Model updating for a large multi-span quasi-periodic viaduct based on free wave characteristics

doi:10.1016/j.jsv.2021.116161

Journal of Sound and Vibration

Volume 506, 18 August 2021, 116161

https://doi.org/10.1016/j.jsv.2021.116161 Get rights and content

Abstract

Quasi-periodic structures, as found in multi-span bridges, multi-bay and multi-storey buildings, are often characterized by a high modal density, even at low frequencies. This clustering of modes poses challenges in operational modal analysis, as well as in finite element model updating, where a pairing of experimental and numerically predicted modes is required. To overcome these difficulties, an alternative method for model updating based on the so-called free wave characteristics has recently been proposed. In this work, this method is applied to calibrate a finite element model of the K032 viaduct of the A11 highway in Bruges, Belgium. Vibration measurements are conducted to experimentally determine the free wave characteristics of the viaduct. In the model updating, the discrepancy between the calculated and identified free wave characteristics is minimized. The results are validated by comparing measured frequency response functions with those obtained from the updated finite element model.

Introduction

Vibration-based finite element model updating is widely used in civil engineering to calibrate numerical models with poorly known parameters [1], [2], [3]. The model updating is usually performed by minimizing the discrepancy between calculated and experimentally identified dynamic data, which generally consist of modal characteristics, including natural frequencies and mode shapes. For repetitive structures, as often found in multi-span bridges and multi-bay and multi-storey buildings, such as parkings and office buildings, mode clustering tends to occur, resulting in closely-spaced natural frequencies characterized by mode shapes with similar wavelength [4], [5]. This makes it difficult to distinguish between these modes in operational modal analysis and consequently poses challenges in model updating, where the matching of measured and calculated modes is required.

As an alternative to model updating based on modal characteristics, the use of free wave characteristics in model updating of one-dimensional periodic structures has been explored in [5]. A one-dimensional periodic structure is composed of identical units that are connected side by side in an identical way [6]. The free wave characteristics of a periodic structure consist of propagation constants and corresponding free waves [7], [8]. As shown by Zhang et al. in [5], these characteristics serve as a valuable alternative for model updating of periodic and quasi-periodic structures.

In the work of Zhang et al. [5], a non-linear least-squares method is proposed to identify free wave characteristics from the measured response of a periodic structure, in extension of the method proposed by Junyi and Balint in [9]. The authors have also explored the use of free wave characteristics in model updating for a simple lab structure. Apart from circumventing the aforementioned challenges concerning mode distinction and mode matching, model updating based on free wave characteristics has the advantage that the measurement of a small number of (consecutive) repetitive units is sufficient, rather than requiring the covarage of the entire structure. In the present paper, the model updating technique proposed in [5] is applied to a finite element model of the K032 viaduct (Fig. 1), a large multi-span quasi-periodic viaduct in Bruges, Belgium. The free wave characteristics used in the model updating are identified from vibration measurements on the bridge deck. It is shown how the model updating leads to a better prediction of the free waves and frequency response functions which were identified from the in situ tests. The resulting model will therefore allow for a more accurate prediction of the dynamic response of the structure as built, which is useful for a validation of the design calculations as well as in case of modifications of the structure or design loads.

The paper is outlined as follows. Section 2 briefly recapitulates the methodology adopted in the model updating. Next, Section 3 presents the K032 viaduct and the measurement setup. Section 4 describes the results of the free wave identification. Section 5 presents the model updating based on free wave characteristics and discusses complementarity with other approaches. Finally, Section 6 concludes the work.

Section snippets

Methodology

Consider the case of an unbounded one-dimensional periodic structure. A conceptual example of such a structure as previously considered in [5] is shown in Fig. 2. As mentioned before, repetitive structures are characterized by mode clustering, which poses challenges in operational modal analysis and model updating. In this paper, the focus is on the use of identified free wave characteristics as an alternative of using identified modal characteristics in model updating for periodic structures.

General information

The K032 viaduct consists of two curved integral bridges in parallel with each other, hereafter referred to as K032L and K032R (Fig. 5). Each bridge has 23 spans, most of them having a length of about 35 m. The total length of the viaduct is about 800 m. The two bridges are connected by a passway at the 19th and 20th span, allowing traffic to pass from one bridge to the other in case of emergency. For the sake of convenience, the global coordinate system $x y z$ is aligned with the Lambert

Free wave identification

As previously mentioned, the signal to noise ratio of the measured accelerations under the considered impact loading is very low. This is especially the case for sensors at a large distance from the location where the loading is applied. The low signal to noise ratio is due to the fact that the bridge is fully exposed to wind loading, leading to continuous low frequency vibrations. As such, it was decided not to use the measured accelerations directly as input for the free wave identification,

Model updating

This section presents the model updating for the K032 viaduct. First, Section 5.1 describes the preliminary unit cell finite element model and its corresponding calculated free waves (in accordance to Section 2.1). Next, Section 5.2 presents the sensitivity analysis that is adopted to select relevant parameters to be determined in the model updating. Section 5.3 presents the results of the model updating. Finally, Section 5.4 discusses complementarity of the presented approach with classical

Conclusions

This paper presents a validation of model updating based on free wave characteristics, using in situ measurements on a multi-span quasi-periodic viaduct in Bruges, Belgium. The free wave characteristics have been identified from FRF measurements on the bridge deck and are used to update a periodic model of the structure. It is found that the model updating leads to a much better prediction of the free wave characteristics, compared to the prediction obtained from an initial model based on the

CRediT authorship contribution statement

Jie Zhang: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Writing - review & editing. Kristof Maes: Validation, Investigation, Writing - original draft, Writing - review & editing. Guido De Roeck: Conceptualization, Methodology, Writing - original draft, Supervision, Funding acquisition. Geert Lombaert: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Supervision, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The research presented in this paper has been performed within the framework of project OT/13/59 “Quantifying and reducing uncertainty in structural dynamics” and project C16/17/008 “Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints”, both funded by the Research Council of KU Leuven. The financial support of KU Leuven is gratefully acknowledged. Kristof Maes is a postdoctoral fellow of the Research Foundation

References (21)

C.-P. Fritzen et al.
Damage detection based on model updating methods
Mech Syst Signal Process
(1998)
J. Zhang et al.
Model updating of periodic structures based on free wave characteristics
J Sound Vib
(2019)
D. Mead
Wave propagation in continuous periodic structures: research contributions from southampton, 1964 – 1995
J Sound Vib
(1996)
L. Brillouin
Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices
(1946)
L. Junyi et al.
An inverse method to determine the dispersion curves of periodic structures based on wave superposition
J Sound Vib
(2015)
D. Mead
A general theory of harmonic wave propagation in linear periodic systems with multiple coupling
J Sound Vib
(1973)
The MathWorks, MATLAB Optimization Toolbox User’s Guide,...
M. Friswell et al.
Finite element model updating in structural dynamics
(1995)
E. Simoen et al.
Dealing with uncertainty in model updating for damage assessment: a review
Mech Syst Signal Process
(2015)
J. Zhang et al.
Optimal sensor placement for multi-setup modal analysis of structures
J Sound Vib
(2017)

There are more references available in the full text version of this article.

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Model updating for a large multi-span quasi-periodic viaduct based on free wave characteristics

Abstract

Introduction

Section snippets

Methodology

General information

Free wave identification

Model updating

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Mech Syst Signal Process

J Sound Vib

J Sound Vib

J Sound Vib

J Sound Vib

Finite element model updating in structural dynamics

Dealing with uncertainty in model updating for damage assessment: a review

Mech Syst Signal Process

Optimal sensor placement for multi-setup modal analysis of structures

J Sound Vib