Shock and Vibration

Volume 2016, Article ID 1346939, 13 pages

http://dx.doi.org/10.1155/2016/1346939

## Damage Identification Investigation of Retaining Wall Structures Based on a Virtual Impulse Response Function

School of Civil Engineering and Architecture, Shaanxi University of Technology, Hanzhong 723000, China

Received 3 May 2016; Revised 13 July 2016; Accepted 31 July 2016

Academic Editor: Emiliano Mucchi

Copyright © 2016 Qian Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To eliminate the influence of excitation on the wavelet packet frequency band energy spectrum (ES), ES is acquired via wavelet packet decomposition of a virtual impulse response function. Based on ES, a character frequency band vector spectrum and damage eigenvector spectrum (DES) are created. Additionally, two damage identification indexes, the energy ratio standard deviation and energy ratio variation coefficient, are proposed. Based on the damage index, an updated damage identification method for retaining wall structures is advanced. The damage state of a retaining wall can be diagnosed through DES, the damage location can be detected through the damage index trend surface, and the damage intensity can be identified by establishing a quantitative relationship between the damage intensity and damage index. To verify the feasibility and validity of this damage identification method, a vibration test on a pile plate retaining wall is performed. Test results demonstrate that it can distinguish whether the retaining wall is damaged, and the location of partial damage within the retaining wall can be easily detected; in addition, the damage intensity of the wall can also be identified validly. Consequently, this damage identification theory and method may be used to identify damage within retaining wall structures.

#### 1. Introduction

Engineering structures or structural components might become damaged due to such factors as variations of their surroundings, load changes, and material degradation. Engineering accidents might occur once this damage accumulates to some extent. In 1951, a steel bridge collapsed suddenly in Quebec, Canada [1]; in 1994, the Seongsu Bridge in Seoul, Korea, collapsed due to a weld break [2], resulting in 32 deaths and 17 injuries. Retaining wall structures, as a type of retaining structure, are often used to support soil and rocks, as well as stabilize slopes. Due to material degradation, variations of surroundings (such as a variation of temperature or humidity), and variations of loads, numerous types of damage (such as cracks or holes) might appear in walls. Once the damage accumulates to a certain extent, accidents might occur. In 2008, an underground wall collapsed in Hangzhou city of Jiangsu province, China, causing 45 deaths, decades of injures, and huge economic losses. In 2014, a retaining wall collapsed in Qingdao city of Shandong province, China, which caused 18 deaths and several injures. Thus, much more attention is now being paid to the detection of damage within structures to avoid or reduce similar accidents in the future.

In 1979, Cawley and Adams detected damage within structures by the natural frequency variation caused by damage [3]. Biswas et al. [4] and Hearn and Testa [5] identified damage in structures and bridges, respectively, by modal variation. In 1998, Doebling et al. noted that damage within structures could be detected from changes in structural characteristics [6]. Thus far, the majority of studies on damage identification have focused on structural components [7–9], building structures [10], bridge structures [11, 12], and other structures [13, 14]. In contrast, damage identification research on retaining wall structures is much rarer.

For a long time, inherent characteristics (such as natural frequencies, modal shapes, modal shape curvature, modal flexibility, and modal strain energy) are used to diagnose damage [15–19]. Damage identification methods, based on inherent characteristics, are widely applied in civil engineering because structural characteristics can be measured easily and conveniently [20]. However, it is difficult to measure these characteristics’ changes in retaining wall structures, which have huge volume and mass. Thus, it is difficult to validly detect damage within retaining wall structures through inherent characteristics. Nevertheless, the damage identification method based on wavelet decomposition of structure dynamic response has better sensitivity and robustness compared with other methods [21], and this approach has been widely used in building structures and bridge structures [22, 23]. However this method has much dependence on external excitation. Thus, damage identification methods based on impulse response functions and virtual impulse response functions are taken into account [24, 25]. These methods also have much better sensitivity and robustness.

In this paper, an updated damage identification method based on a virtual impulse response function is proposed. Additionally, two damage identification indexes are put forward to identify damage within retaining wall structures. By performing a vibration test on a pile plate retaining wall, damage within this wall was detected through the updated damage identification method.

#### 2. Damage Identification Theory and Method

Via wavelet packet decomposition of structural dynamic responses (such as displacement, velocity, and acceleration) caused by constant external excitation, ES (the wavelet packet frequency band energy spectrum) is obtained. Changes in ES can be used to detect damage within structures. However, actual excitation is always changing, and ES that is directly based on structural dynamic responses changes with the change of excitation. Identification indexes based on ES also change with the variation of excitation. Consequently, ES that is directly based on dynamic responses is unable to precisely diagnose damage in structures. However, ES that is based on an impulse response function or frequency response function is able to identify damage because impulse response functions or frequency response functions contain all of the structural dynamic properties. Consequently, ES based on an impulse response function or frequency response function can eliminate the dependence of ES on external excitation.

##### 2.1. IRF (Impulse Response Function)

The motion equation of a structural dynamic system can be expressed aswhere is the mass matrix of the structural dynamic system; is the damping matrix of the structural dynamic system; is the stiffness matrix of the dynamic system; is the displacement matrix of the dynamic system; and is the excitation force matrix of the structural dynamic system.

In modal coordinates, (1) can be rewritten aswhere is the modal matrix; is a vector on the modal axis; and is the th modal shape.

Substituting (2) into (1) and premultiplying (1) by , (3) is obtained: where is the th modal frequency; is the th modal damp ratio; and is the th modal mass.

The solution to (3) can be expressed aswhere .

Finally, where is modal order.

In a structural dynamic system, the response at the th node caused by an external excitation at the th node can be expressed asAfter integration, IRF of the response and excitation can be expressed asNevertheless, IRF can be obtained when only both the excitation and responses are known. Thus, ES based on IRF still has some dependence on the excitation. However, a virtual impulse response function can be acquired when only the responses are known, without consideration of whether the external excitation is known or not.

##### 2.2. VIRF (Virtual Impulse Response Function)

In a multiple-degree-of-freedom structural system, the dynamic response at a reference point can be regarded as a virtual excitation, and the response at a calculating point can be regarded as a virtual response. VIRF can be acquired by virtual excitation and virtual response. The auto-power spectral density of the virtual excitation (the dynamic response at a reference point ) can be expressed aswhere is the Fourier transform of the virtual excitation and is the complex conjugate of the virtual excitation .

The cross-power spectral density between and the virtual response (the response at the calculating point ) can be expressed aswhere is the Fourier transform of the virtual response .

and can be, respectively, expressed as the virtual frequency response function between and can be expressed asafter the inverse Fourier transform, VIRF between and can be expressed as Obviously, the virtual impulse response has no relationship to the excitation; the virtual impulse response function can be acquired when the responses are known. Thus ES based on the virtual impulse response function completely eliminates the influence of the external excitation on ES.

##### 2.3. Updated Damage Identification Method

Wavelet packet decomposition of VIRF between and can be rewritten aswhere is the virtual impulse response function component and is the number of wavelet packet decomposition layers.

###### 2.3.1. ERS (Wavelet Packet Frequency Band Energy Ratio Spectrum)

According to the multiscale decomposition of the structural dynamic system signal [21], changes of the signal components are able to reflect variations of the inherent characteristics in a structural system. Thus, changes of signal components caused by damage can be used to detect damage within structures. In light of wavelet packet damage identification theory, changes of subfrequency band energy are used to identify damage. The subfrequency band energy can be expressed aswhere is the number of signal simple points.

Thus, ES can be expressed asthen, the energy ratio is defined asthus, an energy ratio sequence is created and can be expressed asbecause the energy ratio sequence is messy or without any order, can be sorted from large to small according to values, yielding a new energy ratio sequence : is referred to as ERS.

###### 2.3.2. DES (Wavelet Packet Damage Eigenvector Spectrum)

Due to the interference of measurement noise, it is not possible to completely detect the variation of the energy ratio of every subband. Generally speaking, only those energy ratios that are much larger are easily identified. Thus, the top frequency bands with larger energy ratio are selected to detect damage information. Let :where is the relative cumulative energy ratio and is a threshold. It is via that the top energy ratios in can be determined, and energy ratio changes on these frequency bands can be used to identify damage in the retaining wall. Nevertheless, the effects of residual frequency bands on the identification of damage should not be ignored either. Thus, the energy ratio of residual frequency bands is defined asthe () frequency bands are called character frequency bands. Then the character frequency band vector spectrum is defined ason the basis of , define the energy ratio deviation as where and are the energy ratios of the th character frequency band of undamaged structures and damaged ones, respectively.

Then, define DES asobviously, structures are undamaged when , while structures are damaged when . Thus, it is easy to detect whether retaining wall structures are damaged or not by DES .

###### 2.3.3. Damage Identification Indexes

On the basis of , two damage identification indexes, the energy ratio standard deviation ISD and energy ratio variation coefficient IVC, are proposed, and they are defined asThere is no damage within structures when IVC = 0 or ISD = 0 and there is damage when or . For partial damage locations, the identification index values of the calculating points are different. Generally speaking, the identification index values of calculating points that are located in the damage area will change suddenly. Thus, partial damage locations can be detected by variations of identification indexes. Moreover, the intensity of partial damage within walls can also be identified by establishing a quantitative relationship between damage intensity and identification index. Consequently, this damage identification theory and method can be used to identify damage within retaining wall structures. To verify the feasibility and validity of this damage identification method, a vibration test on a pile plate retaining wall is performed.

#### 3. Vibration Test on a Pile Plate Retaining Wall

A vibration test is performed on a pile plate retaining wall, as shown in Figure 1(a). The height of the wall is 2.2 m, the length is 3.0 m, and the thickness is 0.2 m. Foundation of the wall is anchored in filled soil. The anchored depth is 0.6 m. The retaining wall is a concrete wall (concrete strength grade is C30). Filled soil behind the wall is sand and soil in front of the wall is miscellaneous fill. Material parameters of the wall and filled soil are shown in Table 1. The test equipment includes an exciting hammer, sensors, and a signal collecting system. In this paper, the exciting equipment is a DFC-2 hammer, as shown in Figure 2(a). The sensors are 941B accelerometers which are used to record acceleration signals. The frequency range of this sensor is 0.17~100 Hz; the sensitivity is 0.3 (). These sensors are fixed on the wall by binder, as shown in Figure 2(b). The signal collecting system is the JM3863A wireless vibration test system, as shown in Figure 2(c). The signals collected by the JM3863A wireless vibration test system are transmitted to a computer by the JM1802 gateway, as shown in Figure 2(d).