Modal Superposition Method for the Analysis of Seismic-Wave Amplification

Transcription

1 Bulletin of the Seismological Society of America, Vol. 93, No. 3, pp , June 003 Modal Superposition Method for the Analysis of Seismic-Wave Amplification by Jean-François Semblat, Anne-Marie Duval, and Patrick Dangla Abstract Seismic site effects involve both wave propagation phenomena and vibratory resonance processes. In this article, modal approaches are then considered to investigate site effects in alluvial basins through the latter aspect. Some simplified modal methods are briefly recalled first. For a shallow alluvial site in Nice, France, standard eigenmode estimation is then proposed to make a qualitative analysis and preliminary comparisons with other numerical results obtained by the boundary element method (BEM). The influence of the bedrock on these results is discussed. One considers afterward the modal superposition method to analyze site effects from a quantitative point of view (effective modal mass). For a translational excitation (assumption of rigid base motion) and a frequency range acceptable for such vibratory investigations, a strong amplification appears around 1.6 and 1.8 Hz, giving a very high value of the cumulated effective modal mass (80%). These results are shown to be in good agreement with experimental ones and other numerical ones (BEM). The modal superposition method, accounting for the seismic excitation, then appears as an efficient tool, not only for the vibratory characterization, but also for simplified analyses of site effects in alluvial basins. Site Effects: Vibratory versus Propagative Approaches Seismic site effects are a major issue in the field of earthquake engineering, since the local amplification of the seismic motion can be very large (Bouchon, 1973; Sanchez- Sesma, 1983; Bard, 1985, 1994, 000; Duval et al., 1998; Moeen-Vaziri and Trifunac, 1988; Semblat and Luong, 1998; Sommerville, 1998; Bielak, 1999; Pitilakis et al., 1999; Chávez-García et al., 000; Sanchez-Sesma et al., 000; Semblat et al., 000, 00). These phenomena can strengthen the incident seismic motion and increase the consequences on structures and buildings. Since site effects involve both wave propagation phenomena and vibratory resonance of surface layers, it is possible to consider modal approaches for the analysis. Modal methods are very interesting because they easily lead to some global vibratory features of alluvial basins (Dobry et al., 1976; Bard and Bouchon, 1985; Paolucci, 1999). These methods are generally very efficient, but the eigen properties of the site are mainly qualitative results as far as seismic wave amplification is concerned. In this article, we do not try to directly select some particular, or fundamental, frequencies, but rather estimate a complete set of eigenmodes of the basin. We try afterward to distinguish between these (generally numerous) eigenfrequencies corresponding to strong site effects. This detailed comparison is made possible using a modal superposition method for a specific type of excitation. In the city of Nice, France, many different experimental and numerical researches have been completed to analyze site effects and seismic hazard in the city (Duval, 1996; Semblat et al., 000). For this specific site, numerical studies based on the boundary element method (BEM) allow a complete description of the amplification process (Semblat et al., 000). The main advantage of this method is that it allows an accurate description of the infinite extension of the medium (Dangla, 1988; Bonnet, 1999). Furthermore, it does not involve such drawbacks as numerical dispersion, a typical, but controllable, numerical error in finite-difference or finite-element methods (Ihlenburg and Babuška, 1995; Semblat and Brioist, 000). For the city of Nice, complete BEM results and various comparisons with experimental ones are proposed in Semblat et al., (000). These numerical results will also be considered hereafter. Experimental Investigations and Previous Numerical Results Experimental Amplification Curves The specific site considered is located in the center of Nice (French Riviera). As depicted in Figure 1, the alluvial basin is 000 m wide and its depth is around 60 m in the deepest part (west) down to 30 m in the thinnest part (east). 1144

2 Modal Superposition Method for the Analysis of Seismic-Wave Amplification 1145 West h=64m h=3m East distance (m) Figure 1. Geological profile considered in the center of Nice, France. Many experimental investigations have been performed using different techniques: microtremor measurements and weak seismic event recordings. As shown in previous publications (Duval, 1996; Semblat et al., 000), the main frequencies of the basin are precisely estimated by both experimental methods. Two types of amplification are determined: in the deepest part of the deposit for low frequencies (between 1 and Hz) and in the thinnest part for higher frequencies (above.0 Hz). Considering the horizontal-tovertical spectral ratio from microtremor recordings, it is nevertheless difficult to recover the actual amplification level. In the last section, experimental results from weak event measurements are presented to make comparisons with modal approaches. Motion Amplification Estimated by the BEM To analyze the seismic response of the site, a numerical model based on the BEM (Dangla, 1988; Bonnet, 1999) was previously considered. In these numerical investigations (Semblat et al., 000), the solution of the integral equation is obtained by finite boundary elements discretization and then by collocation, that is, application of the integral equation at each node of the mesh. The dynamic problem is analyzed in two dimensions (plane strain). The computations are performed using the finite-element method (FEM)/BEM code CESAR-LCPC (Humbert, 1989). For the sake of simplicity, the surface alluvial layer is assumed homogeneous. To estimate the influence of the mechanical characteristics of the deposit on the seismic motion amplification, the spectral ratios are computed for three different shear moduli of the alluvial layer: l MPa, l l 1 /3, and l 3 l 1 / (that is, V s1 300, V s 45, and V s3 1 m/sec, respectively, for the shear-wave velocity). As also shown by experiments (Duval, 1996), it leads to a strong site amplification in the deepest part of the basin (west) for low frequencies and in the thinnest part of the deposit (east) above Hz (Semblat et al., 000). Detailed comparisons with modal approaches are proposed in the following. Modal Approaches for the Vibratory Characterization of Geological Structures Various Types of Modal Approaches Modal approaches for the vibration analysis of geological structures are increasingly used (Zhao, 1996; Paolucci 1999). These approaches generally provide the fundamental vibration frequencies of geological structures taking into account their geometry (Bard and Bouchon, 1985; Paolucci, 1999; Semblat et al., 001) or the inhomogeneity of their mechanical features (Dobry et al., 1976; Zhao, 1996; Hadjian, 00). Considering different types of assumptions, it is then possible to estimate the fundamental frequency of a specific alluvial deposit. It is nevertheless not possible to directly compare various frequencies in terms of actual seismic motion amplification. The main assumptions of these various methods are compared hereafter (Table 1). Methods Dedicated to 1D Profiles. Dobry (1976) first proposed complete modal investigations of the vibratory resonance of surface layers. For a homogeneous horizontal layer, a nonhomogeneous one (linear shear modulus variation), or a horizontally multilayered soft deposit, he gave the analytical expressions of the fundamental period of the geological structure. For a horizontal layer of thickness H with shear modulus G varying linearly with depth z, he obtained the expression 0 G G H 1 K K z, (1) H where K G /G H, G 0 and G H being the moduli at the top and base of the layer. To estimate the period T of the layer, Dobry used the characteristics of an equivalent uniform layer, writing T 4H/V eq where Veq G eq/q at equivalent depth z eq given by eq 1 z a K (1 K ), () H H 1 K where a 1 is the first root of the equation J (a )Y (Ka ) J (Ka )Y (a ) 0, (3) J i and Y i being Bessel and Weber s Bessel functions (respectively) of order i. Dobry also proposed a generalization of these results for two-layer and multilayer profiles. He compared seven different approximate methods that quickly give an estimate of the fundamental period of multilayer geological profiles. From this method, other results were derived recently by Hadjian (00) considering an iterative approach to estimate the fundamental period of multilayer profiles. Methods for D or 3D Profiles. The work of Bard and Bouchon (1985) considered D profiles and proposed a simple relation to account for D resonance in geological profiles. Taking into account the velocity contrast in the profile C and its shape ratio h/l (h, depth; l, width), they gave an analytical relation describing the existence condition of D resonances (mainly for sine-shaped valleys) when compared

3 1146 J.-F. Semblat, A.-M. Duval, and P. Dangla Table 1 Various Modal Methods and Main Features of the Corresponding Models Author Homogeneous Heterogeneous 1D D 3D Excitation Dobry et al. (1976) yes yes yes no no no Hadjian (00) yes yes yes no no no Bard and Bouchon (1985) yes no yes yes no no Paolucci (1999) yes yes no yes yes no Present study yes yes (yes) yes (yes) (yes) to 1D resonance combined to lateral propagation. This relation was proposed as follows: h (4) l c C 1 m The modal approach proposed by Paolucci (1999) uses the Rayleigh method and can be considered for any type of D or 3D geological profiles (not only horizontal ones). One previously needs to choose a set of admissible shape functions ŵ k for the basin. The method then consists in minimizing the ratio between the strain energy and the kinetic energy to obtain an estimate of the fundamental frequency x 0 : X ˆr jl (x)ê jl (x)dx x0 min, (5) ŵk q(x)ŵ (x)dx X where ˆr jl and êjl are stress and strain tensors corresponding to ŵ k and q is the mass density. The predictions of Paolucci are found to be very accurate for irregular homogeneous basins, and the error grows up to 0% for nonhomogeneous alluvial deposits. This method was also considered elsewhere to analyze the vibratory features of the shallow alluvial basin depicted in Figure 1 (Semblat et al., 001). Comparison of the Main Features of Various Modal Methods. The main features of the various modal methods presented previously are compared with the present approach in Table 1. For each modal method, Table 1 gives the available features for the corresponding model. The simplified modal procedures proposed by Dobry et al. (1976) and Hadjian (00) deal with both homogeneous and heterogeneous 1D profiles. D profiles are considered in the approach proposed by Bard and Bouchon (1985). Paolucci s method can be used for homogeneous and heterogeneous D as well as 3D profiles. The present study is suitable for any type of profile (1D, D, or 3D) with heterogeneities, and furthermore, the seismic excitation is taken into account in the modal approach. This latter aspect is included under a simplified form considering the assumption of rigid base motion for a specific excitation. As it will be shown in this article, this assumption allows the use of standard modal superposition. k The Proposed Modal Approach Explicit Eigenmodes Estimation and Further Discrimination. The aim of this article is to consider standard modal methods to have fast and simple analysis of site effects. The ultimate goal is not only the vibratory characterization of alluvial deposits but is rather to account for the seismic excitation in the framework of standard modal approaches. We first propose to explicitly compute the eigenmodes of an alluvial basin (regular/irregular, homogeneous/nonhomogeneous, D/3D, etc.) by the finite-element method. It only allows the qualitative analysis of the resonance features of the basin. From this qualitative point of view, some comparisons with the results obtained previously by the BEM will be proposed. To lead the analysis further, we also propose to consider the modal superposition method to account for the seismic excitation and to distinguish between these (sometimes numerous) eigenmodes. This detailed modal approach is explained in the next main section. In the following subsections, we make a qualitative analysis only considering the eigenmodes of the basin (i.e., a vibratory characterization of the deposit). Solution of Eigenvalue Problems by the Finite-Element Method. Considering the finite-element method, it is necessary for both modal approach or wave propagation problems to take care of several drawbacks of this numerical method. For eigenvalue problems, there could be some particular issues to address as lower (or upper) bound values or close eigensolutions (Hu et al., 1998; Lee et al., 1999). Depending on the integration algorithm and the spatial discretization, the numerical wave dispersion should also be controlled for wave propagation models (Ihlenburg and Babuška, 1995; Semblat and Brioist, 000). For undamped systems, the analysis of the eigensolution can be made considering the following expression in modal coordinates (Bisch et al., 1999): 1 1 { } K M{ }, (6) x where x is the eigenfrequency, K and M are the stiffness and mass matrices, and U the real eigenvectors. For proportionally damped systems (Clough and Penzien, 1993; Semblat, 1997), the damped modes can directly be determined from the real undamped modes computed through equation

4 Modal Superposition Method for the Analysis of Seismic-Wave Amplification 1147 (6). For nonclassically damped systems (i.e., nondiagonal damping matrix in the real mode s base), one has to solve a generalized eigenproblem under the typical form (Igusa et al., 1984; Perotti, 1994; Bisch et al., 1999): 1 pw 0 1 pw 1 1, (7) p w K M K C w where K, M, and C are the stiffness, mass, and damping matrices (respectively), and p and w are the complex eigenvalue and eigenvector (respectively). Equation (7) allows the analysis of the modal properties and response of systems with local dampers or materials with variable damping properties (Bisch et al., 1999). If necessary, it is also possible to investigate some types of nonlinearities through modal approaches (Lin and Lim, 1993). The latter approach could be used in the future to investigate, by modal methods, site effects in damped alluvial basins. Eigenfrequencies and Eigenmodes of the Site. Considering the geological profile chosen in the center of Nice (Fig. 1), the eigenfrequencies and eigenmodes of the surface deposit and part of the surrounding bedrock are estimated by the finite-element method. The computations are performed considering different D models to analyze the influence of limit conditions and overall stiffness on the eigen properties. The three models used for the analysis are depicted in Figure : the first model includes the alluvial basin alone, the second one the basin and a part of the elastic bedrock, and the third one the basin and a large part of the bedrock. The corresponding numbers of degrees of freedom for each finite-element model are given in Figure. As shown in the figure, the boundary conditions for each model are as follows: rigid at the base for model 1; no horizontal displacement on lateral boundaries and no vertical displacements at the bottom of models and 3. For the sake of simplicity, the alluvial basin is considered homogeneous; its mass density is q 000 kg/m 3 and its shear-wave velocity V s 300 m/sec (i.e., shear modulus l MPa). For the elastic bedrock, the values are q 300 kg/m 3 and V s 1400 m/ sec. The second and third models depths are then k/5 and 0.7k, respectively (k k(f 1 Hz)). The in-plane modes of vibration of these three models are computed (plane strain) using inverse iteration and subspace method (FEM/BEM code CESAR-LCPC). Figure 3 gives the estimated eigenmodes for the three models around 1.8 Hz. The number of degrees of freedom being different for each model, the number of eigenmodes in the frequency range of the problem are the following: 6 different eigenmodes below Hz for the first model, 19 for the second model, and 47 for the third one. The number of modes for the largest model is very important. As shown in Figure 3, the influence of the boundary conditions is clear when displaying, for a given eigenfrequency, the corresponding eigenmodes for the three models: Figure. Three preliminary models used for the computation of eigenmodes by the finite-element method. The first model shows a stiffer behavior with very small displacements in the thinnest part of the basin. The third model seems to have a much softer one since local modes are even found within the bedrock. This model then appears too large for the vibratory analysis of the deposit since its dimensions are not negligible toward the wavelength (depth 0.7k at 1 Hz). Consequently, it will be considered unsuitable for the further modal investigations. The second model has an intermediate behaviour probably closer to the actual one for an alluvial basin on an elastic (not rigid) bedrock. Figure 3 also shows that there are four resonance areas for models 1 and, but only three areas for model 3. Afterward, these resonance features will be compared with amplification areas from propagative BEM models. For models 1 and, the modal superposition method will finally provide other reasons to allow the choice of the most suitable model. Qualitative Comparisons with Site Effects Computations (BEM). Before the detailed analysis by the modal superposition method, we give some qualitative comparisons between the explicit modal estimation and the numerical results obtained by the BEM (Semblat et al., 000). Figures 4, 5, and 6 give the fifth, fourteenth, and twenty-fifth eigenmodes of the site (second model) of respective eigenfrequencies 1.35, 1.81, and.5 Hz (figures are enlarged by a factor of.75 along the vertical axis). The corresponding displacement along the basin surface (arbitrary scale) is also given in these figures. These modes of vibration are local modes mainly located in the surface alluvial layer, showing that model is suitable for vibratory analysis. As shown in Figures 4 and 5, local modes first appear in the deepest part of the deposit (low eigenfrequencies). For higher modes (Fig. 6), the local (relative) vibratory motion is stronger in the thinnest part of the basin. Experimental studies also lead to these conclusions (Duval, 1996). As shown by the three figures, these qualitative results are in agreement with other numerical ones (BEM) computed around the same frequency value (surface displacement and isovalues). With both methods, the curves even display the number and locations of maximum amplification areas in the basin, that is two areas in the western part of the deposit for 1.4 Hz (Fig. 4), four

5 1148 J.-F. Semblat, A.-M. Duval, and P. Dangla 1st model : mode No.4, f=1.81hz nd model : mode No.14, f=1.8hz 3rd model : mode No.37, f=1.80hz LCPC Figure 3. Eigenmodes (displacements along surface and isovalues) around 1.8 Hz for the three finite-element models considered. mode mode No 5, f=1.35 Hz BEM 1.1 BEM: f=1.40 Hz arbitrary scale BEM : freq.=1.4hz mode n 5: f=1.35 Hz distance (m) Figure 4. Displacement values, along the surface and inside the basin, around 1.4 Hz for both BEM solution (Semblat et al., 000) and modal approach (eigenmode). LCPC areas in this same part for 1.8 Hz (Fig. 5), and five or six areas in the western part and one in the eastern part for. Hz (Fig. 6). From the curves of Figures 4, 5 and 6, it can be noticed that the peaks in eigenmodes and in BEM amplification curves approximately coincide, especially for the main amplification at 1.8 Hz. When compared to the BEM, the modal approach then appears as an interesting mean for the fast and simple qualitative analysis of site effects. At this stage, however, it is not possible to account for the seismic excitation and to know which of these eigenmodes are going to lead to the highest local seismic motion. The key point is now to distinguish between the numerous eigenfrequency values corresponding to the strongest site amplification. Modal Superposition Method for the Analysis of Site Effects Dynamic Response by Modal Superposition Considering the modes of vibration of the site determined in the previous section, we will now try to determine which of the numerous eigenmodes have the largest contribution to the response of the site under a specific excitation. The main goal is to obtain a comparative estimation of site effects by a standard modal method. In the field of structural dynamics, the dynamic response of a structure can be estimated considering its eigenmodes and using a modal superposition method. This method allows the superposition of

6 Modal Superposition Method for the Analysis of Seismic-Wave Amplification 1149 mode arbitrary scale mode No.14, f=1.81 Hz BEM, f=1.8hz BEM BEM: f=1.80 Hz mode n 14: f=1.81 Hz distance (m) Figure 5. Displacement values, along the surface and inside the basin, around 1.8 Hz for both BEM solution (Semblat et al., 000) and modal approach (eigenmode). LCPC mode mode No.5, f=.5 Hz BEM.0 BEM: f=.0 Hz arbitrary scale BEM, f=. Hz mode n 5: f=.5 Hz distance (m) Figure 6. Displacement values, along the surface and inside the basin, around. Hz for both BEM solution (Semblat et al., 000) and modal approach (eigenmode). LCPC the separate modal displacement contributions to determine the whole dynamic response of the structure (Clough and Penzien, 1993; Bisch et al., 1999). It is generally not necessary, in the case of structures, to include all the higherorder modes in the superposition process. For civil engineering structures, it is often sufficient to keep the only 10 first eigenmodes to have a reliable estimation of the response. The finite-element models of the basin depicted in Figure have many degrees of freedom, and the number of eigenmodes in the frequency range of the problem is large. One has to determine the contribution of each mode to the dynamic response of the site. Assumption of Rigid Base Motion for the Whole Basin We would like to apply a standard modal superposition method to account for the seismic excitation and estimate the effective modal masses of the basin. These modal features could then give a fast and simple way to compare the contribution of each modal component to the amplification process. One could not get actual amplification factors but still relative contributions of each spectral component. Considering finite-element model depicted in Figure, one will try to make the analysis within the assumption of rigid base motion. As depicted in Figure 7, the basin is very flat and the part of the elastic bedrock not very large (depth k/5 at 1 Hz). It is then possible to consider a vertical soil column (Fig. 7a) and, from this soil column, build the complete model (Fig. 7e) by multiple symmetries along the axis of largest aspect ratio through steps b, c, d in the figure. Figure 7 shows that this assumption corresponds to a single column (Fig. 7a) with zero horizontal displacements as boundary conditions. These conditions lead to a sum of horizontal displacement components equal to zero for each

7 1150 J.-F. Semblat, A.-M. Duval, and P. Dangla rigid base motion (a) symmetry condition (b) u x A + u = 0 x B u x A u A (c) u x B u B multiple symmetry main symmetry secondary symmetries u=0 x u=0 x (d) u y (e) u x Figure 7. Equivalence with rigid base motion for the whole shallow basin. A B symmetry: ux ux 0. This column could be excited in the vertical direction, ensuring the horizontal boundary conditions would be fulfilled. As shown in Figure 7, the equivalent rigid base motion for the whole model is then assumed as a dynamic excitation in the vertical direction (volumic translational density of force) that is perpendicular to the axis of largest aspect ratio for the basin. The excitation has then to be considered in terms of relative motion since the global rigid motion (center of gravity of the structure) does not lead to any internal strain (Clough and Penzien, 1993). For the whole basin, this approximation with vertical excitation nevertheless allows some horizontal motion since the eigenmodes are in-plane modes. For models 1 and, the effective modal masses for this specific excitation are then computed for the 50 first eigenmodes of the geological profile, that is, between 0.4 and 3.1 Hz. Modal Excitation Factor and Effective Modal Mass In the modal superposition method, there are two widely used concepts for the analysis of the contribution of the different modes of a structure to its whole dynamic response: the modal excitation factor and the effective modal mass. Considering the rigid base motion of a structure, that is translational excitation and small aspect ratio, the modal response {Y n } of the structure in the normal coordinates could be written as (Clough and Penzien, 1993): M {Y } C {Y } K {Y } N { v (t)}, (8) n n n n n n n g where M n, C n, and K n are the generalized mass, damping, and stiffness properties of mode n and N n is the modal excitation factor, such as N n {U n } T M{1}, with {1} the unit translation applied at the base of the structure. The relative displacement vector produced in each mode n is then given by: N n n n M n x n n {Y (t)} {U } V (t), (9) where V n (t) is the modal earthquake response Duhamel integral (Clough and Penzien, 1993). The modal excitation factor N n then expresses the contribution of the mode n to the whole response of the structure. For this type of rigid base motion, it is possible to estimate the contribution of each mode in the direction of the dynamic (or seismic) loading; it is generally known as the effective modal mass of each mode. Considering the kinetic energy of the system, the effective modal mass f n of mode n can be written under the following form: N n M n f. (10) n The effective modal mass f n corresponds to the part of the total mass of the structure responding in the excitation direction in mode n (Clough and Penzien, 1993; Bisch et al., 1999). Theoretically, the sum of the effective modal masses of all the modes is equal to the total mass of the structure. Estimation of Effective Modal Masses for the Shallow Basin Effective Modal Mass for Models 1 and. The values of effective modal mass computed for a vertical translational excitation are given in Figure 8 for models 1 (basin alone) and (intermediate model). These values are very small for frequencies below 1.5 Hz, and they quickly increase above this frequency. The maximum values are reached in model 1 for eigenmode 5 (f Hz) and in model for eigenmodes 8 (f Hz) and 14 (f Hz). For these modes, the percentages of effective modal mass are respectively m eff % for model 1 and for model, m eff 8

8 Modal Superposition Method for the Analysis of Seismic-Wave Amplification 1151 effect. modal mass (%) mode n Hz mode n Hz mode n Hz frequency (Hz) Model No. Model No.1 (basin alone) Figure 8. Effective modal mass versus frequency for models 1 (basin alone) and (intermediate model). 15.1% and m 41 eff 3.8%. For both models, the largest part of the mass of the model responding to the vertical seismic excitation is in the frequency range between 1.6 and 1.9 Hz. The frequency value is larger for model 1 (basin alone) since it is stiffer than model (Fig. 3). As shown in Figure 8, there is also a significant increase of the effective modal mass around a frequency of.6 Hz. It corresponds to wave amplification in the thinnest part of the basin (Fig. 6). However, for large frequencies, the analysis of vibratory resonance is not justified any more since the relative dimensions (toward wavelength) of the model become larger. These results are compared with experimental measurements in the next section. Reliability and Accuracy of the Modal Superposition Method. A good way to characterize the reliability and accuracy of the modal superposition method is to compute the cumulated modal mass, that is, the sum of the effective modal mass for all modes under a given frequency. For finite-element models 1 and, the respective values of cumulated effective modal mass are as follows: Model 1: less than 0% of the total mass is reached at frequency f Hz and only 34% for the secondary peak at. Hz. Model : 6% of the total mass is recovered at 1.85 Hz and up to 7% at. Hz. For model 1, the cumulated modal mass values are very low since the spectral (or modal) content is very poor for this model. It does not seem that the modal superposition method could be properly considered with the basin alone model. For model, the values are rather high, and this is generally considered as a sufficient part of the total mass of the model to give a reliable and accurate estimation of the dynamic response of the structure (Clough and Penzien, 1993). Furthermore, these good results indicate that the assumption of rigid base motion is acceptable for finiteelement model, including a rather small part of the bedrock (around k/5 at 1 Hz). For the geological site studied here, the main modal contributions are found for modes 8 (1.6 Hz) and 14 (1.8 Hz). The first model (basin alone) then appears unsuitable for the use of the modal superposition method to take into account the seismic excitation in the modal approach. The cumulated modal mass for the second model will be compared hereafter with both experimental and BEM results. Modal Superposition versus Experimental Results Since the estimation of the eigenmodes is carried out with plane-strain assumption and with a vertical translational excitation, the numerical results of the modal approach are compared with experimental spectral ratios of the vertical component of displacement. Figure 9 gives three different types of results: 1. The experimental vertical standard spectral ratio for various weak seismic events (Duval, 1996) through mean values and the corresponding standard deviation.. The vertical amplification factor estimated numerically by the BEM (Semblat et al., 000); this is a global value corresponding to the overall amplification factor (Semblat et al., 000) that is the maximum amplification along the whole basin at each frequency (it is not related to a specific location). 3. The cumulated effective modal mass computed by standard modal superposition for model. The percentages of effective modal mass are plotted between 0.001% and 100%. For both numerical approaches, the results are given for three different values of shear modulus for the basin: l MPa (giving V s 300 m/sec as in the previous sections), l 10 MPa, and l 3 90 MPa. The spectral ratio, the BEM amplification factor and the effective modal mass are very small for frequencies below 1 Hz. There is a fast increase of these parameters between 1 and Hz. The three methods (experimental, BEM, and modal superposition) lead to the strongest site response in the same frequency range, which is around 1.8 Hz. Above this frequency, both vertical spectral ratio and effective modal mass are mostly constant. However, it is not possible to analyze the vibratory resonance of the basin for larger frequencies corresponding to shorter wavelengths. A propagative approach is then required. In Figure 9, the influence of shear modulus (or wave velocity) is strong since the effective modal mass increases at lower frequencies for smaller values. The same trend is observed with numerical amplification curves from BEM computations (Semblat et al., 000). Finally, the agreement between seismic measurements on site and modal superposition analysis seems to be satisfactory. Through the effective modal mass, in this specific configuration, modal superposition then appears as an interesting method to easily determine the frequency range cor-

9 115 J.-F. Semblat, A.-M. Duval, and P. Dangla 50 site / reference (vertical) experiments +σ mean σ overall amplification BEM effective modal mass modal superposition frequency (Hz) Figure 9. Comparisons between mean experimental results, overall amplification estimated by BEM, and effective modal mass (modal superposition method). responding to the strongest seismic site effects for D geological profiles. Conclusion A standard modal superposition method is considered to analyze seismic site effects. The specific site considered is a shallow basin located in the city of Nice, France, where experimental investigations from weak events give amplification factors between 10 and 30 for frequencies above 1.0 Hz (Semblat et al., 000). Different modal approaches can be considered to study the eigen properties of an alluvial basin. To use standard modal analysis for alluvial basins, we first considered different models of various sizes to explicitally estimate the eigenmodes of the basin. Afterward, to distinguish between these frequencies toward amplification of seismic motion, we assumed generalized rigid vertical base motion for the basin since it is a shallow one. The analysis of the corresponding effective modal masses for a translational excitation then leads to higher values around actual highest amplification frequencies. From this point of view, comparisons with site/reference spectral ratios determined experimentally and the overall amplification factor estimated previously by the BEM are satisfactory. Finally, the BEM is a good approach for the modelling of wave propagation and excitations sources (wave type, incidence, etc.), the modal approach is rather a method for vibratory resonance analysis through the computation of the eigenfrequencies of a specific geological structure. The results presented here nevertheless show that, for shallow basins and simple excitations, it is possible to account for the seismic loading and distinguish between these eigenmodes

10 Modal Superposition Method for the Analysis of Seismic-Wave Amplification 1153 to determine the frequency range giving the strongest seismic site effects. However, two main issues should be considered in future modal investigations: the influence of damping and nonlinearities (Igusa et al., 1984; Lin and Lim, 1993; Bard and Riepl-Thomas, 000), and the possible interaction with large surface structures in dense urban areas (Guéguen et al., 000). References Bard, P. Y. (1994). Effects of surface geology on ground motion: recent results and remaining issues, Tenth European Conf. on Earthquake Engineering, Vienna, 8 August 3 September 1994, Vol. 1, Bard, P. Y., and M. Bouchon (1985). The two dimensional resonance of sediment filled valleys, Bull. Seism. Soc. Am. 75, Bard, P. Y., and J. Riepl-Thomas (000). Wave propagation in complex geological structures and their effects on strong ground motion, Wave Motion in Earthquake Engineering, E. Kausel and G. 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