Damage Detection for Space Truss Structures Based on Strain
Mode under Ambient Excitation
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Zhao-Dong Xu1 and Ke-Yi Wu2
Abstract: Safety assurance and detection of potential damage for space truss structures have been challenging topics. The two most critical
problems are considered in this paper. One is to develop an effective damage detection method based on strain data under ambient excitation,
and the other is then to optimize the installment of strain sensors owing to numerous structural members in the space truss structures. A method
of damage detection for space truss structures, called the environmental excitation incomplete strain mode (EEISM) method, is proposed. Four
steps are taken in the EEISM method. First, strain mode parameter identification is carried out based on the cross-correlation function of the
strain responses through a combination of the empirical mode decomposition method and the peak amplitude series method. Second, the strain
sensors are located optimally in the space truss structures through sensitive analysis of the strain mode perturbation matrix, which are obtained
by perturbation theory. Third, the modal assurance criterion (MAC) value is applied to locate the damages; that is, the members with the larger
MAC values are defined as the damaged members. Finally, a damage index obtained by solving the perturbation equation is used for damage
quantification. Numerical analysis of a long-span space truss structure including damage location and quantification for single-member and
multimember damages, detection of the various severities of damage, and the effect of the number of sensors is performed to verify the effectiveness of the proposed EEISM method. It is shown from the analysis results that the EEISM method is effective in the location and quantification of damages for single-member and multimember damages. The quantity of the strain sensors has an effect on the damage location and
has no remarkable effect on the damage quantification for the determined damage members. DOI: 10.1061/(ASCE)EM.1943-7889.0000426.
© 2012 American Society of Civil Engineers.
CE Database subject headings: Damage; Strain; Trusses; Excitation.
Author keywords: Damage detection; Strain mode; Space truss structure; Ambient excitation; Incomplete strain mode method.
Introduction
Around the world, long-span space truss structures are often
designed for large-scale public buildings, such as stadiums, gymnasiums, and exhibition halls. To ensure their healthy operational
condition and safety, early identification of damage caused by strong
wind or earthquakes in these structures is of great importance.
Researchers have been committed to finding a reliable damage
detection strategy for long-span space truss structures to address
problems associated with inspection difficulties and the importance
of considering cost.
The displacement mode shape is usually adopted in structural
damage detection (Pandey et al. 1991; Lam et al. 1998; Kaouk et al.
2000). However, the accuracy of damage detection tests for space
truss structures with a large number of members and degrees of
freedom using the complete displacement mode shape is a difficult
task. Kim and Membertkowicz (2001) proposed a two-step
damage detection method for large-scale structures based on an
1
Professor, Key Laboratory of C&PC Structures of the Ministry of
Education, Southeast Univ., Nanjing 210096, China (corresponding author).
E-mail: xuzhdgyq@seu.edu.cn
2
Doctoral Student, Key Laboratory of C&PC Structures of the Ministry
of Education, Southeast Univ., Nanjing 210096, China. E-mail: keyi.wuu@
gmail.com
Note. This manuscript was submitted on June 9, 2009; approved on
February 23, 2012; published online on September 14, 2012. Discussion
period open until March 1, 2013; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Engineering
Mechanics, Vol. 138, No. 10, October 1, 2012. ©ASCE, ISSN 07339399/2012/10-1215–1223/$25.00.
incomplete displacement mode shape. The first step was to determine the approximate damage area using the best matrix correction method, and the second step was to determine damaged
members in the detected damage area using a sensitivity analysis
method. The displacement mode shape would be identified from
the acceleration responses by modal analysis in the time or frequency domain. In a real application, the strain sensors are also
commonly applied as accelerators. Moreover, previous research
has proven that strain responses are more sensitive to local damage.
Yao et al. (1992) observed the strain mode changed during the
damage process of a 5-story frame structure and noticed that the
strain mode was superior to the displacement mode shape in
detecting local damage. Unger et al. (2005) applied the strain mode
in the damage location of a prestressed concrete beam and concluded that the mode shapes were sensitive to the local change of
stiffness nearby the sensor locations.
The objective of this paper is to develop a damage detection
strategy for long-span space truss structures based on the strain
mode. However, two critical problems should also be taken into
consideration when the strain mode is used for damage detection
of space truss structures. First, because space truss structures
are usually large-scale buildings exposed to environmental conditions, the excitation input to the structures can be naturally
considered to be subjected to these environments. Second, the
strain sensors should be installed optimally because of the large
quantity of members in the space truss structures. Therefore, a
damage detection method for space truss structures, named the
environmental excitation incomplete strain mode (EEISM) method,
is proposed. The EEISM method consists of four steps. The first
step is identification of the strain mode parameters. In this step,
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a cross-correlation function of strain responses under ambient excitation is derived. It can be proven that the cross-correlation function of the strain responses under ambient excitation has a similar
equation as the strain impulse response. Then, the strain mode
parameters are determined by combining the empirical mode decomposition method (Huang et al. 1998) and the peak amplitude
series method (Li et al. 2001) based on the cross-correlation function
of the strain responses. Second, based on the perturbation theory, the
optimal location of the strain sensors can be obtained through
sensitivity analysis of the strain mode perturbation matrix (Chen and
Garbat 1980). Third, the modal assurance criterion (MAC) value is
applied to the damage location, and members with larger MAC
values are defined as the damage members. Finally, the damage
index obtained by solving the perturbation equation is used for
damage quantification. Numerical analysis of a long-span space
truss structure, including the damage location and quantification for
single-member and multimember damages and the effect of the
number of sensors, is performed to verify the effectiveness of the
proposed method.
Strain Mode Parameter Identification
the strain responses can be expressed similarly to Eq. (1) as follows
(Tsang 1990):
«ðtÞ ¼
Cross-Correlation Function of the Strain Response
Based on the vibration theory (Clough and Penzien 1993), the
displacement responses of the structure can be expressed as follows
as the linear combination of all mode shapes:
xðtÞ ¼
n
P
fr qr ðtÞ
ð1Þ
n cp fk
P
r r
× expð2zr vnr tÞ sinðvdr tÞ
r¼1 mr vdr
«pk ðtÞ ¼
xi ðtÞ ¼
n
P
fir fkr
r¼1 mr vdr
× expð2zr vnr tÞ sinðvdr tÞ
2x
where fk ðuÞ 5 excitation imposed on point k, and gr ð × Þ can be
defined as
1
× expð2zr vnr tÞ sinðvdr tÞ
ð7Þ
gr ðtÞ ¼
mr vdr
If stationary white noise is imposed on point k, the crosscorrelation function between member p and member q can be
expressed as (Clough and Penzien 1993)
ð8Þ
Rpqk ðtÞ ¼ E «pk ðt þ tÞ«qk ðtÞ
It is known that there is no ideal stationary white noise in nature and
normal ambient excitation can usually be regarded as one kind of
wide-band random process. Research (Clough and Penzien 1993)
shows that Eq. (8) is adaptive to the wide-band random process.
Substituting Eq. (6) into Eq. (8), the cross-correlation function
Rpqk ðtÞ can be rewritten as
n P
n
P
cpr cqs fkr fks
Rpqk ðtÞ ¼
r¼1 s¼1
ðt
2x
ð t1t
2x
gr ðt þ t 2 sÞ gs ðt 2 uÞ E½fk ðsÞfk ðuÞdsdu
ð9Þ
In accordance with the theory of dynamics (Clough and Penzien
1993), the characteristic of white noise can be expressed as
E½ fk ðsÞfk ðuÞ ¼ 2pS0 dðs 2 uÞ
mass of
where vnr and mr 5 rth natural frequency and the rth modal
pffiffiffiffiffiffiffiffiffi
the system without damping, respectively; vdr 5 vnr 1 2 z2r , and
zr 5 rth damping ratio. Substituting Eq. (2) into Eq. (1), the displacement responses of point i under impulse excitation on point
k can be expressed as
ð10Þ
where dð × Þ 5 Dirac function and S0 5 spectral density of white
noise. Substituting Eq. (10) into Eq. (9), the cross-correlation
function Rpqk ðtÞ can be written as
Rpqk ðtÞ ¼
n P
n
P
2pS0 cpr cqs fkr fks
r¼1 s¼1
ðt
ð3Þ
where mode shape fir 5 specific vibration equilibrium state, and the
strain mode is the corresponding strain distribution state. Therefore,
ð5Þ
where vnr , mr , and cPr 5 rth natural frequency, rth modal mass,
and rth strain mode of
member p without system damping, repffiffiffiffiffiffiffiffiffi
spectively; vdr 5 vnr 1 2 z2r ; and zr 5 rth damping ratio. When
excitation is imposed on point k, based on the Duhamel integration, the strain response of member p can be expressed as
follows:
ðt
n
P
p k
cr fr ×
fk ðuÞgr ðt 2 uÞdu
ð6Þ
«pk ðtÞ ¼
where xðtÞ 5 displacement responses, fr 5 rth displacement mode
shape, and qr ðtÞ 5 coordinates associated with the rth mode. If
impulse excitation is imposed on point k, then qr(t) can be written as
ð2Þ
ð4Þ
where cr 5 rth strain mode of the space truss structures. The strain
impulse response of member p under impulse excitation on point k
can be written as (Jones 2005)
r¼1
fkr
qr ðtÞ ¼
× expð2zr vnr tÞ sinðvdr tÞ
mr vdr
cr qr ðtÞ
r¼1
r¼1
The typical method, the natural excitation technique (NExT) (James
et al. 1995) was developed to identify the mode parameters under
ambient excitation based on displacement responses. The basic idea
of NExT can be expressed as follows. The cross-correlation function
of the displacement responses under ambient excitation has an
equation similar to the displacement impulse responses. Thus, the
mode parameters can be identified from the cross-correlation
function based on the classical mode parameter identification
methods in the time domain. The strain mode parameter identification technique is also developed in this paper based on the NExT
idea; i.e., to find a relationship between the cross-correlation
function of the strain responses and the strain impulse response.
n
P
2x
gr ðt þ t 2 uÞ gs ðt 2 uÞdu
Assuming l 5 t 2 u, Eq. (11) can be simplified as
1216 / JOURNAL OF ENGINEERING MECHANICS © ASCE / OCTOBER 2012
J. Eng. Mech., 2012, 138(10): 1215-1223
ð11Þ
Rpqk ðtÞ ¼
n P
n
P
2pS0 cpr cqs fkr fks
r¼1 s¼1
ðx
gr ðl þ tÞ gs ðlÞdl
0
ð12Þ
In accordance with Eq. (7), gr ðl 1 tÞ can be expressed as
intrinsic mode function rpqk ðtÞ corresponding to the first natural
frequency (Huang et al. 1998), and the peak amplitude series method
is used to obtain amplitude of rpqk ðtÞ (Li et al. 2001). This can be
described as
rpqk ðtÞ ¼ Apq expð2j1 vn1 tÞ sinðvd1 t þ gÞ
expð2jr vnr lÞ sinðvdr lÞ
mr vr
expð2jr vnr lÞ cosðvdr lÞ
þ ½expð2jr vnr tÞ sinðvdr tÞ
mr vr
ð13Þ
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gr ðl þ tÞ ¼ ½expð2jr vnr tÞ cosðvdr tÞ
Because gs ðlÞ is independent of t, by substituting Eq. (13) into
Eq. (12), the cross-correlation function Rpqk ðtÞ can be written as
Rpqk ðtÞ ¼
n h
P
Arpqk expð2jr vnr tÞ cosðvdr tÞ
r ¼1
þ
Arpqk
Brpqk
Brpqk
i
expð2jr vnr tÞ sinðvdr tÞ
9
n 2pS0 cp cq fk fk
>
P
Irs
>
r s r s
>
¼
>
=
2
2
m
m
v
J
þ
I
r
s
dr
rs
rs
s¼1
n 2pS0 cp cq fk fk
>
P
Jrs
>
r s r s
>
¼
>
;
2
2
m
m
v
J
þ
I
r
s
dr
rs
rs
s¼1
ð14Þ
ð15Þ
Arpqk ¼
Brpqk
qffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 j21 vn1
ð19Þ
where Td 5 time interval between the adjacent positive (or negative)
amplitude of rpqk ðtÞ. Considering the relationship vn1 5 2pf1 and
Eq. (19), f1 can be written as
f1 ¼
1
qffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Td 1 2 j21
ð20Þ
assuming the time series corresponding to the positive amplitude
of rpqk ðtÞ is Tð1Þ; Tð2Þ; Tð3Þ; . . . ; TðSÞ. Because S is the quantity
of the positive amplitude, Td can be expressed theoretically as
follows:
Td ¼ Tð j þ 1Þ 2 Tð jÞ ð j ¼ 1; 2; :::; S 2 1Þ
where Irs 5 2vdr ðjr vnr 1 js vns Þ
and
Jrs 5 ðv2ds 2 v2dr Þ 1
ðjr vnr 1js vns Þ2 . Assuming tanðgn Þ 5 In =Jn , Eq. (15) can be rewritten as
9
n
2
cpr P
>
2 21=2
>
brs
þ
I
sinðg
Þ
J
rs >
rs
=
mr vdr s¼1 qk rs
p
n
cr P
>
2 21=2
>
¼
brs J 2 þ Irs
cosðgrs Þ >
;
mr vdr s¼1 qk rs
vd1 ¼ 2p ¼
Td
ð16Þ
ð18Þ
ð21Þ
Usually, S cycles of the peak value attenuation curves should be
applied to calculate Td to reduce error
Td ¼
TðSÞ 2 Tð1Þ
S21
ð22Þ
Assuming the positive amplitudes of rpqk ðtÞ are
rð1Þ; rð2Þ; . . . ; rðSÞ, the following equation can be obtained:
rð1Þ
¼ j1 vn1 ½TðSÞ 2 Tð1Þ
ð23Þ
In
rðSÞ
Based on Eqs. (19), (22), and (23), j1 can be expressed as
q k k
where brs
qk 5 2pS0 cs fr fs =ms . By substituting Eq. (16) into Eq.
(14), the cross-correlation function of the strain responses can be
expressed as
p
n
n
2
P
cr P
2 21=2
brs
Rpqk ðtÞ ¼
qk Jrs þ Irs
m
v
r dr s¼1
r¼1
expð2jr vnr tÞ sinðvdr t þ grs Þ
ð17Þ
Comparing Eq. (5) with Eq. (17), it can be found that the crosscorrelation function of the strain responses under white noise
excitation has a similar equation to the strain impulse response.
Therefore, the cross-correlation function of the strain responses can
be expressed as a linear combination of a series of free attenuation
responses. In practical applications, the measured noise data between two different test channels have no correlation. That is, the
cross-correlation function of the measured noise data between two
different test channels equals zero (Halvorsen and Brown 1977);
i.e., when the cross-correlation function of the strain responses is
calculated, the measurement noise can be neglected in theory.
rð1Þ
In
rðSÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j1 ¼
2ffi
rð1Þ
1
In
2pðS 2 1Þ 1 þ
2pðS 2 1Þ rðSÞ
ð24Þ
Substituting Eq. (22) into Eq. (20), f1 can be rewritten as
f1 ¼
S 2 1qffiffiffiffiffiffiffiffiffiffiffiffiffi
½TðSÞ 2 Tð1Þ 1 2 j21
ð25Þ
The following equation is tenable at peak values points of rpqk ðtÞ:
drpqk tj
¼ 0 ½ti ¼ TðjÞ; j ¼ 1; 2; . . . ; S
ð26Þ
dtj
Substituting Eq. (26) in Eq. (18), the following relationship can be
obtained:
qffiffiffiffiffiffiffiffiffiffiffiffiffi
sin vd1 tj þ g ¼ 1 2 j21
tj ¼ TðjÞ;
Strain Mode Parameter Identification
j ¼ 1; 2; . . . ; S
ð27Þ
Based on the cross-correlation function Rpqk ðtÞ given by Eq. (17),
the empirical mode decomposition method is applied to extract
Consequently, rðjÞ can be rewritten as
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J. Eng. Mech., 2012, 138(10): 1215-1223
qffiffiffiffiffiffiffiffiffiffiffiffiffi
rðjÞ ¼ Apq exp½2j1 vn1 TðjÞ 1 2 j21 ðj ¼ 1; 2; . . . ; SÞ
In theory, Apq is independent of j. To improve precision, Eq. (28)
should be rewritten as
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Apq ¼
S
S
P
1 P
1
Aj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi
rð jÞ exp½j1 vn1 Tð jÞ
S j¼1 pq
2 j¼1
S 1 2 j1
Pr1 ¼
ð28Þ
ð29Þ
n
P
s¼1;sr lr
where
rð jÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffi
exp½2j1 vn1 Tð jÞ 1 2 j21
Considering the strain mode of the member is one kind of ratio,
the first strain mode of member p can be expressed as cp1 5 Apq =Aqq
if member q is regarded as the reference member when calculating
the strain mode, and the strain mode of member q is determined
as cq1 5 1. A comparison analysis of the strain mode parameter
identification will be introduced in detail subsequently.
ð33Þ
where n 5 quantity of members, s 5 sth strain mode; and lr and ls 5
rth and sth eigenvalues of the stiffness matrix in Eq. (32). In Eq. (33),
K1« 5 small change of K« when the structure is damaged, and K1« can
be expressed as
K1« ¼
Ajpq ¼
1 cT K1 c c
2 ls s « r s
m
P
cj K «
ð34Þ
j¼1
where cj ð0 # cj # 1Þ 5 reduction coefficient for the strain mode
stiffness of the jth damaged member, which is defined as the damage
index. Substituting Eq. (34) into Eq. (33), the rth strain mode change
can be expressed as
Dcr ¼
¼
n
P
m
1 cT P
cj K « c r c s
s
s¼1;sr lr 2 ls
j¼1
m
P
j¼1
cj
n
P
1
cT K« cr cs
l
2
ls s
r
s¼1;sr
ð35Þ
Optimization of the Sensor Location
Eq. (35) can be rewritten in the form of a matrix
The strain modes of the structure can be obtained by the previously
mentioned technique. However, it is unlikely that a strain sensor will
be installed on every member because of the large number of
members and economic considerations in practical applications.
Here, sensor locations will be determined optimally based on the
sensitivity analysis of the strain mode perturbation matrix, which
can be obtained by the perturbation theory.
Application of the Perturbation Theory in the
Strain Mode
The relationship between the rth strain mode cr and the rth displacement mode Fr can be expressed as (Yam et al. 1996)
cr ¼ TFr
ð30Þ
where cr 5 ½ c1 c2 ⋯ cm T ; Fr 5 ½ f1 f2 ⋯ fn T ;
T 5 transformation matrix from the displacement mode to the strain
mode, which only depends on the geometric conditions of the
structure; n 5 quantity of members; and m 5 number of degrees of
freedom. The rth mode potential energy Ep can be expressed in the
form of the strain mode as follows:
Ep ¼
FTr KFr
¼
cTr K« cr
ð31Þ
where K« 5 axial stiffness of the members, which can be expressed
in the form of a matrix as follows:
3
2
ðEA=lÞ1
0
7
6
...
7
6
7
6
ðEA=lÞj
K« ¼ 6
ð32Þ
7
7
6
5
4
...
0
ðEA=lÞm
where E, A, and l 5 elastic modules, cross area, and length of
the member, respectively, and (EA/l)j 5 strain mode stiffness of
member j.
The first-order perturbation matrix for the rth strain mode can be
expressed as (Chen et al. 1993)
Dcr ¼ Pr1 a
ð36Þ
and Eq. (36) is defined as a perturbation equation, where Pr1 5 firstorder perturbation matrix of the rth strain mode, and a 5 damage
index vector, a 5 ½ c1 c2 ⋯ cm T . The jth column of Pr1
represents the strain mode change of all members when unit damage
ðcj 5 1Þ occurs to the jth member. Eq. (36) shows that the rth strain
mode change Dcr is the linear combination of column vectors in Pr1
when unit damage occurs in various members, and the damage index
cj can be considered as the combination coefficient.
Optimization of the Sensor Location
An analysis method on the optimization of sensor locations is introduced based on the strain modes of space truss structures and the
perturbation theory. The perturbation equation [Eq. (36)] is used to
estimate the quantity of damaged members, and a can be written as
a ¼ ½PTr1 Pr1 21 PTr1 Dcr
ð37Þ
where P0 5 PTr1 Pr1 , which is defined as the Fisher information
matrix. Imamovic and Ewins (1997) applied the Fisher information
matrix to define sensitive members based on the displacement
modes. Here, the Fisher information matrix is applied to define the
sensitive members based on the strain modes. The Fisher information matrix P0 contains the perturbation information of all
members. The best unbiased estimation for a can be obtained when
P0 takes the largest trace value. Various members have various
contributions to their trace values. Therefore, it should be ensured
that the trace values of P0 are as large as possible when the sensor
quantity is determined in order to obtain the best estimation of a.
That is, the sensors on the members with a greater contribution to
their trace values should be retained, while the sensors on the
members with less of a contribution to their trace values should be
discarded; thus, reasonable sensor placement can be obtained. This is
the so-called sensitivity analysis for the strain mode perturbation
matrix. In practice, P0 can be decomposed in accordance with each
test channel
1218 / JOURNAL OF ENGINEERING MECHANICS © ASCE / OCTOBER 2012
J. Eng. Mech., 2012, 138(10): 1215-1223
P0 ¼
m
P
j¼1
T
Pjr1 Pjr1 ¼
m
P
j¼1
Pj0
Calculating the trace value of P0 , the following equation can be
obtained:
tr P0 ¼
m
P
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j¼1
tr Pj0
ð39Þ
Here, the contribution ratio of the damage index (CRDI) is defined as
CRDI ¼ trPj0 =trP0
MACj $ MAC þ gsMAC
ð38Þ
ð40Þ
Therefore, the sensors should be placed on the members with larger
CRDI values to get the maximum trace value of P0 . An example
analysis of the optimal placement of the sensors will be introduced in
detail subsequently.
Damage Location and Quantification
When the optimal placement of sensors is determined, the incomplete strain mode can be obtained by the mode parameter
identification step of the EEISM method described previously. To
date, a limited amount of research has been conducted on the
damage location and quantification of space truss structures using
the incomplete strain mode. Here, the EEISM method is proposed to
identify the damage location and damage quantification of space
truss structures based on the incomplete strain mode.
ð42Þ
where MAC and sMAC 5 average value and the standard deviation
of all MACj , respectively; g 5 guarantee coefficient; and the smaller
g 5 higher identification probability for the damaged members.
Certainly, some members whose MAC values are situated in the
defined scope shown in Eq. (42) are not damaged. These misjudged
members will be eliminated through the damage quantification index.
Damage Quantification
The perturbation equation expressed in Eq. (36) can be used for
estimation of the damage quantity, and the first mode is usually
adopted in calculating damage quantification index a, which is
presented in the following equation:
P11 a ¼ Dc1
ð43Þ
where P11 5 first-order perturbation matrix of the first strain mode and
Dc1 5 change of the first strain mode when the structure is damaged.
Eq. (43) is an overdetermined equation if the number of the damaged
members is less than the number of sensors. The damage quantification index a can be solved by the least-squares method.
Numerical Analysis
To verify the effectiveness of the proposed EEISM method, a numerical analysis of a long-span space truss structure was carried out.
A finite-element model of the space truss structure with a span of 9 3
9 m, 61 nodes, and 200 members shown in Fig. 1. All members with
Damage Location
When the sensor locations are determined, perturbation matrix P11
should be modified such that the rows corresponding to the locations
without sensors are set as zero. Meanwhile, the first strain mode
change vector Dc1 should also be modified; i.e., the values corresponding to the elements without sensors should be set as zero. A
comparison of the degree of similarity between Dc1 and each P11
column (i.e., Pj11 ) is carried out in accordance with the assurance
criterion (Ting et al. 1993), which can be expressed as
MACj ¼
ðDcT1 × Pj11 Þ2
j
ðDcT1 × Dc1 Þ × ðPjT
11 × P11 Þ
ð41Þ
where MACj 5 MAC value of the jth member. The members with
large MAC values will be defined as the damaged members.
Actually, the MAC value can be regarded as the similarity degree
between the tested strain mode change and the theoretical strain
mode change when a special member is damaged; i.e., a larger
MAC value represents a higher degree of similarity, and the
corresponding member can be considered as the damaged
member. When several members are damaged at the same time,
the MAC values between Dc1 and Pj11 that participate in the
combination of Dc1 will be large, while the MAC values between
Dc1 and Pj11 that does not participate in the combination of Dc1
will be small.
The damaged members usually exhibit larger MAC values;
however, they do not always have the largest MAC values. Consequently, a larger MAC value scope must be determined first, as
shown in Eq. (42) (Gonçalves et al. 2007), and the damaged
members are certainly among the members whose MAC values meet
Eq. (42)
Fig. 1. Space truss structure in the numerical analysis
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the same cross section were assumed to be axial tensioncompression members. The stress was considered as fully axial
and uniform along the member, which is a basic assumption for
space truss structures. Although bending deformation actually
existed, it was neglected compared with the axial deformation. In
a real application, bending deformation can also be eliminated by
a compensation strain sensor that can be placed on the symmetric
position of the member. The bearings were set on the four corner
bottom nodes, which are hinge supports in the X, Y, and Z
dimensions. In the finite-element model, there were 40 bottom chord
members with the number ranging from 1 to 40, 60 top chord
members with the number ranging from 41 to 100, and 100 web
members with the number ranging from 101 to 200. The loading was
applied on the intermediate top chord point. In the numerical
analysis, a comparison analysis of the strain mode identification
results was first introduced to observe the efficiency of the strain
mode parameter identification of the EEISM method. Then, the
damage location and quantification analysis followed. Finally, the
damage detection analysis under various numbers of sensors was
carried out. To realize the aforementioned research contents,
12 damage cases were adopted in the numerical analysis (listed in
Table 1). Damage Cases 1–4 were used to verify the damage location
capability when various members were damaged. Member No. 7
(bottom chord), Member No. 63 (top chord), and Member No. 108
(web member) with a stiffness reduction of 20% were selected to
simulate single-member damage. Member No. 11 (bottom chord)
with a stiffness reduction of 15% and Member No. 83 (top chord)
with a stiffness reduction of 20% were supposed to simulate
multimember damage. Cases 5–8 were used to verify the damage
Table 1. Damage Cases in the Numerical Analysis
Damage case Damaged member Damage quantity (%) Sensor quantity
Case
Case
Case
Case
Case
Case
Case
Case
Case
Case
Case
Case
1
2
3
4
5
6
7
8
9
10
11
12
7
63
108
11 and 83
53
53
53
53
34
34
34
34
20
20
20
15 and 20
10
15
20
40
20
20
20
20
50
50
50
50
50
50
50
50
25
50
75
100
quantification ability when various stiffness reductions occurred to
a member; i.e., Member No. 53 with stiffness reductions of 10, 15,
20, and 40%, respectively. Cases 9–12 were used in the damage
detection analysis when various sensor quantities were applied to the
damage detection; i.e., when the sensor quantities were 25, 50, 75,
and 100, and a 20% stiffness reduction occurred to Member No. 34.
The previously defined damaged members are shown as bold lines in
Fig. 1. It must be noted that the structure still remained linear after
slight damage, and the damage was usually simulated by stiffness
deterioration of the members rather than by integral deterioration.
Strain Mode Identification Analysis
Stationary filtering white noise was employed to simulate the ambient excitation imposed on the space truss structure show in Fig. 1.
The strain responses were obtained by finite-element dynamic
analysis, and the strain modes of the members were identified based
on the mode parameter identification step of the EEISM method
described previously. Fig. 2 shows a comparison between the first
strain mode values obtained by the EEISM method and the first strain
mode values obtained by the finite-element mode analysis. In Fig. 2,
the horizontal axis is the member number, and the vertical axis is the
corresponding strain mode value. Here, Member No. 1 is used as the
reference member whose strain mode value is c11 5 1, and the strain
mode values of the other members are the ratios to the strain of the
reference member. As seen in Fig. 2, the first strain mode values
obtained by the EEISM method agree well with the first strain mode
values obtained by the finite-element analysis, and the maximum
error is around 10%. This demonstrates that the EEISM method has
an excellent ability of strain mode identification.
Based on Fig. 2, about 30 members have no obvious amplitude
response for the first strain mode relative to Member No. 1. Here, two
aspects should be noticed. First, the strain mode of the sensitive
member changes greatly. When damage occurs on a member of the
structure, it does not matter if this member has large or small strain
responses. That is, the structure damage will lead to a change of the
strain mode of the sensitive members first. At the same time, the
amplitudes of the strain responses are not necessarily connected to
the sensitive members. In practical applications, it is beneficial to
install the sensors on sensitive members with relatively large strain
responses in order to decrease the relative error in measurement.
Second, sensors are not set on the member with large strain response
intentionally. Based on the damage detection method proposed in
this paper, more attention is paid to the strain mode change of the
sensitive member rather than the strain mode change of the damage
member when some members are damaged.
Fig. 2. Comparison of the strain mode of the space truss structure
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Damage Location Analysis
from Fig. 6 that Member Nos. 11 and 83 have peak MAC values of
0.246 and 0.593, and they can obviously be identified as the
damaged members. It can be concluded from the previous analysis
that the EEISM method has excellent damage location ability for
single damaged members of bottom chord members, top chord
members, and web members. At the same time, the EEISM method
also has excellent damage location ability for multidamaged members.
As described previously, the strain mode parameters can be identified precisely. Then, the change of the incomplete strain mode between the damaged and initial structures (i.e., Dc1 ) should be
obtained with relatively good accuracy. The MAC values that indicate the degree of similarity between Dc1 and each column of
perturbation matrix Pj11 are calculated by Eq. (41), and then the
damages can be located by Eq. (42).
In Damage Case 1, a stiffness reduction of 20% occurring in the
bottom chord of Member No. 7 was assumed. The MAC values of all
members can be calculated by Eq. (41). Fig. 3 shows the MAC
values of all members when Member No. 7 is damaged. It can be
seen that Member No. 7, which has the largest MAC value, can be
easily determined to be a damaged member. In Damage Case 2,
a stiffness reduction of 20% occurring in the bottom chord of
Member No. 63 was assumed. Fig. 4 shows the MAC values of all
members when Member No. 63 is damaged. It can be seen that
Member No. 63 has the largest MAC value of 0.933. Member No. 63
can be determined to be a damaged member. Damage may also occur
in web members, and in Damage Case 3 the stiffness reduction of
20% occurring in the web Member No. 108 was assumed. Fig. 5
shows the MAC values of all members when Member No. 108 is
damaged. It can be seen that Member No. 108 has the largest MAC
value of 0.892, and it can be determined to be a damaged member.
The aforementioned cases all were used in the damage location
analysis of single damaged members. However, damage may also
occur in multimembers. In Damage Case 4 it was assumed that
a stiffness reduction of 15% occurred in Member No. 11 (bottom
chord) and a stiffness reduction of 20% occurred in Member No. 83
(top chord) at the same time. Fig. 6 shows the MAC values of all
members when Member Nos. 11 and 83 are damaged. It can be seen
To observe the damage quantification ability of the EEISM method,
a damage quantification analysis was carried out by assuming that
Member No. 53 had various amounts of damage severity with 10,
15, 20, and 40% stiffness reductions. The MAC values of all
members can be calculated by Eq. (41), and the damage member can
be located by Eq. (42); then, damage index a can be estimated by Eq.
(43). In Damage Cases 5–8, stiffness reductions of 10, 15, 20, and
40%, respectively, occurring to Member No. 53 were assumed. Fig. 7
shows the MAC values of all members in Damage Cases 5–8. As
seen, Member No. 53 has the largest MAC value in every damage
case, and it can easily be determined to be the damaged member.
With an increase of the degree of damage, the MAC value of
Member No. 53 also increased. The MAC values of Member No. 53
were 0.609, 0.642, 0.692, and 0.848 under damage quantities of 10,
15, 20, and 40%, respectively. The damage quantity can be estimated
by Eq. (43). Fig. 8 and Table 2 show comparisons between the real
damage quantity and the detected damage quantity. The detected
damage quantities were 10.16, 15.98, 22.15, and 52.63% under
real damage quantities of 10, 15, 20, and 40%, respectively. The
relative error for estimation of the damage quantity increased
with the increase of the real damage quantity. The reason is that the
Fig. 3. MAC values of Damage Case 1
Fig. 5. MAC values of Damage Case 3
Fig. 4. MAC values of Damage Case 2
Fig. 6. MAC values of Damage Case 4
Damage Quantification Analysis
JOURNAL OF ENGINEERING MECHANICS © ASCE / OCTOBER 2012 / 1221
J. Eng. Mech., 2012, 138(10): 1215-1223
first-order perturbation matrix Pr1 ðr 5 1Þ is adopted in Eq. (33),
where the higher-order terms that have more of an effect on severe
damage are neglected.
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Damage Detection Analysis under Various
Sensor Quantities
Sensors with larger contribution ratios of damage index CRDI calculated by Eq. (40) were chosen as the optimal placement of the
sensors. The number of sensors is usually determined in accordance
with economic considerations and damage identification accuracy
before optimization analysis of sensor placement.
The CRDI values under various numbers of sensor quantities are
shown in Fig. 9. As seen, the CRDI value increases with the increase
of sensor quantities; however, the increasing trend becomes slower
with the increase of the sensor quantity. This means that when the
sensor quantity exceeds a given value, installing more sensors only
slightly benefits the damage information; thus, increasing the sensor
quantity contributes less to the damage detection results when the
sensor quantity exceeds a given value.
To analyze the effect of the sensor quantity on the damaged location and quantification abilities of the EEISM method, damage
cases were set as 9–12, in which it was assumed that sensor
quantities of 25, 50, 75, and 100 corresponding to Cases 9–12 and
Member No. 34 were damaged with a stiffness reduction of 20%, as
shown in Table 3. The MAC values under all cases are shown in Fig.
10. When the sensors quantities were 25 and 50, the MAC values
of Member No. 34 were 0.666 and 0.752, and the MAC values of
Member No. 35 were 0.585 and 0.403, respectively. As seen,
Member Nos. 34 and 35 have larger MAC values in Cases 9 and 10,
and the two members can be detected to be damaged members in
accordance with Eq. (42). Therefore, Member No. 35 was misdiagnosed as the damaged member when the sensor quantity was not
enough. When the sensor quantity was increased to 75 and 100, the
MAC values of Member No. 34 were 0.834 and 0.863, respectively,
while the MAC values of Member No. 35 were 0.221 and 0.097,
which means Member No. 34 had the largest MAC value in Cases 11
and 12, Member No. 34 was detected as the damaged member, and
Member No. 35 was not misdiagnosed. Fig. 11 shows the damage
quantification result of Member No. 34 estimated by Eq. (43) under
Fig. 9. Total CRDI value under a special sensor quantity
Fig. 7. MAC values of Member No. 53
Table 3. Damage Detection for Member No. 34
MAC value
Real damage
quantity (%)
Detected damage
quantity (%)
Damage
case
No. 34 No. 35 No. 34 No. 35
No. 34
No. 35
No. 34
Case
Case
Case
Case
23.00
22.60
22.35
22.23
9.15
8.45
—
—
15.00
13.00
11.75
11.15
9
10
11
12
0.666
0.752
0.834
0.863
0.609
0.642
0.692
0.848
20
20
20
20
0
0
0
0
Fig. 8. Damage quantity detection of Member No. 53
Table 2. Damage Detection for Member No. 53
Damage
case
Case
Case
Case
Case
5
6
7
8
MAC value
Real damage
quantity (%)
Detected damage
quantity (%)
Relative
error (%)
0.609
0.642
0.692
0.848
10
15
20
40
10.16
15.98
22.15
52.63
1.60
6.53
10.75
31.58
Relative
error (%)
Fig. 10. MAC values of Member Nos. 34 and 35
1222 / JOURNAL OF ENGINEERING MECHANICS © ASCE / OCTOBER 2012
J. Eng. Mech., 2012, 138(10): 1215-1223
will only slightly improve the accuracy of the damage
quantification.
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Acknowledgments
This research was supported by the Momentous Research Plan in the
National Natural Science Foundation of China (90915004), Key
Projects in the National Science & Technology Brace Program of
China (2011BAK02B03), and the 333 High-level Talent Project in
Jiangsu Province, which the writers gratefully acknowledge.
References
Fig. 11. Damage quantity detection of Member No. 34
Damage Cases 9–12. As listed in Table 3, the real damage quantity is
20%, and the detected damage quantities are 23.00, 22.60, 22.35,
and 22.23% when the sensors quantities are 25, 50, 75, and 100,
respectively. The relative error between the real damage quantity and
the detected damage quantity is 15.00, 13.00, 11.75, and 11.15%.
This shows the EEISM method has good damage quantification
ability. It can be concluded from the previous analysis that the
damaged location ability of the EEISM method obviously increases
with the increase of the sensor quantity; however, the accuracy of the
damage quantification improves slightly. Especially when the sensor
quantity is large, the sensor quantity has no obvious effect on the
improvement of the accuracy of the damage quantification. Therefore, determining the sensor quantity is required in accordance with
economic considerations and damage identification accuracy. It can
be suggested from the research that the sensor quantities can be
determined as 1/8–3/8 of the total member quantity, which is not
described in detail here for the sake of brevity. Obviously, the
proposed method could detect damage by the incomplete strain
mode, which can save cost and reduce data processing work. In
this paper, the damage detection problem for three or more
members is not taken into consideration, and this problem will be
studied in the future.
Conclusions
A new damage detection method for space truss structures, named
the EEISM method, is proposed in this paper. Numerical analysis of
a long-span space truss structure was performed to verify the effectiveness of the EEISM method. The following conclusions may
be drawn from this study.
1. A similar equation exists between the cross-correlation function of the strain response under environmental excitation and
the strain impulse response. Consequently, the strain mode can
be obtained based on the cross-correlation function of the
strain responses.
2. The EEISM method has good ability in damage location for
single damaged members of bottom chord members, top chord
members, and web members, as well as in damage location for
multidamaged members.
3. The damage index a in the EEISM method can be used to
estimate the damage quantity with good accuracy.
4. The capability of locating damage using the EEISM method will be improved by increasing the sensor quantity but
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