed as U ( x, y, z) 1 ikz Z = u[ x, y, k Z ] e dk Z 2π (2) where u is the 2D displacement for a given value of k z ; Z=z-z S, for a point source at x S,y S,z S. Note that one must respect the classical aliasing limit of Zmax=π/dk Z, where dkz is the resolution of the k Z sampling. In the case of railway-like sources this leads to more realistic solutions than traditional 2D computations and to much faster calculations than full 3D approaches. Implementation in an already existing 2D code is straightforward, the only difficulty being the new singular terms related to the 2.5 D Green function. A semi analytical scheme has been employed, where the singular terms are first identified and subtracted from the Green tensors. Their evaluation is done analytically. The modified -and no longer singular- terms are evaluated with a Gauss integration technique. Several tests have been made to validate MEFISSTO2.5D. 1) the solution for k z =0 should be equal to the 2D solution (cylindrical waves, normal incidence) 2) Applying (2) on the free-field Green functions (displacement or stresses) gives the well known 3D Green solutions, 3) the solution can be tested against solutions obtained with layered media calculations using transfer matrix approaches [3]. An other simple test can be made in the case of a unit force placed in a semi-infinite medium bounded by a rigid surface where the solution is found to be very close to that of two symmetrical forces (with respect to the rigid surface) placed in an infinite domain. 2. APPLICATION TO A BURIED STRUCTURE A simple structure is considered. It is a 3-sided infinite swimming-pool -like structure made of 20 cm-thick concrete buried in a soil characterised with E= N/m 2, η=0.08, υ=0.25, ρ=1500 kg/m 3 (see figure 2). A vertical force of 1 N is placed at point A. Figure 3 shows the 2.5 D (called 3D) vertical displacement Uy along the ABCDE contour (horizontal axis) [A-B (surface, 4 m)-bc (vertical side, 2m)-CD (bottom, 5 m) DE (rear side)] at 20, 31 Hz and 50 Hz as a function of k Z. On each graph the wavenumber k R of the Rayleigh waves is plotted. Clearly, this wavenumber delimits two zones. Above k R the transmission to the

3 paper ID: 510/p.3 structure is strongly reduced. The integration (2) is therefore not significantly affected by higher wavenumbers. Also, the closer to the source, the higher the maximum number to be considered in carrying (2). Figure 4 shows U y (not referenced) along ABCDE as a function of the z coordinate parallel to the structure. The attenuation along z can be seen (point force). In order to simulate an incoherent line source, one must sum the contributions of uncorrelated point excitations along z. Figure 5 compares results, at 50 Hz, for different lengths of summation along z, referenced to calculations made for a 200 m long line. For a 20 m long line, convergence has been reached for the ABC portion but not beyond C for the Ux component of displacement. Convergence is faster for Uy (vertical component). Results are strongly influenced by the values of ground damping. Computations for finite length sources such as tramways need only be carried for the actual source length. Figure 6, shows the transfer function between the rms horizontal (bending) velocity of the vertical foundation and the rms vertical surface velocity averaged between A+1 m and A+2 m, both for a 2D excitation (infinite coherent excitation) and 3D sources (point and incoherent lines of different lengths). The 3D results show higher values, the longer the 3D source the shorter the difference with the 2D results. Results have converged for a 10 m-long line. Next, a 10 cm thick isolating layer made of polystyrene (E= N/m 2, η=0.05, υ=0, ρ=10 kg/m 3 ) is placed on the BC vertical side facing the excitation. Figure 7, similar to Figure 5, shows the resulting vertical displacement. The effect of the isolating layer can easily be seen. Figure 8 compares 2D and 3D (20 m-long line) values of the rms bending displacement Uy (not referenced) along BC, without and with the isolating layer. Figure 9 compares the efficiency of the lining obtained with infinite 2D and finite 3D line sources of different lengths. The efficiency is of the isolating layer is lower with a 3D computation, the longer the line, the larger the difference. The differences are more pronounced for the bending component. 3. CONCLUSION Using 2.5 D expressions for displacement and stress Green functions allows the computation of more realistic situations than a classical BEM code. Little effort is required to modify an existing 2D code. Introducing such an approach in a FEM model can be done using the spectral finite elements developed by S. FINNVENDEN [5]. REFERENCES 1. P. Jean, Boundary and finite elements for 2D soil-structure interaction problems. Acta Acustica.87, (2001) 2. P. Jean, G. Defrance, Y. Gabillet, The importance of source type on the assessment of noise barriers Journal of Sound and Vibration 226(2), (1999) 3. M. Villot, J. Chanut, Vibrational energy analysis of ground/structure interaction in terms of wave type Journal of Sound and Vibration 231, (2000) 4. A.J.B Tadeu, Green s Functions for two-and-a-half-dimensional elastodynamic problems. Journal of Engineering Mechanics (2002) 5. S. Finnvenden, Tyre vibration analysis with conical waveguide finite elements Internoise 2002, Dearborn US. (2002)

4 paper ID: 510/p.4 z k z x y Figure D geometry z A B E x y C D Figure 2. Geometry of buried structure. Figure 3. Vertical displacement Uy along ABCDE as a function of k Z

5 paper ID: 510/p.5 Figure 4. Vertical displacement along ABCDE as a function of position z. No isolating layer. Figure 5. Effect of length of incoherent line source at 50 Hz. Figure 6. rms foundation bending displacement level referred to vertical surface velocity (rms value along 1 m, 1 m after excitation).

6 paper ID: 510/p.6 Figure 7. Vertical displacement along ABCDE as a function of position z. With isolating layer. 2D no lining 3D no lining 2D lining 3D lining Figure 8. Mean displacement along vertical foundation BC Figure 9. Effect of lining on foundation displacement. 2D and 3D calculations.