Earthquake Engineering and Structural Dynamics. DOI: 10.1002/eqe.2454, Final Draft. First published online on June 23, 2014, in press Article available at: http://onlinelibrary.wiley.com/doi/10.1002/eqe.2454/abstract. Wave Dispersion in High-Rise Buildings due to Soil-Structure Interaction Mohammadtaghi Rahmani 1, Mahdi Ebrahimian 2 and Maria I. Todorovska 3 ABSTRACT Nonparametric techniques for estimation of wave dispersion in buildings by seismic interferometry are applied to a simple model of a soil-structure interaction (SSI) system with coupled horizontal and rocking response. The system consists of a viscously damped shearbeam, representing a building, on a rigid foundation embedded in a half-space. The analysis shows that (1) wave propagation through the system is dispersive. The dispersion is characterized by lower phase velocity (softening) in the band containing the fundamental system mode of vibration, and little change in the higher frequency bands, relative to the building shear wave velocity. This mirrors its well known effect on the frequencies of vibration, i.e. reduction for the fundamental mode (softening) and no significant change for the higher modes of vibration, in agreement with the duality of the wave and vibrational nature of structural response. Nevertheless, the phase velocity identified from broader band IRFs is very close to the superstructure shear wave velocity, as found by the earlier study. The analysis reveals that (2) the reason for this apparent paradox is that the latter estimates are biased towards the higher values, representative of the higher frequencies in the band, where the response is less affected by SSI. It is also discussed that (3) bending flexibility and soil flexibility produce similar effects on the phase velocities and frequencies of vibration of a building. 1 Ph.D. Candidate, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email: mrahmani@usc.edu 2 Ph.D. Candidate, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email: mebrahim@usc.edu 3 Research Professor, University of Southern California, Dept. of Civil Eng., Los Angeles, CA 90089-2531, Email: mtodorov@usc.edu 1
1. INTRODUCTION In this short note, we revisit the problem of interferometric identification of a soil-structure interaction (SSI) model, studied earlier in [1], with the objective to (1) investigate the nature of wave dispersion caused by SSI, and (2) explain why the identified wave velocity from time shifts of pulses in impulse response functions is not affected by SSI. The insensitivity to the SSI effects, and consequently to softening of the soil during strong earthquake shaking, is an important argument in favor of using this identification method for structural health monitoring (SHM). We also (3) comment on the feasibility of estimation of the true fixed-base frequency of the superstructure, from recorded horizontal motions only, when the response is affected simultaneously by bending and soil-structure interaction. Todorovska [1] analyzed pulse propagation in a simple linear SSI model, with a shear beam superstructure and with coupled horizontal and rocking response, by measuring the time lag (travel time) between pulses in low-pass filtered impulse response functions (IRF), computed at different locations in the structure. The study demonstrated that, if the IRFs are broader-band (include at least two of the modes of vibration), the pulse travel time, τ, over the height of the structure, H, and, consequently, the pulse velocity c= H / τ are not sensitive to the effects of SSI. In contrast, the transfer-functions (TF) are affected. This finding agreed with reported circumstantial evidence that 1/(4 τ ), used as proxy for the building fixed-base frequency, changed when the structure was damaged and did not change when no damage was observed or was unlikely to have occurred (see Refs. 13, 14 and 20 in the companion paper [2]). A subsequent analysis of 1/(4 τ ) helped explain to what degree the observed wandering of the NS apparent frequency of Millikan library was caused by changes in the structure as opposed to changes in the soil (see Ref. 19 in the companion paper [2]). 2
While this important feature of the identified wave velocity was clearly demonstrated [1], the previous explanations have been only speculative. Benefiting from the nonparametric identification techniques, presented and verified in the companion paper [2], and from the more accurate least squares fitting algorithm, presented in Ref. 6 of the companion paper [2], we explain this feature, as shown further in this short note. 2. METHODOLOGY Without loss of generality, the SSI model consists of a shear beam supported by a semicircular rigid foundation embedded in uniform elastic half-space and excited by horizontal driving motion (see Fig. 1a, left). The foundation has three degrees of freedom, horizontal and vertical translations and rotation in the x-z plane. In the linear solution, the horizontal and rocking motions are coupled, while the vertical motion is uncoupled. The model impedances and derivation of the solution can be found in [3,4]. The model dimensions and material properties were chosen to match approximately the NS response of Millikan Library (a 9-story reinforced concrete moment frame/shear wall building in Pasadena, California) to the Yorba Linda, 2002 earthquake [3]. The shear beam has height H = 44 m, shear wave velocity c S =457.6 m/s (corresponding to fixed-base frequency f 1 = 2.6 Hz) and viscous damping of 1.2%, and the shear wave velocity in the soil of 300 m/s. The SSI model is used to produce observed absolute response of the building, uzω (; ), where z = 0 corresponds to the top, and transfer-function uz (; ω) hz ˆ(,0; ω ) = (1) u(0; ω) 3
from which observed band-pass filtered and regularized IRFs are computed, for virtual source at the roof, as described in the companion paper [2]. The IRFs of the fitted model, which is a fixed base uniform shear beam (see Fig. 1a, right), are calculated in a similar fashion. Fig. 1b shows comparison of the SSI model and fixed-base superstructure IRFs for virtual source at roof level (left) and TFs between the roof and ground level responses (right). The IRFs have been low-pass filtered at 15 Hz, and resemble the observed ones during the Yorba Linda earthquake of 2002, computed over the entire band of the records (0-25 Hz), because the recorded building response is very small beyond 15 Hz (see Ref. 19 in the companion paper [2]). It can be seen that the SSI model and fixed base superstructure TFs differ significantly in the frequency and damping of the first mode, which is shifted towards lower frequencies and is more damped in the SSI model response, due to soil-structure interaction through foundation rocking. In contrast, the time shifts of the pulses are very close. Estimate of the wave velocity c eq is obtained by the waveform inversion algorithm, which minimizes the least squares error between the observed IRFs and the IRFs of the fitted model at ground level within selected time windows (see Ref. 6 in the companion paper [2]). The width of the time windows was chosen to be equal to the width of the central pulse of the virtual source, i.e. 1/(2 f c), where fc = ( fmin + fmax )/2 is the central frequency of the band-pass filter [2]. In addition, estimate res c is obtained from readings of the observed modal frequencies, based on the relationship between the resonant frequencies of a fixed-base uniform shear beam and its shear wave velocity, as explained in the companion paper [2]. This relationship gives c( f1) = 4H f1, c( f2) = 4 H f2 /3, c( f3) = 4 H f3/5 etc., where f i is the frequency of the i-th mode of vibration. 4
3. RESULTS AND ANALYSIS 3.1 Measured Velocities Table 1 summarizes the results of fitting a fixed-base uniform shear beam in the SSI model response. The fitted beam has same height as the superstructure of the SSI model (H =44 m) and 2% damping was assumed. The velocity of the fitted shear beam, c eq, is found by least squares fit of low-pass filtered IRFs in three bands containing the first one, two and three modes of vibration (left; 0-5 Hz, 0-11 Hz and 0-15 Hz), and by least squares fit of band-pass filtered IRFs for three subbands (right; 0-5 Hz, 4.5-10.4 Hz and 10.5-16.5 Hz), and their comparison with the true velocity of the superstructure, c S. In this table, D c eq is the initial value used for the fitting, which is an estimate found directly from time shifts of pulses of the IRFs at ground level. Similarly, Table 2 shows the identified values from the apparent frequencies, c ( f ), for the res i first three modes of vibration and comparison with c S. The results are shown graphically in Figs 2 and 3. Fig. 2 shows the results of fitting IRFs in low-pass bands. Part a) shows comparison of the SSI model and fitted model IRFs for the broadest band (0-15 Hz) and the corresponding TFs. The time windows as well as the frequency band for the fit are shown. It can be seen that, for this band, the fitted IRFs and TFs are very close to those for the fixed base superstructure (see Fig. 1b). In part b), the identified velocities over the low-pass bands, c eq, are compared with the theoretical velocity of the superstructure, c S (the bars indicate the extent of the bands). It can be seen that, over the band that contains only the first mode, which is most affected by SSI, the identified velocity is much smaller than c S, by 28%, while, over the broader bands, which 5
contain also additional mode(s) that are less affected by SSI, the identified velocities are very close to c S, within 7%. Similarly, Fig. 3 shows the results of the identification in three narrower subbands. It can be seen that, for all three subbands, the IRFs of the fitted models are close to the observed IRFs. The TFs are also close, in terms of agreement of the modal frequencies within the subband over which the IRFs were fitted. It can be seen that, in the higher subband, c eq is within 1% from c S, while, in the first subband, it is 28% smaller than c S. Further, the estimates c ( f ) are res i shown, plotted as dots at f res = fi (Table 2). It can be seen that 1 c ( f ) is 36% smaller than c S. 3.2 Wave Dispersion in the Model and Physical Explanation The plot of the identified phase velocities vs. frequency (Fig. 3b) constitutes experimentally measured dispersion for the shear-beam, as part of the SSI system. The dispersion indicates significant softening for frequencies near the fundamental mode of vibration, and no significant change elsewhere, as compared to the beam shear wave velocity. For that reason, as reported earlier in [1], in a band that contains only the fundamental mode of vibration, the pulse travel time and the corresponding estimate of wave velocity are affected by SSI. This effect mirrors the effect of SSI on the resonant frequencies, in agreement with the dual wave and vibrational nature of structural response. The wave dispersion in the shear beam, as part of the SSI system, is caused by the moving boundary condition at its base, and is therefore of geometric nature. The apparent softening is a result of the way elastic waves are transmitted vertically through the beam, in contrast to softening, e.g. due to the material properties or the nature of the deformation of the beam regardless of the boundary conditions. The moving boundary condition is a combination of 6
horizontal and rocking motions, with amplitudes and phase that are frequency dependent, as determined by the coupling of the motions of the three bodies (soil, foundation and structure). Consequently the phase velocity is also frequency dependent. Analysis of amplitude and phase of this SSI model can be found in [5]. 3.3 Qualitative Comparison with Dispersion in Timoshenko Beam Model The pale curve in the background in Figs 2b and 3b represents the phase velocity TB, ph 1 c of the first wave mode of a fixed-base Timoshenko beam model of the NS response of Millikan library [2]. In contrast to the shear beam model, Timoshenko beam model accounts for deformation of a building due to both shear and bending flexibility (see [6] and Ref. 5 in the companion paper [2]). Comparison with the wave velocities of the SSI system and its shear beam superstructure shows that flexibility in flexure, in addition to shear, and SSI affect similarly the phase velocity, both causing softening in the band around the fundamental mode. In terms of phase velocity, the softening is manifested by c( f1) < c( f2), and, in terms of the modal frequencies, it is manifested by increase in the ratio f2 / f 1, which for shear beam superstructure becomes greater than 3. Because of this property, neither the nature nor the severity of the SSI effects for a full-scale building can be assessed solely from observed ratio f 2 / f 1 or estimated ratio c( f2)/ c( f 1). 4. DISCUSSION AND CONCLUSIONS The presented analysis of the SSI model in Fig. 1 showed that (1) wave propagation in a shear-beam building model on flexible soil is dispersive when foundation rocking is present, coupled with the foundation horizontal motion. (2) The identified wave velocity from broaderband IRFs of recorded horizontal response is less affected by SSI because it is biased towards the 7
values representative of the higher frequencies of the band, which are less affected by SSI. For the SSI model in Fig. 1, in which the superstructure is a shear beam, the identified phase velocity is close to the beam shear wave velocity. Further, (3) because the shear beam is lightly damped, and, therefore, the wave dispersion due to material damping is small, extrapolation of the fitted model to lower frequencies predicts closely the actual fixed-base properties in that band, including the fundamental fixed-base frequency. The results confirm that, if a building behaves closely to a uniform shear beam, its fundamental fixed-base frequency can be obtained only from a pair of recorded horizontal response, at base and roof. Otherwise, only a proxy is obtained by fitting a uniform shear beam. It is of interest if a closer proxy to the true value can be obtained by applying the same concept (i.e. fitting a model on a band in which the response is less sensitive to SSI and then extrapolating the model outside that band) with a more realistic model. Exploratory analyses, involving fitting uniform and layered Timoshenko beam models, can be found in recent conference papers [7,8]. ACKNOWLEDGEMENTS This work was in part supported by a grant from the U.S. National Science Foundation (CMMI-0800399). 8
REFERENCES 1. Todorovska MI. Seismic interferometry of a soil-structure interaction model with coupled horizontal and rocking response. Bull. Seism. Soc. Am. 2009, 99(2A):611-625. 2. Ebrahimian M, Rahmani M, Todorovska MI. Nonparametric estimation of wave dispersion in highrise buildings by seismic interferometry. Earthq. Eng. Struct. Dyn. 2014, DOI: 10.1002/eqe.2453 (the companion paper). 3. Todorovska MI, Al Rjoub Y. Effects of rainfall on soil-structure system frequency: examples based on poroelasticity and a comparison with full-scale measurements, Soil Dyn. Earthq. Eng. 2006, 26(6-7):708-717. 4. Todorovska MI, Al Rjoub Y. Environmental effects on measured structural frequencies model prediction of short-term shift during heavy rainfall and comparison with full-scale observations, Struct. Control Health Monit. 2009, 16(4):406-424. 5. Todorovska MI, Trifuanc MD. Analytical Model for Building - Foundation - Soil Interaction: Incident P, SV and Rayleigh Waves. Report No. 90-01, Dept. of Civil Engrg., Univ. of Southern California, Los Angeles, CA, 1990. 6. Boutin C, Hans S, Ibraim E, Roussillon P. In situ experiments and seismic analysis of existing buildings. Part II: Seismic integrity threshold. Earthq. Eng. Struct.Dyn., 2005; 34(12), 1531 1546. 7. Ebrahimian M, Rahmani M, Todorovska MI. Wave method for system identification and health monitoring of buildings extension to fitting Timoshenko beam model, Proceedings of the 10 th National Conference in Earthquake Engineering, Anchorage, Alaska, July 21-25, 2014, Earthquake Engineering Research Institute, Oakland, CA. pp. 10, in press. 8. Ebrahimian M, Todorovska MI. Structural system identification of buildings by a wave method based on a layered Timoshenko beam model, Proc. SPIE 9064, Health Monitoring of Structural and Biological Systems 2014, 90641C (March 9, 2014); doi:10.1117/12.2045219; http://dx.doi.org/10.1117/12.2045219, pp. 13. 9
Table 1 Identified wave velocity in the SSI system from low-pass and band-pass filtered impulse D response functions in different bands ( c eq are estimates obtained directly from pulse travel time and c eq and σ are waveform inversion estimates and the corresponding standard deviation). Comparison with the true velocity of the superstructure c S is also shown. Low pass filtered IRFs, c S = 457.6 m/s Band-pass filtered IRFs, c S = 457.6 m/s Band [Hz] D c eq [m/s] c eq [m/s] σ c eq eq c c S Band [Hz] D c eq [m/s] c eq [m/s] σ c eq 0-5 345.1 328.5 0.7% 0.72 0-5 345.1 328.5 0.7% 0.72 0-11 440 424.3 1.0% 0.93 4.5-10.4 440 454 0.5% 0.99 0-15 440 443 0.9% 0.97 10.5-16.5 463.2 459.4 0.9% 1.00 eq c c S res Table 2 Identified wave velocities, c ( f i ), of the SSI system from the system frequencies, f i, and comparison with the true shear wave velocity of the superstructure, c S Mode No. f i [Hz] res c ( f i ) [m/s] c s [m/s] c res ( fi ) c S 1 1.66 292.2 457.6 0.64 2 7.7 451.7 457.6 0.99 3 13.1 461.1 457.6 1.01 10
Figure Captions Fig. 1 a) Sketches of the soil-structure interaction model used to generate observed response (left) and of the model that is fitted (right). b) Impulse response functions, for virtual source at roof (left), and transfer functions, w.r.t. ground level motion (right) of the SSI model and of its superstructure if it were fixed-base. Fig. 2 Results of identification of the SSI model by fitting low-pass filtered impulse response functions. a) Comparison of the observed and fitted model impulse response functions (left) and transfer-functions (right) for fit in the band 0-15 Hz. b) Comparison of the identified velocities, c eq, in three low-pass bands with the true superstructure velocity, c S. The phase velocity of an approximate Timoshenko beam model of the same building TB, ph c 1 is shown in the background (redrawn from [2]). Fig. 3 Same as Fig. 2 but for identification by fitting band-pass filtered impulse response functions. The dots in part b) show estimates of wave velocity from the SSI model apparent frequencies. 11
Fig. 1 a) Sketches of the soil-structure interaction model used to generate observed response (left) and of the model that is fitted (right). b) Impulse response functions, for virtual source at roof (left), and transfer functions, w.r.t. ground level motion (right) of the SSI model and of its superstructure if it were fixedbase.
Fig. 2 Results of identification of the SSI model by fitting low-pass filtered impulse response functions. a) Comparison of the observed and fitted model impulse response functions (left) and transfer-functions (right) for fit in the band 0-15 Hz. b) Comparison of the identified velocities, c eq, in three low-pass bands with the true superstructure velocity, c S. The phase velocity of an approximate Timoshenko beam TB, ph model of the same building c 1 is shown in the background (redrawn from [2]). 1
Fig. 3 Same as Fig. 2 but for identification by fitting band-pass filtered impulse response functions. The dots in part b) show estimates of wave velocity from the SSI model apparent frequencies. 2