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

Authors: Qian Xu
Journal: Shock and Vibration
 (2016)

---

## 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.


---

## Body

## 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 as(1)
M
U
¨
+
C
U
˙
+
K
U
=
F
t
,where M
∈
R
n
×
n is the mass matrix of the structural dynamic system; C
∈
R
n
×
n is the damping matrix of the structural dynamic system; K
∈
R
n
×
n is the stiffness matrix of the dynamic system; U
∈
R
n
×
1 is the displacement matrix of the dynamic system; and F
∈
R
n
×
1 is the excitation force matrix of the structural dynamic system.In modal coordinates, (1) can be rewritten as(2)
U
t
=
ψ
q
t
=
∑
r
=
1
n
ψ
r
q
r
r
,where ψ is the modal matrix; q
(
t
) is a vector on the modal axis; and ψ
r is the rth modal shape.Substituting (2) into (1) and premultiplying (1) by ψ
T, (3) is obtained: (3)
q
¨
r
t
+
2
ξ
r
ω
r
2
q
r
t
+
ω
r
2
q
r
t
=
1
m
r
ψ
r
T
F
r
t
,where ω
r is the rth modal frequency; ξ
r is the rth modal damp ratio; and m
r is the rth modal mass.The solution to (3) can be expressed as(4)
q
r
r
=
∫
-
x
t
ψ
r
T
F
τ
g
r
t
-
τ
d
τ
g
r
t
=
0
,
t
<
0
1
m
r
ω
r
d
exp
⁡
-
ξ
r
ω
r
t
sin
⁡
ω
r
d
t
,
t
≥
0
,where ω
r
d
=
ω
r
(
1
-
ξ
r
2
)
1
/
2.Finally,(5)
U
t
=
∑
r
=
1
n
ψ
r
·
∫
-
x
t
ψ
r
T
F
τ
g
r
t
-
τ
d
τ
,where n is modal order.In a structural dynamic system, the response at theith node caused by an external excitation at the kth node can be expressed as(6)
U
i
k
t
=
∑
r
=
1
n
ψ
i
r
ψ
k
r
·
∫
-
x
t
F
k
τ
g
r
t
-
τ
d
τ
.After integration, IRF of the response and excitation can be expressed as(7)
U
i
k
t
=
∑
r
=
1
n
ψ
i
r
ψ
k
r
m
r
ω
r
d
exp
⁡
-
ξ
r
ω
r
t
sin
⁡
ω
r
d
t
.Nevertheless, 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 excitationu
j
(
t
) (the dynamic response at a reference point j) can be expressed as(8)
G
u
u
ω
,
j
=
U
∗
ω
,
j
·
U
ω
,
j
,where U
(
ω
,
j
) is the Fourier transform of the virtual excitation u
j
(
t
) and U
∗
(
ω
,
j
) is the complex conjugate of the virtual excitation u
j
(
t
).The cross-power spectral densityG
y
u
(
ω
,
i
,
j
) between u
j
(
t
) and the virtual response u
i
(
t
) (the response at the calculating point i) can be expressed as(9)
G
y
u
ω
,
i
,
j
=
U
∗
ω
,
j
·
U
ω
,
i
,where U
(
ω
,
i
) is the Fourier transform of the virtual response u
i
(
t
).U
(
ω
,
j
) and U
(
ω
,
i
) can be, respectively, expressed as (10)
U
ω
,
j
=
H
ω
,
j
F
ω
,
U
ω
,
i
=
H
ω
,
i
F
ω
;the virtual frequency response function H
y
u
(
ω
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be expressed as(11)
H
y
u
ω
,
i
,
j
=
G
y
u
ω
,
i
,
j
G
u
u
ω
,
j
=
U
∗
ω
,
j
·
U
ω
,
i
U
∗
ω
,
j
·
U
ω
,
i
=
H
∗
ω
,
j
F
∗
ω
·
H
ω
,
i
F
ω
H
∗
ω
,
j
F
∗
ω
·
H
ω
,
j
F
ω
=
H
∗
ω
,
j
·
H
ω
,
i
H
∗
ω
,
j
·
H
ω
,
j
;after the inverse Fourier transform, VIRF H
y
u
(
t
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be expressed as (12)
H
y
u
t
,
i
,
j
=
1
2
π
∫
-
∞
∞
e
-
i
ω
t
H
ω
,
i
,
j
d
ω
.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  VIRFH
y
u
(
t
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be rewritten as(13)
H
y
u
t
,
i
,
j
=
∑
h
g
=
h
1
+
h
2
+
⋯
+
h
2
I
,where h
g is the virtual impulse response function component and I 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 E
g can be expressed as(14)
E
g
=
∑
h
g
,
h
2
h
=
1,2
,
…
,
m
,where m is the number of signal simple points.Thus, ESE
I can be expressed as(15)
E
I
=
E
1
E
2
⋯
E
g
⋯
E
2
I
T
;then, the energy ratio I
g is defined as(16)
I
g
=
E
g
∑
g
=
1
2
I
E
g
×
100
;thus, an energy ratio sequence I
~
G is created and can be expressed as(17)
I
~
G
=
I
~
1
I
~
2
⋯
I
~
g
⋯
I
~
2
I
T
;because the energy ratio sequence I
~
G is messy or without any order, I
~
G can be sorted from large to small according to I
~
g values, yielding a new energy ratio sequence I
G:(18)
I
G
=
I
1
I
2
⋯
I
g
⋯
I
2
I
T
I
g
≥
I
g
+
1
;
I
G 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 topp frequency bands with larger energy ratio are selected to detect damage information. Let ε
p
≤
ε
0:(19)
ε
p
=
∑
q
=
1
p
I
q
q
=
1,2
,
…
,
p
,where ε
p is the relative cumulative energy ratio and ε
0 is a threshold. It is via ε
0 that the top p energy ratios in I
G can be determined, and energy ratio changes on these p 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 as(20)
I
p
+
1
=
100
-
∑
q
=
1
p
I
q
;the (p
+
1) frequency bands are called character frequency bands. Then the character frequency band vector spectrum I
p
+
1 is defined as(21)
I
p
+
1
=
I
1
I
2
⋯
I
p
I
p
+
1
T
;on the basis of I
p
+
1, define the energy ratio deviation D
I
k as (22)
D
I
k
=
I
k
u
-
I
k
d
k
=
1,2
,
…
,
p
+
1
,where I
k
u and I
k
d are the energy ratios of the kth character frequency band of undamaged structures and damaged ones, respectively.Then, define DESD
I
p
+
1 as(23)
D
I
p
+
1
=
DI
1
D
I
2
⋯
D
I
p
D
I
p
+
1
T
;obviously, structures are undamaged when D
I
p
+
1
=
0, while structures are damaged when D
I
p
+
1
≠
0. Thus, it is easy to detect whether retaining wall structures are damaged or not by DES D
I
p
+
1.


#### 2.3.3. Damage Identification Indexes

On the basis ofD
I
p
+
1, two damage identification indexes, the energy ratio standard deviation ISD and energy ratio variation coefficient IVC, are proposed, and they are defined as(24)
ISD
=
∑
k
=
1
p
+
1
D
I
k
2
p
+
1
,
IVC
=
∑
k
=
1
p
+
1
D
I
k
I
k
u
2
.There is no damage within structures when IVC = 0 or ISD = 0 and there is damage when I
V
C
>
0 or I
S
D
>
0. 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.


## 2.1. IRF (Impulse Response Function)

The motion equation of a structural dynamic system can be expressed as(1)
M
U
¨
+
C
U
˙
+
K
U
=
F
t
,where M
∈
R
n
×
n is the mass matrix of the structural dynamic system; C
∈
R
n
×
n is the damping matrix of the structural dynamic system; K
∈
R
n
×
n is the stiffness matrix of the dynamic system; U
∈
R
n
×
1 is the displacement matrix of the dynamic system; and F
∈
R
n
×
1 is the excitation force matrix of the structural dynamic system.In modal coordinates, (1) can be rewritten as(2)
U
t
=
ψ
q
t
=
∑
r
=
1
n
ψ
r
q
r
r
,where ψ is the modal matrix; q
(
t
) is a vector on the modal axis; and ψ
r is the rth modal shape.Substituting (2) into (1) and premultiplying (1) by ψ
T, (3) is obtained: (3)
q
¨
r
t
+
2
ξ
r
ω
r
2
q
r
t
+
ω
r
2
q
r
t
=
1
m
r
ψ
r
T
F
r
t
,where ω
r is the rth modal frequency; ξ
r is the rth modal damp ratio; and m
r is the rth modal mass.The solution to (3) can be expressed as(4)
q
r
r
=
∫
-
x
t
ψ
r
T
F
τ
g
r
t
-
τ
d
τ
g
r
t
=
0
,
t
<
0
1
m
r
ω
r
d
exp
⁡
-
ξ
r
ω
r
t
sin
⁡
ω
r
d
t
,
t
≥
0
,where ω
r
d
=
ω
r
(
1
-
ξ
r
2
)
1
/
2.Finally,(5)
U
t
=
∑
r
=
1
n
ψ
r
·
∫
-
x
t
ψ
r
T
F
τ
g
r
t
-
τ
d
τ
,where n is modal order.In a structural dynamic system, the response at theith node caused by an external excitation at the kth node can be expressed as(6)
U
i
k
t
=
∑
r
=
1
n
ψ
i
r
ψ
k
r
·
∫
-
x
t
F
k
τ
g
r
t
-
τ
d
τ
.After integration, IRF of the response and excitation can be expressed as(7)
U
i
k
t
=
∑
r
=
1
n
ψ
i
r
ψ
k
r
m
r
ω
r
d
exp
⁡
-
ξ
r
ω
r
t
sin
⁡
ω
r
d
t
.Nevertheless, 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 excitationu
j
(
t
) (the dynamic response at a reference point j) can be expressed as(8)
G
u
u
ω
,
j
=
U
∗
ω
,
j
·
U
ω
,
j
,where U
(
ω
,
j
) is the Fourier transform of the virtual excitation u
j
(
t
) and U
∗
(
ω
,
j
) is the complex conjugate of the virtual excitation u
j
(
t
).The cross-power spectral densityG
y
u
(
ω
,
i
,
j
) between u
j
(
t
) and the virtual response u
i
(
t
) (the response at the calculating point i) can be expressed as(9)
G
y
u
ω
,
i
,
j
=
U
∗
ω
,
j
·
U
ω
,
i
,where U
(
ω
,
i
) is the Fourier transform of the virtual response u
i
(
t
).U
(
ω
,
j
) and U
(
ω
,
i
) can be, respectively, expressed as (10)
U
ω
,
j
=
H
ω
,
j
F
ω
,
U
ω
,
i
=
H
ω
,
i
F
ω
;the virtual frequency response function H
y
u
(
ω
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be expressed as(11)
H
y
u
ω
,
i
,
j
=
G
y
u
ω
,
i
,
j
G
u
u
ω
,
j
=
U
∗
ω
,
j
·
U
ω
,
i
U
∗
ω
,
j
·
U
ω
,
i
=
H
∗
ω
,
j
F
∗
ω
·
H
ω
,
i
F
ω
H
∗
ω
,
j
F
∗
ω
·
H
ω
,
j
F
ω
=
H
∗
ω
,
j
·
H
ω
,
i
H
∗
ω
,
j
·
H
ω
,
j
;after the inverse Fourier transform, VIRF H
y
u
(
t
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be expressed as (12)
H
y
u
t
,
i
,
j
=
1
2
π
∫
-
∞
∞
e
-
i
ω
t
H
ω
,
i
,
j
d
ω
.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  VIRFH
y
u
(
t
,
i
,
j
) between u
j
(
t
) and u
i
(
t
) can be rewritten as(13)
H
y
u
t
,
i
,
j
=
∑
h
g
=
h
1
+
h
2
+
⋯
+
h
2
I
,where h
g is the virtual impulse response function component and I 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 E
g can be expressed as(14)
E
g
=
∑
h
g
,
h
2
h
=
1,2
,
…
,
m
,where m is the number of signal simple points.Thus, ESE
I can be expressed as(15)
E
I
=
E
1
E
2
⋯
E
g
⋯
E
2
I
T
;then, the energy ratio I
g is defined as(16)
I
g
=
E
g
∑
g
=
1
2
I
E
g
×
100
;thus, an energy ratio sequence I
~
G is created and can be expressed as(17)
I
~
G
=
I
~
1
I
~
2
⋯
I
~
g
⋯
I
~
2
I
T
;because the energy ratio sequence I
~
G is messy or without any order, I
~
G can be sorted from large to small according to I
~
g values, yielding a new energy ratio sequence I
G:(18)
I
G
=
I
1
I
2
⋯
I
g
⋯
I
2
I
T
I
g
≥
I
g
+
1
;
I
G 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 topp frequency bands with larger energy ratio are selected to detect damage information. Let ε
p
≤
ε
0:(19)
ε
p
=
∑
q
=
1
p
I
q
q
=
1,2
,
…
,
p
,where ε
p is the relative cumulative energy ratio and ε
0 is a threshold. It is via ε
0 that the top p energy ratios in I
G can be determined, and energy ratio changes on these p 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 as(20)
I
p
+
1
=
100
-
∑
q
=
1
p
I
q
;the (p
+
1) frequency bands are called character frequency bands. Then the character frequency band vector spectrum I
p
+
1 is defined as(21)
I
p
+
1
=
I
1
I
2
⋯
I
p
I
p
+
1
T
;on the basis of I
p
+
1, define the energy ratio deviation D
I
k as (22)
D
I
k
=
I
k
u
-
I
k
d
k
=
1,2
,
…
,
p
+
1
,where I
k
u and I
k
d are the energy ratios of the kth character frequency band of undamaged structures and damaged ones, respectively.Then, define DESD
I
p
+
1 as(23)
D
I
p
+
1
=
DI
1
D
I
2
⋯
D
I
p
D
I
p
+
1
T
;obviously, structures are undamaged when D
I
p
+
1
=
0, while structures are damaged when D
I
p
+
1
≠
0. Thus, it is easy to detect whether retaining wall structures are damaged or not by DES D
I
p
+
1.


### 2.3.3. Damage Identification Indexes

On the basis ofD
I
p
+
1, two damage identification indexes, the energy ratio standard deviation ISD and energy ratio variation coefficient IVC, are proposed, and they are defined as(24)
ISD
=
∑
k
=
1
p
+
1
D
I
k
2
p
+
1
,
IVC
=
∑
k
=
1
p
+
1
D
I
k
I
k
u
2
.There is no damage within structures when IVC = 0 or ISD = 0 and there is damage when I
V
C
>
0 or I
S
D
>
0. 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.


## 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 E
g can be expressed as(14)
E
g
=
∑
h
g
,
h
2
h
=
1,2
,
…
,
m
,where m is the number of signal simple points.Thus, ESE
I can be expressed as(15)
E
I
=
E
1
E
2
⋯
E
g
⋯
E
2
I
T
;then, the energy ratio I
g is defined as(16)
I
g
=
E
g
∑
g
=
1
2
I
E
g
×
100
;thus, an energy ratio sequence I
~
G is created and can be expressed as(17)
I
~
G
=
I
~
1
I
~
2
⋯
I
~
g
⋯
I
~
2
I
T
;because the energy ratio sequence I
~
G is messy or without any order, I
~
G can be sorted from large to small according to I
~
g values, yielding a new energy ratio sequence I
G:(18)
I
G
=
I
1
I
2
⋯
I
g
⋯
I
2
I
T
I
g
≥
I
g
+
1
;
I
G 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 topp frequency bands with larger energy ratio are selected to detect damage information. Let ε
p
≤
ε
0:(19)
ε
p
=
∑
q
=
1
p
I
q
q
=
1,2
,
…
,
p
,where ε
p is the relative cumulative energy ratio and ε
0 is a threshold. It is via ε
0 that the top p energy ratios in I
G can be determined, and energy ratio changes on these p 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 as(20)
I
p
+
1
=
100
-
∑
q
=
1
p
I
q
;the (p
+
1) frequency bands are called character frequency bands. Then the character frequency band vector spectrum I
p
+
1 is defined as(21)
I
p
+
1
=
I
1
I
2
⋯
I
p
I
p
+
1
T
;on the basis of I
p
+
1, define the energy ratio deviation D
I
k as (22)
D
I
k
=
I
k
u
-
I
k
d
k
=
1,2
,
…
,
p
+
1
,where I
k
u and I
k
d are the energy ratios of the kth character frequency band of undamaged structures and damaged ones, respectively.Then, define DESD
I
p
+
1 as(23)
D
I
p
+
1
=
DI
1
D
I
2
⋯
D
I
p
D
I
p
+
1
T
;obviously, structures are undamaged when D
I
p
+
1
=
0, while structures are damaged when D
I
p
+
1
≠
0. Thus, it is easy to detect whether retaining wall structures are damaged or not by DES D
I
p
+
1.


## 2.3.3. Damage Identification Indexes

On the basis ofD
I
p
+
1, two damage identification indexes, the energy ratio standard deviation ISD and energy ratio variation coefficient IVC, are proposed, and they are defined as(24)
ISD
=
∑
k
=
1
p
+
1
D
I
k
2
p
+
1
,
IVC
=
∑
k
=
1
p
+
1
D
I
k
I
k
u
2
.There is no damage within structures when IVC = 0 or ISD = 0 and there is damage when I
V
C
>
0 or I
S
D
>
0. 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 Figure1(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 (V
·
s
2
/
m). 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).Table 1
Material parameters of the concrete and filled soil.
Material
Density/(kg/m3)
Elasticity modulus/MPa
Poisson ratio
Cohesive force/kPa
Internal friction angle/°
Concrete
2500
2.11 × 104
0.2
—
—
Filled soil behind the wall
1901
15.4
0.3
0
37.6
Filled soil in front of the wall
2385
2.3 × 104
0.2
460
39.5Figure 1
Pile plate retaining wall.
(a)
Vibration test
(b)
Assignment of sensors (unit: mm)Figure 2
Vibration test equipment.
(a)
DFC-2 exciting hammer
(b)
941B sensor
(c)
JM3863A vibration test system
(d)
JM1802 wireless gatewayIn this paper, the hammer impulse excitation is a type of low strain excitation. Under low strain excitation, both the wall and filled soil vibrate mildly. Thus, it is assumed that the filled soil vibrates together with the wall. The additional dynamic effects of filled soil on the wall can be neglected under low strain excitation. Thus, the filled soil and retaining wall can be regarded as one structure. And the damage identification method is not influenced by filled soil.After doing vibration test for many times, the dynamic responses at measuring points are analyzed by DASP (Data Acquisition and Signal Processing) program to obtain mode shapes. Then normalization of mode shapes is also acquired, as shown in Figure3, where H is height of the wall and L is length. Finally, 25 measuring points are placed on the wall to obtain plenty of mode shapes. On the basis of a modal test, 26 measuring points are assigned on the retaining wall. Points 1 to 25 are calculating points, while point 26 is a reference point. The number of sensor measuring points is as shown in Figure 1(b). The points from I to VI in Figure 1(b) are excitation points.Figure 3
Mode shapes of the retaining wall.
(a)
1st mode shape (ω
1 = 22.41 Hz)
(b)
3rd mode shape (ω
3 = 34.32 Hz)
(c)
4th mode shape (ω
4 = 42.14 Hz)
(d)
5th mode shape (ω
5 = 48.45 Hz)Damage will emerge and accumulate within the retaining wall due to many causes. From a microperspective, the damage is microcracks or holes. To simulate the actual damage within the retaining wall, various holes are drilled, as shown in Figure4. The diameter of a single hole is 0.02 m, and the depth is 0.1 m. All of the holes are drilled in an area 0.25 m × 0.25 m {
L
∈
(
1.375
m
,
1.625
m
)
,
H
∈
(
0.875
m
,
1.125
m
)
}, as shown in Figure 1(b), where L is the length of the retaining wall and H is the height.Figure 4
Partial damage.
(a)
Case1
(b)
Case3
(c)
Case6
(d)
Case8To represent the partial DI (damage intensity), define DI as:(25)
DI
=
N
·
V
h
V
,where N is the number of holes, V
h is the volume of a single hole, and V is the total volume of the partial area (V
=
0.25
m
×
0.25
m
×
0.1
m).When DI = 0, there are no holes within the wall, so there is no damage within the wall. With the increase of the number of holes, the partial DI increases gradually. Eight damage cases are considered to investigate the damage identification of the retaining wall, as listed in Table2. Here, Case 1 is the undamaged case while Cases 2 to 8 are damaged cases.Table 2
Damage cases.
Case
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
DI
0
1.00%
2.00%
4.02%
6.03%
8.04%
10.05%
12.06%In light of damage identification method mentioned above, the damage identification of this retaining wall includes eight steps:(
1
) a virtual impulse response function between the reference point and calculating points should be obtained; (
2
) decomposition of the virtual response function is performed, and ERS is created; (
3
) the wavelet packet character frequency band vector spectrum is established; (
4
) DES is created; (
5
) identification indexes are obtained; (
6
) the damage state of the retaining wall is distinguished; (
7
) there is detection of the location of partial damage within the wall; (
8
) there is quantitative identification of the partial damage intensity.


### 3.1. VIRF

Under the effect of the impulse excitation caused by the hammer, the retaining wall is forced to vibrate. VIRF between the virtual excitation (response at the reference point) and the virtual response (response at the calculating point) is obtained. When the external excitation point is located at point II, VIRF curves of different calculating points are as shown in Figure5. Obviously, VIRF curves of different calculating points are different, and VIRF amplitudes are changed due to the damage within the wall. In particular, VIRF amplitude of a calculating point that is located in damaged area (such as point 12) varies much more greatly. With the increase of the damage intensity, this variation will be even larger.Figure 5
VIRF curve of calculating points.


### 3.2. ERS

There are seven wavelet packet decomposition layers, and the Daubechies 18 function is selected as the wavelet function. Via wavelet packet decomposition of VIRF, 128 subfrequency bands are created. On the basis of the ES, ERS is acquired. When the excitation point is located at point II, ERS of different calculating points are as shown in Figure6. Obviously, the majority of frequency band energy is distributed over a minority of frequency bands. The energy ratio variation caused by damage is nearly undetectable in those frequency bands that have much less energy. Thus, it is unnecessary to use all of the subfrequency bands to identify damage.Figure 6
ERS of calculating points.


### 3.3. CVS (Wavelet Packet Character Frequency Band Vector Spectrum)

Generally speaking, those frequency bands with much larger energy have much greater sensitivity to damage, and they are therefore much more useful in identifying said damage. Because the majority of VIRF energy is distributed over a minority of frequency bands, the minority frequency band energy ratio can be used to replace ERS. Thus, CVS is created. When excitation point is located at point II, CVS of different calculating points are as shown in Figure7.Figure 7
CVS of calculating points.


### 3.4. DES

Based on CVS, DES is created. When the excitation point is located at point II, DES of different calculating points is as shown in Figure8. Damage within the wall causes a variation of the energy ratio deviation D
I
k. With the increase of the damage intensity, DI, D
I
k becomes larger and larger. In addition, different points have different sensitivities for the identification of damage. For instance, points 12 and 25 have a better ability to identify partial damage within the retaining wall, while point 1 has poorer sensitivity to damage.Figure 8
DES of calculating points.


### 3.5. Damage Identification Indexes

Through (24), the damage identification indexes ISD and IVC are obtained. When the external excitation point is located at point II, the damage identification indexes of different calculating points are as listed in Table 3. Under a certain damage intensity, the ISD or IVC of point 12 is the largest in comparison with other points. With the increase of the damage intensity, the indexes’ values increase gradually. Based on the damage location and excitation position, different points have different sensitivities to damage. Point 12, located in the partial area, has the best sensitivity to damage.Table 3
Damage identification index.
DI
ISD
IVC
Point 1
Point 7
Point 12
Point 25
Point 1
Point 7
Point 12
Point 25
0
0
0
0
0
0
0
0
0
1%
0.0058
0.0290
0.2313
0.0263
0.0027
0.0095
0.0925
0.0120
2%
0.0216
0.1905
0.6558
0.1862
0.0123
0.0519
0.2464
0.0543
4.02%
0.0254
0.1801
0.8885
0.2261
0.0133
0.0583
0.3301
0.0580
6.03%
0.0285
0.1895
1.2493
0.2502
0.0137
0.0613
0.4537
0.0587
8.04%
0.0309
0.2216
1.7863
0.2592
0.0135
0.0616
0.6240
0.0574
10.05%
0.0337
0.1882
2.4833
0.2621
0.0130
0.0604
0.8180
0.0556
12.06%
0.0352
0.2192
2.8405
0.2624
0.0127
0.0595
0.9055
0.0551


### 3.6. Distinguishing the Damage State of the Retaining Wall

In light of the damage identification method mentioned in Table3, the retaining wall structure is not damaged when D
I
p
+
1
=
0, while the wall is damaged when D
I
p
+
1
≠
0. From Figure 8, it is easy to distinguish qualitatively whether the wall is damaged or not. There is no damage within the retaining wall when IVC = 0 or ISD = 0, while there is damage when IVC > 0 or ISD > 0. Thus, from Table 3, it is still easy to distinguish quantitatively whether the wall is damaged or not.Via vibration test, responses at the points which are close to excitation point are much larger but the sensitivity of these points is even less. In addition, the sensitivity of points is much less when the distance between measurement points and partial damage is much larger. Only the point which is very close to partial damage has the best sensitivity to damage. Thus, the sensitivity of measurement points is also related with the distance between measurement points and partial damage.


### 3.7. Detecting the Damage Location

25 measuring points are related with 25 ISD or IVC values. Under a certain damage intensity, these twenty-five ISD values can be used to form a surface by MATLAB program, as shown in Figure9. The ISD values change suddenly and reach a peak value in the area {
L
∈
(
1.375
m
,
1.625
m
)
,
H
∈
(
0.875
m
,
1.125
m
)
}, where partial damage is simulated. The same results can be acquired through the IVC presented in Figure 10. Thus, the damage location of the retaining wall can be detected accurately by the position of the ISD (or IVC) peak value on an ISD (or IVC) surface. In addition, the values of ISD or IVC are nonnegative in light of (24). Negative values appear in ISD or IVC surface because of MATLAB program.Figure 9
ISD trend surface.
(a)
Case3
(b)
Case6
(c)
Case8Figure 10
IVC trend surface.
(a)
Case3
(b)
Case6
(c)
Case8


### 3.8. Identifying the Damage Intensity

In this paper, the damage identification includes three key steps: firstly, detecting damage state; secondly, diagnosing damage location; thirdly, identifying damage intensity. Here, the damage intensity is the one of partial damage. After diagnosis of damage location, DI can be identified by obtaining the quantitative relationship between the damage identification index and DI. Here, a single partial damage center is located at point 12; the quantitative relationship between the ISD (or IVC) and DI of point 12 is fitted by the least squared method. The quantitative relationship between the ISD (or IVC) and DI can be expressed as(26)
Y
=
-
0.33
x
5
-
1.52
x
4
+
1.01
x
3
+
2.68
x
2
+
1.82
x
+
0.11
,
Y
=
208
z
5
-
402
z
4
+
240
z
3
-
39
z
2
+
9.77
z
+
0.06
,where Y is the damage intensity, DI, while x and z are the ISD and IVC, respectively.The experimental curve and data-fitting curve are shown in Figures11(a) and 11(b), respectively. It is not difficult to verify that these two curves are nearly superpositions. It is believed that (26) have a better fitting degree to quantitative relationship between DI and the damage identification index. Thus, (26) can be used to identify the intensity of partial damage within the retaining wall.Figure 11
Relationship curve of the damage intensity and the damage identification index.
(a)
DI-ISD
(b)
DI-IVC


## 3.1. VIRF

Under the effect of the impulse excitation caused by the hammer, the retaining wall is forced to vibrate. VIRF between the virtual excitation (response at the reference point) and the virtual response (response at the calculating point) is obtained. When the external excitation point is located at point II, VIRF curves of different calculating points are as shown in Figure5. Obviously, VIRF curves of different calculating points are different, and VIRF amplitudes are changed due to the damage within the wall. In particular, VIRF amplitude of a calculating point that is located in damaged area (such as point 12) varies much more greatly. With the increase of the damage intensity, this variation will be even larger.Figure 5
VIRF curve of calculating points.


## 3.2. ERS

There are seven wavelet packet decomposition layers, and the Daubechies 18 function is selected as the wavelet function. Via wavelet packet decomposition of VIRF, 128 subfrequency bands are created. On the basis of the ES, ERS is acquired. When the excitation point is located at point II, ERS of different calculating points are as shown in Figure6. Obviously, the majority of frequency band energy is distributed over a minority of frequency bands. The energy ratio variation caused by damage is nearly undetectable in those frequency bands that have much less energy. Thus, it is unnecessary to use all of the subfrequency bands to identify damage.Figure 6
ERS of calculating points.


## 3.3. CVS (Wavelet Packet Character Frequency Band Vector Spectrum)

Generally speaking, those frequency bands with much larger energy have much greater sensitivity to damage, and they are therefore much more useful in identifying said damage. Because the majority of VIRF energy is distributed over a minority of frequency bands, the minority frequency band energy ratio can be used to replace ERS. Thus, CVS is created. When excitation point is located at point II, CVS of different calculating points are as shown in Figure7.Figure 7
CVS of calculating points.


## 3.4. DES

Based on CVS, DES is created. When the excitation point is located at point II, DES of different calculating points is as shown in Figure8. Damage within the wall causes a variation of the energy ratio deviation D
I
k. With the increase of the damage intensity, DI, D
I
k becomes larger and larger. In addition, different points have different sensitivities for the identification of damage. For instance, points 12 and 25 have a better ability to identify partial damage within the retaining wall, while point 1 has poorer sensitivity to damage.Figure 8
DES of calculating points.


## 3.5. Damage Identification Indexes

Through (24), the damage identification indexes ISD and IVC are obtained. When the external excitation point is located at point II, the damage identification indexes of different calculating points are as listed in Table 3. Under a certain damage intensity, the ISD or IVC of point 12 is the largest in comparison with other points. With the increase of the damage intensity, the indexes’ values increase gradually. Based on the damage location and excitation position, different points have different sensitivities to damage. Point 12, located in the partial area, has the best sensitivity to damage.Table 3
Damage identification index.
DI
ISD
IVC
Point 1
Point 7
Point 12
Point 25
Point 1
Point 7
Point 12
Point 25
0
0
0
0
0
0
0
0
0
1%
0.0058
0.0290
0.2313
0.0263
0.0027
0.0095
0.0925
0.0120
2%
0.0216
0.1905
0.6558
0.1862
0.0123
0.0519
0.2464
0.0543
4.02%
0.0254
0.1801
0.8885
0.2261
0.0133
0.0583
0.3301
0.0580
6.03%
0.0285
0.1895
1.2493
0.2502
0.0137
0.0613
0.4537
0.0587
8.04%
0.0309
0.2216
1.7863
0.2592
0.0135
0.0616
0.6240
0.0574
10.05%
0.0337
0.1882
2.4833
0.2621
0.0130
0.0604
0.8180
0.0556
12.06%
0.0352
0.2192
2.8405
0.2624
0.0127
0.0595
0.9055
0.0551


## 3.6. Distinguishing the Damage State of the Retaining Wall

In light of the damage identification method mentioned in Table3, the retaining wall structure is not damaged when D
I
p
+
1
=
0, while the wall is damaged when D
I
p
+
1
≠
0. From Figure 8, it is easy to distinguish qualitatively whether the wall is damaged or not. There is no damage within the retaining wall when IVC = 0 or ISD = 0, while there is damage when IVC > 0 or ISD > 0. Thus, from Table 3, it is still easy to distinguish quantitatively whether the wall is damaged or not.Via vibration test, responses at the points which are close to excitation point are much larger but the sensitivity of these points is even less. In addition, the sensitivity of points is much less when the distance between measurement points and partial damage is much larger. Only the point which is very close to partial damage has the best sensitivity to damage. Thus, the sensitivity of measurement points is also related with the distance between measurement points and partial damage.


## 3.7. Detecting the Damage Location

25 measuring points are related with 25 ISD or IVC values. Under a certain damage intensity, these twenty-five ISD values can be used to form a surface by MATLAB program, as shown in Figure9. The ISD values change suddenly and reach a peak value in the area {
L
∈
(
1.375
m
,
1.625
m
)
,
H
∈
(
0.875
m
,
1.125
m
)
}, where partial damage is simulated. The same results can be acquired through the IVC presented in Figure 10. Thus, the damage location of the retaining wall can be detected accurately by the position of the ISD (or IVC) peak value on an ISD (or IVC) surface. In addition, the values of ISD or IVC are nonnegative in light of (24). Negative values appear in ISD or IVC surface because of MATLAB program.Figure 9
ISD trend surface.
(a)
Case3
(b)
Case6
(c)
Case8Figure 10
IVC trend surface.
(a)
Case3
(b)
Case6
(c)
Case8


## 3.8. Identifying the Damage Intensity

In this paper, the damage identification includes three key steps: firstly, detecting damage state; secondly, diagnosing damage location; thirdly, identifying damage intensity. Here, the damage intensity is the one of partial damage. After diagnosis of damage location, DI can be identified by obtaining the quantitative relationship between the damage identification index and DI. Here, a single partial damage center is located at point 12; the quantitative relationship between the ISD (or IVC) and DI of point 12 is fitted by the least squared method. The quantitative relationship between the ISD (or IVC) and DI can be expressed as(26)
Y
=
-
0.33
x
5
-
1.52
x
4
+
1.01
x
3
+
2.68
x
2
+
1.82
x
+
0.11
,
Y
=
208
z
5
-
402
z
4
+
240
z
3
-
39
z
2
+
9.77
z
+
0.06
,where Y is the damage intensity, DI, while x and z are the ISD and IVC, respectively.The experimental curve and data-fitting curve are shown in Figures11(a) and 11(b), respectively. It is not difficult to verify that these two curves are nearly superpositions. It is believed that (26) have a better fitting degree to quantitative relationship between DI and the damage identification index. Thus, (26) can be used to identify the intensity of partial damage within the retaining wall.Figure 11
Relationship curve of the damage intensity and the damage identification index.
(a)
DI-ISD
(b)
DI-IVC


## 4. Discussions

Single damage in a retaining wall is detected through the damage identification method proposed in this paper. In addition, this method is still valid when multidamage exists in the wall. The identification of multidamage is no longer repeated due to the limitation of space. This method is suited for solid-web components or structures, such as beam or plate components, as well as bridges and walls structures. However, damage within truss or frame structures cannot be validly diagnosed via this method. This method can only be used to detect whether the truss of frame structures is damaged or not; the partial damage location and damage intensity cannot be identified through this method. Thus, this damage identification method also has some limitations. Consequently, the method may not be valid when structures change. Additionally, the partial damage location can be precisely identified through this method when sensors are very close to partial damage. However, the partial location cannot be identified precisely when the sensors are far from the partial damage. Essentially, the precise identification of partial damage requires that sensor locations be adjusted repeatedly. Thus, sensor position is the key to valid identification of partial damage locations. Finally, the value of ISD is much larger than the one of IVC, so it is advised that ISD should be the first choice to identify damage when damage intensity or damage range are much less. Both ISD and IVC can be used to detect damage when damage intensity or damage range is much greater.


## 5. Conclusions

ES that is directly based on structural dynamic responses caused by excitation is not suitable for damage identification. However, ES based on the impulse response function of the response and external excitation is able to identify damage. However, the impulse response function can only be calculated when both the response and excitation are known. If either the response or excitation is not known, the impulse response function cannot be acquired. Thus, ES based on the impulse response function is unable to detect damage widely. Nevertheless, a virtual impulse response function can be calculated when the responses are known, regardless of whether the external excitation is known or not. Thus, ES acquired by wavelet packet decomposition of a virtual impulse response function of a virtual excitation and virtual response can be used to identify damage widely.Based on ES, an updated damage identification method and two damage identification indexes are proposed. A vibration test on a pile plate retaining wall demonstrates the method’s sensitivity and that it is easy to distinguish whether the retaining wall is damaged or not via DES or damage identification indexes; the partial damage location can be detected easily; in addition, the damage intensity is able to be validly identified as well. Consequently, this damage identification theory and method, based on a virtual impulse response function, may be used to identify damage within retaining wall structures, which has significance for disaster prevention and mitigation.



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*Source: 1346939-2016-08-25.xml*