Observed non-linear soil-structure interaction from low amplitude earthquakes and forced-vibration recordings Clotaire MICHEL1 , Philippe GUEGUEN2 , Pierino LESTUZZI3 Seismological Service, ETHZ, Sonneggstrasse 5, CH-8092 Zuerich, Switzerland 2 ISTerre, IFSTTAR, Universit´ e Joseph Fourier, CNRS, 1381 rue de la Piscine, F-38041 Grenoble, France 3 Applied Computing and Mechanics Laboratory (IMAC), EPFL, Station 18, CH-1015 Lausanne, Switzerland email:
[email protected],
[email protected],
[email protected] 1 Swiss
ABSTRACT: Quantification of dynamic soil-structure interaction (SSI) effects using recordings in structures is necessary to validate analytical and numerical models. Moreover, the non-linearity of the SSI effects is generally neglected and observations are needed to validate or not this assumption. For this purpose, this paper investigates seismic interferometry on civil engineering structures that it is theoretically able to separate contributions from structural and soil-structure effects in the dynamic properties of structures. The idea is to observe wave travel time in structures, independent of SSI, and relate them to resonance frequencies of the structure. The proposed simplified relationship is investigated for shear wall buildings, including low-rise masonry and high-rise RC structures. A test for a structure without SSI shows that this formula is a good proxy for the structural frequency but is systematically overestimating it due to a shear-wave velocity gradient in the structure. The example of a low-rise masonry house with a shallow foundation on a soft soil shows how much SSI is important in this case and that all the non-linear effects on the fundamental frequency are actually due to SSI. Moreover, the Grenoble City Hall building case shows the limitation of the method in terms of resolution. Moreover, the complexity of such a tall buildings makes the obtained Green’s function difficult to interpret, especially due to torsion. KEY WORDS: Seismic interferometry; Non-linear soil-structure interaction; Earthquake recordings; Forced vibrations; Frequency drop. 1
INTRODUCTION
Dynamic soil-structure interaction (SSI) has extensively been investigated from theoretical (e.g [1]) and observational (e.g [2]) points of view and has been shown to be able to play a major role in the response of buildings to earthquakes. However, few methods are available to ”measure” this effect in situ, which is necessary in order to validate modelling. In particular, the commonly used transfer functions between roof and basement have been shown recently to include SSI effect [3]. More generally, classical modal identification techniques, under ambient vibrations or earthquakes are efficient tools to provide the dynamic properties of the soil-structure system (e.g. [4], [5]) but SSI can hardly be decoupled. Seismic interferometry was recently proposed to extract dynamic properties of structures ([6], [7]) as an alternative tool to classical modal identification techniques. This method allows to estimate the Green’s function between two receivers (e.g. top and bottom) that only depends on traveling waves in between the receivers and not from external effects such as SSI. Todorovska [3] went further and proposed to estimate the structural and the rigid-body fundamental frequencies using simplified equations and applied them to several structures ([8], [9]). Moreover, it has been shown by many authors that the apparent fundamental frequency of structures was decreasing with amplitude, even in the elastic domain and that SSI had probably an impact on this decrease ([10], [11]). For a particular
case, Trifunac et al. [9] used seismic interferometry to follow this evolution under different earthquakes. The objective of this paper is to investigate this method based on seismic interferometry in order to quantify non-linear effects of SSI in the elastic domain by comparison to structural nonlinearities. For that purpose, its assumptions are detailed and their impact is estimated using laboratory tests. It is then applied to a low-rise unreinforced masonry house tested in situ under forced vibrations and to a high-rise reinforced concrete building under weak earthquakes. 2 2.1
TRAVEL TIME OF WAVES IN STRUCTURES Seismic Interferometry
In order to estimate the Green’s function between two receivers, instead of using correlations [12], Snieder and Safak [7] proposed to deconvolve recordings at different locations (transfer functions). They particularly proposed to deconvolve by recordings at the top to suppress the reflections of waves on this boundary and therefore simplify the interpretation of results: the propagation of a unique wave corresponding to the fundamental mode, if dominating, can then be easily followed. Todorovska [3] showed that in the obtained estimation of the Green’s function, only the wave travel times were not affected by SSI, not the oscillatory properties of the transfer functions, as often assumed in the literature. However, on the contrary to a modal analysis method such as Frequency Domain Decomposition used in [4], [5], the Green’s
function is not decomposing normal modes so that the result may be difficult to interpret in terms of modes. For the deconvolution, we used in this paper a water-level method [13] with a ”water level” of 10%. The signals are filtered as little as possible because the broader the signal, the narrower the input function and therefore the more precise the Green’s function [3]. The picking of the maxima in the Green’s function is performed by resampling the obtained time signal. Considering the building as 1-layer over a rigid half-space and propagating shear waves, the fundamental frequency of the building f0 is related to the travel time of waves τ extracted from the Green’s function following: (1)
with fsys the resonance frequency of the whole soil-structure system, i.e. the actually recorded frequency at the building top, and fRB the resonance frequency of the rigid body motion including rocking and relative displacement of the foundation, i.e. soil-structure interaction. The approximation of shear behaviour is not fulfilled in structures with shear walls, on the contrary to steel frames like the Factor building [14] for which the method is adapted. Moreover, the equivalent shear stiffness Ki at story i (altitude Hi ) of a cantilever beam (Young’s modulus E and inertia I) with a fixed base is decreasing when i is increasing following: Ki =
3EI Hi3
2nd Floor
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Figure 1. Ispra PsD test. estimation.
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Example of Green’s function
6 Resonance frequency 1/(4τ) proxy 5
(3)
The shear wave velocity, directly related to the stiffness, is therefore decreasing when the story i is increasing. In this case, 1 [6] showed that the proxy 4τ is over-estimating f0 . This has also been shown in wave propagation in the ground: this formula overestimates the frequencies by 10-15% with a precision of about 30% [15].
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PsD test on a low-rise structure without SSI
In order to show the performance of the method on a real structure with shear walls, it was first tested on data without SSI: a pseudo-dynamic test on a low-rise unreinforced masonry house with concrete floors performed in ELSA laboratory (Ispra, Italy) in the frame of the ESECMaSE project ([16], [11]). Such a structure is clearly behaving with both shear (stiff floors) and flexural (shear walls) components. The ”recorded” signals at the top and bottom of the structures were filtered using a Butterworth low-pass filter at 20 Hz in order to remove highfrequency artifacts. An example of Green’s function estimation is presented on figure 1. It shows the waves propagating up and down in the structure. 1 Figure 2 compares the proxy 4τ to the actually measured resonance frequency for different test amplitude and two
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6 Frequency (Hz)
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The resulting frequency is then related to the structure only and it is then possible to extract the dynamic properties of soilstructure interaction effects through the simplified equation [3] : 1 1 1 = 2+ 2 (2) 2 fsys f0 fRB
1
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1 4τ
Frequency (Hz)
f0 =
different test specimen with different brick type. f0 is measured as the maximum in the power spectral density of the recordings at the top floor. It decreases during the different tests of increasing amplitude due to damage in the structure. The correlation between f0 and the proxy is good but with a systematic overestimation of 10-30% as predicted by Bard and Riepl-Thomas [15]. The wave velocity is 20-40% higher in the ground floor than in the upper floor as quantitatively predicted in the previous section. Therefore, the proposed method cannot be used in absolute values for such kind of buildings, but results of relative shifts, for example as a function of amplitude (e.g. [11]), may be interpreted.
5 4 3 2 1
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Figure 2. Ispra PsD tests with increasing amplitude. 1 Performance of the 4τ proxy for the resonance frequency (top: clay, bottom: calcium silicate specimen)
3 3.1
FORCED VIBRATION TESTS ON A LOW-RISE STRUCTURE SUBJECTED TO SSI Data and interferometry results
The method was then applied to a 2-story masonry structure with a shallow foundation on soft sediments located in Monthey (Switzerland) [11], where forced vibration tests using a hydraulic shaker were performed by EMPA and EPFL (see [11] for more details on the test setup). The tests were performed in the transverse direction using a white noise signal with increasing amplitude. The building was not damaged by the tests. The resonance frequency of the soil-structure system is about 10 Hz under ambient vibrations [11]. The recordings at the bottom, first and second floors were deconvolved by the recordings at the first floor, without filtering (figure 3). This figure shows a wave propagating from top to bottom for positive times that is the result of the shaking at the top floor. The recording at the bottom floor is related to the fundamental frequency of the structure studied in more detail in [11]. It has a velocity of propagation of about 230 m/s which is slower than in RC buildings studied for example in Snieder and Safak [7]. However, at the first floor, the recording is showing a more complex shape with a first wave arrival propagating at a higher velocity (about 500 m/s). It may correspond to the first higher mode of the structure, having a stronger amplitude at this floor.
2nd floor
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Following [3], the rigid body resonance frequency is found to vary from 18.6 to 13.6 Hz between the first and the last test. It means that for the last tests, structure and SSI have comparable frequencies and that, at higher drifts, SSI may become the driving phenomenon. Under strong motion, the observed resonance frequency would then be the rigid body frequency, no more the fundamental frequency of the structure. The relative results, i.e. normalized by the value at the lowest shaking amplitude, are plotted versus inter-story drift amplitude on figure 4. Moreover, the frequency-amplitude relationship extracted from PsD tests in Michel et al. [11] is also reproduced. On the contrary to the soil-structure frequency, the structural frequency is not significantly changing with amplitude, as the relationship derived for this structural type in these amplitude range. It should be stressed that the frequency-amplitude relationship was mostly constrained by shaking at higher amplitudes. The figure shows that the rigid body frequency is decreasing much faster in this case with respect to shaking amplitude, down to nearly 70% of its original value. Moreover, inter-story drift is well explaining this drop since the results are aligned.
c=500 m/s f~25 Hz
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The structural frequency f0 is nearly the same through the different tests with increasing amplitude (Figure 4) at around 11.4 Hz. This value is 17 to 30% higher than the soil-structure system fundamental frequency extracted in [11] (overestimation according to the previous paragraph). Michel et al. [11] proposed an analytical formula to extract f0 value based on Wolf [1]. For this structure, this method gave a f0 value greater than 12.1 Hz (depending on the shear-wave velocity of the ground, poorly known). Both methods seem therefore to overestimate the f0 value but give coherent results.
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Figure 3. Monthey low-rise building. Example of Green’s function estimation. 3.2
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Evolution of the dynamic parameters
In Michel et al. [11], it has been shown that the fundamental frequency of this structure was decreasing faster than predicted from PsD tests studied in the previous paragraph. One of the assumption to explain this was SSI. The objective of this processing here is to validate this hypothesis. Therefore, the structural frequency f0 was computed from the travel time velocity between the top and bottom floor using the 1/4τ proxy. Moreover, the rigid body frequency was then extracted following Todorovska [3].
0.7
Stuctural frequency f0 Rigid body frequency fRB Relationship for URM structure −6
10 Maximum Interstory Drift (m/m)
−5
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Figure 4. Monthey low-rise building. Relative evolution with shaking amplitude of structural fundamental frequency and the rigid body frequency . 4 4.1
SMALL EARTHQUAKE RECORDINGS IN A HIGHRISE STRUCTURE SUBJECTED TO SSI Data and interferometry results
Similarly, the method was applied to the City Hall building of Grenoble (France) that is permanently monitored by the French Accelerometric Network [5]. This structure is a 13story reinforced concrete (RC) shear wall building founded on piles because of a low soil quality. Basic geometrical computations showed that it was subjected to rocking motion,
which is coherent with its geometry. The longitudinal and transverse fundamental frequencies of the soil-structure system are both close to 1.2 Hz under weak earthquakes as found by Fourier transform of the signal (figure 6 top). Moreover, the analytical formulation proposed in Michel et al. [11] shows that the transverse direction should be prone to SSI effects, whereas the longitudinal direction should not. However, the assumptions for the application of this method are not fulfilled (type of foundation). The used dataset is made of 9 weak earthquakes recorded in 2004/2005 with a sufficient signal/noise ratio (see [5] for a description). The permanent instrumentation in the structure includes three 3C sensors at the top and three on the basement allowing to stack the results. The study as a function of the structural drift shows a slight decrease of the soil-structure system frequencies for increasing shaking amplitudes [5]. The objective of this study is to determine if this decrease is due to structural reasons or SSI. The Green’s function between each couple of sensor top/bottom are first estimated and stacked (Figure 5 top). This figure is the example for the Arvieux earthquake, one of the weakest of the dataset. It has been shown that longitudinal, transverse and torsion modes were decoupled under ambient vibrations, but not perfectly [5]. Moreover, the torsional resonance frequency, around 1.4 Hz under ambient vibrations, is also excited during earthquakes so that waves corresponding to the bending and torsion modes are mixed. In the longitudinal direction, this effect is negligible and results are robust. In the transverse direction, during the weakest motions, torsion is activated and propagates through the structure. This can be seen by the different maxima positions depending on the considered couple. Moreover, the obtained function is complex. It shows what is probably a higher mode propagating at high velocity. In order to follow the fundamental mode only, one possibility is to filter the high frequencies, in our case using a Butterworth low pass filter at 5 Hz (Figure 5 centre). It simplifies the obtain function but also widens the input signal and therefore lowers the resolution. Filtering for lower frequencies shifts the results to lower frequencies and leads to wrong interpretations. It is therefore not possible to filter the torsion mode that is too close to the bending mode. Therefore, the torsional motion was removed assuming a rigid body rotation of each story centered on the geometrical centre. This procedure, applied for the sensor at the top and bottom appears to efficiently remove the torsion mode in the frequency domain and provides almost identical signals at the top and at the bottom. This technique is combined with the filtering method described above in order to extract more reliable results. Though these processing efforts, the torsion mode can still be seen in the function showing wide maxima corresponding to both modes in the transverse direction (Figure 5 bottom). Moreover, the picking error is 0.01 s, which corresponds to 5% of the picked delays. Thus, one cannot obtain frequency results with a lower uncertainty, that is higher than the investigated frequency drop (2%). Therefore, it is not possible to determine if the frequency decrease is structural or due to SSI. The obtained velocities are about 260 m/s, which is greater than for the previous studied structure. The resonance
frequencies are plotted with respect to the maximum drift (m/m) of the structure during the event in the considered direction, without torsion motion (figure 6). 4.2
Results on SSI
Figure 6 (centre) represents the structural frequency f0 obtained using the average up-going and down-going travel times for all sensor pairs, with its standard deviation and the 1/4τ proxy. Using this assumption, the longitudinal structural frequency is found to be 1.37 Hz on average, i.e. 18% higher than the soil-structure frequency (overestimated value as shown in the first part). Due to the reasons detailed above, the transverse structural frequency is more scattered with an average value of 1.43 Hz, i.e. 19% higher than the soil-structure frequency. Using this method, it is therefore not clear that transverse frequency is more affected by SSI as predicted using the analytical formulation. The rigid body frequency is then computed following Todorovska [3]. The uncertainties on this estimation are also large. The longitudinal rigid body frequency, is, on average, 2.2 Hz versus 2.4 Hz in the transverse direction, which is not a significant difference, on the contrary to what is predicted by the analytical method. This is clearly due to the foundation type, piles here, that is not accounted for in this analytical method. 5
CONCLUSIONS
Seismic interferometry on civil engineering structures is a new topic that deserve attention because it is theoretically able to separate contributions from structural and soil-structure effects in the dynamic properties of structures. The damping ratio is not investigated here but can also be extracted by this technique [7]. The simplified formula proposed by Todorovska [3] were investigated for shear wall buildings, including lowrise masonry and high-rise RC structures. A test for a structure without SSI showed that this formula was a good proxy for the structural frequency but was systematically overestimating it due to a shear-wave velocity gradient in the structure. The example of a low-rise masonry house with a shallow foundation on a soft soil showed how much SSI was important in this case and that all the non-linear effects on the fundamental frequency were actually due to SSI. Moreover, the City Hall building case showed the limitation of the method in terms of resolution. Moreover, the complexity of such a tall buildings makes the obtained Green’s function difficult to interpret, especially due to torsion. However, this method, based on observations, remains relevant compared to analytical approaches that need strong assumptions. Further developments of the method will include a modelling accounting for velocity variations in the structure and the use of ambient vibrations as proposed by Prieto et al. [14]. ACKNOWLEDGMENTS The work presented in this paper was partially funded by the European Commission through the ESECMaSE project (n. COLLCT-2003-500291), the Foundation for Structural Dynamics and Earthquake Engineering (EMPA-EPFL tests), the Foundation of Swiss Cantonal Insurances (VKF), the Rhone-Alps regional authorities through the Thmatiques Prioritaires program (project
System resonance frequency 0.3 Longitudinal Transverse
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Figure 5. Grenoble City Hall building under Arvieux earthquake. Estimated Green’s function at roof and basement: no filtering (top), low-pass filtering (centre) and low-pass filtering and torsion removal (bottom). VULNERALP) and by the French Research National Agency (ANR) through the RGCU program (project ARVISE ANR-06PRGCU-007-01, http://arvise.grenoble-inp.fr). The Grenoble strong-motion network is operated by the Earth Sciences Institute (ISTerre) for the French Accelerometric Network (RAP).
Figure 6. Grenoble City Hall building. Resonance frequency of the soil-structure system (top), of the structure only (centre) and of the rigid body (bottom). [4]
[5]
[6]
[7]
REFERENCES [1] [2]
[3]
J.-P. Wolf, Dynamic soil-structure interaction, Prentice Hall, Englewood Cliffs, 1985. M.D. Trifunac and M.I. Todorovska, “Recording and interpreting earthquake response of full scale structure,” in NATO Advanced Research Workshop on Strong-Motion Instrumentation for Civil Engineering Structures, Istanbul, Turkey, June 1999, Kluwer. M.I. Todorovska, “Seismic interferometry of a soil-structure interaction model with coupled horizontal and rocking response,” Bulletin of the Seismological Society of America, vol. 99, no. 2A, pp. 611–625, April 2009.
[8]
[9]
C. Michel, P. Gu´eguen, and P.-Y. Bard, “Dynamic parameters of structures extracted from ambient vibration measurements: An aid for the seismic vulnerability assessment of existing buildings in moderate seismic hazard regions,” Soil Dynamics and Earthquake Engineering, vol. 28, no. 8, pp. 593–604, 2008. C. Michel, P. Gu´eguen, S. El Arem, J. Mazars, and P. Kotronis, “Full scale dynamic response of a rc building under weak seismic motions using earthquake recordings, ambient vibrations and modelling,” Earthquake Engineering and Structural Dynamics, vol. 39, no. 4, pp. 419–441, 2010. H. Kawakami and M. Oyunchimeg, “Wave propagation modelling analysis of earthquake records for buildings,” Journal of Asian Architecture and Building Engineering, vol. 3, no. 1, pp. 33–40, May 2004. R. Snieder and E. Safak, “Extracting the building response using seismic interferometry: theory and application to the Millikan Library in Pasadena, California,” Bulletin of the Seismological Society of America, vol. 96, no. 2, pp. 586–598, April 2006. M.I. Todorovska, “Soil-structure system identification of millikan library north–south response during four earthquakes (1970–2002): What caused the observed wandering of the system frequencies?,” Bulletin of the Seismological Society of America, vol. 99, no. 2A, pp. 626–635, April 2009. M.D. Trifunac, M.I. Todorovska, M. I. Manic, and B. Bulajic, “Variability of the fixed-base and soil–structure system frequencies of a building—the case of borik-2 building,” Structural Control and Health Monitoring, vol. 17, pp. 120–151, 2010.
[10] C. Michel and P. Gu´eguen, “Time-frequency analysis of small frequency variations in civil engineering structures under weak and strong motions using a reassignment method,” Structural Health Monitoring, vol. 9, no. 2, pp. 159–171, 2010. [11] C. Michel, B. Zapico, P. Lestuzzi, F. J. Molina, and F. Weber, “Quantification of fundamental frequency drop for unreinforced masonry buildings from dynamic tests,” Earthquake Engineering and Structural Dynamics, 2011. [12] O.I. Lobkis and R. L. Weaver, “On the emergence of the Green’s function in the correlations of a diffuse field,” Journal of the Accoustical Society of America, vol. 110, no. 10, pp. 3011–3017, 2001. [13] R. W. Clayton and R. Wiggins, “Source shape estimation and deconvolution of teleseismic body waves,” Geophysical Journal of the Royal Astronomical Society, vol. 47, pp. 151–177, 1976. [14] G. A. Prieto, J. F. Lawrence, Chung A. I., and M. D. Kohler, “Impulse response of civil structures from ambient noise analysis,” Bulletin of the Seismological Society of America, vol. 100, no. 5A, pp. 2322–2328, October 2010. [15] P.-Y. Bard and J. Riepl-Thomas, Wave Motion in Earthquake Engineering, chapter 2. Wave propagation in complex geological structures and their effects on strong ground motion, Advances in Earthquake Engineering. WIT Press, 1999. [16] Anthoine A., “Definition and design of the test specimen. deliverable d 8.1 of enhanced safety and efficient construction of masonry structures in europe (esecmase),” Tech. Rep., European Commission, Joint Research Center, IPSC, European Laboratory for Structural Assessment (ELSA), Ispra, Italy, 2007.
