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Vibrations in soils: a spectral prediction method
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect ScienceDirect
Procedia Engineering 00 (2017) 000–000 Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 199 (2017) 2675–2680
X International Conference on Structural Dynamics, EURODYN 2017 X International Conference on Structural Dynamics, EURODYN 2017
Vibrations in soils: a spectral prediction method Vibrations in soils: a spectral prediction method
Abdul Karim Jamal Eddinea,a,*, Luca Lenti aa, Jean-Francois Semblat aa Abdul Karim Jamal Eddine *, Luca Lenti , Jean-Francois Semblat a a
Université Paris-Est, GERS, SV, IFSTTAR, F-77447 Marne-la-Vallée, France Université Paris-Est, GERS, SV, IFSTTAR, F-77447 Marne-la-Vallée, France
Abstract Abstract
Dynamic site amplification and soil classification are important issues related to earthquake engineering. While Dynamic amplification and soil classification are important issues related to earthquake While most of thesite studies are dedicated to engineering seismology and earthquake engineering, similar engineering. approaches for soil most of the studies are dedicated to engineering seismology and earthquake engineering, similar approaches for soil classification and site amplification have not been yet fully established in the field of urban vibrations. In this study, classification and site havebeen not been yet fullytoestablished in the field ofthe urban vibrations. this study, several parameters of amplification a given site have investigated synthetically describe geometrical andInmechanical several parameters of athe given site have been investigated to synthetically the geometrical and proxy). mechanical properties that control amplification phenomena (i.e. average velocity describe and velocity gradient-based The properties that control theof amplification phenomena velocity andfrequency velocity gradient-based proxy). with The results of FEM analyses wave propagation have (i.e. beenaverage processed in the domain and analysed results of FEM analyses of wave propagation have been processed in the frequency domain and analysed with respect to the proposed parameters and the distance from the source. Finally, the velocity-gradient-based parameters respect totothe parameters distanceasfrom the source. Finally, the parameters coupled theproposed spectral analysis canand be the considered a coherent methodology forvelocity-gradient-based reliable vibration predictions and coupled to the spectral analysis canvibration be considered as a coherent methodology for reliable vibration predictions and site classification in order to assess sensitivity. site classification in order to assess vibration sensitivity. © 2017 2017 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © © 2017 The under Authors. Published by Ltd. committee of EURODYN 2017. Peer-review responsibility of Elsevier the organizing organizing Peer-review under responsibility of the committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: vibration;gradient;prediction model;spectral method;site classification;wave propagation Keywords: vibration;gradient;prediction model;spectral method;site classification;wave propagation
1. Introduction 1. Introduction Dynamic site amplification is mainly due to surface waves reflected and trapped in weak layers over stiffer layer. Dynamic site amplification is mainly research due to surface waves reflectedand andearthquake trapped inengineering, weak layers the overimportance stiffer layer. While this phenomenon is an important topic in seismology of While this phenomenon is an important research topic in seismology and earthquake engineering, the importance studying such problem for urban vibrations is becoming more and more important with the quick rise of of studying suchinfrastructures. problem for urban vibrations is becoming more and more important with the quick rise of transportation transportation infrastructures. The shear wave velocity, Vs, is usually used to describe the stiffness of materials and is one of the factors that The shear wave velocity, Vs, is usually used to describe the stiffnessFor of materials of the that deeply influence wave propagation, attenuation and amplification. instance, and in is theone field of factors earthquake deeply influence wave propagation, attenuation and amplification. For instance, in the field of earthquake
* Corresponding author. Tel.: +0033-1-81668033; * E-mail Corresponding Tel.: +0033-1-81668033; address:author. [email protected] E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. 1877-7058 2017responsibility The Authors. of Published by Elsevier Ltd. of EURODYN 2017. Peer-review©under the organizing committee Peer-review under responsibility of the organizing committee of EURODYN 2017.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.546
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
2676 2
engineering Vs30 , the harmonic average shear wave velocities in the first 30 m of soil, is usually taken as the main parameter of analysis and classification [1-2]. The doubts about the capability of the single Vs30 proxy to predict site amplification, and therefore precisely estimate the ground displacement, led to the suggestion of alternative proxies for the analysis. Such suggestions are, but not limited to, the use of the soil fundamental frequency f0 along with Vs30 [3-4]. Another suggestion is the use of the gradient of the shear-wave velocity profile factor B30 [5] that describes the variation of soil impedance in the first 30 meters. In the present work, the aim is to combine several parameters in order to provide a spectral approach that serves as a prediction and classification method for vibration levels in multiple sites. Nomenclature Vs B10 Vs3 f0 ANSR
Shear wave velocity Gradient of Vs profile from 0 to 10 m depth Harmonic mean shear-wave velocity in the first 3 m Fundamental resonance frequency of the soil Averaged normalised spectral ratio
2. Problem statement and parameters For the purpose of analyzing the soil behavior under higher frequencies excitation originating from surface sources, the existing parameters used in earthquake engineering are not reliable. Surface waves generated by vibration sources cannot be accurately described considering the shear wave velocity at higher depths, for instanceVs30 . A wavelength of 3m is proposed herein and the factor Vs3 will be used in order to characterize the effective part of soil that controls the effects of surface waves. This wavelength was chosen in order to fit with shallower soil layers, i.e. Vs between 100m/s and 400m/s and for a range of frequencies between 0 and 120Hz. We shall assess the heterogeneity of different sites (i.e. change of seismic impedance with depth) through gradient based proxies for various depths. Gradients Bz are found through a linear regression using the following equation:
log10 Vs ( z ) Bz log10 ( z ) Az
(1)
Where Z is the depth of the soil section considered, A Z is the origin ordinate of the regression and Vs(Z) the shear wave velocity at the given depth. For particular types of soil associated with the intended Vs3 , FEM models of wave propagation in homogeneous soils (half-space cases) and in the same soils above stratified stiffer soils (multilayered cases) have been performed. Displacements induced by a surface impulsive source have been calculated along the surface at different distances from the source for all the models. In particular, the maximum absolute value of the displacement was considered.
Fig. 1. Amplification Factors for Vs=400m/s versus Gradient (a) B20; (b) B15; (c) B10.
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine/ Procedia Engineering 00 (2017) 000–000
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Multiple test-cases yielded to the velocity gradient B10 as the optimal proxy to assess the variation of impedance with depth. Amplification in Fig.1 (Vs3 =400m/s in this case) for instance, represents the amplifications at the surface averaged for the first 20 meters next to the source as a function of the gradient-based parameter. Amplification is calculated as the difference between the total maximum displacement in the multi-layered case and the homogeneous case normalized with the latter. Fig.1 shows how the B10 gradient can be seen as a robust parameter to predict amplification. 3. Computation and data selection For the purpose of the present study, finite element simulations have been performed (CESAR-LCPC software). A surface displacement was applied as the source where a pulse is injected. Absorbing layers were applied in order to avoid spurious waves reflected at the model artificial boundaries [6]. Various simulations have been carried out taking into account four different factors: average velocity Vs3 , depth of the top layer, depth of the bedrock and velocity gradient B10 in the range between 0 and 1. All calculated spectral displacements were normalized by those of the source. 4. Spectral Approach Working with a wide range of vibration, between 0 and 120 Hz, the aim is to study the spectral response of soils at various distances. The approach chosen is to divide the frequencies into bands and divide the distances into three fields: near, mid and far fields. Near field indicates the first twenty meters next to the source. Mid and far fields indicate the second and third twenty meters next to the source respectively. The frequencies between 0 and 120 Hz are divided by bands of 20 Hz. The soils/sites were sorted in accordance with Vs3 and B10 as previously mentioned. For each band and distance segment, values were averaged and then displayed. 4.1. Near-field For Vs3 equal 100m/s, it could be noticed that with an increasing gradient, the dominant frequencies are larger. Similarly the amplitudes associated with peaks are increasing with the gradient. The dominant frequencies are between 0 and 40 Hz with an increasing gradient as shown in Fig.2.a.
Fig. 2. ANSR in the near-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
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Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
For Vs3 equal to 200m/s, and similarly to Vs3 =100m/s, the peak frequency increases with the velocity gradient, likewise for amplitudes as shown in fig.2.b. The frequencies are chosen in the bands between 20 and 60 Hz. Moreover the lower frequencies (0-20Hz) are falling with an increasing gradient, conversely to the higher frequencies. For Vs3 equal to 400m/s, it is harder to achieve higher gradient since it requires very strong material below the surface layer surpassing the stiffness of soils and rocks. Nevertheless the trend continues as for the two previous cases which are ascending higher frequency content with gradient along with larger amplitudes. In this case higher frequencies are dominating the problem (up to 100 Hz) while lower frequencies are declining with higher gradients. From all above, it could be deduced that the lower the gradient, the smaller the influencing frequency and vice versa. Furthermore the correlation between the maximum amplitudes and the gradient is quite clear, i.e. the higher the gradient the larger the amplitude. Inspecting individual cases before averaging and distributing frequencies by bands, for the near field, the peak frequency is always around the value Vs3 ⁄h1 where h1 the depth of the surface layer is. Hence we can conclude that lower gradients are associated to lower frequencies because they are usually associated to deeper layers (larger h1 ) and weaker contrasts (more homogeneous cases). This fact can describe the reason why also larger B10 are associated with higher frequencies. 4.2. Mid-field For Vs3 =100m/s, as previously, the larger the gradient the higher the related frequency. For the midfield, as shown in Fig.3.a, the dominance of the lower frequency band (0-20Hz) is extended to higher gradients (up to B10 =0.7). Moreover the energy carried by the lower frequencies for the smaller gradients are obvious when compared to the near-field, indicating a higher vulnerability for lower frequencies. for Vs3 =200m/s (Fig.3.b), the effect of the 20-40Hz band is starting to become clearer and dominant up to B10 =0.7, similarly to the Vs3 =100m/s. The amplitude and effect of the same band becomes clearer for smaller gradients. For Vs3 =400m/s (Fig.3.c), vibrations in most cases are dominated by 40-60Hz. This frequency range is equivalent to the half of the frequencies dominant in the near-field. Exceptions are for the smallest and largest gradients where 20-40Hz and 60-80Hz bands dominate respectively. High frequencies are associated to smaller wavelengths, in the mid-field a large amount of these waves are filtered through distance and the remaining waves are associated to lower frequencies. This effect is more apparent for smaller gradients. 4.3. Far-field For Vs3 =100m/s (Fig.4.a), while the effects of lower frequencies are still clear, a surge of higher frequencies appears. This surge is not only for B10 =0.9, where the 40-60 Hz band is dominating for the first time. Starting from B10 =0.6, the 20-40Hz band appeared to start dominating again as shown in Fig.4.a. For Vs3 =200m/s (Fig.4.b), the response is totally dominated by the 20-40Hz band and the effects of high frequencies are almost negligible. Similarly to the near and mid-field the smaller the gradient the more it is affected by lower frequencies. For Vs3 =400m/s (Fig.4.c), the peaks are still dominated by the same frequencies as in the mid-field, except for B10 =0.6 which is now dominated by a lower frequency band (40-60Hz). The higher frequencies are clearly less apparent since more small wavelengths are filtered through the increasing distance. The only unexpected change in the above results is the surge of higher frequencies for the Vs3 =100m/s for the higher gradients, especially for B10 =0.9. It was expected that the vibration will be dominated with lower frequencies at farther distances. After inspecting the models where B10 =0.9 and Vs3 =100m/s, it was noticed that it is usually characterized with a huge leap in soil stiffness after one or two weak thin layers. Moreover the inspection showed that the peak in the newly appearing high frequency was always around a value of √3. f0 where f0 is the frequency where the first peak appears which was dominant in the near and mid-field..
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine/ Procedia Engineering 00 (2017) 000–000
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Based on this, two assumptions can be made: (1) either that the probable presence of two weak layers will cause the effect of the second to appear at far distances; (2) either higher modes of excitation are triggered at far distances. Moreover both assumptions could be true and valid.
Fig. 3. ANSR in the mid-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
Fig. 4. ANSR in the far-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
In order to investigate these assumptions, a simple rheological model of a 2DOF system has been studied. The two bodies are assumed of similar properties (in order to assimilate one layer with two degree of freedom or two layers who have negligible differences comparing to the bedrock) over a fully rigid support (assimilating the very strong bedrock). The model is shown in Fig.5. As shown below, writing the frequency equation and then calculating the fundamental and the second mode eigenfrequency, it could be noticed that the frequency of the second mode is at
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
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√3 of the first. Hence, it could be deduced that at a certain distance from the source the top layer will be dominated by the second mode of vibration, in this case for very weak soils with very large gradients.
Fig.5. Rheological model used
4km 16k 2m2 12k 2m2 1 2m 2
4km 16k 2 m 2 12k 2 m 2 2 2m 2
0.5
0.5
k m 3k m
(2)
(3)
5. Conclusion The spectral method offers a tool for sites classification according to their spectral response to surface sources. Moreover it offers an approach to predict vibration attenuation and amplification using proxies which are familiar to engineers such as shear wave velocities and profile gradients (Vs3 and B10 in this case). The spectral analysis has shown that gradients B10 could be associated to certain frequencies. This is because gradients B10 are heavily influenced by layers depth along with a strong contrast between materials. This association is founded on the fact that the fundamental resonance frequency of the soil under surface excitation is strongly dependent on the depth of the first layer (f0 = Vs3 ⁄h1 where h1 is the depth of the surface layer). That means the deeper the first layer or the weaker the contrast, the lower the fundamental resonance frequency. The discrepancy with the case of seismic waves is large since the entire surficial layer is generally excited by seismic motions. Conversely, for surface sources, the response is dominated by surface waves and is thus highly dependent on the distance from the source. Along the distance, the effect of higher frequencies begins to fade in a proportional pattern. This relates to the associated smaller wavelengths, which are filtered through distances. For sites associated to weak surface layers and high gradients, and starting from a critical distance, the second mode of vibration is triggered. It could be concluded that depending on Vs3 and B10 , higher modes of excitation will be triggered from certain distances. References [1] Borcherdt, R. (1994). “Estimates of site-dependent response spectra for design (methodology and justification)”, Earthq. Spectra 10, no. 4, 617–653. [2] Eurocode 8 (2004). Design of structures for earthquake resistance, part 1: General rules, seismic actions and rules for buildings, EN 1998-1, European Committee for Standardization (CEN), Brussels. [3] Luzi, L., R. Puglia, F. Pacor, M. R. Gallipoli, D. Bindi, and M. Mucciarelli(2011). “Proposal for a soil classification based on parameters alternative or complementary to VS30”, Bull. Earthq. Eng. 9, 1977–1898. [4] Cadet, H., P.-Y. Bard, A.-M. Duval, and E. Bertrand (2012). “Site effect assessment using KiK-net data: Part 2—Site amplification prediction equation based on f0 and VSZ”, Bull. Earthq. Eng. 10, no. 2, 451–489. [5] Régnier, J., Bonilla, L.F., Bertrand, E. and Semblat, J.F., (2014). “Influence of the VS profiles beyond 30 m depth on linear site effects: Assessment from the KiK‐net data”. Bulletin of the Seismological Society of America, 104(5), 2337-2348. [6] Semblat, J. F., Lenti, L., & Gandomzadeh, A. (2011). “A simple multi-directional absorbing layer method to simulate elastic wave propagation in unbounded domains”. International Journal for Numerical Methods in Engineering, 85(12), 1543-1563.
ScienceDirect ScienceDirect
Procedia Engineering 00 (2017) 000–000 Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Engineering 199 (2017) 2675–2680
X International Conference on Structural Dynamics, EURODYN 2017 X International Conference on Structural Dynamics, EURODYN 2017
Vibrations in soils: a spectral prediction method Vibrations in soils: a spectral prediction method
Abdul Karim Jamal Eddinea,a,*, Luca Lenti aa, Jean-Francois Semblat aa Abdul Karim Jamal Eddine *, Luca Lenti , Jean-Francois Semblat a a
Université Paris-Est, GERS, SV, IFSTTAR, F-77447 Marne-la-Vallée, France Université Paris-Est, GERS, SV, IFSTTAR, F-77447 Marne-la-Vallée, France
Abstract Abstract
Dynamic site amplification and soil classification are important issues related to earthquake engineering. While Dynamic amplification and soil classification are important issues related to earthquake While most of thesite studies are dedicated to engineering seismology and earthquake engineering, similar engineering. approaches for soil most of the studies are dedicated to engineering seismology and earthquake engineering, similar approaches for soil classification and site amplification have not been yet fully established in the field of urban vibrations. In this study, classification and site havebeen not been yet fullytoestablished in the field ofthe urban vibrations. this study, several parameters of amplification a given site have investigated synthetically describe geometrical andInmechanical several parameters of athe given site have been investigated to synthetically the geometrical and proxy). mechanical properties that control amplification phenomena (i.e. average velocity describe and velocity gradient-based The properties that control theof amplification phenomena velocity andfrequency velocity gradient-based proxy). with The results of FEM analyses wave propagation have (i.e. beenaverage processed in the domain and analysed results of FEM analyses of wave propagation have been processed in the frequency domain and analysed with respect to the proposed parameters and the distance from the source. Finally, the velocity-gradient-based parameters respect totothe parameters distanceasfrom the source. Finally, the parameters coupled theproposed spectral analysis canand be the considered a coherent methodology forvelocity-gradient-based reliable vibration predictions and coupled to the spectral analysis canvibration be considered as a coherent methodology for reliable vibration predictions and site classification in order to assess sensitivity. site classification in order to assess vibration sensitivity. © 2017 2017 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © © 2017 The under Authors. Published by Ltd. committee of EURODYN 2017. Peer-review responsibility of Elsevier the organizing organizing Peer-review under responsibility of the committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: vibration;gradient;prediction model;spectral method;site classification;wave propagation Keywords: vibration;gradient;prediction model;spectral method;site classification;wave propagation
1. Introduction 1. Introduction Dynamic site amplification is mainly due to surface waves reflected and trapped in weak layers over stiffer layer. Dynamic site amplification is mainly research due to surface waves reflectedand andearthquake trapped inengineering, weak layers the overimportance stiffer layer. While this phenomenon is an important topic in seismology of While this phenomenon is an important research topic in seismology and earthquake engineering, the importance studying such problem for urban vibrations is becoming more and more important with the quick rise of of studying suchinfrastructures. problem for urban vibrations is becoming more and more important with the quick rise of transportation transportation infrastructures. The shear wave velocity, Vs, is usually used to describe the stiffness of materials and is one of the factors that The shear wave velocity, Vs, is usually used to describe the stiffnessFor of materials of the that deeply influence wave propagation, attenuation and amplification. instance, and in is theone field of factors earthquake deeply influence wave propagation, attenuation and amplification. For instance, in the field of earthquake
* Corresponding author. Tel.: +0033-1-81668033; * E-mail Corresponding Tel.: +0033-1-81668033; address:author. [email protected] E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. 1877-7058 2017responsibility The Authors. of Published by Elsevier Ltd. of EURODYN 2017. Peer-review©under the organizing committee Peer-review under responsibility of the organizing committee of EURODYN 2017.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.546
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
2676 2
engineering Vs30 , the harmonic average shear wave velocities in the first 30 m of soil, is usually taken as the main parameter of analysis and classification [1-2]. The doubts about the capability of the single Vs30 proxy to predict site amplification, and therefore precisely estimate the ground displacement, led to the suggestion of alternative proxies for the analysis. Such suggestions are, but not limited to, the use of the soil fundamental frequency f0 along with Vs30 [3-4]. Another suggestion is the use of the gradient of the shear-wave velocity profile factor B30 [5] that describes the variation of soil impedance in the first 30 meters. In the present work, the aim is to combine several parameters in order to provide a spectral approach that serves as a prediction and classification method for vibration levels in multiple sites. Nomenclature Vs B10 Vs3 f0 ANSR
Shear wave velocity Gradient of Vs profile from 0 to 10 m depth Harmonic mean shear-wave velocity in the first 3 m Fundamental resonance frequency of the soil Averaged normalised spectral ratio
2. Problem statement and parameters For the purpose of analyzing the soil behavior under higher frequencies excitation originating from surface sources, the existing parameters used in earthquake engineering are not reliable. Surface waves generated by vibration sources cannot be accurately described considering the shear wave velocity at higher depths, for instanceVs30 . A wavelength of 3m is proposed herein and the factor Vs3 will be used in order to characterize the effective part of soil that controls the effects of surface waves. This wavelength was chosen in order to fit with shallower soil layers, i.e. Vs between 100m/s and 400m/s and for a range of frequencies between 0 and 120Hz. We shall assess the heterogeneity of different sites (i.e. change of seismic impedance with depth) through gradient based proxies for various depths. Gradients Bz are found through a linear regression using the following equation:
log10 Vs ( z ) Bz log10 ( z ) Az
(1)
Where Z is the depth of the soil section considered, A Z is the origin ordinate of the regression and Vs(Z) the shear wave velocity at the given depth. For particular types of soil associated with the intended Vs3 , FEM models of wave propagation in homogeneous soils (half-space cases) and in the same soils above stratified stiffer soils (multilayered cases) have been performed. Displacements induced by a surface impulsive source have been calculated along the surface at different distances from the source for all the models. In particular, the maximum absolute value of the displacement was considered.
Fig. 1. Amplification Factors for Vs=400m/s versus Gradient (a) B20; (b) B15; (c) B10.
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine/ Procedia Engineering 00 (2017) 000–000
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Multiple test-cases yielded to the velocity gradient B10 as the optimal proxy to assess the variation of impedance with depth. Amplification in Fig.1 (Vs3 =400m/s in this case) for instance, represents the amplifications at the surface averaged for the first 20 meters next to the source as a function of the gradient-based parameter. Amplification is calculated as the difference between the total maximum displacement in the multi-layered case and the homogeneous case normalized with the latter. Fig.1 shows how the B10 gradient can be seen as a robust parameter to predict amplification. 3. Computation and data selection For the purpose of the present study, finite element simulations have been performed (CESAR-LCPC software). A surface displacement was applied as the source where a pulse is injected. Absorbing layers were applied in order to avoid spurious waves reflected at the model artificial boundaries [6]. Various simulations have been carried out taking into account four different factors: average velocity Vs3 , depth of the top layer, depth of the bedrock and velocity gradient B10 in the range between 0 and 1. All calculated spectral displacements were normalized by those of the source. 4. Spectral Approach Working with a wide range of vibration, between 0 and 120 Hz, the aim is to study the spectral response of soils at various distances. The approach chosen is to divide the frequencies into bands and divide the distances into three fields: near, mid and far fields. Near field indicates the first twenty meters next to the source. Mid and far fields indicate the second and third twenty meters next to the source respectively. The frequencies between 0 and 120 Hz are divided by bands of 20 Hz. The soils/sites were sorted in accordance with Vs3 and B10 as previously mentioned. For each band and distance segment, values were averaged and then displayed. 4.1. Near-field For Vs3 equal 100m/s, it could be noticed that with an increasing gradient, the dominant frequencies are larger. Similarly the amplitudes associated with peaks are increasing with the gradient. The dominant frequencies are between 0 and 40 Hz with an increasing gradient as shown in Fig.2.a.
Fig. 2. ANSR in the near-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
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Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
For Vs3 equal to 200m/s, and similarly to Vs3 =100m/s, the peak frequency increases with the velocity gradient, likewise for amplitudes as shown in fig.2.b. The frequencies are chosen in the bands between 20 and 60 Hz. Moreover the lower frequencies (0-20Hz) are falling with an increasing gradient, conversely to the higher frequencies. For Vs3 equal to 400m/s, it is harder to achieve higher gradient since it requires very strong material below the surface layer surpassing the stiffness of soils and rocks. Nevertheless the trend continues as for the two previous cases which are ascending higher frequency content with gradient along with larger amplitudes. In this case higher frequencies are dominating the problem (up to 100 Hz) while lower frequencies are declining with higher gradients. From all above, it could be deduced that the lower the gradient, the smaller the influencing frequency and vice versa. Furthermore the correlation between the maximum amplitudes and the gradient is quite clear, i.e. the higher the gradient the larger the amplitude. Inspecting individual cases before averaging and distributing frequencies by bands, for the near field, the peak frequency is always around the value Vs3 ⁄h1 where h1 the depth of the surface layer is. Hence we can conclude that lower gradients are associated to lower frequencies because they are usually associated to deeper layers (larger h1 ) and weaker contrasts (more homogeneous cases). This fact can describe the reason why also larger B10 are associated with higher frequencies. 4.2. Mid-field For Vs3 =100m/s, as previously, the larger the gradient the higher the related frequency. For the midfield, as shown in Fig.3.a, the dominance of the lower frequency band (0-20Hz) is extended to higher gradients (up to B10 =0.7). Moreover the energy carried by the lower frequencies for the smaller gradients are obvious when compared to the near-field, indicating a higher vulnerability for lower frequencies. for Vs3 =200m/s (Fig.3.b), the effect of the 20-40Hz band is starting to become clearer and dominant up to B10 =0.7, similarly to the Vs3 =100m/s. The amplitude and effect of the same band becomes clearer for smaller gradients. For Vs3 =400m/s (Fig.3.c), vibrations in most cases are dominated by 40-60Hz. This frequency range is equivalent to the half of the frequencies dominant in the near-field. Exceptions are for the smallest and largest gradients where 20-40Hz and 60-80Hz bands dominate respectively. High frequencies are associated to smaller wavelengths, in the mid-field a large amount of these waves are filtered through distance and the remaining waves are associated to lower frequencies. This effect is more apparent for smaller gradients. 4.3. Far-field For Vs3 =100m/s (Fig.4.a), while the effects of lower frequencies are still clear, a surge of higher frequencies appears. This surge is not only for B10 =0.9, where the 40-60 Hz band is dominating for the first time. Starting from B10 =0.6, the 20-40Hz band appeared to start dominating again as shown in Fig.4.a. For Vs3 =200m/s (Fig.4.b), the response is totally dominated by the 20-40Hz band and the effects of high frequencies are almost negligible. Similarly to the near and mid-field the smaller the gradient the more it is affected by lower frequencies. For Vs3 =400m/s (Fig.4.c), the peaks are still dominated by the same frequencies as in the mid-field, except for B10 =0.6 which is now dominated by a lower frequency band (40-60Hz). The higher frequencies are clearly less apparent since more small wavelengths are filtered through the increasing distance. The only unexpected change in the above results is the surge of higher frequencies for the Vs3 =100m/s for the higher gradients, especially for B10 =0.9. It was expected that the vibration will be dominated with lower frequencies at farther distances. After inspecting the models where B10 =0.9 and Vs3 =100m/s, it was noticed that it is usually characterized with a huge leap in soil stiffness after one or two weak thin layers. Moreover the inspection showed that the peak in the newly appearing high frequency was always around a value of √3. f0 where f0 is the frequency where the first peak appears which was dominant in the near and mid-field..
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine/ Procedia Engineering 00 (2017) 000–000
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Based on this, two assumptions can be made: (1) either that the probable presence of two weak layers will cause the effect of the second to appear at far distances; (2) either higher modes of excitation are triggered at far distances. Moreover both assumptions could be true and valid.
Fig. 3. ANSR in the mid-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
Fig. 4. ANSR in the far-field Spectral representation and distribution (a) Vs3 =100m/s; (b) Vs3 =200m/s; (c) Vs3 =400m/s
In order to investigate these assumptions, a simple rheological model of a 2DOF system has been studied. The two bodies are assumed of similar properties (in order to assimilate one layer with two degree of freedom or two layers who have negligible differences comparing to the bedrock) over a fully rigid support (assimilating the very strong bedrock). The model is shown in Fig.5. As shown below, writing the frequency equation and then calculating the fundamental and the second mode eigenfrequency, it could be noticed that the frequency of the second mode is at
Abdul Karim Jamal Eddine et al. / Procedia Engineering 199 (2017) 2675–2680 Jamal Eddine / Procedia Engineering 00 (2017) 000–000
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√3 of the first. Hence, it could be deduced that at a certain distance from the source the top layer will be dominated by the second mode of vibration, in this case for very weak soils with very large gradients.
Fig.5. Rheological model used
4km 16k 2m2 12k 2m2 1 2m 2
4km 16k 2 m 2 12k 2 m 2 2 2m 2
0.5
0.5
k m 3k m
(2)
(3)
5. Conclusion The spectral method offers a tool for sites classification according to their spectral response to surface sources. Moreover it offers an approach to predict vibration attenuation and amplification using proxies which are familiar to engineers such as shear wave velocities and profile gradients (Vs3 and B10 in this case). The spectral analysis has shown that gradients B10 could be associated to certain frequencies. This is because gradients B10 are heavily influenced by layers depth along with a strong contrast between materials. This association is founded on the fact that the fundamental resonance frequency of the soil under surface excitation is strongly dependent on the depth of the first layer (f0 = Vs3 ⁄h1 where h1 is the depth of the surface layer). That means the deeper the first layer or the weaker the contrast, the lower the fundamental resonance frequency. The discrepancy with the case of seismic waves is large since the entire surficial layer is generally excited by seismic motions. Conversely, for surface sources, the response is dominated by surface waves and is thus highly dependent on the distance from the source. Along the distance, the effect of higher frequencies begins to fade in a proportional pattern. This relates to the associated smaller wavelengths, which are filtered through distances. For sites associated to weak surface layers and high gradients, and starting from a critical distance, the second mode of vibration is triggered. It could be concluded that depending on Vs3 and B10 , higher modes of excitation will be triggered from certain distances. References [1] Borcherdt, R. (1994). “Estimates of site-dependent response spectra for design (methodology and justification)”, Earthq. Spectra 10, no. 4, 617–653. [2] Eurocode 8 (2004). Design of structures for earthquake resistance, part 1: General rules, seismic actions and rules for buildings, EN 1998-1, European Committee for Standardization (CEN), Brussels. [3] Luzi, L., R. Puglia, F. Pacor, M. R. Gallipoli, D. Bindi, and M. Mucciarelli(2011). “Proposal for a soil classification based on parameters alternative or complementary to VS30”, Bull. Earthq. Eng. 9, 1977–1898. [4] Cadet, H., P.-Y. Bard, A.-M. Duval, and E. Bertrand (2012). “Site effect assessment using KiK-net data: Part 2—Site amplification prediction equation based on f0 and VSZ”, Bull. Earthq. Eng. 10, no. 2, 451–489. [5] Régnier, J., Bonilla, L.F., Bertrand, E. and Semblat, J.F., (2014). “Influence of the VS profiles beyond 30 m depth on linear site effects: Assessment from the KiK‐net data”. Bulletin of the Seismological Society of America, 104(5), 2337-2348. [6] Semblat, J. F., Lenti, L., & Gandomzadeh, A. (2011). “A simple multi-directional absorbing layer method to simulate elastic wave propagation in unbounded domains”. International Journal for Numerical Methods in Engineering, 85(12), 1543-1563.