Bayesian model updating of a 20-story office building utilizing operational modal analysis results

Abstract

Most existing buildings are not equipped with long-term monitoring system. For the structural model updating and damage detection of this type of structures, ambient vibration test is popular as artificial excitation is not required. This article presents in detail the full-scale ambient vibration test, operational modal analysis, and model updating of a tall building. To capture the dynamic properties of the target 20-story building with limited number of sensors, a 15-setup ambient vibration test was designed to cover at least three measurement points (each consists of a vertical and two orthogonal horizontal measured degrees of freedom) for each selected floor. The modal parameters of each setup were extracted from the measured acceleration signals using a frequency domain decomposition method and were combined to form the global mode shape through the least-squares method. Due to the regularity of the building, a simple class of shear building models was employed to capture the dynamic characteristics of the building under lateral vibration. The identified modal parameters of the building were employed for the model updating of the shear building model to identify the distribution of inter-story stiffness. Since the “amount” of the measured information is small when compared to the “amount” of required information for identifying the uncertain parameters, the model updating problem is unidentifiable. To handle this problem, the Markov chain Monte Carlo–based Bayesian model updating method is employed in this study. The identified modal parameters revealed interesting features about the dynamic properties of the building. The well-matched results between model-predicted and identified modal parameters show the validity of the shear building model in this case study. This study provides valuable experience in the area of structural model updating and structural health monitoring.

Keywords

ambient vibration test Bayesian model updating Markov chain Monte Carlo simulation operational modal identification system identification

Introduction

Structural health monitoring (SHM) of civil engineering structures is a challenging research area. Efforts, either as practical implementation or theoretical development, have been devoted into this research area to establish SHM systems for existing structures (Behmanesh et al., 2015; Brownjohn et al., 2003; Cheung and Beck, 2009; Hu et al., 2018; Katafygiotis et al., 2000; Lam et al., 2006, 2014, 2017a; Simoen et al., 2013; Yin and Lam, 2010; Yuen and Kuok, 2011; Zhang and Au, 2016). One possible process of SHM includes full-scale vibration test, modal identification, and structural model updating. Full-scale ambient vibration test on civil engineering structures has been carried out for decades and it can be considered as the foundation of SHM since it provides the data for further applications such as system identification (modal identification and model updating) and damage detection. From the literature, most SHM systems were developed for bridges. Based on the best knowledge of the authors, structural model updating of buildings utilizing field test data (Hu et al., 2018; Hu and Yang, 2019; Lam et al., 2017a; Zhang et al., 2016) is not as popular as that in bridges. One notable work is a continuous health monitoring project on a super-tall building using ambient vibration test by Zhang et al. (2016). In this study, ambient vibration tests were implemented during different construction stages of Shanghai tower with a height of 632 m. The corresponding modal properties and their associated uncertainties were identified and investigated using the fast Bayesian fast Fourier transform (FFT) method. Ambient effects such as temperature and humidity are also recorded simultaneously to study the variation of modal properties with the changing environment. Modal analysis is another important step in SHM helping engineers to determine the dynamic properties of structures, and its results can be used for model updating (Behmanesh et al., 2015; Hu et al., 2018; Hu and Yang, 2019; Lam et al., 2014) and subsequently, damage detection (Lam et al., 2006). The aim of modal analysis is to identify modal parameters such as natural frequencies, mode shapes, and modal damping ratios from measured time-domain data such as acceleration responses. The identified modal parameters can be treated as indicators to assess the potential dynamic state of a target structure (e.g. changes in the modal parameters during different service periods can be used to help in the investigation of the potential problems of a structure).

Vibration-based structural model updating is an important component of SHM, and it can be considered as a process of adjusting the uncertain model parameters of a numerical model utilizing a set of measured vibration data so that the discrepancy between the model-predicted responses and the measured ones is minimal. Due to the problem of measurement noise, incomplete measurement, and modeling error, the results of model updating are uncertain in nature. Therefore, the Bayesian probabilistic approach that aims in calculating the posterior probability density function (PDF) of uncertain parameters (Beck, 1989) provides an explicit way to handle the uncertainty associated with the model updating results. Based on past experience, when the model updating problem is locally and globally identifiable, the posterior PDF of uncertain parameters can be well approximated by a Gaussian distribution and a weighted sum of Gaussian distributions, respectively (Beck, 1989). The error of these approximations will be large if the model updating problem is unidentifiable. One possible way to address the unidentifiable model updating problem is to approximate the posterior PDF using samples generated from multi-level Markov chain Monte Carlo (MCMC) simulation (Beck and Au, 2002; Lam et al., 2015). The MCMC-based Bayesian model updating method can be used to efficiently evaluate the posterior marginal PDFs of uncertain parameters without the need to perform high-dimensional numerical integration (Yang et al., 2015).

This article presents a comprehensive work of dynamic field test, modal identification, and MCMC-based Bayesian model updating of a 20-story building. The lower stories of the building are collected to other sub-structures, and it is interesting to explore the dynamic properties of the building. A multi-setup ambient vibration test was designed to cover all desired measurement degrees of freedom (DOFs; at least three measurement points for each selected floor) with only six sensors. Modal parameters were extracted from measurement data using frequency domain decomposition (FDD) and features of the dynamic properties of the building were discussed. In the model updating process, the building was simulated as shear buildings and the nominal values of inter-story stiffness of the building were estimated based on the technical drawings. A case study was specially carried out to demonstrate the capability of MCMC-based Bayesian model updating method in unidentifiable case. Based on the identified modal parameters, the MCMC-based Bayesian method was applied to update the shear building model to identify the inter-story stiffness values of the building. The case study results show that the MCMC-based Bayesian model updating method provides a practical and efficient way to estimate the posterior uncertainty associated with the identified modal parameters. The matching between model-predicted and identified modal parameters confirms the accuracy of the updated model.

Ambient vibration test and experimental results

Ambient vibration test of the tall building

The 20-story office building (see Figure 1) comprised a main tower and a multi-functional hall, which extends out from the building from 4/F to 7/F (Lam et al., 2017b). The building is not an isolated structure; it is connected to an adjacent building from 4/F to 5/F. Measurement was performed in the three staircases and the 19th floor to cover a total of 57 measurement locations (Lam et al., 2017b). Six force-balance tri-axial accelerometers were used in the 15-setup test to obtain measurement data. Figure 2 shows the schematic measurement plan in staircase 1, where Si, for i = 1, …, 6, indicates the sensor ID used in that position. Due to the length of cables, the location of the console was changed from 15/F to 4/F to cover all the sensors during measurement. Figure 3(a) and (b) shows the measured acceleration time history in the same staircase from the sensor at 19/F and 1/F, respectively. It can be found that the amplitude of acceleration at the top of the building (19/F) is approximately twice of that at the bottom (1/F).

Figure 1.

Front view of the 20-story office building (Lam et al., 2017b).

Figure 2.

Measurement plan in staircase 1 (Lam et al., 2017b).

Figure 3.

Time history acceleration from (a) sensor on 19/F and (b) sensor on 1/F.

Modal identification using FDD

FDD is adopted in this article to extract the modal parameters from the measured time-domain data. The basic theory of FDD is briefly reviewed here, and interested readers are redirected to the studies of Brincker et al. (2001) and Brincker and Ventura (2015) for more details. At frequency ω, the power spectrum density (PSD) matrix is written as

P_{y} (i ω) = \bar{V} (i ω) P_{x} (i ω) V^{T} (i ω)

(1)

where P_y(iω) is the n×n PSD matrix of the measured output responses, here n is the number of measured DOFs; i² = −1; P_x(iω) is the m×m PSD matrix of the inputs, here m is the number of inputs; V(iω) is the n×m frequency response function (FRF) matrix; the overbar denotes the complex conjugate and the superscript T denotes the transpose. The FRF matrix is further expressed in the partial fraction form as

V (i ω) = \sum_{j = 1}^{N_{m}} \frac{S_{j}}{i ω - λ_{j}} + \frac{{\bar{S}}_{j}}{i ω - {\bar{λ}}_{j}}

(2)

where N_m is the number of modes, λ_j is the jth pole, and S_j is the residue given by

S_{j} = φ_{j} β_{j}^{T}

(3)

where φ_j is the mode shape of the jth mode and β_j is the modal participation vector of the jth mode. Substituting equation (2) into equation (1) gives the output PSD matrix in the following form

P_{y} (i ω) = \sum_{j = 1}^{N_{m}} \frac{H_{j}}{i ω - λ_{j}} + \frac{{\bar{H}}_{j}}{i ω - {\bar{λ}}_{j}} + \frac{S_{j}}{- i ω - λ_{j}} + \frac{{\bar{S}}_{j}}{- i ω - {\bar{λ}}_{j}}

(4)

where H_j is the jth residue matrix of the output PSD matrix. When the damping is small, it can be shown that the residue matrix is proportional to the mode shape

H_{j} \propto p_{j} φ_{j} φ_{j}^{T}

(5)

where p_j is a constant. The output PSD matrix can then be expressed explicitly as

P_{y} (i ω) \approx \sum_{j = 1}^{N_{m}} \frac{p_{j} φ_{j} φ_{j}^{T}}{i ω - λ_{j}} + \frac{{\bar{p}}_{j} \bar{φ_{j}} \bar{φ_{j}^{T}}}{i ω - {\bar{λ}}_{j}}

(6)

To identify the modal parameters using FDD, the first step is to conduct singular value decomposition (SVD) for output PSD matrices at each frequency step and construct the singular value spectra. The natural frequencies can be identified at the peaks of the curve corresponding to the largest singular values, while the mode shapes can be obtained as the singular vectors of the output PSD matrices at the natural frequencies.

Identified modal parameters

Figure 4 shows the mode shapes of the four identified modes. The global mode shapes were assembled from the partial mode shapes, which were identified from various setups, using the least-squares method (Au, 2011). The basic idea of mode shape assembly is to multiply different partial mode shapes with different scaling factors and then calculate the values of the scaling factor by minimizing the mode shape discrepancy at the reference DOFs in a least-squares sense. To facilitate the discussion, the location at the first floor of staircase 1 is defined as the origin of the x–y coordinate system, with the x and y axes along the east-west (EW) and north-south (NS) directions, respectively. Figure 4(a)–(d) shows the natural frequencies and damping ratios, which were averaged from all the 15 setups.

Figure 4.

Identified mode shape using FDD. (a) First translational mode along NS direction (y-axis); mode 1: 0.91 Hz, 0.58%. (b) First translational mode along EW direction (x-axis); mode 2: 1.08 Hz, 0.55%. (c) Second translational mode along NS direction (y-axis); mode 3: 2.55 Hz, 0.82%. (d) Second translational mode along EW direction (x-axis); mode 4: 3.16 Hz, 1.60%.

Figure 4(a) shows the first identified mode (mode 1) along NS direction (y-axis). This is the fundamental mode of the structure. The natural frequency is less than 1 Hz. The corresponding damping ratio is 0.58%. Apart from translational movement along y-axis, a minor torsional movement can be found in the mode shape. As shown in Figure 1, the left part of the building is supported by a series of very strong shear walls but not for the right part of the building. The difference in lateral stiffness on the left and right parts of the building will induce torsional behavior in the translational mode shape. Figure 4(b) depicts the second identified mode (mode 2) along EW direction (x-axis). The natural frequency of this mode is 1.08 Hz and the damping ratio is 0.55%. A minor torsional behavior can also be found in this mode as in mode 1. Figure 4(c) shows the second translation mode along NS direction (mode 3) with a natural frequency of 2.55 Hz and damping ratio of 0.82%. The second translational movement can be clearly identified by a clear nodal point at 16/F. From the measurement point at 19/F, some torsional vibrations can also be observed. Figure 4(d) shows the second translational mode along EW direction (mode 4) with a natural frequency of 3.16 Hz and damping ratio of 1.6%. It is observed that the damping ratios of the second translational modes are much higher than the corresponding ones of the first translation modes. It is clear from Figure 4 that there is basically no change in the shape of the 19/F in the vibration of all identified modes. Therefore, it can be concluded that the rigid floor diaphragm assumption is valid in the modeling of the target building.

Model updating using shear building model

Bayesian model updating by MCMC simulation

To be self-contained, the basic theory of the MCMC-based Bayesian model updating method is briefly introduced here. Interested readers are redirected to the studies of Beck (2010), Lam et al. (2018), and Yuen et al. (2004) for more detailed formulations and implementations. To appropriately address measurement noise and modeling error, the fractional errors of the natural frequencies and mode shapes are assumed to follow the zero-mean Gaussian distribution with variance σ². The posterior PDFs of the uncertain parameters can be expressed as (Beck, 2010)

p (x | U) = c \exp (- \frac{J (x)}{2 σ^{2}})

(7)

where U denotes the set of identified modal parameters that is the set of measured data in the model updating process, x denotes the uncertain model parameter vector, c is a normalizing constant, and J( x ) is the measure-of-fit function given by

J (x) = \sum_{i = 1}^{n} [{(\frac{{\hat{f}}_{i} - f_{i} (x)}{{\hat{f}}_{i}})}^{2} + (1 - {| \hat{Φ_{i}^{T}} Φ_{i} (x) |}^{2})]

(8)

where the subscript i denotes the mode index, n is the total number of modes to be considered in the model updating process, ${\hat{f}}_{i}$ denotes the measured natural frequency in Hz, $f_{i} (x)$ is the calculated natural frequency whose dependence on uncertain model parameter x is emphasized, and $\hat{Φ_{i}}$ and $Φ_{i} (x)$ denote the measured and calculated mode shapes, respectively.

As the parameter space is usually large and the region of high probability is usually very small, it is impossible to get a set of samples with high probability by a single sampling process. Therefore, a multi-level sampling process is proposed. In between two sampling levels, the bridge PDF is defined to ensure a smooth transition. In the jth sampling level, the bridge PDF is denoted by p_j, and it is constructed according to the posterior PDF in equation (7) as (Lam et al., 2018)

p_{j} = c_{j} \exp (- \frac{J (x)}{2 σ_{j}^{2}})

(9)

where σ_j² controls the size of the region covered by the bridge PDF, and it will gradually decrease from level to level. This ensures that a large region of the parameter space is covered at the beginning to avoid the missing of global minimum. This also ensures the efficient convergence of samples at the end of the sampling process. In each level, sampling is conducted using the Metropolis–Hastings (MH) algorithm (Hastings, 1970; Metropolis et al., 1953) with the proposal PDF constructed by the kernel density (Au and Beck, 1999). The important regions of the posterior PDF are gradually reached as the sampling process goes on. The brief algorithm for MCMC-based Bayesian model updating can be summarized as follows:

Initialize the algorithm parameters such as the sample size N and the changing rate of σ_j² for the bridge PDF in each sampling level. Set the sample level to l and construct the objective function J( x ) based on the identified modal parameters.

In the first sampling level, generate N samples using the MH algorithm with a uniform PDF as the proposed PDF.

In a general level j, the sampling procedure is similar to that in the first level except that the proposal PDF in the MH algorithm is constructed by kernel density estimation based on the samples generated in the previous level j− 1.

Repeat step 3 until the samples in the final level are generated. The posterior marginal PDFs can be efficiently calculated by analytically integrating the kernel density based on the samples in the final sampling level. The posterior uncertainties can then be assessed based on the posterior marginal PDFs.

Nummerical modeling and case studies

As a preliminary study, the lateral vibration of the building is modeled by the class of shear building models. The objective is to identify the inter-story stiffness distribution of the building. Since the shear building model is one-dimensional, it was assumed that the vibrations along the two orthogonal directions were independent, and thus, two individual shear building models were proposed (see Figure 5).

Figure 5.

Shear building model along: (a) NS direction and (b) EW direction.

Based on the identified modal parameters, the vibration of the building is dominated by the lateral vibration modes in the two directions (i.e. NS and EW). To match the two shear building models (one for vibration along NS and the other for vibration along EW), the identified mode shape components along NS (and EW) for each floor are averaged, and the averaged values are compared to the corresonding floor of the shear building model for vibration along NS (and EW) direction.

Based on the technical drawings of the building, the values of inter-story stiffness at a given story was estimated by considering the individual shear walls and columns in that floor. Since the floor plan of G/F to 7/F differs from that of floors above the 7/F, two uncertain model parameters (see Figure 5) were considered for each shear building model. In the model updating process, the stiffness of a given story was calculated by multiplying the nominal values of K_i, for i = 1, 2, by the non-dimensional scaling factors θ_i, for i = 1, 2. In model updating, the nominal values of inter-story stiffness for NS shear building model are K₁ = 8.0 × 10¹⁰ N/m (G/F to 7/F) and K₂ = 6.0 × 10¹⁰ N/m (8/F to 19/F). For EW shear building model, they are K₁ = 9.5 × 10¹⁰ N/m (G/F to 7/F) and K₂ = 7 × 10¹⁰ N/m (8/F to 19/F).

Case study 1: model updating using mode 1 only

To demonstrate the performance of the MCMC-based Bayesian model updating method, a model updating case study using only mode 1 to update the corresponding shear building model was adopted. Figure 6 shows the posterior marginal PDF of the two scaling factors (i.e. the uncertain model parameters). It is clear from the figure that the distributions for both model parameters are very different from Gaussian distribution. Thus, a model updating method, which is based on the assumption that the posterior PDF follows a Gaussian distribution, will result in very misleading result in this case.

Figure 6.

Posterior marginal PDF of NS shear building model for Case study 1: (a) uncertain model parameter θ₁ and (b) uncertain model parameter θ₂.

Furthermore, the regions of high probability are large for both uncertain parameters. This implies that using only one mode in model updating, the uncertainty associated with the model updating result is very large. Next, the MCMC-based Bayesian model updating method is repeated using all four modes.

Case study 2: model updating using all four modes

Based on all four identified translational modes, the two shear building models were updated. The posterior PDF of the uncertain parameters were determined by the generated MCMC samples from the Bayesian model updating (see Figures 7 and 8). The solid lines in Figures 7 and 8 show the posterior marginal PDFs of the scaling factors for NS and EW shear building models, respectively. The posterior marginal PDFs were fitted by Gaussian distributions (the dashed dot lines in Figures 7 and 8). The figures show that the posterior marginal PDFs do not follow a Gaussian distribution. In particular, the two tails are not symmetrical. For discussion purpose, the most probable value (MPV) and the corresponding posterior coefficient of variation (COV) are calculated based on the samples, and they were summarized in Table 1.

Figure 7.

Posterior marginal PDF of NS shear building model for Case study 2: (a) uncertain model parameter θ₁ and (b) uncertain model parameter θ₂.

Figure 8.

Posterior marginal PDF of EW shear building model for Case study 2: (a) uncertain model parameter θ₁ and (b) uncertain model parameter θ₂.

Table 1.

MPVs and posterior COVs of scaling factors for each shear building model.

Uncertain parameter	θ ₁	θ ₂
NS shear building model	1.16 (35.08%)	0.95 (19.94%)
EW shear building model	1.231 (43.94%)	0.96 (20.91%)

It is clear that the MPVs of θ₁ and θ₂ for both NS and EW shear building models are very close to unity implying that the nominal values estimated from technical drawing are, in general, accurate. As for posterior COVs of the scaling factors, it is found that θ₁ of both shear building models are all over 30% (i.e. 35.08% for NS direction and 43.94% for EW direction) and θ₂ for both shear building models are around 20% (i.e. 19.94% for NS direction and 20.91% for EW direction). From the front view of the building in Figure 1 and the elevation view in Figure 2, the floors from 8/F to the roof are relatively consistent, but the floors from G/F to 7/F are not, due to the multi-functional hall. As a result, the level of modeling error for θ₁ (assuming constant inter-story stiffness from G/F to 7/F) is higher than that of modeling error for θ₂ (assuming constant inter-story stiffness from 8/F to the roof).

Table 2 illustrates the discrepancies between the measured and model-predicted natural frequencies (in terms of percentage error) and the mode shapes (in terms of Modal Assurerance Criterion [MAC]). The matching between identified and model-predicted results is generally acceptable for both natural frequency and mode shape. The largest percentage error for natural frequency between measured and model-predicted results is less than 8%. It can be observed that the MAC values for the first translational modes are a bit higher than those of the second translational modes. This matches the intuitive expectation that the vibrational behaviors in higher modes are more complicated and therefore more difficult to fit.

Table 2.

Discrepancies between the measured and model-predicted modal parameters.

Modes	NS-1	EW-1	NS-2	EW-2
Measured (Hz)	0.91	1.08	2.55	3.16
Model-predicted (Hz)	0.93	1.15	2.74	3.28
Percentage error	2.20	6.48	7.45	3.80
MAC (%)	94.3	93.1	88.5	87.6

To demonstrate the accuracy of the updated model, the identified mode shapes (solid line) are plotted together with the model-predicted mode shapes (dashed lines) in Figure 9(a)–(d). Figure 9(a) and (b) focus on the first translational modes, and Figure 9(c) and (d) focus on the second translational modes. The matching between the measured and model-predicted mode shapes is good especially for the first translational modes.

Figure 9.

Comparison between the identified and model-predicted mode shapes. (a) First translational mode along y-axis, (b) first translational mode along x-axis, (c) second translational mode along y-axis, and (d) second translational mode along x-axis.

Conclusion

This article presents a Bayesian structural model updating of 20-story office building utilizing measured dynamic data. The work includes a multiple-setup ambient vibration test, operational modal identification, and Bayesian model updating. A 15-setup ambient vibration test was carried out in the three staircases and the 19/F of the target building to obtain the acceleration time-history data. Four translational modes were identified from the measurement data using the FDD method. In the model updating part, the building was modeled using the class of shear building models. To identify the inter-story stiffness, the MCMC-based Bayesian model updating method was adopted to explicitly address the uncertainty. In the first case study, only mode 1 was emoployed in the model updating process. The results of this case study demonstrate the capability of MCMC-based Bayesian model updating method in the unidentifiable case (due to insufficient data). In the second case study, all four modes were empoloyed in model udpating. By matching the model-predicted and identified modal parameters, it is believed that the updated computer model can reasonably capture the dynamic behavior of the target building. The comprehensive work in this article provides valuable information on the use of ambient test data in structural model updating of full-scale structures using the MCMC-based Bayesian model updating method.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work described in this article was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project no. CityU 11210517 (9042509)).

ORCID iDs

Jun Hu

Mujib Olamide Adeagbo

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