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KSCE Journal of Civil Engineering (0000) 00(0):1-13
DOI 10.1007/s12205-021-0139-1
pISSN 1226-7988, eISSN 1976-3808
www.springer.com/12205
Structural Engineering
Condition Assessment of Stay Cables via Cloud Evidence Fusion
Shuang Sun a, Li Liangb, and Ming Li b
College of Civil Engineering and Architecture, Zhejiang Sci-tech University, Hangzhou 310018, China
School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China
1. Introduction
Bridges are critical transportation infrastructures, and the safety
condition of bridges in their service life significantly affect the
social sustainability and economic development (Liang et al.,
2019). Ever since the last decade, cable-stayed bridges have been
extensively constructed throughout China due to their ability to
cover large spans as well as to guarantee aesthetic requirements.
Stay cables play a vital role in cable-stayed bridges, transmitting
loads from the deck to towers. Cable damage is therefore an
extreme threat to the structural safety of bridges. As outside
components of the structure, cables are subject to a variety of
natural disasters and durability degradations throughout their
lifetime, they are vulnerable to corrosion and fatigue under the
interaction of corrosive environment and cyclic load (Kim, 2001;
Suzumura and Nakamura, 2004). A certain amount of catastrophic
cable-stayed bridge accidents were caused by the cable deterioration,
such as Maracibo Bridge in Venezuela and Haiyin Bridge in
China (Suzumura and Nakamura, 2004; Xu et al., 2011). According
to the statistics of bridge accidents in China, the minimum life
span of cables is only 3 years, and the maximum is 22 years (Li
and Ou, 2016). Cables are required to be replaced more than
once during their service life. The repair fund of Haiyin Bridge
in Guangzhou China cased by cable damage is about 3 million
dollars, and the traffic had to be closed for half a year, which
severely influenced the bridge serviceability (Xu et al., 2011).
Because of these factors, the condition assessment of stay cables
presents a significant challenge to asset managers and maintenance
professionals. Conducting a credible assessment tool of stay
cables is of the utmost importance to ensure their durability and
serviceability, as well as to provide a cost-effective decision-
making for cable management and maintenance.
Three main methods have been developed for the cable
condition assessment, including visual inspection, numerical
simulation, and structural health monitoring (SHM). Cable
surface inspection is a traditional means considered as the first
step to examine the surface condition of cables. It is commonly
utilized to detect the obvious anomalies or damages existing in
polyethylene (PE) pipe, shield, anchorage device and shock
absorber (Li et al., 2019). However, this method is highly labor-
ARTICLE HISTORY ABSTRACT
Received 15 February 2020
Revised 27 June 2020
Accepted 2 November 2020
Published Online 22 January 2021
KEYWORDS
Since the monitoring of cable tension are rather susceptible to environmental influence and
external loads, the condition assessment of stay cables is vitally difficult because of these
uncertainties. In this paper, regarding the health condition of stay cables, a multilevel
assessment framework is presented, which can synthetically combined the evaluation results
from numerical simulation, field monitoring and visual inspection. Based on these methods,
three qualitative and three quantitative indices are selected as the evaluation indices. To
reduce the uncertainties during the assessment procedure, an intelligent methodology based
on cloud model and Dempster-Shafer (D-S) evidence theory is proposed. With the
combination of forward cloud generator and backward cloud generator, the cloud parameters
of in-situ data is transmitted to the cloud model of grade criteria, then the cloud evidence with
relative weights are fused by Dempster combination, the condition grade of the cable is finally
obtained. The Junshan Yangtze River Bridge is adopted to verify the effectiveness of the
proposed methodology. The results show that the uncertainty degree can be obviously
reduced from 55.7% to 6.7%, so that a scientific evaluation of cable conditions can be
obtained. The multilevel assessment framework proposed in this study can serve as an
effective basis for cable replacement and maintenance.
Cable-stayed bridge
Condition assessment
Cloud model
D-S evidence theory
Synthesis weight
CORRESPONDENCE Shuang Sun sophia-sunshuang@163.com College of Civil Engineering and Architecture, Zhejiang Sci-tech University, Hangzhou 310018, China
ⓒ 2021 Korean Society of Civil Engineers
2 S. Sun et al.
intensive, and requires traffic disruption during the inspection.
Besides, the evaluation results of surface inspection are inevitably
subjective and inefficient because of the manual-based involvement
(Sun et al., 2019). Numerical simulation is used to analyze the
mechanical state of stay cables by establishing the finite element
model (FEM). Detailed FEMs of Bridges are key to accurate
assessment of stay cables that requires the application of FEM
updating technique. However, for large structures such as long-
span cable-stayed bridges, FEM updating become more difficult
as multitudes of parameters need to be updated due to more
uncertainties, which results in numerous computations (Han, 2011;
Talebinejad et al., 2011). SHM is regarded as a powerful technique
to identify structural degradation by the in situ monitored data.
The monitored cable acceleration from SHM system can be used
to analyze the cable tension, which is considered as one of the
health indicators for both cables and superstructure of the cable-
stayed bridges (Xiong et al., 2010; Cross et al., 2013; Li et al.,
2014). Unfortunately, the variation of cable tension is not only
influenced by the structural deterioration, but also sensitive to
environmental noise excited by vehicle, temperature, or wind
(Montassar et al., 2015). Therefore, the inaccurate monitoring
data give rise to the difficulties for accurate and effective cable
condition assessment. Condition assessment for stay cables is a
highly complicated process; either of these methods can hardly
be used as the sole evaluation tool directly. Visual inspection,
numerical simulation and SHM should be properly integrated
and make full use of each one's advantages. Meanwhile, the
uncertainties contained in the assessment process would influence
the evaluation accuracy, which are exhibited in two aspects: 1)
randomness, which is often reflected in the monitored data
related to quantitative indices; and 2) fuzziness, which is inherent
in the grade classification criterion and the scores of qualitative
indices.
With the recent advancement in artificial intelligence, the
uncertainties in the cable condition assessment are carried out by
several researchers (Degrauwe et al., 2009; Hassan, 2013; Li et
al., 2014; Liu et al., 2017; Zarbaf et al., 2018; Zhang et al., 2018).
For numerical simulation based method, detailed finite element
model (FEM) comprising elastic modulus, density, and boundary
constraints is difficult to obtain due to parameter uncertainties.
Neural networks, wavelet analysis, genetic algorithm and many
other techniques have been used for FEM updating to identify
the cable damage (Hassan, 2013; Zarbaf et al., 2018; Zhang et
al., 2018). For visual inspection based method, subjective
judgments inevitably contain fuzziness and impreciseness. Liu et
al. (2017) employed fuzzy membership to describe the qualitative
indices obtained from damage investigation. For SHM based
method, the time-varying cable tension force from monitoring
sensors contains randomness, which is caused by vehicle,
temper ature and wind effects. To distinguish the influence of
measurement errors from cable tension variations, Li et al. (2014)
proposed a tension estimation approach based on Kalman filter,
and established real-time identification algorithm. Ren et al.
(2019) evaluate the cable conditions by integrating influence lines
with monitoring data. Degrauwe et al. (2009) presented a
methodology based on fuzzy analysis to investigate the sensitivity of
the structural temperature with respect to the measurement errors
and calculation inaccuracies. Although many studies have been
performed, the proposed methods can only evaluate the cable
conditions by virtue of one single method and cannot deal with
fuzziness and randomness together.
The cloud model (CM) was first proposed by Li in 1995 (Li,
1995), as a new cognitive model, can simultaneously represents
both randomness and fuzziness existing in multi-criteria decision
making problems. It has been applied to weaken the uncertainty
interference in a variety of fields, including intelligent control
(Gao et al., 2017), risk assessment (Zhang et al., 2015), and
natural resources evaluation (Wang et al., 2016; Lu et al., 2017).
Xu et al. (2018) introduced Normal Cloud model in a multi-layer
index system for suspension bridges. D-S evidence theory was
introduced by Dempster and extended by Shafer, as an effective
information fusion method, can diminish the uncertainties during
the evidence fusion process (Certa et al., 2017). With the
Dempster combination rule, multiple pieces of evidence from
different sources are combined, and the accuracy of decision
making can be improved. This paper introduces a cloud evidence
methodology (CEM) for cable condition assessment based on
cloud model and D-S evidence theory. The assessment data by
means of numerical calculation, field monitoring and visual
inspection are effectively integrated in the CEM. Multiple load
cases by numerical calculation are used to establish the grade
criteria of each evaluation index. The in-situ data from monitoring
and inspection are regarded as the quantitative and qualitative
evaluation indices, respectively. The uncertainties in the evaluation
procedures are described by means of cloud model. Furthermore,
D-S theory is applied to fuse various indexes, reduce the
uncertainties, and obtain the safety condition of cables.
2. Evaluation Indices and Critical Thresholds
In cable condition assessment system, the evaluation indices are
divided as qualitative indices from field monitoring and quantitative
indices from visual inspection. For quantitative indices, the
monitored cable tension is not sensitive enough to identity the
cable damage (Li et al., 2018). As a result, its derivative parameters:
cable stress, fatigue stress amplitude and parameter excitation are
extracted as the quantitative indices. According to Standards for
Technical Condition Evaluation of Highway Bridges (JTG/T
H21-2011, 2011), the indices of surface damage detection of
cables, which comprise PE pipe, anchorage device and shock
absorber, are extracted as the qualitative indices.
The critical thresholds of mechanical indices are determined
according to the bridge design codes or the material capacity of
cables (JTG/T D65-01-2007, 2007; JT/T 775-2010, 2010).
2.1 Cable Stress
According to the guidelines for design of highway cable-stayed
bridge (JTG/T D65-01-2007, 2007), the safety coefficient of stay
KSCE Journal of Civil Engineering 3
cables in operation should not be less than 2.5; that means,
the maximum stress of stay cables is restricted within 0.4
times of the characteristic strength, which can be expressed as
follows:
,(1)
where σ
c
is the cable stress (MPa); f
pk
is the characteristic
strength of stay cables.
Thus, if the characteristic strength f
pk
= 1,670 MPa, the critical
threshold of cable stress is 668 MPa.
2.2 Fatigue Stress Amplitude
The cyclic stress may lead to cable fatigue. The difference
between the maximum and minimum stress in a stress cycle is
called fatigue stress amplitude. After the bridge is completed, the
stress variation can hardly be caused by dead load of the bridge.
As a contrast, the cyclic stress is mainly induced by the vehicle
load. The fatigue stress amplitude can be defined as
,(2)
where Δσ
c
is the fatigue stress amplitude of stay cables, mainly
caused by vehicle load (MPa); σ
max
is the maximum cable stress
during a period (MPa); σ
min
is the minimum cable stress during a
period (MPa).
According to the stay cable of parallel steel wires for large-
span cable-stayed bridge (JT/T 775-2010, 2010), the allowable
fatigue stress amplitude of stay cables is restricted within 250
MPa; and thus, the critical threshold of fatigue stress amplitude is
250 MPa.
2.3 Parameter Excitation
Strong transverse vibration of stay cables would still occur even
with no wind or low-speed wind. Relative studies have shown
that in large span cable-stayed bridges, when the ratio of
frequencies of bridge to the frequencies of cables is close to 1 or
2, parameter excitation will be very probable due to the presence
of many low frequencies in the girder and in the stay cables
(Lilien and Pinto, 1994). Parameter excitation would induce
severe damage to the bridge.
If the bending stiffness is ignored, the natural frequency of
cables can be solved by Eq. (3):
,(3)
,(4)
where ω
q
is the bridge frequency (Hz); ω
n
is the cable frequency
(Hz); k is the ratio of frequencies of the bridge to the frequencies
of cables; T is the cable tension (N); L is the cable length (m); m
is the cable mass per unit length (kg/m); n is the mode order, as
the first order mode gives the biggest contribution to the
vibration responses of stay cables, only the first order mode is
considered in the calculation, that is n = 1.
2.4 Qualitative Indices
Some of the concealed damage in cables cannot be detected only
with the above mechanical indices, especially the local corrosion.
Besides, the exterior components of cable wires are easily ages
and cracks because of their long-period exposure under humid
environment (Li et al., 2019). Thus, visual inspection is a vital
aspect of cable assessment, and it always assesses the cable
condition by qualitative evaluation. Here, PE pipe, anchorage
device and shock absorber are taken as the qualitative indices.
PE pipe is an important component which packages the cable
wires. Once the PE pipe is damaged, rainwater or even corrosive
substances will intrude into the interior of the cable via cracks.
Furthermore, the internal steel wires are susceptible to corrosion
and may lead to the cable replacement.
The anchorage device is employed to transmit the force from
the girder to cables, and then to the tower. If the sealing material
in the anchorage area is not tight enough, rainwater can easily
enter the inner of the anchorage, resulting in cable corrosion.
Shock absorber is utilized on bridges to reduce the vibration
amplitude, since the cable-stayed bridge is a large-span structure
with high flexibility and low damping. The intactness of shock
absorber is vital to avoid the structural damage under extreme
vibration.
3. Assessment Methodology
The framework of the cable condition assessment method
proposed in this paper is illustrated in Fig. 1. This framework is
aimed to evaluate the cables comprehensively and effectively by
means of numerical calculation, SHM and visual inspection. A
novel methodology called CEM is presented to deal with the
uncertainties in the assessment framework. The main procedures
are listed below.
1. Select the quantitative and qualitative evaluation indices,
determine their critical thresholds.
2. Classify the condition grades on the basis of calculation
results from multiple load cases, establish the grade criteria
with five cloud models.
3. Transmit the monitoring and inspection data to the backward
and forward cloud generator, determine the membership
degree of each evaluation index.
4. Synthesize the entropy weight and grade weight, consider
both the index effect and grade significance.
5. Fuse all the evaluation indices via evidence theory to
further reduce the uncertainties, determine the final condition
grade of the cable.
3.1 Grade Classification Criteria
In this assessment framework, the cable conditions are classified
into five grades, which are defined as Grade I, Grade II, Grade
III, Grade IV and Grade V from normal to danger. In order to
represent the severity more obviously, these five grades are
marked by five colors, green, yellow, orange, pink, and red,
respectively.
0.4 fσ≤
σσ σΔ= −
nT
Lm
π
ω=
/kωω=
4 S. Sun et al.
For quantitative indices, the minimum value among multiple
load cases by numerical calculation is denoted as min, the critical
line is determined by the critical thresholds. For qualitative
indices, they are scored by experts with hundred-mark system.
Here, Golden Section is utilized as a classification tool to
determine the grade criteria of cable conditions (Lu, 2003).
Golden Section is a number (0.618) which satisfies the equation
. If a line segment be divided into two parts and the
total length is 1, then the most pleasing division is one portion
should be of length 0.618. The condition grade can be continuously
divided by Golden Section as Fig. 2.
3.2 Cloud Model of Grade Criteria
The application of gold section can only realize strong partition
between two adjacent grades. However, the fuzziness in classification
boundary still exists under the influence of cognition uncertainty.
In order to solve that problem, cloud model of the grade criteria
is established to deal with these uncertainties.
The normal cloud model, based on the normal distribution and
Gauss membership, can both model randomness and fuzziness by
means of three parameters: Ex (Expectation), En (Entropy) and
He (Hyper-entropy) (Li et al., 2009).
In the CEM, two kinds of normal cloud generators are utilized
to establish the cloud model of the grade criteria, they are
backward cloud generator and forward cloud generator (Li et al.,
2009). The backward cloud generator is first used to achieve the
parameters (Ex, En, He) of assessment data from monitoring or
inspection. Then, these parameters are transmitted to the forward
r1r–=
Fig. 1.
Flowchart of CEM
Fig. 2.
Grade Criteria Classified by Golden Section
KSCE Journal of Civil Engineering 5
cloud generator to describe the uncertainty degree.
In the backward cloud generator, for a quantitative index, the
three parameters (Ex, En, He) can be acquired based on the
numerical results from various load cases, and they are calculated as
Eq. (5). For a qualitative index, the three parameters can be
acquired based on the score intervals, and they are calculated as
Eq. (6) (Zhang et al., 2015).
,(5)
where A
i
is the numerical result of the ith load case for the
quantitative index A; is the sample mean; S
2
is the sample
deviation; n is the number of load cases.
,(6)
where A
min
and A
max
are the minimum and maximum scores
corresponding to a certain grade for a qualitative index A. Take a
qualitative index for example, as shown in Fig. 2, For Grade II,
A
min
= 95, and A
max
= 100, s is a constant, the corresponding
parameters can be obtained as Ex = 97.5, En = 0.83, and He = s
which should be adjusted manually.
In the forward cloud generator, let U be the universe of
assessment system, x be the assessment data in U, denoted as
. The cloud drop x is a random realization of an index, and
the distribution of x in the universe U satisfies the normal cloud
as Eq. (7), the uncertainty degree of x belonging to the index can
be calculated as Eq. (8) (Zhang et al., 2015):
,(7)
,(8)
where Ex is the mathematical expectation, it is the mean value of
a quantitative index or center score of a qualitative index; En is the
dispersion degree compared with the expectation, it represents the
range of assessment data belonging to a certain grade. Additionally,
En is a measure of the randomness of cloud drops and it is also a
reflection of the fuzziness of qualitative indices. μ(x) is the
uncertainty degree of a cloud drop x belonging to an index.
In the CEM, five normal cloud models are applied to represent
the relationship between evaluation indices and five condition
grades. The five cloud models are one descending half-Cloud,
three full-Clouds and one ascending half-Cloud, as shown in Fig. 3.
Uncertainty degree μ(x) is taken as the membership degree of
each evaluation index.
3.3 Variable Weight Approach
In this assessment system, there are 6 evaluation indices, including 3
quantitative indices and 3 qualitative indices. The contribution of
each index to the final decision making cannot be identical.
Additionally, once the in-situ data of a certain index belong to a
high grade, the index weight should be increased taking into
account the grade significance. Regarding these reasons, we
determine the index weight with the connection of entropy
weight and grade weight, using information entropy and fuzzy
analytical hierarchy (FAH), which can synthetically describe the
objective index contribution and grade significance.
3.3.1 Entropy Weight
Information entropy based approach is used to calculate the
objective index weights of quantitative indices Q
1
− Q
3
. The
concept of information entropy is a measure of uncertainty in a
framework (Shieh and Wu, 2017). The objective index weight is
derived by information entropy through the inherent data of a
framework.
The calculation results from multiple load cases are utilized to
construct a prior matrix , in which the rows are load
cases and the columns are quantitative indices. If the differences
of data in a certain column are larger compared to others, which
indicates a higher disorder degree, the importance of the
corresponding index controlling the prior matrix is more than
other indices. As a consequence, higher disorder indicates higher
1
1
2
Ex A
n
En A A
n
He S En
π
⎧=
⎪
⎪
⎪=× −
⎨
⎪
⎪=−
⎪
⎩
∑
∑
A
()/2
()/6
Ex A A
En A A
He s
=+
⎧
⎪=−
⎨
⎪=
⎩
xU∈
~(, )
~(, )
x N Ex En
En N En He
′
⎧
⎨
⎩
()
( ) exp( )
2
xEx
x
En
µ
−−
=
′
Qq()=
Fig. 3.
Development of Cloud Model of Grade Criteria
6 S. Sun et al.
entropy and higher weights.
The definition of information entropy is expressed as Eq. (9)
(Shieh and Wu, 2017):
,(9)
, (10)
where q
ij
is the value of the jth quantitative index in the ith load
case; f
ij
is the proportion of q
ij
in the whole prior matrix, when q
ij
= 0, f
ij
ln f
ij
= 0; k = 1/ln m, and m is the number of load cases.
Hence, the entropy weight of the jth index in the prior matrix
can be obtained by the following equation:
,(11)
in which , ; ; n is the total of
quantitative indices.
3.3.2 Grade Weight
Grade weight is introduced to modify the weights according to
the condition severity of cables. If an index value approaches to a
higher grade level, which indicates a severe condition of the
cable, then it must be given a higher weight.
The grade severity is defined by the cloud grade coefficient f
J
:
, (12)
, (13)
where f
J
is the cloud grade coefficient; J is the cloud judgment
vector, G is the grade vector, G = [1, 2, 3, 4, 5]
T
;
are the membership degrees of the assessment data to these five
condition grades, respectively, they are calculated from Eq. (8).
To determine the grade weights, FAH is utilized to establish a
fuzzy comparison matrix, the elements in which are relative
strength compared between every two grades (Pan, 2008). The
elements in the matrix are described by decimals 0.1 − 0.9. The
fuzzy comparison matrix of five grades is expressed as
The weights of five grades can be calculated as follows:
, (14)
where ω
k
is the grade weight of the kth Grade, k = 1, 2, 3, 4, 5;
, ; b
ks
is the comparative element in matrix B.
Then, the corresponding grade weights of Grade I to Grade V
are .
The grade weight of quantitative index or qualitative index
can be obtained by linear interpolation approach, as Eq. (15):
, (15)
where ω
2
(j) is the grade weight; f
J
is the cloud grade coefficient,
which can be calculated by Eq. (12).
3.3.3 Synthesis Weight
On the basis of entropy weight and grade weight, the synthesize
weight of evaluation index can be calculated. Synthesis weight
w(j) should be nearly to both entropy weight and grade weight
(Wang, 2002). The minimum entropy solution is employed to
solve this problem, given by Eq. (16):
(16)
Equation (16) is an optimization problem, Lagrange Multiplier is
adopted to derive the final synthesis weight, which is calculated
by Eq. (17):
, (17)
in which w(j) is the synthesis weight; w
1
(j) is the entropy weight;
w
2
(j) is the grade weight; n is the total of evaluation indices.
3.4 Multiple-Criteria Data Fusion
After the determination of grade criteria and synthesize weight,
the evaluation indices with respective weights should be
combined effectively to obtain a reliable conclusion of the cable
condition. The Dempster rule of combination in evidence theory
is applied to combine the weighted indices and minimize the
uncertainties.
In terms of evidence theory, five condition grades constitute
the assessment frame of discernment Θ = {Grade I, Grade II,
Grade III, Grade IV, Grade V}, in which each grade are viewed
as a focal element. The evaluation indices are viewed as
supporting evidence, and the basic probability assignment (BPA)
of evidence is determined by the cloud judgment vector. In
addition, the weight effect of evaluation indices should be
considered during the fusion process.
Considering the weight effect, the Dempster rule of combination
can be expressed as follows:
. (18)
In order to handle the combination rule with relative weights,
a weighted average BPA was proposed in the CEM using the
lnHkff=−
∑
/fq q=
∑
Qq
ij
()
mxn
=
1
() H
wj
nH
−
=
−
∑
0ωj() 1≤≤
() 1jω=
∑
1jn≤≤
=fJG⋅
[, , , , ]Jµµ µµ µ=
,,,,µµ µµ µ
0.5 0.6 0.7 0.8 0.9
0.4 0.5 0.6 0.7 0.8
0.3 0.4 0.5 0.6 0.7
0.2 0.3 0.4 0.5 0.6
0.1 0.2 0.3 0.4 0.5
B
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
.
2
/
25
wb b b==
∑∑∑ ∑
0ω1≤≤
1w=
∑
[0.12,0.16,0.2, 0.24, 0.28]ω=
() /25 0.08jfω=+
min ( )[ln ( ) ln ( )]
()[ln () ln ()]
Fwjwjwj
wj wj w j
=−
+−
∑
∑
,
s.t. () 1; () 0, 1~wj wj j n=>=
∑
.
()
[()()]
() 1
[()()]
wjwj
wj j n
wjwj
=≤≤
∑
mm m m mωω ω ω=⊕ ⊕ ⊕
m
KSCE Journal of Civil Engineering 7
following equation:
. (19)
Then, the combination rule becomes an iterative combination
process with the weighted average BPA. In the frame of
discernment, take the focal element Grade I for example, the
iterative combination can be expressed as follows (Certa, 2017):
, (20)
, (21)
where is the weighted average BPA of n pieces of evidence;
ω
j
is the synthesis weight by Eq. (17); n is the total number of
evaluation indices; and m
j
is the BPA of the jth index belonging
to a certain grade; , are weighted average BPAs with
relation to a and b; m(I) is the BPA with relation to Grade I; K is
the conflict coefficient between two pieces of evidence.
Suppose the number of evaluation indices is n, the combination
process are performed between two pieces of evidence, then, the
times of iterative combination is (n − 1) (Verbert et al., 2017).
After the combination of weighted evaluation indices, the final
grade of the cable condition can be acquired in terms of the
fusion focal element which has the highest probability.
4. Case Study
4.1 Bridge Description
The cable condition assessment method introduced in this paper
is demonstrated using a case study on Junshan Yangtze River
Bridge, which was built across the Yangtze River in Wuhan,
China. It is a cable-stayed bridge with a main span of 460 m and
two side spans of (48+204) m each. Fig. 4 shows a photograph of
the bridge. The streamlined steel box girder is 38.8 m in width
and 3.0 m in height and accommodates six traffic lanes. The two
towers are made of concrete with a total height of 163.5 m. There
are 144 cables made of parallel high-strength galvanized steel
wires. Cables are numbered from WA18-WA1 to WJ1-WJ18
(west side) and EJ18-EJ1 to EA1-EA18 (east side) in one cable
plane. The elevation view of the bridge and cable numbering are
shown in Fig. 5. The bridge was completed in May 2000 and
opened to traffic in December 2001. By the end of 2019, it has
been operational for 18 years.
4.2 Numerical Simulation
In this cable condition assessment system, numerical simulation
is used to determine the grade criteria and entropy index of the
quantitative indices.
The Junshan Yangtze River Bridge is modeled and analyzed
by Midas Civil software. The bridge was simulated by a
longitudinal beam element and many transverse beam elements
representing the full deck. The transverse beams and cables are
linked at the anchorage nodes. Stay cables were modeled by
tension-only truss elements. Beam elements were used for the
towers. The boundary condition of auxiliary piers is sliding
bearing, where the transverse and vertical directions are restricted,
and the node at the bottom of the tower is assumed as fixed
bearing. The bridge beam is connected to the tower using elastic
restraint in such a way that it is fixed in vertical and horizontal
directions. The FEM of the bridge is shown in Fig. 6. The
material parameters of the bridge were modified by modal
properties and force tensions from the field test at the completion
stage.
In order to simulate the different states of the bridge, the
variation regions of influence factors are applied to the FEM to
generate multiple load cases. Steel corrosion ratio, cable corrosion
ratio, vehicle load and temperature are regarded as the influence
mmω=
∑
() ()
Grade
() 1
0Grade
mamb
mK
⎧
⎪≠∅
=⎨−
⎪=∅
⎩
∑
Ⅰ
Ⅰ
Ⅰ
() ()Kmamb=
∑
m
ma()
mb()
Fig. 4.
Overview of Junshan Yangtze River Bridge
Fig. 5.
Elevation View and Cable Numbering
8 S. Sun et al.
factors. The variation regions of each influence factor are listed
in Table 1, in which the vehicle load and temperature are based
on the codes JTG D60-2015 (2015) and JTG/T D65-01-2007
(2007), respectively. From the numerical calculation, 3 × 3 × 4 ×
3 = 108 load cases construct a matrix , where the
rows stands for 108 load cases, and the columns stands for
quantitative indices, in which Q
1
represents cable stress, Q
2
represents fatigue stress amplitude and Q
3
represents parameter
excitation. The envelope of cable tension force from numerical
calculation is illustrated in Fig. 7.
4.3 Tension Monitoring
The studied bridge was equipped with SHM system in 2011.
There are totally 12 kinds of sensors which could continuously
and simultaneously monitor the mechanical responses of the
bridge. Cable tensions were measured by monitoring sensors
installed on each stay cable. The sample frequency is 1/600 Hz.
The cable tensions of Cable WA10 during 10 days are shown in
Fig. 8. Based on the monitored cable tensions, three quantitative
indices could be obtained using Eqs. (1) − (4).
The fatigue stress amplitude is based on the maximum and
minimum of cable tension during a day.
()Qq=
Fig. 6.
FEM of Junshan Yangtze River Bridge
Tab l e 1 .
Variation Region of Influence Factor
Influence factors Variation region Variation
cases
Steel corrosion ratio (%) 3
Cable corrosion ratio (%) 3
Vehicle load ratio (%) 4
Temperature variation ( C) 3
Temperature gradient ( C)
Temperature difference between
cable and beam( C)
Temperature difference between
cable and tower ( C)
Fig. 7.
Envelope Results of Cable Tensions
KSCE Journal of Civil Engineering 9
4.4 Visual Inspection
Visual inspection was conducted to assess three qualitative
indices: PE pipe, anchorage device and shock absorber. Take
Cable WA10 for example, for the PE pipe, there appears to be
slippages from the cable wire (Fig. 9). In addition, there is water
with rust flowed from the anchorage device (Fig. 10). The damping
material in the shock absorber has been exposed, as shown in Fig. 11.
Six experts were invited to give the scores of qualitative
indices. According to the experts' advice, the scores of Cable
WA10 are listed in Table 2.
4.5 Cable Condition Assessment of Junshan Yangtze
River Bridge
The proposed cloud evidence-based method is applied to assess
the cable conditions of Junshan Yangtze River Bridge. It is used
to reduce the uncertainties from numerical simulation, field
monitoring and visual inspection, and obtain the final condition
grade of the cables.
4.5.1 Cloud Model Establishment
Based on the monitoring data during 10 days in Fig. 8, the time
Fig. 8.
Cable Tensions of Cable WA10 for 10-Day
Fig. 9.
PE Pipe
Fig. 10.
Anchorage Device
Fig. 11.
Shock Absorber
Tab l e 2 .
Scores of Qualitative Indices of Cable WA10
Qualitative indices Expert 1 Expert 2 Expert 3 Expert 4 Expert 5 Expert 6
PE pipe 54 65 70 75 66 76
Anchorage device 33 40 35 45 36 48
Shock absorber 66 70 60 65 72 62
10 S. Sun et al.
Fig. 12.
Membership Degree of Evaluation Indices Q-Q: (a) Cloud Model of Q,(b) Cloud Model of Q, (c) Cloud Model of Q, (d) Cloud Model
of Q,(e) Cloud Model of Q, (f) Cloud Model of Q
KSCE Journal of Civil Engineering 11
history of quantitative indices (cable stress, fatigue stress
amplitude and parameter excitation) can be obtained.
The cloud models of grade criteria for each evaluation index
could be established based on the calculation results from
numerical simulation. The time history of quantitative indices
and the qualitative scores were transmitted to the backward cloud
generator; three parameters of cloud model can be obtained
according to Eqs. (7) and (8). Then, the (Ex, En, He) were further
transmitted to the forward cloud generator. Corresponding to the
grade criteria, the membership degree of evaluation indices Q
1
-
Q
6
to five condition grades can be obtained. The results are
shown in Fig. 12.
4.5.2 Weight Calculation
The index weight of each qualitative index is designed as 0.1
based on experts' advice. Then, the total weights of three quantitative
indices are 0.7.
According to the calculation results of 108 load cases, the
objective weights of quantitative indices Q
1
− Q
3
are 0.1989,
0.2614 and 0.2396, which are obtained with the application of
Eqs. (9) − (10). The entropy weight results are shown in the first
column of Table 3. The entropy weight of Q
2
is the highest one
which indicates that the fatigue stress amplitude is more sensitive
than other two quantitative indices.
The membership degree of each evaluation index in Fig. 12 is
substituted into Eqs. (13) − (14), The cloud judgment vector J
and the cloud grade coefficient f
J
are acquired, then the grade
weight ω
2
(j) is further calculated with the application of FAH,
the results are shown in the fourth column of Table 3.
From the results in Table 3, it is noted that the sequence of
evaluation indices by entropy weight w
1
is Q
2
, Q
3
, Q
1
from high
to low, and the following Q
4
Q
5
Q
6
are of the same w
1
. As a
comparison, the sequence of indices by grade weight w
2
is Q
3
Q
5
Q
2
Q
4
Q
6
Q
1
. Index Q
2
(fatigue stress amplitude) has the highest
entropy weight, which indicates that the variation of Q
2
is the
largest in terms of all the simulated states of the bridge. The
grade weight w
2
is based on the severity of the field data. i.e.,
though w
1
of Q
2
is 0.2614, making it the first place, however, w
2
of Q
2
is 0.1697, making it the third place. w
2
of Q
3
is the highest
because its monitoring data is the severest. The w
1
of qualitative
index Q
5
is 0.1, whereas w
2
of Q
5
is 0.1894, making it the second
place, which indicates the condition of anchorage device is
critical, and the subject weight by experts are fairly modified
according to the inspection results. In conclusion, the sequence
of synthesis weight is Q
3
, Q
2
, Q
5
, Q
1
, Q
4
, Q
6
from high to low.
4.5.3 Final Grade Determination
Considering the cloud membership and synthesis weight of each
evaluation index, the average basic probability masses are
generated using the weighted combination rule of Eq. (19). Here,
the result of weighted average BPA is = [0.0002 0.0769
0.2804 0.4433 0.1993]
T
; this is then substituted to Eqs. (20) and
(21) to combine five times. As the fusion results in Fig. 13, by
virtue of evidence theory, the reliability of assessment results is
obviously increased and the confidence interval is reduced. After
combining the information from 6 evaluation indices, the fusion
result is [0 0 0.060 0.933 0.007], the probability of focal element
m(IV) is 93.3% which indicates that the condition of Cable A10
is Grade IV. As a comparison, the assessment result by the
individual index is shown in Fig. 14. It is demonstrated that the
condition interval assessed by an individual index is much larger
m
Tab l e 3 .
Weight Calculation Results of Indices Q1 − Q6
Indices wJfw w
Q0.1989 [0.0000 0.4675 0.5259 0.0066 0.0000] 2.5391 0.1341 0.1591
Q0.2614 [0.0000 0.0000 0.2572 0.7422 0.0006] 3.7434 0.1697 0.2645
Q0.2396 [0.0000 0.0000 0.0501 0.3421 0.6078] 4.5577 0.1938 0.2768
Q0.1 [0.0016 0.0264 0.5616 0.407 0.0034] 3.3842 0.1591 0.0949
Q0.1 [0.0000 0.0000 0.0012 0.723 0.2758] 4.2746 0.1854 0.1105
Q0.1 [0.0000 0.0006 0.6521 0.347 0.0003] 3.347 0.1580 0.0942
Σ111
Fig. 13.
Effect of Combinations on Fusion Probability
Fig. 14.
Assessment Results by Single Index
12 S. Sun et al.
than the fusion results, and the condition grades which the
indices belong to are much more diverse. It is difficult for the
decision maker to obtain the correct conclusion of the cable
condition. The variation of uncertainty degree during the fusion
process is shown in Fig. 15. It depicts that the uncertainty degree
is greatly reduced from 0.557 to 0.067 with application of the
cloud evidence theory, and thus leading to improved assessment
accuracy.
The final condition assessment result of Cable WA10 is Grade
IV, further special inspection must be required to guarantee the
cable safety. For instance, the usage of cable inspection robot for
detailed inspection of dangerous cables; and the repairmen of
damaged components.
5. Conclusions
This paper presented a novel assessment framework for stay
cables based on cloud evidence fusion. This framework combines
numerical simulation, cable tension monitoring, as well as surface
inspection, and the uncertainties existing in these methods are
described by cloud models. Information entropy and FAH are
applied to deflect the relative significance of indices. Through
evidence theory fusion, the uncertainties are greatly reduced and
the condition of the selected cable is finally obtained. The main
conclusions of this study can be drawn as follows.
1. The proposed cable condition assessment method is a
multi-level decision system considering both the quantitative
and qualitative evaluation indices. Combining with the data
from numerical simulation, in-situ monitoring, and visual
inspection, the condition grade of stay cables can be acquired.
2. The propose CEM is based on cloud model, variable
weight and evidence theory. Cloud model can describe
both the randomness and the fuzziness contained in the
assessment process. The synthesis weight is integrated with
entropy weight and grade weight, it can be dynamically
adjusted according to the field data. The evidence theory
can fuse the multiple indices with relative weights, to
reduce the uncertainty degree and obtain a more credible
assessment conclusion.
3. Compared with the assessment results evaluated by an
individual index, the proposed CEM can reduce the uncertainty
degree from 55.7% to 6.7%, which means a more concise
and credible assessment results. The condition grade of the
stay cable becomes obvious with the combination procedure.
Thus, incorrect decision making with individual index can
be avoided, and the proposed CEM can contribute to a
scientific and concise condition assessment system of stay
cables.
4. Through the cloud evidence fusion, the safety status of the
selected cable WA10 is Grade IV, indicating that the
condition of this stay cable reaches the warning line at
present, which is approximately consistent with the actual
situation. This proposed assessment framework may offer a
reference for the cable management and replacement.
Acknowledgments
The authors acknowledge the financial support from the National
Natural Science Foundation of China (No. 51474048) and the
Fundamental Research Funds for the Central Universities (Grant
No. N170104024).
ORCID
Shuang Sun https://orcid.org/0000-0001-6548-6345
Ming Li https://orcid.org/0000-0002-7704-3856
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