Edited by: Babak Moaveni, Tufts University, United States
Reviewed by: Serdar Soyoz, Bogaziçi University, Turkey; Wei Song, University of Alabama, United States
Specialty section: This article was submitted to Structural Sensing, a section of the journal Frontiers in Built Environment
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Seismic exposure of buildings presents difficult engineering challenges. The principles of seismic design involve structures that sustain damage and still protect inhabitants. Precise and accurate knowledge of the residual capacity of damaged structures is essential for informed decision-making regarding clearance for occupancy after major seismic events. Unless structures are permanently monitored, modal properties derived from ambient vibrations are most likely the only source of measurement data that are available. However, such measurement data are linearly elastic and limited to a low number of vibration modes. Structural identification using hysteretic behavior models that exclusively relies on linear measurement data is a complex inverse engineering task that is further complicated by modeling uncertainty. Three structural identification methodologies that involve probabilistic approaches to data interpretation are compared: error-domain model falsification, Bayesian model updating with traditional assumptions as well as modified Bayesian model updating. While noting the assumptions regarding uncertainty definitions, the accuracy and robustness of identification and subsequent predictions are compared. A case study demonstrates limits on non-linear parameter identification performance and identification of potentially wrong prediction ranges for inappropriate model uncertainty distributions.
Earthquakes still pose a major threat to the integrity of existing buildings. Although significant progress has been made on earthquake-resistant design methodologies, large parts of the building stock continue to sustain damage from earthquake actions. Structural damage to buildings after an earthquake is inevitable, especially in the context of design specifications being generally limited to the protection of building occupants rather than guaranteeing structural integrity in regions with low to medium earthquake hazard (Priestley,
Current practice for building assessment after earthquakes exclusively relies on visual inspection. However, large numbers of buildings need inspection in such a context. Combined with a potential need for multiple inspections of the same buildings, important economic losses can result from restricted access leading to loss of business opportunities and needs for provisional housing. Also, visual inspection has been shown to produce subjective assessment results (Marshall et al.,
Evaluation of increased vulnerability during aftershocks of buildings damaged by mainshocks of an earthquake sequence is a known challenge, and it has recently received attention in risk assessment during the design stage (Nazari et al.,
Measurement-based structural identification has the potential to improve upon current assessment techniques and to complement visual inspection. Measurement data can, thus, provide an objective support for post-earthquake decision-making. In addition, identifying parameters of physical models allows engineers to predict structural behavior during future earthquakes and can support the design of retrofitting and strengthening.
Ambient vibrations are an attractive data source in a post-earthquake context as small, sensitive and affordable sensors have emerged. Ambient vibrations are non-destructive and can be measured without any form of actuation and, therefore, ambient vibration measurements are potentially cost-effective and time-efficient. However, a well-known drawback of ambient vibration measurements is the very low amplitude of excitation, resulting in mostly linear responses. In addition, despite controversial results, modal properties derived from ambient vibrations have been shown in the past to be potential indicators of structural damage (Mucciarelli et al.,
Structural identification based on Bayes’ theorem to identify hysteretic models has been proposed (Muto and Beck,
Asgarieh et al. (
In this paper, three structural identification methodologies are compared. The background and underlying assumptions related to Bayesian model updating (BMU) involving traditional and modified likelihood function formulations as well as error-domain model falsification (EDMF) methodologies are described and compared. A methodology to derive non-linear structural parameters from linear modal structural properties is provided. These three structural identification methodologies are then applied to a numerical case study and the robustness and precision regarding parameter identification and behavior extrapolation are compared.
Error-domain model falsification, traditional Bayesian model updating (TBMU), and modified Bayesian model updating (MBMU) are compared. These three methodologies result in populations of solution through taking uncertainties into account. However, the methodologies differ in their initial formulation, starting points, assumptions related to the sources and forms of uncertainty, as well as the implementation of uncertainties.
Error-domain model falsification implements a model falsification strategy that is based on the principles of scientific discovery (Popper,
A population of model instances is created through discrete samples of parameter combinations and, subsequently, used to simulate structural behavior (Raphael and Smith,
Error-domain model falsification explicitly takes into account measurement and modeling uncertainties resulting from multiple sources that are estimated to be biased (Goulet et al.,
Tens of full-scale applications in several countries have confirmed that EDMF is intuitively understood by practicing engineers who need to interpret measurement data and field observations (Smith,
Error-domain model falsification, like any model-based structural identification methodology, relies on the comparison of predictions of model instances with measurements. Intervals of possible parameter values are defined for the parameters
For all
When error estimates, ε_{y} and ε_{g}, are accurate, the true parameter values, θ*, can be obtained through Eq.
However, in open-world applications, modeling error and measurement error cannot be estimated deterministically. Nevertheless, conservative estimates of the uncertainty resulting from model and measurement error can be established through engineering knowledge. Generally, such heuristic estimates of model uncertainties, U_{g}, and measurement uncertainties, U_{y}, take the form of bounded intervals. If there is additional information regarding some uncertainty distributions, more elaborate distributions can be used.
A joint probability density function (PDF), f_{Ui}(u), of the total uncertainty f_{U} is obtained by combining measurement and modeling uncertainties. The total uncertainty f_{U} reflects the engineering estimate of acceptable levels of error on the residual between measurement data and model predictions. Therefore, thresholds are derived from the uncertainty to falsify inappropriate models.
In order to calculate the thresholds, a target identification probability ϕ_{d} is fixed. The probability of false rejection of the correct parameter combination is, thus, given by (1-ϕ_{d}) for correctly estimated uncertainty distributions. The thresholds, T_{low,i} and T_{high,i}, delimitate the shortest interval that has a cumulative probability equal to the fixed target probability as shown in Eq.
The candidate model set (CMS) is defined by all parameter combination instances of the initial model population that verify Eq.
The complete CMS is used to perform behavior predictions. When structural behavior, q_{i}, is predicted, the model uncertainty is added to the prediction range to comply with Eq.
Traditional Bayesian model updating is based on the Bayes theorem of conditional probability. Multiple applications of BMU have been proposed in the past for static and dynamic measurement data (Beck and Katafygiotis,
Based on the Bayes theorem of conditional probability, prior knowledge of model parameters θ is updated using a vector of measurement data
Traditional applications of BMU, also referred to as Bayesian inference, use zero-mean normal (or Gaussian) PDF formulations as likelihood function (see Eq.
Most often, the covariance matrix
The parameter space is sampled using Markov-chain Monte-Carlo (MCMC) sampling (Papadimitriou et al.,
Modified Bayesian model updating is a novel formulation of BMU that avoids relying on zero-mean Gaussian likelihood functions. Relying on likelihood functions that are non-informed (constant probability) distributions is the most obvious difference between TBMU and MBMU. In environmental engineering, relying on uniform likelihood functions is sometimes referred to as generalized likelihood uncertainty estimation (Beven and Binley,
For structures that show non-linear behavior and where ultimate limit states need to be assessed, non-linear structural identification is needed. However, non-linear structural identification of buildings is complicated by the fact that non-destructive tests are limited to the linear range of structural behavior. Unless time-histories are measured during extreme events, such as earthquakes, no measurement data in the non-linear range are available. Therefore, in this paper, modal properties that characterize the linear behavior before and after an earthquake are used to identify parameters of a non-linear behavior model. Hysteretic non-linear time-series simulations are applied to link the building state before and after an earthquake (see Figure
Updated stiffness for a moment-resisting spring undergoing hysteretic non-linear moment-rotation cycles.
Simulation of non-linear time-series requires knowledge of the ground motion parameters that characterized the earthquake. If such knowledge is unavailable due to a coarse seismological network, multiple earthquakes need to be simulated in order to reflect the uniqueness and spatial variability of earthquake signals.
Error-domain model falsification as well as BMU involves multiple model simulations to identify model parameters. Simulating non-linear time-series for multiple model instances (typically thousands of models) is computationally expensive. Therefore, engineers rely on simplified models that are idealized representations of real structures. Lumped mass models or multiple-degree-of-freedom (MDOF) models are simplified models to simulate dynamic behavior of buildings. In such lumped mass models, each floor is represented by a single degree-of-freedom at which the mass of the floor is concentrated and that is linked to other floors by one stiffness element that sums the contribution of all structural elements. In simplified models, non-linear springs are used for lumped plasticity representation. For instance, non-linear hysteretic rotational springs are used at the base to model non-linear behavior of moment-resisting structures.
Damage is observed to often cumulate at the base floor, where shear forces and moments are highest. Reduction in stiffness that is due to structural damage resulting from earthquake actions is, thus, modeled by a reduction of the spring stiffness. A simplified method to update the spring stiffness due to damage is the secant stiffness to the maximum point reached during an earthquake. This simplified stiffness updating provides a lower bound to the damaged spring stiffness (see Figure
Modal parameters that can be used for structural identification are natural frequencies and mode-shapes. While measured and predicted frequencies can be compared directly, the modal assurance criterion (MAC) is used to compare modeled and predicted mode-shapes (Allemang and Brown,
Accuracy and efficiency of the three structural identification techniques are compared through an application using simulated measurements. Although simulated measurements are not realistic with respect to material behavior models, material homogeneity, and boundary conditions, application of structural identification techniques for simulated measurements provides conclusions regarding robustness of parameter identification and non-linear predictions, as the true values are known. However, simulated measurements provide upper bounds to the efficiency of structural identification methodologies.
The baseline model that is considered to be the true structure is a MDOF representation of a four-storey building. Plasticity is lumped into non-linear hysteretic rotational springs at the base and at each floor. The hysteretic behavior model for springs is the modified Takeda model (Takeda et al.,
Description of lumped mass models representing the baseline (true) structure
In reality, models are simplified and approximate representations of reality. Therefore, the model instance g(.) that is used to identify values of the parameter vector θ has a model bias with respect to the true structure. The mass is lumped at the floors and estimated to be 50 t per floor. The structure is assumed to be perfectly moment-resisting, meaning no rotation restraint of the lumped masses is considered. In addition, the plasticity is localized with a non-linear hysteretic spring at the base level. The hysteretic behavior is assumed to be defined by the Gamma-model (Lestuzzi and Badoux,
Parameters θ that are identified for model g(.) are the stiffness EI as well as the parameters of the γ-model governing the behavior of the rotational spring: stiffness (k_{rot}), yield moment (M_{y}), post-yield stiffness (hard), and the gamma-factor (γ). Thus, in total five parameters are identified based on simulated measurements (see Table
True values and prior distributions of the parameters to identify.
Parameter | Unit | Truth | Prior distribution |
---|---|---|---|
Flexural stiffness (EI) | MNm^{2} | 412.5 | 30–90 |
Rotational spring stiffness (k_{rot}) | GNm/rad | 65 | 10–100 |
Base yield moment (M_{y}) | MNm | 1.675 | 0.5–5.0 |
Post-yield rotational spring stiffness (hard) | % | 4.125 | 1–10 |
Gamma (γ) | – | – | 0–0.5 |
The simulated earthquake-aftershock sequence, for which predictions will be performed, are shown in Figure
Simulation of Mainshock and Aftershock sequence using the true model for simulated measurements. For identification, mainshock and aftershock are run for each model instance.
The measurement uncertainty is simulated by randomly adding an instance of the chosen measurement uncertainty to each measured quantity. The measurement error related to natural frequencies follows a normal n(0,1.5%) distribution while measurement error on modal displacements is estimated to be defined by a n(0,5.0%) distribution.
Three identification methodologies are compared in this paper (see
There are two approaches to determine the model uncertainty related to such discrepancies. Either engineering heuristics can help to estimate conservative bounds on the model uncertainty, or measurement data can be used to infer the uncertainty. In the second case, supplementary parameters θ_{i} need to be identified.
A total of eight structural identification applications are compared. The three structural identification methodologies are implemented for three scenarios based on how uncertainties are taken into account. A summary of the eight applications is provided in Table
Eight identification scenarios are used for comparisons.
Identification methodology | Uncertainty scenario |
||
---|---|---|---|
Measurement uncertainty only (S_{1}) | Predefined model uncertainty (S_{2}) | Parametrized uncertainty (S_{3}) | |
Traditional Bayesian model updating (Zero-mean normal) | S_{1,TBMU} | S_{2,TBMU} | S_{3,TBMU} |
Modified Bayesian model updating (uniform) | S_{1,MBMU} | S_{2,MBMU} | S_{3,MBMU} |
Error-domain model falsification (uniform) | S_{1,EDMF} | S_{2,EDMF} | Does not apply |
Two uncertainty scenarios involve predefined uncertainty ranges. First, model error is ignored (Scenarios S_{1} in Table
The third uncertainty scenario involves parametrization of uncertainties. The initial ranges for relative uncertainty SD and mean are 0–100%. EDMF relies on engineering judgment to estimate combined uncertainty; therefore, uncertainty identification is limited to BMU methodologies.
Four separate model uncertainty distributions are derived for: initial frequencies, initial MAC values, post-earthquake frequencies, and post-earthquake MAC values. Model uncertainty distributions are obtained as a uniform distribution between the minimum and the maximum error calculated for the three first modes that are used for identification. Likelihood functions for TBMU and MBMU that are derived from combining predefined measurement and model uncertainties based on Eq.
Likelihood functions for traditional Bayesian model updating (TBMU) (left) and modified Bayesian model updating (MBMU) (right) calculated using true model error values (Scenario S_{2} in Table
Three vibration modes are used for identification. Therefore, in order to limit the number of uncertainty distributions, single distributions are calculated for initial natural frequencies, post-earthquake natural frequencies, initial mode-shapes, and post-earthquake mode-shapes. Therefore, when uncertainties are parametrized for BMU applications, additional parameters need to be identified. For TBMU, SD for four independent zero-mean normal distributions are identified (S_{3,TBMU} in Table
Markov-chain Monte-Carlo sampling is used for the applications of Bayesian model updating (TBMU and MBMU). A Metropolis–Hastings algorithm (Hastings,
Grid sampling is used for EDMF. An initial model population is generated by dividing the parameter space (see Table
The first uncertainty scenario (see Table
Posterior distribution of parameter values obtained using traditional Bayesian model updating (TBMU) ignoring model uncertainty (S_{1,TBMU} in Table
Modified Bayesian model updating fails to provide a starting point for MCMC sampling when model uncertainty is ignored (S_{1,MBMU}). For100,000 randomly selected starting points in the parameter space, no parameter combination provides results that return a likelihood other than 0. Therefore, it is concluded that using MBMU, the model class, g(.), is rejected. Model class rejection indicates either a wrong model, a wrong selection or estimation of model parameters, or an underestimation of uncertainties.
Using EDMF, the initial model population sampled from the parameter space using a regular grid is entirely falsified when model uncertainty is ignores (S_{1,EDMF}). In a similar way to MBMU, this result indicates a misevaluation of the uncertainty or a wrong model class. The capacity to falsify entire model classes increases the robustness of structural identification by reducing the risk of biased results.
The second uncertainty scenario involves predefined uncertainty with correctly estimated measurement and model uncertainty (S_{2} in Table
Identification results for linear parameters resulting from correctly estimated model uncertainty for S_{2,TBMU}
Prediction of modal properties related to the initial state exclusively depend on linear parameters, which are model stiffness and rotational spring stiffness. Due to a small interval of identified values for structural stiffness compared to the initial range (8%), grid sampling for EDMF is re-evaluated using a sequential scheme: based on the results of the coarse grid (120,402 sample, see The Measurement Uncertainty Is Simulated by Randomly Adding an Instance of the Chosen Measurement Uncertainty to Each Measured Quantity. The Measurement Error Related to Natural Frequencies Follows a Normal N(0,1.5%) Distribution While Measurement Error on Modal Displacements Is Estimated to Be Defined by a N(0,5.0%) Distribution. Identification Scenarios), the linear parameters are resampled in the region containing candidate models, thereby increasing the total number of samples to 21,204 (Figure
Identified parameter ranges for EDMF are contained within identified ranges for MBMU. Unlike MBMU, which involves a MCMC sampling scheme, EDMF relies on discrete grid sampling of the parameter space. Therefore, MBMU gives a more refined result of the identified parameter space contour.
While parameter identification is precise for linear parameters, identification of non-linear parameters fails to provide precise results (see Figure
Identification results for non-linear parameters resulting from correctly estimated model uncertainty. Parametric uncertainty cannot be reduced significantly.
The third uncertainty scenario involves parametrizing the likelihood function (S_{3} in Table
Identification results for parametrized uncertainties in traditional Bayesian model updating application (S_{3,TBMU} in Table
For TBMU, the SD of a zero-mean normal distribution is treated as an uncertain parameter. However, the limited number of measurements undermines identifiability of structural parameters in addition to likelihood function parameters. Thus, no reduction in the structural parameter uncertainty is obtained for either linear or non-linear parameters and the results for parameter identification are not reported.
Modified Bayesian model updating relies on uniform likelihood functions with a constant probability inside bounds and zero probability outside the bounds. As MBMU allows biased uncertainty distribution (not centered on 0), parametrizing uncertainty leads to eight additional parameters for scenario S_{3,MBMU} (see Table
Identification of parametrized bounds to the L∞ norm likelihood function implemented in modified Bayesian model updating (S_{3,MBMU} in Table
In this case, 12 measured quantities are available: natural frequencies and mode-shapes of the first three modes before and after the earthquake. Identifying SDs for TBMU adds four likelihood function parameters to the five structural parameters. Identifying bounds for uniform likelihood functions in addition to parameter values would require the identification of a total of 13 parameters. In addition, long computation times, which increase with the number of parameters (higher number of samples are needed to get stable results), a high number of parameters is unidentifiable when limited number of measurements is available. Therefore, in such scenarios, uncertainties need to be identified prior to the identification task.
When such error estimations take the form of bounds on uniform distributions, the identification results may lead to the conclusion that the model class is wrong or the error underestimated. Underestimating the uncertainty can lead to biased identification results when error estimations are taken to be Gaussian, as it is the case using TBMU when model error is ignored. Such biased identification potentially leads to wrong predictions, as shown in the next section.
Identification of parameter values is an intermediate step when the goal is to predict behavior. In this section, identification methodologies are compared with respect to the accuracy and precision of behavior predictions. Predictions are carried out for base moment and top displacement during an aftershock. Since loading conditions differ from measurement conditions (modal properties), extrapolation is needed.
When extrapolation is carried out, the model error potentially differs from the model error that is used for identification. For instance, top displacement depends linearly on stiffness, while model properties depend on the square-root of stiffness. Therefore, model error should be redefined for extrapolation-based prognosis.
Model prediction uncertainty is estimated from true model predictions using Eq.
For identification scenarios that ignore model error (S_{1} in Table
Comparison of predicted absolute maximum base moment during an aftershock for tested identification scenarios. If model errors are ignored (S_{1} in Table
Predictions related to maximum absolute base moment values are presented in Figure
Error-domain model falsification assumes uniform prediction intervals based on bounds that are derived using a target prediction probability of 0.95. For predictions based on BMU, the complete prediction distribution is reported. The prediction results based on the true model error are accurate, as the true values are included in the prediction distributions. Prior parameter distributions without identification result in a prediction interval of 6–68 kNm for base moment. Thus, prediction range resulting from EDMF is reduced to 55% of the initial prediction range. For parametrized uncertainties (Figure
Figure
Comparison of predicted absolute maximum top displacements during an aftershock for tested identification scenarios. If model errors are ignored (S_{1} in Table
Estimating model errors correctly is challenging in full-scale applications. Therefore, the robustness of EDMF results with regard to misevaluated model uncertainty is assessed. As EDMF results are based on grid sampling, changing model uncertainty values does not require new model simulations. EDMF thereby enables the engineer to adapt and change model uncertainties when increased knowledge of the structure is acquired, for instance, through
The influence of model uncertainty levels is first assessed with respect to the number of candidate models that are found. As can be seen in Figure
Size of the candidate model set evaluated for changing levels of model error estimations.
Even more important than identification, the prediction robustness of EDMF with regard to misevaluation of model errors is assessed in Figure
Evolution of the maximum base moment prediction range during an aftershock. No candidate models were found for model uncertainties smaller than 90% of the model uncertainty derived using the true model.
Three structural identification methodologies are compared in terms of parameter identification and behavior prediction. Through varying scenarios for taking model uncertainty into account (no model uncertainty, predefined model uncertainty and parametrized model uncertainty), the data-interpretation methodologies are tested with respect to accuracy of prediction intervals that involve extrapolation from linear modal properties (used for identification) to non-linear time-history predictions. Traditional assumptions, such as zero-mean normal uncertainties are found to be inappropriate in such applications.
Simulated measurements from a simplified structure are used to test three structural identification methodologies. Upper bounds of the usefulness of structural identification using non-linear behavior models from linear measurements are obtained. Model uncertainties, such as ignoring shear contribution, changing hysteretic rules and ignoring springs on upper stories (as shown in Figure
Traditional assumptions for structural identification are shown to be inappropriate. However, future work on full-scale structures and models with increased complexity is needed to validate the accuracy of structural identification methodologies that are based on uninformed (uniform) uncertainty distributions.
Structural identification with non-linear models based on linear measurements is a challenging task. Three identification techniques that provide populations of solutions are reviewed and compared. The following conclusions are drawn:
Uniform probability distributions for combined model and measurement uncertainties are appropriate for situations with unknown and varying uncertainty correlation values. Predictions that involve extrapolation are robust with respect to large, biased and correlated uncertainties. Prediction precision is sacrificed to maintain accuracy.
Traditional Bayesian model updating that relies on non-uniform uncertainty predictions can result in biased prediction ranges. Maximum likelihood estimates should, therefore, be avoided. In addition, unlike MBMU and EDMF, TBMU fails to reject wrong model classes, which can result in wrong identification as well as incorrect predictions.
Adding prediction error is essential for predictions of structural behavior. For prognosis of building responses to actions that differ largely from measurement conditions, prediction uncertainty can be larger than identification uncertainty.
Parametrized uncertainties in MBMU (L∞ norm) provide conservative estimates of upper bounds of the correct model uncertainties. However, small numbers of measurements can undermine identification of parameter values along with parametrized uncertainty distributions.
For parameters that are related to non-linear behavior, the precision of structural identification based on linear measurements (natural frequencies) is understandably lower than for linear parameters. Nevertheless, prediction ranges can be reduced by 30% (for base moments) to 90% (for top displacements) using EDMF.
YR elaborated the application of the methodology to the context of non-linear predictions with linear modal measurement data. PL assisted to the elaboration of the case study and wrote the code for non-linear dynamic simulation. Also, YR wrote the majority of the paper and conducted the research and simulations of the case study. IS was actively involved in developing and adapting the data-interpretation methodology and wrote parts of the contribution. All authors reviewed and accepted the final version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was funded by the Swiss National Science Foundation under Contract No. 200020-169026. The authors thank Dr. Romain Pasquier and Sai Pai for their input and the reviewers for their constructive remarks.