ABSTRACT: Natural frequencies and damping ratios are very important parameters characterizing the dynamic response of buildings. These dynamic characteristics of building structures are observed to vary during different earthquake excitations. To evaluate this variation, an instrumented building was studied. The dynamic properties of the building were ascertained using a time domain subspace state-space system identification technique considering 50 recorded earthquake responses. Relationships between identified natural frequencies and damping ratios, and the peak ground acceleration at the base level of the building and peak response acceleration at the roof level were developed. It was found that response of the building strongly depended on the excitation level of earthquakes. A general trend of decreasing fundamental frequencies and increasing damping ratios were observed with increased level of shaking and response. It is concluded from the investigation that knowledge of variation of dynamic characteristics of building is necessary to better understand its response during earthquakes.
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Natural frequencies and damping ratios are very important parameters which characterize the dynamic response of buildings under dynamic actions such as wind or earthquake excitation. Studies have shown that dynamic characteristics tend to vary with vibration amplitude. Researchers have investigated this phenomenon by adopting different analysis methods. Tamura et al. (1993, 1994) and Tamura & Suganuma (1996) investigated the amplitude dependency of frequency and damping ratios of buildings and towers under wind excitations using Random Decrement Technique. Li et al. (2002, 2003) and Wu et al. (2007) studied amplitude dependant damping of tall buildings under wind excitations by comparing their measured (using an empirical damping model) and predicted response (using a constant damping ratio). They found that the constant values of damping ratio recommended in the standards and used by the structural engineering practitioners for super tall buildings is too high rather than low under serviceability conditions.
In addition to wind-induced vibrations, instrumented buildings present an excellent opportunity to study their response during earthquakes. Celebi (1996) has compared the variation of damping and fundamental periods for five instrumented buildings, using low amplitude tests and strong motion records with the help of power spectrum analysis. Trifunac et al. (2001) investigated the variation in apparent frequency from one earthquake to another by studying five earthquake responses of a building using Fourier analysis. They found that the change of frequency from one earthquake to another is as large as a factor of 3.5 and non-linearity in the response of the foundation soil is the main cause of this change.
These and many other studies (Celebi 2006; Fukuwa et al. 1996; Saito & Yokota 1996; Satake & Yokota 1996) highlight the importance of studying the variation of frequencies and damping ratios of buildings during earthquakes of low, moderate and high intensities. The large variation of predominant frequencies of buildings has significant implications for design codes. The earthquake resistant design starts form evaluating base shear for which building period is an important factor. Damping ratio on the other hand is assumed as a constant parameter at design stage which is observed to be a highly variable parameter under dynamic loadings. Therefore, it is required to acknowledge and incorporate these changes adequately into the design practice. The natural frequencies are usually determined with reasonable accuracy. However, damping ratio is one of the most difficult structural parameter to estimate. It is therefore, necessary to adopt a method which can estimate damping ratio and frequency with reasonable accuracy.
This study will estimate the frequencies and damping ratios using subspace identification technique. For natural input modal analysis, this technique is considered to be the most powerful class of the known system identification techniques in the time domain (Overschee & Moor 1994). The main objective of this paper is to evaluate the variation in dynamic response using 50 recorded earthquakes on a building. Relations between peak ground acceleration (PGA) at the base level of the building and peak response acceleration (PRA) at the roof level are developed with the estimated frequencies and damping ratios. The variation in frequencies and damping ratios in each mode is evaluated. The outcome of this study is expected to further the understanding of dynamic behavior of buildings during earthquakes. For acknowledging and incorporating these changes into the design practice, variety of buildings need to be instrumented and studied under low, moderate and high intensity events so that a comprehensive picture could be developed. This study can be considered as an effort to create an increased awareness of this fact and provide new quantitative data.
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The building under study is instrumented as part of the GeoNet project funded by the Earthquake Commission (EQC) of New Zealand (Baguley & Young 2008). GeoNet is planning to instrument additional structures (buildings and bridges) across New Zealand. Some structures have already been instrumented in the Wellington and Canterbury regions and for some the process is in progress (Baguley & Young 2008; Gledhill et al. 2006).
description of the building, instrumentation and strong motion data
The building under study is located at Lower Hutt approx. 20km north-east of Wellington, New Zealand. This is a four storey reinforced concrete structure with a basement. The structural system consists of beam-column frames with a 229 mm (9â€³) thick reinforced concrete shear core which houses an elevator. The plan of the building is rectangular and frame arrangement is symmetrical but the location of the elevator shaft near the north end makes it unsymmetrical particularly in terms of stiffness distribution as shown in Figure 1. All the exterior beams are 762 Ã- 356 mm (30â€³ Ã- 14â€³) except at roof level where these are 1067 Ã- 356 mm (42â€³ Ã- 14â€³). All the interior beams and columns are 610 Ã- 610 mm (24â€³ Ã- 24â€³). Floors are 127 mm (5â€³) thick reinforced concrete slabs except a small portion of ground floor near the stairs where it is 203 mm (8â€³) thick. The roof comprises of corrugated steel sheets over timber planks supported by steel trusses. The building is resting on separate pad type footings and tie beams of 610 Ã- 356 mm (24â€³ Ã- 14â€³) are provided to join all the footings together. In the basement retaining walls, which are not connected with the columns of the structure are provided at all the four sides of the building.
The building is instrumented with five tri-axial accelerometers. Two accelerometers are fixed at the base level, two at the roof level and one underneath the first floor slab as shown in Figure 2. All the data is stored to a central recording unit and is available online (www.geonet.org.nz). The sensor array is configured to trigger on an event and had recorded many earthquakes of different intensities since its installation in early November 2007 (Baguley & Young 2008).
For this study, 50 earthquakes recorded on the building are selected which had epicenters within 200km from the building.
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Figure 1. A typical floor plan showing the location of stairs and elevator shaft.
Figure 2. Three dimensional sketch of the building showing sensor array marked with sensor numbers and their sensitive axes. Inset shows a planar view marked with sensor locations.
The reason for adopting this was to select earthquakes of such an intensity which can be recorded on the sensors and can shake the modes of interest quite well. The area surrounding the building has not been hit by any strong earthquake since its instrumentation. By selecting such a criterion it was attempted to avoid any earthquakes which have not caused any vibration in the building or have not been able to show its presence on the sensor recordings. Almost all of the recorded 50 earthquakes have Richter magnitude ranging from 3 to 5 except a very few that have more than 5 and the maximum recorded is 5.2. It means that nearly all the earthquakes fall into the category of low intensity except a very few; those can be treated as moderate events.
Examples of the time histories of one of the recorded earthquakes are shown in the Figures 3a, b and 4a, b.
Figure 3. Examples of seismic acceleration time histories from sensor 6: (a) EW-component (b) NS-component.
Figure 4. Examples of seismic acceleration time histories from sensor 4: (a) EW-component (b) NS-component.
Subspace state-space identification
In the time domain modal analysis, subspace state-space identification is considered to be the most powerful identification technique (Overschee & Moor 1994). After sampling of continuous time state space model, the discrete time state space model can be written as:
where A, B, C and D are the discrete state, input, output and control matrices respectively, whereas uk is the excitation vector and xk, ,yk are discrete time state and output vectors respectively. In reality there are always process and measurement noises present so adding these to the above equations result in:
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Here wk and vk are the process and measurement noises respectively.
The data from output yk and/or input xk is assembled in a block Hankel matrix, which is defined as a gathering of a family of matrices that are created by shifting the data matrices.
After this the identification involves two steps. The first step takes projections of certain subspaces calculated from input and output observations (in block Hankel matrix) to estimate the state sequence of the system. This is usually achieved using singular value decomposition (SVD) and QR decomposition. In the second step, a least square problem is solved to estimate the system matrices A, B, C and D. Then the modal parameters, i.e. frequencies and damping ratios, are found by eigenvalue decomposition of the system matrix A. Further details of the identification process can be found elsewhere (Ewins 2000; Overschee & Moor 1996).
Application of subspace state-space identification
The subspace state-space identification technique derives state-space models for linear systems by applying the well-conditioned operations, like SVD to the block Hankel data matrices. In order to determine the proper system order, the trend of the estimated modal parameters in a stabilization chart is observed as the system order increases sequentially. Stability tolerances are chosen based on the variance in frequency and damping ratios among the considered system orders. For this study the following criteria was followed for stability tolerances of each earthquake record:
Standard deviation for frequency variance â‰¤ 0.01Hz
Standard deviation for damping ratio variance â‰¤ 1%.
For the system identification, sensors 6 and 7 were taken as the inputs (excitations) while sensors 3, 4 and 5 as the outputs (responses). Sampling rate was 200Hz and for establishing stabilization diagram, system orders from 60 to 160 were evaluated for each earthquake record.
Table 1. The minimum, maximum, average values and standard deviation of identified frequencies and corresponding damping ratios of the selected 50 earthquakes.
Damping ratios (%)
*This is the standard deviation of the identified frequencies and damping ratios in each of the three modes for 50 recorded earthquakes. Not to be confused with the standard deviation of stability tolerances.
The objective of this research is to study the relationship between PGA, PRA and identified first three modal frequencies and corresponding damping ratios. Based on these relations, the variation in identified parameters is observed and conclusions are drawn in section 5.
The typical first three mode shapes are shown in Figure 5 in planar view. The shape of the first mode shows it to be a translational mode along east-west (EW) direction with a little rotation. The second mode is nearly purely torsional and the third one is a dominant translational mode along north-south (NS) direction coupled with torsion. The shear core present near the north side creates unsymmetrical distribution of stiffness and is a cause of the torsional behavior in all the three modes.
Figure 5. Planar views of the first three mode shapes identified from subspace state-space identification.
In this study, PRA of sensor 4 (at the roof on the north side) and PGA of sensor 6 (at the base on the west side) were considered. The recorded maximum PRA (0.041g) is in the NS direction which is double of that in the EW direction (0.021g). However, there is only one point in that maximum range, while all the other points are below 0.015g (Figs 6a, b). It can be seen that PGA and PRA have good correlations along both EW and NS directions. Therefore, frequencies and damping ratios relations are plotted with one of them only i.e. PRA. The R2 coefficient shown in the plots is used to calculate fit of the regression line. The closer
this value is to 1.0, the better the fit of the regression line.
Table 1 shows the minimum, maximum, average values and standard deviation of the identified frequencies and damping ratios for the selected 50 earthquakes. The standard deviation for the selected 50 events in the first, second and third modal frequencies are 0.10Hz, 0.12Hz and 0.08Hz respectively.
Frequencies are decreasing as the vibration amplitude is increasing and this is observed in all three modes along both EW and NS directions (Figs 7a, b). The best fit regression lines show that the trend is quadratic for all the three modes in EW and NS directions. The first modal frequency has good correlation along EW direction. But the correlations of the three modal frequencies in the NS direction are not good; also the regression lines are influenced by only one point at the end of the curve without which the best trend will be a straight line.
During some events, the first, second or third mode or any two of them were missing in the system identification results which shows that during those particular events these modes did not vibrate strongly enough. It reflects the fact that response of the building is sensitive to the frequency content of the earthquake also. In some events second and third modal frequencies tended to be very close and the minimum difference between these two was found to be 0.03 Hz. This shows the capability of subspace state-space identification technique to identify very close modes.
Damping ratio variation
Identified damping ratios show considerable scatter. Along both EW and NS directions, the first mode damping ratio shows increasing trend with increased level of shaking and its R2 value is reflecting a better correlation than the second and third mode damping ratios in both EW and NS directions (Figs 8a, 9a). However, second mode damping ratios, which correspond to a nearly
Figure 6. PRA of sensor 4 vs. PGA of sensor 6 for 50 earthquake records: (a) EW components, and (b) NS components.
Figure 7. The identified first, second and third modal frequencies for 50 earthquakes vs. PRA: (a) EW components, and (b) NS components.
Figure 8. The identified: (a) first mode damping ratios (b) second mode damping ratios (c) third mode damping ratios vs. EW components of PRA for 50 earthquakes.
Figure 9. The identified: (a) first mode damping ratios (b) second mode damping ratios (c) third mode damping ratios vs. NS components of PRA for 50 earthquakes.
purely torsional mode, does not show any clear trend (Figs 8b, 9b). The third mode is also showing signs of increasing damping with level of shaking with the exception of a few outliers (Figs 8c, 9c). The average values of damping ratio for the selected 50 events for the first, second and third modes are 2.7%, 4.7% and 2.8% respectively (Table 1). The pure torsional mode is showing a higher average damping ratio as compared to the other two coupled translational-torsional modes.
Conclusions and future research
In this study, system identification using subspace state-space technique was used to identify natural frequencies and damping ratios of a four storey reinforced concrete building. This technique is considered as the most powerful technique for the natural input in the time domain. To evaluate the variation in dynamic response the relations among first three natural frequencies and corresponding damping ratios, and PGA and PRA are developed for 50 recorded earthquakes. The main findings of this research are as follows:
Due to unsymmetrical distribution of stiffness the modes are coupled translational-torsional.
PGA and PRA have very good correlation along both EW and NS directions.
Modal frequencies are decreasing and damping ratios are increasing with the increase in the level of excitation except the case of nearly purely torsional mode where damping ratio shows no clear trend. The natural frequencies show a much clearer trend while damping ratios have scattered values.
Subspace state-space identification has the ability to identify very close modes.
Since most of the earthquakes recorded so far are of low to medium intensity, it would be an interesting study to see the response of the building during a strong event.
In some events it is observed that the first one, second or third or any two of the modes are missing. This finding shows that dynamic response of the building is sensitive to the frequency content of the earthquakes and is the focus of future study.
The effect of other factors e.g. temperature on the dynamic characteristics of the building will be explored.