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Structural damage detection using quantile regression

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Abstract

Structural health monitoring is an important emerging engineering discipline in the UK and the world. Structural failure without warning is recognised as a significant hazard in the service life of a structure. Thus there is a need to provide a clear guidance to determine the cut-off line for operation, repair and maintenance. A quantile regression approach has been proposed for structural damage detection using vibration data (accelerations). This method is based on a sequence of quantile autoregressive time series models and the differences between two distributions associated with the residual series of the undamaged and damaged structures are studied at different quantile levels. This new approach is based on the information on damages at any quantile levels, not just at a mean level that is commonly used in the literature. In addition, it does not depend on the distribution of the error term. This is a very useful feature as in practice it can be very difficult to assume a proper distribution for the error term of the model. The performance of the developed method is investigated via extensive simulation studies to detect single-damage and multi-damage scenarios with input and output measurement noise. The proposed method is further substantiated experimentally using an eight-storey steel plane frame model subjected to shaker excitation. Both numerical and experimental results have shown that the proposed method gives reasonably accurate damage identification, including both damage existence and location.

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Abbreviations

\( x_{t} \) :

Time series from an undamaged structure

\( y_{kt} \) :

Time series from the structure after a certain period of service life, at the kth location

\( \tau_{i} \) :

Quantile level

\( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.}}^{\tau } \) :

Quantile AR model of \( x_{t+1} \) conditional on \( x_{t} \)

\( \alpha_{j}^{\tau } \) :

Model parameter of \( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.}}^{\tau } \)

\( e_{t}^{\tau } \) :

Residuals of model \( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.}}^{\tau } \)

\( q_{{y_{kt + 1} \left| {{\mathbf{y}}_{kt} } \right.}}^{\tau } \) :

Quantile AR model of \( y_{kt+1} \) conditional on \( y_{kt} \)

\( \beta_{j}^{\tau } \) :

Model parameter of \( q_{{y_{kt + 1} \left| {{\mathbf{y}}_{kt} } \right.}}^{\tau } \)

\( g_{kt}^{\tau } \) :

Residuals of model \( q_{{y_{kt + 1} \left| {{\mathbf{y}}_{kt} } \right.}}^{\tau } \)

\( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.,{\mathbf{e}}_{t}^{\tau } }}^{\tau } \) :

Quantile AR model of \( x_{t+1} \) conditional on \( x_{t} \) and \( e_{t}^{\tau } \)

\( \alpha_{1j}^{\tau } ,\alpha_{2j}^{\tau } \) :

Model parameters of \( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.,{\mathbf{e}}_{t}^{\tau } }}^{\tau } \)

\( \tilde{e}_{t}^{\tau } \) :

Residuals of model \( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.,{\mathbf{e}}_{t}^{\tau } }}^{\tau } \)

\( \tilde{g}_{kt}^{\tau } \) :

Predicted residuals from the series \( y_{kt} \) using \( g_{kt}^{\tau } \) and model \( q_{{x_{t + 1} \left| {{\mathbf{x}}_{t} } \right.,{\mathbf{e}}_{t}^{\tau } }}^{\tau } \)

\( F_{{\tilde{e}^{\tau } }} \) :

Empirical distribution functions estimated from \( \tilde{e}_{t}^{\tau } \)

\( F_{{\tilde{g}^{\tau } }} \) :

Empirical distribution functions estimated from \( \tilde{g}_{kt}^{\tau } \)

\( D_{k\tau } \) :

Distance measure between \( F_{{\tilde{e}^{\tau } }} \) and \( F_{{\tilde{g}^{\tau } }} \)

\( S_{{k_{1} ,k_{2} }} \) :

Summation of differences in distance measures at the locations \( k_{1} \) and \( k_{2} \)

\( S_{0} \) :

Average value of \( S_{{k_{1} ,k_{2} }} \)

\( m_{k} \) :

Average value of \( D_{k\tau } \)

\( N_{k} \) :

Index number

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Acknowledgments

This research is supported by the Institution of Structural Engineers through Undergraduate Research Grant.

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Authors and Affiliations

  1. Department of Civil Engineering, University of Greenwich, Kent, UK

    Kong Fah Tee & Hua-Peng Chen

  2. College of Business, Economics and Law, Swansea University, Swansea, UK

    Yuzhi Cai

Authors
  1. Kong Fah Tee
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  2. Yuzhi Cai
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  3. Hua-Peng Chen
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Correspondence to Kong Fah Tee.

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Tee, K.F., Cai, Y. & Chen, HP. Structural damage detection using quantile regression. J Civil Struct Health Monit 3, 19–31 (2013). https://doi.org/10.1007/s13349-012-0030-3

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  • DOI: https://doi.org/10.1007/s13349-012-0030-3

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