Canyon Geometry Effects on Seismic SH-Wave scattering using three dimensional BEM

Transcription

1 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) Canyon Geometry Effects on Seismic SH-Wave scattering using three dimensional BEM REZA TARINEJAD* and MOHAMMAD T. AHMADI *Faculty of Engineering Islamic Azad University- Ahar Branch East Azerbaijan Province, Ahar IRAN r_tarinejad@tabrizu.ac.ir Abstract: - Topographic conditions play an important role on the modification of seismic ground motion. In this research, topography effects of canyon sites are analyzed using a three-dimensional boundary element procedure. In this research Effects of canyon geometry on seismic wave scattering are investigated. It is demonstrated that the effect of canyon shape and canyon depth on the topographic amplification is frequency dependent. Deep canyon induces large amplification effect rather than shallow canyon in different frequencies. For high dimensionless frequency the obtained results is more sensitive to increasing of the depth of the canyon in comparison to the low dimensionless frequency. The variation of displacement across the deep canyon is more complicated than the shallow one for the same frequency. Different analyses show that the z-component of displacement is more sensitive to the depth variation and shows large amplification in comparison to the other components. Key-Words: - Scattering, Seismic Wave, Boundary Element Method, Topographic Amplification, Prismatic Canyon, D-model Introduction The effects of surface topography can greatly enlarge the site response exerting an important influence on the distribution of damage observed during earthquakes []. In the last decades the evaluation of topographic effects was done via different approaches. Different analytical and numerical techniques were adopted to deal with topographic effects on the seismic wave scattering problems [(e.g., Trifunac [], Zhang [], Luco [4], Athanasopoulos [5], Assimaki [6], and Geli [7]). In this research the three-dimensional boundary element method is used to study the amplification of elastic waves by a three dimensional canyon. Incident plane harmonic SH-waves are considered. The accuracy of the method is tested through comparison with results of other studies, and effects of canyon geometry are investigated characterized by the P and S wave velocities and c s, respectively. c p Topography An arbitrarily shaped canyon of finite length is considered as illustrated in Figure. Seismic body waves arrive from an arbitrary direction with angles of θ h and θ v respect to the horizontal x- and the vertical z-axes, respectively. The half-space is Fig.. The topographic system considered is an arbitrarily shaped canyon of finite length with the incident waves arriving from an arbitrary direction. ISSN: ISBN:

2 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) To compute the total displacement at the canyon site due to incident body waves, the following four steps should be performed:. Determine the ground motion for free field conditions ( u ff ) for the half-space without the canyon. Closed-form solutions are available for various incident body waves.. Determine the tractions corresponding to the displacements in the previous step.. Apply the opposite of the tractions computed in step at the base of the canyon as a traction boundary condition and determine the scattered displacements ( u s ). 4. Determine the total displacements by superposition of the two displacements obtained in steps and, i.e., u = u + u. u s total In the third step is determined using the multidomain boundary element technique as discussed in the following sections [8]. Background Theory and Boundary Element Method The governing wave equation for an elastic, isotropic and homogeneous body is: u c (. u) c u = b () t in which u denotes the displacement vector, b denotes the body force vector, and c and c are the propagation velocities of compression (P) and shear (S) waves, respectively. The velocities are related to the properties of the medium through:.5.5 c = ( λ + μ / ρ), c = ( μ / ρ) () where λ and μ are the Lame constants and ρ is the mass density [9-]. The boundary element method (BEM) is very effective when dealing with wave propagation problems in infinite media with geometrical irregularities. The main advantage of this method is that discretization is only applied at the boundaries of the physical domain, thus reducing the number of unknown variables significantly in comparison to other methods such as finite element and finite difference techniques. Since the fundamental solutions automatically satisfy the far-field condition, the BEM is especially well-suited to problems involving infinite domains such as a canyon located on a half-space []. The corresponding governing boundary equation for an elastic, isotropic, homogenous body can be ff s obtained using the well-known dynamic reciprocal theorem as: c i u i + p * udγ = u * pdγ () Γ * Γ * where p and u are the fundamental solution for traction and displacement respectively, at a point x when a unit Dirac Delta load is applied at point i. In the BEM the variables u and p are discretized into the values at the so-called collocation nodes. The displacement and traction fields are interpolated over each element using a set of shape functions. The same shape functions are also used to approximate the geometry, i.e. the elements are isoparametric. Dicretiztion of Equation () yields: ne ne * j j d u * Φ Γ = u ΦdΓ p Γj j (4) i i c u + p Γj j= = According to Equation (4) the surface integral is exchanged with a sum of integrals over ne elements. After assembling all equations the following set of equations is obtained: HU=GP (5) in which H is a matrix, U is a displacement vector, G is a matrix, and P is a traction vector, and is the product of the number of elements and number of nodes. The problem has n unknowns that should be obtained by solving Equation (5). 4 Fundamental solutions The fundamental solution in the frequency domain for the displacement is the solution to Eq. () for a harmonic point force with unit amplitude applied at the point y in the l direction, i.e., l ρ b = δ ( r) e (6) where r is the distance between the observation point x and the source point y, the unit vector on l- direction and δ is the Delta Dirac function. In order to find the fundamental displacement solution, the principle of Helmholtz decomposition is used. The fundamental solution expressions are adopted from references [9-]. 5 Numerical Results and Discussion A special-purpose -D computer program (TDASC) was developed to implement the boundary element procedures for an incident plane wave with circular frequency ω. The propagation direction of the wave was defined by angles and corresponding to the angle of the ray (normal to wave front) from the horizontal x- and vertical z-axes (Figure ). The program can be used for canyons of arbitrary shape. The numerical examples of this section are designed ISSN: ISBN:

3 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) to demonstrate the accuracy and efficiency of the method for different cases [8]. 5. Validation study To establish the numerical accuracy of the method, problems involving the scattering of a harmonic plane SH-wave, for which results are available from previous works, are solved using the proposed method. In all cases a semi-circular cross section of radius R cut in a homogenous half space is considered. The semi-cylindrical canyon and a length R of the free field on each side of the canyon and a length 5R along the canyon axis are discretized as shown in Fig.. 5. Effects of canyon geometry In order to investigate the effect of the canyon geometry on the topographic amplification two models of semi-elliptical canyon as illustrated in figure 5 are considered and analyzed. The first model is a shallow canyon with the horizontal large diameter is considered to be R (in the y-direction) and the small diameter is.5r (in the z-direction). The second canyon is a deep canyon with the horizontal small diameter equal to R (in the y-direction) and the large diameter equal to R (in the z-direction). The obtained results are compared with results obtained for the canyon with the semicircular cross section in the same graphs. These models can also be used to investigate the effect of the depth of the canyon due to this fact that the only difference between three models is the depth of points located on the canyon. Fig.. A schematic picture of the descretized semi-cylindrical canyon The model has 4 nine-node boundary elements with 9 nodes along the main boundary. The results obtained by the boundary element (BE) method for a -D SH wave of unit amplitude impinging normal to the canyon axis is compared first with the exact solution by Trifunac []. The results presented in Fig. show the displacement amplitudes around the canyon for the horizontal o angle of θ h = 45 the vertical angles of θ = o and45 o for unit dimensionless frequency v ( Ω = ω R / πc = ). A similar comparison is presented in Figure 4 forθ v = θ h =, which corresponds to a vertically incident SH wave with particle motion perpendicular to the axis of the canyon. For this test case the numerical result obtained by Zhang and Chopra [] is used for the comparison. There is good agreement between results obtained by the BE method and the previous results. The finite length of the canyon and free-field has only a small effect on the computed displacements [8]. (Ux) 5 4 BEM(Tetah=9,Tetav=45) BEM(Tetah=9,Tetav=) Trifunac (Tetah=9,Tetav=) Trifunac (Tetah=9,Tetav=45) Distance(Y/R) Fig.. Displacement amplitudes obtained by the BE method and Trifunac for incident SH-Wave with. Ω = Uy- Zhang & Chopra Uz- Zhang & Chopra Uy- BEM Uz- BEM Fig. 4. Displacement amplitudes obtained by the BE method and Zhang and Chopra for incident SH-Wave with θ h =, θ v = and Ω =. ISSN: ISBN:

4 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) results are shown in the figures 9 and for two different dimensionless frequencies. The same results can be achieved for this kind of geometry. Deep canyon induces large amplification in comparison with the shallow one. On the other hand disturbance of deep canyon on the seismic waves are more than the shallow canyon. This is because of this fact that seismic waves detect deep canyon better than shallow one. For shallow canyon with unit dimensionless frequency it can be seen that the disturbance on the displacement pattern is little and the canyon can not be detect as a scatterer by seismic waves very well. Fig.5: Two elliptical canyons subjected to present analysis, a) Shallow canyon, b) Deep canyon The results obtained for two different circular frequencies for these models are presented in Figures 6 and 7. The effect of canyon shape and canyon depth on the topographic amplification is frequency dependent. The same frequency induces different pattern of displacement variation across the canyon. It is because of this fact that depression on seismic waves induced by the canyon as a scatterer depends on the dimensions of the canyon in comparison with the seismic wave length. So for the deepest one, we are expecting the more disturbances in pattern of displacement variations because of this fact that the depth of canyon is comparable with seismic wave length. On the other hand the less disturbance can be expected for the shallow one because of this fact that the canyon depth is not enough comparable with wave length. Shallow canyon are not detected as a scatterer by seismic waves with low frequencies very well and so less complicated pattern of displacement variation across the canyon is achieved. Results obtained for two different frequencies show that the deep canyon has more amplification effect on the displacement amplitude across the canyon in comparison to shallow canyon. The results obtained for high frequency is more sensitive to increasing of the depth of the canyon in comparison to the low frequency. More analysis has been done on the triangular canyon in order to show the canyon geometry effects. Figure 8 shows two triangular canyons with different angle of slopes and different depths. In order to show the effect of depth in this kind of canyon geometries different analysis for different dimensionless frequencies is done. The Ux, Circular Ux, Elliptic a Ux, Elliptic b Uz, Circular Uz, Elliptic a Uz, Elliptic b - - Uy, Circular Uy, Elliptic a Uy, Elliptic b Figure 6: Comparison of results obtained for three different canyon geometry for ω =. 4 for incident SH-Wave withθ θ = 45. h = V ISSN: ISBN:

5 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) Ux, Circular Ux, Elliptic a Ux, Elliptic b Uz, Circular Uz, Elliptic a Uz, Elliptic b - - Uy, Circular Uy, Elliptic a Uy, Elliptic b - - Figure 7: Comparison of results obtained for three different canyon geometry forω =. 57 for incident SH-Wave with θ h = θ V = 45. The results show that the z-component (along depth) of displacement is more sensitive to the depth variation in comparison to the other components and shows maximum 4 percent increase by two times increasing of the depth for high frequency. Fig.8: Two triangular canyons with different depths, a) angle of slope=45 and L/D=, b) angle of slope=6.5 and L/D= Uz- Tri.b Uz- Tri.a Uy- Tri.b Uy- Tri.a Ux- Tri.b Ux- Tri.a Figure 9: Comparison of results obtained for three different canyon geometry for unit dimensionless frequency ( Ω = ) for incident SH-Wave with θ h θ = 45. = V ISSN: ISBN:

6 Proceedings of the rd IASME / WSEAS International Conference on GEOLOGY and SEISMOLOGY (GES'9) Ux- Tri.a Ux- Tri.b Uz- Tri.a Uz- Tri.b Uy- Tri.a Uy- Tri.b Figure : Comparison of results obtained for three different canyon geometry for unit dimensionless frequency( Ω = )for incident SH-Wave with θ h θ = 45. = V the dimensions of the canyon should be comparable with the seismic wave lengths. Displacement variation and amplification across canyons with different geometry are different. The results show that the depth-direction displacement component is more sensitive and shows more amplification with increasing canyon depth. References: [] Celebi M. Topographical and geological amplifications determined from strong-motion and aftershocks records of the march 985 Chile earthquake, Bulletin of the Seismological Society of America, Vol.77, 987, pp [] Trifunac, M.D. Scattering of plane SHwaves by a semi-cylindrical canyon, Earthquake Engineering and structural Dynamics, Vol., 97, pp [] Zhang, L. and Chopra, A. K., Three- Dimensional Analysis of spatially varying ground motions around a uniform canyon in a homogeneous half-space, Earthquake Engineering and Structural Dynamics, Vol., 99, pp [4] Luco, J.E., Wong, H.L. and De Barros, F.C.P.,Three dimensional response of a cylindrical canyon in a layered half-space, Earthquake Engineering and Structural Dynamics, Vol.9, 99, pp [5] Athanasopoulos, G.A., Pelekis, P.C and Leonidou, E.A., Effects of surface topography on seismic ground response in the Egion (Greece) 5 June 995 earthquake, Soil Dynamics and Earthquake Engineering, Vol.8, 999, pp [6] Assimaki, D., Kausel, E. and Gazetas, G., Topography effects in the 999 Athens earthquake: engineering issues in seismology. 4 Conclusion Proc. th ICSDEE and rd ICEGE, UC Effects of canyon geometry are investigated and the following results obtained: The effect of canyon shape and canyon depth on the topographic amplification is frequency dependent. The amplification of displacement amplitude has direct relation with depth of canyon. Deep canyon induces large amplification effect rather than shallow canyon in different dimensionless frequencies. For high dimensionless frequency the obtained results is more sensitive to increasing of the depth of the canyon in comparison to the low dimensionless frequency. In order to canyon detected as a scatterer Berkeley, Vol., 4, pp.-8. [7] Geli, L., Bard, P.Y. and Jullien, B., The effect of topography on earthquake ground motion: a review and New results, Bulletin of Seismological Society of America, Vol.78(), 988, pp [8] Tarinejad R. Seismic Loading for canyon site structures, Ph.D. Dissertation, Tarbiat Modares University, Tehran, Iran, 8. [9]Dominguez, J., Boundary Elements in Dynamics, Elsevier, 99. []Manolis, G.D. and Beskos, D.E.,Boundary Element Methods in Elastodynamics, Elsevier, 988. ISSN: ISBN: