# Modular Bayesian damage detection for complex civil infrastructure

## Abstract

We address the problem of damage identification in complex civil infrastructure with an integrative modular Bayesian framework. The proposed approach uses multiple response Gaussian processes to build an informative yet computationally affordable probabilistic model, which detects damage through inverse updating. Performance of structural components associated with parameters of the developed model was quantified with a damage metric. Particular emphasis is given to environmental and operational effects, parametric uncertainty and model discrepancy. Additional difficulties due to usage of costly physics-based models and noisy observations are also taken into account. The framework has been used to identify a reduction of a simulated cantilever beam elastic modulus, and anomalous features in main/stay cables and bearings of the Tamar bridge. In the latter case study, displacements, natural frequencies, temperature and traffic monitored throughout one year were used to form a reference baseline, which was compared against a current state, based on one month worth of data. Results suggest that the proposed approach can identify damage with a small error margin, even under the presence of model discrepancy. However, if parameters are sensitive to environmental/operational effects, as observed for the Tamar bridge stay cables, false alarms might occur. Validation with monitored data is also highlighted and supports the feasibility of the proposed approach.

## Keywords

Bayesian inference Damage detection Long suspension bridge Gaussian process Structural health monitoring## 1 Introduction

To be practically useful, a structural health monitoring (SHM) system must be able to identify the performance of complex physical systems. Noteworthy examples of structural failure, such as the 2018 collapse of the Morandi viaduct in Genoa, Italy, justify the development and deployment of SHM in civil infrastructure.

There are two common pathways for damage identification, depending on the type of model that is used to interpret monitored data. These are known as the data-based and physics-based approach. A common ground problem to both pathways, is their coarse misinterpretation of the structural behaviour due to numerous uncertainties. With the advent of powerful statistical sampling techniques, e.g., Markov Chain Monte Carlo methods, various probabilistic inference methods were developed and applied to surpass this challenge.

The Bayesian approach gathered considerable interest [10, 15, 45]. Its conceptual simplicity and consistent treatment of uncertainties contributed to its wide dissemination. Some notable research milestones are attributed to Sohn and Law [36] and Beck et al. [3, 4, 5]. Application of their frameworks for damage identification was illustrated in a seven-story full-scale building [28, 33] and a laboratory reduced scale steel bridge [29]. Overall, two main factors contribute to the methods poor performance. One is the environmental/operational effects—also known as confounding influences—which mask the presence and extent of damage. This is a well known and extensively documented problem [11, 16, 17, 35]. A second factor is large model discrepancy, i.e. the epistemic misfit between predictions and monitored data, caused by modelling assumptions and simplifications. Unless the effects of model discrepancy are small, assuming them as a zero-mean uncorrelated Gaussian [37, 43] (as in the traditional Bayesian methods) results in biased identifications.

For these reasons, the broad influence of environmental variations, and in particular temperature, lead to the development of several temperature-based damage identification methods. Some examples can be seen in Cross et al. [13], Yarnold and Moon [42] or others [26, 41, 44]. State of the art Bayesian methods adopt hierarchical models to account for these variations, such as the frameworks from Huang et al. [20] or Behmanesh et al. [8]. Relatively to model discrepancy, only a handful of studies developed a more functional form, such as a multivariate normal distribution [8] or correlated errors [30, 34]. The model falsification method by Goulet and Smith [18, 19] is an extreme alternative, which does not assume any particular form. Applications are illustrated in benchmark ASCE structures [20], a footbridge in the Tufts university campus [6, 7, 8] and a nine-story building [9]. Results still indicate unidentifiable or potentially biased identifications, and in some of the case studies, certain mode shapes or seasonal fluctuations had to be removed to improve results.

Based on the above remarks, the current paper proposes a hybrid modular Bayesian approach (MBA) to address the damage identification problem. Jesus et al. [22, 23] originally developed and applied the methodology for structural identification (st-id) problems. The MBA explicitly considers environmental/operational effects with multiple response Gaussian processes (mrGp). More importantly, the MBA multivariate normal discrepancy function and ability for data fusion [2] counters many of the problems of traditional Bayesian methods. Thus, the current work highlights the extension and application of the MBA to damage detection. The present paper is divided into a description of the methodology, Sect. 2; an application to a simulated cantilever beam, Sect. 3; and to the Tamar bridge, Sects. 4 and 5. Finally, major conclusions are presented in Sect. 6.

## 2 Modular Bayesian damage detection

### 2.1 Modelling assumptions

The current section gives a short overview of the modelling assumptions underlying the MBA. As commonly seen in Bayesian approaches, the parameters which we wish to learn from, \(\varvec{\theta }\), are treated as random variables. The ‘randomness’ in these variables is associated with our ability to estimate the parameters’ true values \(\varvec{\theta }^*\), and these true values are assumed constant throughout the monitoring period. Bayes’ theorem provides a way to synthesise two sources of information of these parameters, the prior distribution and the likelihood function, into an update posterior distribution.

Given the above assumptions, the next step requires approximating the computer model and the discrepancy function with multiple response Gaussian processes (mrGp) [2, 12]. MrGps efficiently account for the uncertainties of Eq. (1), but require estimation of a set of parameters (known as hyperparameters) from observed and simulated data. The MBA separates such process into two modules, for the computer model and the discrepancy function. A short description of the mrGp formulation will be demonstrated next, to present the hyperparameters and showcase the advantages of the proposed approach.

*q*responses, dependent of

*d*design variables at

*N*sampled time histories. Its description is based on a mean and covariance prior structure. The mean is assumed to have the following form:

*q*-dimensional identity matrix, \(\otimes\) stands for the Kronecker product, \(\varvec{H}\in \mathbb {R}^{N\times p}\) is a regression matrix, and \(\varvec{\beta }\in \mathbb {R}^{p\times q}\) is a matrix of regression coefficients. Matrix \(\varvec{H}\) contains

*N*polynomial functions which are assumed to have a linear form \(h_j(x)=[1\; x_1\ldots x_d]: j=1,\ldots ,p\) of degree \(p=d+1\).

*q*responses that are being approximated and a temporal correlation between the

*N*time histories. The latter matrix \(\varvec{R}\) contains inside each of its entries a correlation function which needs to be assumed. Currently, a linear form has been adopted

*x*to point \(x'\). A final parameter that has to be considered and concludes the description of the mrGp is the variance \(\varvec{\varLambda }\) of the observation error, which can be simply added to Eq. (3) as \(\varvec{\varLambda }\otimes I_N\).

Thus in short, the hyperparameters which have to be estimated are denoted as \(\varvec{\phi }^m=\{\varvec{\beta }_m,\varvec{\varSigma }^2_m,\varvec{\omega }_m\}\) for the model, and \(\varvec{\phi }^\delta =\{\varvec{\beta }_\delta ,\varvec{\varSigma }^2_\delta ,\varvec{\omega }_\delta ,\varvec{\varLambda }\}\) for the discrepancy function error term. Note that when \(m(\varvec{X})=0\) and matrix \(\varvec{R}\) is the identity matrix, the mrGp reduces to a zero-mean uncorrelated Gaussian, which is the common trend of traditional Bayesian frameworks applied in SHM. After estimating the hyperparameters, Bayes’ theorem is applied to obtain the parameter posterior distribution. For additional details of the modelling assumptions, considered uncertainties and application of the methodology for st-id, the reader is referred to previous literature [1, 2, 22, 23, 24].

### 2.2 Damage identification

In this section, the MBA framework is extended for damage identification. In the current context, damage is defined as changes introduced into a structural system which adversely affect its current or future performance. The proposed damage evaluation can be classed as probabilistic, parametric, and supervised. Essentially a reference state (when the structural system is assumed healthy) is established, and subsequently compared against an estimate for a current state. The two elements of Eq. (1) which assess structural change are the structural parameters and the discrepancy function. However, the current work is limited to the analysis of the parameters' influence. Ideally, considered parameters should represent the structural system integrity, e.g. soil permeability, initial strain of prestress cables or stiffness of other key components.

*Task 1*trains a mrGp with simulated data from a computer model, identically to the original MBA. The estimated hyperparameters of this mrGp, \(\phi ^m\), uniquely determine its behaviour. Subsequently,

*Task 2*and

*3*determine information relative to the reference and current state of the structural system, including a trained mrGp of the discrepancy function and the parameters posterior. Each task iterates over module 2 and 3 from the original MBA, with prior and monitored data \(D^e_\mathrm{{r}}\) or \(D^e_\mathrm{{c}}\) inputs for each state. It is recalled that the module 2 of the MBA, approximation of the discrepancy function, requires marginalisation of the computer model mrGp with respect to the parameters prior. Although the priors differ for Tasks 2 and 3, the computer model remains the same as computed in Task 1. Finally, in

*Task 4*the samples of the parameters posteriors are used to propagate the uncertainty to a damage metric DF, defined as follows

Finally, it is worth detailing some aspects of the current framework. Note for example, that during task 2 it is advisable to supply a reference data set \(D^e_\mathrm{{r}}\) which is as informative as possible, i.e. including a large range of environmental and operational variations, ideally in a one year time frame, acquired at the earliest possible stage of the structure life-cycle. In addition, Eq. (5) metric assumes that an *increase* of the parameter is associated with loss of its current or future performance, which might not necessarily be the case, e.g. if the parameter represents the stiffness or area of a structural element. However, a similar metric can be developed for the alternative case. Lastly, note that damage which occurs at a location, other than the one modelled by the identified parameters, would not be readily detected. One possible way to overcome this last limitation is to also analyse the variability of the discrepancy function between the current and reference state, using for example the Kullback–Leibler divergence.

## 3 Numerical example of a cantilever beam

*decrease*of the parameter represents damage.

*F*at its free end, and its tip deflection and rotation are considered as responses for identification (calculated from beam theory). The discrepancy between the model and the actual beam is a rotational spring, of stiffness \(K^*\), located at the fixed end of the beam, as shown in Fig. 2. Additionally, numerical values of the beam properties are shown in Table 1, with

*A*,

*I*and

*L*as the cross-sectional area, second moment of area and length, respectively. The intervals over

*F*and

*E*represent uniform sampling regions for training of the mrGps. For each state, two datasets with 90 and 60 samples have been generated to train the model and discrepancy function mrGps, respectively. The amount of samples were determined on the basis of an evaluation of the mrGps’ accuracy, with partition of 80% training and 20% testing data. Finally, for both the current and reference state, prior information of

*E*is set as a uniform PDF in the same interval as shown above.

Parameters of the cantilever beam

Parameter | Numerical value | Parameter | Numerical value |
---|---|---|---|

\(K^*\) | \(10\times 10^{11}~\mathrm{N\,mm/rad}\) | | \(3000~\mathrm {mm}\) |

\(E^*\) | \(70\times 10^{3}~\mathrm {MPa}\) | | \([20,100]\times 10^{3}~\mathrm {MPa}\) |

| \([1,10]\times 10^{3}~\mathrm {N}\) | | \(6.75\times 10^8~{\mathrm{mm}}^{4}\) |

| \(300\times 300~{\mathrm{mm}}^{2}\) |

## 4 The Tamar bridge reference and current state datasets

The current section frames the application of the MBA for damage detection in the Tamar bridge, highlighting major challenges and relevant details of the two states which are used for health assessment.

The Tamar bridge is a long suspension bridge, which connects Saltash and Plymouth over the Tamar river in south west England. Its construction was finalised in 1961, but its deck has been rebuilt in 1999–2001, with sixteen stay cables added to support the additional loading. Its concrete towers reach an height of 67 m, and it supports three roadway lanes over a span of 335 m. The deck stiffness is increased via supporting steel cables and a truss bridge under the deck. The cables arrangement is subdivided into main, stay and suspension cables. New bearings have also been installed in the Saltash tower, to improve the bridge thermal expansion. Thus, the bridge load history has changed considerably across time, and much research has been developed to understand its behaviour.

Finally, it should be noted that there are other sources of uncertainty which affect the damage identification task and have not been considered, e.g. wind or soil settlements. Up to a certain extent, combining the MBA’s ability to consider model discrepancy and the detailed FE model developed by Westgate [38] accounts for these effects. A detailed analysis of the Tamar bridge FE model discrepancy is shown in Jesus et al. [22].

### 4.1 Analysis of temperature and traffic effects on modal properties and mid-span displacements

In the current section, the monitored data are thoroughly examined, to justify the assumptions detailed in Sect. 2.1 placed over the mrGp of the MBA for damage detection.

Thus, it can be observed that most of the trends in Fig. 8 are linear; the dependency of the mid-span displacements on traffic is not as noticeable as for the other relations; and the LS1a and Northern displacement channels have the highest noise. These factors justify the linear correlations of the mrGp which approximate the discrepancy function. Furthermore, a larger search interval was assumed to estimate the variance hyperparameter \(\varvec{\varLambda }\) of the North mid-span displacement. Usually this estimation is performed by maximisation of the mrGp likelihood function with genetic algorithms.

The dataset for the current state follows similar patterns as the ones shown above. During this period no negative temperatures were registered, and the maximum temperature was \({17.4}\,^{\circ }\hbox {C}\).

### 4.2 Simulation of thermal and traffic effects in the Tamar bridge and mrGp emulation

Although not mentioned previously, it is important that the responses which are used for damage identification (natural frequencies/displacement) are sensitive to changes of the structural components under analysis (cables' initial strain and bearings' stiffness). One way to select appropriate output responses is by performing a sensitivity analysis with the bridge FE model. Westgate and Brownjohn [40] analysis indicates that the Tamar bridge natural frequencies are sensitive to the cables initial strain; justifying the responses chosen in the current publication. Similarly, Westgate et al. [39] has shown that the mid-span displacement is sensitive to the stiffness of the thermal expansion gap bearings. These analyses are based on a complex full-scale FE model of the Tamar bridge which has been developed using ANSYS parametric design language (APDL). The model has approximately 45,000 elements, from which expansion joints are modelled with linear spring elements; truss members with fixed-rotation beam elements; deck/towers with shells and the cables and hangers with uniaxial tension only beam elements. The FE model scale, complexity and computational cost fully justify the use of surrogate modelling.

The adopted traffic model is based on Westgate et al. [39], although our analysis considered only the traffic from the Plymouth to Saltash direction. The effects of traffic are assumed as a set of distributed mass nodes, evenly spread longitudinally across the bridge deck, and asymmetrically in the lateral direction, as shown by the top diagram in Fig. 7. An alternative and more comprehensive vehicular load case could be implemented on the basis of appropriate modelling ratios as presented by Jamali et al. [21].

## 5 Bridge cables/bearings damage identification

The previous sections have detailed the enhancement of the MBA framework for damage detection, field experiment aspects of the Tamar bridge SHM system and its FE model. Now it is possible to present the results of the actual application for each of the structural components under analysis. The validation is based on monitored forces from existent stay cables strain gauge load cells, and the already mentioned Battista’s tests.

Posterior PDF moments for the reference and current health state

Mean | Variance | |||
---|---|---|---|---|

Reference | Current | Reference | Current | |

Main cables | 0.0012 | 0.0012 | 0.25 \(\times\,10^{-6}\) | 0.14 \(\times\,10^{-6}\) |

Stay cables | 0.0024 | 0.0029 | 1.17 \(\times\,10^{-6}\) | 0.13 \(\times\,10^{-6}\) |

Bearings (kN/mm) | 8.3290 | 6.8365 | 5.39 | 4.11 |

### 5.1 Stiffness of bearings arrangements

*E*[DF] = − 17.9% and

*V*[DF] = 37.4%\(^2\). Despite the high uncertainty associated with the current estimate, visible in the PDF of DF, the mean value indicates a lower, and therefore safer, value of stiffness. Such result is in agreement with the conclusions from Battista et al. [14], whose tests were performed 3 months after the time period of the current state (cf. Fig. 6).

### 5.2 Initial strain of main suspension cables

*E*[DF] = − 2.4% and variance

*V*[DF] = 50.3%\(^2\). Once more, the negative expected value indicates an estimate below the reference state for the main cables, and its relatively low value is also reassuring. Although such results are encouraging, unfortunately it is not possible to validate them with any field data, since the Tamar bridge main cables have never been monitored directly.

### 5.3 Initial strain of stay cables

*E*[DF] = 18.60% and

*V*[DF] = 54.5%\(^2\) for the mean and variance, respectively. Similar to the previous example, the identified metric has a considerable uncertainty associated with its identification. However, oppositely to the previous results a non-negligible

*increase*of the initial strain was obtained, which is associated with loss of performance of the structural element. Therefore, it is necessary to investigate the cause of such deviation.

As described in Sect. 2.1, one assumption of the MBA is that the parameters do not change because of environmental/operational effects. Although these effects are considered in the response output \(\varvec{Y}(\varvec{X})\), the same cannot be said for the parameters \(\varvec{\theta }\). This stands in contrast to Behmanesh’s hierarchical Bayes approach, which is able to characterise a parameter’s randomness due to long-term monitoring, i.e. its inherent variability. To assess if the current deviation occurred because of damage, or if it can reasonably be explained by the parameters inherent variability, we consider a dataset from the stay cables, monitored with strain gauge load cells.

*F*is the cable force,

*T*is the temperature in the D062 sensor, and \(\alpha _1\) and \(\alpha _2\) are the coefficients from the polynomial functions. The numerical value of the \(\alpha\) coefficients is displayed in Table 3.

*F*(30) is the cable’s force function at an extreme temperature of 30 \(^\circ\)C, and \(\mu _F\) is the mean value of the cable force. The results are also displayed in the last column of Table 3.

Coefficients of linear polynomial function fitted to stay cable monitored data, and percentual deviation of the stay cable force relative to its mean

Cable | \(\alpha _1\) | \(\alpha _2\) | \(\frac{F(30)-\mu _F}{\mu _F}\)(%) |
---|---|---|---|

S1N | − 5.1 | 993.6 | 10.1 |

S1S | − 12.6 | 1245.6 | |

S2N | − 6.9 | 1136.2 | 12.0 |

S2S | − 6.4 | 1197.8 | 10.5 |

S3N | − 3.4 | 1087.6 | 5.9 |

S4N | − 7.1 | 1478.1 | 9.3 |

S4S | − 9.9 | 1036.7 | |

As it can be noted, two of the analysed stay cables, S1S and S4S, have a deviation due to temperature which is superior to the value identified by the MBA 18.6%, and can therefore reasonably explain the increase of its value. These values are typeset in bold. Furthermore, it should be noted that the data from the current state belong to a colder month (March), which is associated with higher cable force values. In reality, during long-term monitoring almost all parameters often experience some variability. In summary, what the current analysis highlights, is that the MBA performs optimally when external factors do not affect the structural components under evaluation. If instead the components have some inherent variability, the MBA can develop type II errors.

## 6 Conclusions

The current work has highlighted the first implementation of the MBA for damage identification in SHM. The methodology uses supervised learning, comparing information which is separated into a reference and current state. Additionally, to the presentation of the methodology, health assessments of a simulated cantilever beam and the Tamar long suspension bridge have been detailed. Specifically, the beam’s Young’s modulus, and the Tamar bridge’s main, stay cables and its bearings have been examined, and their probability of damage has been computed.

As shown in the simulated example, the MBA is able to detect damage under the presence of model discrepancy and operational variations. The resulting unbiased estimate is attributed to the correlated structure of the mrGp which approximates model discrepancy. However, further investigation is required to reduce the considerable estimation uncertainty, which stems from the assumption of independence between the reference and current states.

Additionally, the MBA is capable of assessing complex civil infrastructure using full-scale FE models, as shown for the Tamar bridge case study. Effects due to environmental and operational effects, noise and model discrepancy were taken into account. Results indicate that the bridge main cables and bearings did not indicate any signs of structural anomalies. In situ tests corroborate the latter conclusion. On the other hand, the bridge stay cables indicated a considerable increase of its initial strain, which was, however, attributed to temperature. Therefore, it is important to note that the MBA does not consider the inherent variability of identified structural parameters.

Lastly, the current study presented a local damage detection through the effect of structural parameters. Although not shown, it should be noted that the MBA formulation also allows to assess damage at a global level (through the model discrepancy). With this work, the authors hope to motivate further developments of the MBA for damage detection, and to enhance the state of the art of the SHM community.

## Notes

### Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) reference number EP/N509796. The leading author gratefully acknowledges the computing resources provided by the Scientific Computing Research Technology Platform of the University of Warwick. The authors would like to thank the two journal reviewers for their time and pertinent comments, which have greatly improved the manuscript.

## References

- 1.Arendt PD, Apley DW, Chen W (2012) Quantification of model uncertainty: calibration, model discrepancy, and identifiability. J Mech Design 134(10):100908. https://doi.org/10.1115/1.4007390 CrossRefGoogle Scholar
- 2.Arendt PD, Apley DW, Chen W, Lamb D, Gorsich D (2012) Improving identifiability in model calibration using multiple responses. J Mech Design 134(10):100909. https://doi.org/10.1115/1.4007573 CrossRefGoogle Scholar
- 3.Beck J, Au SK (2002) Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J Eng Mech 128(4):380–391. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:4(380) CrossRefGoogle Scholar
- 4.Beck J, Katafygiotis L (1998) Updating models and their uncertainties. I: Bayesian statistical framework. J Eng Mech 124(4):455–461. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455) CrossRefGoogle Scholar
- 5.Beck J, Yuen K (2004) Model selection using response measurements: Bayesian probabilistic approach. J Eng Mech 130(2):192–203. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:2(192) CrossRefGoogle Scholar
- 6.Behmanesh I, Moaveni B (2015) Model updating of structures with temperature-dependent properties using a Hierarchical Bayesian framework. In: SHMII-2015Google Scholar
- 7.Behmanesh I, Moaveni B (2015) Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite element model updating. Struct Control Health Monit 22(3):463–483. https://doi.org/10.1002/stc.1684 CrossRefGoogle Scholar
- 8.Behmanesh I, Moaveni B (2016) Accounting for environmental variability, modeling errors, and parameter estimation uncertainties in structural identification. J Sound Vib 374:92–110. https://doi.org/10.1016/j.jsv.2016.03.022 CrossRefGoogle Scholar
- 9.Behmanesh I, Moaveni B, Papadimitriou C (2017) Probabilistic damage identification of a designed 9-story building using modal data in the presence of modeling errors. Eng Struct 131:542–552. https://doi.org/10.1016/j.engstruct.2016.10.033 CrossRefGoogle Scholar
- 10.Ben Abdessalem A, Dervilis N, Wagg D, Worden K (2018) Model selection and parameter estimation in structural dynamics using approximate Bayesian computation. Mech Syst Signal Process 99:306–325. https://doi.org/10.1016/j.ymssp.2017.06.017 CrossRefGoogle Scholar
- 11.Carden EP, Brownjohn JMW (2008) Fuzzy clustering of stability diagrams for vibration-based structural health monitoring. Comput Aided Civil Infrastruct Eng 23(5):360–372. https://doi.org/10.1111/j.1467-8667.2008.00543.x CrossRefGoogle Scholar
- 12.Conti S, O’Hagan A (2010) Bayesian emulation of complex multi-output and dynamic computer models. J Stat Plan Inference 140(3):640–651. https://doi.org/10.1016/j.jspi.2009.08.006 MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Cross EJ, Worden K, Chen Q (2011) Cointegration: a novel approach for the removal of environmental trends in structural health monitoring data. Proc R Soc A Math Phys Eng Sci 467(2133):2712–2732. https://doi.org/10.1098/rspa.2011.0023 CrossRefzbMATHGoogle Scholar
- 14.de Battista N, Brownjohn JM, Tan HP, Koo KY (2015) Measuring and modelling the thermal performance of the Tamar Suspension Bridge using a wireless sensor network. Struct Infrastruct Eng 11(2):176–193. https://doi.org/10.1080/15732479.2013.862727 CrossRefGoogle Scholar
- 15.DiazDelaO F, Garbuno-Inigo A, Au S, Yoshida I (2017) Bayesian updating and model class selection with subset simulation. Comput Methods Appl Mech Eng 317:1102–1121. https://doi.org/10.1016/j.cma.2017.01.006 MathSciNetCrossRefGoogle Scholar
- 16.Farrar CR, Worden K (2007) An introduction to structural health monitoring. Philos Trans R Soc A Math Phys Eng Sci 365(1851):303–315. https://doi.org/10.1098/rsta.2006.1928 CrossRefGoogle Scholar
- 17.Figueiredo E, Radu L, Worden K, Farrar CR (2014) A Bayesian approach based on a Markov-chain Monte Carlo method for damage detection under unknown sources of variability. Eng Struct 80:1–10. https://doi.org/10.1016/j.engstruct.2014.08.042 CrossRefGoogle Scholar
- 18.Goulet JA, Smith IF (2013) Structural identification with systematic errors and unknown uncertainty dependencies. Comput Struct 128:251–258. https://doi.org/10.1016/j.compstruc.2013.07.009 CrossRefGoogle Scholar
- 19.Goulet JA, Smith IFC (2013) Predicting the usefulness of monitoring for identifying the behavior of structures. J Struct Eng 139(10):1716–1727. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000577 CrossRefGoogle Scholar
- 20.Huang Y, Beck JL, Li H (2017) Hierarchical sparse Bayesian learning for structural damage detection: theory, computation and application. Struct Saf 64:37–53. https://doi.org/10.1016/j.strusafe.2016.09.001 CrossRefGoogle Scholar
- 21.Jamali S, Chan THT, Nguyen A, Thambiratnam DP (2018) Modelling techniques for structural evaluation for bridge assessment. J Civil Struct Health Monit 8(2):271–283. https://doi.org/10.1007/s13349-018-0269-4 CrossRefGoogle Scholar
- 22.Jesus A, Brommer P, Westgate R, Koo K, Brownjohn J, Laory I (2018) Bayesian structural identification of a long suspension bridge considering temperature and traffic load effects. Struct Health Monit. https://doi.org/10.1177/1475921718794299 Google Scholar
- 23.Jesus A, Brommer P, Zhu Y, Laory I (2017) Comprehensive Bayesian structural identification using temperature variation. Eng Struct 141:75–82. https://doi.org/10.1016/j.engstruct.2017.01.060 CrossRefGoogle Scholar
- 24.Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc Ser B (Stat Methodol) 63(3):425–464. https://doi.org/10.1111/1467-9868.00294 MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Koo KY, Brownjohn JMW, List DI, Cole R (2013) Structural health monitoring of the Tamar suspension bridge. Struct Control Health Monit 20(4):609–625. https://doi.org/10.1002/stc.1481 CrossRefGoogle Scholar
- 26.Kostić B, Gül M (2017) Vibration-based damage detection of bridges under varying temperature effects using time-series analysis and artificial neural networks. J Bridge Eng 22(10):04017065. https://doi.org/10.1061/(ASCE)BE.1943-5592.0001085 CrossRefGoogle Scholar
- 27.Ku H (1966) Notes on the use of propagation of error formulas. J Res Natl Bureau Stand Sect C Eng Instrum 70C(4):263. https://doi.org/10.6028/jres.070C.025 CrossRefGoogle Scholar
- 28.Moaveni B (2011) System identification study of a 7-story full-scale building slice tested on the UCSD-NEES shake table. J Struct Eng 137(6):705–717. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000300 CrossRefGoogle Scholar
- 29.Ntotsios E, Papadimitriou C, Panetsos P, Karaiskos G, Perros K, Perdikaris PC (2009) Bridge health monitoring system based on vibration measurements. Bull Earthq Eng 7(2):469–483. https://doi.org/10.1007/s10518-008-9067-4 CrossRefGoogle Scholar
- 30.Papadimitriou C, Lombaert G (2012) The effect of prediction error correlation on optimal sensor placement in structural dynamics. Mech Syst Signal Process 28:105–127. https://doi.org/10.1016/j.ymssp.2011.05.019 CrossRefGoogle Scholar
- 31.Peeters B, De Roeck G (1999) Reference-based stochastic subspace identification for output-only modal analysis. Mech Syst Signal Process 13(6):855–878. https://doi.org/10.1006/mssp.1999.1249 CrossRefGoogle Scholar
- 32.Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. Adaptive computation and machine learning. MIT Press, CambridgezbMATHGoogle Scholar
- 33.Simoen E, Moaveni B, Conte J, Lombaert G (2013) Uncertainty quantification in the assessment of progressive damage in a 7-story full-scale building slice. J Eng Mech 139(12):1818–1830. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000610 CrossRefGoogle Scholar
- 34.Simoen E, Papadimitriou C, Lombaert G (2013) On prediction error correlation in Bayesian model updating. J Sound Vib 332(18):4136–4152. https://doi.org/10.1016/j.jsv.2013.03.019 CrossRefGoogle Scholar
- 35.Sohn H (2007) Effects of environmental and operational variability on structural health monitoring. Philos Trans R Soc A Math Phys Eng Sci 365(1851):539–560. https://doi.org/10.1098/rsta.2006.1935 CrossRefGoogle Scholar
- 36.Sohn H, Law KH (1997) A Bayesian probabilistic approach for structure damage detection. Earthquake Engineering & Structural Dynamics 26:1259–1281. https://doi.org/10.1002/(SICI)1096-9845(199712)26:12h1259::AID-EQE709i3.0.CO;2-3 CrossRefGoogle Scholar
- 37.Stull CJ, Earls CJ, Koutsourelakis PS (2011) Model-based structural health monitoring of naval ship hulls. Comput Methods Appl Mech Eng 200(9–12):1137–1149. https://doi.org/10.1016/j.cma.2010.11.018 CrossRefzbMATHGoogle Scholar
- 38.Westgate R (2012) Environmental effects on a suspension bridge’s performance. PhD, SheffieldGoogle Scholar
- 39.Westgate R, Koo KY, Brownjohn J (2015) Effect of solar radiation on suspension bridge performance. J Bridge Eng 20(5):04014077. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000668 CrossRefGoogle Scholar
- 40.Westgate RJ, Brownjohn JMW (2011) Development of a Tamar Bridge finite element model. In: Proulx T (ed) Dynamics of bridges, vol 5. Springer, New York, pp 13–20CrossRefGoogle Scholar
- 41.Xia Q, Cheng Y, Zhang J, Zhu F (2017) In-service condition assessment of a long-span suspension bridge using temperature-induced strain data. J Bridge Eng 22(3):04016124. https://doi.org/10.1061/(ASCE)BE.1943-5592.0001003 CrossRefGoogle Scholar
- 42.Yarnold MT, Moon FL (2015) Temperature-based structural health monitoring baseline for long-span bridges. Eng Struct 86:157–167. https://doi.org/10.1016/j.engstruct.2014.12.042 CrossRefGoogle Scholar
- 43.Zheng W, Yu W (2015) Probabilistic approach to assessing scoured bridge performance and associated uncertainties based on vibration measurements. J Bridge Eng 20(6):04014089. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000683 CrossRefGoogle Scholar
- 44.Zhu Y, Ni YQ, Jesus A, Liu J, Laory I (2018) Thermal strain extraction methodologies for bridge structural condition assessment. Smart Mater Struct. https://doi.org/10.1088/1361-665X/aad5fb Google Scholar
- 45.Zhu YC, Au SK (2018) Bayesian operational modal analysis with asynchronous data, part I: most probable value. Mech Syst Signal Process 98:652–666. https://doi.org/10.1016/j.ymssp.2017.05.027 CrossRefGoogle Scholar

## Copyright information

**OpenAccess**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.