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The degradation of civil engineering structures may lead to a sudden stiffness reduction in a structure and such a sudden damage will cause a discontinuity in the dynamic responses. The detection on structural sudden damage has been actively carried out in this study. The signal singularity of the acceleration responses with sudden stiffness reduction is characterized by the coefficients of continuous wavelet transform with fine scales. A detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the validity of the proposed detection approach for the building at different damage levels. The sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. The effects of both measurement noise intensity and frequency range on the damage detection are numerically investigated.

The wavelet transform is an extension of the traditional Fourier transform with adjustable window location and size. Wavelet analysis combines both time and frequency analysis, which allows it to zoom in on time without any loss of scale resolution. The wavelet transform has recently emerged as a promising tool for structural health monitoring and it is an ideal tool in addressing the issue of time locality of structural damages [

The degradation of civil engineering structures may lead to a sudden stiffness reduction in a structure associated with the events such as weld fracture, column buckling, and brace breakage [

The sudden stiffness loss of structural components may induce the signal discontinuity in the acceleration responses close to the damage location at the damage time instant. It is reported that the time instant and location of the sudden stiffness loss can be detected by using the discrete WT. However, the severity of different sudden damage events cannot be estimated directly by the WT. In this regard, the investigation of detection on sudden damage event of building structures has been actively carried out in this study. The signal feature of the structural acceleration responses of an example building is examined. Three types of dynamic loading, sinusoidal, seismic, and impulse excitations are taken as the inputting excitations. The signal singularity of the acceleration responses with sudden stiffness reduction is characterized by the coefficients of continuous wavelet transform with fine scales. A detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the validity of the proposed detection approach for the building at different damage levels. The sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. The effects of both measurement noise intensity and frequency range on the damage detection are numerically investigated. The made observations demonstrate that the proposed approach can accurately identify the damage events and the damage severity can be estimated by the Lipschitz exponent.

Morlet and Grossmann initially proposed wavelet theory and Meyer developed the mathematical foundations of wavelets. The two America-based researchers Daubechies [

The calculating wavelet coefficients at every possible scale will generate a lot of redundant data. In some practical signal processing cases, the discrete version of the wavelet is often utilized by discretizing the dilation parameter

The signal feature due to a sudden stiffness reduction is firstly investigated by taking a five-story shear building as an example structure (Figure ^{6} kg and 4.0 × 10^{9} N/m, respectively. The Rayleigh damping assumption is adopted to construct the structural damping matrix with the damping ratios in the first two vibration modes being set as 0.05. The original building is supposed to suffer a sudden 20% stiffness reduction in the first story while the horizontal stiffness in other stories remains unchanged. The frequency reduction due to 20% stiffness reduction in the first story is small with a maximum reduction no more than 5% in the first natural frequency. The sinusoidal excitation, seismic excitation, and impulse excitation are utilized, respectively, to calculate the acceleration responses of the example building to examine the signal features due to the sudden stiffness reduction. The seismic excitation used is the first 10 second portion of the El-Centro 1940 earthquake ground acceleration (S-N component) with a peak amplitude 1.0 m/s^{2}. A sinusoidal excitation expressed by (

Elevation of a five-story building model.

Signal discontinuity due to sudden damage.

Seismic excitation

Sinusoidal excitation

Impulse excitation

The acceleration time histories of the first floor under seismic, sinusoidal, and impulse excitations are computed and displayed in Figure

An important property of the WT is the ability to characterize the local regularity of a certain function. To characterize singularity due to sudden stiffness loss, it is necessary to precisely estimate the local singularity of an acceleration signal

For instance, a signal function is not differentiable at

Equation (

When the building structure suffers a sudden stiffness loss during the vibration, the extent of the signal discontinuity in acceleration time histories at the vicinity of the damage varies with different damage severities. Small damage is difficult to be identified using traditional vibration based approaches, but it still introduces some sorts of singularities to the acceleration responses. These singularities may be characterized by using the Lipschitz exponent with the aiding of the wavelet transform. To measure the local regularity of the acceleration signal by utilizing wavelet transform and Lipschitz exponent estimation, the wavelet vanishing moment plays an important role. The wavelet transform estimates the exponent by ignoring the polynomial

Equation (

This local maxima should be a strict local maxima in either the right or left neighbourhood of

Normally, the wavelet with at least

Signal singularity due to sudden stiffness reduction is detected by finding the abscissa where the wavelet modulus maxima converge at fine scales. The discrete wavelet transform (DWT) decomposes signal using discrete scales which cannot provide fine division especially for high frequency components of the original signal. The sudden stiffness loss will cause a high frequency damage signal in the original acceleration time history. Therefore, it is rough to utilize the DWT to obtain the Lipschitz exponent estimation and correlate the damage severity. The CWT as an alternative approach can execute continuous transform between the concerned continuous scale section which make it possible to obtain the accurate maxima line and Lipschitz exponent.

In the signal singularity detection on the acceleration responses with sudden stiffness reduction utilizing CWT, the continuity of the modulus maximum of

The Marr wavelet (

To examine the validity of the proposed detection approach for identifying sudden damage events, the acceleration responses of the example building subjected to the seismic excitation, sinusoidal excitation, and impulse excitation are computed, respectively. The building is subject to a 20% sudden stiffness reduction at times 6.0 s, 6.0 s, and 0.2 s in the first story of the building under seismic excitation, sinusoidal excitation, and impulse excitation, respectively. The time step used in the computation is 0.002 second. Figure

Damage detection under different decomposition scales.

Regarding the building excited by El Centro ground motion, CWT coefficients using large decomposition scales, such as 3.0, fail to detect damage instant while the counterparts using fine scales successfully capture the damage event. If the decomposition scales are relatively small, such as 0.5, CWT coefficients can depict the high frequency components of the original signal and the peak of modulus maxims can be observed at damage instant based on a fine scale interval to depict the damage event. With the increase of decomposition scales (>1.0), however, the frequency components reflected by modulus maxima of CWT coefficients gradually decrease. Thus, middle and high frequency components in the original acceleration responses form some modulus peaks, which make it difficult to detect the sudden damage event.

To examine the structural acceleration responses induced by impulse excitation, one can find that only the CWT coefficients with fine decomposition scale can detect the damage instant. In reality, the sudden stiffness reduction will cause a discontinuity in acceleration responses at damage instant and induce high frequency components into the original response signals. Therefore, one can extract the high frequency components from original acceleration responses using the WT and detect the sudden damage event. The frequency components of acceleration responses of the building subjected to sinusoidal excitation are quite simple and the high frequency signal induced by sudden damage is quite different from other signal components. The CWT coefficients can easily detect the signal singularity and damage event even using coarse decomposition scales. As far as the seismic excited damage building is concerned, the acceleration responses contain more high frequency components than those induced by the sinusoidal excitation. The distinguishing ability in the time-frequency domain under large scale is coarse and it is difficult to capture the sudden damage event under seismic excitations. Under large decomposition scales, it is impossible to distinguish the modulus maxima due to the sudden damage event and the damage detection is not satisfactory.

Figure

Damage detection for each floor under seismic excitation.

The modulus maxima lines for each floor under sinusoidal and impulse excitations with 20% stiffness loss are also investigated, respectively, and the results are not displayed for space limitation. Again, the modulus maximum lines of CWT coefficients appear only at the moment of sudden stiffness reduction at the first floor. Thus, the damage location can be easily captured from the observed maxima line and its distribution along the height of the building. As far as the impulse excited building is concerned, the CWT based detection approach may not give satisfactory results for the building with small damage event (2% sudden stiffness reduction). This is because the signal fluctuates significantly and the energy of damage signal is quite weak.

The parametric study is carried out in this section to investigate the sensitivity of CWT coefficients to damage severity so as to examine the validity of the proposed damage detection approach. The first floor of the example building is supposed to suffer different levels of sudden stiffness reduction but the damage time instants remain unchanged. The CWT coefficients of the first floor of the building subjected to the seismic excitation are displayed in Figure

Detection results for various severities under seismic excitation.

Figure

Modulus maxima line for various severities under seismic excitation.

Decay behaviour along the modulus maxima line under seismic excitation.

It is reported by Mallat [

Variations of Lipschitz exponent with damage severity without noise.

Damage severity | 1% | 2% | 5% | 10% | 20% | 40% |
---|---|---|---|---|---|---|

Seismic excitation | 0.9901 | 0.8008 | 0.6092 | 0.5146 | 0.4558 | 0.4086 |

Sinusoidal excitation | 0.9411 | 0.8202 | 0.6925 | 0.5788 | 0.5144 | 0.4787 |

Impulse excitation | 0.9991 | 0.9734 | 0.9317 | 0.8863 | 0.7005 | 0.5276 |

Decay behaviour along the modulus maximum line under seismic excitation.

To examine the feasibility of the proposed detection approach based on CWT and Lipschitz exponent, the first floor of the five-story building is supposed to suffer 20% sudden stiffness reduction but the sudden reduction occurs at the same time. The detection quality substantially depends on the characteristics of the mother wavelet such as wavelet vanishing moments and supporting length in the time domain. Thus, six different mother wavelets, Haar, Meyer, Morlet, Symlet-2, Daubechie-2, and Coiflet-2, are utilized to examine the effects of properties of mother wavelets on the detection on the structural sudden damage events. The vanishing moments of the Haar, Meyer, Morlet, Symlet, Daubechie-2, and Coiflet-2 wavelets are 1, indefinite, indefinite, 2, 2, and 2, respectively. The damage detection results under seismic excitation using different mother wavelets are shown in Figure

Detection results under seismic excitation using different mother wavelets.

It is also found that the CWT based approach using all six mother wavelets can accurately detect the damage time instant of the building subjected to sinusoidal excitation and impulse excitation. The decay behaviour along the modulus maxima line for the building subjected to 20% sudden stiffness reduction under seismic excitation is displayed in Figure

Decay behaviour along the modulus maxima line using different mother wavelets.

As discussed above, the vanishing moment of a mother wavelet plays an important role in the detection of signal singularity. To this end, the effects of vanishing moments on the detection quality are investigated by using Daubechie and Coiflet wavelets. Figures

Variations of WT coefficients with different vanishing moments.

Decay behaviour along the modulus maxima line with different vanishing moments.

To effect of the noise contamination is a practical issue need to be addressed before applying the proposed approach to health monitoring and damage detection of real structures. Yang et al. (2004) reported that the damage spike identified could be weakened by measurement noise, and strong measurement noise could lead to the failure of damage detection. Hong et al. [

Detection results from contaminated acceleration responses (seismic excitation).

Noise frequency range 0~50 Hz

Noise frequency range 0~100 Hz

Noise frequency range 0~250 Hz

Detection results from contaminated acceleration responses (noise frequency range 0~250 Hz).

Sinusoidal excitation

Impulse excitation

It is worth examining the effects of noise contamination on the magnitude of Lipschitz exponent. The effects of measurement noise on the magnitude of the Lipschitz exponent under seismic excitation are assessed and the results are listed in Table

Noise effects on Lipschitz exponent (seismic excitation).

Noise level | Noise frequency range | ||
---|---|---|---|

0~50 Hz | 0~100 Hz | 0~250 Hz | |

No noise | 0.3252 | 0.3252 | 0.3252 |

2% noise | 0.3249 | 0.3244 | 0.319 |

5% noise | 0.3248 | 0.3204 | 0.3093 |

Variations of Lipschitz exponent with noise intensity under sinusoidal and impulse excitation.

The investigation of detection on sudden damage event of building structures has been carried out in this study. The signal feature of the structural acceleration responses of an example building subjected to sinusoidal, seismic, and impulse excitations due to sudden stiffness reduction is examined. The local signal regularity of the acceleration responses is characterized by the decay of the wavelet transform amplitude across scales. Singularities can be detected by the continuous wavelet transform local maxima at fine scales. In this regard, a detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity.

Extensive numerical simulations have been performed on a five-story shear building to assess the performance of the detection approach based on CWT and Lipschitz exponent with and without noise contamination. The made observations indicate that the detection approach proposed in this study can accurately identify the damage time instant and damage location due to a sudden stiffness reduction in terms of the occurrence time and spatial distribution of coefficient spikes of the CWT. The relationship between Lipschitz exponent and damage severity is different when subjected to different dynamic excitations. The magnitudes of the Lipschitz exponents decrease with the increasing damage severity. The detection quality on the sudden damage even is satisfactory if the noise frequency range is limited. If the noise frequency range is wide enough, the reliability of damage detection quality using the proposed approach gradually decreases with the increase of noise intensity.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful for the financial support from the National Natural Science Foundation of China (51178366), the Technological Project of the Chinese Southern Power Grid Co. Ltd. (K-GD2013-0783), the ESI 1% Project of WUT (No.43, chenbo), and the Natural Science Foundation of Hubei Province (2014CFA026).