# Entropy Analysis for Damage Quantification of Hysteretic Dampers Used as Seismic Protection of Buildings

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## Abstract

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## 1. Introduction

## 2. Web Plastifying Dampers (WPD)

## 3. Description of the Vibration Test on the Dampers

## 4. Vibration Data Analysis: Definition of the RWEE Damage Index

#### 4.1. Wavelet Packet Decomposition and Wavelet Energy

^{N}components at the Nth level, providing flexibility for choosing the way to encode the original signal, so that the reconstruction error is minimal. A schematic representation of the WPT of a time-domain signal f(t) up to the 3rd level of decomposition is presented in Figure 4; it can be obtained, for instance, by summing the signal components ${f}_{1}^{1}\left(t\right)$, ${f}_{3}^{5}\left(t\right)$, ${f}_{3}^{6}\left(t\right)$ and ${f}_{2}^{4}\left(t\right)$. The approximations correspond to the low scale (low frequency components) of the signal, while the details correspond to the high scale (high frequency components) of the signal.

#### 4.2. Wavelet Entropy (WE)

#### 4.3. Relative Wavelet Entropy (RWE): RWEE Damage Index Definition

- Firstly, normalization and scaling must be applied to counteract different excitation levels and environmental conditions and to compensate the off-set and amplitude variability among signals. This normalization is carried out as follows:$$f\left(t\right)=\frac{f{\left(t\right)}_{r}-\overline{f}{\left(t\right)}_{r}}{\sigma \left(f{\left(t\right)}_{r}\right)}$$
- The normalized signal $f\left(t\right)$ is decomposed into multiple sub-signals in various frequency bands using the WPT. Let ${f}_{0}^{1}\left(t\right)$ denote the normalized response signal shown in the first step. Through trial and error tests, it was found that the optimum level of decomposition is 7; thus, 2
^{7}(128 nodes) is the number of sub-signals to be obtained in the proposed procedure. - The energy of each sub-signal is calculated according to the following equation$${E}_{j}^{i}={{\displaystyle \int}}^{\text{}}{\left|{f}_{j}^{i}\left(t\right)\right|}^{2}dt$$Steps 1, 2 and 3 must be carried out in both baseline and inspection phases.
- Finally, the RWEE index is calculated using Equation (14), where ${E}_{j}^{i}$ is the energy of each sub-signal on the level of decomposition j, and ${E}_{jt}$ is the sum of the energy of the all sub-signals of level j, both calculated for the inspection phase. The same parameters calculated for the baseline inspection are ${E}_{j}^{i0}$ and ${E}_{jt}^{0}$, respectively. N is the total number of sub-signals, in the present study 128.$$RWEE=-{\displaystyle \sum}_{i=1}^{N}\left(\frac{{E}_{j}^{i}}{{E}_{jt}}\right)\xb7log\left(\frac{\raisebox{1ex}{${E}_{j}^{i}$}\!\left/ \!\raisebox{-1ex}{${E}_{jt}$}\right.}{\raisebox{1ex}{${E}_{j}^{i0}$}\!\left/ \!\raisebox{-1ex}{${E}_{jt}^{0}$}\right.}\right)$$

## 5. Specimen and Experiment Descriptions

^{2}shaking table of the University of Granada, as shown in Figure 5, and subjected to successive seismic simulations of increasing amplitude. Each specimen underwent one-dimensional dynamic shaking table tests. The acceleration was applied to the table in a horizontal direction, parallel to a plane containing the axis of the dampers and the axis of one (in the case of specimen SD) or two (for specimen FD) columns. Before the test, steel blocks were attached at the top of the RC slab and at the top of the columns of the second story to represent the gravity loads that act on the floors and also to satisfy similitude requirements between prototype and specimen. Pin joint connections were used at the top of the half-columns in the second story, and at the ends of the half-beams of the first floor. The vertical movements of the ends of the half-beams of the first floor were prevented using pin-ended steel bars that connected the ends of the beams with the steel plates on the top of the specimen. The total mass of the specimen (including the additional masses) was 12,450 kg. The tests were conducted controlling the table by a real-time, uniaxial, digital controller able to provide three-variable closed loop control along with adaptive control. Moreover, the tests were conducted in the acceleration control mode in order to accurately reproduce the desired acceleration record in the table. The lateral displacements induced in the test model during each simulation imposed forced axial deformations on the dampers that damaged the I-section steel segments. In each seismic simulation, the shaking table reproduced the ground motion acceleration recorded at Calitri during the Campano Lucano earthquake (1980), scaled in amplitude to different levels. The scaling factors applied to the test model FD were 100%, 200%, 300% and 350% times the acceleration of the original record, and the corresponding peak ground accelerations were 0.16 g, 0.31 g, 0.47 g and 0.54 g (here, g is the acceleration of gravity). The respective levels of damage to the WPDs at the end of each seismic simulation will hereafter be identified as d0, d1, d2 and d3. The scaling factors applied to test model SD were 100%, 200%, 300%, 400%, 500% and 600% times the acceleration of the original record, and the corresponding levels of damage will be identified as d0, d1, d2, d3, d4 and d5 hereafter. The test models and the dampers were instrumented with displacement transducers, strain gauges and accelerometers. For illustrative purposes, Figure 6 shows the histories of acceleration applied to specimens FD and SD.

## 6. Results

## 7. Discussion and Conclusions

^{2}shaking table. The damage accumulated in the I-section of the dampers at the end of each seismic simulation was quantified twice, once with the mechanical damage index ID proposed in past research and extensively calibrated with the results of static and dynamic tests. The main drawback of the ID index is that its calculation requires knowledge of the load-displacement curve endured by the hysteretic damper during the seismic event. The other quantification was provided by the RWEE index, which entails low-cost instrumentation and low computational requirements. Results demonstrate that the RWEE index correlates very well with the ID index. This enables one to assess the damage on hysteretic dampers without resorting to the cumbersome and expensive instrumentation required for in-situ continuous monitoring of dampers.

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Instrumented I-sections in a particular Web Plastifying Damper (WPD) damper; (

**b**) I-section instrumented with two piezoelectric transducers (PZT), one acting as actuator (input signal) and one acting as sensor (output signal); (

**c**) Connection of the PZT transducers to the electronic system; (

**d**) Installation of the dampers in the building.

**Figure 5.**Overview of test models and set-up on the shaking table. (

**a**) Frame with dampers (FD) specimen; (

**b**) Flat-slab with dampers (SD) specimen.

**Figure 7.**Wavelet energy of one I-section of the ground floor of specimen FD. (

**a**) Complete frequency range; (

**b**) Zoom around the 25 kHz natural frequency.

**Figure 8.**Wavelet energy of one I-section of the ground floor of specimen SD. (

**a**) Complete frequency range; (

**b**) Zoom around the 25 kHz natural frequency.

**Figure 9.**The relative wavelet energy entropy index (RWEE) vs. mechanical damage index (ID) for each I-section of the dampers of FD.

Specimen | Location of WPDs | Damage Level | |||||
---|---|---|---|---|---|---|---|

d0 | d1 | d2 | d3 | d4 | d5 | ||

FD | Ground floor | 0.00 | 0.17 | 0.45 | 0.77 | ||

SD | First floor | 0.00 | 0.02 | 0.05 | 0.15 | 0.27 | 0.40 |

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**MDPI and ACS Style**

Suarez, E.; Roldán, A.; Gallego, A.; Benavent-Climent, A. Entropy Analysis for Damage Quantification of Hysteretic Dampers Used as Seismic Protection of Buildings. *Appl. Sci.* **2017**, *7*, 628.
https://doi.org/10.3390/app7060628

**AMA Style**

Suarez E, Roldán A, Gallego A, Benavent-Climent A. Entropy Analysis for Damage Quantification of Hysteretic Dampers Used as Seismic Protection of Buildings. *Applied Sciences*. 2017; 7(6):628.
https://doi.org/10.3390/app7060628

**Chicago/Turabian Style**

Suarez, Elisabet, Andrés Roldán, Antolino Gallego, and Amadeo Benavent-Climent. 2017. "Entropy Analysis for Damage Quantification of Hysteretic Dampers Used as Seismic Protection of Buildings" *Applied Sciences* 7, no. 6: 628.
https://doi.org/10.3390/app7060628