Analytical and Numerical Investigations on the Vertical Seismic Site Response

Transcription

1 Analytical and Numerical Investigations on the Vertical Seismic Site Response Bo Han, Lidija Zdravković, Stavroula Kontoe Department of Civil and Environmental Engineering, Imperial College, London SW7 2AZ ABSTRACT: In this paper, the site response due to the vertical component of the ground motion is analytically and numerically investigated. Firstly, a one dimensional (1D) total-stress analytical solution is studied in frequency domain and compared with the numerical solution obtained from finite element (FE) time-domain analyses, showing a satisfactory agreement. Subsequently, the site response to the vertical ground motion is further investigated, considering solid-fluid interaction, with fully coupled time-domain FE analysis. The undertaken parametric studies show that the predicted dynamic response is strongly affected by the parameters characterising the hydraulic phase, i.e. the soil permeability and loading conditions, both in terms of frequency content and amplification. It is therefore suggested that coupled consolidation analysis is necessary to accurately simulate site response subjected to vertical ground motion. 1 INTRODUCTION During an earthquake, the ground is simultaneously subjected to shaking in both the horizontal and vertical directions. However, common practice for geotechnical earthquake engineering problems is to assess the site response to the horizontal component of the ground motion only. This is attributed to the assumption that the effect of the vertical ground motion component is less significant, due to its smaller magnitude and higher frequency content compared to the horizontal ground motion (Yang and Yan, 2009). When the effect of the vertical ground motion is taken into account for earthquake-resistant design, either simple empirical models for the vertical response spectrum, or empirical ratios between the vertical and horizontal response spectra (V/H), are usually employed. Furthermore the V/H ratios are commonly assumed to be less than 2/3 over the concerned frequency range (UBC, 1997). However, in some recent earthquake events (Papazoglou & Elnashai, 1996; Yang & Sato, 2000; and Bradley, 2012), strong vertical ground motions have been repeatedly observed, which can lead to significant vertical compression damages of engineering structures. Therefore, there is a need for a more systematic and rigorous analysis of soil response that can account for vertical ground motion. In this paper, the vertical site response is firstly investigated by employing a 1D total-stress analytical solution in frequency domain. The results of the 1D analytical solution are then compared to those obtained from FE analyses in time-domain. In order to consider the solid-fluid interaction, the vertical site response is further investigated by fully coupled FE analysis, where the influence of hydraulic-phase parameters on the vertical site response is emphasised through the conducted parametric studies. 2 A TOTAL-STRESS ANALYTICAL SOLUTION 2.1 Formulation Since 1970s, various analytical solutions have been proposed and developed to investigate the site response subjected to different types of motions and soil properties, most of which are based on the frequency domain method. These solutions form the bases of the so called shake-type programs, such as SHAKE (Schnabel et al., 1972) and EERA (Bardet et al., 2000). However, the majority of these programs can only account for the horizontal ground motion, and therefore limited investigation has been concentrated on the site response to vertical ground motion. Therefore, a 1D total-stress analytical solution in frequency domain, capable of simulating the vertical site response, is firstly discussed in this part to investigate the fundamental aspects of the problem. The basic principles of the frequency domain analytical solutions employed in site response analysis can be found in Kramer (1996), which herein are only briefly introduced in Figure 1. The key element

2 of this approach is the transfer function, which determines how much the input motion is amplified by the soil layer at each frequency. Considering a uniform soil layer of isotropic and linear visco-elastic behaviour overlying rigid bedrock, by employing elastic wave propagation theory the transfer function between the top and bottom boundaries for the vertical site response can be derived, as shown in Equation 1 and illustrated in Figure 2. By employing this transfer function, the vertical site response of a soil layer can be simulated by prescribing two parameters into the analytical solution (compressional wave velocity and damping ratio). However, only two extreme loading conditions, undrained and drained, can be reproduced, since the analytical solution is based on the total-stress method, i.e. no distinction between the effective stresses and pore pressures. Figure 1. Schematic graph of the principle of the frequency domain method F 1 cos 2 H v 1 H p v p 2 where ω is the Fourier frequency, H is the soil layer depth, v p is the compressional wave velocity and ξ is the damping ratio of soil subjected to vertical motions. Figure 2. Transfer function between the top and bottom boundaries for the vertical site response of a visco-elastic soil layer (1) visco-elastic soil behaviour and plane strain geometry. The numerical analyses are carried out using the Imperial College Finite Element Program (ICFEP, Potts & Zdravković (1999)), and the FE mesh (consisting of 20 8-noded isoparameteric quadrilateral elements) and boundary conditions are shown in Figure 3. The constant average acceleration time integration method from the Newmark s family of algorithms (Newmark, 1956), is employed for the FE analysis. Vertical accelerations are uniformly prescribed at the bottom boundary of the mesh as input motion. The strong vertical ground motion observed at the Christchurch Cathedral College station (the CCCC station) from the Christchurch Earthquake (New Zealand, 2011), shown in Figures 4 and 5, is chosen for this purpose. The parameters used in FE analysis and analytical solution are listed in Table 1. It should be noted that according to Zienkiewicz (1980), the compressional deformation of saturated porous soil structures is highly dependent on the conditions of soil state (i.e. undrained, drained and transient conditions). The soil constrained modulus under the undrained condition is composed of both the constrained modulus of the soil skeleton and the pore water bulk modulus, while under drained condition, the soil constrained modulus is only affected by the constrained modulus of the soil skeleton (as shown in Equations 2 and 3). Therefore, the soil constrained modulus is highly affected by the loading conditions and the pore water bulk modulus. This phenomenon can be accurately simulated by the two-phase coupled FE formulation by considering the consolidation process. However, herein only the one-phase FE formulation of ICFEP is employed in order to be consistent with the total-stress analytical solution. Therefore, different pore water bulk moduli (2.2E+06 kpa and 0.0 kpa) are employed in the onephase FE analysis to account for the undrained and drained soil behaviour respectively. Moreover, compressional wave velocities of the soil layer under undrained and drained conditions are calculated based on Equations 2 to 5, and implemented in the analytical solution. Finally, for the analytical investigation, 5% material damping is applied under both loading conditions, while for the numerical analysis, Rayleigh damping is employed with a target damping ratio of 5%, where the first and third fundamental frequencies are utilised for the calibration of Rayleigh damping parameters, as suggested by Zerwer et al (2002). 2.2 Comparison with time-domain FE analysis In this section the analytical solution for the vertical site response is compared against the FE analyses performed in time-domain. More specifically, the dynamic response of a soil column subjected to vertical ground motion is simulated by using both an analytical solution and FE analysis, assuming linear

3 D Undrained E1 K f n E D Drained (3) D v Undrained p Undrained (4) (2) Figure 3. Schematic graph of the soil column FE model Figure 4. Acceleration time history of the observed vertical ground motion (at the CCCC station) from the Christchurch Earthquake Figure 5. Acceleration response spectrum of the observed vertical ground motion (at the CCCC station) from the Christchurch Earthquake (5% damping) Table 1. Material properties for the soil column Parameter Value Young's modulus E (kpa) 1.98E+06 Bulk modulus for 2.20E+06/Undrained pore water K f (kpa) 0.0/Drained FE analysis Density ρ (g/cm 3 ) 2.0 Poisson s ratio ν 0.2 Analytical solution Time step Δt (s) Height H (m) 15.0 Compressional wave velocity v p (m/s) /Undrained /Drained Height H (m) 15.0 D v Drained p Drained (5) f v p Undrained (6) H Undrained 4 v p f Drained Drained (7) 4H where D Undrained and D Drained, v p Undrained and v p Drained, f Undrained and f Drained are the constrained moduli, compressional wave velocities and fundamental frequencies for a soil layer under undrained and drained conditions respectively, E is the Young s modulus, K f is the pore water bulk modulus, ν is Poisson s ratio, n is the porosity, ρ is the soil mass density and H is the depth of the soil layer. The comparison of analytical and numerical results is shown in Figures 6 and 7, in terms of acceleration time histories and acceleration response amplification spectra respectively at monitoring point A (see Figure 3). It should be noted that the response spectrum amplification factors are calculated by dividing the response spectra obtained at a point at the top boundary by the ones at a corresponding point at the bottom boundary over the frequency range. Based on Figures 6 and 7, for analyses under both undrained and drained conditions, it can be clearly seen that the vertical site response predicted by the analytical solution compares very well with the numerical results, both in terms of acceleration time histories and acceleration response amplification spectra. It is worthy to mention that different dynamic responses are observed between analyses under the two considered extreme conditions, which basically reflect the influence of loading conditions on the vertical site response for saturated porous materials. In particular, a larger fundamental frequency is observed for the analysis performed under undrained condition. Based on Zienkiewicz s theory (Equations 2 to 7), the fundamental frequencies of the investigated soil column under undrained and drained conditions are calculated and shown in Table 2, where a perfect agreement is observed between the numerically obtained values and those predicted by the analytical solution (Figure 7). The observed differences in vertical site response under the two extreme conditions, essentially imply that the total-stress analytical solution is not sufficient for predicting the vertical site response of a soil layer at any intermediate transient state, when consolidation occurs during the dynamic loading (depending on the range of soil permeability and load-

4 ing duration). Consequently, it is necessary to investigate the vertical site response by using the twophase coupled FE analysis, in order to consider the solid-fluid interaction effect. Figure 6. Comparison of the acceleration time history at point A between analytical and numerical results 3 TWO-PHASE COUPLED NUMERICAL INVESTIGATION In this part, the vertical site response is further investigated by employing fully coupled FE analysis (ICFEP), where parametric studies concerning the variation of hydraulic-phase parameters are conducted. The same FE model, displacement boundary conditions and input motion are employed as the ones used in the previous section. The parameters for the coupled FE analysis are shown in Table 3. Since the two-phase coupled FE formulation is employed for the present analysis, hydraulic boundary conditions need to be defined. As shown in Figure 8, the values of pore water pressure at the top boundary are prescribed as zero, and the degree of freedom of pore water pressure at corresponding nodes of same height along the two lateral boundaries are tied to be identical. The employed hydraulic boundary conditions indicate that the cumulative pore water pressure due to dynamic loading can only be dissipated at the top boundary of the mesh. It should be noted that, neither is Rayleigh damping employed, nor is numerical damping involved in the time integration method (the constant average acceleration method), in order to avoid their influence on the coupled vertical site response. A wide range of soil permeability was parametrically investigated to assess its influence on the vertical site response, as listed in Table 4. Figure 8. Schematic graph of the soil column FE model (Coupled FE analysis) Figure 7. Comparison of the response amplification spectra at point A between analytical and numerical results Table 2. Fundamental frequencies calculated based on Zienkiewicz s theory (1980) Parameter Scenarios Constrained modulus D (kpa) Compressional wave velocity v p (m/s) Fundamental frequency f (Hz) Undrained 8.1E Drained 2.2E Table 3. Material properties for the soil column (Coupled FE analysis) Parameter Value Coupled FE analysis Young's modulus E (kpa) Bulk modulus for pore water K f (kpa) 1.98E E+06 Density ρ (g/cm 3 ) 2.0 Poisson s ratio ν 0.2 Void ratio e Time step Δt (s) Height H (m) 15.0

5 Table 4. Permeability values for parametric studies (Coupled FE analysis) Scenario Case0 Case1 Case2 Case3 Case4 Permeability 1.0E E-7 1.0E-5 1.0E-4 1.0E-3 (m/s) Scenario Case5 Case6 Case7 Case8 Permeability (m/s) 5.0E-3 1.0E-1 1.0E+0 1.0E+2 The resulting vertical site responses at the monitoring point B (Figure 8) for the various values of permeability are compared in Figure 9, expressed as the acceleration response amplification spectra. Results are distinguished in two groups: those involving extreme values of permeability and those adopting permeability values within a range encountered in engineering practice. They are shown in Figures 9a and 9b respectively. For the numerical results involving extreme permeability conditions (case0 and case8), the dynamic responses match those from the analysis performed under purely undrained and drained conditions using one-phase FE formulation respectively. This agreement between the results is expected, since the definitions of the undrained and drained conditions comply with the employment of extremely low and high permeability values respectively. For the numerical results involving a more realistic permeability range, the influence of permeability is emphasised by the varied dynamic responses obtained for the different cases. In particular the following observations can be made when a relatively low permeability is employed (case1 and case2) the results match with those obtained under the undrained condition; as the permeability increases (case3 to case5), the amplification peak factor gradually decreases, but without a significant change in the frequency content. This means that the soil layer maintains the same constrained modulus, but with more damping introduced; when increasingly higher permeability is employed (from case6 to case7), a shift of the fundamental frequency is observed towards the lower frequency range, indicating a lower constrained modulus for the soil column. This phenomenon complies with Zienkiewicz s theory, that the soil constrained modulus is highly affected by the pore water bulk modulus related to different loading conditions (shown in Equations 2 and 3); by further increasing the permeability (case8 in Figure 9a), the amplification factor increases, reaching the peak of the drained dynamic response, indicating less damping involved for the soil column. According to Bardet and Sayed (1993), the change in amplification factors is attributed to the impact of the viscous damping, which is due to the interaction between the solid and the pore fluid phases. It should be noted that neither material damping nor numerical damping are employed for the coupled FE analysis, and therefore the observed viscous damping effect can be entirely attributed to the solid-pore fluid interaction simulated by the twophase coupled FE formulation. Overall, based on the undertaken parametric studies, it is shown that the predicted vertical site response is strongly affected by the parameters characterising the hydraulic phase, i.e. soil permeability and loading conditions, both in terms of frequency content and amplification. It is, therefore, suggested that coupled consolidation analysis is necessary to accurately simulate the vertical site response, in order to consider the effects of solid-fluid interaction. Figure 9. Two-phase coupled vertical site response of a soil column considering a wide permeability range

6 4 CONCLUSIONS The site response to the vertical ground motion is analytically and numerically investigated in this paper. A 1D total-stress analytical solution for vertical site response is firstly studied and compared against time-domain FE analyses, for two extreme loading conditions. It is shown that the vertical site response is significantly affected by the assumed loading conditions (i.e. undrained or drained), highlighting the importance of adopting a coupled-consolidation time-domain formulation for simulating the vertical site response of a soil layer at any intermediate transient state (i.e. when consolidation occurs during the dynamic loading, depending on the range of soil permeability and loading duration). The vertical site response is further investigated with fully coupled FE analysis, considering solidfluid interaction. The undertaken parametric studies show that the predicted response is strongly affected by the parameters characterising the hydraulic phase, i.e. soil permeability and loading conditions, both in terms of frequency content and amplification. In particular, a reduction of the soil constrained modulus is induced by employing higher permeability, reflected by the fundamental frequency shift. Furthermore, the amplification factor change indicates the viscous damping effect due to the interaction between the solid and the pore fluid phases. It is therefore suggested that coupled consolidation analysis is necessary to accurately simulate site response subjected to vertical ground motion. Yang, J. & Sato, T Interpretation of Seismic Vertical Amplification Observed at an Array Site. Bulletin of the Seismological Society of America 90(2): Yang, J. & Yan, X. R Factors affecting site response to multi-directional earthquake loading. Engineering Geology 107(3-4): Zerwer, A., Cascante, G. & Hutchinson, J Parameter Estimation in Finite Element Simulations of Rayleigh Waves. Journal of Geotechnical and Geoenvironmental Engineering 128(3): Zienkiewicz, O. C., Chang, C. T. & BETTESS, P Drained, undrained, consolidating and dynamic behaviour assumptions in soils. Geotechnique 30(4): REFERENCES Bardet, J. P. & Sayed, H Velocity and attenuation of compressional waves in nearly saturated soils. Soil Dynamics and Earthquake Engineering12(7): Bardet, J. P., Ichii, K. & Lin, C. H EERA: A computer program for Equivalent linear Earthquake site Response Analysis of layered soils deposits. University of Southern California, Los Angeles. Bradley, B. 2012, Recorded ground motions from the 22 February 2011 Christchurch earthquake. Proceedings of the Second International Conference on Performance-Based Design in Earthquake Geotechnical Engineering, Taormina, Italy. Kramer Geotechnical earthquake engineering. New Jersey: Prentice Hall. Papazoglou, A. J. and Elnashai, A. S Analytical and field evidence of the damaging effect of vertical earthquake ground motion. Earthquake Engineering and Structural Dynamics 25(2): Potts, D. M. & Zdravkovic, L Finite element analysis in geotechnical engineering: Theory. London: Thomas Telford. Schnabel, P. B., Lysmer, J. & Seed, H. B SHAKE: A Computer Program for earthquake response analysis of horizontally layered sites. Earthquake Engineering Research Centre, University of California, Berkeley. UBC Uniform Building Code. I. C. o. B. Officials. California: Whittier.